LMFDB - Embedded newform 3042.2.b.c.1351.2

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 234)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} +2.00000i q^{5} -2.00000i q^{7} -1.00000i q^{8} +O(q^{10})$	$q+1.00000i q^{2} -1.00000 q^{4} +2.00000i q^{5} -2.00000i q^{7} -1.00000i q^{8} -2.00000 q^{10} -4.00000i q^{11} +2.00000 q^{14} +1.00000 q^{16} +6.00000i q^{19} -2.00000i q^{20} +4.00000 q^{22} -4.00000 q^{23} +1.00000 q^{25} +2.00000i q^{28} -8.00000 q^{29} +2.00000i q^{31} +1.00000i q^{32} +4.00000 q^{35} +6.00000i q^{37} -6.00000 q^{38} +2.00000 q^{40} -6.00000i q^{41} +8.00000 q^{43} +4.00000i q^{44} -4.00000i q^{46} +8.00000i q^{47} +3.00000 q^{49} +1.00000i q^{50} +12.0000 q^{53} +8.00000 q^{55} -2.00000 q^{56} -8.00000i q^{58} +4.00000i q^{59} +10.0000 q^{61} -2.00000 q^{62} -1.00000 q^{64} +2.00000i q^{67} +4.00000i q^{70} +16.0000i q^{71} +14.0000i q^{73} -6.00000 q^{74} -6.00000i q^{76} -8.00000 q^{77} -4.00000 q^{79} +2.00000i q^{80} +6.00000 q^{82} +12.0000i q^{83} +8.00000i q^{86} -4.00000 q^{88} -6.00000i q^{89} +4.00000 q^{92} -8.00000 q^{94} -12.0000 q^{95} +10.0000i q^{97} +3.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4}+O(q^{10}) $ 2 * q - 2 * q^4	$ 2 q - 2 q^{4} - 4 q^{10} + 4 q^{14} + 2 q^{16} + 8 q^{22} - 8 q^{23} + 2 q^{25} - 16 q^{29} + 8 q^{35} - 12 q^{38} + 4 q^{40} + 16 q^{43} + 6 q^{49} + 24 q^{53} + 16 q^{55} - 4 q^{56} + 20 q^{61} - 4 q^{62} - 2 q^{64} - 12 q^{74} - 16 q^{77} - 8 q^{79} + 12 q^{82} - 8 q^{88} + 8 q^{92} - 16 q^{94} - 24 q^{95}+O(q^{100}) $ 2 * q - 2 * q^4 - 4 * q^10 + 4 * q^14 + 2 * q^16 + 8 * q^22 - 8 * q^23 + 2 * q^25 - 16 * q^29 + 8 * q^35 - 12 * q^38 + 4 * q^40 + 16 * q^43 + 6 * q^49 + 24 * q^53 + 16 * q^55 - 4 * q^56 + 20 * q^61 - 4 * q^62 - 2 * q^64 - 12 * q^74 - 16 * q^77 - 8 * q^79 + 12 * q^82 - 8 * q^88 + 8 * q^92 - 16 * q^94 - 24 * 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
$5$	2.00000i	0.894427i	0.894427	+	0.447214i	$0.147584\pi$
$5$	2.00000i	0.894427i	−0.894427	+	0.447214i	$0.852416\pi$
$6$	0	0
$7$	− 2.00000i	− 0.755929i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$	− 2.00000i	− 0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$	− 1.00000i	− 0.353553i
$9$	0	0
$10$	−2.00000	−0.632456
$11$	− 4.00000i	− 1.20605i	−0.797724	−	0.603023i	$-0.793963\pi$
$11$	− 4.00000i	− 1.20605i	0.797724	−	0.603023i	$-0.206037\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	2.00000	0.534522
$15$	0	0
$16$	1.00000	0.250000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	6.00000i	1.37649i	0.725476	+	0.688247i	$0.241620\pi$
$19$	6.00000i	1.37649i	−0.725476	+	0.688247i	$0.758380\pi$
$20$	− 2.00000i	− 0.447214i
$21$	0	0
$22$	4.00000	0.852803
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	2.00000i	0.377964i
$29$	−8.00000	−1.48556	−0.742781	−	0.669534i	$-0.766494\pi$
$29$	−8.00000	−1.48556	−0.742781	+	0.669534i	$0.766494\pi$
$30$	0	0
$31$	2.00000i	0.359211i	0.983739	+	0.179605i	$0.0574821\pi$
$31$	2.00000i	0.359211i	−0.983739	+	0.179605i	$0.942518\pi$
$32$	1.00000i	0.176777i
$33$	0	0
$34$	0	0
$35$	4.00000	0.676123
$36$	0	0
$37$	6.00000i	0.986394i	0.869918	+	0.493197i	$0.164172\pi$
$37$	6.00000i	0.986394i	−0.869918	+	0.493197i	$0.835828\pi$
$38$	−6.00000	−0.973329
$39$	0	0
$40$	2.00000	0.316228
$41$	− 6.00000i	− 0.937043i	−0.883452	−	0.468521i	$-0.844787\pi$
$41$	− 6.00000i	− 0.937043i	0.883452	−	0.468521i	$-0.155213\pi$
$42$	0	0
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	4.00000i	0.603023i
$45$	0	0
$46$	− 4.00000i	− 0.589768i
$47$	8.00000i	1.16692i	0.812142	+	0.583460i	$0.198301\pi$
$47$	8.00000i	1.16692i	−0.812142	+	0.583460i	$0.801699\pi$
$48$	0	0
$49$	3.00000	0.428571
$50$	1.00000i	0.141421i
$51$	0	0
$52$	0	0
$53$	12.0000	1.64833	0.824163	−	0.566352i	$-0.191646\pi$
$53$	12.0000	1.64833	0.824163	+	0.566352i	$0.191646\pi$
$54$	0	0
$55$	8.00000	1.07872
$56$	−2.00000	−0.267261
$57$	0	0
$58$	− 8.00000i	− 1.05045i
$59$	4.00000i	0.520756i	0.965507	+	0.260378i	$0.0838471\pi$
$59$	4.00000i	0.520756i	−0.965507	+	0.260378i	$0.916153\pi$
$60$	0	0
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	−2.00000	−0.254000
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	2.00000i	0.244339i	0.992509	+	0.122169i	$0.0389851\pi$
$67$	2.00000i	0.244339i	−0.992509	+	0.122169i	$0.961015\pi$
$68$	0	0
$69$	0	0
$70$	4.00000i	0.478091i
$71$	16.0000i	1.89885i	0.313993	+	0.949425i	$0.398333\pi$
$71$	16.0000i	1.89885i	−0.313993	+	0.949425i	$0.601667\pi$
$72$	0	0
$73$	14.0000i	1.63858i	0.573382	+	0.819288i	$0.305631\pi$
$73$	14.0000i	1.63858i	−0.573382	+	0.819288i	$0.694369\pi$
$74$	−6.00000	−0.697486
$75$	0	0
$76$	− 6.00000i	− 0.688247i
$77$	−8.00000	−0.911685
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	2.00000i	0.223607i
$81$	0	0
$82$	6.00000	0.662589
$83$	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
$86$	8.00000i	0.862662i
$87$	0	0
$88$	−4.00000	−0.426401
$89$	− 6.00000i	− 0.635999i	−0.948091	−	0.317999i	$-0.896989\pi$
$89$	− 6.00000i	− 0.635999i	0.948091	−	0.317999i	$-0.103011\pi$
$90$	0	0
$91$	0	0
$92$	4.00000	0.417029
$93$	0	0
$94$	−8.00000	−0.825137
$95$	−12.0000	−1.23117
$96$	0	0
$97$	10.0000i	1.01535i	0.861550	+	0.507673i	$0.169494\pi$
$97$	10.0000i	1.01535i	−0.861550	+	0.507673i	$0.830506\pi$
$98$	3.00000i	0.303046i
$99$	0	0
$100$	−1.00000	−0.100000
$101$	16.0000	1.59206	0.796030	−	0.605257i	$-0.206930\pi$
$101$	16.0000	1.59206	0.796030	+	0.605257i	$0.206930\pi$
$102$	0	0
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	0	0
$105$	0	0
$106$	12.0000i	1.16554i
$107$	−20.0000	−1.93347	−0.966736	−	0.255774i	$-0.917670\pi$
$107$	−20.0000	−1.93347	−0.966736	+	0.255774i	$0.917670\pi$
$108$	0	0
$109$	− 10.0000i	− 0.957826i	−0.877862	−	0.478913i	$-0.841031\pi$
$109$	− 10.0000i	− 0.957826i	0.877862	−	0.478913i	$-0.158969\pi$
$110$	8.00000i	0.762770i
$111$	0	0
$112$	− 2.00000i	− 0.188982i
$113$	4.00000	0.376288	0.188144	−	0.982141i	$-0.439753\pi$
$113$	4.00000	0.376288	0.188144	+	0.982141i	$0.439753\pi$
$114$	0	0
$115$	− 8.00000i	− 0.746004i
$116$	8.00000	0.742781
$117$	0	0
$118$	−4.00000	−0.368230
$119$	0	0
$120$	0	0
$121$	−5.00000	−0.454545
$122$	10.0000i	0.905357i
$123$	0	0
$124$	− 2.00000i	− 0.179605i
$125$	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	$0.384500\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	0	0
$130$	0	0
$131$	4.00000	0.349482	0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$132$	0	0
$133$	12.0000	1.04053
$134$	−2.00000	−0.172774
$135$	0	0
$136$	0	0
$137$	− 18.0000i	− 1.53784i	−0.639343	−	0.768922i	$-0.720793\pi$
$137$	− 18.0000i	− 1.53784i	0.639343	−	0.768922i	$-0.279207\pi$
$138$	0	0
$139$	−20.0000	−1.69638	−0.848189	−	0.529694i	$-0.822307\pi$
$139$	−20.0000	−1.69638	−0.848189	+	0.529694i	$0.822307\pi$
$140$	−4.00000	−0.338062
$141$	0	0
$142$	−16.0000	−1.34269
$143$	0	0
$144$	0	0
$145$	− 16.0000i	− 1.32873i
$146$	−14.0000	−1.15865
$147$	0	0
$148$	− 6.00000i	− 0.493197i
$149$	6.00000i	0.491539i	0.969328	+	0.245770i	$0.0790407\pi$
$149$	6.00000i	0.491539i	−0.969328	+	0.245770i	$0.920959\pi$
$150$	0	0
$151$	18.0000i	1.46482i	0.680864	+	0.732410i	$0.261604\pi$
$151$	18.0000i	1.46482i	−0.680864	+	0.732410i	$0.738396\pi$
$152$	6.00000	0.486664
$153$	0	0
$154$	− 8.00000i	− 0.644658i
$155$	−4.00000	−0.321288
$156$	0	0
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	− 4.00000i	− 0.318223i
$159$	0	0
$160$	−2.00000	−0.158114
$161$	8.00000i	0.630488i
$162$	0	0
$163$	− 10.0000i	− 0.783260i	−0.920123	−	0.391630i	$-0.871911\pi$
$163$	− 10.0000i	− 0.783260i	0.920123	−	0.391630i	$-0.128089\pi$
$164$	6.00000i	0.468521i
$165$	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
$170$	0	0
$171$	0	0
$172$	−8.00000	−0.609994
$173$	−16.0000	−1.21646	−0.608229	−	0.793762i	$-0.708120\pi$
$173$	−16.0000	−1.21646	−0.608229	+	0.793762i	$0.708120\pi$
$174$	0	0
$175$	− 2.00000i	− 0.151186i
$176$	− 4.00000i	− 0.301511i
$177$	0	0
$178$	6.00000	0.449719
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	18.0000	1.33793	0.668965	−	0.743294i	$-0.266738\pi$
$181$	18.0000	1.33793	0.668965	+	0.743294i	$0.266738\pi$
$182$	0	0
$183$	0	0
$184$	4.00000i	0.294884i
$185$	−12.0000	−0.882258
$186$	0	0
$187$	0	0
$188$	− 8.00000i	− 0.583460i
$189$	0	0
$190$	− 12.0000i	− 0.870572i
$191$	−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
$193$	2.00000i	0.143963i	0.997406	+	0.0719816i	$0.0229323\pi$
$193$	2.00000i	0.143963i	−0.997406	+	0.0719816i	$0.977068\pi$
$194$	−10.0000	−0.717958
$195$	0	0
$196$	−3.00000	−0.214286
$197$	18.0000i	1.28245i	0.767354	+	0.641223i	$0.221573\pi$
$197$	18.0000i	1.28245i	−0.767354	+	0.641223i	$0.778427\pi$
$198$	0	0
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	− 1.00000i	− 0.0707107i
$201$	0	0
$202$	16.0000i	1.12576i
$203$	16.0000i	1.12298i
$204$	0	0
$205$	12.0000	0.838116
$206$	− 4.00000i	− 0.278693i
$207$	0	0
$208$	0	0
$209$	24.0000	1.66011
$210$	0	0
$211$	8.00000	0.550743	0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000	0.550743	0.275371	+	0.961338i	$0.411199\pi$
$212$	−12.0000	−0.824163
$213$	0	0
$214$	− 20.0000i	− 1.36717i
$215$	16.0000i	1.09119i
$216$	0	0
$217$	4.00000	0.271538
$218$	10.0000	0.677285
$219$	0	0
$220$	−8.00000	−0.539360
$221$	0	0
$222$	0	0
$223$	− 6.00000i	− 0.401790i	−0.979613	−	0.200895i	$-0.935615\pi$
$223$	− 6.00000i	− 0.401790i	0.979613	−	0.200895i	$-0.0643850\pi$
$224$	2.00000	0.133631
$225$	0	0
$226$	4.00000i	0.266076i
$227$	− 12.0000i	− 0.796468i	−0.917284	−	0.398234i	$-0.869623\pi$
$227$	− 12.0000i	− 0.796468i	0.917284	−	0.398234i	$-0.130377\pi$
$228$	0	0
$229$	− 30.0000i	− 1.98246i	−0.132164	−	0.991228i	$-0.542192\pi$
$229$	− 30.0000i	− 1.98246i	0.132164	−	0.991228i	$-0.457808\pi$
$230$	8.00000	0.527504
$231$	0	0
$232$	8.00000i	0.525226i
$233$	−20.0000	−1.31024	−0.655122	−	0.755523i	$-0.727383\pi$
$233$	−20.0000	−1.31024	−0.655122	+	0.755523i	$0.727383\pi$
$234$	0	0
$235$	−16.0000	−1.04372
$236$	− 4.00000i	− 0.260378i
$237$	0	0
$238$	0	0
$239$	24.0000i	1.55243i	0.630468	+	0.776215i	$0.282863\pi$
$239$	24.0000i	1.55243i	−0.630468	+	0.776215i	$0.717137\pi$
$240$	0	0
$241$	6.00000i	0.386494i	0.981150	+	0.193247i	$0.0619019\pi$
$241$	6.00000i	0.386494i	−0.981150	+	0.193247i	$0.938098\pi$
$242$	− 5.00000i	− 0.321412i
$243$	0	0
$244$	−10.0000	−0.640184
$245$	6.00000i	0.383326i
$246$	0	0
$247$	0	0
$248$	2.00000	0.127000
$249$	0	0
$250$	−12.0000	−0.758947
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	16.0000i	1.00591i
$254$	8.00000i	0.501965i
$255$	0	0
$256$	1.00000	0.0625000
$257$	−12.0000	−0.748539	−0.374270	−	0.927320i	$-0.622107\pi$
$257$	−12.0000	−0.748539	−0.374270	+	0.927320i	$0.622107\pi$
$258$	0	0
$259$	12.0000	0.745644
$260$	0	0
$261$	0	0
$262$	4.00000i	0.247121i
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	24.0000i	1.47431i
$266$	12.0000i	0.735767i
$267$	0	0
$268$	− 2.00000i	− 0.122169i
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	22.0000i	1.33640i	0.743980	+	0.668202i	$0.232936\pi$
$271$	22.0000i	1.33640i	−0.743980	+	0.668202i	$0.767064\pi$
$272$	0	0
$273$	0	0
$274$	18.0000	1.08742
$275$	− 4.00000i	− 0.241209i
$276$	0	0
$277$	6.00000	0.360505	0.180253	−	0.983620i	$-0.442309\pi$
$277$	6.00000	0.360505	0.180253	+	0.983620i	$0.442309\pi$
$278$	− 20.0000i	− 1.19952i
$279$	0	0
$280$	− 4.00000i	− 0.239046i
$281$	− 10.0000i	− 0.596550i	−0.954480	−	0.298275i	$-0.903589\pi$
$281$	− 10.0000i	− 0.596550i	0.954480	−	0.298275i	$-0.0964112\pi$
$282$	0	0
$283$	20.0000	1.18888	0.594438	−	0.804141i	$-0.297374\pi$
$283$	20.0000	1.18888	0.594438	+	0.804141i	$0.297374\pi$
$284$	− 16.0000i	− 0.949425i
$285$	0	0
$286$	0	0
$287$	−12.0000	−0.708338
$288$	0	0
$289$	−17.0000	−1.00000
$290$	16.0000	0.939552
$291$	0	0
$292$	− 14.0000i	− 0.819288i
$293$	2.00000i	0.116841i	0.998292	+	0.0584206i	$0.0186065\pi$
$293$	2.00000i	0.116841i	−0.998292	+	0.0584206i	$0.981394\pi$
$294$	0	0
$295$	−8.00000	−0.465778
$296$	6.00000	0.348743
$297$	0	0
$298$	−6.00000	−0.347571
$299$	0	0
$300$	0	0
$301$	− 16.0000i	− 0.922225i
$302$	−18.0000	−1.03578
$303$	0	0
$304$	6.00000i	0.344124i
$305$	20.0000i	1.14520i
$306$	0	0
$307$	2.00000i	0.114146i	0.998370	+	0.0570730i	$0.0181768\pi$
$307$	2.00000i	0.114146i	−0.998370	+	0.0570730i	$0.981823\pi$
$308$	8.00000	0.455842
$309$	0	0
$310$	− 4.00000i	− 0.227185i
$311$	12.0000	0.680458	0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000	0.680458	0.340229	+	0.940343i	$0.389495\pi$
$312$	0	0
$313$	26.0000	1.46961	0.734803	−	0.678280i	$-0.237274\pi$
$313$	26.0000	1.46961	0.734803	+	0.678280i	$0.237274\pi$
$314$	2.00000i	0.112867i
$315$	0	0
$316$	4.00000	0.225018
$317$	− 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
$319$	32.0000i	1.79166i
$320$	− 2.00000i	− 0.111803i
$321$	0	0
$322$	−8.00000	−0.445823
$323$	0	0
$324$	0	0
$325$	0	0
$326$	10.0000	0.553849
$327$	0	0
$328$	−6.00000	−0.331295
$329$	16.0000	0.882109
$330$	0	0
$331$	10.0000i	0.549650i	0.961494	+	0.274825i	$0.0886199\pi$
$331$	10.0000i	0.549650i	−0.961494	+	0.274825i	$0.911380\pi$
$332$	− 12.0000i	− 0.658586i
$333$	0	0
$334$	0	0
$335$	−4.00000	−0.218543
$336$	0	0
$337$	34.0000	1.85210	0.926049	−	0.377403i	$-0.123183\pi$
$337$	34.0000	1.85210	0.926049	+	0.377403i	$0.123183\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	8.00000	0.433224
$342$	0	0
$343$	− 20.0000i	− 1.07990i
$344$	− 8.00000i	− 0.431331i
$345$	0	0
$346$	− 16.0000i	− 0.860165i
$347$	−24.0000	−1.28839	−0.644194	−	0.764862i	$-0.722807\pi$
$347$	−24.0000	−1.28839	−0.644194	+	0.764862i	$0.722807\pi$
$348$	0	0
$349$	22.0000i	1.17763i	0.808267	+	0.588817i	$0.200406\pi$
$349$	22.0000i	1.17763i	−0.808267	+	0.588817i	$0.799594\pi$
$350$	2.00000	0.106904
$351$	0	0
$352$	4.00000	0.213201
$353$	− 14.0000i	− 0.745145i	−0.928003	−	0.372572i	$-0.878476\pi$
$353$	− 14.0000i	− 0.745145i	0.928003	−	0.372572i	$-0.121524\pi$
$354$	0	0
$355$	−32.0000	−1.69838
$356$	6.00000i	0.317999i
$357$	0	0
$358$	− 12.0000i	− 0.634220i
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	−17.0000	−0.894737
$362$	18.0000i	0.946059i
$363$	0	0
$364$	0	0
$365$	−28.0000	−1.46559
$366$	0	0
$367$	8.00000	0.417597	0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000	0.417597	0.208798	+	0.977959i	$0.433045\pi$
$368$	−4.00000	−0.208514
$369$	0	0
$370$	− 12.0000i	− 0.623850i
$371$	− 24.0000i	− 1.24602i
$372$	0	0
$373$	−10.0000	−0.517780	−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000	−0.517780	−0.258890	+	0.965907i	$0.583357\pi$
$374$	0	0
$375$	0	0
$376$	8.00000	0.412568
$377$	0	0
$378$	0	0
$379$	− 10.0000i	− 0.513665i	−0.966456	−	0.256833i	$-0.917321\pi$
$379$	− 10.0000i	− 0.513665i	0.966456	−	0.256833i	$-0.0826790\pi$
$380$	12.0000	0.615587
$381$	0	0
$382$	− 4.00000i	− 0.204658i
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0	0
$385$	− 16.0000i	− 0.815436i
$386$	−2.00000	−0.101797
$387$	0	0
$388$	− 10.0000i	− 0.507673i
$389$	12.0000	0.608424	0.304212	−	0.952604i	$-0.401607\pi$
$389$	12.0000	0.608424	0.304212	+	0.952604i	$0.401607\pi$
$390$	0	0
$391$	0	0
$392$	− 3.00000i	− 0.151523i
$393$	0	0
$394$	−18.0000	−0.906827
$395$	− 8.00000i	− 0.402524i
$396$	0	0
$397$	− 10.0000i	− 0.501886i	−0.968002	−	0.250943i	$-0.919259\pi$
$397$	− 10.0000i	− 0.501886i	0.968002	−	0.250943i	$-0.0807406\pi$
$398$	− 8.00000i	− 0.401004i
$399$	0	0
$400$	1.00000	0.0500000
$401$	− 6.00000i	− 0.299626i	−0.988714	−	0.149813i	$-0.952133\pi$
$401$	− 6.00000i	− 0.299626i	0.988714	−	0.149813i	$-0.0478671\pi$
$402$	0	0
$403$	0	0
$404$	−16.0000	−0.796030
$405$	0	0
$406$	−16.0000	−0.794067
$407$	24.0000	1.18964
$408$	0	0
$409$	26.0000i	1.28562i	0.766027	+	0.642809i	$0.222231\pi$
$409$	26.0000i	1.28562i	−0.766027	+	0.642809i	$0.777769\pi$
$410$	12.0000i	0.592638i
$411$	0	0
$412$	4.00000	0.197066
$413$	8.00000	0.393654
$414$	0	0
$415$	−24.0000	−1.17811
$416$	0	0
$417$	0	0
$418$	24.0000i	1.17388i
$419$	36.0000	1.75872	0.879358	−	0.476162i	$-0.157972\pi$
$419$	36.0000	1.75872	0.879358	+	0.476162i	$0.157972\pi$
$420$	0	0
$421$	− 10.0000i	− 0.487370i	−0.969854	−	0.243685i	$-0.921644\pi$
$421$	− 10.0000i	− 0.487370i	0.969854	−	0.243685i	$-0.0783563\pi$
$422$	8.00000i	0.389434i
$423$	0	0
$424$	− 12.0000i	− 0.582772i
$425$	0	0
$426$	0	0
$427$	− 20.0000i	− 0.967868i
$428$	20.0000	0.966736
$429$	0	0
$430$	−16.0000	−0.771589
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	0	0
$433$	2.00000	0.0961139	0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000	0.0961139	0.0480569	+	0.998845i	$0.484697\pi$
$434$	4.00000i	0.192006i
$435$	0	0
$436$	10.0000i	0.478913i
$437$	− 24.0000i	− 1.14808i
$438$	0	0
$439$	4.00000	0.190910	0.0954548	−	0.995434i	$-0.469569\pi$
$439$	4.00000	0.190910	0.0954548	+	0.995434i	$0.469569\pi$
$440$	− 8.00000i	− 0.381385i
$441$	0	0
$442$	0	0
$443$	36.0000	1.71041	0.855206	−	0.518289i	$-0.173431\pi$
$443$	36.0000	1.71041	0.855206	+	0.518289i	$0.173431\pi$
$444$	0	0
$445$	12.0000	0.568855
$446$	6.00000	0.284108
$447$	0	0
$448$	2.00000i	0.0944911i
$449$	26.0000i	1.22702i	0.789689	+	0.613508i	$0.210242\pi$
$449$	26.0000i	1.22702i	−0.789689	+	0.613508i	$0.789758\pi$
$450$	0	0
$451$	−24.0000	−1.13012
$452$	−4.00000	−0.188144
$453$	0	0
$454$	12.0000	0.563188
$455$	0	0
$456$	0	0
$457$	2.00000i	0.0935561i	0.998905	+	0.0467780i	$0.0148953\pi$
$457$	2.00000i	0.0935561i	−0.998905	+	0.0467780i	$0.985105\pi$
$458$	30.0000	1.40181
$459$	0	0
$460$	8.00000i	0.373002i
$461$	− 30.0000i	− 1.39724i	−0.715493	−	0.698620i	$-0.753798\pi$
$461$	− 30.0000i	− 1.39724i	0.715493	−	0.698620i	$-0.246202\pi$
$462$	0	0
$463$	22.0000i	1.02243i	0.859454	+	0.511213i	$0.170804\pi$
$463$	22.0000i	1.02243i	−0.859454	+	0.511213i	$0.829196\pi$
$464$	−8.00000	−0.371391
$465$	0	0
$466$	− 20.0000i	− 0.926482i
$467$	8.00000	0.370196	0.185098	−	0.982720i	$-0.440740\pi$
$467$	8.00000	0.370196	0.185098	+	0.982720i	$0.440740\pi$
$468$	0	0
$469$	4.00000	0.184703
$470$	− 16.0000i	− 0.738025i
$471$	0	0
$472$	4.00000	0.184115
$473$	− 32.0000i	− 1.47136i
$474$	0	0
$475$	6.00000i	0.275299i
$476$	0	0
$477$	0	0
$478$	−24.0000	−1.09773
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	0	0
$482$	−6.00000	−0.273293
$483$	0	0
$484$	5.00000	0.227273
$485$	−20.0000	−0.908153
$486$	0	0
$487$	− 38.0000i	− 1.72194i	−0.508652	−	0.860972i	$-0.669856\pi$
$487$	− 38.0000i	− 1.72194i	0.508652	−	0.860972i	$-0.330144\pi$
$488$	− 10.0000i	− 0.452679i
$489$	0	0
$490$	−6.00000	−0.271052
$491$	16.0000	0.722070	0.361035	−	0.932552i	$-0.382424\pi$
$491$	16.0000	0.722070	0.361035	+	0.932552i	$0.382424\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	2.00000i	0.0898027i
$497$	32.0000	1.43540
$498$	0	0
$499$	− 22.0000i	− 0.984855i	−0.870353	−	0.492428i	$-0.836110\pi$
$499$	− 22.0000i	− 0.984855i	0.870353	−	0.492428i	$-0.163890\pi$
$500$	− 12.0000i	− 0.536656i
$501$	0	0
$502$	0	0
$503$	32.0000	1.42681	0.713405	−	0.700752i	$-0.247152\pi$
$503$	32.0000	1.42681	0.713405	+	0.700752i	$0.247152\pi$
$504$	0	0
$505$	32.0000i	1.42398i
$506$	−16.0000	−0.711287
$507$	0	0
$508$	−8.00000	−0.354943
$509$	10.0000i	0.443242i	0.975133	+	0.221621i	$0.0711348\pi$
$509$	10.0000i	0.443242i	−0.975133	+	0.221621i	$0.928865\pi$
$510$	0	0
$511$	28.0000	1.23865
$512$	1.00000i	0.0441942i
$513$	0	0
$514$	− 12.0000i	− 0.529297i
$515$	− 8.00000i	− 0.352522i
$516$	0	0
$517$	32.0000	1.40736
$518$	12.0000i	0.527250i
$519$	0	0
$520$	0	0
$521$	−36.0000	−1.57719	−0.788594	−	0.614914i	$-0.789191\pi$
$521$	−36.0000	−1.57719	−0.788594	+	0.614914i	$0.789191\pi$
$522$	0	0
$523$	−12.0000	−0.524723	−0.262362	−	0.964970i	$-0.584501\pi$
$523$	−12.0000	−0.524723	−0.262362	+	0.964970i	$0.584501\pi$
$524$	−4.00000	−0.174741
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−7.00000	−0.304348
$530$	−24.0000	−1.04249
$531$	0	0
$532$	−12.0000	−0.520266
$533$	0	0
$534$	0	0
$535$	− 40.0000i	− 1.72935i
$536$	2.00000	0.0863868
$537$	0	0
$538$	0	0
$539$	− 12.0000i	− 0.516877i
$540$	0	0
$541$	2.00000i	0.0859867i	0.999075	+	0.0429934i	$0.0136894\pi$
$541$	2.00000i	0.0859867i	−0.999075	+	0.0429934i	$0.986311\pi$
$542$	−22.0000	−0.944981
$543$	0	0
$544$	0	0
$545$	20.0000	0.856706
$546$	0	0
$547$	−20.0000	−0.855138	−0.427569	−	0.903983i	$-0.640630\pi$
$547$	−20.0000	−0.855138	−0.427569	+	0.903983i	$0.640630\pi$
$548$	18.0000i	0.768922i
$549$	0	0
$550$	4.00000	0.170561
$551$	− 48.0000i	− 2.04487i
$552$	0	0
$553$	8.00000i	0.340195i
$554$	6.00000i	0.254916i
$555$	0	0
$556$	20.0000	0.848189
$557$	18.0000i	0.762684i	0.924434	+	0.381342i	$0.124538\pi$
$557$	18.0000i	0.762684i	−0.924434	+	0.381342i	$0.875462\pi$
$558$	0	0
$559$	0	0
$560$	4.00000	0.169031
$561$	0	0
$562$	10.0000	0.421825
$563$	20.0000	0.842900	0.421450	−	0.906852i	$-0.361521\pi$
$563$	20.0000	0.842900	0.421450	+	0.906852i	$0.361521\pi$
$564$	0	0
$565$	8.00000i	0.336563i
$566$	20.0000i	0.840663i
$567$	0	0
$568$	16.0000	0.671345
$569$	4.00000	0.167689	0.0838444	−	0.996479i	$-0.473280\pi$
$569$	4.00000	0.167689	0.0838444	+	0.996479i	$0.473280\pi$
$570$	0	0
$571$	−36.0000	−1.50655	−0.753277	−	0.657704i	$-0.771528\pi$
$571$	−36.0000	−1.50655	−0.753277	+	0.657704i	$0.771528\pi$
$572$	0	0
$573$	0	0
$574$	− 12.0000i	− 0.500870i
$575$	−4.00000	−0.166812
$576$	0	0
$577$	30.0000i	1.24892i	0.781058	+	0.624458i	$0.214680\pi$
$577$	30.0000i	1.24892i	−0.781058	+	0.624458i	$0.785320\pi$
$578$	− 17.0000i	− 0.707107i
$579$	0	0
$580$	16.0000i	0.664364i
$581$	24.0000	0.995688
$582$	0	0
$583$	− 48.0000i	− 1.98796i
$584$	14.0000	0.579324
$585$	0	0
$586$	−2.00000	−0.0826192
$587$	− 28.0000i	− 1.15568i	−0.816149	−	0.577842i	$-0.803895\pi$
$587$	− 28.0000i	− 1.15568i	0.816149	−	0.577842i	$-0.196105\pi$
$588$	0	0
$589$	−12.0000	−0.494451
$590$	− 8.00000i	− 0.329355i
$591$	0	0
$592$	6.00000i	0.246598i
$593$	− 38.0000i	− 1.56047i	−0.625485	−	0.780236i	$-0.715099\pi$
$593$	− 38.0000i	− 1.56047i	0.625485	−	0.780236i	$-0.284901\pi$
$594$	0	0
$595$	0	0
$596$	− 6.00000i	− 0.245770i
$597$	0	0
$598$	0	0
$599$	−40.0000	−1.63436	−0.817178	−	0.576386i	$-0.804463\pi$
$599$	−40.0000	−1.63436	−0.817178	+	0.576386i	$0.804463\pi$
$600$	0	0
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	16.0000	0.652111
$603$	0	0
$604$	− 18.0000i	− 0.732410i
$605$	− 10.0000i	− 0.406558i
$606$	0	0
$607$	24.0000	0.974130	0.487065	−	0.873366i	$-0.338067\pi$
$607$	24.0000	0.974130	0.487065	+	0.873366i	$0.338067\pi$
$608$	−6.00000	−0.243332
$609$	0	0
$610$	−20.0000	−0.809776
$611$	0	0
$612$	0	0
$613$	34.0000i	1.37325i	0.727013	+	0.686624i	$0.240908\pi$
$613$	34.0000i	1.37325i	−0.727013	+	0.686624i	$0.759092\pi$
$614$	−2.00000	−0.0807134
$615$	0	0
$616$	8.00000i	0.322329i
$617$	− 14.0000i	− 0.563619i	−0.959470	−	0.281809i	$-0.909065\pi$
$617$	− 14.0000i	− 0.563619i	0.959470	−	0.281809i	$-0.0909346\pi$
$618$	0	0
$619$	34.0000i	1.36658i	0.730149	+	0.683288i	$0.239451\pi$
$619$	34.0000i	1.36658i	−0.730149	+	0.683288i	$0.760549\pi$
$620$	4.00000	0.160644
$621$	0	0
$622$	12.0000i	0.481156i
$623$	−12.0000	−0.480770
$624$	0	0
$625$	−19.0000	−0.760000
$626$	26.0000i	1.03917i
$627$	0	0
$628$	−2.00000	−0.0798087
$629$	0	0
$630$	0	0
$631$	− 14.0000i	− 0.557331i	−0.960388	−	0.278666i	$-0.910108\pi$
$631$	− 14.0000i	− 0.557331i	0.960388	−	0.278666i	$-0.0898921\pi$
$632$	4.00000i	0.159111i
$633$	0	0
$634$	6.00000	0.238290
$635$	16.0000i	0.634941i
$636$	0	0
$637$	0	0
$638$	−32.0000	−1.26689
$639$	0	0
$640$	2.00000	0.0790569
$641$	40.0000	1.57991	0.789953	−	0.613168i	$-0.210105\pi$
$641$	40.0000	1.57991	0.789953	+	0.613168i	$0.210105\pi$
$642$	0	0
$643$	10.0000i	0.394362i	0.980367	+	0.197181i	$0.0631786\pi$
$643$	10.0000i	0.394362i	−0.980367	+	0.197181i	$0.936821\pi$
$644$	− 8.00000i	− 0.315244i
$645$	0	0
$646$	0	0
$647$	−8.00000	−0.314512	−0.157256	−	0.987558i	$-0.550265\pi$
$647$	−8.00000	−0.314512	−0.157256	+	0.987558i	$0.550265\pi$
$648$	0	0
$649$	16.0000	0.628055
$650$	0	0
$651$	0	0
$652$	10.0000i	0.391630i
$653$	0	0		−	1.00000i	$-0.5\pi$
$653$	0	0			1.00000i	$0.5\pi$
$654$	0	0
$655$	8.00000i	0.312586i
$656$	− 6.00000i	− 0.234261i
$657$	0	0
$658$	16.0000i	0.623745i
$659$	−48.0000	−1.86981	−0.934907	−	0.354892i	$-0.884518\pi$
$659$	−48.0000	−1.86981	−0.934907	+	0.354892i	$0.884518\pi$
$660$	0	0
$661$	14.0000i	0.544537i	0.962221	+	0.272268i	$0.0877739\pi$
$661$	14.0000i	0.544537i	−0.962221	+	0.272268i	$0.912226\pi$
$662$	−10.0000	−0.388661
$663$	0	0
$664$	12.0000	0.465690
$665$	24.0000i	0.930680i
$666$	0	0
$667$	32.0000	1.23904
$668$	0	0
$669$	0	0
$670$	− 4.00000i	− 0.154533i
$671$	− 40.0000i	− 1.54418i
$672$	0	0
$673$	−22.0000	−0.848038	−0.424019	−	0.905653i	$-0.639381\pi$
$673$	−22.0000	−0.848038	−0.424019	+	0.905653i	$0.639381\pi$
$674$	34.0000i	1.30963i
$675$	0	0
$676$	0	0
$677$	−12.0000	−0.461197	−0.230599	−	0.973049i	$-0.574068\pi$
$677$	−12.0000	−0.461197	−0.230599	+	0.973049i	$0.574068\pi$
$678$	0	0
$679$	20.0000	0.767530
$680$	0	0
$681$	0	0
$682$	8.00000i	0.306336i
$683$	− 4.00000i	− 0.153056i	−0.997067	−	0.0765279i	$-0.975617\pi$
$683$	− 4.00000i	− 0.153056i	0.997067	−	0.0765279i	$-0.0243834\pi$
$684$	0	0
$685$	36.0000	1.37549
$686$	20.0000	0.763604
$687$	0	0
$688$	8.00000	0.304997
$689$	0	0
$690$	0	0
$691$	26.0000i	0.989087i	0.869153	+	0.494543i	$0.164665\pi$
$691$	26.0000i	0.989087i	−0.869153	+	0.494543i	$0.835335\pi$
$692$	16.0000	0.608229
$693$	0	0
$694$	− 24.0000i	− 0.911028i
$695$	− 40.0000i	− 1.51729i
$696$	0	0
$697$	0	0
$698$	−22.0000	−0.832712
$699$	0	0
$700$	2.00000i	0.0755929i
$701$	12.0000	0.453234	0.226617	−	0.973984i	$-0.427233\pi$
$701$	12.0000	0.453234	0.226617	+	0.973984i	$0.427233\pi$
$702$	0	0
$703$	−36.0000	−1.35777
$704$	4.00000i	0.150756i
$705$	0	0
$706$	14.0000	0.526897
$707$	− 32.0000i	− 1.20348i
$708$	0	0
$709$	− 22.0000i	− 0.826227i	−0.910679	−	0.413114i	$-0.864441\pi$
$709$	− 22.0000i	− 0.826227i	0.910679	−	0.413114i	$-0.135559\pi$
$710$	− 32.0000i	− 1.20094i
$711$	0	0
$712$	−6.00000	−0.224860
$713$	− 8.00000i	− 0.299602i
$714$	0	0
$715$	0	0
$716$	12.0000	0.448461
$717$	0	0
$718$	0	0
$719$	−28.0000	−1.04422	−0.522112	−	0.852877i	$-0.674856\pi$
$719$	−28.0000	−1.04422	−0.522112	+	0.852877i	$0.674856\pi$
$720$	0	0
$721$	8.00000i	0.297936i
$722$	− 17.0000i	− 0.632674i
$723$	0	0
$724$	−18.0000	−0.668965
$725$	−8.00000	−0.297113
$726$	0	0
$727$	8.00000	0.296704	0.148352	−	0.988935i	$-0.452603\pi$
$727$	8.00000	0.296704	0.148352	+	0.988935i	$0.452603\pi$
$728$	0	0
$729$	0	0
$730$	− 28.0000i	− 1.03633i
$731$	0	0
$732$	0	0
$733$	22.0000i	0.812589i	0.913742	+	0.406294i	$0.133179\pi$
$733$	22.0000i	0.812589i	−0.913742	+	0.406294i	$0.866821\pi$
$734$	8.00000i	0.295285i
$735$	0	0
$736$	− 4.00000i	− 0.147442i
$737$	8.00000	0.294684
$738$	0	0
$739$	− 54.0000i	− 1.98642i	−0.116326	−	0.993211i	$-0.537112\pi$
$739$	− 54.0000i	− 1.98642i	0.116326	−	0.993211i	$-0.462888\pi$
$740$	12.0000	0.441129
$741$	0	0
$742$	24.0000	0.881068
$743$	40.0000i	1.46746i	0.679442	+	0.733729i	$0.262222\pi$
$743$	40.0000i	1.46746i	−0.679442	+	0.733729i	$0.737778\pi$
$744$	0	0
$745$	−12.0000	−0.439646
$746$	− 10.0000i	− 0.366126i
$747$	0	0
$748$	0	0
$749$	40.0000i	1.46157i
$750$	0	0
$751$	12.0000	0.437886	0.218943	−	0.975738i	$-0.429739\pi$
$751$	12.0000	0.437886	0.218943	+	0.975738i	$0.429739\pi$
$752$	8.00000i	0.291730i
$753$	0	0
$754$	0	0
$755$	−36.0000	−1.31017
$756$	0	0
$757$	−18.0000	−0.654221	−0.327111	−	0.944986i	$-0.606075\pi$
$757$	−18.0000	−0.654221	−0.327111	+	0.944986i	$0.606075\pi$
$758$	10.0000	0.363216
$759$	0	0
$760$	12.0000i	0.435286i
$761$	30.0000i	1.08750i	0.839248	+	0.543750i	$0.182996\pi$
$761$	30.0000i	1.08750i	−0.839248	+	0.543750i	$0.817004\pi$
$762$	0	0
$763$	−20.0000	−0.724049
$764$	4.00000	0.144715
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	30.0000i	1.08183i	0.841078	+	0.540914i	$0.181921\pi$
$769$	30.0000i	1.08183i	−0.841078	+	0.540914i	$0.818079\pi$
$770$	16.0000	0.576600
$771$	0	0
$772$	− 2.00000i	− 0.0719816i
$773$	14.0000i	0.503545i	0.967786	+	0.251773i	$0.0810135\pi$
$773$	14.0000i	0.503545i	−0.967786	+	0.251773i	$0.918987\pi$
$774$	0	0
$775$	2.00000i	0.0718421i
$776$	10.0000	0.358979
$777$	0	0
$778$	12.0000i	0.430221i
$779$	36.0000	1.28983
$780$	0	0
$781$	64.0000	2.29010
$782$	0	0
$783$	0	0
$784$	3.00000	0.107143
$785$	4.00000i	0.142766i
$786$	0	0
$787$	− 22.0000i	− 0.784215i	−0.919919	−	0.392108i	$-0.871746\pi$
$787$	− 22.0000i	− 0.784215i	0.919919	−	0.392108i	$-0.128254\pi$
$788$	− 18.0000i	− 0.641223i
$789$	0	0
$790$	8.00000	0.284627
$791$	− 8.00000i	− 0.284447i
$792$	0	0
$793$	0	0
$794$	10.0000	0.354887
$795$	0	0
$796$	8.00000	0.283552
$797$	12.0000	0.425062	0.212531	−	0.977154i	$-0.431829\pi$
$797$	12.0000	0.425062	0.212531	+	0.977154i	$0.431829\pi$
$798$	0	0
$799$	0	0
$800$	1.00000i	0.0353553i
$801$	0	0
$802$	6.00000	0.211867
$803$	56.0000	1.97620
$804$	0	0
$805$	−16.0000	−0.563926
$806$	0	0
$807$	0	0
$808$	− 16.0000i	− 0.562878i
$809$	36.0000	1.26569	0.632846	−	0.774277i	$-0.281886\pi$
$809$	36.0000	1.26569	0.632846	+	0.774277i	$0.281886\pi$
$810$	0	0
$811$	− 50.0000i	− 1.75574i	−0.478901	−	0.877869i	$-0.658965\pi$
$811$	− 50.0000i	− 1.75574i	0.478901	−	0.877869i	$-0.341035\pi$
$812$	− 16.0000i	− 0.561490i
$813$	0	0
$814$	24.0000i	0.841200i
$815$	20.0000	0.700569
$816$	0	0
$817$	48.0000i	1.67931i
$818$	−26.0000	−0.909069
$819$	0	0
$820$	−12.0000	−0.419058
$821$	− 34.0000i	− 1.18661i	−0.804978	−	0.593304i	$-0.797823\pi$
$821$	− 34.0000i	− 1.18661i	0.804978	−	0.593304i	$-0.202177\pi$
$822$	0	0
$823$	12.0000	0.418294	0.209147	−	0.977884i	$-0.432931\pi$
$823$	12.0000	0.418294	0.209147	+	0.977884i	$0.432931\pi$
$824$	4.00000i	0.139347i
$825$	0	0
$826$	8.00000i	0.278356i
$827$	− 12.0000i	− 0.417281i	−0.977992	−	0.208640i	$-0.933096\pi$
$827$	− 12.0000i	− 0.417281i	0.977992	−	0.208640i	$-0.0669038\pi$
$828$	0	0
$829$	−46.0000	−1.59765	−0.798823	−	0.601566i	$-0.794544\pi$
$829$	−46.0000	−1.59765	−0.798823	+	0.601566i	$0.794544\pi$
$830$	− 24.0000i	− 0.833052i
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	−24.0000	−0.830057
$837$	0	0
$838$	36.0000i	1.24360i
$839$	16.0000i	0.552381i	0.961103	+	0.276191i	$0.0890721\pi$
$839$	16.0000i	0.552381i	−0.961103	+	0.276191i	$0.910928\pi$
$840$	0	0
$841$	35.0000	1.20690
$842$	10.0000	0.344623
$843$	0	0
$844$	−8.00000	−0.275371
$845$	0	0
$846$	0	0
$847$	10.0000i	0.343604i
$848$	12.0000	0.412082
$849$	0	0
$850$	0	0
$851$	− 24.0000i	− 0.822709i
$852$	0	0
$853$	− 26.0000i	− 0.890223i	−0.895475	−	0.445112i	$-0.853164\pi$
$853$	− 26.0000i	− 0.890223i	0.895475	−	0.445112i	$-0.146836\pi$
$854$	20.0000	0.684386
$855$	0	0
$856$	20.0000i	0.683586i
$857$	28.0000	0.956462	0.478231	−	0.878234i	$-0.341278\pi$
$857$	28.0000	0.956462	0.478231	+	0.878234i	$0.341278\pi$
$858$	0	0
$859$	8.00000	0.272956	0.136478	−	0.990643i	$-0.456422\pi$
$859$	8.00000	0.272956	0.136478	+	0.990643i	$0.456422\pi$
$860$	− 16.0000i	− 0.545595i
$861$	0	0
$862$	0	0
$863$	− 8.00000i	− 0.272323i	−0.990687	−	0.136162i	$-0.956523\pi$
$863$	− 8.00000i	− 0.272323i	0.990687	−	0.136162i	$-0.0434766\pi$
$864$	0	0
$865$	− 32.0000i	− 1.08803i
$866$	2.00000i	0.0679628i
$867$	0	0
$868$	−4.00000	−0.135769
$869$	16.0000i	0.542763i
$870$	0	0
$871$	0	0
$872$	−10.0000	−0.338643
$873$	0	0
$874$	24.0000	0.811812
$875$	24.0000	0.811348
$876$	0	0
$877$	− 6.00000i	− 0.202606i	−0.994856	−	0.101303i	$-0.967699\pi$
$877$	− 6.00000i	− 0.202606i	0.994856	−	0.101303i	$-0.0323011\pi$
$878$	4.00000i	0.134993i
$879$	0	0
$880$	8.00000	0.269680
$881$	−36.0000	−1.21287	−0.606435	−	0.795133i	$-0.707401\pi$
$881$	−36.0000	−1.21287	−0.606435	+	0.795133i	$0.707401\pi$
$882$	0	0
$883$	20.0000	0.673054	0.336527	−	0.941674i	$-0.390748\pi$
$883$	20.0000	0.673054	0.336527	+	0.941674i	$0.390748\pi$
$884$	0	0
$885$	0	0
$886$	36.0000i	1.20944i
$887$	44.0000	1.47738	0.738688	−	0.674048i	$-0.235446\pi$
$887$	44.0000	1.47738	0.738688	+	0.674048i	$0.235446\pi$
$888$	0	0
$889$	− 16.0000i	− 0.536623i
$890$	12.0000i	0.402241i
$891$	0	0
$892$	6.00000i	0.200895i
$893$	−48.0000	−1.60626
$894$	0	0
$895$	− 24.0000i	− 0.802232i
$896$	−2.00000	−0.0668153
$897$	0	0
$898$	−26.0000	−0.867631
$899$	− 16.0000i	− 0.533630i
$900$	0	0
$901$	0	0
$902$	− 24.0000i	− 0.799113i
$903$	0	0
$904$	− 4.00000i	− 0.133038i
$905$	36.0000i	1.19668i
$906$	0	0
$907$	−12.0000	−0.398453	−0.199227	−	0.979953i	$-0.563843\pi$
$907$	−12.0000	−0.398453	−0.199227	+	0.979953i	$0.563843\pi$
$908$	12.0000i	0.398234i
$909$	0	0
$910$	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
$913$	48.0000	1.58857
$914$	−2.00000	−0.0661541
$915$	0	0
$916$	30.0000i	0.991228i
$917$	− 8.00000i	− 0.264183i
$918$	0	0
$919$	12.0000	0.395843	0.197922	−	0.980218i	$-0.436581\pi$
$919$	12.0000	0.395843	0.197922	+	0.980218i	$0.436581\pi$
$920$	−8.00000	−0.263752
$921$	0	0
$922$	30.0000	0.987997
$923$	0	0
$924$	0	0
$925$	6.00000i	0.197279i
$926$	−22.0000	−0.722965
$927$	0	0
$928$	− 8.00000i	− 0.262613i
$929$	− 30.0000i	− 0.984268i	−0.870519	−	0.492134i	$-0.836217\pi$
$929$	− 30.0000i	− 0.984268i	0.870519	−	0.492134i	$-0.163783\pi$
$930$	0	0
$931$	18.0000i	0.589926i
$932$	20.0000	0.655122
$933$	0	0
$934$	8.00000i	0.261768i
$935$	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	$0.336986\pi$
$938$	4.00000i	0.130605i
$939$	0	0
$940$	16.0000	0.521862
$941$	− 26.0000i	− 0.847576i	−0.905761	−	0.423788i	$-0.860700\pi$
$941$	− 26.0000i	− 0.847576i	0.905761	−	0.423788i	$-0.139300\pi$
$942$	0	0
$943$	24.0000i	0.781548i
$944$	4.00000i	0.130189i
$945$	0	0
$946$	32.0000	1.04041
$947$	36.0000i	1.16984i	0.811090	+	0.584921i	$0.198875\pi$
$947$	36.0000i	1.16984i	−0.811090	+	0.584921i	$0.801125\pi$
$948$	0	0
$949$	0	0
$950$	−6.00000	−0.194666
$951$	0	0
$952$	0	0
$953$	−28.0000	−0.907009	−0.453504	−	0.891254i	$-0.649826\pi$
$953$	−28.0000	−0.907009	−0.453504	+	0.891254i	$0.649826\pi$
$954$	0	0
$955$	− 8.00000i	− 0.258874i
$956$	− 24.0000i	− 0.776215i
$957$	0	0
$958$	0	0
$959$	−36.0000	−1.16250
$960$	0	0
$961$	27.0000	0.870968
$962$	0	0
$963$	0	0
$964$	− 6.00000i	− 0.193247i
$965$	−4.00000	−0.128765
$966$	0	0
$967$	58.0000i	1.86515i	0.360971	+	0.932577i	$0.382445\pi$
$967$	58.0000i	1.86515i	−0.360971	+	0.932577i	$0.617555\pi$
$968$	5.00000i	0.160706i
$969$	0	0
$970$	− 20.0000i	− 0.642161i
$971$	48.0000	1.54039	0.770197	−	0.637806i	$-0.220158\pi$
$971$	48.0000	1.54039	0.770197	+	0.637806i	$0.220158\pi$
$972$	0	0
$973$	40.0000i	1.28234i
$974$	38.0000	1.21760
$975$	0	0
$976$	10.0000	0.320092
$977$	− 18.0000i	− 0.575871i	−0.957650	−	0.287936i	$-0.907031\pi$
$977$	− 18.0000i	− 0.575871i	0.957650	−	0.287936i	$-0.0929689\pi$
$978$	0	0
$979$	−24.0000	−0.767043
$980$	− 6.00000i	− 0.191663i
$981$	0	0
$982$	16.0000i	0.510581i
$983$	− 32.0000i	− 1.02064i	−0.859984	−	0.510321i	$-0.829527\pi$
$983$	− 32.0000i	− 1.02064i	0.859984	−	0.510321i	$-0.170473\pi$
$984$	0	0
$985$	−36.0000	−1.14706
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−32.0000	−1.01754
$990$	0	0
$991$	−48.0000	−1.52477	−0.762385	−	0.647124i	$-0.775972\pi$
$991$	−48.0000	−1.52477	−0.762385	+	0.647124i	$0.775972\pi$
$992$	−2.00000	−0.0635001
$993$	0	0
$994$	32.0000i	1.01498i
$995$	− 16.0000i	− 0.507234i
$996$	0	0
$997$	38.0000	1.20347	0.601736	−	0.798695i	$-0.294476\pi$
$997$	38.0000	1.20347	0.601736	+	0.798695i	$0.294476\pi$
$998$	22.0000	0.696398
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3042.2.b.c.1351.2		2
3.2	odd	2		3042.2.b.b.1351.1		2
13.5	odd	4		3042.2.a.o.1.1		1
13.8	odd	4		234.2.a.a.1.1	✓	1
13.12	even	2	inner	3042.2.b.c.1351.1		2
39.5	even	4		3042.2.a.b.1.1		1
39.8	even	4		234.2.a.d.1.1	yes	1
39.38	odd	2		3042.2.b.b.1351.2		2
52.47	even	4		1872.2.a.g.1.1		1
65.8	even	4		5850.2.e.d.5149.2		2
65.34	odd	4		5850.2.a.bv.1.1		1
65.47	even	4		5850.2.e.d.5149.1		2
104.21	odd	4		7488.2.a.bp.1.1		1
104.99	even	4		7488.2.a.bu.1.1		1
117.34	odd	12		2106.2.e.z.1405.1		2
117.47	even	12		2106.2.e.e.1405.1		2
117.86	even	12		2106.2.e.e.703.1		2
117.112	odd	12		2106.2.e.z.703.1		2
156.47	odd	4		1872.2.a.p.1.1		1
195.8	odd	4		5850.2.e.bd.5149.1		2
195.47	odd	4		5850.2.e.bd.5149.2		2
195.164	even	4		5850.2.a.v.1.1		1
312.125	even	4		7488.2.a.j.1.1		1
312.203	odd	4		7488.2.a.s.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
234.2.a.a.1.1	✓	1	13.8	odd	4
234.2.a.d.1.1	yes	1	39.8	even	4
1872.2.a.g.1.1		1	52.47	even	4
1872.2.a.p.1.1		1	156.47	odd	4
2106.2.e.e.703.1		2	117.86	even	12
2106.2.e.e.1405.1		2	117.47	even	12
2106.2.e.z.703.1		2	117.112	odd	12
2106.2.e.z.1405.1		2	117.34	odd	12
3042.2.a.b.1.1		1	39.5	even	4
3042.2.a.o.1.1		1	13.5	odd	4
3042.2.b.b.1351.1		2	3.2	odd	2
3042.2.b.b.1351.2		2	39.38	odd	2
3042.2.b.c.1351.1		2	13.12	even	2	inner
3042.2.b.c.1351.2		2	1.1	even	1	trivial
5850.2.a.v.1.1		1	195.164	even	4
5850.2.a.bv.1.1		1	65.34	odd	4
5850.2.e.d.5149.1		2	65.47	even	4
5850.2.e.d.5149.2		2	65.8	even	4
5850.2.e.bd.5149.1		2	195.8	odd	4
5850.2.e.bd.5149.2		2	195.47	odd	4
7488.2.a.j.1.1		1	312.125	even	4
7488.2.a.s.1.1		1	312.203	odd	4
7488.2.a.bp.1.1		1	104.21	odd	4
7488.2.a.bu.1.1		1	104.99	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} +2.00000i q^{5} -2.00000i q^{7} -1.00000i q^{8} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{4} +2.00000i q^{5} -2.00000i q^{7} -1.00000i q^{8} -2.00000 q^{10} -4.00000i q^{11} +2.00000 q^{14} +1.00000 q^{16} +6.00000i q^{19} -2.00000i q^{20} +4.00000 q^{22} -4.00000 q^{23} +1.00000 q^{25} +2.00000i q^{28} -8.00000 q^{29} +2.00000i q^{31} +1.00000i q^{32} +4.00000 q^{35} +6.00000i q^{37} -6.00000 q^{38} +2.00000 q^{40} -6.00000i q^{41} +8.00000 q^{43} +4.00000i q^{44} -4.00000i q^{46} +8.00000i q^{47} +3.00000 q^{49} +1.00000i q^{50} +12.0000 q^{53} +8.00000 q^{55} -2.00000 q^{56} -8.00000i q^{58} +4.00000i q^{59} +10.0000 q^{61} -2.00000 q^{62} -1.00000 q^{64} +2.00000i q^{67} +4.00000i q^{70} +16.0000i q^{71} +14.0000i q^{73} -6.00000 q^{74} -6.00000i q^{76} -8.00000 q^{77} -4.00000 q^{79} +2.00000i q^{80} +6.00000 q^{82} +12.0000i q^{83} +8.00000i q^{86} -4.00000 q^{88} -6.00000i q^{89} +4.00000 q^{92} -8.00000 q^{94} -12.0000 q^{95} +10.0000i q^{97} +3.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4}+O(q^{10}) \) 2 * q - 2 * q^4	\( 2 q - 2 q^{4} - 4 q^{10} + 4 q^{14} + 2 q^{16} + 8 q^{22} - 8 q^{23} + 2 q^{25} - 16 q^{29} + 8 q^{35} - 12 q^{38} + 4 q^{40} + 16 q^{43} + 6 q^{49} + 24 q^{53} + 16 q^{55} - 4 q^{56} + 20 q^{61} - 4 q^{62} - 2 q^{64} - 12 q^{74} - 16 q^{77} - 8 q^{79} + 12 q^{82} - 8 q^{88} + 8 q^{92} - 16 q^{94} - 24 q^{95}+O(q^{100}) \) 2 * q - 2 * q^4 - 4 * q^10 + 4 * q^14 + 2 * q^16 + 8 * q^22 - 8 * q^23 + 2 * q^25 - 16 * q^29 + 8 * q^35 - 12 * q^38 + 4 * q^40 + 16 * q^43 + 6 * q^49 + 24 * q^53 + 16 * q^55 - 4 * q^56 + 20 * q^61 - 4 * q^62 - 2 * q^64 - 12 * q^74 - 16 * q^77 - 8 * q^79 + 12 * q^82 - 8 * q^88 + 8 * q^92 - 16 * q^94 - 24 * q^95

Introduction

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