LMFDB - Embedded newform 4032.2.v.e.3599.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4032,2,Mod(1583,4032)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4032, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 3, 2, 0]))

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

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

chi := DirichletCharacter("4032.1583");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4032 = 2^{6} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4032.v (of order $4$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$32.1956820950$
Analytic rank:	$0$
Dimension:	$40$
Relative dimension:	$20$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 1008)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			3599.8
Character	$\chi$	$=$	4032.3599
Dual form			4032.2.v.e.1583.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.667815 + 0.667815i) q^{5} -1.00000 q^{7} +O(q^{10})$	$q+(-0.667815 + 0.667815i) q^{5} -1.00000 q^{7} +(1.57551 + 1.57551i) q^{11} +(-1.83034 + 1.83034i) q^{13} -3.40687i q^{17} +(-3.18485 - 3.18485i) q^{19} -0.793288i q^{23} +4.10805i q^{25} +(1.73542 + 1.73542i) q^{29} -3.28367i q^{31} +(0.667815 - 0.667815i) q^{35} +(7.72049 + 7.72049i) q^{37} -7.19799 q^{41} +(5.84265 - 5.84265i) q^{43} -13.0051 q^{47} +1.00000 q^{49} +(-3.34052 + 3.34052i) q^{53} -2.10430 q^{55} +(-7.41533 - 7.41533i) q^{59} +(1.93050 - 1.93050i) q^{61} -2.44465i q^{65} +(-6.38033 - 6.38033i) q^{67} -3.41542i q^{71} -8.13689i q^{73} +(-1.57551 - 1.57551i) q^{77} +0.0502773i q^{79} +(2.29129 - 2.29129i) q^{83} +(2.27516 + 2.27516i) q^{85} +7.18090 q^{89} +(1.83034 - 1.83034i) q^{91} +4.25379 q^{95} +1.49996 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - 40 q^{7}+O(q^{10}) $ 40 * q - 40 * q^7	$ 40 q - 40 q^{7} - 24 q^{13} + 32 q^{19} - 8 q^{37} - 32 q^{43} + 40 q^{49} - 48 q^{55} - 24 q^{61} + 64 q^{85} + 24 q^{91} + 64 q^{97}+O(q^{100}) $ 40 * q - 40 * q^7 - 24 * q^13 + 32 * q^19 - 8 * q^37 - 32 * q^43 + 40 * q^49 - 48 * q^55 - 24 * q^61 + 64 * q^85 + 24 * q^91 + 64 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$127$	$577$	$1793$	$3781$
$\chi(n)$	$-1$	$1$	$-1$	$e\left(\frac{1}{4}\right)$

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
$5$	−0.667815	+	0.667815i	−0.298656	+	0.298656i	−0.840487	−	0.541831i	$-0.817731\pi$
$5$	−0.667815	+	0.667815i	−0.298656	+	0.298656i	0.541831	+	0.840487i	$0.317731\pi$
$6$		0			0
$7$	−1.00000			−0.377964
$7$	−1.00000			−0.377964
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.57551	+	1.57551i	0.475033	+	0.475033i	0.903539	−	0.428506i	$-0.140960\pi$
$11$	1.57551	+	1.57551i	0.475033	+	0.475033i	−0.428506	+	0.903539i	$0.640960\pi$
$12$		0			0
$13$	−1.83034	+	1.83034i	−0.507644	+	0.507644i	−0.913803	−	0.406159i	$-0.866868\pi$
$13$	−1.83034	+	1.83034i	−0.507644	+	0.507644i	0.406159	+	0.913803i	$0.366868\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	3.40687i		−	0.826287i	−0.910666	−	0.413143i	$-0.864431\pi$
$17$		−	3.40687i		−	0.826287i	0.910666	−	0.413143i	$-0.135569\pi$
$18$		0			0
$19$	−3.18485	−	3.18485i	−0.730656	−	0.730656i	0.240094	−	0.970750i	$-0.422822\pi$
$19$	−3.18485	−	3.18485i	−0.730656	−	0.730656i	−0.970750	+	0.240094i	$0.922822\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	0.793288i		−	0.165412i	−0.996574	−	0.0827060i	$-0.973644\pi$
$23$		−	0.793288i		−	0.165412i	0.996574	−	0.0827060i	$-0.0263562\pi$
$24$		0			0
$25$			4.10805i			0.821609i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	1.73542	+	1.73542i	0.322260	+	0.322260i	0.849634	−	0.527373i	$-0.176823\pi$
$29$	1.73542	+	1.73542i	0.322260	+	0.322260i	−0.527373	+	0.849634i	$0.676823\pi$
$30$		0			0
$31$		−	3.28367i		−	0.589765i	−0.955534	−	0.294882i	$-0.904720\pi$
$31$		−	3.28367i		−	0.589765i	0.955534	−	0.294882i	$-0.0952805\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	0.667815	−	0.667815i	0.112881	−	0.112881i
$36$		0			0
$37$	7.72049	+	7.72049i	1.26924	+	1.26924i	0.946481	+	0.322760i	$0.104611\pi$
$37$	7.72049	+	7.72049i	1.26924	+	1.26924i	0.322760	+	0.946481i	$0.395389\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−7.19799			−1.12414			−0.562069	−	0.827091i	$-0.689994\pi$
$41$	−7.19799			−1.12414			−0.562069	+	0.827091i	$0.689994\pi$
$42$		0			0
$43$	5.84265	−	5.84265i	0.890996	−	0.890996i	−0.103621	−	0.994617i	$-0.533043\pi$
$43$	5.84265	−	5.84265i	0.890996	−	0.890996i	0.994617	+	0.103621i	$0.0330430\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−13.0051			−1.89699			−0.948497	−	0.316786i	$-0.897396\pi$
$47$	−13.0051			−1.89699			−0.948497	+	0.316786i	$0.897396\pi$
$48$		0			0
$49$	1.00000			0.142857
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−3.34052	+	3.34052i	−0.458856	+	0.458856i	−0.898280	−	0.439424i	$-0.855183\pi$
$53$	−3.34052	+	3.34052i	−0.458856	+	0.458856i	0.439424	+	0.898280i	$0.355183\pi$
$54$		0			0
$55$	−2.10430			−0.283743
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−7.41533	−	7.41533i	−0.965394	−	0.965394i	0.0340273	−	0.999421i	$-0.489167\pi$
$59$	−7.41533	−	7.41533i	−0.965394	−	0.965394i	−0.999421	+	0.0340273i	$0.989167\pi$
$60$		0			0
$61$	1.93050	−	1.93050i	0.247175	−	0.247175i	−0.572635	−	0.819810i	$-0.694079\pi$
$61$	1.93050	−	1.93050i	0.247175	−	0.247175i	0.819810	+	0.572635i	$0.194079\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	2.44465i		−	0.303222i
$66$		0			0
$67$	−6.38033	−	6.38033i	−0.779481	−	0.779481i	0.200262	−	0.979742i	$-0.435821\pi$
$67$	−6.38033	−	6.38033i	−0.779481	−	0.779481i	−0.979742	+	0.200262i	$0.935821\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	3.41542i		−	0.405336i	−0.979247	−	0.202668i	$-0.935039\pi$
$71$		−	3.41542i		−	0.405336i	0.979247	−	0.202668i	$-0.0649612\pi$
$72$		0			0
$73$		−	8.13689i		−	0.952351i	−0.879350	−	0.476176i	$-0.842023\pi$
$73$		−	8.13689i		−	0.952351i	0.879350	−	0.476176i	$-0.157977\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−1.57551	−	1.57551i	−0.179546	−	0.179546i
$78$		0			0
$79$			0.0502773i			0.00565663i	0.999996	+	0.00282832i	$0.000900282\pi$
$79$			0.0502773i			0.00565663i	−0.999996	+	0.00282832i	$0.999100\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	2.29129	−	2.29129i	0.251502	−	0.251502i	−0.570084	−	0.821586i	$-0.693089\pi$
$83$	2.29129	−	2.29129i	0.251502	−	0.251502i	0.821586	+	0.570084i	$0.193089\pi$
$84$		0			0
$85$	2.27516	+	2.27516i	0.246776	+	0.246776i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	7.18090			0.761174			0.380587	−	0.924745i	$-0.375722\pi$
$89$	7.18090			0.761174			0.380587	+	0.924745i	$0.375722\pi$
$90$		0			0
$91$	1.83034	−	1.83034i	0.191871	−	0.191871i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	4.25379			0.436429
$96$		0			0
$97$	1.49996			0.152298			0.0761490	−	0.997096i	$-0.475738\pi$
$97$	1.49996			0.152298			0.0761490	+	0.997096i	$0.475738\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−6.08005	+	6.08005i	−0.604988	+	0.604988i	−0.941632	−	0.336644i	$-0.890708\pi$
$101$	−6.08005	+	6.08005i	−0.604988	+	0.604988i	0.336644	+	0.941632i	$0.390708\pi$
$102$		0			0
$103$	7.74912			0.763543			0.381772	−	0.924257i	$-0.375314\pi$
$103$	7.74912			0.763543			0.381772	+	0.924257i	$0.375314\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−2.05853	−	2.05853i	−0.199006	−	0.199006i	0.600568	−	0.799574i	$-0.294941\pi$
$107$	−2.05853	−	2.05853i	−0.199006	−	0.199006i	−0.799574	+	0.600568i	$0.794941\pi$
$108$		0			0
$109$	7.13047	−	7.13047i	0.682975	−	0.682975i	−0.277694	−	0.960670i	$-0.589570\pi$
$109$	7.13047	−	7.13047i	0.682975	−	0.682975i	0.960670	+	0.277694i	$0.0895702\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	4.17748i		−	0.392984i	−0.980505	−	0.196492i	$-0.937045\pi$
$113$		−	4.17748i		−	0.392984i	0.980505	−	0.196492i	$-0.0629550\pi$
$114$		0			0
$115$	0.529770	+	0.529770i	0.0494013	+	0.0494013i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			3.40687i			0.312307i
$120$		0			0
$121$		−	6.03555i		−	0.548686i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−6.08249	−	6.08249i	−0.544035	−	0.544035i
$126$		0			0
$127$		−	5.16063i		−	0.457932i	−0.973434	−	0.228966i	$-0.926466\pi$
$127$		−	5.16063i		−	0.457932i	0.973434	−	0.228966i	$-0.0735345\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	1.53587	−	1.53587i	0.134189	−	0.134189i	−0.636822	−	0.771011i	$-0.719751\pi$
$131$	1.53587	−	1.53587i	0.134189	−	0.134189i	0.771011	+	0.636822i	$0.219751\pi$
$132$		0			0
$133$	3.18485	+	3.18485i	0.276162	+	0.276162i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−20.0645			−1.71423			−0.857113	−	0.515128i	$-0.827744\pi$
$137$	−20.0645			−1.71423			−0.857113	+	0.515128i	$0.827744\pi$
$138$		0			0
$139$	3.01947	−	3.01947i	0.256108	−	0.256108i	−0.567361	−	0.823469i	$-0.692036\pi$
$139$	3.01947	−	3.01947i	0.256108	−	0.256108i	0.823469	+	0.567361i	$0.192036\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−5.76742			−0.482296
$144$		0			0
$145$	−2.31789			−0.192490
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−2.11391	+	2.11391i	−0.173178	+	0.173178i	−0.788374	−	0.615196i	$-0.789077\pi$
$149$	−2.11391	+	2.11391i	−0.173178	+	0.173178i	0.615196	+	0.788374i	$0.289077\pi$
$150$		0			0
$151$	−7.08056			−0.576208			−0.288104	−	0.957599i	$-0.593025\pi$
$151$	−7.08056			−0.576208			−0.288104	+	0.957599i	$0.593025\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	2.19289	+	2.19289i	0.176137	+	0.176137i
$156$		0			0
$157$	16.0565	−	16.0565i	1.28144	−	1.28144i	0.341598	−	0.939846i	$-0.389032\pi$
$157$	16.0565	−	16.0565i	1.28144	−	1.28144i	0.939846	−	0.341598i	$-0.110968\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			0.793288i			0.0625199i
$162$		0			0
$163$	−10.9257	−	10.9257i	−0.855764	−	0.855764i	0.135072	−	0.990836i	$-0.456873\pi$
$163$	−10.9257	−	10.9257i	−0.855764	−	0.855764i	−0.990836	+	0.135072i	$0.956873\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	14.8556i		−	1.14956i	−0.818307	−	0.574782i	$-0.805087\pi$
$167$		−	14.8556i		−	1.14956i	0.818307	−	0.574782i	$-0.194913\pi$
$168$		0			0
$169$			6.29974i			0.484595i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−2.49652	−	2.49652i	−0.189807	−	0.189807i	0.605806	−	0.795613i	$-0.292851\pi$
$173$	−2.49652	−	2.49652i	−0.189807	−	0.189807i	−0.795613	+	0.605806i	$0.792851\pi$
$174$		0			0
$175$		−	4.10805i		−	0.310539i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	13.5658	−	13.5658i	1.01396	−	1.01396i	0.0140578	−	0.999901i	$-0.495525\pi$
$179$	13.5658	−	13.5658i	1.01396	−	1.01396i	0.999901	−	0.0140578i	$-0.00447489\pi$
$180$		0			0
$181$	−4.13837	−	4.13837i	−0.307602	−	0.307602i	0.536376	−	0.843979i	$-0.319793\pi$
$181$	−4.13837	−	4.13837i	−0.307602	−	0.307602i	−0.843979	+	0.536376i	$0.819793\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−10.3117			−0.758133
$186$		0			0
$187$	5.36755	−	5.36755i	0.392514	−	0.392514i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−21.9065			−1.58510			−0.792549	−	0.609808i	$-0.791246\pi$
$191$	−21.9065			−1.58510			−0.792549	+	0.609808i	$0.791246\pi$
$192$		0			0
$193$	−21.1429			−1.52190			−0.760950	−	0.648811i	$-0.775267\pi$
$193$	−21.1429			−1.52190			−0.760950	+	0.648811i	$0.775267\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−4.95026	+	4.95026i	−0.352692	+	0.352692i	−0.861110	−	0.508418i	$-0.830230\pi$
$197$	−4.95026	+	4.95026i	−0.352692	+	0.352692i	0.508418	+	0.861110i	$0.330230\pi$
$198$		0			0
$199$	−10.6455			−0.754639			−0.377319	−	0.926083i	$-0.623154\pi$
$199$	−10.6455			−0.754639			−0.377319	+	0.926083i	$0.623154\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−1.73542	−	1.73542i	−0.121803	−	0.121803i
$204$		0			0
$205$	4.80693	−	4.80693i	0.335730	−	0.335730i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	10.0355i		−	0.694172i
$210$		0			0
$211$	−0.563683	−	0.563683i	−0.0388056	−	0.0388056i	0.687438	−	0.726243i	$-0.258735\pi$
$211$	−0.563683	−	0.563683i	−0.0388056	−	0.0388056i	−0.726243	+	0.687438i	$0.758735\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			7.80362i			0.532202i
$216$		0			0
$217$			3.28367i			0.222910i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	6.23571	+	6.23571i	0.419459	+	0.419459i
$222$		0			0
$223$			21.6604i			1.45049i	0.688493	+	0.725243i	$0.258273\pi$
$223$			21.6604i			1.45049i	−0.688493	+	0.725243i	$0.741727\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	2.98117	−	2.98117i	0.197867	−	0.197867i	−0.601218	−	0.799085i	$-0.705318\pi$
$227$	2.98117	−	2.98117i	0.197867	−	0.197867i	0.799085	+	0.601218i	$0.205318\pi$
$228$		0			0
$229$	−10.1998	−	10.1998i	−0.674019	−	0.674019i	0.284621	−	0.958640i	$-0.408132\pi$
$229$	−10.1998	−	10.1998i	−0.674019	−	0.674019i	−0.958640	+	0.284621i	$0.908132\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−16.9408			−1.10983			−0.554914	−	0.831907i	$-0.687249\pi$
$233$	−16.9408			−1.10983			−0.554914	+	0.831907i	$0.687249\pi$
$234$		0			0
$235$	8.68503	−	8.68503i	0.566549	−	0.566549i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−11.7743			−0.761618			−0.380809	−	0.924654i	$-0.624354\pi$
$239$	−11.7743			−0.761618			−0.380809	+	0.924654i	$0.624354\pi$
$240$		0			0
$241$	−1.77496			−0.114335			−0.0571677	−	0.998365i	$-0.518207\pi$
$241$	−1.77496			−0.114335			−0.0571677	+	0.998365i	$0.518207\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−0.667815	+	0.667815i	−0.0426652	+	0.0426652i
$246$		0			0
$247$	11.6587			0.741826
$248$		0			0
$249$		0			0
$250$		0			0
$251$	10.7495	+	10.7495i	0.678503	+	0.678503i	0.959661	−	0.281158i	$-0.0907186\pi$
$251$	10.7495	+	10.7495i	0.678503	+	0.678503i	−0.281158	+	0.959661i	$0.590719\pi$
$252$		0			0
$253$	1.24983	−	1.24983i	0.0785762	−	0.0785762i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	23.2878i		−	1.45265i	−0.687351	−	0.726325i	$-0.741227\pi$
$257$		−	23.2878i		−	1.45265i	0.687351	−	0.726325i	$-0.258773\pi$
$258$		0			0
$259$	−7.72049	−	7.72049i	−0.479728	−	0.479728i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	5.00910i		−	0.308874i	−0.988003	−	0.154437i	$-0.950644\pi$
$263$		−	5.00910i		−	0.308874i	0.988003	−	0.154437i	$-0.0493564\pi$
$264$		0			0
$265$		−	4.46170i		−	0.274080i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−11.1632	−	11.1632i	−0.680633	−	0.680633i	0.279510	−	0.960143i	$-0.409828\pi$
$269$	−11.1632	−	11.1632i	−0.680633	−	0.680633i	−0.960143	+	0.279510i	$0.909828\pi$
$270$		0			0
$271$			16.1939i			0.983706i	0.870678	+	0.491853i	$0.163680\pi$
$271$			16.1939i			0.983706i	−0.870678	+	0.491853i	$0.836320\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−6.47226	+	6.47226i	−0.390292	+	0.390292i
$276$		0			0
$277$	19.0299	+	19.0299i	1.14339	+	1.14339i	0.987825	+	0.155568i	$0.0497207\pi$
$277$	19.0299	+	19.0299i	1.14339	+	1.14339i	0.155568	+	0.987825i	$0.450279\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	14.3974			0.858875			0.429438	−	0.903097i	$-0.358712\pi$
$281$	14.3974			0.858875			0.429438	+	0.903097i	$0.358712\pi$
$282$		0			0
$283$	21.2690	−	21.2690i	1.26431	−	1.26431i	0.315331	−	0.948982i	$-0.397884\pi$
$283$	21.2690	−	21.2690i	1.26431	−	1.26431i	0.948982	−	0.315331i	$-0.102116\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	7.19799			0.424884
$288$		0			0
$289$	5.39325			0.317250
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−18.9536	+	18.9536i	−1.10728	+	1.10728i	−0.113775	+	0.993507i	$0.536294\pi$
$293$	−18.9536	+	18.9536i	−1.10728	+	1.10728i	−0.993507	+	0.113775i	$0.963706\pi$
$294$		0			0
$295$	9.90414			0.576641
$296$		0			0
$297$		0			0
$298$		0			0
$299$	1.45198	+	1.45198i	0.0839704	+	0.0839704i
$300$		0			0
$301$	−5.84265	+	5.84265i	−0.336765	+	0.336765i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			2.57843i			0.147641i
$306$		0			0
$307$	7.90363	+	7.90363i	0.451084	+	0.451084i	0.895714	−	0.444630i	$-0.146665\pi$
$307$	7.90363	+	7.90363i	0.451084	+	0.451084i	−0.444630	+	0.895714i	$0.646665\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	24.8764i		−	1.41061i	−0.708904	−	0.705305i	$-0.750810\pi$
$311$		−	24.8764i		−	1.41061i	0.708904	−	0.705305i	$-0.249190\pi$
$312$		0			0
$313$		−	28.7289i		−	1.62385i	−0.583759	−	0.811927i	$-0.698419\pi$
$313$		−	28.7289i		−	1.62385i	0.583759	−	0.811927i	$-0.301581\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−0.354075	−	0.354075i	−0.0198869	−	0.0198869i	0.697093	−	0.716980i	$-0.254476\pi$
$317$	−0.354075	−	0.354075i	−0.0198869	−	0.0198869i	−0.716980	+	0.697093i	$0.754476\pi$
$318$		0			0
$319$			5.46835i			0.306169i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−10.8504	+	10.8504i	−0.603731	+	0.603731i
$324$		0			0
$325$	−7.51910	−	7.51910i	−0.417085	−	0.417085i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	13.0051			0.716996
$330$		0			0
$331$	11.1492	−	11.1492i	0.612818	−	0.612818i	−0.330862	−	0.943679i	$-0.607339\pi$
$331$	11.1492	−	11.1492i	0.612818	−	0.612818i	0.943679	+	0.330862i	$0.107339\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	8.52176			0.465593
$336$		0			0
$337$	−32.8646			−1.79025			−0.895124	−	0.445817i	$-0.852913\pi$
$337$	−32.8646			−1.79025			−0.895124	+	0.445817i	$0.852913\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	5.17345	−	5.17345i	0.280158	−	0.280158i
$342$		0			0
$343$	−1.00000			−0.0539949
$344$		0			0
$345$		0			0
$346$		0			0
$347$	11.4329	+	11.4329i	0.613751	+	0.613751i	0.943921	−	0.330170i	$-0.107106\pi$
$347$	11.4329	+	11.4329i	0.613751	+	0.613751i	−0.330170	+	0.943921i	$0.607106\pi$
$348$		0			0
$349$	19.4418	−	19.4418i	1.04070	−	1.04070i	0.0415602	−	0.999136i	$-0.486767\pi$
$349$	19.4418	−	19.4418i	1.04070	−	1.04070i	0.999136	−	0.0415602i	$-0.0132328\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	9.70827i		−	0.516719i	−0.966049	−	0.258360i	$-0.916818\pi$
$353$		−	9.70827i		−	0.516719i	0.966049	−	0.258360i	$-0.0831819\pi$
$354$		0			0
$355$	2.28087	+	2.28087i	0.121056	+	0.121056i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			0.535153i			0.0282443i	0.999900	+	0.0141222i	$0.00449537\pi$
$359$			0.535153i			0.0282443i	−0.999900	+	0.0141222i	$0.995505\pi$
$360$		0			0
$361$			1.28659i			0.0677152i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	5.43394	+	5.43394i	0.284426	+	0.284426i
$366$		0			0
$367$			14.3854i			0.750913i	0.926840	+	0.375456i	$0.122514\pi$
$367$			14.3854i			0.750913i	−0.926840	+	0.375456i	$0.877486\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	3.34052	−	3.34052i	0.173431	−	0.173431i
$372$		0			0
$373$	12.7345	+	12.7345i	0.659368	+	0.659368i	0.955231	−	0.295862i	$-0.0956070\pi$
$373$	12.7345	+	12.7345i	0.659368	+	0.659368i	−0.295862	+	0.955231i	$0.595607\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−6.35282			−0.327187
$378$		0			0
$379$	−18.4758	+	18.4758i	−0.949037	+	0.949037i	−0.998763	−	0.0497262i	$-0.984165\pi$
$379$	−18.4758	+	18.4758i	−0.949037	+	0.949037i	0.0497262	+	0.998763i	$0.484165\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−29.0133			−1.48251			−0.741256	−	0.671222i	$-0.765770\pi$
$383$	−29.0133			−1.48251			−0.741256	+	0.671222i	$0.765770\pi$
$384$		0			0
$385$	2.10430			0.107245
$386$		0			0
$387$		0			0
$388$		0			0
$389$	7.75723	−	7.75723i	0.393307	−	0.393307i	−0.482557	−	0.875864i	$-0.660292\pi$
$389$	7.75723	−	7.75723i	0.393307	−	0.393307i	0.875864	+	0.482557i	$0.160292\pi$
$390$		0			0
$391$	−2.70263			−0.136678
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−0.0335759	−	0.0335759i	−0.00168939	−	0.00168939i
$396$		0			0
$397$	13.0436	−	13.0436i	0.654638	−	0.654638i	−0.299468	−	0.954106i	$-0.596809\pi$
$397$	13.0436	−	13.0436i	0.654638	−	0.654638i	0.954106	+	0.299468i	$0.0968093\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			2.31443i			0.115577i	0.998329	+	0.0577887i	$0.0184050\pi$
$401$			2.31443i			0.115577i	−0.998329	+	0.0577887i	$0.981595\pi$
$402$		0			0
$403$	6.01022	+	6.01022i	0.299390	+	0.299390i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			24.3274i			1.20586i
$408$		0			0
$409$			3.87844i			0.191777i	0.995392	+	0.0958883i	$0.0305692\pi$
$409$			3.87844i			0.191777i	−0.995392	+	0.0958883i	$0.969431\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	7.41533	+	7.41533i	0.364884	+	0.364884i
$414$		0			0
$415$			3.06032i			0.150225i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	1.40127	−	1.40127i	0.0684563	−	0.0684563i	−0.672050	−	0.740506i	$-0.734586\pi$
$419$	1.40127	−	1.40127i	0.0684563	−	0.0684563i	0.740506	+	0.672050i	$0.234586\pi$
$420$		0			0
$421$	7.61424	+	7.61424i	0.371095	+	0.371095i	0.867876	−	0.496781i	$-0.165485\pi$
$421$	7.61424	+	7.61424i	0.371095	+	0.371095i	−0.496781	+	0.867876i	$0.665485\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	13.9956			0.678885
$426$		0			0
$427$	−1.93050	+	1.93050i	−0.0934234	+	0.0934234i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	0.496260			0.0239040			0.0119520	−	0.999929i	$-0.496195\pi$
$431$	0.496260			0.0239040			0.0119520	+	0.999929i	$0.496195\pi$
$432$		0			0
$433$	−32.9889			−1.58535			−0.792673	−	0.609647i	$-0.791311\pi$
$433$	−32.9889			−1.58535			−0.792673	+	0.609647i	$0.791311\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−2.52651	+	2.52651i	−0.120859	+	0.120859i
$438$		0			0
$439$	28.1034			1.34130			0.670651	−	0.741773i	$-0.266015\pi$
$439$	28.1034			1.34130			0.670651	+	0.741773i	$0.266015\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−23.8047	−	23.8047i	−1.13099	−	1.13099i	−0.990012	−	0.140981i	$-0.954974\pi$
$443$	−23.8047	−	23.8047i	−1.13099	−	1.13099i	−0.140981	−	0.990012i	$-0.545026\pi$
$444$		0			0
$445$	−4.79552	+	4.79552i	−0.227329	+	0.227329i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	27.3175i		−	1.28919i	−0.764523	−	0.644597i	$-0.777025\pi$
$449$		−	27.3175i		−	1.28919i	0.764523	−	0.644597i	$-0.222975\pi$
$450$		0			0
$451$	−11.3405	−	11.3405i	−0.534003	−	0.534003i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			2.44465i			0.114607i
$456$		0			0
$457$			3.28370i			0.153605i	0.997046	+	0.0768026i	$0.0244711\pi$
$457$			3.28370i			0.153605i	−0.997046	+	0.0768026i	$0.975529\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	21.3723	+	21.3723i	0.995407	+	0.995407i	0.999989	−	0.00458260i	$-0.00145869\pi$
$461$	21.3723	+	21.3723i	0.995407	+	0.995407i	−0.00458260	+	0.999989i	$0.501459\pi$
$462$		0			0
$463$			36.4190i			1.69254i	0.532758	+	0.846268i	$0.321156\pi$
$463$			36.4190i			1.69254i	−0.532758	+	0.846268i	$0.678844\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−22.4979	+	22.4979i	−1.04108	+	1.04108i	−0.0419609	+	0.999119i	$0.513361\pi$
$467$	−22.4979	+	22.4979i	−1.04108	+	1.04108i	−0.999119	+	0.0419609i	$0.986639\pi$
$468$		0			0
$469$	6.38033	+	6.38033i	0.294616	+	0.294616i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	18.4103			0.846505
$474$		0			0
$475$	13.0835	−	13.0835i	0.600313	−	0.600313i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−24.8888			−1.13720			−0.568598	−	0.822616i	$-0.692514\pi$
$479$	−24.8888			−1.13720			−0.568598	+	0.822616i	$0.692514\pi$
$480$		0			0
$481$	−28.2622			−1.28864
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−1.00170	+	1.00170i	−0.0454847	+	0.0454847i
$486$		0			0
$487$	−1.96454			−0.0890217			−0.0445108	−	0.999009i	$-0.514173\pi$
$487$	−1.96454			−0.0890217			−0.0445108	+	0.999009i	$0.514173\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−2.24360	−	2.24360i	−0.101252	−	0.101252i	0.654666	−	0.755918i	$-0.272809\pi$
$491$	−2.24360	−	2.24360i	−0.101252	−	0.101252i	−0.755918	+	0.654666i	$0.772809\pi$
$492$		0			0
$493$	5.91236	−	5.91236i	0.266279	−	0.266279i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			3.41542i			0.153203i
$498$		0			0
$499$	−25.8623	−	25.8623i	−1.15776	−	1.15776i	−0.984957	−	0.172800i	$-0.944718\pi$
$499$	−25.8623	−	25.8623i	−1.15776	−	1.15776i	−0.172800	−	0.984957i	$-0.555282\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			32.7702i			1.46115i	0.682833	+	0.730575i	$0.260748\pi$
$503$			32.7702i			1.46115i	−0.682833	+	0.730575i	$0.739252\pi$
$504$		0			0
$505$		−	8.12070i		−	0.361366i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	1.28132	+	1.28132i	0.0567935	+	0.0567935i	0.734933	−	0.678140i	$-0.237214\pi$
$509$	1.28132	+	1.28132i	0.0567935	+	0.0567935i	−0.678140	+	0.734933i	$0.737214\pi$
$510$		0			0
$511$			8.13689i			0.359955i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−5.17498	+	5.17498i	−0.228037	+	0.228037i
$516$		0			0
$517$	−20.4897	−	20.4897i	−0.901136	−	0.901136i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−25.9473			−1.13677			−0.568386	−	0.822762i	$-0.692432\pi$
$521$	−25.9473			−1.13677			−0.568386	+	0.822762i	$0.692432\pi$
$522$		0			0
$523$	−10.8853	+	10.8853i	−0.475981	+	0.475981i	−0.903844	−	0.427862i	$-0.859267\pi$
$523$	−10.8853	+	10.8853i	−0.475981	+	0.475981i	0.427862	+	0.903844i	$0.359267\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−11.1870			−0.487315
$528$		0			0
$529$	22.3707			0.972639
$530$		0			0
$531$		0			0
$532$		0			0
$533$	13.1747	−	13.1747i	0.570661	−	0.570661i
$534$		0			0
$535$	2.74943			0.118868
$536$		0			0
$537$		0			0
$538$		0			0
$539$	1.57551	+	1.57551i	0.0678619	+	0.0678619i
$540$		0			0
$541$	−11.4352	+	11.4352i	−0.491637	+	0.491637i	−0.908822	−	0.417185i	$-0.863017\pi$
$541$	−11.4352	+	11.4352i	−0.491637	+	0.491637i	0.417185	+	0.908822i	$0.363017\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			9.52368i			0.407949i
$546$		0			0
$547$	−0.00768172	−	0.00768172i	−0.000328447	−	0.000328447i	0.706943	−	0.707271i	$-0.250074\pi$
$547$	−0.00768172	−	0.00768172i	−0.000328447	−	0.000328447i	−0.707271	+	0.706943i	$0.750074\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	11.0541i		−	0.470922i
$552$		0			0
$553$		−	0.0502773i		−	0.00213801i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−0.247153	−	0.247153i	−0.0104722	−	0.0104722i	0.701851	−	0.712324i	$-0.252357\pi$
$557$	−0.247153	−	0.247153i	−0.0104722	−	0.0104722i	−0.712324	+	0.701851i	$0.752357\pi$
$558$		0			0
$559$			21.3880i			0.904617i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−4.42038	+	4.42038i	−0.186297	+	0.186297i	−0.794093	−	0.607796i	$-0.792054\pi$
$563$	−4.42038	+	4.42038i	−0.186297	+	0.186297i	0.607796	+	0.794093i	$0.292054\pi$
$564$		0			0
$565$	2.78978	+	2.78978i	0.117367	+	0.117367i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	24.4012			1.02295			0.511476	−	0.859297i	$-0.329099\pi$
$569$	24.4012			1.02295			0.511476	+	0.859297i	$0.329099\pi$
$570$		0			0
$571$	−8.33750	+	8.33750i	−0.348914	+	0.348914i	−0.859705	−	0.510791i	$-0.829353\pi$
$571$	−8.33750	+	8.33750i	−0.348914	+	0.348914i	0.510791	+	0.859705i	$0.329353\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	3.25886			0.135904
$576$		0			0
$577$	2.92478			0.121760			0.0608800	−	0.998145i	$-0.480609\pi$
$577$	2.92478			0.121760			0.0608800	+	0.998145i	$0.480609\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−2.29129	+	2.29129i	−0.0950589	+	0.0950589i
$582$		0			0
$583$	−10.5260			−0.435944
$584$		0			0
$585$		0			0
$586$		0			0
$587$	16.6434	+	16.6434i	0.686945	+	0.686945i	0.961556	−	0.274611i	$-0.0885490\pi$
$587$	16.6434	+	16.6434i	0.686945	+	0.686945i	−0.274611	+	0.961556i	$0.588549\pi$
$588$		0			0
$589$	−10.4580	+	10.4580i	−0.430915	+	0.430915i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	16.7065i		−	0.686054i	−0.939326	−	0.343027i	$-0.888548\pi$
$593$		−	16.7065i		−	0.686054i	0.939326	−	0.343027i	$-0.111452\pi$
$594$		0			0
$595$	−2.27516	−	2.27516i	−0.0932724	−	0.0932724i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	33.1559i		−	1.35471i	−0.735655	−	0.677356i	$-0.763126\pi$
$599$		−	33.1559i		−	1.35471i	0.735655	−	0.677356i	$-0.236874\pi$
$600$		0			0
$601$			37.5909i			1.53336i	0.642027	+	0.766682i	$0.278094\pi$
$601$			37.5909i			1.53336i	−0.642027	+	0.766682i	$0.721906\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	4.03063	+	4.03063i	0.163869	+	0.163869i
$606$		0			0
$607$		−	0.720055i		−	0.0292261i	−0.999893	−	0.0146131i	$-0.995348\pi$
$607$		−	0.720055i		−	0.0292261i	0.999893	−	0.0146131i	$-0.00465165\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	23.8038	−	23.8038i	0.962998	−	0.962998i
$612$		0			0
$613$	−11.6327	−	11.6327i	−0.469839	−	0.469839i	0.432024	−	0.901862i	$-0.357800\pi$
$613$	−11.6327	−	11.6327i	−0.469839	−	0.469839i	−0.901862	+	0.432024i	$0.857800\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	47.3378			1.90575			0.952873	−	0.303369i	$-0.0981113\pi$
$617$	47.3378			1.90575			0.952873	+	0.303369i	$0.0981113\pi$
$618$		0			0
$619$	10.9927	−	10.9927i	0.441832	−	0.441832i	−0.450795	−	0.892628i	$-0.648859\pi$
$619$	10.9927	−	10.9927i	0.441832	−	0.441832i	0.892628	+	0.450795i	$0.148859\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−7.18090			−0.287697
$624$		0			0
$625$	−12.4163			−0.496651
$626$		0			0
$627$		0			0
$628$		0			0
$629$	26.3027	−	26.3027i	1.04876	−	1.04876i
$630$		0			0
$631$	1.33698			0.0532242			0.0266121	−	0.999646i	$-0.491528\pi$
$631$	1.33698			0.0532242			0.0266121	+	0.999646i	$0.491528\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	3.44635	+	3.44635i	0.136764	+	0.136764i
$636$		0			0
$637$	−1.83034	+	1.83034i	−0.0725206	+	0.0725206i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			26.9298i			1.06367i	0.846850	+	0.531833i	$0.178496\pi$
$641$			26.9298i			1.06367i	−0.846850	+	0.531833i	$0.821504\pi$
$642$		0			0
$643$	29.8299	+	29.8299i	1.17638	+	1.17638i	0.980661	+	0.195716i	$0.0627030\pi$
$643$	29.8299	+	29.8299i	1.17638	+	1.17638i	0.195716	+	0.980661i	$0.437297\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			29.0946i			1.14383i	0.820314	+	0.571914i	$0.193799\pi$
$647$			29.0946i			1.14383i	−0.820314	+	0.571914i	$0.806201\pi$
$648$		0			0
$649$		−	23.3658i		−	0.917189i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−24.8462	−	24.8462i	−0.972308	−	0.972308i	0.0273185	−	0.999627i	$-0.491303\pi$
$653$	−24.8462	−	24.8462i	−0.972308	−	0.972308i	−0.999627	+	0.0273185i	$0.991303\pi$
$654$		0			0
$655$			2.05135i			0.0801530i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	22.2096	−	22.2096i	0.865164	−	0.865164i	−0.126769	−	0.991932i	$-0.540461\pi$
$659$	22.2096	−	22.2096i	0.865164	−	0.865164i	0.991932	+	0.126769i	$0.0404606\pi$
$660$		0			0
$661$	18.7200	+	18.7200i	0.728122	+	0.728122i	0.970245	−	0.242124i	$-0.0778440\pi$
$661$	18.7200	+	18.7200i	0.728122	+	0.728122i	−0.242124	+	0.970245i	$0.577844\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−4.25379			−0.164955
$666$		0			0
$667$	1.37669	−	1.37669i	0.0533057	−	0.0533057i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	6.08303			0.234833
$672$		0			0
$673$	16.1228			0.621488			0.310744	−	0.950494i	$-0.399422\pi$
$673$	16.1228			0.621488			0.310744	+	0.950494i	$0.399422\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	12.0537	−	12.0537i	0.463261	−	0.463261i	−0.436462	−	0.899723i	$-0.643769\pi$
$677$	12.0537	−	12.0537i	0.463261	−	0.463261i	0.899723	+	0.436462i	$0.143769\pi$
$678$		0			0
$679$	−1.49996			−0.0575632
$680$		0			0
$681$		0			0
$682$		0			0
$683$	32.7416	+	32.7416i	1.25282	+	1.25282i	0.954450	+	0.298372i	$0.0964436\pi$
$683$	32.7416	+	32.7416i	1.25282	+	1.25282i	0.298372	+	0.954450i	$0.403556\pi$
$684$		0			0
$685$	13.3994	−	13.3994i	0.511964	−	0.511964i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	12.2285i		−	0.465871i
$690$		0			0
$691$	21.3170	+	21.3170i	0.810936	+	0.810936i	0.984774	−	0.173838i	$-0.0556169\pi$
$691$	21.3170	+	21.3170i	0.810936	+	0.810936i	−0.173838	+	0.984774i	$0.555617\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			4.03290i			0.152977i
$696$		0			0
$697$			24.5226i			0.928860i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	23.9475	+	23.9475i	0.904483	+	0.904483i	0.995820	−	0.0913369i	$-0.0291140\pi$
$701$	23.9475	+	23.9475i	0.904483	+	0.904483i	−0.0913369	+	0.995820i	$0.529114\pi$
$702$		0			0
$703$		−	49.1773i		−	1.85476i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	6.08005	−	6.08005i	0.228664	−	0.228664i
$708$		0			0
$709$	20.3189	+	20.3189i	0.763092	+	0.763092i	0.976880	−	0.213788i	$-0.0685801\pi$
$709$	20.3189	+	20.3189i	0.763092	+	0.763092i	−0.213788	+	0.976880i	$0.568580\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−2.60490			−0.0975542
$714$		0			0
$715$	3.85157	−	3.85157i	0.144041	−	0.144041i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	5.54623			0.206839			0.103420	−	0.994638i	$-0.467022\pi$
$719$	5.54623			0.206839			0.103420	+	0.994638i	$0.467022\pi$
$720$		0			0
$721$	−7.74912			−0.288592
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−7.12920	+	7.12920i	−0.264772	+	0.264772i
$726$		0			0
$727$	−35.2516			−1.30741			−0.653704	−	0.756750i	$-0.726786\pi$
$727$	−35.2516			−1.30741			−0.653704	+	0.756750i	$0.726786\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−19.9051	−	19.9051i	−0.736218	−	0.736218i
$732$		0			0
$733$	−25.7407	+	25.7407i	−0.950753	+	0.950753i	−0.998843	−	0.0480899i	$-0.984687\pi$
$733$	−25.7407	+	25.7407i	−0.950753	+	0.950753i	0.0480899	+	0.998843i	$0.484687\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	20.1045i		−	0.740559i
$738$		0			0
$739$	−8.42027	−	8.42027i	−0.309745	−	0.309745i	0.535066	−	0.844810i	$-0.320287\pi$
$739$	−8.42027	−	8.42027i	−0.309745	−	0.309745i	−0.844810	+	0.535066i	$0.820287\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			37.6054i			1.37961i	0.723996	+	0.689805i	$0.242304\pi$
$743$			37.6054i			1.37961i	−0.723996	+	0.689805i	$0.757696\pi$
$744$		0			0
$745$		−	2.82341i		−	0.103442i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	2.05853	+	2.05853i	0.0752170	+	0.0752170i
$750$		0			0
$751$		−	33.8331i		−	1.23459i	−0.786732	−	0.617294i	$-0.788229\pi$
$751$		−	33.8331i		−	1.23459i	0.786732	−	0.617294i	$-0.211771\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	4.72850	−	4.72850i	0.172088	−	0.172088i
$756$		0			0
$757$	−3.38072	−	3.38072i	−0.122874	−	0.122874i	0.642996	−	0.765870i	$-0.277691\pi$
$757$	−3.38072	−	3.38072i	−0.122874	−	0.122874i	−0.765870	+	0.642996i	$0.777691\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	16.7200			0.606100			0.303050	−	0.952975i	$-0.401995\pi$
$761$	16.7200			0.606100			0.303050	+	0.952975i	$0.401995\pi$
$762$		0			0
$763$	−7.13047	+	7.13047i	−0.258140	+	0.258140i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	27.1451			0.980152
$768$		0			0
$769$	13.6721			0.493030			0.246515	−	0.969139i	$-0.420715\pi$
$769$	13.6721			0.493030			0.246515	+	0.969139i	$0.420715\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−29.3858	+	29.3858i	−1.05694	+	1.05694i	−0.0586568	+	0.998278i	$0.518682\pi$
$773$	−29.3858	+	29.3858i	−1.05694	+	1.05694i	−0.998278	+	0.0586568i	$0.981318\pi$
$774$		0			0
$775$	13.4895			0.484556
$776$		0			0
$777$		0			0
$778$		0			0
$779$	22.9245	+	22.9245i	0.821357	+	0.821357i
$780$		0			0
$781$	5.38103	−	5.38103i	0.192548	−	0.192548i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			21.4455i			0.765422i
$786$		0			0
$787$	−31.3525	−	31.3525i	−1.11760	−	1.11760i	−0.992093	−	0.125504i	$-0.959945\pi$
$787$	−31.3525	−	31.3525i	−1.11760	−	1.11760i	−0.125504	−	0.992093i	$-0.540055\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			4.17748i			0.148534i
$792$		0			0
$793$			7.06693i			0.250954i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	33.0542	+	33.0542i	1.17084	+	1.17084i	0.982010	+	0.188830i	$0.0604695\pi$
$797$	33.0542	+	33.0542i	1.17084	+	1.17084i	0.188830	+	0.982010i	$0.439530\pi$
$798$		0			0
$799$			44.3068i			1.56746i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	12.8197	−	12.8197i	0.452399	−	0.452399i
$804$		0			0
$805$	−0.529770	−	0.529770i	−0.0186719	−	0.0186719i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−21.3611			−0.751016			−0.375508	−	0.926819i	$-0.622532\pi$
$809$	−21.3611			−0.751016			−0.375508	+	0.926819i	$0.622532\pi$
$810$		0			0
$811$	−5.81041	+	5.81041i	−0.204031	+	0.204031i	−0.801725	−	0.597694i	$-0.796084\pi$
$811$	−5.81041	+	5.81041i	−0.204031	+	0.204031i	0.597694	+	0.801725i	$0.296084\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	14.5926			0.511158
$816$		0			0
$817$	−37.2160			−1.30202
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−38.5077	+	38.5077i	−1.34393	+	1.34393i	−0.451816	+	0.892111i	$0.649224\pi$
$821$	−38.5077	+	38.5077i	−1.34393	+	1.34393i	−0.892111	+	0.451816i	$0.850776\pi$
$822$		0			0
$823$	−50.6315			−1.76490			−0.882451	−	0.470404i	$-0.844108\pi$
$823$	−50.6315			−1.76490			−0.882451	+	0.470404i	$0.844108\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	25.9992	+	25.9992i	0.904081	+	0.904081i	0.995786	−	0.0917052i	$-0.0292317\pi$
$827$	25.9992	+	25.9992i	0.904081	+	0.904081i	−0.0917052	+	0.995786i	$0.529232\pi$
$828$		0			0
$829$	5.45810	−	5.45810i	0.189568	−	0.189568i	−0.605941	−	0.795509i	$-0.707203\pi$
$829$	5.45810	−	5.45810i	0.189568	−	0.189568i	0.795509	+	0.605941i	$0.207203\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	3.40687i		−	0.118041i
$834$		0			0
$835$	9.92082	+	9.92082i	0.343324	+	0.343324i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	33.0422i		−	1.14074i	−0.821387	−	0.570372i	$-0.806799\pi$
$839$		−	33.0422i		−	1.14074i	0.821387	−	0.570372i	$-0.193201\pi$
$840$		0			0
$841$		−	22.9766i		−	0.792297i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−4.20706	−	4.20706i	−0.144727	−	0.144727i
$846$		0			0
$847$			6.03555i			0.207384i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	6.12457	−	6.12457i	0.209948	−	0.209948i
$852$		0			0
$853$	−8.05330	−	8.05330i	−0.275740	−	0.275740i	0.555666	−	0.831406i	$-0.312463\pi$
$853$	−8.05330	−	8.05330i	−0.275740	−	0.275740i	−0.831406	+	0.555666i	$0.812463\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−44.6143			−1.52400			−0.761998	−	0.647579i	$-0.775782\pi$
$857$	−44.6143			−1.52400			−0.761998	+	0.647579i	$0.775782\pi$
$858$		0			0
$859$	0.721531	−	0.721531i	0.0246183	−	0.0246183i	−0.694690	−	0.719309i	$-0.744459\pi$
$859$	0.721531	−	0.721531i	0.0246183	−	0.0246183i	0.719309	+	0.694690i	$0.244459\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	26.1045			0.888609			0.444304	−	0.895876i	$-0.353451\pi$
$863$	26.1045			0.888609			0.444304	+	0.895876i	$0.353451\pi$
$864$		0			0
$865$	3.33443			0.113374
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−0.0792122	+	0.0792122i	−0.00268709	+	0.00268709i
$870$		0			0
$871$	23.3563			0.791397
$872$		0			0
$873$		0			0
$874$		0			0
$875$	6.08249	+	6.08249i	0.205626	+	0.205626i
$876$		0			0
$877$	−2.63163	+	2.63163i	−0.0888637	+	0.0888637i	−0.750141	−	0.661278i	$-0.770015\pi$
$877$	−2.63163	+	2.63163i	−0.0888637	+	0.0888637i	0.661278	+	0.750141i	$0.270015\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			6.58849i			0.221972i	0.993822	+	0.110986i	$0.0354009\pi$
$881$			6.58849i			0.221972i	−0.993822	+	0.110986i	$0.964599\pi$
$882$		0			0
$883$	−10.8297	−	10.8297i	−0.364449	−	0.364449i	0.500999	−	0.865448i	$-0.332966\pi$
$883$	−10.8297	−	10.8297i	−0.364449	−	0.364449i	−0.865448	+	0.500999i	$0.832966\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			21.7156i			0.729138i	0.931176	+	0.364569i	$0.118784\pi$
$887$			21.7156i			0.729138i	−0.931176	+	0.364569i	$0.881216\pi$
$888$		0			0
$889$			5.16063i			0.173082i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	41.4195	+	41.4195i	1.38605	+	1.38605i
$894$		0			0
$895$			18.1190i			0.605650i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	5.69856	−	5.69856i	0.190058	−	0.190058i
$900$		0			0
$901$	11.3807	+	11.3807i	0.379146	+	0.379146i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	5.52733			0.183735
$906$		0			0
$907$	−22.7390	+	22.7390i	−0.755035	+	0.755035i	−0.975414	−	0.220379i	$-0.929271\pi$
$907$	−22.7390	+	22.7390i	−0.755035	+	0.755035i	0.220379	+	0.975414i	$0.429271\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	27.1521			0.899589			0.449794	−	0.893132i	$-0.351497\pi$
$911$	27.1521			0.899589			0.449794	+	0.893132i	$0.351497\pi$
$912$		0			0
$913$	7.21991			0.238944
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−1.53587	+	1.53587i	−0.0507189	+	0.0507189i
$918$		0			0
$919$	24.2903			0.801262			0.400631	−	0.916239i	$-0.368791\pi$
$919$	24.2903			0.801262			0.400631	+	0.916239i	$0.368791\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	6.25137	+	6.25137i	0.205766	+	0.205766i
$924$		0			0
$925$	−31.7161	+	31.7161i	−1.04282	+	1.04282i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			39.4028i			1.29276i	0.763014	+	0.646382i	$0.223719\pi$
$929$			39.4028i			1.29276i	−0.763014	+	0.646382i	$0.776281\pi$
$930$		0			0
$931$	−3.18485	−	3.18485i	−0.104379	−	0.104379i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			7.16906i			0.234453i
$936$		0			0
$937$			51.2135i			1.67307i	0.547913	+	0.836536i	$0.315423\pi$
$937$			51.2135i			1.67307i	−0.547913	+	0.836536i	$0.684577\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−28.4201	−	28.4201i	−0.926470	−	0.926470i	0.0710057	−	0.997476i	$-0.477379\pi$
$941$	−28.4201	−	28.4201i	−0.926470	−	0.926470i	−0.997476	+	0.0710057i	$0.977379\pi$
$942$		0			0
$943$			5.71008i			0.185946i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	25.5107	−	25.5107i	0.828985	−	0.828985i	−0.158391	−	0.987376i	$-0.550631\pi$
$947$	25.5107	−	25.5107i	0.828985	−	0.828985i	0.987376	+	0.158391i	$0.0506308\pi$
$948$		0			0
$949$	14.8933	+	14.8933i	0.483455	+	0.483455i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	44.5373			1.44270			0.721352	−	0.692569i	$-0.243521\pi$
$953$	44.5373			1.44270			0.721352	+	0.692569i	$0.243521\pi$
$954$		0			0
$955$	14.6295	−	14.6295i	0.473399	−	0.473399i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	20.0645			0.647917
$960$		0			0
$961$	20.2175			0.652178
$962$		0			0
$963$		0			0
$964$		0			0
$965$	14.1196	−	14.1196i	0.454525	−	0.454525i
$966$		0			0
$967$	2.17126			0.0698231			0.0349116	−	0.999390i	$-0.488885\pi$
$967$	2.17126			0.0698231			0.0349116	+	0.999390i	$0.488885\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−31.9194	−	31.9194i	−1.02434	−	1.02434i	−0.999696	−	0.0246475i	$-0.992154\pi$
$971$	−31.9194	−	31.9194i	−1.02434	−	1.02434i	−0.0246475	−	0.999696i	$-0.507846\pi$
$972$		0			0
$973$	−3.01947	+	3.01947i	−0.0967998	+	0.0967998i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			21.5480i			0.689381i	0.938716	+	0.344691i	$0.112016\pi$
$977$			21.5480i			0.689381i	−0.938716	+	0.344691i	$0.887984\pi$
$978$		0			0
$979$	11.3136	+	11.3136i	0.361583	+	0.361583i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	45.1484i		−	1.44001i	−0.693968	−	0.720006i	$-0.744139\pi$
$983$		−	45.1484i		−	1.44001i	0.693968	−	0.720006i	$-0.255861\pi$
$984$		0			0
$985$		−	6.61172i		−	0.210667i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−4.63490	−	4.63490i	−0.147381	−	0.147381i
$990$		0			0
$991$			12.3053i			0.390892i	0.980714	+	0.195446i	$0.0626154\pi$
$991$			12.3053i			0.390892i	−0.980714	+	0.195446i	$0.937385\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	7.10922	−	7.10922i	0.225377	−	0.225377i
$996$		0			0
$997$	−2.19520	−	2.19520i	−0.0695227	−	0.0695227i	0.671491	−	0.741013i	$-0.265654\pi$
$997$	−2.19520	−	2.19520i	−0.0695227	−	0.0695227i	−0.741013	+	0.671491i	$0.765654\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4032.2.v.e.3599.8		40
3.2	odd	2	inner	4032.2.v.e.3599.13		40
4.3	odd	2		1008.2.v.e.827.8	yes	40
12.11	even	2		1008.2.v.e.827.13	yes	40
16.3	odd	4	inner	4032.2.v.e.1583.13		40
16.13	even	4		1008.2.v.e.323.13	yes	40
48.29	odd	4		1008.2.v.e.323.8	✓	40
48.35	even	4	inner	4032.2.v.e.1583.8		40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1008.2.v.e.323.8	✓	40	48.29	odd	4
1008.2.v.e.323.13	yes	40	16.13	even	4
1008.2.v.e.827.8	yes	40	4.3	odd	2
1008.2.v.e.827.13	yes	40	12.11	even	2
4032.2.v.e.1583.8		40	48.35	even	4	inner
4032.2.v.e.1583.13		40	16.3	odd	4	inner
4032.2.v.e.3599.8		40	1.1	even	1	trivial
4032.2.v.e.3599.13		40	3.2	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.v (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.667815 + 0.667815i) q^{5} -1.00000 q^{7} +O(q^{10})\)	\(q+(-0.667815 + 0.667815i) q^{5} -1.00000 q^{7} +(1.57551 + 1.57551i) q^{11} +(-1.83034 + 1.83034i) q^{13} -3.40687i q^{17} +(-3.18485 - 3.18485i) q^{19} -0.793288i q^{23} +4.10805i q^{25} +(1.73542 + 1.73542i) q^{29} -3.28367i q^{31} +(0.667815 - 0.667815i) q^{35} +(7.72049 + 7.72049i) q^{37} -7.19799 q^{41} +(5.84265 - 5.84265i) q^{43} -13.0051 q^{47} +1.00000 q^{49} +(-3.34052 + 3.34052i) q^{53} -2.10430 q^{55} +(-7.41533 - 7.41533i) q^{59} +(1.93050 - 1.93050i) q^{61} -2.44465i q^{65} +(-6.38033 - 6.38033i) q^{67} -3.41542i q^{71} -8.13689i q^{73} +(-1.57551 - 1.57551i) q^{77} +0.0502773i q^{79} +(2.29129 - 2.29129i) q^{83} +(2.27516 + 2.27516i) q^{85} +7.18090 q^{89} +(1.83034 - 1.83034i) q^{91} +4.25379 q^{95} +1.49996 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 40 q^{7}+O(q^{10}) \) 40 * q - 40 * q^7	\( 40 q - 40 q^{7} - 24 q^{13} + 32 q^{19} - 8 q^{37} - 32 q^{43} + 40 q^{49} - 48 q^{55} - 24 q^{61} + 64 q^{85} + 24 q^{91} + 64 q^{97}+O(q^{100}) \) 40 * q - 40 * q^7 - 24 * q^13 + 32 * q^19 - 8 * q^37 - 32 * q^43 + 40 * q^49 - 48 * q^55 - 24 * q^61 + 64 * q^85 + 24 * q^91 + 64 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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