LMFDB - Embedded newform 4032.2.v.e.1583.16

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			1583.16
Character	$\chi$	$=$	4032.1583
Dual form			4032.2.v.e.3599.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.65702 + 1.65702i) q^{5} -1.00000 q^{7} +O(q^{10})$	$q+(1.65702 + 1.65702i) q^{5} -1.00000 q^{7} +(-0.993222 + 0.993222i) q^{11} +(2.62686 + 2.62686i) q^{13} +2.77728i q^{17} +(-1.56309 + 1.56309i) q^{19} -1.05272i q^{23} +0.491445i q^{25} +(-3.47823 + 3.47823i) q^{29} +1.06124i q^{31} +(-1.65702 - 1.65702i) q^{35} +(-0.0657126 + 0.0657126i) q^{37} -6.31313 q^{41} +(2.38841 + 2.38841i) q^{43} -1.47190 q^{47} +1.00000 q^{49} +(7.63807 + 7.63807i) q^{53} -3.29158 q^{55} +(-4.15490 + 4.15490i) q^{59} +(-7.78123 - 7.78123i) q^{61} +8.70553i q^{65} +(-1.98557 + 1.98557i) q^{67} -13.0855i q^{71} -9.50329i q^{73} +(0.993222 - 0.993222i) q^{77} +9.85051i q^{79} +(1.13002 + 1.13002i) q^{83} +(-4.60202 + 4.60202i) q^{85} -7.04453 q^{89} +(-2.62686 - 2.62686i) q^{91} -5.18016 q^{95} +5.35352 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/4032\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$1793$	$3781$
$\chi(n)$	$-1$	$1$	$-1$	$e\left(\frac{3}{4}\right)$

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
$5$	1.65702	+	1.65702i	0.741043	+	0.741043i	0.972779	−	0.231736i	$-0.0744405\pi$
$5$	1.65702	+	1.65702i	0.741043	+	0.741043i	−0.231736	+	0.972779i	$0.574441\pi$
$6$		0			0
$7$	−1.00000			−0.377964
$7$	−1.00000			−0.377964
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−0.993222	+	0.993222i	−0.299468	+	0.299468i	−0.840805	−	0.541338i	$-0.817918\pi$
$11$	−0.993222	+	0.993222i	−0.299468	+	0.299468i	0.541338	+	0.840805i	$0.317918\pi$
$12$		0			0
$13$	2.62686	+	2.62686i	0.728560	+	0.728560i	0.970333	−	0.241773i	$-0.0777290\pi$
$13$	2.62686	+	2.62686i	0.728560	+	0.728560i	−0.241773	+	0.970333i	$0.577729\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			2.77728i			0.673589i	0.941578	+	0.336795i	$0.109343\pi$
$17$			2.77728i			0.673589i	−0.941578	+	0.336795i	$0.890657\pi$
$18$		0			0
$19$	−1.56309	+	1.56309i	−0.358598	+	0.358598i	−0.863296	−	0.504698i	$-0.831604\pi$
$19$	−1.56309	+	1.56309i	−0.358598	+	0.358598i	0.504698	+	0.863296i	$0.331604\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	1.05272i		−	0.219507i	−0.993959	−	0.109754i	$-0.964994\pi$
$23$		−	1.05272i		−	0.219507i	0.993959	−	0.109754i	$-0.0350062\pi$
$24$		0			0
$25$			0.491445i			0.0982890i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−3.47823	+	3.47823i	−0.645891	+	0.645891i	−0.951997	−	0.306106i	$-0.900974\pi$
$29$	−3.47823	+	3.47823i	−0.645891	+	0.645891i	0.306106	+	0.951997i	$0.400974\pi$
$30$		0			0
$31$			1.06124i			0.190605i	0.995448	+	0.0953026i	$0.0303819\pi$
$31$			1.06124i			0.190605i	−0.995448	+	0.0953026i	$0.969618\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−1.65702	−	1.65702i	−0.280088	−	0.280088i
$36$		0			0
$37$	−0.0657126	+	0.0657126i	−0.0108031	+	0.0108031i	−0.712488	−	0.701685i	$-0.752432\pi$
$37$	−0.0657126	+	0.0657126i	−0.0108031	+	0.0108031i	0.701685	+	0.712488i	$0.252432\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−6.31313			−0.985945			−0.492972	−	0.870045i	$-0.664090\pi$
$41$	−6.31313			−0.985945			−0.492972	+	0.870045i	$0.664090\pi$
$42$		0			0
$43$	2.38841	+	2.38841i	0.364229	+	0.364229i	0.865367	−	0.501138i	$-0.167085\pi$
$43$	2.38841	+	2.38841i	0.364229	+	0.364229i	−0.501138	+	0.865367i	$0.667085\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−1.47190			−0.214699			−0.107349	−	0.994221i	$-0.534236\pi$
$47$	−1.47190			−0.214699			−0.107349	+	0.994221i	$0.534236\pi$
$48$		0			0
$49$	1.00000			0.142857
$50$		0			0
$51$		0			0
$52$		0			0
$53$	7.63807	+	7.63807i	1.04917	+	1.04917i	0.998727	+	0.0504431i	$0.0160634\pi$
$53$	7.63807	+	7.63807i	1.04917	+	1.04917i	0.0504431	+	0.998727i	$0.483937\pi$
$54$		0			0
$55$	−3.29158			−0.443837
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−4.15490	+	4.15490i	−0.540921	+	0.540921i	−0.923799	−	0.382878i	$-0.874933\pi$
$59$	−4.15490	+	4.15490i	−0.540921	+	0.540921i	0.382878	+	0.923799i	$0.374933\pi$
$60$		0			0
$61$	−7.78123	−	7.78123i	−0.996285	−	0.996285i	0.00370838	−	0.999993i	$-0.498820\pi$
$61$	−7.78123	−	7.78123i	−0.996285	−	0.996285i	−0.999993	+	0.00370838i	$0.998820\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			8.70553i			1.07979i
$66$		0			0
$67$	−1.98557	+	1.98557i	−0.242576	+	0.242576i	−0.817915	−	0.575339i	$-0.804870\pi$
$67$	−1.98557	+	1.98557i	−0.242576	+	0.242576i	0.575339	+	0.817915i	$0.304870\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	13.0855i		−	1.55297i	−0.630137	−	0.776484i	$-0.717001\pi$
$71$		−	13.0855i		−	1.55297i	0.630137	−	0.776484i	$-0.282999\pi$
$72$		0			0
$73$		−	9.50329i		−	1.11228i	−0.831090	−	0.556138i	$-0.812283\pi$
$73$		−	9.50329i		−	1.11228i	0.831090	−	0.556138i	$-0.187717\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	0.993222	−	0.993222i	0.113188	−	0.113188i
$78$		0			0
$79$			9.85051i			1.10827i	0.832427	+	0.554134i	$0.186951\pi$
$79$			9.85051i			1.10827i	−0.832427	+	0.554134i	$0.813049\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	1.13002	+	1.13002i	0.124036	+	0.124036i	0.766400	−	0.642364i	$-0.222046\pi$
$83$	1.13002	+	1.13002i	0.124036	+	0.124036i	−0.642364	+	0.766400i	$0.722046\pi$
$84$		0			0
$85$	−4.60202	+	4.60202i	−0.499159	+	0.499159i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−7.04453			−0.746718			−0.373359	−	0.927687i	$-0.621794\pi$
$89$	−7.04453			−0.746718			−0.373359	+	0.927687i	$0.621794\pi$
$90$		0			0
$91$	−2.62686	−	2.62686i	−0.275370	−	0.275370i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−5.18016			−0.531473
$96$		0			0
$97$	5.35352			0.543568			0.271784	−	0.962358i	$-0.412386\pi$
$97$	5.35352			0.543568			0.271784	+	0.962358i	$0.412386\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	5.34591	+	5.34591i	0.531938	+	0.531938i	0.921149	−	0.389211i	$-0.127252\pi$
$101$	5.34591	+	5.34591i	0.531938	+	0.531938i	−0.389211	+	0.921149i	$0.627252\pi$
$102$		0			0
$103$	−3.41355			−0.336347			−0.168173	−	0.985757i	$-0.553787\pi$
$103$	−3.41355			−0.336347			−0.168173	+	0.985757i	$0.553787\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−10.7004	+	10.7004i	−1.03445	+	1.03445i	−0.0350640	+	0.999385i	$0.511164\pi$
$107$	−10.7004	+	10.7004i	−1.03445	+	1.03445i	−0.999385	+	0.0350640i	$0.988836\pi$
$108$		0			0
$109$	−5.36354	−	5.36354i	−0.513734	−	0.513734i	0.401934	−	0.915668i	$-0.368338\pi$
$109$	−5.36354	−	5.36354i	−0.513734	−	0.513734i	−0.915668	+	0.401934i	$0.868338\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			18.2463i			1.71647i	0.513256	+	0.858236i	$0.328439\pi$
$113$			18.2463i			1.71647i	−0.513256	+	0.858236i	$0.671561\pi$
$114$		0			0
$115$	1.74438	−	1.74438i	0.162664	−	0.162664i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	2.77728i		−	0.254593i
$120$		0			0
$121$			9.02702i			0.820638i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	7.47078	−	7.47078i	0.668206	−	0.668206i
$126$		0			0
$127$			0.0998053i			0.00885629i	0.999990	+	0.00442814i	$0.00140953\pi$
$127$			0.0998053i			0.00885629i	−0.999990	+	0.00442814i	$0.998590\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	12.8436	+	12.8436i	1.12215	+	1.12215i	0.991417	+	0.130734i	$0.0417335\pi$
$131$	12.8436	+	12.8436i	1.12215	+	1.12215i	0.130734	+	0.991417i	$0.458266\pi$
$132$		0			0
$133$	1.56309	−	1.56309i	0.135537	−	0.135537i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	3.10918			0.265635			0.132817	−	0.991141i	$-0.457598\pi$
$137$	3.10918			0.265635			0.132817	+	0.991141i	$0.457598\pi$
$138$		0			0
$139$	−10.5794	−	10.5794i	−0.897330	−	0.897330i	0.0978696	−	0.995199i	$-0.468797\pi$
$139$	−10.5794	−	10.5794i	−0.897330	−	0.897330i	−0.995199	+	0.0978696i	$0.968797\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−5.21811			−0.436360
$144$		0			0
$145$	−11.5270			−0.957266
$146$		0			0
$147$		0			0
$148$		0			0
$149$	6.32230	+	6.32230i	0.517943	+	0.517943i	0.916948	−	0.399006i	$-0.130645\pi$
$149$	6.32230	+	6.32230i	0.517943	+	0.517943i	−0.399006	+	0.916948i	$0.630645\pi$
$150$		0			0
$151$	−22.9393			−1.86677			−0.933385	−	0.358877i	$-0.883160\pi$
$151$	−22.9393			−1.86677			−0.933385	+	0.358877i	$0.883160\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−1.75851	+	1.75851i	−0.141247	+	0.141247i
$156$		0			0
$157$	−12.7269	−	12.7269i	−1.01571	−	1.01571i	−0.999875	−	0.0158402i	$-0.994958\pi$
$157$	−12.7269	−	12.7269i	−1.01571	−	1.01571i	−0.0158402	−	0.999875i	$-0.505042\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			1.05272i			0.0829660i
$162$		0			0
$163$	−5.66445	+	5.66445i	−0.443674	+	0.443674i	−0.893245	−	0.449571i	$-0.851577\pi$
$163$	−5.66445	+	5.66445i	−0.443674	+	0.443674i	0.449571	+	0.893245i	$0.351577\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			20.3445i			1.57431i	0.616756	+	0.787154i	$0.288446\pi$
$167$			20.3445i			1.57431i	−0.616756	+	0.787154i	$0.711554\pi$
$168$		0			0
$169$			0.800782i			0.0615986i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	3.94764	−	3.94764i	0.300133	−	0.300133i	−0.540933	−	0.841066i	$-0.681929\pi$
$173$	3.94764	−	3.94764i	0.300133	−	0.300133i	0.841066	+	0.540933i	$0.181929\pi$
$174$		0			0
$175$		−	0.491445i		−	0.0371497i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	12.5761	+	12.5761i	0.939982	+	0.939982i	0.998298	−	0.0583166i	$-0.0185733\pi$
$179$	12.5761	+	12.5761i	0.939982	+	0.939982i	−0.0583166	+	0.998298i	$0.518573\pi$
$180$		0			0
$181$	16.0136	−	16.0136i	1.19028	−	1.19028i	0.213295	−	0.976988i	$-0.431580\pi$
$181$	16.0136	−	16.0136i	1.19028	−	1.19028i	0.976988	−	0.213295i	$-0.0684196\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−0.217775			−0.0160111
$186$		0			0
$187$	−2.75846	−	2.75846i	−0.201718	−	0.201718i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−11.8336			−0.856251			−0.428126	−	0.903719i	$-0.640826\pi$
$191$	−11.8336			−0.856251			−0.428126	+	0.903719i	$0.640826\pi$
$192$		0			0
$193$	21.1021			1.51896			0.759481	−	0.650529i	$-0.225453\pi$
$193$	21.1021			1.51896			0.759481	+	0.650529i	$0.225453\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−5.83308	−	5.83308i	−0.415590	−	0.415590i	0.468091	−	0.883680i	$-0.344942\pi$
$197$	−5.83308	−	5.83308i	−0.415590	−	0.415590i	−0.883680	+	0.468091i	$0.844942\pi$
$198$		0			0
$199$	−23.2865			−1.65074			−0.825369	−	0.564594i	$-0.809033\pi$
$199$	−23.2865			−1.65074			−0.825369	+	0.564594i	$0.809033\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	3.47823	−	3.47823i	0.244124	−	0.244124i
$204$		0			0
$205$	−10.4610	−	10.4610i	−0.730627	−	0.730627i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	3.10500i		−	0.214777i
$210$		0			0
$211$	16.2720	−	16.2720i	1.12021	−	1.12021i	0.128501	−	0.991709i	$-0.458983\pi$
$211$	16.2720	−	16.2720i	1.12021	−	1.12021i	0.991709	−	0.128501i	$-0.0410165\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			7.91529i			0.539818i
$216$		0			0
$217$		−	1.06124i		−	0.0720420i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−7.29553	+	7.29553i	−0.490750	+	0.490750i
$222$		0			0
$223$			22.8221i			1.52828i	0.645051	+	0.764139i	$0.276836\pi$
$223$			22.8221i			1.52828i	−0.645051	+	0.764139i	$0.723164\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	6.09419	+	6.09419i	0.404486	+	0.404486i	0.879810	−	0.475325i	$-0.157669\pi$
$227$	6.09419	+	6.09419i	0.404486	+	0.404486i	−0.475325	+	0.879810i	$0.657669\pi$
$228$		0			0
$229$	−8.57148	+	8.57148i	−0.566420	+	0.566420i	−0.931124	−	0.364704i	$-0.881170\pi$
$229$	−8.57148	+	8.57148i	−0.566420	+	0.566420i	0.364704	+	0.931124i	$0.381170\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−4.90649			−0.321435			−0.160717	−	0.987000i	$-0.551381\pi$
$233$	−4.90649			−0.321435			−0.160717	+	0.987000i	$0.551381\pi$
$234$		0			0
$235$	−2.43897	−	2.43897i	−0.159101	−	0.159101i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	11.1659			0.722260			0.361130	−	0.932515i	$-0.382391\pi$
$239$	11.1659			0.722260			0.361130	+	0.932515i	$0.382391\pi$
$240$		0			0
$241$	−17.8083			−1.14713			−0.573566	−	0.819159i	$-0.694440\pi$
$241$	−17.8083			−1.14713			−0.573566	+	0.819159i	$0.694440\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	1.65702	+	1.65702i	0.105863	+	0.105863i
$246$		0			0
$247$	−8.21206			−0.522521
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−10.7431	+	10.7431i	−0.678099	+	0.678099i	−0.959570	−	0.281471i	$-0.909178\pi$
$251$	−10.7431	+	10.7431i	−0.678099	+	0.678099i	0.281471	+	0.959570i	$0.409178\pi$
$252$		0			0
$253$	1.04559	+	1.04559i	0.0657354	+	0.0657354i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			1.20282i			0.0750300i	0.999296	+	0.0375150i	$0.0119442\pi$
$257$			1.20282i			0.0750300i	−0.999296	+	0.0375150i	$0.988056\pi$
$258$		0			0
$259$	0.0657126	−	0.0657126i	0.00408318	−	0.00408318i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	14.9654i		−	0.922803i	−0.887191	−	0.461402i	$-0.847347\pi$
$263$		−	14.9654i		−	0.922803i	0.887191	−	0.461402i	$-0.152653\pi$
$264$		0			0
$265$			25.3129i			1.55496i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	6.58780	−	6.58780i	0.401665	−	0.401665i	−0.477154	−	0.878819i	$-0.658332\pi$
$269$	6.58780	−	6.58780i	0.401665	−	0.401665i	0.878819	+	0.477154i	$0.158332\pi$
$270$		0			0
$271$		−	27.1436i		−	1.64886i	−0.565965	−	0.824429i	$-0.691496\pi$
$271$		−	27.1436i		−	1.64886i	0.565965	−	0.824429i	$-0.308504\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−0.488114	−	0.488114i	−0.0294344	−	0.0294344i
$276$		0			0
$277$	12.0118	−	12.0118i	0.721720	−	0.721720i	−0.247235	−	0.968955i	$-0.579522\pi$
$277$	12.0118	−	12.0118i	0.721720	−	0.721720i	0.968955	+	0.247235i	$0.0795221\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−8.46830			−0.505176			−0.252588	−	0.967574i	$-0.581282\pi$
$281$	−8.46830			−0.505176			−0.252588	+	0.967574i	$0.581282\pi$
$282$		0			0
$283$	−8.77919	−	8.77919i	−0.521868	−	0.521868i	0.396267	−	0.918135i	$-0.370305\pi$
$283$	−8.77919	−	8.77919i	−0.521868	−	0.521868i	−0.918135	+	0.396267i	$0.870305\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	6.31313			0.372652
$288$		0			0
$289$	9.28671			0.546277
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−9.25965	−	9.25965i	−0.540955	−	0.540955i	0.382854	−	0.923809i	$-0.374941\pi$
$293$	−9.25965	−	9.25965i	−0.540955	−	0.540955i	−0.923809	+	0.382854i	$0.874941\pi$
$294$		0			0
$295$	−13.7695			−0.801692
$296$		0			0
$297$		0			0
$298$		0			0
$299$	2.76535	−	2.76535i	0.159924	−	0.159924i
$300$		0			0
$301$	−2.38841	−	2.38841i	−0.137665	−	0.137665i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	25.7873i		−	1.47658i
$306$		0			0
$307$	9.60065	−	9.60065i	0.547938	−	0.547938i	−0.377906	−	0.925844i	$-0.623356\pi$
$307$	9.60065	−	9.60065i	0.547938	−	0.547938i	0.925844	+	0.377906i	$0.123356\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			23.1840i			1.31464i	0.753611	+	0.657321i	$0.228310\pi$
$311$			23.1840i			1.31464i	−0.753611	+	0.657321i	$0.771690\pi$
$312$		0			0
$313$			27.9361i			1.57904i	0.613723	+	0.789522i	$0.289671\pi$
$313$			27.9361i			1.57904i	−0.613723	+	0.789522i	$0.710329\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−21.5910	+	21.5910i	−1.21267	+	1.21267i	−0.242529	+	0.970144i	$0.577977\pi$
$317$	−21.5910	+	21.5910i	−1.21267	+	1.21267i	−0.970144	+	0.242529i	$0.922023\pi$
$318$		0			0
$319$		−	6.90931i		−	0.386847i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−4.34115	−	4.34115i	−0.241548	−	0.241548i
$324$		0			0
$325$	−1.29096	+	1.29096i	−0.0716094	+	0.0716094i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	1.47190			0.0811485
$330$		0			0
$331$	18.4892	+	18.4892i	1.01626	+	1.01626i	0.999866	+	0.0163917i	$0.00521788\pi$
$331$	18.4892	+	18.4892i	1.01626	+	1.01626i	0.0163917	+	0.999866i	$0.494782\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−6.58027			−0.359519
$336$		0			0
$337$	−8.58189			−0.467486			−0.233743	−	0.972298i	$-0.575097\pi$
$337$	−8.58189			−0.467486			−0.233743	+	0.972298i	$0.575097\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−1.05405	−	1.05405i	−0.0570801	−	0.0570801i
$342$		0			0
$343$	−1.00000			−0.0539949
$344$		0			0
$345$		0			0
$346$		0			0
$347$	19.6238	−	19.6238i	1.05346	−	1.05346i	0.0549718	−	0.998488i	$-0.482493\pi$
$347$	19.6238	−	19.6238i	1.05346	−	1.05346i	0.998488	−	0.0549718i	$-0.0175069\pi$
$348$		0			0
$349$	7.67820	+	7.67820i	0.411004	+	0.411004i	0.882088	−	0.471084i	$-0.156137\pi$
$349$	7.67820	+	7.67820i	0.411004	+	0.411004i	−0.471084	+	0.882088i	$0.656137\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			25.9405i			1.38067i	0.723489	+	0.690336i	$0.242537\pi$
$353$			25.9405i			1.38067i	−0.723489	+	0.690336i	$0.757463\pi$
$354$		0			0
$355$	21.6830	−	21.6830i	1.15082	−	1.15082i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			3.08367i			0.162750i	0.996684	+	0.0813750i	$0.0259312\pi$
$359$			3.08367i			0.162750i	−0.996684	+	0.0813750i	$0.974069\pi$
$360$		0			0
$361$			14.1135i			0.742814i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	15.7472	−	15.7472i	0.824244	−	0.824244i
$366$		0			0
$367$			18.1386i			0.946825i	0.880841	+	0.473412i	$0.156978\pi$
$367$			18.1386i			0.946825i	−0.880841	+	0.473412i	$0.843022\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−7.63807	−	7.63807i	−0.396549	−	0.396549i
$372$		0			0
$373$	−1.70711	+	1.70711i	−0.0883908	+	0.0883908i	−0.749920	−	0.661529i	$-0.769908\pi$
$373$	−1.70711	+	1.70711i	−0.0883908	+	0.0883908i	0.661529	+	0.749920i	$0.269908\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−18.2736			−0.941140
$378$		0			0
$379$	24.2888	+	24.2888i	1.24763	+	1.24763i	0.956762	+	0.290870i	$0.0939448\pi$
$379$	24.2888	+	24.2888i	1.24763	+	1.24763i	0.290870	+	0.956762i	$0.406055\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−10.2655			−0.524544			−0.262272	−	0.964994i	$-0.584472\pi$
$383$	−10.2655			−0.524544			−0.262272	+	0.964994i	$0.584472\pi$
$384$		0			0
$385$	3.29158			0.167755
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−0.924351	−	0.924351i	−0.0468664	−	0.0468664i	0.683285	−	0.730152i	$-0.260551\pi$
$389$	−0.924351	−	0.924351i	−0.0468664	−	0.0468664i	−0.730152	+	0.683285i	$0.760551\pi$
$390$		0			0
$391$	2.92370			0.147858
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−16.3225	+	16.3225i	−0.821275	+	0.821275i
$396$		0			0
$397$	−18.2727	−	18.2727i	−0.917080	−	0.917080i	0.0797361	−	0.996816i	$-0.474592\pi$
$397$	−18.2727	−	18.2727i	−0.917080	−	0.917080i	−0.996816	+	0.0797361i	$0.974592\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			36.9686i			1.84612i	0.384651	+	0.923062i	$0.374322\pi$
$401$			36.9686i			1.84612i	−0.384651	+	0.923062i	$0.625678\pi$
$402$		0			0
$403$	−2.78774	+	2.78774i	−0.138867	+	0.138867i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	0.130534i		−	0.00647035i
$408$		0			0
$409$			4.66305i			0.230573i	0.993332	+	0.115286i	$0.0367786\pi$
$409$			4.66305i			0.230573i	−0.993332	+	0.115286i	$0.963221\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	4.15490	−	4.15490i	0.204449	−	0.204449i
$414$		0			0
$415$			3.74494i			0.183832i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	20.9392	+	20.9392i	1.02294	+	1.02294i	0.999731	+	0.0232142i	$0.00738996\pi$
$419$	20.9392	+	20.9392i	1.02294	+	1.02294i	0.0232142	+	0.999731i	$0.492610\pi$
$420$		0			0
$421$	−18.0980	+	18.0980i	−0.882042	+	0.882042i	−0.993742	−	0.111700i	$-0.964370\pi$
$421$	−18.0980	+	18.0980i	−0.882042	+	0.882042i	0.111700	+	0.993742i	$0.464370\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−1.36488			−0.0662064
$426$		0			0
$427$	7.78123	+	7.78123i	0.376560	+	0.376560i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−30.3329			−1.46108			−0.730541	−	0.682868i	$-0.760732\pi$
$431$	−30.3329			−1.46108			−0.730541	+	0.682868i	$0.760732\pi$
$432$		0			0
$433$	−25.8726			−1.24336			−0.621679	−	0.783272i	$-0.713549\pi$
$433$	−25.8726			−1.24336			−0.621679	+	0.783272i	$0.713549\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	1.64550	+	1.64550i	0.0787150	+	0.0787150i
$438$		0			0
$439$	18.1840			0.867874			0.433937	−	0.900943i	$-0.357124\pi$
$439$	18.1840			0.867874			0.433937	+	0.900943i	$0.357124\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	0.0172574	−	0.0172574i	0.000819922	−	0.000819922i	−0.706697	−	0.707517i	$-0.749815\pi$
$443$	0.0172574	−	0.0172574i	0.000819922	−	0.000819922i	0.707517	+	0.706697i	$0.249815\pi$
$444$		0			0
$445$	−11.6729	−	11.6729i	−0.553350	−	0.553350i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	8.83335i		−	0.416872i	−0.978036	−	0.208436i	$-0.933163\pi$
$449$		−	8.83335i		−	0.416872i	0.978036	−	0.208436i	$-0.0668372\pi$
$450$		0			0
$451$	6.27034	−	6.27034i	0.295259	−	0.295259i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	8.70553i		−	0.408121i
$456$		0			0
$457$			3.53330i			0.165281i	0.996579	+	0.0826404i	$0.0263353\pi$
$457$			3.53330i			0.165281i	−0.996579	+	0.0826404i	$0.973665\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	13.9587	−	13.9587i	0.650123	−	0.650123i	−0.302900	−	0.953022i	$-0.597955\pi$
$461$	13.9587	−	13.9587i	0.650123	−	0.650123i	0.953022	+	0.302900i	$0.0979547\pi$
$462$		0			0
$463$		−	10.6448i		−	0.494705i	−0.968926	−	0.247353i	$-0.920439\pi$
$463$		−	10.6448i		−	0.494705i	0.968926	−	0.247353i	$-0.0795606\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	12.4545	+	12.4545i	0.576325	+	0.576325i	0.933889	−	0.357564i	$-0.116393\pi$
$467$	12.4545	+	12.4545i	0.576325	+	0.576325i	−0.357564	+	0.933889i	$0.616393\pi$
$468$		0			0
$469$	1.98557	−	1.98557i	0.0916852	−	0.0916852i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−4.74444			−0.218149
$474$		0			0
$475$	−0.768175	−	0.768175i	−0.0352463	−	0.0352463i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	22.0204			1.00614			0.503068	−	0.864247i	$-0.332205\pi$
$479$	22.0204			1.00614			0.503068	+	0.864247i	$0.332205\pi$
$480$		0			0
$481$	−0.345236			−0.0157414
$482$		0			0
$483$		0			0
$484$		0			0
$485$	8.87091	+	8.87091i	0.402807	+	0.402807i
$486$		0			0
$487$	23.2822			1.05502			0.527509	−	0.849549i	$-0.323126\pi$
$487$	23.2822			1.05502			0.527509	+	0.849549i	$0.323126\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−0.129520	+	0.129520i	−0.00584515	+	0.00584515i	−0.710023	−	0.704178i	$-0.751316\pi$
$491$	−0.129520	+	0.129520i	−0.00584515	+	0.00584515i	0.704178	+	0.710023i	$0.251316\pi$
$492$		0			0
$493$	−9.66002	−	9.66002i	−0.435065	−	0.435065i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			13.0855i			0.586967i
$498$		0			0
$499$	10.7818	−	10.7818i	0.482661	−	0.482661i	−0.423319	−	0.905980i	$-0.639135\pi$
$499$	10.7818	−	10.7818i	0.482661	−	0.482661i	0.905980	+	0.423319i	$0.139135\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	6.04034i		−	0.269326i	−0.990891	−	0.134663i	$-0.957005\pi$
$503$		−	6.04034i		−	0.269326i	0.990891	−	0.134663i	$-0.0429951\pi$
$504$		0			0
$505$			17.7166i			0.788378i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−9.74993	+	9.74993i	−0.432158	+	0.432158i	−0.889362	−	0.457204i	$-0.848851\pi$
$509$	−9.74993	+	9.74993i	−0.432158	+	0.432158i	0.457204	+	0.889362i	$0.348851\pi$
$510$		0			0
$511$			9.50329i			0.420401i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−5.65632	−	5.65632i	−0.249247	−	0.249247i
$516$		0			0
$517$	1.46192	−	1.46192i	0.0642954	−	0.0642954i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−1.68047			−0.0736225			−0.0368113	−	0.999322i	$-0.511720\pi$
$521$	−1.68047			−0.0736225			−0.0368113	+	0.999322i	$0.511720\pi$
$522$		0			0
$523$	16.9606	+	16.9606i	0.741635	+	0.741635i	0.972892	−	0.231258i	$-0.0742841\pi$
$523$	16.9606	+	16.9606i	0.741635	+	0.741635i	−0.231258	+	0.972892i	$0.574284\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−2.94737			−0.128390
$528$		0			0
$529$	21.8918			0.951816
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−16.5837	−	16.5837i	−0.718320	−	0.718320i
$534$		0			0
$535$	−35.4617			−1.53314
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−0.993222	+	0.993222i	−0.0427811	+	0.0427811i
$540$		0			0
$541$	−5.82266	−	5.82266i	−0.250336	−	0.250336i	0.570773	−	0.821108i	$-0.306644\pi$
$541$	−5.82266	−	5.82266i	−0.250336	−	0.250336i	−0.821108	+	0.570773i	$0.806644\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	17.7750i		−	0.761398i
$546$		0			0
$547$	15.9661	−	15.9661i	0.682662	−	0.682662i	−0.277937	−	0.960599i	$-0.589651\pi$
$547$	15.9661	−	15.9661i	0.682662	−	0.682662i	0.960599	+	0.277937i	$0.0896507\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	10.8736i		−	0.463231i
$552$		0			0
$553$		−	9.85051i		−	0.418886i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	14.1695	−	14.1695i	0.600381	−	0.600381i	−0.340033	−	0.940414i	$-0.610438\pi$
$557$	14.1695	−	14.1695i	0.600381	−	0.600381i	0.940414	+	0.340033i	$0.110438\pi$
$558$		0			0
$559$			12.5480i			0.530725i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	5.96715	+	5.96715i	0.251485	+	0.251485i	0.821579	−	0.570094i	$-0.193093\pi$
$563$	5.96715	+	5.96715i	0.251485	+	0.251485i	−0.570094	+	0.821579i	$0.693093\pi$
$564$		0			0
$565$	−30.2346	+	30.2346i	−1.27198	+	1.27198i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−14.5770			−0.611099			−0.305550	−	0.952176i	$-0.598840\pi$
$569$	−14.5770			−0.611099			−0.305550	+	0.952176i	$0.598840\pi$
$570$		0			0
$571$	1.94424	+	1.94424i	0.0813641	+	0.0813641i	0.746618	−	0.665253i	$-0.231677\pi$
$571$	1.94424	+	1.94424i	0.0813641	+	0.0813641i	−0.665253	+	0.746618i	$0.731677\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	0.517354			0.0215752
$576$		0			0
$577$	−5.60353			−0.233278			−0.116639	−	0.993174i	$-0.537212\pi$
$577$	−5.60353			−0.233278			−0.116639	+	0.993174i	$0.537212\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−1.13002	−	1.13002i	−0.0468812	−	0.0468812i
$582$		0			0
$583$	−15.1726			−0.628385
$584$		0			0
$585$		0			0
$586$		0			0
$587$	15.8040	−	15.8040i	0.652300	−	0.652300i	−0.301246	−	0.953546i	$-0.597403\pi$
$587$	15.8040	−	15.8040i	0.652300	−	0.652300i	0.953546	+	0.301246i	$0.0974026\pi$
$588$		0			0
$589$	−1.65883	−	1.65883i	−0.0683507	−	0.0683507i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	8.08911i		−	0.332180i	−0.986111	−	0.166090i	$-0.946886\pi$
$593$		−	8.08911i		−	0.332180i	0.986111	−	0.166090i	$-0.0531142\pi$
$594$		0			0
$595$	4.60202	−	4.60202i	0.188664	−	0.188664i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	25.5885i		−	1.04552i	−0.852480	−	0.522759i	$-0.824903\pi$
$599$		−	25.5885i		−	1.04552i	0.852480	−	0.522759i	$-0.175097\pi$
$600$		0			0
$601$			26.8368i			1.09470i	0.836905	+	0.547348i	$0.184363\pi$
$601$			26.8368i			1.09470i	−0.836905	+	0.547348i	$0.815637\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−14.9580	+	14.9580i	−0.608128	+	0.608128i
$606$		0			0
$607$		−	17.9052i		−	0.726750i	−0.931643	−	0.363375i	$-0.881624\pi$
$607$		−	17.9052i		−	0.726750i	0.931643	−	0.363375i	$-0.118376\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−3.86648	−	3.86648i	−0.156421	−	0.156421i
$612$		0			0
$613$	13.9653	−	13.9653i	0.564053	−	0.564053i	−0.366403	−	0.930456i	$-0.619411\pi$
$613$	13.9653	−	13.9653i	0.564053	−	0.564053i	0.930456	+	0.366403i	$0.119411\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	6.19844			0.249540			0.124770	−	0.992186i	$-0.460181\pi$
$617$	6.19844			0.249540			0.124770	+	0.992186i	$0.460181\pi$
$618$		0			0
$619$	−26.7837	−	26.7837i	−1.07653	−	1.07653i	−0.996818	−	0.0797087i	$-0.974601\pi$
$619$	−26.7837	−	26.7837i	−1.07653	−	1.07653i	−0.0797087	−	0.996818i	$-0.525399\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	7.04453			0.282233
$624$		0			0
$625$	27.2157			1.08863
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−0.182502	−	0.182502i	−0.00727685	−	0.00727685i
$630$		0			0
$631$	30.5763			1.21723			0.608613	−	0.793468i	$-0.291726\pi$
$631$	30.5763			1.21723			0.608613	+	0.793468i	$0.291726\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−0.165380	+	0.165380i	−0.00656289	+	0.00656289i
$636$		0			0
$637$	2.62686	+	2.62686i	0.104080	+	0.104080i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	43.7713i		−	1.72886i	−0.502752	−	0.864431i	$-0.667679\pi$
$641$		−	43.7713i		−	1.72886i	0.502752	−	0.864431i	$-0.332321\pi$
$642$		0			0
$643$	24.5474	−	24.5474i	0.968053	−	0.968053i	−0.0314518	−	0.999505i	$-0.510013\pi$
$643$	24.5474	−	24.5474i	0.968053	−	0.968053i	0.999505	+	0.0314518i	$0.0100131\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			0.216780i			0.00852248i	0.999991	+	0.00426124i	$0.00135640\pi$
$647$			0.216780i			0.00852248i	−0.999991	+	0.00426124i	$0.998644\pi$
$648$		0			0
$649$		−	8.25347i		−	0.323977i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	12.1867	−	12.1867i	0.476904	−	0.476904i	−0.427236	−	0.904140i	$-0.640513\pi$
$653$	12.1867	−	12.1867i	0.476904	−	0.476904i	0.904140	+	0.427236i	$0.140513\pi$
$654$		0			0
$655$			42.5643i			1.66313i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	14.3745	+	14.3745i	0.559951	+	0.559951i	0.929293	−	0.369343i	$-0.120417\pi$
$659$	14.3745	+	14.3745i	0.559951	+	0.559951i	−0.369343	+	0.929293i	$0.620417\pi$
$660$		0			0
$661$	28.7815	−	28.7815i	1.11947	−	1.11947i	0.127651	−	0.991819i	$-0.459256\pi$
$661$	28.7815	−	28.7815i	1.11947	−	1.11947i	0.991819	−	0.127651i	$-0.0407436\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	5.18016			0.200878
$666$		0			0
$667$	3.66160	+	3.66160i	0.141778	+	0.141778i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	15.4570			0.596710
$672$		0			0
$673$	39.5913			1.52613			0.763066	−	0.646320i	$-0.223693\pi$
$673$	39.5913			1.52613			0.763066	+	0.646320i	$0.223693\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−13.7471	−	13.7471i	−0.528342	−	0.528342i	0.391735	−	0.920078i	$-0.371875\pi$
$677$	−13.7471	−	13.7471i	−0.528342	−	0.528342i	−0.920078	+	0.391735i	$0.871875\pi$
$678$		0			0
$679$	−5.35352			−0.205449
$680$		0			0
$681$		0			0
$682$		0			0
$683$	0.00325507	−	0.00325507i	0.000124552	−	0.000124552i	−0.707045	−	0.707169i	$-0.749972\pi$
$683$	0.00325507	−	0.00325507i	0.000124552	−	0.000124552i	0.707169	+	0.707045i	$0.249972\pi$
$684$		0			0
$685$	5.15197	+	5.15197i	0.196847	+	0.196847i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			40.1283i			1.52877i
$690$		0			0
$691$	17.5751	−	17.5751i	0.668590	−	0.668590i	−0.288800	−	0.957390i	$-0.593256\pi$
$691$	17.5751	−	17.5751i	0.668590	−	0.668590i	0.957390	+	0.288800i	$0.0932562\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	35.0605i		−	1.32992i
$696$		0			0
$697$		−	17.5333i		−	0.664122i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	29.3683	−	29.3683i	1.10923	−	1.10923i	0.115975	−	0.993252i	$-0.463001\pi$
$701$	29.3683	−	29.3683i	1.10923	−	1.10923i	0.993252	−	0.115975i	$-0.0369991\pi$
$702$		0			0
$703$		−	0.205430i		−	0.00774794i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−5.34591	−	5.34591i	−0.201054	−	0.201054i
$708$		0			0
$709$	−21.5493	+	21.5493i	−0.809300	+	0.809300i	−0.984528	−	0.175228i	$-0.943934\pi$
$709$	−21.5493	+	21.5493i	−0.809300	+	0.809300i	0.175228	+	0.984528i	$0.443934\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	1.11719			0.0418393
$714$		0			0
$715$	−8.64652	−	8.64652i	−0.323362	−	0.323362i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	23.1027			0.861585			0.430793	−	0.902451i	$-0.358234\pi$
$719$	23.1027			0.861585			0.430793	+	0.902451i	$0.358234\pi$
$720$		0			0
$721$	3.41355			0.127127
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−1.70936	−	1.70936i	−0.0634840	−	0.0634840i
$726$		0			0
$727$	10.8397			0.402024			0.201012	−	0.979589i	$-0.435577\pi$
$727$	10.8397			0.402024			0.201012	+	0.979589i	$0.435577\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−6.63328	+	6.63328i	−0.245341	+	0.245341i
$732$		0			0
$733$	−5.15354	−	5.15354i	−0.190350	−	0.190350i	0.605497	−	0.795848i	$-0.292974\pi$
$733$	−5.15354	−	5.15354i	−0.190350	−	0.190350i	−0.795848	+	0.605497i	$0.792974\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	3.94423i		−	0.145287i
$738$		0			0
$739$	−12.6864	+	12.6864i	−0.466678	+	0.466678i	−0.900837	−	0.434158i	$-0.857046\pi$
$739$	−12.6864	+	12.6864i	−0.466678	+	0.466678i	0.434158	+	0.900837i	$0.357046\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	24.3109i		−	0.891879i	−0.895063	−	0.445940i	$-0.852870\pi$
$743$		−	24.3109i		−	0.891879i	0.895063	−	0.445940i	$-0.147130\pi$
$744$		0			0
$745$			20.9524i			0.767635i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	10.7004	−	10.7004i	0.390985	−	0.390985i
$750$		0			0
$751$		−	47.8161i		−	1.74483i	−0.488762	−	0.872417i	$-0.662551\pi$
$751$		−	47.8161i		−	1.74483i	0.488762	−	0.872417i	$-0.337449\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−38.0109	−	38.0109i	−1.38336	−	1.38336i
$756$		0			0
$757$	20.6181	−	20.6181i	0.749377	−	0.749377i	−0.224986	−	0.974362i	$-0.572234\pi$
$757$	20.6181	−	20.6181i	0.749377	−	0.749377i	0.974362	+	0.224986i	$0.0722335\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−18.6635			−0.676553			−0.338276	−	0.941047i	$-0.609844\pi$
$761$	−18.6635			−0.676553			−0.338276	+	0.941047i	$0.609844\pi$
$762$		0			0
$763$	5.36354	+	5.36354i	0.194173	+	0.194173i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−21.8287			−0.788187
$768$		0			0
$769$	−13.7015			−0.494090			−0.247045	−	0.969004i	$-0.579459\pi$
$769$	−13.7015			−0.494090			−0.247045	+	0.969004i	$0.579459\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	21.1383	+	21.1383i	0.760294	+	0.760294i	0.976375	−	0.216082i	$-0.0693278\pi$
$773$	21.1383	+	21.1383i	0.760294	+	0.760294i	−0.216082	+	0.976375i	$0.569328\pi$
$774$		0			0
$775$	−0.521543			−0.0187344
$776$		0			0
$777$		0			0
$778$		0			0
$779$	9.86801	−	9.86801i	0.353558	−	0.353558i
$780$		0			0
$781$	12.9969	+	12.9969i	0.465064	+	0.465064i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	42.1774i		−	1.50538i
$786$		0			0
$787$	−25.7503	+	25.7503i	−0.917899	+	0.917899i	−0.996876	−	0.0789778i	$-0.974834\pi$
$787$	−25.7503	+	25.7503i	−0.917899	+	0.917899i	0.0789778	+	0.996876i	$0.474834\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	18.2463i		−	0.648765i
$792$		0			0
$793$		−	40.8804i		−	1.45171i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	13.2613	−	13.2613i	0.469739	−	0.469739i	−0.432091	−	0.901830i	$-0.642224\pi$
$797$	13.2613	−	13.2613i	0.469739	−	0.469739i	0.901830	+	0.432091i	$0.142224\pi$
$798$		0			0
$799$		−	4.08788i		−	0.144619i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	9.43887	+	9.43887i	0.333091	+	0.333091i
$804$		0			0
$805$	−1.74438	+	1.74438i	−0.0614814	+	0.0614814i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	39.0568			1.37316			0.686582	−	0.727053i	$-0.259110\pi$
$809$	39.0568			1.37316			0.686582	+	0.727053i	$0.259110\pi$
$810$		0			0
$811$	−11.8627	−	11.8627i	−0.416554	−	0.416554i	0.467460	−	0.884014i	$-0.345169\pi$
$811$	−11.8627	−	11.8627i	−0.416554	−	0.416554i	−0.884014	+	0.467460i	$0.845169\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−18.7722			−0.657563
$816$		0			0
$817$	−7.46661			−0.261224
$818$		0			0
$819$		0			0
$820$		0			0
$821$	7.89485	+	7.89485i	0.275532	+	0.275532i	0.831323	−	0.555790i	$-0.187584\pi$
$821$	7.89485	+	7.89485i	0.275532	+	0.275532i	−0.555790	+	0.831323i	$0.687584\pi$
$822$		0			0
$823$	23.6159			0.823199			0.411600	−	0.911365i	$-0.364970\pi$
$823$	23.6159			0.823199			0.411600	+	0.911365i	$0.364970\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	2.44372	−	2.44372i	0.0849766	−	0.0849766i	−0.663341	−	0.748317i	$-0.730862\pi$
$827$	2.44372	−	2.44372i	0.0849766	−	0.0849766i	0.748317	+	0.663341i	$0.230862\pi$
$828$		0			0
$829$	29.3029	+	29.3029i	1.01773	+	1.01773i	0.999840	+	0.0178910i	$0.00569520\pi$
$829$	29.3029	+	29.3029i	1.01773	+	1.01773i	0.0178910	+	0.999840i	$0.494305\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			2.77728i			0.0962271i
$834$		0			0
$835$	−33.7114	+	33.7114i	−1.16663	+	1.16663i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	25.2593i		−	0.872047i	−0.899935	−	0.436024i	$-0.856386\pi$
$839$		−	25.2593i		−	0.872047i	0.899935	−	0.436024i	$-0.143614\pi$
$840$		0			0
$841$			4.80385i			0.165650i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−1.32691	+	1.32691i	−0.0456472	+	0.0456472i
$846$		0			0
$847$		−	9.02702i		−	0.310172i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	0.0691771	+	0.0691771i	0.00237136	+	0.00237136i
$852$		0			0
$853$	25.7213	−	25.7213i	0.880680	−	0.880680i	−0.112924	−	0.993604i	$-0.536022\pi$
$853$	25.7213	−	25.7213i	0.880680	−	0.880680i	0.993604	+	0.112924i	$0.0360215\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−46.7116			−1.59564			−0.797819	−	0.602897i	$-0.794013\pi$
$857$	−46.7116			−1.59564			−0.797819	+	0.602897i	$0.794013\pi$
$858$		0			0
$859$	2.01594	+	2.01594i	0.0687829	+	0.0687829i	0.740661	−	0.671878i	$-0.234512\pi$
$859$	2.01594	+	2.01594i	0.0687829	+	0.0687829i	−0.671878	+	0.740661i	$0.734512\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	7.21180			0.245493			0.122746	−	0.992438i	$-0.460830\pi$
$863$	7.21180			0.245493			0.122746	+	0.992438i	$0.460830\pi$
$864$		0			0
$865$	13.0826			0.444823
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−9.78374	−	9.78374i	−0.331891	−	0.331891i
$870$		0			0
$871$	−10.4316			−0.353462
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−7.47078	+	7.47078i	−0.252558	+	0.252558i
$876$		0			0
$877$	−8.12742	−	8.12742i	−0.274443	−	0.274443i	0.556443	−	0.830886i	$-0.312166\pi$
$877$	−8.12742	−	8.12742i	−0.274443	−	0.274443i	−0.830886	+	0.556443i	$0.812166\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	41.1175i		−	1.38529i	−0.721281	−	0.692643i	$-0.756446\pi$
$881$		−	41.1175i		−	1.38529i	0.721281	−	0.692643i	$-0.243554\pi$
$882$		0			0
$883$	−9.38694	+	9.38694i	−0.315896	+	0.315896i	−0.847188	−	0.531293i	$-0.821706\pi$
$883$	−9.38694	+	9.38694i	−0.315896	+	0.315896i	0.531293	+	0.847188i	$0.321706\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	17.8506i		−	0.599366i	−0.954039	−	0.299683i	$-0.903119\pi$
$887$		−	17.8506i		−	0.599366i	0.954039	−	0.299683i	$-0.0968809\pi$
$888$		0			0
$889$		−	0.0998053i		−	0.00334736i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	2.30072	−	2.30072i	0.0769906	−	0.0769906i
$894$		0			0
$895$			41.6777i			1.39313i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−3.69125	−	3.69125i	−0.123110	−	0.123110i
$900$		0			0
$901$	−21.2131	+	21.2131i	−0.706710	+	0.706710i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	53.0698			1.76410
$906$		0			0
$907$	16.6866	+	16.6866i	0.554070	+	0.554070i	0.927613	−	0.373543i	$-0.121857\pi$
$907$	16.6866	+	16.6866i	0.554070	+	0.554070i	−0.373543	+	0.927613i	$0.621857\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	49.1236			1.62754			0.813769	−	0.581189i	$-0.197412\pi$
$911$	49.1236			1.62754			0.813769	+	0.581189i	$0.197412\pi$
$912$		0			0
$913$	−2.24472			−0.0742895
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−12.8436	−	12.8436i	−0.424134	−	0.424134i
$918$		0			0
$919$	−46.5272			−1.53479			−0.767395	−	0.641174i	$-0.778448\pi$
$919$	−46.5272			−1.53479			−0.767395	+	0.641174i	$0.778448\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	34.3739	−	34.3739i	1.13143	−	1.13143i
$924$		0			0
$925$	−0.0322941	−	0.0322941i	−0.00106182	−	0.00106182i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			4.44879i			0.145960i	0.997333	+	0.0729800i	$0.0232509\pi$
$929$			4.44879i			0.145960i	−0.997333	+	0.0729800i	$0.976749\pi$
$930$		0			0
$931$	−1.56309	+	1.56309i	−0.0512283	+	0.0512283i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	9.14165i		−	0.298964i
$936$		0			0
$937$		−	45.1282i		−	1.47428i	−0.675742	−	0.737138i	$-0.736177\pi$
$937$		−	45.1282i		−	1.47428i	0.675742	−	0.737138i	$-0.263823\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−7.54964	+	7.54964i	−0.246111	+	0.246111i	−0.819373	−	0.573261i	$-0.805678\pi$
$941$	−7.54964	+	7.54964i	−0.246111	+	0.246111i	0.573261	+	0.819373i	$0.305678\pi$
$942$		0			0
$943$			6.64596i			0.216422i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	16.1642	+	16.1642i	0.525265	+	0.525265i	0.919157	−	0.393892i	$-0.128872\pi$
$947$	16.1642	+	16.1642i	0.525265	+	0.525265i	−0.393892	+	0.919157i	$0.628872\pi$
$948$		0			0
$949$	24.9638	−	24.9638i	0.810359	−	0.810359i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−19.4691			−0.630667			−0.315334	−	0.948981i	$-0.602116\pi$
$953$	−19.4691			−0.630667			−0.315334	+	0.948981i	$0.602116\pi$
$954$		0			0
$955$	−19.6086	−	19.6086i	−0.634519	−	0.634519i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−3.10918			−0.100401
$960$		0			0
$961$	29.8738			0.963670
$962$		0			0
$963$		0			0
$964$		0			0
$965$	34.9667	+	34.9667i	1.12562	+	1.12562i
$966$		0			0
$967$	47.4224			1.52500			0.762501	−	0.646987i	$-0.223971\pi$
$967$	47.4224			1.52500			0.762501	+	0.646987i	$0.223971\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−26.3134	+	26.3134i	−0.844437	+	0.844437i	−0.989432	−	0.144995i	$-0.953683\pi$
$971$	−26.3134	+	26.3134i	−0.844437	+	0.844437i	0.144995	+	0.989432i	$0.453683\pi$
$972$		0			0
$973$	10.5794	+	10.5794i	0.339159	+	0.339159i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	15.9988i		−	0.511848i	−0.966697	−	0.255924i	$-0.917620\pi$
$977$		−	15.9988i		−	0.511848i	0.966697	−	0.255924i	$-0.0823797\pi$
$978$		0			0
$979$	6.99678	−	6.99678i	0.223618	−	0.223618i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			26.1634i			0.834483i	0.908796	+	0.417242i	$0.137003\pi$
$983$			26.1634i			0.834483i	−0.908796	+	0.417242i	$0.862997\pi$
$984$		0			0
$985$		−	19.3311i		−	0.615939i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	2.51433	−	2.51433i	0.0799509	−	0.0799509i
$990$		0			0
$991$		−	18.2140i		−	0.578587i	−0.957240	−	0.289294i	$-0.906580\pi$
$991$		−	18.2140i		−	0.578587i	0.957240	−	0.289294i	$-0.0934204\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−38.5863	−	38.5863i	−1.22327	−	1.22327i
$996$		0			0
$997$	−10.0322	+	10.0322i	−0.317722	+	0.317722i	−0.847892	−	0.530170i	$-0.822128\pi$
$997$	−10.0322	+	10.0322i	−0.317722	+	0.317722i	0.530170	+	0.847892i	$0.322128\pi$
$998$		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.1583.16		40
3.2	odd	2	inner	4032.2.v.e.1583.5		40
4.3	odd	2		1008.2.v.e.323.15	yes	40
12.11	even	2		1008.2.v.e.323.6	✓	40
16.5	even	4		1008.2.v.e.827.6	yes	40
16.11	odd	4	inner	4032.2.v.e.3599.5		40
48.5	odd	4		1008.2.v.e.827.15	yes	40
48.11	even	4	inner	4032.2.v.e.3599.16		40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1008.2.v.e.323.6	✓	40	12.11	even	2
1008.2.v.e.323.15	yes	40	4.3	odd	2
1008.2.v.e.827.6	yes	40	16.5	even	4
1008.2.v.e.827.15	yes	40	48.5	odd	4
4032.2.v.e.1583.5		40	3.2	odd	2	inner
4032.2.v.e.1583.16		40	1.1	even	1	trivial
4032.2.v.e.3599.5		40	16.11	odd	4	inner
4032.2.v.e.3599.16		40	48.11	even	4	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)

\(f(q)\)	\(=\)	\(q+(1.65702 + 1.65702i) q^{5} -1.00000 q^{7} +O(q^{10})\)	\(q+(1.65702 + 1.65702i) q^{5} -1.00000 q^{7} +(-0.993222 + 0.993222i) q^{11} +(2.62686 + 2.62686i) q^{13} +2.77728i q^{17} +(-1.56309 + 1.56309i) q^{19} -1.05272i q^{23} +0.491445i q^{25} +(-3.47823 + 3.47823i) q^{29} +1.06124i q^{31} +(-1.65702 - 1.65702i) q^{35} +(-0.0657126 + 0.0657126i) q^{37} -6.31313 q^{41} +(2.38841 + 2.38841i) q^{43} -1.47190 q^{47} +1.00000 q^{49} +(7.63807 + 7.63807i) q^{53} -3.29158 q^{55} +(-4.15490 + 4.15490i) q^{59} +(-7.78123 - 7.78123i) q^{61} +8.70553i q^{65} +(-1.98557 + 1.98557i) q^{67} -13.0855i q^{71} -9.50329i q^{73} +(0.993222 - 0.993222i) q^{77} +9.85051i q^{79} +(1.13002 + 1.13002i) q^{83} +(-4.60202 + 4.60202i) q^{85} -7.04453 q^{89} +(-2.62686 - 2.62686i) q^{91} -5.18016 q^{95} +5.35352 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 40 q^{7}+O(q^{10}) \) 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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