LMFDB - Embedded newform 1920.2.k.c.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:	$0$
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:	$\Z[a_1, a_2, a_3]$
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.c.961.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +1.00000i q^{5} -2.00000 q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} +1.00000i q^{5} -2.00000 q^{7} -1.00000 q^{9} +2.00000i q^{11} -2.00000i q^{13} +1.00000 q^{15} +4.00000 q^{17} -4.00000i q^{19} +2.00000i q^{21} -4.00000 q^{23} -1.00000 q^{25} +1.00000i q^{27} -2.00000i q^{29} -4.00000 q^{31} +2.00000 q^{33} -2.00000i q^{35} -2.00000i q^{37} -2.00000 q^{39} +6.00000 q^{41} -4.00000i q^{43} -1.00000i q^{45} -8.00000 q^{47} -3.00000 q^{49} -4.00000i q^{51} -10.0000i q^{53} -2.00000 q^{55} -4.00000 q^{57} -6.00000i q^{59} +2.00000 q^{63} +2.00000 q^{65} -12.0000i q^{67} +4.00000i q^{69} -8.00000 q^{71} -6.00000 q^{73} +1.00000i q^{75} -4.00000i q^{77} -4.00000 q^{79} +1.00000 q^{81} -16.0000i q^{83} +4.00000i q^{85} -2.00000 q^{87} -6.00000 q^{89} +4.00000i q^{91} +4.00000i q^{93} +4.00000 q^{95} -14.0000 q^{97} -2.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 8 q^{17} - 8 q^{23} - 2 q^{25} - 8 q^{31} + 4 q^{33} - 4 q^{39} + 12 q^{41} - 16 q^{47} - 6 q^{49} - 4 q^{55} - 8 q^{57} + 4 q^{63} + 4 q^{65} - 16 q^{71} - 12 q^{73} - 8 q^{79} + 2 q^{81} - 4 q^{87} - 12 q^{89} + 8 q^{95} - 28 q^{97}+O(q^{100}) $ 2 * q - 4 * q^7 - 2 * q^9 + 2 * q^15 + 8 * q^17 - 8 * q^23 - 2 * q^25 - 8 * q^31 + 4 * q^33 - 4 * q^39 + 12 * q^41 - 16 * q^47 - 6 * q^49 - 4 * q^55 - 8 * q^57 + 4 * q^63 + 4 * q^65 - 16 * q^71 - 12 * q^73 - 8 * q^79 + 2 * q^81 - 4 * q^87 - 12 * q^89 + 8 * q^95 - 28 * 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
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	0	0
$19$	− 4.00000i	− 0.917663i	−0.888523	−	0.458831i	$-0.848268\pi$
$19$	− 4.00000i	− 0.917663i	0.888523	−	0.458831i	$-0.151732\pi$
$20$	0	0
$21$	2.00000i	0.436436i
$22$	0	0
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	1.00000i	0.192450i
$28$	0	0
$29$	− 2.00000i	− 0.371391i	−0.982607	−	0.185695i	$-0.940546\pi$
$29$	− 2.00000i	− 0.371391i	0.982607	−	0.185695i	$-0.0594537\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	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
$41$	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
$43$	− 4.00000i	− 0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$	− 4.00000i	− 0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$	0	0
$45$	− 1.00000i	− 0.149071i
$46$	0	0
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	− 4.00000i	− 0.560112i
$52$	0	0
$53$	− 10.0000i	− 1.37361i	−0.726844	−	0.686803i	$-0.759014\pi$
$53$	− 10.0000i	− 1.37361i	0.726844	−	0.686803i	$-0.240986\pi$
$54$	0	0
$55$	−2.00000	−0.269680
$56$	0	0
$57$	−4.00000	−0.529813
$58$	0	0
$59$	− 6.00000i	− 0.781133i	−0.920575	−	0.390567i	$-0.872279\pi$
$59$	− 6.00000i	− 0.781133i	0.920575	−	0.390567i	$-0.127721\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	2.00000	0.251976
$64$	0	0
$65$	2.00000	0.248069
$66$	0	0
$67$	− 12.0000i	− 1.46603i	−0.680211	−	0.733017i	$-0.738112\pi$
$67$	− 12.0000i	− 1.46603i	0.680211	−	0.733017i	$-0.261888\pi$
$68$	0	0
$69$	4.00000i	0.481543i
$70$	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
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	0	0
$75$	1.00000i	0.115470i
$76$	0	0
$77$	− 4.00000i	− 0.455842i
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 16.0000i	− 1.75623i	−0.478451	−	0.878114i	$-0.658802\pi$
$83$	− 16.0000i	− 1.75623i	0.478451	−	0.878114i	$-0.341198\pi$
$84$	0	0
$85$	4.00000i	0.433861i
$86$	0	0
$87$	−2.00000	−0.214423
$88$	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
$93$	4.00000i	0.414781i
$94$	0	0
$95$	4.00000	0.410391
$96$	0	0
$97$	−14.0000	−1.42148	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000	−1.42148	−0.710742	+	0.703452i	$0.751641\pi$
$98$	0	0
$99$	− 2.00000i	− 0.201008i
$100$	0	0
$101$	2.00000i	0.199007i	0.995037	+	0.0995037i	$0.0317255\pi$
$101$	2.00000i	0.199007i	−0.995037	+	0.0995037i	$0.968274\pi$
$102$	0	0
$103$	14.0000	1.37946	0.689730	−	0.724066i	$-0.257729\pi$
$103$	14.0000	1.37946	0.689730	+	0.724066i	$0.257729\pi$
$104$	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
$109$	16.0000i	1.53252i	0.642529	+	0.766261i	$0.277885\pi$
$109$	16.0000i	1.53252i	−0.642529	+	0.766261i	$0.722115\pi$
$110$	0	0
$111$	−2.00000	−0.189832
$112$	0	0
$113$	16.0000	1.50515	0.752577	−	0.658505i	$-0.228811\pi$
$113$	16.0000	1.50515	0.752577	+	0.658505i	$0.228811\pi$
$114$	0	0
$115$	− 4.00000i	− 0.373002i
$116$	0	0
$117$	2.00000i	0.184900i
$118$	0	0
$119$	−8.00000	−0.733359
$120$	0	0
$121$	7.00000	0.636364
$122$	0	0
$123$	− 6.00000i	− 0.541002i
$124$	0	0
$125$	− 1.00000i	− 0.0894427i
$126$	0	0
$127$	−14.0000	−1.24230	−0.621150	−	0.783692i	$-0.713334\pi$
$127$	−14.0000	−1.24230	−0.621150	+	0.783692i	$0.713334\pi$
$128$	0	0
$129$	−4.00000	−0.352180
$130$	0	0
$131$	− 6.00000i	− 0.524222i	−0.965038	−	0.262111i	$-0.915581\pi$
$131$	− 6.00000i	− 0.524222i	0.965038	−	0.262111i	$-0.0844187\pi$
$132$	0	0
$133$	8.00000i	0.693688i
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	0	0
$139$	16.0000i	1.35710i	0.734553	+	0.678551i	$0.237392\pi$
$139$	16.0000i	1.35710i	−0.734553	+	0.678551i	$0.762608\pi$
$140$	0	0
$141$	8.00000i	0.673722i
$142$	0	0
$143$	4.00000	0.334497
$144$	0	0
$145$	2.00000	0.166091
$146$	0	0
$147$	3.00000i	0.247436i
$148$	0	0
$149$	− 18.0000i	− 1.47462i	−0.675556	−	0.737309i	$-0.736096\pi$
$149$	− 18.0000i	− 1.47462i	0.675556	−	0.737309i	$-0.263904\pi$
$150$	0	0
$151$	−24.0000	−1.95309	−0.976546	−	0.215308i	$-0.930924\pi$
$151$	−24.0000	−1.95309	−0.976546	+	0.215308i	$0.930924\pi$
$152$	0	0
$153$	−4.00000	−0.323381
$154$	0	0
$155$	− 4.00000i	− 0.321288i
$156$	0	0
$157$	− 10.0000i	− 0.798087i	−0.916932	−	0.399043i	$-0.869342\pi$
$157$	− 10.0000i	− 0.798087i	0.916932	−	0.399043i	$-0.130658\pi$
$158$	0	0
$159$	−10.0000	−0.793052
$160$	0	0
$161$	8.00000	0.630488
$162$	0	0
$163$	− 20.0000i	− 1.56652i	−0.621694	−	0.783260i	$-0.713555\pi$
$163$	− 20.0000i	− 1.56652i	0.621694	−	0.783260i	$-0.286445\pi$
$164$	0	0
$165$	2.00000i	0.155700i
$166$	0	0
$167$	12.0000	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	4.00000i	0.305888i
$172$	0	0
$173$	6.00000i	0.456172i	0.973641	+	0.228086i	$0.0732467\pi$
$173$	6.00000i	0.456172i	−0.973641	+	0.228086i	$0.926753\pi$
$174$	0	0
$175$	2.00000	0.151186
$176$	0	0
$177$	−6.00000	−0.450988
$178$	0	0
$179$	− 10.0000i	− 0.747435i	−0.927543	−	0.373718i	$-0.878083\pi$
$179$	− 10.0000i	− 0.747435i	0.927543	−	0.373718i	$-0.121917\pi$
$180$	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
$183$	0	0
$184$	0	0
$185$	2.00000	0.147043
$186$	0	0
$187$	8.00000i	0.585018i
$188$	0	0
$189$	− 2.00000i	− 0.145479i
$190$	0	0
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	0	0
$193$	−18.0000	−1.29567	−0.647834	−	0.761781i	$-0.724325\pi$
$193$	−18.0000	−1.29567	−0.647834	+	0.761781i	$0.724325\pi$
$194$	0	0
$195$	− 2.00000i	− 0.143223i
$196$	0	0
$197$	− 6.00000i	− 0.427482i	−0.976890	−	0.213741i	$-0.931435\pi$
$197$	− 6.00000i	− 0.427482i	0.976890	−	0.213741i	$-0.0685649\pi$
$198$	0	0
$199$	8.00000	0.567105	0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000	0.567105	0.283552	+	0.958957i	$0.408487\pi$
$200$	0	0
$201$	−12.0000	−0.846415
$202$	0	0
$203$	4.00000i	0.280745i
$204$	0	0
$205$	6.00000i	0.419058i
$206$	0	0
$207$	4.00000	0.278019
$208$	0	0
$209$	8.00000	0.553372
$210$	0	0
$211$	8.00000i	0.550743i	0.961338	+	0.275371i	$0.0888008\pi$
$211$	8.00000i	0.550743i	−0.961338	+	0.275371i	$0.911199\pi$
$212$	0	0
$213$	8.00000i	0.548151i
$214$	0	0
$215$	4.00000	0.272798
$216$	0	0
$217$	8.00000	0.543075
$218$	0	0
$219$	6.00000i	0.405442i
$220$	0	0
$221$	− 8.00000i	− 0.538138i
$222$	0	0
$223$	2.00000	0.133930	0.0669650	−	0.997755i	$-0.478668\pi$
$223$	2.00000	0.133930	0.0669650	+	0.997755i	$0.478668\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	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
$229$	− 4.00000i	− 0.264327i	−0.991228	−	0.132164i	$-0.957808\pi$
$229$	− 4.00000i	− 0.264327i	0.991228	−	0.132164i	$-0.0421925\pi$
$230$	0	0
$231$	−4.00000	−0.263181
$232$	0	0
$233$	4.00000	0.262049	0.131024	−	0.991379i	$-0.458173\pi$
$233$	4.00000	0.262049	0.131024	+	0.991379i	$0.458173\pi$
$234$	0	0
$235$	− 8.00000i	− 0.521862i
$236$	0	0
$237$	4.00000i	0.259828i
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	−26.0000	−1.67481	−0.837404	−	0.546585i	$-0.815928\pi$
$241$	−26.0000	−1.67481	−0.837404	+	0.546585i	$0.815928\pi$
$242$	0	0
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	− 3.00000i	− 0.191663i
$246$	0	0
$247$	−8.00000	−0.509028
$248$	0	0
$249$	−16.0000	−1.01396
$250$	0	0
$251$	30.0000i	1.89358i	0.321847	+	0.946792i	$0.395696\pi$
$251$	30.0000i	1.89358i	−0.321847	+	0.946792i	$0.604304\pi$
$252$	0	0
$253$	− 8.00000i	− 0.502956i
$254$	0	0
$255$	4.00000	0.250490
$256$	0	0
$257$	8.00000	0.499026	0.249513	−	0.968371i	$-0.419729\pi$
$257$	8.00000	0.499026	0.249513	+	0.968371i	$0.419729\pi$
$258$	0	0
$259$	4.00000i	0.248548i
$260$	0	0
$261$	2.00000i	0.123797i
$262$	0	0
$263$	28.0000	1.72655	0.863277	−	0.504730i	$-0.168408\pi$
$263$	28.0000	1.72655	0.863277	+	0.504730i	$0.168408\pi$
$264$	0	0
$265$	10.0000	0.614295
$266$	0	0
$267$	6.00000i	0.367194i
$268$	0	0
$269$	− 18.0000i	− 1.09748i	−0.835993	−	0.548740i	$-0.815108\pi$
$269$	− 18.0000i	− 1.09748i	0.835993	−	0.548740i	$-0.184892\pi$
$270$	0	0
$271$	24.0000	1.45790	0.728948	−	0.684569i	$-0.240010\pi$
$271$	24.0000	1.45790	0.728948	+	0.684569i	$0.240010\pi$
$272$	0	0
$273$	4.00000	0.242091
$274$	0	0
$275$	− 2.00000i	− 0.120605i
$276$	0	0
$277$	22.0000i	1.32185i	0.750451	+	0.660926i	$0.229836\pi$
$277$	22.0000i	1.32185i	−0.750451	+	0.660926i	$0.770164\pi$
$278$	0	0
$279$	4.00000	0.239474
$280$	0	0
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	0	0
$283$	− 4.00000i	− 0.237775i	−0.992908	−	0.118888i	$-0.962067\pi$
$283$	− 4.00000i	− 0.237775i	0.992908	−	0.118888i	$-0.0379328\pi$
$284$	0	0
$285$	− 4.00000i	− 0.236940i
$286$	0	0
$287$	−12.0000	−0.708338
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	14.0000i	0.820695i
$292$	0	0
$293$	− 6.00000i	− 0.350524i	−0.984522	−	0.175262i	$-0.943923\pi$
$293$	− 6.00000i	− 0.350524i	0.984522	−	0.175262i	$-0.0560772\pi$
$294$	0	0
$295$	6.00000	0.349334
$296$	0	0
$297$	−2.00000	−0.116052
$298$	0	0
$299$	8.00000i	0.462652i
$300$	0	0
$301$	8.00000i	0.461112i
$302$	0	0
$303$	2.00000	0.114897
$304$	0	0
$305$	0	0
$306$	0	0
$307$	12.0000i	0.684876i	0.939540	+	0.342438i	$0.111253\pi$
$307$	12.0000i	0.684876i	−0.939540	+	0.342438i	$0.888747\pi$
$308$	0	0
$309$	− 14.0000i	− 0.796432i
$310$	0	0
$311$	−32.0000	−1.81455	−0.907277	−	0.420534i	$-0.861843\pi$
$311$	−32.0000	−1.81455	−0.907277	+	0.420534i	$0.861843\pi$
$312$	0	0
$313$	2.00000	0.113047	0.0565233	−	0.998401i	$-0.481998\pi$
$313$	2.00000	0.113047	0.0565233	+	0.998401i	$0.481998\pi$
$314$	0	0
$315$	2.00000i	0.112687i
$316$	0	0
$317$	− 34.0000i	− 1.90963i	−0.297200	−	0.954815i	$-0.596053\pi$
$317$	− 34.0000i	− 1.90963i	0.297200	−	0.954815i	$-0.403947\pi$
$318$	0	0
$319$	4.00000	0.223957
$320$	0	0
$321$	8.00000	0.446516
$322$	0	0
$323$	− 16.0000i	− 0.890264i
$324$	0	0
$325$	2.00000i	0.110940i
$326$	0	0
$327$	16.0000	0.884802
$328$	0	0
$329$	16.0000	0.882109
$330$	0	0
$331$	20.0000i	1.09930i	0.835395	+	0.549650i	$0.185239\pi$
$331$	20.0000i	1.09930i	−0.835395	+	0.549650i	$0.814761\pi$
$332$	0	0
$333$	2.00000i	0.109599i
$334$	0	0
$335$	12.0000	0.655630
$336$	0	0
$337$	−10.0000	−0.544735	−0.272367	−	0.962193i	$-0.587807\pi$
$337$	−10.0000	−0.544735	−0.272367	+	0.962193i	$0.587807\pi$
$338$	0	0
$339$	− 16.0000i	− 0.869001i
$340$	0	0
$341$	− 8.00000i	− 0.433224i
$342$	0	0
$343$	20.0000	1.07990
$344$	0	0
$345$	−4.00000	−0.215353
$346$	0	0
$347$	12.0000i	0.644194i	0.946707	+	0.322097i	$0.104388\pi$
$347$	12.0000i	0.644194i	−0.946707	+	0.322097i	$0.895612\pi$
$348$	0	0
$349$	16.0000i	0.856460i	0.903670	+	0.428230i	$0.140863\pi$
$349$	16.0000i	0.856460i	−0.903670	+	0.428230i	$0.859137\pi$
$350$	0	0
$351$	2.00000	0.106752
$352$	0	0
$353$	4.00000	0.212899	0.106449	−	0.994318i	$-0.466052\pi$
$353$	4.00000	0.212899	0.106449	+	0.994318i	$0.466052\pi$
$354$	0	0
$355$	− 8.00000i	− 0.424596i
$356$	0	0
$357$	8.00000i	0.423405i
$358$	0	0
$359$	24.0000	1.26667	0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000	1.26667	0.633336	+	0.773877i	$0.281685\pi$
$360$	0	0
$361$	3.00000	0.157895
$362$	0	0
$363$	− 7.00000i	− 0.367405i
$364$	0	0
$365$	− 6.00000i	− 0.314054i
$366$	0	0
$367$	−6.00000	−0.313197	−0.156599	−	0.987662i	$-0.550053\pi$
$367$	−6.00000	−0.313197	−0.156599	+	0.987662i	$0.550053\pi$
$368$	0	0
$369$	−6.00000	−0.312348
$370$	0	0
$371$	20.0000i	1.03835i
$372$	0	0
$373$	− 14.0000i	− 0.724893i	−0.932005	−	0.362446i	$-0.881942\pi$
$373$	− 14.0000i	− 0.724893i	0.932005	−	0.362446i	$-0.118058\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−4.00000	−0.206010
$378$	0	0
$379$	8.00000i	0.410932i	0.978664	+	0.205466i	$0.0658711\pi$
$379$	8.00000i	0.410932i	−0.978664	+	0.205466i	$0.934129\pi$
$380$	0	0
$381$	14.0000i	0.717242i
$382$	0	0
$383$	32.0000	1.63512	0.817562	−	0.575841i	$-0.195325\pi$
$383$	32.0000	1.63512	0.817562	+	0.575841i	$0.195325\pi$
$384$	0	0
$385$	4.00000	0.203859
$386$	0	0
$387$	4.00000i	0.203331i
$388$	0	0
$389$	− 2.00000i	− 0.101404i	−0.998714	−	0.0507020i	$-0.983854\pi$
$389$	− 2.00000i	− 0.101404i	0.998714	−	0.0507020i	$-0.0161459\pi$
$390$	0	0
$391$	−16.0000	−0.809155
$392$	0	0
$393$	−6.00000	−0.302660
$394$	0	0
$395$	− 4.00000i	− 0.201262i
$396$	0	0
$397$	10.0000i	0.501886i	0.968002	+	0.250943i	$0.0807406\pi$
$397$	10.0000i	0.501886i	−0.968002	+	0.250943i	$0.919259\pi$
$398$	0	0
$399$	8.00000	0.400501
$400$	0	0
$401$	14.0000	0.699127	0.349563	−	0.936913i	$-0.386330\pi$
$401$	14.0000	0.699127	0.349563	+	0.936913i	$0.386330\pi$
$402$	0	0
$403$	8.00000i	0.398508i
$404$	0	0
$405$	1.00000i	0.0496904i
$406$	0	0
$407$	4.00000	0.198273
$408$	0	0
$409$	38.0000	1.87898	0.939490	−	0.342578i	$-0.111300\pi$
$409$	38.0000	1.87898	0.939490	+	0.342578i	$0.111300\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	12.0000i	0.590481i
$414$	0	0
$415$	16.0000	0.785409
$416$	0	0
$417$	16.0000	0.783523
$418$	0	0
$419$	34.0000i	1.66101i	0.557012	+	0.830504i	$0.311948\pi$
$419$	34.0000i	1.66101i	−0.557012	+	0.830504i	$0.688052\pi$
$420$	0	0
$421$	− 20.0000i	− 0.974740i	−0.873195	−	0.487370i	$-0.837956\pi$
$421$	− 20.0000i	− 0.974740i	0.873195	−	0.487370i	$-0.162044\pi$
$422$	0	0
$423$	8.00000	0.388973
$424$	0	0
$425$	−4.00000	−0.194029
$426$	0	0
$427$	0	0
$428$	0	0
$429$	− 4.00000i	− 0.193122i
$430$	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
$433$	−34.0000	−1.63394	−0.816968	−	0.576683i	$-0.804347\pi$
$433$	−34.0000	−1.63394	−0.816968	+	0.576683i	$0.804347\pi$
$434$	0	0
$435$	− 2.00000i	− 0.0958927i
$436$	0	0
$437$	16.0000i	0.765384i
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	3.00000	0.142857
$442$	0	0
$443$	− 12.0000i	− 0.570137i	−0.958507	−	0.285069i	$-0.907984\pi$
$443$	− 12.0000i	− 0.570137i	0.958507	−	0.285069i	$-0.0920164\pi$
$444$	0	0
$445$	− 6.00000i	− 0.284427i
$446$	0	0
$447$	−18.0000	−0.851371
$448$	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
$451$	12.0000i	0.565058i
$452$	0	0
$453$	24.0000i	1.12762i
$454$	0	0
$455$	−4.00000	−0.187523
$456$	0	0
$457$	−2.00000	−0.0935561	−0.0467780	−	0.998905i	$-0.514895\pi$
$457$	−2.00000	−0.0935561	−0.0467780	+	0.998905i	$0.514895\pi$
$458$	0	0
$459$	4.00000i	0.186704i
$460$	0	0
$461$	− 6.00000i	− 0.279448i	−0.990190	−	0.139724i	$-0.955378\pi$
$461$	− 6.00000i	− 0.279448i	0.990190	−	0.139724i	$-0.0446215\pi$
$462$	0	0
$463$	38.0000	1.76601	0.883005	−	0.469364i	$-0.155517\pi$
$463$	38.0000	1.76601	0.883005	+	0.469364i	$0.155517\pi$
$464$	0	0
$465$	−4.00000	−0.185496
$466$	0	0
$467$	24.0000i	1.11059i	0.831654	+	0.555294i	$0.187394\pi$
$467$	24.0000i	1.11059i	−0.831654	+	0.555294i	$0.812606\pi$
$468$	0	0
$469$	24.0000i	1.10822i
$470$	0	0
$471$	−10.0000	−0.460776
$472$	0	0
$473$	8.00000	0.367840
$474$	0	0
$475$	4.00000i	0.183533i
$476$	0	0
$477$	10.0000i	0.457869i
$478$	0	0
$479$	−32.0000	−1.46212	−0.731059	−	0.682315i	$-0.760973\pi$
$479$	−32.0000	−1.46212	−0.731059	+	0.682315i	$0.760973\pi$
$480$	0	0
$481$	−4.00000	−0.182384
$482$	0	0
$483$	− 8.00000i	− 0.364013i
$484$	0	0
$485$	− 14.0000i	− 0.635707i
$486$	0	0
$487$	26.0000	1.17817	0.589086	−	0.808070i	$-0.299488\pi$
$487$	26.0000	1.17817	0.589086	+	0.808070i	$0.299488\pi$
$488$	0	0
$489$	−20.0000	−0.904431
$490$	0	0
$491$	6.00000i	0.270776i	0.990793	+	0.135388i	$0.0432281\pi$
$491$	6.00000i	0.270776i	−0.990793	+	0.135388i	$0.956772\pi$
$492$	0	0
$493$	− 8.00000i	− 0.360302i
$494$	0	0
$495$	2.00000	0.0898933
$496$	0	0
$497$	16.0000	0.717698
$498$	0	0
$499$	− 4.00000i	− 0.179065i	−0.995984	−	0.0895323i	$-0.971463\pi$
$499$	− 4.00000i	− 0.179065i	0.995984	−	0.0895323i	$-0.0285372\pi$
$500$	0	0
$501$	− 12.0000i	− 0.536120i
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	−2.00000	−0.0889988
$506$	0	0
$507$	− 9.00000i	− 0.399704i
$508$	0	0
$509$	6.00000i	0.265945i	0.991120	+	0.132973i	$0.0424523\pi$
$509$	6.00000i	0.265945i	−0.991120	+	0.132973i	$0.957548\pi$
$510$	0	0
$511$	12.0000	0.530849
$512$	0	0
$513$	4.00000	0.176604
$514$	0	0
$515$	14.0000i	0.616914i
$516$	0	0
$517$	− 16.0000i	− 0.703679i
$518$	0	0
$519$	6.00000	0.263371
$520$	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
$523$	− 20.0000i	− 0.874539i	−0.899331	−	0.437269i	$-0.855946\pi$
$523$	− 20.0000i	− 0.874539i	0.899331	−	0.437269i	$-0.144054\pi$
$524$	0	0
$525$	− 2.00000i	− 0.0872872i
$526$	0	0
$527$	−16.0000	−0.696971
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	6.00000i	0.260378i
$532$	0	0
$533$	− 12.0000i	− 0.519778i
$534$	0	0
$535$	−8.00000	−0.345870
$536$	0	0
$537$	−10.0000	−0.431532
$538$	0	0
$539$	− 6.00000i	− 0.258438i
$540$	0	0
$541$	8.00000i	0.343947i	0.985102	+	0.171973i	$0.0550143\pi$
$541$	8.00000i	0.343947i	−0.985102	+	0.171973i	$0.944986\pi$
$542$	0	0
$543$	8.00000	0.343313
$544$	0	0
$545$	−16.0000	−0.685365
$546$	0	0
$547$	36.0000i	1.53925i	0.638497	+	0.769624i	$0.279557\pi$
$547$	36.0000i	1.53925i	−0.638497	+	0.769624i	$0.720443\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−8.00000	−0.340811
$552$	0	0
$553$	8.00000	0.340195
$554$	0	0
$555$	− 2.00000i	− 0.0848953i
$556$	0	0
$557$	− 2.00000i	− 0.0847427i	−0.999102	−	0.0423714i	$-0.986509\pi$
$557$	− 2.00000i	− 0.0847427i	0.999102	−	0.0423714i	$-0.0134913\pi$
$558$	0	0
$559$	−8.00000	−0.338364
$560$	0	0
$561$	8.00000	0.337760
$562$	0	0
$563$	− 12.0000i	− 0.505740i	−0.967500	−	0.252870i	$-0.918626\pi$
$563$	− 12.0000i	− 0.505740i	0.967500	−	0.252870i	$-0.0813744\pi$
$564$	0	0
$565$	16.0000i	0.673125i
$566$	0	0
$567$	−2.00000	−0.0839921
$568$	0	0
$569$	−38.0000	−1.59304	−0.796521	−	0.604610i	$-0.793329\pi$
$569$	−38.0000	−1.59304	−0.796521	+	0.604610i	$0.793329\pi$
$570$	0	0
$571$	20.0000i	0.836974i	0.908223	+	0.418487i	$0.137439\pi$
$571$	20.0000i	0.836974i	−0.908223	+	0.418487i	$0.862561\pi$
$572$	0	0
$573$	8.00000i	0.334205i
$574$	0	0
$575$	4.00000	0.166812
$576$	0	0
$577$	−6.00000	−0.249783	−0.124892	−	0.992170i	$-0.539858\pi$
$577$	−6.00000	−0.249783	−0.124892	+	0.992170i	$0.539858\pi$
$578$	0	0
$579$	18.0000i	0.748054i
$580$	0	0
$581$	32.0000i	1.32758i
$582$	0	0
$583$	20.0000	0.828315
$584$	0	0
$585$	−2.00000	−0.0826898
$586$	0	0
$587$	8.00000i	0.330195i	0.986277	+	0.165098i	$0.0527939\pi$
$587$	8.00000i	0.330195i	−0.986277	+	0.165098i	$0.947206\pi$
$588$	0	0
$589$	16.0000i	0.659269i
$590$	0	0
$591$	−6.00000	−0.246807
$592$	0	0
$593$	24.0000	0.985562	0.492781	−	0.870153i	$-0.335980\pi$
$593$	24.0000	0.985562	0.492781	+	0.870153i	$0.335980\pi$
$594$	0	0
$595$	− 8.00000i	− 0.327968i
$596$	0	0
$597$	− 8.00000i	− 0.327418i
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	−38.0000	−1.55005	−0.775026	−	0.631929i	$-0.782263\pi$
$601$	−38.0000	−1.55005	−0.775026	+	0.631929i	$0.782263\pi$
$602$	0	0
$603$	12.0000i	0.488678i
$604$	0	0
$605$	7.00000i	0.284590i
$606$	0	0
$607$	−34.0000	−1.38002	−0.690009	−	0.723801i	$-0.742393\pi$
$607$	−34.0000	−1.38002	−0.690009	+	0.723801i	$0.742393\pi$
$608$	0	0
$609$	4.00000	0.162088
$610$	0	0
$611$	16.0000i	0.647291i
$612$	0	0
$613$	− 42.0000i	− 1.69636i	−0.529705	−	0.848182i	$-0.677697\pi$
$613$	− 42.0000i	− 1.69636i	0.529705	−	0.848182i	$-0.322303\pi$
$614$	0	0
$615$	6.00000	0.241943
$616$	0	0
$617$	−4.00000	−0.161034	−0.0805170	−	0.996753i	$-0.525657\pi$
$617$	−4.00000	−0.161034	−0.0805170	+	0.996753i	$0.525657\pi$
$618$	0	0
$619$	− 8.00000i	− 0.321547i	−0.986991	−	0.160774i	$-0.948601\pi$
$619$	− 8.00000i	− 0.321547i	0.986991	−	0.160774i	$-0.0513989\pi$
$620$	0	0
$621$	− 4.00000i	− 0.160514i
$622$	0	0
$623$	12.0000	0.480770
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	− 8.00000i	− 0.319489i
$628$	0	0
$629$	− 8.00000i	− 0.318981i
$630$	0	0
$631$	−44.0000	−1.75161	−0.875806	−	0.482663i	$-0.839670\pi$
$631$	−44.0000	−1.75161	−0.875806	+	0.482663i	$0.839670\pi$
$632$	0	0
$633$	8.00000	0.317971
$634$	0	0
$635$	− 14.0000i	− 0.555573i
$636$	0	0
$637$	6.00000i	0.237729i
$638$	0	0
$639$	8.00000	0.316475
$640$	0	0
$641$	18.0000	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$642$	0	0
$643$	− 12.0000i	− 0.473234i	−0.971603	−	0.236617i	$-0.923961\pi$
$643$	− 12.0000i	− 0.473234i	0.971603	−	0.236617i	$-0.0760386\pi$
$644$	0	0
$645$	− 4.00000i	− 0.157500i
$646$	0	0
$647$	24.0000	0.943537	0.471769	−	0.881722i	$-0.343616\pi$
$647$	24.0000	0.943537	0.471769	+	0.881722i	$0.343616\pi$
$648$	0	0
$649$	12.0000	0.471041
$650$	0	0
$651$	− 8.00000i	− 0.313545i
$652$	0	0
$653$	34.0000i	1.33052i	0.746611	+	0.665261i	$0.231680\pi$
$653$	34.0000i	1.33052i	−0.746611	+	0.665261i	$0.768320\pi$
$654$	0	0
$655$	6.00000	0.234439
$656$	0	0
$657$	6.00000	0.234082
$658$	0	0
$659$	18.0000i	0.701180i	0.936529	+	0.350590i	$0.114019\pi$
$659$	18.0000i	0.701180i	−0.936529	+	0.350590i	$0.885981\pi$
$660$	0	0
$661$	− 32.0000i	− 1.24466i	−0.782757	−	0.622328i	$-0.786187\pi$
$661$	− 32.0000i	− 1.24466i	0.782757	−	0.622328i	$-0.213813\pi$
$662$	0	0
$663$	−8.00000	−0.310694
$664$	0	0
$665$	−8.00000	−0.310227
$666$	0	0
$667$	8.00000i	0.309761i
$668$	0	0
$669$	− 2.00000i	− 0.0773245i
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−34.0000	−1.31060	−0.655302	−	0.755367i	$-0.727459\pi$
$673$	−34.0000	−1.31060	−0.655302	+	0.755367i	$0.727459\pi$
$674$	0	0
$675$	− 1.00000i	− 0.0384900i
$676$	0	0
$677$	22.0000i	0.845529i	0.906240	+	0.422764i	$0.138940\pi$
$677$	22.0000i	0.845529i	−0.906240	+	0.422764i	$0.861060\pi$
$678$	0	0
$679$	28.0000	1.07454
$680$	0	0
$681$	−4.00000	−0.153280
$682$	0	0
$683$	− 40.0000i	− 1.53056i	−0.643699	−	0.765279i	$-0.722601\pi$
$683$	− 40.0000i	− 1.53056i	0.643699	−	0.765279i	$-0.277399\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−4.00000	−0.152610
$688$	0	0
$689$	−20.0000	−0.761939
$690$	0	0
$691$	20.0000i	0.760836i	0.924815	+	0.380418i	$0.124220\pi$
$691$	20.0000i	0.760836i	−0.924815	+	0.380418i	$0.875780\pi$
$692$	0	0
$693$	4.00000i	0.151947i
$694$	0	0
$695$	−16.0000	−0.606915
$696$	0	0
$697$	24.0000	0.909065
$698$	0	0
$699$	− 4.00000i	− 0.151294i
$700$	0	0
$701$	− 34.0000i	− 1.28416i	−0.766637	−	0.642081i	$-0.778071\pi$
$701$	− 34.0000i	− 1.28416i	0.766637	−	0.642081i	$-0.221929\pi$
$702$	0	0
$703$	−8.00000	−0.301726
$704$	0	0
$705$	−8.00000	−0.301297
$706$	0	0
$707$	− 4.00000i	− 0.150435i
$708$	0	0
$709$	52.0000i	1.95290i	0.215742	+	0.976450i	$0.430783\pi$
$709$	52.0000i	1.95290i	−0.215742	+	0.976450i	$0.569217\pi$
$710$	0	0
$711$	4.00000	0.150012
$712$	0	0
$713$	16.0000	0.599205
$714$	0	0
$715$	4.00000i	0.149592i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	−28.0000	−1.04277
$722$	0	0
$723$	26.0000i	0.966950i
$724$	0	0
$725$	2.00000i	0.0742781i
$726$	0	0
$727$	26.0000	0.964287	0.482143	−	0.876092i	$-0.339858\pi$
$727$	26.0000	0.964287	0.482143	+	0.876092i	$0.339858\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	− 16.0000i	− 0.591781i
$732$	0	0
$733$	− 46.0000i	− 1.69905i	−0.527549	−	0.849524i	$-0.676889\pi$
$733$	− 46.0000i	− 1.69905i	0.527549	−	0.849524i	$-0.323111\pi$
$734$	0	0
$735$	−3.00000	−0.110657
$736$	0	0
$737$	24.0000	0.884051
$738$	0	0
$739$	4.00000i	0.147142i	0.997290	+	0.0735712i	$0.0234396\pi$
$739$	4.00000i	0.147142i	−0.997290	+	0.0735712i	$0.976560\pi$
$740$	0	0
$741$	8.00000i	0.293887i
$742$	0	0
$743$	32.0000	1.17397	0.586983	−	0.809599i	$-0.300316\pi$
$743$	32.0000	1.17397	0.586983	+	0.809599i	$0.300316\pi$
$744$	0	0
$745$	18.0000	0.659469
$746$	0	0
$747$	16.0000i	0.585409i
$748$	0	0
$749$	− 16.0000i	− 0.584627i
$750$	0	0
$751$	−28.0000	−1.02173	−0.510867	−	0.859660i	$-0.670676\pi$
$751$	−28.0000	−1.02173	−0.510867	+	0.859660i	$0.670676\pi$
$752$	0	0
$753$	30.0000	1.09326
$754$	0	0
$755$	− 24.0000i	− 0.873449i
$756$	0	0
$757$	6.00000i	0.218074i	0.994038	+	0.109037i	$0.0347767\pi$
$757$	6.00000i	0.218074i	−0.994038	+	0.109037i	$0.965223\pi$
$758$	0	0
$759$	−8.00000	−0.290382
$760$	0	0
$761$	−14.0000	−0.507500	−0.253750	−	0.967270i	$-0.581664\pi$
$761$	−14.0000	−0.507500	−0.253750	+	0.967270i	$0.581664\pi$
$762$	0	0
$763$	− 32.0000i	− 1.15848i
$764$	0	0
$765$	− 4.00000i	− 0.144620i
$766$	0	0
$767$	−12.0000	−0.433295
$768$	0	0
$769$	34.0000	1.22607	0.613036	−	0.790055i	$-0.289948\pi$
$769$	34.0000	1.22607	0.613036	+	0.790055i	$0.289948\pi$
$770$	0	0
$771$	− 8.00000i	− 0.288113i
$772$	0	0
$773$	10.0000i	0.359675i	0.983696	+	0.179838i	$0.0575572\pi$
$773$	10.0000i	0.359675i	−0.983696	+	0.179838i	$0.942443\pi$
$774$	0	0
$775$	4.00000	0.143684
$776$	0	0
$777$	4.00000	0.143499
$778$	0	0
$779$	− 24.0000i	− 0.859889i
$780$	0	0
$781$	− 16.0000i	− 0.572525i
$782$	0	0
$783$	2.00000	0.0714742
$784$	0	0
$785$	10.0000	0.356915
$786$	0	0
$787$	− 12.0000i	− 0.427754i	−0.976861	−	0.213877i	$-0.931391\pi$
$787$	− 12.0000i	− 0.427754i	0.976861	−	0.213877i	$-0.0686091\pi$
$788$	0	0
$789$	− 28.0000i	− 0.996826i
$790$	0	0
$791$	−32.0000	−1.13779
$792$	0	0
$793$	0	0
$794$	0	0
$795$	− 10.0000i	− 0.354663i
$796$	0	0
$797$	38.0000i	1.34603i	0.739629	+	0.673015i	$0.235001\pi$
$797$	38.0000i	1.34603i	−0.739629	+	0.673015i	$0.764999\pi$
$798$	0	0
$799$	−32.0000	−1.13208
$800$	0	0
$801$	6.00000	0.212000
$802$	0	0
$803$	− 12.0000i	− 0.423471i
$804$	0	0
$805$	8.00000i	0.281963i
$806$	0	0
$807$	−18.0000	−0.633630
$808$	0	0
$809$	−34.0000	−1.19538	−0.597688	−	0.801729i	$-0.703914\pi$
$809$	−34.0000	−1.19538	−0.597688	+	0.801729i	$0.703914\pi$
$810$	0	0
$811$	− 40.0000i	− 1.40459i	−0.711886	−	0.702295i	$-0.752159\pi$
$811$	− 40.0000i	− 1.40459i	0.711886	−	0.702295i	$-0.247841\pi$
$812$	0	0
$813$	− 24.0000i	− 0.841717i
$814$	0	0
$815$	20.0000	0.700569
$816$	0	0
$817$	−16.0000	−0.559769
$818$	0	0
$819$	− 4.00000i	− 0.139771i
$820$	0	0
$821$	18.0000i	0.628204i	0.949389	+	0.314102i	$0.101703\pi$
$821$	18.0000i	0.628204i	−0.949389	+	0.314102i	$0.898297\pi$
$822$	0	0
$823$	−38.0000	−1.32460	−0.662298	−	0.749240i	$-0.730419\pi$
$823$	−38.0000	−1.32460	−0.662298	+	0.749240i	$0.730419\pi$
$824$	0	0
$825$	−2.00000	−0.0696311
$826$	0	0
$827$	− 28.0000i	− 0.973655i	−0.873498	−	0.486828i	$-0.838154\pi$
$827$	− 28.0000i	− 0.973655i	0.873498	−	0.486828i	$-0.161846\pi$
$828$	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
$831$	22.0000	0.763172
$832$	0	0
$833$	−12.0000	−0.415775
$834$	0	0
$835$	12.0000i	0.415277i
$836$	0	0
$837$	− 4.00000i	− 0.138260i
$838$	0	0
$839$	24.0000	0.828572	0.414286	−	0.910147i	$-0.364031\pi$
$839$	24.0000	0.828572	0.414286	+	0.910147i	$0.364031\pi$
$840$	0	0
$841$	25.0000	0.862069
$842$	0	0
$843$	− 18.0000i	− 0.619953i
$844$	0	0
$845$	9.00000i	0.309609i
$846$	0	0
$847$	−14.0000	−0.481046
$848$	0	0
$849$	−4.00000	−0.137280
$850$	0	0
$851$	8.00000i	0.274236i
$852$	0	0
$853$	46.0000i	1.57501i	0.616308	+	0.787505i	$0.288628\pi$
$853$	46.0000i	1.57501i	−0.616308	+	0.787505i	$0.711372\pi$
$854$	0	0
$855$	−4.00000	−0.136797
$856$	0	0
$857$	20.0000	0.683187	0.341593	−	0.939848i	$-0.389033\pi$
$857$	20.0000	0.683187	0.341593	+	0.939848i	$0.389033\pi$
$858$	0	0
$859$	− 56.0000i	− 1.91070i	−0.295484	−	0.955348i	$-0.595481\pi$
$859$	− 56.0000i	− 1.91070i	0.295484	−	0.955348i	$-0.404519\pi$
$860$	0	0
$861$	12.0000i	0.408959i
$862$	0	0
$863$	−44.0000	−1.49778	−0.748889	−	0.662696i	$-0.769412\pi$
$863$	−44.0000	−1.49778	−0.748889	+	0.662696i	$0.769412\pi$
$864$	0	0
$865$	−6.00000	−0.204006
$866$	0	0
$867$	1.00000i	0.0339618i
$868$	0	0
$869$	− 8.00000i	− 0.271381i
$870$	0	0
$871$	−24.0000	−0.813209
$872$	0	0
$873$	14.0000	0.473828
$874$	0	0
$875$	2.00000i	0.0676123i
$876$	0	0
$877$	− 2.00000i	− 0.0675352i	−0.999430	−	0.0337676i	$-0.989249\pi$
$877$	− 2.00000i	− 0.0675352i	0.999430	−	0.0337676i	$-0.0107506\pi$
$878$	0	0
$879$	−6.00000	−0.202375
$880$	0	0
$881$	−2.00000	−0.0673817	−0.0336909	−	0.999432i	$-0.510726\pi$
$881$	−2.00000	−0.0673817	−0.0336909	+	0.999432i	$0.510726\pi$
$882$	0	0
$883$	28.0000i	0.942275i	0.882060	+	0.471138i	$0.156156\pi$
$883$	28.0000i	0.942275i	−0.882060	+	0.471138i	$0.843844\pi$
$884$	0	0
$885$	− 6.00000i	− 0.201688i
$886$	0	0
$887$	−36.0000	−1.20876	−0.604381	−	0.796696i	$-0.706579\pi$
$887$	−36.0000	−1.20876	−0.604381	+	0.796696i	$0.706579\pi$
$888$	0	0
$889$	28.0000	0.939090
$890$	0	0
$891$	2.00000i	0.0670025i
$892$	0	0
$893$	32.0000i	1.07084i
$894$	0	0
$895$	10.0000	0.334263
$896$	0	0
$897$	8.00000	0.267112
$898$	0	0
$899$	8.00000i	0.266815i
$900$	0	0
$901$	− 40.0000i	− 1.33259i
$902$	0	0
$903$	8.00000	0.266223
$904$	0	0
$905$	−8.00000	−0.265929
$906$	0	0
$907$	− 28.0000i	− 0.929725i	−0.885383	−	0.464862i	$-0.846104\pi$
$907$	− 28.0000i	− 0.929725i	0.885383	−	0.464862i	$-0.153896\pi$
$908$	0	0
$909$	− 2.00000i	− 0.0663358i
$910$	0	0
$911$	48.0000	1.59031	0.795155	−	0.606406i	$-0.207389\pi$
$911$	48.0000	1.59031	0.795155	+	0.606406i	$0.207389\pi$
$912$	0	0
$913$	32.0000	1.05905
$914$	0	0
$915$	0	0
$916$	0	0
$917$	12.0000i	0.396275i
$918$	0	0
$919$	−36.0000	−1.18753	−0.593765	−	0.804638i	$-0.702359\pi$
$919$	−36.0000	−1.18753	−0.593765	+	0.804638i	$0.702359\pi$
$920$	0	0
$921$	12.0000	0.395413
$922$	0	0
$923$	16.0000i	0.526646i
$924$	0	0
$925$	2.00000i	0.0657596i
$926$	0	0
$927$	−14.0000	−0.459820
$928$	0	0
$929$	−34.0000	−1.11550	−0.557752	−	0.830008i	$-0.688336\pi$
$929$	−34.0000	−1.11550	−0.557752	+	0.830008i	$0.688336\pi$
$930$	0	0
$931$	12.0000i	0.393284i
$932$	0	0
$933$	32.0000i	1.04763i
$934$	0	0
$935$	−8.00000	−0.261628
$936$	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
$939$	− 2.00000i	− 0.0652675i
$940$	0	0
$941$	− 38.0000i	− 1.23876i	−0.785090	−	0.619382i	$-0.787383\pi$
$941$	− 38.0000i	− 1.23876i	0.785090	−	0.619382i	$-0.212617\pi$
$942$	0	0
$943$	−24.0000	−0.781548
$944$	0	0
$945$	2.00000	0.0650600
$946$	0	0
$947$	− 12.0000i	− 0.389948i	−0.980808	−	0.194974i	$-0.937538\pi$
$947$	− 12.0000i	− 0.389948i	0.980808	−	0.194974i	$-0.0624622\pi$
$948$	0	0
$949$	12.0000i	0.389536i
$950$	0	0
$951$	−34.0000	−1.10253
$952$	0	0
$953$	−24.0000	−0.777436	−0.388718	−	0.921357i	$-0.627082\pi$
$953$	−24.0000	−0.777436	−0.388718	+	0.921357i	$0.627082\pi$
$954$	0	0
$955$	− 8.00000i	− 0.258874i
$956$	0	0
$957$	− 4.00000i	− 0.129302i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	− 8.00000i	− 0.257796i
$964$	0	0
$965$	− 18.0000i	− 0.579441i
$966$	0	0
$967$	54.0000	1.73652	0.868261	−	0.496107i	$-0.165238\pi$
$967$	54.0000	1.73652	0.868261	+	0.496107i	$0.165238\pi$
$968$	0	0
$969$	−16.0000	−0.513994
$970$	0	0
$971$	− 26.0000i	− 0.834380i	−0.908819	−	0.417190i	$-0.863015\pi$
$971$	− 26.0000i	− 0.834380i	0.908819	−	0.417190i	$-0.136985\pi$
$972$	0	0
$973$	− 32.0000i	− 1.02587i
$974$	0	0
$975$	2.00000	0.0640513
$976$	0	0
$977$	−36.0000	−1.15174	−0.575871	−	0.817541i	$-0.695337\pi$
$977$	−36.0000	−1.15174	−0.575871	+	0.817541i	$0.695337\pi$
$978$	0	0
$979$	− 12.0000i	− 0.383522i
$980$	0	0
$981$	− 16.0000i	− 0.510841i
$982$	0	0
$983$	8.00000	0.255160	0.127580	−	0.991828i	$-0.459279\pi$
$983$	8.00000	0.255160	0.127580	+	0.991828i	$0.459279\pi$
$984$	0	0
$985$	6.00000	0.191176
$986$	0	0
$987$	− 16.0000i	− 0.509286i
$988$	0	0
$989$	16.0000i	0.508770i
$990$	0	0
$991$	16.0000	0.508257	0.254128	−	0.967170i	$-0.418211\pi$
$991$	16.0000	0.508257	0.254128	+	0.967170i	$0.418211\pi$
$992$	0	0
$993$	20.0000	0.634681
$994$	0	0
$995$	8.00000i	0.253617i
$996$	0	0
$997$	− 62.0000i	− 1.96356i	−0.190022	−	0.981780i	$-0.560856\pi$
$997$	− 62.0000i	− 1.96356i	0.190022	−	0.981780i	$-0.439144\pi$
$998$	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.c.961.1	✓	2
3.2	odd	2		5760.2.k.d.2881.1		2
4.3	odd	2		1920.2.k.f.961.2	yes	2
8.3	odd	2		1920.2.k.f.961.1	yes	2
8.5	even	2	inner	1920.2.k.c.961.2	yes	2
12.11	even	2		5760.2.k.g.2881.1		2
16.3	odd	4		3840.2.a.h.1.1		1
16.5	even	4		3840.2.a.f.1.1		1
16.11	odd	4		3840.2.a.p.1.1		1
16.13	even	4		3840.2.a.bb.1.1		1
24.5	odd	2		5760.2.k.d.2881.2		2
24.11	even	2		5760.2.k.g.2881.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1920.2.k.c.961.1	✓	2	1.1	even	1	trivial
1920.2.k.c.961.2	yes	2	8.5	even	2	inner
1920.2.k.f.961.1	yes	2	8.3	odd	2
1920.2.k.f.961.2	yes	2	4.3	odd	2
3840.2.a.f.1.1		1	16.5	even	4
3840.2.a.h.1.1		1	16.3	odd	4
3840.2.a.p.1.1		1	16.11	odd	4
3840.2.a.bb.1.1		1	16.13	even	4
5760.2.k.d.2881.1		2	3.2	odd	2
5760.2.k.d.2881.2		2	24.5	odd	2
5760.2.k.g.2881.1		2	12.11	even	2
5760.2.k.g.2881.2		2	24.11	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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +1.00000i q^{5} -2.00000 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} +1.00000i q^{5} -2.00000 q^{7} -1.00000 q^{9} +2.00000i q^{11} -2.00000i q^{13} +1.00000 q^{15} +4.00000 q^{17} -4.00000i q^{19} +2.00000i q^{21} -4.00000 q^{23} -1.00000 q^{25} +1.00000i q^{27} -2.00000i q^{29} -4.00000 q^{31} +2.00000 q^{33} -2.00000i q^{35} -2.00000i q^{37} -2.00000 q^{39} +6.00000 q^{41} -4.00000i q^{43} -1.00000i q^{45} -8.00000 q^{47} -3.00000 q^{49} -4.00000i q^{51} -10.0000i q^{53} -2.00000 q^{55} -4.00000 q^{57} -6.00000i q^{59} +2.00000 q^{63} +2.00000 q^{65} -12.0000i q^{67} +4.00000i q^{69} -8.00000 q^{71} -6.00000 q^{73} +1.00000i q^{75} -4.00000i q^{77} -4.00000 q^{79} +1.00000 q^{81} -16.0000i q^{83} +4.00000i q^{85} -2.00000 q^{87} -6.00000 q^{89} +4.00000i q^{91} +4.00000i q^{93} +4.00000 q^{95} -14.0000 q^{97} -2.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{7} - 2 q^{9}+O(q^{10}) \) 2 * q - 4 * q^7 - 2 * q^9	\( 2 q - 4 q^{7} - 2 q^{9} + 2 q^{15} + 8 q^{17} - 8 q^{23} - 2 q^{25} - 8 q^{31} + 4 q^{33} - 4 q^{39} + 12 q^{41} - 16 q^{47} - 6 q^{49} - 4 q^{55} - 8 q^{57} + 4 q^{63} + 4 q^{65} - 16 q^{71} - 12 q^{73} - 8 q^{79} + 2 q^{81} - 4 q^{87} - 12 q^{89} + 8 q^{95} - 28 q^{97}+O(q^{100}) \) 2 * q - 4 * q^7 - 2 * q^9 + 2 * q^15 + 8 * q^17 - 8 * q^23 - 2 * q^25 - 8 * q^31 + 4 * q^33 - 4 * q^39 + 12 * q^41 - 16 * q^47 - 6 * q^49 - 4 * q^55 - 8 * q^57 + 4 * q^63 + 4 * q^65 - 16 * q^71 - 12 * q^73 - 8 * q^79 + 2 * q^81 - 4 * q^87 - 12 * q^89 + 8 * q^95 - 28 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more