LMFDB - Embedded newform 8100.2.d.o.649.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8100,2,Mod(649,8100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8100, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8100.649");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8100 = 2^{2} \cdot 3^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8100.d (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:	$64.6788256372$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.5089536.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 16x^{2} - 24x + 18 $ x^6 - 2x^5 + 2x^4 + 2x^3 + 16x^2 - 24*x + 18
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 180)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.5
Root			$1.66044 + 1.66044i$ of defining polynomial
Character	$\chi$	$=$	8100.649
Dual form			8100.2.d.o.649.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.83502i q^{7} +1.70739 q^{11} +3.70739i q^{13} +1.70739i q^{17} -0.292611 q^{19} +5.83502i q^{23} +8.67004 q^{29} +0.292611 q^{31} -11.9627i q^{37} +6.96265 q^{41} +3.70739i q^{43} +1.87237i q^{47} -7.70739 q^{49} +11.6700i q^{53} +11.6700 q^{59} +14.9627 q^{61} -9.54241i q^{67} -15.9627 q^{71} +8.00000i q^{73} +6.54787i q^{77} -2.00000 q^{79} +11.8350i q^{83} -3.00000 q^{89} -14.2179 q^{91} +3.67004i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 12 q^{19} - 6 q^{29} + 12 q^{31} - 6 q^{41} - 36 q^{49} + 12 q^{59} + 42 q^{61} - 48 q^{71} - 12 q^{79} - 18 q^{89}+O(q^{100}) $ 6 * q - 12 * q^19 - 6 * q^29 + 12 * q^31 - 6 * q^41 - 36 * q^49 + 12 * q^59 + 42 * q^61 - 48 * q^71 - 12 * q^79 - 18 * q^89

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/8100\mathbb{Z}\right)^\times$.

$n$	$4051$	$6401$	$7777$
$\chi(n)$	$1$	$1$	$-1$

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
$7$	3.83502i	1.44950i	0.689011	+	0.724751i	$0.258045\pi$
$7$	3.83502i	1.44950i	−0.689011	+	0.724751i	$0.741955\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.70739	0.514797	0.257399	−	0.966305i	$-0.417135\pi$
$11$	1.70739	0.514797	0.257399	+	0.966305i	$0.417135\pi$
$12$	0	0
$13$	3.70739i	1.02824i	0.857717	+	0.514122i	$0.171882\pi$
$13$	3.70739i	1.02824i	−0.857717	+	0.514122i	$0.828118\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.70739i	0.414103i	0.978330	+	0.207051i	$0.0663867\pi$
$17$	1.70739i	0.414103i	−0.978330	+	0.207051i	$0.933613\pi$
$18$	0	0
$19$	−0.292611	−0.0671295	−0.0335647	−	0.999437i	$-0.510686\pi$
$19$	−0.292611	−0.0671295	−0.0335647	+	0.999437i	$0.510686\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	5.83502i	1.21669i	0.793674	+	0.608343i	$0.208165\pi$
$23$	5.83502i	1.21669i	−0.793674	+	0.608343i	$0.791835\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	8.67004	1.60999	0.804993	−	0.593284i	$-0.202169\pi$
$29$	8.67004	1.60999	0.804993	+	0.593284i	$0.202169\pi$
$30$	0	0
$31$	0.292611	0.0525544	0.0262772	−	0.999655i	$-0.491635\pi$
$31$	0.292611	0.0525544	0.0262772	+	0.999655i	$0.491635\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 11.9627i	− 1.96665i	−0.181862	−	0.983324i	$-0.558212\pi$
$37$	− 11.9627i	− 1.96665i	0.181862	−	0.983324i	$-0.441788\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	6.96265	1.08738	0.543692	−	0.839285i	$-0.317026\pi$
$41$	6.96265	1.08738	0.543692	+	0.839285i	$0.317026\pi$
$42$	0	0
$43$	3.70739i	0.565372i	0.959213	+	0.282686i	$0.0912254\pi$
$43$	3.70739i	0.565372i	−0.959213	+	0.282686i	$0.908775\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.87237i	0.273113i	0.990632	+	0.136556i	$0.0436035\pi$
$47$	1.87237i	0.273113i	−0.990632	+	0.136556i	$0.956396\pi$
$48$	0	0
$49$	−7.70739	−1.10106
$50$	0	0
$51$	0	0
$52$	0	0
$53$	11.6700i	1.60300i	0.597992	+	0.801502i	$0.295965\pi$
$53$	11.6700i	1.60300i	−0.597992	+	0.801502i	$0.704035\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	11.6700	1.51931	0.759655	−	0.650326i	$-0.225368\pi$
$59$	11.6700	1.51931	0.759655	+	0.650326i	$0.225368\pi$
$60$	0	0
$61$	14.9627	1.91577	0.957886	−	0.287150i	$-0.0927077\pi$
$61$	14.9627	1.91577	0.957886	+	0.287150i	$0.0927077\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 9.54241i	− 1.16579i	−0.812547	−	0.582896i	$-0.801920\pi$
$67$	− 9.54241i	− 1.16579i	0.812547	−	0.582896i	$-0.198080\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−15.9627	−1.89442	−0.947209	−	0.320616i	$-0.896110\pi$
$71$	−15.9627	−1.89442	−0.947209	+	0.320616i	$0.896110\pi$
$72$	0	0
$73$	8.00000i	0.936329i	0.883641	+	0.468165i	$0.155085\pi$
$73$	8.00000i	0.936329i	−0.883641	+	0.468165i	$0.844915\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	6.54787i	0.746200i
$78$	0	0
$79$	−2.00000	−0.225018	−0.112509	−	0.993651i	$-0.535889\pi$
$79$	−2.00000	−0.225018	−0.112509	+	0.993651i	$0.535889\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	11.8350i	1.29906i	0.760335	+	0.649531i	$0.225035\pi$
$83$	11.8350i	1.29906i	−0.760335	+	0.649531i	$0.774965\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−3.00000	−0.317999	−0.159000	−	0.987279i	$-0.550827\pi$
$89$	−3.00000	−0.317999	−0.159000	+	0.987279i	$0.550827\pi$
$90$	0	0
$91$	−14.2179	−1.49044
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	3.67004i	0.372636i	0.982489	+	0.186318i	$0.0596555\pi$
$97$	3.67004i	0.372636i	−0.982489	+	0.186318i	$0.940344\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−0.329957	−0.0328320	−0.0164160	−	0.999865i	$-0.505226\pi$
$101$	−0.329957	−0.0328320	−0.0164160	+	0.999865i	$0.505226\pi$
$102$	0	0
$103$	− 3.12217i	− 0.307636i	−0.988099	−	0.153818i	$-0.950843\pi$
$103$	− 3.12217i	− 0.307636i	0.988099	−	0.153818i	$-0.0491570\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 14.9198i	− 1.44236i	−0.692750	−	0.721178i	$-0.743601\pi$
$107$	− 14.9198i	− 1.44236i	0.692750	−	0.721178i	$-0.256399\pi$
$108$	0	0
$109$	−6.70739	−0.642451	−0.321226	−	0.947003i	$-0.604095\pi$
$109$	−6.70739	−0.642451	−0.321226	+	0.947003i	$0.604095\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 5.67004i	− 0.533393i	−0.963781	−	0.266696i	$-0.914068\pi$
$113$	− 5.67004i	− 0.533393i	0.963781	−	0.266696i	$-0.0859321\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−6.54787	−0.600243
$120$	0	0
$121$	−8.08482	−0.734984
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	5.54241i	0.491809i	0.969294	+	0.245905i	$0.0790850\pi$
$127$	5.54241i	0.491809i	−0.969294	+	0.245905i	$0.920915\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−12.8296	−1.12092	−0.560462	−	0.828180i	$-0.689376\pi$
$131$	−12.8296	−1.12092	−0.560462	+	0.828180i	$0.689376\pi$
$132$	0	0
$133$	− 1.12217i	− 0.0973043i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	9.08482i	0.776169i	0.921624	+	0.388084i	$0.126863\pi$
$137$	9.08482i	0.776169i	−0.921624	+	0.388084i	$0.873137\pi$
$138$	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
$141$	0	0
$142$	0	0
$143$	6.32996i	0.529338i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−8.12217	−0.665394	−0.332697	−	0.943034i	$-0.607959\pi$
$149$	−8.12217	−0.665394	−0.332697	+	0.943034i	$0.607959\pi$
$150$	0	0
$151$	21.0475	1.71282	0.856410	−	0.516297i	$-0.172690\pi$
$151$	21.0475	1.71282	0.856410	+	0.516297i	$0.172690\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0.255264i	0.0203723i	0.999948	+	0.0101861i	$0.00324241\pi$
$157$	0.255264i	0.0203723i	−0.999948	+	0.0101861i	$0.996758\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−22.3774	−1.76359
$162$	0	0
$163$	− 11.7074i	− 0.916994i	−0.888696	−	0.458497i	$-0.848388\pi$
$163$	− 11.7074i	− 0.916994i	0.888696	−	0.458497i	$-0.151612\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	16.1276i	1.24799i	0.781427	+	0.623997i	$0.214492\pi$
$167$	16.1276i	1.24799i	−0.781427	+	0.623997i	$0.785508\pi$
$168$	0	0
$169$	−0.744736	−0.0572874
$170$	0	0
$171$	0	0
$172$	0	0
$173$	8.58522i	0.652722i	0.945245	+	0.326361i	$0.105823\pi$
$173$	8.58522i	0.652722i	−0.945245	+	0.326361i	$0.894177\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−7.37743	−0.551415	−0.275708	−	0.961242i	$-0.588912\pi$
$179$	−7.37743	−0.551415	−0.275708	+	0.961242i	$0.588912\pi$
$180$	0	0
$181$	17.4996	1.30074	0.650368	−	0.759620i	$-0.274615\pi$
$181$	17.4996	1.30074	0.650368	+	0.759620i	$0.274615\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	2.91518i	0.213179i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−6.54787	−0.473788	−0.236894	−	0.971536i	$-0.576129\pi$
$191$	−6.54787	−0.473788	−0.236894	+	0.971536i	$0.576129\pi$
$192$	0	0
$193$	7.12217i	0.512665i	0.966589	+	0.256332i	$0.0825142\pi$
$193$	7.12217i	0.512665i	−0.966589	+	0.256332i	$0.917486\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 17.6700i	− 1.25894i	−0.777025	−	0.629469i	$-0.783272\pi$
$197$	− 17.6700i	− 1.25894i	0.777025	−	0.629469i	$-0.216728\pi$
$198$	0	0
$199$	−11.6327	−0.824620	−0.412310	−	0.911044i	$-0.635278\pi$
$199$	−11.6327	−0.824620	−0.412310	+	0.911044i	$0.635278\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	33.2498i	2.33368i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−0.499600	−0.0345581
$210$	0	0
$211$	5.63270	0.387771	0.193885	−	0.981024i	$-0.437891\pi$
$211$	5.63270	0.387771	0.193885	+	0.981024i	$0.437891\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	1.12217i	0.0761777i
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−6.32996	−0.425799
$222$	0	0
$223$	− 8.12763i	− 0.544266i	−0.962260	−	0.272133i	$-0.912271\pi$
$223$	− 8.12763i	− 0.544266i	0.962260	−	0.272133i	$-0.0877292\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	9.96265i	0.661245i	0.943763	+	0.330622i	$0.107259\pi$
$227$	9.96265i	0.661245i	−0.943763	+	0.330622i	$0.892741\pi$
$228$	0	0
$229$	−1.25526	−0.0829502	−0.0414751	−	0.999140i	$-0.513206\pi$
$229$	−1.25526	−0.0829502	−0.0414751	+	0.999140i	$0.513206\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 3.96265i	− 0.259602i	−0.991540	−	0.129801i	$-0.958566\pi$
$233$	− 3.96265i	− 0.259602i	0.991540	−	0.129801i	$-0.0414339\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−5.67004	−0.366765	−0.183382	−	0.983042i	$-0.558705\pi$
$239$	−5.67004	−0.366765	−0.183382	+	0.983042i	$0.558705\pi$
$240$	0	0
$241$	−16.6327	−1.07141	−0.535703	−	0.844406i	$-0.679953\pi$
$241$	−16.6327	−1.07141	−0.535703	+	0.844406i	$0.679953\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 1.08482i	− 0.0690255i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−13.3774	−0.844376	−0.422188	−	0.906508i	$-0.638738\pi$
$251$	−13.3774	−0.844376	−0.422188	+	0.906508i	$0.638738\pi$
$252$	0	0
$253$	9.96265i	0.626347i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0.829557i	0.0517464i	0.999665	+	0.0258732i	$0.00823661\pi$
$257$	0.829557i	0.0517464i	−0.999665	+	0.0258732i	$0.991763\pi$
$258$	0	0
$259$	45.8770	2.85066
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 12.2179i	− 0.753389i	−0.926338	−	0.376695i	$-0.877061\pi$
$263$	− 12.2179i	− 0.753389i	0.926338	−	0.376695i	$-0.122939\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	17.5369	1.06925	0.534623	−	0.845091i	$-0.320454\pi$
$269$	17.5369	1.06925	0.534623	+	0.845091i	$0.320454\pi$
$270$	0	0
$271$	−15.1222	−0.918606	−0.459303	−	0.888280i	$-0.651901\pi$
$271$	−15.1222	−0.918606	−0.459303	+	0.888280i	$0.651901\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	5.70739i	0.342924i	0.985191	+	0.171462i	$0.0548490\pi$
$277$	5.70739i	0.342924i	−0.985191	+	0.171462i	$0.945151\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−7.29261	−0.435041	−0.217520	−	0.976056i	$-0.569797\pi$
$281$	−7.29261	−0.435041	−0.217520	+	0.976056i	$0.569797\pi$
$282$	0	0
$283$	− 10.1650i	− 0.604245i	−0.953269	−	0.302123i	$-0.902305\pi$
$283$	− 10.1650i	− 0.604245i	0.953269	−	0.302123i	$-0.0976952\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	26.7019i	1.57616i
$288$	0	0
$289$	14.0848	0.828519
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 13.0475i	− 0.762242i	−0.924525	−	0.381121i	$-0.875538\pi$
$293$	− 13.0475i	− 0.762242i	0.924525	−	0.381121i	$-0.124462\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−21.6327	−1.25105
$300$	0	0
$301$	−14.2179	−0.819507
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 1.50506i	− 0.0858986i	−0.999077	−	0.0429493i	$-0.986325\pi$
$307$	− 1.50506i	− 0.0858986i	0.999077	−	0.0429493i	$-0.0136754\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	5.45213	0.309162	0.154581	−	0.987980i	$-0.450597\pi$
$311$	5.45213	0.309162	0.154581	+	0.987980i	$0.450597\pi$
$312$	0	0
$313$	− 10.5479i	− 0.596201i	−0.954534	−	0.298101i	$-0.903647\pi$
$313$	− 10.5479i	− 0.596201i	0.954534	−	0.298101i	$-0.0963531\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	25.3774i	1.42534i	0.701500	+	0.712669i	$0.252514\pi$
$317$	25.3774i	1.42534i	−0.701500	+	0.712669i	$0.747486\pi$
$318$	0	0
$319$	14.8031	0.828817
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 0.499600i	− 0.0277985i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−7.18057	−0.395878
$330$	0	0
$331$	−31.0848	−1.70858	−0.854288	−	0.519800i	$-0.826007\pi$
$331$	−31.0848	−1.70858	−0.854288	+	0.519800i	$0.826007\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 10.2553i	− 0.558640i	−0.960198	−	0.279320i	$-0.909891\pi$
$337$	− 10.2553i	− 0.558640i	0.960198	−	0.279320i	$-0.0901090\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0.499600	0.0270549
$342$	0	0
$343$	− 2.71285i	− 0.146480i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 7.04748i	− 0.378328i	−0.981945	−	0.189164i	$-0.939422\pi$
$347$	− 7.04748i	− 0.378328i	0.981945	−	0.189164i	$-0.0605778\pi$
$348$	0	0
$349$	−29.1696	−1.56142	−0.780708	−	0.624897i	$-0.785141\pi$
$349$	−29.1696	−1.56142	−0.780708	+	0.624897i	$0.785141\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	24.0000i	1.27739i	0.769460	+	0.638696i	$0.220526\pi$
$353$	24.0000i	1.27739i	−0.769460	+	0.638696i	$0.779474\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−28.7922	−1.51959	−0.759797	−	0.650160i	$-0.774702\pi$
$359$	−28.7922	−1.51959	−0.759797	+	0.650160i	$0.774702\pi$
$360$	0	0
$361$	−18.9144	−0.995494
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	11.0475i	0.576674i	0.957529	+	0.288337i	$0.0931023\pi$
$367$	11.0475i	0.576674i	−0.957529	+	0.288337i	$0.906898\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−44.7549	−2.32356
$372$	0	0
$373$	− 15.6700i	− 0.811364i	−0.914014	−	0.405682i	$-0.867034\pi$
$373$	− 15.6700i	− 0.811364i	0.914014	−	0.405682i	$-0.132966\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	32.1432i	1.65546i
$378$	0	0
$379$	6.58522	0.338260	0.169130	−	0.985594i	$-0.445904\pi$
$379$	6.58522	0.338260	0.169130	+	0.985594i	$0.445904\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	12.2179i	0.624306i	0.950032	+	0.312153i	$0.101050\pi$
$383$	12.2179i	0.624306i	−0.950032	+	0.312153i	$0.898950\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	27.7175	1.40533	0.702667	−	0.711519i	$-0.251992\pi$
$389$	27.7175	1.40533	0.702667	+	0.711519i	$0.251992\pi$
$390$	0	0
$391$	−9.96265	−0.503833
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	26.5105i	1.33053i	0.746609	+	0.665263i	$0.231680\pi$
$397$	26.5105i	1.33053i	−0.746609	+	0.665263i	$0.768320\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−16.8405	−0.840974	−0.420487	−	0.907299i	$-0.638141\pi$
$401$	−16.8405	−0.840974	−0.420487	+	0.907299i	$0.638141\pi$
$402$	0	0
$403$	1.08482i	0.0540388i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 20.4249i	− 1.01243i
$408$	0	0
$409$	−15.0957	−0.746437	−0.373218	−	0.927744i	$-0.621746\pi$
$409$	−15.0957	−0.746437	−0.373218	+	0.927744i	$0.621746\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	44.7549i	2.20224i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−2.36730	−0.115650	−0.0578252	−	0.998327i	$-0.518417\pi$
$419$	−2.36730	−0.115650	−0.0578252	+	0.998327i	$0.518417\pi$
$420$	0	0
$421$	9.92531	0.483730	0.241865	−	0.970310i	$-0.422241\pi$
$421$	9.92531	0.483730	0.241865	+	0.970310i	$0.422241\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	57.3821i	2.77691i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	13.9253	0.670758	0.335379	−	0.942083i	$-0.391136\pi$
$431$	13.9253	0.670758	0.335379	+	0.942083i	$0.391136\pi$
$432$	0	0
$433$	− 2.29261i	− 0.110176i	−0.998482	−	0.0550879i	$-0.982456\pi$
$433$	− 2.29261i	− 0.110176i	0.998482	−	0.0550879i	$-0.0175439\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 1.70739i	− 0.0816755i
$438$	0	0
$439$	−25.0101	−1.19367	−0.596834	−	0.802365i	$-0.703575\pi$
$439$	−25.0101	−1.19367	−0.596834	+	0.802365i	$0.703575\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	19.8724i	0.944165i	0.881554	+	0.472082i	$0.156498\pi$
$443$	19.8724i	0.944165i	−0.881554	+	0.472082i	$0.843502\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	32.2553	1.52222	0.761110	−	0.648623i	$-0.224655\pi$
$449$	32.2553	1.52222	0.761110	+	0.648623i	$0.224655\pi$
$450$	0	0
$451$	11.8880	0.559782
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 19.8880i	− 0.930320i	−0.885227	−	0.465160i	$-0.845997\pi$
$457$	− 19.8880i	− 0.930320i	0.885227	−	0.465160i	$-0.154003\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	21.7658	1.01373	0.506867	−	0.862024i	$-0.330804\pi$
$461$	21.7658	1.01373	0.506867	+	0.862024i	$0.330804\pi$
$462$	0	0
$463$	17.9627i	0.834795i	0.908724	+	0.417398i	$0.137058\pi$
$463$	17.9627i	0.834795i	−0.908724	+	0.417398i	$0.862942\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	19.0475i	0.881412i	0.897651	+	0.440706i	$0.145272\pi$
$467$	19.0475i	0.881412i	−0.897651	+	0.440706i	$0.854728\pi$
$468$	0	0
$469$	36.5953	1.68982
$470$	0	0
$471$	0	0
$472$	0	0
$473$	6.32996i	0.291052i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	20.7549	0.948314	0.474157	−	0.880440i	$-0.342753\pi$
$479$	20.7549	0.948314	0.474157	+	0.880440i	$0.342753\pi$
$480$	0	0
$481$	44.3502	2.02220
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 4.86690i	− 0.220540i	−0.993902	−	0.110270i	$-0.964828\pi$
$487$	− 4.86690i	− 0.220540i	0.993902	−	0.110270i	$-0.0351716\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−38.2553	−1.72644	−0.863218	−	0.504831i	$-0.831555\pi$
$491$	−38.2553	−1.72644	−0.863218	+	0.504831i	$0.831555\pi$
$492$	0	0
$493$	14.8031i	0.666700i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 61.2171i	− 2.74596i
$498$	0	0
$499$	26.8405	1.20155	0.600773	−	0.799420i	$-0.294860\pi$
$499$	26.8405	1.20155	0.600773	+	0.799420i	$0.294860\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	18.8825i	0.841929i	0.907077	+	0.420964i	$0.138308\pi$
$503$	18.8825i	0.841929i	−0.907077	+	0.420964i	$0.861692\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	33.1696	1.47022	0.735109	−	0.677949i	$-0.237131\pi$
$509$	33.1696	1.47022	0.735109	+	0.677949i	$0.237131\pi$
$510$	0	0
$511$	−30.6802	−1.35721
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	3.19686i	0.140598i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	38.0101	1.66525	0.832627	−	0.553834i	$-0.186836\pi$
$521$	38.0101	1.66525	0.832627	+	0.553834i	$0.186836\pi$
$522$	0	0
$523$	− 37.4677i	− 1.63835i	−0.573544	−	0.819174i	$-0.694432\pi$
$523$	− 37.4677i	− 1.63835i	0.573544	−	0.819174i	$-0.305568\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0.499600i	0.0217629i
$528$	0	0
$529$	−11.0475	−0.480325
$530$	0	0
$531$	0	0
$532$	0	0
$533$	25.8133i	1.11810i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−13.1595	−0.566820
$540$	0	0
$541$	31.7549	1.36525	0.682624	−	0.730770i	$-0.260839\pi$
$541$	31.7549	1.36525	0.682624	+	0.730770i	$0.260839\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 13.5051i	− 0.577435i	−0.957414	−	0.288717i	$-0.906771\pi$
$547$	− 13.5051i	− 0.577435i	0.957414	−	0.288717i	$-0.0932288\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−2.53695	−0.108078
$552$	0	0
$553$	− 7.67004i	− 0.326163i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	6.00000i	0.254228i	0.991888	+	0.127114i	$0.0405714\pi$
$557$	6.00000i	0.254228i	−0.991888	+	0.127114i	$0.959429\pi$
$558$	0	0
$559$	−13.7447	−0.581340
$560$	0	0
$561$	0	0
$562$	0	0
$563$	31.2125i	1.31545i	0.753259	+	0.657724i	$0.228481\pi$
$563$	31.2125i	1.31545i	−0.753259	+	0.657724i	$0.771519\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−12.8296	−0.537843	−0.268922	−	0.963162i	$-0.586667\pi$
$569$	−12.8296	−0.537843	−0.268922	+	0.963162i	$0.586667\pi$
$570$	0	0
$571$	−15.3401	−0.641963	−0.320981	−	0.947086i	$-0.604013\pi$
$571$	−15.3401	−0.641963	−0.320981	+	0.947086i	$0.604013\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 33.3292i	− 1.38751i	−0.720211	−	0.693755i	$-0.755955\pi$
$577$	− 33.3292i	− 1.38751i	0.720211	−	0.693755i	$-0.244045\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−45.3876	−1.88299
$582$	0	0
$583$	19.9253i	0.825222i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	41.5051i	1.71310i	0.516066	+	0.856549i	$0.327396\pi$
$587$	41.5051i	1.71310i	−0.516066	+	0.856549i	$0.672604\pi$
$588$	0	0
$589$	−0.0856210	−0.00352795
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 17.0584i	− 0.700505i	−0.936655	−	0.350252i	$-0.886096\pi$
$593$	− 17.0584i	− 0.700505i	0.936655	−	0.350252i	$-0.113904\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	25.9253	1.05928	0.529640	−	0.848223i	$-0.322327\pi$
$599$	25.9253	1.05928	0.529640	+	0.848223i	$0.322327\pi$
$600$	0	0
$601$	−11.0957	−0.452605	−0.226303	−	0.974057i	$-0.572664\pi$
$601$	−11.0957	−0.452605	−0.226303	+	0.974057i	$0.572664\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 26.3829i	− 1.07085i	−0.844583	−	0.535424i	$-0.820152\pi$
$607$	− 26.3829i	− 1.07085i	0.844583	−	0.535424i	$-0.179848\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−6.94160	−0.280827
$612$	0	0
$613$	2.87783i	0.116235i	0.998310	+	0.0581173i	$0.0185097\pi$
$613$	2.87783i	0.116235i	−0.998310	+	0.0581173i	$0.981490\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 10.8405i	− 0.436422i	−0.975902	−	0.218211i	$-0.929978\pi$
$617$	− 10.8405i	− 0.436422i	0.975902	−	0.218211i	$-0.0700220\pi$
$618$	0	0
$619$	−8.87783	−0.356830	−0.178415	−	0.983955i	$-0.557097\pi$
$619$	−8.87783	−0.356830	−0.178415	+	0.983955i	$0.557097\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 11.5051i	− 0.460941i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	20.4249	0.814394
$630$	0	0
$631$	14.3300	0.570467	0.285233	−	0.958458i	$-0.407929\pi$
$631$	14.3300	0.570467	0.285233	+	0.958458i	$0.407929\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 28.5743i	− 1.13215i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−46.0475	−1.81877	−0.909383	−	0.415960i	$-0.863446\pi$
$641$	−46.0475	−1.81877	−0.909383	+	0.415960i	$0.863446\pi$
$642$	0	0
$643$	3.54241i	0.139699i	0.997558	+	0.0698495i	$0.0222519\pi$
$643$	3.54241i	0.139699i	−0.997558	+	0.0698495i	$0.977748\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	16.1276i	0.634043i	0.948418	+	0.317021i	$0.102683\pi$
$647$	16.1276i	0.634043i	−0.948418	+	0.317021i	$0.897317\pi$
$648$	0	0
$649$	19.9253	0.782137
$650$	0	0
$651$	0	0
$652$	0	0
$653$	25.5471i	0.999734i	0.866102	+	0.499867i	$0.166618\pi$
$653$	25.5471i	0.999734i	−0.866102	+	0.499867i	$0.833382\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−36.4996	−1.42182	−0.710911	−	0.703282i	$-0.751717\pi$
$659$	−36.4996	−1.42182	−0.710911	+	0.703282i	$0.751717\pi$
$660$	0	0
$661$	−7.68097	−0.298755	−0.149378	−	0.988780i	$-0.547727\pi$
$661$	−7.68097	−0.298755	−0.149378	+	0.988780i	$0.547727\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	50.5899i	1.95885i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	25.5471	0.986234
$672$	0	0
$673$	10.0373i	0.386911i	0.981109	+	0.193456i	$0.0619696\pi$
$673$	10.0373i	0.386911i	−0.981109	+	0.193456i	$0.938030\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 29.4521i	− 1.13194i	−0.824427	−	0.565969i	$-0.808502\pi$
$677$	− 29.4521i	− 1.13194i	0.824427	−	0.565969i	$-0.191498\pi$
$678$	0	0
$679$	−14.0747	−0.540137
$680$	0	0
$681$	0	0
$682$	0	0
$683$	24.2179i	0.926673i	0.886182	+	0.463336i	$0.153348\pi$
$683$	24.2179i	0.926673i	−0.886182	+	0.463336i	$0.846652\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−43.2654	−1.64828
$690$	0	0
$691$	0.244336	0.00929499	0.00464749	−	0.999989i	$-0.498521\pi$
$691$	0.244336	0.00929499	0.00464749	+	0.999989i	$0.498521\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	11.8880i	0.450289i
$698$	0	0
$699$	0	0
$700$	0	0
$701$	7.79221	0.294308	0.147154	−	0.989114i	$-0.452989\pi$
$701$	7.79221	0.294308	0.147154	+	0.989114i	$0.452989\pi$
$702$	0	0
$703$	3.50040i	0.132020i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 1.26539i	− 0.0475900i
$708$	0	0
$709$	24.8397	0.932874	0.466437	−	0.884554i	$-0.345537\pi$
$709$	24.8397	0.932874	0.466437	+	0.884554i	$0.345537\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	1.70739i	0.0639422i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	35.0101	1.30566	0.652829	−	0.757506i	$-0.273582\pi$
$719$	35.0101	1.30566	0.652829	+	0.757506i	$0.273582\pi$
$720$	0	0
$721$	11.9736	0.445919
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 22.3082i	− 0.827365i	−0.910421	−	0.413683i	$-0.864242\pi$
$727$	− 22.3082i	− 0.827365i	0.910421	−	0.413683i	$-0.135758\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−6.32996	−0.234122
$732$	0	0
$733$	− 16.1696i	− 0.597239i	−0.954372	−	0.298620i	$-0.903474\pi$
$733$	− 16.1696i	− 0.597239i	0.954372	−	0.298620i	$-0.0965262\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 16.2926i	− 0.600146i
$738$	0	0
$739$	−38.9354	−1.43226	−0.716132	−	0.697965i	$-0.754089\pi$
$739$	−38.9354	−1.43226	−0.716132	+	0.697965i	$0.754089\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	5.33542i	0.195738i	0.995199	+	0.0978688i	$0.0312026\pi$
$743$	5.33542i	0.195738i	−0.995199	+	0.0978688i	$0.968797\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	57.2179	2.09070
$750$	0	0
$751$	34.2070	1.24823	0.624115	−	0.781332i	$-0.285460\pi$
$751$	34.2070	1.24823	0.624115	+	0.781332i	$0.285460\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 0.952525i	− 0.0346201i	−0.999850	−	0.0173101i	$-0.994490\pi$
$757$	− 0.952525i	− 0.0346201i	0.999850	−	0.0173101i	$-0.00551024\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−16.5953	−0.601581	−0.300790	−	0.953690i	$-0.597250\pi$
$761$	−16.5953	−0.601581	−0.300790	+	0.953690i	$0.597250\pi$
$762$	0	0
$763$	− 25.7230i	− 0.931234i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	43.2654i	1.56222i
$768$	0	0
$769$	−6.59535	−0.237834	−0.118917	−	0.992904i	$-0.537942\pi$
$769$	−6.59535	−0.237834	−0.118917	+	0.992904i	$0.537942\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 18.8296i	− 0.677252i	−0.940921	−	0.338626i	$-0.890038\pi$
$773$	− 18.8296i	− 0.677252i	0.940921	−	0.338626i	$-0.109962\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−2.03735	−0.0729955
$780$	0	0
$781$	−27.2545	−0.975241
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 7.12217i	− 0.253878i	−0.991911	−	0.126939i	$-0.959485\pi$
$787$	− 7.12217i	− 0.253878i	0.991911	−	0.126939i	$-0.0405152\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	21.7447	0.773154
$792$	0	0
$793$	55.4724i	1.96988i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 48.7175i	− 1.72566i	−0.505492	−	0.862832i	$-0.668689\pi$
$797$	− 48.7175i	− 1.72566i	0.505492	−	0.862832i	$-0.331311\pi$
$798$	0	0
$799$	−3.19686	−0.113097
$800$	0	0
$801$	0	0
$802$	0	0
$803$	13.6591i	0.482020i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	19.1595	0.673613	0.336806	−	0.941574i	$-0.390653\pi$
$809$	19.1595	0.673613	0.336806	+	0.941574i	$0.390653\pi$
$810$	0	0
$811$	13.3401	0.468434	0.234217	−	0.972184i	$-0.424747\pi$
$811$	13.3401	0.468434	0.234217	+	0.972184i	$0.424747\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 1.08482i	− 0.0379531i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1.62257	0.0566280	0.0283140	−	0.999599i	$-0.490986\pi$
$821$	1.62257	0.0566280	0.0283140	+	0.999599i	$0.490986\pi$
$822$	0	0
$823$	26.0529i	0.908148i	0.890964	+	0.454074i	$0.150030\pi$
$823$	26.0529i	0.908148i	−0.890964	+	0.454074i	$0.849970\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	37.4786i	1.30326i	0.758537	+	0.651630i	$0.225915\pi$
$827$	37.4786i	1.30326i	−0.758537	+	0.651630i	$0.774085\pi$
$828$	0	0
$829$	−32.9627	−1.14484	−0.572420	−	0.819960i	$-0.693995\pi$
$829$	−32.9627	−1.14484	−0.572420	+	0.819960i	$0.693995\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 13.1595i	− 0.455950i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−12.8778	−0.444592	−0.222296	−	0.974979i	$-0.571355\pi$
$839$	−12.8778	−0.444592	−0.222296	+	0.974979i	$0.571355\pi$
$840$	0	0
$841$	46.1696	1.59206
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 31.0055i	− 1.06536i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	69.8023	2.39279
$852$	0	0
$853$	− 44.1323i	− 1.51106i	−0.655113	−	0.755531i	$-0.727379\pi$
$853$	− 44.1323i	− 1.51106i	0.655113	−	0.755531i	$-0.272621\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 15.6327i	− 0.534003i	−0.963696	−	0.267001i	$-0.913967\pi$
$857$	− 15.6327i	− 0.534003i	0.963696	−	0.267001i	$-0.0860328\pi$
$858$	0	0
$859$	−7.88796	−0.269134	−0.134567	−	0.990905i	$-0.542964\pi$
$859$	−7.88796	−0.269134	−0.134567	+	0.990905i	$0.542964\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 9.53148i	− 0.324455i	−0.986753	−	0.162228i	$-0.948132\pi$
$863$	− 9.53148i	− 0.324455i	0.986753	−	0.162228i	$-0.0518679\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−3.41478	−0.115838
$870$	0	0
$871$	35.3774	1.19872
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 37.9517i	− 1.28154i	−0.767733	−	0.640769i	$-0.778615\pi$
$877$	− 37.9517i	− 1.28154i	0.767733	−	0.640769i	$-0.221385\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−23.2070	−0.781863	−0.390932	−	0.920420i	$-0.627847\pi$
$881$	−23.2070	−0.781863	−0.390932	+	0.920420i	$0.627847\pi$
$882$	0	0
$883$	30.9572i	1.04179i	0.853620	+	0.520896i	$0.174402\pi$
$883$	30.9572i	1.04179i	−0.853620	+	0.520896i	$0.825598\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	37.8770i	1.27179i	0.771777	+	0.635893i	$0.219368\pi$
$887$	37.8770i	1.27179i	−0.771777	+	0.635893i	$0.780632\pi$
$888$	0	0
$889$	−21.2553	−0.712879
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 0.547875i	− 0.0183339i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	2.53695	0.0846119
$900$	0	0
$901$	−19.9253	−0.663808
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 32.3829i	− 1.07526i	−0.843182	−	0.537628i	$-0.819320\pi$
$907$	− 32.3829i	− 1.07526i	0.843182	−	0.537628i	$-0.180680\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	45.2545	1.49935	0.749674	−	0.661808i	$-0.230210\pi$
$911$	45.2545	1.49935	0.749674	+	0.661808i	$0.230210\pi$
$912$	0	0
$913$	20.2070i	0.668754i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 49.2016i	− 1.62478i
$918$	0	0
$919$	−21.9253	−0.723249	−0.361625	−	0.932324i	$-0.617778\pi$
$919$	−21.9253	−0.723249	−0.361625	+	0.932324i	$0.617778\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 59.1798i	− 1.94793i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−13.2654	−0.435223	−0.217612	−	0.976035i	$-0.569827\pi$
$929$	−13.2654	−0.435223	−0.217612	+	0.976035i	$0.569827\pi$
$930$	0	0
$931$	2.25526	0.0739133
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	40.4996i	1.32306i	0.749917	+	0.661532i	$0.230093\pi$
$937$	40.4996i	1.32306i	−0.749917	+	0.661532i	$0.769907\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−2.67004	−0.0870409	−0.0435205	−	0.999053i	$-0.513857\pi$
$941$	−2.67004	−0.0870409	−0.0435205	+	0.999053i	$0.513857\pi$
$942$	0	0
$943$	40.6272i	1.32300i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	36.8825i	1.19852i	0.800554	+	0.599260i	$0.204538\pi$
$947$	36.8825i	1.19852i	−0.800554	+	0.599260i	$0.795462\pi$
$948$	0	0
$949$	−29.6591	−0.962776
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 12.7175i	− 0.411961i	−0.978556	−	0.205980i	$-0.933962\pi$
$953$	− 12.7175i	− 0.411961i	0.978556	−	0.205980i	$-0.0660383\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−34.8405	−1.12506
$960$	0	0
$961$	−30.9144	−0.997238
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 42.3455i	− 1.36174i	−0.732404	−	0.680871i	$-0.761602\pi$
$967$	− 42.3455i	− 1.36174i	0.732404	−	0.680871i	$-0.238398\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−5.38836	−0.172921	−0.0864604	−	0.996255i	$-0.527556\pi$
$971$	−5.38836	−0.172921	−0.0864604	+	0.996255i	$0.527556\pi$
$972$	0	0
$973$	15.3401i	0.491781i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	28.1323i	0.900032i	0.893021	+	0.450016i	$0.148582\pi$
$977$	28.1323i	0.900032i	−0.893021	+	0.450016i	$0.851418\pi$
$978$	0	0
$979$	−5.12217	−0.163705
$980$	0	0
$981$	0	0
$982$	0	0
$983$	25.7603i	0.821627i	0.911719	+	0.410813i	$0.134755\pi$
$983$	25.7603i	0.821627i	−0.911719	+	0.410813i	$0.865245\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−21.6327	−0.687880
$990$	0	0
$991$	23.0848	0.733314	0.366657	−	0.930356i	$-0.380502\pi$
$991$	23.0848	0.733314	0.366657	+	0.930356i	$0.380502\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	32.8405i	1.04007i	0.854145	+	0.520034i	$0.174081\pi$
$997$	32.8405i	1.04007i	−0.854145	+	0.520034i	$0.825919\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8100.2.d.o.649.5		6
3.2	odd	2		8100.2.d.p.649.5		6
5.2	odd	4		1620.2.a.j.1.1		3
5.3	odd	4		8100.2.a.u.1.3		3
5.4	even	2	inner	8100.2.d.o.649.2		6
9.2	odd	6		900.2.s.c.49.1		12
9.4	even	3		2700.2.s.c.2449.2		12
9.5	odd	6		900.2.s.c.349.6		12
9.7	even	3		2700.2.s.c.1549.5		12
15.2	even	4		1620.2.a.i.1.1		3
15.8	even	4		8100.2.a.v.1.3		3
15.14	odd	2		8100.2.d.p.649.2		6
20.7	even	4		6480.2.a.bw.1.3		3
45.2	even	12		180.2.i.b.121.2	yes	6
45.4	even	6		2700.2.s.c.2449.5		12
45.7	odd	12		540.2.i.b.361.3		6
45.13	odd	12		2700.2.i.c.1801.1		6
45.14	odd	6		900.2.s.c.349.1		12
45.22	odd	12		540.2.i.b.181.3		6
45.23	even	12		900.2.i.c.601.2		6
45.29	odd	6		900.2.s.c.49.6		12
45.32	even	12		180.2.i.b.61.2	✓	6
45.34	even	6		2700.2.s.c.1549.2		12
45.38	even	12		900.2.i.c.301.2		6
45.43	odd	12		2700.2.i.c.901.1		6
60.47	odd	4		6480.2.a.bt.1.3		3
180.7	even	12		2160.2.q.i.1441.1		6
180.47	odd	12		720.2.q.k.481.2		6
180.67	even	12		2160.2.q.i.721.1		6
180.167	odd	12		720.2.q.k.241.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
180.2.i.b.61.2	✓	6	45.32	even	12
180.2.i.b.121.2	yes	6	45.2	even	12
540.2.i.b.181.3		6	45.22	odd	12
540.2.i.b.361.3		6	45.7	odd	12
720.2.q.k.241.2		6	180.167	odd	12
720.2.q.k.481.2		6	180.47	odd	12
900.2.i.c.301.2		6	45.38	even	12
900.2.i.c.601.2		6	45.23	even	12
900.2.s.c.49.1		12	9.2	odd	6
900.2.s.c.49.6		12	45.29	odd	6
900.2.s.c.349.1		12	45.14	odd	6
900.2.s.c.349.6		12	9.5	odd	6
1620.2.a.i.1.1		3	15.2	even	4
1620.2.a.j.1.1		3	5.2	odd	4
2160.2.q.i.721.1		6	180.67	even	12
2160.2.q.i.1441.1		6	180.7	even	12
2700.2.i.c.901.1		6	45.43	odd	12
2700.2.i.c.1801.1		6	45.13	odd	12
2700.2.s.c.1549.2		12	45.34	even	6
2700.2.s.c.1549.5		12	9.7	even	3
2700.2.s.c.2449.2		12	9.4	even	3
2700.2.s.c.2449.5		12	45.4	even	6
6480.2.a.bt.1.3		3	60.47	odd	4
6480.2.a.bw.1.3		3	20.7	even	4
8100.2.a.u.1.3		3	5.3	odd	4
8100.2.a.v.1.3		3	15.8	even	4
8100.2.d.o.649.2		6	5.4	even	2	inner
8100.2.d.o.649.5		6	1.1	even	1	trivial
8100.2.d.p.649.2		6	15.14	odd	2
8100.2.d.p.649.5		6	3.2	odd	2

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:	\( N \)	\(=\)	\( 8100 = 2^{2} \cdot 3^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8100.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.83502i q^{7} +1.70739 q^{11} +3.70739i q^{13} +1.70739i q^{17} -0.292611 q^{19} +5.83502i q^{23} +8.67004 q^{29} +0.292611 q^{31} -11.9627i q^{37} +6.96265 q^{41} +3.70739i q^{43} +1.87237i q^{47} -7.70739 q^{49} +11.6700i q^{53} +11.6700 q^{59} +14.9627 q^{61} -9.54241i q^{67} -15.9627 q^{71} +8.00000i q^{73} +6.54787i q^{77} -2.00000 q^{79} +11.8350i q^{83} -3.00000 q^{89} -14.2179 q^{91} +3.67004i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 12 q^{19} - 6 q^{29} + 12 q^{31} - 6 q^{41} - 36 q^{49} + 12 q^{59} + 42 q^{61} - 48 q^{71} - 12 q^{79} - 18 q^{89}+O(q^{100}) \) 6 * q - 12 * q^19 - 6 * q^29 + 12 * q^31 - 6 * q^41 - 36 * q^49 + 12 * q^59 + 42 * q^61 - 48 * q^71 - 12 * q^79 - 18 * q^89

Introduction

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