LMFDB - Embedded newform 4032.2.j.d.2591.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4032,2,Mod(2591,4032)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4032, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 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("4032.2591");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4032 = 2^{6} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4032.j (of order $2$, degree $1$, 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:	$32.1956820950$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2591.3
Root			$0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	4032.2591
Dual form			4032.2.j.d.2591.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.82843 q^{5} -1.00000i q^{7} +O(q^{10})$	$q+2.82843 q^{5} -1.00000i q^{7} -1.41421i q^{11} +2.00000i q^{13} +2.82843i q^{17} +4.00000 q^{19} +1.41421 q^{23} +3.00000 q^{25} +1.41421 q^{29} -2.82843i q^{35} +10.0000i q^{37} +5.65685i q^{41} +2.00000 q^{43} +2.82843 q^{47} -1.00000 q^{49} +1.41421 q^{53} -4.00000i q^{55} -8.48528i q^{59} -6.00000i q^{61} +5.65685i q^{65} +4.00000 q^{67} +12.7279 q^{71} +10.0000 q^{73} -1.41421 q^{77} -6.00000i q^{79} +5.65685i q^{83} +8.00000i q^{85} +11.3137i q^{89} +2.00000 q^{91} +11.3137 q^{95} +2.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q + 16 q^{19} + 12 q^{25} + 8 q^{43} - 4 q^{49} + 16 q^{67} + 40 q^{73} + 8 q^{91} + 8 q^{97}+O(q^{100}) $ 4 * q + 16 * q^19 + 12 * q^25 + 8 * q^43 - 4 * q^49 + 16 * q^67 + 40 * q^73 + 8 * q^91 + 8 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4032\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$1793$	$3781$
$\chi(n)$	$-1$	$1$	$-1$	$-1$

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
$5$	2.82843	1.26491	0.632456	−	0.774597i	$-0.282047\pi$
$5$	2.82843	1.26491	0.632456	+	0.774597i	$0.282047\pi$
$6$	0	0
$7$	− 1.00000i	− 0.377964i
$7$	− 1.00000i	− 0.377964i
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 1.41421i	− 0.426401i	−0.977008	−	0.213201i	$-0.931611\pi$
$11$	− 1.41421i	− 0.426401i	0.977008	−	0.213201i	$-0.0683888\pi$
$12$	0	0
$13$	2.00000i	0.554700i	0.960769	+	0.277350i	$0.0894562\pi$
$13$	2.00000i	0.554700i	−0.960769	+	0.277350i	$0.910544\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.82843i	0.685994i	0.939336	+	0.342997i	$0.111442\pi$
$17$	2.82843i	0.685994i	−0.939336	+	0.342997i	$0.888558\pi$
$18$	0	0
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.41421	0.294884	0.147442	−	0.989071i	$-0.452896\pi$
$23$	1.41421	0.294884	0.147442	+	0.989071i	$0.452896\pi$
$24$	0	0
$25$	3.00000	0.600000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	1.41421	0.262613	0.131306	−	0.991342i	$-0.458083\pi$
$29$	1.41421	0.262613	0.131306	+	0.991342i	$0.458083\pi$
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 2.82843i	− 0.478091i
$36$	0	0
$37$	10.0000i	1.64399i	0.569495	+	0.821995i	$0.307139\pi$
$37$	10.0000i	1.64399i	−0.569495	+	0.821995i	$0.692861\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	5.65685i	0.883452i	0.897150	+	0.441726i	$0.145634\pi$
$41$	5.65685i	0.883452i	−0.897150	+	0.441726i	$0.854366\pi$
$42$	0	0
$43$	2.00000	0.304997	0.152499	−	0.988304i	$-0.451268\pi$
$43$	2.00000	0.304997	0.152499	+	0.988304i	$0.451268\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	2.82843	0.412568	0.206284	−	0.978492i	$-0.433863\pi$
$47$	2.82843	0.412568	0.206284	+	0.978492i	$0.433863\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	1.41421	0.194257	0.0971286	−	0.995272i	$-0.469034\pi$
$53$	1.41421	0.194257	0.0971286	+	0.995272i	$0.469034\pi$
$54$	0	0
$55$	− 4.00000i	− 0.539360i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 8.48528i	− 1.10469i	−0.833616	−	0.552345i	$-0.813733\pi$
$59$	− 8.48528i	− 1.10469i	0.833616	−	0.552345i	$-0.186267\pi$
$60$	0	0
$61$	− 6.00000i	− 0.768221i	−0.923287	−	0.384111i	$-0.874508\pi$
$61$	− 6.00000i	− 0.768221i	0.923287	−	0.384111i	$-0.125492\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	5.65685i	0.701646i
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	12.7279	1.51053	0.755263	−	0.655422i	$-0.227509\pi$
$71$	12.7279	1.51053	0.755263	+	0.655422i	$0.227509\pi$
$72$	0	0
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1.41421	−0.161165
$78$	0	0
$79$	− 6.00000i	− 0.675053i	−0.941316	−	0.337526i	$-0.890410\pi$
$79$	− 6.00000i	− 0.675053i	0.941316	−	0.337526i	$-0.109590\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	5.65685i	0.620920i	0.950586	+	0.310460i	$0.100483\pi$
$83$	5.65685i	0.620920i	−0.950586	+	0.310460i	$0.899517\pi$
$84$	0	0
$85$	8.00000i	0.867722i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	11.3137i	1.19925i	0.800281	+	0.599625i	$0.204684\pi$
$89$	11.3137i	1.19925i	−0.800281	+	0.599625i	$0.795316\pi$
$90$	0	0
$91$	2.00000	0.209657
$92$	0	0
$93$	0	0
$94$	0	0
$95$	11.3137	1.16076
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−5.65685	−0.562878	−0.281439	−	0.959579i	$-0.590812\pi$
$101$	−5.65685	−0.562878	−0.281439	+	0.959579i	$0.590812\pi$
$102$	0	0
$103$	− 8.00000i	− 0.788263i	−0.919054	−	0.394132i	$-0.871045\pi$
$103$	− 8.00000i	− 0.788263i	0.919054	−	0.394132i	$-0.128955\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	18.3848i	1.77732i	0.458563	+	0.888662i	$0.348364\pi$
$107$	18.3848i	1.77732i	−0.458563	+	0.888662i	$0.651636\pi$
$108$	0	0
$109$	− 14.0000i	− 1.34096i	−0.741929	−	0.670478i	$-0.766089\pi$
$109$	− 14.0000i	− 1.34096i	0.741929	−	0.670478i	$-0.233911\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 1.41421i	− 0.133038i	−0.997785	−	0.0665190i	$-0.978811\pi$
$113$	− 1.41421i	− 0.133038i	0.997785	−	0.0665190i	$-0.0211893\pi$
$114$	0	0
$115$	4.00000	0.373002
$116$	0	0
$117$	0	0
$118$	0	0
$119$	2.82843	0.259281
$120$	0	0
$121$	9.00000	0.818182
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−5.65685	−0.505964
$126$	0	0
$127$	− 6.00000i	− 0.532414i	−0.963916	−	0.266207i	$-0.914230\pi$
$127$	− 6.00000i	− 0.532414i	0.963916	−	0.266207i	$-0.0857705\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 11.3137i	− 0.988483i	−0.869325	−	0.494242i	$-0.835446\pi$
$131$	− 11.3137i	− 0.988483i	0.869325	−	0.494242i	$-0.164554\pi$
$132$	0	0
$133$	− 4.00000i	− 0.346844i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	7.07107i	0.604122i	0.953289	+	0.302061i	$0.0976746\pi$
$137$	7.07107i	0.604122i	−0.953289	+	0.302061i	$0.902325\pi$
$138$	0	0
$139$	−20.0000	−1.69638	−0.848189	−	0.529694i	$-0.822307\pi$
$139$	−20.0000	−1.69638	−0.848189	+	0.529694i	$0.822307\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	2.82843	0.236525
$144$	0	0
$145$	4.00000	0.332182
$146$	0	0
$147$	0	0
$148$	0	0
$149$	18.3848	1.50614	0.753070	−	0.657941i	$-0.228572\pi$
$149$	18.3848	1.50614	0.753070	+	0.657941i	$0.228572\pi$
$150$	0	0
$151$	− 16.0000i	− 1.30206i	−0.759051	−	0.651031i	$-0.774337\pi$
$151$	− 16.0000i	− 1.30206i	0.759051	−	0.651031i	$-0.225663\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 22.0000i	− 1.75579i	−0.478852	−	0.877896i	$-0.658947\pi$
$157$	− 22.0000i	− 1.75579i	0.478852	−	0.877896i	$-0.341053\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 1.41421i	− 0.111456i
$162$	0	0
$163$	−20.0000	−1.56652	−0.783260	−	0.621694i	$-0.786445\pi$
$163$	−20.0000	−1.56652	−0.783260	+	0.621694i	$0.786445\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	19.7990	1.53209	0.766046	−	0.642786i	$-0.222221\pi$
$167$	19.7990	1.53209	0.766046	+	0.642786i	$0.222221\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−16.9706	−1.29025	−0.645124	−	0.764078i	$-0.723194\pi$
$173$	−16.9706	−1.29025	−0.645124	+	0.764078i	$0.723194\pi$
$174$	0	0
$175$	− 3.00000i	− 0.226779i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	9.89949i	0.739923i	0.929047	+	0.369961i	$0.120629\pi$
$179$	9.89949i	0.739923i	−0.929047	+	0.369961i	$0.879371\pi$
$180$	0	0
$181$	6.00000i	0.445976i	0.974821	+	0.222988i	$0.0715812\pi$
$181$	6.00000i	0.445976i	−0.974821	+	0.222988i	$0.928419\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	28.2843i	2.07950i
$186$	0	0
$187$	4.00000	0.292509
$188$	0	0
$189$	0	0
$190$	0	0
$191$	4.24264	0.306987	0.153493	−	0.988150i	$-0.450948\pi$
$191$	4.24264	0.306987	0.153493	+	0.988150i	$0.450948\pi$
$192$	0	0
$193$	−4.00000	−0.287926	−0.143963	−	0.989583i	$-0.545985\pi$
$193$	−4.00000	−0.287926	−0.143963	+	0.989583i	$0.545985\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−4.24264	−0.302276	−0.151138	−	0.988513i	$-0.548294\pi$
$197$	−4.24264	−0.302276	−0.151138	+	0.988513i	$0.548294\pi$
$198$	0	0
$199$	8.00000i	0.567105i	0.958957	+	0.283552i	$0.0915130\pi$
$199$	8.00000i	0.567105i	−0.958957	+	0.283552i	$0.908487\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 1.41421i	− 0.0992583i
$204$	0	0
$205$	16.0000i	1.11749i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 5.65685i	− 0.391293i
$210$	0	0
$211$	−22.0000	−1.51454	−0.757271	−	0.653101i	$-0.773468\pi$
$211$	−22.0000	−1.51454	−0.757271	+	0.653101i	$0.773468\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	5.65685	0.385794
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−5.65685	−0.380521
$222$	0	0
$223$	− 16.0000i	− 1.07144i	−0.844396	−	0.535720i	$-0.820040\pi$
$223$	− 16.0000i	− 1.07144i	0.844396	−	0.535720i	$-0.179960\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2.82843i	0.187729i	0.995585	+	0.0938647i	$0.0299221\pi$
$227$	2.82843i	0.187729i	−0.995585	+	0.0938647i	$0.970078\pi$
$228$	0	0
$229$	− 10.0000i	− 0.660819i	−0.943838	−	0.330409i	$-0.892813\pi$
$229$	− 10.0000i	− 0.660819i	0.943838	−	0.330409i	$-0.107187\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	26.8701i	1.76032i	0.474681	+	0.880158i	$0.342563\pi$
$233$	26.8701i	1.76032i	−0.474681	+	0.880158i	$0.657437\pi$
$234$	0	0
$235$	8.00000	0.521862
$236$	0	0
$237$	0	0
$238$	0	0
$239$	9.89949	0.640345	0.320173	−	0.947359i	$-0.396259\pi$
$239$	9.89949	0.640345	0.320173	+	0.947359i	$0.396259\pi$
$240$	0	0
$241$	14.0000	0.901819	0.450910	−	0.892570i	$-0.351100\pi$
$241$	14.0000	0.901819	0.450910	+	0.892570i	$0.351100\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−2.82843	−0.180702
$246$	0	0
$247$	8.00000i	0.509028i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	8.48528i	0.535586i	0.963476	+	0.267793i	$0.0862944\pi$
$251$	8.48528i	0.535586i	−0.963476	+	0.267793i	$0.913706\pi$
$252$	0	0
$253$	− 2.00000i	− 0.125739i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	5.65685i	0.352865i	0.984313	+	0.176432i	$0.0564557\pi$
$257$	5.65685i	0.352865i	−0.984313	+	0.176432i	$0.943544\pi$
$258$	0	0
$259$	10.0000	0.621370
$260$	0	0
$261$	0	0
$262$	0	0
$263$	15.5563	0.959246	0.479623	−	0.877475i	$-0.340774\pi$
$263$	15.5563	0.959246	0.479623	+	0.877475i	$0.340774\pi$
$264$	0	0
$265$	4.00000	0.245718
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−19.7990	−1.20717	−0.603583	−	0.797300i	$-0.706261\pi$
$269$	−19.7990	−1.20717	−0.603583	+	0.797300i	$0.706261\pi$
$270$	0	0
$271$	8.00000i	0.485965i	0.970031	+	0.242983i	$0.0781258\pi$
$271$	8.00000i	0.485965i	−0.970031	+	0.242983i	$0.921874\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 4.24264i	− 0.255841i
$276$	0	0
$277$	− 4.00000i	− 0.240337i	−0.992754	−	0.120168i	$-0.961657\pi$
$277$	− 4.00000i	− 0.240337i	0.992754	−	0.120168i	$-0.0383434\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	9.89949i	0.590554i	0.955412	+	0.295277i	$0.0954120\pi$
$281$	9.89949i	0.590554i	−0.955412	+	0.295277i	$0.904588\pi$
$282$	0	0
$283$	24.0000	1.42665	0.713326	−	0.700832i	$-0.247188\pi$
$283$	24.0000	1.42665	0.713326	+	0.700832i	$0.247188\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	5.65685	0.333914
$288$	0	0
$289$	9.00000	0.529412
$290$	0	0
$291$	0	0
$292$	0	0
$293$	22.6274	1.32191	0.660954	−	0.750426i	$-0.270152\pi$
$293$	22.6274	1.32191	0.660954	+	0.750426i	$0.270152\pi$
$294$	0	0
$295$	− 24.0000i	− 1.39733i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	2.82843i	0.163572i
$300$	0	0
$301$	− 2.00000i	− 0.115278i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 16.9706i	− 0.971732i
$306$	0	0
$307$	0	0		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−25.4558	−1.44347	−0.721734	−	0.692170i	$-0.756655\pi$
$311$	−25.4558	−1.44347	−0.721734	+	0.692170i	$0.756655\pi$
$312$	0	0
$313$	30.0000	1.69570	0.847850	−	0.530236i	$-0.177897\pi$
$313$	30.0000	1.69570	0.847850	+	0.530236i	$0.177897\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	15.5563	0.873732	0.436866	−	0.899527i	$-0.356088\pi$
$317$	15.5563	0.873732	0.436866	+	0.899527i	$0.356088\pi$
$318$	0	0
$319$	− 2.00000i	− 0.111979i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	11.3137i	0.629512i
$324$	0	0
$325$	6.00000i	0.332820i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 2.82843i	− 0.155936i
$330$	0	0
$331$	−30.0000	−1.64895	−0.824475	−	0.565899i	$-0.808529\pi$
$331$	−30.0000	−1.64895	−0.824475	+	0.565899i	$0.808529\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	11.3137	0.618134
$336$	0	0
$337$	−28.0000	−1.52526	−0.762629	−	0.646837i	$-0.776092\pi$
$337$	−28.0000	−1.52526	−0.762629	+	0.646837i	$0.776092\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	1.00000i	0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 26.8701i	− 1.44246i	−0.692696	−	0.721230i	$-0.743577\pi$
$347$	− 26.8701i	− 1.44246i	0.692696	−	0.721230i	$-0.256423\pi$
$348$	0	0
$349$	10.0000i	0.535288i	0.963518	+	0.267644i	$0.0862451\pi$
$349$	10.0000i	0.535288i	−0.963518	+	0.267644i	$0.913755\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	19.7990i	1.05379i	0.849929	+	0.526897i	$0.176645\pi$
$353$	19.7990i	1.05379i	−0.849929	+	0.526897i	$0.823355\pi$
$354$	0	0
$355$	36.0000	1.91068
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−15.5563	−0.821033	−0.410516	−	0.911853i	$-0.634652\pi$
$359$	−15.5563	−0.821033	−0.410516	+	0.911853i	$0.634652\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	0	0
$363$	0	0
$364$	0	0
$365$	28.2843	1.48047
$366$	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
$371$	− 1.41421i	− 0.0734223i
$372$	0	0
$373$	16.0000i	0.828449i	0.910175	+	0.414224i	$0.135947\pi$
$373$	16.0000i	0.828449i	−0.910175	+	0.414224i	$0.864053\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2.82843i	0.145671i
$378$	0	0
$379$	30.0000	1.54100	0.770498	−	0.637442i	$-0.220007\pi$
$379$	30.0000	1.54100	0.770498	+	0.637442i	$0.220007\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−11.3137	−0.578103	−0.289052	−	0.957313i	$-0.593340\pi$
$383$	−11.3137	−0.578103	−0.289052	+	0.957313i	$0.593340\pi$
$384$	0	0
$385$	−4.00000	−0.203859
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−24.0416	−1.21896	−0.609480	−	0.792802i	$-0.708622\pi$
$389$	−24.0416	−1.21896	−0.609480	+	0.792802i	$0.708622\pi$
$390$	0	0
$391$	4.00000i	0.202289i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 16.9706i	− 0.853882i
$396$	0	0
$397$	− 2.00000i	− 0.100377i	−0.998740	−	0.0501886i	$-0.984018\pi$
$397$	− 2.00000i	− 0.100377i	0.998740	−	0.0501886i	$-0.0159822\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 7.07107i	− 0.353112i	−0.984291	−	0.176556i	$-0.943504\pi$
$401$	− 7.07107i	− 0.353112i	0.984291	−	0.176556i	$-0.0564957\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	14.1421	0.701000
$408$	0	0
$409$	18.0000	0.890043	0.445021	−	0.895520i	$-0.353196\pi$
$409$	18.0000	0.890043	0.445021	+	0.895520i	$0.353196\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−8.48528	−0.417533
$414$	0	0
$415$	16.0000i	0.785409i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 14.1421i	− 0.690889i	−0.938439	−	0.345444i	$-0.887728\pi$
$419$	− 14.1421i	− 0.690889i	0.938439	−	0.345444i	$-0.112272\pi$
$420$	0	0
$421$	4.00000i	0.194948i	0.995238	+	0.0974740i	$0.0310763\pi$
$421$	4.00000i	0.194948i	−0.995238	+	0.0974740i	$0.968924\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	8.48528i	0.411597i
$426$	0	0
$427$	−6.00000	−0.290360
$428$	0	0
$429$	0	0
$430$	0	0
$431$	24.0416	1.15804	0.579022	−	0.815312i	$-0.303434\pi$
$431$	24.0416	1.15804	0.579022	+	0.815312i	$0.303434\pi$
$432$	0	0
$433$	−18.0000	−0.865025	−0.432512	−	0.901628i	$-0.642373\pi$
$433$	−18.0000	−0.865025	−0.432512	+	0.901628i	$0.642373\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	5.65685	0.270604
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	4.24264i	0.201574i	0.994908	+	0.100787i	$0.0321361\pi$
$443$	4.24264i	0.201574i	−0.994908	+	0.100787i	$0.967864\pi$
$444$	0	0
$445$	32.0000i	1.51695i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 32.5269i	− 1.53504i	−0.641025	−	0.767520i	$-0.721491\pi$
$449$	− 32.5269i	− 1.53504i	0.641025	−	0.767520i	$-0.278509\pi$
$450$	0	0
$451$	8.00000	0.376705
$452$	0	0
$453$	0	0
$454$	0	0
$455$	5.65685	0.265197
$456$	0	0
$457$	24.0000	1.12267	0.561336	−	0.827588i	$-0.310287\pi$
$457$	24.0000	1.12267	0.561336	+	0.827588i	$0.310287\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	22.6274	1.05386	0.526932	−	0.849907i	$-0.323342\pi$
$461$	22.6274	1.05386	0.526932	+	0.849907i	$0.323342\pi$
$462$	0	0
$463$	− 14.0000i	− 0.650635i	−0.945605	−	0.325318i	$-0.894529\pi$
$463$	− 14.0000i	− 0.650635i	0.945605	−	0.325318i	$-0.105471\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	2.82843i	0.130884i	0.997856	+	0.0654420i	$0.0208457\pi$
$467$	2.82843i	0.130884i	−0.997856	+	0.0654420i	$0.979154\pi$
$468$	0	0
$469$	− 4.00000i	− 0.184703i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 2.82843i	− 0.130051i
$474$	0	0
$475$	12.0000	0.550598
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−36.7696	−1.68004	−0.840022	−	0.542553i	$-0.817458\pi$
$479$	−36.7696	−1.68004	−0.840022	+	0.542553i	$0.817458\pi$
$480$	0	0
$481$	−20.0000	−0.911922
$482$	0	0
$483$	0	0
$484$	0	0
$485$	5.65685	0.256865
$486$	0	0
$487$	− 8.00000i	− 0.362515i	−0.983436	−	0.181257i	$-0.941983\pi$
$487$	− 8.00000i	− 0.362515i	0.983436	−	0.181257i	$-0.0580167\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 15.5563i	− 0.702048i	−0.936366	−	0.351024i	$-0.885834\pi$
$491$	− 15.5563i	− 0.702048i	0.936366	−	0.351024i	$-0.114166\pi$
$492$	0	0
$493$	4.00000i	0.180151i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 12.7279i	− 0.570925i
$498$	0	0
$499$	2.00000	0.0895323	0.0447661	−	0.998997i	$-0.485746\pi$
$499$	2.00000	0.0895323	0.0447661	+	0.998997i	$0.485746\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	−16.0000	−0.711991
$506$	0	0
$507$	0	0
$508$	0	0
$509$	2.82843	0.125368	0.0626839	−	0.998033i	$-0.480034\pi$
$509$	2.82843	0.125368	0.0626839	+	0.998033i	$0.480034\pi$
$510$	0	0
$511$	− 10.0000i	− 0.442374i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 22.6274i	− 0.997083i
$516$	0	0
$517$	− 4.00000i	− 0.175920i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	36.7696i	1.61090i	0.592661	+	0.805452i	$0.298077\pi$
$521$	36.7696i	1.61090i	−0.592661	+	0.805452i	$0.701923\pi$
$522$	0	0
$523$	−32.0000	−1.39926	−0.699631	−	0.714504i	$-0.746652\pi$
$523$	−32.0000	−1.39926	−0.699631	+	0.714504i	$0.746652\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−21.0000	−0.913043
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−11.3137	−0.490051
$534$	0	0
$535$	52.0000i	2.24816i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1.41421i	0.0609145i
$540$	0	0
$541$	4.00000i	0.171973i	0.996296	+	0.0859867i	$0.0274043\pi$
$541$	4.00000i	0.171973i	−0.996296	+	0.0859867i	$0.972596\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 39.5980i	− 1.69619i
$546$	0	0
$547$	−28.0000	−1.19719	−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000	−1.19719	−0.598597	+	0.801050i	$0.704275\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	5.65685	0.240990
$552$	0	0
$553$	−6.00000	−0.255146
$554$	0	0
$555$	0	0
$556$	0	0
$557$	15.5563	0.659144	0.329572	−	0.944131i	$-0.393096\pi$
$557$	15.5563	0.659144	0.329572	+	0.944131i	$0.393096\pi$
$558$	0	0
$559$	4.00000i	0.169182i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	14.1421i	0.596020i	0.954563	+	0.298010i	$0.0963229\pi$
$563$	14.1421i	0.596020i	−0.954563	+	0.298010i	$0.903677\pi$
$564$	0	0
$565$	− 4.00000i	− 0.168281i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	12.7279i	0.533582i	0.963754	+	0.266791i	$0.0859634\pi$
$569$	12.7279i	0.533582i	−0.963754	+	0.266791i	$0.914037\pi$
$570$	0	0
$571$	12.0000	0.502184	0.251092	−	0.967963i	$-0.419210\pi$
$571$	12.0000	0.502184	0.251092	+	0.967963i	$0.419210\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	4.24264	0.176930
$576$	0	0
$577$	2.00000	0.0832611	0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000	0.0832611	0.0416305	+	0.999133i	$0.486745\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	5.65685	0.234686
$582$	0	0
$583$	− 2.00000i	− 0.0828315i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	31.1127i	1.28416i	0.766638	+	0.642079i	$0.221928\pi$
$587$	31.1127i	1.28416i	−0.766638	+	0.642079i	$0.778072\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 19.7990i	− 0.813047i	−0.913640	−	0.406524i	$-0.866741\pi$
$593$	− 19.7990i	− 0.813047i	0.913640	−	0.406524i	$-0.133259\pi$
$594$	0	0
$595$	8.00000	0.327968
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−24.0416	−0.982314	−0.491157	−	0.871071i	$-0.663426\pi$
$599$	−24.0416	−0.982314	−0.491157	+	0.871071i	$0.663426\pi$
$600$	0	0
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	25.4558	1.03493
$606$	0	0
$607$	28.0000i	1.13648i	0.822861	+	0.568242i	$0.192376\pi$
$607$	28.0000i	1.13648i	−0.822861	+	0.568242i	$0.807624\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	5.65685i	0.228852i
$612$	0	0
$613$	− 20.0000i	− 0.807792i	−0.914805	−	0.403896i	$-0.867656\pi$
$613$	− 20.0000i	− 0.807792i	0.914805	−	0.403896i	$-0.132344\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 26.8701i	− 1.08175i	−0.841104	−	0.540874i	$-0.818094\pi$
$617$	− 26.8701i	− 1.08175i	0.841104	−	0.540874i	$-0.181906\pi$
$618$	0	0
$619$	−4.00000	−0.160774	−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000	−0.160774	−0.0803868	+	0.996764i	$0.525616\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	11.3137	0.453274
$624$	0	0
$625$	−31.0000	−1.24000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−28.2843	−1.12777
$630$	0	0
$631$	6.00000i	0.238856i	0.992843	+	0.119428i	$0.0381061\pi$
$631$	6.00000i	0.238856i	−0.992843	+	0.119428i	$0.961894\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 16.9706i	− 0.673456i
$636$	0	0
$637$	− 2.00000i	− 0.0792429i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 43.8406i	− 1.73160i	−0.500390	−	0.865800i	$-0.666810\pi$
$641$	− 43.8406i	− 1.73160i	0.500390	−	0.865800i	$-0.333190\pi$
$642$	0	0
$643$	16.0000	0.630978	0.315489	−	0.948929i	$-0.397831\pi$
$643$	16.0000	0.630978	0.315489	+	0.948929i	$0.397831\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−36.7696	−1.44556	−0.722780	−	0.691078i	$-0.757136\pi$
$647$	−36.7696	−1.44556	−0.722780	+	0.691078i	$0.757136\pi$
$648$	0	0
$649$	−12.0000	−0.471041
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−26.8701	−1.05151	−0.525753	−	0.850637i	$-0.676216\pi$
$653$	−26.8701	−1.05151	−0.525753	+	0.850637i	$0.676216\pi$
$654$	0	0
$655$	− 32.0000i	− 1.25034i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	41.0122i	1.59761i	0.601591	+	0.798804i	$0.294534\pi$
$659$	41.0122i	1.59761i	−0.601591	+	0.798804i	$0.705466\pi$
$660$	0	0
$661$	− 18.0000i	− 0.700119i	−0.936727	−	0.350059i	$-0.886161\pi$
$661$	− 18.0000i	− 0.700119i	0.936727	−	0.350059i	$-0.113839\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 11.3137i	− 0.438727i
$666$	0	0
$667$	2.00000	0.0774403
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−8.48528	−0.327571
$672$	0	0
$673$	−8.00000	−0.308377	−0.154189	−	0.988041i	$-0.549276\pi$
$673$	−8.00000	−0.308377	−0.154189	+	0.988041i	$0.549276\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−22.6274	−0.869642	−0.434821	−	0.900517i	$-0.643188\pi$
$677$	−22.6274	−0.869642	−0.434821	+	0.900517i	$0.643188\pi$
$678$	0	0
$679$	− 2.00000i	− 0.0767530i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	41.0122i	1.56929i	0.619947	+	0.784644i	$0.287154\pi$
$683$	41.0122i	1.56929i	−0.619947	+	0.784644i	$0.712846\pi$
$684$	0	0
$685$	20.0000i	0.764161i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	2.82843i	0.107754i
$690$	0	0
$691$	−8.00000	−0.304334	−0.152167	−	0.988355i	$-0.548625\pi$
$691$	−8.00000	−0.304334	−0.152167	+	0.988355i	$0.548625\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−56.5685	−2.14577
$696$	0	0
$697$	−16.0000	−0.606043
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−29.6985	−1.12170	−0.560848	−	0.827919i	$-0.689525\pi$
$701$	−29.6985	−1.12170	−0.560848	+	0.827919i	$0.689525\pi$
$702$	0	0
$703$	40.0000i	1.50863i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	5.65685i	0.212748i
$708$	0	0
$709$	− 30.0000i	− 1.12667i	−0.826227	−	0.563337i	$-0.809517\pi$
$709$	− 30.0000i	− 1.12667i	0.826227	−	0.563337i	$-0.190483\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	8.00000	0.299183
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−45.2548	−1.68772	−0.843860	−	0.536563i	$-0.819722\pi$
$719$	−45.2548	−1.68772	−0.843860	+	0.536563i	$0.819722\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	0	0
$723$	0	0
$724$	0	0
$725$	4.24264	0.157568
$726$	0	0
$727$	− 16.0000i	− 0.593407i	−0.954970	−	0.296704i	$-0.904113\pi$
$727$	− 16.0000i	− 0.593407i	0.954970	−	0.296704i	$-0.0958873\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	5.65685i	0.209226i
$732$	0	0
$733$	14.0000i	0.517102i	0.965998	+	0.258551i	$0.0832450\pi$
$733$	14.0000i	0.517102i	−0.965998	+	0.258551i	$0.916755\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 5.65685i	− 0.208373i
$738$	0	0
$739$	−28.0000	−1.03000	−0.514998	−	0.857191i	$-0.672207\pi$
$739$	−28.0000	−1.03000	−0.514998	+	0.857191i	$0.672207\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	35.3553	1.29706	0.648531	−	0.761188i	$-0.275384\pi$
$743$	35.3553	1.29706	0.648531	+	0.761188i	$0.275384\pi$
$744$	0	0
$745$	52.0000	1.90513
$746$	0	0
$747$	0	0
$748$	0	0
$749$	18.3848	0.671765
$750$	0	0
$751$	16.0000i	0.583848i	0.956441	+	0.291924i	$0.0942955\pi$
$751$	16.0000i	0.583848i	−0.956441	+	0.291924i	$0.905705\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 45.2548i	− 1.64699i
$756$	0	0
$757$	− 34.0000i	− 1.23575i	−0.786276	−	0.617876i	$-0.787994\pi$
$757$	− 34.0000i	− 1.23575i	0.786276	−	0.617876i	$-0.212006\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	28.2843i	1.02530i	0.858596	+	0.512652i	$0.171337\pi$
$761$	28.2843i	1.02530i	−0.858596	+	0.512652i	$0.828663\pi$
$762$	0	0
$763$	−14.0000	−0.506834
$764$	0	0
$765$	0	0
$766$	0	0
$767$	16.9706	0.612772
$768$	0	0
$769$	−38.0000	−1.37032	−0.685158	−	0.728395i	$-0.740267\pi$
$769$	−38.0000	−1.37032	−0.685158	+	0.728395i	$0.740267\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−5.65685	−0.203463	−0.101731	−	0.994812i	$-0.532438\pi$
$773$	−5.65685	−0.203463	−0.101731	+	0.994812i	$0.532438\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	22.6274i	0.810711i
$780$	0	0
$781$	− 18.0000i	− 0.644091i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 62.2254i	− 2.22092i
$786$	0	0
$787$	−48.0000	−1.71102	−0.855508	−	0.517790i	$-0.826755\pi$
$787$	−48.0000	−1.71102	−0.855508	+	0.517790i	$0.826755\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−1.41421	−0.0502836
$792$	0	0
$793$	12.0000	0.426132
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−11.3137	−0.400752	−0.200376	−	0.979719i	$-0.564216\pi$
$797$	−11.3137	−0.400752	−0.200376	+	0.979719i	$0.564216\pi$
$798$	0	0
$799$	8.00000i	0.283020i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 14.1421i	− 0.499065i
$804$	0	0
$805$	− 4.00000i	− 0.140981i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 26.8701i	− 0.944701i	−0.881411	−	0.472350i	$-0.843406\pi$
$809$	− 26.8701i	− 0.944701i	0.881411	−	0.472350i	$-0.156594\pi$
$810$	0	0
$811$	−12.0000	−0.421377	−0.210688	−	0.977553i	$-0.567571\pi$
$811$	−12.0000	−0.421377	−0.210688	+	0.977553i	$0.567571\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−56.5685	−1.98151
$816$	0	0
$817$	8.00000	0.279885
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−41.0122	−1.43134	−0.715668	−	0.698441i	$-0.753877\pi$
$821$	−41.0122	−1.43134	−0.715668	+	0.698441i	$0.753877\pi$
$822$	0	0
$823$	42.0000i	1.46403i	0.681290	+	0.732014i	$0.261419\pi$
$823$	42.0000i	1.46403i	−0.681290	+	0.732014i	$0.738581\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 46.6690i	− 1.62284i	−0.584462	−	0.811421i	$-0.698695\pi$
$827$	− 46.6690i	− 1.62284i	0.584462	−	0.811421i	$-0.301305\pi$
$828$	0	0
$829$	− 2.00000i	− 0.0694629i	−0.999397	−	0.0347314i	$-0.988942\pi$
$829$	− 2.00000i	− 0.0694629i	0.999397	−	0.0347314i	$-0.0110576\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 2.82843i	− 0.0979992i
$834$	0	0
$835$	56.0000	1.93796
$836$	0	0
$837$	0	0
$838$	0	0
$839$	2.82843	0.0976481	0.0488241	−	0.998807i	$-0.484453\pi$
$839$	2.82843	0.0976481	0.0488241	+	0.998807i	$0.484453\pi$
$840$	0	0
$841$	−27.0000	−0.931034
$842$	0	0
$843$	0	0
$844$	0	0
$845$	25.4558	0.875708
$846$	0	0
$847$	− 9.00000i	− 0.309244i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	14.1421i	0.484786i
$852$	0	0
$853$	− 22.0000i	− 0.753266i	−0.926363	−	0.376633i	$-0.877082\pi$
$853$	− 22.0000i	− 0.753266i	0.926363	−	0.376633i	$-0.122918\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	28.2843i	0.966172i	0.875573	+	0.483086i	$0.160484\pi$
$857$	28.2843i	0.966172i	−0.875573	+	0.483086i	$0.839516\pi$
$858$	0	0
$859$	−40.0000	−1.36478	−0.682391	−	0.730987i	$-0.739060\pi$
$859$	−40.0000	−1.36478	−0.682391	+	0.730987i	$0.739060\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−35.3553	−1.20351	−0.601755	−	0.798681i	$-0.705532\pi$
$863$	−35.3553	−1.20351	−0.601755	+	0.798681i	$0.705532\pi$
$864$	0	0
$865$	−48.0000	−1.63205
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−8.48528	−0.287843
$870$	0	0
$871$	8.00000i	0.271070i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	5.65685i	0.191237i
$876$	0	0
$877$	− 28.0000i	− 0.945493i	−0.881199	−	0.472746i	$-0.843263\pi$
$877$	− 28.0000i	− 0.945493i	0.881199	−	0.472746i	$-0.156737\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 25.4558i	− 0.857629i	−0.903393	−	0.428815i	$-0.858931\pi$
$881$	− 25.4558i	− 0.857629i	0.903393	−	0.428815i	$-0.141069\pi$
$882$	0	0
$883$	−18.0000	−0.605748	−0.302874	−	0.953031i	$-0.597946\pi$
$883$	−18.0000	−0.605748	−0.302874	+	0.953031i	$0.597946\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−19.7990	−0.664785	−0.332393	−	0.943141i	$-0.607856\pi$
$887$	−19.7990	−0.664785	−0.332393	+	0.943141i	$0.607856\pi$
$888$	0	0
$889$	−6.00000	−0.201234
$890$	0	0
$891$	0	0
$892$	0	0
$893$	11.3137	0.378599
$894$	0	0
$895$	28.0000i	0.935937i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	4.00000i	0.133259i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	16.9706i	0.564121i
$906$	0	0
$907$	−10.0000	−0.332045	−0.166022	−	0.986122i	$-0.553092\pi$
$907$	−10.0000	−0.332045	−0.166022	+	0.986122i	$0.553092\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	9.89949	0.327985	0.163992	−	0.986462i	$-0.447563\pi$
$911$	9.89949	0.327985	0.163992	+	0.986462i	$0.447563\pi$
$912$	0	0
$913$	8.00000	0.264761
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−11.3137	−0.373612
$918$	0	0
$919$	56.0000i	1.84727i	0.383274	+	0.923635i	$0.374797\pi$
$919$	56.0000i	1.84727i	−0.383274	+	0.923635i	$0.625203\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	25.4558i	0.837889i
$924$	0	0
$925$	30.0000i	0.986394i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 16.9706i	− 0.556786i	−0.960467	−	0.278393i	$-0.910198\pi$
$929$	− 16.9706i	− 0.556786i	0.960467	−	0.278393i	$-0.0898018\pi$
$930$	0	0
$931$	−4.00000	−0.131095
$932$	0	0
$933$	0	0
$934$	0	0
$935$	11.3137	0.369998
$936$	0	0
$937$	−42.0000	−1.37208	−0.686040	−	0.727564i	$-0.740653\pi$
$937$	−42.0000	−1.37208	−0.686040	+	0.727564i	$0.740653\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	8.48528	0.276612	0.138306	−	0.990390i	$-0.455834\pi$
$941$	8.48528	0.276612	0.138306	+	0.990390i	$0.455834\pi$
$942$	0	0
$943$	8.00000i	0.260516i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 29.6985i	− 0.965071i	−0.875876	−	0.482536i	$-0.839716\pi$
$947$	− 29.6985i	− 0.965071i	0.875876	−	0.482536i	$-0.160284\pi$
$948$	0	0
$949$	20.0000i	0.649227i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 12.7279i	− 0.412298i	−0.978521	−	0.206149i	$-0.933907\pi$
$953$	− 12.7279i	− 0.412298i	0.978521	−	0.206149i	$-0.0660931\pi$
$954$	0	0
$955$	12.0000	0.388311
$956$	0	0
$957$	0	0
$958$	0	0
$959$	7.07107	0.228337
$960$	0	0
$961$	31.0000	1.00000
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−11.3137	−0.364201
$966$	0	0
$967$	− 10.0000i	− 0.321578i	−0.986989	−	0.160789i	$-0.948596\pi$
$967$	− 10.0000i	− 0.321578i	0.986989	−	0.160789i	$-0.0514039\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	56.5685i	1.81537i	0.419651	+	0.907685i	$0.362152\pi$
$971$	56.5685i	1.81537i	−0.419651	+	0.907685i	$0.637848\pi$
$972$	0	0
$973$	20.0000i	0.641171i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 46.6690i	− 1.49308i	−0.665343	−	0.746538i	$-0.731715\pi$
$977$	− 46.6690i	− 1.49308i	0.665343	−	0.746538i	$-0.268285\pi$
$978$	0	0
$979$	16.0000	0.511362
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−56.5685	−1.80426	−0.902128	−	0.431468i	$-0.857996\pi$
$983$	−56.5685	−1.80426	−0.902128	+	0.431468i	$0.857996\pi$
$984$	0	0
$985$	−12.0000	−0.382352
$986$	0	0
$987$	0	0
$988$	0	0
$989$	2.82843	0.0899388
$990$	0	0
$991$	− 16.0000i	− 0.508257i	−0.967170	−	0.254128i	$-0.918211\pi$
$991$	− 16.0000i	− 0.508257i	0.967170	−	0.254128i	$-0.0817886\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	22.6274i	0.717337i
$996$	0	0
$997$	34.0000i	1.07679i	0.842692	+	0.538395i	$0.180969\pi$
$997$	34.0000i	1.07679i	−0.842692	+	0.538395i	$0.819031\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4032.2.j.d.2591.3	yes	4
3.2	odd	2	inner	4032.2.j.d.2591.1	yes	4
4.3	odd	2		4032.2.j.a.2591.4	yes	4
8.3	odd	2	inner	4032.2.j.d.2591.2	yes	4
8.5	even	2		4032.2.j.a.2591.1	✓	4
12.11	even	2		4032.2.j.a.2591.2	yes	4
24.5	odd	2		4032.2.j.a.2591.3	yes	4
24.11	even	2	inner	4032.2.j.d.2591.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4032.2.j.a.2591.1	✓	4	8.5	even	2
4032.2.j.a.2591.2	yes	4	12.11	even	2
4032.2.j.a.2591.3	yes	4	24.5	odd	2
4032.2.j.a.2591.4	yes	4	4.3	odd	2
4032.2.j.d.2591.1	yes	4	3.2	odd	2	inner
4032.2.j.d.2591.2	yes	4	8.3	odd	2	inner
4032.2.j.d.2591.3	yes	4	1.1	even	1	trivial
4032.2.j.d.2591.4	yes	4	24.11	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:	\( N \)	\(=\)	\( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4032.j (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.82843 q^{5} -1.00000i q^{7} +O(q^{10})\)	\(q+2.82843 q^{5} -1.00000i q^{7} -1.41421i q^{11} +2.00000i q^{13} +2.82843i q^{17} +4.00000 q^{19} +1.41421 q^{23} +3.00000 q^{25} +1.41421 q^{29} -2.82843i q^{35} +10.0000i q^{37} +5.65685i q^{41} +2.00000 q^{43} +2.82843 q^{47} -1.00000 q^{49} +1.41421 q^{53} -4.00000i q^{55} -8.48528i q^{59} -6.00000i q^{61} +5.65685i q^{65} +4.00000 q^{67} +12.7279 q^{71} +10.0000 q^{73} -1.41421 q^{77} -6.00000i q^{79} +5.65685i q^{83} +8.00000i q^{85} +11.3137i q^{89} +2.00000 q^{91} +11.3137 q^{95} +2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 16 q^{19} + 12 q^{25} + 8 q^{43} - 4 q^{49} + 16 q^{67} + 40 q^{73} + 8 q^{91} + 8 q^{97}+O(q^{100}) \) 4 * q + 16 * q^19 + 12 * q^25 + 8 * q^43 - 4 * q^49 + 16 * q^67 + 40 * q^73 + 8 * q^91 + 8 * q^97

Introduction

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