LMFDB - Embedded newform 448.8.a.t.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [448,8,Mod(1,448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(448, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("448.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 448 = 2^{6} \cdot 7 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	448.a (trivial)

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:	yes
Analytic conductor:	$139.948491417$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{865}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 216 $ x^2 - x - 216
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 7)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-14.2054$ of defining polynomial
Character	$\chi$	$=$	448.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+76.4109 q^{3} -312.054 q^{5} +343.000 q^{7} +3651.62 q^{9} +O(q^{10})$	$q+76.4109 q^{3} -312.054 q^{5} +343.000 q^{7} +3651.62 q^{9} -283.831 q^{11} +11703.2 q^{13} -23844.4 q^{15} +4020.56 q^{17} +3787.73 q^{19} +26208.9 q^{21} +2582.14 q^{23} +19253.0 q^{25} +111913. q^{27} +127617. q^{29} +159037. q^{31} -21687.8 q^{33} -107035. q^{35} -583390. q^{37} +894252. q^{39} +18577.8 q^{41} -323000. q^{43} -1.13951e6 q^{45} +1923.22 q^{47} +117649. q^{49} +307215. q^{51} -1.35319e6 q^{53} +88570.8 q^{55} +289424. q^{57} +868620. q^{59} +1.27874e6 q^{61} +1.25251e6 q^{63} -3.65203e6 q^{65} +584595. q^{67} +197304. q^{69} +5.67464e6 q^{71} +4.97301e6 q^{73} +1.47114e6 q^{75} -97354.1 q^{77} -870364. q^{79} +565282. q^{81} +3.87968e6 q^{83} -1.25463e6 q^{85} +9.75133e6 q^{87} +1.02269e7 q^{89} +4.01420e6 q^{91} +1.21521e7 q^{93} -1.18198e6 q^{95} +3.98924e6 q^{97} -1.03644e6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 94 q^{3} - 330 q^{5} + 686 q^{7} + 1774 q^{9}+O(q^{10}) $ 2 * q + 94 * q^3 - 330 * q^5 + 686 * q^7 + 1774 * q^9	$ 2 q + 94 q^{3} - 330 q^{5} + 686 q^{7} + 1774 q^{9} + 2844 q^{11} - 2534 q^{13} - 24160 q^{15} - 1488 q^{17} + 32810 q^{19} + 32242 q^{21} + 6576 q^{23} - 58550 q^{25} + 40420 q^{27} - 20640 q^{29} + 391836 q^{31} + 33328 q^{33} - 113190 q^{35} - 367392 q^{37} + 643832 q^{39} + 734664 q^{41} - 480476 q^{43} - 1105810 q^{45} + 1089108 q^{47} + 235298 q^{49} + 210324 q^{51} - 2858844 q^{53} + 32440 q^{55} + 799900 q^{57} + 160170 q^{59} + 864646 q^{61} + 608482 q^{63} - 3396540 q^{65} - 328648 q^{67} + 267552 q^{69} + 7500216 q^{71} + 4301244 q^{73} + 102650 q^{75} + 975492 q^{77} + 6408440 q^{79} + 3414142 q^{81} + 11659074 q^{83} - 1155780 q^{85} + 7143620 q^{87} + 9772260 q^{89} - 869162 q^{91} + 16246872 q^{93} - 1702800 q^{95} + 10762752 q^{97} - 6909332 q^{99}+O(q^{100}) $ 2 * q + 94 * q^3 - 330 * q^5 + 686 * q^7 + 1774 * q^9 + 2844 * q^11 - 2534 * q^13 - 24160 * q^15 - 1488 * q^17 + 32810 * q^19 + 32242 * q^21 + 6576 * q^23 - 58550 * q^25 + 40420 * q^27 - 20640 * q^29 + 391836 * q^31 + 33328 * q^33 - 113190 * q^35 - 367392 * q^37 + 643832 * q^39 + 734664 * q^41 - 480476 * q^43 - 1105810 * q^45 + 1089108 * q^47 + 235298 * q^49 + 210324 * q^51 - 2858844 * q^53 + 32440 * q^55 + 799900 * q^57 + 160170 * q^59 + 864646 * q^61 + 608482 * q^63 - 3396540 * q^65 - 328648 * q^67 + 267552 * q^69 + 7500216 * q^71 + 4301244 * q^73 + 102650 * q^75 + 975492 * q^77 + 6408440 * q^79 + 3414142 * q^81 + 11659074 * q^83 - 1155780 * q^85 + 7143620 * q^87 + 9772260 * q^89 - 869162 * q^91 + 16246872 * q^93 - 1702800 * q^95 + 10762752 * q^97 - 6909332 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	0	0
$2$	0	0
$3$	76.4109	1.63392	0.816960	−	0.576694i	$-0.195658\pi$
$3$	76.4109	1.63392	0.816960	+	0.576694i	$0.195658\pi$
$4$	0	0
$5$	−312.054	−1.11644	−0.558220	−	0.829693i	$-0.688515\pi$
$5$	−312.054	−1.11644	−0.558220	+	0.829693i	$0.688515\pi$
$6$	0	0
$7$	343.000	0.377964
$7$	343.000	0.377964
$8$	0	0
$9$	3651.62	1.66969
$10$	0	0
$11$	−283.831	−0.0642963	−0.0321481	−	0.999483i	$-0.510235\pi$
$11$	−283.831	−0.0642963	−0.0321481	+	0.999483i	$0.510235\pi$
$12$	0	0
$13$	11703.2	1.47742	0.738708	−	0.674025i	$-0.235436\pi$
$13$	11703.2	1.47742	0.738708	+	0.674025i	$0.235436\pi$
$14$	0	0
$15$	−23844.4	−1.82417
$16$	0	0
$17$	4020.56	0.198479	0.0992397	−	0.995064i	$-0.468359\pi$
$17$	4020.56	0.198479	0.0992397	+	0.995064i	$0.468359\pi$
$18$	0	0
$19$	3787.73	0.126690	0.0633449	−	0.997992i	$-0.479823\pi$
$19$	3787.73	0.126690	0.0633449	+	0.997992i	$0.479823\pi$
$20$	0	0
$21$	26208.9	0.617564
$22$	0	0
$23$	2582.14	0.0442519	0.0221260	−	0.999755i	$-0.492957\pi$
$23$	2582.14	0.0442519	0.0221260	+	0.999755i	$0.492957\pi$
$24$	0	0
$25$	19253.0	0.246438
$26$	0	0
$27$	111913.	1.09423
$28$	0	0
$29$	127617.	0.971663	0.485832	−	0.874052i	$-0.338517\pi$
$29$	127617.	0.971663	0.485832	+	0.874052i	$0.338517\pi$
$30$	0	0
$31$	159037.	0.958808	0.479404	−	0.877594i	$-0.340853\pi$
$31$	159037.	0.958808	0.479404	+	0.877594i	$0.340853\pi$
$32$	0	0
$33$	−21687.8	−0.105055
$34$	0	0
$35$	−107035.	−0.421975
$36$	0	0
$37$	−583390.	−1.89345	−0.946723	−	0.322049i	$-0.895628\pi$
$37$	−583390.	−1.89345	−0.946723	+	0.322049i	$0.895628\pi$
$38$	0	0
$39$	894252.	2.41398
$40$	0	0
$41$	18577.8	0.0420969	0.0210484	−	0.999778i	$-0.493300\pi$
$41$	18577.8	0.0420969	0.0210484	+	0.999778i	$0.493300\pi$
$42$	0	0
$43$	−323000.	−0.619531	−0.309766	−	0.950813i	$-0.600251\pi$
$43$	−323000.	−0.619531	−0.309766	+	0.950813i	$0.600251\pi$
$44$	0	0
$45$	−1.13951e6	−1.86411
$46$	0	0
$47$	1923.22	0.00270201	0.00135100	−	0.999999i	$-0.499570\pi$
$47$	1923.22	0.00270201	0.00135100	+	0.999999i	$0.499570\pi$
$48$	0	0
$49$	117649.	0.142857
$50$	0	0
$51$	307215.	0.324300
$52$	0	0
$53$	−1.35319e6	−1.24851	−0.624256	−	0.781220i	$-0.714598\pi$
$53$	−1.35319e6	−1.24851	−0.624256	+	0.781220i	$0.714598\pi$
$54$	0	0
$55$	88570.8	0.0717829
$56$	0	0
$57$	289424.	0.207001
$58$	0	0
$59$	868620.	0.550615	0.275307	−	0.961356i	$-0.411220\pi$
$59$	868620.	0.550615	0.275307	+	0.961356i	$0.411220\pi$
$60$	0	0
$61$	1.27874e6	0.721319	0.360660	−	0.932698i	$-0.382552\pi$
$61$	1.27874e6	0.721319	0.360660	+	0.932698i	$0.382552\pi$
$62$	0	0
$63$	1.25251e6	0.631085
$64$	0	0
$65$	−3.65203e6	−1.64945
$66$	0	0
$67$	584595.	0.237461	0.118731	−	0.992926i	$-0.462117\pi$
$67$	584595.	0.237461	0.118731	+	0.992926i	$0.462117\pi$
$68$	0	0
$69$	197304.	0.0723041
$70$	0	0
$71$	5.67464e6	1.88163	0.940814	−	0.338922i	$-0.110062\pi$
$71$	5.67464e6	1.88163	0.940814	+	0.338922i	$0.110062\pi$
$72$	0	0
$73$	4.97301e6	1.49620	0.748099	−	0.663587i	$-0.230967\pi$
$73$	4.97301e6	1.49620	0.748099	+	0.663587i	$0.230967\pi$
$74$	0	0
$75$	1.47114e6	0.402660
$76$	0	0
$77$	−97354.1	−0.0243017
$78$	0	0
$79$	−870364.	−0.198612	−0.0993061	−	0.995057i	$-0.531662\pi$
$79$	−870364.	−0.198612	−0.0993061	+	0.995057i	$0.531662\pi$
$80$	0	0
$81$	565282.	0.118186
$82$	0	0
$83$	3.87968e6	0.744771	0.372386	−	0.928078i	$-0.378540\pi$
$83$	3.87968e6	0.744771	0.372386	+	0.928078i	$0.378540\pi$
$84$	0	0
$85$	−1.25463e6	−0.221590
$86$	0	0
$87$	9.75133e6	1.58762
$88$	0	0
$89$	1.02269e7	1.53773	0.768864	−	0.639412i	$-0.220822\pi$
$89$	1.02269e7	1.53773	0.768864	+	0.639412i	$0.220822\pi$
$90$	0	0
$91$	4.01420e6	0.558411
$92$	0	0
$93$	1.21521e7	1.56662
$94$	0	0
$95$	−1.18198e6	−0.141442
$96$	0	0
$97$	3.98924e6	0.443802	0.221901	−	0.975069i	$-0.428774\pi$
$97$	3.98924e6	0.443802	0.221901	+	0.975069i	$0.428774\pi$
$98$	0	0
$99$	−1.03644e6	−0.107355
$100$	0	0
$101$	61325.9	0.00592270	0.00296135	−	0.999996i	$-0.499057\pi$
$101$	61325.9	0.00592270	0.00296135	+	0.999996i	$0.499057\pi$
$102$	0	0
$103$	1.86031e7	1.67747	0.838736	−	0.544539i	$-0.183295\pi$
$103$	1.86031e7	1.67747	0.838736	+	0.544539i	$0.183295\pi$
$104$	0	0
$105$	−8.17861e6	−0.689473
$106$	0	0
$107$	6.70845e6	0.529394	0.264697	−	0.964332i	$-0.414728\pi$
$107$	6.70845e6	0.529394	0.264697	+	0.964332i	$0.414728\pi$
$108$	0	0
$109$	−1.34334e7	−0.993555	−0.496778	−	0.867878i	$-0.665484\pi$
$109$	−1.34334e7	−0.993555	−0.496778	+	0.867878i	$0.665484\pi$
$110$	0	0
$111$	−4.45773e7	−3.09374
$112$	0	0
$113$	2.59915e7	1.69456	0.847280	−	0.531147i	$-0.178239\pi$
$113$	2.59915e7	1.69456	0.847280	+	0.531147i	$0.178239\pi$
$114$	0	0
$115$	−805768.	−0.0494046
$116$	0	0
$117$	4.27357e7	2.46683
$118$	0	0
$119$	1.37905e6	0.0750182
$120$	0	0
$121$	−1.94066e7	−0.995866
$122$	0	0
$123$	1.41954e6	0.0687829
$124$	0	0
$125$	1.83713e7	0.841307
$126$	0	0
$127$	3.26332e7	1.41367	0.706833	−	0.707380i	$-0.250123\pi$
$127$	3.26332e7	1.41367	0.706833	+	0.707380i	$0.250123\pi$
$128$	0	0
$129$	−2.46807e7	−1.01226
$130$	0	0
$131$	4.40185e7	1.71075	0.855373	−	0.518013i	$-0.173328\pi$
$131$	4.40185e7	1.71075	0.855373	+	0.518013i	$0.173328\pi$
$132$	0	0
$133$	1.29919e6	0.0478842
$134$	0	0
$135$	−3.49230e7	−1.22164
$136$	0	0
$137$	7.19777e6	0.239153	0.119576	−	0.992825i	$-0.461846\pi$
$137$	7.19777e6	0.239153	0.119576	+	0.992825i	$0.461846\pi$
$138$	0	0
$139$	−2.50479e7	−0.791079	−0.395539	−	0.918449i	$-0.629442\pi$
$139$	−2.50479e7	−0.791079	−0.395539	+	0.918449i	$0.629442\pi$
$140$	0	0
$141$	146955.	0.00441487
$142$	0	0
$143$	−3.32173e6	−0.0949923
$144$	0	0
$145$	−3.98235e7	−1.08480
$146$	0	0
$147$	8.98966e6	0.233417
$148$	0	0
$149$	38001.6	0.000941130 0	0.000470565	−	1.00000i	$-0.499850\pi$
$149$	38001.6	0.000941130 0	0.000470565		1.00000i	$0.499850\pi$
$150$	0	0
$151$	−8.26640e6	−0.195388	−0.0976939	−	0.995217i	$-0.531147\pi$
$151$	−8.26640e6	−0.195388	−0.0976939	+	0.995217i	$0.531147\pi$
$152$	0	0
$153$	1.46816e7	0.331400
$154$	0	0
$155$	−4.96281e7	−1.07045
$156$	0	0
$157$	−4.00357e7	−0.825656	−0.412828	−	0.910809i	$-0.635459\pi$
$157$	−4.00357e7	−0.825656	−0.412828	+	0.910809i	$0.635459\pi$
$158$	0	0
$159$	−1.03398e8	−2.03997
$160$	0	0
$161$	885674.	0.0167257
$162$	0	0
$163$	−7.43567e7	−1.34482	−0.672409	−	0.740180i	$-0.734740\pi$
$163$	−7.43567e7	−1.34482	−0.672409	+	0.740180i	$0.734740\pi$
$164$	0	0
$165$	6.76777e6	0.117288
$166$	0	0
$167$	−6.14875e7	−1.02160	−0.510798	−	0.859701i	$-0.670650\pi$
$167$	−6.14875e7	−1.02160	−0.510798	+	0.859701i	$0.670650\pi$
$168$	0	0
$169$	7.42164e7	1.18276
$170$	0	0
$171$	1.38314e7	0.211533
$172$	0	0
$173$	1.75888e7	0.258271	0.129135	−	0.991627i	$-0.458780\pi$
$173$	1.75888e7	0.258271	0.129135	+	0.991627i	$0.458780\pi$
$174$	0	0
$175$	6.60376e6	0.0931447
$176$	0	0
$177$	6.63720e7	0.899661
$178$	0	0
$179$	−1.07789e8	−1.40472	−0.702360	−	0.711822i	$-0.747870\pi$
$179$	−1.07789e8	−1.40472	−0.702360	+	0.711822i	$0.747870\pi$
$180$	0	0
$181$	7.06656e7	0.885794	0.442897	−	0.896572i	$-0.353951\pi$
$181$	7.06656e7	0.885794	0.442897	+	0.896572i	$0.353951\pi$
$182$	0	0
$183$	9.77096e7	1.17858
$184$	0	0
$185$	1.82049e8	2.11392
$186$	0	0
$187$	−1.14116e6	−0.0127615
$188$	0	0
$189$	3.83862e7	0.413579
$190$	0	0
$191$	7.26186e7	0.754103	0.377052	−	0.926192i	$-0.376938\pi$
$191$	7.26186e7	0.754103	0.377052	+	0.926192i	$0.376938\pi$
$192$	0	0
$193$	3.46942e6	0.0347382	0.0173691	−	0.999849i	$-0.494471\pi$
$193$	3.46942e6	0.0347382	0.0173691	+	0.999849i	$0.494471\pi$
$194$	0	0
$195$	−2.79055e8	−2.69506
$196$	0	0
$197$	−6.22429e7	−0.580040	−0.290020	−	0.957021i	$-0.593662\pi$
$197$	−6.22429e7	−0.580040	−0.290020	+	0.957021i	$0.593662\pi$
$198$	0	0
$199$	−6.07528e7	−0.546488	−0.273244	−	0.961945i	$-0.588097\pi$
$199$	−6.07528e7	−0.546488	−0.273244	+	0.961945i	$0.588097\pi$
$200$	0	0
$201$	4.46694e7	0.387993
$202$	0	0
$203$	4.37726e7	0.367254
$204$	0	0
$205$	−5.79727e6	−0.0469986
$206$	0	0
$207$	9.42900e6	0.0738872
$208$	0	0
$209$	−1.07508e6	−0.00814568
$210$	0	0
$211$	1.46897e8	1.07653	0.538264	−	0.842776i	$-0.319080\pi$
$211$	1.46897e8	1.07653	0.538264	+	0.842776i	$0.319080\pi$
$212$	0	0
$213$	4.33604e8	3.07443
$214$	0	0
$215$	1.00794e8	0.691669
$216$	0	0
$217$	5.45496e7	0.362395
$218$	0	0
$219$	3.79992e8	2.44467
$220$	0	0
$221$	4.70534e7	0.293237
$222$	0	0
$223$	−1.41501e8	−0.854461	−0.427230	−	0.904143i	$-0.640511\pi$
$223$	−1.41501e8	−0.854461	−0.427230	+	0.904143i	$0.640511\pi$
$224$	0	0
$225$	7.03045e7	0.411476
$226$	0	0
$227$	3.40504e7	0.193211	0.0966054	−	0.995323i	$-0.469202\pi$
$227$	3.40504e7	0.193211	0.0966054	+	0.995323i	$0.469202\pi$
$228$	0	0
$229$	−1.56200e8	−0.859524	−0.429762	−	0.902942i	$-0.641403\pi$
$229$	−1.56200e8	−0.859524	−0.429762	+	0.902942i	$0.641403\pi$
$230$	0	0
$231$	−7.43891e6	−0.0397070
$232$	0	0
$233$	1.91857e8	0.993645	0.496822	−	0.867852i	$-0.334500\pi$
$233$	1.91857e8	0.993645	0.496822	+	0.867852i	$0.334500\pi$
$234$	0	0
$235$	−600150.	−0.00301663
$236$	0	0
$237$	−6.65053e7	−0.324516
$238$	0	0
$239$	3.07750e8	1.45816	0.729081	−	0.684427i	$-0.239948\pi$
$239$	3.07750e8	1.45816	0.729081	+	0.684427i	$0.239948\pi$
$240$	0	0
$241$	−2.14502e8	−0.987124	−0.493562	−	0.869711i	$-0.664305\pi$
$241$	−2.14502e8	−0.987124	−0.493562	+	0.869711i	$0.664305\pi$
$242$	0	0
$243$	−2.01560e8	−0.901121
$244$	0	0
$245$	−3.67129e7	−0.159491
$246$	0	0
$247$	4.43286e7	0.187174
$248$	0	0
$249$	2.96450e8	1.21690
$250$	0	0
$251$	−1.60485e8	−0.640585	−0.320293	−	0.947319i	$-0.603781\pi$
$251$	−1.60485e8	−0.640585	−0.320293	+	0.947319i	$0.603781\pi$
$252$	0	0
$253$	−732891.	−0.00284523
$254$	0	0
$255$	−9.58677e7	−0.362061
$256$	0	0
$257$	1.34126e7	0.0492887	0.0246444	−	0.999696i	$-0.492155\pi$
$257$	1.34126e7	0.0492887	0.0246444	+	0.999696i	$0.492155\pi$
$258$	0	0
$259$	−2.00103e8	−0.715655
$260$	0	0
$261$	4.66009e8	1.62238
$262$	0	0
$263$	1.49544e8	0.506900	0.253450	−	0.967348i	$-0.418435\pi$
$263$	1.49544e8	0.506900	0.253450	+	0.967348i	$0.418435\pi$
$264$	0	0
$265$	4.22269e8	1.39389
$266$	0	0
$267$	7.81447e8	2.51252
$268$	0	0
$269$	2.02126e8	0.633125	0.316562	−	0.948572i	$-0.397471\pi$
$269$	2.02126e8	0.633125	0.316562	+	0.948572i	$0.397471\pi$
$270$	0	0
$271$	−1.59103e7	−0.0485608	−0.0242804	−	0.999705i	$-0.507729\pi$
$271$	−1.59103e7	−0.0485608	−0.0242804	+	0.999705i	$0.507729\pi$
$272$	0	0
$273$	3.06728e8	0.912399
$274$	0	0
$275$	−5.46459e6	−0.0158450
$276$	0	0
$277$	2.83666e8	0.801915	0.400958	−	0.916097i	$-0.368677\pi$
$277$	2.83666e8	0.801915	0.400958	+	0.916097i	$0.368677\pi$
$278$	0	0
$279$	5.80742e8	1.60092
$280$	0	0
$281$	1.42010e8	0.381811	0.190905	−	0.981608i	$-0.438858\pi$
$281$	1.42010e8	0.381811	0.190905	+	0.981608i	$0.438858\pi$
$282$	0	0
$283$	2.00581e8	0.526063	0.263031	−	0.964787i	$-0.415278\pi$
$283$	2.00581e8	0.526063	0.263031	+	0.964787i	$0.415278\pi$
$284$	0	0
$285$	−9.03160e7	−0.231104
$286$	0	0
$287$	6.37217e6	0.0159111
$288$	0	0
$289$	−3.94174e8	−0.960606
$290$	0	0
$291$	3.04821e8	0.725137
$292$	0	0
$293$	5.15499e8	1.19727	0.598633	−	0.801023i	$-0.295711\pi$
$293$	5.15499e8	1.19727	0.598633	+	0.801023i	$0.295711\pi$
$294$	0	0
$295$	−2.71057e8	−0.614728
$296$	0	0
$297$	−3.17644e7	−0.0703548
$298$	0	0
$299$	3.02193e7	0.0653785
$300$	0	0
$301$	−1.10789e8	−0.234161
$302$	0	0
$303$	4.68597e6	0.00967721
$304$	0	0
$305$	−3.99036e8	−0.805309
$306$	0	0
$307$	7.65707e8	1.51035	0.755176	−	0.655522i	$-0.227552\pi$
$307$	7.65707e8	1.51035	0.755176	+	0.655522i	$0.227552\pi$
$308$	0	0
$309$	1.42148e9	2.74085
$310$	0	0
$311$	−1.59867e8	−0.301367	−0.150684	−	0.988582i	$-0.548147\pi$
$311$	−1.59867e8	−0.301367	−0.150684	+	0.988582i	$0.548147\pi$
$312$	0	0
$313$	−2.07759e8	−0.382961	−0.191480	−	0.981496i	$-0.561329\pi$
$313$	−2.07759e8	−0.382961	−0.191480	+	0.981496i	$0.561329\pi$
$314$	0	0
$315$	−3.90850e8	−0.704569
$316$	0	0
$317$	−5.47705e7	−0.0965693	−0.0482847	−	0.998834i	$-0.515375\pi$
$317$	−5.47705e7	−0.0965693	−0.0482847	+	0.998834i	$0.515375\pi$
$318$	0	0
$319$	−3.62217e7	−0.0624743
$320$	0	0
$321$	5.12598e8	0.864987
$322$	0	0
$323$	1.52288e7	0.0251453
$324$	0	0
$325$	2.25321e8	0.364091
$326$	0	0
$327$	−1.02646e9	−1.62339
$328$	0	0
$329$	659665.	0.00102126
$330$	0	0
$331$	−2.41167e8	−0.365527	−0.182763	−	0.983157i	$-0.558504\pi$
$331$	−2.41167e8	−0.365527	−0.182763	+	0.983157i	$0.558504\pi$
$332$	0	0
$333$	−2.13032e9	−3.16148
$334$	0	0
$335$	−1.82425e8	−0.265111
$336$	0	0
$337$	−7.24589e8	−1.03130	−0.515652	−	0.856798i	$-0.672450\pi$
$337$	−7.24589e8	−1.03130	−0.515652	+	0.856798i	$0.672450\pi$
$338$	0	0
$339$	1.98603e9	2.76877
$340$	0	0
$341$	−4.51396e7	−0.0616478
$342$	0	0
$343$	4.03536e7	0.0539949
$344$	0	0
$345$	−6.15694e7	−0.0807232
$346$	0	0
$347$	−9.93073e8	−1.27593	−0.637966	−	0.770064i	$-0.720224\pi$
$347$	−9.93073e8	−1.27593	−0.637966	+	0.770064i	$0.720224\pi$
$348$	0	0
$349$	−5.45485e8	−0.686901	−0.343450	−	0.939171i	$-0.611596\pi$
$349$	−5.45485e8	−0.686901	−0.343450	+	0.939171i	$0.611596\pi$
$350$	0	0
$351$	1.30974e9	1.61663
$352$	0	0
$353$	−3.78963e8	−0.458549	−0.229274	−	0.973362i	$-0.573635\pi$
$353$	−3.78963e8	−0.458549	−0.229274	+	0.973362i	$0.573635\pi$
$354$	0	0
$355$	−1.77080e9	−2.10073
$356$	0	0
$357$	1.05375e8	0.122574
$358$	0	0
$359$	−1.23744e9	−1.41155	−0.705773	−	0.708438i	$-0.749400\pi$
$359$	−1.23744e9	−1.41155	−0.705773	+	0.708438i	$0.749400\pi$
$360$	0	0
$361$	−8.79525e8	−0.983950
$362$	0	0
$363$	−1.48288e9	−1.62717
$364$	0	0
$365$	−1.55185e9	−1.67042
$366$	0	0
$367$	−8.13553e8	−0.859122	−0.429561	−	0.903038i	$-0.641332\pi$
$367$	−8.13553e8	−0.859122	−0.429561	+	0.903038i	$0.641332\pi$
$368$	0	0
$369$	6.78390e7	0.0702889
$370$	0	0
$371$	−4.64144e8	−0.471893
$372$	0	0
$373$	−7.99709e8	−0.797905	−0.398952	−	0.916972i	$-0.630626\pi$
$373$	−7.99709e8	−0.797905	−0.398952	+	0.916972i	$0.630626\pi$
$374$	0	0
$375$	1.40377e9	1.37463
$376$	0	0
$377$	1.49353e9	1.43555
$378$	0	0
$379$	−1.27251e9	−1.20067	−0.600334	−	0.799749i	$-0.704966\pi$
$379$	−1.27251e9	−1.20067	−0.600334	+	0.799749i	$0.704966\pi$
$380$	0	0
$381$	2.49353e9	2.30982
$382$	0	0
$383$	3.29059e8	0.299280	0.149640	−	0.988741i	$-0.452188\pi$
$383$	3.29059e8	0.299280	0.149640	+	0.988741i	$0.452188\pi$
$384$	0	0
$385$	3.03798e7	0.0271314
$386$	0	0
$387$	−1.17948e9	−1.03443
$388$	0	0
$389$	−9.37771e8	−0.807743	−0.403872	−	0.914816i	$-0.632336\pi$
$389$	−9.37771e8	−0.807743	−0.403872	+	0.914816i	$0.632336\pi$
$390$	0	0
$391$	1.03817e7	0.00878310
$392$	0	0
$393$	3.36349e9	2.79522
$394$	0	0
$395$	2.71601e8	0.221739
$396$	0	0
$397$	1.00184e9	0.803588	0.401794	−	0.915730i	$-0.368387\pi$
$397$	1.00184e9	0.803588	0.401794	+	0.915730i	$0.368387\pi$
$398$	0	0
$399$	9.92724e7	0.0782390
$400$	0	0
$401$	−1.31207e9	−1.01614	−0.508068	−	0.861317i	$-0.669640\pi$
$401$	−1.31207e9	−1.01614	−0.508068	+	0.861317i	$0.669640\pi$
$402$	0	0
$403$	1.86124e9	1.41656
$404$	0	0
$405$	−1.76399e8	−0.131948
$406$	0	0
$407$	1.65584e8	0.121741
$408$	0	0
$409$	8.61243e8	0.622435	0.311217	−	0.950339i	$-0.399263\pi$
$409$	8.61243e8	0.622435	0.311217	+	0.950339i	$0.399263\pi$
$410$	0	0
$411$	5.49988e8	0.390757
$412$	0	0
$413$	2.97937e8	0.208113
$414$	0	0
$415$	−1.21067e9	−0.831492
$416$	0	0
$417$	−1.91393e9	−1.29256
$418$	0	0
$419$	4.12403e8	0.273888	0.136944	−	0.990579i	$-0.456272\pi$
$419$	4.12403e8	0.273888	0.136944	+	0.990579i	$0.456272\pi$
$420$	0	0
$421$	9.13691e8	0.596777	0.298389	−	0.954444i	$-0.403551\pi$
$421$	9.13691e8	0.596777	0.298389	+	0.954444i	$0.403551\pi$
$422$	0	0
$423$	7.02288e6	0.00451153
$424$	0	0
$425$	7.74077e7	0.0489129
$426$	0	0
$427$	4.38607e8	0.272633
$428$	0	0
$429$	−2.53817e8	−0.155210
$430$	0	0
$431$	6.54101e8	0.393527	0.196764	−	0.980451i	$-0.436957\pi$
$431$	6.54101e8	0.393527	0.196764	+	0.980451i	$0.436957\pi$
$432$	0	0
$433$	2.14647e9	1.27062	0.635312	−	0.772256i	$-0.280872\pi$
$433$	2.14647e9	1.27062	0.635312	+	0.772256i	$0.280872\pi$
$434$	0	0
$435$	−3.04295e9	−1.77248
$436$	0	0
$437$	9.78045e6	0.00560627
$438$	0	0
$439$	−3.11978e9	−1.75994	−0.879970	−	0.475029i	$-0.842438\pi$
$439$	−3.11978e9	−1.75994	−0.879970	+	0.475029i	$0.842438\pi$
$440$	0	0
$441$	4.29610e8	0.238528
$442$	0	0
$443$	−1.99049e9	−1.08780	−0.543898	−	0.839151i	$-0.683052\pi$
$443$	−1.99049e9	−1.08780	−0.543898	+	0.839151i	$0.683052\pi$
$444$	0	0
$445$	−3.19135e9	−1.71678
$446$	0	0
$447$	2.90373e6	0.00153773
$448$	0	0
$449$	1.83855e9	0.958547	0.479274	−	0.877666i	$-0.340900\pi$
$449$	1.83855e9	0.958547	0.479274	+	0.877666i	$0.340900\pi$
$450$	0	0
$451$	−5.27295e6	−0.00270667
$452$	0	0
$453$	−6.31643e8	−0.319248
$454$	0	0
$455$	−1.25265e9	−0.623432
$456$	0	0
$457$	−3.51474e9	−1.72261	−0.861304	−	0.508091i	$-0.830351\pi$
$457$	−3.51474e9	−1.72261	−0.861304	+	0.508091i	$0.830351\pi$
$458$	0	0
$459$	4.49954e8	0.217182
$460$	0	0
$461$	2.98808e9	1.42049	0.710247	−	0.703953i	$-0.248583\pi$
$461$	2.98808e9	1.42049	0.710247	+	0.703953i	$0.248583\pi$
$462$	0	0
$463$	1.77331e9	0.830330	0.415165	−	0.909746i	$-0.363724\pi$
$463$	1.77331e9	0.830330	0.415165	+	0.909746i	$0.363724\pi$
$464$	0	0
$465$	−3.79213e9	−1.74903
$466$	0	0
$467$	1.97100e9	0.895524	0.447762	−	0.894153i	$-0.352221\pi$
$467$	1.97100e9	0.895524	0.447762	+	0.894153i	$0.352221\pi$
$468$	0	0
$469$	2.00516e8	0.0897520
$470$	0	0
$471$	−3.05917e9	−1.34906
$472$	0	0
$473$	9.16775e7	0.0398335
$474$	0	0
$475$	7.29250e7	0.0312212
$476$	0	0
$477$	−4.94134e9	−2.08464
$478$	0	0
$479$	2.43401e9	1.01193	0.505963	−	0.862555i	$-0.331137\pi$
$479$	2.43401e9	1.01193	0.505963	+	0.862555i	$0.331137\pi$
$480$	0	0
$481$	−6.82753e9	−2.79741
$482$	0	0
$483$	6.76751e7	0.0273284
$484$	0	0
$485$	−1.24486e9	−0.495478
$486$	0	0
$487$	1.38401e9	0.542983	0.271492	−	0.962441i	$-0.412483\pi$
$487$	1.38401e9	0.542983	0.271492	+	0.962441i	$0.412483\pi$
$488$	0	0
$489$	−5.68166e9	−2.19732
$490$	0	0
$491$	2.40004e9	0.915025	0.457512	−	0.889203i	$-0.348741\pi$
$491$	2.40004e9	0.915025	0.457512	+	0.889203i	$0.348741\pi$
$492$	0	0
$493$	5.13092e8	0.192855
$494$	0	0
$495$	3.23427e8	0.119856
$496$	0	0
$497$	1.94640e9	0.711189
$498$	0	0
$499$	1.27349e9	0.458823	0.229411	−	0.973330i	$-0.426320\pi$
$499$	1.27349e9	0.458823	0.229411	+	0.973330i	$0.426320\pi$
$500$	0	0
$501$	−4.69831e9	−1.66921
$502$	0	0
$503$	2.82372e9	0.989313	0.494657	−	0.869089i	$-0.335294\pi$
$503$	2.82372e9	0.989313	0.494657	+	0.869089i	$0.335294\pi$
$504$	0	0
$505$	−1.91370e7	−0.00661233
$506$	0	0
$507$	5.67094e9	1.93253
$508$	0	0
$509$	−3.73735e9	−1.25618	−0.628090	−	0.778141i	$-0.716163\pi$
$509$	−3.73735e9	−1.25618	−0.628090	+	0.778141i	$0.716163\pi$
$510$	0	0
$511$	1.70574e9	0.565510
$512$	0	0
$513$	4.23897e8	0.138628
$514$	0	0
$515$	−5.80518e9	−1.87280
$516$	0	0
$517$	−545870.	−0.000173729 0
$518$	0	0
$519$	1.34398e9	0.421994
$520$	0	0
$521$	3.50738e9	1.08655	0.543276	−	0.839554i	$-0.317184\pi$
$521$	3.50738e9	1.08655	0.543276	+	0.839554i	$0.317184\pi$
$522$	0	0
$523$	3.36763e9	1.02936	0.514682	−	0.857381i	$-0.327910\pi$
$523$	3.36763e9	1.02936	0.514682	+	0.857381i	$0.327910\pi$
$524$	0	0
$525$	5.04599e8	0.152191
$526$	0	0
$527$	6.39417e8	0.190304
$528$	0	0
$529$	−3.39816e9	−0.998042
$530$	0	0
$531$	3.17187e9	0.919359
$532$	0	0
$533$	2.17419e8	0.0621946
$534$	0	0
$535$	−2.09340e9	−0.591036
$536$	0	0
$537$	−8.23627e9	−2.29520
$538$	0	0
$539$	−3.33925e7	−0.00918518
$540$	0	0
$541$	4.50424e9	1.22301	0.611506	−	0.791239i	$-0.290564\pi$
$541$	4.50424e9	1.22301	0.611506	+	0.791239i	$0.290564\pi$
$542$	0	0
$543$	5.39962e9	1.44732
$544$	0	0
$545$	4.19194e9	1.10924
$546$	0	0
$547$	−2.31422e9	−0.604574	−0.302287	−	0.953217i	$-0.597750\pi$
$547$	−2.31422e9	−0.604574	−0.302287	+	0.953217i	$0.597750\pi$
$548$	0	0
$549$	4.66947e9	1.20438
$550$	0	0
$551$	4.83379e8	0.123100
$552$	0	0
$553$	−2.98535e8	−0.0750684
$554$	0	0
$555$	1.39106e10	3.45397
$556$	0	0
$557$	−5.61006e9	−1.37554	−0.687771	−	0.725927i	$-0.741411\pi$
$557$	−5.61006e9	−1.37554	−0.687771	+	0.725927i	$0.741411\pi$
$558$	0	0
$559$	−3.78014e9	−0.915306
$560$	0	0
$561$	−8.71971e7	−0.0208513
$562$	0	0
$563$	−4.02844e9	−0.951388	−0.475694	−	0.879611i	$-0.657803\pi$
$563$	−4.02844e9	−0.951388	−0.475694	+	0.879611i	$0.657803\pi$
$564$	0	0
$565$	−8.11076e9	−1.89187
$566$	0	0
$567$	1.93892e8	0.0446702
$568$	0	0
$569$	1.35716e9	0.308843	0.154421	−	0.988005i	$-0.450649\pi$
$569$	1.35716e9	0.308843	0.154421	+	0.988005i	$0.450649\pi$
$570$	0	0
$571$	3.08298e9	0.693017	0.346509	−	0.938047i	$-0.387367\pi$
$571$	3.08298e9	0.693017	0.346509	+	0.938047i	$0.387367\pi$
$572$	0	0
$573$	5.54885e9	1.23214
$574$	0	0
$575$	4.97138e7	0.0109053
$576$	0	0
$577$	2.33045e9	0.505039	0.252519	−	0.967592i	$-0.418741\pi$
$577$	2.33045e9	0.505039	0.252519	+	0.967592i	$0.418741\pi$
$578$	0	0
$579$	2.65102e8	0.0567594
$580$	0	0
$581$	1.33073e9	0.281497
$582$	0	0
$583$	3.84077e8	0.0802747
$584$	0	0
$585$	−1.33359e10	−2.75407
$586$	0	0
$587$	−6.77557e9	−1.38265	−0.691325	−	0.722544i	$-0.742973\pi$
$587$	−6.77557e9	−1.38265	−0.691325	+	0.722544i	$0.742973\pi$
$588$	0	0
$589$	6.02389e8	0.121471
$590$	0	0
$591$	−4.75603e9	−0.947739
$592$	0	0
$593$	7.09723e9	1.39765	0.698824	−	0.715294i	$-0.253707\pi$
$593$	7.09723e9	1.39765	0.698824	+	0.715294i	$0.253707\pi$
$594$	0	0
$595$	−4.30340e8	−0.0837533
$596$	0	0
$597$	−4.64218e9	−0.892918
$598$	0	0
$599$	−5.44979e9	−1.03606	−0.518032	−	0.855361i	$-0.673335\pi$
$599$	−5.44979e9	−1.03606	−0.518032	+	0.855361i	$0.673335\pi$
$600$	0	0
$601$	7.81035e8	0.146761	0.0733804	−	0.997304i	$-0.476621\pi$
$601$	7.81035e8	0.146761	0.0733804	+	0.997304i	$0.476621\pi$
$602$	0	0
$603$	2.13472e9	0.396488
$604$	0	0
$605$	6.05592e9	1.11182
$606$	0	0
$607$	3.70105e9	0.671683	0.335841	−	0.941919i	$-0.390979\pi$
$607$	3.70105e9	0.671683	0.335841	+	0.941919i	$0.390979\pi$
$608$	0	0
$609$	3.34471e9	0.600064
$610$	0	0
$611$	2.25078e7	0.00399199
$612$	0	0
$613$	1.01586e8	0.0178125	0.00890623	−	0.999960i	$-0.497165\pi$
$613$	1.01586e8	0.0178125	0.00890623	+	0.999960i	$0.497165\pi$
$614$	0	0
$615$	−4.42975e8	−0.0767920
$616$	0	0
$617$	−2.48425e9	−0.425791	−0.212896	−	0.977075i	$-0.568289\pi$
$617$	−2.48425e9	−0.425791	−0.212896	+	0.977075i	$0.568289\pi$
$618$	0	0
$619$	−6.89160e9	−1.16789	−0.583946	−	0.811792i	$-0.698492\pi$
$619$	−6.89160e9	−1.16789	−0.583946	+	0.811792i	$0.698492\pi$
$620$	0	0
$621$	2.88975e8	0.0484217
$622$	0	0
$623$	3.50783e9	0.581206
$624$	0	0
$625$	−7.23698e9	−1.18571
$626$	0	0
$627$	−8.21475e7	−0.0133094
$628$	0	0
$629$	−2.34556e9	−0.375810
$630$	0	0
$631$	−1.97310e9	−0.312641	−0.156321	−	0.987706i	$-0.549963\pi$
$631$	−1.97310e9	−0.312641	−0.156321	+	0.987706i	$0.549963\pi$
$632$	0	0
$633$	1.12246e10	1.75896
$634$	0	0
$635$	−1.01833e10	−1.57827
$636$	0	0
$637$	1.37687e9	0.211059
$638$	0	0
$639$	2.07216e10	3.14175
$640$	0	0
$641$	3.79593e9	0.569266	0.284633	−	0.958637i	$-0.408128\pi$
$641$	3.79593e9	0.569266	0.284633	+	0.958637i	$0.408128\pi$
$642$	0	0
$643$	−1.07326e10	−1.59208	−0.796042	−	0.605241i	$-0.793077\pi$
$643$	−1.07326e10	−1.59208	−0.796042	+	0.605241i	$0.793077\pi$
$644$	0	0
$645$	7.70173e9	1.13013
$646$	0	0
$647$	−5.28221e9	−0.766744	−0.383372	−	0.923594i	$-0.625237\pi$
$647$	−5.28221e9	−0.766744	−0.383372	+	0.923594i	$0.625237\pi$
$648$	0	0
$649$	−2.46541e8	−0.0354025
$650$	0	0
$651$	4.16818e9	0.592125
$652$	0	0
$653$	−9.46338e8	−0.133000	−0.0664998	−	0.997786i	$-0.521183\pi$
$653$	−9.46338e8	−0.133000	−0.0664998	+	0.997786i	$0.521183\pi$
$654$	0	0
$655$	−1.37362e10	−1.90994
$656$	0	0
$657$	1.81595e10	2.49819
$658$	0	0
$659$	−4.19089e9	−0.570436	−0.285218	−	0.958463i	$-0.592066\pi$
$659$	−4.19089e9	−0.570436	−0.285218	+	0.958463i	$0.592066\pi$
$660$	0	0
$661$	−9.39353e9	−1.26510	−0.632548	−	0.774521i	$-0.717991\pi$
$661$	−9.39353e9	−1.26510	−0.632548	+	0.774521i	$0.717991\pi$
$662$	0	0
$663$	3.59540e9	0.479126
$664$	0	0
$665$	−4.05419e8	−0.0534599
$666$	0	0
$667$	3.29525e8	0.0429980
$668$	0	0
$669$	−1.08122e10	−1.39612
$670$	0	0
$671$	−3.62946e8	−0.0463781
$672$	0	0
$673$	9.73932e9	1.23162	0.615809	−	0.787895i	$-0.288829\pi$
$673$	9.73932e9	1.23162	0.615809	+	0.787895i	$0.288829\pi$
$674$	0	0
$675$	2.15466e9	0.269659
$676$	0	0
$677$	−7.93402e9	−0.982727	−0.491364	−	0.870955i	$-0.663501\pi$
$677$	−7.93402e9	−0.982727	−0.491364	+	0.870955i	$0.663501\pi$
$678$	0	0
$679$	1.36831e9	0.167741
$680$	0	0
$681$	2.60182e9	0.315691
$682$	0	0
$683$	−1.11816e10	−1.34286	−0.671432	−	0.741067i	$-0.734320\pi$
$683$	−1.11816e10	−1.34286	−0.671432	+	0.741067i	$0.734320\pi$
$684$	0	0
$685$	−2.24609e9	−0.267000
$686$	0	0
$687$	−1.19354e10	−1.40439
$688$	0	0
$689$	−1.58366e10	−1.84457
$690$	0	0
$691$	−7.53867e9	−0.869204	−0.434602	−	0.900623i	$-0.643111\pi$
$691$	−7.53867e9	−0.869204	−0.434602	+	0.900623i	$0.643111\pi$
$692$	0	0
$693$	−3.55500e8	−0.0405764
$694$	0	0
$695$	7.81631e9	0.883192
$696$	0	0
$697$	7.46930e7	0.00835536
$698$	0	0
$699$	1.46599e10	1.62354
$700$	0	0
$701$	1.21953e10	1.33714	0.668572	−	0.743647i	$-0.266906\pi$
$701$	1.21953e10	1.33714	0.668572	+	0.743647i	$0.266906\pi$
$702$	0	0
$703$	−2.20972e9	−0.239880
$704$	0	0
$705$	−4.58580e7	−0.00492893
$706$	0	0
$707$	2.10348e7	0.00223857
$708$	0	0
$709$	−1.22512e10	−1.29097	−0.645484	−	0.763774i	$-0.723344\pi$
$709$	−1.22512e10	−1.29097	−0.645484	+	0.763774i	$0.723344\pi$
$710$	0	0
$711$	−3.17824e9	−0.331622
$712$	0	0
$713$	4.10655e8	0.0424291
$714$	0	0
$715$	1.03656e9	0.106053
$716$	0	0
$717$	2.35155e10	2.38252
$718$	0	0
$719$	−8.02518e9	−0.805200	−0.402600	−	0.915376i	$-0.631893\pi$
$719$	−8.02518e9	−0.805200	−0.402600	+	0.915376i	$0.631893\pi$
$720$	0	0
$721$	6.38086e9	0.634024
$722$	0	0
$723$	−1.63903e10	−1.61288
$724$	0	0
$725$	2.45701e9	0.239455
$726$	0	0
$727$	−3.03346e9	−0.292797	−0.146399	−	0.989226i	$-0.546768\pi$
$727$	−3.03346e9	−0.292797	−0.146399	+	0.989226i	$0.546768\pi$
$728$	0	0
$729$	−1.66377e10	−1.59055
$730$	0	0
$731$	−1.29864e9	−0.122964
$732$	0	0
$733$	1.96893e10	1.84657	0.923285	−	0.384116i	$-0.125494\pi$
$733$	1.96893e10	1.84657	0.923285	+	0.384116i	$0.125494\pi$
$734$	0	0
$735$	−2.80526e9	−0.260596
$736$	0	0
$737$	−1.65926e8	−0.0152679
$738$	0	0
$739$	5.69524e9	0.519106	0.259553	−	0.965729i	$-0.416425\pi$
$739$	5.69524e9	0.519106	0.259553	+	0.965729i	$0.416425\pi$
$740$	0	0
$741$	3.38719e9	0.305827
$742$	0	0
$743$	−2.15270e10	−1.92540	−0.962702	−	0.270563i	$-0.912790\pi$
$743$	−2.15270e10	−1.92540	−0.962702	+	0.270563i	$0.912790\pi$
$744$	0	0
$745$	−1.18586e7	−0.00105071
$746$	0	0
$747$	1.41671e10	1.24354
$748$	0	0
$749$	2.30100e9	0.200092
$750$	0	0
$751$	−1.14509e10	−0.986507	−0.493254	−	0.869886i	$-0.664193\pi$
$751$	−1.14509e10	−0.986507	−0.493254	+	0.869886i	$0.664193\pi$
$752$	0	0
$753$	−1.22628e10	−1.04666
$754$	0	0
$755$	2.57957e9	0.218139
$756$	0	0
$757$	1.65342e10	1.38531	0.692655	−	0.721269i	$-0.256441\pi$
$757$	1.65342e10	1.38531	0.692655	+	0.721269i	$0.256441\pi$
$758$	0	0
$759$	−5.60009e7	−0.00464888
$760$	0	0
$761$	9.30606e9	0.765455	0.382728	−	0.923861i	$-0.374985\pi$
$761$	9.30606e9	0.765455	0.382728	+	0.923861i	$0.374985\pi$
$762$	0	0
$763$	−4.60764e9	−0.375529
$764$	0	0
$765$	−4.58145e9	−0.369988
$766$	0	0
$767$	1.01656e10	0.813487
$768$	0	0
$769$	4.85159e9	0.384717	0.192359	−	0.981325i	$-0.438386\pi$
$769$	4.85159e9	0.384717	0.192359	+	0.981325i	$0.438386\pi$
$770$	0	0
$771$	1.02487e9	0.0805339
$772$	0	0
$773$	−2.04661e10	−1.59370	−0.796852	−	0.604175i	$-0.793503\pi$
$773$	−2.04661e10	−1.59370	−0.796852	+	0.604175i	$0.793503\pi$
$774$	0	0
$775$	3.06193e9	0.236287
$776$	0	0
$777$	−1.52900e10	−1.16932
$778$	0	0
$779$	7.03676e7	0.00533324
$780$	0	0
$781$	−1.61064e9	−0.120982
$782$	0	0
$783$	1.42820e10	1.06322
$784$	0	0
$785$	1.24933e10	0.921795
$786$	0	0
$787$	−1.69075e10	−1.23642	−0.618211	−	0.786012i	$-0.712142\pi$
$787$	−1.69075e10	−1.23642	−0.618211	+	0.786012i	$0.712142\pi$
$788$	0	0
$789$	1.14268e10	0.828235
$790$	0	0
$791$	8.91508e9	0.640483
$792$	0	0
$793$	1.49653e10	1.06569
$794$	0	0
$795$	3.22659e10	2.27750
$796$	0	0
$797$	1.04945e10	0.734273	0.367136	−	0.930167i	$-0.380338\pi$
$797$	1.04945e10	0.734273	0.367136	+	0.930167i	$0.380338\pi$
$798$	0	0
$799$	7.73243e6	0.000536293 0
$800$	0	0
$801$	3.73448e10	2.56754
$802$	0	0
$803$	−1.41149e9	−0.0961999
$804$	0	0
$805$	−2.76378e8	−0.0186732
$806$	0	0
$807$	1.54446e10	1.03448
$808$	0	0
$809$	−1.97682e10	−1.31265	−0.656324	−	0.754479i	$-0.727889\pi$
$809$	−1.97682e10	−1.31265	−0.656324	+	0.754479i	$0.727889\pi$
$810$	0	0
$811$	1.15699e10	0.761653	0.380826	−	0.924647i	$-0.375640\pi$
$811$	1.15699e10	0.761653	0.380826	+	0.924647i	$0.375640\pi$
$812$	0	0
$813$	−1.21572e9	−0.0793445
$814$	0	0
$815$	2.32033e10	1.50141
$816$	0	0
$817$	−1.22344e9	−0.0784883
$818$	0	0
$819$	1.46583e10	0.932376
$820$	0	0
$821$	8.26784e9	0.521424	0.260712	−	0.965417i	$-0.416043\pi$
$821$	8.26784e9	0.521424	0.260712	+	0.965417i	$0.416043\pi$
$822$	0	0
$823$	−7.82205e9	−0.489126	−0.244563	−	0.969633i	$-0.578645\pi$
$823$	−7.82205e9	−0.489126	−0.244563	+	0.969633i	$0.578645\pi$
$824$	0	0
$825$	−4.17554e8	−0.0258895
$826$	0	0
$827$	−6.32518e9	−0.388869	−0.194435	−	0.980915i	$-0.562287\pi$
$827$	−6.32518e9	−0.388869	−0.194435	+	0.980915i	$0.562287\pi$
$828$	0	0
$829$	2.15011e8	0.0131075	0.00655376	−	0.999979i	$-0.497914\pi$
$829$	2.15011e8	0.0131075	0.00655376	+	0.999979i	$0.497914\pi$
$830$	0	0
$831$	2.16752e10	1.31027
$832$	0	0
$833$	4.73015e8	0.0283542
$834$	0	0
$835$	1.91874e10	1.14055
$836$	0	0
$837$	1.77983e10	1.04915
$838$	0	0
$839$	−2.80409e10	−1.63917	−0.819587	−	0.572955i	$-0.805797\pi$
$839$	−2.80409e10	−1.63917	−0.819587	+	0.572955i	$0.805797\pi$
$840$	0	0
$841$	−9.63768e8	−0.0558710
$842$	0	0
$843$	1.08511e10	0.623848
$844$	0	0
$845$	−2.31595e10	−1.32048
$846$	0	0
$847$	−6.65647e9	−0.376402
$848$	0	0
$849$	1.53266e10	0.859545
$850$	0	0
$851$	−1.50639e9	−0.0837886
$852$	0	0
$853$	3.24393e10	1.78957	0.894786	−	0.446495i	$-0.147328\pi$
$853$	3.24393e10	1.78957	0.894786	+	0.446495i	$0.147328\pi$
$854$	0	0
$855$	−4.31614e9	−0.236164
$856$	0	0
$857$	1.73118e10	0.939525	0.469762	−	0.882793i	$-0.344340\pi$
$857$	1.73118e10	0.939525	0.469762	+	0.882793i	$0.344340\pi$
$858$	0	0
$859$	7.48729e9	0.403040	0.201520	−	0.979484i	$-0.435412\pi$
$859$	7.48729e9	0.403040	0.201520	+	0.979484i	$0.435412\pi$
$860$	0	0
$861$	4.86903e8	0.0259975
$862$	0	0
$863$	−2.23774e10	−1.18514	−0.592572	−	0.805517i	$-0.701887\pi$
$863$	−2.23774e10	−1.18514	−0.592572	+	0.805517i	$0.701887\pi$
$864$	0	0
$865$	−5.48867e9	−0.288344
$866$	0	0
$867$	−3.01192e10	−1.56955
$868$	0	0
$869$	2.47036e8	0.0127700
$870$	0	0
$871$	6.84163e9	0.350829
$872$	0	0
$873$	1.45672e10	0.741014
$874$	0	0
$875$	6.30135e9	0.317984
$876$	0	0
$877$	2.95004e10	1.47683	0.738414	−	0.674348i	$-0.235575\pi$
$877$	2.95004e10	1.47683	0.738414	+	0.674348i	$0.235575\pi$
$878$	0	0
$879$	3.93897e10	1.95624
$880$	0	0
$881$	−2.42854e10	−1.19655	−0.598273	−	0.801293i	$-0.704146\pi$
$881$	−2.42854e10	−1.19655	−0.598273	+	0.801293i	$0.704146\pi$
$882$	0	0
$883$	−2.14449e10	−1.04824	−0.524120	−	0.851644i	$-0.675606\pi$
$883$	−2.14449e10	−1.04824	−0.524120	+	0.851644i	$0.675606\pi$
$884$	0	0
$885$	−2.07117e10	−1.00442
$886$	0	0
$887$	−3.53723e10	−1.70189	−0.850943	−	0.525258i	$-0.823969\pi$
$887$	−3.53723e10	−1.70189	−0.850943	+	0.525258i	$0.823969\pi$
$888$	0	0
$889$	1.11932e10	0.534316
$890$	0	0
$891$	−1.60445e8	−0.00759894
$892$	0	0
$893$	7.28464e6	0.000342317 0
$894$	0	0
$895$	3.36361e10	1.56829
$896$	0	0
$897$	2.30908e9	0.106823
$898$	0	0
$899$	2.02958e10	0.931638
$900$	0	0
$901$	−5.44058e9	−0.247804
$902$	0	0
$903$	−8.46549e9	−0.382600
$904$	0	0
$905$	−2.20515e10	−0.988936
$906$	0	0
$907$	1.03301e10	0.459703	0.229852	−	0.973226i	$-0.426176\pi$
$907$	1.03301e10	0.459703	0.229852	+	0.973226i	$0.426176\pi$
$908$	0	0
$909$	2.23939e8	0.00988910
$910$	0	0
$911$	7.15628e9	0.313598	0.156799	−	0.987631i	$-0.449883\pi$
$911$	7.15628e9	0.313598	0.156799	+	0.987631i	$0.449883\pi$
$912$	0	0
$913$	−1.10118e9	−0.0478860
$914$	0	0
$915$	−3.04907e10	−1.31581
$916$	0	0
$917$	1.50983e10	0.646601
$918$	0	0
$919$	−1.78768e10	−0.759776	−0.379888	−	0.925032i	$-0.624038\pi$
$919$	−1.78768e10	−0.759776	−0.379888	+	0.925032i	$0.624038\pi$
$920$	0	0
$921$	5.85083e10	2.46779
$922$	0	0
$923$	6.64114e10	2.77995
$924$	0	0
$925$	−1.12320e10	−0.466617
$926$	0	0
$927$	6.79315e10	2.80086
$928$	0	0
$929$	−8.30589e9	−0.339884	−0.169942	−	0.985454i	$-0.554358\pi$
$929$	−8.30589e9	−0.339884	−0.169942	+	0.985454i	$0.554358\pi$
$930$	0	0
$931$	4.45623e8	0.0180985
$932$	0	0
$933$	−1.22155e10	−0.492410
$934$	0	0
$935$	3.56104e8	0.0142474
$936$	0	0
$937$	−1.89534e10	−0.752660	−0.376330	−	0.926486i	$-0.622814\pi$
$937$	−1.89534e10	−0.752660	−0.376330	+	0.926486i	$0.622814\pi$
$938$	0	0
$939$	−1.58750e10	−0.625727
$940$	0	0
$941$	−1.85991e9	−0.0727659	−0.0363830	−	0.999338i	$-0.511584\pi$
$941$	−1.85991e9	−0.0727659	−0.0363830	+	0.999338i	$0.511584\pi$
$942$	0	0
$943$	4.79703e7	0.00186287
$944$	0	0
$945$	−1.19786e10	−0.461736
$946$	0	0
$947$	−2.17737e10	−0.833121	−0.416561	−	0.909108i	$-0.636765\pi$
$947$	−2.17737e10	−0.833121	−0.416561	+	0.909108i	$0.636765\pi$
$948$	0	0
$949$	5.82001e10	2.21051
$950$	0	0
$951$	−4.18506e9	−0.157787
$952$	0	0
$953$	2.45189e10	0.917646	0.458823	−	0.888528i	$-0.348271\pi$
$953$	2.45189e10	0.917646	0.458823	+	0.888528i	$0.348271\pi$
$954$	0	0
$955$	−2.26609e10	−0.841911
$956$	0	0
$957$	−2.76773e9	−0.102078
$958$	0	0
$959$	2.46883e9	0.0903913
$960$	0	0
$961$	−2.21993e9	−0.0806875
$962$	0	0
$963$	2.44967e10	0.883926
$964$	0	0
$965$	−1.08265e9	−0.0387831
$966$	0	0
$967$	−3.75735e10	−1.33625	−0.668127	−	0.744047i	$-0.732904\pi$
$967$	−3.75735e10	−1.33625	−0.668127	+	0.744047i	$0.732904\pi$
$968$	0	0
$969$	1.16365e9	0.0410854
$970$	0	0
$971$	−3.61803e10	−1.26825	−0.634125	−	0.773231i	$-0.718639\pi$
$971$	−3.61803e10	−1.26825	−0.634125	+	0.773231i	$0.718639\pi$
$972$	0	0
$973$	−8.59143e9	−0.299000
$974$	0	0
$975$	1.72170e10	0.594896
$976$	0	0
$977$	4.30992e10	1.47856	0.739279	−	0.673399i	$-0.235166\pi$
$977$	4.30992e10	1.47856	0.739279	+	0.673399i	$0.235166\pi$
$978$	0	0
$979$	−2.90272e9	−0.0988701
$980$	0	0
$981$	−4.90536e10	−1.65893
$982$	0	0
$983$	2.72338e10	0.914472	0.457236	−	0.889345i	$-0.348839\pi$
$983$	2.72338e10	0.914472	0.457236	+	0.889345i	$0.348839\pi$
$984$	0	0
$985$	1.94232e10	0.647580
$986$	0	0
$987$	5.04056e7	0.00166866
$988$	0	0
$989$	−8.34031e8	−0.0274155
$990$	0	0
$991$	4.11594e10	1.34342	0.671709	−	0.740815i	$-0.265561\pi$
$991$	4.11594e10	1.34342	0.671709	+	0.740815i	$0.265561\pi$
$992$	0	0
$993$	−1.84278e10	−0.597242
$994$	0	0
$995$	1.89582e10	0.610121
$996$	0	0
$997$	−4.76679e10	−1.52333	−0.761663	−	0.647973i	$-0.775617\pi$
$997$	−4.76679e10	−1.52333	−0.761663	+	0.647973i	$0.775617\pi$
$998$	0	0
$999$	−6.52890e10	−2.07186

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.8.a.t.1.2		2
4.3	odd	2		448.8.a.k.1.1		2
8.3	odd	2		7.8.a.b.1.1	✓	2
8.5	even	2		112.8.a.f.1.1		2
24.11	even	2		63.8.a.e.1.2		2
40.3	even	4		175.8.b.b.99.4		4
40.19	odd	2		175.8.a.c.1.2		2
40.27	even	4		175.8.b.b.99.1		4
56.3	even	6		49.8.c.f.30.2		4
56.11	odd	6		49.8.c.e.30.2		4
56.19	even	6		49.8.c.f.18.2		4
56.27	even	2		49.8.a.c.1.1		2
56.51	odd	6		49.8.c.e.18.2		4
168.83	odd	2		441.8.a.l.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7.8.a.b.1.1	✓	2	8.3	odd	2
49.8.a.c.1.1		2	56.27	even	2
49.8.c.e.18.2		4	56.51	odd	6
49.8.c.e.30.2		4	56.11	odd	6
49.8.c.f.18.2		4	56.19	even	6
49.8.c.f.30.2		4	56.3	even	6
63.8.a.e.1.2		2	24.11	even	2
112.8.a.f.1.1		2	8.5	even	2
175.8.a.c.1.2		2	40.19	odd	2
175.8.b.b.99.1		4	40.27	even	4
175.8.b.b.99.4		4	40.3	even	4
441.8.a.l.1.2		2	168.83	odd	2
448.8.a.k.1.1		2	4.3	odd	2
448.8.a.t.1.2		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	448.a (trivial)

\(f(q)\)	\(=\)	\(q+76.4109 q^{3} -312.054 q^{5} +343.000 q^{7} +3651.62 q^{9} +O(q^{10})\)	\(q+76.4109 q^{3} -312.054 q^{5} +343.000 q^{7} +3651.62 q^{9} -283.831 q^{11} +11703.2 q^{13} -23844.4 q^{15} +4020.56 q^{17} +3787.73 q^{19} +26208.9 q^{21} +2582.14 q^{23} +19253.0 q^{25} +111913. q^{27} +127617. q^{29} +159037. q^{31} -21687.8 q^{33} -107035. q^{35} -583390. q^{37} +894252. q^{39} +18577.8 q^{41} -323000. q^{43} -1.13951e6 q^{45} +1923.22 q^{47} +117649. q^{49} +307215. q^{51} -1.35319e6 q^{53} +88570.8 q^{55} +289424. q^{57} +868620. q^{59} +1.27874e6 q^{61} +1.25251e6 q^{63} -3.65203e6 q^{65} +584595. q^{67} +197304. q^{69} +5.67464e6 q^{71} +4.97301e6 q^{73} +1.47114e6 q^{75} -97354.1 q^{77} -870364. q^{79} +565282. q^{81} +3.87968e6 q^{83} -1.25463e6 q^{85} +9.75133e6 q^{87} +1.02269e7 q^{89} +4.01420e6 q^{91} +1.21521e7 q^{93} -1.18198e6 q^{95} +3.98924e6 q^{97} -1.03644e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 94 q^{3} - 330 q^{5} + 686 q^{7} + 1774 q^{9}+O(q^{10}) \) 2 * q + 94 * q^3 - 330 * q^5 + 686 * q^7 + 1774 * q^9	\( 2 q + 94 q^{3} - 330 q^{5} + 686 q^{7} + 1774 q^{9} + 2844 q^{11} - 2534 q^{13} - 24160 q^{15} - 1488 q^{17} + 32810 q^{19} + 32242 q^{21} + 6576 q^{23} - 58550 q^{25} + 40420 q^{27} - 20640 q^{29} + 391836 q^{31} + 33328 q^{33} - 113190 q^{35} - 367392 q^{37} + 643832 q^{39} + 734664 q^{41} - 480476 q^{43} - 1105810 q^{45} + 1089108 q^{47} + 235298 q^{49} + 210324 q^{51} - 2858844 q^{53} + 32440 q^{55} + 799900 q^{57} + 160170 q^{59} + 864646 q^{61} + 608482 q^{63} - 3396540 q^{65} - 328648 q^{67} + 267552 q^{69} + 7500216 q^{71} + 4301244 q^{73} + 102650 q^{75} + 975492 q^{77} + 6408440 q^{79} + 3414142 q^{81} + 11659074 q^{83} - 1155780 q^{85} + 7143620 q^{87} + 9772260 q^{89} - 869162 q^{91} + 16246872 q^{93} - 1702800 q^{95} + 10762752 q^{97} - 6909332 q^{99}+O(q^{100}) \) 2 * q + 94 * q^3 - 330 * q^5 + 686 * q^7 + 1774 * q^9 + 2844 * q^11 - 2534 * q^13 - 24160 * q^15 - 1488 * q^17 + 32810 * q^19 + 32242 * q^21 + 6576 * q^23 - 58550 * q^25 + 40420 * q^27 - 20640 * q^29 + 391836 * q^31 + 33328 * q^33 - 113190 * q^35 - 367392 * q^37 + 643832 * q^39 + 734664 * q^41 - 480476 * q^43 - 1105810 * q^45 + 1089108 * q^47 + 235298 * q^49 + 210324 * q^51 - 2858844 * q^53 + 32440 * q^55 + 799900 * q^57 + 160170 * q^59 + 864646 * q^61 + 608482 * q^63 - 3396540 * q^65 - 328648 * q^67 + 267552 * q^69 + 7500216 * q^71 + 4301244 * q^73 + 102650 * q^75 + 975492 * q^77 + 6408440 * q^79 + 3414142 * q^81 + 11659074 * q^83 - 1155780 * q^85 + 7143620 * q^87 + 9772260 * q^89 - 869162 * q^91 + 16246872 * q^93 - 1702800 * q^95 + 10762752 * q^97 - 6909332 * q^99

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