LMFDB - Embedded newform 400.10.a.r.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,10,Mod(1,400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(400, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("400.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 400 = 2^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	400.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:	$206.014334466$
Analytic rank:	$1$
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:	no (minimal twist has level 40)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$6.78233$ of defining polynomial
Character	$\chi$	$=$	400.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+216.776 q^{3} +1662.09 q^{7} +27308.8 q^{9} +O(q^{10})$	$q+216.776 q^{3} +1662.09 q^{7} +27308.8 q^{9} -34372.0 q^{11} -162675. q^{13} -91796.1 q^{17} +536227. q^{19} +360300. q^{21} +234899. q^{23} +1.65309e6 q^{27} +7.25706e6 q^{29} -8.23051e6 q^{31} -7.45102e6 q^{33} +1.56850e7 q^{37} -3.52641e7 q^{39} -1.67318e7 q^{41} +9.36208e6 q^{43} -2.69410e7 q^{47} -3.75911e7 q^{49} -1.98992e7 q^{51} -3.41105e7 q^{53} +1.16241e8 q^{57} -1.12155e8 q^{59} +5.49531e6 q^{61} +4.53896e7 q^{63} -2.40686e8 q^{67} +5.09204e7 q^{69} +2.58502e8 q^{71} +3.53583e8 q^{73} -5.71292e7 q^{77} -3.66190e8 q^{79} -1.79169e8 q^{81} -9.74953e6 q^{83} +1.57316e9 q^{87} -1.14201e9 q^{89} -2.70380e8 q^{91} -1.78418e9 q^{93} -1.25555e9 q^{97} -9.38657e8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 108 q^{3} - 908 q^{7} + 19458 q^{9}+O(q^{10}) $ 2 * q + 108 * q^3 - 908 * q^7 + 19458 * q^9	$ 2 q + 108 q^{3} - 908 q^{7} + 19458 q^{9} - 25120 q^{11} - 146948 q^{13} - 169268 q^{17} + 25480 q^{19} + 639864 q^{21} + 1782748 q^{23} + 4648104 q^{27} + 7323340 q^{29} - 10677272 q^{31} - 8457408 q^{33} + 5750460 q^{37} - 36974808 q^{39} - 7795764 q^{41} + 16770524 q^{43} + 15393892 q^{47} - 71339334 q^{49} - 11472120 q^{51} + 58529292 q^{53} + 171798192 q^{57} - 59618264 q^{59} - 188163772 q^{61} + 65566836 q^{63} - 105998252 q^{67} - 117448344 q^{69} + 168665592 q^{71} + 390212412 q^{73} - 80907584 q^{77} - 466946256 q^{79} - 350427222 q^{81} + 329535164 q^{83} + 1565947656 q^{87} - 108334860 q^{89} - 310800616 q^{91} - 1518028560 q^{93} - 2043058628 q^{97} - 1011292704 q^{99}+O(q^{100}) $ 2 * q + 108 * q^3 - 908 * q^7 + 19458 * q^9 - 25120 * q^11 - 146948 * q^13 - 169268 * q^17 + 25480 * q^19 + 639864 * q^21 + 1782748 * q^23 + 4648104 * q^27 + 7323340 * q^29 - 10677272 * q^31 - 8457408 * q^33 + 5750460 * q^37 - 36974808 * q^39 - 7795764 * q^41 + 16770524 * q^43 + 15393892 * q^47 - 71339334 * q^49 - 11472120 * q^51 + 58529292 * q^53 + 171798192 * q^57 - 59618264 * q^59 - 188163772 * q^61 + 65566836 * q^63 - 105998252 * q^67 - 117448344 * q^69 + 168665592 * q^71 + 390212412 * q^73 - 80907584 * q^77 - 466946256 * q^79 - 350427222 * q^81 + 329535164 * q^83 + 1565947656 * q^87 - 108334860 * q^89 - 310800616 * q^91 - 1518028560 * q^93 - 2043058628 * q^97 - 1011292704 * 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
$3$	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
$5$	0	0
$5$	0	0
$6$	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	$-0.615152\pi$
$11$	−34372.0	−0.707844	−0.353922	+	0.935275i	$0.615152\pi$
$12$	0	0
$13$	−162675.	−1.57971	−0.789853	−	0.613296i	$-0.789843\pi$
$13$	−162675.	−1.57971	−0.789853	+	0.613296i	$0.789843\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−91796.1	−0.266566	−0.133283	−	0.991078i	$-0.542552\pi$
$17$	−91796.1	−0.266566	−0.133283	+	0.991078i	$0.542552\pi$
$18$	0	0
$19$	536227.	0.943969	0.471985	−	0.881607i	$-0.343538\pi$
$19$	536227.	0.943969	0.471985	+	0.881607i	$0.343538\pi$
$20$	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
$25$	0	0
$26$	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	$-0.795343\pi$
$31$	−8.23051e6	−1.60066	−0.800330	+	0.599559i	$0.795343\pi$
$32$	0	0
$33$	−7.45102e6	−1.09371
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.56850e7	1.37587	0.687936	−	0.725771i	$-0.258517\pi$
$37$	1.56850e7	1.37587	0.687936	+	0.725771i	$0.258517\pi$
$38$	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
$45$	0	0
$46$	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	$-0.595954\pi$
$53$	−3.41105e7	−0.593809	−0.296905	+	0.954907i	$0.595954\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1.16241e8	1.45856
$58$	0	0
$59$	−1.12155e8	−1.20499	−0.602496	−	0.798122i	$-0.705827\pi$
$59$	−1.12155e8	−1.20499	−0.602496	+	0.798122i	$0.705827\pi$
$60$	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
$65$	0	0
$66$	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	$-0.293720\pi$
$71$	2.58502e8	1.20726	0.603631	+	0.797264i	$0.293720\pi$
$72$	0	0
$73$	3.53583e8	1.45726	0.728632	−	0.684905i	$-0.240156\pi$
$73$	3.53583e8	1.45726	0.728632	+	0.684905i	$0.240156\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−5.71292e7	−0.185204
$78$	0	0
$79$	−3.66190e8	−1.05775	−0.528876	−	0.848699i	$-0.677386\pi$
$79$	−3.66190e8	−1.05775	−0.528876	+	0.848699i	$0.677386\pi$
$80$	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
$85$	0	0
$86$	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
$95$	0	0
$96$	0	0
$97$	−1.25555e9	−1.44000	−0.719999	−	0.693975i	$-0.755858\pi$
$97$	−1.25555e9	−1.44000	−0.719999	+	0.693975i	$0.755858\pi$
$98$	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
$105$	0	0
$106$	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	$-0.608316\pi$
$113$	−1.15695e9	−0.667513	−0.333757	+	0.942659i	$0.608316\pi$
$114$	0	0
$115$	0	0
$116$	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
$125$	0	0
$126$	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	$-0.250830\pi$
$131$	4.75445e9	1.41052	0.705261	+	0.708948i	$0.250830\pi$
$132$	0	0
$133$	8.91256e8	0.246985
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1.57706e9	−0.382478	−0.191239	−	0.981543i	$-0.561251\pi$
$137$	−1.57706e9	−0.382478	−0.191239	+	0.981543i	$0.561251\pi$
$138$	0	0
$139$	−3.40723e9	−0.774167	−0.387084	−	0.922045i	$-0.626518\pi$
$139$	−3.40723e9	−0.774167	−0.387084	+	0.922045i	$0.626518\pi$
$140$	0	0
$141$	−5.84015e9	−1.24434
$142$	0	0
$143$	5.59147e9	1.11818
$144$	0	0
$145$	0	0
$146$	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	$-0.456422\pi$
$151$	1.74377e9	0.272957	0.136478	+	0.990643i	$0.456422\pi$
$152$	0	0
$153$	−2.50684e9	−0.369841
$154$	0	0
$155$	0	0
$156$	0	0
$157$	6.13142e9	0.805402	0.402701	−	0.915332i	$-0.368071\pi$
$157$	6.13142e9	0.805402	0.402701	+	0.915332i	$0.368071\pi$
$158$	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
$165$	0	0
$166$	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	$-0.543566\pi$
$173$	−3.21498e9	−0.272880	−0.136440	+	0.990648i	$0.543566\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−2.43125e10	−1.86187
$178$	0	0
$179$	−2.10234e10	−1.53061	−0.765305	−	0.643667i	$-0.777412\pi$
$179$	−2.10234e10	−1.53061	−0.765305	+	0.643667i	$0.777412\pi$
$180$	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
$185$	0	0
$186$	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	$-0.742588\pi$
$191$	−2.53988e10	−1.38090	−0.690451	+	0.723380i	$0.742588\pi$
$192$	0	0
$193$	1.70167e10	0.882809	0.441404	−	0.897308i	$-0.354480\pi$
$193$	1.70167e10	0.882809	0.441404	+	0.897308i	$0.354480\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−4.38218e9	−0.207297	−0.103648	−	0.994614i	$-0.533052\pi$
$197$	−4.38218e9	−0.207297	−0.103648	+	0.994614i	$0.533052\pi$
$198$	0	0
$199$	1.37237e10	0.620343	0.310171	−	0.950681i	$-0.399614\pi$
$199$	1.37237e10	0.620343	0.310171	+	0.950681i	$0.399614\pi$
$200$	0	0
$201$	−5.21749e10	−2.25465
$202$	0	0
$203$	1.20619e10	0.498520
$204$	0	0
$205$	0	0
$206$	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	$-0.432405\pi$
$211$	1.21366e10	0.421526	0.210763	+	0.977537i	$0.432405\pi$
$212$	0	0
$213$	5.60370e10	1.86538
$214$	0	0
$215$	0	0
$216$	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
$225$	0	0
$226$	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	$-0.407002\pi$
$233$	2.59156e10	0.576049	0.288024	+	0.957623i	$0.407002\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−7.93811e10	−1.63437
$238$	0	0
$239$	−5.76903e10	−1.14370	−0.571850	−	0.820358i	$-0.693774\pi$
$239$	−5.76903e10	−1.14370	−0.571850	+	0.820358i	$0.693774\pi$
$240$	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
$245$	0	0
$246$	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	$-0.332071\pi$
$251$	6.33143e10	1.00686	0.503431	+	0.864035i	$0.332071\pi$
$252$	0	0
$253$	−8.07393e9	−0.123892
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−4.21910e10	−0.603283	−0.301642	−	0.953421i	$-0.597535\pi$
$257$	−4.21910e10	−0.603283	−0.301642	+	0.953421i	$0.597535\pi$
$258$	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
$265$	0	0
$266$	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	$-0.595210\pi$
$271$	−5.23276e10	−0.589344	−0.294672	+	0.955598i	$0.595210\pi$
$272$	0	0
$273$	−5.86119e10	−0.638637
$274$	0	0
$275$	0	0
$276$	0	0
$277$	5.78954e10	0.590860	0.295430	−	0.955364i	$-0.404537\pi$
$277$	5.78954e10	0.590860	0.295430	+	0.955364i	$0.404537\pi$
$278$	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
$285$	0	0
$286$	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	$-0.783037\pi$
$293$	−1.95934e11	−1.55312	−0.776561	+	0.630043i	$0.783037\pi$
$294$	0	0
$295$	0	0
$296$	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
$305$	0	0
$306$	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	$-0.548339\pi$
$311$	−4.99151e10	−0.302559	−0.151280	+	0.988491i	$0.548339\pi$
$312$	0	0
$313$	1.19272e11	0.702408	0.351204	−	0.936299i	$-0.385772\pi$
$313$	1.19272e11	0.702408	0.351204	+	0.936299i	$0.385772\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2.85879e11	1.59007	0.795034	−	0.606565i	$-0.207453\pi$
$317$	2.85879e11	1.59007	0.795034	+	0.606565i	$0.207453\pi$
$318$	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
$325$	0	0
$326$	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	$-0.338643\pi$
$331$	2.12047e11	0.970973	0.485486	+	0.874244i	$0.338643\pi$
$332$	0	0
$333$	4.28340e11	1.90893
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−2.00430e11	−0.846502	−0.423251	−	0.906012i	$-0.639111\pi$
$337$	−2.00430e11	−0.846502	−0.423251	+	0.906012i	$0.639111\pi$
$338$	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
$345$	0	0
$346$	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	$-0.786225\pi$
$353$	−4.56755e11	−1.56566	−0.782830	+	0.622235i	$0.786225\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−3.30742e10	−0.107766
$358$	0	0
$359$	1.54059e11	0.489509	0.244755	−	0.969585i	$-0.421293\pi$
$359$	1.54059e11	0.489509	0.244755	+	0.969585i	$0.421293\pi$
$360$	0	0
$361$	−3.51479e10	−0.108922
$362$	0	0
$363$	−2.55040e11	−0.770954
$364$	0	0
$365$	0	0
$366$	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	$-0.427805\pi$
$373$	1.68130e11	0.449734	0.224867	+	0.974389i	$0.427805\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−1.18054e12	−3.00986
$378$	0	0
$379$	1.00434e11	0.250038	0.125019	−	0.992154i	$-0.460101\pi$
$379$	1.00434e11	0.250038	0.125019	+	0.992154i	$0.460101\pi$
$380$	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
$385$	0	0
$386$	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
$395$	0	0
$396$	0	0
$397$	4.70667e11	0.950948	0.475474	−	0.879730i	$-0.342277\pi$
$397$	4.70667e11	0.950948	0.475474	+	0.879730i	$0.342277\pi$
$398$	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
$405$	0	0
$406$	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
$415$	0	0
$416$	0	0
$417$	−7.38605e11	−1.19619
$418$	0	0
$419$	−7.01758e11	−1.11231	−0.556153	−	0.831080i	$-0.687723\pi$
$419$	−7.01758e11	−1.11231	−0.556153	+	0.831080i	$0.687723\pi$
$420$	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
$425$	0	0
$426$	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	$-0.390566\pi$
$431$	4.82937e11	0.674129	0.337064	+	0.941482i	$0.390566\pi$
$432$	0	0
$433$	−7.71175e11	−1.05428	−0.527142	−	0.849777i	$-0.676736\pi$
$433$	−7.71175e11	−1.05428	−0.527142	+	0.849777i	$0.676736\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.25959e11	0.165220
$438$	0	0
$439$	6.62301e11	0.851070	0.425535	−	0.904942i	$-0.360086\pi$
$439$	6.62301e11	0.851070	0.425535	+	0.904942i	$0.360086\pi$
$440$	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
$445$	0	0
$446$	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
$455$	0	0
$456$	0	0
$457$	8.28175e11	0.888176	0.444088	−	0.895983i	$-0.353528\pi$
$457$	8.28175e11	0.888176	0.444088	+	0.895983i	$0.353528\pi$
$458$	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
$465$	0	0
$466$	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
$475$	0	0
$476$	0	0
$477$	−9.31518e11	−0.823869
$478$	0	0
$479$	1.38500e12	1.20210	0.601049	−	0.799212i	$-0.294750\pi$
$479$	1.38500e12	1.20210	0.601049	+	0.799212i	$0.294750\pi$
$480$	0	0
$481$	−2.55157e12	−2.17347
$482$	0	0
$483$	8.46341e10	0.0707593
$484$	0	0
$485$	0	0
$486$	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	$-0.407081\pi$
$491$	7.41251e11	0.575571	0.287785	+	0.957695i	$0.407081\pi$
$492$	0	0
$493$	−6.66171e11	−0.507895
$494$	0	0
$495$	0	0
$496$	0	0
$497$	4.29653e11	0.315874
$498$	0	0
$499$	−1.51393e12	−1.09308	−0.546542	−	0.837432i	$-0.684056\pi$
$499$	−1.51393e12	−1.09308	−0.546542	+	0.837432i	$0.684056\pi$
$500$	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
$505$	0	0
$506$	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
$515$	0	0
$516$	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
$525$	0	0
$526$	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
$535$	0	0
$536$	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
$545$	0	0
$546$	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
$555$	0	0
$556$	0	0
$557$	3.15166e12	1.38737	0.693683	−	0.720280i	$-0.255987\pi$
$557$	3.15166e12	1.38737	0.693683	+	0.720280i	$0.255987\pi$
$558$	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
$565$	0	0
$566$	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	$-0.687248\pi$
$571$	−2.81914e12	−1.10982	−0.554911	+	0.831909i	$0.687248\pi$
$572$	0	0
$573$	−5.50584e12	−2.13367
$574$	0	0
$575$	0	0
$576$	0	0
$577$	2.80815e11	0.105470	0.0527350	−	0.998609i	$-0.483206\pi$
$577$	2.80815e11	0.105470	0.0527350	+	0.998609i	$0.483206\pi$
$578$	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
$585$	0	0
$586$	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	$-0.796175\pi$
$593$	−4.82940e12	−1.60379	−0.801895	+	0.597465i	$0.796175\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	2.97496e12	0.958511
$598$	0	0
$599$	−1.72589e12	−0.547763	−0.273881	−	0.961763i	$-0.588308\pi$
$599$	−1.72589e12	−0.547763	−0.273881	+	0.961763i	$0.588308\pi$
$600$	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
$605$	0	0
$606$	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	$-0.437895\pi$
$613$	1.35557e12	0.387747	0.193874	+	0.981026i	$0.437895\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	4.35935e12	1.21098	0.605492	−	0.795852i	$-0.292977\pi$
$617$	4.35935e12	1.21098	0.605492	+	0.795852i	$0.292977\pi$
$618$	0	0
$619$	−1.67780e12	−0.459338	−0.229669	−	0.973269i	$-0.573764\pi$
$619$	−1.67780e12	−0.459338	−0.229669	+	0.973269i	$0.573764\pi$
$620$	0	0
$621$	3.88309e11	0.104777
$622$	0	0
$623$	−1.89813e12	−0.504811
$624$	0	0
$625$	0	0
$626$	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	$-0.673264\pi$
$631$	−4.12439e12	−1.03568	−0.517842	+	0.855476i	$0.673264\pi$
$632$	0	0
$633$	2.63091e12	0.651313
$634$	0	0
$635$	0	0
$636$	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
$645$	0	0
$646$	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	$-0.564690\pi$
$653$	−1.87557e12	−0.403668	−0.201834	+	0.979420i	$0.564690\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	9.65592e12	2.02185
$658$	0	0
$659$	2.02453e11	0.0418158	0.0209079	−	0.999781i	$-0.493344\pi$
$659$	2.02453e11	0.0418158	0.0209079	+	0.999781i	$0.493344\pi$
$660$	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
$665$	0	0
$666$	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	$-0.483138\pi$
$673$	5.63582e11	0.105898	0.0529491	+	0.998597i	$0.483138\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−8.43518e11	−0.154328	−0.0771641	−	0.997018i	$-0.524587\pi$
$677$	−8.43518e11	−0.154328	−0.0771641	+	0.997018i	$0.524587\pi$
$678$	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
$685$	0	0
$686$	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	$-0.462228\pi$
$691$	1.41900e12	0.236773	0.118386	+	0.992968i	$0.462228\pi$
$692$	0	0
$693$	−1.56013e12	−0.256957
$694$	0	0
$695$	0	0
$696$	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
$705$	0	0
$706$	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
$715$	0	0
$716$	0	0
$717$	−1.25059e13	−1.76717
$718$	0	0
$719$	8.40313e12	1.17263	0.586315	−	0.810083i	$-0.300578\pi$
$719$	8.40313e12	1.17263	0.586315	+	0.810083i	$0.300578\pi$
$720$	0	0
$721$	−3.48260e12	−0.479949
$722$	0	0
$723$	−9.95833e12	−1.35539
$724$	0	0
$725$	0	0
$726$	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	$-0.530759\pi$
$733$	−1.50813e12	−0.192962	−0.0964810	+	0.995335i	$0.530759\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	8.27285e12	1.03288
$738$	0	0
$739$	1.48289e13	1.82898	0.914492	−	0.404603i	$-0.132590\pi$
$739$	1.48289e13	1.82898	0.914492	+	0.404603i	$0.132590\pi$
$740$	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
$745$	0	0
$746$	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	$-0.378329\pi$
$751$	6.50309e12	0.746002	0.373001	+	0.927831i	$0.378329\pi$
$752$	0	0
$753$	1.37250e13	1.55573
$754$	0	0
$755$	0	0
$756$	0	0
$757$	1.11069e11	0.0122931	0.00614653	−	0.999981i	$-0.498043\pi$
$757$	1.11069e11	0.0122931	0.00614653	+	0.999981i	$0.498043\pi$
$758$	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
$765$	0	0
$766$	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	$-0.168153\pi$
$773$	1.71471e13	1.72736	0.863681	+	0.504038i	$0.168153\pi$
$774$	0	0
$775$	0	0
$776$	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
$785$	0	0
$786$	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
$795$	0	0
$796$	0	0
$797$	−9.75044e12	−0.855976	−0.427988	−	0.903784i	$-0.640777\pi$
$797$	−9.75044e12	−0.855976	−0.427988	+	0.903784i	$0.640777\pi$
$798$	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
$805$	0	0
$806$	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	$-0.501414\pi$
$811$	−1.09433e11	−0.00888287	−0.00444143	+	0.999990i	$0.501414\pi$
$812$	0	0
$813$	−1.13434e13	−0.910614
$814$	0	0
$815$	0	0
$816$	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
$825$	0	0
$826$	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
$835$	0	0
$836$	0	0
$837$	−1.36058e13	−0.958206
$838$	0	0
$839$	4.34425e12	0.302681	0.151341	−	0.988482i	$-0.451641\pi$
$839$	4.34425e12	0.302681	0.151341	+	0.988482i	$0.451641\pi$
$840$	0	0
$841$	3.81578e13	2.63028
$842$	0	0
$843$	1.58509e13	1.08101
$844$	0	0
$845$	0	0
$846$	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	$-0.357757\pi$
$853$	1.33638e13	0.864290	0.432145	+	0.901804i	$0.357757\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−2.60497e13	−1.64964	−0.824819	−	0.565397i	$-0.808723\pi$
$857$	−2.60497e13	−1.64964	−0.824819	+	0.565397i	$0.808723\pi$
$858$	0	0
$859$	−2.01187e12	−0.126076	−0.0630379	−	0.998011i	$-0.520079\pi$
$859$	−2.01187e12	−0.126076	−0.0630379	+	0.998011i	$0.520079\pi$
$860$	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
$865$	0	0
$866$	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
$875$	0	0
$876$	0	0
$877$	−1.83122e13	−1.04531	−0.522653	−	0.852545i	$-0.675058\pi$
$877$	−1.83122e13	−1.04531	−0.522653	+	0.852545i	$0.675058\pi$
$878$	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
$885$	0	0
$886$	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
$895$	0	0
$896$	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
$905$	0	0
$906$	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	$-0.558889\pi$
$911$	−7.64836e12	−0.367905	−0.183952	+	0.982935i	$0.558889\pi$
$912$	0	0
$913$	3.35110e11	0.0159613
$914$	0	0
$915$	0	0
$916$	0	0
$917$	7.90232e12	0.369056
$918$	0	0
$919$	−3.19451e13	−1.47735	−0.738676	−	0.674060i	$-0.764549\pi$
$919$	−3.19451e13	−1.47735	−0.738676	+	0.674060i	$0.764549\pi$
$920$	0	0
$921$	−4.97639e13	−2.27901
$922$	0	0
$923$	−4.20519e13	−1.90712
$924$	0	0
$925$	0	0
$926$	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
$935$	0	0
$936$	0	0
$937$	1.10891e13	0.469966	0.234983	−	0.971999i	$-0.424497\pi$
$937$	1.10891e13	0.469966	0.234983	+	0.971999i	$0.424497\pi$
$938$	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
$945$	0	0
$946$	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	$-0.481229\pi$
$953$	3.00142e12	0.117872	0.0589358	+	0.998262i	$0.481229\pi$
$954$	0	0
$955$	0	0
$956$	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
$965$	0	0
$966$	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	$-0.516304\pi$
$971$	−2.83646e12	−0.102398	−0.0511988	+	0.998688i	$0.516304\pi$
$972$	0	0
$973$	−5.66311e12	−0.202557
$974$	0	0
$975$	0	0
$976$	0	0
$977$	9.66226e12	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$977$	9.66226e12	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$978$	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
$985$	0	0
$986$	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	$-0.665443\pi$
$991$	−3.01597e13	−0.993335	−0.496668	+	0.867941i	$0.665443\pi$
$992$	0	0
$993$	4.59668e13	1.50028
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−1.98318e13	−0.635675	−0.317837	−	0.948145i	$-0.602957\pi$
$997$	−1.98318e13	−0.635675	−0.317837	+	0.948145i	$0.602957\pi$
$998$	0	0
$999$	2.59288e13	0.823640

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.10.a.c.1.2	✓	2	20.19	odd	2
80.10.a.g.1.1		2	5.4	even	2
200.10.a.c.1.1		2	4.3	odd	2
200.10.c.d.49.1		4	20.3	even	4
200.10.c.d.49.4		4	20.7	even	4
320.10.a.n.1.1		2	40.19	odd	2
320.10.a.q.1.2		2	40.29	even	2
360.10.a.g.1.2		2	60.59	even	2
400.10.a.r.1.2		2	1.1	even	1	trivial
400.10.c.m.49.1		4	5.2	odd	4
400.10.c.m.49.4		4	5.3	odd	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	400.a (trivial)

\(f(q)\)	\(=\)	\(q+216.776 q^{3} +1662.09 q^{7} +27308.8 q^{9} +O(q^{10})\)	\(q+216.776 q^{3} +1662.09 q^{7} +27308.8 q^{9} -34372.0 q^{11} -162675. q^{13} -91796.1 q^{17} +536227. q^{19} +360300. q^{21} +234899. q^{23} +1.65309e6 q^{27} +7.25706e6 q^{29} -8.23051e6 q^{31} -7.45102e6 q^{33} +1.56850e7 q^{37} -3.52641e7 q^{39} -1.67318e7 q^{41} +9.36208e6 q^{43} -2.69410e7 q^{47} -3.75911e7 q^{49} -1.98992e7 q^{51} -3.41105e7 q^{53} +1.16241e8 q^{57} -1.12155e8 q^{59} +5.49531e6 q^{61} +4.53896e7 q^{63} -2.40686e8 q^{67} +5.09204e7 q^{69} +2.58502e8 q^{71} +3.53583e8 q^{73} -5.71292e7 q^{77} -3.66190e8 q^{79} -1.79169e8 q^{81} -9.74953e6 q^{83} +1.57316e9 q^{87} -1.14201e9 q^{89} -2.70380e8 q^{91} -1.78418e9 q^{93} -1.25555e9 q^{97} -9.38657e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 108 q^{3} - 908 q^{7} + 19458 q^{9}+O(q^{10}) \) 2 * q + 108 * q^3 - 908 * q^7 + 19458 * q^9	\( 2 q + 108 q^{3} - 908 q^{7} + 19458 q^{9} - 25120 q^{11} - 146948 q^{13} - 169268 q^{17} + 25480 q^{19} + 639864 q^{21} + 1782748 q^{23} + 4648104 q^{27} + 7323340 q^{29} - 10677272 q^{31} - 8457408 q^{33} + 5750460 q^{37} - 36974808 q^{39} - 7795764 q^{41} + 16770524 q^{43} + 15393892 q^{47} - 71339334 q^{49} - 11472120 q^{51} + 58529292 q^{53} + 171798192 q^{57} - 59618264 q^{59} - 188163772 q^{61} + 65566836 q^{63} - 105998252 q^{67} - 117448344 q^{69} + 168665592 q^{71} + 390212412 q^{73} - 80907584 q^{77} - 466946256 q^{79} - 350427222 q^{81} + 329535164 q^{83} + 1565947656 q^{87} - 108334860 q^{89} - 310800616 q^{91} - 1518028560 q^{93} - 2043058628 q^{97} - 1011292704 q^{99}+O(q^{100}) \) 2 * q + 108 * q^3 - 908 * q^7 + 19458 * q^9 - 25120 * q^11 - 146948 * q^13 - 169268 * q^17 + 25480 * q^19 + 639864 * q^21 + 1782748 * q^23 + 4648104 * q^27 + 7323340 * q^29 - 10677272 * q^31 - 8457408 * q^33 + 5750460 * q^37 - 36974808 * q^39 - 7795764 * q^41 + 16770524 * q^43 + 15393892 * q^47 - 71339334 * q^49 - 11472120 * q^51 + 58529292 * q^53 + 171798192 * q^57 - 59618264 * q^59 - 188163772 * q^61 + 65566836 * q^63 - 105998252 * q^67 - 117448344 * q^69 + 168665592 * q^71 + 390212412 * q^73 - 80907584 * q^77 - 466946256 * q^79 - 350427222 * q^81 + 329535164 * q^83 + 1565947656 * q^87 - 108334860 * q^89 - 310800616 * q^91 - 1518028560 * q^93 - 2043058628 * q^97 - 1011292704 * q^99

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