LMFDB - Embedded newform 2816.2.c.m.1409.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2816,2,Mod(1409,2816)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2816, 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("2816.1409");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2816 = 2^{8} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2816.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$22.4858732092$
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:	no (minimal twist has level 352)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1409.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	2816.1409
Dual form			2816.2.c.m.1409.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} -1.00000i q^{5} +4.00000 q^{7} +2.00000 q^{9} +1.00000i q^{11} -2.00000i q^{13} -1.00000 q^{15} +2.00000i q^{19} -4.00000i q^{21} +9.00000 q^{23} +4.00000 q^{25} -5.00000i q^{27} +4.00000i q^{29} -5.00000 q^{31} +1.00000 q^{33} -4.00000i q^{35} +9.00000i q^{37} -2.00000 q^{39} -2.00000 q^{41} -6.00000i q^{43} -2.00000i q^{45} +4.00000 q^{47} +9.00000 q^{49} +6.00000i q^{53} +1.00000 q^{55} +2.00000 q^{57} -5.00000i q^{59} +8.00000 q^{63} -2.00000 q^{65} +13.0000i q^{67} -9.00000i q^{69} -1.00000 q^{71} -14.0000 q^{73} -4.00000i q^{75} +4.00000i q^{77} +10.0000 q^{79} +1.00000 q^{81} -14.0000i q^{83} +4.00000 q^{87} +13.0000 q^{89} -8.00000i q^{91} +5.00000i q^{93} +2.00000 q^{95} -19.0000 q^{97} +2.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 8 q^{7} + 4 q^{9} - 2 q^{15} + 18 q^{23} + 8 q^{25} - 10 q^{31} + 2 q^{33} - 4 q^{39} - 4 q^{41} + 8 q^{47} + 18 q^{49} + 2 q^{55} + 4 q^{57} + 16 q^{63} - 4 q^{65} - 2 q^{71} - 28 q^{73} + 20 q^{79}+ \cdots - 38 q^{97}+O(q^{100}) $ 2 * q + 8 * q^7 + 4 * q^9 - 2 * q^15 + 18 * q^23 + 8 * q^25 - 10 * q^31 + 2 * q^33 - 4 * q^39 - 4 * q^41 + 8 * q^47 + 18 * q^49 + 2 * q^55 + 4 * q^57 + 16 * q^63 - 4 * q^65 - 2 * q^71 - 28 * q^73 + 20 * q^79 + 2 * q^81 + 8 * q^87 + 26 * q^89 + 4 * q^95 - 38 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2816\mathbb{Z}\right)^\times$.

$n$	$1025$	$1541$	$2047$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	− 1.00000i	− 0.577350i	−0.957427	−	0.288675i	$-0.906785\pi$
$3$	− 1.00000i	− 0.577350i	0.957427	−	0.288675i	$-0.0932147\pi$
$4$	0	0
$5$	− 1.00000i	− 0.447214i	−0.974679	−	0.223607i	$-0.928217\pi$
$5$	− 1.00000i	− 0.447214i	0.974679	−	0.223607i	$-0.0717831\pi$
$6$	0	0
$7$	4.00000	1.51186	0.755929	−	0.654654i	$-0.227186\pi$
$7$	4.00000	1.51186	0.755929	+	0.654654i	$0.227186\pi$
$8$	0	0
$9$	2.00000	0.666667
$10$	0	0
$11$	1.00000i	0.301511i
$11$	1.00000i	0.301511i
$12$	0	0
$13$	− 2.00000i	− 0.554700i	−0.960769	−	0.277350i	$-0.910544\pi$
$13$	− 2.00000i	− 0.554700i	0.960769	−	0.277350i	$-0.0894562\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	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
$21$	− 4.00000i	− 0.872872i
$22$	0	0
$23$	9.00000	1.87663	0.938315	−	0.345782i	$-0.112386\pi$
$23$	9.00000	1.87663	0.938315	+	0.345782i	$0.112386\pi$
$24$	0	0
$25$	4.00000	0.800000
$26$	0	0
$27$	− 5.00000i	− 0.962250i
$28$	0	0
$29$	4.00000i	0.742781i	0.928477	+	0.371391i	$0.121119\pi$
$29$	4.00000i	0.742781i	−0.928477	+	0.371391i	$0.878881\pi$
$30$	0	0
$31$	−5.00000	−0.898027	−0.449013	−	0.893525i	$-0.648224\pi$
$31$	−5.00000	−0.898027	−0.449013	+	0.893525i	$0.648224\pi$
$32$	0	0
$33$	1.00000	0.174078
$34$	0	0
$35$	− 4.00000i	− 0.676123i
$36$	0	0
$37$	9.00000i	1.47959i	0.672832	+	0.739795i	$0.265078\pi$
$37$	9.00000i	1.47959i	−0.672832	+	0.739795i	$0.734922\pi$
$38$	0	0
$39$	−2.00000	−0.320256
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	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
$45$	− 2.00000i	− 0.298142i
$46$	0	0
$47$	4.00000	0.583460	0.291730	−	0.956501i	$-0.405769\pi$
$47$	4.00000	0.583460	0.291730	+	0.956501i	$0.405769\pi$
$48$	0	0
$49$	9.00000	1.28571
$50$	0	0
$51$	0	0
$52$	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
$55$	1.00000	0.134840
$56$	0	0
$57$	2.00000	0.264906
$58$	0	0
$59$	− 5.00000i	− 0.650945i	−0.945552	−	0.325472i	$-0.894477\pi$
$59$	− 5.00000i	− 0.650945i	0.945552	−	0.325472i	$-0.105523\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	8.00000	1.00791
$64$	0	0
$65$	−2.00000	−0.248069
$66$	0	0
$67$	13.0000i	1.58820i	0.607785	+	0.794101i	$0.292058\pi$
$67$	13.0000i	1.58820i	−0.607785	+	0.794101i	$0.707942\pi$
$68$	0	0
$69$	− 9.00000i	− 1.08347i
$70$	0	0
$71$	−1.00000	−0.118678	−0.0593391	−	0.998238i	$-0.518899\pi$
$71$	−1.00000	−0.118678	−0.0593391	+	0.998238i	$0.518899\pi$
$72$	0	0
$73$	−14.0000	−1.63858	−0.819288	−	0.573382i	$-0.805631\pi$
$73$	−14.0000	−1.63858	−0.819288	+	0.573382i	$0.805631\pi$
$74$	0	0
$75$	− 4.00000i	− 0.461880i
$76$	0	0
$77$	4.00000i	0.455842i
$78$	0	0
$79$	10.0000	1.12509	0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000	1.12509	0.562544	+	0.826767i	$0.309823\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 14.0000i	− 1.53670i	−0.640030	−	0.768350i	$-0.721078\pi$
$83$	− 14.0000i	− 1.53670i	0.640030	−	0.768350i	$-0.278922\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	4.00000	0.428845
$88$	0	0
$89$	13.0000	1.37800	0.688999	−	0.724763i	$-0.258051\pi$
$89$	13.0000	1.37800	0.688999	+	0.724763i	$0.258051\pi$
$90$	0	0
$91$	− 8.00000i	− 0.838628i
$92$	0	0
$93$	5.00000i	0.518476i
$94$	0	0
$95$	2.00000	0.205196
$96$	0	0
$97$	−19.0000	−1.92916	−0.964579	−	0.263795i	$-0.915026\pi$
$97$	−19.0000	−1.92916	−0.964579	+	0.263795i	$0.915026\pi$
$98$	0	0
$99$	2.00000i	0.201008i
$100$	0	0
$101$	12.0000i	1.19404i	0.802225	+	0.597022i	$0.203650\pi$
$101$	12.0000i	1.19404i	−0.802225	+	0.597022i	$0.796350\pi$
$102$	0	0
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	0	0
$105$	−4.00000	−0.390360
$106$	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
$109$	10.0000i	0.957826i	0.877862	+	0.478913i	$0.158969\pi$
$109$	10.0000i	0.957826i	−0.877862	+	0.478913i	$0.841031\pi$
$110$	0	0
$111$	9.00000	0.854242
$112$	0	0
$113$	−7.00000	−0.658505	−0.329252	−	0.944242i	$-0.606797\pi$
$113$	−7.00000	−0.658505	−0.329252	+	0.944242i	$0.606797\pi$
$114$	0	0
$115$	− 9.00000i	− 0.839254i
$116$	0	0
$117$	− 4.00000i	− 0.369800i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−1.00000	−0.0909091
$122$	0	0
$123$	2.00000i	0.180334i
$124$	0	0
$125$	− 9.00000i	− 0.804984i
$126$	0	0
$127$	−2.00000	−0.177471	−0.0887357	−	0.996055i	$-0.528283\pi$
$127$	−2.00000	−0.177471	−0.0887357	+	0.996055i	$0.528283\pi$
$128$	0	0
$129$	−6.00000	−0.528271
$130$	0	0
$131$	− 16.0000i	− 1.39793i	−0.715158	−	0.698963i	$-0.753645\pi$
$131$	− 16.0000i	− 1.39793i	0.715158	−	0.698963i	$-0.246355\pi$
$132$	0	0
$133$	8.00000i	0.693688i
$134$	0	0
$135$	−5.00000	−0.430331
$136$	0	0
$137$	−13.0000	−1.11066	−0.555332	−	0.831628i	$-0.687409\pi$
$137$	−13.0000	−1.11066	−0.555332	+	0.831628i	$0.687409\pi$
$138$	0	0
$139$	− 16.0000i	− 1.35710i	−0.734553	−	0.678551i	$-0.762608\pi$
$139$	− 16.0000i	− 1.35710i	0.734553	−	0.678551i	$-0.237392\pi$
$140$	0	0
$141$	− 4.00000i	− 0.336861i
$142$	0	0
$143$	2.00000	0.167248
$144$	0	0
$145$	4.00000	0.332182
$146$	0	0
$147$	− 9.00000i	− 0.742307i
$148$	0	0
$149$	− 12.0000i	− 0.983078i	−0.870855	−	0.491539i	$-0.836434\pi$
$149$	− 12.0000i	− 0.983078i	0.870855	−	0.491539i	$-0.163566\pi$
$150$	0	0
$151$	14.0000	1.13930	0.569652	−	0.821886i	$-0.307078\pi$
$151$	14.0000	1.13930	0.569652	+	0.821886i	$0.307078\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	5.00000i	0.401610i
$156$	0	0
$157$	13.0000i	1.03751i	0.854922	+	0.518756i	$0.173605\pi$
$157$	13.0000i	1.03751i	−0.854922	+	0.518756i	$0.826395\pi$
$158$	0	0
$159$	6.00000	0.475831
$160$	0	0
$161$	36.0000	2.83720
$162$	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
$165$	− 1.00000i	− 0.0778499i
$166$	0	0
$167$	18.0000	1.39288	0.696441	−	0.717614i	$-0.254766\pi$
$167$	18.0000	1.39288	0.696441	+	0.717614i	$0.254766\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	4.00000i	0.305888i
$172$	0	0
$173$	− 22.0000i	− 1.67263i	−0.548250	−	0.836315i	$-0.684706\pi$
$173$	− 22.0000i	− 1.67263i	0.548250	−	0.836315i	$-0.315294\pi$
$174$	0	0
$175$	16.0000	1.20949
$176$	0	0
$177$	−5.00000	−0.375823
$178$	0	0
$179$	− 11.0000i	− 0.822179i	−0.911595	−	0.411089i	$-0.865148\pi$
$179$	− 11.0000i	− 0.822179i	0.911595	−	0.411089i	$-0.134852\pi$
$180$	0	0
$181$	− 11.0000i	− 0.817624i	−0.912619	−	0.408812i	$-0.865943\pi$
$181$	− 11.0000i	− 0.817624i	0.912619	−	0.408812i	$-0.134057\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	9.00000	0.661693
$186$	0	0
$187$	0	0
$188$	0	0
$189$	− 20.0000i	− 1.45479i
$190$	0	0
$191$	−15.0000	−1.08536	−0.542681	−	0.839939i	$-0.682591\pi$
$191$	−15.0000	−1.08536	−0.542681	+	0.839939i	$0.682591\pi$
$192$	0	0
$193$	−12.0000	−0.863779	−0.431889	−	0.901927i	$-0.642153\pi$
$193$	−12.0000	−0.863779	−0.431889	+	0.901927i	$0.642153\pi$
$194$	0	0
$195$	2.00000i	0.143223i
$196$	0	0
$197$	− 22.0000i	− 1.56744i	−0.621117	−	0.783718i	$-0.713321\pi$
$197$	− 22.0000i	− 1.56744i	0.621117	−	0.783718i	$-0.286679\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	13.0000	0.916949
$202$	0	0
$203$	16.0000i	1.12298i
$204$	0	0
$205$	2.00000i	0.139686i
$206$	0	0
$207$	18.0000	1.25109
$208$	0	0
$209$	−2.00000	−0.138343
$210$	0	0
$211$	12.0000i	0.826114i	0.910705	+	0.413057i	$0.135539\pi$
$211$	12.0000i	0.826114i	−0.910705	+	0.413057i	$0.864461\pi$
$212$	0	0
$213$	1.00000i	0.0685189i
$214$	0	0
$215$	−6.00000	−0.409197
$216$	0	0
$217$	−20.0000	−1.35769
$218$	0	0
$219$	14.0000i	0.946032i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−25.0000	−1.67412	−0.837062	−	0.547108i	$-0.815729\pi$
$223$	−25.0000	−1.67412	−0.837062	+	0.547108i	$0.815729\pi$
$224$	0	0
$225$	8.00000	0.533333
$226$	0	0
$227$	− 24.0000i	− 1.59294i	−0.604681	−	0.796468i	$-0.706699\pi$
$227$	− 24.0000i	− 1.59294i	0.604681	−	0.796468i	$-0.293301\pi$
$228$	0	0
$229$	9.00000i	0.594737i	0.954763	+	0.297368i	$0.0961089\pi$
$229$	9.00000i	0.594737i	−0.954763	+	0.297368i	$0.903891\pi$
$230$	0	0
$231$	4.00000	0.263181
$232$	0	0
$233$	12.0000	0.786146	0.393073	−	0.919507i	$-0.371412\pi$
$233$	12.0000	0.786146	0.393073	+	0.919507i	$0.371412\pi$
$234$	0	0
$235$	− 4.00000i	− 0.260931i
$236$	0	0
$237$	− 10.0000i	− 0.649570i
$238$	0	0
$239$	18.0000	1.16432	0.582162	−	0.813073i	$-0.302207\pi$
$239$	18.0000	1.16432	0.582162	+	0.813073i	$0.302207\pi$
$240$	0	0
$241$	−4.00000	−0.257663	−0.128831	−	0.991667i	$-0.541123\pi$
$241$	−4.00000	−0.257663	−0.128831	+	0.991667i	$0.541123\pi$
$242$	0	0
$243$	− 16.0000i	− 1.02640i
$244$	0	0
$245$	− 9.00000i	− 0.574989i
$246$	0	0
$247$	4.00000	0.254514
$248$	0	0
$249$	−14.0000	−0.887214
$250$	0	0
$251$	− 17.0000i	− 1.07303i	−0.843891	−	0.536515i	$-0.819740\pi$
$251$	− 17.0000i	− 1.07303i	0.843891	−	0.536515i	$-0.180260\pi$
$252$	0	0
$253$	9.00000i	0.565825i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	0	0
$259$	36.0000i	2.23693i
$260$	0	0
$261$	8.00000i	0.495188i
$262$	0	0
$263$	−4.00000	−0.246651	−0.123325	−	0.992366i	$-0.539356\pi$
$263$	−4.00000	−0.246651	−0.123325	+	0.992366i	$0.539356\pi$
$264$	0	0
$265$	6.00000	0.368577
$266$	0	0
$267$	− 13.0000i	− 0.795587i
$268$	0	0
$269$	14.0000i	0.853595i	0.904347	+	0.426798i	$0.140358\pi$
$269$	14.0000i	0.853595i	−0.904347	+	0.426798i	$0.859642\pi$
$270$	0	0
$271$	−10.0000	−0.607457	−0.303728	−	0.952759i	$-0.598232\pi$
$271$	−10.0000	−0.607457	−0.303728	+	0.952759i	$0.598232\pi$
$272$	0	0
$273$	−8.00000	−0.484182
$274$	0	0
$275$	4.00000i	0.241209i
$276$	0	0
$277$	14.0000i	0.841178i	0.907251	+	0.420589i	$0.138177\pi$
$277$	14.0000i	0.841178i	−0.907251	+	0.420589i	$0.861823\pi$
$278$	0	0
$279$	−10.0000	−0.598684
$280$	0	0
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	0	0
$283$	8.00000i	0.475551i	0.971320	+	0.237775i	$0.0764182\pi$
$283$	8.00000i	0.475551i	−0.971320	+	0.237775i	$0.923582\pi$
$284$	0	0
$285$	− 2.00000i	− 0.118470i
$286$	0	0
$287$	−8.00000	−0.472225
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	19.0000i	1.11380i
$292$	0	0
$293$	10.0000i	0.584206i	0.956387	+	0.292103i	$0.0943550\pi$
$293$	10.0000i	0.584206i	−0.956387	+	0.292103i	$0.905645\pi$
$294$	0	0
$295$	−5.00000	−0.291111
$296$	0	0
$297$	5.00000	0.290129
$298$	0	0
$299$	− 18.0000i	− 1.04097i
$300$	0	0
$301$	− 24.0000i	− 1.38334i
$302$	0	0
$303$	12.0000	0.689382
$304$	0	0
$305$	0	0
$306$	0	0
$307$	34.0000i	1.94048i	0.242140	+	0.970241i	$0.422151\pi$
$307$	34.0000i	1.94048i	−0.242140	+	0.970241i	$0.577849\pi$
$308$	0	0
$309$	8.00000i	0.455104i
$310$	0	0
$311$	16.0000	0.907277	0.453638	−	0.891186i	$-0.350126\pi$
$311$	16.0000	0.907277	0.453638	+	0.891186i	$0.350126\pi$
$312$	0	0
$313$	−7.00000	−0.395663	−0.197832	−	0.980236i	$-0.563390\pi$
$313$	−7.00000	−0.395663	−0.197832	+	0.980236i	$0.563390\pi$
$314$	0	0
$315$	− 8.00000i	− 0.450749i
$316$	0	0
$317$	9.00000i	0.505490i	0.967533	+	0.252745i	$0.0813334\pi$
$317$	9.00000i	0.505490i	−0.967533	+	0.252745i	$0.918667\pi$
$318$	0	0
$319$	−4.00000	−0.223957
$320$	0	0
$321$	8.00000	0.446516
$322$	0	0
$323$	0	0
$324$	0	0
$325$	− 8.00000i	− 0.443760i
$326$	0	0
$327$	10.0000	0.553001
$328$	0	0
$329$	16.0000	0.882109
$330$	0	0
$331$	13.0000i	0.714545i	0.934000	+	0.357272i	$0.116293\pi$
$331$	13.0000i	0.714545i	−0.934000	+	0.357272i	$0.883707\pi$
$332$	0	0
$333$	18.0000i	0.986394i
$334$	0	0
$335$	13.0000	0.710266
$336$	0	0
$337$	12.0000	0.653682	0.326841	−	0.945079i	$-0.394016\pi$
$337$	12.0000	0.653682	0.326841	+	0.945079i	$0.394016\pi$
$338$	0	0
$339$	7.00000i	0.380188i
$340$	0	0
$341$	− 5.00000i	− 0.270765i
$342$	0	0
$343$	8.00000	0.431959
$344$	0	0
$345$	−9.00000	−0.484544
$346$	0	0
$347$	− 10.0000i	− 0.536828i	−0.963304	−	0.268414i	$-0.913500\pi$
$347$	− 10.0000i	− 0.536828i	0.963304	−	0.268414i	$-0.0864995\pi$
$348$	0	0
$349$	8.00000i	0.428230i	0.976808	+	0.214115i	$0.0686868\pi$
$349$	8.00000i	0.428230i	−0.976808	+	0.214115i	$0.931313\pi$
$350$	0	0
$351$	−10.0000	−0.533761
$352$	0	0
$353$	3.00000	0.159674	0.0798369	−	0.996808i	$-0.474560\pi$
$353$	3.00000	0.159674	0.0798369	+	0.996808i	$0.474560\pi$
$354$	0	0
$355$	1.00000i	0.0530745i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−22.0000	−1.16112	−0.580558	−	0.814219i	$-0.697165\pi$
$359$	−22.0000	−1.16112	−0.580558	+	0.814219i	$0.697165\pi$
$360$	0	0
$361$	15.0000	0.789474
$362$	0	0
$363$	1.00000i	0.0524864i
$364$	0	0
$365$	14.0000i	0.732793i
$366$	0	0
$367$	−13.0000	−0.678594	−0.339297	−	0.940679i	$-0.610189\pi$
$367$	−13.0000	−0.678594	−0.339297	+	0.940679i	$0.610189\pi$
$368$	0	0
$369$	−4.00000	−0.208232
$370$	0	0
$371$	24.0000i	1.24602i
$372$	0	0
$373$	− 32.0000i	− 1.65690i	−0.560065	−	0.828449i	$-0.689224\pi$
$373$	− 32.0000i	− 1.65690i	0.560065	−	0.828449i	$-0.310776\pi$
$374$	0	0
$375$	−9.00000	−0.464758
$376$	0	0
$377$	8.00000	0.412021
$378$	0	0
$379$	− 7.00000i	− 0.359566i	−0.983706	−	0.179783i	$-0.942460\pi$
$379$	− 7.00000i	− 0.359566i	0.983706	−	0.179783i	$-0.0575395\pi$
$380$	0	0
$381$	2.00000i	0.102463i
$382$	0	0
$383$	−9.00000	−0.459879	−0.229939	−	0.973205i	$-0.573853\pi$
$383$	−9.00000	−0.459879	−0.229939	+	0.973205i	$0.573853\pi$
$384$	0	0
$385$	4.00000	0.203859
$386$	0	0
$387$	− 12.0000i	− 0.609994i
$388$	0	0
$389$	19.0000i	0.963338i	0.876353	+	0.481669i	$0.159969\pi$
$389$	19.0000i	0.963338i	−0.876353	+	0.481669i	$0.840031\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−16.0000	−0.807093
$394$	0	0
$395$	− 10.0000i	− 0.503155i
$396$	0	0
$397$	22.0000i	1.10415i	0.833795	+	0.552074i	$0.186163\pi$
$397$	22.0000i	1.10415i	−0.833795	+	0.552074i	$0.813837\pi$
$398$	0	0
$399$	8.00000	0.400501
$400$	0	0
$401$	−30.0000	−1.49813	−0.749064	−	0.662497i	$-0.769497\pi$
$401$	−30.0000	−1.49813	−0.749064	+	0.662497i	$0.769497\pi$
$402$	0	0
$403$	10.0000i	0.498135i
$404$	0	0
$405$	− 1.00000i	− 0.0496904i
$406$	0	0
$407$	−9.00000	−0.446113
$408$	0	0
$409$	−18.0000	−0.890043	−0.445021	−	0.895520i	$-0.646804\pi$
$409$	−18.0000	−0.890043	−0.445021	+	0.895520i	$0.646804\pi$
$410$	0	0
$411$	13.0000i	0.641243i
$412$	0	0
$413$	− 20.0000i	− 0.984136i
$414$	0	0
$415$	−14.0000	−0.687233
$416$	0	0
$417$	−16.0000	−0.783523
$418$	0	0
$419$	24.0000i	1.17248i	0.810139	+	0.586238i	$0.199392\pi$
$419$	24.0000i	1.17248i	−0.810139	+	0.586238i	$0.800608\pi$
$420$	0	0
$421$	2.00000i	0.0974740i	0.998812	+	0.0487370i	$0.0155196\pi$
$421$	2.00000i	0.0974740i	−0.998812	+	0.0487370i	$0.984480\pi$
$422$	0	0
$423$	8.00000	0.388973
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	− 2.00000i	− 0.0965609i
$430$	0	0
$431$	8.00000	0.385346	0.192673	−	0.981263i	$-0.438284\pi$
$431$	8.00000	0.385346	0.192673	+	0.981263i	$0.438284\pi$
$432$	0	0
$433$	5.00000	0.240285	0.120142	−	0.992757i	$-0.461665\pi$
$433$	5.00000	0.240285	0.120142	+	0.992757i	$0.461665\pi$
$434$	0	0
$435$	− 4.00000i	− 0.191785i
$436$	0	0
$437$	18.0000i	0.861057i
$438$	0	0
$439$	20.0000	0.954548	0.477274	−	0.878755i	$-0.341625\pi$
$439$	20.0000	0.954548	0.477274	+	0.878755i	$0.341625\pi$
$440$	0	0
$441$	18.0000	0.857143
$442$	0	0
$443$	11.0000i	0.522626i	0.965254	+	0.261313i	$0.0841554\pi$
$443$	11.0000i	0.522626i	−0.965254	+	0.261313i	$0.915845\pi$
$444$	0	0
$445$	− 13.0000i	− 0.616259i
$446$	0	0
$447$	−12.0000	−0.567581
$448$	0	0
$449$	−9.00000	−0.424736	−0.212368	−	0.977190i	$-0.568118\pi$
$449$	−9.00000	−0.424736	−0.212368	+	0.977190i	$0.568118\pi$
$450$	0	0
$451$	− 2.00000i	− 0.0941763i
$452$	0	0
$453$	− 14.0000i	− 0.657777i
$454$	0	0
$455$	−8.00000	−0.375046
$456$	0	0
$457$	0	0		−	1.00000i	$-0.5\pi$
$457$	0	0			1.00000i	$0.5\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	5.00000	0.232370	0.116185	−	0.993228i	$-0.462933\pi$
$463$	5.00000	0.232370	0.116185	+	0.993228i	$0.462933\pi$
$464$	0	0
$465$	5.00000	0.231869
$466$	0	0
$467$	− 19.0000i	− 0.879215i	−0.898190	−	0.439608i	$-0.855118\pi$
$467$	− 19.0000i	− 0.879215i	0.898190	−	0.439608i	$-0.144882\pi$
$468$	0	0
$469$	52.0000i	2.40114i
$470$	0	0
$471$	13.0000	0.599008
$472$	0	0
$473$	6.00000	0.275880
$474$	0	0
$475$	8.00000i	0.367065i
$476$	0	0
$477$	12.0000i	0.549442i
$478$	0	0
$479$	−38.0000	−1.73626	−0.868132	−	0.496333i	$-0.834679\pi$
$479$	−38.0000	−1.73626	−0.868132	+	0.496333i	$0.834679\pi$
$480$	0	0
$481$	18.0000	0.820729
$482$	0	0
$483$	− 36.0000i	− 1.63806i
$484$	0	0
$485$	19.0000i	0.862746i
$486$	0	0
$487$	−31.0000	−1.40474	−0.702372	−	0.711810i	$-0.747876\pi$
$487$	−31.0000	−1.40474	−0.702372	+	0.711810i	$0.747876\pi$
$488$	0	0
$489$	−20.0000	−0.904431
$490$	0	0
$491$	8.00000i	0.361035i	0.983572	+	0.180517i	$0.0577772\pi$
$491$	8.00000i	0.361035i	−0.983572	+	0.180517i	$0.942223\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	2.00000	0.0898933
$496$	0	0
$497$	−4.00000	−0.179425
$498$	0	0
$499$	28.0000i	1.25345i	0.779240	+	0.626726i	$0.215605\pi$
$499$	28.0000i	1.25345i	−0.779240	+	0.626726i	$0.784395\pi$
$500$	0	0
$501$	− 18.0000i	− 0.804181i
$502$	0	0
$503$	6.00000	0.267527	0.133763	−	0.991013i	$-0.457294\pi$
$503$	6.00000	0.267527	0.133763	+	0.991013i	$0.457294\pi$
$504$	0	0
$505$	12.0000	0.533993
$506$	0	0
$507$	− 9.00000i	− 0.399704i
$508$	0	0
$509$	23.0000i	1.01946i	0.860335	+	0.509729i	$0.170254\pi$
$509$	23.0000i	1.01946i	−0.860335	+	0.509729i	$0.829746\pi$
$510$	0	0
$511$	−56.0000	−2.47729
$512$	0	0
$513$	10.0000	0.441511
$514$	0	0
$515$	8.00000i	0.352522i
$516$	0	0
$517$	4.00000i	0.175920i
$518$	0	0
$519$	−22.0000	−0.965693
$520$	0	0
$521$	−17.0000	−0.744784	−0.372392	−	0.928076i	$-0.621462\pi$
$521$	−17.0000	−0.744784	−0.372392	+	0.928076i	$0.621462\pi$
$522$	0	0
$523$	42.0000i	1.83653i	0.395964	+	0.918266i	$0.370410\pi$
$523$	42.0000i	1.83653i	−0.395964	+	0.918266i	$0.629590\pi$
$524$	0	0
$525$	− 16.0000i	− 0.698297i
$526$	0	0
$527$	0	0
$528$	0	0
$529$	58.0000	2.52174
$530$	0	0
$531$	− 10.0000i	− 0.433963i
$532$	0	0
$533$	4.00000i	0.173259i
$534$	0	0
$535$	8.00000	0.345870
$536$	0	0
$537$	−11.0000	−0.474685
$538$	0	0
$539$	9.00000i	0.387657i
$540$	0	0
$541$	− 10.0000i	− 0.429934i	−0.976621	−	0.214967i	$-0.931036\pi$
$541$	− 10.0000i	− 0.429934i	0.976621	−	0.214967i	$-0.0689643\pi$
$542$	0	0
$543$	−11.0000	−0.472055
$544$	0	0
$545$	10.0000	0.428353
$546$	0	0
$547$	− 20.0000i	− 0.855138i	−0.903983	−	0.427569i	$-0.859370\pi$
$547$	− 20.0000i	− 0.855138i	0.903983	−	0.427569i	$-0.140630\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−8.00000	−0.340811
$552$	0	0
$553$	40.0000	1.70097
$554$	0	0
$555$	− 9.00000i	− 0.382029i
$556$	0	0
$557$	− 4.00000i	− 0.169485i	−0.996403	−	0.0847427i	$-0.972993\pi$
$557$	− 4.00000i	− 0.169485i	0.996403	−	0.0847427i	$-0.0270068\pi$
$558$	0	0
$559$	−12.0000	−0.507546
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 24.0000i	− 1.01148i	−0.862686	−	0.505740i	$-0.831220\pi$
$563$	− 24.0000i	− 1.01148i	0.862686	−	0.505740i	$-0.168780\pi$
$564$	0	0
$565$	7.00000i	0.294492i
$566$	0	0
$567$	4.00000	0.167984
$568$	0	0
$569$	−4.00000	−0.167689	−0.0838444	−	0.996479i	$-0.526720\pi$
$569$	−4.00000	−0.167689	−0.0838444	+	0.996479i	$0.526720\pi$
$570$	0	0
$571$	− 4.00000i	− 0.167395i	−0.996491	−	0.0836974i	$-0.973327\pi$
$571$	− 4.00000i	− 0.167395i	0.996491	−	0.0836974i	$-0.0266729\pi$
$572$	0	0
$573$	15.0000i	0.626634i
$574$	0	0
$575$	36.0000	1.50130
$576$	0	0
$577$	−11.0000	−0.457936	−0.228968	−	0.973434i	$-0.573535\pi$
$577$	−11.0000	−0.457936	−0.228968	+	0.973434i	$0.573535\pi$
$578$	0	0
$579$	12.0000i	0.498703i
$580$	0	0
$581$	− 56.0000i	− 2.32327i
$582$	0	0
$583$	−6.00000	−0.248495
$584$	0	0
$585$	−4.00000	−0.165380
$586$	0	0
$587$	4.00000i	0.165098i	0.996587	+	0.0825488i	$0.0263060\pi$
$587$	4.00000i	0.165098i	−0.996587	+	0.0825488i	$0.973694\pi$
$588$	0	0
$589$	− 10.0000i	− 0.412043i
$590$	0	0
$591$	−22.0000	−0.904959
$592$	0	0
$593$	−12.0000	−0.492781	−0.246390	−	0.969171i	$-0.579245\pi$
$593$	−12.0000	−0.492781	−0.246390	+	0.969171i	$0.579245\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	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
$601$	40.0000	1.63163	0.815817	−	0.578310i	$-0.196288\pi$
$601$	40.0000	1.63163	0.815817	+	0.578310i	$0.196288\pi$
$602$	0	0
$603$	26.0000i	1.05880i
$604$	0	0
$605$	1.00000i	0.0406558i
$606$	0	0
$607$	−2.00000	−0.0811775	−0.0405887	−	0.999176i	$-0.512923\pi$
$607$	−2.00000	−0.0811775	−0.0405887	+	0.999176i	$0.512923\pi$
$608$	0	0
$609$	16.0000	0.648353
$610$	0	0
$611$	− 8.00000i	− 0.323645i
$612$	0	0
$613$	− 32.0000i	− 1.29247i	−0.763139	−	0.646234i	$-0.776343\pi$
$613$	− 32.0000i	− 1.29247i	0.763139	−	0.646234i	$-0.223657\pi$
$614$	0	0
$615$	2.00000	0.0806478
$616$	0	0
$617$	26.0000	1.04672	0.523360	−	0.852111i	$-0.324678\pi$
$617$	26.0000	1.04672	0.523360	+	0.852111i	$0.324678\pi$
$618$	0	0
$619$	37.0000i	1.48716i	0.668649	+	0.743578i	$0.266873\pi$
$619$	37.0000i	1.48716i	−0.668649	+	0.743578i	$0.733127\pi$
$620$	0	0
$621$	− 45.0000i	− 1.80579i
$622$	0	0
$623$	52.0000	2.08334
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	2.00000i	0.0798723i
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−19.0000	−0.756378	−0.378189	−	0.925728i	$-0.623453\pi$
$631$	−19.0000	−0.756378	−0.378189	+	0.925728i	$0.623453\pi$
$632$	0	0
$633$	12.0000	0.476957
$634$	0	0
$635$	2.00000i	0.0793676i
$636$	0	0
$637$	− 18.0000i	− 0.713186i
$638$	0	0
$639$	−2.00000	−0.0791188
$640$	0	0
$641$	−5.00000	−0.197488	−0.0987441	−	0.995113i	$-0.531483\pi$
$641$	−5.00000	−0.197488	−0.0987441	+	0.995113i	$0.531483\pi$
$642$	0	0
$643$	41.0000i	1.61688i	0.588577	+	0.808441i	$0.299688\pi$
$643$	41.0000i	1.61688i	−0.588577	+	0.808441i	$0.700312\pi$
$644$	0	0
$645$	6.00000i	0.236250i
$646$	0	0
$647$	7.00000	0.275198	0.137599	−	0.990488i	$-0.456061\pi$
$647$	7.00000	0.275198	0.137599	+	0.990488i	$0.456061\pi$
$648$	0	0
$649$	5.00000	0.196267
$650$	0	0
$651$	20.0000i	0.783862i
$652$	0	0
$653$	39.0000i	1.52619i	0.646288	+	0.763094i	$0.276321\pi$
$653$	39.0000i	1.52619i	−0.646288	+	0.763094i	$0.723679\pi$
$654$	0	0
$655$	−16.0000	−0.625172
$656$	0	0
$657$	−28.0000	−1.09238
$658$	0	0
$659$	30.0000i	1.16863i	0.811525	+	0.584317i	$0.198638\pi$
$659$	30.0000i	1.16863i	−0.811525	+	0.584317i	$0.801362\pi$
$660$	0	0
$661$	− 17.0000i	− 0.661223i	−0.943767	−	0.330612i	$-0.892745\pi$
$661$	− 17.0000i	− 0.661223i	0.943767	−	0.330612i	$-0.107255\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	8.00000	0.310227
$666$	0	0
$667$	36.0000i	1.39393i
$668$	0	0
$669$	25.0000i	0.966556i
$670$	0	0
$671$	0	0
$672$	0	0
$673$	22.0000	0.848038	0.424019	−	0.905653i	$-0.360619\pi$
$673$	22.0000	0.848038	0.424019	+	0.905653i	$0.360619\pi$
$674$	0	0
$675$	− 20.0000i	− 0.769800i
$676$	0	0
$677$	12.0000i	0.461197i	0.973049	+	0.230599i	$0.0740685\pi$
$677$	12.0000i	0.461197i	−0.973049	+	0.230599i	$0.925932\pi$
$678$	0	0
$679$	−76.0000	−2.91661
$680$	0	0
$681$	−24.0000	−0.919682
$682$	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
$685$	13.0000i	0.496704i
$686$	0	0
$687$	9.00000	0.343371
$688$	0	0
$689$	12.0000	0.457164
$690$	0	0
$691$	5.00000i	0.190209i	0.995467	+	0.0951045i	$0.0303185\pi$
$691$	5.00000i	0.190209i	−0.995467	+	0.0951045i	$0.969681\pi$
$692$	0	0
$693$	8.00000i	0.303895i
$694$	0	0
$695$	−16.0000	−0.606915
$696$	0	0
$697$	0	0
$698$	0	0
$699$	− 12.0000i	− 0.453882i
$700$	0	0
$701$	− 20.0000i	− 0.755390i	−0.925930	−	0.377695i	$-0.876717\pi$
$701$	− 20.0000i	− 0.755390i	0.925930	−	0.377695i	$-0.123283\pi$
$702$	0	0
$703$	−18.0000	−0.678883
$704$	0	0
$705$	−4.00000	−0.150649
$706$	0	0
$707$	48.0000i	1.80523i
$708$	0	0
$709$	− 7.00000i	− 0.262891i	−0.991323	−	0.131445i	$-0.958038\pi$
$709$	− 7.00000i	− 0.262891i	0.991323	−	0.131445i	$-0.0419618\pi$
$710$	0	0
$711$	20.0000	0.750059
$712$	0	0
$713$	−45.0000	−1.68526
$714$	0	0
$715$	− 2.00000i	− 0.0747958i
$716$	0	0
$717$	− 18.0000i	− 0.672222i
$718$	0	0
$719$	39.0000	1.45445	0.727227	−	0.686397i	$-0.240809\pi$
$719$	39.0000	1.45445	0.727227	+	0.686397i	$0.240809\pi$
$720$	0	0
$721$	−32.0000	−1.19174
$722$	0	0
$723$	4.00000i	0.148762i
$724$	0	0
$725$	16.0000i	0.594225i
$726$	0	0
$727$	1.00000	0.0370879	0.0185440	−	0.999828i	$-0.494097\pi$
$727$	1.00000	0.0370879	0.0185440	+	0.999828i	$0.494097\pi$
$728$	0	0
$729$	−13.0000	−0.481481
$730$	0	0
$731$	0	0
$732$	0	0
$733$	− 24.0000i	− 0.886460i	−0.896408	−	0.443230i	$-0.853832\pi$
$733$	− 24.0000i	− 0.886460i	0.896408	−	0.443230i	$-0.146168\pi$
$734$	0	0
$735$	−9.00000	−0.331970
$736$	0	0
$737$	−13.0000	−0.478861
$738$	0	0
$739$	2.00000i	0.0735712i	0.999323	+	0.0367856i	$0.0117119\pi$
$739$	2.00000i	0.0735712i	−0.999323	+	0.0367856i	$0.988288\pi$
$740$	0	0
$741$	− 4.00000i	− 0.146944i
$742$	0	0
$743$	−44.0000	−1.61420	−0.807102	−	0.590412i	$-0.798965\pi$
$743$	−44.0000	−1.61420	−0.807102	+	0.590412i	$0.798965\pi$
$744$	0	0
$745$	−12.0000	−0.439646
$746$	0	0
$747$	− 28.0000i	− 1.02447i
$748$	0	0
$749$	32.0000i	1.16925i
$750$	0	0
$751$	1.00000	0.0364905	0.0182453	−	0.999834i	$-0.494192\pi$
$751$	1.00000	0.0364905	0.0182453	+	0.999834i	$0.494192\pi$
$752$	0	0
$753$	−17.0000	−0.619514
$754$	0	0
$755$	− 14.0000i	− 0.509512i
$756$	0	0
$757$	10.0000i	0.363456i	0.983349	+	0.181728i	$0.0581691\pi$
$757$	10.0000i	0.363456i	−0.983349	+	0.181728i	$0.941831\pi$
$758$	0	0
$759$	9.00000	0.326679
$760$	0	0
$761$	−18.0000	−0.652499	−0.326250	−	0.945284i	$-0.605785\pi$
$761$	−18.0000	−0.652499	−0.326250	+	0.945284i	$0.605785\pi$
$762$	0	0
$763$	40.0000i	1.44810i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−10.0000	−0.361079
$768$	0	0
$769$	−50.0000	−1.80305	−0.901523	−	0.432731i	$-0.857550\pi$
$769$	−50.0000	−1.80305	−0.901523	+	0.432731i	$0.857550\pi$
$770$	0	0
$771$	− 6.00000i	− 0.216085i
$772$	0	0
$773$	14.0000i	0.503545i	0.967786	+	0.251773i	$0.0810135\pi$
$773$	14.0000i	0.503545i	−0.967786	+	0.251773i	$0.918987\pi$
$774$	0	0
$775$	−20.0000	−0.718421
$776$	0	0
$777$	36.0000	1.29149
$778$	0	0
$779$	− 4.00000i	− 0.143315i
$780$	0	0
$781$	− 1.00000i	− 0.0357828i
$782$	0	0
$783$	20.0000	0.714742
$784$	0	0
$785$	13.0000	0.463990
$786$	0	0
$787$	4.00000i	0.142585i	0.997455	+	0.0712923i	$0.0227123\pi$
$787$	4.00000i	0.142585i	−0.997455	+	0.0712923i	$0.977288\pi$
$788$	0	0
$789$	4.00000i	0.142404i
$790$	0	0
$791$	−28.0000	−0.995565
$792$	0	0
$793$	0	0
$794$	0	0
$795$	− 6.00000i	− 0.212798i
$796$	0	0
$797$	− 23.0000i	− 0.814702i	−0.913272	−	0.407351i	$-0.866453\pi$
$797$	− 23.0000i	− 0.814702i	0.913272	−	0.407351i	$-0.133547\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	26.0000	0.918665
$802$	0	0
$803$	− 14.0000i	− 0.494049i
$804$	0	0
$805$	− 36.0000i	− 1.26883i
$806$	0	0
$807$	14.0000	0.492823
$808$	0	0
$809$	−18.0000	−0.632846	−0.316423	−	0.948618i	$-0.602482\pi$
$809$	−18.0000	−0.632846	−0.316423	+	0.948618i	$0.602482\pi$
$810$	0	0
$811$	− 46.0000i	− 1.61528i	−0.589677	−	0.807639i	$-0.700745\pi$
$811$	− 46.0000i	− 1.61528i	0.589677	−	0.807639i	$-0.299255\pi$
$812$	0	0
$813$	10.0000i	0.350715i
$814$	0	0
$815$	−20.0000	−0.700569
$816$	0	0
$817$	12.0000	0.419827
$818$	0	0
$819$	− 16.0000i	− 0.559085i
$820$	0	0
$821$	42.0000i	1.46581i	0.680331	+	0.732905i	$0.261836\pi$
$821$	42.0000i	1.46581i	−0.680331	+	0.732905i	$0.738164\pi$
$822$	0	0
$823$	−27.0000	−0.941161	−0.470580	−	0.882357i	$-0.655955\pi$
$823$	−27.0000	−0.941161	−0.470580	+	0.882357i	$0.655955\pi$
$824$	0	0
$825$	4.00000	0.139262
$826$	0	0
$827$	− 34.0000i	− 1.18230i	−0.806563	−	0.591148i	$-0.798675\pi$
$827$	− 34.0000i	− 1.18230i	0.806563	−	0.591148i	$-0.201325\pi$
$828$	0	0
$829$	− 11.0000i	− 0.382046i	−0.981586	−	0.191023i	$-0.938820\pi$
$829$	− 11.0000i	− 0.382046i	0.981586	−	0.191023i	$-0.0611805\pi$
$830$	0	0
$831$	14.0000	0.485655
$832$	0	0
$833$	0	0
$834$	0	0
$835$	− 18.0000i	− 0.622916i
$836$	0	0
$837$	25.0000i	0.864126i
$838$	0	0
$839$	21.0000	0.725001	0.362500	−	0.931984i	$-0.381923\pi$
$839$	21.0000	0.725001	0.362500	+	0.931984i	$0.381923\pi$
$840$	0	0
$841$	13.0000	0.448276
$842$	0	0
$843$	18.0000i	0.619953i
$844$	0	0
$845$	− 9.00000i	− 0.309609i
$846$	0	0
$847$	−4.00000	−0.137442
$848$	0	0
$849$	8.00000	0.274559
$850$	0	0
$851$	81.0000i	2.77664i
$852$	0	0
$853$	− 46.0000i	− 1.57501i	−0.616308	−	0.787505i	$-0.711372\pi$
$853$	− 46.0000i	− 1.57501i	0.616308	−	0.787505i	$-0.288628\pi$
$854$	0	0
$855$	4.00000	0.136797
$856$	0	0
$857$	−16.0000	−0.546550	−0.273275	−	0.961936i	$-0.588107\pi$
$857$	−16.0000	−0.546550	−0.273275	+	0.961936i	$0.588107\pi$
$858$	0	0
$859$	23.0000i	0.784750i	0.919805	+	0.392375i	$0.128346\pi$
$859$	23.0000i	0.784750i	−0.919805	+	0.392375i	$0.871654\pi$
$860$	0	0
$861$	8.00000i	0.272639i
$862$	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
$865$	−22.0000	−0.748022
$866$	0	0
$867$	17.0000i	0.577350i
$868$	0	0
$869$	10.0000i	0.339227i
$870$	0	0
$871$	26.0000	0.880976
$872$	0	0
$873$	−38.0000	−1.28611
$874$	0	0
$875$	− 36.0000i	− 1.21702i
$876$	0	0
$877$	42.0000i	1.41824i	0.705088	+	0.709120i	$0.250907\pi$
$877$	42.0000i	1.41824i	−0.705088	+	0.709120i	$0.749093\pi$
$878$	0	0
$879$	10.0000	0.337292
$880$	0	0
$881$	21.0000	0.707508	0.353754	−	0.935339i	$-0.384905\pi$
$881$	21.0000	0.707508	0.353754	+	0.935339i	$0.384905\pi$
$882$	0	0
$883$	− 4.00000i	− 0.134611i	−0.997732	−	0.0673054i	$-0.978560\pi$
$883$	− 4.00000i	− 0.134611i	0.997732	−	0.0673054i	$-0.0214402\pi$
$884$	0	0
$885$	5.00000i	0.168073i
$886$	0	0
$887$	42.0000	1.41022	0.705111	−	0.709097i	$-0.250897\pi$
$887$	42.0000	1.41022	0.705111	+	0.709097i	$0.250897\pi$
$888$	0	0
$889$	−8.00000	−0.268311
$890$	0	0
$891$	1.00000i	0.0335013i
$892$	0	0
$893$	8.00000i	0.267710i
$894$	0	0
$895$	−11.0000	−0.367689
$896$	0	0
$897$	−18.0000	−0.601003
$898$	0	0
$899$	− 20.0000i	− 0.667037i
$900$	0	0
$901$	0	0
$902$	0	0
$903$	−24.0000	−0.798670
$904$	0	0
$905$	−11.0000	−0.365652
$906$	0	0
$907$	12.0000i	0.398453i	0.979953	+	0.199227i	$0.0638430\pi$
$907$	12.0000i	0.398453i	−0.979953	+	0.199227i	$0.936157\pi$
$908$	0	0
$909$	24.0000i	0.796030i
$910$	0	0
$911$	24.0000	0.795155	0.397578	−	0.917568i	$-0.369851\pi$
$911$	24.0000	0.795155	0.397578	+	0.917568i	$0.369851\pi$
$912$	0	0
$913$	14.0000	0.463332
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 64.0000i	− 2.11347i
$918$	0	0
$919$	34.0000	1.12156	0.560778	−	0.827966i	$-0.310502\pi$
$919$	34.0000	1.12156	0.560778	+	0.827966i	$0.310502\pi$
$920$	0	0
$921$	34.0000	1.12034
$922$	0	0
$923$	2.00000i	0.0658308i
$924$	0	0
$925$	36.0000i	1.18367i
$926$	0	0
$927$	−16.0000	−0.525509
$928$	0	0
$929$	−46.0000	−1.50921	−0.754606	−	0.656179i	$-0.772172\pi$
$929$	−46.0000	−1.50921	−0.754606	+	0.656179i	$0.772172\pi$
$930$	0	0
$931$	18.0000i	0.589926i
$932$	0	0
$933$	− 16.0000i	− 0.523816i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−50.0000	−1.63343	−0.816714	−	0.577042i	$-0.804207\pi$
$937$	−50.0000	−1.63343	−0.816714	+	0.577042i	$0.804207\pi$
$938$	0	0
$939$	7.00000i	0.228436i
$940$	0	0
$941$	− 18.0000i	− 0.586783i	−0.955992	−	0.293392i	$-0.905216\pi$
$941$	− 18.0000i	− 0.586783i	0.955992	−	0.293392i	$-0.0947840\pi$
$942$	0	0
$943$	−18.0000	−0.586161
$944$	0	0
$945$	−20.0000	−0.650600
$946$	0	0
$947$	− 27.0000i	− 0.877382i	−0.898638	−	0.438691i	$-0.855442\pi$
$947$	− 27.0000i	− 0.877382i	0.898638	−	0.438691i	$-0.144558\pi$
$948$	0	0
$949$	28.0000i	0.908918i
$950$	0	0
$951$	9.00000	0.291845
$952$	0	0
$953$	36.0000	1.16615	0.583077	−	0.812417i	$-0.301849\pi$
$953$	36.0000	1.16615	0.583077	+	0.812417i	$0.301849\pi$
$954$	0	0
$955$	15.0000i	0.485389i
$956$	0	0
$957$	4.00000i	0.129302i
$958$	0	0
$959$	−52.0000	−1.67917
$960$	0	0
$961$	−6.00000	−0.193548
$962$	0	0
$963$	16.0000i	0.515593i
$964$	0	0
$965$	12.0000i	0.386294i
$966$	0	0
$967$	−4.00000	−0.128631	−0.0643157	−	0.997930i	$-0.520486\pi$
$967$	−4.00000	−0.128631	−0.0643157	+	0.997930i	$0.520486\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	13.0000i	0.417190i	0.978002	+	0.208595i	$0.0668890\pi$
$971$	13.0000i	0.417190i	−0.978002	+	0.208595i	$0.933111\pi$
$972$	0	0
$973$	− 64.0000i	− 2.05175i
$974$	0	0
$975$	−8.00000	−0.256205
$976$	0	0
$977$	17.0000	0.543878	0.271939	−	0.962314i	$-0.412335\pi$
$977$	17.0000	0.543878	0.271939	+	0.962314i	$0.412335\pi$
$978$	0	0
$979$	13.0000i	0.415482i
$980$	0	0
$981$	20.0000i	0.638551i
$982$	0	0
$983$	49.0000	1.56286	0.781429	−	0.623995i	$-0.214491\pi$
$983$	49.0000	1.56286	0.781429	+	0.623995i	$0.214491\pi$
$984$	0	0
$985$	−22.0000	−0.700978
$986$	0	0
$987$	− 16.0000i	− 0.509286i
$988$	0	0
$989$	− 54.0000i	− 1.71710i
$990$	0	0
$991$	−32.0000	−1.01651	−0.508257	−	0.861206i	$-0.669710\pi$
$991$	−32.0000	−1.01651	−0.508257	+	0.861206i	$0.669710\pi$
$992$	0	0
$993$	13.0000	0.412543
$994$	0	0
$995$	0	0
$996$	0	0
$997$	28.0000i	0.886769i	0.896332	+	0.443384i	$0.146222\pi$
$997$	28.0000i	0.886769i	−0.896332	+	0.443384i	$0.853778\pi$
$998$	0	0
$999$	45.0000	1.42374

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2816.2.c.m.1409.1		2
4.3	odd	2		2816.2.c.a.1409.2		2
8.3	odd	2		2816.2.c.a.1409.1		2
8.5	even	2	inner	2816.2.c.m.1409.2		2
16.3	odd	4		704.2.a.d.1.1		1
16.5	even	4		352.2.a.c.1.1	✓	1
16.11	odd	4		352.2.a.e.1.1	yes	1
16.13	even	4		704.2.a.g.1.1		1
48.5	odd	4		3168.2.a.g.1.1		1
48.11	even	4		3168.2.a.j.1.1		1
48.29	odd	4		6336.2.a.bq.1.1		1
48.35	even	4		6336.2.a.bv.1.1		1
80.59	odd	4		8800.2.a.i.1.1		1
80.69	even	4		8800.2.a.t.1.1		1
176.21	odd	4		3872.2.a.e.1.1		1
176.43	even	4		3872.2.a.j.1.1		1
176.109	odd	4		7744.2.a.y.1.1		1
176.131	even	4		7744.2.a.i.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
352.2.a.c.1.1	✓	1	16.5	even	4
352.2.a.e.1.1	yes	1	16.11	odd	4
704.2.a.d.1.1		1	16.3	odd	4
704.2.a.g.1.1		1	16.13	even	4
2816.2.c.a.1409.1		2	8.3	odd	2
2816.2.c.a.1409.2		2	4.3	odd	2
2816.2.c.m.1409.1		2	1.1	even	1	trivial
2816.2.c.m.1409.2		2	8.5	even	2	inner
3168.2.a.g.1.1		1	48.5	odd	4
3168.2.a.j.1.1		1	48.11	even	4
3872.2.a.e.1.1		1	176.21	odd	4
3872.2.a.j.1.1		1	176.43	even	4
6336.2.a.bq.1.1		1	48.29	odd	4
6336.2.a.bv.1.1		1	48.35	even	4
7744.2.a.i.1.1		1	176.131	even	4
7744.2.a.y.1.1		1	176.109	odd	4
8800.2.a.i.1.1		1	80.59	odd	4
8800.2.a.t.1.1		1	80.69	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:	\( N \)	\(=\)	\( 2816 = 2^{8} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2816.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} -1.00000i q^{5} +4.00000 q^{7} +2.00000 q^{9} +1.00000i q^{11} -2.00000i q^{13} -1.00000 q^{15} +2.00000i q^{19} -4.00000i q^{21} +9.00000 q^{23} +4.00000 q^{25} -5.00000i q^{27} +4.00000i q^{29} -5.00000 q^{31} +1.00000 q^{33} -4.00000i q^{35} +9.00000i q^{37} -2.00000 q^{39} -2.00000 q^{41} -6.00000i q^{43} -2.00000i q^{45} +4.00000 q^{47} +9.00000 q^{49} +6.00000i q^{53} +1.00000 q^{55} +2.00000 q^{57} -5.00000i q^{59} +8.00000 q^{63} -2.00000 q^{65} +13.0000i q^{67} -9.00000i q^{69} -1.00000 q^{71} -14.0000 q^{73} -4.00000i q^{75} +4.00000i q^{77} +10.0000 q^{79} +1.00000 q^{81} -14.0000i q^{83} +4.00000 q^{87} +13.0000 q^{89} -8.00000i q^{91} +5.00000i q^{93} +2.00000 q^{95} -19.0000 q^{97} +2.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 8 q^{7} + 4 q^{9} - 2 q^{15} + 18 q^{23} + 8 q^{25} - 10 q^{31} + 2 q^{33} - 4 q^{39} - 4 q^{41} + 8 q^{47} + 18 q^{49} + 2 q^{55} + 4 q^{57} + 16 q^{63} - 4 q^{65} - 2 q^{71} - 28 q^{73} + 20 q^{79}+ \cdots - 38 q^{97}+O(q^{100}) \) 2 * q + 8 * q^7 + 4 * q^9 - 2 * q^15 + 18 * q^23 + 8 * q^25 - 10 * q^31 + 2 * q^33 - 4 * q^39 - 4 * q^41 + 8 * q^47 + 18 * q^49 + 2 * q^55 + 4 * q^57 + 16 * q^63 - 4 * q^65 - 2 * q^71 - 28 * q^73 + 20 * q^79 + 2 * q^81 + 8 * q^87 + 26 * q^89 + 4 * q^95 - 38 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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