LMFDB - Embedded newform 588.2.be.a.257.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [588,2,Mod(5,588)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(588, base_ring=CyclotomicField(42)) chi = DirichletCharacter(H, H._module([0, 21, 29])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("588.5"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 588 = 2^{2} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	588.be (of order $42$, degree $12$, minimal)

Newform invariants

comment:select newform

sage:traces = [12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$4.69520363885$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\Q(\zeta_{21})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{11} + x^{9} - x^{8} + x^{6} - x^{4} + x^{3} - x + 1 $ x^12 - x^11 + x^9 - x^8 + x^6 - x^4 + x^3 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{42}]$

Embedding invariants

Embedding label			257.1
Root			$0.955573 + 0.294755i$ of defining polynomial
Character	$\chi$	$=$	588.257
Dual form			588.2.be.a.437.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.17809 - 1.26968i) q^{3} +(2.62818 + 0.304447i) q^{7} +(-0.224190 + 2.99161i) q^{9} +(1.61335 - 3.35015i) q^{13} +(6.38151 + 3.68437i) q^{19} +(-2.70969 - 3.69562i) q^{21} +(-4.13119 - 2.81660i) q^{25} +(4.06252 - 3.23975i) q^{27} +(6.51271 - 3.76012i) q^{31} +(-3.98985 - 10.1660i) q^{37} +(-6.15430 + 1.89835i) q^{39} +(2.43907 - 10.6863i) q^{43} +(6.81462 + 1.60028i) q^{49} +(-2.84004 - 12.4430i) q^{57} +(6.16795 - 2.42074i) q^{61} +(-1.50000 + 7.79423i) q^{63} +(8.15511 + 14.1251i) q^{67} +(-6.87882 + 10.0894i) q^{73} +(1.29074 + 8.56353i) q^{75} +(-7.33092 + 12.6975i) q^{79} +(-8.89948 - 1.34138i) q^{81} +(5.26011 - 8.31361i) q^{91} +(-12.4467 - 3.83931i) q^{93} -6.40982i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 3 q^{3} + 5 q^{7} - 3 q^{9} + 15 q^{19} + 6 q^{21} - 5 q^{25} + 3 q^{31} - 4 q^{37} - 30 q^{39} - 26 q^{43} - 11 q^{49} + 30 q^{57} + 19 q^{61} - 18 q^{63} + 5 q^{67} + 3 q^{73} - 15 q^{75} + 17 q^{79}+ \cdots + 3 q^{93}+O(q^{100}) $ 12 * q - 3 * q^3 + 5 * q^7 - 3 * q^9 + 15 * q^19 + 6 * q^21 - 5 * q^25 + 3 * q^31 - 4 * q^37 - 30 * q^39 - 26 * q^43 - 11 * q^49 + 30 * q^57 + 19 * q^61 - 18 * q^63 + 5 * q^67 + 3 * q^73 - 15 * q^75 + 17 * q^79 + 9 * q^81 + 9 * q^91 + 3 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/588\mathbb{Z}\right)^\times$.

$n$	$197$	$295$	$493$
$\chi(n)$	$-1$	$1$	$e\left(\frac{11}{42}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−1.17809	−	1.26968i	−0.680173	−	0.733052i
$3$	−1.17809	−	1.26968i	−0.680173	−	0.733052i
$4$		0			0
$5$		0			0		0.294755	−	0.955573i	$-0.404762\pi$
$5$		0			0		−0.294755	+	0.955573i	$0.595238\pi$
$6$		0			0
$7$	2.62818	+	0.304447i	0.993357	+	0.115070i
$7$	2.62818	+	0.304447i	0.993357	+	0.115070i
$8$		0			0
$9$	−0.224190	+	2.99161i	−0.0747301	+	0.997204i
$10$		0			0
$11$		0			0		−0.0747301	−	0.997204i	$-0.523810\pi$
$11$		0			0		0.0747301	+	0.997204i	$0.476190\pi$
$12$		0			0
$13$	1.61335	−	3.35015i	0.447462	−	0.929165i	−0.548220	−	0.836334i	$-0.684694\pi$
$13$	1.61335	−	3.35015i	0.447462	−	0.929165i	0.995682	−	0.0928304i	$-0.0295914\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		0.149042	−	0.988831i	$-0.452381\pi$
$17$		0			0		−0.149042	+	0.988831i	$0.547619\pi$
$18$		0			0
$19$	6.38151	+	3.68437i	1.46402	+	0.845252i	0.999194	−	0.0401503i	$-0.0127837\pi$
$19$	6.38151	+	3.68437i	1.46402	+	0.845252i	0.464826	+	0.885402i	$0.346117\pi$
$20$		0			0
$21$	−2.70969	−	3.69562i	−0.591302	−	0.806450i
$22$		0			0
$23$		0			0		0.988831	−	0.149042i	$-0.0476190\pi$
$23$		0			0		−0.988831	+	0.149042i	$0.952381\pi$
$24$		0			0
$25$	−4.13119	−	2.81660i	−0.826239	−	0.563320i
$26$		0			0
$27$	4.06252	−	3.23975i	0.781831	−	0.623490i
$28$		0			0
$29$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$29$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$30$		0			0
$31$	6.51271	−	3.76012i	1.16972	−	0.675337i	0.216104	−	0.976370i	$-0.430665\pi$
$31$	6.51271	−	3.76012i	1.16972	−	0.675337i	0.953613	+	0.301034i	$0.0973317\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−3.98985	−	10.1660i	−0.655928	−	1.67128i	−0.738127	−	0.674661i	$-0.764290\pi$
$37$	−3.98985	−	10.1660i	−0.655928	−	1.67128i	0.0821995	−	0.996616i	$-0.473806\pi$
$38$		0			0
$39$	−6.15430	+	1.89835i	−0.985477	+	0.303980i
$40$		0			0
$41$		0			0		0.974928	−	0.222521i	$-0.0714286\pi$
$41$		0			0		−0.974928	+	0.222521i	$0.928571\pi$
$42$		0			0
$43$	2.43907	−	10.6863i	0.371955	−	1.62964i	−0.349328	−	0.937000i	$-0.613590\pi$
$43$	2.43907	−	10.6863i	0.371955	−	1.62964i	0.721283	−	0.692640i	$-0.243553\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.563320	−	0.826239i	$-0.690476\pi$
$47$		0			0		0.563320	+	0.826239i	$0.309524\pi$
$48$		0			0
$49$	6.81462	+	1.60028i	0.973518	+	0.228612i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		0.365341	−	0.930874i	$-0.380952\pi$
$53$		0			0		−0.365341	+	0.930874i	$0.619048\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−2.84004	−	12.4430i	−0.376173	−	1.64812i
$58$		0			0
$59$		0			0		−0.294755	−	0.955573i	$-0.595238\pi$
$59$		0			0		0.294755	+	0.955573i	$0.404762\pi$
$60$		0			0
$61$	6.16795	−	2.42074i	0.789725	−	0.309944i	0.0640184	−	0.997949i	$-0.479608\pi$
$61$	6.16795	−	2.42074i	0.789725	−	0.309944i	0.725706	+	0.688005i	$0.241513\pi$
$62$		0			0
$63$	−1.50000	+	7.79423i	−0.188982	+	0.981981i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	8.15511	+	14.1251i	0.996306	+	1.72565i	0.572525	+	0.819887i	$0.305964\pi$
$67$	8.15511	+	14.1251i	0.996306	+	1.72565i	0.423780	+	0.905765i	$0.360703\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$71$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$72$		0			0
$73$	−6.87882	+	10.0894i	−0.805106	+	1.18087i	0.175422	+	0.984493i	$0.443871\pi$
$73$	−6.87882	+	10.0894i	−0.805106	+	1.18087i	−0.980528	+	0.196380i	$0.937081\pi$
$74$		0			0
$75$	1.29074	+	8.56353i	0.149042	+	0.988831i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−7.33092	+	12.6975i	−0.824793	+	1.42858i	0.0772843	+	0.997009i	$0.475375\pi$
$79$	−7.33092	+	12.6975i	−0.824793	+	1.42858i	−0.902077	+	0.431574i	$0.857958\pi$
$80$		0			0
$81$	−8.89948	−	1.34138i	−0.988831	−	0.149042i
$82$		0			0
$83$		0			0		−0.433884	−	0.900969i	$-0.642857\pi$
$83$		0			0		0.433884	+	0.900969i	$0.357143\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.997204	−	0.0747301i	$-0.976190\pi$
$89$		0			0		0.997204	+	0.0747301i	$0.0238095\pi$
$90$		0			0
$91$	5.26011	−	8.31361i	0.551409	−	0.871503i
$92$		0			0
$93$	−12.4467	−	3.83931i	−1.29067	−	0.398118i
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	6.40982i		−	0.650818i	−0.945573	−	0.325409i	$-0.894498\pi$
$97$		−	6.40982i		−	0.650818i	0.945573	−	0.325409i	$-0.105502\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		−0.680173	−	0.733052i	$-0.738095\pi$
$101$		0			0		0.680173	+	0.733052i	$0.261905\pi$
$102$		0			0
$103$	−5.80232	+	18.8106i	−0.571719	+	1.85347i	−0.0528713	+	0.998601i	$0.516837\pi$
$103$	−5.80232	+	18.8106i	−0.571719	+	1.85347i	−0.518848	+	0.854867i	$0.673639\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.0747301	−	0.997204i	$-0.476190\pi$
$107$		0			0		−0.0747301	+	0.997204i	$0.523810\pi$
$108$		0			0
$109$	0.315004	+	4.20344i	0.0301719	+	0.402616i	0.991759	+	0.128117i	$0.0408932\pi$
$109$	0.315004	+	4.20344i	0.0301719	+	0.402616i	−0.961587	+	0.274500i	$0.911488\pi$
$110$		0			0
$111$	−8.20715	+	17.0423i	−0.778989	+	1.61759i
$112$		0			0
$113$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$113$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	9.66065	+	5.57758i	0.893128	+	0.515647i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−10.8771	+	1.63946i	−0.988831	+	0.149042i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	6.65300	−	8.34260i	0.590358	−	0.740286i	−0.393482	−	0.919332i	$-0.628730\pi$
$127$	6.65300	−	8.34260i	0.590358	−	0.740286i	0.983841	+	0.179046i	$0.0573012\pi$
$128$		0			0
$129$	−16.4416	+	9.49258i	−1.44760	+	0.835775i
$130$		0			0
$131$		0			0		0.680173	−	0.733052i	$-0.261905\pi$
$131$		0			0		−0.680173	+	0.733052i	$0.738095\pi$
$132$		0			0
$133$	15.6500	+	11.6260i	1.35703	+	1.00810i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.955573	−	0.294755i	$-0.0952381\pi$
$137$		0			0		−0.955573	+	0.294755i	$0.904762\pi$
$138$		0			0
$139$	20.7339	−	4.73237i	1.75862	−	0.401395i	0.783207	−	0.621761i	$-0.213582\pi$
$139$	20.7339	−	4.73237i	1.75862	−	0.401395i	0.975417	+	0.220366i	$0.0707252\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−5.99641	−	10.5377i	−0.494576	−	0.869134i
$148$		0			0
$149$		0			0		0.826239	−	0.563320i	$-0.190476\pi$
$149$		0			0		−0.826239	+	0.563320i	$0.809524\pi$
$150$		0			0
$151$	−5.16044	+	13.1486i	−0.419951	+	1.07002i	0.551794	+	0.833981i	$0.313944\pi$
$151$	−5.16044	+	13.1486i	−0.419951	+	1.07002i	−0.971744	+	0.236036i	$0.924152\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−2.65986	−	8.62307i	−0.212280	−	0.688196i	−0.997609	−	0.0691164i	$-0.977982\pi$
$157$	−2.65986	−	8.62307i	−0.212280	−	0.688196i	0.785328	−	0.619080i	$-0.212494\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−16.0249	−	14.8690i	−1.25517	−	1.16463i	−0.979076	−	0.203497i	$-0.934769\pi$
$163$	−16.0249	−	14.8690i	−1.25517	−	1.16463i	−0.276094	−	0.961131i	$-0.589040\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.781831	−	0.623490i	$-0.785714\pi$
$167$		0			0		0.781831	+	0.623490i	$0.214286\pi$
$168$		0			0
$169$	−0.515253	−	0.646106i	−0.0396348	−	0.0497005i
$170$		0			0
$171$	−12.4529	+	18.2650i	−0.952295	+	1.39676i
$172$		0			0
$173$		0			0		−0.149042	−	0.988831i	$-0.547619\pi$
$173$		0			0		0.149042	+	0.988831i	$0.452381\pi$
$174$		0			0
$175$	−10.0000	−	8.66025i	−0.755929	−	0.654654i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.988831	−	0.149042i	$-0.952381\pi$
$179$		0			0		0.988831	+	0.149042i	$0.0476190\pi$
$180$		0			0
$181$	6.19761	+	12.8695i	0.460665	+	0.956580i	0.993866	+	0.110592i	$0.0352746\pi$
$181$	6.19761	+	12.8695i	0.460665	+	0.956580i	−0.533201	+	0.845989i	$0.679011\pi$
$182$		0			0
$183$	−10.3400	−	4.97948i	−0.764354	−	0.368094i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	11.6633	−	7.27781i	0.848383	−	0.529383i
$190$		0			0
$191$		0			0		−0.955573	−	0.294755i	$-0.904762\pi$
$191$		0			0		0.955573	+	0.294755i	$0.0952381\pi$
$192$		0			0
$193$	−14.8924	+	13.8181i	−1.07198	+	0.994650i	−0.999996	−	0.00275594i	$-0.999123\pi$
$193$	−14.8924	+	13.8181i	−1.07198	+	0.994650i	−0.0719816	+	0.997406i	$0.522932\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$	−6.14039	−	6.61777i	−0.435281	−	0.469121i	0.476871	−	0.878973i	$-0.341771\pi$
$199$	−6.14039	−	6.61777i	−0.435281	−	0.469121i	−0.912152	+	0.409852i	$0.865580\pi$
$200$		0			0
$201$	8.32688	−	26.9951i	0.587333	−	1.90409i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−26.0689	+	12.5541i	−1.79465	+	0.864260i	−0.858354	+	0.513058i	$0.828513\pi$
$211$	−26.0689	+	12.5541i	−1.79465	+	0.864260i	−0.936300	+	0.351202i	$0.885773\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	18.2613	−	7.89947i	1.23966	−	0.536251i
$218$		0			0
$219$	20.9142	−	3.15231i	1.41325	−	0.213013i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	22.1812	−	17.6889i	1.48536	−	1.18454i	0.547851	−	0.836576i	$-0.315446\pi$
$223$	22.1812	−	17.6889i	1.48536	−	1.18454i	0.937509	−	0.347960i	$-0.113126\pi$
$224$		0			0
$225$	9.35235	−	11.7275i	0.623490	−	0.781831i
$226$		0			0
$227$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$227$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$228$		0			0
$229$	−13.8655	+	14.9434i	−0.916257	+	0.987491i	−0.999959	−	0.00900024i	$-0.997135\pi$
$229$	−13.8655	+	14.9434i	−0.916257	+	0.987491i	0.0837021	+	0.996491i	$0.473326\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.365341	−	0.930874i	$-0.619048\pi$
$233$		0			0		0.365341	+	0.930874i	$0.380952\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	24.7583	−	5.65093i	1.60823	−	0.367067i
$238$		0			0
$239$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$239$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$240$		0			0
$241$	−18.4459	−	7.23948i	−1.18820	−	0.466336i	−0.312843	−	0.949805i	$-0.601281\pi$
$241$	−18.4459	−	7.23948i	−1.18820	−	0.466336i	−0.875361	+	0.483469i	$0.839377\pi$
$242$		0			0
$243$	8.78129	+	12.8798i	0.563320	+	0.826239i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	22.6388	−	15.4349i	1.44047	−	0.982097i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.974928	−	0.222521i	$-0.928571\pi$
$251$		0			0		0.974928	+	0.222521i	$0.0714286\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0		0.930874	−	0.365341i	$-0.119048\pi$
$257$		0			0		−0.930874	+	0.365341i	$0.880952\pi$
$258$		0			0
$259$	−7.39103	−	27.9327i	−0.459256	−	1.73565i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$263$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		0.563320	−	0.826239i	$-0.309524\pi$
$269$		0			0		−0.563320	+	0.826239i	$0.690476\pi$
$270$		0			0
$271$	3.49412	+	23.1820i	0.212253	+	1.40820i	0.800408	+	0.599456i	$0.204616\pi$
$271$	3.49412	+	23.1820i	0.212253	+	1.40820i	−0.588155	+	0.808748i	$0.700146\pi$
$272$		0			0
$273$	−16.7525	+	3.11554i	−1.01391	+	0.188561i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	32.8257	+	4.94768i	1.97230	+	0.297277i	0.997115	+	0.0759082i	$0.0241856\pi$
$277$	32.8257	+	4.94768i	1.97230	+	0.297277i	0.975190	+	0.221369i	$0.0710525\pi$
$278$		0			0
$279$	9.78872	+	20.3265i	0.586035	+	1.21691i
$280$		0			0
$281$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$281$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$282$		0			0
$283$	5.49148	−	0.411530i	0.326435	−	0.0244629i	0.0894950	−	0.995987i	$-0.471475\pi$
$283$	5.49148	−	0.411530i	0.326435	−	0.0244629i	0.236940	+	0.971524i	$0.423856\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−16.2447	−	5.01084i	−0.955573	−	0.294755i
$290$		0			0
$291$	−8.13844	+	7.55137i	−0.477084	+	0.442669i
$292$		0			0
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	9.66372	−	27.3428i	0.557007	−	1.57601i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−7.93361	+	16.4743i	−0.452795	+	0.940239i	0.542194	+	0.840254i	$0.317594\pi$
$307$	−7.93361	+	16.4743i	−0.452795	+	0.940239i	−0.994989	+	0.0999855i	$0.968120\pi$
$308$		0			0
$309$	30.7192	−	14.7936i	1.74756	−	0.841579i
$310$		0			0
$311$		0			0		0.149042	−	0.988831i	$-0.452381\pi$
$311$		0			0		−0.149042	+	0.988831i	$0.547619\pi$
$312$		0			0
$313$	−8.96734	−	5.17730i	−0.506864	−	0.292638i	0.224680	−	0.974433i	$-0.427866\pi$
$313$	−8.96734	−	5.17730i	−0.506864	−	0.292638i	−0.731544	+	0.681795i	$0.761200\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		0.988831	−	0.149042i	$-0.0476190\pi$
$317$		0			0		−0.988831	+	0.149042i	$0.952381\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−16.1011	+	9.29597i	−0.893128	+	0.515647i
$326$		0			0
$327$	4.96593	−	5.35200i	0.274617	−	0.295966i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−13.2241	−	33.6945i	−0.726863	−	1.85202i	−0.458481	−	0.888704i	$-0.651606\pi$
$331$	−13.2241	−	33.6945i	−0.726863	−	1.85202i	−0.268382	−	0.963312i	$-0.586489\pi$
$332$		0			0
$333$	31.3072	−	9.65698i	1.71562	−	0.529199i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−6.32076	+	27.6930i	−0.344314	+	1.50854i	0.445552	+	0.895256i	$0.353008\pi$
$337$	−6.32076	+	27.6930i	−0.344314	+	1.50854i	−0.789865	+	0.613280i	$0.789850\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	17.4228	+	6.28052i	0.940744	+	0.339116i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.365341	−	0.930874i	$-0.380952\pi$
$347$		0			0		−0.365341	+	0.930874i	$0.619048\pi$
$348$		0			0
$349$	−36.3500	−	8.29666i	−1.94577	−	0.444110i	−0.987279	−	0.158998i	$-0.949174\pi$
$349$	−36.3500	−	8.29666i	−1.94577	−	0.444110i	−0.958494	−	0.285112i	$-0.907969\pi$
$350$		0			0
$351$	−4.29939	−	18.8369i	−0.229485	−	1.00544i
$352$		0			0
$353$		0			0		−0.294755	−	0.955573i	$-0.595238\pi$
$353$		0			0		0.294755	+	0.955573i	$0.404762\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.733052	−	0.680173i	$-0.761905\pi$
$359$		0			0		0.733052	+	0.680173i	$0.238095\pi$
$360$		0			0
$361$	17.6491	+	30.5692i	0.928902	+	1.60890i
$362$		0			0
$363$	14.8959	+	11.8791i	0.781831	+	0.623490i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	19.8476	−	29.1110i	1.03603	−	1.51958i	0.193383	−	0.981123i	$-0.438054\pi$
$367$	19.8476	−	29.1110i	1.03603	−	1.51958i	0.842651	−	0.538460i	$-0.180994\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−4.87429	+	8.44251i	−0.252381	+	0.437137i	−0.964181	−	0.265246i	$-0.914547\pi$
$373$	−4.87429	+	8.44251i	−0.252381	+	0.437137i	0.711800	+	0.702382i	$0.247880\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−13.3843	−	6.44555i	−0.687506	−	0.331085i	0.0573090	−	0.998356i	$-0.481748\pi$
$379$	−13.3843	−	6.44555i	−0.687506	−	0.331085i	−0.744815	+	0.667271i	$0.767462\pi$
$380$		0			0
$381$	−18.4303	+	1.38116i	−0.944213	+	0.0707590i
$382$		0			0
$383$		0			0		−0.997204	−	0.0747301i	$-0.976190\pi$
$383$		0			0		0.997204	+	0.0747301i	$0.0238095\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	31.4223	+	9.69251i	1.59729	+	0.492698i
$388$		0			0
$389$		0			0		0.733052	−	0.680173i	$-0.238095\pi$
$389$		0			0		−0.733052	+	0.680173i	$0.761905\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	7.55160	−	24.4817i	0.379004	−	1.22870i	−0.542409	−	0.840115i	$-0.682488\pi$
$397$	7.55160	−	24.4817i	0.379004	−	1.22870i	0.921413	−	0.388586i	$-0.127036\pi$
$398$		0			0
$399$	−3.67588	−	33.5671i	−0.184024	−	1.68046i
$400$		0			0
$401$		0			0		0.0747301	−	0.997204i	$-0.476190\pi$
$401$		0			0		−0.0747301	+	0.997204i	$0.523810\pi$
$402$		0			0
$403$	−2.08969	−	27.8849i	−0.104095	−	1.38905i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−6.02660	+	39.9839i	−0.297996	+	1.97708i	−0.124932	+	0.992165i	$0.539871\pi$
$409$	−6.02660	+	39.9839i	−0.297996	+	1.97708i	−0.173064	+	0.984911i	$0.555367\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−30.4351	−	20.7503i	−1.49041	−	1.01615i
$418$		0			0
$419$		0			0		0.781831	−	0.623490i	$-0.214286\pi$
$419$		0			0		−0.781831	+	0.623490i	$0.785714\pi$
$420$		0			0
$421$	−4.63548	+	5.81271i	−0.225920	+	0.283294i	−0.881853	−	0.471525i	$-0.843704\pi$
$421$	−4.63548	+	5.81271i	−0.225920	+	0.283294i	0.655933	+	0.754819i	$0.272275\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	16.9474	−	4.48432i	0.820144	−	0.217011i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.955573	−	0.294755i	$-0.0952381\pi$
$431$		0			0		−0.955573	+	0.294755i	$0.904762\pi$
$432$		0			0
$433$	32.3866	−	7.39203i	1.55640	−	0.355238i	0.644159	−	0.764892i	$-0.277208\pi$
$433$	32.3866	−	7.39203i	1.55640	−	0.355238i	0.912241	+	0.409654i	$0.134351\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−19.2855	−	28.2867i	−0.920449	−	1.35005i	−0.936682	−	0.350182i	$-0.886120\pi$
$439$	−19.2855	−	28.2867i	−0.920449	−	1.35005i	0.0162329	−	0.999868i	$-0.494833\pi$
$440$		0			0
$441$	−6.31520	+	20.0279i	−0.300724	+	0.953711i
$442$		0			0
$443$		0			0		0.826239	−	0.563320i	$-0.190476\pi$
$443$		0			0		−0.826239	+	0.563320i	$0.809524\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$449$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	22.7740	−	8.93815i	1.07002	−	0.419951i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−31.0441	−	28.8047i	−1.45218	−	1.34743i	−0.819604	−	0.572930i	$-0.805807\pi$
$457$	−31.0441	−	28.8047i	−1.45218	−	1.34743i	−0.632577	−	0.774498i	$-0.718003\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.781831	−	0.623490i	$-0.785714\pi$
$461$		0			0		0.781831	+	0.623490i	$0.214286\pi$
$462$		0			0
$463$	4.91296	+	6.16066i	0.228325	+	0.286310i	0.882776	−	0.469794i	$-0.155672\pi$
$463$	4.91296	+	6.16066i	0.228325	+	0.286310i	−0.654451	+	0.756104i	$0.727100\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.149042	−	0.988831i	$-0.547619\pi$
$467$		0			0		0.149042	+	0.988831i	$0.452381\pi$
$468$		0			0
$469$	17.1327	+	39.6060i	0.791116	+	1.82883i
$470$		0			0
$471$	−7.81500	+	13.5360i	−0.360096	+	0.623705i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−15.9859	−	33.1950i	−0.733482	−	1.52309i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.997204	−	0.0747301i	$-0.0238095\pi$
$479$		0			0		−0.997204	+	0.0747301i	$0.976190\pi$
$480$		0			0
$481$	−40.4946	−	3.03465i	−1.84639	−	0.138368i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−31.5156	+	29.2422i	−1.42811	+	1.32509i	−0.560203	+	0.828355i	$0.689277\pi$
$487$	−31.5156	+	29.2422i	−1.42811	+	1.32509i	−0.867903	+	0.496734i	$0.834533\pi$
$488$		0			0
$489$			37.8636i			1.71225i
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−3.04246	+	40.5988i	−0.136199	+	1.81745i	0.342547	+	0.939501i	$0.388710\pi$
$499$	−3.04246	+	40.5988i	−0.136199	+	1.81745i	−0.478746	+	0.877953i	$0.658909\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.433884	−	0.900969i	$-0.357143\pi$
$503$		0			0		−0.433884	+	0.900969i	$0.642857\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−0.213334	+	1.41538i	−0.00947451	+	0.0628593i
$508$		0			0
$509$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$509$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$510$		0			0
$511$	−21.1505	+	24.4224i	−0.935641	+	1.08039i
$512$		0			0
$513$	37.8614	−	5.70669i	1.67162	−	0.251957i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$521$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$522$		0			0
$523$	−29.4773	+	31.7690i	−1.28895	+	1.38916i	−0.410121	+	0.912031i	$0.634513\pi$
$523$	−29.4773	+	31.7690i	−1.28895	+	1.38916i	−0.878833	+	0.477130i	$0.841677\pi$
$524$		0			0
$525$	0.785159	+	22.8994i	0.0342672	+	0.999413i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	21.9782	−	6.77937i	0.955573	−	0.294755i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−28.1782	+	19.2116i	−1.21148	+	0.825970i	−0.988847	−	0.148933i	$-0.952416\pi$
$541$	−28.1782	+	19.2116i	−1.21148	+	0.825970i	−0.222628	+	0.974903i	$0.571464\pi$
$542$		0			0
$543$	9.03878	−	23.0304i	0.387891	−	0.988331i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	1.33094	+	5.83123i	0.0569068	+	0.249325i	0.995378	−	0.0960317i	$-0.0306150\pi$
$547$	1.33094	+	5.83123i	0.0569068	+	0.249325i	−0.938471	+	0.345357i	$0.887758\pi$
$548$		0			0
$549$	5.85912	+	18.9948i	0.250061	+	0.810679i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−23.1327	+	31.1395i	−0.983702	+	1.32418i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$557$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$558$		0			0
$559$	−31.8655	−	25.4119i	−1.34777	−	1.07481i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.563320	−	0.826239i	$-0.309524\pi$
$563$		0			0		−0.563320	+	0.826239i	$0.690476\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−22.9810	−	6.23481i	−0.965112	−	0.261837i
$568$		0			0
$569$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$569$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$570$		0			0
$571$	−22.2815	−	3.35839i	−0.932451	−	0.140544i	−0.334790	−	0.942293i	$-0.608665\pi$
$571$	−22.2815	−	3.35839i	−0.932451	−	0.140544i	−0.597662	+	0.801749i	$0.703903\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−0.317044	+	0.0237591i	−0.0131987	+	0.000989106i	−0.0813278	−	0.996687i	$-0.525916\pi$
$577$	−0.317044	+	0.0237591i	−0.0131987	+	0.000989106i	0.0681291	+	0.997677i	$0.478297\pi$
$578$		0			0
$579$	35.0893	+	2.62958i	1.45826	+	0.109281i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$		0			0
$589$	55.4146			2.28332
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		0.294755	−	0.955573i	$-0.404762\pi$
$593$		0			0		−0.294755	+	0.955573i	$0.595238\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−1.16851	+	15.5927i	−0.0478240	+	0.638167i
$598$		0			0
$599$		0			0		−0.0747301	−	0.997204i	$-0.523810\pi$
$599$		0			0		0.0747301	+	0.997204i	$0.476190\pi$
$600$		0			0
$601$	−18.7877	+	39.0131i	−0.766367	+	1.59138i	0.0394563	+	0.999221i	$0.487437\pi$
$601$	−18.7877	+	39.0131i	−0.766367	+	1.59138i	−0.805823	+	0.592156i	$0.798277\pi$
$602$		0			0
$603$	−44.0850	+	21.2302i	−1.79528	+	0.864562i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	34.5000	+	19.9186i	1.40031	+	0.808470i	0.994424	−	0.105453i	$-0.0336291\pi$
$607$	34.5000	+	19.9186i	1.40031	+	0.808470i	0.405887	+	0.913923i	$0.366962\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	26.1768	+	17.8471i	1.05727	+	0.720836i	0.961566	−	0.274574i	$-0.0885368\pi$
$613$	26.1768	+	17.8471i	1.05727	+	0.720836i	0.0957060	+	0.995410i	$0.469489\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$617$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$618$		0			0
$619$	40.5000	−	23.3827i	1.62783	−	0.939829i	0.643094	−	0.765787i	$-0.277650\pi$
$619$	40.5000	−	23.3827i	1.62783	−	0.939829i	0.984738	−	0.174042i	$-0.0556830\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	9.13353	+	23.2718i	0.365341	+	0.930874i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−10.3232	+	45.2288i	−0.410959	+	1.80053i	0.168695	+	0.985668i	$0.446045\pi$
$631$	−10.3232	+	45.2288i	−0.410959	+	1.80053i	−0.579654	+	0.814862i	$0.696812\pi$
$632$		0			0
$633$	46.6513	+	18.3093i	1.85422	+	0.727728i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	16.3555	−	20.2482i	0.648030	−	0.802263i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.365341	−	0.930874i	$-0.380952\pi$
$641$		0			0		−0.365341	+	0.930874i	$0.619048\pi$
$642$		0			0
$643$	−48.8061	−	11.1397i	−1.92472	−	0.439306i	−0.997974	−	0.0636258i	$-0.979734\pi$
$643$	−48.8061	−	11.1397i	−1.92472	−	0.439306i	−0.926750	−	0.375680i	$-0.877409\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.294755	−	0.955573i	$-0.595238\pi$
$647$		0			0		0.294755	+	0.955573i	$0.404762\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−31.5434	−	13.8798i	−1.23628	−	0.543991i
$652$		0			0
$653$		0			0		−0.733052	−	0.680173i	$-0.761905\pi$
$653$		0			0		0.733052	+	0.680173i	$0.238095\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−28.6413	−	22.8407i	−1.11741	−	0.891101i
$658$		0			0
$659$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$659$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$660$		0			0
$661$	28.2953	−	41.5015i	1.10056	−	1.61422i	0.373526	−	0.927620i	$-0.378149\pi$
$661$	28.2953	−	41.5015i	1.10056	−	1.61422i	0.727033	−	0.686603i	$-0.240899\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−48.5908	−	7.32388i	−1.87863	−	0.283158i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−41.5143	−	19.9922i	−1.60026	−	0.770643i	−0.600673	−	0.799495i	$-0.705101\pi$
$673$	−41.5143	−	19.9922i	−1.60026	−	0.770643i	−0.999584	+	0.0288517i	$0.990815\pi$
$674$		0			0
$675$	−25.9081	+	1.94154i	−0.997204	+	0.0747301i
$676$		0			0
$677$		0			0		−0.997204	−	0.0747301i	$-0.976190\pi$
$677$		0			0		0.997204	+	0.0747301i	$0.0238095\pi$
$678$		0			0
$679$	1.95145	−	16.8461i	0.0748899	−	0.646495i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.733052	−	0.680173i	$-0.238095\pi$
$683$		0			0		−0.733052	+	0.680173i	$0.761905\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	35.3083			1.34710
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−9.70009	+	31.4469i	−0.369009	+	1.19630i	0.560706	+	0.828015i	$0.310530\pi$
$691$	−9.70009	+	31.4469i	−0.369009	+	1.19630i	−0.929714	+	0.368282i	$0.879946\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$701$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$702$		0			0
$703$	11.9939	−	79.5744i	0.452359	−	3.00121i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	40.4071	−	6.09040i	1.51752	−	0.228730i	0.663188	−	0.748453i	$-0.269203\pi$
$709$	40.4071	−	6.09040i	1.51752	−	0.228730i	0.854335	+	0.519723i	$0.173965\pi$
$710$		0			0
$711$	−36.3425	−	24.7779i	−1.36295	−	0.929245i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.680173	−	0.733052i	$-0.261905\pi$
$719$		0			0		−0.680173	+	0.733052i	$0.738095\pi$
$720$		0			0
$721$	−20.9764	+	47.6712i	−0.781201	+	1.77537i
$722$		0			0
$723$	12.5391	+	31.9492i	0.466336	+	1.18820i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	52.3474	−	11.9479i	1.94146	−	0.443125i	0.950096	−	0.311958i	$-0.100985\pi$
$727$	52.3474	−	11.9479i	1.94146	−	0.443125i	0.991360	−	0.131167i	$-0.0418725\pi$
$728$		0			0
$729$	6.00807	−	26.3231i	0.222521	−	0.974928i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−6.19255	−	9.08280i	−0.228727	−	0.335481i	0.694669	−	0.719330i	$-0.255551\pi$
$733$	−6.19255	−	9.08280i	−0.228727	−	0.335481i	−0.923396	+	0.383849i	$0.874598\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	17.1972	−	43.8178i	0.632609	−	1.61186i	−0.149529	−	0.988757i	$-0.547776\pi$
$739$	17.1972	−	43.8178i	0.632609	−	1.61186i	0.782138	−	0.623105i	$-0.214129\pi$
$740$		0			0
$741$	−46.2680	−	10.5604i	−1.69970	−	0.387945i
$742$		0			0
$743$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$743$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	9.81271	+	9.10487i	0.358071	+	0.332241i	0.838610	−	0.544732i	$-0.183369\pi$
$751$	9.81271	+	9.10487i	0.358071	+	0.332241i	−0.480539	+	0.876973i	$0.659559\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	33.0868	+	41.4895i	1.20256	+	1.50796i	0.808098	+	0.589048i	$0.200497\pi$
$757$	33.0868	+	41.4895i	1.20256	+	1.50796i	0.394460	+	0.918913i	$0.370932\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.149042	−	0.988831i	$-0.547619\pi$
$761$		0			0		0.149042	+	0.988831i	$0.452381\pi$
$762$		0			0
$763$	−0.451840	+	11.1433i	−0.0163577	+	0.403414i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−12.7756	−	26.5289i	−0.460702	−	0.956657i	−0.993860	−	0.110642i	$-0.964709\pi$
$769$	−12.7756	−	26.5289i	−0.460702	−	0.956657i	0.533159	−	0.846015i	$-0.321005\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		0.997204	−	0.0747301i	$-0.0238095\pi$
$773$		0			0		−0.997204	+	0.0747301i	$0.976190\pi$
$774$		0			0
$775$	−37.4960	−	2.80994i	−1.34690	−	0.100936i
$776$		0			0
$777$	−26.7583	+	42.2916i	−0.959950	+	1.51720i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	28.3106	+	30.5116i	1.00917	+	1.08762i	0.996146	+	0.0877066i	$0.0279538\pi$
$787$	28.3106	+	30.5116i	1.00917	+	1.08762i	0.0130192	+	0.999915i	$0.495856\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	1.84120	−	24.5691i	0.0653828	−	0.872473i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		0.433884	−	0.900969i	$-0.357143\pi$
$797$		0			0		−0.433884	+	0.900969i	$0.642857\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		−0.826239	−	0.563320i	$-0.809524\pi$
$809$		0			0		0.826239	+	0.563320i	$0.190476\pi$
$810$		0			0
$811$	40.8769	−	32.5982i	1.43538	−	1.14468i	0.470369	−	0.882470i	$-0.344121\pi$
$811$	40.8769	−	32.5982i	1.43538	−	1.14468i	0.965011	−	0.262208i	$-0.0844506\pi$
$812$		0			0
$813$	25.3173	−	31.7469i	0.887918	−	1.11341i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	54.9371	−	59.2081i	1.92201	−	2.07143i
$818$		0			0
$819$	23.6918	+	17.6000i	0.827859	+	0.614995i
$820$		0			0
$821$		0			0		−0.365341	−	0.930874i	$-0.619048\pi$
$821$		0			0		0.365341	+	0.930874i	$0.380952\pi$
$822$		0			0
$823$	34.7153	−	10.7082i	1.21010	−	0.373266i	0.376907	−	0.926251i	$-0.376988\pi$
$823$	34.7153	−	10.7082i	1.21010	−	0.373266i	0.833191	+	0.552986i	$0.186512\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$827$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$828$		0			0
$829$	51.6411	+	20.2677i	1.79357	+	0.703925i	0.994748	+	0.102358i	$0.0326389\pi$
$829$	51.6411	+	20.2677i	1.79357	+	0.703925i	0.798823	+	0.601566i	$0.205456\pi$
$830$		0			0
$831$	−32.3898	−	47.5071i	−1.12359	−	1.64800i
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	14.2762	−	36.3751i	0.493456	−	1.25731i
$838$		0			0
$839$		0			0		−0.974928	−	0.222521i	$-0.928571\pi$
$839$		0			0		0.974928	+	0.222521i	$0.0714286\pi$
$840$		0			0
$841$	−6.45311	−	28.2729i	−0.222521	−	0.974928i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−29.0862	+	0.997286i	−0.999413	+	0.0342672i
$848$		0			0
$849$	−6.99199	−	6.48762i	−0.239965	−	0.222655i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−45.6242	−	36.3841i	−1.56214	−	1.24577i	−0.808552	−	0.588425i	$-0.799748\pi$
$853$	−45.6242	−	36.3841i	−1.56214	−	1.24577i	−0.753592	−	0.657343i	$-0.771680\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.563320	−	0.826239i	$-0.309524\pi$
$857$		0			0		−0.563320	+	0.826239i	$0.690476\pi$
$858$		0			0
$859$	−5.94719	−	39.4570i	−0.202916	−	1.34626i	−0.825809	−	0.563950i	$-0.809281\pi$
$859$	−5.94719	−	39.4570i	−0.202916	−	1.34626i	0.622893	−	0.782307i	$-0.285957\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$863$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	12.7756	+	26.5289i	0.433884	+	0.900969i
$868$		0			0
$869$		0			0
$870$		0			0
$871$	60.4781	−	4.53221i	2.04922	−	0.153568i
$872$		0			0
$873$	19.1757	+	1.43702i	0.648999	+	0.0486357i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	9.16461	+	2.82691i	0.309467	+	0.0954579i	0.445599	−	0.895233i	$-0.352991\pi$
$877$	9.16461	+	2.82691i	0.309467	+	0.0954579i	−0.136132	+	0.990691i	$0.543467\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$	−34.1908			−1.15061			−0.575306	−	0.817939i	$-0.695117\pi$
$883$	−34.1908			−1.15061			−0.575306	+	0.817939i	$0.695117\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.294755	−	0.955573i	$-0.404762\pi$
$887$		0			0		−0.294755	+	0.955573i	$0.595238\pi$
$888$		0			0
$889$	20.0251	−	19.9003i	0.671622	−	0.667436i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$	−46.1015	+	19.9426i	−1.53416	+	0.663647i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−49.5351	−	33.7725i	−1.64479	−	1.12140i	−0.876682	−	0.481070i	$-0.840248\pi$
$907$	−49.5351	−	33.7725i	−1.64479	−	1.12140i	−0.768104	−	0.640325i	$-0.778800\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$911$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	14.7964	+	37.7006i	0.488088	+	1.24363i	0.936773	+	0.349938i	$0.113797\pi$
$919$	14.7964	+	37.7006i	0.488088	+	1.24363i	−0.448685	+	0.893690i	$0.648107\pi$
$920$		0			0
$921$	30.2637	−	9.33512i	0.997223	−	0.307603i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−12.1506	+	53.2355i	−0.399511	+	1.75037i
$926$		0			0
$927$	−54.9733	−	21.5754i	−1.80556	−	0.708630i
$928$		0			0
$929$		0			0		−0.563320	−	0.826239i	$-0.690476\pi$
$929$		0			0		0.563320	+	0.826239i	$0.309524\pi$
$930$		0			0
$931$	37.5916	+	35.3198i	1.23201	+	1.15756i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−21.2391	−	4.84769i	−0.693852	−	0.158367i	−0.138971	−	0.990297i	$-0.544379\pi$
$937$	−21.2391	−	4.84769i	−0.693852	−	0.158367i	−0.554882	+	0.831929i	$0.687236\pi$
$938$		0			0
$939$	3.99084	+	17.4850i	0.130236	+	0.570602i
$940$		0			0
$941$		0			0		−0.294755	−	0.955573i	$-0.595238\pi$
$941$		0			0		0.294755	+	0.955573i	$0.404762\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.733052	−	0.680173i	$-0.761905\pi$
$947$		0			0		0.733052	+	0.680173i	$0.238095\pi$
$948$		0			0
$949$	22.7030	+	39.3228i	0.736971	+	1.27647i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$953$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	12.7769	−	22.1303i	0.412159	−	0.713880i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	51.5629	+	24.8314i	1.65815	+	0.798523i	0.998914	+	0.0465864i	$0.0148343\pi$
$967$	51.5629	+	24.8314i	1.65815	+	0.798523i	0.659236	+	0.751936i	$0.270880\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.997204	−	0.0747301i	$-0.976190\pi$
$971$		0			0		0.997204	+	0.0747301i	$0.0238095\pi$
$972$		0			0
$973$	55.9331	−	6.12513i	1.79313	−	0.196363i
$974$		0			0
$975$	30.7715	+	9.49176i	0.985477	+	0.303980i
$976$		0			0
$977$		0			0		0.733052	−	0.680173i	$-0.238095\pi$
$977$		0			0		−0.733052	+	0.680173i	$0.761905\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−12.6457			−0.403745
$982$		0			0
$983$		0			0		−0.680173	−	0.733052i	$-0.738095\pi$
$983$		0			0		0.680173	+	0.733052i	$0.261905\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−4.61254	−	61.5501i	−0.146522	−	1.95520i	−0.270011	−	0.962857i	$-0.587027\pi$
$991$	−4.61254	−	61.5501i	−0.146522	−	1.95520i	0.123489	−	0.992346i	$-0.460592\pi$
$992$		0			0
$993$	−27.2021	+	56.4857i	−0.863232	+	1.79252i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	3.35593	−	22.2652i	0.106283	−	0.705145i	−0.870698	−	0.491819i	$-0.836332\pi$
$997$	3.35593	−	22.2652i	0.106283	−	0.705145i	0.976981	−	0.213326i	$-0.0684297\pi$
$998$		0			0
$999$	−49.1441	−	28.3733i	−1.55485	−	0.897693i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	588.2.be.a.257.1	✓	12
3.2	odd	2	CM	588.2.be.a.257.1	✓	12
49.45	odd	42	inner	588.2.be.a.437.1	yes	12
147.143	even	42	inner	588.2.be.a.437.1	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
588.2.be.a.257.1	✓	12	1.1	even	1	trivial
588.2.be.a.257.1	✓	12	3.2	odd	2	CM
588.2.be.a.437.1	yes	12	49.45	odd	42	inner
588.2.be.a.437.1	yes	12	147.143	even	42	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	588.be (of order \(42\), degree \(12\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.17809 - 1.26968i) q^{3} +(2.62818 + 0.304447i) q^{7} +(-0.224190 + 2.99161i) q^{9} +(1.61335 - 3.35015i) q^{13} +(6.38151 + 3.68437i) q^{19} +(-2.70969 - 3.69562i) q^{21} +(-4.13119 - 2.81660i) q^{25} +(4.06252 - 3.23975i) q^{27} +(6.51271 - 3.76012i) q^{31} +(-3.98985 - 10.1660i) q^{37} +(-6.15430 + 1.89835i) q^{39} +(2.43907 - 10.6863i) q^{43} +(6.81462 + 1.60028i) q^{49} +(-2.84004 - 12.4430i) q^{57} +(6.16795 - 2.42074i) q^{61} +(-1.50000 + 7.79423i) q^{63} +(8.15511 + 14.1251i) q^{67} +(-6.87882 + 10.0894i) q^{73} +(1.29074 + 8.56353i) q^{75} +(-7.33092 + 12.6975i) q^{79} +(-8.89948 - 1.34138i) q^{81} +(5.26011 - 8.31361i) q^{91} +(-12.4467 - 3.83931i) q^{93} -6.40982i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 3 q^{3} + 5 q^{7} - 3 q^{9} + 15 q^{19} + 6 q^{21} - 5 q^{25} + 3 q^{31} - 4 q^{37} - 30 q^{39} - 26 q^{43} - 11 q^{49} + 30 q^{57} + 19 q^{61} - 18 q^{63} + 5 q^{67} + 3 q^{73} - 15 q^{75} + 17 q^{79}+ \cdots + 3 q^{93}+O(q^{100}) \) 12 * q - 3 * q^3 + 5 * q^7 - 3 * q^9 + 15 * q^19 + 6 * q^21 - 5 * q^25 + 3 * q^31 - 4 * q^37 - 30 * q^39 - 26 * q^43 - 11 * q^49 + 30 * q^57 + 19 * q^61 - 18 * q^63 + 5 * q^67 + 3 * q^73 - 15 * q^75 + 17 * q^79 + 9 * q^81 + 9 * q^91 + 3 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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