LMFDB - Embedded newform 504.2.s.a.361.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [504,2,Mod(289,504)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(504, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))

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

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

chi := DirichletCharacter("504.289");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 504 = 2^{3} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	504.s (of order $3$, 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:	$4.02446026187$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			361.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	504.361
Dual form			504.2.s.a.289.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.00000 - 3.46410i) q^{5} +(2.50000 + 0.866025i) q^{7} +O(q^{10})$	$q+(-2.00000 - 3.46410i) q^{5} +(2.50000 + 0.866025i) q^{7} -3.00000 q^{13} +(2.00000 - 3.46410i) q^{17} +(-3.50000 - 6.06218i) q^{19} +(-2.00000 - 3.46410i) q^{23} +(-5.50000 + 9.52628i) q^{25} -8.00000 q^{29} +(2.50000 - 4.33013i) q^{31} +(-2.00000 - 10.3923i) q^{35} +(-1.50000 - 2.59808i) q^{37} +8.00000 q^{41} +11.0000 q^{43} +(-2.00000 - 3.46410i) q^{47} +(5.50000 + 4.33013i) q^{49} +(-2.00000 + 3.46410i) q^{53} +(-6.00000 + 10.3923i) q^{59} +(1.00000 + 1.73205i) q^{61} +(6.00000 + 10.3923i) q^{65} +(1.50000 - 2.59808i) q^{67} +12.0000 q^{71} +(-0.500000 + 0.866025i) q^{73} +(-0.500000 - 0.866025i) q^{79} +12.0000 q^{83} -16.0000 q^{85} +(-4.00000 - 6.92820i) q^{89} +(-7.50000 - 2.59808i) q^{91} +(-14.0000 + 24.2487i) q^{95} -2.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{5} + 5 q^{7}+O(q^{10}) $ 2 * q - 4 * q^5 + 5 * q^7	$ 2 q - 4 q^{5} + 5 q^{7} - 6 q^{13} + 4 q^{17} - 7 q^{19} - 4 q^{23} - 11 q^{25} - 16 q^{29} + 5 q^{31} - 4 q^{35} - 3 q^{37} + 16 q^{41} + 22 q^{43} - 4 q^{47} + 11 q^{49} - 4 q^{53} - 12 q^{59} + 2 q^{61} + 12 q^{65} + 3 q^{67} + 24 q^{71} - q^{73} - q^{79} + 24 q^{83} - 32 q^{85} - 8 q^{89} - 15 q^{91} - 28 q^{95} - 4 q^{97}+O(q^{100}) $ 2 * q - 4 * q^5 + 5 * q^7 - 6 * q^13 + 4 * q^17 - 7 * q^19 - 4 * q^23 - 11 * q^25 - 16 * q^29 + 5 * q^31 - 4 * q^35 - 3 * q^37 + 16 * q^41 + 22 * q^43 - 4 * q^47 + 11 * q^49 - 4 * q^53 - 12 * q^59 + 2 * q^61 + 12 * q^65 + 3 * q^67 + 24 * q^71 - q^73 - q^79 + 24 * q^83 - 32 * q^85 - 8 * q^89 - 15 * q^91 - 28 * q^95 - 4 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/504\mathbb{Z}\right)^\times$.

$n$	$73$	$127$	$253$	$281$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$	$1$	$1$

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
$5$	−2.00000	−	3.46410i	−0.894427	−	1.54919i	−0.834512	−	0.550990i	$-0.814250\pi$
$5$	−2.00000	−	3.46410i	−0.894427	−	1.54919i	−0.0599153	−	0.998203i	$-0.519083\pi$
$6$		0			0
$7$	2.50000	+	0.866025i	0.944911	+	0.327327i
$7$	2.50000	+	0.866025i	0.944911	+	0.327327i
$8$		0			0
$9$		0			0
$10$		0			0
$11$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$11$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$12$		0			0
$13$	−3.00000			−0.832050			−0.416025	−	0.909353i	$-0.636577\pi$
$13$	−3.00000			−0.832050			−0.416025	+	0.909353i	$0.636577\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.00000	−	3.46410i	0.485071	−	0.840168i	−0.514782	−	0.857321i	$-0.672127\pi$
$17$	2.00000	−	3.46410i	0.485071	−	0.840168i	0.999853	+	0.0171533i	$0.00546033\pi$
$18$		0			0
$19$	−3.50000	−	6.06218i	−0.802955	−	1.39076i	−0.917663	−	0.397360i	$-0.869927\pi$
$19$	−3.50000	−	6.06218i	−0.802955	−	1.39076i	0.114708	−	0.993399i	$-0.463407\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−2.00000	−	3.46410i	−0.417029	−	0.722315i	0.578610	−	0.815604i	$-0.303595\pi$
$23$	−2.00000	−	3.46410i	−0.417029	−	0.722315i	−0.995639	+	0.0932891i	$0.970262\pi$
$24$		0			0
$25$	−5.50000	+	9.52628i	−1.10000	+	1.90526i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−8.00000			−1.48556			−0.742781	−	0.669534i	$-0.766494\pi$
$29$	−8.00000			−1.48556			−0.742781	+	0.669534i	$0.766494\pi$
$30$		0			0
$31$	2.50000	−	4.33013i	0.449013	−	0.777714i	−0.549309	−	0.835619i	$-0.685109\pi$
$31$	2.50000	−	4.33013i	0.449013	−	0.777714i	0.998322	+	0.0579057i	$0.0184423\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−2.00000	−	10.3923i	−0.338062	−	1.75662i
$36$		0			0
$37$	−1.50000	−	2.59808i	−0.246598	−	0.427121i	0.715981	−	0.698119i	$-0.245980\pi$
$37$	−1.50000	−	2.59808i	−0.246598	−	0.427121i	−0.962580	+	0.270998i	$0.912646\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	8.00000			1.24939			0.624695	−	0.780869i	$-0.285223\pi$
$41$	8.00000			1.24939			0.624695	+	0.780869i	$0.285223\pi$
$42$		0			0
$43$	11.0000			1.67748			0.838742	−	0.544529i	$-0.183292\pi$
$43$	11.0000			1.67748			0.838742	+	0.544529i	$0.183292\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−2.00000	−	3.46410i	−0.291730	−	0.505291i	0.682489	−	0.730896i	$-0.260898\pi$
$47$	−2.00000	−	3.46410i	−0.291730	−	0.505291i	−0.974219	+	0.225605i	$0.927564\pi$
$48$		0			0
$49$	5.50000	+	4.33013i	0.785714	+	0.618590i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−2.00000	+	3.46410i	−0.274721	+	0.475831i	−0.970065	−	0.242846i	$-0.921919\pi$
$53$	−2.00000	+	3.46410i	−0.274721	+	0.475831i	0.695344	+	0.718677i	$0.255252\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−6.00000	+	10.3923i	−0.781133	+	1.35296i	0.150148	+	0.988663i	$0.452025\pi$
$59$	−6.00000	+	10.3923i	−0.781133	+	1.35296i	−0.931282	+	0.364299i	$0.881308\pi$
$60$		0			0
$61$	1.00000	+	1.73205i	0.128037	+	0.221766i	0.922916	−	0.385002i	$-0.125799\pi$
$61$	1.00000	+	1.73205i	0.128037	+	0.221766i	−0.794879	+	0.606768i	$0.792466\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	6.00000	+	10.3923i	0.744208	+	1.28901i
$66$		0			0
$67$	1.50000	−	2.59808i	0.183254	−	0.317406i	−0.759733	−	0.650236i	$-0.774670\pi$
$67$	1.50000	−	2.59808i	0.183254	−	0.317406i	0.942987	+	0.332830i	$0.108004\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	12.0000			1.42414			0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000			1.42414			0.712069	+	0.702109i	$0.247758\pi$
$72$		0			0
$73$	−0.500000	+	0.866025i	−0.0585206	+	0.101361i	−0.893801	−	0.448463i	$-0.851972\pi$
$73$	−0.500000	+	0.866025i	−0.0585206	+	0.101361i	0.835281	+	0.549823i	$0.185305\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−0.500000	−	0.866025i	−0.0562544	−	0.0974355i	0.836527	−	0.547926i	$-0.184582\pi$
$79$	−0.500000	−	0.866025i	−0.0562544	−	0.0974355i	−0.892781	+	0.450490i	$0.851249\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	12.0000			1.31717			0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000			1.31717			0.658586	+	0.752506i	$0.271155\pi$
$84$		0			0
$85$	−16.0000			−1.73544
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−4.00000	−	6.92820i	−0.423999	−	0.734388i	0.572327	−	0.820025i	$-0.306041\pi$
$89$	−4.00000	−	6.92820i	−0.423999	−	0.734388i	−0.996326	+	0.0856373i	$0.972707\pi$
$90$		0			0
$91$	−7.50000	−	2.59808i	−0.786214	−	0.272352i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−14.0000	+	24.2487i	−1.43637	+	2.48787i
$96$		0			0
$97$	−2.00000			−0.203069			−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000			−0.203069			−0.101535	+	0.994832i	$0.532375\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$101$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$102$		0			0
$103$	−1.50000	−	2.59808i	−0.147799	−	0.255996i	0.782614	−	0.622507i	$-0.213886\pi$
$103$	−1.50000	−	2.59808i	−0.147799	−	0.255996i	−0.930414	+	0.366511i	$0.880552\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	6.00000	+	10.3923i	0.580042	+	1.00466i	0.995474	+	0.0950377i	$0.0302972\pi$
$107$	6.00000	+	10.3923i	0.580042	+	1.00466i	−0.415432	+	0.909624i	$0.636370\pi$
$108$		0			0
$109$	5.50000	−	9.52628i	0.526804	−	0.912452i	−0.472708	−	0.881219i	$-0.656723\pi$
$109$	5.50000	−	9.52628i	0.526804	−	0.912452i	0.999512	−	0.0312328i	$-0.00994332\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−12.0000			−1.12887			−0.564433	−	0.825479i	$-0.690905\pi$
$113$	−12.0000			−1.12887			−0.564433	+	0.825479i	$0.690905\pi$
$114$		0			0
$115$	−8.00000	+	13.8564i	−0.746004	+	1.29212i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	8.00000	−	6.92820i	0.733359	−	0.635107i
$120$		0			0
$121$	5.50000	+	9.52628i	0.500000	+	0.866025i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	24.0000			2.14663
$126$		0			0
$127$	−1.00000			−0.0887357			−0.0443678	−	0.999015i	$-0.514127\pi$
$127$	−1.00000			−0.0887357			−0.0443678	+	0.999015i	$0.514127\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	4.00000	+	6.92820i	0.349482	+	0.605320i	0.986157	−	0.165812i	$-0.0530244\pi$
$131$	4.00000	+	6.92820i	0.349482	+	0.605320i	−0.636676	+	0.771132i	$0.719691\pi$
$132$		0			0
$133$	−3.50000	−	18.1865i	−0.303488	−	1.57697i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−8.00000	+	13.8564i	−0.683486	+	1.18383i	0.290424	+	0.956898i	$0.406204\pi$
$137$	−8.00000	+	13.8564i	−0.683486	+	1.18383i	−0.973910	+	0.226935i	$0.927130\pi$
$138$		0			0
$139$	15.0000			1.27228			0.636142	−	0.771572i	$-0.280529\pi$
$139$	15.0000			1.27228			0.636142	+	0.771572i	$0.280529\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	16.0000	+	27.7128i	1.32873	+	2.30142i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	8.00000	+	13.8564i	0.655386	+	1.13516i	0.981797	+	0.189933i	$0.0608272\pi$
$149$	8.00000	+	13.8564i	0.655386	+	1.13516i	−0.326411	+	0.945228i	$0.605840\pi$
$150$		0			0
$151$	10.0000	−	17.3205i	0.813788	−	1.40952i	−0.0964061	−	0.995342i	$-0.530735\pi$
$151$	10.0000	−	17.3205i	0.813788	−	1.40952i	0.910195	−	0.414181i	$-0.135932\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−20.0000			−1.60644
$156$		0			0
$157$	9.00000	−	15.5885i	0.718278	−	1.24409i	−0.243403	−	0.969925i	$-0.578264\pi$
$157$	9.00000	−	15.5885i	0.718278	−	1.24409i	0.961681	−	0.274169i	$-0.0884028\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−2.00000	−	10.3923i	−0.157622	−	0.819028i
$162$		0			0
$163$	4.00000	+	6.92820i	0.313304	+	0.542659i	0.979076	−	0.203497i	$-0.0652307\pi$
$163$	4.00000	+	6.92820i	0.313304	+	0.542659i	−0.665771	+	0.746156i	$0.731897\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	12.0000			0.928588			0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000			0.928588			0.464294	+	0.885681i	$0.346308\pi$
$168$		0			0
$169$	−4.00000			−0.307692
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−2.00000	−	3.46410i	−0.152057	−	0.263371i	0.779926	−	0.625871i	$-0.215256\pi$
$173$	−2.00000	−	3.46410i	−0.152057	−	0.263371i	−0.931984	+	0.362500i	$0.881923\pi$
$174$		0			0
$175$	−22.0000	+	19.0526i	−1.66304	+	1.44024i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	10.0000	−	17.3205i	0.747435	−	1.29460i	−0.201613	−	0.979465i	$-0.564618\pi$
$179$	10.0000	−	17.3205i	0.747435	−	1.29460i	0.949048	−	0.315130i	$-0.102048\pi$
$180$		0			0
$181$	5.00000			0.371647			0.185824	−	0.982583i	$-0.440505\pi$
$181$	5.00000			0.371647			0.185824	+	0.982583i	$0.440505\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−6.00000	+	10.3923i	−0.441129	+	0.764057i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	2.00000	+	3.46410i	0.144715	+	0.250654i	0.929267	−	0.369410i	$-0.120440\pi$
$191$	2.00000	+	3.46410i	0.144715	+	0.250654i	−0.784552	+	0.620063i	$0.787107\pi$
$192$		0			0
$193$	4.50000	−	7.79423i	0.323917	−	0.561041i	−0.657376	−	0.753563i	$-0.728333\pi$
$193$	4.50000	−	7.79423i	0.323917	−	0.561041i	0.981293	+	0.192522i	$0.0616668\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	12.0000			0.854965			0.427482	−	0.904024i	$-0.359401\pi$
$197$	12.0000			0.854965			0.427482	+	0.904024i	$0.359401\pi$
$198$		0			0
$199$	−10.0000	+	17.3205i	−0.708881	+	1.22782i	0.256391	+	0.966573i	$0.417466\pi$
$199$	−10.0000	+	17.3205i	−0.708881	+	1.22782i	−0.965272	+	0.261245i	$0.915867\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−20.0000	−	6.92820i	−1.40372	−	0.486265i
$204$		0			0
$205$	−16.0000	−	27.7128i	−1.11749	−	1.93555i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−16.0000			−1.10149			−0.550743	−	0.834675i	$-0.685655\pi$
$211$	−16.0000			−1.10149			−0.550743	+	0.834675i	$0.685655\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−22.0000	−	38.1051i	−1.50039	−	2.59875i
$216$		0			0
$217$	10.0000	−	8.66025i	0.678844	−	0.587896i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−6.00000	+	10.3923i	−0.403604	+	0.699062i
$222$		0			0
$223$	−12.0000			−0.803579			−0.401790	−	0.915732i	$-0.631612\pi$
$223$	−12.0000			−0.803579			−0.401790	+	0.915732i	$0.631612\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	4.00000	−	6.92820i	0.265489	−	0.459841i	−0.702202	−	0.711977i	$-0.747800\pi$
$227$	4.00000	−	6.92820i	0.265489	−	0.459841i	0.967692	+	0.252136i	$0.0811332\pi$
$228$		0			0
$229$	−2.50000	−	4.33013i	−0.165205	−	0.286143i	0.771523	−	0.636201i	$-0.219495\pi$
$229$	−2.50000	−	4.33013i	−0.165205	−	0.286143i	−0.936728	+	0.350058i	$0.886162\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−14.0000	−	24.2487i	−0.917170	−	1.58859i	−0.803692	−	0.595045i	$-0.797134\pi$
$233$	−14.0000	−	24.2487i	−0.917170	−	1.58859i	−0.113478	−	0.993540i	$-0.536199\pi$
$234$		0			0
$235$	−8.00000	+	13.8564i	−0.521862	+	0.903892i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−8.00000			−0.517477			−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000			−0.517477			−0.258738	+	0.965947i	$0.583307\pi$
$240$		0			0
$241$	1.00000	−	1.73205i	0.0644157	−	0.111571i	−0.832019	−	0.554747i	$-0.812815\pi$
$241$	1.00000	−	1.73205i	0.0644157	−	0.111571i	0.896435	+	0.443176i	$0.146148\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	4.00000	−	27.7128i	0.255551	−	1.77051i
$246$		0			0
$247$	10.5000	+	18.1865i	0.668099	+	1.15718i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	20.0000			1.26239			0.631194	−	0.775625i	$-0.282565\pi$
$251$	20.0000			1.26239			0.631194	+	0.775625i	$0.282565\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−6.00000	−	10.3923i	−0.374270	−	0.648254i	0.615948	−	0.787787i	$-0.288773\pi$
$257$	−6.00000	−	10.3923i	−0.374270	−	0.648254i	−0.990217	+	0.139533i	$0.955440\pi$
$258$		0			0
$259$	−1.50000	−	7.79423i	−0.0932055	−	0.484310i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−12.0000	+	20.7846i	−0.739952	+	1.28163i	0.212565	+	0.977147i	$0.431818\pi$
$263$	−12.0000	+	20.7846i	−0.739952	+	1.28163i	−0.952517	+	0.304487i	$0.901515\pi$
$264$		0			0
$265$	16.0000			0.982872
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$269$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$270$		0			0
$271$	−10.0000	−	17.3205i	−0.607457	−	1.05215i	−0.991658	−	0.128897i	$-0.958856\pi$
$271$	−10.0000	−	17.3205i	−0.607457	−	1.05215i	0.384201	−	0.923249i	$-0.374477\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−6.50000	+	11.2583i	−0.390547	+	0.676448i	−0.992522	−	0.122068i	$-0.961047\pi$
$277$	−6.50000	+	11.2583i	−0.390547	+	0.676448i	0.601975	+	0.798515i	$0.294381\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	12.0000			0.715860			0.357930	−	0.933748i	$-0.383483\pi$
$281$	12.0000			0.715860			0.357930	+	0.933748i	$0.383483\pi$
$282$		0			0
$283$	0.500000	−	0.866025i	0.0297219	−	0.0514799i	−0.850782	−	0.525519i	$-0.823871\pi$
$283$	0.500000	−	0.866025i	0.0297219	−	0.0514799i	0.880504	+	0.474039i	$0.157204\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	20.0000	+	6.92820i	1.18056	+	0.408959i
$288$		0			0
$289$	0.500000	+	0.866025i	0.0294118	+	0.0509427i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−24.0000			−1.40209			−0.701047	−	0.713115i	$-0.747284\pi$
$293$	−24.0000			−1.40209			−0.701047	+	0.713115i	$0.747284\pi$
$294$		0			0
$295$	48.0000			2.79467
$296$		0			0
$297$		0			0
$298$		0			0
$299$	6.00000	+	10.3923i	0.346989	+	0.601003i
$300$		0			0
$301$	27.5000	+	9.52628i	1.58507	+	0.549086i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	4.00000	−	6.92820i	0.229039	−	0.396708i
$306$		0			0
$307$	−11.0000			−0.627803			−0.313902	−	0.949456i	$-0.601636\pi$
$307$	−11.0000			−0.627803			−0.313902	+	0.949456i	$0.601636\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	12.0000	−	20.7846i	0.680458	−	1.17859i	−0.294384	−	0.955687i	$-0.595114\pi$
$311$	12.0000	−	20.7846i	0.680458	−	1.17859i	0.974841	−	0.222900i	$-0.0715523\pi$
$312$		0			0
$313$	6.50000	+	11.2583i	0.367402	+	0.636358i	0.989158	−	0.146852i	$-0.0469141\pi$
$313$	6.50000	+	11.2583i	0.367402	+	0.636358i	−0.621757	+	0.783210i	$0.713581\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−12.0000	−	20.7846i	−0.673987	−	1.16738i	−0.976764	−	0.214318i	$-0.931247\pi$
$317$	−12.0000	−	20.7846i	−0.673987	−	1.16738i	0.302777	−	0.953062i	$-0.402086\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−28.0000			−1.55796
$324$		0			0
$325$	16.5000	−	28.5788i	0.915255	−	1.58527i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−2.00000	−	10.3923i	−0.110264	−	0.572946i
$330$		0			0
$331$	3.50000	+	6.06218i	0.192377	+	0.333207i	0.946038	−	0.324057i	$-0.105047\pi$
$331$	3.50000	+	6.06218i	0.192377	+	0.333207i	−0.753660	+	0.657264i	$0.771714\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−12.0000			−0.655630
$336$		0			0
$337$	−19.0000			−1.03500			−0.517498	−	0.855684i	$-0.673136\pi$
$337$	−19.0000			−1.03500			−0.517498	+	0.855684i	$0.673136\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	10.0000	+	15.5885i	0.539949	+	0.841698i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	6.00000	−	10.3923i	0.322097	−	0.557888i	−0.658824	−	0.752297i	$-0.728946\pi$
$347$	6.00000	−	10.3923i	0.322097	−	0.557888i	0.980921	+	0.194409i	$0.0622790\pi$
$348$		0			0
$349$	−34.0000			−1.81998			−0.909989	−	0.414632i	$-0.863910\pi$
$349$	−34.0000			−1.81998			−0.909989	+	0.414632i	$0.863910\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−6.00000	+	10.3923i	−0.319348	+	0.553127i	−0.980352	−	0.197256i	$-0.936797\pi$
$353$	−6.00000	+	10.3923i	−0.319348	+	0.553127i	0.661004	+	0.750382i	$0.270130\pi$
$354$		0			0
$355$	−24.0000	−	41.5692i	−1.27379	−	2.20627i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	12.0000	+	20.7846i	0.633336	+	1.09697i	0.986865	+	0.161546i	$0.0516481\pi$
$359$	12.0000	+	20.7846i	0.633336	+	1.09697i	−0.353529	+	0.935423i	$0.615019\pi$
$360$		0			0
$361$	−15.0000	+	25.9808i	−0.789474	+	1.36741i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	4.00000			0.209370
$366$		0			0
$367$	−8.50000	+	14.7224i	−0.443696	+	0.768505i	−0.997960	−	0.0638362i	$-0.979666\pi$
$367$	−8.50000	+	14.7224i	−0.443696	+	0.768505i	0.554264	+	0.832341i	$0.313000\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−8.00000	+	6.92820i	−0.415339	+	0.359694i
$372$		0			0
$373$	0.500000	+	0.866025i	0.0258890	+	0.0448411i	0.878680	−	0.477412i	$-0.158425\pi$
$373$	0.500000	+	0.866025i	0.0258890	+	0.0448411i	−0.852791	+	0.522253i	$0.825092\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	24.0000			1.23606
$378$		0			0
$379$	21.0000			1.07870			0.539349	−	0.842082i	$-0.318670\pi$
$379$	21.0000			1.07870			0.539349	+	0.842082i	$0.318670\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−6.00000	−	10.3923i	−0.306586	−	0.531022i	0.671027	−	0.741433i	$-0.265853\pi$
$383$	−6.00000	−	10.3923i	−0.306586	−	0.531022i	−0.977613	+	0.210411i	$0.932520\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	2.00000	−	3.46410i	0.101404	−	0.175637i	−0.810859	−	0.585241i	$-0.801000\pi$
$389$	2.00000	−	3.46410i	0.101404	−	0.175637i	0.912263	+	0.409604i	$0.134333\pi$
$390$		0			0
$391$	−16.0000			−0.809155
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−2.00000	+	3.46410i	−0.100631	+	0.174298i
$396$		0			0
$397$	−3.50000	−	6.06218i	−0.175660	−	0.304252i	0.764730	−	0.644351i	$-0.222873\pi$
$397$	−3.50000	−	6.06218i	−0.175660	−	0.304252i	−0.940389	+	0.340099i	$0.889539\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	18.0000	+	31.1769i	0.898877	+	1.55690i	0.828932	+	0.559350i	$0.188949\pi$
$401$	18.0000	+	31.1769i	0.898877	+	1.55690i	0.0699455	+	0.997551i	$0.477717\pi$
$402$		0			0
$403$	−7.50000	+	12.9904i	−0.373602	+	0.647097i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	7.50000	−	12.9904i	0.370851	−	0.642333i	−0.618846	−	0.785513i	$-0.712399\pi$
$409$	7.50000	−	12.9904i	0.370851	−	0.642333i	0.989697	+	0.143180i	$0.0457327\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−24.0000	+	20.7846i	−1.18096	+	1.02274i
$414$		0			0
$415$	−24.0000	−	41.5692i	−1.17811	−	2.04055i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	16.0000			0.781651			0.390826	−	0.920465i	$-0.372190\pi$
$419$	16.0000			0.781651			0.390826	+	0.920465i	$0.372190\pi$
$420$		0			0
$421$	1.00000			0.0487370			0.0243685	−	0.999703i	$-0.492242\pi$
$421$	1.00000			0.0487370			0.0243685	+	0.999703i	$0.492242\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	22.0000	+	38.1051i	1.06716	+	1.84837i
$426$		0			0
$427$	1.00000	+	5.19615i	0.0483934	+	0.251459i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	6.00000	−	10.3923i	0.289010	−	0.500580i	−0.684564	−	0.728953i	$-0.740007\pi$
$431$	6.00000	−	10.3923i	0.289010	−	0.500580i	0.973574	+	0.228373i	$0.0733406\pi$
$432$		0			0
$433$	−5.00000			−0.240285			−0.120142	−	0.992757i	$-0.538335\pi$
$433$	−5.00000			−0.240285			−0.120142	+	0.992757i	$0.538335\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−14.0000	+	24.2487i	−0.669711	+	1.15997i
$438$		0			0
$439$	6.00000	+	10.3923i	0.286364	+	0.495998i	0.972939	−	0.231062i	$-0.0742199\pi$
$439$	6.00000	+	10.3923i	0.286364	+	0.495998i	−0.686575	+	0.727059i	$0.740887\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−2.00000	−	3.46410i	−0.0950229	−	0.164584i	0.814595	−	0.580030i	$-0.196959\pi$
$443$	−2.00000	−	3.46410i	−0.0950229	−	0.164584i	−0.909618	+	0.415445i	$0.863626\pi$
$444$		0			0
$445$	−16.0000	+	27.7128i	−0.758473	+	1.31371i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	20.0000			0.943858			0.471929	−	0.881636i	$-0.343558\pi$
$449$	20.0000			0.943858			0.471929	+	0.881636i	$0.343558\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$	6.00000	+	31.1769i	0.281284	+	1.46160i
$456$		0			0
$457$	−0.500000	−	0.866025i	−0.0233890	−	0.0405110i	0.854094	−	0.520119i	$-0.174112\pi$
$457$	−0.500000	−	0.866025i	−0.0233890	−	0.0405110i	−0.877483	+	0.479608i	$0.840779\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$	5.00000			0.232370			0.116185	−	0.993228i	$-0.462933\pi$
$463$	5.00000			0.232370			0.116185	+	0.993228i	$0.462933\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	6.00000	+	10.3923i	0.277647	+	0.480899i	0.970799	−	0.239892i	$-0.0771121\pi$
$467$	6.00000	+	10.3923i	0.277647	+	0.480899i	−0.693153	+	0.720791i	$0.743779\pi$
$468$		0			0
$469$	6.00000	−	5.19615i	0.277054	−	0.239936i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	77.0000			3.53300
$476$		0			0
$477$		0			0
$478$		0			0
$479$	2.00000	−	3.46410i	0.0913823	−	0.158279i	−0.816711	−	0.577047i	$-0.804205\pi$
$479$	2.00000	−	3.46410i	0.0913823	−	0.158279i	0.908093	+	0.418769i	$0.137538\pi$
$480$		0			0
$481$	4.50000	+	7.79423i	0.205182	+	0.355386i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	4.00000	+	6.92820i	0.181631	+	0.314594i
$486$		0			0
$487$	17.5000	−	30.3109i	0.793001	−	1.37352i	−0.131100	−	0.991369i	$-0.541851\pi$
$487$	17.5000	−	30.3109i	0.793001	−	1.37352i	0.924101	−	0.382148i	$-0.124816\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	24.0000			1.08310			0.541552	−	0.840667i	$-0.317837\pi$
$491$	24.0000			1.08310			0.541552	+	0.840667i	$0.317837\pi$
$492$		0			0
$493$	−16.0000	+	27.7128i	−0.720604	+	1.24812i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	30.0000	+	10.3923i	1.34568	+	0.466159i
$498$		0			0
$499$	14.5000	+	25.1147i	0.649109	+	1.12429i	0.983336	+	0.181797i	$0.0581915\pi$
$499$	14.5000	+	25.1147i	0.649109	+	1.12429i	−0.334227	+	0.942493i	$0.608475\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−40.0000			−1.78351			−0.891756	−	0.452517i	$-0.850526\pi$
$503$	−40.0000			−1.78351			−0.891756	+	0.452517i	$0.850526\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	6.00000	+	10.3923i	0.265945	+	0.460631i	0.967811	−	0.251679i	$-0.0809826\pi$
$509$	6.00000	+	10.3923i	0.265945	+	0.460631i	−0.701866	+	0.712309i	$0.747649\pi$
$510$		0			0
$511$	−2.00000	+	1.73205i	−0.0884748	+	0.0766214i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−6.00000	+	10.3923i	−0.264392	+	0.457940i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	12.0000	−	20.7846i	0.525730	−	0.910590i	−0.473821	−	0.880621i	$-0.657126\pi$
$521$	12.0000	−	20.7846i	0.525730	−	0.910590i	0.999551	−	0.0299693i	$-0.00954094\pi$
$522$		0			0
$523$	−14.5000	−	25.1147i	−0.634041	−	1.09819i	−0.986718	−	0.162446i	$-0.948062\pi$
$523$	−14.5000	−	25.1147i	−0.634041	−	1.09819i	0.352677	−	0.935745i	$-0.385272\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−10.0000	−	17.3205i	−0.435607	−	0.754493i
$528$		0			0
$529$	3.50000	−	6.06218i	0.152174	−	0.263573i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−24.0000			−1.03956
$534$		0			0
$535$	24.0000	−	41.5692i	1.03761	−	1.79719i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	11.5000	+	19.9186i	0.494424	+	0.856367i	0.999979	−	0.00642713i	$-0.00204583\pi$
$541$	11.5000	+	19.9186i	0.494424	+	0.856367i	−0.505556	+	0.862794i	$0.668712\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−44.0000			−1.88475
$546$		0			0
$547$	−8.00000			−0.342055			−0.171028	−	0.985266i	$-0.554709\pi$
$547$	−8.00000			−0.342055			−0.171028	+	0.985266i	$0.554709\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	28.0000	+	48.4974i	1.19284	+	2.06606i
$552$		0			0
$553$	−0.500000	−	2.59808i	−0.0212622	−	0.110481i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−4.00000	+	6.92820i	−0.169485	+	0.293557i	−0.938239	−	0.345988i	$-0.887544\pi$
$557$	−4.00000	+	6.92820i	−0.169485	+	0.293557i	0.768754	+	0.639545i	$0.220877\pi$
$558$		0			0
$559$	−33.0000			−1.39575
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−10.0000	+	17.3205i	−0.421450	+	0.729972i	−0.996082	−	0.0884397i	$-0.971812\pi$
$563$	−10.0000	+	17.3205i	−0.421450	+	0.729972i	0.574632	+	0.818412i	$0.305145\pi$
$564$		0			0
$565$	24.0000	+	41.5692i	1.00969	+	1.74883i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−12.0000	−	20.7846i	−0.503066	−	0.871336i	−0.999994	−	0.00354413i	$-0.998872\pi$
$569$	−12.0000	−	20.7846i	−0.503066	−	0.871336i	0.496928	−	0.867792i	$-0.334461\pi$
$570$		0			0
$571$	12.5000	−	21.6506i	0.523109	−	0.906051i	−0.476530	−	0.879158i	$-0.658105\pi$
$571$	12.5000	−	21.6506i	0.523109	−	0.906051i	0.999638	−	0.0268925i	$-0.00856117\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	44.0000			1.83493
$576$		0			0
$577$	−13.5000	+	23.3827i	−0.562012	+	0.973434i	0.435308	+	0.900281i	$0.356639\pi$
$577$	−13.5000	+	23.3827i	−0.562012	+	0.973434i	−0.997321	+	0.0731526i	$0.976694\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	30.0000	+	10.3923i	1.24461	+	0.431145i
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−16.0000			−0.660391			−0.330195	−	0.943913i	$-0.607115\pi$
$587$	−16.0000			−0.660391			−0.330195	+	0.943913i	$0.607115\pi$
$588$		0			0
$589$	−35.0000			−1.44215
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−12.0000	−	20.7846i	−0.492781	−	0.853522i	0.507184	−	0.861838i	$-0.330686\pi$
$593$	−12.0000	−	20.7846i	−0.492781	−	0.853522i	−0.999965	+	0.00831589i	$0.997353\pi$
$594$		0			0
$595$	−40.0000	−	13.8564i	−1.63984	−	0.568057i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	8.00000	−	13.8564i	0.326871	−	0.566157i	−0.655018	−	0.755613i	$-0.727339\pi$
$599$	8.00000	−	13.8564i	0.326871	−	0.566157i	0.981889	+	0.189456i	$0.0606724\pi$
$600$		0			0
$601$	−41.0000			−1.67242			−0.836212	−	0.548406i	$-0.815235\pi$
$601$	−41.0000			−1.67242			−0.836212	+	0.548406i	$0.815235\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	22.0000	−	38.1051i	0.894427	−	1.54919i
$606$		0			0
$607$	17.5000	+	30.3109i	0.710303	+	1.23028i	0.964743	+	0.263193i	$0.0847754\pi$
$607$	17.5000	+	30.3109i	0.710303	+	1.23028i	−0.254440	+	0.967088i	$0.581891\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	6.00000	+	10.3923i	0.242734	+	0.420428i
$612$		0			0
$613$	13.0000	−	22.5167i	0.525065	−	0.909439i	−0.474509	−	0.880251i	$-0.657374\pi$
$613$	13.0000	−	22.5167i	0.525065	−	0.909439i	0.999574	−	0.0291886i	$-0.00929235\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0			−	1.00000i	$-0.5\pi$
$617$		0			0				1.00000i	$0.5\pi$
$618$		0			0
$619$	−4.50000	+	7.79423i	−0.180870	+	0.313276i	−0.942177	−	0.335115i	$-0.891225\pi$
$619$	−4.50000	+	7.79423i	−0.180870	+	0.313276i	0.761307	+	0.648392i	$0.224558\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−4.00000	−	20.7846i	−0.160257	−	0.832718i
$624$		0			0
$625$	−20.5000	−	35.5070i	−0.820000	−	1.42028i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−12.0000			−0.478471
$630$		0			0
$631$	−4.00000			−0.159237			−0.0796187	−	0.996825i	$-0.525370\pi$
$631$	−4.00000			−0.159237			−0.0796187	+	0.996825i	$0.525370\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	2.00000	+	3.46410i	0.0793676	+	0.137469i
$636$		0			0
$637$	−16.5000	−	12.9904i	−0.653754	−	0.514698i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	14.0000	−	24.2487i	0.552967	−	0.957767i	−0.445092	−	0.895485i	$-0.646829\pi$
$641$	14.0000	−	24.2487i	0.552967	−	0.957767i	0.998059	−	0.0622816i	$-0.0198377\pi$
$642$		0			0
$643$	7.00000			0.276053			0.138027	−	0.990429i	$-0.455924\pi$
$643$	7.00000			0.276053			0.138027	+	0.990429i	$0.455924\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−24.0000	+	41.5692i	−0.943537	+	1.63425i	−0.184884	+	0.982760i	$0.559191\pi$
$647$	−24.0000	+	41.5692i	−0.943537	+	1.63425i	−0.758654	+	0.651494i	$0.774142\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−8.00000	−	13.8564i	−0.313064	−	0.542243i	0.665960	−	0.745988i	$-0.268022\pi$
$653$	−8.00000	−	13.8564i	−0.313064	−	0.542243i	−0.979024	+	0.203744i	$0.934689\pi$
$654$		0			0
$655$	16.0000	−	27.7128i	0.625172	−	1.08283i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	4.00000			0.155818			0.0779089	−	0.996960i	$-0.475176\pi$
$659$	4.00000			0.155818			0.0779089	+	0.996960i	$0.475176\pi$
$660$		0			0
$661$	−17.5000	+	30.3109i	−0.680671	+	1.17896i	0.294105	+	0.955773i	$0.404978\pi$
$661$	−17.5000	+	30.3109i	−0.680671	+	1.17896i	−0.974776	+	0.223184i	$0.928355\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−56.0000	+	48.4974i	−2.17159	+	1.88065i
$666$		0			0
$667$	16.0000	+	27.7128i	0.619522	+	1.07304i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−29.0000			−1.11787			−0.558934	−	0.829212i	$-0.688789\pi$
$673$	−29.0000			−1.11787			−0.558934	+	0.829212i	$0.688789\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$677$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$678$		0			0
$679$	−5.00000	−	1.73205i	−0.191882	−	0.0664700i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	18.0000	−	31.1769i	0.688751	−	1.19295i	−0.283491	−	0.958975i	$-0.591493\pi$
$683$	18.0000	−	31.1769i	0.688751	−	1.19295i	0.972242	−	0.233977i	$-0.0751739\pi$
$684$		0			0
$685$	64.0000			2.44531
$686$		0			0
$687$		0			0
$688$		0			0
$689$	6.00000	−	10.3923i	0.228582	−	0.395915i
$690$		0			0
$691$	−8.50000	−	14.7224i	−0.323355	−	0.560068i	0.657823	−	0.753173i	$-0.271478\pi$
$691$	−8.50000	−	14.7224i	−0.323355	−	0.560068i	−0.981178	+	0.193105i	$0.938144\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−30.0000	−	51.9615i	−1.13796	−	1.97101i
$696$		0			0
$697$	16.0000	−	27.7128i	0.606043	−	1.04970i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	20.0000			0.755390			0.377695	−	0.925930i	$-0.376717\pi$
$701$	20.0000			0.755390			0.377695	+	0.925930i	$0.376717\pi$
$702$		0			0
$703$	−10.5000	+	18.1865i	−0.396015	+	0.685918i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	19.0000	+	32.9090i	0.713560	+	1.23592i	0.963512	+	0.267664i	$0.0862517\pi$
$709$	19.0000	+	32.9090i	0.713560	+	1.23592i	−0.249952	+	0.968258i	$0.580415\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−20.0000			−0.749006
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$719$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$720$		0			0
$721$	−1.50000	−	7.79423i	−0.0558629	−	0.290272i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	44.0000	−	76.2102i	1.63412	−	2.83038i
$726$		0			0
$727$	5.00000			0.185440			0.0927199	−	0.995692i	$-0.470444\pi$
$727$	5.00000			0.185440			0.0927199	+	0.995692i	$0.470444\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	22.0000	−	38.1051i	0.813699	−	1.40937i
$732$		0			0
$733$	17.5000	+	30.3109i	0.646377	+	1.11956i	0.983982	+	0.178270i	$0.0570501\pi$
$733$	17.5000	+	30.3109i	0.646377	+	1.11956i	−0.337604	+	0.941288i	$0.609617\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−9.50000	+	16.4545i	−0.349463	+	0.605288i	−0.986154	−	0.165831i	$-0.946969\pi$
$739$	−9.50000	+	16.4545i	−0.349463	+	0.605288i	0.636691	+	0.771119i	$0.280303\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−8.00000			−0.293492			−0.146746	−	0.989174i	$-0.546880\pi$
$743$	−8.00000			−0.293492			−0.146746	+	0.989174i	$0.546880\pi$
$744$		0			0
$745$	32.0000	−	55.4256i	1.17239	−	2.03064i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	6.00000	+	31.1769i	0.219235	+	1.13918i
$750$		0			0
$751$	−5.50000	−	9.52628i	−0.200698	−	0.347619i	0.748056	−	0.663636i	$-0.230988\pi$
$751$	−5.50000	−	9.52628i	−0.200698	−	0.347619i	−0.948753	+	0.316017i	$0.897654\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−80.0000			−2.91150
$756$		0			0
$757$	10.0000			0.363456			0.181728	−	0.983349i	$-0.441831\pi$
$757$	10.0000			0.363456			0.181728	+	0.983349i	$0.441831\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−6.00000	−	10.3923i	−0.217500	−	0.376721i	0.736543	−	0.676391i	$-0.236457\pi$
$761$	−6.00000	−	10.3923i	−0.217500	−	0.376721i	−0.954043	+	0.299670i	$0.903123\pi$
$762$		0			0
$763$	22.0000	−	19.0526i	0.796453	−	0.689749i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	18.0000	−	31.1769i	0.649942	−	1.12573i
$768$		0			0
$769$	31.0000			1.11789			0.558944	−	0.829205i	$-0.311207\pi$
$769$	31.0000			1.11789			0.558944	+	0.829205i	$0.311207\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−4.00000	+	6.92820i	−0.143870	+	0.249190i	−0.928951	−	0.370203i	$-0.879288\pi$
$773$	−4.00000	+	6.92820i	−0.143870	+	0.249190i	0.785081	+	0.619393i	$0.212621\pi$
$774$		0			0
$775$	27.5000	+	47.6314i	0.987829	+	1.71097i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−28.0000	−	48.4974i	−1.00320	−	1.73760i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−72.0000			−2.56979
$786$		0			0
$787$	−20.0000	+	34.6410i	−0.712923	+	1.23482i	0.250832	+	0.968031i	$0.419296\pi$
$787$	−20.0000	+	34.6410i	−0.712923	+	1.23482i	−0.963755	+	0.266788i	$0.914038\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−30.0000	−	10.3923i	−1.06668	−	0.369508i
$792$		0			0
$793$	−3.00000	−	5.19615i	−0.106533	−	0.184521i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	12.0000			0.425062			0.212531	−	0.977154i	$-0.431829\pi$
$797$	12.0000			0.425062			0.212531	+	0.977154i	$0.431829\pi$
$798$		0			0
$799$	−16.0000			−0.566039
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$	−32.0000	+	27.7128i	−1.12785	+	0.976748i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−6.00000	+	10.3923i	−0.210949	+	0.365374i	−0.952012	−	0.306062i	$-0.900989\pi$
$809$	−6.00000	+	10.3923i	−0.210949	+	0.365374i	0.741063	+	0.671436i	$0.234322\pi$
$810$		0			0
$811$	24.0000			0.842754			0.421377	−	0.906886i	$-0.361547\pi$
$811$	24.0000			0.842754			0.421377	+	0.906886i	$0.361547\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	16.0000	−	27.7128i	0.560456	−	0.970737i
$816$		0			0
$817$	−38.5000	−	66.6840i	−1.34694	−	2.33298i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	18.0000	+	31.1769i	0.628204	+	1.08808i	0.987912	+	0.155017i	$0.0495431\pi$
$821$	18.0000	+	31.1769i	0.628204	+	1.08808i	−0.359708	+	0.933065i	$0.617124\pi$
$822$		0			0
$823$	2.00000	−	3.46410i	0.0697156	−	0.120751i	−0.829060	−	0.559159i	$-0.811124\pi$
$823$	2.00000	−	3.46410i	0.0697156	−	0.120751i	0.898776	+	0.438408i	$0.144457\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	36.0000			1.25184			0.625921	−	0.779886i	$-0.284723\pi$
$827$	36.0000			1.25184			0.625921	+	0.779886i	$0.284723\pi$
$828$		0			0
$829$	9.50000	−	16.4545i	0.329949	−	0.571488i	−0.652553	−	0.757743i	$-0.726302\pi$
$829$	9.50000	−	16.4545i	0.329949	−	0.571488i	0.982501	+	0.186256i	$0.0596352\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	26.0000	−	10.3923i	0.900847	−	0.360072i
$834$		0			0
$835$	−24.0000	−	41.5692i	−0.830554	−	1.43856i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	35.0000			1.20690
$842$		0			0
$843$		0			0
$844$		0			0
$845$	8.00000	+	13.8564i	0.275208	+	0.476675i
$846$		0			0
$847$	5.50000	+	28.5788i	0.188982	+	0.981981i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−6.00000	+	10.3923i	−0.205677	+	0.356244i
$852$		0			0
$853$	−17.0000			−0.582069			−0.291034	−	0.956713i	$-0.593999\pi$
$853$	−17.0000			−0.582069			−0.291034	+	0.956713i	$0.593999\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	24.0000	−	41.5692i	0.819824	−	1.41998i	−0.0859870	−	0.996296i	$-0.527404\pi$
$857$	24.0000	−	41.5692i	0.819824	−	1.41998i	0.905811	−	0.423681i	$-0.139262\pi$
$858$		0			0
$859$	4.00000	+	6.92820i	0.136478	+	0.236387i	0.926161	−	0.377128i	$-0.123088\pi$
$859$	4.00000	+	6.92820i	0.136478	+	0.236387i	−0.789683	+	0.613515i	$0.789755\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	4.00000	+	6.92820i	0.136162	+	0.235839i	0.926041	−	0.377424i	$-0.123190\pi$
$863$	4.00000	+	6.92820i	0.136162	+	0.235839i	−0.789879	+	0.613263i	$0.789857\pi$
$864$		0			0
$865$	−8.00000	+	13.8564i	−0.272008	+	0.471132i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$	−4.50000	+	7.79423i	−0.152477	+	0.264097i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	60.0000	+	20.7846i	2.02837	+	0.702648i
$876$		0			0
$877$	−7.00000	−	12.1244i	−0.236373	−	0.409410i	0.723298	−	0.690536i	$-0.242625\pi$
$877$	−7.00000	−	12.1244i	−0.236373	−	0.409410i	−0.959671	+	0.281126i	$0.909292\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−28.0000			−0.943344			−0.471672	−	0.881774i	$-0.656349\pi$
$881$	−28.0000			−0.943344			−0.471672	+	0.881774i	$0.656349\pi$
$882$		0			0
$883$	−47.0000			−1.58168			−0.790838	−	0.612026i	$-0.790355\pi$
$883$	−47.0000			−1.58168			−0.790838	+	0.612026i	$0.790355\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	12.0000	+	20.7846i	0.402921	+	0.697879i	0.994077	−	0.108678i	$-0.0346618\pi$
$887$	12.0000	+	20.7846i	0.402921	+	0.697879i	−0.591156	+	0.806557i	$0.701328\pi$
$888$		0			0
$889$	−2.50000	−	0.866025i	−0.0838473	−	0.0290456i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−14.0000	+	24.2487i	−0.468492	+	0.811452i
$894$		0			0
$895$	−80.0000			−2.67411
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−20.0000	+	34.6410i	−0.667037	+	1.15534i
$900$		0			0
$901$	8.00000	+	13.8564i	0.266519	+	0.461624i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−10.0000	−	17.3205i	−0.332411	−	0.575753i
$906$		0			0
$907$	−26.5000	+	45.8993i	−0.879918	+	1.52406i	−0.0284883	+	0.999594i	$0.509069\pi$
$907$	−26.5000	+	45.8993i	−0.879918	+	1.52406i	−0.851430	+	0.524469i	$0.824264\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−4.00000			−0.132526			−0.0662630	−	0.997802i	$-0.521108\pi$
$911$	−4.00000			−0.132526			−0.0662630	+	0.997802i	$0.521108\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$	4.00000	+	20.7846i	0.132092	+	0.686368i
$918$		0			0
$919$	5.50000	+	9.52628i	0.181428	+	0.314243i	0.942367	−	0.334581i	$-0.108595\pi$
$919$	5.50000	+	9.52628i	0.181428	+	0.314243i	−0.760939	+	0.648824i	$0.775261\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−36.0000			−1.18495
$924$		0			0
$925$	33.0000			1.08503
$926$		0			0
$927$		0			0
$928$		0			0
$929$	14.0000	+	24.2487i	0.459325	+	0.795574i	0.998925	−	0.0463469i	$-0.0147580\pi$
$929$	14.0000	+	24.2487i	0.459325	+	0.795574i	−0.539600	+	0.841921i	$0.681425\pi$
$930$		0			0
$931$	7.00000	−	48.4974i	0.229416	−	1.58944i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−45.0000			−1.47009			−0.735043	−	0.678021i	$-0.762838\pi$
$937$	−45.0000			−1.47009			−0.735043	+	0.678021i	$0.762838\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	18.0000	−	31.1769i	0.586783	−	1.01634i	−0.407867	−	0.913041i	$-0.633727\pi$
$941$	18.0000	−	31.1769i	0.586783	−	1.01634i	0.994651	−	0.103297i	$-0.0329393\pi$
$942$		0			0
$943$	−16.0000	−	27.7128i	−0.521032	−	0.902453i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−4.00000	−	6.92820i	−0.129983	−	0.225136i	0.793687	−	0.608326i	$-0.208159\pi$
$947$	−4.00000	−	6.92820i	−0.129983	−	0.225136i	−0.923670	+	0.383190i	$0.874825\pi$
$948$		0			0
$949$	1.50000	−	2.59808i	0.0486921	−	0.0843371i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	20.0000			0.647864			0.323932	−	0.946080i	$-0.394995\pi$
$953$	20.0000			0.647864			0.323932	+	0.946080i	$0.394995\pi$
$954$		0			0
$955$	8.00000	−	13.8564i	0.258874	−	0.448383i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−32.0000	+	27.7128i	−1.03333	+	0.894893i
$960$		0			0
$961$	3.00000	+	5.19615i	0.0967742	+	0.167618i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−36.0000			−1.15888
$966$		0			0
$967$	57.0000			1.83300			0.916498	−	0.400039i	$-0.131003\pi$
$967$	57.0000			1.83300			0.916498	+	0.400039i	$0.131003\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−10.0000	−	17.3205i	−0.320915	−	0.555842i	0.659762	−	0.751475i	$-0.270657\pi$
$971$	−10.0000	−	17.3205i	−0.320915	−	0.555842i	−0.980677	+	0.195633i	$0.937324\pi$
$972$		0			0
$973$	37.5000	+	12.9904i	1.20219	+	0.416452i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	24.0000	−	41.5692i	0.767828	−	1.32992i	−0.170910	−	0.985287i	$-0.554671\pi$
$977$	24.0000	−	41.5692i	0.767828	−	1.32992i	0.938738	−	0.344631i	$-0.111996\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−6.00000	+	10.3923i	−0.191370	+	0.331463i	−0.945705	−	0.325027i	$-0.894626\pi$
$983$	−6.00000	+	10.3923i	−0.191370	+	0.331463i	0.754334	+	0.656490i	$0.227960\pi$
$984$		0			0
$985$	−24.0000	−	41.5692i	−0.764704	−	1.32451i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−22.0000	−	38.1051i	−0.699559	−	1.21167i
$990$		0			0
$991$	−1.50000	+	2.59808i	−0.0476491	+	0.0825306i	−0.888866	−	0.458167i	$-0.848506\pi$
$991$	−1.50000	+	2.59808i	−0.0476491	+	0.0825306i	0.841217	+	0.540697i	$0.181840\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	80.0000			2.53617
$996$		0			0
$997$	14.5000	−	25.1147i	0.459220	−	0.795392i	−0.539700	−	0.841857i	$-0.681462\pi$
$997$	14.5000	−	25.1147i	0.459220	−	0.795392i	0.998920	+	0.0464655i	$0.0147958\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	504.2.s.a.361.1	yes	2
3.2	odd	2		504.2.s.h.361.1	yes	2
4.3	odd	2		1008.2.s.a.865.1		2
7.2	even	3	inner	504.2.s.a.289.1	✓	2
7.3	odd	6		3528.2.a.c.1.1		1
7.4	even	3		3528.2.a.z.1.1		1
7.5	odd	6		3528.2.s.bb.3313.1		2
7.6	odd	2		3528.2.s.bb.361.1		2
12.11	even	2		1008.2.s.q.865.1		2
21.2	odd	6		504.2.s.h.289.1	yes	2
21.5	even	6		3528.2.s.b.3313.1		2
21.11	odd	6		3528.2.a.a.1.1		1
21.17	even	6		3528.2.a.ba.1.1		1
21.20	even	2		3528.2.s.b.361.1		2
28.3	even	6		7056.2.a.d.1.1		1
28.11	odd	6		7056.2.a.cb.1.1		1
28.23	odd	6		1008.2.s.a.289.1		2
84.11	even	6		7056.2.a.b.1.1		1
84.23	even	6		1008.2.s.q.289.1		2
84.59	odd	6		7056.2.a.cc.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.2.s.a.289.1	✓	2	7.2	even	3	inner
504.2.s.a.361.1	yes	2	1.1	even	1	trivial
504.2.s.h.289.1	yes	2	21.2	odd	6
504.2.s.h.361.1	yes	2	3.2	odd	2
1008.2.s.a.289.1		2	28.23	odd	6
1008.2.s.a.865.1		2	4.3	odd	2
1008.2.s.q.289.1		2	84.23	even	6
1008.2.s.q.865.1		2	12.11	even	2
3528.2.a.a.1.1		1	21.11	odd	6
3528.2.a.c.1.1		1	7.3	odd	6
3528.2.a.z.1.1		1	7.4	even	3
3528.2.a.ba.1.1		1	21.17	even	6
3528.2.s.b.361.1		2	21.20	even	2
3528.2.s.b.3313.1		2	21.5	even	6
3528.2.s.bb.361.1		2	7.6	odd	2
3528.2.s.bb.3313.1		2	7.5	odd	6
7056.2.a.b.1.1		1	84.11	even	6
7056.2.a.d.1.1		1	28.3	even	6
7056.2.a.cb.1.1		1	28.11	odd	6
7056.2.a.cc.1.1		1	84.59	odd	6

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:	\( N \)	\(=\)	\( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	504.s (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.00000 - 3.46410i) q^{5} +(2.50000 + 0.866025i) q^{7} +O(q^{10})\)	\(q+(-2.00000 - 3.46410i) q^{5} +(2.50000 + 0.866025i) q^{7} -3.00000 q^{13} +(2.00000 - 3.46410i) q^{17} +(-3.50000 - 6.06218i) q^{19} +(-2.00000 - 3.46410i) q^{23} +(-5.50000 + 9.52628i) q^{25} -8.00000 q^{29} +(2.50000 - 4.33013i) q^{31} +(-2.00000 - 10.3923i) q^{35} +(-1.50000 - 2.59808i) q^{37} +8.00000 q^{41} +11.0000 q^{43} +(-2.00000 - 3.46410i) q^{47} +(5.50000 + 4.33013i) q^{49} +(-2.00000 + 3.46410i) q^{53} +(-6.00000 + 10.3923i) q^{59} +(1.00000 + 1.73205i) q^{61} +(6.00000 + 10.3923i) q^{65} +(1.50000 - 2.59808i) q^{67} +12.0000 q^{71} +(-0.500000 + 0.866025i) q^{73} +(-0.500000 - 0.866025i) q^{79} +12.0000 q^{83} -16.0000 q^{85} +(-4.00000 - 6.92820i) q^{89} +(-7.50000 - 2.59808i) q^{91} +(-14.0000 + 24.2487i) q^{95} -2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{5} + 5 q^{7}+O(q^{10}) \) 2 * q - 4 * q^5 + 5 * q^7	\( 2 q - 4 q^{5} + 5 q^{7} - 6 q^{13} + 4 q^{17} - 7 q^{19} - 4 q^{23} - 11 q^{25} - 16 q^{29} + 5 q^{31} - 4 q^{35} - 3 q^{37} + 16 q^{41} + 22 q^{43} - 4 q^{47} + 11 q^{49} - 4 q^{53} - 12 q^{59} + 2 q^{61} + 12 q^{65} + 3 q^{67} + 24 q^{71} - q^{73} - q^{79} + 24 q^{83} - 32 q^{85} - 8 q^{89} - 15 q^{91} - 28 q^{95} - 4 q^{97}+O(q^{100}) \) 2 * q - 4 * q^5 + 5 * q^7 - 6 * q^13 + 4 * q^17 - 7 * q^19 - 4 * q^23 - 11 * q^25 - 16 * q^29 + 5 * q^31 - 4 * q^35 - 3 * q^37 + 16 * q^41 + 22 * q^43 - 4 * q^47 + 11 * q^49 - 4 * q^53 - 12 * q^59 + 2 * q^61 + 12 * q^65 + 3 * q^67 + 24 * q^71 - q^73 - q^79 + 24 * q^83 - 32 * q^85 - 8 * q^89 - 15 * q^91 - 28 * q^95 - 4 * q^97

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