LMFDB - Embedded newform 3042.2.b.a.1351.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3042,2,Mod(1351,3042)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3042, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3042.1351");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3042 = 2 \cdot 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3042.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$24.2904922949$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 26)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1351.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	3042.1351
Dual form			3042.2.b.a.1351.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} -3.00000i q^{5} -1.00000i q^{7} +1.00000i q^{8} +O(q^{10})$	$q-1.00000i q^{2} -1.00000 q^{4} -3.00000i q^{5} -1.00000i q^{7} +1.00000i q^{8} -3.00000 q^{10} -6.00000i q^{11} -1.00000 q^{14} +1.00000 q^{16} -3.00000 q^{17} -2.00000i q^{19} +3.00000i q^{20} -6.00000 q^{22} -4.00000 q^{25} +1.00000i q^{28} -6.00000 q^{29} +4.00000i q^{31} -1.00000i q^{32} +3.00000i q^{34} -3.00000 q^{35} -7.00000i q^{37} -2.00000 q^{38} +3.00000 q^{40} +1.00000 q^{43} +6.00000i q^{44} -3.00000i q^{47} +6.00000 q^{49} +4.00000i q^{50} -18.0000 q^{55} +1.00000 q^{56} +6.00000i q^{58} +6.00000i q^{59} +8.00000 q^{61} +4.00000 q^{62} -1.00000 q^{64} -14.0000i q^{67} +3.00000 q^{68} +3.00000i q^{70} -3.00000i q^{71} +2.00000i q^{73} -7.00000 q^{74} +2.00000i q^{76} -6.00000 q^{77} +8.00000 q^{79} -3.00000i q^{80} +12.0000i q^{83} +9.00000i q^{85} -1.00000i q^{86} +6.00000 q^{88} +6.00000i q^{89} -3.00000 q^{94} -6.00000 q^{95} +10.0000i q^{97} -6.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4}+O(q^{10}) $ 2 * q - 2 * q^4	$ 2 q - 2 q^{4} - 6 q^{10} - 2 q^{14} + 2 q^{16} - 6 q^{17} - 12 q^{22} - 8 q^{25} - 12 q^{29} - 6 q^{35} - 4 q^{38} + 6 q^{40} + 2 q^{43} + 12 q^{49} - 36 q^{55} + 2 q^{56} + 16 q^{61} + 8 q^{62} - 2 q^{64} + 6 q^{68} - 14 q^{74} - 12 q^{77} + 16 q^{79} + 12 q^{88} - 6 q^{94} - 12 q^{95}+O(q^{100}) $ 2 * q - 2 * q^4 - 6 * q^10 - 2 * q^14 + 2 * q^16 - 6 * q^17 - 12 * q^22 - 8 * q^25 - 12 * q^29 - 6 * q^35 - 4 * q^38 + 6 * q^40 + 2 * q^43 + 12 * q^49 - 36 * q^55 + 2 * q^56 + 16 * q^61 + 8 * q^62 - 2 * q^64 + 6 * q^68 - 14 * q^74 - 12 * q^77 + 16 * q^79 + 12 * q^88 - 6 * q^94 - 12 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3042\mathbb{Z}\right)^\times$.

$n$	$677$	$847$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 1.00000i	− 0.707107i
$2$	− 1.00000i	− 0.707107i
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	− 3.00000i	− 1.34164i	−0.741620	−	0.670820i	$-0.765942\pi$
$5$	− 3.00000i	− 1.34164i	0.741620	−	0.670820i	$-0.234058\pi$
$6$	0	0
$7$	− 1.00000i	− 0.377964i	−0.981981	−	0.188982i	$-0.939481\pi$
$7$	− 1.00000i	− 0.377964i	0.981981	−	0.188982i	$-0.0605189\pi$
$8$	1.00000i	0.353553i
$9$	0	0
$10$	−3.00000	−0.948683
$11$	− 6.00000i	− 1.80907i	−0.426401	−	0.904534i	$-0.640219\pi$
$11$	− 6.00000i	− 1.80907i	0.426401	−	0.904534i	$-0.359781\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	0	0
$19$	− 2.00000i	− 0.458831i	−0.973329	−	0.229416i	$-0.926318\pi$
$19$	− 2.00000i	− 0.458831i	0.973329	−	0.229416i	$-0.0736815\pi$
$20$	3.00000i	0.670820i
$21$	0	0
$22$	−6.00000	−1.27920
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	0	0
$27$	0	0
$28$	1.00000i	0.188982i
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	4.00000i	0.718421i	0.933257	+	0.359211i	$0.116954\pi$
$31$	4.00000i	0.718421i	−0.933257	+	0.359211i	$0.883046\pi$
$32$	− 1.00000i	− 0.176777i
$33$	0	0
$34$	3.00000i	0.514496i
$35$	−3.00000	−0.507093
$36$	0	0
$37$	− 7.00000i	− 1.15079i	−0.817875	−	0.575396i	$-0.804848\pi$
$37$	− 7.00000i	− 1.15079i	0.817875	−	0.575396i	$-0.195152\pi$
$38$	−2.00000	−0.324443
$39$	0	0
$40$	3.00000	0.474342
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	1.00000	0.152499	0.0762493	−	0.997089i	$-0.475706\pi$
$43$	1.00000	0.152499	0.0762493	+	0.997089i	$0.475706\pi$
$44$	6.00000i	0.904534i
$45$	0	0
$46$	0	0
$47$	− 3.00000i	− 0.437595i	−0.975770	−	0.218797i	$-0.929787\pi$
$47$	− 3.00000i	− 0.437595i	0.975770	−	0.218797i	$-0.0702134\pi$
$48$	0	0
$49$	6.00000	0.857143
$50$	4.00000i	0.565685i
$51$	0	0
$52$	0	0
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	−18.0000	−2.42712
$56$	1.00000	0.133631
$57$	0	0
$58$	6.00000i	0.787839i
$59$	6.00000i	0.781133i	0.920575	+	0.390567i	$0.127721\pi$
$59$	6.00000i	0.781133i	−0.920575	+	0.390567i	$0.872279\pi$
$60$	0	0
$61$	8.00000	1.02430	0.512148	−	0.858898i	$-0.328850\pi$
$61$	8.00000	1.02430	0.512148	+	0.858898i	$0.328850\pi$
$62$	4.00000	0.508001
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	− 14.0000i	− 1.71037i	−0.518321	−	0.855186i	$-0.673443\pi$
$67$	− 14.0000i	− 1.71037i	0.518321	−	0.855186i	$-0.326557\pi$
$68$	3.00000	0.363803
$69$	0	0
$70$	3.00000i	0.358569i
$71$	− 3.00000i	− 0.356034i	−0.984027	−	0.178017i	$-0.943032\pi$
$71$	− 3.00000i	− 0.356034i	0.984027	−	0.178017i	$-0.0569683\pi$
$72$	0	0
$73$	2.00000i	0.234082i	0.993127	+	0.117041i	$0.0373409\pi$
$73$	2.00000i	0.234082i	−0.993127	+	0.117041i	$0.962659\pi$
$74$	−7.00000	−0.813733
$75$	0	0
$76$	2.00000i	0.229416i
$77$	−6.00000	−0.683763
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	− 3.00000i	− 0.335410i
$81$	0	0
$82$	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
$85$	9.00000i	0.976187i
$86$	− 1.00000i	− 0.107833i
$87$	0	0
$88$	6.00000	0.639602
$89$	6.00000i	0.635999i	0.948091	+	0.317999i	$0.103011\pi$
$89$	6.00000i	0.635999i	−0.948091	+	0.317999i	$0.896989\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	−3.00000	−0.309426
$95$	−6.00000	−0.615587
$96$	0	0
$97$	10.0000i	1.01535i	0.861550	+	0.507673i	$0.169494\pi$
$97$	10.0000i	1.01535i	−0.861550	+	0.507673i	$0.830506\pi$
$98$	− 6.00000i	− 0.606092i
$99$	0	0
$100$	4.00000	0.400000
$101$	−12.0000	−1.19404	−0.597022	−	0.802225i	$-0.703650\pi$
$101$	−12.0000	−1.19404	−0.597022	+	0.802225i	$0.703650\pi$
$102$	0	0
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	0	0
$109$	7.00000i	0.670478i	0.942133	+	0.335239i	$0.108817\pi$
$109$	7.00000i	0.670478i	−0.942133	+	0.335239i	$0.891183\pi$
$110$	18.0000i	1.71623i
$111$	0	0
$112$	− 1.00000i	− 0.0944911i
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	0	0
$116$	6.00000	0.557086
$117$	0	0
$118$	6.00000	0.552345
$119$	3.00000i	0.275010i
$120$	0	0
$121$	−25.0000	−2.27273
$122$	− 8.00000i	− 0.724286i
$123$	0	0
$124$	− 4.00000i	− 0.359211i
$125$	− 3.00000i	− 0.268328i
$126$	0	0
$127$	−20.0000	−1.77471	−0.887357	−	0.461084i	$-0.847461\pi$
$127$	−20.0000	−1.77471	−0.887357	+	0.461084i	$0.847461\pi$
$128$	1.00000i	0.0883883i
$129$	0	0
$130$	0	0
$131$	21.0000	1.83478	0.917389	−	0.397991i	$-0.130293\pi$
$131$	21.0000	1.83478	0.917389	+	0.397991i	$0.130293\pi$
$132$	0	0
$133$	−2.00000	−0.173422
$134$	−14.0000	−1.20942
$135$	0	0
$136$	− 3.00000i	− 0.257248i
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	−13.0000	−1.10265	−0.551323	−	0.834292i	$-0.685877\pi$
$139$	−13.0000	−1.10265	−0.551323	+	0.834292i	$0.685877\pi$
$140$	3.00000	0.253546
$141$	0	0
$142$	−3.00000	−0.251754
$143$	0	0
$144$	0	0
$145$	18.0000i	1.49482i
$146$	2.00000	0.165521
$147$	0	0
$148$	7.00000i	0.575396i
$149$	− 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
$151$	17.0000i	1.38344i	0.722166	+	0.691720i	$0.243147\pi$
$151$	17.0000i	1.38344i	−0.722166	+	0.691720i	$0.756853\pi$
$152$	2.00000	0.162221
$153$	0	0
$154$	6.00000i	0.483494i
$155$	12.0000	0.963863
$156$	0	0
$157$	14.0000	1.11732	0.558661	−	0.829396i	$-0.311315\pi$
$157$	14.0000	1.11732	0.558661	+	0.829396i	$0.311315\pi$
$158$	− 8.00000i	− 0.636446i
$159$	0	0
$160$	−3.00000	−0.237171
$161$	0	0
$162$	0	0
$163$	− 16.0000i	− 1.25322i	−0.779334	−	0.626608i	$-0.784443\pi$
$163$	− 16.0000i	− 1.25322i	0.779334	−	0.626608i	$-0.215557\pi$
$164$	0	0
$165$	0	0
$166$	12.0000	0.931381
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	0	0
$170$	9.00000	0.690268
$171$	0	0
$172$	−1.00000	−0.0762493
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	4.00000i	0.302372i
$176$	− 6.00000i	− 0.452267i
$177$	0	0
$178$	6.00000	0.449719
$179$	3.00000	0.224231	0.112115	−	0.993695i	$-0.464237\pi$
$179$	3.00000	0.224231	0.112115	+	0.993695i	$0.464237\pi$
$180$	0	0
$181$	−20.0000	−1.48659	−0.743294	−	0.668965i	$-0.766738\pi$
$181$	−20.0000	−1.48659	−0.743294	+	0.668965i	$0.766738\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−21.0000	−1.54395
$186$	0	0
$187$	18.0000i	1.31629i
$188$	3.00000i	0.218797i
$189$	0	0
$190$	6.00000i	0.435286i
$191$	18.0000	1.30243	0.651217	−	0.758891i	$-0.274259\pi$
$191$	18.0000	1.30243	0.651217	+	0.758891i	$0.274259\pi$
$192$	0	0
$193$	− 4.00000i	− 0.287926i	−0.989583	−	0.143963i	$-0.954015\pi$
$193$	− 4.00000i	− 0.287926i	0.989583	−	0.143963i	$-0.0459847\pi$
$194$	10.0000	0.717958
$195$	0	0
$196$	−6.00000	−0.428571
$197$	3.00000i	0.213741i	0.994273	+	0.106871i	$0.0340831\pi$
$197$	3.00000i	0.213741i	−0.994273	+	0.106871i	$0.965917\pi$
$198$	0	0
$199$	−2.00000	−0.141776	−0.0708881	−	0.997484i	$-0.522583\pi$
$199$	−2.00000	−0.141776	−0.0708881	+	0.997484i	$0.522583\pi$
$200$	− 4.00000i	− 0.282843i
$201$	0	0
$202$	12.0000i	0.844317i
$203$	6.00000i	0.421117i
$204$	0	0
$205$	0	0
$206$	− 4.00000i	− 0.278693i
$207$	0	0
$208$	0	0
$209$	−12.0000	−0.830057
$210$	0	0
$211$	−13.0000	−0.894957	−0.447478	−	0.894295i	$-0.647678\pi$
$211$	−13.0000	−0.894957	−0.447478	+	0.894295i	$0.647678\pi$
$212$	0	0
$213$	0	0
$214$	12.0000i	0.820303i
$215$	− 3.00000i	− 0.204598i
$216$	0	0
$217$	4.00000	0.271538
$218$	7.00000	0.474100
$219$	0	0
$220$	18.0000	1.21356
$221$	0	0
$222$	0	0
$223$	19.0000i	1.27233i	0.771551	+	0.636167i	$0.219481\pi$
$223$	19.0000i	1.27233i	−0.771551	+	0.636167i	$0.780519\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	− 6.00000i	− 0.399114i
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	0	0
$229$	− 13.0000i	− 0.859064i	−0.903052	−	0.429532i	$-0.858679\pi$
$229$	− 13.0000i	− 0.859064i	0.903052	−	0.429532i	$-0.141321\pi$
$230$	0	0
$231$	0	0
$232$	− 6.00000i	− 0.393919i
$233$	−27.0000	−1.76883	−0.884414	−	0.466702i	$-0.845442\pi$
$233$	−27.0000	−1.76883	−0.884414	+	0.466702i	$0.845442\pi$
$234$	0	0
$235$	−9.00000	−0.587095
$236$	− 6.00000i	− 0.390567i
$237$	0	0
$238$	3.00000	0.194461
$239$	15.0000i	0.970269i	0.874439	+	0.485135i	$0.161229\pi$
$239$	15.0000i	0.970269i	−0.874439	+	0.485135i	$0.838771\pi$
$240$	0	0
$241$	− 10.0000i	− 0.644157i	−0.946713	−	0.322078i	$-0.895619\pi$
$241$	− 10.0000i	− 0.644157i	0.946713	−	0.322078i	$-0.104381\pi$
$242$	25.0000i	1.60706i
$243$	0	0
$244$	−8.00000	−0.512148
$245$	− 18.0000i	− 1.14998i
$246$	0	0
$247$	0	0
$248$	−4.00000	−0.254000
$249$	0	0
$250$	−3.00000	−0.189737
$251$	24.0000	1.51487	0.757433	−	0.652913i	$-0.226453\pi$
$251$	24.0000	1.51487	0.757433	+	0.652913i	$0.226453\pi$
$252$	0	0
$253$	0	0
$254$	20.0000i	1.25491i
$255$	0	0
$256$	1.00000	0.0625000
$257$	9.00000	0.561405	0.280702	−	0.959795i	$-0.409433\pi$
$257$	9.00000	0.561405	0.280702	+	0.959795i	$0.409433\pi$
$258$	0	0
$259$	−7.00000	−0.434959
$260$	0	0
$261$	0	0
$262$	− 21.0000i	− 1.29738i
$263$	12.0000	0.739952	0.369976	−	0.929041i	$-0.379366\pi$
$263$	12.0000	0.739952	0.369976	+	0.929041i	$0.379366\pi$
$264$	0	0
$265$	0	0
$266$	2.00000i	0.122628i
$267$	0	0
$268$	14.0000i	0.855186i
$269$	−24.0000	−1.46331	−0.731653	−	0.681677i	$-0.761251\pi$
$269$	−24.0000	−1.46331	−0.731653	+	0.681677i	$0.761251\pi$
$270$	0	0
$271$	11.0000i	0.668202i	0.942537	+	0.334101i	$0.108433\pi$
$271$	11.0000i	0.668202i	−0.942537	+	0.334101i	$0.891567\pi$
$272$	−3.00000	−0.181902
$273$	0	0
$274$	0	0
$275$	24.0000i	1.44725i
$276$	0	0
$277$	28.0000	1.68236	0.841178	−	0.540758i	$-0.181862\pi$
$277$	28.0000	1.68236	0.841178	+	0.540758i	$0.181862\pi$
$278$	13.0000i	0.779688i
$279$	0	0
$280$	− 3.00000i	− 0.179284i
$281$	6.00000i	0.357930i	0.983855	+	0.178965i	$0.0572749\pi$
$281$	6.00000i	0.357930i	−0.983855	+	0.178965i	$0.942725\pi$
$282$	0	0
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	3.00000i	0.178017i
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−8.00000	−0.470588
$290$	18.0000	1.05700
$291$	0	0
$292$	− 2.00000i	− 0.117041i
$293$	− 21.0000i	− 1.22683i	−0.789760	−	0.613417i	$-0.789795\pi$
$293$	− 21.0000i	− 1.22683i	0.789760	−	0.613417i	$-0.210205\pi$
$294$	0	0
$295$	18.0000	1.04800
$296$	7.00000	0.406867
$297$	0	0
$298$	−6.00000	−0.347571
$299$	0	0
$300$	0	0
$301$	− 1.00000i	− 0.0576390i
$302$	17.0000	0.978240
$303$	0	0
$304$	− 2.00000i	− 0.114708i
$305$	− 24.0000i	− 1.37424i
$306$	0	0
$307$	2.00000i	0.114146i	0.998370	+	0.0570730i	$0.0181768\pi$
$307$	2.00000i	0.114146i	−0.998370	+	0.0570730i	$0.981823\pi$
$308$	6.00000	0.341882
$309$	0	0
$310$	− 12.0000i	− 0.681554i
$311$	−30.0000	−1.70114	−0.850572	−	0.525859i	$-0.823744\pi$
$311$	−30.0000	−1.70114	−0.850572	+	0.525859i	$0.823744\pi$
$312$	0	0
$313$	−1.00000	−0.0565233	−0.0282617	−	0.999601i	$-0.508997\pi$
$313$	−1.00000	−0.0565233	−0.0282617	+	0.999601i	$0.508997\pi$
$314$	− 14.0000i	− 0.790066i
$315$	0	0
$316$	−8.00000	−0.450035
$317$	− 6.00000i	− 0.336994i	−0.985702	−	0.168497i	$-0.946109\pi$
$317$	− 6.00000i	− 0.336994i	0.985702	−	0.168497i	$-0.0538913\pi$
$318$	0	0
$319$	36.0000i	2.01561i
$320$	3.00000i	0.167705i
$321$	0	0
$322$	0	0
$323$	6.00000i	0.333849i
$324$	0	0
$325$	0	0
$326$	−16.0000	−0.886158
$327$	0	0
$328$	0	0
$329$	−3.00000	−0.165395
$330$	0	0
$331$	− 8.00000i	− 0.439720i	−0.975531	−	0.219860i	$-0.929440\pi$
$331$	− 8.00000i	− 0.439720i	0.975531	−	0.219860i	$-0.0705600\pi$
$332$	− 12.0000i	− 0.658586i
$333$	0	0
$334$	0	0
$335$	−42.0000	−2.29471
$336$	0	0
$337$	−23.0000	−1.25289	−0.626445	−	0.779466i	$-0.715491\pi$
$337$	−23.0000	−1.25289	−0.626445	+	0.779466i	$0.715491\pi$
$338$	0	0
$339$	0	0
$340$	− 9.00000i	− 0.488094i
$341$	24.0000	1.29967
$342$	0	0
$343$	− 13.0000i	− 0.701934i
$344$	1.00000i	0.0539164i
$345$	0	0
$346$	0	0
$347$	−3.00000	−0.161048	−0.0805242	−	0.996753i	$-0.525659\pi$
$347$	−3.00000	−0.161048	−0.0805242	+	0.996753i	$0.525659\pi$
$348$	0	0
$349$	− 19.0000i	− 1.01705i	−0.861048	−	0.508523i	$-0.830192\pi$
$349$	− 19.0000i	− 1.01705i	0.861048	−	0.508523i	$-0.169808\pi$
$350$	4.00000	0.213809
$351$	0	0
$352$	−6.00000	−0.319801
$353$	24.0000i	1.27739i	0.769460	+	0.638696i	$0.220526\pi$
$353$	24.0000i	1.27739i	−0.769460	+	0.638696i	$0.779474\pi$
$354$	0	0
$355$	−9.00000	−0.477670
$356$	− 6.00000i	− 0.317999i
$357$	0	0
$358$	− 3.00000i	− 0.158555i
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	15.0000	0.789474
$362$	20.0000i	1.05118i
$363$	0	0
$364$	0	0
$365$	6.00000	0.314054
$366$	0	0
$367$	26.0000	1.35719	0.678594	−	0.734513i	$-0.262589\pi$
$367$	26.0000	1.35719	0.678594	+	0.734513i	$0.262589\pi$
$368$	0	0
$369$	0	0
$370$	21.0000i	1.09174i
$371$	0	0
$372$	0	0
$373$	−4.00000	−0.207112	−0.103556	−	0.994624i	$-0.533022\pi$
$373$	−4.00000	−0.207112	−0.103556	+	0.994624i	$0.533022\pi$
$374$	18.0000	0.930758
$375$	0	0
$376$	3.00000	0.154713
$377$	0	0
$378$	0	0
$379$	− 20.0000i	− 1.02733i	−0.857991	−	0.513665i	$-0.828287\pi$
$379$	− 20.0000i	− 1.02733i	0.857991	−	0.513665i	$-0.171713\pi$
$380$	6.00000	0.307794
$381$	0	0
$382$	− 18.0000i	− 0.920960i
$383$	21.0000i	1.07305i	0.843884	+	0.536525i	$0.180263\pi$
$383$	21.0000i	1.07305i	−0.843884	+	0.536525i	$0.819737\pi$
$384$	0	0
$385$	18.0000i	0.917365i
$386$	−4.00000	−0.203595
$387$	0	0
$388$	− 10.0000i	− 0.507673i
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	0	0
$392$	6.00000i	0.303046i
$393$	0	0
$394$	3.00000	0.151138
$395$	− 24.0000i	− 1.20757i
$396$	0	0
$397$	− 34.0000i	− 1.70641i	−0.521575	−	0.853206i	$-0.674655\pi$
$397$	− 34.0000i	− 1.70641i	0.521575	−	0.853206i	$-0.325345\pi$
$398$	2.00000i	0.100251i
$399$	0	0
$400$	−4.00000	−0.200000
$401$	− 36.0000i	− 1.79775i	−0.438201	−	0.898877i	$-0.644384\pi$
$401$	− 36.0000i	− 1.79775i	0.438201	−	0.898877i	$-0.355616\pi$
$402$	0	0
$403$	0	0
$404$	12.0000	0.597022
$405$	0	0
$406$	6.00000	0.297775
$407$	−42.0000	−2.08186
$408$	0	0
$409$	− 32.0000i	− 1.58230i	−0.611623	−	0.791149i	$-0.709483\pi$
$409$	− 32.0000i	− 1.58230i	0.611623	−	0.791149i	$-0.290517\pi$
$410$	0	0
$411$	0	0
$412$	−4.00000	−0.197066
$413$	6.00000	0.295241
$414$	0	0
$415$	36.0000	1.76717
$416$	0	0
$417$	0	0
$418$	12.0000i	0.586939i
$419$	−9.00000	−0.439679	−0.219839	−	0.975536i	$-0.570553\pi$
$419$	−9.00000	−0.439679	−0.219839	+	0.975536i	$0.570553\pi$
$420$	0	0
$421$	− 17.0000i	− 0.828529i	−0.910156	−	0.414265i	$-0.864039\pi$
$421$	− 17.0000i	− 0.828529i	0.910156	−	0.414265i	$-0.135961\pi$
$422$	13.0000i	0.632830i
$423$	0	0
$424$	0	0
$425$	12.0000	0.582086
$426$	0	0
$427$	− 8.00000i	− 0.387147i
$428$	12.0000	0.580042
$429$	0	0
$430$	−3.00000	−0.144673
$431$	− 33.0000i	− 1.58955i	−0.606902	−	0.794777i	$-0.707588\pi$
$431$	− 33.0000i	− 1.58955i	0.606902	−	0.794777i	$-0.292412\pi$
$432$	0	0
$433$	25.0000	1.20142	0.600712	−	0.799466i	$-0.294884\pi$
$433$	25.0000	1.20142	0.600712	+	0.799466i	$0.294884\pi$
$434$	− 4.00000i	− 0.192006i
$435$	0	0
$436$	− 7.00000i	− 0.335239i
$437$	0	0
$438$	0	0
$439$	−26.0000	−1.24091	−0.620456	−	0.784241i	$-0.713053\pi$
$439$	−26.0000	−1.24091	−0.620456	+	0.784241i	$0.713053\pi$
$440$	− 18.0000i	− 0.858116i
$441$	0	0
$442$	0	0
$443$	−21.0000	−0.997740	−0.498870	−	0.866677i	$-0.666252\pi$
$443$	−21.0000	−0.997740	−0.498870	+	0.866677i	$0.666252\pi$
$444$	0	0
$445$	18.0000	0.853282
$446$	19.0000	0.899676
$447$	0	0
$448$	1.00000i	0.0472456i
$449$	− 6.00000i	− 0.283158i	−0.989927	−	0.141579i	$-0.954782\pi$
$449$	− 6.00000i	− 0.283158i	0.989927	−	0.141579i	$-0.0452178\pi$
$450$	0	0
$451$	0	0
$452$	−6.00000	−0.282216
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	10.0000i	0.467780i	0.972263	+	0.233890i	$0.0751456\pi$
$457$	10.0000i	0.467780i	−0.972263	+	0.233890i	$0.924854\pi$
$458$	−13.0000	−0.607450
$459$	0	0
$460$	0	0
$461$	9.00000i	0.419172i	0.977790	+	0.209586i	$0.0672116\pi$
$461$	9.00000i	0.419172i	−0.977790	+	0.209586i	$0.932788\pi$
$462$	0	0
$463$	− 40.0000i	− 1.85896i	−0.368875	−	0.929479i	$-0.620257\pi$
$463$	− 40.0000i	− 1.85896i	0.368875	−	0.929479i	$-0.379743\pi$
$464$	−6.00000	−0.278543
$465$	0	0
$466$	27.0000i	1.25075i
$467$	36.0000	1.66588	0.832941	−	0.553362i	$-0.186655\pi$
$467$	36.0000	1.66588	0.832941	+	0.553362i	$0.186655\pi$
$468$	0	0
$469$	−14.0000	−0.646460
$470$	9.00000i	0.415139i
$471$	0	0
$472$	−6.00000	−0.276172
$473$	− 6.00000i	− 0.275880i
$474$	0	0
$475$	8.00000i	0.367065i
$476$	− 3.00000i	− 0.137505i
$477$	0	0
$478$	15.0000	0.686084
$479$	21.0000i	0.959514i	0.877401	+	0.479757i	$0.159275\pi$
$479$	21.0000i	0.959514i	−0.877401	+	0.479757i	$0.840725\pi$
$480$	0	0
$481$	0	0
$482$	−10.0000	−0.455488
$483$	0	0
$484$	25.0000	1.13636
$485$	30.0000	1.36223
$486$	0	0
$487$	16.0000i	0.725029i	0.931978	+	0.362515i	$0.118082\pi$
$487$	16.0000i	0.725029i	−0.931978	+	0.362515i	$0.881918\pi$
$488$	8.00000i	0.362143i
$489$	0	0
$490$	−18.0000	−0.813157
$491$	−9.00000	−0.406164	−0.203082	−	0.979162i	$-0.565096\pi$
$491$	−9.00000	−0.406164	−0.203082	+	0.979162i	$0.565096\pi$
$492$	0	0
$493$	18.0000	0.810679
$494$	0	0
$495$	0	0
$496$	4.00000i	0.179605i
$497$	−3.00000	−0.134568
$498$	0	0
$499$	40.0000i	1.79065i	0.445418	+	0.895323i	$0.353055\pi$
$499$	40.0000i	1.79065i	−0.445418	+	0.895323i	$0.646945\pi$
$500$	3.00000i	0.134164i
$501$	0	0
$502$	− 24.0000i	− 1.07117i
$503$	30.0000	1.33763	0.668817	−	0.743427i	$-0.266801\pi$
$503$	30.0000	1.33763	0.668817	+	0.743427i	$0.266801\pi$
$504$	0	0
$505$	36.0000i	1.60198i
$506$	0	0
$507$	0	0
$508$	20.0000	0.887357
$509$	− 18.0000i	− 0.797836i	−0.916987	−	0.398918i	$-0.869386\pi$
$509$	− 18.0000i	− 0.797836i	0.916987	−	0.398918i	$-0.130614\pi$
$510$	0	0
$511$	2.00000	0.0884748
$512$	− 1.00000i	− 0.0441942i
$513$	0	0
$514$	− 9.00000i	− 0.396973i
$515$	− 12.0000i	− 0.528783i
$516$	0	0
$517$	−18.0000	−0.791639
$518$	7.00000i	0.307562i
$519$	0	0
$520$	0	0
$521$	9.00000	0.394297	0.197149	−	0.980374i	$-0.436832\pi$
$521$	9.00000	0.394297	0.197149	+	0.980374i	$0.436832\pi$
$522$	0	0
$523$	20.0000	0.874539	0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000	0.874539	0.437269	+	0.899331i	$0.355946\pi$
$524$	−21.0000	−0.917389
$525$	0	0
$526$	− 12.0000i	− 0.523225i
$527$	− 12.0000i	− 0.522728i
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	0	0
$532$	2.00000	0.0867110
$533$	0	0
$534$	0	0
$535$	36.0000i	1.55642i
$536$	14.0000	0.604708
$537$	0	0
$538$	24.0000i	1.03471i
$539$	− 36.0000i	− 1.55063i
$540$	0	0
$541$	11.0000i	0.472927i	0.971640	+	0.236463i	$0.0759884\pi$
$541$	11.0000i	0.472927i	−0.971640	+	0.236463i	$0.924012\pi$
$542$	11.0000	0.472490
$543$	0	0
$544$	3.00000i	0.128624i
$545$	21.0000	0.899541
$546$	0	0
$547$	17.0000	0.726868	0.363434	−	0.931620i	$-0.381604\pi$
$547$	17.0000	0.726868	0.363434	+	0.931620i	$0.381604\pi$
$548$	0	0
$549$	0	0
$550$	24.0000	1.02336
$551$	12.0000i	0.511217i
$552$	0	0
$553$	− 8.00000i	− 0.340195i
$554$	− 28.0000i	− 1.18961i
$555$	0	0
$556$	13.0000	0.551323
$557$	− 3.00000i	− 0.127114i	−0.997978	−	0.0635570i	$-0.979756\pi$
$557$	− 3.00000i	− 0.127114i	0.997978	−	0.0635570i	$-0.0202445\pi$
$558$	0	0
$559$	0	0
$560$	−3.00000	−0.126773
$561$	0	0
$562$	6.00000	0.253095
$563$	39.0000	1.64365	0.821827	−	0.569737i	$-0.192955\pi$
$563$	39.0000	1.64365	0.821827	+	0.569737i	$0.192955\pi$
$564$	0	0
$565$	− 18.0000i	− 0.757266i
$566$	− 4.00000i	− 0.168133i
$567$	0	0
$568$	3.00000	0.125877
$569$	15.0000	0.628833	0.314416	−	0.949285i	$-0.398191\pi$
$569$	15.0000	0.628833	0.314416	+	0.949285i	$0.398191\pi$
$570$	0	0
$571$	−5.00000	−0.209243	−0.104622	−	0.994512i	$-0.533363\pi$
$571$	−5.00000	−0.209243	−0.104622	+	0.994512i	$0.533363\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 38.0000i	− 1.58196i	−0.611842	−	0.790980i	$-0.709571\pi$
$577$	− 38.0000i	− 1.58196i	0.611842	−	0.790980i	$-0.290429\pi$
$578$	8.00000i	0.332756i
$579$	0	0
$580$	− 18.0000i	− 0.747409i
$581$	12.0000	0.497844
$582$	0	0
$583$	0	0
$584$	−2.00000	−0.0827606
$585$	0	0
$586$	−21.0000	−0.867502
$587$	24.0000i	0.990586i	0.868726	+	0.495293i	$0.164939\pi$
$587$	24.0000i	0.990586i	−0.868726	+	0.495293i	$0.835061\pi$
$588$	0	0
$589$	8.00000	0.329634
$590$	− 18.0000i	− 0.741048i
$591$	0	0
$592$	− 7.00000i	− 0.287698i
$593$	− 18.0000i	− 0.739171i	−0.929197	−	0.369586i	$-0.879500\pi$
$593$	− 18.0000i	− 0.739171i	0.929197	−	0.369586i	$-0.120500\pi$
$594$	0	0
$595$	9.00000	0.368964
$596$	6.00000i	0.245770i
$597$	0	0
$598$	0	0
$599$	−6.00000	−0.245153	−0.122577	−	0.992459i	$-0.539116\pi$
$599$	−6.00000	−0.245153	−0.122577	+	0.992459i	$0.539116\pi$
$600$	0	0
$601$	−19.0000	−0.775026	−0.387513	−	0.921864i	$-0.626666\pi$
$601$	−19.0000	−0.775026	−0.387513	+	0.921864i	$0.626666\pi$
$602$	−1.00000	−0.0407570
$603$	0	0
$604$	− 17.0000i	− 0.691720i
$605$	75.0000i	3.04918i
$606$	0	0
$607$	14.0000	0.568242	0.284121	−	0.958788i	$-0.408298\pi$
$607$	14.0000	0.568242	0.284121	+	0.958788i	$0.408298\pi$
$608$	−2.00000	−0.0811107
$609$	0	0
$610$	−24.0000	−0.971732
$611$	0	0
$612$	0	0
$613$	− 38.0000i	− 1.53481i	−0.641165	−	0.767403i	$-0.721549\pi$
$613$	− 38.0000i	− 1.53481i	0.641165	−	0.767403i	$-0.278451\pi$
$614$	2.00000	0.0807134
$615$	0	0
$616$	− 6.00000i	− 0.241747i
$617$	− 24.0000i	− 0.966204i	−0.875564	−	0.483102i	$-0.839510\pi$
$617$	− 24.0000i	− 0.966204i	0.875564	−	0.483102i	$-0.160490\pi$
$618$	0	0
$619$	− 28.0000i	− 1.12542i	−0.826656	−	0.562708i	$-0.809760\pi$
$619$	− 28.0000i	− 1.12542i	0.826656	−	0.562708i	$-0.190240\pi$
$620$	−12.0000	−0.481932
$621$	0	0
$622$	30.0000i	1.20289i
$623$	6.00000	0.240385
$624$	0	0
$625$	−29.0000	−1.16000
$626$	1.00000i	0.0399680i
$627$	0	0
$628$	−14.0000	−0.558661
$629$	21.0000i	0.837325i
$630$	0	0
$631$	29.0000i	1.15447i	0.816577	+	0.577236i	$0.195869\pi$
$631$	29.0000i	1.15447i	−0.816577	+	0.577236i	$0.804131\pi$
$632$	8.00000i	0.318223i
$633$	0	0
$634$	−6.00000	−0.238290
$635$	60.0000i	2.38103i
$636$	0	0
$637$	0	0
$638$	36.0000	1.42525
$639$	0	0
$640$	3.00000	0.118585
$641$	−18.0000	−0.710957	−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000	−0.710957	−0.355479	+	0.934684i	$0.615682\pi$
$642$	0	0
$643$	− 14.0000i	− 0.552106i	−0.961142	−	0.276053i	$-0.910973\pi$
$643$	− 14.0000i	− 0.552106i	0.961142	−	0.276053i	$-0.0890266\pi$
$644$	0	0
$645$	0	0
$646$	6.00000	0.236067
$647$	−6.00000	−0.235884	−0.117942	−	0.993020i	$-0.537630\pi$
$647$	−6.00000	−0.235884	−0.117942	+	0.993020i	$0.537630\pi$
$648$	0	0
$649$	36.0000	1.41312
$650$	0	0
$651$	0	0
$652$	16.0000i	0.626608i
$653$	0	0		−	1.00000i	$-0.5\pi$
$653$	0	0			1.00000i	$0.5\pi$
$654$	0	0
$655$	− 63.0000i	− 2.46161i
$656$	0	0
$657$	0	0
$658$	3.00000i	0.116952i
$659$	36.0000	1.40236	0.701180	−	0.712984i	$-0.252657\pi$
$659$	36.0000	1.40236	0.701180	+	0.712984i	$0.252657\pi$
$660$	0	0
$661$	− 22.0000i	− 0.855701i	−0.903850	−	0.427850i	$-0.859271\pi$
$661$	− 22.0000i	− 0.855701i	0.903850	−	0.427850i	$-0.140729\pi$
$662$	−8.00000	−0.310929
$663$	0	0
$664$	−12.0000	−0.465690
$665$	6.00000i	0.232670i
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	42.0000i	1.62260i
$671$	− 48.0000i	− 1.85302i
$672$	0	0
$673$	19.0000	0.732396	0.366198	−	0.930537i	$-0.380659\pi$
$673$	19.0000	0.732396	0.366198	+	0.930537i	$0.380659\pi$
$674$	23.0000i	0.885927i
$675$	0	0
$676$	0	0
$677$	−48.0000	−1.84479	−0.922395	−	0.386248i	$-0.873771\pi$
$677$	−48.0000	−1.84479	−0.922395	+	0.386248i	$0.873771\pi$
$678$	0	0
$679$	10.0000	0.383765
$680$	−9.00000	−0.345134
$681$	0	0
$682$	− 24.0000i	− 0.919007i
$683$	− 24.0000i	− 0.918334i	−0.888350	−	0.459167i	$-0.848148\pi$
$683$	− 24.0000i	− 0.918334i	0.888350	−	0.459167i	$-0.151852\pi$
$684$	0	0
$685$	0	0
$686$	−13.0000	−0.496342
$687$	0	0
$688$	1.00000	0.0381246
$689$	0	0
$690$	0	0
$691$	− 8.00000i	− 0.304334i	−0.988355	−	0.152167i	$-0.951375\pi$
$691$	− 8.00000i	− 0.304334i	0.988355	−	0.152167i	$-0.0486252\pi$
$692$	0	0
$693$	0	0
$694$	3.00000i	0.113878i
$695$	39.0000i	1.47935i
$696$	0	0
$697$	0	0
$698$	−19.0000	−0.719161
$699$	0	0
$700$	− 4.00000i	− 0.151186i
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	0	0
$703$	−14.0000	−0.528020
$704$	6.00000i	0.226134i
$705$	0	0
$706$	24.0000	0.903252
$707$	12.0000i	0.451306i
$708$	0	0
$709$	26.0000i	0.976450i	0.872718	+	0.488225i	$0.162356\pi$
$709$	26.0000i	0.976450i	−0.872718	+	0.488225i	$0.837644\pi$
$710$	9.00000i	0.337764i
$711$	0	0
$712$	−6.00000	−0.224860
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−3.00000	−0.112115
$717$	0	0
$718$	0	0
$719$	6.00000	0.223762	0.111881	−	0.993722i	$-0.464312\pi$
$719$	6.00000	0.223762	0.111881	+	0.993722i	$0.464312\pi$
$720$	0	0
$721$	− 4.00000i	− 0.148968i
$722$	− 15.0000i	− 0.558242i
$723$	0	0
$724$	20.0000	0.743294
$725$	24.0000	0.891338
$726$	0	0
$727$	10.0000	0.370879	0.185440	−	0.982656i	$-0.440629\pi$
$727$	10.0000	0.370879	0.185440	+	0.982656i	$0.440629\pi$
$728$	0	0
$729$	0	0
$730$	− 6.00000i	− 0.222070i
$731$	−3.00000	−0.110959
$732$	0	0
$733$	− 23.0000i	− 0.849524i	−0.905305	−	0.424762i	$-0.860358\pi$
$733$	− 23.0000i	− 0.849524i	0.905305	−	0.424762i	$-0.139642\pi$
$734$	− 26.0000i	− 0.959678i
$735$	0	0
$736$	0	0
$737$	−84.0000	−3.09418
$738$	0	0
$739$	20.0000i	0.735712i	0.929883	+	0.367856i	$0.119908\pi$
$739$	20.0000i	0.735712i	−0.929883	+	0.367856i	$0.880092\pi$
$740$	21.0000	0.771975
$741$	0	0
$742$	0	0
$743$	− 9.00000i	− 0.330178i	−0.986279	−	0.165089i	$-0.947209\pi$
$743$	− 9.00000i	− 0.330178i	0.986279	−	0.165089i	$-0.0527911\pi$
$744$	0	0
$745$	−18.0000	−0.659469
$746$	4.00000i	0.146450i
$747$	0	0
$748$	− 18.0000i	− 0.658145i
$749$	12.0000i	0.438470i
$750$	0	0
$751$	40.0000	1.45962	0.729810	−	0.683650i	$-0.239608\pi$
$751$	40.0000	1.45962	0.729810	+	0.683650i	$0.239608\pi$
$752$	− 3.00000i	− 0.109399i
$753$	0	0
$754$	0	0
$755$	51.0000	1.85608
$756$	0	0
$757$	−16.0000	−0.581530	−0.290765	−	0.956795i	$-0.593910\pi$
$757$	−16.0000	−0.581530	−0.290765	+	0.956795i	$0.593910\pi$
$758$	−20.0000	−0.726433
$759$	0	0
$760$	− 6.00000i	− 0.217643i
$761$	6.00000i	0.217500i	0.994069	+	0.108750i	$0.0346848\pi$
$761$	6.00000i	0.217500i	−0.994069	+	0.108750i	$0.965315\pi$
$762$	0	0
$763$	7.00000	0.253417
$764$	−18.0000	−0.651217
$765$	0	0
$766$	21.0000	0.758761
$767$	0	0
$768$	0	0
$769$	− 32.0000i	− 1.15395i	−0.816762	−	0.576975i	$-0.804233\pi$
$769$	− 32.0000i	− 1.15395i	0.816762	−	0.576975i	$-0.195767\pi$
$770$	18.0000	0.648675
$771$	0	0
$772$	4.00000i	0.143963i
$773$	− 39.0000i	− 1.40273i	−0.712801	−	0.701366i	$-0.752574\pi$
$773$	− 39.0000i	− 1.40273i	0.712801	−	0.701366i	$-0.247426\pi$
$774$	0	0
$775$	− 16.0000i	− 0.574737i
$776$	−10.0000	−0.358979
$777$	0	0
$778$	6.00000i	0.215110i
$779$	0	0
$780$	0	0
$781$	−18.0000	−0.644091
$782$	0	0
$783$	0	0
$784$	6.00000	0.214286
$785$	− 42.0000i	− 1.49904i
$786$	0	0
$787$	− 40.0000i	− 1.42585i	−0.701242	−	0.712923i	$-0.747371\pi$
$787$	− 40.0000i	− 1.42585i	0.701242	−	0.712923i	$-0.252629\pi$
$788$	− 3.00000i	− 0.106871i
$789$	0	0
$790$	−24.0000	−0.853882
$791$	− 6.00000i	− 0.213335i
$792$	0	0
$793$	0	0
$794$	−34.0000	−1.20661
$795$	0	0
$796$	2.00000	0.0708881
$797$	−42.0000	−1.48772	−0.743858	−	0.668338i	$-0.767006\pi$
$797$	−42.0000	−1.48772	−0.743858	+	0.668338i	$0.767006\pi$
$798$	0	0
$799$	9.00000i	0.318397i
$800$	4.00000i	0.141421i
$801$	0	0
$802$	−36.0000	−1.27120
$803$	12.0000	0.423471
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	− 12.0000i	− 0.422159i
$809$	33.0000	1.16022	0.580109	−	0.814539i	$-0.303010\pi$
$809$	33.0000	1.16022	0.580109	+	0.814539i	$0.303010\pi$
$810$	0	0
$811$	− 20.0000i	− 0.702295i	−0.936320	−	0.351147i	$-0.885792\pi$
$811$	− 20.0000i	− 0.702295i	0.936320	−	0.351147i	$-0.114208\pi$
$812$	− 6.00000i	− 0.210559i
$813$	0	0
$814$	42.0000i	1.47210i
$815$	−48.0000	−1.68137
$816$	0	0
$817$	− 2.00000i	− 0.0699711i
$818$	−32.0000	−1.11885
$819$	0	0
$820$	0	0
$821$	− 3.00000i	− 0.104701i	−0.998629	−	0.0523504i	$-0.983329\pi$
$821$	− 3.00000i	− 0.104701i	0.998629	−	0.0523504i	$-0.0166713\pi$
$822$	0	0
$823$	−14.0000	−0.488009	−0.244005	−	0.969774i	$-0.578461\pi$
$823$	−14.0000	−0.488009	−0.244005	+	0.969774i	$0.578461\pi$
$824$	4.00000i	0.139347i
$825$	0	0
$826$	− 6.00000i	− 0.208767i
$827$	− 18.0000i	− 0.625921i	−0.949766	−	0.312961i	$-0.898679\pi$
$827$	− 18.0000i	− 0.625921i	0.949766	−	0.312961i	$-0.101321\pi$
$828$	0	0
$829$	−38.0000	−1.31979	−0.659897	−	0.751356i	$-0.729400\pi$
$829$	−38.0000	−1.31979	−0.659897	+	0.751356i	$0.729400\pi$
$830$	− 36.0000i	− 1.24958i
$831$	0	0
$832$	0	0
$833$	−18.0000	−0.623663
$834$	0	0
$835$	0	0
$836$	12.0000	0.415029
$837$	0	0
$838$	9.00000i	0.310900i
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	−17.0000	−0.585859
$843$	0	0
$844$	13.0000	0.447478
$845$	0	0
$846$	0	0
$847$	25.0000i	0.859010i
$848$	0	0
$849$	0	0
$850$	− 12.0000i	− 0.411597i
$851$	0	0
$852$	0	0
$853$	− 37.0000i	− 1.26686i	−0.773802	−	0.633428i	$-0.781647\pi$
$853$	− 37.0000i	− 1.26686i	0.773802	−	0.633428i	$-0.218353\pi$
$854$	−8.00000	−0.273754
$855$	0	0
$856$	− 12.0000i	− 0.410152i
$857$	−42.0000	−1.43469	−0.717346	−	0.696717i	$-0.754643\pi$
$857$	−42.0000	−1.43469	−0.717346	+	0.696717i	$0.754643\pi$
$858$	0	0
$859$	−4.00000	−0.136478	−0.0682391	−	0.997669i	$-0.521738\pi$
$859$	−4.00000	−0.136478	−0.0682391	+	0.997669i	$0.521738\pi$
$860$	3.00000i	0.102299i
$861$	0	0
$862$	−33.0000	−1.12398
$863$	− 45.0000i	− 1.53182i	−0.642949	−	0.765909i	$-0.722289\pi$
$863$	− 45.0000i	− 1.53182i	0.642949	−	0.765909i	$-0.277711\pi$
$864$	0	0
$865$	0	0
$866$	− 25.0000i	− 0.849535i
$867$	0	0
$868$	−4.00000	−0.135769
$869$	− 48.0000i	− 1.62829i
$870$	0	0
$871$	0	0
$872$	−7.00000	−0.237050
$873$	0	0
$874$	0	0
$875$	−3.00000	−0.101419
$876$	0	0
$877$	13.0000i	0.438979i	0.975615	+	0.219489i	$0.0704391\pi$
$877$	13.0000i	0.438979i	−0.975615	+	0.219489i	$0.929561\pi$
$878$	26.0000i	0.877457i
$879$	0	0
$880$	−18.0000	−0.606780
$881$	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
$883$	−29.0000	−0.975928	−0.487964	−	0.872864i	$-0.662260\pi$
$883$	−29.0000	−0.975928	−0.487964	+	0.872864i	$0.662260\pi$
$884$	0	0
$885$	0	0
$886$	21.0000i	0.705509i
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	20.0000i	0.670778i
$890$	− 18.0000i	− 0.603361i
$891$	0	0
$892$	− 19.0000i	− 0.636167i
$893$	−6.00000	−0.200782
$894$	0	0
$895$	− 9.00000i	− 0.300837i
$896$	1.00000	0.0334077
$897$	0	0
$898$	−6.00000	−0.200223
$899$	− 24.0000i	− 0.800445i
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	6.00000i	0.199557i
$905$	60.0000i	1.99447i
$906$	0	0
$907$	37.0000	1.22856	0.614282	−	0.789086i	$-0.289446\pi$
$907$	37.0000	1.22856	0.614282	+	0.789086i	$0.289446\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−30.0000	−0.993944	−0.496972	−	0.867766i	$-0.665555\pi$
$911$	−30.0000	−0.993944	−0.496972	+	0.867766i	$0.665555\pi$
$912$	0	0
$913$	72.0000	2.38285
$914$	10.0000	0.330771
$915$	0	0
$916$	13.0000i	0.429532i
$917$	− 21.0000i	− 0.693481i
$918$	0	0
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	0	0
$921$	0	0
$922$	9.00000	0.296399
$923$	0	0
$924$	0	0
$925$	28.0000i	0.920634i
$926$	−40.0000	−1.31448
$927$	0	0
$928$	6.00000i	0.196960i
$929$	36.0000i	1.18112i	0.806993	+	0.590561i	$0.201093\pi$
$929$	36.0000i	1.18112i	−0.806993	+	0.590561i	$0.798907\pi$
$930$	0	0
$931$	− 12.0000i	− 0.393284i
$932$	27.0000	0.884414
$933$	0	0
$934$	− 36.0000i	− 1.17796i
$935$	54.0000	1.76599
$936$	0	0
$937$	−34.0000	−1.11073	−0.555366	−	0.831606i	$-0.687422\pi$
$937$	−34.0000	−1.11073	−0.555366	+	0.831606i	$0.687422\pi$
$938$	14.0000i	0.457116i
$939$	0	0
$940$	9.00000	0.293548
$941$	− 21.0000i	− 0.684580i	−0.939594	−	0.342290i	$-0.888797\pi$
$941$	− 21.0000i	− 0.684580i	0.939594	−	0.342290i	$-0.111203\pi$
$942$	0	0
$943$	0	0
$944$	6.00000i	0.195283i
$945$	0	0
$946$	−6.00000	−0.195077
$947$	− 6.00000i	− 0.194974i	−0.995237	−	0.0974869i	$-0.968920\pi$
$947$	− 6.00000i	− 0.194974i	0.995237	−	0.0974869i	$-0.0310804\pi$
$948$	0	0
$949$	0	0
$950$	8.00000	0.259554
$951$	0	0
$952$	−3.00000	−0.0972306
$953$	15.0000	0.485898	0.242949	−	0.970039i	$-0.421885\pi$
$953$	15.0000	0.485898	0.242949	+	0.970039i	$0.421885\pi$
$954$	0	0
$955$	− 54.0000i	− 1.74740i
$956$	− 15.0000i	− 0.485135i
$957$	0	0
$958$	21.0000	0.678479
$959$	0	0
$960$	0	0
$961$	15.0000	0.483871
$962$	0	0
$963$	0	0
$964$	10.0000i	0.322078i
$965$	−12.0000	−0.386294
$966$	0	0
$967$	31.0000i	0.996893i	0.866921	+	0.498446i	$0.166096\pi$
$967$	31.0000i	0.996893i	−0.866921	+	0.498446i	$0.833904\pi$
$968$	− 25.0000i	− 0.803530i
$969$	0	0
$970$	− 30.0000i	− 0.963242i
$971$	3.00000	0.0962746	0.0481373	−	0.998841i	$-0.484672\pi$
$971$	3.00000	0.0962746	0.0481373	+	0.998841i	$0.484672\pi$
$972$	0	0
$973$	13.0000i	0.416761i
$974$	16.0000	0.512673
$975$	0	0
$976$	8.00000	0.256074
$977$	− 54.0000i	− 1.72761i	−0.503824	−	0.863807i	$-0.668074\pi$
$977$	− 54.0000i	− 1.72761i	0.503824	−	0.863807i	$-0.331926\pi$
$978$	0	0
$979$	36.0000	1.15056
$980$	18.0000i	0.574989i
$981$	0	0
$982$	9.00000i	0.287202i
$983$	− 39.0000i	− 1.24391i	−0.783054	−	0.621953i	$-0.786339\pi$
$983$	− 39.0000i	− 1.24391i	0.783054	−	0.621953i	$-0.213661\pi$
$984$	0	0
$985$	9.00000	0.286764
$986$	− 18.0000i	− 0.573237i
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	2.00000	0.0635321	0.0317660	−	0.999495i	$-0.489887\pi$
$991$	2.00000	0.0635321	0.0317660	+	0.999495i	$0.489887\pi$
$992$	4.00000	0.127000
$993$	0	0
$994$	3.00000i	0.0951542i
$995$	6.00000i	0.190213i
$996$	0	0
$997$	−46.0000	−1.45683	−0.728417	−	0.685134i	$-0.759744\pi$
$997$	−46.0000	−1.45683	−0.728417	+	0.685134i	$0.759744\pi$
$998$	40.0000	1.26618
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3042.2.b.a.1351.1		2
3.2	odd	2		338.2.b.c.337.2		2
12.11	even	2		2704.2.f.d.337.2		2
13.5	odd	4		3042.2.a.a.1.1		1
13.8	odd	4		234.2.a.e.1.1		1
13.12	even	2	inner	3042.2.b.a.1351.2		2
39.2	even	12		338.2.c.a.191.1		2
39.5	even	4		338.2.a.f.1.1		1
39.8	even	4		26.2.a.a.1.1	✓	1
39.11	even	12		338.2.c.d.191.1		2
39.17	odd	6		338.2.e.a.23.1		4
39.20	even	12		338.2.c.d.315.1		2
39.23	odd	6		338.2.e.a.147.2		4
39.29	odd	6		338.2.e.a.147.1		4
39.32	even	12		338.2.c.a.315.1		2
39.35	odd	6		338.2.e.a.23.2		4
39.38	odd	2		338.2.b.c.337.1		2
52.47	even	4		1872.2.a.q.1.1		1
65.8	even	4		5850.2.e.a.5149.1		2
65.34	odd	4		5850.2.a.p.1.1		1
65.47	even	4		5850.2.e.a.5149.2		2
104.21	odd	4		7488.2.a.g.1.1		1
104.99	even	4		7488.2.a.h.1.1		1
117.34	odd	12		2106.2.e.b.1405.1		2
117.47	even	12		2106.2.e.ba.1405.1		2
117.86	even	12		2106.2.e.ba.703.1		2
117.112	odd	12		2106.2.e.b.703.1		2
156.47	odd	4		208.2.a.a.1.1		1
156.83	odd	4		2704.2.a.f.1.1		1
156.155	even	2		2704.2.f.d.337.1		2
195.8	odd	4		650.2.b.d.599.2		2
195.44	even	4		8450.2.a.c.1.1		1
195.47	odd	4		650.2.b.d.599.1		2
195.164	even	4		650.2.a.j.1.1		1
273.47	odd	12		1274.2.f.r.1145.1		2
273.86	even	12		1274.2.f.p.1145.1		2
273.125	odd	4		1274.2.a.d.1.1		1
273.164	odd	12		1274.2.f.r.79.1		2
273.242	even	12		1274.2.f.p.79.1		2
312.125	even	4		832.2.a.d.1.1		1
312.203	odd	4		832.2.a.i.1.1		1
429.164	odd	4		3146.2.a.n.1.1		1
624.125	even	4		3328.2.b.m.1665.2		2
624.203	odd	4		3328.2.b.j.1665.2		2
624.437	even	4		3328.2.b.m.1665.1		2
624.515	odd	4		3328.2.b.j.1665.1		2
663.203	even	4		7514.2.a.c.1.1		1
741.398	odd	4		9386.2.a.j.1.1		1
780.359	odd	4		5200.2.a.x.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
26.2.a.a.1.1	✓	1	39.8	even	4
208.2.a.a.1.1		1	156.47	odd	4
234.2.a.e.1.1		1	13.8	odd	4
338.2.a.f.1.1		1	39.5	even	4
338.2.b.c.337.1		2	39.38	odd	2
338.2.b.c.337.2		2	3.2	odd	2
338.2.c.a.191.1		2	39.2	even	12
338.2.c.a.315.1		2	39.32	even	12
338.2.c.d.191.1		2	39.11	even	12
338.2.c.d.315.1		2	39.20	even	12
338.2.e.a.23.1		4	39.17	odd	6
338.2.e.a.23.2		4	39.35	odd	6
338.2.e.a.147.1		4	39.29	odd	6
338.2.e.a.147.2		4	39.23	odd	6
650.2.a.j.1.1		1	195.164	even	4
650.2.b.d.599.1		2	195.47	odd	4
650.2.b.d.599.2		2	195.8	odd	4
832.2.a.d.1.1		1	312.125	even	4
832.2.a.i.1.1		1	312.203	odd	4
1274.2.a.d.1.1		1	273.125	odd	4
1274.2.f.p.79.1		2	273.242	even	12
1274.2.f.p.1145.1		2	273.86	even	12
1274.2.f.r.79.1		2	273.164	odd	12
1274.2.f.r.1145.1		2	273.47	odd	12
1872.2.a.q.1.1		1	52.47	even	4
2106.2.e.b.703.1		2	117.112	odd	12
2106.2.e.b.1405.1		2	117.34	odd	12
2106.2.e.ba.703.1		2	117.86	even	12
2106.2.e.ba.1405.1		2	117.47	even	12
2704.2.a.f.1.1		1	156.83	odd	4
2704.2.f.d.337.1		2	156.155	even	2
2704.2.f.d.337.2		2	12.11	even	2
3042.2.a.a.1.1		1	13.5	odd	4
3042.2.b.a.1351.1		2	1.1	even	1	trivial
3042.2.b.a.1351.2		2	13.12	even	2	inner
3146.2.a.n.1.1		1	429.164	odd	4
3328.2.b.j.1665.1		2	624.515	odd	4
3328.2.b.j.1665.2		2	624.203	odd	4
3328.2.b.m.1665.1		2	624.437	even	4
3328.2.b.m.1665.2		2	624.125	even	4
5200.2.a.x.1.1		1	780.359	odd	4
5850.2.a.p.1.1		1	65.34	odd	4
5850.2.e.a.5149.1		2	65.8	even	4
5850.2.e.a.5149.2		2	65.47	even	4
7488.2.a.g.1.1		1	104.21	odd	4
7488.2.a.h.1.1		1	104.99	even	4
7514.2.a.c.1.1		1	663.203	even	4
8450.2.a.c.1.1		1	195.44	even	4
9386.2.a.j.1.1		1	741.398	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:	\( N \)	\(=\)	\( 3042 = 2 \cdot 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3042.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} -3.00000i q^{5} -1.00000i q^{7} +1.00000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{4} -3.00000i q^{5} -1.00000i q^{7} +1.00000i q^{8} -3.00000 q^{10} -6.00000i q^{11} -1.00000 q^{14} +1.00000 q^{16} -3.00000 q^{17} -2.00000i q^{19} +3.00000i q^{20} -6.00000 q^{22} -4.00000 q^{25} +1.00000i q^{28} -6.00000 q^{29} +4.00000i q^{31} -1.00000i q^{32} +3.00000i q^{34} -3.00000 q^{35} -7.00000i q^{37} -2.00000 q^{38} +3.00000 q^{40} +1.00000 q^{43} +6.00000i q^{44} -3.00000i q^{47} +6.00000 q^{49} +4.00000i q^{50} -18.0000 q^{55} +1.00000 q^{56} +6.00000i q^{58} +6.00000i q^{59} +8.00000 q^{61} +4.00000 q^{62} -1.00000 q^{64} -14.0000i q^{67} +3.00000 q^{68} +3.00000i q^{70} -3.00000i q^{71} +2.00000i q^{73} -7.00000 q^{74} +2.00000i q^{76} -6.00000 q^{77} +8.00000 q^{79} -3.00000i q^{80} +12.0000i q^{83} +9.00000i q^{85} -1.00000i q^{86} +6.00000 q^{88} +6.00000i q^{89} -3.00000 q^{94} -6.00000 q^{95} +10.0000i q^{97} -6.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4}+O(q^{10}) \) 2 * q - 2 * q^4	\( 2 q - 2 q^{4} - 6 q^{10} - 2 q^{14} + 2 q^{16} - 6 q^{17} - 12 q^{22} - 8 q^{25} - 12 q^{29} - 6 q^{35} - 4 q^{38} + 6 q^{40} + 2 q^{43} + 12 q^{49} - 36 q^{55} + 2 q^{56} + 16 q^{61} + 8 q^{62} - 2 q^{64} + 6 q^{68} - 14 q^{74} - 12 q^{77} + 16 q^{79} + 12 q^{88} - 6 q^{94} - 12 q^{95}+O(q^{100}) \) 2 * q - 2 * q^4 - 6 * q^10 - 2 * q^14 + 2 * q^16 - 6 * q^17 - 12 * q^22 - 8 * q^25 - 12 * q^29 - 6 * q^35 - 4 * q^38 + 6 * q^40 + 2 * q^43 + 12 * q^49 - 36 * q^55 + 2 * q^56 + 16 * q^61 + 8 * q^62 - 2 * q^64 + 6 * q^68 - 14 * q^74 - 12 * q^77 + 16 * q^79 + 12 * q^88 - 6 * q^94 - 12 * q^95

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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