LMFDB - Embedded newform 1920.2.k.b.961.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1920,2,Mod(961,1920)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1920, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1920.961");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1920 = 2^{7} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1920.k (of order $2$, degree $1$, 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:	$15.3312771881$
Analytic rank:	$1$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			961.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	1920.961
Dual form			1920.2.k.b.961.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} -1.00000i q^{5} -2.00000 q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} -1.00000i q^{5} -2.00000 q^{7} -1.00000 q^{9} +2.00000i q^{11} -2.00000i q^{13} -1.00000 q^{15} +2.00000i q^{21} -1.00000 q^{25} +1.00000i q^{27} -6.00000i q^{29} -8.00000 q^{31} +2.00000 q^{33} +2.00000i q^{35} -2.00000i q^{37} -2.00000 q^{39} -10.0000 q^{41} +12.0000i q^{43} +1.00000i q^{45} -4.00000 q^{47} -3.00000 q^{49} +2.00000i q^{53} +2.00000 q^{55} +10.0000i q^{59} +8.00000i q^{61} +2.00000 q^{63} -2.00000 q^{65} -4.00000i q^{67} -8.00000 q^{71} +10.0000 q^{73} +1.00000i q^{75} -4.00000i q^{77} -8.00000 q^{79} +1.00000 q^{81} +8.00000i q^{83} -6.00000 q^{87} -6.00000 q^{89} +4.00000i q^{91} +8.00000i q^{93} +10.0000 q^{97} -2.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{7} - 2 q^{9}+O(q^{10}) $ 2 * q - 4 * q^7 - 2 * q^9	$ 2 q - 4 q^{7} - 2 q^{9} - 2 q^{15} - 2 q^{25} - 16 q^{31} + 4 q^{33} - 4 q^{39} - 20 q^{41} - 8 q^{47} - 6 q^{49} + 4 q^{55} + 4 q^{63} - 4 q^{65} - 16 q^{71} + 20 q^{73} - 16 q^{79} + 2 q^{81} - 12 q^{87} - 12 q^{89} + 20 q^{97}+O(q^{100}) $ 2 * q - 4 * q^7 - 2 * q^9 - 2 * q^15 - 2 * q^25 - 16 * q^31 + 4 * q^33 - 4 * q^39 - 20 * q^41 - 8 * q^47 - 6 * q^49 + 4 * q^55 + 4 * q^63 - 4 * q^65 - 16 * q^71 + 20 * q^73 - 16 * q^79 + 2 * q^81 - 12 * q^87 - 12 * q^89 + 20 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1920\mathbb{Z}\right)^\times$.

$n$	$511$	$641$	$901$	$1537$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	0	0
$5$	− 1.00000i	− 0.447214i
$5$	− 1.00000i	− 0.447214i
$6$	0	0
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	2.00000i	0.603023i	0.953463	+	0.301511i	$0.0974911\pi$
$11$	2.00000i	0.603023i	−0.953463	+	0.301511i	$0.902509\pi$
$12$	0	0
$13$	− 2.00000i	− 0.554700i	−0.960769	−	0.277350i	$-0.910544\pi$
$13$	− 2.00000i	− 0.554700i	0.960769	−	0.277350i	$-0.0894562\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0	0
$21$	2.00000i	0.436436i
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	1.00000i	0.192450i
$28$	0	0
$29$	− 6.00000i	− 1.11417i	−0.830455	−	0.557086i	$-0.811919\pi$
$29$	− 6.00000i	− 1.11417i	0.830455	−	0.557086i	$-0.188081\pi$
$30$	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
$33$	2.00000	0.348155
$34$	0	0
$35$	2.00000i	0.338062i
$36$	0	0
$37$	− 2.00000i	− 0.328798i	−0.986394	−	0.164399i	$-0.947432\pi$
$37$	− 2.00000i	− 0.328798i	0.986394	−	0.164399i	$-0.0525685\pi$
$38$	0	0
$39$	−2.00000	−0.320256
$40$	0	0
$41$	−10.0000	−1.56174	−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000	−1.56174	−0.780869	+	0.624695i	$0.785223\pi$
$42$	0	0
$43$	12.0000i	1.82998i	0.403473	+	0.914991i	$0.367803\pi$
$43$	12.0000i	1.82998i	−0.403473	+	0.914991i	$0.632197\pi$
$44$	0	0
$45$	1.00000i	0.149071i
$46$	0	0
$47$	−4.00000	−0.583460	−0.291730	−	0.956501i	$-0.594231\pi$
$47$	−4.00000	−0.583460	−0.291730	+	0.956501i	$0.594231\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	0	0
$52$	0	0
$53$	2.00000i	0.274721i	0.990521	+	0.137361i	$0.0438619\pi$
$53$	2.00000i	0.274721i	−0.990521	+	0.137361i	$0.956138\pi$
$54$	0	0
$55$	2.00000	0.269680
$56$	0	0
$57$	0	0
$58$	0	0
$59$	10.0000i	1.30189i	0.759125	+	0.650945i	$0.225627\pi$
$59$	10.0000i	1.30189i	−0.759125	+	0.650945i	$0.774373\pi$
$60$	0	0
$61$	8.00000i	1.02430i	0.858898	+	0.512148i	$0.171150\pi$
$61$	8.00000i	1.02430i	−0.858898	+	0.512148i	$0.828850\pi$
$62$	0	0
$63$	2.00000	0.251976
$64$	0	0
$65$	−2.00000	−0.248069
$66$	0	0
$67$	− 4.00000i	− 0.488678i	−0.969690	−	0.244339i	$-0.921429\pi$
$67$	− 4.00000i	− 0.488678i	0.969690	−	0.244339i	$-0.0785709\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	0	0
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	0	0
$75$	1.00000i	0.115470i
$76$	0	0
$77$	− 4.00000i	− 0.455842i
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	8.00000i	0.878114i	0.898459	+	0.439057i	$0.144687\pi$
$83$	8.00000i	0.878114i	−0.898459	+	0.439057i	$0.855313\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−6.00000	−0.643268
$88$	0	0
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	4.00000i	0.419314i
$92$	0	0
$93$	8.00000i	0.829561i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	0	0
$99$	− 2.00000i	− 0.201008i
$100$	0	0
$101$	− 2.00000i	− 0.199007i	−0.995037	−	0.0995037i	$-0.968274\pi$
$101$	− 2.00000i	− 0.199007i	0.995037	−	0.0995037i	$-0.0317255\pi$
$102$	0	0
$103$	−2.00000	−0.197066	−0.0985329	−	0.995134i	$-0.531415\pi$
$103$	−2.00000	−0.197066	−0.0985329	+	0.995134i	$0.531415\pi$
$104$	0	0
$105$	2.00000	0.195180
$106$	0	0
$107$	8.00000i	0.773389i	0.922208	+	0.386695i	$0.126383\pi$
$107$	8.00000i	0.773389i	−0.922208	+	0.386695i	$0.873617\pi$
$108$	0	0
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	−2.00000	−0.189832
$112$	0	0
$113$	−4.00000	−0.376288	−0.188144	−	0.982141i	$-0.560247\pi$
$113$	−4.00000	−0.376288	−0.188144	+	0.982141i	$0.560247\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	2.00000i	0.184900i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	7.00000	0.636364
$122$	0	0
$123$	10.0000i	0.901670i
$124$	0	0
$125$	1.00000i	0.0894427i
$126$	0	0
$127$	2.00000	0.177471	0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000	0.177471	0.0887357	+	0.996055i	$0.471717\pi$
$128$	0	0
$129$	12.0000	1.05654
$130$	0	0
$131$	− 14.0000i	− 1.22319i	−0.791173	−	0.611593i	$-0.790529\pi$
$131$	− 14.0000i	− 1.22319i	0.791173	−	0.611593i	$-0.209471\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	−4.00000	−0.341743	−0.170872	−	0.985293i	$-0.554658\pi$
$137$	−4.00000	−0.341743	−0.170872	+	0.985293i	$0.554658\pi$
$138$	0	0
$139$	12.0000i	1.01783i	0.860818	+	0.508913i	$0.169953\pi$
$139$	12.0000i	1.01783i	−0.860818	+	0.508913i	$0.830047\pi$
$140$	0	0
$141$	4.00000i	0.336861i
$142$	0	0
$143$	4.00000	0.334497
$144$	0	0
$145$	−6.00000	−0.498273
$146$	0	0
$147$	3.00000i	0.247436i
$148$	0	0
$149$	− 6.00000i	− 0.491539i	−0.969328	−	0.245770i	$-0.920959\pi$
$149$	− 6.00000i	− 0.491539i	0.969328	−	0.245770i	$-0.0790407\pi$
$150$	0	0
$151$	4.00000	0.325515	0.162758	−	0.986666i	$-0.447961\pi$
$151$	4.00000	0.325515	0.162758	+	0.986666i	$0.447961\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	8.00000i	0.642575i
$156$	0	0
$157$	6.00000i	0.478852i	0.970915	+	0.239426i	$0.0769593\pi$
$157$	6.00000i	0.478852i	−0.970915	+	0.239426i	$0.923041\pi$
$158$	0	0
$159$	2.00000	0.158610
$160$	0	0
$161$	0	0
$162$	0	0
$163$	4.00000i	0.313304i	0.987654	+	0.156652i	$0.0500701\pi$
$163$	4.00000i	0.313304i	−0.987654	+	0.156652i	$0.949930\pi$
$164$	0	0
$165$	− 2.00000i	− 0.155700i
$166$	0	0
$167$	−24.0000	−1.85718	−0.928588	−	0.371113i	$-0.878976\pi$
$167$	−24.0000	−1.85718	−0.928588	+	0.371113i	$0.878976\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 14.0000i	− 1.06440i	−0.846619	−	0.532200i	$-0.821365\pi$
$173$	− 14.0000i	− 1.06440i	0.846619	−	0.532200i	$-0.178635\pi$
$174$	0	0
$175$	2.00000	0.151186
$176$	0	0
$177$	10.0000	0.751646
$178$	0	0
$179$	− 18.0000i	− 1.34538i	−0.739923	−	0.672692i	$-0.765138\pi$
$179$	− 18.0000i	− 1.34538i	0.739923	−	0.672692i	$-0.234862\pi$
$180$	0	0
$181$	8.00000i	0.594635i	0.954779	+	0.297318i	$0.0960920\pi$
$181$	8.00000i	0.594635i	−0.954779	+	0.297318i	$0.903908\pi$
$182$	0	0
$183$	8.00000	0.591377
$184$	0	0
$185$	−2.00000	−0.147043
$186$	0	0
$187$	0	0
$188$	0	0
$189$	− 2.00000i	− 0.145479i
$190$	0	0
$191$	−24.0000	−1.73658	−0.868290	−	0.496058i	$-0.834780\pi$
$191$	−24.0000	−1.73658	−0.868290	+	0.496058i	$0.834780\pi$
$192$	0	0
$193$	−10.0000	−0.719816	−0.359908	−	0.932988i	$-0.617192\pi$
$193$	−10.0000	−0.719816	−0.359908	+	0.932988i	$0.617192\pi$
$194$	0	0
$195$	2.00000i	0.143223i
$196$	0	0
$197$	− 26.0000i	− 1.85242i	−0.377004	−	0.926212i	$-0.623046\pi$
$197$	− 26.0000i	− 1.85242i	0.377004	−	0.926212i	$-0.376954\pi$
$198$	0	0
$199$	−20.0000	−1.41776	−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000	−1.41776	−0.708881	+	0.705328i	$0.750800\pi$
$200$	0	0
$201$	−4.00000	−0.282138
$202$	0	0
$203$	12.0000i	0.842235i
$204$	0	0
$205$	10.0000i	0.698430i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	− 20.0000i	− 1.37686i	−0.725304	−	0.688428i	$-0.758301\pi$
$211$	− 20.0000i	− 1.37686i	0.725304	−	0.688428i	$-0.241699\pi$
$212$	0	0
$213$	8.00000i	0.548151i
$214$	0	0
$215$	12.0000	0.818393
$216$	0	0
$217$	16.0000	1.08615
$218$	0	0
$219$	− 10.0000i	− 0.675737i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	26.0000	1.74109	0.870544	−	0.492090i	$-0.163767\pi$
$223$	26.0000	1.74109	0.870544	+	0.492090i	$0.163767\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	− 12.0000i	− 0.796468i	−0.917284	−	0.398234i	$-0.869623\pi$
$227$	− 12.0000i	− 0.796468i	0.917284	−	0.398234i	$-0.130377\pi$
$228$	0	0
$229$	28.0000i	1.85029i	0.379611	+	0.925146i	$0.376058\pi$
$229$	28.0000i	1.85029i	−0.379611	+	0.925146i	$0.623942\pi$
$230$	0	0
$231$	−4.00000	−0.263181
$232$	0	0
$233$	−16.0000	−1.04819	−0.524097	−	0.851658i	$-0.675597\pi$
$233$	−16.0000	−1.04819	−0.524097	+	0.851658i	$0.675597\pi$
$234$	0	0
$235$	4.00000i	0.260931i
$236$	0	0
$237$	8.00000i	0.519656i
$238$	0	0
$239$	8.00000	0.517477	0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000	0.517477	0.258738	+	0.965947i	$0.416693\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	0	0
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	3.00000i	0.191663i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	8.00000	0.506979
$250$	0	0
$251$	− 10.0000i	− 0.631194i	−0.948893	−	0.315597i	$-0.897795\pi$
$251$	− 10.0000i	− 0.631194i	0.948893	−	0.315597i	$-0.102205\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−12.0000	−0.748539	−0.374270	−	0.927320i	$-0.622107\pi$
$257$	−12.0000	−0.748539	−0.374270	+	0.927320i	$0.622107\pi$
$258$	0	0
$259$	4.00000i	0.248548i
$260$	0	0
$261$	6.00000i	0.371391i
$262$	0	0
$263$	16.0000	0.986602	0.493301	−	0.869859i	$-0.335790\pi$
$263$	16.0000	0.986602	0.493301	+	0.869859i	$0.335790\pi$
$264$	0	0
$265$	2.00000	0.122859
$266$	0	0
$267$	6.00000i	0.367194i
$268$	0	0
$269$	− 22.0000i	− 1.34136i	−0.741745	−	0.670682i	$-0.766002\pi$
$269$	− 22.0000i	− 1.34136i	0.741745	−	0.670682i	$-0.233998\pi$
$270$	0	0
$271$	−28.0000	−1.70088	−0.850439	−	0.526073i	$-0.823664\pi$
$271$	−28.0000	−1.70088	−0.850439	+	0.526073i	$0.823664\pi$
$272$	0	0
$273$	4.00000	0.242091
$274$	0	0
$275$	− 2.00000i	− 0.120605i
$276$	0	0
$277$	− 2.00000i	− 0.120168i	−0.998193	−	0.0600842i	$-0.980863\pi$
$277$	− 2.00000i	− 0.120168i	0.998193	−	0.0600842i	$-0.0191369\pi$
$278$	0	0
$279$	8.00000	0.478947
$280$	0	0
$281$	−22.0000	−1.31241	−0.656205	−	0.754583i	$-0.727839\pi$
$281$	−22.0000	−1.31241	−0.656205	+	0.754583i	$0.727839\pi$
$282$	0	0
$283$	20.0000i	1.18888i	0.804141	+	0.594438i	$0.202626\pi$
$283$	20.0000i	1.18888i	−0.804141	+	0.594438i	$0.797374\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	20.0000	1.18056
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	− 10.0000i	− 0.586210i
$292$	0	0
$293$	6.00000i	0.350524i	0.984522	+	0.175262i	$0.0560772\pi$
$293$	6.00000i	0.350524i	−0.984522	+	0.175262i	$0.943923\pi$
$294$	0	0
$295$	10.0000	0.582223
$296$	0	0
$297$	−2.00000	−0.116052
$298$	0	0
$299$	0	0
$300$	0	0
$301$	− 24.0000i	− 1.38334i
$302$	0	0
$303$	−2.00000	−0.114897
$304$	0	0
$305$	8.00000	0.458079
$306$	0	0
$307$	− 28.0000i	− 1.59804i	−0.601302	−	0.799022i	$-0.705351\pi$
$307$	− 28.0000i	− 1.59804i	0.601302	−	0.799022i	$-0.294649\pi$
$308$	0	0
$309$	2.00000i	0.113776i
$310$	0	0
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	0	0
$313$	26.0000	1.46961	0.734803	−	0.678280i	$-0.237274\pi$
$313$	26.0000	1.46961	0.734803	+	0.678280i	$0.237274\pi$
$314$	0	0
$315$	− 2.00000i	− 0.112687i
$316$	0	0
$317$	2.00000i	0.112331i	0.998421	+	0.0561656i	$0.0178875\pi$
$317$	2.00000i	0.112331i	−0.998421	+	0.0561656i	$0.982113\pi$
$318$	0	0
$319$	12.0000	0.671871
$320$	0	0
$321$	8.00000	0.446516
$322$	0	0
$323$	0	0
$324$	0	0
$325$	2.00000i	0.110940i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	8.00000	0.441054
$330$	0	0
$331$	− 16.0000i	− 0.879440i	−0.898135	−	0.439720i	$-0.855078\pi$
$331$	− 16.0000i	− 0.879440i	0.898135	−	0.439720i	$-0.144922\pi$
$332$	0	0
$333$	2.00000i	0.109599i
$334$	0	0
$335$	−4.00000	−0.218543
$336$	0	0
$337$	−26.0000	−1.41631	−0.708155	−	0.706057i	$-0.750472\pi$
$337$	−26.0000	−1.41631	−0.708155	+	0.706057i	$0.750472\pi$
$338$	0	0
$339$	4.00000i	0.217250i
$340$	0	0
$341$	− 16.0000i	− 0.866449i
$342$	0	0
$343$	20.0000	1.07990
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 12.0000i	− 0.644194i	−0.946707	−	0.322097i	$-0.895612\pi$
$347$	− 12.0000i	− 0.644194i	0.946707	−	0.322097i	$-0.104388\pi$
$348$	0	0
$349$	24.0000i	1.28469i	0.766415	+	0.642345i	$0.222038\pi$
$349$	24.0000i	1.28469i	−0.766415	+	0.642345i	$0.777962\pi$
$350$	0	0
$351$	2.00000	0.106752
$352$	0	0
$353$	−24.0000	−1.27739	−0.638696	−	0.769460i	$-0.720526\pi$
$353$	−24.0000	−1.27739	−0.638696	+	0.769460i	$0.720526\pi$
$354$	0	0
$355$	8.00000i	0.424596i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−8.00000	−0.422224	−0.211112	−	0.977462i	$-0.567708\pi$
$359$	−8.00000	−0.422224	−0.211112	+	0.977462i	$0.567708\pi$
$360$	0	0
$361$	19.0000	1.00000
$362$	0	0
$363$	− 7.00000i	− 0.367405i
$364$	0	0
$365$	− 10.0000i	− 0.523424i
$366$	0	0
$367$	18.0000	0.939592	0.469796	−	0.882775i	$-0.344327\pi$
$367$	18.0000	0.939592	0.469796	+	0.882775i	$0.344327\pi$
$368$	0	0
$369$	10.0000	0.520579
$370$	0	0
$371$	− 4.00000i	− 0.207670i
$372$	0	0
$373$	− 30.0000i	− 1.55334i	−0.629907	−	0.776671i	$-0.716907\pi$
$373$	− 30.0000i	− 1.55334i	0.629907	−	0.776671i	$-0.283093\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	−12.0000	−0.618031
$378$	0	0
$379$	− 28.0000i	− 1.43826i	−0.694874	−	0.719132i	$-0.744540\pi$
$379$	− 28.0000i	− 1.43826i	0.694874	−	0.719132i	$-0.255460\pi$
$380$	0	0
$381$	− 2.00000i	− 0.102463i
$382$	0	0
$383$	28.0000	1.43073	0.715367	−	0.698749i	$-0.246260\pi$
$383$	28.0000	1.43073	0.715367	+	0.698749i	$0.246260\pi$
$384$	0	0
$385$	−4.00000	−0.203859
$386$	0	0
$387$	− 12.0000i	− 0.609994i
$388$	0	0
$389$	− 14.0000i	− 0.709828i	−0.934899	−	0.354914i	$-0.884510\pi$
$389$	− 14.0000i	− 0.709828i	0.934899	−	0.354914i	$-0.115490\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−14.0000	−0.706207
$394$	0	0
$395$	8.00000i	0.402524i
$396$	0	0
$397$	2.00000i	0.100377i	0.998740	+	0.0501886i	$0.0159822\pi$
$397$	2.00000i	0.100377i	−0.998740	+	0.0501886i	$0.984018\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−10.0000	−0.499376	−0.249688	−	0.968326i	$-0.580328\pi$
$401$	−10.0000	−0.499376	−0.249688	+	0.968326i	$0.580328\pi$
$402$	0	0
$403$	16.0000i	0.797017i
$404$	0	0
$405$	− 1.00000i	− 0.0496904i
$406$	0	0
$407$	4.00000	0.198273
$408$	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
$411$	4.00000i	0.197305i
$412$	0	0
$413$	− 20.0000i	− 0.984136i
$414$	0	0
$415$	8.00000	0.392705
$416$	0	0
$417$	12.0000	0.587643
$418$	0	0
$419$	− 6.00000i	− 0.293119i	−0.989202	−	0.146560i	$-0.953180\pi$
$419$	− 6.00000i	− 0.293119i	0.989202	−	0.146560i	$-0.0468200\pi$
$420$	0	0
$421$	12.0000i	0.584844i	0.956289	+	0.292422i	$0.0944612\pi$
$421$	12.0000i	0.584844i	−0.956289	+	0.292422i	$0.905539\pi$
$422$	0	0
$423$	4.00000	0.194487
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 16.0000i	− 0.774294i
$428$	0	0
$429$	− 4.00000i	− 0.193122i
$430$	0	0
$431$	−24.0000	−1.15604	−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−24.0000	−1.15604	−0.578020	+	0.816023i	$0.696174\pi$
$432$	0	0
$433$	30.0000	1.44171	0.720854	−	0.693087i	$-0.243750\pi$
$433$	30.0000	1.44171	0.720854	+	0.693087i	$0.243750\pi$
$434$	0	0
$435$	6.00000i	0.287678i
$436$	0	0
$437$	0	0
$438$	0	0
$439$	20.0000	0.954548	0.477274	−	0.878755i	$-0.341625\pi$
$439$	20.0000	0.954548	0.477274	+	0.878755i	$0.341625\pi$
$440$	0	0
$441$	3.00000	0.142857
$442$	0	0
$443$	12.0000i	0.570137i	0.958507	+	0.285069i	$0.0920164\pi$
$443$	12.0000i	0.570137i	−0.958507	+	0.285069i	$0.907984\pi$
$444$	0	0
$445$	6.00000i	0.284427i
$446$	0	0
$447$	−6.00000	−0.283790
$448$	0	0
$449$	14.0000	0.660701	0.330350	−	0.943858i	$-0.392833\pi$
$449$	14.0000	0.660701	0.330350	+	0.943858i	$0.392833\pi$
$450$	0	0
$451$	− 20.0000i	− 0.941763i
$452$	0	0
$453$	− 4.00000i	− 0.187936i
$454$	0	0
$455$	4.00000	0.187523
$456$	0	0
$457$	22.0000	1.02912	0.514558	−	0.857455i	$-0.327956\pi$
$457$	22.0000	1.02912	0.514558	+	0.857455i	$0.327956\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	14.0000i	0.652045i	0.945362	+	0.326023i	$0.105709\pi$
$461$	14.0000i	0.652045i	−0.945362	+	0.326023i	$0.894291\pi$
$462$	0	0
$463$	14.0000	0.650635	0.325318	−	0.945605i	$-0.394529\pi$
$463$	14.0000	0.650635	0.325318	+	0.945605i	$0.394529\pi$
$464$	0	0
$465$	8.00000	0.370991
$466$	0	0
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	0	0
$469$	8.00000i	0.369406i
$470$	0	0
$471$	6.00000	0.276465
$472$	0	0
$473$	−24.0000	−1.10352
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 2.00000i	− 0.0915737i
$478$	0	0
$479$	16.0000	0.731059	0.365529	−	0.930800i	$-0.380888\pi$
$479$	16.0000	0.731059	0.365529	+	0.930800i	$0.380888\pi$
$480$	0	0
$481$	−4.00000	−0.182384
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 10.0000i	− 0.454077i
$486$	0	0
$487$	−38.0000	−1.72194	−0.860972	−	0.508652i	$-0.830144\pi$
$487$	−38.0000	−1.72194	−0.860972	+	0.508652i	$0.830144\pi$
$488$	0	0
$489$	4.00000	0.180886
$490$	0	0
$491$	− 42.0000i	− 1.89543i	−0.319113	−	0.947717i	$-0.603385\pi$
$491$	− 42.0000i	− 1.89543i	0.319113	−	0.947717i	$-0.396615\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−2.00000	−0.0898933
$496$	0	0
$497$	16.0000	0.717698
$498$	0	0
$499$	− 16.0000i	− 0.716258i	−0.933672	−	0.358129i	$-0.883415\pi$
$499$	− 16.0000i	− 0.716258i	0.933672	−	0.358129i	$-0.116585\pi$
$500$	0	0
$501$	24.0000i	1.07224i
$502$	0	0
$503$	44.0000	1.96186	0.980932	−	0.194354i	$-0.0622609\pi$
$503$	44.0000	1.96186	0.980932	+	0.194354i	$0.0622609\pi$
$504$	0	0
$505$	−2.00000	−0.0889988
$506$	0	0
$507$	− 9.00000i	− 0.399704i
$508$	0	0
$509$	10.0000i	0.443242i	0.975133	+	0.221621i	$0.0711348\pi$
$509$	10.0000i	0.443242i	−0.975133	+	0.221621i	$0.928865\pi$
$510$	0	0
$511$	−20.0000	−0.884748
$512$	0	0
$513$	0	0
$514$	0	0
$515$	2.00000i	0.0881305i
$516$	0	0
$517$	− 8.00000i	− 0.351840i
$518$	0	0
$519$	−14.0000	−0.614532
$520$	0	0
$521$	2.00000	0.0876216	0.0438108	−	0.999040i	$-0.486050\pi$
$521$	2.00000	0.0876216	0.0438108	+	0.999040i	$0.486050\pi$
$522$	0	0
$523$	− 4.00000i	− 0.174908i	−0.996169	−	0.0874539i	$-0.972127\pi$
$523$	− 4.00000i	− 0.174908i	0.996169	−	0.0874539i	$-0.0278730\pi$
$524$	0	0
$525$	− 2.00000i	− 0.0872872i
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	− 10.0000i	− 0.433963i
$532$	0	0
$533$	20.0000i	0.866296i
$534$	0	0
$535$	8.00000	0.345870
$536$	0	0
$537$	−18.0000	−0.776757
$538$	0	0
$539$	− 6.00000i	− 0.258438i
$540$	0	0
$541$	− 8.00000i	− 0.343947i	−0.985102	−	0.171973i	$-0.944986\pi$
$541$	− 8.00000i	− 0.343947i	0.985102	−	0.171973i	$-0.0550143\pi$
$542$	0	0
$543$	8.00000	0.343313
$544$	0	0
$545$	0	0
$546$	0	0
$547$	4.00000i	0.171028i	0.996337	+	0.0855138i	$0.0272532\pi$
$547$	4.00000i	0.171028i	−0.996337	+	0.0855138i	$0.972747\pi$
$548$	0	0
$549$	− 8.00000i	− 0.341432i
$550$	0	0
$551$	0	0
$552$	0	0
$553$	16.0000	0.680389
$554$	0	0
$555$	2.00000i	0.0848953i
$556$	0	0
$557$	42.0000i	1.77960i	0.456354	+	0.889799i	$0.349155\pi$
$557$	42.0000i	1.77960i	−0.456354	+	0.889799i	$0.650845\pi$
$558$	0	0
$559$	24.0000	1.01509
$560$	0	0
$561$	0	0
$562$	0	0
$563$	44.0000i	1.85438i	0.374593	+	0.927189i	$0.377783\pi$
$563$	44.0000i	1.85438i	−0.374593	+	0.927189i	$0.622217\pi$
$564$	0	0
$565$	4.00000i	0.168281i
$566$	0	0
$567$	−2.00000	−0.0839921
$568$	0	0
$569$	−14.0000	−0.586911	−0.293455	−	0.955973i	$-0.594805\pi$
$569$	−14.0000	−0.586911	−0.293455	+	0.955973i	$0.594805\pi$
$570$	0	0
$571$	− 16.0000i	− 0.669579i	−0.942293	−	0.334790i	$-0.891335\pi$
$571$	− 16.0000i	− 0.669579i	0.942293	−	0.334790i	$-0.108665\pi$
$572$	0	0
$573$	24.0000i	1.00261i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−14.0000	−0.582828	−0.291414	−	0.956597i	$-0.594126\pi$
$577$	−14.0000	−0.582828	−0.291414	+	0.956597i	$0.594126\pi$
$578$	0	0
$579$	10.0000i	0.415586i
$580$	0	0
$581$	− 16.0000i	− 0.663792i
$582$	0	0
$583$	−4.00000	−0.165663
$584$	0	0
$585$	2.00000	0.0826898
$586$	0	0
$587$	− 24.0000i	− 0.990586i	−0.868726	−	0.495293i	$-0.835061\pi$
$587$	− 24.0000i	− 0.990586i	0.868726	−	0.495293i	$-0.164939\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−26.0000	−1.06950
$592$	0	0
$593$	36.0000	1.47834	0.739171	−	0.673517i	$-0.235217\pi$
$593$	36.0000	1.47834	0.739171	+	0.673517i	$0.235217\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	20.0000i	0.818546i
$598$	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
$601$	42.0000	1.71322	0.856608	−	0.515968i	$-0.172568\pi$
$601$	42.0000	1.71322	0.856608	+	0.515968i	$0.172568\pi$
$602$	0	0
$603$	4.00000i	0.162893i
$604$	0	0
$605$	− 7.00000i	− 0.284590i
$606$	0	0
$607$	14.0000	0.568242	0.284121	−	0.958788i	$-0.408298\pi$
$607$	14.0000	0.568242	0.284121	+	0.958788i	$0.408298\pi$
$608$	0	0
$609$	12.0000	0.486265
$610$	0	0
$611$	8.00000i	0.323645i
$612$	0	0
$613$	14.0000i	0.565455i	0.959200	+	0.282727i	$0.0912392\pi$
$613$	14.0000i	0.565455i	−0.959200	+	0.282727i	$0.908761\pi$
$614$	0	0
$615$	10.0000	0.403239
$616$	0	0
$617$	−24.0000	−0.966204	−0.483102	−	0.875564i	$-0.660490\pi$
$617$	−24.0000	−0.966204	−0.483102	+	0.875564i	$0.660490\pi$
$618$	0	0
$619$	4.00000i	0.160774i	0.996764	+	0.0803868i	$0.0256155\pi$
$619$	4.00000i	0.160774i	−0.996764	+	0.0803868i	$0.974384\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	12.0000	0.480770
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	40.0000	1.59237	0.796187	−	0.605050i	$-0.206847\pi$
$631$	40.0000	1.59237	0.796187	+	0.605050i	$0.206847\pi$
$632$	0	0
$633$	−20.0000	−0.794929
$634$	0	0
$635$	− 2.00000i	− 0.0793676i
$636$	0	0
$637$	6.00000i	0.237729i
$638$	0	0
$639$	8.00000	0.316475
$640$	0	0
$641$	10.0000	0.394976	0.197488	−	0.980305i	$-0.436722\pi$
$641$	10.0000	0.394976	0.197488	+	0.980305i	$0.436722\pi$
$642$	0	0
$643$	− 4.00000i	− 0.157745i	−0.996885	−	0.0788723i	$-0.974868\pi$
$643$	− 4.00000i	− 0.157745i	0.996885	−	0.0788723i	$-0.0251319\pi$
$644$	0	0
$645$	− 12.0000i	− 0.472500i
$646$	0	0
$647$	−36.0000	−1.41531	−0.707653	−	0.706560i	$-0.750246\pi$
$647$	−36.0000	−1.41531	−0.707653	+	0.706560i	$0.750246\pi$
$648$	0	0
$649$	−20.0000	−0.785069
$650$	0	0
$651$	− 16.0000i	− 0.627089i
$652$	0	0
$653$	14.0000i	0.547862i	0.961749	+	0.273931i	$0.0883240\pi$
$653$	14.0000i	0.547862i	−0.961749	+	0.273931i	$0.911676\pi$
$654$	0	0
$655$	−14.0000	−0.547025
$656$	0	0
$657$	−10.0000	−0.390137
$658$	0	0
$659$	− 14.0000i	− 0.545363i	−0.962104	−	0.272681i	$-0.912090\pi$
$659$	− 14.0000i	− 0.545363i	0.962104	−	0.272681i	$-0.0879105\pi$
$660$	0	0
$661$	− 40.0000i	− 1.55582i	−0.628376	−	0.777910i	$-0.716280\pi$
$661$	− 40.0000i	− 1.55582i	0.628376	−	0.777910i	$-0.283720\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	− 26.0000i	− 1.00522i
$670$	0	0
$671$	−16.0000	−0.617673
$672$	0	0
$673$	14.0000	0.539660	0.269830	−	0.962908i	$-0.413032\pi$
$673$	14.0000	0.539660	0.269830	+	0.962908i	$0.413032\pi$
$674$	0	0
$675$	− 1.00000i	− 0.0384900i
$676$	0	0
$677$	− 6.00000i	− 0.230599i	−0.993331	−	0.115299i	$-0.963217\pi$
$677$	− 6.00000i	− 0.230599i	0.993331	−	0.115299i	$-0.0367827\pi$
$678$	0	0
$679$	−20.0000	−0.767530
$680$	0	0
$681$	−12.0000	−0.459841
$682$	0	0
$683$	− 16.0000i	− 0.612223i	−0.951996	−	0.306111i	$-0.900972\pi$
$683$	− 16.0000i	− 0.612223i	0.951996	−	0.306111i	$-0.0990280\pi$
$684$	0	0
$685$	4.00000i	0.152832i
$686$	0	0
$687$	28.0000	1.06827
$688$	0	0
$689$	4.00000	0.152388
$690$	0	0
$691$	16.0000i	0.608669i	0.952565	+	0.304334i	$0.0984340\pi$
$691$	16.0000i	0.608669i	−0.952565	+	0.304334i	$0.901566\pi$
$692$	0	0
$693$	4.00000i	0.151947i
$694$	0	0
$695$	12.0000	0.455186
$696$	0	0
$697$	0	0
$698$	0	0
$699$	16.0000i	0.605176i
$700$	0	0
$701$	2.00000i	0.0755390i	0.999286	+	0.0377695i	$0.0120253\pi$
$701$	2.00000i	0.0755390i	−0.999286	+	0.0377695i	$0.987975\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	4.00000	0.150649
$706$	0	0
$707$	4.00000i	0.150435i
$708$	0	0
$709$	36.0000i	1.35201i	0.736898	+	0.676004i	$0.236290\pi$
$709$	36.0000i	1.35201i	−0.736898	+	0.676004i	$0.763710\pi$
$710$	0	0
$711$	8.00000	0.300023
$712$	0	0
$713$	0	0
$714$	0	0
$715$	− 4.00000i	− 0.149592i
$716$	0	0
$717$	− 8.00000i	− 0.298765i
$718$	0	0
$719$	−24.0000	−0.895049	−0.447524	−	0.894272i	$-0.647694\pi$
$719$	−24.0000	−0.895049	−0.447524	+	0.894272i	$0.647694\pi$
$720$	0	0
$721$	4.00000	0.148968
$722$	0	0
$723$	10.0000i	0.371904i
$724$	0	0
$725$	6.00000i	0.222834i
$726$	0	0
$727$	−22.0000	−0.815935	−0.407967	−	0.912996i	$-0.633762\pi$
$727$	−22.0000	−0.815935	−0.407967	+	0.912996i	$0.633762\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	0	0
$732$	0	0
$733$	− 22.0000i	− 0.812589i	−0.913742	−	0.406294i	$-0.866821\pi$
$733$	− 22.0000i	− 0.812589i	0.913742	−	0.406294i	$-0.133179\pi$
$734$	0	0
$735$	3.00000	0.110657
$736$	0	0
$737$	8.00000	0.294684
$738$	0	0
$739$	40.0000i	1.47142i	0.677295	+	0.735712i	$0.263152\pi$
$739$	40.0000i	1.47142i	−0.677295	+	0.735712i	$0.736848\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−36.0000	−1.32071	−0.660356	−	0.750953i	$-0.729595\pi$
$743$	−36.0000	−1.32071	−0.660356	+	0.750953i	$0.729595\pi$
$744$	0	0
$745$	−6.00000	−0.219823
$746$	0	0
$747$	− 8.00000i	− 0.292705i
$748$	0	0
$749$	− 16.0000i	− 0.584627i
$750$	0	0
$751$	−48.0000	−1.75154	−0.875772	−	0.482724i	$-0.839647\pi$
$751$	−48.0000	−1.75154	−0.875772	+	0.482724i	$0.839647\pi$
$752$	0	0
$753$	−10.0000	−0.364420
$754$	0	0
$755$	− 4.00000i	− 0.145575i
$756$	0	0
$757$	14.0000i	0.508839i	0.967094	+	0.254419i	$0.0818843\pi$
$757$	14.0000i	0.508839i	−0.967094	+	0.254419i	$0.918116\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	34.0000	1.23250	0.616250	−	0.787551i	$-0.288651\pi$
$761$	34.0000	1.23250	0.616250	+	0.787551i	$0.288651\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	20.0000	0.722158
$768$	0	0
$769$	−46.0000	−1.65880	−0.829401	−	0.558653i	$-0.811318\pi$
$769$	−46.0000	−1.65880	−0.829401	+	0.558653i	$0.811318\pi$
$770$	0	0
$771$	12.0000i	0.432169i
$772$	0	0
$773$	38.0000i	1.36677i	0.730061	+	0.683383i	$0.239492\pi$
$773$	38.0000i	1.36677i	−0.730061	+	0.683383i	$0.760508\pi$
$774$	0	0
$775$	8.00000	0.287368
$776$	0	0
$777$	4.00000	0.143499
$778$	0	0
$779$	0	0
$780$	0	0
$781$	− 16.0000i	− 0.572525i
$782$	0	0
$783$	6.00000	0.214423
$784$	0	0
$785$	6.00000	0.214149
$786$	0	0
$787$	52.0000i	1.85360i	0.375555	+	0.926800i	$0.377452\pi$
$787$	52.0000i	1.85360i	−0.375555	+	0.926800i	$0.622548\pi$
$788$	0	0
$789$	− 16.0000i	− 0.569615i
$790$	0	0
$791$	8.00000	0.284447
$792$	0	0
$793$	16.0000	0.568177
$794$	0	0
$795$	− 2.00000i	− 0.0709327i
$796$	0	0
$797$	− 14.0000i	− 0.495905i	−0.968772	−	0.247953i	$-0.920242\pi$
$797$	− 14.0000i	− 0.495905i	0.968772	−	0.247953i	$-0.0797578\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	6.00000	0.212000
$802$	0	0
$803$	20.0000i	0.705785i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−22.0000	−0.774437
$808$	0	0
$809$	6.00000	0.210949	0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000	0.210949	0.105474	+	0.994422i	$0.466364\pi$
$810$	0	0
$811$	− 4.00000i	− 0.140459i	−0.997531	−	0.0702295i	$-0.977627\pi$
$811$	− 4.00000i	− 0.140459i	0.997531	−	0.0702295i	$-0.0223732\pi$
$812$	0	0
$813$	28.0000i	0.982003i
$814$	0	0
$815$	4.00000	0.140114
$816$	0	0
$817$	0	0
$818$	0	0
$819$	− 4.00000i	− 0.139771i
$820$	0	0
$821$	6.00000i	0.209401i	0.994504	+	0.104701i	$0.0333885\pi$
$821$	6.00000i	0.209401i	−0.994504	+	0.104701i	$0.966612\pi$
$822$	0	0
$823$	34.0000	1.18517	0.592583	−	0.805510i	$-0.298108\pi$
$823$	34.0000	1.18517	0.592583	+	0.805510i	$0.298108\pi$
$824$	0	0
$825$	−2.00000	−0.0696311
$826$	0	0
$827$	− 12.0000i	− 0.417281i	−0.977992	−	0.208640i	$-0.933096\pi$
$827$	− 12.0000i	− 0.417281i	0.977992	−	0.208640i	$-0.0669038\pi$
$828$	0	0
$829$	16.0000i	0.555703i	0.960624	+	0.277851i	$0.0896223\pi$
$829$	16.0000i	0.555703i	−0.960624	+	0.277851i	$0.910378\pi$
$830$	0	0
$831$	−2.00000	−0.0693792
$832$	0	0
$833$	0	0
$834$	0	0
$835$	24.0000i	0.830554i
$836$	0	0
$837$	− 8.00000i	− 0.276520i
$838$	0	0
$839$	−24.0000	−0.828572	−0.414286	−	0.910147i	$-0.635969\pi$
$839$	−24.0000	−0.828572	−0.414286	+	0.910147i	$0.635969\pi$
$840$	0	0
$841$	−7.00000	−0.241379
$842$	0	0
$843$	22.0000i	0.757720i
$844$	0	0
$845$	− 9.00000i	− 0.309609i
$846$	0	0
$847$	−14.0000	−0.481046
$848$	0	0
$849$	20.0000	0.686398
$850$	0	0
$851$	0	0
$852$	0	0
$853$	− 42.0000i	− 1.43805i	−0.694983	−	0.719026i	$-0.744588\pi$
$853$	− 42.0000i	− 1.43805i	0.694983	−	0.719026i	$-0.255412\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	24.0000	0.819824	0.409912	−	0.912125i	$-0.365559\pi$
$857$	24.0000	0.819824	0.409912	+	0.912125i	$0.365559\pi$
$858$	0	0
$859$	− 12.0000i	− 0.409435i	−0.978821	−	0.204717i	$-0.934372\pi$
$859$	− 12.0000i	− 0.409435i	0.978821	−	0.204717i	$-0.0656275\pi$
$860$	0	0
$861$	− 20.0000i	− 0.681598i
$862$	0	0
$863$	8.00000	0.272323	0.136162	−	0.990687i	$-0.456523\pi$
$863$	8.00000	0.272323	0.136162	+	0.990687i	$0.456523\pi$
$864$	0	0
$865$	−14.0000	−0.476014
$866$	0	0
$867$	17.0000i	0.577350i
$868$	0	0
$869$	− 16.0000i	− 0.542763i
$870$	0	0
$871$	−8.00000	−0.271070
$872$	0	0
$873$	−10.0000	−0.338449
$874$	0	0
$875$	− 2.00000i	− 0.0676123i
$876$	0	0
$877$	− 34.0000i	− 1.14810i	−0.818821	−	0.574049i	$-0.805372\pi$
$877$	− 34.0000i	− 1.14810i	0.818821	−	0.574049i	$-0.194628\pi$
$878$	0	0
$879$	6.00000	0.202375
$880$	0	0
$881$	14.0000	0.471672	0.235836	−	0.971793i	$-0.424217\pi$
$881$	14.0000	0.471672	0.235836	+	0.971793i	$0.424217\pi$
$882$	0	0
$883$	36.0000i	1.21150i	0.795656	+	0.605748i	$0.207126\pi$
$883$	36.0000i	1.21150i	−0.795656	+	0.605748i	$0.792874\pi$
$884$	0	0
$885$	− 10.0000i	− 0.336146i
$886$	0	0
$887$	−8.00000	−0.268614	−0.134307	−	0.990940i	$-0.542881\pi$
$887$	−8.00000	−0.268614	−0.134307	+	0.990940i	$0.542881\pi$
$888$	0	0
$889$	−4.00000	−0.134156
$890$	0	0
$891$	2.00000i	0.0670025i
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−18.0000	−0.601674
$896$	0	0
$897$	0	0
$898$	0	0
$899$	48.0000i	1.60089i
$900$	0	0
$901$	0	0
$902$	0	0
$903$	−24.0000	−0.798670
$904$	0	0
$905$	8.00000	0.265929
$906$	0	0
$907$	− 4.00000i	− 0.132818i	−0.997792	−	0.0664089i	$-0.978846\pi$
$907$	− 4.00000i	− 0.132818i	0.997792	−	0.0664089i	$-0.0211542\pi$
$908$	0	0
$909$	2.00000i	0.0663358i
$910$	0	0
$911$	−24.0000	−0.795155	−0.397578	−	0.917568i	$-0.630149\pi$
$911$	−24.0000	−0.795155	−0.397578	+	0.917568i	$0.630149\pi$
$912$	0	0
$913$	−16.0000	−0.529523
$914$	0	0
$915$	− 8.00000i	− 0.264472i
$916$	0	0
$917$	28.0000i	0.924641i
$918$	0	0
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	0	0
$921$	−28.0000	−0.922631
$922$	0	0
$923$	16.0000i	0.526646i
$924$	0	0
$925$	2.00000i	0.0657596i
$926$	0	0
$927$	2.00000	0.0656886
$928$	0	0
$929$	30.0000	0.984268	0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000	0.984268	0.492134	+	0.870519i	$0.336217\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	− 24.0000i	− 0.785725i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−2.00000	−0.0653372	−0.0326686	−	0.999466i	$-0.510401\pi$
$937$	−2.00000	−0.0653372	−0.0326686	+	0.999466i	$0.510401\pi$
$938$	0	0
$939$	− 26.0000i	− 0.848478i
$940$	0	0
$941$	54.0000i	1.76035i	0.474650	+	0.880175i	$0.342575\pi$
$941$	54.0000i	1.76035i	−0.474650	+	0.880175i	$0.657425\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−2.00000	−0.0650600
$946$	0	0
$947$	20.0000i	0.649913i	0.945729	+	0.324956i	$0.105350\pi$
$947$	20.0000i	0.649913i	−0.945729	+	0.324956i	$0.894650\pi$
$948$	0	0
$949$	− 20.0000i	− 0.649227i
$950$	0	0
$951$	2.00000	0.0648544
$952$	0	0
$953$	−44.0000	−1.42530	−0.712650	−	0.701520i	$-0.752505\pi$
$953$	−44.0000	−1.42530	−0.712650	+	0.701520i	$0.752505\pi$
$954$	0	0
$955$	24.0000i	0.776622i
$956$	0	0
$957$	− 12.0000i	− 0.387905i
$958$	0	0
$959$	8.00000	0.258333
$960$	0	0
$961$	33.0000	1.06452
$962$	0	0
$963$	− 8.00000i	− 0.257796i
$964$	0	0
$965$	10.0000i	0.321911i
$966$	0	0
$967$	−34.0000	−1.09337	−0.546683	−	0.837340i	$-0.684110\pi$
$967$	−34.0000	−1.09337	−0.546683	+	0.837340i	$0.684110\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	30.0000i	0.962746i	0.876516	+	0.481373i	$0.159862\pi$
$971$	30.0000i	0.962746i	−0.876516	+	0.481373i	$0.840138\pi$
$972$	0	0
$973$	− 24.0000i	− 0.769405i
$974$	0	0
$975$	2.00000	0.0640513
$976$	0	0
$977$	−24.0000	−0.767828	−0.383914	−	0.923369i	$-0.625424\pi$
$977$	−24.0000	−0.767828	−0.383914	+	0.923369i	$0.625424\pi$
$978$	0	0
$979$	− 12.0000i	− 0.383522i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−28.0000	−0.893061	−0.446531	−	0.894768i	$-0.647341\pi$
$983$	−28.0000	−0.893061	−0.446531	+	0.894768i	$0.647341\pi$
$984$	0	0
$985$	−26.0000	−0.828429
$986$	0	0
$987$	− 8.00000i	− 0.254643i
$988$	0	0
$989$	0	0
$990$	0	0
$991$	4.00000	0.127064	0.0635321	−	0.997980i	$-0.479763\pi$
$991$	4.00000	0.127064	0.0635321	+	0.997980i	$0.479763\pi$
$992$	0	0
$993$	−16.0000	−0.507745
$994$	0	0
$995$	20.0000i	0.634043i
$996$	0	0
$997$	42.0000i	1.33015i	0.746775	+	0.665077i	$0.231601\pi$
$997$	42.0000i	1.33015i	−0.746775	+	0.665077i	$0.768399\pi$
$998$	0	0
$999$	2.00000	0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1920.2.k.b.961.1	✓	2
3.2	odd	2		5760.2.k.e.2881.2		2
4.3	odd	2		1920.2.k.h.961.2	yes	2
8.3	odd	2		1920.2.k.h.961.1	yes	2
8.5	even	2	inner	1920.2.k.b.961.2	yes	2
12.11	even	2		5760.2.k.i.2881.2		2
16.3	odd	4		3840.2.a.c.1.1		1
16.5	even	4		3840.2.a.l.1.1		1
16.11	odd	4		3840.2.a.y.1.1		1
16.13	even	4		3840.2.a.t.1.1		1
24.5	odd	2		5760.2.k.e.2881.1		2
24.11	even	2		5760.2.k.i.2881.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1920.2.k.b.961.1	✓	2	1.1	even	1	trivial
1920.2.k.b.961.2	yes	2	8.5	even	2	inner
1920.2.k.h.961.1	yes	2	8.3	odd	2
1920.2.k.h.961.2	yes	2	4.3	odd	2
3840.2.a.c.1.1		1	16.3	odd	4
3840.2.a.l.1.1		1	16.5	even	4
3840.2.a.t.1.1		1	16.13	even	4
3840.2.a.y.1.1		1	16.11	odd	4
5760.2.k.e.2881.1		2	24.5	odd	2
5760.2.k.e.2881.2		2	3.2	odd	2
5760.2.k.i.2881.1		2	24.11	even	2
5760.2.k.i.2881.2		2	12.11	even	2

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:	\( N \)	\(=\)	\( 1920 = 2^{7} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1920.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} -1.00000i q^{5} -2.00000 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} -1.00000i q^{5} -2.00000 q^{7} -1.00000 q^{9} +2.00000i q^{11} -2.00000i q^{13} -1.00000 q^{15} +2.00000i q^{21} -1.00000 q^{25} +1.00000i q^{27} -6.00000i q^{29} -8.00000 q^{31} +2.00000 q^{33} +2.00000i q^{35} -2.00000i q^{37} -2.00000 q^{39} -10.0000 q^{41} +12.0000i q^{43} +1.00000i q^{45} -4.00000 q^{47} -3.00000 q^{49} +2.00000i q^{53} +2.00000 q^{55} +10.0000i q^{59} +8.00000i q^{61} +2.00000 q^{63} -2.00000 q^{65} -4.00000i q^{67} -8.00000 q^{71} +10.0000 q^{73} +1.00000i q^{75} -4.00000i q^{77} -8.00000 q^{79} +1.00000 q^{81} +8.00000i q^{83} -6.00000 q^{87} -6.00000 q^{89} +4.00000i q^{91} +8.00000i q^{93} +10.0000 q^{97} -2.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{7} - 2 q^{9}+O(q^{10}) \) 2 * q - 4 * q^7 - 2 * q^9	\( 2 q - 4 q^{7} - 2 q^{9} - 2 q^{15} - 2 q^{25} - 16 q^{31} + 4 q^{33} - 4 q^{39} - 20 q^{41} - 8 q^{47} - 6 q^{49} + 4 q^{55} + 4 q^{63} - 4 q^{65} - 16 q^{71} + 20 q^{73} - 16 q^{79} + 2 q^{81} - 12 q^{87} - 12 q^{89} + 20 q^{97}+O(q^{100}) \) 2 * q - 4 * q^7 - 2 * q^9 - 2 * q^15 - 2 * q^25 - 16 * q^31 + 4 * q^33 - 4 * q^39 - 20 * q^41 - 8 * q^47 - 6 * q^49 + 4 * q^55 + 4 * q^63 - 4 * q^65 - 16 * q^71 + 20 * q^73 - 16 * q^79 + 2 * q^81 - 12 * q^87 - 12 * q^89 + 20 * q^97

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