LMFDB - Embedded newform 5400.2.f.s.649.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5400,2,Mod(649,5400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5400, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("5400.649");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5400 = 2^{3} \cdot 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5400.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:	$43.1192170915$
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:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1080)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	5400.649
Dual form			5400.2.f.s.649.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.00000i q^{7} +O(q^{10})$	$q-4.00000i q^{7} +2.00000 q^{11} -4.00000i q^{13} +1.00000i q^{17} +5.00000 q^{19} -5.00000i q^{23} -8.00000 q^{29} +7.00000 q^{31} -6.00000i q^{37} +6.00000 q^{41} +2.00000i q^{43} +8.00000i q^{47} -9.00000 q^{49} -9.00000i q^{53} -4.00000 q^{59} +13.0000 q^{61} -10.0000i q^{67} -6.00000 q^{71} +6.00000i q^{73} -8.00000i q^{77} -9.00000 q^{79} +17.0000i q^{83} +6.00000 q^{89} -16.0000 q^{91} -8.00000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q+O(q^{10}) $ 2 * q	$ 2 q + 4 q^{11} + 10 q^{19} - 16 q^{29} + 14 q^{31} + 12 q^{41} - 18 q^{49} - 8 q^{59} + 26 q^{61} - 12 q^{71} - 18 q^{79} + 12 q^{89} - 32 q^{91}+O(q^{100}) $ 2 * q + 4 * q^11 + 10 * q^19 - 16 * q^29 + 14 * q^31 + 12 * q^41 - 18 * q^49 - 8 * q^59 + 26 * q^61 - 12 * q^71 - 18 * q^79 + 12 * q^89 - 32 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5400\mathbb{Z}\right)^\times$.

$n$	$1001$	$1351$	$2377$	$2701$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 4.00000i	− 1.51186i	−0.654654	−	0.755929i	$-0.727186\pi$
$7$	− 4.00000i	− 1.51186i	0.654654	−	0.755929i	$-0.272814\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	− 4.00000i	− 1.10940i	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	− 4.00000i	− 1.10940i	0.832050	−	0.554700i	$-0.187167\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.00000i	0.242536i	0.992620	+	0.121268i	$0.0386960\pi$
$17$	1.00000i	0.242536i	−0.992620	+	0.121268i	$0.961304\pi$
$18$	0	0
$19$	5.00000	1.14708	0.573539	−	0.819178i	$-0.305570\pi$
$19$	5.00000	1.14708	0.573539	+	0.819178i	$0.305570\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 5.00000i	− 1.04257i	−0.853382	−	0.521286i	$-0.825452\pi$
$23$	− 5.00000i	− 1.04257i	0.853382	−	0.521286i	$-0.174548\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−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
$31$	7.00000	1.25724	0.628619	−	0.777714i	$-0.283621\pi$
$31$	7.00000	1.25724	0.628619	+	0.777714i	$0.283621\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 6.00000i	− 0.986394i	−0.869918	−	0.493197i	$-0.835828\pi$
$37$	− 6.00000i	− 0.986394i	0.869918	−	0.493197i	$-0.164172\pi$
$38$	0	0
$39$	0	0
$40$	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
$43$	2.00000i	0.304997i	0.988304	+	0.152499i	$0.0487319\pi$
$43$	2.00000i	0.304997i	−0.988304	+	0.152499i	$0.951268\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	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
$49$	−9.00000	−1.28571
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 9.00000i	− 1.23625i	−0.786082	−	0.618123i	$-0.787894\pi$
$53$	− 9.00000i	− 1.23625i	0.786082	−	0.618123i	$-0.212106\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	13.0000	1.66448	0.832240	−	0.554416i	$-0.187058\pi$
$61$	13.0000	1.66448	0.832240	+	0.554416i	$0.187058\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 10.0000i	− 1.22169i	−0.791748	−	0.610847i	$-0.790829\pi$
$67$	− 10.0000i	− 1.22169i	0.791748	−	0.610847i	$-0.209171\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	0	0
$73$	6.00000i	0.702247i	0.936329	+	0.351123i	$0.114200\pi$
$73$	6.00000i	0.702247i	−0.936329	+	0.351123i	$0.885800\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 8.00000i	− 0.911685i
$78$	0	0
$79$	−9.00000	−1.01258	−0.506290	−	0.862364i	$-0.668983\pi$
$79$	−9.00000	−1.01258	−0.506290	+	0.862364i	$0.668983\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	17.0000i	1.86599i	0.359886	+	0.932996i	$0.382816\pi$
$83$	17.0000i	1.86599i	−0.359886	+	0.932996i	$0.617184\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	−16.0000	−1.67726
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 8.00000i	− 0.812277i	−0.913812	−	0.406138i	$-0.866875\pi$
$97$	− 8.00000i	− 0.812277i	0.913812	−	0.406138i	$-0.133125\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	12.0000	1.19404	0.597022	−	0.802225i	$-0.296350\pi$
$101$	12.0000	1.19404	0.597022	+	0.802225i	$0.296350\pi$
$102$	0	0
$103$	− 4.00000i	− 0.394132i	−0.980390	−	0.197066i	$-0.936859\pi$
$103$	− 4.00000i	− 0.394132i	0.980390	−	0.197066i	$-0.0631413\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	12.0000i	1.16008i	0.814587	+	0.580042i	$0.196964\pi$
$107$	12.0000i	1.16008i	−0.814587	+	0.580042i	$0.803036\pi$
$108$	0	0
$109$	−17.0000	−1.62830	−0.814152	−	0.580651i	$-0.802798\pi$
$109$	−17.0000	−1.62830	−0.814152	+	0.580651i	$0.802798\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 6.00000i	− 0.564433i	−0.959351	−	0.282216i	$-0.908930\pi$
$113$	− 6.00000i	− 0.564433i	0.959351	−	0.282216i	$-0.0910696\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	4.00000	0.366679
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 18.0000i	− 1.59724i	−0.601834	−	0.798621i	$-0.705563\pi$
$127$	− 18.0000i	− 1.59724i	0.601834	−	0.798621i	$-0.294437\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	6.00000	0.524222	0.262111	−	0.965038i	$-0.415581\pi$
$131$	6.00000	0.524222	0.262111	+	0.965038i	$0.415581\pi$
$132$	0	0
$133$	− 20.0000i	− 1.73422i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	11.0000i	0.939793i	0.882721	+	0.469897i	$0.155709\pi$
$137$	11.0000i	0.939793i	−0.882721	+	0.469897i	$0.844291\pi$
$138$	0	0
$139$	−12.0000	−1.01783	−0.508913	−	0.860818i	$-0.669953\pi$
$139$	−12.0000	−1.01783	−0.508913	+	0.860818i	$0.669953\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 8.00000i	− 0.668994i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	14.0000	1.14692	0.573462	−	0.819232i	$-0.305600\pi$
$149$	14.0000	1.14692	0.573462	+	0.819232i	$0.305600\pi$
$150$	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
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	22.0000i	1.75579i	0.478852	+	0.877896i	$0.341053\pi$
$157$	22.0000i	1.75579i	−0.478852	+	0.877896i	$0.658947\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−20.0000	−1.57622
$162$	0	0
$163$	2.00000i	0.156652i	0.996928	+	0.0783260i	$0.0249575\pi$
$163$	2.00000i	0.156652i	−0.996928	+	0.0783260i	$0.975042\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 15.0000i	− 1.16073i	−0.814355	−	0.580367i	$-0.802909\pi$
$167$	− 15.0000i	− 1.16073i	0.814355	−	0.580367i	$-0.197091\pi$
$168$	0	0
$169$	−3.00000	−0.230769
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 7.00000i	− 0.532200i	−0.963945	−	0.266100i	$-0.914265\pi$
$173$	− 7.00000i	− 0.532200i	0.963945	−	0.266100i	$-0.0857352\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−16.0000	−1.19590	−0.597948	−	0.801535i	$-0.704017\pi$
$179$	−16.0000	−1.19590	−0.597948	+	0.801535i	$0.704017\pi$
$180$	0	0
$181$	−7.00000	−0.520306	−0.260153	−	0.965567i	$-0.583773\pi$
$181$	−7.00000	−0.520306	−0.260153	+	0.965567i	$0.583773\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	2.00000i	0.146254i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−26.0000	−1.88129	−0.940647	−	0.339387i	$-0.889781\pi$
$191$	−26.0000	−1.88129	−0.940647	+	0.339387i	$0.889781\pi$
$192$	0	0
$193$	− 16.0000i	− 1.15171i	−0.817554	−	0.575853i	$-0.804670\pi$
$193$	− 16.0000i	− 1.15171i	0.817554	−	0.575853i	$-0.195330\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 25.0000i	− 1.78118i	−0.454811	−	0.890588i	$-0.650293\pi$
$197$	− 25.0000i	− 1.78118i	0.454811	−	0.890588i	$-0.349707\pi$
$198$	0	0
$199$	4.00000	0.283552	0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000	0.283552	0.141776	+	0.989899i	$0.454719\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	32.0000i	2.24596i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	10.0000	0.691714
$210$	0	0
$211$	17.0000	1.17033	0.585164	−	0.810915i	$-0.301030\pi$
$211$	17.0000	1.17033	0.585164	+	0.810915i	$0.301030\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 28.0000i	− 1.90076i
$218$	0	0
$219$	0	0
$220$	0	0
$221$	4.00000	0.269069
$222$	0	0
$223$	− 2.00000i	− 0.133930i	−0.997755	−	0.0669650i	$-0.978668\pi$
$223$	− 2.00000i	− 0.133930i	0.997755	−	0.0669650i	$-0.0213316\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 9.00000i	− 0.597351i	−0.954355	−	0.298675i	$-0.903455\pi$
$227$	− 9.00000i	− 0.597351i	0.954355	−	0.298675i	$-0.0965448\pi$
$228$	0	0
$229$	−19.0000	−1.25556	−0.627778	−	0.778393i	$-0.716035\pi$
$229$	−19.0000	−1.25556	−0.627778	+	0.778393i	$0.716035\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	6.00000i	0.393073i	0.980497	+	0.196537i	$0.0629694\pi$
$233$	6.00000i	0.393073i	−0.980497	+	0.196537i	$0.937031\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−6.00000	−0.388108	−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000	−0.388108	−0.194054	+	0.980991i	$0.562164\pi$
$240$	0	0
$241$	−1.00000	−0.0644157	−0.0322078	−	0.999481i	$-0.510254\pi$
$241$	−1.00000	−0.0644157	−0.0322078	+	0.999481i	$0.510254\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 20.0000i	− 1.27257i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	0	0
$253$	− 10.0000i	− 0.628695i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3.00000i	0.187135i	0.995613	+	0.0935674i	$0.0298271\pi$
$257$	3.00000i	0.187135i	−0.995613	+	0.0935674i	$0.970173\pi$
$258$	0	0
$259$	−24.0000	−1.49129
$260$	0	0
$261$	0	0
$262$	0	0
$263$	12.0000i	0.739952i	0.929041	+	0.369976i	$0.120634\pi$
$263$	12.0000i	0.739952i	−0.929041	+	0.369976i	$0.879366\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	26.0000	1.58525	0.792624	−	0.609711i	$-0.208714\pi$
$269$	26.0000	1.58525	0.792624	+	0.609711i	$0.208714\pi$
$270$	0	0
$271$	15.0000	0.911185	0.455593	−	0.890188i	$-0.349427\pi$
$271$	15.0000	0.911185	0.455593	+	0.890188i	$0.349427\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 22.0000i	− 1.32185i	−0.750451	−	0.660926i	$-0.770164\pi$
$277$	− 22.0000i	− 1.32185i	0.750451	−	0.660926i	$-0.229836\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−16.0000	−0.954480	−0.477240	−	0.878773i	$-0.658363\pi$
$281$	−16.0000	−0.954480	−0.477240	+	0.878773i	$0.658363\pi$
$282$	0	0
$283$	− 14.0000i	− 0.832214i	−0.909316	−	0.416107i	$-0.863394\pi$
$283$	− 14.0000i	− 0.832214i	0.909316	−	0.416107i	$-0.136606\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 24.0000i	− 1.41668i
$288$	0	0
$289$	16.0000	0.941176
$290$	0	0
$291$	0	0
$292$	0	0
$293$	1.00000i	0.0584206i	0.999573	+	0.0292103i	$0.00929925\pi$
$293$	1.00000i	0.0584206i	−0.999573	+	0.0292103i	$0.990701\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−20.0000	−1.15663
$300$	0	0
$301$	8.00000	0.461112
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	10.0000i	0.570730i	0.958419	+	0.285365i	$0.0921148\pi$
$307$	10.0000i	0.570730i	−0.958419	+	0.285365i	$0.907885\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	16.0000	0.907277	0.453638	−	0.891186i	$-0.350126\pi$
$311$	16.0000	0.907277	0.453638	+	0.891186i	$0.350126\pi$
$312$	0	0
$313$	28.0000i	1.58265i	0.611393	+	0.791327i	$0.290609\pi$
$313$	28.0000i	1.58265i	−0.611393	+	0.791327i	$0.709391\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	33.0000i	1.85346i	0.375722	+	0.926732i	$0.377395\pi$
$317$	33.0000i	1.85346i	−0.375722	+	0.926732i	$0.622605\pi$
$318$	0	0
$319$	−16.0000	−0.895828
$320$	0	0
$321$	0	0
$322$	0	0
$323$	5.00000i	0.278207i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	32.0000	1.76422
$330$	0	0
$331$	0	0		−	1.00000i	$-0.5\pi$
$331$	0	0			1.00000i	$0.5\pi$
$332$	0	0
$333$	0	0
$334$	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
$339$	0	0
$340$	0	0
$341$	14.0000	0.758143
$342$	0	0
$343$	8.00000i	0.431959i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 20.0000i	− 1.07366i	−0.843692	−	0.536828i	$-0.819622\pi$
$347$	− 20.0000i	− 1.07366i	0.843692	−	0.536828i	$-0.180378\pi$
$348$	0	0
$349$	−9.00000	−0.481759	−0.240879	−	0.970555i	$-0.577436\pi$
$349$	−9.00000	−0.481759	−0.240879	+	0.970555i	$0.577436\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 34.0000i	− 1.80964i	−0.425797	−	0.904819i	$-0.640006\pi$
$353$	− 34.0000i	− 1.80964i	0.425797	−	0.904819i	$-0.359994\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−30.0000	−1.58334	−0.791670	−	0.610949i	$-0.790788\pi$
$359$	−30.0000	−1.58334	−0.791670	+	0.610949i	$0.790788\pi$
$360$	0	0
$361$	6.00000	0.315789
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 4.00000i	− 0.208798i	−0.994535	−	0.104399i	$-0.966708\pi$
$367$	− 4.00000i	− 0.208798i	0.994535	−	0.104399i	$-0.0332919\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−36.0000	−1.86903
$372$	0	0
$373$	20.0000i	1.03556i	0.855514	+	0.517780i	$0.173242\pi$
$373$	20.0000i	1.03556i	−0.855514	+	0.517780i	$0.826758\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	32.0000i	1.64808i
$378$	0	0
$379$	−13.0000	−0.667765	−0.333883	−	0.942615i	$-0.608359\pi$
$379$	−13.0000	−0.667765	−0.333883	+	0.942615i	$0.608359\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 21.0000i	− 1.07305i	−0.843884	−	0.536525i	$-0.819737\pi$
$383$	− 21.0000i	− 1.07305i	0.843884	−	0.536525i	$-0.180263\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	5.00000	0.252861
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 32.0000i	− 1.60603i	−0.595956	−	0.803017i	$-0.703227\pi$
$397$	− 32.0000i	− 1.60603i	0.595956	−	0.803017i	$-0.296773\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	30.0000	1.49813	0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000	1.49813	0.749064	+	0.662497i	$0.230503\pi$
$402$	0	0
$403$	− 28.0000i	− 1.39478i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 12.0000i	− 0.594818i
$408$	0	0
$409$	23.0000	1.13728	0.568638	−	0.822588i	$-0.307470\pi$
$409$	23.0000	1.13728	0.568638	+	0.822588i	$0.307470\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	16.0000i	0.787309i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	6.00000	0.293119	0.146560	−	0.989202i	$-0.453180\pi$
$419$	6.00000	0.293119	0.146560	+	0.989202i	$0.453180\pi$
$420$	0	0
$421$	15.0000	0.731055	0.365528	−	0.930800i	$-0.380889\pi$
$421$	15.0000	0.731055	0.365528	+	0.930800i	$0.380889\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 52.0000i	− 2.51646i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−20.0000	−0.963366	−0.481683	−	0.876346i	$-0.659974\pi$
$431$	−20.0000	−0.963366	−0.481683	+	0.876346i	$0.659974\pi$
$432$	0	0
$433$	− 20.0000i	− 0.961139i	−0.876957	−	0.480569i	$-0.840430\pi$
$433$	− 20.0000i	− 0.961139i	0.876957	−	0.480569i	$-0.159570\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 25.0000i	− 1.19591i
$438$	0	0
$439$	5.00000	0.238637	0.119318	−	0.992856i	$-0.461929\pi$
$439$	5.00000	0.238637	0.119318	+	0.992856i	$0.461929\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 3.00000i	− 0.142534i	−0.997457	−	0.0712672i	$-0.977296\pi$
$443$	− 3.00000i	− 0.142534i	0.997457	−	0.0712672i	$-0.0227043\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	0	0		−	1.00000i	$-0.5\pi$
$449$	0	0			1.00000i	$0.5\pi$
$450$	0	0
$451$	12.0000	0.565058
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 8.00000i	− 0.374224i	−0.982339	−	0.187112i	$-0.940087\pi$
$457$	− 8.00000i	− 0.374224i	0.982339	−	0.187112i	$-0.0599128\pi$
$458$	0	0
$459$	0	0
$460$	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
$463$	− 6.00000i	− 0.278844i	−0.990233	−	0.139422i	$-0.955476\pi$
$463$	− 6.00000i	− 0.278844i	0.990233	−	0.139422i	$-0.0445244\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	31.0000i	1.43451i	0.696811	+	0.717254i	$0.254601\pi$
$467$	31.0000i	1.43451i	−0.696811	+	0.717254i	$0.745399\pi$
$468$	0	0
$469$	−40.0000	−1.84703
$470$	0	0
$471$	0	0
$472$	0	0
$473$	4.00000i	0.183920i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−26.0000	−1.18797	−0.593985	−	0.804476i	$-0.702446\pi$
$479$	−26.0000	−1.18797	−0.593985	+	0.804476i	$0.702446\pi$
$480$	0	0
$481$	−24.0000	−1.09431
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	34.0000i	1.54069i	0.637629	+	0.770344i	$0.279915\pi$
$487$	34.0000i	1.54069i	−0.637629	+	0.770344i	$0.720085\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	30.0000	1.35388	0.676941	−	0.736038i	$-0.263305\pi$
$491$	30.0000	1.35388	0.676941	+	0.736038i	$0.263305\pi$
$492$	0	0
$493$	− 8.00000i	− 0.360302i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	24.0000i	1.07655i
$498$	0	0
$499$	1.00000	0.0447661	0.0223831	−	0.999749i	$-0.492875\pi$
$499$	1.00000	0.0447661	0.0223831	+	0.999749i	$0.492875\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	27.0000i	1.20387i	0.798545	+	0.601935i	$0.205603\pi$
$503$	27.0000i	1.20387i	−0.798545	+	0.601935i	$0.794397\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	28.0000	1.24108	0.620539	−	0.784176i	$-0.286914\pi$
$509$	28.0000	1.24108	0.620539	+	0.784176i	$0.286914\pi$
$510$	0	0
$511$	24.0000	1.06170
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	16.0000i	0.703679i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	14.0000	0.613351	0.306676	−	0.951814i	$-0.400783\pi$
$521$	14.0000	0.613351	0.306676	+	0.951814i	$0.400783\pi$
$522$	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
$527$	7.00000i	0.304925i
$528$	0	0
$529$	−2.00000	−0.0869565
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 24.0000i	− 1.03956i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−18.0000	−0.775315
$540$	0	0
$541$	22.0000	0.945854	0.472927	−	0.881102i	$-0.343197\pi$
$541$	22.0000	0.945854	0.472927	+	0.881102i	$0.343197\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 10.0000i	− 0.427569i	−0.976881	−	0.213785i	$-0.931421\pi$
$547$	− 10.0000i	− 0.427569i	0.976881	−	0.213785i	$-0.0685791\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−40.0000	−1.70406
$552$	0	0
$553$	36.0000i	1.53088i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	22.0000i	0.932170i	0.884740	+	0.466085i	$0.154336\pi$
$557$	22.0000i	0.932170i	−0.884740	+	0.466085i	$0.845664\pi$
$558$	0	0
$559$	8.00000	0.338364
$560$	0	0
$561$	0	0
$562$	0	0
$563$	4.00000i	0.168580i	0.996441	+	0.0842900i	$0.0268622\pi$
$563$	4.00000i	0.168580i	−0.996441	+	0.0842900i	$0.973138\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−28.0000	−1.17382	−0.586911	−	0.809652i	$-0.699656\pi$
$569$	−28.0000	−1.17382	−0.586911	+	0.809652i	$0.699656\pi$
$570$	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
$577$	− 10.0000i	− 0.416305i	−0.978096	−	0.208153i	$-0.933255\pi$
$577$	− 10.0000i	− 0.416305i	0.978096	−	0.208153i	$-0.0667451\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	68.0000	2.82112
$582$	0	0
$583$	− 18.0000i	− 0.745484i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	33.0000i	1.36206i	0.732257	+	0.681028i	$0.238467\pi$
$587$	33.0000i	1.36206i	−0.732257	+	0.681028i	$0.761533\pi$
$588$	0	0
$589$	35.0000	1.44215
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 15.0000i	− 0.615976i	−0.951390	−	0.307988i	$-0.900344\pi$
$593$	− 15.0000i	− 0.615976i	0.951390	−	0.307988i	$-0.0996557\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−36.0000	−1.47092	−0.735460	−	0.677568i	$-0.763034\pi$
$599$	−36.0000	−1.47092	−0.735460	+	0.677568i	$0.763034\pi$
$600$	0	0
$601$	−47.0000	−1.91717	−0.958585	−	0.284807i	$-0.908071\pi$
$601$	−47.0000	−1.91717	−0.958585	+	0.284807i	$0.908071\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	30.0000i	1.21766i	0.793300	+	0.608831i	$0.208361\pi$
$607$	30.0000i	1.21766i	−0.793300	+	0.608831i	$0.791639\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	32.0000	1.29458
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 27.0000i	− 1.08698i	−0.839416	−	0.543490i	$-0.817103\pi$
$617$	− 27.0000i	− 1.08698i	0.839416	−	0.543490i	$-0.182897\pi$
$618$	0	0
$619$	28.0000	1.12542	0.562708	−	0.826656i	$-0.309760\pi$
$619$	28.0000	1.12542	0.562708	+	0.826656i	$0.309760\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 24.0000i	− 0.961540i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	6.00000	0.239236
$630$	0	0
$631$	1.00000	0.0398094	0.0199047	−	0.999802i	$-0.493664\pi$
$631$	1.00000	0.0398094	0.0199047	+	0.999802i	$0.493664\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	36.0000i	1.42637i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−48.0000	−1.89589	−0.947943	−	0.318440i	$-0.896841\pi$
$641$	−48.0000	−1.89589	−0.947943	+	0.318440i	$0.896841\pi$
$642$	0	0
$643$	− 12.0000i	− 0.473234i	−0.971603	−	0.236617i	$-0.923961\pi$
$643$	− 12.0000i	− 0.473234i	0.971603	−	0.236617i	$-0.0760386\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	33.0000i	1.29736i	0.761060	+	0.648682i	$0.224679\pi$
$647$	33.0000i	1.29736i	−0.761060	+	0.648682i	$0.775321\pi$
$648$	0	0
$649$	−8.00000	−0.314027
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 25.0000i	− 0.978326i	−0.872192	−	0.489163i	$-0.837302\pi$
$653$	− 25.0000i	− 0.978326i	0.872192	−	0.489163i	$-0.162698\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	−18.0000	−0.700119	−0.350059	−	0.936727i	$-0.613839\pi$
$661$	−18.0000	−0.700119	−0.350059	+	0.936727i	$0.613839\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	40.0000i	1.54881i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	26.0000	1.00372
$672$	0	0
$673$	36.0000i	1.38770i	0.720121	+	0.693849i	$0.244086\pi$
$673$	36.0000i	1.38770i	−0.720121	+	0.693849i	$0.755914\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 34.0000i	− 1.30673i	−0.757045	−	0.653363i	$-0.773358\pi$
$677$	− 34.0000i	− 1.30673i	0.757045	−	0.653363i	$-0.226642\pi$
$678$	0	0
$679$	−32.0000	−1.22805
$680$	0	0
$681$	0	0
$682$	0	0
$683$	9.00000i	0.344375i	0.985064	+	0.172188i	$0.0550836\pi$
$683$	9.00000i	0.344375i	−0.985064	+	0.172188i	$0.944916\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−36.0000	−1.37149
$690$	0	0
$691$	17.0000	0.646710	0.323355	−	0.946278i	$-0.395189\pi$
$691$	17.0000	0.646710	0.323355	+	0.946278i	$0.395189\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	6.00000i	0.227266i
$698$	0	0
$699$	0	0
$700$	0	0
$701$	10.0000	0.377695	0.188847	−	0.982006i	$-0.439525\pi$
$701$	10.0000	0.377695	0.188847	+	0.982006i	$0.439525\pi$
$702$	0	0
$703$	− 30.0000i	− 1.13147i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 48.0000i	− 1.80523i
$708$	0	0
$709$	46.0000	1.72757	0.863783	−	0.503864i	$-0.168089\pi$
$709$	46.0000	1.72757	0.863783	+	0.503864i	$0.168089\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 35.0000i	− 1.31076i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	34.0000	1.26799	0.633993	−	0.773339i	$-0.281415\pi$
$719$	34.0000	1.26799	0.633993	+	0.773339i	$0.281415\pi$
$720$	0	0
$721$	−16.0000	−0.595871
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 4.00000i	− 0.148352i	−0.997245	−	0.0741759i	$-0.976367\pi$
$727$	− 4.00000i	− 0.148352i	0.997245	−	0.0741759i	$-0.0236326\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−2.00000	−0.0739727
$732$	0	0
$733$	− 24.0000i	− 0.886460i	−0.896408	−	0.443230i	$-0.853832\pi$
$733$	− 24.0000i	− 0.886460i	0.896408	−	0.443230i	$-0.146168\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 20.0000i	− 0.736709i
$738$	0	0
$739$	−9.00000	−0.331070	−0.165535	−	0.986204i	$-0.552935\pi$
$739$	−9.00000	−0.331070	−0.165535	+	0.986204i	$0.552935\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	48.0000i	1.76095i	0.474093	+	0.880475i	$0.342776\pi$
$743$	48.0000i	1.76095i	−0.474093	+	0.880475i	$0.657224\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	48.0000	1.75388
$750$	0	0
$751$	−39.0000	−1.42313	−0.711565	−	0.702620i	$-0.752013\pi$
$751$	−39.0000	−1.42313	−0.711565	+	0.702620i	$0.752013\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	38.0000i	1.38113i	0.723269	+	0.690567i	$0.242639\pi$
$757$	38.0000i	1.38113i	−0.723269	+	0.690567i	$0.757361\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−24.0000	−0.869999	−0.435000	−	0.900431i	$-0.643252\pi$
$761$	−24.0000	−0.869999	−0.435000	+	0.900431i	$0.643252\pi$
$762$	0	0
$763$	68.0000i	2.46177i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	16.0000i	0.577727i
$768$	0	0
$769$	35.0000	1.26213	0.631066	−	0.775729i	$-0.282618\pi$
$769$	35.0000	1.26213	0.631066	+	0.775729i	$0.282618\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 33.0000i	− 1.18693i	−0.804861	−	0.593464i	$-0.797760\pi$
$773$	− 33.0000i	− 1.18693i	0.804861	−	0.593464i	$-0.202240\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	30.0000	1.07486
$780$	0	0
$781$	−12.0000	−0.429394
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	38.0000i	1.35455i	0.735728	+	0.677277i	$0.236840\pi$
$787$	38.0000i	1.35455i	−0.735728	+	0.677277i	$0.763160\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−24.0000	−0.853342
$792$	0	0
$793$	− 52.0000i	− 1.84657i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 21.0000i	− 0.743858i	−0.928261	−	0.371929i	$-0.878696\pi$
$797$	− 21.0000i	− 0.743858i	0.928261	−	0.371929i	$-0.121304\pi$
$798$	0	0
$799$	−8.00000	−0.283020
$800$	0	0
$801$	0	0
$802$	0	0
$803$	12.0000i	0.423471i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	18.0000	0.632846	0.316423	−	0.948618i	$-0.397518\pi$
$809$	18.0000	0.632846	0.316423	+	0.948618i	$0.397518\pi$
$810$	0	0
$811$	−28.0000	−0.983213	−0.491606	−	0.870817i	$-0.663590\pi$
$811$	−28.0000	−0.983213	−0.491606	+	0.870817i	$0.663590\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	10.0000i	0.349856i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	2.00000	0.0698005	0.0349002	−	0.999391i	$-0.488889\pi$
$821$	2.00000	0.0698005	0.0349002	+	0.999391i	$0.488889\pi$
$822$	0	0
$823$	50.0000i	1.74289i	0.490493	+	0.871445i	$0.336817\pi$
$823$	50.0000i	1.74289i	−0.490493	+	0.871445i	$0.663183\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 3.00000i	− 0.104320i	−0.998639	−	0.0521601i	$-0.983389\pi$
$827$	− 3.00000i	− 0.104320i	0.998639	−	0.0521601i	$-0.0166106\pi$
$828$	0	0
$829$	−34.0000	−1.18087	−0.590434	−	0.807086i	$-0.701044\pi$
$829$	−34.0000	−1.18087	−0.590434	+	0.807086i	$0.701044\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 9.00000i	− 0.311832i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−28.0000	−0.966667	−0.483334	−	0.875436i	$-0.660574\pi$
$839$	−28.0000	−0.966667	−0.483334	+	0.875436i	$0.660574\pi$
$840$	0	0
$841$	35.0000	1.20690
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	28.0000i	0.962091i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−30.0000	−1.02839
$852$	0	0
$853$	− 32.0000i	− 1.09566i	−0.836590	−	0.547830i	$-0.815454\pi$
$853$	− 32.0000i	− 1.09566i	0.836590	−	0.547830i	$-0.184546\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	45.0000i	1.53717i	0.639747	+	0.768585i	$0.279039\pi$
$857$	45.0000i	1.53717i	−0.639747	+	0.768585i	$0.720961\pi$
$858$	0	0
$859$	31.0000	1.05771	0.528853	−	0.848713i	$-0.322622\pi$
$859$	31.0000	1.05771	0.528853	+	0.848713i	$0.322622\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	39.0000i	1.32758i	0.747921	+	0.663788i	$0.231052\pi$
$863$	39.0000i	1.32758i	−0.747921	+	0.663788i	$0.768948\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−18.0000	−0.610608
$870$	0	0
$871$	−40.0000	−1.35535
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 14.0000i	− 0.472746i	−0.971662	−	0.236373i	$-0.924041\pi$
$877$	− 14.0000i	− 0.472746i	0.971662	−	0.236373i	$-0.0759588\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−24.0000	−0.808581	−0.404290	−	0.914631i	$-0.632481\pi$
$881$	−24.0000	−0.808581	−0.404290	+	0.914631i	$0.632481\pi$
$882$	0	0
$883$	4.00000i	0.134611i	0.997732	+	0.0673054i	$0.0214402\pi$
$883$	4.00000i	0.134611i	−0.997732	+	0.0673054i	$0.978560\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	17.0000i	0.570804i	0.958408	+	0.285402i	$0.0921271\pi$
$887$	17.0000i	0.570804i	−0.958408	+	0.285402i	$0.907873\pi$
$888$	0	0
$889$	−72.0000	−2.41480
$890$	0	0
$891$	0	0
$892$	0	0
$893$	40.0000i	1.33855i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−56.0000	−1.86770
$900$	0	0
$901$	9.00000	0.299833
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 12.0000i	− 0.398453i	−0.979953	−	0.199227i	$-0.936157\pi$
$907$	− 12.0000i	− 0.398453i	0.979953	−	0.199227i	$-0.0638430\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	32.0000	1.06021	0.530104	−	0.847933i	$-0.322153\pi$
$911$	32.0000	1.06021	0.530104	+	0.847933i	$0.322153\pi$
$912$	0	0
$913$	34.0000i	1.12524i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 24.0000i	− 0.792550i
$918$	0	0
$919$	40.0000	1.31948	0.659739	−	0.751495i	$-0.270667\pi$
$919$	40.0000	1.31948	0.659739	+	0.751495i	$0.270667\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	24.0000i	0.789970i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−38.0000	−1.24674	−0.623370	−	0.781927i	$-0.714237\pi$
$929$	−38.0000	−1.24674	−0.623370	+	0.781927i	$0.714237\pi$
$930$	0	0
$931$	−45.0000	−1.47482
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	28.0000i	0.914720i	0.889282	+	0.457360i	$0.151205\pi$
$937$	28.0000i	0.914720i	−0.889282	+	0.457360i	$0.848795\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	38.0000	1.23876	0.619382	−	0.785090i	$-0.287383\pi$
$941$	38.0000	1.23876	0.619382	+	0.785090i	$0.287383\pi$
$942$	0	0
$943$	− 30.0000i	− 0.976934i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	11.0000i	0.357452i	0.983899	+	0.178726i	$0.0571976\pi$
$947$	11.0000i	0.357452i	−0.983899	+	0.178726i	$0.942802\pi$
$948$	0	0
$949$	24.0000	0.779073
$950$	0	0
$951$	0	0
$952$	0	0
$953$	6.00000i	0.194359i	0.995267	+	0.0971795i	$0.0309821\pi$
$953$	6.00000i	0.194359i	−0.995267	+	0.0971795i	$0.969018\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	44.0000	1.42083
$960$	0	0
$961$	18.0000	0.580645
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 42.0000i	− 1.35063i	−0.737530	−	0.675314i	$-0.764008\pi$
$967$	− 42.0000i	− 1.35063i	0.737530	−	0.675314i	$-0.235992\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−26.0000	−0.834380	−0.417190	−	0.908819i	$-0.636985\pi$
$971$	−26.0000	−0.834380	−0.417190	+	0.908819i	$0.636985\pi$
$972$	0	0
$973$	48.0000i	1.53881i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 38.0000i	− 1.21573i	−0.794041	−	0.607864i	$-0.792027\pi$
$977$	− 38.0000i	− 1.21573i	0.794041	−	0.607864i	$-0.207973\pi$
$978$	0	0
$979$	12.0000	0.383522
$980$	0	0
$981$	0	0
$982$	0	0
$983$	27.0000i	0.861166i	0.902551	+	0.430583i	$0.141692\pi$
$983$	27.0000i	0.861166i	−0.902551	+	0.430583i	$0.858308\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	10.0000	0.317982
$990$	0	0
$991$	5.00000	0.158830	0.0794151	−	0.996842i	$-0.474695\pi$
$991$	5.00000	0.158830	0.0794151	+	0.996842i	$0.474695\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	42.0000i	1.33015i	0.746775	+	0.665077i	$0.231601\pi$
$997$	42.0000i	1.33015i	−0.746775	+	0.665077i	$0.768399\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5400.2.f.s.649.1		2
3.2	odd	2		5400.2.f.l.649.1		2
5.2	odd	4		5400.2.a.bu.1.1		1
5.3	odd	4		1080.2.a.a.1.1	✓	1
5.4	even	2	inner	5400.2.f.s.649.2		2
15.2	even	4		5400.2.a.br.1.1		1
15.8	even	4		1080.2.a.g.1.1	yes	1
15.14	odd	2		5400.2.f.l.649.2		2
20.3	even	4		2160.2.a.l.1.1		1
40.3	even	4		8640.2.a.cg.1.1		1
40.13	odd	4		8640.2.a.bf.1.1		1
45.13	odd	12		3240.2.q.w.2161.1		2
45.23	even	12		3240.2.q.l.2161.1		2
45.38	even	12		3240.2.q.l.1081.1		2
45.43	odd	12		3240.2.q.w.1081.1		2
60.23	odd	4		2160.2.a.w.1.1		1
120.53	even	4		8640.2.a.a.1.1		1
120.83	odd	4		8640.2.a.bd.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1080.2.a.a.1.1	✓	1	5.3	odd	4
1080.2.a.g.1.1	yes	1	15.8	even	4
2160.2.a.l.1.1		1	20.3	even	4
2160.2.a.w.1.1		1	60.23	odd	4
3240.2.q.l.1081.1		2	45.38	even	12
3240.2.q.l.2161.1		2	45.23	even	12
3240.2.q.w.1081.1		2	45.43	odd	12
3240.2.q.w.2161.1		2	45.13	odd	12
5400.2.a.br.1.1		1	15.2	even	4
5400.2.a.bu.1.1		1	5.2	odd	4
5400.2.f.l.649.1		2	3.2	odd	2
5400.2.f.l.649.2		2	15.14	odd	2
5400.2.f.s.649.1		2	1.1	even	1	trivial
5400.2.f.s.649.2		2	5.4	even	2	inner
8640.2.a.a.1.1		1	120.53	even	4
8640.2.a.bd.1.1		1	120.83	odd	4
8640.2.a.bf.1.1		1	40.13	odd	4
8640.2.a.cg.1.1		1	40.3	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:	\( N \)	\(=\)	\( 5400 = 2^{3} \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5400.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-4.00000i q^{7} +O(q^{10})\)	\(q-4.00000i q^{7} +2.00000 q^{11} -4.00000i q^{13} +1.00000i q^{17} +5.00000 q^{19} -5.00000i q^{23} -8.00000 q^{29} +7.00000 q^{31} -6.00000i q^{37} +6.00000 q^{41} +2.00000i q^{43} +8.00000i q^{47} -9.00000 q^{49} -9.00000i q^{53} -4.00000 q^{59} +13.0000 q^{61} -10.0000i q^{67} -6.00000 q^{71} +6.00000i q^{73} -8.00000i q^{77} -9.00000 q^{79} +17.0000i q^{83} +6.00000 q^{89} -16.0000 q^{91} -8.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q+O(q^{10}) \) 2 * q	\( 2 q + 4 q^{11} + 10 q^{19} - 16 q^{29} + 14 q^{31} + 12 q^{41} - 18 q^{49} - 8 q^{59} + 26 q^{61} - 12 q^{71} - 18 q^{79} + 12 q^{89} - 32 q^{91}+O(q^{100}) \) 2 * q + 4 * q^11 + 10 * q^19 - 16 * q^29 + 14 * q^31 + 12 * q^41 - 18 * q^49 - 8 * q^59 + 26 * q^61 - 12 * q^71 - 18 * q^79 + 12 * q^89 - 32 * q^91

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