LMFDB - Embedded newform 4608.2.k.bl.1153.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4608,2,Mod(1153,4608)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4608, base_ring=CyclotomicField(4))

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

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

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

chi := DirichletCharacter("4608.1153");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4608 = 2^{9} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4608.k (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:	$36.7950652514$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 24x^{14} + 192x^{12} + 672x^{10} + 1092x^{8} + 880x^{6} + 352x^{4} + 64x^{2} + 4 $ x^16 + 24x^14 + 192x^12 + 672x^10 + 1092x^8 + 880x^6 + 352x^4 + 64*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{23} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1153.6
Root			$2.13875i$ of defining polynomial
Character	$\chi$	$=$	4608.1153
Dual form			4608.2.k.bl.3457.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.25928 - 1.25928i) q^{5} +3.29066i q^{7} +O(q^{10})$	$q+(1.25928 - 1.25928i) q^{5} +3.29066i q^{7} +(1.08239 - 1.08239i) q^{11} +(-0.414214 - 0.414214i) q^{13} +6.08034 q^{17} +(1.36303 + 1.36303i) q^{19} +5.22625i q^{23} +1.82843i q^{25} +(-4.82106 - 4.82106i) q^{29} +6.01673 q^{31} +(4.14386 + 4.14386i) q^{35} +(-2.41421 + 2.41421i) q^{37} -1.04322i q^{41} +(-7.94435 + 7.94435i) q^{43} -0.896683 q^{47} -3.82843 q^{49} +(4.82106 - 4.82106i) q^{53} -2.72607i q^{55} +(7.39104 - 7.39104i) q^{59} +(3.24264 + 3.24264i) q^{61} -1.04322 q^{65} +(-6.58132 - 6.58132i) q^{67} +14.7821i q^{71} -6.48528i q^{73} +(3.56178 + 3.56178i) q^{77} -3.29066 q^{79} +(9.37011 + 9.37011i) q^{83} +(7.65685 - 7.65685i) q^{85} +7.12356i q^{89} +(1.36303 - 1.36303i) q^{91} +3.43289 q^{95} +7.31371 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q + 16 q^{13} - 16 q^{37} - 16 q^{49} - 16 q^{61} + 32 q^{85} - 64 q^{97}+O(q^{100}) $ 16 * q + 16 * q^13 - 16 * q^37 - 16 * q^49 - 16 * q^61 + 32 * q^85 - 64 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4608\mathbb{Z}\right)^\times$.

$n$	$2053$	$3583$	$4097$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$1$	$1$

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
$5$	1.25928	−	1.25928i	0.563167	−	0.563167i	−0.367039	−	0.930206i	$-0.619628\pi$
$5$	1.25928	−	1.25928i	0.563167	−	0.563167i	0.930206	+	0.367039i	$0.119628\pi$
$6$		0			0
$7$			3.29066i			1.24375i	0.783116	+	0.621876i	$0.213629\pi$
$7$			3.29066i			1.24375i	−0.783116	+	0.621876i	$0.786371\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.08239	−	1.08239i	0.326354	−	0.326354i	−0.524845	−	0.851198i	$-0.675877\pi$
$11$	1.08239	−	1.08239i	0.326354	−	0.326354i	0.851198	+	0.524845i	$0.175877\pi$
$12$		0			0
$13$	−0.414214	−	0.414214i	−0.114882	−	0.114882i	0.647329	−	0.762211i	$-0.275886\pi$
$13$	−0.414214	−	0.414214i	−0.114882	−	0.114882i	−0.762211	+	0.647329i	$0.775886\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	6.08034			1.47470			0.737350	−	0.675511i	$-0.236077\pi$
$17$	6.08034			1.47470			0.737350	+	0.675511i	$0.236077\pi$
$18$		0			0
$19$	1.36303	+	1.36303i	0.312702	+	0.312702i	0.845955	−	0.533254i	$-0.179031\pi$
$19$	1.36303	+	1.36303i	0.312702	+	0.312702i	−0.533254	+	0.845955i	$0.679031\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			5.22625i			1.08975i	0.838518	+	0.544874i	$0.183423\pi$
$23$			5.22625i			1.08975i	−0.838518	+	0.544874i	$0.816577\pi$
$24$		0			0
$25$			1.82843i			0.365685i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−4.82106	−	4.82106i	−0.895249	−	0.895249i	0.0997625	−	0.995011i	$-0.468192\pi$
$29$	−4.82106	−	4.82106i	−0.895249	−	0.895249i	−0.995011	+	0.0997625i	$0.968192\pi$
$30$		0			0
$31$	6.01673			1.08064			0.540318	−	0.841461i	$-0.318304\pi$
$31$	6.01673			1.08064			0.540318	+	0.841461i	$0.318304\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	4.14386	+	4.14386i	0.700440	+	0.700440i
$36$		0			0
$37$	−2.41421	+	2.41421i	−0.396894	+	0.396894i	−0.877136	−	0.480242i	$-0.840549\pi$
$37$	−2.41421	+	2.41421i	−0.396894	+	0.396894i	0.480242	+	0.877136i	$0.340549\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	1.04322i		−	0.162924i	−0.996676	−	0.0814619i	$-0.974041\pi$
$41$		−	1.04322i		−	0.162924i	0.996676	−	0.0814619i	$-0.0259589\pi$
$42$		0			0
$43$	−7.94435	+	7.94435i	−1.21150	+	1.21150i	−0.240969	+	0.970533i	$0.577465\pi$
$43$	−7.94435	+	7.94435i	−1.21150	+	1.21150i	−0.970533	+	0.240969i	$0.922535\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−0.896683			−0.130795			−0.0653973	−	0.997859i	$-0.520831\pi$
$47$	−0.896683			−0.130795			−0.0653973	+	0.997859i	$0.520831\pi$
$48$		0			0
$49$	−3.82843			−0.546918
$50$		0			0
$51$		0			0
$52$		0			0
$53$	4.82106	−	4.82106i	0.662224	−	0.662224i	−0.293680	−	0.955904i	$-0.594880\pi$
$53$	4.82106	−	4.82106i	0.662224	−	0.662224i	0.955904	+	0.293680i	$0.0948800\pi$
$54$		0			0
$55$		−	2.72607i		−	0.367583i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	7.39104	−	7.39104i	0.962231	−	0.962231i	−0.0370813	−	0.999312i	$-0.511806\pi$
$59$	7.39104	−	7.39104i	0.962231	−	0.962231i	0.999312	+	0.0370813i	$0.0118060\pi$
$60$		0			0
$61$	3.24264	+	3.24264i	0.415178	+	0.415178i	0.883538	−	0.468360i	$-0.155155\pi$
$61$	3.24264	+	3.24264i	0.415178	+	0.415178i	−0.468360	+	0.883538i	$0.655155\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−1.04322			−0.129396
$66$		0			0
$67$	−6.58132	−	6.58132i	−0.804036	−	0.804036i	0.179688	−	0.983724i	$-0.442491\pi$
$67$	−6.58132	−	6.58132i	−0.804036	−	0.804036i	−0.983724	+	0.179688i	$0.942491\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			14.7821i			1.75431i	0.480208	+	0.877155i	$0.340561\pi$
$71$			14.7821i			1.75431i	−0.480208	+	0.877155i	$0.659439\pi$
$72$		0			0
$73$		−	6.48528i		−	0.759045i	−0.925183	−	0.379522i	$-0.876088\pi$
$73$		−	6.48528i		−	0.759045i	0.925183	−	0.379522i	$-0.123912\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	3.56178	+	3.56178i	0.405903	+	0.405903i
$78$		0			0
$79$	−3.29066			−0.370228			−0.185114	−	0.982717i	$-0.559265\pi$
$79$	−3.29066			−0.370228			−0.185114	+	0.982717i	$0.559265\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	9.37011	+	9.37011i	1.02850	+	1.02850i	0.999582	+	0.0289217i	$0.00920734\pi$
$83$	9.37011	+	9.37011i	1.02850	+	1.02850i	0.0289217	+	0.999582i	$0.490793\pi$
$84$		0			0
$85$	7.65685	−	7.65685i	0.830502	−	0.830502i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			7.12356i			0.755096i	0.925990	+	0.377548i	$0.123233\pi$
$89$			7.12356i			0.755096i	−0.925990	+	0.377548i	$0.876767\pi$
$90$		0			0
$91$	1.36303	−	1.36303i	0.142885	−	0.142885i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	3.43289			0.352207
$96$		0			0
$97$	7.31371			0.742595			0.371297	−	0.928514i	$-0.378913\pi$
$97$	7.31371			0.742595			0.371297	+	0.928514i	$0.378913\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−10.9014	+	10.9014i	−1.08473	+	1.08473i	−0.0886692	+	0.996061i	$0.528261\pi$
$101$	−10.9014	+	10.9014i	−1.08473	+	1.08473i	−0.996061	+	0.0886692i	$0.971739\pi$
$102$		0			0
$103$			6.01673i			0.592846i	0.955057	+	0.296423i	$0.0957938\pi$
$103$			6.01673i			0.592846i	−0.955057	+	0.296423i	$0.904206\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	9.55582	−	9.55582i	0.923796	−	0.923796i	−0.0734992	−	0.997295i	$-0.523417\pi$
$107$	9.55582	−	9.55582i	0.923796	−	0.923796i	0.997295	+	0.0734992i	$0.0234166\pi$
$108$		0			0
$109$	−9.24264	−	9.24264i	−0.885284	−	0.885284i	0.108781	−	0.994066i	$-0.465305\pi$
$109$	−9.24264	−	9.24264i	−0.885284	−	0.885284i	−0.994066	+	0.108781i	$0.965305\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−10.0742			−0.947705			−0.473852	−	0.880604i	$-0.657137\pi$
$113$	−10.0742			−0.947705			−0.473852	+	0.880604i	$0.657137\pi$
$114$		0			0
$115$	6.58132	+	6.58132i	0.613711	+	0.613711i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			20.0083i			1.83416i
$120$		0			0
$121$			8.65685i			0.786987i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	8.59890	+	8.59890i	0.769109	+	0.769109i
$126$		0			0
$127$	0.564588			0.0500990			0.0250495	−	0.999686i	$-0.492026\pi$
$127$	0.564588			0.0500990			0.0250495	+	0.999686i	$0.492026\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	8.28772	+	8.28772i	0.724101	+	0.724101i	0.969438	−	0.245337i	$-0.0788985\pi$
$131$	8.28772	+	8.28772i	0.724101	+	0.724101i	−0.245337	+	0.969438i	$0.578898\pi$
$132$		0			0
$133$	−4.48528	+	4.48528i	−0.388923	+	0.388923i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	16.1546i		−	1.38018i	−0.723724	−	0.690090i	$-0.757571\pi$
$137$		−	16.1546i		−	1.38018i	0.723724	−	0.690090i	$-0.242429\pi$
$138$		0			0
$139$	13.1626	−	13.1626i	1.11644	−	1.11644i	0.124180	−	0.992260i	$-0.460370\pi$
$139$	13.1626	−	13.1626i	1.11644	−	1.11644i	0.992260	−	0.124180i	$-0.0396300\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−0.896683			−0.0749844
$144$		0			0
$145$	−12.1421			−1.00835
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−3.77784	+	3.77784i	−0.309493	+	0.309493i	−0.844713	−	0.535220i	$-0.820229\pi$
$149$	−3.77784	+	3.77784i	−0.309493	+	0.309493i	0.535220	+	0.844713i	$0.320229\pi$
$150$		0			0
$151$		−	0.564588i		−	0.0459455i	−0.999736	−	0.0229727i	$-0.992687\pi$
$151$		−	0.564588i		−	0.0459455i	0.999736	−	0.0229727i	$-0.00731309\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	7.57675	−	7.57675i	0.608579	−	0.608579i
$156$		0			0
$157$	10.0711	+	10.0711i	0.803759	+	0.803759i	0.983681	−	0.179922i	$-0.0575846\pi$
$157$	10.0711	+	10.0711i	0.803759	+	0.803759i	−0.179922	+	0.983681i	$0.557585\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−17.1978			−1.35538
$162$		0			0
$163$	11.7996	+	11.7996i	0.924216	+	0.924216i	0.997324	−	0.0731084i	$-0.0232919\pi$
$163$	11.7996	+	11.7996i	0.924216	+	0.924216i	−0.0731084	+	0.997324i	$0.523292\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			21.8017i			1.68707i	0.537078	+	0.843533i	$0.319528\pi$
$167$			21.8017i			1.68707i	−0.537078	+	0.843533i	$0.680472\pi$
$168$		0			0
$169$		−	12.6569i		−	0.973604i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	2.30250	+	2.30250i	0.175056	+	0.175056i	0.789197	−	0.614141i	$-0.210497\pi$
$173$	2.30250	+	2.30250i	0.175056	+	0.175056i	−0.614141	+	0.789197i	$0.710497\pi$
$174$		0			0
$175$	−6.01673			−0.454822
$176$		0			0
$177$		0			0
$178$		0			0
$179$	11.3492	+	11.3492i	0.848278	+	0.848278i	0.989918	−	0.141640i	$-0.0452376\pi$
$179$	11.3492	+	11.3492i	0.848278	+	0.848278i	−0.141640	+	0.989918i	$0.545238\pi$
$180$		0			0
$181$	0.899495	−	0.899495i	0.0668589	−	0.0668589i	−0.672887	−	0.739746i	$-0.734946\pi$
$181$	0.899495	−	0.899495i	0.0668589	−	0.0668589i	0.739746	+	0.672887i	$0.234946\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			6.08034i			0.447036i
$186$		0			0
$187$	6.58132	−	6.58132i	0.481273	−	0.481273i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−11.3492			−0.821198			−0.410599	−	0.911816i	$-0.634680\pi$
$191$	−11.3492			−0.821198			−0.410599	+	0.911816i	$0.634680\pi$
$192$		0			0
$193$	12.4853			0.898710			0.449355	−	0.893353i	$-0.351654\pi$
$193$	12.4853			0.898710			0.449355	+	0.893353i	$0.351654\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	2.30250	−	2.30250i	0.164046	−	0.164046i	−0.620310	−	0.784357i	$-0.712993\pi$
$197$	2.30250	−	2.30250i	0.164046	−	0.164046i	0.784357	+	0.620310i	$0.212993\pi$
$198$		0			0
$199$		−	19.1794i		−	1.35959i	−0.733403	−	0.679794i	$-0.762069\pi$
$199$		−	19.1794i		−	1.35959i	0.733403	−	0.679794i	$-0.237931\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	15.8645	−	15.8645i	1.11347	−	1.11347i
$204$		0			0
$205$	−1.31371	−	1.31371i	−0.0917534	−	0.0917534i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	2.95068			0.204103
$210$		0			0
$211$	−18.6148	−	18.6148i	−1.28149	−	1.28149i	−0.939818	−	0.341676i	$-0.889006\pi$
$211$	−18.6148	−	18.6148i	−1.28149	−	1.28149i	−0.341676	−	0.939818i	$-0.610994\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			20.0083i			1.36456i
$216$		0			0
$217$			19.7990i			1.34404i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−2.51856	−	2.51856i	−0.169417	−	0.169417i
$222$		0			0
$223$	−21.9054			−1.46690			−0.733448	−	0.679746i	$-0.762090\pi$
$223$	−21.9054			−1.46690			−0.733448	+	0.679746i	$0.762090\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	11.5349	+	11.5349i	0.765598	+	0.765598i	0.977328	−	0.211730i	$-0.0679098\pi$
$227$	11.5349	+	11.5349i	0.765598	+	0.765598i	−0.211730	+	0.977328i	$0.567910\pi$
$228$		0			0
$229$	−17.7279	+	17.7279i	−1.17149	+	1.17149i	−0.189640	+	0.981854i	$0.560732\pi$
$229$	−17.7279	+	17.7279i	−1.17149	+	1.17149i	−0.981854	+	0.189640i	$0.939268\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			27.2720i			1.78665i	0.449410	+	0.893326i	$0.351634\pi$
$233$			27.2720i			1.78665i	−0.449410	+	0.893326i	$0.648366\pi$
$234$		0			0
$235$	−1.12918	+	1.12918i	−0.0736593	+	0.0736593i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	13.8854			0.898171			0.449086	−	0.893489i	$-0.351750\pi$
$239$	13.8854			0.898171			0.449086	+	0.893489i	$0.351750\pi$
$240$		0			0
$241$	−10.1421			−0.653312			−0.326656	−	0.945143i	$-0.605922\pi$
$241$	−10.1421			−0.653312			−0.326656	+	0.945143i	$0.605922\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−4.82106	+	4.82106i	−0.308006	+	0.308006i
$246$		0			0
$247$		−	1.12918i		−	0.0718477i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	5.41196	−	5.41196i	0.341600	−	0.341600i	−0.515369	−	0.856969i	$-0.672345\pi$
$251$	5.41196	−	5.41196i	0.341600	−	0.341600i	0.856969	+	0.515369i	$0.172345\pi$
$252$		0			0
$253$	5.65685	+	5.65685i	0.355643	+	0.355643i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	17.1978			1.07277			0.536385	−	0.843974i	$-0.319790\pi$
$257$	17.1978			1.07277			0.536385	+	0.843974i	$0.319790\pi$
$258$		0			0
$259$	−7.94435	−	7.94435i	−0.493638	−	0.493638i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			19.1116i			1.17847i	0.807960	+	0.589237i	$0.200572\pi$
$263$			19.1116i			1.17847i	−0.807960	+	0.589237i	$0.799428\pi$
$264$		0			0
$265$		−	12.1421i		−	0.745885i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−3.34572	−	3.34572i	−0.203992	−	0.203992i	0.597716	−	0.801708i	$-0.296075\pi$
$269$	−3.34572	−	3.34572i	−0.203992	−	0.203992i	−0.801708	+	0.597716i	$0.796075\pi$
$270$		0			0
$271$	16.4533			0.999466			0.499733	−	0.866179i	$-0.333431\pi$
$271$	16.4533			0.999466			0.499733	+	0.866179i	$0.333431\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.97908	+	1.97908i	0.119343	+	0.119343i
$276$		0			0
$277$	20.0711	−	20.0711i	1.20595	−	1.20595i	0.233627	−	0.972326i	$-0.424940\pi$
$277$	20.0711	−	20.0711i	1.20595	−	1.20595i	0.972326	−	0.233627i	$-0.0750596\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	31.4449i		−	1.87585i	−0.346842	−	0.937924i	$-0.612746\pi$
$281$		−	31.4449i		−	1.87585i	0.346842	−	0.937924i	$-0.387254\pi$
$282$		0			0
$283$	−12.0335	+	12.0335i	−0.715315	+	0.715315i	−0.967642	−	0.252327i	$-0.918804\pi$
$283$	−12.0335	+	12.0335i	−0.715315	+	0.715315i	0.252327	+	0.967642i	$0.418804\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	3.43289			0.202637
$288$		0			0
$289$	19.9706			1.17474
$290$		0			0
$291$		0			0
$292$		0			0
$293$	19.5003	−	19.5003i	1.13922	−	1.13922i	0.150630	−	0.988590i	$-0.451870\pi$
$293$	19.5003	−	19.5003i	1.13922	−	1.13922i	0.988590	−	0.150630i	$-0.0481302\pi$
$294$		0			0
$295$		−	18.6148i		−	1.08379i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	2.16478	−	2.16478i	0.125193	−	0.125193i
$300$		0			0
$301$	−26.1421	−	26.1421i	−1.50681	−	1.50681i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	8.16679			0.467629
$306$		0			0
$307$	6.58132	+	6.58132i	0.375615	+	0.375615i	0.869518	−	0.493902i	$-0.164430\pi$
$307$	6.58132	+	6.58132i	0.375615	+	0.375615i	−0.493902	+	0.869518i	$0.664430\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			10.4525i			0.592707i	0.955078	+	0.296354i	$0.0957707\pi$
$311$			10.4525i			0.592707i	−0.955078	+	0.296354i	$0.904229\pi$
$312$		0			0
$313$			10.1421i			0.573267i	0.958040	+	0.286634i	$0.0925363\pi$
$313$			10.1421i			0.573267i	−0.958040	+	0.286634i	$0.907464\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−18.4571	−	18.4571i	−1.03665	−	1.03665i	−0.999302	−	0.0373510i	$-0.988108\pi$
$317$	−18.4571	−	18.4571i	−1.03665	−	1.03665i	−0.0373510	−	0.999302i	$-0.511892\pi$
$318$		0			0
$319$	−10.4366			−0.584335
$320$		0			0
$321$		0			0
$322$		0			0
$323$	8.28772	+	8.28772i	0.461141	+	0.461141i
$324$		0			0
$325$	0.757359	−	0.757359i	0.0420107	−	0.0420107i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	2.95068i		−	0.162676i
$330$		0			0
$331$	13.1626	−	13.1626i	0.723484	−	0.723484i	−0.245829	−	0.969313i	$-0.579060\pi$
$331$	13.1626	−	13.1626i	0.723484	−	0.723484i	0.969313	+	0.245829i	$0.0790603\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−16.5754			−0.905613
$336$		0			0
$337$	−14.4853			−0.789064			−0.394532	−	0.918882i	$-0.629093\pi$
$337$	−14.4853			−0.789064			−0.394532	+	0.918882i	$0.629093\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	6.51246	−	6.51246i	0.352669	−	0.352669i
$342$		0			0
$343$			10.4366i			0.563521i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	17.6578	−	17.6578i	0.947922	−	0.947922i	−0.0507875	−	0.998709i	$-0.516173\pi$
$347$	17.6578	−	17.6578i	0.947922	−	0.947922i	0.998709	+	0.0507875i	$0.0161731\pi$
$348$		0			0
$349$	15.3848	+	15.3848i	0.823528	+	0.823528i	0.986612	−	0.163084i	$-0.0521442\pi$
$349$	15.3848	+	15.3848i	0.823528	+	0.823528i	−0.163084	+	0.986612i	$0.552144\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	21.3707			1.13745			0.568724	−	0.822529i	$-0.307437\pi$
$353$	21.3707			1.13745			0.568724	+	0.822529i	$0.307437\pi$
$354$		0			0
$355$	18.6148	+	18.6148i	0.987969	+	0.987969i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			4.32957i			0.228506i	0.993452	+	0.114253i	$0.0364474\pi$
$359$			4.32957i			0.228506i	−0.993452	+	0.114253i	$0.963553\pi$
$360$		0			0
$361$		−	15.2843i		−	0.804435i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−8.16679	−	8.16679i	−0.427469	−	0.427469i
$366$		0			0
$367$	18.0502			0.942212			0.471106	−	0.882077i	$-0.343855\pi$
$367$	18.0502			0.942212			0.471106	+	0.882077i	$0.343855\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	15.8645	+	15.8645i	0.823642	+	0.823642i
$372$		0			0
$373$	3.72792	−	3.72792i	0.193024	−	0.193024i	−0.603977	−	0.797002i	$-0.706418\pi$
$373$	3.72792	−	3.72792i	0.193024	−	0.193024i	0.797002	+	0.603977i	$0.206418\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			3.99390i			0.205696i
$378$		0			0
$379$	−5.21828	+	5.21828i	−0.268045	+	0.268045i	−0.828312	−	0.560267i	$-0.810699\pi$
$379$	−5.21828	+	5.21828i	−0.268045	+	0.268045i	0.560267	+	0.828312i	$0.310699\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	20.0083			1.02238			0.511189	−	0.859468i	$-0.329205\pi$
$383$	20.0083			1.02238			0.511189	+	0.859468i	$0.329205\pi$
$384$		0			0
$385$	8.97056			0.457182
$386$		0			0
$387$		0			0
$388$		0			0
$389$	0.216058	−	0.216058i	0.0109546	−	0.0109546i	−0.701608	−	0.712563i	$-0.747534\pi$
$389$	0.216058	−	0.216058i	0.0109546	−	0.0109546i	0.712563	+	0.701608i	$0.247534\pi$
$390$		0			0
$391$			31.7774i			1.60705i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−4.14386	+	4.14386i	−0.208500	+	0.208500i
$396$		0			0
$397$	11.5858	+	11.5858i	0.581474	+	0.581474i	0.935308	−	0.353834i	$-0.115122\pi$
$397$	11.5858	+	11.5858i	0.581474	+	0.581474i	−0.353834	+	0.935308i	$0.615122\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	14.0681			0.702529			0.351265	−	0.936276i	$-0.385752\pi$
$401$	14.0681			0.702529			0.351265	+	0.936276i	$0.385752\pi$
$402$		0			0
$403$	−2.49221	−	2.49221i	−0.124146	−	0.124146i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			5.22625i			0.259056i
$408$		0			0
$409$			1.65685i			0.0819262i	0.999161	+	0.0409631i	$0.0130426\pi$
$409$			1.65685i			0.0819262i	−0.999161	+	0.0409631i	$0.986957\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	24.3214	+	24.3214i	1.19678	+	1.19678i
$414$		0			0
$415$	23.5992			1.15844
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−9.74153	−	9.74153i	−0.475905	−	0.475905i	0.427914	−	0.903819i	$-0.359248\pi$
$419$	−9.74153	−	9.74153i	−0.475905	−	0.475905i	−0.903819	+	0.427914i	$0.859248\pi$
$420$		0			0
$421$	−20.0711	+	20.0711i	−0.978204	+	0.978204i	−0.999767	−	0.0215635i	$-0.993136\pi$
$421$	−20.0711	+	20.0711i	−0.978204	+	0.978204i	0.0215635	+	0.999767i	$0.493136\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			11.1175i			0.539276i
$426$		0			0
$427$	−10.6704	+	10.6704i	−0.516378	+	0.516378i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−15.6788			−0.755219			−0.377610	−	0.925965i	$-0.623254\pi$
$431$	−15.6788			−0.755219			−0.377610	+	0.925965i	$0.623254\pi$
$432$		0			0
$433$	−28.9706			−1.39224			−0.696118	−	0.717927i	$-0.745091\pi$
$433$	−28.9706			−1.39224			−0.696118	+	0.717927i	$0.745091\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−7.12356	+	7.12356i	−0.340766	+	0.340766i
$438$		0			0
$439$			32.3420i			1.54360i	0.635866	+	0.771799i	$0.280643\pi$
$439$			32.3420i			1.54360i	−0.635866	+	0.771799i	$0.719357\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−9.37011	+	9.37011i	−0.445187	+	0.445187i	−0.893751	−	0.448564i	$-0.851936\pi$
$443$	−9.37011	+	9.37011i	−0.445187	+	0.445187i	0.448564	+	0.893751i	$0.351936\pi$
$444$		0			0
$445$	8.97056	+	8.97056i	0.425245	+	0.425245i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	11.1175			0.524666			0.262333	−	0.964977i	$-0.415508\pi$
$449$	11.1175			0.524666			0.262333	+	0.964977i	$0.415508\pi$
$450$		0			0
$451$	−1.12918	−	1.12918i	−0.0531708	−	0.0531708i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	3.43289i		−	0.160936i
$456$		0			0
$457$		−	15.3137i		−	0.716345i	−0.933655	−	0.358173i	$-0.883400\pi$
$457$		−	15.3137i		−	0.716345i	0.933655	−	0.358173i	$-0.116600\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−17.4139	−	17.4139i	−0.811044	−	0.811044i	0.173746	−	0.984790i	$-0.444413\pi$
$461$	−17.4139	−	17.4139i	−0.811044	−	0.811044i	−0.984790	+	0.173746i	$0.944413\pi$
$462$		0			0
$463$	−26.8898			−1.24968			−0.624839	−	0.780754i	$-0.714835\pi$
$463$	−26.8898			−1.24968			−0.624839	+	0.780754i	$0.714835\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−7.20533	−	7.20533i	−0.333423	−	0.333423i	0.520462	−	0.853885i	$-0.325760\pi$
$467$	−7.20533	−	7.20533i	−0.333423	−	0.333423i	−0.853885	+	0.520462i	$0.825760\pi$
$468$		0			0
$469$	21.6569	−	21.6569i	1.00002	−	1.00002i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			17.1978i			0.790756i
$474$		0			0
$475$	−2.49221	+	2.49221i	−0.114350	+	0.114350i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−12.9887			−0.593469			−0.296735	−	0.954960i	$-0.595898\pi$
$479$	−12.9887			−0.593469			−0.296735	+	0.954960i	$0.595898\pi$
$480$		0			0
$481$	2.00000			0.0911922
$482$		0			0
$483$		0			0
$484$		0			0
$485$	9.21001	−	9.21001i	0.418205	−	0.418205i
$486$		0			0
$487$			4.41983i			0.200282i	0.994973	+	0.100141i	$0.0319293\pi$
$487$			4.41983i			0.200282i	−0.994973	+	0.100141i	$0.968071\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	2.16478	−	2.16478i	0.0976954	−	0.0976954i	−0.656570	−	0.754265i	$-0.727993\pi$
$491$	2.16478	−	2.16478i	0.0976954	−	0.0976954i	0.754265	+	0.656570i	$0.227993\pi$
$492$		0			0
$493$	−29.3137	−	29.3137i	−1.32022	−	1.32022i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−48.6427			−2.18193
$498$		0			0
$499$	−6.58132	−	6.58132i	−0.294620	−	0.294620i	0.544282	−	0.838902i	$-0.316802\pi$
$499$	−6.58132	−	6.58132i	−0.294620	−	0.294620i	−0.838902	+	0.544282i	$0.816802\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	4.32957i		−	0.193046i	−0.995331	−	0.0965230i	$-0.969228\pi$
$503$		−	4.32957i		−	0.193046i	0.995331	−	0.0965230i	$-0.0307721\pi$
$504$		0			0
$505$			27.4558i			1.22177i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	9.85818	+	9.85818i	0.436956	+	0.436956i	0.890986	−	0.454030i	$-0.150014\pi$
$509$	9.85818	+	9.85818i	0.436956	+	0.436956i	−0.454030	+	0.890986i	$0.650014\pi$
$510$		0			0
$511$	21.3408			0.944063
$512$		0			0
$513$		0			0
$514$		0			0
$515$	7.57675	+	7.57675i	0.333871	+	0.333871i
$516$		0			0
$517$	−0.970563	+	0.970563i	−0.0426853	+	0.0426853i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	21.1917i		−	0.928425i	−0.885724	−	0.464213i	$-0.846337\pi$
$521$		−	21.1917i		−	0.928425i	0.885724	−	0.464213i	$-0.153663\pi$
$522$		0			0
$523$	−14.5257	+	14.5257i	−0.635163	+	0.635163i	−0.949358	−	0.314195i	$-0.898265\pi$
$523$	−14.5257	+	14.5257i	−0.635163	+	0.635163i	0.314195	+	0.949358i	$0.398265\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	36.5838			1.59361
$528$		0			0
$529$	−4.31371			−0.187553
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−0.432117	+	0.432117i	−0.0187170	+	0.0187170i
$534$		0			0
$535$		−	24.0669i		−	1.04050i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−4.14386	+	4.14386i	−0.178489	+	0.178489i
$540$		0			0
$541$	4.75736	+	4.75736i	0.204535	+	0.204535i	0.801940	−	0.597405i	$-0.203801\pi$
$541$	4.75736	+	4.75736i	0.204535	+	0.204535i	−0.597405	+	0.801940i	$0.703801\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−23.2781			−0.997126
$546$		0			0
$547$	−23.8331	−	23.8331i	−1.01903	−	1.01903i	−0.999815	−	0.0192122i	$-0.993884\pi$
$547$	−23.8331	−	23.8331i	−1.01903	−	1.01903i	−0.0192122	−	0.999815i	$-0.506116\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	13.1426i		−	0.559892i
$552$		0			0
$553$		−	10.8284i		−	0.460472i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−1.25928	−	1.25928i	−0.0533574	−	0.0533574i	0.679925	−	0.733282i	$-0.262012\pi$
$557$	−1.25928	−	1.25928i	−0.0533574	−	0.0533574i	−0.733282	+	0.679925i	$0.762012\pi$
$558$		0			0
$559$	6.58132			0.278360
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−28.4818	−	28.4818i	−1.20036	−	1.20036i	−0.974056	−	0.226307i	$-0.927335\pi$
$563$	−28.4818	−	28.4818i	−1.20036	−	1.20036i	−0.226307	−	0.974056i	$-0.572665\pi$
$564$		0			0
$565$	−12.6863	+	12.6863i	−0.533716	+	0.533716i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	13.2039i		−	0.553537i	−0.960937	−	0.276768i	$-0.910737\pi$
$569$		−	13.2039i		−	0.553537i	0.960937	−	0.276768i	$-0.0892634\pi$
$570$		0			0
$571$	−12.0335	+	12.0335i	−0.503584	+	0.503584i	−0.912550	−	0.408965i	$-0.865890\pi$
$571$	−12.0335	+	12.0335i	−0.503584	+	0.503584i	0.408965	+	0.912550i	$0.365890\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−9.55582			−0.398505
$576$		0			0
$577$	37.1127			1.54502			0.772511	−	0.635001i	$-0.219001\pi$
$577$	37.1127			1.54502			0.772511	+	0.635001i	$0.219001\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−30.8338	+	30.8338i	−1.27920	+	1.27920i
$582$		0			0
$583$		−	10.4366i		−	0.432238i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−10.4525	+	10.4525i	−0.431421	+	0.431421i	−0.889112	−	0.457691i	$-0.848677\pi$
$587$	−10.4525	+	10.4525i	−0.431421	+	0.431421i	0.457691	+	0.889112i	$0.348677\pi$
$588$		0			0
$589$	8.20101	+	8.20101i	0.337917	+	0.337917i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−10.0742			−0.413699			−0.206850	−	0.978373i	$-0.566321\pi$
$593$	−10.0742			−0.413699			−0.206850	+	0.978373i	$0.566321\pi$
$594$		0			0
$595$	25.1961	+	25.1961i	1.03294	+	1.03294i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	13.8854i		−	0.567342i	−0.958922	−	0.283671i	$-0.908448\pi$
$599$		−	13.8854i		−	0.567342i	0.958922	−	0.283671i	$-0.0915523\pi$
$600$		0			0
$601$			20.8284i			0.849609i	0.905285	+	0.424805i	$0.139657\pi$
$601$			20.8284i			0.849609i	−0.905285	+	0.424805i	$0.860343\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	10.9014	+	10.9014i	0.443205	+	0.443205i
$606$		0			0
$607$	−1.69376			−0.0687477			−0.0343739	−	0.999409i	$-0.510944\pi$
$607$	−1.69376			−0.0687477			−0.0343739	+	0.999409i	$0.510944\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	0.371418	+	0.371418i	0.0150260	+	0.0150260i
$612$		0			0
$613$	−14.0711	+	14.0711i	−0.568325	+	0.568325i	−0.931659	−	0.363334i	$-0.881638\pi$
$613$	−14.0711	+	14.0711i	−0.568325	+	0.568325i	0.363334	+	0.931659i	$0.381638\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$		0			0
$619$	−18.6148	+	18.6148i	−0.748191	+	0.748191i	−0.974139	−	0.225948i	$-0.927452\pi$
$619$	−18.6148	+	18.6148i	−0.748191	+	0.748191i	0.225948	+	0.974139i	$0.427452\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−23.4412			−0.939152
$624$		0			0
$625$	12.5147			0.500589
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−14.6792	+	14.6792i	−0.585300	+	0.585300i
$630$		0			0
$631$		−	41.6494i		−	1.65804i	−0.559222	−	0.829018i	$-0.688900\pi$
$631$		−	41.6494i		−	1.65804i	0.559222	−	0.829018i	$-0.311100\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	0.710974	−	0.710974i	0.0282141	−	0.0282141i
$636$		0			0
$637$	1.58579	+	1.58579i	0.0628311	+	0.0628311i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−6.08034			−0.240159			−0.120080	−	0.992764i	$-0.538315\pi$
$641$	−6.08034			−0.240159			−0.120080	+	0.992764i	$0.538315\pi$
$642$		0			0
$643$	−4.08910	−	4.08910i	−0.161259	−	0.161259i	0.621866	−	0.783124i	$-0.286375\pi$
$643$	−4.08910	−	4.08910i	−0.161259	−	0.161259i	−0.783124	+	0.621866i	$0.786375\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	32.9970i		−	1.29725i	−0.761109	−	0.648624i	$-0.775345\pi$
$647$		−	32.9970i		−	1.29725i	0.761109	−	0.648624i	$-0.224655\pi$
$648$		0			0
$649$		−	16.0000i		−	0.628055i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	27.0560	+	27.0560i	1.05878	+	1.05878i	0.998161	+	0.0606218i	$0.0193084\pi$
$653$	27.0560	+	27.0560i	1.05878	+	1.05878i	0.0606218	+	0.998161i	$0.480692\pi$
$654$		0			0
$655$	20.8731			0.815580
$656$		0			0
$657$		0			0
$658$		0			0
$659$	2.16478	+	2.16478i	0.0843280	+	0.0843280i	0.748013	−	0.663685i	$-0.231008\pi$
$659$	2.16478	+	2.16478i	0.0843280	+	0.0843280i	−0.663685	+	0.748013i	$0.731008\pi$
$660$		0			0
$661$	28.5563	−	28.5563i	1.11071	−	1.11071i	0.117659	−	0.993054i	$-0.462461\pi$
$661$	28.5563	−	28.5563i	1.11071	−	1.11071i	0.993054	−	0.117659i	$-0.0375390\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			11.2965i			0.438058i
$666$		0			0
$667$	25.1961	−	25.1961i	0.975596	−	0.975596i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	7.01962			0.270989
$672$		0			0
$673$	6.34315			0.244510			0.122255	−	0.992499i	$-0.460987\pi$
$673$	6.34315			0.244510			0.122255	+	0.992499i	$0.460987\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−10.4693	+	10.4693i	−0.402367	+	0.402367i	−0.879067	−	0.476699i	$-0.841833\pi$
$677$	−10.4693	+	10.4693i	−0.402367	+	0.402367i	0.476699	+	0.879067i	$0.341833\pi$
$678$		0			0
$679$			24.0669i			0.923603i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−11.9063	+	11.9063i	−0.455582	+	0.455582i	−0.897202	−	0.441620i	$-0.854404\pi$
$683$	−11.9063	+	11.9063i	−0.455582	+	0.455582i	0.441620	+	0.897202i	$0.354404\pi$
$684$		0			0
$685$	−20.3431	−	20.3431i	−0.777272	−	0.777272i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−3.99390			−0.152155
$690$		0			0
$691$	1.36303	+	1.36303i	0.0518523	+	0.0518523i	0.732557	−	0.680705i	$-0.238327\pi$
$691$	1.36303	+	1.36303i	0.0518523	+	0.0518523i	−0.680705	+	0.732557i	$0.738327\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	33.1509i		−	1.25748i
$696$		0			0
$697$		−	6.34315i		−	0.240264i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−13.8521	−	13.8521i	−0.523186	−	0.523186i	0.395346	−	0.918532i	$-0.370625\pi$
$701$	−13.8521	−	13.8521i	−0.523186	−	0.523186i	−0.918532	+	0.395346i	$0.870625\pi$
$702$		0			0
$703$	−6.58132			−0.248219
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−35.8728	−	35.8728i	−1.34914	−	1.34914i
$708$		0			0
$709$	4.75736	−	4.75736i	0.178666	−	0.178666i	−0.612108	−	0.790774i	$-0.709678\pi$
$709$	4.75736	−	4.75736i	0.178666	−	0.178666i	0.790774	+	0.612108i	$0.209678\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			31.4449i			1.17762i
$714$		0			0
$715$	−1.12918	+	1.12918i	−0.0422288	+	0.0422288i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	6.12293			0.228347			0.114173	−	0.993461i	$-0.463578\pi$
$719$	6.12293			0.228347			0.114173	+	0.993461i	$0.463578\pi$
$720$		0			0
$721$	−19.7990			−0.737353
$722$		0			0
$723$		0			0
$724$		0			0
$725$	8.81496	−	8.81496i	0.327379	−	0.327379i
$726$		0			0
$727$			11.4689i			0.425357i	0.977122	+	0.212678i	$0.0682187\pi$
$727$			11.4689i			0.425357i	−0.977122	+	0.212678i	$0.931781\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−48.3044	+	48.3044i	−1.78660	+	1.78660i
$732$		0			0
$733$	20.8995	+	20.8995i	0.771940	+	0.771940i	0.978446	−	0.206505i	$-0.0662090\pi$
$733$	20.8995	+	20.8995i	0.771940	+	0.771940i	−0.206505	+	0.978446i	$0.566209\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−14.2471			−0.524800
$738$		0			0
$739$	−12.0335	−	12.0335i	−0.442658	−	0.442658i	0.450247	−	0.892904i	$-0.351336\pi$
$739$	−12.0335	−	12.0335i	−0.442658	−	0.442658i	−0.892904	+	0.450247i	$0.851336\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	13.8854i		−	0.509406i	−0.967019	−	0.254703i	$-0.918022\pi$
$743$		−	13.8854i		−	0.509406i	0.967019	−	0.254703i	$-0.0819776\pi$
$744$		0			0
$745$			9.51472i			0.348592i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	31.4449	+	31.4449i	1.14897	+	1.14897i
$750$		0			0
$751$	−52.5537			−1.91771			−0.958855	−	0.283896i	$-0.908373\pi$
$751$	−52.5537			−1.91771			−0.958855	+	0.283896i	$0.908373\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−0.710974	−	0.710974i	−0.0258750	−	0.0258750i
$756$		0			0
$757$	−35.3848	+	35.3848i	−1.28608	+	1.28608i	−0.348934	+	0.937147i	$0.613456\pi$
$757$	−35.3848	+	35.3848i	−1.28608	+	1.28608i	−0.937147	+	0.348934i	$0.886544\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	21.1917i		−	0.768199i	−0.923292	−	0.384099i	$-0.874512\pi$
$761$		−	21.1917i		−	0.768199i	0.923292	−	0.384099i	$-0.125488\pi$
$762$		0			0
$763$	30.4144	−	30.4144i	1.10107	−	1.10107i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−6.12293			−0.221086
$768$		0			0
$769$	−25.1716			−0.907710			−0.453855	−	0.891076i	$-0.649952\pi$
$769$	−25.1716			−0.907710			−0.453855	+	0.891076i	$0.649952\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	23.4942	−	23.4942i	0.845028	−	0.845028i	−0.144480	−	0.989508i	$-0.546151\pi$
$773$	23.4942	−	23.4942i	0.845028	−	0.845028i	0.989508	+	0.144480i	$0.0461509\pi$
$774$		0			0
$775$			11.0011i			0.395173i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	1.42195	−	1.42195i	0.0509466	−	0.0509466i
$780$		0			0
$781$	16.0000	+	16.0000i	0.572525	+	0.572525i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	25.3646			0.905301
$786$		0			0
$787$	5.21828	+	5.21828i	0.186012	+	0.186012i	0.793969	−	0.607958i	$-0.208011\pi$
$787$	5.21828	+	5.21828i	0.186012	+	0.186012i	−0.607958	+	0.793969i	$0.708011\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	33.1509i		−	1.17871i
$792$		0			0
$793$		−	2.68629i		−	0.0953930i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−20.5435	−	20.5435i	−0.727689	−	0.727689i	0.242470	−	0.970159i	$-0.422042\pi$
$797$	−20.5435	−	20.5435i	−0.727689	−	0.727689i	−0.970159	+	0.242470i	$0.922042\pi$
$798$		0			0
$799$	−5.45214			−0.192883
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−7.01962	−	7.01962i	−0.247717	−	0.247717i
$804$		0			0
$805$	−21.6569	+	21.6569i	−0.763304	+	0.763304i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	27.4510i		−	0.965127i	−0.875861	−	0.482563i	$-0.839706\pi$
$809$		−	27.4510i		−	0.965127i	0.875861	−	0.482563i	$-0.160294\pi$
$810$		0			0
$811$	34.2696	−	34.2696i	1.20337	−	1.20337i	0.230233	−	0.973135i	$-0.426051\pi$
$811$	34.2696	−	34.2696i	1.20337	−	1.20337i	0.973135	−	0.230233i	$-0.0739489\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	29.7180			1.04098
$816$		0			0
$817$	−21.6569			−0.757677
$818$		0			0
$819$		0			0
$820$		0			0
$821$	15.5064	−	15.5064i	0.541177	−	0.541177i	−0.382697	−	0.923874i	$-0.625005\pi$
$821$	15.5064	−	15.5064i	0.541177	−	0.541177i	0.923874	+	0.382697i	$0.125005\pi$
$822$		0			0
$823$			3.29066i			0.114705i	0.998354	+	0.0573526i	$0.0182659\pi$
$823$			3.29066i			0.114705i	−0.998354	+	0.0573526i	$0.981734\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−1.26810	+	1.26810i	−0.0440962	+	0.0440962i	−0.728811	−	0.684715i	$-0.759927\pi$
$827$	−1.26810	+	1.26810i	−0.0440962	+	0.0440962i	0.684715	+	0.728811i	$0.259927\pi$
$828$		0			0
$829$	−6.75736	−	6.75736i	−0.234693	−	0.234693i	0.579955	−	0.814648i	$-0.303070\pi$
$829$	−6.75736	−	6.75736i	−0.234693	−	0.234693i	−0.814648	+	0.579955i	$0.803070\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−23.2781			−0.806540
$834$		0			0
$835$	27.4544	+	27.4544i	0.950100	+	0.950100i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	31.3575i		−	1.08258i	−0.840835	−	0.541291i	$-0.817936\pi$
$839$		−	31.3575i		−	1.08258i	0.840835	−	0.541291i	$-0.182064\pi$
$840$		0			0
$841$			17.4853i			0.602941i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−15.9385	−	15.9385i	−0.548302	−	0.548302i
$846$		0			0
$847$	−28.4867			−0.978816
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−12.6173	−	12.6173i	−0.432515	−	0.432515i
$852$		0			0
$853$	35.7279	−	35.7279i	1.22330	−	1.22330i	0.256849	−	0.966451i	$-0.417316\pi$
$853$	35.7279	−	35.7279i	1.22330	−	1.22330i	0.966451	−	0.256849i	$-0.0826844\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			3.99390i			0.136429i	0.997671	+	0.0682145i	$0.0217302\pi$
$857$			3.99390i			0.136429i	−0.997671	+	0.0682145i	$0.978270\pi$
$858$		0			0
$859$	11.7996	−	11.7996i	0.402597	−	0.402597i	−0.476550	−	0.879147i	$-0.658113\pi$
$859$	11.7996	−	11.7996i	0.402597	−	0.402597i	0.879147	+	0.476550i	$0.158113\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−7.76245			−0.264237			−0.132119	−	0.991234i	$-0.542178\pi$
$863$	−7.76245			−0.264237			−0.132119	+	0.991234i	$0.542178\pi$
$864$		0			0
$865$	5.79899			0.197172
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−3.56178	+	3.56178i	−0.120825	+	0.120825i
$870$		0			0
$871$			5.45214i			0.184739i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−28.2960	+	28.2960i	−0.956581	+	0.956581i
$876$		0			0
$877$	−29.5858	−	29.5858i	−0.999041	−	0.999041i	0.000958519	−	1.00000i	$-0.499695\pi$
$877$	−29.5858	−	29.5858i	−0.999041	−	0.999041i	−1.00000		0.000958519i	$0.999695\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	51.5934			1.73823			0.869113	−	0.494613i	$-0.164690\pi$
$881$	51.5934			1.73823			0.869113	+	0.494613i	$0.164690\pi$
$882$		0			0
$883$	−10.6704	−	10.6704i	−0.359088	−	0.359088i	0.504389	−	0.863477i	$-0.331718\pi$
$883$	−10.6704	−	10.6704i	−0.359088	−	0.359088i	−0.863477	+	0.504389i	$0.831718\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	42.5529i		−	1.42878i	−0.699745	−	0.714392i	$-0.746703\pi$
$887$		−	42.5529i		−	1.42878i	0.699745	−	0.714392i	$-0.253297\pi$
$888$		0			0
$889$			1.85786i			0.0623108i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−1.22221	−	1.22221i	−0.0408997	−	0.0408997i
$894$		0			0
$895$	28.5836			0.955445
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−29.0070	−	29.0070i	−0.967438	−	0.967438i
$900$		0			0
$901$	29.3137	−	29.3137i	0.976581	−	0.976581i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	2.26543i		−	0.0753055i
$906$		0			0
$907$	−17.2517	+	17.2517i	−0.572834	+	0.572834i	−0.932919	−	0.360085i	$-0.882748\pi$
$907$	−17.2517	+	17.2517i	−0.572834	+	0.572834i	0.360085	+	0.932919i	$0.382748\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−53.9020			−1.78585			−0.892927	−	0.450201i	$-0.851352\pi$
$911$	−53.9020			−1.78585			−0.892927	+	0.450201i	$0.851352\pi$
$912$		0			0
$913$	20.2843			0.671311
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−27.2720	+	27.2720i	−0.900602	+	0.900602i
$918$		0			0
$919$		−	12.1303i		−	0.400142i	−0.979781	−	0.200071i	$-0.935883\pi$
$919$		−	12.1303i		−	0.400142i	0.979781	−	0.200071i	$-0.0641174\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	6.12293	−	6.12293i	0.201539	−	0.201539i
$924$		0			0
$925$	−4.41421	−	4.41421i	−0.145138	−	0.145138i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	23.2781			0.763731			0.381866	−	0.924218i	$-0.375282\pi$
$929$	23.2781			0.763731			0.381866	+	0.924218i	$0.375282\pi$
$930$		0			0
$931$	−5.21828	−	5.21828i	−0.171022	−	0.171022i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	16.5754i		−	0.542075i
$936$		0			0
$937$		−	51.9411i		−	1.69684i	−0.529322	−	0.848421i	$-0.677553\pi$
$937$		−	51.9411i		−	1.69684i	0.529322	−	0.848421i	$-0.322447\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−34.6117	−	34.6117i	−1.12831	−	1.12831i	−0.990452	−	0.137856i	$-0.955979\pi$
$941$	−34.6117	−	34.6117i	−1.12831	−	1.12831i	−0.137856	−	0.990452i	$-0.544021\pi$
$942$		0			0
$943$	5.45214			0.177546
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−21.8017	−	21.8017i	−0.708460	−	0.708460i	0.257751	−	0.966211i	$-0.417018\pi$
$947$	−21.8017	−	21.8017i	−0.708460	−	0.708460i	−0.966211	+	0.257751i	$0.917018\pi$
$948$		0			0
$949$	−2.68629	+	2.68629i	−0.0872007	+	0.0872007i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			35.4388i			1.14798i	0.818864	+	0.573988i	$0.194604\pi$
$953$			35.4388i			1.14798i	−0.818864	+	0.573988i	$0.805396\pi$
$954$		0			0
$955$	−14.2918	+	14.2918i	−0.462472	+	0.462472i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	53.1592			1.71660
$960$		0			0
$961$	5.20101			0.167775
$962$		0			0
$963$		0			0
$964$		0			0
$965$	15.7225	−	15.7225i	0.506124	−	0.506124i
$966$		0			0
$967$		−	49.3599i		−	1.58731i	−0.608371	−	0.793653i	$-0.708177\pi$
$967$		−	49.3599i		−	1.58731i	0.608371	−	0.793653i	$-0.291823\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−32.4399	+	32.4399i	−1.04105	+	1.04105i	−0.0419253	+	0.999121i	$0.513349\pi$
$971$	−32.4399	+	32.4399i	−1.04105	+	1.04105i	−0.999121	+	0.0419253i	$0.986651\pi$
$972$		0			0
$973$	43.3137	+	43.3137i	1.38857	+	1.38857i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	25.3646			0.811485			0.405743	−	0.913987i	$-0.367013\pi$
$977$	25.3646			0.811485			0.405743	+	0.913987i	$0.367013\pi$
$978$		0			0
$979$	7.71049	+	7.71049i	0.246428	+	0.246428i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	41.8100i		−	1.33353i	−0.745267	−	0.666766i	$-0.767678\pi$
$983$		−	41.8100i		−	1.33353i	0.745267	−	0.666766i	$-0.232322\pi$
$984$		0			0
$985$		−	5.79899i		−	0.184771i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−41.5192	−	41.5192i	−1.32023	−	1.32023i
$990$		0			0
$991$	−24.1638			−0.767588			−0.383794	−	0.923419i	$-0.625383\pi$
$991$	−24.1638			−0.767588			−0.383794	+	0.923419i	$0.625383\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−24.1522	−	24.1522i	−0.765676	−	0.765676i
$996$		0			0
$997$	−7.10051	+	7.10051i	−0.224875	+	0.224875i	−0.810548	−	0.585673i	$-0.800830\pi$
$997$	−7.10051	+	7.10051i	−0.224875	+	0.224875i	0.585673	+	0.810548i	$0.300830\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4608.2.k.bl.1153.6	yes	16
3.2	odd	2	inner	4608.2.k.bl.1153.4	yes	16
4.3	odd	2	inner	4608.2.k.bl.1153.5	yes	16
8.3	odd	2		4608.2.k.bk.1153.3	✓	16
8.5	even	2		4608.2.k.bk.1153.4	yes	16
12.11	even	2	inner	4608.2.k.bl.1153.3	yes	16
16.3	odd	4		4608.2.k.bk.3457.4	yes	16
16.5	even	4	inner	4608.2.k.bl.3457.5	yes	16
16.11	odd	4	inner	4608.2.k.bl.3457.6	yes	16
16.13	even	4		4608.2.k.bk.3457.3	yes	16
24.5	odd	2		4608.2.k.bk.1153.6	yes	16
24.11	even	2		4608.2.k.bk.1153.5	yes	16
32.5	even	8		9216.2.a.br.1.5		8
32.11	odd	8		9216.2.a.bs.1.4		8
32.21	even	8		9216.2.a.bs.1.3		8
32.27	odd	8		9216.2.a.br.1.6		8
48.5	odd	4	inner	4608.2.k.bl.3457.3	yes	16
48.11	even	4	inner	4608.2.k.bl.3457.4	yes	16
48.29	odd	4		4608.2.k.bk.3457.5	yes	16
48.35	even	4		4608.2.k.bk.3457.6	yes	16
96.5	odd	8		9216.2.a.br.1.3		8
96.11	even	8		9216.2.a.bs.1.6		8
96.53	odd	8		9216.2.a.bs.1.5		8
96.59	even	8		9216.2.a.br.1.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4608.2.k.bk.1153.3	✓	16	8.3	odd	2
4608.2.k.bk.1153.4	yes	16	8.5	even	2
4608.2.k.bk.1153.5	yes	16	24.11	even	2
4608.2.k.bk.1153.6	yes	16	24.5	odd	2
4608.2.k.bk.3457.3	yes	16	16.13	even	4
4608.2.k.bk.3457.4	yes	16	16.3	odd	4
4608.2.k.bk.3457.5	yes	16	48.29	odd	4
4608.2.k.bk.3457.6	yes	16	48.35	even	4
4608.2.k.bl.1153.3	yes	16	12.11	even	2	inner
4608.2.k.bl.1153.4	yes	16	3.2	odd	2	inner
4608.2.k.bl.1153.5	yes	16	4.3	odd	2	inner
4608.2.k.bl.1153.6	yes	16	1.1	even	1	trivial
4608.2.k.bl.3457.3	yes	16	48.5	odd	4	inner
4608.2.k.bl.3457.4	yes	16	48.11	even	4	inner
4608.2.k.bl.3457.5	yes	16	16.5	even	4	inner
4608.2.k.bl.3457.6	yes	16	16.11	odd	4	inner
9216.2.a.br.1.3		8	96.5	odd	8
9216.2.a.br.1.4		8	96.59	even	8
9216.2.a.br.1.5		8	32.5	even	8
9216.2.a.br.1.6		8	32.27	odd	8
9216.2.a.bs.1.3		8	32.21	even	8
9216.2.a.bs.1.4		8	32.11	odd	8
9216.2.a.bs.1.5		8	96.53	odd	8
9216.2.a.bs.1.6		8	96.11	even	8

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:	\( N \)	\(=\)	\( 4608 = 2^{9} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4608.k (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.25928 - 1.25928i) q^{5} +3.29066i q^{7} +O(q^{10})\)	\(q+(1.25928 - 1.25928i) q^{5} +3.29066i q^{7} +(1.08239 - 1.08239i) q^{11} +(-0.414214 - 0.414214i) q^{13} +6.08034 q^{17} +(1.36303 + 1.36303i) q^{19} +5.22625i q^{23} +1.82843i q^{25} +(-4.82106 - 4.82106i) q^{29} +6.01673 q^{31} +(4.14386 + 4.14386i) q^{35} +(-2.41421 + 2.41421i) q^{37} -1.04322i q^{41} +(-7.94435 + 7.94435i) q^{43} -0.896683 q^{47} -3.82843 q^{49} +(4.82106 - 4.82106i) q^{53} -2.72607i q^{55} +(7.39104 - 7.39104i) q^{59} +(3.24264 + 3.24264i) q^{61} -1.04322 q^{65} +(-6.58132 - 6.58132i) q^{67} +14.7821i q^{71} -6.48528i q^{73} +(3.56178 + 3.56178i) q^{77} -3.29066 q^{79} +(9.37011 + 9.37011i) q^{83} +(7.65685 - 7.65685i) q^{85} +7.12356i q^{89} +(1.36303 - 1.36303i) q^{91} +3.43289 q^{95} +7.31371 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q + 16 q^{13} - 16 q^{37} - 16 q^{49} - 16 q^{61} + 32 q^{85} - 64 q^{97}+O(q^{100}) \) 16 * q + 16 * q^13 - 16 * q^37 - 16 * q^49 - 16 * q^61 + 32 * q^85 - 64 * q^97

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