LMFDB - Embedded newform 552.2.b.a.413.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [552,2,Mod(413,552)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(552, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("552.413");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 552 = 2^{3} \cdot 3 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	552.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:	$4.40774219157$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.8869743.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{3} + 8 $ x^6 - 3*x^3 + 8
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			413.5
Root			$1.33454 - 0.467979i$ of defining polynomial
Character	$\chi$	$=$	552.413
Dual form			552.2.b.a.413.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.33454 - 0.467979i) q^{2} +(-0.227452 - 1.71705i) q^{3} +(1.56199 - 1.24907i) q^{4} +(-1.10709 - 2.18503i) q^{6} +(1.50000 - 2.39792i) q^{8} +(-2.89653 + 0.781094i) q^{9} +O(q^{10})$	\(q+(1.33454 - 0.467979i) q^{2} +(-0.227452 - 1.71705i) q^{3} +(1.56199 - 1.24907i) q^{4} +(-1.10709 - 2.18503i) q^{6} +(1.50000 - 2.39792i) q^{8} +(-2.89653 + 0.781094i) q^{9} +(-2.50000 - 2.39792i) q^{12} -5.30602i q^{13} +(0.879635 - 3.90208i) q^{16} +(-3.50000 + 2.39792i) q^{18} +4.79583i q^{23} +(-4.45852 - 2.03017i) q^{24} +5.00000 q^{25} +(-2.48310 - 7.08109i) q^{26} +(2.00000 + 4.79583i) q^{27} +6.70287 q^{29} -11.1312 q^{31} +(-0.652183 - 5.61913i) q^{32} +(-3.54871 + 4.83804i) q^{36} +(-9.11071 + 1.20686i) q^{39} +12.1742i q^{41} +(2.24435 + 6.40023i) q^{46} -6.55848i q^{47} +(-6.90015 - 0.622843i) q^{48} +7.00000 q^{49} +(6.67270 - 2.33989i) q^{50} +(-6.62760 - 8.28796i) q^{52} +(4.91343 + 5.46427i) q^{54} +(8.94525 - 3.13680i) q^{58} +12.0000 q^{59} +(-14.8551 + 5.20918i) q^{62} +(-3.50000 - 7.19375i) q^{64} +(8.23469 - 1.09082i) q^{69} +14.0461i q^{71} +(-2.47180 + 8.11728i) q^{72} +7.61268 q^{73} +(-1.13726 - 8.58526i) q^{75} +(-11.5938 + 5.87423i) q^{78} +(7.77979 - 4.52492i) q^{81} +(5.69728 + 16.2470i) q^{82} +(-1.52458 - 11.5092i) q^{87} +(5.99034 + 7.49105i) q^{92} +(2.53182 + 19.1129i) q^{93} +(-3.06923 - 8.75255i) q^{94} +(-9.50000 + 2.39792i) q^{96} +(9.34178 - 3.27585i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 9 q^{8}+O(q^{10}) $ 6 * q + 9 * q^8	$ 6 q + 9 q^{8} - 15 q^{12} - 21 q^{18} + 30 q^{25} - 27 q^{26} + 12 q^{27} - 24 q^{39} + 42 q^{49} + 3 q^{52} + 15 q^{58} + 72 q^{59} - 45 q^{62} - 21 q^{64} - 51 q^{78} + 33 q^{82} - 48 q^{87} + 6 q^{93} + 39 q^{94} - 57 q^{96}+O(q^{100}) $ 6 * q + 9 * q^8 - 15 * q^12 - 21 * q^18 + 30 * q^25 - 27 * q^26 + 12 * q^27 - 24 * q^39 + 42 * q^49 + 3 * q^52 + 15 * q^58 + 72 * q^59 - 45 * q^62 - 21 * q^64 - 51 * q^78 + 33 * q^82 - 48 * q^87 + 6 * q^93 + 39 * q^94 - 57 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/552\mathbb{Z}\right)^\times$.

$n$	$97$	$185$	$277$	$415$
$\chi(n)$	$-1$	$-1$	$-1$	$1$

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 $

$2$	1.33454	−	0.467979i	0.943662	−	0.330911i
$2$	1.33454	−	0.467979i	0.943662	−	0.330911i
$3$	−0.227452	−	1.71705i	−0.131319	−	0.991340i
$3$	−0.227452	−	1.71705i	−0.131319	−	0.991340i
$4$	1.56199	−	1.24907i	0.780996	−	0.624536i
$5$		0			0		1.00000			$0$
$5$		0			0		−1.00000			$\pi$
$6$	−1.10709	−	2.18503i	−0.451967	−	0.892035i
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$	1.50000	−	2.39792i	0.530330	−	0.847791i
$9$	−2.89653	+	0.781094i	−0.965510	+	0.260365i
$10$		0			0
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$	−2.50000	−	2.39792i	−0.721688	−	0.692219i
$13$		−	5.30602i		−	1.47162i	−0.677185	−	0.735812i	$-0.736801\pi$
$13$		−	5.30602i		−	1.47162i	0.677185	−	0.735812i	$-0.263199\pi$
$14$		0			0
$15$		0			0
$16$	0.879635	−	3.90208i	0.219909	−	0.975520i
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$	−3.50000	+	2.39792i	−0.824958	+	0.565194i
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			4.79583i			1.00000i
$23$			4.79583i			1.00000i
$24$	−4.45852	−	2.03017i	−0.910092	−	0.414406i
$25$	5.00000			1.00000
$26$	−2.48310	−	7.08109i	−0.486977	−	1.38872i
$27$	2.00000	+	4.79583i	0.384900	+	0.922958i
$28$		0			0
$29$	6.70287			1.24469			0.622346	−	0.782742i	$-0.286180\pi$
$29$	6.70287			1.24469			0.622346	+	0.782742i	$0.286180\pi$
$30$		0			0
$31$	−11.1312			−1.99923			−0.999613	−	0.0278144i	$-0.991145\pi$
$31$	−11.1312			−1.99923			−0.999613	+	0.0278144i	$0.991145\pi$
$32$	−0.652183	−	5.61913i	−0.115291	−	0.993332i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−3.54871	+	4.83804i	−0.591452	+	0.806340i
$37$		0			0			−	1.00000i	$-0.5\pi$
$37$		0			0				1.00000i	$0.5\pi$
$38$		0			0
$39$	−9.11071	+	1.20686i	−1.45888	+	0.193253i
$40$		0			0
$41$			12.1742i			1.90129i	0.310274	+	0.950647i	$0.399579\pi$
$41$			12.1742i			1.90129i	−0.310274	+	0.950647i	$0.600421\pi$
$42$		0			0
$43$		0			0			−	1.00000i	$-0.5\pi$
$43$		0			0				1.00000i	$0.5\pi$
$44$		0			0
$45$		0			0
$46$	2.24435	+	6.40023i	0.330911	+	0.943662i
$47$		−	6.55848i		−	0.956652i	−0.878182	−	0.478326i	$-0.841244\pi$
$47$		−	6.55848i		−	0.956652i	0.878182	−	0.478326i	$-0.158756\pi$
$48$	−6.90015	−	0.622843i	−0.995951	−	0.0898996i
$49$	7.00000			1.00000
$50$	6.67270	−	2.33989i	0.943662	−	0.330911i
$51$		0			0
$52$	−6.62760	−	8.28796i	−0.919083	−	1.14933i
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$	4.91343	+	5.46427i	0.668633	+	0.743593i
$55$		0			0
$56$		0			0
$57$		0			0
$58$	8.94525	−	3.13680i	1.17457	−	0.411882i
$59$	12.0000			1.56227			0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000			1.56227			0.781133	+	0.624364i	$0.214642\pi$
$60$		0			0
$61$		0			0			−	1.00000i	$-0.5\pi$
$61$		0			0				1.00000i	$0.5\pi$
$62$	−14.8551	+	5.20918i	−1.88659	+	0.661566i
$63$		0			0
$64$	−3.50000	−	7.19375i	−0.437500	−	0.899218i
$65$		0			0
$66$		0			0
$67$		0			0			−	1.00000i	$-0.5\pi$
$67$		0			0				1.00000i	$0.5\pi$
$68$		0			0
$69$	8.23469	−	1.09082i	0.991340	−	0.131319i
$70$		0			0
$71$			14.0461i			1.66697i	0.552542	+	0.833485i	$0.313658\pi$
$71$			14.0461i			1.66697i	−0.552542	+	0.833485i	$0.686342\pi$
$72$	−2.47180	+	8.11728i	−0.291304	+	0.956630i
$73$	7.61268			0.890997			0.445498	−	0.895283i	$-0.353027\pi$
$73$	7.61268			0.890997			0.445498	+	0.895283i	$0.353027\pi$
$74$		0			0
$75$	−1.13726	−	8.58526i	−0.131319	−	0.991340i
$76$		0			0
$77$		0			0
$78$	−11.5938	+	5.87423i	−1.31274	+	0.665125i
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$		0			0
$81$	7.77979	−	4.52492i	0.864421	−	0.502769i
$82$	5.69728	+	16.2470i	0.629159	+	1.79418i
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−1.52458	−	11.5092i	−0.163452	−	1.23391i
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$		0			0
$92$	5.99034	+	7.49105i	0.624536	+	0.780996i
$93$	2.53182	+	19.1129i	0.262537	+	1.98191i
$94$	−3.06923	−	8.75255i	−0.316567	−	0.902756i
$95$		0			0
$96$	−9.50000	+	2.39792i	−0.969590	+	0.244736i
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$	9.34178	−	3.27585i	0.943662	−	0.330911i
$99$		0			0
$100$	7.80996	−	6.24536i	0.780996	−	0.624536i
$101$	−6.00000			−0.597022			−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000			−0.597022			−0.298511	+	0.954406i	$0.596490\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$	−12.7234	−	7.95903i	−1.24763	−	0.780447i
$105$		0			0
$106$		0			0
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$	9.11432	+	4.99290i	0.877026	+	0.480442i
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$		0			0
$115$		0			0
$116$	10.4698	−	8.37237i	0.972099	−	0.777355i
$117$	4.14450	+	15.3690i	0.383159	+	1.42087i
$118$	16.0145	−	5.61575i	1.47425	−	0.516971i
$119$		0			0
$120$		0			0
$121$	11.0000			1.00000
$122$		0			0
$123$	20.9038	−	2.76905i	1.88483	−	0.249677i
$124$	−17.3869	+	13.9037i	−1.56139	+	1.24859i
$125$		0			0
$126$		0			0
$127$	8.40180			0.745539			0.372769	−	0.927924i	$-0.378408\pi$
$127$	8.40180			0.745539			0.372769	+	0.927924i	$0.378408\pi$
$128$	−8.03741	−	7.96241i	−0.710413	−	0.703785i
$129$		0			0
$130$		0			0
$131$	−22.7173			−1.98482			−0.992412	−	0.122958i	$-0.960762\pi$
$131$	−22.7173			−1.98482			−0.992412	+	0.122958i	$0.960762\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0			−	1.00000i	$-0.5\pi$
$137$		0			0				1.00000i	$0.5\pi$
$138$	10.4790	−	5.30941i	0.892035	−	0.451967i
$139$			10.9218i			0.926372i	0.886261	+	0.463186i	$0.153294\pi$
$139$			10.9218i			0.926372i	−0.886261	+	0.463186i	$0.846706\pi$
$140$		0			0
$141$	−11.2612	+	1.49174i	−0.948368	+	0.125627i
$142$	6.57330	+	18.7451i	0.551619	+	1.57306i
$143$		0			0
$144$	0.500000	+	11.9896i	0.0416667	+	0.999132i
$145$		0			0
$146$	10.1594	−	3.56257i	0.840800	−	0.294841i
$147$	−1.59216	−	12.0194i	−0.131319	−	0.991340i
$148$		0			0
$149$		0			0		1.00000			$0$
$149$		0			0		−1.00000			$\pi$
$150$	−5.53544	−	10.9252i	−0.451967	−	0.892035i
$151$	−13.8606			−1.12796			−0.563982	−	0.825787i	$-0.690731\pi$
$151$	−13.8606			−1.12796			−0.563982	+	0.825787i	$0.690731\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−12.7234	+	13.2650i	−1.01869	+	1.06205i
$157$		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
$162$	8.26486	−	9.67947i	0.649349	−	0.760491i
$163$			9.68285i			0.758420i	0.925311	+	0.379210i	$0.123804\pi$
$163$			9.68285i			0.758420i	−0.925311	+	0.379210i	$0.876196\pi$
$164$	15.2065	+	19.0160i	1.18743	+	1.48490i
$165$		0			0
$166$		0			0
$167$		−	9.59166i		−	0.742225i	−0.928588	−	0.371113i	$-0.878976\pi$
$167$		−	9.59166i		−	0.742225i	0.928588	−	0.371113i	$-0.121024\pi$
$168$		0			0
$169$	−15.1538			−1.16568
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−18.0000			−1.36851			−0.684257	−	0.729241i	$-0.739873\pi$
$173$	−18.0000			−1.36851			−0.684257	+	0.729241i	$0.739873\pi$
$174$	−7.42066	−	14.6460i	−0.562559	−	1.11031i
$175$		0			0
$176$		0			0
$177$	−2.72942	−	20.6046i	−0.205156	−	1.54874i
$178$		0			0
$179$	−19.9879			−1.49397			−0.746984	−	0.664842i	$-0.768499\pi$
$179$	−19.9879			−1.49397			−0.746984	+	0.664842i	$0.768499\pi$
$180$		0			0
$181$		0			0			−	1.00000i	$-0.5\pi$
$181$		0			0				1.00000i	$0.5\pi$
$182$		0			0
$183$		0			0
$184$	11.5000	+	7.19375i	0.847791	+	0.530330i
$185$		0			0
$186$	12.3232	+	24.3221i	0.903583	+	1.78338i
$187$		0			0
$188$	−8.19201	−	10.2443i	−0.597464	−	0.747141i
$189$		0			0
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$	−11.5560	+	7.64591i	−0.833979	+	0.551796i
$193$	27.1457			1.95399			0.976995	−	0.213262i	$-0.0684089\pi$
$193$	27.1457			1.95399			0.976995	+	0.213262i	$0.0684089\pi$
$194$		0			0
$195$		0			0
$196$	10.9339	−	8.74351i	0.780996	−	0.624536i
$197$	−28.0555			−1.99887			−0.999436	−	0.0335834i	$-0.989308\pi$
$197$	−28.0555			−1.99887			−0.999436	+	0.0335834i	$0.989308\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$	7.50000	−	11.9896i	0.530330	−	0.847791i
$201$		0			0
$202$	−8.00724	+	2.80787i	−0.563387	+	0.197561i
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−3.74599	−	13.8913i	−0.260365	−	0.965510i
$208$	−20.7045	−	4.66736i	−1.43560	−	0.323623i
$209$		0			0
$210$		0			0
$211$		−	28.7750i		−	1.98095i	−0.137686	−	0.990476i	$-0.543966\pi$
$211$		−	28.7750i		−	1.98095i	0.137686	−	0.990476i	$-0.456034\pi$
$212$		0			0
$213$	24.1179	−	3.19482i	1.65253	−	0.218906i
$214$		0			0
$215$		0			0
$216$	14.5000	+	2.39792i	0.986600	+	0.163158i
$217$		0			0
$218$		0			0
$219$	−1.73152	−	13.0714i	−0.117005	−	0.883281i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−8.00000			−0.535720			−0.267860	−	0.963458i	$-0.586316\pi$
$223$	−8.00000			−0.535720			−0.267860	+	0.963458i	$0.586316\pi$
$224$		0			0
$225$	−14.4827	+	3.90547i	−0.965510	+	0.260365i
$226$		0			0
$227$		0			0		1.00000			$0$
$227$		0			0		−1.00000			$\pi$
$228$		0			0
$229$		0			0			−	1.00000i	$-0.5\pi$
$229$		0			0				1.00000i	$0.5\pi$
$230$		0			0
$231$		0			0
$232$	10.0543	−	16.0729i	0.660098	−	1.05524i
$233$			29.0350i			1.90215i	0.308965	+	0.951073i	$0.400017\pi$
$233$			29.0350i			1.90215i	−0.308965	+	0.951073i	$0.599983\pi$
$234$	12.7234	+	18.5711i	0.831754	+	1.21403i
$235$		0			0
$236$	18.7439	−	14.9889i	1.22012	−	0.975692i
$237$		0			0
$238$		0			0
$239$			27.1631i			1.75703i	0.477711	+	0.878517i	$0.341467\pi$
$239$			27.1631i			1.75703i	−0.477711	+	0.878517i	$0.658533\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$	14.6799	−	5.14777i	0.943662	−	0.330911i
$243$	−9.53906	−	12.3291i	−0.611931	−	0.790911i
$244$		0			0
$245$		0			0
$246$	26.6010	−	13.4779i	1.69602	−	0.859322i
$247$		0			0
$248$	−16.6968	+	26.6917i	−1.06025	+	1.69493i
$249$		0			0
$250$		0			0
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$		0			0
$254$	11.2125	−	3.93186i	0.703537	−	0.246707i
$255$		0			0
$256$	−14.4525	−	6.86482i	−0.903280	−	0.429051i
$257$		−	19.6619i		−	1.22647i	−0.789899	−	0.613237i	$-0.789867\pi$
$257$		−	19.6619i		−	1.22647i	0.789899	−	0.613237i	$-0.210133\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−19.4151	+	5.23557i	−1.20176	+	0.324074i
$262$	−30.3172	+	10.6312i	−1.87300	+	0.656800i
$263$		0			0			−	1.00000i	$-0.5\pi$
$263$		0			0				1.00000i	$0.5\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−1.24402			−0.0758493			−0.0379247	−	0.999281i	$-0.512075\pi$
$269$	−1.24402			−0.0758493			−0.0379247	+	0.999281i	$0.512075\pi$
$270$		0			0
$271$	−16.0000			−0.971931			−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000			−0.971931			−0.485965	+	0.873978i	$0.661532\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$	11.5000	−	11.9896i	0.692219	−	0.721688i
$277$		−	14.6791i		−	0.881984i	−0.897511	−	0.440992i	$-0.854627\pi$
$277$		−	14.6791i		−	0.881984i	0.897511	−	0.440992i	$-0.145373\pi$
$278$	5.11115	+	14.5755i	0.306547	+	0.874182i
$279$	32.2419	−	8.69453i	1.93027	−	0.520528i
$280$		0			0
$281$		0			0			−	1.00000i	$-0.5\pi$
$281$		0			0				1.00000i	$0.5\pi$
$282$	−14.3305	+	7.26081i	−0.853367	+	0.432375i
$283$		0			0			−	1.00000i	$-0.5\pi$
$283$		0			0				1.00000i	$0.5\pi$
$284$	17.5446	+	21.9400i	1.04108	+	1.30190i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	6.27814	+	15.7666i	0.369943	+	0.929054i
$289$	−17.0000			−1.00000
$290$		0			0
$291$		0			0
$292$	11.8909	−	9.50879i	0.695865	−	0.556460i
$293$		0			0		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$	−7.74961	−	15.2952i	−0.451967	−	0.892035i
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	25.4468			1.47162
$300$	−12.5000	−	11.9896i	−0.721688	−	0.692219i
$301$		0			0
$302$	−18.4976	+	6.48649i	−1.06442	+	0.373255i
$303$	1.36471	+	10.3023i	0.0784007	+	0.591852i
$304$		0			0
$305$		0			0
$306$		0			0
$307$			28.7750i			1.64228i	0.570730	+	0.821138i	$0.306660\pi$
$307$			28.7750i			1.64228i	−0.570730	+	0.821138i	$0.693340\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	34.6508i		−	1.96486i	−0.186621	−	0.982432i	$-0.559754\pi$
$311$		−	34.6508i		−	1.96486i	0.186621	−	0.982432i	$-0.440246\pi$
$312$	−10.7721	+	23.6570i	−0.609850	+	1.33931i
$313$		0			0		1.00000			$0$
$313$		0			0		−1.00000			$\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	30.0000			1.68497			0.842484	−	0.538721i	$-0.181092\pi$
$317$	30.0000			1.68497			0.842484	+	0.538721i	$0.181092\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	6.50000	−	16.7854i	0.361111	−	0.932523i
$325$		−	26.5301i		−	1.47162i
$326$	4.53137	+	12.9221i	0.250969	+	0.715692i
$327$		0			0
$328$	29.1928	+	18.2613i	1.61190	+	1.00831i
$329$		0			0
$330$		0			0
$331$			31.5264i			1.73285i	0.499310	+	0.866423i	$0.333587\pi$
$331$			31.5264i			1.73285i	−0.499310	+	0.866423i	$0.666413\pi$
$332$		0			0
$333$		0			0
$334$	−4.48870	−	12.8005i	−0.245610	−	0.700410i
$335$		0			0
$336$		0			0
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$	−20.2234	+	7.09167i	−1.10001	+	0.385736i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−24.0217	+	8.42362i	−1.29142	+	0.452857i
$347$	−36.0000			−1.93258			−0.966291	−	0.257454i	$-0.917117\pi$
$347$	−36.0000			−1.93258			−0.966291	+	0.257454i	$0.917117\pi$
$348$	−16.7572	−	16.0729i	−0.898279	−	0.861599i
$349$		−	35.9032i		−	1.92186i	−0.276800	−	0.960928i	$-0.589274\pi$
$349$		−	35.9032i		−	1.92186i	0.276800	−	0.960928i	$-0.410726\pi$
$350$		0			0
$351$	25.4468	−	10.6120i	1.35825	−	0.566429i
$352$		0			0
$353$		−	21.5473i		−	1.14685i	−0.819258	−	0.573425i	$-0.805614\pi$
$353$		−	21.5473i		−	1.14685i	0.819258	−	0.573425i	$-0.194386\pi$
$354$	−13.2850	−	26.2204i	−0.706092	−	1.39360i
$355$		0			0
$356$		0			0
$357$		0			0
$358$	−26.6747	+	9.35392i	−1.40980	+	0.494370i
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	−19.0000			−1.00000
$362$		0			0
$363$	−2.50197	−	18.8876i	−0.131319	−	0.991340i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		1.00000			$0$
$367$		0			0		−1.00000			$\pi$
$368$	18.7137	+	4.21858i	0.975520	+	0.219909i
$369$	−9.50921	−	35.2630i	−0.495030	−	1.83572i
$370$		0			0
$371$		0			0
$372$	27.8281	+	26.6917i	1.44282	+	1.38390i
$373$		0			0			−	1.00000i	$-0.5\pi$
$373$		0			0				1.00000i	$0.5\pi$
$374$		0			0
$375$		0			0
$376$	−15.7267	−	9.83772i	−0.811041	−	0.507341i
$377$		−	35.5656i		−	1.83172i
$378$		0			0
$379$		0			0			−	1.00000i	$-0.5\pi$
$379$		0			0				1.00000i	$0.5\pi$
$380$		0			0
$381$	−1.91101	−	14.4263i	−0.0979038	−	0.739083i
$382$		0			0
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$	−11.8437	+	15.6117i	−0.604399	+	0.796682i
$385$		0			0
$386$	36.2270	−	12.7036i	1.84391	−	0.646597i
$387$		0			0
$388$		0			0
$389$		0			0		1.00000			$0$
$389$		0			0		−1.00000			$\pi$
$390$		0			0
$391$		0			0
$392$	10.5000	−	16.7854i	0.530330	−	0.847791i
$393$	5.16711	+	39.0068i	0.260646	+	1.96764i
$394$	−37.4412	+	13.1294i	−1.88626	+	0.661449i
$395$		0			0
$396$		0			0
$397$		−	4.06711i		−	0.204122i	−0.994778	−	0.102061i	$-0.967456\pi$
$397$		−	4.06711i		−	0.204122i	0.994778	−	0.102061i	$-0.0325437\pi$
$398$		0			0
$399$		0			0
$400$	4.39818	−	19.5104i	0.219909	−	0.975520i
$401$		0			0			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$		0			0
$403$			59.0625i			2.94211i
$404$	−9.37195	+	7.49444i	−0.466272	+	0.372862i
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−36.9122			−1.82519			−0.912595	−	0.408864i	$-0.865925\pi$
$409$	−36.9122			−1.82519			−0.912595	+	0.408864i	$0.865925\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	−11.5000	−	16.7854i	−0.565194	−	0.824958i
$415$		0			0
$416$	−29.8152	+	3.46050i	−1.46181	+	0.169665i
$417$	18.7532	−	2.48418i	0.918350	−	0.121651i
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$		0			0			−	1.00000i	$-0.5\pi$
$421$		0			0				1.00000i	$0.5\pi$
$422$	−13.4661	−	38.4014i	−0.655519	−	1.86935i
$423$	5.12279	+	18.9968i	0.249078	+	0.923658i
$424$		0			0
$425$		0			0
$426$	30.6912	−	15.5503i	1.48700	−	0.753415i
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$	20.4730	−	3.58558i	0.985008	−	0.172511i
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	−8.42790	−	16.6339i	−0.402701	−	0.794800i
$439$	5.67237			0.270728			0.135364	−	0.990796i	$-0.456780\pi$
$439$	5.67237			0.270728			0.135364	+	0.990796i	$0.456780\pi$
$440$		0			0
$441$	−20.2757	+	5.46766i	−0.965510	+	0.260365i
$442$		0			0
$443$	17.2585			0.819976			0.409988	−	0.912091i	$-0.365533\pi$
$443$	17.2585			0.819976			0.409988	+	0.912091i	$0.365533\pi$
$444$		0			0
$445$		0			0
$446$	−10.6763	+	3.74383i	−0.505538	+	0.177276i
$447$		0			0
$448$		0			0
$449$			38.3667i			1.81063i	0.424736	+	0.905317i	$0.360367\pi$
$449$			38.3667i			1.81063i	−0.424736	+	0.905317i	$0.639633\pi$
$450$	−17.5000	+	11.9896i	−0.824958	+	0.565194i
$451$		0			0
$452$		0			0
$453$	3.15263	+	23.7994i	0.148124	+	1.11820i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		1.00000			$0$
$457$		0			0		−1.00000			$\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	36.0024			1.67680			0.838399	−	0.545056i	$-0.183492\pi$
$461$	36.0024			1.67680			0.838399	+	0.545056i	$0.183492\pi$
$462$		0			0
$463$	32.0000			1.48717			0.743583	−	0.668644i	$-0.233125\pi$
$463$	32.0000			1.48717			0.743583	+	0.668644i	$0.233125\pi$
$464$	5.89608	−	26.1551i	0.273719	−	1.21422i
$465$		0			0
$466$	13.5878	+	38.7484i	0.629441	+	1.79498i
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$	25.6707	+	18.8295i	1.18663	+	0.870396i
$469$		0			0
$470$		0			0
$471$		0			0
$472$	18.0000	−	28.7750i	0.828517	−	1.32448i
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$	12.7118	+	36.2502i	0.581422	+	1.65805i
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	17.1819	−	13.7398i	0.780996	−	0.624536i
$485$		0			0
$486$	−18.5000	−	11.9896i	−0.839177	−	0.543858i
$487$	16.5901			0.751768			0.375884	−	0.926667i	$-0.377339\pi$
$487$	16.5901			0.751768			0.375884	+	0.926667i	$0.377339\pi$
$488$		0			0
$489$	16.6260	−	2.20238i	0.751852	−	0.0995953i
$490$		0			0
$491$	44.0700			1.98885			0.994425	−	0.105445i	$-0.0336267\pi$
$491$	44.0700			1.98885			0.994425	+	0.105445i	$0.0336267\pi$
$492$	29.1928	−	30.4356i	1.31611	−	1.37214i
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−9.79142	+	43.4349i	−0.439648	+	1.95029i
$497$		0			0
$498$		0			0
$499$		−	11.5412i		−	0.516656i	−0.966057	−	0.258328i	$-0.916828\pi$
$499$		−	11.5412i		−	0.516656i	0.966057	−	0.258328i	$-0.0831715\pi$
$500$		0			0
$501$	−16.4694	+	2.18164i	−0.735798	+	0.0974686i
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	3.44677	+	26.0199i	0.153076	+	1.15558i
$508$	13.1235	−	10.4945i	0.582263	−	0.465616i
$509$	−41.4612			−1.83774			−0.918869	−	0.394564i	$-0.870896\pi$
$509$	−41.4612			−1.83774			−0.918869	+	0.394564i	$0.870896\pi$
$510$		0			0
$511$		0			0
$512$	−22.5000	−	2.39792i	−0.994369	−	0.105974i
$513$		0			0
$514$	−9.20135	−	26.2396i	−0.405854	−	1.15738i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$	4.09414	+	30.9069i	0.179713	+	1.35666i
$520$		0			0
$521$		0			0			−	1.00000i	$-0.5\pi$
$521$		0			0				1.00000i	$0.5\pi$
$522$	−23.4600	+	16.0729i	−1.02682	+	0.703493i
$523$		0			0			−	1.00000i	$-0.5\pi$
$523$		0			0				1.00000i	$0.5\pi$
$524$	−35.4843	+	28.3756i	−1.55014	+	1.23959i
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−23.0000			−1.00000
$530$		0			0
$531$	−34.7584	+	9.37312i	−1.50838	+	0.406759i
$532$		0			0
$533$	64.5967			2.79799
$534$		0			0
$535$		0			0
$536$		0			0
$537$	4.54629	+	34.3203i	0.196187	+	1.48103i
$538$	−1.66020	+	0.582176i	−0.0715761	+	0.0250994i
$539$		0			0
$540$		0			0
$541$			46.5153i			1.99985i	0.0123734	+	0.999923i	$0.496061\pi$
$541$			46.5153i			1.99985i	−0.0123734	+	0.999923i	$0.503939\pi$
$542$	−21.3526	+	7.48766i	−0.917174	+	0.321623i
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$			9.06340i			0.387523i	0.981049	+	0.193761i	$0.0620688\pi$
$547$			9.06340i			0.387523i	−0.981049	+	0.193761i	$0.937931\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$	9.73634	−	21.3823i	0.414406	−	0.910092i
$553$		0			0
$554$	−6.86953	−	19.5899i	−0.291858	−	0.832295i
$555$		0			0
$556$	13.6421	+	17.0597i	0.578553	+	0.723493i
$557$		0			0		1.00000			$0$
$557$		0			0		−1.00000			$\pi$
$558$	38.9593	−	26.6917i	1.64928	−	1.12995i
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		1.00000			$0$
$563$		0			0		−1.00000			$\pi$
$564$	−15.7267	+	16.3962i	−0.662213	+	0.690404i
$565$		0			0
$566$		0			0
$567$		0			0
$568$	33.6815	+	21.0692i	1.41324	+	0.884044i
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$		0			0
$571$		0			0			−	1.00000i	$-0.5\pi$
$571$		0			0				1.00000i	$0.5\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			23.9792i			1.00000i
$576$	15.7568	+	18.1031i	0.656535	+	0.754295i
$577$	−32.6045			−1.35734			−0.678672	−	0.734441i	$-0.737444\pi$
$577$	−32.6045			−1.35734			−0.678672	+	0.734441i	$0.737444\pi$
$578$	−22.6872	+	7.95564i	−0.943662	+	0.330911i
$579$	−6.17434	−	46.6106i	−0.256597	−	1.93707i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$	11.4190	−	18.2546i	0.472522	−	0.755379i
$585$		0			0
$586$		0			0
$587$	41.3406			1.70631			0.853154	−	0.521660i	$-0.174687\pi$
$587$	41.3406			1.70631			0.853154	+	0.521660i	$0.174687\pi$
$588$	−17.5000	−	16.7854i	−0.721688	−	0.692219i
$589$		0			0
$590$		0			0
$591$	6.38128	+	48.1727i	0.262491	+	1.98156i
$592$		0			0
$593$			38.3667i			1.57553i	0.615976	+	0.787765i	$0.288762\pi$
$593$			38.3667i			1.57553i	−0.615976	+	0.787765i	$0.711238\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$	33.9597	−	11.9085i	1.38872	−	0.486977i
$599$			9.59166i			0.391905i	0.980613	+	0.195952i	$0.0627798\pi$
$599$			9.59166i			0.391905i	−0.980613	+	0.195952i	$0.937220\pi$
$600$	−22.2926	−	10.1508i	−0.910092	−	0.414406i
$601$	42.3711			1.72835			0.864176	−	0.503190i	$-0.167841\pi$
$601$	42.3711			1.72835			0.864176	+	0.503190i	$0.167841\pi$
$602$		0			0
$603$		0			0
$604$	−21.6502	+	17.3130i	−0.880935	+	0.704454i
$605$		0			0
$606$	6.64252	+	13.1102i	0.269834	+	0.532565i
$607$	40.0000			1.62355			0.811775	−	0.583970i	$-0.198502\pi$
$607$	40.0000			1.62355			0.811775	+	0.583970i	$0.198502\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−34.7994			−1.40783
$612$		0			0
$613$		0			0			−	1.00000i	$-0.5\pi$
$613$		0			0				1.00000i	$0.5\pi$
$614$	13.4661	+	38.4014i	0.543447	+	1.54975i
$615$		0			0
$616$		0			0
$617$		0			0			−	1.00000i	$-0.5\pi$
$617$		0			0				1.00000i	$0.5\pi$
$618$		0			0
$619$		0			0			−	1.00000i	$-0.5\pi$
$619$		0			0				1.00000i	$0.5\pi$
$620$		0			0
$621$	−23.0000	+	9.59166i	−0.922958	+	0.384900i
$622$	−16.2158	−	46.2428i	−0.650195	−	1.85417i
$623$		0			0
$624$	−3.30482	+	36.6123i	−0.132298	+	1.46567i
$625$	25.0000			1.00000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$	−49.4081	+	6.54493i	−1.96380	+	0.260138i
$634$	40.0362	−	14.0394i	1.59004	−	0.557574i
$635$		0			0
$636$		0			0
$637$		−	37.1421i		−	1.47162i
$638$		0			0
$639$	−10.9714	−	40.6851i	−0.434020	−	1.60948i
$640$		0			0
$641$		0			0			−	1.00000i	$-0.5\pi$
$641$		0			0				1.00000i	$0.5\pi$
$642$		0			0
$643$		0			0			−	1.00000i	$-0.5\pi$
$643$		0			0				1.00000i	$0.5\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			47.7677i			1.87794i	0.343996	+	0.938971i	$0.388219\pi$
$647$			47.7677i			1.87794i	−0.343996	+	0.938971i	$0.611781\pi$
$648$	0.819290	−	25.4427i	0.0321848	−	0.999482i
$649$		0			0
$650$	−12.4155	−	35.4055i	−0.486977	−	1.38872i
$651$		0			0
$652$	12.0946	+	15.1245i	0.473661	+	0.592322i
$653$	−12.1617			−0.475925			−0.237962	−	0.971274i	$-0.576480\pi$
$653$	−12.1617			−0.475925			−0.237962	+	0.971274i	$0.576480\pi$
$654$		0			0
$655$		0			0
$656$	47.5048	+	10.7089i	1.85475	+	0.418111i
$657$	−22.0504	+	5.94622i	−0.860267	+	0.231984i
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$		0			0			−	1.00000i	$-0.5\pi$
$661$		0			0				1.00000i	$0.5\pi$
$662$	14.7537	+	42.0732i	0.573418	+	1.63522i
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$			32.1458i			1.24469i
$668$	−11.9807	−	14.9821i	−0.463547	−	0.579675i
$669$	1.81962	+	13.7364i	0.0703504	+	0.531080i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−21.6868			−0.835966			−0.417983	−	0.908455i	$-0.637263\pi$
$673$	−21.6868			−0.835966			−0.417983	+	0.908455i	$0.637263\pi$
$674$		0			0
$675$	10.0000	+	23.9792i	0.384900	+	0.922958i
$676$	−23.6702	+	18.9282i	−0.910391	+	0.728009i
$677$		0			0		1.00000			$0$
$677$		0			0		−1.00000			$\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	28.1762			1.07813			0.539066	−	0.842263i	$-0.318777\pi$
$683$	28.1762			1.07813			0.539066	+	0.842263i	$0.318777\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$		−	28.7750i		−	1.09465i	−0.836919	−	0.547326i	$-0.815646\pi$
$691$		−	28.7750i		−	1.09465i	0.836919	−	0.547326i	$-0.184354\pi$
$692$	−28.1158	+	22.4833i	−1.06880	+	0.854687i
$693$		0			0
$694$	−48.0434	+	16.8472i	−1.82370	+	0.639512i
$695$		0			0
$696$	−29.8849	−	13.6079i	−1.13278	−	0.515808i
$697$		0			0
$698$	−16.8019	−	47.9143i	−0.635963	−	1.81358i
$699$	49.8546	−	6.60407i	1.88567	−	0.249789i
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$	28.9935	−	26.0707i	1.09429	−	0.983976i
$703$		0			0
$704$		0			0
$705$		0			0
$706$	−10.0837	−	28.7558i	−0.379505	−	1.08224i
$707$		0			0
$708$	−30.0000	−	28.7750i	−1.12747	−	1.08143i
$709$		0			0			−	1.00000i	$-0.5\pi$
$709$		0			0				1.00000i	$0.5\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	53.3835i		−	1.99923i
$714$		0			0
$715$		0			0
$716$	−31.2210	+	24.9664i	−1.16678	+	0.933037i
$717$	46.6404	−	6.17830i	1.74182	−	0.230733i
$718$		0			0
$719$		−	47.9583i		−	1.78854i	−0.447524	−	0.894272i	$-0.647694\pi$
$719$		−	47.9583i		−	1.78854i	0.447524	−	0.894272i	$-0.352306\pi$
$720$		0			0
$721$		0			0
$722$	−25.3563	+	8.89160i	−0.943662	+	0.330911i
$723$		0			0
$724$		0			0
$725$	33.5144			1.24469
$726$	−12.1780	−	24.0353i	−0.451967	−	0.892035i
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$	−19.0000	+	19.1833i	−0.703704	+	0.710494i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0			−	1.00000i	$-0.5\pi$
$733$		0			0				1.00000i	$0.5\pi$
$734$		0			0
$735$		0			0
$736$	26.9484	−	3.12776i	0.993332	−	0.115291i
$737$		0			0
$738$	−29.1928	−	42.6098i	−1.07460	−	1.56849i
$739$		−	52.1310i		−	1.91767i	−0.283964	−	0.958835i	$-0.591650\pi$
$739$		−	52.1310i		−	1.91767i	0.283964	−	0.958835i	$-0.408350\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0			−	1.00000i	$-0.5\pi$
$743$		0			0				1.00000i	$0.5\pi$
$744$	49.6288	+	22.5982i	1.81948	+	0.828491i
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$	−25.5917	−	5.76907i	−0.933234	−	0.210376i
$753$		0			0
$754$	−16.6439	−	47.4636i	−0.606136	−	1.72852i
$755$		0			0
$756$		0			0
$757$		0			0			−	1.00000i	$-0.5\pi$
$757$		0			0				1.00000i	$0.5\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	44.0103i		−	1.59537i	−0.603072	−	0.797687i	$-0.706057\pi$
$761$		−	44.0103i		−	1.59537i	0.603072	−	0.797687i	$-0.293943\pi$
$762$	−9.30152	−	18.3582i	−0.336959	−	0.665047i
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	63.6722i		−	2.29907i
$768$	−8.50000	+	26.3771i	−0.306717	+	0.951801i
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$	−33.7605	+	4.47214i	−1.21585	+	0.161060i
$772$	42.4013	−	33.9069i	1.52606	−	1.22034i
$773$		0			0		1.00000			$0$
$773$		0			0		−1.00000			$\pi$
$774$		0			0
$775$	−55.6561			−1.99923
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$	13.4057	+	32.1458i	0.479082	+	1.14880i
$784$	6.15745	−	27.3146i	0.219909	−	0.975520i
$785$		0			0
$786$	25.1501	+	49.6381i	0.897074	+	1.77053i
$787$		0			0			−	1.00000i	$-0.5\pi$
$787$		0			0				1.00000i	$0.5\pi$
$788$	−43.8225	+	35.0434i	−1.56111	+	1.24837i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$	−1.90332	−	5.42771i	−0.0675463	−	0.192622i
$795$		0			0
$796$		0			0
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$		0			0
$799$		0			0
$800$	−3.26092	−	28.0957i	−0.115291	−	0.993332i
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$	27.6400	+	78.8212i	0.973577	+	2.77636i
$807$	0.282955	+	2.13605i	0.00996049	+	0.0751925i
$808$	−9.00000	+	14.3875i	−0.316619	+	0.506150i
$809$		−	38.3667i		−	1.34890i	−0.738321	−	0.674450i	$-0.764381\pi$
$809$		−	38.3667i		−	1.34890i	0.738321	−	0.674450i	$-0.235619\pi$
$810$		0			0
$811$		−	29.6680i		−	1.04178i	−0.853622	−	0.520892i	$-0.825599\pi$
$811$		−	29.6680i		−	1.04178i	0.853622	−	0.520892i	$-0.174401\pi$
$812$		0			0
$813$	3.63923	+	27.4728i	0.127633	+	0.963514i
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	−49.2608	+	17.2741i	−1.72236	+	0.603976i
$819$		0			0
$820$		0			0
$821$	−54.0000			−1.88461			−0.942306	−	0.334751i	$-0.891348\pi$
$821$	−54.0000			−1.88461			−0.942306	+	0.334751i	$0.891348\pi$
$822$		0			0
$823$	52.9267			1.84491			0.922454	−	0.386107i	$-0.126180\pi$
$823$	52.9267			1.84491			0.922454	+	0.386107i	$0.126180\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$	−23.2024	−	17.0190i	−0.806340	−	0.591452i
$829$			57.5500i			1.99879i	0.0347314	+	0.999397i	$0.488942\pi$
$829$			57.5500i			1.99879i	−0.0347314	+	0.999397i	$0.511058\pi$
$830$		0			0
$831$	−25.2048	+	3.33880i	−0.874346	+	0.115822i
$832$	−38.1702	+	18.5711i	−1.32331	+	0.643836i
$833$		0			0
$834$	23.8644	−	12.0913i	0.826356	−	0.418689i
$835$		0			0
$836$		0			0
$837$	−22.2624	−	53.3835i	−0.769503	−	1.84520i
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	15.9285			0.549258
$842$		0			0
$843$		0			0
$844$	−35.9420	−	44.9463i	−1.23718	−	1.54712i
$845$		0			0
$846$	15.7267	+	22.9547i	0.540694	+	0.789198i
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$	33.6815	−	35.1153i	1.15391	−	1.20303i
$853$			57.5500i			1.97047i	0.171197	+	0.985237i	$0.445237\pi$
$853$			57.5500i			1.97047i	−0.171197	+	0.985237i	$0.554763\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			38.4081i			1.31200i	0.754762	+	0.655998i	$0.227752\pi$
$857$			38.4081i			1.31200i	−0.754762	+	0.655998i	$0.772248\pi$
$858$		0			0
$859$		−	50.8921i		−	1.73642i	−0.496201	−	0.868208i	$-0.665272\pi$
$859$		−	50.8921i		−	1.73642i	0.496201	−	0.868208i	$-0.334728\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	12.1878i		−	0.414877i	−0.978248	−	0.207438i	$-0.933487\pi$
$863$		−	12.1878i		−	0.414877i	0.978248	−	0.207438i	$-0.0665126\pi$
$864$	25.6440	−	14.3660i	0.872428	−	0.488742i
$865$		0			0
$866$		0			0
$867$	3.86668	+	29.1899i	0.131319	+	0.991340i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$	−19.0317	−	18.2546i	−0.643021	−	0.616764i
$877$		−	57.5500i		−	1.94332i	−0.236373	−	0.971662i	$-0.575959\pi$
$877$		−	57.5500i		−	1.94332i	0.236373	−	0.971662i	$-0.424041\pi$
$878$	7.57000	−	2.65455i	0.255475	−	0.0895867i
$879$		0			0
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$	−24.5000	+	16.7854i	−0.824958	+	0.565194i
$883$		−	28.7750i		−	0.968355i	−0.874970	−	0.484178i	$-0.839119\pi$
$883$		−	28.7750i		−	0.968355i	0.874970	−	0.484178i	$-0.160881\pi$
$884$		0			0
$885$		0			0
$886$	23.0321	−	8.07661i	0.773780	−	0.271339i
$887$		−	29.0215i		−	0.974445i	−0.873278	−	0.487223i	$-0.838010\pi$
$887$		−	29.0215i		−	0.974445i	0.873278	−	0.487223i	$-0.161990\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$	−12.4959	+	9.99258i	−0.418395	+	0.334576i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−5.78792	−	43.6934i	−0.193253	−	1.45888i
$898$	17.9548	+	51.2018i	0.599159	+	1.70863i
$899$	−74.6111			−2.48842
$900$	−17.7436	+	24.1902i	−0.591452	+	0.806340i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$	15.3449	+	30.2859i	0.509802	+	1.00618i
$907$		0			0			−	1.00000i	$-0.5\pi$
$907$		0			0				1.00000i	$0.5\pi$
$908$		0			0
$909$	17.3792	−	4.68656i	0.576431	−	0.155443i
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$	49.4081	−	6.54493i	1.62805	−	0.215663i
$922$	48.0466	−	16.8484i	1.58233	−	0.554871i
$923$	74.5291			2.45315
$924$		0			0
$925$		0			0
$926$	42.7053	−	14.9753i	1.40338	−	0.492120i
$927$		0			0
$928$	−4.37150	−	37.6643i	−0.143502	−	1.23639i
$929$		−	10.2888i		−	0.337563i	−0.985653	−	0.168782i	$-0.946017\pi$
$929$		−	10.2888i		−	0.337563i	0.985653	−	0.168782i	$-0.0539833\pi$
$930$		0			0
$931$		0			0
$932$	36.2668	+	45.3524i	1.18796	+	1.48557i
$933$	−59.4971	+	7.88139i	−1.94785	+	0.258025i
$934$		0			0
$935$		0			0
$936$	43.0704	+	13.1154i	1.40780	+	0.428691i
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		1.00000			$0$
$941$		0			0		−1.00000			$\pi$
$942$		0			0
$943$	−58.3855			−1.90129
$944$	10.5556	−	46.8250i	0.343556	−	1.52402i
$945$		0			0
$946$		0			0
$947$	−49.5288			−1.60947			−0.804735	−	0.593634i	$-0.797693\pi$
$947$	−49.5288			−1.60947			−0.804735	+	0.593634i	$0.797693\pi$
$948$		0			0
$949$		−	40.3930i		−	1.31121i
$950$		0			0
$951$	−6.82356	−	51.5115i	−0.221269	−	1.67038i
$952$		0			0
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$		0			0
$955$		0			0
$956$	33.9287	+	42.4285i	1.09733	+	1.37224i
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	92.9041			2.99691
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−2.94295			−0.0946388			−0.0473194	−	0.998880i	$-0.515068\pi$
$967$	−2.94295			−0.0946388			−0.0473194	+	0.998880i	$0.515068\pi$
$968$	16.5000	−	26.3771i	0.530330	−	0.847791i
$969$		0			0
$970$		0			0
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$	−30.2999	−	7.34296i	−0.971868	−	0.235525i
$973$		0			0
$974$	22.1401	−	7.76380i	0.709415	−	0.248768i
$975$	−45.5535	+	6.03432i	−1.45888	+	0.193253i
$976$		0			0
$977$		0			0			−	1.00000i	$-0.5\pi$
$977$		0			0				1.00000i	$0.5\pi$
$978$	21.1573	−	10.7198i	0.676537	−	0.342780i
$979$		0			0
$980$		0			0
$981$		0			0
$982$	58.8131	−	20.6238i	1.87680	−	0.658132i
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$	24.7157	−	54.2791i	0.787908	−	1.73035i
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−56.0000			−1.77890			−0.889449	−	0.457034i	$-0.848912\pi$
$991$	−56.0000			−1.77890			−0.889449	+	0.457034i	$0.848912\pi$
$992$	7.25960	+	62.5478i	0.230492	+	1.98589i
$993$	54.1324	−	7.17074i	1.71784	−	0.227557i
$994$		0			0
$995$		0			0
$996$		0			0
$997$		−	57.5500i		−	1.82263i	−0.411714	−	0.911313i	$-0.635070\pi$
$997$		−	57.5500i		−	1.82263i	0.411714	−	0.911313i	$-0.364930\pi$
$998$	−5.40105	−	15.4022i	−0.170967	−	0.487549i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	552.2.b.a.413.5	✓	6
3.2	odd	2		552.2.b.b.413.2	yes	6
4.3	odd	2		2208.2.b.a.689.4		6
8.3	odd	2		2208.2.b.b.689.3		6
8.5	even	2		552.2.b.b.413.1	yes	6
12.11	even	2		2208.2.b.b.689.4		6
23.22	odd	2	CM	552.2.b.a.413.5	✓	6
24.5	odd	2	inner	552.2.b.a.413.6	yes	6
24.11	even	2		2208.2.b.a.689.3		6
69.68	even	2		552.2.b.b.413.2	yes	6
92.91	even	2		2208.2.b.a.689.4		6
184.45	odd	2		552.2.b.b.413.1	yes	6
184.91	even	2		2208.2.b.b.689.3		6
276.275	odd	2		2208.2.b.b.689.4		6
552.275	odd	2		2208.2.b.a.689.3		6
552.413	even	2	inner	552.2.b.a.413.6	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
552.2.b.a.413.5	✓	6	1.1	even	1	trivial
552.2.b.a.413.5	✓	6	23.22	odd	2	CM
552.2.b.a.413.6	yes	6	24.5	odd	2	inner
552.2.b.a.413.6	yes	6	552.413	even	2	inner
552.2.b.b.413.1	yes	6	8.5	even	2
552.2.b.b.413.1	yes	6	184.45	odd	2
552.2.b.b.413.2	yes	6	3.2	odd	2
552.2.b.b.413.2	yes	6	69.68	even	2
2208.2.b.a.689.3		6	24.11	even	2
2208.2.b.a.689.3		6	552.275	odd	2
2208.2.b.a.689.4		6	4.3	odd	2
2208.2.b.a.689.4		6	92.91	even	2
2208.2.b.b.689.3		6	8.3	odd	2
2208.2.b.b.689.3		6	184.91	even	2
2208.2.b.b.689.4		6	12.11	even	2
2208.2.b.b.689.4		6	276.275	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:	\( N \)	\(=\)	\( 552 = 2^{3} \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	552.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(1.33454 - 0.467979i) q^{2} +(-0.227452 - 1.71705i) q^{3} +(1.56199 - 1.24907i) q^{4} +(-1.10709 - 2.18503i) q^{6} +(1.50000 - 2.39792i) q^{8} +(-2.89653 + 0.781094i) q^{9} +O(q^{10})\)	\(q+(1.33454 - 0.467979i) q^{2} +(-0.227452 - 1.71705i) q^{3} +(1.56199 - 1.24907i) q^{4} +(-1.10709 - 2.18503i) q^{6} +(1.50000 - 2.39792i) q^{8} +(-2.89653 + 0.781094i) q^{9} +(-2.50000 - 2.39792i) q^{12} -5.30602i q^{13} +(0.879635 - 3.90208i) q^{16} +(-3.50000 + 2.39792i) q^{18} +4.79583i q^{23} +(-4.45852 - 2.03017i) q^{24} +5.00000 q^{25} +(-2.48310 - 7.08109i) q^{26} +(2.00000 + 4.79583i) q^{27} +6.70287 q^{29} -11.1312 q^{31} +(-0.652183 - 5.61913i) q^{32} +(-3.54871 + 4.83804i) q^{36} +(-9.11071 + 1.20686i) q^{39} +12.1742i q^{41} +(2.24435 + 6.40023i) q^{46} -6.55848i q^{47} +(-6.90015 - 0.622843i) q^{48} +7.00000 q^{49} +(6.67270 - 2.33989i) q^{50} +(-6.62760 - 8.28796i) q^{52} +(4.91343 + 5.46427i) q^{54} +(8.94525 - 3.13680i) q^{58} +12.0000 q^{59} +(-14.8551 + 5.20918i) q^{62} +(-3.50000 - 7.19375i) q^{64} +(8.23469 - 1.09082i) q^{69} +14.0461i q^{71} +(-2.47180 + 8.11728i) q^{72} +7.61268 q^{73} +(-1.13726 - 8.58526i) q^{75} +(-11.5938 + 5.87423i) q^{78} +(7.77979 - 4.52492i) q^{81} +(5.69728 + 16.2470i) q^{82} +(-1.52458 - 11.5092i) q^{87} +(5.99034 + 7.49105i) q^{92} +(2.53182 + 19.1129i) q^{93} +(-3.06923 - 8.75255i) q^{94} +(-9.50000 + 2.39792i) q^{96} +(9.34178 - 3.27585i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 9 q^{8}+O(q^{10}) \) 6 * q + 9 * q^8	\( 6 q + 9 q^{8} - 15 q^{12} - 21 q^{18} + 30 q^{25} - 27 q^{26} + 12 q^{27} - 24 q^{39} + 42 q^{49} + 3 q^{52} + 15 q^{58} + 72 q^{59} - 45 q^{62} - 21 q^{64} - 51 q^{78} + 33 q^{82} - 48 q^{87} + 6 q^{93} + 39 q^{94} - 57 q^{96}+O(q^{100}) \) 6 * q + 9 * q^8 - 15 * q^12 - 21 * q^18 + 30 * q^25 - 27 * q^26 + 12 * q^27 - 24 * q^39 + 42 * q^49 + 3 * q^52 + 15 * q^58 + 72 * q^59 - 45 * q^62 - 21 * q^64 - 51 * q^78 + 33 * q^82 - 48 * q^87 + 6 * q^93 + 39 * q^94 - 57 * q^96

Introduction

L-functions

Modular forms

Varieties

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Database

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