LMFDB - Embedded newform 4000.1.cq.a.33.1

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			33.1
Root			$-0.368125 - 0.929776i$ of defining polynomial
Character	$\chi$	$=$	4000.33
Dual form			4000.1.cq.a.1697.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.982287 - 0.187381i) q^{5} +(-0.368125 + 0.929776i) q^{9} +O(q^{10})$	$q+(0.982287 - 0.187381i) q^{5} +(-0.368125 + 0.929776i) q^{9} +(0.269058 + 0.621757i) q^{13} +(0.512091 - 1.76263i) q^{17} +(0.929776 - 0.368125i) q^{25} +(-0.211645 - 1.67534i) q^{29} +(1.46409 + 0.327263i) q^{37} +(-0.0604991 + 0.961606i) q^{41} +(-0.187381 + 0.982287i) q^{45} +(0.951057 + 0.309017i) q^{49} +(-0.415230 + 1.15334i) q^{53} +(0.123357 + 1.96070i) q^{61} +(0.380798 + 0.560327i) q^{65} +(0.00591203 - 0.0625427i) q^{73} +(-0.728969 - 0.684547i) q^{81} +(0.172737 - 1.82736i) q^{85} +(0.383238 - 0.317042i) q^{89} +(1.22037 - 1.57330i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/4000\mathbb{Z}\right)^\times$.

$n$	$1377$	$2501$	$2751$
$\chi(n)$	$e\left(\frac{83}{100}\right)$	$1$	$1$

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		−0.562083	−	0.827081i	$-0.690000\pi$
$3$		0			0		0.562083	+	0.827081i	$0.310000\pi$
$4$		0			0
$5$	0.982287	−	0.187381i	0.982287	−	0.187381i
$5$	0.982287	−	0.187381i	0.982287	−	0.187381i
$6$		0			0
$7$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$7$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$8$		0			0
$9$	−0.368125	+	0.929776i	−0.368125	+	0.929776i
$10$		0			0
$11$		0			0		0.248690	−	0.968583i	$-0.420000\pi$
$11$		0			0		−0.248690	+	0.968583i	$0.580000\pi$
$12$		0			0
$13$	0.269058	+	0.621757i	0.269058	+	0.621757i	0.998027	−	0.0627905i	$-0.0200000\pi$
$13$	0.269058	+	0.621757i	0.269058	+	0.621757i	−0.728969	+	0.684547i	$0.760000\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	0.512091	−	1.76263i	0.512091	−	1.76263i	−0.125333	−	0.992115i	$-0.540000\pi$
$17$	0.512091	−	1.76263i	0.512091	−	1.76263i	0.637424	−	0.770513i	$-0.280000\pi$
$18$		0			0
$19$		0			0		−0.187381	−	0.982287i	$-0.560000\pi$
$19$		0			0		0.187381	+	0.982287i	$0.440000\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		−0.750111	−	0.661312i	$-0.770000\pi$
$23$		0			0		0.750111	+	0.661312i	$0.230000\pi$
$24$		0			0
$25$	0.929776	−	0.368125i	0.929776	−	0.368125i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−0.211645	−	1.67534i	−0.211645	−	1.67534i	−0.637424	−	0.770513i	$-0.720000\pi$
$29$	−0.211645	−	1.67534i	−0.211645	−	1.67534i	0.425779	−	0.904827i	$-0.360000\pi$
$30$		0			0
$31$		0			0		−0.481754	−	0.876307i	$-0.660000\pi$
$31$		0			0		0.481754	+	0.876307i	$0.340000\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	1.46409	+	0.327263i	1.46409	+	0.327263i	0.876307	−	0.481754i	$-0.160000\pi$
$37$	1.46409	+	0.327263i	1.46409	+	0.327263i	0.587785	+	0.809017i	$0.300000\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−0.0604991	+	0.961606i	−0.0604991	+	0.961606i	0.844328	+	0.535827i	$0.180000\pi$
$41$	−0.0604991	+	0.961606i	−0.0604991	+	0.961606i	−0.904827	+	0.425779i	$0.860000\pi$
$42$		0			0
$43$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$43$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$44$		0			0
$45$	−0.187381	+	0.982287i	−0.187381	+	0.982287i
$46$		0			0
$47$		0			0		−0.999507	−	0.0314108i	$-0.990000\pi$
$47$		0			0		0.999507	+	0.0314108i	$0.0100000\pi$
$48$		0			0
$49$	0.951057	+	0.309017i	0.951057	+	0.309017i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−0.415230	+	1.15334i	−0.415230	+	1.15334i	0.535827	+	0.844328i	$0.320000\pi$
$53$	−0.415230	+	1.15334i	−0.415230	+	1.15334i	−0.951057	+	0.309017i	$0.900000\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		0.637424	−	0.770513i	$-0.280000\pi$
$59$		0			0		−0.637424	+	0.770513i	$0.720000\pi$
$60$		0			0
$61$	0.123357	+	1.96070i	0.123357	+	1.96070i	0.248690	+	0.968583i	$0.420000\pi$
$61$	0.123357	+	1.96070i	0.123357	+	1.96070i	−0.125333	+	0.992115i	$0.540000\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	0.380798	+	0.560327i	0.380798	+	0.560327i
$66$		0			0
$67$		0			0		−0.612907	−	0.790155i	$-0.710000\pi$
$67$		0			0		0.612907	+	0.790155i	$0.290000\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.684547	−	0.728969i	$-0.740000\pi$
$71$		0			0		0.684547	+	0.728969i	$0.260000\pi$
$72$		0			0
$73$	0.00591203	−	0.0625427i	0.00591203	−	0.0625427i	−0.992115	−	0.125333i	$-0.960000\pi$
$73$	0.00591203	−	0.0625427i	0.00591203	−	0.0625427i	0.998027	+	0.0627905i	$0.0200000\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		0.187381	−	0.982287i	$-0.440000\pi$
$79$		0			0		−0.187381	+	0.982287i	$0.560000\pi$
$80$		0			0
$81$	−0.728969	−	0.684547i	−0.728969	−	0.684547i
$82$		0			0
$83$		0			0		−0.827081	−	0.562083i	$-0.810000\pi$
$83$		0			0		0.827081	+	0.562083i	$0.190000\pi$
$84$		0			0
$85$	0.172737	−	1.82736i	0.172737	−	1.82736i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	0.383238	−	0.317042i	0.383238	−	0.317042i	−0.425779	−	0.904827i	$-0.640000\pi$
$89$	0.383238	−	0.317042i	0.383238	−	0.317042i	0.809017	+	0.587785i	$0.200000\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	1.22037	−	1.57330i	1.22037	−	1.57330i	0.535827	−	0.844328i	$-0.320000\pi$
$97$	1.22037	−	1.57330i	1.22037	−	1.57330i	0.684547	−	0.728969i	$-0.260000\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−1.03137	+	0.749337i	−1.03137	+	0.749337i	−0.968583	−	0.248690i	$-0.920000\pi$
$101$	−1.03137	+	0.749337i	−1.03137	+	0.749337i	−0.0627905	+	0.998027i	$0.520000\pi$
$102$		0			0
$103$		0			0		0.960294	−	0.278991i	$-0.0900000\pi$
$103$		0			0		−0.960294	+	0.278991i	$0.910000\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$107$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$108$		0			0
$109$	−1.23098	+	0.781202i	−1.23098	+	0.781202i	−0.982287	−	0.187381i	$-0.940000\pi$
$109$	−1.23098	+	0.781202i	−1.23098	+	0.781202i	−0.248690	+	0.968583i	$0.580000\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	1.71384	−	0.162006i	1.71384	−	0.162006i	0.809017	−	0.587785i	$-0.200000\pi$
$113$	1.71384	−	0.162006i	1.71384	−	0.162006i	0.904827	+	0.425779i	$0.140000\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−0.677142	+	0.0212800i	−0.677142	+	0.0212800i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.876307	−	0.481754i	−0.876307	−	0.481754i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	0.844328	−	0.535827i	0.844328	−	0.535827i
$126$		0			0
$127$		0			0		−0.509041	−	0.860742i	$-0.670000\pi$
$127$		0			0		0.509041	+	0.860742i	$0.330000\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		−0.904827	−	0.425779i	$-0.860000\pi$
$131$		0			0		0.904827	+	0.425779i	$0.140000\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	0.105793	+	1.11918i	0.105793	+	1.11918i	0.876307	+	0.481754i	$0.160000\pi$
$137$	0.105793	+	1.11918i	0.105793	+	1.11918i	−0.770513	+	0.637424i	$0.780000\pi$
$138$		0			0
$139$		0			0		0.929776	−	0.368125i	$-0.120000\pi$
$139$		0			0		−0.929776	+	0.368125i	$0.880000\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	−0.521823	−	1.60601i	−0.521823	−	1.60601i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1.16630	−	1.60528i	−1.16630	−	1.60528i	−0.684547	−	0.728969i	$-0.740000\pi$
$149$	−1.16630	−	1.60528i	−1.16630	−	1.60528i	−0.481754	−	0.876307i	$-0.660000\pi$
$150$		0			0
$151$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$151$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$152$		0			0
$153$	1.45034	+	1.12500i	1.45034	+	1.12500i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−0.907118	−	0.462200i	−0.907118	−	0.462200i	−0.0627905	−	0.998027i	$-0.520000\pi$
$157$	−0.907118	−	0.462200i	−0.907118	−	0.462200i	−0.844328	+	0.535827i	$0.820000\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$		0			0		−0.661312	−	0.750111i	$-0.730000\pi$
$163$		0			0		0.661312	+	0.750111i	$0.270000\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.562083	−	0.827081i	$-0.310000\pi$
$167$		0			0		−0.562083	+	0.827081i	$0.690000\pi$
$168$		0			0
$169$	0.370358	−	0.394391i	0.370358	−	0.394391i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−0.172737	−	0.0747498i	−0.172737	−	0.0747498i	0.309017	−	0.951057i	$-0.400000\pi$
$173$	−0.172737	−	0.0747498i	−0.172737	−	0.0747498i	−0.481754	+	0.876307i	$0.660000\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.728969	−	0.684547i	$-0.240000\pi$
$179$		0			0		−0.728969	+	0.684547i	$0.760000\pi$
$180$		0			0
$181$	−1.92189	−	0.242791i	−1.92189	−	0.242791i	−0.929776	−	0.368125i	$-0.880000\pi$
$181$	−1.92189	−	0.242791i	−1.92189	−	0.242791i	−0.992115	+	0.125333i	$0.960000\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	1.49948	+	0.0471231i	1.49948	+	0.0471231i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.770513	−	0.637424i	$-0.780000\pi$
$191$		0			0		0.770513	+	0.637424i	$0.220000\pi$
$192$		0			0
$193$	1.16967	+	1.16967i	1.16967	+	1.16967i	0.982287	+	0.187381i	$0.0600000\pi$
$193$	1.16967	+	1.16967i	1.16967	+	1.16967i	0.187381	+	0.982287i	$0.440000\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−1.80704	−	0.650576i	−1.80704	−	0.650576i	−0.998027	−	0.0627905i	$-0.980000\pi$
$197$	−1.80704	−	0.650576i	−1.80704	−	0.650576i	−0.809017	−	0.587785i	$-0.800000\pi$
$198$		0			0
$199$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$199$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	0.120759	+	0.955910i	0.120759	+	0.955910i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		−0.844328	−	0.535827i	$-0.820000\pi$
$211$		0			0		0.844328	+	0.535827i	$0.180000\pi$
$212$		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
$221$	1.23371	−	0.155854i	1.23371	−	0.155854i
$222$		0			0
$223$		0			0		−0.218143	−	0.975917i	$-0.570000\pi$
$223$		0			0		0.218143	+	0.975917i	$0.430000\pi$
$224$		0			0
$225$			1.00000i			1.00000i
$226$		0			0
$227$		0			0		0.661312	−	0.750111i	$-0.270000\pi$
$227$		0			0		−0.661312	+	0.750111i	$0.730000\pi$
$228$		0			0
$229$	−0.226810	+	0.106729i	−0.226810	+	0.106729i	−0.535827	−	0.844328i	$-0.680000\pi$
$229$	−0.226810	+	0.106729i	−0.226810	+	0.106729i	0.309017	+	0.951057i	$0.400000\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−0.535827	−	0.155672i	−0.535827	−	0.155672i		−	1.00000i	$-0.5\pi$
$233$	−0.535827	−	0.155672i	−0.535827	−	0.155672i	−0.535827	+	0.844328i	$0.680000\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		−0.968583	−	0.248690i	$-0.920000\pi$
$239$		0			0		0.968583	+	0.248690i	$0.0800000\pi$
$240$		0			0
$241$	−1.85588	−	0.734796i	−1.85588	−	0.734796i	−0.951057	−	0.309017i	$-0.900000\pi$
$241$	−1.85588	−	0.734796i	−1.85588	−	0.734796i	−0.904827	−	0.425779i	$-0.860000\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	0.992115	+	0.125333i	0.992115	+	0.125333i
$246$		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
$257$	0.312715	−	1.97440i	0.312715	−	1.97440i	0.125333	−	0.992115i	$-0.460000\pi$
$257$	0.312715	−	1.97440i	0.312715	−	1.97440i	0.187381	−	0.982287i	$-0.440000\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	1.63560	+	0.419952i	1.63560	+	0.419952i
$262$		0			0
$263$		0			0		0.917755	−	0.397148i	$-0.130000\pi$
$263$		0			0		−0.917755	+	0.397148i	$0.870000\pi$
$264$		0			0
$265$	−0.191760	+	1.21072i	−0.191760	+	1.21072i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	0.368125	−	0.0702235i	0.368125	−	0.0702235i		−	1.00000i	$-0.5\pi$
$269$	0.368125	−	0.0702235i	0.368125	−	0.0702235i	0.368125	+	0.929776i	$0.380000\pi$
$270$		0			0
$271$		0			0		0.904827	−	0.425779i	$-0.140000\pi$
$271$		0			0		−0.904827	+	0.425779i	$0.860000\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	0.425779	+	1.90483i	0.425779	+	1.90483i	0.425779	+	0.904827i	$0.360000\pi$
$277$	0.425779	+	1.90483i	0.425779	+	1.90483i			1.00000i	$0.5\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−0.219661	+	0.120759i	−0.219661	+	0.120759i	−0.587785	−	0.809017i	$-0.700000\pi$
$281$	−0.219661	+	0.120759i	−0.219661	+	0.120759i	0.368125	+	0.929776i	$0.380000\pi$
$282$		0			0
$283$		0			0		−0.338738	−	0.940881i	$-0.610000\pi$
$283$		0			0		0.338738	+	0.940881i	$0.390000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−2.00029	−	1.26942i	−2.00029	−	1.26942i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	0.388734	−	0.198070i	0.388734	−	0.198070i	−0.248690	−	0.968583i	$-0.580000\pi$
$293$	0.388734	−	0.198070i	0.388734	−	0.198070i	0.637424	+	0.770513i	$0.280000\pi$
$294$		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
$305$	0.488570	+	1.90285i	0.488570	+	1.90285i
$306$		0			0
$307$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$307$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.998027	−	0.0627905i	$-0.0200000\pi$
$311$		0			0		−0.998027	+	0.0627905i	$0.980000\pi$
$312$		0			0
$313$	1.70029	+	1.00555i	1.70029	+	1.00555i	0.929776	+	0.368125i	$0.120000\pi$
$313$	1.70029	+	1.00555i	1.70029	+	1.00555i	0.770513	+	0.637424i	$0.220000\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−0.717446	+	0.556508i	−0.717446	+	0.556508i	−0.904827	−	0.425779i	$-0.860000\pi$
$317$	−0.717446	+	0.556508i	−0.717446	+	0.556508i	0.187381	+	0.982287i	$0.440000\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	0.479048	+	0.479048i	0.479048	+	0.479048i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		0.684547	−	0.728969i	$-0.260000\pi$
$331$		0			0		−0.684547	+	0.728969i	$0.740000\pi$
$332$		0			0
$333$	−0.843250	+	1.24080i	−0.843250	+	1.24080i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−1.17847	−	1.33671i	−1.17847	−	1.33671i	−0.929776	−	0.368125i	$-0.880000\pi$
$337$	−1.17847	−	1.33671i	−1.17847	−	1.33671i	−0.248690	−	0.968583i	$-0.580000\pi$
$338$		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		−0.790155	−	0.612907i	$-0.790000\pi$
$347$		0			0		0.790155	+	0.612907i	$0.210000\pi$
$348$		0			0
$349$	−1.17325	+	1.61484i	−1.17325	+	1.61484i	−0.535827	+	0.844328i	$0.680000\pi$
$349$	−1.17325	+	1.61484i	−1.17325	+	1.61484i	−0.637424	+	0.770513i	$0.720000\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−0.0872876	−	0.300446i	−0.0872876	−	0.300446i	0.904827	−	0.425779i	$-0.140000\pi$
$353$	−0.0872876	−	0.300446i	−0.0872876	−	0.300446i	−0.992115	+	0.125333i	$0.960000\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.535827	−	0.844328i	$-0.680000\pi$
$359$		0			0		0.535827	+	0.844328i	$0.320000\pi$
$360$		0			0
$361$	−0.929776	+	0.368125i	−0.929776	+	0.368125i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−0.00591203	−	0.0625427i	−0.00591203	−	0.0625427i
$366$		0			0
$367$		0			0		−0.0314108	−	0.999507i	$-0.510000\pi$
$367$		0			0		0.0314108	+	0.999507i	$0.490000\pi$
$368$		0			0
$369$	−0.871808	−	0.410241i	−0.871808	−	0.410241i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−0.159263	−	0.269299i	−0.159263	−	0.269299i	0.770513	−	0.637424i	$-0.220000\pi$
$373$	−0.159263	−	0.269299i	−0.159263	−	0.269299i	−0.929776	+	0.368125i	$0.880000\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	0.984709	−	0.582355i	0.984709	−	0.582355i
$378$		0			0
$379$		0			0		−0.876307	−	0.481754i	$-0.840000\pi$
$379$		0			0		0.876307	+	0.481754i	$0.160000\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.999507	−	0.0314108i	$-0.0100000\pi$
$383$		0			0		−0.999507	+	0.0314108i	$0.990000\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−0.645180	−	1.62954i	−0.645180	−	1.62954i	−0.770513	−	0.637424i	$-0.780000\pi$
$389$	−0.645180	−	1.62954i	−0.645180	−	1.62954i	0.125333	−	0.992115i	$-0.460000\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	1.71126	−	0.497166i	1.71126	−	0.497166i	0.728969	−	0.684547i	$-0.240000\pi$
$397$	1.71126	−	0.497166i	1.71126	−	0.497166i	0.982287	+	0.187381i	$0.0600000\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	0.303189	+	0.220280i	0.303189	+	0.220280i	0.728969	−	0.684547i	$-0.240000\pi$
$401$	0.303189	+	0.220280i	0.303189	+	0.220280i	−0.425779	+	0.904827i	$0.640000\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−0.844328	−	0.535827i	−0.844328	−	0.535827i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−0.0312307	−	0.121636i	−0.0312307	−	0.121636i	0.951057	−	0.309017i	$-0.100000\pi$
$409$	−0.0312307	−	0.121636i	−0.0312307	−	0.121636i	−0.982287	+	0.187381i	$0.940000\pi$
$410$		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		−0.728969	−	0.684547i	$-0.760000\pi$
$419$		0			0		0.728969	+	0.684547i	$0.240000\pi$
$420$		0			0
$421$	−0.288760	+	1.51373i	−0.288760	+	1.51373i	0.481754	+	0.876307i	$0.340000\pi$
$421$	−0.288760	+	1.51373i	−0.288760	+	1.51373i	−0.770513	+	0.637424i	$0.780000\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−0.172737	−	1.82736i	−0.172737	−	1.82736i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.125333	−	0.992115i	$-0.460000\pi$
$431$		0			0		−0.125333	+	0.992115i	$0.540000\pi$
$432$		0			0
$433$	−1.17714	−	1.51756i	−1.17714	−	1.51756i	−0.809017	−	0.587785i	$-0.800000\pi$
$433$	−1.17714	−	1.51756i	−1.17714	−	1.51756i	−0.368125	−	0.929776i	$-0.620000\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		−0.0627905	−	0.998027i	$-0.520000\pi$
$439$		0			0		0.0627905	+	0.998027i	$0.480000\pi$
$440$		0			0
$441$	−0.637424	+	0.770513i	−0.637424	+	0.770513i
$442$		0			0
$443$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$443$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$444$		0			0
$445$	0.317042	−	0.383238i	0.317042	−	0.383238i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−1.84235	+	0.598617i	−1.84235	+	0.598617i	−0.844328	+	0.535827i	$0.820000\pi$
$449$	−1.84235	+	0.598617i	−1.84235	+	0.598617i	−0.998027	+	0.0627905i	$0.980000\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−0.412215	−	0.809017i	−0.412215	−	0.809017i	0.587785	−	0.809017i	$-0.300000\pi$
$457$	−0.412215	−	0.809017i	−0.412215	−	0.809017i	−1.00000			$\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	0.996398	−	1.57007i	0.996398	−	1.57007i	0.187381	−	0.982287i	$-0.440000\pi$
$461$	0.996398	−	1.57007i	0.996398	−	1.57007i	0.809017	−	0.587785i	$-0.200000\pi$
$462$		0			0
$463$		0			0		−0.975917	−	0.218143i	$-0.930000\pi$
$463$		0			0		0.975917	+	0.218143i	$0.0700000\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.940881	−	0.338738i	$-0.110000\pi$
$467$		0			0		−0.940881	+	0.338738i	$0.890000\pi$
$468$		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
$477$	−0.919497	−	0.810645i	−0.919497	−	0.810645i
$478$		0			0
$479$		0			0		−0.425779	−	0.904827i	$-0.640000\pi$
$479$		0			0		0.425779	+	0.904827i	$0.360000\pi$
$480$		0			0
$481$	0.190448	+	0.998362i	0.190448	+	0.998362i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	0.903951	−	1.77410i	0.903951	−	1.77410i
$486$		0			0
$487$		0			0		−0.397148	−	0.917755i	$-0.630000\pi$
$487$		0			0		0.397148	+	0.917755i	$0.370000\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		0.368125	−	0.929776i	$-0.380000\pi$
$491$		0			0		−0.368125	+	0.929776i	$0.620000\pi$
$492$		0			0
$493$	−3.06138	−	0.484875i	−3.06138	−	0.484875i
$494$		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		−0.562083	−	0.827081i	$-0.690000\pi$
$503$		0			0		0.562083	+	0.827081i	$0.310000\pi$
$504$		0			0
$505$	−0.872693	+	0.929324i	−0.872693	+	0.929324i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−0.271031	+	0.684547i	−0.271031	+	0.684547i	0.728969	+	0.684547i	$0.240000\pi$
$509$	−0.271031	+	0.684547i	−0.271031	+	0.684547i	−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
$521$	−0.844844	−	1.79538i	−0.844844	−	1.79538i	−0.535827	−	0.844328i	$-0.680000\pi$
$521$	−0.844844	−	1.79538i	−0.844844	−	1.79538i	−0.309017	−	0.951057i	$-0.600000\pi$
$522$		0			0
$523$		0			0		−0.750111	−	0.661312i	$-0.770000\pi$
$523$		0			0		0.750111	+	0.661312i	$0.230000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	0.125333	+	0.992115i	0.125333	+	0.992115i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−0.614163	+	0.221112i	−0.614163	+	0.221112i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−0.0672897	+	1.06954i	−0.0672897	+	1.06954i	0.809017	+	0.587785i	$0.200000\pi$
$541$	−0.0672897	+	1.06954i	−0.0672897	+	1.06954i	−0.876307	+	0.481754i	$0.840000\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−1.06279	+	0.998027i	−1.06279	+	0.998027i
$546$		0			0
$547$		0			0		−0.999507	−	0.0314108i	$-0.990000\pi$
$547$		0			0		0.999507	+	0.0314108i	$0.0100000\pi$
$548$		0			0
$549$	−1.86842	−	0.607087i	−1.86842	−	0.607087i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	1.21727	−	1.21727i	1.21727	−	1.21727i	0.248690	−	0.968583i	$-0.420000\pi$
$557$	1.21727	−	1.21727i	1.21727	−	1.21727i	0.968583	−	0.248690i	$-0.0800000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.509041	−	0.860742i	$-0.330000\pi$
$563$		0			0		−0.509041	+	0.860742i	$0.670000\pi$
$564$		0			0
$565$	1.65313	−	0.480279i	1.65313	−	0.480279i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−0.233064	+	1.84489i	−0.233064	+	1.84489i	0.248690	+	0.968583i	$0.420000\pi$
$569$	−0.233064	+	1.84489i	−0.233064	+	1.84489i	−0.481754	+	0.876307i	$0.660000\pi$
$570$		0			0
$571$		0			0		−0.684547	−	0.728969i	$-0.740000\pi$
$571$		0			0		0.684547	+	0.728969i	$0.260000\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−0.360603	+	0.833304i	−0.360603	+	0.833304i	0.637424	+	0.770513i	$0.280000\pi$
$577$	−0.360603	+	0.833304i	−0.360603	+	0.833304i	−0.998027	+	0.0627905i	$0.980000\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$	−0.661160	+	0.147787i	−0.661160	+	0.147787i
$586$		0			0
$587$		0			0		0.750111	−	0.661312i	$-0.230000\pi$
$587$		0			0		−0.750111	+	0.661312i	$0.770000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−0.886114	+	1.73910i	−0.886114	+	1.73910i	−0.248690	+	0.968583i	$0.580000\pi$
$593$	−0.886114	+	1.73910i	−0.886114	+	1.73910i	−0.637424	+	0.770513i	$0.720000\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$599$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$600$		0			0
$601$	−1.46404	+	1.06369i	−1.46404	+	1.06369i	−0.481754	+	0.876307i	$0.660000\pi$
$601$	−1.46404	+	1.06369i	−1.46404	+	1.06369i	−0.982287	+	0.187381i	$0.940000\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−0.951057	−	0.309017i	−0.951057	−	0.309017i
$606$		0			0
$607$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$607$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−1.31675	+	0.124470i	−1.31675	+	0.124470i	−0.728969	−	0.684547i	$-0.760000\pi$
$613$	−1.31675	+	0.124470i	−1.31675	+	0.124470i	−0.587785	+	0.809017i	$0.700000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−1.91964	+	0.0603271i	−1.91964	+	0.0603271i	−0.968583	−	0.248690i	$-0.920000\pi$
$617$	−1.91964	+	0.0603271i	−1.91964	+	0.0603271i	−0.951057	+	0.309017i	$0.900000\pi$
$618$		0			0
$619$		0			0		0.425779	−	0.904827i	$-0.360000\pi$
$619$		0			0		−0.425779	+	0.904827i	$0.640000\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.728969	−	0.684547i	0.728969	−	0.684547i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	1.32659	−	2.41306i	1.32659	−	2.41306i
$630$		0			0
$631$		0			0		−0.904827	−	0.425779i	$-0.860000\pi$
$631$		0			0		0.904827	+	0.425779i	$0.140000\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	0.0637561	+	0.674469i	0.0637561	+	0.674469i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−0.629902	−	0.992567i	−0.629902	−	0.992567i	−0.998027	−	0.0627905i	$-0.980000\pi$
$641$	−0.629902	−	0.992567i	−0.629902	−	0.992567i	0.368125	−	0.929776i	$-0.380000\pi$
$642$		0			0
$643$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$643$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.278991	−	0.960294i	$-0.590000\pi$
$647$		0			0		0.278991	+	0.960294i	$0.410000\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	1.57330	+	1.22037i	1.57330	+	1.22037i	0.844328	+	0.535827i	$0.180000\pi$
$653$	1.57330	+	1.22037i	1.57330	+	1.22037i	0.728969	+	0.684547i	$0.240000\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	0.0559744	+	0.0285204i	0.0559744	+	0.0285204i
$658$		0			0
$659$		0			0		0.968583	−	0.248690i	$-0.0800000\pi$
$659$		0			0		−0.968583	+	0.248690i	$0.920000\pi$
$660$		0			0
$661$	−0.749337	−	0.905793i	−0.749337	−	0.905793i	0.248690	−	0.968583i	$-0.420000\pi$
$661$	−0.749337	−	0.905793i	−0.749337	−	0.905793i	−0.998027	+	0.0627905i	$0.980000\pi$
$662$		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
$673$	0.400404	+	0.173270i	0.400404	+	0.173270i	0.587785	−	0.809017i	$-0.300000\pi$
$673$	0.400404	+	0.173270i	0.400404	+	0.173270i	−0.187381	+	0.982287i	$0.560000\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−1.77410	−	0.167702i	−1.77410	−	0.167702i	−0.844328	−	0.535827i	$-0.820000\pi$
$677$	−1.77410	−	0.167702i	−1.77410	−	0.167702i	−0.929776	+	0.368125i	$0.880000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.790155	−	0.612907i	$-0.210000\pi$
$683$		0			0		−0.790155	+	0.612907i	$0.790000\pi$
$684$		0			0
$685$	0.313633	+	1.07953i	0.313633	+	1.07953i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−0.828821	+	0.0521450i	−0.828821	+	0.0521450i
$690$		0			0
$691$		0			0		−0.770513	−	0.637424i	$-0.780000\pi$
$691$		0			0		0.770513	+	0.637424i	$0.220000\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	1.66397	+	0.599067i	1.66397	+	0.599067i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−0.559214	+	1.72108i	−0.559214	+	1.72108i	0.125333	+	0.992115i	$0.460000\pi$
$701$	−0.559214	+	1.72108i	−0.559214	+	1.72108i	−0.684547	+	0.728969i	$0.740000\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	1.80608	+	0.113629i	1.80608	+	0.113629i	0.929776	−	0.368125i	$-0.120000\pi$
$709$	1.80608	+	0.113629i	1.80608	+	0.113629i	0.876307	+	0.481754i	$0.160000\pi$
$710$		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		0.876307	−	0.481754i	$-0.160000\pi$
$719$		0			0		−0.876307	+	0.481754i	$0.840000\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−0.813516	−	1.47978i	−0.813516	−	1.47978i
$726$		0			0
$727$		0			0		0.661312	−	0.750111i	$-0.270000\pi$
$727$		0			0		−0.661312	+	0.750111i	$0.730000\pi$
$728$		0			0
$729$	0.904827	−	0.425779i	0.904827	−	0.425779i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−0.300446	−	0.0872876i	−0.300446	−	0.0872876i	0.125333	−	0.992115i	$-0.460000\pi$
$733$	−0.300446	−	0.0872876i	−0.300446	−	0.0872876i	−0.425779	+	0.904827i	$0.640000\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		−0.968583	−	0.248690i	$-0.920000\pi$
$739$		0			0		0.968583	+	0.248690i	$0.0800000\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$743$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$744$		0			0
$745$	−1.44644	−	1.35830i	−1.44644	−	1.35830i
$746$		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
$757$	0.175858	−	1.11033i	0.175858	−	1.11033i	−0.728969	−	0.684547i	$-0.760000\pi$
$757$	0.175858	−	1.11033i	0.175858	−	1.11033i	0.904827	−	0.425779i	$-0.140000\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	1.41213	+	0.362574i	1.41213	+	0.362574i	0.876307	−	0.481754i	$-0.160000\pi$
$761$	1.41213	+	0.362574i	1.41213	+	0.362574i	0.535827	+	0.844328i	$0.320000\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	1.63545	+	0.833304i	1.63545	+	0.833304i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	1.86842	−	0.356420i	1.86842	−	0.356420i	0.876307	−	0.481754i	$-0.160000\pi$
$769$	1.86842	−	0.356420i	1.86842	−	0.356420i	0.992115	+	0.125333i	$0.0400000\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	1.32197	−	1.49948i	1.32197	−	1.49948i	0.637424	−	0.770513i	$-0.280000\pi$
$773$	1.32197	−	1.49948i	1.32197	−	1.49948i	0.684547	−	0.728969i	$-0.260000\pi$
$774$		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
$785$	−0.977659	−	0.284036i	−0.977659	−	0.284036i
$786$		0			0
$787$		0			0		0.218143	−	0.975917i	$-0.430000\pi$
$787$		0			0		−0.218143	+	0.975917i	$0.570000\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−1.18589	+	0.604239i	−1.18589	+	0.604239i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−0.0175266	+	0.557707i	−0.0175266	+	0.557707i	0.951057	+	0.309017i	$0.100000\pi$
$797$	−0.0175266	+	0.557707i	−0.0175266	+	0.557707i	−0.968583	+	0.248690i	$0.920000\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	0.153699	+	0.473036i	0.153699	+	0.473036i
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−0.567290	−	0.469303i	−0.567290	−	0.469303i	0.309017	−	0.951057i	$-0.400000\pi$
$809$	−0.567290	−	0.469303i	−0.567290	−	0.469303i	−0.876307	+	0.481754i	$0.840000\pi$
$810$		0			0
$811$		0			0		0.998027	−	0.0627905i	$-0.0200000\pi$
$811$		0			0		−0.998027	+	0.0627905i	$0.980000\pi$
$812$		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
$821$	0.856954	−	0.804733i	0.856954	−	0.804733i	−0.125333	−	0.992115i	$-0.540000\pi$
$821$	0.856954	−	0.804733i	0.856954	−	0.804733i	0.982287	+	0.187381i	$0.0600000\pi$
$822$		0			0
$823$		0			0		−0.995562	−	0.0941083i	$-0.970000\pi$
$823$		0			0		0.995562	+	0.0941083i	$0.0300000\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.917755	−	0.397148i	$-0.870000\pi$
$827$		0			0		0.917755	+	0.397148i	$0.130000\pi$
$828$		0			0
$829$	−0.836475	−	0.159566i	−0.836475	−	0.159566i	−0.248690	−	0.968583i	$-0.580000\pi$
$829$	−0.836475	−	0.159566i	−0.836475	−	0.159566i	−0.587785	+	0.809017i	$0.700000\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	1.03171	−	1.51811i	1.03171	−	1.51811i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.637424	−	0.770513i	$-0.720000\pi$
$839$		0			0		0.637424	+	0.770513i	$0.280000\pi$
$840$		0			0
$841$	−1.79339	+	0.460464i	−1.79339	+	0.460464i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	0.289896	−	0.456804i	0.289896	−	0.456804i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	0.440892	+	1.51756i	0.440892	+	1.51756i	0.809017	+	0.587785i	$0.200000\pi$
$853$	0.440892	+	1.51756i	0.440892	+	1.51756i	−0.368125	+	0.929776i	$0.620000\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1.76007	+	0.278768i	−1.76007	+	0.278768i	−0.951057	−	0.309017i	$-0.900000\pi$
$857$	−1.76007	+	0.278768i	−1.76007	+	0.278768i	−0.809017	+	0.587785i	$0.800000\pi$
$858$		0			0
$859$		0			0		−0.535827	−	0.844328i	$-0.680000\pi$
$859$		0			0		0.535827	+	0.844328i	$0.320000\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.0941083	−	0.995562i	$-0.530000\pi$
$863$		0			0		0.0941083	+	0.995562i	$0.470000\pi$
$864$		0			0
$865$	−0.183684	−	0.0410582i	−0.183684	−	0.0410582i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	1.01356	+	1.71384i	1.01356	+	1.71384i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−0.0540731	+	0.0319788i	−0.0540731	+	0.0319788i	−0.535827	−	0.844328i	$-0.680000\pi$
$877$	−0.0540731	+	0.0319788i	−0.0540731	+	0.0319788i	0.481754	+	0.876307i	$0.340000\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	0.500534	−	1.06369i	0.500534	−	1.06369i	−0.481754	−	0.876307i	$-0.660000\pi$
$881$	0.500534	−	1.06369i	0.500534	−	1.06369i	0.982287	−	0.187381i	$-0.0600000\pi$
$882$		0			0
$883$		0			0		0.999507	−	0.0314108i	$-0.0100000\pi$
$883$		0			0		−0.999507	+	0.0314108i	$0.990000\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.995562	−	0.0941083i	$-0.0300000\pi$
$887$		0			0		−0.995562	+	0.0941083i	$0.970000\pi$
$888$		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
$901$	1.82028	+	1.32251i	1.82028	+	1.32251i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−1.93334	+	0.121636i	−1.93334	+	0.121636i
$906$		0			0
$907$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$907$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$908$		0			0
$909$	−0.317042	−	1.23480i	−0.317042	−	1.23480i
$910$		0			0
$911$		0			0		0.770513	−	0.637424i	$-0.220000\pi$
$911$		0			0		−0.770513	+	0.637424i	$0.780000\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		−0.728969	−	0.684547i	$-0.760000\pi$
$919$		0			0		0.728969	+	0.684547i	$0.240000\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	1.48175	−	0.234686i	1.48175	−	0.234686i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	0.872693	+	0.929324i	0.872693	+	0.929324i	0.998027	−	0.0627905i	$-0.0200000\pi$
$929$	0.872693	+	0.929324i	0.872693	+	0.929324i	−0.125333	+	0.992115i	$0.540000\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	0.673270	−	1.13844i	0.673270	−	1.13844i	−0.309017	−	0.951057i	$-0.600000\pi$
$937$	0.673270	−	1.13844i	0.673270	−	1.13844i	0.982287	−	0.187381i	$-0.0600000\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−0.683098	+	0.825723i	−0.683098	+	0.825723i	−0.992115	−	0.125333i	$-0.960000\pi$
$941$	−0.683098	+	0.825723i	−0.683098	+	0.825723i	0.309017	+	0.951057i	$0.400000\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.338738	−	0.940881i	$-0.390000\pi$
$947$		0			0		−0.338738	+	0.940881i	$0.610000\pi$
$948$		0			0
$949$	0.0404770	−	0.0131518i	0.0404770	−	0.0131518i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	0.793904	+	0.0249494i	0.793904	+	0.0249494i	0.425779	−	0.904827i	$-0.360000\pi$
$953$	0.793904	+	0.0249494i	0.793904	+	0.0249494i	0.368125	+	0.929776i	$0.380000\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−0.535827	+	0.844328i	−0.535827	+	0.844328i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	1.36812	+	0.929776i	1.36812	+	0.929776i
$966$		0			0
$967$		0			0		0.940881	−	0.338738i	$-0.110000\pi$
$967$		0			0		−0.940881	+	0.338738i	$0.890000\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.125333	−	0.992115i	$-0.540000\pi$
$971$		0			0		0.125333	+	0.992115i	$0.460000\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1.49948	+	1.32197i	1.49948	+	1.32197i	0.770513	+	0.637424i	$0.220000\pi$
$977$	1.49948	+	1.32197i	1.49948	+	1.32197i	0.728969	+	0.684547i	$0.240000\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−0.273190	−	1.43211i	−0.273190	−	1.43211i
$982$		0			0
$983$		0			0		0.278991	−	0.960294i	$-0.410000\pi$
$983$		0			0		−0.278991	+	0.960294i	$0.590000\pi$
$984$		0			0
$985$	−1.89694	−	0.300446i	−1.89694	−	0.300446i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		0.368125	−	0.929776i	$-0.380000\pi$
$991$		0			0		−0.368125	+	0.929776i	$0.620000\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	0.510361	+	0.750973i	0.510361	+	0.750973i	0.992115	−	0.125333i	$-0.0400000\pi$
$997$	0.510361	+	0.750973i	0.510361	+	0.750973i	−0.481754	+	0.876307i	$0.660000\pi$
$998$		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.33.1	✓	40
4.3	odd	2	CM	4000.1.cq.a.33.1	✓	40
125.72	odd	100	inner	4000.1.cq.a.1697.1	yes	40
500.447	even	100	inner	4000.1.cq.a.1697.1	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4000.1.cq.a.33.1	✓	40	1.1	even	1	trivial
4000.1.cq.a.33.1	✓	40	4.3	odd	2	CM
4000.1.cq.a.1697.1	yes	40	125.72	odd	100	inner
4000.1.cq.a.1697.1	yes	40	500.447	even	100	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:	\( N \)	\(=\)	\( 4000 = 2^{5} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	4000.cq (of order \(100\), degree \(40\), minimal)

\(f(q)\)	\(=\)	\(q+(0.982287 - 0.187381i) q^{5} +(-0.368125 + 0.929776i) q^{9} +O(q^{10})\)	\(q+(0.982287 - 0.187381i) q^{5} +(-0.368125 + 0.929776i) q^{9} +(0.269058 + 0.621757i) q^{13} +(0.512091 - 1.76263i) q^{17} +(0.929776 - 0.368125i) q^{25} +(-0.211645 - 1.67534i) q^{29} +(1.46409 + 0.327263i) q^{37} +(-0.0604991 + 0.961606i) q^{41} +(-0.187381 + 0.982287i) q^{45} +(0.951057 + 0.309017i) q^{49} +(-0.415230 + 1.15334i) q^{53} +(0.123357 + 1.96070i) q^{61} +(0.380798 + 0.560327i) q^{65} +(0.00591203 - 0.0625427i) q^{73} +(-0.728969 - 0.684547i) q^{81} +(0.172737 - 1.82736i) q^{85} +(0.383238 - 0.317042i) q^{89} +(1.22037 - 1.57330i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q+O(q^{10}) \) 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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