LMFDB - Embedded newform 6900.2.f.c.6349.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6900,2,Mod(6349,6900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6900, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6900.6349");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6900.f (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:	$55.0967773947$
Analytic rank:	$1$
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 1380)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			6349.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	6900.6349
Dual form			6900.2.f.c.6349.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} -1.00000 q^{9} -6.00000i q^{13} -2.00000i q^{17} -6.00000 q^{19} -1.00000i q^{23} +1.00000i q^{27} -6.00000 q^{29} -4.00000 q^{31} -4.00000i q^{37} -6.00000 q^{39} +6.00000 q^{41} +4.00000i q^{43} +4.00000i q^{47} +7.00000 q^{49} -2.00000 q^{51} +6.00000i q^{53} +6.00000i q^{57} -14.0000 q^{59} +6.00000 q^{61} +4.00000i q^{67} -1.00000 q^{69} -2.00000 q^{71} +10.0000i q^{73} +14.0000 q^{79} +1.00000 q^{81} +4.00000i q^{83} +6.00000i q^{87} -18.0000 q^{89} +4.00000i q^{93} -12.0000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^9	$ 2 q - 2 q^{9} - 12 q^{19} - 12 q^{29} - 8 q^{31} - 12 q^{39} + 12 q^{41} + 14 q^{49} - 4 q^{51} - 28 q^{59} + 12 q^{61} - 2 q^{69} - 4 q^{71} + 28 q^{79} + 2 q^{81} - 36 q^{89}+O(q^{100}) $ 2 * q - 2 * q^9 - 12 * q^19 - 12 * q^29 - 8 * q^31 - 12 * q^39 + 12 * q^41 + 14 * q^49 - 4 * q^51 - 28 * q^59 + 12 * q^61 - 2 * q^69 - 4 * q^71 + 28 * q^79 + 2 * q^81 - 36 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/6900\mathbb{Z}\right)^\times$.

$n$	$277$	$1201$	$3451$	$4601$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$5$	0	0
$6$	0	0
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	− 6.00000i	− 1.66410i	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	− 6.00000i	− 1.66410i	0.554700	−	0.832050i	$-0.312833\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 2.00000i	− 0.485071i	−0.970143	−	0.242536i	$-0.922021\pi$
$17$	− 2.00000i	− 0.485071i	0.970143	−	0.242536i	$-0.0779791\pi$
$18$	0	0
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 1.00000i	− 0.208514i
$23$	− 1.00000i	− 0.208514i
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000i	0.192450i
$28$	0	0
$29$	−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
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 4.00000i	− 0.657596i	−0.944400	−	0.328798i	$-0.893356\pi$
$37$	− 4.00000i	− 0.657596i	0.944400	−	0.328798i	$-0.106644\pi$
$38$	0	0
$39$	−6.00000	−0.960769
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	4.00000i	0.609994i	0.952353	+	0.304997i	$0.0986555\pi$
$43$	4.00000i	0.609994i	−0.952353	+	0.304997i	$0.901344\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.00000i	0.583460i	0.956501	+	0.291730i	$0.0942309\pi$
$47$	4.00000i	0.583460i	−0.956501	+	0.291730i	$0.905769\pi$
$48$	0	0
$49$	7.00000	1.00000
$50$	0	0
$51$	−2.00000	−0.280056
$52$	0	0
$53$	6.00000i	0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$	6.00000i	0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	6.00000i	0.794719i
$58$	0	0
$59$	−14.0000	−1.82264	−0.911322	−	0.411693i	$-0.864937\pi$
$59$	−14.0000	−1.82264	−0.911322	+	0.411693i	$0.864937\pi$
$60$	0	0
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	0	0
$63$	0	0
$64$	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
$69$	−1.00000	−0.120386
$70$	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
$73$	10.0000i	1.17041i	0.810885	+	0.585206i	$0.198986\pi$
$73$	10.0000i	1.17041i	−0.810885	+	0.585206i	$0.801014\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	14.0000	1.57512	0.787562	−	0.616236i	$-0.211343\pi$
$79$	14.0000	1.57512	0.787562	+	0.616236i	$0.211343\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	4.00000i	0.439057i	0.975606	+	0.219529i	$0.0704519\pi$
$83$	4.00000i	0.439057i	−0.975606	+	0.219529i	$0.929548\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	6.00000i	0.643268i
$88$	0	0
$89$	−18.0000	−1.90800	−0.953998	−	0.299813i	$-0.903076\pi$
$89$	−18.0000	−1.90800	−0.953998	+	0.299813i	$0.903076\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	4.00000i	0.414781i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 12.0000i	− 1.21842i	−0.793011	−	0.609208i	$-0.791488\pi$
$97$	− 12.0000i	− 1.21842i	0.793011	−	0.609208i	$-0.208512\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	18.0000	1.79107	0.895533	−	0.444994i	$-0.146794\pi$
$101$	18.0000	1.79107	0.895533	+	0.444994i	$0.146794\pi$
$102$	0	0
$103$	4.00000i	0.394132i	0.980390	+	0.197066i	$0.0631413\pi$
$103$	4.00000i	0.394132i	−0.980390	+	0.197066i	$0.936859\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 12.0000i	− 1.16008i	−0.814587	−	0.580042i	$-0.803036\pi$
$107$	− 12.0000i	− 1.16008i	0.814587	−	0.580042i	$-0.196964\pi$
$108$	0	0
$109$	−18.0000	−1.72409	−0.862044	−	0.506834i	$-0.830816\pi$
$109$	−18.0000	−1.72409	−0.862044	+	0.506834i	$0.830816\pi$
$110$	0	0
$111$	−4.00000	−0.379663
$112$	0	0
$113$	6.00000i	0.564433i	0.959351	+	0.282216i	$0.0910696\pi$
$113$	6.00000i	0.564433i	−0.959351	+	0.282216i	$0.908930\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	6.00000i	0.554700i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	− 6.00000i	− 0.541002i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	8.00000i	0.709885i	0.934888	+	0.354943i	$0.115500\pi$
$127$	8.00000i	0.709885i	−0.934888	+	0.354943i	$0.884500\pi$
$128$	0	0
$129$	4.00000	0.352180
$130$	0	0
$131$	10.0000	0.873704	0.436852	−	0.899533i	$-0.356093\pi$
$131$	10.0000	0.873704	0.436852	+	0.899533i	$0.356093\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	18.0000i	1.53784i	0.639343	+	0.768922i	$0.279207\pi$
$137$	18.0000i	1.53784i	−0.639343	+	0.768922i	$0.720793\pi$
$138$	0	0
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	4.00000	0.336861
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 7.00000i	− 0.577350i
$148$	0	0
$149$	18.0000	1.47462	0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000	1.47462	0.737309	+	0.675556i	$0.236096\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	0	0
$153$	2.00000i	0.161690i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	6.00000	0.475831
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 12.0000i	− 0.939913i	−0.882690	−	0.469956i	$-0.844270\pi$
$163$	− 12.0000i	− 0.939913i	0.882690	−	0.469956i	$-0.155730\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	16.0000i	1.23812i	0.785345	+	0.619059i	$0.212486\pi$
$167$	16.0000i	1.23812i	−0.785345	+	0.619059i	$0.787514\pi$
$168$	0	0
$169$	−23.0000	−1.76923
$170$	0	0
$171$	6.00000	0.458831
$172$	0	0
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	14.0000i	1.05230i
$178$	0	0
$179$	−14.0000	−1.04641	−0.523205	−	0.852207i	$-0.675264\pi$
$179$	−14.0000	−1.04641	−0.523205	+	0.852207i	$0.675264\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	0	0
$183$	− 6.00000i	− 0.443533i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	− 18.0000i	− 1.29567i	−0.761781	−	0.647834i	$-0.775675\pi$
$193$	− 18.0000i	− 1.29567i	0.761781	−	0.647834i	$-0.224325\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	8.00000i	0.569976i	0.958531	+	0.284988i	$0.0919897\pi$
$197$	8.00000i	0.569976i	−0.958531	+	0.284988i	$0.908010\pi$
$198$	0	0
$199$	−2.00000	−0.141776	−0.0708881	−	0.997484i	$-0.522583\pi$
$199$	−2.00000	−0.141776	−0.0708881	+	0.997484i	$0.522583\pi$
$200$	0	0
$201$	4.00000	0.282138
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1.00000i	0.0695048i
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−8.00000	−0.550743	−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000	−0.550743	−0.275371	+	0.961338i	$0.588801\pi$
$212$	0	0
$213$	2.00000i	0.137038i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	10.0000	0.675737
$220$	0	0
$221$	−12.0000	−0.807207
$222$	0	0
$223$	24.0000i	1.60716i	0.595198	+	0.803579i	$0.297074\pi$
$223$	24.0000i	1.60716i	−0.595198	+	0.803579i	$0.702926\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	12.0000i	0.796468i	0.917284	+	0.398234i	$0.130377\pi$
$227$	12.0000i	0.796468i	−0.917284	+	0.398234i	$0.869623\pi$
$228$	0	0
$229$	−30.0000	−1.98246	−0.991228	−	0.132164i	$-0.957808\pi$
$229$	−30.0000	−1.98246	−0.991228	+	0.132164i	$0.957808\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	8.00000i	0.524097i	0.965055	+	0.262049i	$0.0843981\pi$
$233$	8.00000i	0.524097i	−0.965055	+	0.262049i	$0.915602\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 14.0000i	− 0.909398i
$238$	0	0
$239$	−14.0000	−0.905585	−0.452792	−	0.891616i	$-0.649572\pi$
$239$	−14.0000	−0.905585	−0.452792	+	0.891616i	$0.649572\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	0	0
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	36.0000i	2.29063i
$248$	0	0
$249$	4.00000	0.253490
$250$	0	0
$251$	−4.00000	−0.252478	−0.126239	−	0.992000i	$-0.540291\pi$
$251$	−4.00000	−0.252478	−0.126239	+	0.992000i	$0.540291\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 12.0000i	− 0.748539i	−0.927320	−	0.374270i	$-0.877893\pi$
$257$	− 12.0000i	− 0.748539i	0.927320	−	0.374270i	$-0.122107\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	6.00000	0.371391
$262$	0	0
$263$	16.0000i	0.986602i	0.869859	+	0.493301i	$0.164210\pi$
$263$	16.0000i	0.986602i	−0.869859	+	0.493301i	$0.835790\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	18.0000i	1.10158i
$268$	0	0
$269$	−10.0000	−0.609711	−0.304855	−	0.952399i	$-0.598608\pi$
$269$	−10.0000	−0.609711	−0.304855	+	0.952399i	$0.598608\pi$
$270$	0	0
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	10.0000i	0.600842i	0.953807	+	0.300421i	$0.0971271\pi$
$277$	10.0000i	0.600842i	−0.953807	+	0.300421i	$0.902873\pi$
$278$	0	0
$279$	4.00000	0.239474
$280$	0	0
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	0	0
$283$	− 24.0000i	− 1.42665i	−0.700832	−	0.713326i	$-0.747188\pi$
$283$	− 24.0000i	− 1.42665i	0.700832	−	0.713326i	$-0.252812\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	13.0000	0.764706
$290$	0	0
$291$	−12.0000	−0.703452
$292$	0	0
$293$	− 18.0000i	− 1.05157i	−0.850617	−	0.525786i	$-0.823771\pi$
$293$	− 18.0000i	− 1.05157i	0.850617	−	0.525786i	$-0.176229\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−6.00000	−0.346989
$300$	0	0
$301$	0	0
$302$	0	0
$303$	− 18.0000i	− 1.03407i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	28.0000i	1.59804i	0.601302	+	0.799022i	$0.294649\pi$
$307$	28.0000i	1.59804i	−0.601302	+	0.799022i	$0.705351\pi$
$308$	0	0
$309$	4.00000	0.227552
$310$	0	0
$311$	18.0000	1.02069	0.510343	−	0.859971i	$-0.329518\pi$
$311$	18.0000	1.02069	0.510343	+	0.859971i	$0.329518\pi$
$312$	0	0
$313$	− 24.0000i	− 1.35656i	−0.734803	−	0.678280i	$-0.762726\pi$
$313$	− 24.0000i	− 1.35656i	0.734803	−	0.678280i	$-0.237274\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 4.00000i	− 0.224662i	−0.993671	−	0.112331i	$-0.964168\pi$
$317$	− 4.00000i	− 0.224662i	0.993671	−	0.112331i	$-0.0358318\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−12.0000	−0.669775
$322$	0	0
$323$	12.0000i	0.667698i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	18.0000i	0.995402i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−20.0000	−1.09930	−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000	−1.09930	−0.549650	+	0.835395i	$0.685239\pi$
$332$	0	0
$333$	4.00000i	0.219199i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 20.0000i	− 1.08947i	−0.838608	−	0.544735i	$-0.816630\pi$
$337$	− 20.0000i	− 1.08947i	0.838608	−	0.544735i	$-0.183370\pi$
$338$	0	0
$339$	6.00000	0.325875
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 12.0000i	− 0.644194i	−0.946707	−	0.322097i	$-0.895612\pi$
$347$	− 12.0000i	− 0.644194i	0.946707	−	0.322097i	$-0.104388\pi$
$348$	0	0
$349$	−34.0000	−1.81998	−0.909989	−	0.414632i	$-0.863910\pi$
$349$	−34.0000	−1.81998	−0.909989	+	0.414632i	$0.863910\pi$
$350$	0	0
$351$	6.00000	0.320256
$352$	0	0
$353$	12.0000i	0.638696i	0.947638	+	0.319348i	$0.103464\pi$
$353$	12.0000i	0.638696i	−0.947638	+	0.319348i	$0.896536\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	20.0000	1.05556	0.527780	−	0.849381i	$-0.323025\pi$
$359$	20.0000	1.05556	0.527780	+	0.849381i	$0.323025\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	0	0
$363$	11.0000i	0.577350i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	12.0000i	0.626395i	0.949688	+	0.313197i	$0.101400\pi$
$367$	12.0000i	0.626395i	−0.949688	+	0.313197i	$0.898600\pi$
$368$	0	0
$369$	−6.00000	−0.312348
$370$	0	0
$371$	0	0
$372$	0	0
$373$	8.00000i	0.414224i	0.978317	+	0.207112i	$0.0664065\pi$
$373$	8.00000i	0.414224i	−0.978317	+	0.207112i	$0.933593\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	36.0000i	1.85409i
$378$	0	0
$379$	−6.00000	−0.308199	−0.154100	−	0.988055i	$-0.549248\pi$
$379$	−6.00000	−0.308199	−0.154100	+	0.988055i	$0.549248\pi$
$380$	0	0
$381$	8.00000	0.409852
$382$	0	0
$383$	− 8.00000i	− 0.408781i	−0.978889	−	0.204390i	$-0.934479\pi$
$383$	− 8.00000i	− 0.408781i	0.978889	−	0.204390i	$-0.0655212\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 4.00000i	− 0.203331i
$388$	0	0
$389$	10.0000	0.507020	0.253510	−	0.967333i	$-0.418415\pi$
$389$	10.0000	0.507020	0.253510	+	0.967333i	$0.418415\pi$
$390$	0	0
$391$	−2.00000	−0.101144
$392$	0	0
$393$	− 10.0000i	− 0.504433i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	18.0000i	0.903394i	0.892171	+	0.451697i	$0.149181\pi$
$397$	18.0000i	0.903394i	−0.892171	+	0.451697i	$0.850819\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	38.0000	1.89763	0.948815	−	0.315833i	$-0.102284\pi$
$401$	38.0000	1.89763	0.948815	+	0.315833i	$0.102284\pi$
$402$	0	0
$403$	24.0000i	1.19553i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	2.00000	0.0988936	0.0494468	−	0.998777i	$-0.484254\pi$
$409$	2.00000	0.0988936	0.0494468	+	0.998777i	$0.484254\pi$
$410$	0	0
$411$	18.0000	0.887875
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 4.00000i	− 0.195881i
$418$	0	0
$419$	20.0000	0.977064	0.488532	−	0.872546i	$-0.337533\pi$
$419$	20.0000	0.977064	0.488532	+	0.872546i	$0.337533\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	0	0
$423$	− 4.00000i	− 0.194487i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−8.00000	−0.385346	−0.192673	−	0.981263i	$-0.561716\pi$
$431$	−8.00000	−0.385346	−0.192673	+	0.981263i	$0.561716\pi$
$432$	0	0
$433$	28.0000i	1.34559i	0.739827	+	0.672797i	$0.234907\pi$
$433$	28.0000i	1.34559i	−0.739827	+	0.672797i	$0.765093\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	6.00000i	0.287019i
$438$	0	0
$439$	16.0000	0.763638	0.381819	−	0.924237i	$-0.375298\pi$
$439$	16.0000	0.763638	0.381819	+	0.924237i	$0.375298\pi$
$440$	0	0
$441$	−7.00000	−0.333333
$442$	0	0
$443$	8.00000i	0.380091i	0.981775	+	0.190046i	$0.0608636\pi$
$443$	8.00000i	0.380091i	−0.981775	+	0.190046i	$0.939136\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 18.0000i	− 0.851371i
$448$	0	0
$449$	14.0000	0.660701	0.330350	−	0.943858i	$-0.392833\pi$
$449$	14.0000	0.660701	0.330350	+	0.943858i	$0.392833\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	16.0000i	0.751746i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 4.00000i	− 0.187112i	−0.995614	−	0.0935561i	$-0.970177\pi$
$457$	− 4.00000i	− 0.187112i	0.995614	−	0.0935561i	$-0.0298234\pi$
$458$	0	0
$459$	2.00000	0.0933520
$460$	0	0
$461$	−6.00000	−0.279448	−0.139724	−	0.990190i	$-0.544622\pi$
$461$	−6.00000	−0.279448	−0.139724	+	0.990190i	$0.544622\pi$
$462$	0	0
$463$	32.0000i	1.48717i	0.668644	+	0.743583i	$0.266875\pi$
$463$	32.0000i	1.48717i	−0.668644	+	0.743583i	$0.733125\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	28.0000i	1.29569i	0.761774	+	0.647843i	$0.224329\pi$
$467$	28.0000i	1.29569i	−0.761774	+	0.647843i	$0.775671\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 6.00000i	− 0.274721i
$478$	0	0
$479$	12.0000	0.548294	0.274147	−	0.961688i	$-0.411605\pi$
$479$	12.0000	0.548294	0.274147	+	0.961688i	$0.411605\pi$
$480$	0	0
$481$	−24.0000	−1.09431
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 16.0000i	− 0.725029i	−0.931978	−	0.362515i	$-0.881918\pi$
$487$	− 16.0000i	− 0.725029i	0.931978	−	0.362515i	$-0.118082\pi$
$488$	0	0
$489$	−12.0000	−0.542659
$490$	0	0
$491$	6.00000	0.270776	0.135388	−	0.990793i	$-0.456772\pi$
$491$	6.00000	0.270776	0.135388	+	0.990793i	$0.456772\pi$
$492$	0	0
$493$	12.0000i	0.540453i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−28.0000	−1.25345	−0.626726	−	0.779240i	$-0.715605\pi$
$499$	−28.0000	−1.25345	−0.626726	+	0.779240i	$0.715605\pi$
$500$	0	0
$501$	16.0000	0.714827
$502$	0	0
$503$	16.0000i	0.713405i	0.934218	+	0.356702i	$0.116099\pi$
$503$	16.0000i	0.713405i	−0.934218	+	0.356702i	$0.883901\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	23.0000i	1.02147i
$508$	0	0
$509$	−22.0000	−0.975133	−0.487566	−	0.873086i	$-0.662115\pi$
$509$	−22.0000	−0.975133	−0.487566	+	0.873086i	$0.662115\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	− 6.00000i	− 0.264906i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	22.0000	0.963837	0.481919	−	0.876216i	$-0.339940\pi$
$521$	22.0000	0.963837	0.481919	+	0.876216i	$0.339940\pi$
$522$	0	0
$523$	8.00000i	0.349816i	0.984585	+	0.174908i	$0.0559627\pi$
$523$	8.00000i	0.349816i	−0.984585	+	0.174908i	$0.944037\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	8.00000i	0.348485i
$528$	0	0
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	14.0000	0.607548
$532$	0	0
$533$	− 36.0000i	− 1.55933i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	14.0000i	0.604145i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	2.00000	0.0859867	0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000	0.0859867	0.0429934	+	0.999075i	$0.486311\pi$
$542$	0	0
$543$	− 2.00000i	− 0.0858282i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 28.0000i	− 1.19719i	−0.801050	−	0.598597i	$-0.795725\pi$
$547$	− 28.0000i	− 1.19719i	0.801050	−	0.598597i	$-0.204275\pi$
$548$	0	0
$549$	−6.00000	−0.256074
$550$	0	0
$551$	36.0000	1.53365
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 22.0000i	− 0.932170i	−0.884740	−	0.466085i	$-0.845664\pi$
$557$	− 22.0000i	− 0.932170i	0.884740	−	0.466085i	$-0.154336\pi$
$558$	0	0
$559$	24.0000	1.01509
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 28.0000i	− 1.18006i	−0.807382	−	0.590030i	$-0.799116\pi$
$563$	− 28.0000i	− 1.18006i	0.807382	−	0.590030i	$-0.200884\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−10.0000	−0.419222	−0.209611	−	0.977785i	$-0.567220\pi$
$569$	−10.0000	−0.419222	−0.209611	+	0.977785i	$0.567220\pi$
$570$	0	0
$571$	−10.0000	−0.418487	−0.209243	−	0.977864i	$-0.567100\pi$
$571$	−10.0000	−0.418487	−0.209243	+	0.977864i	$0.567100\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 14.0000i	− 0.582828i	−0.956597	−	0.291414i	$-0.905874\pi$
$577$	− 14.0000i	− 0.582828i	0.956597	−	0.291414i	$-0.0941257\pi$
$578$	0	0
$579$	−18.0000	−0.748054
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	36.0000i	1.48588i	0.669359	+	0.742940i	$0.266569\pi$
$587$	36.0000i	1.48588i	−0.669359	+	0.742940i	$0.733431\pi$
$588$	0	0
$589$	24.0000	0.988903
$590$	0	0
$591$	8.00000	0.329076
$592$	0	0
$593$	36.0000i	1.47834i	0.673517	+	0.739171i	$0.264783\pi$
$593$	36.0000i	1.47834i	−0.673517	+	0.739171i	$0.735217\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	2.00000i	0.0818546i
$598$	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
$601$	22.0000	0.897399	0.448699	−	0.893683i	$-0.351887\pi$
$601$	22.0000	0.897399	0.448699	+	0.893683i	$0.351887\pi$
$602$	0	0
$603$	− 4.00000i	− 0.162893i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 32.0000i	− 1.29884i	−0.760430	−	0.649420i	$-0.775012\pi$
$607$	− 32.0000i	− 1.29884i	0.760430	−	0.649420i	$-0.224988\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	24.0000	0.970936
$612$	0	0
$613$	− 16.0000i	− 0.646234i	−0.946359	−	0.323117i	$-0.895269\pi$
$613$	− 16.0000i	− 0.646234i	0.946359	−	0.323117i	$-0.104731\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 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
$619$	10.0000	0.401934	0.200967	−	0.979598i	$-0.435592\pi$
$619$	10.0000	0.401934	0.200967	+	0.979598i	$0.435592\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−8.00000	−0.318981
$630$	0	0
$631$	−22.0000	−0.875806	−0.437903	−	0.899022i	$-0.644279\pi$
$631$	−22.0000	−0.875806	−0.437903	+	0.899022i	$0.644279\pi$
$632$	0	0
$633$	8.00000i	0.317971i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 42.0000i	− 1.66410i
$638$	0	0
$639$	2.00000	0.0791188
$640$	0	0
$641$	−46.0000	−1.81689	−0.908445	−	0.418004i	$-0.862730\pi$
$641$	−46.0000	−1.81689	−0.908445	+	0.418004i	$0.862730\pi$
$642$	0	0
$643$	44.0000i	1.73519i	0.497271	+	0.867595i	$0.334335\pi$
$643$	44.0000i	1.73519i	−0.497271	+	0.867595i	$0.665665\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 12.0000i	− 0.471769i	−0.971781	−	0.235884i	$-0.924201\pi$
$647$	− 12.0000i	− 0.471769i	0.971781	−	0.235884i	$-0.0757987\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 24.0000i	− 0.939193i	−0.882881	−	0.469596i	$-0.844399\pi$
$653$	− 24.0000i	− 0.939193i	0.882881	−	0.469596i	$-0.155601\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 10.0000i	− 0.390137i
$658$	0	0
$659$	−24.0000	−0.934907	−0.467454	−	0.884018i	$-0.654829\pi$
$659$	−24.0000	−0.934907	−0.467454	+	0.884018i	$0.654829\pi$
$660$	0	0
$661$	−34.0000	−1.32245	−0.661223	−	0.750189i	$-0.729962\pi$
$661$	−34.0000	−1.32245	−0.661223	+	0.750189i	$0.729962\pi$
$662$	0	0
$663$	12.0000i	0.466041i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	6.00000i	0.232321i
$668$	0	0
$669$	24.0000	0.927894
$670$	0	0
$671$	0	0
$672$	0	0
$673$	− 6.00000i	− 0.231283i	−0.993291	−	0.115642i	$-0.963108\pi$
$673$	− 6.00000i	− 0.231283i	0.993291	−	0.115642i	$-0.0368924\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	6.00000i	0.230599i	0.993331	+	0.115299i	$0.0367827\pi$
$677$	6.00000i	0.230599i	−0.993331	+	0.115299i	$0.963217\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	12.0000	0.459841
$682$	0	0
$683$	44.0000i	1.68361i	0.539779	+	0.841807i	$0.318508\pi$
$683$	44.0000i	1.68361i	−0.539779	+	0.841807i	$0.681492\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	30.0000i	1.14457i
$688$	0	0
$689$	36.0000	1.37149
$690$	0	0
$691$	48.0000	1.82601	0.913003	−	0.407953i	$-0.133757\pi$
$691$	48.0000	1.82601	0.913003	+	0.407953i	$0.133757\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 12.0000i	− 0.454532i
$698$	0	0
$699$	8.00000	0.302588
$700$	0	0
$701$	−18.0000	−0.679851	−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000	−0.679851	−0.339925	+	0.940452i	$0.610402\pi$
$702$	0	0
$703$	24.0000i	0.905177i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−38.0000	−1.42712	−0.713560	−	0.700594i	$-0.752918\pi$
$709$	−38.0000	−1.42712	−0.713560	+	0.700594i	$0.752918\pi$
$710$	0	0
$711$	−14.0000	−0.525041
$712$	0	0
$713$	4.00000i	0.149801i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	14.0000i	0.522840i
$718$	0	0
$719$	−10.0000	−0.372937	−0.186469	−	0.982461i	$-0.559704\pi$
$719$	−10.0000	−0.372937	−0.186469	+	0.982461i	$0.559704\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	10.0000i	0.371904i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 8.00000i	− 0.296704i	−0.988935	−	0.148352i	$-0.952603\pi$
$727$	− 8.00000i	− 0.296704i	0.988935	−	0.148352i	$-0.0473968\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	8.00000	0.295891
$732$	0	0
$733$	− 16.0000i	− 0.590973i	−0.955347	−	0.295487i	$-0.904518\pi$
$733$	− 16.0000i	− 0.590973i	0.955347	−	0.295487i	$-0.0954818\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	0	0
$741$	36.0000	1.32249
$742$	0	0
$743$	16.0000i	0.586983i	0.955962	+	0.293492i	$0.0948173\pi$
$743$	16.0000i	0.586983i	−0.955962	+	0.293492i	$0.905183\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 4.00000i	− 0.146352i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−30.0000	−1.09472	−0.547358	−	0.836899i	$-0.684366\pi$
$751$	−30.0000	−1.09472	−0.547358	+	0.836899i	$0.684366\pi$
$752$	0	0
$753$	4.00000i	0.145768i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 4.00000i	− 0.145382i	−0.997354	−	0.0726912i	$-0.976841\pi$
$757$	− 4.00000i	− 0.145382i	0.997354	−	0.0726912i	$-0.0231588\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−30.0000	−1.08750	−0.543750	−	0.839248i	$-0.682996\pi$
$761$	−30.0000	−1.08750	−0.543750	+	0.839248i	$0.682996\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	84.0000i	3.03306i
$768$	0	0
$769$	−30.0000	−1.08183	−0.540914	−	0.841078i	$-0.681921\pi$
$769$	−30.0000	−1.08183	−0.540914	+	0.841078i	$0.681921\pi$
$770$	0	0
$771$	−12.0000	−0.432169
$772$	0	0
$773$	− 46.0000i	− 1.65451i	−0.561830	−	0.827253i	$-0.689903\pi$
$773$	− 46.0000i	− 1.65451i	0.561830	−	0.827253i	$-0.310097\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−36.0000	−1.28983
$780$	0	0
$781$	0	0
$782$	0	0
$783$	− 6.00000i	− 0.214423i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 28.0000i	− 0.998092i	−0.866575	−	0.499046i	$-0.833684\pi$
$787$	− 28.0000i	− 0.998092i	0.866575	−	0.499046i	$-0.166316\pi$
$788$	0	0
$789$	16.0000	0.569615
$790$	0	0
$791$	0	0
$792$	0	0
$793$	− 36.0000i	− 1.27840i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 30.0000i	− 1.06265i	−0.847167	−	0.531327i	$-0.821693\pi$
$797$	− 30.0000i	− 1.06265i	0.847167	−	0.531327i	$-0.178307\pi$
$798$	0	0
$799$	8.00000	0.283020
$800$	0	0
$801$	18.0000	0.635999
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	10.0000i	0.352017i
$808$	0	0
$809$	42.0000	1.47664	0.738321	−	0.674450i	$-0.235619\pi$
$809$	42.0000	1.47664	0.738321	+	0.674450i	$0.235619\pi$
$810$	0	0
$811$	−44.0000	−1.54505	−0.772524	−	0.634985i	$-0.781006\pi$
$811$	−44.0000	−1.54505	−0.772524	+	0.634985i	$0.781006\pi$
$812$	0	0
$813$	8.00000i	0.280572i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 24.0000i	− 0.839654i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	38.0000	1.32621	0.663105	−	0.748527i	$-0.269238\pi$
$821$	38.0000	1.32621	0.663105	+	0.748527i	$0.269238\pi$
$822$	0	0
$823$	− 40.0000i	− 1.39431i	−0.716919	−	0.697156i	$-0.754448\pi$
$823$	− 40.0000i	− 1.39431i	0.716919	−	0.697156i	$-0.245552\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 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
$829$	−2.00000	−0.0694629	−0.0347314	−	0.999397i	$-0.511058\pi$
$829$	−2.00000	−0.0694629	−0.0347314	+	0.999397i	$0.511058\pi$
$830$	0	0
$831$	10.0000	0.346896
$832$	0	0
$833$	− 14.0000i	− 0.485071i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 4.00000i	− 0.138260i
$838$	0	0
$839$	−24.0000	−0.828572	−0.414286	−	0.910147i	$-0.635969\pi$
$839$	−24.0000	−0.828572	−0.414286	+	0.910147i	$0.635969\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	10.0000i	0.344418i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−24.0000	−0.823678
$850$	0	0
$851$	−4.00000	−0.137118
$852$	0	0
$853$	46.0000i	1.57501i	0.616308	+	0.787505i	$0.288628\pi$
$853$	46.0000i	1.57501i	−0.616308	+	0.787505i	$0.711372\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	32.0000i	1.09310i	0.837427	+	0.546550i	$0.184059\pi$
$857$	32.0000i	1.09310i	−0.837427	+	0.546550i	$0.815941\pi$
$858$	0	0
$859$	−8.00000	−0.272956	−0.136478	−	0.990643i	$-0.543578\pi$
$859$	−8.00000	−0.272956	−0.136478	+	0.990643i	$0.543578\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 48.0000i	− 1.63394i	−0.576681	−	0.816970i	$-0.695652\pi$
$863$	− 48.0000i	− 1.63394i	0.576681	−	0.816970i	$-0.304348\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 13.0000i	− 0.441503i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	24.0000	0.813209
$872$	0	0
$873$	12.0000i	0.406138i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 2.00000i	− 0.0675352i	−0.999430	−	0.0337676i	$-0.989249\pi$
$877$	− 2.00000i	− 0.0675352i	0.999430	−	0.0337676i	$-0.0107506\pi$
$878$	0	0
$879$	−18.0000	−0.607125
$880$	0	0
$881$	14.0000	0.471672	0.235836	−	0.971793i	$-0.424217\pi$
$881$	14.0000	0.471672	0.235836	+	0.971793i	$0.424217\pi$
$882$	0	0
$883$	36.0000i	1.21150i	0.795656	+	0.605748i	$0.207126\pi$
$883$	36.0000i	1.21150i	−0.795656	+	0.605748i	$0.792874\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	32.0000i	1.07445i	0.843437	+	0.537227i	$0.180528\pi$
$887$	32.0000i	1.07445i	−0.843437	+	0.537227i	$0.819472\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 24.0000i	− 0.803129i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	6.00000i	0.200334i
$898$	0	0
$899$	24.0000	0.800445
$900$	0	0
$901$	12.0000	0.399778
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 8.00000i	− 0.265636i	−0.991140	−	0.132818i	$-0.957597\pi$
$907$	− 8.00000i	− 0.265636i	0.991140	−	0.132818i	$-0.0424025\pi$
$908$	0	0
$909$	−18.0000	−0.597022
$910$	0	0
$911$	−44.0000	−1.45779	−0.728893	−	0.684628i	$-0.759965\pi$
$911$	−44.0000	−1.45779	−0.728893	+	0.684628i	$0.759965\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−38.0000	−1.25350	−0.626752	−	0.779219i	$-0.715616\pi$
$919$	−38.0000	−1.25350	−0.626752	+	0.779219i	$0.715616\pi$
$920$	0	0
$921$	28.0000	0.922631
$922$	0	0
$923$	12.0000i	0.394985i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 4.00000i	− 0.131377i
$928$	0	0
$929$	−10.0000	−0.328089	−0.164045	−	0.986453i	$-0.552454\pi$
$929$	−10.0000	−0.328089	−0.164045	+	0.986453i	$0.552454\pi$
$930$	0	0
$931$	−42.0000	−1.37649
$932$	0	0
$933$	− 18.0000i	− 0.589294i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 32.0000i	− 1.04539i	−0.852518	−	0.522697i	$-0.824926\pi$
$937$	− 32.0000i	− 1.04539i	0.852518	−	0.522697i	$-0.175074\pi$
$938$	0	0
$939$	−24.0000	−0.783210
$940$	0	0
$941$	6.00000	0.195594	0.0977972	−	0.995206i	$-0.468820\pi$
$941$	6.00000	0.195594	0.0977972	+	0.995206i	$0.468820\pi$
$942$	0	0
$943$	− 6.00000i	− 0.195387i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 32.0000i	− 1.03986i	−0.854209	−	0.519930i	$-0.825958\pi$
$947$	− 32.0000i	− 1.03986i	0.854209	−	0.519930i	$-0.174042\pi$
$948$	0	0
$949$	60.0000	1.94768
$950$	0	0
$951$	−4.00000	−0.129709
$952$	0	0
$953$	− 6.00000i	− 0.194359i	−0.995267	−	0.0971795i	$-0.969018\pi$
$953$	− 6.00000i	− 0.194359i	0.995267	−	0.0971795i	$-0.0309821\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	12.0000i	0.386695i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	8.00000i	0.257263i	0.991692	+	0.128631i	$0.0410584\pi$
$967$	8.00000i	0.257263i	−0.991692	+	0.128631i	$0.958942\pi$
$968$	0	0
$969$	12.0000	0.385496
$970$	0	0
$971$	−20.0000	−0.641831	−0.320915	−	0.947108i	$-0.603990\pi$
$971$	−20.0000	−0.641831	−0.320915	+	0.947108i	$0.603990\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	34.0000i	1.08776i	0.839164	+	0.543878i	$0.183045\pi$
$977$	34.0000i	1.08776i	−0.839164	+	0.543878i	$0.816955\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	18.0000	0.574696
$982$	0	0
$983$	24.0000i	0.765481i	0.923856	+	0.382741i	$0.125020\pi$
$983$	24.0000i	0.765481i	−0.923856	+	0.382741i	$0.874980\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	4.00000	0.127193
$990$	0	0
$991$	−20.0000	−0.635321	−0.317660	−	0.948205i	$-0.602897\pi$
$991$	−20.0000	−0.635321	−0.317660	+	0.948205i	$0.602897\pi$
$992$	0	0
$993$	20.0000i	0.634681i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 50.0000i	− 1.58352i	−0.610835	−	0.791758i	$-0.709166\pi$
$997$	− 50.0000i	− 1.58352i	0.610835	−	0.791758i	$-0.290834\pi$
$998$	0	0
$999$	4.00000	0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6900.2.f.c.6349.1		2
5.2	odd	4		1380.2.a.b.1.1	✓	1
5.3	odd	4		6900.2.a.g.1.1		1
5.4	even	2	inner	6900.2.f.c.6349.2		2
15.2	even	4		4140.2.a.j.1.1		1
20.7	even	4		5520.2.a.w.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1380.2.a.b.1.1	✓	1	5.2	odd	4
4140.2.a.j.1.1		1	15.2	even	4
5520.2.a.w.1.1		1	20.7	even	4
6900.2.a.g.1.1		1	5.3	odd	4
6900.2.f.c.6349.1		2	1.1	even	1	trivial
6900.2.f.c.6349.2		2	5.4	even	2	inner

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:	\( N \)	\(=\)	\( 6900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6900.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} -1.00000 q^{9} -6.00000i q^{13} -2.00000i q^{17} -6.00000 q^{19} -1.00000i q^{23} +1.00000i q^{27} -6.00000 q^{29} -4.00000 q^{31} -4.00000i q^{37} -6.00000 q^{39} +6.00000 q^{41} +4.00000i q^{43} +4.00000i q^{47} +7.00000 q^{49} -2.00000 q^{51} +6.00000i q^{53} +6.00000i q^{57} -14.0000 q^{59} +6.00000 q^{61} +4.00000i q^{67} -1.00000 q^{69} -2.00000 q^{71} +10.0000i q^{73} +14.0000 q^{79} +1.00000 q^{81} +4.00000i q^{83} +6.00000i q^{87} -18.0000 q^{89} +4.00000i q^{93} -12.0000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^9	\( 2 q - 2 q^{9} - 12 q^{19} - 12 q^{29} - 8 q^{31} - 12 q^{39} + 12 q^{41} + 14 q^{49} - 4 q^{51} - 28 q^{59} + 12 q^{61} - 2 q^{69} - 4 q^{71} + 28 q^{79} + 2 q^{81} - 36 q^{89}+O(q^{100}) \) 2 * q - 2 * q^9 - 12 * q^19 - 12 * q^29 - 8 * q^31 - 12 * q^39 + 12 * q^41 + 14 * q^49 - 4 * q^51 - 28 * q^59 + 12 * q^61 - 2 * q^69 - 4 * q^71 + 28 * q^79 + 2 * q^81 - 36 * q^89

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