LMFDB - Embedded newform 4032.2.c.j.2017.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4032,2,Mod(2017,4032)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4032, 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("4032.2017");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4032 = 2^{6} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4032.c (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:	$32.1956820950$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1344)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2017.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	4032.2017
Dual form			4032.2.c.j.2017.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000i q^{5} +1.00000 q^{7} +O(q^{10})$	$q+2.00000i q^{5} +1.00000 q^{7} +2.00000i q^{11} +6.00000i q^{13} +6.00000 q^{17} +8.00000i q^{19} -8.00000 q^{23} +1.00000 q^{25} +8.00000 q^{31} +2.00000i q^{35} -4.00000i q^{37} -6.00000 q^{41} -6.00000i q^{43} -12.0000 q^{47} +1.00000 q^{49} -12.0000i q^{53} -4.00000 q^{55} +4.00000i q^{59} +10.0000i q^{61} -12.0000 q^{65} -2.00000i q^{67} +6.00000 q^{73} +2.00000i q^{77} -8.00000i q^{83} +12.0000i q^{85} +6.00000 q^{89} +6.00000i q^{91} -16.0000 q^{95} -10.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{7}+O(q^{10}) $ 2 * q + 2 * q^7	$ 2 q + 2 q^{7} + 12 q^{17} - 16 q^{23} + 2 q^{25} + 16 q^{31} - 12 q^{41} - 24 q^{47} + 2 q^{49} - 8 q^{55} - 24 q^{65} + 12 q^{73} + 12 q^{89} - 32 q^{95} - 20 q^{97}+O(q^{100}) $ 2 * q + 2 * q^7 + 12 * q^17 - 16 * q^23 + 2 * q^25 + 16 * q^31 - 12 * q^41 - 24 * q^47 + 2 * q^49 - 8 * q^55 - 24 * q^65 + 12 * q^73 + 12 * q^89 - 32 * q^95 - 20 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4032\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$1793$	$3781$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	2.00000i	0.894427i	0.894427	+	0.447214i	$0.147584\pi$
$5$	2.00000i	0.894427i	−0.894427	+	0.447214i	$0.852416\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0	0
$10$	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
$13$	6.00000i	1.66410i	0.554700	+	0.832050i	$0.312833\pi$
$13$	6.00000i	1.66410i	−0.554700	+	0.832050i	$0.687167\pi$
$14$	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
$19$	8.00000i	1.83533i	0.397360	+	0.917663i	$0.369927\pi$
$19$	8.00000i	1.83533i	−0.397360	+	0.917663i	$0.630073\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−8.00000	−1.66812	−0.834058	−	0.551677i	$-0.813988\pi$
$23$	−8.00000	−1.66812	−0.834058	+	0.551677i	$0.813988\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.00000i	0.338062i
$36$	0	0
$37$	− 4.00000i	− 0.657596i	−0.944400	−	0.328798i	$-0.893356\pi$
$37$	− 4.00000i	− 0.657596i	0.944400	−	0.328798i	$-0.106644\pi$
$38$	0	0
$39$	0	0
$40$	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
$43$	− 6.00000i	− 0.914991i	−0.889212	−	0.457496i	$-0.848747\pi$
$43$	− 6.00000i	− 0.914991i	0.889212	−	0.457496i	$-0.151253\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−12.0000	−1.75038	−0.875190	−	0.483779i	$-0.839264\pi$
$47$	−12.0000	−1.75038	−0.875190	+	0.483779i	$0.839264\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 12.0000i	− 1.64833i	−0.566352	−	0.824163i	$-0.691646\pi$
$53$	− 12.0000i	− 1.64833i	0.566352	−	0.824163i	$-0.308354\pi$
$54$	0	0
$55$	−4.00000	−0.539360
$56$	0	0
$57$	0	0
$58$	0	0
$59$	4.00000i	0.520756i	0.965507	+	0.260378i	$0.0838471\pi$
$59$	4.00000i	0.520756i	−0.965507	+	0.260378i	$0.916153\pi$
$60$	0	0
$61$	10.0000i	1.28037i	0.768221	+	0.640184i	$0.221142\pi$
$61$	10.0000i	1.28037i	−0.768221	+	0.640184i	$0.778858\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−12.0000	−1.48842
$66$	0	0
$67$	− 2.00000i	− 0.244339i	−0.992509	−	0.122169i	$-0.961015\pi$
$67$	− 2.00000i	− 0.244339i	0.992509	−	0.122169i	$-0.0389851\pi$
$68$	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
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.00000i	0.227921i
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 8.00000i	− 0.878114i	−0.898459	−	0.439057i	$-0.855313\pi$
$83$	− 8.00000i	− 0.878114i	0.898459	−	0.439057i	$-0.144687\pi$
$84$	0	0
$85$	12.0000i	1.30158i
$86$	0	0
$87$	0	0
$88$	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
$91$	6.00000i	0.628971i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−16.0000	−1.64157
$96$	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
$101$	18.0000i	1.79107i	0.444994	+	0.895533i	$0.353206\pi$
$101$	18.0000i	1.79107i	−0.444994	+	0.895533i	$0.646794\pi$
$102$	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
$107$	− 10.0000i	− 0.966736i	−0.875417	−	0.483368i	$-0.839413\pi$
$107$	− 10.0000i	− 0.966736i	0.875417	−	0.483368i	$-0.160587\pi$
$108$	0	0
$109$	4.00000i	0.383131i	0.981480	+	0.191565i	$0.0613564\pi$
$109$	4.00000i	0.383131i	−0.981480	+	0.191565i	$0.938644\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	18.0000	1.69330	0.846649	−	0.532152i	$-0.178617\pi$
$113$	18.0000	1.69330	0.846649	+	0.532152i	$0.178617\pi$
$114$	0	0
$115$	− 16.0000i	− 1.49201i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	6.00000	0.550019
$120$	0	0
$121$	7.00000	0.636364
$122$	0	0
$123$	0	0
$124$	0	0
$125$	12.0000i	1.07331i
$126$	0	0
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 8.00000i	− 0.698963i	−0.936943	−	0.349482i	$-0.886358\pi$
$131$	− 8.00000i	− 0.698963i	0.936943	−	0.349482i	$-0.113642\pi$
$132$	0	0
$133$	8.00000i	0.693688i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2.00000	−0.170872	−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000	−0.170872	−0.0854358	+	0.996344i	$0.527228\pi$
$138$	0	0
$139$	8.00000i	0.678551i	0.940687	+	0.339276i	$0.110182\pi$
$139$	8.00000i	0.678551i	−0.940687	+	0.339276i	$0.889818\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−12.0000	−1.00349
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 4.00000i	− 0.327693i	−0.986486	−	0.163846i	$-0.947610\pi$
$149$	− 4.00000i	− 0.327693i	0.986486	−	0.163846i	$-0.0523901\pi$
$150$	0	0
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	16.0000i	1.28515i
$156$	0	0
$157$	14.0000i	1.11732i	0.829396	+	0.558661i	$0.188685\pi$
$157$	14.0000i	1.11732i	−0.829396	+	0.558661i	$0.811315\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−8.00000	−0.630488
$162$	0	0
$163$	2.00000i	0.156652i	0.996928	+	0.0783260i	$0.0249575\pi$
$163$	2.00000i	0.156652i	−0.996928	+	0.0783260i	$0.975042\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−8.00000	−0.619059	−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000	−0.619059	−0.309529	+	0.950890i	$0.600171\pi$
$168$	0	0
$169$	−23.0000	−1.76923
$170$	0	0
$171$	0	0
$172$	0	0
$173$	2.00000i	0.152057i	0.997106	+	0.0760286i	$0.0242240\pi$
$173$	2.00000i	0.152057i	−0.997106	+	0.0760286i	$0.975776\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	0	0
$178$	0	0
$179$	14.0000i	1.04641i	0.852207	+	0.523205i	$0.175264\pi$
$179$	14.0000i	1.04641i	−0.852207	+	0.523205i	$0.824736\pi$
$180$	0	0
$181$	− 14.0000i	− 1.04061i	−0.853980	−	0.520306i	$-0.825818\pi$
$181$	− 14.0000i	− 1.04061i	0.853980	−	0.520306i	$-0.174182\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	8.00000	0.588172
$186$	0	0
$187$	12.0000i	0.877527i
$188$	0	0
$189$	0	0
$190$	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
$193$	−2.00000	−0.143963	−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000	−0.143963	−0.0719816	+	0.997406i	$0.522932\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	12.0000i	0.854965i	0.904024	+	0.427482i	$0.140599\pi$
$197$	12.0000i	0.854965i	−0.904024	+	0.427482i	$0.859401\pi$
$198$	0	0
$199$	4.00000	0.283552	0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000	0.283552	0.141776	+	0.989899i	$0.454719\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	− 12.0000i	− 0.838116i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−16.0000	−1.10674
$210$	0	0
$211$	− 10.0000i	− 0.688428i	−0.938891	−	0.344214i	$-0.888145\pi$
$211$	− 10.0000i	− 0.688428i	0.938891	−	0.344214i	$-0.111855\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	12.0000	0.818393
$216$	0	0
$217$	8.00000	0.543075
$218$	0	0
$219$	0	0
$220$	0	0
$221$	36.0000i	2.42162i
$222$	0	0
$223$	4.00000	0.267860	0.133930	−	0.990991i	$-0.457240\pi$
$223$	4.00000	0.267860	0.133930	+	0.990991i	$0.457240\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 16.0000i	− 1.06196i	−0.847385	−	0.530979i	$-0.821824\pi$
$227$	− 16.0000i	− 1.06196i	0.847385	−	0.530979i	$-0.178176\pi$
$228$	0	0
$229$	− 14.0000i	− 0.925146i	−0.886581	−	0.462573i	$-0.846926\pi$
$229$	− 14.0000i	− 0.925146i	0.886581	−	0.462573i	$-0.153074\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	22.0000	1.44127	0.720634	−	0.693316i	$-0.243851\pi$
$233$	22.0000	1.44127	0.720634	+	0.693316i	$0.243851\pi$
$234$	0	0
$235$	− 24.0000i	− 1.56559i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−16.0000	−1.03495	−0.517477	−	0.855697i	$-0.673129\pi$
$239$	−16.0000	−1.03495	−0.517477	+	0.855697i	$0.673129\pi$
$240$	0	0
$241$	−18.0000	−1.15948	−0.579741	−	0.814801i	$-0.696846\pi$
$241$	−18.0000	−1.15948	−0.579741	+	0.814801i	$0.696846\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	2.00000i	0.127775i
$246$	0	0
$247$	−48.0000	−3.05417
$248$	0	0
$249$	0	0
$250$	0	0
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	0	0
$253$	− 16.0000i	− 1.00591i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−22.0000	−1.37232	−0.686161	−	0.727450i	$-0.740706\pi$
$257$	−22.0000	−1.37232	−0.686161	+	0.727450i	$0.740706\pi$
$258$	0	0
$259$	− 4.00000i	− 0.248548i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	8.00000	0.493301	0.246651	−	0.969104i	$-0.420670\pi$
$263$	8.00000	0.493301	0.246651	+	0.969104i	$0.420670\pi$
$264$	0	0
$265$	24.0000	1.47431
$266$	0	0
$267$	0	0
$268$	0	0
$269$	18.0000i	1.09748i	0.835993	+	0.548740i	$0.184892\pi$
$269$	18.0000i	1.09748i	−0.835993	+	0.548740i	$0.815108\pi$
$270$	0	0
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	2.00000i	0.120605i
$276$	0	0
$277$	8.00000i	0.480673i	0.970690	+	0.240337i	$0.0772579\pi$
$277$	8.00000i	0.480673i	−0.970690	+	0.240337i	$0.922742\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	0	0
$283$	− 12.0000i	− 0.713326i	−0.934233	−	0.356663i	$-0.883914\pi$
$283$	− 12.0000i	− 0.713326i	0.934233	−	0.356663i	$-0.116086\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−6.00000	−0.354169
$288$	0	0
$289$	19.0000	1.11765
$290$	0	0
$291$	0	0
$292$	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
$295$	−8.00000	−0.465778
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 48.0000i	− 2.77591i
$300$	0	0
$301$	− 6.00000i	− 0.345834i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−20.0000	−1.14520
$306$	0	0
$307$	16.0000i	0.913168i	0.889680	+	0.456584i	$0.150927\pi$
$307$	16.0000i	0.913168i	−0.889680	+	0.456584i	$0.849073\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	28.0000	1.58773	0.793867	−	0.608091i	$-0.208065\pi$
$311$	28.0000	1.58773	0.793867	+	0.608091i	$0.208065\pi$
$312$	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
$315$	0	0
$316$	0	0
$317$	20.0000i	1.12331i	0.827371	+	0.561656i	$0.189836\pi$
$317$	20.0000i	1.12331i	−0.827371	+	0.561656i	$0.810164\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	48.0000i	2.67079i
$324$	0	0
$325$	6.00000i	0.332820i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−12.0000	−0.661581
$330$	0	0
$331$	10.0000i	0.549650i	0.961494	+	0.274825i	$0.0886199\pi$
$331$	10.0000i	0.549650i	−0.961494	+	0.274825i	$0.911380\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	4.00000	0.218543
$336$	0	0
$337$	−34.0000	−1.85210	−0.926049	−	0.377403i	$-0.876817\pi$
$337$	−34.0000	−1.85210	−0.926049	+	0.377403i	$0.876817\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	16.0000i	0.866449i
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	18.0000i	0.966291i	0.875540	+	0.483145i	$0.160506\pi$
$347$	18.0000i	0.966291i	−0.875540	+	0.483145i	$0.839494\pi$
$348$	0	0
$349$	6.00000i	0.321173i	0.987022	+	0.160586i	$0.0513385\pi$
$349$	6.00000i	0.321173i	−0.987022	+	0.160586i	$0.948662\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	34.0000	1.80964	0.904819	−	0.425797i	$-0.140006\pi$
$353$	34.0000	1.80964	0.904819	+	0.425797i	$0.140006\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−16.0000	−0.844448	−0.422224	−	0.906492i	$-0.638750\pi$
$359$	−16.0000	−0.844448	−0.422224	+	0.906492i	$0.638750\pi$
$360$	0	0
$361$	−45.0000	−2.36842
$362$	0	0
$363$	0	0
$364$	0	0
$365$	12.0000i	0.628109i
$366$	0	0
$367$	−36.0000	−1.87918	−0.939592	−	0.342296i	$-0.888796\pi$
$367$	−36.0000	−1.87918	−0.939592	+	0.342296i	$0.888796\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 12.0000i	− 0.623009i
$372$	0	0
$373$	8.00000i	0.414224i	0.978317	+	0.207112i	$0.0664065\pi$
$373$	8.00000i	0.414224i	−0.978317	+	0.207112i	$0.933593\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	10.0000i	0.513665i	0.966456	+	0.256833i	$0.0826790\pi$
$379$	10.0000i	0.513665i	−0.966456	+	0.256833i	$0.917321\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	12.0000	0.613171	0.306586	−	0.951843i	$-0.400813\pi$
$383$	12.0000	0.613171	0.306586	+	0.951843i	$0.400813\pi$
$384$	0	0
$385$	−4.00000	−0.203859
$386$	0	0
$387$	0	0
$388$	0	0
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	−48.0000	−2.42746
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	14.0000i	0.702640i	0.936255	+	0.351320i	$0.114267\pi$
$397$	14.0000i	0.702640i	−0.936255	+	0.351320i	$0.885733\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	10.0000	0.499376	0.249688	−	0.968326i	$-0.419672\pi$
$401$	10.0000	0.499376	0.249688	+	0.968326i	$0.419672\pi$
$402$	0	0
$403$	48.0000i	2.39105i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	8.00000	0.396545
$408$	0	0
$409$	10.0000	0.494468	0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000	0.494468	0.247234	+	0.968956i	$0.420478\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	4.00000i	0.196827i
$414$	0	0
$415$	16.0000	0.785409
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 4.00000i	− 0.195413i	−0.995215	−	0.0977064i	$-0.968849\pi$
$419$	− 4.00000i	− 0.195413i	0.995215	−	0.0977064i	$-0.0311506\pi$
$420$	0	0
$421$	− 40.0000i	− 1.94948i	−0.223341	−	0.974740i	$-0.571696\pi$
$421$	− 40.0000i	− 1.94948i	0.223341	−	0.974740i	$-0.428304\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	6.00000	0.291043
$426$	0	0
$427$	10.0000i	0.483934i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	16.0000	0.770693	0.385346	−	0.922772i	$-0.374082\pi$
$431$	16.0000	0.770693	0.385346	+	0.922772i	$0.374082\pi$
$432$	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
$435$	0	0
$436$	0	0
$437$	− 64.0000i	− 3.06154i
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 2.00000i	− 0.0950229i	−0.998871	−	0.0475114i	$-0.984871\pi$
$443$	− 2.00000i	− 0.0950229i	0.998871	−	0.0475114i	$-0.0151291\pi$
$444$	0	0
$445$	12.0000i	0.568855i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	30.0000	1.41579	0.707894	−	0.706319i	$-0.249646\pi$
$449$	30.0000	1.41579	0.707894	+	0.706319i	$0.249646\pi$
$450$	0	0
$451$	− 12.0000i	− 0.565058i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−12.0000	−0.562569
$456$	0	0
$457$	30.0000	1.40334	0.701670	−	0.712502i	$-0.252438\pi$
$457$	30.0000	1.40334	0.701670	+	0.712502i	$0.252438\pi$
$458$	0	0
$459$	0	0
$460$	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
$463$	16.0000	0.743583	0.371792	−	0.928316i	$-0.378744\pi$
$463$	16.0000	0.743583	0.371792	+	0.928316i	$0.378744\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	16.0000i	0.740392i	0.928954	+	0.370196i	$0.120709\pi$
$467$	16.0000i	0.740392i	−0.928954	+	0.370196i	$0.879291\pi$
$468$	0	0
$469$	− 2.00000i	− 0.0923514i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	12.0000	0.551761
$474$	0	0
$475$	8.00000i	0.367065i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−20.0000	−0.913823	−0.456912	−	0.889512i	$-0.651044\pi$
$479$	−20.0000	−0.913823	−0.456912	+	0.889512i	$0.651044\pi$
$480$	0	0
$481$	24.0000	1.09431
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 20.0000i	− 0.908153i
$486$	0	0
$487$	32.0000	1.45006	0.725029	−	0.688718i	$-0.241826\pi$
$487$	32.0000	1.45006	0.725029	+	0.688718i	$0.241826\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	30.0000i	1.35388i	0.736038	+	0.676941i	$0.236695\pi$
$491$	30.0000i	1.35388i	−0.736038	+	0.676941i	$0.763305\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	− 10.0000i	− 0.447661i	−0.974628	−	0.223831i	$-0.928144\pi$
$499$	− 10.0000i	− 0.447661i	0.974628	−	0.223831i	$-0.0718563\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−28.0000	−1.24846	−0.624229	−	0.781241i	$-0.714587\pi$
$503$	−28.0000	−1.24846	−0.624229	+	0.781241i	$0.714587\pi$
$504$	0	0
$505$	−36.0000	−1.60198
$506$	0	0
$507$	0	0
$508$	0	0
$509$	18.0000i	0.797836i	0.916987	+	0.398918i	$0.130614\pi$
$509$	18.0000i	0.797836i	−0.916987	+	0.398918i	$0.869386\pi$
$510$	0	0
$511$	6.00000	0.265424
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 8.00000i	− 0.352522i
$516$	0	0
$517$	− 24.0000i	− 1.05552i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	18.0000	0.788594	0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000	0.788594	0.394297	+	0.918983i	$0.370988\pi$
$522$	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
$525$	0	0
$526$	0	0
$527$	48.0000	2.09091
$528$	0	0
$529$	41.0000	1.78261
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 36.0000i	− 1.55933i
$534$	0	0
$535$	20.0000	0.864675
$536$	0	0
$537$	0	0
$538$	0	0
$539$	2.00000i	0.0861461i
$540$	0	0
$541$	− 16.0000i	− 0.687894i	−0.938989	−	0.343947i	$-0.888236\pi$
$541$	− 16.0000i	− 0.687894i	0.938989	−	0.343947i	$-0.111764\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−8.00000	−0.342682
$546$	0	0
$547$	− 38.0000i	− 1.62476i	−0.583127	−	0.812381i	$-0.698171\pi$
$547$	− 38.0000i	− 1.62476i	0.583127	−	0.812381i	$-0.301829\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	12.0000i	0.508456i	0.967144	+	0.254228i	$0.0818214\pi$
$557$	12.0000i	0.508456i	−0.967144	+	0.254228i	$0.918179\pi$
$558$	0	0
$559$	36.0000	1.52264
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 28.0000i	− 1.18006i	−0.807382	−	0.590030i	$-0.799116\pi$
$563$	− 28.0000i	− 1.18006i	0.807382	−	0.590030i	$-0.200884\pi$
$564$	0	0
$565$	36.0000i	1.51453i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−22.0000	−0.922288	−0.461144	−	0.887325i	$-0.652561\pi$
$569$	−22.0000	−0.922288	−0.461144	+	0.887325i	$0.652561\pi$
$570$	0	0
$571$	2.00000i	0.0836974i	0.999124	+	0.0418487i	$0.0133247\pi$
$571$	2.00000i	0.0836974i	−0.999124	+	0.0418487i	$0.986675\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−8.00000	−0.333623
$576$	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
$579$	0	0
$580$	0	0
$581$	− 8.00000i	− 0.331896i
$582$	0	0
$583$	24.0000	0.993978
$584$	0	0
$585$	0	0
$586$	0	0
$587$	24.0000i	0.990586i	0.868726	+	0.495293i	$0.164939\pi$
$587$	24.0000i	0.990586i	−0.868726	+	0.495293i	$0.835061\pi$
$588$	0	0
$589$	64.0000i	2.63707i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−22.0000	−0.903432	−0.451716	−	0.892162i	$-0.649188\pi$
$593$	−22.0000	−0.903432	−0.451716	+	0.892162i	$0.649188\pi$
$594$	0	0
$595$	12.0000i	0.491952i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	32.0000	1.30748	0.653742	−	0.756717i	$-0.273198\pi$
$599$	32.0000	1.30748	0.653742	+	0.756717i	$0.273198\pi$
$600$	0	0
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	14.0000i	0.569181i
$606$	0	0
$607$	−28.0000	−1.13648	−0.568242	−	0.822861i	$-0.692376\pi$
$607$	−28.0000	−1.13648	−0.568242	+	0.822861i	$0.692376\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 72.0000i	− 2.91281i
$612$	0	0
$613$	− 28.0000i	− 1.13091i	−0.824779	−	0.565455i	$-0.808701\pi$
$613$	− 28.0000i	− 1.13091i	0.824779	−	0.565455i	$-0.191299\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−30.0000	−1.20775	−0.603877	−	0.797077i	$-0.706378\pi$
$617$	−30.0000	−1.20775	−0.603877	+	0.797077i	$0.706378\pi$
$618$	0	0
$619$	− 20.0000i	− 0.803868i	−0.915669	−	0.401934i	$-0.868338\pi$
$619$	− 20.0000i	− 0.803868i	0.915669	−	0.401934i	$-0.131662\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	6.00000	0.240385
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 24.0000i	− 0.956943i
$630$	0	0
$631$	24.0000	0.955425	0.477712	−	0.878516i	$-0.341466\pi$
$631$	24.0000	0.955425	0.477712	+	0.878516i	$0.341466\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 16.0000i	− 0.634941i
$636$	0	0
$637$	6.00000i	0.237729i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−6.00000	−0.236986	−0.118493	−	0.992955i	$-0.537806\pi$
$641$	−6.00000	−0.236986	−0.118493	+	0.992955i	$0.537806\pi$
$642$	0	0
$643$	28.0000i	1.10421i	0.833774	+	0.552106i	$0.186176\pi$
$643$	28.0000i	1.10421i	−0.833774	+	0.552106i	$0.813824\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	8.00000	0.314512	0.157256	−	0.987558i	$-0.449735\pi$
$647$	8.00000	0.314512	0.157256	+	0.987558i	$0.449735\pi$
$648$	0	0
$649$	−8.00000	−0.314027
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 16.0000i	− 0.626128i	−0.949732	−	0.313064i	$-0.898644\pi$
$653$	− 16.0000i	− 0.626128i	0.949732	−	0.313064i	$-0.101356\pi$
$654$	0	0
$655$	16.0000	0.625172
$656$	0	0
$657$	0	0
$658$	0	0
$659$	42.0000i	1.63609i	0.575156	+	0.818044i	$0.304941\pi$
$659$	42.0000i	1.63609i	−0.575156	+	0.818044i	$0.695059\pi$
$660$	0	0
$661$	6.00000i	0.233373i	0.993169	+	0.116686i	$0.0372273\pi$
$661$	6.00000i	0.233373i	−0.993169	+	0.116686i	$0.962773\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−16.0000	−0.620453
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−20.0000	−0.772091
$672$	0	0
$673$	−14.0000	−0.539660	−0.269830	−	0.962908i	$-0.586968\pi$
$673$	−14.0000	−0.539660	−0.269830	+	0.962908i	$0.586968\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 26.0000i	− 0.999261i	−0.866239	−	0.499631i	$-0.833469\pi$
$677$	− 26.0000i	− 0.999261i	0.866239	−	0.499631i	$-0.166531\pi$
$678$	0	0
$679$	−10.0000	−0.383765
$680$	0	0
$681$	0	0
$682$	0	0
$683$	30.0000i	1.14792i	0.818884	+	0.573959i	$0.194593\pi$
$683$	30.0000i	1.14792i	−0.818884	+	0.573959i	$0.805407\pi$
$684$	0	0
$685$	− 4.00000i	− 0.152832i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	72.0000	2.74298
$690$	0	0
$691$	24.0000i	0.913003i	0.889723	+	0.456502i	$0.150898\pi$
$691$	24.0000i	0.913003i	−0.889723	+	0.456502i	$0.849102\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−16.0000	−0.606915
$696$	0	0
$697$	−36.0000	−1.36360
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 24.0000i	− 0.906467i	−0.891392	−	0.453234i	$-0.850270\pi$
$701$	− 24.0000i	− 0.906467i	0.891392	−	0.453234i	$-0.149730\pi$
$702$	0	0
$703$	32.0000	1.20690
$704$	0	0
$705$	0	0
$706$	0	0
$707$	18.0000i	0.676960i
$708$	0	0
$709$	4.00000i	0.150223i	0.997175	+	0.0751116i	$0.0239313\pi$
$709$	4.00000i	0.150223i	−0.997175	+	0.0751116i	$0.976069\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−64.0000	−2.39682
$714$	0	0
$715$	− 24.0000i	− 0.897549i
$716$	0	0
$717$	0	0
$718$	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
$723$	0	0
$724$	0	0
$725$	0	0
$726$	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
$729$	0	0
$730$	0	0
$731$	− 36.0000i	− 1.33151i
$732$	0	0
$733$	2.00000i	0.0738717i	0.999318	+	0.0369358i	$0.0117597\pi$
$733$	2.00000i	0.0738717i	−0.999318	+	0.0369358i	$0.988240\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	4.00000	0.147342
$738$	0	0
$739$	− 38.0000i	− 1.39785i	−0.715194	−	0.698926i	$-0.753662\pi$
$739$	− 38.0000i	− 1.39785i	0.715194	−	0.698926i	$-0.246338\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−16.0000	−0.586983	−0.293492	−	0.955962i	$-0.594817\pi$
$743$	−16.0000	−0.586983	−0.293492	+	0.955962i	$0.594817\pi$
$744$	0	0
$745$	8.00000	0.293097
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 10.0000i	− 0.365392i
$750$	0	0
$751$	8.00000	0.291924	0.145962	−	0.989290i	$-0.453372\pi$
$751$	8.00000	0.291924	0.145962	+	0.989290i	$0.453372\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 16.0000i	− 0.582300i
$756$	0	0
$757$	− 20.0000i	− 0.726912i	−0.931611	−	0.363456i	$-0.881597\pi$
$757$	− 20.0000i	− 0.726912i	0.931611	−	0.363456i	$-0.118403\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−6.00000	−0.217500	−0.108750	−	0.994069i	$-0.534685\pi$
$761$	−6.00000	−0.217500	−0.108750	+	0.994069i	$0.534685\pi$
$762$	0	0
$763$	4.00000i	0.144810i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−24.0000	−0.866590
$768$	0	0
$769$	−26.0000	−0.937584	−0.468792	−	0.883309i	$-0.655311\pi$
$769$	−26.0000	−0.937584	−0.468792	+	0.883309i	$0.655311\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 14.0000i	− 0.503545i	−0.967786	−	0.251773i	$-0.918987\pi$
$773$	− 14.0000i	− 0.503545i	0.967786	−	0.251773i	$-0.0810135\pi$
$774$	0	0
$775$	8.00000	0.287368
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 48.0000i	− 1.71978i
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−28.0000	−0.999363
$786$	0	0
$787$	− 16.0000i	− 0.570338i	−0.958477	−	0.285169i	$-0.907950\pi$
$787$	− 16.0000i	− 0.570338i	0.958477	−	0.285169i	$-0.0920498\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	18.0000	0.640006
$792$	0	0
$793$	−60.0000	−2.13066
$794$	0	0
$795$	0	0
$796$	0	0
$797$	2.00000i	0.0708436i	0.999372	+	0.0354218i	$0.0112775\pi$
$797$	2.00000i	0.0708436i	−0.999372	+	0.0354218i	$0.988723\pi$
$798$	0	0
$799$	−72.0000	−2.54718
$800$	0	0
$801$	0	0
$802$	0	0
$803$	12.0000i	0.423471i
$804$	0	0
$805$	− 16.0000i	− 0.563926i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	10.0000	0.351581	0.175791	−	0.984428i	$-0.443752\pi$
$809$	10.0000	0.351581	0.175791	+	0.984428i	$0.443752\pi$
$810$	0	0
$811$	4.00000i	0.140459i	0.997531	+	0.0702295i	$0.0223732\pi$
$811$	4.00000i	0.140459i	−0.997531	+	0.0702295i	$0.977627\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−4.00000	−0.140114
$816$	0	0
$817$	48.0000	1.67931
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 36.0000i	− 1.25641i	−0.778048	−	0.628204i	$-0.783790\pi$
$821$	− 36.0000i	− 1.25641i	0.778048	−	0.628204i	$-0.216210\pi$
$822$	0	0
$823$	16.0000	0.557725	0.278862	−	0.960331i	$-0.410043\pi$
$823$	16.0000	0.557725	0.278862	+	0.960331i	$0.410043\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 18.0000i	− 0.625921i	−0.949766	−	0.312961i	$-0.898679\pi$
$827$	− 18.0000i	− 0.625921i	0.949766	−	0.312961i	$-0.101321\pi$
$828$	0	0
$829$	− 46.0000i	− 1.59765i	−0.601566	−	0.798823i	$-0.705456\pi$
$829$	− 46.0000i	− 1.59765i	0.601566	−	0.798823i	$-0.294544\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	6.00000	0.207888
$834$	0	0
$835$	− 16.0000i	− 0.553703i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	48.0000	1.65714	0.828572	−	0.559883i	$-0.189154\pi$
$839$	48.0000	1.65714	0.828572	+	0.559883i	$0.189154\pi$
$840$	0	0
$841$	29.0000	1.00000
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 46.0000i	− 1.58245i
$846$	0	0
$847$	7.00000	0.240523
$848$	0	0
$849$	0	0
$850$	0	0
$851$	32.0000i	1.09695i
$852$	0	0
$853$	42.0000i	1.43805i	0.694983	+	0.719026i	$0.255412\pi$
$853$	42.0000i	1.43805i	−0.694983	+	0.719026i	$0.744588\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	58.0000	1.98124	0.990621	−	0.136637i	$-0.0436295\pi$
$857$	58.0000	1.98124	0.990621	+	0.136637i	$0.0436295\pi$
$858$	0	0
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	16.0000	0.544646	0.272323	−	0.962206i	$-0.412208\pi$
$863$	16.0000	0.544646	0.272323	+	0.962206i	$0.412208\pi$
$864$	0	0
$865$	−4.00000	−0.136004
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	12.0000	0.406604
$872$	0	0
$873$	0	0
$874$	0	0
$875$	12.0000i	0.405674i
$876$	0	0
$877$	36.0000i	1.21563i	0.794077	+	0.607817i	$0.207955\pi$
$877$	36.0000i	1.21563i	−0.794077	+	0.607817i	$0.792045\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	10.0000	0.336909	0.168454	−	0.985709i	$-0.446122\pi$
$881$	10.0000	0.336909	0.168454	+	0.985709i	$0.446122\pi$
$882$	0	0
$883$	− 2.00000i	− 0.0673054i	−0.999434	−	0.0336527i	$-0.989286\pi$
$883$	− 2.00000i	− 0.0673054i	0.999434	−	0.0336527i	$-0.0107140\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	8.00000	0.268614	0.134307	−	0.990940i	$-0.457119\pi$
$887$	8.00000	0.268614	0.134307	+	0.990940i	$0.457119\pi$
$888$	0	0
$889$	−8.00000	−0.268311
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 96.0000i	− 3.21252i
$894$	0	0
$895$	−28.0000	−0.935937
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	− 72.0000i	− 2.39867i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	28.0000	0.930751
$906$	0	0
$907$	54.0000i	1.79304i	0.443003	+	0.896520i	$0.353913\pi$
$907$	54.0000i	1.79304i	−0.443003	+	0.896520i	$0.646087\pi$
$908$	0	0
$909$	0	0
$910$	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
$913$	16.0000	0.529523
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 8.00000i	− 0.264183i
$918$	0	0
$919$	32.0000	1.05558	0.527791	−	0.849374i	$-0.323020\pi$
$919$	32.0000	1.05558	0.527791	+	0.849374i	$0.323020\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	− 4.00000i	− 0.131519i
$926$	0	0
$927$	0	0
$928$	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
$931$	8.00000i	0.262189i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−24.0000	−0.784884
$936$	0	0
$937$	22.0000	0.718709	0.359354	−	0.933201i	$-0.382997\pi$
$937$	22.0000	0.718709	0.359354	+	0.933201i	$0.382997\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	14.0000i	0.456387i	0.973616	+	0.228193i	$0.0732819\pi$
$941$	14.0000i	0.456387i	−0.973616	+	0.228193i	$0.926718\pi$
$942$	0	0
$943$	48.0000	1.56310
$944$	0	0
$945$	0	0
$946$	0	0
$947$	26.0000i	0.844886i	0.906389	+	0.422443i	$0.138827\pi$
$947$	26.0000i	0.844886i	−0.906389	+	0.422443i	$0.861173\pi$
$948$	0	0
$949$	36.0000i	1.16861i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	6.00000	0.194359	0.0971795	−	0.995267i	$-0.469018\pi$
$953$	6.00000	0.194359	0.0971795	+	0.995267i	$0.469018\pi$
$954$	0	0
$955$	− 16.0000i	− 0.517748i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−2.00000	−0.0645834
$960$	0	0
$961$	33.0000	1.06452
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 4.00000i	− 0.128765i
$966$	0	0
$967$	56.0000	1.80084	0.900419	−	0.435023i	$-0.143260\pi$
$967$	56.0000	1.80084	0.900419	+	0.435023i	$0.143260\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	20.0000i	0.641831i	0.947108	+	0.320915i	$0.103990\pi$
$971$	20.0000i	0.641831i	−0.947108	+	0.320915i	$0.896010\pi$
$972$	0	0
$973$	8.00000i	0.256468i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−50.0000	−1.59964	−0.799821	−	0.600239i	$-0.795072\pi$
$977$	−50.0000	−1.59964	−0.799821	+	0.600239i	$0.795072\pi$
$978$	0	0
$979$	12.0000i	0.383522i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−8.00000	−0.255160	−0.127580	−	0.991828i	$-0.540721\pi$
$983$	−8.00000	−0.255160	−0.127580	+	0.991828i	$0.540721\pi$
$984$	0	0
$985$	−24.0000	−0.764704
$986$	0	0
$987$	0	0
$988$	0	0
$989$	48.0000i	1.52631i
$990$	0	0
$991$	−56.0000	−1.77890	−0.889449	−	0.457034i	$-0.848912\pi$
$991$	−56.0000	−1.77890	−0.889449	+	0.457034i	$0.848912\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	8.00000i	0.253617i
$996$	0	0
$997$	− 34.0000i	− 1.07679i	−0.842692	−	0.538395i	$-0.819031\pi$
$997$	− 34.0000i	− 1.07679i	0.842692	−	0.538395i	$-0.180969\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4032.2.c.j.2017.2		2
3.2	odd	2		1344.2.c.d.673.2	yes	2
4.3	odd	2		4032.2.c.e.2017.2		2
8.3	odd	2		4032.2.c.e.2017.1		2
8.5	even	2	inner	4032.2.c.j.2017.1		2
12.11	even	2		1344.2.c.a.673.1	✓	2
24.5	odd	2		1344.2.c.d.673.1	yes	2
24.11	even	2		1344.2.c.a.673.2	yes	2
48.5	odd	4		5376.2.a.k.1.1		1
48.11	even	4		5376.2.a.f.1.1		1
48.29	odd	4		5376.2.a.a.1.1		1
48.35	even	4		5376.2.a.h.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1344.2.c.a.673.1	✓	2	12.11	even	2
1344.2.c.a.673.2	yes	2	24.11	even	2
1344.2.c.d.673.1	yes	2	24.5	odd	2
1344.2.c.d.673.2	yes	2	3.2	odd	2
4032.2.c.e.2017.1		2	8.3	odd	2
4032.2.c.e.2017.2		2	4.3	odd	2
4032.2.c.j.2017.1		2	8.5	even	2	inner
4032.2.c.j.2017.2		2	1.1	even	1	trivial
5376.2.a.a.1.1		1	48.29	odd	4
5376.2.a.f.1.1		1	48.11	even	4
5376.2.a.h.1.1		1	48.35	even	4
5376.2.a.k.1.1		1	48.5	odd	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:	\( N \)	\(=\)	\( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4032.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.00000i q^{5} +1.00000 q^{7} +O(q^{10})\)	\(q+2.00000i q^{5} +1.00000 q^{7} +2.00000i q^{11} +6.00000i q^{13} +6.00000 q^{17} +8.00000i q^{19} -8.00000 q^{23} +1.00000 q^{25} +8.00000 q^{31} +2.00000i q^{35} -4.00000i q^{37} -6.00000 q^{41} -6.00000i q^{43} -12.0000 q^{47} +1.00000 q^{49} -12.0000i q^{53} -4.00000 q^{55} +4.00000i q^{59} +10.0000i q^{61} -12.0000 q^{65} -2.00000i q^{67} +6.00000 q^{73} +2.00000i q^{77} -8.00000i q^{83} +12.0000i q^{85} +6.00000 q^{89} +6.00000i q^{91} -16.0000 q^{95} -10.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{7}+O(q^{10}) \) 2 * q + 2 * q^7	\( 2 q + 2 q^{7} + 12 q^{17} - 16 q^{23} + 2 q^{25} + 16 q^{31} - 12 q^{41} - 24 q^{47} + 2 q^{49} - 8 q^{55} - 24 q^{65} + 12 q^{73} + 12 q^{89} - 32 q^{95} - 20 q^{97}+O(q^{100}) \) 2 * q + 2 * q^7 + 12 * q^17 - 16 * q^23 + 2 * q^25 + 16 * q^31 - 12 * q^41 - 24 * q^47 + 2 * q^49 - 8 * q^55 - 24 * q^65 + 12 * q^73 + 12 * q^89 - 32 * q^95 - 20 * q^97

Introduction

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