LMFDB - Embedded newform 552.2.b.a.413.1

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			413.1
Root			$-1.07255 - 0.921756i$ of defining polynomial
Character	$\chi$	$=$	552.413
Dual form			552.2.b.a.413.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.07255 - 0.921756i) q^{2} +(-1.37328 + 1.05550i) q^{3} +(0.300733 + 1.97726i) q^{4} +(2.44584 + 0.133749i) q^{6} +(1.50000 - 2.39792i) q^{8} +(0.771819 - 2.89902i) q^{9} +O(q^{10})$	\(q+(-1.07255 - 0.921756i) q^{2} +(-1.37328 + 1.05550i) q^{3} +(0.300733 + 1.97726i) q^{4} +(2.44584 + 0.133749i) q^{6} +(1.50000 - 2.39792i) q^{8} +(0.771819 - 2.89902i) q^{9} +(-2.50000 - 2.39792i) q^{12} -1.57601i q^{13} +(-3.81912 + 1.18925i) q^{16} +(-3.50000 + 2.39792i) q^{18} +4.79583i q^{23} +(0.471086 + 4.87628i) q^{24} +5.00000 q^{25} +(-1.45270 + 1.69035i) q^{26} +(2.00000 + 4.79583i) q^{27} +3.94950 q^{29} +5.83384 q^{31} +(5.19240 + 2.24476i) q^{32} +(5.96422 + 0.654257i) q^{36} +(1.66349 + 2.16431i) q^{39} -2.64601i q^{41} +(4.42059 - 5.14378i) q^{46} +13.7071i q^{47} +(3.98947 - 5.66428i) q^{48} +7.00000 q^{49} +(-5.36276 - 4.60878i) q^{50} +(3.11619 - 0.473959i) q^{52} +(2.27548 - 6.98729i) q^{54} +(-4.23604 - 3.64047i) q^{58} +12.0000 q^{59} +(-6.25710 - 5.37738i) q^{62} +(-3.50000 - 7.19375i) q^{64} +(-5.06202 - 6.58604i) q^{69} +1.04102i q^{71} +(-5.79387 - 6.19928i) q^{72} +9.44264 q^{73} +(-6.86642 + 5.27752i) q^{75} +(0.210791 - 3.85467i) q^{78} +(-7.80859 - 4.47503i) q^{81} +(-2.43897 + 2.83798i) q^{82} +(-5.42378 + 4.16872i) q^{87} +(-9.48261 + 1.44226i) q^{92} +(-8.01152 + 6.15765i) q^{93} +(12.6346 - 14.7015i) q^{94} +(-9.50000 + 2.39792i) q^{96} +(-7.50786 - 6.45229i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.07255	−	0.921756i	−0.758408	−	0.651780i
$2$	−1.07255	−	0.921756i	−0.758408	−	0.651780i
$3$	−1.37328	+	1.05550i	−0.792866	+	0.609396i
$3$	−1.37328	+	1.05550i	−0.792866	+	0.609396i
$4$	0.300733	+	1.97726i	0.150366	+	0.988630i
$5$		0			0		1.00000			$0$
$5$		0			0		−1.00000			$\pi$
$6$	2.44584	+	0.133749i	0.998508	+	0.0546029i
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$	1.50000	−	2.39792i	0.530330	−	0.847791i
$9$	0.771819	−	2.89902i	0.257273	−	0.966339i
$10$		0			0
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$	−2.50000	−	2.39792i	−0.721688	−	0.692219i
$13$		−	1.57601i		−	0.437107i	−0.975825	−	0.218554i	$-0.929866\pi$
$13$		−	1.57601i		−	0.437107i	0.975825	−	0.218554i	$-0.0701339\pi$
$14$		0			0
$15$		0			0
$16$	−3.81912	+	1.18925i	−0.954780	+	0.297314i
$17$		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
$24$	0.471086	+	4.87628i	0.0961600	+	0.995366i
$25$	5.00000			1.00000
$26$	−1.45270	+	1.69035i	−0.284898	+	0.331506i
$27$	2.00000	+	4.79583i	0.384900	+	0.922958i
$28$		0			0
$29$	3.94950			0.733404			0.366702	−	0.930339i	$-0.380487\pi$
$29$	3.94950			0.733404			0.366702	+	0.930339i	$0.380487\pi$
$30$		0			0
$31$	5.83384			1.04779			0.523895	−	0.851783i	$-0.324479\pi$
$31$	5.83384			1.04779			0.523895	+	0.851783i	$0.324479\pi$
$32$	5.19240	+	2.24476i	0.917896	+	0.396821i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	5.96422	+	0.654257i	0.994037	+	0.109043i
$37$		0			0			−	1.00000i	$-0.5\pi$
$37$		0			0				1.00000i	$0.5\pi$
$38$		0			0
$39$	1.66349	+	2.16431i	0.266372	+	0.346568i
$40$		0			0
$41$		−	2.64601i		−	0.413237i	−0.978422	−	0.206618i	$-0.933754\pi$
$41$		−	2.64601i		−	0.413237i	0.978422	−	0.206618i	$-0.0662459\pi$
$42$		0			0
$43$		0			0			−	1.00000i	$-0.5\pi$
$43$		0			0				1.00000i	$0.5\pi$
$44$		0			0
$45$		0			0
$46$	4.42059	−	5.14378i	0.651780	−	0.758408i
$47$			13.7071i			1.99938i	0.0248485	+	0.999691i	$0.492090\pi$
$47$			13.7071i			1.99938i	−0.0248485	+	0.999691i	$0.507910\pi$
$48$	3.98947	−	5.66428i	0.575831	−	0.817569i
$49$	7.00000			1.00000
$50$	−5.36276	−	4.60878i	−0.758408	−	0.651780i
$51$		0			0
$52$	3.11619	−	0.473959i	0.432138	−	0.0657263i
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$	2.27548	−	6.98729i	0.309654	−	0.950849i
$55$		0			0
$56$		0			0
$57$		0			0
$58$	−4.23604	−	3.64047i	−0.556219	−	0.478018i
$59$	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	$0.5\pi$
$62$	−6.25710	−	5.37738i	−0.794652	−	0.682928i
$63$		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
$69$	−5.06202	−	6.58604i	−0.609396	−	0.792866i
$70$		0			0
$71$			1.04102i			0.123546i	0.998090	+	0.0617729i	$0.0196755\pi$
$71$			1.04102i			0.123546i	−0.998090	+	0.0617729i	$0.980325\pi$
$72$	−5.79387	−	6.19928i	−0.682814	−	0.730592i
$73$	9.44264			1.10518			0.552588	−	0.833454i	$-0.313640\pi$
$73$	9.44264			1.10518			0.552588	+	0.833454i	$0.313640\pi$
$74$		0			0
$75$	−6.86642	+	5.27752i	−0.792866	+	0.609396i
$76$		0			0
$77$		0			0
$78$	0.210791	−	3.85467i	0.0238673	−	0.436455i
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$		0			0
$81$	−7.80859	−	4.47503i	−0.867621	−	0.497225i
$82$	−2.43897	+	2.83798i	−0.269339	+	0.313402i
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−5.42378	+	4.16872i	−0.581491	+	0.446933i
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$		0			0
$92$	−9.48261	+	1.44226i	−0.988630	+	0.150366i
$93$	−8.01152	+	6.15765i	−0.830756	+	0.638519i
$94$	12.6346	−	14.7015i	1.30316	−	1.51635i
$95$		0			0
$96$	−9.50000	+	2.39792i	−0.969590	+	0.244736i
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$	−7.50786	−	6.45229i	−0.758408	−	0.651780i
$99$		0			0
$100$	1.50366	+	9.88630i	0.150366	+	0.988630i
$101$	−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			$\pi$
$104$	−3.77915	−	2.36402i	−0.370576	−	0.231811i
$105$		0			0
$106$		0			0
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$	−8.88114	+	5.39679i	−0.854588	+	0.519306i
$109$		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
$116$	1.18774	+	7.80919i	0.110279	+	0.725065i
$117$	−4.56889	−	1.21640i	−0.422394	−	0.112456i
$118$	−12.8706	−	11.0611i	−1.18484	−	1.01825i
$119$		0			0
$120$		0			0
$121$	11.0000			1.00000
$122$		0			0
$123$	2.79287	+	3.63372i	0.251825	+	0.327641i
$124$	1.75443	+	11.5350i	0.157552	+	1.03588i
$125$		0			0
$126$		0			0
$127$	−22.3133			−1.97998			−0.989990	−	0.141134i	$-0.954925\pi$
$127$	−22.3133			−1.97998			−0.989990	+	0.141134i	$0.954925\pi$
$128$	−2.87695	+	10.9418i	−0.254289	+	0.967128i
$129$		0			0
$130$		0			0
$131$	8.92112			0.779442			0.389721	−	0.920933i	$-0.372572\pi$
$131$	8.92112			0.779442			0.389721	+	0.920933i	$0.372572\pi$
$132$		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	$0.5\pi$
$138$	−0.641439	+	11.7298i	−0.0546029	+	0.998508i
$139$			12.6371i			1.07186i	0.844261	+	0.535932i	$0.180040\pi$
$139$			12.6371i			1.07186i	−0.844261	+	0.535932i	$0.819960\pi$
$140$		0			0
$141$	−14.4679	−	18.8237i	−1.21842	−	1.58524i
$142$	0.959562	−	1.11654i	0.0805247	−	0.0936982i
$143$		0			0
$144$	0.500000	+	11.9896i	0.0416667	+	0.999132i
$145$		0			0
$146$	−10.1277	−	8.70380i	−0.838175	−	0.720332i
$147$	−9.61299	+	7.38853i	−0.792866	+	0.609396i
$148$		0			0
$149$		0			0		1.00000			$0$
$149$		0			0		−1.00000			$\pi$
$150$	12.2292	+	0.668746i	0.998508	+	0.0546029i
$151$	−10.6456			−0.866324			−0.433162	−	0.901316i	$-0.642602\pi$
$151$	−10.6456			−0.866324			−0.433162	+	0.901316i	$0.642602\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−3.77915	+	3.94003i	−0.302574	+	0.315455i
$157$		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
$162$	4.25023	+	11.9973i	0.333930	+	0.942598i
$163$		−	25.3031i		−	1.98189i	−0.134250	−	0.990947i	$-0.542863\pi$
$163$		−	25.3031i		−	1.98189i	0.134250	−	0.990947i	$-0.457137\pi$
$164$	5.23185	−	0.795741i	0.408538	−	0.0621369i
$165$		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
$169$	10.5162			0.808937
$170$		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	$0.739873\pi$
$174$	9.65983	+	0.528243i	0.732309	+	0.0400460i
$175$		0			0
$176$		0			0
$177$	−16.4794	+	12.6661i	−1.23867	+	0.952039i
$178$		0			0
$179$	25.4005			1.89852			0.949262	−	0.314486i	$-0.101832\pi$
$179$	25.4005			1.89852			0.949262	+	0.314486i	$0.101832\pi$
$180$		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
$186$	14.2686	+	0.780272i	1.04623	+	0.0572123i
$187$		0			0
$188$	−27.1025	+	4.12217i	−1.97665	+	0.300640i
$189$		0			0
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$	12.3995	+	6.18479i	0.894859	+	0.446349i
$193$	−18.7045			−1.34638			−0.673188	−	0.739471i	$-0.735076\pi$
$193$	−18.7045			−1.34638			−0.673188	+	0.739471i	$0.735076\pi$
$194$		0			0
$195$		0			0
$196$	2.10513	+	13.8408i	0.150366	+	0.988630i
$197$	13.2113			0.941268			0.470634	−	0.882329i	$-0.344025\pi$
$197$	13.2113			0.941268			0.470634	+	0.882329i	$0.344025\pi$
$198$		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
$202$	6.43531	+	5.53053i	0.452787	+	0.389127i
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	13.9032	+	3.70151i	0.966339	+	0.257273i
$208$	1.87428	+	6.01898i	0.129958	+	0.417341i
$209$		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
$213$	−1.09880	−	1.42961i	−0.0752884	−	0.0979553i
$214$		0			0
$215$		0			0
$216$	14.5000	+	2.39792i	0.986600	+	0.163158i
$217$		0			0
$218$		0			0
$219$	−12.9674	+	9.96675i	−0.876257	+	0.673490i
$220$		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
$225$	3.85909	−	14.4951i	0.257273	−	0.966339i
$226$		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
$232$	5.92425	−	9.47057i	0.388946	−	0.621773i
$233$		−	22.6861i		−	1.48622i	−0.669171	−	0.743108i	$-0.733351\pi$
$233$		−	22.6861i		−	1.48622i	0.669171	−	0.743108i	$-0.266649\pi$
$234$	3.77915	+	5.51605i	0.247051	+	0.360595i
$235$		0			0
$236$	3.60879	+	23.7271i	0.234912	+	1.54450i
$237$		0			0
$238$		0			0
$239$		−	26.3731i		−	1.70594i	−0.521963	−	0.852968i	$-0.674800\pi$
$239$		−	26.3731i		−	1.70594i	0.521963	−	0.852968i	$-0.325200\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$	−11.7981	−	10.1393i	−0.758408	−	0.651780i
$243$	15.4468	−	2.09652i	0.990915	−	0.134492i
$244$		0			0
$245$		0			0
$246$	0.353901	−	6.47170i	0.0225639	−	0.412620i
$247$		0			0
$248$	8.75076	−	13.9891i	0.555674	−	0.888306i
$249$		0			0
$250$		0			0
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$		0			0
$254$	23.9321	+	20.5674i	1.50163	+	1.29051i
$255$		0			0
$256$	13.1713	−	9.08381i	0.823209	−	0.567738i
$257$		−	12.1021i		−	0.754907i	−0.926028	−	0.377454i	$-0.876800\pi$
$257$		−	12.1021i		−	0.754907i	0.926028	−	0.377454i	$-0.123200\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	3.04830	−	11.4497i	0.188685	−	0.708716i
$262$	−9.56836	−	8.22309i	−0.591135	−	0.508024i
$263$		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
$269$	29.0093			1.76873			0.884365	−	0.466797i	$-0.154592\pi$
$269$	29.0093			1.76873			0.884365	+	0.466797i	$0.154592\pi$
$270$		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
$277$			33.2122i			1.99553i	0.0668451	+	0.997763i	$0.478707\pi$
$277$			33.2122i			1.99553i	−0.0668451	+	0.997763i	$0.521293\pi$
$278$	11.6483	−	13.5539i	0.698619	−	0.812910i
$279$	4.50267	−	16.9124i	0.269568	−	1.01252i
$280$		0			0
$281$		0			0			−	1.00000i	$-0.5\pi$
$281$		0			0				1.00000i	$0.5\pi$
$282$	−1.83331	+	33.5253i	−0.109172	+	1.99640i
$283$		0			0			−	1.00000i	$-0.5\pi$
$283$		0			0				1.00000i	$0.5\pi$
$284$	−2.05836	+	0.313068i	−0.122141	+	0.0185771i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	10.5152	−	13.3203i	0.619613	−	0.784907i
$289$	−17.0000			−1.00000
$290$		0			0
$291$		0			0
$292$	2.83971	+	18.6706i	0.166181	+	1.09261i
$293$		0			0		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$	17.1208	+	0.936245i	0.998508	+	0.0546029i
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	7.55829			0.437107
$300$	−12.5000	−	11.9896i	−0.721688	−	0.692219i
$301$		0			0
$302$	11.4179	+	9.81261i	0.657027	+	0.564652i
$303$	8.23970	−	6.33303i	0.473359	−	0.363823i
$304$		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
$311$			11.6250i			0.659196i	0.944121	+	0.329598i	$0.106913\pi$
$311$			11.6250i			0.659196i	−0.944121	+	0.329598i	$0.893087\pi$
$312$	7.68508	−	0.742437i	0.435082	−	0.0420322i
$313$		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
$325$		−	7.88006i		−	0.437107i
$326$	−23.3233	+	27.1389i	−1.29176	+	1.50309i
$327$		0			0
$328$	−6.34490	−	3.96901i	−0.350339	−	0.219152i
$329$		0			0
$330$		0			0
$331$		−	0.0289779i		−	0.00159277i	−1.00000		0.000796384i	$-0.999747\pi$
$331$		−	0.0289779i		−	0.00159277i	1.00000		0.000796384i	$-0.000253497\pi$
$332$		0			0
$333$		0			0
$334$	−8.84117	+	10.2876i	−0.483767	+	0.562910i
$335$		0			0
$336$		0			0
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$	−11.2791	−	9.69335i	−0.613505	−	0.527249i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	19.3059	+	16.5916i	1.03789	+	0.891970i
$347$	−36.0000			−1.93258			−0.966291	−	0.257454i	$-0.917117\pi$
$347$	−36.0000			−1.93258			−0.966291	+	0.257454i	$0.917117\pi$
$348$	−9.87375	−	9.47057i	−0.529288	−	0.507676i
$349$			26.9081i			1.44036i	0.693788	+	0.720180i	$0.255941\pi$
$349$			26.9081i			1.44036i	−0.693788	+	0.720180i	$0.744059\pi$
$350$		0			0
$351$	7.55829	−	3.15203i	0.403432	−	0.168243i
$352$		0			0
$353$			37.4342i			1.99242i	0.0869714	+	0.996211i	$0.472281\pi$
$353$			37.4342i			1.99242i	−0.0869714	+	0.996211i	$0.527719\pi$
$354$	29.3500	+	1.60499i	1.55994	+	0.0853043i
$355$		0			0
$356$		0			0
$357$		0			0
$358$	−27.2434	−	23.4131i	−1.43986	−	1.23742i
$359$		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
$363$	−15.1061	+	11.6106i	−0.792866	+	0.609396i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		1.00000			$0$
$367$		0			0		−1.00000			$\pi$
$368$	−5.70346	−	18.3159i	−0.297314	−	0.954780i
$369$	−7.67082	−	2.04224i	−0.399327	−	0.106315i
$370$		0			0
$371$		0			0
$372$	−14.5846	−	13.9891i	−0.756177	−	0.725299i
$373$		0			0			−	1.00000i	$-0.5\pi$
$373$		0			0				1.00000i	$0.5\pi$
$374$		0			0
$375$		0			0
$376$	32.8684	+	20.5606i	1.69506	+	1.06033i
$377$		−	6.22446i		−	0.320576i
$378$		0			0
$379$		0			0			−	1.00000i	$-0.5\pi$
$379$		0			0				1.00000i	$0.5\pi$
$380$		0			0
$381$	30.6424	−	23.5517i	1.56986	−	1.20659i
$382$		0			0
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$	−7.59827	−	18.0628i	−0.387747	−	0.921766i
$385$		0			0
$386$	20.0615	+	17.2409i	1.02110	+	0.877541i
$387$		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
$393$	−12.2512	+	9.41628i	−0.617993	+	0.474989i
$394$	−14.1698	−	12.1776i	−0.713865	−	0.613499i
$395$		0			0
$396$		0			0
$397$			36.3642i			1.82507i	0.409002	+	0.912534i	$0.365877\pi$
$397$			36.3642i			1.82507i	−0.409002	+	0.912534i	$0.634123\pi$
$398$		0			0
$399$		0			0
$400$	−19.0956	+	5.94627i	−0.954780	+	0.297314i
$401$		0			0			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$		0			0
$403$		−	9.19421i		−	0.457996i
$404$	−1.80440	−	11.8636i	−0.0897721	−	0.590234i
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	32.7780			1.62077			0.810384	−	0.585899i	$-0.199258\pi$
$409$	32.7780			1.62077			0.810384	+	0.585899i	$0.199258\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	−11.5000	−	16.7854i	−0.565194	−	0.824958i
$415$		0			0
$416$	3.53777	−	8.18330i	0.173453	−	0.401219i
$417$	−13.3385	−	17.3543i	−0.653189	−	0.849844i
$418$		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	$0.5\pi$
$422$	−26.5235	+	30.8627i	−1.29114	+	1.50237i
$423$	39.7370	+	10.5794i	1.93208	+	0.514387i
$424$		0			0
$425$		0			0
$426$	−0.139235	+	2.54615i	−0.00674596	+	0.123362i
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$	−13.3417	−	15.9373i	−0.641903	−	0.766786i
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	23.0951	+	1.26295i	1.10353	+	0.0603459i
$439$	−38.7927			−1.85147			−0.925736	−	0.378170i	$-0.876554\pi$
$439$	−38.7927			−1.85147			−0.925736	+	0.378170i	$0.876554\pi$
$440$		0			0
$441$	5.40273	−	20.2931i	0.257273	−	0.966339i
$442$		0			0
$443$	−41.8799			−1.98978			−0.994888	−	0.100985i	$-0.967800\pi$
$443$	−41.8799			−1.98978			−0.994888	+	0.100985i	$0.967800\pi$
$444$		0			0
$445$		0			0
$446$	8.58041	+	7.37405i	0.406294	+	0.349171i
$447$		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
$453$	14.6194	−	11.2364i	0.686879	−	0.527934i
$454$		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
$461$	−38.2711			−1.78246			−0.891232	−	0.453547i	$-0.850158\pi$
$461$	−38.2711			−1.78246			−0.891232	+	0.453547i	$0.850158\pi$
$462$		0			0
$463$	32.0000			1.48717			0.743583	−	0.668644i	$-0.233125\pi$
$463$	32.0000			1.48717			0.743583	+	0.668644i	$0.233125\pi$
$464$	−15.0836	+	4.69696i	−0.700239	+	0.218051i
$465$		0			0
$466$	−20.9111	+	24.3320i	−0.968686	+	1.12716i
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$	1.03112	−	9.39969i	0.0476634	−	0.434501i
$469$		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
$478$	−24.3096	+	28.2865i	−1.11189	+	1.29380i
$479$		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
$484$	3.30806	+	21.7499i	0.150366	+	0.988630i
$485$		0			0
$486$	−18.5000	−	11.9896i	−0.839177	−	0.543858i
$487$	27.1250			1.22915			0.614575	−	0.788858i	$-0.289328\pi$
$487$	27.1250			1.22915			0.614575	+	0.788858i	$0.289328\pi$
$488$		0			0
$489$	26.7076	+	34.7484i	1.20776	+	1.57138i
$490$		0			0
$491$	−26.0819			−1.17706			−0.588531	−	0.808475i	$-0.700293\pi$
$491$	−26.0819			−1.17706			−0.588531	+	0.808475i	$0.700293\pi$
$492$	−6.34490	+	6.61502i	−0.286050	+	0.298228i
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−22.2801	+	6.93792i	−1.00041	+	0.311522i
$497$		0			0
$498$		0			0
$499$		−	31.6072i		−	1.41493i	−0.706747	−	0.707466i	$-0.749838\pi$
$499$		−	31.6072i		−	1.41493i	0.706747	−	0.707466i	$-0.250162\pi$
$500$		0			0
$501$	10.1240	+	13.1721i	0.452309	+	0.588485i
$502$		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
$507$	−14.4417	+	11.0999i	−0.641379	+	0.492963i
$508$	−6.71033	−	44.1191i	−0.297723	−	1.95747i
$509$	5.31232			0.235465			0.117732	−	0.993045i	$-0.462438\pi$
$509$	5.31232			0.235465			0.117732	+	0.993045i	$0.462438\pi$
$510$		0			0
$511$		0			0
$512$	−22.5000	−	2.39792i	−0.994369	−	0.105974i
$513$		0			0
$514$	−11.1552	+	12.9801i	−0.492033	+	0.572528i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$	24.7191	−	18.9991i	1.08505	−	0.833967i
$520$		0			0
$521$		0			0			−	1.00000i	$-0.5\pi$
$521$		0			0				1.00000i	$0.5\pi$
$522$	−13.8232	+	9.47057i	−0.605027	+	0.414515i
$523$		0			0			−	1.00000i	$-0.5\pi$
$523$		0			0				1.00000i	$0.5\pi$
$524$	2.68287	+	17.6394i	0.117202	+	0.770580i
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−23.0000			−1.00000
$530$		0			0
$531$	9.26182	−	34.7882i	0.401929	−	1.50968i
$532$		0			0
$533$	−4.17014			−0.180629
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−34.8821	+	26.8104i	−1.50528	+	1.15695i
$538$	−31.1140	−	26.7395i	−1.34142	−	1.15282i
$539$		0			0
$540$		0			0
$541$		−	23.7561i		−	1.02135i	−0.859772	−	0.510677i	$-0.829395\pi$
$541$		−	23.7561i		−	1.02135i	0.859772	−	0.510677i	$-0.170605\pi$
$542$	17.1608	+	14.7481i	0.737120	+	0.633485i
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$		−	44.2733i		−	1.89299i	−0.322722	−	0.946494i	$-0.604598\pi$
$547$		−	44.2733i		−	1.89299i	0.322722	−	0.946494i	$-0.395402\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$	−23.3858	+	2.25925i	−0.995366	+	0.0961600i
$553$		0			0
$554$	30.6135	−	35.6218i	1.30064	−	1.51342i
$555$		0			0
$556$	−24.9868	+	3.80038i	−1.05968	+	0.161172i
$557$		0			0		1.00000			$0$
$557$		0			0		−1.00000			$\pi$
$558$	−20.4184	+	13.9891i	−0.864382	+	0.592204i
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		1.00000			$0$
$563$		0			0		−1.00000			$\pi$
$564$	32.8684	−	34.2677i	1.38401	−	1.44293i
$565$		0			0
$566$		0			0
$567$		0			0
$568$	2.49627	+	1.56152i	0.104741	+	0.0655201i
$569$		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
$576$	−23.5562	+	4.59429i	−0.981507	+	0.191429i
$577$	−14.2544			−0.593417			−0.296708	−	0.954968i	$-0.595889\pi$
$577$	−14.2544			−0.593417			−0.296708	+	0.954968i	$0.595889\pi$
$578$	18.2334	+	15.6698i	0.758408	+	0.651780i
$579$	25.6865	−	19.7426i	1.06750	−	0.820476i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$	14.1640	−	22.6426i	0.586109	−	0.936959i
$585$		0			0
$586$		0			0
$587$	−42.5613			−1.75669			−0.878347	−	0.478023i	$-0.841354\pi$
$587$	−42.5613			−1.75669			−0.878347	+	0.478023i	$0.841354\pi$
$588$	−17.5000	−	16.7854i	−0.721688	−	0.692219i
$589$		0			0
$590$		0			0
$591$	−18.1429	+	13.9446i	−0.746299	+	0.573605i
$592$		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
$598$	−8.10666	−	6.96690i	−0.331506	−	0.284898i
$599$			9.59166i			0.391905i	0.980613	+	0.195952i	$0.0627798\pi$
$599$			9.59166i			0.391905i	−0.980613	+	0.195952i	$0.937220\pi$
$600$	2.35543	+	24.3814i	0.0961600	+	0.995366i
$601$	0.180813			0.00737552			0.00368776	−	0.999993i	$-0.498826\pi$
$601$	0.180813			0.00737552			0.00368776	+	0.999993i	$0.498826\pi$
$602$		0			0
$603$		0			0
$604$	−3.20147	−	21.0491i	−0.130266	−	0.856474i
$605$		0			0
$606$	−14.6750	−	0.802495i	−0.596132	−	0.0325991i
$607$	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
$611$	21.6025			0.873945
$612$		0			0
$613$		0			0			−	1.00000i	$-0.5\pi$
$613$		0			0				1.00000i	$0.5\pi$
$614$	26.5235	−	30.8627i	1.07040	−	1.24552i
$615$		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
$622$	10.7155	−	12.4685i	0.429650	−	0.499940i
$623$		0			0
$624$	−8.92698	−	6.28746i	−0.357365	−	0.251700i
$625$	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
$633$	30.3721	+	39.5162i	1.20718	+	1.57063i
$634$	−32.1765	−	27.6527i	−1.27789	−	1.09823i
$635$		0			0
$636$		0			0
$637$		−	11.0321i		−	0.437107i
$638$		0			0
$639$	3.01792	+	0.803476i	0.119387	+	0.0317850i
$640$		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
$647$		−	39.0392i		−	1.53479i	−0.641175	−	0.767395i	$-0.721553\pi$
$647$		−	39.0392i		−	1.53479i	0.641175	−	0.767395i	$-0.278447\pi$
$648$	−22.4436	+	12.0118i	−0.881669	+	0.471868i
$649$		0			0
$650$	−7.26349	+	8.45177i	−0.284898	+	0.331506i
$651$		0			0
$652$	50.0309	−	7.60948i	1.95936	−	0.298010i
$653$	−36.9083			−1.44433			−0.722167	−	0.691719i	$-0.756854\pi$
$653$	−36.9083			−1.44433			−0.722167	+	0.691719i	$0.756854\pi$
$654$		0			0
$655$		0			0
$656$	3.14677	+	10.1054i	0.122861	+	0.394550i
$657$	7.28800	−	27.3744i	0.284332	−	1.06798i
$658$		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	$0.5\pi$
$662$	−0.0267105	+	0.0310802i	−0.00103813	+	0.00120797i
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$			18.9411i			0.733404i
$668$	18.9652	−	2.88453i	0.733786	−	0.111606i
$669$	10.9863	−	8.44404i	0.424754	−	0.326465i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	51.6633			1.99147			0.995737	−	0.0922433i	$-0.0294037\pi$
$673$	51.6633			1.99147			0.995737	+	0.0922433i	$0.0294037\pi$
$674$		0			0
$675$	10.0000	+	23.9792i	0.384900	+	0.922958i
$676$	3.16256	+	20.7932i	0.121637	+	0.799740i
$677$		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
$683$	24.0377			0.919777			0.459889	−	0.887977i	$-0.347889\pi$
$683$	24.0377			0.919777			0.459889	+	0.887977i	$0.347889\pi$
$684$		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	$-0.184354\pi$
$692$	−5.41319	−	35.5907i	−0.205779	−	1.35296i
$693$		0			0
$694$	38.6118	+	33.1832i	1.46569	+	1.25962i
$695$		0			0
$696$	1.86055	+	19.2589i	0.0705241	+	0.730005i
$697$		0			0
$698$	24.8027	−	28.8604i	0.938797	−	1.09238i
$699$	23.9453	+	31.1545i	0.905695	+	1.17837i
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$	−11.0121	−	3.58619i	−0.415623	−	0.135352i
$703$		0			0
$704$		0			0
$705$		0			0
$706$	34.5052	−	40.1501i	1.29862	−	1.51107i
$707$		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
$713$			27.9781i			1.04779i
$714$		0			0
$715$		0			0
$716$	7.63877	+	50.2235i	0.285474	+	1.87694i
$717$	27.8370	+	36.2178i	1.03959	+	1.35258i
$718$		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
$722$	20.3785	+	17.5134i	0.758408	+	0.651780i
$723$		0			0
$724$		0			0
$725$	19.7475			0.733404
$726$	26.9042	+	1.47124i	0.998508	+	0.0546029i
$727$		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
$736$	−10.7655	+	24.9019i	−0.396821	+	0.917896i
$737$		0			0
$738$	6.34490	+	9.26102i	0.233559	+	0.340903i
$739$			12.6950i			0.466994i	0.972357	+	0.233497i	$0.0750170\pi$
$739$			12.6950i			0.466994i	−0.972357	+	0.233497i	$0.924983\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0			−	1.00000i	$-0.5\pi$
$743$		0			0				1.00000i	$0.5\pi$
$744$	2.74824	+	28.4474i	0.100755	+	1.04293i
$745$		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			$\pi$
$752$	−16.3012	−	52.3490i	−0.594443	−	1.90897i
$753$		0			0
$754$	−5.73743	+	6.67606i	−0.208945	+	0.243128i
$755$		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
$761$		−	6.81007i		−	0.246865i	−0.992353	−	0.123432i	$-0.960610\pi$
$761$		−	6.81007i		−	0.246865i	0.992353	−	0.123432i	$-0.0393902\pi$
$762$	−54.5745	−	2.98438i	−1.97703	−	0.108113i
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	18.9122i		−	0.682878i
$768$	−8.50000	+	26.3771i	−0.306717	+	0.951801i
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$	12.7738	+	16.6196i	0.460038	+	0.598540i
$772$	−5.62504	−	36.9836i	−0.202450	−	1.33107i
$773$		0			0		1.00000			$0$
$773$		0			0		−1.00000			$\pi$
$774$		0			0
$775$	29.1692			1.04779
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$	7.89900	+	18.9411i	0.282287	+	0.676901i
$784$	−26.7338	+	8.32478i	−0.954780	+	0.297314i
$785$		0			0
$786$	21.8196	+	1.19319i	0.778279	+	0.0425598i
$787$		0			0			−	1.00000i	$-0.5\pi$
$787$		0			0				1.00000i	$0.5\pi$
$788$	3.97308	+	26.1222i	0.141535	+	0.930566i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$	33.5189	−	39.0025i	1.18954	−	1.38415i
$795$		0			0
$796$		0			0
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$		0			0
$799$		0			0
$800$	25.9620	+	11.2238i	0.917896	+	0.396821i
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$	−8.47482	+	9.86126i	−0.298513	+	0.347348i
$807$	−39.8380	+	30.6195i	−1.40237	+	1.07786i
$808$	−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
$811$			56.9393i			1.99941i	0.0242949	+	0.999705i	$0.492266\pi$
$811$			56.9393i			1.99941i	−0.0242949	+	0.999705i	$0.507734\pi$
$812$		0			0
$813$	21.9725	−	16.8881i	0.770611	−	0.592291i
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	−35.1561	−	30.2133i	−1.22920	−	1.05638i
$819$		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
$823$	−45.6486			−1.59121			−0.795605	−	0.605815i	$-0.792847\pi$
$823$	−45.6486			−1.59121			−0.795605	+	0.605815i	$0.792847\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$	−3.13771	+	28.6034i	−0.109043	+	0.994037i
$829$			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
$831$	−35.0556	−	45.6098i	−1.21607	−	1.58219i
$832$	−11.3374	+	5.51605i	−0.393055	+	0.191234i
$833$		0			0
$834$	−1.69020	+	30.9082i	−0.0585268	+	1.07026i
$835$		0			0
$836$		0			0
$837$	11.6677	+	27.9781i	0.403294	+	0.967066i
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	−13.4015			−0.462119
$842$		0			0
$843$		0			0
$844$	56.8957	−	8.65358i	1.95843	−	0.297869i
$845$		0			0
$846$	−32.8684	−	47.9748i	−1.13004	−	1.64941i
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$	2.49627	−	2.60254i	0.0855208	−	0.0891616i
$853$			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
$857$		−	57.4743i		−	1.96328i	−0.190730	−	0.981642i	$-0.561086\pi$
$857$		−	57.4743i		−	1.96328i	0.190730	−	0.981642i	$-0.438914\pi$
$858$		0			0
$859$			50.6353i			1.72765i	0.503790	+	0.863826i	$0.331939\pi$
$859$			50.6353i			1.72765i	−0.503790	+	0.863826i	$0.668061\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			55.8693i			1.90181i	0.309477	+	0.950907i	$0.399846\pi$
$863$			55.8693i			1.90181i	−0.309477	+	0.950907i	$0.600154\pi$
$864$	−0.380679	+	29.3914i	−0.0129510	+	0.999916i
$865$		0			0
$866$		0			0
$867$	23.3458	−	17.9436i	0.792866	−	0.609396i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$	−23.6066	−	22.6426i	−0.797593	−	0.765024i
$877$		−	57.5500i		−	1.94332i	−0.236373	−	0.971662i	$-0.575959\pi$
$877$		−	57.5500i		−	1.94332i	0.236373	−	0.971662i	$-0.424041\pi$
$878$	41.6071	+	35.7574i	1.40417	+	1.20675i
$879$		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
$886$	44.9184	+	38.6031i	1.50906	+	1.29690i
$887$		−	30.5372i		−	1.02534i	−0.858586	−	0.512669i	$-0.828657\pi$
$887$		−	30.5372i		−	1.02534i	0.858586	−	0.512669i	$-0.171343\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$	−2.40586	−	15.8181i	−0.0805542	−	0.529629i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−10.3797	+	7.97782i	−0.346568	+	0.266372i
$898$	35.3647	−	41.1502i	1.18013	−	1.37320i
$899$	23.0408			0.768452
$900$	29.8211	+	3.27129i	0.994037	+	0.109043i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$	−26.0373	−	1.42384i	−0.865032	−	0.0473038i
$907$		0			0			−	1.00000i	$-0.5\pi$
$907$		0			0				1.00000i	$0.5\pi$
$908$		0			0
$909$	−4.63091	+	17.3941i	−0.153598	+	0.576926i
$910$		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
$921$	−30.3721	−	39.5162i	−1.00080	−	1.30210i
$922$	41.0478	+	35.2766i	1.35184	+	1.16177i
$923$	1.64065			0.0540028
$924$		0			0
$925$		0			0
$926$	−34.3216	−	29.4962i	−1.12788	−	0.969305i
$927$		0			0
$928$	20.5074	+	8.86567i	0.673188	+	0.291030i
$929$		−	46.8903i		−	1.53842i	−0.638996	−	0.769210i	$-0.720650\pi$
$929$		−	46.8903i		−	1.53842i	0.638996	−	0.769210i	$-0.279350\pi$
$930$		0			0
$931$		0			0
$932$	44.8564	−	6.82246i	1.46932	−	0.223477i
$933$	−12.2703	−	15.9645i	−0.401711	−	0.522654i
$934$		0			0
$935$		0			0
$936$	−9.77015	+	9.13121i	−0.319347	+	0.298463i
$937$		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
$943$	12.6898			0.413237
$944$	−45.8294	+	14.2711i	−1.49162	+	0.464483i
$945$		0			0
$946$		0			0
$947$	−6.87688			−0.223469			−0.111734	−	0.993738i	$-0.535641\pi$
$947$	−6.87688			−0.223469			−0.111734	+	0.993738i	$0.535641\pi$
$948$		0			0
$949$		−	14.8817i		−	0.483081i
$950$		0			0
$951$	−41.1985	+	31.6651i	−1.33595	+	1.02681i
$952$		0			0
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$		0			0
$955$		0			0
$956$	52.1466	−	7.93127i	1.68654	−	0.256515i
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	3.03372			0.0978619
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	55.2721			1.77743			0.888715	−	0.458460i	$-0.151599\pi$
$967$	55.2721			1.77743			0.888715	+	0.458460i	$0.151599\pi$
$968$	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			$\pi$
$972$	8.79074	+	29.9119i	0.281963	+	0.959425i
$973$		0			0
$974$	−29.0929	−	25.0026i	−0.932198	−	0.801135i
$975$	8.31745	+	10.8216i	0.266372	+	0.346568i
$976$		0			0
$977$		0			0			−	1.00000i	$-0.5\pi$
$977$		0			0				1.00000i	$0.5\pi$
$978$	3.38428	−	61.8873i	0.108217	−	1.97894i
$979$		0			0
$980$		0			0
$981$		0			0
$982$	27.9742	+	24.0412i	0.892693	+	0.767185i
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$	12.9027	−	1.24650i	0.411322	−	0.0397368i
$985$		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	$0.848912\pi$
$992$	30.2917	+	13.0956i	0.961761	+	0.415785i
$993$	0.0305863	+	0.0397948i	0.000970626	+	0.00126285i
$994$		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	$-0.364930\pi$
$998$	−29.1341	+	33.9003i	−0.922224	+	1.07310i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	552.2.b.a.413.1	✓	6
3.2	odd	2		552.2.b.b.413.6	yes	6
4.3	odd	2		2208.2.b.a.689.5		6
8.3	odd	2		2208.2.b.b.689.2		6
8.5	even	2		552.2.b.b.413.5	yes	6
12.11	even	2		2208.2.b.b.689.1		6
23.22	odd	2	CM	552.2.b.a.413.1	✓	6
24.5	odd	2	inner	552.2.b.a.413.2	yes	6
24.11	even	2		2208.2.b.a.689.6		6
69.68	even	2		552.2.b.b.413.6	yes	6
92.91	even	2		2208.2.b.a.689.5		6
184.45	odd	2		552.2.b.b.413.5	yes	6
184.91	even	2		2208.2.b.b.689.2		6
276.275	odd	2		2208.2.b.b.689.1		6
552.275	odd	2		2208.2.b.a.689.6		6
552.413	even	2	inner	552.2.b.a.413.2	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
552.2.b.a.413.1	✓	6	1.1	even	1	trivial
552.2.b.a.413.1	✓	6	23.22	odd	2	CM
552.2.b.a.413.2	yes	6	24.5	odd	2	inner
552.2.b.a.413.2	yes	6	552.413	even	2	inner
552.2.b.b.413.5	yes	6	8.5	even	2
552.2.b.b.413.5	yes	6	184.45	odd	2
552.2.b.b.413.6	yes	6	3.2	odd	2
552.2.b.b.413.6	yes	6	69.68	even	2
2208.2.b.a.689.5		6	4.3	odd	2
2208.2.b.a.689.5		6	92.91	even	2
2208.2.b.a.689.6		6	24.11	even	2
2208.2.b.a.689.6		6	552.275	odd	2
2208.2.b.b.689.1		6	12.11	even	2
2208.2.b.b.689.1		6	276.275	odd	2
2208.2.b.b.689.2		6	8.3	odd	2
2208.2.b.b.689.2		6	184.91	even	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)

\(f(q)\)	\(=\)	\(q+(-1.07255 - 0.921756i) q^{2} +(-1.37328 + 1.05550i) q^{3} +(0.300733 + 1.97726i) q^{4} +(2.44584 + 0.133749i) q^{6} +(1.50000 - 2.39792i) q^{8} +(0.771819 - 2.89902i) q^{9} +O(q^{10})\)	\(q+(-1.07255 - 0.921756i) q^{2} +(-1.37328 + 1.05550i) q^{3} +(0.300733 + 1.97726i) q^{4} +(2.44584 + 0.133749i) q^{6} +(1.50000 - 2.39792i) q^{8} +(0.771819 - 2.89902i) q^{9} +(-2.50000 - 2.39792i) q^{12} -1.57601i q^{13} +(-3.81912 + 1.18925i) q^{16} +(-3.50000 + 2.39792i) q^{18} +4.79583i q^{23} +(0.471086 + 4.87628i) q^{24} +5.00000 q^{25} +(-1.45270 + 1.69035i) q^{26} +(2.00000 + 4.79583i) q^{27} +3.94950 q^{29} +5.83384 q^{31} +(5.19240 + 2.24476i) q^{32} +(5.96422 + 0.654257i) q^{36} +(1.66349 + 2.16431i) q^{39} -2.64601i q^{41} +(4.42059 - 5.14378i) q^{46} +13.7071i q^{47} +(3.98947 - 5.66428i) q^{48} +7.00000 q^{49} +(-5.36276 - 4.60878i) q^{50} +(3.11619 - 0.473959i) q^{52} +(2.27548 - 6.98729i) q^{54} +(-4.23604 - 3.64047i) q^{58} +12.0000 q^{59} +(-6.25710 - 5.37738i) q^{62} +(-3.50000 - 7.19375i) q^{64} +(-5.06202 - 6.58604i) q^{69} +1.04102i q^{71} +(-5.79387 - 6.19928i) q^{72} +9.44264 q^{73} +(-6.86642 + 5.27752i) q^{75} +(0.210791 - 3.85467i) q^{78} +(-7.80859 - 4.47503i) q^{81} +(-2.43897 + 2.83798i) q^{82} +(-5.42378 + 4.16872i) q^{87} +(-9.48261 + 1.44226i) q^{92} +(-8.01152 + 6.15765i) q^{93} +(12.6346 - 14.7015i) q^{94} +(-9.50000 + 2.39792i) q^{96} +(-7.50786 - 6.45229i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 9 q^{8}+O(q^{10}) \) 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

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Database

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