LMFDB - Embedded newform 40.10.a.c.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [40,10,Mod(1,40)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(40, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 10, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("40.1"); S:= CuspForms(chi, 10); N := Newforms(S);

Level:	$ N $	$=$	$ 40 = 2^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	40.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,108] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$20.6014334466$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{46}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 46 $ x^2 - 46
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{3}\cdot 3 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$6.78233$ of defining polynomial
Character	$\chi$	$=$	40.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+216.776 q^{3} -625.000 q^{5} +1662.09 q^{7} +27308.8 q^{9} +34372.0 q^{11} +162675. q^{13} -135485. q^{15} +91796.1 q^{17} -536227. q^{19} +360300. q^{21} +234899. q^{23} +390625. q^{25} +1.65309e6 q^{27} +7.25706e6 q^{29} +8.23051e6 q^{31} +7.45102e6 q^{33} -1.03880e6 q^{35} -1.56850e7 q^{37} +3.52641e7 q^{39} -1.67318e7 q^{41} +9.36208e6 q^{43} -1.70680e7 q^{45} -2.69410e7 q^{47} -3.75911e7 q^{49} +1.98992e7 q^{51} +3.41105e7 q^{53} -2.14825e7 q^{55} -1.16241e8 q^{57} +1.12155e8 q^{59} +5.49531e6 q^{61} +4.53896e7 q^{63} -1.01672e8 q^{65} -2.40686e8 q^{67} +5.09204e7 q^{69} -2.58502e8 q^{71} -3.53583e8 q^{73} +8.46781e7 q^{75} +5.71292e7 q^{77} +3.66190e8 q^{79} -1.79169e8 q^{81} -9.74953e6 q^{83} -5.73726e7 q^{85} +1.57316e9 q^{87} -1.14201e9 q^{89} +2.70380e8 q^{91} +1.78418e9 q^{93} +3.35142e8 q^{95} +1.25555e9 q^{97} +9.38657e8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 108 q^{3} - 1250 q^{5} - 908 q^{7} + 19458 q^{9} + 25120 q^{11} + 146948 q^{13} - 67500 q^{15} + 169268 q^{17} - 25480 q^{19} + 639864 q^{21} + 1782748 q^{23} + 781250 q^{25} + 4648104 q^{27} + 7323340 q^{29}+ \cdots + 1011292704 q^{99}+O(q^{100}) $ 2 * q + 108 * q^3 - 1250 * q^5 - 908 * q^7 + 19458 * q^9 + 25120 * q^11 + 146948 * q^13 - 67500 * q^15 + 169268 * q^17 - 25480 * q^19 + 639864 * q^21 + 1782748 * q^23 + 781250 * q^25 + 4648104 * q^27 + 7323340 * q^29 + 10677272 * q^31 + 8457408 * q^33 + 567500 * q^35 - 5750460 * q^37 + 36974808 * q^39 - 7795764 * q^41 + 16770524 * q^43 - 12161250 * q^45 + 15393892 * q^47 - 71339334 * q^49 + 11472120 * q^51 - 58529292 * q^53 - 15700000 * q^55 - 171798192 * q^57 + 59618264 * q^59 - 188163772 * q^61 + 65566836 * q^63 - 91842500 * q^65 - 105998252 * q^67 - 117448344 * q^69 - 168665592 * q^71 - 390212412 * q^73 + 42187500 * q^75 + 80907584 * q^77 + 466946256 * q^79 - 350427222 * q^81 + 329535164 * q^83 - 105792500 * q^85 + 1565947656 * q^87 - 108334860 * q^89 + 310800616 * q^91 + 1518028560 * q^93 + 15925000 * q^95 + 2043058628 * q^97 + 1011292704 * q^99

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
$3$	216.776	1.54513	0.772566	−	0.634935i	$-0.218973\pi$
$3$	216.776	1.54513	0.772566	+	0.634935i	$0.218973\pi$
$4$	0	0
$5$	−625.000	−0.447214
$5$	−625.000	−0.447214
$6$	0	0
$7$	1662.09	0.261645	0.130823	−	0.991406i	$-0.458238\pi$
$7$	1662.09	0.261645	0.130823	+	0.991406i	$0.458238\pi$
$8$	0	0
$9$	27308.8	1.38743
$10$	0	0
$11$	34372.0	0.707844	0.353922	−	0.935275i	$-0.384848\pi$
$11$	34372.0	0.707844	0.353922	+	0.935275i	$0.384848\pi$
$12$	0	0
$13$	162675.	1.57971	0.789853	−	0.613296i	$-0.210157\pi$
$13$	162675.	1.57971	0.789853	+	0.613296i	$0.210157\pi$
$14$	0	0
$15$	−135485.	−0.691004
$16$	0	0
$17$	91796.1	0.266566	0.133283	−	0.991078i	$-0.457448\pi$
$17$	91796.1	0.266566	0.133283	+	0.991078i	$0.457448\pi$
$18$	0	0
$19$	−536227.	−0.943969	−0.471985	−	0.881607i	$-0.656462\pi$
$19$	−536227.	−0.943969	−0.471985	+	0.881607i	$0.656462\pi$
$20$	0	0
$21$	360300.	0.404276
$22$	0	0
$23$	234899.	0.175027	0.0875136	−	0.996163i	$-0.472108\pi$
$23$	234899.	0.175027	0.0875136	+	0.996163i	$0.472108\pi$
$24$	0	0
$25$	390625.	0.200000
$26$	0	0
$27$	1.65309e6	0.598631
$28$	0	0
$29$	7.25706e6	1.90533	0.952665	−	0.304023i	$-0.0983301\pi$
$29$	7.25706e6	1.90533	0.952665	+	0.304023i	$0.0983301\pi$
$30$	0	0
$31$	8.23051e6	1.60066	0.800330	−	0.599559i	$-0.204657\pi$
$31$	8.23051e6	1.60066	0.800330	+	0.599559i	$0.204657\pi$
$32$	0	0
$33$	7.45102e6	1.09371
$34$	0	0
$35$	−1.03880e6	−0.117011
$36$	0	0
$37$	−1.56850e7	−1.37587	−0.687936	−	0.725771i	$-0.741483\pi$
$37$	−1.56850e7	−1.37587	−0.687936	+	0.725771i	$0.741483\pi$
$38$	0	0
$39$	3.52641e7	2.44085
$40$	0	0
$41$	−1.67318e7	−0.924730	−0.462365	−	0.886690i	$-0.652999\pi$
$41$	−1.67318e7	−0.924730	−0.462365	+	0.886690i	$0.652999\pi$
$42$	0	0
$43$	9.36208e6	0.417604	0.208802	−	0.977958i	$-0.433044\pi$
$43$	9.36208e6	0.417604	0.208802	+	0.977958i	$0.433044\pi$
$44$	0	0
$45$	−1.70680e7	−0.620478
$46$	0	0
$47$	−2.69410e7	−0.805328	−0.402664	−	0.915348i	$-0.631916\pi$
$47$	−2.69410e7	−0.805328	−0.402664	+	0.915348i	$0.631916\pi$
$48$	0	0
$49$	−3.75911e7	−0.931542
$50$	0	0
$51$	1.98992e7	0.411879
$52$	0	0
$53$	3.41105e7	0.593809	0.296905	−	0.954907i	$-0.404046\pi$
$53$	3.41105e7	0.593809	0.296905	+	0.954907i	$0.404046\pi$
$54$	0	0
$55$	−2.14825e7	−0.316557
$56$	0	0
$57$	−1.16241e8	−1.45856
$58$	0	0
$59$	1.12155e8	1.20499	0.602496	−	0.798122i	$-0.294173\pi$
$59$	1.12155e8	1.20499	0.602496	+	0.798122i	$0.294173\pi$
$60$	0	0
$61$	5.49531e6	0.0508168	0.0254084	−	0.999677i	$-0.491911\pi$
$61$	5.49531e6	0.0508168	0.0254084	+	0.999677i	$0.491911\pi$
$62$	0	0
$63$	4.53896e7	0.363014
$64$	0	0
$65$	−1.01672e8	−0.706466
$66$	0	0
$67$	−2.40686e8	−1.45920	−0.729598	−	0.683876i	$-0.760293\pi$
$67$	−2.40686e8	−1.45920	−0.729598	+	0.683876i	$0.760293\pi$
$68$	0	0
$69$	5.09204e7	0.270440
$70$	0	0
$71$	−2.58502e8	−1.20726	−0.603631	−	0.797264i	$-0.706280\pi$
$71$	−2.58502e8	−1.20726	−0.603631	+	0.797264i	$0.706280\pi$
$72$	0	0
$73$	−3.53583e8	−1.45726	−0.728632	−	0.684905i	$-0.759844\pi$
$73$	−3.53583e8	−1.45726	−0.728632	+	0.684905i	$0.759844\pi$
$74$	0	0
$75$	8.46781e7	0.309026
$76$	0	0
$77$	5.71292e7	0.185204
$78$	0	0
$79$	3.66190e8	1.05775	0.528876	−	0.848699i	$-0.322614\pi$
$79$	3.66190e8	1.05775	0.528876	+	0.848699i	$0.322614\pi$
$80$	0	0
$81$	−1.79169e8	−0.462467
$82$	0	0
$83$	−9.74953e6	−0.0225493	−0.0112746	−	0.999936i	$-0.503589\pi$
$83$	−9.74953e6	−0.0225493	−0.0112746	+	0.999936i	$0.503589\pi$
$84$	0	0
$85$	−5.73726e7	−0.119212
$86$	0	0
$87$	1.57316e9	2.94398
$88$	0	0
$89$	−1.14201e9	−1.92937	−0.964687	−	0.263397i	$-0.915157\pi$
$89$	−1.14201e9	−1.92937	−0.964687	+	0.263397i	$0.915157\pi$
$90$	0	0
$91$	2.70380e8	0.413322
$92$	0	0
$93$	1.78418e9	2.47323
$94$	0	0
$95$	3.35142e8	0.422156
$96$	0	0
$97$	1.25555e9	1.44000	0.719999	−	0.693975i	$-0.244142\pi$
$97$	1.25555e9	1.44000	0.719999	+	0.693975i	$0.244142\pi$
$98$	0	0
$99$	9.38657e8	0.982084
$100$	0	0
$101$	2.63563e8	0.252022	0.126011	−	0.992029i	$-0.459783\pi$
$101$	2.63563e8	0.252022	0.126011	+	0.992029i	$0.459783\pi$
$102$	0	0
$103$	−2.09532e9	−1.83435	−0.917176	−	0.398482i	$-0.869537\pi$
$103$	−2.09532e9	−1.83435	−0.917176	+	0.398482i	$0.869537\pi$
$104$	0	0
$105$	−2.25188e8	−0.180798
$106$	0	0
$107$	−3.30509e8	−0.243757	−0.121878	−	0.992545i	$-0.538892\pi$
$107$	−3.30509e8	−0.243757	−0.121878	+	0.992545i	$0.538892\pi$
$108$	0	0
$109$	−1.65293e9	−1.12160	−0.560798	−	0.827953i	$-0.689505\pi$
$109$	−1.65293e9	−1.12160	−0.560798	+	0.827953i	$0.689505\pi$
$110$	0	0
$111$	−3.40014e9	−2.12590
$112$	0	0
$113$	1.15695e9	0.667513	0.333757	−	0.942659i	$-0.391684\pi$
$113$	1.15695e9	0.667513	0.333757	+	0.942659i	$0.391684\pi$
$114$	0	0
$115$	−1.46812e8	−0.0782745
$116$	0	0
$117$	4.44246e9	2.19173
$118$	0	0
$119$	1.52573e8	0.0697456
$120$	0	0
$121$	−1.17652e9	−0.498957
$122$	0	0
$123$	−3.62705e9	−1.42883
$124$	0	0
$125$	−2.44141e8	−0.0894427
$126$	0	0
$127$	1.56129e9	0.532559	0.266280	−	0.963896i	$-0.414206\pi$
$127$	1.56129e9	0.532559	0.266280	+	0.963896i	$0.414206\pi$
$128$	0	0
$129$	2.02947e9	0.645253
$130$	0	0
$131$	−4.75445e9	−1.41052	−0.705261	−	0.708948i	$-0.749170\pi$
$131$	−4.75445e9	−1.41052	−0.705261	+	0.708948i	$0.749170\pi$
$132$	0	0
$133$	−8.91256e8	−0.246985
$134$	0	0
$135$	−1.03318e9	−0.267716
$136$	0	0
$137$	1.57706e9	0.382478	0.191239	−	0.981543i	$-0.438749\pi$
$137$	1.57706e9	0.382478	0.191239	+	0.981543i	$0.438749\pi$
$138$	0	0
$139$	3.40723e9	0.774167	0.387084	−	0.922045i	$-0.373482\pi$
$139$	3.40723e9	0.774167	0.387084	+	0.922045i	$0.373482\pi$
$140$	0	0
$141$	−5.84015e9	−1.24434
$142$	0	0
$143$	5.59147e9	1.11818
$144$	0	0
$145$	−4.53567e9	−0.852089
$146$	0	0
$147$	−8.14884e9	−1.43935
$148$	0	0
$149$	−4.20713e8	−0.0699274	−0.0349637	−	0.999389i	$-0.511132\pi$
$149$	−4.20713e8	−0.0699274	−0.0349637	+	0.999389i	$0.511132\pi$
$150$	0	0
$151$	−1.74377e9	−0.272957	−0.136478	−	0.990643i	$-0.543578\pi$
$151$	−1.74377e9	−0.272957	−0.136478	+	0.990643i	$0.543578\pi$
$152$	0	0
$153$	2.50684e9	0.369841
$154$	0	0
$155$	−5.14407e9	−0.715837
$156$	0	0
$157$	−6.13142e9	−0.805402	−0.402701	−	0.915332i	$-0.631929\pi$
$157$	−6.13142e9	−0.805402	−0.402701	+	0.915332i	$0.631929\pi$
$158$	0	0
$159$	7.39434e9	0.917513
$160$	0	0
$161$	3.90422e8	0.0457950
$162$	0	0
$163$	1.07953e10	1.19782	0.598910	−	0.800816i	$-0.295601\pi$
$163$	1.07953e10	1.19782	0.598910	+	0.800816i	$0.295601\pi$
$164$	0	0
$165$	−4.65689e9	−0.489123
$166$	0	0
$167$	1.55417e10	1.54623	0.773115	−	0.634266i	$-0.218698\pi$
$167$	1.55417e10	1.54623	0.773115	+	0.634266i	$0.218698\pi$
$168$	0	0
$169$	1.58587e10	1.49547
$170$	0	0
$171$	−1.46437e10	−1.30969
$172$	0	0
$173$	3.21498e9	0.272880	0.136440	−	0.990648i	$-0.456434\pi$
$173$	3.21498e9	0.272880	0.136440	+	0.990648i	$0.456434\pi$
$174$	0	0
$175$	6.49253e8	0.0523290
$176$	0	0
$177$	2.43125e10	1.86187
$178$	0	0
$179$	2.10234e10	1.53061	0.765305	−	0.643667i	$-0.222588\pi$
$179$	2.10234e10	1.53061	0.765305	+	0.643667i	$0.222588\pi$
$180$	0	0
$181$	−1.54514e10	−1.07008	−0.535038	−	0.844828i	$-0.679703\pi$
$181$	−1.54514e10	−1.07008	−0.535038	+	0.844828i	$0.679703\pi$
$182$	0	0
$183$	1.19125e9	0.0785187
$184$	0	0
$185$	9.80315e9	0.615309
$186$	0	0
$187$	3.15521e9	0.188687
$188$	0	0
$189$	2.74758e9	0.156629
$190$	0	0
$191$	2.53988e10	1.38090	0.690451	−	0.723380i	$-0.257412\pi$
$191$	2.53988e10	1.38090	0.690451	+	0.723380i	$0.257412\pi$
$192$	0	0
$193$	−1.70167e10	−0.882809	−0.441404	−	0.897308i	$-0.645520\pi$
$193$	−1.70167e10	−0.882809	−0.441404	+	0.897308i	$0.645520\pi$
$194$	0	0
$195$	−2.20400e10	−1.09158
$196$	0	0
$197$	4.38218e9	0.207297	0.103648	−	0.994614i	$-0.466948\pi$
$197$	4.38218e9	0.207297	0.103648	+	0.994614i	$0.466948\pi$
$198$	0	0
$199$	−1.37237e10	−0.620343	−0.310171	−	0.950681i	$-0.600386\pi$
$199$	−1.37237e10	−0.620343	−0.310171	+	0.950681i	$0.600386\pi$
$200$	0	0
$201$	−5.21749e10	−2.25465
$202$	0	0
$203$	1.20619e10	0.498520
$204$	0	0
$205$	1.04574e10	0.413552
$206$	0	0
$207$	6.41480e9	0.242838
$208$	0	0
$209$	−1.84312e10	−0.668183
$210$	0	0
$211$	−1.21366e10	−0.421526	−0.210763	−	0.977537i	$-0.567595\pi$
$211$	−1.21366e10	−0.421526	−0.210763	+	0.977537i	$0.567595\pi$
$212$	0	0
$213$	−5.60370e10	−1.86538
$214$	0	0
$215$	−5.85130e9	−0.186758
$216$	0	0
$217$	1.36798e10	0.418805
$218$	0	0
$219$	−7.66483e10	−2.25166
$220$	0	0
$221$	1.49330e10	0.421095
$222$	0	0
$223$	6.20297e10	1.67968	0.839842	−	0.542830i	$-0.182647\pi$
$223$	6.20297e10	1.67968	0.839842	+	0.542830i	$0.182647\pi$
$224$	0	0
$225$	1.06675e10	0.277486
$226$	0	0
$227$	3.51959e10	0.879783	0.439891	−	0.898051i	$-0.355017\pi$
$227$	3.51959e10	0.879783	0.439891	+	0.898051i	$0.355017\pi$
$228$	0	0
$229$	−3.83973e10	−0.922659	−0.461329	−	0.887229i	$-0.652627\pi$
$229$	−3.83973e10	−0.922659	−0.461329	+	0.887229i	$0.652627\pi$
$230$	0	0
$231$	1.23842e10	0.286164
$232$	0	0
$233$	−2.59156e10	−0.576049	−0.288024	−	0.957623i	$-0.592998\pi$
$233$	−2.59156e10	−0.576049	−0.288024	+	0.957623i	$0.592998\pi$
$234$	0	0
$235$	1.68381e10	0.360153
$236$	0	0
$237$	7.93811e10	1.63437
$238$	0	0
$239$	5.76903e10	1.14370	0.571850	−	0.820358i	$-0.306226\pi$
$239$	5.76903e10	1.14370	0.571850	+	0.820358i	$0.306226\pi$
$240$	0	0
$241$	−4.59383e10	−0.877200	−0.438600	−	0.898682i	$-0.644525\pi$
$241$	−4.59383e10	−0.877200	−0.438600	+	0.898682i	$0.644525\pi$
$242$	0	0
$243$	−7.13773e10	−1.31320
$244$	0	0
$245$	2.34944e10	0.416598
$246$	0	0
$247$	−8.72309e10	−1.49119
$248$	0	0
$249$	−2.11346e9	−0.0348416
$250$	0	0
$251$	−6.33143e10	−1.00686	−0.503431	−	0.864035i	$-0.667929\pi$
$251$	−6.33143e10	−1.00686	−0.503431	+	0.864035i	$0.667929\pi$
$252$	0	0
$253$	8.07393e9	0.123892
$254$	0	0
$255$	−1.24370e10	−0.184198
$256$	0	0
$257$	4.21910e10	0.603283	0.301642	−	0.953421i	$-0.402465\pi$
$257$	4.21910e10	0.603283	0.301642	+	0.953421i	$0.402465\pi$
$258$	0	0
$259$	−2.60699e10	−0.359990
$260$	0	0
$261$	1.98182e11	2.64351
$262$	0	0
$263$	5.01270e10	0.646057	0.323028	−	0.946389i	$-0.395299\pi$
$263$	5.01270e10	0.646057	0.323028	+	0.946389i	$0.395299\pi$
$264$	0	0
$265$	−2.13191e10	−0.265560
$266$	0	0
$267$	−2.47561e11	−2.98114
$268$	0	0
$269$	7.21552e10	0.840199	0.420099	−	0.907478i	$-0.361995\pi$
$269$	7.21552e10	0.840199	0.420099	+	0.907478i	$0.361995\pi$
$270$	0	0
$271$	5.23276e10	0.589344	0.294672	−	0.955598i	$-0.404790\pi$
$271$	5.23276e10	0.589344	0.294672	+	0.955598i	$0.404790\pi$
$272$	0	0
$273$	5.86119e10	0.638637
$274$	0	0
$275$	1.34266e10	0.141569
$276$	0	0
$277$	−5.78954e10	−0.590860	−0.295430	−	0.955364i	$-0.595463\pi$
$277$	−5.78954e10	−0.590860	−0.295430	+	0.955364i	$0.595463\pi$
$278$	0	0
$279$	2.24765e11	2.22081
$280$	0	0
$281$	7.31211e10	0.699623	0.349812	−	0.936820i	$-0.386246\pi$
$281$	7.31211e10	0.699623	0.349812	+	0.936820i	$0.386246\pi$
$282$	0	0
$283$	−5.12368e10	−0.474835	−0.237418	−	0.971408i	$-0.576301\pi$
$283$	−5.12368e10	−0.474835	−0.237418	+	0.971408i	$0.576301\pi$
$284$	0	0
$285$	7.26507e10	0.652286
$286$	0	0
$287$	−2.78097e10	−0.241951
$288$	0	0
$289$	−1.10161e11	−0.928943
$290$	0	0
$291$	2.72173e11	2.22499
$292$	0	0
$293$	1.95934e11	1.55312	0.776561	−	0.630043i	$-0.216963\pi$
$293$	1.95934e11	1.55312	0.776561	+	0.630043i	$0.216963\pi$
$294$	0	0
$295$	−7.00968e10	−0.538889
$296$	0	0
$297$	5.68200e10	0.423737
$298$	0	0
$299$	3.82122e10	0.276491
$300$	0	0
$301$	1.55606e10	0.109264
$302$	0	0
$303$	5.71341e10	0.389407
$304$	0	0
$305$	−3.43457e9	−0.0227260
$306$	0	0
$307$	−2.29564e11	−1.47496	−0.737481	−	0.675368i	$-0.763985\pi$
$307$	−2.29564e11	−1.47496	−0.737481	+	0.675368i	$0.763985\pi$
$308$	0	0
$309$	−4.54215e11	−2.83431
$310$	0	0
$311$	4.99151e10	0.302559	0.151280	−	0.988491i	$-0.451661\pi$
$311$	4.99151e10	0.302559	0.151280	+	0.988491i	$0.451661\pi$
$312$	0	0
$313$	−1.19272e11	−0.702408	−0.351204	−	0.936299i	$-0.614228\pi$
$313$	−1.19272e11	−0.702408	−0.351204	+	0.936299i	$0.614228\pi$
$314$	0	0
$315$	−2.83685e10	−0.162345
$316$	0	0
$317$	−2.85879e11	−1.59007	−0.795034	−	0.606565i	$-0.792547\pi$
$317$	−2.85879e11	−1.59007	−0.795034	+	0.606565i	$0.792547\pi$
$318$	0	0
$319$	2.49440e11	1.34868
$320$	0	0
$321$	−7.16464e10	−0.376636
$322$	0	0
$323$	−4.92236e10	−0.251630
$324$	0	0
$325$	6.35450e10	0.315941
$326$	0	0
$327$	−3.58316e11	−1.73301
$328$	0	0
$329$	−4.47782e10	−0.210710
$330$	0	0
$331$	−2.12047e11	−0.970973	−0.485486	−	0.874244i	$-0.661357\pi$
$331$	−2.12047e11	−0.970973	−0.485486	+	0.874244i	$0.661357\pi$
$332$	0	0
$333$	−4.28340e11	−1.90893
$334$	0	0
$335$	1.50429e11	0.652573
$336$	0	0
$337$	2.00430e11	0.846502	0.423251	−	0.906012i	$-0.360889\pi$
$337$	2.00430e11	0.846502	0.423251	+	0.906012i	$0.360889\pi$
$338$	0	0
$339$	2.50798e11	1.03140
$340$	0	0
$341$	2.82899e11	1.13302
$342$	0	0
$343$	−1.29551e11	−0.505378
$344$	0	0
$345$	−3.18252e10	−0.120944
$346$	0	0
$347$	−9.86210e10	−0.365163	−0.182582	−	0.983191i	$-0.558445\pi$
$347$	−9.86210e10	−0.365163	−0.182582	+	0.983191i	$0.558445\pi$
$348$	0	0
$349$	−4.79143e11	−1.72882	−0.864411	−	0.502785i	$-0.832309\pi$
$349$	−4.79143e11	−1.72882	−0.864411	+	0.502785i	$0.832309\pi$
$350$	0	0
$351$	2.68917e11	0.945662
$352$	0	0
$353$	4.56755e11	1.56566	0.782830	−	0.622235i	$-0.213775\pi$
$353$	4.56755e11	1.56566	0.782830	+	0.622235i	$0.213775\pi$
$354$	0	0
$355$	1.61564e11	0.539904
$356$	0	0
$357$	3.30742e10	0.107766
$358$	0	0
$359$	−1.54059e11	−0.489509	−0.244755	−	0.969585i	$-0.578707\pi$
$359$	−1.54059e11	−0.489509	−0.244755	+	0.969585i	$0.578707\pi$
$360$	0	0
$361$	−3.51479e10	−0.108922
$362$	0	0
$363$	−2.55040e11	−0.770954
$364$	0	0
$365$	2.20989e11	0.651708
$366$	0	0
$367$	3.42604e11	0.985815	0.492908	−	0.870082i	$-0.335934\pi$
$367$	3.42604e11	0.985815	0.492908	+	0.870082i	$0.335934\pi$
$368$	0	0
$369$	−4.56925e11	−1.28300
$370$	0	0
$371$	5.66947e10	0.155367
$372$	0	0
$373$	−1.68130e11	−0.449734	−0.224867	−	0.974389i	$-0.572195\pi$
$373$	−1.68130e11	−0.449734	−0.224867	+	0.974389i	$0.572195\pi$
$374$	0	0
$375$	−5.29238e10	−0.138201
$376$	0	0
$377$	1.18054e12	3.00986
$378$	0	0
$379$	−1.00434e11	−0.250038	−0.125019	−	0.992154i	$-0.539899\pi$
$379$	−1.00434e11	−0.250038	−0.125019	+	0.992154i	$0.539899\pi$
$380$	0	0
$381$	3.38451e11	0.822874
$382$	0	0
$383$	5.07533e11	1.20523	0.602615	−	0.798032i	$-0.294126\pi$
$383$	5.07533e11	1.20523	0.602615	+	0.798032i	$0.294126\pi$
$384$	0	0
$385$	−3.57058e10	−0.0828257
$386$	0	0
$387$	2.55667e11	0.579396
$388$	0	0
$389$	−1.26566e11	−0.280249	−0.140125	−	0.990134i	$-0.544750\pi$
$389$	−1.26566e11	−0.280249	−0.140125	+	0.990134i	$0.544750\pi$
$390$	0	0
$391$	2.15628e10	0.0466562
$392$	0	0
$393$	−1.03065e12	−2.17944
$394$	0	0
$395$	−2.28868e11	−0.473041
$396$	0	0
$397$	−4.70667e11	−0.950948	−0.475474	−	0.879730i	$-0.657723\pi$
$397$	−4.70667e11	−0.950948	−0.475474	+	0.879730i	$0.657723\pi$
$398$	0	0
$399$	−1.93203e11	−0.381624
$400$	0	0
$401$	−9.20810e11	−1.77836	−0.889182	−	0.457554i	$-0.848726\pi$
$401$	−9.20810e11	−1.77836	−0.889182	+	0.457554i	$0.848726\pi$
$402$	0	0
$403$	1.33890e12	2.52857
$404$	0	0
$405$	1.11981e11	0.206821
$406$	0	0
$407$	−5.39126e11	−0.973902
$408$	0	0
$409$	8.23952e11	1.45595	0.727976	−	0.685603i	$-0.240461\pi$
$409$	8.23952e11	1.45595	0.727976	+	0.685603i	$0.240461\pi$
$410$	0	0
$411$	3.41870e11	0.590979
$412$	0	0
$413$	1.86411e11	0.315280
$414$	0	0
$415$	6.09345e9	0.0100843
$416$	0	0
$417$	7.38605e11	1.19619
$418$	0	0
$419$	7.01758e11	1.11231	0.556153	−	0.831080i	$-0.312277\pi$
$419$	7.01758e11	1.11231	0.556153	+	0.831080i	$0.312277\pi$
$420$	0	0
$421$	1.22669e11	0.190311	0.0951555	−	0.995462i	$-0.469665\pi$
$421$	1.22669e11	0.190311	0.0951555	+	0.995462i	$0.469665\pi$
$422$	0	0
$423$	−7.35725e11	−1.11734
$424$	0	0
$425$	3.58579e10	0.0533131
$426$	0	0
$427$	9.13368e9	0.0132960
$428$	0	0
$429$	1.21210e12	1.72774
$430$	0	0
$431$	−4.82937e11	−0.674129	−0.337064	−	0.941482i	$-0.609434\pi$
$431$	−4.82937e11	−0.674129	−0.337064	+	0.941482i	$0.609434\pi$
$432$	0	0
$433$	7.71175e11	1.05428	0.527142	−	0.849777i	$-0.323264\pi$
$433$	7.71175e11	1.05428	0.527142	+	0.849777i	$0.323264\pi$
$434$	0	0
$435$	−9.83223e11	−1.31659
$436$	0	0
$437$	−1.25959e11	−0.165220
$438$	0	0
$439$	−6.62301e11	−0.851070	−0.425535	−	0.904942i	$-0.639914\pi$
$439$	−6.62301e11	−0.851070	−0.425535	+	0.904942i	$0.639914\pi$
$440$	0	0
$441$	−1.02657e12	−1.29245
$442$	0	0
$443$	−3.76774e11	−0.464798	−0.232399	−	0.972621i	$-0.574657\pi$
$443$	−3.76774e11	−0.464798	−0.232399	+	0.972621i	$0.574657\pi$
$444$	0	0
$445$	7.13759e11	0.862843
$446$	0	0
$447$	−9.12004e10	−0.108047
$448$	0	0
$449$	5.77774e11	0.670888	0.335444	−	0.942060i	$-0.391114\pi$
$449$	5.77774e11	0.670888	0.335444	+	0.942060i	$0.391114\pi$
$450$	0	0
$451$	−5.75105e11	−0.654564
$452$	0	0
$453$	−3.78008e11	−0.421754
$454$	0	0
$455$	−1.68988e11	−0.184843
$456$	0	0
$457$	−8.28175e11	−0.888176	−0.444088	−	0.895983i	$-0.646472\pi$
$457$	−8.28175e11	−0.888176	−0.444088	+	0.895983i	$0.646472\pi$
$458$	0	0
$459$	1.51747e11	0.159575
$460$	0	0
$461$	2.77554e11	0.286216	0.143108	−	0.989707i	$-0.454290\pi$
$461$	2.77554e11	0.286216	0.143108	+	0.989707i	$0.454290\pi$
$462$	0	0
$463$	−1.01088e11	−0.102232	−0.0511159	−	0.998693i	$-0.516278\pi$
$463$	−1.01088e11	−0.102232	−0.0511159	+	0.998693i	$0.516278\pi$
$464$	0	0
$465$	−1.11511e12	−1.10606
$466$	0	0
$467$	−1.30742e12	−1.27201	−0.636004	−	0.771686i	$-0.719414\pi$
$467$	−1.30742e12	−1.27201	−0.636004	+	0.771686i	$0.719414\pi$
$468$	0	0
$469$	−4.00041e11	−0.381792
$470$	0	0
$471$	−1.32914e12	−1.24445
$472$	0	0
$473$	3.21793e11	0.295598
$474$	0	0
$475$	−2.09464e11	−0.188794
$476$	0	0
$477$	9.31518e11	0.823869
$478$	0	0
$479$	−1.38500e12	−1.20210	−0.601049	−	0.799212i	$-0.705250\pi$
$479$	−1.38500e12	−1.20210	−0.601049	+	0.799212i	$0.705250\pi$
$480$	0	0
$481$	−2.55157e12	−2.17347
$482$	0	0
$483$	8.46341e10	0.0707593
$484$	0	0
$485$	−7.84720e11	−0.643987
$486$	0	0
$487$	9.54305e11	0.768789	0.384394	−	0.923169i	$-0.374410\pi$
$487$	9.54305e11	0.768789	0.384394	+	0.923169i	$0.374410\pi$
$488$	0	0
$489$	2.34017e12	1.85079
$490$	0	0
$491$	−7.41251e11	−0.575571	−0.287785	−	0.957695i	$-0.592919\pi$
$491$	−7.41251e11	−0.575571	−0.287785	+	0.957695i	$0.592919\pi$
$492$	0	0
$493$	6.66171e11	0.507895
$494$	0	0
$495$	−5.86661e11	−0.439201
$496$	0	0
$497$	−4.29653e11	−0.315874
$498$	0	0
$499$	1.51393e12	1.09308	0.546542	−	0.837432i	$-0.315944\pi$
$499$	1.51393e12	1.09308	0.546542	+	0.837432i	$0.315944\pi$
$500$	0	0
$501$	3.36906e12	2.38913
$502$	0	0
$503$	1.58191e12	1.10186	0.550929	−	0.834552i	$-0.314274\pi$
$503$	1.58191e12	1.10186	0.550929	+	0.834552i	$0.314274\pi$
$504$	0	0
$505$	−1.64727e11	−0.112708
$506$	0	0
$507$	3.43779e12	2.31070
$508$	0	0
$509$	−2.29553e11	−0.151584	−0.0757918	−	0.997124i	$-0.524148\pi$
$509$	−2.29553e11	−0.151584	−0.0757918	+	0.997124i	$0.524148\pi$
$510$	0	0
$511$	−5.87686e11	−0.381286
$512$	0	0
$513$	−8.86432e11	−0.565089
$514$	0	0
$515$	1.30957e12	0.820347
$516$	0	0
$517$	−9.26014e11	−0.570046
$518$	0	0
$519$	6.96931e11	0.421635
$520$	0	0
$521$	−1.73274e12	−1.03030	−0.515149	−	0.857101i	$-0.672263\pi$
$521$	−1.73274e12	−1.03030	−0.515149	+	0.857101i	$0.672263\pi$
$522$	0	0
$523$	2.45691e12	1.43592	0.717961	−	0.696083i	$-0.245075\pi$
$523$	2.45691e12	1.43592	0.717961	+	0.696083i	$0.245075\pi$
$524$	0	0
$525$	1.40742e11	0.0808552
$526$	0	0
$527$	7.55529e11	0.426681
$528$	0	0
$529$	−1.74598e12	−0.969366
$530$	0	0
$531$	3.06281e12	1.67184
$532$	0	0
$533$	−2.72185e12	−1.46080
$534$	0	0
$535$	2.06568e11	0.109011
$536$	0	0
$537$	4.55737e12	2.36499
$538$	0	0
$539$	−1.29208e12	−0.659386
$540$	0	0
$541$	−2.10080e12	−1.05438	−0.527190	−	0.849748i	$-0.676754\pi$
$541$	−2.10080e12	−1.05438	−0.527190	+	0.849748i	$0.676754\pi$
$542$	0	0
$543$	−3.34949e12	−1.65341
$544$	0	0
$545$	1.03308e12	0.501593
$546$	0	0
$547$	3.58621e12	1.71275	0.856373	−	0.516357i	$-0.172712\pi$
$547$	3.58621e12	1.71275	0.856373	+	0.516357i	$0.172712\pi$
$548$	0	0
$549$	1.50070e11	0.0705048
$550$	0	0
$551$	−3.89144e12	−1.79857
$552$	0	0
$553$	6.08639e11	0.276756
$554$	0	0
$555$	2.12509e12	0.950733
$556$	0	0
$557$	−3.15166e12	−1.38737	−0.693683	−	0.720280i	$-0.744013\pi$
$557$	−3.15166e12	−1.38737	−0.693683	+	0.720280i	$0.744013\pi$
$558$	0	0
$559$	1.52298e12	0.659691
$560$	0	0
$561$	6.83975e11	0.291546
$562$	0	0
$563$	−3.70986e12	−1.55621	−0.778107	−	0.628131i	$-0.783820\pi$
$563$	−3.70986e12	−1.55621	−0.778107	+	0.628131i	$0.783820\pi$
$564$	0	0
$565$	−7.23091e11	−0.298521
$566$	0	0
$567$	−2.97795e11	−0.121002
$568$	0	0
$569$	−1.52350e12	−0.609308	−0.304654	−	0.952463i	$-0.598541\pi$
$569$	−1.52350e12	−0.609308	−0.304654	+	0.952463i	$0.598541\pi$
$570$	0	0
$571$	2.81914e12	1.10982	0.554911	−	0.831909i	$-0.312752\pi$
$571$	2.81914e12	1.10982	0.554911	+	0.831909i	$0.312752\pi$
$572$	0	0
$573$	5.50584e12	2.13367
$574$	0	0
$575$	9.17573e10	0.0350054
$576$	0	0
$577$	−2.80815e11	−0.105470	−0.0527350	−	0.998609i	$-0.516794\pi$
$577$	−2.80815e11	−0.105470	−0.0527350	+	0.998609i	$0.516794\pi$
$578$	0	0
$579$	−3.68880e12	−1.36406
$580$	0	0
$581$	−1.62046e10	−0.00589990
$582$	0	0
$583$	1.17245e12	0.420324
$584$	0	0
$585$	−2.77654e12	−0.980173
$586$	0	0
$587$	1.71929e12	0.597691	0.298845	−	0.954302i	$-0.403399\pi$
$587$	1.71929e12	0.597691	0.298845	+	0.954302i	$0.403399\pi$
$588$	0	0
$589$	−4.41343e12	−1.51097
$590$	0	0
$591$	9.49952e11	0.320301
$592$	0	0
$593$	4.82940e12	1.60379	0.801895	−	0.597465i	$-0.203825\pi$
$593$	4.82940e12	1.60379	0.801895	+	0.597465i	$0.203825\pi$
$594$	0	0
$595$	−9.53582e10	−0.0311912
$596$	0	0
$597$	−2.97496e12	−0.958511
$598$	0	0
$599$	1.72589e12	0.547763	0.273881	−	0.961763i	$-0.411692\pi$
$599$	1.72589e12	0.547763	0.273881	+	0.961763i	$0.411692\pi$
$600$	0	0
$601$	5.06692e12	1.58420	0.792098	−	0.610394i	$-0.208989\pi$
$601$	5.06692e12	1.58420	0.792098	+	0.610394i	$0.208989\pi$
$602$	0	0
$603$	−6.57284e12	−2.02453
$604$	0	0
$605$	7.35322e11	0.223140
$606$	0	0
$607$	−1.57706e12	−0.471520	−0.235760	−	0.971811i	$-0.575758\pi$
$607$	−1.57706e12	−0.471520	−0.235760	+	0.971811i	$0.575758\pi$
$608$	0	0
$609$	2.61472e12	0.770279
$610$	0	0
$611$	−4.38263e12	−1.27218
$612$	0	0
$613$	−1.35557e12	−0.387747	−0.193874	−	0.981026i	$-0.562105\pi$
$613$	−1.35557e12	−0.387747	−0.193874	+	0.981026i	$0.562105\pi$
$614$	0	0
$615$	2.26691e12	0.638992
$616$	0	0
$617$	−4.35935e12	−1.21098	−0.605492	−	0.795852i	$-0.707023\pi$
$617$	−4.35935e12	−1.21098	−0.605492	+	0.795852i	$0.707023\pi$
$618$	0	0
$619$	1.67780e12	0.459338	0.229669	−	0.973269i	$-0.426236\pi$
$619$	1.67780e12	0.459338	0.229669	+	0.973269i	$0.426236\pi$
$620$	0	0
$621$	3.88309e11	0.104777
$622$	0	0
$623$	−1.89813e12	−0.504811
$624$	0	0
$625$	1.52588e11	0.0400000
$626$	0	0
$627$	−3.99544e12	−1.03243
$628$	0	0
$629$	−1.43983e12	−0.366760
$630$	0	0
$631$	4.12439e12	1.03568	0.517842	−	0.855476i	$-0.326736\pi$
$631$	4.12439e12	1.03568	0.517842	+	0.855476i	$0.326736\pi$
$632$	0	0
$633$	−2.63091e12	−0.651313
$634$	0	0
$635$	−9.75808e11	−0.238168
$636$	0	0
$637$	−6.11514e12	−1.47156
$638$	0	0
$639$	−7.05938e12	−1.67499
$640$	0	0
$641$	2.54925e12	0.596420	0.298210	−	0.954500i	$-0.403610\pi$
$641$	2.54925e12	0.596420	0.298210	+	0.954500i	$0.403610\pi$
$642$	0	0
$643$	−1.49409e11	−0.0344688	−0.0172344	−	0.999851i	$-0.505486\pi$
$643$	−1.49409e11	−0.0344688	−0.0172344	+	0.999851i	$0.505486\pi$
$644$	0	0
$645$	−1.26842e12	−0.288566
$646$	0	0
$647$	−7.88552e11	−0.176913	−0.0884567	−	0.996080i	$-0.528194\pi$
$647$	−7.88552e11	−0.176913	−0.0884567	+	0.996080i	$0.528194\pi$
$648$	0	0
$649$	3.85498e12	0.852946
$650$	0	0
$651$	2.96546e12	0.647109
$652$	0	0
$653$	1.87557e12	0.403668	0.201834	−	0.979420i	$-0.435310\pi$
$653$	1.87557e12	0.403668	0.201834	+	0.979420i	$0.435310\pi$
$654$	0	0
$655$	2.97153e12	0.630805
$656$	0	0
$657$	−9.65592e12	−2.02185
$658$	0	0
$659$	−2.02453e11	−0.0418158	−0.0209079	−	0.999781i	$-0.506656\pi$
$659$	−2.02453e11	−0.0418158	−0.0209079	+	0.999781i	$0.506656\pi$
$660$	0	0
$661$	2.41063e12	0.491160	0.245580	−	0.969376i	$-0.421022\pi$
$661$	2.41063e12	0.491160	0.245580	+	0.969376i	$0.421022\pi$
$662$	0	0
$663$	3.23711e12	0.650648
$664$	0	0
$665$	5.57035e11	0.110455
$666$	0	0
$667$	1.70468e12	0.333484
$668$	0	0
$669$	1.34465e13	2.59533
$670$	0	0
$671$	1.88885e11	0.0359704
$672$	0	0
$673$	−5.63582e11	−0.105898	−0.0529491	−	0.998597i	$-0.516862\pi$
$673$	−5.63582e11	−0.105898	−0.0529491	+	0.998597i	$0.516862\pi$
$674$	0	0
$675$	6.45738e11	0.119726
$676$	0	0
$677$	8.43518e11	0.154328	0.0771641	−	0.997018i	$-0.475413\pi$
$677$	8.43518e11	0.154328	0.0771641	+	0.997018i	$0.475413\pi$
$678$	0	0
$679$	2.08684e12	0.376768
$680$	0	0
$681$	7.62962e12	1.35938
$682$	0	0
$683$	5.43927e12	0.956418	0.478209	−	0.878246i	$-0.341286\pi$
$683$	5.43927e12	0.956418	0.478209	+	0.878246i	$0.341286\pi$
$684$	0	0
$685$	−9.85665e11	−0.171050
$686$	0	0
$687$	−8.32361e12	−1.42563
$688$	0	0
$689$	5.54894e12	0.938044
$690$	0	0
$691$	−1.41900e12	−0.236773	−0.118386	−	0.992968i	$-0.537772\pi$
$691$	−1.41900e12	−0.236773	−0.118386	+	0.992968i	$0.537772\pi$
$692$	0	0
$693$	1.56013e12	0.256957
$694$	0	0
$695$	−2.12952e12	−0.346218
$696$	0	0
$697$	−1.53591e12	−0.246501
$698$	0	0
$699$	−5.61787e12	−0.890071
$700$	0	0
$701$	9.10116e12	1.42353	0.711763	−	0.702419i	$-0.247897\pi$
$701$	9.10116e12	1.42353	0.711763	+	0.702419i	$0.247897\pi$
$702$	0	0
$703$	8.41075e12	1.29878
$704$	0	0
$705$	3.65009e12	0.556484
$706$	0	0
$707$	4.38064e11	0.0659402
$708$	0	0
$709$	3.00283e12	0.446296	0.223148	−	0.974785i	$-0.428367\pi$
$709$	3.00283e12	0.446296	0.223148	+	0.974785i	$0.428367\pi$
$710$	0	0
$711$	1.00002e13	1.46756
$712$	0	0
$713$	1.93334e12	0.280159
$714$	0	0
$715$	−3.49467e12	−0.500068
$716$	0	0
$717$	1.25059e13	1.76717
$718$	0	0
$719$	−8.40313e12	−1.17263	−0.586315	−	0.810083i	$-0.699422\pi$
$719$	−8.40313e12	−1.17263	−0.586315	+	0.810083i	$0.699422\pi$
$720$	0	0
$721$	−3.48260e12	−0.479949
$722$	0	0
$723$	−9.95833e12	−1.35539
$724$	0	0
$725$	2.83479e12	0.381066
$726$	0	0
$727$	6.70907e12	0.890753	0.445377	−	0.895343i	$-0.353070\pi$
$727$	6.70907e12	0.890753	0.445377	+	0.895343i	$0.353070\pi$
$728$	0	0
$729$	−1.19463e13	−1.56660
$730$	0	0
$731$	8.59403e11	0.111319
$732$	0	0
$733$	1.50813e12	0.192962	0.0964810	−	0.995335i	$-0.469241\pi$
$733$	1.50813e12	0.192962	0.0964810	+	0.995335i	$0.469241\pi$
$734$	0	0
$735$	5.09302e12	0.643699
$736$	0	0
$737$	−8.27285e12	−1.03288
$738$	0	0
$739$	−1.48289e13	−1.82898	−0.914492	−	0.404603i	$-0.867410\pi$
$739$	−1.48289e13	−1.82898	−0.914492	+	0.404603i	$0.867410\pi$
$740$	0	0
$741$	−1.89096e13	−2.30409
$742$	0	0
$743$	−1.91312e12	−0.230299	−0.115150	−	0.993348i	$-0.536735\pi$
$743$	−1.91312e12	−0.230299	−0.115150	+	0.993348i	$0.536735\pi$
$744$	0	0
$745$	2.62945e11	0.0312725
$746$	0	0
$747$	−2.66248e11	−0.0312855
$748$	0	0
$749$	−5.49335e11	−0.0637777
$750$	0	0
$751$	−6.50309e12	−0.746002	−0.373001	−	0.927831i	$-0.621671\pi$
$751$	−6.50309e12	−0.746002	−0.373001	+	0.927831i	$0.621671\pi$
$752$	0	0
$753$	−1.37250e13	−1.55573
$754$	0	0
$755$	1.08986e12	0.122070
$756$	0	0
$757$	−1.11069e11	−0.0122931	−0.00614653	−	0.999981i	$-0.501957\pi$
$757$	−1.11069e11	−0.0122931	−0.00614653	+	0.999981i	$0.501957\pi$
$758$	0	0
$759$	1.75023e12	0.191429
$760$	0	0
$761$	1.21294e13	1.31102	0.655510	−	0.755186i	$-0.272454\pi$
$761$	1.21294e13	1.31102	0.655510	+	0.755186i	$0.272454\pi$
$762$	0	0
$763$	−2.74732e12	−0.293460
$764$	0	0
$765$	−1.56678e12	−0.165398
$766$	0	0
$767$	1.82448e13	1.90353
$768$	0	0
$769$	−1.19313e13	−1.23033	−0.615164	−	0.788399i	$-0.710910\pi$
$769$	−1.19313e13	−1.23033	−0.615164	+	0.788399i	$0.710910\pi$
$770$	0	0
$771$	9.14600e12	0.932152
$772$	0	0
$773$	−1.71471e13	−1.72736	−0.863681	−	0.504038i	$-0.831847\pi$
$773$	−1.71471e13	−1.72736	−0.863681	+	0.504038i	$0.831847\pi$
$774$	0	0
$775$	3.21504e12	0.320132
$776$	0	0
$777$	−5.65133e12	−0.556232
$778$	0	0
$779$	8.97204e12	0.872917
$780$	0	0
$781$	−8.88523e12	−0.854553
$782$	0	0
$783$	1.19966e13	1.14059
$784$	0	0
$785$	3.83214e12	0.360187
$786$	0	0
$787$	−2.12129e11	−0.0197112	−0.00985561	−	0.999951i	$-0.503137\pi$
$787$	−2.12129e11	−0.0197112	−0.00985561	+	0.999951i	$0.503137\pi$
$788$	0	0
$789$	1.08663e13	0.998242
$790$	0	0
$791$	1.92294e12	0.174652
$792$	0	0
$793$	8.93950e11	0.0802757
$794$	0	0
$795$	−4.62146e12	−0.410325
$796$	0	0
$797$	9.75044e12	0.855976	0.427988	−	0.903784i	$-0.359223\pi$
$797$	9.75044e12	0.855976	0.427988	+	0.903784i	$0.359223\pi$
$798$	0	0
$799$	−2.47308e12	−0.214673
$800$	0	0
$801$	−3.11870e13	−2.67687
$802$	0	0
$803$	−1.21533e13	−1.03152
$804$	0	0
$805$	−2.44014e11	−0.0204801
$806$	0	0
$807$	1.56415e13	1.29822
$808$	0	0
$809$	1.00162e13	0.822120	0.411060	−	0.911608i	$-0.365159\pi$
$809$	1.00162e13	0.822120	0.411060	+	0.911608i	$0.365159\pi$
$810$	0	0
$811$	1.09433e11	0.00888287	0.00444143	−	0.999990i	$-0.498586\pi$
$811$	1.09433e11	0.00888287	0.00444143	+	0.999990i	$0.498586\pi$
$812$	0	0
$813$	1.13434e13	0.910614
$814$	0	0
$815$	−6.74708e12	−0.535681
$816$	0	0
$817$	−5.02020e12	−0.394205
$818$	0	0
$819$	7.38376e12	0.573456
$820$	0	0
$821$	−2.10914e13	−1.62017	−0.810086	−	0.586312i	$-0.800579\pi$
$821$	−2.10914e13	−1.62017	−0.810086	+	0.586312i	$0.800579\pi$
$822$	0	0
$823$	6.07029e12	0.461222	0.230611	−	0.973046i	$-0.425927\pi$
$823$	6.07029e12	0.461222	0.230611	+	0.973046i	$0.425927\pi$
$824$	0	0
$825$	2.91055e12	0.218742
$826$	0	0
$827$	−1.09879e13	−0.816846	−0.408423	−	0.912793i	$-0.633921\pi$
$827$	−1.09879e13	−0.816846	−0.408423	+	0.912793i	$0.633921\pi$
$828$	0	0
$829$	−1.34601e13	−0.989811	−0.494906	−	0.868947i	$-0.664797\pi$
$829$	−1.34601e13	−0.989811	−0.494906	+	0.868947i	$0.664797\pi$
$830$	0	0
$831$	−1.25503e13	−0.912957
$832$	0	0
$833$	−3.45072e12	−0.248317
$834$	0	0
$835$	−9.71355e12	−0.691495
$836$	0	0
$837$	1.36058e13	0.958206
$838$	0	0
$839$	−4.34425e12	−0.302681	−0.151341	−	0.988482i	$-0.548359\pi$
$839$	−4.34425e12	−0.302681	−0.151341	+	0.988482i	$0.548359\pi$
$840$	0	0
$841$	3.81578e13	2.63028
$842$	0	0
$843$	1.58509e13	1.08101
$844$	0	0
$845$	−9.91170e12	−0.668795
$846$	0	0
$847$	−1.95547e12	−0.130550
$848$	0	0
$849$	−1.11069e13	−0.733683
$850$	0	0
$851$	−3.68440e12	−0.240815
$852$	0	0
$853$	−1.33638e13	−0.864290	−0.432145	−	0.901804i	$-0.642243\pi$
$853$	−1.33638e13	−0.864290	−0.432145	+	0.901804i	$0.642243\pi$
$854$	0	0
$855$	9.15233e12	0.585712
$856$	0	0
$857$	2.60497e13	1.64964	0.824819	−	0.565397i	$-0.191277\pi$
$857$	2.60497e13	1.64964	0.824819	+	0.565397i	$0.191277\pi$
$858$	0	0
$859$	2.01187e12	0.126076	0.0630379	−	0.998011i	$-0.479921\pi$
$859$	2.01187e12	0.126076	0.0630379	+	0.998011i	$0.479921\pi$
$860$	0	0
$861$	−6.02847e12	−0.373846
$862$	0	0
$863$	2.36870e13	1.45366	0.726828	−	0.686819i	$-0.240994\pi$
$863$	2.36870e13	1.45366	0.726828	+	0.686819i	$0.240994\pi$
$864$	0	0
$865$	−2.00936e12	−0.122036
$866$	0	0
$867$	−2.38803e13	−1.43534
$868$	0	0
$869$	1.25867e13	0.748723
$870$	0	0
$871$	−3.91536e13	−2.30510
$872$	0	0
$873$	3.42876e13	1.99790
$874$	0	0
$875$	−4.05783e11	−0.0234022
$876$	0	0
$877$	1.83122e13	1.04531	0.522653	−	0.852545i	$-0.324942\pi$
$877$	1.83122e13	1.04531	0.522653	+	0.852545i	$0.324942\pi$
$878$	0	0
$879$	4.24738e13	2.39978
$880$	0	0
$881$	3.34824e13	1.87251	0.936256	−	0.351320i	$-0.114267\pi$
$881$	3.34824e13	1.87251	0.936256	+	0.351320i	$0.114267\pi$
$882$	0	0
$883$	−1.73556e13	−0.960763	−0.480382	−	0.877060i	$-0.659502\pi$
$883$	−1.73556e13	−0.960763	−0.480382	+	0.877060i	$0.659502\pi$
$884$	0	0
$885$	−1.51953e13	−0.832654
$886$	0	0
$887$	−1.34047e13	−0.727109	−0.363555	−	0.931573i	$-0.618437\pi$
$887$	−1.34047e13	−0.727109	−0.363555	+	0.931573i	$0.618437\pi$
$888$	0	0
$889$	2.59501e12	0.139341
$890$	0	0
$891$	−6.15839e12	−0.327354
$892$	0	0
$893$	1.44465e13	0.760204
$894$	0	0
$895$	−1.31396e13	−0.684510
$896$	0	0
$897$	8.28348e12	0.427216
$898$	0	0
$899$	5.97294e13	3.04979
$900$	0	0
$901$	3.13122e12	0.158289
$902$	0	0
$903$	3.37316e12	0.168827
$904$	0	0
$905$	9.65713e12	0.478552
$906$	0	0
$907$	9.01114e12	0.442127	0.221064	−	0.975259i	$-0.429047\pi$
$907$	9.01114e12	0.442127	0.221064	+	0.975259i	$0.429047\pi$
$908$	0	0
$909$	7.19758e12	0.349663
$910$	0	0
$911$	7.64836e12	0.367905	0.183952	−	0.982935i	$-0.441111\pi$
$911$	7.64836e12	0.367905	0.183952	+	0.982935i	$0.441111\pi$
$912$	0	0
$913$	−3.35110e11	−0.0159613
$914$	0	0
$915$	−7.44531e11	−0.0351146
$916$	0	0
$917$	−7.90232e12	−0.369056
$918$	0	0
$919$	3.19451e13	1.47735	0.738676	−	0.674060i	$-0.235451\pi$
$919$	3.19451e13	1.47735	0.738676	+	0.674060i	$0.235451\pi$
$920$	0	0
$921$	−4.97639e13	−2.27901
$922$	0	0
$923$	−4.20519e13	−1.90712
$924$	0	0
$925$	−6.12697e12	−0.275174
$926$	0	0
$927$	−5.72207e13	−2.54504
$928$	0	0
$929$	−1.36029e13	−0.599186	−0.299593	−	0.954067i	$-0.596851\pi$
$929$	−1.36029e13	−0.599186	−0.299593	+	0.954067i	$0.596851\pi$
$930$	0	0
$931$	2.01574e13	0.879347
$932$	0	0
$933$	1.08204e13	0.467494
$934$	0	0
$935$	−1.97201e12	−0.0843833
$936$	0	0
$937$	−1.10891e13	−0.469966	−0.234983	−	0.971999i	$-0.575503\pi$
$937$	−1.10891e13	−0.469966	−0.234983	+	0.971999i	$0.575503\pi$
$938$	0	0
$939$	−2.58553e13	−1.08531
$940$	0	0
$941$	−2.46259e13	−1.02385	−0.511927	−	0.859029i	$-0.671068\pi$
$941$	−2.46259e13	−1.02385	−0.511927	+	0.859029i	$0.671068\pi$
$942$	0	0
$943$	−3.93028e12	−0.161853
$944$	0	0
$945$	−1.71724e12	−0.0700466
$946$	0	0
$947$	4.82162e13	1.94813	0.974065	−	0.226267i	$-0.0726521\pi$
$947$	4.82162e13	1.94813	0.974065	+	0.226267i	$0.0726521\pi$
$948$	0	0
$949$	−5.75192e13	−2.30205
$950$	0	0
$951$	−6.19717e13	−2.45686
$952$	0	0
$953$	−3.00142e12	−0.117872	−0.0589358	−	0.998262i	$-0.518771\pi$
$953$	−3.00142e12	−0.117872	−0.0589358	+	0.998262i	$0.518771\pi$
$954$	0	0
$955$	−1.58742e13	−0.617558
$956$	0	0
$957$	5.40725e13	2.08388
$958$	0	0
$959$	2.62122e12	0.100074
$960$	0	0
$961$	4.13017e13	1.56211
$962$	0	0
$963$	−9.02581e12	−0.338195
$964$	0	0
$965$	1.06354e13	0.394804
$966$	0	0
$967$	−1.36956e13	−0.503690	−0.251845	−	0.967768i	$-0.581037\pi$
$967$	−1.36956e13	−0.503690	−0.251845	+	0.967768i	$0.581037\pi$
$968$	0	0
$969$	−1.06705e13	−0.388801
$970$	0	0
$971$	2.83646e12	0.102398	0.0511988	−	0.998688i	$-0.483696\pi$
$971$	2.83646e12	0.102398	0.0511988	+	0.998688i	$0.483696\pi$
$972$	0	0
$973$	5.66311e12	0.202557
$974$	0	0
$975$	1.37750e13	0.488171
$976$	0	0
$977$	−9.66226e12	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$977$	−9.66226e12	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$978$	0	0
$979$	−3.92533e13	−1.36570
$980$	0	0
$981$	−4.51396e13	−1.55614
$982$	0	0
$983$	−1.05400e13	−0.360039	−0.180019	−	0.983663i	$-0.557616\pi$
$983$	−1.05400e13	−0.360039	−0.180019	+	0.983663i	$0.557616\pi$
$984$	0	0
$985$	−2.73886e12	−0.0927059
$986$	0	0
$987$	−9.70684e12	−0.325575
$988$	0	0
$989$	2.19914e12	0.0730920
$990$	0	0
$991$	3.01597e13	0.993335	0.496668	−	0.867941i	$-0.334557\pi$
$991$	3.01597e13	0.993335	0.496668	+	0.867941i	$0.334557\pi$
$992$	0	0
$993$	−4.59668e13	−1.50028
$994$	0	0
$995$	8.57730e12	0.277426
$996$	0	0
$997$	1.98318e13	0.635675	0.317837	−	0.948145i	$-0.397043\pi$
$997$	1.98318e13	0.635675	0.317837	+	0.948145i	$0.397043\pi$
$998$	0	0
$999$	−2.59288e13	−0.823640

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	40.10.a.c.1.2	✓	2
3.2	odd	2		360.10.a.g.1.2		2
4.3	odd	2		80.10.a.g.1.1		2
5.2	odd	4		200.10.c.d.49.1		4
5.3	odd	4		200.10.c.d.49.4		4
5.4	even	2		200.10.a.c.1.1		2
8.3	odd	2		320.10.a.q.1.2		2
8.5	even	2		320.10.a.n.1.1		2
20.3	even	4		400.10.c.m.49.1		4
20.7	even	4		400.10.c.m.49.4		4
20.19	odd	2		400.10.a.r.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.10.a.c.1.2	✓	2	1.1	even	1	trivial
80.10.a.g.1.1		2	4.3	odd	2
200.10.a.c.1.1		2	5.4	even	2
200.10.c.d.49.1		4	5.2	odd	4
200.10.c.d.49.4		4	5.3	odd	4
320.10.a.n.1.1		2	8.5	even	2
320.10.a.q.1.2		2	8.3	odd	2
360.10.a.g.1.2		2	3.2	odd	2
400.10.a.r.1.2		2	20.19	odd	2
400.10.c.m.49.1		4	20.3	even	4
400.10.c.m.49.4		4	20.7	even	4

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 40 = 2^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	40.a (trivial)

\(f(q)\)	\(=\)	\(q+216.776 q^{3} -625.000 q^{5} +1662.09 q^{7} +27308.8 q^{9} +34372.0 q^{11} +162675. q^{13} -135485. q^{15} +91796.1 q^{17} -536227. q^{19} +360300. q^{21} +234899. q^{23} +390625. q^{25} +1.65309e6 q^{27} +7.25706e6 q^{29} +8.23051e6 q^{31} +7.45102e6 q^{33} -1.03880e6 q^{35} -1.56850e7 q^{37} +3.52641e7 q^{39} -1.67318e7 q^{41} +9.36208e6 q^{43} -1.70680e7 q^{45} -2.69410e7 q^{47} -3.75911e7 q^{49} +1.98992e7 q^{51} +3.41105e7 q^{53} -2.14825e7 q^{55} -1.16241e8 q^{57} +1.12155e8 q^{59} +5.49531e6 q^{61} +4.53896e7 q^{63} -1.01672e8 q^{65} -2.40686e8 q^{67} +5.09204e7 q^{69} -2.58502e8 q^{71} -3.53583e8 q^{73} +8.46781e7 q^{75} +5.71292e7 q^{77} +3.66190e8 q^{79} -1.79169e8 q^{81} -9.74953e6 q^{83} -5.73726e7 q^{85} +1.57316e9 q^{87} -1.14201e9 q^{89} +2.70380e8 q^{91} +1.78418e9 q^{93} +3.35142e8 q^{95} +1.25555e9 q^{97} +9.38657e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 108 q^{3} - 1250 q^{5} - 908 q^{7} + 19458 q^{9} + 25120 q^{11} + 146948 q^{13} - 67500 q^{15} + 169268 q^{17} - 25480 q^{19} + 639864 q^{21} + 1782748 q^{23} + 781250 q^{25} + 4648104 q^{27} + 7323340 q^{29}+ \cdots + 1011292704 q^{99}+O(q^{100}) \) 2 * q + 108 * q^3 - 1250 * q^5 - 908 * q^7 + 19458 * q^9 + 25120 * q^11 + 146948 * q^13 - 67500 * q^15 + 169268 * q^17 - 25480 * q^19 + 639864 * q^21 + 1782748 * q^23 + 781250 * q^25 + 4648104 * q^27 + 7323340 * q^29 + 10677272 * q^31 + 8457408 * q^33 + 567500 * q^35 - 5750460 * q^37 + 36974808 * q^39 - 7795764 * q^41 + 16770524 * q^43 - 12161250 * q^45 + 15393892 * q^47 - 71339334 * q^49 + 11472120 * q^51 - 58529292 * q^53 - 15700000 * q^55 - 171798192 * q^57 + 59618264 * q^59 - 188163772 * q^61 + 65566836 * q^63 - 91842500 * q^65 - 105998252 * q^67 - 117448344 * q^69 - 168665592 * q^71 - 390212412 * q^73 + 42187500 * q^75 + 80907584 * q^77 + 466946256 * q^79 - 350427222 * q^81 + 329535164 * q^83 - 105792500 * q^85 + 1565947656 * q^87 - 108334860 * q^89 + 310800616 * q^91 + 1518028560 * q^93 + 15925000 * q^95 + 2043058628 * q^97 + 1011292704 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more