LMFDB - Embedded newform 4032.2.c.q.2017.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4032,2,Mod(2017,4032)]

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([0, 1, 0, 0]))

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

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

chi := DirichletCharacter("4032.2017");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4032 = 2^{6} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4032.c (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:	$8$
Coefficient field:	8.0.897122304.10
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 8x^{6} + 51x^{4} - 104x^{2} + 169 $ x^8 - 8x^6 + 51x^4 - 104*x^2 + 169
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2017.8
Root			$1.30421 - 0.752986i$ of defining polynomial
Character	$\chi$	$=$	4032.2017
Dual form			4032.2.c.q.2017.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.11439i q^{5} -1.00000 q^{7} +O(q^{10})$	$q+4.11439i q^{5} -1.00000 q^{7} -4.11439i q^{11} +5.46410i q^{13} -1.10245 q^{17} -3.46410i q^{19} -7.12633 q^{23} -11.9282 q^{25} +6.02388i q^{29} +2.00000 q^{31} -4.11439i q^{35} -11.4641i q^{37} +1.10245 q^{41} -8.92820i q^{43} -6.02388 q^{47} +1.00000 q^{49} +16.9282 q^{55} -6.02388i q^{59} -5.46410i q^{61} -22.4814 q^{65} +10.0000i q^{67} -9.33123 q^{71} -0.928203 q^{73} +4.11439i q^{77} -2.92820 q^{79} +14.2527i q^{83} -4.53590i q^{85} +15.3551 q^{89} -5.46410i q^{91} +14.2527 q^{95} -4.92820 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{7}+O(q^{10}) $ 8 * q - 8 * q^7	$ 8 q - 8 q^{7} - 40 q^{25} + 16 q^{31} + 8 q^{49} + 80 q^{55} + 48 q^{73} + 32 q^{79} + 16 q^{97}+O(q^{100}) $ 8 * q - 8 * q^7 - 40 * q^25 + 16 * q^31 + 8 * q^49 + 80 * q^55 + 48 * q^73 + 32 * q^79 + 16 * 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
$5$	4.11439i	1.84001i	0.391905	+	0.920006i	$0.371816\pi$
$5$	4.11439i	1.84001i	−0.391905	+	0.920006i	$0.628184\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 4.11439i	− 1.24054i	−0.784390	−	0.620268i	$-0.787024\pi$
$11$	− 4.11439i	− 1.24054i	0.784390	−	0.620268i	$-0.212976\pi$
$12$	0	0
$13$	5.46410i	1.51547i	0.652563	+	0.757735i	$0.273694\pi$
$13$	5.46410i	1.51547i	−0.652563	+	0.757735i	$0.726306\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.10245	−0.267383	−0.133691	−	0.991023i	$-0.542683\pi$
$17$	−1.10245	−0.267383	−0.133691	+	0.991023i	$0.542683\pi$
$18$	0	0
$19$	− 3.46410i	− 0.794719i	−0.917663	−	0.397360i	$-0.869927\pi$
$19$	− 3.46410i	− 0.794719i	0.917663	−	0.397360i	$-0.130073\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−7.12633	−1.48594	−0.742971	−	0.669323i	$-0.766584\pi$
$23$	−7.12633	−1.48594	−0.742971	+	0.669323i	$0.766584\pi$
$24$	0	0
$25$	−11.9282	−2.38564
$26$	0	0
$27$	0	0
$28$	0	0
$29$	6.02388i	1.11861i	0.828963	+	0.559304i	$0.188931\pi$
$29$	6.02388i	1.11861i	−0.828963	+	0.559304i	$0.811069\pi$
$30$	0	0
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 4.11439i	− 0.695459i
$36$	0	0
$37$	− 11.4641i	− 1.88469i	−0.334648	−	0.942343i	$-0.608617\pi$
$37$	− 11.4641i	− 1.88469i	0.334648	−	0.942343i	$-0.391383\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	1.10245	0.172173	0.0860867	−	0.996288i	$-0.472564\pi$
$41$	1.10245	0.172173	0.0860867	+	0.996288i	$0.472564\pi$
$42$	0	0
$43$	− 8.92820i	− 1.36154i	−0.732498	−	0.680769i	$-0.761646\pi$
$43$	− 8.92820i	− 1.36154i	0.732498	−	0.680769i	$-0.238354\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−6.02388	−0.878674	−0.439337	−	0.898322i	$-0.644787\pi$
$47$	−6.02388	−0.878674	−0.439337	+	0.898322i	$0.644787\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	16.9282	2.28260
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 6.02388i	− 0.784243i	−0.919913	−	0.392121i	$-0.871741\pi$
$59$	− 6.02388i	− 0.784243i	0.919913	−	0.392121i	$-0.128259\pi$
$60$	0	0
$61$	− 5.46410i	− 0.699607i	−0.936823	−	0.349803i	$-0.886248\pi$
$61$	− 5.46410i	− 0.699607i	0.936823	−	0.349803i	$-0.113752\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−22.4814	−2.78848
$66$	0	0
$67$	10.0000i	1.22169i	0.791748	+	0.610847i	$0.209171\pi$
$67$	10.0000i	1.22169i	−0.791748	+	0.610847i	$0.790829\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−9.33123	−1.10741	−0.553706	−	0.832712i	$-0.686787\pi$
$71$	−9.33123	−1.10741	−0.553706	+	0.832712i	$0.686787\pi$
$72$	0	0
$73$	−0.928203	−0.108638	−0.0543190	−	0.998524i	$-0.517299\pi$
$73$	−0.928203	−0.108638	−0.0543190	+	0.998524i	$0.517299\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4.11439i	0.468878i
$78$	0	0
$79$	−2.92820	−0.329449	−0.164724	−	0.986340i	$-0.552673\pi$
$79$	−2.92820	−0.329449	−0.164724	+	0.986340i	$0.552673\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	14.2527i	1.56443i	0.623007	+	0.782217i	$0.285911\pi$
$83$	14.2527i	1.56443i	−0.623007	+	0.782217i	$0.714089\pi$
$84$	0	0
$85$	− 4.53590i	− 0.491987i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	15.3551	1.62764	0.813819	−	0.581118i	$-0.197385\pi$
$89$	15.3551	1.62764	0.813819	+	0.581118i	$0.197385\pi$
$90$	0	0
$91$	− 5.46410i	− 0.572793i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	14.2527	1.46229
$96$	0	0
$97$	−4.92820	−0.500383	−0.250192	−	0.968196i	$-0.580494\pi$
$97$	−4.92820	−0.500383	−0.250192	+	0.968196i	$0.580494\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	4.11439i	0.409397i	0.978825	+	0.204699i	$0.0656214\pi$
$101$	4.11439i	0.409397i	−0.978825	+	0.204699i	$0.934379\pi$
$102$	0	0
$103$	−8.92820	−0.879722	−0.439861	−	0.898066i	$-0.644972\pi$
$103$	−8.92820	−0.879722	−0.439861	+	0.898066i	$0.644972\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	16.1622i	1.56245i	0.624246	+	0.781227i	$0.285406\pi$
$107$	16.1622i	1.56245i	−0.624246	+	0.781227i	$0.714594\pi$
$108$	0	0
$109$	− 4.00000i	− 0.383131i	−0.981480	−	0.191565i	$-0.938644\pi$
$109$	− 4.00000i	− 0.383131i	0.981480	−	0.191565i	$-0.0613564\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−8.22878	−0.774098	−0.387049	−	0.922059i	$-0.626506\pi$
$113$	−8.22878	−0.774098	−0.387049	+	0.922059i	$0.626506\pi$
$114$	0	0
$115$	− 29.3205i	− 2.73415i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1.10245	0.101061
$120$	0	0
$121$	−5.92820	−0.538928
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 28.5053i	− 2.54959i
$126$	0	0
$127$	14.9282	1.32466	0.662332	−	0.749211i	$-0.269567\pi$
$127$	14.9282	1.32466	0.662332	+	0.749211i	$0.269567\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 14.2527i	− 1.24526i	−0.782516	−	0.622631i	$-0.786064\pi$
$131$	− 14.2527i	− 1.24526i	0.782516	−	0.622631i	$-0.213936\pi$
$132$	0	0
$133$	3.46410i	0.300376i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	14.2527	1.21769	0.608844	−	0.793290i	$-0.291634\pi$
$137$	14.2527	1.21769	0.608844	+	0.793290i	$0.291634\pi$
$138$	0	0
$139$	− 17.8564i	− 1.51456i	−0.653090	−	0.757280i	$-0.726528\pi$
$139$	− 17.8564i	− 1.51456i	0.653090	−	0.757280i	$-0.273472\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	22.4814	1.87999
$144$	0	0
$145$	−24.7846	−2.05825
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 16.4576i	− 1.34826i	−0.738615	−	0.674128i	$-0.764520\pi$
$149$	− 16.4576i	− 1.34826i	0.738615	−	0.674128i	$-0.235480\pi$
$150$	0	0
$151$	4.00000	0.325515	0.162758	−	0.986666i	$-0.447961\pi$
$151$	4.00000	0.325515	0.162758	+	0.986666i	$0.447961\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	8.22878i	0.660951i
$156$	0	0
$157$	− 8.39230i	− 0.669779i	−0.942257	−	0.334889i	$-0.891301\pi$
$157$	− 8.39230i	− 0.669779i	0.942257	−	0.334889i	$-0.108699\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	7.12633	0.561634
$162$	0	0
$163$	− 15.8564i	− 1.24197i	−0.783823	−	0.620985i	$-0.786733\pi$
$163$	− 15.8564i	− 1.24197i	0.783823	−	0.620985i	$-0.213267\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	14.2527	1.10290	0.551452	−	0.834207i	$-0.314074\pi$
$167$	14.2527	1.10290	0.551452	+	0.834207i	$0.314074\pi$
$168$	0	0
$169$	−16.8564	−1.29665
$170$	0	0
$171$	0	0
$172$	0	0
$173$	12.3432i	0.938434i	0.883083	+	0.469217i	$0.155464\pi$
$173$	12.3432i	0.938434i	−0.883083	+	0.469217i	$0.844536\pi$
$174$	0	0
$175$	11.9282	0.901687
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0.295400i	0.0220792i	0.999939	+	0.0110396i	$0.00351409\pi$
$179$	0.295400i	0.0220792i	−0.999939	+	0.0110396i	$0.996486\pi$
$180$	0	0
$181$	− 14.5359i	− 1.08044i	−0.841522	−	0.540222i	$-0.818340\pi$
$181$	− 14.5359i	− 1.08044i	0.841522	−	0.540222i	$-0.181660\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	47.1678	3.46784
$186$	0	0
$187$	4.53590i	0.331698i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−25.7888	−1.86601	−0.933006	−	0.359862i	$-0.882824\pi$
$191$	−25.7888	−1.86601	−0.933006	+	0.359862i	$0.882824\pi$
$192$	0	0
$193$	−14.9282	−1.07456	−0.537278	−	0.843405i	$-0.680547\pi$
$193$	−14.9282	−1.07456	−0.537278	+	0.843405i	$0.680547\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 14.2527i	− 1.01546i	−0.861516	−	0.507730i	$-0.830485\pi$
$197$	− 14.2527i	− 1.01546i	0.861516	−	0.507730i	$-0.169515\pi$
$198$	0	0
$199$	−16.7846	−1.18983	−0.594915	−	0.803789i	$-0.702814\pi$
$199$	−16.7846	−1.18983	−0.594915	+	0.803789i	$0.702814\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 6.02388i	− 0.422794i
$204$	0	0
$205$	4.53590i	0.316801i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−14.2527	−0.985877
$210$	0	0
$211$	10.0000i	0.688428i	0.938891	+	0.344214i	$0.111855\pi$
$211$	10.0000i	0.688428i	−0.938891	+	0.344214i	$0.888145\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	36.7341	2.50525
$216$	0	0
$217$	−2.00000	−0.135769
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 6.02388i	− 0.405210i
$222$	0	0
$223$	−18.9282	−1.26753	−0.633763	−	0.773527i	$-0.718491\pi$
$223$	−18.9282	−1.26753	−0.633763	+	0.773527i	$0.718491\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 24.6863i	− 1.63849i	−0.573444	−	0.819245i	$-0.694393\pi$
$227$	− 24.6863i	− 1.63849i	0.573444	−	0.819245i	$-0.305607\pi$
$228$	0	0
$229$	− 17.4641i	− 1.15406i	−0.816723	−	0.577030i	$-0.804212\pi$
$229$	− 17.4641i	− 1.15406i	0.816723	−	0.577030i	$-0.195788\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−18.0717	−1.18391	−0.591957	−	0.805970i	$-0.701644\pi$
$233$	−18.0717	−1.18391	−0.591957	+	0.805970i	$0.701644\pi$
$234$	0	0
$235$	− 24.7846i	− 1.61677i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	11.5361	0.746210	0.373105	−	0.927789i	$-0.378293\pi$
$239$	11.5361	0.746210	0.373105	+	0.927789i	$0.378293\pi$
$240$	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
$245$	4.11439i	0.262859i
$246$	0	0
$247$	18.9282	1.20437
$248$	0	0
$249$	0	0
$250$	0	0
$251$	24.6863i	1.55819i	0.626907	+	0.779094i	$0.284321\pi$
$251$	24.6863i	1.55819i	−0.626907	+	0.779094i	$0.715679\pi$
$252$	0	0
$253$	29.3205i	1.84336i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−3.30734	−0.206306	−0.103153	−	0.994665i	$-0.532893\pi$
$257$	−3.30734	−0.206306	−0.103153	+	0.994665i	$0.532893\pi$
$258$	0	0
$259$	11.4641i	0.712345i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−4.92144	−0.303469	−0.151734	−	0.988421i	$-0.548486\pi$
$263$	−4.92144	−0.303469	−0.151734	+	0.988421i	$0.548486\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	12.3432i	0.752576i	0.926503	+	0.376288i	$0.122800\pi$
$269$	12.3432i	0.752576i	−0.926503	+	0.376288i	$0.877200\pi$
$270$	0	0
$271$	28.9282	1.75726	0.878632	−	0.477500i	$-0.158457\pi$
$271$	28.9282	1.75726	0.878632	+	0.477500i	$0.158457\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	49.0773i	2.95947i
$276$	0	0
$277$	− 9.60770i	− 0.577270i	−0.957439	−	0.288635i	$-0.906799\pi$
$277$	− 9.60770i	− 0.577270i	0.957439	−	0.288635i	$-0.0932015\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−22.4814	−1.34113	−0.670565	−	0.741851i	$-0.733948\pi$
$281$	−22.4814	−1.34113	−0.670565	+	0.741851i	$0.733948\pi$
$282$	0	0
$283$	6.39230i	0.379983i	0.981786	+	0.189992i	$0.0608461\pi$
$283$	6.39230i	0.379983i	−0.981786	+	0.189992i	$0.939154\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1.10245	−0.0650754
$288$	0	0
$289$	−15.7846	−0.928506
$290$	0	0
$291$	0	0
$292$	0	0
$293$	7.93338i	0.463473i	0.972779	+	0.231736i	$0.0744407\pi$
$293$	7.93338i	0.463473i	−0.972779	+	0.231736i	$0.925559\pi$
$294$	0	0
$295$	24.7846	1.44302
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 38.9390i	− 2.25190i
$300$	0	0
$301$	8.92820i	0.514613i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	22.4814	1.28728
$306$	0	0
$307$	− 2.39230i	− 0.136536i	−0.997667	−	0.0682680i	$-0.978253\pi$
$307$	− 2.39230i	− 0.136536i	0.997667	−	0.0682680i	$-0.0217473\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−6.02388	−0.341583	−0.170792	−	0.985307i	$-0.554632\pi$
$311$	−6.02388	−0.341583	−0.170792	+	0.985307i	$0.554632\pi$
$312$	0	0
$313$	12.9282	0.730745	0.365373	−	0.930861i	$-0.380942\pi$
$313$	12.9282	0.730745	0.365373	+	0.930861i	$0.380942\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 4.40979i	− 0.247678i	−0.992302	−	0.123839i	$-0.960479\pi$
$317$	− 4.40979i	− 0.247678i	0.992302	−	0.123839i	$-0.0395207\pi$
$318$	0	0
$319$	24.7846	1.38767
$320$	0	0
$321$	0	0
$322$	0	0
$323$	3.81899i	0.212494i
$324$	0	0
$325$	− 65.1769i	− 3.61536i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	6.02388	0.332108
$330$	0	0
$331$	− 6.00000i	− 0.329790i	−0.986311	−	0.164895i	$-0.947272\pi$
$331$	− 6.00000i	− 0.329790i	0.986311	−	0.164895i	$-0.0527285\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−41.1439	−2.24793
$336$	0	0
$337$	−18.0000	−0.980522	−0.490261	−	0.871576i	$-0.663099\pi$
$337$	−18.0000	−0.980522	−0.490261	+	0.871576i	$0.663099\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 8.22878i	− 0.445613i
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 8.52418i	− 0.457602i	−0.973473	−	0.228801i	$-0.926520\pi$
$347$	− 8.52418i	− 0.457602i	0.973473	−	0.228801i	$-0.0734805\pi$
$348$	0	0
$349$	6.53590i	0.349859i	0.984581	+	0.174929i	$0.0559697\pi$
$349$	6.53590i	0.349859i	−0.984581	+	0.174929i	$0.944030\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−17.5600	−0.934625	−0.467312	−	0.884092i	$-0.654778\pi$
$353$	−17.5600	−0.934625	−0.467312	+	0.884092i	$0.654778\pi$
$354$	0	0
$355$	− 38.3923i	− 2.03765i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−7.12633	−0.376113	−0.188057	−	0.982158i	$-0.560219\pi$
$359$	−7.12633	−0.376113	−0.188057	+	0.982158i	$0.560219\pi$
$360$	0	0
$361$	7.00000	0.368421
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 3.81899i	− 0.199895i
$366$	0	0
$367$	−8.78461	−0.458553	−0.229276	−	0.973361i	$-0.573636\pi$
$367$	−8.78461	−0.458553	−0.229276	+	0.973361i	$0.573636\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	− 6.92820i	− 0.358729i	−0.983783	−	0.179364i	$-0.942596\pi$
$373$	− 6.92820i	− 0.358729i	0.983783	−	0.179364i	$-0.0574041\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−32.9151	−1.69521
$378$	0	0
$379$	0.143594i	0.00737590i	0.999993	+	0.00368795i	$0.00117391\pi$
$379$	0.143594i	0.00737590i	−0.999993	+	0.00368795i	$0.998826\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−24.6863	−1.26141	−0.630706	−	0.776021i	$-0.717235\pi$
$383$	−24.6863	−1.26141	−0.630706	+	0.776021i	$0.717235\pi$
$384$	0	0
$385$	−16.9282	−0.862741
$386$	0	0
$387$	0	0
$388$	0	0
$389$	38.9390i	1.97429i	0.159840	+	0.987143i	$0.448902\pi$
$389$	38.9390i	1.97429i	−0.159840	+	0.987143i	$0.551098\pi$
$390$	0	0
$391$	7.85641	0.397316
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 12.0478i	− 0.606189i
$396$	0	0
$397$	25.4641i	1.27801i	0.769204	+	0.639003i	$0.220653\pi$
$397$	25.4641i	1.27801i	−0.769204	+	0.639003i	$0.779347\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−30.7102	−1.53360	−0.766798	−	0.641889i	$-0.778151\pi$
$401$	−30.7102	−1.53360	−0.766798	+	0.641889i	$0.778151\pi$
$402$	0	0
$403$	10.9282i	0.544373i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−47.1678	−2.33802
$408$	0	0
$409$	−4.92820	−0.243684	−0.121842	−	0.992550i	$-0.538880\pi$
$409$	−4.92820	−0.243684	−0.121842	+	0.992550i	$0.538880\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	6.02388i	0.296416i
$414$	0	0
$415$	−58.6410	−2.87857
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 26.8912i	− 1.31372i	−0.754011	−	0.656861i	$-0.771884\pi$
$419$	− 26.8912i	− 1.31372i	0.754011	−	0.656861i	$-0.228116\pi$
$420$	0	0
$421$	5.32051i	0.259306i	0.991559	+	0.129653i	$0.0413863\pi$
$421$	5.32051i	0.259306i	−0.991559	+	0.129653i	$0.958614\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	13.1502	0.637879
$426$	0	0
$427$	5.46410i	0.264426i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−4.92144	−0.237057	−0.118529	−	0.992951i	$-0.537818\pi$
$431$	−4.92144	−0.237057	−0.118529	+	0.992951i	$0.537818\pi$
$432$	0	0
$433$	−35.8564	−1.72315	−0.861574	−	0.507631i	$-0.830521\pi$
$433$	−35.8564	−1.72315	−0.861574	+	0.507631i	$0.830521\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	24.6863i	1.18091i
$438$	0	0
$439$	−13.8564	−0.661330	−0.330665	−	0.943748i	$-0.607273\pi$
$439$	−13.8564	−0.661330	−0.330665	+	0.943748i	$0.607273\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	12.3432i	0.586442i	0.956045	+	0.293221i	$0.0947271\pi$
$443$	12.3432i	0.586442i	−0.956045	+	0.293221i	$0.905273\pi$
$444$	0	0
$445$	63.1769i	2.99487i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	29.0961	1.37313	0.686566	−	0.727068i	$-0.259117\pi$
$449$	29.0961	1.37313	0.686566	+	0.727068i	$0.259117\pi$
$450$	0	0
$451$	− 4.53590i	− 0.213587i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	22.4814	1.05395
$456$	0	0
$457$	23.8564	1.11596	0.557978	−	0.829856i	$-0.311577\pi$
$457$	23.8564	1.11596	0.557978	+	0.829856i	$0.311577\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	28.2099i	1.31387i	0.753948	+	0.656934i	$0.228147\pi$
$461$	28.2099i	1.31387i	−0.753948	+	0.656934i	$0.771853\pi$
$462$	0	0
$463$	−16.0000	−0.743583	−0.371792	−	0.928316i	$-0.621256\pi$
$463$	−16.0000	−0.743583	−0.371792	+	0.928316i	$0.621256\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	12.6386i	0.584843i	0.956289	+	0.292422i	$0.0944611\pi$
$467$	12.6386i	0.584843i	−0.956289	+	0.292422i	$0.905539\pi$
$468$	0	0
$469$	− 10.0000i	− 0.461757i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−36.7341	−1.68904
$474$	0	0
$475$	41.3205i	1.89591i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−1.61410	−0.0737499	−0.0368749	−	0.999320i	$-0.511740\pi$
$479$	−1.61410	−0.0737499	−0.0368749	+	0.999320i	$0.511740\pi$
$480$	0	0
$481$	62.6410	2.85618
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 20.2765i	− 0.920711i
$486$	0	0
$487$	16.7846	0.760583	0.380292	−	0.924867i	$-0.375824\pi$
$487$	16.7846	0.760583	0.380292	+	0.924867i	$0.375824\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 12.3432i	− 0.557039i	−0.960431	−	0.278520i	$-0.910156\pi$
$491$	− 12.3432i	− 0.557039i	0.960431	−	0.278520i	$-0.0898438\pi$
$492$	0	0
$493$	− 6.64102i	− 0.299096i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	9.33123	0.418563
$498$	0	0
$499$	− 3.07180i	− 0.137513i	−0.997633	−	0.0687563i	$-0.978097\pi$
$499$	− 3.07180i	− 0.137513i	0.997633	−	0.0687563i	$-0.0219031\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−3.81899	−0.170280	−0.0851402	−	0.996369i	$-0.527134\pi$
$503$	−3.81899	−0.170280	−0.0851402	+	0.996369i	$0.527134\pi$
$504$	0	0
$505$	−16.9282	−0.753295
$506$	0	0
$507$	0	0
$508$	0	0
$509$	16.1622i	0.716375i	0.933650	+	0.358188i	$0.116605\pi$
$509$	16.1622i	0.716375i	−0.933650	+	0.358188i	$0.883395\pi$
$510$	0	0
$511$	0.928203	0.0410613
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 36.7341i	− 1.61870i
$516$	0	0
$517$	24.7846i	1.09003i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−3.30734	−0.144897	−0.0724486	−	0.997372i	$-0.523081\pi$
$521$	−3.30734	−0.144897	−0.0724486	+	0.997372i	$0.523081\pi$
$522$	0	0
$523$	8.00000i	0.349816i	0.984585	+	0.174908i	$0.0559627\pi$
$523$	8.00000i	0.349816i	−0.984585	+	0.174908i	$0.944037\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−2.20489	−0.0960467
$528$	0	0
$529$	27.7846	1.20803
$530$	0	0
$531$	0	0
$532$	0	0
$533$	6.02388i	0.260923i
$534$	0	0
$535$	−66.4974	−2.87493
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 4.11439i	− 0.177219i
$540$	0	0
$541$	− 27.1769i	− 1.16843i	−0.811600	−	0.584213i	$-0.801403\pi$
$541$	− 27.1769i	− 1.16843i	0.811600	−	0.584213i	$-0.198597\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	16.4576	0.704964
$546$	0	0
$547$	4.92820i	0.210715i	0.994434	+	0.105357i	$0.0335986\pi$
$547$	4.92820i	0.210715i	−0.994434	+	0.105357i	$0.966401\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	20.8673	0.888979
$552$	0	0
$553$	2.92820	0.124520
$554$	0	0
$555$	0	0
$556$	0	0
$557$	12.0478i	0.510480i	0.966878	+	0.255240i	$0.0821545\pi$
$557$	12.0478i	0.510480i	−0.966878	+	0.255240i	$0.917845\pi$
$558$	0	0
$559$	48.7846	2.06337
$560$	0	0
$561$	0	0
$562$	0	0
$563$	10.4337i	0.439727i	0.975531	+	0.219863i	$0.0705612\pi$
$563$	10.4337i	0.439727i	−0.975531	+	0.219863i	$0.929439\pi$
$564$	0	0
$565$	− 33.8564i	− 1.42435i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	1.61410	0.0676664	0.0338332	−	0.999427i	$-0.489229\pi$
$569$	1.61410	0.0676664	0.0338332	+	0.999427i	$0.489229\pi$
$570$	0	0
$571$	14.7846i	0.618717i	0.950946	+	0.309358i	$0.100114\pi$
$571$	14.7846i	0.618717i	−0.950946	+	0.309358i	$0.899886\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	85.0043	3.54493
$576$	0	0
$577$	3.07180	0.127881	0.0639403	−	0.997954i	$-0.479633\pi$
$577$	3.07180	0.127881	0.0639403	+	0.997954i	$0.479633\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 14.2527i	− 0.591300i
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 8.22878i	− 0.339638i	−0.985475	−	0.169819i	$-0.945682\pi$
$587$	− 8.22878i	− 0.339638i	0.985475	−	0.169819i	$-0.0543183\pi$
$588$	0	0
$589$	− 6.92820i	− 0.285472i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	34.0176	1.39693	0.698467	−	0.715642i	$-0.253866\pi$
$593$	34.0176	1.39693	0.698467	+	0.715642i	$0.253866\pi$
$594$	0	0
$595$	4.53590i	0.185954i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−13.7410	−0.561443	−0.280721	−	0.959789i	$-0.590574\pi$
$599$	−13.7410	−0.561443	−0.280721	+	0.959789i	$0.590574\pi$
$600$	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
$605$	− 24.3909i	− 0.991633i
$606$	0	0
$607$	−42.9282	−1.74240	−0.871201	−	0.490926i	$-0.836658\pi$
$607$	−42.9282	−1.74240	−0.871201	+	0.490926i	$0.836658\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 32.9151i	− 1.33160i
$612$	0	0
$613$	23.7128i	0.957751i	0.877883	+	0.478876i	$0.158956\pi$
$613$	23.7128i	0.957751i	−0.877883	+	0.478876i	$0.841044\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−2.20489	−0.0887657	−0.0443829	−	0.999015i	$-0.514132\pi$
$617$	−2.20489	−0.0887657	−0.0443829	+	0.999015i	$0.514132\pi$
$618$	0	0
$619$	− 32.0000i	− 1.28619i	−0.765787	−	0.643094i	$-0.777650\pi$
$619$	− 32.0000i	− 1.28619i	0.765787	−	0.643094i	$-0.222350\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−15.3551	−0.615190
$624$	0	0
$625$	57.6410	2.30564
$626$	0	0
$627$	0	0
$628$	0	0
$629$	12.6386i	0.503933i
$630$	0	0
$631$	28.7846	1.14590	0.572949	−	0.819591i	$-0.305799\pi$
$631$	28.7846	1.14590	0.572949	+	0.819591i	$0.305799\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	61.4204i	2.43740i
$636$	0	0
$637$	5.46410i	0.216496i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	30.7102	1.21298	0.606490	−	0.795091i	$-0.292577\pi$
$641$	30.7102	1.21298	0.606490	+	0.795091i	$0.292577\pi$
$642$	0	0
$643$	42.1051i	1.66046i	0.557418	+	0.830232i	$0.311792\pi$
$643$	42.1051i	1.66046i	−0.557418	+	0.830232i	$0.688208\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−35.1200	−1.38071	−0.690355	−	0.723471i	$-0.742546\pi$
$647$	−35.1200	−1.38071	−0.690355	+	0.723471i	$0.742546\pi$
$648$	0	0
$649$	−24.7846	−0.972881
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 32.3243i	− 1.26495i	−0.774581	−	0.632474i	$-0.782039\pi$
$653$	− 32.3243i	− 1.26495i	0.774581	−	0.632474i	$-0.217961\pi$
$654$	0	0
$655$	58.6410	2.29129
$656$	0	0
$657$	0	0
$658$	0	0
$659$	28.8007i	1.12192i	0.827844	+	0.560959i	$0.189567\pi$
$659$	28.8007i	1.12192i	−0.827844	+	0.560959i	$0.810433\pi$
$660$	0	0
$661$	− 13.1769i	− 0.512523i	−0.966608	−	0.256261i	$-0.917509\pi$
$661$	− 13.1769i	− 0.512523i	0.966608	−	0.256261i	$-0.0824908\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−14.2527	−0.552695
$666$	0	0
$667$	− 42.9282i	− 1.66219i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−22.4814	−0.867887
$672$	0	0
$673$	34.6410	1.33531	0.667657	−	0.744469i	$-0.267297\pi$
$673$	34.6410	1.33531	0.667657	+	0.744469i	$0.267297\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	8.52418i	0.327611i	0.986493	+	0.163805i	$0.0523769\pi$
$677$	8.52418i	0.327611i	−0.986493	+	0.163805i	$0.947623\pi$
$678$	0	0
$679$	4.92820	0.189127
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 4.11439i	− 0.157433i	−0.996897	−	0.0787164i	$-0.974918\pi$
$683$	− 4.11439i	− 0.157433i	0.996897	−	0.0787164i	$-0.0250821\pi$
$684$	0	0
$685$	58.6410i	2.24056i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	7.71281i	0.293409i	0.989180	+	0.146705i	$0.0468667\pi$
$691$	7.71281i	0.293409i	−0.989180	+	0.146705i	$0.953133\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	73.4682	2.78681
$696$	0	0
$697$	−1.21539	−0.0460362
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 24.6863i	− 0.932390i	−0.884682	−	0.466195i	$-0.845624\pi$
$701$	− 24.6863i	− 0.932390i	0.884682	−	0.466195i	$-0.154376\pi$
$702$	0	0
$703$	−39.7128	−1.49780
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 4.11439i	− 0.154738i
$708$	0	0
$709$	− 41.8564i	− 1.57195i	−0.618258	−	0.785975i	$-0.712161\pi$
$709$	− 41.8564i	− 1.57195i	0.618258	−	0.785975i	$-0.287839\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−14.2527	−0.533766
$714$	0	0
$715$	92.4974i	3.45921i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−0.590800	−0.0220331	−0.0110166	−	0.999939i	$-0.503507\pi$
$719$	−0.590800	−0.0220331	−0.0110166	+	0.999939i	$0.503507\pi$
$720$	0	0
$721$	8.92820	0.332504
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 71.8541i	− 2.66860i
$726$	0	0
$727$	−3.07180	−0.113927	−0.0569633	−	0.998376i	$-0.518142\pi$
$727$	−3.07180	−0.113927	−0.0569633	+	0.998376i	$0.518142\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	9.84287i	0.364052i
$732$	0	0
$733$	1.75129i	0.0646853i	0.999477	+	0.0323427i	$0.0102968\pi$
$733$	1.75129i	0.0646853i	−0.999477	+	0.0323427i	$0.989703\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	41.1439	1.51555
$738$	0	0
$739$	− 5.71281i	− 0.210149i	−0.994464	−	0.105075i	$-0.966492\pi$
$739$	− 5.71281i	− 0.210149i	0.994464	−	0.105075i	$-0.0335081\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−44.4512	−1.63076	−0.815379	−	0.578928i	$-0.803471\pi$
$743$	−44.4512	−1.63076	−0.815379	+	0.578928i	$0.803471\pi$
$744$	0	0
$745$	67.7128	2.48081
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 16.1622i	− 0.590552i
$750$	0	0
$751$	−27.7128	−1.01125	−0.505627	−	0.862752i	$-0.668739\pi$
$751$	−27.7128	−1.01125	−0.505627	+	0.862752i	$0.668739\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	16.4576i	0.598952i
$756$	0	0
$757$	7.71281i	0.280327i	0.990128	+	0.140163i	$0.0447628\pi$
$757$	7.71281i	0.280327i	−0.990128	+	0.140163i	$0.955237\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−25.1980	−0.913426	−0.456713	−	0.889614i	$-0.650973\pi$
$761$	−25.1980	−0.913426	−0.456713	+	0.889614i	$0.650973\pi$
$762$	0	0
$763$	4.00000i	0.144810i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	32.9151	1.18850
$768$	0	0
$769$	51.5692	1.85963	0.929817	−	0.368023i	$-0.119965\pi$
$769$	51.5692	1.85963	0.929817	+	0.368023i	$0.119965\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 33.2105i	− 1.19450i	−0.802055	−	0.597250i	$-0.796260\pi$
$773$	− 33.2105i	− 1.19450i	0.802055	−	0.597250i	$-0.203740\pi$
$774$	0	0
$775$	−23.8564	−0.856947
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 3.81899i	− 0.136830i
$780$	0	0
$781$	38.3923i	1.37378i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	34.5292	1.23240
$786$	0	0
$787$	45.5692i	1.62437i	0.583402	+	0.812184i	$0.301721\pi$
$787$	45.5692i	1.62437i	−0.583402	+	0.812184i	$0.698279\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	8.22878	0.292582
$792$	0	0
$793$	29.8564	1.06023
$794$	0	0
$795$	0	0
$796$	0	0
$797$	16.7530i	0.593420i	0.954968	+	0.296710i	$0.0958895\pi$
$797$	16.7530i	0.593420i	−0.954968	+	0.296710i	$0.904110\pi$
$798$	0	0
$799$	6.64102	0.234942
$800$	0	0
$801$	0	0
$802$	0	0
$803$	3.81899i	0.134769i
$804$	0	0
$805$	29.3205i	1.03341i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−45.5537	−1.60158	−0.800791	−	0.598944i	$-0.795587\pi$
$809$	−45.5537	−1.60158	−0.800791	+	0.598944i	$0.795587\pi$
$810$	0	0
$811$	37.8564i	1.32932i	0.747147	+	0.664659i	$0.231423\pi$
$811$	37.8564i	1.32932i	−0.747147	+	0.664659i	$0.768577\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	65.2394	2.28524
$816$	0	0
$817$	−30.9282	−1.08204
$818$	0	0
$819$	0	0
$820$	0	0
$821$	28.5053i	0.994843i	0.867509	+	0.497421i	$0.165720\pi$
$821$	28.5053i	0.994843i	−0.867509	+	0.497421i	$0.834280\pi$
$822$	0	0
$823$	−14.1436	−0.493015	−0.246507	−	0.969141i	$-0.579283\pi$
$823$	−14.1436	−0.493015	−0.246507	+	0.969141i	$0.579283\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	33.2105i	1.15484i	0.816446	+	0.577421i	$0.195941\pi$
$827$	33.2105i	1.15484i	−0.816446	+	0.577421i	$0.804059\pi$
$828$	0	0
$829$	− 13.1769i	− 0.457653i	−0.973467	−	0.228827i	$-0.926511\pi$
$829$	− 13.1769i	− 0.457653i	0.973467	−	0.228827i	$-0.0734889\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−1.10245	−0.0381975
$834$	0	0
$835$	58.6410i	2.02936i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	35.1200	1.21248	0.606239	−	0.795283i	$-0.292678\pi$
$839$	35.1200	1.21248	0.606239	+	0.795283i	$0.292678\pi$
$840$	0	0
$841$	−7.28719	−0.251282
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 69.3538i	− 2.38584i
$846$	0	0
$847$	5.92820	0.203695
$848$	0	0
$849$	0	0
$850$	0	0
$851$	81.6970i	2.80054i
$852$	0	0
$853$	− 28.3923i	− 0.972134i	−0.873922	−	0.486067i	$-0.838431\pi$
$853$	− 28.3923i	− 0.972134i	0.873922	−	0.486067i	$-0.161569\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−37.2458	−1.27229	−0.636145	−	0.771569i	$-0.719472\pi$
$857$	−37.2458	−1.27229	−0.636145	+	0.771569i	$0.719472\pi$
$858$	0	0
$859$	33.3205i	1.13688i	0.822724	+	0.568441i	$0.192453\pi$
$859$	33.3205i	1.13688i	−0.822724	+	0.568441i	$0.807547\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−14.7643	−0.502583	−0.251292	−	0.967911i	$-0.580855\pi$
$863$	−14.7643	−0.502583	−0.251292	+	0.967911i	$0.580855\pi$
$864$	0	0
$865$	−50.7846	−1.72673
$866$	0	0
$867$	0	0
$868$	0	0
$869$	12.0478i	0.408693i
$870$	0	0
$871$	−54.6410	−1.85144
$872$	0	0
$873$	0	0
$874$	0	0
$875$	28.5053i	0.963656i
$876$	0	0
$877$	44.0000i	1.48577i	0.669417	+	0.742887i	$0.266544\pi$
$877$	44.0000i	1.48577i	−0.669417	+	0.742887i	$0.733456\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	46.0653	1.55198	0.775990	−	0.630745i	$-0.217251\pi$
$881$	46.0653	1.55198	0.775990	+	0.630745i	$0.217251\pi$
$882$	0	0
$883$	− 24.9282i	− 0.838901i	−0.907778	−	0.419450i	$-0.862223\pi$
$883$	− 24.9282i	− 0.838901i	0.907778	−	0.419450i	$-0.137777\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−21.8906	−0.735016	−0.367508	−	0.930020i	$-0.619789\pi$
$887$	−21.8906	−0.735016	−0.367508	+	0.930020i	$0.619789\pi$
$888$	0	0
$889$	−14.9282	−0.500676
$890$	0	0
$891$	0	0
$892$	0	0
$893$	20.8673i	0.698299i
$894$	0	0
$895$	−1.21539	−0.0406260
$896$	0	0
$897$	0	0
$898$	0	0
$899$	12.0478i	0.401816i
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	59.8064	1.98803
$906$	0	0
$907$	44.6410i	1.48228i	0.671350	+	0.741140i	$0.265715\pi$
$907$	44.6410i	1.48228i	−0.671350	+	0.741140i	$0.734285\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	52.0892	1.72579	0.862896	−	0.505381i	$-0.168648\pi$
$911$	52.0892	1.72579	0.862896	+	0.505381i	$0.168648\pi$
$912$	0	0
$913$	58.6410	1.94073
$914$	0	0
$915$	0	0
$916$	0	0
$917$	14.2527i	0.470664i
$918$	0	0
$919$	7.21539	0.238014	0.119007	−	0.992893i	$-0.462029\pi$
$919$	7.21539	0.238014	0.119007	+	0.992893i	$0.462029\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 50.9868i	− 1.67825i
$924$	0	0
$925$	136.746i	4.49619i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−13.1502	−0.431445	−0.215722	−	0.976455i	$-0.569211\pi$
$929$	−13.1502	−0.431445	−0.215722	+	0.976455i	$0.569211\pi$
$930$	0	0
$931$	− 3.46410i	− 0.113531i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−18.6625	−0.610328
$936$	0	0
$937$	16.9282	0.553020	0.276510	−	0.961011i	$-0.410822\pi$
$937$	16.9282	0.553020	0.276510	+	0.961011i	$0.410822\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	28.2099i	0.919617i	0.888018	+	0.459809i	$0.152082\pi$
$941$	28.2099i	0.919617i	−0.888018	+	0.459809i	$0.847918\pi$
$942$	0	0
$943$	−7.85641	−0.255840
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 52.8963i	− 1.71890i	−0.511222	−	0.859449i	$-0.670807\pi$
$947$	− 52.8963i	− 1.71890i	0.511222	−	0.859449i	$-0.329193\pi$
$948$	0	0
$949$	− 5.07180i	− 0.164637i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−20.8673	−0.675960	−0.337980	−	0.941153i	$-0.609744\pi$
$953$	−20.8673	−0.675960	−0.337980	+	0.941153i	$0.609744\pi$
$954$	0	0
$955$	− 106.105i	− 3.43348i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−14.2527	−0.460243
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 61.4204i	− 1.97719i
$966$	0	0
$967$	4.78461	0.153863	0.0769313	−	0.997036i	$-0.475488\pi$
$967$	4.78461	0.153863	0.0769313	+	0.997036i	$0.475488\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 20.8673i	− 0.669665i	−0.942278	−	0.334833i	$-0.891320\pi$
$971$	− 20.8673i	− 0.669665i	0.942278	−	0.334833i	$-0.108680\pi$
$972$	0	0
$973$	17.8564i	0.572450i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−46.5770	−1.49013	−0.745065	−	0.666992i	$-0.767581\pi$
$977$	−46.5770	−1.49013	−0.745065	+	0.666992i	$0.767581\pi$
$978$	0	0
$979$	− 63.1769i	− 2.01914i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−25.2771	−0.806216	−0.403108	−	0.915153i	$-0.632070\pi$
$983$	−25.2771	−0.806216	−0.403108	+	0.915153i	$0.632070\pi$
$984$	0	0
$985$	58.6410	1.86846
$986$	0	0
$987$	0	0
$988$	0	0
$989$	63.6253i	2.02317i
$990$	0	0
$991$	−52.0000	−1.65183	−0.825917	−	0.563791i	$-0.809342\pi$
$991$	−52.0000	−1.65183	−0.825917	+	0.563791i	$0.809342\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 69.0584i	− 2.18930i
$996$	0	0
$997$	− 59.0333i	− 1.86960i	−0.355169	−	0.934802i	$-0.615577\pi$
$997$	− 59.0333i	− 1.86960i	0.355169	−	0.934802i	$-0.384423\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4032.2.c.q.2017.8	yes	8
3.2	odd	2	inner	4032.2.c.q.2017.2	yes	8
4.3	odd	2		4032.2.c.r.2017.8	yes	8
8.3	odd	2		4032.2.c.r.2017.1	yes	8
8.5	even	2	inner	4032.2.c.q.2017.1	✓	8
12.11	even	2		4032.2.c.r.2017.2	yes	8
24.5	odd	2	inner	4032.2.c.q.2017.7	yes	8
24.11	even	2		4032.2.c.r.2017.7	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4032.2.c.q.2017.1	✓	8	8.5	even	2	inner
4032.2.c.q.2017.2	yes	8	3.2	odd	2	inner
4032.2.c.q.2017.7	yes	8	24.5	odd	2	inner
4032.2.c.q.2017.8	yes	8	1.1	even	1	trivial
4032.2.c.r.2017.1	yes	8	8.3	odd	2
4032.2.c.r.2017.2	yes	8	12.11	even	2
4032.2.c.r.2017.7	yes	8	24.11	even	2
4032.2.c.r.2017.8	yes	8	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4032.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+4.11439i q^{5} -1.00000 q^{7} +O(q^{10})\)	\(q+4.11439i q^{5} -1.00000 q^{7} -4.11439i q^{11} +5.46410i q^{13} -1.10245 q^{17} -3.46410i q^{19} -7.12633 q^{23} -11.9282 q^{25} +6.02388i q^{29} +2.00000 q^{31} -4.11439i q^{35} -11.4641i q^{37} +1.10245 q^{41} -8.92820i q^{43} -6.02388 q^{47} +1.00000 q^{49} +16.9282 q^{55} -6.02388i q^{59} -5.46410i q^{61} -22.4814 q^{65} +10.0000i q^{67} -9.33123 q^{71} -0.928203 q^{73} +4.11439i q^{77} -2.92820 q^{79} +14.2527i q^{83} -4.53590i q^{85} +15.3551 q^{89} -5.46410i q^{91} +14.2527 q^{95} -4.92820 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{7}+O(q^{10}) \) 8 * q - 8 * q^7	\( 8 q - 8 q^{7} - 40 q^{25} + 16 q^{31} + 8 q^{49} + 80 q^{55} + 48 q^{73} + 32 q^{79} + 16 q^{97}+O(q^{100}) \) 8 * q - 8 * q^7 - 40 * q^25 + 16 * q^31 + 8 * q^49 + 80 * q^55 + 48 * q^73 + 32 * q^79 + 16 * q^97

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