LMFDB - Embedded newform 4032.2.h.c.575.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4032,2,Mod(575,4032)]

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, 0, 1, 0]))

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

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

chi := DirichletCharacter("4032.575");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4032 = 2^{6} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4032.h (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:	$32.1956820950$
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:	no (minimal twist has level 2016)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			575.3
Root			$0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	4032.575
Dual form			4032.2.h.c.575.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.82843i q^{5} -1.00000i q^{7} +O(q^{10})$	$q+2.82843i q^{5} -1.00000i q^{7} +1.41421 q^{11} +2.00000 q^{13} -7.07107 q^{23} -3.00000 q^{25} +9.89949i q^{29} +4.00000i q^{31} +2.82843 q^{35} +2.82843i q^{41} -4.00000i q^{43} -2.82843 q^{47} -1.00000 q^{49} +4.24264i q^{53} +4.00000i q^{55} -2.82843 q^{59} +14.0000 q^{61} +5.65685i q^{65} -10.0000i q^{67} +7.07107 q^{71} -6.00000 q^{73} -1.41421i q^{77} +14.0000i q^{79} +11.3137 q^{83} +8.48528i q^{89} -2.00000i q^{91} -14.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q + 8 q^{13} - 12 q^{25} - 4 q^{49} + 56 q^{61} - 24 q^{73} - 56 q^{97}+O(q^{100}) $ 4 * q + 8 * q^13 - 12 * q^25 - 4 * q^49 + 56 * q^61 - 24 * q^73 - 56 * 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
$5$	2.82843i	1.26491i	0.774597	+	0.632456i	$0.217953\pi$
$5$	2.82843i	1.26491i	−0.774597	+	0.632456i	$0.782047\pi$
$6$	0	0
$7$	− 1.00000i	− 0.377964i
$7$	− 1.00000i	− 0.377964i
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.41421	0.426401	0.213201	−	0.977008i	$-0.431611\pi$
$11$	1.41421	0.426401	0.213201	+	0.977008i	$0.431611\pi$
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	0	0
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−7.07107	−1.47442	−0.737210	−	0.675664i	$-0.763857\pi$
$23$	−7.07107	−1.47442	−0.737210	+	0.675664i	$0.763857\pi$
$24$	0	0
$25$	−3.00000	−0.600000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	9.89949i	1.83829i	0.393919	+	0.919145i	$0.371119\pi$
$29$	9.89949i	1.83829i	−0.393919	+	0.919145i	$0.628881\pi$
$30$	0	0
$31$	4.00000i	0.718421i	0.933257	+	0.359211i	$0.116954\pi$
$31$	4.00000i	0.718421i	−0.933257	+	0.359211i	$0.883046\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.82843	0.478091
$36$	0	0
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	2.82843i	0.441726i	0.975305	+	0.220863i	$0.0708874\pi$
$41$	2.82843i	0.441726i	−0.975305	+	0.220863i	$0.929113\pi$
$42$	0	0
$43$	− 4.00000i	− 0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$	− 4.00000i	− 0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−2.82843	−0.412568	−0.206284	−	0.978492i	$-0.566137\pi$
$47$	−2.82843	−0.412568	−0.206284	+	0.978492i	$0.566137\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	4.24264i	0.582772i	0.956606	+	0.291386i	$0.0941163\pi$
$53$	4.24264i	0.582772i	−0.956606	+	0.291386i	$0.905884\pi$
$54$	0	0
$55$	4.00000i	0.539360i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−2.82843	−0.368230	−0.184115	−	0.982905i	$-0.558942\pi$
$59$	−2.82843	−0.368230	−0.184115	+	0.982905i	$0.558942\pi$
$60$	0	0
$61$	14.0000	1.79252	0.896258	−	0.443533i	$-0.146275\pi$
$61$	14.0000	1.79252	0.896258	+	0.443533i	$0.146275\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	5.65685i	0.701646i
$66$	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
$71$	7.07107	0.839181	0.419591	−	0.907713i	$-0.362174\pi$
$71$	7.07107	0.839181	0.419591	+	0.907713i	$0.362174\pi$
$72$	0	0
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 1.41421i	− 0.161165i
$78$	0	0
$79$	14.0000i	1.57512i	0.616236	+	0.787562i	$0.288657\pi$
$79$	14.0000i	1.57512i	−0.616236	+	0.787562i	$0.711343\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	11.3137	1.24184	0.620920	−	0.783874i	$-0.286759\pi$
$83$	11.3137	1.24184	0.620920	+	0.783874i	$0.286759\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	8.48528i	0.899438i	0.893170	+	0.449719i	$0.148476\pi$
$89$	8.48528i	0.899438i	−0.893170	+	0.449719i	$0.851524\pi$
$90$	0	0
$91$	− 2.00000i	− 0.209657i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−14.0000	−1.42148	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000	−1.42148	−0.710742	+	0.703452i	$0.751641\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	11.3137i	1.12576i	0.826540	+	0.562878i	$0.190306\pi$
$101$	11.3137i	1.12576i	−0.826540	+	0.562878i	$0.809694\pi$
$102$	0	0
$103$	− 16.0000i	− 1.57653i	−0.615338	−	0.788263i	$-0.710980\pi$
$103$	− 16.0000i	− 1.57653i	0.615338	−	0.788263i	$-0.289020\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−4.24264	−0.410152	−0.205076	−	0.978746i	$-0.565744\pi$
$107$	−4.24264	−0.410152	−0.205076	+	0.978746i	$0.565744\pi$
$108$	0	0
$109$	−12.0000	−1.14939	−0.574696	−	0.818367i	$-0.694880\pi$
$109$	−12.0000	−1.14939	−0.574696	+	0.818367i	$0.694880\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 7.07107i	− 0.665190i	−0.943070	−	0.332595i	$-0.892076\pi$
$113$	− 7.07107i	− 0.665190i	0.943070	−	0.332595i	$-0.107924\pi$
$114$	0	0
$115$	− 20.0000i	− 1.86501i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−9.00000	−0.818182
$122$	0	0
$123$	0	0
$124$	0	0
$125$	5.65685i	0.505964i
$126$	0	0
$127$	18.0000i	1.59724i	0.601834	+	0.798621i	$0.294437\pi$
$127$	18.0000i	1.59724i	−0.601834	+	0.798621i	$0.705563\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	5.65685	0.494242	0.247121	−	0.968985i	$-0.420516\pi$
$131$	5.65685	0.494242	0.247121	+	0.968985i	$0.420516\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 1.41421i	− 0.120824i	−0.998174	−	0.0604122i	$-0.980758\pi$
$137$	− 1.41421i	− 0.120824i	0.998174	−	0.0604122i	$-0.0192415\pi$
$138$	0	0
$139$	16.0000i	1.35710i	0.734553	+	0.678551i	$0.237392\pi$
$139$	16.0000i	1.35710i	−0.734553	+	0.678551i	$0.762608\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	2.82843	0.236525
$144$	0	0
$145$	−28.0000	−2.32527
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1.41421i	0.115857i	0.998321	+	0.0579284i	$0.0184495\pi$
$149$	1.41421i	0.115857i	−0.998321	+	0.0579284i	$0.981550\pi$
$150$	0	0
$151$	8.00000i	0.651031i	0.945537	+	0.325515i	$0.105538\pi$
$151$	8.00000i	0.651031i	−0.945537	+	0.325515i	$0.894462\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−11.3137	−0.908739
$156$	0	0
$157$	−14.0000	−1.11732	−0.558661	−	0.829396i	$-0.688685\pi$
$157$	−14.0000	−1.11732	−0.558661	+	0.829396i	$0.688685\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	7.07107i	0.557278i
$162$	0	0
$163$	6.00000i	0.469956i	0.972001	+	0.234978i	$0.0755019\pi$
$163$	6.00000i	0.469956i	−0.972001	+	0.234978i	$0.924498\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−14.1421	−1.09435	−0.547176	−	0.837018i	$-0.684297\pi$
$167$	−14.1421	−1.09435	−0.547176	+	0.837018i	$0.684297\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	0	0
$172$	0	0
$173$	11.3137i	0.860165i	0.902790	+	0.430083i	$0.141516\pi$
$173$	11.3137i	0.860165i	−0.902790	+	0.430083i	$0.858484\pi$
$174$	0	0
$175$	3.00000i	0.226779i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	12.7279	0.951330	0.475665	−	0.879627i	$-0.342208\pi$
$179$	12.7279	0.951330	0.475665	+	0.879627i	$0.342208\pi$
$180$	0	0
$181$	−6.00000	−0.445976	−0.222988	−	0.974821i	$-0.571581\pi$
$181$	−6.00000	−0.445976	−0.222988	+	0.974821i	$0.571581\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−24.0416	−1.73959	−0.869796	−	0.493412i	$-0.835749\pi$
$191$	−24.0416	−1.73959	−0.869796	+	0.493412i	$0.835749\pi$
$192$	0	0
$193$	4.00000	0.287926	0.143963	−	0.989583i	$-0.454015\pi$
$193$	4.00000	0.287926	0.143963	+	0.989583i	$0.454015\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	18.3848i	1.30986i	0.755689	+	0.654931i	$0.227302\pi$
$197$	18.3848i	1.30986i	−0.755689	+	0.654931i	$0.772698\pi$
$198$	0	0
$199$	− 20.0000i	− 1.41776i	−0.705328	−	0.708881i	$-0.749200\pi$
$199$	− 20.0000i	− 1.41776i	0.705328	−	0.708881i	$-0.250800\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	9.89949	0.694808
$204$	0	0
$205$	−8.00000	−0.558744
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	20.0000i	1.37686i	0.725304	+	0.688428i	$0.241699\pi$
$211$	20.0000i	1.37686i	−0.725304	+	0.688428i	$0.758301\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	11.3137	0.771589
$216$	0	0
$217$	4.00000	0.271538
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	4.00000i	0.267860i	0.990991	+	0.133930i	$0.0427597\pi$
$223$	4.00000i	0.267860i	−0.990991	+	0.133930i	$0.957240\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	14.1421	0.938647	0.469323	−	0.883026i	$-0.344498\pi$
$227$	14.1421	0.938647	0.469323	+	0.883026i	$0.344498\pi$
$228$	0	0
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	18.3848i	1.20443i	0.798335	+	0.602213i	$0.205714\pi$
$233$	18.3848i	1.20443i	−0.798335	+	0.602213i	$0.794286\pi$
$234$	0	0
$235$	− 8.00000i	− 0.521862i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	18.3848	1.18921	0.594606	−	0.804017i	$-0.297308\pi$
$239$	18.3848	1.18921	0.594606	+	0.804017i	$0.297308\pi$
$240$	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
$245$	− 2.82843i	− 0.180702i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−25.4558	−1.60676	−0.803379	−	0.595468i	$-0.796967\pi$
$251$	−25.4558	−1.60676	−0.803379	+	0.595468i	$0.796967\pi$
$252$	0	0
$253$	−10.0000	−0.628695
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 19.7990i	− 1.23503i	−0.786560	−	0.617514i	$-0.788140\pi$
$257$	− 19.7990i	− 1.23503i	0.786560	−	0.617514i	$-0.211860\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−1.41421	−0.0872041	−0.0436021	−	0.999049i	$-0.513883\pi$
$263$	−1.41421	−0.0872041	−0.0436021	+	0.999049i	$0.513883\pi$
$264$	0	0
$265$	−12.0000	−0.737154
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 8.48528i	− 0.517357i	−0.965964	−	0.258678i	$-0.916713\pi$
$269$	− 8.48528i	− 0.517357i	0.965964	−	0.258678i	$-0.0832870\pi$
$270$	0	0
$271$	− 24.0000i	− 1.45790i	−0.684569	−	0.728948i	$-0.740010\pi$
$271$	− 24.0000i	− 1.45790i	0.684569	−	0.728948i	$-0.259990\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−4.24264	−0.255841
$276$	0	0
$277$	−2.00000	−0.120168	−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000	−0.120168	−0.0600842	+	0.998193i	$0.519137\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 9.89949i	− 0.590554i	−0.955412	−	0.295277i	$-0.904588\pi$
$281$	− 9.89949i	− 0.590554i	0.955412	−	0.295277i	$-0.0954120\pi$
$282$	0	0
$283$	− 32.0000i	− 1.90220i	−0.308879	−	0.951101i	$-0.599954\pi$
$283$	− 32.0000i	− 1.90220i	0.308879	−	0.951101i	$-0.400046\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.82843	0.166957
$288$	0	0
$289$	17.0000	1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 5.65685i	− 0.330477i	−0.986254	−	0.165238i	$-0.947161\pi$
$293$	− 5.65685i	− 0.330477i	0.986254	−	0.165238i	$-0.0528394\pi$
$294$	0	0
$295$	− 8.00000i	− 0.465778i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−14.1421	−0.817861
$300$	0	0
$301$	−4.00000	−0.230556
$302$	0	0
$303$	0	0
$304$	0	0
$305$	39.5980i	2.26737i
$306$	0	0
$307$	12.0000i	0.684876i	0.939540	+	0.342438i	$0.111253\pi$
$307$	12.0000i	0.684876i	−0.939540	+	0.342438i	$0.888747\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−19.7990	−1.12270	−0.561349	−	0.827579i	$-0.689717\pi$
$311$	−19.7990	−1.12270	−0.561349	+	0.827579i	$0.689717\pi$
$312$	0	0
$313$	−10.0000	−0.565233	−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000	−0.565233	−0.282617	+	0.959233i	$0.591202\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	9.89949i	0.556011i	0.960579	+	0.278006i	$0.0896734\pi$
$317$	9.89949i	0.556011i	−0.960579	+	0.278006i	$0.910327\pi$
$318$	0	0
$319$	14.0000i	0.783850i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−6.00000	−0.332820
$326$	0	0
$327$	0	0
$328$	0	0
$329$	2.82843i	0.155936i
$330$	0	0
$331$	4.00000i	0.219860i	0.993939	+	0.109930i	$0.0350627\pi$
$331$	4.00000i	0.219860i	−0.993939	+	0.109930i	$0.964937\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	28.2843	1.54533
$336$	0	0
$337$	−4.00000	−0.217894	−0.108947	−	0.994048i	$-0.534748\pi$
$337$	−4.00000	−0.217894	−0.108947	+	0.994048i	$0.534748\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	5.65685i	0.306336i
$342$	0	0
$343$	1.00000i	0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	7.07107	0.379595	0.189797	−	0.981823i	$-0.439217\pi$
$347$	7.07107	0.379595	0.189797	+	0.981823i	$0.439217\pi$
$348$	0	0
$349$	10.0000	0.535288	0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000	0.535288	0.267644	+	0.963518i	$0.413755\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	28.2843i	1.50542i	0.658352	+	0.752710i	$0.271254\pi$
$353$	28.2843i	1.50542i	−0.658352	+	0.752710i	$0.728746\pi$
$354$	0	0
$355$	20.0000i	1.06149i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−12.7279	−0.671754	−0.335877	−	0.941906i	$-0.609033\pi$
$359$	−12.7279	−0.671754	−0.335877	+	0.941906i	$0.609033\pi$
$360$	0	0
$361$	19.0000	1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 16.9706i	− 0.888280i
$366$	0	0
$367$	− 16.0000i	− 0.835193i	−0.908633	−	0.417597i	$-0.862873\pi$
$367$	− 16.0000i	− 0.835193i	0.908633	−	0.417597i	$-0.137127\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	4.24264	0.220267
$372$	0	0
$373$	−10.0000	−0.517780	−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000	−0.517780	−0.258890	+	0.965907i	$0.583357\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	19.7990i	1.01970i
$378$	0	0
$379$	20.0000i	1.02733i	0.857991	+	0.513665i	$0.171713\pi$
$379$	20.0000i	1.02733i	−0.857991	+	0.513665i	$0.828287\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	28.2843	1.44526	0.722629	−	0.691236i	$-0.242933\pi$
$383$	28.2843	1.44526	0.722629	+	0.691236i	$0.242933\pi$
$384$	0	0
$385$	4.00000	0.203859
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 9.89949i	− 0.501924i	−0.967997	−	0.250962i	$-0.919253\pi$
$389$	− 9.89949i	− 0.501924i	0.967997	−	0.250962i	$-0.0807470\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−39.5980	−1.99239
$396$	0	0
$397$	30.0000	1.50566	0.752828	−	0.658217i	$-0.228689\pi$
$397$	30.0000	1.50566	0.752828	+	0.658217i	$0.228689\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 1.41421i	− 0.0706225i	−0.999376	−	0.0353112i	$-0.988758\pi$
$401$	− 1.41421i	− 0.0706225i	0.999376	−	0.0353112i	$-0.0112422\pi$
$402$	0	0
$403$	8.00000i	0.398508i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	14.0000	0.692255	0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000	0.692255	0.346128	+	0.938187i	$0.387496\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	2.82843i	0.139178i
$414$	0	0
$415$	32.0000i	1.57082i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	2.82843	0.138178	0.0690889	−	0.997611i	$-0.477991\pi$
$419$	2.82843	0.138178	0.0690889	+	0.997611i	$0.477991\pi$
$420$	0	0
$421$	26.0000	1.26716	0.633581	−	0.773676i	$-0.281584\pi$
$421$	26.0000	1.26716	0.633581	+	0.773676i	$0.281584\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 14.0000i	− 0.677507i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	38.1838	1.83925	0.919624	−	0.392801i	$-0.128494\pi$
$431$	38.1838	1.83925	0.919624	+	0.392801i	$0.128494\pi$
$432$	0	0
$433$	18.0000	0.865025	0.432512	−	0.901628i	$-0.357627\pi$
$433$	18.0000	0.865025	0.432512	+	0.901628i	$0.357627\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	− 40.0000i	− 1.90910i	−0.298057	−	0.954548i	$-0.596339\pi$
$439$	− 40.0000i	− 1.90910i	0.298057	−	0.954548i	$-0.403661\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−26.8701	−1.27663	−0.638317	−	0.769773i	$-0.720369\pi$
$443$	−26.8701	−1.27663	−0.638317	+	0.769773i	$0.720369\pi$
$444$	0	0
$445$	−24.0000	−1.13771
$446$	0	0
$447$	0	0
$448$	0	0
$449$	38.1838i	1.80200i	0.433816	+	0.901002i	$0.357167\pi$
$449$	38.1838i	1.80200i	−0.433816	+	0.901002i	$0.642833\pi$
$450$	0	0
$451$	4.00000i	0.188353i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	5.65685	0.265197
$456$	0	0
$457$	40.0000	1.87112	0.935561	−	0.353166i	$-0.114895\pi$
$457$	40.0000	1.87112	0.935561	+	0.353166i	$0.114895\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	22.6274i	1.05386i	0.849907	+	0.526932i	$0.176658\pi$
$461$	22.6274i	1.05386i	−0.849907	+	0.526932i	$0.823342\pi$
$462$	0	0
$463$	− 26.0000i	− 1.20832i	−0.796862	−	0.604161i	$-0.793508\pi$
$463$	− 26.0000i	− 1.20832i	0.796862	−	0.604161i	$-0.206492\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−25.4558	−1.17796	−0.588978	−	0.808149i	$-0.700470\pi$
$467$	−25.4558	−1.17796	−0.588978	+	0.808149i	$0.700470\pi$
$468$	0	0
$469$	−10.0000	−0.461757
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 5.65685i	− 0.260102i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−31.1127	−1.42158	−0.710788	−	0.703407i	$-0.751661\pi$
$479$	−31.1127	−1.42158	−0.710788	+	0.703407i	$0.751661\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 39.5980i	− 1.79805i
$486$	0	0
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	26.8701	1.21263	0.606314	−	0.795225i	$-0.292647\pi$
$491$	26.8701	1.21263	0.606314	+	0.795225i	$0.292647\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 7.07107i	− 0.317181i
$498$	0	0
$499$	4.00000i	0.179065i	0.995984	+	0.0895323i	$0.0285372\pi$
$499$	4.00000i	0.179065i	−0.995984	+	0.0895323i	$0.971463\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	5.65685	0.252227	0.126113	−	0.992016i	$-0.459750\pi$
$503$	5.65685	0.252227	0.126113	+	0.992016i	$0.459750\pi$
$504$	0	0
$505$	−32.0000	−1.42398
$506$	0	0
$507$	0	0
$508$	0	0
$509$	14.1421i	0.626839i	0.949615	+	0.313420i	$0.101475\pi$
$509$	14.1421i	0.626839i	−0.949615	+	0.313420i	$0.898525\pi$
$510$	0	0
$511$	6.00000i	0.265424i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	45.2548	1.99417
$516$	0	0
$517$	−4.00000	−0.175920
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 5.65685i	− 0.247831i	−0.992293	−	0.123916i	$-0.960455\pi$
$521$	− 5.65685i	− 0.247831i	0.992293	−	0.123916i	$-0.0395452\pi$
$522$	0	0
$523$	16.0000i	0.699631i	0.936819	+	0.349816i	$0.113756\pi$
$523$	16.0000i	0.699631i	−0.936819	+	0.349816i	$0.886244\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	27.0000	1.17391
$530$	0	0
$531$	0	0
$532$	0	0
$533$	5.65685i	0.245026i
$534$	0	0
$535$	− 12.0000i	− 0.518805i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−1.41421	−0.0609145
$540$	0	0
$541$	−34.0000	−1.46177	−0.730887	−	0.682498i	$-0.760893\pi$
$541$	−34.0000	−1.46177	−0.730887	+	0.682498i	$0.760893\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 33.9411i	− 1.45388i
$546$	0	0
$547$	34.0000i	1.45374i	0.686778	+	0.726868i	$0.259025\pi$
$547$	34.0000i	1.45374i	−0.686778	+	0.726868i	$0.740975\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	14.0000	0.595341
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 21.2132i	− 0.898832i	−0.893323	−	0.449416i	$-0.851632\pi$
$557$	− 21.2132i	− 0.898832i	0.893323	−	0.449416i	$-0.148368\pi$
$558$	0	0
$559$	− 8.00000i	− 0.338364i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−14.1421	−0.596020	−0.298010	−	0.954563i	$-0.596323\pi$
$563$	−14.1421	−0.596020	−0.298010	+	0.954563i	$0.596323\pi$
$564$	0	0
$565$	20.0000	0.841406
$566$	0	0
$567$	0	0
$568$	0	0
$569$	1.41421i	0.0592869i	0.999561	+	0.0296435i	$0.00943719\pi$
$569$	1.41421i	0.0592869i	−0.999561	+	0.0296435i	$0.990563\pi$
$570$	0	0
$571$	− 10.0000i	− 0.418487i	−0.977864	−	0.209243i	$-0.932900\pi$
$571$	− 10.0000i	− 0.418487i	0.977864	−	0.209243i	$-0.0671001\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	21.2132	0.884652
$576$	0	0
$577$	−6.00000	−0.249783	−0.124892	−	0.992170i	$-0.539858\pi$
$577$	−6.00000	−0.249783	−0.124892	+	0.992170i	$0.539858\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 11.3137i	− 0.469372i
$582$	0	0
$583$	6.00000i	0.248495i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	19.7990	0.817192	0.408596	−	0.912715i	$-0.366019\pi$
$587$	19.7990	0.817192	0.408596	+	0.912715i	$0.366019\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 33.9411i	− 1.39379i	−0.717171	−	0.696897i	$-0.754563\pi$
$593$	− 33.9411i	− 1.39379i	0.717171	−	0.696897i	$-0.245437\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−18.3848	−0.751182	−0.375591	−	0.926786i	$-0.622560\pi$
$599$	−18.3848	−0.751182	−0.375591	+	0.926786i	$0.622560\pi$
$600$	0	0
$601$	46.0000	1.87638	0.938190	−	0.346122i	$-0.112502\pi$
$601$	46.0000	1.87638	0.938190	+	0.346122i	$0.112502\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 25.4558i	− 1.03493i
$606$	0	0
$607$	8.00000i	0.324710i	0.986732	+	0.162355i	$0.0519090\pi$
$607$	8.00000i	0.324710i	−0.986732	+	0.162355i	$0.948091\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−5.65685	−0.228852
$612$	0	0
$613$	−6.00000	−0.242338	−0.121169	−	0.992632i	$-0.538664\pi$
$613$	−6.00000	−0.242338	−0.121169	+	0.992632i	$0.538664\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 9.89949i	− 0.398539i	−0.979945	−	0.199269i	$-0.936143\pi$
$617$	− 9.89949i	− 0.398539i	0.979945	−	0.199269i	$-0.0638569\pi$
$618$	0	0
$619$	4.00000i	0.160774i	0.996764	+	0.0803868i	$0.0256155\pi$
$619$	4.00000i	0.160774i	−0.996764	+	0.0803868i	$0.974384\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	8.48528	0.339956
$624$	0	0
$625$	−31.0000	−1.24000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	− 6.00000i	− 0.238856i	−0.992843	−	0.119428i	$-0.961894\pi$
$631$	− 6.00000i	− 0.238856i	0.992843	−	0.119428i	$-0.0381061\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−50.9117	−2.02037
$636$	0	0
$637$	−2.00000	−0.0792429
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 15.5563i	− 0.614439i	−0.951639	−	0.307219i	$-0.900601\pi$
$641$	− 15.5563i	− 0.614439i	0.951639	−	0.307219i	$-0.0993986\pi$
$642$	0	0
$643$	32.0000i	1.26196i	0.775800	+	0.630978i	$0.217346\pi$
$643$	32.0000i	1.26196i	−0.775800	+	0.630978i	$0.782654\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−42.4264	−1.66795	−0.833977	−	0.551799i	$-0.813942\pi$
$647$	−42.4264	−1.66795	−0.833977	+	0.551799i	$0.813942\pi$
$648$	0	0
$649$	−4.00000	−0.157014
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 1.41421i	− 0.0553425i	−0.999617	−	0.0276712i	$-0.991191\pi$
$653$	− 1.41421i	− 0.0553425i	0.999617	−	0.0276712i	$-0.00880915\pi$
$654$	0	0
$655$	16.0000i	0.625172i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	26.8701	1.04671	0.523354	−	0.852115i	$-0.324680\pi$
$659$	26.8701	1.04671	0.523354	+	0.852115i	$0.324680\pi$
$660$	0	0
$661$	−2.00000	−0.0777910	−0.0388955	−	0.999243i	$-0.512384\pi$
$661$	−2.00000	−0.0777910	−0.0388955	+	0.999243i	$0.512384\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 70.0000i	− 2.71041i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	19.7990	0.764332
$672$	0	0
$673$	40.0000	1.54189	0.770943	−	0.636904i	$-0.219785\pi$
$673$	40.0000	1.54189	0.770943	+	0.636904i	$0.219785\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	45.2548i	1.73928i	0.493682	+	0.869642i	$0.335651\pi$
$677$	45.2548i	1.73928i	−0.493682	+	0.869642i	$0.664349\pi$
$678$	0	0
$679$	14.0000i	0.537271i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	38.1838	1.46106	0.730531	−	0.682880i	$-0.239273\pi$
$683$	38.1838	1.46106	0.730531	+	0.682880i	$0.239273\pi$
$684$	0	0
$685$	4.00000	0.152832
$686$	0	0
$687$	0	0
$688$	0	0
$689$	8.48528i	0.323263i
$690$	0	0
$691$	20.0000i	0.760836i	0.924815	+	0.380418i	$0.124220\pi$
$691$	20.0000i	0.760836i	−0.924815	+	0.380418i	$0.875780\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−45.2548	−1.71661
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 46.6690i	− 1.76267i	−0.472496	−	0.881333i	$-0.656647\pi$
$701$	− 46.6690i	− 1.76267i	0.472496	−	0.881333i	$-0.343353\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	11.3137	0.425496
$708$	0	0
$709$	−20.0000	−0.751116	−0.375558	−	0.926799i	$-0.622549\pi$
$709$	−20.0000	−0.751116	−0.375558	+	0.926799i	$0.622549\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 28.2843i	− 1.05925i
$714$	0	0
$715$	8.00000i	0.299183i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	50.9117	1.89869	0.949343	−	0.314242i	$-0.101750\pi$
$719$	50.9117	1.89869	0.949343	+	0.314242i	$0.101750\pi$
$720$	0	0
$721$	−16.0000	−0.595871
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 29.6985i	− 1.10297i
$726$	0	0
$727$	8.00000i	0.296704i	0.988935	+	0.148352i	$0.0473968\pi$
$727$	8.00000i	0.296704i	−0.988935	+	0.148352i	$0.952603\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	26.0000	0.960332	0.480166	−	0.877178i	$-0.340576\pi$
$733$	26.0000	0.960332	0.480166	+	0.877178i	$0.340576\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 14.1421i	− 0.520932i
$738$	0	0
$739$	34.0000i	1.25071i	0.780340	+	0.625355i	$0.215046\pi$
$739$	34.0000i	1.25071i	−0.780340	+	0.625355i	$0.784954\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−18.3848	−0.674472	−0.337236	−	0.941420i	$-0.609492\pi$
$743$	−18.3848	−0.674472	−0.337236	+	0.941420i	$0.609492\pi$
$744$	0	0
$745$	−4.00000	−0.146549
$746$	0	0
$747$	0	0
$748$	0	0
$749$	4.24264i	0.155023i
$750$	0	0
$751$	− 40.0000i	− 1.45962i	−0.683650	−	0.729810i	$-0.739608\pi$
$751$	− 40.0000i	− 1.45962i	0.683650	−	0.729810i	$-0.260392\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−22.6274	−0.823496
$756$	0	0
$757$	52.0000	1.88997	0.944986	−	0.327111i	$-0.106075\pi$
$757$	52.0000	1.88997	0.944986	+	0.327111i	$0.106075\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 31.1127i	− 1.12783i	−0.825831	−	0.563917i	$-0.809294\pi$
$761$	− 31.1127i	− 1.12783i	0.825831	−	0.563917i	$-0.190706\pi$
$762$	0	0
$763$	12.0000i	0.434429i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−5.65685	−0.204257
$768$	0	0
$769$	−34.0000	−1.22607	−0.613036	−	0.790055i	$-0.710052\pi$
$769$	−34.0000	−1.22607	−0.613036	+	0.790055i	$0.710052\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 50.9117i	− 1.83117i	−0.402129	−	0.915583i	$-0.631730\pi$
$773$	− 50.9117i	− 1.83117i	0.402129	−	0.915583i	$-0.368270\pi$
$774$	0	0
$775$	− 12.0000i	− 0.431053i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	10.0000	0.357828
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 39.5980i	− 1.41331i
$786$	0	0
$787$	− 44.0000i	− 1.56843i	−0.620489	−	0.784215i	$-0.713066\pi$
$787$	− 44.0000i	− 1.56843i	0.620489	−	0.784215i	$-0.286934\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−7.07107	−0.251418
$792$	0	0
$793$	28.0000	0.994309
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 11.3137i	− 0.400752i	−0.979719	−	0.200376i	$-0.935784\pi$
$797$	− 11.3137i	− 0.400752i	0.979719	−	0.200376i	$-0.0642164\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−8.48528	−0.299439
$804$	0	0
$805$	−20.0000	−0.704907
$806$	0	0
$807$	0	0
$808$	0	0
$809$	12.7279i	0.447490i	0.974648	+	0.223745i	$0.0718283\pi$
$809$	12.7279i	0.447490i	−0.974648	+	0.223745i	$0.928172\pi$
$810$	0	0
$811$	− 36.0000i	− 1.26413i	−0.774915	−	0.632065i	$-0.782207\pi$
$811$	− 36.0000i	− 1.26413i	0.774915	−	0.632065i	$-0.217793\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−16.9706	−0.594453
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 32.5269i	− 1.13520i	−0.823305	−	0.567599i	$-0.807873\pi$
$821$	− 32.5269i	− 1.13520i	0.823305	−	0.567599i	$-0.192127\pi$
$822$	0	0
$823$	46.0000i	1.60346i	0.597687	+	0.801730i	$0.296087\pi$
$823$	46.0000i	1.60346i	−0.597687	+	0.801730i	$0.703913\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−12.7279	−0.442593	−0.221297	−	0.975207i	$-0.571029\pi$
$827$	−12.7279	−0.442593	−0.221297	+	0.975207i	$0.571029\pi$
$828$	0	0
$829$	−6.00000	−0.208389	−0.104194	−	0.994557i	$-0.533226\pi$
$829$	−6.00000	−0.208389	−0.104194	+	0.994557i	$0.533226\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	− 40.0000i	− 1.38426i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	31.1127	1.07413	0.537065	−	0.843541i	$-0.319533\pi$
$839$	31.1127	1.07413	0.537065	+	0.843541i	$0.319533\pi$
$840$	0	0
$841$	−69.0000	−2.37931
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 25.4558i	− 0.875708i
$846$	0	0
$847$	9.00000i	0.309244i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	14.0000	0.479351	0.239675	−	0.970853i	$-0.422959\pi$
$853$	14.0000	0.479351	0.239675	+	0.970853i	$0.422959\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 14.1421i	− 0.483086i	−0.970390	−	0.241543i	$-0.922346\pi$
$857$	− 14.1421i	− 0.483086i	0.970390	−	0.241543i	$-0.0776535\pi$
$858$	0	0
$859$	− 28.0000i	− 0.955348i	−0.878537	−	0.477674i	$-0.841480\pi$
$859$	− 28.0000i	− 0.955348i	0.878537	−	0.477674i	$-0.158520\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	38.1838	1.29979	0.649895	−	0.760024i	$-0.274813\pi$
$863$	38.1838	1.29979	0.649895	+	0.760024i	$0.274813\pi$
$864$	0	0
$865$	−32.0000	−1.08803
$866$	0	0
$867$	0	0
$868$	0	0
$869$	19.7990i	0.671635i
$870$	0	0
$871$	− 20.0000i	− 0.677674i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	5.65685	0.191237
$876$	0	0
$877$	38.0000	1.28317	0.641584	−	0.767052i	$-0.278277\pi$
$877$	38.0000	1.28317	0.641584	+	0.767052i	$0.278277\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	0	0
$883$	36.0000i	1.21150i	0.795656	+	0.605748i	$0.207126\pi$
$883$	36.0000i	1.21150i	−0.795656	+	0.605748i	$0.792874\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−36.7696	−1.23460	−0.617300	−	0.786728i	$-0.711774\pi$
$887$	−36.7696	−1.23460	−0.617300	+	0.786728i	$0.711774\pi$
$888$	0	0
$889$	18.0000	0.603701
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	36.0000i	1.20335i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−39.5980	−1.32067
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 16.9706i	− 0.564121i
$906$	0	0
$907$	− 28.0000i	− 0.929725i	−0.885383	−	0.464862i	$-0.846104\pi$
$907$	− 28.0000i	− 0.929725i	0.885383	−	0.464862i	$-0.153896\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	41.0122	1.35879	0.679397	−	0.733771i	$-0.262241\pi$
$911$	41.0122	1.35879	0.679397	+	0.733771i	$0.262241\pi$
$912$	0	0
$913$	16.0000	0.529523
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 5.65685i	− 0.186806i
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	14.1421	0.465494
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 31.1127i	− 1.02077i	−0.859945	−	0.510387i	$-0.829502\pi$
$929$	− 31.1127i	− 1.02077i	0.859945	−	0.510387i	$-0.170498\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−30.0000	−0.980057	−0.490029	−	0.871706i	$-0.663014\pi$
$937$	−30.0000	−0.980057	−0.490029	+	0.871706i	$0.663014\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	19.7990i	0.645429i	0.946496	+	0.322714i	$0.104595\pi$
$941$	19.7990i	0.645429i	−0.946496	+	0.322714i	$0.895405\pi$
$942$	0	0
$943$	− 20.0000i	− 0.651290i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	52.3259	1.70036	0.850182	−	0.526489i	$-0.176492\pi$
$947$	52.3259	1.70036	0.850182	+	0.526489i	$0.176492\pi$
$948$	0	0
$949$	−12.0000	−0.389536
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 21.2132i	− 0.687163i	−0.939123	−	0.343582i	$-0.888360\pi$
$953$	− 21.2132i	− 0.687163i	0.939123	−	0.343582i	$-0.111640\pi$
$954$	0	0
$955$	− 68.0000i	− 2.20043i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−1.41421	−0.0456673
$960$	0	0
$961$	15.0000	0.483871
$962$	0	0
$963$	0	0
$964$	0	0
$965$	11.3137i	0.364201i
$966$	0	0
$967$	22.0000i	0.707472i	0.935345	+	0.353736i	$0.115089\pi$
$967$	22.0000i	0.707472i	−0.935345	+	0.353736i	$0.884911\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−5.65685	−0.181537	−0.0907685	−	0.995872i	$-0.528932\pi$
$971$	−5.65685	−0.181537	−0.0907685	+	0.995872i	$0.528932\pi$
$972$	0	0
$973$	16.0000	0.512936
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1.41421i	0.0452447i	0.999744	+	0.0226224i	$0.00720153\pi$
$977$	1.41421i	0.0452447i	−0.999744	+	0.0226224i	$0.992798\pi$
$978$	0	0
$979$	12.0000i	0.383522i
$980$	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
$985$	−52.0000	−1.65686
$986$	0	0
$987$	0	0
$988$	0	0
$989$	28.2843i	0.899388i
$990$	0	0
$991$	48.0000i	1.52477i	0.647124	+	0.762385i	$0.275972\pi$
$991$	48.0000i	1.52477i	−0.647124	+	0.762385i	$0.724028\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	56.5685	1.79334
$996$	0	0
$997$	58.0000	1.83688	0.918439	−	0.395562i	$-0.129450\pi$
$997$	58.0000	1.83688	0.918439	+	0.395562i	$0.129450\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4032.2.h.c.575.3		4
3.2	odd	2	inner	4032.2.h.c.575.1		4
4.3	odd	2	inner	4032.2.h.c.575.4		4
8.3	odd	2		2016.2.h.a.575.2	yes	4
8.5	even	2		2016.2.h.a.575.1	✓	4
12.11	even	2	inner	4032.2.h.c.575.2		4
24.5	odd	2		2016.2.h.a.575.3	yes	4
24.11	even	2		2016.2.h.a.575.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2016.2.h.a.575.1	✓	4	8.5	even	2
2016.2.h.a.575.2	yes	4	8.3	odd	2
2016.2.h.a.575.3	yes	4	24.5	odd	2
2016.2.h.a.575.4	yes	4	24.11	even	2
4032.2.h.c.575.1		4	3.2	odd	2	inner
4032.2.h.c.575.2		4	12.11	even	2	inner
4032.2.h.c.575.3		4	1.1	even	1	trivial
4032.2.h.c.575.4		4	4.3	odd	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.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.82843i q^{5} -1.00000i q^{7} +O(q^{10})\)	\(q+2.82843i q^{5} -1.00000i q^{7} +1.41421 q^{11} +2.00000 q^{13} -7.07107 q^{23} -3.00000 q^{25} +9.89949i q^{29} +4.00000i q^{31} +2.82843 q^{35} +2.82843i q^{41} -4.00000i q^{43} -2.82843 q^{47} -1.00000 q^{49} +4.24264i q^{53} +4.00000i q^{55} -2.82843 q^{59} +14.0000 q^{61} +5.65685i q^{65} -10.0000i q^{67} +7.07107 q^{71} -6.00000 q^{73} -1.41421i q^{77} +14.0000i q^{79} +11.3137 q^{83} +8.48528i q^{89} -2.00000i q^{91} -14.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 8 q^{13} - 12 q^{25} - 4 q^{49} + 56 q^{61} - 24 q^{73} - 56 q^{97}+O(q^{100}) \) 4 * q + 8 * q^13 - 12 * q^25 - 4 * q^49 + 56 * q^61 - 24 * q^73 - 56 * q^97

Introduction

L-functions

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