LMFDB - Embedded newform 552.2.b.b.413.3

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			413.3
Root			$-0.261988 + 1.38973i$ of defining polynomial
Character	$\chi$	$=$	552.413
Dual form			552.2.b.b.413.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.261988 - 1.38973i) q^{2} +(-1.60074 + 0.661546i) q^{3} +(-1.86272 - 0.728188i) q^{4} +(0.500000 + 2.39792i) q^{6} +(-1.50000 + 2.39792i) q^{8} +(2.12471 - 2.11792i) q^{9} +O(q^{10})$	\(q+(0.261988 - 1.38973i) q^{2} +(-1.60074 + 0.661546i) q^{3} +(-1.86272 - 0.728188i) q^{4} +(0.500000 + 2.39792i) q^{6} +(-1.50000 + 2.39792i) q^{8} +(2.12471 - 2.11792i) q^{9} +(3.46346 - 0.0666417i) q^{12} +6.88203i q^{13} +(2.93948 + 2.71283i) q^{16} +(-2.38670 - 3.50766i) q^{18} -4.79583i q^{23} +(0.814772 - 4.83075i) q^{24} +5.00000 q^{25} +(9.56420 + 1.80301i) q^{26} +(-2.00000 + 4.79583i) q^{27} +10.6524 q^{29} +5.29738 q^{31} +(4.54022 - 3.37437i) q^{32} +(-5.50000 + 2.39792i) q^{36} +(-4.55278 - 11.0163i) q^{39} +9.52822i q^{41} +(-6.66493 - 1.25645i) q^{46} +7.14860i q^{47} +(-6.50000 - 2.39792i) q^{48} +7.00000 q^{49} +(1.30994 - 6.94867i) q^{50} +(5.01141 - 12.8193i) q^{52} +(6.14096 + 4.03592i) q^{54} +(2.79080 - 14.8040i) q^{58} -12.0000 q^{59} +(1.38785 - 7.36195i) q^{62} +(-3.50000 - 7.19375i) q^{64} +(3.17267 + 7.67686i) q^{69} +15.0872i q^{71} +(1.89153 + 8.27177i) q^{72} -17.0553 q^{73} +(-8.00368 + 3.30773i) q^{75} +(-16.5025 + 3.44102i) q^{78} +(0.0288070 - 8.99995i) q^{81} +(13.2417 + 2.49628i) q^{82} +(-17.0516 + 7.04704i) q^{87} +(-3.49227 + 8.93331i) q^{92} +(-8.47970 + 3.50446i) q^{93} +(9.93465 + 1.87285i) q^{94} +(-5.03539 + 8.40505i) q^{96} +(1.83392 - 9.72814i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.261988	−	1.38973i	0.185254	−	0.982691i
$2$	0.261988	−	1.38973i	0.185254	−	0.982691i
$3$	−1.60074	+	0.661546i	−0.924185	+	0.381944i
$3$	−1.60074	+	0.661546i	−0.924185	+	0.381944i
$4$	−1.86272	−	0.728188i	−0.931362	−	0.364094i
$5$		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
$9$	2.12471	−	2.11792i	0.708238	−	0.705974i
$10$		0			0
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$	3.46346	−	0.0666417i	0.999815	−	0.0192378i
$13$			6.88203i			1.90873i	0.298639	+	0.954366i	$0.403467\pi$
$13$			6.88203i			1.90873i	−0.298639	+	0.954366i	$0.596533\pi$
$14$		0			0
$15$		0			0
$16$	2.93948	+	2.71283i	0.734871	+	0.678207i
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$	−2.38670	−	3.50766i	−0.562551	−	0.826763i
$19$		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
$24$	0.814772	−	4.83075i	0.166315	−	0.986073i
$25$	5.00000			1.00000
$26$	9.56420	+	1.80301i	1.87569	+	0.353600i
$27$	−2.00000	+	4.79583i	−0.384900	+	0.922958i
$28$		0			0
$29$	10.6524			1.97810			0.989048	−	0.147596i	$-0.0471536\pi$
$29$	10.6524			1.97810			0.989048	+	0.147596i	$0.0471536\pi$
$30$		0			0
$31$	5.29738			0.951437			0.475719	−	0.879598i	$-0.342188\pi$
$31$	5.29738			0.951437			0.475719	+	0.879598i	$0.342188\pi$
$32$	4.54022	−	3.37437i	0.802605	−	0.596511i
$33$		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
$39$	−4.55278	−	11.0163i	−0.729029	−	1.76402i
$40$		0			0
$41$			9.52822i			1.48806i	0.668148	+	0.744029i	$0.267087\pi$
$41$			9.52822i			1.48806i	−0.668148	+	0.744029i	$0.732913\pi$
$42$		0			0
$43$		0			0			−	1.00000i	$-0.5\pi$
$43$		0			0				1.00000i	$0.5\pi$
$44$		0			0
$45$		0			0
$46$	−6.66493	−	1.25645i	−0.982691	−	0.185254i
$47$			7.14860i			1.04273i	0.853334	+	0.521365i	$0.174577\pi$
$47$			7.14860i			1.04273i	−0.853334	+	0.521365i	$0.825423\pi$
$48$	−6.50000	−	2.39792i	−0.938194	−	0.346109i
$49$	7.00000			1.00000
$50$	1.30994	−	6.94867i	0.185254	−	0.982691i
$51$		0			0
$52$	5.01141	−	12.8193i	0.694958	−	1.77772i
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$	6.14096	+	4.03592i	0.835678	+	0.549219i
$55$		0			0
$56$		0			0
$57$		0			0
$58$	2.79080	−	14.8040i	0.366449	−	1.94386i
$59$	−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	$0.5\pi$
$62$	1.38785	−	7.36195i	0.176257	−	0.934968i
$63$		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
$69$	3.17267	+	7.67686i	0.381944	+	0.924185i
$70$		0			0
$71$			15.0872i			1.79052i	0.445548	+	0.895258i	$0.353009\pi$
$71$			15.0872i			1.79052i	−0.445548	+	0.895258i	$0.646991\pi$
$72$	1.89153	+	8.27177i	0.222919	+	0.974837i
$73$	−17.0553			−1.99617			−0.998087	−	0.0618285i	$-0.980307\pi$
$73$	−17.0553			−1.99617			−0.998087	+	0.0618285i	$0.980307\pi$
$74$		0			0
$75$	−8.00368	+	3.30773i	−0.924185	+	0.381944i
$76$		0			0
$77$		0			0
$78$	−16.5025	+	3.44102i	−1.86854	+	0.389618i
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$		0			0
$81$	0.0288070	−	8.99995i	0.00320078	−	0.999995i
$82$	13.2417	+	2.49628i	1.46230	+	0.275668i
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−17.0516	+	7.04704i	−1.82813	+	0.755522i
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$		0			0
$92$	−3.49227	+	8.93331i	−0.364094	+	0.931362i
$93$	−8.47970	+	3.50446i	−0.879304	+	0.363396i
$94$	9.93465	+	1.87285i	1.02468	+	0.193170i
$95$		0			0
$96$	−5.03539	+	8.40505i	−0.513922	+	0.857837i
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$	1.83392	−	9.72814i	0.185254	−	0.982691i
$99$		0			0
$100$	−9.31362	−	3.64094i	−0.931362	−	0.364094i
$101$	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			$\pi$
$104$	−16.5025	−	10.3230i	−1.61821	−	1.01226i
$105$		0			0
$106$		0			0
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$	7.21772	−	7.47694i	0.694525	−	0.719469i
$109$		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
$116$	−19.8424	−	7.75693i	−1.84232	−	0.720213i
$117$	14.5756	+	14.6223i	1.34752	+	1.35184i
$118$	−3.14386	+	16.6768i	−0.289416	+	1.53523i
$119$		0			0
$120$		0			0
$121$	11.0000			1.00000
$122$		0			0
$123$	−6.30336	−	15.2522i	−0.568355	−	1.37524i
$124$	−9.86755	−	3.85749i	−0.886133	−	0.346413i
$125$		0			0
$126$		0			0
$127$	13.9115			1.23444			0.617221	−	0.786790i	$-0.288258\pi$
$127$	13.9115			1.23444			0.617221	+	0.786790i	$0.288258\pi$
$128$	−10.9144	+	2.97939i	−0.964702	+	0.263344i
$129$		0			0
$130$		0			0
$131$	−13.7962			−1.20538			−0.602691	−	0.797975i	$-0.705905\pi$
$131$	−13.7962			−1.20538			−0.602691	+	0.797975i	$0.705905\pi$
$132$		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
$139$		−	23.5588i		−	1.99824i	−0.0419997	−	0.999118i	$-0.513373\pi$
$139$		−	23.5588i		−	1.99824i	0.0419997	−	0.999118i	$-0.486627\pi$
$140$		0			0
$141$	−4.72913	−	11.4430i	−0.398265	−	0.963676i
$142$	20.9671	+	3.95266i	1.75952	+	0.331700i
$143$		0			0
$144$	11.9911	−	0.461621i	0.999260	−	0.0384685i
$145$		0			0
$146$	−4.46829	+	23.7024i	−0.369798	+	1.96162i
$147$	−11.2052	+	4.63083i	−0.924185	+	0.381944i
$148$		0			0
$149$		0			0		1.00000			$0$
$149$		0			0		−1.00000			$\pi$
$150$	2.50000	+	11.9896i	0.204124	+	0.978945i
$151$	24.5062			1.99429			0.997144	−	0.0755288i	$-0.0240645\pi$
$151$	24.5062			1.99429			0.997144	+	0.0755288i	$0.0240645\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	0.458630	+	23.8356i	0.0367198	+	1.90838i
$157$		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
$163$			15.6203i			1.22348i	0.791061	+	0.611738i	$0.209529\pi$
$163$			15.6203i			1.22348i	−0.791061	+	0.611738i	$0.790471\pi$
$164$	6.93833	−	17.7484i	0.541793	−	1.38592i
$165$		0			0
$166$		0			0
$167$			9.59166i			0.742225i	0.928588	+	0.371113i	$0.121024\pi$
$167$			9.59166i			0.742225i	−0.928588	+	0.371113i	$0.878976\pi$
$168$		0			0
$169$	−34.3624			−2.64326
$170$		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	$0.260127\pi$
$174$	5.32618	+	25.5435i	0.403777	+	1.93645i
$175$		0			0
$176$		0			0
$177$	19.2088	−	7.93856i	1.44382	−	0.596699i
$178$		0			0
$179$	5.41261			0.404557			0.202279	−	0.979328i	$-0.435165\pi$
$179$	5.41261			0.404557			0.202279	+	0.979328i	$0.435165\pi$
$180$		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
$186$	2.64869	+	12.7027i	0.194211	+	0.931405i
$187$		0			0
$188$	5.20552	−	13.3159i	0.379652	−	0.971159i
$189$		0			0
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$	10.3616	+	9.19988i	0.747782	+	0.663944i
$193$	−8.44124			−0.607613			−0.303807	−	0.952734i	$-0.598258\pi$
$193$	−8.44124			−0.607613			−0.303807	+	0.952734i	$0.598258\pi$
$194$		0			0
$195$		0			0
$196$	−13.0391	−	5.09732i	−0.931362	−	0.364094i
$197$	−14.8442			−1.05760			−0.528802	−	0.848745i	$-0.677359\pi$
$197$	−14.8442			−1.05760			−0.528802	+	0.848745i	$0.677359\pi$
$198$		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
$202$	1.57193	−	8.33841i	0.110601	−	0.586688i
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−10.1572	−	10.1898i	−0.705974	−	0.708238i
$208$	−18.6698	+	20.2296i	−1.29452	+	1.40267i
$209$		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
$213$	−9.98085	−	24.1506i	−0.683877	−	1.65477i
$214$		0			0
$215$		0			0
$216$	−8.50000	−	11.9896i	−0.578352	−	0.815788i
$217$		0			0
$218$		0			0
$219$	27.3011	−	11.2829i	1.84483	−	0.762427i
$220$		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
$225$	10.6236	−	10.5896i	0.708238	−	0.705974i
$226$		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
$232$	−15.9786	+	25.5435i	−1.04904	+	1.67701i
$233$			6.34890i			0.415930i	0.978136	+	0.207965i	$0.0666840\pi$
$233$			6.34890i			0.415930i	−0.978136	+	0.207965i	$0.933316\pi$
$234$	24.1398	−	16.4253i	1.57807	−	1.07376i
$235$		0			0
$236$	22.3527	+	8.73826i	1.45504	+	0.568812i
$237$		0			0
$238$		0			0
$239$			0.789960i			0.0510982i	0.999674	+	0.0255491i	$0.00813342\pi$
$239$			0.789960i			0.0510982i	−0.999674	+	0.0255491i	$0.991867\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$	2.88187	−	15.2871i	0.185254	−	0.982691i
$243$	5.90778	+	14.4256i	0.378984	+	0.925403i
$244$		0			0
$245$		0			0
$246$	−22.8479	+	4.76411i	−1.45673	+	0.303748i
$247$		0			0
$248$	−7.94607	+	12.7027i	−0.504576	+	0.806620i
$249$		0			0
$250$		0			0
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$		0			0
$254$	3.64464	−	19.3332i	0.228685	−	1.21307i
$255$		0			0
$256$	1.28113	+	15.9486i	0.0800709	+	0.996789i
$257$		−	31.7640i		−	1.98138i	−0.136130	−	0.990691i	$-0.543466\pi$
$257$		−	31.7640i		−	1.98138i	0.136130	−	0.990691i	$-0.456534\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	22.6332	−	22.5609i	1.40096	−	1.39648i
$262$	−3.61445	+	19.1731i	−0.223301	+	1.18452i
$263$		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
$269$	27.7653			1.69288			0.846440	−	0.532484i	$-0.178741\pi$
$269$	27.7653			1.69288			0.846440	+	0.532484i	$0.178741\pi$
$270$		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
$276$	−0.319602	−	16.6102i	−0.0192378	−	0.999815i
$277$		−	18.5330i		−	1.11354i	−0.830666	−	0.556771i	$-0.812040\pi$
$277$		−	18.5330i		−	1.11354i	0.830666	−	0.556771i	$-0.187960\pi$
$278$	−32.7405	−	6.17214i	−1.96365	−	0.370180i
$279$	11.2554	−	11.2194i	0.673843	−	0.671690i
$280$		0			0
$281$		0			0			−	1.00000i	$-0.5\pi$
$281$		0			0				1.00000i	$0.5\pi$
$282$	−17.1417	+	3.57430i	−1.02078	+	0.212846i
$283$		0			0			−	1.00000i	$-0.5\pi$
$283$		0			0				1.00000i	$0.5\pi$
$284$	10.9863	−	28.1032i	0.651916	−	1.66762i
$285$		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
$292$	31.7693	+	12.4195i	1.85916	+	0.726795i
$293$		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
$299$	33.0051			1.90873
$300$	17.3173	−	0.333208i	0.999815	−	0.0192378i
$301$		0			0
$302$	6.42034	−	34.0571i	0.369449	−	1.95977i
$303$	−9.60442	+	3.96928i	−0.551759	+	0.228029i
$304$		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
$311$		−	23.0257i		−	1.30567i	−0.757501	−	0.652834i	$-0.773580\pi$
$311$		−	23.0257i		−	1.30567i	0.757501	−	0.652834i	$-0.226420\pi$
$312$	33.2454	+	5.60728i	1.88215	+	0.317450i
$313$		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
$324$	−6.60732	+	16.7435i	−0.367073	+	0.930192i
$325$			34.4102i			1.90873i
$326$	21.7081	+	4.09233i	1.20230	+	0.226653i
$327$		0			0
$328$	−22.8479	−	14.2923i	−1.26156	−	0.789162i
$329$		0			0
$330$		0			0
$331$		−	31.4974i		−	1.73125i	−0.500690	−	0.865627i	$-0.666920\pi$
$331$		−	31.4974i		−	1.73125i	0.500690	−	0.865627i	$-0.333080\pi$
$332$		0			0
$333$		0			0
$334$	13.3299	+	2.51290i	0.729378	+	0.137500i
$335$		0			0
$336$		0			0
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$	−9.00253	+	47.7546i	−0.489673	+	2.59751i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	4.71579	−	25.0152i	0.253522	−	1.34483i
$347$	36.0000			1.93258			0.966291	−	0.257454i	$-0.0828835\pi$
$347$	36.0000			1.93258			0.966291	+	0.257454i	$0.0828835\pi$
$348$	36.8941	−	0.709892i	1.97773	−	0.0380542i
$349$			8.99508i			0.481496i	0.970588	+	0.240748i	$0.0773927\pi$
$349$			8.99508i			0.481496i	−0.970588	+	0.240748i	$0.922607\pi$
$350$		0			0
$351$	−33.0051	−	13.7641i	−1.76168	−	0.734671i
$352$		0			0
$353$			15.8869i			0.845572i	0.906230	+	0.422786i	$0.138948\pi$
$353$			15.8869i			0.845572i	−0.906230	+	0.422786i	$0.861052\pi$
$354$	−6.00000	−	28.7750i	−0.318896	−	1.52937i
$355$		0			0
$356$		0			0
$357$		0			0
$358$	1.41804	−	7.52209i	0.0749457	−	0.397555i
$359$		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
$363$	−17.6081	+	7.27701i	−0.924185	+	0.381944i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		1.00000			$0$
$367$		0			0		−1.00000			$\pi$
$368$	13.0103	−	14.0973i	0.678207	−	0.734871i
$369$	20.1800	+	20.2447i	1.05053	+	1.05390i
$370$		0			0
$371$		0			0
$372$	18.3473	−	0.353026i	0.951261	−	0.0183035i
$373$		0			0			−	1.00000i	$-0.5\pi$
$373$		0			0				1.00000i	$0.5\pi$
$374$		0			0
$375$		0			0
$376$	−17.1417	−	10.7229i	−0.884018	−	0.552991i
$377$			73.3099i			3.77565i
$378$		0			0
$379$		0			0			−	1.00000i	$-0.5\pi$
$379$		0			0				1.00000i	$0.5\pi$
$380$		0			0
$381$	−22.2686	+	9.20307i	−1.14085	+	0.471488i
$382$		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
$386$	−2.21150	+	11.7311i	−0.112563	+	0.597096i
$387$		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
$393$	22.0841	−	9.12685i	1.11400	−	0.460388i
$394$	−3.88900	+	20.6295i	−0.195925	+	1.03930i
$395$		0			0
$396$		0			0
$397$		−	32.2971i		−	1.62095i	−0.585777	−	0.810473i	$-0.699210\pi$
$397$		−	32.2971i		−	1.62095i	0.585777	−	0.810473i	$-0.300790\pi$
$398$		0			0
$399$		0			0
$400$	14.6974	+	13.5641i	0.734871	+	0.678207i
$401$		0			0			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$		0			0
$403$			36.4567i			1.81604i
$404$	−11.1763	−	4.36913i	−0.556044	−	0.217372i
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	4.13420			0.204423			0.102211	−	0.994763i	$-0.467408\pi$
$409$	4.13420			0.204423			0.102211	+	0.994763i	$0.467408\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	−16.8221	+	11.4462i	−0.826763	+	0.562551i
$415$		0			0
$416$	23.2225	+	31.2459i	1.13858	+	1.53196i
$417$	15.5853	+	37.7115i	0.763214	+	1.84674i
$418$		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	$0.5\pi$
$422$	−39.9896	−	7.53871i	−1.94666	−	0.366979i
$423$	15.1402	+	15.1887i	0.736141	+	0.738501i
$424$		0			0
$425$		0			0
$426$	−36.1777	+	7.54358i	−1.75282	+	0.365488i
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$	−18.8892	+	8.67162i	−0.908809	+	0.417213i
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	−8.52766	−	40.8972i	−0.407467	−	1.95414i
$439$	33.1203			1.58075			0.790373	−	0.612626i	$-0.209887\pi$
$439$	33.1203			1.58075			0.790373	+	0.612626i	$0.209887\pi$
$440$		0			0
$441$	14.8730	−	14.8255i	0.708238	−	0.705974i
$442$		0			0
$443$	−24.6214			−1.16980			−0.584900	−	0.811105i	$-0.698866\pi$
$443$	−24.6214			−1.16980			−0.584900	+	0.811105i	$0.698866\pi$
$444$		0			0
$445$		0			0
$446$	−2.09591	+	11.1179i	−0.0992440	+	0.526447i
$447$		0			0
$448$		0			0
$449$		−	38.3667i		−	1.81063i	−0.424736	−	0.905317i	$-0.639633\pi$
$449$		−	38.3667i		−	1.81063i	0.424736	−	0.905317i	$-0.360367\pi$
$450$	−11.9335	−	17.5383i	−0.562551	−	0.826763i
$451$		0			0
$452$		0			0
$453$	−39.2280	+	16.2120i	−1.84309	+	0.761706i
$454$		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
$461$	−2.26875			−0.105666			−0.0528331	−	0.998603i	$-0.516825\pi$
$461$	−2.26875			−0.105666			−0.0528331	+	0.998603i	$0.516825\pi$
$462$		0			0
$463$	32.0000			1.48717			0.743583	−	0.668644i	$-0.233125\pi$
$463$	32.0000			1.48717			0.743583	+	0.668644i	$0.233125\pi$
$464$	31.3125	+	28.8980i	1.45365	+	1.34156i
$465$		0			0
$466$	8.82328	+	1.66334i	0.408731	+	0.0770526i
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$	−16.5025	−	37.8512i	−0.762830	−	1.74967i
$469$		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
$478$	1.09783	+	0.206960i	0.0502138	+	0.00946614i
$479$		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
$484$	−20.4900	−	8.01007i	−0.931362	−	0.364094i
$485$		0			0
$486$	21.5955	−	4.43090i	0.979593	−	0.200990i
$487$	−43.7150			−1.98092			−0.990459	−	0.137808i	$-0.955994\pi$
$487$	−43.7150			−1.98092			−0.990459	+	0.137808i	$0.955994\pi$
$488$		0			0
$489$	−10.3335	−	25.0040i	−0.467299	−	1.13072i
$490$		0			0
$491$	17.9880			0.811789			0.405894	−	0.913920i	$-0.366960\pi$
$491$	17.9880			0.811789			0.405894	+	0.913920i	$0.366960\pi$
$492$	0.634976	+	33.0006i	0.0286269	+	1.48778i
$493$		0			0
$494$		0			0
$495$		0			0
$496$	15.5716	+	14.3709i	0.699184	+	0.645271i
$497$		0			0
$498$		0			0
$499$			43.1484i			1.93159i	0.259310	+	0.965794i	$0.416505\pi$
$499$			43.1484i			1.93159i	−0.259310	+	0.965794i	$0.583495\pi$
$500$		0			0
$501$	−6.34533	−	15.3537i	−0.283488	−	0.685954i
$502$		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
$507$	55.0051	−	22.7323i	2.44286	−	1.00958i
$508$	−25.9132	−	10.1302i	−1.14971	−	0.449453i
$509$	−36.1489			−1.60227			−0.801136	−	0.598482i	$-0.795771\pi$
$509$	−36.1489			−1.60227			−0.801136	+	0.598482i	$0.795771\pi$
$510$		0			0
$511$		0			0
$512$	22.5000	+	2.39792i	0.994369	+	0.105974i
$513$		0			0
$514$	−44.1435	−	8.32179i	−1.94709	−	0.367058i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$	−28.8133	+	11.9078i	−1.26476	+	0.522696i
$520$		0			0
$521$		0			0			−	1.00000i	$-0.5\pi$
$521$		0			0				1.00000i	$0.5\pi$
$522$	−25.4240	−	37.3649i	−1.11278	−	1.63542i
$523$		0			0			−	1.00000i	$-0.5\pi$
$523$		0			0				1.00000i	$0.5\pi$
$524$	25.6986	+	10.0462i	1.12265	+	0.438872i
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−23.0000			−1.00000
$530$		0			0
$531$	−25.4966	+	25.4151i	−1.10646	+	1.10292i
$532$		0			0
$533$	−65.5735			−2.84030
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−8.66415	+	3.58069i	−0.373886	+	0.154518i
$538$	7.27418	−	38.5864i	0.313612	−	1.66358i
$539$		0			0
$540$		0			0
$541$		−	22.7591i		−	0.978492i	−0.872146	−	0.489246i	$-0.837272\pi$
$541$		−	22.7591i		−	0.978492i	0.872146	−	0.489246i	$-0.162728\pi$
$542$	−4.19181	+	22.2358i	−0.180054	+	0.955107i
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$			35.2099i			1.50546i	0.658327	+	0.752732i	$0.271265\pi$
$547$			35.2099i			1.50546i	−0.658327	+	0.752732i	$0.728735\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$	−23.1675	−	3.90751i	−0.986073	−	0.166315i
$553$		0			0
$554$	−25.7560	−	4.85544i	−1.09427	−	0.206288i
$555$		0			0
$556$	−17.1553	+	43.8836i	−0.727546	+	1.86108i
$557$		0			0		1.00000			$0$
$557$		0			0		−1.00000			$\pi$
$558$	−12.6433	−	18.5814i	−0.535232	−	0.786613i
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		1.00000			$0$
$563$		0			0		−1.00000			$\pi$
$564$	0.476394	+	24.7589i	0.0200598	+	1.04254i
$565$		0			0
$566$		0			0
$567$		0			0
$568$	−36.1777	−	22.6307i	−1.51798	−	0.949564i
$569$		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
$576$	−22.6723	−	7.87192i	−0.944679	−	0.327996i
$577$	46.8589			1.95076			0.975381	−	0.220527i	$-0.0707777\pi$
$577$	46.8589			1.95076			0.975381	+	0.220527i	$0.0707777\pi$
$578$	−4.45380	+	23.6255i	−0.185254	+	0.982691i
$579$	13.5122	−	5.58427i	0.561548	−	0.232074i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$	25.5830	−	40.8972i	1.05863	−	1.69234i
$585$		0			0
$586$		0			0
$587$	−1.22079			−0.0503876			−0.0251938	−	0.999683i	$-0.508020\pi$
$587$	−1.22079			−0.0503876			−0.0251938	+	0.999683i	$0.508020\pi$
$588$	24.2442	−	0.466492i	0.999815	−	0.0192378i
$589$		0			0
$590$		0			0
$591$	23.7616	−	9.82012i	0.977422	−	0.403946i
$592$		0			0
$593$		−	38.3667i		−	1.57553i	−0.615976	−	0.787765i	$-0.711238\pi$
$593$		−	38.3667i		−	1.57553i	0.615976	−	0.787765i	$-0.288762\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$	8.64694	−	45.8683i	0.353600	−	1.87569i
$599$		−	9.59166i		−	0.391905i	−0.980613	−	0.195952i	$-0.937220\pi$
$599$		−	9.59166i		−	0.391905i	0.980613	−	0.195952i	$-0.0627798\pi$
$600$	4.07386	−	24.1538i	0.166315	−	0.986073i
$601$	−42.5519			−1.73573			−0.867863	−	0.496803i	$-0.834507\pi$
$601$	−42.5519			−1.73573			−0.867863	+	0.496803i	$0.834507\pi$
$602$		0			0
$603$		0			0
$604$	−45.6483	−	17.8451i	−1.85740	−	0.726108i
$605$		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
$611$	−49.1969			−1.99029
$612$		0			0
$613$		0			0			−	1.00000i	$-0.5\pi$
$613$		0			0				1.00000i	$0.5\pi$
$614$	39.9896	+	7.53871i	1.61385	+	0.304237i
$615$		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
$622$	−31.9996	−	6.03247i	−1.28307	−	0.241880i
$623$		0			0
$624$	16.5025	−	44.7332i	0.660630	−	1.79076i
$625$	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
$633$	19.0360	+	46.0612i	0.756613	+	1.83077i
$634$	−7.85965	+	41.6920i	−0.312146	+	1.65580i
$635$		0			0
$636$		0			0
$637$			48.1742i			1.90873i
$638$		0			0
$639$	31.9534	+	32.0559i	1.26406	+	1.26811i
$640$		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
$647$			8.72852i			0.343153i	0.985171	+	0.171577i	$0.0548861\pi$
$647$			8.72852i			0.343153i	−0.985171	+	0.171577i	$0.945114\pi$
$648$	21.5379	+	13.5690i	0.846089	+	0.533041i
$649$		0			0
$650$	47.8210	+	9.01505i	1.87569	+	0.353600i
$651$		0			0
$652$	11.3745	−	29.0963i	0.445460	−	1.13950i
$653$	−49.0700			−1.92026			−0.960129	−	0.279556i	$-0.909813\pi$
$653$	−49.0700			−1.92026			−0.960129	+	0.279556i	$0.909813\pi$
$654$		0			0
$655$		0			0
$656$	−25.8484	+	28.0080i	−1.00921	+	1.09353i
$657$	−36.2376	+	36.1218i	−1.41377	+	1.40925i
$658$		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	$0.5\pi$
$662$	−43.7730	−	8.25195i	−1.70129	−	0.320721i
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		−	51.0870i		−	1.97810i
$668$	6.98454	−	17.8666i	0.270240	−	0.691280i
$669$	12.8059	−	5.29237i	0.495104	−	0.204615i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−29.9764			−1.15551			−0.577753	−	0.816211i	$-0.696070\pi$
$673$	−29.9764			−1.15551			−0.577753	+	0.816211i	$0.696070\pi$
$674$		0			0
$675$	−10.0000	+	23.9792i	−0.384900	+	0.922958i
$676$	64.0076	+	25.0223i	2.46183	+	0.962395i
$677$		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
$683$	52.2139			1.99791			0.998955	−	0.0457131i	$-0.0145560\pi$
$683$	52.2139			1.99791			0.998955	+	0.0457131i	$0.0145560\pi$
$684$		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	$-0.184354\pi$
$692$	−33.5290	−	13.1074i	−1.27458	−	0.498268i
$693$		0			0
$694$	9.43158	−	50.0304i	0.358018	−	1.89913i
$695$		0			0
$696$	8.67925	−	51.4589i	0.328986	−	1.95055i
$697$		0			0
$698$	12.5008	+	2.35661i	0.473161	+	0.0891988i
$699$	−4.20009	−	10.1629i	−0.158862	−	0.384397i
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$	−27.7753	+	42.2623i	−1.04831	+	1.59509i
$703$		0			0
$704$		0			0
$705$		0			0
$706$	22.0785	+	4.16217i	0.830936	+	0.156645i
$707$		0			0
$708$	−41.5615	+	0.799700i	−1.56198	+	0.0300546i
$709$		0			0			−	1.00000i	$-0.5\pi$
$709$		0			0				1.00000i	$0.5\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	25.4053i		−	0.951437i
$714$		0			0
$715$		0			0
$716$	−10.0822	−	3.94140i	−0.376789	−	0.147297i
$717$	−0.522595	−	1.26452i	−0.0195167	−	0.0472243i
$718$		0			0
$719$			47.9583i			1.78854i	0.447524	+	0.894272i	$0.352306\pi$
$719$			47.9583i			1.78854i	−0.447524	+	0.894272i	$0.647694\pi$
$720$		0			0
$721$		0			0
$722$	−4.97778	+	26.4050i	−0.185254	+	0.982691i
$723$		0			0
$724$		0			0
$725$	53.2618			1.97810
$726$	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
$736$	−16.1829	−	21.7741i	−0.596511	−	0.802605i
$737$		0			0
$738$	33.4217	−	22.7410i	1.23027	−	0.837108i
$739$			39.4360i			1.45068i	0.688393	+	0.725338i	$0.258316\pi$
$739$			39.4360i			1.45068i	−0.688393	+	0.725338i	$0.741684\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0			−	1.00000i	$-0.5\pi$
$743$		0			0				1.00000i	$0.5\pi$
$744$	4.31615	−	25.5903i	0.158238	−	0.938186i
$745$		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			$\pi$
$752$	−19.3929	+	21.0132i	−0.707187	+	0.766272i
$753$		0			0
$754$	101.881	+	19.2063i	3.71030	+	0.699454i
$755$		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
$761$		−	50.8204i		−	1.84224i	−0.389281	−	0.921119i	$-0.627276\pi$
$761$		−	50.8204i		−	1.84224i	0.389281	−	0.921119i	$-0.372724\pi$
$762$	6.95573	+	33.3585i	0.251979	+	1.20845i
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	82.5844i		−	2.98195i
$768$	−12.6015	−	24.6820i	−0.454718	−	0.890635i
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$	21.0133	+	50.8457i	0.756777	+	1.83116i
$772$	15.7237	+	6.14681i	0.565908	+	0.221228i
$773$		0			0		1.00000			$0$
$773$		0			0		−1.00000			$\pi$
$774$		0			0
$775$	26.4869			0.951437
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$	−21.3047	+	51.0870i	−0.761369	+	1.82570i
$784$	20.5764	+	18.9898i	0.734871	+	0.678207i
$785$		0			0
$786$	−6.89811	−	33.0822i	−0.246048	−	1.18000i
$787$		0			0			−	1.00000i	$-0.5\pi$
$787$		0			0				1.00000i	$0.5\pi$
$788$	27.6506	+	10.8094i	0.985012	+	0.385067i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$	−44.8844	−	8.46146i	−1.59289	−	0.300286i
$795$		0			0
$796$		0			0
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$		0			0
$799$		0			0
$800$	22.7011	−	16.8719i	0.802605	−	0.596511i
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$	50.6652	+	9.55123i	1.78460	+	0.336428i
$807$	−44.4449	+	18.3680i	−1.56454	+	0.646585i
$808$	−9.00000	+	14.3875i	−0.316619	+	0.506150i
$809$			38.3667i			1.34890i	0.738321	+	0.674450i	$0.235619\pi$
$809$			38.3667i			1.34890i	−0.738321	+	0.674450i	$0.764381\pi$
$810$		0			0
$811$		−	27.2713i		−	0.957625i	−0.877917	−	0.478812i	$-0.841067\pi$
$811$		−	27.2713i		−	0.957625i	0.877917	−	0.478812i	$-0.158933\pi$
$812$		0			0
$813$	25.6118	−	10.5847i	0.898244	−	0.371223i
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	1.08311	−	5.74544i	0.0378701	−	0.200884i
$819$		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
$823$	−7.27806			−0.253697			−0.126849	−	0.991922i	$-0.540486\pi$
$823$	−7.27806			−0.253697			−0.126849	+	0.991922i	$0.540486\pi$
$824$		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	$0.488942\pi$
$829$			57.5500i			1.99879i	−0.0347314	+	0.999397i	$0.511058\pi$
$830$		0			0
$831$	12.2605	+	29.6665i	0.425311	+	1.02912i
$832$	49.5076	−	24.0871i	1.71637	−	0.835070i
$833$		0			0
$834$	56.4921	−	11.7794i	1.95616	−	0.407888i
$835$		0			0
$836$		0			0
$837$	−10.5948	+	25.4053i	−0.366208	+	0.878137i
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	84.4730			2.91286
$842$		0			0
$843$		0			0
$844$	−20.9536	+	53.5999i	−0.721253	+	1.84498i
$845$		0			0
$846$	25.0748	−	17.0616i	0.862090	−	0.586589i
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$	1.00543	+	52.2538i	0.0344456	+	1.79018i
$853$			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
$857$		−	19.0662i		−	0.651288i	−0.945492	−	0.325644i	$-0.894419\pi$
$857$		−	19.0662i		−	0.651288i	0.945492	−	0.325644i	$-0.105581\pi$
$858$		0			0
$859$			0.256826i			0.00876280i	0.999990	+	0.00438140i	$0.00139465\pi$
$859$			0.256826i			0.00876280i	−0.999990	+	0.00438140i	$0.998605\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			43.6815i			1.48694i	0.668771	+	0.743469i	$0.266821\pi$
$863$			43.6815i			1.48694i	−0.668771	+	0.743469i	$0.733179\pi$
$864$	7.10249	+	28.5229i	0.241632	+	0.970368i
$865$		0			0
$866$		0			0
$867$	27.2125	−	11.2463i	0.924185	−	0.381944i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$	−59.0704	+	1.13659i	−1.99580	+	0.0384020i
$877$		−	57.5500i		−	1.94332i	−0.236373	−	0.971662i	$-0.575959\pi$
$877$		−	57.5500i		−	1.94332i	0.236373	−	0.971662i	$-0.424041\pi$
$878$	8.67713	−	46.0284i	0.292839	−	1.55338i
$879$		0			0
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$	−16.7069	−	24.5536i	−0.562551	−	0.826763i
$883$		−	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
$886$	−6.45053	+	34.2173i	−0.216710	+	1.14955i
$887$		−	59.5587i		−	1.99978i	−0.0146917	−	0.999892i	$-0.504677\pi$
$887$		−	59.5587i		−	1.99978i	0.0146917	−	0.999892i	$-0.495323\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$	14.9018	+	5.82550i	0.498949	+	0.195052i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−52.8324	+	21.8344i	−1.76402	+	0.729029i
$898$	−53.3195	−	10.0516i	−1.77929	−	0.335427i
$899$	56.4296			1.88203
$900$	−27.5000	+	11.9896i	−0.916667	+	0.399653i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$	12.2531	+	58.7638i	0.407082	+	1.95230i
$907$		0			0			−	1.00000i	$-0.5\pi$
$907$		0			0				1.00000i	$0.5\pi$
$908$		0			0
$909$	12.7483	−	12.7075i	0.422834	−	0.421482i
$910$		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
$921$	−19.0360	−	46.0612i	−0.627257	−	1.51777i
$922$	−0.594385	+	3.15296i	−0.0195750	+	0.103837i
$923$	−103.830			−3.41762
$924$		0			0
$925$		0			0
$926$	8.38362	−	44.4715i	0.275503	−	1.46142i
$927$		0			0
$928$	48.3641	−	35.9451i	1.58763	−	1.17996i
$929$		−	57.1790i		−	1.87598i	−0.346658	−	0.937992i	$-0.612683\pi$
$929$		−	57.1790i		−	1.87598i	0.346658	−	0.937992i	$-0.387317\pi$
$930$		0			0
$931$		0			0
$932$	4.62319	−	11.8262i	0.151438	−	0.387382i
$933$	15.2326	+	36.8581i	0.498692	+	1.20668i
$934$		0			0
$935$		0			0
$936$	−56.9266	+	13.0176i	−1.86070	+	0.425493i
$937$		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
$943$	45.6957			1.48806
$944$	−35.2738	−	32.5539i	−1.14806	−	1.05954i
$945$		0			0
$946$		0			0
$947$	−56.4057			−1.83294			−0.916470	−	0.400104i	$-0.868974\pi$
$947$	−56.4057			−1.83294			−0.916470	+	0.400104i	$0.868974\pi$
$948$		0			0
$949$		−	117.375i		−	3.81016i
$950$		0			0
$951$	48.0221	−	19.8464i	1.55722	−	0.643563i
$952$		0			0
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$		0			0
$955$		0			0
$956$	0.575239	−	1.47148i	0.0186046	−	0.0475910i
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−2.93779			−0.0947673
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−52.3291			−1.68279			−0.841396	−	0.540420i	$-0.818265\pi$
$967$	−52.3291			−1.68279			−0.841396	+	0.540420i	$0.818265\pi$
$968$	−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
$974$	−11.4528	+	60.7523i	−0.366972	+	1.94663i
$975$	−22.7639	−	55.0816i	−0.729029	−	1.76402i
$976$		0			0
$977$		0			0			−	1.00000i	$-0.5\pi$
$977$		0			0				1.00000i	$0.5\pi$
$978$	−37.4561	+	7.81014i	−1.19772	+	0.249741i
$979$		0			0
$980$		0			0
$981$		0			0
$982$	4.71265	−	24.9986i	0.150387	−	0.797737i
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$	46.0284	+	7.76332i	1.46733	+	0.247486i
$985$		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	$0.848912\pi$
$992$	24.0513	−	17.8753i	0.763628	−	0.567542i
$993$	20.8370	+	50.4190i	0.661242	+	1.60000i
$994$		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	$-0.364930\pi$
$998$	59.9648	+	11.3044i	1.89815	+	0.357834i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	552.2.b.b.413.3	yes	6
3.2	odd	2		552.2.b.a.413.4	yes	6
4.3	odd	2		2208.2.b.b.689.5		6
8.3	odd	2		2208.2.b.a.689.2		6
8.5	even	2		552.2.b.a.413.3	✓	6
12.11	even	2		2208.2.b.a.689.1		6
23.22	odd	2	CM	552.2.b.b.413.3	yes	6
24.5	odd	2	inner	552.2.b.b.413.4	yes	6
24.11	even	2		2208.2.b.b.689.6		6
69.68	even	2		552.2.b.a.413.4	yes	6
92.91	even	2		2208.2.b.b.689.5		6
184.45	odd	2		552.2.b.a.413.3	✓	6
184.91	even	2		2208.2.b.a.689.2		6
276.275	odd	2		2208.2.b.a.689.1		6
552.275	odd	2		2208.2.b.b.689.6		6
552.413	even	2	inner	552.2.b.b.413.4	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
552.2.b.a.413.3	✓	6	8.5	even	2
552.2.b.a.413.3	✓	6	184.45	odd	2
552.2.b.a.413.4	yes	6	3.2	odd	2
552.2.b.a.413.4	yes	6	69.68	even	2
552.2.b.b.413.3	yes	6	1.1	even	1	trivial
552.2.b.b.413.3	yes	6	23.22	odd	2	CM
552.2.b.b.413.4	yes	6	24.5	odd	2	inner
552.2.b.b.413.4	yes	6	552.413	even	2	inner
2208.2.b.a.689.1		6	12.11	even	2
2208.2.b.a.689.1		6	276.275	odd	2
2208.2.b.a.689.2		6	8.3	odd	2
2208.2.b.a.689.2		6	184.91	even	2
2208.2.b.b.689.5		6	4.3	odd	2
2208.2.b.b.689.5		6	92.91	even	2
2208.2.b.b.689.6		6	24.11	even	2
2208.2.b.b.689.6		6	552.275	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)

\(f(q)\)	\(=\)	\(q+(0.261988 - 1.38973i) q^{2} +(-1.60074 + 0.661546i) q^{3} +(-1.86272 - 0.728188i) q^{4} +(0.500000 + 2.39792i) q^{6} +(-1.50000 + 2.39792i) q^{8} +(2.12471 - 2.11792i) q^{9} +O(q^{10})\)	\(q+(0.261988 - 1.38973i) q^{2} +(-1.60074 + 0.661546i) q^{3} +(-1.86272 - 0.728188i) q^{4} +(0.500000 + 2.39792i) q^{6} +(-1.50000 + 2.39792i) q^{8} +(2.12471 - 2.11792i) q^{9} +(3.46346 - 0.0666417i) q^{12} +6.88203i q^{13} +(2.93948 + 2.71283i) q^{16} +(-2.38670 - 3.50766i) q^{18} -4.79583i q^{23} +(0.814772 - 4.83075i) q^{24} +5.00000 q^{25} +(9.56420 + 1.80301i) q^{26} +(-2.00000 + 4.79583i) q^{27} +10.6524 q^{29} +5.29738 q^{31} +(4.54022 - 3.37437i) q^{32} +(-5.50000 + 2.39792i) q^{36} +(-4.55278 - 11.0163i) q^{39} +9.52822i q^{41} +(-6.66493 - 1.25645i) q^{46} +7.14860i q^{47} +(-6.50000 - 2.39792i) q^{48} +7.00000 q^{49} +(1.30994 - 6.94867i) q^{50} +(5.01141 - 12.8193i) q^{52} +(6.14096 + 4.03592i) q^{54} +(2.79080 - 14.8040i) q^{58} -12.0000 q^{59} +(1.38785 - 7.36195i) q^{62} +(-3.50000 - 7.19375i) q^{64} +(3.17267 + 7.67686i) q^{69} +15.0872i q^{71} +(1.89153 + 8.27177i) q^{72} -17.0553 q^{73} +(-8.00368 + 3.30773i) q^{75} +(-16.5025 + 3.44102i) q^{78} +(0.0288070 - 8.99995i) q^{81} +(13.2417 + 2.49628i) q^{82} +(-17.0516 + 7.04704i) q^{87} +(-3.49227 + 8.93331i) q^{92} +(-8.47970 + 3.50446i) q^{93} +(9.93465 + 1.87285i) q^{94} +(-5.03539 + 8.40505i) q^{96} +(1.83392 - 9.72814i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 3 q^{6} - 9 q^{8}+O(q^{10}) \) 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

Varieties

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Database

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