LMFDB - Embedded newform 2900.2.c.i.349.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2900,2,Mod(349,2900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2900, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2900.349");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2900 = 2^{2} \cdot 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2900.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:	$23.1566165862$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 18x^{8} + 99x^{6} + 163x^{4} + 75x^{2} + 9 $ x^10 + 18x^8 + 99x^6 + 163x^4 + 75x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			349.4
Root			$-0.672645i$ of defining polynomial
Character	$\chi$	$=$	2900.349
Dual form			2900.2.c.i.349.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.672645i q^{3} +3.37701i q^{7} +2.54755 q^{9} +2.11471 q^{11} +1.91246i q^{13} -5.53382i q^{17} +7.99382 q^{19} +2.27153 q^{21} +3.27602i q^{23} -3.73153i q^{27} +1.00000 q^{29} -10.2833 q^{31} -1.42245i q^{33} +1.37246i q^{37} +1.28640 q^{39} -1.81453 q^{41} +1.60338i q^{43} -0.327355i q^{47} -4.40418 q^{49} -3.72230 q^{51} +4.18848i q^{53} -5.37701i q^{57} +3.30908 q^{59} +7.00306 q^{61} +8.60309i q^{63} +6.57443i q^{67} +2.20360 q^{69} +15.5005 q^{71} +5.43283i q^{73} +7.14140i q^{77} -2.72819 q^{79} +5.13265 q^{81} +14.9969i q^{83} -0.672645i q^{87} -12.6048 q^{89} -6.45838 q^{91} +6.91700i q^{93} +0.851941i q^{97} +5.38734 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 6 q^{9} + 8 q^{11} + 8 q^{19} + 6 q^{21} + 10 q^{29} + 6 q^{31} + 36 q^{39} + 10 q^{41} + 6 q^{49} + 18 q^{51} + 46 q^{59} + 10 q^{61} + 60 q^{69} + 46 q^{71} - 8 q^{79} + 18 q^{81} + 12 q^{89} + 54 q^{91}+ \cdots + 72 q^{99}+O(q^{100}) $ 10 * q - 6 * q^9 + 8 * q^11 + 8 * q^19 + 6 * q^21 + 10 * q^29 + 6 * q^31 + 36 * q^39 + 10 * q^41 + 6 * q^49 + 18 * q^51 + 46 * q^59 + 10 * q^61 + 60 * q^69 + 46 * q^71 - 8 * q^79 + 18 * q^81 + 12 * q^89 + 54 * q^91 + 72 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2900\mathbb{Z}\right)^\times$.

$n$	$901$	$1277$	$1451$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	− 0.672645i	− 0.388352i	−0.980967	−	0.194176i	$-0.937797\pi$
$3$	− 0.672645i	− 0.388352i	0.980967	−	0.194176i	$-0.0622033\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.37701i	1.27639i	0.769875	+	0.638194i	$0.220318\pi$
$7$	3.37701i	1.27639i	−0.769875	+	0.638194i	$0.779682\pi$
$8$	0	0
$9$	2.54755	0.849183
$10$	0	0
$11$	2.11471	0.637610	0.318805	−	0.947820i	$-0.396718\pi$
$11$	2.11471	0.637610	0.318805	+	0.947820i	$0.396718\pi$
$12$	0	0
$13$	1.91246i	0.530420i	0.964191	+	0.265210i	$0.0854412\pi$
$13$	1.91246i	0.530420i	−0.964191	+	0.265210i	$0.914559\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 5.53382i	− 1.34215i	−0.741390	−	0.671074i	$-0.765833\pi$
$17$	− 5.53382i	− 1.34215i	0.741390	−	0.671074i	$-0.234167\pi$
$18$	0	0
$19$	7.99382	1.83391	0.916955	−	0.398992i	$-0.130640\pi$
$19$	7.99382	1.83391	0.916955	+	0.398992i	$0.130640\pi$
$20$	0	0
$21$	2.27153	0.495688
$22$	0	0
$23$	3.27602i	0.683098i	0.939864	+	0.341549i	$0.110951\pi$
$23$	3.27602i	0.683098i	−0.939864	+	0.341549i	$0.889049\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 3.73153i	− 0.718134i
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−10.2833	−1.84693	−0.923466	−	0.383679i	$-0.874657\pi$
$31$	−10.2833	−1.84693	−0.923466	+	0.383679i	$0.874657\pi$
$32$	0	0
$33$	− 1.42245i	− 0.247617i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.37246i	0.225631i	0.993616	+	0.112815i	$0.0359869\pi$
$37$	1.37246i	0.225631i	−0.993616	+	0.112815i	$0.964013\pi$
$38$	0	0
$39$	1.28640	0.205989
$40$	0	0
$41$	−1.81453	−0.283382	−0.141691	−	0.989911i	$-0.545254\pi$
$41$	−1.81453	−0.283382	−0.141691	+	0.989911i	$0.545254\pi$
$42$	0	0
$43$	1.60338i	0.244513i	0.992499	+	0.122256i	$0.0390130\pi$
$43$	1.60338i	0.244513i	−0.992499	+	0.122256i	$0.960987\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 0.327355i	− 0.0477496i	−0.999715	−	0.0238748i	$-0.992400\pi$
$47$	− 0.327355i	− 0.0477496i	0.999715	−	0.0238748i	$-0.00760031\pi$
$48$	0	0
$49$	−4.40418	−0.629168
$50$	0	0
$51$	−3.72230	−0.521226
$52$	0	0
$53$	4.18848i	0.575332i	0.957731	+	0.287666i	$0.0928792\pi$
$53$	4.18848i	0.575332i	−0.957731	+	0.287666i	$0.907121\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 5.37701i	− 0.712202i
$58$	0	0
$59$	3.30908	0.430805	0.215403	−	0.976525i	$-0.430894\pi$
$59$	3.30908	0.430805	0.215403	+	0.976525i	$0.430894\pi$
$60$	0	0
$61$	7.00306	0.896650	0.448325	−	0.893871i	$-0.352021\pi$
$61$	7.00306	0.896650	0.448325	+	0.893871i	$0.352021\pi$
$62$	0	0
$63$	8.60309i	1.08389i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	6.57443i	0.803195i	0.915816	+	0.401597i	$0.131545\pi$
$67$	6.57443i	0.803195i	−0.915816	+	0.401597i	$0.868455\pi$
$68$	0	0
$69$	2.20360	0.265282
$70$	0	0
$71$	15.5005	1.83957	0.919784	−	0.392425i	$-0.128364\pi$
$71$	15.5005	1.83957	0.919784	+	0.392425i	$0.128364\pi$
$72$	0	0
$73$	5.43283i	0.635865i	0.948113	+	0.317933i	$0.102989\pi$
$73$	5.43283i	0.635865i	−0.948113	+	0.317933i	$0.897011\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	7.14140i	0.813838i
$78$	0	0
$79$	−2.72819	−0.306945	−0.153472	−	0.988153i	$-0.549046\pi$
$79$	−2.72819	−0.306945	−0.153472	+	0.988153i	$0.549046\pi$
$80$	0	0
$81$	5.13265	0.570294
$82$	0	0
$83$	14.9969i	1.64612i	0.567953	+	0.823061i	$0.307735\pi$
$83$	14.9969i	1.64612i	−0.567953	+	0.823061i	$0.692265\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 0.672645i	− 0.0721151i
$88$	0	0
$89$	−12.6048	−1.33610	−0.668051	−	0.744116i	$-0.732871\pi$
$89$	−12.6048	−1.33610	−0.668051	+	0.744116i	$0.732871\pi$
$90$	0	0
$91$	−6.45838	−0.677022
$92$	0	0
$93$	6.91700i	0.717260i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	0.851941i	0.0865015i	0.999064	+	0.0432508i	$0.0137714\pi$
$97$	0.851941i	0.0865015i	−0.999064	+	0.0432508i	$0.986229\pi$
$98$	0	0
$99$	5.38734	0.541448
$100$	0	0
$101$	17.4522	1.73656	0.868279	−	0.496075i	$-0.165226\pi$
$101$	17.4522	1.73656	0.868279	+	0.496075i	$0.165226\pi$
$102$	0	0
$103$	10.6951i	1.05382i	0.849920	+	0.526911i	$0.176650\pi$
$103$	10.6951i	1.05382i	−0.849920	+	0.526911i	$0.823350\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 20.5895i	− 1.99047i	−0.0975276	−	0.995233i	$-0.531093\pi$
$107$	− 20.5895i	− 1.99047i	0.0975276	−	0.995233i	$-0.468907\pi$
$108$	0	0
$109$	−0.972490	−0.0931477	−0.0465738	−	0.998915i	$-0.514830\pi$
$109$	−0.972490	−0.0931477	−0.0465738	+	0.998915i	$0.514830\pi$
$110$	0	0
$111$	0.923178	0.0876242
$112$	0	0
$113$	− 12.7192i	− 1.19652i	−0.801302	−	0.598260i	$-0.795859\pi$
$113$	− 12.7192i	− 1.19652i	0.801302	−	0.598260i	$-0.204141\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	4.87207i	0.450423i
$118$	0	0
$119$	18.6878	1.71310
$120$	0	0
$121$	−6.52799	−0.593453
$122$	0	0
$123$	1.22053i	0.110052i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	15.2036i	1.34910i	0.738227	+	0.674552i	$0.235663\pi$
$127$	15.2036i	1.34910i	−0.738227	+	0.674552i	$0.764337\pi$
$128$	0	0
$129$	1.07850	0.0949569
$130$	0	0
$131$	6.03755	0.527503	0.263752	−	0.964591i	$-0.415040\pi$
$131$	6.03755	0.527503	0.263752	+	0.964591i	$0.415040\pi$
$132$	0	0
$133$	26.9952i	2.34078i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 15.4901i	− 1.32341i	−0.749765	−	0.661704i	$-0.769834\pi$
$137$	− 15.4901i	− 1.32341i	0.749765	−	0.661704i	$-0.230166\pi$
$138$	0	0
$139$	−2.09639	−0.177813	−0.0889065	−	0.996040i	$-0.528337\pi$
$139$	−2.09639	−0.177813	−0.0889065	+	0.996040i	$0.528337\pi$
$140$	0	0
$141$	−0.220194	−0.0185437
$142$	0	0
$143$	4.04430i	0.338201i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	2.96245i	0.244339i
$148$	0	0
$149$	−5.07653	−0.415886	−0.207943	−	0.978141i	$-0.566677\pi$
$149$	−5.07653	−0.415886	−0.207943	+	0.978141i	$0.566677\pi$
$150$	0	0
$151$	12.6412	1.02872	0.514361	−	0.857574i	$-0.328029\pi$
$151$	12.6412	1.02872	0.514361	+	0.857574i	$0.328029\pi$
$152$	0	0
$153$	− 14.0977i	− 1.13973i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 19.7495i	− 1.57618i	−0.615561	−	0.788090i	$-0.711070\pi$
$157$	− 19.7495i	− 1.57618i	0.615561	−	0.788090i	$-0.288930\pi$
$158$	0	0
$159$	2.81736	0.223431
$160$	0	0
$161$	−11.0631	−0.871898
$162$	0	0
$163$	− 19.9484i	− 1.56248i	−0.624230	−	0.781240i	$-0.714587\pi$
$163$	− 19.9484i	− 1.56248i	0.624230	−	0.781240i	$-0.285413\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 2.34529i	− 0.181484i	−0.995874	−	0.0907420i	$-0.971076\pi$
$167$	− 2.34529i	− 0.181484i	0.995874	−	0.0907420i	$-0.0289239\pi$
$168$	0	0
$169$	9.34251	0.718655
$170$	0	0
$171$	20.3647	1.55732
$172$	0	0
$173$	14.8187i	1.12665i	0.826236	+	0.563324i	$0.190478\pi$
$173$	14.8187i	1.12665i	−0.826236	+	0.563324i	$0.809522\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 2.22584i	− 0.167304i
$178$	0	0
$179$	6.12381	0.457715	0.228857	−	0.973460i	$-0.426501\pi$
$179$	6.12381	0.457715	0.228857	+	0.973460i	$0.426501\pi$
$180$	0	0
$181$	−18.4853	−1.37400	−0.686999	−	0.726658i	$-0.741072\pi$
$181$	−18.4853	−1.37400	−0.686999	+	0.726658i	$0.741072\pi$
$182$	0	0
$183$	− 4.71057i	− 0.348216i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 11.7024i	− 0.855768i
$188$	0	0
$189$	12.6014	0.916618
$190$	0	0
$191$	12.1585	0.879757	0.439878	−	0.898057i	$-0.355022\pi$
$191$	12.1585	0.879757	0.439878	+	0.898057i	$0.355022\pi$
$192$	0	0
$193$	19.6332i	1.41323i	0.707600	+	0.706614i	$0.249778\pi$
$193$	19.6332i	1.41323i	−0.707600	+	0.706614i	$0.750222\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	17.0656i	1.21588i	0.793985	+	0.607938i	$0.208003\pi$
$197$	17.0656i	1.21588i	−0.793985	+	0.607938i	$0.791997\pi$
$198$	0	0
$199$	22.8668	1.62098	0.810491	−	0.585751i	$-0.199200\pi$
$199$	22.8668	1.62098	0.810491	+	0.585751i	$0.199200\pi$
$200$	0	0
$201$	4.42226	0.311922
$202$	0	0
$203$	3.37701i	0.237019i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	8.34582i	0.580075i
$208$	0	0
$209$	16.9047	1.16932
$210$	0	0
$211$	18.2332	1.25522	0.627611	−	0.778527i	$-0.284033\pi$
$211$	18.2332	1.25522	0.627611	+	0.778527i	$0.284033\pi$
$212$	0	0
$213$	− 10.4263i	− 0.714400i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 34.7267i	− 2.35740i
$218$	0	0
$219$	3.65437	0.246939
$220$	0	0
$221$	10.5832	0.711902
$222$	0	0
$223$	3.50379i	0.234631i	0.993095	+	0.117315i	$0.0374288\pi$
$223$	3.50379i	0.234631i	−0.993095	+	0.117315i	$0.962571\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	13.0723i	0.867641i	0.900999	+	0.433820i	$0.142835\pi$
$227$	13.0723i	0.867641i	−0.900999	+	0.433820i	$0.857165\pi$
$228$	0	0
$229$	−15.7072	−1.03796	−0.518981	−	0.854786i	$-0.673689\pi$
$229$	−15.7072	−1.03796	−0.518981	+	0.854786i	$0.673689\pi$
$230$	0	0
$231$	4.80363	0.316056
$232$	0	0
$233$	− 5.96096i	− 0.390515i	−0.980752	−	0.195258i	$-0.937446\pi$
$233$	− 5.96096i	− 0.390515i	0.980752	−	0.195258i	$-0.0625543\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	1.83510i	0.119203i
$238$	0	0
$239$	5.20661	0.336787	0.168394	−	0.985720i	$-0.446142\pi$
$239$	5.20661	0.336787	0.168394	+	0.985720i	$0.446142\pi$
$240$	0	0
$241$	−9.12261	−0.587639	−0.293819	−	0.955861i	$-0.594926\pi$
$241$	−9.12261	−0.587639	−0.293819	+	0.955861i	$0.594926\pi$
$242$	0	0
$243$	− 14.6470i	− 0.939608i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	15.2878i	0.972742i
$248$	0	0
$249$	10.0876	0.639274
$250$	0	0
$251$	−23.8338	−1.50437	−0.752187	−	0.658950i	$-0.771001\pi$
$251$	−23.8338	−1.50437	−0.752187	+	0.658950i	$0.771001\pi$
$252$	0	0
$253$	6.92785i	0.435550i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	8.79497i	0.548615i	0.961642	+	0.274307i	$0.0884486\pi$
$257$	8.79497i	0.548615i	−0.961642	+	0.274307i	$0.911551\pi$
$258$	0	0
$259$	−4.63481	−0.287993
$260$	0	0
$261$	2.54755	0.157689
$262$	0	0
$263$	− 17.0631i	− 1.05216i	−0.850436	−	0.526079i	$-0.823662\pi$
$263$	− 17.0631i	− 1.05216i	0.850436	−	0.526079i	$-0.176338\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	8.47853i	0.518877i
$268$	0	0
$269$	−18.1534	−1.10683	−0.553415	−	0.832906i	$-0.686676\pi$
$269$	−18.1534	−1.10683	−0.553415	+	0.832906i	$0.686676\pi$
$270$	0	0
$271$	−13.2003	−0.801861	−0.400930	−	0.916109i	$-0.631313\pi$
$271$	−13.2003	−0.801861	−0.400930	+	0.916109i	$0.631313\pi$
$272$	0	0
$273$	4.34420i	0.262923i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 15.7679i	− 0.947403i	−0.880685	−	0.473702i	$-0.842917\pi$
$277$	− 15.7679i	− 0.947403i	0.880685	−	0.473702i	$-0.157083\pi$
$278$	0	0
$279$	−26.1972	−1.56838
$280$	0	0
$281$	−1.11500	−0.0665154	−0.0332577	−	0.999447i	$-0.510588\pi$
$281$	−1.11500	−0.0665154	−0.0332577	+	0.999447i	$0.510588\pi$
$282$	0	0
$283$	17.7462i	1.05490i	0.849586	+	0.527450i	$0.176852\pi$
$283$	17.7462i	1.05490i	−0.849586	+	0.527450i	$0.823148\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 6.12768i	− 0.361705i
$288$	0	0
$289$	−13.6232	−0.801363
$290$	0	0
$291$	0.573054	0.0335930
$292$	0	0
$293$	− 1.44336i	− 0.0843218i	−0.999111	−	0.0421609i	$-0.986576\pi$
$293$	− 1.44336i	− 0.0843218i	0.999111	−	0.0421609i	$-0.0134242\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 7.89112i	− 0.457889i
$298$	0	0
$299$	−6.26525	−0.362328
$300$	0	0
$301$	−5.41461	−0.312093
$302$	0	0
$303$	− 11.7391i	− 0.674396i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 5.34395i	− 0.304995i	−0.988304	−	0.152498i	$-0.951268\pi$
$307$	− 5.34395i	− 0.304995i	0.988304	−	0.152498i	$-0.0487316\pi$
$308$	0	0
$309$	7.19403	0.409254
$310$	0	0
$311$	24.6634	1.39854	0.699268	−	0.714860i	$-0.253509\pi$
$311$	24.6634	1.39854	0.699268	+	0.714860i	$0.253509\pi$
$312$	0	0
$313$	33.9786i	1.92058i	0.278998	+	0.960292i	$0.409998\pi$
$313$	33.9786i	1.92058i	−0.278998	+	0.960292i	$0.590002\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 10.0876i	− 0.566575i	−0.959035	−	0.283287i	$-0.908575\pi$
$317$	− 10.0876i	− 0.566575i	0.959035	−	0.283287i	$-0.0914251\pi$
$318$	0	0
$319$	2.11471	0.118401
$320$	0	0
$321$	−13.8495	−0.773001
$322$	0	0
$323$	− 44.2364i	− 2.46138i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0.654141i	0.0361741i
$328$	0	0
$329$	1.10548	0.0609471
$330$	0	0
$331$	−16.3891	−0.900824	−0.450412	−	0.892821i	$-0.648723\pi$
$331$	−16.3891	−0.900824	−0.450412	+	0.892821i	$0.648723\pi$
$332$	0	0
$333$	3.49641i	0.191602i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 18.3782i	− 1.00112i	−0.865700	−	0.500562i	$-0.833127\pi$
$337$	− 18.3782i	− 1.00112i	0.865700	−	0.500562i	$-0.166873\pi$
$338$	0	0
$339$	−8.55549	−0.464671
$340$	0	0
$341$	−21.7462	−1.17762
$342$	0	0
$343$	8.76611i	0.473326i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	14.6029i	0.783925i	0.919981	+	0.391962i	$0.128204\pi$
$347$	14.6029i	0.783925i	−0.919981	+	0.391962i	$0.871796\pi$
$348$	0	0
$349$	−21.4716	−1.14935	−0.574675	−	0.818382i	$-0.694871\pi$
$349$	−21.4716	−1.14935	−0.574675	+	0.818382i	$0.694871\pi$
$350$	0	0
$351$	7.13639	0.380912
$352$	0	0
$353$	22.9674i	1.22243i	0.791465	+	0.611214i	$0.209319\pi$
$353$	22.9674i	1.22243i	−0.791465	+	0.611214i	$0.790681\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 12.5702i	− 0.665287i
$358$	0	0
$359$	−1.62824	−0.0859351	−0.0429675	−	0.999076i	$-0.513681\pi$
$359$	−1.62824	−0.0859351	−0.0429675	+	0.999076i	$0.513681\pi$
$360$	0	0
$361$	44.9012	2.36322
$362$	0	0
$363$	4.39102i	0.230469i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 2.19451i	− 0.114552i	−0.998358	−	0.0572761i	$-0.981758\pi$
$367$	− 2.19451i	− 0.114552i	0.998358	−	0.0572761i	$-0.0182415\pi$
$368$	0	0
$369$	−4.62260	−0.240643
$370$	0	0
$371$	−14.1445	−0.734347
$372$	0	0
$373$	− 6.00186i	− 0.310764i	−0.987854	−	0.155382i	$-0.950339\pi$
$373$	− 6.00186i	− 0.310764i	0.987854	−	0.155382i	$-0.0496609\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.91246i	0.0984965i
$378$	0	0
$379$	0.966315	0.0496363	0.0248181	−	0.999692i	$-0.492099\pi$
$379$	0.966315	0.0496363	0.0248181	+	0.999692i	$0.492099\pi$
$380$	0	0
$381$	10.2267	0.523927
$382$	0	0
$383$	11.6799i	0.596816i	0.954438	+	0.298408i	$0.0964555\pi$
$383$	11.6799i	0.596816i	−0.954438	+	0.298408i	$0.903544\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	4.08468i	0.207636i
$388$	0	0
$389$	2.55610	0.129600	0.0647998	−	0.997898i	$-0.479359\pi$
$389$	2.55610	0.129600	0.0647998	+	0.997898i	$0.479359\pi$
$390$	0	0
$391$	18.1289	0.916819
$392$	0	0
$393$	− 4.06113i	− 0.204857i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	9.08996i	0.456212i	0.973636	+	0.228106i	$0.0732533\pi$
$397$	9.08996i	0.456212i	−0.973636	+	0.228106i	$0.926747\pi$
$398$	0	0
$399$	18.1582	0.909047
$400$	0	0
$401$	4.58696	0.229062	0.114531	−	0.993420i	$-0.463464\pi$
$401$	4.58696	0.229062	0.114531	+	0.993420i	$0.463464\pi$
$402$	0	0
$403$	− 19.6663i	− 0.979650i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	2.90236i	0.143865i
$408$	0	0
$409$	10.5214	0.520248	0.260124	−	0.965575i	$-0.416237\pi$
$409$	10.5214	0.520248	0.260124	+	0.965575i	$0.416237\pi$
$410$	0	0
$411$	−10.4193	−0.513948
$412$	0	0
$413$	11.1748i	0.549875i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	1.41012i	0.0690540i
$418$	0	0
$419$	−5.75688	−0.281242	−0.140621	−	0.990064i	$-0.544910\pi$
$419$	−5.75688	−0.281242	−0.140621	+	0.990064i	$0.544910\pi$
$420$	0	0
$421$	−29.0307	−1.41487	−0.707436	−	0.706778i	$-0.750148\pi$
$421$	−29.0307	−1.41487	−0.707436	+	0.706778i	$0.750148\pi$
$422$	0	0
$423$	− 0.833952i	− 0.0405482i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	23.6494i	1.14447i
$428$	0	0
$429$	2.72038	0.131341
$430$	0	0
$431$	−6.64217	−0.319942	−0.159971	−	0.987122i	$-0.551140\pi$
$431$	−6.64217	−0.319942	−0.159971	+	0.987122i	$0.551140\pi$
$432$	0	0
$433$	− 23.9576i	− 1.15133i	−0.817687	−	0.575663i	$-0.804744\pi$
$433$	− 23.9576i	− 1.15133i	0.817687	−	0.575663i	$-0.195256\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	26.1879i	1.25274i
$438$	0	0
$439$	14.7719	0.705022	0.352511	−	0.935808i	$-0.385328\pi$
$439$	14.7719	0.705022	0.352511	+	0.935808i	$0.385328\pi$
$440$	0	0
$441$	−11.2199	−0.534279
$442$	0	0
$443$	6.01027i	0.285557i	0.989755	+	0.142778i	$0.0456036\pi$
$443$	6.01027i	0.285557i	−0.989755	+	0.142778i	$0.954396\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	3.41471i	0.161510i
$448$	0	0
$449$	−13.0960	−0.618037	−0.309018	−	0.951056i	$-0.600000\pi$
$449$	−13.0960	−0.618037	−0.309018	+	0.951056i	$0.600000\pi$
$450$	0	0
$451$	−3.83721	−0.180687
$452$	0	0
$453$	− 8.50301i	− 0.399506i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	30.1969i	1.41255i	0.707938	+	0.706275i	$0.249626\pi$
$457$	30.1969i	1.41255i	−0.707938	+	0.706275i	$0.750374\pi$
$458$	0	0
$459$	−20.6496	−0.963842
$460$	0	0
$461$	−8.78242	−0.409038	−0.204519	−	0.978863i	$-0.565563\pi$
$461$	−8.78242	−0.409038	−0.204519	+	0.978863i	$0.565563\pi$
$462$	0	0
$463$	− 33.7175i	− 1.56699i	−0.621401	−	0.783493i	$-0.713436\pi$
$463$	− 33.7175i	− 1.56699i	0.621401	−	0.783493i	$-0.286564\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	19.9838i	0.924739i	0.886687	+	0.462370i	$0.153001\pi$
$467$	19.9838i	0.924739i	−0.886687	+	0.462370i	$0.846999\pi$
$468$	0	0
$469$	−22.2019	−1.02519
$470$	0	0
$471$	−13.2844	−0.612112
$472$	0	0
$473$	3.39068i	0.155904i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	10.6703i	0.488562i
$478$	0	0
$479$	15.2093	0.694930	0.347465	−	0.937693i	$-0.387043\pi$
$479$	15.2093	0.694930	0.347465	+	0.937693i	$0.387043\pi$
$480$	0	0
$481$	−2.62477	−0.119679
$482$	0	0
$483$	7.44157i	0.338603i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 25.9309i	− 1.17504i	−0.809210	−	0.587520i	$-0.800104\pi$
$487$	− 25.9309i	− 1.17504i	0.809210	−	0.587520i	$-0.199896\pi$
$488$	0	0
$489$	−13.4182	−0.606792
$490$	0	0
$491$	43.0917	1.94470	0.972351	−	0.233523i	$-0.0750254\pi$
$491$	43.0917	1.94470	0.972351	+	0.233523i	$0.0750254\pi$
$492$	0	0
$493$	− 5.53382i	− 0.249231i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	52.3452i	2.34800i
$498$	0	0
$499$	−15.1581	−0.678568	−0.339284	−	0.940684i	$-0.610185\pi$
$499$	−15.1581	−0.678568	−0.339284	+	0.940684i	$0.610185\pi$
$500$	0	0
$501$	−1.57755	−0.0704796
$502$	0	0
$503$	17.7730i	0.792461i	0.918151	+	0.396230i	$0.129682\pi$
$503$	17.7730i	0.792461i	−0.918151	+	0.396230i	$0.870318\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 6.28420i	− 0.279091i
$508$	0	0
$509$	−13.7177	−0.608029	−0.304014	−	0.952667i	$-0.598327\pi$
$509$	−13.7177	−0.608029	−0.304014	+	0.952667i	$0.598327\pi$
$510$	0	0
$511$	−18.3467	−0.811611
$512$	0	0
$513$	− 29.8292i	− 1.31699i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 0.692262i	− 0.0304456i
$518$	0	0
$519$	9.96775	0.437536
$520$	0	0
$521$	−28.4327	−1.24566	−0.622830	−	0.782357i	$-0.714017\pi$
$521$	−28.4327	−1.24566	−0.622830	+	0.782357i	$0.714017\pi$
$522$	0	0
$523$	− 16.0837i	− 0.703289i	−0.936134	−	0.351645i	$-0.885623\pi$
$523$	− 16.0837i	− 0.703289i	0.936134	−	0.351645i	$-0.114377\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	56.9059i	2.47886i
$528$	0	0
$529$	12.2677	0.533378
$530$	0	0
$531$	8.43004	0.365833
$532$	0	0
$533$	− 3.47021i	− 0.150311i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 4.11915i	− 0.177754i
$538$	0	0
$539$	−9.31357	−0.401164
$540$	0	0
$541$	−26.5985	−1.14356	−0.571779	−	0.820408i	$-0.693747\pi$
$541$	−26.5985	−1.14356	−0.571779	+	0.820408i	$0.693747\pi$
$542$	0	0
$543$	12.4340i	0.533595i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 10.6965i	− 0.457350i	−0.973503	−	0.228675i	$-0.926561\pi$
$547$	− 10.6965i	− 0.457350i	0.973503	−	0.228675i	$-0.0734392\pi$
$548$	0	0
$549$	17.8406	0.761420
$550$	0	0
$551$	7.99382	0.340548
$552$	0	0
$553$	− 9.21310i	− 0.391781i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 40.1771i	− 1.70236i	−0.524876	−	0.851179i	$-0.675888\pi$
$557$	− 40.1771i	− 1.70236i	0.524876	−	0.851179i	$-0.324112\pi$
$558$	0	0
$559$	−3.06639	−0.129694
$560$	0	0
$561$	−7.87159	−0.332339
$562$	0	0
$563$	− 22.8080i	− 0.961243i	−0.876928	−	0.480622i	$-0.840411\pi$
$563$	− 22.8080i	− 0.961243i	0.876928	−	0.480622i	$-0.159589\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	17.3330i	0.727917i
$568$	0	0
$569$	0.554721	0.0232551	0.0116275	−	0.999932i	$-0.496299\pi$
$569$	0.554721	0.0232551	0.0116275	+	0.999932i	$0.496299\pi$
$570$	0	0
$571$	−10.8977	−0.456055	−0.228028	−	0.973655i	$-0.573228\pi$
$571$	−10.8977	−0.456055	−0.228028	+	0.973655i	$0.573228\pi$
$572$	0	0
$573$	− 8.17834i	− 0.341655i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 36.1882i	− 1.50653i	−0.657715	−	0.753267i	$-0.728477\pi$
$577$	− 36.1882i	− 1.50653i	0.657715	−	0.753267i	$-0.271523\pi$
$578$	0	0
$579$	13.2062	0.548829
$580$	0	0
$581$	−50.6446	−2.10109
$582$	0	0
$583$	8.85743i	0.366837i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	10.6509i	0.439609i	0.975544	+	0.219805i	$0.0705420\pi$
$587$	10.6509i	0.439609i	−0.975544	+	0.219805i	$0.929458\pi$
$588$	0	0
$589$	−82.2028	−3.38711
$590$	0	0
$591$	11.4791	0.472187
$592$	0	0
$593$	− 43.0193i	− 1.76659i	−0.468815	−	0.883296i	$-0.655319\pi$
$593$	− 43.0193i	− 1.76659i	0.468815	−	0.883296i	$-0.344681\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 15.3812i	− 0.629511i
$598$	0	0
$599$	4.72552	0.193080	0.0965398	−	0.995329i	$-0.469223\pi$
$599$	4.72552	0.193080	0.0965398	+	0.995329i	$0.469223\pi$
$600$	0	0
$601$	−24.7076	−1.00784	−0.503921	−	0.863750i	$-0.668110\pi$
$601$	−24.7076	−1.00784	−0.503921	+	0.863750i	$0.668110\pi$
$602$	0	0
$603$	16.7487i	0.682059i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 15.0370i	− 0.610331i	−0.952299	−	0.305166i	$-0.901288\pi$
$607$	− 15.0370i	− 0.610331i	0.952299	−	0.305166i	$-0.0987119\pi$
$608$	0	0
$609$	2.27153	0.0920469
$610$	0	0
$611$	0.626052	0.0253273
$612$	0	0
$613$	22.1098i	0.893007i	0.894782	+	0.446504i	$0.147331\pi$
$613$	22.1098i	0.893007i	−0.894782	+	0.446504i	$0.852669\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 32.9264i	− 1.32557i	−0.748811	−	0.662783i	$-0.769375\pi$
$617$	− 32.9264i	− 1.32557i	0.748811	−	0.662783i	$-0.230625\pi$
$618$	0	0
$619$	2.81002	0.112944	0.0564721	−	0.998404i	$-0.482015\pi$
$619$	2.81002	0.112944	0.0564721	+	0.998404i	$0.482015\pi$
$620$	0	0
$621$	12.2246	0.490555
$622$	0	0
$623$	− 42.5663i	− 1.70538i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 11.3708i	− 0.454107i
$628$	0	0
$629$	7.59495	0.302830
$630$	0	0
$631$	5.73420	0.228275	0.114137	−	0.993465i	$-0.463590\pi$
$631$	5.73420	0.228275	0.114137	+	0.993465i	$0.463590\pi$
$632$	0	0
$633$	− 12.2644i	− 0.487468i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 8.42279i	− 0.333723i
$638$	0	0
$639$	39.4882	1.56213
$640$	0	0
$641$	−12.3873	−0.489270	−0.244635	−	0.969615i	$-0.578668\pi$
$641$	−12.3873	−0.489270	−0.244635	+	0.969615i	$0.578668\pi$
$642$	0	0
$643$	− 46.2926i	− 1.82560i	−0.408404	−	0.912801i	$-0.633915\pi$
$643$	− 46.2926i	− 1.82560i	0.408404	−	0.912801i	$-0.366085\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 47.7387i	− 1.87680i	−0.345551	−	0.938400i	$-0.612308\pi$
$647$	− 47.7387i	− 1.87680i	0.345551	−	0.938400i	$-0.387692\pi$
$648$	0	0
$649$	6.99776	0.274686
$650$	0	0
$651$	−23.3588	−0.915502
$652$	0	0
$653$	0.227460i	0.00890118i	0.999990	+	0.00445059i	$0.00141667\pi$
$653$	0.227460i	0.00890118i	−0.999990	+	0.00445059i	$0.998583\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	13.8404i	0.539966i
$658$	0	0
$659$	19.8569	0.773513	0.386757	−	0.922182i	$-0.373595\pi$
$659$	19.8569	0.773513	0.386757	+	0.922182i	$0.373595\pi$
$660$	0	0
$661$	−25.7959	−1.00334	−0.501671	−	0.865058i	$-0.667281\pi$
$661$	−25.7959	−1.00334	−0.501671	+	0.865058i	$0.667281\pi$
$662$	0	0
$663$	− 7.11873i	− 0.276468i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	3.27602i	0.126848i
$668$	0	0
$669$	2.35680	0.0911193
$670$	0	0
$671$	14.8095	0.571713
$672$	0	0
$673$	− 39.4615i	− 1.52113i	−0.649263	−	0.760564i	$-0.724923\pi$
$673$	− 39.4615i	− 1.52113i	0.649263	−	0.760564i	$-0.275077\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.31980i	0.0507241i	0.999678	+	0.0253621i	$0.00807386\pi$
$677$	1.31980i	0.0507241i	−0.999678	+	0.0253621i	$0.991926\pi$
$678$	0	0
$679$	−2.87701	−0.110410
$680$	0	0
$681$	8.79304	0.336950
$682$	0	0
$683$	− 6.48925i	− 0.248304i	−0.992263	−	0.124152i	$-0.960379\pi$
$683$	− 6.48925i	− 0.248304i	0.992263	−	0.124152i	$-0.0396211\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	10.5654i	0.403095i
$688$	0	0
$689$	−8.01028	−0.305167
$690$	0	0
$691$	−18.6676	−0.710150	−0.355075	−	0.934838i	$-0.615545\pi$
$691$	−18.6676	−0.710150	−0.355075	+	0.934838i	$0.615545\pi$
$692$	0	0
$693$	18.1931i	0.691098i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	10.0413i	0.380340i
$698$	0	0
$699$	−4.00961	−0.151657
$700$	0	0
$701$	7.90767	0.298669	0.149334	−	0.988787i	$-0.452287\pi$
$701$	7.90767	0.298669	0.149334	+	0.988787i	$0.452287\pi$
$702$	0	0
$703$	10.9712i	0.413787i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	58.9362i	2.21652i
$708$	0	0
$709$	−24.1817	−0.908163	−0.454082	−	0.890960i	$-0.650033\pi$
$709$	−24.1817	−0.908163	−0.454082	+	0.890960i	$0.650033\pi$
$710$	0	0
$711$	−6.95019	−0.260652
$712$	0	0
$713$	− 33.6883i	− 1.26164i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 3.50220i	− 0.130792i
$718$	0	0
$719$	−43.1372	−1.60875	−0.804374	−	0.594123i	$-0.797499\pi$
$719$	−43.1372	−1.60875	−0.804374	+	0.594123i	$0.797499\pi$
$720$	0	0
$721$	−36.1175	−1.34509
$722$	0	0
$723$	6.13628i	0.228211i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 40.6603i	− 1.50801i	−0.656870	−	0.754004i	$-0.728120\pi$
$727$	− 40.6603i	− 1.50801i	0.656870	−	0.754004i	$-0.271880\pi$
$728$	0	0
$729$	5.54568	0.205396
$730$	0	0
$731$	8.87280	0.328172
$732$	0	0
$733$	− 24.1828i	− 0.893211i	−0.894731	−	0.446606i	$-0.852633\pi$
$733$	− 24.1828i	− 0.893211i	0.894731	−	0.446606i	$-0.147367\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	13.9030i	0.512125i
$738$	0	0
$739$	−30.4007	−1.11831	−0.559153	−	0.829064i	$-0.688874\pi$
$739$	−30.4007	−1.11831	−0.559153	+	0.829064i	$0.688874\pi$
$740$	0	0
$741$	10.2833	0.377766
$742$	0	0
$743$	24.6273i	0.903488i	0.892148	+	0.451744i	$0.149198\pi$
$743$	24.6273i	0.903488i	−0.892148	+	0.451744i	$0.850802\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	38.2053i	1.39786i
$748$	0	0
$749$	69.5310	2.54061
$750$	0	0
$751$	−4.60339	−0.167980	−0.0839901	−	0.996467i	$-0.526766\pi$
$751$	−4.60339	−0.167980	−0.0839901	+	0.996467i	$0.526766\pi$
$752$	0	0
$753$	16.0317i	0.584226i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 36.2417i	− 1.31723i	−0.752482	−	0.658613i	$-0.771144\pi$
$757$	− 36.2417i	− 1.31723i	0.752482	−	0.658613i	$-0.228856\pi$
$758$	0	0
$759$	4.65998	0.169147
$760$	0	0
$761$	31.3704	1.13718	0.568588	−	0.822623i	$-0.307490\pi$
$761$	31.3704	1.13718	0.568588	+	0.822623i	$0.307490\pi$
$762$	0	0
$763$	− 3.28411i	− 0.118893i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	6.32847i	0.228508i
$768$	0	0
$769$	1.33146	0.0480135	0.0240068	−	0.999712i	$-0.492358\pi$
$769$	1.33146	0.0480135	0.0240068	+	0.999712i	$0.492358\pi$
$770$	0	0
$771$	5.91589	0.213056
$772$	0	0
$773$	− 32.9499i	− 1.18513i	−0.805524	−	0.592563i	$-0.798116\pi$
$773$	− 32.9499i	− 1.18513i	0.805524	−	0.592563i	$-0.201884\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	3.11758i	0.111843i
$778$	0	0
$779$	−14.5050	−0.519696
$780$	0	0
$781$	32.7791	1.17293
$782$	0	0
$783$	− 3.73153i	− 0.133354i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 9.27094i	− 0.330473i	−0.986254	−	0.165237i	$-0.947161\pi$
$787$	− 9.27094i	− 0.330473i	0.986254	−	0.165237i	$-0.0528388\pi$
$788$	0	0
$789$	−11.4774	−0.408607
$790$	0	0
$791$	42.9528	1.52722
$792$	0	0
$793$	13.3930i	0.475601i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	34.1758i	1.21057i	0.796009	+	0.605285i	$0.206941\pi$
$797$	34.1758i	1.21057i	−0.796009	+	0.605285i	$0.793059\pi$
$798$	0	0
$799$	−1.81152	−0.0640871
$800$	0	0
$801$	−32.1112	−1.13459
$802$	0	0
$803$	11.4889i	0.405434i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	12.2108i	0.429839i
$808$	0	0
$809$	−20.0891	−0.706294	−0.353147	−	0.935568i	$-0.614889\pi$
$809$	−20.0891	−0.706294	−0.353147	+	0.935568i	$0.614889\pi$
$810$	0	0
$811$	−8.35300	−0.293314	−0.146657	−	0.989187i	$-0.546851\pi$
$811$	−8.35300	−0.293314	−0.146657	+	0.989187i	$0.546851\pi$
$812$	0	0
$813$	8.87911i	0.311404i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	12.8171i	0.448414i
$818$	0	0
$819$	−16.4530	−0.574915
$820$	0	0
$821$	−5.83477	−0.203635	−0.101817	−	0.994803i	$-0.532466\pi$
$821$	−5.83477	−0.203635	−0.101817	+	0.994803i	$0.532466\pi$
$822$	0	0
$823$	0.662009i	0.0230762i	0.999933	+	0.0115381i	$0.00367277\pi$
$823$	0.662009i	0.0230762i	−0.999933	+	0.0115381i	$0.996327\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	42.7516i	1.48662i	0.668947	+	0.743310i	$0.266745\pi$
$827$	42.7516i	1.48662i	−0.668947	+	0.743310i	$0.733255\pi$
$828$	0	0
$829$	−24.2593	−0.842562	−0.421281	−	0.906930i	$-0.638419\pi$
$829$	−24.2593	−0.842562	−0.421281	+	0.906930i	$0.638419\pi$
$830$	0	0
$831$	−10.6062	−0.367926
$832$	0	0
$833$	24.3719i	0.844437i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	38.3724i	1.32634i
$838$	0	0
$839$	16.5618	0.571777	0.285889	−	0.958263i	$-0.407711\pi$
$839$	16.5618	0.571777	0.285889	+	0.958263i	$0.407711\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	0.750000i	0.0258314i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 22.0451i	− 0.757477i
$848$	0	0
$849$	11.9369	0.409672
$850$	0	0
$851$	−4.49621	−0.154128
$852$	0	0
$853$	2.34257i	0.0802081i	0.999196	+	0.0401041i	$0.0127689\pi$
$853$	2.34257i	0.0802081i	−0.999196	+	0.0401041i	$0.987231\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 55.3020i	− 1.88908i	−0.328397	−	0.944540i	$-0.606508\pi$
$857$	− 55.3020i	− 1.88908i	0.328397	−	0.944540i	$-0.393492\pi$
$858$	0	0
$859$	−44.9751	−1.53453	−0.767266	−	0.641329i	$-0.778383\pi$
$859$	−44.9751	−1.53453	−0.767266	+	0.641329i	$0.778383\pi$
$860$	0	0
$861$	−4.12175	−0.140469
$862$	0	0
$863$	48.3195i	1.64481i	0.568900	+	0.822407i	$0.307369\pi$
$863$	48.3195i	1.64481i	−0.568900	+	0.822407i	$0.692631\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	9.16356i	0.311211i
$868$	0	0
$869$	−5.76933	−0.195711
$870$	0	0
$871$	−12.5733	−0.426030
$872$	0	0
$873$	2.17036i	0.0734556i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 56.1140i	− 1.89483i	−0.320001	−	0.947417i	$-0.603683\pi$
$877$	− 56.1140i	− 1.89483i	0.320001	−	0.947417i	$-0.396317\pi$
$878$	0	0
$879$	−0.970867	−0.0327465
$880$	0	0
$881$	58.1133	1.95789	0.978944	−	0.204129i	$-0.0654361\pi$
$881$	58.1133	1.95789	0.978944	+	0.204129i	$0.0654361\pi$
$882$	0	0
$883$	− 21.9911i	− 0.740061i	−0.929020	−	0.370030i	$-0.879347\pi$
$883$	− 21.9911i	− 0.740061i	0.929020	−	0.370030i	$-0.120653\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	13.1035i	0.439973i	0.975503	+	0.219986i	$0.0706013\pi$
$887$	13.1035i	0.439973i	−0.975503	+	0.219986i	$0.929399\pi$
$888$	0	0
$889$	−51.3428	−1.72198
$890$	0	0
$891$	10.8541	0.363626
$892$	0	0
$893$	− 2.61682i	− 0.0875685i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	4.21429i	0.140711i
$898$	0	0
$899$	−10.2833	−0.342967
$900$	0	0
$901$	23.1783	0.772180
$902$	0	0
$903$	3.64211i	0.121202i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	51.7491i	1.71830i	0.511724	+	0.859150i	$0.329007\pi$
$907$	51.7491i	1.71830i	−0.511724	+	0.859150i	$0.670993\pi$
$908$	0	0
$909$	44.4603	1.47466
$910$	0	0
$911$	−28.9974	−0.960726	−0.480363	−	0.877070i	$-0.659495\pi$
$911$	−28.9974	−0.960726	−0.480363	+	0.877070i	$0.659495\pi$
$912$	0	0
$913$	31.7141i	1.04958i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	20.3889i	0.673299i
$918$	0	0
$919$	−25.0943	−0.827784	−0.413892	−	0.910326i	$-0.635831\pi$
$919$	−25.0943	−0.827784	−0.413892	+	0.910326i	$0.635831\pi$
$920$	0	0
$921$	−3.59458	−0.118445
$922$	0	0
$923$	29.6440i	0.975743i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	27.2464i	0.894888i
$928$	0	0
$929$	−50.3675	−1.65250	−0.826251	−	0.563301i	$-0.809531\pi$
$929$	−50.3675	−1.65250	−0.826251	+	0.563301i	$0.809531\pi$
$930$	0	0
$931$	−35.2062	−1.15384
$932$	0	0
$933$	− 16.5897i	− 0.543124i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 39.2746i	− 1.28304i	−0.767104	−	0.641522i	$-0.778303\pi$
$937$	− 39.2746i	− 1.28304i	0.767104	−	0.641522i	$-0.221697\pi$
$938$	0	0
$939$	22.8555	0.745862
$940$	0	0
$941$	4.60332	0.150064	0.0750320	−	0.997181i	$-0.476094\pi$
$941$	4.60332	0.150064	0.0750320	+	0.997181i	$0.476094\pi$
$942$	0	0
$943$	− 5.94443i	− 0.193577i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.40704i	0.0457226i	0.999739	+	0.0228613i	$0.00727762\pi$
$947$	1.40704i	0.0457226i	−0.999739	+	0.0228613i	$0.992722\pi$
$948$	0	0
$949$	−10.3901	−0.337275
$950$	0	0
$951$	−6.78536	−0.220030
$952$	0	0
$953$	− 45.4121i	− 1.47104i	−0.677502	−	0.735521i	$-0.736938\pi$
$953$	− 45.4121i	− 1.47104i	0.677502	−	0.735521i	$-0.263062\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 1.42245i	− 0.0459813i
$958$	0	0
$959$	52.3102	1.68918
$960$	0	0
$961$	74.7460	2.41116
$962$	0	0
$963$	− 52.4528i	− 1.69027i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	12.6658i	0.407305i	0.979043	+	0.203652i	$0.0652813\pi$
$967$	12.6658i	0.407305i	−0.979043	+	0.203652i	$0.934719\pi$
$968$	0	0
$969$	−29.7554	−0.955881
$970$	0	0
$971$	−24.5447	−0.787676	−0.393838	−	0.919180i	$-0.628853\pi$
$971$	−24.5447	−0.787676	−0.393838	+	0.919180i	$0.628853\pi$
$972$	0	0
$973$	− 7.07951i	− 0.226959i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 25.3008i	− 0.809444i	−0.914440	−	0.404722i	$-0.867368\pi$
$977$	− 25.3008i	− 0.809444i	0.914440	−	0.404722i	$-0.132632\pi$
$978$	0	0
$979$	−26.6554	−0.851912
$980$	0	0
$981$	−2.47747	−0.0790994
$982$	0	0
$983$	− 25.1822i	− 0.803188i	−0.915818	−	0.401594i	$-0.868456\pi$
$983$	− 25.1822i	− 0.803188i	0.915818	−	0.401594i	$-0.131544\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 0.743596i	− 0.0236689i
$988$	0	0
$989$	−5.25269	−0.167026
$990$	0	0
$991$	13.0936	0.415933	0.207967	−	0.978136i	$-0.433315\pi$
$991$	13.0936	0.415933	0.207967	+	0.978136i	$0.433315\pi$
$992$	0	0
$993$	11.0240i	0.349837i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 31.0809i	− 0.984341i	−0.870499	−	0.492171i	$-0.836204\pi$
$997$	− 31.0809i	− 0.984341i	0.870499	−	0.492171i	$-0.163796\pi$
$998$	0	0
$999$	5.12138	0.162033

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2900.2.c.i.349.4		10
5.2	odd	4		2900.2.a.k.1.2	yes	5
5.3	odd	4		2900.2.a.i.1.4	✓	5
5.4	even	2	inner	2900.2.c.i.349.7		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2900.2.a.i.1.4	✓	5	5.3	odd	4
2900.2.a.k.1.2	yes	5	5.2	odd	4
2900.2.c.i.349.4		10	1.1	even	1	trivial
2900.2.c.i.349.7		10	5.4	even	2	inner

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:	\( N \)	\(=\)	\( 2900 = 2^{2} \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2900.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.672645i q^{3} +3.37701i q^{7} +2.54755 q^{9} +2.11471 q^{11} +1.91246i q^{13} -5.53382i q^{17} +7.99382 q^{19} +2.27153 q^{21} +3.27602i q^{23} -3.73153i q^{27} +1.00000 q^{29} -10.2833 q^{31} -1.42245i q^{33} +1.37246i q^{37} +1.28640 q^{39} -1.81453 q^{41} +1.60338i q^{43} -0.327355i q^{47} -4.40418 q^{49} -3.72230 q^{51} +4.18848i q^{53} -5.37701i q^{57} +3.30908 q^{59} +7.00306 q^{61} +8.60309i q^{63} +6.57443i q^{67} +2.20360 q^{69} +15.5005 q^{71} +5.43283i q^{73} +7.14140i q^{77} -2.72819 q^{79} +5.13265 q^{81} +14.9969i q^{83} -0.672645i q^{87} -12.6048 q^{89} -6.45838 q^{91} +6.91700i q^{93} +0.851941i q^{97} +5.38734 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 6 q^{9} + 8 q^{11} + 8 q^{19} + 6 q^{21} + 10 q^{29} + 6 q^{31} + 36 q^{39} + 10 q^{41} + 6 q^{49} + 18 q^{51} + 46 q^{59} + 10 q^{61} + 60 q^{69} + 46 q^{71} - 8 q^{79} + 18 q^{81} + 12 q^{89} + 54 q^{91}+ \cdots + 72 q^{99}+O(q^{100}) \) 10 * q - 6 * q^9 + 8 * q^11 + 8 * q^19 + 6 * q^21 + 10 * q^29 + 6 * q^31 + 36 * q^39 + 10 * q^41 + 6 * q^49 + 18 * q^51 + 46 * q^59 + 10 * q^61 + 60 * q^69 + 46 * q^71 - 8 * q^79 + 18 * q^81 + 12 * q^89 + 54 * q^91 + 72 * q^99

Introduction

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