LMFDB - Embedded newform 342.8.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [342,8,Mod(1,342)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(342, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("342.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 342 = 2 \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	342.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:	$106.835678716$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 114)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	342.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-8.00000 q^{2} +64.0000 q^{4} +47.0000 q^{5} +405.000 q^{7} -512.000 q^{8} +O(q^{10})$

\(q-8.00000 q^{2} +64.0000 q^{4} +47.0000 q^{5} +405.000 q^{7} -512.000 q^{8} -376.000 q^{10} +5789.00 q^{11} -2686.00 q^{13} -3240.00 q^{14} +4096.00 q^{16} -22167.0 q^{17} -6859.00 q^{19} +3008.00 q^{20} -46312.0 q^{22} -12772.0 q^{23} -75916.0 q^{25} +21488.0 q^{26} +25920.0 q^{28} +207538. q^{29} -22106.0 q^{31} -32768.0 q^{32} +177336. q^{34} +19035.0 q^{35} -550160. q^{37} +54872.0 q^{38} -24064.0 q^{40} +206800. q^{41} -565547. q^{43} +370496. q^{44} +102176. q^{46} -176953. q^{47} -659518. q^{49} +607328. q^{50} -171904. q^{52} +717230. q^{53} +272083. q^{55} -207360. q^{56} -1.66030e6 q^{58} -193968. q^{59} +2.28582e6 q^{61} +176848. q^{62} +262144. q^{64} -126242. q^{65} -3.37306e6 q^{67} -1.41869e6 q^{68} -152280. q^{70} -110068. q^{71} +2.64009e6 q^{73} +4.40128e6 q^{74} -438976. q^{76} +2.34454e6 q^{77} +4.87090e6 q^{79} +192512. q^{80} -1.65440e6 q^{82} +5.99200e6 q^{83} -1.04185e6 q^{85} +4.52438e6 q^{86} -2.96397e6 q^{88} -3.07867e6 q^{89} -1.08783e6 q^{91} -817408. q^{92} +1.41562e6 q^{94} -322373. q^{95} +682750. q^{97} +5.27614e6 q^{98} +O(q^{100})\)

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 $

$2$	−8.00000	−0.707107
$2$	−8.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	64.0000	0.500000
$5$	47.0000	0.168152	0.0840762	−	0.996459i	$-0.473206\pi$
$5$	47.0000	0.168152	0.0840762	+	0.996459i	$0.473206\pi$
$6$	0	0
$7$	405.000	0.446285	0.223142	−	0.974786i	$-0.428369\pi$
$7$	405.000	0.446285	0.223142	+	0.974786i	$0.428369\pi$
$8$	−512.000	−0.353553
$9$	0	0
$10$	−376.000	−0.118902
$11$	5789.00	1.31138	0.655691	−	0.755029i	$-0.272377\pi$
$11$	5789.00	1.31138	0.655691	+	0.755029i	$0.272377\pi$
$12$	0	0
$13$	−2686.00	−0.339082	−0.169541	−	0.985523i	$-0.554228\pi$
$13$	−2686.00	−0.339082	−0.169541	+	0.985523i	$0.554228\pi$
$14$	−3240.00	−0.315571
$15$	0	0
$16$	4096.00	0.250000
$17$	−22167.0	−1.09430	−0.547149	−	0.837035i	$-0.684287\pi$
$17$	−22167.0	−1.09430	−0.547149	+	0.837035i	$0.684287\pi$
$18$	0	0
$19$	−6859.00	−0.229416
$19$	−6859.00	−0.229416
$20$	3008.00	0.0840762
$21$	0	0
$22$	−46312.0	−0.927287
$23$	−12772.0	−0.218883	−0.109441	−	0.993993i	$-0.534906\pi$
$23$	−12772.0	−0.218883	−0.109441	+	0.993993i	$0.534906\pi$
$24$	0	0
$25$	−75916.0	−0.971725
$26$	21488.0	0.239767
$27$	0	0
$28$	25920.0	0.223142
$29$	207538.	1.58017	0.790086	−	0.612995i	$-0.210036\pi$
$29$	207538.	1.58017	0.790086	+	0.612995i	$0.210036\pi$
$30$	0	0
$31$	−22106.0	−0.133274	−0.0666368	−	0.997777i	$-0.521227\pi$
$31$	−22106.0	−0.133274	−0.0666368	+	0.997777i	$0.521227\pi$
$32$	−32768.0	−0.176777
$33$	0	0
$34$	177336.	0.773786
$35$	19035.0	0.0750438
$36$	0	0
$37$	−550160.	−1.78560	−0.892798	−	0.450458i	$-0.851261\pi$
$37$	−550160.	−1.78560	−0.892798	+	0.450458i	$0.851261\pi$
$38$	54872.0	0.162221
$39$	0	0
$40$	−24064.0	−0.0594508
$41$	206800.	0.468605	0.234303	−	0.972164i	$-0.424719\pi$
$41$	206800.	0.468605	0.234303	+	0.972164i	$0.424719\pi$
$42$	0	0
$43$	−565547.	−1.08475	−0.542374	−	0.840137i	$-0.682475\pi$
$43$	−565547.	−1.08475	−0.542374	+	0.840137i	$0.682475\pi$
$44$	370496.	0.655691
$45$	0	0
$46$	102176.	0.154773
$47$	−176953.	−0.248608	−0.124304	−	0.992244i	$-0.539670\pi$
$47$	−176953.	−0.248608	−0.124304	+	0.992244i	$0.539670\pi$
$48$	0	0
$49$	−659518.	−0.800830
$50$	607328.	0.687113
$51$	0	0
$52$	−171904.	−0.169541
$53$	717230.	0.661748	0.330874	−	0.943675i	$-0.392656\pi$
$53$	717230.	0.661748	0.330874	+	0.943675i	$0.392656\pi$
$54$	0	0
$55$	272083.	0.220512
$56$	−207360.	−0.157785
$57$	0	0
$58$	−1.66030e6	−1.11735
$59$	−193968.	−0.122956	−0.0614778	−	0.998108i	$-0.519581\pi$
$59$	−193968.	−0.122956	−0.0614778	+	0.998108i	$0.519581\pi$
$60$	0	0
$61$	2.28582e6	1.28940	0.644700	−	0.764436i	$-0.276982\pi$
$61$	2.28582e6	1.28940	0.644700	+	0.764436i	$0.276982\pi$
$62$	176848.	0.0942387
$63$	0	0
$64$	262144.	0.125000
$65$	−126242.	−0.0570174
$66$	0	0
$67$	−3.37306e6	−1.37013	−0.685065	−	0.728482i	$-0.740226\pi$
$67$	−3.37306e6	−1.37013	−0.685065	+	0.728482i	$0.740226\pi$
$68$	−1.41869e6	−0.547149
$69$	0	0
$70$	−152280.	−0.0530640
$71$	−110068.	−0.0364970	−0.0182485	−	0.999833i	$-0.505809\pi$
$71$	−110068.	−0.0364970	−0.0182485	+	0.999833i	$0.505809\pi$
$72$	0	0
$73$	2.64009e6	0.794309	0.397154	−	0.917752i	$-0.369998\pi$
$73$	2.64009e6	0.794309	0.397154	+	0.917752i	$0.369998\pi$
$74$	4.40128e6	1.26261
$75$	0	0
$76$	−438976.	−0.114708
$77$	2.34454e6	0.585249
$78$	0	0
$79$	4.87090e6	1.11151	0.555757	−	0.831345i	$-0.312429\pi$
$79$	4.87090e6	1.11151	0.555757	+	0.831345i	$0.312429\pi$
$80$	192512.	0.0420381
$81$	0	0
$82$	−1.65440e6	−0.331354
$83$	5.99200e6	1.15027	0.575133	−	0.818060i	$-0.304950\pi$
$83$	5.99200e6	1.15027	0.575133	+	0.818060i	$0.304950\pi$
$84$	0	0
$85$	−1.04185e6	−0.184009
$86$	4.52438e6	0.767033
$87$	0	0
$88$	−2.96397e6	−0.463643
$89$	−3.07867e6	−0.462911	−0.231456	−	0.972845i	$-0.574349\pi$
$89$	−3.07867e6	−0.462911	−0.231456	+	0.972845i	$0.574349\pi$
$90$	0	0
$91$	−1.08783e6	−0.151327
$92$	−817408.	−0.109441
$93$	0	0
$94$	1.41562e6	0.175793
$95$	−322373.	−0.0385768
$96$	0	0
$97$	682750.	0.0759557	0.0379779	−	0.999279i	$-0.487908\pi$
$97$	682750.	0.0759557	0.0379779	+	0.999279i	$0.487908\pi$
$98$	5.27614e6	0.566272
$99$	0	0
$100$	−4.85862e6	−0.485862
$101$	7.67788e6	0.741509	0.370754	−	0.928731i	$-0.379099\pi$
$101$	7.67788e6	0.741509	0.370754	+	0.928731i	$0.379099\pi$
$102$	0	0
$103$	−1.79123e7	−1.61518	−0.807592	−	0.589742i	$-0.799229\pi$
$103$	−1.79123e7	−1.61518	−0.807592	+	0.589742i	$0.799229\pi$
$104$	1.37523e6	0.119883
$105$	0	0
$106$	−5.73784e6	−0.467927
$107$	−2.25525e7	−1.77972	−0.889859	−	0.456235i	$-0.849198\pi$
$107$	−2.25525e7	−1.77972	−0.889859	+	0.456235i	$0.849198\pi$
$108$	0	0
$109$	1.55690e7	1.15151	0.575754	−	0.817623i	$-0.304709\pi$
$109$	1.55690e7	1.15151	0.575754	+	0.817623i	$0.304709\pi$
$110$	−2.17666e6	−0.155925
$111$	0	0
$112$	1.65888e6	0.111571
$113$	1.96353e6	0.128015	0.0640077	−	0.997949i	$-0.479612\pi$
$113$	1.96353e6	0.128015	0.0640077	+	0.997949i	$0.479612\pi$
$114$	0	0
$115$	−600284.	−0.0368056
$116$	1.32824e7	0.790086
$117$	0	0
$118$	1.55174e6	0.0869427
$119$	−8.97764e6	−0.488368
$120$	0	0
$121$	1.40253e7	0.719722
$122$	−1.82866e7	−0.911743
$123$	0	0
$124$	−1.41478e6	−0.0666368
$125$	−7.23993e6	−0.331550
$126$	0	0
$127$	2.44391e6	0.105870	0.0529350	−	0.998598i	$-0.483142\pi$
$127$	2.44391e6	0.105870	0.0529350	+	0.998598i	$0.483142\pi$
$128$	−2.09715e6	−0.0883883
$129$	0	0
$130$	1.00994e6	0.0403174
$131$	−3.32146e7	−1.29086	−0.645430	−	0.763820i	$-0.723322\pi$
$131$	−3.32146e7	−1.29086	−0.645430	+	0.763820i	$0.723322\pi$
$132$	0	0
$133$	−2.77790e6	−0.102385
$134$	2.69844e7	0.968828
$135$	0	0
$136$	1.13495e7	0.386893
$137$	−3.83662e7	−1.27475	−0.637377	−	0.770552i	$-0.719981\pi$
$137$	−3.83662e7	−1.27475	−0.637377	+	0.770552i	$0.719981\pi$
$138$	0	0
$139$	−1.15521e7	−0.364846	−0.182423	−	0.983220i	$-0.558394\pi$
$139$	−1.15521e7	−0.364846	−0.182423	+	0.983220i	$0.558394\pi$
$140$	1.21824e6	0.0375219
$141$	0	0
$142$	880544.	0.0258073
$143$	−1.55493e7	−0.444665
$144$	0	0
$145$	9.75429e6	0.265710
$146$	−2.11207e7	−0.561661
$147$	0	0
$148$	−3.52102e7	−0.892798
$149$	2.75761e6	0.0682937	0.0341469	−	0.999417i	$-0.489129\pi$
$149$	2.75761e6	0.0682937	0.0341469	+	0.999417i	$0.489129\pi$
$150$	0	0
$151$	5.96131e7	1.40904	0.704518	−	0.709686i	$-0.251163\pi$
$151$	5.96131e7	1.40904	0.704518	+	0.709686i	$0.251163\pi$
$152$	3.51181e6	0.0811107
$153$	0	0
$154$	−1.87564e7	−0.413834
$155$	−1.03898e6	−0.0224103
$156$	0	0
$157$	−6.39589e7	−1.31902	−0.659511	−	0.751695i	$-0.729237\pi$
$157$	−6.39589e7	−1.31902	−0.659511	+	0.751695i	$0.729237\pi$
$158$	−3.89672e7	−0.785959
$159$	0	0
$160$	−1.54010e6	−0.0297254
$161$	−5.17266e6	−0.0976840
$162$	0	0
$163$	−6.83608e6	−0.123638	−0.0618188	−	0.998087i	$-0.519690\pi$
$163$	−6.83608e6	−0.123638	−0.0618188	+	0.998087i	$0.519690\pi$
$164$	1.32352e7	0.234303
$165$	0	0
$166$	−4.79360e7	−0.813361
$167$	−1.07101e8	−1.77946	−0.889729	−	0.456489i	$-0.849107\pi$
$167$	−1.07101e8	−1.77946	−0.889729	+	0.456489i	$0.849107\pi$
$168$	0	0
$169$	−5.55339e7	−0.885024
$170$	8.33479e6	0.130114
$171$	0	0
$172$	−3.61950e7	−0.542374
$173$	8.70178e7	1.27775	0.638876	−	0.769310i	$-0.279400\pi$
$173$	8.70178e7	1.27775	0.638876	+	0.769310i	$0.279400\pi$
$174$	0	0
$175$	−3.07460e7	−0.433666
$176$	2.37117e7	0.327845
$177$	0	0
$178$	2.46293e7	0.327328
$179$	−3.52042e7	−0.458784	−0.229392	−	0.973334i	$-0.573674\pi$
$179$	−3.52042e7	−0.458784	−0.229392	+	0.973334i	$0.573674\pi$
$180$	0	0
$181$	−1.00787e8	−1.26337	−0.631685	−	0.775226i	$-0.717636\pi$
$181$	−1.00787e8	−1.26337	−0.631685	+	0.775226i	$0.717636\pi$
$182$	8.70264e6	0.107004
$183$	0	0
$184$	6.53926e6	0.0773867
$185$	−2.58575e7	−0.300252
$186$	0	0
$187$	−1.28325e8	−1.43504
$188$	−1.13250e7	−0.124304
$189$	0	0
$190$	2.57898e6	0.0272779
$191$	−1.25924e8	−1.30765	−0.653827	−	0.756644i	$-0.726837\pi$
$191$	−1.25924e8	−1.30765	−0.653827	+	0.756644i	$0.726837\pi$
$192$	0	0
$193$	−1.74405e8	−1.74626	−0.873130	−	0.487487i	$-0.837914\pi$
$193$	−1.74405e8	−1.74626	−0.873130	+	0.487487i	$0.837914\pi$
$194$	−5.46200e6	−0.0537088
$195$	0	0
$196$	−4.22092e7	−0.400415
$197$	−7.00999e7	−0.653259	−0.326630	−	0.945152i	$-0.605913\pi$
$197$	−7.00999e7	−0.653259	−0.326630	+	0.945152i	$0.605913\pi$
$198$	0	0
$199$	−1.03652e8	−0.932376	−0.466188	−	0.884686i	$-0.654373\pi$
$199$	−1.03652e8	−0.932376	−0.466188	+	0.884686i	$0.654373\pi$
$200$	3.88690e7	0.343557
$201$	0	0
$202$	−6.14230e7	−0.524326
$203$	8.40529e7	0.705207
$204$	0	0
$205$	9.71960e6	0.0787970
$206$	1.43299e8	1.14211
$207$	0	0
$208$	−1.10019e7	−0.0847704
$209$	−3.97068e7	−0.300852
$210$	0	0
$211$	1.11182e8	0.814788	0.407394	−	0.913252i	$-0.366438\pi$
$211$	1.11182e8	0.814788	0.407394	+	0.913252i	$0.366438\pi$
$212$	4.59027e7	0.330874
$213$	0	0
$214$	1.80420e8	1.25845
$215$	−2.65807e7	−0.182403
$216$	0	0
$217$	−8.95293e6	−0.0594780
$218$	−1.24552e8	−0.814239
$219$	0	0
$220$	1.74133e7	0.110256
$221$	5.95406e7	0.371056
$222$	0	0
$223$	−1.13226e8	−0.683719	−0.341859	−	0.939751i	$-0.611057\pi$
$223$	−1.13226e8	−0.683719	−0.341859	+	0.939751i	$0.611057\pi$
$224$	−1.32710e7	−0.0788927
$225$	0	0
$226$	−1.57082e7	−0.0905205
$227$	1.50354e8	0.853145	0.426573	−	0.904453i	$-0.359721\pi$
$227$	1.50354e8	0.853145	0.426573	+	0.904453i	$0.359721\pi$
$228$	0	0
$229$	−9.19277e7	−0.505850	−0.252925	−	0.967486i	$-0.581393\pi$
$229$	−9.19277e7	−0.505850	−0.252925	+	0.967486i	$0.581393\pi$
$230$	4.80227e6	0.0260255
$231$	0	0
$232$	−1.06259e8	−0.558676
$233$	−3.00383e7	−0.155571	−0.0777857	−	0.996970i	$-0.524785\pi$
$233$	−3.00383e7	−0.155571	−0.0777857	+	0.996970i	$0.524785\pi$
$234$	0	0
$235$	−8.31679e6	−0.0418040
$236$	−1.24140e7	−0.0614778
$237$	0	0
$238$	7.18211e7	0.345329
$239$	2.84664e8	1.34878	0.674388	−	0.738377i	$-0.264407\pi$
$239$	2.84664e8	1.34878	0.674388	+	0.738377i	$0.264407\pi$
$240$	0	0
$241$	−2.10262e8	−0.967609	−0.483805	−	0.875176i	$-0.660746\pi$
$241$	−2.10262e8	−0.967609	−0.483805	+	0.875176i	$0.660746\pi$
$242$	−1.12203e8	−0.508920
$243$	0	0
$244$	1.46292e8	0.644700
$245$	−3.09973e7	−0.134661
$246$	0	0
$247$	1.84233e7	0.0777907
$248$	1.13183e7	0.0471193
$249$	0	0
$250$	5.79194e7	0.234441
$251$	−2.72044e8	−1.08588	−0.542939	−	0.839772i	$-0.682689\pi$
$251$	−2.72044e8	−1.08588	−0.542939	+	0.839772i	$0.682689\pi$
$252$	0	0
$253$	−7.39371e7	−0.287039
$254$	−1.95513e7	−0.0748614
$255$	0	0
$256$	1.67772e7	0.0625000
$257$	−1.87633e8	−0.689513	−0.344756	−	0.938692i	$-0.612038\pi$
$257$	−1.87633e8	−0.689513	−0.344756	+	0.938692i	$0.612038\pi$
$258$	0	0
$259$	−2.22815e8	−0.796884
$260$	−8.07949e6	−0.0285087
$261$	0	0
$262$	2.65716e8	0.912775
$263$	−1.71217e8	−0.580364	−0.290182	−	0.956971i	$-0.593716\pi$
$263$	−1.71217e8	−0.580364	−0.290182	+	0.956971i	$0.593716\pi$
$264$	0	0
$265$	3.37098e7	0.111275
$266$	2.22232e7	0.0723969
$267$	0	0
$268$	−2.15876e8	−0.685065
$269$	3.29403e8	1.03180	0.515899	−	0.856650i	$-0.327458\pi$
$269$	3.29403e8	1.03180	0.515899	+	0.856650i	$0.327458\pi$
$270$	0	0
$271$	−3.33163e8	−1.01687	−0.508434	−	0.861101i	$-0.669775\pi$
$271$	−3.33163e8	−1.01687	−0.508434	+	0.861101i	$0.669775\pi$
$272$	−9.07960e7	−0.273575
$273$	0	0
$274$	3.06929e8	0.901388
$275$	−4.39478e8	−1.27430
$276$	0	0
$277$	2.79160e8	0.789177	0.394589	−	0.918858i	$-0.370887\pi$
$277$	2.79160e8	0.789177	0.394589	+	0.918858i	$0.370887\pi$
$278$	9.24170e7	0.257985
$279$	0	0
$280$	−9.74592e6	−0.0265320
$281$	−1.55752e8	−0.418756	−0.209378	−	0.977835i	$-0.567144\pi$
$281$	−1.55752e8	−0.418756	−0.209378	+	0.977835i	$0.567144\pi$
$282$	0	0
$283$	4.07437e8	1.06858	0.534292	−	0.845300i	$-0.320578\pi$
$283$	4.07437e8	1.06858	0.534292	+	0.845300i	$0.320578\pi$
$284$	−7.04435e6	−0.0182485
$285$	0	0
$286$	1.24394e8	0.314426
$287$	8.37540e7	0.209131
$288$	0	0
$289$	8.10372e7	0.197489
$290$	−7.80343e7	−0.187885
$291$	0	0
$292$	1.68966e8	0.397154
$293$	−2.37989e8	−0.552738	−0.276369	−	0.961052i	$-0.589131\pi$
$293$	−2.37989e8	−0.552738	−0.276369	+	0.961052i	$0.589131\pi$
$294$	0	0
$295$	−9.11650e6	−0.0206753
$296$	2.81682e8	0.631303
$297$	0	0
$298$	−2.20609e7	−0.0482910
$299$	3.43056e7	0.0742191
$300$	0	0
$301$	−2.29047e8	−0.484107
$302$	−4.76905e8	−0.996339
$303$	0	0
$304$	−2.80945e7	−0.0573539
$305$	1.07433e8	0.216815
$306$	0	0
$307$	8.12671e8	1.60299	0.801494	−	0.598002i	$-0.204039\pi$
$307$	8.12671e8	1.60299	0.801494	+	0.598002i	$0.204039\pi$
$308$	1.50051e8	0.292625
$309$	0	0
$310$	8.31186e6	0.0158465
$311$	−3.39048e7	−0.0639146	−0.0319573	−	0.999489i	$-0.510174\pi$
$311$	−3.39048e7	−0.0639146	−0.0319573	+	0.999489i	$0.510174\pi$
$312$	0	0
$313$	−6.63210e8	−1.22249	−0.611246	−	0.791441i	$-0.709331\pi$
$313$	−6.63210e8	−1.22249	−0.611246	+	0.791441i	$0.709331\pi$
$314$	5.11671e8	0.932689
$315$	0	0
$316$	3.11738e8	0.555757
$317$	9.38878e8	1.65540	0.827698	−	0.561174i	$-0.189650\pi$
$317$	9.38878e8	1.65540	0.827698	+	0.561174i	$0.189650\pi$
$318$	0	0
$319$	1.20144e9	2.07221
$320$	1.23208e7	0.0210190
$321$	0	0
$322$	4.13813e7	0.0690730
$323$	1.52043e8	0.251049
$324$	0	0
$325$	2.03910e8	0.329494
$326$	5.46886e7	0.0874250
$327$	0	0
$328$	−1.05882e8	−0.165677
$329$	−7.16660e7	−0.110950
$330$	0	0
$331$	5.49170e7	0.0832356	0.0416178	−	0.999134i	$-0.486749\pi$
$331$	5.49170e7	0.0832356	0.0416178	+	0.999134i	$0.486749\pi$
$332$	3.83488e8	0.575133
$333$	0	0
$334$	8.56812e8	1.25827
$335$	−1.58534e8	−0.230391
$336$	0	0
$337$	−1.96331e8	−0.279437	−0.139719	−	0.990191i	$-0.544620\pi$
$337$	−1.96331e8	−0.279437	−0.139719	+	0.990191i	$0.544620\pi$
$338$	4.44271e8	0.625806
$339$	0	0
$340$	−6.66783e7	−0.0920044
$341$	−1.27972e8	−0.174773
$342$	0	0
$343$	−6.00640e8	−0.803683
$344$	2.89560e8	0.383517
$345$	0	0
$346$	−6.96142e8	−0.903507
$347$	1.95752e8	0.251509	0.125754	−	0.992061i	$-0.459865\pi$
$347$	1.95752e8	0.251509	0.125754	+	0.992061i	$0.459865\pi$
$348$	0	0
$349$	−2.87515e7	−0.0362052	−0.0181026	−	0.999836i	$-0.505763\pi$
$349$	−2.87515e7	−0.0362052	−0.0181026	+	0.999836i	$0.505763\pi$
$350$	2.45968e8	0.306648
$351$	0	0
$352$	−1.89694e8	−0.231822
$353$	−1.29097e9	−1.56209	−0.781044	−	0.624476i	$-0.785312\pi$
$353$	−1.29097e9	−1.56209	−0.781044	+	0.624476i	$0.785312\pi$
$354$	0	0
$355$	−5.17320e6	−0.00613705
$356$	−1.97035e8	−0.231456
$357$	0	0
$358$	2.81633e8	0.324410
$359$	−7.85526e8	−0.896046	−0.448023	−	0.894022i	$-0.647872\pi$
$359$	−7.85526e8	−0.896046	−0.448023	+	0.894022i	$0.647872\pi$
$360$	0	0
$361$	4.70459e7	0.0526316
$362$	8.06297e8	0.893337
$363$	0	0
$364$	−6.96211e7	−0.0756635
$365$	1.24084e8	0.133565
$366$	0	0
$367$	8.27544e8	0.873896	0.436948	−	0.899487i	$-0.356059\pi$
$367$	8.27544e8	0.873896	0.436948	+	0.899487i	$0.356059\pi$
$368$	−5.23141e7	−0.0547207
$369$	0	0
$370$	2.06860e8	0.212310
$371$	2.90478e8	0.295328
$372$	0	0
$373$	1.05698e9	1.05459	0.527297	−	0.849681i	$-0.323205\pi$
$373$	1.05698e9	1.05459	0.527297	+	0.849681i	$0.323205\pi$
$374$	1.02660e9	1.01473
$375$	0	0
$376$	9.05999e7	0.0878963
$377$	−5.57447e8	−0.535808
$378$	0	0
$379$	−3.39419e8	−0.320257	−0.160128	−	0.987096i	$-0.551191\pi$
$379$	−3.39419e8	−0.320257	−0.160128	+	0.987096i	$0.551191\pi$
$380$	−2.06319e7	−0.0192884
$381$	0	0
$382$	1.00739e9	0.924651
$383$	1.12239e9	1.02082	0.510409	−	0.859932i	$-0.329494\pi$
$383$	1.12239e9	1.02082	0.510409	+	0.859932i	$0.329494\pi$
$384$	0	0
$385$	1.10194e8	0.0984111
$386$	1.39524e9	1.23479
$387$	0	0
$388$	4.36960e7	0.0379779
$389$	−3.44125e8	−0.296410	−0.148205	−	0.988957i	$-0.547349\pi$
$389$	−3.44125e8	−0.296410	−0.148205	+	0.988957i	$0.547349\pi$
$390$	0	0
$391$	2.83117e8	0.239523
$392$	3.37673e8	0.283136
$393$	0	0
$394$	5.60799e8	0.461924
$395$	2.28932e8	0.186904
$396$	0	0
$397$	3.21642e8	0.257992	0.128996	−	0.991645i	$-0.458825\pi$
$397$	3.21642e8	0.257992	0.128996	+	0.991645i	$0.458825\pi$
$398$	8.29215e8	0.659290
$399$	0	0
$400$	−3.10952e8	−0.242931
$401$	1.47300e9	1.14077	0.570385	−	0.821378i	$-0.306794\pi$
$401$	1.47300e9	1.14077	0.570385	+	0.821378i	$0.306794\pi$
$402$	0	0
$403$	5.93767e7	0.0451906
$404$	4.91384e8	0.370754
$405$	0	0
$406$	−6.72423e8	−0.498657
$407$	−3.18488e9	−2.34160
$408$	0	0
$409$	−5.01533e8	−0.362467	−0.181233	−	0.983440i	$-0.558009\pi$
$409$	−5.01533e8	−0.362467	−0.181233	+	0.983440i	$0.558009\pi$
$410$	−7.77568e7	−0.0557179
$411$	0	0
$412$	−1.14639e9	−0.807592
$413$	−7.85570e7	−0.0548732
$414$	0	0
$415$	2.81624e8	0.193420
$416$	8.80148e7	0.0599417
$417$	0	0
$418$	3.17654e8	0.212734
$419$	1.14307e9	0.759145	0.379572	−	0.925162i	$-0.376071\pi$
$419$	1.14307e9	0.759145	0.379572	+	0.925162i	$0.376071\pi$
$420$	0	0
$421$	−6.88681e8	−0.449812	−0.224906	−	0.974381i	$-0.572207\pi$
$421$	−6.88681e8	−0.449812	−0.224906	+	0.974381i	$0.572207\pi$
$422$	−8.89454e8	−0.576142
$423$	0	0
$424$	−3.67222e8	−0.233963
$425$	1.68283e9	1.06336
$426$	0	0
$427$	9.25757e8	0.575439
$428$	−1.44336e9	−0.889859
$429$	0	0
$430$	2.12646e8	0.128978
$431$	−1.27854e9	−0.769206	−0.384603	−	0.923082i	$-0.625662\pi$
$431$	−1.27854e9	−0.769206	−0.384603	+	0.923082i	$0.625662\pi$
$432$	0	0
$433$	−2.35255e8	−0.139261	−0.0696307	−	0.997573i	$-0.522182\pi$
$433$	−2.35255e8	−0.139261	−0.0696307	+	0.997573i	$0.522182\pi$
$434$	7.16234e7	0.0420573
$435$	0	0
$436$	9.96413e8	0.575754
$437$	8.76031e7	0.0502151
$438$	0	0
$439$	−3.43092e8	−0.193546	−0.0967732	−	0.995306i	$-0.530852\pi$
$439$	−3.43092e8	−0.193546	−0.0967732	+	0.995306i	$0.530852\pi$
$440$	−1.39306e8	−0.0779627
$441$	0	0
$442$	−4.76324e8	−0.262377
$443$	−2.22955e9	−1.21844	−0.609220	−	0.793002i	$-0.708517\pi$
$443$	−2.22955e9	−1.21844	−0.609220	+	0.793002i	$0.708517\pi$
$444$	0	0
$445$	−1.44697e8	−0.0778396
$446$	9.05805e8	0.483462
$447$	0	0
$448$	1.06168e8	0.0557856
$449$	2.43306e9	1.26850	0.634249	−	0.773129i	$-0.281309\pi$
$449$	2.43306e9	1.26850	0.634249	+	0.773129i	$0.281309\pi$
$450$	0	0
$451$	1.19717e9	0.614520
$452$	1.25666e8	0.0640077
$453$	0	0
$454$	−1.20283e9	−0.603265
$455$	−5.11280e7	−0.0254460
$456$	0	0
$457$	1.70135e9	0.833849	0.416925	−	0.908941i	$-0.363108\pi$
$457$	1.70135e9	0.833849	0.416925	+	0.908941i	$0.363108\pi$
$458$	7.35421e8	0.357690
$459$	0	0
$460$	−3.84182e7	−0.0184028
$461$	−5.13279e8	−0.244006	−0.122003	−	0.992530i	$-0.538932\pi$
$461$	−5.13279e8	−0.244006	−0.122003	+	0.992530i	$0.538932\pi$
$462$	0	0
$463$	−2.75445e8	−0.128974	−0.0644869	−	0.997919i	$-0.520541\pi$
$463$	−2.75445e8	−0.128974	−0.0644869	+	0.997919i	$0.520541\pi$
$464$	8.50076e8	0.395043
$465$	0	0
$466$	2.40307e8	0.110006
$467$	−2.23875e9	−1.01718	−0.508589	−	0.861009i	$-0.669833\pi$
$467$	−2.23875e9	−1.01718	−0.508589	+	0.861009i	$0.669833\pi$
$468$	0	0
$469$	−1.36609e9	−0.611468
$470$	6.65343e7	0.0295599
$471$	0	0
$472$	9.93116e7	0.0434713
$473$	−3.27395e9	−1.42252
$474$	0	0
$475$	5.20708e8	0.222929
$476$	−5.74569e8	−0.244184
$477$	0	0
$478$	−2.27731e9	−0.953729
$479$	6.95463e8	0.289134	0.144567	−	0.989495i	$-0.453821\pi$
$479$	6.95463e8	0.289134	0.144567	+	0.989495i	$0.453821\pi$
$480$	0	0
$481$	1.47773e9	0.605463
$482$	1.68209e9	0.684203
$483$	0	0
$484$	8.97622e8	0.359861
$485$	3.20892e7	0.0127721
$486$	0	0
$487$	−1.71141e9	−0.671432	−0.335716	−	0.941963i	$-0.608978\pi$
$487$	−1.71141e9	−0.671432	−0.335716	+	0.941963i	$0.608978\pi$
$488$	−1.17034e9	−0.455871
$489$	0	0
$490$	2.47979e8	0.0952200
$491$	−1.02267e9	−0.389897	−0.194949	−	0.980813i	$-0.562454\pi$
$491$	−1.02267e9	−0.389897	−0.194949	+	0.980813i	$0.562454\pi$
$492$	0	0
$493$	−4.60049e9	−1.72918
$494$	−1.47386e8	−0.0550063
$495$	0	0
$496$	−9.05462e7	−0.0333184
$497$	−4.45775e7	−0.0162880
$498$	0	0
$499$	1.90293e9	0.685601	0.342800	−	0.939408i	$-0.388625\pi$
$499$	1.90293e9	0.685601	0.342800	+	0.939408i	$0.388625\pi$
$500$	−4.63355e8	−0.165775
$501$	0	0
$502$	2.17635e9	0.767832
$503$	1.45377e9	0.509341	0.254670	−	0.967028i	$-0.418033\pi$
$503$	1.45377e9	0.509341	0.254670	+	0.967028i	$0.418033\pi$
$504$	0	0
$505$	3.60860e8	0.124686
$506$	5.91497e8	0.202967
$507$	0	0
$508$	1.56410e8	0.0529350
$509$	−3.33356e9	−1.12046	−0.560230	−	0.828337i	$-0.689287\pi$
$509$	−3.33356e9	−1.12046	−0.560230	+	0.828337i	$0.689287\pi$
$510$	0	0
$511$	1.06924e9	0.354488
$512$	−1.34218e8	−0.0441942
$513$	0	0
$514$	1.50106e9	0.487559
$515$	−8.41880e8	−0.271597
$516$	0	0
$517$	−1.02438e9	−0.326020
$518$	1.78252e9	0.563482
$519$	0	0
$520$	6.46359e7	0.0201587
$521$	4.02480e9	1.24685	0.623423	−	0.781885i	$-0.285742\pi$
$521$	4.02480e9	1.24685	0.623423	+	0.781885i	$0.285742\pi$
$522$	0	0
$523$	2.17345e9	0.664344	0.332172	−	0.943219i	$-0.392219\pi$
$523$	2.17345e9	0.664344	0.332172	+	0.943219i	$0.392219\pi$
$524$	−2.12573e9	−0.645430
$525$	0	0
$526$	1.36973e9	0.410379
$527$	4.90024e8	0.145841
$528$	0	0
$529$	−3.24170e9	−0.952090
$530$	−2.69678e8	−0.0786830
$531$	0	0
$532$	−1.77785e8	−0.0511924
$533$	−5.55465e8	−0.158895
$534$	0	0
$535$	−1.05997e9	−0.299264
$536$	1.72700e9	0.484414
$537$	0	0
$538$	−2.63522e9	−0.729591
$539$	−3.81795e9	−1.05019
$540$	0	0
$541$	−5.51388e9	−1.49715	−0.748577	−	0.663048i	$-0.769263\pi$
$541$	−5.51388e9	−1.49715	−0.748577	+	0.663048i	$0.769263\pi$
$542$	2.66531e9	0.719035
$543$	0	0
$544$	7.26368e8	0.193446
$545$	7.31741e8	0.193629
$546$	0	0
$547$	5.10480e9	1.33359	0.666796	−	0.745240i	$-0.267665\pi$
$547$	5.10480e9	1.33359	0.666796	+	0.745240i	$0.267665\pi$
$548$	−2.45544e9	−0.637377
$549$	0	0
$550$	3.51582e9	0.901068
$551$	−1.42350e9	−0.362517
$552$	0	0
$553$	1.97272e9	0.496051
$554$	−2.23328e9	−0.558033
$555$	0	0
$556$	−7.39336e8	−0.182423
$557$	−3.97367e9	−0.974313	−0.487156	−	0.873315i	$-0.661966\pi$
$557$	−3.97367e9	−0.974313	−0.487156	+	0.873315i	$0.661966\pi$
$558$	0	0
$559$	1.51906e9	0.367818
$560$	7.79674e7	0.0187609
$561$	0	0
$562$	1.24602e9	0.296106
$563$	−5.32406e7	−0.0125737	−0.00628685	−	0.999980i	$-0.502001\pi$
$563$	−5.32406e7	−0.0125737	−0.00628685	+	0.999980i	$0.502001\pi$
$564$	0	0
$565$	9.22857e7	0.0215261
$566$	−3.25950e9	−0.755602
$567$	0	0
$568$	5.63548e7	0.0129036
$569$	−3.32310e9	−0.756224	−0.378112	−	0.925760i	$-0.623427\pi$
$569$	−3.32310e9	−0.756224	−0.378112	+	0.925760i	$0.623427\pi$
$570$	0	0
$571$	−6.35409e9	−1.42832	−0.714162	−	0.699980i	$-0.753192\pi$
$571$	−6.35409e9	−1.42832	−0.714162	+	0.699980i	$0.753192\pi$
$572$	−9.95152e8	−0.222333
$573$	0	0
$574$	−6.70032e8	−0.147878
$575$	9.69599e8	0.212694
$576$	0	0
$577$	1.06285e9	0.230334	0.115167	−	0.993346i	$-0.463260\pi$
$577$	1.06285e9	0.230334	0.115167	+	0.993346i	$0.463260\pi$
$578$	−6.48298e8	−0.139646
$579$	0	0
$580$	6.24274e8	0.132855
$581$	2.42676e9	0.513346
$582$	0	0
$583$	4.15204e9	0.867805
$584$	−1.35173e9	−0.280830
$585$	0	0
$586$	1.90391e9	0.390845
$587$	7.41543e9	1.51322	0.756611	−	0.653865i	$-0.226853\pi$
$587$	7.41543e9	1.51322	0.756611	+	0.653865i	$0.226853\pi$
$588$	0	0
$589$	1.51625e8	0.0305751
$590$	7.29320e7	0.0146196
$591$	0	0
$592$	−2.25346e9	−0.446399
$593$	3.62669e9	0.714199	0.357100	−	0.934066i	$-0.383766\pi$
$593$	3.62669e9	0.714199	0.357100	+	0.934066i	$0.383766\pi$
$594$	0	0
$595$	−4.21949e8	−0.0821203
$596$	1.76487e8	0.0341469
$597$	0	0
$598$	−2.74445e8	−0.0524808
$599$	−7.20953e9	−1.37061	−0.685304	−	0.728257i	$-0.740331\pi$
$599$	−7.20953e9	−1.37061	−0.685304	+	0.728257i	$0.740331\pi$
$600$	0	0
$601$	−5.31825e9	−0.999328	−0.499664	−	0.866219i	$-0.666543\pi$
$601$	−5.31825e9	−0.999328	−0.499664	+	0.866219i	$0.666543\pi$
$602$	1.83237e9	0.342315
$603$	0	0
$604$	3.81524e9	0.704518
$605$	6.59191e8	0.121023
$606$	0	0
$607$	−7.90514e9	−1.43466	−0.717330	−	0.696734i	$-0.754636\pi$
$607$	−7.90514e9	−1.43466	−0.717330	+	0.696734i	$0.754636\pi$
$608$	2.24756e8	0.0405554
$609$	0	0
$610$	−8.59468e8	−0.153312
$611$	4.75296e8	0.0842985
$612$	0	0
$613$	−8.31774e9	−1.45846	−0.729229	−	0.684270i	$-0.760121\pi$
$613$	−8.31774e9	−1.45846	−0.729229	+	0.684270i	$0.760121\pi$
$614$	−6.50137e9	−1.13348
$615$	0	0
$616$	−1.20041e9	−0.206917
$617$	1.66772e9	0.285841	0.142921	−	0.989734i	$-0.454351\pi$
$617$	1.66772e9	0.285841	0.142921	+	0.989734i	$0.454351\pi$
$618$	0	0
$619$	−7.25881e8	−0.123012	−0.0615061	−	0.998107i	$-0.519590\pi$
$619$	−7.25881e8	−0.123012	−0.0615061	+	0.998107i	$0.519590\pi$
$620$	−6.64948e7	−0.0112051
$621$	0	0
$622$	2.71239e8	0.0451945
$623$	−1.24686e9	−0.206590
$624$	0	0
$625$	5.59066e9	0.915974
$626$	5.30568e9	0.864432
$627$	0	0
$628$	−4.09337e9	−0.659511
$629$	1.21954e10	1.95397
$630$	0	0
$631$	1.19248e8	0.0188951	0.00944754	−	0.999955i	$-0.496993\pi$
$631$	1.19248e8	0.0188951	0.00944754	+	0.999955i	$0.496993\pi$
$632$	−2.49390e9	−0.392979
$633$	0	0
$634$	−7.51102e9	−1.17054
$635$	1.14864e8	0.0178023
$636$	0	0
$637$	1.77147e9	0.271547
$638$	−9.61150e9	−1.46527
$639$	0	0
$640$	−9.85661e7	−0.0148627
$641$	2.06745e9	0.310050	0.155025	−	0.987911i	$-0.450454\pi$
$641$	2.06745e9	0.310050	0.155025	+	0.987911i	$0.450454\pi$
$642$	0	0
$643$	7.06077e9	1.04740	0.523701	−	0.851902i	$-0.324551\pi$
$643$	7.06077e9	1.04740	0.523701	+	0.851902i	$0.324551\pi$
$644$	−3.31050e8	−0.0488420
$645$	0	0
$646$	−1.21635e9	−0.177519
$647$	−8.75594e8	−0.127098	−0.0635489	−	0.997979i	$-0.520242\pi$
$647$	−8.75594e8	−0.127098	−0.0635489	+	0.997979i	$0.520242\pi$
$648$	0	0
$649$	−1.12288e9	−0.161242
$650$	−1.63128e9	−0.232987
$651$	0	0
$652$	−4.37509e8	−0.0618188
$653$	1.38371e10	1.94469	0.972344	−	0.233554i	$-0.0750357\pi$
$653$	1.38371e10	1.94469	0.972344	+	0.233554i	$0.0750357\pi$
$654$	0	0
$655$	−1.56108e9	−0.217061
$656$	8.47053e8	0.117151
$657$	0	0
$658$	5.73328e8	0.0784535
$659$	2.59664e9	0.353437	0.176719	−	0.984261i	$-0.443452\pi$
$659$	2.59664e9	0.353437	0.176719	+	0.984261i	$0.443452\pi$
$660$	0	0
$661$	−3.39825e8	−0.0457668	−0.0228834	−	0.999738i	$-0.507285\pi$
$661$	−3.39825e8	−0.0457668	−0.0228834	+	0.999738i	$0.507285\pi$
$662$	−4.39336e8	−0.0588565
$663$	0	0
$664$	−3.06790e9	−0.406680
$665$	−1.30561e8	−0.0172162
$666$	0	0
$667$	−2.65068e9	−0.345873
$668$	−6.85449e9	−0.889729
$669$	0	0
$670$	1.26827e9	0.162911
$671$	1.32326e10	1.69089
$672$	0	0
$673$	−1.12882e10	−1.42749	−0.713745	−	0.700405i	$-0.753003\pi$
$673$	−1.12882e10	−1.42749	−0.713745	+	0.700405i	$0.753003\pi$
$674$	1.57065e9	0.197592
$675$	0	0
$676$	−3.55417e9	−0.442512
$677$	4.58827e9	0.568315	0.284157	−	0.958778i	$-0.408286\pi$
$677$	4.58827e9	0.568315	0.284157	+	0.958778i	$0.408286\pi$
$678$	0	0
$679$	2.76514e8	0.0338979
$680$	5.33427e8	0.0650569
$681$	0	0
$682$	1.02377e9	0.123583
$683$	−7.97420e8	−0.0957667	−0.0478833	−	0.998853i	$-0.515248\pi$
$683$	−7.97420e8	−0.0957667	−0.0478833	+	0.998853i	$0.515248\pi$
$684$	0	0
$685$	−1.80321e9	−0.214353
$686$	4.80512e9	0.568289
$687$	0	0
$688$	−2.31648e9	−0.271187
$689$	−1.92648e9	−0.224387
$690$	0	0
$691$	−1.32565e10	−1.52846	−0.764232	−	0.644942i	$-0.776882\pi$
$691$	−1.32565e10	−1.52846	−0.764232	+	0.644942i	$0.776882\pi$
$692$	5.56914e9	0.638876
$693$	0	0
$694$	−1.56602e9	−0.177844
$695$	−5.42950e8	−0.0613498
$696$	0	0
$697$	−4.58414e9	−0.512794
$698$	2.30012e8	0.0256009
$699$	0	0
$700$	−1.96774e9	−0.216833
$701$	7.50045e9	0.822384	0.411192	−	0.911549i	$-0.365113\pi$
$701$	7.50045e9	0.822384	0.411192	+	0.911549i	$0.365113\pi$
$702$	0	0
$703$	3.77355e9	0.409644
$704$	1.51755e9	0.163923
$705$	0	0
$706$	1.03278e10	1.10456
$707$	3.10954e9	0.330924
$708$	0	0
$709$	−6.17807e9	−0.651015	−0.325508	−	0.945539i	$-0.605535\pi$
$709$	−6.17807e9	−0.651015	−0.325508	+	0.945539i	$0.605535\pi$
$710$	4.13856e7	0.00433955
$711$	0	0
$712$	1.57628e9	0.163664
$713$	2.82338e8	0.0291713
$714$	0	0
$715$	−7.30815e8	−0.0747715
$716$	−2.25307e9	−0.229392
$717$	0	0
$718$	6.28421e9	0.633600
$719$	−3.92953e9	−0.394266	−0.197133	−	0.980377i	$-0.563163\pi$
$719$	−3.92953e9	−0.394266	−0.197133	+	0.980377i	$0.563163\pi$
$720$	0	0
$721$	−7.25450e9	−0.720832
$722$	−3.76367e8	−0.0372161
$723$	0	0
$724$	−6.45038e9	−0.631685
$725$	−1.57555e10	−1.53549
$726$	0	0
$727$	−1.25997e10	−1.21616	−0.608078	−	0.793878i	$-0.708059\pi$
$727$	−1.25997e10	−1.21616	−0.608078	+	0.793878i	$0.708059\pi$
$728$	5.56969e8	0.0535021
$729$	0	0
$730$	−9.92675e8	−0.0944446
$731$	1.25365e10	1.18704
$732$	0	0
$733$	9.94131e9	0.932352	0.466176	−	0.884692i	$-0.345631\pi$
$733$	9.94131e9	0.932352	0.466176	+	0.884692i	$0.345631\pi$
$734$	−6.62035e9	−0.617938
$735$	0	0
$736$	4.18513e8	0.0386934
$737$	−1.95266e10	−1.79676
$738$	0	0
$739$	1.01991e10	0.929619	0.464810	−	0.885411i	$-0.346123\pi$
$739$	1.01991e10	0.929619	0.464810	+	0.885411i	$0.346123\pi$
$740$	−1.65488e9	−0.150126
$741$	0	0
$742$	−2.32383e9	−0.208829
$743$	−9.87905e9	−0.883598	−0.441799	−	0.897114i	$-0.645659\pi$
$743$	−9.87905e9	−0.883598	−0.441799	+	0.897114i	$0.645659\pi$
$744$	0	0
$745$	1.29608e8	0.0114837
$746$	−8.45583e9	−0.745711
$747$	0	0
$748$	−8.21278e9	−0.717521
$749$	−9.13376e9	−0.794261
$750$	0	0
$751$	1.79143e10	1.54333	0.771667	−	0.636027i	$-0.219423\pi$
$751$	1.79143e10	1.54333	0.771667	+	0.636027i	$0.219423\pi$
$752$	−7.24799e8	−0.0621520
$753$	0	0
$754$	4.45958e9	0.378873
$755$	2.80181e9	0.236933
$756$	0	0
$757$	4.16109e9	0.348636	0.174318	−	0.984689i	$-0.444228\pi$
$757$	4.16109e9	0.348636	0.174318	+	0.984689i	$0.444228\pi$
$758$	2.71535e9	0.226456
$759$	0	0
$760$	1.65055e8	0.0136390
$761$	−2.13922e10	−1.75958	−0.879788	−	0.475365i	$-0.842316\pi$
$761$	−2.13922e10	−1.75958	−0.879788	+	0.475365i	$0.842316\pi$
$762$	0	0
$763$	6.30543e9	0.513900
$764$	−8.05915e9	−0.653827
$765$	0	0
$766$	−8.97912e9	−0.721827
$767$	5.20998e8	0.0416920
$768$	0	0
$769$	2.12703e10	1.68667	0.843337	−	0.537384i	$-0.180588\pi$
$769$	2.12703e10	1.68667	0.843337	+	0.537384i	$0.180588\pi$
$770$	−8.81549e8	−0.0695871
$771$	0	0
$772$	−1.11619e10	−0.873130
$773$	−1.57603e10	−1.22726	−0.613629	−	0.789594i	$-0.710291\pi$
$773$	−1.57603e10	−1.22726	−0.613629	+	0.789594i	$0.710291\pi$
$774$	0	0
$775$	1.67820e9	0.129505
$776$	−3.49568e8	−0.0268544
$777$	0	0
$778$	2.75300e9	0.209593
$779$	−1.41844e9	−0.107505
$780$	0	0
$781$	−6.37184e8	−0.0478615
$782$	−2.26494e9	−0.169368
$783$	0	0
$784$	−2.70139e9	−0.200208
$785$	−3.00607e9	−0.221797
$786$	0	0
$787$	2.03896e10	1.49106	0.745532	−	0.666470i	$-0.232196\pi$
$787$	2.03896e10	1.49106	0.745532	+	0.666470i	$0.232196\pi$
$788$	−4.48639e9	−0.326630
$789$	0	0
$790$	−1.83146e9	−0.132161
$791$	7.95228e8	0.0571313
$792$	0	0
$793$	−6.13971e9	−0.437212
$794$	−2.57314e9	−0.182428
$795$	0	0
$796$	−6.63372e9	−0.466188
$797$	1.13761e10	0.795959	0.397980	−	0.917394i	$-0.369711\pi$
$797$	1.13761e10	0.795959	0.397980	+	0.917394i	$0.369711\pi$
$798$	0	0
$799$	3.92252e9	0.272051
$800$	2.48762e9	0.171778
$801$	0	0
$802$	−1.17840e10	−0.806646
$803$	1.52835e10	1.04164
$804$	0	0
$805$	−2.43115e8	−0.0164258
$806$	−4.75014e8	−0.0319546
$807$	0	0
$808$	−3.93107e9	−0.262163
$809$	−1.85306e10	−1.23046	−0.615232	−	0.788346i	$-0.710938\pi$
$809$	−1.85306e10	−1.23046	−0.615232	+	0.788346i	$0.710938\pi$
$810$	0	0
$811$	−4.42227e9	−0.291120	−0.145560	−	0.989349i	$-0.546498\pi$
$811$	−4.42227e9	−0.291120	−0.145560	+	0.989349i	$0.546498\pi$
$812$	5.37938e9	0.352603
$813$	0	0
$814$	2.54790e10	1.65576
$815$	−3.21296e8	−0.0207899
$816$	0	0
$817$	3.87909e9	0.248858
$818$	4.01227e9	0.256303
$819$	0	0
$820$	6.22054e8	0.0393985
$821$	−1.34681e9	−0.0849385	−0.0424692	−	0.999098i	$-0.513522\pi$
$821$	−1.34681e9	−0.0849385	−0.0424692	+	0.999098i	$0.513522\pi$
$822$	0	0
$823$	8.76278e9	0.547952	0.273976	−	0.961737i	$-0.411661\pi$
$823$	8.76278e9	0.547952	0.273976	+	0.961737i	$0.411661\pi$
$824$	9.17112e9	0.571054
$825$	0	0
$826$	6.28456e8	0.0388012
$827$	−2.11561e10	−1.30067	−0.650333	−	0.759649i	$-0.725371\pi$
$827$	−2.11561e10	−1.30067	−0.650333	+	0.759649i	$0.725371\pi$
$828$	0	0
$829$	2.96669e10	1.80855	0.904276	−	0.426947i	$-0.140411\pi$
$829$	2.96669e10	1.80855	0.904276	+	0.426947i	$0.140411\pi$
$830$	−2.25299e9	−0.136768
$831$	0	0
$832$	−7.04119e8	−0.0423852
$833$	1.46195e10	0.876347
$834$	0	0
$835$	−5.03377e9	−0.299220
$836$	−2.54123e9	−0.150426
$837$	0	0
$838$	−9.14458e9	−0.536797
$839$	9.75674e9	0.570345	0.285173	−	0.958476i	$-0.407949\pi$
$839$	9.75674e9	0.570345	0.285173	+	0.958476i	$0.407949\pi$
$840$	0	0
$841$	2.58221e10	1.49695
$842$	5.50945e9	0.318065
$843$	0	0
$844$	7.11563e9	0.407394
$845$	−2.61009e9	−0.148819
$846$	0	0
$847$	5.68027e9	0.321201
$848$	2.93777e9	0.165437
$849$	0	0
$850$	−1.34626e10	−0.751907
$851$	7.02664e9	0.390836
$852$	0	0
$853$	2.61418e10	1.44216	0.721082	−	0.692850i	$-0.243645\pi$
$853$	2.61418e10	1.44216	0.721082	+	0.692850i	$0.243645\pi$
$854$	−7.40605e9	−0.406897
$855$	0	0
$856$	1.15469e10	0.629226
$857$	3.22214e10	1.74868	0.874341	−	0.485311i	$-0.161294\pi$
$857$	3.22214e10	1.74868	0.874341	+	0.485311i	$0.161294\pi$
$858$	0	0
$859$	−2.19107e10	−1.17945	−0.589727	−	0.807603i	$-0.700765\pi$
$859$	−2.19107e10	−1.17945	−0.589727	+	0.807603i	$0.700765\pi$
$860$	−1.70117e9	−0.0912015
$861$	0	0
$862$	1.02283e10	0.543911
$863$	1.50947e10	0.799440	0.399720	−	0.916637i	$-0.369107\pi$
$863$	1.50947e10	0.799440	0.399720	+	0.916637i	$0.369107\pi$
$864$	0	0
$865$	4.08984e9	0.214857
$866$	1.88204e9	0.0984726
$867$	0	0
$868$	−5.72988e8	−0.0297390
$869$	2.81977e10	1.45762
$870$	0	0
$871$	9.06003e9	0.464586
$872$	−7.97131e9	−0.407119
$873$	0	0
$874$	−7.00825e8	−0.0355075
$875$	−2.93217e9	−0.147966
$876$	0	0
$877$	−2.36643e10	−1.18467	−0.592333	−	0.805694i	$-0.701793\pi$
$877$	−2.36643e10	−1.18467	−0.592333	+	0.805694i	$0.701793\pi$
$878$	2.74474e9	0.136858
$879$	0	0
$880$	1.11445e9	0.0551280
$881$	−1.03279e10	−0.508857	−0.254428	−	0.967092i	$-0.581887\pi$
$881$	−1.03279e10	−0.508857	−0.254428	+	0.967092i	$0.581887\pi$
$882$	0	0
$883$	1.49309e10	0.729833	0.364916	−	0.931040i	$-0.381098\pi$
$883$	1.49309e10	0.729833	0.364916	+	0.931040i	$0.381098\pi$
$884$	3.81060e9	0.185528
$885$	0	0
$886$	1.78364e10	0.861567
$887$	2.68807e10	1.29333	0.646663	−	0.762776i	$-0.276164\pi$
$887$	2.68807e10	1.29333	0.646663	+	0.762776i	$0.276164\pi$
$888$	0	0
$889$	9.89785e8	0.0472481
$890$	1.15758e9	0.0550409
$891$	0	0
$892$	−7.24644e9	−0.341859
$893$	1.21372e9	0.0570346
$894$	0	0
$895$	−1.65460e9	−0.0771456
$896$	−8.49347e8	−0.0394464
$897$	0	0
$898$	−1.94644e10	−0.896964
$899$	−4.58784e9	−0.210595
$900$	0	0
$901$	−1.58988e10	−0.724150
$902$	−9.57732e9	−0.434531
$903$	0	0
$904$	−1.00533e9	−0.0452603
$905$	−4.73700e9	−0.212438
$906$	0	0
$907$	−2.97093e10	−1.32211	−0.661054	−	0.750339i	$-0.729890\pi$
$907$	−2.97093e10	−1.32211	−0.661054	+	0.750339i	$0.729890\pi$
$908$	9.62262e9	0.426573
$909$	0	0
$910$	4.09024e8	0.0179930
$911$	3.32322e10	1.45628	0.728141	−	0.685428i	$-0.240385\pi$
$911$	3.32322e10	1.45628	0.728141	+	0.685428i	$0.240385\pi$
$912$	0	0
$913$	3.46877e10	1.50844
$914$	−1.36108e10	−0.589621
$915$	0	0
$916$	−5.88337e9	−0.252925
$917$	−1.34519e10	−0.576091
$918$	0	0
$919$	−1.06486e10	−0.452573	−0.226287	−	0.974061i	$-0.572659\pi$
$919$	−1.06486e10	−0.452573	−0.226287	+	0.974061i	$0.572659\pi$
$920$	3.07345e8	0.0130128
$921$	0	0
$922$	4.10623e9	0.172538
$923$	2.95643e8	0.0123755
$924$	0	0
$925$	4.17659e10	1.73511
$926$	2.20356e9	0.0911983
$927$	0	0
$928$	−6.80061e9	−0.279338
$929$	−4.22749e10	−1.72993	−0.864963	−	0.501836i	$-0.832658\pi$
$929$	−4.22749e10	−1.72993	−0.864963	+	0.501836i	$0.832658\pi$
$930$	0	0
$931$	4.52363e9	0.183723
$932$	−1.92245e9	−0.0777857
$933$	0	0
$934$	1.79100e10	0.719253
$935$	−6.03126e9	−0.241306
$936$	0	0
$937$	5.94141e9	0.235940	0.117970	−	0.993017i	$-0.462361\pi$
$937$	5.94141e9	0.235940	0.117970	+	0.993017i	$0.462361\pi$
$938$	1.09287e10	0.432373
$939$	0	0
$940$	−5.32275e8	−0.0209020
$941$	−2.03236e10	−0.795126	−0.397563	−	0.917575i	$-0.630144\pi$
$941$	−2.03236e10	−0.795126	−0.397563	+	0.917575i	$0.630144\pi$
$942$	0	0
$943$	−2.64125e9	−0.102570
$944$	−7.94493e8	−0.0307389
$945$	0	0
$946$	2.61916e10	1.00587
$947$	−4.77133e9	−0.182564	−0.0912819	−	0.995825i	$-0.529096\pi$
$947$	−4.77133e9	−0.182564	−0.0912819	+	0.995825i	$0.529096\pi$
$948$	0	0
$949$	−7.09129e9	−0.269335
$950$	−4.16566e9	−0.157635
$951$	0	0
$952$	4.59655e9	0.172664
$953$	4.56589e10	1.70884	0.854418	−	0.519586i	$-0.173914\pi$
$953$	4.56589e10	1.70884	0.854418	+	0.519586i	$0.173914\pi$
$954$	0	0
$955$	−5.91844e9	−0.219885
$956$	1.82185e10	0.674388
$957$	0	0
$958$	−5.56370e9	−0.204449
$959$	−1.55383e10	−0.568903
$960$	0	0
$961$	−2.70239e10	−0.982238
$962$	−1.18218e10	−0.428127
$963$	0	0
$964$	−1.34567e10	−0.483805
$965$	−8.19705e9	−0.293638
$966$	0	0
$967$	−1.01197e10	−0.359895	−0.179947	−	0.983676i	$-0.557593\pi$
$967$	−1.01197e10	−0.359895	−0.179947	+	0.983676i	$0.557593\pi$
$968$	−7.18098e9	−0.254460
$969$	0	0
$970$	−2.56714e8	−0.00903126
$971$	4.55205e10	1.59566	0.797828	−	0.602884i	$-0.205982\pi$
$971$	4.55205e10	1.59566	0.797828	+	0.602884i	$0.205982\pi$
$972$	0	0
$973$	−4.67861e9	−0.162825
$974$	1.36913e10	0.474774
$975$	0	0
$976$	9.36271e9	0.322350
$977$	4.25111e10	1.45838	0.729191	−	0.684311i	$-0.239897\pi$
$977$	4.25111e10	1.45838	0.729191	+	0.684311i	$0.239897\pi$
$978$	0	0
$979$	−1.78224e10	−0.607053
$980$	−1.98383e9	−0.0673307
$981$	0	0
$982$	8.18136e9	0.275699
$983$	4.87857e10	1.63816	0.819079	−	0.573681i	$-0.194485\pi$
$983$	4.87857e10	1.63816	0.819079	+	0.573681i	$0.194485\pi$
$984$	0	0
$985$	−3.29470e9	−0.109847
$986$	3.68040e10	1.22272
$987$	0	0
$988$	1.17909e9	0.0388953
$989$	7.22317e9	0.237433
$990$	0	0
$991$	5.46555e10	1.78392	0.891962	−	0.452111i	$-0.149329\pi$
$991$	5.46555e10	1.78392	0.891962	+	0.452111i	$0.149329\pi$
$992$	7.24369e8	0.0235597
$993$	0	0
$994$	3.56620e8	0.0115174
$995$	−4.87164e9	−0.156781
$996$	0	0
$997$	1.31485e10	0.420187	0.210093	−	0.977681i	$-0.432623\pi$
$997$	1.31485e10	0.420187	0.210093	+	0.977681i	$0.432623\pi$
$998$	−1.52235e10	−0.484793
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	342.8.a.a.1.1		1
3.2	odd	2		114.8.a.d.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.8.a.d.1.1	✓	1	3.2	odd	2
342.8.a.a.1.1		1	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:	\( N \)	\(=\)	\( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	342.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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