LMFDB - Embedded newform 200.12.c.a.49.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [200,12,Mod(49,200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(200, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 12, names="a")

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

chi := DirichletCharacter("200.49");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 200 = 2^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	200.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:	$153.668636112$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	200.49
Dual form			200.12.c.a.49.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+520.000i q^{3} -15148.0i q^{7} -93253.0 q^{9} +369324. q^{11} +877926. i q^{13} -3.28871e6i q^{17} -1.56080e6 q^{19} +7.87696e6 q^{21} -1.28994e7i q^{23} +4.36249e7i q^{27} -4.63225e7 q^{29} -5.94275e7 q^{31} +1.92048e8i q^{33} -6.66072e8i q^{37} -4.56522e8 q^{39} +6.74573e8 q^{41} +1.18838e9i q^{43} -2.26148e9i q^{47} +1.74786e9 q^{49} +1.71013e9 q^{51} -3.09924e9i q^{53} -8.11614e8i q^{57} +1.38382e9 q^{59} -1.23091e9 q^{61} +1.41260e9i q^{63} -1.75593e10i q^{67} +6.70767e9 q^{69} -2.78727e9 q^{71} -2.12373e10i q^{73} -5.59452e9i q^{77} +4.30659e10 q^{79} -3.92044e10 q^{81} -2.15524e10i q^{83} -2.40877e10i q^{87} +9.21446e10 q^{89} +1.32988e10 q^{91} -3.09023e10i q^{93} +8.04846e10i q^{97} -3.44406e10 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 186506 q^{9} + 738648 q^{11} - 3121592 q^{19} + 15753920 q^{21} - 92645004 q^{29} - 118855056 q^{31} - 913043040 q^{39} + 1349146388 q^{41} + 3495729678 q^{49} + 3420254240 q^{51} + 2767639992 q^{59}+ \cdots - 68881141944 q^{99}+O(q^{100}) $ 2 * q - 186506 * q^9 + 738648 * q^11 - 3121592 * q^19 + 15753920 * q^21 - 92645004 * q^29 - 118855056 * q^31 - 913043040 * q^39 + 1349146388 * q^41 + 3495729678 * q^49 + 3420254240 * q^51 + 2767639992 * q^59 - 2461813044 * q^61 + 13415346880 * q^69 - 5574545760 * q^71 + 86131783360 * q^79 - 78408853582 * q^81 + 184289105708 * q^89 + 26597646096 * q^91 - 68881141944 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/200\mathbb{Z}\right)^\times$.

$n$	$101$	$151$	$177$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	520.000i	1.23548i	0.786382	+	0.617741i	$0.211952\pi$
$3$	520.000i	1.23548i	−0.786382	+	0.617741i	$0.788048\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 15148.0i	− 0.340656i	−0.985387	−	0.170328i	$-0.945517\pi$
$7$	− 15148.0i	− 0.340656i	0.985387	−	0.170328i	$-0.0544827\pi$
$8$	0	0
$9$	−93253.0	−0.526416
$10$	0	0
$11$	369324.	0.691429	0.345715	−	0.938340i	$-0.387637\pi$
$11$	369324.	0.691429	0.345715	+	0.938340i	$0.387637\pi$
$12$	0	0
$13$	877926.i	0.655797i	0.944713	+	0.327899i	$0.106340\pi$
$13$	877926.i	0.655797i	−0.944713	+	0.327899i	$0.893660\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 3.28871e6i	− 0.561767i	−0.959742	−	0.280883i	$-0.909373\pi$
$17$	− 3.28871e6i	− 0.561767i	0.959742	−	0.280883i	$-0.0906274\pi$
$18$	0	0
$19$	−1.56080e6	−0.144611	−0.0723055	−	0.997383i	$-0.523036\pi$
$19$	−1.56080e6	−0.144611	−0.0723055	+	0.997383i	$0.523036\pi$
$20$	0	0
$21$	7.87696e6	0.420874
$22$	0	0
$23$	− 1.28994e7i	− 0.417893i	−0.977927	−	0.208947i	$-0.932996\pi$
$23$	− 1.28994e7i	− 0.417893i	0.977927	−	0.208947i	$-0.0670035\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	4.36249e7i	0.585105i
$28$	0	0
$29$	−4.63225e7	−0.419375	−0.209688	−	0.977768i	$-0.567245\pi$
$29$	−4.63225e7	−0.419375	−0.209688	+	0.977768i	$0.567245\pi$
$30$	0	0
$31$	−5.94275e7	−0.372819	−0.186410	−	0.982472i	$-0.559685\pi$
$31$	−5.94275e7	−0.372819	−0.186410	+	0.982472i	$0.559685\pi$
$32$	0	0
$33$	1.92048e8i	0.854248i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 6.66072e8i	− 1.57911i	−0.613681	−	0.789554i	$-0.710312\pi$
$37$	− 6.66072e8i	− 1.57911i	0.613681	−	0.789554i	$-0.289688\pi$
$38$	0	0
$39$	−4.56522e8	−0.810226
$40$	0	0
$41$	6.74573e8	0.909322	0.454661	−	0.890664i	$-0.349760\pi$
$41$	6.74573e8	0.909322	0.454661	+	0.890664i	$0.349760\pi$
$42$	0	0
$43$	1.18838e9i	1.23276i	0.787448	+	0.616381i	$0.211402\pi$
$43$	1.18838e9i	1.23276i	−0.787448	+	0.616381i	$0.788598\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 2.26148e9i	− 1.43832i	−0.694846	−	0.719159i	$-0.744527\pi$
$47$	− 2.26148e9i	− 1.43832i	0.694846	−	0.719159i	$-0.255473\pi$
$48$	0	0
$49$	1.74786e9	0.883953
$50$	0	0
$51$	1.71013e9	0.694053
$52$	0	0
$53$	− 3.09924e9i	− 1.01798i	−0.860773	−	0.508988i	$-0.830020\pi$
$53$	− 3.09924e9i	− 1.01798i	0.860773	−	0.508988i	$-0.169980\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 8.11614e8i	− 0.178664i
$58$	0	0
$59$	1.38382e9	0.251996	0.125998	−	0.992030i	$-0.459787\pi$
$59$	1.38382e9	0.251996	0.125998	+	0.992030i	$0.459787\pi$
$60$	0	0
$61$	−1.23091e9	−0.186600	−0.0932999	−	0.995638i	$-0.529742\pi$
$61$	−1.23091e9	−0.186600	−0.0932999	+	0.995638i	$0.529742\pi$
$62$	0	0
$63$	1.41260e9i	0.179327i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 1.75593e10i	− 1.58890i	−0.607332	−	0.794448i	$-0.707760\pi$
$67$	− 1.75593e10i	− 1.58890i	0.607332	−	0.794448i	$-0.292240\pi$
$68$	0	0
$69$	6.70767e9	0.516300
$70$	0	0
$71$	−2.78727e9	−0.183341	−0.0916703	−	0.995789i	$-0.529221\pi$
$71$	−2.78727e9	−0.183341	−0.0916703	+	0.995789i	$0.529221\pi$
$72$	0	0
$73$	− 2.12373e10i	− 1.19901i	−0.800370	−	0.599506i	$-0.795364\pi$
$73$	− 2.12373e10i	− 1.19901i	0.800370	−	0.599506i	$-0.204636\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 5.59452e9i	− 0.235540i
$78$	0	0
$79$	4.30659e10	1.57465	0.787326	−	0.616537i	$-0.211465\pi$
$79$	4.30659e10	1.57465	0.787326	+	0.616537i	$0.211465\pi$
$80$	0	0
$81$	−3.92044e10	−1.24930
$82$	0	0
$83$	− 2.15524e10i	− 0.600574i	−0.953849	−	0.300287i	$-0.902918\pi$
$83$	− 2.15524e10i	− 0.600574i	0.953849	−	0.300287i	$-0.0970824\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 2.40877e10i	− 0.518131i
$88$	0	0
$89$	9.21446e10	1.74914	0.874570	−	0.484899i	$-0.161144\pi$
$89$	9.21446e10	1.74914	0.874570	+	0.484899i	$0.161144\pi$
$90$	0	0
$91$	1.32988e10	0.223401
$92$	0	0
$93$	− 3.09023e10i	− 0.460612i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	8.04846e10i	0.951630i	0.879545	+	0.475815i	$0.157847\pi$
$97$	8.04846e10i	0.951630i	−0.879545	+	0.475815i	$0.842153\pi$
$98$	0	0
$99$	−3.44406e10	−0.363979
$100$	0	0
$101$	−2.70977e9	−0.0256545	−0.0128273	−	0.999918i	$-0.504083\pi$
$101$	−2.70977e9	−0.0256545	−0.0128273	+	0.999918i	$0.504083\pi$
$102$	0	0
$103$	1.46761e11i	1.24740i	0.781664	+	0.623700i	$0.214371\pi$
$103$	1.46761e11i	1.24740i	−0.781664	+	0.623700i	$0.785629\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 8.43362e10i	− 0.581304i	−0.956829	−	0.290652i	$-0.906128\pi$
$107$	− 8.43362e10i	− 0.581304i	0.956829	−	0.290652i	$-0.0938722\pi$
$108$	0	0
$109$	−1.78870e11	−1.11351	−0.556754	−	0.830678i	$-0.687953\pi$
$109$	−1.78870e11	−1.11351	−0.556754	+	0.830678i	$0.687953\pi$
$110$	0	0
$111$	3.46358e11	1.95096
$112$	0	0
$113$	− 1.97874e11i	− 1.01032i	−0.863026	−	0.505159i	$-0.831434\pi$
$113$	− 1.97874e11i	− 1.01032i	0.863026	−	0.505159i	$-0.168566\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 8.18692e10i	− 0.345222i
$118$	0	0
$119$	−4.98173e10	−0.191369
$120$	0	0
$121$	−1.48911e11	−0.521926
$122$	0	0
$123$	3.50778e11i	1.12345i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	4.77982e11i	1.28378i	0.766796	+	0.641891i	$0.221850\pi$
$127$	4.77982e11i	1.28378i	−0.766796	+	0.641891i	$0.778150\pi$
$128$	0	0
$129$	−6.17959e11	−1.52306
$130$	0	0
$131$	8.74294e10	0.198000	0.0990001	−	0.995087i	$-0.468436\pi$
$131$	8.74294e10	0.198000	0.0990001	+	0.995087i	$0.468436\pi$
$132$	0	0
$133$	2.36429e10i	0.0492626i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	7.55489e11i	1.33741i	0.743527	+	0.668705i	$0.233151\pi$
$137$	7.55489e11i	1.33741i	−0.743527	+	0.668705i	$0.766849\pi$
$138$	0	0
$139$	7.97051e11	1.30288	0.651440	−	0.758700i	$-0.274165\pi$
$139$	7.97051e11	1.30288	0.651440	+	0.758700i	$0.274165\pi$
$140$	0	0
$141$	1.17597e12	1.77702
$142$	0	0
$143$	3.24239e11i	0.453437i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	9.08890e11i	1.09211i
$148$	0	0
$149$	−1.28254e12	−1.43069	−0.715347	−	0.698770i	$-0.753731\pi$
$149$	−1.28254e12	−1.43069	−0.715347	+	0.698770i	$0.753731\pi$
$150$	0	0
$151$	2.19927e11	0.227985	0.113992	−	0.993482i	$-0.463636\pi$
$151$	2.19927e11	0.227985	0.113992	+	0.993482i	$0.463636\pi$
$152$	0	0
$153$	3.06682e11i	0.295723i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	1.16452e12i	0.974316i	0.873314	+	0.487158i	$0.161966\pi$
$157$	1.16452e12i	0.974316i	−0.873314	+	0.487158i	$0.838034\pi$
$158$	0	0
$159$	1.61160e12	1.25769
$160$	0	0
$161$	−1.95400e11	−0.142358
$162$	0	0
$163$	− 7.69042e11i	− 0.523502i	−0.965135	−	0.261751i	$-0.915700\pi$
$163$	− 7.69042e11i	− 0.523502i	0.965135	−	0.261751i	$-0.0842999\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.29682e12i	0.772572i	0.922379	+	0.386286i	$0.126242\pi$
$167$	1.29682e12i	0.772572i	−0.922379	+	0.386286i	$0.873758\pi$
$168$	0	0
$169$	1.02141e12	0.569930
$170$	0	0
$171$	1.45549e11	0.0761255
$172$	0	0
$173$	− 3.03670e12i	− 1.48987i	−0.667138	−	0.744935i	$-0.732481\pi$
$173$	− 3.03670e12i	− 1.48987i	0.667138	−	0.744935i	$-0.267519\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	7.19586e11i	0.311336i
$178$	0	0
$179$	2.78170e12	1.13141	0.565704	−	0.824608i	$-0.308604\pi$
$179$	2.78170e12	1.13141	0.565704	+	0.824608i	$0.308604\pi$
$180$	0	0
$181$	4.29653e11	0.164394	0.0821970	−	0.996616i	$-0.473806\pi$
$181$	4.29653e11	0.164394	0.0821970	+	0.996616i	$0.473806\pi$
$182$	0	0
$183$	− 6.40071e11i	− 0.230541i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 1.21460e12i	− 0.388422i
$188$	0	0
$189$	6.60830e11	0.199319
$190$	0	0
$191$	5.79538e12	1.64967	0.824837	−	0.565370i	$-0.191267\pi$
$191$	5.79538e12	1.64967	0.824837	+	0.565370i	$0.191267\pi$
$192$	0	0
$193$	1.38281e12i	0.371705i	0.982578	+	0.185853i	$0.0595047\pi$
$193$	1.38281e12i	0.371705i	−0.982578	+	0.185853i	$0.940495\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 1.44241e12i	− 0.346358i	−0.984890	−	0.173179i	$-0.944596\pi$
$197$	− 1.44241e12i	− 0.346358i	0.984890	−	0.173179i	$-0.0554039\pi$
$198$	0	0
$199$	1.39119e12	0.316006	0.158003	−	0.987439i	$-0.449494\pi$
$199$	1.39119e12	0.316006	0.158003	+	0.987439i	$0.449494\pi$
$200$	0	0
$201$	9.13083e12	1.96305
$202$	0	0
$203$	7.01693e11i	0.142863i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1.20291e12i	0.219986i
$208$	0	0
$209$	−5.76439e11	−0.0999883
$210$	0	0
$211$	6.37749e12	1.04978	0.524888	−	0.851172i	$-0.324107\pi$
$211$	6.37749e12	1.04978	0.524888	+	0.851172i	$0.324107\pi$
$212$	0	0
$213$	− 1.44938e12i	− 0.226514i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	9.00208e11i	0.127003i
$218$	0	0
$219$	1.10434e13	1.48136
$220$	0	0
$221$	2.88724e12	0.368405
$222$	0	0
$223$	2.78487e12i	0.338165i	0.985602	+	0.169082i	$0.0540804\pi$
$223$	2.78487e12i	0.338165i	−0.985602	+	0.169082i	$0.945920\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1.98933e11i	0.0219061i	0.999940	+	0.0109530i	$0.00348653\pi$
$227$	1.98933e11i	0.0219061i	−0.999940	+	0.0109530i	$0.996513\pi$
$228$	0	0
$229$	1.08817e13	1.14183	0.570917	−	0.821008i	$-0.306588\pi$
$229$	1.08817e13	1.14183	0.570917	+	0.821008i	$0.306588\pi$
$230$	0	0
$231$	2.90915e12	0.291005
$232$	0	0
$233$	3.07714e12i	0.293555i	0.989170	+	0.146777i	$0.0468901\pi$
$233$	3.07714e12i	0.293555i	−0.989170	+	0.146777i	$0.953110\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	2.23943e13i	1.94545i
$238$	0	0
$239$	−1.87849e13	−1.55819	−0.779096	−	0.626905i	$-0.784321\pi$
$239$	−1.87849e13	−1.55819	−0.779096	+	0.626905i	$0.784321\pi$
$240$	0	0
$241$	5.78763e12	0.458571	0.229286	−	0.973359i	$-0.426361\pi$
$241$	5.78763e12	0.458571	0.229286	+	0.973359i	$0.426361\pi$
$242$	0	0
$243$	− 1.26583e13i	− 0.958386i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 1.37026e12i	− 0.0948355i
$248$	0	0
$249$	1.12073e13	0.741998
$250$	0	0
$251$	1.41941e13	0.899297	0.449649	−	0.893206i	$-0.351549\pi$
$251$	1.41941e13	0.899297	0.449649	+	0.893206i	$0.351549\pi$
$252$	0	0
$253$	− 4.76405e12i	− 0.288944i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2.78713e13i	1.55069i	0.631538	+	0.775345i	$0.282424\pi$
$257$	2.78713e13i	1.55069i	−0.631538	+	0.775345i	$0.717576\pi$
$258$	0	0
$259$	−1.00897e13	−0.537933
$260$	0	0
$261$	4.31971e12	0.220766
$262$	0	0
$263$	7.67167e12i	0.375953i	0.982174	+	0.187976i	$0.0601928\pi$
$263$	7.67167e12i	0.375953i	−0.982174	+	0.187976i	$0.939807\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	4.79152e13i	2.16103i
$268$	0	0
$269$	−1.49027e12	−0.0645100	−0.0322550	−	0.999480i	$-0.510269\pi$
$269$	−1.49027e12	−0.0645100	−0.0322550	+	0.999480i	$0.510269\pi$
$270$	0	0
$271$	1.92607e13	0.800461	0.400231	−	0.916414i	$-0.368930\pi$
$271$	1.92607e13	0.800461	0.400231	+	0.916414i	$0.368930\pi$
$272$	0	0
$273$	6.91539e12i	0.276008i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	3.27599e13i	1.20699i	0.797366	+	0.603496i	$0.206226\pi$
$277$	3.27599e13i	1.20699i	−0.797366	+	0.603496i	$0.793774\pi$
$278$	0	0
$279$	5.54180e12	0.196258
$280$	0	0
$281$	2.97179e13	1.01189	0.505945	−	0.862566i	$-0.331144\pi$
$281$	2.97179e13	1.01189	0.505945	+	0.862566i	$0.331144\pi$
$282$	0	0
$283$	− 4.71296e13i	− 1.54336i	−0.636009	−	0.771681i	$-0.719416\pi$
$283$	− 4.71296e13i	− 1.54336i	0.636009	−	0.771681i	$-0.280584\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 1.02184e13i	− 0.309766i
$288$	0	0
$289$	2.34563e13	0.684418
$290$	0	0
$291$	−4.18520e13	−1.17572
$292$	0	0
$293$	5.66352e13i	1.53220i	0.642723	+	0.766099i	$0.277805\pi$
$293$	5.66352e13i	1.53220i	−0.642723	+	0.766099i	$0.722195\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1.61117e13i	0.404558i
$298$	0	0
$299$	1.13247e13	0.274053
$300$	0	0
$301$	1.80016e13	0.419948
$302$	0	0
$303$	− 1.40908e12i	− 0.0316957i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 1.69885e13i	− 0.355545i	−0.984072	−	0.177773i	$-0.943111\pi$
$307$	− 1.69885e13i	− 0.355545i	0.984072	−	0.177773i	$-0.0568891\pi$
$308$	0	0
$309$	−7.63157e13	−1.54114
$310$	0	0
$311$	3.16551e13	0.616967	0.308483	−	0.951230i	$-0.400179\pi$
$311$	3.16551e13	0.616967	0.308483	+	0.951230i	$0.400179\pi$
$312$	0	0
$313$	1.68954e13i	0.317889i	0.987288	+	0.158944i	$0.0508091\pi$
$313$	1.68954e13i	0.317889i	−0.987288	+	0.158944i	$0.949191\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 8.33034e13i	− 1.46163i	−0.682577	−	0.730814i	$-0.739141\pi$
$317$	− 8.33034e13i	− 1.46163i	0.682577	−	0.730814i	$-0.260859\pi$
$318$	0	0
$319$	−1.71080e13	−0.289968
$320$	0	0
$321$	4.38548e13	0.718191
$322$	0	0
$323$	5.13300e12i	0.0812376i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 9.30127e13i	− 1.37572i
$328$	0	0
$329$	−3.42569e13	−0.489972
$330$	0	0
$331$	−8.02352e13	−1.10997	−0.554984	−	0.831861i	$-0.687276\pi$
$331$	−8.02352e13	−1.10997	−0.554984	+	0.831861i	$0.687276\pi$
$332$	0	0
$333$	6.21132e13i	0.831268i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	4.75182e13i	0.595518i	0.954641	+	0.297759i	$0.0962392\pi$
$337$	4.75182e13i	0.595518i	−0.954641	+	0.297759i	$0.903761\pi$
$338$	0	0
$339$	1.02895e14	1.24823
$340$	0	0
$341$	−2.19480e13	−0.257778
$342$	0	0
$343$	− 5.64292e13i	− 0.641780i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 6.60009e13i	− 0.704268i	−0.935950	−	0.352134i	$-0.885456\pi$
$347$	− 6.60009e13i	− 0.704268i	0.935950	−	0.352134i	$-0.114544\pi$
$348$	0	0
$349$	2.58510e12	0.0267262	0.0133631	−	0.999911i	$-0.495746\pi$
$349$	2.58510e12	0.0267262	0.0133631	+	0.999911i	$0.495746\pi$
$350$	0	0
$351$	−3.82994e13	−0.383710
$352$	0	0
$353$	− 7.12598e13i	− 0.691965i	−0.938241	−	0.345982i	$-0.887546\pi$
$353$	− 7.12598e13i	− 0.691965i	0.938241	−	0.345982i	$-0.112454\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 2.59050e13i	− 0.236433i
$358$	0	0
$359$	1.14430e14	1.01279	0.506394	−	0.862302i	$-0.330978\pi$
$359$	1.14430e14	1.01279	0.506394	+	0.862302i	$0.330978\pi$
$360$	0	0
$361$	−1.14054e14	−0.979088
$362$	0	0
$363$	− 7.74340e13i	− 0.644830i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 2.25740e14i	− 1.76988i	−0.465701	−	0.884942i	$-0.654198\pi$
$367$	− 2.25740e14i	− 1.76988i	0.465701	−	0.884942i	$-0.345802\pi$
$368$	0	0
$369$	−6.29060e13	−0.478682
$370$	0	0
$371$	−4.69473e13	−0.346780
$372$	0	0
$373$	− 1.32058e14i	− 0.947037i	−0.880784	−	0.473519i	$-0.842984\pi$
$373$	− 1.32058e14i	− 0.947037i	0.880784	−	0.473519i	$-0.157016\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 4.06677e13i	− 0.275025i
$378$	0	0
$379$	2.71021e14	1.78028	0.890139	−	0.455689i	$-0.150607\pi$
$379$	2.71021e14	1.78028	0.890139	+	0.455689i	$0.150607\pi$
$380$	0	0
$381$	−2.48551e14	−1.58609
$382$	0	0
$383$	1.84806e14i	1.14584i	0.819612	+	0.572918i	$0.194189\pi$
$383$	1.84806e14i	1.14584i	−0.819612	+	0.572918i	$0.805811\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 1.10820e14i	− 0.648946i
$388$	0	0
$389$	−2.84280e14	−1.61817	−0.809085	−	0.587692i	$-0.800037\pi$
$389$	−2.84280e14	−1.61817	−0.809085	+	0.587692i	$0.800037\pi$
$390$	0	0
$391$	−4.24222e13	−0.234759
$392$	0	0
$393$	4.54633e13i	0.244626i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	8.03668e13i	0.409005i	0.978866	+	0.204503i	$0.0655577\pi$
$397$	8.03668e13i	0.409005i	−0.978866	+	0.204503i	$0.934442\pi$
$398$	0	0
$399$	−1.22943e13	−0.0608631
$400$	0	0
$401$	5.04840e13	0.243142	0.121571	−	0.992583i	$-0.461207\pi$
$401$	5.04840e13	0.243142	0.121571	+	0.992583i	$0.461207\pi$
$402$	0	0
$403$	− 5.21730e13i	− 0.244494i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 2.45996e14i	− 1.09184i
$408$	0	0
$409$	−2.45253e14	−1.05959	−0.529793	−	0.848127i	$-0.677730\pi$
$409$	−2.45253e14	−1.05959	−0.529793	+	0.848127i	$0.677730\pi$
$410$	0	0
$411$	−3.92854e14	−1.65235
$412$	0	0
$413$	− 2.09621e13i	− 0.0858439i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	4.14466e14i	1.60969i
$418$	0	0
$419$	−1.18961e14	−0.450015	−0.225007	−	0.974357i	$-0.572241\pi$
$419$	−1.18961e14	−0.450015	−0.225007	+	0.974357i	$0.572241\pi$
$420$	0	0
$421$	−4.75617e14	−1.75269	−0.876346	−	0.481682i	$-0.840026\pi$
$421$	−4.75617e14	−1.75269	−0.876346	+	0.481682i	$0.840026\pi$
$422$	0	0
$423$	2.10890e14i	0.757153i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	1.86458e13i	0.0635663i
$428$	0	0
$429$	−1.68604e14	−0.560214
$430$	0	0
$431$	3.79194e14	1.22811	0.614054	−	0.789264i	$-0.289538\pi$
$431$	3.79194e14	1.22811	0.614054	+	0.789264i	$0.289538\pi$
$432$	0	0
$433$	1.03919e14i	0.328104i	0.986452	+	0.164052i	$0.0524565\pi$
$433$	1.03919e14i	0.328104i	−0.986452	+	0.164052i	$0.947543\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2.01333e13i	0.0604320i
$438$	0	0
$439$	−1.28892e14	−0.377286	−0.188643	−	0.982046i	$-0.560409\pi$
$439$	−1.28892e14	−0.377286	−0.188643	+	0.982046i	$0.560409\pi$
$440$	0	0
$441$	−1.62994e14	−0.465327
$442$	0	0
$443$	1.09197e14i	0.304081i	0.988374	+	0.152040i	$0.0485844\pi$
$443$	1.09197e14i	0.304081i	−0.988374	+	0.152040i	$0.951416\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 6.66921e14i	− 1.76760i
$448$	0	0
$449$	−6.90500e14	−1.78570	−0.892851	−	0.450353i	$-0.851298\pi$
$449$	−6.90500e14	−1.78570	−0.892851	+	0.450353i	$0.851298\pi$
$450$	0	0
$451$	2.49136e14	0.628732
$452$	0	0
$453$	1.14362e14i	0.281671i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	4.26165e14i	1.00009i	0.865999	+	0.500045i	$0.166683\pi$
$457$	4.26165e14i	1.00009i	−0.865999	+	0.500045i	$0.833317\pi$
$458$	0	0
$459$	1.43469e14	0.328692
$460$	0	0
$461$	2.05565e14	0.459827	0.229914	−	0.973211i	$-0.426156\pi$
$461$	2.05565e14	0.459827	0.229914	+	0.973211i	$0.426156\pi$
$462$	0	0
$463$	1.75036e14i	0.382323i	0.981559	+	0.191162i	$0.0612255\pi$
$463$	1.75036e14i	0.382323i	−0.981559	+	0.191162i	$0.938775\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 7.90745e14i	− 1.64738i	−0.567040	−	0.823690i	$-0.691912\pi$
$467$	− 7.90745e14i	− 1.64738i	0.567040	−	0.823690i	$-0.308088\pi$
$468$	0	0
$469$	−2.65988e14	−0.541267
$470$	0	0
$471$	−6.05552e14	−1.20375
$472$	0	0
$473$	4.38898e14i	0.852368i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	2.89013e14i	0.535879i
$478$	0	0
$479$	−1.46975e14	−0.266317	−0.133158	−	0.991095i	$-0.542512\pi$
$479$	−1.46975e14	−0.266317	−0.133158	+	0.991095i	$0.542512\pi$
$480$	0	0
$481$	5.84762e14	1.03557
$482$	0	0
$483$	− 1.01608e14i	− 0.175881i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	3.93392e12i	0.00650754i	0.999995	+	0.00325377i	$0.00103571\pi$
$487$	3.93392e12i	0.00650754i	−0.999995	+	0.00325377i	$0.998964\pi$
$488$	0	0
$489$	3.99902e14	0.646777
$490$	0	0
$491$	−5.17934e14	−0.819079	−0.409540	−	0.912292i	$-0.634311\pi$
$491$	−5.17934e14	−0.819079	−0.409540	+	0.912292i	$0.634311\pi$
$492$	0	0
$493$	1.52341e14i	0.235591i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	4.22216e13i	0.0624561i
$498$	0	0
$499$	6.73314e14	0.974237	0.487118	−	0.873336i	$-0.338048\pi$
$499$	6.73314e14	0.974237	0.487118	+	0.873336i	$0.338048\pi$
$500$	0	0
$501$	−6.74347e14	−0.954499
$502$	0	0
$503$	− 1.36762e15i	− 1.89383i	−0.321477	−	0.946917i	$-0.604179\pi$
$503$	− 1.36762e15i	− 1.89383i	0.321477	−	0.946917i	$-0.395821\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	5.31131e14i	0.704139i
$508$	0	0
$509$	−3.36149e14	−0.436098	−0.218049	−	0.975938i	$-0.569969\pi$
$509$	−3.36149e14	−0.436098	−0.218049	+	0.975938i	$0.569969\pi$
$510$	0	0
$511$	−3.21703e14	−0.408451
$512$	0	0
$513$	− 6.80895e13i	− 0.0846126i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 8.35219e14i	− 0.994495i
$518$	0	0
$519$	1.57908e15	1.84071
$520$	0	0
$521$	1.44444e15	1.64851	0.824255	−	0.566218i	$-0.191594\pi$
$521$	1.44444e15	1.64851	0.824255	+	0.566218i	$0.191594\pi$
$522$	0	0
$523$	− 1.49597e15i	− 1.67172i	−0.548946	−	0.835858i	$-0.684971\pi$
$523$	− 1.49597e15i	− 1.67172i	0.548946	−	0.835858i	$-0.315029\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.95440e14i	0.209437i
$528$	0	0
$529$	7.86416e14	0.825365
$530$	0	0
$531$	−1.29045e14	−0.132655
$532$	0	0
$533$	5.92225e14i	0.596331i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	1.44649e15i	1.39783i
$538$	0	0
$539$	6.45528e14	0.611191
$540$	0	0
$541$	1.04365e14	0.0968214	0.0484107	−	0.998828i	$-0.484584\pi$
$541$	1.04365e14	0.0968214	0.0484107	+	0.998828i	$0.484584\pi$
$542$	0	0
$543$	2.23420e14i	0.203106i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 1.03877e15i	− 0.906963i	−0.891266	−	0.453482i	$-0.850182\pi$
$547$	− 1.03877e15i	− 0.906963i	0.891266	−	0.453482i	$-0.149818\pi$
$548$	0	0
$549$	1.14786e14	0.0982291
$550$	0	0
$551$	7.23000e13	0.0606463
$552$	0	0
$553$	− 6.52362e14i	− 0.536415i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 1.35863e15i	− 1.07374i	−0.843666	−	0.536868i	$-0.819607\pi$
$557$	− 1.35863e15i	− 1.07374i	0.843666	−	0.536868i	$-0.180393\pi$
$558$	0	0
$559$	−1.04331e15	−0.808442
$560$	0	0
$561$	6.31591e14	0.479888
$562$	0	0
$563$	− 1.65689e14i	− 0.123452i	−0.998093	−	0.0617258i	$-0.980340\pi$
$563$	− 1.65689e14i	− 0.123452i	0.998093	−	0.0617258i	$-0.0196604\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	5.93869e14i	0.425582i
$568$	0	0
$569$	1.40376e15	0.986675	0.493338	−	0.869838i	$-0.335777\pi$
$569$	1.40376e15	0.986675	0.493338	+	0.869838i	$0.335777\pi$
$570$	0	0
$571$	2.03413e15	1.40243	0.701213	−	0.712952i	$-0.252642\pi$
$571$	2.03413e15	1.40243	0.701213	+	0.712952i	$0.252642\pi$
$572$	0	0
$573$	3.01360e15i	2.03814i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	2.30956e15i	1.50336i	0.659528	+	0.751680i	$0.270756\pi$
$577$	2.30956e15i	1.50336i	−0.659528	+	0.751680i	$0.729244\pi$
$578$	0	0
$579$	−7.19063e14	−0.459235
$580$	0	0
$581$	−3.26476e14	−0.204589
$582$	0	0
$583$	− 1.14462e15i	− 0.703859i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	3.72490e13i	0.0220600i	0.999939	+	0.0110300i	$0.00351103\pi$
$587$	3.72490e13i	0.0220600i	−0.999939	+	0.0110300i	$0.996489\pi$
$588$	0	0
$589$	9.27542e13	0.0539138
$590$	0	0
$591$	7.50054e14	0.427919
$592$	0	0
$593$	− 1.15171e15i	− 0.644973i	−0.946574	−	0.322487i	$-0.895481\pi$
$593$	− 1.15171e15i	− 0.644973i	0.946574	−	0.322487i	$-0.104519\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	7.23421e14i	0.390420i
$598$	0	0
$599$	1.08569e15	0.575254	0.287627	−	0.957743i	$-0.407134\pi$
$599$	1.08569e15	0.575254	0.287627	+	0.957743i	$0.407134\pi$
$600$	0	0
$601$	3.16442e15	1.64621	0.823103	−	0.567891i	$-0.192241\pi$
$601$	3.16442e15	1.64621	0.823103	+	0.567891i	$0.192241\pi$
$602$	0	0
$603$	1.63746e15i	0.836420i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	9.07115e14i	0.446812i	0.974726	+	0.223406i	$0.0717175\pi$
$607$	9.07115e14i	0.446812i	−0.974726	+	0.223406i	$0.928282\pi$
$608$	0	0
$609$	−3.64880e14	−0.176504
$610$	0	0
$611$	1.98541e15	0.943245
$612$	0	0
$613$	− 6.98859e14i	− 0.326105i	−0.986617	−	0.163052i	$-0.947866\pi$
$613$	− 6.98859e14i	− 0.326105i	0.986617	−	0.163052i	$-0.0521339\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	2.96416e15i	1.33455i	0.744813	+	0.667273i	$0.232538\pi$
$617$	2.96416e15i	1.33455i	−0.744813	+	0.667273i	$0.767462\pi$
$618$	0	0
$619$	−2.55190e15	−1.12867	−0.564333	−	0.825547i	$-0.690867\pi$
$619$	−2.55190e15	−1.12867	−0.564333	+	0.825547i	$0.690867\pi$
$620$	0	0
$621$	5.62734e14	0.244511
$622$	0	0
$623$	− 1.39581e15i	− 0.595855i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 2.99748e14i	− 0.123534i
$628$	0	0
$629$	−2.19052e15	−0.887090
$630$	0	0
$631$	2.01160e15	0.800533	0.400266	−	0.916399i	$-0.368918\pi$
$631$	2.01160e15	0.800533	0.400266	+	0.916399i	$0.368918\pi$
$632$	0	0
$633$	3.31629e15i	1.29698i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	1.53450e15i	0.579694i
$638$	0	0
$639$	2.59922e14	0.0965134
$640$	0	0
$641$	3.63571e15	1.32700	0.663498	−	0.748178i	$-0.269071\pi$
$641$	3.63571e15	1.32700	0.663498	+	0.748178i	$0.269071\pi$
$642$	0	0
$643$	3.75494e15i	1.34723i	0.739081	+	0.673617i	$0.235260\pi$
$643$	3.75494e15i	1.34723i	−0.739081	+	0.673617i	$0.764740\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	3.24138e15i	1.12397i	0.827146	+	0.561987i	$0.189963\pi$
$647$	3.24138e15i	1.12397i	−0.827146	+	0.561987i	$0.810037\pi$
$648$	0	0
$649$	5.11078e14	0.174237
$650$	0	0
$651$	−4.68108e14	−0.156910
$652$	0	0
$653$	− 5.23724e15i	− 1.72616i	−0.505071	−	0.863078i	$-0.668534\pi$
$653$	− 5.23724e15i	− 1.72616i	0.505071	−	0.863078i	$-0.331466\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	1.98044e15i	0.631179i
$658$	0	0
$659$	1.83448e15	0.574969	0.287484	−	0.957785i	$-0.407181\pi$
$659$	1.83448e15	0.574969	0.287484	+	0.957785i	$0.407181\pi$
$660$	0	0
$661$	−4.91377e15	−1.51463	−0.757315	−	0.653050i	$-0.773489\pi$
$661$	−4.91377e15	−1.51463	−0.757315	+	0.653050i	$0.773489\pi$
$662$	0	0
$663$	1.50137e15i	0.455158i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	5.97531e14i	0.175254i
$668$	0	0
$669$	−1.44813e15	−0.417796
$670$	0	0
$671$	−4.54603e14	−0.129021
$672$	0	0
$673$	− 6.37904e15i	− 1.78103i	−0.454950	−	0.890517i	$-0.650343\pi$
$673$	− 6.37904e15i	− 1.78103i	0.454950	−	0.890517i	$-0.349657\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	2.49224e15i	0.673521i	0.941590	+	0.336761i	$0.109331\pi$
$677$	2.49224e15i	0.673521i	−0.941590	+	0.336761i	$0.890669\pi$
$678$	0	0
$679$	1.21918e15	0.324179
$680$	0	0
$681$	−1.03445e14	−0.0270646
$682$	0	0
$683$	1.44353e15i	0.371630i	0.982585	+	0.185815i	$0.0594926\pi$
$683$	1.44353e15i	0.371630i	−0.982585	+	0.185815i	$0.940507\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	5.65850e15i	1.41071i
$688$	0	0
$689$	2.72090e15	0.667586
$690$	0	0
$691$	2.66415e15	0.643322	0.321661	−	0.946855i	$-0.395759\pi$
$691$	2.66415e15	0.643322	0.321661	+	0.946855i	$0.395759\pi$
$692$	0	0
$693$	5.21706e14i	0.123992i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 2.21847e15i	− 0.510827i
$698$	0	0
$699$	−1.60011e15	−0.362682
$700$	0	0
$701$	−6.99888e15	−1.56163	−0.780817	−	0.624760i	$-0.785197\pi$
$701$	−6.99888e15	−1.56163	−0.780817	+	0.624760i	$0.785197\pi$
$702$	0	0
$703$	1.03960e15i	0.228356i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	4.10475e13i	0.00873937i
$708$	0	0
$709$	3.20041e15	0.670889	0.335445	−	0.942060i	$-0.391113\pi$
$709$	3.20041e15	0.670889	0.335445	+	0.942060i	$0.391113\pi$
$710$	0	0
$711$	−4.01602e15	−0.828922
$712$	0	0
$713$	7.66578e14i	0.155799i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 9.76816e15i	− 1.92512i
$718$	0	0
$719$	5.98965e15	1.16250	0.581250	−	0.813725i	$-0.302564\pi$
$719$	5.98965e15	1.16250	0.581250	+	0.813725i	$0.302564\pi$
$720$	0	0
$721$	2.22314e15	0.424935
$722$	0	0
$723$	3.00957e15i	0.566557i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 8.51869e15i	− 1.55573i	−0.628432	−	0.777864i	$-0.716303\pi$
$727$	− 8.51869e15i	− 1.55573i	0.628432	−	0.777864i	$-0.283697\pi$
$728$	0	0
$729$	−3.62638e14	−0.0652337
$730$	0	0
$731$	3.90824e15	0.692525
$732$	0	0
$733$	− 4.77143e15i	− 0.832868i	−0.909166	−	0.416434i	$-0.863280\pi$
$733$	− 4.77143e15i	− 0.832868i	0.909166	−	0.416434i	$-0.136720\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 6.48507e15i	− 1.09861i
$738$	0	0
$739$	−5.43737e15	−0.907496	−0.453748	−	0.891130i	$-0.649913\pi$
$739$	−5.43737e15	−0.907496	−0.453748	+	0.891130i	$0.649913\pi$
$740$	0	0
$741$	7.12537e14	0.117168
$742$	0	0
$743$	− 5.35180e15i	− 0.867084i	−0.901133	−	0.433542i	$-0.857264\pi$
$743$	− 5.35180e15i	− 0.867084i	0.901133	−	0.433542i	$-0.142736\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	2.00983e15i	0.316152i
$748$	0	0
$749$	−1.27753e15	−0.198025
$750$	0	0
$751$	1.75117e15	0.267490	0.133745	−	0.991016i	$-0.457300\pi$
$751$	1.75117e15	0.267490	0.133745	+	0.991016i	$0.457300\pi$
$752$	0	0
$753$	7.38095e15i	1.11107i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 1.12765e16i	− 1.64872i	−0.566065	−	0.824361i	$-0.691535\pi$
$757$	− 1.12765e16i	− 1.64872i	0.566065	−	0.824361i	$-0.308465\pi$
$758$	0	0
$759$	2.47730e15	0.356985
$760$	0	0
$761$	2.83502e14	0.0402662	0.0201331	−	0.999797i	$-0.493591\pi$
$761$	2.83502e14	0.0402662	0.0201331	+	0.999797i	$0.493591\pi$
$762$	0	0
$763$	2.70953e15i	0.379323i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.21489e15i	0.165258i
$768$	0	0
$769$	−1.53743e15	−0.206158	−0.103079	−	0.994673i	$-0.532869\pi$
$769$	−1.53743e15	−0.206158	−0.103079	+	0.994673i	$0.532869\pi$
$770$	0	0
$771$	−1.44931e16	−1.91585
$772$	0	0
$773$	− 2.13194e15i	− 0.277835i	−0.990304	−	0.138917i	$-0.955638\pi$
$773$	− 2.13194e15i	− 0.277835i	0.990304	−	0.138917i	$-0.0443623\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 5.24663e15i	− 0.664606i
$778$	0	0
$779$	−1.05287e15	−0.131498
$780$	0	0
$781$	−1.02941e15	−0.126767
$782$	0	0
$783$	− 2.02081e15i	− 0.245379i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 2.42829e15i	− 0.286707i	−0.989672	−	0.143354i	$-0.954211\pi$
$787$	− 2.42829e15i	− 0.286707i	0.989672	−	0.143354i	$-0.0457886\pi$
$788$	0	0
$789$	−3.98927e15	−0.464483
$790$	0	0
$791$	−2.99740e15	−0.344171
$792$	0	0
$793$	− 1.08064e15i	− 0.122372i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 1.44628e16i	− 1.59305i	−0.604603	−	0.796527i	$-0.706668\pi$
$797$	− 1.44628e16i	− 1.59305i	0.604603	−	0.796527i	$-0.293332\pi$
$798$	0	0
$799$	−7.43735e15	−0.807999
$800$	0	0
$801$	−8.59276e15	−0.920775
$802$	0	0
$803$	− 7.84344e15i	− 0.829032i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 7.74939e14i	− 0.0797009i
$808$	0	0
$809$	−2.77511e14	−0.0281555	−0.0140777	−	0.999901i	$-0.504481\pi$
$809$	−2.77511e14	−0.0281555	−0.0140777	+	0.999901i	$0.504481\pi$
$810$	0	0
$811$	3.50430e15	0.350741	0.175370	−	0.984503i	$-0.443888\pi$
$811$	3.50430e15	0.350741	0.175370	+	0.984503i	$0.443888\pi$
$812$	0	0
$813$	1.00155e16i	0.988955i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 1.85482e15i	− 0.178271i
$818$	0	0
$819$	−1.24016e15	−0.117602
$820$	0	0
$821$	−5.16560e15	−0.483318	−0.241659	−	0.970361i	$-0.577692\pi$
$821$	−5.16560e15	−0.483318	−0.241659	+	0.970361i	$0.577692\pi$
$822$	0	0
$823$	1.18043e16i	1.08978i	0.838507	+	0.544891i	$0.183429\pi$
$823$	1.18043e16i	1.08978i	−0.838507	+	0.544891i	$0.816571\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.09512e16i	0.984419i	0.870477	+	0.492210i	$0.163811\pi$
$827$	1.09512e16i	0.984419i	−0.870477	+	0.492210i	$0.836189\pi$
$828$	0	0
$829$	3.27381e15	0.290404	0.145202	−	0.989402i	$-0.453617\pi$
$829$	3.27381e15	0.290404	0.145202	+	0.989402i	$0.453617\pi$
$830$	0	0
$831$	−1.70352e16	−1.49122
$832$	0	0
$833$	− 5.74821e15i	− 0.496576i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 2.59252e15i	− 0.218138i
$838$	0	0
$839$	−2.15644e16	−1.79080	−0.895399	−	0.445266i	$-0.853109\pi$
$839$	−2.15644e16	−1.79080	−0.895399	+	0.445266i	$0.853109\pi$
$840$	0	0
$841$	−1.00547e16	−0.824124
$842$	0	0
$843$	1.54533e16i	1.25017i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	2.25571e15i	0.177797i
$848$	0	0
$849$	2.45074e16	1.90680
$850$	0	0
$851$	−8.59191e15	−0.659899
$852$	0	0
$853$	− 2.56475e15i	− 0.194458i	−0.995262	−	0.0972290i	$-0.969002\pi$
$853$	− 2.56475e15i	− 0.194458i	0.995262	−	0.0972290i	$-0.0309979\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 1.87155e15i	− 0.138295i	−0.997606	−	0.0691476i	$-0.977972\pi$
$857$	− 1.87155e15i	− 0.138295i	0.997606	−	0.0691476i	$-0.0220280\pi$
$858$	0	0
$859$	−1.62278e16	−1.18385	−0.591924	−	0.805994i	$-0.701631\pi$
$859$	−1.62278e16	−1.18385	−0.591924	+	0.805994i	$0.701631\pi$
$860$	0	0
$861$	5.31359e15	0.382711
$862$	0	0
$863$	− 4.59848e15i	− 0.327005i	−0.986543	−	0.163503i	$-0.947721\pi$
$863$	− 4.59848e15i	− 0.327005i	0.986543	−	0.163503i	$-0.0522792\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	1.21973e16i	0.845586i
$868$	0	0
$869$	1.59053e16	1.08876
$870$	0	0
$871$	1.54158e16	1.04199
$872$	0	0
$873$	− 7.50543e15i	− 0.500953i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	1.16270e16i	0.756778i	0.925647	+	0.378389i	$0.123522\pi$
$877$	1.16270e16i	0.756778i	−0.925647	+	0.378389i	$0.876478\pi$
$878$	0	0
$879$	−2.94503e16	−1.89300
$880$	0	0
$881$	−7.39605e15	−0.469497	−0.234748	−	0.972056i	$-0.575427\pi$
$881$	−7.39605e15	−0.469497	−0.234748	+	0.972056i	$0.575427\pi$
$882$	0	0
$883$	− 1.37331e16i	− 0.860961i	−0.902600	−	0.430480i	$-0.858344\pi$
$883$	− 1.37331e16i	− 0.860961i	0.902600	−	0.430480i	$-0.141656\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1.05140e16i	0.642966i	0.946915	+	0.321483i	$0.104181\pi$
$887$	1.05140e16i	0.642966i	−0.946915	+	0.321483i	$0.895819\pi$
$888$	0	0
$889$	7.24047e15	0.437328
$890$	0	0
$891$	−1.44791e16	−0.863804
$892$	0	0
$893$	3.52971e15i	0.207997i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	5.88884e15i	0.338588i
$898$	0	0
$899$	2.75283e15	0.156351
$900$	0	0
$901$	−1.01925e16	−0.571865
$902$	0	0
$903$	9.36084e15i	0.518838i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	1.03279e15i	0.0558691i	0.999610	+	0.0279346i	$0.00889300\pi$
$907$	1.03279e15i	0.0558691i	−0.999610	+	0.0279346i	$0.991107\pi$
$908$	0	0
$909$	2.52694e14	0.0135050
$910$	0	0
$911$	−3.30501e15	−0.174510	−0.0872552	−	0.996186i	$-0.527810\pi$
$911$	−3.30501e15	−0.174510	−0.0872552	+	0.996186i	$0.527810\pi$
$912$	0	0
$913$	− 7.95982e15i	− 0.415254i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 1.32438e15i	− 0.0674499i
$918$	0	0
$919$	−1.60824e16	−0.809313	−0.404657	−	0.914469i	$-0.632609\pi$
$919$	−1.60824e16	−0.809313	−0.404657	+	0.914469i	$0.632609\pi$
$920$	0	0
$921$	8.83403e15	0.439269
$922$	0	0
$923$	− 2.44702e15i	− 0.120234i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 1.36859e16i	− 0.656652i
$928$	0	0
$929$	−2.02897e16	−0.962031	−0.481016	−	0.876712i	$-0.659732\pi$
$929$	−2.02897e16	−0.962031	−0.481016	+	0.876712i	$0.659732\pi$
$930$	0	0
$931$	−2.72806e15	−0.127829
$932$	0	0
$933$	1.64607e16i	0.762251i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 2.85837e16i	− 1.29286i	−0.762975	−	0.646428i	$-0.776262\pi$
$937$	− 2.85837e16i	− 1.29286i	0.762975	−	0.646428i	$-0.223738\pi$
$938$	0	0
$939$	−8.78562e15	−0.392746
$940$	0	0
$941$	2.69594e16	1.19115	0.595577	−	0.803298i	$-0.296923\pi$
$941$	2.69594e16	1.19115	0.595577	+	0.803298i	$0.296923\pi$
$942$	0	0
$943$	− 8.70157e15i	− 0.380000i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	5.57448e15i	0.237837i	0.992904	+	0.118918i	$0.0379427\pi$
$947$	5.57448e15i	0.237837i	−0.992904	+	0.118918i	$0.962057\pi$
$948$	0	0
$949$	1.86448e16	0.786309
$950$	0	0
$951$	4.33178e16	1.80581
$952$	0	0
$953$	3.11592e16i	1.28403i	0.766691	+	0.642016i	$0.221902\pi$
$953$	3.11592e16i	1.28403i	−0.766691	+	0.642016i	$0.778098\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 8.89617e15i	− 0.358251i
$958$	0	0
$959$	1.14441e16	0.455597
$960$	0	0
$961$	−2.18768e16	−0.861006
$962$	0	0
$963$	7.86461e15i	0.306008i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 5.46769e15i	− 0.207950i	−0.994580	−	0.103975i	$-0.966844\pi$
$967$	− 5.46769e15i	− 0.207950i	0.994580	−	0.103975i	$-0.0331561\pi$
$968$	0	0
$969$	−2.66916e15	−0.100368
$970$	0	0
$971$	8.64934e15	0.321571	0.160786	−	0.986989i	$-0.448597\pi$
$971$	8.64934e15	0.321571	0.160786	+	0.986989i	$0.448597\pi$
$972$	0	0
$973$	− 1.20737e16i	− 0.443834i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 3.43815e16i	− 1.23568i	−0.786305	−	0.617839i	$-0.788009\pi$
$977$	− 3.43815e16i	− 1.23568i	0.786305	−	0.617839i	$-0.211991\pi$
$978$	0	0
$979$	3.40312e16	1.20941
$980$	0	0
$981$	1.66802e16	0.586168
$982$	0	0
$983$	8.23928e15i	0.286315i	0.989700	+	0.143158i	$0.0457256\pi$
$983$	8.23928e15i	0.286315i	−0.989700	+	0.143158i	$0.954274\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 1.78136e16i	− 0.605351i
$988$	0	0
$989$	1.53294e16	0.515163
$990$	0	0
$991$	2.66218e16	0.884775	0.442387	−	0.896824i	$-0.354132\pi$
$991$	2.66218e16	0.884775	0.442387	+	0.896824i	$0.354132\pi$
$992$	0	0
$993$	− 4.17223e16i	− 1.37135i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	5.72816e16i	1.84158i	0.390055	+	0.920792i	$0.372456\pi$
$997$	5.72816e16i	1.84158i	−0.390055	+	0.920792i	$0.627544\pi$
$998$	0	0
$999$	2.90573e16	0.923944

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	200.12.c.a.49.2		2
5.2	odd	4		40.12.a.a.1.1	✓	1
5.3	odd	4		200.12.a.a.1.1		1
5.4	even	2	inner	200.12.c.a.49.1		2
20.7	even	4		80.12.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.12.a.a.1.1	✓	1	5.2	odd	4
80.12.a.b.1.1		1	20.7	even	4
200.12.a.a.1.1		1	5.3	odd	4
200.12.c.a.49.1		2	5.4	even	2	inner
200.12.c.a.49.2		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 200 = 2^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	200.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+520.000i q^{3} -15148.0i q^{7} -93253.0 q^{9} +369324. q^{11} +877926. i q^{13} -3.28871e6i q^{17} -1.56080e6 q^{19} +7.87696e6 q^{21} -1.28994e7i q^{23} +4.36249e7i q^{27} -4.63225e7 q^{29} -5.94275e7 q^{31} +1.92048e8i q^{33} -6.66072e8i q^{37} -4.56522e8 q^{39} +6.74573e8 q^{41} +1.18838e9i q^{43} -2.26148e9i q^{47} +1.74786e9 q^{49} +1.71013e9 q^{51} -3.09924e9i q^{53} -8.11614e8i q^{57} +1.38382e9 q^{59} -1.23091e9 q^{61} +1.41260e9i q^{63} -1.75593e10i q^{67} +6.70767e9 q^{69} -2.78727e9 q^{71} -2.12373e10i q^{73} -5.59452e9i q^{77} +4.30659e10 q^{79} -3.92044e10 q^{81} -2.15524e10i q^{83} -2.40877e10i q^{87} +9.21446e10 q^{89} +1.32988e10 q^{91} -3.09023e10i q^{93} +8.04846e10i q^{97} -3.44406e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 186506 q^{9} + 738648 q^{11} - 3121592 q^{19} + 15753920 q^{21} - 92645004 q^{29} - 118855056 q^{31} - 913043040 q^{39} + 1349146388 q^{41} + 3495729678 q^{49} + 3420254240 q^{51} + 2767639992 q^{59}+ \cdots - 68881141944 q^{99}+O(q^{100}) \) 2 * q - 186506 * q^9 + 738648 * q^11 - 3121592 * q^19 + 15753920 * q^21 - 92645004 * q^29 - 118855056 * q^31 - 913043040 * q^39 + 1349146388 * q^41 + 3495729678 * q^49 + 3420254240 * q^51 + 2767639992 * q^59 - 2461813044 * q^61 + 13415346880 * q^69 - 5574545760 * q^71 + 86131783360 * q^79 - 78408853582 * q^81 + 184289105708 * q^89 + 26597646096 * q^91 - 68881141944 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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