LMFDB - Embedded newform 140.6.a.d.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [140,6,Mod(1,140)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(140, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("140.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 140 = 2^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	140.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:	$22.4537347738$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\mathbb{Q}[x]/(x^{3} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 499x - 210 $ x^3 - 499*x - 210
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$22.5458$ of defining polynomial
Character	$\chi$	$=$	140.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+24.5458 q^{3} +25.0000 q^{5} +49.0000 q^{7} +359.498 q^{9} +O(q^{10})$	$q+24.5458 q^{3} +25.0000 q^{5} +49.0000 q^{7} +359.498 q^{9} +90.2356 q^{11} -14.4413 q^{13} +613.646 q^{15} +407.384 q^{17} +2289.19 q^{19} +1202.75 q^{21} -505.976 q^{23} +625.000 q^{25} +2859.53 q^{27} -3164.56 q^{29} -6231.43 q^{31} +2214.91 q^{33} +1225.00 q^{35} +5388.70 q^{37} -354.474 q^{39} +11764.3 q^{41} -5824.58 q^{43} +8987.44 q^{45} +7349.38 q^{47} +2401.00 q^{49} +9999.57 q^{51} +14971.6 q^{53} +2255.89 q^{55} +56190.1 q^{57} -47153.5 q^{59} -4289.61 q^{61} +17615.4 q^{63} -361.033 q^{65} +48898.3 q^{67} -12419.6 q^{69} +58567.6 q^{71} -1599.07 q^{73} +15341.1 q^{75} +4421.54 q^{77} -79121.1 q^{79} -17168.4 q^{81} -71472.1 q^{83} +10184.6 q^{85} -77676.9 q^{87} -77165.9 q^{89} -707.624 q^{91} -152956. q^{93} +57229.8 q^{95} +11559.6 q^{97} +32439.5 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 6 q^{3} + 75 q^{5} + 147 q^{7} + 281 q^{9}+O(q^{10}) $ 3 * q + 6 * q^3 + 75 * q^5 + 147 * q^7 + 281 * q^9	$ 3 q + 6 q^{3} + 75 q^{5} + 147 q^{7} + 281 q^{9} - 14 q^{11} - 4 q^{13} + 150 q^{15} + 44 q^{17} + 2328 q^{19} + 294 q^{21} + 3676 q^{23} + 1875 q^{25} + 3726 q^{27} + 4092 q^{29} + 5888 q^{31} + 11318 q^{33} + 3675 q^{35} + 11378 q^{37} + 13702 q^{39} + 11450 q^{41} + 18544 q^{43} + 7025 q^{45} + 21754 q^{47} + 7203 q^{49} + 31762 q^{51} + 7494 q^{53} - 350 q^{55} + 7332 q^{57} + 12388 q^{59} + 27182 q^{61} + 13769 q^{63} - 100 q^{65} - 7676 q^{67} - 62140 q^{69} + 81992 q^{71} + 230 q^{73} + 3750 q^{75} - 686 q^{77} - 15926 q^{79} - 32101 q^{81} - 86100 q^{83} + 1100 q^{85} - 85098 q^{87} - 95710 q^{89} - 196 q^{91} - 229472 q^{93} + 58200 q^{95} - 176188 q^{97} - 114368 q^{99}+O(q^{100}) $ 3 * q + 6 * q^3 + 75 * q^5 + 147 * q^7 + 281 * q^9 - 14 * q^11 - 4 * q^13 + 150 * q^15 + 44 * q^17 + 2328 * q^19 + 294 * q^21 + 3676 * q^23 + 1875 * q^25 + 3726 * q^27 + 4092 * q^29 + 5888 * q^31 + 11318 * q^33 + 3675 * q^35 + 11378 * q^37 + 13702 * q^39 + 11450 * q^41 + 18544 * q^43 + 7025 * q^45 + 21754 * q^47 + 7203 * q^49 + 31762 * q^51 + 7494 * q^53 - 350 * q^55 + 7332 * q^57 + 12388 * q^59 + 27182 * q^61 + 13769 * q^63 - 100 * q^65 - 7676 * q^67 - 62140 * q^69 + 81992 * q^71 + 230 * q^73 + 3750 * q^75 - 686 * q^77 - 15926 * q^79 - 32101 * q^81 - 86100 * q^83 + 1100 * q^85 - 85098 * q^87 - 95710 * q^89 - 196 * q^91 - 229472 * q^93 + 58200 * q^95 - 176188 * q^97 - 114368 * 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
$3$	24.5458	1.57462	0.787308	−	0.616560i	$-0.211474\pi$
$3$	24.5458	1.57462	0.787308	+	0.616560i	$0.211474\pi$
$4$	0	0
$5$	25.0000	0.447214
$5$	25.0000	0.447214
$6$	0	0
$7$	49.0000	0.377964
$7$	49.0000	0.377964
$8$	0	0
$9$	359.498	1.47941
$10$	0	0
$11$	90.2356	0.224852	0.112426	−	0.993660i	$-0.464138\pi$
$11$	90.2356	0.224852	0.112426	+	0.993660i	$0.464138\pi$
$12$	0	0
$13$	−14.4413	−0.0237000	−0.0118500	−	0.999930i	$-0.503772\pi$
$13$	−14.4413	−0.0237000	−0.0118500	+	0.999930i	$0.503772\pi$
$14$	0	0
$15$	613.646	0.704190
$16$	0	0
$17$	407.384	0.341886	0.170943	−	0.985281i	$-0.445319\pi$
$17$	407.384	0.341886	0.170943	+	0.985281i	$0.445319\pi$
$18$	0	0
$19$	2289.19	1.45478	0.727391	−	0.686223i	$-0.240732\pi$
$19$	2289.19	1.45478	0.727391	+	0.686223i	$0.240732\pi$
$20$	0	0
$21$	1202.75	0.595149
$22$	0	0
$23$	−505.976	−0.199439	−0.0997197	−	0.995016i	$-0.531795\pi$
$23$	−505.976	−0.199439	−0.0997197	+	0.995016i	$0.531795\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	2859.53	0.754893
$28$	0	0
$29$	−3164.56	−0.698745	−0.349373	−	0.936984i	$-0.613605\pi$
$29$	−3164.56	−0.698745	−0.349373	+	0.936984i	$0.613605\pi$
$30$	0	0
$31$	−6231.43	−1.16462	−0.582309	−	0.812968i	$-0.697851\pi$
$31$	−6231.43	−1.16462	−0.582309	+	0.812968i	$0.697851\pi$
$32$	0	0
$33$	2214.91	0.354055
$34$	0	0
$35$	1225.00	0.169031
$36$	0	0
$37$	5388.70	0.647113	0.323556	−	0.946209i	$-0.395121\pi$
$37$	5388.70	0.647113	0.323556	+	0.946209i	$0.395121\pi$
$38$	0	0
$39$	−354.474	−0.0373184
$40$	0	0
$41$	11764.3	1.09297	0.546484	−	0.837469i	$-0.315966\pi$
$41$	11764.3	1.09297	0.546484	+	0.837469i	$0.315966\pi$
$42$	0	0
$43$	−5824.58	−0.480390	−0.240195	−	0.970725i	$-0.577211\pi$
$43$	−5824.58	−0.480390	−0.240195	+	0.970725i	$0.577211\pi$
$44$	0	0
$45$	8987.44	0.661614
$46$	0	0
$47$	7349.38	0.485295	0.242648	−	0.970114i	$-0.421984\pi$
$47$	7349.38	0.485295	0.242648	+	0.970114i	$0.421984\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	9999.57	0.538339
$52$	0	0
$53$	14971.6	0.732114	0.366057	−	0.930592i	$-0.380707\pi$
$53$	14971.6	0.732114	0.366057	+	0.930592i	$0.380707\pi$
$54$	0	0
$55$	2255.89	0.100557
$56$	0	0
$57$	56190.1	2.29072
$58$	0	0
$59$	−47153.5	−1.76353	−0.881767	−	0.471685i	$-0.843646\pi$
$59$	−47153.5	−1.76353	−0.881767	+	0.471685i	$0.843646\pi$
$60$	0	0
$61$	−4289.61	−0.147602	−0.0738011	−	0.997273i	$-0.523513\pi$
$61$	−4289.61	−0.147602	−0.0738011	+	0.997273i	$0.523513\pi$
$62$	0	0
$63$	17615.4	0.559166
$64$	0	0
$65$	−361.033	−0.0105990
$66$	0	0
$67$	48898.3	1.33078	0.665390	−	0.746496i	$-0.268265\pi$
$67$	48898.3	1.33078	0.665390	+	0.746496i	$0.268265\pi$
$68$	0	0
$69$	−12419.6	−0.314040
$70$	0	0
$71$	58567.6	1.37883	0.689417	−	0.724365i	$-0.257867\pi$
$71$	58567.6	1.37883	0.689417	+	0.724365i	$0.257867\pi$
$72$	0	0
$73$	−1599.07	−0.0351206	−0.0175603	−	0.999846i	$-0.505590\pi$
$73$	−1599.07	−0.0351206	−0.0175603	+	0.999846i	$0.505590\pi$
$74$	0	0
$75$	15341.1	0.314923
$76$	0	0
$77$	4421.54	0.0849859
$78$	0	0
$79$	−79121.1	−1.42634	−0.713172	−	0.700989i	$-0.752742\pi$
$79$	−79121.1	−1.42634	−0.713172	+	0.700989i	$0.752742\pi$
$80$	0	0
$81$	−17168.4	−0.290748
$82$	0	0
$83$	−71472.1	−1.13878	−0.569392	−	0.822066i	$-0.692821\pi$
$83$	−71472.1	−1.13878	−0.569392	+	0.822066i	$0.692821\pi$
$84$	0	0
$85$	10184.6	0.152896
$86$	0	0
$87$	−77676.9	−1.10026
$88$	0	0
$89$	−77165.9	−1.03264	−0.516322	−	0.856395i	$-0.672699\pi$
$89$	−77165.9	−1.03264	−0.516322	+	0.856395i	$0.672699\pi$
$90$	0	0
$91$	−707.624	−0.00895775
$92$	0	0
$93$	−152956.	−1.83383
$94$	0	0
$95$	57229.8	0.650599
$96$	0	0
$97$	11559.6	0.124743	0.0623713	−	0.998053i	$-0.480134\pi$
$97$	11559.6	0.124743	0.0623713	+	0.998053i	$0.480134\pi$
$98$	0	0
$99$	32439.5	0.332649
$100$	0	0
$101$	−171423.	−1.67211	−0.836057	−	0.548642i	$-0.815145\pi$
$101$	−171423.	−1.67211	−0.836057	+	0.548642i	$0.815145\pi$
$102$	0	0
$103$	−182469.	−1.69471	−0.847357	−	0.531024i	$-0.821807\pi$
$103$	−182469.	−1.69471	−0.847357	+	0.531024i	$0.821807\pi$
$104$	0	0
$105$	30068.6	0.266159
$106$	0	0
$107$	−76696.7	−0.647616	−0.323808	−	0.946123i	$-0.604963\pi$
$107$	−76696.7	−0.647616	−0.323808	+	0.946123i	$0.604963\pi$
$108$	0	0
$109$	−13146.0	−0.105981	−0.0529904	−	0.998595i	$-0.516875\pi$
$109$	−13146.0	−0.105981	−0.0529904	+	0.998595i	$0.516875\pi$
$110$	0	0
$111$	132270.	1.01895
$112$	0	0
$113$	5996.97	0.0441811	0.0220905	−	0.999756i	$-0.492968\pi$
$113$	5996.97	0.0441811	0.0220905	+	0.999756i	$0.492968\pi$
$114$	0	0
$115$	−12649.4	−0.0891920
$116$	0	0
$117$	−5191.61	−0.0350621
$118$	0	0
$119$	19961.8	0.129221
$120$	0	0
$121$	−152909.	−0.949442
$122$	0	0
$123$	288765.	1.72101
$124$	0	0
$125$	15625.0	0.0894427
$126$	0	0
$127$	−109534.	−0.602613	−0.301306	−	0.953527i	$-0.597423\pi$
$127$	−109534.	−0.602613	−0.301306	+	0.953527i	$0.597423\pi$
$128$	0	0
$129$	−142969.	−0.756429
$130$	0	0
$131$	385186.	1.96107	0.980533	−	0.196354i	$-0.0629101\pi$
$131$	385186.	1.96107	0.980533	+	0.196354i	$0.0629101\pi$
$132$	0	0
$133$	112170.	0.549856
$134$	0	0
$135$	71488.3	0.337599
$136$	0	0
$137$	−315816.	−1.43758	−0.718791	−	0.695226i	$-0.755304\pi$
$137$	−315816.	−1.43758	−0.718791	+	0.695226i	$0.755304\pi$
$138$	0	0
$139$	95433.5	0.418952	0.209476	−	0.977814i	$-0.432824\pi$
$139$	95433.5	0.418952	0.209476	+	0.977814i	$0.432824\pi$
$140$	0	0
$141$	180397.	0.764154
$142$	0	0
$143$	−1303.12	−0.00532898
$144$	0	0
$145$	−79114.1	−0.312488
$146$	0	0
$147$	58934.5	0.224945
$148$	0	0
$149$	−151387.	−0.558629	−0.279315	−	0.960200i	$-0.590107\pi$
$149$	−151387.	−0.558629	−0.279315	+	0.960200i	$0.590107\pi$
$150$	0	0
$151$	300641.	1.07301	0.536507	−	0.843896i	$-0.319743\pi$
$151$	300641.	1.07301	0.536507	+	0.843896i	$0.319743\pi$
$152$	0	0
$153$	146453.	0.505791
$154$	0	0
$155$	−155786.	−0.520833
$156$	0	0
$157$	−306110.	−0.991125	−0.495563	−	0.868572i	$-0.665038\pi$
$157$	−306110.	−0.991125	−0.495563	+	0.868572i	$0.665038\pi$
$158$	0	0
$159$	367491.	1.15280
$160$	0	0
$161$	−24792.8	−0.0753810
$162$	0	0
$163$	373022.	1.09968	0.549839	−	0.835271i	$-0.314689\pi$
$163$	373022.	1.09968	0.549839	+	0.835271i	$0.314689\pi$
$164$	0	0
$165$	55372.7	0.158338
$166$	0	0
$167$	−317441.	−0.880789	−0.440394	−	0.897804i	$-0.645161\pi$
$167$	−317441.	−0.880789	−0.440394	+	0.897804i	$0.645161\pi$
$168$	0	0
$169$	−371084.	−0.999438
$170$	0	0
$171$	822959.	2.15223
$172$	0	0
$173$	−354815.	−0.901335	−0.450668	−	0.892692i	$-0.648814\pi$
$173$	−354815.	−0.901335	−0.450668	+	0.892692i	$0.648814\pi$
$174$	0	0
$175$	30625.0	0.0755929
$176$	0	0
$177$	−1.15742e6	−2.77689
$178$	0	0
$179$	356294.	0.831144	0.415572	−	0.909560i	$-0.363581\pi$
$179$	356294.	0.831144	0.415572	+	0.909560i	$0.363581\pi$
$180$	0	0
$181$	9772.82	0.0221729	0.0110865	−	0.999939i	$-0.496471\pi$
$181$	9772.82	0.0221729	0.0110865	+	0.999939i	$0.496471\pi$
$182$	0	0
$183$	−105292.	−0.232417
$184$	0	0
$185$	134718.	0.289398
$186$	0	0
$187$	36760.5	0.0768736
$188$	0	0
$189$	140117.	0.285323
$190$	0	0
$191$	−302267.	−0.599525	−0.299762	−	0.954014i	$-0.596907\pi$
$191$	−302267.	−0.599525	−0.299762	+	0.954014i	$0.596907\pi$
$192$	0	0
$193$	267712.	0.517339	0.258669	−	0.965966i	$-0.416716\pi$
$193$	267712.	0.517339	0.258669	+	0.965966i	$0.416716\pi$
$194$	0	0
$195$	−8861.84	−0.0166893
$196$	0	0
$197$	−151930.	−0.278920	−0.139460	−	0.990228i	$-0.544537\pi$
$197$	−151930.	−0.278920	−0.139460	+	0.990228i	$0.544537\pi$
$198$	0	0
$199$	825686.	1.47803	0.739013	−	0.673691i	$-0.235292\pi$
$199$	825686.	1.47803	0.739013	+	0.673691i	$0.235292\pi$
$200$	0	0
$201$	1.20025e6	2.09547
$202$	0	0
$203$	−155064.	−0.264101
$204$	0	0
$205$	294108.	0.488790
$206$	0	0
$207$	−181897.	−0.295053
$208$	0	0
$209$	206567.	0.327110
$210$	0	0
$211$	972289.	1.50345	0.751726	−	0.659476i	$-0.229222\pi$
$211$	972289.	1.50345	0.751726	+	0.659476i	$0.229222\pi$
$212$	0	0
$213$	1.43759e6	2.17113
$214$	0	0
$215$	−145615.	−0.214837
$216$	0	0
$217$	−305340.	−0.440184
$218$	0	0
$219$	−39250.6	−0.0553014
$220$	0	0
$221$	−5883.15	−0.00810269
$222$	0	0
$223$	−1.35209e6	−1.82073	−0.910363	−	0.413811i	$-0.864197\pi$
$223$	−1.35209e6	−1.82073	−0.910363	+	0.413811i	$0.864197\pi$
$224$	0	0
$225$	224686.	0.295883
$226$	0	0
$227$	876401.	1.12885	0.564427	−	0.825483i	$-0.309097\pi$
$227$	876401.	1.12885	0.564427	+	0.825483i	$0.309097\pi$
$228$	0	0
$229$	1.10402e6	1.39119	0.695597	−	0.718433i	$-0.255140\pi$
$229$	1.10402e6	1.39119	0.695597	+	0.718433i	$0.255140\pi$
$230$	0	0
$231$	108530.	0.133820
$232$	0	0
$233$	81858.7	0.0987814	0.0493907	−	0.998780i	$-0.484272\pi$
$233$	81858.7	0.0987814	0.0493907	+	0.998780i	$0.484272\pi$
$234$	0	0
$235$	183735.	0.217031
$236$	0	0
$237$	−1.94209e6	−2.24594
$238$	0	0
$239$	−1.45239e6	−1.64471	−0.822354	−	0.568977i	$-0.807339\pi$
$239$	−1.45239e6	−1.64471	−0.822354	+	0.568977i	$0.807339\pi$
$240$	0	0
$241$	1.11432e6	1.23586	0.617929	−	0.786234i	$-0.287972\pi$
$241$	1.11432e6	1.23586	0.617929	+	0.786234i	$0.287972\pi$
$242$	0	0
$243$	−1.11628e6	−1.21271
$244$	0	0
$245$	60025.0	0.0638877
$246$	0	0
$247$	−33058.9	−0.0344783
$248$	0	0
$249$	−1.75434e6	−1.79315
$250$	0	0
$251$	−157265.	−0.157560	−0.0787801	−	0.996892i	$-0.525102\pi$
$251$	−157265.	−0.157560	−0.0787801	+	0.996892i	$0.525102\pi$
$252$	0	0
$253$	−45657.1	−0.0448443
$254$	0	0
$255$	249989.	0.240752
$256$	0	0
$257$	655000.	0.618598	0.309299	−	0.950965i	$-0.399906\pi$
$257$	655000.	0.618598	0.309299	+	0.950965i	$0.399906\pi$
$258$	0	0
$259$	264046.	0.244586
$260$	0	0
$261$	−1.13765e6	−1.03373
$262$	0	0
$263$	−498514.	−0.444414	−0.222207	−	0.975000i	$-0.571326\pi$
$263$	−498514.	−0.444414	−0.222207	+	0.975000i	$0.571326\pi$
$264$	0	0
$265$	374290.	0.327411
$266$	0	0
$267$	−1.89410e6	−1.62602
$268$	0	0
$269$	−1.00547e6	−0.847202	−0.423601	−	0.905849i	$-0.639234\pi$
$269$	−1.00547e6	−0.847202	−0.423601	+	0.905849i	$0.639234\pi$
$270$	0	0
$271$	93641.4	0.0774541	0.0387271	−	0.999250i	$-0.487670\pi$
$271$	93641.4	0.0774541	0.0387271	+	0.999250i	$0.487670\pi$
$272$	0	0
$273$	−17369.2	−0.0141050
$274$	0	0
$275$	56397.2	0.0449703
$276$	0	0
$277$	12211.8	0.00956271	0.00478135	−	0.999989i	$-0.498478\pi$
$277$	12211.8	0.00956271	0.00478135	+	0.999989i	$0.498478\pi$
$278$	0	0
$279$	−2.24018e6	−1.72295
$280$	0	0
$281$	−389424.	−0.294210	−0.147105	−	0.989121i	$-0.546996\pi$
$281$	−389424.	−0.294210	−0.147105	+	0.989121i	$0.546996\pi$
$282$	0	0
$283$	1.23856e6	0.919283	0.459642	−	0.888105i	$-0.347978\pi$
$283$	1.23856e6	0.919283	0.459642	+	0.888105i	$0.347978\pi$
$284$	0	0
$285$	1.40475e6	1.02444
$286$	0	0
$287$	576452.	0.413103
$288$	0	0
$289$	−1.25390e6	−0.883114
$290$	0	0
$291$	283741.	0.196422
$292$	0	0
$293$	−1.36915e6	−0.931716	−0.465858	−	0.884860i	$-0.654254\pi$
$293$	−1.36915e6	−0.931716	−0.465858	+	0.884860i	$0.654254\pi$
$294$	0	0
$295$	−1.17884e6	−0.788677
$296$	0	0
$297$	258032.	0.169739
$298$	0	0
$299$	7306.96	0.00472671
$300$	0	0
$301$	−285404.	−0.181570
$302$	0	0
$303$	−4.20772e6	−2.63294
$304$	0	0
$305$	−107240.	−0.0660097
$306$	0	0
$307$	2.15266e6	1.30356	0.651778	−	0.758409i	$-0.274023\pi$
$307$	2.15266e6	1.30356	0.651778	+	0.758409i	$0.274023\pi$
$308$	0	0
$309$	−4.47885e6	−2.66852
$310$	0	0
$311$	2.58326e6	1.51449	0.757245	−	0.653131i	$-0.226545\pi$
$311$	2.58326e6	1.51449	0.757245	+	0.653131i	$0.226545\pi$
$312$	0	0
$313$	3.07474e6	1.77398	0.886988	−	0.461793i	$-0.152794\pi$
$313$	3.07474e6	1.77398	0.886988	+	0.461793i	$0.152794\pi$
$314$	0	0
$315$	440385.	0.250067
$316$	0	0
$317$	−1.04850e6	−0.586031	−0.293015	−	0.956108i	$-0.594659\pi$
$317$	−1.04850e6	−0.586031	−0.293015	+	0.956108i	$0.594659\pi$
$318$	0	0
$319$	−285556.	−0.157114
$320$	0	0
$321$	−1.88258e6	−1.01975
$322$	0	0
$323$	932579.	0.497370
$324$	0	0
$325$	−9025.81	−0.00474000
$326$	0	0
$327$	−322680.	−0.166879
$328$	0	0
$329$	360120.	0.183424
$330$	0	0
$331$	2.26689e6	1.13726	0.568631	−	0.822592i	$-0.307473\pi$
$331$	2.26689e6	1.13726	0.568631	+	0.822592i	$0.307473\pi$
$332$	0	0
$333$	1.93723e6	0.957348
$334$	0	0
$335$	1.22246e6	0.595143
$336$	0	0
$337$	2.78963e6	1.33805	0.669024	−	0.743240i	$-0.266712\pi$
$337$	2.78963e6	1.33805	0.669024	+	0.743240i	$0.266712\pi$
$338$	0	0
$339$	147201.	0.0695682
$340$	0	0
$341$	−562297.	−0.261866
$342$	0	0
$343$	117649.	0.0539949
$344$	0	0
$345$	−310490.	−0.140443
$346$	0	0
$347$	1.74629e6	0.778562	0.389281	−	0.921119i	$-0.372723\pi$
$347$	1.74629e6	0.778562	0.389281	+	0.921119i	$0.372723\pi$
$348$	0	0
$349$	3.21671e6	1.41367	0.706836	−	0.707377i	$-0.250122\pi$
$349$	3.21671e6	1.41367	0.706836	+	0.707377i	$0.250122\pi$
$350$	0	0
$351$	−41295.4	−0.0178910
$352$	0	0
$353$	507518.	0.216778	0.108389	−	0.994109i	$-0.465431\pi$
$353$	507518.	0.216778	0.108389	+	0.994109i	$0.465431\pi$
$354$	0	0
$355$	1.46419e6	0.616633
$356$	0	0
$357$	489979.	0.203473
$358$	0	0
$359$	804256.	0.329350	0.164675	−	0.986348i	$-0.447342\pi$
$359$	804256.	0.329350	0.164675	+	0.986348i	$0.447342\pi$
$360$	0	0
$361$	2.76430e6	1.11639
$362$	0	0
$363$	−3.75327e6	−1.49501
$364$	0	0
$365$	−39976.8	−0.0157064
$366$	0	0
$367$	1.60921e6	0.623658	0.311829	−	0.950138i	$-0.399058\pi$
$367$	1.60921e6	0.623658	0.311829	+	0.950138i	$0.399058\pi$
$368$	0	0
$369$	4.22925e6	1.61695
$370$	0	0
$371$	733609.	0.276713
$372$	0	0
$373$	−5.19298e6	−1.93261	−0.966306	−	0.257397i	$-0.917135\pi$
$373$	−5.19298e6	−1.93261	−0.966306	+	0.257397i	$0.917135\pi$
$374$	0	0
$375$	383529.	0.140838
$376$	0	0
$377$	45700.4	0.0165603
$378$	0	0
$379$	−114187.	−0.0408337	−0.0204169	−	0.999792i	$-0.506499\pi$
$379$	−114187.	−0.0408337	−0.0204169	+	0.999792i	$0.506499\pi$
$380$	0	0
$381$	−2.68859e6	−0.948884
$382$	0	0
$383$	−3.03725e6	−1.05799	−0.528997	−	0.848624i	$-0.677432\pi$
$383$	−3.03725e6	−1.05799	−0.528997	+	0.848624i	$0.677432\pi$
$384$	0	0
$385$	110539.	0.0380069
$386$	0	0
$387$	−2.09392e6	−0.710695
$388$	0	0
$389$	1.75485e6	0.587985	0.293993	−	0.955808i	$-0.405016\pi$
$389$	1.75485e6	0.587985	0.293993	+	0.955808i	$0.405016\pi$
$390$	0	0
$391$	−206127.	−0.0681855
$392$	0	0
$393$	9.45471e6	3.08793
$394$	0	0
$395$	−1.97803e6	−0.637881
$396$	0	0
$397$	−411623.	−0.131076	−0.0655380	−	0.997850i	$-0.520876\pi$
$397$	−411623.	−0.131076	−0.0655380	+	0.997850i	$0.520876\pi$
$398$	0	0
$399$	2.75332e6	0.865812
$400$	0	0
$401$	−3.34807e6	−1.03976	−0.519881	−	0.854238i	$-0.674024\pi$
$401$	−3.34807e6	−1.03976	−0.519881	+	0.854238i	$0.674024\pi$
$402$	0	0
$403$	89990.0	0.0276014
$404$	0	0
$405$	−429209.	−0.130026
$406$	0	0
$407$	486253.	0.145504
$408$	0	0
$409$	−1.67438e6	−0.494933	−0.247467	−	0.968896i	$-0.579598\pi$
$409$	−1.67438e6	−0.494933	−0.247467	+	0.968896i	$0.579598\pi$
$410$	0	0
$411$	−7.75197e6	−2.26364
$412$	0	0
$413$	−2.31052e6	−0.666553
$414$	0	0
$415$	−1.78680e6	−0.509279
$416$	0	0
$417$	2.34249e6	0.659688
$418$	0	0
$419$	2.61117e6	0.726608	0.363304	−	0.931671i	$-0.381649\pi$
$419$	2.61117e6	0.726608	0.363304	+	0.931671i	$0.381649\pi$
$420$	0	0
$421$	3.20146e6	0.880324	0.440162	−	0.897918i	$-0.354921\pi$
$421$	3.20146e6	0.880324	0.440162	+	0.897918i	$0.354921\pi$
$422$	0	0
$423$	2.64209e6	0.717953
$424$	0	0
$425$	254615.	0.0683772
$426$	0	0
$427$	−210191.	−0.0557884
$428$	0	0
$429$	−31986.1	−0.00839109
$430$	0	0
$431$	5.57601e6	1.44587	0.722937	−	0.690914i	$-0.242792\pi$
$431$	5.57601e6	1.44587	0.722937	+	0.690914i	$0.242792\pi$
$432$	0	0
$433$	5.12039e6	1.31245	0.656226	−	0.754564i	$-0.272152\pi$
$433$	5.12039e6	1.31245	0.656226	+	0.754564i	$0.272152\pi$
$434$	0	0
$435$	−1.94192e6	−0.492049
$436$	0	0
$437$	−1.15828e6	−0.290141
$438$	0	0
$439$	3.83173e6	0.948930	0.474465	−	0.880274i	$-0.342642\pi$
$439$	3.83173e6	0.948930	0.474465	+	0.880274i	$0.342642\pi$
$440$	0	0
$441$	863154.	0.211345
$442$	0	0
$443$	−2.70686e6	−0.655325	−0.327662	−	0.944795i	$-0.606261\pi$
$443$	−2.70686e6	−0.655325	−0.327662	+	0.944795i	$0.606261\pi$
$444$	0	0
$445$	−1.92915e6	−0.461812
$446$	0	0
$447$	−3.71592e6	−0.879626
$448$	0	0
$449$	−7.55860e6	−1.76940	−0.884699	−	0.466164i	$-0.845636\pi$
$449$	−7.55860e6	−1.76940	−0.884699	+	0.466164i	$0.845636\pi$
$450$	0	0
$451$	1.06156e6	0.245756
$452$	0	0
$453$	7.37948e6	1.68959
$454$	0	0
$455$	−17690.6	−0.00400603
$456$	0	0
$457$	−2.67340e6	−0.598790	−0.299395	−	0.954129i	$-0.596785\pi$
$457$	−2.67340e6	−0.598790	−0.299395	+	0.954129i	$0.596785\pi$
$458$	0	0
$459$	1.16493e6	0.258087
$460$	0	0
$461$	−792744.	−0.173732	−0.0868661	−	0.996220i	$-0.527685\pi$
$461$	−792744.	−0.173732	−0.0868661	+	0.996220i	$0.527685\pi$
$462$	0	0
$463$	6.72717e6	1.45841	0.729206	−	0.684294i	$-0.239890\pi$
$463$	6.72717e6	1.45841	0.729206	+	0.684294i	$0.239890\pi$
$464$	0	0
$465$	−3.82389e6	−0.820112
$466$	0	0
$467$	−7.79719e6	−1.65442	−0.827211	−	0.561892i	$-0.810074\pi$
$467$	−7.79719e6	−1.65442	−0.827211	+	0.561892i	$0.810074\pi$
$468$	0	0
$469$	2.39601e6	0.502988
$470$	0	0
$471$	−7.51373e6	−1.56064
$472$	0	0
$473$	−525584.	−0.108016
$474$	0	0
$475$	1.43074e6	0.290957
$476$	0	0
$477$	5.38226e6	1.08310
$478$	0	0
$479$	−2.99000e6	−0.595433	−0.297717	−	0.954654i	$-0.596225\pi$
$479$	−2.99000e6	−0.595433	−0.297717	+	0.954654i	$0.596225\pi$
$480$	0	0
$481$	−77819.9	−0.0153366
$482$	0	0
$483$	−608561.	−0.118696
$484$	0	0
$485$	288991.	0.0557866
$486$	0	0
$487$	1.01844e7	1.94587	0.972934	−	0.231082i	$-0.0742264\pi$
$487$	1.01844e7	1.94587	0.972934	+	0.231082i	$0.0742264\pi$
$488$	0	0
$489$	9.15614e6	1.73157
$490$	0	0
$491$	−2.36159e6	−0.442079	−0.221040	−	0.975265i	$-0.570945\pi$
$491$	−2.36159e6	−0.442079	−0.221040	+	0.975265i	$0.570945\pi$
$492$	0	0
$493$	−1.28919e6	−0.238891
$494$	0	0
$495$	810987.	0.148765
$496$	0	0
$497$	2.86981e6	0.521150
$498$	0	0
$499$	1.04692e7	1.88219	0.941096	−	0.338141i	$-0.109798\pi$
$499$	1.04692e7	1.88219	0.941096	+	0.338141i	$0.109798\pi$
$500$	0	0
$501$	−7.79185e6	−1.38690
$502$	0	0
$503$	−5.05909e6	−0.891565	−0.445782	−	0.895141i	$-0.647074\pi$
$503$	−5.05909e6	−0.891565	−0.445782	+	0.895141i	$0.647074\pi$
$504$	0	0
$505$	−4.28558e6	−0.747792
$506$	0	0
$507$	−9.10858e6	−1.57373
$508$	0	0
$509$	8.22380e6	1.40695	0.703474	−	0.710721i	$-0.251631\pi$
$509$	8.22380e6	1.40695	0.703474	+	0.710721i	$0.251631\pi$
$510$	0	0
$511$	−78354.6	−0.0132743
$512$	0	0
$513$	6.54602e6	1.09821
$514$	0	0
$515$	−4.56173e6	−0.757899
$516$	0	0
$517$	663176.	0.109119
$518$	0	0
$519$	−8.70922e6	−1.41926
$520$	0	0
$521$	1.91440e6	0.308985	0.154493	−	0.987994i	$-0.450626\pi$
$521$	1.91440e6	0.308985	0.154493	+	0.987994i	$0.450626\pi$
$522$	0	0
$523$	−1.13982e7	−1.82215	−0.911073	−	0.412245i	$-0.864745\pi$
$523$	−1.13982e7	−1.82215	−0.911073	+	0.412245i	$0.864745\pi$
$524$	0	0
$525$	751716.	0.119030
$526$	0	0
$527$	−2.53858e6	−0.398167
$528$	0	0
$529$	−6.18033e6	−0.960224
$530$	0	0
$531$	−1.69516e7	−2.60900
$532$	0	0
$533$	−169892.	−0.0259033
$534$	0	0
$535$	−1.91742e6	−0.289623
$536$	0	0
$537$	8.74554e6	1.30873
$538$	0	0
$539$	216656.	0.0321217
$540$	0	0
$541$	−2.18345e6	−0.320738	−0.160369	−	0.987057i	$-0.551268\pi$
$541$	−2.18345e6	−0.320738	−0.160369	+	0.987057i	$0.551268\pi$
$542$	0	0
$543$	239882.	0.0349139
$544$	0	0
$545$	−328650.	−0.0473961
$546$	0	0
$547$	−3.45056e6	−0.493084	−0.246542	−	0.969132i	$-0.579294\pi$
$547$	−3.45056e6	−0.493084	−0.246542	+	0.969132i	$0.579294\pi$
$548$	0	0
$549$	−1.54210e6	−0.218365
$550$	0	0
$551$	−7.24430e6	−1.01652
$552$	0	0
$553$	−3.87693e6	−0.539108
$554$	0	0
$555$	3.30675e6	0.455690
$556$	0	0
$557$	−1.17331e7	−1.60241	−0.801204	−	0.598391i	$-0.795807\pi$
$557$	−1.17331e7	−1.60241	−0.801204	+	0.598391i	$0.795807\pi$
$558$	0	0
$559$	84114.5	0.0113852
$560$	0	0
$561$	902317.	0.121046
$562$	0	0
$563$	−604258.	−0.0803436	−0.0401718	−	0.999193i	$-0.512791\pi$
$563$	−604258.	−0.0803436	−0.0401718	+	0.999193i	$0.512791\pi$
$564$	0	0
$565$	149924.	0.0197584
$566$	0	0
$567$	−841250.	−0.109892
$568$	0	0
$569$	1.22306e6	0.158368	0.0791842	−	0.996860i	$-0.474768\pi$
$569$	1.22306e6	0.158368	0.0791842	+	0.996860i	$0.474768\pi$
$570$	0	0
$571$	−6.34785e6	−0.814772	−0.407386	−	0.913256i	$-0.633560\pi$
$571$	−6.34785e6	−0.814772	−0.407386	+	0.913256i	$0.633560\pi$
$572$	0	0
$573$	−7.41939e6	−0.944021
$574$	0	0
$575$	−316235.	−0.0398879
$576$	0	0
$577$	5.02896e6	0.628838	0.314419	−	0.949284i	$-0.398190\pi$
$577$	5.02896e6	0.628838	0.314419	+	0.949284i	$0.398190\pi$
$578$	0	0
$579$	6.57122e6	0.814610
$580$	0	0
$581$	−3.50213e6	−0.430420
$582$	0	0
$583$	1.35097e6	0.164617
$584$	0	0
$585$	−129790.	−0.0156802
$586$	0	0
$587$	1.24089e7	1.48641	0.743206	−	0.669063i	$-0.233304\pi$
$587$	1.24089e7	1.48641	0.743206	+	0.669063i	$0.233304\pi$
$588$	0	0
$589$	−1.42649e7	−1.69427
$590$	0	0
$591$	−3.72926e6	−0.439191
$592$	0	0
$593$	−1.60941e7	−1.87945	−0.939725	−	0.341932i	$-0.888919\pi$
$593$	−1.60941e7	−1.87945	−0.939725	+	0.341932i	$0.888919\pi$
$594$	0	0
$595$	499045.	0.0577893
$596$	0	0
$597$	2.02671e7	2.32732
$598$	0	0
$599$	2.42343e6	0.275971	0.137986	−	0.990434i	$-0.455937\pi$
$599$	2.42343e6	0.275971	0.137986	+	0.990434i	$0.455937\pi$
$600$	0	0
$601$	7.84941e6	0.886443	0.443222	−	0.896412i	$-0.353835\pi$
$601$	7.84941e6	0.886443	0.443222	+	0.896412i	$0.353835\pi$
$602$	0	0
$603$	1.75788e7	1.96878
$604$	0	0
$605$	−3.82271e6	−0.424603
$606$	0	0
$607$	6.84175e6	0.753695	0.376847	−	0.926275i	$-0.377008\pi$
$607$	6.84175e6	0.753695	0.376847	+	0.926275i	$0.377008\pi$
$608$	0	0
$609$	−3.80617e6	−0.415858
$610$	0	0
$611$	−106135.	−0.0115015
$612$	0	0
$613$	−1.16723e6	−0.125460	−0.0627302	−	0.998031i	$-0.519981\pi$
$613$	−1.16723e6	−0.125460	−0.0627302	+	0.998031i	$0.519981\pi$
$614$	0	0
$615$	7.21913e6	0.769657
$616$	0	0
$617$	8.05925e6	0.852279	0.426140	−	0.904657i	$-0.359873\pi$
$617$	8.05925e6	0.852279	0.426140	+	0.904657i	$0.359873\pi$
$618$	0	0
$619$	5.10346e6	0.535350	0.267675	−	0.963509i	$-0.413745\pi$
$619$	5.10346e6	0.535350	0.267675	+	0.963509i	$0.413745\pi$
$620$	0	0
$621$	−1.44686e6	−0.150555
$622$	0	0
$623$	−3.78113e6	−0.390303
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	5.07035e6	0.515073
$628$	0	0
$629$	2.19527e6	0.221239
$630$	0	0
$631$	6.01731e6	0.601630	0.300815	−	0.953683i	$-0.402741\pi$
$631$	6.01731e6	0.601630	0.300815	+	0.953683i	$0.402741\pi$
$632$	0	0
$633$	2.38656e7	2.36736
$634$	0	0
$635$	−2.73834e6	−0.269497
$636$	0	0
$637$	−34673.6	−0.00338571
$638$	0	0
$639$	2.10549e7	2.03987
$640$	0	0
$641$	6.28323e6	0.604002	0.302001	−	0.953308i	$-0.402345\pi$
$641$	6.28323e6	0.604002	0.302001	+	0.953308i	$0.402345\pi$
$642$	0	0
$643$	−7.65315e6	−0.729983	−0.364991	−	0.931011i	$-0.618928\pi$
$643$	−7.65315e6	−0.729983	−0.364991	+	0.931011i	$0.618928\pi$
$644$	0	0
$645$	−3.57423e6	−0.338285
$646$	0	0
$647$	2.00172e7	1.87994	0.939969	−	0.341260i	$-0.110854\pi$
$647$	2.00172e7	1.87994	0.939969	+	0.341260i	$0.110854\pi$
$648$	0	0
$649$	−4.25492e6	−0.396534
$650$	0	0
$651$	−7.49483e6	−0.693121
$652$	0	0
$653$	−2.94459e6	−0.270236	−0.135118	−	0.990830i	$-0.543141\pi$
$653$	−2.94459e6	−0.270236	−0.135118	+	0.990830i	$0.543141\pi$
$654$	0	0
$655$	9.62965e6	0.877015
$656$	0	0
$657$	−574863.	−0.0519579
$658$	0	0
$659$	−6.11116e6	−0.548164	−0.274082	−	0.961706i	$-0.588374\pi$
$659$	−6.11116e6	−0.548164	−0.274082	+	0.961706i	$0.588374\pi$
$660$	0	0
$661$	6.81315e6	0.606518	0.303259	−	0.952908i	$-0.401925\pi$
$661$	6.81315e6	0.606518	0.303259	+	0.952908i	$0.401925\pi$
$662$	0	0
$663$	−144407.	−0.0127586
$664$	0	0
$665$	2.80426e6	0.245903
$666$	0	0
$667$	1.60120e6	0.139357
$668$	0	0
$669$	−3.31882e7	−2.86694
$670$	0	0
$671$	−387075.	−0.0331886
$672$	0	0
$673$	−1.02171e7	−0.869540	−0.434770	−	0.900541i	$-0.643170\pi$
$673$	−1.02171e7	−0.869540	−0.434770	+	0.900541i	$0.643170\pi$
$674$	0	0
$675$	1.78721e6	0.150979
$676$	0	0
$677$	3.09917e6	0.259881	0.129940	−	0.991522i	$-0.458521\pi$
$677$	3.09917e6	0.259881	0.129940	+	0.991522i	$0.458521\pi$
$678$	0	0
$679$	566422.	0.0471483
$680$	0	0
$681$	2.15120e7	1.77751
$682$	0	0
$683$	1.08480e7	0.889814	0.444907	−	0.895577i	$-0.353237\pi$
$683$	1.08480e7	0.889814	0.444907	+	0.895577i	$0.353237\pi$
$684$	0	0
$685$	−7.89540e6	−0.642906
$686$	0	0
$687$	2.70990e7	2.19059
$688$	0	0
$689$	−216210.	−0.0173511
$690$	0	0
$691$	1.27049e6	0.101222	0.0506112	−	0.998718i	$-0.483883\pi$
$691$	1.27049e6	0.101222	0.0506112	+	0.998718i	$0.483883\pi$
$692$	0	0
$693$	1.58953e6	0.125729
$694$	0	0
$695$	2.38584e6	0.187361
$696$	0	0
$697$	4.79260e6	0.373671
$698$	0	0
$699$	2.00929e6	0.155543
$700$	0	0
$701$	2.34177e6	0.179990	0.0899952	−	0.995942i	$-0.471315\pi$
$701$	2.34177e6	0.179990	0.0899952	+	0.995942i	$0.471315\pi$
$702$	0	0
$703$	1.23358e7	0.941408
$704$	0	0
$705$	4.50992e6	0.341740
$706$	0	0
$707$	−8.39973e6	−0.632000
$708$	0	0
$709$	−7.30501e6	−0.545764	−0.272882	−	0.962048i	$-0.587977\pi$
$709$	−7.30501e6	−0.545764	−0.272882	+	0.962048i	$0.587977\pi$
$710$	0	0
$711$	−2.84438e7	−2.11015
$712$	0	0
$713$	3.15296e6	0.232271
$714$	0	0
$715$	−32578.0	−0.00238319
$716$	0	0
$717$	−3.56501e7	−2.58978
$718$	0	0
$719$	1.26261e7	0.910850	0.455425	−	0.890274i	$-0.349487\pi$
$719$	1.26261e7	0.910850	0.455425	+	0.890274i	$0.349487\pi$
$720$	0	0
$721$	−8.94098e6	−0.640541
$722$	0	0
$723$	2.73520e7	1.94600
$724$	0	0
$725$	−1.97785e6	−0.139749
$726$	0	0
$727$	1.43022e7	1.00361	0.501806	−	0.864980i	$-0.332669\pi$
$727$	1.43022e7	1.00361	0.501806	+	0.864980i	$0.332669\pi$
$728$	0	0
$729$	−2.32281e7	−1.61880
$730$	0	0
$731$	−2.37284e6	−0.164238
$732$	0	0
$733$	−2.32911e7	−1.60114	−0.800572	−	0.599237i	$-0.795471\pi$
$733$	−2.32911e7	−1.60114	−0.800572	+	0.599237i	$0.795471\pi$
$734$	0	0
$735$	1.47336e6	0.100599
$736$	0	0
$737$	4.41236e6	0.299228
$738$	0	0
$739$	−1.00700e7	−0.678293	−0.339146	−	0.940734i	$-0.610138\pi$
$739$	−1.00700e7	−0.678293	−0.339146	+	0.940734i	$0.610138\pi$
$740$	0	0
$741$	−811458.	−0.0542901
$742$	0	0
$743$	−1.66744e7	−1.10810	−0.554049	−	0.832484i	$-0.686918\pi$
$743$	−1.66744e7	−1.10810	−0.554049	+	0.832484i	$0.686918\pi$
$744$	0	0
$745$	−3.78468e6	−0.249826
$746$	0	0
$747$	−2.56940e7	−1.68473
$748$	0	0
$749$	−3.75814e6	−0.244776
$750$	0	0
$751$	39313.6	0.00254356	0.00127178	−	0.999999i	$-0.499595\pi$
$751$	39313.6	0.00254356	0.00127178	+	0.999999i	$0.499595\pi$
$752$	0	0
$753$	−3.86019e6	−0.248097
$754$	0	0
$755$	7.51602e6	0.479867
$756$	0	0
$757$	−4.77082e6	−0.302589	−0.151295	−	0.988489i	$-0.548344\pi$
$757$	−4.77082e6	−0.302589	−0.151295	+	0.988489i	$0.548344\pi$
$758$	0	0
$759$	−1.12069e6	−0.0706125
$760$	0	0
$761$	1.58376e7	0.991350	0.495675	−	0.868508i	$-0.334921\pi$
$761$	1.58376e7	0.991350	0.495675	+	0.868508i	$0.334921\pi$
$762$	0	0
$763$	−644154.	−0.0400570
$764$	0	0
$765$	3.66134e6	0.226197
$766$	0	0
$767$	680958.	0.0417957
$768$	0	0
$769$	1.80265e7	1.09925	0.549624	−	0.835412i	$-0.314771\pi$
$769$	1.80265e7	1.09925	0.549624	+	0.835412i	$0.314771\pi$
$770$	0	0
$771$	1.60775e7	0.974055
$772$	0	0
$773$	−7.60103e6	−0.457534	−0.228767	−	0.973481i	$-0.573469\pi$
$773$	−7.60103e6	−0.457534	−0.228767	+	0.973481i	$0.573469\pi$
$774$	0	0
$775$	−3.89464e6	−0.232924
$776$	0	0
$777$	6.48124e6	0.385128
$778$	0	0
$779$	2.69308e7	1.59003
$780$	0	0
$781$	5.28488e6	0.310033
$782$	0	0
$783$	−9.04917e6	−0.527478
$784$	0	0
$785$	−7.65275e6	−0.443245
$786$	0	0
$787$	−3.13160e7	−1.80231	−0.901155	−	0.433498i	$-0.857279\pi$
$787$	−3.13160e7	−1.80231	−0.901155	+	0.433498i	$0.857279\pi$
$788$	0	0
$789$	−1.22364e7	−0.699781
$790$	0	0
$791$	293852.	0.0166989
$792$	0	0
$793$	61947.5	0.00349817
$794$	0	0
$795$	9.18726e6	0.515547
$796$	0	0
$797$	7.51589e6	0.419117	0.209558	−	0.977796i	$-0.432797\pi$
$797$	7.51589e6	0.419117	0.209558	+	0.977796i	$0.432797\pi$
$798$	0	0
$799$	2.99402e6	0.165916
$800$	0	0
$801$	−2.77410e7	−1.52771
$802$	0	0
$803$	−144293.	−0.00789691
$804$	0	0
$805$	−619821.	−0.0337114
$806$	0	0
$807$	−2.46800e7	−1.33402
$808$	0	0
$809$	3.23042e7	1.73535	0.867676	−	0.497130i	$-0.165613\pi$
$809$	3.23042e7	1.73535	0.867676	+	0.497130i	$0.165613\pi$
$810$	0	0
$811$	3.53053e7	1.88490	0.942448	−	0.334354i	$-0.108518\pi$
$811$	3.53053e7	1.88490	0.942448	+	0.334354i	$0.108518\pi$
$812$	0	0
$813$	2.29851e6	0.121960
$814$	0	0
$815$	9.32555e6	0.491791
$816$	0	0
$817$	−1.33336e7	−0.698862
$818$	0	0
$819$	−254389.	−0.0132522
$820$	0	0
$821$	1.06861e7	0.553299	0.276650	−	0.960971i	$-0.410776\pi$
$821$	1.06861e7	0.553299	0.276650	+	0.960971i	$0.410776\pi$
$822$	0	0
$823$	−1.14692e7	−0.590246	−0.295123	−	0.955459i	$-0.595361\pi$
$823$	−1.14692e7	−0.590246	−0.295123	+	0.955459i	$0.595361\pi$
$824$	0	0
$825$	1.38432e6	0.0708110
$826$	0	0
$827$	−3.29781e6	−0.167673	−0.0838363	−	0.996480i	$-0.526717\pi$
$827$	−3.29781e6	−0.167673	−0.0838363	+	0.996480i	$0.526717\pi$
$828$	0	0
$829$	−7.63786e6	−0.385998	−0.192999	−	0.981199i	$-0.561821\pi$
$829$	−7.63786e6	−0.385998	−0.192999	+	0.981199i	$0.561821\pi$
$830$	0	0
$831$	299749.	0.0150576
$832$	0	0
$833$	978128.	0.0488408
$834$	0	0
$835$	−7.93603e6	−0.393901
$836$	0	0
$837$	−1.78190e7	−0.879162
$838$	0	0
$839$	2.15181e6	0.105536	0.0527678	−	0.998607i	$-0.483196\pi$
$839$	2.15181e6	0.105536	0.0527678	+	0.998607i	$0.483196\pi$
$840$	0	0
$841$	−1.04967e7	−0.511755
$842$	0	0
$843$	−9.55874e6	−0.463267
$844$	0	0
$845$	−9.27711e6	−0.446962
$846$	0	0
$847$	−7.49252e6	−0.358855
$848$	0	0
$849$	3.04014e7	1.44752
$850$	0	0
$851$	−2.72656e6	−0.129060
$852$	0	0
$853$	6.77711e6	0.318913	0.159456	−	0.987205i	$-0.449026\pi$
$853$	6.77711e6	0.318913	0.159456	+	0.987205i	$0.449026\pi$
$854$	0	0
$855$	2.05740e7	0.962505
$856$	0	0
$857$	−2.76782e7	−1.28732	−0.643658	−	0.765313i	$-0.722584\pi$
$857$	−2.76782e7	−1.28732	−0.643658	+	0.765313i	$0.722584\pi$
$858$	0	0
$859$	9.64843e6	0.446143	0.223071	−	0.974802i	$-0.428392\pi$
$859$	9.64843e6	0.446143	0.223071	+	0.974802i	$0.428392\pi$
$860$	0	0
$861$	1.41495e7	0.650479
$862$	0	0
$863$	−1.44710e7	−0.661413	−0.330707	−	0.943734i	$-0.607287\pi$
$863$	−1.44710e7	−0.661413	−0.330707	+	0.943734i	$0.607287\pi$
$864$	0	0
$865$	−8.87037e6	−0.403089
$866$	0	0
$867$	−3.07779e7	−1.39057
$868$	0	0
$869$	−7.13954e6	−0.320716
$870$	0	0
$871$	−706154.	−0.0315395
$872$	0	0
$873$	4.15566e6	0.184546
$874$	0	0
$875$	765625.	0.0338062
$876$	0	0
$877$	−1.65012e7	−0.724463	−0.362231	−	0.932088i	$-0.617985\pi$
$877$	−1.65012e7	−0.724463	−0.362231	+	0.932088i	$0.617985\pi$
$878$	0	0
$879$	−3.36070e7	−1.46709
$880$	0	0
$881$	2.14305e7	0.930236	0.465118	−	0.885249i	$-0.346012\pi$
$881$	2.14305e7	0.930236	0.465118	+	0.885249i	$0.346012\pi$
$882$	0	0
$883$	2.06482e7	0.891213	0.445606	−	0.895229i	$-0.352988\pi$
$883$	2.06482e7	0.891213	0.445606	+	0.895229i	$0.352988\pi$
$884$	0	0
$885$	−2.89355e7	−1.24186
$886$	0	0
$887$	−2.33225e7	−0.995326	−0.497663	−	0.867370i	$-0.665808\pi$
$887$	−2.33225e7	−0.995326	−0.497663	+	0.867370i	$0.665808\pi$
$888$	0	0
$889$	−5.36715e6	−0.227766
$890$	0	0
$891$	−1.54920e6	−0.0653751
$892$	0	0
$893$	1.68241e7	0.705999
$894$	0	0
$895$	8.90736e6	0.371699
$896$	0	0
$897$	179355.	0.00744275
$898$	0	0
$899$	1.97198e7	0.813772
$900$	0	0
$901$	6.09919e6	0.250300
$902$	0	0
$903$	−7.00549e6	−0.285903
$904$	0	0
$905$	244320.	0.00991604
$906$	0	0
$907$	3.53172e6	0.142550	0.0712752	−	0.997457i	$-0.477293\pi$
$907$	3.53172e6	0.142550	0.0712752	+	0.997457i	$0.477293\pi$
$908$	0	0
$909$	−6.16262e7	−2.47375
$910$	0	0
$911$	4.09852e7	1.63618	0.818091	−	0.575089i	$-0.195033\pi$
$911$	4.09852e7	1.63618	0.818091	+	0.575089i	$0.195033\pi$
$912$	0	0
$913$	−6.44932e6	−0.256057
$914$	0	0
$915$	−2.63230e6	−0.103940
$916$	0	0
$917$	1.88741e7	0.741213
$918$	0	0
$919$	−4.53948e7	−1.77304	−0.886518	−	0.462693i	$-0.846883\pi$
$919$	−4.53948e7	−1.77304	−0.886518	+	0.462693i	$0.846883\pi$
$920$	0	0
$921$	5.28389e7	2.05260
$922$	0	0
$923$	−845793.	−0.0326783
$924$	0	0
$925$	3.36794e6	0.129423
$926$	0	0
$927$	−6.55972e7	−2.50718
$928$	0	0
$929$	−3.26640e7	−1.24174	−0.620869	−	0.783914i	$-0.713220\pi$
$929$	−3.26640e7	−1.24174	−0.620869	+	0.783914i	$0.713220\pi$
$930$	0	0
$931$	5.49635e6	0.207826
$932$	0	0
$933$	6.34082e7	2.38474
$934$	0	0
$935$	919012.	0.0343789
$936$	0	0
$937$	1.27944e7	0.476068	0.238034	−	0.971257i	$-0.423497\pi$
$937$	1.27944e7	0.476068	0.238034	+	0.971257i	$0.423497\pi$
$938$	0	0
$939$	7.54720e7	2.79333
$940$	0	0
$941$	1.71929e7	0.632957	0.316479	−	0.948600i	$-0.397499\pi$
$941$	1.71929e7	0.632957	0.316479	+	0.948600i	$0.397499\pi$
$942$	0	0
$943$	−5.95248e6	−0.217981
$944$	0	0
$945$	3.50293e6	0.127600
$946$	0	0
$947$	−4.33455e7	−1.57061	−0.785307	−	0.619107i	$-0.787495\pi$
$947$	−4.33455e7	−1.57061	−0.785307	+	0.619107i	$0.787495\pi$
$948$	0	0
$949$	23092.7	0.000832356 0
$950$	0	0
$951$	−2.57363e7	−0.922773
$952$	0	0
$953$	−6.85588e6	−0.244529	−0.122265	−	0.992498i	$-0.539016\pi$
$953$	−6.85588e6	−0.244529	−0.122265	+	0.992498i	$0.539016\pi$
$954$	0	0
$955$	−7.55667e6	−0.268116
$956$	0	0
$957$	−7.00922e6	−0.247394
$958$	0	0
$959$	−1.54750e7	−0.543355
$960$	0	0
$961$	1.02016e7	0.356335
$962$	0	0
$963$	−2.75723e7	−0.958092
$964$	0	0
$965$	6.69281e6	0.231361
$966$	0	0
$967$	1.81761e7	0.625080	0.312540	−	0.949905i	$-0.398820\pi$
$967$	1.81761e7	0.625080	0.312540	+	0.949905i	$0.398820\pi$
$968$	0	0
$969$	2.28909e7	0.783166
$970$	0	0
$971$	−2.09710e7	−0.713790	−0.356895	−	0.934144i	$-0.616165\pi$
$971$	−2.09710e7	−0.713790	−0.356895	+	0.934144i	$0.616165\pi$
$972$	0	0
$973$	4.67624e6	0.158349
$974$	0	0
$975$	−221546.	−0.00746367
$976$	0	0
$977$	2.72323e7	0.912742	0.456371	−	0.889790i	$-0.349149\pi$
$977$	2.72323e7	0.912742	0.456371	+	0.889790i	$0.349149\pi$
$978$	0	0
$979$	−6.96311e6	−0.232192
$980$	0	0
$981$	−4.72596e6	−0.156790
$982$	0	0
$983$	−3.97759e7	−1.31291	−0.656457	−	0.754363i	$-0.727946\pi$
$983$	−3.97759e7	−1.31291	−0.656457	+	0.754363i	$0.727946\pi$
$984$	0	0
$985$	−3.79826e6	−0.124737
$986$	0	0
$987$	8.83944e6	0.288823
$988$	0	0
$989$	2.94710e6	0.0958086
$990$	0	0
$991$	−9.12673e6	−0.295210	−0.147605	−	0.989046i	$-0.547156\pi$
$991$	−9.12673e6	−0.295210	−0.147605	+	0.989046i	$0.547156\pi$
$992$	0	0
$993$	5.56427e7	1.79075
$994$	0	0
$995$	2.06421e7	0.660994
$996$	0	0
$997$	−1.68299e7	−0.536221	−0.268110	−	0.963388i	$-0.586399\pi$
$997$	−1.68299e7	−0.536221	−0.268110	+	0.963388i	$0.586399\pi$
$998$	0	0
$999$	1.54092e7	0.488501

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	140.6.a.d.1.3	✓	3
4.3	odd	2		560.6.a.t.1.1		3
5.2	odd	4		700.6.e.g.449.1		6
5.3	odd	4		700.6.e.g.449.6		6
5.4	even	2		700.6.a.i.1.1		3
7.6	odd	2		980.6.a.h.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.6.a.d.1.3	✓	3	1.1	even	1	trivial
560.6.a.t.1.1		3	4.3	odd	2
700.6.a.i.1.1		3	5.4	even	2
700.6.e.g.449.1		6	5.2	odd	4
700.6.e.g.449.6		6	5.3	odd	4
980.6.a.h.1.1		3	7.6	odd	2

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:	\( N \)	\(=\)	\( 140 = 2^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	140.a (trivial)

\(f(q)\)	\(=\)	\(q+24.5458 q^{3} +25.0000 q^{5} +49.0000 q^{7} +359.498 q^{9} +O(q^{10})\)	\(q+24.5458 q^{3} +25.0000 q^{5} +49.0000 q^{7} +359.498 q^{9} +90.2356 q^{11} -14.4413 q^{13} +613.646 q^{15} +407.384 q^{17} +2289.19 q^{19} +1202.75 q^{21} -505.976 q^{23} +625.000 q^{25} +2859.53 q^{27} -3164.56 q^{29} -6231.43 q^{31} +2214.91 q^{33} +1225.00 q^{35} +5388.70 q^{37} -354.474 q^{39} +11764.3 q^{41} -5824.58 q^{43} +8987.44 q^{45} +7349.38 q^{47} +2401.00 q^{49} +9999.57 q^{51} +14971.6 q^{53} +2255.89 q^{55} +56190.1 q^{57} -47153.5 q^{59} -4289.61 q^{61} +17615.4 q^{63} -361.033 q^{65} +48898.3 q^{67} -12419.6 q^{69} +58567.6 q^{71} -1599.07 q^{73} +15341.1 q^{75} +4421.54 q^{77} -79121.1 q^{79} -17168.4 q^{81} -71472.1 q^{83} +10184.6 q^{85} -77676.9 q^{87} -77165.9 q^{89} -707.624 q^{91} -152956. q^{93} +57229.8 q^{95} +11559.6 q^{97} +32439.5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 6 q^{3} + 75 q^{5} + 147 q^{7} + 281 q^{9}+O(q^{10}) \) 3 * q + 6 * q^3 + 75 * q^5 + 147 * q^7 + 281 * q^9	\( 3 q + 6 q^{3} + 75 q^{5} + 147 q^{7} + 281 q^{9} - 14 q^{11} - 4 q^{13} + 150 q^{15} + 44 q^{17} + 2328 q^{19} + 294 q^{21} + 3676 q^{23} + 1875 q^{25} + 3726 q^{27} + 4092 q^{29} + 5888 q^{31} + 11318 q^{33} + 3675 q^{35} + 11378 q^{37} + 13702 q^{39} + 11450 q^{41} + 18544 q^{43} + 7025 q^{45} + 21754 q^{47} + 7203 q^{49} + 31762 q^{51} + 7494 q^{53} - 350 q^{55} + 7332 q^{57} + 12388 q^{59} + 27182 q^{61} + 13769 q^{63} - 100 q^{65} - 7676 q^{67} - 62140 q^{69} + 81992 q^{71} + 230 q^{73} + 3750 q^{75} - 686 q^{77} - 15926 q^{79} - 32101 q^{81} - 86100 q^{83} + 1100 q^{85} - 85098 q^{87} - 95710 q^{89} - 196 q^{91} - 229472 q^{93} + 58200 q^{95} - 176188 q^{97} - 114368 q^{99}+O(q^{100}) \) 3 * q + 6 * q^3 + 75 * q^5 + 147 * q^7 + 281 * q^9 - 14 * q^11 - 4 * q^13 + 150 * q^15 + 44 * q^17 + 2328 * q^19 + 294 * q^21 + 3676 * q^23 + 1875 * q^25 + 3726 * q^27 + 4092 * q^29 + 5888 * q^31 + 11318 * q^33 + 3675 * q^35 + 11378 * q^37 + 13702 * q^39 + 11450 * q^41 + 18544 * q^43 + 7025 * q^45 + 21754 * q^47 + 7203 * q^49 + 31762 * q^51 + 7494 * q^53 - 350 * q^55 + 7332 * q^57 + 12388 * q^59 + 27182 * q^61 + 13769 * q^63 - 100 * q^65 - 7676 * q^67 - 62140 * q^69 + 81992 * q^71 + 230 * q^73 + 3750 * q^75 - 686 * q^77 - 15926 * q^79 - 32101 * q^81 - 86100 * q^83 + 1100 * q^85 - 85098 * q^87 - 95710 * q^89 - 196 * q^91 - 229472 * q^93 + 58200 * q^95 - 176188 * q^97 - 114368 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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