LMFDB - Embedded newform 975.2.o.e.476.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [975,2,Mod(476,975)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(975, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 0, 1]))

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

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

chi := DirichletCharacter("975.476");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 975 = 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	975.o (of order $4$, degree $2$, 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:	$7.78541419707$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label			476.2
Root			$0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	975.476
Dual form			975.2.o.e.551.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205 q^{3} +2.00000i q^{4} +(-3.36603 + 3.36603i) q^{7} +3.00000 q^{9} +3.46410i q^{12} +(2.59808 + 2.50000i) q^{13} -4.00000 q^{16} +(-3.83013 - 3.83013i) q^{19} +(-5.83013 + 5.83013i) q^{21} +5.19615 q^{27} +(-6.73205 - 6.73205i) q^{28} +(0.830127 + 0.830127i) q^{31} +6.00000i q^{36} +(-8.46410 + 8.46410i) q^{37} +(4.50000 + 4.33013i) q^{39} +12.1244i q^{43} -6.92820 q^{48} -15.6603i q^{49} +(-5.00000 + 5.19615i) q^{52} +(-6.63397 - 6.63397i) q^{57} +8.66025 q^{61} +(-10.0981 + 10.0981i) q^{63} -8.00000i q^{64} +(5.29423 + 5.29423i) q^{67} +(11.9282 - 11.9282i) q^{73} +(7.66025 - 7.66025i) q^{76} +17.3205 q^{79} +9.00000 q^{81} +(-11.6603 - 11.6603i) q^{84} +(-17.1603 + 0.330127i) q^{91} +(1.43782 + 1.43782i) q^{93} +(-7.02628 - 7.02628i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 10 q^{7} + 12 q^{9} - 16 q^{16} + 2 q^{19} - 6 q^{21} - 20 q^{28} - 14 q^{31} - 20 q^{37} + 18 q^{39} - 20 q^{52} - 30 q^{57} - 30 q^{63} - 10 q^{67} + 20 q^{73} - 4 q^{76} + 36 q^{81} - 12 q^{84}+ \cdots + 10 q^{97}+O(q^{100}) $ 4 * q - 10 * q^7 + 12 * q^9 - 16 * q^16 + 2 * q^19 - 6 * q^21 - 20 * q^28 - 14 * q^31 - 20 * q^37 + 18 * q^39 - 20 * q^52 - 30 * q^57 - 30 * q^63 - 10 * q^67 + 20 * q^73 - 4 * q^76 + 36 * q^81 - 12 * q^84 - 34 * q^91 + 30 * q^93 + 10 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/975\mathbb{Z}\right)^\times$.

$n$	$301$	$326$	$352$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$-1$	$1$

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		−0.707107	−	0.707107i	$-0.750000\pi$
$2$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$3$	1.73205			1.00000
$3$	1.73205			1.00000
$4$			2.00000i			1.00000i
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−3.36603	+	3.36603i	−1.27224	+	1.27224i	−0.327327	+	0.944911i	$0.606148\pi$
$7$	−3.36603	+	3.36603i	−1.27224	+	1.27224i	−0.944911	+	0.327327i	$0.893852\pi$
$8$		0			0
$9$	3.00000			1.00000
$10$		0			0
$11$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$11$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$12$			3.46410i			1.00000i
$13$	2.59808	+	2.50000i	0.720577	+	0.693375i
$13$	2.59808	+	2.50000i	0.720577	+	0.693375i
$14$		0			0
$15$		0			0
$16$	−4.00000			−1.00000
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$		0			0
$19$	−3.83013	−	3.83013i	−0.878691	−	0.878691i	0.114708	−	0.993399i	$-0.463407\pi$
$19$	−3.83013	−	3.83013i	−0.878691	−	0.878691i	−0.993399	+	0.114708i	$0.963407\pi$
$20$		0			0
$21$	−5.83013	+	5.83013i	−1.27224	+	1.27224i
$22$		0			0
$23$		0			0			−	1.00000i	$-0.5\pi$
$23$		0			0				1.00000i	$0.5\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	5.19615			1.00000
$28$	−6.73205	−	6.73205i	−1.27224	−	1.27224i
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	0.830127	+	0.830127i	0.149095	+	0.149095i	0.777714	−	0.628619i	$-0.216379\pi$
$31$	0.830127	+	0.830127i	0.149095	+	0.149095i	−0.628619	+	0.777714i	$0.716379\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$			6.00000i			1.00000i
$37$	−8.46410	+	8.46410i	−1.39149	+	1.39149i	−0.569495	+	0.821995i	$0.692861\pi$
$37$	−8.46410	+	8.46410i	−1.39149	+	1.39149i	−0.821995	+	0.569495i	$0.807139\pi$
$38$		0			0
$39$	4.50000	+	4.33013i	0.720577	+	0.693375i
$40$		0			0
$41$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$41$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$42$		0			0
$43$			12.1244i			1.84895i	0.381246	+	0.924473i	$0.375495\pi$
$43$			12.1244i			1.84895i	−0.381246	+	0.924473i	$0.624505\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$47$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$48$	−6.92820			−1.00000
$49$		−	15.6603i		−	2.23718i
$50$		0			0
$51$		0			0
$52$	−5.00000	+	5.19615i	−0.693375	+	0.720577i
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−6.63397	−	6.63397i	−0.878691	−	0.878691i
$58$		0			0
$59$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$59$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$60$		0			0
$61$	8.66025			1.10883			0.554416	−	0.832240i	$-0.312942\pi$
$61$	8.66025			1.10883			0.554416	+	0.832240i	$0.312942\pi$
$62$		0			0
$63$	−10.0981	+	10.0981i	−1.27224	+	1.27224i
$64$		−	8.00000i		−	1.00000i
$65$		0			0
$66$		0			0
$67$	5.29423	+	5.29423i	0.646793	+	0.646793i	0.952217	−	0.305424i	$-0.0987981\pi$
$67$	5.29423	+	5.29423i	0.646793	+	0.646793i	−0.305424	+	0.952217i	$0.598798\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$71$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$72$		0			0
$73$	11.9282	−	11.9282i	1.39609	−	1.39609i	0.585206	−	0.810885i	$-0.301014\pi$
$73$	11.9282	−	11.9282i	1.39609	−	1.39609i	0.810885	−	0.585206i	$-0.198986\pi$
$74$		0			0
$75$		0			0
$76$	7.66025	−	7.66025i	0.878691	−	0.878691i
$77$		0			0
$78$		0			0
$79$	17.3205			1.94871			0.974355	−	0.225018i	$-0.0722440\pi$
$79$	17.3205			1.94871			0.974355	+	0.225018i	$0.0722440\pi$
$80$		0			0
$81$	9.00000			1.00000
$82$		0			0
$83$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$83$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$84$	−11.6603	−	11.6603i	−1.27224	−	1.27224i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$89$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$90$		0			0
$91$	−17.1603	+	0.330127i	−1.79888	+	0.0346067i
$92$		0			0
$93$	1.43782	+	1.43782i	0.149095	+	0.149095i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−7.02628	−	7.02628i	−0.713411	−	0.713411i	0.253837	−	0.967247i	$-0.418307\pi$
$97$	−7.02628	−	7.02628i	−0.713411	−	0.713411i	−0.967247	+	0.253837i	$0.918307\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$		0			0
$103$		−	3.46410i		−	0.341328i	−0.985329	−	0.170664i	$-0.945409\pi$
$103$		−	3.46410i		−	0.341328i	0.985329	−	0.170664i	$-0.0545913\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$			10.3923i			1.00000i
$109$	13.8301	+	13.8301i	1.32469	+	1.32469i	0.909935	+	0.414751i	$0.136131\pi$
$109$	13.8301	+	13.8301i	1.32469	+	1.32469i	0.414751	+	0.909935i	$0.363869\pi$
$110$		0			0
$111$	−14.6603	+	14.6603i	−1.39149	+	1.39149i
$112$	13.4641	−	13.4641i	1.27224	−	1.27224i
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	7.79423	+	7.50000i	0.720577	+	0.693375i
$118$		0			0
$119$		0			0
$120$		0			0
$121$		−	11.0000i		−	1.00000i
$122$		0			0
$123$		0			0
$124$	−1.66025	+	1.66025i	−0.149095	+	0.149095i
$125$		0			0
$126$		0			0
$127$			20.0000i			1.77471i	0.461084	+	0.887357i	$0.347461\pi$
$127$			20.0000i			1.77471i	−0.461084	+	0.887357i	$0.652539\pi$
$128$		0			0
$129$			21.0000i			1.84895i
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$	25.7846			2.23581
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$137$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$138$		0			0
$139$	−16.0000			−1.35710			−0.678551	−	0.734553i	$-0.737392\pi$
$139$	−16.0000			−1.35710			−0.678551	+	0.734553i	$0.737392\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−12.0000			−1.00000
$145$		0			0
$146$		0			0
$147$		−	27.1244i		−	2.23718i
$148$	−16.9282	−	16.9282i	−1.39149	−	1.39149i
$149$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$149$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$150$		0			0
$151$	7.16987	−	7.16987i	0.583476	−	0.583476i	−0.352381	−	0.935857i	$-0.614628\pi$
$151$	7.16987	−	7.16987i	0.583476	−	0.583476i	0.935857	+	0.352381i	$0.114628\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−8.66025	+	9.00000i	−0.693375	+	0.720577i
$157$	25.0000			1.99522			0.997609	−	0.0691164i	$-0.0220180\pi$
$157$	25.0000			1.99522			0.997609	+	0.0691164i	$0.0220180\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−9.90192	+	9.90192i	−0.775579	+	0.775579i	−0.979076	−	0.203497i	$-0.934769\pi$
$163$	−9.90192	+	9.90192i	−0.775579	+	0.775579i	0.203497	+	0.979076i	$0.434769\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$167$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$168$		0			0
$169$	0.500000	+	12.9904i	0.0384615	+	0.999260i
$170$		0			0
$171$	−11.4904	−	11.4904i	−0.878691	−	0.878691i
$172$	−24.2487			−1.84895
$173$		0			0			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$		−	25.9808i		−	1.93113i	−0.260153	−	0.965567i	$-0.583773\pi$
$181$		−	25.9808i		−	1.93113i	0.260153	−	0.965567i	$-0.416227\pi$
$182$		0			0
$183$	15.0000			1.10883
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−17.4904	+	17.4904i	−1.27224	+	1.27224i
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$		−	13.8564i		−	1.00000i
$193$	18.5622	−	18.5622i	1.33613	−	1.33613i	0.436365	−	0.899770i	$-0.356266\pi$
$193$	18.5622	−	18.5622i	1.33613	−	1.33613i	0.899770	−	0.436365i	$-0.143734\pi$
$194$		0			0
$195$		0			0
$196$	31.3205			2.23718
$197$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$197$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$198$		0			0
$199$			11.0000i			0.779769i	0.920864	+	0.389885i	$0.127485\pi$
$199$			11.0000i			0.779769i	−0.920864	+	0.389885i	$0.872515\pi$
$200$		0			0
$201$	9.16987	+	9.16987i	0.646793	+	0.646793i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−10.3923	−	10.0000i	−0.720577	−	0.693375i
$209$		0			0
$210$		0			0
$211$	25.9808			1.78859			0.894295	−	0.447478i	$-0.147678\pi$
$211$	25.9808			1.78859			0.894295	+	0.447478i	$0.147678\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−5.58846			−0.379369
$218$		0			0
$219$	20.6603	−	20.6603i	1.39609	−	1.39609i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−17.2224	−	17.2224i	−1.15330	−	1.15330i	−0.985887	−	0.167412i	$-0.946459\pi$
$223$	−17.2224	−	17.2224i	−1.15330	−	1.15330i	−0.167412	−	0.985887i	$-0.553541\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$227$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$228$	13.2679	−	13.2679i	0.878691	−	0.878691i
$229$	−18.8301	+	18.8301i	−1.24433	+	1.24433i	−0.286143	+	0.958187i	$0.592373\pi$
$229$	−18.8301	+	18.8301i	−1.24433	+	1.24433i	−0.958187	+	0.286143i	$0.907627\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0			−	1.00000i	$-0.5\pi$
$233$		0			0				1.00000i	$0.5\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	30.0000			1.94871
$238$		0			0
$239$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$239$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$240$		0			0
$241$	−4.49038	+	4.49038i	−0.289251	+	0.289251i	−0.836784	−	0.547533i	$-0.815567\pi$
$241$	−4.49038	+	4.49038i	−0.289251	+	0.289251i	0.547533	+	0.836784i	$0.315567\pi$
$242$		0			0
$243$	15.5885			1.00000
$244$			17.3205i			1.10883i
$245$		0			0
$246$		0			0
$247$	−0.375644	−	19.5263i	−0.0239017	−	1.24243i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$	−20.1962	−	20.1962i	−1.27224	−	1.27224i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	16.0000			1.00000
$257$		0			0			−	1.00000i	$-0.5\pi$
$257$		0			0				1.00000i	$0.5\pi$
$258$		0			0
$259$		−	56.9808i		−	3.54061i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		1.00000			$0$
$263$		0			0		−1.00000			$\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−10.5885	+	10.5885i	−0.646793	+	0.646793i
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$	−5.33975	+	5.33975i	−0.324366	+	0.324366i	−0.850439	−	0.526073i	$-0.823664\pi$
$271$	−5.33975	+	5.33975i	−0.324366	+	0.324366i	0.526073	+	0.850439i	$0.323664\pi$
$272$		0			0
$273$	−29.7224	+	0.571797i	−1.79888	+	0.0346067i
$274$		0			0
$275$		0			0
$276$		0			0
$277$		−	32.9090i		−	1.97731i	−0.150210	−	0.988654i	$-0.547995\pi$
$277$		−	32.9090i		−	1.97731i	0.150210	−	0.988654i	$-0.452005\pi$
$278$		0			0
$279$	2.49038	+	2.49038i	0.149095	+	0.149095i
$280$		0			0
$281$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$281$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$282$		0			0
$283$			25.0000i			1.48610i	0.669238	+	0.743048i	$0.266621\pi$
$283$			25.0000i			1.48610i	−0.669238	+	0.743048i	$0.733379\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−17.0000			−1.00000
$290$		0			0
$291$	−12.1699	−	12.1699i	−0.713411	−	0.713411i
$292$	23.8564	+	23.8564i	1.39609	+	1.39609i
$293$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$293$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−40.8109	−	40.8109i	−2.35230	−	2.35230i
$302$		0			0
$303$		0			0
$304$	15.3205	+	15.3205i	0.878691	+	0.878691i
$305$		0			0
$306$		0			0
$307$	18.3660	−	18.3660i	1.04820	−	1.04820i	0.0494267	−	0.998778i	$-0.484261\pi$
$307$	18.3660	−	18.3660i	1.04820	−	1.04820i	0.998778	−	0.0494267i	$-0.0157394\pi$
$308$		0			0
$309$		−	6.00000i		−	0.341328i
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$	−5.19615			−0.293704			−0.146852	−	0.989158i	$-0.546914\pi$
$313$	−5.19615			−0.293704			−0.146852	+	0.989158i	$0.546914\pi$
$314$		0			0
$315$		0			0
$316$			34.6410i			1.94871i
$317$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$317$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$			18.0000i			1.00000i
$325$		0			0
$326$		0			0
$327$	23.9545	+	23.9545i	1.32469	+	1.32469i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−24.6603	−	24.6603i	−1.35545	−	1.35545i	−0.879440	−	0.476011i	$-0.842082\pi$
$331$	−24.6603	−	24.6603i	−1.35545	−	1.35545i	−0.476011	−	0.879440i	$-0.657918\pi$
$332$		0			0
$333$	−25.3923	+	25.3923i	−1.39149	+	1.39149i
$334$		0			0
$335$		0			0
$336$	23.3205	−	23.3205i	1.27224	−	1.27224i
$337$		−	5.00000i		−	0.272367i	−0.990684	−	0.136184i	$-0.956516\pi$
$337$		−	5.00000i		−	0.272367i	0.990684	−	0.136184i	$-0.0434837\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	29.1506	+	29.1506i	1.57399	+	1.57399i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		1.00000			$0$
$347$		0			0		−1.00000			$\pi$
$348$		0			0
$349$	10.3205	−	10.3205i	0.552444	−	0.552444i	−0.374701	−	0.927146i	$-0.622255\pi$
$349$	10.3205	−	10.3205i	0.552444	−	0.552444i	0.927146	+	0.374701i	$0.122255\pi$
$350$		0			0
$351$	13.5000	+	12.9904i	0.720577	+	0.693375i
$352$		0			0
$353$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$353$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$359$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$360$		0			0
$361$			10.3397i			0.544197i
$362$		0			0
$363$		−	19.0526i		−	1.00000i
$364$	−0.660254	−	34.3205i	−0.0346067	−	1.79888i
$365$		0			0
$366$		0			0
$367$	−35.0000			−1.82699			−0.913493	−	0.406855i	$-0.866625\pi$
$367$	−35.0000			−1.82699			−0.913493	+	0.406855i	$0.866625\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−2.87564	+	2.87564i	−0.149095	+	0.149095i
$373$	29.4449			1.52460			0.762299	−	0.647225i	$-0.224071\pi$
$373$	29.4449			1.52460			0.762299	+	0.647225i	$0.224071\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	1.50962	+	1.50962i	0.0775439	+	0.0775439i	0.744815	−	0.667271i	$-0.232538\pi$
$379$	1.50962	+	1.50962i	0.0775439	+	0.0775439i	−0.667271	+	0.744815i	$0.732538\pi$
$380$		0			0
$381$			34.6410i			1.77471i
$382$		0			0
$383$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$383$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$			36.3731i			1.84895i
$388$	14.0526	−	14.0526i	0.713411	−	0.713411i
$389$		0			0			−	1.00000i	$-0.5\pi$
$389$		0			0				1.00000i	$0.5\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−7.97372	+	7.97372i	−0.400190	+	0.400190i	−0.878300	−	0.478110i	$-0.841322\pi$
$397$	−7.97372	+	7.97372i	−0.400190	+	0.400190i	0.478110	+	0.878300i	$0.341322\pi$
$398$		0			0
$399$	44.6603			2.23581
$400$		0			0
$401$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$401$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$402$		0			0
$403$	0.0814157	+	4.23205i	0.00405561	+	0.210813i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	28.4904	+	28.4904i	1.40876	+	1.40876i	0.766426	+	0.642333i	$0.222033\pi$
$409$	28.4904	+	28.4904i	1.40876	+	1.40876i	0.642333	+	0.766426i	$0.277967\pi$
$410$		0			0
$411$		0			0
$412$	6.92820			0.341328
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−27.7128			−1.35710
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$	28.3205	+	28.3205i	1.38026	+	1.38026i	0.844150	+	0.536107i	$0.180106\pi$
$421$	28.3205	+	28.3205i	1.38026	+	1.38026i	0.536107	+	0.844150i	$0.319894\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−29.1506	+	29.1506i	−1.41070	+	1.41070i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$431$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$432$	−20.7846			−1.00000
$433$		−	35.0000i		−	1.68199i	−0.541041	−	0.840996i	$-0.681970\pi$
$433$		−	35.0000i		−	1.68199i	0.541041	−	0.840996i	$-0.318030\pi$
$434$		0			0
$435$		0			0
$436$	−27.6603	+	27.6603i	−1.32469	+	1.32469i
$437$		0			0
$438$		0			0
$439$		−	8.66025i		−	0.413331i	−0.978412	−	0.206666i	$-0.933739\pi$
$439$		−	8.66025i		−	0.413331i	0.978412	−	0.206666i	$-0.0662612\pi$
$440$		0			0
$441$		−	46.9808i		−	2.23718i
$442$		0			0
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$	−29.3205	−	29.3205i	−1.39149	−	1.39149i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	26.9282	+	26.9282i	1.27224	+	1.27224i
$449$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$449$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	12.4186	−	12.4186i	0.583476	−	0.583476i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	15.7846	+	15.7846i	0.738373	+	0.738373i	0.972263	−	0.233890i	$-0.0751456\pi$
$457$	15.7846	+	15.7846i	0.738373	+	0.738373i	−0.233890	+	0.972263i	$0.575146\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$461$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$462$		0			0
$463$	−9.05256	+	9.05256i	−0.420708	+	0.420708i	−0.885448	−	0.464739i	$-0.846148\pi$
$463$	−9.05256	+	9.05256i	−0.420708	+	0.420708i	0.464739	+	0.885448i	$0.346148\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$	−15.0000	+	15.5885i	−0.693375	+	0.720577i
$469$	−35.6410			−1.64575
$470$		0			0
$471$	43.3013			1.99522
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$479$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$480$		0			0
$481$	−43.1506	+	0.830127i	−1.96750	+	0.0378505i
$482$		0			0
$483$		0			0
$484$	22.0000			1.00000
$485$		0			0
$486$		0			0
$487$	5.68653	+	5.68653i	0.257681	+	0.257681i	0.824110	−	0.566429i	$-0.191675\pi$
$487$	5.68653	+	5.68653i	0.257681	+	0.257681i	−0.566429	+	0.824110i	$0.691675\pi$
$488$		0			0
$489$	−17.1506	+	17.1506i	−0.775579	+	0.775579i
$490$		0			0
$491$		0			0			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−3.32051	−	3.32051i	−0.149095	−	0.149095i
$497$		0			0
$498$		0			0
$499$	−16.1506	−	16.1506i	−0.723002	−	0.723002i	0.246214	−	0.969216i	$-0.420813\pi$
$499$	−16.1506	−	16.1506i	−0.723002	−	0.723002i	−0.969216	+	0.246214i	$0.920813\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	0.866025	+	22.5000i	0.0384615	+	0.999260i
$508$	−40.0000			−1.77471
$509$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$509$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$510$		0			0
$511$			80.3013i			3.55232i
$512$		0			0
$513$	−19.9019	−	19.9019i	−0.878691	−	0.878691i
$514$		0			0
$515$		0			0
$516$	−42.0000			−1.84895
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$	−35.0000			−1.53044			−0.765222	−	0.643767i	$-0.777371\pi$
$523$	−35.0000			−1.53044			−0.765222	+	0.643767i	$0.777371\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−23.0000			−1.00000
$530$		0			0
$531$		0			0
$532$			51.5692i			2.23581i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−30.1506	+	30.1506i	−1.29628	+	1.29628i	−0.365444	+	0.930834i	$0.619083\pi$
$541$	−30.1506	+	30.1506i	−1.29628	+	1.29628i	−0.930834	+	0.365444i	$0.880917\pi$
$542$		0			0
$543$		−	45.0000i		−	1.93113i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−40.0000			−1.71028			−0.855138	−	0.518400i	$-0.826528\pi$
$547$	−40.0000			−1.71028			−0.855138	+	0.518400i	$0.826528\pi$
$548$		0			0
$549$	25.9808			1.10883
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−58.3013	+	58.3013i	−2.47922	+	2.47922i
$554$		0			0
$555$		0			0
$556$		−	32.0000i		−	1.35710i
$557$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$557$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$558$		0			0
$559$	−30.3109	+	31.5000i	−1.28201	+	1.33231i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0			−	1.00000i	$-0.5\pi$
$563$		0			0				1.00000i	$0.5\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−30.2942	+	30.2942i	−1.27224	+	1.27224i
$568$		0			0
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$		0			0
$571$			47.0000i			1.96689i	0.181210	+	0.983444i	$0.441999\pi$
$571$			47.0000i			1.96689i	−0.181210	+	0.983444i	$0.558001\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		−	24.0000i		−	1.00000i
$577$	1.04552	+	1.04552i	0.0435255	+	0.0435255i	0.728535	−	0.685009i	$-0.240202\pi$
$577$	1.04552	+	1.04552i	0.0435255	+	0.0435255i	−0.685009	+	0.728535i	$0.740202\pi$
$578$		0			0
$579$	32.1506	−	32.1506i	1.33613	−	1.33613i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$587$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$588$	54.2487			2.23718
$589$		−	6.35898i		−	0.262017i
$590$		0			0
$591$		0			0
$592$	33.8564	−	33.8564i	1.39149	−	1.39149i
$593$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$593$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$			19.0526i			0.779769i
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$		0			0
$601$	43.3013			1.76630			0.883148	−	0.469095i	$-0.155420\pi$
$601$	43.3013			1.76630			0.883148	+	0.469095i	$0.155420\pi$
$602$		0			0
$603$	15.8827	+	15.8827i	0.646793	+	0.646793i
$604$	14.3397	+	14.3397i	0.583476	+	0.583476i
$605$		0			0
$606$		0			0
$607$	−20.0000			−0.811775			−0.405887	−	0.913923i	$-0.633038\pi$
$607$	−20.0000			−0.811775			−0.405887	+	0.913923i	$0.633038\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−29.2487	−	29.2487i	−1.18134	−	1.18134i	−0.979396	−	0.201948i	$-0.935273\pi$
$613$	−29.2487	−	29.2487i	−1.18134	−	1.18134i	−0.201948	−	0.979396i	$-0.564727\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$617$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$618$		0			0
$619$	20.1699	−	20.1699i	0.810696	−	0.810696i	−0.174042	−	0.984738i	$-0.555683\pi$
$619$	20.1699	−	20.1699i	0.810696	−	0.810696i	0.984738	+	0.174042i	$0.0556830\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	−18.0000	−	17.3205i	−0.720577	−	0.693375i
$625$		0			0
$626$		0			0
$627$		0			0
$628$			50.0000i			1.99522i
$629$		0			0
$630$		0			0
$631$	8.50962	−	8.50962i	0.338763	−	0.338763i	−0.517139	−	0.855901i	$-0.673003\pi$
$631$	8.50962	−	8.50962i	0.338763	−	0.338763i	0.855901	+	0.517139i	$0.173003\pi$
$632$		0			0
$633$	45.0000			1.78859
$634$		0			0
$635$		0			0
$636$		0			0
$637$	39.1506	−	40.6865i	1.55120	−	1.61206i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0			−	1.00000i	$-0.5\pi$
$641$		0			0				1.00000i	$0.5\pi$
$642$		0			0
$643$	4.41154	+	4.41154i	0.173974	+	0.173974i	0.788723	−	0.614749i	$-0.210743\pi$
$643$	4.41154	+	4.41154i	0.173974	+	0.173974i	−0.614749	+	0.788723i	$0.710743\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−9.67949			−0.379369
$652$	−19.8038	−	19.8038i	−0.775579	−	0.775579i
$653$		0			0		1.00000			$0$
$653$		0			0		−1.00000			$\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	35.7846	−	35.7846i	1.39609	−	1.39609i
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$	36.3205	−	36.3205i	1.41270	−	1.41270i	0.673690	−	0.739014i	$-0.264708\pi$
$661$	36.3205	−	36.3205i	1.41270	−	1.41270i	0.739014	−	0.673690i	$-0.235292\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−29.8301	−	29.8301i	−1.15330	−	1.15330i
$670$		0			0
$671$		0			0
$672$		0			0
$673$			13.8564i			0.534125i	0.963679	+	0.267063i	$0.0860531\pi$
$673$			13.8564i			0.534125i	−0.963679	+	0.267063i	$0.913947\pi$
$674$		0			0
$675$		0			0
$676$	−25.9808	+	1.00000i	−0.999260	+	0.0384615i
$677$		0			0		1.00000			$0$
$677$		0			0		−1.00000			$\pi$
$678$		0			0
$679$	47.3013			1.81526
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$683$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$684$	22.9808	−	22.9808i	0.878691	−	0.878691i
$685$		0			0
$686$		0			0
$687$	−32.6147	+	32.6147i	−1.24433	+	1.24433i
$688$		−	48.4974i		−	1.84895i
$689$		0			0
$690$		0			0
$691$	−21.9808	−	21.9808i	−0.836188	−	0.836188i	0.152167	−	0.988355i	$-0.451375\pi$
$691$	−21.9808	−	21.9808i	−0.836188	−	0.836188i	−0.988355	+	0.152167i	$0.951375\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$	64.8372			2.44538
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	6.15064	−	6.15064i	0.230992	−	0.230992i	−0.582115	−	0.813107i	$-0.697775\pi$
$709$	6.15064	−	6.15064i	0.230992	−	0.230992i	0.813107	+	0.582115i	$0.197775\pi$
$710$		0			0
$711$	51.9615			1.94871
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$	11.6603	+	11.6603i	0.434251	+	0.434251i
$722$		0			0
$723$	−7.77757	+	7.77757i	−0.289251	+	0.289251i
$724$	51.9615			1.93113
$725$		0			0
$726$		0			0
$727$		−	5.00000i		−	0.185440i	−0.995692	−	0.0927199i	$-0.970444\pi$
$727$		−	5.00000i		−	0.185440i	0.995692	−	0.0927199i	$-0.0295561\pi$
$728$		0			0
$729$	27.0000			1.00000
$730$		0			0
$731$		0			0
$732$			30.0000i			1.10883i
$733$	14.6077	+	14.6077i	0.539548	+	0.539548i	0.923396	−	0.383849i	$-0.125402\pi$
$733$	14.6077	+	14.6077i	0.539548	+	0.539548i	−0.383849	+	0.923396i	$0.625402\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−17.9808	+	17.9808i	−0.661433	+	0.661433i	−0.955718	−	0.294285i	$-0.904919\pi$
$739$	−17.9808	+	17.9808i	−0.661433	+	0.661433i	0.294285	+	0.955718i	$0.404919\pi$
$740$		0			0
$741$	−0.650635	−	33.8205i	−0.0239017	−	1.24243i
$742$		0			0
$743$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$743$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$			17.3205i			0.632034i	0.948753	+	0.316017i	$0.102346\pi$
$751$			17.3205i			0.632034i	−0.948753	+	0.316017i	$0.897654\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	−34.9808	−	34.9808i	−1.27224	−	1.27224i
$757$	−1.73205			−0.0629525			−0.0314762	−	0.999505i	$-0.510021\pi$
$757$	−1.73205			−0.0629525			−0.0314762	+	0.999505i	$0.510021\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$761$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$762$		0			0
$763$	−93.1051			−3.37063
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	27.7128			1.00000
$769$	−37.4904	−	37.4904i	−1.35194	−	1.35194i	−0.883493	−	0.468445i	$-0.844814\pi$
$769$	−37.4904	−	37.4904i	−1.35194	−	1.35194i	−0.468445	−	0.883493i	$-0.655186\pi$
$770$		0			0
$771$		0			0
$772$	37.1244	+	37.1244i	1.33613	+	1.33613i
$773$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$773$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		−	98.6936i		−	3.54061i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$			62.6410i			2.23718i
$785$		0			0
$786$		0			0
$787$	12.6147	−	12.6147i	0.449667	−	0.449667i	−0.445577	−	0.895244i	$-0.647001\pi$
$787$	12.6147	−	12.6147i	0.449667	−	0.449667i	0.895244	+	0.445577i	$0.147001\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	22.5000	+	21.6506i	0.798998	+	0.768837i
$794$		0			0
$795$		0			0
$796$	−22.0000			−0.779769
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	−18.3397	+	18.3397i	−0.646793	+	0.646793i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$		0			0
$811$	−3.15064	−	3.15064i	−0.110634	−	0.110634i	0.649623	−	0.760257i	$-0.274927\pi$
$811$	−3.15064	−	3.15064i	−0.110634	−	0.110634i	−0.760257	+	0.649623i	$0.774927\pi$
$812$		0			0
$813$	−9.24871	+	9.24871i	−0.324366	+	0.324366i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	46.4378	−	46.4378i	1.62465	−	1.62465i
$818$		0			0
$819$	−51.4808	+	0.990381i	−1.79888	+	0.0346067i
$820$		0			0
$821$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$821$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$822$		0			0
$823$			57.1577i			1.99239i	0.0871445	+	0.996196i	$0.472226\pi$
$823$			57.1577i			1.99239i	−0.0871445	+	0.996196i	$0.527774\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$827$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$828$		0			0
$829$			46.0000i			1.59765i	0.601566	+	0.798823i	$0.294544\pi$
$829$			46.0000i			1.59765i	−0.601566	+	0.798823i	$0.705456\pi$
$830$		0			0
$831$		−	57.0000i		−	1.97731i
$832$	20.0000	−	20.7846i	0.693375	−	0.720577i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	4.31347	+	4.31347i	0.149095	+	0.149095i
$838$		0			0
$839$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$839$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$840$		0			0
$841$	29.0000			1.00000
$842$		0			0
$843$		0			0
$844$			51.9615i			1.78859i
$845$		0			0
$846$		0			0
$847$	37.0263	+	37.0263i	1.27224	+	1.27224i
$848$		0			0
$849$			43.3013i			1.48610i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	40.8827	−	40.8827i	1.39980	−	1.39980i	0.599189	−	0.800608i	$-0.295490\pi$
$853$	40.8827	−	40.8827i	1.39980	−	1.39980i	0.800608	−	0.599189i	$-0.204510\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0			−	1.00000i	$-0.5\pi$
$857$		0			0				1.00000i	$0.5\pi$
$858$		0			0
$859$	17.3205			0.590968			0.295484	−	0.955348i	$-0.404519\pi$
$859$	17.3205			0.590968			0.295484	+	0.955348i	$0.404519\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$863$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−29.4449			−1.00000
$868$		−	11.1769i		−	0.379369i
$869$		0			0
$870$		0			0
$871$	0.519238	+	26.9904i	0.0175937	+	0.914534i
$872$		0			0
$873$	−21.0788	−	21.0788i	−0.713411	−	0.713411i
$874$		0			0
$875$		0			0
$876$	41.3205	+	41.3205i	1.39609	+	1.39609i
$877$	−39.3468	−	39.3468i	−1.32865	−	1.32865i	−0.906552	−	0.422095i	$-0.861295\pi$
$877$	−39.3468	−	39.3468i	−1.32865	−	1.32865i	−0.422095	−	0.906552i	$-0.638705\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$		−	55.0000i		−	1.85090i	−0.378873	−	0.925449i	$-0.623688\pi$
$883$		−	55.0000i		−	1.85090i	0.378873	−	0.925449i	$-0.376312\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		1.00000			$0$
$887$		0			0		−1.00000			$\pi$
$888$		0			0
$889$	−67.3205	−	67.3205i	−2.25786	−	2.25786i
$890$		0			0
$891$		0			0
$892$	34.4449	−	34.4449i	1.15330	−	1.15330i
$893$		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			0
$902$		0			0
$903$	−70.6865	−	70.6865i	−2.35230	−	2.35230i
$904$		0			0
$905$		0			0
$906$		0			0
$907$		−	40.0000i		−	1.32818i	−0.747653	−	0.664089i	$-0.768820\pi$
$907$		−	40.0000i		−	1.32818i	0.747653	−	0.664089i	$-0.231180\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$	26.5359	+	26.5359i	0.878691	+	0.878691i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−37.6603	−	37.6603i	−1.24433	−	1.24433i
$917$		0			0
$918$		0			0
$919$	−60.6218			−1.99973			−0.999864	−	0.0164935i	$-0.994750\pi$
$919$	−60.6218			−1.99973			−0.999864	+	0.0164935i	$0.994750\pi$
$920$		0			0
$921$	31.8109	−	31.8109i	1.04820	−	1.04820i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		−	10.3923i		−	0.341328i
$928$		0			0
$929$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$929$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$930$		0			0
$931$	−59.9808	+	59.9808i	−1.96579	+	1.96579i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−50.2295			−1.64093			−0.820463	−	0.571700i	$-0.806284\pi$
$937$	−50.2295			−1.64093			−0.820463	+	0.571700i	$0.806284\pi$
$938$		0			0
$939$	−9.00000			−0.293704
$940$		0			0
$941$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$941$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$947$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$948$			60.0000i			1.94871i
$949$	60.8109	−	1.16987i	1.97400	−	0.0379757i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$		−	29.6218i		−	0.955541i
$962$		0			0
$963$		0			0
$964$	−8.98076	−	8.98076i	−0.289251	−	0.289251i
$965$		0			0
$966$		0			0
$967$	−39.4449	−	39.4449i	−1.26846	−	1.26846i	−0.946883	−	0.321578i	$-0.895787\pi$
$967$	−39.4449	−	39.4449i	−1.26846	−	1.26846i	−0.321578	−	0.946883i	$-0.604213\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$			31.1769i			1.00000i
$973$	53.8564	−	53.8564i	1.72656	−	1.72656i
$974$		0			0
$975$		0			0
$976$	−34.6410			−1.10883
$977$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$977$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	41.4904	+	41.4904i	1.32469	+	1.32469i
$982$		0			0
$983$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$983$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	39.0526	−	0.751289i	1.24243	−	0.0239017i
$989$		0			0
$990$		0			0
$991$	17.0000			0.540023			0.270011	−	0.962857i	$-0.412973\pi$
$991$	17.0000			0.540023			0.270011	+	0.962857i	$0.412973\pi$
$992$		0			0
$993$	−42.7128	−	42.7128i	−1.35545	−	1.35545i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	10.0000			0.316703			0.158352	−	0.987383i	$-0.449382\pi$
$997$	10.0000			0.316703			0.158352	+	0.987383i	$0.449382\pi$
$998$		0			0
$999$	−43.9808	+	43.9808i	−1.39149	+	1.39149i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	975.2.o.e.476.2	✓	4
3.2	odd	2	CM	975.2.o.e.476.2	✓	4
5.2	odd	4		975.2.n.e.749.1		4
5.3	odd	4		975.2.n.j.749.2		4
5.4	even	2		975.2.o.h.476.1	yes	4
13.5	odd	4	inner	975.2.o.e.551.2	yes	4
15.2	even	4		975.2.n.e.749.1		4
15.8	even	4		975.2.n.j.749.2		4
15.14	odd	2		975.2.o.h.476.1	yes	4
39.5	even	4	inner	975.2.o.e.551.2	yes	4
65.18	even	4		975.2.n.e.824.2		4
65.44	odd	4		975.2.o.h.551.1	yes	4
65.57	even	4		975.2.n.j.824.1		4
195.44	even	4		975.2.o.h.551.1	yes	4
195.83	odd	4		975.2.n.e.824.2		4
195.122	odd	4		975.2.n.j.824.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
975.2.n.e.749.1		4	5.2	odd	4
975.2.n.e.749.1		4	15.2	even	4
975.2.n.e.824.2		4	65.18	even	4
975.2.n.e.824.2		4	195.83	odd	4
975.2.n.j.749.2		4	5.3	odd	4
975.2.n.j.749.2		4	15.8	even	4
975.2.n.j.824.1		4	65.57	even	4
975.2.n.j.824.1		4	195.122	odd	4
975.2.o.e.476.2	✓	4	1.1	even	1	trivial
975.2.o.e.476.2	✓	4	3.2	odd	2	CM
975.2.o.e.551.2	yes	4	13.5	odd	4	inner
975.2.o.e.551.2	yes	4	39.5	even	4	inner
975.2.o.h.476.1	yes	4	5.4	even	2
975.2.o.h.476.1	yes	4	15.14	odd	2
975.2.o.h.551.1	yes	4	65.44	odd	4
975.2.o.h.551.1	yes	4	195.44	even	4

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:	\( N \)	\(=\)	\( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	975.o (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+1.73205 q^{3} +2.00000i q^{4} +(-3.36603 + 3.36603i) q^{7} +3.00000 q^{9} +3.46410i q^{12} +(2.59808 + 2.50000i) q^{13} -4.00000 q^{16} +(-3.83013 - 3.83013i) q^{19} +(-5.83013 + 5.83013i) q^{21} +5.19615 q^{27} +(-6.73205 - 6.73205i) q^{28} +(0.830127 + 0.830127i) q^{31} +6.00000i q^{36} +(-8.46410 + 8.46410i) q^{37} +(4.50000 + 4.33013i) q^{39} +12.1244i q^{43} -6.92820 q^{48} -15.6603i q^{49} +(-5.00000 + 5.19615i) q^{52} +(-6.63397 - 6.63397i) q^{57} +8.66025 q^{61} +(-10.0981 + 10.0981i) q^{63} -8.00000i q^{64} +(5.29423 + 5.29423i) q^{67} +(11.9282 - 11.9282i) q^{73} +(7.66025 - 7.66025i) q^{76} +17.3205 q^{79} +9.00000 q^{81} +(-11.6603 - 11.6603i) q^{84} +(-17.1603 + 0.330127i) q^{91} +(1.43782 + 1.43782i) q^{93} +(-7.02628 - 7.02628i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 10 q^{7} + 12 q^{9} - 16 q^{16} + 2 q^{19} - 6 q^{21} - 20 q^{28} - 14 q^{31} - 20 q^{37} + 18 q^{39} - 20 q^{52} - 30 q^{57} - 30 q^{63} - 10 q^{67} + 20 q^{73} - 4 q^{76} + 36 q^{81} - 12 q^{84}+ \cdots + 10 q^{97}+O(q^{100}) \) 4 * q - 10 * q^7 + 12 * q^9 - 16 * q^16 + 2 * q^19 - 6 * q^21 - 20 * q^28 - 14 * q^31 - 20 * q^37 + 18 * q^39 - 20 * q^52 - 30 * q^57 - 30 * q^63 - 10 * q^67 + 20 * q^73 - 4 * q^76 + 36 * q^81 - 12 * q^84 - 34 * q^91 + 30 * q^93 + 10 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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