LMFDB - Embedded newform 5376.2.c.bd.2689.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5376,2,Mod(2689,5376)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5376, 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("5376.2689");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5376 = 2^{8} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5376.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:	$42.9275761266$
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 168)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2689.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	5376.2689
Dual form			5376.2.c.bd.2689.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} -2.00000i q^{5} +1.00000 q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} -2.00000i q^{5} +1.00000 q^{7} -1.00000 q^{9} -2.00000i q^{13} +2.00000 q^{15} +6.00000 q^{17} -4.00000i q^{19} +1.00000i q^{21} +4.00000 q^{23} +1.00000 q^{25} -1.00000i q^{27} +6.00000i q^{29} -8.00000 q^{31} -2.00000i q^{35} +10.0000i q^{37} +2.00000 q^{39} +10.0000 q^{41} -12.0000i q^{43} +2.00000i q^{45} -8.00000 q^{47} +1.00000 q^{49} +6.00000i q^{51} -6.00000i q^{53} +4.00000 q^{57} -4.00000i q^{59} -10.0000i q^{61} -1.00000 q^{63} -4.00000 q^{65} +12.0000i q^{67} +4.00000i q^{69} -4.00000 q^{71} -2.00000 q^{73} +1.00000i q^{75} +8.00000 q^{79} +1.00000 q^{81} +4.00000i q^{83} -12.0000i q^{85} -6.00000 q^{87} -6.00000 q^{89} -2.00000i q^{91} -8.00000i q^{93} -8.00000 q^{95} +10.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{7} - 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^7 - 2 * q^9	$ 2 q + 2 q^{7} - 2 q^{9} + 4 q^{15} + 12 q^{17} + 8 q^{23} + 2 q^{25} - 16 q^{31} + 4 q^{39} + 20 q^{41} - 16 q^{47} + 2 q^{49} + 8 q^{57} - 2 q^{63} - 8 q^{65} - 8 q^{71} - 4 q^{73} + 16 q^{79} + 2 q^{81} - 12 q^{87} - 12 q^{89} - 16 q^{95} + 20 q^{97}+O(q^{100}) $ 2 * q + 2 * q^7 - 2 * q^9 + 4 * q^15 + 12 * q^17 + 8 * q^23 + 2 * q^25 - 16 * q^31 + 4 * q^39 + 20 * q^41 - 16 * q^47 + 2 * q^49 + 8 * q^57 - 2 * q^63 - 8 * q^65 - 8 * q^71 - 4 * q^73 + 16 * q^79 + 2 * q^81 - 12 * q^87 - 12 * q^89 - 16 * q^95 + 20 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5376\mathbb{Z}\right)^\times$.

$n$	$1793$	$2815$	$4609$	$5125$
$\chi(n)$	$1$	$1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	0	0
$5$	− 2.00000i	− 0.894427i	−0.894427	−	0.447214i	$-0.852416\pi$
$5$	− 2.00000i	− 0.894427i	0.894427	−	0.447214i	$-0.147584\pi$
$6$	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
$13$	− 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
$15$	2.00000	0.516398
$16$	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
$19$	− 4.00000i	− 0.917663i	−0.888523	−	0.458831i	$-0.848268\pi$
$19$	− 4.00000i	− 0.917663i	0.888523	−	0.458831i	$-0.151732\pi$
$20$	0	0
$21$	1.00000i	0.218218i
$22$	0	0
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	6.00000i	1.11417i	0.830455	+	0.557086i	$0.188081\pi$
$29$	6.00000i	1.11417i	−0.830455	+	0.557086i	$0.811919\pi$
$30$	0	0
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 2.00000i	− 0.338062i
$36$	0	0
$37$	10.0000i	1.64399i	0.569495	+	0.821995i	$0.307139\pi$
$37$	10.0000i	1.64399i	−0.569495	+	0.821995i	$0.692861\pi$
$38$	0	0
$39$	2.00000	0.320256
$40$	0	0
$41$	10.0000	1.56174	0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000	1.56174	0.780869	+	0.624695i	$0.214777\pi$
$42$	0	0
$43$	− 12.0000i	− 1.82998i	−0.403473	−	0.914991i	$-0.632197\pi$
$43$	− 12.0000i	− 1.82998i	0.403473	−	0.914991i	$-0.367803\pi$
$44$	0	0
$45$	2.00000i	0.298142i
$46$	0	0
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	6.00000i	0.840168i
$52$	0	0
$53$	− 6.00000i	− 0.824163i	−0.911147	−	0.412082i	$-0.864802\pi$
$53$	− 6.00000i	− 0.824163i	0.911147	−	0.412082i	$-0.135198\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	4.00000	0.529813
$58$	0	0
$59$	− 4.00000i	− 0.520756i	−0.965507	−	0.260378i	$-0.916153\pi$
$59$	− 4.00000i	− 0.520756i	0.965507	−	0.260378i	$-0.0838471\pi$
$60$	0	0
$61$	− 10.0000i	− 1.28037i	−0.768221	−	0.640184i	$-0.778858\pi$
$61$	− 10.0000i	− 1.28037i	0.768221	−	0.640184i	$-0.221142\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	−4.00000	−0.496139
$66$	0	0
$67$	12.0000i	1.46603i	0.680211	+	0.733017i	$0.261888\pi$
$67$	12.0000i	1.46603i	−0.680211	+	0.733017i	$0.738112\pi$
$68$	0	0
$69$	4.00000i	0.481543i
$70$	0	0
$71$	−4.00000	−0.474713	−0.237356	−	0.971423i	$-0.576281\pi$
$71$	−4.00000	−0.474713	−0.237356	+	0.971423i	$0.576281\pi$
$72$	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
$75$	1.00000i	0.115470i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	4.00000i	0.439057i	0.975606	+	0.219529i	$0.0704519\pi$
$83$	4.00000i	0.439057i	−0.975606	+	0.219529i	$0.929548\pi$
$84$	0	0
$85$	− 12.0000i	− 1.30158i
$86$	0	0
$87$	−6.00000	−0.643268
$88$	0	0
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	− 2.00000i	− 0.209657i
$92$	0	0
$93$	− 8.00000i	− 0.829561i
$94$	0	0
$95$	−8.00000	−0.820783
$96$	0	0
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 10.0000i	− 0.995037i	−0.867453	−	0.497519i	$-0.834245\pi$
$101$	− 10.0000i	− 0.995037i	0.867453	−	0.497519i	$-0.165755\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	2.00000	0.195180
$106$	0	0
$107$	− 16.0000i	− 1.54678i	−0.633932	−	0.773389i	$-0.718560\pi$
$107$	− 16.0000i	− 1.54678i	0.633932	−	0.773389i	$-0.281440\pi$
$108$	0	0
$109$	14.0000i	1.34096i	0.741929	+	0.670478i	$0.233911\pi$
$109$	14.0000i	1.34096i	−0.741929	+	0.670478i	$0.766089\pi$
$110$	0	0
$111$	−10.0000	−0.949158
$112$	0	0
$113$	−14.0000	−1.31701	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000	−1.31701	−0.658505	+	0.752577i	$0.728811\pi$
$114$	0	0
$115$	− 8.00000i	− 0.746004i
$116$	0	0
$117$	2.00000i	0.184900i
$118$	0	0
$119$	6.00000	0.550019
$120$	0	0
$121$	11.0000	1.00000
$122$	0	0
$123$	10.0000i	0.901670i
$124$	0	0
$125$	− 12.0000i	− 1.07331i
$126$	0	0
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	0	0
$129$	12.0000	1.05654
$130$	0	0
$131$	− 12.0000i	− 1.04844i	−0.851581	−	0.524222i	$-0.824356\pi$
$131$	− 12.0000i	− 1.04844i	0.851581	−	0.524222i	$-0.175644\pi$
$132$	0	0
$133$	− 4.00000i	− 0.346844i
$134$	0	0
$135$	−2.00000	−0.172133
$136$	0	0
$137$	14.0000	1.19610	0.598050	−	0.801459i	$-0.295942\pi$
$137$	14.0000	1.19610	0.598050	+	0.801459i	$0.295942\pi$
$138$	0	0
$139$	4.00000i	0.339276i	0.985506	+	0.169638i	$0.0542598\pi$
$139$	4.00000i	0.339276i	−0.985506	+	0.169638i	$0.945740\pi$
$140$	0	0
$141$	− 8.00000i	− 0.673722i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	12.0000	0.996546
$146$	0	0
$147$	1.00000i	0.0824786i
$148$	0	0
$149$	− 6.00000i	− 0.491539i	−0.969328	−	0.245770i	$-0.920959\pi$
$149$	− 6.00000i	− 0.491539i	0.969328	−	0.245770i	$-0.0790407\pi$
$150$	0	0
$151$	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
$155$	16.0000i	1.28515i
$156$	0	0
$157$	− 10.0000i	− 0.798087i	−0.916932	−	0.399043i	$-0.869342\pi$
$157$	− 10.0000i	− 0.798087i	0.916932	−	0.399043i	$-0.130658\pi$
$158$	0	0
$159$	6.00000	0.475831
$160$	0	0
$161$	4.00000	0.315244
$162$	0	0
$163$	− 4.00000i	− 0.313304i	−0.987654	−	0.156652i	$-0.949930\pi$
$163$	− 4.00000i	− 0.313304i	0.987654	−	0.156652i	$-0.0500701\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−16.0000	−1.23812	−0.619059	−	0.785345i	$-0.712486\pi$
$167$	−16.0000	−1.23812	−0.619059	+	0.785345i	$0.712486\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	4.00000i	0.305888i
$172$	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
$177$	4.00000	0.300658
$178$	0	0
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	0	0
$181$	− 6.00000i	− 0.445976i	−0.974821	−	0.222988i	$-0.928419\pi$
$181$	− 6.00000i	− 0.445976i	0.974821	−	0.222988i	$-0.0715812\pi$
$182$	0	0
$183$	10.0000	0.739221
$184$	0	0
$185$	20.0000	1.47043
$186$	0	0
$187$	0	0
$188$	0	0
$189$	− 1.00000i	− 0.0727393i
$190$	0	0
$191$	−4.00000	−0.289430	−0.144715	−	0.989473i	$-0.546227\pi$
$191$	−4.00000	−0.289430	−0.144715	+	0.989473i	$0.546227\pi$
$192$	0	0
$193$	18.0000	1.29567	0.647834	−	0.761781i	$-0.275675\pi$
$193$	18.0000	1.29567	0.647834	+	0.761781i	$0.275675\pi$
$194$	0	0
$195$	− 4.00000i	− 0.286446i
$196$	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
$199$	24.0000	1.70131	0.850657	−	0.525720i	$-0.176204\pi$
$199$	24.0000	1.70131	0.850657	+	0.525720i	$0.176204\pi$
$200$	0	0
$201$	−12.0000	−0.846415
$202$	0	0
$203$	6.00000i	0.421117i
$204$	0	0
$205$	− 20.0000i	− 1.39686i
$206$	0	0
$207$	−4.00000	−0.278019
$208$	0	0
$209$	0	0
$210$	0	0
$211$	− 12.0000i	− 0.826114i	−0.910705	−	0.413057i	$-0.864461\pi$
$211$	− 12.0000i	− 0.826114i	0.910705	−	0.413057i	$-0.135539\pi$
$212$	0	0
$213$	− 4.00000i	− 0.274075i
$214$	0	0
$215$	−24.0000	−1.63679
$216$	0	0
$217$	−8.00000	−0.543075
$218$	0	0
$219$	− 2.00000i	− 0.135147i
$220$	0	0
$221$	− 12.0000i	− 0.807207i
$222$	0	0
$223$	16.0000	1.07144	0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000	1.07144	0.535720	+	0.844396i	$0.320040\pi$
$224$	0	0
$225$	−1.00000	−0.0666667
$226$	0	0
$227$	12.0000i	0.796468i	0.917284	+	0.398234i	$0.130377\pi$
$227$	12.0000i	0.796468i	−0.917284	+	0.398234i	$0.869623\pi$
$228$	0	0
$229$	− 6.00000i	− 0.396491i	−0.980152	−	0.198246i	$-0.936476\pi$
$229$	− 6.00000i	− 0.396491i	0.980152	−	0.198246i	$-0.0635244\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−18.0000	−1.17922	−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000	−1.17922	−0.589610	+	0.807688i	$0.700718\pi$
$234$	0	0
$235$	16.0000i	1.04372i
$236$	0	0
$237$	8.00000i	0.519656i
$238$	0	0
$239$	−12.0000	−0.776215	−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000	−0.776215	−0.388108	+	0.921614i	$0.626871\pi$
$240$	0	0
$241$	−22.0000	−1.41714	−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000	−1.41714	−0.708572	+	0.705638i	$0.750660\pi$
$242$	0	0
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	− 2.00000i	− 0.127775i
$246$	0	0
$247$	−8.00000	−0.509028
$248$	0	0
$249$	−4.00000	−0.253490
$250$	0	0
$251$	− 20.0000i	− 1.26239i	−0.775625	−	0.631194i	$-0.782565\pi$
$251$	− 20.0000i	− 1.26239i	0.775625	−	0.631194i	$-0.217435\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	12.0000	0.751469
$256$	0	0
$257$	14.0000	0.873296	0.436648	−	0.899632i	$-0.356166\pi$
$257$	14.0000	0.873296	0.436648	+	0.899632i	$0.356166\pi$
$258$	0	0
$259$	10.0000i	0.621370i
$260$	0	0
$261$	− 6.00000i	− 0.371391i
$262$	0	0
$263$	−28.0000	−1.72655	−0.863277	−	0.504730i	$-0.831592\pi$
$263$	−28.0000	−1.72655	−0.863277	+	0.504730i	$0.831592\pi$
$264$	0	0
$265$	−12.0000	−0.737154
$266$	0	0
$267$	− 6.00000i	− 0.367194i
$268$	0	0
$269$	10.0000i	0.609711i	0.952399	+	0.304855i	$0.0986081\pi$
$269$	10.0000i	0.609711i	−0.952399	+	0.304855i	$0.901392\pi$
$270$	0	0
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	0	0
$273$	2.00000	0.121046
$274$	0	0
$275$	0	0
$276$	0	0
$277$	10.0000i	0.600842i	0.953807	+	0.300421i	$0.0971271\pi$
$277$	10.0000i	0.600842i	−0.953807	+	0.300421i	$0.902873\pi$
$278$	0	0
$279$	8.00000	0.478947
$280$	0	0
$281$	30.0000	1.78965	0.894825	−	0.446417i	$-0.147300\pi$
$281$	30.0000	1.78965	0.894825	+	0.446417i	$0.147300\pi$
$282$	0	0
$283$	12.0000i	0.713326i	0.934233	+	0.356663i	$0.116086\pi$
$283$	12.0000i	0.713326i	−0.934233	+	0.356663i	$0.883914\pi$
$284$	0	0
$285$	− 8.00000i	− 0.473879i
$286$	0	0
$287$	10.0000	0.590281
$288$	0	0
$289$	19.0000	1.11765
$290$	0	0
$291$	10.0000i	0.586210i
$292$	0	0
$293$	22.0000i	1.28525i	0.766179	+	0.642627i	$0.222155\pi$
$293$	22.0000i	1.28525i	−0.766179	+	0.642627i	$0.777845\pi$
$294$	0	0
$295$	−8.00000	−0.465778
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 8.00000i	− 0.462652i
$300$	0	0
$301$	− 12.0000i	− 0.691669i
$302$	0	0
$303$	10.0000	0.574485
$304$	0	0
$305$	−20.0000	−1.14520
$306$	0	0
$307$	− 4.00000i	− 0.228292i	−0.993464	−	0.114146i	$-0.963587\pi$
$307$	− 4.00000i	− 0.228292i	0.993464	−	0.114146i	$-0.0364132\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	6.00000	0.339140	0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000	0.339140	0.169570	+	0.985518i	$0.445762\pi$
$314$	0	0
$315$	2.00000i	0.112687i
$316$	0	0
$317$	− 2.00000i	− 0.112331i	−0.998421	−	0.0561656i	$-0.982113\pi$
$317$	− 2.00000i	− 0.112331i	0.998421	−	0.0561656i	$-0.0178875\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	16.0000	0.893033
$322$	0	0
$323$	− 24.0000i	− 1.33540i
$324$	0	0
$325$	− 2.00000i	− 0.110940i
$326$	0	0
$327$	−14.0000	−0.774202
$328$	0	0
$329$	−8.00000	−0.441054
$330$	0	0
$331$	20.0000i	1.09930i	0.835395	+	0.549650i	$0.185239\pi$
$331$	20.0000i	1.09930i	−0.835395	+	0.549650i	$0.814761\pi$
$332$	0	0
$333$	− 10.0000i	− 0.547997i
$334$	0	0
$335$	24.0000	1.31126
$336$	0	0
$337$	−30.0000	−1.63420	−0.817102	−	0.576493i	$-0.804421\pi$
$337$	−30.0000	−1.63420	−0.817102	+	0.576493i	$0.804421\pi$
$338$	0	0
$339$	− 14.0000i	− 0.760376i
$340$	0	0
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	8.00000	0.430706
$346$	0	0
$347$	8.00000i	0.429463i	0.976673	+	0.214731i	$0.0688876\pi$
$347$	8.00000i	0.429463i	−0.976673	+	0.214731i	$0.931112\pi$
$348$	0	0
$349$	− 10.0000i	− 0.535288i	−0.963518	−	0.267644i	$-0.913755\pi$
$349$	− 10.0000i	− 0.535288i	0.963518	−	0.267644i	$-0.0862451\pi$
$350$	0	0
$351$	−2.00000	−0.106752
$352$	0	0
$353$	−10.0000	−0.532246	−0.266123	−	0.963939i	$-0.585743\pi$
$353$	−10.0000	−0.532246	−0.266123	+	0.963939i	$0.585743\pi$
$354$	0	0
$355$	8.00000i	0.424596i
$356$	0	0
$357$	6.00000i	0.317554i
$358$	0	0
$359$	12.0000	0.633336	0.316668	−	0.948536i	$-0.397436\pi$
$359$	12.0000	0.633336	0.316668	+	0.948536i	$0.397436\pi$
$360$	0	0
$361$	3.00000	0.157895
$362$	0	0
$363$	11.0000i	0.577350i
$364$	0	0
$365$	4.00000i	0.209370i
$366$	0	0
$367$	32.0000	1.67039	0.835193	−	0.549957i	$-0.185356\pi$
$367$	32.0000	1.67039	0.835193	+	0.549957i	$0.185356\pi$
$368$	0	0
$369$	−10.0000	−0.520579
$370$	0	0
$371$	− 6.00000i	− 0.311504i
$372$	0	0
$373$	− 22.0000i	− 1.13912i	−0.821951	−	0.569558i	$-0.807114\pi$
$373$	− 22.0000i	− 1.13912i	0.821951	−	0.569558i	$-0.192886\pi$
$374$	0	0
$375$	12.0000	0.619677
$376$	0	0
$377$	12.0000	0.618031
$378$	0	0
$379$	4.00000i	0.205466i	0.994709	+	0.102733i	$0.0327588\pi$
$379$	4.00000i	0.205466i	−0.994709	+	0.102733i	$0.967241\pi$
$380$	0	0
$381$	8.00000i	0.409852i
$382$	0	0
$383$	−16.0000	−0.817562	−0.408781	−	0.912633i	$-0.634046\pi$
$383$	−16.0000	−0.817562	−0.408781	+	0.912633i	$0.634046\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	12.0000i	0.609994i
$388$	0	0
$389$	− 14.0000i	− 0.709828i	−0.934899	−	0.354914i	$-0.884510\pi$
$389$	− 14.0000i	− 0.709828i	0.934899	−	0.354914i	$-0.115490\pi$
$390$	0	0
$391$	24.0000	1.21373
$392$	0	0
$393$	12.0000	0.605320
$394$	0	0
$395$	− 16.0000i	− 0.805047i
$396$	0	0
$397$	− 26.0000i	− 1.30490i	−0.757831	−	0.652451i	$-0.773741\pi$
$397$	− 26.0000i	− 1.30490i	0.757831	−	0.652451i	$-0.226259\pi$
$398$	0	0
$399$	4.00000	0.200250
$400$	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
$403$	16.0000i	0.797017i
$404$	0	0
$405$	− 2.00000i	− 0.0993808i
$406$	0	0
$407$	0	0
$408$	0	0
$409$	14.0000	0.692255	0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000	0.692255	0.346128	+	0.938187i	$0.387496\pi$
$410$	0	0
$411$	14.0000i	0.690569i
$412$	0	0
$413$	− 4.00000i	− 0.196827i
$414$	0	0
$415$	8.00000	0.392705
$416$	0	0
$417$	−4.00000	−0.195881
$418$	0	0
$419$	− 36.0000i	− 1.75872i	−0.476162	−	0.879358i	$-0.657972\pi$
$419$	− 36.0000i	− 1.75872i	0.476162	−	0.879358i	$-0.342028\pi$
$420$	0	0
$421$	− 6.00000i	− 0.292422i	−0.989253	−	0.146211i	$-0.953292\pi$
$421$	− 6.00000i	− 0.292422i	0.989253	−	0.146211i	$-0.0467079\pi$
$422$	0	0
$423$	8.00000	0.388973
$424$	0	0
$425$	6.00000	0.291043
$426$	0	0
$427$	− 10.0000i	− 0.483934i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	12.0000	0.578020	0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000	0.578020	0.289010	+	0.957326i	$0.406674\pi$
$432$	0	0
$433$	34.0000	1.63394	0.816968	−	0.576683i	$-0.195653\pi$
$433$	34.0000	1.63394	0.816968	+	0.576683i	$0.195653\pi$
$434$	0	0
$435$	12.0000i	0.575356i
$436$	0	0
$437$	− 16.0000i	− 0.765384i
$438$	0	0
$439$	−24.0000	−1.14546	−0.572729	−	0.819745i	$-0.694115\pi$
$439$	−24.0000	−1.14546	−0.572729	+	0.819745i	$0.694115\pi$
$440$	0	0
$441$	−1.00000	−0.0476190
$442$	0	0
$443$	40.0000i	1.90046i	0.311553	+	0.950229i	$0.399151\pi$
$443$	40.0000i	1.90046i	−0.311553	+	0.950229i	$0.600849\pi$
$444$	0	0
$445$	12.0000i	0.568855i
$446$	0	0
$447$	6.00000	0.283790
$448$	0	0
$449$	34.0000	1.60456	0.802280	−	0.596948i	$-0.203620\pi$
$449$	34.0000	1.60456	0.802280	+	0.596948i	$0.203620\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	8.00000i	0.375873i
$454$	0	0
$455$	−4.00000	−0.187523
$456$	0	0
$457$	−10.0000	−0.467780	−0.233890	−	0.972263i	$-0.575146\pi$
$457$	−10.0000	−0.467780	−0.233890	+	0.972263i	$0.575146\pi$
$458$	0	0
$459$	− 6.00000i	− 0.280056i
$460$	0	0
$461$	18.0000i	0.838344i	0.907907	+	0.419172i	$0.137680\pi$
$461$	18.0000i	0.838344i	−0.907907	+	0.419172i	$0.862320\pi$
$462$	0	0
$463$	8.00000	0.371792	0.185896	−	0.982569i	$-0.440481\pi$
$463$	8.00000	0.371792	0.185896	+	0.982569i	$0.440481\pi$
$464$	0	0
$465$	−16.0000	−0.741982
$466$	0	0
$467$	− 4.00000i	− 0.185098i	−0.995708	−	0.0925490i	$-0.970499\pi$
$467$	− 4.00000i	− 0.185098i	0.995708	−	0.0925490i	$-0.0295015\pi$
$468$	0	0
$469$	12.0000i	0.554109i
$470$	0	0
$471$	10.0000	0.460776
$472$	0	0
$473$	0	0
$474$	0	0
$475$	− 4.00000i	− 0.183533i
$476$	0	0
$477$	6.00000i	0.274721i
$478$	0	0
$479$	−24.0000	−1.09659	−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000	−1.09659	−0.548294	+	0.836286i	$0.684723\pi$
$480$	0	0
$481$	20.0000	0.911922
$482$	0	0
$483$	4.00000i	0.182006i
$484$	0	0
$485$	− 20.0000i	− 0.908153i
$486$	0	0
$487$	8.00000	0.362515	0.181257	−	0.983436i	$-0.441983\pi$
$487$	8.00000	0.362515	0.181257	+	0.983436i	$0.441983\pi$
$488$	0	0
$489$	4.00000	0.180886
$490$	0	0
$491$	− 24.0000i	− 1.08310i	−0.840667	−	0.541552i	$-0.817837\pi$
$491$	− 24.0000i	− 1.08310i	0.840667	−	0.541552i	$-0.182163\pi$
$492$	0	0
$493$	36.0000i	1.62136i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−4.00000	−0.179425
$498$	0	0
$499$	4.00000i	0.179065i	0.995984	+	0.0895323i	$0.0285372\pi$
$499$	4.00000i	0.179065i	−0.995984	+	0.0895323i	$0.971463\pi$
$500$	0	0
$501$	− 16.0000i	− 0.714827i
$502$	0	0
$503$	24.0000	1.07011	0.535054	−	0.844818i	$-0.320291\pi$
$503$	24.0000	1.07011	0.535054	+	0.844818i	$0.320291\pi$
$504$	0	0
$505$	−20.0000	−0.889988
$506$	0	0
$507$	9.00000i	0.399704i
$508$	0	0
$509$	− 6.00000i	− 0.265945i	−0.991120	−	0.132973i	$-0.957548\pi$
$509$	− 6.00000i	− 0.265945i	0.991120	−	0.132973i	$-0.0424523\pi$
$510$	0	0
$511$	−2.00000	−0.0884748
$512$	0	0
$513$	−4.00000	−0.176604
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−2.00000	−0.0877903
$520$	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
$523$	− 12.0000i	− 0.524723i	−0.964970	−	0.262362i	$-0.915499\pi$
$523$	− 12.0000i	− 0.524723i	0.964970	−	0.262362i	$-0.0845013\pi$
$524$	0	0
$525$	1.00000i	0.0436436i
$526$	0	0
$527$	−48.0000	−2.09091
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	4.00000i	0.173585i
$532$	0	0
$533$	− 20.0000i	− 0.866296i
$534$	0	0
$535$	−32.0000	−1.38348
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	− 2.00000i	− 0.0859867i	−0.999075	−	0.0429934i	$-0.986311\pi$
$541$	− 2.00000i	− 0.0859867i	0.999075	−	0.0429934i	$-0.0136894\pi$
$542$	0	0
$543$	6.00000	0.257485
$544$	0	0
$545$	28.0000	1.19939
$546$	0	0
$547$	28.0000i	1.19719i	0.801050	+	0.598597i	$0.204275\pi$
$547$	28.0000i	1.19719i	−0.801050	+	0.598597i	$0.795725\pi$
$548$	0	0
$549$	10.0000i	0.426790i
$550$	0	0
$551$	24.0000	1.02243
$552$	0	0
$553$	8.00000	0.340195
$554$	0	0
$555$	20.0000i	0.848953i
$556$	0	0
$557$	− 18.0000i	− 0.762684i	−0.924434	−	0.381342i	$-0.875462\pi$
$557$	− 18.0000i	− 0.762684i	0.924434	−	0.381342i	$-0.124538\pi$
$558$	0	0
$559$	−24.0000	−1.01509
$560$	0	0
$561$	0	0
$562$	0	0
$563$	12.0000i	0.505740i	0.967500	+	0.252870i	$0.0813744\pi$
$563$	12.0000i	0.505740i	−0.967500	+	0.252870i	$0.918626\pi$
$564$	0	0
$565$	28.0000i	1.17797i
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	−18.0000	−0.754599	−0.377300	−	0.926091i	$-0.623147\pi$
$569$	−18.0000	−0.754599	−0.377300	+	0.926091i	$0.623147\pi$
$570$	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
$573$	− 4.00000i	− 0.167102i
$574$	0	0
$575$	4.00000	0.166812
$576$	0	0
$577$	18.0000	0.749350	0.374675	−	0.927156i	$-0.377754\pi$
$577$	18.0000	0.749350	0.374675	+	0.927156i	$0.377754\pi$
$578$	0	0
$579$	18.0000i	0.748054i
$580$	0	0
$581$	4.00000i	0.165948i
$582$	0	0
$583$	0	0
$584$	0	0
$585$	4.00000	0.165380
$586$	0	0
$587$	− 36.0000i	− 1.48588i	−0.669359	−	0.742940i	$-0.733431\pi$
$587$	− 36.0000i	− 1.48588i	0.669359	−	0.742940i	$-0.266569\pi$
$588$	0	0
$589$	32.0000i	1.31854i
$590$	0	0
$591$	22.0000	0.904959
$592$	0	0
$593$	6.00000	0.246390	0.123195	−	0.992382i	$-0.460686\pi$
$593$	6.00000	0.246390	0.123195	+	0.992382i	$0.460686\pi$
$594$	0	0
$595$	− 12.0000i	− 0.491952i
$596$	0	0
$597$	24.0000i	0.982255i
$598$	0	0
$599$	−36.0000	−1.47092	−0.735460	−	0.677568i	$-0.763034\pi$
$599$	−36.0000	−1.47092	−0.735460	+	0.677568i	$0.763034\pi$
$600$	0	0
$601$	22.0000	0.897399	0.448699	−	0.893683i	$-0.351887\pi$
$601$	22.0000	0.897399	0.448699	+	0.893683i	$0.351887\pi$
$602$	0	0
$603$	− 12.0000i	− 0.488678i
$604$	0	0
$605$	− 22.0000i	− 0.894427i
$606$	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
$609$	−6.00000	−0.243132
$610$	0	0
$611$	16.0000i	0.647291i
$612$	0	0
$613$	10.0000i	0.403896i	0.979396	+	0.201948i	$0.0647272\pi$
$613$	10.0000i	0.403896i	−0.979396	+	0.201948i	$0.935273\pi$
$614$	0	0
$615$	20.0000	0.806478
$616$	0	0
$617$	−18.0000	−0.724653	−0.362326	−	0.932051i	$-0.618017\pi$
$617$	−18.0000	−0.724653	−0.362326	+	0.932051i	$0.618017\pi$
$618$	0	0
$619$	− 12.0000i	− 0.482321i	−0.970485	−	0.241160i	$-0.922472\pi$
$619$	− 12.0000i	− 0.482321i	0.970485	−	0.241160i	$-0.0775280\pi$
$620$	0	0
$621$	− 4.00000i	− 0.160514i
$622$	0	0
$623$	−6.00000	−0.240385
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	60.0000i	2.39236i
$630$	0	0
$631$	32.0000	1.27390	0.636950	−	0.770905i	$-0.280196\pi$
$631$	32.0000	1.27390	0.636950	+	0.770905i	$0.280196\pi$
$632$	0	0
$633$	12.0000	0.476957
$634$	0	0
$635$	− 16.0000i	− 0.634941i
$636$	0	0
$637$	− 2.00000i	− 0.0792429i
$638$	0	0
$639$	4.00000	0.158238
$640$	0	0
$641$	26.0000	1.02694	0.513469	−	0.858108i	$-0.328360\pi$
$641$	26.0000	1.02694	0.513469	+	0.858108i	$0.328360\pi$
$642$	0	0
$643$	− 20.0000i	− 0.788723i	−0.918955	−	0.394362i	$-0.870966\pi$
$643$	− 20.0000i	− 0.788723i	0.918955	−	0.394362i	$-0.129034\pi$
$644$	0	0
$645$	− 24.0000i	− 0.944999i
$646$	0	0
$647$	32.0000	1.25805	0.629025	−	0.777385i	$-0.283454\pi$
$647$	32.0000	1.25805	0.629025	+	0.777385i	$0.283454\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	− 8.00000i	− 0.313545i
$652$	0	0
$653$	6.00000i	0.234798i	0.993085	+	0.117399i	$0.0374557\pi$
$653$	6.00000i	0.234798i	−0.993085	+	0.117399i	$0.962544\pi$
$654$	0	0
$655$	−24.0000	−0.937758
$656$	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
$661$	− 14.0000i	− 0.544537i	−0.962221	−	0.272268i	$-0.912226\pi$
$661$	− 14.0000i	− 0.544537i	0.962221	−	0.272268i	$-0.0877739\pi$
$662$	0	0
$663$	12.0000	0.466041
$664$	0	0
$665$	−8.00000	−0.310227
$666$	0	0
$667$	24.0000i	0.929284i
$668$	0	0
$669$	16.0000i	0.618596i
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−46.0000	−1.77317	−0.886585	−	0.462566i	$-0.846929\pi$
$673$	−46.0000	−1.77317	−0.886585	+	0.462566i	$0.846929\pi$
$674$	0	0
$675$	− 1.00000i	− 0.0384900i
$676$	0	0
$677$	38.0000i	1.46046i	0.683202	+	0.730229i	$0.260587\pi$
$677$	38.0000i	1.46046i	−0.683202	+	0.730229i	$0.739413\pi$
$678$	0	0
$679$	10.0000	0.383765
$680$	0	0
$681$	−12.0000	−0.459841
$682$	0	0
$683$	40.0000i	1.53056i	0.643699	+	0.765279i	$0.277399\pi$
$683$	40.0000i	1.53056i	−0.643699	+	0.765279i	$0.722601\pi$
$684$	0	0
$685$	− 28.0000i	− 1.06983i
$686$	0	0
$687$	6.00000	0.228914
$688$	0	0
$689$	−12.0000	−0.457164
$690$	0	0
$691$	36.0000i	1.36950i	0.728776	+	0.684752i	$0.240090\pi$
$691$	36.0000i	1.36950i	−0.728776	+	0.684752i	$0.759910\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	8.00000	0.303457
$696$	0	0
$697$	60.0000	2.27266
$698$	0	0
$699$	− 18.0000i	− 0.680823i
$700$	0	0
$701$	38.0000i	1.43524i	0.696435	+	0.717620i	$0.254769\pi$
$701$	38.0000i	1.43524i	−0.696435	+	0.717620i	$0.745231\pi$
$702$	0	0
$703$	40.0000	1.50863
$704$	0	0
$705$	−16.0000	−0.602595
$706$	0	0
$707$	− 10.0000i	− 0.376089i
$708$	0	0
$709$	− 6.00000i	− 0.225335i	−0.993633	−	0.112667i	$-0.964061\pi$
$709$	− 6.00000i	− 0.225335i	0.993633	−	0.112667i	$-0.0359394\pi$
$710$	0	0
$711$	−8.00000	−0.300023
$712$	0	0
$713$	−32.0000	−1.19841
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 12.0000i	− 0.448148i
$718$	0	0
$719$	−16.0000	−0.596699	−0.298350	−	0.954457i	$-0.596436\pi$
$719$	−16.0000	−0.596699	−0.298350	+	0.954457i	$0.596436\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	− 22.0000i	− 0.818189i
$724$	0	0
$725$	6.00000i	0.222834i
$726$	0	0
$727$	16.0000	0.593407	0.296704	−	0.954970i	$-0.404113\pi$
$727$	16.0000	0.593407	0.296704	+	0.954970i	$0.404113\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	− 72.0000i	− 2.66302i
$732$	0	0
$733$	14.0000i	0.517102i	0.965998	+	0.258551i	$0.0832450\pi$
$733$	14.0000i	0.517102i	−0.965998	+	0.258551i	$0.916755\pi$
$734$	0	0
$735$	2.00000	0.0737711
$736$	0	0
$737$	0	0
$738$	0	0
$739$	28.0000i	1.03000i	0.857191	+	0.514998i	$0.172207\pi$
$739$	28.0000i	1.03000i	−0.857191	+	0.514998i	$0.827793\pi$
$740$	0	0
$741$	− 8.00000i	− 0.293887i
$742$	0	0
$743$	−52.0000	−1.90769	−0.953847	−	0.300291i	$-0.902916\pi$
$743$	−52.0000	−1.90769	−0.953847	+	0.300291i	$0.902916\pi$
$744$	0	0
$745$	−12.0000	−0.439646
$746$	0	0
$747$	− 4.00000i	− 0.146352i
$748$	0	0
$749$	− 16.0000i	− 0.584627i
$750$	0	0
$751$	0	0		−	1.00000i	$-0.5\pi$
$751$	0	0			1.00000i	$0.5\pi$
$752$	0	0
$753$	20.0000	0.728841
$754$	0	0
$755$	− 16.0000i	− 0.582300i
$756$	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
$759$	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
$763$	14.0000i	0.506834i
$764$	0	0
$765$	12.0000i	0.433861i
$766$	0	0
$767$	−8.00000	−0.288863
$768$	0	0
$769$	−14.0000	−0.504853	−0.252426	−	0.967616i	$-0.581229\pi$
$769$	−14.0000	−0.504853	−0.252426	+	0.967616i	$0.581229\pi$
$770$	0	0
$771$	14.0000i	0.504198i
$772$	0	0
$773$	38.0000i	1.36677i	0.730061	+	0.683383i	$0.239492\pi$
$773$	38.0000i	1.36677i	−0.730061	+	0.683383i	$0.760508\pi$
$774$	0	0
$775$	−8.00000	−0.287368
$776$	0	0
$777$	−10.0000	−0.358748
$778$	0	0
$779$	− 40.0000i	− 1.43315i
$780$	0	0
$781$	0	0
$782$	0	0
$783$	6.00000	0.214423
$784$	0	0
$785$	−20.0000	−0.713831
$786$	0	0
$787$	20.0000i	0.712923i	0.934310	+	0.356462i	$0.116017\pi$
$787$	20.0000i	0.712923i	−0.934310	+	0.356462i	$0.883983\pi$
$788$	0	0
$789$	− 28.0000i	− 0.996826i
$790$	0	0
$791$	−14.0000	−0.497783
$792$	0	0
$793$	−20.0000	−0.710221
$794$	0	0
$795$	− 12.0000i	− 0.425596i
$796$	0	0
$797$	18.0000i	0.637593i	0.947823	+	0.318796i	$0.103279\pi$
$797$	18.0000i	0.637593i	−0.947823	+	0.318796i	$0.896721\pi$
$798$	0	0
$799$	−48.0000	−1.69812
$800$	0	0
$801$	6.00000	0.212000
$802$	0	0
$803$	0	0
$804$	0	0
$805$	− 8.00000i	− 0.281963i
$806$	0	0
$807$	−10.0000	−0.352017
$808$	0	0
$809$	−26.0000	−0.914111	−0.457056	−	0.889438i	$-0.651096\pi$
$809$	−26.0000	−0.914111	−0.457056	+	0.889438i	$0.651096\pi$
$810$	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
$813$	− 8.00000i	− 0.280572i
$814$	0	0
$815$	−8.00000	−0.280228
$816$	0	0
$817$	−48.0000	−1.67931
$818$	0	0
$819$	2.00000i	0.0698857i
$820$	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
$823$	−32.0000	−1.11545	−0.557725	−	0.830026i	$-0.688326\pi$
$823$	−32.0000	−1.11545	−0.557725	+	0.830026i	$0.688326\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 16.0000i	− 0.556375i	−0.960527	−	0.278187i	$-0.910266\pi$
$827$	− 16.0000i	− 0.556375i	0.960527	−	0.278187i	$-0.0897336\pi$
$828$	0	0
$829$	14.0000i	0.486240i	0.969996	+	0.243120i	$0.0781709\pi$
$829$	14.0000i	0.486240i	−0.969996	+	0.243120i	$0.921829\pi$
$830$	0	0
$831$	−10.0000	−0.346896
$832$	0	0
$833$	6.00000	0.207888
$834$	0	0
$835$	32.0000i	1.10741i
$836$	0	0
$837$	8.00000i	0.276520i
$838$	0	0
$839$	−48.0000	−1.65714	−0.828572	−	0.559883i	$-0.810846\pi$
$839$	−48.0000	−1.65714	−0.828572	+	0.559883i	$0.810846\pi$
$840$	0	0
$841$	−7.00000	−0.241379
$842$	0	0
$843$	30.0000i	1.03325i
$844$	0	0
$845$	− 18.0000i	− 0.619219i
$846$	0	0
$847$	11.0000	0.377964
$848$	0	0
$849$	−12.0000	−0.411839
$850$	0	0
$851$	40.0000i	1.37118i
$852$	0	0
$853$	− 14.0000i	− 0.479351i	−0.970853	−	0.239675i	$-0.922959\pi$
$853$	− 14.0000i	− 0.479351i	0.970853	−	0.239675i	$-0.0770410\pi$
$854$	0	0
$855$	8.00000	0.273594
$856$	0	0
$857$	−38.0000	−1.29806	−0.649028	−	0.760765i	$-0.724824\pi$
$857$	−38.0000	−1.29806	−0.649028	+	0.760765i	$0.724824\pi$
$858$	0	0
$859$	44.0000i	1.50126i	0.660722	+	0.750630i	$0.270250\pi$
$859$	44.0000i	1.50126i	−0.660722	+	0.750630i	$0.729750\pi$
$860$	0	0
$861$	10.0000i	0.340799i
$862$	0	0
$863$	4.00000	0.136162	0.0680808	−	0.997680i	$-0.478312\pi$
$863$	4.00000	0.136162	0.0680808	+	0.997680i	$0.478312\pi$
$864$	0	0
$865$	4.00000	0.136004
$866$	0	0
$867$	19.0000i	0.645274i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	24.0000	0.813209
$872$	0	0
$873$	−10.0000	−0.338449
$874$	0	0
$875$	− 12.0000i	− 0.405674i
$876$	0	0
$877$	− 2.00000i	− 0.0675352i	−0.999430	−	0.0337676i	$-0.989249\pi$
$877$	− 2.00000i	− 0.0675352i	0.999430	−	0.0337676i	$-0.0107506\pi$
$878$	0	0
$879$	−22.0000	−0.742042
$880$	0	0
$881$	−42.0000	−1.41502	−0.707508	−	0.706705i	$-0.750181\pi$
$881$	−42.0000	−1.41502	−0.707508	+	0.706705i	$0.750181\pi$
$882$	0	0
$883$	4.00000i	0.134611i	0.997732	+	0.0673054i	$0.0214402\pi$
$883$	4.00000i	0.134611i	−0.997732	+	0.0673054i	$0.978560\pi$
$884$	0	0
$885$	− 8.00000i	− 0.268917i
$886$	0	0
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	8.00000	0.268311
$890$	0	0
$891$	0	0
$892$	0	0
$893$	32.0000i	1.07084i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	8.00000	0.267112
$898$	0	0
$899$	− 48.0000i	− 1.60089i
$900$	0	0
$901$	− 36.0000i	− 1.19933i
$902$	0	0
$903$	12.0000	0.399335
$904$	0	0
$905$	−12.0000	−0.398893
$906$	0	0
$907$	52.0000i	1.72663i	0.504664	+	0.863316i	$0.331616\pi$
$907$	52.0000i	1.72663i	−0.504664	+	0.863316i	$0.668384\pi$
$908$	0	0
$909$	10.0000i	0.331679i
$910$	0	0
$911$	12.0000	0.397578	0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000	0.397578	0.198789	+	0.980042i	$0.436299\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	− 20.0000i	− 0.661180i
$916$	0	0
$917$	− 12.0000i	− 0.396275i
$918$	0	0
$919$	−8.00000	−0.263896	−0.131948	−	0.991257i	$-0.542123\pi$
$919$	−8.00000	−0.263896	−0.131948	+	0.991257i	$0.542123\pi$
$920$	0	0
$921$	4.00000	0.131804
$922$	0	0
$923$	8.00000i	0.263323i
$924$	0	0
$925$	10.0000i	0.328798i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	30.0000	0.984268	0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000	0.984268	0.492134	+	0.870519i	$0.336217\pi$
$930$	0	0
$931$	− 4.00000i	− 0.131095i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−10.0000	−0.326686	−0.163343	−	0.986569i	$-0.552228\pi$
$937$	−10.0000	−0.326686	−0.163343	+	0.986569i	$0.552228\pi$
$938$	0	0
$939$	6.00000i	0.195803i
$940$	0	0
$941$	− 22.0000i	− 0.717180i	−0.933495	−	0.358590i	$-0.883258\pi$
$941$	− 22.0000i	− 0.717180i	0.933495	−	0.358590i	$-0.116742\pi$
$942$	0	0
$943$	40.0000	1.30258
$944$	0	0
$945$	−2.00000	−0.0650600
$946$	0	0
$947$	− 24.0000i	− 0.779895i	−0.920837	−	0.389948i	$-0.872493\pi$
$947$	− 24.0000i	− 0.779895i	0.920837	−	0.389948i	$-0.127507\pi$
$948$	0	0
$949$	4.00000i	0.129845i
$950$	0	0
$951$	2.00000	0.0648544
$952$	0	0
$953$	38.0000	1.23094	0.615470	−	0.788160i	$-0.288966\pi$
$953$	38.0000	1.23094	0.615470	+	0.788160i	$0.288966\pi$
$954$	0	0
$955$	8.00000i	0.258874i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	14.0000	0.452084
$960$	0	0
$961$	33.0000	1.06452
$962$	0	0
$963$	16.0000i	0.515593i
$964$	0	0
$965$	− 36.0000i	− 1.15888i
$966$	0	0
$967$	48.0000	1.54358	0.771788	−	0.635880i	$-0.219363\pi$
$967$	48.0000	1.54358	0.771788	+	0.635880i	$0.219363\pi$
$968$	0	0
$969$	24.0000	0.770991
$970$	0	0
$971$	− 28.0000i	− 0.898563i	−0.893390	−	0.449281i	$-0.851680\pi$
$971$	− 28.0000i	− 0.898563i	0.893390	−	0.449281i	$-0.148320\pi$
$972$	0	0
$973$	4.00000i	0.128234i
$974$	0	0
$975$	2.00000	0.0640513
$976$	0	0
$977$	−22.0000	−0.703842	−0.351921	−	0.936030i	$-0.614471\pi$
$977$	−22.0000	−0.703842	−0.351921	+	0.936030i	$0.614471\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	− 14.0000i	− 0.446986i
$982$	0	0
$983$	40.0000	1.27580	0.637901	−	0.770118i	$-0.279803\pi$
$983$	40.0000	1.27580	0.637901	+	0.770118i	$0.279803\pi$
$984$	0	0
$985$	−44.0000	−1.40196
$986$	0	0
$987$	− 8.00000i	− 0.254643i
$988$	0	0
$989$	− 48.0000i	− 1.52631i
$990$	0	0
$991$	−48.0000	−1.52477	−0.762385	−	0.647124i	$-0.775972\pi$
$991$	−48.0000	−1.52477	−0.762385	+	0.647124i	$0.775972\pi$
$992$	0	0
$993$	−20.0000	−0.634681
$994$	0	0
$995$	− 48.0000i	− 1.52170i
$996$	0	0
$997$	2.00000i	0.0633406i	0.999498	+	0.0316703i	$0.0100827\pi$
$997$	2.00000i	0.0633406i	−0.999498	+	0.0316703i	$0.989917\pi$
$998$	0	0
$999$	10.0000	0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5376.2.c.bd.2689.2		2
4.3	odd	2		5376.2.c.f.2689.1		2
8.3	odd	2		5376.2.c.f.2689.2		2
8.5	even	2	inner	5376.2.c.bd.2689.1		2
16.3	odd	4		1344.2.a.n.1.1		1
16.5	even	4		168.2.a.b.1.1	✓	1
16.11	odd	4		336.2.a.c.1.1		1
16.13	even	4		1344.2.a.c.1.1		1
48.5	odd	4		504.2.a.b.1.1		1
48.11	even	4		1008.2.a.e.1.1		1
48.29	odd	4		4032.2.a.be.1.1		1
48.35	even	4		4032.2.a.bj.1.1		1
80.37	odd	4		4200.2.t.m.1849.1		2
80.53	odd	4		4200.2.t.m.1849.2		2
80.59	odd	4		8400.2.a.bx.1.1		1
80.69	even	4		4200.2.a.i.1.1		1
112.5	odd	12		1176.2.q.j.361.1		2
112.11	odd	12		2352.2.q.o.961.1		2
112.13	odd	4		9408.2.a.cy.1.1		1
112.27	even	4		2352.2.a.q.1.1		1
112.37	even	12		1176.2.q.b.361.1		2
112.53	even	12		1176.2.q.b.961.1		2
112.59	even	12		2352.2.q.j.961.1		2
112.69	odd	4		1176.2.a.a.1.1		1
112.75	even	12		2352.2.q.j.1537.1		2
112.83	even	4		9408.2.a.bc.1.1		1
112.101	odd	12		1176.2.q.j.961.1		2
112.107	odd	12		2352.2.q.o.1537.1		2
336.5	even	12		3528.2.s.h.361.1		2
336.53	odd	12		3528.2.s.v.3313.1		2
336.101	even	12		3528.2.s.h.3313.1		2
336.149	odd	12		3528.2.s.v.361.1		2
336.251	odd	4		7056.2.a.br.1.1		1
336.293	even	4		3528.2.a.w.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.2.a.b.1.1	✓	1	16.5	even	4
336.2.a.c.1.1		1	16.11	odd	4
504.2.a.b.1.1		1	48.5	odd	4
1008.2.a.e.1.1		1	48.11	even	4
1176.2.a.a.1.1		1	112.69	odd	4
1176.2.q.b.361.1		2	112.37	even	12
1176.2.q.b.961.1		2	112.53	even	12
1176.2.q.j.361.1		2	112.5	odd	12
1176.2.q.j.961.1		2	112.101	odd	12
1344.2.a.c.1.1		1	16.13	even	4
1344.2.a.n.1.1		1	16.3	odd	4
2352.2.a.q.1.1		1	112.27	even	4
2352.2.q.j.961.1		2	112.59	even	12
2352.2.q.j.1537.1		2	112.75	even	12
2352.2.q.o.961.1		2	112.11	odd	12
2352.2.q.o.1537.1		2	112.107	odd	12
3528.2.a.w.1.1		1	336.293	even	4
3528.2.s.h.361.1		2	336.5	even	12
3528.2.s.h.3313.1		2	336.101	even	12
3528.2.s.v.361.1		2	336.149	odd	12
3528.2.s.v.3313.1		2	336.53	odd	12
4032.2.a.be.1.1		1	48.29	odd	4
4032.2.a.bj.1.1		1	48.35	even	4
4200.2.a.i.1.1		1	80.69	even	4
4200.2.t.m.1849.1		2	80.37	odd	4
4200.2.t.m.1849.2		2	80.53	odd	4
5376.2.c.f.2689.1		2	4.3	odd	2
5376.2.c.f.2689.2		2	8.3	odd	2
5376.2.c.bd.2689.1		2	8.5	even	2	inner
5376.2.c.bd.2689.2		2	1.1	even	1	trivial
7056.2.a.br.1.1		1	336.251	odd	4
8400.2.a.bx.1.1		1	80.59	odd	4
9408.2.a.bc.1.1		1	112.83	even	4
9408.2.a.cy.1.1		1	112.13	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:	\( N \)	\(=\)	\( 5376 = 2^{8} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5376.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} -2.00000i q^{5} +1.00000 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} -2.00000i q^{5} +1.00000 q^{7} -1.00000 q^{9} -2.00000i q^{13} +2.00000 q^{15} +6.00000 q^{17} -4.00000i q^{19} +1.00000i q^{21} +4.00000 q^{23} +1.00000 q^{25} -1.00000i q^{27} +6.00000i q^{29} -8.00000 q^{31} -2.00000i q^{35} +10.0000i q^{37} +2.00000 q^{39} +10.0000 q^{41} -12.0000i q^{43} +2.00000i q^{45} -8.00000 q^{47} +1.00000 q^{49} +6.00000i q^{51} -6.00000i q^{53} +4.00000 q^{57} -4.00000i q^{59} -10.0000i q^{61} -1.00000 q^{63} -4.00000 q^{65} +12.0000i q^{67} +4.00000i q^{69} -4.00000 q^{71} -2.00000 q^{73} +1.00000i q^{75} +8.00000 q^{79} +1.00000 q^{81} +4.00000i q^{83} -12.0000i q^{85} -6.00000 q^{87} -6.00000 q^{89} -2.00000i q^{91} -8.00000i q^{93} -8.00000 q^{95} +10.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{7} - 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^7 - 2 * q^9	\( 2 q + 2 q^{7} - 2 q^{9} + 4 q^{15} + 12 q^{17} + 8 q^{23} + 2 q^{25} - 16 q^{31} + 4 q^{39} + 20 q^{41} - 16 q^{47} + 2 q^{49} + 8 q^{57} - 2 q^{63} - 8 q^{65} - 8 q^{71} - 4 q^{73} + 16 q^{79} + 2 q^{81} - 12 q^{87} - 12 q^{89} - 16 q^{95} + 20 q^{97}+O(q^{100}) \) 2 * q + 2 * q^7 - 2 * q^9 + 4 * q^15 + 12 * q^17 + 8 * q^23 + 2 * q^25 - 16 * q^31 + 4 * q^39 + 20 * q^41 - 16 * q^47 + 2 * q^49 + 8 * q^57 - 2 * q^63 - 8 * q^65 - 8 * q^71 - 4 * q^73 + 16 * q^79 + 2 * q^81 - 12 * q^87 - 12 * q^89 - 16 * q^95 + 20 * q^97

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