LMFDB - Embedded newform 1512.2.s.j.1297.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1512,2,Mod(865,1512)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1512, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))

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

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

chi := DirichletCharacter("1512.865");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1512 = 2^{3} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1512.s (of order $3$, degree $2$, 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:	$12.0733807856$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			1297.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1512.1297
Dual form			1512.2.s.j.865.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.00000 - 3.46410i) q^{5} +(2.00000 + 1.73205i) q^{7} +O(q^{10})$	$q+(2.00000 - 3.46410i) q^{5} +(2.00000 + 1.73205i) q^{7} +(-1.00000 - 1.73205i) q^{11} +5.00000 q^{13} +(-3.00000 - 5.19615i) q^{17} +(2.00000 - 3.46410i) q^{19} +(-3.00000 + 5.19615i) q^{23} +(-5.50000 - 9.52628i) q^{25} +6.00000 q^{29} +(3.50000 + 6.06218i) q^{31} +(10.0000 - 3.46410i) q^{35} +(-3.50000 + 6.06218i) q^{37} +2.00000 q^{41} -7.00000 q^{43} +(1.00000 - 1.73205i) q^{47} +(1.00000 + 6.92820i) q^{49} +(-3.00000 - 5.19615i) q^{53} -8.00000 q^{55} +(-3.00000 - 5.19615i) q^{59} +(4.50000 - 7.79423i) q^{61} +(10.0000 - 17.3205i) q^{65} +(3.50000 + 6.06218i) q^{67} -8.00000 q^{71} +(-5.00000 - 8.66025i) q^{73} +(1.00000 - 5.19615i) q^{77} +(-0.500000 + 0.866025i) q^{79} +14.0000 q^{83} -24.0000 q^{85} +(-6.00000 + 10.3923i) q^{89} +(10.0000 + 8.66025i) q^{91} +(-8.00000 - 13.8564i) q^{95} -15.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{5} + 4 q^{7}+O(q^{10}) $ 2 * q + 4 * q^5 + 4 * q^7	$ 2 q + 4 q^{5} + 4 q^{7} - 2 q^{11} + 10 q^{13} - 6 q^{17} + 4 q^{19} - 6 q^{23} - 11 q^{25} + 12 q^{29} + 7 q^{31} + 20 q^{35} - 7 q^{37} + 4 q^{41} - 14 q^{43} + 2 q^{47} + 2 q^{49} - 6 q^{53} - 16 q^{55} - 6 q^{59} + 9 q^{61} + 20 q^{65} + 7 q^{67} - 16 q^{71} - 10 q^{73} + 2 q^{77} - q^{79} + 28 q^{83} - 48 q^{85} - 12 q^{89} + 20 q^{91} - 16 q^{95} - 30 q^{97}+O(q^{100}) $ 2 * q + 4 * q^5 + 4 * q^7 - 2 * q^11 + 10 * q^13 - 6 * q^17 + 4 * q^19 - 6 * q^23 - 11 * q^25 + 12 * q^29 + 7 * q^31 + 20 * q^35 - 7 * q^37 + 4 * q^41 - 14 * q^43 + 2 * q^47 + 2 * q^49 - 6 * q^53 - 16 * q^55 - 6 * q^59 + 9 * q^61 + 20 * q^65 + 7 * q^67 - 16 * q^71 - 10 * q^73 + 2 * q^77 - q^79 + 28 * q^83 - 48 * q^85 - 12 * q^89 + 20 * q^91 - 16 * q^95 - 30 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1512\mathbb{Z}\right)^\times$.

$n$	$757$	$785$	$1081$	$1135$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{3}\right)$	$1$

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			0
$3$		0			0
$4$		0			0
$5$	2.00000	−	3.46410i	0.894427	−	1.54919i	0.0599153	−	0.998203i	$-0.480917\pi$
$5$	2.00000	−	3.46410i	0.894427	−	1.54919i	0.834512	−	0.550990i	$-0.185750\pi$
$6$		0			0
$7$	2.00000	+	1.73205i	0.755929	+	0.654654i
$7$	2.00000	+	1.73205i	0.755929	+	0.654654i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.00000	−	1.73205i	−0.301511	−	0.522233i	0.674967	−	0.737848i	$-0.264158\pi$
$11$	−1.00000	−	1.73205i	−0.301511	−	0.522233i	−0.976478	+	0.215615i	$0.930824\pi$
$12$		0			0
$13$	5.00000			1.38675			0.693375	−	0.720577i	$-0.256123\pi$
$13$	5.00000			1.38675			0.693375	+	0.720577i	$0.256123\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−3.00000	−	5.19615i	−0.727607	−	1.26025i	−0.957892	−	0.287129i	$-0.907299\pi$
$17$	−3.00000	−	5.19615i	−0.727607	−	1.26025i	0.230285	−	0.973123i	$-0.426034\pi$
$18$		0			0
$19$	2.00000	−	3.46410i	0.458831	−	0.794719i	−0.540068	−	0.841621i	$-0.681602\pi$
$19$	2.00000	−	3.46410i	0.458831	−	0.794719i	0.998899	+	0.0469020i	$0.0149348\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−3.00000	+	5.19615i	−0.625543	+	1.08347i	0.362892	+	0.931831i	$0.381789\pi$
$23$	−3.00000	+	5.19615i	−0.625543	+	1.08347i	−0.988436	+	0.151642i	$0.951544\pi$
$24$		0			0
$25$	−5.50000	−	9.52628i	−1.10000	−	1.90526i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	6.00000			1.11417			0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000			1.11417			0.557086	+	0.830455i	$0.311919\pi$
$30$		0			0
$31$	3.50000	+	6.06218i	0.628619	+	1.08880i	0.987829	+	0.155543i	$0.0497126\pi$
$31$	3.50000	+	6.06218i	0.628619	+	1.08880i	−0.359211	+	0.933257i	$0.616954\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	10.0000	−	3.46410i	1.69031	−	0.585540i
$36$		0			0
$37$	−3.50000	+	6.06218i	−0.575396	+	0.996616i	0.420602	+	0.907245i	$0.361819\pi$
$37$	−3.50000	+	6.06218i	−0.575396	+	0.996616i	−0.995998	+	0.0893706i	$0.971514\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	2.00000			0.312348			0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000			0.312348			0.156174	+	0.987730i	$0.450084\pi$
$42$		0			0
$43$	−7.00000			−1.06749			−0.533745	−	0.845645i	$-0.679216\pi$
$43$	−7.00000			−1.06749			−0.533745	+	0.845645i	$0.679216\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	1.00000	−	1.73205i	0.145865	−	0.252646i	−0.783830	−	0.620975i	$-0.786737\pi$
$47$	1.00000	−	1.73205i	0.145865	−	0.252646i	0.929695	+	0.368329i	$0.120070\pi$
$48$		0			0
$49$	1.00000	+	6.92820i	0.142857	+	0.989743i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−3.00000	−	5.19615i	−0.412082	−	0.713746i	0.583036	−	0.812447i	$-0.301865\pi$
$53$	−3.00000	−	5.19615i	−0.412082	−	0.713746i	−0.995117	+	0.0987002i	$0.968532\pi$
$54$		0			0
$55$	−8.00000			−1.07872
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−3.00000	−	5.19615i	−0.390567	−	0.676481i	0.601958	−	0.798528i	$-0.294388\pi$
$59$	−3.00000	−	5.19615i	−0.390567	−	0.676481i	−0.992524	+	0.122047i	$0.961054\pi$
$60$		0			0
$61$	4.50000	−	7.79423i	0.576166	−	0.997949i	−0.419748	−	0.907641i	$-0.637882\pi$
$61$	4.50000	−	7.79423i	0.576166	−	0.997949i	0.995914	−	0.0903080i	$-0.0287851\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	10.0000	−	17.3205i	1.24035	−	2.14834i
$66$		0			0
$67$	3.50000	+	6.06218i	0.427593	+	0.740613i	0.996659	−	0.0816792i	$-0.0260283\pi$
$67$	3.50000	+	6.06218i	0.427593	+	0.740613i	−0.569066	+	0.822292i	$0.692695\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−8.00000			−0.949425			−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000			−0.949425			−0.474713	+	0.880141i	$0.657448\pi$
$72$		0			0
$73$	−5.00000	−	8.66025i	−0.585206	−	1.01361i	−0.994850	−	0.101361i	$-0.967680\pi$
$73$	−5.00000	−	8.66025i	−0.585206	−	1.01361i	0.409644	−	0.912245i	$-0.365653\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	1.00000	−	5.19615i	0.113961	−	0.592157i
$78$		0			0
$79$	−0.500000	+	0.866025i	−0.0562544	+	0.0974355i	−0.892781	−	0.450490i	$-0.851249\pi$
$79$	−0.500000	+	0.866025i	−0.0562544	+	0.0974355i	0.836527	+	0.547926i	$0.184582\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	14.0000			1.53670			0.768350	−	0.640030i	$-0.221078\pi$
$83$	14.0000			1.53670			0.768350	+	0.640030i	$0.221078\pi$
$84$		0			0
$85$	−24.0000			−2.60317
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−6.00000	+	10.3923i	−0.635999	+	1.10158i	0.350304	+	0.936636i	$0.386078\pi$
$89$	−6.00000	+	10.3923i	−0.635999	+	1.10158i	−0.986303	+	0.164946i	$0.947255\pi$
$90$		0			0
$91$	10.0000	+	8.66025i	1.04828	+	0.907841i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−8.00000	−	13.8564i	−0.820783	−	1.42164i
$96$		0			0
$97$	−15.0000			−1.52302			−0.761510	−	0.648154i	$-0.775541\pi$
$97$	−15.0000			−1.52302			−0.761510	+	0.648154i	$0.775541\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	8.00000	+	13.8564i	0.796030	+	1.37876i	0.922183	+	0.386753i	$0.126403\pi$
$101$	8.00000	+	13.8564i	0.796030	+	1.37876i	−0.126153	+	0.992011i	$0.540263\pi$
$102$		0			0
$103$	6.50000	−	11.2583i	0.640464	−	1.10932i	−0.344865	−	0.938652i	$-0.612075\pi$
$103$	6.50000	−	11.2583i	0.640464	−	1.10932i	0.985329	−	0.170664i	$-0.0545913\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	3.00000	−	5.19615i	0.290021	−	0.502331i	−0.683793	−	0.729676i	$-0.739671\pi$
$107$	3.00000	−	5.19615i	0.290021	−	0.502331i	0.973814	+	0.227345i	$0.0730044\pi$
$108$		0			0
$109$	−2.50000	−	4.33013i	−0.239457	−	0.414751i	0.721102	−	0.692829i	$-0.243636\pi$
$109$	−2.50000	−	4.33013i	−0.239457	−	0.414751i	−0.960558	+	0.278078i	$0.910303\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	8.00000			0.752577			0.376288	−	0.926503i	$-0.377200\pi$
$113$	8.00000			0.752577			0.376288	+	0.926503i	$0.377200\pi$
$114$		0			0
$115$	12.0000	+	20.7846i	1.11901	+	1.93817i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	3.00000	−	15.5885i	0.275010	−	1.42899i
$120$		0			0
$121$	3.50000	−	6.06218i	0.318182	−	0.551107i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−24.0000			−2.14663
$126$		0			0
$127$	13.0000			1.15356			0.576782	−	0.816898i	$-0.304308\pi$
$127$	13.0000			1.15356			0.576782	+	0.816898i	$0.304308\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	2.00000	−	3.46410i	0.174741	−	0.302660i	−0.765331	−	0.643637i	$-0.777425\pi$
$131$	2.00000	−	3.46410i	0.174741	−	0.302660i	0.940072	+	0.340977i	$0.110758\pi$
$132$		0			0
$133$	10.0000	−	3.46410i	0.867110	−	0.300376i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−3.00000	−	5.19615i	−0.256307	−	0.443937i	0.708942	−	0.705266i	$-0.249173\pi$
$137$	−3.00000	−	5.19615i	−0.256307	−	0.443937i	−0.965250	+	0.261329i	$0.915839\pi$
$138$		0			0
$139$	7.00000			0.593732			0.296866	−	0.954919i	$-0.404058\pi$
$139$	7.00000			0.593732			0.296866	+	0.954919i	$0.404058\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−5.00000	−	8.66025i	−0.418121	−	0.724207i
$144$		0			0
$145$	12.0000	−	20.7846i	0.996546	−	1.72607i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$149$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$150$		0			0
$151$	4.50000	+	7.79423i	0.366205	+	0.634285i	0.988969	−	0.148124i	$-0.0473236\pi$
$151$	4.50000	+	7.79423i	0.366205	+	0.634285i	−0.622764	+	0.782410i	$0.713990\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	28.0000			2.24901
$156$		0			0
$157$	3.00000	+	5.19615i	0.239426	+	0.414698i	0.960550	−	0.278108i	$-0.0897074\pi$
$157$	3.00000	+	5.19615i	0.239426	+	0.414698i	−0.721124	+	0.692806i	$0.756374\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−15.0000	+	5.19615i	−1.18217	+	0.409514i
$162$		0			0
$163$	−10.5000	+	18.1865i	−0.822423	+	1.42448i	0.0814491	+	0.996678i	$0.474045\pi$
$163$	−10.5000	+	18.1865i	−0.822423	+	1.42448i	−0.903873	+	0.427802i	$0.859288\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−2.00000			−0.154765			−0.0773823	−	0.997001i	$-0.524656\pi$
$167$	−2.00000			−0.154765			−0.0773823	+	0.997001i	$0.524656\pi$
$168$		0			0
$169$	12.0000			0.923077
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$173$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$174$		0			0
$175$	5.50000	−	28.5788i	0.415761	−	2.16036i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	2.00000	+	3.46410i	0.149487	+	0.258919i	0.931038	−	0.364922i	$-0.118904\pi$
$179$	2.00000	+	3.46410i	0.149487	+	0.258919i	−0.781551	+	0.623841i	$0.785571\pi$
$180$		0			0
$181$	−2.00000			−0.148659			−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000			−0.148659			−0.0743294	+	0.997234i	$0.523682\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	14.0000	+	24.2487i	1.02930	+	1.78280i
$186$		0			0
$187$	−6.00000	+	10.3923i	−0.438763	+	0.759961i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−10.0000	+	17.3205i	−0.723575	+	1.25327i	0.235983	+	0.971757i	$0.424169\pi$
$191$	−10.0000	+	17.3205i	−0.723575	+	1.25327i	−0.959558	+	0.281511i	$0.909164\pi$
$192$		0			0
$193$	−3.50000	−	6.06218i	−0.251936	−	0.436365i	0.712123	−	0.702055i	$-0.247734\pi$
$193$	−3.50000	−	6.06218i	−0.251936	−	0.436365i	−0.964059	+	0.265689i	$0.914400\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	12.0000			0.854965			0.427482	−	0.904024i	$-0.359401\pi$
$197$	12.0000			0.854965			0.427482	+	0.904024i	$0.359401\pi$
$198$		0			0
$199$	1.50000	+	2.59808i	0.106332	+	0.184173i	0.914282	−	0.405079i	$-0.132756\pi$
$199$	1.50000	+	2.59808i	0.106332	+	0.184173i	−0.807950	+	0.589252i	$0.799423\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	12.0000	+	10.3923i	0.842235	+	0.729397i
$204$		0			0
$205$	4.00000	−	6.92820i	0.279372	−	0.483887i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−8.00000			−0.553372
$210$		0			0
$211$	−27.0000			−1.85876			−0.929378	−	0.369129i	$-0.879656\pi$
$211$	−27.0000			−1.85876			−0.929378	+	0.369129i	$0.879656\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−14.0000	+	24.2487i	−0.954792	+	1.65375i
$216$		0			0
$217$	−3.50000	+	18.1865i	−0.237595	+	1.23458i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−15.0000	−	25.9808i	−1.00901	−	1.74766i
$222$		0			0
$223$	24.0000			1.60716			0.803579	−	0.595198i	$-0.202926\pi$
$223$	24.0000			1.60716			0.803579	+	0.595198i	$0.202926\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−5.00000	−	8.66025i	−0.331862	−	0.574801i	0.651015	−	0.759065i	$-0.274343\pi$
$227$	−5.00000	−	8.66025i	−0.331862	−	0.574801i	−0.982877	+	0.184263i	$0.941010\pi$
$228$		0			0
$229$	−6.50000	+	11.2583i	−0.429532	+	0.743971i	−0.996832	−	0.0795401i	$-0.974655\pi$
$229$	−6.50000	+	11.2583i	−0.429532	+	0.743971i	0.567300	+	0.823511i	$0.307988\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−4.00000	+	6.92820i	−0.262049	+	0.453882i	−0.966786	−	0.255586i	$-0.917731\pi$
$233$	−4.00000	+	6.92820i	−0.262049	+	0.453882i	0.704737	+	0.709468i	$0.251065\pi$
$234$		0			0
$235$	−4.00000	−	6.92820i	−0.260931	−	0.451946i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−4.00000			−0.258738			−0.129369	−	0.991596i	$-0.541295\pi$
$239$	−4.00000			−0.258738			−0.129369	+	0.991596i	$0.541295\pi$
$240$		0			0
$241$	−0.500000	−	0.866025i	−0.0322078	−	0.0557856i	0.849472	−	0.527633i	$-0.176921\pi$
$241$	−0.500000	−	0.866025i	−0.0322078	−	0.0557856i	−0.881680	+	0.471848i	$0.843587\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	26.0000	+	10.3923i	1.66108	+	0.663940i
$246$		0			0
$247$	10.0000	−	17.3205i	0.636285	−	1.10208i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−30.0000			−1.89358			−0.946792	−	0.321847i	$-0.895696\pi$
$251$	−30.0000			−1.89358			−0.946792	+	0.321847i	$0.895696\pi$
$252$		0			0
$253$	12.0000			0.754434
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−7.00000	+	12.1244i	−0.436648	+	0.756297i	−0.997429	−	0.0716680i	$-0.977168\pi$
$257$	−7.00000	+	12.1244i	−0.436648	+	0.756297i	0.560781	+	0.827964i	$0.310501\pi$
$258$		0			0
$259$	−17.5000	+	6.06218i	−1.08740	+	0.376685i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	3.00000	+	5.19615i	0.184988	+	0.320408i	0.943572	−	0.331166i	$-0.107442\pi$
$263$	3.00000	+	5.19615i	0.184988	+	0.320408i	−0.758585	+	0.651575i	$0.774109\pi$
$264$		0			0
$265$	−24.0000			−1.47431
$266$		0			0
$267$		0			0
$268$		0			0
$269$	16.0000	+	27.7128i	0.975537	+	1.68968i	0.678151	+	0.734923i	$0.262782\pi$
$269$	16.0000	+	27.7128i	0.975537	+	1.68968i	0.297386	+	0.954757i	$0.403885\pi$
$270$		0			0
$271$	−7.50000	+	12.9904i	−0.455593	+	0.789109i	−0.998722	−	0.0505395i	$-0.983906\pi$
$271$	−7.50000	+	12.9904i	−0.455593	+	0.789109i	0.543130	+	0.839649i	$0.317239\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−11.0000	+	19.0526i	−0.663325	+	1.14891i
$276$		0			0
$277$	−0.500000	−	0.866025i	−0.0300421	−	0.0520344i	0.850613	−	0.525792i	$-0.176231\pi$
$277$	−0.500000	−	0.866025i	−0.0300421	−	0.0520344i	−0.880656	+	0.473757i	$0.842897\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	24.0000			1.43172			0.715860	−	0.698244i	$-0.246035\pi$
$281$	24.0000			1.43172			0.715860	+	0.698244i	$0.246035\pi$
$282$		0			0
$283$	5.50000	+	9.52628i	0.326941	+	0.566279i	0.981903	−	0.189383i	$-0.0606488\pi$
$283$	5.50000	+	9.52628i	0.326941	+	0.566279i	−0.654962	+	0.755662i	$0.727315\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	4.00000	+	3.46410i	0.236113	+	0.204479i
$288$		0			0
$289$	−9.50000	+	16.4545i	−0.558824	+	0.967911i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−30.0000			−1.75262			−0.876309	−	0.481749i	$-0.840002\pi$
$293$	−30.0000			−1.75262			−0.876309	+	0.481749i	$0.840002\pi$
$294$		0			0
$295$	−24.0000			−1.39733
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−15.0000	+	25.9808i	−0.867472	+	1.50251i
$300$		0			0
$301$	−14.0000	−	12.1244i	−0.806947	−	0.698836i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−18.0000	−	31.1769i	−1.03068	−	1.78518i
$306$		0			0
$307$	7.00000			0.399511			0.199756	−	0.979846i	$-0.435985\pi$
$307$	7.00000			0.399511			0.199756	+	0.979846i	$0.435985\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	8.00000	+	13.8564i	0.453638	+	0.785725i	0.998609	−	0.0527306i	$-0.0167924\pi$
$311$	8.00000	+	13.8564i	0.453638	+	0.785725i	−0.544970	+	0.838455i	$0.683459\pi$
$312$		0			0
$313$	−5.00000	+	8.66025i	−0.282617	+	0.489506i	−0.972028	−	0.234863i	$-0.924536\pi$
$313$	−5.00000	+	8.66025i	−0.282617	+	0.489506i	0.689412	+	0.724370i	$0.257869\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	3.00000	−	5.19615i	0.168497	−	0.291845i	−0.769395	−	0.638774i	$-0.779442\pi$
$317$	3.00000	−	5.19615i	0.168497	−	0.291845i	0.937892	+	0.346929i	$0.112775\pi$
$318$		0			0
$319$	−6.00000	−	10.3923i	−0.335936	−	0.581857i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−24.0000			−1.33540
$324$		0			0
$325$	−27.5000	−	47.6314i	−1.52543	−	2.64211i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	5.00000	−	1.73205i	0.275659	−	0.0954911i
$330$		0			0
$331$	2.00000	−	3.46410i	0.109930	−	0.190404i	−0.805812	−	0.592172i	$-0.798271\pi$
$331$	2.00000	−	3.46410i	0.109930	−	0.190404i	0.915742	+	0.401768i	$0.131604\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	28.0000			1.52980
$336$		0			0
$337$	26.0000			1.41631			0.708155	−	0.706057i	$-0.249528\pi$
$337$	26.0000			1.41631			0.708155	+	0.706057i	$0.249528\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	7.00000	−	12.1244i	0.379071	−	0.656571i
$342$		0			0
$343$	−10.0000	+	15.5885i	−0.539949	+	0.841698i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$347$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$348$		0			0
$349$	15.0000			0.802932			0.401466	−	0.915874i	$-0.368501\pi$
$349$	15.0000			0.802932			0.401466	+	0.915874i	$0.368501\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−14.0000	−	24.2487i	−0.745145	−	1.29063i	−0.950127	−	0.311863i	$-0.899047\pi$
$353$	−14.0000	−	24.2487i	−0.745145	−	1.29063i	0.204982	−	0.978766i	$-0.434286\pi$
$354$		0			0
$355$	−16.0000	+	27.7128i	−0.849192	+	1.47084i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−3.00000	+	5.19615i	−0.158334	+	0.274242i	−0.934268	−	0.356572i	$-0.883946\pi$
$359$	−3.00000	+	5.19615i	−0.158334	+	0.274242i	0.775934	+	0.630814i	$0.217279\pi$
$360$		0			0
$361$	1.50000	+	2.59808i	0.0789474	+	0.136741i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−40.0000			−2.09370
$366$		0			0
$367$	−14.0000	−	24.2487i	−0.730794	−	1.26577i	−0.956544	−	0.291587i	$-0.905817\pi$
$367$	−14.0000	−	24.2487i	−0.730794	−	1.26577i	0.225750	−	0.974185i	$-0.427517\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	3.00000	−	15.5885i	0.155752	−	0.809312i
$372$		0			0
$373$	−5.00000	+	8.66025i	−0.258890	+	0.448411i	−0.965945	−	0.258748i	$-0.916690\pi$
$373$	−5.00000	+	8.66025i	−0.258890	+	0.448411i	0.707055	+	0.707159i	$0.250023\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	30.0000			1.54508
$378$		0			0
$379$	17.0000			0.873231			0.436616	−	0.899648i	$-0.356177\pi$
$379$	17.0000			0.873231			0.436616	+	0.899648i	$0.356177\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−15.0000	+	25.9808i	−0.766464	+	1.32755i	0.173005	+	0.984921i	$0.444652\pi$
$383$	−15.0000	+	25.9808i	−0.766464	+	1.32755i	−0.939469	+	0.342634i	$0.888681\pi$
$384$		0			0
$385$	−16.0000	−	13.8564i	−0.815436	−	0.706188i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	10.0000	+	17.3205i	0.507020	+	0.878185i	0.999967	+	0.00812520i	$0.00258636\pi$
$389$	10.0000	+	17.3205i	0.507020	+	0.878185i	−0.492947	+	0.870059i	$0.664080\pi$
$390$		0			0
$391$	36.0000			1.82060
$392$		0			0
$393$		0			0
$394$		0			0
$395$	2.00000	+	3.46410i	0.100631	+	0.174298i
$396$		0			0
$397$	2.50000	−	4.33013i	0.125471	−	0.217323i	−0.796446	−	0.604710i	$-0.793289\pi$
$397$	2.50000	−	4.33013i	0.125471	−	0.217323i	0.921917	+	0.387387i	$0.126622\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−9.00000	+	15.5885i	−0.449439	+	0.778450i	−0.998350	−	0.0574304i	$-0.981709\pi$
$401$	−9.00000	+	15.5885i	−0.449439	+	0.778450i	0.548911	+	0.835881i	$0.315043\pi$
$402$		0			0
$403$	17.5000	+	30.3109i	0.871737	+	1.50989i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	14.0000			0.693954
$408$		0			0
$409$	17.5000	+	30.3109i	0.865319	+	1.49878i	0.866730	+	0.498778i	$0.166218\pi$
$409$	17.5000	+	30.3109i	0.865319	+	1.49878i	−0.00141047	+	0.999999i	$0.500449\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	3.00000	−	15.5885i	0.147620	−	0.767058i
$414$		0			0
$415$	28.0000	−	48.4974i	1.37447	−	2.38064i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−4.00000			−0.195413			−0.0977064	−	0.995215i	$-0.531151\pi$
$419$	−4.00000			−0.195413			−0.0977064	+	0.995215i	$0.531151\pi$
$420$		0			0
$421$	22.0000			1.07221			0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000			1.07221			0.536107	+	0.844150i	$0.319894\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−33.0000	+	57.1577i	−1.60074	+	2.77255i
$426$		0			0
$427$	22.5000	−	7.79423i	1.08885	−	0.377189i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−5.00000	−	8.66025i	−0.240842	−	0.417150i	0.720113	−	0.693857i	$-0.244090\pi$
$431$	−5.00000	−	8.66025i	−0.240842	−	0.417150i	−0.960954	+	0.276707i	$0.910757\pi$
$432$		0			0
$433$	7.00000			0.336399			0.168199	−	0.985753i	$-0.446205\pi$
$433$	7.00000			0.336399			0.168199	+	0.985753i	$0.446205\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	12.0000	+	20.7846i	0.574038	+	0.994263i
$438$		0			0
$439$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$439$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−12.0000	+	20.7846i	−0.570137	+	0.987507i	0.426414	+	0.904528i	$0.359777\pi$
$443$	−12.0000	+	20.7846i	−0.570137	+	0.987507i	−0.996551	+	0.0829786i	$0.973557\pi$
$444$		0			0
$445$	24.0000	+	41.5692i	1.13771	+	1.97057i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	28.0000			1.32140			0.660701	−	0.750649i	$-0.270259\pi$
$449$	28.0000			1.32140			0.660701	+	0.750649i	$0.270259\pi$
$450$		0			0
$451$	−2.00000	−	3.46410i	−0.0941763	−	0.163118i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	50.0000	−	17.3205i	2.34404	−	0.811998i
$456$		0			0
$457$	5.50000	−	9.52628i	0.257279	−	0.445621i	−0.708233	−	0.705979i	$-0.750507\pi$
$457$	5.50000	−	9.52628i	0.257279	−	0.445621i	0.965512	+	0.260358i	$0.0838407\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−6.00000			−0.279448			−0.139724	−	0.990190i	$-0.544622\pi$
$461$	−6.00000			−0.279448			−0.139724	+	0.990190i	$0.544622\pi$
$462$		0			0
$463$	8.00000			0.371792			0.185896	−	0.982569i	$-0.440481\pi$
$463$	8.00000			0.371792			0.185896	+	0.982569i	$0.440481\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−20.0000	+	34.6410i	−0.925490	+	1.60300i	−0.134718	+	0.990884i	$0.543013\pi$
$467$	−20.0000	+	34.6410i	−0.925490	+	1.60300i	−0.790772	+	0.612111i	$0.790321\pi$
$468$		0			0
$469$	−3.50000	+	18.1865i	−0.161615	+	0.839776i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	7.00000	+	12.1244i	0.321860	+	0.557478i
$474$		0			0
$475$	−44.0000			−2.01886
$476$		0			0
$477$		0			0
$478$		0			0
$479$	12.0000	+	20.7846i	0.548294	+	0.949673i	0.998392	+	0.0566937i	$0.0180558\pi$
$479$	12.0000	+	20.7846i	0.548294	+	0.949673i	−0.450098	+	0.892979i	$0.648611\pi$
$480$		0			0
$481$	−17.5000	+	30.3109i	−0.797931	+	1.38206i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−30.0000	+	51.9615i	−1.36223	+	2.35945i
$486$		0			0
$487$	−4.00000	−	6.92820i	−0.181257	−	0.313947i	0.761052	−	0.648691i	$-0.224683\pi$
$487$	−4.00000	−	6.92820i	−0.181257	−	0.313947i	−0.942309	+	0.334744i	$0.891350\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−6.00000			−0.270776			−0.135388	−	0.990793i	$-0.543228\pi$
$491$	−6.00000			−0.270776			−0.135388	+	0.990793i	$0.543228\pi$
$492$		0			0
$493$	−18.0000	−	31.1769i	−0.810679	−	1.40414i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−16.0000	−	13.8564i	−0.717698	−	0.621545i
$498$		0			0
$499$	15.5000	−	26.8468i	0.693875	−	1.20183i	−0.276683	−	0.960961i	$-0.589235\pi$
$499$	15.5000	−	26.8468i	0.693875	−	1.20183i	0.970558	−	0.240866i	$-0.0774314\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−26.0000			−1.15928			−0.579641	−	0.814872i	$-0.696807\pi$
$503$	−26.0000			−1.15928			−0.579641	+	0.814872i	$0.696807\pi$
$504$		0			0
$505$	64.0000			2.84796
$506$		0			0
$507$		0			0
$508$		0			0
$509$	19.0000	−	32.9090i	0.842160	−	1.45866i	−0.0459045	−	0.998946i	$-0.514617\pi$
$509$	19.0000	−	32.9090i	0.842160	−	1.45866i	0.888065	−	0.459718i	$-0.152050\pi$
$510$		0			0
$511$	5.00000	−	25.9808i	0.221187	−	1.14932i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−26.0000	−	45.0333i	−1.14570	−	1.98441i
$516$		0			0
$517$	−4.00000			−0.175920
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−3.00000	−	5.19615i	−0.131432	−	0.227648i	0.792797	−	0.609486i	$-0.208624\pi$
$521$	−3.00000	−	5.19615i	−0.131432	−	0.227648i	−0.924229	+	0.381839i	$0.875291\pi$
$522$		0			0
$523$	−12.5000	+	21.6506i	−0.546587	+	0.946716i	0.451918	+	0.892059i	$0.350740\pi$
$523$	−12.5000	+	21.6506i	−0.546587	+	0.946716i	−0.998505	+	0.0546569i	$0.982594\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	21.0000	−	36.3731i	0.914774	−	1.58444i
$528$		0			0
$529$	−6.50000	−	11.2583i	−0.282609	−	0.489493i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	10.0000			0.433148
$534$		0			0
$535$	−12.0000	−	20.7846i	−0.518805	−	0.898597i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	11.0000	−	8.66025i	0.473804	−	0.373024i
$540$		0			0
$541$	−19.0000	+	32.9090i	−0.816874	+	1.41487i	0.0911008	+	0.995842i	$0.470961\pi$
$541$	−19.0000	+	32.9090i	−0.816874	+	1.41487i	−0.907975	+	0.419025i	$0.862372\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−20.0000			−0.856706
$546$		0			0
$547$	−3.00000			−0.128271			−0.0641354	−	0.997941i	$-0.520429\pi$
$547$	−3.00000			−0.128271			−0.0641354	+	0.997941i	$0.520429\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	12.0000	−	20.7846i	0.511217	−	0.885454i
$552$		0			0
$553$	−2.50000	+	0.866025i	−0.106311	+	0.0368271i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−10.0000	−	17.3205i	−0.423714	−	0.733893i	0.572586	−	0.819845i	$-0.305940\pi$
$557$	−10.0000	−	17.3205i	−0.423714	−	0.733893i	−0.996299	+	0.0859514i	$0.972607\pi$
$558$		0			0
$559$	−35.0000			−1.48034
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−7.00000	−	12.1244i	−0.295015	−	0.510981i	0.679974	−	0.733237i	$-0.261991\pi$
$563$	−7.00000	−	12.1244i	−0.295015	−	0.510981i	−0.974988	+	0.222256i	$0.928658\pi$
$564$		0			0
$565$	16.0000	−	27.7128i	0.673125	−	1.16589i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	20.0000	−	34.6410i	0.838444	−	1.45223i	−0.0527519	−	0.998608i	$-0.516799\pi$
$569$	20.0000	−	34.6410i	0.838444	−	1.45223i	0.891196	−	0.453619i	$-0.149867\pi$
$570$		0			0
$571$	−8.00000	−	13.8564i	−0.334790	−	0.579873i	0.648655	−	0.761083i	$-0.275332\pi$
$571$	−8.00000	−	13.8564i	−0.334790	−	0.579873i	−0.983444	+	0.181210i	$0.941999\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	66.0000			2.75239
$576$		0			0
$577$	−3.50000	−	6.06218i	−0.145707	−	0.252372i	0.783930	−	0.620850i	$-0.213212\pi$
$577$	−3.50000	−	6.06218i	−0.145707	−	0.252372i	−0.929636	+	0.368478i	$0.879879\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	28.0000	+	24.2487i	1.16164	+	1.00601i
$582$		0			0
$583$	−6.00000	+	10.3923i	−0.248495	+	0.430405i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−36.0000			−1.48588			−0.742940	−	0.669359i	$-0.766569\pi$
$587$	−36.0000			−1.48588			−0.742940	+	0.669359i	$0.766569\pi$
$588$		0			0
$589$	28.0000			1.15372
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−20.0000	+	34.6410i	−0.821302	+	1.42254i	0.0834118	+	0.996515i	$0.473418\pi$
$593$	−20.0000	+	34.6410i	−0.821302	+	1.42254i	−0.904713	+	0.426021i	$0.859915\pi$
$594$		0			0
$595$	−48.0000	−	41.5692i	−1.96781	−	1.70417i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−6.00000	−	10.3923i	−0.245153	−	0.424618i	0.717021	−	0.697051i	$-0.245505\pi$
$599$	−6.00000	−	10.3923i	−0.245153	−	0.424618i	−0.962175	+	0.272433i	$0.912172\pi$
$600$		0			0
$601$	23.0000			0.938190			0.469095	−	0.883148i	$-0.344580\pi$
$601$	23.0000			0.938190			0.469095	+	0.883148i	$0.344580\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−14.0000	−	24.2487i	−0.569181	−	0.985850i
$606$		0			0
$607$	−4.00000	+	6.92820i	−0.162355	+	0.281207i	−0.935713	−	0.352763i	$-0.885242\pi$
$607$	−4.00000	+	6.92820i	−0.162355	+	0.281207i	0.773358	+	0.633970i	$0.218576\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	5.00000	−	8.66025i	0.202278	−	0.350356i
$612$		0			0
$613$	10.5000	+	18.1865i	0.424091	+	0.734547i	0.996335	−	0.0855362i	$-0.0272603\pi$
$613$	10.5000	+	18.1865i	0.424091	+	0.734547i	−0.572244	+	0.820083i	$0.693927\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	34.0000			1.36879			0.684394	−	0.729112i	$-0.260067\pi$
$617$	34.0000			1.36879			0.684394	+	0.729112i	$0.260067\pi$
$618$		0			0
$619$	11.5000	+	19.9186i	0.462224	+	0.800595i	0.999071	−	0.0430838i	$-0.0137183\pi$
$619$	11.5000	+	19.9186i	0.462224	+	0.800595i	−0.536847	+	0.843679i	$0.680385\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−30.0000	+	10.3923i	−1.20192	+	0.416359i
$624$		0			0
$625$	−20.5000	+	35.5070i	−0.820000	+	1.42028i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	42.0000			1.67465
$630$		0			0
$631$	−9.00000			−0.358284			−0.179142	−	0.983823i	$-0.557332\pi$
$631$	−9.00000			−0.358284			−0.179142	+	0.983823i	$0.557332\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	26.0000	−	45.0333i	1.03178	−	1.78709i
$636$		0			0
$637$	5.00000	+	34.6410i	0.198107	+	1.37253i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	3.00000	+	5.19615i	0.118493	+	0.205236i	0.919171	−	0.393860i	$-0.128860\pi$
$641$	3.00000	+	5.19615i	0.118493	+	0.205236i	−0.800678	+	0.599095i	$0.795527\pi$
$642$		0			0
$643$	1.00000			0.0394362			0.0197181	−	0.999806i	$-0.493723\pi$
$643$	1.00000			0.0394362			0.0197181	+	0.999806i	$0.493723\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−14.0000	−	24.2487i	−0.550397	−	0.953315i	−0.998246	−	0.0592060i	$-0.981143\pi$
$647$	−14.0000	−	24.2487i	−0.550397	−	0.953315i	0.447849	−	0.894109i	$-0.352190\pi$
$648$		0			0
$649$	−6.00000	+	10.3923i	−0.235521	+	0.407934i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−5.00000	+	8.66025i	−0.195665	+	0.338902i	−0.947118	−	0.320884i	$-0.896020\pi$
$653$	−5.00000	+	8.66025i	−0.195665	+	0.338902i	0.751453	+	0.659786i	$0.229353\pi$
$654$		0			0
$655$	−8.00000	−	13.8564i	−0.312586	−	0.541415i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	36.0000			1.40236			0.701180	−	0.712984i	$-0.252657\pi$
$659$	36.0000			1.40236			0.701180	+	0.712984i	$0.252657\pi$
$660$		0			0
$661$	−1.00000	−	1.73205i	−0.0388955	−	0.0673690i	0.845922	−	0.533306i	$-0.179051\pi$
$661$	−1.00000	−	1.73205i	−0.0388955	−	0.0673690i	−0.884818	+	0.465937i	$0.845717\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	8.00000	−	41.5692i	0.310227	−	1.61199i
$666$		0			0
$667$	−18.0000	+	31.1769i	−0.696963	+	1.20717i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−18.0000			−0.694882
$672$		0			0
$673$	−26.0000			−1.00223			−0.501113	−	0.865382i	$-0.667076\pi$
$673$	−26.0000			−1.00223			−0.501113	+	0.865382i	$0.667076\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−18.0000	+	31.1769i	−0.691796	+	1.19823i	0.279453	+	0.960159i	$0.409847\pi$
$677$	−18.0000	+	31.1769i	−0.691796	+	1.19823i	−0.971249	+	0.238067i	$0.923486\pi$
$678$		0			0
$679$	−30.0000	−	25.9808i	−1.15129	−	0.997050i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−12.0000	−	20.7846i	−0.459167	−	0.795301i	0.539750	−	0.841825i	$-0.318519\pi$
$683$	−12.0000	−	20.7846i	−0.459167	−	0.795301i	−0.998917	+	0.0465244i	$0.985185\pi$
$684$		0			0
$685$	−24.0000			−0.916993
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−15.0000	−	25.9808i	−0.571454	−	0.989788i
$690$		0			0
$691$	−15.5000	+	26.8468i	−0.589648	+	1.02130i	0.404631	+	0.914480i	$0.367400\pi$
$691$	−15.5000	+	26.8468i	−0.589648	+	1.02130i	−0.994278	+	0.106820i	$0.965933\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	14.0000	−	24.2487i	0.531050	−	0.919806i
$696$		0			0
$697$	−6.00000	−	10.3923i	−0.227266	−	0.393637i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−24.0000			−0.906467			−0.453234	−	0.891392i	$-0.649730\pi$
$701$	−24.0000			−0.906467			−0.453234	+	0.891392i	$0.649730\pi$
$702$		0			0
$703$	14.0000	+	24.2487i	0.528020	+	0.914557i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−8.00000	+	41.5692i	−0.300871	+	1.56337i
$708$		0			0
$709$	−4.50000	+	7.79423i	−0.169001	+	0.292718i	−0.938069	−	0.346449i	$-0.887387\pi$
$709$	−4.50000	+	7.79423i	−0.169001	+	0.292718i	0.769068	+	0.639167i	$0.220721\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−42.0000			−1.57291
$714$		0			0
$715$	−40.0000			−1.49592
$716$		0			0
$717$		0			0
$718$		0			0
$719$	20.0000	−	34.6410i	0.745874	−	1.29189i	−0.203911	−	0.978989i	$-0.565365\pi$
$719$	20.0000	−	34.6410i	0.745874	−	1.29189i	0.949785	−	0.312903i	$-0.101301\pi$
$720$		0			0
$721$	32.5000	−	11.2583i	1.21036	−	0.419282i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−33.0000	−	57.1577i	−1.22559	−	2.12278i
$726$		0			0
$727$	19.0000			0.704671			0.352335	−	0.935874i	$-0.385388\pi$
$727$	19.0000			0.704671			0.352335	+	0.935874i	$0.385388\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	21.0000	+	36.3731i	0.776713	+	1.34531i
$732$		0			0
$733$	15.5000	−	26.8468i	0.572506	−	0.991609i	−0.423802	−	0.905755i	$-0.639305\pi$
$733$	15.5000	−	26.8468i	0.572506	−	0.991609i	0.996308	−	0.0858539i	$-0.0273618\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	7.00000	−	12.1244i	0.257848	−	0.446606i
$738$		0			0
$739$	11.5000	+	19.9186i	0.423034	+	0.732717i	0.996235	−	0.0866983i	$-0.0276316\pi$
$739$	11.5000	+	19.9186i	0.423034	+	0.732717i	−0.573200	+	0.819415i	$0.694298\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−34.0000			−1.24734			−0.623670	−	0.781688i	$-0.714359\pi$
$743$	−34.0000			−1.24734			−0.623670	+	0.781688i	$0.714359\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	15.0000	−	5.19615i	0.548088	−	0.189863i
$750$		0			0
$751$	16.0000	−	27.7128i	0.583848	−	1.01125i	−0.411170	−	0.911559i	$-0.634880\pi$
$751$	16.0000	−	27.7128i	0.583848	−	1.01125i	0.995018	−	0.0996961i	$-0.0317870\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	36.0000			1.31017
$756$		0			0
$757$	−11.0000			−0.399802			−0.199901	−	0.979816i	$-0.564062\pi$
$757$	−11.0000			−0.399802			−0.199901	+	0.979816i	$0.564062\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$761$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$762$		0			0
$763$	2.50000	−	12.9904i	0.0905061	−	0.470283i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−15.0000	−	25.9808i	−0.541619	−	0.938111i
$768$		0			0
$769$	−10.0000			−0.360609			−0.180305	−	0.983611i	$-0.557708\pi$
$769$	−10.0000			−0.360609			−0.180305	+	0.983611i	$0.557708\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	8.00000	+	13.8564i	0.287740	+	0.498380i	0.973270	−	0.229664i	$-0.0737628\pi$
$773$	8.00000	+	13.8564i	0.287740	+	0.498380i	−0.685530	+	0.728044i	$0.740429\pi$
$774$		0			0
$775$	38.5000	−	66.6840i	1.38296	−	2.39536i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	4.00000	−	6.92820i	0.143315	−	0.248229i
$780$		0			0
$781$	8.00000	+	13.8564i	0.286263	+	0.495821i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	24.0000			0.856597
$786$		0			0
$787$	−11.5000	−	19.9186i	−0.409931	−	0.710021i	0.584951	−	0.811069i	$-0.301114\pi$
$787$	−11.5000	−	19.9186i	−0.409931	−	0.710021i	−0.994882	+	0.101048i	$0.967780\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	16.0000	+	13.8564i	0.568895	+	0.492677i
$792$		0			0
$793$	22.5000	−	38.9711i	0.798998	−	1.38391i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	42.0000			1.48772			0.743858	−	0.668338i	$-0.232994\pi$
$797$	42.0000			1.48772			0.743858	+	0.668338i	$0.232994\pi$
$798$		0			0
$799$	−12.0000			−0.424529
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−10.0000	+	17.3205i	−0.352892	+	0.611227i
$804$		0			0
$805$	−12.0000	+	62.3538i	−0.422944	+	2.19768i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	5.00000	+	8.66025i	0.175791	+	0.304478i	0.940435	−	0.339975i	$-0.110418\pi$
$809$	5.00000	+	8.66025i	0.175791	+	0.304478i	−0.764644	+	0.644453i	$0.777085\pi$
$810$		0			0
$811$	16.0000			0.561836			0.280918	−	0.959732i	$-0.409361\pi$
$811$	16.0000			0.561836			0.280918	+	0.959732i	$0.409361\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	42.0000	+	72.7461i	1.47120	+	2.54819i
$816$		0			0
$817$	−14.0000	+	24.2487i	−0.489798	+	0.848355i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	11.0000	−	19.0526i	0.383903	−	0.664939i	−0.607714	−	0.794156i	$-0.707913\pi$
$821$	11.0000	−	19.0526i	0.383903	−	0.664939i	0.991616	+	0.129217i	$0.0412465\pi$
$822$		0			0
$823$	−7.50000	−	12.9904i	−0.261434	−	0.452816i	0.705190	−	0.709019i	$-0.250862\pi$
$823$	−7.50000	−	12.9904i	−0.261434	−	0.452816i	−0.966623	+	0.256203i	$0.917529\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	20.0000			0.695468			0.347734	−	0.937593i	$-0.386951\pi$
$827$	20.0000			0.695468			0.347734	+	0.937593i	$0.386951\pi$
$828$		0			0
$829$	5.00000	+	8.66025i	0.173657	+	0.300783i	0.939696	−	0.342012i	$-0.111108\pi$
$829$	5.00000	+	8.66025i	0.173657	+	0.300783i	−0.766039	+	0.642795i	$0.777775\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	33.0000	−	25.9808i	1.14338	−	0.900180i
$834$		0			0
$835$	−4.00000	+	6.92820i	−0.138426	+	0.239760i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−6.00000			−0.207143			−0.103572	−	0.994622i	$-0.533027\pi$
$839$	−6.00000			−0.207143			−0.103572	+	0.994622i	$0.533027\pi$
$840$		0			0
$841$	7.00000			0.241379
$842$		0			0
$843$		0			0
$844$		0			0
$845$	24.0000	−	41.5692i	0.825625	−	1.43002i
$846$		0			0
$847$	17.5000	−	6.06218i	0.601307	−	0.208299i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−21.0000	−	36.3731i	−0.719871	−	1.24685i
$852$		0			0
$853$	26.0000			0.890223			0.445112	−	0.895475i	$-0.353164\pi$
$853$	26.0000			0.890223			0.445112	+	0.895475i	$0.353164\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−12.0000	−	20.7846i	−0.409912	−	0.709989i	0.584967	−	0.811057i	$-0.301107\pi$
$857$	−12.0000	−	20.7846i	−0.409912	−	0.709989i	−0.994880	+	0.101068i	$0.967774\pi$
$858$		0			0
$859$	15.5000	−	26.8468i	0.528853	−	0.916001i	−0.470581	−	0.882357i	$-0.655956\pi$
$859$	15.5000	−	26.8468i	0.528853	−	0.916001i	0.999434	−	0.0336436i	$-0.0107111\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	22.0000	−	38.1051i	0.748889	−	1.29711i	−0.199467	−	0.979905i	$-0.563921\pi$
$863$	22.0000	−	38.1051i	0.748889	−	1.29711i	0.948356	−	0.317209i	$-0.102746\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	2.00000			0.0678454
$870$		0			0
$871$	17.5000	+	30.3109i	0.592965	+	1.02705i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−48.0000	−	41.5692i	−1.62270	−	1.40530i
$876$		0			0
$877$	10.5000	−	18.1865i	0.354560	−	0.614116i	−0.632483	−	0.774574i	$-0.717964\pi$
$877$	10.5000	−	18.1865i	0.354560	−	0.614116i	0.987043	+	0.160459i	$0.0512974\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	18.0000			0.606435			0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000			0.606435			0.303218	+	0.952921i	$0.401939\pi$
$882$		0			0
$883$	−56.0000			−1.88455			−0.942275	−	0.334840i	$-0.891318\pi$
$883$	−56.0000			−1.88455			−0.942275	+	0.334840i	$0.891318\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	5.00000	−	8.66025i	0.167884	−	0.290783i	−0.769792	−	0.638295i	$-0.779640\pi$
$887$	5.00000	−	8.66025i	0.167884	−	0.290783i	0.937676	+	0.347512i	$0.112973\pi$
$888$		0			0
$889$	26.0000	+	22.5167i	0.872012	+	0.755185i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−4.00000	−	6.92820i	−0.133855	−	0.231843i
$894$		0			0
$895$	16.0000			0.534821
$896$		0			0
$897$		0			0
$898$		0			0
$899$	21.0000	+	36.3731i	0.700389	+	1.21311i
$900$		0			0
$901$	−18.0000	+	31.1769i	−0.599667	+	1.03865i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−4.00000	+	6.92820i	−0.132964	+	0.230301i
$906$		0			0
$907$	−2.50000	−	4.33013i	−0.0830111	−	0.143780i	0.821531	−	0.570164i	$-0.193120\pi$
$907$	−2.50000	−	4.33013i	−0.0830111	−	0.143780i	−0.904542	+	0.426385i	$0.859787\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−42.0000			−1.39152			−0.695761	−	0.718273i	$-0.744933\pi$
$911$	−42.0000			−1.39152			−0.695761	+	0.718273i	$0.744933\pi$
$912$		0			0
$913$	−14.0000	−	24.2487i	−0.463332	−	0.802515i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	10.0000	−	3.46410i	0.330229	−	0.114395i
$918$		0			0
$919$	24.5000	−	42.4352i	0.808180	−	1.39981i	−0.105942	−	0.994372i	$-0.533786\pi$
$919$	24.5000	−	42.4352i	0.808180	−	1.39981i	0.914123	−	0.405437i	$-0.132881\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−40.0000			−1.31662
$924$		0			0
$925$	77.0000			2.53174
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−2.00000	+	3.46410i	−0.0656179	+	0.113653i	−0.896968	−	0.442096i	$-0.854235\pi$
$929$	−2.00000	+	3.46410i	−0.0656179	+	0.113653i	0.831350	+	0.555749i	$0.187569\pi$
$930$		0			0
$931$	26.0000	+	10.3923i	0.852116	+	0.340594i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	24.0000	+	41.5692i	0.784884	+	1.35946i
$936$		0			0
$937$	43.0000			1.40475			0.702374	−	0.711808i	$-0.252123\pi$
$937$	43.0000			1.40475			0.702374	+	0.711808i	$0.252123\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	3.00000	+	5.19615i	0.0977972	+	0.169390i	0.910773	−	0.412908i	$-0.135487\pi$
$941$	3.00000	+	5.19615i	0.0977972	+	0.169390i	−0.812975	+	0.582298i	$0.802154\pi$
$942$		0			0
$943$	−6.00000	+	10.3923i	−0.195387	+	0.338420i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−11.0000	+	19.0526i	−0.357452	+	0.619125i	−0.987534	−	0.157403i	$-0.949688\pi$
$947$	−11.0000	+	19.0526i	−0.357452	+	0.619125i	0.630082	+	0.776528i	$0.283021\pi$
$948$		0			0
$949$	−25.0000	−	43.3013i	−0.811534	−	1.40562i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	6.00000			0.194359			0.0971795	−	0.995267i	$-0.469018\pi$
$953$	6.00000			0.194359			0.0971795	+	0.995267i	$0.469018\pi$
$954$		0			0
$955$	40.0000	+	69.2820i	1.29437	+	2.24191i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	3.00000	−	15.5885i	0.0968751	−	0.503378i
$960$		0			0
$961$	−9.00000	+	15.5885i	−0.290323	+	0.502853i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−28.0000			−0.901352
$966$		0			0
$967$	−43.0000			−1.38279			−0.691393	−	0.722478i	$-0.743003\pi$
$967$	−43.0000			−1.38279			−0.691393	+	0.722478i	$0.743003\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	6.00000	−	10.3923i	0.192549	−	0.333505i	−0.753545	−	0.657396i	$-0.771658\pi$
$971$	6.00000	−	10.3923i	0.192549	−	0.333505i	0.946094	+	0.323891i	$0.104991\pi$
$972$		0			0
$973$	14.0000	+	12.1244i	0.448819	+	0.388689i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−1.00000	−	1.73205i	−0.0319928	−	0.0554132i	0.849586	−	0.527451i	$-0.176852\pi$
$977$	−1.00000	−	1.73205i	−0.0319928	−	0.0554132i	−0.881579	+	0.472037i	$0.843519\pi$
$978$		0			0
$979$	24.0000			0.767043
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−18.0000	−	31.1769i	−0.574111	−	0.994389i	−0.996138	−	0.0878058i	$-0.972015\pi$
$983$	−18.0000	−	31.1769i	−0.574111	−	0.994389i	0.422027	−	0.906583i	$-0.361319\pi$
$984$		0			0
$985$	24.0000	−	41.5692i	0.764704	−	1.32451i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	21.0000	−	36.3731i	0.667761	−	1.15660i
$990$		0			0
$991$	−17.5000	−	30.3109i	−0.555906	−	0.962857i	−0.997832	−	0.0658059i	$-0.979038\pi$
$991$	−17.5000	−	30.3109i	−0.555906	−	0.962857i	0.441927	−	0.897051i	$-0.354295\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	12.0000			0.380426
$996$		0			0
$997$	8.50000	+	14.7224i	0.269198	+	0.466264i	0.968655	−	0.248410i	$-0.0799082\pi$
$997$	8.50000	+	14.7224i	0.269198	+	0.466264i	−0.699457	+	0.714675i	$0.746575\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1512.2.s.j.1297.1	yes	2
3.2	odd	2		1512.2.s.a.1297.1	yes	2
7.4	even	3	inner	1512.2.s.j.865.1	yes	2
21.11	odd	6		1512.2.s.a.865.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1512.2.s.a.865.1	✓	2	21.11	odd	6
1512.2.s.a.1297.1	yes	2	3.2	odd	2
1512.2.s.j.865.1	yes	2	7.4	even	3	inner
1512.2.s.j.1297.1	yes	2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1512 = 2^{3} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1512.s (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(2.00000 - 3.46410i) q^{5} +(2.00000 + 1.73205i) q^{7} +O(q^{10})\)	\(q+(2.00000 - 3.46410i) q^{5} +(2.00000 + 1.73205i) q^{7} +(-1.00000 - 1.73205i) q^{11} +5.00000 q^{13} +(-3.00000 - 5.19615i) q^{17} +(2.00000 - 3.46410i) q^{19} +(-3.00000 + 5.19615i) q^{23} +(-5.50000 - 9.52628i) q^{25} +6.00000 q^{29} +(3.50000 + 6.06218i) q^{31} +(10.0000 - 3.46410i) q^{35} +(-3.50000 + 6.06218i) q^{37} +2.00000 q^{41} -7.00000 q^{43} +(1.00000 - 1.73205i) q^{47} +(1.00000 + 6.92820i) q^{49} +(-3.00000 - 5.19615i) q^{53} -8.00000 q^{55} +(-3.00000 - 5.19615i) q^{59} +(4.50000 - 7.79423i) q^{61} +(10.0000 - 17.3205i) q^{65} +(3.50000 + 6.06218i) q^{67} -8.00000 q^{71} +(-5.00000 - 8.66025i) q^{73} +(1.00000 - 5.19615i) q^{77} +(-0.500000 + 0.866025i) q^{79} +14.0000 q^{83} -24.0000 q^{85} +(-6.00000 + 10.3923i) q^{89} +(10.0000 + 8.66025i) q^{91} +(-8.00000 - 13.8564i) q^{95} -15.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{5} + 4 q^{7}+O(q^{10}) \) 2 * q + 4 * q^5 + 4 * q^7	\( 2 q + 4 q^{5} + 4 q^{7} - 2 q^{11} + 10 q^{13} - 6 q^{17} + 4 q^{19} - 6 q^{23} - 11 q^{25} + 12 q^{29} + 7 q^{31} + 20 q^{35} - 7 q^{37} + 4 q^{41} - 14 q^{43} + 2 q^{47} + 2 q^{49} - 6 q^{53} - 16 q^{55} - 6 q^{59} + 9 q^{61} + 20 q^{65} + 7 q^{67} - 16 q^{71} - 10 q^{73} + 2 q^{77} - q^{79} + 28 q^{83} - 48 q^{85} - 12 q^{89} + 20 q^{91} - 16 q^{95} - 30 q^{97}+O(q^{100}) \) 2 * q + 4 * q^5 + 4 * q^7 - 2 * q^11 + 10 * q^13 - 6 * q^17 + 4 * q^19 - 6 * q^23 - 11 * q^25 + 12 * q^29 + 7 * q^31 + 20 * q^35 - 7 * q^37 + 4 * q^41 - 14 * q^43 + 2 * q^47 + 2 * q^49 - 6 * q^53 - 16 * q^55 - 6 * q^59 + 9 * q^61 + 20 * q^65 + 7 * q^67 - 16 * q^71 - 10 * q^73 + 2 * q^77 - q^79 + 28 * q^83 - 48 * q^85 - 12 * q^89 + 20 * q^91 - 16 * q^95 - 30 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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