LMFDB - Embedded newform 756.2.t.d.269.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [756,2,Mod(269,756)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(756, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 1]))

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

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

chi := DirichletCharacter("756.269");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 756 = 2^{2} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	756.t (of order $6$, 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:	$6.03669039281$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			269.1
Root			$-0.707107 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	756.269
Dual form			756.2.t.d.593.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.12132 - 3.67423i) q^{5} +(-2.50000 + 0.866025i) q^{7} +O(q^{10})$	$q+(-2.12132 - 3.67423i) q^{5} +(-2.50000 + 0.866025i) q^{7} +3.46410i q^{13} +(-2.12132 + 3.67423i) q^{17} +(-1.50000 + 0.866025i) q^{19} +(6.36396 - 3.67423i) q^{23} +(-6.50000 + 11.2583i) q^{25} +7.34847i q^{29} +(-1.50000 - 0.866025i) q^{31} +(8.48528 + 7.34847i) q^{35} +(4.00000 + 6.92820i) q^{37} -4.24264 q^{41} -5.00000 q^{43} +(-2.12132 - 3.67423i) q^{47} +(5.50000 - 4.33013i) q^{49} +(-6.36396 - 3.67423i) q^{53} +(-4.24264 + 7.34847i) q^{59} +(-4.50000 + 2.59808i) q^{61} +(12.7279 - 7.34847i) q^{65} +(-1.00000 + 1.73205i) q^{67} -14.6969i q^{71} +(-7.50000 - 4.33013i) q^{73} +(2.00000 + 3.46410i) q^{79} -16.9706 q^{83} +18.0000 q^{85} +(-6.36396 - 11.0227i) q^{89} +(-3.00000 - 8.66025i) q^{91} +(6.36396 + 3.67423i) q^{95} +15.5885i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 10 q^{7}+O(q^{10}) $ 4 * q - 10 * q^7	$ 4 q - 10 q^{7} - 6 q^{19} - 26 q^{25} - 6 q^{31} + 16 q^{37} - 20 q^{43} + 22 q^{49} - 18 q^{61} - 4 q^{67} - 30 q^{73} + 8 q^{79} + 72 q^{85} - 12 q^{91}+O(q^{100}) $ 4 * q - 10 * q^7 - 6 * q^19 - 26 * q^25 - 6 * q^31 + 16 * q^37 - 20 * q^43 + 22 * q^49 - 18 * q^61 - 4 * q^67 - 30 * q^73 + 8 * q^79 + 72 * q^85 - 12 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/756\mathbb{Z}\right)^\times$.

$n$	$29$	$325$	$379$
$\chi(n)$	$-1$	$e\left(\frac{1}{6}\right)$	$1$

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
$5$	−2.12132	−	3.67423i	−0.948683	−	1.64317i	−0.748203	−	0.663470i	$-0.769083\pi$
$5$	−2.12132	−	3.67423i	−0.948683	−	1.64317i	−0.200480	−	0.979698i	$-0.564250\pi$
$6$		0			0
$7$	−2.50000	+	0.866025i	−0.944911	+	0.327327i
$7$	−2.50000	+	0.866025i	−0.944911	+	0.327327i
$8$		0			0
$9$		0			0
$10$		0			0
$11$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$11$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$12$		0			0
$13$			3.46410i			0.960769i	0.877058	+	0.480384i	$0.159503\pi$
$13$			3.46410i			0.960769i	−0.877058	+	0.480384i	$0.840497\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−2.12132	+	3.67423i	−0.514496	+	0.891133i	0.485363	+	0.874313i	$0.338688\pi$
$17$	−2.12132	+	3.67423i	−0.514496	+	0.891133i	−0.999859	+	0.0168199i	$0.994646\pi$
$18$		0			0
$19$	−1.50000	+	0.866025i	−0.344124	+	0.198680i	−0.662094	−	0.749421i	$-0.730332\pi$
$19$	−1.50000	+	0.866025i	−0.344124	+	0.198680i	0.317970	+	0.948101i	$0.396999\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	6.36396	−	3.67423i	1.32698	−	0.766131i	0.342147	−	0.939647i	$-0.388846\pi$
$23$	6.36396	−	3.67423i	1.32698	−	0.766131i	0.984831	+	0.173516i	$0.0555127\pi$
$24$		0			0
$25$	−6.50000	+	11.2583i	−1.30000	+	2.25167i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			7.34847i			1.36458i	0.731083	+	0.682288i	$0.239015\pi$
$29$			7.34847i			1.36458i	−0.731083	+	0.682288i	$0.760985\pi$
$30$		0			0
$31$	−1.50000	−	0.866025i	−0.269408	−	0.155543i	0.359211	−	0.933257i	$-0.383046\pi$
$31$	−1.50000	−	0.866025i	−0.269408	−	0.155543i	−0.628619	+	0.777714i	$0.716379\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	8.48528	+	7.34847i	1.43427	+	1.24212i
$36$		0			0
$37$	4.00000	+	6.92820i	0.657596	+	1.13899i	0.981236	+	0.192809i	$0.0617599\pi$
$37$	4.00000	+	6.92820i	0.657596	+	1.13899i	−0.323640	+	0.946180i	$0.604907\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−4.24264			−0.662589			−0.331295	−	0.943527i	$-0.607485\pi$
$41$	−4.24264			−0.662589			−0.331295	+	0.943527i	$0.607485\pi$
$42$		0			0
$43$	−5.00000			−0.762493			−0.381246	−	0.924473i	$-0.624505\pi$
$43$	−5.00000			−0.762493			−0.381246	+	0.924473i	$0.624505\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−2.12132	−	3.67423i	−0.309426	−	0.535942i	0.668811	−	0.743433i	$-0.266804\pi$
$47$	−2.12132	−	3.67423i	−0.309426	−	0.535942i	−0.978237	+	0.207491i	$0.933470\pi$
$48$		0			0
$49$	5.50000	−	4.33013i	0.785714	−	0.618590i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−6.36396	−	3.67423i	−0.874157	−	0.504695i	−0.00542976	−	0.999985i	$-0.501728\pi$
$53$	−6.36396	−	3.67423i	−0.874157	−	0.504695i	−0.868728	+	0.495290i	$0.835062\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−4.24264	+	7.34847i	−0.552345	+	0.956689i	0.445760	+	0.895152i	$0.352933\pi$
$59$	−4.24264	+	7.34847i	−0.552345	+	0.956689i	−0.998105	+	0.0615367i	$0.980400\pi$
$60$		0			0
$61$	−4.50000	+	2.59808i	−0.576166	+	0.332650i	−0.759608	−	0.650381i	$-0.774609\pi$
$61$	−4.50000	+	2.59808i	−0.576166	+	0.332650i	0.183442	+	0.983030i	$0.441276\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	12.7279	−	7.34847i	1.57870	−	0.911465i
$66$		0			0
$67$	−1.00000	+	1.73205i	−0.122169	+	0.211604i	−0.920623	−	0.390453i	$-0.872318\pi$
$67$	−1.00000	+	1.73205i	−0.122169	+	0.211604i	0.798454	+	0.602056i	$0.205652\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	14.6969i		−	1.74421i	−0.489323	−	0.872103i	$-0.662756\pi$
$71$		−	14.6969i		−	1.74421i	0.489323	−	0.872103i	$-0.337244\pi$
$72$		0			0
$73$	−7.50000	−	4.33013i	−0.877809	−	0.506803i	−0.00787336	−	0.999969i	$-0.502506\pi$
$73$	−7.50000	−	4.33013i	−0.877809	−	0.506803i	−0.869935	+	0.493166i	$0.835840\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	2.00000	+	3.46410i	0.225018	+	0.389742i	0.956325	−	0.292306i	$-0.0944227\pi$
$79$	2.00000	+	3.46410i	0.225018	+	0.389742i	−0.731307	+	0.682048i	$0.761089\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−16.9706			−1.86276			−0.931381	−	0.364047i	$-0.881395\pi$
$83$	−16.9706			−1.86276			−0.931381	+	0.364047i	$0.881395\pi$
$84$		0			0
$85$	18.0000			1.95237
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−6.36396	−	11.0227i	−0.674579	−	1.16840i	−0.976592	−	0.215101i	$-0.930992\pi$
$89$	−6.36396	−	11.0227i	−0.674579	−	1.16840i	0.302013	−	0.953304i	$-0.402341\pi$
$90$		0			0
$91$	−3.00000	−	8.66025i	−0.314485	−	0.907841i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	6.36396	+	3.67423i	0.652929	+	0.376969i
$96$		0			0
$97$			15.5885i			1.58277i	0.611319	+	0.791384i	$0.290639\pi$
$97$			15.5885i			1.58277i	−0.611319	+	0.791384i	$0.709361\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$101$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$102$		0			0
$103$	6.00000	−	3.46410i	0.591198	−	0.341328i	−0.174373	−	0.984680i	$-0.555790\pi$
$103$	6.00000	−	3.46410i	0.591198	−	0.341328i	0.765571	+	0.643352i	$0.222457\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−12.7279	+	7.34847i	−1.23045	+	0.710403i	−0.967125	−	0.254302i	$-0.918154\pi$
$107$	−12.7279	+	7.34847i	−1.23045	+	0.710403i	−0.263330	+	0.964706i	$0.584821\pi$
$108$		0			0
$109$	−0.500000	+	0.866025i	−0.0478913	+	0.0829502i	−0.888977	−	0.457951i	$-0.848583\pi$
$109$	−0.500000	+	0.866025i	−0.0478913	+	0.0829502i	0.841086	+	0.540901i	$0.181917\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			14.6969i			1.38257i	0.722581	+	0.691286i	$0.242955\pi$
$113$			14.6969i			1.38257i	−0.722581	+	0.691286i	$0.757045\pi$
$114$		0			0
$115$	−27.0000	−	15.5885i	−2.51776	−	1.45363i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	2.12132	−	11.0227i	0.194461	−	1.01045i
$120$		0			0
$121$	−5.50000	−	9.52628i	−0.500000	−	0.866025i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	33.9411			3.03579
$126$		0			0
$127$	17.0000			1.50851			0.754253	−	0.656584i	$-0.227999\pi$
$127$	17.0000			1.50851			0.754253	+	0.656584i	$0.227999\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	6.36396	+	11.0227i	0.556022	+	0.963058i	0.997823	+	0.0659452i	$0.0210063\pi$
$131$	6.36396	+	11.0227i	0.556022	+	0.963058i	−0.441801	+	0.897113i	$0.645660\pi$
$132$		0			0
$133$	3.00000	−	3.46410i	0.260133	−	0.300376i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	6.36396	+	3.67423i	0.543710	+	0.313911i	0.746581	−	0.665294i	$-0.231694\pi$
$137$	6.36396	+	3.67423i	0.543710	+	0.313911i	−0.202871	+	0.979205i	$0.565027\pi$
$138$		0			0
$139$		−	13.8564i		−	1.17529i	−0.809121	−	0.587643i	$-0.800056\pi$
$139$		−	13.8564i		−	1.17529i	0.809121	−	0.587643i	$-0.199944\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	27.0000	−	15.5885i	2.24223	−	1.29455i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−6.36396	+	3.67423i	−0.521356	+	0.301005i	−0.737489	−	0.675359i	$-0.763989\pi$
$149$	−6.36396	+	3.67423i	−0.521356	+	0.301005i	0.216133	+	0.976364i	$0.430655\pi$
$150$		0			0
$151$	−0.500000	+	0.866025i	−0.0406894	+	0.0704761i	−0.885653	−	0.464348i	$-0.846289\pi$
$151$	−0.500000	+	0.866025i	−0.0406894	+	0.0704761i	0.844963	+	0.534824i	$0.179622\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			7.34847i			0.590243i
$156$		0			0
$157$	9.00000	+	5.19615i	0.718278	+	0.414698i	0.814119	−	0.580699i	$-0.197221\pi$
$157$	9.00000	+	5.19615i	0.718278	+	0.414698i	−0.0958404	+	0.995397i	$0.530554\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−12.7279	+	14.6969i	−1.00310	+	1.15828i
$162$		0			0
$163$	3.50000	+	6.06218i	0.274141	+	0.474826i	0.969918	−	0.243432i	$-0.0782731\pi$
$163$	3.50000	+	6.06218i	0.274141	+	0.474826i	−0.695777	+	0.718258i	$0.744940\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−8.48528			−0.656611			−0.328305	−	0.944572i	$-0.606478\pi$
$167$	−8.48528			−0.656611			−0.328305	+	0.944572i	$0.606478\pi$
$168$		0			0
$169$	1.00000			0.0769231
$170$		0			0
$171$		0			0
$172$		0			0
$173$	6.36396	+	11.0227i	0.483843	+	0.838041i	0.999828	−	0.0185571i	$-0.00590724\pi$
$173$	6.36396	+	11.0227i	0.483843	+	0.838041i	−0.515985	+	0.856598i	$0.672574\pi$
$174$		0			0
$175$	6.50000	−	33.7750i	0.491354	−	2.55315i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−6.36396	−	3.67423i	−0.475665	−	0.274625i	0.242943	−	0.970040i	$-0.421887\pi$
$179$	−6.36396	−	3.67423i	−0.475665	−	0.274625i	−0.718608	+	0.695415i	$0.755220\pi$
$180$		0			0
$181$		−	19.0526i		−	1.41617i	−0.706129	−	0.708083i	$-0.749560\pi$
$181$		−	19.0526i		−	1.41617i	0.706129	−	0.708083i	$-0.250440\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	16.9706	−	29.3939i	1.24770	−	2.16108i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−6.36396	+	3.67423i	−0.460480	+	0.265858i	−0.712246	−	0.701930i	$-0.752322\pi$
$191$	−6.36396	+	3.67423i	−0.460480	+	0.265858i	0.251766	+	0.967788i	$0.418989\pi$
$192$		0			0
$193$	−8.00000	+	13.8564i	−0.575853	+	0.997406i	0.420096	+	0.907480i	$0.361996\pi$
$193$	−8.00000	+	13.8564i	−0.575853	+	0.997406i	−0.995948	+	0.0899262i	$0.971337\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	14.6969i		−	1.04711i	−0.851991	−	0.523557i	$-0.824605\pi$
$197$		−	14.6969i		−	1.04711i	0.851991	−	0.523557i	$-0.175395\pi$
$198$		0			0
$199$	−13.5000	−	7.79423i	−0.956990	−	0.552518i	−0.0617444	−	0.998092i	$-0.519666\pi$
$199$	−13.5000	−	7.79423i	−0.956990	−	0.552518i	−0.895245	+	0.445574i	$0.853000\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−6.36396	−	18.3712i	−0.446663	−	1.28940i
$204$		0			0
$205$	9.00000	+	15.5885i	0.628587	+	1.08875i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	13.0000			0.894957			0.447478	−	0.894295i	$-0.352322\pi$
$211$	13.0000			0.894957			0.447478	+	0.894295i	$0.352322\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	10.6066	+	18.3712i	0.723364	+	1.25290i
$216$		0			0
$217$	4.50000	+	0.866025i	0.305480	+	0.0587896i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−12.7279	−	7.34847i	−0.856173	−	0.494312i
$222$		0			0
$223$		−	20.7846i		−	1.39184i	−0.718119	−	0.695920i	$-0.754997\pi$
$223$		−	20.7846i		−	1.39184i	0.718119	−	0.695920i	$-0.245003\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	6.36396	−	11.0227i	0.422391	−	0.731603i	−0.573782	−	0.819008i	$-0.694524\pi$
$227$	6.36396	−	11.0227i	0.422391	−	0.731603i	0.996173	+	0.0874056i	$0.0278576\pi$
$228$		0			0
$229$	−7.50000	+	4.33013i	−0.495614	+	0.286143i	−0.726900	−	0.686743i	$-0.759040\pi$
$229$	−7.50000	+	4.33013i	−0.495614	+	0.286143i	0.231287	+	0.972886i	$0.425707\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−19.0919	+	11.0227i	−1.25075	+	0.722121i	−0.971259	−	0.238027i	$-0.923499\pi$
$233$	−19.0919	+	11.0227i	−1.25075	+	0.722121i	−0.279492	+	0.960148i	$0.590166\pi$
$234$		0			0
$235$	−9.00000	+	15.5885i	−0.587095	+	1.01688i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	22.0454i		−	1.42600i	−0.701165	−	0.712999i	$-0.747336\pi$
$239$		−	22.0454i		−	1.42600i	0.701165	−	0.712999i	$-0.252664\pi$
$240$		0			0
$241$	−4.50000	−	2.59808i	−0.289870	−	0.167357i	0.348013	−	0.937490i	$-0.386857\pi$
$241$	−4.50000	−	2.59808i	−0.289870	−	0.167357i	−0.637883	+	0.770133i	$0.720190\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−27.5772	−	11.0227i	−1.76184	−	0.704215i
$246$		0			0
$247$	−3.00000	−	5.19615i	−0.190885	−	0.330623i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	4.24264			0.267793			0.133897	−	0.990995i	$-0.457251\pi$
$251$	4.24264			0.267793			0.133897	+	0.990995i	$0.457251\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$257$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$258$		0			0
$259$	−16.0000	−	13.8564i	−0.994192	−	0.860995i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	12.7279	+	7.34847i	0.784837	+	0.453126i	0.838142	−	0.545452i	$-0.183642\pi$
$263$	12.7279	+	7.34847i	0.784837	+	0.453126i	−0.0533046	+	0.998578i	$0.516975\pi$
$264$		0			0
$265$			31.1769i			1.91518i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−4.24264	+	7.34847i	−0.258678	+	0.448044i	−0.965888	−	0.258960i	$-0.916620\pi$
$269$	−4.24264	+	7.34847i	−0.258678	+	0.448044i	0.707210	+	0.707004i	$0.249954\pi$
$270$		0			0
$271$	4.50000	−	2.59808i	0.273356	−	0.157822i	−0.357056	−	0.934083i	$-0.616219\pi$
$271$	4.50000	−	2.59808i	0.273356	−	0.157822i	0.630412	+	0.776261i	$0.282886\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−11.5000	+	19.9186i	−0.690968	+	1.19679i	0.280553	+	0.959839i	$0.409482\pi$
$277$	−11.5000	+	19.9186i	−0.690968	+	1.19679i	−0.971521	+	0.236953i	$0.923851\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$	−7.50000	−	4.33013i	−0.445829	−	0.257399i	0.260238	−	0.965544i	$-0.416199\pi$
$283$	−7.50000	−	4.33013i	−0.445829	−	0.257399i	−0.706067	+	0.708145i	$0.749532\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	10.6066	−	3.67423i	0.626088	−	0.216883i
$288$		0			0
$289$	−0.500000	−	0.866025i	−0.0294118	−	0.0509427i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	8.48528			0.495715			0.247858	−	0.968796i	$-0.420273\pi$
$293$	8.48528			0.495715			0.247858	+	0.968796i	$0.420273\pi$
$294$		0			0
$295$	36.0000			2.09600
$296$		0			0
$297$		0			0
$298$		0			0
$299$	12.7279	+	22.0454i	0.736075	+	1.27492i
$300$		0			0
$301$	12.5000	−	4.33013i	0.720488	−	0.249584i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	19.0919	+	11.0227i	1.09320	+	0.631158i
$306$		0			0
$307$			32.9090i			1.87821i	0.343625	+	0.939107i	$0.388345\pi$
$307$			32.9090i			1.87821i	−0.343625	+	0.939107i	$0.611655\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−8.48528	+	14.6969i	−0.481156	+	0.833387i	−0.999766	−	0.0216240i	$-0.993116\pi$
$311$	−8.48528	+	14.6969i	−0.481156	+	0.833387i	0.518610	+	0.855011i	$0.326450\pi$
$312$		0			0
$313$	28.5000	−	16.4545i	1.61092	−	0.930062i	0.621757	−	0.783210i	$-0.286419\pi$
$313$	28.5000	−	16.4545i	1.61092	−	0.930062i	0.989158	−	0.146852i	$-0.0469141\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−12.7279	+	7.34847i	−0.714871	+	0.412731i	−0.812862	−	0.582456i	$-0.802092\pi$
$317$	−12.7279	+	7.34847i	−0.714871	+	0.412731i	0.0979908	+	0.995187i	$0.468758\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	7.34847i		−	0.408880i
$324$		0			0
$325$	−39.0000	−	22.5167i	−2.16333	−	1.24900i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	8.48528	+	7.34847i	0.467809	+	0.405134i
$330$		0			0
$331$	2.50000	+	4.33013i	0.137412	+	0.238005i	0.926516	−	0.376254i	$-0.122788\pi$
$331$	2.50000	+	4.33013i	0.137412	+	0.238005i	−0.789104	+	0.614260i	$0.789455\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	8.48528			0.463600
$336$		0			0
$337$	7.00000			0.381314			0.190657	−	0.981657i	$-0.438938\pi$
$337$	7.00000			0.381314			0.190657	+	0.981657i	$0.438938\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−10.0000	+	15.5885i	−0.539949	+	0.841698i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	12.7279	+	7.34847i	0.683271	+	0.394486i	0.801086	−	0.598549i	$-0.204256\pi$
$347$	12.7279	+	7.34847i	0.683271	+	0.394486i	−0.117816	+	0.993035i	$0.537589\pi$
$348$		0			0
$349$			5.19615i			0.278144i	0.990282	+	0.139072i	$0.0444119\pi$
$349$			5.19615i			0.278144i	−0.990282	+	0.139072i	$0.955588\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	4.24264	−	7.34847i	0.225813	−	0.391120i	−0.730750	−	0.682645i	$-0.760830\pi$
$353$	4.24264	−	7.34847i	0.225813	−	0.391120i	0.956563	+	0.291526i	$0.0941628\pi$
$354$		0			0
$355$	−54.0000	+	31.1769i	−2.86602	+	1.65470i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$359$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$360$		0			0
$361$	−8.00000	+	13.8564i	−0.421053	+	0.729285i
$362$		0			0
$363$		0			0
$364$		0			0
$365$			36.7423i			1.92318i
$366$		0			0
$367$	16.5000	+	9.52628i	0.861293	+	0.497268i	0.864445	−	0.502727i	$-0.167670\pi$
$367$	16.5000	+	9.52628i	0.861293	+	0.497268i	−0.00315207	+	0.999995i	$0.501003\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	19.0919	+	3.67423i	0.991201	+	0.190757i
$372$		0			0
$373$	−5.50000	−	9.52628i	−0.284779	−	0.493252i	0.687776	−	0.725923i	$-0.258587\pi$
$373$	−5.50000	−	9.52628i	−0.284779	−	0.493252i	−0.972556	+	0.232671i	$0.925254\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−25.4558			−1.31104
$378$		0			0
$379$	−26.0000			−1.33553			−0.667765	−	0.744372i	$-0.732749\pi$
$379$	−26.0000			−1.33553			−0.667765	+	0.744372i	$0.732749\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−4.24264	−	7.34847i	−0.216789	−	0.375489i	0.737036	−	0.675854i	$-0.236225\pi$
$383$	−4.24264	−	7.34847i	−0.216789	−	0.375489i	−0.953824	+	0.300365i	$0.902892\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−19.0919	−	11.0227i	−0.967997	−	0.558873i	−0.0693719	−	0.997591i	$-0.522100\pi$
$389$	−19.0919	−	11.0227i	−0.967997	−	0.558873i	−0.898625	+	0.438718i	$0.855433\pi$
$390$		0			0
$391$			31.1769i			1.57668i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	8.48528	−	14.6969i	0.426941	−	0.739483i
$396$		0			0
$397$	7.50000	−	4.33013i	0.376414	−	0.217323i	−0.299843	−	0.953989i	$-0.596934\pi$
$397$	7.50000	−	4.33013i	0.376414	−	0.217323i	0.676257	+	0.736666i	$0.263601\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$401$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$402$		0			0
$403$	3.00000	−	5.19615i	0.149441	−	0.258839i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−15.0000	−	8.66025i	−0.741702	−	0.428222i	0.0809857	−	0.996715i	$-0.474193\pi$
$409$	−15.0000	−	8.66025i	−0.741702	−	0.428222i	−0.822688	+	0.568493i	$0.807527\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	4.24264	−	22.0454i	0.208767	−	1.08478i
$414$		0			0
$415$	36.0000	+	62.3538i	1.76717	+	3.06083i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	4.24264			0.207267			0.103633	−	0.994616i	$-0.466953\pi$
$419$	4.24264			0.207267			0.103633	+	0.994616i	$0.466953\pi$
$420$		0			0
$421$	−13.0000			−0.633581			−0.316791	−	0.948495i	$-0.602605\pi$
$421$	−13.0000			−0.633581			−0.316791	+	0.948495i	$0.602605\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−27.5772	−	47.7650i	−1.33769	−	2.31695i
$426$		0			0
$427$	9.00000	−	10.3923i	0.435541	−	0.502919i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−25.4558	−	14.6969i	−1.22616	−	0.707927i	−0.259939	−	0.965625i	$-0.583703\pi$
$431$	−25.4558	−	14.6969i	−1.22616	−	0.707927i	−0.966225	+	0.257698i	$0.917036\pi$
$432$		0			0
$433$		−	12.1244i		−	0.582659i	−0.956623	−	0.291330i	$-0.905902\pi$
$433$		−	12.1244i		−	0.582659i	0.956623	−	0.291330i	$-0.0940977\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−6.36396	+	11.0227i	−0.304430	+	0.527287i
$438$		0			0
$439$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$439$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	19.0919	−	11.0227i	0.907083	−	0.523704i	0.0275914	−	0.999619i	$-0.491216\pi$
$443$	19.0919	−	11.0227i	0.907083	−	0.523704i	0.879491	+	0.475915i	$0.157883\pi$
$444$		0			0
$445$	−27.0000	+	46.7654i	−1.27992	+	2.21689i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			7.34847i			0.346796i	0.984852	+	0.173398i	$0.0554746\pi$
$449$			7.34847i			0.346796i	−0.984852	+	0.173398i	$0.944525\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−25.4558	+	29.3939i	−1.19339	+	1.37801i
$456$		0			0
$457$	11.5000	+	19.9186i	0.537947	+	0.931752i	0.999014	+	0.0443868i	$0.0141334\pi$
$457$	11.5000	+	19.9186i	0.537947	+	0.931752i	−0.461067	+	0.887365i	$0.652533\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	8.48528			0.395199			0.197599	−	0.980283i	$-0.436685\pi$
$461$	8.48528			0.395199			0.197599	+	0.980283i	$0.436685\pi$
$462$		0			0
$463$	−7.00000			−0.325318			−0.162659	−	0.986682i	$-0.552007\pi$
$463$	−7.00000			−0.325318			−0.162659	+	0.986682i	$0.552007\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−12.7279	−	22.0454i	−0.588978	−	1.02014i	−0.994367	−	0.105995i	$-0.966197\pi$
$467$	−12.7279	−	22.0454i	−0.588978	−	1.02014i	0.405389	−	0.914144i	$-0.367136\pi$
$468$		0			0
$469$	1.00000	−	5.19615i	0.0461757	−	0.239936i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		−	22.5167i		−	1.03314i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−10.6066	+	18.3712i	−0.484628	+	0.839400i	−0.999844	−	0.0176600i	$-0.994378\pi$
$479$	−10.6066	+	18.3712i	−0.484628	+	0.839400i	0.515216	+	0.857060i	$0.327712\pi$
$480$		0			0
$481$	−24.0000	+	13.8564i	−1.09431	+	0.631798i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	57.2756	−	33.0681i	2.60075	−	1.50155i
$486$		0			0
$487$	−0.500000	+	0.866025i	−0.0226572	+	0.0392434i	−0.877132	−	0.480250i	$-0.840546\pi$
$487$	−0.500000	+	0.866025i	−0.0226572	+	0.0392434i	0.854475	+	0.519493i	$0.173879\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	29.3939i		−	1.32653i	−0.748386	−	0.663264i	$-0.769171\pi$
$491$		−	29.3939i		−	1.32653i	0.748386	−	0.663264i	$-0.230829\pi$
$492$		0			0
$493$	−27.0000	−	15.5885i	−1.21602	−	0.702069i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	12.7279	+	36.7423i	0.570925	+	1.64812i
$498$		0			0
$499$	−12.5000	−	21.6506i	−0.559577	−	0.969216i	−0.997532	−	0.0702185i	$-0.977630\pi$
$499$	−12.5000	−	21.6506i	−0.559577	−	0.969216i	0.437955	−	0.898997i	$-0.355703\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−29.6985			−1.32419			−0.662095	−	0.749420i	$-0.730332\pi$
$503$	−29.6985			−1.32419			−0.662095	+	0.749420i	$0.730332\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	8.48528	+	14.6969i	0.376103	+	0.651430i	0.990492	−	0.137574i	$-0.0439304\pi$
$509$	8.48528	+	14.6969i	0.376103	+	0.651430i	−0.614388	+	0.789004i	$0.710597\pi$
$510$		0			0
$511$	22.5000	+	4.33013i	0.995341	+	0.191554i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−25.4558	−	14.6969i	−1.12172	−	0.647624i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−4.24264	+	7.34847i	−0.185873	+	0.321942i	−0.943871	−	0.330316i	$-0.892845\pi$
$521$	−4.24264	+	7.34847i	−0.185873	+	0.321942i	0.757997	+	0.652258i	$0.226178\pi$
$522$		0			0
$523$	3.00000	−	1.73205i	0.131181	−	0.0757373i	−0.432973	−	0.901407i	$-0.642536\pi$
$523$	3.00000	−	1.73205i	0.131181	−	0.0757373i	0.564154	+	0.825669i	$0.309202\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	6.36396	−	3.67423i	0.277218	−	0.160052i
$528$		0			0
$529$	15.5000	−	26.8468i	0.673913	−	1.16725i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	14.6969i		−	0.636595i
$534$		0			0
$535$	54.0000	+	31.1769i	2.33462	+	1.34790i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−17.5000	−	30.3109i	−0.752384	−	1.30317i	−0.946664	−	0.322221i	$-0.895571\pi$
$541$	−17.5000	−	30.3109i	−0.752384	−	1.30317i	0.194281	−	0.980946i	$-0.437763\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	4.24264			0.181735
$546$		0			0
$547$	17.0000			0.726868			0.363434	−	0.931620i	$-0.381604\pi$
$547$	17.0000			0.726868			0.363434	+	0.931620i	$0.381604\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−6.36396	−	11.0227i	−0.271114	−	0.469583i
$552$		0			0
$553$	−8.00000	−	6.92820i	−0.340195	−	0.294617i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	19.0919	+	11.0227i	0.808949	+	0.467047i	0.846591	−	0.532244i	$-0.178651\pi$
$557$	19.0919	+	11.0227i	0.808949	+	0.467047i	−0.0376417	+	0.999291i	$0.511985\pi$
$558$		0			0
$559$		−	17.3205i		−	0.732579i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−19.0919	+	33.0681i	−0.804627	+	1.39365i	0.111916	+	0.993718i	$0.464301\pi$
$563$	−19.0919	+	33.0681i	−0.804627	+	1.39365i	−0.916543	+	0.399937i	$0.869032\pi$
$564$		0			0
$565$	54.0000	−	31.1769i	2.27180	−	1.31162i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	12.7279	−	7.34847i	0.533582	−	0.308064i	−0.208892	−	0.977939i	$-0.566986\pi$
$569$	12.7279	−	7.34847i	0.533582	−	0.308064i	0.742474	+	0.669875i	$0.233652\pi$
$570$		0			0
$571$	−5.50000	+	9.52628i	−0.230168	+	0.398662i	−0.957857	−	0.287244i	$-0.907261\pi$
$571$	−5.50000	+	9.52628i	−0.230168	+	0.398662i	0.727690	+	0.685907i	$0.240594\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			95.5301i			3.98388i
$576$		0			0
$577$	3.00000	+	1.73205i	0.124892	+	0.0721062i	0.561144	−	0.827718i	$-0.310361\pi$
$577$	3.00000	+	1.73205i	0.124892	+	0.0721062i	−0.436253	+	0.899824i	$0.643695\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	42.4264	−	14.6969i	1.76014	−	0.609732i
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	12.7279			0.525338			0.262669	−	0.964886i	$-0.415397\pi$
$587$	12.7279			0.525338			0.262669	+	0.964886i	$0.415397\pi$
$588$		0			0
$589$	3.00000			0.123613
$590$		0			0
$591$		0			0
$592$		0			0
$593$	8.48528	+	14.6969i	0.348449	+	0.603531i	0.985974	−	0.166898i	$-0.0533752\pi$
$593$	8.48528	+	14.6969i	0.348449	+	0.603531i	−0.637525	+	0.770429i	$0.720042\pi$
$594$		0			0
$595$	−45.0000	+	15.5885i	−1.84482	+	0.639064i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	19.0919	+	11.0227i	0.780073	+	0.450375i	0.836456	−	0.548034i	$-0.184623\pi$
$599$	19.0919	+	11.0227i	0.780073	+	0.450375i	−0.0563830	+	0.998409i	$0.517957\pi$
$600$		0			0
$601$			12.1244i			0.494563i	0.968944	+	0.247281i	$0.0795372\pi$
$601$			12.1244i			0.494563i	−0.968944	+	0.247281i	$0.920463\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−23.3345	+	40.4166i	−0.948683	+	1.64317i
$606$		0			0
$607$	−7.50000	+	4.33013i	−0.304416	+	0.175754i	−0.644425	−	0.764668i	$-0.722903\pi$
$607$	−7.50000	+	4.33013i	−0.304416	+	0.175754i	0.340009	+	0.940422i	$0.389570\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	12.7279	−	7.34847i	0.514917	−	0.297287i
$612$		0			0
$613$	3.50000	−	6.06218i	0.141364	−	0.244849i	−0.786647	−	0.617403i	$-0.788185\pi$
$613$	3.50000	−	6.06218i	0.141364	−	0.244849i	0.928010	+	0.372554i	$0.121518\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	29.3939i		−	1.18335i	−0.806176	−	0.591676i	$-0.798466\pi$
$617$		−	29.3939i		−	1.18335i	0.806176	−	0.591676i	$-0.201534\pi$
$618$		0			0
$619$	−12.0000	−	6.92820i	−0.482321	−	0.278468i	0.239062	−	0.971004i	$-0.423160\pi$
$619$	−12.0000	−	6.92820i	−0.482321	−	0.278468i	−0.721383	+	0.692536i	$0.756493\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	25.4558	+	22.0454i	1.01987	+	0.883231i
$624$		0			0
$625$	−39.5000	−	68.4160i	−1.58000	−	2.73664i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−33.9411			−1.35332
$630$		0			0
$631$	29.0000			1.15447			0.577236	−	0.816577i	$-0.304131\pi$
$631$	29.0000			1.15447			0.577236	+	0.816577i	$0.304131\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−36.0624	−	62.4620i	−1.43109	−	2.47873i
$636$		0			0
$637$	15.0000	+	19.0526i	0.594322	+	0.754890i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	31.8198	+	18.3712i	1.25681	+	0.725618i	0.972452	−	0.233102i	$-0.0748875\pi$
$641$	31.8198	+	18.3712i	1.25681	+	0.725618i	0.284354	+	0.958719i	$0.408221\pi$
$642$		0			0
$643$			19.0526i			0.751360i	0.926750	+	0.375680i	$0.122591\pi$
$643$			19.0526i			0.751360i	−0.926750	+	0.375680i	$0.877409\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	12.7279	−	22.0454i	0.500386	−	0.866694i	−0.499614	−	0.866248i	$-0.666525\pi$
$647$	12.7279	−	22.0454i	0.500386	−	0.866694i	1.00000		0.000446060i	$-0.000141985\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	6.36396	−	3.67423i	0.249041	−	0.143784i	−0.370284	−	0.928919i	$-0.620740\pi$
$653$	6.36396	−	3.67423i	0.249041	−	0.143784i	0.619325	+	0.785135i	$0.287406\pi$
$654$		0			0
$655$	27.0000	−	46.7654i	1.05498	−	1.82727i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	22.0454i		−	0.858767i	−0.903122	−	0.429384i	$-0.858731\pi$
$659$		−	22.0454i		−	0.858767i	0.903122	−	0.429384i	$-0.141269\pi$
$660$		0			0
$661$	25.5000	+	14.7224i	0.991835	+	0.572636i	0.905822	−	0.423658i	$-0.139254\pi$
$661$	25.5000	+	14.7224i	0.991835	+	0.572636i	0.0860127	+	0.996294i	$0.472587\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−19.0919	−	3.67423i	−0.740351	−	0.142481i
$666$		0			0
$667$	27.0000	+	46.7654i	1.04544	+	1.81076i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−5.00000			−0.192736			−0.0963679	−	0.995346i	$-0.530723\pi$
$673$	−5.00000			−0.192736			−0.0963679	+	0.995346i	$0.530723\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	12.7279	+	22.0454i	0.489174	+	0.847274i	0.999922	−	0.0124562i	$-0.00396505\pi$
$677$	12.7279	+	22.0454i	0.489174	+	0.847274i	−0.510749	+	0.859730i	$0.670632\pi$
$678$		0			0
$679$	−13.5000	−	38.9711i	−0.518082	−	1.49558i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	12.7279	+	7.34847i	0.487020	+	0.281181i	0.723338	−	0.690495i	$-0.242607\pi$
$683$	12.7279	+	7.34847i	0.487020	+	0.281181i	−0.236317	+	0.971676i	$0.575940\pi$
$684$		0			0
$685$		−	31.1769i		−	1.19121i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	12.7279	−	22.0454i	0.484895	−	0.839863i
$690$		0			0
$691$	−1.50000	+	0.866025i	−0.0570627	+	0.0329452i	−0.528260	−	0.849083i	$-0.677155\pi$
$691$	−1.50000	+	0.866025i	−0.0570627	+	0.0329452i	0.471197	+	0.882028i	$0.343822\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−50.9117	+	29.3939i	−1.93119	+	1.11497i
$696$		0			0
$697$	9.00000	−	15.5885i	0.340899	−	0.590455i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			7.34847i			0.277548i	0.990324	+	0.138774i	$0.0443161\pi$
$701$			7.34847i			0.277548i	−0.990324	+	0.138774i	$0.955684\pi$
$702$		0			0
$703$	−12.0000	−	6.92820i	−0.452589	−	0.261302i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−18.5000	−	32.0429i	−0.694782	−	1.20340i	−0.970254	−	0.242089i	$-0.922167\pi$
$709$	−18.5000	−	32.0429i	−0.694782	−	1.20340i	0.275472	−	0.961309i	$-0.411166\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−12.7279			−0.476664
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−4.24264	−	7.34847i	−0.158224	−	0.274052i	0.776004	−	0.630727i	$-0.217243\pi$
$719$	−4.24264	−	7.34847i	−0.158224	−	0.274052i	−0.934228	+	0.356676i	$0.883910\pi$
$720$		0			0
$721$	−12.0000	+	13.8564i	−0.446903	+	0.516040i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−82.7315	−	47.7650i	−3.07257	−	1.77395i
$726$		0			0
$727$		−	22.5167i		−	0.835097i	−0.908655	−	0.417548i	$-0.862889\pi$
$727$		−	22.5167i		−	0.835097i	0.908655	−	0.417548i	$-0.137111\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	10.6066	−	18.3712i	0.392299	−	0.679482i
$732$		0			0
$733$	−1.50000	+	0.866025i	−0.0554038	+	0.0319874i	−0.527446	−	0.849589i	$-0.676850\pi$
$733$	−1.50000	+	0.866025i	−0.0554038	+	0.0319874i	0.472042	+	0.881576i	$0.343517\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	18.5000	−	32.0429i	0.680534	−	1.17872i	−0.294285	−	0.955718i	$-0.595081\pi$
$739$	18.5000	−	32.0429i	0.680534	−	1.17872i	0.974818	−	0.223001i	$-0.0715853\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	7.34847i		−	0.269589i	−0.990874	−	0.134795i	$-0.956963\pi$
$743$		−	7.34847i		−	0.269589i	0.990874	−	0.134795i	$-0.0430375\pi$
$744$		0			0
$745$	27.0000	+	15.5885i	0.989203	+	0.571117i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	25.4558	−	29.3939i	0.930136	−	1.07403i
$750$		0			0
$751$	5.50000	+	9.52628i	0.200698	+	0.347619i	0.948753	−	0.316017i	$-0.102346\pi$
$751$	5.50000	+	9.52628i	0.200698	+	0.347619i	−0.748056	+	0.663636i	$0.769012\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	4.24264			0.154406
$756$		0			0
$757$	−47.0000			−1.70824			−0.854122	−	0.520073i	$-0.825905\pi$
$757$	−47.0000			−1.70824			−0.854122	+	0.520073i	$0.825905\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	8.48528	+	14.6969i	0.307591	+	0.532764i	0.977835	−	0.209377i	$-0.0671437\pi$
$761$	8.48528	+	14.6969i	0.307591	+	0.532764i	−0.670244	+	0.742141i	$0.733810\pi$
$762$		0			0
$763$	0.500000	−	2.59808i	0.0181012	−	0.0940567i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−25.4558	−	14.6969i	−0.919157	−	0.530676i
$768$		0			0
$769$		−	43.3013i		−	1.56148i	−0.624854	−	0.780742i	$-0.714841\pi$
$769$		−	43.3013i		−	1.56148i	0.624854	−	0.780742i	$-0.285159\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−14.8492	+	25.7196i	−0.534090	+	0.925071i	0.465117	+	0.885249i	$0.346012\pi$
$773$	−14.8492	+	25.7196i	−0.534090	+	0.925071i	−0.999207	+	0.0398218i	$0.987321\pi$
$774$		0			0
$775$	19.5000	−	11.2583i	0.700461	−	0.404411i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	6.36396	−	3.67423i	0.228013	−	0.131643i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	44.0908i		−	1.57367i
$786$		0			0
$787$	22.5000	+	12.9904i	0.802038	+	0.463057i	0.844183	−	0.536054i	$-0.180086\pi$
$787$	22.5000	+	12.9904i	0.802038	+	0.463057i	−0.0421450	+	0.999112i	$0.513419\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−12.7279	−	36.7423i	−0.452553	−	1.30641i
$792$		0			0
$793$	−9.00000	−	15.5885i	−0.319599	−	0.553562i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$	18.0000			0.636794
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$	81.0000	+	15.5885i	2.85487	+	0.549421i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	25.4558	+	14.6969i	0.894980	+	0.516717i	0.875568	−	0.483095i	$-0.160487\pi$
$809$	25.4558	+	14.6969i	0.894980	+	0.516717i	0.0194117	+	0.999812i	$0.493821\pi$
$810$		0			0
$811$		−	20.7846i		−	0.729846i	−0.931038	−	0.364923i	$-0.881095\pi$
$811$		−	20.7846i		−	0.729846i	0.931038	−	0.364923i	$-0.118905\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	14.8492	−	25.7196i	0.520146	−	0.900920i
$816$		0			0
$817$	7.50000	−	4.33013i	0.262392	−	0.151492i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−25.4558	+	14.6969i	−0.888415	+	0.512927i	−0.873424	−	0.486961i	$-0.838105\pi$
$821$	−25.4558	+	14.6969i	−0.888415	+	0.512927i	−0.0149913	+	0.999888i	$0.504772\pi$
$822$		0			0
$823$	15.5000	−	26.8468i	0.540296	−	0.935820i	−0.458591	−	0.888648i	$-0.651646\pi$
$823$	15.5000	−	26.8468i	0.540296	−	0.935820i	0.998887	−	0.0471726i	$-0.0150211\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			29.3939i			1.02213i	0.859543	+	0.511063i	$0.170748\pi$
$827$			29.3939i			1.02213i	−0.859543	+	0.511063i	$0.829252\pi$
$828$		0			0
$829$	−1.50000	−	0.866025i	−0.0520972	−	0.0300783i	0.473725	−	0.880673i	$-0.342909\pi$
$829$	−1.50000	−	0.866025i	−0.0520972	−	0.0300783i	−0.525822	+	0.850594i	$0.676242\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	4.24264	+	29.3939i	0.146999	+	1.01844i
$834$		0			0
$835$	18.0000	+	31.1769i	0.622916	+	1.07892i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	42.4264			1.46472			0.732361	−	0.680916i	$-0.238418\pi$
$839$	42.4264			1.46472			0.732361	+	0.680916i	$0.238418\pi$
$840$		0			0
$841$	−25.0000			−0.862069
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−2.12132	−	3.67423i	−0.0729756	−	0.126398i
$846$		0			0
$847$	22.0000	+	19.0526i	0.755929	+	0.654654i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	50.9117	+	29.3939i	1.74523	+	1.00761i
$852$		0			0
$853$		−	19.0526i		−	0.652347i	−0.945310	−	0.326174i	$-0.894241\pi$
$853$		−	19.0526i		−	0.652347i	0.945310	−	0.326174i	$-0.105759\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	4.24264	−	7.34847i	0.144926	−	0.251019i	−0.784419	−	0.620231i	$-0.787039\pi$
$857$	4.24264	−	7.34847i	0.144926	−	0.251019i	0.929345	+	0.369212i	$0.120372\pi$
$858$		0			0
$859$	−31.5000	+	18.1865i	−1.07477	+	0.620517i	−0.929480	−	0.368873i	$-0.879744\pi$
$859$	−31.5000	+	18.1865i	−1.07477	+	0.620517i	−0.145286	+	0.989390i	$0.546410\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	44.5477	−	25.7196i	1.51642	−	0.875507i	0.516608	−	0.856222i	$-0.327194\pi$
$863$	44.5477	−	25.7196i	1.51642	−	0.875507i	0.999814	−	0.0192849i	$-0.00613896\pi$
$864$		0			0
$865$	27.0000	−	46.7654i	0.918028	−	1.59007i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$	−6.00000	−	3.46410i	−0.203302	−	0.117377i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−84.8528	+	29.3939i	−2.86855	+	0.993694i
$876$		0			0
$877$	−11.5000	−	19.9186i	−0.388327	−	0.672603i	0.603897	−	0.797062i	$-0.293614\pi$
$877$	−11.5000	−	19.9186i	−0.388327	−	0.672603i	−0.992225	+	0.124459i	$0.960280\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−38.1838			−1.28644			−0.643222	−	0.765680i	$-0.722403\pi$
$881$	−38.1838			−1.28644			−0.643222	+	0.765680i	$0.722403\pi$
$882$		0			0
$883$	−35.0000			−1.17784			−0.588922	−	0.808190i	$-0.700447\pi$
$883$	−35.0000			−1.17784			−0.588922	+	0.808190i	$0.700447\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	29.6985	+	51.4393i	0.997178	+	1.72716i	0.563610	+	0.826041i	$0.309412\pi$
$887$	29.6985	+	51.4393i	0.997178	+	1.72716i	0.433568	+	0.901121i	$0.357254\pi$
$888$		0			0
$889$	−42.5000	+	14.7224i	−1.42540	+	0.493775i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	6.36396	+	3.67423i	0.212962	+	0.122954i
$894$		0			0
$895$			31.1769i			1.04213i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	6.36396	−	11.0227i	0.212250	−	0.367628i
$900$		0			0
$901$	27.0000	−	15.5885i	0.899500	−	0.519327i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−70.0036	+	40.4166i	−2.32700	+	1.34349i
$906$		0			0
$907$	−20.0000	+	34.6410i	−0.664089	+	1.15024i	0.315442	+	0.948945i	$0.397847\pi$
$907$	−20.0000	+	34.6410i	−0.664089	+	1.15024i	−0.979531	+	0.201291i	$0.935486\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			22.0454i			0.730397i	0.930930	+	0.365198i	$0.118999\pi$
$911$			22.0454i			0.730397i	−0.930930	+	0.365198i	$0.881001\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−25.4558	−	22.0454i	−0.840626	−	0.728003i
$918$		0			0
$919$	8.50000	+	14.7224i	0.280389	+	0.485648i	0.971481	−	0.237119i	$-0.0762032\pi$
$919$	8.50000	+	14.7224i	0.280389	+	0.485648i	−0.691091	+	0.722767i	$0.742870\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	50.9117			1.67578
$924$		0			0
$925$	−104.000			−3.41950
$926$		0			0
$927$		0			0
$928$		0			0
$929$	23.3345	+	40.4166i	0.765581	+	1.32603i	0.939939	+	0.341343i	$0.110882\pi$
$929$	23.3345	+	40.4166i	0.765581	+	1.32603i	−0.174358	+	0.984682i	$0.555785\pi$
$930$		0			0
$931$	−4.50000	+	11.2583i	−0.147482	+	0.368977i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$			38.1051i			1.24484i	0.782683	+	0.622420i	$0.213850\pi$
$937$			38.1051i			1.24484i	−0.782683	+	0.622420i	$0.786150\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−4.24264	+	7.34847i	−0.138306	+	0.239553i	−0.926856	−	0.375418i	$-0.877499\pi$
$941$	−4.24264	+	7.34847i	−0.138306	+	0.239553i	0.788549	+	0.614971i	$0.210832\pi$
$942$		0			0
$943$	−27.0000	+	15.5885i	−0.879241	+	0.507630i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	31.8198	−	18.3712i	1.03400	−	0.596983i	0.115875	−	0.993264i	$-0.463033\pi$
$947$	31.8198	−	18.3712i	1.03400	−	0.596983i	0.918129	+	0.396281i	$0.129699\pi$
$948$		0			0
$949$	15.0000	−	25.9808i	0.486921	−	0.843371i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	51.4393i		−	1.66628i	−0.553061	−	0.833141i	$-0.686540\pi$
$953$		−	51.4393i		−	1.66628i	0.553061	−	0.833141i	$-0.313460\pi$
$954$		0			0
$955$	27.0000	+	15.5885i	0.873699	+	0.504431i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−19.0919	−	3.67423i	−0.616509	−	0.118647i
$960$		0			0
$961$	−14.0000	−	24.2487i	−0.451613	−	0.782216i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	67.8823			2.18521
$966$		0			0
$967$	38.0000			1.22200			0.610999	−	0.791632i	$-0.290768\pi$
$967$	38.0000			1.22200			0.610999	+	0.791632i	$0.290768\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	27.5772	+	47.7650i	0.884993	+	1.53285i	0.845721	+	0.533625i	$0.179171\pi$
$971$	27.5772	+	47.7650i	0.884993	+	1.53285i	0.0392720	+	0.999229i	$0.487496\pi$
$972$		0			0
$973$	12.0000	+	34.6410i	0.384702	+	1.11054i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	12.7279	+	7.34847i	0.407202	+	0.235098i	0.689587	−	0.724203i	$-0.257792\pi$
$977$	12.7279	+	7.34847i	0.407202	+	0.235098i	−0.282385	+	0.959301i	$0.591125\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$	21.2132	−	36.7423i	0.676596	−	1.17190i	−0.299404	−	0.954127i	$-0.596788\pi$
$983$	21.2132	−	36.7423i	0.676596	−	1.17190i	0.976000	−	0.217772i	$-0.0698790\pi$
$984$		0			0
$985$	−54.0000	+	31.1769i	−1.72058	+	0.993379i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−31.8198	+	18.3712i	−1.01181	+	0.584169i
$990$		0			0
$991$	13.0000	−	22.5167i	0.412959	−	0.715265i	−0.582253	−	0.813008i	$-0.697829\pi$
$991$	13.0000	−	22.5167i	0.412959	−	0.715265i	0.995212	+	0.0977423i	$0.0311621\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			66.1362i			2.09666i
$996$		0			0
$997$	46.5000	+	26.8468i	1.47267	+	0.850246i	0.999527	−	0.0307397i	$-0.00978629\pi$
$997$	46.5000	+	26.8468i	1.47267	+	0.850246i	0.473142	+	0.880986i	$0.343120\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	756.2.t.d.269.1	✓	4
3.2	odd	2	inner	756.2.t.d.269.2	yes	4
7.3	odd	6		5292.2.f.d.2645.1		4
7.4	even	3		5292.2.f.d.2645.4		4
7.5	odd	6	inner	756.2.t.d.593.2	yes	4
9.2	odd	6		2268.2.bm.h.1025.1		4
9.4	even	3		2268.2.w.g.269.1		4
9.5	odd	6		2268.2.w.g.269.2		4
9.7	even	3		2268.2.bm.h.1025.2		4
21.5	even	6	inner	756.2.t.d.593.1	yes	4
21.11	odd	6		5292.2.f.d.2645.2		4
21.17	even	6		5292.2.f.d.2645.3		4
63.5	even	6		2268.2.bm.h.593.2		4
63.40	odd	6		2268.2.bm.h.593.1		4
63.47	even	6		2268.2.w.g.1349.1		4
63.61	odd	6		2268.2.w.g.1349.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
756.2.t.d.269.1	✓	4	1.1	even	1	trivial
756.2.t.d.269.2	yes	4	3.2	odd	2	inner
756.2.t.d.593.1	yes	4	21.5	even	6	inner
756.2.t.d.593.2	yes	4	7.5	odd	6	inner
2268.2.w.g.269.1		4	9.4	even	3
2268.2.w.g.269.2		4	9.5	odd	6
2268.2.w.g.1349.1		4	63.47	even	6
2268.2.w.g.1349.2		4	63.61	odd	6
2268.2.bm.h.593.1		4	63.40	odd	6
2268.2.bm.h.593.2		4	63.5	even	6
2268.2.bm.h.1025.1		4	9.2	odd	6
2268.2.bm.h.1025.2		4	9.7	even	3
5292.2.f.d.2645.1		4	7.3	odd	6
5292.2.f.d.2645.2		4	21.11	odd	6
5292.2.f.d.2645.3		4	21.17	even	6
5292.2.f.d.2645.4		4	7.4	even	3

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:	\( N \)	\(=\)	\( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	756.t (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.12132 - 3.67423i) q^{5} +(-2.50000 + 0.866025i) q^{7} +O(q^{10})\)	\(q+(-2.12132 - 3.67423i) q^{5} +(-2.50000 + 0.866025i) q^{7} +3.46410i q^{13} +(-2.12132 + 3.67423i) q^{17} +(-1.50000 + 0.866025i) q^{19} +(6.36396 - 3.67423i) q^{23} +(-6.50000 + 11.2583i) q^{25} +7.34847i q^{29} +(-1.50000 - 0.866025i) q^{31} +(8.48528 + 7.34847i) q^{35} +(4.00000 + 6.92820i) q^{37} -4.24264 q^{41} -5.00000 q^{43} +(-2.12132 - 3.67423i) q^{47} +(5.50000 - 4.33013i) q^{49} +(-6.36396 - 3.67423i) q^{53} +(-4.24264 + 7.34847i) q^{59} +(-4.50000 + 2.59808i) q^{61} +(12.7279 - 7.34847i) q^{65} +(-1.00000 + 1.73205i) q^{67} -14.6969i q^{71} +(-7.50000 - 4.33013i) q^{73} +(2.00000 + 3.46410i) q^{79} -16.9706 q^{83} +18.0000 q^{85} +(-6.36396 - 11.0227i) q^{89} +(-3.00000 - 8.66025i) q^{91} +(6.36396 + 3.67423i) q^{95} +15.5885i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 10 q^{7}+O(q^{10}) \) 4 * q - 10 * q^7	\( 4 q - 10 q^{7} - 6 q^{19} - 26 q^{25} - 6 q^{31} + 16 q^{37} - 20 q^{43} + 22 q^{49} - 18 q^{61} - 4 q^{67} - 30 q^{73} + 8 q^{79} + 72 q^{85} - 12 q^{91}+O(q^{100}) \) 4 * q - 10 * q^7 - 6 * q^19 - 26 * q^25 - 6 * q^31 + 16 * q^37 - 20 * q^43 + 22 * q^49 - 18 * q^61 - 4 * q^67 - 30 * q^73 + 8 * q^79 + 72 * q^85 - 12 * q^91

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