LMFDB - Embedded newform 1232.2.q.f.529.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1232,2,Mod(177,1232)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1232, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2, 0]))

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

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

chi := DirichletCharacter("1232.177");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1232 = 2^{4} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1232.q (of order $3$, degree $2$, not 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:	$9.83756952902$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 2x^{2} + 4 $ x^4 + 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 154)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			529.1
Root			$-0.707107 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	1232.529
Dual form			1232.2.q.f.177.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.20711 + 2.09077i) q^{3} +(0.292893 + 0.507306i) q^{5} +(-1.62132 + 2.09077i) q^{7} +(-1.41421 - 2.44949i) q^{9} +O(q^{10})$	\(q+(-1.20711 + 2.09077i) q^{3} +(0.292893 + 0.507306i) q^{5} +(-1.62132 + 2.09077i) q^{7} +(-1.41421 - 2.44949i) q^{9} +(0.500000 - 0.866025i) q^{11} -3.82843 q^{13} -1.41421 q^{15} +(-1.82843 + 3.16693i) q^{17} +(-0.292893 - 0.507306i) q^{19} +(-2.41421 - 5.91359i) q^{21} +(-3.12132 - 5.40629i) q^{23} +(2.32843 - 4.03295i) q^{25} -0.414214 q^{27} +2.65685 q^{29} +(-2.00000 + 3.46410i) q^{31} +(1.20711 + 2.09077i) q^{33} +(-1.53553 - 0.210133i) q^{35} +(4.70711 + 8.15295i) q^{37} +(4.62132 - 8.00436i) q^{39} -5.41421 q^{41} +5.65685 q^{43} +(0.828427 - 1.43488i) q^{45} +(-5.24264 - 9.08052i) q^{47} +(-1.74264 - 6.77962i) q^{49} +(-4.41421 - 7.64564i) q^{51} +(-3.94975 + 6.84116i) q^{53} +0.585786 q^{55} +1.41421 q^{57} +(-2.79289 + 4.83743i) q^{59} +(-5.91421 - 10.2437i) q^{61} +(7.41421 + 1.01461i) q^{63} +(-1.12132 - 1.94218i) q^{65} +(1.37868 - 2.38794i) q^{67} +15.0711 q^{69} +11.0711 q^{71} +(4.70711 - 8.15295i) q^{73} +(5.62132 + 9.73641i) q^{75} +(1.00000 + 2.44949i) q^{77} +(-6.62132 - 11.4685i) q^{79} +(4.74264 - 8.21449i) q^{81} +12.1421 q^{83} -2.14214 q^{85} +(-3.20711 + 5.55487i) q^{87} +(-6.24264 - 10.8126i) q^{89} +(6.20711 - 8.00436i) q^{91} +(-4.82843 - 8.36308i) q^{93} +(0.171573 - 0.297173i) q^{95} -3.82843 q^{97} -2.82843 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{3} + 4 q^{5} + 2 q^{7}+O(q^{10}) $ 4 * q - 2 * q^3 + 4 * q^5 + 2 * q^7	$ 4 q - 2 q^{3} + 4 q^{5} + 2 q^{7} + 2 q^{11} - 4 q^{13} + 4 q^{17} - 4 q^{19} - 4 q^{21} - 4 q^{23} - 2 q^{25} + 4 q^{27} - 12 q^{29} - 8 q^{31} + 2 q^{33} + 8 q^{35} + 16 q^{37} + 10 q^{39} - 16 q^{41} - 8 q^{45} - 4 q^{47} + 10 q^{49} - 12 q^{51} + 4 q^{53} + 8 q^{55} - 14 q^{59} - 18 q^{61} + 24 q^{63} + 4 q^{65} + 14 q^{67} + 32 q^{69} + 16 q^{71} + 16 q^{73} + 14 q^{75} + 4 q^{77} - 18 q^{79} + 2 q^{81} - 8 q^{83} + 48 q^{85} - 10 q^{87} - 8 q^{89} + 22 q^{91} - 8 q^{93} + 12 q^{95} - 4 q^{97}+O(q^{100}) $ 4 * q - 2 * q^3 + 4 * q^5 + 2 * q^7 + 2 * q^11 - 4 * q^13 + 4 * q^17 - 4 * q^19 - 4 * q^21 - 4 * q^23 - 2 * q^25 + 4 * q^27 - 12 * q^29 - 8 * q^31 + 2 * q^33 + 8 * q^35 + 16 * q^37 + 10 * q^39 - 16 * q^41 - 8 * q^45 - 4 * q^47 + 10 * q^49 - 12 * q^51 + 4 * q^53 + 8 * q^55 - 14 * q^59 - 18 * q^61 + 24 * q^63 + 4 * q^65 + 14 * q^67 + 32 * q^69 + 16 * q^71 + 16 * q^73 + 14 * q^75 + 4 * q^77 - 18 * q^79 + 2 * q^81 - 8 * q^83 + 48 * q^85 - 10 * q^87 - 8 * q^89 + 22 * q^91 - 8 * q^93 + 12 * q^95 - 4 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1232\mathbb{Z}\right)^\times$.

$n$	$309$	$353$	$463$	$673$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$1$	$1$

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			0
$2$		0			0
$3$	−1.20711	+	2.09077i	−0.696923	+	1.20711i	0.272605	+	0.962126i	$0.412115\pi$
$3$	−1.20711	+	2.09077i	−0.696923	+	1.20711i	−0.969528	+	0.244981i	$0.921218\pi$
$4$		0			0
$5$	0.292893	+	0.507306i	0.130986	+	0.226874i	0.924057	−	0.382255i	$-0.124852\pi$
$5$	0.292893	+	0.507306i	0.130986	+	0.226874i	−0.793071	+	0.609129i	$0.791519\pi$
$6$		0			0
$7$	−1.62132	+	2.09077i	−0.612801	+	0.790237i
$7$	−1.62132	+	2.09077i	−0.612801	+	0.790237i
$8$		0			0
$9$	−1.41421	−	2.44949i	−0.471405	−	0.816497i
$10$		0			0
$11$	0.500000	−	0.866025i	0.150756	−	0.261116i
$11$	0.500000	−	0.866025i	0.150756	−	0.261116i
$12$		0			0
$13$	−3.82843			−1.06181			−0.530907	−	0.847430i	$-0.678149\pi$
$13$	−3.82843			−1.06181			−0.530907	+	0.847430i	$0.678149\pi$
$14$		0			0
$15$	−1.41421			−0.365148
$16$		0			0
$17$	−1.82843	+	3.16693i	−0.443459	+	0.768093i	−0.997943	−	0.0641009i	$-0.979582\pi$
$17$	−1.82843	+	3.16693i	−0.443459	+	0.768093i	0.554485	+	0.832194i	$0.312915\pi$
$18$		0			0
$19$	−0.292893	−	0.507306i	−0.0671943	−	0.116384i	0.830471	−	0.557062i	$-0.188071\pi$
$19$	−0.292893	−	0.507306i	−0.0671943	−	0.116384i	−0.897665	+	0.440678i	$0.854738\pi$
$20$		0			0
$21$	−2.41421	−	5.91359i	−0.526825	−	1.29045i
$22$		0			0
$23$	−3.12132	−	5.40629i	−0.650840	−	1.12729i	−0.982919	−	0.184037i	$-0.941083\pi$
$23$	−3.12132	−	5.40629i	−0.650840	−	1.12729i	0.332079	−	0.943252i	$-0.392250\pi$
$24$		0			0
$25$	2.32843	−	4.03295i	0.465685	−	0.806591i
$26$		0			0
$27$	−0.414214			−0.0797154
$28$		0			0
$29$	2.65685			0.493365			0.246683	−	0.969096i	$-0.420659\pi$
$29$	2.65685			0.493365			0.246683	+	0.969096i	$0.420659\pi$
$30$		0			0
$31$	−2.00000	+	3.46410i	−0.359211	+	0.622171i	−0.987829	−	0.155543i	$-0.950287\pi$
$31$	−2.00000	+	3.46410i	−0.359211	+	0.622171i	0.628619	+	0.777714i	$0.283621\pi$
$32$		0			0
$33$	1.20711	+	2.09077i	0.210130	+	0.363956i
$34$		0			0
$35$	−1.53553	−	0.210133i	−0.259553	−	0.0355190i
$36$		0			0
$37$	4.70711	+	8.15295i	0.773844	+	1.34034i	0.935442	+	0.353480i	$0.115002\pi$
$37$	4.70711	+	8.15295i	0.773844	+	1.34034i	−0.161599	+	0.986857i	$0.551665\pi$
$38$		0			0
$39$	4.62132	−	8.00436i	0.740003	−	1.28172i
$40$		0			0
$41$	−5.41421			−0.845558			−0.422779	−	0.906233i	$-0.638945\pi$
$41$	−5.41421			−0.845558			−0.422779	+	0.906233i	$0.638945\pi$
$42$		0			0
$43$	5.65685			0.862662			0.431331	−	0.902194i	$-0.358044\pi$
$43$	5.65685			0.862662			0.431331	+	0.902194i	$0.358044\pi$
$44$		0			0
$45$	0.828427	−	1.43488i	0.123495	−	0.213899i
$46$		0			0
$47$	−5.24264	−	9.08052i	−0.764718	−	1.32453i	−0.940396	−	0.340082i	$-0.889545\pi$
$47$	−5.24264	−	9.08052i	−0.764718	−	1.32453i	0.175678	−	0.984448i	$-0.443788\pi$
$48$		0			0
$49$	−1.74264	−	6.77962i	−0.248949	−	0.968517i
$50$		0			0
$51$	−4.41421	−	7.64564i	−0.618114	−	1.07060i
$52$		0			0
$53$	−3.94975	+	6.84116i	−0.542540	+	0.939706i	0.456218	+	0.889868i	$0.349204\pi$
$53$	−3.94975	+	6.84116i	−0.542540	+	0.939706i	−0.998757	+	0.0498379i	$0.984130\pi$
$54$		0			0
$55$	0.585786			0.0789874
$56$		0			0
$57$	1.41421			0.187317
$58$		0			0
$59$	−2.79289	+	4.83743i	−0.363604	+	0.629780i	−0.988551	−	0.150887i	$-0.951787\pi$
$59$	−2.79289	+	4.83743i	−0.363604	+	0.629780i	0.624947	+	0.780667i	$0.285120\pi$
$60$		0			0
$61$	−5.91421	−	10.2437i	−0.757237	−	1.31157i	−0.944254	−	0.329217i	$-0.893215\pi$
$61$	−5.91421	−	10.2437i	−0.757237	−	1.31157i	0.187017	−	0.982357i	$-0.440118\pi$
$62$		0			0
$63$	7.41421	+	1.01461i	0.934103	+	0.127829i
$64$		0			0
$65$	−1.12132	−	1.94218i	−0.139083	−	0.240898i
$66$		0			0
$67$	1.37868	−	2.38794i	0.168433	−	0.291734i	−0.769436	−	0.638723i	$-0.779463\pi$
$67$	1.37868	−	2.38794i	0.168433	−	0.291734i	0.937869	+	0.346990i	$0.112796\pi$
$68$		0			0
$69$	15.0711			1.81434
$70$		0			0
$71$	11.0711			1.31389			0.656947	−	0.753937i	$-0.271848\pi$
$71$	11.0711			1.31389			0.656947	+	0.753937i	$0.271848\pi$
$72$		0			0
$73$	4.70711	−	8.15295i	0.550925	−	0.954230i	−0.447283	−	0.894393i	$-0.647608\pi$
$73$	4.70711	−	8.15295i	0.550925	−	0.954230i	0.998208	−	0.0598379i	$-0.0190584\pi$
$74$		0			0
$75$	5.62132	+	9.73641i	0.649094	+	1.12426i
$76$		0			0
$77$	1.00000	+	2.44949i	0.113961	+	0.279145i
$78$		0			0
$79$	−6.62132	−	11.4685i	−0.744957	−	1.29030i	−0.950215	−	0.311595i	$-0.899137\pi$
$79$	−6.62132	−	11.4685i	−0.744957	−	1.29030i	0.205258	−	0.978708i	$-0.434197\pi$
$80$		0			0
$81$	4.74264	−	8.21449i	0.526960	−	0.912722i
$82$		0			0
$83$	12.1421			1.33277			0.666386	−	0.745607i	$-0.267840\pi$
$83$	12.1421			1.33277			0.666386	+	0.745607i	$0.267840\pi$
$84$		0			0
$85$	−2.14214			−0.232347
$86$		0			0
$87$	−3.20711	+	5.55487i	−0.343838	+	0.595545i
$88$		0			0
$89$	−6.24264	−	10.8126i	−0.661719	−	1.14613i	−0.980164	−	0.198189i	$-0.936494\pi$
$89$	−6.24264	−	10.8126i	−0.661719	−	1.14613i	0.318445	−	0.947941i	$-0.396839\pi$
$90$		0			0
$91$	6.20711	−	8.00436i	0.650682	−	0.839085i
$92$		0			0
$93$	−4.82843	−	8.36308i	−0.500685	−	0.867211i
$94$		0			0
$95$	0.171573	−	0.297173i	0.0176030	−	0.0304893i
$96$		0			0
$97$	−3.82843			−0.388718			−0.194359	−	0.980930i	$-0.562263\pi$
$97$	−3.82843			−0.388718			−0.194359	+	0.980930i	$0.562263\pi$
$98$		0			0
$99$	−2.82843			−0.284268
$100$		0			0
$101$	−3.08579	+	5.34474i	−0.307047	+	0.531821i	−0.977715	−	0.209936i	$-0.932674\pi$
$101$	−3.08579	+	5.34474i	−0.307047	+	0.531821i	0.670668	+	0.741758i	$0.266008\pi$
$102$		0			0
$103$	6.70711	+	11.6170i	0.660871	+	1.14466i	0.980387	+	0.197081i	$0.0631462\pi$
$103$	6.70711	+	11.6170i	0.660871	+	1.14466i	−0.319516	+	0.947581i	$0.603520\pi$
$104$		0			0
$105$	2.29289	−	2.95680i	0.223763	−	0.288554i
$106$		0			0
$107$	−1.53553	−	2.65962i	−0.148446	−	0.257115i	0.782207	−	0.623018i	$-0.214094\pi$
$107$	−1.53553	−	2.65962i	−0.148446	−	0.257115i	−0.930653	+	0.365903i	$0.880760\pi$
$108$		0			0
$109$	−8.24264	+	14.2767i	−0.789502	+	1.36746i	0.136771	+	0.990603i	$0.456328\pi$
$109$	−8.24264	+	14.2767i	−0.789502	+	1.36746i	−0.926272	+	0.376854i	$0.877006\pi$
$110$		0			0
$111$	−22.7279			−2.15724
$112$		0			0
$113$	−8.17157			−0.768717			−0.384358	−	0.923184i	$-0.625577\pi$
$113$	−8.17157			−0.768717			−0.384358	+	0.923184i	$0.625577\pi$
$114$		0			0
$115$	1.82843	−	3.16693i	0.170502	−	0.295318i
$116$		0			0
$117$	5.41421	+	9.37769i	0.500544	+	0.866968i
$118$		0			0
$119$	−3.65685	−	8.95743i	−0.335223	−	0.821126i
$120$		0			0
$121$	−0.500000	−	0.866025i	−0.0454545	−	0.0787296i
$122$		0			0
$123$	6.53553	−	11.3199i	0.589289	−	1.02068i
$124$		0			0
$125$	5.65685			0.505964
$126$		0			0
$127$	−15.7279			−1.39563			−0.697814	−	0.716279i	$-0.745844\pi$
$127$	−15.7279			−1.39563			−0.697814	+	0.716279i	$0.745844\pi$
$128$		0			0
$129$	−6.82843	+	11.8272i	−0.601209	+	1.04133i
$130$		0			0
$131$	−0.292893	−	0.507306i	−0.0255902	−	0.0443235i	0.852947	−	0.521998i	$-0.174813\pi$
$131$	−0.292893	−	0.507306i	−0.0255902	−	0.0443235i	−0.878537	+	0.477674i	$0.841480\pi$
$132$		0			0
$133$	1.53553	+	0.210133i	0.133148	+	0.0182208i
$134$		0			0
$135$	−0.121320	−	0.210133i	−0.0104416	−	0.0180854i
$136$		0			0
$137$	8.32843	−	14.4253i	0.711546	−	1.23243i	−0.252731	−	0.967537i	$-0.581329\pi$
$137$	8.32843	−	14.4253i	0.711546	−	1.23243i	0.964277	−	0.264897i	$-0.0853378\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$	25.3137			2.13180
$142$		0			0
$143$	−1.91421	+	3.31552i	−0.160075	+	0.277257i
$144$		0			0
$145$	0.778175	+	1.34784i	0.0646239	+	0.111932i
$146$		0			0
$147$	16.2782	+	4.54026i	1.34260	+	0.374474i
$148$		0			0
$149$	−8.82843	−	15.2913i	−0.723253	−	1.25271i	−0.959689	−	0.281064i	$-0.909313\pi$
$149$	−8.82843	−	15.2913i	−0.723253	−	1.25271i	0.236436	−	0.971647i	$-0.424021\pi$
$150$		0			0
$151$	−7.86396	+	13.6208i	−0.639960	+	1.10844i	0.345481	+	0.938426i	$0.387716\pi$
$151$	−7.86396	+	13.6208i	−0.639960	+	1.10844i	−0.985441	+	0.170018i	$0.945618\pi$
$152$		0			0
$153$	10.3431			0.836194
$154$		0			0
$155$	−2.34315			−0.188206
$156$		0			0
$157$	−8.82843	+	15.2913i	−0.704585	+	1.22038i	0.262256	+	0.964998i	$0.415534\pi$
$157$	−8.82843	+	15.2913i	−0.704585	+	1.22038i	−0.966841	+	0.255379i	$0.917800\pi$
$158$		0			0
$159$	−9.53553	−	16.5160i	−0.756217	−	1.30981i
$160$		0			0
$161$	16.3640	+	2.23936i	1.28966	+	0.176486i
$162$		0			0
$163$	4.86396	+	8.42463i	0.380975	+	0.659868i	0.991202	−	0.132359i	$-0.0422551\pi$
$163$	4.86396	+	8.42463i	0.380975	+	0.659868i	−0.610227	+	0.792227i	$0.708922\pi$
$164$		0			0
$165$	−0.707107	+	1.22474i	−0.0550482	+	0.0953463i
$166$		0			0
$167$	−13.7279			−1.06230			−0.531149	−	0.847278i	$-0.678240\pi$
$167$	−13.7279			−1.06230			−0.531149	+	0.847278i	$0.678240\pi$
$168$		0			0
$169$	1.65685			0.127450
$170$		0			0
$171$	−0.828427	+	1.43488i	−0.0633514	+	0.109728i
$172$		0			0
$173$	4.91421	+	8.51167i	0.373621	+	0.647130i	0.990120	−	0.140226i	$-0.0447828\pi$
$173$	4.91421	+	8.51167i	0.373621	+	0.647130i	−0.616499	+	0.787356i	$0.711449\pi$
$174$		0			0
$175$	4.65685	+	11.4069i	0.352025	+	0.862282i
$176$		0			0
$177$	−6.74264	−	11.6786i	−0.506808	−	0.877817i
$178$		0			0
$179$	0.449747	−	0.778985i	0.0336157	−	0.0582241i	−0.848728	−	0.528829i	$-0.822631\pi$
$179$	0.449747	−	0.778985i	0.0336157	−	0.0582241i	0.882344	+	0.470605i	$0.155964\pi$
$180$		0			0
$181$	−7.65685			−0.569129			−0.284565	−	0.958657i	$-0.591849\pi$
$181$	−7.65685			−0.569129			−0.284565	+	0.958657i	$0.591849\pi$
$182$		0			0
$183$	28.5563			2.11095
$184$		0			0
$185$	−2.75736	+	4.77589i	−0.202725	+	0.351130i
$186$		0			0
$187$	1.82843	+	3.16693i	0.133708	+	0.231589i
$188$		0			0
$189$	0.671573	−	0.866025i	0.0488497	−	0.0629941i
$190$		0			0
$191$	−3.58579	−	6.21076i	−0.259458	−	0.449395i	0.706639	−	0.707575i	$-0.250211\pi$
$191$	−3.58579	−	6.21076i	−0.259458	−	0.449395i	−0.966097	+	0.258180i	$0.916877\pi$
$192$		0			0
$193$	−10.9497	+	18.9655i	−0.788180	+	1.36517i	0.138901	+	0.990306i	$0.455643\pi$
$193$	−10.9497	+	18.9655i	−0.788180	+	1.36517i	−0.927081	+	0.374862i	$0.877690\pi$
$194$		0			0
$195$	5.41421			0.387720
$196$		0			0
$197$	−0.514719			−0.0366722			−0.0183361	−	0.999832i	$-0.505837\pi$
$197$	−0.514719			−0.0366722			−0.0183361	+	0.999832i	$0.505837\pi$
$198$		0			0
$199$	0.0502525	−	0.0870399i	0.00356231	−	0.00617010i	−0.864239	−	0.503082i	$-0.832199\pi$
$199$	0.0502525	−	0.0870399i	0.00356231	−	0.00617010i	0.867801	+	0.496912i	$0.165533\pi$
$200$		0			0
$201$	3.32843	+	5.76500i	0.234769	+	0.406632i
$202$		0			0
$203$	−4.30761	+	5.55487i	−0.302335	+	0.389876i
$204$		0			0
$205$	−1.58579	−	2.74666i	−0.110756	−	0.191835i
$206$		0			0
$207$	−8.82843	+	15.2913i	−0.613618	+	1.06282i
$208$		0			0
$209$	−0.585786			−0.0405197
$210$		0			0
$211$	−7.41421			−0.510416			−0.255208	−	0.966886i	$-0.582144\pi$
$211$	−7.41421			−0.510416			−0.255208	+	0.966886i	$0.582144\pi$
$212$		0			0
$213$	−13.3640	+	23.1471i	−0.915684	+	1.58601i
$214$		0			0
$215$	1.65685	+	2.86976i	0.112997	+	0.195716i
$216$		0			0
$217$	−4.00000	−	9.79796i	−0.271538	−	0.665129i
$218$		0			0
$219$	11.3640	+	19.6830i	0.767905	+	1.33005i
$220$		0			0
$221$	7.00000	−	12.1244i	0.470871	−	0.815572i
$222$		0			0
$223$	−8.58579			−0.574947			−0.287473	−	0.957789i	$-0.592815\pi$
$223$	−8.58579			−0.574947			−0.287473	+	0.957789i	$0.592815\pi$
$224$		0			0
$225$	−13.1716			−0.878105
$226$		0			0
$227$	−14.4142	+	24.9662i	−0.956705	+	1.65706i	−0.226288	+	0.974061i	$0.572659\pi$
$227$	−14.4142	+	24.9662i	−0.956705	+	1.65706i	−0.730417	+	0.683001i	$0.760674\pi$
$228$		0			0
$229$	−11.6569	−	20.1903i	−0.770307	−	1.33421i	−0.937395	−	0.348268i	$-0.886770\pi$
$229$	−11.6569	−	20.1903i	−0.770307	−	1.33421i	0.167088	−	0.985942i	$-0.446564\pi$
$230$		0			0
$231$	−6.32843	−	0.866025i	−0.416380	−	0.0569803i
$232$		0			0
$233$	−0.707107	−	1.22474i	−0.0463241	−	0.0802357i	0.841934	−	0.539581i	$-0.181417\pi$
$233$	−0.707107	−	1.22474i	−0.0463241	−	0.0802357i	−0.888258	+	0.459345i	$0.848084\pi$
$234$		0			0
$235$	3.07107	−	5.31925i	0.200334	−	0.346989i
$236$		0			0
$237$	31.9706			2.07671
$238$		0			0
$239$	−20.2132			−1.30748			−0.653742	−	0.756718i	$-0.726802\pi$
$239$	−20.2132			−1.30748			−0.653742	+	0.756718i	$0.726802\pi$
$240$		0			0
$241$	−6.12132	+	10.6024i	−0.394309	+	0.682963i	−0.993013	−	0.118007i	$-0.962349\pi$
$241$	−6.12132	+	10.6024i	−0.394309	+	0.682963i	0.598704	+	0.800971i	$0.295683\pi$
$242$		0			0
$243$	10.8284	+	18.7554i	0.694644	+	1.20316i
$244$		0			0
$245$	2.92893	−	2.86976i	0.187123	−	0.183342i
$246$		0			0
$247$	1.12132	+	1.94218i	0.0713479	+	0.123578i
$248$		0			0
$249$	−14.6569	+	25.3864i	−0.928840	+	1.60880i
$250$		0			0
$251$	26.1421			1.65008			0.825038	−	0.565077i	$-0.191153\pi$
$251$	26.1421			1.65008			0.825038	+	0.565077i	$0.191153\pi$
$252$		0			0
$253$	−6.24264			−0.392471
$254$		0			0
$255$	2.58579	−	4.47871i	0.161928	−	0.280468i
$256$		0			0
$257$	12.5711	+	21.7737i	0.784162	+	1.35821i	0.929499	+	0.368826i	$0.120240\pi$
$257$	12.5711	+	21.7737i	0.784162	+	1.35821i	−0.145337	+	0.989382i	$0.546427\pi$
$258$		0			0
$259$	−24.6777	−	3.37706i	−1.53340	−	0.209840i
$260$		0			0
$261$	−3.75736	−	6.50794i	−0.232575	−	0.402831i
$262$		0			0
$263$	−8.52082	+	14.7585i	−0.525416	+	0.910047i	0.474146	+	0.880446i	$0.342757\pi$
$263$	−8.52082	+	14.7585i	−0.525416	+	0.910047i	−0.999562	+	0.0296008i	$0.990576\pi$
$264$		0			0
$265$	−4.62742			−0.284260
$266$		0			0
$267$	30.1421			1.84467
$268$		0			0
$269$	−1.17157	+	2.02922i	−0.0714321	+	0.123724i	−0.899529	−	0.436861i	$-0.856090\pi$
$269$	−1.17157	+	2.02922i	−0.0714321	+	0.123724i	0.828097	+	0.560585i	$0.189424\pi$
$270$		0			0
$271$	2.27817	+	3.94591i	0.138389	+	0.239697i	0.926887	−	0.375340i	$-0.122474\pi$
$271$	2.27817	+	3.94591i	0.138389	+	0.239697i	−0.788498	+	0.615038i	$0.789141\pi$
$272$		0			0
$273$	9.24264	+	22.6398i	0.559390	+	1.37022i
$274$		0			0
$275$	−2.32843	−	4.03295i	−0.140409	−	0.243196i
$276$		0			0
$277$	−0.914214	+	1.58346i	−0.0549298	+	0.0951412i	−0.892183	−	0.451675i	$-0.850827\pi$
$277$	−0.914214	+	1.58346i	−0.0549298	+	0.0951412i	0.837253	+	0.546816i	$0.184160\pi$
$278$		0			0
$279$	11.3137			0.677334
$280$		0			0
$281$	8.72792			0.520664			0.260332	−	0.965519i	$-0.416168\pi$
$281$	8.72792			0.520664			0.260332	+	0.965519i	$0.416168\pi$
$282$		0			0
$283$	−11.7071	+	20.2773i	−0.695915	+	1.20536i	0.273956	+	0.961742i	$0.411668\pi$
$283$	−11.7071	+	20.2773i	−0.695915	+	1.20536i	−0.969871	+	0.243618i	$0.921666\pi$
$284$		0			0
$285$	0.414214	+	0.717439i	0.0245359	+	0.0424974i
$286$		0			0
$287$	8.77817	−	11.3199i	0.518159	−	0.668191i
$288$		0			0
$289$	1.81371	+	3.14144i	0.106689	+	0.184790i
$290$		0			0
$291$	4.62132	−	8.00436i	0.270907	−	0.469224i
$292$		0			0
$293$	−10.8284			−0.632603			−0.316302	−	0.948659i	$-0.602441\pi$
$293$	−10.8284			−0.632603			−0.316302	+	0.948659i	$0.602441\pi$
$294$		0			0
$295$	−3.27208			−0.190508
$296$		0			0
$297$	−0.207107	+	0.358719i	−0.0120176	+	0.0208150i
$298$		0			0
$299$	11.9497	+	20.6976i	0.691072	+	1.19697i
$300$		0			0
$301$	−9.17157	+	11.8272i	−0.528641	+	0.681707i
$302$		0			0
$303$	−7.44975	−	12.9033i	−0.427977	−	0.741278i
$304$		0			0
$305$	3.46447	−	6.00063i	0.198375	−	0.343595i
$306$		0			0
$307$	−9.89949			−0.564994			−0.282497	−	0.959268i	$-0.591163\pi$
$307$	−9.89949			−0.564994			−0.282497	+	0.959268i	$0.591163\pi$
$308$		0			0
$309$	−32.3848			−1.84231
$310$		0			0
$311$	8.36396	−	14.4868i	0.474277	−	0.821471i	−0.525289	−	0.850924i	$-0.676043\pi$
$311$	8.36396	−	14.4868i	0.474277	−	0.821471i	0.999566	+	0.0294522i	$0.00937629\pi$
$312$		0			0
$313$	10.3284	+	17.8894i	0.583797	+	1.01117i	0.995024	+	0.0996335i	$0.0317670\pi$
$313$	10.3284	+	17.8894i	0.583797	+	1.01117i	−0.411227	+	0.911533i	$0.634900\pi$
$314$		0			0
$315$	1.65685	+	4.05845i	0.0933532	+	0.228668i
$316$		0			0
$317$	−4.34315	−	7.52255i	−0.243935	−	0.422508i	0.717896	−	0.696150i	$-0.245105\pi$
$317$	−4.34315	−	7.52255i	−0.243935	−	0.422508i	−0.961832	+	0.273642i	$0.911772\pi$
$318$		0			0
$319$	1.32843	−	2.30090i	0.0743776	−	0.128826i
$320$		0			0
$321$	7.41421			0.413821
$322$		0			0
$323$	2.14214			0.119192
$324$		0			0
$325$	−8.91421	+	15.4399i	−0.494472	+	0.856450i
$326$		0			0
$327$	−19.8995	−	34.4669i	−1.10044	−	1.90603i
$328$		0			0
$329$	27.4853	+	3.76127i	1.51531	+	0.207366i
$330$		0			0
$331$	−12.0355	−	20.8462i	−0.661533	−	1.14581i	−0.980213	−	0.197946i	$-0.936573\pi$
$331$	−12.0355	−	20.8462i	−0.661533	−	1.14581i	0.318680	−	0.947862i	$-0.396760\pi$
$332$		0			0
$333$	13.3137	−	23.0600i	0.729587	−	1.26368i
$334$		0			0
$335$	1.61522			0.0882491
$336$		0			0
$337$	−28.2426			−1.53847			−0.769237	−	0.638963i	$-0.779364\pi$
$337$	−28.2426			−1.53847			−0.769237	+	0.638963i	$0.779364\pi$
$338$		0			0
$339$	9.86396	−	17.0849i	0.535737	−	0.927923i
$340$		0			0
$341$	2.00000	+	3.46410i	0.108306	+	0.187592i
$342$		0			0
$343$	17.0000	+	7.34847i	0.917914	+	0.396780i
$344$		0			0
$345$	4.41421	+	7.64564i	0.237653	+	0.411628i
$346$		0			0
$347$	8.70711	−	15.0812i	0.467422	−	0.809599i	−0.531885	−	0.846816i	$-0.678516\pi$
$347$	8.70711	−	15.0812i	0.467422	−	0.809599i	0.999307	+	0.0372179i	$0.0118495\pi$
$348$		0			0
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$		0			0
$351$	1.58579			0.0846430
$352$		0			0
$353$	1.34315	−	2.32640i	0.0714884	−	0.123822i	−0.828065	−	0.560632i	$-0.810558\pi$
$353$	1.34315	−	2.32640i	0.0714884	−	0.123822i	0.899554	+	0.436810i	$0.143892\pi$
$354$		0			0
$355$	3.24264	+	5.61642i	0.172101	+	0.298089i
$356$		0			0
$357$	23.1421	+	3.16693i	1.22481	+	0.167612i
$358$		0			0
$359$	9.62132	+	16.6646i	0.507794	+	0.879525i	0.999959	+	0.00902308i	$0.00287218\pi$
$359$	9.62132	+	16.6646i	0.507794	+	0.879525i	−0.492165	+	0.870502i	$0.663794\pi$
$360$		0			0
$361$	9.32843	−	16.1573i	0.490970	−	0.850385i
$362$		0			0
$363$	2.41421			0.126713
$364$		0			0
$365$	5.51472			0.288654
$366$		0			0
$367$	−5.36396	+	9.29065i	−0.279996	+	0.484968i	−0.971384	−	0.237516i	$-0.923667\pi$
$367$	−5.36396	+	9.29065i	−0.279996	+	0.484968i	0.691387	+	0.722485i	$0.257000\pi$
$368$		0			0
$369$	7.65685	+	13.2621i	0.398600	+	0.690395i
$370$		0			0
$371$	−7.89949	−	19.3497i	−0.410121	−	1.00459i
$372$		0			0
$373$	9.98528	+	17.2950i	0.517018	+	0.895502i	0.999805	+	0.0197638i	$0.00629142\pi$
$373$	9.98528	+	17.2950i	0.517018	+	0.895502i	−0.482786	+	0.875738i	$0.660375\pi$
$374$		0			0
$375$	−6.82843	+	11.8272i	−0.352618	+	0.610753i
$376$		0			0
$377$	−10.1716			−0.523863
$378$		0			0
$379$	−27.8701			−1.43159			−0.715794	−	0.698311i	$-0.753935\pi$
$379$	−27.8701			−1.43159			−0.715794	+	0.698311i	$0.753935\pi$
$380$		0			0
$381$	18.9853	−	32.8835i	0.972645	−	1.68467i
$382$		0			0
$383$	−3.19239	−	5.52938i	−0.163123	−	0.282538i	0.772864	−	0.634572i	$-0.218824\pi$
$383$	−3.19239	−	5.52938i	−0.163123	−	0.282538i	−0.935987	+	0.352034i	$0.885490\pi$
$384$		0			0
$385$	−0.949747	+	1.22474i	−0.0484036	+	0.0624188i
$386$		0			0
$387$	−8.00000	−	13.8564i	−0.406663	−	0.704361i
$388$		0			0
$389$	11.3640	−	19.6830i	0.576176	−	0.997966i	−0.419737	−	0.907646i	$-0.637878\pi$
$389$	11.3640	−	19.6830i	0.576176	−	0.997966i	0.995913	−	0.0903199i	$-0.0287889\pi$
$390$		0			0
$391$	22.8284			1.15448
$392$		0			0
$393$	1.41421			0.0713376
$394$		0			0
$395$	3.87868	−	6.71807i	0.195158	−	0.338023i
$396$		0			0
$397$	−11.0000	−	19.0526i	−0.552074	−	0.956221i	−0.998125	−	0.0612128i	$-0.980503\pi$
$397$	−11.0000	−	19.0526i	−0.552074	−	0.956221i	0.446051	−	0.895008i	$-0.352830\pi$
$398$		0			0
$399$	−2.29289	+	2.95680i	−0.114788	+	0.148025i
$400$		0			0
$401$	−9.15685	−	15.8601i	−0.457271	−	0.792017i	0.541544	−	0.840672i	$-0.317840\pi$
$401$	−9.15685	−	15.8601i	−0.457271	−	0.792017i	−0.998816	+	0.0486549i	$0.984507\pi$
$402$		0			0
$403$	7.65685	−	13.2621i	0.381415	−	0.660630i
$404$		0			0
$405$	5.55635			0.276097
$406$		0			0
$407$	9.41421			0.466645
$408$		0			0
$409$	−1.36396	+	2.36245i	−0.0674435	+	0.116816i	−0.897775	−	0.440454i	$-0.854818\pi$
$409$	−1.36396	+	2.36245i	−0.0674435	+	0.116816i	0.830332	+	0.557269i	$0.188151\pi$
$410$		0			0
$411$	20.1066	+	34.8257i	0.991786	+	1.71782i
$412$		0			0
$413$	−5.58579	−	13.6823i	−0.274859	−	0.673263i
$414$		0			0
$415$	3.55635	+	6.15978i	0.174574	+	0.302372i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	26.1421			1.27713			0.638563	−	0.769569i	$-0.279529\pi$
$419$	26.1421			1.27713			0.638563	+	0.769569i	$0.279529\pi$
$420$		0			0
$421$	0.686292			0.0334478			0.0167239	−	0.999860i	$-0.494676\pi$
$421$	0.686292			0.0334478			0.0167239	+	0.999860i	$0.494676\pi$
$422$		0			0
$423$	−14.8284	+	25.6836i	−0.720983	+	1.24878i
$424$		0			0
$425$	8.51472	+	14.7479i	0.413025	+	0.715379i
$426$		0			0
$427$	31.0061	+	4.24309i	1.50049	+	0.205337i
$428$		0			0
$429$	−4.62132	−	8.00436i	−0.223119	−	0.386454i
$430$		0			0
$431$	−8.79289	+	15.2297i	−0.423539	+	0.733591i	−0.996283	−	0.0861437i	$-0.972546\pi$
$431$	−8.79289	+	15.2297i	−0.423539	+	0.733591i	0.572744	+	0.819734i	$0.305879\pi$
$432$		0			0
$433$	26.1421			1.25631			0.628155	−	0.778088i	$-0.283810\pi$
$433$	26.1421			1.25631			0.628155	+	0.778088i	$0.283810\pi$
$434$		0			0
$435$	−3.75736			−0.180152
$436$		0			0
$437$	−1.82843	+	3.16693i	−0.0874655	+	0.151495i
$438$		0			0
$439$	−13.6924	−	23.7159i	−0.653502	−	1.13190i	−0.982267	−	0.187487i	$-0.939966\pi$
$439$	−13.6924	−	23.7159i	−0.653502	−	1.13190i	0.328765	−	0.944412i	$-0.393368\pi$
$440$		0			0
$441$	−14.1421	+	13.8564i	−0.673435	+	0.659829i
$442$		0			0
$443$	6.31371	+	10.9357i	0.299973	+	0.519569i	0.976130	−	0.217189i	$-0.0696889\pi$
$443$	6.31371	+	10.9357i	0.299973	+	0.519569i	−0.676156	+	0.736758i	$0.736356\pi$
$444$		0			0
$445$	3.65685	−	6.33386i	0.173352	−	0.300254i
$446$		0			0
$447$	42.6274			2.01621
$448$		0			0
$449$	22.3431			1.05444			0.527219	−	0.849729i	$-0.323235\pi$
$449$	22.3431			1.05444			0.527219	+	0.849729i	$0.323235\pi$
$450$		0			0
$451$	−2.70711	+	4.68885i	−0.127473	+	0.220789i
$452$		0			0
$453$	−18.9853	−	32.8835i	−0.892006	−	1.54500i
$454$		0			0
$455$	5.87868	+	0.804479i	0.275597	+	0.0377146i
$456$		0			0
$457$	5.82843	+	10.0951i	0.272642	+	0.472230i	0.969538	−	0.244943i	$-0.0787691\pi$
$457$	5.82843	+	10.0951i	0.272642	+	0.472230i	−0.696895	+	0.717173i	$0.745436\pi$
$458$		0			0
$459$	0.757359	−	1.31178i	0.0353505	−	0.0612289i
$460$		0			0
$461$	−8.31371			−0.387208			−0.193604	−	0.981080i	$-0.562018\pi$
$461$	−8.31371			−0.387208			−0.193604	+	0.981080i	$0.562018\pi$
$462$		0			0
$463$	−12.8284			−0.596188			−0.298094	−	0.954537i	$-0.596351\pi$
$463$	−12.8284			−0.596188			−0.298094	+	0.954537i	$0.596351\pi$
$464$		0			0
$465$	2.82843	−	4.89898i	0.131165	−	0.227185i
$466$		0			0
$467$	−17.0000	−	29.4449i	−0.786666	−	1.36255i	−0.927999	−	0.372584i	$-0.878472\pi$
$467$	−17.0000	−	29.4449i	−0.786666	−	1.36255i	0.141332	−	0.989962i	$-0.454861\pi$
$468$		0			0
$469$	2.75736	+	6.75412i	0.127323	+	0.311876i
$470$		0			0
$471$	−21.3137	−	36.9164i	−0.982084	−	1.70102i
$472$		0			0
$473$	2.82843	−	4.89898i	0.130051	−	0.225255i
$474$		0			0
$475$	−2.72792			−0.125166
$476$		0			0
$477$	22.3431			1.02302
$478$		0			0
$479$	−5.96447	+	10.3308i	−0.272523	+	0.472024i	−0.969507	−	0.245062i	$-0.921192\pi$
$479$	−5.96447	+	10.3308i	−0.272523	+	0.472024i	0.696984	+	0.717087i	$0.254525\pi$
$480$		0			0
$481$	−18.0208	−	31.2130i	−0.821678	−	1.42319i
$482$		0			0
$483$	−24.4350	+	31.5101i	−1.11183	+	1.43376i
$484$		0			0
$485$	−1.12132	−	1.94218i	−0.0509165	−	0.0881900i
$486$		0			0
$487$	−4.82843	+	8.36308i	−0.218797	+	0.378967i	−0.954440	−	0.298402i	$-0.903547\pi$
$487$	−4.82843	+	8.36308i	−0.218797	+	0.378967i	0.735644	+	0.677369i	$0.236880\pi$
$488$		0			0
$489$	−23.4853			−1.06204
$490$		0			0
$491$	−19.1716			−0.865201			−0.432600	−	0.901586i	$-0.642404\pi$
$491$	−19.1716			−0.865201			−0.432600	+	0.901586i	$0.642404\pi$
$492$		0			0
$493$	−4.85786	+	8.41407i	−0.218787	+	0.378951i
$494$		0			0
$495$	−0.828427	−	1.43488i	−0.0372350	−	0.0644930i
$496$		0			0
$497$	−17.9497	+	23.1471i	−0.805156	+	1.03829i
$498$		0			0
$499$	11.0711	+	19.1757i	0.495609	+	0.858420i	0.999987	−	0.00506282i	$-0.00161155\pi$
$499$	11.0711	+	19.1757i	0.495609	+	0.858420i	−0.504378	+	0.863483i	$0.668278\pi$
$500$		0			0
$501$	16.5711	−	28.7019i	0.740341	−	1.28231i
$502$		0			0
$503$	−4.21320			−0.187857			−0.0939287	−	0.995579i	$-0.529943\pi$
$503$	−4.21320			−0.187857			−0.0939287	+	0.995579i	$0.529943\pi$
$504$		0			0
$505$	−3.61522			−0.160875
$506$		0			0
$507$	−2.00000	+	3.46410i	−0.0888231	+	0.153846i
$508$		0			0
$509$	−4.65685	−	8.06591i	−0.206411	−	0.357515i	0.744170	−	0.667990i	$-0.232845\pi$
$509$	−4.65685	−	8.06591i	−0.206411	−	0.357515i	−0.950582	+	0.310475i	$0.899512\pi$
$510$		0			0
$511$	9.41421	+	23.0600i	0.416460	+	1.02012i
$512$		0			0
$513$	0.121320	+	0.210133i	0.00535642	+	0.00927760i
$514$		0			0
$515$	−3.92893	+	6.80511i	−0.173129	+	0.299869i
$516$		0			0
$517$	−10.4853			−0.461142
$518$		0			0
$519$	−23.7279			−1.04154
$520$		0			0
$521$	14.1421	−	24.4949i	0.619578	−	1.07314i	−0.369984	−	0.929038i	$-0.620637\pi$
$521$	14.1421	−	24.4949i	0.619578	−	1.07314i	0.989563	−	0.144103i	$-0.0460297\pi$
$522$		0			0
$523$	−6.36396	−	11.0227i	−0.278277	−	0.481989i	0.692680	−	0.721245i	$-0.256430\pi$
$523$	−6.36396	−	11.0227i	−0.278277	−	0.481989i	−0.970957	+	0.239256i	$0.923097\pi$
$524$		0			0
$525$	−29.4706	−	4.03295i	−1.28620	−	0.176013i
$526$		0			0
$527$	−7.31371	−	12.6677i	−0.318590	−	0.551814i
$528$		0			0
$529$	−7.98528	+	13.8309i	−0.347186	+	0.601344i
$530$		0			0
$531$	15.7990			0.685618
$532$		0			0
$533$	20.7279			0.897826
$534$		0			0
$535$	0.899495	−	1.55797i	0.0388886	−	0.0673570i
$536$		0			0
$537$	1.08579	+	1.88064i	0.0468551	+	0.0811555i
$538$		0			0
$539$	−6.74264	−	1.88064i	−0.290426	−	0.0810048i
$540$		0			0
$541$	6.42893	+	11.1352i	0.276401	+	0.478741i	0.970488	−	0.241151i	$-0.0775248\pi$
$541$	6.42893	+	11.1352i	0.276401	+	0.478741i	−0.694086	+	0.719892i	$0.744191\pi$
$542$		0			0
$543$	9.24264	−	16.0087i	0.396640	−	0.687000i
$544$		0			0
$545$	−9.65685			−0.413654
$546$		0			0
$547$	−34.8701			−1.49094			−0.745468	−	0.666541i	$-0.767774\pi$
$547$	−34.8701			−1.49094			−0.745468	+	0.666541i	$0.767774\pi$
$548$		0			0
$549$	−16.7279	+	28.9736i	−0.713930	+	1.23656i
$550$		0			0
$551$	−0.778175	−	1.34784i	−0.0331514	−	0.0574198i
$552$		0			0
$553$	34.7132	+	4.75039i	1.47616	+	0.202007i
$554$		0			0
$555$	−6.65685	−	11.5300i	−0.282568	−	0.489422i
$556$		0			0
$557$	−3.75736	+	6.50794i	−0.159204	+	0.275750i	−0.934582	−	0.355748i	$-0.884226\pi$
$557$	−3.75736	+	6.50794i	−0.159204	+	0.275750i	0.775378	+	0.631498i	$0.217560\pi$
$558$		0			0
$559$	−21.6569			−0.915987
$560$		0			0
$561$	−8.82843			−0.372736
$562$		0			0
$563$	4.53553	−	7.85578i	0.191150	−	0.331081i	−0.754482	−	0.656321i	$-0.772112\pi$
$563$	4.53553	−	7.85578i	0.191150	−	0.331081i	0.945632	+	0.325240i	$0.105445\pi$
$564$		0			0
$565$	−2.39340	−	4.14549i	−0.100691	−	0.174402i
$566$		0			0
$567$	9.48528	+	23.2341i	0.398344	+	0.975740i
$568$		0			0
$569$	2.00000	+	3.46410i	0.0838444	+	0.145223i	0.904898	−	0.425628i	$-0.139947\pi$
$569$	2.00000	+	3.46410i	0.0838444	+	0.145223i	−0.821054	+	0.570851i	$0.806613\pi$
$570$		0			0
$571$	13.1924	−	22.8499i	0.552084	−	0.956238i	−0.446040	−	0.895013i	$-0.647166\pi$
$571$	13.1924	−	22.8499i	0.552084	−	0.956238i	0.998124	−	0.0612248i	$-0.0195007\pi$
$572$		0			0
$573$	17.3137			0.723291
$574$		0			0
$575$	−29.0711			−1.21235
$576$		0			0
$577$	4.84315	−	8.38857i	0.201623	−	0.349221i	−0.747429	−	0.664342i	$-0.768712\pi$
$577$	4.84315	−	8.38857i	0.201623	−	0.349221i	0.949051	+	0.315121i	$0.102045\pi$
$578$		0			0
$579$	−26.4350	−	45.7868i	−1.09860	−	1.90284i
$580$		0			0
$581$	−19.6863	+	25.3864i	−0.816725	+	1.05321i
$582$		0			0
$583$	3.94975	+	6.84116i	0.163582	+	0.283332i
$584$		0			0
$585$	−3.17157	+	5.49333i	−0.131128	+	0.227121i
$586$		0			0
$587$	25.1005			1.03601			0.518004	−	0.855378i	$-0.326675\pi$
$587$	25.1005			1.03601			0.518004	+	0.855378i	$0.326675\pi$
$588$		0			0
$589$	2.34315			0.0965476
$590$		0			0
$591$	0.621320	−	1.07616i	0.0255577	−	0.0442672i
$592$		0			0
$593$	−11.8492	−	20.5235i	−0.486590	−	0.842799i	0.513291	−	0.858215i	$-0.328426\pi$
$593$	−11.8492	−	20.5235i	−0.486590	−	0.842799i	−0.999881	+	0.0154159i	$0.995093\pi$
$594$		0			0
$595$	3.47309	−	4.47871i	0.142383	−	0.183609i
$596$		0			0
$597$	0.121320	+	0.210133i	0.00496531	+	0.00860017i
$598$		0			0
$599$	21.3137	−	36.9164i	0.870855	−	1.50836i	0.00974040	−	0.999953i	$-0.496899\pi$
$599$	21.3137	−	36.9164i	0.870855	−	1.50836i	0.861114	−	0.508412i	$-0.169767\pi$
$600$		0			0
$601$	31.9411			1.30291			0.651453	−	0.758689i	$-0.274160\pi$
$601$	31.9411			1.30291			0.651453	+	0.758689i	$0.274160\pi$
$602$		0			0
$603$	−7.79899			−0.317599
$604$		0			0
$605$	0.292893	−	0.507306i	0.0119078	−	0.0206249i
$606$		0			0
$607$	−21.4853	−	37.2136i	−0.872061	−	1.51045i	−0.859861	−	0.510528i	$-0.829450\pi$
$607$	−21.4853	−	37.2136i	−0.872061	−	1.51045i	−0.0121994	−	0.999926i	$-0.503883\pi$
$608$		0			0
$609$	−6.41421	−	15.7116i	−0.259917	−	0.636664i
$610$		0			0
$611$	20.0711	+	34.7641i	0.811988	+	1.40641i
$612$		0			0
$613$	−14.3137	+	24.7921i	−0.578125	+	1.00134i	0.417569	+	0.908645i	$0.362882\pi$
$613$	−14.3137	+	24.7921i	−0.578125	+	1.00134i	−0.995694	+	0.0926971i	$0.970451\pi$
$614$		0			0
$615$	7.65685			0.308754
$616$		0			0
$617$	−41.9706			−1.68967			−0.844836	−	0.535026i	$-0.820302\pi$
$617$	−41.9706			−1.68967			−0.844836	+	0.535026i	$0.820302\pi$
$618$		0			0
$619$	−10.9706	+	19.0016i	−0.440944	+	0.763738i	−0.997760	−	0.0668984i	$-0.978690\pi$
$619$	−10.9706	+	19.0016i	−0.440944	+	0.763738i	0.556816	+	0.830636i	$0.312023\pi$
$620$		0			0
$621$	1.29289	+	2.23936i	0.0518820	+	0.0898623i
$622$		0			0
$623$	32.7279	+	4.47871i	1.31122	+	0.179436i
$624$		0			0
$625$	−9.98528	−	17.2950i	−0.399411	−	0.691801i
$626$		0			0
$627$	0.707107	−	1.22474i	0.0282391	−	0.0489116i
$628$		0			0
$629$	−34.4264			−1.37267
$630$		0			0
$631$	−23.2721			−0.926447			−0.463223	−	0.886242i	$-0.653307\pi$
$631$	−23.2721			−0.926447			−0.463223	+	0.886242i	$0.653307\pi$
$632$		0			0
$633$	8.94975	−	15.5014i	0.355721	−	0.616126i
$634$		0			0
$635$	−4.60660	−	7.97887i	−0.182807	−	0.316632i
$636$		0			0
$637$	6.67157	+	25.9553i	0.264337	+	1.02839i
$638$		0			0
$639$	−15.6569	−	27.1185i	−0.619376	−	1.07279i
$640$		0			0
$641$	−7.64214	+	13.2366i	−0.301846	+	0.522813i	−0.976554	−	0.215272i	$-0.930936\pi$
$641$	−7.64214	+	13.2366i	−0.301846	+	0.522813i	0.674708	+	0.738085i	$0.264270\pi$
$642$		0			0
$643$	−1.58579			−0.0625373			−0.0312687	−	0.999511i	$-0.509955\pi$
$643$	−1.58579			−0.0625373			−0.0312687	+	0.999511i	$0.509955\pi$
$644$		0			0
$645$	−8.00000			−0.315000
$646$		0			0
$647$	15.0919	−	26.1399i	0.593323	−	1.02767i	−0.400458	−	0.916315i	$-0.631149\pi$
$647$	15.0919	−	26.1399i	0.593323	−	1.02767i	0.993781	−	0.111351i	$-0.0355177\pi$
$648$		0			0
$649$	2.79289	+	4.83743i	0.109631	+	0.189886i
$650$		0			0
$651$	25.3137	+	3.46410i	0.992122	+	0.135769i
$652$		0			0
$653$	9.19239	+	15.9217i	0.359726	+	0.623064i	0.987915	−	0.154997i	$-0.0495369\pi$
$653$	9.19239	+	15.9217i	0.359726	+	0.623064i	−0.628189	+	0.778061i	$0.716204\pi$
$654$		0			0
$655$	0.171573	−	0.297173i	0.00670391	−	0.0116115i
$656$		0			0
$657$	−26.6274			−1.03883
$658$		0			0
$659$	12.0000			0.467454			0.233727	−	0.972302i	$-0.424908\pi$
$659$	12.0000			0.467454			0.233727	+	0.972302i	$0.424908\pi$
$660$		0			0
$661$	−9.48528	+	16.4290i	−0.368935	+	0.639014i	−0.989399	−	0.145221i	$-0.953611\pi$
$661$	−9.48528	+	16.4290i	−0.368935	+	0.639014i	0.620465	+	0.784234i	$0.286944\pi$
$662$		0			0
$663$	16.8995	+	29.2708i	0.656322	+	1.13678i
$664$		0			0
$665$	0.343146	+	0.840532i	0.0133066	+	0.0325944i
$666$		0			0
$667$	−8.29289	−	14.3637i	−0.321102	−	0.556165i
$668$		0			0
$669$	10.3640	−	17.9509i	0.400694	−	0.694022i
$670$		0			0
$671$	−11.8284			−0.456631
$672$		0			0
$673$	5.55635			0.214182			0.107091	−	0.994249i	$-0.465846\pi$
$673$	5.55635			0.214182			0.107091	+	0.994249i	$0.465846\pi$
$674$		0			0
$675$	−0.964466	+	1.67050i	−0.0371223	+	0.0642977i
$676$		0			0
$677$	17.6569	+	30.5826i	0.678608	+	1.17538i	0.975400	+	0.220441i	$0.0707498\pi$
$677$	17.6569	+	30.5826i	0.678608	+	1.17538i	−0.296792	+	0.954942i	$0.595917\pi$
$678$		0			0
$679$	6.20711	−	8.00436i	0.238207	−	0.307179i
$680$		0			0
$681$	−34.7990	−	60.2736i	−1.33350	−	2.30969i
$682$		0			0
$683$	6.79289	−	11.7656i	0.259923	−	0.450200i	−0.706298	−	0.707914i	$-0.749636\pi$
$683$	6.79289	−	11.7656i	0.259923	−	0.450200i	0.966221	+	0.257715i	$0.0829695\pi$
$684$		0			0
$685$	9.75736			0.372810
$686$		0			0
$687$	56.2843			2.14738
$688$		0			0
$689$	15.1213	−	26.1909i	0.576076	−	0.997794i
$690$		0			0
$691$	14.0355	+	24.3103i	0.533937	+	0.924806i	0.999214	+	0.0396407i	$0.0126213\pi$
$691$	14.0355	+	24.3103i	0.533937	+	0.924806i	−0.465277	+	0.885165i	$0.654045\pi$
$692$		0			0
$693$	4.58579	−	5.91359i	0.174200	−	0.224639i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	9.89949	−	17.1464i	0.374970	−	0.649467i
$698$		0			0
$699$	3.41421			0.129137
$700$		0			0
$701$	−26.1127			−0.986263			−0.493132	−	0.869955i	$-0.664148\pi$
$701$	−26.1127			−0.986263			−0.493132	+	0.869955i	$0.664148\pi$
$702$		0			0
$703$	2.75736	−	4.77589i	0.103996	−	0.180126i
$704$		0			0
$705$	7.41421	+	12.8418i	0.279235	+	0.483650i
$706$		0			0
$707$	−6.17157	−	15.1172i	−0.232106	−	0.568541i
$708$		0			0
$709$	6.02082	+	10.4284i	0.226116	+	0.391645i	0.956654	−	0.291228i	$-0.0940637\pi$
$709$	6.02082	+	10.4284i	0.226116	+	0.391645i	−0.730537	+	0.682873i	$0.760730\pi$
$710$		0			0
$711$	−18.7279	+	32.4377i	−0.702352	+	1.21651i
$712$		0			0
$713$	24.9706			0.935155
$714$		0			0
$715$	−2.24264			−0.0838700
$716$		0			0
$717$	24.3995	−	42.2612i	0.911216	−	1.57827i
$718$		0			0
$719$	−0.757359	−	1.31178i	−0.0282447	−	0.0489213i	0.851558	−	0.524261i	$-0.175658\pi$
$719$	−0.757359	−	1.31178i	−0.0282447	−	0.0489213i	−0.879802	+	0.475340i	$0.842325\pi$
$720$		0			0
$721$	−35.1630	−	4.81194i	−1.30954	−	0.179206i
$722$		0			0
$723$	−14.7782	−	25.5965i	−0.549606	−	0.951946i
$724$		0			0
$725$	6.18629	−	10.7150i	0.229753	−	0.397944i
$726$		0			0
$727$	−36.4264			−1.35098			−0.675490	−	0.737369i	$-0.736068\pi$
$727$	−36.4264			−1.35098			−0.675490	+	0.737369i	$0.736068\pi$
$728$		0			0
$729$	−23.8284			−0.882534
$730$		0			0
$731$	−10.3431	+	17.9149i	−0.382555	+	0.662605i
$732$		0			0
$733$	6.50000	+	11.2583i	0.240083	+	0.415836i	0.960738	−	0.277458i	$-0.0894920\pi$
$733$	6.50000	+	11.2583i	0.240083	+	0.415836i	−0.720655	+	0.693294i	$0.756159\pi$
$734$		0			0
$735$	2.46447	+	9.58783i	0.0909032	+	0.353652i
$736$		0			0
$737$	−1.37868	−	2.38794i	−0.0507843	−	0.0879610i
$738$		0			0
$739$	−26.2132	+	45.4026i	−0.964268	+	1.67016i	−0.252700	+	0.967545i	$0.581319\pi$
$739$	−26.2132	+	45.4026i	−0.964268	+	1.67016i	−0.711568	+	0.702617i	$0.752015\pi$
$740$		0			0
$741$	−5.41421			−0.198896
$742$		0			0
$743$	13.3137			0.488433			0.244216	−	0.969721i	$-0.421469\pi$
$743$	13.3137			0.488433			0.244216	+	0.969721i	$0.421469\pi$
$744$		0			0
$745$	5.17157	−	8.95743i	0.189472	−	0.328175i
$746$		0			0
$747$	−17.1716	−	29.7420i	−0.628275	−	1.08820i
$748$		0			0
$749$	8.05025	+	1.10165i	0.294150	+	0.0402535i
$750$		0			0
$751$	22.8284	+	39.5400i	0.833021	+	1.44283i	0.895631	+	0.444797i	$0.146724\pi$
$751$	22.8284	+	39.5400i	0.833021	+	1.44283i	−0.0626103	+	0.998038i	$0.519943\pi$
$752$		0			0
$753$	−31.5563	+	54.6572i	−1.14998	+	1.99182i
$754$		0			0
$755$	−9.21320			−0.335303
$756$		0			0
$757$	8.34315			0.303237			0.151618	−	0.988439i	$-0.451552\pi$
$757$	8.34315			0.303237			0.151618	+	0.988439i	$0.451552\pi$
$758$		0			0
$759$	7.53553	−	13.0519i	0.273523	−	0.473755i
$760$		0			0
$761$	−15.4853	−	26.8213i	−0.561341	−	0.972271i	−0.997380	−	0.0723433i	$-0.976952\pi$
$761$	−15.4853	−	26.8213i	−0.561341	−	0.972271i	0.436039	−	0.899928i	$-0.356381\pi$
$762$		0			0
$763$	−16.4853	−	40.3805i	−0.596807	−	1.46187i
$764$		0			0
$765$	3.02944	+	5.24714i	0.109530	+	0.189711i
$766$		0			0
$767$	10.6924	−	18.5198i	0.386080	−	0.668710i
$768$		0			0
$769$	22.9706			0.828340			0.414170	−	0.910200i	$-0.364072\pi$
$769$	22.9706			0.828340			0.414170	+	0.910200i	$0.364072\pi$
$770$		0			0
$771$	−60.6985			−2.18600
$772$		0			0
$773$	10.9706	−	19.0016i	0.394584	−	0.683439i	−0.598464	−	0.801150i	$-0.704222\pi$
$773$	10.9706	−	19.0016i	0.394584	−	0.683439i	0.993048	+	0.117710i	$0.0375555\pi$
$774$		0			0
$775$	9.31371	+	16.1318i	0.334558	+	0.579472i
$776$		0			0
$777$	36.8492	−	47.5189i	1.32196	−	1.70473i
$778$		0			0
$779$	1.58579	+	2.74666i	0.0568167	+	0.0984094i
$780$		0			0
$781$	5.53553	−	9.58783i	0.198077	−	0.343079i
$782$		0			0
$783$	−1.10051			−0.0393288
$784$		0			0
$785$	−10.3431			−0.369163
$786$		0			0
$787$	6.77817	−	11.7401i	0.241616	−	0.418491i	−0.719559	−	0.694431i	$-0.755656\pi$
$787$	6.77817	−	11.7401i	0.241616	−	0.418491i	0.961175	+	0.275941i	$0.0889893\pi$
$788$		0			0
$789$	−20.5711	−	35.6301i	−0.732349	−	1.26847i
$790$		0			0
$791$	13.2487	−	17.0849i	0.471071

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:	\( N \)	\(=\)	\( 1232 = 2^{4} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1232.q (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.20711 + 2.09077i) q^{3} +(0.292893 + 0.507306i) q^{5} +(-1.62132 + 2.09077i) q^{7} +(-1.41421 - 2.44949i) q^{9} +O(q^{10})\)	\(q+(-1.20711 + 2.09077i) q^{3} +(0.292893 + 0.507306i) q^{5} +(-1.62132 + 2.09077i) q^{7} +(-1.41421 - 2.44949i) q^{9} +(0.500000 - 0.866025i) q^{11} -3.82843 q^{13} -1.41421 q^{15} +(-1.82843 + 3.16693i) q^{17} +(-0.292893 - 0.507306i) q^{19} +(-2.41421 - 5.91359i) q^{21} +(-3.12132 - 5.40629i) q^{23} +(2.32843 - 4.03295i) q^{25} -0.414214 q^{27} +2.65685 q^{29} +(-2.00000 + 3.46410i) q^{31} +(1.20711 + 2.09077i) q^{33} +(-1.53553 - 0.210133i) q^{35} +(4.70711 + 8.15295i) q^{37} +(4.62132 - 8.00436i) q^{39} -5.41421 q^{41} +5.65685 q^{43} +(0.828427 - 1.43488i) q^{45} +(-5.24264 - 9.08052i) q^{47} +(-1.74264 - 6.77962i) q^{49} +(-4.41421 - 7.64564i) q^{51} +(-3.94975 + 6.84116i) q^{53} +0.585786 q^{55} +1.41421 q^{57} +(-2.79289 + 4.83743i) q^{59} +(-5.91421 - 10.2437i) q^{61} +(7.41421 + 1.01461i) q^{63} +(-1.12132 - 1.94218i) q^{65} +(1.37868 - 2.38794i) q^{67} +15.0711 q^{69} +11.0711 q^{71} +(4.70711 - 8.15295i) q^{73} +(5.62132 + 9.73641i) q^{75} +(1.00000 + 2.44949i) q^{77} +(-6.62132 - 11.4685i) q^{79} +(4.74264 - 8.21449i) q^{81} +12.1421 q^{83} -2.14214 q^{85} +(-3.20711 + 5.55487i) q^{87} +(-6.24264 - 10.8126i) q^{89} +(6.20711 - 8.00436i) q^{91} +(-4.82843 - 8.36308i) q^{93} +(0.171573 - 0.297173i) q^{95} -3.82843 q^{97} -2.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{3} + 4 q^{5} + 2 q^{7}+O(q^{10}) \) 4 * q - 2 * q^3 + 4 * q^5 + 2 * q^7	\( 4 q - 2 q^{3} + 4 q^{5} + 2 q^{7} + 2 q^{11} - 4 q^{13} + 4 q^{17} - 4 q^{19} - 4 q^{21} - 4 q^{23} - 2 q^{25} + 4 q^{27} - 12 q^{29} - 8 q^{31} + 2 q^{33} + 8 q^{35} + 16 q^{37} + 10 q^{39} - 16 q^{41} - 8 q^{45} - 4 q^{47} + 10 q^{49} - 12 q^{51} + 4 q^{53} + 8 q^{55} - 14 q^{59} - 18 q^{61} + 24 q^{63} + 4 q^{65} + 14 q^{67} + 32 q^{69} + 16 q^{71} + 16 q^{73} + 14 q^{75} + 4 q^{77} - 18 q^{79} + 2 q^{81} - 8 q^{83} + 48 q^{85} - 10 q^{87} - 8 q^{89} + 22 q^{91} - 8 q^{93} + 12 q^{95} - 4 q^{97}+O(q^{100}) \) 4 * q - 2 * q^3 + 4 * q^5 + 2 * q^7 + 2 * q^11 - 4 * q^13 + 4 * q^17 - 4 * q^19 - 4 * q^21 - 4 * q^23 - 2 * q^25 + 4 * q^27 - 12 * q^29 - 8 * q^31 + 2 * q^33 + 8 * q^35 + 16 * q^37 + 10 * q^39 - 16 * q^41 - 8 * q^45 - 4 * q^47 + 10 * q^49 - 12 * q^51 + 4 * q^53 + 8 * q^55 - 14 * q^59 - 18 * q^61 + 24 * q^63 + 4 * q^65 + 14 * q^67 + 32 * q^69 + 16 * q^71 + 16 * q^73 + 14 * q^75 + 4 * q^77 - 18 * q^79 + 2 * q^81 - 8 * q^83 + 48 * q^85 - 10 * q^87 - 8 * q^89 + 22 * q^91 - 8 * q^93 + 12 * q^95 - 4 * q^97

Introduction

L-functions

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