LMFDB - Embedded newform 4032.2.p.f.1567.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4032,2,Mod(1567,4032)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

chi = DirichletCharacter(H, H._module([1, 1, 0, 1]))

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

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

chi := DirichletCharacter("4032.1567");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4032 = 2^{6} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4032.p (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:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	no (minimal twist has level 448)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1567.2
Root			$0.965926 + 0.258819i$ of defining polynomial
Character	$\chi$	$=$	4032.1567
Dual form			4032.2.p.f.1567.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{5} +(-2.44949 + 1.00000i) q^{7} +O(q^{10})$	$q-1.41421 q^{5} +(-2.44949 + 1.00000i) q^{7} -3.46410 q^{11} +4.24264 q^{13} +4.89898i q^{17} -4.24264i q^{19} -3.00000 q^{25} -4.89898 q^{31} +(3.46410 - 1.41421i) q^{35} +6.92820i q^{37} -9.79796i q^{41} +3.46410 q^{43} -4.89898 q^{47} +(5.00000 - 4.89898i) q^{49} +13.8564i q^{53} +4.89898 q^{55} -7.07107i q^{59} -4.24264 q^{61} -6.00000 q^{65} +10.3923 q^{67} -6.00000i q^{71} +4.89898i q^{73} +(8.48528 - 3.46410i) q^{77} +14.0000i q^{79} -9.89949i q^{83} -6.92820i q^{85} -14.6969i q^{89} +(-10.3923 + 4.24264i) q^{91} +6.00000i q^{95} -4.89898i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 24 q^{25} + 40 q^{49} - 48 q^{65}+O(q^{100}) $ 8 * q - 24 * q^25 + 40 * q^49 - 48 * q^65

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
$5$	−1.41421			−0.632456			−0.316228	−	0.948683i	$-0.602416\pi$
$5$	−1.41421			−0.632456			−0.316228	+	0.948683i	$0.602416\pi$
$6$		0			0
$7$	−2.44949	+	1.00000i	−0.925820	+	0.377964i
$7$	−2.44949	+	1.00000i	−0.925820	+	0.377964i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−3.46410			−1.04447			−0.522233	−	0.852803i	$-0.674901\pi$
$11$	−3.46410			−1.04447			−0.522233	+	0.852803i	$0.674901\pi$
$12$		0			0
$13$	4.24264			1.17670			0.588348	−	0.808608i	$-0.299778\pi$
$13$	4.24264			1.17670			0.588348	+	0.808608i	$0.299778\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			4.89898i			1.18818i	0.804400	+	0.594089i	$0.202487\pi$
$17$			4.89898i			1.18818i	−0.804400	+	0.594089i	$0.797513\pi$
$18$		0			0
$19$		−	4.24264i		−	0.973329i	−0.873589	−	0.486664i	$-0.838214\pi$
$19$		−	4.24264i		−	0.973329i	0.873589	−	0.486664i	$-0.161786\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$		0			0
$25$	−3.00000			−0.600000
$26$		0			0
$27$		0			0
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	−4.89898			−0.879883			−0.439941	−	0.898027i	$-0.645001\pi$
$31$	−4.89898			−0.879883			−0.439941	+	0.898027i	$0.645001\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	3.46410	−	1.41421i	0.585540	−	0.239046i
$36$		0			0
$37$			6.92820i			1.13899i	0.821995	+	0.569495i	$0.192861\pi$
$37$			6.92820i			1.13899i	−0.821995	+	0.569495i	$0.807139\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	9.79796i		−	1.53018i	−0.643921	−	0.765092i	$-0.722693\pi$
$41$		−	9.79796i		−	1.53018i	0.643921	−	0.765092i	$-0.277307\pi$
$42$		0			0
$43$	3.46410			0.528271			0.264135	−	0.964486i	$-0.414913\pi$
$43$	3.46410			0.528271			0.264135	+	0.964486i	$0.414913\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−4.89898			−0.714590			−0.357295	−	0.933992i	$-0.616301\pi$
$47$	−4.89898			−0.714590			−0.357295	+	0.933992i	$0.616301\pi$
$48$		0			0
$49$	5.00000	−	4.89898i	0.714286	−	0.699854i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			13.8564i			1.90332i	0.307148	+	0.951662i	$0.400625\pi$
$53$			13.8564i			1.90332i	−0.307148	+	0.951662i	$0.599375\pi$
$54$		0			0
$55$	4.89898			0.660578
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	7.07107i		−	0.920575i	−0.887770	−	0.460287i	$-0.847746\pi$
$59$		−	7.07107i		−	0.920575i	0.887770	−	0.460287i	$-0.152254\pi$
$60$		0			0
$61$	−4.24264			−0.543214			−0.271607	−	0.962408i	$-0.587555\pi$
$61$	−4.24264			−0.543214			−0.271607	+	0.962408i	$0.587555\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−6.00000			−0.744208
$66$		0			0
$67$	10.3923			1.26962			0.634811	−	0.772667i	$-0.281078\pi$
$67$	10.3923			1.26962			0.634811	+	0.772667i	$0.281078\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	6.00000i		−	0.712069i	−0.934473	−	0.356034i	$-0.884129\pi$
$71$		−	6.00000i		−	0.712069i	0.934473	−	0.356034i	$-0.115871\pi$
$72$		0			0
$73$			4.89898i			0.573382i	0.958023	+	0.286691i	$0.0925553\pi$
$73$			4.89898i			0.573382i	−0.958023	+	0.286691i	$0.907445\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	8.48528	−	3.46410i	0.966988	−	0.394771i
$78$		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
$83$		−	9.89949i		−	1.08661i	−0.839535	−	0.543305i	$-0.817173\pi$
$83$		−	9.89949i		−	1.08661i	0.839535	−	0.543305i	$-0.182827\pi$
$84$		0			0
$85$		−	6.92820i		−	0.751469i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	14.6969i		−	1.55787i	−0.627103	−	0.778936i	$-0.715760\pi$
$89$		−	14.6969i		−	1.55787i	0.627103	−	0.778936i	$-0.284240\pi$
$90$		0			0
$91$	−10.3923	+	4.24264i	−1.08941	+	0.444750i
$92$		0			0
$93$		0			0
$94$		0			0
$95$			6.00000i			0.615587i
$96$		0			0
$97$		−	4.89898i		−	0.497416i	−0.968579	−	0.248708i	$-0.919994\pi$
$97$		−	4.89898i		−	0.497416i	0.968579	−	0.248708i	$-0.0800060\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	18.3848			1.82935			0.914677	−	0.404186i	$-0.132445\pi$
$101$	18.3848			1.82935			0.914677	+	0.404186i	$0.132445\pi$
$102$		0			0
$103$		0			0			−	1.00000i	$-0.5\pi$
$103$		0			0				1.00000i	$0.5\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	17.3205			1.67444			0.837218	−	0.546869i	$-0.184180\pi$
$107$	17.3205			1.67444			0.837218	+	0.546869i	$0.184180\pi$
$108$		0			0
$109$		−	13.8564i		−	1.32720i	−0.748086	−	0.663602i	$-0.769027\pi$
$109$		−	13.8564i		−	1.32720i	0.748086	−	0.663602i	$-0.230973\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−4.89898	−	12.0000i	−0.449089	−	1.10004i
$120$		0			0
$121$	1.00000			0.0909091
$122$		0			0
$123$		0			0
$124$		0			0
$125$	11.3137			1.01193
$126$		0			0
$127$		−	16.0000i		−	1.41977i	−0.704317	−	0.709885i	$-0.748747\pi$
$127$		−	16.0000i		−	1.41977i	0.704317	−	0.709885i	$-0.251253\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	15.5563i		−	1.35916i	−0.733599	−	0.679582i	$-0.762161\pi$
$131$		−	15.5563i		−	1.35916i	0.733599	−	0.679582i	$-0.237839\pi$
$132$		0			0
$133$	4.24264	+	10.3923i	0.367884	+	0.901127i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	18.0000			1.53784			0.768922	−	0.639343i	$-0.220793\pi$
$137$	18.0000			1.53784			0.768922	+	0.639343i	$0.220793\pi$
$138$		0			0
$139$		−	4.24264i		−	0.359856i	−0.983680	−	0.179928i	$-0.942414\pi$
$139$		−	4.24264i		−	0.359856i	0.983680	−	0.179928i	$-0.0575865\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−14.6969			−1.22902
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	6.92820i		−	0.567581i	−0.958886	−	0.283790i	$-0.908408\pi$
$149$		−	6.92820i		−	0.567581i	0.958886	−	0.283790i	$-0.0915919\pi$
$150$		0			0
$151$		−	8.00000i		−	0.651031i	−0.945537	−	0.325515i	$-0.894462\pi$
$151$		−	8.00000i		−	0.651031i	0.945537	−	0.325515i	$-0.105538\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	6.92820			0.556487
$156$		0			0
$157$	21.2132			1.69300			0.846499	−	0.532390i	$-0.178706\pi$
$157$	21.2132			1.69300			0.846499	+	0.532390i	$0.178706\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−3.46410			−0.271329			−0.135665	−	0.990755i	$-0.543317\pi$
$163$	−3.46410			−0.271329			−0.135665	+	0.990755i	$0.543317\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	9.79796			0.758189			0.379094	−	0.925358i	$-0.376236\pi$
$167$	9.79796			0.758189			0.379094	+	0.925358i	$0.376236\pi$
$168$		0			0
$169$	5.00000			0.384615
$170$		0			0
$171$		0			0
$172$		0			0
$173$	7.07107			0.537603			0.268802	−	0.963196i	$-0.413372\pi$
$173$	7.07107			0.537603			0.268802	+	0.963196i	$0.413372\pi$
$174$		0			0
$175$	7.34847	−	3.00000i	0.555492	−	0.226779i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−17.3205			−1.29460			−0.647298	−	0.762237i	$-0.724101\pi$
$179$	−17.3205			−1.29460			−0.647298	+	0.762237i	$0.724101\pi$
$180$		0			0
$181$	−4.24264			−0.315353			−0.157676	−	0.987491i	$-0.550400\pi$
$181$	−4.24264			−0.315353			−0.157676	+	0.987491i	$0.550400\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	9.79796i		−	0.720360i
$186$		0			0
$187$		−	16.9706i		−	1.24101i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	18.0000i		−	1.30243i	−0.758891	−	0.651217i	$-0.774259\pi$
$191$		−	18.0000i		−	1.30243i	0.758891	−	0.651217i	$-0.225741\pi$
$192$		0			0
$193$	−4.00000			−0.287926			−0.143963	−	0.989583i	$-0.545985\pi$
$193$	−4.00000			−0.287926			−0.143963	+	0.989583i	$0.545985\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	6.92820i		−	0.493614i	−0.969065	−	0.246807i	$-0.920619\pi$
$197$		−	6.92820i		−	0.493614i	0.969065	−	0.246807i	$-0.0793814\pi$
$198$		0			0
$199$	−9.79796			−0.694559			−0.347279	−	0.937762i	$-0.612894\pi$
$199$	−9.79796			−0.694559			−0.347279	+	0.937762i	$0.612894\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$			13.8564i			0.967773i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			14.6969i			1.01661i
$210$		0			0
$211$	10.3923			0.715436			0.357718	−	0.933830i	$-0.383555\pi$
$211$	10.3923			0.715436			0.357718	+	0.933830i	$0.383555\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−4.89898			−0.334108
$216$		0			0
$217$	12.0000	−	4.89898i	0.814613	−	0.332564i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			20.7846i			1.39812i
$222$		0			0
$223$	−29.3939			−1.96836			−0.984180	−	0.177173i	$-0.943305\pi$
$223$	−29.3939			−1.96836			−0.984180	+	0.177173i	$0.943305\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			9.89949i			0.657053i	0.944495	+	0.328526i	$0.106552\pi$
$227$			9.89949i			0.657053i	−0.944495	+	0.328526i	$0.893448\pi$
$228$		0			0
$229$	21.2132			1.40181			0.700904	−	0.713256i	$-0.252780\pi$
$229$	21.2132			1.40181			0.700904	+	0.713256i	$0.252780\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−6.00000			−0.393073			−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000			−0.393073			−0.196537	+	0.980497i	$0.562969\pi$
$234$		0			0
$235$	6.92820			0.451946
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$			14.6969i			0.946713i	0.880871	+	0.473357i	$0.156958\pi$
$241$			14.6969i			0.946713i	−0.880871	+	0.473357i	$0.843042\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−7.07107	+	6.92820i	−0.451754	+	0.442627i
$246$		0			0
$247$		−	18.0000i		−	1.14531i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	15.5563i		−	0.981908i	−0.871185	−	0.490954i	$-0.836648\pi$
$251$		−	15.5563i		−	0.981908i	0.871185	−	0.490954i	$-0.163352\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0		1.00000			$0$
$257$		0			0		−1.00000			$\pi$
$258$		0			0
$259$	−6.92820	−	16.9706i	−0.430498	−	1.05450i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			18.0000i			1.10993i	0.831875	+	0.554964i	$0.187268\pi$
$263$			18.0000i			1.10993i	−0.831875	+	0.554964i	$0.812732\pi$
$264$		0			0
$265$		−	19.5959i		−	1.20377i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−9.89949			−0.603583			−0.301791	−	0.953374i	$-0.597585\pi$
$269$	−9.89949			−0.603583			−0.301791	+	0.953374i	$0.597585\pi$
$270$		0			0
$271$	19.5959			1.19037			0.595184	−	0.803590i	$-0.297079\pi$
$271$	19.5959			1.19037			0.595184	+	0.803590i	$0.297079\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	10.3923			0.626680
$276$		0			0
$277$			13.8564i			0.832551i	0.909239	+	0.416275i	$0.136665\pi$
$277$			13.8564i			0.832551i	−0.909239	+	0.416275i	$0.863335\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	6.00000			0.357930			0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000			0.357930			0.178965	+	0.983855i	$0.442725\pi$
$282$		0			0
$283$		−	12.7279i		−	0.756596i	−0.925684	−	0.378298i	$-0.876509\pi$
$283$		−	12.7279i		−	0.756596i	0.925684	−	0.378298i	$-0.123491\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	9.79796	+	24.0000i	0.578355	+	1.41668i
$288$		0			0
$289$	−7.00000			−0.411765
$290$		0			0
$291$		0			0
$292$		0			0
$293$	7.07107			0.413096			0.206548	−	0.978436i	$-0.433777\pi$
$293$	7.07107			0.413096			0.206548	+	0.978436i	$0.433777\pi$
$294$		0			0
$295$			10.0000i			0.582223i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−8.48528	+	3.46410i	−0.489083	+	0.199667i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	6.00000			0.343559
$306$		0			0
$307$			4.24264i			0.242140i	0.992644	+	0.121070i	$0.0386326\pi$
$307$			4.24264i			0.242140i	−0.992644	+	0.121070i	$0.961367\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−24.4949			−1.38898			−0.694489	−	0.719503i	$-0.744370\pi$
$311$	−24.4949			−1.38898			−0.694489	+	0.719503i	$0.744370\pi$
$312$		0			0
$313$		−	19.5959i		−	1.10763i	−0.832641	−	0.553813i	$-0.813172\pi$
$313$		−	19.5959i		−	1.10763i	0.832641	−	0.553813i	$-0.186828\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	6.92820i		−	0.389127i	−0.980890	−	0.194563i	$-0.937671\pi$
$317$		−	6.92820i		−	0.389127i	0.980890	−	0.194563i	$-0.0623290\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$	20.7846			1.15649
$324$		0			0
$325$	−12.7279			−0.706018
$326$		0			0
$327$		0			0
$328$		0			0
$329$	12.0000	−	4.89898i	0.661581	−	0.270089i
$330$		0			0
$331$	−3.46410			−0.190404			−0.0952021	−	0.995458i	$-0.530350\pi$
$331$	−3.46410			−0.190404			−0.0952021	+	0.995458i	$0.530350\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−14.6969			−0.802980
$336$		0			0
$337$	−16.0000			−0.871576			−0.435788	−	0.900049i	$-0.643530\pi$
$337$	−16.0000			−0.871576			−0.435788	+	0.900049i	$0.643530\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	16.9706			0.919007
$342$		0			0
$343$	−7.34847	+	17.0000i	−0.396780	+	0.917914i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−3.46410			−0.185963			−0.0929814	−	0.995668i	$-0.529640\pi$
$347$	−3.46410			−0.185963			−0.0929814	+	0.995668i	$0.529640\pi$
$348$		0			0
$349$	−4.24264			−0.227103			−0.113552	−	0.993532i	$-0.536223\pi$
$349$	−4.24264			−0.227103			−0.113552	+	0.993532i	$0.536223\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	9.79796i		−	0.521493i	−0.965407	−	0.260746i	$-0.916031\pi$
$353$		−	9.79796i		−	0.521493i	0.965407	−	0.260746i	$-0.0839686\pi$
$354$		0			0
$355$			8.48528i			0.450352i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			24.0000i			1.26667i	0.773877	+	0.633336i	$0.218315\pi$
$359$			24.0000i			1.26667i	−0.773877	+	0.633336i	$0.781685\pi$
$360$		0			0
$361$	1.00000			0.0526316
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	6.92820i		−	0.362639i
$366$		0			0
$367$	9.79796			0.511449			0.255725	−	0.966750i	$-0.417686\pi$
$367$	9.79796			0.511449			0.255725	+	0.966750i	$0.417686\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−13.8564	−	33.9411i	−0.719389	−	1.76214i
$372$		0			0
$373$		−	13.8564i		−	0.717458i	−0.933442	−	0.358729i	$-0.883210\pi$
$373$		−	13.8564i		−	0.717458i	0.933442	−	0.358729i	$-0.116790\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−3.46410			−0.177939			−0.0889695	−	0.996034i	$-0.528357\pi$
$379$	−3.46410			−0.177939			−0.0889695	+	0.996034i	$0.528357\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−4.89898			−0.250326			−0.125163	−	0.992136i	$-0.539945\pi$
$383$	−4.89898			−0.250326			−0.125163	+	0.992136i	$0.539945\pi$
$384$		0			0
$385$	−12.0000	+	4.89898i	−0.611577	+	0.249675i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			6.92820i			0.351274i	0.984455	+	0.175637i	$0.0561985\pi$
$389$			6.92820i			0.351274i	−0.984455	+	0.175637i	$0.943802\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	19.7990i		−	0.996195i
$396$		0			0
$397$	−4.24264			−0.212932			−0.106466	−	0.994316i	$-0.533954\pi$
$397$	−4.24264			−0.212932			−0.106466	+	0.994316i	$0.533954\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	12.0000			0.599251			0.299626	−	0.954057i	$-0.403138\pi$
$401$	12.0000			0.599251			0.299626	+	0.954057i	$0.403138\pi$
$402$		0			0
$403$	−20.7846			−1.03536
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	24.0000i		−	1.18964i
$408$		0			0
$409$		−	19.5959i		−	0.968956i	−0.874804	−	0.484478i	$-0.839010\pi$
$409$		−	19.5959i		−	0.968956i	0.874804	−	0.484478i	$-0.160990\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	7.07107	+	17.3205i	0.347945	+	0.852286i
$414$		0			0
$415$			14.0000i			0.687233i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			7.07107i			0.345444i	0.984971	+	0.172722i	$0.0552563\pi$
$419$			7.07107i			0.345444i	−0.984971	+	0.172722i	$0.944744\pi$
$420$		0			0
$421$		−	20.7846i		−	1.01298i	−0.862246	−	0.506490i	$-0.830943\pi$
$421$		−	20.7846i		−	1.01298i	0.862246	−	0.506490i	$-0.169057\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	14.6969i		−	0.712906i
$426$		0			0
$427$	10.3923	−	4.24264i	0.502919	−	0.205316i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	24.0000i		−	1.15604i	−0.816023	−	0.578020i	$-0.803826\pi$
$431$		−	24.0000i		−	1.15604i	0.816023	−	0.578020i	$-0.196174\pi$
$432$		0			0
$433$			34.2929i			1.64801i	0.566583	+	0.824005i	$0.308265\pi$
$433$			34.2929i			1.64801i	−0.566583	+	0.824005i	$0.691735\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	24.4949			1.16908			0.584539	−	0.811366i	$-0.301275\pi$
$439$	24.4949			1.16908			0.584539	+	0.811366i	$0.301275\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−10.3923			−0.493753			−0.246877	−	0.969047i	$-0.579404\pi$
$443$	−10.3923			−0.493753			−0.246877	+	0.969047i	$0.579404\pi$
$444$		0			0
$445$			20.7846i			0.985285i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−30.0000			−1.41579			−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000			−1.41579			−0.707894	+	0.706319i	$0.750354\pi$
$450$		0			0
$451$			33.9411i			1.59823i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	14.6969	−	6.00000i	0.689003	−	0.281284i
$456$		0			0
$457$	16.0000			0.748448			0.374224	−	0.927338i	$-0.377909\pi$
$457$	16.0000			0.748448			0.374224	+	0.927338i	$0.377909\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	15.5563			0.724531			0.362266	−	0.932075i	$-0.382003\pi$
$461$	15.5563			0.724531			0.362266	+	0.932075i	$0.382003\pi$
$462$		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
$467$			26.8701i			1.24340i	0.783256	+	0.621699i	$0.213557\pi$
$467$			26.8701i			1.24340i	−0.783256	+	0.621699i	$0.786443\pi$
$468$		0			0
$469$	−25.4558	+	10.3923i	−1.17544	+	0.479872i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−12.0000			−0.551761
$474$		0			0
$475$			12.7279i			0.583997i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	34.2929			1.56688			0.783440	−	0.621467i	$-0.213463\pi$
$479$	34.2929			1.56688			0.783440	+	0.621467i	$0.213463\pi$
$480$		0			0
$481$			29.3939i			1.34025i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			6.92820i			0.314594i
$486$		0			0
$487$			16.0000i			0.725029i	0.931978	+	0.362515i	$0.118082\pi$
$487$			16.0000i			0.725029i	−0.931978	+	0.362515i	$0.881918\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	38.1051			1.71966			0.859830	−	0.510581i	$-0.170569\pi$
$491$	38.1051			1.71966			0.859830	+	0.510581i	$0.170569\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	6.00000	+	14.6969i	0.269137	+	0.659248i
$498$		0			0
$499$	24.2487			1.08552			0.542761	−	0.839887i	$-0.317379\pi$
$499$	24.2487			1.08552			0.542761	+	0.839887i	$0.317379\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−24.4949			−1.09217			−0.546087	−	0.837729i	$-0.683883\pi$
$503$	−24.4949			−1.09217			−0.546087	+	0.837729i	$0.683883\pi$
$504$		0			0
$505$	−26.0000			−1.15698
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−24.0416			−1.06563			−0.532813	−	0.846233i	$-0.678865\pi$
$509$	−24.0416			−1.06563			−0.532813	+	0.846233i	$0.678865\pi$
$510$		0			0
$511$	−4.89898	−	12.0000i	−0.216718	−	0.530849i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	16.9706			0.746364
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	9.79796i		−	0.429256i	−0.976696	−	0.214628i	$-0.931146\pi$
$521$		−	9.79796i		−	0.429256i	0.976696	−	0.214628i	$-0.0688540\pi$
$522$		0			0
$523$			12.7279i			0.556553i	0.960501	+	0.278277i	$0.0897632\pi$
$523$			12.7279i			0.556553i	−0.960501	+	0.278277i	$0.910237\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	24.0000i		−	1.04546i
$528$		0			0
$529$	23.0000			1.00000
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	41.5692i		−	1.80056i
$534$		0			0
$535$	−24.4949			−1.05901
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−17.3205	+	16.9706i	−0.746047	+	0.730974i
$540$		0			0
$541$		−	34.6410i		−	1.48933i	−0.667436	−	0.744667i	$-0.732608\pi$
$541$		−	34.6410i		−	1.48933i	0.667436	−	0.744667i	$-0.267392\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			19.5959i			0.839397i
$546$		0			0
$547$	24.2487			1.03680			0.518400	−	0.855138i	$-0.326528\pi$
$547$	24.2487			1.03680			0.518400	+	0.855138i	$0.326528\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−14.0000	−	34.2929i	−0.595341	−	1.45828i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	27.7128i		−	1.17423i	−0.809504	−	0.587115i	$-0.800264\pi$
$557$		−	27.7128i		−	1.17423i	0.809504	−	0.587115i	$-0.199736\pi$
$558$		0			0
$559$	14.6969			0.621614
$560$		0			0
$561$		0			0
$562$		0			0
$563$			15.5563i			0.655622i	0.944743	+	0.327811i	$0.106311\pi$
$563$			15.5563i			0.655622i	−0.944743	+	0.327811i	$0.893689\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	24.0000			1.00613			0.503066	−	0.864248i	$-0.332205\pi$
$569$	24.0000			1.00613			0.503066	+	0.864248i	$0.332205\pi$
$570$		0			0
$571$	−38.1051			−1.59465			−0.797325	−	0.603550i	$-0.793752\pi$
$571$	−38.1051			−1.59465			−0.797325	+	0.603550i	$0.793752\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	9.89949	+	24.2487i	0.410700	+	1.00601i
$582$		0			0
$583$		−	48.0000i		−	1.98796i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			32.5269i			1.34253i	0.741218	+	0.671265i	$0.234249\pi$
$587$			32.5269i			1.34253i	−0.741218	+	0.671265i	$0.765751\pi$
$588$		0			0
$589$			20.7846i			0.856415i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			9.79796i			0.402354i	0.979555	+	0.201177i	$0.0644766\pi$
$593$			9.79796i			0.402354i	−0.979555	+	0.201177i	$0.935523\pi$
$594$		0			0
$595$	6.92820	+	16.9706i	0.284029	+	0.695725i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			30.0000i			1.22577i	0.790173	+	0.612883i	$0.209990\pi$
$599$			30.0000i			1.22577i	−0.790173	+	0.612883i	$0.790010\pi$
$600$		0			0
$601$			4.89898i			0.199834i	0.994996	+	0.0999168i	$0.0318577\pi$
$601$			4.89898i			0.199834i	−0.994996	+	0.0999168i	$0.968142\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−1.41421			−0.0574960
$606$		0			0
$607$	−29.3939			−1.19306			−0.596530	−	0.802591i	$-0.703454\pi$
$607$	−29.3939			−1.19306			−0.596530	+	0.802591i	$0.703454\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−20.7846			−0.840855
$612$		0			0
$613$			20.7846i			0.839482i	0.907644	+	0.419741i	$0.137879\pi$
$613$			20.7846i			0.839482i	−0.907644	+	0.419741i	$0.862121\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−36.0000			−1.44931			−0.724653	−	0.689114i	$-0.758000\pi$
$617$	−36.0000			−1.44931			−0.724653	+	0.689114i	$0.758000\pi$
$618$		0			0
$619$		−	12.7279i		−	0.511578i	−0.966733	−	0.255789i	$-0.917665\pi$
$619$		−	12.7279i		−	0.511578i	0.966733	−	0.255789i	$-0.0823353\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	14.6969	+	36.0000i	0.588820	+	1.44231i
$624$		0			0
$625$	−1.00000			−0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−33.9411			−1.35332
$630$		0			0
$631$			34.0000i			1.35352i	0.736204	+	0.676759i	$0.236616\pi$
$631$			34.0000i			1.35352i	−0.736204	+	0.676759i	$0.763384\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$			22.6274i			0.897942i
$636$		0			0
$637$	21.2132	−	20.7846i	0.840498	−	0.823516i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−12.0000			−0.473972			−0.236986	−	0.971513i	$-0.576159\pi$
$641$	−12.0000			−0.473972			−0.236986	+	0.971513i	$0.576159\pi$
$642$		0			0
$643$		−	38.1838i		−	1.50582i	−0.658123	−	0.752910i	$-0.728649\pi$
$643$		−	38.1838i		−	1.50582i	0.658123	−	0.752910i	$-0.271351\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	29.3939			1.15559			0.577796	−	0.816181i	$-0.303913\pi$
$647$	29.3939			1.15559			0.577796	+	0.816181i	$0.303913\pi$
$648$		0			0
$649$			24.4949i			0.961509i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			6.92820i			0.271122i	0.990769	+	0.135561i	$0.0432836\pi$
$653$			6.92820i			0.271122i	−0.990769	+	0.135561i	$0.956716\pi$
$654$		0			0
$655$			22.0000i			0.859611i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	24.2487			0.944596			0.472298	−	0.881439i	$-0.343425\pi$
$659$	24.2487			0.944596			0.472298	+	0.881439i	$0.343425\pi$
$660$		0			0
$661$	21.2132			0.825098			0.412549	−	0.910935i	$-0.364639\pi$
$661$	21.2132			0.825098			0.412549	+	0.910935i	$0.364639\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−6.00000	−	14.6969i	−0.232670	−	0.569923i
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$	14.6969			0.567369
$672$		0			0
$673$	2.00000			0.0770943			0.0385472	−	0.999257i	$-0.487727\pi$
$673$	2.00000			0.0770943			0.0385472	+	0.999257i	$0.487727\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−41.0122			−1.57623			−0.788113	−	0.615530i	$-0.788942\pi$
$677$	−41.0122			−1.57623			−0.788113	+	0.615530i	$0.788942\pi$
$678$		0			0
$679$	4.89898	+	12.0000i	0.188006	+	0.460518i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−10.3923			−0.397650			−0.198825	−	0.980035i	$-0.563713\pi$
$683$	−10.3923			−0.397650			−0.198825	+	0.980035i	$0.563713\pi$
$684$		0			0
$685$	−25.4558			−0.972618
$686$		0			0
$687$		0			0
$688$		0			0
$689$			58.7878i			2.23964i
$690$		0			0
$691$		−	12.7279i		−	0.484193i	−0.970252	−	0.242096i	$-0.922165\pi$
$691$		−	12.7279i		−	0.484193i	0.970252	−	0.242096i	$-0.0778351\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			6.00000i			0.227593i
$696$		0			0
$697$	48.0000			1.81813
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	34.6410i		−	1.30837i	−0.756333	−	0.654187i	$-0.773011\pi$
$701$		−	34.6410i		−	1.30837i	0.756333	−	0.654187i	$-0.226989\pi$
$702$		0			0
$703$	29.3939			1.10861
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−45.0333	+	18.3848i	−1.69365	+	0.691431i
$708$		0			0
$709$		−	27.7128i		−	1.04078i	−0.853930	−	0.520388i	$-0.825787\pi$
$709$		−	27.7128i		−	1.04078i	0.853930	−	0.520388i	$-0.174213\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$	20.7846			0.777300
$716$		0			0
$717$		0			0
$718$		0			0
$719$	4.89898			0.182701			0.0913506	−	0.995819i	$-0.470882\pi$
$719$	4.89898			0.182701			0.0913506	+	0.995819i	$0.470882\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−19.5959			−0.726772			−0.363386	−	0.931639i	$-0.618379\pi$
$727$	−19.5959			−0.726772			−0.363386	+	0.931639i	$0.618379\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			16.9706i			0.627679i
$732$		0			0
$733$	12.7279			0.470117			0.235058	−	0.971981i	$-0.424472\pi$
$733$	12.7279			0.470117			0.235058	+	0.971981i	$0.424472\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−36.0000			−1.32608
$738$		0			0
$739$	−31.1769			−1.14686			−0.573431	−	0.819254i	$-0.694388\pi$
$739$	−31.1769			−1.14686			−0.573431	+	0.819254i	$0.694388\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	24.0000i		−	0.880475i	−0.897881	−	0.440237i	$-0.854894\pi$
$743$		−	24.0000i		−	0.880475i	0.897881	−	0.440237i	$-0.145106\pi$
$744$		0			0
$745$			9.79796i			0.358969i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−42.4264	+	17.3205i	−1.55023	+	0.632878i
$750$		0			0
$751$			8.00000i			0.291924i	0.989290	+	0.145962i	$0.0466277\pi$
$751$			8.00000i			0.291924i	−0.989290	+	0.145962i	$0.953372\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			11.3137i			0.411748i
$756$		0			0
$757$		−	34.6410i		−	1.25905i	−0.776981	−	0.629525i	$-0.783250\pi$
$757$		−	34.6410i		−	1.25905i	0.776981	−	0.629525i	$-0.216750\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	19.5959i		−	0.710351i	−0.934800	−	0.355176i	$-0.884421\pi$
$761$		−	19.5959i		−	0.710351i	0.934800	−	0.355176i	$-0.115579\pi$
$762$		0			0
$763$	13.8564	+	33.9411i	0.501636	+	1.22875i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	30.0000i		−	1.08324i
$768$		0			0
$769$		−	24.4949i		−	0.883309i	−0.897185	−	0.441654i	$-0.854392\pi$
$769$		−	24.4949i		−	0.883309i	0.897185	−	0.441654i	$-0.145608\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−35.3553			−1.27164			−0.635822	−	0.771836i	$-0.719339\pi$
$773$	−35.3553			−1.27164			−0.635822	+	0.771836i	$0.719339\pi$
$774$		0			0
$775$	14.6969			0.527930
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−41.5692			−1.48937
$780$		0			0
$781$			20.7846i			0.743732i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−30.0000			−1.07075
$786$		0			0
$787$		−	21.2132i		−	0.756169i	−0.925771	−	0.378085i	$-0.876583\pi$
$787$		−	21.2132i		−	0.756169i	0.925771	−	0.378085i	$-0.123417\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−18.0000			−0.639199
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−15.5563			−0.551034			−0.275517	−	0.961296i	$-0.588849\pi$
$797$	−15.5563			−0.551034			−0.275517	+	0.961296i	$0.588849\pi$
$798$		0			0
$799$		−	24.0000i		−	0.849059i
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	16.9706i		−	0.598878i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	48.0000			1.68759			0.843795	−	0.536666i	$-0.180316\pi$
$809$	48.0000			1.68759			0.843795	+	0.536666i	$0.180316\pi$
$810$		0			0
$811$			55.1543i			1.93673i	0.249537	+	0.968365i	$0.419722\pi$
$811$			55.1543i			1.93673i	−0.249537	+	0.968365i	$0.580278\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	4.89898			0.171604
$816$		0			0
$817$		−	14.6969i		−	0.514181i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		1.00000			$0$
$821$		0			0		−1.00000			$\pi$
$822$		0			0
$823$			22.0000i			0.766872i	0.923567	+	0.383436i	$0.125259\pi$
$823$			22.0000i			0.766872i	−0.923567	+	0.383436i	$0.874741\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	10.3923			0.361376			0.180688	−	0.983540i	$-0.442168\pi$
$827$	10.3923			0.361376			0.180688	+	0.983540i	$0.442168\pi$
$828$		0			0
$829$	−55.1543			−1.91559			−0.957795	−	0.287454i	$-0.907191\pi$
$829$	−55.1543			−1.91559			−0.957795	+	0.287454i	$0.907191\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	24.0000	+	24.4949i	0.831551	+	0.848698i
$834$		0			0
$835$	−13.8564			−0.479521
$836$		0			0
$837$		0			0
$838$		0			0
$839$	48.9898			1.69132			0.845658	−	0.533726i	$-0.179208\pi$
$839$	48.9898			1.69132			0.845658	+	0.533726i	$0.179208\pi$
$840$		0			0
$841$	29.0000			1.00000
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−7.07107			−0.243252
$846$		0			0
$847$	−2.44949	+	1.00000i	−0.0841655	+	0.0343604i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	4.24264			0.145265			0.0726326	−	0.997359i	$-0.476860\pi$
$853$	4.24264			0.145265			0.0726326	+	0.997359i	$0.476860\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	39.1918i		−	1.33877i	−0.742917	−	0.669384i	$-0.766558\pi$
$857$		−	39.1918i		−	1.33877i	0.742917	−	0.669384i	$-0.233442\pi$
$858$		0			0
$859$		−	29.6985i		−	1.01330i	−0.862152	−	0.506650i	$-0.830884\pi$
$859$		−	29.6985i		−	1.01330i	0.862152	−	0.506650i	$-0.169116\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			6.00000i			0.204242i	0.994772	+	0.102121i	$0.0325630\pi$
$863$			6.00000i			0.204242i	−0.994772	+	0.102121i	$0.967437\pi$
$864$		0			0
$865$	−10.0000			−0.340010
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	48.4974i		−	1.64516i
$870$		0			0
$871$	44.0908			1.49396
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−27.7128	+	11.3137i	−0.936864	+	0.382473i
$876$		0			0
$877$			6.92820i			0.233949i	0.993135	+	0.116974i	$0.0373195\pi$
$877$			6.92820i			0.233949i	−0.993135	+	0.116974i	$0.962680\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			29.3939i			0.990305i	0.868806	+	0.495152i	$0.164888\pi$
$881$			29.3939i			0.990305i	−0.868806	+	0.495152i	$0.835112\pi$
$882$		0			0
$883$	−24.2487			−0.816034			−0.408017	−	0.912974i	$-0.633780\pi$
$883$	−24.2487			−0.816034			−0.408017	+	0.912974i	$0.633780\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−19.5959			−0.657967			−0.328983	−	0.944336i	$-0.606706\pi$
$887$	−19.5959			−0.657967			−0.328983	+	0.944336i	$0.606706\pi$
$888$		0			0
$889$	16.0000	+	39.1918i	0.536623	+	1.31445i
$890$		0			0
$891$		0			0
$892$		0			0
$893$			20.7846i			0.695530i
$894$		0			0
$895$	24.4949			0.818774
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$	−67.8823			−2.26149
$902$		0			0
$903$		0			0
$904$		0			0
$905$	6.00000			0.199447
$906$		0			0
$907$	−17.3205			−0.575118			−0.287559	−	0.957763i	$-0.592844\pi$
$907$	−17.3205			−0.575118			−0.287559	+	0.957763i	$0.592844\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$			34.2929i			1.13493i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	15.5563	+	38.1051i	0.513716	+	1.25834i
$918$		0			0
$919$		−	34.0000i		−	1.12156i	−0.827966	−	0.560778i	$-0.810502\pi$
$919$		−	34.0000i		−	1.12156i	0.827966	−	0.560778i	$-0.189498\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	25.4558i		−	0.837889i
$924$		0			0
$925$		−	20.7846i		−	0.683394i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			53.8888i			1.76803i	0.467455	+	0.884017i	$0.345171\pi$
$929$			53.8888i			1.76803i	−0.467455	+	0.884017i	$0.654829\pi$
$930$		0			0
$931$	−20.7846	−	21.2132i	−0.681188	−	0.695235i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			24.0000i			0.784884i
$936$		0			0
$937$			4.89898i			0.160043i	0.996793	+	0.0800213i	$0.0254988\pi$
$937$			4.89898i			0.160043i	−0.996793	+	0.0800213i	$0.974501\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	1.41421			0.0461020			0.0230510	−	0.999734i	$-0.492662\pi$
$941$	1.41421			0.0461020			0.0230510	+	0.999734i	$0.492662\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	24.2487			0.787977			0.393989	−	0.919115i	$-0.371095\pi$
$947$	24.2487			0.787977			0.393989	+	0.919115i	$0.371095\pi$
$948$		0			0
$949$			20.7846i			0.674697i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	30.0000			0.971795			0.485898	−	0.874016i	$-0.338493\pi$
$953$	30.0000			0.971795			0.485898	+	0.874016i	$0.338493\pi$
$954$		0			0
$955$			25.4558i			0.823732i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−44.0908	+	18.0000i	−1.42377	+	0.581250i
$960$		0			0
$961$	−7.00000			−0.225806
$962$		0			0
$963$		0			0
$964$		0			0
$965$	5.65685			0.182101
$966$		0			0
$967$		−	8.00000i		−	0.257263i	−0.991692	−	0.128631i	$-0.958942\pi$
$967$		−	8.00000i		−	0.257263i	0.991692	−	0.128631i	$-0.0410584\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	7.07107i		−	0.226921i	−0.993542	−	0.113461i	$-0.963806\pi$
$971$		−	7.07107i		−	0.226921i	0.993542	−	0.113461i	$-0.0361936\pi$
$972$		0			0
$973$	4.24264	+	10.3923i	0.136013	+	0.333162i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−30.0000			−0.959785			−0.479893	−	0.877327i	$-0.659324\pi$
$977$	−30.0000			−0.959785			−0.479893	+	0.877327i	$0.659324\pi$
$978$		0			0
$979$			50.9117i			1.62714i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−9.79796			−0.312506			−0.156253	−	0.987717i	$-0.549942\pi$
$983$	−9.79796			−0.312506			−0.156253	+	0.987717i	$0.549942\pi$
$984$		0			0
$985$			9.79796i			0.312189i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		−	2.00000i		−	0.0635321i	−0.999495	−	0.0317660i	$-0.989887\pi$
$991$		−	2.00000i		−	0.0635321i	0.999495	−	0.0317660i	$-0.0101131\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	13.8564			0.439278
$996$		0			0
$997$	−4.24264			−0.134366			−0.0671829	−	0.997741i	$-0.521401\pi$
$997$	−4.24264			−0.134366			−0.0671829	+	0.997741i	$0.521401\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4032.2.p.f.1567.2		8
3.2	odd	2		448.2.e.b.223.3	yes	8
4.3	odd	2	inner	4032.2.p.f.1567.3		8
7.6	odd	2	inner	4032.2.p.f.1567.8		8
8.3	odd	2	inner	4032.2.p.f.1567.7		8
8.5	even	2	inner	4032.2.p.f.1567.6		8
12.11	even	2		448.2.e.b.223.8	yes	8
21.20	even	2		448.2.e.b.223.6	yes	8
24.5	odd	2		448.2.e.b.223.5	yes	8
24.11	even	2		448.2.e.b.223.2	yes	8
28.27	even	2	inner	4032.2.p.f.1567.5		8
48.5	odd	4		1792.2.f.j.1791.4		8
48.11	even	4		1792.2.f.j.1791.7		8
48.29	odd	4		1792.2.f.j.1791.6		8
48.35	even	4		1792.2.f.j.1791.1		8
56.13	odd	2	inner	4032.2.p.f.1567.4		8
56.27	even	2	inner	4032.2.p.f.1567.1		8
84.83	odd	2		448.2.e.b.223.1	✓	8
168.83	odd	2		448.2.e.b.223.7	yes	8
168.125	even	2		448.2.e.b.223.4	yes	8
336.83	odd	4		1792.2.f.j.1791.8		8
336.125	even	4		1792.2.f.j.1791.3		8
336.251	odd	4		1792.2.f.j.1791.2		8
336.293	even	4		1792.2.f.j.1791.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
448.2.e.b.223.1	✓	8	84.83	odd	2
448.2.e.b.223.2	yes	8	24.11	even	2
448.2.e.b.223.3	yes	8	3.2	odd	2
448.2.e.b.223.4	yes	8	168.125	even	2
448.2.e.b.223.5	yes	8	24.5	odd	2
448.2.e.b.223.6	yes	8	21.20	even	2
448.2.e.b.223.7	yes	8	168.83	odd	2
448.2.e.b.223.8	yes	8	12.11	even	2
1792.2.f.j.1791.1		8	48.35	even	4
1792.2.f.j.1791.2		8	336.251	odd	4
1792.2.f.j.1791.3		8	336.125	even	4
1792.2.f.j.1791.4		8	48.5	odd	4
1792.2.f.j.1791.5		8	336.293	even	4
1792.2.f.j.1791.6		8	48.29	odd	4
1792.2.f.j.1791.7		8	48.11	even	4
1792.2.f.j.1791.8		8	336.83	odd	4
4032.2.p.f.1567.1		8	56.27	even	2	inner
4032.2.p.f.1567.2		8	1.1	even	1	trivial
4032.2.p.f.1567.3		8	4.3	odd	2	inner
4032.2.p.f.1567.4		8	56.13	odd	2	inner
4032.2.p.f.1567.5		8	28.27	even	2	inner
4032.2.p.f.1567.6		8	8.5	even	2	inner
4032.2.p.f.1567.7		8	8.3	odd	2	inner
4032.2.p.f.1567.8		8	7.6	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.p (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421 q^{5} +(-2.44949 + 1.00000i) q^{7} +O(q^{10})\)	\(q-1.41421 q^{5} +(-2.44949 + 1.00000i) q^{7} -3.46410 q^{11} +4.24264 q^{13} +4.89898i q^{17} -4.24264i q^{19} -3.00000 q^{25} -4.89898 q^{31} +(3.46410 - 1.41421i) q^{35} +6.92820i q^{37} -9.79796i q^{41} +3.46410 q^{43} -4.89898 q^{47} +(5.00000 - 4.89898i) q^{49} +13.8564i q^{53} +4.89898 q^{55} -7.07107i q^{59} -4.24264 q^{61} -6.00000 q^{65} +10.3923 q^{67} -6.00000i q^{71} +4.89898i q^{73} +(8.48528 - 3.46410i) q^{77} +14.0000i q^{79} -9.89949i q^{83} -6.92820i q^{85} -14.6969i q^{89} +(-10.3923 + 4.24264i) q^{91} +6.00000i q^{95} -4.89898i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 24 q^{25} + 40 q^{49} - 48 q^{65}+O(q^{100}) \) 8 * q - 24 * q^25 + 40 * q^49 - 48 * q^65

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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