LMFDB - Embedded newform 5376.2.c.j.2689.1

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 672)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} -1.00000 q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} -1.00000 q^{7} -1.00000 q^{9} +2.00000i q^{11} +2.00000i q^{13} +4.00000 q^{17} -4.00000i q^{19} +1.00000i q^{21} -6.00000 q^{23} +5.00000 q^{25} +1.00000i q^{27} +2.00000i q^{29} +2.00000 q^{33} -6.00000i q^{37} +2.00000 q^{39} -8.00000 q^{41} +8.00000i q^{43} +4.00000 q^{47} +1.00000 q^{49} -4.00000i q^{51} -6.00000i q^{53} -4.00000 q^{57} +14.0000i q^{61} +1.00000 q^{63} +4.00000i q^{67} +6.00000i q^{69} -2.00000 q^{71} +2.00000 q^{73} -5.00000i q^{75} -2.00000i q^{77} -4.00000 q^{79} +1.00000 q^{81} +12.0000i q^{83} +2.00000 q^{87} -2.00000i q^{91} +6.00000 q^{97} -2.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 8 q^{17} - 12 q^{23} + 10 q^{25} + 4 q^{33} + 4 q^{39} - 16 q^{41} + 8 q^{47} + 2 q^{49} - 8 q^{57} + 2 q^{63} - 4 q^{71} + 4 q^{73} - 8 q^{79} + 2 q^{81} + 4 q^{87} + 12 q^{97}+O(q^{100}) $ 2 * q - 2 * q^7 - 2 * q^9 + 8 * q^17 - 12 * q^23 + 10 * q^25 + 4 * q^33 + 4 * q^39 - 16 * q^41 + 8 * q^47 + 2 * q^49 - 8 * q^57 + 2 * q^63 - 4 * q^71 + 4 * q^73 - 8 * q^79 + 2 * q^81 + 4 * q^87 + 12 * 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
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	2.00000i	0.603023i	0.953463	+	0.301511i	$0.0974911\pi$
$11$	2.00000i	0.603023i	−0.953463	+	0.301511i	$0.902509\pi$
$12$	0	0
$13$	2.00000i	0.554700i	0.960769	+	0.277350i	$0.0894562\pi$
$13$	2.00000i	0.554700i	−0.960769	+	0.277350i	$0.910544\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	0	0
$19$	− 4.00000i	− 0.917663i	−0.888523	−	0.458831i	$-0.848268\pi$
$19$	− 4.00000i	− 0.917663i	0.888523	−	0.458831i	$-0.151732\pi$
$20$	0	0
$21$	1.00000i	0.218218i
$22$	0	0
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	0	0
$25$	5.00000	1.00000
$26$	0	0
$27$	1.00000i	0.192450i
$28$	0	0
$29$	2.00000i	0.371391i	0.982607	+	0.185695i	$0.0594537\pi$
$29$	2.00000i	0.371391i	−0.982607	+	0.185695i	$0.940546\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	2.00000	0.348155
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 6.00000i	− 0.986394i	−0.869918	−	0.493197i	$-0.835828\pi$
$37$	− 6.00000i	− 0.986394i	0.869918	−	0.493197i	$-0.164172\pi$
$38$	0	0
$39$	2.00000	0.320256
$40$	0	0
$41$	−8.00000	−1.24939	−0.624695	−	0.780869i	$-0.714777\pi$
$41$	−8.00000	−1.24939	−0.624695	+	0.780869i	$0.714777\pi$
$42$	0	0
$43$	8.00000i	1.21999i	0.792406	+	0.609994i	$0.208828\pi$
$43$	8.00000i	1.21999i	−0.792406	+	0.609994i	$0.791172\pi$
$44$	0	0
$45$	0	0
$46$	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
$49$	1.00000	0.142857
$50$	0	0
$51$	− 4.00000i	− 0.560112i
$52$	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
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	14.0000i	1.79252i	0.443533	+	0.896258i	$0.353725\pi$
$61$	14.0000i	1.79252i	−0.443533	+	0.896258i	$0.646275\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	0	0
$66$	0	0
$67$	4.00000i	0.488678i	0.969690	+	0.244339i	$0.0785709\pi$
$67$	4.00000i	0.488678i	−0.969690	+	0.244339i	$0.921429\pi$
$68$	0	0
$69$	6.00000i	0.722315i
$70$	0	0
$71$	−2.00000	−0.237356	−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000	−0.237356	−0.118678	+	0.992933i	$0.537866\pi$
$72$	0	0
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	0	0
$75$	− 5.00000i	− 0.577350i
$76$	0	0
$77$	− 2.00000i	− 0.227921i
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	12.0000i	1.31717i	0.752506	+	0.658586i	$0.228845\pi$
$83$	12.0000i	1.31717i	−0.752506	+	0.658586i	$0.771155\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	2.00000	0.214423
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	− 2.00000i	− 0.209657i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	6.00000	0.609208	0.304604	−	0.952479i	$-0.401476\pi$
$97$	6.00000	0.609208	0.304604	+	0.952479i	$0.401476\pi$
$98$	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
$105$	0	0
$106$	0	0
$107$	− 6.00000i	− 0.580042i	−0.957020	−	0.290021i	$-0.906338\pi$
$107$	− 6.00000i	− 0.580042i	0.957020	−	0.290021i	$-0.0936623\pi$
$108$	0	0
$109$	18.0000i	1.72409i	0.506834	+	0.862044i	$0.330816\pi$
$109$	18.0000i	1.72409i	−0.506834	+	0.862044i	$0.669184\pi$
$110$	0	0
$111$	−6.00000	−0.569495
$112$	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
$115$	0	0
$116$	0	0
$117$	− 2.00000i	− 0.184900i
$118$	0	0
$119$	−4.00000	−0.366679
$120$	0	0
$121$	7.00000	0.636364
$122$	0	0
$123$	8.00000i	0.721336i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−12.0000	−1.06483	−0.532414	−	0.846484i	$-0.678715\pi$
$127$	−12.0000	−1.06483	−0.532414	+	0.846484i	$0.678715\pi$
$128$	0	0
$129$	8.00000	0.704361
$130$	0	0
$131$	20.0000i	1.74741i	0.486458	+	0.873704i	$0.338289\pi$
$131$	20.0000i	1.74741i	−0.486458	+	0.873704i	$0.661711\pi$
$132$	0	0
$133$	4.00000i	0.346844i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	18.0000	1.53784	0.768922	−	0.639343i	$-0.220793\pi$
$137$	18.0000	1.53784	0.768922	+	0.639343i	$0.220793\pi$
$138$	0	0
$139$	− 12.0000i	− 1.01783i	−0.860818	−	0.508913i	$-0.830047\pi$
$139$	− 12.0000i	− 1.01783i	0.860818	−	0.508913i	$-0.169953\pi$
$140$	0	0
$141$	− 4.00000i	− 0.336861i
$142$	0	0
$143$	−4.00000	−0.334497
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 1.00000i	− 0.0824786i
$148$	0	0
$149$	10.0000i	0.819232i	0.912258	+	0.409616i	$0.134337\pi$
$149$	10.0000i	0.819232i	−0.912258	+	0.409616i	$0.865663\pi$
$150$	0	0
$151$	16.0000	1.30206	0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000	1.30206	0.651031	+	0.759051i	$0.274337\pi$
$152$	0	0
$153$	−4.00000	−0.323381
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 2.00000i	− 0.159617i	−0.996810	−	0.0798087i	$-0.974569\pi$
$157$	− 2.00000i	− 0.159617i	0.996810	−	0.0798087i	$-0.0254309\pi$
$158$	0	0
$159$	−6.00000	−0.475831
$160$	0	0
$161$	6.00000	0.472866
$162$	0	0
$163$	4.00000i	0.313304i	0.987654	+	0.156652i	$0.0500701\pi$
$163$	4.00000i	0.313304i	−0.987654	+	0.156652i	$0.949930\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	4.00000i	0.305888i
$172$	0	0
$173$	12.0000i	0.912343i	0.889892	+	0.456172i	$0.150780\pi$
$173$	12.0000i	0.912343i	−0.889892	+	0.456172i	$0.849220\pi$
$174$	0	0
$175$	−5.00000	−0.377964
$176$	0	0
$177$	0	0
$178$	0	0
$179$	10.0000i	0.747435i	0.927543	+	0.373718i	$0.121917\pi$
$179$	10.0000i	0.747435i	−0.927543	+	0.373718i	$0.878083\pi$
$180$	0	0
$181$	14.0000i	1.04061i	0.853980	+	0.520306i	$0.174182\pi$
$181$	14.0000i	1.04061i	−0.853980	+	0.520306i	$0.825818\pi$
$182$	0	0
$183$	14.0000	1.03491
$184$	0	0
$185$	0	0
$186$	0	0
$187$	8.00000i	0.585018i
$188$	0	0
$189$	− 1.00000i	− 0.0727393i
$190$	0	0
$191$	10.0000	0.723575	0.361787	−	0.932261i	$-0.382167\pi$
$191$	10.0000	0.723575	0.361787	+	0.932261i	$0.382167\pi$
$192$	0	0
$193$	−26.0000	−1.87152	−0.935760	−	0.352636i	$-0.885285\pi$
$193$	−26.0000	−1.87152	−0.935760	+	0.352636i	$0.885285\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2.00000i	0.142494i	0.997459	+	0.0712470i	$0.0226979\pi$
$197$	2.00000i	0.142494i	−0.997459	+	0.0712470i	$0.977302\pi$
$198$	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
$201$	4.00000	0.282138
$202$	0	0
$203$	− 2.00000i	− 0.140372i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	6.00000	0.417029
$208$	0	0
$209$	8.00000	0.553372
$210$	0	0
$211$	− 24.0000i	− 1.65223i	−0.563503	−	0.826114i	$-0.690547\pi$
$211$	− 24.0000i	− 1.65223i	0.563503	−	0.826114i	$-0.309453\pi$
$212$	0	0
$213$	2.00000i	0.137038i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	− 2.00000i	− 0.135147i
$220$	0	0
$221$	8.00000i	0.538138i
$222$	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
$225$	−5.00000	−0.333333
$226$	0	0
$227$	− 8.00000i	− 0.530979i	−0.964114	−	0.265489i	$-0.914466\pi$
$227$	− 8.00000i	− 0.530979i	0.964114	−	0.265489i	$-0.0855335\pi$
$228$	0	0
$229$	− 2.00000i	− 0.132164i	−0.997814	−	0.0660819i	$-0.978950\pi$
$229$	− 2.00000i	− 0.132164i	0.997814	−	0.0660819i	$-0.0210498\pi$
$230$	0	0
$231$	−2.00000	−0.131590
$232$	0	0
$233$	26.0000	1.70332	0.851658	−	0.524097i	$-0.175597\pi$
$233$	26.0000	1.70332	0.851658	+	0.524097i	$0.175597\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	4.00000i	0.259828i
$238$	0	0
$239$	−26.0000	−1.68180	−0.840900	−	0.541190i	$-0.817974\pi$
$239$	−26.0000	−1.68180	−0.840900	+	0.541190i	$0.817974\pi$
$240$	0	0
$241$	22.0000	1.41714	0.708572	−	0.705638i	$-0.249340\pi$
$241$	22.0000	1.41714	0.708572	+	0.705638i	$0.249340\pi$
$242$	0	0
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	8.00000	0.509028
$248$	0	0
$249$	12.0000	0.760469
$250$	0	0
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	0	0
$253$	− 12.0000i	− 0.754434i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−8.00000	−0.499026	−0.249513	−	0.968371i	$-0.580271\pi$
$257$	−8.00000	−0.499026	−0.249513	+	0.968371i	$0.580271\pi$
$258$	0	0
$259$	6.00000i	0.372822i
$260$	0	0
$261$	− 2.00000i	− 0.123797i
$262$	0	0
$263$	−14.0000	−0.863277	−0.431638	−	0.902047i	$-0.642064\pi$
$263$	−14.0000	−0.863277	−0.431638	+	0.902047i	$0.642064\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 16.0000i	− 0.975537i	−0.872973	−	0.487769i	$-0.837811\pi$
$269$	− 16.0000i	− 0.975537i	0.872973	−	0.487769i	$-0.162189\pi$
$270$	0	0
$271$	32.0000	1.94386	0.971931	−	0.235267i	$-0.0755965\pi$
$271$	32.0000	1.94386	0.971931	+	0.235267i	$0.0755965\pi$
$272$	0	0
$273$	−2.00000	−0.121046
$274$	0	0
$275$	10.0000i	0.603023i
$276$	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
$279$	0	0
$280$	0	0
$281$	2.00000	0.119310	0.0596550	−	0.998219i	$-0.481000\pi$
$281$	2.00000	0.119310	0.0596550	+	0.998219i	$0.481000\pi$
$282$	0	0
$283$	4.00000i	0.237775i	0.992908	+	0.118888i	$0.0379328\pi$
$283$	4.00000i	0.237775i	−0.992908	+	0.118888i	$0.962067\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	8.00000	0.472225
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	− 6.00000i	− 0.351726i
$292$	0	0
$293$	28.0000i	1.63578i	0.575376	+	0.817889i	$0.304856\pi$
$293$	28.0000i	1.63578i	−0.575376	+	0.817889i	$0.695144\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−2.00000	−0.116052
$298$	0	0
$299$	− 12.0000i	− 0.693978i
$300$	0	0
$301$	− 8.00000i	− 0.461112i
$302$	0	0
$303$	12.0000	0.689382
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 28.0000i	− 1.59804i	−0.601302	−	0.799022i	$-0.705351\pi$
$307$	− 28.0000i	− 1.59804i	0.601302	−	0.799022i	$-0.294649\pi$
$308$	0	0
$309$	8.00000i	0.455104i
$310$	0	0
$311$	−28.0000	−1.58773	−0.793867	−	0.608091i	$-0.791935\pi$
$311$	−28.0000	−1.58773	−0.793867	+	0.608091i	$0.791935\pi$
$312$	0	0
$313$	14.0000	0.791327	0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000	0.791327	0.395663	+	0.918396i	$0.370515\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	14.0000i	0.786318i	0.919470	+	0.393159i	$0.128618\pi$
$317$	14.0000i	0.786318i	−0.919470	+	0.393159i	$0.871382\pi$
$318$	0	0
$319$	−4.00000	−0.223957
$320$	0	0
$321$	−6.00000	−0.334887
$322$	0	0
$323$	− 16.0000i	− 0.890264i
$324$	0	0
$325$	10.0000i	0.554700i
$326$	0	0
$327$	18.0000	0.995402
$328$	0	0
$329$	−4.00000	−0.220527
$330$	0	0
$331$	− 16.0000i	− 0.879440i	−0.898135	−	0.439720i	$-0.855078\pi$
$331$	− 16.0000i	− 0.879440i	0.898135	−	0.439720i	$-0.144922\pi$
$332$	0	0
$333$	6.00000i	0.328798i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	26.0000	1.41631	0.708155	−	0.706057i	$-0.249528\pi$
$337$	26.0000	1.41631	0.708155	+	0.706057i	$0.249528\pi$
$338$	0	0
$339$	− 18.0000i	− 0.977626i
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	30.0000i	1.61048i	0.592946	+	0.805242i	$0.297965\pi$
$347$	30.0000i	1.61048i	−0.592946	+	0.805242i	$0.702035\pi$
$348$	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
$353$	−36.0000	−1.91609	−0.958043	−	0.286623i	$-0.907467\pi$
$353$	−36.0000	−1.91609	−0.958043	+	0.286623i	$0.907467\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	4.00000i	0.211702i
$358$	0	0
$359$	−10.0000	−0.527780	−0.263890	−	0.964553i	$-0.585006\pi$
$359$	−10.0000	−0.527780	−0.263890	+	0.964553i	$0.585006\pi$
$360$	0	0
$361$	3.00000	0.157895
$362$	0	0
$363$	− 7.00000i	− 0.367405i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−8.00000	−0.417597	−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000	−0.417597	−0.208798	+	0.977959i	$0.566955\pi$
$368$	0	0
$369$	8.00000	0.416463
$370$	0	0
$371$	6.00000i	0.311504i
$372$	0	0
$373$	14.0000i	0.724893i	0.932005	+	0.362446i	$0.118058\pi$
$373$	14.0000i	0.724893i	−0.932005	+	0.362446i	$0.881942\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−4.00000	−0.206010
$378$	0	0
$379$	16.0000i	0.821865i	0.911666	+	0.410932i	$0.134797\pi$
$379$	16.0000i	0.821865i	−0.911666	+	0.410932i	$0.865203\pi$
$380$	0	0
$381$	12.0000i	0.614779i
$382$	0	0
$383$	8.00000	0.408781	0.204390	−	0.978889i	$-0.434479\pi$
$383$	8.00000	0.408781	0.204390	+	0.978889i	$0.434479\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 8.00000i	− 0.406663i
$388$	0	0
$389$	− 26.0000i	− 1.31825i	−0.752032	−	0.659126i	$-0.770926\pi$
$389$	− 26.0000i	− 1.31825i	0.752032	−	0.659126i	$-0.229074\pi$
$390$	0	0
$391$	−24.0000	−1.21373
$392$	0	0
$393$	20.0000	1.00887
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 18.0000i	− 0.903394i	−0.892171	−	0.451697i	$-0.850819\pi$
$397$	− 18.0000i	− 0.903394i	0.892171	−	0.451697i	$-0.149181\pi$
$398$	0	0
$399$	4.00000	0.200250
$400$	0	0
$401$	14.0000	0.699127	0.349563	−	0.936913i	$-0.386330\pi$
$401$	14.0000	0.699127	0.349563	+	0.936913i	$0.386330\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	12.0000	0.594818
$408$	0	0
$409$	−14.0000	−0.692255	−0.346128	−	0.938187i	$-0.612504\pi$
$409$	−14.0000	−0.692255	−0.346128	+	0.938187i	$0.612504\pi$
$410$	0	0
$411$	− 18.0000i	− 0.887875i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−12.0000	−0.587643
$418$	0	0
$419$	8.00000i	0.390826i	0.980721	+	0.195413i	$0.0626047\pi$
$419$	8.00000i	0.390826i	−0.980721	+	0.195413i	$0.937395\pi$
$420$	0	0
$421$	10.0000i	0.487370i	0.969854	+	0.243685i	$0.0783563\pi$
$421$	10.0000i	0.487370i	−0.969854	+	0.243685i	$0.921644\pi$
$422$	0	0
$423$	−4.00000	−0.194487
$424$	0	0
$425$	20.0000	0.970143
$426$	0	0
$427$	− 14.0000i	− 0.677507i
$428$	0	0
$429$	4.00000i	0.193122i
$430$	0	0
$431$	18.0000	0.867029	0.433515	−	0.901146i	$-0.357273\pi$
$431$	18.0000	0.867029	0.433515	+	0.901146i	$0.357273\pi$
$432$	0	0
$433$	2.00000	0.0961139	0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000	0.0961139	0.0480569	+	0.998845i	$0.484697\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	24.0000i	1.14808i
$438$	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
$443$	22.0000i	1.04525i	0.852562	+	0.522626i	$0.175047\pi$
$443$	22.0000i	1.04525i	−0.852562	+	0.522626i	$0.824953\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	10.0000	0.472984
$448$	0	0
$449$	−6.00000	−0.283158	−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000	−0.283158	−0.141579	+	0.989927i	$0.545218\pi$
$450$	0	0
$451$	− 16.0000i	− 0.753411i
$452$	0	0
$453$	− 16.0000i	− 0.751746i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−10.0000	−0.467780	−0.233890	−	0.972263i	$-0.575146\pi$
$457$	−10.0000	−0.467780	−0.233890	+	0.972263i	$0.575146\pi$
$458$	0	0
$459$	4.00000i	0.186704i
$460$	0	0
$461$	36.0000i	1.67669i	0.545142	+	0.838344i	$0.316476\pi$
$461$	36.0000i	1.67669i	−0.545142	+	0.838344i	$0.683524\pi$
$462$	0	0
$463$	4.00000	0.185896	0.0929479	−	0.995671i	$-0.470371\pi$
$463$	4.00000	0.185896	0.0929479	+	0.995671i	$0.470371\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	32.0000i	1.48078i	0.672176	+	0.740392i	$0.265360\pi$
$467$	32.0000i	1.48078i	−0.672176	+	0.740392i	$0.734640\pi$
$468$	0	0
$469$	− 4.00000i	− 0.184703i
$470$	0	0
$471$	−2.00000	−0.0921551
$472$	0	0
$473$	−16.0000	−0.735681
$474$	0	0
$475$	− 20.0000i	− 0.917663i
$476$	0	0
$477$	6.00000i	0.274721i
$478$	0	0
$479$	−36.0000	−1.64488	−0.822441	−	0.568850i	$-0.807388\pi$
$479$	−36.0000	−1.64488	−0.822441	+	0.568850i	$0.807388\pi$
$480$	0	0
$481$	12.0000	0.547153
$482$	0	0
$483$	− 6.00000i	− 0.273009i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	40.0000	1.81257	0.906287	−	0.422664i	$-0.138905\pi$
$487$	40.0000	1.81257	0.906287	+	0.422664i	$0.138905\pi$
$488$	0	0
$489$	4.00000	0.180886
$490$	0	0
$491$	6.00000i	0.270776i	0.990793	+	0.135388i	$0.0432281\pi$
$491$	6.00000i	0.270776i	−0.990793	+	0.135388i	$0.956772\pi$
$492$	0	0
$493$	8.00000i	0.360302i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	2.00000	0.0897123
$498$	0	0
$499$	40.0000i	1.79065i	0.445418	+	0.895323i	$0.353055\pi$
$499$	40.0000i	1.79065i	−0.445418	+	0.895323i	$0.646945\pi$
$500$	0	0
$501$	12.0000i	0.536120i
$502$	0	0
$503$	−24.0000	−1.07011	−0.535054	−	0.844818i	$-0.679709\pi$
$503$	−24.0000	−1.07011	−0.535054	+	0.844818i	$0.679709\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 9.00000i	− 0.399704i
$508$	0	0
$509$	− 16.0000i	− 0.709188i	−0.935020	−	0.354594i	$-0.884619\pi$
$509$	− 16.0000i	− 0.709188i	0.935020	−	0.354594i	$-0.115381\pi$
$510$	0	0
$511$	−2.00000	−0.0884748
$512$	0	0
$513$	4.00000	0.176604
$514$	0	0
$515$	0	0
$516$	0	0
$517$	8.00000i	0.351840i
$518$	0	0
$519$	12.0000	0.526742
$520$	0	0
$521$	12.0000	0.525730	0.262865	−	0.964833i	$-0.415333\pi$
$521$	12.0000	0.525730	0.262865	+	0.964833i	$0.415333\pi$
$522$	0	0
$523$	28.0000i	1.22435i	0.790721	+	0.612177i	$0.209706\pi$
$523$	28.0000i	1.22435i	−0.790721	+	0.612177i	$0.790294\pi$
$524$	0	0
$525$	5.00000i	0.218218i
$526$	0	0
$527$	0	0
$528$	0	0
$529$	13.0000	0.565217
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 16.0000i	− 0.693037i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	10.0000	0.431532
$538$	0	0
$539$	2.00000i	0.0861461i
$540$	0	0
$541$	14.0000i	0.601907i	0.953639	+	0.300954i	$0.0973049\pi$
$541$	14.0000i	0.601907i	−0.953639	+	0.300954i	$0.902695\pi$
$542$	0	0
$543$	14.0000	0.600798
$544$	0	0
$545$	0	0
$546$	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
$549$	− 14.0000i	− 0.597505i
$550$	0	0
$551$	8.00000	0.340811
$552$	0	0
$553$	4.00000	0.170097
$554$	0	0
$555$	0	0
$556$	0	0
$557$	22.0000i	0.932170i	0.884740	+	0.466085i	$0.154336\pi$
$557$	22.0000i	0.932170i	−0.884740	+	0.466085i	$0.845664\pi$
$558$	0	0
$559$	−16.0000	−0.676728
$560$	0	0
$561$	8.00000	0.337760
$562$	0	0
$563$	− 16.0000i	− 0.674320i	−0.941447	−	0.337160i	$-0.890534\pi$
$563$	− 16.0000i	− 0.674320i	0.941447	−	0.337160i	$-0.109466\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	10.0000	0.419222	0.209611	−	0.977785i	$-0.432780\pi$
$569$	10.0000	0.419222	0.209611	+	0.977785i	$0.432780\pi$
$570$	0	0
$571$	− 20.0000i	− 0.836974i	−0.908223	−	0.418487i	$-0.862561\pi$
$571$	− 20.0000i	− 0.836974i	0.908223	−	0.418487i	$-0.137439\pi$
$572$	0	0
$573$	− 10.0000i	− 0.417756i
$574$	0	0
$575$	−30.0000	−1.25109
$576$	0	0
$577$	10.0000	0.416305	0.208153	−	0.978096i	$-0.433255\pi$
$577$	10.0000	0.416305	0.208153	+	0.978096i	$0.433255\pi$
$578$	0	0
$579$	26.0000i	1.08052i
$580$	0	0
$581$	− 12.0000i	− 0.497844i
$582$	0	0
$583$	12.0000	0.496989
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 24.0000i	− 0.990586i	−0.868726	−	0.495293i	$-0.835061\pi$
$587$	− 24.0000i	− 0.990586i	0.868726	−	0.495293i	$-0.164939\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	2.00000	0.0822690
$592$	0	0
$593$	−4.00000	−0.164260	−0.0821302	−	0.996622i	$-0.526172\pi$
$593$	−4.00000	−0.164260	−0.0821302	+	0.996622i	$0.526172\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 24.0000i	− 0.982255i
$598$	0	0
$599$	38.0000	1.55264	0.776319	−	0.630340i	$-0.217085\pi$
$599$	38.0000	1.55264	0.776319	+	0.630340i	$0.217085\pi$
$600$	0	0
$601$	−18.0000	−0.734235	−0.367118	−	0.930175i	$-0.619655\pi$
$601$	−18.0000	−0.734235	−0.367118	+	0.930175i	$0.619655\pi$
$602$	0	0
$603$	− 4.00000i	− 0.162893i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	40.0000	1.62355	0.811775	−	0.583970i	$-0.198502\pi$
$607$	40.0000	1.62355	0.811775	+	0.583970i	$0.198502\pi$
$608$	0	0
$609$	−2.00000	−0.0810441
$610$	0	0
$611$	8.00000i	0.323645i
$612$	0	0
$613$	22.0000i	0.888572i	0.895885	+	0.444286i	$0.146543\pi$
$613$	22.0000i	0.888572i	−0.895885	+	0.444286i	$0.853457\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−22.0000	−0.885687	−0.442843	−	0.896599i	$-0.646030\pi$
$617$	−22.0000	−0.885687	−0.442843	+	0.896599i	$0.646030\pi$
$618$	0	0
$619$	− 28.0000i	− 1.12542i	−0.826656	−	0.562708i	$-0.809760\pi$
$619$	− 28.0000i	− 1.12542i	0.826656	−	0.562708i	$-0.190240\pi$
$620$	0	0
$621$	− 6.00000i	− 0.240772i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	− 8.00000i	− 0.319489i
$628$	0	0
$629$	− 24.0000i	− 0.956943i
$630$	0	0
$631$	20.0000	0.796187	0.398094	−	0.917345i	$-0.369672\pi$
$631$	20.0000	0.796187	0.398094	+	0.917345i	$0.369672\pi$
$632$	0	0
$633$	−24.0000	−0.953914
$634$	0	0
$635$	0	0
$636$	0	0
$637$	2.00000i	0.0792429i
$638$	0	0
$639$	2.00000	0.0791188
$640$	0	0
$641$	22.0000	0.868948	0.434474	−	0.900684i	$-0.356934\pi$
$641$	22.0000	0.868948	0.434474	+	0.900684i	$0.356934\pi$
$642$	0	0
$643$	4.00000i	0.157745i	0.996885	+	0.0788723i	$0.0251319\pi$
$643$	4.00000i	0.157745i	−0.996885	+	0.0788723i	$0.974868\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	12.0000	0.471769	0.235884	−	0.971781i	$-0.424201\pi$
$647$	12.0000	0.471769	0.235884	+	0.971781i	$0.424201\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 46.0000i	− 1.80012i	−0.435767	−	0.900060i	$-0.643523\pi$
$653$	− 46.0000i	− 1.80012i	0.435767	−	0.900060i	$-0.356477\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−2.00000	−0.0780274
$658$	0	0
$659$	6.00000i	0.233727i	0.993148	+	0.116863i	$0.0372840\pi$
$659$	6.00000i	0.233727i	−0.993148	+	0.116863i	$0.962716\pi$
$660$	0	0
$661$	− 30.0000i	− 1.16686i	−0.812162	−	0.583432i	$-0.801709\pi$
$661$	− 30.0000i	− 1.16686i	0.812162	−	0.583432i	$-0.198291\pi$
$662$	0	0
$663$	8.00000	0.310694
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 12.0000i	− 0.464642i
$668$	0	0
$669$	− 16.0000i	− 0.618596i
$670$	0	0
$671$	−28.0000	−1.08093
$672$	0	0
$673$	−34.0000	−1.31060	−0.655302	−	0.755367i	$-0.727459\pi$
$673$	−34.0000	−1.31060	−0.655302	+	0.755367i	$0.727459\pi$
$674$	0	0
$675$	5.00000i	0.192450i
$676$	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
$679$	−6.00000	−0.230259
$680$	0	0
$681$	−8.00000	−0.306561
$682$	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
$685$	0	0
$686$	0	0
$687$	−2.00000	−0.0763048
$688$	0	0
$689$	12.0000	0.457164
$690$	0	0
$691$	− 20.0000i	− 0.760836i	−0.924815	−	0.380418i	$-0.875780\pi$
$691$	− 20.0000i	− 0.760836i	0.924815	−	0.380418i	$-0.124220\pi$
$692$	0	0
$693$	2.00000i	0.0759737i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−32.0000	−1.21209
$698$	0	0
$699$	− 26.0000i	− 0.983410i
$700$	0	0
$701$	26.0000i	0.982006i	0.871158	+	0.491003i	$0.163370\pi$
$701$	26.0000i	0.982006i	−0.871158	+	0.491003i	$0.836630\pi$
$702$	0	0
$703$	−24.0000	−0.905177
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 12.0000i	− 0.451306i
$708$	0	0
$709$	− 26.0000i	− 0.976450i	−0.872718	−	0.488225i	$-0.837644\pi$
$709$	− 26.0000i	− 0.976450i	0.872718	−	0.488225i	$-0.162356\pi$
$710$	0	0
$711$	4.00000	0.150012
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	26.0000i	0.970988i
$718$	0	0
$719$	−48.0000	−1.79010	−0.895049	−	0.445968i	$-0.852860\pi$
$719$	−48.0000	−1.79010	−0.895049	+	0.445968i	$0.852860\pi$
$720$	0	0
$721$	8.00000	0.297936
$722$	0	0
$723$	− 22.0000i	− 0.818189i
$724$	0	0
$725$	10.0000i	0.371391i
$726$	0	0
$727$	−8.00000	−0.296704	−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000	−0.296704	−0.148352	+	0.988935i	$0.547397\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	32.0000i	1.18356i
$732$	0	0
$733$	− 6.00000i	− 0.221615i	−0.993842	−	0.110808i	$-0.964656\pi$
$733$	− 6.00000i	− 0.221615i	0.993842	−	0.110808i	$-0.0353437\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−8.00000	−0.294684
$738$	0	0
$739$	36.0000i	1.32428i	0.749380	+	0.662141i	$0.230352\pi$
$739$	36.0000i	1.32428i	−0.749380	+	0.662141i	$0.769648\pi$
$740$	0	0
$741$	− 8.00000i	− 0.293887i
$742$	0	0
$743$	6.00000	0.220119	0.110059	−	0.993925i	$-0.464896\pi$
$743$	6.00000	0.220119	0.110059	+	0.993925i	$0.464896\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 12.0000i	− 0.439057i
$748$	0	0
$749$	6.00000i	0.219235i
$750$	0	0
$751$	16.0000	0.583848	0.291924	−	0.956441i	$-0.405705\pi$
$751$	16.0000	0.583848	0.291924	+	0.956441i	$0.405705\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 10.0000i	− 0.363456i	−0.983349	−	0.181728i	$-0.941831\pi$
$757$	− 10.0000i	− 0.363456i	0.983349	−	0.181728i	$-0.0581691\pi$
$758$	0	0
$759$	−12.0000	−0.435572
$760$	0	0
$761$	−40.0000	−1.45000	−0.724999	−	0.688749i	$-0.758160\pi$
$761$	−40.0000	−1.45000	−0.724999	+	0.688749i	$0.758160\pi$
$762$	0	0
$763$	− 18.0000i	− 0.651644i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−46.0000	−1.65880	−0.829401	−	0.558653i	$-0.811318\pi$
$769$	−46.0000	−1.65880	−0.829401	+	0.558653i	$0.811318\pi$
$770$	0	0
$771$	8.00000i	0.288113i
$772$	0	0
$773$	− 12.0000i	− 0.431610i	−0.976436	−	0.215805i	$-0.930762\pi$
$773$	− 12.0000i	− 0.431610i	0.976436	−	0.215805i	$-0.0692376\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	6.00000	0.215249
$778$	0	0
$779$	32.0000i	1.14652i
$780$	0	0
$781$	− 4.00000i	− 0.143131i
$782$	0	0
$783$	−2.00000	−0.0714742
$784$	0	0
$785$	0	0
$786$	0	0
$787$	28.0000i	0.998092i	0.866575	+	0.499046i	$0.166316\pi$
$787$	28.0000i	0.998092i	−0.866575	+	0.499046i	$0.833684\pi$
$788$	0	0
$789$	14.0000i	0.498413i
$790$	0	0
$791$	−18.0000	−0.640006
$792$	0	0
$793$	−28.0000	−0.994309
$794$	0	0
$795$	0	0
$796$	0	0
$797$	12.0000i	0.425062i	0.977154	+	0.212531i	$0.0681706\pi$
$797$	12.0000i	0.425062i	−0.977154	+	0.212531i	$0.931829\pi$
$798$	0	0
$799$	16.0000	0.566039
$800$	0	0
$801$	0	0
$802$	0	0
$803$	4.00000i	0.141157i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−16.0000	−0.563227
$808$	0	0
$809$	−50.0000	−1.75791	−0.878953	−	0.476908i	$-0.841757\pi$
$809$	−50.0000	−1.75791	−0.878953	+	0.476908i	$0.841757\pi$
$810$	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
$813$	− 32.0000i	− 1.12229i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	32.0000	1.11954
$818$	0	0
$819$	2.00000i	0.0698857i
$820$	0	0
$821$	− 30.0000i	− 1.04701i	−0.852023	−	0.523504i	$-0.824625\pi$
$821$	− 30.0000i	− 1.04701i	0.852023	−	0.523504i	$-0.175375\pi$
$822$	0	0
$823$	−4.00000	−0.139431	−0.0697156	−	0.997567i	$-0.522209\pi$
$823$	−4.00000	−0.139431	−0.0697156	+	0.997567i	$0.522209\pi$
$824$	0	0
$825$	10.0000	0.348155
$826$	0	0
$827$	42.0000i	1.46048i	0.683189	+	0.730242i	$0.260592\pi$
$827$	42.0000i	1.46048i	−0.683189	+	0.730242i	$0.739408\pi$
$828$	0	0
$829$	50.0000i	1.73657i	0.496064	+	0.868286i	$0.334778\pi$
$829$	50.0000i	1.73657i	−0.496064	+	0.868286i	$0.665222\pi$
$830$	0	0
$831$	10.0000	0.346896
$832$	0	0
$833$	4.00000	0.138592
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−4.00000	−0.138095	−0.0690477	−	0.997613i	$-0.521996\pi$
$839$	−4.00000	−0.138095	−0.0690477	+	0.997613i	$0.521996\pi$
$840$	0	0
$841$	25.0000	0.862069
$842$	0	0
$843$	− 2.00000i	− 0.0688837i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−7.00000	−0.240523
$848$	0	0
$849$	4.00000	0.137280
$850$	0	0
$851$	36.0000i	1.23406i
$852$	0	0
$853$	− 30.0000i	− 1.02718i	−0.858036	−	0.513590i	$-0.828315\pi$
$853$	− 30.0000i	− 1.02718i	0.858036	−	0.513590i	$-0.171685\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	40.0000	1.36637	0.683187	−	0.730243i	$-0.260593\pi$
$857$	40.0000	1.36637	0.683187	+	0.730243i	$0.260593\pi$
$858$	0	0
$859$	4.00000i	0.136478i	0.997669	+	0.0682391i	$0.0217381\pi$
$859$	4.00000i	0.136478i	−0.997669	+	0.0682391i	$0.978262\pi$
$860$	0	0
$861$	− 8.00000i	− 0.272639i
$862$	0	0
$863$	6.00000	0.204242	0.102121	−	0.994772i	$-0.467437\pi$
$863$	6.00000	0.204242	0.102121	+	0.994772i	$0.467437\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	1.00000i	0.0339618i
$868$	0	0
$869$	− 8.00000i	− 0.271381i
$870$	0	0
$871$	−8.00000	−0.271070
$872$	0	0
$873$	−6.00000	−0.203069
$874$	0	0
$875$	0	0
$876$	0	0
$877$	2.00000i	0.0675352i	0.999430	+	0.0337676i	$0.0107506\pi$
$877$	2.00000i	0.0675352i	−0.999430	+	0.0337676i	$0.989249\pi$
$878$	0	0
$879$	28.0000	0.944417
$880$	0	0
$881$	−12.0000	−0.404290	−0.202145	−	0.979356i	$-0.564791\pi$
$881$	−12.0000	−0.404290	−0.202145	+	0.979356i	$0.564791\pi$
$882$	0	0
$883$	− 56.0000i	− 1.88455i	−0.334840	−	0.942275i	$-0.608682\pi$
$883$	− 56.0000i	− 1.88455i	0.334840	−	0.942275i	$-0.391318\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−12.0000	−0.402921	−0.201460	−	0.979497i	$-0.564569\pi$
$887$	−12.0000	−0.402921	−0.201460	+	0.979497i	$0.564569\pi$
$888$	0	0
$889$	12.0000	0.402467
$890$	0	0
$891$	2.00000i	0.0670025i
$892$	0	0
$893$	− 16.0000i	− 0.535420i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−12.0000	−0.400668
$898$	0	0
$899$	0	0
$900$	0	0
$901$	− 24.0000i	− 0.799556i
$902$	0	0
$903$	−8.00000	−0.266223
$904$	0	0
$905$	0	0
$906$	0	0
$907$	16.0000i	0.531271i	0.964073	+	0.265636i	$0.0855818\pi$
$907$	16.0000i	0.531271i	−0.964073	+	0.265636i	$0.914418\pi$
$908$	0	0
$909$	− 12.0000i	− 0.398015i
$910$	0	0
$911$	−14.0000	−0.463841	−0.231920	−	0.972735i	$-0.574501\pi$
$911$	−14.0000	−0.463841	−0.231920	+	0.972735i	$0.574501\pi$
$912$	0	0
$913$	−24.0000	−0.794284
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 20.0000i	− 0.660458i
$918$	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
$921$	−28.0000	−0.922631
$922$	0	0
$923$	− 4.00000i	− 0.131662i
$924$	0	0
$925$	− 30.0000i	− 0.986394i
$926$	0	0
$927$	8.00000	0.262754
$928$	0	0
$929$	−40.0000	−1.31236	−0.656179	−	0.754606i	$-0.727828\pi$
$929$	−40.0000	−1.31236	−0.656179	+	0.754606i	$0.727828\pi$
$930$	0	0
$931$	− 4.00000i	− 0.131095i
$932$	0	0
$933$	28.0000i	0.916679i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	30.0000	0.980057	0.490029	−	0.871706i	$-0.336986\pi$
$937$	30.0000	0.980057	0.490029	+	0.871706i	$0.336986\pi$
$938$	0	0
$939$	− 14.0000i	− 0.456873i
$940$	0	0
$941$	− 56.0000i	− 1.82555i	−0.408465	−	0.912774i	$-0.633936\pi$
$941$	− 56.0000i	− 1.82555i	0.408465	−	0.912774i	$-0.366064\pi$
$942$	0	0
$943$	48.0000	1.56310
$944$	0	0
$945$	0	0
$946$	0	0
$947$	14.0000i	0.454939i	0.973785	+	0.227469i	$0.0730452\pi$
$947$	14.0000i	0.454939i	−0.973785	+	0.227469i	$0.926955\pi$
$948$	0	0
$949$	4.00000i	0.129845i
$950$	0	0
$951$	14.0000	0.453981
$952$	0	0
$953$	−26.0000	−0.842223	−0.421111	−	0.907009i	$-0.638360\pi$
$953$	−26.0000	−0.842223	−0.421111	+	0.907009i	$0.638360\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	4.00000i	0.129302i
$958$	0	0
$959$	−18.0000	−0.581250
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	6.00000i	0.193347i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−52.0000	−1.67221	−0.836104	−	0.548572i	$-0.815172\pi$
$967$	−52.0000	−1.67221	−0.836104	+	0.548572i	$0.815172\pi$
$968$	0	0
$969$	−16.0000	−0.513994
$970$	0	0
$971$	28.0000i	0.898563i	0.893390	+	0.449281i	$0.148320\pi$
$971$	28.0000i	0.898563i	−0.893390	+	0.449281i	$0.851680\pi$
$972$	0	0
$973$	12.0000i	0.384702i
$974$	0	0
$975$	10.0000	0.320256
$976$	0	0
$977$	54.0000	1.72761	0.863807	−	0.503824i	$-0.168074\pi$
$977$	54.0000	1.72761	0.863807	+	0.503824i	$0.168074\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	− 18.0000i	− 0.574696i
$982$	0	0
$983$	−32.0000	−1.02064	−0.510321	−	0.859984i	$-0.670473\pi$
$983$	−32.0000	−1.02064	−0.510321	+	0.859984i	$0.670473\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	4.00000i	0.127321i
$988$	0	0
$989$	− 48.0000i	− 1.52631i
$990$	0	0
$991$	16.0000	0.508257	0.254128	−	0.967170i	$-0.418211\pi$
$991$	16.0000	0.508257	0.254128	+	0.967170i	$0.418211\pi$
$992$	0	0
$993$	−16.0000	−0.507745
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 14.0000i	− 0.443384i	−0.975117	−	0.221692i	$-0.928842\pi$
$997$	− 14.0000i	− 0.443384i	0.975117	−	0.221692i	$-0.0711580\pi$
$998$	0	0
$999$	6.00000	0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5376.2.c.j.2689.1		2
4.3	odd	2		5376.2.c.y.2689.2		2
8.3	odd	2		5376.2.c.y.2689.1		2
8.5	even	2	inner	5376.2.c.j.2689.2		2
16.3	odd	4		672.2.a.c.1.1	✓	1
16.5	even	4		1344.2.a.e.1.1		1
16.11	odd	4		1344.2.a.p.1.1		1
16.13	even	4		672.2.a.g.1.1	yes	1
48.5	odd	4		4032.2.a.x.1.1		1
48.11	even	4		4032.2.a.q.1.1		1
48.29	odd	4		2016.2.a.i.1.1		1
48.35	even	4		2016.2.a.d.1.1		1
112.13	odd	4		4704.2.a.k.1.1		1
112.27	even	4		9408.2.a.w.1.1		1
112.69	odd	4		9408.2.a.ch.1.1		1
112.83	even	4		4704.2.a.z.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.2.a.c.1.1	✓	1	16.3	odd	4
672.2.a.g.1.1	yes	1	16.13	even	4
1344.2.a.e.1.1		1	16.5	even	4
1344.2.a.p.1.1		1	16.11	odd	4
2016.2.a.d.1.1		1	48.35	even	4
2016.2.a.i.1.1		1	48.29	odd	4
4032.2.a.q.1.1		1	48.11	even	4
4032.2.a.x.1.1		1	48.5	odd	4
4704.2.a.k.1.1		1	112.13	odd	4
4704.2.a.z.1.1		1	112.83	even	4
5376.2.c.j.2689.1		2	1.1	even	1	trivial
5376.2.c.j.2689.2		2	8.5	even	2	inner
5376.2.c.y.2689.1		2	8.3	odd	2
5376.2.c.y.2689.2		2	4.3	odd	2
9408.2.a.w.1.1		1	112.27	even	4
9408.2.a.ch.1.1		1	112.69	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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} -1.00000 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} -1.00000 q^{7} -1.00000 q^{9} +2.00000i q^{11} +2.00000i q^{13} +4.00000 q^{17} -4.00000i q^{19} +1.00000i q^{21} -6.00000 q^{23} +5.00000 q^{25} +1.00000i q^{27} +2.00000i q^{29} +2.00000 q^{33} -6.00000i q^{37} +2.00000 q^{39} -8.00000 q^{41} +8.00000i q^{43} +4.00000 q^{47} +1.00000 q^{49} -4.00000i q^{51} -6.00000i q^{53} -4.00000 q^{57} +14.0000i q^{61} +1.00000 q^{63} +4.00000i q^{67} +6.00000i q^{69} -2.00000 q^{71} +2.00000 q^{73} -5.00000i q^{75} -2.00000i q^{77} -4.00000 q^{79} +1.00000 q^{81} +12.0000i q^{83} +2.00000 q^{87} -2.00000i q^{91} +6.00000 q^{97} -2.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{7} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^7 - 2 * q^9	\( 2 q - 2 q^{7} - 2 q^{9} + 8 q^{17} - 12 q^{23} + 10 q^{25} + 4 q^{33} + 4 q^{39} - 16 q^{41} + 8 q^{47} + 2 q^{49} - 8 q^{57} + 2 q^{63} - 4 q^{71} + 4 q^{73} - 8 q^{79} + 2 q^{81} + 4 q^{87} + 12 q^{97}+O(q^{100}) \) 2 * q - 2 * q^7 - 2 * q^9 + 8 * q^17 - 12 * q^23 + 10 * q^25 + 4 * q^33 + 4 * q^39 - 16 * q^41 + 8 * q^47 + 2 * q^49 - 8 * q^57 + 2 * q^63 - 4 * q^71 + 4 * q^73 - 8 * q^79 + 2 * q^81 + 4 * q^87 + 12 * q^97

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