LMFDB - Embedded newform 2304.2.l.c.1727.3

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			1727.3
Root			$-0.965926 - 0.258819i$ of defining polynomial
Character	$\chi$	$=$	2304.1727
Dual form			2304.2.l.c.575.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.517638 + 0.517638i) q^{5} -1.41421 q^{7} +O(q^{10})$	$q+(0.517638 + 0.517638i) q^{5} -1.41421 q^{7} +(-2.73205 + 2.73205i) q^{11} +(-1.73205 - 1.73205i) q^{13} -0.378937i q^{17} +(0.378937 - 0.378937i) q^{19} -3.46410i q^{23} -4.46410i q^{25} +(4.76028 - 4.76028i) q^{29} -0.656339i q^{31} +(-0.732051 - 0.732051i) q^{35} +(4.46410 - 4.46410i) q^{37} +10.1769 q^{41} +(6.31319 + 6.31319i) q^{43} +10.3923 q^{47} -5.00000 q^{49} +(4.00240 + 4.00240i) q^{53} -2.82843 q^{55} +(-4.92820 + 4.92820i) q^{59} +(-3.00000 - 3.00000i) q^{61} -1.79315i q^{65} +(1.03528 - 1.03528i) q^{67} -14.0000i q^{71} -8.92820i q^{73} +(3.86370 - 3.86370i) q^{77} -11.9700i q^{79} +(7.26795 + 7.26795i) q^{83} +(0.196152 - 0.196152i) q^{85} +13.2827 q^{89} +(2.44949 + 2.44949i) q^{91} +0.392305 q^{95} +2.39230 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times$.

$n$	$1279$	$1793$	$2053$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{3}{4}\right)$

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
$5$	0.517638	+	0.517638i	0.231495	+	0.231495i	0.813316	−	0.581822i	$-0.197660\pi$
$5$	0.517638	+	0.517638i	0.231495	+	0.231495i	−0.581822	+	0.813316i	$0.697660\pi$
$6$		0			0
$7$	−1.41421			−0.534522			−0.267261	−	0.963624i	$-0.586119\pi$
$7$	−1.41421			−0.534522			−0.267261	+	0.963624i	$0.586119\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2.73205	+	2.73205i	−0.823744	+	0.823744i	−0.986643	−	0.162899i	$-0.947916\pi$
$11$	−2.73205	+	2.73205i	−0.823744	+	0.823744i	0.162899	+	0.986643i	$0.447916\pi$
$12$		0			0
$13$	−1.73205	−	1.73205i	−0.480384	−	0.480384i	0.424870	−	0.905254i	$-0.360320\pi$
$13$	−1.73205	−	1.73205i	−0.480384	−	0.480384i	−0.905254	+	0.424870i	$0.860320\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	0.378937i		−	0.0919058i	−0.998944	−	0.0459529i	$-0.985368\pi$
$17$		−	0.378937i		−	0.0919058i	0.998944	−	0.0459529i	$-0.0146324\pi$
$18$		0			0
$19$	0.378937	−	0.378937i	0.0869342	−	0.0869342i	−0.662302	−	0.749237i	$-0.730421\pi$
$19$	0.378937	−	0.378937i	0.0869342	−	0.0869342i	0.749237	+	0.662302i	$0.230421\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	3.46410i		−	0.722315i	−0.932505	−	0.361158i	$-0.882382\pi$
$23$		−	3.46410i		−	0.722315i	0.932505	−	0.361158i	$-0.117618\pi$
$24$		0			0
$25$		−	4.46410i		−	0.892820i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	4.76028	−	4.76028i	0.883962	−	0.883962i	−0.109973	−	0.993935i	$-0.535076\pi$
$29$	4.76028	−	4.76028i	0.883962	−	0.883962i	0.993935	+	0.109973i	$0.0350764\pi$
$30$		0			0
$31$		−	0.656339i		−	0.117882i	−0.998261	−	0.0589410i	$-0.981228\pi$
$31$		−	0.656339i		−	0.117882i	0.998261	−	0.0589410i	$-0.0187724\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−0.732051	−	0.732051i	−0.123739	−	0.123739i
$36$		0			0
$37$	4.46410	−	4.46410i	0.733894	−	0.733894i	−0.237495	−	0.971389i	$-0.576326\pi$
$37$	4.46410	−	4.46410i	0.733894	−	0.733894i	0.971389	+	0.237495i	$0.0763263\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	10.1769			1.58936			0.794682	−	0.607026i	$-0.207638\pi$
$41$	10.1769			1.58936			0.794682	+	0.607026i	$0.207638\pi$
$42$		0			0
$43$	6.31319	+	6.31319i	0.962753	+	0.962753i	0.999331	−	0.0365779i	$-0.0116457\pi$
$43$	6.31319	+	6.31319i	0.962753	+	0.962753i	−0.0365779	+	0.999331i	$0.511646\pi$
$44$		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
$53$	4.00240	+	4.00240i	0.549772	+	0.549772i	0.926375	−	0.376602i	$-0.122908\pi$
$53$	4.00240	+	4.00240i	0.549772	+	0.549772i	−0.376602	+	0.926375i	$0.622908\pi$
$54$		0			0
$55$	−2.82843			−0.381385
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−4.92820	+	4.92820i	−0.641597	+	0.641597i	−0.950948	−	0.309351i	$-0.899888\pi$
$59$	−4.92820	+	4.92820i	−0.641597	+	0.641597i	0.309351	+	0.950948i	$0.399888\pi$
$60$		0			0
$61$	−3.00000	−	3.00000i	−0.384111	−	0.384111i	0.488470	−	0.872581i	$-0.337555\pi$
$61$	−3.00000	−	3.00000i	−0.384111	−	0.384111i	−0.872581	+	0.488470i	$0.837555\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	1.79315i		−	0.222413i
$66$		0			0
$67$	1.03528	−	1.03528i	0.126479	−	0.126479i	−0.641034	−	0.767513i	$-0.721494\pi$
$67$	1.03528	−	1.03528i	0.126479	−	0.126479i	0.767513	+	0.641034i	$0.221494\pi$
$68$		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
$73$		−	8.92820i		−	1.04497i	−0.852649	−	0.522484i	$-0.825006\pi$
$73$		−	8.92820i		−	1.04497i	0.852649	−	0.522484i	$-0.174994\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	3.86370	−	3.86370i	0.440310	−	0.440310i
$78$		0			0
$79$		−	11.9700i		−	1.34674i	−0.739308	−	0.673368i	$-0.764847\pi$
$79$		−	11.9700i		−	1.34674i	0.739308	−	0.673368i	$-0.235153\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	7.26795	+	7.26795i	0.797761	+	0.797761i	0.982742	−	0.184981i	$-0.0592224\pi$
$83$	7.26795	+	7.26795i	0.797761	+	0.797761i	−0.184981	+	0.982742i	$0.559222\pi$
$84$		0			0
$85$	0.196152	−	0.196152i	0.0212757	−	0.0212757i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	13.2827			1.40797			0.703983	−	0.710217i	$-0.251403\pi$
$89$	13.2827			1.40797			0.703983	+	0.710217i	$0.251403\pi$
$90$		0			0
$91$	2.44949	+	2.44949i	0.256776	+	0.256776i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	0.392305			0.0402496
$96$		0			0
$97$	2.39230			0.242902			0.121451	−	0.992597i	$-0.461245\pi$
$97$	2.39230			0.242902			0.121451	+	0.992597i	$0.461245\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	9.65926	+	9.65926i	0.961132	+	0.961132i	0.999272	−	0.0381403i	$-0.0121434\pi$
$101$	9.65926	+	9.65926i	0.961132	+	0.961132i	−0.0381403	+	0.999272i	$0.512143\pi$
$102$		0			0
$103$	−18.9396			−1.86617			−0.933086	−	0.359653i	$-0.882895\pi$
$103$	−18.9396			−1.86617			−0.933086	+	0.359653i	$0.882895\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−13.4641	+	13.4641i	−1.30162	+	1.30162i	−0.374327	+	0.927297i	$0.622126\pi$
$107$	−13.4641	+	13.4641i	−1.30162	+	1.30162i	−0.927297	+	0.374327i	$0.877874\pi$
$108$		0			0
$109$	−4.26795	−	4.26795i	−0.408795	−	0.408795i	0.472523	−	0.881318i	$-0.343343\pi$
$109$	−4.26795	−	4.26795i	−0.408795	−	0.408795i	−0.881318	+	0.472523i	$0.843343\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	4.24264i		−	0.399114i	−0.979886	−	0.199557i	$-0.936050\pi$
$113$		−	4.24264i		−	0.399114i	0.979886	−	0.199557i	$-0.0639503\pi$
$114$		0			0
$115$	1.79315	−	1.79315i	0.167212	−	0.167212i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			0.535898i			0.0491257i
$120$		0			0
$121$		−	3.92820i		−	0.357109i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	4.89898	−	4.89898i	0.438178	−	0.438178i
$126$		0			0
$127$		−	5.55532i		−	0.492955i	−0.969149	−	0.246477i	$-0.920727\pi$
$127$		−	5.55532i		−	0.492955i	0.969149	−	0.246477i	$-0.0792731\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−6.92820	−	6.92820i	−0.605320	−	0.605320i	0.336399	−	0.941719i	$-0.390791\pi$
$131$	−6.92820	−	6.92820i	−0.605320	−	0.605320i	−0.941719	+	0.336399i	$0.890791\pi$
$132$		0			0
$133$	−0.535898	+	0.535898i	−0.0464683	+	0.0464683i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	8.10634			0.692572			0.346286	−	0.938129i	$-0.387443\pi$
$137$	8.10634			0.692572			0.346286	+	0.938129i	$0.387443\pi$
$138$		0			0
$139$	8.76268	+	8.76268i	0.743241	+	0.743241i	0.973200	−	0.229959i	$-0.0738593\pi$
$139$	8.76268	+	8.76268i	0.743241	+	0.743241i	−0.229959	+	0.973200i	$0.573859\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	9.46410			0.791428
$144$		0			0
$145$	4.92820			0.409265
$146$		0			0
$147$		0			0
$148$		0			0
$149$	1.55291	+	1.55291i	0.127220	+	0.127220i	0.767850	−	0.640630i	$-0.221327\pi$
$149$	1.55291	+	1.55291i	0.127220	+	0.127220i	−0.640630	+	0.767850i	$0.721327\pi$
$150$		0			0
$151$	9.14162			0.743934			0.371967	−	0.928246i	$-0.378683\pi$
$151$	9.14162			0.743934			0.371967	+	0.928246i	$0.378683\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	0.339746	−	0.339746i	0.0272891	−	0.0272891i
$156$		0			0
$157$	−9.92820	−	9.92820i	−0.792357	−	0.792357i	0.189520	−	0.981877i	$-0.439307\pi$
$157$	−9.92820	−	9.92820i	−0.792357	−	0.792357i	−0.981877	+	0.189520i	$0.939307\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			4.89898i			0.386094i
$162$		0			0
$163$	−9.14162	+	9.14162i	−0.716027	+	0.716027i	−0.967789	−	0.251762i	$-0.918990\pi$
$163$	−9.14162	+	9.14162i	−0.716027	+	0.716027i	0.251762	+	0.967789i	$0.418990\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	13.4641i		−	1.04188i	−0.853592	−	0.520942i	$-0.825581\pi$
$167$		−	13.4641i		−	1.04188i	0.853592	−	0.520942i	$-0.174419\pi$
$168$		0			0
$169$		−	7.00000i		−	0.538462i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	12.8666	−	12.8666i	0.978231	−	0.978231i	−0.0215368	−	0.999768i	$-0.506856\pi$
$173$	12.8666	−	12.8666i	0.978231	−	0.978231i	0.999768	+	0.0215368i	$0.00685592\pi$
$174$		0			0
$175$			6.31319i			0.477233i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	6.00000	+	6.00000i	0.448461	+	0.448461i	0.894843	−	0.446382i	$-0.147288\pi$
$179$	6.00000	+	6.00000i	0.448461	+	0.448461i	−0.446382	+	0.894843i	$0.647288\pi$
$180$		0			0
$181$	11.1962	−	11.1962i	0.832203	−	0.832203i	−0.155614	−	0.987818i	$-0.549736\pi$
$181$	11.1962	−	11.1962i	0.832203	−	0.832203i	0.987818	+	0.155614i	$0.0497357\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	4.62158			0.339785
$186$		0			0
$187$	1.03528	+	1.03528i	0.0757069	+	0.0757069i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−6.53590			−0.472921			−0.236461	−	0.971641i	$-0.575987\pi$
$191$	−6.53590			−0.472921			−0.236461	+	0.971641i	$0.575987\pi$
$192$		0			0
$193$	−18.7846			−1.35215			−0.676073	−	0.736835i	$-0.736320\pi$
$193$	−18.7846			−1.35215			−0.676073	+	0.736835i	$0.736320\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−4.48288	−	4.48288i	−0.319392	−	0.319392i	0.529142	−	0.848533i	$-0.322514\pi$
$197$	−4.48288	−	4.48288i	−0.319392	−	0.319392i	−0.848533	+	0.529142i	$0.822514\pi$
$198$		0			0
$199$	−0.101536			−0.00719769			−0.00359885	−	0.999994i	$-0.501146\pi$
$199$	−0.101536			−0.00719769			−0.00359885	+	0.999994i	$0.501146\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−6.73205	+	6.73205i	−0.472497	+	0.472497i
$204$		0			0
$205$	5.26795	+	5.26795i	0.367930	+	0.367930i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			2.07055i			0.143223i
$210$		0			0
$211$	15.1774	−	15.1774i	1.04486	−	1.04486i	0.0459106	−	0.998946i	$-0.485381\pi$
$211$	15.1774	−	15.1774i	1.04486	−	1.04486i	0.998946	−	0.0459106i	$-0.0146189\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			6.53590i			0.445745i
$216$		0			0
$217$			0.928203i			0.0630105i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−0.656339	+	0.656339i	−0.0441501	+	0.0441501i
$222$		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
$227$	−11.2679	−	11.2679i	−0.747880	−	0.747880i	0.226201	−	0.974081i	$-0.427369\pi$
$227$	−11.2679	−	11.2679i	−0.747880	−	0.747880i	−0.974081	+	0.226201i	$0.927369\pi$
$228$		0			0
$229$	−3.19615	+	3.19615i	−0.211208	+	0.211208i	−0.804780	−	0.593573i	$-0.797717\pi$
$229$	−3.19615	+	3.19615i	−0.211208	+	0.211208i	0.593573	+	0.804780i	$0.297717\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−10.6574			−0.698188			−0.349094	−	0.937088i	$-0.613511\pi$
$233$	−10.6574			−0.698188			−0.349094	+	0.937088i	$0.613511\pi$
$234$		0			0
$235$	5.37945	+	5.37945i	0.350917	+	0.350917i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	14.5359			0.940249			0.470125	−	0.882600i	$-0.344209\pi$
$239$	14.5359			0.940249			0.470125	+	0.882600i	$0.344209\pi$
$240$		0			0
$241$	13.4641			0.867299			0.433650	−	0.901082i	$-0.357226\pi$
$241$	13.4641			0.867299			0.433650	+	0.901082i	$0.357226\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−2.58819	−	2.58819i	−0.165353	−	0.165353i
$246$		0			0
$247$	−1.31268			−0.0835237
$248$		0			0
$249$		0			0
$250$		0			0
$251$	9.66025	−	9.66025i	0.609750	−	0.609750i	−0.333131	−	0.942881i	$-0.608105\pi$
$251$	9.66025	−	9.66025i	0.609750	−	0.609750i	0.942881	+	0.333131i	$0.108105\pi$
$252$		0			0
$253$	9.46410	+	9.46410i	0.595003	+	0.595003i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	2.17209i		−	0.135491i	−0.997703	−	0.0677456i	$-0.978419\pi$
$257$		−	2.17209i		−	0.135491i	0.997703	−	0.0677456i	$-0.0215806\pi$
$258$		0			0
$259$	−6.31319	+	6.31319i	−0.392283	+	0.392283i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	10.9282i		−	0.673862i	−0.941529	−	0.336931i	$-0.890611\pi$
$263$		−	10.9282i		−	0.673862i	0.941529	−	0.336931i	$-0.109389\pi$
$264$		0			0
$265$			4.14359i			0.254539i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−14.2808	+	14.2808i	−0.870718	+	0.870718i	−0.992551	−	0.121833i	$-0.961123\pi$
$269$	−14.2808	+	14.2808i	−0.870718	+	0.870718i	0.121833	+	0.992551i	$0.461123\pi$
$270$		0			0
$271$		−	22.5259i		−	1.36835i	−0.729318	−	0.684175i	$-0.760162\pi$
$271$		−	22.5259i		−	1.36835i	0.729318	−	0.684175i	$-0.239838\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	12.1962	+	12.1962i	0.735456	+	0.735456i
$276$		0			0
$277$	0.803848	−	0.803848i	0.0482985	−	0.0482985i	−0.682545	−	0.730844i	$-0.739127\pi$
$277$	0.803848	−	0.803848i	0.0482985	−	0.0482985i	0.730844	+	0.682545i	$0.239127\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−0.656339			−0.0391539			−0.0195769	−	0.999808i	$-0.506232\pi$
$281$	−0.656339			−0.0391539			−0.0195769	+	0.999808i	$0.506232\pi$
$282$		0			0
$283$	−19.7990	−	19.7990i	−1.17693	−	1.17693i	−0.980523	−	0.196405i	$-0.937073\pi$
$283$	−19.7990	−	19.7990i	−1.17693	−	1.17693i	−0.196405	−	0.980523i	$-0.562927\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−14.3923			−0.849551
$288$		0			0
$289$	16.8564			0.991553
$290$		0			0
$291$		0			0
$292$		0			0
$293$	1.65445	+	1.65445i	0.0966540	+	0.0966540i	0.753780	−	0.657126i	$-0.228228\pi$
$293$	1.65445	+	1.65445i	0.0966540	+	0.0966540i	−0.657126	+	0.753780i	$0.728228\pi$
$294$		0			0
$295$	−5.10205			−0.297053
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−6.00000	+	6.00000i	−0.346989	+	0.346989i
$300$		0			0
$301$	−8.92820	−	8.92820i	−0.514613	−	0.514613i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	3.10583i		−	0.177839i
$306$		0			0
$307$	−3.86370	+	3.86370i	−0.220513	+	0.220513i	−0.808715	−	0.588201i	$-0.799836\pi$
$307$	−3.86370	+	3.86370i	−0.220513	+	0.220513i	0.588201	+	0.808715i	$0.299836\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			29.8564i			1.69300i	0.532388	+	0.846501i	$0.321295\pi$
$311$			29.8564i			1.69300i	−0.532388	+	0.846501i	$0.678705\pi$
$312$		0			0
$313$		−	2.00000i		−	0.113047i	−0.998401	−	0.0565233i	$-0.981998\pi$
$313$		−	2.00000i		−	0.113047i	0.998401	−	0.0565233i	$-0.0180015\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−19.1798	+	19.1798i	−1.07725	+	1.07725i	−0.0804904	+	0.996755i	$0.525649\pi$
$317$	−19.1798	+	19.1798i	−1.07725	+	1.07725i	−0.996755	+	0.0804904i	$0.974351\pi$
$318$		0			0
$319$			26.0106i			1.45632i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−0.143594	−	0.143594i	−0.00798976	−	0.00798976i
$324$		0			0
$325$	−7.73205	+	7.73205i	−0.428897	+	0.428897i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−14.6969			−0.810268
$330$		0			0
$331$	−8.00481	−	8.00481i	−0.439984	−	0.439984i	0.452022	−	0.892007i	$-0.350703\pi$
$331$	−8.00481	−	8.00481i	−0.439984	−	0.439984i	−0.892007	+	0.452022i	$0.850703\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	1.07180			0.0585585
$336$		0			0
$337$	−18.7846			−1.02326			−0.511631	−	0.859205i	$-0.670959\pi$
$337$	−18.7846			−1.02326			−0.511631	+	0.859205i	$0.670959\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	1.79315	+	1.79315i	0.0971046	+	0.0971046i
$342$		0			0
$343$	16.9706			0.916324
$344$		0			0
$345$		0			0
$346$		0			0
$347$	25.1244	−	25.1244i	1.34875	−	1.34875i	0.461721	−	0.887025i	$-0.347232\pi$
$347$	25.1244	−	25.1244i	1.34875	−	1.34875i	0.887025	−	0.461721i	$-0.152768\pi$
$348$		0			0
$349$	7.39230	+	7.39230i	0.395701	+	0.395701i	0.876714	−	0.481013i	$-0.159731\pi$
$349$	7.39230	+	7.39230i	0.395701	+	0.395701i	−0.481013	+	0.876714i	$0.659731\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	17.4238i		−	0.927377i	−0.885998	−	0.463688i	$-0.846526\pi$
$353$		−	17.4238i		−	0.927377i	0.885998	−	0.463688i	$-0.153474\pi$
$354$		0			0
$355$	7.24693	−	7.24693i	0.384627	−	0.384627i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			3.85641i			0.203533i	0.994808	+	0.101767i	$0.0324495\pi$
$359$			3.85641i			0.203533i	−0.994808	+	0.101767i	$0.967550\pi$
$360$		0			0
$361$			18.7128i			0.984885i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	4.62158	−	4.62158i	0.241904	−	0.241904i
$366$		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
$371$	−5.66025	−	5.66025i	−0.293866	−	0.293866i
$372$		0			0
$373$	−22.8564	+	22.8564i	−1.18346	+	1.18346i	−0.204618	+	0.978842i	$0.565595\pi$
$373$	−22.8564	+	22.8564i	−1.18346	+	1.18346i	−0.978842	+	0.204618i	$0.934405\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−16.4901			−0.849283
$378$		0			0
$379$	−8.10634	−	8.10634i	−0.416395	−	0.416395i	0.467564	−	0.883959i	$-0.345132\pi$
$379$	−8.10634	−	8.10634i	−0.416395	−	0.416395i	−0.883959	+	0.467564i	$0.845132\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−25.4641			−1.30115			−0.650577	−	0.759440i	$-0.725473\pi$
$383$	−25.4641			−1.30115			−0.650577	+	0.759440i	$0.725473\pi$
$384$		0			0
$385$	4.00000			0.203859
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−23.6999	−	23.6999i	−1.20163	−	1.20163i	−0.973670	−	0.227960i	$-0.926794\pi$
$389$	−23.6999	−	23.6999i	−1.20163	−	1.20163i	−0.227960	−	0.973670i	$-0.573206\pi$
$390$		0			0
$391$	−1.31268			−0.0663850
$392$		0			0
$393$		0			0
$394$		0			0
$395$	6.19615	−	6.19615i	0.311762	−	0.311762i
$396$		0			0
$397$	16.3205	+	16.3205i	0.819103	+	0.819103i	0.985978	−	0.166875i	$-0.0533677\pi$
$397$	16.3205	+	16.3205i	0.819103	+	0.819103i	−0.166875	+	0.985978i	$0.553368\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			17.3495i			0.866393i	0.901300	+	0.433196i	$0.142614\pi$
$401$			17.3495i			0.866393i	−0.901300	+	0.433196i	$0.857386\pi$
$402$		0			0
$403$	−1.13681	+	1.13681i	−0.0566286	+	0.0566286i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			24.3923i			1.20908i
$408$		0			0
$409$			14.3923i			0.711654i	0.934552	+	0.355827i	$0.115801\pi$
$409$			14.3923i			0.711654i	−0.934552	+	0.355827i	$0.884199\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	6.96953	−	6.96953i	0.342948	−	0.342948i
$414$		0			0
$415$			7.52433i			0.369355i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−7.26795	−	7.26795i	−0.355063	−	0.355063i	0.506927	−	0.861989i	$-0.330781\pi$
$419$	−7.26795	−	7.26795i	−0.355063	−	0.355063i	−0.861989	+	0.506927i	$0.830781\pi$
$420$		0			0
$421$	17.7321	−	17.7321i	0.864207	−	0.864207i	−0.127616	−	0.991824i	$-0.540733\pi$
$421$	17.7321	−	17.7321i	0.864207	−	0.864207i	0.991824	+	0.127616i	$0.0407326\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−1.69161			−0.0820554
$426$		0			0
$427$	4.24264	+	4.24264i	0.205316	+	0.205316i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	2.67949			0.129067			0.0645333	−	0.997916i	$-0.479444\pi$
$431$	2.67949			0.129067			0.0645333	+	0.997916i	$0.479444\pi$
$432$		0			0
$433$	−1.07180			−0.0515073			−0.0257536	−	0.999668i	$-0.508199\pi$
$433$	−1.07180			−0.0515073			−0.0257536	+	0.999668i	$0.508199\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−1.31268	−	1.31268i	−0.0627939	−	0.0627939i
$438$		0			0
$439$	29.6985			1.41743			0.708716	−	0.705494i	$-0.249275\pi$
$439$	29.6985			1.41743			0.708716	+	0.705494i	$0.249275\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	26.5885	−	26.5885i	1.26326	−	1.26326i	0.313750	−	0.949506i	$-0.398415\pi$
$443$	26.5885	−	26.5885i	1.26326	−	1.26326i	0.949506	−	0.313750i	$-0.101585\pi$
$444$		0			0
$445$	6.87564	+	6.87564i	0.325937	+	0.325937i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			18.6622i			0.880723i	0.897821	+	0.440361i	$0.145150\pi$
$449$			18.6622i			0.880723i	−0.897821	+	0.440361i	$0.854850\pi$
$450$		0			0
$451$	−27.8038	+	27.8038i	−1.30923	+	1.30923i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			2.53590i			0.118885i
$456$		0			0
$457$			21.3205i			0.997331i	0.866794	+	0.498666i	$0.166176\pi$
$457$			21.3205i			0.997331i	−0.866794	+	0.498666i	$0.833824\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−25.2156	+	25.2156i	−1.17441	+	1.17441i	−0.193260	+	0.981147i	$0.561906\pi$
$461$	−25.2156	+	25.2156i	−1.17441	+	1.17441i	−0.981147	+	0.193260i	$0.938094\pi$
$462$		0			0
$463$			35.1523i			1.63366i	0.576875	+	0.816832i	$0.304272\pi$
$463$			35.1523i			1.63366i	−0.576875	+	0.816832i	$0.695728\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	4.33975	+	4.33975i	0.200819	+	0.200819i	0.800351	−	0.599532i	$-0.204646\pi$
$467$	4.33975	+	4.33975i	0.200819	+	0.200819i	−0.599532	+	0.800351i	$0.704646\pi$
$468$		0			0
$469$	−1.46410	+	1.46410i	−0.0676059	+	0.0676059i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−34.4959			−1.58612
$474$		0			0
$475$	−1.69161	−	1.69161i	−0.0776166	−	0.0776166i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−30.7846			−1.40658			−0.703292	−	0.710901i	$-0.748288\pi$
$479$	−30.7846			−1.40658			−0.703292	+	0.710901i	$0.748288\pi$
$480$		0			0
$481$	−15.4641			−0.705102
$482$		0			0
$483$		0			0
$484$		0			0
$485$	1.23835	+	1.23835i	0.0562305	+	0.0562305i
$486$		0			0
$487$	25.9091			1.17405			0.587027	−	0.809567i	$-0.300298\pi$
$487$	25.9091			1.17405			0.587027	+	0.809567i	$0.300298\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−0.143594	+	0.143594i	−0.00648029	+	0.00648029i	−0.710340	−	0.703859i	$-0.751459\pi$
$491$	−0.143594	+	0.143594i	−0.00648029	+	0.00648029i	0.703859	+	0.710340i	$0.251459\pi$
$492$		0			0
$493$	−1.80385	−	1.80385i	−0.0812412	−	0.0812412i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			19.7990i			0.888106i
$498$		0			0
$499$	27.3233	−	27.3233i	1.22316	−	1.22316i	0.256658	−	0.966502i	$-0.417379\pi$
$499$	27.3233	−	27.3233i	1.22316	−	1.22316i	0.966502	−	0.256658i	$-0.0826213\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			11.0718i			0.493667i	0.969058	+	0.246834i	$0.0793901\pi$
$503$			11.0718i			0.493667i	−0.969058	+	0.246834i	$0.920610\pi$
$504$		0			0
$505$			10.0000i			0.444994i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−11.1750	+	11.1750i	−0.495324	+	0.495324i	−0.909979	−	0.414655i	$-0.863902\pi$
$509$	−11.1750	+	11.1750i	−0.495324	+	0.495324i	0.414655	+	0.909979i	$0.363902\pi$
$510$		0			0
$511$			12.6264i			0.558558i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−9.80385	−	9.80385i	−0.432009	−	0.432009i
$516$		0			0
$517$	−28.3923	+	28.3923i	−1.24869	+	1.24869i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−21.2875			−0.932624			−0.466312	−	0.884620i	$-0.654418\pi$
$521$	−21.2875			−0.932624			−0.466312	+	0.884620i	$0.654418\pi$
$522$		0			0
$523$	1.13681	+	1.13681i	0.0497093	+	0.0497093i	0.731524	−	0.681815i	$-0.238809\pi$
$523$	1.13681	+	1.13681i	0.0497093	+	0.0497093i	−0.681815	+	0.731524i	$0.738809\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−0.248711			−0.0108340
$528$		0			0
$529$	11.0000			0.478261
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−17.6269	−	17.6269i	−0.763506	−	0.763506i
$534$		0			0
$535$	−13.9391			−0.602638
$536$		0			0
$537$		0			0
$538$		0			0
$539$	13.6603	−	13.6603i	0.588389	−	0.588389i
$540$		0			0
$541$	11.0526	+	11.0526i	0.475187	+	0.475187i	0.903588	−	0.428402i	$-0.140923\pi$
$541$	11.0526	+	11.0526i	0.475187	+	0.475187i	−0.428402	+	0.903588i	$0.640923\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	4.41851i		−	0.189268i
$546$		0			0
$547$	−16.1112	+	16.1112i	−0.688863	+	0.688863i	−0.961981	−	0.273118i	$-0.911945\pi$
$547$	−16.1112	+	16.1112i	−0.688863	+	0.688863i	0.273118	+	0.961981i	$0.411945\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	3.60770i		−	0.153693i
$552$		0			0
$553$			16.9282i			0.719860i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−1.83032	+	1.83032i	−0.0775530	+	0.0775530i	−0.744819	−	0.667266i	$-0.767464\pi$
$557$	−1.83032	+	1.83032i	−0.0775530	+	0.0775530i	0.667266	+	0.744819i	$0.267464\pi$
$558$		0			0
$559$		−	21.8695i		−	0.924983i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	18.0526	+	18.0526i	0.760825	+	0.760825i	0.976471	−	0.215647i	$-0.0691860\pi$
$563$	18.0526	+	18.0526i	0.760825	+	0.760825i	−0.215647	+	0.976471i	$0.569186\pi$
$564$		0			0
$565$	2.19615	−	2.19615i	0.0923928	−	0.0923928i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−15.0759			−0.632014			−0.316007	−	0.948757i	$-0.602342\pi$
$569$	−15.0759			−0.632014			−0.316007	+	0.948757i	$0.602342\pi$
$570$		0			0
$571$	−19.7990	−	19.7990i	−0.828562	−	0.828562i	0.158756	−	0.987318i	$-0.449252\pi$
$571$	−19.7990	−	19.7990i	−0.828562	−	0.828562i	−0.987318	+	0.158756i	$0.949252\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−15.4641			−0.644898
$576$		0			0
$577$	38.9282			1.62060			0.810301	−	0.586014i	$-0.199304\pi$
$577$	38.9282			1.62060			0.810301	+	0.586014i	$0.199304\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−10.2784	−	10.2784i	−0.426421	−	0.426421i
$582$		0			0
$583$	−21.8695			−0.905744
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−3.07180	+	3.07180i	−0.126787	+	0.126787i	−0.767653	−	0.640866i	$-0.778575\pi$
$587$	−3.07180	+	3.07180i	−0.126787	+	0.126787i	0.640866	+	0.767653i	$0.278575\pi$
$588$		0			0
$589$	−0.248711	−	0.248711i	−0.0102480	−	0.0102480i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	32.3238i		−	1.32738i	−0.748007	−	0.663691i	$-0.768989\pi$
$593$		−	32.3238i		−	1.32738i	0.748007	−	0.663691i	$-0.231011\pi$
$594$		0			0
$595$	−0.277401	+	0.277401i	−0.0113723	+	0.0113723i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			33.3205i			1.36144i	0.732544	+	0.680720i	$0.238333\pi$
$599$			33.3205i			1.36144i	−0.732544	+	0.680720i	$0.761667\pi$
$600$		0			0
$601$			12.7846i			0.521495i	0.965407	+	0.260748i	$0.0839690\pi$
$601$			12.7846i			0.521495i	−0.965407	+	0.260748i	$0.916031\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	2.03339	−	2.03339i	0.0826690	−	0.0826690i
$606$		0			0
$607$			17.6269i			0.715454i	0.933826	+	0.357727i	$0.116448\pi$
$607$			17.6269i			0.715454i	−0.933826	+	0.357727i	$0.883552\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−18.0000	−	18.0000i	−0.728202	−	0.728202i
$612$		0			0
$613$	12.6077	−	12.6077i	0.509220	−	0.509220i	−0.405067	−	0.914287i	$-0.632752\pi$
$613$	12.6077	−	12.6077i	0.509220	−	0.509220i	0.914287	+	0.405067i	$0.132752\pi$
$614$		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
$619$	−20.3538	−	20.3538i	−0.818088	−	0.818088i	0.167743	−	0.985831i	$-0.446352\pi$
$619$	−20.3538	−	20.3538i	−0.818088	−	0.818088i	−0.985831	+	0.167743i	$0.946352\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−18.7846			−0.752589
$624$		0			0
$625$	−17.2487			−0.689948
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−1.69161	−	1.69161i	−0.0674491	−	0.0674491i
$630$		0			0
$631$	29.2923			1.16611			0.583055	−	0.812433i	$-0.301857\pi$
$631$	29.2923			1.16611			0.583055	+	0.812433i	$0.301857\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	2.87564	−	2.87564i	0.114116	−	0.114116i
$636$		0			0
$637$	8.66025	+	8.66025i	0.343132	+	0.343132i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			11.4896i			0.453811i	0.973917	+	0.226905i	$0.0728608\pi$
$641$			11.4896i			0.453811i	−0.973917	+	0.226905i	$0.927139\pi$
$642$		0			0
$643$	25.9091	−	25.9091i	1.02176	−	1.02176i	0.0219976	−	0.999758i	$-0.492997\pi$
$643$	25.9091	−	25.9091i	1.02176	−	1.02176i	0.999758	−	0.0219976i	$-0.00700261\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			18.3923i			0.723076i	0.932357	+	0.361538i	$0.117748\pi$
$647$			18.3923i			0.723076i	−0.932357	+	0.361538i	$0.882252\pi$
$648$		0			0
$649$		−	26.9282i		−	1.05702i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−20.0392	+	20.0392i	−0.784196	+	0.784196i	−0.980536	−	0.196340i	$-0.937094\pi$
$653$	−20.0392	+	20.0392i	−0.784196	+	0.784196i	0.196340	+	0.980536i	$0.437094\pi$
$654$		0			0
$655$		−	7.17260i		−	0.280257i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	6.39230	+	6.39230i	0.249009	+	0.249009i	0.820564	−	0.571555i	$-0.193660\pi$
$659$	6.39230	+	6.39230i	0.249009	+	0.249009i	−0.571555	+	0.820564i	$0.693660\pi$
$660$		0			0
$661$	−21.7846	+	21.7846i	−0.847323	+	0.847323i	−0.989798	−	0.142475i	$-0.954494\pi$
$661$	−21.7846	+	21.7846i	−0.847323	+	0.847323i	0.142475	+	0.989798i	$0.454494\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−0.554803			−0.0215143
$666$		0			0
$667$	−16.4901	−	16.4901i	−0.638499	−	0.638499i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	16.3923			0.632818
$672$		0			0
$673$	−8.78461			−0.338622			−0.169311	−	0.985563i	$-0.554154\pi$
$673$	−8.78461			−0.338622			−0.169311	+	0.985563i	$0.554154\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	31.9077	+	31.9077i	1.22631	+	1.22631i	0.965348	+	0.260966i	$0.0840410\pi$
$677$	31.9077	+	31.9077i	1.22631	+	1.22631i	0.260966	+	0.965348i	$0.415959\pi$
$678$		0			0
$679$	−3.38323			−0.129836
$680$		0			0
$681$		0			0
$682$		0			0
$683$	11.2679	−	11.2679i	0.431156	−	0.431156i	−0.457865	−	0.889022i	$-0.651386\pi$
$683$	11.2679	−	11.2679i	0.431156	−	0.431156i	0.889022	+	0.457865i	$0.151386\pi$
$684$		0			0
$685$	4.19615	+	4.19615i	0.160327	+	0.160327i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	13.8647i		−	0.528204i
$690$		0			0
$691$	−23.8386	+	23.8386i	−0.906862	+	0.906862i	−0.996018	−	0.0891562i	$-0.971583\pi$
$691$	−23.8386	+	23.8386i	−0.906862	+	0.906862i	0.0891562	+	0.996018i	$0.471583\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			9.07180i			0.344113i
$696$		0			0
$697$		−	3.85641i		−	0.146072i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−16.9062	+	16.9062i	−0.638538	+	0.638538i	−0.950195	−	0.311657i	$-0.899116\pi$
$701$	−16.9062	+	16.9062i	−0.638538	+	0.638538i	0.311657	+	0.950195i	$0.399116\pi$
$702$		0			0
$703$		−	3.38323i		−	0.127601i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−13.6603	−	13.6603i	−0.513747	−	0.513747i
$708$		0			0
$709$	−5.19615	+	5.19615i	−0.195146	+	0.195146i	−0.797915	−	0.602770i	$-0.794064\pi$
$709$	−5.19615	+	5.19615i	−0.195146	+	0.195146i	0.602770	+	0.797915i	$0.294064\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−2.27362			−0.0851479
$714$		0			0
$715$	4.89898	+	4.89898i	0.183211	+	0.183211i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	25.7128			0.958926			0.479463	−	0.877562i	$-0.340831\pi$
$719$	25.7128			0.958926			0.479463	+	0.877562i	$0.340831\pi$
$720$		0			0
$721$	26.7846			0.997511
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−21.2504	−	21.2504i	−0.789219	−	0.789219i
$726$		0			0
$727$	18.9396			0.702430			0.351215	−	0.936295i	$-0.385769\pi$
$727$	18.9396			0.702430			0.351215	+	0.936295i	$0.385769\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	2.39230	−	2.39230i	0.0884826	−	0.0884826i
$732$		0			0
$733$	−37.4449	−	37.4449i	−1.38306	−	1.38306i	−0.839136	−	0.543921i	$-0.816939\pi$
$733$	−37.4449	−	37.4449i	−1.38306	−	1.38306i	−0.543921	−	0.839136i	$-0.683061\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			5.65685i			0.208373i
$738$		0			0
$739$	−12.6264	+	12.6264i	−0.464469	+	0.464469i	−0.900117	−	0.435648i	$-0.856519\pi$
$739$	−12.6264	+	12.6264i	−0.464469	+	0.464469i	0.435648	+	0.900117i	$0.356519\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	40.1051i		−	1.47131i	−0.677354	−	0.735657i	$-0.736874\pi$
$743$		−	40.1051i		−	1.47131i	0.677354	−	0.735657i	$-0.263126\pi$
$744$		0			0
$745$			1.60770i			0.0589014i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	19.0411	−	19.0411i	0.695747	−	0.695747i
$750$		0			0
$751$			14.7985i			0.540004i	0.962860	+	0.270002i	$0.0870244\pi$
$751$			14.7985i			0.540004i	−0.962860	+	0.270002i	$0.912976\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	4.73205	+	4.73205i	0.172217	+	0.172217i
$756$		0			0
$757$	34.6603	−	34.6603i	1.25975	−	1.25975i	0.308536	−	0.951213i	$-0.400161\pi$
$757$	34.6603	−	34.6603i	1.25975	−	1.25975i	0.951213	−	0.308536i	$-0.0998389\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	17.1464			0.621558			0.310779	−	0.950482i	$-0.399410\pi$
$761$	17.1464			0.621558			0.310779	+	0.950482i	$0.399410\pi$
$762$		0			0
$763$	6.03579	+	6.03579i	0.218510	+	0.218510i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	17.0718			0.616427
$768$		0			0
$769$	−45.7128			−1.64845			−0.824223	−	0.566265i	$-0.808388\pi$
$769$	−45.7128			−1.64845			−0.824223	+	0.566265i	$0.808388\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−34.3572	−	34.3572i	−1.23574	−	1.23574i	−0.961724	−	0.274019i	$-0.911647\pi$
$773$	−34.3572	−	34.3572i	−1.23574	−	1.23574i	−0.274019	−	0.961724i	$-0.588353\pi$
$774$		0			0
$775$	−2.92996			−0.105247
$776$		0			0
$777$		0			0
$778$		0			0
$779$	3.85641	−	3.85641i	0.138170	−	0.138170i
$780$		0			0
$781$	38.2487	+	38.2487i	1.36865	+	1.36865i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	10.2784i		−	0.366853i
$786$		0			0
$787$	24.3190	−	24.3190i	0.866880	−	0.866880i	−0.125246	−	0.992126i	$-0.539972\pi$
$787$	24.3190	−	24.3190i	0.866880	−	0.866880i	0.992126	+	0.125246i	$0.0399720\pi$
$788$		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
$797$	9.76079	−	9.76079i	0.345745	−	0.345745i	−0.512777	−	0.858522i	$-0.671383\pi$
$797$	9.76079	−	9.76079i	0.345745	−	0.345745i	0.858522	+	0.512777i	$0.171383\pi$
$798$		0			0
$799$		−	3.93803i		−	0.139318i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	24.3923	+	24.3923i	0.860786	+	0.860786i
$804$		0			0
$805$	−2.53590	+	2.53590i	−0.0893787	+	0.0893787i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	12.4505			0.437737			0.218868	−	0.975754i	$-0.429763\pi$
$809$	12.4505			0.437737			0.218868	+	0.975754i	$0.429763\pi$
$810$		0			0
$811$	21.4906	+	21.4906i	0.754637	+	0.754637i	0.975341	−	0.220704i	$-0.0708354\pi$
$811$	21.4906	+	21.4906i	0.754637	+	0.754637i	−0.220704	+	0.975341i	$0.570835\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−9.46410			−0.331513
$816$		0			0
$817$	4.78461			0.167392
$818$		0			0
$819$		0			0
$820$		0			0
$821$	14.5582	+	14.5582i	0.508086	+	0.508086i	0.913939	−	0.405853i	$-0.133025\pi$
$821$	14.5582	+	14.5582i	0.508086	+	0.508086i	−0.405853	+	0.913939i	$0.633025\pi$
$822$		0			0
$823$	−17.4238			−0.607357			−0.303678	−	0.952775i	$-0.598215\pi$
$823$	−17.4238			−0.607357			−0.303678	+	0.952775i	$0.598215\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−4.39230	+	4.39230i	−0.152735	+	0.152735i	−0.779339	−	0.626603i	$-0.784445\pi$
$827$	−4.39230	+	4.39230i	−0.152735	+	0.152735i	0.626603	+	0.779339i	$0.284445\pi$
$828$		0			0
$829$	15.1962	+	15.1962i	0.527784	+	0.527784i	0.919911	−	0.392127i	$-0.128261\pi$
$829$	15.1962	+	15.1962i	0.527784	+	0.527784i	−0.392127	+	0.919911i	$0.628261\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			1.89469i			0.0656470i
$834$		0			0
$835$	6.96953	−	6.96953i	0.241191	−	0.241191i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	9.21539i		−	0.318151i	−0.987266	−	0.159075i	$-0.949149\pi$
$839$		−	9.21539i		−	0.318151i	0.987266	−	0.159075i	$-0.0508513\pi$
$840$		0			0
$841$		−	16.3205i		−	0.562776i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	3.62347	−	3.62347i	0.124651	−	0.124651i
$846$		0			0
$847$			5.55532i			0.190883i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−15.4641	−	15.4641i	−0.530103	−	0.530103i
$852$		0			0
$853$	5.78461	−	5.78461i	0.198061	−	0.198061i	−0.601107	−	0.799168i	$-0.705274\pi$
$853$	5.78461	−	5.78461i	0.198061	−	0.198061i	0.799168	+	0.601107i	$0.205274\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−9.62209			−0.328684			−0.164342	−	0.986403i	$-0.552550\pi$
$857$	−9.62209			−0.328684			−0.164342	+	0.986403i	$0.552550\pi$
$858$		0			0
$859$	17.4238	+	17.4238i	0.594493	+	0.594493i	0.938842	−	0.344349i	$-0.111900\pi$
$859$	17.4238	+	17.4238i	0.594493	+	0.594493i	−0.344349	+	0.938842i	$0.611900\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−32.3923			−1.10265			−0.551323	−	0.834292i	$-0.685877\pi$
$863$	−32.3923			−1.10265			−0.551323	+	0.834292i	$0.685877\pi$
$864$		0			0
$865$	13.3205			0.452911
$866$		0			0
$867$		0			0
$868$		0			0
$869$	32.7028	+	32.7028i	1.10937	+	1.10937i
$870$		0			0
$871$	−3.58630			−0.121517
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−6.92820	+	6.92820i	−0.234216	+	0.234216i
$876$		0			0
$877$	−18.4641	−	18.4641i	−0.623488	−	0.623488i	0.322933	−	0.946422i	$-0.395331\pi$
$877$	−18.4641	−	18.4641i	−0.623488	−	0.623488i	−0.946422	+	0.322933i	$0.895331\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			27.4249i			0.923967i	0.886889	+	0.461984i	$0.152862\pi$
$881$			27.4249i			0.923967i	−0.886889	+	0.461984i	$0.847138\pi$
$882$		0			0
$883$	8.58682	−	8.58682i	0.288969	−	0.288969i	−0.547703	−	0.836673i	$-0.684498\pi$
$883$	8.58682	−	8.58682i	0.288969	−	0.288969i	0.836673	+	0.547703i	$0.184498\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	33.0718i		−	1.11044i	−0.831703	−	0.555221i	$-0.812634\pi$
$887$		−	33.0718i		−	1.11044i	0.831703	−	0.555221i	$-0.187366\pi$
$888$		0			0
$889$			7.85641i			0.263495i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	3.93803	−	3.93803i	0.131781	−	0.131781i
$894$		0			0
$895$			6.21166i			0.207633i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−3.12436	−	3.12436i	−0.104203	−	0.104203i
$900$		0			0
$901$	1.51666	−	1.51666i	0.0505273	−	0.0505273i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	11.5911			0.385302
$906$		0			0
$907$	8.38375	+	8.38375i	0.278378	+	0.278378i	0.832461	−	0.554083i	$-0.186931\pi$
$907$	8.38375	+	8.38375i	0.278378	+	0.278378i	−0.554083	+	0.832461i	$0.686931\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	49.1769			1.62930			0.814652	−	0.579950i	$-0.196928\pi$
$911$	49.1769			1.62930			0.814652	+	0.579950i	$0.196928\pi$
$912$		0			0
$913$	−39.7128			−1.31430
$914$		0			0
$915$		0			0
$916$		0			0
$917$	9.79796	+	9.79796i	0.323557	+	0.323557i
$918$		0			0
$919$	−2.92996			−0.0966506			−0.0483253	−	0.998832i	$-0.515388\pi$
$919$	−2.92996			−0.0966506			−0.0483253	+	0.998832i	$0.515388\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−24.2487	+	24.2487i	−0.798156	+	0.798156i
$924$		0			0
$925$	−19.9282	−	19.9282i	−0.655235	−	0.655235i
$926$		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
$931$	−1.89469	+	1.89469i	−0.0620959	+	0.0620959i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			1.07180i			0.0350515i
$936$		0			0
$937$		−	30.0000i		−	0.980057i	−0.871706	−	0.490029i	$-0.836986\pi$
$937$		−	30.0000i		−	0.980057i	0.871706	−	0.490029i	$-0.163014\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	17.2108	−	17.2108i	0.561056	−	0.561056i	−0.368551	−	0.929607i	$-0.620146\pi$
$941$	17.2108	−	17.2108i	0.561056	−	0.561056i	0.929607	+	0.368551i	$0.120146\pi$
$942$		0			0
$943$		−	35.2538i		−	1.14802i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	9.60770	+	9.60770i	0.312208	+	0.312208i	0.845765	−	0.533556i	$-0.179145\pi$
$947$	9.60770	+	9.60770i	0.312208	+	0.312208i	−0.533556	+	0.845765i	$0.679145\pi$
$948$		0			0
$949$	−15.4641	+	15.4641i	−0.501986	+	0.501986i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	38.2581			1.23930			0.619651	−	0.784878i	$-0.287274\pi$
$953$	38.2581			1.23930			0.619651	+	0.784878i	$0.287274\pi$
$954$		0			0
$955$	−3.38323	−	3.38323i	−0.109479	−	0.109479i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−11.4641			−0.370195
$960$		0			0
$961$	30.5692			0.986104
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−9.72363	−	9.72363i	−0.313015	−	0.313015i
$966$		0			0
$967$	−29.2923			−0.941978			−0.470989	−	0.882139i	$-0.656103\pi$
$967$	−29.2923			−0.941978			−0.470989	+	0.882139i	$0.656103\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−34.9808	+	34.9808i	−1.12259	+	1.12259i	−0.131235	+	0.991351i	$0.541894\pi$
$971$	−34.9808	+	34.9808i	−1.12259	+	1.12259i	−0.991351	+	0.131235i	$0.958106\pi$
$972$		0			0
$973$	−12.3923	−	12.3923i	−0.397279	−	0.397279i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			13.0053i			0.416077i	0.978121	+	0.208039i	$0.0667080\pi$
$977$			13.0053i			0.416077i	−0.978121	+	0.208039i	$0.933292\pi$
$978$		0			0
$979$	−36.2891	+	36.2891i	−1.15980	+	1.15980i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	30.6410i		−	0.977297i	−0.872481	−	0.488648i	$-0.837490\pi$
$983$		−	30.6410i		−	0.977297i	0.872481	−	0.488648i	$-0.162510\pi$
$984$		0			0
$985$		−	4.64102i		−	0.147875i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	21.8695	−	21.8695i	0.695411	−	0.695411i
$990$		0			0
$991$		−	45.7081i		−	1.45197i	−0.687713	−	0.725983i	$-0.741385\pi$
$991$		−	45.7081i		−	1.45197i	0.687713	−	0.725983i	$-0.258615\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−0.0525589	−	0.0525589i	−0.00166623	−	0.00166623i
$996$		0			0
$997$	21.0000	−	21.0000i	0.665077	−	0.665077i	−0.291496	−	0.956572i	$-0.594153\pi$
$997$	21.0000	−	21.0000i	0.665077	−	0.665077i	0.956572	+	0.291496i	$0.0941528\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2304.2.l.c.1727.3	yes	8
3.2	odd	2		2304.2.l.g.1727.2	yes	8
4.3	odd	2		2304.2.l.g.1727.3	yes	8
8.3	odd	2		2304.2.l.b.1727.2	yes	8
8.5	even	2		2304.2.l.f.1727.2	yes	8
12.11	even	2	inner	2304.2.l.c.1727.2	yes	8
16.3	odd	4		2304.2.l.b.575.3	yes	8
16.5	even	4	inner	2304.2.l.c.575.2	yes	8
16.11	odd	4		2304.2.l.g.575.2	yes	8
16.13	even	4		2304.2.l.f.575.3	yes	8
24.5	odd	2		2304.2.l.b.1727.3	yes	8
24.11	even	2		2304.2.l.f.1727.3	yes	8
48.5	odd	4		2304.2.l.g.575.3	yes	8
48.11	even	4	inner	2304.2.l.c.575.3	yes	8
48.29	odd	4		2304.2.l.b.575.2	✓	8
48.35	even	4		2304.2.l.f.575.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2304.2.l.b.575.2	✓	8	48.29	odd	4
2304.2.l.b.575.3	yes	8	16.3	odd	4
2304.2.l.b.1727.2	yes	8	8.3	odd	2
2304.2.l.b.1727.3	yes	8	24.5	odd	2
2304.2.l.c.575.2	yes	8	16.5	even	4	inner
2304.2.l.c.575.3	yes	8	48.11	even	4	inner
2304.2.l.c.1727.2	yes	8	12.11	even	2	inner
2304.2.l.c.1727.3	yes	8	1.1	even	1	trivial
2304.2.l.f.575.2	yes	8	48.35	even	4
2304.2.l.f.575.3	yes	8	16.13	even	4
2304.2.l.f.1727.2	yes	8	8.5	even	2
2304.2.l.f.1727.3	yes	8	24.11	even	2
2304.2.l.g.575.2	yes	8	16.11	odd	4
2304.2.l.g.575.3	yes	8	48.5	odd	4
2304.2.l.g.1727.2	yes	8	3.2	odd	2
2304.2.l.g.1727.3	yes	8	4.3	odd	2

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)

\(f(q)\)	\(=\)	\(q+(0.517638 + 0.517638i) q^{5} -1.41421 q^{7} +O(q^{10})\)	\(q+(0.517638 + 0.517638i) q^{5} -1.41421 q^{7} +(-2.73205 + 2.73205i) q^{11} +(-1.73205 - 1.73205i) q^{13} -0.378937i q^{17} +(0.378937 - 0.378937i) q^{19} -3.46410i q^{23} -4.46410i q^{25} +(4.76028 - 4.76028i) q^{29} -0.656339i q^{31} +(-0.732051 - 0.732051i) q^{35} +(4.46410 - 4.46410i) q^{37} +10.1769 q^{41} +(6.31319 + 6.31319i) q^{43} +10.3923 q^{47} -5.00000 q^{49} +(4.00240 + 4.00240i) q^{53} -2.82843 q^{55} +(-4.92820 + 4.92820i) q^{59} +(-3.00000 - 3.00000i) q^{61} -1.79315i q^{65} +(1.03528 - 1.03528i) q^{67} -14.0000i q^{71} -8.92820i q^{73} +(3.86370 - 3.86370i) q^{77} -11.9700i q^{79} +(7.26795 + 7.26795i) q^{83} +(0.196152 - 0.196152i) q^{85} +13.2827 q^{89} +(2.44949 + 2.44949i) q^{91} +0.392305 q^{95} +2.39230 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 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

Related objects

Downloads

Learn more