LMFDB - Embedded newform 600.2.w.i.293.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [600,2,Mod(293,600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(600, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 2, 2, 3]))

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

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

chi := DirichletCharacter("600.293");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 600 = 2^{3} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	600.w (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:	$4.79102412128$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	8.0.3317760000.5
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 7x^{4} + 16 $ x^8 - 7*x^4 + 16
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label			293.1
Root			$-1.40294 - 0.178197i$ of defining polynomial
Character	$\chi$	$=$	600.293
Dual form			600.2.w.i.557.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.40294 - 0.178197i) q^{2} +(-1.22474 - 1.22474i) q^{3} +(1.93649 + 0.500000i) q^{4} +(1.50000 + 1.93649i) q^{6} +(-2.62769 - 1.04655i) q^{8} +3.00000i q^{9} +O(q^{10})$	\(q+(-1.40294 - 0.178197i) q^{2} +(-1.22474 - 1.22474i) q^{3} +(1.93649 + 0.500000i) q^{4} +(1.50000 + 1.93649i) q^{6} +(-2.62769 - 1.04655i) q^{8} +3.00000i q^{9} +(-1.75934 - 2.98408i) q^{12} +(3.50000 + 1.93649i) q^{16} +(3.16228 + 3.16228i) q^{17} +(0.534591 - 4.20883i) q^{18} -7.74597 q^{19} +(-6.32456 + 6.32456i) q^{23} +(1.93649 + 4.50000i) q^{24} +(3.67423 - 3.67423i) q^{27} +8.00000 q^{31} +(-4.56522 - 3.34047i) q^{32} +(-3.87298 - 5.00000i) q^{34} +(-1.50000 + 5.80948i) q^{36} +(10.8671 + 1.38031i) q^{38} +(10.0000 - 7.74597i) q^{46} +(6.32456 + 6.32456i) q^{47} +(-1.91490 - 6.65832i) q^{48} +7.00000i q^{49} -7.74597i q^{51} +(9.79796 + 9.79796i) q^{53} +(-5.80948 + 4.50000i) q^{54} +(9.48683 + 9.48683i) q^{57} +15.4919i q^{61} +(-11.2235 - 1.42558i) q^{62} +(5.80948 + 5.50000i) q^{64} +(4.54259 + 7.70486i) q^{68} +15.4919 q^{69} +(3.13964 - 7.88306i) q^{72} +(-15.0000 - 3.87298i) q^{76} -16.0000i q^{79} -9.00000 q^{81} +(2.44949 + 2.44949i) q^{83} +(-15.4097 + 9.08517i) q^{92} +(-9.79796 - 9.79796i) q^{93} +(-7.74597 - 10.0000i) q^{94} +(1.50000 + 9.68246i) q^{96} +(1.24738 - 9.82059i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 12 q^{6}+O(q^{10}) $ 8 * q + 12 * q^6	$ 8 q + 12 q^{6} + 28 q^{16} + 64 q^{31} - 12 q^{36} + 80 q^{46} - 120 q^{76} - 72 q^{81} + 12 q^{96}+O(q^{100}) $ 8 * q + 12 * q^6 + 28 * q^16 + 64 * q^31 - 12 * q^36 + 80 * q^46 - 120 * q^76 - 72 * q^81 + 12 * q^96

Display 100 coefficients 10 coefficients

Character values

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

$n$	$151$	$301$	$401$	$577$
$\chi(n)$	$1$	$-1$	$-1$	$e\left(\frac{3}{4}\right)$

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 $

$2$	−1.40294	−	0.178197i	−0.992030	−	0.126004i
$2$	−1.40294	−	0.178197i	−0.992030	−	0.126004i
$3$	−1.22474	−	1.22474i	−0.707107	−	0.707107i
$3$	−1.22474	−	1.22474i	−0.707107	−	0.707107i
$4$	1.93649	+	0.500000i	0.968246	+	0.250000i
$5$		0			0
$5$		0			0
$6$	1.50000	+	1.93649i	0.612372	+	0.790569i
$7$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$7$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$8$	−2.62769	−	1.04655i	−0.929028	−	0.370011i
$9$			3.00000i			1.00000i
$10$		0			0
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$	−1.75934	−	2.98408i	−0.507877	−	0.861430i
$13$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$13$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$14$		0			0
$15$		0			0
$16$	3.50000	+	1.93649i	0.875000	+	0.484123i
$17$	3.16228	+	3.16228i	0.766965	+	0.766965i	0.977571	−	0.210606i	$-0.0675437\pi$
$17$	3.16228	+	3.16228i	0.766965	+	0.766965i	−0.210606	+	0.977571i	$0.567544\pi$
$18$	0.534591	−	4.20883i	0.126004	−	0.992030i
$19$	−7.74597			−1.77705			−0.888523	−	0.458831i	$-0.848268\pi$
$19$	−7.74597			−1.77705			−0.888523	+	0.458831i	$0.848268\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−6.32456	+	6.32456i	−1.31876	+	1.31876i	−0.404004	+	0.914757i	$0.632382\pi$
$23$	−6.32456	+	6.32456i	−1.31876	+	1.31876i	−0.914757	+	0.404004i	$0.867618\pi$
$24$	1.93649	+	4.50000i	0.395285	+	0.918559i
$25$		0			0
$26$		0			0
$27$	3.67423	−	3.67423i	0.707107	−	0.707107i
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	8.00000			1.43684			0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000			1.43684			0.718421	+	0.695608i	$0.244865\pi$
$32$	−4.56522	−	3.34047i	−0.807024	−	0.590518i
$33$		0			0
$34$	−3.87298	−	5.00000i	−0.664211	−	0.857493i
$35$		0			0
$36$	−1.50000	+	5.80948i	−0.250000	+	0.968246i
$37$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$37$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$38$	10.8671	+	1.38031i	1.76288	+	0.223916i
$39$		0			0
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$43$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$44$		0			0
$45$		0			0
$46$	10.0000	−	7.74597i	1.47442	−	1.14208i
$47$	6.32456	+	6.32456i	0.922531	+	0.922531i	0.997208	−	0.0746766i	$-0.0237924\pi$
$47$	6.32456	+	6.32456i	0.922531	+	0.922531i	−0.0746766	+	0.997208i	$0.523792\pi$
$48$	−1.91490	−	6.65832i	−0.276392	−	0.961045i
$49$			7.00000i			1.00000i
$50$		0			0
$51$		−	7.74597i		−	1.08465i
$52$		0			0
$53$	9.79796	+	9.79796i	1.34585	+	1.34585i	0.890113	+	0.455740i	$0.150625\pi$
$53$	9.79796	+	9.79796i	1.34585	+	1.34585i	0.455740	+	0.890113i	$0.349375\pi$
$54$	−5.80948	+	4.50000i	−0.790569	+	0.612372i
$55$		0			0
$56$		0			0
$57$	9.48683	+	9.48683i	1.25656	+	1.25656i
$58$		0			0
$59$		0			0		1.00000			$0$
$59$		0			0		−1.00000			$\pi$
$60$		0			0
$61$			15.4919i			1.98354i	0.128037	+	0.991769i	$0.459132\pi$
$61$			15.4919i			1.98354i	−0.128037	+	0.991769i	$0.540868\pi$
$62$	−11.2235	−	1.42558i	−1.42539	−	0.181048i
$63$		0			0
$64$	5.80948	+	5.50000i	0.726184	+	0.687500i
$65$		0			0
$66$		0			0
$67$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$67$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$68$	4.54259	+	7.70486i	0.550869	+	0.934352i
$69$	15.4919			1.86501
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	3.13964	−	7.88306i	0.370011	−	0.929028i
$73$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$73$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$74$		0			0
$75$		0			0
$76$	−15.0000	−	3.87298i	−1.72062	−	0.444262i
$77$		0			0
$78$		0			0
$79$		−	16.0000i		−	1.80014i	−0.435745	−	0.900070i	$-0.643515\pi$
$79$		−	16.0000i		−	1.80014i	0.435745	−	0.900070i	$-0.356485\pi$
$80$		0			0
$81$	−9.00000			−1.00000
$82$		0			0
$83$	2.44949	+	2.44949i	0.268866	+	0.268866i	0.828643	−	0.559777i	$-0.189113\pi$
$83$	2.44949	+	2.44949i	0.268866	+	0.268866i	−0.559777	+	0.828643i	$0.689113\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$		0			0
$92$	−15.4097	+	9.08517i	−1.60658	+	0.947195i
$93$	−9.79796	−	9.79796i	−1.01600	−	1.01600i
$94$	−7.74597	−	10.0000i	−0.798935	−	1.03142i
$95$		0			0
$96$	1.50000	+	9.68246i	0.153093	+	0.988212i
$97$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$97$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$98$	1.24738	−	9.82059i	0.126004	−	0.992030i
$99$		0			0
$100$		0			0
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$	−1.38031	+	10.8671i	−0.136671	+	1.07601i
$103$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$103$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$104$		0			0
$105$		0			0
$106$	−12.0000	−	15.4919i	−1.16554	−	1.50471i
$107$	−7.34847	+	7.34847i	−0.710403	+	0.710403i	−0.966620	−	0.256216i	$-0.917524\pi$
$107$	−7.34847	+	7.34847i	−0.710403	+	0.710403i	0.256216	+	0.966620i	$0.417524\pi$
$108$	8.95224	−	5.27801i	0.861430	−	0.507877i
$109$	15.4919			1.48386			0.741929	−	0.670478i	$-0.233911\pi$
$109$	15.4919			1.48386			0.741929	+	0.670478i	$0.233911\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−3.16228	+	3.16228i	−0.297482	+	0.297482i	−0.840027	−	0.542545i	$-0.817461\pi$
$113$	−3.16228	+	3.16228i	−0.297482	+	0.297482i	0.542545	+	0.840027i	$0.317461\pi$
$114$	−11.6190	−	15.0000i	−1.08821	−	1.40488i
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−11.0000			−1.00000
$122$	2.76062	−	21.7343i	0.249934	−	1.96773i
$123$		0			0
$124$	15.4919	+	4.00000i	1.39122	+	0.359211i
$125$		0			0
$126$		0			0
$127$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$127$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$128$	−7.17027	−	8.75141i	−0.633769	−	0.773523i
$129$		0			0
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	−5.00000	−	11.6190i	−0.428746	−	0.996317i
$137$	−15.8114	−	15.8114i	−1.35086	−	1.35086i	−0.884703	−	0.466155i	$-0.845639\pi$
$137$	−15.8114	−	15.8114i	−1.35086	−	1.35086i	−0.466155	−	0.884703i	$-0.654361\pi$
$138$	−21.7343	−	2.76062i	−1.85014	−	0.234999i
$139$	−23.2379			−1.97101			−0.985506	−	0.169638i	$-0.945740\pi$
$139$	−23.2379			−1.97101			−0.985506	+	0.169638i	$0.945740\pi$
$140$		0			0
$141$		−	15.4919i		−	1.30466i
$142$		0			0
$143$		0			0
$144$	−5.80948	+	10.5000i	−0.484123	+	0.875000i
$145$		0			0
$146$		0			0
$147$	8.57321	−	8.57321i	0.707107	−	0.707107i
$148$		0			0
$149$		0			0		1.00000			$0$
$149$		0			0		−1.00000			$\pi$
$150$		0			0
$151$	−8.00000			−0.651031			−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000			−0.651031			−0.325515	+	0.945537i	$0.605538\pi$
$152$	20.3540	+	8.10653i	1.65093	+	0.657526i
$153$	−9.48683	+	9.48683i	−0.766965	+	0.766965i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$157$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$158$	−2.85115	+	22.4471i	−0.226825	+	1.78579i
$159$		−	24.0000i		−	1.90332i
$160$		0			0
$161$		0			0
$162$	12.6265	+	1.60377i	0.992030	+	0.126004i
$163$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$163$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$164$		0			0
$165$		0			0
$166$	−3.00000	−	3.87298i	−0.232845	−	0.300602i
$167$	6.32456	+	6.32456i	0.489409	+	0.489409i	0.908120	−	0.418711i	$-0.137518\pi$
$167$	6.32456	+	6.32456i	0.489409	+	0.489409i	−0.418711	+	0.908120i	$0.637518\pi$
$168$		0			0
$169$			13.0000i			1.00000i
$170$		0			0
$171$		−	23.2379i		−	1.77705i
$172$		0			0
$173$	−9.79796	−	9.79796i	−0.744925	−	0.744925i	0.228596	−	0.973521i	$-0.426586\pi$
$173$	−9.79796	−	9.79796i	−0.744925	−	0.744925i	−0.973521	+	0.228596i	$0.926586\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$		0			0
$181$			15.4919i			1.15151i	0.817624	+	0.575753i	$0.195291\pi$
$181$			15.4919i			1.15151i	−0.817624	+	0.575753i	$0.804709\pi$
$182$		0			0
$183$	18.9737	−	18.9737i	1.40257	−	1.40257i
$184$	23.2379	−	10.0000i	1.71312	−	0.737210i
$185$		0			0
$186$	12.0000	+	15.4919i	0.879883	+	1.13592i
$187$		0			0
$188$	9.08517	+	15.4097i	0.662604	+	1.12387i
$189$		0			0
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$	−0.379028	−	13.8512i	−0.0273540	−	0.999626i
$193$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$193$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$194$		0			0
$195$		0			0
$196$	−3.50000	+	13.5554i	−0.250000	+	0.968246i
$197$	19.5959	−	19.5959i	1.39615	−	1.39615i	0.585424	−	0.810727i	$-0.300928\pi$
$197$	19.5959	−	19.5959i	1.39615	−	1.39615i	0.810727	−	0.585424i	$-0.199072\pi$
$198$		0			0
$199$			16.0000i			1.13421i	0.823646	+	0.567105i	$0.191937\pi$
$199$			16.0000i			1.13421i	−0.823646	+	0.567105i	$0.808063\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$	3.87298	−	15.0000i	0.271163	−	1.05021i
$205$		0			0
$206$		0			0
$207$	−18.9737	−	18.9737i	−1.31876	−	1.31876i
$208$		0			0
$209$		0			0
$210$		0			0
$211$			7.74597i			0.533254i	0.963800	+	0.266627i	$0.0859092\pi$
$211$			7.74597i			0.533254i	−0.963800	+	0.266627i	$0.914091\pi$
$212$	14.0747	+	23.8726i	0.966653	+	1.63958i
$213$		0			0
$214$	11.6190	−	9.00000i	0.794255	−	0.615227i
$215$		0			0
$216$	−13.5000	+	5.80948i	−0.918559	+	0.395285i
$217$		0			0
$218$	−21.7343	−	2.76062i	−1.47203	−	0.186972i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$223$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$224$		0			0
$225$		0			0
$226$	5.00000	−	3.87298i	0.332595	−	0.257627i
$227$	−17.1464	+	17.1464i	−1.13805	+	1.13805i	−0.149249	+	0.988800i	$0.547686\pi$
$227$	−17.1464	+	17.1464i	−1.13805	+	1.13805i	−0.988800	+	0.149249i	$0.952314\pi$
$228$	13.6278	+	23.1146i	0.902520	+	1.53080i
$229$	−15.4919			−1.02374			−0.511868	−	0.859064i	$-0.671046\pi$
$229$	−15.4919			−1.02374			−0.511868	+	0.859064i	$0.671046\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	15.8114	−	15.8114i	1.03584	−	1.03584i	0.0365050	−	0.999333i	$-0.488378\pi$
$233$	15.8114	−	15.8114i	1.03584	−	1.03584i	0.999333	−	0.0365050i	$-0.0116225\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−19.5959	+	19.5959i	−1.27289	+	1.27289i
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	−2.00000			−0.128831			−0.0644157	−	0.997923i	$-0.520518\pi$
$241$	−2.00000			−0.128831			−0.0644157	+	0.997923i	$0.520518\pi$
$242$	15.4324	+	1.96017i	0.992030	+	0.126004i
$243$	11.0227	+	11.0227i	0.707107	+	0.707107i
$244$	−7.74597	+	30.0000i	−0.495885	+	1.92055i
$245$		0			0
$246$		0			0
$247$		0			0
$248$	−21.0215	−	8.37238i	−1.33487	−	0.531647i
$249$		−	6.00000i		−	0.380235i
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	8.50000	+	13.5554i	0.531250	+	0.847215i
$257$	−22.1359	−	22.1359i	−1.38080	−	1.38080i	−0.843196	−	0.537606i	$-0.819329\pi$
$257$	−22.1359	−	22.1359i	−1.38080	−	1.38080i	−0.537606	−	0.843196i	$-0.680671\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	6.32456	−	6.32456i	0.389989	−	0.389989i	−0.484695	−	0.874683i	$-0.661069\pi$
$263$	6.32456	−	6.32456i	0.389989	−	0.389989i	0.874683	+	0.484695i	$0.161069\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$	32.0000			1.94386			0.971931	−	0.235267i	$-0.0755965\pi$
$271$	32.0000			1.94386			0.971931	+	0.235267i	$0.0755965\pi$
$272$	4.94425	+	17.1917i	0.299789	+	1.04240i
$273$		0			0
$274$	19.3649	+	25.0000i	1.16988	+	1.51031i
$275$		0			0
$276$	30.0000	+	7.74597i	1.80579	+	0.466252i
$277$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$277$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$278$	32.6014	+	4.14092i	1.95530	+	0.248356i
$279$			24.0000i			1.43684i
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$	−2.76062	+	21.7343i	−0.164392	+	1.29426i
$283$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$283$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	10.0214	−	13.6957i	0.590518	−	0.807024i
$289$			3.00000i			0.176471i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	9.79796	+	9.79796i	0.572403	+	0.572403i	0.932799	−	0.360396i	$-0.117359\pi$
$293$	9.79796	+	9.79796i	0.572403	+	0.572403i	−0.360396	+	0.932799i	$0.617359\pi$
$294$	−13.5554	+	10.5000i	−0.790569	+	0.612372i
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$	11.2235	+	1.42558i	0.645842	+	0.0820327i
$303$		0			0
$304$	−27.1109	−	15.0000i	−1.55492	−	0.860309i
$305$		0			0
$306$	15.0000	−	11.6190i	0.857493	−	0.664211i
$307$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$307$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$313$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$314$		0			0
$315$		0			0
$316$	8.00000	−	30.9839i	0.450035	−	1.74298i
$317$	−19.5959	+	19.5959i	−1.10062	+	1.10062i	−0.106280	+	0.994336i	$0.533894\pi$
$317$	−19.5959	+	19.5959i	−1.10062	+	1.10062i	−0.994336	+	0.106280i	$0.966106\pi$
$318$	−4.27673	+	33.6706i	−0.239827	+	1.88815i
$319$		0			0
$320$		0			0
$321$	18.0000			1.00466
$322$		0			0
$323$	−24.4949	−	24.4949i	−1.36293	−	1.36293i
$324$	−17.4284	−	4.50000i	−0.968246	−	0.250000i
$325$		0			0
$326$		0			0
$327$	−18.9737	−	18.9737i	−1.04925	−	1.04925i
$328$		0			0
$329$		0			0
$330$		0			0
$331$			23.2379i			1.27727i	0.769510	+	0.638635i	$0.220501\pi$
$331$			23.2379i			1.27727i	−0.769510	+	0.638635i	$0.779499\pi$
$332$	3.51867	+	5.96816i	0.193112	+	0.327545i
$333$		0			0
$334$	−7.74597	−	10.0000i	−0.423840	−	0.547176i
$335$		0			0
$336$		0			0
$337$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$337$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$338$	2.31656	−	18.2382i	0.126004	−	0.992030i
$339$	7.74597			0.420703
$340$		0			0
$341$		0			0
$342$	−4.14092	+	32.6014i	−0.223916	+	1.76288i
$343$		0			0
$344$		0			0
$345$		0			0
$346$	12.0000	+	15.4919i	0.645124	+	0.832851i
$347$	7.34847	−	7.34847i	0.394486	−	0.394486i	−0.481797	−	0.876283i	$-0.660016\pi$
$347$	7.34847	−	7.34847i	0.394486	−	0.394486i	0.876283	+	0.481797i	$0.160016\pi$
$348$		0			0
$349$	−15.4919			−0.829264			−0.414632	−	0.909989i	$-0.636090\pi$
$349$	−15.4919			−0.829264			−0.414632	+	0.909989i	$0.636090\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	22.1359	−	22.1359i	1.17818	−	1.17818i	0.197969	−	0.980208i	$-0.436565\pi$
$353$	22.1359	−	22.1359i	1.17818	−	1.17818i	0.980208	−	0.197969i	$-0.0634346\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	41.0000			2.15789
$362$	2.76062	−	21.7343i	0.145095	−	1.14233i
$363$	13.4722	+	13.4722i	0.707107	+	0.707107i
$364$		0			0
$365$		0			0
$366$	−30.0000	+	23.2379i	−1.56813	+	1.21466i
$367$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$367$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$368$	−34.3834	+	9.88849i	−1.79236	+	0.515473i
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−14.0747	−	23.8726i	−0.729738	−	1.23774i
$373$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$373$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$374$		0			0
$375$		0			0
$376$	−10.0000	−	23.2379i	−0.515711	−	1.19840i
$377$		0			0
$378$		0			0
$379$	38.7298			1.98942			0.994709	−	0.102733i	$-0.0327588\pi$
$379$	38.7298			1.98942			0.994709	+	0.102733i	$0.0327588\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−6.32456	+	6.32456i	−0.323170	+	0.323170i	−0.849982	−	0.526812i	$-0.823387\pi$
$383$	−6.32456	+	6.32456i	−0.323170	+	0.323170i	0.526812	+	0.849982i	$0.323387\pi$
$384$	−1.93649	+	19.5000i	−0.0988212	+	0.995105i
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0		1.00000			$0$
$389$		0			0		−1.00000			$\pi$
$390$		0			0
$391$	−40.0000			−2.02289
$392$	7.32584	−	18.3938i	0.370011	−	0.929028i
$393$		0			0
$394$	−30.9839	+	24.0000i	−1.56094	+	1.20910i
$395$		0			0
$396$		0			0
$397$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$397$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$398$	2.85115	−	22.4471i	0.142915	−	1.12517i
$399$		0			0
$400$		0			0
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$	−8.10653	+	20.3540i	−0.401333	+	1.00767i
$409$		−	26.0000i		−	1.28562i	−0.766027	−	0.642809i	$-0.777769\pi$
$409$		−	26.0000i		−	1.28562i	0.766027	−	0.642809i	$-0.222231\pi$
$410$		0			0
$411$			38.7298i			1.91040i
$412$		0			0
$413$		0			0
$414$	23.2379	+	30.0000i	1.14208	+	1.47442i
$415$		0			0
$416$		0			0
$417$	28.4605	+	28.4605i	1.39372	+	1.39372i
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$		−	15.4919i		−	0.755031i	−0.926003	−	0.377515i	$-0.876779\pi$
$421$		−	15.4919i		−	0.755031i	0.926003	−	0.377515i	$-0.123221\pi$
$422$	1.38031	−	10.8671i	0.0671923	−	0.529004i
$423$	−18.9737	+	18.9737i	−0.922531	+	0.922531i
$424$	−15.4919	−	36.0000i	−0.752355	−	1.74831i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	−17.9045	+	10.5560i	−0.865446	+	0.510244i
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$	19.9749	−	5.74470i	0.961045	−	0.276392i
$433$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$433$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$434$		0			0
$435$		0			0
$436$	30.0000	+	7.74597i	1.43674	+	0.370965i
$437$	48.9898	−	48.9898i	2.34350	−	2.34350i
$438$		0			0
$439$			16.0000i			0.763638i	0.924237	+	0.381819i	$0.124702\pi$
$439$			16.0000i			0.763638i	−0.924237	+	0.381819i	$0.875298\pi$
$440$		0			0
$441$	−21.0000			−1.00000
$442$		0			0
$443$	−26.9444	−	26.9444i	−1.28017	−	1.28017i	−0.940572	−	0.339595i	$-0.889710\pi$
$443$	−26.9444	−	26.9444i	−1.28017	−	1.28017i	−0.339595	−	0.940572i	$-0.610290\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$		0			0
$451$		0			0
$452$	−7.70486	+	4.54259i	−0.362406	+	0.213665i
$453$	9.79796	+	9.79796i	0.460348	+	0.460348i
$454$	27.1109	−	21.0000i	1.27238	−	0.985579i
$455$		0			0
$456$	−15.0000	−	34.8569i	−0.702439	−	1.63232i
$457$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$457$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$458$	21.7343	+	2.76062i	1.01558	+	0.128995i
$459$	23.2379			1.08465
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$463$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$464$		0			0
$465$		0			0
$466$	−25.0000	+	19.3649i	−1.15810	+	0.897062i
$467$	17.1464	−	17.1464i	0.793442	−	0.793442i	−0.188610	−	0.982052i	$-0.560398\pi$
$467$	17.1464	−	17.1464i	0.793442	−	0.793442i	0.982052	+	0.188610i	$0.0603982\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$	30.9839	−	24.0000i	1.42314	−	1.10236i
$475$		0			0
$476$		0			0
$477$	−29.3939	+	29.3939i	−1.34585	+	1.34585i
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$		0			0
$482$	2.80588	+	0.356394i	0.127805	+	0.0162333i
$483$		0			0
$484$	−21.3014	−	5.50000i	−0.968246	−	0.250000i
$485$		0			0
$486$	−13.5000	−	17.4284i	−0.612372	−	0.790569i
$487$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$487$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$488$	16.2131	−	40.7079i	0.733930	−	1.84276i
$489$		0			0
$490$		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
$496$	28.0000	+	15.4919i	1.25724	+	0.695608i
$497$		0			0
$498$	−1.06918	+	8.41765i	−0.0479112	+	0.377204i
$499$	7.74597			0.346757			0.173379	−	0.984855i	$-0.444532\pi$
$499$	7.74597			0.346757			0.173379	+	0.984855i	$0.444532\pi$
$500$		0			0
$501$		−	15.4919i		−	0.692129i
$502$		0			0
$503$	−31.6228	+	31.6228i	−1.40999	+	1.40999i	−0.650386	+	0.759604i	$0.725393\pi$
$503$	−31.6228	+	31.6228i	−1.40999	+	1.40999i	−0.759604	+	0.650386i	$0.774607\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	15.9217	−	15.9217i	0.707107	−	0.707107i
$508$		0			0
$509$		0			0		1.00000			$0$
$509$		0			0		−1.00000			$\pi$
$510$		0			0
$511$		0			0
$512$	−9.50947	−	20.5322i	−0.420263	−	0.907402i
$513$	−28.4605	+	28.4605i	−1.25656	+	1.25656i
$514$	27.1109	+	35.0000i	1.19581	+	1.54378i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$			24.0000i			1.05348i
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$523$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$524$		0			0
$525$		0			0
$526$	−10.0000	+	7.74597i	−0.436021	+	0.337740i
$527$	25.2982	+	25.2982i	1.10201	+	1.10201i
$528$		0			0
$529$		−	57.0000i		−	2.47826i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$		−	46.4758i		−	1.99815i	−0.0429934	−	0.999075i	$-0.513689\pi$
$541$		−	46.4758i		−	1.99815i	0.0429934	−	0.999075i	$-0.486311\pi$
$542$	−44.8941	−	5.70230i	−1.92837	−	0.244935i
$543$	18.9737	−	18.9737i	0.814238	−	0.814238i
$544$	−3.87298	−	25.0000i	−0.166053	−	1.07187i
$545$		0			0
$546$		0			0
$547$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$547$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$548$	−22.7129	−	38.5243i	−0.970248	−	1.64568i
$549$	−46.4758			−1.98354
$550$		0			0
$551$		0			0
$552$	−40.7079	−	16.2131i	−1.73265	−	0.690073i
$553$		0			0
$554$		0			0
$555$		0			0
$556$	−45.0000	−	11.6190i	−1.90843	−	0.492753i
$557$	−29.3939	+	29.3939i	−1.24546	+	1.24546i	−0.287754	+	0.957704i	$0.592909\pi$
$557$	−29.3939	+	29.3939i	−1.24546	+	1.24546i	−0.957704	+	0.287754i	$0.907091\pi$
$558$	4.27673	−	33.6706i	0.181048	−	1.42539i
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	22.0454	+	22.0454i	0.929103	+	0.929103i	0.997648	−	0.0685449i	$-0.0218356\pi$
$563$	22.0454	+	22.0454i	0.929103	+	0.929103i	−0.0685449	+	0.997648i	$0.521836\pi$
$564$	7.74597	−	30.0000i	0.326164	−	1.26323i
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$		0			0
$571$		−	38.7298i		−	1.62079i	−0.585882	−	0.810397i	$-0.699252\pi$
$571$		−	38.7298i		−	1.62079i	0.585882	−	0.810397i	$-0.300748\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−16.5000	+	17.4284i	−0.687500	+	0.726184i
$577$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$577$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$578$	0.534591	−	4.20883i	0.0222361	−	0.175064i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$	−12.0000	−	15.4919i	−0.495715	−	0.639966i
$587$	−31.8434	+	31.8434i	−1.31432	+	1.31432i	−0.396116	+	0.918201i	$0.629642\pi$
$587$	−31.8434	+	31.8434i	−1.31432	+	1.31432i	−0.918201	+	0.396116i	$0.870358\pi$
$588$	20.8886	−	12.3154i	0.861430	−	0.507877i
$589$	−61.9677			−2.55334
$590$		0			0
$591$	−48.0000			−1.97446
$592$		0			0
$593$	3.16228	−	3.16228i	0.129859	−	0.129859i	−0.639190	−	0.769049i	$-0.720730\pi$
$593$	3.16228	−	3.16228i	0.129859	−	0.129859i	0.769049	+	0.639190i	$0.220730\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	19.5959	−	19.5959i	0.802008	−	0.802008i
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	−38.0000			−1.55005			−0.775026	−	0.631929i	$-0.782263\pi$
$601$	−38.0000			−1.55005			−0.775026	+	0.631929i	$0.782263\pi$
$602$		0			0
$603$		0			0
$604$	−15.4919	−	4.00000i	−0.630358	−	0.162758i
$605$		0			0
$606$		0			0
$607$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$607$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$608$	35.3620	+	25.8752i	1.43412	+	1.04938i
$609$		0			0
$610$		0			0
$611$		0			0
$612$	−23.1146	+	13.6278i	−0.934352	+	0.550869i
$613$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$613$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	34.7851	+	34.7851i	1.40039	+	1.40039i	0.798810	+	0.601584i	$0.205464\pi$
$617$	34.7851	+	34.7851i	1.40039	+	1.40039i	0.601584	+	0.798810i	$0.294536\pi$
$618$		0			0
$619$	23.2379			0.934010			0.467005	−	0.884255i	$-0.345333\pi$
$619$	23.2379			0.934010			0.467005	+	0.884255i	$0.345333\pi$
$620$		0			0
$621$			46.4758i			1.86501i
$622$		0			0
$623$		0			0
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−32.0000			−1.27390			−0.636950	−	0.770905i	$-0.719804\pi$
$631$	−32.0000			−1.27390			−0.636950	+	0.770905i	$0.719804\pi$
$632$	−16.7448	+	42.0430i	−0.666071	+	1.67238i
$633$	9.48683	−	9.48683i	0.377068	−	0.377068i
$634$	30.9839	−	24.0000i	1.23053	−	0.953162i
$635$		0			0
$636$	12.0000	−	46.4758i	0.475831	−	1.84289i
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$	−25.2530	−	3.20755i	−0.996655	−	0.126592i
$643$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$643$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$644$		0			0
$645$		0			0
$646$	30.0000	+	38.7298i	1.18033	+	1.52380i
$647$	31.6228	+	31.6228i	1.24322	+	1.24322i	0.958658	+	0.284562i	$0.0918482\pi$
$647$	31.6228	+	31.6228i	1.24322	+	1.24322i	0.284562	+	0.958658i	$0.408152\pi$
$648$	23.6492	+	9.41893i	0.929028	+	0.370011i
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−9.79796	−	9.79796i	−0.383424	−	0.383424i	0.488910	−	0.872334i	$-0.337395\pi$
$653$	−9.79796	−	9.79796i	−0.383424	−	0.383424i	−0.872334	+	0.488910i	$0.837395\pi$
$654$	23.2379	+	30.0000i	0.908674	+	1.17309i
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$		−	46.4758i		−	1.80770i	−0.427850	−	0.903850i	$-0.640729\pi$
$661$		−	46.4758i		−	1.80770i	0.427850	−	0.903850i	$-0.359271\pi$
$662$	4.14092	−	32.6014i	0.160942	−	1.26709i
$663$		0			0
$664$	−3.87298	−	9.00000i	−0.150301	−	0.349268i
$665$		0			0
$666$		0			0
$667$		0			0
$668$	9.08517	+	15.4097i	0.351516	+	0.596220i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$673$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$674$		0			0
$675$		0			0
$676$	−6.50000	+	25.1744i	−0.250000	+	0.968246i
$677$	29.3939	−	29.3939i	1.12970	−	1.12970i	0.139473	−	0.990226i	$-0.455459\pi$
$677$	29.3939	−	29.3939i	1.12970	−	1.12970i	0.990226	−	0.139473i	$-0.0445407\pi$
$678$	−10.8671	−	1.38031i	−0.417350	−	0.0530104i
$679$		0			0
$680$		0			0
$681$	42.0000			1.60944
$682$		0			0
$683$	26.9444	+	26.9444i	1.03100	+	1.03100i	0.999504	+	0.0314944i	$0.0100266\pi$
$683$	26.9444	+	26.9444i	1.03100	+	1.03100i	0.0314944	+	0.999504i	$0.489973\pi$
$684$	11.6190	−	45.0000i	0.444262	−	1.72062i
$685$		0			0
$686$		0			0
$687$	18.9737	+	18.9737i	0.723891	+	0.723891i
$688$		0			0
$689$		0			0
$690$		0			0
$691$			7.74597i			0.294670i	0.989087	+	0.147335i	$0.0470696\pi$
$691$			7.74597i			0.294670i	−0.989087	+	0.147335i	$0.952930\pi$
$692$	−14.0747	−	23.8726i	−0.535039	−	0.907502i
$693$		0			0
$694$	−11.6190	+	9.00000i	−0.441049	+	0.341635i
$695$		0			0
$696$		0			0
$697$		0			0
$698$	21.7343	+	2.76062i	0.822655	+	0.104491i
$699$	−38.7298			−1.46490
$700$		0			0
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	−35.0000	+	27.1109i	−1.31724	+	1.02033i
$707$		0			0
$708$		0			0
$709$	46.4758			1.74544			0.872718	−	0.488225i	$-0.162356\pi$
$709$	46.4758			1.74544			0.872718	+	0.488225i	$0.162356\pi$
$710$		0			0
$711$	48.0000			1.80014
$712$		0			0
$713$	−50.5964	+	50.5964i	−1.89485	+	1.89485i
$714$		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
$721$		0			0
$722$	−57.5206	−	7.30608i	−2.14070	−	0.271904i
$723$	2.44949	+	2.44949i	0.0910975	+	0.0910975i
$724$	−7.74597	+	30.0000i	−0.287877	+	1.11494i
$725$		0			0
$726$	−16.5000	−	21.3014i	−0.612372	−	0.790569i
$727$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$727$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$728$		0			0
$729$		−	27.0000i		−	1.00000i
$730$		0			0
$731$		0			0
$732$	46.2292	−	27.2555i	1.70868	−	1.00739i
$733$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$733$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$734$		0			0
$735$		0			0
$736$	50.0000	−	7.74597i	1.84302	−	0.285520i
$737$		0			0
$738$		0			0
$739$	54.2218			1.99458			0.997290	−	0.0735712i	$-0.0234396\pi$
$739$	54.2218			1.99458			0.997290	+	0.0735712i	$0.0234396\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	31.6228	−	31.6228i	1.16013	−	1.16013i	0.175680	−	0.984447i	$-0.443788\pi$
$743$	31.6228	−	31.6228i	1.16013	−	1.16013i	0.984447	−	0.175680i	$-0.0562123\pi$
$744$	15.4919	+	36.0000i	0.567962	+	1.31982i
$745$		0			0
$746$		0			0
$747$	−7.34847	+	7.34847i	−0.268866	+	0.268866i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	8.00000			0.291924			0.145962	−	0.989290i	$-0.453372\pi$
$751$	8.00000			0.291924			0.145962	+	0.989290i	$0.453372\pi$
$752$	9.88849	+	34.3834i	0.360596	+	1.25383i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$757$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$758$	−54.3357	−	6.90154i	−1.97356	−	0.250675i
$759$		0			0
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$	10.0000	−	7.74597i	0.361315	−	0.279873i
$767$		0			0
$768$	6.19163	−	27.0123i	0.223421	−	0.974722i
$769$			46.0000i			1.65880i	0.558653	+	0.829401i	$0.311318\pi$
$769$			46.0000i			1.65880i	−0.558653	+	0.829401i	$0.688682\pi$
$770$		0			0
$771$			54.2218i			1.95275i
$772$		0			0
$773$	39.1918	+	39.1918i	1.40963	+	1.40963i	0.761686	+	0.647947i	$0.224372\pi$
$773$	39.1918	+	39.1918i	1.40963	+	1.40963i	0.647947	+	0.761686i	$0.275628\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$	56.1177	+	7.12788i	2.00676	+	0.254892i
$783$		0			0
$784$	−13.5554	+	24.5000i	−0.484123	+	0.875000i
$785$		0			0
$786$		0			0
$787$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$787$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$788$	47.7453	−	28.1494i	1.70086	−	1.00278i
$789$	−15.4919			−0.551527
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$	−8.00000	+	30.9839i	−0.283552	+	1.09819i
$797$	−19.5959	+	19.5959i	−0.694123	+	0.694123i	−0.963136	−	0.269013i	$-0.913302\pi$
$797$	−19.5959	+	19.5959i	−0.694123	+	0.694123i	0.269013	+	0.963136i	$0.413302\pi$
$798$		0			0
$799$			40.0000i			1.41510i
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0			−	1.00000i	$-0.5\pi$
$809$		0			0				1.00000i	$0.5\pi$
$810$		0			0
$811$			23.2379i			0.815993i	0.912983	+	0.407997i	$0.133772\pi$
$811$			23.2379i			0.815993i	−0.912983	+	0.407997i	$0.866228\pi$
$812$		0			0
$813$	−39.1918	−	39.1918i	−1.37452	−	1.37452i
$814$		0			0
$815$		0			0
$816$	15.0000	−	27.1109i	0.525105	−	0.949071i
$817$		0			0
$818$	−4.63312	+	36.4765i	−0.161993	+	1.27537i
$819$		0			0
$820$		0			0
$821$		0			0			−	1.00000i	$-0.5\pi$
$821$		0			0				1.00000i	$0.5\pi$
$822$	6.90154	−	54.3357i	0.240719	−	1.89518i
$823$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$823$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	31.8434	−	31.8434i	1.10730	−	1.10730i	0.113799	−	0.993504i	$-0.463698\pi$
$827$	31.8434	−	31.8434i	1.10730	−	1.10730i	0.993504	−	0.113799i	$-0.0363018\pi$
$828$	−27.2555	−	46.2292i	−0.947195	−	1.60658i
$829$	46.4758			1.61417			0.807086	−	0.590434i	$-0.201044\pi$
$829$	46.4758			1.61417			0.807086	+	0.590434i	$0.201044\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−22.1359	+	22.1359i	−0.766965	+	0.766965i
$834$	−34.8569	−	45.0000i	−1.20699	−	1.55822i
$835$		0			0
$836$		0			0
$837$	29.3939	−	29.3939i	1.01600	−	1.01600i
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	29.0000			1.00000
$842$	−2.76062	+	21.7343i	−0.0951371	+	0.749013i
$843$		0			0
$844$	−3.87298	+	15.0000i	−0.133314	+	0.516321i
$845$		0			0
$846$	30.0000	−	23.2379i	1.03142	−	0.798935i
$847$		0			0
$848$	15.3192	+	53.2665i	0.526063	+	1.82918i
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$853$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$854$		0			0
$855$		0			0
$856$	27.0000	−	11.6190i	0.922841	−	0.397128i
$857$	−41.1096	−	41.1096i	−1.40428	−	1.40428i	−0.785812	−	0.618466i	$-0.787755\pi$
$857$	−41.1096	−	41.1096i	−1.40428	−	1.40428i	−0.618466	−	0.785812i	$-0.712245\pi$
$858$		0			0
$859$	−38.7298			−1.32144			−0.660722	−	0.750630i	$-0.729750\pi$
$859$	−38.7298			−1.32144			−0.660722	+	0.750630i	$0.729750\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−31.6228	+	31.6228i	−1.07645	+	1.07645i	−0.0796271	+	0.996825i	$0.525373\pi$
$863$	−31.6228	+	31.6228i	−1.07645	+	1.07645i	−0.996825	+	0.0796271i	$0.974627\pi$
$864$	−29.0474	+	4.50000i	−0.988212	+	0.153093i
$865$		0			0
$866$		0			0
$867$	3.67423	−	3.67423i	0.124784	−	0.124784i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	−40.7079	−	16.2131i	−1.37855	−	0.549043i
$873$		0			0
$874$	−77.4597	+	60.0000i	−2.62011	+	2.02953i
$875$		0			0
$876$		0			0
$877$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$877$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$878$	2.85115	−	22.4471i	0.0962217	−	0.757552i
$879$		−	24.0000i		−	0.809500i
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	29.4618	+	3.74214i	0.992030	+	0.126004i
$883$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$883$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$884$		0			0
$885$		0			0
$886$	33.0000	+	42.6028i	1.10866	+	1.43127i
$887$	−6.32456	−	6.32456i	−0.212358	−	0.212358i	0.592911	−	0.805268i	$-0.297979\pi$
$887$	−6.32456	−	6.32456i	−0.212358	−	0.212358i	−0.805268	+	0.592911i	$0.797979\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−48.9898	−	48.9898i	−1.63938	−	1.63938i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$			61.9677i			2.06444i
$902$		0			0
$903$		0			0
$904$	11.6190	−	5.00000i	0.386441	−	0.166298i
$905$		0			0
$906$	−12.0000	−	15.4919i	−0.398673	−	0.514685i
$907$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$907$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$908$	−41.7771	+	24.6307i	−1.38642	+	0.817399i
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$	14.8327	+	51.5751i	0.491161	+	1.70782i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−30.0000	−	7.74597i	−0.991228	−	0.255934i
$917$		0			0
$918$	−32.6014	−	4.14092i	−1.07601	−	0.136671i
$919$		−	56.0000i		−	1.84727i	−0.383274	−	0.923635i	$-0.625203\pi$
$919$		−	56.0000i		−	1.84727i	0.383274	−	0.923635i	$-0.374797\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0			−	1.00000i	$-0.5\pi$
$929$		0			0				1.00000i	$0.5\pi$
$930$		0			0
$931$		−	54.2218i		−	1.77705i
$932$	38.5243	−	22.7129i	1.26191	−	0.743987i
$933$		0			0
$934$	−27.1109	+	21.0000i	−0.887095	+	0.687141i
$935$		0			0
$936$		0			0
$937$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$937$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0			−	1.00000i	$-0.5\pi$
$941$		0			0				1.00000i	$0.5\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−41.6413	+	41.6413i	−1.35316	+	1.35316i	−0.471060	+	0.882101i	$0.656129\pi$
$947$	−41.6413	+	41.6413i	−1.35316	+	1.35316i	−0.882101	+	0.471060i	$0.843871\pi$
$948$	−47.7453	+	28.1494i	−1.55069	+	0.914249i
$949$		0			0
$950$		0			0
$951$	48.0000			1.55651
$952$		0			0
$953$	41.1096	−	41.1096i	1.33167	−	1.33167i	0.427795	−	0.903876i	$-0.359290\pi$
$953$	41.1096	−	41.1096i	1.33167	−	1.33167i	0.903876	−	0.427795i	$-0.140710\pi$
$954$	46.4758	−	36.0000i	1.50471	−	1.16554i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	33.0000			1.06452
$962$		0			0
$963$	−22.0454	−	22.0454i	−0.710403	−	0.710403i
$964$	−3.87298	−	1.00000i	−0.124740	−	0.0322078i
$965$		0			0
$966$		0			0
$967$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$967$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$968$	28.9046	+	11.5120i	0.929028	+	0.370011i
$969$			60.0000i			1.92748i
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$	15.8340	+	26.8567i	0.507877	+	0.861430i
$973$		0			0
$974$		0			0
$975$		0			0
$976$	−30.0000	+	54.2218i	−0.960277	+	1.73560i
$977$	−3.16228	−	3.16228i	−0.101170	−	0.101170i	0.654710	−	0.755880i	$-0.272791\pi$
$977$	−3.16228	−	3.16228i	−0.101170	−	0.101170i	−0.755880	+	0.654710i	$0.772791\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$			46.4758i			1.48386i
$982$		0			0
$983$	44.2719	−	44.2719i	1.41205	−	1.41205i	0.666964	−	0.745090i	$-0.267594\pi$
$983$	44.2719	−	44.2719i	1.41205	−	1.41205i	0.745090	−	0.666964i	$-0.232406\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	32.0000			1.01651			0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000			1.01651			0.508257	+	0.861206i	$0.330290\pi$
$992$	−36.5218	−	26.7238i	−1.15957	−	0.848481i
$993$	28.4605	−	28.4605i	0.903167	−	0.903167i
$994$		0			0
$995$		0			0
$996$	3.00000	−	11.6190i	0.0950586	−	0.368161i
$997$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$997$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$998$	−10.8671	−	1.38031i	−0.343993	−	0.0436929i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	600.2.w.i.293.1	✓	8
3.2	odd	2	inner	600.2.w.i.293.4	yes	8
5.2	odd	4	inner	600.2.w.i.557.3	yes	8
5.3	odd	4	inner	600.2.w.i.557.2	yes	8
5.4	even	2	inner	600.2.w.i.293.4	yes	8
8.5	even	2	inner	600.2.w.i.293.2	yes	8
15.2	even	4	inner	600.2.w.i.557.2	yes	8
15.8	even	4	inner	600.2.w.i.557.3	yes	8
15.14	odd	2	CM	600.2.w.i.293.1	✓	8
24.5	odd	2	inner	600.2.w.i.293.3	yes	8
40.13	odd	4	inner	600.2.w.i.557.1	yes	8
40.29	even	2	inner	600.2.w.i.293.3	yes	8
40.37	odd	4	inner	600.2.w.i.557.4	yes	8
120.29	odd	2	inner	600.2.w.i.293.2	yes	8
120.53	even	4	inner	600.2.w.i.557.4	yes	8
120.77	even	4	inner	600.2.w.i.557.1	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
600.2.w.i.293.1	✓	8	1.1	even	1	trivial
600.2.w.i.293.1	✓	8	15.14	odd	2	CM
600.2.w.i.293.2	yes	8	8.5	even	2	inner
600.2.w.i.293.2	yes	8	120.29	odd	2	inner
600.2.w.i.293.3	yes	8	24.5	odd	2	inner
600.2.w.i.293.3	yes	8	40.29	even	2	inner
600.2.w.i.293.4	yes	8	3.2	odd	2	inner
600.2.w.i.293.4	yes	8	5.4	even	2	inner
600.2.w.i.557.1	yes	8	40.13	odd	4	inner
600.2.w.i.557.1	yes	8	120.77	even	4	inner
600.2.w.i.557.2	yes	8	5.3	odd	4	inner
600.2.w.i.557.2	yes	8	15.2	even	4	inner
600.2.w.i.557.3	yes	8	5.2	odd	4	inner
600.2.w.i.557.3	yes	8	15.8	even	4	inner
600.2.w.i.557.4	yes	8	40.37	odd	4	inner
600.2.w.i.557.4	yes	8	120.53	even	4	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	600.w (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.40294 - 0.178197i) q^{2} +(-1.22474 - 1.22474i) q^{3} +(1.93649 + 0.500000i) q^{4} +(1.50000 + 1.93649i) q^{6} +(-2.62769 - 1.04655i) q^{8} +3.00000i q^{9} +O(q^{10})\)	\(q+(-1.40294 - 0.178197i) q^{2} +(-1.22474 - 1.22474i) q^{3} +(1.93649 + 0.500000i) q^{4} +(1.50000 + 1.93649i) q^{6} +(-2.62769 - 1.04655i) q^{8} +3.00000i q^{9} +(-1.75934 - 2.98408i) q^{12} +(3.50000 + 1.93649i) q^{16} +(3.16228 + 3.16228i) q^{17} +(0.534591 - 4.20883i) q^{18} -7.74597 q^{19} +(-6.32456 + 6.32456i) q^{23} +(1.93649 + 4.50000i) q^{24} +(3.67423 - 3.67423i) q^{27} +8.00000 q^{31} +(-4.56522 - 3.34047i) q^{32} +(-3.87298 - 5.00000i) q^{34} +(-1.50000 + 5.80948i) q^{36} +(10.8671 + 1.38031i) q^{38} +(10.0000 - 7.74597i) q^{46} +(6.32456 + 6.32456i) q^{47} +(-1.91490 - 6.65832i) q^{48} +7.00000i q^{49} -7.74597i q^{51} +(9.79796 + 9.79796i) q^{53} +(-5.80948 + 4.50000i) q^{54} +(9.48683 + 9.48683i) q^{57} +15.4919i q^{61} +(-11.2235 - 1.42558i) q^{62} +(5.80948 + 5.50000i) q^{64} +(4.54259 + 7.70486i) q^{68} +15.4919 q^{69} +(3.13964 - 7.88306i) q^{72} +(-15.0000 - 3.87298i) q^{76} -16.0000i q^{79} -9.00000 q^{81} +(2.44949 + 2.44949i) q^{83} +(-15.4097 + 9.08517i) q^{92} +(-9.79796 - 9.79796i) q^{93} +(-7.74597 - 10.0000i) q^{94} +(1.50000 + 9.68246i) q^{96} +(1.24738 - 9.82059i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 12 q^{6}+O(q^{10}) \) 8 * q + 12 * q^6	\( 8 q + 12 q^{6} + 28 q^{16} + 64 q^{31} - 12 q^{36} + 80 q^{46} - 120 q^{76} - 72 q^{81} + 12 q^{96}+O(q^{100}) \) 8 * q + 12 * q^6 + 28 * q^16 + 64 * q^31 - 12 * q^36 + 80 * q^46 - 120 * q^76 - 72 * q^81 + 12 * q^96

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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