LMFDB - Embedded newform 1900.2.c.c.1749.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1900,2,Mod(1749,1900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1900, 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("1900.1749");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1900 = 2^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1900.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:	$15.1715763840$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 380)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1749.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1900.1749
Dual form			1900.2.c.c.1749.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000i q^{7} +3.00000 q^{9} -4.00000 q^{11} +4.00000i q^{13} +6.00000i q^{17} -1.00000 q^{19} +2.00000i q^{23} +6.00000 q^{29} -8.00000 q^{31} +4.00000i q^{37} +6.00000 q^{41} +6.00000i q^{43} +6.00000i q^{47} +3.00000 q^{49} -8.00000i q^{53} +12.0000 q^{59} +6.00000 q^{61} -6.00000i q^{63} +10.0000i q^{73} +8.00000i q^{77} +8.00000 q^{79} +9.00000 q^{81} -14.0000i q^{83} -14.0000 q^{89} +8.00000 q^{91} +16.0000i q^{97} -12.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{9} - 8 q^{11} - 2 q^{19} + 12 q^{29} - 16 q^{31} + 12 q^{41} + 6 q^{49} + 24 q^{59} + 12 q^{61} + 16 q^{79} + 18 q^{81} - 28 q^{89} + 16 q^{91} - 24 q^{99}+O(q^{100}) $ 2 * q + 6 * q^9 - 8 * q^11 - 2 * q^19 + 12 * q^29 - 16 * q^31 + 12 * q^41 + 6 * q^49 + 24 * q^59 + 12 * q^61 + 16 * q^79 + 18 * q^81 - 28 * q^89 + 16 * q^91 - 24 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times$.

$n$	$77$	$401$	$951$
$\chi(n)$	$-1$	$1$	$1$

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	0	1.00000			$0$
$3$	0	0	−1.00000			$\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 2.00000i	− 0.755929i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$	− 2.00000i	− 0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$	0	0
$9$	3.00000	1.00000
$10$	0	0
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	0	0
$13$	4.00000i	1.10940i	0.832050	+	0.554700i	$0.187167\pi$
$13$	4.00000i	1.10940i	−0.832050	+	0.554700i	$0.812833\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.00000i	1.45521i	0.685994	+	0.727607i	$0.259367\pi$
$17$	6.00000i	1.45521i	−0.685994	+	0.727607i	$0.740633\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.00000i	0.417029i	0.978019	+	0.208514i	$0.0668628\pi$
$23$	2.00000i	0.417029i	−0.978019	+	0.208514i	$0.933137\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	4.00000i	0.657596i	0.944400	+	0.328798i	$0.106644\pi$
$37$	4.00000i	0.657596i	−0.944400	+	0.328798i	$0.893356\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	6.00000i	0.914991i	0.889212	+	0.457496i	$0.151253\pi$
$43$	6.00000i	0.914991i	−0.889212	+	0.457496i	$0.848747\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.00000i	0.875190i	0.899172	+	0.437595i	$0.144170\pi$
$47$	6.00000i	0.875190i	−0.899172	+	0.437595i	$0.855830\pi$
$48$	0	0
$49$	3.00000	0.428571
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 8.00000i	− 1.09888i	−0.835532	−	0.549442i	$-0.814840\pi$
$53$	− 8.00000i	− 1.09888i	0.835532	−	0.549442i	$-0.185160\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	0	0
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	0	0
$63$	− 6.00000i	− 0.755929i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	10.0000i	1.17041i	0.810885	+	0.585206i	$0.198986\pi$
$73$	10.0000i	1.17041i	−0.810885	+	0.585206i	$0.801014\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	8.00000i	0.911685i
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	− 14.0000i	− 1.53670i	−0.640030	−	0.768350i	$-0.721078\pi$
$83$	− 14.0000i	− 1.53670i	0.640030	−	0.768350i	$-0.278922\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−14.0000	−1.48400	−0.741999	−	0.670402i	$-0.766122\pi$
$89$	−14.0000	−1.48400	−0.741999	+	0.670402i	$0.766122\pi$
$90$	0	0
$91$	8.00000	0.838628
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	16.0000i	1.62455i	0.583272	+	0.812277i	$0.301772\pi$
$97$	16.0000i	1.62455i	−0.583272	+	0.812277i	$0.698228\pi$
$98$	0	0
$99$	−12.0000	−1.20605
$100$	0	0
$101$	−14.0000	−1.39305	−0.696526	−	0.717532i	$-0.745272\pi$
$101$	−14.0000	−1.39305	−0.696526	+	0.717532i	$0.745272\pi$
$102$	0	0
$103$	20.0000i	1.97066i	0.170664	+	0.985329i	$0.445409\pi$
$103$	20.0000i	1.97066i	−0.170664	+	0.985329i	$0.554591\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 4.00000i	− 0.386695i	−0.981130	−	0.193347i	$-0.938066\pi$
$107$	− 4.00000i	− 0.386695i	0.981130	−	0.193347i	$-0.0619344\pi$
$108$	0	0
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	12.0000i	1.10940i
$118$	0	0
$119$	12.0000	1.10004
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	12.0000i	1.06483i	0.846484	+	0.532414i	$0.178715\pi$
$127$	12.0000i	1.06483i	−0.846484	+	0.532414i	$0.821285\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−8.00000	−0.698963	−0.349482	−	0.936943i	$-0.613642\pi$
$131$	−8.00000	−0.698963	−0.349482	+	0.936943i	$0.613642\pi$
$132$	0	0
$133$	2.00000i	0.173422i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	6.00000i	0.512615i	0.966595	+	0.256307i	$0.0825059\pi$
$137$	6.00000i	0.512615i	−0.966595	+	0.256307i	$0.917494\pi$
$138$	0	0
$139$	8.00000	0.678551	0.339276	−	0.940687i	$-0.389818\pi$
$139$	8.00000	0.678551	0.339276	+	0.940687i	$0.389818\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 16.0000i	− 1.33799i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	10.0000	0.819232	0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000	0.819232	0.409616	+	0.912258i	$0.365663\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	0	0
$153$	18.0000i	1.45521i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 18.0000i	− 1.43656i	−0.695756	−	0.718278i	$-0.744931\pi$
$157$	− 18.0000i	− 1.43656i	0.695756	−	0.718278i	$-0.255069\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	4.00000	0.315244
$162$	0	0
$163$	14.0000i	1.09656i	0.836293	+	0.548282i	$0.184718\pi$
$163$	14.0000i	1.09656i	−0.836293	+	0.548282i	$0.815282\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	16.0000i	1.23812i	0.785345	+	0.619059i	$0.212486\pi$
$167$	16.0000i	1.23812i	−0.785345	+	0.619059i	$0.787514\pi$
$168$	0	0
$169$	−3.00000	−0.230769
$170$	0	0
$171$	−3.00000	−0.229416
$172$	0	0
$173$	8.00000i	0.608229i	0.952636	+	0.304114i	$0.0983605\pi$
$173$	8.00000i	0.608229i	−0.952636	+	0.304114i	$0.901639\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	0	0
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 24.0000i	− 1.75505i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	0	0
$193$	− 16.0000i	− 1.15171i	−0.817554	−	0.575853i	$-0.804670\pi$
$193$	− 16.0000i	− 1.15171i	0.817554	−	0.575853i	$-0.195330\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 22.0000i	− 1.56744i	−0.621117	−	0.783718i	$-0.713321\pi$
$197$	− 22.0000i	− 1.56744i	0.621117	−	0.783718i	$-0.286679\pi$
$198$	0	0
$199$	24.0000	1.70131	0.850657	−	0.525720i	$-0.176204\pi$
$199$	24.0000	1.70131	0.850657	+	0.525720i	$0.176204\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 12.0000i	− 0.842235i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	6.00000i	0.417029i
$208$	0	0
$209$	4.00000	0.276686
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	16.0000i	1.08615i
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−24.0000	−1.61441
$222$	0	0
$223$	− 4.00000i	− 0.267860i	−0.990991	−	0.133930i	$-0.957240\pi$
$223$	− 4.00000i	− 0.267860i	0.990991	−	0.133930i	$-0.0427597\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 4.00000i	− 0.265489i	−0.991150	−	0.132745i	$-0.957621\pi$
$227$	− 4.00000i	− 0.265489i	0.991150	−	0.132745i	$-0.0423790\pi$
$228$	0	0
$229$	6.00000	0.396491	0.198246	−	0.980152i	$-0.436476\pi$
$229$	6.00000	0.396491	0.198246	+	0.980152i	$0.436476\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	14.0000i	0.917170i	0.888650	+	0.458585i	$0.151644\pi$
$233$	14.0000i	0.917170i	−0.888650	+	0.458585i	$0.848356\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−16.0000	−1.03495	−0.517477	−	0.855697i	$-0.673129\pi$
$239$	−16.0000	−1.03495	−0.517477	+	0.855697i	$0.673129\pi$
$240$	0	0
$241$	6.00000	0.386494	0.193247	−	0.981150i	$-0.438098\pi$
$241$	6.00000	0.386494	0.193247	+	0.981150i	$0.438098\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 4.00000i	− 0.254514i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	− 8.00000i	− 0.502956i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 20.0000i	− 1.24757i	−0.781598	−	0.623783i	$-0.785595\pi$
$257$	− 20.0000i	− 1.24757i	0.781598	−	0.623783i	$-0.214405\pi$
$258$	0	0
$259$	8.00000	0.497096
$260$	0	0
$261$	18.0000	1.11417
$262$	0	0
$263$	− 14.0000i	− 0.863277i	−0.902047	−	0.431638i	$-0.857936\pi$
$263$	− 14.0000i	− 0.863277i	0.902047	−	0.431638i	$-0.142064\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−14.0000	−0.853595	−0.426798	−	0.904347i	$-0.640358\pi$
$269$	−14.0000	−0.853595	−0.426798	+	0.904347i	$0.640358\pi$
$270$	0	0
$271$	−24.0000	−1.45790	−0.728948	−	0.684569i	$-0.759990\pi$
$271$	−24.0000	−1.45790	−0.728948	+	0.684569i	$0.759990\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 26.0000i	− 1.56219i	−0.624413	−	0.781094i	$-0.714662\pi$
$277$	− 26.0000i	− 1.56219i	0.624413	−	0.781094i	$-0.285338\pi$
$278$	0	0
$279$	−24.0000	−1.43684
$280$	0	0
$281$	30.0000	1.78965	0.894825	−	0.446417i	$-0.147300\pi$
$281$	30.0000	1.78965	0.894825	+	0.446417i	$0.147300\pi$
$282$	0	0
$283$	− 22.0000i	− 1.30776i	−0.756596	−	0.653882i	$-0.773139\pi$
$283$	− 22.0000i	− 1.30776i	0.756596	−	0.653882i	$-0.226861\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 12.0000i	− 0.708338i
$288$	0	0
$289$	−19.0000	−1.11765
$290$	0	0
$291$	0	0
$292$	0	0
$293$	24.0000i	1.40209i	0.713115	+	0.701047i	$0.247284\pi$
$293$	24.0000i	1.40209i	−0.713115	+	0.701047i	$0.752716\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−8.00000	−0.462652
$300$	0	0
$301$	12.0000	0.691669
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	4.00000i	0.228292i	0.993464	+	0.114146i	$0.0364132\pi$
$307$	4.00000i	0.228292i	−0.993464	+	0.114146i	$0.963587\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−8.00000	−0.453638	−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000	−0.453638	−0.226819	+	0.973937i	$0.572833\pi$
$312$	0	0
$313$	22.0000i	1.24351i	0.783210	+	0.621757i	$0.213581\pi$
$313$	22.0000i	1.24351i	−0.783210	+	0.621757i	$0.786419\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 12.0000i	− 0.673987i	−0.941507	−	0.336994i	$-0.890590\pi$
$317$	− 12.0000i	− 0.673987i	0.941507	−	0.336994i	$-0.109410\pi$
$318$	0	0
$319$	−24.0000	−1.34374
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 6.00000i	− 0.333849i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	12.0000	0.661581
$330$	0	0
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	0	0
$333$	12.0000i	0.657596i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	24.0000i	1.30736i	0.756770	+	0.653682i	$0.226776\pi$
$337$	24.0000i	1.30736i	−0.756770	+	0.653682i	$0.773224\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	32.0000	1.73290
$342$	0	0
$343$	− 20.0000i	− 1.07990i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 6.00000i	− 0.322097i	−0.986947	−	0.161048i	$-0.948512\pi$
$347$	− 6.00000i	− 0.322097i	0.986947	−	0.161048i	$-0.0514875\pi$
$348$	0	0
$349$	−14.0000	−0.749403	−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000	−0.749403	−0.374701	+	0.927146i	$0.622255\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	2.00000i	0.106449i	0.998583	+	0.0532246i	$0.0169499\pi$
$353$	2.00000i	0.106449i	−0.998583	+	0.0532246i	$0.983050\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−4.00000	−0.211112	−0.105556	−	0.994413i	$-0.533662\pi$
$359$	−4.00000	−0.211112	−0.105556	+	0.994413i	$0.533662\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 10.0000i	− 0.521996i	−0.965339	−	0.260998i	$-0.915948\pi$
$367$	− 10.0000i	− 0.521996i	0.965339	−	0.260998i	$-0.0840516\pi$
$368$	0	0
$369$	18.0000	0.937043
$370$	0	0
$371$	−16.0000	−0.830679
$372$	0	0
$373$	8.00000i	0.414224i	0.978317	+	0.207112i	$0.0664065\pi$
$373$	8.00000i	0.414224i	−0.978317	+	0.207112i	$0.933593\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	24.0000i	1.23606i
$378$	0	0
$379$	−12.0000	−0.616399	−0.308199	−	0.951322i	$-0.599726\pi$
$379$	−12.0000	−0.616399	−0.308199	+	0.951322i	$0.599726\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 32.0000i	− 1.63512i	−0.575841	−	0.817562i	$-0.695325\pi$
$383$	− 32.0000i	− 1.63512i	0.575841	−	0.817562i	$-0.304675\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	18.0000i	0.914991i
$388$	0	0
$389$	−26.0000	−1.31825	−0.659126	−	0.752032i	$-0.729074\pi$
$389$	−26.0000	−1.31825	−0.659126	+	0.752032i	$0.729074\pi$
$390$	0	0
$391$	−12.0000	−0.606866
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 10.0000i	− 0.501886i	−0.968002	−	0.250943i	$-0.919259\pi$
$397$	− 10.0000i	− 0.501886i	0.968002	−	0.250943i	$-0.0807406\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−30.0000	−1.49813	−0.749064	−	0.662497i	$-0.769497\pi$
$401$	−30.0000	−1.49813	−0.749064	+	0.662497i	$0.769497\pi$
$402$	0	0
$403$	− 32.0000i	− 1.59403i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 16.0000i	− 0.793091i
$408$	0	0
$409$	−10.0000	−0.494468	−0.247234	−	0.968956i	$-0.579522\pi$
$409$	−10.0000	−0.494468	−0.247234	+	0.968956i	$0.579522\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 24.0000i	− 1.18096i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	4.00000	0.195413	0.0977064	−	0.995215i	$-0.468849\pi$
$419$	4.00000	0.195413	0.0977064	+	0.995215i	$0.468849\pi$
$420$	0	0
$421$	10.0000	0.487370	0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000	0.487370	0.243685	+	0.969854i	$0.421644\pi$
$422$	0	0
$423$	18.0000i	0.875190i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 12.0000i	− 0.580721i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	24.0000	1.15604	0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000	1.15604	0.578020	+	0.816023i	$0.303826\pi$
$432$	0	0
$433$	16.0000i	0.768911i	0.923144	+	0.384455i	$0.125611\pi$
$433$	16.0000i	0.768911i	−0.923144	+	0.384455i	$0.874389\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 2.00000i	− 0.0956730i
$438$	0	0
$439$	−8.00000	−0.381819	−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000	−0.381819	−0.190910	+	0.981608i	$0.561144\pi$
$440$	0	0
$441$	9.00000	0.428571
$442$	0	0
$443$	− 30.0000i	− 1.42534i	−0.701498	−	0.712672i	$-0.747485\pi$
$443$	− 30.0000i	− 1.42534i	0.701498	−	0.712672i	$-0.252515\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−2.00000	−0.0943858	−0.0471929	−	0.998886i	$-0.515028\pi$
$449$	−2.00000	−0.0943858	−0.0471929	+	0.998886i	$0.515028\pi$
$450$	0	0
$451$	−24.0000	−1.13012
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	10.0000i	0.467780i	0.972263	+	0.233890i	$0.0751456\pi$
$457$	10.0000i	0.467780i	−0.972263	+	0.233890i	$0.924854\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−18.0000	−0.838344	−0.419172	−	0.907907i	$-0.637680\pi$
$461$	−18.0000	−0.838344	−0.419172	+	0.907907i	$0.637680\pi$
$462$	0	0
$463$	− 22.0000i	− 1.02243i	−0.859454	−	0.511213i	$-0.829196\pi$
$463$	− 22.0000i	− 1.02243i	0.859454	−	0.511213i	$-0.170804\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 14.0000i	− 0.647843i	−0.946084	−	0.323921i	$-0.894999\pi$
$467$	− 14.0000i	− 0.647843i	0.946084	−	0.323921i	$-0.105001\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 24.0000i	− 1.10352i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 24.0000i	− 1.09888i
$478$	0	0
$479$	12.0000	0.548294	0.274147	−	0.961688i	$-0.411605\pi$
$479$	12.0000	0.548294	0.274147	+	0.961688i	$0.411605\pi$
$480$	0	0
$481$	−16.0000	−0.729537
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 32.0000i	− 1.45006i	−0.688718	−	0.725029i	$-0.741826\pi$
$487$	− 32.0000i	− 1.45006i	0.688718	−	0.725029i	$-0.258174\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−24.0000	−1.08310	−0.541552	−	0.840667i	$-0.682163\pi$
$491$	−24.0000	−1.08310	−0.541552	+	0.840667i	$0.682163\pi$
$492$	0	0
$493$	36.0000i	1.62136i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−24.0000	−1.07439	−0.537194	−	0.843459i	$-0.680516\pi$
$499$	−24.0000	−1.07439	−0.537194	+	0.843459i	$0.680516\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 6.00000i	− 0.267527i	−0.991013	−	0.133763i	$-0.957294\pi$
$503$	− 6.00000i	− 0.267527i	0.991013	−	0.133763i	$-0.0427062\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−26.0000	−1.15243	−0.576215	−	0.817298i	$-0.695471\pi$
$509$	−26.0000	−1.15243	−0.576215	+	0.817298i	$0.695471\pi$
$510$	0	0
$511$	20.0000	0.884748
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 24.0000i	− 1.05552i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−30.0000	−1.31432	−0.657162	−	0.753749i	$-0.728243\pi$
$521$	−30.0000	−1.31432	−0.657162	+	0.753749i	$0.728243\pi$
$522$	0	0
$523$	36.0000i	1.57417i	0.616844	+	0.787085i	$0.288411\pi$
$523$	36.0000i	1.57417i	−0.616844	+	0.787085i	$0.711589\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 48.0000i	− 2.09091i
$528$	0	0
$529$	19.0000	0.826087
$530$	0	0
$531$	36.0000	1.56227
$532$	0	0
$533$	24.0000i	1.03956i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−12.0000	−0.516877
$540$	0	0
$541$	34.0000	1.46177	0.730887	−	0.682498i	$-0.239107\pi$
$541$	34.0000	1.46177	0.730887	+	0.682498i	$0.239107\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	20.0000i	0.855138i	0.903983	+	0.427569i	$0.140630\pi$
$547$	20.0000i	0.855138i	−0.903983	+	0.427569i	$0.859370\pi$
$548$	0	0
$549$	18.0000	0.768221
$550$	0	0
$551$	−6.00000	−0.255609
$552$	0	0
$553$	− 16.0000i	− 0.680389i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 18.0000i	− 0.762684i	−0.924434	−	0.381342i	$-0.875462\pi$
$557$	− 18.0000i	− 0.762684i	0.924434	−	0.381342i	$-0.124538\pi$
$558$	0	0
$559$	−24.0000	−1.01509
$560$	0	0
$561$	0	0
$562$	0	0
$563$	4.00000i	0.168580i	0.996441	+	0.0842900i	$0.0268622\pi$
$563$	4.00000i	0.168580i	−0.996441	+	0.0842900i	$0.973138\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 18.0000i	− 0.755929i
$568$	0	0
$569$	30.0000	1.25767	0.628833	−	0.777541i	$-0.283533\pi$
$569$	30.0000	1.25767	0.628833	+	0.777541i	$0.283533\pi$
$570$	0	0
$571$	36.0000	1.50655	0.753277	−	0.657704i	$-0.228472\pi$
$571$	36.0000	1.50655	0.753277	+	0.657704i	$0.228472\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	2.00000i	0.0832611i	0.999133	+	0.0416305i	$0.0132552\pi$
$577$	2.00000i	0.0832611i	−0.999133	+	0.0416305i	$0.986745\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−28.0000	−1.16164
$582$	0	0
$583$	32.0000i	1.32530i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 6.00000i	− 0.247647i	−0.992304	−	0.123823i	$-0.960484\pi$
$587$	− 6.00000i	− 0.247647i	0.992304	−	0.123823i	$-0.0395156\pi$
$588$	0	0
$589$	8.00000	0.329634
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 42.0000i	− 1.72473i	−0.506284	−	0.862367i	$-0.668981\pi$
$593$	− 42.0000i	− 1.72473i	0.506284	−	0.862367i	$-0.331019\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−16.0000	−0.653742	−0.326871	−	0.945069i	$-0.605994\pi$
$599$	−16.0000	−0.653742	−0.326871	+	0.945069i	$0.605994\pi$
$600$	0	0
$601$	30.0000	1.22373	0.611863	−	0.790964i	$-0.290420\pi$
$601$	30.0000	1.22373	0.611863	+	0.790964i	$0.290420\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	16.0000i	0.649420i	0.945814	+	0.324710i	$0.105267\pi$
$607$	16.0000i	0.649420i	−0.945814	+	0.324710i	$0.894733\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−24.0000	−0.970936
$612$	0	0
$613$	2.00000i	0.0807792i	0.999184	+	0.0403896i	$0.0128599\pi$
$613$	2.00000i	0.0807792i	−0.999184	+	0.0403896i	$0.987140\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 10.0000i	− 0.402585i	−0.979531	−	0.201292i	$-0.935486\pi$
$617$	− 10.0000i	− 0.402585i	0.979531	−	0.201292i	$-0.0645141\pi$
$618$	0	0
$619$	48.0000	1.92928	0.964641	−	0.263566i	$-0.0848986\pi$
$619$	48.0000	1.92928	0.964641	+	0.263566i	$0.0848986\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	28.0000i	1.12180i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−24.0000	−0.956943
$630$	0	0
$631$	−40.0000	−1.59237	−0.796187	−	0.605050i	$-0.793153\pi$
$631$	−40.0000	−1.59237	−0.796187	+	0.605050i	$0.793153\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	12.0000i	0.475457i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	14.0000	0.552967	0.276483	−	0.961019i	$-0.410831\pi$
$641$	14.0000	0.552967	0.276483	+	0.961019i	$0.410831\pi$
$642$	0	0
$643$	34.0000i	1.34083i	0.741987	+	0.670415i	$0.233884\pi$
$643$	34.0000i	1.34083i	−0.741987	+	0.670415i	$0.766116\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 6.00000i	− 0.235884i	−0.993020	−	0.117942i	$-0.962370\pi$
$647$	− 6.00000i	− 0.235884i	0.993020	−	0.117942i	$-0.0376297\pi$
$648$	0	0
$649$	−48.0000	−1.88416
$650$	0	0
$651$	0	0
$652$	0	0
$653$	10.0000i	0.391330i	0.980671	+	0.195665i	$0.0626866\pi$
$653$	10.0000i	0.391330i	−0.980671	+	0.195665i	$0.937313\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	30.0000i	1.17041i
$658$	0	0
$659$	36.0000	1.40236	0.701180	−	0.712984i	$-0.252657\pi$
$659$	36.0000	1.40236	0.701180	+	0.712984i	$0.252657\pi$
$660$	0	0
$661$	42.0000	1.63361	0.816805	−	0.576913i	$-0.195743\pi$
$661$	42.0000	1.63361	0.816805	+	0.576913i	$0.195743\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	12.0000i	0.464642i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−24.0000	−0.926510
$672$	0	0
$673$	− 28.0000i	− 1.07932i	−0.841883	−	0.539660i	$-0.818553\pi$
$673$	− 28.0000i	− 1.07932i	0.841883	−	0.539660i	$-0.181447\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0	0
$679$	32.0000	1.22805
$680$	0	0
$681$	0	0
$682$	0	0
$683$	44.0000i	1.68361i	0.539779	+	0.841807i	$0.318508\pi$
$683$	44.0000i	1.68361i	−0.539779	+	0.841807i	$0.681492\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	32.0000	1.21910
$690$	0	0
$691$	−36.0000	−1.36950	−0.684752	−	0.728776i	$-0.740090\pi$
$691$	−36.0000	−1.36950	−0.684752	+	0.728776i	$0.740090\pi$
$692$	0	0
$693$	24.0000i	0.911685i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	36.0000i	1.36360i
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−18.0000	−0.679851	−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000	−0.679851	−0.339925	+	0.940452i	$0.610402\pi$
$702$	0	0
$703$	− 4.00000i	− 0.150863i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	28.0000i	1.05305i
$708$	0	0
$709$	10.0000	0.375558	0.187779	−	0.982211i	$-0.439871\pi$
$709$	10.0000	0.375558	0.187779	+	0.982211i	$0.439871\pi$
$710$	0	0
$711$	24.0000	0.900070
$712$	0	0
$713$	− 16.0000i	− 0.599205i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	12.0000	0.447524	0.223762	−	0.974644i	$-0.428166\pi$
$719$	12.0000	0.447524	0.223762	+	0.974644i	$0.428166\pi$
$720$	0	0
$721$	40.0000	1.48968
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	14.0000i	0.519231i	0.965712	+	0.259616i	$0.0835959\pi$
$727$	14.0000i	0.519231i	−0.965712	+	0.259616i	$0.916404\pi$
$728$	0	0
$729$	27.0000	1.00000
$730$	0	0
$731$	−36.0000	−1.33151
$732$	0	0
$733$	− 42.0000i	− 1.55131i	−0.631160	−	0.775653i	$-0.717421\pi$
$733$	− 42.0000i	− 1.55131i	0.631160	−	0.775653i	$-0.282579\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−44.0000	−1.61857	−0.809283	−	0.587419i	$-0.800144\pi$
$739$	−44.0000	−1.61857	−0.809283	+	0.587419i	$0.800144\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 24.0000i	− 0.880475i	−0.897881	−	0.440237i	$-0.854894\pi$
$743$	− 24.0000i	− 0.880475i	0.897881	−	0.440237i	$-0.145106\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 42.0000i	− 1.53670i
$748$	0	0
$749$	−8.00000	−0.292314
$750$	0	0
$751$	40.0000	1.45962	0.729810	−	0.683650i	$-0.239608\pi$
$751$	40.0000	1.45962	0.729810	+	0.683650i	$0.239608\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	42.0000i	1.52652i	0.646094	+	0.763258i	$0.276401\pi$
$757$	42.0000i	1.52652i	−0.646094	+	0.763258i	$0.723599\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−10.0000	−0.362500	−0.181250	−	0.983437i	$-0.558014\pi$
$761$	−10.0000	−0.362500	−0.181250	+	0.983437i	$0.558014\pi$
$762$	0	0
$763$	− 4.00000i	− 0.144810i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	48.0000i	1.73318i
$768$	0	0
$769$	2.00000	0.0721218	0.0360609	−	0.999350i	$-0.488519\pi$
$769$	2.00000	0.0721218	0.0360609	+	0.999350i	$0.488519\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 36.0000i	− 1.29483i	−0.762138	−	0.647415i	$-0.775850\pi$
$773$	− 36.0000i	− 1.29483i	0.762138	−	0.647415i	$-0.224150\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−6.00000	−0.214972
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	12.0000i	0.427754i	0.976861	+	0.213877i	$0.0686091\pi$
$787$	12.0000i	0.427754i	−0.976861	+	0.213877i	$0.931391\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	24.0000i	0.852265i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	16.0000i	0.566749i	0.959009	+	0.283375i	$0.0914540\pi$
$797$	16.0000i	0.566749i	−0.959009	+	0.283375i	$0.908546\pi$
$798$	0	0
$799$	−36.0000	−1.27359
$800$	0	0
$801$	−42.0000	−1.48400
$802$	0	0
$803$	− 40.0000i	− 1.41157i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−10.0000	−0.351581	−0.175791	−	0.984428i	$-0.556248\pi$
$809$	−10.0000	−0.351581	−0.175791	+	0.984428i	$0.556248\pi$
$810$	0	0
$811$	28.0000	0.983213	0.491606	−	0.870817i	$-0.336410\pi$
$811$	28.0000	0.983213	0.491606	+	0.870817i	$0.336410\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 6.00000i	− 0.209913i
$818$	0	0
$819$	24.0000	0.838628
$820$	0	0
$821$	−22.0000	−0.767805	−0.383903	−	0.923374i	$-0.625420\pi$
$821$	−22.0000	−0.767805	−0.383903	+	0.923374i	$0.625420\pi$
$822$	0	0
$823$	− 50.0000i	− 1.74289i	−0.490493	−	0.871445i	$-0.663183\pi$
$823$	− 50.0000i	− 1.74289i	0.490493	−	0.871445i	$-0.336817\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	24.0000i	0.834562i	0.908778	+	0.417281i	$0.137017\pi$
$827$	24.0000i	0.834562i	−0.908778	+	0.417281i	$0.862983\pi$
$828$	0	0
$829$	−14.0000	−0.486240	−0.243120	−	0.969996i	$-0.578171\pi$
$829$	−14.0000	−0.486240	−0.243120	+	0.969996i	$0.578171\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	18.0000i	0.623663i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	40.0000	1.38095	0.690477	−	0.723355i	$-0.257401\pi$
$839$	40.0000	1.38095	0.690477	+	0.723355i	$0.257401\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 10.0000i	− 0.343604i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−8.00000	−0.274236
$852$	0	0
$853$	− 18.0000i	− 0.616308i	−0.951336	−	0.308154i	$-0.900289\pi$
$853$	− 18.0000i	− 0.616308i	0.951336	−	0.308154i	$-0.0997113\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 52.0000i	− 1.77629i	−0.459567	−	0.888143i	$-0.651995\pi$
$857$	− 52.0000i	− 1.77629i	0.459567	−	0.888143i	$-0.348005\pi$
$858$	0	0
$859$	28.0000	0.955348	0.477674	−	0.878537i	$-0.341480\pi$
$859$	28.0000	0.955348	0.477674	+	0.878537i	$0.341480\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	40.0000i	1.36162i	0.732462	+	0.680808i	$0.238371\pi$
$863$	40.0000i	1.36162i	−0.732462	+	0.680808i	$0.761629\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−32.0000	−1.08553
$870$	0	0
$871$	0	0
$872$	0	0
$873$	48.0000i	1.62455i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 24.0000i	− 0.810422i	−0.914223	−	0.405211i	$-0.867198\pi$
$877$	− 24.0000i	− 0.810422i	0.914223	−	0.405211i	$-0.132802\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	10.0000	0.336909	0.168454	−	0.985709i	$-0.446122\pi$
$881$	10.0000	0.336909	0.168454	+	0.985709i	$0.446122\pi$
$882$	0	0
$883$	6.00000i	0.201916i	0.994891	+	0.100958i	$0.0321908\pi$
$883$	6.00000i	0.201916i	−0.994891	+	0.100958i	$0.967809\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 4.00000i	− 0.134307i	−0.997743	−	0.0671534i	$-0.978608\pi$
$887$	− 4.00000i	− 0.134307i	0.997743	−	0.0671534i	$-0.0213917\pi$
$888$	0	0
$889$	24.0000	0.804934
$890$	0	0
$891$	−36.0000	−1.20605
$892$	0	0
$893$	− 6.00000i	− 0.200782i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−48.0000	−1.60089
$900$	0	0
$901$	48.0000	1.59911
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	44.0000i	1.46100i	0.682915	+	0.730498i	$0.260712\pi$
$907$	44.0000i	1.46100i	−0.682915	+	0.730498i	$0.739288\pi$
$908$	0	0
$909$	−42.0000	−1.39305
$910$	0	0
$911$	24.0000	0.795155	0.397578	−	0.917568i	$-0.369851\pi$
$911$	24.0000	0.795155	0.397578	+	0.917568i	$0.369851\pi$
$912$	0	0
$913$	56.0000i	1.85333i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	16.0000i	0.528367i
$918$	0	0
$919$	32.0000	1.05558	0.527791	−	0.849374i	$-0.323020\pi$
$919$	32.0000	1.05558	0.527791	+	0.849374i	$0.323020\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	60.0000i	1.97066i
$928$	0	0
$929$	42.0000	1.37798	0.688988	−	0.724773i	$-0.258055\pi$
$929$	42.0000	1.37798	0.688988	+	0.724773i	$0.258055\pi$
$930$	0	0
$931$	−3.00000	−0.0983210
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	38.0000i	1.24141i	0.784046	+	0.620703i	$0.213153\pi$
$937$	38.0000i	1.24141i	−0.784046	+	0.620703i	$0.786847\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−18.0000	−0.586783	−0.293392	−	0.955992i	$-0.594784\pi$
$941$	−18.0000	−0.586783	−0.293392	+	0.955992i	$0.594784\pi$
$942$	0	0
$943$	12.0000i	0.390774i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	18.0000i	0.584921i	0.956278	+	0.292461i	$0.0944741\pi$
$947$	18.0000i	0.584921i	−0.956278	+	0.292461i	$0.905526\pi$
$948$	0	0
$949$	−40.0000	−1.29845
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 60.0000i	− 1.94359i	−0.235826	−	0.971795i	$-0.575780\pi$
$953$	− 60.0000i	− 1.94359i	0.235826	−	0.971795i	$-0.424220\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	12.0000	0.387500
$960$	0	0
$961$	33.0000	1.06452
$962$	0	0
$963$	− 12.0000i	− 0.386695i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 14.0000i	− 0.450210i	−0.974335	−	0.225105i	$-0.927728\pi$
$967$	− 14.0000i	− 0.450210i	0.974335	−	0.225105i	$-0.0722725\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	4.00000	0.128366	0.0641831	−	0.997938i	$-0.479556\pi$
$971$	4.00000	0.128366	0.0641831	+	0.997938i	$0.479556\pi$
$972$	0	0
$973$	− 16.0000i	− 0.512936i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	4.00000i	0.127971i	0.997951	+	0.0639857i	$0.0203812\pi$
$977$	4.00000i	0.127971i	−0.997951	+	0.0639857i	$0.979619\pi$
$978$	0	0
$979$	56.0000	1.78977
$980$	0	0
$981$	6.00000	0.191565
$982$	0	0
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−12.0000	−0.381578
$990$	0	0
$991$	−48.0000	−1.52477	−0.762385	−	0.647124i	$-0.775972\pi$
$991$	−48.0000	−1.52477	−0.762385	+	0.647124i	$0.775972\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	22.0000i	0.696747i	0.937356	+	0.348373i	$0.113266\pi$
$997$	22.0000i	0.696747i	−0.937356	+	0.348373i	$0.886734\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1900.2.c.c.1749.1		2
5.2	odd	4		1900.2.a.c.1.1		1
5.3	odd	4		380.2.a.a.1.1	✓	1
5.4	even	2	inner	1900.2.c.c.1749.2		2
15.8	even	4		3420.2.a.d.1.1		1
20.3	even	4		1520.2.a.e.1.1		1
20.7	even	4		7600.2.a.j.1.1		1
40.3	even	4		6080.2.a.n.1.1		1
40.13	odd	4		6080.2.a.m.1.1		1
95.18	even	4		7220.2.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
380.2.a.a.1.1	✓	1	5.3	odd	4
1520.2.a.e.1.1		1	20.3	even	4
1900.2.a.c.1.1		1	5.2	odd	4
1900.2.c.c.1749.1		2	1.1	even	1	trivial
1900.2.c.c.1749.2		2	5.4	even	2	inner
3420.2.a.d.1.1		1	15.8	even	4
6080.2.a.m.1.1		1	40.13	odd	4
6080.2.a.n.1.1		1	40.3	even	4
7220.2.a.d.1.1		1	95.18	even	4
7600.2.a.j.1.1		1	20.7	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1900.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.00000i q^{7} +3.00000 q^{9} -4.00000 q^{11} +4.00000i q^{13} +6.00000i q^{17} -1.00000 q^{19} +2.00000i q^{23} +6.00000 q^{29} -8.00000 q^{31} +4.00000i q^{37} +6.00000 q^{41} +6.00000i q^{43} +6.00000i q^{47} +3.00000 q^{49} -8.00000i q^{53} +12.0000 q^{59} +6.00000 q^{61} -6.00000i q^{63} +10.0000i q^{73} +8.00000i q^{77} +8.00000 q^{79} +9.00000 q^{81} -14.0000i q^{83} -14.0000 q^{89} +8.00000 q^{91} +16.0000i q^{97} -12.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{9} - 8 q^{11} - 2 q^{19} + 12 q^{29} - 16 q^{31} + 12 q^{41} + 6 q^{49} + 24 q^{59} + 12 q^{61} + 16 q^{79} + 18 q^{81} - 28 q^{89} + 16 q^{91} - 24 q^{99}+O(q^{100}) \) 2 * q + 6 * q^9 - 8 * q^11 - 2 * q^19 + 12 * q^29 - 16 * q^31 + 12 * q^41 + 6 * q^49 + 24 * q^59 + 12 * q^61 + 16 * q^79 + 18 * q^81 - 28 * q^89 + 16 * q^91 - 24 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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