LMFDB - Embedded newform 91.8.c.a.64.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [91,8,Mod(64,91)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(91, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 8, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("91.64");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 91 = 7 \cdot 13 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	91.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$28.4270373191$
Analytic rank:	$0$
Dimension:	$50$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			64.16
Character	$\chi$	$=$	91.64
Dual form			91.8.c.a.64.35

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-10.6365i q^{2} +74.1121 q^{3} +14.8657 q^{4} -114.172i q^{5} -788.291i q^{6} -343.000i q^{7} -1519.59i q^{8} +3305.61 q^{9} +O(q^{10})$	\(q-10.6365i q^{2} +74.1121 q^{3} +14.8657 q^{4} -114.172i q^{5} -788.291i q^{6} -343.000i q^{7} -1519.59i q^{8} +3305.61 q^{9} -1214.39 q^{10} -168.503i q^{11} +1101.73 q^{12} +(-6264.78 - 4847.79i) q^{13} -3648.31 q^{14} -8461.56i q^{15} -14260.2 q^{16} -13866.5 q^{17} -35160.0i q^{18} +939.578i q^{19} -1697.25i q^{20} -25420.5i q^{21} -1792.28 q^{22} +53368.8 q^{23} -112620. i q^{24} +65089.7 q^{25} +(-51563.3 + 66635.1i) q^{26} +82902.6 q^{27} -5098.93i q^{28} +143613. q^{29} -90001.1 q^{30} -68551.5i q^{31} -42828.8i q^{32} -12488.1i q^{33} +147491. i q^{34} -39161.1 q^{35} +49140.1 q^{36} -68398.1i q^{37} +9993.78 q^{38} +(-464296. - 359280. i) q^{39} -173495. q^{40} +335866. i q^{41} -270384. q^{42} +249039. q^{43} -2504.92i q^{44} -377410. i q^{45} -567656. i q^{46} +614022. i q^{47} -1.05685e6 q^{48} -117649. q^{49} -692324. i q^{50} -1.02768e6 q^{51} +(-93130.2 - 72065.6i) q^{52} -1.14162e6 q^{53} -881791. i q^{54} -19238.4 q^{55} -521218. q^{56} +69634.1i q^{57} -1.52754e6i q^{58} -2.21793e6i q^{59} -125787. i q^{60} +594732. q^{61} -729145. q^{62} -1.13382e6i q^{63} -2.28085e6 q^{64} +(-553483. + 715265. i) q^{65} -132830. q^{66} +2.83650e6i q^{67} -206136. q^{68} +3.95528e6 q^{69} +416536. i q^{70} +4.83746e6i q^{71} -5.02316e6i q^{72} -287868. i q^{73} -727514. q^{74} +4.82393e6 q^{75} +13967.5i q^{76} -57796.6 q^{77} +(-3.82147e6 + 4.93847e6i) q^{78} +7.55449e6 q^{79} +1.62812e6i q^{80} -1.08528e6 q^{81} +3.57242e6 q^{82} +1.78074e6i q^{83} -377893. i q^{84} +1.58318e6i q^{85} -2.64890e6i q^{86} +1.06435e7 q^{87} -256055. q^{88} -44704.8i q^{89} -4.01430e6 q^{90} +(-1.66279e6 + 2.14882e6i) q^{91} +793364. q^{92} -5.08050e6i q^{93} +6.53102e6 q^{94} +107274. q^{95} -3.17413e6i q^{96} -2.21623e6i q^{97} +1.25137e6i q^{98} -557006. i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 50 q - 3328 q^{4} + 40514 q^{9}+O(q^{10}) $ 50 * q - 3328 * q^4 + 40514 * q^9	$ 50 q - 3328 q^{4} + 40514 q^{9} + 5320 q^{10} + 8700 q^{12} + 17044 q^{13} + 10976 q^{14} + 228808 q^{16} + 33664 q^{17} + 70228 q^{22} - 75042 q^{23} - 664772 q^{25} + 78276 q^{26} - 661404 q^{27} + 135778 q^{29} + 994888 q^{30} + 372498 q^{35} - 3549604 q^{36} + 338468 q^{38} - 973080 q^{39} + 79316 q^{40} + 296352 q^{42} - 53618 q^{43} + 1400384 q^{48} - 5882450 q^{49} - 2182360 q^{51} - 6982340 q^{52} + 2841746 q^{53} + 6871356 q^{55} - 2107392 q^{56} + 1773716 q^{61} - 6969608 q^{62} - 9449120 q^{64} - 7901430 q^{65} - 11755548 q^{66} + 11829980 q^{68} + 3564460 q^{69} + 45595884 q^{74} - 7220964 q^{75} + 186592 q^{77} - 8093012 q^{78} - 21257822 q^{79} + 53034530 q^{81} + 10907568 q^{82} + 14135000 q^{87} - 51594780 q^{88} - 61226356 q^{90} - 8096858 q^{91} - 11200212 q^{92} + 80667028 q^{94} + 30430066 q^{95}+O(q^{100}) $ 50 * q - 3328 * q^4 + 40514 * q^9 + 5320 * q^10 + 8700 * q^12 + 17044 * q^13 + 10976 * q^14 + 228808 * q^16 + 33664 * q^17 + 70228 * q^22 - 75042 * q^23 - 664772 * q^25 + 78276 * q^26 - 661404 * q^27 + 135778 * q^29 + 994888 * q^30 + 372498 * q^35 - 3549604 * q^36 + 338468 * q^38 - 973080 * q^39 + 79316 * q^40 + 296352 * q^42 - 53618 * q^43 + 1400384 * q^48 - 5882450 * q^49 - 2182360 * q^51 - 6982340 * q^52 + 2841746 * q^53 + 6871356 * q^55 - 2107392 * q^56 + 1773716 * q^61 - 6969608 * q^62 - 9449120 * q^64 - 7901430 * q^65 - 11755548 * q^66 + 11829980 * q^68 + 3564460 * q^69 + 45595884 * q^74 - 7220964 * q^75 + 186592 * q^77 - 8093012 * q^78 - 21257822 * q^79 + 53034530 * q^81 + 10907568 * q^82 + 14135000 * q^87 - 51594780 * q^88 - 61226356 * q^90 - 8096858 * q^91 - 11200212 * q^92 + 80667028 * q^94 + 30430066 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/91\mathbb{Z}\right)^\times$.

$n$	$15$	$66$
$\chi(n)$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		−	10.6365i		−	0.940139i	−0.882629	−	0.470070i	$-0.844229\pi$
$2$		−	10.6365i		−	0.940139i	0.882629	−	0.470070i	$-0.155771\pi$
$3$	74.1121			1.58477			0.792383	−	0.610024i	$-0.208840\pi$
$3$	74.1121			1.58477			0.792383	+	0.610024i	$0.208840\pi$
$4$	14.8657			0.116138
$5$		−	114.172i		−	0.408476i	−0.978921	−	0.204238i	$-0.934528\pi$
$5$		−	114.172i		−	0.408476i	0.978921	−	0.204238i	$-0.0654716\pi$
$6$		−	788.291i		−	1.48990i
$7$		−	343.000i		−	0.377964i
$7$		−	343.000i		−	0.377964i
$8$		−	1519.59i		−	1.04933i
$9$	3305.61			1.51148
$10$	−1214.39			−0.384024
$11$		−	168.503i		−	0.0381710i	−0.999818	−	0.0190855i	$-0.993925\pi$
$11$		−	168.503i		−	0.0381710i	0.999818	−	0.0190855i	$-0.00607548\pi$
$12$	1101.73			0.184052
$13$	−6264.78	−	4847.79i	−0.790868	−	0.611986i
$13$	−6264.78	−	4847.79i	−0.790868	−	0.611986i
$14$	−3648.31			−0.355339
$15$		−	8461.56i		−	0.647338i
$16$	−14260.2			−0.870374
$17$	−13866.5			−0.684537			−0.342269	−	0.939602i	$-0.611195\pi$
$17$	−13866.5			−0.684537			−0.342269	+	0.939602i	$0.611195\pi$
$18$		−	35160.0i		−	1.42100i
$19$			939.578i			0.0314264i	0.999877	+	0.0157132i	$0.00500188\pi$
$19$			939.578i			0.0314264i	−0.999877	+	0.0157132i	$0.994998\pi$
$20$		−	1697.25i		−	0.0474396i
$21$		−	25420.5i		−	0.598985i
$22$	−1792.28			−0.0358861
$23$	53368.8			0.914619			0.457310	−	0.889308i	$-0.348813\pi$
$23$	53368.8			0.914619			0.457310	+	0.889308i	$0.348813\pi$
$24$		−	112620.i		−	1.66293i
$25$	65089.7			0.833148
$26$	−51563.3	+	66635.1i	−0.575352	+	0.743526i
$27$	82902.6			0.810579
$28$		−	5098.93i		−	0.0438961i
$29$	143613.			1.09346			0.546729	−	0.837310i	$-0.315873\pi$
$29$	143613.			1.09346			0.546729	+	0.837310i	$0.315873\pi$
$30$	−90001.1			−0.608588
$31$		−	68551.5i		−	0.413286i	−0.978416	−	0.206643i	$-0.933746\pi$
$31$		−	68551.5i		−	0.413286i	0.978416	−	0.206643i	$-0.0662539\pi$
$32$		−	42828.8i		−	0.231053i
$33$		−	12488.1i		−	0.0604921i
$34$			147491.i			0.643560i
$35$	−39161.1			−0.154389
$36$	49140.1			0.175541
$37$		−	68398.1i		−	0.221992i	−0.993821	−	0.110996i	$-0.964596\pi$
$37$		−	68398.1i		−	0.221992i	0.993821	−	0.110996i	$-0.0354041\pi$
$38$	9993.78			0.0295452
$39$	−464296.	−	359280.i	−1.25334	−	0.969855i
$40$	−173495.			−0.428624
$41$			335866.i			0.761066i	0.924767	+	0.380533i	$0.124259\pi$
$41$			335866.i			0.761066i	−0.924767	+	0.380533i	$0.875741\pi$
$42$	−270384.			−0.563129
$43$	249039.			0.477671			0.238835	−	0.971060i	$-0.423234\pi$
$43$	249039.			0.477671			0.238835	+	0.971060i	$0.423234\pi$
$44$		−	2504.92i		−	0.00443311i
$45$		−	377410.i		−	0.617403i
$46$		−	567656.i		−	0.859869i
$47$			614022.i			0.862663i	0.902193	+	0.431332i	$0.141956\pi$
$47$			614022.i			0.862663i	−0.902193	+	0.431332i	$0.858044\pi$
$48$	−1.05685e6			−1.37934
$49$	−117649.			−0.142857
$50$		−	692324.i		−	0.783275i
$51$	−1.02768e6			−1.08483
$52$	−93130.2	−	72065.6i	−0.0918500	−	0.0710750i
$53$	−1.14162e6			−1.05331			−0.526655	−	0.850079i	$-0.676554\pi$
$53$	−1.14162e6			−1.05331			−0.526655	+	0.850079i	$0.676554\pi$
$54$		−	881791.i		−	0.762057i
$55$	−19238.4			−0.0155919
$56$	−521218.			−0.396608
$57$			69634.1i			0.0498035i
$58$		−	1.52754e6i		−	1.02800i
$59$		−	2.21793e6i		−	1.40594i	−0.711220	−	0.702969i	$-0.751857\pi$
$59$		−	2.21793e6i		−	1.40594i	0.711220	−	0.702969i	$-0.248143\pi$
$60$		−	125787.i		−	0.0751806i
$61$	594732.			0.335480			0.167740	−	0.985831i	$-0.446353\pi$
$61$	594732.			0.335480			0.167740	+	0.985831i	$0.446353\pi$
$62$	−729145.			−0.388547
$63$		−	1.13382e6i		−	0.571286i
$64$	−2.28085e6			−1.08760
$65$	−553483.	+	715265.i	−0.249982	+	0.323050i
$66$	−132830.			−0.0568710
$67$			2.83650e6i			1.15218i	0.817386	+	0.576091i	$0.195423\pi$
$67$			2.83650e6i			1.15218i	−0.817386	+	0.576091i	$0.804577\pi$
$68$	−206136.			−0.0795009
$69$	3.95528e6			1.44946
$70$			416536.i			0.145147i
$71$			4.83746e6i			1.60403i	0.597301	+	0.802017i	$0.296240\pi$
$71$			4.83746e6i			1.60403i	−0.597301	+	0.802017i	$0.703760\pi$
$72$		−	5.02316e6i		−	1.58604i
$73$		−	287868.i		−	0.0866090i	−0.999062	−	0.0433045i	$-0.986211\pi$
$73$		−	287868.i		−	0.0866090i	0.999062	−	0.0433045i	$-0.0137886\pi$
$74$	−727514.			−0.208704
$75$	4.82393e6			1.32034
$76$			13967.5i			0.00364981i
$77$	−57796.6			−0.0144273
$78$	−3.82147e6	+	4.93847e6i	−0.911799	+	1.17831i
$79$	7.55449e6			1.72389			0.861947	−	0.506999i	$-0.169245\pi$
$79$	7.55449e6			1.72389			0.861947	+	0.506999i	$0.169245\pi$
$80$			1.62812e6i			0.355526i
$81$	−1.08528e6			−0.226905
$82$	3.57242e6			0.715508
$83$			1.78074e6i			0.341844i	0.985285	+	0.170922i	$0.0546745\pi$
$83$			1.78074e6i			0.341844i	−0.985285	+	0.170922i	$0.945325\pi$
$84$		−	377893.i		−	0.0695650i
$85$			1.58318e6i			0.279617i
$86$		−	2.64890e6i		−	0.449077i
$87$	1.06435e7			1.73287
$88$	−256055.			−0.0400538
$89$		−	44704.8i		−	0.00672186i	−0.999994	−	0.00336093i	$-0.998930\pi$
$89$		−	44704.8i		−	0.00672186i	0.999994	−	0.00336093i	$-0.00106982\pi$
$90$	−4.01430e6			−0.580445
$91$	−1.66279e6	+	2.14882e6i	−0.231309	+	0.298920i
$92$	793364.			0.106222
$93$		−	5.08050e6i		−	0.654962i
$94$	6.53102e6			0.811024
$95$	107274.			0.0128369
$96$		−	3.17413e6i		−	0.366164i
$97$		−	2.21623e6i		−	0.246555i	−0.992372	−	0.123277i	$-0.960659\pi$
$97$		−	2.21623e6i		−	0.246555i	0.992372	−	0.123277i	$-0.0393405\pi$
$98$			1.25137e6i			0.134306i
$99$		−	557006.i		−	0.0576948i
$100$	967602.			0.0967602
$101$	1.79703e7			1.73552			0.867759	−	0.496985i	$-0.165559\pi$
$101$	1.79703e7			1.73552			0.867759	+	0.496985i	$0.165559\pi$
$102$			1.09309e7i			1.01989i
$103$	−9.59752e6			−0.865424			−0.432712	−	0.901532i	$-0.642443\pi$
$103$	−9.59752e6			−0.865424			−0.432712	+	0.901532i	$0.642443\pi$
$104$	−7.36663e6	+	9.51987e6i	−0.642173	+	0.829878i
$105$	−2.90232e6			−0.244671
$106$			1.21428e7i			0.990258i
$107$	−4.58926e6			−0.362159			−0.181080	−	0.983468i	$-0.557959\pi$
$107$	−4.58926e6			−0.362159			−0.181080	+	0.983468i	$0.557959\pi$
$108$	1.23240e6			0.0941391
$109$			6.60089e6i			0.488213i	0.969748	+	0.244107i	$0.0784947\pi$
$109$			6.60089e6i			0.488213i	−0.969748	+	0.244107i	$0.921505\pi$
$110$			204629.i			0.0146586i
$111$		−	5.06913e6i		−	0.351806i
$112$			4.89125e6i			0.328970i
$113$	−6.27265e6			−0.408956			−0.204478	−	0.978871i	$-0.565550\pi$
$113$	−6.27265e6			−0.408956			−0.204478	+	0.978871i	$0.565550\pi$
$114$	740661.			0.0468223
$115$		−	6.09325e6i		−	0.373600i
$116$	2.13491e6			0.126992
$117$	−2.07089e7	−	1.60249e7i	−1.19538	−	0.925006i
$118$	−2.35910e7			−1.32178
$119$			4.75622e6i			0.258731i
$120$	−1.28581e7			−0.679268
$121$	1.94588e7			0.998543
$122$		−	6.32584e6i		−	0.315398i
$123$			2.48917e7i			1.20611i
$124$		−	1.01906e6i		−	0.0479983i
$125$		−	1.63512e7i		−	0.748796i
$126$	−1.20599e7			−0.537089
$127$	2.21045e7			0.957562			0.478781	−	0.877934i	$-0.341079\pi$
$127$	2.21045e7			0.957562			0.478781	+	0.877934i	$0.341079\pi$
$128$			1.87781e7i			0.791439i
$129$	1.84569e7			0.756996
$130$	7.60789e6	+	5.88711e6i	0.303712	+	0.235017i
$131$	−1.13398e6			−0.0440712			−0.0220356	−	0.999757i	$-0.507015\pi$
$131$	−1.13398e6			−0.0440712			−0.0220356	+	0.999757i	$0.507015\pi$
$132$		−	185645.i		−	0.00702544i
$133$	322275.			0.0118781
$134$	3.01703e7			1.08321
$135$		−	9.46519e6i		−	0.331102i
$136$			2.10714e7i			0.718302i
$137$			4.75044e7i			1.57838i	0.614149	+	0.789190i	$0.289499\pi$
$137$			4.75044e7i			1.57838i	−0.614149	+	0.789190i	$0.710501\pi$
$138$		−	4.20702e7i		−	1.36269i
$139$	4.24329e6			0.134014			0.0670071	−	0.997752i	$-0.478655\pi$
$139$	4.24329e6			0.134014			0.0670071	+	0.997752i	$0.478655\pi$
$140$	−582157.			−0.0179305
$141$			4.55065e7i			1.36712i
$142$	5.14535e7			1.50802
$143$	−816868.	+	1.05564e6i	−0.0233602	+	0.0301883i
$144$	−4.71387e7			−1.31555
$145$		−	1.63967e7i		−	0.446651i
$146$	−3.06189e6			−0.0814245
$147$	−8.71922e6			−0.226395
$148$		−	1.01678e6i		−	0.0257818i
$149$			2.61339e7i			0.647221i	0.946190	+	0.323610i	$0.104897\pi$
$149$			2.61339e7i			0.647221i	−0.946190	+	0.323610i	$0.895103\pi$
$150$		−	5.13096e7i		−	1.24131i
$151$		−	5.05842e7i		−	1.19563i	−0.801635	−	0.597813i	$-0.796036\pi$
$151$		−	5.05842e7i		−	1.19563i	0.801635	−	0.597813i	$-0.203964\pi$
$152$	1.42777e6			0.0329766
$153$	−4.58374e7			−1.03467
$154$			614752.i			0.0135637i
$155$	−7.82669e6			−0.168817
$156$	−6.90208e6	−	5.34094e6i	−0.145561	−	0.112637i
$157$	4.67559e7			0.964246			0.482123	−	0.876104i	$-0.339866\pi$
$157$	4.67559e7			0.964246			0.482123	+	0.876104i	$0.339866\pi$
$158$		−	8.03531e7i		−	1.62070i
$159$	−8.46080e7			−1.66925
$160$	−4.88987e6			−0.0943794
$161$		−	1.83055e7i		−	0.345694i
$162$			1.15435e7i			0.213322i
$163$		−	6.34544e7i		−	1.14764i	−0.818982	−	0.573820i	$-0.805461\pi$
$163$		−	6.34544e7i		−	1.14764i	0.818982	−	0.573820i	$-0.194539\pi$
$164$			4.99287e6i			0.0883888i
$165$	−1.42580e6			−0.0247096
$166$	1.89408e7			0.321381
$167$		−	1.67374e7i		−	0.278087i	−0.990286	−	0.139044i	$-0.955597\pi$
$167$		−	1.67374e7i		−	0.278087i	0.990286	−	0.139044i	$-0.0444028\pi$
$168$	−3.86286e7			−0.628530
$169$	1.57464e7	+	6.07406e7i	0.250945	+	0.968001i
$170$	1.68394e7			0.262879
$171$			3.10588e6i			0.0475005i
$172$	3.70214e6			0.0554758
$173$	1.53038e7			0.224718			0.112359	−	0.993668i	$-0.464159\pi$
$173$	1.53038e7			0.224718			0.112359	+	0.993668i	$0.464159\pi$
$174$		−	1.13209e8i		−	1.62914i
$175$		−	2.23258e7i		−	0.314900i
$176$			2.40289e6i			0.0332231i
$177$		−	1.64376e8i		−	2.22808i
$178$	−475501.			−0.00631949
$179$	−6.53950e7			−0.852234			−0.426117	−	0.904668i	$-0.640119\pi$
$179$	−6.53950e7			−0.852234			−0.426117	+	0.904668i	$0.640119\pi$
$180$		−	5.61045e6i		−	0.0717041i
$181$	1.20588e8			1.51157			0.755786	−	0.654819i	$-0.227255\pi$
$181$	1.20588e8			1.51157			0.755786	+	0.654819i	$0.227255\pi$
$182$	2.28558e7	+	1.76862e7i	0.281027	+	0.217463i
$183$	4.40768e7			0.531657
$184$		−	8.10985e7i		−	0.959733i
$185$	−7.80918e6			−0.0906785
$186$	−5.40385e7			−0.615755
$187$			2.33656e6i			0.0261295i
$188$			9.12785e6i			0.100188i
$189$		−	2.84356e7i		−	0.306370i
$190$		−	1.14101e6i		−	0.0120685i
$191$	−1.31600e8			−1.36659			−0.683295	−	0.730143i	$-0.739454\pi$
$191$	−1.31600e8			−1.36659			−0.683295	+	0.730143i	$0.739454\pi$
$192$	−1.69039e8			−1.72358
$193$			1.78021e8i			1.78246i	0.453547	+	0.891232i	$0.350158\pi$
$193$			1.78021e8i			1.78246i	−0.453547	+	0.891232i	$0.649842\pi$
$194$	−2.35728e7			−0.231796
$195$	−4.10199e7	+	5.30098e7i	−0.396162	+	0.511959i
$196$	−1.74893e6			−0.0165912
$197$			1.34869e8i			1.25685i	0.777872	+	0.628423i	$0.216299\pi$
$197$			1.34869e8i			1.25685i	−0.777872	+	0.628423i	$0.783701\pi$
$198$	−5.92458e6			−0.0542412
$199$	−1.24159e8			−1.11684			−0.558422	−	0.829557i	$-0.688593\pi$
$199$	−1.24159e8			−1.11684			−0.558422	+	0.829557i	$0.688593\pi$
$200$		−	9.89093e7i		−	0.874243i
$201$			2.10219e8i			1.82594i
$202$		−	1.91140e8i		−	1.63163i
$203$		−	4.92594e7i		−	0.413288i
$204$	−1.52772e7			−0.125990
$205$	3.83466e7			0.310877
$206$			1.02084e8i			0.813619i
$207$	1.76417e8			1.38243
$208$	8.93371e7	+	6.91304e7i	0.688351	+	0.532657i
$209$	158322.			0.00119958
$210$			3.08704e7i			0.230025i
$211$	1.24541e8			0.912694			0.456347	−	0.889802i	$-0.349158\pi$
$211$	1.24541e8			0.912694			0.456347	+	0.889802i	$0.349158\pi$
$212$	−1.69710e7			−0.122329
$213$			3.58515e8i			2.54202i
$214$			4.88135e7i			0.340480i
$215$		−	2.84334e7i		−	0.195117i
$216$		−	1.25978e8i		−	0.850561i
$217$	−2.35132e7			−0.156207
$218$	7.02101e7			0.458988
$219$		−	2.13345e7i		−	0.137255i
$220$	−285992.			−0.00181082
$221$	8.68709e7	+	6.72220e7i	0.541379	+	0.418927i
$222$	−5.39176e7			−0.330746
$223$			2.18755e8i			1.32097i	0.750841	+	0.660483i	$0.229648\pi$
$223$			2.18755e8i			1.32097i	−0.750841	+	0.660483i	$0.770352\pi$
$224$	−1.46903e7			−0.0873297
$225$	2.15161e8			1.25929
$226$			6.67188e7i			0.384475i
$227$			2.86203e8i			1.62399i	0.583662	+	0.811997i	$0.301619\pi$
$227$			2.86203e8i			1.62399i	−0.583662	+	0.811997i	$0.698381\pi$
$228$			1.03516e6i			0.00578409i
$229$			1.07096e7i			0.0589319i	0.999566	+	0.0294659i	$0.00938065\pi$
$229$			1.07096e7i			0.0589319i	−0.999566	+	0.0294659i	$0.990619\pi$
$230$	−6.48106e7			−0.351236
$231$	−4.28343e6			−0.0228639
$232$		−	2.18233e8i		−	1.14739i
$233$	5.09436e7			0.263842			0.131921	−	0.991260i	$-0.457885\pi$
$233$	5.09436e7			0.263842			0.131921	+	0.991260i	$0.457885\pi$
$234$	−1.70448e8	+	2.20270e8i	−0.869635	+	1.12383i
$235$	7.01044e7			0.352377
$236$		−	3.29711e7i		−	0.163283i
$237$	5.59880e8			2.73197
$238$	5.05894e7			0.243243
$239$		−	2.13201e8i		−	1.01018i	−0.863068	−	0.505088i	$-0.831460\pi$
$239$		−	2.13201e8i		−	1.01018i	0.863068	−	0.505088i	$-0.168540\pi$
$240$			1.20664e8i			0.563426i
$241$		−	2.59990e8i		−	1.19646i	−0.801325	−	0.598229i	$-0.795871\pi$
$241$		−	2.59990e8i		−	1.19646i	0.801325	−	0.598229i	$-0.204129\pi$
$242$		−	2.06973e8i		−	0.938769i
$243$	−2.61740e8			−1.17017
$244$	8.84109e6			0.0389620
$245$			1.34323e7i			0.0583537i
$246$	2.64760e8			1.13391
$247$	4.55487e6	−	5.88625e6i	0.0192326	−	0.0248542i
$248$	−1.04170e8			−0.433672
$249$			1.31975e8i			0.541742i
$250$	−1.73919e8			−0.703973
$251$	−9.99560e7			−0.398980			−0.199490	−	0.979900i	$-0.563929\pi$
$251$	−9.99560e7			−0.398980			−0.199490	+	0.979900i	$0.563929\pi$
$252$		−	1.68551e7i		−	0.0663481i
$253$		−	8.99283e6i		−	0.0349120i
$254$		−	2.35113e8i		−	0.900242i
$255$			1.17333e8i			0.443127i
$256$	−9.22164e7			−0.343533
$257$	−4.27055e8			−1.56934			−0.784672	−	0.619912i	$-0.787168\pi$
$257$	−4.27055e8			−1.56934			−0.784672	+	0.619912i	$0.787168\pi$
$258$		−	1.96316e8i		−	0.711682i
$259$	−2.34605e7			−0.0839052
$260$	−8.22791e6	+	1.06329e7i	−0.0290324	+	0.0375185i
$261$	4.74730e8			1.65274
$262$			1.20615e7i			0.0414331i
$263$	9.72487e7			0.329639			0.164820	−	0.986324i	$-0.447296\pi$
$263$	9.72487e7			0.329639			0.164820	+	0.986324i	$0.447296\pi$
$264$	−1.89768e7			−0.0634759
$265$			1.30342e8i			0.430252i
$266$		−	3.42787e6i		−	0.0111670i
$267$		−	3.31317e6i		−	0.0106526i
$268$			4.21665e7i			0.133812i
$269$	−4.32940e8			−1.35611			−0.678054	−	0.735012i	$-0.737176\pi$
$269$	−4.32940e8			−1.35611			−0.678054	+	0.735012i	$0.737176\pi$
$270$	−1.00676e8			−0.311282
$271$			4.05794e7i			0.123855i	0.998081	+	0.0619274i	$0.0197247\pi$
$271$			4.05794e7i			0.123855i	−0.998081	+	0.0619274i	$0.980275\pi$
$272$	1.97740e8			0.595803
$273$	−1.23233e8	+	1.59254e8i	−0.366571	+	0.473718i
$274$	5.05279e8			1.48390
$275$		−	1.09678e7i		−	0.0318021i
$276$	5.87979e7			0.168337
$277$	2.03959e8			0.576585			0.288293	−	0.957542i	$-0.406912\pi$
$277$	2.03959e8			0.576585			0.288293	+	0.957542i	$0.406912\pi$
$278$		−	4.51336e7i		−	0.125992i
$279$		−	2.26604e8i		−	0.624675i
$280$			5.95087e7i			0.162005i
$281$		−	4.93316e8i		−	1.32633i	−0.748472	−	0.663167i	$-0.769212\pi$
$281$		−	4.93316e8i		−	1.32633i	0.748472	−	0.663167i	$-0.230788\pi$
$282$	4.84028e8			1.28528
$283$	−8.20090e7			−0.215084			−0.107542	−	0.994201i	$-0.534298\pi$
$283$	−8.20090e7			−0.215084			−0.107542	+	0.994201i	$0.534298\pi$
$284$			7.19122e7i			0.186290i
$285$	7.95030e6			0.0203435
$286$	1.12282e7	+	8.68859e6i	0.0283812	+	0.0219618i
$287$	1.15202e8			0.287656
$288$		−	1.41575e8i		−	0.349232i
$289$	−2.18058e8			−0.531409
$290$	−1.74403e8			−0.419914
$291$		−	1.64249e8i		−	0.390732i
$292$		−	4.27935e6i		−	0.0100586i
$293$			3.80546e8i			0.883833i	0.897056	+	0.441916i	$0.145701\pi$
$293$			3.80546e8i			0.883833i	−0.897056	+	0.441916i	$0.854299\pi$
$294$			9.27417e7i			0.212843i
$295$	−2.53227e8			−0.574292
$296$	−1.03937e8			−0.232942
$297$		−	1.39694e7i		−	0.0309406i
$298$	2.77972e8			0.608478
$299$	−3.34344e8	−	2.58721e8i	−0.723343	−	0.559735i
$300$	7.17111e7			0.153342
$301$		−	8.54205e7i		−	0.180543i
$302$	−5.38037e8			−1.12406
$303$	1.33181e9			2.75039
$304$		−	1.33986e7i		−	0.0273527i
$305$		−	6.79019e7i		−	0.137035i
$306$			4.87548e8i			0.972729i
$307$		−	4.93863e8i		−	0.974141i	−0.873363	−	0.487071i	$-0.838065\pi$
$307$		−	4.93863e8i		−	0.974141i	0.873363	−	0.487071i	$-0.161935\pi$
$308$	−859186.			−0.00167556
$309$	−7.11293e8			−1.37149
$310$			8.32483e7i			0.158712i
$311$	−3.75398e8			−0.707669			−0.353834	−	0.935308i	$-0.615122\pi$
$311$	−3.75398e8			−0.707669			−0.353834	+	0.935308i	$0.615122\pi$
$312$	−5.45956e8	+	7.05538e8i	−1.01769	+	1.31516i
$313$	1.80861e8			0.333380			0.166690	−	0.986009i	$-0.446692\pi$
$313$	1.80861e8			0.333380			0.166690	+	0.986009i	$0.446692\pi$
$314$		−	4.97317e8i		−	0.906525i
$315$	−1.29451e8			−0.233357
$316$	1.12303e8			0.200210
$317$		−	5.53386e6i		−	0.00975709i	−0.999988	−	0.00487855i	$-0.998447\pi$
$317$		−	5.53386e6i		−	0.00975709i	0.999988	−	0.00487855i	$-0.00155290\pi$
$318$			8.99929e8i			1.56933i
$319$		−	2.41993e7i		−	0.0417384i
$320$			2.60410e8i			0.444256i
$321$	−3.40120e8			−0.573937
$322$	−1.94706e8			−0.325000
$323$		−	1.30287e7i		−	0.0215126i
$324$	−1.61334e7			−0.0263523
$325$	−4.07772e8	−	3.15541e8i	−0.658910	−	0.509875i
$326$	−6.74931e8			−1.07894
$327$			4.89206e8i			0.773704i
$328$	5.10377e8			0.798606
$329$	2.10610e8			0.326056
$330$			1.51655e7i			0.0232304i
$331$		−	8.24612e8i		−	1.24983i	−0.780692	−	0.624916i	$-0.785133\pi$
$331$		−	8.24612e8i		−	1.24983i	0.780692	−	0.624916i	$-0.214867\pi$
$332$			2.64719e7i			0.0397011i
$333$		−	2.26097e8i		−	0.335537i
$334$	−1.78027e8			−0.261441
$335$	3.23850e8			0.470638
$336$			3.62501e8i			0.521341i
$337$	−8.08632e8			−1.15092			−0.575461	−	0.817829i	$-0.695177\pi$
$337$	−8.08632e8			−1.15092			−0.575461	+	0.817829i	$0.695177\pi$
$338$	6.46066e8	−	1.67486e8i	0.910056	−	0.235924i
$339$	−4.64879e8			−0.648099
$340$			2.35350e7i			0.0324742i
$341$	−1.15511e7			−0.0157756
$342$	3.30356e7			0.0446571
$343$			4.03536e7i			0.0539949i
$344$		−	3.78437e8i		−	0.501232i
$345$		−	4.51584e8i		−	0.592068i
$346$		−	1.62778e8i		−	0.211266i
$347$	−1.04620e9			−1.34419			−0.672096	−	0.740464i	$-0.734606\pi$
$347$	−1.04620e9			−1.34419			−0.672096	+	0.740464i	$0.734606\pi$
$348$	1.58223e8			0.201253
$349$		−	3.82112e7i		−	0.0481173i	−0.999711	−	0.0240587i	$-0.992341\pi$
$349$		−	3.82112e7i		−	0.0481173i	0.999711	−	0.0240587i	$-0.00765885\pi$
$350$	−2.37467e8			−0.296050
$351$	−5.19367e8	−	4.01894e8i	−0.641061	−	0.496063i
$352$	−7.21679e6			−0.00881952
$353$		−	1.43647e9i		−	1.73814i	−0.494693	−	0.869068i	$-0.664719\pi$
$353$		−	1.43647e9i		−	1.73814i	0.494693	−	0.869068i	$-0.335281\pi$
$354$	−1.74838e9			−2.09471
$355$	5.52305e8			0.655209
$356$		−	664568.i		−	0.000780665i
$357$			3.52494e8i			0.410027i
$358$			6.95571e8i			0.801218i
$359$		−	1.64722e9i		−	1.87897i	−0.342587	−	0.939486i	$-0.611303\pi$
$359$		−	1.64722e9i		−	1.87897i	0.342587	−	0.939486i	$-0.388697\pi$
$360$	−5.73506e8			−0.647857
$361$	8.92989e8			0.999012
$362$		−	1.28263e9i		−	1.42109i
$363$	1.44213e9			1.58246
$364$	−2.47185e7	+	3.19437e7i	−0.0268638	+	0.0347160i
$365$	−3.28666e7			−0.0353777
$366$		−	4.68822e8i		−	0.499832i
$367$	−1.12072e8			−0.118349			−0.0591746	−	0.998248i	$-0.518847\pi$
$367$	−1.12072e8			−0.118349			−0.0591746	+	0.998248i	$0.518847\pi$
$368$	−7.61051e8			−0.796061
$369$			1.11024e9i			1.15034i
$370$			8.30620e7i			0.0852504i
$371$			3.91576e8i			0.398114i
$372$		−	7.55250e7i		−	0.0760660i
$373$	1.04570e9			1.04334			0.521672	−	0.853146i	$-0.325308\pi$
$373$	1.04570e9			1.04334			0.521672	+	0.853146i	$0.325308\pi$
$374$	2.48527e7			0.0245654
$375$		−	1.21182e9i		−	1.18667i
$376$	9.33059e8			0.905215
$377$	−8.99707e8	−	6.96207e8i	−0.864781	−	0.669182i
$378$	−3.02454e8			−0.288030
$379$		−	5.92649e8i		−	0.559191i	−0.960118	−	0.279595i	$-0.909800\pi$
$379$		−	5.92649e8i		−	0.559191i	0.960118	−	0.279595i	$-0.0902003\pi$
$380$	1.59470e6			0.00149086
$381$	1.63821e9			1.51751
$382$			1.39976e9i			1.28478i
$383$			6.33559e8i			0.576224i	0.957597	+	0.288112i	$0.0930275\pi$
$383$			6.33559e8i			0.576224i	−0.957597	+	0.288112i	$0.906972\pi$
$384$			1.39169e9i			1.25424i
$385$			6.59878e6i			0.00589320i
$386$	1.89351e9			1.67576
$387$	8.23228e8			0.721991
$388$		−	3.29458e7i		−	0.0286344i
$389$	5.46010e8			0.470302			0.235151	−	0.971959i	$-0.424442\pi$
$389$	5.46010e8			0.470302			0.235151	+	0.971959i	$0.424442\pi$
$390$	5.63837e8	+	4.36306e8i	0.481313	+	0.372448i
$391$	−7.40041e8			−0.626091
$392$			1.78778e8i			0.149904i
$393$	−8.40415e7			−0.0698425
$394$	1.43453e9			1.18161
$395$		−	8.62515e8i		−	0.704169i
$396$		−	8.28028e6i		−	0.00670057i
$397$			2.44912e9i			1.96446i	0.187682	+	0.982230i	$0.439903\pi$
$397$			2.44912e9i			1.96446i	−0.187682	+	0.982230i	$0.560097\pi$
$398$			1.32061e9i			1.04999i
$399$	2.38845e7			0.0188240
$400$	−9.28192e8			−0.725150
$401$		−	1.08739e9i		−	0.842135i	−0.907029	−	0.421068i	$-0.861655\pi$
$401$		−	1.08739e9i		−	0.842135i	0.907029	−	0.421068i	$-0.138345\pi$
$402$	2.23599e9			1.71664
$403$	−3.32323e8	+	4.29460e8i	−0.252926	+	0.326855i
$404$	2.67140e8			0.201560
$405$			1.23909e8i			0.0926850i
$406$	−5.23946e8			−0.388549
$407$	−1.15253e7			−0.00847368
$408$			1.56165e9i			1.13834i
$409$		−	1.56254e9i		−	1.12927i	−0.825340	−	0.564636i	$-0.809017\pi$
$409$		−	1.56254e9i		−	1.12927i	0.825340	−	0.564636i	$-0.190983\pi$
$410$		−	4.07872e8i		−	0.292268i
$411$			3.52065e9i			2.50136i
$412$	−1.42674e8			−0.100509
$413$	−7.60751e8			−0.531395
$414$		−	1.87645e9i		−	1.29968i
$415$	2.03311e8			0.139635
$416$	−2.07625e8	+	2.68313e8i	−0.141401	+	0.182732i
$417$	3.14479e8			0.212381
$418$		−	1.68399e6i		−	0.00112777i
$419$	−1.47452e8			−0.0979269			−0.0489635	−	0.998801i	$-0.515592\pi$
$419$	−1.47452e8			−0.0979269			−0.0489635	+	0.998801i	$0.515592\pi$
$420$	−4.31449e7			−0.0284156
$421$			2.22049e9i			1.45031i	0.688583	+	0.725157i	$0.258233\pi$
$421$			2.22049e9i			1.45031i	−0.688583	+	0.725157i	$0.741767\pi$
$422$		−	1.32468e9i		−	0.858060i
$423$			2.02972e9i			1.30390i
$424$			1.73479e9i			1.10526i
$425$	−9.02569e8			−0.570320
$426$	3.81333e9			2.38985
$427$		−	2.03993e8i		−	0.126800i
$428$	−6.82225e7			−0.0420605
$429$	−6.05398e7	+	7.82355e7i	−0.0370204	+	0.0478413i
$430$	−3.02431e8			−0.183437
$431$			7.23149e8i			0.435068i	0.976053	+	0.217534i	$0.0698013\pi$
$431$			7.23149e8i			0.435068i	−0.976053	+	0.217534i	$0.930199\pi$
$432$	−1.18221e9			−0.705506
$433$	−2.25714e9			−1.33614			−0.668068	−	0.744100i	$-0.732879\pi$
$433$	−2.25714e9			−1.33614			−0.668068	+	0.744100i	$0.732879\pi$
$434$			2.50097e8i			0.146857i
$435$		−	1.21519e9i		−	0.707837i
$436$			9.81267e7i			0.0567002i
$437$			5.01442e7i			0.0287432i
$438$	−2.26924e8			−0.129039
$439$	−2.06827e9			−1.16676			−0.583379	−	0.812200i	$-0.698270\pi$
$439$	−2.06827e9			−1.16676			−0.583379	+	0.812200i	$0.698270\pi$
$440$			2.92344e7i			0.0163610i
$441$	−3.88902e8			−0.215926
$442$	7.15005e8	−	9.23999e8i	0.393850	−	0.508971i
$443$	2.90341e9			1.58670			0.793352	−	0.608763i	$-0.208334\pi$
$443$	2.90341e9			1.58670			0.793352	+	0.608763i	$0.208334\pi$
$444$		−	7.53561e7i		−	0.0408581i
$445$	−5.10406e6			−0.00274572
$446$	2.32678e9			1.24189
$447$			1.93684e9i			1.02569i
$448$			7.82333e8i			0.411072i
$449$			4.14567e8i			0.216139i	0.994143	+	0.108069i	$0.0344669\pi$
$449$			4.14567e8i			0.216139i	−0.994143	+	0.108069i	$0.965533\pi$
$450$		−	2.28855e9i		−	1.18391i
$451$	5.65945e7			0.0290507
$452$	−9.32472e7			−0.0474954
$453$		−	3.74890e9i		−	1.89479i
$454$	3.04419e9			1.52678
$455$	2.45336e8	+	1.89845e8i	0.122102	+	0.0944841i
$456$	1.05815e8			0.0522601
$457$			3.42388e9i			1.67807i	0.544074	+	0.839037i	$0.316881\pi$
$457$			3.42388e9i			1.67807i	−0.544074	+	0.839037i	$0.683119\pi$
$458$	1.13913e8			0.0554041
$459$	−1.14957e9			−0.554871
$460$		−	9.05803e7i		−	0.0433892i
$461$		−	9.83763e8i		−	0.467667i	−0.972277	−	0.233834i	$-0.924873\pi$
$461$		−	9.83763e8i		−	0.467667i	0.972277	−	0.233834i	$-0.0751271\pi$
$462$			4.55606e7i			0.0214952i
$463$			2.67163e9i			1.25096i	0.780241	+	0.625479i	$0.215096\pi$
$463$			2.67163e9i			1.25096i	−0.780241	+	0.625479i	$0.784904\pi$
$464$	−2.04796e9			−0.951717
$465$	−5.80053e8			−0.267536
$466$		−	5.41859e8i		−	0.248048i
$467$	−3.87627e9			−1.76118			−0.880592	−	0.473876i	$-0.842855\pi$
$467$	−3.87627e9			−1.76118			−0.880592	+	0.473876i	$0.842855\pi$
$468$	−3.07852e8	−	2.38221e8i	−0.138830	−	0.107428i
$469$	9.72920e8			0.435484
$470$		−	7.45662e8i		−	0.331283i
$471$	3.46518e9			1.52810
$472$	−3.37034e9			−1.47529
$473$		−	4.19640e7i		−	0.0182332i
$474$		−	5.95514e9i		−	2.56843i
$475$			6.11568e7i			0.0261829i
$476$			7.07045e7i			0.0300485i
$477$	−3.77375e9			−1.59206
$478$	−2.26771e9			−0.949707
$479$		−	2.79394e9i		−	1.16156i	−0.814059	−	0.580782i	$-0.802747\pi$
$479$		−	2.79394e9i		−	1.16156i	0.814059	−	0.580782i	$-0.197253\pi$
$480$	−3.62399e8			−0.149569
$481$	−3.31579e8	+	4.28499e8i	−0.135856	+	0.175567i
$482$	−2.76538e9			−1.12484
$483$		−	1.35666e9i		−	0.547843i
$484$	2.89268e8			0.115969
$485$	−2.53032e8			−0.100712
$486$			2.78399e9i			1.10012i
$487$			3.38002e9i			1.32607i	0.748586	+	0.663037i	$0.230733\pi$
$487$			3.38002e9i			1.32607i	−0.748586	+	0.663037i	$0.769267\pi$
$488$		−	9.03746e8i		−	0.352028i
$489$		−	4.70274e9i		−	1.81874i
$490$	1.42872e8			0.0548606
$491$	−1.71924e9			−0.655467			−0.327733	−	0.944770i	$-0.606285\pi$
$491$	−1.71924e9			−0.655467			−0.327733	+	0.944770i	$0.606285\pi$
$492$			3.70033e8i			0.140075i
$493$	−1.99142e9			−0.748513
$494$	−6.26089e7	−	4.84477e7i	−0.0233664	−	0.0180813i
$495$	−6.35947e7			−0.0235669
$496$			9.77558e8i			0.359713i
$497$	1.65925e9			0.606268
$498$	1.40374e9			0.509313
$499$			5.68127e8i			0.204688i	0.994749	+	0.102344i	$0.0326343\pi$
$499$			5.68127e8i			0.204688i	−0.994749	+	0.102344i	$0.967366\pi$
$500$		−	2.43071e8i		−	0.0869638i
$501$		−	1.24045e9i		−	0.440703i
$502$			1.06318e9i			0.375097i
$503$	9.68057e8			0.339167			0.169583	−	0.985516i	$-0.445758\pi$
$503$	9.68057e8			0.339167			0.169583	+	0.985516i	$0.445758\pi$
$504$	−1.72294e9			−0.599465
$505$		−	2.05171e9i		−	0.708917i
$506$	−9.56518e7			−0.0328221
$507$	1.16700e9	+	4.50162e9i	0.397689	+	1.53406i
$508$	3.28598e8			0.111209
$509$		−	4.17738e9i		−	1.40408i	−0.712138	−	0.702040i	$-0.752273\pi$
$509$		−	4.17738e9i		−	1.40408i	0.712138	−	0.702040i	$-0.247727\pi$
$510$	1.24800e9			0.416601
$511$	−9.87386e7			−0.0327351
$512$			3.38446e9i			1.11441i
$513$			7.78935e7i			0.0254736i
$514$			4.54235e9i			1.47540i
$515$			1.09577e9i			0.353505i
$516$	2.74374e8			0.0879161
$517$	1.03465e8			0.0329288
$518$			2.49537e8i			0.0788826i
$519$	1.13420e9			0.356126
$520$	1.08691e9	+	8.41065e8i	0.338985	+	0.262312i
$521$	−1.35766e9			−0.420590			−0.210295	−	0.977638i	$-0.567442\pi$
$521$	−1.35766e9			−0.420590			−0.210295	+	0.977638i	$0.567442\pi$
$522$		−	5.04945e9i		−	1.55381i
$523$	1.40191e9			0.428513			0.214256	−	0.976777i	$-0.431267\pi$
$523$	1.40191e9			0.428513			0.214256	+	0.976777i	$0.431267\pi$
$524$	−1.68573e7			−0.00511835
$525$		−	1.65461e9i		−	0.499043i
$526$		−	1.03438e9i		−	0.309907i
$527$			9.50572e8i			0.282910i
$528$			1.78083e8i			0.0526508i
$529$	−5.56592e8			−0.163472
$530$	1.38637e9			0.404496
$531$		−	7.33162e9i		−	2.12505i
$532$	4.79084e6			0.00137950
$533$	1.62821e9	−	2.10413e9i	0.465762	−	0.601903i
$534$	−3.52404e7			−0.0100149
$535$			5.23967e8i			0.147933i
$536$	4.31030e9			1.20901
$537$	−4.84656e9			−1.35059
$538$			4.60495e9i			1.27493i
$539$			1.98242e7i			0.00545301i
$540$		−	1.40707e8i		−	0.0384535i
$541$		−	5.51413e9i		−	1.49722i	−0.663009	−	0.748611i	$-0.730721\pi$
$541$		−	5.51413e9i		−	1.49722i	0.663009	−	0.748611i	$-0.269279\pi$
$542$	4.31621e8			0.116441
$543$	8.93703e9			2.39549
$544$			5.93887e8i			0.158164i
$545$	7.53639e8			0.199423
$546$	1.69390e9	+	1.31076e9i	0.445361	+	0.344628i
$547$	5.90154e9			1.54174			0.770868	−	0.636995i	$-0.219823\pi$
$547$	5.90154e9			1.54174			0.770868	+	0.636995i	$0.219823\pi$
$548$			7.06185e8i			0.183310i
$549$	1.96595e9			0.507072
$550$	−1.16659e8			−0.0298984
$551$			1.34936e8i			0.0343635i
$552$		−	6.01038e9i		−	1.52095i
$553$		−	2.59119e9i		−	0.651571i
$554$		−	2.16940e9i		−	0.542070i
$555$	−5.78755e8			−0.143704
$556$	6.30794e7			0.0155642
$557$		−	2.67890e9i		−	0.656845i	−0.944531	−	0.328423i	$-0.893483\pi$
$557$		−	2.67890e9i		−	0.656845i	0.944531	−	0.328423i	$-0.106517\pi$
$558$	−2.41027e9			−0.587281
$559$	−1.56018e9	−	1.20729e9i	−0.377775	−	0.292328i
$560$	5.58446e8			0.134376
$561$			1.73167e8i			0.0414091i
$562$	−5.24713e9			−1.24694
$563$	−5.49367e9			−1.29743			−0.648714	−	0.761032i	$-0.724693\pi$
$563$	−5.49367e9			−1.29743			−0.648714	+	0.761032i	$0.724693\pi$
$564$			6.76485e8i			0.158775i
$565$			7.16163e8i			0.167048i
$566$			8.72285e8i			0.202209i
$567$			3.72250e8i			0.0857619i
$568$	7.35094e9			1.68315
$569$	−7.37667e9			−1.67868			−0.839339	−	0.543608i	$-0.817058\pi$
$569$	−7.37667e9			−1.67868			−0.839339	+	0.543608i	$0.817058\pi$
$570$		−	8.45630e7i		−	0.0191258i
$571$	6.64618e8			0.149398			0.0746991	−	0.997206i	$-0.476200\pi$
$571$	6.64618e8			0.149398			0.0746991	+	0.997206i	$0.476200\pi$
$572$	−1.21433e7	+	1.56928e7i	−0.00271300	+	0.00350601i
$573$	−9.75314e9			−2.16572
$574$		−	1.22534e9i		−	0.270437i
$575$	3.47376e9			0.762013
$576$	−7.53961e9			−1.64388
$577$		−	7.99768e9i		−	1.73320i	−0.499004	−	0.866600i	$-0.666301\pi$
$577$		−	7.99768e9i		−	1.73320i	0.499004	−	0.866600i	$-0.333699\pi$
$578$			2.31936e9i			0.499598i
$579$			1.31935e10i			2.82479i
$580$		−	2.43748e8i		−	0.0518732i
$581$	6.10794e8			0.129205
$582$	−1.74703e9			−0.367342
$583$			1.92367e8i			0.0402059i
$584$	−4.37440e8			−0.0908810
$585$	−1.82960e9	+	2.36439e9i	−0.377843	+	0.488285i
$586$	4.04766e9			0.830926
$587$			8.32357e8i			0.169854i	0.996387	+	0.0849271i	$0.0270657\pi$
$587$			8.32357e8i			0.169854i	−0.996387	+	0.0849271i	$0.972934\pi$
$588$	−1.29617e8			−0.0262931
$589$	6.44094e7			0.0129881
$590$			2.69344e9i			0.539914i
$591$			9.99547e9i			1.99181i
$592$			9.75371e8i			0.193216i
$593$			4.31448e8i			0.0849643i	0.999097	+	0.0424822i	$0.0135266\pi$
$593$			4.31448e8i			0.0849643i	−0.999097	+	0.0424822i	$0.986473\pi$
$594$	−1.48585e8			−0.0290885
$595$	5.43030e8			0.105685
$596$			3.88498e8i			0.0751670i
$597$	−9.20170e9			−1.76994
$598$	−2.75187e9	+	3.55624e9i	−0.526228	+	0.680044i
$599$	−3.24380e9			−0.616680			−0.308340	−	0.951276i	$-0.599773\pi$
$599$	−3.24380e9			−0.616680			−0.308340	+	0.951276i	$0.599773\pi$
$600$		−	7.33038e9i		−	1.38547i
$601$	−5.48153e8			−0.103001			−0.0515005	−	0.998673i	$-0.516400\pi$
$601$	−5.48153e8			−0.103001			−0.0515005	+	0.998673i	$0.516400\pi$
$602$	−9.08572e8			−0.169735
$603$			9.37636e9i			1.74150i
$604$		−	7.51969e8i		−	0.138858i
$605$		−	2.22166e9i		−	0.407880i
$606$		−	1.41658e10i		−	2.58575i
$607$	4.50570e9			0.817715			0.408858	−	0.912598i	$-0.365927\pi$
$607$	4.50570e9			0.817715			0.408858	+	0.912598i	$0.365927\pi$
$608$	4.02410e7			0.00726116
$609$		−	3.65072e9i		−	0.654965i
$610$	−7.22236e8			−0.128832
$611$	2.97665e9	−	3.84671e9i	0.527938	−	0.682253i
$612$	−6.81404e8			−0.120164
$613$		−	6.76566e9i		−	1.18631i	−0.805088	−	0.593156i	$-0.797882\pi$
$613$		−	6.76566e9i		−	1.18631i	0.805088	−	0.593156i	$-0.202118\pi$
$614$	−5.25295e9			−0.915828
$615$	2.84195e9			0.492667
$616$			8.78269e7i			0.0151389i
$617$			5.30692e9i			0.909587i	0.890597	+	0.454794i	$0.150287\pi$
$617$			5.30692e9i			0.909587i	−0.890597	+	0.454794i	$0.849713\pi$
$618$			7.56564e9i			1.28940i
$619$			1.00376e9i			0.170103i	0.996377	+	0.0850514i	$0.0271055\pi$
$619$			1.00376e9i			0.170103i	−0.996377	+	0.0850514i	$0.972895\pi$
$620$	−1.16349e8			−0.0196061
$621$	4.42442e9			0.741371
$622$			3.99290e9i			0.665307i
$623$	−1.53338e7			−0.00254063
$624$	6.62096e9	+	5.12340e9i	1.09087	+	0.844136i
$625$	3.21828e9			0.527283
$626$		−	1.92372e9i		−	0.313423i
$627$	1.17336e7			0.00190105
$628$	6.95058e8			0.111986
$629$			9.48445e8i			0.151962i
$630$			1.37691e9i			0.219388i
$631$			2.54822e9i			0.403771i	0.979409	+	0.201885i	$0.0647068\pi$
$631$			2.54822e9i			0.403771i	−0.979409	+	0.201885i	$0.935293\pi$
$632$		−	1.14797e10i		−	1.80893i
$633$	9.23004e9			1.44641
$634$	−5.88606e7			−0.00917303
$635$		−	2.52372e9i		−	0.391141i
$636$	−1.25775e9			−0.193864
$637$	7.37045e8	+	5.70337e8i	0.112981	+	0.0874266i
$638$	−2.57395e8			−0.0392399
$639$			1.59908e10i			2.42447i
$640$	2.14394e9			0.323283
$641$	1.08219e10			1.62293			0.811464	−	0.584402i	$-0.198671\pi$
$641$	1.08219e10			1.62293			0.811464	+	0.584402i	$0.198671\pi$
$642$			3.61767e9i			0.539581i
$643$			4.66742e9i			0.692370i	0.938166	+	0.346185i	$0.112523\pi$
$643$			4.66742e9i			0.692370i	−0.938166	+	0.346185i	$0.887477\pi$
$644$		−	2.72124e8i		−	0.0401482i
$645$		−	2.10726e9i		−	0.309214i
$646$	−1.38579e8			−0.0202248
$647$	2.34242e9			0.340016			0.170008	−	0.985443i	$-0.445621\pi$
$647$	2.34242e9			0.340016			0.170008	+	0.985443i	$0.445621\pi$
$648$			1.64917e9i			0.238097i
$649$	−3.73729e8			−0.0536661
$650$	−3.35624e9	+	4.33726e9i	−0.479354	+	0.619467i
$651$	−1.74261e9			−0.247552
$652$		−	9.43293e8i		−	0.133285i
$653$	−5.28003e9			−0.742062			−0.371031	−	0.928620i	$-0.620996\pi$
$653$	−5.28003e9			−0.742062			−0.371031	+	0.928620i	$0.620996\pi$
$654$	5.20342e9			0.727389
$655$			1.29469e8i			0.0180020i
$656$		−	4.78951e9i		−	0.662412i
$657$		−	9.51579e8i		−	0.130908i
$658$		−	2.24014e9i		−	0.306538i
$659$	−2.08085e9			−0.283231			−0.141616	−	0.989922i	$-0.545230\pi$
$659$	−2.08085e9			−0.283231			−0.141616	+	0.989922i	$0.545230\pi$
$660$	−2.11955e7			−0.00286972
$661$		−	6.15792e9i		−	0.829333i	−0.909973	−	0.414667i	$-0.863898\pi$
$661$		−	6.15792e9i		−	0.829333i	0.909973	−	0.414667i	$-0.136102\pi$
$662$	−8.77096e9			−1.17502
$663$	6.43819e9	+	4.98197e9i	0.857958	+	0.663902i
$664$	2.70599e9			0.358705
$665$		−	3.67949e7i		−	0.00485190i
$666$	−2.40488e9			−0.315452
$667$	7.66448e9			1.00010
$668$		−	2.48813e8i		−	0.0322965i
$669$			1.62124e10i			2.09342i
$670$		−	3.44462e9i		−	0.442465i
$671$		−	1.00214e8i		−	0.0128056i
$672$	−1.08873e9			−0.138397
$673$	−6.20876e8			−0.0785149			−0.0392574	−	0.999229i	$-0.512499\pi$
$673$	−6.20876e8			−0.0785149			−0.0392574	+	0.999229i	$0.512499\pi$
$674$			8.60098e9i			1.08203i
$675$	5.39610e9			0.675332
$676$	2.34082e8	+	9.02951e8i	0.0291443	+	0.112422i
$677$	3.03237e9			0.375596			0.187798	−	0.982208i	$-0.439865\pi$
$677$	3.03237e9			0.375596			0.187798	+	0.982208i	$0.439865\pi$
$678$			4.94467e9i			0.609303i
$679$	−7.60167e8			−0.0931890
$680$	2.40577e9			0.293409
$681$			2.12111e10i			2.57365i
$682$			1.22863e8i			0.0148312i
$683$			8.30120e9i			0.996938i	0.866908	+	0.498469i	$0.166104\pi$
$683$			8.30120e9i			0.996938i	−0.866908	+	0.498469i	$0.833896\pi$
$684$			4.61710e7i			0.00551662i
$685$	5.42369e9			0.644730
$686$	4.29220e8			0.0507627
$687$			7.93713e8i			0.0933932i
$688$	−3.55135e9			−0.415752
$689$	7.15200e9	+	5.53433e9i	0.833030	+	0.644611i
$690$	−4.80325e9			−0.556626
$691$		−	1.28925e10i		−	1.48650i	−0.669013	−	0.743251i	$-0.733283\pi$
$691$		−	1.28925e10i		−	1.48650i	0.669013	−	0.743251i	$-0.266717\pi$
$692$	2.27501e8			0.0260983
$693$	−1.91053e8			−0.0218066
$694$			1.11279e10i			1.26373i
$695$		−	4.84467e8i		−	0.0547416i
$696$		−	1.61737e10i		−	1.81835i
$697$		−	4.65730e9i		−	0.520978i
$698$	−4.06432e8			−0.0452370
$699$	3.77554e9			0.418128
$700$		−	3.31888e8i		−	0.0365719i
$701$	−8.77448e9			−0.962074			−0.481037	−	0.876700i	$-0.659740\pi$
$701$	−8.77448e9			−0.962074			−0.481037	+	0.876700i	$0.659740\pi$
$702$	−4.27473e9	+	5.52423e9i	−0.466368	+	0.602687i
$703$	6.42653e7			0.00697643
$704$			3.84331e8i			0.0415147i
$705$	5.19559e9			0.558435
$706$	−1.52789e10			−1.63409
$707$		−	6.16380e9i		−	0.655964i
$708$		−	2.44356e9i		−	0.258765i
$709$		−	9.70583e9i		−	1.02275i	−0.859357	−	0.511377i	$-0.829136\pi$
$709$		−	9.70583e9i		−	1.02275i	0.859357	−	0.511377i	$-0.170864\pi$
$710$		−	5.87457e9i		−	0.615988i
$711$	2.49722e10			2.60563
$712$	−6.79328e7			−0.00705342
$713$		−	3.65851e9i		−	0.377999i
$714$	3.74929e9			0.385483
$715$	1.20525e8	+	9.32638e7i	0.0123312	+	0.00954205i
$716$	−9.72141e8			−0.0989768
$717$		−	1.58008e10i		−	1.60089i
$718$	−1.75206e10			−1.76650
$719$	1.03347e10			1.03692			0.518462	−	0.855101i	$-0.326505\pi$
$719$	1.03347e10			1.03692			0.518462	+	0.855101i	$0.326505\pi$
$720$			5.38194e9i			0.537372i
$721$			3.29195e9i			0.327099i
$722$		−	9.49824e9i		−	0.939211i
$723$		−	1.92684e10i		−	1.89610i
$724$	1.79262e9			0.175551
$725$	9.34775e9			0.911012
$726$		−	1.53392e10i		−	1.48773i
$727$	−4.90286e9			−0.473238			−0.236619	−	0.971603i	$-0.576039\pi$
$727$	−4.90286e9			−0.473238			−0.236619	+	0.971603i	$0.576039\pi$
$728$	3.26532e9	+	2.52675e9i	0.313664	+	0.242719i
$729$	−1.70246e10			−1.62754
$730$			3.49584e8i			0.0332599i
$731$	−3.45332e9			−0.326983
$732$	6.55232e8			0.0617457
$733$		−	1.93237e10i		−	1.81229i	−0.422971	−	0.906143i	$-0.639013\pi$
$733$		−	1.93237e10i		−	1.81229i	0.422971	−	0.906143i	$-0.360987\pi$
$734$			1.19205e9i			0.111265i
$735$			9.95494e8i			0.0924769i
$736$		−	2.28572e9i		−	0.211325i
$737$	4.77960e8			0.0439800
$738$	1.18090e10			1.08148
$739$		−	6.38280e9i		−	0.581776i	−0.956757	−	0.290888i	$-0.906049\pi$
$739$		−	6.38280e9i		−	0.581776i	0.956757	−	0.290888i	$-0.0939507\pi$
$740$	−1.16089e8			−0.0105312
$741$	3.37571e8	−	4.36243e8i	0.0304791	−	0.0393880i
$742$	4.16498e9			0.374282
$743$			4.29657e9i			0.384291i	0.981366	+	0.192146i	$0.0615446\pi$
$743$			4.29657e9i			0.384291i	−0.981366	+	0.192146i	$0.938455\pi$
$744$	−7.72025e9			−0.687268
$745$	2.98377e9			0.264374
$746$		−	1.11226e10i		−	0.980889i
$747$			5.88644e9i			0.516690i
$748$			3.47345e7i			0.00303463i
$749$			1.57412e9i			0.136883i
$750$	−1.28895e10			−1.11563
$751$	1.05440e10			0.908372			0.454186	−	0.890907i	$-0.349930\pi$
$751$	1.05440e10			0.908372			0.454186	+	0.890907i	$0.349930\pi$
$752$		−	8.75608e9i		−	0.750840i
$753$	−7.40796e9			−0.632290
$754$	−7.40518e9	+	9.56970e9i	−0.629124	+	0.813015i
$755$	−5.77532e9			−0.488384
$756$		−	4.22715e8i		−	0.0355812i
$757$	−8.92453e9			−0.747739			−0.373869	−	0.927481i	$-0.621969\pi$
$757$	−8.92453e9			−0.747739			−0.373869	+	0.927481i	$0.621969\pi$
$758$	−6.30369e9			−0.525717
$759$		−	6.66478e8i		−	0.0553273i
$760$		−	1.63012e8i		−	0.0134701i
$761$		−	6.93435e9i		−	0.570374i	−0.958472	−	0.285187i	$-0.907944\pi$
$761$		−	6.93435e9i		−	0.570374i	0.958472	−	0.285187i	$-0.0920556\pi$
$762$		−	1.74248e10i		−	1.42667i
$763$	2.26410e9			0.184527
$764$	−1.95632e9			−0.158713
$765$			5.23337e9i			0.422636i
$766$	6.73882e9			0.541731
$767$	−1.07521e10	+	1.38949e10i	−0.860415	+	1.11191i
$768$	−6.83436e9			−0.544419
$769$			2.35979e10i			1.87124i	0.353003	+	0.935622i	$0.385161\pi$
$769$			2.35979e10i			1.87124i	−0.353003	+	0.935622i	$0.614839\pi$
$770$	7.01877e7			0.00554043
$771$	−3.16500e10			−2.48704
$772$			2.64640e9i			0.207012i
$773$			2.11113e10i			1.64395i	0.569526	+	0.821973i	$0.307127\pi$
$773$			2.11113e10i			1.64395i	−0.569526	+	0.821973i	$0.692873\pi$
$774$		−	8.75623e9i		−	0.678772i
$775$		−	4.46199e9i		−	0.344328i
$776$	−3.36775e9			−0.258716
$777$	−1.73871e9			−0.132970
$778$		−	5.80761e9i		−	0.442149i
$779$	−3.15572e8			−0.0239176
$780$	−6.09788e8	+	7.88027e8i	−0.0460095	+	0.0594580i
$781$	8.15129e8			0.0612277
$782$			7.87142e9i			0.588613i
$783$	1.19059e10			0.886334
$784$	1.67770e9			0.124339
$785$		−	5.33823e9i		−	0.393871i
$786$			8.93904e8i			0.0656617i
$787$		−	7.08462e9i		−	0.518090i	−0.965865	−	0.259045i	$-0.916592\pi$
$787$		−	7.08462e9i		−	0.518090i	0.965865	−	0.259045i	$-0.0834078\pi$
$788$			2.00493e9i			0.145968i
$789$	7.20731e9			0.522401
$790$	−9.17410e9			−0.662016
$791$			2.15152e9i			0.154571i
$792$	−8.46419e8			−0.0605406
$793$	−3.72586e9	−	2.88313e9i	−0.265321	−	0.205309i
$794$	2.60500e10			1.84687
$795$			9.65989e9i			0.681848i
$796$	−1.84571e9			−0.129708
$797$	−1.42527e10			−0.997224			−0.498612	−	0.866825i	$-0.666157\pi$
$797$	−1.42527e10			−0.997224			−0.498612	+	0.866825i	$0.666157\pi$
$798$		−	2.54047e8i		−	0.0176971i
$799$		−	8.51436e9i		−	0.590525i
$800$		−	2.78771e9i		−	0.192501i
$801$		−	1.47777e8i		−	0.0101600i
$802$	−1.15660e10			−0.791724
$803$	−4.85067e7			−0.00330596
$804$			3.12505e9i			0.212061i
$805$	−2.08998e9			−0.141207
$806$	4.56793e9	+	3.53474e9i	0.307289	+	0.237785i
$807$	−3.20861e10			−2.14911
$808$		−	2.73073e10i		−	1.82112i
$809$	−1.14294e10			−0.758931			−0.379465	−	0.925206i	$-0.623892\pi$
$809$	−1.14294e10			−0.758931			−0.379465	+	0.925206i	$0.623892\pi$
$810$	1.31795e9			0.0871369
$811$			1.12064e10i			0.737726i	0.929484	+	0.368863i	$0.120253\pi$
$811$			1.12064e10i			0.737726i	−0.929484	+	0.368863i	$0.879747\pi$
$812$		−	7.32275e8i		−	0.0479985i
$813$			3.00742e9i			0.196281i
$814$			1.22588e8i			0.00796644i
$815$	−7.24475e9			−0.468783
$816$	1.46549e10			0.944208
$817$			2.33992e8i			0.0150115i
$818$	−1.66199e10			−1.06167
$819$	−5.49654e9	+	7.10316e9i	−0.349620	+	0.451812i
$820$	5.70048e8			0.0361047
$821$			2.48187e10i			1.56523i	0.622508	+	0.782614i	$0.286114\pi$
$821$			2.48187e10i			1.56523i	−0.622508	+	0.782614i	$0.713886\pi$
$822$	3.74473e10			2.35163
$823$	7.28413e9			0.455490			0.227745	−	0.973721i	$-0.426865\pi$
$823$	7.28413e9			0.455490			0.227745	+	0.973721i	$0.426865\pi$
$824$			1.45843e10i			0.908111i
$825$		−	8.12849e8i		−	0.0503989i
$826$			8.09170e9i			0.499585i
$827$		−	2.00853e10i		−	1.23483i	−0.786637	−	0.617416i	$-0.788179\pi$
$827$		−	2.00853e10i		−	1.23483i	0.786637	−	0.617416i	$-0.211821\pi$
$828$	2.62255e9			0.160553
$829$	−1.39818e10			−0.852357			−0.426179	−	0.904639i	$-0.640140\pi$
$829$	−1.39818e10			−0.852357			−0.426179	+	0.904639i	$0.640140\pi$
$830$		−	2.16251e9i		−	0.131276i
$831$	1.51158e10			0.913752
$832$	1.42890e10	+	1.10571e10i	0.860145	+	0.665594i
$833$	1.63138e9			0.0977910
$834$		−	3.34495e9i		−	0.199668i
$835$	−1.91095e9			−0.113592
$836$	2.35356e6			0.000139317
$837$		−	5.68310e9i		−	0.335001i
$838$			1.56837e9i			0.0920649i
$839$			3.26654e10i			1.90951i	0.297400	+	0.954753i	$0.403881\pi$
$839$			3.26654e10i			1.90951i	−0.297400	+	0.954753i	$0.596119\pi$
$840$			4.41032e9i			0.256739i
$841$	3.37495e9			0.195651
$842$	2.36182e10			1.36350
$843$		−	3.65607e10i		−	2.10193i
$844$	1.85139e9			0.105999
$845$	6.93491e9	−	1.79781e9i	0.395405	−	0.102505i
$846$	2.15890e10			1.22585
$847$		−	6.67436e9i		−	0.377414i
$848$	1.62797e10			0.916774
$849$	−6.07786e9			−0.340858
$850$			9.60014e9i			0.536181i
$851$		−	3.65033e9i		−	0.203038i
$852$			5.32957e9i			0.295225i
$853$		−	3.25362e10i		−	1.79492i	−0.441094	−	0.897461i	$-0.645409\pi$
$853$		−	3.25362e10i		−	1.79492i	0.441094	−	0.897461i	$-0.354591\pi$
$854$	−2.16976e9			−0.119209
$855$	3.54606e8			0.0194028
$856$			6.97377e9i			0.380023i
$857$	−2.02467e10			−1.09880			−0.549402	−	0.835558i	$-0.685145\pi$
$857$	−2.02467e10			−1.09880			−0.549402	+	0.835558i	$0.685145\pi$
$858$	8.32149e8	+	6.43930e8i	0.0449775	+	0.0348043i
$859$	5.76797e9			0.310489			0.155245	−	0.987876i	$-0.450383\pi$
$859$	5.76797e9			0.310489			0.155245	+	0.987876i	$0.450383\pi$
$860$		−	4.22682e8i		−	0.0226605i
$861$	8.53787e9			0.455867
$862$	7.69175e9			0.409025
$863$			1.64757e10i			0.872583i	0.899805	+	0.436291i	$0.143708\pi$
$863$			1.64757e10i			0.872583i	−0.899805	+	0.436291i	$0.856292\pi$
$864$		−	3.55062e9i		−	0.187286i
$865$		−	1.74727e9i		−	0.0917919i
$866$			2.40080e10i			1.25615i
$867$	−1.61607e10			−0.842159
$868$	−3.49539e8			−0.0181416
$869$		−	1.27296e9i		−	0.0658028i
$870$	−1.29254e10			−0.665466
$871$	1.37507e10	−	1.77701e10i	0.705120	−	0.911224i
$872$	1.00306e10			0.512295
$873$		−	7.32599e9i		−	0.372663i
$874$	5.33357e8			0.0270226
$875$	−5.60845e9			−0.283018
$876$		−	3.17152e8i		−	0.0159405i
$877$		−	1.07079e9i		−	0.0536051i	−0.999641	−	0.0268026i	$-0.991467\pi$
$877$		−	1.07079e9i		−	0.0536051i	0.999641	−	0.0268026i	$-0.00853255\pi$
$878$			2.19990e10i			1.09692i
$879$			2.82031e10i			1.40067i
$880$	2.74344e8			0.0135708
$881$	−3.49434e10			−1.72167			−0.860835	−	0.508884i	$-0.830058\pi$
$881$	−3.49434e10			−1.72167			−0.860835	+	0.508884i	$0.830058\pi$
$882$			4.13654e9i			0.203000i
$883$	1.19689e10			0.585050			0.292525	−	0.956258i	$-0.405505\pi$
$883$	1.19689e10			0.585050			0.292525	+	0.956258i	$0.405505\pi$
$884$	1.29139e9	+	9.99301e8i	0.0628747	+	0.0486534i
$885$	−1.87672e10			−0.910118
$886$		−	3.08821e10i		−	1.49172i
$887$	1.35467e10			0.651778			0.325889	−	0.945408i	$-0.394336\pi$
$887$	1.35467e10			0.651778			0.325889	+	0.945408i	$0.394336\pi$
$888$	−7.70298e9			−0.369159
$889$		−	7.58183e9i		−	0.361925i
$890$			5.42891e7i			0.00258136i
$891$			1.82873e8i			0.00866119i
$892$			3.25195e9i			0.153414i
$893$	−5.76921e8			−0.0271104
$894$	2.06011e10			0.964294
$895$			7.46630e9i			0.348117i
$896$	6.44090e9			0.299136
$897$	−2.47790e10	−	1.91744e10i	−1.14633	−	0.887048i
$898$	4.40953e9			0.203201
$899$		−	9.84492e9i		−	0.451911i
$900$	3.19852e9			0.146251
$901$	1.58303e10			0.721030
$902$		−	6.01965e8i		−	0.0273117i
$903$		−	6.33070e9i		−	0.286118i
$904$			9.53183e9i			0.429128i
$905$		−	1.37678e10i		−	0.617440i
$906$	−3.98751e10			−1.78136
$907$	−3.66478e10			−1.63088			−0.815440	−	0.578841i	$-0.803505\pi$
$907$	−3.66478e10			−1.63088			−0.815440	+	0.578841i	$0.803505\pi$
$908$			4.25461e9i			0.188608i
$909$	5.94026e10			2.62320
$910$	2.01928e9	−	2.60951e9i	0.0888282	−	0.114792i
$911$	1.14959e10			0.503768			0.251884	−	0.967757i	$-0.418950\pi$
$911$	1.14959e10			0.503768			0.251884	+	0.967757i	$0.418950\pi$
$912$		−	9.92997e8i		−	0.0433477i
$913$	3.00061e8			0.0130485
$914$	3.64179e10			1.57762
$915$		−	5.03236e9i		−	0.217169i
$916$			1.59206e8i			0.00684423i
$917$			3.88954e8i			0.0166574i
$918$			1.22274e10i			0.521656i
$919$	1.73638e10			0.737973			0.368986	−	0.929435i	$-0.379705\pi$
$919$	1.73638e10			0.737973			0.368986	+	0.929435i	$0.379705\pi$
$920$	−9.25921e9			−0.392028
$921$		−	3.66012e10i		−	1.54379i
$922$	−1.04638e10			−0.439673
$923$	2.34510e10	−	3.03057e10i	0.981647	−	1.26858i
$924$	−6.36761e7			−0.00265537
$925$		−	4.45201e9i		−	0.184952i
$926$	2.84167e10			1.17607
$927$	−3.17257e10			−1.30807
$928$		−	6.15079e9i		−	0.252646i
$929$			1.54963e10i			0.634124i	0.948405	+	0.317062i	$0.102696\pi$
$929$			1.54963e10i			0.634124i	−0.948405	+	0.317062i	$0.897304\pi$
$930$			6.16971e9i			0.251521i
$931$		−	1.10540e8i		−	0.00448949i
$932$	7.57311e8			0.0306421
$933$	−2.78215e10			−1.12149
$934$			4.12298e10i			1.65576i
$935$	2.66770e8			0.0106733
$936$	−2.43512e10	+	3.14690e10i	−0.970632	+	1.25435i
$937$	4.82696e10			1.91684			0.958418	−	0.285367i	$-0.0921154\pi$
$937$	4.82696e10			1.91684			0.958418	+	0.285367i	$0.0921154\pi$
$938$		−	1.03484e10i		−	0.409415i
$939$	1.34040e10			0.528329
$940$	1.04215e9			0.0409244
$941$			1.94033e10i			0.759121i	0.925167	+	0.379561i	$0.123925\pi$
$941$			1.94033e10i			0.759121i	−0.925167	+	0.379561i	$0.876075\pi$
$942$		−	3.68573e10i		−	1.43663i
$943$			1.79248e10i			0.696086i
$944$			3.16282e10i			1.22369i
$945$	−3.24656e9			−0.125145
$946$	−4.46348e8			−0.0171417
$947$		−	1.91176e9i		−	0.0731492i	−0.999331	−	0.0365746i	$-0.988355\pi$
$947$		−	1.91176e9i		−	0.0731492i	0.999331	−	0.0365746i	$-0.0116447\pi$
$948$	8.32299e9			0.317286
$949$	−1.39552e9	+	1.80343e9i	−0.0530035	+	0.0684963i
$950$	6.50492e8			0.0246155
$951$		−	4.10126e8i		−	0.0154627i
$952$	7.22749e9			0.271493
$953$	−3.79969e9			−0.142208			−0.0711039	−	0.997469i	$-0.522652\pi$
$953$	−3.79969e9			−0.142208			−0.0711039	+	0.997469i	$0.522652\pi$
$954$			4.01394e10i			1.49676i
$955$			1.50251e10i			0.558219i
$956$		−	3.16938e9i		−	0.117320i
$957$		−	1.79347e9i		−	0.0661456i
$958$	−2.97177e10			−1.09203
$959$	1.62940e10			0.596572
$960$			1.92996e10i			0.704042i
$961$	2.28133e10			0.829195
$962$	4.55771e9	+	3.52683e9i	0.165057	+	0.127724i
$963$	−1.51703e10			−0.547397
$964$		−	3.86493e9i		−	0.138954i
$965$	2.03251e10			0.728093
$966$	−1.44301e10			−0.515049
$967$			7.56523e9i			0.269048i	0.990910	+	0.134524i	$0.0429505\pi$
$967$			7.56523e9i			0.269048i	−0.990910	+	0.134524i	$0.957050\pi$
$968$		−	2.95693e10i		−	1.04780i
$969$		−	9.65585e8i		−	0.0340924i
$970$			2.69137e9i			0.0946830i
$971$	4.16079e10			1.45851			0.729253	−	0.684244i	$-0.239868\pi$
$971$	4.16079e10			1.45851			0.729253	+	0.684244i	$0.239868\pi$
$972$	−3.89095e9			−0.135901
$973$		−	1.45545e9i		−	0.0506526i
$974$	3.59515e10			1.24670
$975$	−3.02209e10	−	2.33854e10i	−1.04422	−	0.808032i
$976$	−8.48100e9			−0.291993
$977$			2.54746e10i			0.873928i	0.899479	+	0.436964i	$0.143946\pi$
$977$			2.54746e10i			0.873928i	−0.899479	+	0.436964i	$0.856054\pi$
$978$	−5.00206e10			−1.70987
$979$	−7.53291e6			−0.000256580
$980$			1.99680e8i			0.00677708i
$981$			2.18200e10i			0.737925i
$982$			1.82866e10i			0.616230i
$983$			5.37049e10i			1.80334i	0.432429	+	0.901668i	$0.357657\pi$
$983$			5.37049e10i			1.80334i	−0.432429	+	0.901668i	$0.642343\pi$
$984$	3.78251e10			1.26560
$985$	1.53984e10			0.513391
$986$			2.11817e10i			0.703706i
$987$	1.56087e10			0.516723
$988$	6.77113e7	−	8.75031e7i	0.00223363	−	0.00288652i
$989$	1.32909e10			0.436887
$990$			6.76423e8i			0.0221562i
$991$	−4.32240e10			−1.41080			−0.705402	−	0.708807i	$-0.749234\pi$
$991$	−4.32240e10			−1.41080			−0.705402	+	0.708807i	$0.749234\pi$
$992$	−2.93598e9			−0.0954909
$993$		−	6.11138e10i		−	1.98069i
$994$		−	1.76486e10i		−	0.569976i
$995$			1.41755e10i			0.456204i
$996$			1.96189e9i			0.0629169i
$997$	−2.88373e10			−0.921556			−0.460778	−	0.887515i	$-0.652430\pi$
$997$	−2.88373e10			−0.921556			−0.460778	+	0.887515i	$0.652430\pi$
$998$	6.04286e9			0.192436
$999$		−	5.67038e9i		−	0.179942i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.8.c.a.64.16	✓	50
13.12	even	2	inner	91.8.c.a.64.35	yes	50

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.8.c.a.64.16	✓	50	1.1	even	1	trivial
91.8.c.a.64.35	yes	50	13.12	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 91 = 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	91.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-10.6365i q^{2} +74.1121 q^{3} +14.8657 q^{4} -114.172i q^{5} -788.291i q^{6} -343.000i q^{7} -1519.59i q^{8} +3305.61 q^{9} +O(q^{10})\)	\(q-10.6365i q^{2} +74.1121 q^{3} +14.8657 q^{4} -114.172i q^{5} -788.291i q^{6} -343.000i q^{7} -1519.59i q^{8} +3305.61 q^{9} -1214.39 q^{10} -168.503i q^{11} +1101.73 q^{12} +(-6264.78 - 4847.79i) q^{13} -3648.31 q^{14} -8461.56i q^{15} -14260.2 q^{16} -13866.5 q^{17} -35160.0i q^{18} +939.578i q^{19} -1697.25i q^{20} -25420.5i q^{21} -1792.28 q^{22} +53368.8 q^{23} -112620. i q^{24} +65089.7 q^{25} +(-51563.3 + 66635.1i) q^{26} +82902.6 q^{27} -5098.93i q^{28} +143613. q^{29} -90001.1 q^{30} -68551.5i q^{31} -42828.8i q^{32} -12488.1i q^{33} +147491. i q^{34} -39161.1 q^{35} +49140.1 q^{36} -68398.1i q^{37} +9993.78 q^{38} +(-464296. - 359280. i) q^{39} -173495. q^{40} +335866. i q^{41} -270384. q^{42} +249039. q^{43} -2504.92i q^{44} -377410. i q^{45} -567656. i q^{46} +614022. i q^{47} -1.05685e6 q^{48} -117649. q^{49} -692324. i q^{50} -1.02768e6 q^{51} +(-93130.2 - 72065.6i) q^{52} -1.14162e6 q^{53} -881791. i q^{54} -19238.4 q^{55} -521218. q^{56} +69634.1i q^{57} -1.52754e6i q^{58} -2.21793e6i q^{59} -125787. i q^{60} +594732. q^{61} -729145. q^{62} -1.13382e6i q^{63} -2.28085e6 q^{64} +(-553483. + 715265. i) q^{65} -132830. q^{66} +2.83650e6i q^{67} -206136. q^{68} +3.95528e6 q^{69} +416536. i q^{70} +4.83746e6i q^{71} -5.02316e6i q^{72} -287868. i q^{73} -727514. q^{74} +4.82393e6 q^{75} +13967.5i q^{76} -57796.6 q^{77} +(-3.82147e6 + 4.93847e6i) q^{78} +7.55449e6 q^{79} +1.62812e6i q^{80} -1.08528e6 q^{81} +3.57242e6 q^{82} +1.78074e6i q^{83} -377893. i q^{84} +1.58318e6i q^{85} -2.64890e6i q^{86} +1.06435e7 q^{87} -256055. q^{88} -44704.8i q^{89} -4.01430e6 q^{90} +(-1.66279e6 + 2.14882e6i) q^{91} +793364. q^{92} -5.08050e6i q^{93} +6.53102e6 q^{94} +107274. q^{95} -3.17413e6i q^{96} -2.21623e6i q^{97} +1.25137e6i q^{98} -557006. i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q - 3328 q^{4} + 40514 q^{9}+O(q^{10}) \) 50 * q - 3328 * q^4 + 40514 * q^9	\( 50 q - 3328 q^{4} + 40514 q^{9} + 5320 q^{10} + 8700 q^{12} + 17044 q^{13} + 10976 q^{14} + 228808 q^{16} + 33664 q^{17} + 70228 q^{22} - 75042 q^{23} - 664772 q^{25} + 78276 q^{26} - 661404 q^{27} + 135778 q^{29} + 994888 q^{30} + 372498 q^{35} - 3549604 q^{36} + 338468 q^{38} - 973080 q^{39} + 79316 q^{40} + 296352 q^{42} - 53618 q^{43} + 1400384 q^{48} - 5882450 q^{49} - 2182360 q^{51} - 6982340 q^{52} + 2841746 q^{53} + 6871356 q^{55} - 2107392 q^{56} + 1773716 q^{61} - 6969608 q^{62} - 9449120 q^{64} - 7901430 q^{65} - 11755548 q^{66} + 11829980 q^{68} + 3564460 q^{69} + 45595884 q^{74} - 7220964 q^{75} + 186592 q^{77} - 8093012 q^{78} - 21257822 q^{79} + 53034530 q^{81} + 10907568 q^{82} + 14135000 q^{87} - 51594780 q^{88} - 61226356 q^{90} - 8096858 q^{91} - 11200212 q^{92} + 80667028 q^{94} + 30430066 q^{95}+O(q^{100}) \) 50 * q - 3328 * q^4 + 40514 * q^9 + 5320 * q^10 + 8700 * q^12 + 17044 * q^13 + 10976 * q^14 + 228808 * q^16 + 33664 * q^17 + 70228 * q^22 - 75042 * q^23 - 664772 * q^25 + 78276 * q^26 - 661404 * q^27 + 135778 * q^29 + 994888 * q^30 + 372498 * q^35 - 3549604 * q^36 + 338468 * q^38 - 973080 * q^39 + 79316 * q^40 + 296352 * q^42 - 53618 * q^43 + 1400384 * q^48 - 5882450 * q^49 - 2182360 * q^51 - 6982340 * q^52 + 2841746 * q^53 + 6871356 * q^55 - 2107392 * q^56 + 1773716 * q^61 - 6969608 * q^62 - 9449120 * q^64 - 7901430 * q^65 - 11755548 * q^66 + 11829980 * q^68 + 3564460 * q^69 + 45595884 * q^74 - 7220964 * q^75 + 186592 * q^77 - 8093012 * q^78 - 21257822 * q^79 + 53034530 * q^81 + 10907568 * q^82 + 14135000 * q^87 - 51594780 * q^88 - 61226356 * q^90 - 8096858 * q^91 - 11200212 * q^92 + 80667028 * q^94 + 30430066 * q^95

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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