LMFDB - Embedded newform 600.2.m.b.299.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [600,2,Mod(299,600)]

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, 1]))

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

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

chi := DirichletCharacter("600.299");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 600 = 2^{3} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	600.m (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$4.79102412128$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2}\cdot 5^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			299.3
Root			$-0.258819 - 0.965926i$ of defining polynomial
Character	$\chi$	$=$	600.299
Dual form			600.2.m.b.299.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{2} +(1.57313 - 0.724745i) q^{3} +2.00000 q^{4} +(-2.22474 + 1.02494i) q^{6} -2.82843 q^{8} +(1.94949 - 2.28024i) q^{9} +O(q^{10})$	$q-1.41421 q^{2} +(1.57313 - 0.724745i) q^{3} +2.00000 q^{4} +(-2.22474 + 1.02494i) q^{6} -2.82843 q^{8} +(1.94949 - 2.28024i) q^{9} +6.61037i q^{11} +(3.14626 - 1.44949i) q^{12} +4.00000 q^{16} +2.36773 q^{17} +(-2.75699 + 3.22474i) q^{18} +8.34847 q^{19} -9.34847i q^{22} +(-4.44949 + 2.04989i) q^{24} +(1.41421 - 5.00000i) q^{27} -5.65685 q^{32} +(4.79083 + 10.3990i) q^{33} -3.34847 q^{34} +(3.89898 - 4.56048i) q^{36} -11.8065 q^{38} +0.460702i q^{41} -10.0000i q^{43} +13.2207i q^{44} +(6.29253 - 2.89898i) q^{48} -7.00000 q^{49} +(3.72474 - 1.71600i) q^{51} +(-2.00000 + 7.07107i) q^{54} +(13.1332 - 6.05051i) q^{57} -14.1421i q^{59} +8.00000 q^{64} +(-6.77526 - 14.7064i) q^{66} +14.3485i q^{67} +4.73545 q^{68} +(-5.51399 + 6.44949i) q^{72} +13.6969i q^{73} +16.6969 q^{76} +(-1.39898 - 8.89060i) q^{81} -0.651531i q^{82} +14.1742 q^{83} +14.1421i q^{86} -18.6969i q^{88} -12.7600i q^{89} +(-8.89898 + 4.09978i) q^{96} +10.0000i q^{97} +9.89949 q^{98} +(15.0732 + 12.8868i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 16 q^{4} - 8 q^{6} - 4 q^{9}+O(q^{10}) $ 8 * q + 16 * q^4 - 8 * q^6 - 4 * q^9	$ 8 q + 16 q^{4} - 8 q^{6} - 4 q^{9} + 32 q^{16} + 8 q^{19} - 16 q^{24} + 32 q^{34} - 8 q^{36} - 56 q^{49} + 20 q^{51} - 16 q^{54} + 64 q^{64} - 64 q^{66} + 16 q^{76} + 28 q^{81} - 32 q^{96} + 52 q^{99}+O(q^{100}) $ 8 * q + 16 * q^4 - 8 * q^6 - 4 * q^9 + 32 * q^16 + 8 * q^19 - 16 * q^24 + 32 * q^34 - 8 * q^36 - 56 * q^49 + 20 * q^51 - 16 * q^54 + 64 * q^64 - 64 * q^66 + 16 * q^76 + 28 * q^81 - 32 * q^96 + 52 * 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 $

$2$	−1.41421			−1.00000
$2$	−1.41421			−1.00000
$3$	1.57313	−	0.724745i	0.908248	−	0.418432i
$3$	1.57313	−	0.724745i	0.908248	−	0.418432i
$4$	2.00000			1.00000
$5$		0			0
$5$		0			0
$6$	−2.22474	+	1.02494i	−0.908248	+	0.418432i
$7$		0			0			−	1.00000i	$-0.5\pi$
$7$		0			0				1.00000i	$0.5\pi$
$8$	−2.82843			−1.00000
$9$	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	$0.526448\pi$
$12$	3.14626	−	1.44949i	0.908248	−	0.418432i
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$		0			0
$15$		0			0
$16$	4.00000			1.00000
$17$	2.36773			0.574258			0.287129	−	0.957892i	$-0.407299\pi$
$17$	2.36773			0.574258			0.287129	+	0.957892i	$0.407299\pi$
$18$	−2.75699	+	3.22474i	−0.649830	+	0.760080i
$19$	8.34847			1.91527			0.957635	−	0.287984i	$-0.0929851\pi$
$19$	8.34847			1.91527			0.957635	+	0.287984i	$0.0929851\pi$
$20$		0			0
$21$		0			0
$22$		−	9.34847i		−	1.99310i
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$	−4.44949	+	2.04989i	−0.908248	+	0.418432i
$25$		0			0
$26$		0			0
$27$	1.41421	−	5.00000i	0.272166	−	0.962250i
$28$		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			$\pi$
$32$	−5.65685			−1.00000
$33$	4.79083	+	10.3990i	0.833976	+	1.81023i
$34$	−3.34847			−0.574258
$35$		0			0
$36$	3.89898	−	4.56048i	0.649830	−	0.760080i
$37$		0			0			−	1.00000i	$-0.5\pi$
$37$		0			0				1.00000i	$0.5\pi$
$38$	−11.8065			−1.91527
$39$		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
$43$		−	10.0000i		−	1.52499i	−0.646997	−	0.762493i	$-0.723975\pi$
$43$		−	10.0000i		−	1.52499i	0.646997	−	0.762493i	$-0.276025\pi$
$44$			13.2207i			1.99310i
$45$		0			0
$46$		0			0
$47$		0			0		1.00000			$0$
$47$		0			0		−1.00000			$\pi$
$48$	6.29253	−	2.89898i	0.908248	−	0.418432i
$49$	−7.00000			−1.00000
$50$		0			0
$51$	3.72474	−	1.71600i	0.521569	−	0.240288i
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$	−2.00000	+	7.07107i	−0.272166	+	0.962250i
$55$		0			0
$56$		0			0
$57$	13.1332	−	6.05051i	1.73954	−	0.801410i
$58$		0			0
$59$		−	14.1421i		−	1.84115i	−0.390567	−	0.920575i	$-0.627721\pi$
$59$		−	14.1421i		−	1.84115i	0.390567	−	0.920575i	$-0.372279\pi$
$60$		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
$67$			14.3485i			1.75294i	0.481452	+	0.876472i	$0.340109\pi$
$67$			14.3485i			1.75294i	−0.481452	+	0.876472i	$0.659891\pi$
$68$	4.73545			0.574258
$69$		0			0
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$	−5.51399	+	6.44949i	−0.649830	+	0.760080i
$73$			13.6969i			1.60311i	0.597924	+	0.801553i	$0.295992\pi$
$73$			13.6969i			1.60311i	−0.597924	+	0.801553i	$0.704008\pi$
$74$		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
$82$		−	0.651531i		−	0.0719495i
$83$	14.1742			1.55583			0.777913	−	0.628372i	$-0.216279\pi$
$83$	14.1742			1.55583			0.777913	+	0.628372i	$0.216279\pi$
$84$		0			0
$85$		0			0
$86$			14.1421i			1.52499i
$87$		0			0
$88$		−	18.6969i		−	1.99310i
$89$		−	12.7600i		−	1.35256i	−0.736644	−	0.676280i	$-0.763591\pi$
$89$		−	12.7600i		−	1.35256i	0.736644	−	0.676280i	$-0.236409\pi$
$90$		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
$97$			10.0000i			1.01535i	0.861550	+	0.507673i	$0.169494\pi$
$97$			10.0000i			1.01535i	−0.861550	+	0.507673i	$0.830506\pi$
$98$	9.89949			1.00000
$99$	15.0732	+	12.8868i	1.51492	+	1.29518i
$100$		0			0
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$	−5.26758	+	2.42679i	−0.521569	+	0.240288i
$103$		0			0			−	1.00000i	$-0.5\pi$
$103$		0			0				1.00000i	$0.5\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	4.70334			0.454689			0.227345	−	0.973814i	$-0.426996\pi$
$107$	4.70334			0.454689			0.227345	+	0.973814i	$0.426996\pi$
$108$	2.82843	−	10.0000i	0.272166	−	0.962250i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−21.2453			−1.99859			−0.999295	−	0.0375328i	$-0.988050\pi$
$113$	−21.2453			−1.99859			−0.999295	+	0.0375328i	$0.988050\pi$
$114$	−18.5732	+	8.55671i	−1.73954	+	0.801410i
$115$		0			0
$116$		0			0
$117$		0			0
$118$			20.0000i			1.84115i
$119$		0			0
$120$		0			0
$121$	−32.6969			−2.97245
$122$		0			0
$123$	0.333891	+	0.724745i	0.0301060	+	0.0653480i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0			−	1.00000i	$-0.5\pi$
$127$		0			0				1.00000i	$0.5\pi$
$128$	−11.3137			−1.00000
$129$	−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	$0.788023\pi$
$132$	9.58166	+	20.7980i	0.833976	+	1.81023i
$133$		0			0
$134$		−	20.2918i		−	1.75294i
$135$		0			0
$136$	−6.69694			−0.574258
$137$	−16.5099			−1.41053			−0.705266	−	0.708942i	$-0.749173\pi$
$137$	−16.5099			−1.41053			−0.705266	+	0.708942i	$0.749173\pi$
$138$		0			0
$139$	−18.3485			−1.55630			−0.778148	−	0.628080i	$-0.783841\pi$
$139$	−18.3485			−1.55630			−0.778148	+	0.628080i	$0.783841\pi$
$140$		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
$147$	−11.0119	+	5.07321i	−0.908248	+	0.418432i
$148$		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			$\pi$
$152$	−23.6130			−1.91527
$153$	4.61586	−	5.39898i	0.373170	−	0.436482i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0			−	1.00000i	$-0.5\pi$
$157$		0			0				1.00000i	$0.5\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	1.97846	+	12.5732i	0.155442	+	0.987845i
$163$		−	23.0454i		−	1.80506i	−0.430632	−	0.902528i	$-0.641709\pi$
$163$		−	23.0454i		−	1.80506i	0.430632	−	0.902528i	$-0.358291\pi$
$164$			0.921404i			0.0719495i
$165$		0			0
$166$	−20.0454			−1.55583
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$	−13.0000			−1.00000
$170$		0			0
$171$	16.2753	−	19.0365i	1.24460	−	1.45576i
$172$		−	20.0000i		−	1.52499i
$173$		0			0		1.00000			$0$
$173$		0			0		−1.00000			$\pi$
$174$		0			0
$175$		0			0
$176$			26.4415i			1.99310i
$177$	−10.2494	−	22.2474i	−0.770395	−	1.67222i
$178$			18.0454i			1.35256i
$179$			5.68896i			0.425213i	0.977138	+	0.212607i	$0.0681952\pi$
$179$			5.68896i			0.425213i	−0.977138	+	0.212607i	$0.931805\pi$
$180$		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
$187$			15.6515i			1.14455i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$	12.5851	−	5.79796i	0.908248	−	0.418432i
$193$		−	3.69694i		−	0.266111i	−0.991109	−	0.133056i	$-0.957521\pi$
$193$		−	3.69694i		−	0.266111i	0.991109	−	0.133056i	$-0.0424789\pi$
$194$		−	14.1421i		−	1.01535i
$195$		0			0
$196$	−14.0000			−1.00000
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$	−21.3167	−	18.2247i	−1.51492	−	1.29518i
$199$		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			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$		0			0
$225$		0			0
$226$	30.0454			1.99859
$227$	−2.82843			−0.187729			−0.0938647	−	0.995585i	$-0.529922\pi$
$227$	−2.82843			−0.187729			−0.0938647	+	0.995585i	$0.529922\pi$
$228$	26.2665	−	12.1010i	1.73954	−	0.801410i
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−5.65685			−0.370593			−0.185296	−	0.982683i	$-0.559325\pi$
$233$	−5.65685			−0.370593			−0.185296	+	0.982683i	$0.559325\pi$
$234$		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	$0.482594\pi$
$242$	46.2405			2.97245
$243$	−8.64420	−	12.9722i	−0.554526	−	0.832167i
$244$		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
$257$	−11.3137			−0.705730			−0.352865	−	0.935674i	$-0.614792\pi$
$257$	−11.3137			−0.705730			−0.352865	+	0.935674i	$0.614792\pi$
$258$	10.2494	+	22.2474i	0.638102	+	1.38507i
$259$		0			0
$260$		0			0
$261$		0			0
$262$		−	20.0000i		−	1.23560i
$263$		0			0		1.00000			$0$
$263$		0			0		−1.00000			$\pi$
$264$	−13.5505	−	29.4128i	−0.833976	−	1.81023i
$265$		0			0
$266$		0			0
$267$	−9.24777	−	20.0732i	−0.565954	−	1.22846i
$268$			28.6969i			1.75294i
$269$		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			$\pi$
$272$	9.47090			0.574258
$273$		0			0
$274$	23.3485			1.41053
$275$		0			0
$276$		0			0
$277$		0			0			−	1.00000i	$-0.5\pi$
$277$		0			0				1.00000i	$0.5\pi$
$278$	25.9487			1.55630
$279$		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
$283$			33.0454i			1.96435i	0.187980	+	0.982173i	$0.439806\pi$
$283$			33.0454i			1.96435i	−0.187980	+	0.982173i	$0.560194\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−11.0280	+	12.8990i	−0.649830	+	0.760080i
$289$	−11.3939			−0.670228
$290$		0			0
$291$	7.24745	+	15.7313i	0.424853	+	0.922186i
$292$			27.3939i			1.60311i
$293$		0			0		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$	15.5732	−	7.17461i	0.908248	−	0.418432i
$295$		0			0
$296$		0			0
$297$	33.0518	+	9.34847i	1.91786	+	0.542453i
$298$		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
$307$		−	24.3485i		−	1.38964i	−0.719183	−	0.694820i	$-0.755484\pi$
$307$		−	24.3485i		−	1.38964i	0.719183	−	0.694820i	$-0.244516\pi$
$308$		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
$313$		−	10.0000i		−	0.565233i	−0.959233	−	0.282617i	$-0.908798\pi$
$313$		−	10.0000i		−	0.565233i	0.959233	−	0.282617i	$-0.0912024\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		1.00000			$0$
$317$		0			0		−1.00000			$\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	7.39898	−	3.40872i	0.412971	−	0.190256i
$322$		0			0
$323$	19.7669			1.09986
$324$	−2.79796	−	17.7812i	−0.155442	−	0.987845i
$325$		0			0
$326$			32.5911i			1.80506i
$327$		0			0
$328$		−	1.30306i		−	0.0719495i
$329$		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	$0.913301\pi$
$332$	28.3485			1.55583
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$		−	22.3939i		−	1.21987i	−0.792451	−	0.609936i	$-0.791195\pi$
$337$		−	22.3939i		−	1.21987i	0.792451	−	0.609936i	$-0.208805\pi$
$338$	18.3848			1.00000
$339$	−33.4217	+	15.3974i	−1.81522	+	0.836274i
$340$		0			0
$341$		0			0
$342$	−23.0167	+	26.9217i	−1.24460	+	1.45576i
$343$		0			0
$344$			28.2843i			1.52499i
$345$		0			0
$346$		0			0
$347$	23.5809			1.26589			0.632945	−	0.774197i	$-0.281846\pi$
$347$	23.5809			1.26589			0.632945	+	0.774197i	$0.281846\pi$
$348$		0			0
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$		0			0
$351$		0			0
$352$		−	37.3939i		−	1.99310i
$353$	−22.6274			−1.20434			−0.602168	−	0.798369i	$-0.705696\pi$
$353$	−22.6274			−1.20434			−0.602168	+	0.798369i	$0.705696\pi$
$354$	14.4949	+	31.4626i	0.770395	+	1.67222i
$355$		0			0
$356$		−	25.5201i		−	1.35256i
$357$		0			0
$358$		−	8.04541i		−	0.425213i
$359$		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
$363$	−51.4366	+	23.6969i	−2.69972	+	1.24377i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0			−	1.00000i	$-0.5\pi$
$367$		0			0				1.00000i	$0.5\pi$
$368$		0			0
$369$	1.05051	+	0.898133i	0.0546874	+	0.0467550i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0			−	1.00000i	$-0.5\pi$
$373$		0			0				1.00000i	$0.5\pi$
$374$		−	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	$-0.403264\pi$
$379$	11.6515			0.598499			0.299249	+	0.954175i	$0.403264\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		1.00000			$0$
$383$		0			0		−1.00000			$\pi$
$384$	−17.7980	+	8.19955i	−0.908248	+	0.418432i
$385$		0			0
$386$			5.22826i			0.266111i
$387$	−22.8024	−	19.4949i	−1.15911	−	0.990981i
$388$			20.0000i			1.01535i
$389$		0			0			−	1.00000i	$-0.5\pi$
$389$		0			0				1.00000i	$0.5\pi$
$390$		0			0
$391$		0			0
$392$	19.7990			1.00000
$393$	10.2494	+	22.2474i	0.517016	+	1.12224i
$394$		0			0
$395$		0			0
$396$	30.1464	+	25.7737i	1.51492	+	1.29518i
$397$		0			0			−	1.00000i	$-0.5\pi$
$397$		0			0				1.00000i	$0.5\pi$
$398$		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	$-0.118800\pi$
$402$	−14.7064	−	31.9217i	−0.733487	−	1.59211i
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$	−10.5352	+	4.85357i	−0.521569	+	0.240288i
$409$	18.3939			0.909519			0.454759	−	0.890614i	$-0.349725\pi$
$409$	18.3939			0.909519			0.454759	+	0.890614i	$0.349725\pi$
$410$		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
$417$	−28.8646	+	13.2980i	−1.41350	+	0.651204i
$418$		−	78.0454i		−	3.81733i
$419$		−	33.9732i		−	1.65970i	−0.557986	−	0.829851i	$-0.688426\pi$
$419$		−	33.9732i		−	1.65970i	0.557986	−	0.829851i	$-0.311574\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$	−21.2774			−1.03577
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$	9.40669			0.454689
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$	5.65685	−	20.0000i	0.272166	−	0.962250i
$433$		−	33.6969i		−	1.61937i	−0.586864	−	0.809686i	$-0.699638\pi$
$433$		−	33.6969i		−	1.61937i	0.586864	−	0.809686i	$-0.300362\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	−14.0386	−	30.4722i	−0.670790	−	1.45602i
$439$		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
$443$	−37.7873			−1.79533			−0.897664	−	0.440681i	$-0.854737\pi$
$443$	−37.7873			−1.79533			−0.897664	+	0.440681i	$0.854737\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$			39.2015i			1.85003i	0.379927	+	0.925016i	$0.375949\pi$
$449$			39.2015i			1.85003i	−0.379927	+	0.925016i	$0.624051\pi$
$450$		0			0
$451$	−3.04541			−0.143403
$452$	−42.4906			−1.99859
$453$		0			0
$454$	4.00000			0.187729
$455$		0			0
$456$	−37.1464	+	17.1134i	−1.73954	+	0.801410i
$457$			42.3939i			1.98310i	0.129718	+	0.991551i	$0.458593\pi$
$457$			42.3939i			1.98310i	−0.129718	+	0.991551i	$0.541407\pi$
$458$		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			−	1.00000i	$-0.5\pi$
$463$		0			0				1.00000i	$0.5\pi$
$464$		0			0
$465$		0			0
$466$	8.00000			0.370593
$467$	31.1127			1.43972			0.719862	−	0.694117i	$-0.244205\pi$
$467$	31.1127			1.43972			0.719862	+	0.694117i	$0.244205\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$			40.0000i			1.84115i
$473$	66.1037			3.03945
$474$		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
$482$	−2.39983			−0.109309
$483$		0			0
$484$	−65.3939			−2.97245
$485$		0			0
$486$	12.2247	+	18.3455i	0.554526	+	0.832167i
$487$		0			0			−	1.00000i	$-0.5\pi$
$487$		0			0				1.00000i	$0.5\pi$
$488$		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	$0.896615\pi$
$492$	0.667783	+	1.44949i	0.0301060	+	0.0653480i
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$	−31.5341	+	14.5278i	−1.41308	+	0.651007i
$499$	−14.0000			−0.626726			−0.313363	−	0.949633i	$-0.601456\pi$
$499$	−14.0000			−0.626726			−0.313363	+	0.949633i	$0.601456\pi$
$500$		0			0
$501$		0			0
$502$			29.3485i			1.30989i
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−20.4507	+	9.42168i	−0.908248	+	0.418432i
$508$		0			0
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$		0			0
$512$	−22.6274			−1.00000
$513$	11.8065	−	41.7423i	0.521271	−	1.84297i
$514$	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
$523$			3.04541i			0.133166i	0.997781	+	0.0665832i	$0.0212098\pi$
$523$			3.04541i			0.133166i	−0.997781	+	0.0665832i	$0.978790\pi$
$524$			28.2843i			1.23560i
$525$		0			0
$526$		0			0
$527$		0			0
$528$	19.1633	+	41.5959i	0.833976	+	1.81023i
$529$	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
$537$	4.12305	+	8.94949i	0.177923	+	0.386199i
$538$		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
$547$		−	15.6515i		−	0.669211i	−0.942358	−	0.334606i	$-0.891397\pi$
$547$		−	15.6515i		−	0.669211i	0.942358	−	0.334606i	$-0.108603\pi$
$548$	−33.0197			−1.41053
$549$		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		1.00000			$0$
$557$		0			0		−1.00000			$\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	11.3434	+	24.6219i	0.478917	+	1.03954i
$562$			40.0000i			1.68730i
$563$	36.7696			1.54965			0.774826	−	0.632175i	$-0.217837\pi$
$563$	36.7696			1.54965			0.774826	+	0.632175i	$0.217837\pi$
$564$		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	$-0.823964\pi$
$569$		−	25.0594i		−	1.05054i	0.850935	−	0.525271i	$-0.176036\pi$
$570$		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
$577$			12.3939i			0.515964i	0.966150	+	0.257982i	$0.0830575\pi$
$577$			12.3939i			0.515964i	−0.966150	+	0.257982i	$0.916942\pi$
$578$	16.1134			0.670228
$579$	−2.67934	−	5.81577i	−0.111349	−	0.241695i
$580$		0			0
$581$		0			0
$582$	−10.2494	−	22.2474i	−0.424853	−	0.922186i
$583$		0			0
$584$		−	38.7408i		−	1.60311i
$585$		0			0
$586$		0			0
$587$	−18.8455			−0.777836			−0.388918	−	0.921272i	$-0.627151\pi$
$587$	−18.8455			−0.777836			−0.388918	+	0.921272i	$0.627151\pi$
$588$	−22.0239	+	10.1464i	−0.908248	+	0.418432i
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−7.03896			−0.289055			−0.144528	−	0.989501i	$-0.546166\pi$
$593$	−7.03896			−0.289055			−0.144528	+	0.989501i	$0.546166\pi$
$594$	−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
$603$	32.7179	+	27.9722i	1.33238	+	1.13912i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0			−	1.00000i	$-0.5\pi$
$607$		0			0				1.00000i	$0.5\pi$
$608$	−47.2261			−1.91527
$609$		0			0
$610$		0			0
$611$		0			0
$612$	9.23171	−	10.7980i	0.373170	−	0.436482i
$613$		0			0			−	1.00000i	$-0.5\pi$
$613$		0			0				1.00000i	$0.5\pi$
$614$			34.4339i			1.38964i
$615$		0			0
$616$		0			0
$617$	39.5980			1.59415			0.797077	−	0.603877i	$-0.206378\pi$
$617$	39.5980			1.59415			0.797077	+	0.603877i	$0.206378\pi$
$618$		0			0
$619$	−26.0000			−1.04503			−0.522514	−	0.852631i	$-0.675006\pi$
$619$	−26.0000			−1.04503			−0.522514	+	0.852631i	$0.675006\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$		0			0
$626$			14.1421i			0.565233i
$627$	39.9961	+	86.8156i	1.59729	+	3.46708i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$	23.6684	−	10.9041i	0.940735	−	0.433398i
$634$		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	$-0.188654\pi$
$642$	−10.4637	+	4.82066i	−0.412971	+	0.190256i
$643$			50.0000i			1.97181i	0.167313	+	0.985904i	$0.446491\pi$
$643$			50.0000i			1.97181i	−0.167313	+	0.985904i	$0.553509\pi$
$644$		0			0
$645$		0			0
$646$	−27.9546			−1.09986
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$	3.95691	+	25.1464i	0.155442	+	0.987845i
$649$	93.4847			3.66960
$650$		0			0
$651$		0			0
$652$		−	46.0908i		−	1.80506i
$653$		0			0		1.00000			$0$
$653$		0			0		−1.00000			$\pi$
$654$		0			0
$655$		0			0
$656$			1.84281i			0.0719495i
$657$	31.2323	+	26.7020i	1.21849	+	1.04175i
$658$		0			0
$659$			8.45317i			0.329289i	0.986353	+	0.164644i	$0.0526477\pi$
$659$			8.45317i			0.329289i	−0.986353	+	0.164644i	$0.947352\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$	49.5617			1.92627
$663$		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
$673$		−	10.0000i		−	0.385472i	−0.981251	−	0.192736i	$-0.938264\pi$
$673$		−	10.0000i		−	0.385472i	0.981251	−	0.192736i	$-0.0617360\pi$
$674$			31.6697i			1.21987i
$675$		0			0
$676$	−26.0000			−1.00000
$677$		0			0		1.00000			$0$
$677$		0			0		−1.00000			$\pi$
$678$	47.2654	−	21.7753i	1.81522	−	0.836274i
$679$		0			0
$680$		0			0
$681$	−4.44949	+	2.04989i	−0.170505	+	0.0785519i
$682$		0			0
$683$	51.9294			1.98702			0.993512	−	0.113728i	$-0.0362792\pi$
$683$	51.9294			1.98702			0.993512	+	0.113728i	$0.0362792\pi$
$684$	32.5505	−	38.0730i	1.24460	−	1.45576i
$685$		0			0
$686$		0			0
$687$		0			0
$688$		−	40.0000i		−	1.52499i
$689$		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
$697$			1.09082i			0.0413176i
$698$		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
$708$	−20.4989	−	44.4949i	−0.770395	−	1.67222i
$709$		0			0		1.00000			$0$
$709$		0			0		−1.00000			$\pi$
$710$		0			0
$711$		0			0
$712$			36.0908i			1.35256i
$713$		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
$722$	−71.6963			−2.66826
$723$	2.66951	−	1.22985i	0.0992801	−	0.0457385i
$724$		0			0
$725$		0			0
$726$	72.7423	−	33.5125i	2.69972	−	1.24377i
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$		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			−	1.00000i	$-0.5\pi$
$733$		0			0				1.00000i	$0.5\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−94.8486			−3.49379
$738$	−1.48565	−	1.27015i	−0.0546874	−	0.0467550i
$739$	34.0000			1.25071			0.625355	−	0.780340i	$-0.284954\pi$
$739$	34.0000			1.25071			0.625355	+	0.780340i	$0.284954\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	27.6325	−	32.3207i	1.01102	−	1.18255i
$748$			31.3031i			1.14455i
$749$		0			0
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$		0			0
$753$	−15.0403	−	32.6464i	−0.548098	−	1.18970i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0			−	1.00000i	$-0.5\pi$
$757$		0			0				1.00000i	$0.5\pi$
$758$	−16.4778			−0.598499
$759$		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
$768$	25.1701	−	11.5959i	0.908248	−	0.418432i
$769$	−55.0908			−1.98663			−0.993313	−	0.115454i	$-0.963168\pi$
$769$	−55.0908			−1.98663			−0.993313	+	0.115454i	$0.963168\pi$
$770$		0			0
$771$	−17.7980	+	8.19955i	−0.640978	+	0.295300i
$772$		−	7.39388i		−	0.266111i
$773$		0			0		1.00000			$0$
$773$		0			0		−1.00000			$\pi$
$774$	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
$787$		−	50.0000i		−	1.78231i	−0.453701	−	0.891154i	$-0.649897\pi$
$787$		−	50.0000i		−	1.78231i	0.453701	−	0.891154i	$-0.350103\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$	−42.6335	−	36.4495i	−1.51492	−	1.29518i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	−29.0959	−	24.8755i	−1.02805	−	0.878934i
$802$			20.6515i			0.729231i
$803$	−90.5418			−3.19515
$804$	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	$-0.533636\pi$
$809$		−	56.5685i		−	1.98884i	0.105474	−	0.994422i	$-0.466364\pi$
$810$		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
$817$		−	83.4847i		−	2.92076i
$818$	−26.0129			−0.909519
$819$		0			0
$820$		0			0
$821$		0			0			−	1.00000i	$-0.5\pi$
$821$		0			0				1.00000i	$0.5\pi$
$822$	36.7302	−	16.9217i	1.28111	−	0.590211i
$823$		0			0			−	1.00000i	$-0.5\pi$
$823$		0			0				1.00000i	$0.5\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	56.6649			1.97043			0.985215	−	0.171321i	$-0.0548036\pi$
$827$	56.6649			1.97043			0.985215	+	0.171321i	$0.0548036\pi$
$828$		0			0
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−16.5741			−0.574258
$834$	40.8207	−	18.8062i	1.41350	−	0.651204i
$835$		0			0
$836$			110.373i			3.81733i
$837$		0			0
$838$			48.0454i			1.65970i
$839$		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
$843$	−20.4989	−	44.4949i	−0.706019	−	1.53249i
$844$	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			−	1.00000i	$-0.5\pi$
$853$		0			0				1.00000i	$0.5\pi$
$854$		0			0
$855$		0			0
$856$	−13.3031			−0.454689
$857$	35.4517			1.21101			0.605503	−	0.795843i	$-0.292972\pi$
$857$	35.4517			1.21101			0.605503	+	0.795843i	$0.292972\pi$
$858$		0			0
$859$	−21.6515			−0.738741			−0.369370	−	0.929282i	$-0.620427\pi$
$859$	−21.6515			−0.738741			−0.369370	+	0.929282i	$0.620427\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$	−8.00000	+	28.2843i	−0.272166	+	0.962250i
$865$		0			0
$866$			47.6547i			1.61937i
$867$	−17.9241	+	8.25765i	−0.608733	+	0.280445i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	22.8024	+	19.4949i	0.771744	+	0.659802i
$874$		0			0
$875$		0			0
$876$	19.8536	+	43.0942i	0.670790	+	1.45602i
$877$		0			0			−	1.00000i	$-0.5\pi$
$877$		0			0				1.00000i	$0.5\pi$
$878$		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	$0.598061\pi$
$882$	19.2990	−	22.5732i	0.649830	−	0.760080i
$883$			50.4393i			1.69742i	0.528861	+	0.848709i	$0.322619\pi$
$883$			50.4393i			1.69742i	−0.528861	+	0.848709i	$0.677381\pi$
$884$		0			0
$885$		0			0
$886$	53.4393			1.79533
$887$		0			0		1.00000			$0$
$887$		0			0		−1.00000			$\pi$
$888$		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
$898$		−	55.4393i		−	1.85003i
$899$		0			0
$900$		0			0
$901$		0			0
$902$	4.30686			0.143403
$903$		0			0
$904$	60.0908			1.99859
$905$		0			0
$906$		0			0
$907$			10.0000i			0.332045i	0.986122	+	0.166022i	$0.0530924\pi$
$907$			10.0000i			0.332045i	−0.986122	+	0.166022i	$0.946908\pi$
$908$	−5.65685			−0.187729
$909$		0			0
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$	52.5330	−	24.2020i	1.73954	−	0.801410i
$913$			93.6969i			3.10092i
$914$		−	59.9540i		−	1.98310i
$915$		0			0
$916$		0			0
$917$		0			0
$918$	−4.73545	+	16.7423i	−0.156293	+	0.552580i
$919$		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	$0.153582\pi$
$929$			28.2843i			0.927977i	−0.885841	+	0.463988i	$0.846418\pi$
$930$		0			0
$931$	−58.4393			−1.91527
$932$	−11.3137			−0.370593
$933$		0			0
$934$	−44.0000			−1.43972
$935$		0			0
$936$		0			0
$937$		−	61.0908i		−	1.99575i	−0.0651578	−	0.997875i	$-0.520755\pi$
$937$		−	61.0908i		−	1.99575i	0.0651578	−	0.997875i	$-0.479245\pi$
$938$		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
$947$	−53.7401			−1.74632			−0.873160	−	0.487435i	$-0.837933\pi$
$947$	−53.7401			−1.74632			−0.873160	+	0.487435i	$0.837933\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−59.0005			−1.91121			−0.955607	−	0.294646i	$-0.904798\pi$
$953$	−59.0005			−1.91121			−0.955607	+	0.294646i	$0.904798\pi$
$954$		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
$963$	9.16912	−	10.7247i	0.295471	−	0.345600i
$964$	3.39388			0.109309
$965$		0			0
$966$		0			0
$967$		0			0			−	1.00000i	$-0.5\pi$
$967$		0			0				1.00000i	$0.5\pi$
$968$	92.4809			2.97245
$969$	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	$0.833049\pi$
$972$	−17.2884	−	25.9444i	−0.554526	−	0.832167i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	25.9165			0.829144			0.414572	−	0.910017i	$-0.363931\pi$
$977$	25.9165			0.829144			0.414572	+	0.910017i	$0.363931\pi$
$978$	23.6203	+	51.2702i	0.755292	+	1.63944i
$979$	84.3485			2.69579
$980$		0			0
$981$		0			0
$982$		−	20.0000i		−	0.638226i
$983$		0			0		1.00000			$0$
$983$		0			0		−1.00000			$\pi$
$984$	−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
$993$	−55.1311	+	25.3990i	−1.74953	+	0.806012i
$994$		0			0
$995$		0			0
$996$	44.5959	−	20.5454i	1.41308	−	0.651007i
$997$		0			0			−	1.00000i	$-0.5\pi$
$997$		0			0				1.00000i	$0.5\pi$
$998$	19.7990			0.626726
$999$		0			0

Twists

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

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

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.m (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} +(1.57313 - 0.724745i) q^{3} +2.00000 q^{4} +(-2.22474 + 1.02494i) q^{6} -2.82843 q^{8} +(1.94949 - 2.28024i) q^{9} +O(q^{10})\)	\(q-1.41421 q^{2} +(1.57313 - 0.724745i) q^{3} +2.00000 q^{4} +(-2.22474 + 1.02494i) q^{6} -2.82843 q^{8} +(1.94949 - 2.28024i) q^{9} +6.61037i q^{11} +(3.14626 - 1.44949i) q^{12} +4.00000 q^{16} +2.36773 q^{17} +(-2.75699 + 3.22474i) q^{18} +8.34847 q^{19} -9.34847i q^{22} +(-4.44949 + 2.04989i) q^{24} +(1.41421 - 5.00000i) q^{27} -5.65685 q^{32} +(4.79083 + 10.3990i) q^{33} -3.34847 q^{34} +(3.89898 - 4.56048i) q^{36} -11.8065 q^{38} +0.460702i q^{41} -10.0000i q^{43} +13.2207i q^{44} +(6.29253 - 2.89898i) q^{48} -7.00000 q^{49} +(3.72474 - 1.71600i) q^{51} +(-2.00000 + 7.07107i) q^{54} +(13.1332 - 6.05051i) q^{57} -14.1421i q^{59} +8.00000 q^{64} +(-6.77526 - 14.7064i) q^{66} +14.3485i q^{67} +4.73545 q^{68} +(-5.51399 + 6.44949i) q^{72} +13.6969i q^{73} +16.6969 q^{76} +(-1.39898 - 8.89060i) q^{81} -0.651531i q^{82} +14.1742 q^{83} +14.1421i q^{86} -18.6969i q^{88} -12.7600i q^{89} +(-8.89898 + 4.09978i) q^{96} +10.0000i q^{97} +9.89949 q^{98} +(15.0732 + 12.8868i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{4} - 8 q^{6} - 4 q^{9}+O(q^{10}) \) 8 * q + 16 * q^4 - 8 * q^6 - 4 * q^9	\( 8 q + 16 q^{4} - 8 q^{6} - 4 q^{9} + 32 q^{16} + 8 q^{19} - 16 q^{24} + 32 q^{34} - 8 q^{36} - 56 q^{49} + 20 q^{51} - 16 q^{54} + 64 q^{64} - 64 q^{66} + 16 q^{76} + 28 q^{81} - 32 q^{96} + 52 q^{99}+O(q^{100}) \) 8 * q + 16 * q^4 - 8 * q^6 - 4 * q^9 + 32 * q^16 + 8 * q^19 - 16 * q^24 + 32 * q^34 - 8 * q^36 - 56 * q^49 + 20 * q^51 - 16 * q^54 + 64 * q^64 - 64 * q^66 + 16 * q^76 + 28 * q^81 - 32 * q^96 + 52 * q^99

Introduction

L-functions

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Database

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