LMFDB - Embedded newform 1792.2.b.g.897.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1792,2,Mod(897,1792)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1792, 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("1792.897");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1792 = 2^{8} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1792.b (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:	$14.3091920422$
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:	$ 2 $
Twist minimal:	no (minimal twist has level 14)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			897.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	1792.897
Dual form			1792.2.b.g.897.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000i q^{3} +1.00000 q^{7} -1.00000 q^{9} +4.00000i q^{13} +6.00000 q^{17} +2.00000i q^{19} -2.00000i q^{21} +5.00000 q^{25} -4.00000i q^{27} +6.00000i q^{29} +4.00000 q^{31} +2.00000i q^{37} +8.00000 q^{39} -6.00000 q^{41} -8.00000i q^{43} +12.0000 q^{47} +1.00000 q^{49} -12.0000i q^{51} +6.00000i q^{53} +4.00000 q^{57} +6.00000i q^{59} -8.00000i q^{61} -1.00000 q^{63} -4.00000i q^{67} -2.00000 q^{73} -10.0000i q^{75} -8.00000 q^{79} -11.0000 q^{81} -6.00000i q^{83} +12.0000 q^{87} +6.00000 q^{89} +4.00000i q^{91} -8.00000i q^{93} -10.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{7} - 2 q^{9} + 12 q^{17} + 10 q^{25} + 8 q^{31} + 16 q^{39} - 12 q^{41} + 24 q^{47} + 2 q^{49} + 8 q^{57} - 2 q^{63} - 4 q^{73} - 16 q^{79} - 22 q^{81} + 24 q^{87} + 12 q^{89} - 20 q^{97}+O(q^{100}) $ 2 * q + 2 * q^7 - 2 * q^9 + 12 * q^17 + 10 * q^25 + 8 * q^31 + 16 * q^39 - 12 * q^41 + 24 * q^47 + 2 * q^49 + 8 * q^57 - 2 * q^63 - 4 * q^73 - 16 * q^79 - 22 * q^81 + 24 * q^87 + 12 * q^89 - 20 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1792\mathbb{Z}\right)^\times$.

$n$	$1023$	$1025$	$1541$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	− 2.00000i	− 1.15470i	−0.816497	−	0.577350i	$-0.804087\pi$
$3$	− 2.00000i	− 1.15470i	0.816497	−	0.577350i	$-0.195913\pi$
$4$	0	0
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	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
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	0	0
$19$	2.00000i	0.458831i	0.973329	+	0.229416i	$0.0736815\pi$
$19$	2.00000i	0.458831i	−0.973329	+	0.229416i	$0.926318\pi$
$20$	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
$25$	5.00000	1.00000
$26$	0	0
$27$	− 4.00000i	− 0.769800i
$28$	0	0
$29$	6.00000i	1.11417i	0.830455	+	0.557086i	$0.188081\pi$
$29$	6.00000i	1.11417i	−0.830455	+	0.557086i	$0.811919\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2.00000i	0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$	2.00000i	0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$	0	0
$39$	8.00000	1.28103
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	− 8.00000i	− 1.21999i	−0.792406	−	0.609994i	$-0.791172\pi$
$43$	− 8.00000i	− 1.21999i	0.792406	−	0.609994i	$-0.208828\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	12.0000	1.75038	0.875190	−	0.483779i	$-0.160736\pi$
$47$	12.0000	1.75038	0.875190	+	0.483779i	$0.160736\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	− 12.0000i	− 1.68034i
$52$	0	0
$53$	6.00000i	0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$	6.00000i	0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	4.00000	0.529813
$58$	0	0
$59$	6.00000i	0.781133i	0.920575	+	0.390567i	$0.127721\pi$
$59$	6.00000i	0.781133i	−0.920575	+	0.390567i	$0.872279\pi$
$60$	0	0
$61$	− 8.00000i	− 1.02430i	−0.858898	−	0.512148i	$-0.828850\pi$
$61$	− 8.00000i	− 1.02430i	0.858898	−	0.512148i	$-0.171150\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	0	0
$66$	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
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	0	0
$75$	− 10.0000i	− 1.15470i
$76$	0	0
$77$	0	0
$78$	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
$81$	−11.0000	−1.22222
$82$	0	0
$83$	− 6.00000i	− 0.658586i	−0.944228	−	0.329293i	$-0.893190\pi$
$83$	− 6.00000i	− 0.658586i	0.944228	−	0.329293i	$-0.106810\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	12.0000	1.28654
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	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	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 12.0000i	− 1.16008i	−0.814587	−	0.580042i	$-0.803036\pi$
$107$	− 12.0000i	− 1.16008i	0.814587	−	0.580042i	$-0.196964\pi$
$108$	0	0
$109$	− 2.00000i	− 0.191565i	−0.995402	−	0.0957826i	$-0.969465\pi$
$109$	− 2.00000i	− 0.191565i	0.995402	−	0.0957826i	$-0.0305354\pi$
$110$	0	0
$111$	4.00000	0.379663
$112$	0	0
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 4.00000i	− 0.369800i
$118$	0	0
$119$	6.00000	0.550019
$120$	0	0
$121$	11.0000	1.00000
$122$	0	0
$123$	12.0000i	1.08200i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	16.0000	1.41977	0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000	1.41977	0.709885	+	0.704317i	$0.248747\pi$
$128$	0	0
$129$	−16.0000	−1.40872
$130$	0	0
$131$	18.0000i	1.57267i	0.617802	+	0.786334i	$0.288023\pi$
$131$	18.0000i	1.57267i	−0.617802	+	0.786334i	$0.711977\pi$
$132$	0	0
$133$	2.00000i	0.173422i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−18.0000	−1.53784	−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000	−1.53784	−0.768922	+	0.639343i	$0.779207\pi$
$138$	0	0
$139$	− 14.0000i	− 1.18746i	−0.804663	−	0.593732i	$-0.797654\pi$
$139$	− 14.0000i	− 1.18746i	0.804663	−	0.593732i	$-0.202346\pi$
$140$	0	0
$141$	− 24.0000i	− 2.02116i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 2.00000i	− 0.164957i
$148$	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
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	0	0
$153$	−6.00000	−0.485071
$154$	0	0
$155$	0	0
$156$	0	0
$157$	4.00000i	0.319235i	0.987179	+	0.159617i	$0.0510260\pi$
$157$	4.00000i	0.319235i	−0.987179	+	0.159617i	$0.948974\pi$
$158$	0	0
$159$	12.0000	0.951662
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 16.0000i	− 1.25322i	−0.779334	−	0.626608i	$-0.784443\pi$
$163$	− 16.0000i	− 1.25322i	0.779334	−	0.626608i	$-0.215557\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	0	0
$169$	−3.00000	−0.230769
$170$	0	0
$171$	− 2.00000i	− 0.152944i
$172$	0	0
$173$	12.0000i	0.912343i	0.889892	+	0.456172i	$0.150780\pi$
$173$	12.0000i	0.912343i	−0.889892	+	0.456172i	$0.849220\pi$
$174$	0	0
$175$	5.00000	0.377964
$176$	0	0
$177$	12.0000	0.901975
$178$	0	0
$179$	− 12.0000i	− 0.896922i	−0.893802	−	0.448461i	$-0.851972\pi$
$179$	− 12.0000i	− 0.896922i	0.893802	−	0.448461i	$-0.148028\pi$
$180$	0	0
$181$	20.0000i	1.48659i	0.668965	+	0.743294i	$0.266738\pi$
$181$	20.0000i	1.48659i	−0.668965	+	0.743294i	$0.733262\pi$
$182$	0	0
$183$	−16.0000	−1.18275
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	− 4.00000i	− 0.290957i
$190$	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
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 18.0000i	− 1.28245i	−0.767354	−	0.641223i	$-0.778427\pi$
$197$	− 18.0000i	− 1.28245i	0.767354	−	0.641223i	$-0.221573\pi$
$198$	0	0
$199$	20.0000	1.41776	0.708881	−	0.705328i	$-0.249200\pi$
$199$	20.0000	1.41776	0.708881	+	0.705328i	$0.249200\pi$
$200$	0	0
$201$	−8.00000	−0.564276
$202$	0	0
$203$	6.00000i	0.421117i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	− 4.00000i	− 0.275371i	−0.990476	−	0.137686i	$-0.956034\pi$
$211$	− 4.00000i	− 0.275371i	0.990476	−	0.137686i	$-0.0439664\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	4.00000	0.271538
$218$	0	0
$219$	4.00000i	0.270295i
$220$	0	0
$221$	24.0000i	1.61441i
$222$	0	0
$223$	−8.00000	−0.535720	−0.267860	−	0.963458i	$-0.586316\pi$
$223$	−8.00000	−0.535720	−0.267860	+	0.963458i	$0.586316\pi$
$224$	0	0
$225$	−5.00000	−0.333333
$226$	0	0
$227$	18.0000i	1.19470i	0.801980	+	0.597351i	$0.203780\pi$
$227$	18.0000i	1.19470i	−0.801980	+	0.597351i	$0.796220\pi$
$228$	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
$231$	0	0
$232$	0	0
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	16.0000i	1.03931i
$238$	0	0
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	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
$243$	10.0000i	0.641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−8.00000	−0.509028
$248$	0	0
$249$	−12.0000	−0.760469
$250$	0	0
$251$	18.0000i	1.13615i	0.822977	+	0.568075i	$0.192312\pi$
$251$	18.0000i	1.13615i	−0.822977	+	0.568075i	$0.807688\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	0	0
$259$	2.00000i	0.124274i
$260$	0	0
$261$	− 6.00000i	− 0.371391i
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 12.0000i	− 0.734388i
$268$	0	0
$269$	12.0000i	0.731653i	0.930683	+	0.365826i	$0.119214\pi$
$269$	12.0000i	0.731653i	−0.930683	+	0.365826i	$0.880786\pi$
$270$	0	0
$271$	16.0000	0.971931	0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000	0.971931	0.485965	+	0.873978i	$0.338468\pi$
$272$	0	0
$273$	8.00000	0.484182
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 10.0000i	− 0.600842i	−0.953807	−	0.300421i	$-0.902873\pi$
$277$	− 10.0000i	− 0.600842i	0.953807	−	0.300421i	$-0.0971271\pi$
$278$	0	0
$279$	−4.00000	−0.239474
$280$	0	0
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	0	0
$283$	22.0000i	1.30776i	0.756596	+	0.653882i	$0.226861\pi$
$283$	22.0000i	1.30776i	−0.756596	+	0.653882i	$0.773139\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−6.00000	−0.354169
$288$	0	0
$289$	19.0000	1.11765
$290$	0	0
$291$	20.0000i	1.17242i
$292$	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
$299$	0	0
$300$	0	0
$301$	− 8.00000i	− 0.461112i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	2.00000i	0.114146i	0.998370	+	0.0570730i	$0.0181768\pi$
$307$	2.00000i	0.114146i	−0.998370	+	0.0570730i	$0.981823\pi$
$308$	0	0
$309$	8.00000i	0.455104i
$310$	0	0
$311$	−24.0000	−1.36092	−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000	−1.36092	−0.680458	+	0.732787i	$0.738219\pi$
$312$	0	0
$313$	10.0000	0.565233	0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000	0.565233	0.282617	+	0.959233i	$0.408798\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 6.00000i	− 0.336994i	−0.985702	−	0.168497i	$-0.946109\pi$
$317$	− 6.00000i	− 0.336994i	0.985702	−	0.168497i	$-0.0538913\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−24.0000	−1.33955
$322$	0	0
$323$	12.0000i	0.667698i
$324$	0	0
$325$	20.0000i	1.10940i
$326$	0	0
$327$	−4.00000	−0.221201
$328$	0	0
$329$	12.0000	0.661581
$330$	0	0
$331$	− 8.00000i	− 0.439720i	−0.975531	−	0.219860i	$-0.929440\pi$
$331$	− 8.00000i	− 0.439720i	0.975531	−	0.219860i	$-0.0705600\pi$
$332$	0	0
$333$	− 2.00000i	− 0.109599i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	14.0000	0.762629	0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000	0.762629	0.381314	+	0.924445i	$0.375472\pi$
$338$	0	0
$339$	− 12.0000i	− 0.651751i
$340$	0	0
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	24.0000i	1.28839i	0.764862	+	0.644194i	$0.222807\pi$
$347$	24.0000i	1.28839i	−0.764862	+	0.644194i	$0.777193\pi$
$348$	0	0
$349$	28.0000i	1.49881i	0.662114	+	0.749403i	$0.269659\pi$
$349$	28.0000i	1.49881i	−0.662114	+	0.749403i	$0.730341\pi$
$350$	0	0
$351$	16.0000	0.854017
$352$	0	0
$353$	18.0000	0.958043	0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000	0.958043	0.479022	+	0.877803i	$0.340992\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 12.0000i	− 0.635107i
$358$	0	0
$359$	−24.0000	−1.26667	−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000	−1.26667	−0.633336	+	0.773877i	$0.718315\pi$
$360$	0	0
$361$	15.0000	0.789474
$362$	0	0
$363$	− 22.0000i	− 1.15470i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−8.00000	−0.417597	−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000	−0.417597	−0.208798	+	0.977959i	$0.566955\pi$
$368$	0	0
$369$	6.00000	0.312348
$370$	0	0
$371$	6.00000i	0.311504i
$372$	0	0
$373$	14.0000i	0.724893i	0.932005	+	0.362446i	$0.118058\pi$
$373$	14.0000i	0.724893i	−0.932005	+	0.362446i	$0.881942\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−24.0000	−1.23606
$378$	0	0
$379$	16.0000i	0.821865i	0.911666	+	0.410932i	$0.134797\pi$
$379$	16.0000i	0.821865i	−0.911666	+	0.410932i	$0.865203\pi$
$380$	0	0
$381$	− 32.0000i	− 1.63941i
$382$	0	0
$383$	−36.0000	−1.83951	−0.919757	−	0.392488i	$-0.871614\pi$
$383$	−36.0000	−1.83951	−0.919757	+	0.392488i	$0.871614\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	8.00000i	0.406663i
$388$	0	0
$389$	18.0000i	0.912636i	0.889817	+	0.456318i	$0.150832\pi$
$389$	18.0000i	0.912636i	−0.889817	+	0.456318i	$0.849168\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	36.0000	1.81596
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 20.0000i	− 1.00377i	−0.864934	−	0.501886i	$-0.832640\pi$
$397$	− 20.0000i	− 1.00377i	0.864934	−	0.501886i	$-0.167360\pi$
$398$	0	0
$399$	4.00000	0.200250
$400$	0	0
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	0	0
$403$	16.0000i	0.797017i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−14.0000	−0.692255	−0.346128	−	0.938187i	$-0.612504\pi$
$409$	−14.0000	−0.692255	−0.346128	+	0.938187i	$0.612504\pi$
$410$	0	0
$411$	36.0000i	1.77575i
$412$	0	0
$413$	6.00000i	0.295241i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−28.0000	−1.37117
$418$	0	0
$419$	6.00000i	0.293119i	0.989202	+	0.146560i	$0.0468200\pi$
$419$	6.00000i	0.293119i	−0.989202	+	0.146560i	$0.953180\pi$
$420$	0	0
$421$	− 10.0000i	− 0.487370i	−0.969854	−	0.243685i	$-0.921644\pi$
$421$	− 10.0000i	− 0.487370i	0.969854	−	0.243685i	$-0.0783563\pi$
$422$	0	0
$423$	−12.0000	−0.583460
$424$	0	0
$425$	30.0000	1.45521
$426$	0	0
$427$	− 8.00000i	− 0.387147i
$428$	0	0
$429$	0	0
$430$	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
$433$	−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
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	8.00000	0.381819	0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000	0.381819	0.190910	+	0.981608i	$0.438856\pi$
$440$	0	0
$441$	−1.00000	−0.0476190
$442$	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
$445$	0	0
$446$	0	0
$447$	−36.0000	−1.70274
$448$	0	0
$449$	18.0000	0.849473	0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000	0.849473	0.424736	+	0.905317i	$0.360367\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	− 16.0000i	− 0.751746i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	10.0000	0.467780	0.233890	−	0.972263i	$-0.424854\pi$
$457$	10.0000	0.467780	0.233890	+	0.972263i	$0.424854\pi$
$458$	0	0
$459$	− 24.0000i	− 1.12022i
$460$	0	0
$461$	− 12.0000i	− 0.558896i	−0.960161	−	0.279448i	$-0.909849\pi$
$461$	− 12.0000i	− 0.558896i	0.960161	−	0.279448i	$-0.0901514\pi$
$462$	0	0
$463$	−32.0000	−1.48717	−0.743583	−	0.668644i	$-0.766875\pi$
$463$	−32.0000	−1.48717	−0.743583	+	0.668644i	$0.766875\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 6.00000i	− 0.277647i	−0.990317	−	0.138823i	$-0.955668\pi$
$467$	− 6.00000i	− 0.277647i	0.990317	−	0.138823i	$-0.0443321\pi$
$468$	0	0
$469$	− 4.00000i	− 0.184703i
$470$	0	0
$471$	8.00000	0.368621
$472$	0	0
$473$	0	0
$474$	0	0
$475$	10.0000i	0.458831i
$476$	0	0
$477$	− 6.00000i	− 0.274721i
$478$	0	0
$479$	36.0000	1.64488	0.822441	−	0.568850i	$-0.192612\pi$
$479$	36.0000	1.64488	0.822441	+	0.568850i	$0.192612\pi$
$480$	0	0
$481$	−8.00000	−0.364769
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−16.0000	−0.725029	−0.362515	−	0.931978i	$-0.618082\pi$
$487$	−16.0000	−0.725029	−0.362515	+	0.931978i	$0.618082\pi$
$488$	0	0
$489$	−32.0000	−1.44709
$490$	0	0
$491$	12.0000i	0.541552i	0.962642	+	0.270776i	$0.0872803\pi$
$491$	12.0000i	0.541552i	−0.962642	+	0.270776i	$0.912720\pi$
$492$	0	0
$493$	36.0000i	1.62136i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	− 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
$501$	24.0000i	1.07224i
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	6.00000i	0.266469i
$508$	0	0
$509$	− 36.0000i	− 1.59567i	−0.602875	−	0.797836i	$-0.705978\pi$
$509$	− 36.0000i	− 1.59567i	0.602875	−	0.797836i	$-0.294022\pi$
$510$	0	0
$511$	−2.00000	−0.0884748
$512$	0	0
$513$	8.00000	0.353209
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	24.0000	1.05348
$520$	0	0
$521$	−6.00000	−0.262865	−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000	−0.262865	−0.131432	+	0.991325i	$0.541958\pi$
$522$	0	0
$523$	− 2.00000i	− 0.0874539i	−0.999044	−	0.0437269i	$-0.986077\pi$
$523$	− 2.00000i	− 0.0874539i	0.999044	−	0.0437269i	$-0.0139232\pi$
$524$	0	0
$525$	− 10.0000i	− 0.436436i
$526$	0	0
$527$	24.0000	1.04546
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	− 6.00000i	− 0.260378i
$532$	0	0
$533$	− 24.0000i	− 1.03956i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−24.0000	−1.03568
$538$	0	0
$539$	0	0
$540$	0	0
$541$	− 38.0000i	− 1.63375i	−0.576816	−	0.816874i	$-0.695705\pi$
$541$	− 38.0000i	− 1.63375i	0.576816	−	0.816874i	$-0.304295\pi$
$542$	0	0
$543$	40.0000	1.71656
$544$	0	0
$545$	0	0
$546$	0	0
$547$	8.00000i	0.342055i	0.985266	+	0.171028i	$0.0547087\pi$
$547$	8.00000i	0.342055i	−0.985266	+	0.171028i	$0.945291\pi$
$548$	0	0
$549$	8.00000i	0.341432i
$550$	0	0
$551$	−12.0000	−0.511217
$552$	0	0
$553$	−8.00000	−0.340195
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 6.00000i	− 0.254228i	−0.991888	−	0.127114i	$-0.959429\pi$
$557$	− 6.00000i	− 0.254228i	0.991888	−	0.127114i	$-0.0405714\pi$
$558$	0	0
$559$	32.0000	1.35346
$560$	0	0
$561$	0	0
$562$	0	0
$563$	30.0000i	1.26435i	0.774826	+	0.632175i	$0.217837\pi$
$563$	30.0000i	1.26435i	−0.774826	+	0.632175i	$0.782163\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−11.0000	−0.461957
$568$	0	0
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	0	0
$571$	− 32.0000i	− 1.33916i	−0.742741	−	0.669579i	$-0.766474\pi$
$571$	− 32.0000i	− 1.33916i	0.742741	−	0.669579i	$-0.233526\pi$
$572$	0	0
$573$	48.0000i	2.00523i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	2.00000	0.0832611	0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000	0.0832611	0.0416305	+	0.999133i	$0.486745\pi$
$578$	0	0
$579$	− 28.0000i	− 1.16364i
$580$	0	0
$581$	− 6.00000i	− 0.248922i
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	42.0000i	1.73353i	0.498721	+	0.866763i	$0.333803\pi$
$587$	42.0000i	1.73353i	−0.498721	+	0.866763i	$0.666197\pi$
$588$	0	0
$589$	8.00000i	0.329634i
$590$	0	0
$591$	−36.0000	−1.48084
$592$	0	0
$593$	−6.00000	−0.246390	−0.123195	−	0.992382i	$-0.539314\pi$
$593$	−6.00000	−0.246390	−0.123195	+	0.992382i	$0.539314\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 40.0000i	− 1.63709i
$598$	0	0
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	0	0
$601$	−26.0000	−1.06056	−0.530281	−	0.847822i	$-0.677914\pi$
$601$	−26.0000	−1.06056	−0.530281	+	0.847822i	$0.677914\pi$
$602$	0	0
$603$	4.00000i	0.162893i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−32.0000	−1.29884	−0.649420	−	0.760430i	$-0.724988\pi$
$607$	−32.0000	−1.29884	−0.649420	+	0.760430i	$0.724988\pi$
$608$	0	0
$609$	12.0000	0.486265
$610$	0	0
$611$	48.0000i	1.94187i
$612$	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
$617$	−6.00000	−0.241551	−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000	−0.241551	−0.120775	+	0.992680i	$0.538538\pi$
$618$	0	0
$619$	− 26.0000i	− 1.04503i	−0.852631	−	0.522514i	$-0.824994\pi$
$619$	− 26.0000i	− 1.04503i	0.852631	−	0.522514i	$-0.175006\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	6.00000	0.240385
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	12.0000i	0.478471i
$630$	0	0
$631$	−16.0000	−0.636950	−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000	−0.636950	−0.318475	+	0.947931i	$0.603171\pi$
$632$	0	0
$633$	−8.00000	−0.317971
$634$	0	0
$635$	0	0
$636$	0	0
$637$	4.00000i	0.158486i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−18.0000	−0.710957	−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000	−0.710957	−0.355479	+	0.934684i	$0.615682\pi$
$642$	0	0
$643$	14.0000i	0.552106i	0.961142	+	0.276053i	$0.0890266\pi$
$643$	14.0000i	0.552106i	−0.961142	+	0.276053i	$0.910973\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−12.0000	−0.471769	−0.235884	−	0.971781i	$-0.575799\pi$
$647$	−12.0000	−0.471769	−0.235884	+	0.971781i	$0.575799\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	− 8.00000i	− 0.313545i
$652$	0	0
$653$	− 18.0000i	− 0.704394i	−0.935926	−	0.352197i	$-0.885435\pi$
$653$	− 18.0000i	− 0.704394i	0.935926	−	0.352197i	$-0.114565\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	2.00000	0.0780274
$658$	0	0
$659$	− 24.0000i	− 0.934907i	−0.884018	−	0.467454i	$-0.845171\pi$
$659$	− 24.0000i	− 0.934907i	0.884018	−	0.467454i	$-0.154829\pi$
$660$	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
$663$	48.0000	1.86417
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	16.0000i	0.618596i
$670$	0	0
$671$	0	0
$672$	0	0
$673$	26.0000	1.00223	0.501113	−	0.865382i	$-0.332924\pi$
$673$	26.0000	1.00223	0.501113	+	0.865382i	$0.332924\pi$
$674$	0	0
$675$	− 20.0000i	− 0.769800i
$676$	0	0
$677$	− 12.0000i	− 0.461197i	−0.973049	−	0.230599i	$-0.925932\pi$
$677$	− 12.0000i	− 0.461197i	0.973049	−	0.230599i	$-0.0740685\pi$
$678$	0	0
$679$	−10.0000	−0.383765
$680$	0	0
$681$	36.0000	1.37952
$682$	0	0
$683$	12.0000i	0.459167i	0.973289	+	0.229584i	$0.0737364\pi$
$683$	12.0000i	0.459167i	−0.973289	+	0.229584i	$0.926264\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−8.00000	−0.305219
$688$	0	0
$689$	−24.0000	−0.914327
$690$	0	0
$691$	− 46.0000i	− 1.74992i	−0.484193	−	0.874961i	$-0.660887\pi$
$691$	− 46.0000i	− 1.74992i	0.484193	−	0.874961i	$-0.339113\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−36.0000	−1.36360
$698$	0	0
$699$	− 12.0000i	− 0.453882i
$700$	0	0
$701$	− 18.0000i	− 0.679851i	−0.940452	−	0.339925i	$-0.889598\pi$
$701$	− 18.0000i	− 0.679851i	0.940452	−	0.339925i	$-0.110402\pi$
$702$	0	0
$703$	−4.00000	−0.150863
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	− 46.0000i	− 1.72757i	−0.503864	−	0.863783i	$-0.668089\pi$
$709$	− 46.0000i	− 1.72757i	0.503864	−	0.863783i	$-0.331911\pi$
$710$	0	0
$711$	8.00000	0.300023
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	48.0000i	1.79259i
$718$	0	0
$719$	−12.0000	−0.447524	−0.223762	−	0.974644i	$-0.571834\pi$
$719$	−12.0000	−0.447524	−0.223762	+	0.974644i	$0.571834\pi$
$720$	0	0
$721$	−4.00000	−0.148968
$722$	0	0
$723$	20.0000i	0.743808i
$724$	0	0
$725$	30.0000i	1.11417i
$726$	0	0
$727$	44.0000	1.63187	0.815935	−	0.578144i	$-0.196223\pi$
$727$	44.0000	1.63187	0.815935	+	0.578144i	$0.196223\pi$
$728$	0	0
$729$	−13.0000	−0.481481
$730$	0	0
$731$	− 48.0000i	− 1.77534i
$732$	0	0
$733$	40.0000i	1.47743i	0.674016	+	0.738717i	$0.264568\pi$
$733$	40.0000i	1.47743i	−0.674016	+	0.738717i	$0.735432\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	− 16.0000i	− 0.588570i	−0.955718	−	0.294285i	$-0.904919\pi$
$739$	− 16.0000i	− 0.588570i	0.955718	−	0.294285i	$-0.0950814\pi$
$740$	0	0
$741$	16.0000i	0.587775i
$742$	0	0
$743$	24.0000	0.880475	0.440237	−	0.897881i	$-0.354894\pi$
$743$	24.0000	0.880475	0.440237	+	0.897881i	$0.354894\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	6.00000i	0.219529i
$748$	0	0
$749$	− 12.0000i	− 0.438470i
$750$	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
$753$	36.0000	1.31191
$754$	0	0
$755$	0	0
$756$	0	0
$757$	2.00000i	0.0726912i	0.999339	+	0.0363456i	$0.0115717\pi$
$757$	2.00000i	0.0726912i	−0.999339	+	0.0363456i	$0.988428\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	18.0000	0.652499	0.326250	−	0.945284i	$-0.394215\pi$
$761$	18.0000	0.652499	0.326250	+	0.945284i	$0.394215\pi$
$762$	0	0
$763$	− 2.00000i	− 0.0724049i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−24.0000	−0.866590
$768$	0	0
$769$	14.0000	0.504853	0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000	0.504853	0.252426	+	0.967616i	$0.418771\pi$
$770$	0	0
$771$	− 36.0000i	− 1.29651i
$772$	0	0
$773$	24.0000i	0.863220i	0.902060	+	0.431610i	$0.142054\pi$
$773$	24.0000i	0.863220i	−0.902060	+	0.431610i	$0.857946\pi$
$774$	0	0
$775$	20.0000	0.718421
$776$	0	0
$777$	4.00000	0.143499
$778$	0	0
$779$	− 12.0000i	− 0.429945i
$780$	0	0
$781$	0	0
$782$	0	0
$783$	24.0000	0.857690
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 22.0000i	− 0.784215i	−0.919919	−	0.392108i	$-0.871746\pi$
$787$	− 22.0000i	− 0.784215i	0.919919	−	0.392108i	$-0.128254\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	6.00000	0.213335
$792$	0	0
$793$	32.0000	1.13635
$794$	0	0
$795$	0	0
$796$	0	0
$797$	12.0000i	0.425062i	0.977154	+	0.212531i	$0.0681706\pi$
$797$	12.0000i	0.425062i	−0.977154	+	0.212531i	$0.931829\pi$
$798$	0	0
$799$	72.0000	2.54718
$800$	0	0
$801$	−6.00000	−0.212000
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	24.0000	0.844840
$808$	0	0
$809$	−6.00000	−0.210949	−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000	−0.210949	−0.105474	+	0.994422i	$0.533636\pi$
$810$	0	0
$811$	− 2.00000i	− 0.0702295i	−0.999383	−	0.0351147i	$-0.988820\pi$
$811$	− 2.00000i	− 0.0702295i	0.999383	−	0.0351147i	$-0.0111797\pi$
$812$	0	0
$813$	− 32.0000i	− 1.12229i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	16.0000	0.559769
$818$	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
$823$	−40.0000	−1.39431	−0.697156	−	0.716919i	$-0.745552\pi$
$823$	−40.0000	−1.39431	−0.697156	+	0.716919i	$0.745552\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	36.0000i	1.25184i	0.779886	+	0.625921i	$0.215277\pi$
$827$	36.0000i	1.25184i	−0.779886	+	0.625921i	$0.784723\pi$
$828$	0	0
$829$	− 56.0000i	− 1.94496i	−0.232986	−	0.972480i	$-0.574849\pi$
$829$	− 56.0000i	− 1.94496i	0.232986	−	0.972480i	$-0.425151\pi$
$830$	0	0
$831$	−20.0000	−0.693792
$832$	0	0
$833$	6.00000	0.207888
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 16.0000i	− 0.553041i
$838$	0	0
$839$	12.0000	0.414286	0.207143	−	0.978311i	$-0.433583\pi$
$839$	12.0000	0.414286	0.207143	+	0.978311i	$0.433583\pi$
$840$	0	0
$841$	−7.00000	−0.241379
$842$	0	0
$843$	− 12.0000i	− 0.413302i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	11.0000	0.377964
$848$	0	0
$849$	44.0000	1.51008
$850$	0	0
$851$	0	0
$852$	0	0
$853$	44.0000i	1.50653i	0.657716	+	0.753266i	$0.271523\pi$
$853$	44.0000i	1.50653i	−0.657716	+	0.753266i	$0.728477\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	18.0000	0.614868	0.307434	−	0.951569i	$-0.400530\pi$
$857$	18.0000	0.614868	0.307434	+	0.951569i	$0.400530\pi$
$858$	0	0
$859$	− 14.0000i	− 0.477674i	−0.971060	−	0.238837i	$-0.923234\pi$
$859$	− 14.0000i	− 0.477674i	0.971060	−	0.238837i	$-0.0767661\pi$
$860$	0	0
$861$	12.0000i	0.408959i
$862$	0	0
$863$	24.0000	0.816970	0.408485	−	0.912765i	$-0.366057\pi$
$863$	24.0000	0.816970	0.408485	+	0.912765i	$0.366057\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 38.0000i	− 1.29055i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	16.0000	0.542139
$872$	0	0
$873$	10.0000	0.338449
$874$	0	0
$875$	0	0
$876$	0	0
$877$	22.0000i	0.742887i	0.928456	+	0.371444i	$0.121137\pi$
$877$	22.0000i	0.742887i	−0.928456	+	0.371444i	$0.878863\pi$
$878$	0	0
$879$	48.0000	1.61900
$880$	0	0
$881$	−54.0000	−1.81931	−0.909653	−	0.415369i	$-0.863653\pi$
$881$	−54.0000	−1.81931	−0.909653	+	0.415369i	$0.863653\pi$
$882$	0	0
$883$	20.0000i	0.673054i	0.941674	+	0.336527i	$0.109252\pi$
$883$	20.0000i	0.673054i	−0.941674	+	0.336527i	$0.890748\pi$
$884$	0	0
$885$	0	0
$886$	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
$889$	16.0000	0.536623
$890$	0	0
$891$	0	0
$892$	0	0
$893$	24.0000i	0.803129i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	24.0000i	0.800445i
$900$	0	0
$901$	36.0000i	1.19933i
$902$	0	0
$903$	−16.0000	−0.532447
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 44.0000i	− 1.46100i	−0.682915	−	0.730498i	$-0.739288\pi$
$907$	− 44.0000i	− 1.46100i	0.682915	−	0.730498i	$-0.260712\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−48.0000	−1.59031	−0.795155	−	0.606406i	$-0.792611\pi$
$911$	−48.0000	−1.59031	−0.795155	+	0.606406i	$0.792611\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	18.0000i	0.594412i
$918$	0	0
$919$	56.0000	1.84727	0.923635	−	0.383274i	$-0.125203\pi$
$919$	56.0000	1.84727	0.923635	+	0.383274i	$0.125203\pi$
$920$	0	0
$921$	4.00000	0.131804
$922$	0	0
$923$	0	0
$924$	0	0
$925$	10.0000i	0.328798i
$926$	0	0
$927$	4.00000	0.131377
$928$	0	0
$929$	6.00000	0.196854	0.0984268	−	0.995144i	$-0.468619\pi$
$929$	6.00000	0.196854	0.0984268	+	0.995144i	$0.468619\pi$
$930$	0	0
$931$	2.00000i	0.0655474i
$932$	0	0
$933$	48.0000i	1.57145i
$934$	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
$939$	− 20.0000i	− 0.652675i
$940$	0	0
$941$	24.0000i	0.782378i	0.920310	+	0.391189i	$0.127936\pi$
$941$	24.0000i	0.782378i	−0.920310	+	0.391189i	$0.872064\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	24.0000i	0.779895i	0.920837	+	0.389948i	$0.127507\pi$
$947$	24.0000i	0.779895i	−0.920837	+	0.389948i	$0.872493\pi$
$948$	0	0
$949$	− 8.00000i	− 0.259691i
$950$	0	0
$951$	−12.0000	−0.389127
$952$	0	0
$953$	54.0000	1.74923	0.874616	−	0.484817i	$-0.161114\pi$
$953$	54.0000	1.74923	0.874616	+	0.484817i	$0.161114\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−18.0000	−0.581250
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	12.0000i	0.386695i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	32.0000	1.02905	0.514525	−	0.857475i	$-0.327968\pi$
$967$	32.0000	1.02905	0.514525	+	0.857475i	$0.327968\pi$
$968$	0	0
$969$	24.0000	0.770991
$970$	0	0
$971$	6.00000i	0.192549i	0.995355	+	0.0962746i	$0.0306927\pi$
$971$	6.00000i	0.192549i	−0.995355	+	0.0962746i	$0.969307\pi$
$972$	0	0
$973$	− 14.0000i	− 0.448819i
$974$	0	0
$975$	40.0000	1.28103
$976$	0	0
$977$	−6.00000	−0.191957	−0.0959785	−	0.995383i	$-0.530598\pi$
$977$	−6.00000	−0.191957	−0.0959785	+	0.995383i	$0.530598\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	2.00000i	0.0638551i
$982$	0	0
$983$	−36.0000	−1.14822	−0.574111	−	0.818778i	$-0.694652\pi$
$983$	−36.0000	−1.14822	−0.574111	+	0.818778i	$0.694652\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 24.0000i	− 0.763928i
$988$	0	0
$989$	0	0
$990$	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
$993$	−16.0000	−0.507745
$994$	0	0
$995$	0	0
$996$	0	0
$997$	8.00000i	0.253363i	0.991943	+	0.126681i	$0.0404325\pi$
$997$	8.00000i	0.253363i	−0.991943	+	0.126681i	$0.959567\pi$
$998$	0	0
$999$	8.00000	0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1792.2.b.g.897.1		2
4.3	odd	2		1792.2.b.c.897.2		2
8.3	odd	2		1792.2.b.c.897.1		2
8.5	even	2	inner	1792.2.b.g.897.2		2
16.3	odd	4		14.2.a.a.1.1	✓	1
16.5	even	4		448.2.a.a.1.1		1
16.11	odd	4		448.2.a.g.1.1		1
16.13	even	4		112.2.a.c.1.1		1
48.5	odd	4		4032.2.a.r.1.1		1
48.11	even	4		4032.2.a.w.1.1		1
48.29	odd	4		1008.2.a.h.1.1		1
48.35	even	4		126.2.a.b.1.1		1
80.3	even	4		350.2.c.d.99.2		2
80.13	odd	4		2800.2.g.h.449.2		2
80.19	odd	4		350.2.a.f.1.1		1
80.29	even	4		2800.2.a.g.1.1		1
80.67	even	4		350.2.c.d.99.1		2
80.77	odd	4		2800.2.g.h.449.1		2
112.3	even	12		98.2.c.a.79.1		2
112.13	odd	4		784.2.a.b.1.1		1
112.19	even	12		98.2.c.a.67.1		2
112.27	even	4		3136.2.a.e.1.1		1
112.45	odd	12		784.2.i.i.177.1		2
112.51	odd	12		98.2.c.b.67.1		2
112.61	odd	12		784.2.i.i.753.1		2
112.67	odd	12		98.2.c.b.79.1		2
112.69	odd	4		3136.2.a.z.1.1		1
112.83	even	4		98.2.a.a.1.1		1
112.93	even	12		784.2.i.c.753.1		2
112.109	even	12		784.2.i.c.177.1		2
144.67	odd	12		1134.2.f.l.379.1		2
144.83	even	12		1134.2.f.f.757.1		2
144.115	odd	12		1134.2.f.l.757.1		2
144.131	even	12		1134.2.f.f.379.1		2
176.131	even	4		1694.2.a.e.1.1		1
208.51	odd	4		2366.2.a.j.1.1		1
208.83	even	4		2366.2.d.b.337.2		2
208.99	even	4		2366.2.d.b.337.1		2
240.83	odd	4		3150.2.g.j.2899.1		2
240.179	even	4		3150.2.a.i.1.1		1
240.227	odd	4		3150.2.g.j.2899.2		2
272.67	odd	4		4046.2.a.f.1.1		1
304.227	even	4		5054.2.a.c.1.1		1
336.83	odd	4		882.2.a.i.1.1		1
336.125	even	4		7056.2.a.bd.1.1		1
336.131	odd	12		882.2.g.d.361.1		2
336.179	even	12		882.2.g.c.667.1		2
336.227	odd	12		882.2.g.d.667.1		2
336.275	even	12		882.2.g.c.361.1		2
368.275	even	4		7406.2.a.a.1.1		1
560.83	odd	4		2450.2.c.c.99.2		2
560.307	odd	4		2450.2.c.c.99.1		2
560.419	even	4		2450.2.a.t.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.2.a.a.1.1	✓	1	16.3	odd	4
98.2.a.a.1.1		1	112.83	even	4
98.2.c.a.67.1		2	112.19	even	12
98.2.c.a.79.1		2	112.3	even	12
98.2.c.b.67.1		2	112.51	odd	12
98.2.c.b.79.1		2	112.67	odd	12
112.2.a.c.1.1		1	16.13	even	4
126.2.a.b.1.1		1	48.35	even	4
350.2.a.f.1.1		1	80.19	odd	4
350.2.c.d.99.1		2	80.67	even	4
350.2.c.d.99.2		2	80.3	even	4
448.2.a.a.1.1		1	16.5	even	4
448.2.a.g.1.1		1	16.11	odd	4
784.2.a.b.1.1		1	112.13	odd	4
784.2.i.c.177.1		2	112.109	even	12
784.2.i.c.753.1		2	112.93	even	12
784.2.i.i.177.1		2	112.45	odd	12
784.2.i.i.753.1		2	112.61	odd	12
882.2.a.i.1.1		1	336.83	odd	4
882.2.g.c.361.1		2	336.275	even	12
882.2.g.c.667.1		2	336.179	even	12
882.2.g.d.361.1		2	336.131	odd	12
882.2.g.d.667.1		2	336.227	odd	12
1008.2.a.h.1.1		1	48.29	odd	4
1134.2.f.f.379.1		2	144.131	even	12
1134.2.f.f.757.1		2	144.83	even	12
1134.2.f.l.379.1		2	144.67	odd	12
1134.2.f.l.757.1		2	144.115	odd	12
1694.2.a.e.1.1		1	176.131	even	4
1792.2.b.c.897.1		2	8.3	odd	2
1792.2.b.c.897.2		2	4.3	odd	2
1792.2.b.g.897.1		2	1.1	even	1	trivial
1792.2.b.g.897.2		2	8.5	even	2	inner
2366.2.a.j.1.1		1	208.51	odd	4
2366.2.d.b.337.1		2	208.99	even	4
2366.2.d.b.337.2		2	208.83	even	4
2450.2.a.t.1.1		1	560.419	even	4
2450.2.c.c.99.1		2	560.307	odd	4
2450.2.c.c.99.2		2	560.83	odd	4
2800.2.a.g.1.1		1	80.29	even	4
2800.2.g.h.449.1		2	80.77	odd	4
2800.2.g.h.449.2		2	80.13	odd	4
3136.2.a.e.1.1		1	112.27	even	4
3136.2.a.z.1.1		1	112.69	odd	4
3150.2.a.i.1.1		1	240.179	even	4
3150.2.g.j.2899.1		2	240.83	odd	4
3150.2.g.j.2899.2		2	240.227	odd	4
4032.2.a.r.1.1		1	48.5	odd	4
4032.2.a.w.1.1		1	48.11	even	4
4046.2.a.f.1.1		1	272.67	odd	4
5054.2.a.c.1.1		1	304.227	even	4
7056.2.a.bd.1.1		1	336.125	even	4
7406.2.a.a.1.1		1	368.275	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:	\( N \)	\(=\)	\( 1792 = 2^{8} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1792.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.00000i q^{3} +1.00000 q^{7} -1.00000 q^{9} +4.00000i q^{13} +6.00000 q^{17} +2.00000i q^{19} -2.00000i q^{21} +5.00000 q^{25} -4.00000i q^{27} +6.00000i q^{29} +4.00000 q^{31} +2.00000i q^{37} +8.00000 q^{39} -6.00000 q^{41} -8.00000i q^{43} +12.0000 q^{47} +1.00000 q^{49} -12.0000i q^{51} +6.00000i q^{53} +4.00000 q^{57} +6.00000i q^{59} -8.00000i q^{61} -1.00000 q^{63} -4.00000i q^{67} -2.00000 q^{73} -10.0000i q^{75} -8.00000 q^{79} -11.0000 q^{81} -6.00000i q^{83} +12.0000 q^{87} +6.00000 q^{89} +4.00000i q^{91} -8.00000i q^{93} -10.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{7} - 2 q^{9} + 12 q^{17} + 10 q^{25} + 8 q^{31} + 16 q^{39} - 12 q^{41} + 24 q^{47} + 2 q^{49} + 8 q^{57} - 2 q^{63} - 4 q^{73} - 16 q^{79} - 22 q^{81} + 24 q^{87} + 12 q^{89} - 20 q^{97}+O(q^{100}) \) 2 * q + 2 * q^7 - 2 * q^9 + 12 * q^17 + 10 * q^25 + 8 * q^31 + 16 * q^39 - 12 * q^41 + 24 * q^47 + 2 * q^49 + 8 * q^57 - 2 * q^63 - 4 * q^73 - 16 * q^79 - 22 * q^81 + 24 * q^87 + 12 * q^89 - 20 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more