LMFDB - Embedded newform 552.2.b.b.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			$-1.07255 + 0.921756i$ of defining polynomial
Character	$\chi$	$=$	552.413
Dual form			552.2.b.b.413.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.07255 - 0.921756i) q^{2} +(1.37328 - 1.05550i) q^{3} +(0.300733 - 1.97726i) q^{4} +(0.500000 - 2.39792i) q^{6} +(-1.50000 - 2.39792i) q^{8} +(0.771819 - 2.89902i) q^{9} +O(q^{10})$	\(q+(1.07255 - 0.921756i) q^{2} +(1.37328 - 1.05550i) q^{3} +(0.300733 - 1.97726i) q^{4} +(0.500000 - 2.39792i) q^{6} +(-1.50000 - 2.39792i) q^{8} +(0.771819 - 2.89902i) q^{9} +(-1.67402 - 3.03277i) q^{12} +1.57601i q^{13} +(-3.81912 - 1.18925i) q^{16} +(-1.84437 - 3.82077i) q^{18} +4.79583i q^{23} +(-4.59094 - 1.70976i) q^{24} +5.00000 q^{25} +(1.45270 + 1.69035i) q^{26} +(-2.00000 - 4.79583i) q^{27} -3.94950 q^{29} +5.83384 q^{31} +(-5.19240 + 2.24476i) q^{32} +(-5.50000 - 2.39792i) q^{36} +(1.66349 + 2.16431i) q^{39} -2.64601i q^{41} +(4.42059 + 5.14378i) q^{46} +13.7071i q^{47} +(-6.50000 + 2.39792i) q^{48} +7.00000 q^{49} +(5.36276 - 4.60878i) q^{50} +(3.11619 + 0.473959i) q^{52} +(-6.56569 - 3.30026i) q^{54} +(-4.23604 + 3.64047i) q^{58} -12.0000 q^{59} +(6.25710 - 5.37738i) q^{62} +(-3.50000 + 7.19375i) q^{64} +(5.06202 + 6.58604i) q^{69} +1.04102i q^{71} +(-8.10932 + 2.49777i) q^{72} +9.44264 q^{73} +(6.86642 - 5.27752i) q^{75} +(3.77915 + 0.788006i) q^{78} +(-7.80859 - 4.47503i) q^{81} +(-2.43897 - 2.83798i) q^{82} +(-5.42378 + 4.16872i) q^{87} +(9.48261 + 1.44226i) q^{92} +(8.01152 - 6.15765i) q^{93} +(12.6346 + 14.7015i) q^{94} +(-4.76129 + 8.56330i) q^{96} +(7.50786 - 6.45229i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 3 q^{6} - 9 q^{8}+O(q^{10}) $ 6 * q + 3 * q^6 - 9 * q^8	$ 6 q + 3 q^{6} - 9 q^{8} + 30 q^{25} + 27 q^{26} - 12 q^{27} - 33 q^{36} - 24 q^{39} - 39 q^{48} + 42 q^{49} + 3 q^{52} + 15 q^{58} - 72 q^{59} + 45 q^{62} - 21 q^{64} + 33 q^{82} - 48 q^{87} - 6 q^{93} + 39 q^{94}+O(q^{100}) $ 6 * q + 3 * q^6 - 9 * q^8 + 30 * q^25 + 27 * q^26 - 12 * q^27 - 33 * q^36 - 24 * q^39 - 39 * q^48 + 42 * q^49 + 3 * q^52 + 15 * q^58 - 72 * q^59 + 45 * q^62 - 21 * q^64 + 33 * q^82 - 48 * q^87 - 6 * q^93 + 39 * q^94

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			$\pi$
$6$	0.500000	−	2.39792i	0.204124	−	0.978945i
$7$		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			$\pi$
$12$	−1.67402	−	3.03277i	−0.483247	−	0.875484i
$13$			1.57601i			0.437107i	0.975825	+	0.218554i	$0.0701339\pi$
$13$			1.57601i			0.437107i	−0.975825	+	0.218554i	$0.929866\pi$
$14$		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	$0.5\pi$
$18$	−1.84437	−	3.82077i	−0.434722	−	0.900565i
$19$		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
$24$	−4.59094	−	1.70976i	−0.937121	−	0.349004i
$25$	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	$-0.619513\pi$
$29$	−3.94950			−0.733404			−0.366702	+	0.930339i	$0.619513\pi$
$30$		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
$36$	−5.50000	−	2.39792i	−0.916667	−	0.399653i
$37$		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	$0.507910\pi$
$48$	−6.50000	+	2.39792i	−0.938194	+	0.346109i
$49$	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			$\pi$
$54$	−6.56569	−	3.30026i	−0.893477	−	0.449109i
$55$		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	$-0.785358\pi$
$59$	−12.0000			−1.56227			−0.781133	+	0.624364i	$0.785358\pi$
$60$		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	$0.980325\pi$
$72$	−8.10932	+	2.49777i	−0.955693	+	0.294365i
$73$	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
$78$	3.77915	+	0.788006i	0.427904	+	0.0892242i
$79$		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
$96$	−4.76129	+	8.56330i	−0.485947	+	0.873988i
$97$		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	$-0.403510\pi$
$101$	6.00000			0.597022			0.298511	+	0.954406i	$0.403510\pi$
$102$		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			$\pi$
$108$	−10.0841	+	2.51226i	−0.970341	+	0.241742i
$109$		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	$-0.627428\pi$
$131$	−8.92112			−0.779442			−0.389721	+	0.920933i	$0.627428\pi$
$132$		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	$0.5\pi$
$138$	11.5000	+	2.39792i	0.978945	+	0.204124i
$139$		−	12.6371i		−	1.07186i	−0.844261	−	0.535932i	$-0.819960\pi$
$139$		−	12.6371i		−	1.07186i	0.844261	−	0.535932i	$-0.180040\pi$
$140$		0			0
$141$	14.4679	+	18.8237i	1.21842	+	1.58524i
$142$	0.959562	+	1.11654i	0.0805247	+	0.0936982i
$143$		0			0
$144$	−6.39533	+	10.1538i	−0.532945	+	0.846150i
$145$		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			$\pi$
$150$	2.50000	−	11.9896i	0.204124	−	0.978945i
$151$	−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
$156$	4.77968	−	2.63827i	0.382681	−	0.211231i
$157$		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
$162$	−12.5000	+	2.39792i	−0.982093	+	0.188398i
$163$			25.3031i			1.98189i	0.134250	+	0.990947i	$0.457137\pi$
$163$			25.3031i			1.98189i	−0.134250	+	0.990947i	$0.542863\pi$
$164$	−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	$-0.260127\pi$
$173$	18.0000			1.36851			0.684257	+	0.729241i	$0.260127\pi$
$174$	−1.97475	+	9.47057i	−0.149705	+	0.717962i
$175$		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	$-0.898168\pi$
$179$	−25.4005			−1.89852			−0.949262	+	0.314486i	$0.898168\pi$
$180$		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
$186$	2.91692	−	13.9891i	0.213879	−	1.02573i
$187$		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	$0.5\pi$
$192$	2.78654	+	13.5733i	0.201101	+	0.979570i
$193$	−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	$-0.655975\pi$
$197$	−13.2113			−0.941268			−0.470634	+	0.882329i	$0.655975\pi$
$198$		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	$0.456034\pi$
$211$			28.7750i			1.98095i	−0.137686	+	0.990476i	$0.543966\pi$
$212$		0			0
$213$	1.09880	+	1.42961i	0.0752884	+	0.0979553i
$214$		0			0
$215$		0			0
$216$	−8.50000	+	11.9896i	−0.578352	+	0.815788i
$217$		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	$-0.266649\pi$
$234$	6.02159	−	2.90675i	0.393643	−	0.190020i
$235$		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
$246$	−6.34490	−	1.32300i	−0.404536	−	0.0843516i
$247$		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	$-0.845408\pi$
$269$	−29.0093			−1.76873			−0.884365	+	0.466797i	$0.845408\pi$
$270$		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
$276$	14.5446	−	8.02830i	0.875484	−	0.483247i
$277$		−	33.2122i		−	1.99553i	−0.0668451	−	0.997763i	$-0.521293\pi$
$277$		−	33.2122i		−	1.99553i	0.0668451	−	0.997763i	$-0.478707\pi$
$278$	−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	$0.5\pi$
$282$	32.8684	+	6.85354i	1.95729	+	0.408122i
$283$		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
$288$	2.50000	+	16.7854i	0.147314	+	0.989090i
$289$	−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			$\pi$
$294$	3.50000	−	16.7854i	0.204124	−	0.978945i
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−7.55829			−0.437107
$300$	−8.37008	−	15.1638i	−0.483247	−	0.875484i
$301$		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	$-0.693340\pi$
$307$		−	28.7750i		−	1.64228i	0.570730	−	0.821138i	$-0.306660\pi$
$308$		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	$0.893087\pi$
$312$	2.69461	−	7.23538i	0.152552	−	0.409623i
$313$		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	$-0.818908\pi$
$317$	−30.0000			−1.68497			−0.842484	+	0.538721i	$0.818908\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−11.1966	+	14.0938i	−0.622033	+	0.782991i
$325$			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	$0.000253497\pi$
$331$			0.0289779i			0.00159277i	−1.00000		0.000796384i	$0.999747\pi$
$332$		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	$-0.0828835\pi$
$347$	36.0000			1.93258			0.966291	+	0.257454i	$0.0828835\pi$
$348$	6.61153	+	11.9779i	0.354415	+	0.642083i
$349$		−	26.9081i		−	1.44036i	−0.693788	−	0.720180i	$-0.744059\pi$
$349$		−	26.9081i		−	1.44036i	0.693788	−	0.720180i	$-0.255941\pi$
$350$		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	$0.527719\pi$
$354$	−6.00000	+	28.7750i	−0.318896	+	1.52937i
$355$		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
$372$	−9.76595	−	17.6927i	−0.506341	−	0.917323i
$373$		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	$0.5\pi$
$384$	15.5000	+	11.9896i	0.790981	+	0.611841i
$385$		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	$-0.634123\pi$
$397$		−	36.3642i		−	1.82507i	0.409002	−	0.912534i	$-0.365877\pi$
$398$		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
$414$	18.3238	−	8.84529i	0.900565	−	0.434722i
$415$		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
$426$	2.49627	+	0.520508i	0.120945	+	0.0252187i
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$	1.93478	+	20.6944i	0.0930870	+	0.995658i
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	4.72132	−	22.6426i	0.225593	−	1.08191i
$439$	−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	$-0.0321996\pi$
$443$	41.8799			1.98978			0.994888	+	0.100985i	$0.0321996\pi$
$444$		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	$0.639633\pi$
$450$	−9.22185	−	19.1039i	−0.434722	−	0.900565i
$451$		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	$-0.149842\pi$
$461$	38.2711			1.78246			0.891232	+	0.453547i	$0.149842\pi$
$462$		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			$\pi$
$468$	3.77915	−	8.66807i	0.174691	−	0.400682i
$469$		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
$486$	−14.6350	+	16.4868i	−0.663859	+	0.747858i
$487$	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	$-0.299707\pi$
$491$	26.0819			1.17706			0.588531	+	0.808475i	$0.299707\pi$
$492$	−8.02472	+	4.42946i	−0.361782	+	0.199695i
$493$		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	$0.250162\pi$
$499$			31.6072i			1.41493i	−0.706747	+	0.707466i	$0.749838\pi$
$500$		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	$-0.537562\pi$
$509$	−5.31232			−0.235465			−0.117732	+	0.993045i	$0.537562\pi$
$510$		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	$0.5\pi$
$522$	7.28434	+	15.0901i	0.318827	+	0.660477i
$523$		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	$0.170605\pi$
$541$			23.7561i			1.02135i	−0.859772	+	0.510677i	$0.829395\pi$
$542$	−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	$0.395402\pi$
$547$			44.2733i			1.89299i	−0.322722	+	0.946494i	$0.604598\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$	8.19973	−	22.0174i	0.349004	−	0.937121i
$553$		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			$\pi$
$558$	−10.7598	−	22.2898i	−0.455497	−	0.943602i
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		1.00000			$0$
$563$		0			0		−1.00000			$\pi$
$564$	41.5703	−	22.9459i	1.75043	−	0.966196i
$565$		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
$576$	18.1534	+	15.6988i	0.756393	+	0.654118i
$577$	−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	$-0.158646\pi$
$587$	42.5613			1.75669			0.878347	+	0.478023i	$0.158646\pi$
$588$	−11.7181	−	21.2294i	−0.483247	−	0.875484i
$589$		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	$0.937220\pi$
$600$	−22.9547	−	8.54881i	−0.937121	−	0.349004i
$601$	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
$606$	3.00000	−	14.3875i	0.121867	−	0.584452i
$607$	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
$624$	−3.77915	−	10.2441i	−0.151287	−	0.410092i
$625$	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	$-0.278447\pi$
$648$	0.982145	+	25.4369i	0.0385823	+	0.999255i
$649$		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	$-0.243146\pi$
$653$	36.9083			1.44433			0.722167	+	0.691719i	$0.243146\pi$
$654$		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	$-0.652111\pi$
$683$	−24.0377			−0.919777			−0.459889	+	0.887977i	$0.652111\pi$
$684$		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	$0.184354\pi$
$691$			28.7750i			1.09465i	−0.836919	+	0.547326i	$0.815646\pi$
$692$	5.41319	−	35.5907i	0.205779	−	1.35296i
$693$		0			0
$694$	38.6118	−	33.1832i	1.46569	−	1.25962i
$695$		0			0
$696$	18.1319	+	6.75270i	0.687288	+	0.255961i
$697$		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			$\pi$
$702$	5.20126	−	10.3476i	0.196309	−	0.390545i
$703$		0			0
$704$		0			0
$705$		0			0
$706$	34.5052	+	40.1501i	1.29862	+	1.51107i
$707$		0			0
$708$	20.0882	+	36.3932i	0.754961	+	1.36774i
$709$		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
$726$	5.50000	−	26.3771i	0.204124	−	0.978945i
$727$		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
$738$	−10.1098	+	4.88022i	−0.372146	+	0.179643i
$739$		−	12.6950i		−	0.466994i	−0.972357	−	0.233497i	$-0.924983\pi$
$739$		−	12.6950i		−	0.466994i	0.972357	−	0.233497i	$-0.0750170\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0			−	1.00000i	$-0.5\pi$
$743$		0			0				1.00000i	$0.5\pi$
$744$	−26.7828	−	9.97448i	−0.981905	−	0.365682i
$745$		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	$-0.0393902\pi$
$762$	−11.1566	+	53.5053i	−0.404162	+	1.93829i
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	18.9122i		−	0.682878i
$768$	27.6760	−	1.42777i	0.998672	−	0.0515204i
$769$		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
$786$	−4.46056	+	21.3921i	−0.159103	+	0.763030i
$787$		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	$-0.507734\pi$
$811$		−	56.9393i		−	1.99941i	0.0242949	−	0.999705i	$-0.492266\pi$
$812$		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	$-0.108652\pi$
$821$	54.0000			1.88461			0.942306	+	0.334751i	$0.108652\pi$
$822$		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			$\pi$
$828$	11.5000	−	26.3771i	0.399653	−	0.916667i
$829$		−	57.5500i		−	1.99879i	−0.0347314	−	0.999397i	$-0.511058\pi$
$829$		−	57.5500i		−	1.99879i	0.0347314	−	0.999397i	$-0.488942\pi$
$830$		0			0
$831$	−35.0556	−	45.6098i	−1.21607	−	1.58219i
$832$	−11.3374	−	5.51605i	−0.393055	−	0.191234i
$833$		0			0
$834$	−30.3027	−	6.31854i	−1.04930	−	0.218793i
$835$		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
$846$	52.3716	−	25.2809i	1.80057	−	0.869176i
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$	3.15716	−	1.74268i	0.108162	−	0.0597032i
$853$		−	57.5500i		−	1.97047i	−0.171197	−	0.985237i	$-0.554763\pi$
$853$		−	57.5500i		−	1.97047i	0.171197	−	0.985237i	$-0.445237\pi$
$854$		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	$-0.668061\pi$
$859$		−	50.6353i		−	1.72765i	0.503790	−	0.863826i	$-0.331939\pi$
$860$		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	$0.600154\pi$
$864$	21.1503	+	20.4124i	0.719548	+	0.694443i
$865$		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
$876$	−15.8071	−	28.6373i	−0.534073	−	0.967565i
$877$			57.5500i			1.94332i	0.236373	+	0.971662i	$0.424041\pi$
$877$			57.5500i			1.94332i	−0.236373	+	0.971662i	$0.575959\pi$
$878$	−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	$0.5\pi$
$882$	−12.9106	−	26.7454i	−0.434722	−	0.900565i
$883$			28.7750i			0.968355i	0.874970	+	0.484178i	$0.160881\pi$
$883$			28.7750i			0.968355i	−0.874970	+	0.484178i	$0.839119\pi$
$884$		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
$900$	−27.5000	−	11.9896i	−0.916667	−	0.399653i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$	−5.32278	+	25.5272i	−0.176838	+	0.848084i
$907$		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
$936$	−3.93652	−	12.7804i	−0.128669	−	0.417740i
$937$		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	$-0.464359\pi$
$947$	6.87688			0.223469			0.111734	+	0.993738i	$0.464359\pi$
$948$		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			$\pi$
$972$	−0.500000	+	31.1729i	−0.0160375	+	0.999871i
$973$		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	$0.5\pi$
$978$	60.6748	+	12.6516i	1.94017	+	0.404553i
$979$		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	$0.5\pi$
$984$	−4.52404	+	12.1477i	−0.144221	+	0.387253i
$985$		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	$0.364930\pi$
$997$			57.5500i			1.82263i	−0.411714	+	0.911313i	$0.635070\pi$
$998$	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.b.413.5	yes	6
3.2	odd	2		552.2.b.a.413.2	yes	6
4.3	odd	2		2208.2.b.b.689.2		6
8.3	odd	2		2208.2.b.a.689.5		6
8.5	even	2		552.2.b.a.413.1	✓	6
12.11	even	2		2208.2.b.a.689.6		6
23.22	odd	2	CM	552.2.b.b.413.5	yes	6
24.5	odd	2	inner	552.2.b.b.413.6	yes	6
24.11	even	2		2208.2.b.b.689.1		6
69.68	even	2		552.2.b.a.413.2	yes	6
92.91	even	2		2208.2.b.b.689.2		6
184.45	odd	2		552.2.b.a.413.1	✓	6
184.91	even	2		2208.2.b.a.689.5		6
276.275	odd	2		2208.2.b.a.689.6		6
552.275	odd	2		2208.2.b.b.689.1		6
552.413	even	2	inner	552.2.b.b.413.6	yes	6

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

\(f(q)\)	\(=\)	\(q+(1.07255 - 0.921756i) q^{2} +(1.37328 - 1.05550i) q^{3} +(0.300733 - 1.97726i) q^{4} +(0.500000 - 2.39792i) q^{6} +(-1.50000 - 2.39792i) q^{8} +(0.771819 - 2.89902i) q^{9} +O(q^{10})\)	\(q+(1.07255 - 0.921756i) q^{2} +(1.37328 - 1.05550i) q^{3} +(0.300733 - 1.97726i) q^{4} +(0.500000 - 2.39792i) q^{6} +(-1.50000 - 2.39792i) q^{8} +(0.771819 - 2.89902i) q^{9} +(-1.67402 - 3.03277i) q^{12} +1.57601i q^{13} +(-3.81912 - 1.18925i) q^{16} +(-1.84437 - 3.82077i) q^{18} +4.79583i q^{23} +(-4.59094 - 1.70976i) q^{24} +5.00000 q^{25} +(1.45270 + 1.69035i) q^{26} +(-2.00000 - 4.79583i) q^{27} -3.94950 q^{29} +5.83384 q^{31} +(-5.19240 + 2.24476i) q^{32} +(-5.50000 - 2.39792i) q^{36} +(1.66349 + 2.16431i) q^{39} -2.64601i q^{41} +(4.42059 + 5.14378i) q^{46} +13.7071i q^{47} +(-6.50000 + 2.39792i) q^{48} +7.00000 q^{49} +(5.36276 - 4.60878i) q^{50} +(3.11619 + 0.473959i) q^{52} +(-6.56569 - 3.30026i) q^{54} +(-4.23604 + 3.64047i) q^{58} -12.0000 q^{59} +(6.25710 - 5.37738i) q^{62} +(-3.50000 + 7.19375i) q^{64} +(5.06202 + 6.58604i) q^{69} +1.04102i q^{71} +(-8.10932 + 2.49777i) q^{72} +9.44264 q^{73} +(6.86642 - 5.27752i) q^{75} +(3.77915 + 0.788006i) q^{78} +(-7.80859 - 4.47503i) q^{81} +(-2.43897 - 2.83798i) q^{82} +(-5.42378 + 4.16872i) q^{87} +(9.48261 + 1.44226i) q^{92} +(8.01152 - 6.15765i) q^{93} +(12.6346 + 14.7015i) q^{94} +(-4.76129 + 8.56330i) q^{96} +(7.50786 - 6.45229i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 3 q^{6} - 9 q^{8}+O(q^{10}) \) 6 * q + 3 * q^6 - 9 * q^8	\( 6 q + 3 q^{6} - 9 q^{8} + 30 q^{25} + 27 q^{26} - 12 q^{27} - 33 q^{36} - 24 q^{39} - 39 q^{48} + 42 q^{49} + 3 q^{52} + 15 q^{58} - 72 q^{59} + 45 q^{62} - 21 q^{64} + 33 q^{82} - 48 q^{87} - 6 q^{93} + 39 q^{94}+O(q^{100}) \) 6 * q + 3 * q^6 - 9 * q^8 + 30 * q^25 + 27 * q^26 - 12 * q^27 - 33 * q^36 - 24 * q^39 - 39 * q^48 + 42 * q^49 + 3 * q^52 + 15 * q^58 - 72 * q^59 + 45 * q^62 - 21 * q^64 + 33 * q^82 - 48 * q^87 - 6 * q^93 + 39 * q^94

Introduction

L-functions

Modular forms

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Database

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