LMFDB - Embedded newform 2016.2.h.b.575.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2016,2,Mod(575,2016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2016, 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("2016.575");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2016 = 2^{5} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2016.h (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:	$16.0978410475$
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:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			575.1
Root			$-0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	2016.575
Dual form			2016.2.h.b.575.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{7} +O(q^{10})$	$q-1.00000i q^{7} -4.24264 q^{11} -2.00000 q^{13} -2.82843i q^{17} +4.00000i q^{19} +1.41421 q^{23} +5.00000 q^{25} +7.07107i q^{29} +8.00000i q^{31} +8.00000 q^{37} +5.65685i q^{41} +4.00000i q^{43} -2.82843 q^{47} -1.00000 q^{49} +7.07107i q^{53} +2.82843 q^{59} -2.00000 q^{61} -2.00000i q^{67} -12.7279 q^{71} +14.0000 q^{73} +4.24264i q^{77} +2.00000i q^{79} -5.65685 q^{83} -11.3137i q^{89} +2.00000i q^{91} +14.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 8 q^{13} + 20 q^{25} + 32 q^{37} - 4 q^{49} - 8 q^{61} + 56 q^{73} + 56 q^{97}+O(q^{100}) $ 4 * q - 8 * q^13 + 20 * q^25 + 32 * q^37 - 4 * q^49 - 8 * q^61 + 56 * q^73 + 56 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2016\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$1765$	$1793$
$\chi(n)$	$-1$	$1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	0	0
$7$	− 1.00000i	− 0.377964i
$7$	− 1.00000i	− 0.377964i
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−4.24264	−1.27920	−0.639602	−	0.768706i	$-0.720901\pi$
$11$	−4.24264	−1.27920	−0.639602	+	0.768706i	$0.720901\pi$
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 2.82843i	− 0.685994i	−0.939336	−	0.342997i	$-0.888558\pi$
$17$	− 2.82843i	− 0.685994i	0.939336	−	0.342997i	$-0.111442\pi$
$18$	0	0
$19$	4.00000i	0.917663i	0.888523	+	0.458831i	$0.151732\pi$
$19$	4.00000i	0.917663i	−0.888523	+	0.458831i	$0.848268\pi$
$20$	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
$25$	5.00000	1.00000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	7.07107i	1.31306i	0.754298	+	0.656532i	$0.227977\pi$
$29$	7.07107i	1.31306i	−0.754298	+	0.656532i	$0.772023\pi$
$30$	0	0
$31$	8.00000i	1.43684i	0.695608	+	0.718421i	$0.255135\pi$
$31$	8.00000i	1.43684i	−0.695608	+	0.718421i	$0.744865\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	8.00000	1.31519	0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000	1.31519	0.657596	+	0.753371i	$0.271573\pi$
$38$	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
$43$	4.00000i	0.609994i	0.952353	+	0.304997i	$0.0986555\pi$
$43$	4.00000i	0.609994i	−0.952353	+	0.304997i	$0.901344\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−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
$53$	7.07107i	0.971286i	0.874157	+	0.485643i	$0.161414\pi$
$53$	7.07107i	0.971286i	−0.874157	+	0.485643i	$0.838586\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	2.82843	0.368230	0.184115	−	0.982905i	$-0.441058\pi$
$59$	2.82843	0.368230	0.184115	+	0.982905i	$0.441058\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 2.00000i	− 0.244339i	−0.992509	−	0.122169i	$-0.961015\pi$
$67$	− 2.00000i	− 0.244339i	0.992509	−	0.122169i	$-0.0389851\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−12.7279	−1.51053	−0.755263	−	0.655422i	$-0.772491\pi$
$71$	−12.7279	−1.51053	−0.755263	+	0.655422i	$0.772491\pi$
$72$	0	0
$73$	14.0000	1.63858	0.819288	−	0.573382i	$-0.194369\pi$
$73$	14.0000	1.63858	0.819288	+	0.573382i	$0.194369\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4.24264i	0.483494i
$78$	0	0
$79$	2.00000i	0.225018i	0.993651	+	0.112509i	$0.0358886\pi$
$79$	2.00000i	0.225018i	−0.993651	+	0.112509i	$0.964111\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−5.65685	−0.620920	−0.310460	−	0.950586i	$-0.600483\pi$
$83$	−5.65685	−0.620920	−0.310460	+	0.950586i	$0.600483\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 11.3137i	− 1.19925i	−0.800281	−	0.599625i	$-0.795316\pi$
$89$	− 11.3137i	− 1.19925i	0.800281	−	0.599625i	$-0.204684\pi$
$90$	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	$-0.248359\pi$
$97$	14.0000	1.42148	0.710742	+	0.703452i	$0.248359\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	8.48528i	0.844317i	0.906522	+	0.422159i	$0.138727\pi$
$101$	8.48528i	0.844317i	−0.906522	+	0.422159i	$0.861273\pi$
$102$	0	0
$103$	12.0000i	1.18240i	0.806527	+	0.591198i	$0.201345\pi$
$103$	12.0000i	1.18240i	−0.806527	+	0.591198i	$0.798655\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.41421	0.136717	0.0683586	−	0.997661i	$-0.478224\pi$
$107$	1.41421	0.136717	0.0683586	+	0.997661i	$0.478224\pi$
$108$	0	0
$109$	−4.00000	−0.383131	−0.191565	−	0.981480i	$-0.561356\pi$
$109$	−4.00000	−0.383131	−0.191565	+	0.981480i	$0.561356\pi$
$110$	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
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−2.82843	−0.259281
$120$	0	0
$121$	7.00000	0.636364
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	6.00000i	0.532414i	0.963916	+	0.266207i	$0.0857705\pi$
$127$	6.00000i	0.532414i	−0.963916	+	0.266207i	$0.914230\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−16.9706	−1.48272	−0.741362	−	0.671105i	$-0.765820\pi$
$131$	−16.9706	−1.48272	−0.741362	+	0.671105i	$0.765820\pi$
$132$	0	0
$133$	4.00000	0.346844
$134$	0	0
$135$	0	0
$136$	0	0
$137$	4.24264i	0.362473i	0.983440	+	0.181237i	$0.0580100\pi$
$137$	4.24264i	0.362473i	−0.983440	+	0.181237i	$0.941990\pi$
$138$	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
$143$	8.48528	0.709575
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	21.2132i	1.73785i	0.494941	+	0.868927i	$0.335190\pi$
$149$	21.2132i	1.73785i	−0.494941	+	0.868927i	$0.664810\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 1.41421i	− 0.111456i
$162$	0	0
$163$	− 2.00000i	− 0.156652i	−0.996928	−	0.0783260i	$-0.975042\pi$
$163$	− 2.00000i	− 0.156652i	0.996928	−	0.0783260i	$-0.0249575\pi$
$164$	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
$173$	14.1421i	1.07521i	0.843198	+	0.537603i	$0.180670\pi$
$173$	14.1421i	1.07521i	−0.843198	+	0.537603i	$0.819330\pi$
$174$	0	0
$175$	− 5.00000i	− 0.377964i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	18.3848	1.37414	0.687071	−	0.726590i	$-0.258896\pi$
$179$	18.3848	1.37414	0.687071	+	0.726590i	$0.258896\pi$
$180$	0	0
$181$	−18.0000	−1.33793	−0.668965	−	0.743294i	$-0.733262\pi$
$181$	−18.0000	−1.33793	−0.668965	+	0.743294i	$0.733262\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	12.0000i	0.877527i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	7.07107	0.511645	0.255822	−	0.966724i	$-0.417654\pi$
$191$	7.07107	0.511645	0.255822	+	0.966724i	$0.417654\pi$
$192$	0	0
$193$	20.0000	1.43963	0.719816	−	0.694165i	$-0.244226\pi$
$193$	20.0000	1.43963	0.719816	+	0.694165i	$0.244226\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 7.07107i	− 0.503793i	−0.967754	−	0.251896i	$-0.918946\pi$
$197$	− 7.07107i	− 0.503793i	0.967754	−	0.251896i	$-0.0810542\pi$
$198$	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
$203$	7.07107	0.496292
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 16.9706i	− 1.17388i
$210$	0	0
$211$	4.00000i	0.275371i	0.990476	+	0.137686i	$0.0439664\pi$
$211$	4.00000i	0.275371i	−0.990476	+	0.137686i	$0.956034\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	8.00000	0.543075
$218$	0	0
$219$	0	0
$220$	0	0
$221$	5.65685i	0.380521i
$222$	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
$227$	−25.4558	−1.68956	−0.844782	−	0.535111i	$-0.820270\pi$
$227$	−25.4558	−1.68956	−0.844782	+	0.535111i	$0.820270\pi$
$228$	0	0
$229$	−18.0000	−1.18947	−0.594737	−	0.803921i	$-0.702744\pi$
$229$	−18.0000	−1.18947	−0.594737	+	0.803921i	$0.702744\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 9.89949i	− 0.648537i	−0.945965	−	0.324269i	$-0.894882\pi$
$233$	− 9.89949i	− 0.648537i	0.945965	−	0.324269i	$-0.105118\pi$
$234$	0	0
$235$	0	0
$236$	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
$241$	−6.00000	−0.386494	−0.193247	−	0.981150i	$-0.561902\pi$
$241$	−6.00000	−0.386494	−0.193247	+	0.981150i	$0.561902\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 8.00000i	− 0.509028i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	19.7990	1.24970	0.624851	−	0.780744i	$-0.285160\pi$
$251$	19.7990	1.24970	0.624851	+	0.780744i	$0.285160\pi$
$252$	0	0
$253$	−6.00000	−0.377217
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 11.3137i	− 0.705730i	−0.935674	−	0.352865i	$-0.885208\pi$
$257$	− 11.3137i	− 0.705730i	0.935674	−	0.352865i	$-0.114792\pi$
$258$	0	0
$259$	− 8.00000i	− 0.497096i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−4.24264	−0.261612	−0.130806	−	0.991408i	$-0.541757\pi$
$263$	−4.24264	−0.261612	−0.130806	+	0.991408i	$0.541757\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	5.65685i	0.344904i	0.985018	+	0.172452i	$0.0551690\pi$
$269$	5.65685i	0.344904i	−0.985018	+	0.172452i	$0.944831\pi$
$270$	0	0
$271$	20.0000i	1.21491i	0.794353	+	0.607457i	$0.207810\pi$
$271$	20.0000i	1.21491i	−0.794353	+	0.607457i	$0.792190\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−21.2132	−1.27920
$276$	0	0
$277$	18.0000	1.08152	0.540758	−	0.841178i	$-0.318138\pi$
$277$	18.0000	1.08152	0.540758	+	0.841178i	$0.318138\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 15.5563i	− 0.928014i	−0.885832	−	0.464007i	$-0.846411\pi$
$281$	− 15.5563i	− 0.928014i	0.885832	−	0.464007i	$-0.153589\pi$
$282$	0	0
$283$	− 28.0000i	− 1.66443i	−0.554455	−	0.832214i	$-0.687073\pi$
$283$	− 28.0000i	− 1.66443i	0.554455	−	0.832214i	$-0.312927\pi$
$284$	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
$293$	2.82843i	0.165238i	0.996581	+	0.0826192i	$0.0263285\pi$
$293$	2.82843i	0.165238i	−0.996581	+	0.0826192i	$0.973671\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−2.82843	−0.163572
$300$	0	0
$301$	4.00000	0.230556
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 16.0000i	− 0.913168i	−0.889680	−	0.456584i	$-0.849073\pi$
$307$	− 16.0000i	− 0.913168i	0.889680	−	0.456584i	$-0.150927\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	14.1421	0.801927	0.400963	−	0.916094i	$-0.368675\pi$
$311$	14.1421	0.801927	0.400963	+	0.916094i	$0.368675\pi$
$312$	0	0
$313$	−34.0000	−1.92179	−0.960897	−	0.276907i	$-0.910691\pi$
$313$	−34.0000	−1.92179	−0.960897	+	0.276907i	$0.910691\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.41421i	0.0794301i	0.999211	+	0.0397151i	$0.0126450\pi$
$317$	1.41421i	0.0794301i	−0.999211	+	0.0397151i	$0.987355\pi$
$318$	0	0
$319$	− 30.0000i	− 1.67968i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	11.3137	0.629512
$324$	0	0
$325$	−10.0000	−0.554700
$326$	0	0
$327$	0	0
$328$	0	0
$329$	2.82843i	0.155936i
$330$	0	0
$331$	− 28.0000i	− 1.53902i	−0.638635	−	0.769510i	$-0.720501\pi$
$331$	− 28.0000i	− 1.53902i	0.638635	−	0.769510i	$-0.279499\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	4.00000	0.217894	0.108947	−	0.994048i	$-0.465252\pi$
$337$	4.00000	0.217894	0.108947	+	0.994048i	$0.465252\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 33.9411i	− 1.83801i
$342$	0	0
$343$	1.00000i	0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	24.0416	1.29062	0.645311	−	0.763920i	$-0.276728\pi$
$347$	24.0416	1.29062	0.645311	+	0.763920i	$0.276728\pi$
$348$	0	0
$349$	2.00000	0.107058	0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000	0.107058	0.0535288	+	0.998566i	$0.482953\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	25.4558i	1.35488i	0.735579	+	0.677439i	$0.236910\pi$
$353$	25.4558i	1.35488i	−0.735579	+	0.677439i	$0.763090\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	18.3848	0.970311	0.485156	−	0.874428i	$-0.338763\pi$
$359$	18.3848	0.970311	0.485156	+	0.874428i	$0.338763\pi$
$360$	0	0
$361$	3.00000	0.157895
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	24.0000i	1.25279i	0.779506	+	0.626395i	$0.215470\pi$
$367$	24.0000i	1.25279i	−0.779506	+	0.626395i	$0.784530\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	7.07107	0.367112
$372$	0	0
$373$	−22.0000	−1.13912	−0.569558	−	0.821951i	$-0.692886\pi$
$373$	−22.0000	−1.13912	−0.569558	+	0.821951i	$0.692886\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 14.1421i	− 0.728357i
$378$	0	0
$379$	36.0000i	1.84920i	0.380945	+	0.924598i	$0.375599\pi$
$379$	36.0000i	1.84920i	−0.380945	+	0.924598i	$0.624401\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−16.9706	−0.867155	−0.433578	−	0.901116i	$-0.642749\pi$
$383$	−16.9706	−0.867155	−0.433578	+	0.901116i	$0.642749\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	4.24264i	0.215110i	0.994199	+	0.107555i	$0.0343022\pi$
$389$	4.24264i	0.215110i	−0.994199	+	0.107555i	$0.965698\pi$
$390$	0	0
$391$	− 4.00000i	− 0.202289i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−34.0000	−1.70641	−0.853206	−	0.521575i	$-0.825345\pi$
$397$	−34.0000	−1.70641	−0.853206	+	0.521575i	$0.825345\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	4.24264i	0.211867i	0.994373	+	0.105934i	$0.0337831\pi$
$401$	4.24264i	0.211867i	−0.994373	+	0.105934i	$0.966217\pi$
$402$	0	0
$403$	− 16.0000i	− 0.797017i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−33.9411	−1.68240
$408$	0	0
$409$	2.00000	0.0988936	0.0494468	−	0.998777i	$-0.484254\pi$
$409$	2.00000	0.0988936	0.0494468	+	0.998777i	$0.484254\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 2.82843i	− 0.139178i
$414$	0	0
$415$	0	0
$416$	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	$-0.718416\pi$
$421$	−26.0000	−1.26716	−0.633581	+	0.773676i	$0.718416\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 14.1421i	− 0.685994i
$426$	0	0
$427$	2.00000i	0.0967868i
$428$	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
$433$	−38.0000	−1.82616	−0.913082	−	0.407777i	$-0.866304\pi$
$433$	−38.0000	−1.82616	−0.913082	+	0.407777i	$0.866304\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	5.65685i	0.270604i
$438$	0	0
$439$	− 8.00000i	− 0.381819i	−0.981608	−	0.190910i	$-0.938856\pi$
$439$	− 8.00000i	− 0.381819i	0.981608	−	0.190910i	$-0.0611437\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	35.3553	1.67978	0.839891	−	0.542754i	$-0.182619\pi$
$443$	35.3553	1.67978	0.839891	+	0.542754i	$0.182619\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 18.3848i	− 0.867631i	−0.901002	−	0.433816i	$-0.857167\pi$
$449$	− 18.3848i	− 0.867631i	0.901002	−	0.433816i	$-0.142833\pi$
$450$	0	0
$451$	− 24.0000i	− 1.13012i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	8.00000	0.374224	0.187112	−	0.982339i	$-0.440087\pi$
$457$	8.00000	0.374224	0.187112	+	0.982339i	$0.440087\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	2.82843i	0.131733i	0.997828	+	0.0658665i	$0.0209811\pi$
$461$	2.82843i	0.131733i	−0.997828	+	0.0658665i	$0.979019\pi$
$462$	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
$467$	−14.1421	−0.654420	−0.327210	−	0.944952i	$-0.606108\pi$
$467$	−14.1421	−0.654420	−0.327210	+	0.944952i	$0.606108\pi$
$468$	0	0
$469$	−2.00000	−0.0923514
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 16.9706i	− 0.780307i
$474$	0	0
$475$	20.0000i	0.917663i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	42.4264	1.93851	0.969256	−	0.246054i	$-0.0791342\pi$
$479$	42.4264	1.93851	0.969256	+	0.246054i	$0.0791342\pi$
$480$	0	0
$481$	−16.0000	−0.729537
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 40.0000i	− 1.81257i	−0.422664	−	0.906287i	$-0.638905\pi$
$487$	− 40.0000i	− 1.81257i	0.422664	−	0.906287i	$-0.361095\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−1.41421	−0.0638226	−0.0319113	−	0.999491i	$-0.510159\pi$
$491$	−1.41421	−0.0638226	−0.0319113	+	0.999491i	$0.510159\pi$
$492$	0	0
$493$	20.0000	0.900755
$494$	0	0
$495$	0	0
$496$	0	0
$497$	12.7279i	0.570925i
$498$	0	0
$499$	− 36.0000i	− 1.61158i	−0.592200	−	0.805791i	$-0.701741\pi$
$499$	− 36.0000i	− 1.61158i	0.592200	−	0.805791i	$-0.298259\pi$
$500$	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
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 39.5980i	− 1.75515i	−0.479440	−	0.877575i	$-0.659160\pi$
$509$	− 39.5980i	− 1.75515i	0.479440	−	0.877575i	$-0.340840\pi$
$510$	0	0
$511$	− 14.0000i	− 0.619324i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	12.0000	0.527759
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 31.1127i	− 1.36307i	−0.731785	−	0.681536i	$-0.761312\pi$
$521$	− 31.1127i	− 1.36307i	0.731785	−	0.681536i	$-0.238688\pi$
$522$	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
$527$	22.6274	0.985666
$528$	0	0
$529$	−21.0000	−0.913043
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 11.3137i	− 0.490051i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	4.24264	0.182743
$540$	0	0
$541$	34.0000	1.46177	0.730887	−	0.682498i	$-0.239107\pi$
$541$	34.0000	1.46177	0.730887	+	0.682498i	$0.239107\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	2.00000i	0.0855138i	0.999086	+	0.0427569i	$0.0136141\pi$
$547$	2.00000i	0.0855138i	−0.999086	+	0.0427569i	$0.986386\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−28.2843	−1.20495
$552$	0	0
$553$	2.00000	0.0850487
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 1.41421i	− 0.0599222i	−0.999551	−	0.0299611i	$-0.990462\pi$
$557$	− 1.41421i	− 0.0599222i	0.999551	−	0.0299611i	$-0.00953833\pi$
$558$	0	0
$559$	− 8.00000i	− 0.338364i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−19.7990	−0.834428	−0.417214	−	0.908808i	$-0.636993\pi$
$563$	−19.7990	−0.834428	−0.417214	+	0.908808i	$0.636993\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	41.0122i	1.71932i	0.510866	+	0.859660i	$0.329325\pi$
$569$	41.0122i	1.71932i	−0.510866	+	0.859660i	$0.670675\pi$
$570$	0	0
$571$	22.0000i	0.920671i	0.887745	+	0.460336i	$0.152271\pi$
$571$	22.0000i	0.920671i	−0.887745	+	0.460336i	$0.847729\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	7.07107	0.294884
$576$	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
$581$	5.65685i	0.234686i
$582$	0	0
$583$	− 30.0000i	− 1.24247i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−19.7990	−0.817192	−0.408596	−	0.912715i	$-0.633981\pi$
$587$	−19.7990	−0.817192	−0.408596	+	0.912715i	$0.633981\pi$
$588$	0	0
$589$	−32.0000	−1.31854
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 14.1421i	− 0.580748i	−0.956913	−	0.290374i	$-0.906220\pi$
$593$	− 14.1421i	− 0.580748i	0.956913	−	0.290374i	$-0.0937797\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−43.8406	−1.79128	−0.895640	−	0.444781i	$-0.853282\pi$
$599$	−43.8406	−1.79128	−0.895640	+	0.444781i	$0.853282\pi$
$600$	0	0
$601$	−26.0000	−1.06056	−0.530281	−	0.847822i	$-0.677914\pi$
$601$	−26.0000	−1.06056	−0.530281	+	0.847822i	$0.677914\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	48.0000i	1.94826i	0.225989	+	0.974130i	$0.427439\pi$
$607$	48.0000i	1.94826i	−0.225989	+	0.974130i	$0.572561\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	5.65685	0.228852
$612$	0	0
$613$	14.0000	0.565455	0.282727	−	0.959200i	$-0.408761\pi$
$613$	14.0000	0.565455	0.282727	+	0.959200i	$0.408761\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	41.0122i	1.65109i	0.564337	+	0.825544i	$0.309132\pi$
$617$	41.0122i	1.65109i	−0.564337	+	0.825544i	$0.690868\pi$
$618$	0	0
$619$	− 36.0000i	− 1.44696i	−0.690344	−	0.723481i	$-0.742541\pi$
$619$	− 36.0000i	− 1.44696i	0.690344	−	0.723481i	$-0.257459\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−11.3137	−0.453274
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 22.6274i	− 0.902214i
$630$	0	0
$631$	− 18.0000i	− 0.716569i	−0.933613	−	0.358284i	$-0.883362\pi$
$631$	− 18.0000i	− 0.716569i	0.933613	−	0.358284i	$-0.116638\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	2.00000	0.0792429
$638$	0	0
$639$	0	0
$640$	0	0
$641$	46.6690i	1.84332i	0.388003	+	0.921658i	$0.373165\pi$
$641$	46.6690i	1.84332i	−0.388003	+	0.921658i	$0.626835\pi$
$642$	0	0
$643$	4.00000i	0.157745i	0.996885	+	0.0788723i	$0.0251319\pi$
$643$	4.00000i	0.157745i	−0.996885	+	0.0788723i	$0.974868\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	48.0833	1.89035	0.945174	−	0.326567i	$-0.105892\pi$
$647$	48.0833	1.89035	0.945174	+	0.326567i	$0.105892\pi$
$648$	0	0
$649$	−12.0000	−0.471041
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 26.8701i	− 1.05151i	−0.850637	−	0.525753i	$-0.823784\pi$
$653$	− 26.8701i	− 1.05151i	0.850637	−	0.525753i	$-0.176216\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−1.41421	−0.0550899	−0.0275450	−	0.999621i	$-0.508769\pi$
$659$	−1.41421	−0.0550899	−0.0275450	+	0.999621i	$0.508769\pi$
$660$	0	0
$661$	22.0000	0.855701	0.427850	−	0.903850i	$-0.359271\pi$
$661$	22.0000	0.855701	0.427850	+	0.903850i	$0.359271\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	10.0000i	0.387202i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	8.48528	0.327571
$672$	0	0
$673$	−40.0000	−1.54189	−0.770943	−	0.636904i	$-0.780215\pi$
$673$	−40.0000	−1.54189	−0.770943	+	0.636904i	$0.780215\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	2.82843i	0.108705i	0.998522	+	0.0543526i	$0.0173095\pi$
$677$	2.82843i	0.108705i	−0.998522	+	0.0543526i	$0.982690\pi$
$678$	0	0
$679$	− 14.0000i	− 0.537271i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	43.8406	1.67751	0.838757	−	0.544505i	$-0.183283\pi$
$683$	43.8406	1.67751	0.838757	+	0.544505i	$0.183283\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 14.1421i	− 0.538772i
$690$	0	0
$691$	28.0000i	1.06517i	0.846376	+	0.532585i	$0.178779\pi$
$691$	28.0000i	1.06517i	−0.846376	+	0.532585i	$0.821221\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	16.0000	0.606043
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 38.1838i	− 1.44218i	−0.692841	−	0.721090i	$-0.743641\pi$
$701$	− 38.1838i	− 1.44218i	0.692841	−	0.721090i	$-0.256359\pi$
$702$	0	0
$703$	32.0000i	1.20690i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	8.48528	0.319122
$708$	0	0
$709$	20.0000	0.751116	0.375558	−	0.926799i	$-0.377451\pi$
$709$	20.0000	0.751116	0.375558	+	0.926799i	$0.377451\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	11.3137i	0.423702i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	39.5980	1.47676	0.738378	−	0.674387i	$-0.235592\pi$
$719$	39.5980	1.47676	0.738378	+	0.674387i	$0.235592\pi$
$720$	0	0
$721$	12.0000	0.446903
$722$	0	0
$723$	0	0
$724$	0	0
$725$	35.3553i	1.31306i
$726$	0	0
$727$	4.00000i	0.148352i	0.997245	+	0.0741759i	$0.0236326\pi$
$727$	4.00000i	0.148352i	−0.997245	+	0.0741759i	$0.976367\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	11.3137	0.418453
$732$	0	0
$733$	−50.0000	−1.84679	−0.923396	−	0.383849i	$-0.874598\pi$
$733$	−50.0000	−1.84679	−0.923396	+	0.383849i	$0.874598\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	8.48528i	0.312559i
$738$	0	0
$739$	− 38.0000i	− 1.39785i	−0.715194	−	0.698926i	$-0.753662\pi$
$739$	− 38.0000i	− 1.39785i	0.715194	−	0.698926i	$-0.246338\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−43.8406	−1.60836	−0.804178	−	0.594388i	$-0.797394\pi$
$743$	−43.8406	−1.60836	−0.804178	+	0.594388i	$0.797394\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 1.41421i	− 0.0516742i
$750$	0	0
$751$	24.0000i	0.875772i	0.899030	+	0.437886i	$0.144273\pi$
$751$	24.0000i	0.875772i	−0.899030	+	0.437886i	$0.855727\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	12.0000	0.436147	0.218074	−	0.975932i	$-0.430023\pi$
$757$	12.0000	0.436147	0.218074	+	0.975932i	$0.430023\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	33.9411i	1.23036i	0.788385	+	0.615182i	$0.210918\pi$
$761$	33.9411i	1.23036i	−0.788385	+	0.615182i	$0.789082\pi$
$762$	0	0
$763$	4.00000i	0.144810i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−5.65685	−0.204257
$768$	0	0
$769$	−2.00000	−0.0721218	−0.0360609	−	0.999350i	$-0.511481\pi$
$769$	−2.00000	−0.0721218	−0.0360609	+	0.999350i	$0.511481\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 36.7696i	− 1.32251i	−0.750162	−	0.661254i	$-0.770024\pi$
$773$	− 36.7696i	− 1.32251i	0.750162	−	0.661254i	$-0.229976\pi$
$774$	0	0
$775$	40.0000i	1.43684i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−22.6274	−0.810711
$780$	0	0
$781$	54.0000	1.93227
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	4.00000i	0.142585i	0.997455	+	0.0712923i	$0.0227123\pi$
$787$	4.00000i	0.142585i	−0.997455	+	0.0712923i	$0.977288\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−7.07107	−0.251418
$792$	0	0
$793$	4.00000	0.142044
$794$	0	0
$795$	0	0
$796$	0	0
$797$	14.1421i	0.500940i	0.968124	+	0.250470i	$0.0805852\pi$
$797$	14.1421i	0.500940i	−0.968124	+	0.250470i	$0.919415\pi$
$798$	0	0
$799$	8.00000i	0.283020i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−59.3970	−2.09607
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	1.41421i	0.0497211i	0.999691	+	0.0248606i	$0.00791417\pi$
$809$	1.41421i	0.0497211i	−0.999691	+	0.0248606i	$0.992086\pi$
$810$	0	0
$811$	− 12.0000i	− 0.421377i	−0.977553	−	0.210688i	$-0.932429\pi$
$811$	− 12.0000i	− 0.421377i	0.977553	−	0.210688i	$-0.0675706\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−16.0000	−0.559769
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 1.41421i	− 0.0493564i	−0.999695	−	0.0246782i	$-0.992144\pi$
$821$	− 1.41421i	− 0.0493564i	0.999695	−	0.0246782i	$-0.00785611\pi$
$822$	0	0
$823$	− 14.0000i	− 0.488009i	−0.969774	−	0.244005i	$-0.921539\pi$
$823$	− 14.0000i	− 0.488009i	0.969774	−	0.244005i	$-0.0784612\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	38.1838	1.32778	0.663890	−	0.747830i	$-0.268904\pi$
$827$	38.1838	1.32778	0.663890	+	0.747830i	$0.268904\pi$
$828$	0	0
$829$	−2.00000	−0.0694629	−0.0347314	−	0.999397i	$-0.511058\pi$
$829$	−2.00000	−0.0694629	−0.0347314	+	0.999397i	$0.511058\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	2.82843i	0.0979992i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−53.7401	−1.85531	−0.927657	−	0.373432i	$-0.878181\pi$
$839$	−53.7401	−1.85531	−0.927657	+	0.373432i	$0.878181\pi$
$840$	0	0
$841$	−21.0000	−0.724138
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 7.00000i	− 0.240523i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	11.3137	0.387829
$852$	0	0
$853$	38.0000	1.30110	0.650548	−	0.759465i	$-0.274539\pi$
$853$	38.0000	1.30110	0.650548	+	0.759465i	$0.274539\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	50.9117i	1.73911i	0.493836	+	0.869555i	$0.335594\pi$
$857$	50.9117i	1.73911i	−0.493836	+	0.869555i	$0.664406\pi$
$858$	0	0
$859$	24.0000i	0.818869i	0.912339	+	0.409435i	$0.134274\pi$
$859$	24.0000i	0.818869i	−0.912339	+	0.409435i	$0.865726\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	35.3553	1.20351	0.601755	−	0.798681i	$-0.294468\pi$
$863$	35.3553	1.20351	0.601755	+	0.798681i	$0.294468\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 8.48528i	− 0.287843i
$870$	0	0
$871$	4.00000i	0.135535i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	18.0000	0.607817	0.303908	−	0.952701i	$-0.401708\pi$
$877$	18.0000	0.607817	0.303908	+	0.952701i	$0.401708\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	8.48528i	0.285876i	0.989732	+	0.142938i	$0.0456550\pi$
$881$	8.48528i	0.285876i	−0.989732	+	0.142938i	$0.954345\pi$
$882$	0	0
$883$	12.0000i	0.403832i	0.979403	+	0.201916i	$0.0647168\pi$
$883$	12.0000i	0.403832i	−0.979403	+	0.201916i	$0.935283\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−14.1421	−0.474846	−0.237423	−	0.971406i	$-0.576303\pi$
$887$	−14.1421	−0.474846	−0.237423	+	0.971406i	$0.576303\pi$
$888$	0	0
$889$	6.00000	0.201234
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 11.3137i	− 0.378599i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−56.5685	−1.88667
$900$	0	0
$901$	20.0000	0.666297
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	12.0000i	0.398453i	0.979953	+	0.199227i	$0.0638430\pi$
$907$	12.0000i	0.398453i	−0.979953	+	0.199227i	$0.936157\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−57.9828	−1.92105	−0.960527	−	0.278186i	$-0.910267\pi$
$911$	−57.9828	−1.92105	−0.960527	+	0.278186i	$0.910267\pi$
$912$	0	0
$913$	24.0000	0.794284
$914$	0	0
$915$	0	0
$916$	0	0
$917$	16.9706i	0.560417i
$918$	0	0
$919$	8.00000i	0.263896i	0.991257	+	0.131948i	$0.0421231\pi$
$919$	8.00000i	0.263896i	−0.991257	+	0.131948i	$0.957877\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	25.4558	0.837889
$924$	0	0
$925$	40.0000	1.31519
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 56.5685i	− 1.85595i	−0.372638	−	0.927977i	$-0.621547\pi$
$929$	− 56.5685i	− 1.85595i	0.372638	−	0.927977i	$-0.378453\pi$
$930$	0	0
$931$	− 4.00000i	− 0.131095i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	18.0000	0.588034	0.294017	−	0.955800i	$-0.405008\pi$
$937$	18.0000	0.588034	0.294017	+	0.955800i	$0.405008\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	5.65685i	0.184408i	0.995740	+	0.0922041i	$0.0293912\pi$
$941$	5.65685i	0.184408i	−0.995740	+	0.0922041i	$0.970609\pi$
$942$	0	0
$943$	8.00000i	0.260516i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−43.8406	−1.42463	−0.712314	−	0.701861i	$-0.752353\pi$
$947$	−43.8406	−1.42463	−0.712314	+	0.701861i	$0.752353\pi$
$948$	0	0
$949$	−28.0000	−0.908918
$950$	0	0
$951$	0	0
$952$	0	0
$953$	12.7279i	0.412298i	0.978521	+	0.206149i	$0.0660931\pi$
$953$	12.7279i	0.412298i	−0.978521	+	0.206149i	$0.933907\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	4.24264	0.137002
$960$	0	0
$961$	−33.0000	−1.06452
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	50.0000i	1.60789i	0.594703	+	0.803946i	$0.297270\pi$
$967$	50.0000i	1.60789i	−0.594703	+	0.803946i	$0.702730\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−22.6274	−0.726148	−0.363074	−	0.931760i	$-0.618273\pi$
$971$	−22.6274	−0.726148	−0.363074	+	0.931760i	$0.618273\pi$
$972$	0	0
$973$	16.0000	0.512936
$974$	0	0
$975$	0	0
$976$	0	0
$977$	41.0122i	1.31210i	0.754719	+	0.656048i	$0.227773\pi$
$977$	41.0122i	1.31210i	−0.754719	+	0.656048i	$0.772227\pi$
$978$	0	0
$979$	48.0000i	1.53409i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	28.2843	0.902128	0.451064	−	0.892492i	$-0.351045\pi$
$983$	28.2843	0.902128	0.451064	+	0.892492i	$0.351045\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	5.65685i	0.179878i
$990$	0	0
$991$	− 8.00000i	− 0.254128i	−0.991894	−	0.127064i	$-0.959445\pi$
$991$	− 8.00000i	− 0.254128i	0.991894	−	0.127064i	$-0.0405554\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	42.0000	1.33015	0.665077	−	0.746775i	$-0.268399\pi$
$997$	42.0000	1.33015	0.665077	+	0.746775i	$0.268399\pi$
$998$	0	0
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2016.2.h.b.575.1	✓	4	1.1	even	1	trivial
2016.2.h.b.575.2	yes	4	3.2	odd	2	inner
2016.2.h.b.575.3	yes	4	12.11	even	2	inner
2016.2.h.b.575.4	yes	4	4.3	odd	2	inner
4032.2.h.d.575.1		4	24.5	odd	2
4032.2.h.d.575.2		4	8.5	even	2
4032.2.h.d.575.3		4	8.3	odd	2
4032.2.h.d.575.4		4	24.11	even	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:	\( N \)	\(=\)	\( 2016 = 2^{5} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2016.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{7} +O(q^{10})\)	\(q-1.00000i q^{7} -4.24264 q^{11} -2.00000 q^{13} -2.82843i q^{17} +4.00000i q^{19} +1.41421 q^{23} +5.00000 q^{25} +7.07107i q^{29} +8.00000i q^{31} +8.00000 q^{37} +5.65685i q^{41} +4.00000i q^{43} -2.82843 q^{47} -1.00000 q^{49} +7.07107i q^{53} +2.82843 q^{59} -2.00000 q^{61} -2.00000i q^{67} -12.7279 q^{71} +14.0000 q^{73} +4.24264i q^{77} +2.00000i q^{79} -5.65685 q^{83} -11.3137i q^{89} +2.00000i q^{91} +14.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 8 q^{13} + 20 q^{25} + 32 q^{37} - 4 q^{49} - 8 q^{61} + 56 q^{73} + 56 q^{97}+O(q^{100}) \) 4 * q - 8 * q^13 + 20 * q^25 + 32 * q^37 - 4 * q^49 - 8 * q^61 + 56 * q^73 + 56 * q^97

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