LMFDB - Embedded newform 2592.2.p.g.2159.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2592,2,Mod(431,2592)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2592, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 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("2592.431");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2592 = 2^{5} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2592.p (of order $6$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$20.6972242039$
Analytic rank:	$0$
Dimension:	$48$
Relative dimension:	$24$ over $\Q(\zeta_{6})$
Twist minimal:	no (minimal twist has level 648)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			2159.14
Character	$\chi$	$=$	2592.2159
Dual form			2592.2.p.g.431.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.474833 - 0.822435i) q^{5} +(-4.07898 + 2.35500i) q^{7} +O(q^{10})$	$q+(0.474833 - 0.822435i) q^{5} +(-4.07898 + 2.35500i) q^{7} +(3.72007 - 2.14778i) q^{11} +(-3.52349 - 2.03429i) q^{13} +1.19178i q^{17} +3.17693 q^{19} +(0.375325 - 0.650083i) q^{23} +(2.04907 + 3.54909i) q^{25} +(3.87181 + 6.70617i) q^{29} +(-0.496917 - 0.286895i) q^{31} +4.47293i q^{35} -4.87320i q^{37} +(-8.45135 - 4.87939i) q^{41} +(-2.65770 - 4.60327i) q^{43} +(-4.91012 - 8.50458i) q^{47} +(7.59204 - 13.1498i) q^{49} +0.877682 q^{53} -4.07935i q^{55} +(-1.51936 - 0.877204i) q^{59} +(8.59196 - 4.96057i) q^{61} +(-3.34614 + 1.93189i) q^{65} +(5.48335 - 9.49744i) q^{67} +9.91048 q^{71} +8.74944 q^{73} +(-10.1160 + 17.5215i) q^{77} +(4.97571 - 2.87273i) q^{79} +(10.8283 - 6.25173i) q^{83} +(0.980161 + 0.565896i) q^{85} +10.1440i q^{89} +19.1630 q^{91} +(1.50851 - 2.61282i) q^{95} +(-3.91511 - 6.78118i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q+O(q^{10}) $ 48 * q	$ 48 q - 24 q^{25} + 24 q^{49} + 48 q^{67}+O(q^{100}) $ 48 * q - 24 * q^25 + 24 * q^49 + 48 * q^67

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2592\mathbb{Z}\right)^\times$.

$n$	$325$	$1217$	$2431$
$\chi(n)$	$-1$	$e\left(\frac{5}{6}\right)$	$-1$

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
$5$	0.474833	−	0.822435i	0.212352	−	0.367804i	−0.740098	−	0.672499i	$-0.765221\pi$
$5$	0.474833	−	0.822435i	0.212352	−	0.367804i	0.952450	+	0.304695i	$0.0985544\pi$
$6$		0			0
$7$	−4.07898	+	2.35500i	−1.54171	+	0.890106i	−0.542978	+	0.839747i	$0.682703\pi$
$7$	−4.07898	+	2.35500i	−1.54171	+	0.890106i	−0.998731	+	0.0503587i	$0.983964\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	3.72007	−	2.14778i	1.12164	−	0.647581i	0.179823	−	0.983699i	$-0.442448\pi$
$11$	3.72007	−	2.14778i	1.12164	−	0.647581i	0.941820	+	0.336118i	$0.109114\pi$
$12$		0			0
$13$	−3.52349	−	2.03429i	−0.977240	−	0.564210i	−0.0758039	−	0.997123i	$-0.524152\pi$
$13$	−3.52349	−	2.03429i	−0.977240	−	0.564210i	−0.901436	+	0.432913i	$0.857486\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			1.19178i			0.289049i	0.989501	+	0.144524i	$0.0461652\pi$
$17$			1.19178i			0.289049i	−0.989501	+	0.144524i	$0.953835\pi$
$18$		0			0
$19$	3.17693			0.728837			0.364418	−	0.931235i	$-0.381268\pi$
$19$	3.17693			0.728837			0.364418	+	0.931235i	$0.381268\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	0.375325	−	0.650083i	0.0782608	−	0.135552i	−0.824239	−	0.566242i	$-0.808397\pi$
$23$	0.375325	−	0.650083i	0.0782608	−	0.135552i	0.902500	+	0.430691i	$0.141730\pi$
$24$		0			0
$25$	2.04907	+	3.54909i	0.409813	+	0.709818i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	3.87181	+	6.70617i	0.718977	+	1.24530i	0.961405	+	0.275136i	$0.0887227\pi$
$29$	3.87181	+	6.70617i	0.718977	+	1.24530i	−0.242428	+	0.970169i	$0.577944\pi$
$30$		0			0
$31$	−0.496917	−	0.286895i	−0.0892490	−	0.0515279i	0.454711	−	0.890639i	$-0.349742\pi$
$31$	−0.496917	−	0.286895i	−0.0892490	−	0.0515279i	−0.543960	+	0.839111i	$0.683076\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$			4.47293i			0.756062i
$36$		0			0
$37$		−	4.87320i		−	0.801148i	−0.916264	−	0.400574i	$-0.868811\pi$
$37$		−	4.87320i		−	0.801148i	0.916264	−	0.400574i	$-0.131189\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−8.45135	−	4.87939i	−1.31988	−	0.762032i	−0.336170	−	0.941801i	$-0.609132\pi$
$41$	−8.45135	−	4.87939i	−1.31988	−	0.762032i	−0.983709	+	0.179769i	$0.942465\pi$
$42$		0			0
$43$	−2.65770	−	4.60327i	−0.405295	−	0.701992i	0.589060	−	0.808089i	$-0.299498\pi$
$43$	−2.65770	−	4.60327i	−0.405295	−	0.701992i	−0.994356	+	0.106097i	$0.966165\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−4.91012	−	8.50458i	−0.716214	−	1.24052i	−0.962489	−	0.271320i	$-0.912540\pi$
$47$	−4.91012	−	8.50458i	−0.716214	−	1.24052i	0.246275	−	0.969200i	$-0.420793\pi$
$48$		0			0
$49$	7.59204	−	13.1498i	1.08458	−	1.87854i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	0.877682			0.120559			0.0602794	−	0.998182i	$-0.480801\pi$
$53$	0.877682			0.120559			0.0602794	+	0.998182i	$0.480801\pi$
$54$		0			0
$55$		−	4.07935i		−	0.550060i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−1.51936	−	0.877204i	−0.197804	−	0.114202i	0.397827	−	0.917461i	$-0.369765\pi$
$59$	−1.51936	−	0.877204i	−0.197804	−	0.114202i	−0.595631	+	0.803258i	$0.703098\pi$
$60$		0			0
$61$	8.59196	−	4.96057i	1.10009	−	0.635136i	0.163844	−	0.986486i	$-0.447611\pi$
$61$	8.59196	−	4.96057i	1.10009	−	0.635136i	0.936244	+	0.351350i	$0.114277\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−3.34614	+	1.93189i	−0.415037	+	0.239622i
$66$		0			0
$67$	5.48335	−	9.49744i	0.669898	−	1.16030i	−0.308034	−	0.951375i	$-0.599671\pi$
$67$	5.48335	−	9.49744i	0.669898	−	1.16030i	0.977932	−	0.208922i	$-0.0669954\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	9.91048			1.17616			0.588079	−	0.808804i	$-0.299884\pi$
$71$	9.91048			1.17616			0.588079	+	0.808804i	$0.299884\pi$
$72$		0			0
$73$	8.74944			1.02404			0.512022	−	0.858972i	$-0.328897\pi$
$73$	8.74944			1.02404			0.512022	+	0.858972i	$0.328897\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−10.1160	+	17.5215i	−1.15283	+	1.99676i
$78$		0			0
$79$	4.97571	−	2.87273i	0.559811	−	0.323207i	−0.193259	−	0.981148i	$-0.561906\pi$
$79$	4.97571	−	2.87273i	0.559811	−	0.323207i	0.753070	+	0.657941i	$0.228572\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	10.8283	−	6.25173i	1.18856	−	0.686217i	0.230583	−	0.973053i	$-0.425937\pi$
$83$	10.8283	−	6.25173i	1.18856	−	0.686217i	0.957980	+	0.286836i	$0.0926034\pi$
$84$		0			0
$85$	0.980161	+	0.565896i	0.106313	+	0.0613800i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			10.1440i			1.07527i	0.843179	+	0.537633i	$0.180681\pi$
$89$			10.1440i			1.07527i	−0.843179	+	0.537633i	$0.819319\pi$
$90$		0			0
$91$	19.1630			2.00882
$92$		0			0
$93$		0			0
$94$		0			0
$95$	1.50851	−	2.61282i	0.154770	−	0.268069i
$96$		0			0
$97$	−3.91511	−	6.78118i	−0.397520	−	0.688524i	0.595900	−	0.803059i	$-0.296796\pi$
$97$	−3.91511	−	6.78118i	−0.397520	−	0.688524i	−0.993419	+	0.114535i	$0.963462\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−8.92656	−	15.4613i	−0.888226	−	1.53845i	−0.841971	−	0.539522i	$-0.818605\pi$
$101$	−8.92656	−	15.4613i	−0.888226	−	1.53845i	−0.0462544	−	0.998930i	$-0.514728\pi$
$102$		0			0
$103$	2.12533	+	1.22706i	0.209415	+	0.120906i	0.601040	−	0.799219i	$-0.294753\pi$
$103$	2.12533	+	1.22706i	0.209415	+	0.120906i	−0.391624	+	0.920125i	$0.628087\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			0.283208i			0.0273788i	0.999906	+	0.0136894i	$0.00435760\pi$
$107$			0.283208i			0.0273788i	−0.999906	+	0.0136894i	$0.995642\pi$
$108$		0			0
$109$			9.74716i			0.933609i	0.884361	+	0.466805i	$0.154595\pi$
$109$			9.74716i			0.933609i	−0.884361	+	0.466805i	$0.845405\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−4.09400	−	2.36367i	−0.385131	−	0.222356i	0.294917	−	0.955523i	$-0.404708\pi$
$113$	−4.09400	−	2.36367i	−0.385131	−	0.222356i	−0.680048	+	0.733167i	$0.738041\pi$
$114$		0			0
$115$	−0.356434	−	0.617362i	−0.0332376	−	0.0575693i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−2.80664	−	4.86124i	−0.257284	−	0.445629i
$120$		0			0
$121$	3.72594	−	6.45351i	0.338721	−	0.586683i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	8.64019			0.772802
$126$		0			0
$127$		−	13.8042i		−	1.22492i	−0.790500	−	0.612462i	$-0.790179\pi$
$127$		−	13.8042i		−	1.22492i	0.790500	−	0.612462i	$-0.209821\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	4.10031	+	2.36731i	0.358245	+	0.206833i	0.668311	−	0.743882i	$-0.267018\pi$
$131$	4.10031	+	2.36731i	0.358245	+	0.206833i	−0.310065	+	0.950715i	$0.600351\pi$
$132$		0			0
$133$	−12.9586	+	7.48166i	−1.12365	+	0.648742i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	13.4704	−	7.77716i	1.15086	−	0.664448i	0.201762	−	0.979435i	$-0.435333\pi$
$137$	13.4704	−	7.77716i	1.15086	−	0.664448i	0.949096	+	0.314987i	$0.102000\pi$
$138$		0			0
$139$	3.94490	−	6.83276i	0.334602	−	0.579547i	−0.648806	−	0.760953i	$-0.724732\pi$
$139$	3.94490	−	6.83276i	0.334602	−	0.579547i	0.983408	+	0.181406i	$0.0580649\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−17.4768			−1.46148
$144$		0			0
$145$	7.35385			0.610704
$146$		0			0
$147$		0			0
$148$		0			0
$149$	8.00344	−	13.8624i	0.655667	−	1.13565i	−0.326059	−	0.945350i	$-0.605721\pi$
$149$	8.00344	−	13.8624i	0.655667	−	1.13565i	0.981726	−	0.190300i	$-0.0609459\pi$
$150$		0			0
$151$	1.57918	−	0.911739i	0.128512	−	0.0741962i	−0.434366	−	0.900736i	$-0.643028\pi$
$151$	1.57918	−	0.911739i	0.128512	−	0.0741962i	0.562878	+	0.826540i	$0.309694\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−0.471905	+	0.272455i	−0.0379044	+	0.0218841i
$156$		0			0
$157$	−17.7438	−	10.2444i	−1.41611	−	0.817589i	−0.420152	−	0.907454i	$-0.638023\pi$
$157$	−17.7438	−	10.2444i	−1.41611	−	0.817589i	−0.995954	+	0.0898651i	$0.971356\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			3.53556i			0.278641i
$162$		0			0
$163$	7.59374			0.594787			0.297394	−	0.954755i	$-0.403883\pi$
$163$	7.59374			0.594787			0.297394	+	0.954755i	$0.403883\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−2.80664	+	4.86124i	−0.217184	+	0.376174i	−0.953946	−	0.299978i	$-0.903021\pi$
$167$	−2.80664	+	4.86124i	−0.217184	+	0.376174i	0.736762	+	0.676152i	$0.236354\pi$
$168$		0			0
$169$	1.77664	+	3.07723i	0.136665	+	0.236710i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	6.31622	+	10.9400i	0.480214	+	0.831754i	0.999742	−	0.0226986i	$-0.00722582\pi$
$173$	6.31622	+	10.9400i	0.480214	+	0.831754i	−0.519529	+	0.854453i	$0.673892\pi$
$174$		0			0
$175$	−16.7162	−	9.65110i	−1.26363	−	0.729555i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			20.5282i			1.53435i	0.641439	+	0.767174i	$0.278338\pi$
$179$			20.5282i			1.53435i	−0.641439	+	0.767174i	$0.721662\pi$
$180$		0			0
$181$		−	9.96062i		−	0.740367i	−0.928959	−	0.370184i	$-0.879295\pi$
$181$		−	9.96062i		−	0.740367i	0.928959	−	0.370184i	$-0.120705\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−4.00789	−	2.31395i	−0.294666	−	0.170125i
$186$		0			0
$187$	2.55968	+	4.43350i	0.187182	+	0.324210i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	11.4999	+	19.9185i	0.832105	+	1.44125i	0.896366	+	0.443316i	$0.146198\pi$
$191$	11.4999	+	19.9185i	0.832105	+	1.44125i	−0.0642603	+	0.997933i	$0.520469\pi$
$192$		0			0
$193$	0.500000	−	0.866025i	0.0359908	−	0.0623379i	−0.847469	−	0.530845i	$-0.821875\pi$
$193$	0.500000	−	0.866025i	0.0359908	−	0.0623379i	0.883460	+	0.468507i	$0.155208\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	0.519639			0.0370228			0.0185114	−	0.999829i	$-0.494107\pi$
$197$	0.519639			0.0370228			0.0185114	+	0.999829i	$0.494107\pi$
$198$		0			0
$199$			0.630646i			0.0447053i	0.999750	+	0.0223527i	$0.00711567\pi$
$199$			0.630646i			0.0447053i	−0.999750	+	0.0223527i	$0.992884\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−31.5861	−	18.2362i	−2.21691	−	1.27993i
$204$		0			0
$205$	−8.02596	+	4.63379i	−0.560557	+	0.323638i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	11.8184	−	6.82335i	0.817495	−	0.471981i
$210$		0			0
$211$	4.95897	−	8.58918i	0.341389	−	0.591304i	−0.643302	−	0.765613i	$-0.722436\pi$
$211$	4.95897	−	8.58918i	0.341389	−	0.591304i	0.984691	+	0.174309i	$0.0557692\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−5.04785			−0.344261
$216$		0			0
$217$	2.70255			0.183461
$218$		0			0
$219$		0			0
$220$		0			0
$221$	2.42442	−	4.19922i	0.163084	−	0.282470i
$222$		0			0
$223$	−6.15507	+	3.55363i	−0.412174	+	0.237969i	−0.691723	−	0.722162i	$-0.743148\pi$
$223$	−6.15507	+	3.55363i	−0.412174	+	0.237969i	0.279549	+	0.960131i	$0.409815\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−10.7925	+	6.23105i	−0.716323	+	0.413569i	−0.813398	−	0.581708i	$-0.802385\pi$
$227$	−10.7925	+	6.23105i	−0.716323	+	0.413569i	0.0970749	+	0.995277i	$0.469051\pi$
$228$		0			0
$229$	8.45996	+	4.88436i	0.559050	+	0.322768i	0.752764	−	0.658290i	$-0.228720\pi$
$229$	8.45996	+	4.88436i	0.559050	+	0.322768i	−0.193714	+	0.981058i	$0.562053\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			4.66388i			0.305541i	0.988262	+	0.152771i	$0.0488196\pi$
$233$			4.66388i			0.305541i	−0.988262	+	0.152771i	$0.951180\pi$
$234$		0			0
$235$	−9.32595			−0.608358
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−1.71266	+	2.96641i	−0.110783	+	0.191881i	−0.916086	−	0.400982i	$-0.868669\pi$
$239$	−1.71266	+	2.96641i	−0.110783	+	0.191881i	0.805303	+	0.592863i	$0.202002\pi$
$240$		0			0
$241$	−0.627860	−	1.08748i	−0.0404440	−	0.0700510i	0.845095	−	0.534616i	$-0.179544\pi$
$241$	−0.627860	−	1.08748i	−0.0404440	−	0.0700510i	−0.885539	+	0.464565i	$0.846211\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−7.20990	−	12.4879i	−0.460624	−	0.797824i
$246$		0			0
$247$	−11.1939	−	6.46278i	−0.712248	−	0.411217i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	20.2340i		−	1.27716i	−0.769556	−	0.638579i	$-0.779523\pi$
$251$		−	20.2340i		−	1.27716i	0.769556	−	0.638579i	$-0.220477\pi$
$252$		0			0
$253$		−	3.22447i		−	0.202721i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	1.01857	+	0.588069i	0.0635364	+	0.0366828i	0.531432	−	0.847101i	$-0.321654\pi$
$257$	1.01857	+	0.588069i	0.0635364	+	0.0366828i	−0.467895	+	0.883784i	$0.654988\pi$
$258$		0			0
$259$	11.4764	+	19.8777i	0.713107	+	1.23514i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	3.51217	+	6.08325i	0.216570	+	0.375109i	0.953757	−	0.300579i	$-0.0971799\pi$
$263$	3.51217	+	6.08325i	0.216570	+	0.375109i	−0.737187	+	0.675688i	$0.763847\pi$
$264$		0			0
$265$	0.416752	−	0.721836i	0.0256009	−	0.0443420i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−11.4172			−0.696119			−0.348060	−	0.937472i	$-0.613159\pi$
$269$	−11.4172			−0.696119			−0.348060	+	0.937472i	$0.613159\pi$
$270$		0			0
$271$			10.9905i			0.667625i	0.942639	+	0.333813i	$0.108335\pi$
$271$			10.9905i			0.667625i	−0.942639	+	0.333813i	$0.891665\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	15.2453	+	8.80190i	0.919328	+	0.530775i
$276$		0			0
$277$	3.11917	−	1.80085i	0.187413	−	0.108203i	−0.403358	−	0.915042i	$-0.632157\pi$
$277$	3.11917	−	1.80085i	0.187413	−	0.108203i	0.590771	+	0.806839i	$0.298824\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−10.6296	+	6.13703i	−0.634111	+	0.366104i	−0.782343	−	0.622848i	$-0.785975\pi$
$281$	−10.6296	+	6.13703i	−0.634111	+	0.366104i	0.148231	+	0.988953i	$0.452642\pi$
$282$		0			0
$283$	−1.27237	+	2.20381i	−0.0756344	+	0.131003i	−0.901362	−	0.433066i	$-0.857432\pi$
$283$	−1.27237	+	2.20381i	−0.0756344	+	0.131003i	0.825728	+	0.564069i	$0.190765\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	45.9638			2.71316
$288$		0			0
$289$	15.5797			0.916451
$290$		0			0
$291$		0			0
$292$		0			0
$293$	0.929672	−	1.61024i	0.0543120	−	0.0940712i	−0.837591	−	0.546298i	$-0.816037\pi$
$293$	0.929672	−	1.61024i	0.0543120	−	0.0940712i	0.891903	+	0.452226i	$0.149370\pi$
$294$		0			0
$295$	−1.44289	+	0.833051i	−0.0840081	+	0.0485021i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−2.64491	+	1.52704i	−0.152959	+	0.0883109i
$300$		0			0
$301$	21.6814	+	12.5178i	1.24969	+	0.721512i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	9.42177i		−	0.539489i
$306$		0			0
$307$	−29.0809			−1.65973			−0.829867	−	0.557961i	$-0.811584\pi$
$307$	−29.0809			−1.65973			−0.829867	+	0.557961i	$0.811584\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−9.36300	+	16.2172i	−0.530927	+	0.919593i	0.468421	+	0.883505i	$0.344823\pi$
$311$	−9.36300	+	16.2172i	−0.530927	+	0.919593i	−0.999349	+	0.0360877i	$0.988510\pi$
$312$		0			0
$313$	−8.31967	−	14.4101i	−0.470256	−	0.814507i	0.529166	−	0.848519i	$-0.322505\pi$
$313$	−8.31967	−	14.4101i	−0.470256	−	0.814507i	−0.999421	+	0.0340117i	$0.989172\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−6.55605	−	11.3554i	−0.368224	−	0.637783i	0.621064	−	0.783760i	$-0.286701\pi$
$317$	−6.55605	−	11.3554i	−0.368224	−	0.637783i	−0.989288	+	0.145977i	$0.953368\pi$
$318$		0			0
$319$	28.8068	+	16.6316i	1.61287	+	0.931191i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			3.78619i			0.210669i
$324$		0			0
$325$		−	16.6736i		−	0.924883i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	40.0565	+	23.1266i	2.20839	+	1.27501i
$330$		0			0
$331$	−16.3976	−	28.4014i	−0.901292	−	1.56108i	−0.825818	−	0.563937i	$-0.809286\pi$
$331$	−16.3976	−	28.4014i	−0.901292	−	1.56108i	−0.0754743	−	0.997148i	$-0.524047\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−5.20735	−	9.01940i	−0.284508	−	0.492782i
$336$		0			0
$337$	−14.1351	+	24.4827i	−0.769986	+	1.33366i	0.167583	+	0.985858i	$0.446404\pi$
$337$	−14.1351	+	24.4827i	−0.769986	+	1.33366i	−0.937570	+	0.347798i	$0.886930\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−2.46475			−0.133474
$342$		0			0
$343$			38.5470i			2.08134i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−5.92687	−	3.42188i	−0.318171	−	0.183696i	0.332406	−	0.943136i	$-0.392140\pi$
$347$	−5.92687	−	3.42188i	−0.318171	−	0.183696i	−0.650577	+	0.759440i	$0.725473\pi$
$348$		0			0
$349$	12.5091	−	7.22213i	0.669596	−	0.386592i	−0.126327	−	0.991989i	$-0.540319\pi$
$349$	12.5091	−	7.22213i	0.669596	−	0.386592i	0.795924	+	0.605397i	$0.206986\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	13.7547	−	7.94128i	0.732088	−	0.422671i	−0.0870973	−	0.996200i	$-0.527759\pi$
$353$	13.7547	−	7.94128i	0.732088	−	0.422671i	0.819186	+	0.573528i	$0.194426\pi$
$354$		0			0
$355$	4.70582	−	8.15073i	0.249759	−	0.432596i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−3.84545			−0.202955			−0.101477	−	0.994838i	$-0.532357\pi$
$359$	−3.84545			−0.202955			−0.101477	+	0.994838i	$0.532357\pi$
$360$		0			0
$361$	−8.90714			−0.468797
$362$		0			0
$363$		0			0
$364$		0			0
$365$	4.15452	−	7.19584i	0.217458	−	0.376648i
$366$		0			0
$367$	23.8803	−	13.7873i	1.24654	−	0.719692i	0.276124	−	0.961122i	$-0.410950\pi$
$367$	23.8803	−	13.7873i	1.24654	−	0.719692i	0.970418	+	0.241430i	$0.0776165\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−3.58004	+	2.06694i	−0.185867	+	0.107310i
$372$		0			0
$373$	7.00779	+	4.04595i	0.362849	+	0.209491i	0.670330	−	0.742063i	$-0.266153\pi$
$373$	7.00779	+	4.04595i	0.362849	+	0.209491i	−0.307481	+	0.951554i	$0.599486\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	31.5055i		−	1.62261i
$378$		0			0
$379$	17.9143			0.920195			0.460098	−	0.887868i	$-0.347814\pi$
$379$	17.9143			0.920195			0.460098	+	0.887868i	$0.347814\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−2.88757	+	5.00142i	−0.147548	+	0.255561i	−0.930321	−	0.366747i	$-0.880471\pi$
$383$	−2.88757	+	5.00142i	−0.147548	+	0.255561i	0.782773	+	0.622308i	$0.213805\pi$
$384$		0			0
$385$	9.60687	+	16.6396i	0.489611	+	0.848032i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	11.0163	+	19.0807i	0.558547	+	0.967432i	0.997618	+	0.0689799i	$0.0219744\pi$
$389$	11.0163	+	19.0807i	0.558547	+	0.967432i	−0.439071	+	0.898453i	$0.644692\pi$
$390$		0			0
$391$	0.774755	+	0.447305i	0.0391810	+	0.0226212i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	5.45626i		−	0.274534i
$396$		0			0
$397$		−	26.3815i		−	1.32405i	−0.749481	−	0.662026i	$-0.769697\pi$
$397$		−	26.3815i		−	1.32405i	0.749481	−	0.662026i	$-0.230303\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	1.11822	+	0.645607i	0.0558414	+	0.0322401i	0.527661	−	0.849455i	$-0.323069\pi$
$401$	1.11822	+	0.645607i	0.0558414	+	0.0322401i	−0.471819	+	0.881695i	$0.656403\pi$
$402$		0			0
$403$	1.16725	+	2.02174i	0.0581451	+	0.100710i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−10.4666	−	18.1286i	−0.518808	−	0.898602i
$408$		0			0
$409$	0.772426	−	1.33788i	0.0381940	−	0.0661539i	−0.846297	−	0.532712i	$-0.821173\pi$
$409$	0.772426	−	1.33788i	0.0381940	−	0.0661539i	0.884490	+	0.466558i	$0.154506\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	8.26326			0.406608
$414$		0			0
$415$		−	11.8741i		−	0.582878i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	12.9309	+	7.46567i	0.631717	+	0.364722i	0.781417	−	0.624010i	$-0.214497\pi$
$419$	12.9309	+	7.46567i	0.631717	+	0.364722i	−0.149700	+	0.988732i	$0.547831\pi$
$420$		0			0
$421$	12.8947	−	7.44478i	0.628451	−	0.362836i	−0.151701	−	0.988426i	$-0.548475\pi$
$421$	12.8947	−	7.44478i	0.628451	−	0.362836i	0.780152	+	0.625590i	$0.215142\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−4.22973	+	2.44204i	−0.205172	+	0.118456i
$426$		0			0
$427$	−23.3643	+	40.4681i	−1.13068	+	1.95839i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−30.8436			−1.48568			−0.742842	−	0.669467i	$-0.766523\pi$
$431$	−30.8436			−1.48568			−0.742842	+	0.669467i	$0.766523\pi$
$432$		0			0
$433$	−29.1892			−1.40275			−0.701373	−	0.712795i	$-0.747429\pi$
$433$	−29.1892			−1.40275			−0.701373	+	0.712795i	$0.747429\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	1.19238	−	2.06527i	0.0570393	−	0.0987950i
$438$		0			0
$439$	−1.08226	+	0.624843i	−0.0516535	+	0.0298221i	−0.525604	−	0.850729i	$-0.676161\pi$
$439$	−1.08226	+	0.624843i	−0.0516535	+	0.0298221i	0.473951	+	0.880551i	$0.342827\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−23.7521	+	13.7133i	−1.12850	+	0.651538i	−0.943556	−	0.331212i	$-0.892542\pi$
$443$	−23.7521	+	13.7133i	−1.12850	+	0.651538i	−0.184940	+	0.982750i	$0.559209\pi$
$444$		0			0
$445$	8.34282	+	4.81673i	0.395487	+	0.228335i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			16.2552i			0.767132i	0.923513	+	0.383566i	$0.125304\pi$
$449$			16.2552i			0.767132i	−0.923513	+	0.383566i	$0.874696\pi$
$450$		0			0
$451$	−41.9194			−1.97391
$452$		0			0
$453$		0			0
$454$		0			0
$455$	9.09921	−	15.7603i	0.426578	−	0.738854i
$456$		0			0
$457$	−9.44307	−	16.3559i	−0.441728	−	0.765096i	0.556090	−	0.831122i	$-0.312301\pi$
$457$	−9.44307	−	16.3559i	−0.441728	−	0.765096i	−0.997818	+	0.0660267i	$0.978968\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−9.65240	−	16.7184i	−0.449557	−	0.778655i	0.548800	−	0.835954i	$-0.315085\pi$
$461$	−9.65240	−	16.7184i	−0.449557	−	0.778655i	−0.998357	+	0.0572981i	$0.981751\pi$
$462$		0			0
$463$	−2.02823	−	1.17100i	−0.0942598	−	0.0544209i	0.452129	−	0.891952i	$-0.350665\pi$
$463$	−2.02823	−	1.17100i	−0.0942598	−	0.0544209i	−0.546389	+	0.837532i	$0.683998\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			12.2034i			0.564708i	0.959310	+	0.282354i	$0.0911153\pi$
$467$			12.2034i			0.564708i	−0.959310	+	0.282354i	$0.908885\pi$
$468$		0			0
$469$			51.6531i			2.38512i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−19.7736	−	11.4163i	−0.909193	−	0.524923i
$474$		0			0
$475$	6.50974	+	11.2752i	0.298687	+	0.517341i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	17.7237	+	30.6983i	0.809815	+	1.40264i	0.912992	+	0.407978i	$0.133766\pi$
$479$	17.7237	+	30.6983i	0.809815	+	1.40264i	−0.103177	+	0.994663i	$0.532901\pi$
$480$		0			0
$481$	−9.91348	+	17.1706i	−0.452016	+	0.782914i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−7.43610			−0.337656
$486$		0			0
$487$		−	18.7853i		−	0.851242i	−0.904902	−	0.425621i	$-0.860056\pi$
$487$		−	18.7853i		−	0.851242i	0.904902	−	0.425621i	$-0.139944\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−1.06466	−	0.614681i	−0.0480474	−	0.0277402i	0.475784	−	0.879562i	$-0.342164\pi$
$491$	−1.06466	−	0.614681i	−0.0480474	−	0.0277402i	−0.523831	+	0.851822i	$0.675498\pi$
$492$		0			0
$493$	−7.99227	+	4.61434i	−0.359954	+	0.207819i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−40.4246	+	23.3392i	−1.81329	+	1.04690i
$498$		0			0
$499$	−1.73661	+	3.00789i	−0.0777413	+	0.134652i	−0.902275	−	0.431161i	$-0.858104\pi$
$499$	−1.73661	+	3.00789i	−0.0777413	+	0.134652i	0.824534	+	0.565813i	$0.191437\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−31.5709			−1.40768			−0.703838	−	0.710360i	$-0.748532\pi$
$503$	−31.5709			−1.40768			−0.703838	+	0.710360i	$0.748532\pi$
$504$		0			0
$505$	−16.9545			−0.754465
$506$		0			0
$507$		0			0
$508$		0			0
$509$	15.7887	−	27.3468i	0.699820	−	1.21212i	−0.268709	−	0.963222i	$-0.586597\pi$
$509$	15.7887	−	27.3468i	0.699820	−	1.21212i	0.968529	−	0.248902i	$-0.0800698\pi$
$510$		0			0
$511$	−35.6888	+	20.6049i	−1.57878	+	0.911508i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	2.01836	−	1.16530i	0.0889394	−	0.0513492i
$516$		0			0
$517$	−36.5319	−	21.0917i	−1.60667	−	0.927613i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	8.99232i		−	0.393961i	−0.980407	−	0.196980i	$-0.936886\pi$
$521$		−	8.99232i		−	0.393961i	0.980407	−	0.196980i	$-0.0631136\pi$
$522$		0			0
$523$	−2.78978			−0.121988			−0.0609942	−	0.998138i	$-0.519427\pi$
$523$	−2.78978			−0.121988			−0.0609942	+	0.998138i	$0.519427\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	0.341916	−	0.592215i	0.0148941	−	0.0257973i
$528$		0			0
$529$	11.2183	+	19.4306i	0.487751	+	0.844809i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	19.8521	+	34.3849i	0.859892	+	1.48938i
$534$		0			0
$535$	0.232920	+	0.134477i	0.0100700	+	0.00581393i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	65.2242i		−	2.80940i
$540$		0			0
$541$			38.5456i			1.65721i	0.559836	+	0.828603i	$0.310864\pi$
$541$			38.5456i			1.65721i	−0.559836	+	0.828603i	$0.689136\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	8.01641	+	4.62828i	0.343385	+	0.198254i
$546$		0			0
$547$	10.8890	+	18.8604i	0.465581	+	0.806411i	0.999228	−	0.0392972i	$-0.0125119\pi$
$547$	10.8890	+	18.8604i	0.465581	+	0.806411i	−0.533646	+	0.845708i	$0.679179\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	12.3005	+	21.3050i	0.524017	+	0.907624i
$552$		0			0
$553$	−13.5305	+	23.4356i	−0.575377	+	0.996582i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−22.9871			−0.973995			−0.486998	−	0.873403i	$-0.661908\pi$
$557$	−22.9871			−0.973995			−0.486998	+	0.873403i	$0.661908\pi$
$558$		0			0
$559$			21.6261i			0.914686i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	35.7358	+	20.6321i	1.50608	+	0.869537i	0.999975	+	0.00706728i	$0.00224960\pi$
$563$	35.7358	+	20.6321i	1.50608	+	0.869537i	0.506108	+	0.862470i	$0.331084\pi$
$564$		0			0
$565$	−3.88793	+	2.24470i	−0.163567	+	0.0944352i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−24.6609	+	14.2380i	−1.03384	+	0.596888i	−0.918082	−	0.396390i	$-0.870263\pi$
$569$	−24.6609	+	14.2380i	−1.03384	+	0.596888i	−0.115757	+	0.993278i	$0.536930\pi$
$570$		0			0
$571$	−17.7603	+	30.7617i	−0.743245	+	1.28734i	0.207765	+	0.978179i	$0.433381\pi$
$571$	−17.7603	+	30.7617i	−0.743245	+	1.28734i	−0.951010	+	0.309159i	$0.899952\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	3.07627			0.128289
$576$		0			0
$577$	−30.5676			−1.27255			−0.636273	−	0.771464i	$-0.719525\pi$
$577$	−30.5676			−1.27255			−0.636273	+	0.771464i	$0.719525\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−29.4457	+	51.0014i	−1.22161	+	2.11589i
$582$		0			0
$583$	3.26504	−	1.88507i	0.135224	−	0.0780716i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	16.1608	−	9.33044i	0.667028	−	0.385109i	−0.127922	−	0.991784i	$-0.540831\pi$
$587$	16.1608	−	9.33044i	0.667028	−	0.385109i	0.794949	+	0.606676i	$0.207497\pi$
$588$		0			0
$589$	−1.57867	−	0.911445i	−0.0650479	−	0.0375554i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	0.676531i		−	0.0277818i	−0.999904	−	0.0138909i	$-0.995578\pi$
$593$		−	0.676531i		−	0.0277818i	0.999904	−	0.0138909i	$-0.00442175\pi$
$594$		0			0
$595$	−5.33074			−0.218539
$596$		0			0
$597$		0			0
$598$		0			0
$599$	18.8685	−	32.6813i	0.770947	−	1.33532i	−0.166097	−	0.986109i	$-0.553116\pi$
$599$	18.8685	−	32.6813i	0.770947	−	1.33532i	0.937044	−	0.349211i	$-0.113550\pi$
$600$		0			0
$601$	11.5562	+	20.0159i	0.471386	+	0.816464i	0.999464	−	0.0327315i	$-0.0104206\pi$
$601$	11.5562	+	20.0159i	0.471386	+	0.816464i	−0.528078	+	0.849196i	$0.677087\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−3.53839	−	6.12868i	−0.143856	−	0.249166i
$606$		0			0
$607$	−2.04207	−	1.17899i	−0.0828851	−	0.0478537i	0.457985	−	0.888960i	$-0.348571\pi$
$607$	−2.04207	−	1.17899i	−0.0828851	−	0.0478537i	−0.540870	+	0.841106i	$0.681905\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			39.9543i			1.61638i
$612$		0			0
$613$			23.7808i			0.960497i	0.877132	+	0.480249i	$0.159454\pi$
$613$			23.7808i			0.960497i	−0.877132	+	0.480249i	$0.840546\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−3.17492	−	1.83304i	−0.127817	−	0.0737954i	0.434728	−	0.900562i	$-0.356845\pi$
$617$	−3.17492	−	1.83304i	−0.127817	−	0.0737954i	−0.562545	+	0.826766i	$0.690178\pi$
$618$		0			0
$619$	−8.79359	−	15.2309i	−0.353444	−	0.612183i	0.633406	−	0.773820i	$-0.281656\pi$
$619$	−8.79359	−	15.2309i	−0.353444	−	0.612183i	−0.986850	+	0.161636i	$0.948323\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−23.8892	−	41.3773i	−0.957101	−	1.65775i
$624$		0			0
$625$	−6.14269	+	10.6394i	−0.245708	+	0.425578i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	5.80777			0.231571
$630$		0			0
$631$		−	21.9248i		−	0.872811i	−0.899750	−	0.436405i	$-0.856251\pi$
$631$		−	21.9248i		−	0.872811i	0.899750	−	0.436405i	$-0.143749\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−11.3530	−	6.55468i	−0.450532	−	0.260115i
$636$		0			0
$637$	−53.5009	+	30.8888i	−2.11978	+	1.22386i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	18.0562	−	10.4248i	0.713178	−	0.411754i	−0.0990584	−	0.995082i	$-0.531583\pi$
$641$	18.0562	−	10.4248i	0.713178	−	0.411754i	0.812237	+	0.583328i	$0.198250\pi$
$642$		0			0
$643$	−12.8540	+	22.2638i	−0.506914	+	0.878000i	0.493054	+	0.869998i	$0.335880\pi$
$643$	−12.8540	+	22.2638i	−0.506914	+	0.878000i	−0.999968	+	0.00800162i	$0.997453\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	26.0025			1.02226			0.511131	−	0.859503i	$-0.329227\pi$
$647$	26.0025			1.02226			0.511131	+	0.859503i	$0.329227\pi$
$648$		0			0
$649$	−7.53617			−0.295821
$650$		0			0
$651$		0			0
$652$		0			0
$653$	10.7670	−	18.6490i	0.421346	−	0.729792i	−0.574726	−	0.818346i	$-0.694891\pi$
$653$	10.7670	−	18.6490i	0.421346	−	0.729792i	0.996071	+	0.0885542i	$0.0282246\pi$
$654$		0			0
$655$	3.89392	−	2.24816i	0.152148	−	0.0878428i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−11.5947	+	6.69419i	−0.451665	+	0.260769i	−0.708533	−	0.705678i	$-0.750643\pi$
$659$	−11.5947	+	6.69419i	−0.451665	+	0.260769i	0.256868	+	0.966446i	$0.417309\pi$
$660$		0			0
$661$	14.4644	+	8.35103i	0.562600	+	0.324818i	0.754189	−	0.656658i	$-0.228030\pi$
$661$	14.4644	+	8.35103i	0.562600	+	0.324818i	−0.191588	+	0.981475i	$0.561364\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			14.2102i			0.551046i
$666$		0			0
$667$	5.81275			0.225071
$668$		0			0
$669$		0			0
$670$		0			0
$671$	21.3085	−	36.9073i	0.822604	−	1.42479i
$672$		0			0
$673$	−5.47889	−	9.48971i	−0.211196	−	0.365802i	0.740893	−	0.671623i	$-0.234402\pi$
$673$	−5.47889	−	9.48971i	−0.211196	−	0.365802i	−0.952089	+	0.305821i	$0.901069\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	11.2742	+	19.5275i	0.433302	+	0.750501i	0.997155	−	0.0753738i	$-0.0240150\pi$
$677$	11.2742	+	19.5275i	0.433302	+	0.750501i	−0.563853	+	0.825875i	$0.690682\pi$
$678$		0			0
$679$	31.9393	+	18.4402i	1.22572	+	0.707669i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	36.4666i		−	1.39536i	−0.716412	−	0.697678i	$-0.754217\pi$
$683$		−	36.4666i		−	1.39536i	0.716412	−	0.697678i	$-0.245783\pi$
$684$		0			0
$685$		−	14.7714i		−	0.564387i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−3.09250	−	1.78546i	−0.117815	−	0.0680205i
$690$		0			0
$691$	2.50258	+	4.33459i	0.0952025	+	0.164896i	0.909693	−	0.415281i	$-0.136317\pi$
$691$	2.50258	+	4.33459i	0.0952025	+	0.164896i	−0.814491	+	0.580177i	$0.802983\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−3.74634	−	6.48884i	−0.142107	−	0.246136i
$696$		0			0
$697$	5.81515	−	10.0721i	0.220265	−	0.381509i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	11.4972			0.434243			0.217122	−	0.976145i	$-0.430333\pi$
$701$	11.4972			0.434243			0.217122	+	0.976145i	$0.430333\pi$
$702$		0			0
$703$		−	15.4818i		−	0.583907i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	72.8225	+	42.0441i	2.73877	+	1.58123i
$708$		0			0
$709$	29.7367	−	17.1685i	1.11679	−	0.644776i	0.176207	−	0.984353i	$-0.443617\pi$
$709$	29.7367	−	17.1685i	1.11679	−	0.644776i	0.940578	+	0.339577i	$0.110284\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−0.373011	+	0.215358i	−0.0139694	+	0.00806523i
$714$		0			0
$715$	−8.29857	+	14.3735i	−0.310349	+	0.537540i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−40.5660			−1.51286			−0.756429	−	0.654076i	$-0.773058\pi$
$719$	−40.5660			−1.51286			−0.756429	+	0.654076i	$0.773058\pi$
$720$		0			0
$721$	−11.5589			−0.430477
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−15.8672	+	27.4828i	−0.589293	+	1.02069i
$726$		0			0
$727$	33.7590	−	19.4908i	1.25205	−	0.722873i	0.280535	−	0.959844i	$-0.409488\pi$
$727$	33.7590	−	19.4908i	1.25205	−	0.722873i	0.971517	+	0.236971i	$0.0761547\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	5.48608	−	3.16739i	0.202910	−	0.117150i
$732$		0			0
$733$	31.5208	+	18.1986i	1.16425	+	0.672179i	0.952319	−	0.305105i	$-0.0986917\pi$
$733$	31.5208	+	18.1986i	1.16425	+	0.672179i	0.211930	+	0.977285i	$0.432025\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	47.1082i		−	1.73525i
$738$		0			0
$739$	19.8657			0.730770			0.365385	−	0.930857i	$-0.380937\pi$
$739$	19.8657			0.730770			0.365385	+	0.930857i	$0.380937\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−6.13689	+	10.6294i	−0.225141	+	0.389955i	−0.956362	−	0.292186i	$-0.905618\pi$
$743$	−6.13689	+	10.6294i	−0.225141	+	0.389955i	0.731221	+	0.682141i	$0.238951\pi$
$744$		0			0
$745$	−7.60060	−	13.1646i	−0.278464	−	0.482314i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−0.666955	−	1.15520i	−0.0243700	−	0.0422101i
$750$		0			0
$751$	−4.29326	−	2.47872i	−0.156663	−	0.0904496i	0.419619	−	0.907700i	$-0.362164\pi$
$751$	−4.29326	−	2.47872i	−0.156663	−	0.0904496i	−0.576282	+	0.817251i	$0.695497\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	1.73169i		−	0.0630228i
$756$		0			0
$757$			0.135856i			0.00493778i	0.999997	+	0.00246889i	$0.000785872\pi$
$757$			0.135856i			0.00493778i	−0.999997	+	0.00246889i	$0.999214\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−1.85960	−	1.07364i	−0.0674106	−	0.0389195i	0.465916	−	0.884829i	$-0.345725\pi$
$761$	−1.85960	−	1.07364i	−0.0674106	−	0.0389195i	−0.533326	+	0.845910i	$0.679058\pi$
$762$		0			0
$763$	−22.9546	−	39.7585i	−0.831011	−	1.43935i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	3.56897	+	6.18164i	0.128868	+	0.223206i
$768$		0			0
$769$	6.85585	−	11.8747i	0.247228	−	0.428212i	−0.715527	−	0.698585i	$-0.753814\pi$
$769$	6.85585	−	11.8747i	0.247228	−	0.428212i	0.962756	+	0.270373i	$0.0871469\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	21.2269			0.763477			0.381738	−	0.924270i	$-0.375326\pi$
$773$	21.2269			0.763477			0.381738	+	0.924270i	$0.375326\pi$
$774$		0			0
$775$		−	2.35147i		−	0.0844673i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−26.8493	−	15.5015i	−0.961976	−	0.555397i
$780$		0			0
$781$	36.8677	−	21.2856i	1.31923	−	0.761657i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−16.8506	+	9.72872i	−0.601425	+	0.347233i
$786$		0			0
$787$	10.6033	−	18.3654i	0.377966	−	0.654657i	−0.612800	−	0.790238i	$-0.709957\pi$
$787$	10.6033	−	18.3654i	0.377966	−	0.654657i	0.990766	+	0.135581i	$0.0432902\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	22.2658			0.791680
$792$		0			0
$793$	−40.3649			−1.43340
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−17.6435	+	30.5595i	−0.624967	+	1.08247i	0.363581	+	0.931563i	$0.381554\pi$
$797$	−17.6435	+	30.5595i	−0.624967	+	1.08247i	−0.988547	+	0.150911i	$0.951779\pi$
$798$		0			0
$799$	10.1356	−	5.85178i	0.358571	−	0.207021i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	32.5485	−	18.7919i	1.14861	−	0.663151i
$804$		0			0
$805$	2.90777	+	1.67880i	0.102485	+	0.0591700i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	26.8216i		−	0.942996i	−0.881867	−	0.471498i	$-0.843714\pi$
$809$		−	26.8216i		−	0.942996i	0.881867	−	0.471498i	$-0.156286\pi$
$810$		0			0
$811$	−7.68860			−0.269983			−0.134992	−	0.990847i	$-0.543101\pi$
$811$	−7.68860			−0.269983			−0.134992	+	0.990847i	$0.543101\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	3.60576	−	6.24535i	0.126304	−	0.218765i
$816$		0			0
$817$	−8.44332	−	14.6243i	−0.295394	−	0.511638i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	12.3690	+	21.4237i	0.431680	+	0.747692i	0.997018	−	0.0771675i	$-0.0245876\pi$
$821$	12.3690	+	21.4237i	0.431680	+	0.747692i	−0.565338	+	0.824859i	$0.691254\pi$
$822$		0			0
$823$	27.2907	+	15.7563i	0.951294	+	0.549230i	0.893483	−	0.449098i	$-0.148255\pi$
$823$	27.2907	+	15.7563i	0.951294	+	0.549230i	0.0578112	+	0.998328i	$0.481588\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			15.9087i			0.553200i	0.960985	+	0.276600i	$0.0892077\pi$
$827$			15.9087i			0.553200i	−0.960985	+	0.276600i	$0.910792\pi$
$828$		0			0
$829$		−	40.2678i		−	1.39856i	−0.714848	−	0.699280i	$-0.753504\pi$
$829$		−	40.2678i		−	1.39856i	0.714848	−	0.699280i	$-0.246496\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	15.6717	+	9.04803i	0.542991	+	0.313496i
$834$		0			0
$835$	2.66537	+	4.61655i	0.0922389	+	0.159762i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	2.66890	+	4.62267i	0.0921408	+	0.159592i	0.908412	−	0.418077i	$-0.137296\pi$
$839$	2.66890	+	4.62267i	0.0921408	+	0.159592i	−0.816271	+	0.577669i	$0.803962\pi$
$840$		0			0
$841$	−15.4818	+	26.8153i	−0.533856	+	0.924666i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	3.37443			0.116084
$846$		0			0
$847$			35.0983i			1.20599i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−3.16798	−	1.82903i	−0.108597	−	0.0626985i
$852$		0			0
$853$	28.1392	−	16.2462i	0.963468	−	0.556258i	0.0662290	−	0.997804i	$-0.478903\pi$
$853$	28.1392	−	16.2462i	0.963468	−	0.556258i	0.897239	+	0.441546i	$0.145570\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−21.1136	+	12.1899i	−0.721227	+	0.416400i	−0.815204	−	0.579174i	$-0.803375\pi$
$857$	−21.1136	+	12.1899i	−0.721227	+	0.416400i	0.0939774	+	0.995574i	$0.470042\pi$
$858$		0			0
$859$	−6.04812	+	10.4757i	−0.206359	+	0.357425i	−0.950565	−	0.310526i	$-0.899495\pi$
$859$	−6.04812	+	10.4757i	−0.206359	+	0.357425i	0.744206	+	0.667951i	$0.232828\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	39.4505			1.34291			0.671456	−	0.741044i	$-0.265669\pi$
$863$	39.4505			1.34291			0.671456	+	0.741044i	$0.265669\pi$
$864$		0			0
$865$	11.9966			0.407897
$866$		0			0
$867$		0			0
$868$		0			0
$869$	12.3400	−	21.3735i	0.418605	−	0.725046i
$870$		0			0
$871$	−38.6410	+	22.3094i	−1.30930	+	0.755925i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−35.2431	+	20.3476i	−1.19144	+	0.687876i
$876$		0			0
$877$	5.96635	+	3.44467i	0.201469	+	0.116318i	0.597341	−	0.801988i	$-0.296224\pi$
$877$	5.96635	+	3.44467i	0.201469	+	0.116318i	−0.395871	+	0.918306i	$0.629557\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			44.3192i			1.49315i	0.665300	+	0.746576i	$0.268304\pi$
$881$			44.3192i			1.49315i	−0.665300	+	0.746576i	$0.731696\pi$
$882$		0			0
$883$	−48.0579			−1.61728			−0.808639	−	0.588305i	$-0.799795\pi$
$883$	−48.0579			−1.61728			−0.808639	+	0.588305i	$0.799795\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−13.5146	+	23.4079i	−0.453775	+	0.785961i	−0.998617	−	0.0525774i	$-0.983256\pi$
$887$	−13.5146	+	23.4079i	−0.453775	+	0.785961i	0.544842	+	0.838539i	$0.316590\pi$
$888$		0			0
$889$	32.5088	+	56.3070i	1.09031	+	1.88848i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−15.5991	−	27.0184i	−0.522004	−	0.904137i
$894$		0			0
$895$	16.8831	+	9.74746i	0.564340	+	0.325822i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	4.44322i		−	0.148190i
$900$		0			0
$901$			1.04600i			0.0348474i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−8.19196	−	4.72963i	−0.272310	−	0.157218i
$906$		0			0
$907$	2.71996	+	4.71111i	0.0903149	+	0.156430i	0.907644	−	0.419742i	$-0.137879\pi$
$907$	2.71996	+	4.71111i	0.0903149	+	0.156430i	−0.817329	+	0.576172i	$0.804546\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−19.7758	−	34.2528i	−0.655203	−	1.13484i	−0.981843	−	0.189696i	$-0.939250\pi$
$911$	−19.7758	−	34.2528i	−0.655203	−	1.13484i	0.326640	−	0.945149i	$-0.394083\pi$
$912$		0			0
$913$	26.8547	−	46.5137i	0.888762	−	1.53938i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−22.3001			−0.736414
$918$		0			0
$919$		−	22.7057i		−	0.748991i	−0.927229	−	0.374495i	$-0.877816\pi$
$919$		−	22.7057i		−	0.748991i	0.927229	−	0.374495i	$-0.122184\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−34.9194	−	20.1608i	−1.14939	−	0.663599i
$924$		0			0
$925$	17.2954	−	9.98551i	0.568669	−	0.328321i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−36.0663	+	20.8229i	−1.18330	+	0.683176i	−0.956775	−	0.290830i	$-0.906068\pi$
$929$	−36.0663	+	20.8229i	−1.18330	+	0.683176i	−0.226521	+	0.974006i	$0.572735\pi$
$930$		0			0
$931$	24.1194	−	41.7760i	0.790480	−	1.36915i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	4.86169			0.158994
$936$		0			0
$937$	54.4421			1.77855			0.889274	−	0.457376i	$-0.151211\pi$
$937$	54.4421			1.77855			0.889274	+	0.457376i	$0.151211\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	21.8217	−	37.7963i	0.711368	−	1.23213i	−0.252976	−	0.967473i	$-0.581409\pi$
$941$	21.8217	−	37.7963i	0.711368	−	1.23213i	0.964344	−	0.264653i	$-0.0852573\pi$
$942$		0			0
$943$	−6.34401	+	3.66272i	−0.206589	+	0.119274i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	11.5858	−	6.68905i	0.376487	−	0.217365i	−0.299802	−	0.954002i	$-0.596920\pi$
$947$	11.5858	−	6.68905i	0.376487	−	0.217365i	0.676289	+	0.736637i	$0.263587\pi$
$948$		0			0
$949$	−30.8285	−	17.7989i	−1.00074	−	0.577775i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			36.5239i			1.18313i	0.806259	+	0.591563i	$0.201489\pi$
$953$			36.5239i			1.18313i	−0.806259	+	0.591563i	$0.798511\pi$
$954$		0			0
$955$	21.8422			0.706796
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−36.6304	+	63.4458i	−1.18286	+	2.04877i
$960$		0			0
$961$	−15.3354	−	26.5617i	−0.494690	−	0.856828i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−0.474833	−	0.822435i	−0.0152854	−	0.0264751i
$966$		0			0
$967$	−19.4015	−	11.2015i	−0.623911	−	0.360215i	0.154479	−	0.987996i	$-0.450630\pi$
$967$	−19.4015	−	11.2015i	−0.623911	−	0.360215i	−0.778390	+	0.627781i	$0.783963\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	33.7841i		−	1.08418i	−0.840320	−	0.542091i	$-0.817633\pi$
$971$		−	33.7841i		−	1.08418i	0.840320	−	0.542091i	$-0.182367\pi$
$972$		0			0
$973$			37.1609i			1.19132i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−12.6710	−	7.31560i	−0.405381	−	0.234047i	0.283422	−	0.958995i	$-0.408530\pi$
$977$	−12.6710	−	7.31560i	−0.405381	−	0.234047i	−0.688803	+	0.724948i	$0.741864\pi$
$978$		0			0
$979$	21.7872	+	37.7365i	0.696322	+	1.20606i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	18.8379	+	32.6283i	0.600837	+	1.04068i	0.992695	+	0.120654i	$0.0384992\pi$
$983$	18.8379	+	32.6283i	0.600837	+	1.04068i	−0.391858	+	0.920026i	$0.628168\pi$
$984$		0			0
$985$	0.246742	−	0.427370i	0.00786185	−	0.0136171i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−3.99001			−0.126875
$990$		0			0
$991$			25.7352i			0.817504i	0.912645	+	0.408752i	$0.134036\pi$
$991$			25.7352i			0.817504i	−0.912645	+	0.408752i	$0.865964\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	0.518666	+	0.299452i	0.0164428	+	0.00949326i
$996$		0			0
$997$	−33.4015	+	19.2844i	−1.05784	+	0.610743i	−0.924833	−	0.380373i	$-0.875796\pi$
$997$	−33.4015	+	19.2844i	−1.05784	+	0.610743i	−0.133004	+	0.991115i	$0.542462\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2592.2.p.g.2159.14		48
3.2	odd	2	inner	2592.2.p.g.2159.12		48
4.3	odd	2		648.2.l.g.539.23		48
8.3	odd	2	inner	2592.2.p.g.2159.11		48
8.5	even	2		648.2.l.g.539.18		48
9.2	odd	6	inner	2592.2.p.g.431.11		48
9.4	even	3		2592.2.f.c.1295.12		24
9.5	odd	6		2592.2.f.c.1295.14		24
9.7	even	3	inner	2592.2.p.g.431.13		48
12.11	even	2		648.2.l.g.539.2		48
24.5	odd	2		648.2.l.g.539.7		48
24.11	even	2	inner	2592.2.p.g.2159.13		48
36.7	odd	6		648.2.l.g.107.7		48
36.11	even	6		648.2.l.g.107.18		48
36.23	even	6		648.2.f.c.323.15	yes	24
36.31	odd	6		648.2.f.c.323.10	yes	24
72.5	odd	6		648.2.f.c.323.9	✓	24
72.11	even	6	inner	2592.2.p.g.431.14		48
72.13	even	6		648.2.f.c.323.16	yes	24
72.29	odd	6		648.2.l.g.107.23		48
72.43	odd	6	inner	2592.2.p.g.431.12		48
72.59	even	6		2592.2.f.c.1295.11		24
72.61	even	6		648.2.l.g.107.2		48
72.67	odd	6		2592.2.f.c.1295.13		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.2.f.c.323.9	✓	24	72.5	odd	6
648.2.f.c.323.10	yes	24	36.31	odd	6
648.2.f.c.323.15	yes	24	36.23	even	6
648.2.f.c.323.16	yes	24	72.13	even	6
648.2.l.g.107.2		48	72.61	even	6
648.2.l.g.107.7		48	36.7	odd	6
648.2.l.g.107.18		48	36.11	even	6
648.2.l.g.107.23		48	72.29	odd	6
648.2.l.g.539.2		48	12.11	even	2
648.2.l.g.539.7		48	24.5	odd	2
648.2.l.g.539.18		48	8.5	even	2
648.2.l.g.539.23		48	4.3	odd	2
2592.2.f.c.1295.11		24	72.59	even	6
2592.2.f.c.1295.12		24	9.4	even	3
2592.2.f.c.1295.13		24	72.67	odd	6
2592.2.f.c.1295.14		24	9.5	odd	6
2592.2.p.g.431.11		48	9.2	odd	6	inner
2592.2.p.g.431.12		48	72.43	odd	6	inner
2592.2.p.g.431.13		48	9.7	even	3	inner
2592.2.p.g.431.14		48	72.11	even	6	inner
2592.2.p.g.2159.11		48	8.3	odd	2	inner
2592.2.p.g.2159.12		48	3.2	odd	2	inner
2592.2.p.g.2159.13		48	24.11	even	2	inner
2592.2.p.g.2159.14		48	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2592 = 2^{5} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2592.p (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.474833 - 0.822435i) q^{5} +(-4.07898 + 2.35500i) q^{7} +O(q^{10})\)	\(q+(0.474833 - 0.822435i) q^{5} +(-4.07898 + 2.35500i) q^{7} +(3.72007 - 2.14778i) q^{11} +(-3.52349 - 2.03429i) q^{13} +1.19178i q^{17} +3.17693 q^{19} +(0.375325 - 0.650083i) q^{23} +(2.04907 + 3.54909i) q^{25} +(3.87181 + 6.70617i) q^{29} +(-0.496917 - 0.286895i) q^{31} +4.47293i q^{35} -4.87320i q^{37} +(-8.45135 - 4.87939i) q^{41} +(-2.65770 - 4.60327i) q^{43} +(-4.91012 - 8.50458i) q^{47} +(7.59204 - 13.1498i) q^{49} +0.877682 q^{53} -4.07935i q^{55} +(-1.51936 - 0.877204i) q^{59} +(8.59196 - 4.96057i) q^{61} +(-3.34614 + 1.93189i) q^{65} +(5.48335 - 9.49744i) q^{67} +9.91048 q^{71} +8.74944 q^{73} +(-10.1160 + 17.5215i) q^{77} +(4.97571 - 2.87273i) q^{79} +(10.8283 - 6.25173i) q^{83} +(0.980161 + 0.565896i) q^{85} +10.1440i q^{89} +19.1630 q^{91} +(1.50851 - 2.61282i) q^{95} +(-3.91511 - 6.78118i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q+O(q^{10}) \) 48 * q	\( 48 q - 24 q^{25} + 24 q^{49} + 48 q^{67}+O(q^{100}) \) 48 * q - 24 * q^25 + 24 * q^49 + 48 * q^67

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