LMFDB - Embedded newform 400.6.c.l.49.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,6,Mod(49,400)]

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, 1]))

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

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

chi := DirichletCharacter("400.49");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 400 = 2^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	400.c (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$64.1535279252$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{129})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 65x^{2} + 1024 $ x^4 + 65*x^2 + 1024
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.3
Root			$5.17891i$ of defining polynomial
Character	$\chi$	$=$	400.49
Dual form			400.6.c.l.49.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+16.7156i q^{3} -94.1469i q^{7} -36.4124 q^{9} +O(q^{10})$	$q+16.7156i q^{3} -94.1469i q^{7} -36.4124 q^{9} -143.706 q^{11} -421.412i q^{13} +1982.11i q^{17} -1317.76 q^{19} +1573.73 q^{21} -4020.02i q^{23} +3453.24i q^{27} +6417.28 q^{29} +2350.64 q^{31} -2402.14i q^{33} -7876.58i q^{37} +7044.18 q^{39} +15081.6 q^{41} +1141.40i q^{43} +21557.3i q^{47} +7943.36 q^{49} -33132.3 q^{51} -9560.44i q^{53} -22027.2i q^{57} +42740.7 q^{59} +32132.1 q^{61} +3428.11i q^{63} +30371.4i q^{67} +67197.2 q^{69} -36006.7 q^{71} +63438.0i q^{73} +13529.5i q^{77} -89922.8 q^{79} -66571.4 q^{81} +38211.2i q^{83} +107269. i q^{87} -5745.69 q^{89} -39674.7 q^{91} +39292.5i q^{93} +178780. i q^{97} +5232.69 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 1236 q^{9}+O(q^{10}) $ 4 * q - 1236 * q^9	$ 4 q - 1236 q^{9} - 1120 q^{11} - 2000 q^{19} + 5568 q^{21} - 2680 q^{29} + 4496 q^{31} - 41424 q^{39} + 46152 q^{41} + 45948 q^{49} - 171600 q^{51} + 125168 q^{59} + 28216 q^{61} + 224448 q^{69} - 94416 q^{71} - 131808 q^{79} + 142596 q^{81} + 110040 q^{89} + 2128 q^{91} + 494688 q^{99}+O(q^{100}) $ 4 * q - 1236 * q^9 - 1120 * q^11 - 2000 * q^19 + 5568 * q^21 - 2680 * q^29 + 4496 * q^31 - 41424 * q^39 + 46152 * q^41 + 45948 * q^49 - 171600 * q^51 + 125168 * q^59 + 28216 * q^61 + 224448 * q^69 - 94416 * q^71 - 131808 * q^79 + 142596 * q^81 + 110040 * q^89 + 2128 * q^91 + 494688 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/400\mathbb{Z}\right)^\times$.

$n$	$101$	$177$	$351$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	16.7156i	1.07231i	0.844120	+	0.536154i	$0.180123\pi$
$3$	16.7156i	1.07231i	−0.844120	+	0.536154i	$0.819877\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 94.1469i	− 0.726208i	−0.931749	−	0.363104i	$-0.881717\pi$
$7$	− 94.1469i	− 0.726208i	0.931749	−	0.363104i	$-0.118283\pi$
$8$	0	0
$9$	−36.4124	−0.149845
$10$	0	0
$11$	−143.706	−0.358091	−0.179046	−	0.983841i	$-0.557301\pi$
$11$	−143.706	−0.358091	−0.179046	+	0.983841i	$0.557301\pi$
$12$	0	0
$13$	− 421.412i	− 0.691590i	−0.938310	−	0.345795i	$-0.887609\pi$
$13$	− 421.412i	− 0.691590i	0.938310	−	0.345795i	$-0.112391\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1982.11i	1.66344i	0.555198	+	0.831718i	$0.312642\pi$
$17$	1982.11i	1.66344i	−0.555198	+	0.831718i	$0.687358\pi$
$18$	0	0
$19$	−1317.76	−0.837439	−0.418720	−	0.908116i	$-0.637521\pi$
$19$	−1317.76	−0.837439	−0.418720	+	0.908116i	$0.637521\pi$
$20$	0	0
$21$	1573.73	0.778719
$22$	0	0
$23$	− 4020.02i	− 1.58456i	−0.610157	−	0.792280i	$-0.708894\pi$
$23$	− 4020.02i	− 1.58456i	0.610157	−	0.792280i	$-0.291106\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	3453.24i	0.911628i
$28$	0	0
$29$	6417.28	1.41695	0.708477	−	0.705734i	$-0.249383\pi$
$29$	6417.28	1.41695	0.708477	+	0.705734i	$0.249383\pi$
$30$	0	0
$31$	2350.64	0.439322	0.219661	−	0.975576i	$-0.429505\pi$
$31$	2350.64	0.439322	0.219661	+	0.975576i	$0.429505\pi$
$32$	0	0
$33$	− 2402.14i	− 0.383984i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 7876.58i	− 0.945874i	−0.881096	−	0.472937i	$-0.843194\pi$
$37$	− 7876.58i	− 0.945874i	0.881096	−	0.472937i	$-0.156806\pi$
$38$	0	0
$39$	7044.18	0.741598
$40$	0	0
$41$	15081.6	1.40116	0.700582	−	0.713572i	$-0.252924\pi$
$41$	15081.6	1.40116	0.700582	+	0.713572i	$0.252924\pi$
$42$	0	0
$43$	1141.40i	0.0941384i	0.998892	+	0.0470692i	$0.0149881\pi$
$43$	1141.40i	0.0941384i	−0.998892	+	0.0470692i	$0.985012\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	21557.3i	1.42348i	0.702445	+	0.711738i	$0.252092\pi$
$47$	21557.3i	1.42348i	−0.702445	+	0.711738i	$0.747908\pi$
$48$	0	0
$49$	7943.36	0.472622
$50$	0	0
$51$	−33132.3	−1.78372
$52$	0	0
$53$	− 9560.44i	− 0.467507i	−0.972296	−	0.233754i	$-0.924899\pi$
$53$	− 9560.44i	− 0.467507i	0.972296	−	0.233754i	$-0.0751009\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 22027.2i	− 0.897993i
$58$	0	0
$59$	42740.7	1.59850	0.799248	−	0.601002i	$-0.205232\pi$
$59$	42740.7	1.59850	0.799248	+	0.601002i	$0.205232\pi$
$60$	0	0
$61$	32132.1	1.10564	0.552820	−	0.833301i	$-0.313552\pi$
$61$	32132.1	1.10564	0.552820	+	0.833301i	$0.313552\pi$
$62$	0	0
$63$	3428.11i	0.108819i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	30371.4i	0.826567i	0.910602	+	0.413283i	$0.135618\pi$
$67$	30371.4i	0.826567i	−0.910602	+	0.413283i	$0.864382\pi$
$68$	0	0
$69$	67197.2	1.69914
$70$	0	0
$71$	−36006.7	−0.847692	−0.423846	−	0.905734i	$-0.639320\pi$
$71$	−36006.7	−0.847692	−0.423846	+	0.905734i	$0.639320\pi$
$72$	0	0
$73$	63438.0i	1.39329i	0.717414	+	0.696647i	$0.245326\pi$
$73$	63438.0i	1.39329i	−0.717414	+	0.696647i	$0.754674\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	13529.5i	0.260049i
$78$	0	0
$79$	−89922.8	−1.62107	−0.810536	−	0.585689i	$-0.800824\pi$
$79$	−89922.8	−1.62107	−0.810536	+	0.585689i	$0.800824\pi$
$80$	0	0
$81$	−66571.4	−1.12739
$82$	0	0
$83$	38211.2i	0.608829i	0.952540	+	0.304414i	$0.0984608\pi$
$83$	38211.2i	0.608829i	−0.952540	+	0.304414i	$0.901539\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	107269.i	1.51941i
$88$	0	0
$89$	−5745.69	−0.0768895	−0.0384447	−	0.999261i	$-0.512240\pi$
$89$	−5745.69	−0.0768895	−0.0384447	+	0.999261i	$0.512240\pi$
$90$	0	0
$91$	−39674.7	−0.502238
$92$	0	0
$93$	39292.5i	0.471088i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	178780.i	1.92926i	0.263613	+	0.964629i	$0.415086\pi$
$97$	178780.i	1.92926i	−0.263613	+	0.964629i	$0.584914\pi$
$98$	0	0
$99$	5232.69	0.0536583
$100$	0	0
$101$	152223.	1.48483	0.742417	−	0.669938i	$-0.233679\pi$
$101$	152223.	1.48483	0.742417	+	0.669938i	$0.233679\pi$
$102$	0	0
$103$	− 35830.1i	− 0.332778i	−0.986060	−	0.166389i	$-0.946789\pi$
$103$	− 35830.1i	− 0.332778i	0.986060	−	0.166389i	$-0.0532108\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 70030.2i	− 0.591324i	−0.955293	−	0.295662i	$-0.904460\pi$
$107$	− 70030.2i	− 0.591324i	0.955293	−	0.295662i	$-0.0955403\pi$
$108$	0	0
$109$	−38466.9	−0.310114	−0.155057	−	0.987906i	$-0.549556\pi$
$109$	−38466.9	−0.310114	−0.155057	+	0.987906i	$0.549556\pi$
$110$	0	0
$111$	131662.	1.01427
$112$	0	0
$113$	− 39951.6i	− 0.294333i	−0.989112	−	0.147166i	$-0.952985\pi$
$113$	− 39951.6i	− 0.294333i	0.989112	−	0.147166i	$-0.0470153\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	15344.6i	0.103632i
$118$	0	0
$119$	186610.	1.20800
$120$	0	0
$121$	−140400.	−0.871771
$122$	0	0
$123$	252099.i	1.50248i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	233239.i	1.28319i	0.767043	+	0.641596i	$0.221727\pi$
$127$	233239.i	1.28319i	−0.767043	+	0.641596i	$0.778273\pi$
$128$	0	0
$129$	−19079.2	−0.100945
$130$	0	0
$131$	−55237.0	−0.281224	−0.140612	−	0.990065i	$-0.544907\pi$
$131$	−55237.0	−0.281224	−0.140612	+	0.990065i	$0.544907\pi$
$132$	0	0
$133$	124063.i	0.608155i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	261520.i	1.19043i	0.803566	+	0.595215i	$0.202933\pi$
$137$	261520.i	1.19043i	−0.803566	+	0.595215i	$0.797067\pi$
$138$	0	0
$139$	293366.	1.28787	0.643935	−	0.765080i	$-0.277301\pi$
$139$	293366.	1.28787	0.643935	+	0.765080i	$0.277301\pi$
$140$	0	0
$141$	−360345.	−1.52641
$142$	0	0
$143$	60559.6i	0.247653i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	132778.i	0.506797i
$148$	0	0
$149$	304505.	1.12364	0.561822	−	0.827258i	$-0.310101\pi$
$149$	304505.	1.12364	0.561822	+	0.827258i	$0.310101\pi$
$150$	0	0
$151$	337909.	1.20603	0.603015	−	0.797730i	$-0.293966\pi$
$151$	337909.	1.20603	0.603015	+	0.797730i	$0.293966\pi$
$152$	0	0
$153$	− 72173.5i	− 0.249258i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	68385.9i	0.221420i	0.993853	+	0.110710i	$0.0353125\pi$
$157$	68385.9i	0.221420i	−0.993853	+	0.110710i	$0.964687\pi$
$158$	0	0
$159$	159809.	0.501312
$160$	0	0
$161$	−378473.	−1.15072
$162$	0	0
$163$	− 404471.i	− 1.19239i	−0.802840	−	0.596195i	$-0.796678\pi$
$163$	− 404471.i	− 1.19239i	0.802840	−	0.596195i	$-0.203322\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 411733.i	− 1.14242i	−0.820805	−	0.571209i	$-0.806475\pi$
$167$	− 411733.i	− 1.14242i	0.820805	−	0.571209i	$-0.193525\pi$
$168$	0	0
$169$	193705.	0.521703
$170$	0	0
$171$	47982.9	0.125486
$172$	0	0
$173$	162247.i	0.412155i	0.978536	+	0.206077i	$0.0660698\pi$
$173$	162247.i	0.412155i	−0.978536	+	0.206077i	$0.933930\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	714438.i	1.71408i
$178$	0	0
$179$	384396.	0.896698	0.448349	−	0.893859i	$-0.352012\pi$
$179$	384396.	0.896698	0.448349	+	0.893859i	$0.352012\pi$
$180$	0	0
$181$	579219.	1.31415	0.657077	−	0.753823i	$-0.271792\pi$
$181$	579219.	1.31415	0.657077	+	0.753823i	$0.271792\pi$
$182$	0	0
$183$	537108.i	1.18559i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 284842.i	− 0.595662i
$188$	0	0
$189$	325112.	0.662031
$190$	0	0
$191$	−142455.	−0.282550	−0.141275	−	0.989970i	$-0.545120\pi$
$191$	−142455.	−0.282550	−0.141275	+	0.989970i	$0.545120\pi$
$192$	0	0
$193$	− 267272.i	− 0.516487i	−0.966080	−	0.258244i	$-0.916856\pi$
$193$	− 267272.i	− 0.516487i	0.966080	−	0.258244i	$-0.0831438\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	14189.5i	0.0260496i	0.999915	+	0.0130248i	$0.00414604\pi$
$197$	14189.5i	0.0260496i	−0.999915	+	0.0130248i	$0.995854\pi$
$198$	0	0
$199$	169198.	0.302875	0.151437	−	0.988467i	$-0.451610\pi$
$199$	169198.	0.302875	0.151437	+	0.988467i	$0.451610\pi$
$200$	0	0
$201$	−507677.	−0.886335
$202$	0	0
$203$	− 604167.i	− 1.02900i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	146379.i	0.237439i
$208$	0	0
$209$	189371.	0.299880
$210$	0	0
$211$	407591.	0.630259	0.315129	−	0.949049i	$-0.397952\pi$
$211$	407591.	0.630259	0.315129	+	0.949049i	$0.397952\pi$
$212$	0	0
$213$	− 601875.i	− 0.908987i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 221306.i	− 0.319039i
$218$	0	0
$219$	−1.06041e6	−1.49404
$220$	0	0
$221$	835287.	1.15042
$222$	0	0
$223$	− 103743.i	− 0.139700i	−0.997558	−	0.0698500i	$-0.977748\pi$
$223$	− 103743.i	− 0.139700i	0.997558	−	0.0698500i	$-0.0222521\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	803726.i	1.03524i	0.855609	+	0.517622i	$0.173183\pi$
$227$	803726.i	1.03524i	−0.855609	+	0.517622i	$0.826817\pi$
$228$	0	0
$229$	−1.35955e6	−1.71319	−0.856596	−	0.515988i	$-0.827425\pi$
$229$	−1.35955e6	−1.71319	−0.856596	+	0.515988i	$0.827425\pi$
$230$	0	0
$231$	−226154.	−0.278852
$232$	0	0
$233$	622827.i	0.751584i	0.926704	+	0.375792i	$0.122629\pi$
$233$	622827.i	0.751584i	−0.926704	+	0.375792i	$0.877371\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 1.50312e6i	− 1.73829i
$238$	0	0
$239$	−642843.	−0.727964	−0.363982	−	0.931406i	$-0.618583\pi$
$239$	−642843.	−0.727964	−0.363982	+	0.931406i	$0.618583\pi$
$240$	0	0
$241$	1.11814e6	1.24009	0.620046	−	0.784566i	$-0.287114\pi$
$241$	1.11814e6	1.24009	0.620046	+	0.784566i	$0.287114\pi$
$242$	0	0
$243$	− 273644.i	− 0.297283i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	555322.i	0.579165i
$248$	0	0
$249$	−638724.	−0.652852
$250$	0	0
$251$	−731067.	−0.732442	−0.366221	−	0.930528i	$-0.619349\pi$
$251$	−731067.	−0.732442	−0.366221	+	0.930528i	$0.619349\pi$
$252$	0	0
$253$	577702.i	0.567418i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 1.76622e6i	− 1.66806i	−0.551720	−	0.834030i	$-0.686028\pi$
$257$	− 1.76622e6i	− 1.66806i	0.551720	−	0.834030i	$-0.313972\pi$
$258$	0	0
$259$	−741555.	−0.686901
$260$	0	0
$261$	−233668.	−0.212324
$262$	0	0
$263$	1.00926e6i	0.899735i	0.893095	+	0.449868i	$0.148529\pi$
$263$	1.00926e6i	0.899735i	−0.893095	+	0.449868i	$0.851471\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 96042.8i	− 0.0824492i
$268$	0	0
$269$	775681.	0.653585	0.326793	−	0.945096i	$-0.394032\pi$
$269$	775681.	0.653585	0.326793	+	0.945096i	$0.394032\pi$
$270$	0	0
$271$	−1.21395e6	−1.00410	−0.502052	−	0.864837i	$-0.667422\pi$
$271$	−1.21395e6	−1.00410	−0.502052	+	0.864837i	$0.667422\pi$
$272$	0	0
$273$	− 663187.i	− 0.538554i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 218505.i	− 0.171104i	−0.996334	−	0.0855522i	$-0.972735\pi$
$277$	− 218505.i	− 0.171104i	0.996334	−	0.0855522i	$-0.0272654\pi$
$278$	0	0
$279$	−85592.6	−0.0658303
$280$	0	0
$281$	−316219.	−0.238903	−0.119452	−	0.992840i	$-0.538114\pi$
$281$	−316219.	−0.238903	−0.119452	+	0.992840i	$0.538114\pi$
$282$	0	0
$283$	− 927934.i	− 0.688733i	−0.938835	−	0.344366i	$-0.888094\pi$
$283$	− 927934.i	− 0.688733i	0.938835	−	0.344366i	$-0.111906\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 1.41989e6i	− 1.01754i
$288$	0	0
$289$	−2.50892e6	−1.76702
$290$	0	0
$291$	−2.98842e6	−2.06876
$292$	0	0
$293$	− 262992.i	− 0.178967i	−0.995988	−	0.0894835i	$-0.971478\pi$
$293$	− 262992.i	− 0.178967i	0.995988	−	0.0894835i	$-0.0285216\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 496252.i	− 0.326446i
$298$	0	0
$299$	−1.69409e6	−1.09587
$300$	0	0
$301$	107459.	0.0683640
$302$	0	0
$303$	2.54451e6i	1.59220i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 2.15704e6i	− 1.30621i	−0.757269	−	0.653104i	$-0.773467\pi$
$307$	− 2.15704e6i	− 1.30621i	0.757269	−	0.653104i	$-0.226533\pi$
$308$	0	0
$309$	598923.	0.356841
$310$	0	0
$311$	2.33078e6	1.36647	0.683235	−	0.730199i	$-0.260573\pi$
$311$	2.33078e6	1.36647	0.683235	+	0.730199i	$0.260573\pi$
$312$	0	0
$313$	1.12235e6i	0.647539i	0.946136	+	0.323769i	$0.104950\pi$
$313$	1.12235e6i	0.647539i	−0.946136	+	0.323769i	$0.895050\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 1.27955e6i	− 0.715169i	−0.933881	−	0.357584i	$-0.883600\pi$
$317$	− 1.27955e6i	− 0.715169i	0.933881	−	0.357584i	$-0.116400\pi$
$318$	0	0
$319$	−922203.	−0.507399
$320$	0	0
$321$	1.17060e6	0.634082
$322$	0	0
$323$	− 2.61196e6i	− 1.39303i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 642998.i	− 0.332537i
$328$	0	0
$329$	2.02956e6	1.03374
$330$	0	0
$331$	2.09571e6	1.05138	0.525691	−	0.850675i	$-0.323807\pi$
$331$	2.09571e6	1.05138	0.525691	+	0.850675i	$0.323807\pi$
$332$	0	0
$333$	286805.i	0.141735i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 571289.i	− 0.274019i	−0.990570	−	0.137010i	$-0.956251\pi$
$337$	− 571289.i	− 0.274019i	0.990570	−	0.137010i	$-0.0437491\pi$
$338$	0	0
$339$	667817.	0.315615
$340$	0	0
$341$	−337802.	−0.157317
$342$	0	0
$343$	− 2.33017e6i	− 1.06943i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 2.24067e6i	− 0.998973i	−0.866322	−	0.499486i	$-0.833522\pi$
$347$	− 2.24067e6i	− 0.998973i	0.866322	−	0.499486i	$-0.166478\pi$
$348$	0	0
$349$	−130448.	−0.0573290	−0.0286645	−	0.999589i	$-0.509125\pi$
$349$	−130448.	−0.0573290	−0.0286645	+	0.999589i	$0.509125\pi$
$350$	0	0
$351$	1.45524e6	0.630473
$352$	0	0
$353$	− 1.95452e6i	− 0.834839i	−0.908714	−	0.417420i	$-0.862935\pi$
$353$	− 1.95452e6i	− 0.834839i	0.908714	−	0.417420i	$-0.137065\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	3.11930e6i	1.29535i
$358$	0	0
$359$	−850543.	−0.348305	−0.174153	−	0.984719i	$-0.555719\pi$
$359$	−850543.	−0.348305	−0.174153	+	0.984719i	$0.555719\pi$
$360$	0	0
$361$	−739600.	−0.298696
$362$	0	0
$363$	− 2.34687e6i	− 0.934807i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	1.50673e6i	0.583941i	0.956427	+	0.291970i	$0.0943109\pi$
$367$	1.50673e6i	0.583941i	−0.956427	+	0.291970i	$0.905689\pi$
$368$	0	0
$369$	−549159.	−0.209958
$370$	0	0
$371$	−900086.	−0.339507
$372$	0	0
$373$	− 2.90602e6i	− 1.08150i	−0.841184	−	0.540749i	$-0.818141\pi$
$373$	− 2.90602e6i	− 1.08150i	0.841184	−	0.540749i	$-0.181859\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 2.70432e6i	− 0.979952i
$378$	0	0
$379$	5.16710e6	1.84777	0.923887	−	0.382665i	$-0.124994\pi$
$379$	5.16710e6	1.84777	0.923887	+	0.382665i	$0.124994\pi$
$380$	0	0
$381$	−3.89874e6	−1.37598
$382$	0	0
$383$	3.85088e6i	1.34142i	0.741721	+	0.670708i	$0.234010\pi$
$383$	3.85088e6i	1.34142i	−0.741721	+	0.670708i	$0.765990\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 41561.1i	− 0.0141062i
$388$	0	0
$389$	−5.49855e6	−1.84236	−0.921179	−	0.389138i	$-0.872773\pi$
$389$	−5.49855e6	−1.84236	−0.921179	+	0.389138i	$0.872773\pi$
$390$	0	0
$391$	7.96814e6	2.63582
$392$	0	0
$393$	− 923321.i	− 0.301558i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	1.71425e6i	0.545881i	0.962031	+	0.272941i	$0.0879963\pi$
$397$	1.71425e6i	0.545881i	−0.962031	+	0.272941i	$0.912004\pi$
$398$	0	0
$399$	−2.07380e6	−0.652130
$400$	0	0
$401$	3.63329e6	1.12834	0.564169	−	0.825660i	$-0.309197\pi$
$401$	3.63329e6	1.12834	0.564169	+	0.825660i	$0.309197\pi$
$402$	0	0
$403$	− 990591.i	− 0.303831i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.13191e6i	0.338709i
$408$	0	0
$409$	1.38246e6	0.408642	0.204321	−	0.978904i	$-0.434501\pi$
$409$	1.38246e6	0.408642	0.204321	+	0.978904i	$0.434501\pi$
$410$	0	0
$411$	−4.37148e6	−1.27651
$412$	0	0
$413$	− 4.02390e6i	− 1.16084i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	4.90379e6i	1.38099i
$418$	0	0
$419$	2.55435e6	0.710795	0.355398	−	0.934715i	$-0.384345\pi$
$419$	2.55435e6	0.710795	0.355398	+	0.934715i	$0.384345\pi$
$420$	0	0
$421$	−1.68110e6	−0.462263	−0.231132	−	0.972923i	$-0.574243\pi$
$421$	−1.68110e6	−0.462263	−0.231132	+	0.972923i	$0.574243\pi$
$422$	0	0
$423$	− 784954.i	− 0.213301i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 3.02513e6i	− 0.802925i
$428$	0	0
$429$	−1.01229e6	−0.265560
$430$	0	0
$431$	2.32369e6	0.602539	0.301269	−	0.953539i	$-0.402590\pi$
$431$	2.32369e6	0.602539	0.301269	+	0.953539i	$0.402590\pi$
$432$	0	0
$433$	4.06439e6i	1.04178i	0.853624	+	0.520890i	$0.174400\pi$
$433$	4.06439e6i	1.04178i	−0.853624	+	0.520890i	$0.825600\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	5.29744e6i	1.32697i
$438$	0	0
$439$	6.39272e6	1.58316	0.791579	−	0.611066i	$-0.209259\pi$
$439$	6.39272e6	1.58316	0.791579	+	0.611066i	$0.209259\pi$
$440$	0	0
$441$	−289237.	−0.0708202
$442$	0	0
$443$	3.23515e6i	0.783222i	0.920131	+	0.391611i	$0.128082\pi$
$443$	3.23515e6i	0.783222i	−0.920131	+	0.391611i	$0.871918\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	5.08999e6i	1.20489i
$448$	0	0
$449$	2.74812e6	0.643310	0.321655	−	0.946857i	$-0.395761\pi$
$449$	2.74812e6	0.643310	0.321655	+	0.946857i	$0.395761\pi$
$450$	0	0
$451$	−2.16732e6	−0.501745
$452$	0	0
$453$	5.64837e6i	1.29324i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 5.37326e6i	− 1.20350i	−0.798684	−	0.601751i	$-0.794470\pi$
$457$	− 5.37326e6i	− 1.20350i	0.798684	−	0.601751i	$-0.205530\pi$
$458$	0	0
$459$	−6.84472e6	−1.51644
$460$	0	0
$461$	−8.19928e6	−1.79690	−0.898449	−	0.439078i	$-0.855305\pi$
$461$	−8.19928e6	−1.79690	−0.898449	+	0.439078i	$0.855305\pi$
$462$	0	0
$463$	3.76963e6i	0.817233i	0.912706	+	0.408617i	$0.133989\pi$
$463$	3.76963e6i	0.817233i	−0.912706	+	0.408617i	$0.866011\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 6.03412e6i	− 1.28033i	−0.768237	−	0.640165i	$-0.778866\pi$
$467$	− 6.03412e6i	− 1.28033i	0.768237	−	0.640165i	$-0.221134\pi$
$468$	0	0
$469$	2.85937e6	0.600259
$470$	0	0
$471$	−1.14311e6	−0.237431
$472$	0	0
$473$	− 164026.i	− 0.0337101i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	348119.i	0.0700537i
$478$	0	0
$479$	−5.26406e6	−1.04829	−0.524146	−	0.851629i	$-0.675615\pi$
$479$	−5.26406e6	−1.04829	−0.524146	+	0.851629i	$0.675615\pi$
$480$	0	0
$481$	−3.31929e6	−0.654157
$482$	0	0
$483$	− 6.32641e6i	− 1.23393i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	2.19979e6i	0.420300i	0.977669	+	0.210150i	$0.0673952\pi$
$487$	2.19979e6i	0.420300i	−0.977669	+	0.210150i	$0.932605\pi$
$488$	0	0
$489$	6.76099e6	1.27861
$490$	0	0
$491$	−6.47314e6	−1.21174	−0.605872	−	0.795562i	$-0.707176\pi$
$491$	−6.47314e6	−1.21174	−0.605872	+	0.795562i	$0.707176\pi$
$492$	0	0
$493$	1.27198e7i	2.35701i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	3.38992e6i	0.615600i
$498$	0	0
$499$	8.62090e6	1.54989	0.774946	−	0.632028i	$-0.217777\pi$
$499$	8.62090e6	1.54989	0.774946	+	0.632028i	$0.217777\pi$
$500$	0	0
$501$	6.88238e6	1.22502
$502$	0	0
$503$	8.10557e6i	1.42845i	0.699918	+	0.714223i	$0.253220\pi$
$503$	8.10557e6i	1.42845i	−0.699918	+	0.714223i	$0.746780\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	3.23789e6i	0.559426i
$508$	0	0
$509$	6.39610e6	1.09426	0.547131	−	0.837047i	$-0.315720\pi$
$509$	6.39610e6	1.09426	0.547131	+	0.837047i	$0.315720\pi$
$510$	0	0
$511$	5.97250e6	1.01182
$512$	0	0
$513$	− 4.55055e6i	− 0.763433i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 3.09792e6i	− 0.509735i
$518$	0	0
$519$	−2.71205e6	−0.441957
$520$	0	0
$521$	7.53647e6	1.21639	0.608197	−	0.793786i	$-0.291893\pi$
$521$	7.53647e6	1.21639	0.608197	+	0.793786i	$0.291893\pi$
$522$	0	0
$523$	1.87780e6i	0.300189i	0.988672	+	0.150095i	$0.0479578\pi$
$523$	1.87780e6i	0.300189i	−0.988672	+	0.150095i	$0.952042\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4.65924e6i	0.730784i
$528$	0	0
$529$	−9.72424e6	−1.51083
$530$	0	0
$531$	−1.55629e6	−0.239527
$532$	0	0
$533$	− 6.35559e6i	− 0.969031i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	6.42542e6i	0.961537i
$538$	0	0
$539$	−1.14151e6	−0.169242
$540$	0	0
$541$	2.44482e6	0.359131	0.179566	−	0.983746i	$-0.442531\pi$
$541$	2.44482e6	0.359131	0.179566	+	0.983746i	$0.442531\pi$
$542$	0	0
$543$	9.68202e6i	1.40918i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	1.09485e7i	1.56453i	0.622944	+	0.782266i	$0.285936\pi$
$547$	1.09485e7i	1.56453i	−0.622944	+	0.782266i	$0.714064\pi$
$548$	0	0
$549$	−1.17001e6	−0.165675
$550$	0	0
$551$	−8.45645e6	−1.18661
$552$	0	0
$553$	8.46595e6i	1.17723i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 768938.i	− 0.105015i	−0.998621	−	0.0525077i	$-0.983279\pi$
$557$	− 768938.i	− 0.105015i	0.998621	−	0.0525077i	$-0.0167214\pi$
$558$	0	0
$559$	481000.	0.0651052
$560$	0	0
$561$	4.76131e6	0.638733
$562$	0	0
$563$	− 1.47386e7i	− 1.95968i	−0.199792	−	0.979838i	$-0.564026\pi$
$563$	− 1.47386e7i	− 1.95968i	0.199792	−	0.979838i	$-0.435974\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	6.26749e6i	0.818721i
$568$	0	0
$569$	3.33696e6	0.432086	0.216043	−	0.976384i	$-0.430685\pi$
$569$	3.33696e6	0.432086	0.216043	+	0.976384i	$0.430685\pi$
$570$	0	0
$571$	−1.54413e7	−1.98195	−0.990974	−	0.134051i	$-0.957201\pi$
$571$	−1.54413e7	−1.98195	−0.990974	+	0.134051i	$0.957201\pi$
$572$	0	0
$573$	− 2.38123e6i	− 0.302981i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 4.05636e6i	− 0.507221i	−0.967306	−	0.253610i	$-0.918382\pi$
$577$	− 4.05636e6i	− 0.507221i	0.967306	−	0.253610i	$-0.0816181\pi$
$578$	0	0
$579$	4.46762e6	0.553834
$580$	0	0
$581$	3.59746e6	0.442136
$582$	0	0
$583$	1.37389e6i	0.167410i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1.22567e7i	1.46818i	0.679052	+	0.734090i	$0.262391\pi$
$587$	1.22567e7i	1.46818i	−0.679052	+	0.734090i	$0.737609\pi$
$588$	0	0
$589$	−3.09759e6	−0.367905
$590$	0	0
$591$	−237186.	−0.0279332
$592$	0	0
$593$	− 8.58993e6i	− 1.00312i	−0.865123	−	0.501560i	$-0.832760\pi$
$593$	− 8.58993e6i	− 1.00312i	0.865123	−	0.501560i	$-0.167240\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	2.82825e6i	0.324775i
$598$	0	0
$599$	8.85849e6	1.00877	0.504386	−	0.863479i	$-0.331719\pi$
$599$	8.85849e6	1.00877	0.504386	+	0.863479i	$0.331719\pi$
$600$	0	0
$601$	−8.50579e6	−0.960569	−0.480284	−	0.877113i	$-0.659467\pi$
$601$	−8.50579e6	−0.960569	−0.480284	+	0.877113i	$0.659467\pi$
$602$	0	0
$603$	− 1.10590e6i	− 0.123857i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	3.15481e6i	0.347537i	0.984787	+	0.173768i	$0.0555944\pi$
$607$	3.15481e6i	0.347537i	−0.984787	+	0.173768i	$0.944406\pi$
$608$	0	0
$609$	1.00990e7	1.10341
$610$	0	0
$611$	9.08453e6	0.984463
$612$	0	0
$613$	− 3.20689e6i	− 0.344694i	−0.985036	−	0.172347i	$-0.944865\pi$
$613$	− 3.20689e6i	− 0.344694i	0.985036	−	0.172347i	$-0.0551350\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 1.31916e6i	− 0.139504i	−0.997564	−	0.0697519i	$-0.977779\pi$
$617$	− 1.31916e6i	− 0.139504i	0.997564	−	0.0697519i	$-0.0222207\pi$
$618$	0	0
$619$	−3.36061e6	−0.352526	−0.176263	−	0.984343i	$-0.556401\pi$
$619$	−3.36061e6	−0.352526	−0.176263	+	0.984343i	$0.556401\pi$
$620$	0	0
$621$	1.38821e7	1.44453
$622$	0	0
$623$	540939.i	0.0558377i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	3.16545e6i	0.321563i
$628$	0	0
$629$	1.56123e7	1.57340
$630$	0	0
$631$	1.46547e7	1.46522	0.732610	−	0.680648i	$-0.238302\pi$
$631$	1.46547e7	1.46522	0.732610	+	0.680648i	$0.238302\pi$
$632$	0	0
$633$	6.81315e6i	0.675832i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 3.34743e6i	− 0.326861i
$638$	0	0
$639$	1.31109e6	0.127023
$640$	0	0
$641$	−5.67545e6	−0.545576	−0.272788	−	0.962074i	$-0.587946\pi$
$641$	−5.67545e6	−0.545576	−0.272788	+	0.962074i	$0.587946\pi$
$642$	0	0
$643$	− 1.81422e7i	− 1.73047i	−0.501370	−	0.865233i	$-0.667170\pi$
$643$	− 1.81422e7i	− 1.73047i	0.501370	−	0.865233i	$-0.332830\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 472390.i	− 0.0443649i	−0.999754	−	0.0221825i	$-0.992939\pi$
$647$	− 472390.i	− 0.0443649i	0.999754	−	0.0221825i	$-0.00706148\pi$
$648$	0	0
$649$	−6.14210e6	−0.572407
$650$	0	0
$651$	3.69927e6	0.342108
$652$	0	0
$653$	− 9.79442e6i	− 0.898867i	−0.893314	−	0.449434i	$-0.851626\pi$
$653$	− 9.79442e6i	− 0.898867i	0.893314	−	0.449434i	$-0.148374\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 2.30993e6i	− 0.208778i
$658$	0	0
$659$	−9.83251e6	−0.881964	−0.440982	−	0.897516i	$-0.645370\pi$
$659$	−9.83251e6	−0.881964	−0.440982	+	0.897516i	$0.645370\pi$
$660$	0	0
$661$	1.84120e6	0.163907	0.0819535	−	0.996636i	$-0.473884\pi$
$661$	1.84120e6	0.163907	0.0819535	+	0.996636i	$0.473884\pi$
$662$	0	0
$663$	1.39624e7i	1.23360i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 2.57976e7i	− 2.24525i
$668$	0	0
$669$	1.73413e6	0.149802
$670$	0	0
$671$	−4.61758e6	−0.395920
$672$	0	0
$673$	1.02990e7i	0.876510i	0.898851	+	0.438255i	$0.144403\pi$
$673$	1.02990e7i	0.876510i	−0.898851	+	0.438255i	$0.855597\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 3.63461e6i	− 0.304779i	−0.988320	−	0.152390i	$-0.951303\pi$
$677$	− 3.63461e6i	− 0.304779i	0.988320	−	0.152390i	$-0.0486969\pi$
$678$	0	0
$679$	1.68316e7	1.40104
$680$	0	0
$681$	−1.34348e7	−1.11010
$682$	0	0
$683$	− 9.78443e6i	− 0.802571i	−0.915953	−	0.401286i	$-0.868563\pi$
$683$	− 9.78443e6i	− 0.802571i	0.915953	−	0.401286i	$-0.131437\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 2.27257e7i	− 1.83707i
$688$	0	0
$689$	−4.02889e6	−0.323323
$690$	0	0
$691$	−4.67949e6	−0.372824	−0.186412	−	0.982472i	$-0.559686\pi$
$691$	−4.67949e6	−0.372824	−0.186412	+	0.982472i	$0.559686\pi$
$692$	0	0
$693$	− 492641.i	− 0.0389671i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	2.98935e7i	2.33075i
$698$	0	0
$699$	−1.04110e7	−0.805930
$700$	0	0
$701$	−565147.	−0.0434376	−0.0217188	−	0.999764i	$-0.506914\pi$
$701$	−565147.	−0.0434376	−0.0217188	+	0.999764i	$0.506914\pi$
$702$	0	0
$703$	1.03795e7i	0.792112i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 1.43314e7i	− 1.07830i
$708$	0	0
$709$	1.00907e6	0.0753889	0.0376944	−	0.999289i	$-0.487999\pi$
$709$	1.00907e6	0.0753889	0.0376944	+	0.999289i	$0.487999\pi$
$710$	0	0
$711$	3.27431e6	0.242910
$712$	0	0
$713$	− 9.44964e6i	− 0.696132i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 1.07455e7i	− 0.780602i
$718$	0	0
$719$	−6.02329e6	−0.434522	−0.217261	−	0.976114i	$-0.569712\pi$
$719$	−6.02329e6	−0.434522	−0.217261	+	0.976114i	$0.569712\pi$
$720$	0	0
$721$	−3.37329e6	−0.241666
$722$	0	0
$723$	1.86904e7i	1.32976i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	2.30828e6i	0.161977i	0.996715	+	0.0809883i	$0.0258076\pi$
$727$	2.30828e6i	0.161977i	−0.996715	+	0.0809883i	$0.974192\pi$
$728$	0	0
$729$	−1.16027e7	−0.808612
$730$	0	0
$731$	−2.26238e6	−0.156593
$732$	0	0
$733$	1.65224e7i	1.13583i	0.823088	+	0.567914i	$0.192249\pi$
$733$	1.65224e7i	1.13583i	−0.823088	+	0.567914i	$0.807751\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 4.36456e6i	− 0.295986i
$738$	0	0
$739$	2.41051e7	1.62367	0.811834	−	0.583888i	$-0.198469\pi$
$739$	2.41051e7	1.62367	0.811834	+	0.583888i	$0.198469\pi$
$740$	0	0
$741$	−9.28255e6	−0.621043
$742$	0	0
$743$	− 2.54593e7i	− 1.69190i	−0.533262	−	0.845950i	$-0.679034\pi$
$743$	− 2.54593e7i	− 1.69190i	0.533262	−	0.845950i	$-0.320966\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 1.39136e6i	− 0.0912301i
$748$	0	0
$749$	−6.59312e6	−0.429424
$750$	0	0
$751$	1.25073e7	0.809215	0.404608	−	0.914490i	$-0.367408\pi$
$751$	1.25073e7	0.809215	0.404608	+	0.914490i	$0.367408\pi$
$752$	0	0
$753$	− 1.22203e7i	− 0.785404i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 5.34914e6i	− 0.339269i	−0.985507	−	0.169635i	$-0.945741\pi$
$757$	− 5.34914e6i	− 0.339269i	0.985507	−	0.169635i	$-0.0542588\pi$
$758$	0	0
$759$	−9.65666e6	−0.608447
$760$	0	0
$761$	−1.80369e7	−1.12901	−0.564507	−	0.825428i	$-0.690934\pi$
$761$	−1.80369e7	−1.12901	−0.564507	+	0.825428i	$0.690934\pi$
$762$	0	0
$763$	3.62154e6i	0.225207i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 1.80115e7i	− 1.10550i
$768$	0	0
$769$	−2.41928e7	−1.47526	−0.737632	−	0.675203i	$-0.764056\pi$
$769$	−2.41928e7	−1.47526	−0.737632	+	0.675203i	$0.764056\pi$
$770$	0	0
$771$	2.95234e7	1.78867
$772$	0	0
$773$	3.22531e6i	0.194143i	0.995277	+	0.0970716i	$0.0309476\pi$
$773$	3.22531e6i	0.194143i	−0.995277	+	0.0970716i	$0.969052\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 1.23956e7i	− 0.736570i
$778$	0	0
$779$	−1.98740e7	−1.17339
$780$	0	0
$781$	5.17439e6	0.303551
$782$	0	0
$783$	2.21604e7i	1.29174i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	1.12227e6i	0.0645896i	0.999478	+	0.0322948i	$0.0102815\pi$
$787$	1.12227e6i	0.0645896i	−0.999478	+	0.0322948i	$0.989718\pi$
$788$	0	0
$789$	−1.68705e7	−0.964793
$790$	0	0
$791$	−3.76132e6	−0.213747
$792$	0	0
$793$	− 1.35408e7i	− 0.764650i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 1.48763e7i	− 0.829560i	−0.909922	−	0.414780i	$-0.863859\pi$
$797$	− 1.48763e7i	− 0.829560i	0.909922	−	0.414780i	$-0.136141\pi$
$798$	0	0
$799$	−4.27291e7	−2.36786
$800$	0	0
$801$	209214.	0.0115215
$802$	0	0
$803$	− 9.11644e6i	− 0.498926i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	1.29660e7i	0.700845i
$808$	0	0
$809$	−1.86436e7	−1.00152	−0.500759	−	0.865587i	$-0.666946\pi$
$809$	−1.86436e7	−1.00152	−0.500759	+	0.865587i	$0.666946\pi$
$810$	0	0
$811$	−4.50701e6	−0.240623	−0.120311	−	0.992736i	$-0.538389\pi$
$811$	−4.50701e6	−0.240623	−0.120311	+	0.992736i	$0.538389\pi$
$812$	0	0
$813$	− 2.02920e7i	− 1.07671i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 1.50409e6i	− 0.0788352i
$818$	0	0
$819$	1.44465e6	0.0752580
$820$	0	0
$821$	−2.26560e7	−1.17307	−0.586537	−	0.809922i	$-0.699509\pi$
$821$	−2.26560e7	−1.17307	−0.586537	+	0.809922i	$0.699509\pi$
$822$	0	0
$823$	− 2.76806e7i	− 1.42454i	−0.701903	−	0.712272i	$-0.747666\pi$
$823$	− 2.76806e7i	− 1.42454i	0.701903	−	0.712272i	$-0.252334\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	7.42673e6i	0.377602i	0.982015	+	0.188801i	$0.0604600\pi$
$827$	7.42673e6i	0.377602i	−0.982015	+	0.188801i	$0.939540\pi$
$828$	0	0
$829$	−3.41770e7	−1.72722	−0.863610	−	0.504160i	$-0.831802\pi$
$829$	−3.41770e7	−1.72722	−0.863610	+	0.504160i	$0.831802\pi$
$830$	0	0
$831$	3.65244e6	0.183477
$832$	0	0
$833$	1.57446e7i	0.786177i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	8.11734e6i	0.400498i
$838$	0	0
$839$	155990.	0.00765051	0.00382526	−	0.999993i	$-0.498782\pi$
$839$	155990.	0.00765051	0.00382526	+	0.999993i	$0.498782\pi$
$840$	0	0
$841$	2.06703e7	1.00776
$842$	0	0
$843$	− 5.28580e6i	− 0.256178i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	1.32182e7i	0.633087i
$848$	0	0
$849$	1.55110e7	0.738534
$850$	0	0
$851$	−3.16640e7	−1.49879
$852$	0	0
$853$	2.52778e7i	1.18951i	0.803908	+	0.594754i	$0.202750\pi$
$853$	2.52778e7i	1.18951i	−0.803908	+	0.594754i	$0.797250\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1.40854e6i	0.0655112i	0.999463	+	0.0327556i	$0.0104283\pi$
$857$	1.40854e6i	0.0655112i	−0.999463	+	0.0327556i	$0.989572\pi$
$858$	0	0
$859$	−246158.	−0.0113823	−0.00569116	−	0.999984i	$-0.501812\pi$
$859$	−246158.	−0.0113823	−0.00569116	+	0.999984i	$0.501812\pi$
$860$	0	0
$861$	2.37344e7	1.09111
$862$	0	0
$863$	1.17019e7i	0.534849i	0.963579	+	0.267424i	$0.0861725\pi$
$863$	1.17019e7i	0.534849i	−0.963579	+	0.267424i	$0.913827\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 4.19381e7i	− 1.89479i
$868$	0	0
$869$	1.29225e7	0.580492
$870$	0	0
$871$	1.27989e7	0.571646
$872$	0	0
$873$	− 6.50982e6i	− 0.289090i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 7.85174e6i	− 0.344720i	−0.985034	−	0.172360i	$-0.944861\pi$
$877$	− 7.85174e6i	− 0.344720i	0.985034	−	0.172360i	$-0.0551393\pi$
$878$	0	0
$879$	4.39607e6	0.191908
$880$	0	0
$881$	105461.	0.00457777	0.00228888	−	0.999997i	$-0.499271\pi$
$881$	105461.	0.00457777	0.00228888	+	0.999997i	$0.499271\pi$
$882$	0	0
$883$	1.43760e7i	0.620491i	0.950656	+	0.310245i	$0.100411\pi$
$883$	1.43760e7i	0.620491i	−0.950656	+	0.310245i	$0.899589\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 2.71725e7i	− 1.15963i	−0.814748	−	0.579816i	$-0.803125\pi$
$887$	− 2.71725e7i	− 1.15963i	0.814748	−	0.579816i	$-0.196875\pi$
$888$	0	0
$889$	2.19587e7	0.931864
$890$	0	0
$891$	9.56672e6	0.403709
$892$	0	0
$893$	− 2.84075e7i	− 1.19208i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 2.83177e7i	− 1.17511i
$898$	0	0
$899$	1.50847e7	0.622499
$900$	0	0
$901$	1.89499e7	0.777668
$902$	0	0
$903$	1.79625e6i	0.0733073i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	5.25290e6i	0.212022i	0.994365	+	0.106011i	$0.0338079\pi$
$907$	5.25290e6i	0.212022i	−0.994365	+	0.106011i	$0.966192\pi$
$908$	0	0
$909$	−5.54282e6	−0.222495
$910$	0	0
$911$	−2.68057e7	−1.07012	−0.535059	−	0.844815i	$-0.679711\pi$
$911$	−2.68057e7	−1.07012	−0.535059	+	0.844815i	$0.679711\pi$
$912$	0	0
$913$	− 5.49118e6i	− 0.218016i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	5.20039e6i	0.204227i
$918$	0	0
$919$	−1.25954e7	−0.491951	−0.245976	−	0.969276i	$-0.579108\pi$
$919$	−1.25954e7	−0.491951	−0.245976	+	0.969276i	$0.579108\pi$
$920$	0	0
$921$	3.60563e7	1.40066
$922$	0	0
$923$	1.51737e7i	0.586255i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	1.30466e6i	0.0498652i
$928$	0	0
$929$	3.23322e7	1.22912	0.614562	−	0.788868i	$-0.289333\pi$
$929$	3.23322e7	1.22912	0.614562	+	0.788868i	$0.289333\pi$
$930$	0	0
$931$	−1.04675e7	−0.395792
$932$	0	0
$933$	3.89604e7i	1.46528i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	2.04882e7i	0.762350i	0.924503	+	0.381175i	$0.124481\pi$
$937$	2.04882e7i	0.762350i	−0.924503	+	0.381175i	$0.875519\pi$
$938$	0	0
$939$	−1.87607e7	−0.694361
$940$	0	0
$941$	−5.87135e6	−0.216154	−0.108077	−	0.994143i	$-0.534469\pi$
$941$	−5.87135e6	−0.216154	−0.108077	+	0.994143i	$0.534469\pi$
$942$	0	0
$943$	− 6.06285e7i	− 2.22023i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.86718e7i	0.676567i	0.941044	+	0.338284i	$0.109846\pi$
$947$	1.86718e7i	0.676567i	−0.941044	+	0.338284i	$0.890154\pi$
$948$	0	0
$949$	2.67336e7	0.963588
$950$	0	0
$951$	2.13885e7	0.766882
$952$	0	0
$953$	− 1.55896e7i	− 0.556037i	−0.960576	−	0.278018i	$-0.910322\pi$
$953$	− 1.55896e7i	− 0.556037i	0.960576	−	0.278018i	$-0.0896776\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 1.54152e7i	− 0.544088i
$958$	0	0
$959$	2.46213e7	0.864500
$960$	0	0
$961$	−2.31036e7	−0.806996
$962$	0	0
$963$	2.54997e6i	0.0886071i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	3.24491e7i	1.11593i	0.829865	+	0.557964i	$0.188417\pi$
$967$	3.24491e7i	1.11593i	−0.829865	+	0.557964i	$0.811583\pi$
$968$	0	0
$969$	4.36605e7	1.49375
$970$	0	0
$971$	−3.73661e7	−1.27183	−0.635916	−	0.771758i	$-0.719378\pi$
$971$	−3.73661e7	−1.27183	−0.635916	+	0.771758i	$0.719378\pi$
$972$	0	0
$973$	− 2.76195e7i	− 0.935261i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 5.23874e7i	− 1.75586i	−0.478787	−	0.877931i	$-0.658923\pi$
$977$	− 5.23874e7i	− 1.75586i	0.478787	−	0.877931i	$-0.341077\pi$
$978$	0	0
$979$	825691.	0.0275335
$980$	0	0
$981$	1.40067e6	0.0464690
$982$	0	0
$983$	− 4.75140e7i	− 1.56833i	−0.620551	−	0.784166i	$-0.713091\pi$
$983$	− 4.75140e7i	− 1.56833i	0.620551	−	0.784166i	$-0.286909\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	3.39253e7i	1.10849i
$988$	0	0
$989$	4.58846e6	0.149168
$990$	0	0
$991$	−2.47278e7	−0.799835	−0.399918	−	0.916551i	$-0.630961\pi$
$991$	−2.47278e7	−0.799835	−0.399918	+	0.916551i	$0.630961\pi$
$992$	0	0
$993$	3.50311e7i	1.12741i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	1.25139e7i	0.398709i	0.979927	+	0.199355i	$0.0638845\pi$
$997$	1.25139e7i	0.398709i	−0.979927	+	0.199355i	$0.936115\pi$
$998$	0	0
$999$	2.71997e7	0.862285

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.6.c.l.49.3		4
4.3	odd	2		200.6.c.e.49.2		4
5.2	odd	4		400.6.a.q.1.2		2
5.3	odd	4		80.6.a.i.1.1		2
5.4	even	2	inner	400.6.c.l.49.2		4
15.8	even	4		720.6.a.z.1.1		2
20.3	even	4		40.6.a.d.1.2	✓	2
20.7	even	4		200.6.a.g.1.1		2
20.19	odd	2		200.6.c.e.49.3		4
40.3	even	4		320.6.a.w.1.1		2
40.13	odd	4		320.6.a.q.1.2		2
60.23	odd	4		360.6.a.l.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.6.a.d.1.2	✓	2	20.3	even	4
80.6.a.i.1.1		2	5.3	odd	4
200.6.a.g.1.1		2	20.7	even	4
200.6.c.e.49.2		4	4.3	odd	2
200.6.c.e.49.3		4	20.19	odd	2
320.6.a.q.1.2		2	40.13	odd	4
320.6.a.w.1.1		2	40.3	even	4
360.6.a.l.1.2		2	60.23	odd	4
400.6.a.q.1.2		2	5.2	odd	4
400.6.c.l.49.2		4	5.4	even	2	inner
400.6.c.l.49.3		4	1.1	even	1	trivial
720.6.a.z.1.1		2	15.8	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	400.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+16.7156i q^{3} -94.1469i q^{7} -36.4124 q^{9} +O(q^{10})\)	\(q+16.7156i q^{3} -94.1469i q^{7} -36.4124 q^{9} -143.706 q^{11} -421.412i q^{13} +1982.11i q^{17} -1317.76 q^{19} +1573.73 q^{21} -4020.02i q^{23} +3453.24i q^{27} +6417.28 q^{29} +2350.64 q^{31} -2402.14i q^{33} -7876.58i q^{37} +7044.18 q^{39} +15081.6 q^{41} +1141.40i q^{43} +21557.3i q^{47} +7943.36 q^{49} -33132.3 q^{51} -9560.44i q^{53} -22027.2i q^{57} +42740.7 q^{59} +32132.1 q^{61} +3428.11i q^{63} +30371.4i q^{67} +67197.2 q^{69} -36006.7 q^{71} +63438.0i q^{73} +13529.5i q^{77} -89922.8 q^{79} -66571.4 q^{81} +38211.2i q^{83} +107269. i q^{87} -5745.69 q^{89} -39674.7 q^{91} +39292.5i q^{93} +178780. i q^{97} +5232.69 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 1236 q^{9}+O(q^{10}) \) 4 * q - 1236 * q^9	\( 4 q - 1236 q^{9} - 1120 q^{11} - 2000 q^{19} + 5568 q^{21} - 2680 q^{29} + 4496 q^{31} - 41424 q^{39} + 46152 q^{41} + 45948 q^{49} - 171600 q^{51} + 125168 q^{59} + 28216 q^{61} + 224448 q^{69} - 94416 q^{71} - 131808 q^{79} + 142596 q^{81} + 110040 q^{89} + 2128 q^{91} + 494688 q^{99}+O(q^{100}) \) 4 * q - 1236 * q^9 - 1120 * q^11 - 2000 * q^19 + 5568 * q^21 - 2680 * q^29 + 4496 * q^31 - 41424 * q^39 + 46152 * q^41 + 45948 * q^49 - 171600 * q^51 + 125168 * q^59 + 28216 * q^61 + 224448 * q^69 - 94416 * q^71 - 131808 * q^79 + 142596 * q^81 + 110040 * q^89 + 2128 * q^91 + 494688 * q^99

Introduction

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