LMFDB - Embedded newform 2304.2.l.g.575.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2304,2,Mod(575,2304)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2304, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 1, 2]))

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

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

chi := DirichletCharacter("2304.575");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2304 = 2^{8} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2304.l (of order $4$, 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:	$18.3975326257$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			575.1
Root			$0.258819 - 0.965926i$ of defining polynomial
Character	$\chi$	$=$	2304.575
Dual form			2304.2.l.g.1727.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.93185 + 1.93185i) q^{5} +1.41421 q^{7} +O(q^{10})$	$q+(-1.93185 + 1.93185i) q^{5} +1.41421 q^{7} +(-0.732051 - 0.732051i) q^{11} +(1.73205 - 1.73205i) q^{13} +5.27792i q^{17} +(-5.27792 - 5.27792i) q^{19} +3.46410i q^{23} -2.46410i q^{25} +(2.31079 + 2.31079i) q^{29} +9.14162i q^{31} +(-2.73205 + 2.73205i) q^{35} +(-2.46410 - 2.46410i) q^{37} -4.52004 q^{41} +(3.48477 - 3.48477i) q^{43} +10.3923 q^{47} -5.00000 q^{49} +(-8.24504 + 8.24504i) q^{53} +2.82843 q^{55} +(-8.92820 - 8.92820i) q^{59} +(-3.00000 + 3.00000i) q^{61} +6.69213i q^{65} +(3.86370 + 3.86370i) q^{67} -14.0000i q^{71} -4.92820i q^{73} +(-1.03528 - 1.03528i) q^{77} -2.17209i q^{79} +(-10.7321 + 10.7321i) q^{83} +(-10.1962 - 10.1962i) q^{85} -16.1112 q^{89} +(2.44949 - 2.44949i) q^{91} +20.3923 q^{95} -18.3923 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 8 q^{11} - 8 q^{35} + 8 q^{37} - 40 q^{49} - 16 q^{59} - 24 q^{61} - 72 q^{83} - 40 q^{85} + 80 q^{95} - 64 q^{97}+O(q^{100}) $ 8 * q + 8 * q^11 - 8 * q^35 + 8 * q^37 - 40 * q^49 - 16 * q^59 - 24 * q^61 - 72 * q^83 - 40 * q^85 + 80 * q^95 - 64 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times$.

$n$	$1279$	$1793$	$2053$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{1}{4}\right)$

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
$5$	−1.93185	+	1.93185i	−0.863950	+	0.863950i	−0.991794	−	0.127844i	$-0.959194\pi$
$5$	−1.93185	+	1.93185i	−0.863950	+	0.863950i	0.127844	+	0.991794i	$0.459194\pi$
$6$		0			0
$7$	1.41421			0.534522			0.267261	−	0.963624i	$-0.413881\pi$
$7$	1.41421			0.534522			0.267261	+	0.963624i	$0.413881\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−0.732051	−	0.732051i	−0.220722	−	0.220722i	0.588081	−	0.808802i	$-0.299884\pi$
$11$	−0.732051	−	0.732051i	−0.220722	−	0.220722i	−0.808802	+	0.588081i	$0.799884\pi$
$12$		0			0
$13$	1.73205	−	1.73205i	0.480384	−	0.480384i	−0.424870	−	0.905254i	$-0.639680\pi$
$13$	1.73205	−	1.73205i	0.480384	−	0.480384i	0.905254	+	0.424870i	$0.139680\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			5.27792i			1.28008i	0.768340	+	0.640041i	$0.221083\pi$
$17$			5.27792i			1.28008i	−0.768340	+	0.640041i	$0.778917\pi$
$18$		0			0
$19$	−5.27792	−	5.27792i	−1.21084	−	1.21084i	−0.970752	−	0.240085i	$-0.922825\pi$
$19$	−5.27792	−	5.27792i	−1.21084	−	1.21084i	−0.240085	−	0.970752i	$-0.577175\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			3.46410i			0.722315i	0.932505	+	0.361158i	$0.117618\pi$
$23$			3.46410i			0.722315i	−0.932505	+	0.361158i	$0.882382\pi$
$24$		0			0
$25$		−	2.46410i		−	0.492820i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	2.31079	+	2.31079i	0.429103	+	0.429103i	0.888323	−	0.459220i	$-0.151871\pi$
$29$	2.31079	+	2.31079i	0.429103	+	0.429103i	−0.459220	+	0.888323i	$0.651871\pi$
$30$		0			0
$31$			9.14162i			1.64188i	0.571012	+	0.820942i	$0.306551\pi$
$31$			9.14162i			1.64188i	−0.571012	+	0.820942i	$0.693449\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−2.73205	+	2.73205i	−0.461801	+	0.461801i
$36$		0			0
$37$	−2.46410	−	2.46410i	−0.405096	−	0.405096i	0.474929	−	0.880024i	$-0.342474\pi$
$37$	−2.46410	−	2.46410i	−0.405096	−	0.405096i	−0.880024	+	0.474929i	$0.842474\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−4.52004			−0.705912			−0.352956	−	0.935640i	$-0.614823\pi$
$41$	−4.52004			−0.705912			−0.352956	+	0.935640i	$0.614823\pi$
$42$		0			0
$43$	3.48477	−	3.48477i	0.531422	−	0.531422i	−0.389574	−	0.920995i	$-0.627378\pi$
$43$	3.48477	−	3.48477i	0.531422	−	0.531422i	0.920995	+	0.389574i	$0.127378\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	10.3923			1.51587			0.757937	−	0.652328i	$-0.226208\pi$
$47$	10.3923			1.51587			0.757937	+	0.652328i	$0.226208\pi$
$48$		0			0
$49$	−5.00000			−0.714286
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−8.24504	+	8.24504i	−1.13254	+	1.13254i	−0.142791	+	0.989753i	$0.545608\pi$
$53$	−8.24504	+	8.24504i	−1.13254	+	1.13254i	−0.989753	+	0.142791i	$0.954392\pi$
$54$		0			0
$55$	2.82843			0.381385
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−8.92820	−	8.92820i	−1.16235	−	1.16235i	−0.983959	−	0.178394i	$-0.942910\pi$
$59$	−8.92820	−	8.92820i	−1.16235	−	1.16235i	−0.178394	−	0.983959i	$-0.557090\pi$
$60$		0			0
$61$	−3.00000	+	3.00000i	−0.384111	+	0.384111i	−0.872581	−	0.488470i	$-0.837555\pi$
$61$	−3.00000	+	3.00000i	−0.384111	+	0.384111i	0.488470	+	0.872581i	$0.337555\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			6.69213i			0.830057i
$66$		0			0
$67$	3.86370	+	3.86370i	0.472026	+	0.472026i	0.902570	−	0.430543i	$-0.141678\pi$
$67$	3.86370	+	3.86370i	0.472026	+	0.472026i	−0.430543	+	0.902570i	$0.641678\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	14.0000i		−	1.66149i	−0.556650	−	0.830747i	$-0.687914\pi$
$71$		−	14.0000i		−	1.66149i	0.556650	−	0.830747i	$-0.312086\pi$
$72$		0			0
$73$		−	4.92820i		−	0.576803i	−0.957510	−	0.288401i	$-0.906876\pi$
$73$		−	4.92820i		−	0.576803i	0.957510	−	0.288401i	$-0.0931237\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−1.03528	−	1.03528i	−0.117981	−	0.117981i
$78$		0			0
$79$		−	2.17209i		−	0.244379i	−0.992507	−	0.122190i	$-0.961008\pi$
$79$		−	2.17209i		−	0.244379i	0.992507	−	0.122190i	$-0.0389916\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−10.7321	+	10.7321i	−1.17800	+	1.17800i	−0.197741	+	0.980254i	$0.563361\pi$
$83$	−10.7321	+	10.7321i	−1.17800	+	1.17800i	−0.980254	+	0.197741i	$0.936639\pi$
$84$		0			0
$85$	−10.1962	−	10.1962i	−1.10593	−	1.10593i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−16.1112			−1.70778			−0.853889	−	0.520455i	$-0.825763\pi$
$89$	−16.1112			−1.70778			−0.853889	+	0.520455i	$0.825763\pi$
$90$		0			0
$91$	2.44949	−	2.44949i	0.256776	−	0.256776i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	20.3923			2.09221
$96$		0			0
$97$	−18.3923			−1.86746			−0.933728	−	0.357984i	$-0.883464\pi$
$97$	−18.3923			−1.86746			−0.933728	+	0.357984i	$0.883464\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−2.58819	+	2.58819i	−0.257535	+	0.257535i	−0.824051	−	0.566516i	$-0.808291\pi$
$101$	−2.58819	+	2.58819i	−0.257535	+	0.257535i	0.566516	+	0.824051i	$0.308291\pi$
$102$		0			0
$103$	−10.4543			−1.03009			−0.515046	−	0.857162i	$-0.672225\pi$
$103$	−10.4543			−1.03009			−0.515046	+	0.857162i	$0.672225\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	6.53590	+	6.53590i	0.631849	+	0.631849i	0.948532	−	0.316682i	$-0.102569\pi$
$107$	6.53590	+	6.53590i	0.631849	+	0.631849i	−0.316682	+	0.948532i	$0.602569\pi$
$108$		0			0
$109$	−7.73205	+	7.73205i	−0.740596	+	0.740596i	−0.972693	−	0.232097i	$-0.925441\pi$
$109$	−7.73205	+	7.73205i	−0.740596	+	0.740596i	0.232097	+	0.972693i	$0.425441\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			4.24264i			0.399114i	0.979886	+	0.199557i	$0.0639503\pi$
$113$			4.24264i			0.399114i	−0.979886	+	0.199557i	$0.936050\pi$
$114$		0			0
$115$	−6.69213	−	6.69213i	−0.624044	−	0.624044i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			7.46410i			0.684233i
$120$		0			0
$121$		−	9.92820i		−	0.902564i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−4.89898	−	4.89898i	−0.438178	−	0.438178i
$126$		0			0
$127$			14.0406i			1.24590i	0.782261	+	0.622951i	$0.214066\pi$
$127$			14.0406i			1.24590i	−0.782261	+	0.622951i	$0.785934\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−6.92820	+	6.92820i	−0.605320	+	0.605320i	−0.941719	−	0.336399i	$-0.890791\pi$
$131$	−6.92820	+	6.92820i	−0.605320	+	0.605320i	0.336399	+	0.941719i	$0.390791\pi$
$132$		0			0
$133$	−7.46410	−	7.46410i	−0.647220	−	0.647220i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	3.20736			0.274024			0.137012	−	0.990569i	$-0.456250\pi$
$137$	3.20736			0.274024			0.137012	+	0.990569i	$0.456250\pi$
$138$		0			0
$139$	5.93426	−	5.93426i	0.503337	−	0.503337i	−0.409136	−	0.912473i	$-0.634170\pi$
$139$	5.93426	−	5.93426i	0.503337	−	0.503337i	0.912473	+	0.409136i	$0.134170\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−2.53590			−0.212062
$144$		0			0
$145$	−8.92820			−0.741447
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−5.79555	+	5.79555i	−0.474790	+	0.474790i	−0.903461	−	0.428671i	$-0.858982\pi$
$149$	−5.79555	+	5.79555i	−0.474790	+	0.474790i	0.428671	+	0.903461i	$0.358982\pi$
$150$		0			0
$151$	0.656339			0.0534121			0.0267060	−	0.999643i	$-0.491498\pi$
$151$	0.656339			0.0534121			0.0267060	+	0.999643i	$0.491498\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−17.6603	−	17.6603i	−1.41851	−	1.41851i
$156$		0			0
$157$	3.92820	−	3.92820i	0.313505	−	0.313505i	−0.532761	−	0.846266i	$-0.678846\pi$
$157$	3.92820	−	3.92820i	0.313505	−	0.313505i	0.846266	+	0.532761i	$0.178846\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			4.89898i			0.386094i
$162$		0			0
$163$	−0.656339	−	0.656339i	−0.0514084	−	0.0514084i	0.680935	−	0.732344i	$-0.261574\pi$
$163$	−0.656339	−	0.656339i	−0.0514084	−	0.0514084i	−0.732344	+	0.680935i	$0.761574\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	6.53590i		−	0.505763i	−0.967497	−	0.252882i	$-0.918622\pi$
$167$		−	6.53590i		−	0.505763i	0.967497	−	0.252882i	$-0.0813783\pi$
$168$		0			0
$169$			7.00000i			0.538462i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	5.51815	+	5.51815i	0.419537	+	0.419537i	0.885044	−	0.465507i	$-0.154128\pi$
$173$	5.51815	+	5.51815i	0.419537	+	0.419537i	−0.465507	+	0.885044i	$0.654128\pi$
$174$		0			0
$175$		−	3.48477i		−	0.263424i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−6.00000	+	6.00000i	−0.448461	+	0.448461i	−0.894843	−	0.446382i	$-0.852712\pi$
$179$	−6.00000	+	6.00000i	−0.448461	+	0.448461i	0.446382	+	0.894843i	$0.352712\pi$
$180$		0			0
$181$	0.803848	+	0.803848i	0.0597495	+	0.0597495i	0.736350	−	0.676601i	$-0.236548\pi$
$181$	0.803848	+	0.803848i	0.0597495	+	0.0597495i	−0.676601	+	0.736350i	$0.736548\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	9.52056			0.699965
$186$		0			0
$187$	3.86370	−	3.86370i	0.282542	−	0.282542i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	13.4641			0.974228			0.487114	−	0.873338i	$-0.338050\pi$
$191$	13.4641			0.974228			0.487114	+	0.873338i	$0.338050\pi$
$192$		0			0
$193$	22.7846			1.64007			0.820036	−	0.572312i	$-0.193953\pi$
$193$	22.7846			1.64007			0.820036	+	0.572312i	$0.193953\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−16.7303	+	16.7303i	−1.19199	+	1.19199i	−0.215478	+	0.976509i	$0.569131\pi$
$197$	−16.7303	+	16.7303i	−1.19199	+	1.19199i	−0.976509	+	0.215478i	$0.930869\pi$
$198$		0			0
$199$	19.6975			1.39632			0.698158	−	0.715944i	$-0.254003\pi$
$199$	19.6975			1.39632			0.698158	+	0.715944i	$0.254003\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	3.26795	+	3.26795i	0.229365	+	0.229365i
$204$		0			0
$205$	8.73205	−	8.73205i	0.609873	−	0.609873i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			7.72741i			0.534516i
$210$		0			0
$211$	−10.2784	−	10.2784i	−0.707596	−	0.707596i	0.258433	−	0.966029i	$-0.416794\pi$
$211$	−10.2784	−	10.2784i	−0.707596	−	0.707596i	−0.966029	+	0.258433i	$0.916794\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			13.4641i			0.918244i
$216$		0			0
$217$			12.9282i			0.877624i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	9.14162	+	9.14162i	0.614932	+	0.614932i
$222$		0			0
$223$			12.7279i			0.852325i	0.904647	+	0.426162i	$0.140135\pi$
$223$			12.7279i			0.852325i	−0.904647	+	0.426162i	$0.859865\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	14.7321	−	14.7321i	0.977801	−	0.977801i	−0.0219583	−	0.999759i	$-0.506990\pi$
$227$	14.7321	−	14.7321i	0.977801	−	0.977801i	0.999759	+	0.0219583i	$0.00699012\pi$
$228$		0			0
$229$	7.19615	+	7.19615i	0.475535	+	0.475535i	0.903700	−	0.428165i	$-0.140840\pi$
$229$	7.19615	+	7.19615i	0.475535	+	0.475535i	−0.428165	+	0.903700i	$0.640840\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−20.4553			−1.34007			−0.670037	−	0.742328i	$-0.733722\pi$
$233$	−20.4553			−1.34007			−0.670037	+	0.742328i	$0.733722\pi$
$234$		0			0
$235$	−20.0764	+	20.0764i	−1.30964	+	1.30964i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−21.4641			−1.38840			−0.694199	−	0.719783i	$-0.744241\pi$
$239$	−21.4641			−1.38840			−0.694199	+	0.719783i	$0.744241\pi$
$240$		0			0
$241$	6.53590			0.421014			0.210507	−	0.977592i	$-0.432488\pi$
$241$	6.53590			0.421014			0.210507	+	0.977592i	$0.432488\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	9.65926	−	9.65926i	0.617107	−	0.617107i
$246$		0			0
$247$	−18.2832			−1.16333
$248$		0			0
$249$		0			0
$250$		0			0
$251$	7.66025	+	7.66025i	0.483511	+	0.483511i	0.906251	−	0.422740i	$-0.138932\pi$
$251$	7.66025	+	7.66025i	0.483511	+	0.483511i	−0.422740	+	0.906251i	$0.638932\pi$
$252$		0			0
$253$	2.53590	−	2.53590i	0.159431	−	0.159431i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			11.9700i			0.746671i	0.927696	+	0.373336i	$0.121786\pi$
$257$			11.9700i			0.746671i	−0.927696	+	0.373336i	$0.878214\pi$
$258$		0			0
$259$	−3.48477	−	3.48477i	−0.216533	−	0.216533i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			2.92820i			0.180561i	0.995916	+	0.0902804i	$0.0287763\pi$
$263$			2.92820i			0.180561i	−0.995916	+	0.0902804i	$0.971224\pi$
$264$		0			0
$265$		−	31.8564i		−	1.95692i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−6.93237	−	6.93237i	−0.422674	−	0.422674i	0.463449	−	0.886123i	$-0.346612\pi$
$269$	−6.93237	−	6.93237i	−0.422674	−	0.422674i	−0.886123	+	0.463449i	$0.846612\pi$
$270$		0			0
$271$		−	2.92996i		−	0.177983i	−0.996032	−	0.0889913i	$-0.971636\pi$
$271$		−	2.92996i		−	0.177983i	0.996032	−	0.0889913i	$-0.0283643\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1.80385	+	1.80385i	−0.108776	+	0.108776i
$276$		0			0
$277$	11.1962	+	11.1962i	0.672712	+	0.672712i	0.958340	−	0.285629i	$-0.0922024\pi$
$277$	11.1962	+	11.1962i	0.672712	+	0.672712i	−0.285629	+	0.958340i	$0.592202\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	9.14162			0.545343			0.272672	−	0.962107i	$-0.412093\pi$
$281$	9.14162			0.545343			0.272672	+	0.962107i	$0.412093\pi$
$282$		0			0
$283$	19.7990	−	19.7990i	1.17693	−	1.17693i	0.196405	−	0.980523i	$-0.437073\pi$
$283$	19.7990	−	19.7990i	1.17693	−	1.17693i	0.980523	−	0.196405i	$-0.0629267\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−6.39230			−0.377326
$288$		0			0
$289$	−10.8564			−0.638612
$290$		0			0
$291$		0			0
$292$		0			0
$293$	13.9019	−	13.9019i	0.812158	−	0.812158i	−0.172799	−	0.984957i	$-0.555281\pi$
$293$	13.9019	−	13.9019i	0.812158	−	0.812158i	0.984957	+	0.172799i	$0.0552812\pi$
$294$		0			0
$295$	34.4959			2.00843
$296$		0			0
$297$		0			0
$298$		0			0
$299$	6.00000	+	6.00000i	0.346989	+	0.346989i
$300$		0			0
$301$	4.92820	−	4.92820i	0.284057	−	0.284057i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	11.5911i		−	0.663705i
$306$		0			0
$307$	−1.03528	−	1.03528i	−0.0590863	−	0.0590863i	0.676946	−	0.736033i	$-0.263303\pi$
$307$	−1.03528	−	1.03528i	−0.0590863	−	0.0590863i	−0.736033	+	0.676946i	$0.763303\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			2.14359i			0.121552i	0.998151	+	0.0607760i	$0.0193575\pi$
$311$			2.14359i			0.121552i	−0.998151	+	0.0607760i	$0.980642\pi$
$312$		0			0
$313$			2.00000i			0.113047i	0.998401	+	0.0565233i	$0.0180015\pi$
$313$			2.00000i			0.113047i	−0.998401	+	0.0565233i	$0.981998\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−2.03339	−	2.03339i	−0.114206	−	0.114206i	0.647694	−	0.761901i	$-0.275733\pi$
$317$	−2.03339	−	2.03339i	−0.114206	−	0.114206i	−0.761901	+	0.647694i	$0.775733\pi$
$318$		0			0
$319$		−	3.38323i		−	0.189425i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	27.8564	−	27.8564i	1.54997	−	1.54997i
$324$		0			0
$325$	−4.26795	−	4.26795i	−0.236743	−	0.236743i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	14.6969			0.810268
$330$		0			0
$331$	−16.4901	+	16.4901i	−0.906377	+	0.906377i	−0.995978	−	0.0896005i	$-0.971441\pi$
$331$	−16.4901	+	16.4901i	−0.906377	+	0.906377i	0.0896005	+	0.995978i	$0.471441\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−14.9282			−0.815615
$336$		0			0
$337$	22.7846			1.24116			0.620578	−	0.784144i	$-0.286898\pi$
$337$	22.7846			1.24116			0.620578	+	0.784144i	$0.286898\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	6.69213	−	6.69213i	0.362399	−	0.362399i
$342$		0			0
$343$	−16.9706			−0.916324
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−0.875644	−	0.875644i	−0.0470071	−	0.0470071i	0.683213	−	0.730220i	$-0.260582\pi$
$347$	−0.875644	−	0.875644i	−0.0470071	−	0.0470071i	−0.730220	+	0.683213i	$0.760582\pi$
$348$		0			0
$349$	−13.3923	+	13.3923i	−0.716874	+	0.716874i	−0.967964	−	0.251090i	$-0.919211\pi$
$349$	−13.3923	+	13.3923i	−0.716874	+	0.716874i	0.251090	+	0.967964i	$0.419211\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	31.5660i		−	1.68009i	−0.542519	−	0.840043i	$-0.682529\pi$
$353$		−	31.5660i		−	1.68009i	0.542519	−	0.840043i	$-0.317471\pi$
$354$		0			0
$355$	27.0459	+	27.0459i	1.43545	+	1.43545i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	23.8564i		−	1.25909i	−0.776963	−	0.629546i	$-0.783241\pi$
$359$		−	23.8564i		−	1.25909i	0.776963	−	0.629546i	$-0.216759\pi$
$360$		0			0
$361$			36.7128i			1.93225i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	9.52056	+	9.52056i	0.498329	+	0.498329i
$366$		0			0
$367$		−	18.3848i		−	0.959678i	−0.877357	−	0.479839i	$-0.840695\pi$
$367$		−	18.3848i		−	0.959678i	0.877357	−	0.479839i	$-0.159305\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−11.6603	+	11.6603i	−0.605370	+	0.605370i
$372$		0			0
$373$	4.85641	+	4.85641i	0.251455	+	0.251455i	0.821567	−	0.570112i	$-0.193100\pi$
$373$	4.85641	+	4.85641i	0.251455	+	0.251455i	−0.570112	+	0.821567i	$0.693100\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	8.00481			0.412269
$378$		0			0
$379$	3.20736	−	3.20736i	0.164751	−	0.164751i	−0.619917	−	0.784668i	$-0.712834\pi$
$379$	3.20736	−	3.20736i	0.164751	−	0.164751i	0.784668	+	0.619917i	$0.212834\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	18.5359			0.947140			0.473570	−	0.880756i	$-0.342965\pi$
$383$	18.5359			0.947140			0.473570	+	0.880756i	$0.342965\pi$
$384$		0			0
$385$	4.00000			0.203859
$386$		0			0
$387$		0			0
$388$		0			0
$389$	8.14351	−	8.14351i	0.412892	−	0.412892i	−0.469853	−	0.882745i	$-0.655693\pi$
$389$	8.14351	−	8.14351i	0.412892	−	0.412892i	0.882745	+	0.469853i	$0.155693\pi$
$390$		0			0
$391$	−18.2832			−0.924623
$392$		0			0
$393$		0			0
$394$		0			0
$395$	4.19615	+	4.19615i	0.211131	+	0.211131i
$396$		0			0
$397$	−18.3205	+	18.3205i	−0.919480	+	0.919480i	−0.996991	−	0.0775115i	$-0.975303\pi$
$397$	−18.3205	+	18.3205i	−0.919480	+	0.919480i	0.0775115	+	0.996991i	$0.475303\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	22.2485i		−	1.11104i	−0.831504	−	0.555518i	$-0.812520\pi$
$401$		−	22.2485i		−	1.11104i	0.831504	−	0.555518i	$-0.187480\pi$
$402$		0			0
$403$	15.8338	+	15.8338i	0.788735	+	0.788735i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			3.60770i			0.178827i
$408$		0			0
$409$			6.39230i			0.316079i	0.987433	+	0.158040i	$0.0505174\pi$
$409$			6.39230i			0.316079i	−0.987433	+	0.158040i	$0.949483\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−12.6264	−	12.6264i	−0.621304	−	0.621304i
$414$		0			0
$415$		−	41.4655i		−	2.03546i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	10.7321	−	10.7321i	0.524295	−	0.524295i	−0.394571	−	0.918866i	$-0.629107\pi$
$419$	10.7321	−	10.7321i	0.524295	−	0.524295i	0.918866	+	0.394571i	$0.129107\pi$
$420$		0			0
$421$	14.2679	+	14.2679i	0.695377	+	0.695377i	0.963410	−	0.268033i	$-0.0863735\pi$
$421$	14.2679	+	14.2679i	0.695377	+	0.695377i	−0.268033	+	0.963410i	$0.586373\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	13.0053			0.630851
$426$		0			0
$427$	−4.24264	+	4.24264i	−0.205316	+	0.205316i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−37.3205			−1.79767			−0.898833	−	0.438292i	$-0.855584\pi$
$431$	−37.3205			−1.79767			−0.898833	+	0.438292i	$0.855584\pi$
$432$		0			0
$433$	−14.9282			−0.717404			−0.358702	−	0.933452i	$-0.616781\pi$
$433$	−14.9282			−0.717404			−0.358702	+	0.933452i	$0.616781\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	18.2832	−	18.2832i	0.874606	−	0.874606i
$438$		0			0
$439$	−29.6985			−1.41743			−0.708716	−	0.705494i	$-0.750725\pi$
$439$	−29.6985			−1.41743			−0.708716	+	0.705494i	$0.750725\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	4.58846	+	4.58846i	0.218004	+	0.218004i	0.807657	−	0.589653i	$-0.200735\pi$
$443$	4.58846	+	4.58846i	0.218004	+	0.218004i	−0.589653	+	0.807657i	$0.700735\pi$
$444$		0			0
$445$	31.1244	−	31.1244i	1.47544	−	1.47544i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	3.96524i		−	0.187131i	−0.995613	−	0.0935656i	$-0.970174\pi$
$449$		−	3.96524i		−	0.187131i	0.995613	−	0.0935656i	$-0.0298265\pi$
$450$		0			0
$451$	3.30890	+	3.30890i	0.155810	+	0.155810i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			9.46410i			0.443684i
$456$		0			0
$457$			13.3205i			0.623107i	0.950229	+	0.311554i	$0.100849\pi$
$457$			13.3205i			0.623107i	−0.950229	+	0.311554i	$0.899151\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−12.9682	−	12.9682i	−0.603987	−	0.603987i	0.337381	−	0.941368i	$-0.390459\pi$
$461$	−12.9682	−	12.9682i	−0.603987	−	0.603987i	−0.941368	+	0.337381i	$0.890459\pi$
$462$		0			0
$463$		−	4.03957i		−	0.187735i	−0.995585	−	0.0938673i	$-0.970077\pi$
$463$		−	4.03957i		−	0.187735i	0.995585	−	0.0938673i	$-0.0299230\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−21.6603	+	21.6603i	−1.00232	+	1.00232i	−0.00231971	+	0.999997i	$0.500738\pi$
$467$	−21.6603	+	21.6603i	−1.00232	+	1.00232i	−0.999997	+	0.00231971i	$0.999262\pi$
$468$		0			0
$469$	5.46410	+	5.46410i	0.252309	+	0.252309i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−5.10205			−0.234593
$474$		0			0
$475$	−13.0053	+	13.0053i	−0.596725	+	0.596725i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−10.7846			−0.492761			−0.246381	−	0.969173i	$-0.579241\pi$
$479$	−10.7846			−0.492761			−0.246381	+	0.969173i	$0.579241\pi$
$480$		0			0
$481$	−8.53590			−0.389203
$482$		0			0
$483$		0			0
$484$		0			0
$485$	35.5312	−	35.5312i	1.61339	−	1.61339i
$486$		0			0
$487$	23.0807			1.04589			0.522943	−	0.852368i	$-0.324834\pi$
$487$	23.0807			1.04589			0.522943	+	0.852368i	$0.324834\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	27.8564	+	27.8564i	1.25714	+	1.25714i	0.952451	+	0.304691i	$0.0985531\pi$
$491$	27.8564	+	27.8564i	1.25714	+	1.25714i	0.304691	+	0.952451i	$0.401447\pi$
$492$		0			0
$493$	−12.1962	+	12.1962i	−0.549287	+	0.549287i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	19.7990i		−	0.888106i
$498$		0			0
$499$	21.6665	+	21.6665i	0.969924	+	0.969924i	0.999561	−	0.0296363i	$-0.00943492\pi$
$499$	21.6665	+	21.6665i	0.969924	+	0.969924i	−0.0296363	+	0.999561i	$0.509435\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			24.9282i			1.11149i	0.831352	+	0.555747i	$0.187568\pi$
$503$			24.9282i			1.11149i	−0.831352	+	0.555747i	$0.812432\pi$
$504$		0			0
$505$		−	10.0000i		−	0.444994i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−18.5235	−	18.5235i	−0.821039	−	0.821039i	0.165218	−	0.986257i	$-0.447167\pi$
$509$	−18.5235	−	18.5235i	−0.821039	−	0.821039i	−0.986257	+	0.165218i	$0.947167\pi$
$510$		0			0
$511$		−	6.96953i		−	0.308314i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	20.1962	−	20.1962i	0.889949	−	0.889949i
$516$		0			0
$517$	−7.60770	−	7.60770i	−0.334586	−	0.334586i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	32.6012			1.42829			0.714143	−	0.700000i	$-0.246817\pi$
$521$	32.6012			1.42829			0.714143	+	0.700000i	$0.246817\pi$
$522$		0			0
$523$	−15.8338	+	15.8338i	−0.692362	+	0.692362i	−0.962751	−	0.270389i	$-0.912848\pi$
$523$	−15.8338	+	15.8338i	−0.692362	+	0.692362i	0.270389	+	0.962751i	$0.412848\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−48.2487			−2.10175
$528$		0			0
$529$	11.0000			0.478261
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−7.82894	+	7.82894i	−0.339109	+	0.339109i
$534$		0			0
$535$	−25.2528			−1.09177
$536$		0			0
$537$		0			0
$538$		0			0
$539$	3.66025	+	3.66025i	0.157658	+	0.157658i
$540$		0			0
$541$	−27.0526	+	27.0526i	−1.16308	+	1.16308i	−0.179283	+	0.983798i	$0.557378\pi$
$541$	−27.0526	+	27.0526i	−1.16308	+	1.16308i	−0.983798	+	0.179283i	$0.942622\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	29.8744i		−	1.27968i
$546$		0			0
$547$	−13.2827	−	13.2827i	−0.567928	−	0.567928i	0.363619	−	0.931548i	$-0.381541\pi$
$547$	−13.2827	−	13.2827i	−0.567928	−	0.567928i	−0.931548	+	0.363619i	$0.881541\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	24.3923i		−	1.03915i
$552$		0			0
$553$		−	3.07180i		−	0.130626i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	20.2151	+	20.2151i	0.856541	+	0.856541i	0.990929	−	0.134388i	$-0.0429068\pi$
$557$	20.2151	+	20.2151i	0.856541	+	0.856541i	−0.134388	+	0.990929i	$0.542907\pi$
$558$		0			0
$559$		−	12.0716i		−	0.510574i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	20.0526	−	20.0526i	0.845115	−	0.845115i	−0.144404	−	0.989519i	$-0.546127\pi$
$563$	20.0526	−	20.0526i	0.845115	−	0.845115i	0.989519	+	0.144404i	$0.0461265\pi$
$564$		0			0
$565$	−8.19615	−	8.19615i	−0.344815	−	0.344815i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	9.41902			0.394866			0.197433	−	0.980316i	$-0.436740\pi$
$569$	9.41902			0.394866			0.197433	+	0.980316i	$0.436740\pi$
$570$		0			0
$571$	19.7990	−	19.7990i	0.828562	−	0.828562i	−0.158756	−	0.987318i	$-0.550748\pi$
$571$	19.7990	−	19.7990i	0.828562	−	0.828562i	0.987318	+	0.158756i	$0.0507483\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	8.53590			0.355972
$576$		0			0
$577$	25.0718			1.04375			0.521876	−	0.853021i	$-0.325232\pi$
$577$	25.0718			1.04375			0.521876	+	0.853021i	$0.325232\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−15.1774	+	15.1774i	−0.629665	+	0.629665i
$582$		0			0
$583$	12.0716			0.499954
$584$		0			0
$585$		0			0
$586$		0			0
$587$	16.9282	+	16.9282i	0.698702	+	0.698702i	0.964130	−	0.265429i	$-0.0855135\pi$
$587$	16.9282	+	16.9282i	0.698702	+	0.698702i	−0.265429	+	0.964130i	$0.585514\pi$
$588$		0			0
$589$	48.2487	−	48.2487i	1.98805	−	1.98805i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	6.86800i		−	0.282035i	−0.990007	−	0.141017i	$-0.954963\pi$
$593$		−	6.86800i		−	0.282035i	0.990007	−	0.141017i	$-0.0450373\pi$
$594$		0			0
$595$	−14.4195	−	14.4195i	−0.591143	−	0.591143i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	1.32051i		−	0.0539545i	−0.999636	−	0.0269772i	$-0.991412\pi$
$599$		−	1.32051i		−	0.0539545i	0.999636	−	0.0269772i	$-0.00858817\pi$
$600$		0			0
$601$			28.7846i			1.17415i	0.809533	+	0.587074i	$0.199720\pi$
$601$			28.7846i			1.17415i	−0.809533	+	0.587074i	$0.800280\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	19.1798	+	19.1798i	0.779770	+	0.779770i
$606$		0			0
$607$			7.82894i			0.317767i	0.987297	+	0.158883i	$0.0507894\pi$
$607$			7.82894i			0.317767i	−0.987297	+	0.158883i	$0.949211\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	18.0000	−	18.0000i	0.728202	−	0.728202i
$612$		0			0
$613$	33.3923	+	33.3923i	1.34870	+	1.34870i	0.887073	+	0.461630i	$0.152735\pi$
$613$	33.3923	+	33.3923i	1.34870	+	1.34870i	0.461630	+	0.887073i	$0.347265\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−7.07107			−0.284670			−0.142335	−	0.989819i	$-0.545461\pi$
$617$	−7.07107			−0.284670			−0.142335	+	0.989819i	$0.545461\pi$
$618$		0			0
$619$	−9.04008	+	9.04008i	−0.363352	+	0.363352i	−0.865045	−	0.501694i	$-0.832710\pi$
$619$	−9.04008	+	9.04008i	−0.363352	+	0.363352i	0.501694	+	0.865045i	$0.332710\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−22.7846			−0.912846
$624$		0			0
$625$	31.2487			1.24995
$626$		0			0
$627$		0			0
$628$		0			0
$629$	13.0053	−	13.0053i	0.518556	−	0.518556i
$630$		0			0
$631$	49.0913			1.95430			0.977148	−	0.212562i	$-0.0681809\pi$
$631$	49.0913			1.95430			0.977148	+	0.212562i	$0.0681809\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−27.1244	−	27.1244i	−1.07640	−	1.07640i
$636$		0			0
$637$	−8.66025	+	8.66025i	−0.343132	+	0.343132i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			22.8033i			0.900676i	0.892858	+	0.450338i	$0.148696\pi$
$641$			22.8033i			0.900676i	−0.892858	+	0.450338i	$0.851304\pi$
$642$		0			0
$643$	23.0807	+	23.0807i	0.910213	+	0.910213i	0.996289	−	0.0860753i	$-0.0274326\pi$
$643$	23.0807	+	23.0807i	0.910213	+	0.910213i	−0.0860753	+	0.996289i	$0.527433\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	2.39230i		−	0.0940512i	−0.998894	−	0.0470256i	$-0.985026\pi$
$647$		−	2.39230i		−	0.0940512i	0.998894	−	0.0470256i	$-0.0149742\pi$
$648$		0			0
$649$			13.0718i			0.513113i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−32.2867	−	32.2867i	−1.26348	−	1.26348i	−0.949397	−	0.314078i	$-0.898305\pi$
$653$	−32.2867	−	32.2867i	−1.26348	−	1.26348i	−0.314078	−	0.949397i	$-0.601695\pi$
$654$		0			0
$655$		−	26.7685i		−	1.04593i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	14.3923	−	14.3923i	0.560645	−	0.560645i	−0.368846	−	0.929491i	$-0.620247\pi$
$659$	14.3923	−	14.3923i	0.560645	−	0.560645i	0.929491	+	0.368846i	$0.120247\pi$
$660$		0			0
$661$	19.7846	+	19.7846i	0.769532	+	0.769532i	0.978024	−	0.208492i	$-0.0668555\pi$
$661$	19.7846	+	19.7846i	0.769532	+	0.769532i	−0.208492	+	0.978024i	$0.566856\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	28.8391			1.11833
$666$		0			0
$667$	−8.00481	+	8.00481i	−0.309947	+	0.309947i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	4.39230			0.169563
$672$		0			0
$673$	32.7846			1.26375			0.631877	−	0.775069i	$-0.282285\pi$
$673$	32.7846			1.26375			0.631877	+	0.775069i	$0.282285\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	14.7613	−	14.7613i	0.567323	−	0.567323i	−0.364054	−	0.931378i	$-0.618608\pi$
$677$	14.7613	−	14.7613i	0.567323	−	0.567323i	0.931378	+	0.364054i	$0.118608\pi$
$678$		0			0
$679$	−26.0106			−0.998197
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−14.7321	−	14.7321i	−0.563706	−	0.563706i	0.366652	−	0.930358i	$-0.380504\pi$
$683$	−14.7321	−	14.7321i	−0.563706	−	0.563706i	−0.930358	+	0.366652i	$0.880504\pi$
$684$		0			0
$685$	−6.19615	+	6.19615i	−0.236743	+	0.236743i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			28.5617i			1.08811i
$690$		0			0
$691$	−15.3533	−	15.3533i	−0.584066	−	0.584066i	0.351952	−	0.936018i	$-0.385518\pi$
$691$	−15.3533	−	15.3533i	−0.584066	−	0.584066i	−0.936018	+	0.351952i	$0.885518\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			22.9282i			0.869716i
$696$		0			0
$697$		−	23.8564i		−	0.903626i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	29.6341	+	29.6341i	1.11927	+	1.11927i	0.991849	+	0.127416i	$0.0406682\pi$
$701$	29.6341	+	29.6341i	1.11927	+	1.11927i	0.127416	+	0.991849i	$0.459332\pi$
$702$		0			0
$703$			26.0106i			0.981010i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−3.66025	+	3.66025i	−0.137658	+	0.137658i
$708$		0			0
$709$	5.19615	+	5.19615i	0.195146	+	0.195146i	0.797915	−	0.602770i	$-0.205936\pi$
$709$	5.19615	+	5.19615i	0.195146	+	0.195146i	−0.602770	+	0.797915i	$0.705936\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−31.6675			−1.18596
$714$		0			0
$715$	4.89898	−	4.89898i	0.183211	−	0.183211i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	29.7128			1.10810			0.554050	−	0.832483i	$-0.313081\pi$
$719$	29.7128			1.10810			0.554050	+	0.832483i	$0.313081\pi$
$720$		0			0
$721$	−14.7846			−0.550608
$722$		0			0
$723$		0			0
$724$		0			0
$725$	5.69402	−	5.69402i	0.211471	−	0.211471i
$726$		0			0
$727$	10.4543			0.387728			0.193864	−	0.981028i	$-0.437898\pi$
$727$	10.4543			0.387728			0.193864	+	0.981028i	$0.437898\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	18.3923	+	18.3923i	0.680264	+	0.680264i
$732$		0			0
$733$	21.4449	−	21.4449i	0.792084	−	0.792084i	−0.189749	−	0.981833i	$-0.560767\pi$
$733$	21.4449	−	21.4449i	0.792084	−	0.792084i	0.981833	+	0.189749i	$0.0607673\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	5.65685i		−	0.208373i
$738$		0			0
$739$	−6.96953	−	6.96953i	−0.256378	−	0.256378i	0.567201	−	0.823579i	$-0.308026\pi$
$739$	−6.96953	−	6.96953i	−0.256378	−	0.256378i	−0.823579	+	0.567201i	$0.808026\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			36.1051i			1.32457i	0.749253	+	0.662284i	$0.230413\pi$
$743$			36.1051i			1.32457i	−0.749253	+	0.662284i	$0.769587\pi$
$744$		0			0
$745$		−	22.3923i		−	0.820391i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	9.24316	+	9.24316i	0.337738	+	0.337738i
$750$		0			0
$751$			5.00052i			0.182471i	0.995829	+	0.0912357i	$0.0290817\pi$
$751$			5.00052i			0.182471i	−0.995829	+	0.0912357i	$0.970918\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−1.26795	+	1.26795i	−0.0461454	+	0.0461454i
$756$		0			0
$757$	17.3397	+	17.3397i	0.630224	+	0.630224i	0.948124	−	0.317900i	$-0.102978\pi$
$757$	17.3397	+	17.3397i	0.630224	+	0.630224i	−0.317900	+	0.948124i	$0.602978\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−17.1464			−0.621558			−0.310779	−	0.950482i	$-0.600590\pi$
$761$	−17.1464			−0.621558			−0.310779	+	0.950482i	$0.600590\pi$
$762$		0			0
$763$	−10.9348	+	10.9348i	−0.395865	+	0.395865i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−30.9282			−1.11675
$768$		0			0
$769$	9.71281			0.350253			0.175126	−	0.984546i	$-0.443967\pi$
$769$	9.71281			0.350253			0.175126	+	0.984546i	$0.443967\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−12.3118	+	12.3118i	−0.442825	+	0.442825i	−0.892960	−	0.450135i	$-0.851376\pi$
$773$	−12.3118	+	12.3118i	−0.442825	+	0.442825i	0.450135	+	0.892960i	$0.351376\pi$
$774$		0			0
$775$	22.5259			0.809154
$776$		0			0
$777$		0			0
$778$		0			0
$779$	23.8564	+	23.8564i	0.854744	+	0.854744i
$780$		0			0
$781$	−10.2487	+	10.2487i	−0.366728	+	0.366728i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			15.1774i			0.541705i
$786$		0			0
$787$	−9.62209	−	9.62209i	−0.342991	−	0.342991i	0.514500	−	0.857490i	$-0.327978\pi$
$787$	−9.62209	−	9.62209i	−0.342991	−	0.342991i	−0.857490	+	0.514500i	$0.827978\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			6.00000i			0.213335i
$792$		0			0
$793$			10.3923i			0.369042i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	17.1093	+	17.1093i	0.606041	+	0.606041i	0.941909	−	0.335868i	$-0.109030\pi$
$797$	17.1093	+	17.1093i	0.606041	+	0.606041i	−0.335868	+	0.941909i	$0.609030\pi$
$798$		0			0
$799$			54.8497i			1.94044i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−3.60770	+	3.60770i	−0.127313	+	0.127313i
$804$		0			0
$805$	−9.46410	−	9.46410i	−0.333566	−	0.333566i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	27.1475			0.954454			0.477227	−	0.878780i	$-0.341642\pi$
$809$	27.1475			0.954454			0.477227	+	0.878780i	$0.341642\pi$
$810$		0			0
$811$	−6.79367	+	6.79367i	−0.238558	+	0.238558i	−0.816253	−	0.577695i	$-0.803952\pi$
$811$	−6.79367	+	6.79367i	−0.238558	+	0.238558i	0.577695	+	0.816253i	$0.303952\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	2.53590			0.0888286
$816$		0			0
$817$	−36.7846			−1.28693
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−7.48717	+	7.48717i	−0.261304	+	0.261304i	−0.825584	−	0.564280i	$-0.809154\pi$
$821$	−7.48717	+	7.48717i	−0.261304	+	0.261304i	0.564280	+	0.825584i	$0.309154\pi$
$822$		0			0
$823$	−31.5660			−1.10032			−0.550160	−	0.835059i	$-0.685433\pi$
$823$	−31.5660			−1.10032			−0.550160	+	0.835059i	$0.685433\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−16.3923	−	16.3923i	−0.570016	−	0.570016i	0.362117	−	0.932133i	$-0.382054\pi$
$827$	−16.3923	−	16.3923i	−0.570016	−	0.570016i	−0.932133	+	0.362117i	$0.882054\pi$
$828$		0			0
$829$	4.80385	−	4.80385i	0.166845	−	0.166845i	−0.618746	−	0.785591i	$-0.712359\pi$
$829$	4.80385	−	4.80385i	0.166845	−	0.166845i	0.785591	+	0.618746i	$0.212359\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	26.3896i		−	0.914345i
$834$		0			0
$835$	12.6264	+	12.6264i	0.436954	+	0.436954i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	50.7846i		−	1.75328i	−0.481147	−	0.876640i	$-0.659780\pi$
$839$		−	50.7846i		−	1.75328i	0.481147	−	0.876640i	$-0.340220\pi$
$840$		0			0
$841$		−	18.3205i		−	0.631742i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−13.5230	−	13.5230i	−0.465204	−	0.465204i
$846$		0			0
$847$		−	14.0406i		−	0.482441i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	8.53590	−	8.53590i	0.292607	−	0.292607i
$852$		0			0
$853$	−35.7846	−	35.7846i	−1.22524	−	1.22524i	−0.965744	−	0.259498i	$-0.916443\pi$
$853$	−35.7846	−	35.7846i	−1.22524	−	1.22524i	−0.259498	−	0.965744i	$-0.583557\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−24.3190			−0.830722			−0.415361	−	0.909657i	$-0.636345\pi$
$857$	−24.3190			−0.830722			−0.415361	+	0.909657i	$0.636345\pi$
$858$		0			0
$859$	31.5660	−	31.5660i	1.07702	−	1.07702i	0.0802414	−	0.996775i	$-0.474431\pi$
$859$	31.5660	−	31.5660i	1.07702	−	1.07702i	0.996775	−	0.0802414i	$-0.0255691\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	11.6077			0.395131			0.197565	−	0.980290i	$-0.436697\pi$
$863$	11.6077			0.395131			0.197565	+	0.980290i	$0.436697\pi$
$864$		0			0
$865$	−21.3205			−0.724919
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−1.59008	+	1.59008i	−0.0539397	+	0.0539397i
$870$		0			0
$871$	13.3843			0.453508
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−6.92820	−	6.92820i	−0.234216	−	0.234216i
$876$		0			0
$877$	−11.5359	+	11.5359i	−0.389540	+	0.389540i	−0.874523	−	0.484984i	$-0.838826\pi$
$877$	−11.5359	+	11.5359i	−0.389540	+	0.389540i	0.484984	+	0.874523i	$0.338826\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			1.96902i			0.0663378i	0.999450	+	0.0331689i	$0.0105599\pi$
$881$			1.96902i			0.0663378i	−0.999450	+	0.0331689i	$0.989440\pi$
$882$		0			0
$883$	−28.1827	−	28.1827i	−0.948425	−	0.948425i	0.0503091	−	0.998734i	$-0.483979\pi$
$883$	−28.1827	−	28.1827i	−0.948425	−	0.948425i	−0.998734	+	0.0503091i	$0.983979\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	46.9282i		−	1.57569i	−0.615870	−	0.787847i	$-0.711196\pi$
$887$		−	46.9282i		−	1.57569i	0.615870	−	0.787847i	$-0.288804\pi$
$888$		0			0
$889$			19.8564i			0.665962i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−54.8497	−	54.8497i	−1.83548	−	1.83548i
$894$		0			0
$895$		−	23.1822i		−	0.774896i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−21.1244	+	21.1244i	−0.704537	+	0.704537i
$900$		0			0
$901$	−43.5167	−	43.5167i	−1.44975	−	1.44975i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−3.10583			−0.103241
$906$		0			0
$907$	11.2122	−	11.2122i	0.372294	−	0.372294i	−0.496018	−	0.868312i	$-0.665205\pi$
$907$	11.2122	−	11.2122i	0.372294	−	0.372294i	0.868312	+	0.496018i	$0.165205\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	13.1769			0.436571			0.218285	−	0.975885i	$-0.429954\pi$
$911$	13.1769			0.436571			0.218285	+	0.975885i	$0.429954\pi$
$912$		0			0
$913$	15.7128			0.520018
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−9.79796	+	9.79796i	−0.323557	+	0.323557i
$918$		0			0
$919$	22.5259			0.743060			0.371530	−	0.928421i	$-0.378833\pi$
$919$	22.5259			0.743060			0.371530	+	0.928421i	$0.378833\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−24.2487	−	24.2487i	−0.798156	−	0.798156i
$924$		0			0
$925$	−6.07180	+	6.07180i	−0.199639	+	0.199639i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			17.1464i			0.562556i	0.959626	+	0.281278i	$0.0907583\pi$
$929$			17.1464i			0.562556i	−0.959626	+	0.281278i	$0.909242\pi$
$930$		0			0
$931$	26.3896	+	26.3896i	0.864884	+	0.864884i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			14.9282i			0.488204i
$936$		0			0
$937$			30.0000i			0.980057i	0.871706	+	0.490029i	$0.163014\pi$
$937$			30.0000i			0.980057i	−0.871706	+	0.490029i	$0.836986\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	29.4582	+	29.4582i	0.960311	+	0.960311i	0.999242	−	0.0389305i	$-0.0123951\pi$
$941$	29.4582	+	29.4582i	0.960311	+	0.960311i	−0.0389305	+	0.999242i	$0.512395\pi$
$942$		0			0
$943$		−	15.6579i		−	0.509891i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−30.3923	+	30.3923i	−0.987617	+	0.987617i	−0.999924	−	0.0123071i	$-0.996082\pi$
$947$	−30.3923	+	30.3923i	−0.987617	+	0.987617i	0.0123071	+	0.999924i	$0.496082\pi$
$948$		0			0
$949$	−8.53590	−	8.53590i	−0.277087	−	0.277087i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−15.6307			−0.506327			−0.253164	−	0.967423i	$-0.581471\pi$
$953$	−15.6307			−0.506327			−0.253164	+	0.967423i	$0.581471\pi$
$954$		0			0
$955$	−26.0106	+	26.0106i	−0.841685	+	0.841685i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	4.53590			0.146472
$960$		0			0
$961$	−52.5692			−1.69578
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−44.0165	+	44.0165i	−1.41694	+	1.41694i
$966$		0			0
$967$	−49.0913			−1.57867			−0.789335	−	0.613962i	$-0.789575\pi$
$967$	−49.0913			−1.57867			−0.789335	+	0.613962i	$0.789575\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−16.9808	−	16.9808i	−0.544939	−	0.544939i	0.380034	−	0.924973i	$-0.375912\pi$
$971$	−16.9808	−	16.9808i	−0.544939	−	0.544939i	−0.924973	+	0.380034i	$0.875912\pi$
$972$		0			0
$973$	8.39230	−	8.39230i	0.269045	−	0.269045i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			1.69161i			0.0541196i	0.999634	+	0.0270598i	$0.00861445\pi$
$977$			1.69161i			0.0541196i	−0.999634	+	0.0270598i	$0.991386\pi$
$978$		0			0
$979$	11.7942	+	11.7942i	0.376944	+	0.376944i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			38.6410i			1.23246i	0.787567	+	0.616229i	$0.211340\pi$
$983$			38.6410i			1.23246i	−0.787567	+	0.616229i	$0.788660\pi$
$984$		0			0
$985$		−	64.6410i		−	2.05963i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	12.0716	+	12.0716i	0.383854	+	0.383854i
$990$		0			0
$991$			3.28169i			0.104246i	0.998641	+	0.0521232i	$0.0165989\pi$
$991$			3.28169i			0.104246i	−0.998641	+	0.0521232i	$0.983401\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−38.0526	+	38.0526i	−1.20635	+	1.20635i
$996$		0			0
$997$	21.0000	+	21.0000i	0.665077	+	0.665077i	0.956572	−	0.291496i	$-0.0941528\pi$
$997$	21.0000	+	21.0000i	0.665077	+	0.665077i	−0.291496	+	0.956572i	$0.594153\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2304.2.l.g.575.1	yes	8
3.2	odd	2		2304.2.l.c.575.4	yes	8
4.3	odd	2		2304.2.l.c.575.1	yes	8
8.3	odd	2		2304.2.l.f.575.4	yes	8
8.5	even	2		2304.2.l.b.575.4	yes	8
12.11	even	2	inner	2304.2.l.g.575.4	yes	8
16.3	odd	4		2304.2.l.c.1727.4	yes	8
16.5	even	4		2304.2.l.b.1727.1	yes	8
16.11	odd	4		2304.2.l.f.1727.1	yes	8
16.13	even	4	inner	2304.2.l.g.1727.4	yes	8
24.5	odd	2		2304.2.l.f.575.1	yes	8
24.11	even	2		2304.2.l.b.575.1	✓	8
48.5	odd	4		2304.2.l.f.1727.4	yes	8
48.11	even	4		2304.2.l.b.1727.4	yes	8
48.29	odd	4		2304.2.l.c.1727.1	yes	8
48.35	even	4	inner	2304.2.l.g.1727.1	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2304.2.l.b.575.1	✓	8	24.11	even	2
2304.2.l.b.575.4	yes	8	8.5	even	2
2304.2.l.b.1727.1	yes	8	16.5	even	4
2304.2.l.b.1727.4	yes	8	48.11	even	4
2304.2.l.c.575.1	yes	8	4.3	odd	2
2304.2.l.c.575.4	yes	8	3.2	odd	2
2304.2.l.c.1727.1	yes	8	48.29	odd	4
2304.2.l.c.1727.4	yes	8	16.3	odd	4
2304.2.l.f.575.1	yes	8	24.5	odd	2
2304.2.l.f.575.4	yes	8	8.3	odd	2
2304.2.l.f.1727.1	yes	8	16.11	odd	4
2304.2.l.f.1727.4	yes	8	48.5	odd	4
2304.2.l.g.575.1	yes	8	1.1	even	1	trivial
2304.2.l.g.575.4	yes	8	12.11	even	2	inner
2304.2.l.g.1727.1	yes	8	48.35	even	4	inner
2304.2.l.g.1727.4	yes	8	16.13	even	4	inner

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:	\( N \)	\(=\)	\( 2304 = 2^{8} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2304.l (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.93185 + 1.93185i) q^{5} +1.41421 q^{7} +O(q^{10})\)	\(q+(-1.93185 + 1.93185i) q^{5} +1.41421 q^{7} +(-0.732051 - 0.732051i) q^{11} +(1.73205 - 1.73205i) q^{13} +5.27792i q^{17} +(-5.27792 - 5.27792i) q^{19} +3.46410i q^{23} -2.46410i q^{25} +(2.31079 + 2.31079i) q^{29} +9.14162i q^{31} +(-2.73205 + 2.73205i) q^{35} +(-2.46410 - 2.46410i) q^{37} -4.52004 q^{41} +(3.48477 - 3.48477i) q^{43} +10.3923 q^{47} -5.00000 q^{49} +(-8.24504 + 8.24504i) q^{53} +2.82843 q^{55} +(-8.92820 - 8.92820i) q^{59} +(-3.00000 + 3.00000i) q^{61} +6.69213i q^{65} +(3.86370 + 3.86370i) q^{67} -14.0000i q^{71} -4.92820i q^{73} +(-1.03528 - 1.03528i) q^{77} -2.17209i q^{79} +(-10.7321 + 10.7321i) q^{83} +(-10.1962 - 10.1962i) q^{85} -16.1112 q^{89} +(2.44949 - 2.44949i) q^{91} +20.3923 q^{95} -18.3923 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 8 q^{11} - 8 q^{35} + 8 q^{37} - 40 q^{49} - 16 q^{59} - 24 q^{61} - 72 q^{83} - 40 q^{85} + 80 q^{95} - 64 q^{97}+O(q^{100}) \) 8 * q + 8 * q^11 - 8 * q^35 + 8 * q^37 - 40 * q^49 - 16 * q^59 - 24 * q^61 - 72 * q^83 - 40 * q^85 + 80 * q^95 - 64 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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