LMFDB - Embedded newform 1260.2.c.a.811.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1260,2,Mod(811,1260)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1260, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1260.811");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1260 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1260.c (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:	$10.0611506547$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 140)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			811.1
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1260.811
Dual form			1260.2.c.a.811.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.366025 - 1.36603i) q^{2} +(-1.73205 + 1.00000i) q^{4} -1.00000i q^{5} +(-1.73205 + 2.00000i) q^{7} +(2.00000 + 2.00000i) q^{8} +O(q^{10})$	\(q+(-0.366025 - 1.36603i) q^{2} +(-1.73205 + 1.00000i) q^{4} -1.00000i q^{5} +(-1.73205 + 2.00000i) q^{7} +(2.00000 + 2.00000i) q^{8} +(-1.36603 + 0.366025i) q^{10} -0.267949i q^{11} +0.464102i q^{13} +(3.36603 + 1.63397i) q^{14} +(2.00000 - 3.46410i) q^{16} -6.46410i q^{17} +6.00000 q^{19} +(1.00000 + 1.73205i) q^{20} +(-0.366025 + 0.0980762i) q^{22} -1.46410i q^{23} -1.00000 q^{25} +(0.633975 - 0.169873i) q^{26} +(1.00000 - 5.19615i) q^{28} -7.92820 q^{29} -6.00000 q^{31} +(-5.46410 - 1.46410i) q^{32} +(-8.83013 + 2.36603i) q^{34} +(2.00000 + 1.73205i) q^{35} -9.46410 q^{37} +(-2.19615 - 8.19615i) q^{38} +(2.00000 - 2.00000i) q^{40} -3.46410i q^{41} +2.00000i q^{43} +(0.267949 + 0.464102i) q^{44} +(-2.00000 + 0.535898i) q^{46} -1.73205 q^{47} +(-1.00000 - 6.92820i) q^{49} +(0.366025 + 1.36603i) q^{50} +(-0.464102 - 0.803848i) q^{52} -2.00000 q^{53} -0.267949 q^{55} +(-7.46410 + 0.535898i) q^{56} +(2.90192 + 10.8301i) q^{58} +3.46410 q^{59} -9.46410i q^{61} +(2.19615 + 8.19615i) q^{62} +8.00000i q^{64} +0.464102 q^{65} -3.46410i q^{67} +(6.46410 + 11.1962i) q^{68} +(1.63397 - 3.36603i) q^{70} -7.46410i q^{71} -12.9282i q^{73} +(3.46410 + 12.9282i) q^{74} +(-10.3923 + 6.00000i) q^{76} +(0.535898 + 0.464102i) q^{77} +14.6603i q^{79} +(-3.46410 - 2.00000i) q^{80} +(-4.73205 + 1.26795i) q^{82} -15.4641 q^{83} -6.46410 q^{85} +(2.73205 - 0.732051i) q^{86} +(0.535898 - 0.535898i) q^{88} +2.53590i q^{89} +(-0.928203 - 0.803848i) q^{91} +(1.46410 + 2.53590i) q^{92} +(0.633975 + 2.36603i) q^{94} -6.00000i q^{95} +13.3923i q^{97} +(-9.09808 + 3.90192i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} + 8 q^{8}+O(q^{10}) $ 4 * q + 2 * q^2 + 8 * q^8	$ 4 q + 2 q^{2} + 8 q^{8} - 2 q^{10} + 10 q^{14} + 8 q^{16} + 24 q^{19} + 4 q^{20} + 2 q^{22} - 4 q^{25} + 6 q^{26} + 4 q^{28} - 4 q^{29} - 24 q^{31} - 8 q^{32} - 18 q^{34} + 8 q^{35} - 24 q^{37} + 12 q^{38} + 8 q^{40} + 8 q^{44} - 8 q^{46} - 4 q^{49} - 2 q^{50} + 12 q^{52} - 8 q^{53} - 8 q^{55} - 16 q^{56} + 22 q^{58} - 12 q^{62} - 12 q^{65} + 12 q^{68} + 10 q^{70} + 16 q^{77} - 12 q^{82} - 48 q^{83} - 12 q^{85} + 4 q^{86} + 16 q^{88} + 24 q^{91} - 8 q^{92} + 6 q^{94} - 26 q^{98}+O(q^{100}) $ 4 * q + 2 * q^2 + 8 * q^8 - 2 * q^10 + 10 * q^14 + 8 * q^16 + 24 * q^19 + 4 * q^20 + 2 * q^22 - 4 * q^25 + 6 * q^26 + 4 * q^28 - 4 * q^29 - 24 * q^31 - 8 * q^32 - 18 * q^34 + 8 * q^35 - 24 * q^37 + 12 * q^38 + 8 * q^40 + 8 * q^44 - 8 * q^46 - 4 * q^49 - 2 * q^50 + 12 * q^52 - 8 * q^53 - 8 * q^55 - 16 * q^56 + 22 * q^58 - 12 * q^62 - 12 * q^65 + 12 * q^68 + 10 * q^70 + 16 * q^77 - 12 * q^82 - 48 * q^83 - 12 * q^85 + 4 * q^86 + 16 * q^88 + 24 * q^91 - 8 * q^92 + 6 * q^94 - 26 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1260\mathbb{Z}\right)^\times$.

$n$	$281$	$631$	$757$	$1081$
$\chi(n)$	$1$	$-1$	$1$	$-1$

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 $

$2$	−0.366025	−	1.36603i	−0.258819	−	0.965926i
$2$	−0.366025	−	1.36603i	−0.258819	−	0.965926i
$3$		0			0
$3$		0			0
$4$	−1.73205	+	1.00000i	−0.866025	+	0.500000i
$5$		−	1.00000i		−	0.447214i
$5$		−	1.00000i		−	0.447214i
$6$		0			0
$7$	−1.73205	+	2.00000i	−0.654654	+	0.755929i
$7$	−1.73205	+	2.00000i	−0.654654	+	0.755929i
$8$	2.00000	+	2.00000i	0.707107	+	0.707107i
$9$		0			0
$10$	−1.36603	+	0.366025i	−0.431975	+	0.115747i
$11$		−	0.267949i		−	0.0807897i	−0.999184	−	0.0403949i	$-0.987138\pi$
$11$		−	0.267949i		−	0.0807897i	0.999184	−	0.0403949i	$-0.0128616\pi$
$12$		0			0
$13$			0.464102i			0.128719i	0.997927	+	0.0643593i	$0.0205004\pi$
$13$			0.464102i			0.128719i	−0.997927	+	0.0643593i	$0.979500\pi$
$14$	3.36603	+	1.63397i	0.899608	+	0.436698i
$15$		0			0
$16$	2.00000	−	3.46410i	0.500000	−	0.866025i
$17$		−	6.46410i		−	1.56777i	−0.620903	−	0.783887i	$-0.713234\pi$
$17$		−	6.46410i		−	1.56777i	0.620903	−	0.783887i	$-0.286766\pi$
$18$		0			0
$19$	6.00000			1.37649			0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000			1.37649			0.688247	+	0.725476i	$0.258380\pi$
$20$	1.00000	+	1.73205i	0.223607	+	0.387298i
$21$		0			0
$22$	−0.366025	+	0.0980762i	−0.0780369	+	0.0209099i
$23$		−	1.46410i		−	0.305286i	−0.988281	−	0.152643i	$-0.951221\pi$
$23$		−	1.46410i		−	0.305286i	0.988281	−	0.152643i	$-0.0487785\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$	0.633975	−	0.169873i	0.124333	−	0.0333148i
$27$		0			0
$28$	1.00000	−	5.19615i	0.188982	−	0.981981i
$29$	−7.92820			−1.47223			−0.736115	−	0.676856i	$-0.763342\pi$
$29$	−7.92820			−1.47223			−0.736115	+	0.676856i	$0.763342\pi$
$30$		0			0
$31$	−6.00000			−1.07763			−0.538816	−	0.842424i	$-0.681128\pi$
$31$	−6.00000			−1.07763			−0.538816	+	0.842424i	$0.681128\pi$
$32$	−5.46410	−	1.46410i	−0.965926	−	0.258819i
$33$		0			0
$34$	−8.83013	+	2.36603i	−1.51435	+	0.405770i
$35$	2.00000	+	1.73205i	0.338062	+	0.292770i
$36$		0			0
$37$	−9.46410			−1.55589			−0.777944	−	0.628333i	$-0.783737\pi$
$37$	−9.46410			−1.55589			−0.777944	+	0.628333i	$0.783737\pi$
$38$	−2.19615	−	8.19615i	−0.356263	−	1.32959i
$39$		0			0
$40$	2.00000	−	2.00000i	0.316228	−	0.316228i
$41$		−	3.46410i		−	0.541002i	−0.962720	−	0.270501i	$-0.912811\pi$
$41$		−	3.46410i		−	0.541002i	0.962720	−	0.270501i	$-0.0871893\pi$
$42$		0			0
$43$			2.00000i			0.304997i	0.988304	+	0.152499i	$0.0487319\pi$
$43$			2.00000i			0.304997i	−0.988304	+	0.152499i	$0.951268\pi$
$44$	0.267949	+	0.464102i	0.0403949	+	0.0699660i
$45$		0			0
$46$	−2.00000	+	0.535898i	−0.294884	+	0.0790139i
$47$	−1.73205			−0.252646			−0.126323	−	0.991989i	$-0.540318\pi$
$47$	−1.73205			−0.252646			−0.126323	+	0.991989i	$0.540318\pi$
$48$		0			0
$49$	−1.00000	−	6.92820i	−0.142857	−	0.989743i
$50$	0.366025	+	1.36603i	0.0517638	+	0.193185i
$51$		0			0
$52$	−0.464102	−	0.803848i	−0.0643593	−	0.111474i
$53$	−2.00000			−0.274721			−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000			−0.274721			−0.137361	+	0.990521i	$0.543862\pi$
$54$		0			0
$55$	−0.267949			−0.0361303
$56$	−7.46410	+	0.535898i	−0.997433	+	0.0716124i
$57$		0			0
$58$	2.90192	+	10.8301i	0.381041	+	1.42207i
$59$	3.46410			0.450988			0.225494	−	0.974245i	$-0.427600\pi$
$59$	3.46410			0.450988			0.225494	+	0.974245i	$0.427600\pi$
$60$		0			0
$61$		−	9.46410i		−	1.21175i	−0.795558	−	0.605877i	$-0.792822\pi$
$61$		−	9.46410i		−	1.21175i	0.795558	−	0.605877i	$-0.207178\pi$
$62$	2.19615	+	8.19615i	0.278912	+	1.04091i
$63$		0			0
$64$			8.00000i			1.00000i
$65$	0.464102			0.0575647
$66$		0			0
$67$		−	3.46410i		−	0.423207i	−0.977356	−	0.211604i	$-0.932131\pi$
$67$		−	3.46410i		−	0.423207i	0.977356	−	0.211604i	$-0.0678686\pi$
$68$	6.46410	+	11.1962i	0.783887	+	1.35773i
$69$		0			0
$70$	1.63397	−	3.36603i	0.195297	−	0.402317i
$71$		−	7.46410i		−	0.885826i	−0.896565	−	0.442913i	$-0.853945\pi$
$71$		−	7.46410i		−	0.885826i	0.896565	−	0.442913i	$-0.146055\pi$
$72$		0			0
$73$		−	12.9282i		−	1.51313i	−0.653917	−	0.756566i	$-0.726876\pi$
$73$		−	12.9282i		−	1.51313i	0.653917	−	0.756566i	$-0.273124\pi$
$74$	3.46410	+	12.9282i	0.402694	+	1.50287i
$75$		0			0
$76$	−10.3923	+	6.00000i	−1.19208	+	0.688247i
$77$	0.535898	+	0.464102i	0.0610713	+	0.0528893i
$78$		0			0
$79$			14.6603i			1.64941i	0.565565	+	0.824704i	$0.308658\pi$
$79$			14.6603i			1.64941i	−0.565565	+	0.824704i	$0.691342\pi$
$80$	−3.46410	−	2.00000i	−0.387298	−	0.223607i
$81$		0			0
$82$	−4.73205	+	1.26795i	−0.522568	+	0.140022i
$83$	−15.4641			−1.69741			−0.848703	−	0.528870i	$-0.822616\pi$
$83$	−15.4641			−1.69741			−0.848703	+	0.528870i	$0.822616\pi$
$84$		0			0
$85$	−6.46410			−0.701130
$86$	2.73205	−	0.732051i	0.294605	−	0.0789391i
$87$		0			0
$88$	0.535898	−	0.535898i	0.0571270	−	0.0571270i
$89$			2.53590i			0.268805i	0.990927	+	0.134402i	$0.0429115\pi$
$89$			2.53590i			0.268805i	−0.990927	+	0.134402i	$0.957089\pi$
$90$		0			0
$91$	−0.928203	−	0.803848i	−0.0973021	−	0.0842661i
$92$	1.46410	+	2.53590i	0.152643	+	0.264386i
$93$		0			0
$94$	0.633975	+	2.36603i	0.0653895	+	0.244037i
$95$		−	6.00000i		−	0.615587i
$96$		0			0
$97$			13.3923i			1.35978i	0.733313	+	0.679891i	$0.237973\pi$
$97$			13.3923i			1.35978i	−0.733313	+	0.679891i	$0.762027\pi$
$98$	−9.09808	+	3.90192i	−0.919044	+	0.394154i
$99$		0			0
$100$	1.73205	−	1.00000i	0.173205	−	0.100000i
$101$		−	15.4641i		−	1.53874i	−0.638806	−	0.769368i	$-0.720571\pi$
$101$		−	15.4641i		−	1.53874i	0.638806	−	0.769368i	$-0.279429\pi$
$102$		0			0
$103$	−6.80385			−0.670403			−0.335202	−	0.942146i	$-0.608804\pi$
$103$	−6.80385			−0.670403			−0.335202	+	0.942146i	$0.608804\pi$
$104$	−0.928203	+	0.928203i	−0.0910178	+	0.0910178i
$105$		0			0
$106$	0.732051	+	2.73205i	0.0711031	+	0.265360i
$107$			2.39230i			0.231273i	0.993292	+	0.115636i	$0.0368907\pi$
$107$			2.39230i			0.231273i	−0.993292	+	0.115636i	$0.963109\pi$
$108$		0			0
$109$	2.07180			0.198442			0.0992211	−	0.995065i	$-0.468365\pi$
$109$	2.07180			0.198442			0.0992211	+	0.995065i	$0.468365\pi$
$110$	0.0980762	+	0.366025i	0.00935120	+	0.0348992i
$111$		0			0
$112$	3.46410	+	10.0000i	0.327327	+	0.944911i
$113$	−5.46410			−0.514019			−0.257010	−	0.966409i	$-0.582737\pi$
$113$	−5.46410			−0.514019			−0.257010	+	0.966409i	$0.582737\pi$
$114$		0			0
$115$	−1.46410			−0.136528
$116$	13.7321	−	7.92820i	1.27499	−	0.736115i
$117$		0			0
$118$	−1.26795	−	4.73205i	−0.116724	−	0.435621i
$119$	12.9282	+	11.1962i	1.18513	+	1.02635i
$120$		0			0
$121$	10.9282			0.993473
$122$	−12.9282	+	3.46410i	−1.17046	+	0.313625i
$123$		0			0
$124$	10.3923	−	6.00000i	0.933257	−	0.538816i
$125$			1.00000i			0.0894427i
$126$		0			0
$127$		−	15.4641i		−	1.37222i	−0.727499	−	0.686109i	$-0.759317\pi$
$127$		−	15.4641i		−	1.37222i	0.727499	−	0.686109i	$-0.240683\pi$
$128$	10.9282	−	2.92820i	0.965926	−	0.258819i
$129$		0			0
$130$	−0.169873	−	0.633975i	−0.0148988	−	0.0556033i
$131$	−2.53590			−0.221562			−0.110781	−	0.993845i	$-0.535335\pi$
$131$	−2.53590			−0.221562			−0.110781	+	0.993845i	$0.535335\pi$
$132$		0			0
$133$	−10.3923	+	12.0000i	−0.901127	+	1.04053i
$134$	−4.73205	+	1.26795i	−0.408787	+	0.109534i
$135$		0			0
$136$	12.9282	−	12.9282i	1.10858	−	1.10858i
$137$	−20.3923			−1.74223			−0.871116	−	0.491077i	$-0.836603\pi$
$137$	−20.3923			−1.74223			−0.871116	+	0.491077i	$0.836603\pi$
$138$		0			0
$139$	6.92820			0.587643			0.293821	−	0.955860i	$-0.405073\pi$
$139$	6.92820			0.587643			0.293821	+	0.955860i	$0.405073\pi$
$140$	−5.19615	−	1.00000i	−0.439155	−	0.0845154i
$141$		0			0
$142$	−10.1962	+	2.73205i	−0.855642	+	0.229269i
$143$	0.124356			0.0103991
$144$		0			0
$145$			7.92820i			0.658401i
$146$	−17.6603	+	4.73205i	−1.46157	+	0.391627i
$147$		0			0
$148$	16.3923	−	9.46410i	1.34744	−	0.777944i
$149$	−3.07180			−0.251651			−0.125826	−	0.992052i	$-0.540158\pi$
$149$	−3.07180			−0.251651			−0.125826	+	0.992052i	$0.540158\pi$
$150$		0			0
$151$			15.1962i			1.23665i	0.785924	+	0.618323i	$0.212188\pi$
$151$			15.1962i			1.23665i	−0.785924	+	0.618323i	$0.787812\pi$
$152$	12.0000	+	12.0000i	0.973329	+	0.973329i
$153$		0			0
$154$	0.437822	−	0.901924i	0.0352807	−	0.0726791i
$155$			6.00000i			0.481932i
$156$		0			0
$157$		−	7.85641i		−	0.627009i	−0.949587	−	0.313505i	$-0.898497\pi$
$157$		−	7.85641i		−	0.627009i	0.949587	−	0.313505i	$-0.101503\pi$
$158$	20.0263	−	5.36603i	1.59321	−	0.426898i
$159$		0			0
$160$	−1.46410	+	5.46410i	−0.115747	+	0.431975i
$161$	2.92820	+	2.53590i	0.230775	+	0.199857i
$162$		0			0
$163$		−	20.7846i		−	1.62798i	−0.580881	−	0.813988i	$-0.697292\pi$
$163$		−	20.7846i		−	1.62798i	0.580881	−	0.813988i	$-0.302708\pi$
$164$	3.46410	+	6.00000i	0.270501	+	0.468521i
$165$		0			0
$166$	5.66025	+	21.1244i	0.439321	+	1.63957i
$167$	5.19615			0.402090			0.201045	−	0.979582i	$-0.435566\pi$
$167$	5.19615			0.402090			0.201045	+	0.979582i	$0.435566\pi$
$168$		0			0
$169$	12.7846			0.983432
$170$	2.36603	+	8.83013i	0.181466	+	0.677240i
$171$		0			0
$172$	−2.00000	−	3.46410i	−0.152499	−	0.264135i
$173$		−	14.3205i		−	1.08877i	−0.838836	−	0.544384i	$-0.816763\pi$
$173$		−	14.3205i		−	1.08877i	0.838836	−	0.544384i	$-0.183237\pi$
$174$		0			0
$175$	1.73205	−	2.00000i	0.130931	−	0.151186i
$176$	−0.928203	−	0.535898i	−0.0699660	−	0.0403949i
$177$		0			0
$178$	3.46410	−	0.928203i	0.259645	−	0.0695718i
$179$			6.39230i			0.477783i	0.971046	+	0.238892i	$0.0767841\pi$
$179$			6.39230i			0.477783i	−0.971046	+	0.238892i	$0.923216\pi$
$180$		0			0
$181$			0.928203i			0.0689928i	0.999405	+	0.0344964i	$0.0109827\pi$
$181$			0.928203i			0.0689928i	−0.999405	+	0.0344964i	$0.989017\pi$
$182$	−0.758330	+	1.56218i	−0.0562112	+	0.115796i
$183$		0			0
$184$	2.92820	−	2.92820i	0.215870	−	0.215870i
$185$			9.46410i			0.695815i
$186$		0			0
$187$	−1.73205			−0.126660
$188$	3.00000	−	1.73205i	0.218797	−	0.126323i
$189$		0			0
$190$	−8.19615	+	2.19615i	−0.594611	+	0.159326i
$191$		−	7.19615i		−	0.520695i	−0.965515	−	0.260348i	$-0.916163\pi$
$191$		−	7.19615i		−	0.520695i	0.965515	−	0.260348i	$-0.0838372\pi$
$192$		0			0
$193$	−9.46410			−0.681241			−0.340620	−	0.940201i	$-0.610637\pi$
$193$	−9.46410			−0.681241			−0.340620	+	0.940201i	$0.610637\pi$
$194$	18.2942	−	4.90192i	1.31345	−	0.351938i
$195$		0			0
$196$	8.66025	+	11.0000i	0.618590	+	0.785714i
$197$	13.3205			0.949047			0.474523	−	0.880243i	$-0.342620\pi$
$197$	13.3205			0.949047			0.474523	+	0.880243i	$0.342620\pi$
$198$		0			0
$199$	3.46410			0.245564			0.122782	−	0.992434i	$-0.460818\pi$
$199$	3.46410			0.245564			0.122782	+	0.992434i	$0.460818\pi$
$200$	−2.00000	−	2.00000i	−0.141421	−	0.141421i
$201$		0			0
$202$	−21.1244	+	5.66025i	−1.48630	+	0.398254i
$203$	13.7321	−	15.8564i	0.963801	−	1.11290i
$204$		0			0
$205$	−3.46410			−0.241943
$206$	2.49038	+	9.29423i	0.173513	+	0.647560i
$207$		0			0
$208$	1.60770	+	0.928203i	0.111474	+	0.0643593i
$209$		−	1.60770i		−	0.111207i
$210$		0			0
$211$			3.19615i			0.220032i	0.993930	+	0.110016i	$0.0350902\pi$
$211$			3.19615i			0.220032i	−0.993930	+	0.110016i	$0.964910\pi$
$212$	3.46410	−	2.00000i	0.237915	−	0.137361i
$213$		0			0
$214$	3.26795	−	0.875644i	0.223392	−	0.0598578i
$215$	2.00000			0.136399
$216$		0			0
$217$	10.3923	−	12.0000i	0.705476	−	0.814613i
$218$	−0.758330	−	2.83013i	−0.0513606	−	0.191680i
$219$		0			0
$220$	0.464102	−	0.267949i	0.0312897	−	0.0180651i
$221$	3.00000			0.201802
$222$		0			0
$223$	−13.7321			−0.919566			−0.459783	−	0.888031i	$-0.652073\pi$
$223$	−13.7321			−0.919566			−0.459783	+	0.888031i	$0.652073\pi$
$224$	12.3923	−	8.39230i	0.827996	−	0.560734i
$225$		0			0
$226$	2.00000	+	7.46410i	0.133038	+	0.496505i
$227$	−20.6603			−1.37127			−0.685635	−	0.727946i	$-0.740475\pi$
$227$	−20.6603			−1.37127			−0.685635	+	0.727946i	$0.740475\pi$
$228$		0			0
$229$			8.53590i			0.564068i	0.959404	+	0.282034i	$0.0910091\pi$
$229$			8.53590i			0.564068i	−0.959404	+	0.282034i	$0.908991\pi$
$230$	0.535898	+	2.00000i	0.0353361	+	0.131876i
$231$		0			0
$232$	−15.8564	−	15.8564i	−1.04102	−	1.04102i
$233$	9.07180			0.594313			0.297157	−	0.954829i	$-0.403962\pi$
$233$	9.07180			0.594313			0.297157	+	0.954829i	$0.403962\pi$
$234$		0			0
$235$			1.73205i			0.112987i
$236$	−6.00000	+	3.46410i	−0.390567	+	0.225494i
$237$		0			0
$238$	10.5622	−	21.7583i	0.684644	−	1.41038i
$239$			23.9808i			1.55119i	0.631233	+	0.775593i	$0.282549\pi$
$239$			23.9808i			1.55119i	−0.631233	+	0.775593i	$0.717451\pi$
$240$		0			0
$241$		−	16.3923i		−	1.05592i	−0.849269	−	0.527961i	$-0.822957\pi$
$241$		−	16.3923i		−	1.05592i	0.849269	−	0.527961i	$-0.177043\pi$
$242$	−4.00000	−	14.9282i	−0.257130	−	0.959621i
$243$		0			0
$244$	9.46410	+	16.3923i	0.605877	+	1.04941i
$245$	−6.92820	+	1.00000i	−0.442627	+	0.0638877i
$246$		0			0
$247$			2.78461i			0.177180i
$248$	−12.0000	−	12.0000i	−0.762001	−	0.762001i
$249$		0			0
$250$	1.36603	−	0.366025i	0.0863950	−	0.0231495i
$251$	25.8564			1.63204			0.816021	−	0.578022i	$-0.196175\pi$
$251$	25.8564			1.63204			0.816021	+	0.578022i	$0.196175\pi$
$252$		0			0
$253$	−0.392305			−0.0246640
$254$	−21.1244	+	5.66025i	−1.32546	+	0.355156i
$255$		0			0
$256$	−8.00000	−	13.8564i	−0.500000	−	0.866025i
$257$			6.00000i			0.374270i	0.982334	+	0.187135i	$0.0599201\pi$
$257$			6.00000i			0.374270i	−0.982334	+	0.187135i	$0.940080\pi$
$258$		0			0
$259$	16.3923	−	18.9282i	1.01857	−	1.17614i
$260$	−0.803848	+	0.464102i	−0.0498525	+	0.0287824i
$261$		0			0
$262$	0.928203	+	3.46410i	0.0573446	+	0.214013i
$263$			11.4641i			0.706907i	0.935452	+	0.353453i	$0.114993\pi$
$263$			11.4641i			0.706907i	−0.935452	+	0.353453i	$0.885007\pi$
$264$		0			0
$265$			2.00000i			0.122859i
$266$	20.1962	+	9.80385i	1.23831	+	0.601112i
$267$		0			0
$268$	3.46410	+	6.00000i	0.211604	+	0.366508i
$269$		−	12.0000i		−	0.731653i	−0.930683	−	0.365826i	$-0.880786\pi$
$269$		−	12.0000i		−	0.731653i	0.930683	−	0.365826i	$-0.119214\pi$
$270$		0			0
$271$	9.46410			0.574903			0.287452	−	0.957795i	$-0.407192\pi$
$271$	9.46410			0.574903			0.287452	+	0.957795i	$0.407192\pi$
$272$	−22.3923	−	12.9282i	−1.35773	−	0.783887i
$273$		0			0
$274$	7.46410	+	27.8564i	0.450923	+	1.68287i
$275$			0.267949i			0.0161579i
$276$		0			0
$277$	16.7846			1.00849			0.504245	−	0.863561i	$-0.331771\pi$
$277$	16.7846			1.00849			0.504245	+	0.863561i	$0.331771\pi$
$278$	−2.53590	−	9.46410i	−0.152093	−	0.567619i
$279$		0			0
$280$	0.535898	+	7.46410i	0.0320261	+	0.446065i
$281$	7.92820			0.472957			0.236478	−	0.971637i	$-0.424007\pi$
$281$	7.92820			0.472957			0.236478	+	0.971637i	$0.424007\pi$
$282$		0			0
$283$	12.1244			0.720718			0.360359	−	0.932814i	$-0.382654\pi$
$283$	12.1244			0.720718			0.360359	+	0.932814i	$0.382654\pi$
$284$	7.46410	+	12.9282i	0.442913	+	0.767148i
$285$		0			0
$286$	−0.0455173	−	0.169873i	−0.00269150	−	0.0100448i
$287$	6.92820	+	6.00000i	0.408959	+	0.354169i
$288$		0			0
$289$	−24.7846			−1.45792
$290$	10.8301	−	2.90192i	0.635967	−	0.170407i
$291$		0			0
$292$	12.9282	+	22.3923i	0.756566	+	1.31041i
$293$			20.3205i			1.18714i	0.804784	+	0.593568i	$0.202281\pi$
$293$			20.3205i			1.18714i	−0.804784	+	0.593568i	$0.797719\pi$
$294$		0			0
$295$		−	3.46410i		−	0.201688i
$296$	−18.9282	−	18.9282i	−1.10018	−	1.10018i
$297$		0			0
$298$	1.12436	+	4.19615i	0.0651322	+	0.243077i
$299$	0.679492			0.0392960
$300$		0			0
$301$	−4.00000	−	3.46410i	−0.230556	−	0.199667i
$302$	20.7583	−	5.56218i	1.19451	−	0.320067i
$303$		0			0
$304$	12.0000	−	20.7846i	0.688247	−	1.19208i
$305$	−9.46410			−0.541913
$306$		0			0
$307$	1.73205			0.0988534			0.0494267	−	0.998778i	$-0.484261\pi$
$307$	1.73205			0.0988534			0.0494267	+	0.998778i	$0.484261\pi$
$308$	−1.39230	−	0.267949i	−0.0793339	−	0.0152678i
$309$		0			0
$310$	8.19615	−	2.19615i	0.465510	−	0.124733i
$311$	−7.85641			−0.445496			−0.222748	−	0.974876i	$-0.571503\pi$
$311$	−7.85641			−0.445496			−0.222748	+	0.974876i	$0.571503\pi$
$312$		0			0
$313$			17.5359i			0.991188i	0.868554	+	0.495594i	$0.165050\pi$
$313$			17.5359i			0.991188i	−0.868554	+	0.495594i	$0.834950\pi$
$314$	−10.7321	+	2.87564i	−0.605645	+	0.162282i
$315$		0			0
$316$	−14.6603	−	25.3923i	−0.824704	−	1.42843i
$317$	3.07180			0.172529			0.0862646	−	0.996272i	$-0.472507\pi$
$317$	3.07180			0.172529			0.0862646	+	0.996272i	$0.472507\pi$
$318$		0			0
$319$			2.12436i			0.118941i
$320$	8.00000			0.447214
$321$		0			0
$322$	2.39230	−	4.92820i	0.133318	−	0.274638i
$323$		−	38.7846i		−	2.15803i
$324$		0			0
$325$		−	0.464102i		−	0.0257437i
$326$	−28.3923	+	7.60770i	−1.57250	+	0.421351i
$327$		0			0
$328$	6.92820	−	6.92820i	0.382546	−	0.382546i
$329$	3.00000	−	3.46410i	0.165395	−	0.190982i
$330$		0			0
$331$		−	26.3923i		−	1.45065i	−0.688405	−	0.725326i	$-0.741689\pi$
$331$		−	26.3923i		−	1.45065i	0.688405	−	0.725326i	$-0.258311\pi$
$332$	26.7846	−	15.4641i	1.47000	−	0.848703i
$333$		0			0
$334$	−1.90192	−	7.09808i	−0.104069	−	0.388389i
$335$	−3.46410			−0.189264
$336$		0			0
$337$		0			0			−	1.00000i	$-0.5\pi$
$337$		0			0				1.00000i	$0.5\pi$
$338$	−4.67949	−	17.4641i	−0.254531	−	0.949922i
$339$		0			0
$340$	11.1962	−	6.46410i	0.607197	−	0.350565i
$341$			1.60770i			0.0870616i
$342$		0			0
$343$	15.5885	+	10.0000i	0.841698	+	0.539949i
$344$	−4.00000	+	4.00000i	−0.215666	+	0.215666i
$345$		0			0
$346$	−19.5622	+	5.24167i	−1.05167	+	0.281794i
$347$		−	34.2487i		−	1.83857i	−0.393596	−	0.919284i	$-0.628769\pi$
$347$		−	34.2487i		−	1.83857i	0.393596	−	0.919284i	$-0.371231\pi$
$348$		0			0
$349$			5.32051i			0.284800i	0.989809	+	0.142400i	$0.0454820\pi$
$349$			5.32051i			0.284800i	−0.989809	+	0.142400i	$0.954518\pi$
$350$	−3.36603	−	1.63397i	−0.179922	−	0.0873396i
$351$		0			0
$352$	−0.392305	+	1.46410i	−0.0209099	+	0.0780369i
$353$		−	7.39230i		−	0.393453i	−0.980458	−	0.196726i	$-0.936969\pi$
$353$		−	7.39230i		−	0.393453i	0.980458	−	0.196726i	$-0.0630311\pi$
$354$		0			0
$355$	−7.46410			−0.396153
$356$	−2.53590	−	4.39230i	−0.134402	−	0.232792i
$357$		0			0
$358$	8.73205	−	2.33975i	0.461503	−	0.123659i
$359$		−	25.3205i		−	1.33637i	−0.743997	−	0.668183i	$-0.767072\pi$
$359$		−	25.3205i		−	1.33637i	0.743997	−	0.668183i	$-0.232928\pi$
$360$		0			0
$361$	17.0000			0.894737
$362$	1.26795	−	0.339746i	0.0666419	−	0.0178567i
$363$		0			0
$364$	2.41154	+	0.464102i	0.126399	+	0.0243255i
$365$	−12.9282			−0.676693
$366$		0			0
$367$	11.8756			0.619904			0.309952	−	0.950752i	$-0.399687\pi$
$367$	11.8756			0.619904			0.309952	+	0.950752i	$0.399687\pi$
$368$	−5.07180	−	2.92820i	−0.264386	−	0.152643i
$369$		0			0
$370$	12.9282	−	3.46410i	0.672105	−	0.180090i
$371$	3.46410	−	4.00000i	0.179847	−	0.207670i
$372$		0			0
$373$	−3.60770			−0.186799			−0.0933997	−	0.995629i	$-0.529773\pi$
$373$	−3.60770			−0.186799			−0.0933997	+	0.995629i	$0.529773\pi$
$374$	0.633975	+	2.36603i	0.0327820	+	0.122344i
$375$		0			0
$376$	−3.46410	−	3.46410i	−0.178647	−	0.178647i
$377$		−	3.67949i		−	0.189503i
$378$		0			0
$379$			5.60770i			0.288048i	0.989574	+	0.144024i	$0.0460042\pi$
$379$			5.60770i			0.288048i	−0.989574	+	0.144024i	$0.953996\pi$
$380$	6.00000	+	10.3923i	0.307794	+	0.533114i
$381$		0			0
$382$	−9.83013	+	2.63397i	−0.502953	+	0.134766i
$383$	−27.4641			−1.40335			−0.701675	−	0.712497i	$-0.747564\pi$
$383$	−27.4641			−1.40335			−0.701675	+	0.712497i	$0.747564\pi$
$384$		0			0
$385$	0.464102	−	0.535898i	0.0236528	−	0.0273119i
$386$	3.46410	+	12.9282i	0.176318	+	0.658028i
$387$		0			0
$388$	−13.3923	−	23.1962i	−0.679891	−	1.17761i
$389$	20.8564			1.05746			0.528731	−	0.848790i	$-0.322668\pi$
$389$	20.8564			1.05746			0.528731	+	0.848790i	$0.322668\pi$
$390$		0			0
$391$	−9.46410			−0.478620
$392$	11.8564	−	15.8564i	0.598839	−	0.800869i
$393$		0			0
$394$	−4.87564	−	18.1962i	−0.245631	−	0.916709i
$395$	14.6603			0.737637
$396$		0			0
$397$		−	12.4641i		−	0.625555i	−0.949826	−	0.312778i	$-0.898741\pi$
$397$		−	12.4641i		−	0.625555i	0.949826	−	0.312778i	$-0.101259\pi$
$398$	−1.26795	−	4.73205i	−0.0635566	−	0.237196i
$399$		0			0
$400$	−2.00000	+	3.46410i	−0.100000	+	0.173205i
$401$	10.0718			0.502962			0.251481	−	0.967862i	$-0.419082\pi$
$401$	10.0718			0.502962			0.251481	+	0.967862i	$0.419082\pi$
$402$		0			0
$403$		−	2.78461i		−	0.138711i
$404$	15.4641	+	26.7846i	0.769368	+	1.33258i
$405$		0			0
$406$	−26.6865	−	12.9545i	−1.32443	−	0.642920i
$407$			2.53590i			0.125700i
$408$		0			0
$409$		−	4.14359i		−	0.204888i	−0.994739	−	0.102444i	$-0.967334\pi$
$409$		−	4.14359i		−	0.204888i	0.994739	−	0.102444i	$-0.0326662\pi$
$410$	1.26795	+	4.73205i	0.0626195	+	0.233699i
$411$		0			0
$412$	11.7846	−	6.80385i	0.580586	−	0.335202i
$413$	−6.00000	+	6.92820i	−0.295241	+	0.340915i
$414$		0			0
$415$			15.4641i			0.759103i
$416$	0.679492	−	2.53590i	0.0333148	−	0.124333i
$417$		0			0
$418$	−2.19615	+	0.588457i	−0.107417	+	0.0287824i
$419$	24.2487			1.18463			0.592314	−	0.805708i	$-0.298215\pi$
$419$	24.2487			1.18463			0.592314	+	0.805708i	$0.298215\pi$
$420$		0			0
$421$	19.0000			0.926003			0.463002	−	0.886357i	$-0.346772\pi$
$421$	19.0000			0.926003			0.463002	+	0.886357i	$0.346772\pi$
$422$	4.36603	−	1.16987i	0.212535	−	0.0569485i
$423$		0			0
$424$	−4.00000	−	4.00000i	−0.194257	−	0.194257i
$425$			6.46410i			0.313555i
$426$		0			0
$427$	18.9282	+	16.3923i	0.916000	+	0.793279i
$428$	−2.39230	−	4.14359i	−0.115636	−	0.200288i
$429$		0			0
$430$	−0.732051	−	2.73205i	−0.0353026	−	0.131751i
$431$			13.5885i			0.654533i	0.944932	+	0.327266i	$0.106127\pi$
$431$			13.5885i			0.654533i	−0.944932	+	0.327266i	$0.893873\pi$
$432$		0			0
$433$			31.8564i			1.53092i	0.643483	+	0.765461i	$0.277489\pi$
$433$			31.8564i			1.53092i	−0.643483	+	0.765461i	$0.722511\pi$
$434$	−20.1962	−	9.80385i	−0.969446	−	0.470600i
$435$		0			0
$436$	−3.58846	+	2.07180i	−0.171856	+	0.0992211i
$437$		−	8.78461i		−	0.420225i
$438$		0			0
$439$	−39.7128			−1.89539			−0.947695	−	0.319179i	$-0.896593\pi$
$439$	−39.7128			−1.89539			−0.947695	+	0.319179i	$0.896593\pi$
$440$	−0.535898	−	0.535898i	−0.0255480	−	0.0255480i
$441$		0			0
$442$	−1.09808	−	4.09808i	−0.0522302	−	0.194926i
$443$			26.0000i			1.23530i	0.786454	+	0.617649i	$0.211915\pi$
$443$			26.0000i			1.23530i	−0.786454	+	0.617649i	$0.788085\pi$
$444$		0			0
$445$	2.53590			0.120213
$446$	5.02628	+	18.7583i	0.238001	+	0.888233i
$447$		0			0
$448$	−16.0000	−	13.8564i	−0.755929	−	0.654654i
$449$	−11.9282			−0.562927			−0.281463	−	0.959572i	$-0.590820\pi$
$449$	−11.9282			−0.562927			−0.281463	+	0.959572i	$0.590820\pi$
$450$		0			0
$451$	−0.928203			−0.0437074
$452$	9.46410	−	5.46410i	0.445154	−	0.257010i
$453$		0			0
$454$	7.56218	+	28.2224i	0.354911	+	1.32454i
$455$	−0.803848	+	0.928203i	−0.0376850	+	0.0435148i
$456$		0			0
$457$	20.5359			0.960629			0.480314	−	0.877096i	$-0.340523\pi$
$457$	20.5359			0.960629			0.480314	+	0.877096i	$0.340523\pi$
$458$	11.6603	−	3.12436i	0.544848	−	0.145992i
$459$		0			0
$460$	2.53590	−	1.46410i	0.118237	−	0.0682641i
$461$			27.7128i			1.29071i	0.763881	+	0.645357i	$0.223291\pi$
$461$			27.7128i			1.29071i	−0.763881	+	0.645357i	$0.776709\pi$
$462$		0			0
$463$		−	16.3923i		−	0.761815i	−0.924613	−	0.380908i	$-0.875612\pi$
$463$		−	16.3923i		−	0.761815i	0.924613	−	0.380908i	$-0.124388\pi$
$464$	−15.8564	+	27.4641i	−0.736115	+	1.27499i
$465$		0			0
$466$	−3.32051	−	12.3923i	−0.153820	−	0.574062i
$467$	−22.5167			−1.04195			−0.520973	−	0.853573i	$-0.674431\pi$
$467$	−22.5167			−1.04195			−0.520973	+	0.853573i	$0.674431\pi$
$468$		0			0
$469$	6.92820	+	6.00000i	0.319915	+	0.277054i
$470$	2.36603	−	0.633975i	0.109137	−	0.0292431i
$471$		0			0
$472$	6.92820	+	6.92820i	0.318896	+	0.318896i
$473$	0.535898			0.0246406
$474$		0			0
$475$	−6.00000			−0.275299
$476$	−33.5885	−	6.46410i	−1.53952	−	0.296282i
$477$		0			0
$478$	32.7583	−	8.77757i	1.49833	−	0.401477i
$479$	−25.1769			−1.15036			−0.575181	−	0.818026i	$-0.695068\pi$
$479$	−25.1769			−1.15036			−0.575181	+	0.818026i	$0.695068\pi$
$480$		0			0
$481$		−	4.39230i		−	0.200272i
$482$	−22.3923	+	6.00000i	−1.01994	+	0.273293i
$483$		0			0
$484$	−18.9282	+	10.9282i	−0.860373	+	0.496737i
$485$	13.3923			0.608113
$486$		0			0
$487$		−	12.7846i		−	0.579326i	−0.957129	−	0.289663i	$-0.906457\pi$
$487$		−	12.7846i		−	0.579326i	0.957129	−	0.289663i	$-0.0935432\pi$
$488$	18.9282	−	18.9282i	0.856840	−	0.856840i
$489$		0			0
$490$	3.90192	+	9.09808i	0.176271	+	0.411009i
$491$			9.87564i			0.445682i	0.974855	+	0.222841i	$0.0715330\pi$
$491$			9.87564i			0.445682i	−0.974855	+	0.222841i	$0.928467\pi$
$492$		0			0
$493$			51.2487i			2.30813i
$494$	3.80385	−	1.01924i	0.171143	−	0.0458577i
$495$		0			0
$496$	−12.0000	+	20.7846i	−0.538816	+	0.933257i
$497$	14.9282	+	12.9282i	0.669621	+	0.579909i
$498$		0			0
$499$			25.5885i			1.14550i	0.819731	+	0.572748i	$0.194123\pi$
$499$			25.5885i			1.14550i	−0.819731	+	0.572748i	$0.805877\pi$
$500$	−1.00000	−	1.73205i	−0.0447214	−	0.0774597i
$501$		0			0
$502$	−9.46410	−	35.3205i	−0.422404	−	1.57643i
$503$	15.5885			0.695055			0.347527	−	0.937670i	$-0.387021\pi$
$503$	15.5885			0.695055			0.347527	+	0.937670i	$0.387021\pi$
$504$		0			0
$505$	−15.4641			−0.688143
$506$	0.143594	+	0.535898i	0.00638351	+	0.0238236i
$507$		0			0
$508$	15.4641	+	26.7846i	0.686109	+	1.18837i
$509$			25.8564i			1.14607i	0.819533	+	0.573033i	$0.194233\pi$
$509$			25.8564i			1.14607i	−0.819533	+	0.573033i	$0.805767\pi$
$510$		0			0
$511$	25.8564	+	22.3923i	1.14382	+	0.990577i
$512$	−16.0000	+	16.0000i	−0.707107	+	0.707107i
$513$		0			0
$514$	8.19615	−	2.19615i	0.361517	−	0.0968681i
$515$			6.80385i			0.299813i
$516$		0			0
$517$			0.464102i			0.0204112i
$518$	−31.8564	−	15.4641i	−1.39969	−	0.679454i
$519$		0			0
$520$	0.928203	+	0.928203i	0.0407044	+	0.0407044i
$521$			13.6077i			0.596164i	0.954540	+	0.298082i	$0.0963469\pi$
$521$			13.6077i			0.596164i	−0.954540	+	0.298082i	$0.903653\pi$
$522$		0			0
$523$	−24.2487			−1.06032			−0.530161	−	0.847897i	$-0.677869\pi$
$523$	−24.2487			−1.06032			−0.530161	+	0.847897i	$0.677869\pi$
$524$	4.39230	−	2.53590i	0.191879	−	0.110781i
$525$		0			0
$526$	15.6603	−	4.19615i	0.682820	−	0.182961i
$527$			38.7846i			1.68948i
$528$		0			0
$529$	20.8564			0.906800
$530$	2.73205	−	0.732051i	0.118673	−	0.0317983i
$531$		0			0
$532$	6.00000	−	31.1769i	0.260133	−	1.35169i
$533$	1.60770			0.0696370
$534$		0			0
$535$	2.39230			0.103428
$536$	6.92820	−	6.92820i	0.299253	−	0.299253i
$537$		0			0
$538$	−16.3923	+	4.39230i	−0.706722	+	0.189366i
$539$	−1.85641	+	0.267949i	−0.0799611	+	0.0115414i
$540$		0			0
$541$	7.78461			0.334687			0.167343	−	0.985899i	$-0.446481\pi$
$541$	7.78461			0.334687			0.167343	+	0.985899i	$0.446481\pi$
$542$	−3.46410	−	12.9282i	−0.148796	−	0.555314i
$543$		0			0
$544$	−9.46410	+	35.3205i	−0.405770	+	1.51435i
$545$		−	2.07180i		−	0.0887460i
$546$		0			0
$547$			21.4641i			0.917739i	0.888504	+	0.458869i	$0.151745\pi$
$547$			21.4641i			0.917739i	−0.888504	+	0.458869i	$0.848255\pi$
$548$	35.3205	−	20.3923i	1.50882	−	0.871116i
$549$		0			0
$550$	0.366025	−	0.0980762i	0.0156074	−	0.00418198i
$551$	−47.5692			−2.02652
$552$		0			0
$553$	−29.3205	−	25.3923i	−1.24683	−	1.07979i
$554$	−6.14359	−	22.9282i	−0.261016	−	0.974126i
$555$		0			0
$556$	−12.0000	+	6.92820i	−0.508913	+	0.293821i
$557$	21.8564			0.926086			0.463043	−	0.886336i	$-0.346758\pi$
$557$	21.8564			0.926086			0.463043	+	0.886336i	$0.346758\pi$
$558$		0			0
$559$	−0.928203			−0.0392588
$560$	10.0000	−	3.46410i	0.422577	−	0.146385i
$561$		0			0
$562$	−2.90192	−	10.8301i	−0.122410	−	0.456841i
$563$	−19.1769			−0.808211			−0.404105	−	0.914712i	$-0.632417\pi$
$563$	−19.1769			−0.808211			−0.404105	+	0.914712i	$0.632417\pi$
$564$		0			0
$565$			5.46410i			0.229876i
$566$	−4.43782	−	16.5622i	−0.186536	−	0.696160i
$567$		0			0
$568$	14.9282	−	14.9282i	0.626373	−	0.626373i
$569$	−7.07180			−0.296465			−0.148233	−	0.988953i	$-0.547358\pi$
$569$	−7.07180			−0.296465			−0.148233	+	0.988953i	$0.547358\pi$
$570$		0			0
$571$		−	6.67949i		−	0.279528i	−0.990185	−	0.139764i	$-0.955366\pi$
$571$		−	6.67949i		−	0.279528i	0.990185	−	0.139764i	$-0.0446344\pi$
$572$	−0.215390	+	0.124356i	−0.00900592	+	0.00519957i
$573$		0			0
$574$	5.66025	−	11.6603i	0.236254	−	0.486690i
$575$			1.46410i			0.0610573i
$576$		0			0
$577$		−	19.3923i		−	0.807312i	−0.914911	−	0.403656i	$-0.867739\pi$
$577$		−	19.3923i		−	0.807312i	0.914911	−	0.403656i	$-0.132261\pi$
$578$	9.07180	+	33.8564i	0.377337	+	1.40824i
$579$		0			0
$580$	−7.92820	−	13.7321i	−0.329201	−	0.570192i
$581$	26.7846	−	30.9282i	1.11121	−	1.28312i
$582$		0			0
$583$			0.535898i			0.0221946i
$584$	25.8564	−	25.8564i	1.06995	−	1.06995i
$585$		0			0
$586$	27.7583	−	7.43782i	1.14669	−	0.307254i
$587$	20.5359			0.847607			0.423804	−	0.905754i	$-0.360695\pi$
$587$	20.5359			0.847607			0.423804	+	0.905754i	$0.360695\pi$
$588$		0			0
$589$	−36.0000			−1.48335
$590$	−4.73205	+	1.26795i	−0.194815	+	0.0522006i
$591$		0			0
$592$	−18.9282	+	32.7846i	−0.777944	+	1.34744i
$593$			30.4641i			1.25101i	0.780220	+	0.625505i	$0.215107\pi$
$593$			30.4641i			1.25101i	−0.780220	+	0.625505i	$0.784893\pi$
$594$		0			0
$595$	11.1962	−	12.9282i	0.458997	−	0.530005i
$596$	5.32051	−	3.07180i	0.217937	−	0.125826i
$597$		0			0
$598$	−0.248711	−	0.928203i	−0.0101706	−	0.0379571i
$599$			10.1244i			0.413670i	0.978376	+	0.206835i	$0.0663163\pi$
$599$			10.1244i			0.413670i	−0.978376	+	0.206835i	$0.933684\pi$
$600$		0			0
$601$			14.7846i			0.603077i	0.953454	+	0.301538i	$0.0975001\pi$
$601$			14.7846i			0.603077i	−0.953454	+	0.301538i	$0.902500\pi$
$602$	−3.26795	+	6.73205i	−0.133192	+	0.274378i
$603$		0			0
$604$	−15.1962	−	26.3205i	−0.618323	−	1.07097i
$605$		−	10.9282i		−	0.444295i
$606$		0			0
$607$	−29.1962			−1.18504			−0.592518	−	0.805557i	$-0.701866\pi$
$607$	−29.1962			−1.18504			−0.592518	+	0.805557i	$0.701866\pi$
$608$	−32.7846	−	8.78461i	−1.32959	−	0.356263i
$609$		0			0
$610$	3.46410	+	12.9282i	0.140257	+	0.523448i
$611$		−	0.803848i		−	0.0325202i
$612$		0			0
$613$	−10.0000			−0.403896			−0.201948	−	0.979396i	$-0.564727\pi$
$613$	−10.0000			−0.403896			−0.201948	+	0.979396i	$0.564727\pi$
$614$	−0.633975	−	2.36603i	−0.0255851	−	0.0954850i
$615$		0			0
$616$	0.143594	+	2.00000i	0.00578555	+	0.0805823i
$617$	22.9282			0.923055			0.461527	−	0.887126i	$-0.347302\pi$
$617$	22.9282			0.923055			0.461527	+	0.887126i	$0.347302\pi$
$618$		0			0
$619$	0.679492			0.0273111			0.0136555	−	0.999907i	$-0.495653\pi$
$619$	0.679492			0.0273111			0.0136555	+	0.999907i	$0.495653\pi$
$620$	−6.00000	−	10.3923i	−0.240966	−	0.417365i
$621$		0			0
$622$	2.87564	+	10.7321i	0.115303	+	0.430316i
$623$	−5.07180	−	4.39230i	−0.203197	−	0.175974i
$624$		0			0
$625$	1.00000			0.0400000
$626$	23.9545	−	6.41858i	0.957414	−	0.256538i
$627$		0			0
$628$	7.85641	+	13.6077i	0.313505	+	0.543006i
$629$			61.1769i			2.43928i
$630$		0			0
$631$			38.9090i			1.54894i	0.632610	+	0.774471i	$0.281984\pi$
$631$			38.9090i			1.54894i	−0.632610	+	0.774471i	$0.718016\pi$
$632$	−29.3205	+	29.3205i	−1.16631	+	1.16631i
$633$		0			0
$634$	−1.12436	−	4.19615i	−0.0446539	−	0.166651i
$635$	−15.4641			−0.613674
$636$		0			0
$637$	3.21539	−	0.464102i	0.127398	−	0.0183884i
$638$	2.90192	−	0.777568i	0.114888	−	0.0307842i
$639$		0			0
$640$	−2.92820	−	10.9282i	−0.115747	−	0.431975i
$641$	−8.92820			−0.352643			−0.176321	−	0.984333i	$-0.556420\pi$
$641$	−8.92820			−0.352643			−0.176321	+	0.984333i	$0.556420\pi$
$642$		0			0
$643$	7.05256			0.278126			0.139063	−	0.990284i	$-0.455591\pi$
$643$	7.05256			0.278126			0.139063	+	0.990284i	$0.455591\pi$
$644$	−7.60770	−	1.46410i	−0.299785	−	0.0576937i
$645$		0			0
$646$	−52.9808	+	14.1962i	−2.08450	+	0.558540i
$647$	10.3923			0.408564			0.204282	−	0.978912i	$-0.434514\pi$
$647$	10.3923			0.408564			0.204282	+	0.978912i	$0.434514\pi$
$648$		0			0
$649$		−	0.928203i		−	0.0364352i
$650$	−0.633975	+	0.169873i	−0.0248665	+	0.00666297i
$651$		0			0
$652$	20.7846	+	36.0000i	0.813988	+	1.40987i
$653$	−17.6077			−0.689042			−0.344521	−	0.938779i	$-0.611959\pi$
$653$	−17.6077			−0.689042			−0.344521	+	0.938779i	$0.611959\pi$
$654$		0			0
$655$			2.53590i			0.0990857i
$656$	−12.0000	−	6.92820i	−0.468521	−	0.270501i
$657$		0			0
$658$	−5.83013	−	2.83013i	−0.227282	−	0.110330i
$659$			31.1962i			1.21523i	0.794232	+	0.607615i	$0.207874\pi$
$659$			31.1962i			1.21523i	−0.794232	+	0.607615i	$0.792126\pi$
$660$		0			0
$661$		−	39.7128i		−	1.54465i	−0.635228	−	0.772325i	$-0.719094\pi$
$661$		−	39.7128i		−	1.54465i	0.635228	−	0.772325i	$-0.280906\pi$
$662$	−36.0526	+	9.66025i	−1.40122	+	0.375456i
$663$		0			0
$664$	−30.9282	−	30.9282i	−1.20025	−	1.20025i
$665$	12.0000	+	10.3923i	0.465340	+	0.402996i
$666$		0			0
$667$			11.6077i			0.449452i
$668$	−9.00000	+	5.19615i	−0.348220	+	0.201045i
$669$		0			0
$670$	1.26795	+	4.73205i	0.0489852	+	0.182815i
$671$	−2.53590			−0.0978973
$672$		0			0
$673$	−13.1769			−0.507933			−0.253966	−	0.967213i	$-0.581735\pi$
$673$	−13.1769			−0.507933			−0.253966	+	0.967213i	$0.581735\pi$
$674$		0			0
$675$		0			0
$676$	−22.1436	+	12.7846i	−0.851677	+	0.491716i
$677$		−	25.3923i		−	0.975906i	−0.872870	−	0.487953i	$-0.837744\pi$
$677$		−	25.3923i		−	0.975906i	0.872870	−	0.487953i	$-0.162256\pi$
$678$		0			0
$679$	−26.7846	−	23.1962i	−1.02790	−	0.890187i
$680$	−12.9282	−	12.9282i	−0.495774	−	0.495774i
$681$		0			0
$682$	2.19615	−	0.588457i	0.0840950	−	0.0225332i
$683$		−	7.32051i		−	0.280111i	−0.990144	−	0.140056i	$-0.955272\pi$
$683$		−	7.32051i		−	0.280111i	0.990144	−	0.140056i	$-0.0447282\pi$
$684$		0			0
$685$			20.3923i			0.779150i
$686$	7.95448	−	24.9545i	0.303704	−	0.952767i
$687$		0			0
$688$	6.92820	+	4.00000i	0.264135	+	0.152499i
$689$		−	0.928203i		−	0.0353617i
$690$		0			0
$691$	−22.6410			−0.861305			−0.430652	−	0.902518i	$-0.641717\pi$
$691$	−22.6410			−0.861305			−0.430652	+	0.902518i	$0.641717\pi$
$692$	14.3205	+	24.8038i	0.544384	+	0.942901i
$693$		0			0
$694$	−46.7846	+	12.5359i	−1.77592	+	0.475856i
$695$		−	6.92820i		−	0.262802i
$696$		0			0
$697$	−22.3923			−0.848169
$698$	7.26795	−	1.94744i	0.275096	−	0.0737117i
$699$		0			0
$700$	−1.00000	+	5.19615i	−0.0377964	+	0.196396i
$701$	37.7846			1.42711			0.713553	−	0.700602i	$-0.247085\pi$
$701$	37.7846			1.42711			0.713553	+	0.700602i	$0.247085\pi$
$702$		0			0
$703$	−56.7846			−2.14167
$704$	2.14359			0.0807897
$705$		0			0
$706$	−10.0981	+	2.70577i	−0.380046	+	0.101833i
$707$	30.9282	+	26.7846i	1.16317	+	1.00734i
$708$		0			0
$709$	21.0000			0.788672			0.394336	−	0.918966i	$-0.370975\pi$
$709$	21.0000			0.788672			0.394336	+	0.918966i	$0.370975\pi$
$710$	2.73205	+	10.1962i	0.102532	+	0.382655i
$711$		0			0
$712$	−5.07180	+	5.07180i	−0.190074	+	0.190074i
$713$			8.78461i			0.328986i
$714$		0			0
$715$		−	0.124356i		−	0.00465064i
$716$	−6.39230	−	11.0718i	−0.238892	−	0.413772i
$717$		0			0
$718$	−34.5885	+	9.26795i	−1.29083	+	0.345877i
$719$	−38.5359			−1.43715			−0.718573	−	0.695451i	$-0.755205\pi$
$719$	−38.5359			−1.43715			−0.718573	+	0.695451i	$0.755205\pi$
$720$		0			0
$721$	11.7846	−	13.6077i	0.438882	−	0.506777i
$722$	−6.22243	−	23.2224i	−0.231575	−	0.864249i
$723$		0			0
$724$	−0.928203	−	1.60770i	−0.0344964	−	0.0597495i
$725$	7.92820			0.294446
$726$		0			0
$727$	−13.6077			−0.504681			−0.252341	−	0.967638i	$-0.581200\pi$
$727$	−13.6077			−0.504681			−0.252341	+	0.967638i	$0.581200\pi$
$728$	−0.248711	−	3.46410i	−0.00921785	−	0.128388i
$729$		0			0
$730$	4.73205	+	17.6603i	0.175141	+	0.653635i
$731$	12.9282			0.478167
$732$		0			0
$733$		−	11.5359i		−	0.426088i	−0.977043	−	0.213044i	$-0.931662\pi$
$733$		−	11.5359i		−	0.426088i	0.977043	−	0.213044i	$-0.0683378\pi$
$734$	−4.34679	−	16.2224i	−0.160443	−	0.598781i
$735$		0			0
$736$	−2.14359	+	8.00000i	−0.0790139	+	0.294884i
$737$	−0.928203			−0.0341908
$738$		0			0
$739$		−	4.26795i		−	0.156999i	−0.996914	−	0.0784995i	$-0.974987\pi$
$739$		−	4.26795i		−	0.156999i	0.996914	−	0.0784995i	$-0.0250129\pi$
$740$	−9.46410	−	16.3923i	−0.347907	−	0.602593i
$741$		0			0
$742$	−6.73205	−	3.26795i	−0.247141	−	0.119970i
$743$		−	30.3923i		−	1.11499i	−0.830182	−	0.557493i	$-0.811763\pi$
$743$		−	30.3923i		−	1.11499i	0.830182	−	0.557493i	$-0.188237\pi$
$744$		0			0
$745$			3.07180i			0.112542i
$746$	1.32051	+	4.92820i	0.0483472	+	0.180434i
$747$		0			0
$748$	3.00000	−	1.73205i	0.109691	−	0.0633300i
$749$	−4.78461	−	4.14359i	−0.174826	−	0.151404i
$750$		0			0
$751$		−	25.5885i		−	0.933736i	−0.884327	−	0.466868i	$-0.845382\pi$
$751$		−	25.5885i		−	0.933736i	0.884327	−	0.466868i	$-0.154618\pi$
$752$	−3.46410	+	6.00000i	−0.126323	+	0.218797i
$753$		0			0
$754$	−5.02628	+	1.34679i	−0.183046	+	0.0490471i
$755$	15.1962			0.553045
$756$		0			0
$757$	37.8564			1.37591			0.687957	−	0.725751i	$-0.258508\pi$
$757$	37.8564			1.37591			0.687957	+	0.725751i	$0.258508\pi$
$758$	7.66025	−	2.05256i	0.278233	−	0.0745523i
$759$		0			0
$760$	12.0000	−	12.0000i	0.435286	−	0.435286i
$761$		−	42.2487i		−	1.53151i	−0.643130	−	0.765757i	$-0.722364\pi$
$761$		−	42.2487i		−	1.53151i	0.643130	−	0.765757i	$-0.277636\pi$
$762$		0			0
$763$	−3.58846	+	4.14359i	−0.129911	+	0.150008i
$764$	7.19615	+	12.4641i	0.260348	+	0.450935i
$765$		0			0
$766$	10.0526	+	37.5167i	0.363214	+	1.35553i
$767$			1.60770i			0.0580505i
$768$		0			0
$769$		−	18.0000i		−	0.649097i	−0.945869	−	0.324548i	$-0.894788\pi$
$769$		−	18.0000i		−	0.649097i	0.945869	−	0.324548i	$-0.105212\pi$
$770$	−0.901924	−	0.437822i	−0.0325031	−	0.0157780i
$771$		0			0
$772$	16.3923	−	9.46410i	0.589972	−	0.340620i
$773$		−	5.53590i		−	0.199112i	−0.995032	−	0.0995562i	$-0.968258\pi$
$773$		−	5.53590i		−	0.199112i	0.995032	−	0.0995562i	$-0.0317423\pi$
$774$		0			0
$775$	6.00000			0.215526
$776$	−26.7846	+	26.7846i	−0.961511	+	0.961511i
$777$		0			0
$778$	−7.63397	−	28.4904i	−0.273691	−	1.02143i
$779$		−	20.7846i		−	0.744686i
$780$		0			0
$781$	−2.00000			−0.0715656
$782$	3.46410	+	12.9282i	0.123876	+	0.462312i
$783$		0			0
$784$	−26.0000	−	10.3923i	−0.928571	−	0.371154i

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:	\( N \)	\(=\)	\( 1260 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1260.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.366025 - 1.36603i) q^{2} +(-1.73205 + 1.00000i) q^{4} -1.00000i q^{5} +(-1.73205 + 2.00000i) q^{7} +(2.00000 + 2.00000i) q^{8} +O(q^{10})\)	\(q+(-0.366025 - 1.36603i) q^{2} +(-1.73205 + 1.00000i) q^{4} -1.00000i q^{5} +(-1.73205 + 2.00000i) q^{7} +(2.00000 + 2.00000i) q^{8} +(-1.36603 + 0.366025i) q^{10} -0.267949i q^{11} +0.464102i q^{13} +(3.36603 + 1.63397i) q^{14} +(2.00000 - 3.46410i) q^{16} -6.46410i q^{17} +6.00000 q^{19} +(1.00000 + 1.73205i) q^{20} +(-0.366025 + 0.0980762i) q^{22} -1.46410i q^{23} -1.00000 q^{25} +(0.633975 - 0.169873i) q^{26} +(1.00000 - 5.19615i) q^{28} -7.92820 q^{29} -6.00000 q^{31} +(-5.46410 - 1.46410i) q^{32} +(-8.83013 + 2.36603i) q^{34} +(2.00000 + 1.73205i) q^{35} -9.46410 q^{37} +(-2.19615 - 8.19615i) q^{38} +(2.00000 - 2.00000i) q^{40} -3.46410i q^{41} +2.00000i q^{43} +(0.267949 + 0.464102i) q^{44} +(-2.00000 + 0.535898i) q^{46} -1.73205 q^{47} +(-1.00000 - 6.92820i) q^{49} +(0.366025 + 1.36603i) q^{50} +(-0.464102 - 0.803848i) q^{52} -2.00000 q^{53} -0.267949 q^{55} +(-7.46410 + 0.535898i) q^{56} +(2.90192 + 10.8301i) q^{58} +3.46410 q^{59} -9.46410i q^{61} +(2.19615 + 8.19615i) q^{62} +8.00000i q^{64} +0.464102 q^{65} -3.46410i q^{67} +(6.46410 + 11.1962i) q^{68} +(1.63397 - 3.36603i) q^{70} -7.46410i q^{71} -12.9282i q^{73} +(3.46410 + 12.9282i) q^{74} +(-10.3923 + 6.00000i) q^{76} +(0.535898 + 0.464102i) q^{77} +14.6603i q^{79} +(-3.46410 - 2.00000i) q^{80} +(-4.73205 + 1.26795i) q^{82} -15.4641 q^{83} -6.46410 q^{85} +(2.73205 - 0.732051i) q^{86} +(0.535898 - 0.535898i) q^{88} +2.53590i q^{89} +(-0.928203 - 0.803848i) q^{91} +(1.46410 + 2.53590i) q^{92} +(0.633975 + 2.36603i) q^{94} -6.00000i q^{95} +13.3923i q^{97} +(-9.09808 + 3.90192i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} + 8 q^{8}+O(q^{10}) \) 4 * q + 2 * q^2 + 8 * q^8	\( 4 q + 2 q^{2} + 8 q^{8} - 2 q^{10} + 10 q^{14} + 8 q^{16} + 24 q^{19} + 4 q^{20} + 2 q^{22} - 4 q^{25} + 6 q^{26} + 4 q^{28} - 4 q^{29} - 24 q^{31} - 8 q^{32} - 18 q^{34} + 8 q^{35} - 24 q^{37} + 12 q^{38} + 8 q^{40} + 8 q^{44} - 8 q^{46} - 4 q^{49} - 2 q^{50} + 12 q^{52} - 8 q^{53} - 8 q^{55} - 16 q^{56} + 22 q^{58} - 12 q^{62} - 12 q^{65} + 12 q^{68} + 10 q^{70} + 16 q^{77} - 12 q^{82} - 48 q^{83} - 12 q^{85} + 4 q^{86} + 16 q^{88} + 24 q^{91} - 8 q^{92} + 6 q^{94} - 26 q^{98}+O(q^{100}) \) 4 * q + 2 * q^2 + 8 * q^8 - 2 * q^10 + 10 * q^14 + 8 * q^16 + 24 * q^19 + 4 * q^20 + 2 * q^22 - 4 * q^25 + 6 * q^26 + 4 * q^28 - 4 * q^29 - 24 * q^31 - 8 * q^32 - 18 * q^34 + 8 * q^35 - 24 * q^37 + 12 * q^38 + 8 * q^40 + 8 * q^44 - 8 * q^46 - 4 * q^49 - 2 * q^50 + 12 * q^52 - 8 * q^53 - 8 * q^55 - 16 * q^56 + 22 * q^58 - 12 * q^62 - 12 * q^65 + 12 * q^68 + 10 * q^70 + 16 * q^77 - 12 * q^82 - 48 * q^83 - 12 * q^85 + 4 * q^86 + 16 * q^88 + 24 * q^91 - 8 * q^92 + 6 * q^94 - 26 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more