LMFDB - Embedded newform 1232.2.q.f.529.2

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			529.2
Root			$0.707107 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	1232.529
Dual form			1232.2.q.f.177.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.207107 - 0.358719i) q^{3} +(1.70711 + 2.95680i) q^{5} +(2.62132 - 0.358719i) q^{7} +(1.41421 + 2.44949i) q^{9} +O(q^{10})$	\(q+(0.207107 - 0.358719i) q^{3} +(1.70711 + 2.95680i) q^{5} +(2.62132 - 0.358719i) q^{7} +(1.41421 + 2.44949i) q^{9} +(0.500000 - 0.866025i) q^{11} +1.82843 q^{13} +1.41421 q^{15} +(3.82843 - 6.63103i) q^{17} +(-1.70711 - 2.95680i) q^{19} +(0.414214 - 1.01461i) q^{21} +(1.12132 + 1.94218i) q^{23} +(-3.32843 + 5.76500i) q^{25} +2.41421 q^{27} -8.65685 q^{29} +(-2.00000 + 3.46410i) q^{31} +(-0.207107 - 0.358719i) q^{33} +(5.53553 + 7.13834i) q^{35} +(3.29289 + 5.70346i) q^{37} +(0.378680 - 0.655892i) q^{39} -2.58579 q^{41} -5.65685 q^{43} +(-4.82843 + 8.36308i) q^{45} +(3.24264 + 5.61642i) q^{47} +(6.74264 - 1.88064i) q^{49} +(-1.58579 - 2.74666i) q^{51} +(5.94975 - 10.3053i) q^{53} +3.41421 q^{55} -1.41421 q^{57} +(-4.20711 + 7.28692i) q^{59} +(-3.08579 - 5.34474i) q^{61} +(4.58579 + 5.91359i) q^{63} +(3.12132 + 5.40629i) q^{65} +(5.62132 - 9.73641i) q^{67} +0.928932 q^{69} -3.07107 q^{71} +(3.29289 - 5.70346i) q^{73} +(1.37868 + 2.38794i) q^{75} +(1.00000 - 2.44949i) q^{77} +(-2.37868 - 4.11999i) q^{79} +(-3.74264 + 6.48244i) q^{81} -16.1421 q^{83} +26.1421 q^{85} +(-1.79289 + 3.10538i) q^{87} +(2.24264 + 3.88437i) q^{89} +(4.79289 - 0.655892i) q^{91} +(0.828427 + 1.43488i) q^{93} +(5.82843 - 10.0951i) q^{95} +1.82843 q^{97} +2.82843 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	0.207107	−	0.358719i	0.119573	−	0.207107i	−0.800025	−	0.599966i	$-0.795181\pi$
$3$	0.207107	−	0.358719i	0.119573	−	0.207107i	0.919599	+	0.392859i	$0.128514\pi$
$4$		0			0
$5$	1.70711	+	2.95680i	0.763441	+	1.32232i	0.941067	+	0.338221i	$0.109825\pi$
$5$	1.70711	+	2.95680i	0.763441	+	1.32232i	−0.177625	+	0.984098i	$0.556842\pi$
$6$		0			0
$7$	2.62132	−	0.358719i	0.990766	−	0.135583i
$7$	2.62132	−	0.358719i	0.990766	−	0.135583i
$8$		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
$13$	1.82843			0.507114			0.253557	−	0.967320i	$-0.418399\pi$
$13$	1.82843			0.507114			0.253557	+	0.967320i	$0.418399\pi$
$14$		0			0
$15$	1.41421			0.365148
$16$		0			0
$17$	3.82843	−	6.63103i	0.928530	−	1.60826i	0.142747	−	0.989759i	$-0.454407\pi$
$17$	3.82843	−	6.63103i	0.928530	−	1.60826i	0.785783	−	0.618502i	$-0.212260\pi$
$18$		0			0
$19$	−1.70711	−	2.95680i	−0.391637	−	0.678335i	0.601028	−	0.799228i	$-0.294758\pi$
$19$	−1.70711	−	2.95680i	−0.391637	−	0.678335i	−0.992666	+	0.120892i	$0.961424\pi$
$20$		0			0
$21$	0.414214	−	1.01461i	0.0903888	−	0.221406i
$22$		0			0
$23$	1.12132	+	1.94218i	0.233811	+	0.404973i	0.958927	−	0.283654i	$-0.0915468\pi$
$23$	1.12132	+	1.94218i	0.233811	+	0.404973i	−0.725115	+	0.688628i	$0.758213\pi$
$24$		0			0
$25$	−3.32843	+	5.76500i	−0.665685	+	1.15300i
$26$		0			0
$27$	2.41421			0.464616
$28$		0			0
$29$	−8.65685			−1.60754			−0.803769	−	0.594942i	$-0.797175\pi$
$29$	−8.65685			−1.60754			−0.803769	+	0.594942i	$0.797175\pi$
$30$		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
$33$	−0.207107	−	0.358719i	−0.0360527	−	0.0624450i
$34$		0			0
$35$	5.53553	+	7.13834i	0.935676	+	1.20660i
$36$		0			0
$37$	3.29289	+	5.70346i	0.541348	+	0.937643i	0.998827	+	0.0484222i	$0.0154193\pi$
$37$	3.29289	+	5.70346i	0.541348	+	0.937643i	−0.457479	+	0.889221i	$0.651247\pi$
$38$		0			0
$39$	0.378680	−	0.655892i	0.0606373	−	0.105027i
$40$		0			0
$41$	−2.58579			−0.403832			−0.201916	−	0.979403i	$-0.564717\pi$
$41$	−2.58579			−0.403832			−0.201916	+	0.979403i	$0.564717\pi$
$42$		0			0
$43$	−5.65685			−0.862662			−0.431331	−	0.902194i	$-0.641956\pi$
$43$	−5.65685			−0.862662			−0.431331	+	0.902194i	$0.641956\pi$
$44$		0			0
$45$	−4.82843	+	8.36308i	−0.719779	+	1.24669i
$46$		0			0
$47$	3.24264	+	5.61642i	0.472988	+	0.819239i	0.999522	−	0.0309151i	$-0.00984215\pi$
$47$	3.24264	+	5.61642i	0.472988	+	0.819239i	−0.526534	+	0.850154i	$0.676509\pi$
$48$		0			0
$49$	6.74264	−	1.88064i	0.963234	−	0.268662i
$50$		0			0
$51$	−1.58579	−	2.74666i	−0.222055	−	0.384610i
$52$		0			0
$53$	5.94975	−	10.3053i	0.817261	−	1.41554i	−0.0904325	−	0.995903i	$-0.528825\pi$
$53$	5.94975	−	10.3053i	0.817261	−	1.41554i	0.907693	−	0.419634i	$-0.137842\pi$
$54$		0			0
$55$	3.41421			0.460372
$56$		0			0
$57$	−1.41421			−0.187317
$58$		0			0
$59$	−4.20711	+	7.28692i	−0.547719	+	0.948677i	0.450712	+	0.892670i	$0.351170\pi$
$59$	−4.20711	+	7.28692i	−0.547719	+	0.948677i	−0.998430	+	0.0560070i	$0.982163\pi$
$60$		0			0
$61$	−3.08579	−	5.34474i	−0.395094	−	0.684324i	0.598019	−	0.801482i	$-0.295955\pi$
$61$	−3.08579	−	5.34474i	−0.395094	−	0.684324i	−0.993113	+	0.117158i	$0.962621\pi$
$62$		0			0
$63$	4.58579	+	5.91359i	0.577755	+	0.745042i
$64$		0			0
$65$	3.12132	+	5.40629i	0.387152	+	0.670567i
$66$		0			0
$67$	5.62132	−	9.73641i	0.686754	−	1.18949i	−0.286129	−	0.958191i	$-0.592368\pi$
$67$	5.62132	−	9.73641i	0.686754	−	1.18949i	0.972882	−	0.231301i	$-0.0742982\pi$
$68$		0			0
$69$	0.928932			0.111830
$70$		0			0
$71$	−3.07107			−0.364469			−0.182234	−	0.983255i	$-0.558333\pi$
$71$	−3.07107			−0.364469			−0.182234	+	0.983255i	$0.558333\pi$
$72$		0			0
$73$	3.29289	−	5.70346i	0.385404	−	0.667539i	−0.606421	−	0.795144i	$-0.707395\pi$
$73$	3.29289	−	5.70346i	0.385404	−	0.667539i	0.991825	+	0.127604i	$0.0407288\pi$
$74$		0			0
$75$	1.37868	+	2.38794i	0.159196	+	0.275736i
$76$		0			0
$77$	1.00000	−	2.44949i	0.113961	−	0.279145i
$78$		0			0
$79$	−2.37868	−	4.11999i	−0.267622	−	0.463536i	0.700625	−	0.713530i	$-0.252905\pi$
$79$	−2.37868	−	4.11999i	−0.267622	−	0.463536i	−0.968247	+	0.249994i	$0.919571\pi$
$80$		0			0
$81$	−3.74264	+	6.48244i	−0.415849	+	0.720272i
$82$		0			0
$83$	−16.1421			−1.77183			−0.885915	−	0.463848i	$-0.846468\pi$
$83$	−16.1421			−1.77183			−0.885915	+	0.463848i	$0.846468\pi$
$84$		0			0
$85$	26.1421			2.83551
$86$		0			0
$87$	−1.79289	+	3.10538i	−0.192218	+	0.332932i
$88$		0			0
$89$	2.24264	+	3.88437i	0.237719	+	0.411742i	0.960060	−	0.279796i	$-0.0902668\pi$
$89$	2.24264	+	3.88437i	0.237719	+	0.411742i	−0.722340	+	0.691538i	$0.756933\pi$
$90$		0			0
$91$	4.79289	−	0.655892i	0.502432	−	0.0687562i
$92$		0			0
$93$	0.828427	+	1.43488i	0.0859039	+	0.148790i
$94$		0			0
$95$	5.82843	−	10.0951i	0.597984	−	1.03574i
$96$		0			0
$97$	1.82843			0.185649			0.0928243	−	0.995683i	$-0.470411\pi$
$97$	1.82843			0.185649			0.0928243	+	0.995683i	$0.470411\pi$
$98$		0			0
$99$	2.82843			0.284268
$100$		0			0
$101$	−5.91421	+	10.2437i	−0.588486	+	1.01929i	0.405945	+	0.913898i	$0.366943\pi$
$101$	−5.91421	+	10.2437i	−0.588486	+	1.01929i	−0.994431	+	0.105390i	$0.966391\pi$
$102$		0			0
$103$	5.29289	+	9.16756i	0.521524	+	0.903307i	0.999687	+	0.0250350i	$0.00796973\pi$
$103$	5.29289	+	9.16756i	0.521524	+	0.903307i	−0.478162	+	0.878271i	$0.658697\pi$
$104$		0			0
$105$	3.70711	−	0.507306i	0.361777	−	0.0495080i
$106$		0			0
$107$	5.53553	+	9.58783i	0.535140	+	0.926890i	0.999157	+	0.0410635i	$0.0130746\pi$
$107$	5.53553	+	9.58783i	0.535140	+	0.926890i	−0.464016	+	0.885827i	$0.653592\pi$
$108$		0			0
$109$	0.242641	−	0.420266i	0.0232408	−	0.0402542i	−0.854171	−	0.519992i	$-0.825935\pi$
$109$	0.242641	−	0.420266i	0.0232408	−	0.0402542i	0.877412	+	0.479738i	$0.159268\pi$
$110$		0			0
$111$	2.72792			0.258923
$112$		0			0
$113$	−13.8284			−1.30087			−0.650434	−	0.759562i	$-0.725413\pi$
$113$	−13.8284			−1.30087			−0.650434	+	0.759562i	$0.725413\pi$
$114$		0			0
$115$	−3.82843	+	6.63103i	−0.357003	+	0.618347i
$116$		0			0
$117$	2.58579	+	4.47871i	0.239056	+	0.414057i
$118$		0			0
$119$	7.65685	−	18.7554i	0.701903	−	1.71930i
$120$		0			0
$121$	−0.500000	−	0.866025i	−0.0454545	−	0.0787296i
$122$		0			0
$123$	−0.535534	+	0.927572i	−0.0482875	+	0.0836363i
$124$		0			0
$125$	−5.65685			−0.505964
$126$		0			0
$127$	9.72792			0.863213			0.431607	−	0.902062i	$-0.357947\pi$
$127$	9.72792			0.863213			0.431607	+	0.902062i	$0.357947\pi$
$128$		0			0
$129$	−1.17157	+	2.02922i	−0.103151	+	0.178663i
$130$		0			0
$131$	−1.70711	−	2.95680i	−0.149151	−	0.258336i	0.781763	−	0.623575i	$-0.214321\pi$
$131$	−1.70711	−	2.95680i	−0.149151	−	0.258336i	−0.930914	+	0.365239i	$0.880987\pi$
$132$		0			0
$133$	−5.53553	−	7.13834i	−0.479992	−	0.618972i
$134$		0			0
$135$	4.12132	+	7.13834i	0.354707	+	0.614370i
$136$		0			0
$137$	2.67157	−	4.62730i	0.228248	−	0.395337i	−0.729041	−	0.684470i	$-0.760034\pi$
$137$	2.67157	−	4.62730i	0.228248	−	0.395337i	0.957289	+	0.289133i	$0.0933670\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$	2.68629			0.226227
$142$		0			0
$143$	0.914214	−	1.58346i	0.0764504	−	0.132416i
$144$		0			0
$145$	−14.7782	−	25.5965i	−1.22726	−	2.12568i
$146$		0			0
$147$	0.721825	−	2.80821i	0.0595352	−	0.231617i
$148$		0			0
$149$	−3.17157	−	5.49333i	−0.259825	−	0.450031i	0.706370	−	0.707843i	$-0.250332\pi$
$149$	−3.17157	−	5.49333i	−0.259825	−	0.450031i	−0.966195	+	0.257812i	$0.916998\pi$
$150$		0			0
$151$	4.86396	−	8.42463i	0.395824	−	0.685586i	−0.597382	−	0.801957i	$-0.703793\pi$
$151$	4.86396	−	8.42463i	0.395824	−	0.685586i	0.993206	+	0.116370i	$0.0371259\pi$
$152$		0			0
$153$	21.6569			1.75085
$154$		0			0
$155$	−13.6569			−1.09694
$156$		0			0
$157$	−3.17157	+	5.49333i	−0.253119	+	0.438415i	−0.964383	−	0.264510i	$-0.914790\pi$
$157$	−3.17157	+	5.49333i	−0.253119	+	0.438415i	0.711264	+	0.702925i	$0.248123\pi$
$158$		0			0
$159$	−2.46447	−	4.26858i	−0.195445	−	0.338520i
$160$		0			0
$161$	3.63604	+	4.68885i	0.286560	+	0.369533i
$162$		0			0
$163$	−7.86396	−	13.6208i	−0.615953	−	1.06686i	−0.990217	−	0.139539i	$-0.955438\pi$
$163$	−7.86396	−	13.6208i	−0.615953	−	1.06686i	0.374264	−	0.927322i	$-0.377896\pi$
$164$		0			0
$165$	0.707107	−	1.22474i	0.0550482	−	0.0953463i
$166$		0			0
$167$	11.7279			0.907534			0.453767	−	0.891120i	$-0.350080\pi$
$167$	11.7279			0.907534			0.453767	+	0.891120i	$0.350080\pi$
$168$		0			0
$169$	−9.65685			−0.742835
$170$		0			0
$171$	4.82843	−	8.36308i	0.369239	−	0.639541i
$172$		0			0
$173$	2.08579	+	3.61269i	0.158579	+	0.274668i	0.934357	−	0.356339i	$-0.115975\pi$
$173$	2.08579	+	3.61269i	0.158579	+	0.274668i	−0.775777	+	0.631007i	$0.782642\pi$
$174$		0			0
$175$	−6.65685	+	16.3059i	−0.503211	+	1.23261i
$176$		0			0
$177$	1.74264	+	3.01834i	0.130985	+	0.226872i
$178$		0			0
$179$	−9.44975	+	16.3674i	−0.706307	+	1.22336i	0.259910	+	0.965633i	$0.416307\pi$
$179$	−9.44975	+	16.3674i	−0.706307	+	1.22336i	−0.966218	+	0.257727i	$0.917026\pi$
$180$		0			0
$181$	3.65685			0.271812			0.135906	−	0.990722i	$-0.456606\pi$
$181$	3.65685			0.271812			0.135906	+	0.990722i	$0.456606\pi$
$182$		0			0
$183$	−2.55635			−0.188971
$184$		0			0
$185$	−11.2426	+	19.4728i	−0.826575	+	1.43167i
$186$		0			0
$187$	−3.82843	−	6.63103i	−0.279962	−	0.484909i
$188$		0			0
$189$	6.32843	−	0.866025i	0.460325	−	0.0629941i
$190$		0			0
$191$	−6.41421	−	11.1097i	−0.464116	−	0.803873i	0.535045	−	0.844824i	$-0.320295\pi$
$191$	−6.41421	−	11.1097i	−0.464116	−	0.803873i	−0.999161	+	0.0409507i	$0.986961\pi$
$192$		0			0
$193$	−1.05025	+	1.81909i	−0.0755988	+	0.130941i	−0.901347	−	0.433099i	$-0.857420\pi$
$193$	−1.05025	+	1.81909i	−0.0755988	+	0.130941i	0.825748	+	0.564040i	$0.190754\pi$
$194$		0			0
$195$	2.58579			0.185172
$196$		0			0
$197$	−17.4853			−1.24577			−0.622887	−	0.782312i	$-0.714041\pi$
$197$	−17.4853			−1.24577			−0.622887	+	0.782312i	$0.714041\pi$
$198$		0			0
$199$	9.94975	−	17.2335i	0.705319	−	1.22165i	−0.261257	−	0.965269i	$-0.584137\pi$
$199$	9.94975	−	17.2335i	0.705319	−	1.22165i	0.966576	−	0.256379i	$-0.0825295\pi$
$200$		0			0
$201$	−2.32843	−	4.03295i	−0.164235	−	0.284463i
$202$		0			0
$203$	−22.6924	+	3.10538i	−1.59269	+	0.217955i
$204$		0			0
$205$	−4.41421	−	7.64564i	−0.308302	−	0.533995i
$206$		0			0
$207$	−3.17157	+	5.49333i	−0.220440	+	0.381813i
$208$		0			0
$209$	−3.41421			−0.236166
$210$		0			0
$211$	−4.58579			−0.315699			−0.157849	−	0.987463i	$-0.550456\pi$
$211$	−4.58579			−0.315699			−0.157849	+	0.987463i	$0.550456\pi$
$212$		0			0
$213$	−0.636039	+	1.10165i	−0.0435807	+	0.0754839i
$214$		0			0
$215$	−9.65685	−	16.7262i	−0.658592	−	1.14071i
$216$		0			0
$217$	−4.00000	+	9.79796i	−0.271538	+	0.665129i
$218$		0			0
$219$	−1.36396	−	2.36245i	−0.0921679	−	0.159640i
$220$		0			0
$221$	7.00000	−	12.1244i	0.470871	−	0.815572i
$222$		0			0
$223$	−11.4142			−0.764352			−0.382176	−	0.924089i	$-0.624825\pi$
$223$	−11.4142			−0.764352			−0.382176	+	0.924089i	$0.624825\pi$
$224$		0			0
$225$	−18.8284			−1.25523
$226$		0			0
$227$	−11.5858	+	20.0672i	−0.768976	+	1.33190i	0.169143	+	0.985591i	$0.445900\pi$
$227$	−11.5858	+	20.0672i	−0.768976	+	1.33190i	−0.938119	+	0.346313i	$0.887433\pi$
$228$		0			0
$229$	−0.343146	−	0.594346i	−0.0226757	−	0.0392755i	0.854465	−	0.519509i	$-0.173885\pi$
$229$	−0.343146	−	0.594346i	−0.0226757	−	0.0392755i	−0.877141	+	0.480234i	$0.840552\pi$
$230$		0			0
$231$	−0.671573	−	0.866025i	−0.0441863	−	0.0569803i
$232$		0			0
$233$	0.707107	+	1.22474i	0.0463241	+	0.0802357i	0.888258	−	0.459345i	$-0.151916\pi$
$233$	0.707107	+	1.22474i	0.0463241	+	0.0802357i	−0.841934	+	0.539581i	$0.818583\pi$
$234$		0			0
$235$	−11.0711	+	19.1757i	−0.722197	+	1.25088i
$236$		0			0
$237$	−1.97056			−0.128002
$238$		0			0
$239$	22.2132			1.43685			0.718426	−	0.695603i	$-0.244863\pi$
$239$	22.2132			1.43685			0.718426	+	0.695603i	$0.244863\pi$
$240$		0			0
$241$	−1.87868	+	3.25397i	−0.121016	+	0.209607i	−0.920169	−	0.391522i	$-0.871949\pi$
$241$	−1.87868	+	3.25397i	−0.121016	+	0.209607i	0.799152	+	0.601129i	$0.205282\pi$
$242$		0			0
$243$	5.17157	+	8.95743i	0.331757	+	0.574619i
$244$		0			0
$245$	17.0711	+	16.7262i	1.09063	+	1.06860i
$246$		0			0
$247$	−3.12132	−	5.40629i	−0.198605	−	0.343994i
$248$		0			0
$249$	−3.34315	+	5.79050i	−0.211863	+	0.366958i
$250$		0			0
$251$	−2.14214			−0.135210			−0.0676052	−	0.997712i	$-0.521536\pi$
$251$	−2.14214			−0.135210			−0.0676052	+	0.997712i	$0.521536\pi$
$252$		0			0
$253$	2.24264			0.140994
$254$		0			0
$255$	5.41421	−	9.37769i	0.339051	−	0.587254i
$256$		0			0
$257$	−1.57107	−	2.72117i	−0.0980005	−	0.169742i	0.812856	−	0.582464i	$-0.197911\pi$
$257$	−1.57107	−	2.72117i	−0.0980005	−	0.169742i	−0.910857	+	0.412722i	$0.864578\pi$
$258$		0			0
$259$	10.6777	+	13.7694i	0.663478	+	0.855587i
$260$		0			0
$261$	−12.2426	−	21.2049i	−0.757800	−	1.31255i
$262$		0			0
$263$	15.5208	−	26.8828i	0.957054	−	1.65767i	0.227459	−	0.973788i	$-0.426958\pi$
$263$	15.5208	−	26.8828i	0.957054	−	1.65767i	0.729595	−	0.683879i	$-0.239709\pi$
$264$		0			0
$265$	40.6274			2.49572
$266$		0			0
$267$	1.85786			0.113699
$268$		0			0
$269$	−6.82843	+	11.8272i	−0.416337	+	0.721116i	−0.995568	−	0.0940473i	$-0.970020\pi$
$269$	−6.82843	+	11.8272i	−0.416337	+	0.721116i	0.579231	+	0.815163i	$0.303353\pi$
$270$		0			0
$271$	−13.2782	−	22.9985i	−0.806592	−	1.39706i	−0.915211	−	0.402974i	$-0.867976\pi$
$271$	−13.2782	−	22.9985i	−0.806592	−	1.39706i	0.108620	−	0.994083i	$-0.465357\pi$
$272$		0			0
$273$	0.757359	−	1.85514i	0.0458375	−	0.112278i
$274$		0			0
$275$	3.32843	+	5.76500i	0.200712	+	0.347643i
$276$		0			0
$277$	1.91421	−	3.31552i	0.115014	−	0.199210i	−0.802771	−	0.596287i	$-0.796642\pi$
$277$	1.91421	−	3.31552i	0.115014	−	0.199210i	0.917785	+	0.397077i	$0.129975\pi$
$278$		0			0
$279$	−11.3137			−0.677334
$280$		0			0
$281$	−16.7279			−0.997904			−0.498952	−	0.866630i	$-0.666282\pi$
$281$	−16.7279			−0.997904			−0.498952	+	0.866630i	$0.666282\pi$
$282$		0			0
$283$	−10.2929	+	17.8278i	−0.611849	+	1.05975i	0.379080	+	0.925364i	$0.376241\pi$
$283$	−10.2929	+	17.8278i	−0.611849	+	1.05975i	−0.990929	+	0.134389i	$0.957093\pi$
$284$		0			0
$285$	−2.41421	−	4.18154i	−0.143006	−	0.247693i
$286$		0			0
$287$	−6.77817	+	0.927572i	−0.400103	+	0.0547528i
$288$		0			0
$289$	−20.8137	−	36.0504i	−1.22434	−	2.12061i
$290$		0			0
$291$	0.378680	−	0.655892i	0.0221986	−	0.0384491i
$292$		0			0
$293$	−5.17157			−0.302127			−0.151063	−	0.988524i	$-0.548270\pi$
$293$	−5.17157			−0.302127			−0.151063	+	0.988524i	$0.548270\pi$
$294$		0			0
$295$	−28.7279			−1.67260
$296$		0			0
$297$	1.20711	−	2.09077i	0.0700434	−	0.121319i
$298$		0			0
$299$	2.05025	+	3.55114i	0.118569	+	0.205368i
$300$		0			0
$301$	−14.8284	+	2.02922i	−0.854696	+	0.116963i
$302$		0			0
$303$	2.44975	+	4.24309i	0.140734	+	0.243759i
$304$		0			0
$305$	10.5355	−	18.2481i	0.603263	−	1.04488i
$306$		0			0
$307$	9.89949			0.564994			0.282497	−	0.959268i	$-0.408837\pi$
$307$	9.89949			0.564994			0.282497	+	0.959268i	$0.408837\pi$
$308$		0			0
$309$	4.38478			0.249441
$310$		0			0
$311$	−4.36396	+	7.55860i	−0.247458	+	0.428609i	−0.962820	−	0.270145i	$-0.912928\pi$
$311$	−4.36396	+	7.55860i	−0.247458	+	0.428609i	0.715362	+	0.698754i	$0.246262\pi$
$312$		0			0
$313$	4.67157	+	8.09140i	0.264053	+	0.457353i	0.967315	−	0.253578i	$-0.0816073\pi$
$313$	4.67157	+	8.09140i	0.264053	+	0.457353i	−0.703262	+	0.710931i	$0.748274\pi$
$314$		0			0
$315$	−9.65685	+	23.6544i	−0.544102	+	1.33277i
$316$		0			0
$317$	−15.6569	−	27.1185i	−0.879377	−	1.52312i	−0.852026	−	0.523499i	$-0.824626\pi$
$317$	−15.6569	−	27.1185i	−0.879377	−	1.52312i	−0.0273502	−	0.999626i	$-0.508707\pi$
$318$		0			0
$319$	−4.32843	+	7.49706i	−0.242345	+	0.419755i
$320$		0			0
$321$	4.58579			0.255954
$322$		0			0
$323$	−26.1421			−1.45459
$324$		0			0
$325$	−6.08579	+	10.5409i	−0.337579	+	0.584703i
$326$		0			0
$327$	−0.100505	−	0.174080i	−0.00555794	−	0.00962664i
$328$		0			0
$329$	10.5147	+	13.5592i	0.579695	+	0.747545i
$330$		0			0
$331$	−4.96447	−	8.59871i	−0.272872	−	0.472628i	0.696724	−	0.717339i	$-0.254640\pi$
$331$	−4.96447	−	8.59871i	−0.272872	−	0.472628i	−0.969596	+	0.244711i	$0.921307\pi$
$332$		0			0
$333$	−9.31371	+	16.1318i	−0.510388	+	0.884018i
$334$		0			0
$335$	38.3848			2.09718
$336$		0			0
$337$	−19.7574			−1.07625			−0.538126	−	0.842864i	$-0.680868\pi$
$337$	−19.7574			−1.07625			−0.538126	+	0.842864i	$0.680868\pi$
$338$		0			0
$339$	−2.86396	+	4.96053i	−0.155549	+	0.269419i
$340$		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
$345$	1.58579	+	2.74666i	0.0853759	+	0.147875i
$346$		0			0
$347$	7.29289	−	12.6317i	0.391503	−	0.678103i	−0.601145	−	0.799140i	$-0.705289\pi$
$347$	7.29289	−	12.6317i	0.391503	−	0.678103i	0.992648	+	0.121037i	$0.0386219\pi$
$348$		0			0
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$		0			0
$351$	4.41421			0.235613
$352$		0			0
$353$	12.6569	−	21.9223i	0.673656	−	1.16681i	−0.303203	−	0.952926i	$-0.598056\pi$
$353$	12.6569	−	21.9223i	0.673656	−	1.16681i	0.976860	−	0.213881i	$-0.0686105\pi$
$354$		0			0
$355$	−5.24264	−	9.08052i	−0.278250	−	0.481944i
$356$		0			0
$357$	−5.14214	−	6.63103i	−0.272151	−	0.350951i
$358$		0			0
$359$	5.37868	+	9.31615i	0.283876	+	0.491687i	0.972336	−	0.233587i	$-0.0750463\pi$
$359$	5.37868	+	9.31615i	0.283876	+	0.491687i	−0.688460	+	0.725274i	$0.741713\pi$
$360$		0			0
$361$	3.67157	−	6.35935i	0.193241	−	0.334703i
$362$		0			0
$363$	−0.414214			−0.0217406
$364$		0			0
$365$	22.4853			1.17693
$366$		0			0
$367$	7.36396	−	12.7548i	0.384396	−	0.665793i	−0.607290	−	0.794481i	$-0.707743\pi$
$367$	7.36396	−	12.7548i	0.384396	−	0.665793i	0.991685	+	0.128688i	$0.0410765\pi$
$368$		0			0
$369$	−3.65685	−	6.33386i	−0.190368	−	0.329727i
$370$		0			0
$371$	11.8995	−	29.1477i	0.617791	−	1.51327i
$372$		0			0
$373$	−6.98528	−	12.0989i	−0.361684	−	0.626455i	0.626554	−	0.779378i	$-0.284465\pi$
$373$	−6.98528	−	12.0989i	−0.361684	−	0.626455i	−0.988238	+	0.152923i	$0.951131\pi$
$374$		0			0
$375$	−1.17157	+	2.02922i	−0.0604998	+	0.104789i
$376$		0			0
$377$	−15.8284			−0.815205
$378$		0			0
$379$	25.8701			1.32886			0.664428	−	0.747352i	$-0.268675\pi$
$379$	25.8701			1.32886			0.664428	+	0.747352i	$0.268675\pi$
$380$		0			0
$381$	2.01472	−	3.48960i	0.103217	−	0.178777i
$382$		0			0
$383$	15.1924	+	26.3140i	0.776295	+	1.34458i	0.934064	+	0.357106i	$0.116236\pi$
$383$	15.1924	+	26.3140i	0.776295	+	1.34458i	−0.157769	+	0.987476i	$0.550430\pi$
$384$		0			0
$385$	8.94975	−	1.22474i	0.456121	−	0.0624188i
$386$		0			0
$387$	−8.00000	−	13.8564i	−0.406663	−	0.704361i
$388$		0			0
$389$	−1.36396	+	2.36245i	−0.0691556	+	0.119781i	−0.898530	−	0.438912i	$-0.855364\pi$
$389$	−1.36396	+	2.36245i	−0.0691556	+	0.119781i	0.829374	+	0.558693i	$0.188697\pi$
$390$		0			0
$391$	17.1716			0.868404
$392$		0			0
$393$	−1.41421			−0.0713376
$394$		0			0
$395$	8.12132	−	14.0665i	0.408628	−	0.707764i
$396$		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
$399$	−3.70711	+	0.507306i	−0.185587	+	0.0253971i
$400$		0			0
$401$	2.15685	+	3.73578i	0.107708	+	0.186556i	0.914841	−	0.403813i	$-0.132315\pi$
$401$	2.15685	+	3.73578i	0.107708	+	0.186556i	−0.807133	+	0.590369i	$0.798982\pi$
$402$		0			0
$403$	−3.65685	+	6.33386i	−0.182161	+	0.315512i
$404$		0			0
$405$	−25.5563			−1.26991
$406$		0			0
$407$	6.58579			0.326445
$408$		0			0
$409$	11.3640	−	19.6830i	0.561912	−	0.973260i	−0.435418	−	0.900228i	$-0.643399\pi$
$409$	11.3640	−	19.6830i	0.561912	−	0.973260i	0.997330	−	0.0730312i	$-0.0232673\pi$
$410$		0			0
$411$	−1.10660	−	1.91669i	−0.0545846	−	0.0945434i
$412$		0			0
$413$	−8.41421	+	20.6105i	−0.414036	+	1.01418i
$414$		0			0
$415$	−27.5563	−	47.7290i	−1.35269	−	2.34292i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−2.14214			−0.104650			−0.0523251	−	0.998630i	$-0.516663\pi$
$419$	−2.14214			−0.104650			−0.0523251	+	0.998630i	$0.516663\pi$
$420$		0			0
$421$	23.3137			1.13624			0.568120	−	0.822946i	$-0.307671\pi$
$421$	23.3137			1.13624			0.568120	+	0.822946i	$0.307671\pi$
$422$		0			0
$423$	−9.17157	+	15.8856i	−0.445937	+	0.772386i
$424$		0			0
$425$	25.4853	+	44.1418i	1.23622	+	2.14119i
$426$		0			0
$427$	−10.0061	−	12.9033i	−0.484229	−	0.624436i
$428$		0			0
$429$	−0.378680	−	0.655892i	−0.0182828	−	0.0316668i
$430$		0			0
$431$	−10.2071	+	17.6792i	−0.491659	+	0.851578i	−0.999954	−	0.00960469i	$-0.996943\pi$
$431$	−10.2071	+	17.6792i	−0.491659	+	0.851578i	0.508295	+	0.861183i	$0.330276\pi$
$432$		0			0
$433$	−2.14214			−0.102944			−0.0514722	−	0.998674i	$-0.516391\pi$
$433$	−2.14214			−0.102944			−0.0514722	+	0.998674i	$0.516391\pi$
$434$		0			0
$435$	−12.2426			−0.586990
$436$		0			0
$437$	3.82843	−	6.63103i	0.183139	−	0.317205i
$438$		0			0
$439$	4.69239	+	8.12745i	0.223955	+	0.387902i	0.956006	−	0.293349i	$-0.0947696\pi$
$439$	4.69239	+	8.12745i	0.223955	+	0.387902i	−0.732050	+	0.681251i	$0.761436\pi$
$440$		0			0
$441$	14.1421	+	13.8564i	0.673435	+	0.659829i
$442$		0			0
$443$	−16.3137	−	28.2562i	−0.775088	−	1.34249i	−0.934745	−	0.355319i	$-0.884372\pi$
$443$	−16.3137	−	28.2562i	−0.775088	−	1.34249i	0.159658	−	0.987172i	$-0.448961\pi$
$444$		0			0
$445$	−7.65685	+	13.2621i	−0.362970	+	0.628682i
$446$		0			0
$447$	−2.62742			−0.124273
$448$		0			0
$449$	33.6569			1.58837			0.794183	−	0.607679i	$-0.207899\pi$
$449$	33.6569			1.58837			0.794183	+	0.607679i	$0.207899\pi$
$450$		0			0
$451$	−1.29289	+	2.23936i	−0.0608800	+	0.105447i
$452$		0			0
$453$	−2.01472	−	3.48960i	−0.0946597	−	0.163955i
$454$		0			0
$455$	10.1213	+	13.0519i	0.474495	+	0.611884i
$456$		0			0
$457$	0.171573	+	0.297173i	0.00802584	+	0.0139012i	0.870010	−	0.493033i	$-0.164112\pi$
$457$	0.171573	+	0.297173i	0.00802584	+	0.0139012i	−0.861985	+	0.506934i	$0.830779\pi$
$458$		0			0
$459$	9.24264	−	16.0087i	0.431410	−	0.747223i
$460$		0			0
$461$	14.3137			0.666656			0.333328	−	0.942811i	$-0.391828\pi$
$461$	14.3137			0.666656			0.333328	+	0.942811i	$0.391828\pi$
$462$		0			0
$463$	−7.17157			−0.333291			−0.166646	−	0.986017i	$-0.553294\pi$
$463$	−7.17157			−0.333291			−0.166646	+	0.986017i	$0.553294\pi$
$464$		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
$469$	11.2426	−	27.5387i	0.519137	−	1.27162i
$470$		0			0
$471$	1.31371	+	2.27541i	0.0605325	+	0.104845i
$472$		0			0
$473$	−2.82843	+	4.89898i	−0.130051	+	0.225255i
$474$		0			0
$475$	22.7279			1.04283
$476$		0			0
$477$	33.6569			1.54104
$478$		0			0
$479$	−13.0355	+	22.5782i	−0.595609	+	1.03162i	0.397852	+	0.917450i	$0.369756\pi$
$479$	−13.0355	+	22.5782i	−0.595609	+	1.03162i	−0.993461	+	0.114175i	$0.963578\pi$
$480$		0			0
$481$	6.02082	+	10.4284i	0.274526	+	0.475492i
$482$		0			0
$483$	2.43503	−	0.333226i	0.110798	−	0.0151623i
$484$		0			0
$485$	3.12132	+	5.40629i	0.141732	+	0.245487i
$486$		0			0
$487$	0.828427	−	1.43488i	0.0375396	−	0.0650205i	−0.846645	−	0.532158i	$-0.821381\pi$
$487$	0.828427	−	1.43488i	0.0375396	−	0.0650205i	0.884185	+	0.467137i	$0.154715\pi$
$488$		0			0
$489$	−6.51472			−0.294606
$490$		0			0
$491$	−24.8284			−1.12049			−0.560246	−	0.828327i	$-0.689293\pi$
$491$	−24.8284			−1.12049			−0.560246	+	0.828327i	$0.689293\pi$
$492$		0			0
$493$	−33.1421	+	57.4039i	−1.49265	+	2.58534i
$494$		0			0
$495$	4.82843	+	8.36308i	0.217022	+	0.375893i
$496$		0			0
$497$	−8.05025	+	1.10165i	−0.361103	+	0.0494158i
$498$		0			0
$499$	−3.07107	−	5.31925i	−0.137480	−	0.238122i	0.789062	−	0.614313i	$-0.210567\pi$
$499$	−3.07107	−	5.31925i	−0.137480	−	0.238122i	−0.926542	+	0.376191i	$0.877234\pi$
$500$		0			0
$501$	2.42893	−	4.20703i	0.108517	−	0.187956i
$502$		0			0
$503$	38.2132			1.70384			0.851921	−	0.523670i	$-0.175437\pi$
$503$	38.2132			1.70384			0.851921	+	0.523670i	$0.175437\pi$
$504$		0			0
$505$	−40.3848			−1.79710
$506$		0			0
$507$	−2.00000	+	3.46410i	−0.0888231	+	0.153846i
$508$		0			0
$509$	6.65685	+	11.5300i	0.295060	+	0.511059i	0.974999	−	0.222210i	$-0.0713271\pi$
$509$	6.65685	+	11.5300i	0.295060	+	0.511059i	−0.679939	+	0.733269i	$0.737994\pi$
$510$		0			0
$511$	6.58579	−	16.1318i	0.291338	−	0.713630i
$512$		0			0
$513$	−4.12132	−	7.13834i	−0.181961	−	0.315165i
$514$		0			0
$515$	−18.0711	+	31.3000i	−0.796306	+	1.37924i
$516$		0			0
$517$	6.48528			0.285222
$518$		0			0
$519$	1.72792			0.0758474
$520$		0			0
$521$	−14.1421	+	24.4949i	−0.619578	+	1.07314i	0.369984	+	0.929038i	$0.379363\pi$
$521$	−14.1421	+	24.4949i	−0.619578	+	1.07314i	−0.989563	+	0.144103i	$0.953970\pi$
$522$		0			0
$523$	6.36396	+	11.0227i	0.278277	+	0.481989i	0.970957	−	0.239256i	$-0.0769035\pi$
$523$	6.36396	+	11.0227i	0.278277	+	0.481989i	−0.692680	+	0.721245i	$0.743570\pi$
$524$		0			0
$525$	4.47056	+	5.76500i	0.195111	+	0.251605i
$526$		0			0
$527$	15.3137	+	26.5241i	0.667076	+	1.15541i
$528$		0			0
$529$	8.98528	−	15.5630i	0.390664	−	0.676651i
$530$		0			0
$531$	−23.7990			−1.03279
$532$		0			0
$533$	−4.72792			−0.204789
$534$		0			0
$535$	−18.8995	+	32.7349i	−0.817096	+	1.41525i
$536$		0			0
$537$	3.91421	+	6.77962i	0.168911	+	0.292562i
$538$		0			0
$539$	1.74264	−	6.77962i	0.0750608	−	0.292019i
$540$		0			0
$541$	20.5711	+	35.6301i	0.884419	+	1.53186i	0.846378	+	0.532583i	$0.178779\pi$
$541$	20.5711	+	35.6301i	0.884419	+	1.53186i	0.0380415	+	0.999276i	$0.487888\pi$
$542$		0			0
$543$	0.757359	−	1.31178i	0.0325014	−	0.0562941i
$544$		0			0
$545$	1.65685			0.0709718
$546$		0			0
$547$	18.8701			0.806825			0.403413	−	0.915018i	$-0.367824\pi$
$547$	18.8701			0.806825			0.403413	+	0.915018i	$0.367824\pi$
$548$		0			0
$549$	8.72792	−	15.1172i	0.372499	−	0.645187i
$550$		0			0
$551$	14.7782	+	25.5965i	0.629571	+	1.09045i
$552$		0			0
$553$	−7.71320	−	9.94655i	−0.327999	−	0.422970i
$554$		0			0
$555$	4.65685	+	8.06591i	0.197672	+	0.342379i
$556$		0			0
$557$	−12.2426	+	21.2049i	−0.518737	+	0.898479i	0.481026	+	0.876707i	$0.340264\pi$
$557$	−12.2426	+	21.2049i	−0.518737	+	0.898479i	−0.999763	+	0.0217729i	$0.993069\pi$
$558$		0			0
$559$	−10.3431			−0.437468
$560$		0			0
$561$	−3.17157			−0.133904
$562$		0			0
$563$	−2.53553	+	4.39167i	−0.106860	+	0.185087i	−0.914497	−	0.404594i	$-0.867413\pi$
$563$	−2.53553	+	4.39167i	−0.106860	+	0.185087i	0.807637	+	0.589681i	$0.200746\pi$
$564$		0			0
$565$	−23.6066	−	40.8878i	−0.993137	−	1.72016i
$566$		0			0
$567$	−7.48528	+	18.3351i	−0.314352	+	0.770003i
$568$		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
$571$	−5.19239	+	8.99348i	−0.217295	+	0.376365i	−0.953980	−	0.299870i	$-0.903057\pi$
$571$	−5.19239	+	8.99348i	−0.217295	+	0.376365i	0.736685	+	0.676236i	$0.236390\pi$
$572$		0			0
$573$	−5.31371			−0.221983
$574$		0			0
$575$	−14.9289			−0.622580
$576$		0			0
$577$	16.1569	−	27.9845i	0.672619	−	1.16501i	−0.304540	−	0.952499i	$-0.598503\pi$
$577$	16.1569	−	27.9845i	0.672619	−	1.16501i	0.977159	−	0.212510i	$-0.0681639\pi$
$578$		0			0
$579$	0.435029	+	0.753492i	0.0180792	+	0.0313141i
$580$		0			0
$581$	−42.3137	+	5.79050i	−1.75547	+	0.240230i
$582$		0			0
$583$	−5.94975	−	10.3053i	−0.246413	−	0.426800i
$584$		0			0
$585$	−8.82843	+	15.2913i	−0.365011	+	0.632217i
$586$		0			0
$587$	44.8995			1.85320			0.926600	−	0.376048i	$-0.122717\pi$
$587$	44.8995			1.85320			0.926600	+	0.376048i	$0.122717\pi$
$588$		0			0
$589$	13.6569			0.562721
$590$		0			0
$591$	−3.62132	+	6.27231i	−0.148961	+	0.258008i
$592$		0			0
$593$	17.8492	+	30.9158i	0.732981	+	1.26956i	0.955604	+	0.294655i	$0.0952047\pi$
$593$	17.8492	+	30.9158i	0.732981	+	1.26956i	−0.222623	+	0.974905i	$0.571462\pi$
$594$		0			0
$595$	68.5269	−	9.37769i	2.80933	−	0.384448i
$596$		0			0
$597$	−4.12132	−	7.13834i	−0.168674	−	0.292153i
$598$		0			0
$599$	−1.31371	+	2.27541i	−0.0536767	+	0.0929707i	−0.891615	−	0.452794i	$-0.850427\pi$
$599$	−1.31371	+	2.27541i	−0.0536767	+	0.0929707i	0.837939	+	0.545765i	$0.183761\pi$
$600$		0			0
$601$	−35.9411			−1.46607			−0.733035	−	0.680191i	$-0.761897\pi$
$601$	−35.9411			−1.46607			−0.733035	+	0.680191i	$0.761897\pi$
$602$		0			0
$603$	31.7990			1.29495
$604$		0			0
$605$	1.70711	−	2.95680i	0.0694038	−	0.120211i
$606$		0			0
$607$	−4.51472	−	7.81972i	−0.183247	−	0.317393i	0.759738	−	0.650230i	$-0.225327\pi$
$607$	−4.51472	−	7.81972i	−0.183247	−	0.317393i	−0.942984	+	0.332837i	$0.891994\pi$
$608$		0			0
$609$	−3.58579	+	8.78335i	−0.145303	+	0.355919i
$610$		0			0
$611$	5.92893	+	10.2692i	0.239859	+	0.415448i
$612$		0			0
$613$	8.31371	−	14.3998i	0.335788	−	0.581601i	−0.647848	−	0.761769i	$-0.724331\pi$
$613$	8.31371	−	14.3998i	0.335788	−	0.581601i	0.983636	+	0.180168i	$0.0576643\pi$
$614$		0			0
$615$	−3.65685			−0.147459
$616$		0			0
$617$	−8.02944			−0.323253			−0.161626	−	0.986852i	$-0.551674\pi$
$617$	−8.02944			−0.323253			−0.161626	+	0.986852i	$0.551674\pi$
$618$		0			0
$619$	22.9706	−	39.7862i	0.923265	−	1.59914i	0.128937	−	0.991653i	$-0.458844\pi$
$619$	22.9706	−	39.7862i	0.923265	−	1.59914i	0.794328	−	0.607489i	$-0.207823\pi$
$620$		0			0
$621$	2.70711	+	4.68885i	0.108632	+	0.188157i
$622$		0			0
$623$	7.27208	+	9.37769i	0.291350	+	0.375709i
$624$		0			0
$625$	6.98528	+	12.0989i	0.279411	+	0.483954i
$626$		0			0
$627$	−0.707107	+	1.22474i	−0.0282391	+	0.0489116i
$628$		0			0
$629$	50.4264			2.01063
$630$		0			0
$631$	−48.7279			−1.93983			−0.969914	−	0.243448i	$-0.921722\pi$
$631$	−48.7279			−1.93983			−0.969914	+	0.243448i	$0.921722\pi$
$632$		0			0
$633$	−0.949747	+	1.64501i	−0.0377491	+	0.0653833i
$634$		0			0
$635$	16.6066	+	28.7635i	0.659013	+	1.14144i
$636$		0			0
$637$	12.3284	−	3.43861i	0.488470	−	0.136243i
$638$		0			0
$639$	−4.34315	−	7.52255i	−0.171812	−	0.297587i
$640$		0			0
$641$	20.6421	−	35.7532i	0.815315	−	1.41217i	−0.0937859	−	0.995592i	$-0.529897\pi$
$641$	20.6421	−	35.7532i	0.815315	−	1.41217i	0.909101	−	0.416575i	$-0.136770\pi$
$642$		0			0
$643$	−4.41421			−0.174080			−0.0870398	−	0.996205i	$-0.527741\pi$
$643$	−4.41421			−0.174080			−0.0870398	+	0.996205i	$0.527741\pi$
$644$		0			0
$645$	−8.00000			−0.315000
$646$		0			0
$647$	−23.0919	+	39.9963i	−0.907836	+	1.57242i	−0.0907706	+	0.995872i	$0.528933\pi$
$647$	−23.0919	+	39.9963i	−0.907836	+	1.57242i	−0.817065	+	0.576546i	$0.804400\pi$
$648$		0			0
$649$	4.20711	+	7.28692i	0.165143	+	0.286037i
$650$		0			0
$651$	2.68629	+	3.46410i	0.105284	+	0.135769i
$652$		0			0
$653$	−9.19239	−	15.9217i	−0.359726	−	0.623064i	0.628189	−	0.778061i	$-0.283796\pi$
$653$	−9.19239	−	15.9217i	−0.359726	−	0.623064i	−0.987915	+	0.154997i	$0.950463\pi$
$654$		0			0
$655$	5.82843	−	10.0951i	0.227735	−	0.394449i
$656$		0			0
$657$	18.6274			0.726725
$658$		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
$661$	7.48528	−	12.9649i	0.291144	−	0.504276i	−0.682937	−	0.730478i	$-0.739298\pi$
$661$	7.48528	−	12.9649i	0.291144	−	0.504276i	0.974080	+	0.226202i	$0.0726309\pi$
$662$		0			0
$663$	−2.89949	−	5.02207i	−0.112607	−	0.195041i
$664$		0			0
$665$	11.6569	−	28.5533i	0.452033	−	1.10725i
$666$		0			0
$667$	−9.70711	−	16.8132i	−0.375861	−	0.651010i
$668$		0			0
$669$	−2.36396	+	4.09450i	−0.0913960	+	0.158303i
$670$		0			0
$671$	−6.17157			−0.238251
$672$		0			0
$673$	−25.5563			−0.985125			−0.492562	−	0.870277i	$-0.663940\pi$
$673$	−25.5563			−0.985125			−0.492562	+	0.870277i	$0.663940\pi$
$674$		0			0
$675$	−8.03553	+	13.9180i	−0.309288	+	0.535702i
$676$		0			0
$677$	6.34315	+	10.9867i	0.243787	+	0.422251i	0.961790	−	0.273789i	$-0.0882769\pi$
$677$	6.34315	+	10.9867i	0.243787	+	0.422251i	−0.718003	+	0.696040i	$0.754944\pi$
$678$		0			0
$679$	4.79289	−	0.655892i	0.183934	−	0.0251708i
$680$		0			0
$681$	4.79899	+	8.31209i	0.183898	+	0.318520i
$682$		0			0
$683$	8.20711	−	14.2151i	0.314036	−	0.543927i	−0.665196	−	0.746669i	$-0.731652\pi$
$683$	8.20711	−	14.2151i	0.314036	−	0.543927i	0.979232	+	0.202742i	$0.0649853\pi$
$684$		0			0
$685$	18.2426			0.697015
$686$		0			0
$687$	−0.284271			−0.0108456
$688$		0			0
$689$	10.8787	−	18.8424i	0.414445	−	0.717839i
$690$		0			0
$691$	6.96447	+	12.0628i	0.264941	+	0.458891i	0.967548	−	0.252687i	$-0.0813143\pi$
$691$	6.96447	+	12.0628i	0.264941	+	0.458891i	−0.702607	+	0.711578i	$0.747981\pi$
$692$		0			0
$693$	7.41421	−	1.01461i	0.281643	−	0.0385419i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−9.89949	+	17.1464i	−0.374970	+	0.649467i
$698$		0			0
$699$	0.585786			0.0221565
$700$		0			0
$701$	36.1127			1.36396			0.681979	−	0.731372i	$-0.261120\pi$
$701$	36.1127			1.36396			0.681979	+	0.731372i	$0.261120\pi$
$702$		0			0
$703$	11.2426	−	19.4728i	0.424024	−	0.734432i
$704$		0			0
$705$	4.58579	+	7.94282i	0.172711	+	0.299144i
$706$		0			0
$707$	−11.8284	+	28.9736i	−0.444854	+	1.08966i
$708$		0			0
$709$	−18.0208	−	31.2130i	−0.676786	−	1.17223i	−0.975943	−	0.218024i	$-0.930039\pi$
$709$	−18.0208	−	31.2130i	−0.676786	−	1.17223i	0.299158	−	0.954204i	$-0.403294\pi$
$710$		0			0
$711$	6.72792	−	11.6531i	0.252317	−	0.437026i
$712$		0			0
$713$	−8.97056			−0.335950
$714$		0			0
$715$	6.24264			0.233462
$716$		0			0
$717$	4.60051	−	7.96831i	0.171809	−	0.297582i
$718$		0			0
$719$	−9.24264	−	16.0087i	−0.344692	−	0.597025i	0.640605	−	0.767870i	$-0.278683\pi$
$719$	−9.24264	−	16.0087i	−0.344692	−	0.597025i	−0.985298	+	0.170846i	$0.945350\pi$
$720$		0			0
$721$	17.1630	+	22.1324i	0.639182	+	0.824255i
$722$		0			0
$723$	0.778175	+	1.34784i	0.0289406	+	0.0501266i
$724$		0			0
$725$	28.8137	−	49.9068i	1.07011	−	1.85349i
$726$		0			0
$727$	48.4264			1.79604			0.898018	−	0.439959i	$-0.145007\pi$
$727$	48.4264			1.79604			0.898018	+	0.439959i	$0.145007\pi$
$728$		0			0
$729$	−18.1716			−0.673021
$730$		0			0
$731$	−21.6569	+	37.5108i	−0.801008	+	1.38739i
$732$		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
$735$	9.53553	−	2.65962i	0.351723	−	0.0981017i
$736$		0			0
$737$	−5.62132	−	9.73641i	−0.207064	−	0.358645i
$738$		0			0
$739$	16.2132	−	28.0821i	0.596412	−	1.03302i	−0.396934	−	0.917847i	$-0.629926\pi$
$739$	16.2132	−	28.0821i	0.596412	−	1.03302i	0.993346	−	0.115169i	$-0.0367410\pi$
$740$		0			0
$741$	−2.58579			−0.0949912
$742$		0			0
$743$	−9.31371			−0.341687			−0.170843	−	0.985298i	$-0.554649\pi$
$743$	−9.31371			−0.341687			−0.170843	+	0.985298i	$0.554649\pi$
$744$		0			0
$745$	10.8284	−	18.7554i	0.396723	−	0.687144i
$746$		0			0
$747$	−22.8284	−	39.5400i	−0.835248	−	1.44669i
$748$		0			0
$749$	17.9497	+	23.1471i	0.655869	+	0.845775i
$750$		0			0
$751$	17.1716	+	29.7420i	0.626600	+	1.08530i	0.988229	+	0.152980i	$0.0488872\pi$
$751$	17.1716	+	29.7420i	0.626600	+	1.08530i	−0.361630	+	0.932322i	$0.617780\pi$
$752$		0			0
$753$	−0.443651	+	0.768426i	−0.0161675	+	0.0280030i
$754$		0			0
$755$	33.2132			1.20875
$756$		0			0
$757$	19.6569			0.714441			0.357220	−	0.934020i	$-0.383725\pi$
$757$	19.6569			0.714441			0.357220	+	0.934020i	$0.383725\pi$
$758$		0			0
$759$	0.464466	−	0.804479i	0.0168591	−	0.0292007i
$760$		0			0
$761$	1.48528	+	2.57258i	0.0538414	+	0.0932561i	0.891690	−	0.452647i	$-0.149520\pi$
$761$	1.48528	+	2.57258i	0.0538414	+	0.0932561i	−0.837849	+	0.545903i	$0.816187\pi$
$762$		0			0
$763$	0.485281	−	1.18869i	0.0175684	−	0.0430335i
$764$		0			0
$765$	36.9706	+	64.0349i	1.33667	+	2.31519i
$766$		0			0
$767$	−7.69239	+	13.3236i	−0.277756	+	0.481088i
$768$		0			0
$769$	−10.9706			−0.395609			−0.197804	−	0.980242i	$-0.563381\pi$
$769$	−10.9706			−0.395609			−0.197804	+	0.980242i	$0.563381\pi$
$770$		0			0
$771$	−1.30152			−0.0468729
$772$		0			0
$773$	−22.9706	+	39.7862i	−0.826194	+	1.43101i	0.0748099	+	0.997198i	$0.476165\pi$
$773$	−22.9706	+	39.7862i	−0.826194	+	1.43101i	−0.901004	+	0.433812i	$0.857168\pi$
$774$		0			0
$775$	−13.3137	−	23.0600i	−0.478243	−	0.828340i
$776$		0			0
$777$	7.15076	−	0.978559i	0.256532	−	0.0351056i
$778$		0			0
$779$	4.41421	+	7.64564i	0.158156	+	0.273934i
$780$		0			0
$781$	−1.53553	+	2.65962i	−0.0549457	+	0.0951688i
$782$		0			0
$783$	−20.8995			−0.746887
$784$		0			0
$785$	−21.6569			−0.772966
$786$		0			0
$787$	−8.77817	+	15.2042i	−0.312908	+	0.541973i	−0.978991	−	0.203905i	$-0.934637\pi$
$787$	−8.77817	+	15.2042i	−0.312908	+	0.541973i	0.666082	+	0.745878i	$0.267970\pi$
$788$		0			0
$789$	−6.42893	−	11.1352i	−0.228876	−	0.396425i
$790$		0			0
$791$	−36.2487	+	4.96053i	−1.28886	+	0.176376i

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)

\(f(q)\)	\(=\)	\(q+(0.207107 - 0.358719i) q^{3} +(1.70711 + 2.95680i) q^{5} +(2.62132 - 0.358719i) q^{7} +(1.41421 + 2.44949i) q^{9} +O(q^{10})\)	\(q+(0.207107 - 0.358719i) q^{3} +(1.70711 + 2.95680i) q^{5} +(2.62132 - 0.358719i) q^{7} +(1.41421 + 2.44949i) q^{9} +(0.500000 - 0.866025i) q^{11} +1.82843 q^{13} +1.41421 q^{15} +(3.82843 - 6.63103i) q^{17} +(-1.70711 - 2.95680i) q^{19} +(0.414214 - 1.01461i) q^{21} +(1.12132 + 1.94218i) q^{23} +(-3.32843 + 5.76500i) q^{25} +2.41421 q^{27} -8.65685 q^{29} +(-2.00000 + 3.46410i) q^{31} +(-0.207107 - 0.358719i) q^{33} +(5.53553 + 7.13834i) q^{35} +(3.29289 + 5.70346i) q^{37} +(0.378680 - 0.655892i) q^{39} -2.58579 q^{41} -5.65685 q^{43} +(-4.82843 + 8.36308i) q^{45} +(3.24264 + 5.61642i) q^{47} +(6.74264 - 1.88064i) q^{49} +(-1.58579 - 2.74666i) q^{51} +(5.94975 - 10.3053i) q^{53} +3.41421 q^{55} -1.41421 q^{57} +(-4.20711 + 7.28692i) q^{59} +(-3.08579 - 5.34474i) q^{61} +(4.58579 + 5.91359i) q^{63} +(3.12132 + 5.40629i) q^{65} +(5.62132 - 9.73641i) q^{67} +0.928932 q^{69} -3.07107 q^{71} +(3.29289 - 5.70346i) q^{73} +(1.37868 + 2.38794i) q^{75} +(1.00000 - 2.44949i) q^{77} +(-2.37868 - 4.11999i) q^{79} +(-3.74264 + 6.48244i) q^{81} -16.1421 q^{83} +26.1421 q^{85} +(-1.79289 + 3.10538i) q^{87} +(2.24264 + 3.88437i) q^{89} +(4.79289 - 0.655892i) q^{91} +(0.828427 + 1.43488i) q^{93} +(5.82843 - 10.0951i) q^{95} +1.82843 q^{97} +2.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{3} + 4 q^{5} + 2 q^{7}+O(q^{10}) \) 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

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more