LMFDB - Embedded newform 600.2.b.b.251.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [600,2,Mod(251,600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(600, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("600.251");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 600 = 2^{3} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	600.b (of order $2$, degree $1$, 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:	$4.79102412128$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{2} + 4 $ x^4 - 2*x^2 + 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			251.4
Root			$-1.22474 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	600.251
Dual form			600.2.b.b.251.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421i q^{2} +(0.724745 + 1.57313i) q^{3} -2.00000 q^{4} +(-2.22474 + 1.02494i) q^{6} -2.82843i q^{8} +(-1.94949 + 2.28024i) q^{9} +O(q^{10})$	$q+1.41421i q^{2} +(0.724745 + 1.57313i) q^{3} -2.00000 q^{4} +(-2.22474 + 1.02494i) q^{6} -2.82843i q^{8} +(-1.94949 + 2.28024i) q^{9} +6.61037i q^{11} +(-1.44949 - 3.14626i) q^{12} +4.00000 q^{16} -2.36773i q^{17} +(-3.22474 - 2.75699i) q^{18} -8.34847 q^{19} -9.34847 q^{22} +(4.44949 - 2.04989i) q^{24} +(-5.00000 - 1.41421i) q^{27} +5.65685i q^{32} +(-10.3990 + 4.79083i) q^{33} +3.34847 q^{34} +(3.89898 - 4.56048i) q^{36} -11.8065i q^{38} +0.460702i q^{41} +10.0000 q^{43} -13.2207i q^{44} +(2.89898 + 6.29253i) q^{48} +7.00000 q^{49} +(3.72474 - 1.71600i) q^{51} +(2.00000 - 7.07107i) q^{54} +(-6.05051 - 13.1332i) q^{57} +14.1421i q^{59} -8.00000 q^{64} +(-6.77526 - 14.7064i) q^{66} +14.3485 q^{67} +4.73545i q^{68} +(6.44949 + 5.51399i) q^{72} -13.6969 q^{73} +16.6969 q^{76} +(-1.39898 - 8.89060i) q^{81} -0.651531 q^{82} +14.1742i q^{83} +14.1421i q^{86} +18.6969 q^{88} +12.7600i q^{89} +(-8.89898 + 4.09978i) q^{96} +10.0000 q^{97} +9.89949i q^{98} +(-15.0732 - 12.8868i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{3} - 8 q^{4} - 4 q^{6} + 2 q^{9}+O(q^{10}) $ 4 * q - 2 * q^3 - 8 * q^4 - 4 * q^6 + 2 * q^9	$ 4 q - 2 q^{3} - 8 q^{4} - 4 q^{6} + 2 q^{9} + 4 q^{12} + 16 q^{16} - 8 q^{18} - 4 q^{19} - 8 q^{22} + 8 q^{24} - 20 q^{27} - 22 q^{33} - 16 q^{34} - 4 q^{36} + 40 q^{43} - 8 q^{48} + 28 q^{49} + 10 q^{51} + 8 q^{54} - 34 q^{57} - 32 q^{64} - 32 q^{66} + 28 q^{67} + 16 q^{72} + 4 q^{73} + 8 q^{76} + 14 q^{81} - 32 q^{82} + 16 q^{88} - 16 q^{96} + 40 q^{97} - 26 q^{99}+O(q^{100}) $ 4 * q - 2 * q^3 - 8 * q^4 - 4 * q^6 + 2 * q^9 + 4 * q^12 + 16 * q^16 - 8 * q^18 - 4 * q^19 - 8 * q^22 + 8 * q^24 - 20 * q^27 - 22 * q^33 - 16 * q^34 - 4 * q^36 + 40 * q^43 - 8 * q^48 + 28 * q^49 + 10 * q^51 + 8 * q^54 - 34 * q^57 - 32 * q^64 - 32 * q^66 + 28 * q^67 + 16 * q^72 + 4 * q^73 + 8 * q^76 + 14 * q^81 - 32 * q^82 + 16 * q^88 - 16 * q^96 + 40 * q^97 - 26 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/600\mathbb{Z}\right)^\times$.

$n$	$151$	$301$	$401$	$577$
$\chi(n)$	$-1$	$-1$	$-1$	$1$

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 $

$2$			1.41421i			1.00000i
$2$			1.41421i			1.00000i
$3$	0.724745	+	1.57313i	0.418432	+	0.908248i
$3$	0.724745	+	1.57313i	0.418432	+	0.908248i
$4$	−2.00000			−1.00000
$5$		0			0
$5$		0			0
$6$	−2.22474	+	1.02494i	−0.908248	+	0.418432i
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$		−	2.82843i		−	1.00000i
$9$	−1.94949	+	2.28024i	−0.649830	+	0.760080i
$10$		0			0
$11$			6.61037i			1.99310i	0.0829925	+	0.996550i	$0.473552\pi$
$11$			6.61037i			1.99310i	−0.0829925	+	0.996550i	$0.526448\pi$
$12$	−1.44949	−	3.14626i	−0.418432	−	0.908248i
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$		0			0
$15$		0			0
$16$	4.00000			1.00000
$17$		−	2.36773i		−	0.574258i	−0.957892	−	0.287129i	$-0.907299\pi$
$17$		−	2.36773i		−	0.574258i	0.957892	−	0.287129i	$-0.0927008\pi$
$18$	−3.22474	−	2.75699i	−0.760080	−	0.649830i
$19$	−8.34847			−1.91527			−0.957635	−	0.287984i	$-0.907015\pi$
$19$	−8.34847			−1.91527			−0.957635	+	0.287984i	$0.907015\pi$
$20$		0			0
$21$		0			0
$22$	−9.34847			−1.99310
$23$		0			0			−	1.00000i	$-0.5\pi$
$23$		0			0				1.00000i	$0.5\pi$
$24$	4.44949	−	2.04989i	0.908248	−	0.418432i
$25$		0			0
$26$		0			0
$27$	−5.00000	−	1.41421i	−0.962250	−	0.272166i
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$			5.65685i			1.00000i
$33$	−10.3990	+	4.79083i	−1.81023	+	0.833976i
$34$	3.34847			0.574258
$35$		0			0
$36$	3.89898	−	4.56048i	0.649830	−	0.760080i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		−	11.8065i		−	1.91527i
$39$		0			0
$40$		0			0
$41$			0.460702i			0.0719495i	0.999353	+	0.0359748i	$0.0114536\pi$
$41$			0.460702i			0.0719495i	−0.999353	+	0.0359748i	$0.988546\pi$
$42$		0			0
$43$	10.0000			1.52499			0.762493	−	0.646997i	$-0.223975\pi$
$43$	10.0000			1.52499			0.762493	+	0.646997i	$0.223975\pi$
$44$		−	13.2207i		−	1.99310i
$45$		0			0
$46$		0			0
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$	2.89898	+	6.29253i	0.418432	+	0.908248i
$49$	7.00000			1.00000
$50$		0			0
$51$	3.72474	−	1.71600i	0.521569	−	0.240288i
$52$		0			0
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$	2.00000	−	7.07107i	0.272166	−	0.962250i
$55$		0			0
$56$		0			0
$57$	−6.05051	−	13.1332i	−0.801410	−	1.73954i
$58$		0			0
$59$			14.1421i			1.84115i	0.390567	+	0.920575i	$0.372279\pi$
$59$			14.1421i			1.84115i	−0.390567	+	0.920575i	$0.627721\pi$
$60$		0			0
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$		0			0
$63$		0			0
$64$	−8.00000			−1.00000
$65$		0			0
$66$	−6.77526	−	14.7064i	−0.833976	−	1.81023i
$67$	14.3485			1.75294			0.876472	−	0.481452i	$-0.159891\pi$
$67$	14.3485			1.75294			0.876472	+	0.481452i	$0.159891\pi$
$68$			4.73545i			0.574258i
$69$		0			0
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$	6.44949	+	5.51399i	0.760080	+	0.649830i
$73$	−13.6969			−1.60311			−0.801553	−	0.597924i	$-0.795992\pi$
$73$	−13.6969			−1.60311			−0.801553	+	0.597924i	$0.795992\pi$
$74$		0			0
$75$		0			0
$76$	16.6969			1.91527
$77$		0			0
$78$		0			0
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$		0			0
$81$	−1.39898	−	8.89060i	−0.155442	−	0.987845i
$82$	−0.651531			−0.0719495
$83$			14.1742i			1.55583i	0.628372	+	0.777913i	$0.283721\pi$
$83$			14.1742i			1.55583i	−0.628372	+	0.777913i	$0.716279\pi$
$84$		0			0
$85$		0			0
$86$			14.1421i			1.52499i
$87$		0			0
$88$	18.6969			1.99310
$89$			12.7600i			1.35256i	0.736644	+	0.676280i	$0.236409\pi$
$89$			12.7600i			1.35256i	−0.736644	+	0.676280i	$0.763591\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$	−8.89898	+	4.09978i	−0.908248	+	0.418432i
$97$	10.0000			1.01535			0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000			1.01535			0.507673	+	0.861550i	$0.330506\pi$
$98$			9.89949i			1.00000i
$99$	−15.0732	−	12.8868i	−1.51492	−	1.29518i
$100$		0			0
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$	2.42679	+	5.26758i	0.240288	+	0.521569i
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	4.70334i		−	0.454689i	−0.973814	−	0.227345i	$-0.926996\pi$
$107$		−	4.70334i		−	0.454689i	0.973814	−	0.227345i	$-0.0730044\pi$
$108$	10.0000	+	2.82843i	0.962250	+	0.272166i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	21.2453i		−	1.99859i	−0.0375328	−	0.999295i	$-0.511950\pi$
$113$		−	21.2453i		−	1.99859i	0.0375328	−	0.999295i	$-0.488050\pi$
$114$	18.5732	−	8.55671i	1.73954	−	0.801410i
$115$		0			0
$116$		0			0
$117$		0			0
$118$	−20.0000			−1.84115
$119$		0			0
$120$		0			0
$121$	−32.6969			−2.97245
$122$		0			0
$123$	−0.724745	+	0.333891i	−0.0653480	+	0.0301060i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$		−	11.3137i		−	1.00000i
$129$	7.24745	+	15.7313i	0.638102	+	1.38507i
$130$		0			0
$131$			14.1421i			1.23560i	0.786334	+	0.617802i	$0.211977\pi$
$131$			14.1421i			1.23560i	−0.786334	+	0.617802i	$0.788023\pi$
$132$	20.7980	−	9.58166i	1.81023	−	0.833976i
$133$		0			0
$134$			20.2918i			1.75294i
$135$		0			0
$136$	−6.69694			−0.574258
$137$			16.5099i			1.41053i	0.708942	+	0.705266i	$0.249173\pi$
$137$			16.5099i			1.41053i	−0.708942	+	0.705266i	$0.750827\pi$
$138$		0			0
$139$	18.3485			1.55630			0.778148	−	0.628080i	$-0.216159\pi$
$139$	18.3485			1.55630			0.778148	+	0.628080i	$0.216159\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−7.79796	+	9.12096i	−0.649830	+	0.760080i
$145$		0			0
$146$		−	19.3704i		−	1.60311i
$147$	5.07321	+	11.0119i	0.418432	+	0.908248i
$148$		0			0
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$			23.6130i			1.91527i
$153$	5.39898	+	4.61586i	0.436482	+	0.373170i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		1.00000			$0$
$157$		0			0		−1.00000			$\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	12.5732	−	1.97846i	0.987845	−	0.155442i
$163$	23.0454			1.80506			0.902528	−	0.430632i	$-0.141709\pi$
$163$	23.0454			1.80506			0.902528	+	0.430632i	$0.141709\pi$
$164$		−	0.921404i		−	0.0719495i
$165$		0			0
$166$	−20.0454			−1.55583
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$		0			0
$169$	13.0000			1.00000
$170$		0			0
$171$	16.2753	−	19.0365i	1.24460	−	1.45576i
$172$	−20.0000			−1.52499
$173$		0			0			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$		0			0
$175$		0			0
$176$			26.4415i			1.99310i
$177$	−22.2474	+	10.2494i	−1.67222	+	0.770395i
$178$	−18.0454			−1.35256
$179$		−	5.68896i		−	0.425213i	−0.977138	−	0.212607i	$-0.931805\pi$
$179$		−	5.68896i		−	0.425213i	0.977138	−	0.212607i	$-0.0681952\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	15.6515			1.14455
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$	−5.79796	−	12.5851i	−0.418432	−	0.908248i
$193$	3.69694			0.266111			0.133056	−	0.991109i	$-0.457521\pi$
$193$	3.69694			0.266111			0.133056	+	0.991109i	$0.457521\pi$
$194$			14.1421i			1.01535i
$195$		0			0
$196$	−14.0000			−1.00000
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$	18.2247	−	21.3167i	1.29518	−	1.51492i
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$		0			0
$201$	10.3990	+	22.5720i	0.733487	+	1.59211i
$202$		0			0
$203$		0			0
$204$	−7.44949	+	3.43199i	−0.521569	+	0.240288i
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	55.1864i		−	3.81733i
$210$		0			0
$211$	15.0454			1.03577			0.517884	−	0.855451i	$-0.326720\pi$
$211$	15.0454			1.03577			0.517884	+	0.855451i	$0.326720\pi$
$212$		0			0
$213$		0			0
$214$	6.65153			0.454689
$215$		0			0
$216$	−4.00000	+	14.1421i	−0.272166	+	0.962250i
$217$		0			0
$218$		0			0
$219$	−9.92679	−	21.5471i	−0.670790	−	1.45602i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$		0			0
$225$		0			0
$226$	30.0454			1.99859
$227$			2.82843i			0.187729i	0.995585	+	0.0938647i	$0.0299221\pi$
$227$			2.82843i			0.187729i	−0.995585	+	0.0938647i	$0.970078\pi$
$228$	12.1010	+	26.2665i	0.801410	+	1.73954i
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	5.65685i		−	0.370593i	−0.982683	−	0.185296i	$-0.940675\pi$
$233$		−	5.65685i		−	0.370593i	0.982683	−	0.185296i	$-0.0593245\pi$
$234$		0			0
$235$		0			0
$236$		−	28.2843i		−	1.84115i
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	1.69694			0.109309			0.0546547	−	0.998505i	$-0.482594\pi$
$241$	1.69694			0.109309			0.0546547	+	0.998505i	$0.482594\pi$
$242$		−	46.2405i		−	2.97245i
$243$	12.9722	−	8.64420i	0.832167	−	0.554526i
$244$		0			0
$245$		0			0
$246$	−0.472194	−	1.02494i	−0.0301060	−	0.0653480i
$247$		0			0
$248$		0			0
$249$	−22.2980	+	10.2727i	−1.41308	+	0.651007i
$250$		0			0
$251$		−	20.7525i		−	1.30989i	−0.755678	−	0.654943i	$-0.772693\pi$
$251$		−	20.7525i		−	1.30989i	0.755678	−	0.654943i	$-0.227307\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	16.0000			1.00000
$257$			11.3137i			0.705730i	0.935674	+	0.352865i	$0.114792\pi$
$257$			11.3137i			0.705730i	−0.935674	+	0.352865i	$0.885208\pi$
$258$	−22.2474	+	10.2494i	−1.38507	+	0.638102i
$259$		0			0
$260$		0			0
$261$		0			0
$262$	−20.0000			−1.23560
$263$		0			0			−	1.00000i	$-0.5\pi$
$263$		0			0				1.00000i	$0.5\pi$
$264$	13.5505	+	29.4128i	0.833976	+	1.81023i
$265$		0			0
$266$		0			0
$267$	−20.0732	+	9.24777i	−1.22846	+	0.565954i
$268$	−28.6969			−1.75294
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		−	9.47090i		−	0.574258i
$273$		0			0
$274$	−23.3485			−1.41053
$275$		0			0
$276$		0			0
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$			25.9487i			1.55630i
$279$		0			0
$280$		0			0
$281$		−	28.2843i		−	1.68730i	−0.536895	−	0.843649i	$-0.680403\pi$
$281$		−	28.2843i		−	1.68730i	0.536895	−	0.843649i	$-0.319597\pi$
$282$		0			0
$283$	−33.0454			−1.96435			−0.982173	−	0.187980i	$-0.939806\pi$
$283$	−33.0454			−1.96435			−0.982173	+	0.187980i	$0.939806\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−12.8990	−	11.0280i	−0.760080	−	0.649830i
$289$	11.3939			0.670228
$290$		0			0
$291$	7.24745	+	15.7313i	0.424853	+	0.922186i
$292$	27.3939			1.60311
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$	−15.5732	+	7.17461i	−0.908248	+	0.418432i
$295$		0			0
$296$		0			0
$297$	9.34847	−	33.0518i	0.542453	−	1.91786i
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$	−33.3939			−1.91527
$305$		0			0
$306$	−6.52781	+	7.63531i	−0.373170	+	0.436482i
$307$	−24.3485			−1.38964			−0.694820	−	0.719183i	$-0.744516\pi$
$307$	−24.3485			−1.38964			−0.694820	+	0.719183i	$0.744516\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$	10.0000			0.565233			0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000			0.565233			0.282617	+	0.959233i	$0.408798\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0			−	1.00000i	$-0.5\pi$
$317$		0			0				1.00000i	$0.5\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	7.39898	−	3.40872i	0.412971	−	0.190256i
$322$		0			0
$323$			19.7669i			1.09986i
$324$	2.79796	+	17.7812i	0.155442	+	0.987845i
$325$		0			0
$326$			32.5911i			1.80506i
$327$		0			0
$328$	1.30306			0.0719495
$329$		0			0
$330$		0			0
$331$	−35.0454			−1.92627			−0.963135	−	0.269019i	$-0.913301\pi$
$331$	−35.0454			−1.92627			−0.963135	+	0.269019i	$0.913301\pi$
$332$		−	28.3485i		−	1.55583i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−22.3939			−1.21987			−0.609936	−	0.792451i	$-0.708805\pi$
$337$	−22.3939			−1.21987			−0.609936	+	0.792451i	$0.708805\pi$
$338$			18.3848i			1.00000i
$339$	33.4217	−	15.3974i	1.81522	−	0.836274i
$340$		0			0
$341$		0			0
$342$	26.9217	+	23.0167i	1.45576	+	1.24460i
$343$		0			0
$344$		−	28.2843i		−	1.52499i
$345$		0			0
$346$		0			0
$347$		−	23.5809i		−	1.26589i	−0.774197	−	0.632945i	$-0.781846\pi$
$347$		−	23.5809i		−	1.26589i	0.774197	−	0.632945i	$-0.218154\pi$
$348$		0			0
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$		0			0
$351$		0			0
$352$	−37.3939			−1.99310
$353$		−	22.6274i		−	1.20434i	−0.798369	−	0.602168i	$-0.794304\pi$
$353$		−	22.6274i		−	1.20434i	0.798369	−	0.602168i	$-0.205696\pi$
$354$	−14.4949	−	31.4626i	−0.770395	−	1.67222i
$355$		0			0
$356$		−	25.5201i		−	1.35256i
$357$		0			0
$358$	8.04541			0.425213
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	50.6969			2.66826
$362$		0			0
$363$	−23.6969	−	51.4366i	−1.24377	−	2.69972i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		1.00000			$0$
$367$		0			0		−1.00000			$\pi$
$368$		0			0
$369$	−1.05051	−	0.898133i	−0.0546874	−	0.0467550i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$			22.1346i			1.14455i
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−11.6515			−0.598499			−0.299249	−	0.954175i	$-0.596736\pi$
$379$	−11.6515			−0.598499			−0.299249	+	0.954175i	$0.596736\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$	17.7980	−	8.19955i	0.908248	−	0.418432i
$385$		0			0
$386$			5.22826i			0.266111i
$387$	−19.4949	+	22.8024i	−0.990981	+	1.15911i
$388$	−20.0000			−1.01535
$389$		0			0			−	1.00000i	$-0.5\pi$
$389$		0			0				1.00000i	$0.5\pi$
$390$		0			0
$391$		0			0
$392$		−	19.7990i		−	1.00000i
$393$	−22.2474	+	10.2494i	−1.12224	+	0.517016i
$394$		0			0
$395$		0			0
$396$	30.1464	+	25.7737i	1.51492	+	1.29518i
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	14.6028i		−	0.729231i	−0.931158	−	0.364615i	$-0.881200\pi$
$401$		−	14.6028i		−	0.729231i	0.931158	−	0.364615i	$-0.118800\pi$
$402$	−31.9217	+	14.7064i	−1.59211	+	0.733487i
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$	−4.85357	−	10.5352i	−0.240288	−	0.521569i
$409$	−18.3939			−0.909519			−0.454759	−	0.890614i	$-0.650275\pi$
$409$	−18.3939			−0.909519			−0.454759	+	0.890614i	$0.650275\pi$
$410$		0			0
$411$	−25.9722	+	11.9654i	−1.28111	+	0.590211i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	13.2980	+	28.8646i	0.651204	+	1.41350i
$418$	78.0454			3.81733
$419$			33.9732i			1.65970i	0.557986	+	0.829851i	$0.311574\pi$
$419$			33.9732i			1.65970i	−0.557986	+	0.829851i	$0.688426\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$			21.2774i			1.03577i
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$			9.40669i			0.454689i
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$	−20.0000	−	5.65685i	−0.962250	−	0.272166i
$433$	33.6969			1.61937			0.809686	−	0.586864i	$-0.199638\pi$
$433$	33.6969			1.61937			0.809686	+	0.586864i	$0.199638\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	30.4722	−	14.0386i	1.45602	−	0.670790i
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$		0			0
$441$	−13.6464	+	15.9617i	−0.649830	+	0.760080i
$442$		0			0
$443$		−	37.7873i		−	1.79533i	−0.440681	−	0.897664i	$-0.645263\pi$
$443$		−	37.7873i		−	1.79533i	0.440681	−	0.897664i	$-0.354737\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	39.2015i		−	1.85003i	−0.379927	−	0.925016i	$-0.624051\pi$
$449$		−	39.2015i		−	1.85003i	0.379927	−	0.925016i	$-0.375949\pi$
$450$		0			0
$451$	−3.04541			−0.143403
$452$			42.4906i			1.99859i
$453$		0			0
$454$	−4.00000			−0.187729
$455$		0			0
$456$	−37.1464	+	17.1134i	−1.73954	+	0.801410i
$457$	42.3939			1.98310			0.991551	−	0.129718i	$-0.0414071\pi$
$457$	42.3939			1.98310			0.991551	+	0.129718i	$0.0414071\pi$
$458$		0			0
$459$	−3.34847	+	11.8386i	−0.156293	+	0.552580i
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		0			0
$465$		0			0
$466$	8.00000			0.370593
$467$		−	31.1127i		−	1.43972i	−0.694117	−	0.719862i	$-0.744205\pi$
$467$		−	31.1127i		−	1.43972i	0.694117	−	0.719862i	$-0.255795\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$	40.0000			1.84115
$473$			66.1037i			3.03945i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$		0			0
$482$			2.39983i			0.109309i
$483$		0			0
$484$	65.3939			2.97245
$485$		0			0
$486$	12.2247	+	18.3455i	0.554526	+	0.832167i
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$		0			0
$489$	16.7020	+	36.2535i	0.755292	+	1.63944i
$490$		0			0
$491$			14.1421i			0.638226i	0.947717	+	0.319113i	$0.103385\pi$
$491$			14.1421i			0.638226i	−0.947717	+	0.319113i	$0.896615\pi$
$492$	1.44949	−	0.667783i	0.0653480	−	0.0301060i
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$	−14.5278	−	31.5341i	−0.651007	−	1.41308i
$499$	14.0000			0.626726			0.313363	−	0.949633i	$-0.398544\pi$
$499$	14.0000			0.626726			0.313363	+	0.949633i	$0.398544\pi$
$500$		0			0
$501$		0			0
$502$	29.3485			1.30989
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	9.42168	+	20.4507i	0.418432	+	0.908248i
$508$		0			0
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$		0			0
$512$			22.6274i			1.00000i
$513$	41.7423	+	11.8065i	1.84297	+	0.521271i
$514$	−16.0000			−0.705730
$515$		0			0
$516$	−14.4949	−	31.4626i	−0.638102	−	1.38507i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$			27.8236i			1.21897i	0.792797	+	0.609486i	$0.208624\pi$
$521$			27.8236i			1.21897i	−0.792797	+	0.609486i	$0.791376\pi$
$522$		0			0
$523$	−3.04541			−0.133166			−0.0665832	−	0.997781i	$-0.521210\pi$
$523$	−3.04541			−0.133166			−0.0665832	+	0.997781i	$0.521210\pi$
$524$		−	28.2843i		−	1.23560i
$525$		0			0
$526$		0			0
$527$		0			0
$528$	−41.5959	+	19.1633i	−1.81023	+	0.833976i
$529$	−23.0000			−1.00000
$530$		0			0
$531$	−32.2474	−	27.5699i	−1.39942	−	1.19643i
$532$		0			0
$533$		0			0
$534$	−13.0783	−	28.3878i	−0.565954	−	1.22846i
$535$		0			0
$536$		−	40.5836i		−	1.75294i
$537$	8.94949	−	4.12305i	0.386199	−	0.177923i
$538$		0			0
$539$			46.2726i			1.99310i
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$		0			0
$544$	13.3939			0.574258
$545$		0			0
$546$		0			0
$547$	−15.6515			−0.669211			−0.334606	−	0.942358i	$-0.608603\pi$
$547$	−15.6515			−0.669211			−0.334606	+	0.942358i	$0.608603\pi$
$548$		−	33.0197i		−	1.41053i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	−36.6969			−1.55630
$557$		0			0			−	1.00000i	$-0.5\pi$
$557$		0			0				1.00000i	$0.5\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	11.3434	+	24.6219i	0.478917	+	1.03954i
$562$	40.0000			1.68730
$563$			36.7696i			1.54965i	0.632175	+	0.774826i	$0.282163\pi$
$563$			36.7696i			1.54965i	−0.632175	+	0.774826i	$0.717837\pi$
$564$		0			0
$565$		0			0
$566$		−	46.7333i		−	1.96435i
$567$		0			0
$568$		0			0
$569$			25.0594i			1.05054i	0.850935	+	0.525271i	$0.176036\pi$
$569$			25.0594i			1.05054i	−0.850935	+	0.525271i	$0.823964\pi$
$570$		0			0
$571$	−22.0000			−0.920671			−0.460336	−	0.887745i	$-0.652271\pi$
$571$	−22.0000			−0.920671			−0.460336	+	0.887745i	$0.652271\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	15.5959	−	18.2419i	0.649830	−	0.760080i
$577$	12.3939			0.515964			0.257982	−	0.966150i	$-0.416942\pi$
$577$	12.3939			0.515964			0.257982	+	0.966150i	$0.416942\pi$
$578$			16.1134i			0.670228i
$579$	2.67934	+	5.81577i	0.111349	+	0.241695i
$580$		0			0
$581$		0			0
$582$	−22.2474	+	10.2494i	−0.922186	+	0.424853i
$583$		0			0
$584$			38.7408i			1.60311i
$585$		0			0
$586$		0			0
$587$			18.8455i			0.777836i	0.921272	+	0.388918i	$0.127151\pi$
$587$			18.8455i			0.777836i	−0.921272	+	0.388918i	$0.872849\pi$
$588$	−10.1464	−	22.0239i	−0.418432	−	0.908248i
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	7.03896i		−	0.289055i	−0.989501	−	0.144528i	$-0.953834\pi$
$593$		−	7.03896i		−	0.289055i	0.989501	−	0.144528i	$-0.0461663\pi$
$594$	46.7423	+	13.2207i	1.91786	+	0.542453i
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	8.30306			0.338689			0.169344	−	0.985557i	$-0.445835\pi$
$601$	8.30306			0.338689			0.169344	+	0.985557i	$0.445835\pi$
$602$		0			0
$603$	−27.9722	+	32.7179i	−1.13912	+	1.33238i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		1.00000			$0$
$607$		0			0		−1.00000			$\pi$
$608$		−	47.2261i		−	1.91527i
$609$		0			0
$610$		0			0
$611$		0			0
$612$	−10.7980	−	9.23171i	−0.436482	−	0.373170i
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$		−	34.4339i		−	1.38964i
$615$		0			0
$616$		0			0
$617$		−	39.5980i		−	1.59415i	−0.603877	−	0.797077i	$-0.706378\pi$
$617$		−	39.5980i		−	1.59415i	0.603877	−	0.797077i	$-0.293622\pi$
$618$		0			0
$619$	26.0000			1.04503			0.522514	−	0.852631i	$-0.324994\pi$
$619$	26.0000			1.04503			0.522514	+	0.852631i	$0.324994\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$		0			0
$626$			14.1421i			0.565233i
$627$	86.8156	−	39.9961i	3.46708	−	1.59729i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$	10.9041	+	23.6684i	0.433398	+	0.940735i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	28.2843i		−	1.11716i	−0.829450	−	0.558581i	$-0.811346\pi$
$641$		−	28.2843i		−	1.11716i	0.829450	−	0.558581i	$-0.188654\pi$
$642$	4.82066	+	10.4637i	0.190256	+	0.412971i
$643$	−50.0000			−1.97181			−0.985904	−	0.167313i	$-0.946491\pi$
$643$	−50.0000			−1.97181			−0.985904	+	0.167313i	$0.946491\pi$
$644$		0			0
$645$		0			0
$646$	−27.9546			−1.09986
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$	−25.1464	+	3.95691i	−0.987845	+	0.155442i
$649$	−93.4847			−3.66960
$650$		0			0
$651$		0			0
$652$	−46.0908			−1.80506
$653$		0			0			−	1.00000i	$-0.5\pi$
$653$		0			0				1.00000i	$0.5\pi$
$654$		0			0
$655$		0			0
$656$			1.84281i			0.0719495i
$657$	26.7020	−	31.2323i	1.04175	−	1.21849i
$658$		0			0
$659$		−	8.45317i		−	0.329289i	−0.986353	−	0.164644i	$-0.947352\pi$
$659$		−	8.45317i		−	0.329289i	0.986353	−	0.164644i	$-0.0526477\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$		−	49.5617i		−	1.92627i
$663$		0			0
$664$	40.0908			1.55583
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	10.0000			0.385472			0.192736	−	0.981251i	$-0.438264\pi$
$673$	10.0000			0.385472			0.192736	+	0.981251i	$0.438264\pi$
$674$		−	31.6697i		−	1.21987i
$675$		0			0
$676$	−26.0000			−1.00000
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$	21.7753	+	47.2654i	0.836274	+	1.81522i
$679$		0			0
$680$		0			0
$681$	−4.44949	+	2.04989i	−0.170505	+	0.0785519i
$682$		0			0
$683$			51.9294i			1.98702i	0.113728	+	0.993512i	$0.463721\pi$
$683$			51.9294i			1.98702i	−0.113728	+	0.993512i	$0.536279\pi$
$684$	−32.5505	+	38.0730i	−1.24460	+	1.45576i
$685$		0			0
$686$		0			0
$687$		0			0
$688$	40.0000			1.52499
$689$		0			0
$690$		0			0
$691$	45.0454			1.71361			0.856804	−	0.515642i	$-0.172447\pi$
$691$	45.0454			1.71361			0.856804	+	0.515642i	$0.172447\pi$
$692$		0			0
$693$		0			0
$694$	33.3485			1.26589
$695$		0			0
$696$		0			0
$697$	1.09082			0.0413176
$698$		0			0
$699$	8.89898	−	4.09978i	0.336590	−	0.155068i
$700$		0			0
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$		0			0
$704$		−	52.8829i		−	1.99310i
$705$		0			0
$706$	32.0000			1.20434
$707$		0			0
$708$	44.4949	−	20.4989i	1.67222	−	0.770395i
$709$		0			0		1.00000			$0$
$709$		0			0		−1.00000			$\pi$
$710$		0			0
$711$		0			0
$712$	36.0908			1.35256
$713$		0			0
$714$		0			0
$715$		0			0
$716$			11.3779i			0.425213i
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$		0			0
$722$			71.6963i			2.66826i
$723$	1.22985	+	2.66951i	0.0457385	+	0.0992801i
$724$		0			0
$725$		0			0
$726$	72.7423	−	33.5125i	2.69972	−	1.24377i
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$	23.0000	+	14.1421i	0.851852	+	0.523783i
$730$		0			0
$731$		−	23.6773i		−	0.875735i
$732$		0			0
$733$		0			0		1.00000			$0$
$733$		0			0		−1.00000			$\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			94.8486i			3.49379i
$738$	1.27015	−	1.48565i	0.0467550	−	0.0546874i
$739$	−34.0000			−1.25071			−0.625355	−	0.780340i	$-0.715046\pi$
$739$	−34.0000			−1.25071			−0.625355	+	0.780340i	$0.715046\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0			−	1.00000i	$-0.5\pi$
$743$		0			0				1.00000i	$0.5\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−32.3207	−	27.6325i	−1.18255	−	1.01102i
$748$	−31.3031			−1.14455
$749$		0			0
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$		0			0
$753$	32.6464	−	15.0403i	1.18970	−	0.548098i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$		−	16.4778i		−	0.598499i
$759$		0			0
$760$		0			0
$761$			52.4222i			1.90030i	0.311787	+	0.950152i	$0.399073\pi$
$761$			52.4222i			1.90030i	−0.311787	+	0.950152i	$0.600927\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	11.5959	+	25.1701i	0.418432	+	0.908248i
$769$	55.0908			1.98663			0.993313	−	0.115454i	$-0.0368323\pi$
$769$	55.0908			1.98663			0.993313	+	0.115454i	$0.0368323\pi$
$770$		0			0
$771$	−17.7980	+	8.19955i	−0.640978	+	0.295300i
$772$	−7.39388			−0.266111
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$	−32.2474	−	27.5699i	−1.15911	−	0.990981i
$775$		0			0
$776$		−	28.2843i		−	1.01535i
$777$		0			0
$778$		0			0
$779$		−	3.84616i		−	0.137803i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	28.0000			1.00000
$785$		0			0
$786$	−14.4949	−	31.4626i	−0.517016	−	1.12224i
$787$	−50.0000			−1.78231			−0.891154	−	0.453701i	$-0.850103\pi$
$787$	−50.0000			−1.78231			−0.891154	+	0.453701i	$0.850103\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$	−36.4495	+	42.6335i	−1.29518	+	1.51492i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	−29.0959	−	24.8755i	−1.02805	−	0.878934i
$802$	20.6515			0.729231
$803$		−	90.5418i		−	3.19515i
$804$	−20.7980	−	45.1441i	−0.733487	−	1.59211i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$			56.5685i			1.98884i	0.105474	+	0.994422i	$0.466364\pi$
$809$			56.5685i			1.98884i	−0.105474	+	0.994422i	$0.533636\pi$
$810$		0			0
$811$	38.0000			1.33436			0.667180	−	0.744896i	$-0.267501\pi$
$811$	38.0000			1.33436			0.667180	+	0.744896i	$0.267501\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$	14.8990	−	6.86399i	0.521569	−	0.240288i
$817$	−83.4847			−2.92076
$818$		−	26.0129i		−	0.909519i
$819$		0			0
$820$		0			0
$821$		0			0			−	1.00000i	$-0.5\pi$
$821$		0			0				1.00000i	$0.5\pi$
$822$	−16.9217	−	36.7302i	−0.590211	−	1.28111i
$823$		0			0		1.00000			$0$
$823$		0			0		−1.00000			$\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	56.6649i		−	1.97043i	−0.171321	−	0.985215i	$-0.554804\pi$
$827$		−	56.6649i		−	1.97043i	0.171321	−	0.985215i	$-0.445196\pi$
$828$		0			0
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	16.5741i		−	0.574258i
$834$	−40.8207	+	18.8062i	−1.41350	+	0.651204i
$835$		0			0
$836$			110.373i			3.81733i
$837$		0			0
$838$	−48.0454			−1.65970
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	−29.0000			−1.00000
$842$		0			0
$843$	44.4949	−	20.4989i	1.53249	−	0.706019i
$844$	−30.0908			−1.03577
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−23.9495	−	51.9848i	−0.821944	−	1.78411i
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		1.00000			$0$
$853$		0			0		−1.00000			$\pi$
$854$		0			0
$855$		0			0
$856$	−13.3031			−0.454689
$857$		−	35.4517i		−	1.21101i	−0.795843	−	0.605503i	$-0.792972\pi$
$857$		−	35.4517i		−	1.21101i	0.795843	−	0.605503i	$-0.207028\pi$
$858$		0			0
$859$	21.6515			0.738741			0.369370	−	0.929282i	$-0.379573\pi$
$859$	21.6515			0.738741			0.369370	+	0.929282i	$0.379573\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$	8.00000	−	28.2843i	0.272166	−	0.962250i
$865$		0			0
$866$			47.6547i			1.61937i
$867$	8.25765	+	17.9241i	0.280445	+	0.608733i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−19.4949	+	22.8024i	−0.659802	+	0.771744i
$874$		0			0
$875$		0			0
$876$	19.8536	+	43.0942i	0.670790	+	1.45602i
$877$		0			0		1.00000			$0$
$877$		0			0		−1.00000			$\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			56.5685i			1.90584i	0.303218	+	0.952921i	$0.401939\pi$
$881$			56.5685i			1.90584i	−0.303218	+	0.952921i	$0.598061\pi$
$882$	−22.5732	−	19.2990i	−0.760080	−	0.649830i
$883$	−50.4393			−1.69742			−0.848709	−	0.528861i	$-0.822619\pi$
$883$	−50.4393			−1.69742			−0.848709	+	0.528861i	$0.822619\pi$
$884$		0			0
$885$		0			0
$886$	53.4393			1.79533
$887$		0			0			−	1.00000i	$-0.5\pi$
$887$		0			0				1.00000i	$0.5\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	58.7702	−	9.24777i	1.96887	−	0.309812i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	55.4393			1.85003
$899$		0			0
$900$		0			0
$901$		0			0
$902$		−	4.30686i		−	0.143403i
$903$		0			0
$904$	−60.0908			−1.99859
$905$		0			0
$906$		0			0
$907$	10.0000			0.332045			0.166022	−	0.986122i	$-0.446908\pi$
$907$	10.0000			0.332045			0.166022	+	0.986122i	$0.446908\pi$
$908$		−	5.65685i		−	0.187729i
$909$		0			0
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$	−24.2020	−	52.5330i	−0.801410	−	1.73954i
$913$	−93.6969			−3.10092
$914$			59.9540i			1.98310i
$915$		0			0
$916$		0			0
$917$		0			0
$918$	−16.7423	−	4.73545i	−0.552580	−	0.156293i
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$	−17.6464	−	38.3034i	−0.581470	−	1.26214i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	28.2843i		−	0.927977i	−0.885841	−	0.463988i	$-0.846418\pi$
$929$		−	28.2843i		−	0.927977i	0.885841	−	0.463988i	$-0.153582\pi$
$930$		0			0
$931$	−58.4393			−1.91527
$932$			11.3137i			0.370593i
$933$		0			0
$934$	44.0000			1.43972
$935$		0			0
$936$		0			0
$937$	−61.0908			−1.99575			−0.997875	−	0.0651578i	$-0.979245\pi$
$937$	−61.0908			−1.99575			−0.997875	+	0.0651578i	$0.979245\pi$
$938$		0			0
$939$	7.24745	+	15.7313i	0.236512	+	0.513372i
$940$		0			0
$941$		0			0			−	1.00000i	$-0.5\pi$
$941$		0			0				1.00000i	$0.5\pi$
$942$		0			0
$943$		0			0
$944$			56.5685i			1.84115i
$945$		0			0
$946$	−93.4847			−3.03945
$947$			53.7401i			1.74632i	0.487435	+	0.873160i	$0.337933\pi$
$947$			53.7401i			1.74632i	−0.487435	+	0.873160i	$0.662067\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	59.0005i		−	1.91121i	−0.294646	−	0.955607i	$-0.595202\pi$
$953$		−	59.0005i		−	1.91121i	0.294646	−	0.955607i	$-0.404798\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	31.0000			1.00000
$962$		0			0
$963$	10.7247	+	9.16912i	0.345600	+	0.295471i
$964$	−3.39388			−0.109309
$965$		0			0
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$			92.4809i			2.97245i
$969$	−31.0959	+	14.3259i	−0.998945	+	0.460216i
$970$		0			0
$971$			31.2090i			1.00155i	0.865579	+	0.500773i	$0.166951\pi$
$971$			31.2090i			1.00155i	−0.865579	+	0.500773i	$0.833049\pi$
$972$	−25.9444	+	17.2884i	−0.832167	+	0.554526i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	25.9165i		−	0.829144i	−0.910017	−	0.414572i	$-0.863931\pi$
$977$		−	25.9165i		−	0.829144i	0.910017	−	0.414572i	$-0.136069\pi$
$978$	−51.2702	+	23.6203i	−1.63944	+	0.755292i
$979$	−84.3485			−2.69579
$980$		0			0
$981$		0			0
$982$	−20.0000			−0.638226
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$	0.944387	+	2.04989i	0.0301060	+	0.0653480i
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		0			0
$993$	−25.3990	−	55.1311i	−0.806012	−	1.74953i
$994$		0			0
$995$		0			0
$996$	44.5959	−	20.5454i	1.41308	−	0.651007i
$997$		0			0		1.00000			$0$
$997$		0			0		−1.00000			$\pi$
$998$			19.7990i			0.626726i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	600.2.b.b.251.4	yes	4
3.2	odd	2	inner	600.2.b.b.251.2	✓	4
4.3	odd	2		2400.2.b.d.2351.1		4
5.2	odd	4		600.2.m.b.299.3		8
5.3	odd	4		600.2.m.b.299.6		8
5.4	even	2		600.2.b.d.251.1	yes	4
8.3	odd	2	CM	600.2.b.b.251.4	yes	4
8.5	even	2		2400.2.b.d.2351.1		4
12.11	even	2		2400.2.b.d.2351.2		4
15.2	even	4		600.2.m.b.299.5		8
15.8	even	4		600.2.m.b.299.4		8
15.14	odd	2		600.2.b.d.251.3	yes	4
20.3	even	4		2400.2.m.b.1199.7		8
20.7	even	4		2400.2.m.b.1199.2		8
20.19	odd	2		2400.2.b.b.2351.4		4
24.5	odd	2		2400.2.b.d.2351.2		4
24.11	even	2	inner	600.2.b.b.251.2	✓	4
40.3	even	4		600.2.m.b.299.6		8
40.13	odd	4		2400.2.m.b.1199.7		8
40.19	odd	2		600.2.b.d.251.1	yes	4
40.27	even	4		600.2.m.b.299.3		8
40.29	even	2		2400.2.b.b.2351.4		4
40.37	odd	4		2400.2.m.b.1199.2		8
60.23	odd	4		2400.2.m.b.1199.1		8
60.47	odd	4		2400.2.m.b.1199.8		8
60.59	even	2		2400.2.b.b.2351.3		4
120.29	odd	2		2400.2.b.b.2351.3		4
120.53	even	4		2400.2.m.b.1199.1		8
120.59	even	2		600.2.b.d.251.3	yes	4
120.77	even	4		2400.2.m.b.1199.8		8
120.83	odd	4		600.2.m.b.299.4		8
120.107	odd	4		600.2.m.b.299.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
600.2.b.b.251.2	✓	4	3.2	odd	2	inner
600.2.b.b.251.2	✓	4	24.11	even	2	inner
600.2.b.b.251.4	yes	4	1.1	even	1	trivial
600.2.b.b.251.4	yes	4	8.3	odd	2	CM
600.2.b.d.251.1	yes	4	5.4	even	2
600.2.b.d.251.1	yes	4	40.19	odd	2
600.2.b.d.251.3	yes	4	15.14	odd	2
600.2.b.d.251.3	yes	4	120.59	even	2
600.2.m.b.299.3		8	5.2	odd	4
600.2.m.b.299.3		8	40.27	even	4
600.2.m.b.299.4		8	15.8	even	4
600.2.m.b.299.4		8	120.83	odd	4
600.2.m.b.299.5		8	15.2	even	4
600.2.m.b.299.5		8	120.107	odd	4
600.2.m.b.299.6		8	5.3	odd	4
600.2.m.b.299.6		8	40.3	even	4
2400.2.b.b.2351.3		4	60.59	even	2
2400.2.b.b.2351.3		4	120.29	odd	2
2400.2.b.b.2351.4		4	20.19	odd	2
2400.2.b.b.2351.4		4	40.29	even	2
2400.2.b.d.2351.1		4	4.3	odd	2
2400.2.b.d.2351.1		4	8.5	even	2
2400.2.b.d.2351.2		4	12.11	even	2
2400.2.b.d.2351.2		4	24.5	odd	2
2400.2.m.b.1199.1		8	60.23	odd	4
2400.2.m.b.1199.1		8	120.53	even	4
2400.2.m.b.1199.2		8	20.7	even	4
2400.2.m.b.1199.2		8	40.37	odd	4
2400.2.m.b.1199.7		8	20.3	even	4
2400.2.m.b.1199.7		8	40.13	odd	4
2400.2.m.b.1199.8		8	60.47	odd	4
2400.2.m.b.1199.8		8	120.77	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:	\( N \)	\(=\)	\( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	600.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} +(0.724745 + 1.57313i) q^{3} -2.00000 q^{4} +(-2.22474 + 1.02494i) q^{6} -2.82843i q^{8} +(-1.94949 + 2.28024i) q^{9} +O(q^{10})\)	\(q+1.41421i q^{2} +(0.724745 + 1.57313i) q^{3} -2.00000 q^{4} +(-2.22474 + 1.02494i) q^{6} -2.82843i q^{8} +(-1.94949 + 2.28024i) q^{9} +6.61037i q^{11} +(-1.44949 - 3.14626i) q^{12} +4.00000 q^{16} -2.36773i q^{17} +(-3.22474 - 2.75699i) q^{18} -8.34847 q^{19} -9.34847 q^{22} +(4.44949 - 2.04989i) q^{24} +(-5.00000 - 1.41421i) q^{27} +5.65685i q^{32} +(-10.3990 + 4.79083i) q^{33} +3.34847 q^{34} +(3.89898 - 4.56048i) q^{36} -11.8065i q^{38} +0.460702i q^{41} +10.0000 q^{43} -13.2207i q^{44} +(2.89898 + 6.29253i) q^{48} +7.00000 q^{49} +(3.72474 - 1.71600i) q^{51} +(2.00000 - 7.07107i) q^{54} +(-6.05051 - 13.1332i) q^{57} +14.1421i q^{59} -8.00000 q^{64} +(-6.77526 - 14.7064i) q^{66} +14.3485 q^{67} +4.73545i q^{68} +(6.44949 + 5.51399i) q^{72} -13.6969 q^{73} +16.6969 q^{76} +(-1.39898 - 8.89060i) q^{81} -0.651531 q^{82} +14.1742i q^{83} +14.1421i q^{86} +18.6969 q^{88} +12.7600i q^{89} +(-8.89898 + 4.09978i) q^{96} +10.0000 q^{97} +9.89949i q^{98} +(-15.0732 - 12.8868i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{3} - 8 q^{4} - 4 q^{6} + 2 q^{9}+O(q^{10}) \) 4 * q - 2 * q^3 - 8 * q^4 - 4 * q^6 + 2 * q^9	\( 4 q - 2 q^{3} - 8 q^{4} - 4 q^{6} + 2 q^{9} + 4 q^{12} + 16 q^{16} - 8 q^{18} - 4 q^{19} - 8 q^{22} + 8 q^{24} - 20 q^{27} - 22 q^{33} - 16 q^{34} - 4 q^{36} + 40 q^{43} - 8 q^{48} + 28 q^{49} + 10 q^{51} + 8 q^{54} - 34 q^{57} - 32 q^{64} - 32 q^{66} + 28 q^{67} + 16 q^{72} + 4 q^{73} + 8 q^{76} + 14 q^{81} - 32 q^{82} + 16 q^{88} - 16 q^{96} + 40 q^{97} - 26 q^{99}+O(q^{100}) \) 4 * q - 2 * q^3 - 8 * q^4 - 4 * q^6 + 2 * q^9 + 4 * q^12 + 16 * q^16 - 8 * q^18 - 4 * q^19 - 8 * q^22 + 8 * q^24 - 20 * q^27 - 22 * q^33 - 16 * q^34 - 4 * q^36 + 40 * q^43 - 8 * q^48 + 28 * q^49 + 10 * q^51 + 8 * q^54 - 34 * q^57 - 32 * q^64 - 32 * q^66 + 28 * q^67 + 16 * q^72 + 4 * q^73 + 8 * q^76 + 14 * q^81 - 32 * q^82 + 16 * q^88 - 16 * q^96 + 40 * q^97 - 26 * q^99

Introduction

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