LMFDB - Embedded newform 4000.1.cq.a.1377.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4000,1,Mod(33,4000)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4000, base_ring=CyclotomicField(100))

chi = DirichletCharacter(H, H._module([0, 0, 83]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("4000.33");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 4000 = 2^{5} \cdot 5^{3} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	4000.cq (of order $100$, degree $40$, 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:	$1.99626005053$
Analytic rank:	$0$
Dimension:	$40$
Coefficient field:	$\Q(\zeta_{100})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{40} - x^{30} + x^{20} - x^{10} + 1 $ x^40 - x^30 + x^20 - x^10 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{100}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{100} - \cdots)$

Embedding invariants

Embedding label			1377.1
Root			$-0.904827 + 0.425779i$ of defining polynomial
Character	$\chi$	$=$	4000.1377
Dual form			4000.1.cq.a.1313.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.844328 - 0.535827i) q^{5} +(-0.904827 - 0.425779i) q^{9} +O(q^{10})$	$q+(-0.844328 - 0.535827i) q^{5} +(-0.904827 - 0.425779i) q^{9} +(-0.344863 - 0.957895i) q^{13} +(-1.24443 + 1.41153i) q^{17} +(0.425779 + 0.904827i) q^{25} +(-0.0922765 + 0.233064i) q^{29} +(1.01385 + 1.30704i) q^{37} +(0.374023 + 1.96070i) q^{41} +(0.535827 + 0.844328i) q^{45} +(-0.587785 + 0.809017i) q^{49} +(-0.404329 + 0.683684i) q^{53} +(0.316423 - 1.65875i) q^{61} +(-0.222088 + 0.993564i) q^{65} +(-1.91206 + 0.555506i) q^{73} +(0.637424 + 0.770513i) q^{81} +(1.80704 - 0.524995i) q^{85} +(0.659566 - 1.19975i) q^{89} +(-0.221601 - 0.512091i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q+O(q^{10}) $ 40 * q	$ 40 q + 10 q^{85} + 10 q^{89}+O(q^{100}) $ 40 * q + 10 * q^85 + 10 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4000\mathbb{Z}\right)^\times$.

$n$	$1377$	$2501$	$2751$
$\chi(n)$	$e\left(\frac{1}{100}\right)$	$1$	$1$

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		0.218143	−	0.975917i	$-0.430000\pi$
$3$		0			0		−0.218143	+	0.975917i	$0.570000\pi$
$4$		0			0
$5$	−0.844328	−	0.535827i	−0.844328	−	0.535827i
$5$	−0.844328	−	0.535827i	−0.844328	−	0.535827i
$6$		0			0
$7$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$7$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$8$		0			0
$9$	−0.904827	−	0.425779i	−0.904827	−	0.425779i
$10$		0			0
$11$		0			0		−0.684547	−	0.728969i	$-0.740000\pi$
$11$		0			0		0.684547	+	0.728969i	$0.260000\pi$
$12$		0			0
$13$	−0.344863	−	0.957895i	−0.344863	−	0.957895i	−0.982287	−	0.187381i	$-0.940000\pi$
$13$	−0.344863	−	0.957895i	−0.344863	−	0.957895i	0.637424	−	0.770513i	$-0.280000\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.24443	+	1.41153i	−1.24443	+	1.41153i	−0.368125	+	0.929776i	$0.620000\pi$
$17$	−1.24443	+	1.41153i	−1.24443	+	1.41153i	−0.876307	+	0.481754i	$0.840000\pi$
$18$		0			0
$19$		0			0		0.535827	−	0.844328i	$-0.320000\pi$
$19$		0			0		−0.535827	+	0.844328i	$0.680000\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		0.827081	−	0.562083i	$-0.190000\pi$
$23$		0			0		−0.827081	+	0.562083i	$0.810000\pi$
$24$		0			0
$25$	0.425779	+	0.904827i	0.425779	+	0.904827i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−0.0922765	+	0.233064i	−0.0922765	+	0.233064i	−0.968583	−	0.248690i	$-0.920000\pi$
$29$	−0.0922765	+	0.233064i	−0.0922765	+	0.233064i	0.876307	+	0.481754i	$0.160000\pi$
$30$		0			0
$31$		0			0		0.998027	−	0.0627905i	$-0.0200000\pi$
$31$		0			0		−0.998027	+	0.0627905i	$0.980000\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	1.01385	+	1.30704i	1.01385	+	1.30704i	0.951057	+	0.309017i	$0.100000\pi$
$37$	1.01385	+	1.30704i	1.01385	+	1.30704i	0.0627905	+	0.998027i	$0.480000\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	0.374023	+	1.96070i	0.374023	+	1.96070i	0.248690	+	0.968583i	$0.420000\pi$
$41$	0.374023	+	1.96070i	0.374023	+	1.96070i	0.125333	+	0.992115i	$0.460000\pi$
$42$		0			0
$43$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$43$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$44$		0			0
$45$	0.535827	+	0.844328i	0.535827	+	0.844328i
$46$		0			0
$47$		0			0		−0.0941083	−	0.995562i	$-0.530000\pi$
$47$		0			0		0.0941083	+	0.995562i	$0.470000\pi$
$48$		0			0
$49$	−0.587785	+	0.809017i	−0.587785	+	0.809017i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−0.404329	+	0.683684i	−0.404329	+	0.683684i	−0.992115	−	0.125333i	$-0.960000\pi$
$53$	−0.404329	+	0.683684i	−0.404329	+	0.683684i	0.587785	+	0.809017i	$0.300000\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		0.876307	−	0.481754i	$-0.160000\pi$
$59$		0			0		−0.876307	+	0.481754i	$0.840000\pi$
$60$		0			0
$61$	0.316423	−	1.65875i	0.316423	−	1.65875i	−0.368125	−	0.929776i	$-0.620000\pi$
$61$	0.316423	−	1.65875i	0.316423	−	1.65875i	0.684547	−	0.728969i	$-0.260000\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−0.222088	+	0.993564i	−0.222088	+	0.993564i
$66$		0			0
$67$		0			0		0.397148	−	0.917755i	$-0.370000\pi$
$67$		0			0		−0.397148	+	0.917755i	$0.630000\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.770513	−	0.637424i	$-0.780000\pi$
$71$		0			0		0.770513	+	0.637424i	$0.220000\pi$
$72$		0			0
$73$	−1.91206	+	0.555506i	−1.91206	+	0.555506i	−0.929776	+	0.368125i	$0.880000\pi$
$73$	−1.91206	+	0.555506i	−1.91206	+	0.555506i	−0.982287	+	0.187381i	$0.940000\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		−0.535827	−	0.844328i	$-0.680000\pi$
$79$		0			0		0.535827	+	0.844328i	$0.320000\pi$
$80$		0			0
$81$	0.637424	+	0.770513i	0.637424	+	0.770513i
$82$		0			0
$83$		0			0		0.975917	−	0.218143i	$-0.0700000\pi$
$83$		0			0		−0.975917	+	0.218143i	$0.930000\pi$
$84$		0			0
$85$	1.80704	−	0.524995i	1.80704	−	0.524995i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	0.659566	−	1.19975i	0.659566	−	1.19975i	−0.309017	−	0.951057i	$-0.600000\pi$
$89$	0.659566	−	1.19975i	0.659566	−	1.19975i	0.968583	−	0.248690i	$-0.0800000\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−0.221601	−	0.512091i	−0.221601	−	0.512091i	0.770513	−	0.637424i	$-0.220000\pi$
$97$	−0.221601	−	0.512091i	−0.221601	−	0.512091i	−0.992115	+	0.125333i	$0.960000\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−0.541587	+	1.66683i	−0.541587	+	1.66683i	0.187381	+	0.982287i	$0.440000\pi$
$101$	−0.541587	+	1.66683i	−0.541587	+	1.66683i	−0.728969	+	0.684547i	$0.760000\pi$
$102$		0			0
$103$		0			0		0.750111	−	0.661312i	$-0.230000\pi$
$103$		0			0		−0.750111	+	0.661312i	$0.770000\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$107$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$108$		0			0
$109$	0.159781	−	1.26480i	0.159781	−	1.26480i	−0.684547	−	0.728969i	$-0.740000\pi$
$109$	0.159781	−	1.26480i	0.159781	−	1.26480i	0.844328	−	0.535827i	$-0.180000\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−0.557707	+	1.91964i	−0.557707	+	1.91964i	−0.248690	+	0.968583i	$0.580000\pi$
$113$	−0.557707	+	1.91964i	−0.557707	+	1.91964i	−0.309017	+	0.951057i	$0.600000\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−0.0958101	+	1.01356i	−0.0958101	+	1.01356i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.0627905	+	0.998027i	−0.0627905	+	0.998027i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	0.125333	−	0.992115i	0.125333	−	0.992115i
$126$		0			0
$127$		0			0		0.0314108	−	0.999507i	$-0.490000\pi$
$127$		0			0		−0.0314108	+	0.999507i	$0.510000\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		0.248690	−	0.968583i	$-0.420000\pi$
$131$		0			0		−0.248690	+	0.968583i	$0.580000\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−0.418963	−	0.121720i	−0.418963	−	0.121720i	0.0627905	−	0.998027i	$-0.480000\pi$
$137$	−0.418963	−	0.121720i	−0.418963	−	0.121720i	−0.481754	+	0.876307i	$0.660000\pi$
$138$		0			0
$139$		0			0		−0.425779	−	0.904827i	$-0.640000\pi$
$139$		0			0		0.425779	+	0.904827i	$0.360000\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	0.202793	−	0.147338i	0.202793	−	0.147338i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1.76854	−	0.574633i	−1.76854	−	0.574633i	−0.770513	−	0.637424i	$-0.780000\pi$
$149$	−1.76854	−	0.574633i	−1.76854	−	0.574633i	−0.998027	+	0.0627905i	$0.980000\pi$
$150$		0			0
$151$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$151$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$152$		0			0
$153$	1.72700	−	0.747338i	1.72700	−	0.747338i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	0.0620481	+	0.00982745i	0.0620481	+	0.00982745i	0.187381	−	0.982287i	$-0.440000\pi$
$157$	0.0620481	+	0.00982745i	0.0620481	+	0.00982745i	−0.125333	+	0.992115i	$0.540000\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$		0			0		0.562083	−	0.827081i	$-0.310000\pi$
$163$		0			0		−0.562083	+	0.827081i	$0.690000\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.218143	−	0.975917i	$-0.570000\pi$
$167$		0			0		0.218143	+	0.975917i	$0.430000\pi$
$168$		0			0
$169$	−0.0281181	+	0.0232613i	−0.0281181	+	0.0232613i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1.80704	−	0.650576i	−1.80704	−	0.650576i	−0.998027	−	0.0627905i	$-0.980000\pi$
$173$	−1.80704	−	0.650576i	−1.80704	−	0.650576i	−0.809017	−	0.587785i	$-0.800000\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.637424	−	0.770513i	$-0.280000\pi$
$179$		0			0		−0.637424	+	0.770513i	$0.720000\pi$
$180$		0			0
$181$	−1.35556	+	0.536702i	−1.35556	+	0.536702i	−0.929776	−	0.368125i	$-0.880000\pi$
$181$	−1.35556	+	0.536702i	−1.35556	+	0.536702i	−0.425779	+	0.904827i	$0.640000\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−0.155670	−	1.64682i	−0.155670	−	1.64682i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.481754	−	0.876307i	$-0.660000\pi$
$191$		0			0		0.481754	+	0.876307i	$0.340000\pi$
$192$		0			0
$193$	−1.38015	+	1.38015i	−1.38015	+	1.38015i	−0.535827	+	0.844328i	$0.680000\pi$
$193$	−1.38015	+	1.38015i	−1.38015	+	1.38015i	−0.844328	+	0.535827i	$0.820000\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	1.29130	+	0.763675i	1.29130	+	0.763675i	0.982287	−	0.187381i	$-0.0600000\pi$
$197$	1.29130	+	0.763675i	1.29130	+	0.763675i	0.309017	+	0.951057i	$0.400000\pi$
$198$		0			0
$199$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$199$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	0.734796	−	1.85588i	0.734796	−	1.85588i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		−0.125333	−	0.992115i	$-0.540000\pi$
$211$		0			0		0.125333	+	0.992115i	$0.460000\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	1.78126	+	0.705249i	1.78126	+	0.705249i
$222$		0			0
$223$		0			0		−0.790155	−	0.612907i	$-0.790000\pi$
$223$		0			0		0.790155	+	0.612907i	$0.210000\pi$
$224$		0			0
$225$		−	1.00000i		−	1.00000i
$226$		0			0
$227$		0			0		−0.562083	−	0.827081i	$-0.690000\pi$
$227$		0			0		0.562083	+	0.827081i	$0.310000\pi$
$228$		0			0
$229$	0.183098	+	0.713118i	0.183098	+	0.713118i	0.992115	+	0.125333i	$0.0400000\pi$
$229$	0.183098	+	0.713118i	0.183098	+	0.713118i	−0.809017	+	0.587785i	$0.800000\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	0.992115	+	0.874667i	0.992115	+	0.874667i	0.992115	−	0.125333i	$-0.0400000\pi$
$233$	0.992115	+	0.874667i	0.992115	+	0.874667i			1.00000i	$0.5\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		0.728969	−	0.684547i	$-0.240000\pi$
$239$		0			0		−0.728969	+	0.684547i	$0.760000\pi$
$240$		0			0
$241$	0.836475	−	1.77760i	0.836475	−	1.77760i	0.248690	−	0.968583i	$-0.420000\pi$
$241$	0.836475	−	1.77760i	0.836475	−	1.77760i	0.587785	−	0.809017i	$-0.300000\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	0.929776	−	0.368125i	0.929776	−	0.368125i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−0.167702	+	0.0854486i	−0.167702	+	0.0854486i	−0.535827	−	0.844328i	$-0.680000\pi$
$257$	−0.167702	+	0.0854486i	−0.167702	+	0.0854486i	0.368125	+	0.929776i	$0.380000\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	0.182728	−	0.171593i	0.182728	−	0.171593i
$262$		0			0
$263$		0			0		0.940881	−	0.338738i	$-0.110000\pi$
$263$		0			0		−0.940881	+	0.338738i	$0.890000\pi$
$264$		0			0
$265$	0.707723	−	0.360603i	0.707723	−	0.360603i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	0.904827	+	0.574221i	0.904827	+	0.574221i	0.904827	−	0.425779i	$-0.140000\pi$
$269$	0.904827	+	0.574221i	0.904827	+	0.574221i			1.00000i	$0.5\pi$
$270$		0			0
$271$		0			0		−0.248690	−	0.968583i	$-0.580000\pi$
$271$		0			0		0.248690	+	0.968583i	$0.420000\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−0.968583	−	0.751310i	−0.968583	−	0.751310i		−	1.00000i	$-0.5\pi$
$277$	−0.968583	−	0.751310i	−0.968583	−	0.751310i	−0.968583	+	0.248690i	$0.920000\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−0.0462295	−	0.734796i	−0.0462295	−	0.734796i	−0.951057	−	0.309017i	$-0.900000\pi$
$281$	−0.0462295	−	0.734796i	−0.0462295	−	0.734796i	0.904827	−	0.425779i	$-0.140000\pi$
$282$		0			0
$283$		0			0		−0.509041	−	0.860742i	$-0.670000\pi$
$283$		0			0		0.509041	+	0.860742i	$0.330000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−0.318475	−	2.52099i	−0.318475	−	2.52099i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−1.56085	+	0.247215i	−1.56085	+	0.247215i	−0.876307	−	0.481754i	$-0.840000\pi$
$293$	−1.56085	+	0.247215i	−1.56085	+	0.247215i	−0.684547	+	0.728969i	$0.740000\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−1.15596	+	1.23098i	−1.15596	+	1.23098i
$306$		0			0
$307$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$307$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.982287	−	0.187381i	$-0.940000\pi$
$311$		0			0		0.982287	+	0.187381i	$0.0600000\pi$
$312$		0			0
$313$	0.907533	−	0.0285204i	0.907533	−	0.0285204i	0.425779	−	0.904827i	$-0.360000\pi$
$313$	0.907533	−	0.0285204i	0.907533	−	0.0285204i	0.481754	+	0.876307i	$0.340000\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−0.287137	−	0.124255i	−0.287137	−	0.124255i	0.248690	−	0.968583i	$-0.420000\pi$
$317$	−0.287137	−	0.124255i	−0.287137	−	0.124255i	−0.535827	+	0.844328i	$0.680000\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	0.719893	−	0.719893i	0.719893	−	0.719893i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		0.770513	−	0.637424i	$-0.220000\pi$
$331$		0			0		−0.770513	+	0.637424i	$0.780000\pi$
$332$		0			0
$333$	−0.360844	−	1.61432i	−0.360844	−	1.61432i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−1.11033	+	1.63380i	−1.11033	+	1.63380i	−0.425779	+	0.904827i	$0.640000\pi$
$337$	−1.11033	+	1.63380i	−1.11033	+	1.63380i	−0.684547	+	0.728969i	$0.740000\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.917755	−	0.397148i	$-0.130000\pi$
$347$		0			0		−0.917755	+	0.397148i	$0.870000\pi$
$348$		0			0
$349$	1.86842	−	0.607087i	1.86842	−	0.607087i	0.876307	−	0.481754i	$-0.160000\pi$
$349$	1.86842	−	0.607087i	1.86842	−	0.607087i	0.992115	−	0.125333i	$-0.0400000\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−1.17847	−	1.33671i	−1.17847	−	1.33671i	−0.929776	−	0.368125i	$-0.880000\pi$
$353$	−1.17847	−	1.33671i	−1.17847	−	1.33671i	−0.248690	−	0.968583i	$-0.580000\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.992115	−	0.125333i	$-0.960000\pi$
$359$		0			0		0.992115	+	0.125333i	$0.0400000\pi$
$360$		0			0
$361$	−0.425779	−	0.904827i	−0.425779	−	0.904827i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	1.91206	+	0.555506i	1.91206	+	0.555506i
$366$		0			0
$367$		0			0		−0.995562	−	0.0941083i	$-0.970000\pi$
$367$		0			0		0.995562	+	0.0941083i	$0.0300000\pi$
$368$		0			0
$369$	0.496398	−	1.93334i	0.496398	−	1.93334i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	0.0559744	−	1.78113i	0.0559744	−	1.78113i	−0.425779	−	0.904827i	$-0.640000\pi$
$373$	0.0559744	−	1.78113i	0.0559744	−	1.78113i	0.481754	−	0.876307i	$-0.340000\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	0.255073	+	0.00801600i	0.255073	+	0.00801600i
$378$		0			0
$379$		0			0		0.0627905	−	0.998027i	$-0.480000\pi$
$379$		0			0		−0.0627905	+	0.998027i	$0.520000\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.0941083	−	0.995562i	$-0.470000\pi$
$383$		0			0		−0.0941083	+	0.995562i	$0.530000\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−0.113629	+	0.0534698i	−0.113629	+	0.0534698i	−0.481754	−	0.876307i	$-0.660000\pi$
$389$	−0.113629	+	0.0534698i	−0.113629	+	0.0534698i	0.368125	+	0.929776i	$0.380000\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−1.48175	+	1.30634i	−1.48175	+	1.30634i	−0.637424	+	0.770513i	$0.720000\pi$
$397$	−1.48175	+	1.30634i	−1.48175	+	1.30634i	−0.844328	+	0.535827i	$0.820000\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	0.331159	+	1.01920i	0.331159	+	1.01920i	0.968583	+	0.248690i	$0.0800000\pi$
$401$	0.331159	+	1.01920i	0.331159	+	1.01920i	−0.637424	+	0.770513i	$0.720000\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−0.125333	−	0.992115i	−0.125333	−	0.992115i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	0.256543	−	0.273190i	0.256543	−	0.273190i	−0.587785	−	0.809017i	$-0.700000\pi$
$409$	0.256543	−	0.273190i	0.256543	−	0.273190i	0.844328	+	0.535827i	$0.180000\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		−0.637424	−	0.770513i	$-0.720000\pi$
$419$		0			0		0.637424	+	0.770513i	$0.280000\pi$
$420$		0			0
$421$	0.516273	+	0.813516i	0.516273	+	0.813516i	0.998027	−	0.0627905i	$-0.0200000\pi$
$421$	0.516273	+	0.813516i	0.516273	+	0.813516i	−0.481754	+	0.876307i	$0.660000\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−1.80704	−	0.524995i	−1.80704	−	0.524995i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.368125	−	0.929776i	$-0.620000\pi$
$431$		0			0		0.368125	+	0.929776i	$0.380000\pi$
$432$		0			0
$433$	−0.595810	+	1.37684i	−0.595810	+	1.37684i	0.309017	+	0.951057i	$0.400000\pi$
$433$	−0.595810	+	1.37684i	−0.595810	+	1.37684i	−0.904827	+	0.425779i	$0.860000\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		0.187381	−	0.982287i	$-0.440000\pi$
$439$		0			0		−0.187381	+	0.982287i	$0.560000\pi$
$440$		0			0
$441$	0.876307	−	0.481754i	0.876307	−	0.481754i
$442$		0			0
$443$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$443$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$444$		0			0
$445$	−1.19975	+	0.659566i	−1.19975	+	0.659566i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	0.856954	+	1.17950i	0.856954	+	1.17950i	0.982287	+	0.187381i	$0.0600000\pi$
$449$	0.856954	+	1.17950i	0.856954	+	1.17950i	−0.125333	+	0.992115i	$0.540000\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−0.0489435	−	0.309017i	−0.0489435	−	0.309017i	0.951057	−	0.309017i	$-0.100000\pi$
$457$	−0.0489435	−	0.309017i	−0.0489435	−	0.309017i	−1.00000			$\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−0.844844	+	0.106729i	−0.844844	+	0.106729i	−0.535827	−	0.844328i	$-0.680000\pi$
$461$	−0.844844	+	0.106729i	−0.844844	+	0.106729i	−0.309017	+	0.951057i	$0.600000\pi$
$462$		0			0
$463$		0			0		−0.612907	−	0.790155i	$-0.710000\pi$
$463$		0			0		0.612907	+	0.790155i	$0.290000\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.860742	−	0.509041i	$-0.170000\pi$
$467$		0			0		−0.860742	+	0.509041i	$0.830000\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	0.656947	−	0.446460i	0.656947	−	0.446460i
$478$		0			0
$479$		0			0		0.968583	−	0.248690i	$-0.0800000\pi$
$479$		0			0		−0.968583	+	0.248690i	$0.920000\pi$
$480$		0			0
$481$	0.902371	−	1.42191i	0.902371	−	1.42191i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−0.0872876	+	0.551113i	−0.0872876	+	0.551113i
$486$		0			0
$487$		0			0		−0.338738	−	0.940881i	$-0.610000\pi$
$487$		0			0		0.338738	+	0.940881i	$0.390000\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		−0.904827	−	0.425779i	$-0.860000\pi$
$491$		0			0		0.904827	+	0.425779i	$0.140000\pi$
$492$		0			0
$493$	−0.214145	−	0.420283i	−0.214145	−	0.420283i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		1.00000			$0$
$499$		0			0		−1.00000			$\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.218143	−	0.975917i	$-0.430000\pi$
$503$		0			0		−0.218143	+	0.975917i	$0.570000\pi$
$504$		0			0
$505$	1.35041	−	1.11716i	1.35041	−	1.11716i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−1.63742	−	0.770513i	−1.63742	−	0.770513i	−0.637424	−	0.770513i	$-0.720000\pi$
$509$	−1.63742	−	0.770513i	−1.63742	−	0.770513i	−1.00000			$\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	1.80113	−	0.462452i	1.80113	−	0.462452i	0.809017	−	0.587785i	$-0.200000\pi$
$521$	1.80113	−	0.462452i	1.80113	−	0.462452i	0.992115	+	0.125333i	$0.0400000\pi$
$522$		0			0
$523$		0			0		0.827081	−	0.562083i	$-0.190000\pi$
$523$		0			0		−0.827081	+	0.562083i	$0.810000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	0.368125	−	0.929776i	0.368125	−	0.929776i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	1.74915	−	1.03445i	1.74915	−	1.03445i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−0.371808	−	1.94908i	−0.371808	−	1.94908i	−0.309017	−	0.951057i	$-0.600000\pi$
$541$	−0.371808	−	1.94908i	−0.371808	−	1.94908i	−0.0627905	−	0.998027i	$-0.520000\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−0.812619	+	0.982287i	−0.812619	+	0.982287i
$546$		0			0
$547$		0			0		−0.0941083	−	0.995562i	$-0.530000\pi$
$547$		0			0		0.0941083	+	0.995562i	$0.470000\pi$
$548$		0			0
$549$	−0.992567	+	1.36615i	−0.992567	+	1.36615i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	1.41352	+	1.41352i	1.41352	+	1.41352i	0.728969	+	0.684547i	$0.240000\pi$
$557$	1.41352	+	1.41352i	1.41352	+	1.41352i	0.684547	+	0.728969i	$0.260000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.0314108	−	0.999507i	$-0.510000\pi$
$563$		0			0		0.0314108	+	0.999507i	$0.490000\pi$
$564$		0			0
$565$	1.49948	−	1.32197i	1.49948	−	1.32197i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−0.313480	−	0.791759i	−0.313480	−	0.791759i	−0.998027	−	0.0627905i	$-0.980000\pi$
$569$	−0.313480	−	0.791759i	−0.313480	−	0.791759i	0.684547	−	0.728969i	$-0.260000\pi$
$570$		0			0
$571$		0			0		−0.770513	−	0.637424i	$-0.780000\pi$
$571$		0			0		0.770513	+	0.637424i	$0.220000\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	0.105981	−	0.294372i	0.105981	−	0.294372i	−0.876307	−	0.481754i	$-0.840000\pi$
$577$	0.105981	−	0.294372i	0.105981	−	0.294372i	0.982287	+	0.187381i	$0.0600000\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$	0.623990	−	0.804443i	0.623990	−	0.804443i
$586$		0			0
$587$		0			0		−0.827081	−	0.562083i	$-0.810000\pi$
$587$		0			0		0.827081	+	0.562083i	$0.190000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	0.191760	−	1.21072i	0.191760	−	1.21072i	−0.684547	−	0.728969i	$-0.740000\pi$
$593$	0.191760	−	1.21072i	0.191760	−	1.21072i	0.876307	−	0.481754i	$-0.160000\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$599$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$600$		0			0
$601$	−0.153699	+	0.473036i	−0.153699	+	0.473036i	−0.998027	−	0.0627905i	$-0.980000\pi$
$601$	−0.153699	+	0.473036i	−0.153699	+	0.473036i	0.844328	+	0.535827i	$0.180000\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	0.587785	−	0.809017i	0.587785	−	0.809017i
$606$		0			0
$607$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$607$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−0.313633	+	1.07953i	−0.313633	+	1.07953i	0.637424	+	0.770513i	$0.280000\pi$
$613$	−0.313633	+	1.07953i	−0.313633	+	1.07953i	−0.951057	+	0.309017i	$0.900000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−0.141183	+	1.49356i	−0.141183	+	1.49356i	0.587785	+	0.809017i	$0.300000\pi$
$617$	−0.141183	+	1.49356i	−0.141183	+	1.49356i	−0.728969	+	0.684547i	$0.760000\pi$
$618$		0			0
$619$		0			0		−0.968583	−	0.248690i	$-0.920000\pi$
$619$		0			0		0.968583	+	0.248690i	$0.0800000\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.637424	+	0.770513i	−0.637424	+	0.770513i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−3.10659	−	0.195450i	−3.10659	−	0.195450i
$630$		0			0
$631$		0			0		0.248690	−	0.968583i	$-0.420000\pi$
$631$		0			0		−0.248690	+	0.968583i	$0.580000\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	0.977659	+	0.284036i	0.977659	+	0.284036i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	1.88711	+	0.238398i	1.88711	+	0.238398i	0.982287	−	0.187381i	$-0.0600000\pi$
$641$	1.88711	+	0.238398i	1.88711	+	0.238398i	0.904827	+	0.425779i	$0.140000\pi$
$642$		0			0
$643$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$643$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.661312	−	0.750111i	$-0.730000\pi$
$647$		0			0		0.661312	+	0.750111i	$0.270000\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−0.512091	+	0.221601i	−0.512091	+	0.221601i	−0.637424	−	0.770513i	$-0.720000\pi$
$653$	−0.512091	+	0.221601i	−0.512091	+	0.221601i	0.125333	+	0.992115i	$0.460000\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	1.96661	+	0.311480i	1.96661	+	0.311480i
$658$		0			0
$659$		0			0		−0.728969	−	0.684547i	$-0.760000\pi$
$659$		0			0		0.728969	+	0.684547i	$0.240000\pi$
$660$		0			0
$661$	1.66683	+	0.916350i	1.66683	+	0.916350i	0.982287	+	0.187381i	$0.0600000\pi$
$661$	1.66683	+	0.916350i	1.66683	+	0.916350i	0.684547	+	0.728969i	$0.260000\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	1.48688	+	0.535311i	1.48688	+	0.535311i	0.951057	−	0.309017i	$-0.100000\pi$
$673$	1.48688	+	0.535311i	1.48688	+	0.535311i	0.535827	+	0.844328i	$0.320000\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−0.551113	−	1.89694i	−0.551113	−	1.89694i	−0.425779	−	0.904827i	$-0.640000\pi$
$677$	−0.551113	−	1.89694i	−0.551113	−	1.89694i	−0.125333	−	0.992115i	$-0.540000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.917755	−	0.397148i	$-0.870000\pi$
$683$		0			0		0.917755	+	0.397148i	$0.130000\pi$
$684$		0			0
$685$	0.288521	+	0.327263i	0.288521	+	0.327263i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	0.794335	+	0.151528i	0.794335	+	0.151528i
$690$		0			0
$691$		0			0		−0.481754	−	0.876307i	$-0.660000\pi$
$691$		0			0		0.481754	+	0.876307i	$0.340000\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−3.23303	−	1.91201i	−3.23303	−	1.91201i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−0.402389	−	0.292352i	−0.402389	−	0.292352i	0.368125	−	0.929776i	$-0.380000\pi$
$701$	−0.402389	−	0.292352i	−0.402389	−	0.292352i	−0.770513	+	0.637424i	$0.780000\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	0.488570	−	0.0931997i	0.488570	−	0.0931997i	0.0627905	−	0.998027i	$-0.480000\pi$
$709$	0.488570	−	0.0931997i	0.488570	−	0.0931997i	0.425779	+	0.904827i	$0.360000\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.0627905	−	0.998027i	$-0.520000\pi$
$719$		0			0		0.0627905	+	0.998027i	$0.480000\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−0.250172	+	0.0157395i	−0.250172	+	0.0157395i
$726$		0			0
$727$		0			0		−0.562083	−	0.827081i	$-0.690000\pi$
$727$		0			0		0.562083	+	0.827081i	$0.310000\pi$
$728$		0			0
$729$	−0.248690	−	0.968583i	−0.248690	−	0.968583i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	1.33671	+	1.17847i	1.33671	+	1.17847i	0.968583	+	0.248690i	$0.0800000\pi$
$733$	1.33671	+	1.17847i	1.33671	+	1.17847i	0.368125	+	0.929776i	$0.380000\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		0.728969	−	0.684547i	$-0.240000\pi$
$739$		0			0		−0.728969	+	0.684547i	$0.760000\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$743$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$744$		0			0
$745$	1.18532	+	1.43281i	1.18532	+	1.43281i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0			−	1.00000i	$-0.5\pi$
$751$		0			0				1.00000i	$0.5\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	0.388734	−	0.198070i	0.388734	−	0.198070i	−0.248690	−	0.968583i	$-0.580000\pi$
$757$	0.388734	−	0.198070i	0.388734	−	0.198070i	0.637424	+	0.770513i	$0.280000\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−0.929324	+	0.872693i	−0.929324	+	0.872693i	−0.992115	−	0.125333i	$-0.960000\pi$
$761$	−0.929324	+	0.872693i	−0.929324	+	0.872693i	0.0627905	+	0.998027i	$0.480000\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−1.85859	−	0.294372i	−1.85859	−	0.294372i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	0.992567	+	0.629902i	0.992567	+	0.629902i	0.929776	−	0.368125i	$-0.120000\pi$
$769$	0.992567	+	0.629902i	0.992567	+	0.629902i	0.0627905	+	0.998027i	$0.480000\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−0.105793	−	0.155670i	−0.105793	−	0.155670i	0.770513	−	0.637424i	$-0.220000\pi$
$773$	−0.105793	−	0.155670i	−0.105793	−	0.155670i	−0.876307	+	0.481754i	$0.840000\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−0.0471231	−	0.0415446i	−0.0471231	−	0.0415446i
$786$		0			0
$787$		0			0		0.790155	−	0.612907i	$-0.210000\pi$
$787$		0			0		−0.790155	+	0.612907i	$0.790000\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−1.69803	+	0.268941i	−1.69803	+	0.268941i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−1.31675	+	0.124470i	−1.31675	+	0.124470i	−0.728969	−	0.684547i	$-0.760000\pi$
$797$	−1.31675	+	0.124470i	−1.31675	+	0.124470i	−0.587785	+	0.809017i	$0.700000\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	−1.10762	+	0.804733i	−1.10762	+	0.804733i
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−0.871808	−	1.58581i	−0.871808	−	1.58581i	−0.809017	−	0.587785i	$-0.800000\pi$
$809$	−0.871808	−	1.58581i	−0.871808	−	1.58581i	−0.0627905	−	0.998027i	$-0.520000\pi$
$810$		0			0
$811$		0			0		−0.982287	−	0.187381i	$-0.940000\pi$
$811$		0			0		0.982287	+	0.187381i	$0.0600000\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−1.21245	+	1.46560i	−1.21245	+	1.46560i	−0.368125	+	0.929776i	$0.620000\pi$
$821$	−1.21245	+	1.46560i	−1.21245	+	1.46560i	−0.844328	+	0.535827i	$0.820000\pi$
$822$		0			0
$823$		0			0		−0.278991	−	0.960294i	$-0.590000\pi$
$823$		0			0		0.278991	+	0.960294i	$0.410000\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.940881	−	0.338738i	$-0.890000\pi$
$827$		0			0		0.940881	+	0.338738i	$0.110000\pi$
$828$		0			0
$829$	−1.63560	+	1.03799i	−1.63560	+	1.03799i	−0.684547	+	0.728969i	$0.740000\pi$
$829$	−1.63560	+	1.03799i	−1.63560	+	1.03799i	−0.951057	+	0.309017i	$0.900000\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−0.410494	−	1.83644i	−0.410494	−	1.83644i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.876307	−	0.481754i	$-0.840000\pi$
$839$		0			0		0.876307	+	0.481754i	$0.160000\pi$
$840$		0			0
$841$	0.683165	+	0.641534i	0.683165	+	0.641534i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	0.0362049	−	0.00457374i	0.0362049	−	0.00457374i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−1.21384	−	1.37684i	−1.21384	−	1.37684i	−0.904827	−	0.425779i	$-0.860000\pi$
$853$	−1.21384	−	1.37684i	−1.21384	−	1.37684i	−0.309017	−	0.951057i	$-0.600000\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	0.896802	−	1.76007i	0.896802	−	1.76007i	0.309017	−	0.951057i	$-0.400000\pi$
$857$	0.896802	−	1.76007i	0.896802	−	1.76007i	0.587785	−	0.809017i	$-0.300000\pi$
$858$		0			0
$859$		0			0		−0.992115	−	0.125333i	$-0.960000\pi$
$859$		0			0		0.992115	+	0.125333i	$0.0400000\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.960294	−	0.278991i	$-0.910000\pi$
$863$		0			0		0.960294	+	0.278991i	$0.0900000\pi$
$864$		0			0
$865$	1.17714	+	1.51756i	1.17714	+	1.51756i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−0.0175266	+	0.557707i	−0.0175266	+	0.557707i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	1.99014	+	0.0625427i	1.99014	+	0.0625427i	0.998027	−	0.0627905i	$-0.0200000\pi$
$877$	1.99014	+	0.0625427i	1.99014	+	0.0625427i	0.992115	+	0.125333i	$0.0400000\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−1.84235	−	0.473036i	−1.84235	−	0.473036i	−0.844328	−	0.535827i	$-0.820000\pi$
$881$	−1.84235	−	0.473036i	−1.84235	−	0.473036i	−0.998027	+	0.0627905i	$0.980000\pi$
$882$		0			0
$883$		0			0		0.0941083	−	0.995562i	$-0.470000\pi$
$883$		0			0		−0.0941083	+	0.995562i	$0.530000\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.278991	−	0.960294i	$-0.410000\pi$
$887$		0			0		−0.278991	+	0.960294i	$0.590000\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$	−0.461880	−	1.42152i	−0.461880	−	1.42152i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	1.43211	+	0.273190i	1.43211	+	0.273190i
$906$		0			0
$907$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$907$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$908$		0			0
$909$	1.19975	−	1.27760i	1.19975	−	1.27760i
$910$		0			0
$911$		0			0		0.481754	−	0.876307i	$-0.340000\pi$
$911$		0			0		−0.481754	+	0.876307i	$0.660000\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		−0.637424	−	0.770513i	$-0.720000\pi$
$919$		0			0		0.637424	+	0.770513i	$0.280000\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−0.750973	+	1.47387i	−0.750973	+	1.47387i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−1.35041	−	1.11716i	−1.35041	−	1.11716i	−0.982287	−	0.187381i	$-0.940000\pi$
$929$	−1.35041	−	1.11716i	−1.35041	−	1.11716i	−0.368125	−	0.929776i	$-0.620000\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−0.0353109	−	1.12361i	−0.0353109	−	1.12361i	−0.844328	−	0.535827i	$-0.820000\pi$
$937$	−0.0353109	−	1.12361i	−0.0353109	−	1.12361i	0.809017	−	0.587785i	$-0.200000\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−1.73879	+	0.955910i	−1.73879	+	0.955910i	−0.809017	+	0.587785i	$0.800000\pi$
$941$	−1.73879	+	0.955910i	−1.73879	+	0.955910i	−0.929776	+	0.368125i	$0.880000\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.509041	−	0.860742i	$-0.330000\pi$
$947$		0			0		−0.509041	+	0.860742i	$0.670000\pi$
$948$		0			0
$949$	1.19152	+	1.63998i	1.19152	+	1.63998i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−0.0637561	−	0.674469i	−0.0637561	−	0.674469i	−0.968583	−	0.248690i	$-0.920000\pi$
$953$	−0.0637561	−	0.674469i	−0.0637561	−	0.674469i	0.904827	−	0.425779i	$-0.140000\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	0.992115	−	0.125333i	0.992115	−	0.125333i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	1.90483	−	0.425779i	1.90483	−	0.425779i
$966$		0			0
$967$		0			0		0.860742	−	0.509041i	$-0.170000\pi$
$967$		0			0		−0.860742	+	0.509041i	$0.830000\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.368125	−	0.929776i	$-0.380000\pi$
$971$		0			0		−0.368125	+	0.929776i	$0.620000\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−0.155670	+	0.105793i	−0.155670	+	0.105793i	−0.637424	−	0.770513i	$-0.720000\pi$
$977$	−0.155670	+	0.105793i	−0.155670	+	0.105793i	0.481754	+	0.876307i	$0.340000\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−0.683098	+	1.07639i	−0.683098	+	1.07639i
$982$		0			0
$983$		0			0		0.661312	−	0.750111i	$-0.270000\pi$
$983$		0			0		−0.661312	+	0.750111i	$0.730000\pi$
$984$		0			0
$985$	−0.681087	−	1.33671i	−0.681087	−	1.33671i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		−0.904827	−	0.425779i	$-0.860000\pi$
$991$		0			0		0.904827	+	0.425779i	$0.140000\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−0.0682502	+	0.305334i	−0.0682502	+	0.305334i	−0.998027	−	0.0627905i	$-0.980000\pi$
$997$	−0.0682502	+	0.305334i	−0.0682502	+	0.305334i	0.929776	+	0.368125i	$0.120000\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4000.1.cq.a.1377.1	yes	40
4.3	odd	2	CM	4000.1.cq.a.1377.1	yes	40
125.63	odd	100	inner	4000.1.cq.a.1313.1	✓	40
500.63	even	100	inner	4000.1.cq.a.1313.1	✓	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4000.1.cq.a.1313.1	✓	40	125.63	odd	100	inner
4000.1.cq.a.1313.1	✓	40	500.63	even	100	inner
4000.1.cq.a.1377.1	yes	40	1.1	even	1	trivial
4000.1.cq.a.1377.1	yes	40	4.3	odd	2	CM

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:	\( N \)	\(=\)	\( 4000 = 2^{5} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	4000.cq (of order \(100\), degree \(40\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.844328 - 0.535827i) q^{5} +(-0.904827 - 0.425779i) q^{9} +O(q^{10})\)	\(q+(-0.844328 - 0.535827i) q^{5} +(-0.904827 - 0.425779i) q^{9} +(-0.344863 - 0.957895i) q^{13} +(-1.24443 + 1.41153i) q^{17} +(0.425779 + 0.904827i) q^{25} +(-0.0922765 + 0.233064i) q^{29} +(1.01385 + 1.30704i) q^{37} +(0.374023 + 1.96070i) q^{41} +(0.535827 + 0.844328i) q^{45} +(-0.587785 + 0.809017i) q^{49} +(-0.404329 + 0.683684i) q^{53} +(0.316423 - 1.65875i) q^{61} +(-0.222088 + 0.993564i) q^{65} +(-1.91206 + 0.555506i) q^{73} +(0.637424 + 0.770513i) q^{81} +(1.80704 - 0.524995i) q^{85} +(0.659566 - 1.19975i) q^{89} +(-0.221601 - 0.512091i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q+O(q^{10}) \) 40 * q	\( 40 q + 10 q^{85} + 10 q^{89}+O(q^{100}) \) 40 * q + 10 * q^85 + 10 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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