LMFDB - Embedded newform 2736.2.s.t.577.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2736,2,Mod(577,2736)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2736, 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("2736.577");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2736 = 2^{4} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2736.s (of order $3$, degree $2$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$21.8470699930$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{2} + 4 $ x^4 - 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 228)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			577.1
Root			$1.22474 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	2736.577
Dual form			2736.2.s.t.1873.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.22474 - 2.12132i) q^{5} -1.44949 q^{7} +O(q^{10})$	$q+(-1.22474 - 2.12132i) q^{5} -1.44949 q^{7} +2.44949 q^{11} +(0.500000 - 0.866025i) q^{13} +(-2.44949 - 4.24264i) q^{17} +(-1.00000 + 4.24264i) q^{19} +(1.77526 - 3.07483i) q^{23} +(-0.500000 + 0.866025i) q^{25} +(2.44949 - 4.24264i) q^{29} +4.55051 q^{31} +(1.77526 + 3.07483i) q^{35} -10.7980 q^{37} +(0.724745 + 1.25529i) q^{43} +(-5.44949 + 9.43879i) q^{47} -4.89898 q^{49} +(1.22474 - 2.12132i) q^{53} +(-3.00000 - 5.19615i) q^{55} +(-3.67423 - 6.36396i) q^{59} +(5.94949 - 10.3048i) q^{61} -2.44949 q^{65} +(-4.72474 + 8.18350i) q^{67} +(-3.00000 - 5.19615i) q^{71} +(1.05051 + 1.81954i) q^{73} -3.55051 q^{77} +(-1.17423 - 2.03383i) q^{79} -7.10102 q^{83} +(-6.00000 + 10.3923i) q^{85} +(-4.22474 + 7.31747i) q^{89} +(-0.724745 + 1.25529i) q^{91} +(10.2247 - 3.07483i) q^{95} +(9.34847 + 16.1920i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{7}+O(q^{10}) $ 4 * q + 4 * q^7	$ 4 q + 4 q^{7} + 2 q^{13} - 4 q^{19} + 12 q^{23} - 2 q^{25} + 28 q^{31} + 12 q^{35} - 4 q^{37} - 2 q^{43} - 12 q^{47} - 12 q^{55} + 14 q^{61} - 14 q^{67} - 12 q^{71} + 14 q^{73} - 24 q^{77} + 10 q^{79} - 48 q^{83} - 24 q^{85} - 12 q^{89} + 2 q^{91} + 36 q^{95} + 8 q^{97}+O(q^{100}) $ 4 * q + 4 * q^7 + 2 * q^13 - 4 * q^19 + 12 * q^23 - 2 * q^25 + 28 * q^31 + 12 * q^35 - 4 * q^37 - 2 * q^43 - 12 * q^47 - 12 * q^55 + 14 * q^61 - 14 * q^67 - 12 * q^71 + 14 * q^73 - 24 * q^77 + 10 * q^79 - 48 * q^83 - 24 * q^85 - 12 * q^89 + 2 * q^91 + 36 * q^95 + 8 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times$.

$n$	$1009$	$1217$	$1711$	$2053$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$1$	$1$	$1$

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
$5$	−1.22474	−	2.12132i	−0.547723	−	0.948683i	−0.998430	−	0.0560116i	$-0.982162\pi$
$5$	−1.22474	−	2.12132i	−0.547723	−	0.948683i	0.450708	−	0.892672i	$-0.351172\pi$
$6$		0			0
$7$	−1.44949			−0.547856			−0.273928	−	0.961750i	$-0.588323\pi$
$7$	−1.44949			−0.547856			−0.273928	+	0.961750i	$0.588323\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.44949			0.738549			0.369274	−	0.929320i	$-0.379606\pi$
$11$	2.44949			0.738549			0.369274	+	0.929320i	$0.379606\pi$
$12$		0			0
$13$	0.500000	−	0.866025i	0.138675	−	0.240192i	−0.788320	−	0.615265i	$-0.789049\pi$
$13$	0.500000	−	0.866025i	0.138675	−	0.240192i	0.926995	+	0.375073i	$0.122382\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−2.44949	−	4.24264i	−0.594089	−	1.02899i	−0.993675	−	0.112296i	$-0.964180\pi$
$17$	−2.44949	−	4.24264i	−0.594089	−	1.02899i	0.399586	−	0.916696i	$-0.369154\pi$
$18$		0			0
$19$	−1.00000	+	4.24264i	−0.229416	+	0.973329i
$19$	−1.00000	+	4.24264i	−0.229416	+	0.973329i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	1.77526	−	3.07483i	0.370166	−	0.641147i	−0.619425	−	0.785056i	$-0.712634\pi$
$23$	1.77526	−	3.07483i	0.370166	−	0.641147i	0.989591	+	0.143909i	$0.0459674\pi$
$24$		0			0
$25$	−0.500000	+	0.866025i	−0.100000	+	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	2.44949	−	4.24264i	0.454859	−	0.787839i	−0.543821	−	0.839201i	$-0.683023\pi$
$29$	2.44949	−	4.24264i	0.454859	−	0.787839i	0.998680	+	0.0513625i	$0.0163564\pi$
$30$		0			0
$31$	4.55051			0.817296			0.408648	−	0.912692i	$-0.366000\pi$
$31$	4.55051			0.817296			0.408648	+	0.912692i	$0.366000\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1.77526	+	3.07483i	0.300073	+	0.519741i
$36$		0			0
$37$	−10.7980			−1.77517			−0.887587	−	0.460641i	$-0.847620\pi$
$37$	−10.7980			−1.77517			−0.887587	+	0.460641i	$0.847620\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$41$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$42$		0			0
$43$	0.724745	+	1.25529i	0.110523	+	0.191431i	0.915981	−	0.401221i	$-0.131414\pi$
$43$	0.724745	+	1.25529i	0.110523	+	0.191431i	−0.805459	+	0.592652i	$0.798081\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−5.44949	+	9.43879i	−0.794890	+	1.37679i	0.128019	+	0.991772i	$0.459138\pi$
$47$	−5.44949	+	9.43879i	−0.794890	+	1.37679i	−0.922909	+	0.385018i	$0.874195\pi$
$48$		0			0
$49$	−4.89898			−0.699854
$50$		0			0
$51$		0			0
$52$		0			0
$53$	1.22474	−	2.12132i	0.168232	−	0.291386i	−0.769567	−	0.638567i	$-0.779528\pi$
$53$	1.22474	−	2.12132i	0.168232	−	0.291386i	0.937798	+	0.347181i	$0.112861\pi$
$54$		0			0
$55$	−3.00000	−	5.19615i	−0.404520	−	0.700649i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−3.67423	−	6.36396i	−0.478345	−	0.828517i	0.521347	−	0.853345i	$-0.325430\pi$
$59$	−3.67423	−	6.36396i	−0.478345	−	0.828517i	−0.999692	+	0.0248275i	$0.992096\pi$
$60$		0			0
$61$	5.94949	−	10.3048i	0.761754	−	1.31940i	−0.180192	−	0.983632i	$-0.557672\pi$
$61$	5.94949	−	10.3048i	0.761754	−	1.31940i	0.941946	−	0.335765i	$-0.108995\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−2.44949			−0.303822
$66$		0			0
$67$	−4.72474	+	8.18350i	−0.577219	+	0.999773i	0.418577	+	0.908181i	$0.362529\pi$
$67$	−4.72474	+	8.18350i	−0.577219	+	0.999773i	−0.995797	+	0.0915922i	$0.970804\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−3.00000	−	5.19615i	−0.356034	−	0.616670i	0.631260	−	0.775571i	$-0.282538\pi$
$71$	−3.00000	−	5.19615i	−0.356034	−	0.616670i	−0.987294	+	0.158901i	$0.949205\pi$
$72$		0			0
$73$	1.05051	+	1.81954i	0.122953	+	0.212961i	0.920931	−	0.389726i	$-0.127430\pi$
$73$	1.05051	+	1.81954i	0.122953	+	0.212961i	−0.797978	+	0.602687i	$0.794097\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−3.55051			−0.404618
$78$		0			0
$79$	−1.17423	−	2.03383i	−0.132112	−	0.228824i	0.792379	−	0.610030i	$-0.208842\pi$
$79$	−1.17423	−	2.03383i	−0.132112	−	0.228824i	−0.924490	+	0.381205i	$0.875509\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−7.10102			−0.779438			−0.389719	−	0.920934i	$-0.627428\pi$
$83$	−7.10102			−0.779438			−0.389719	+	0.920934i	$0.627428\pi$
$84$		0			0
$85$	−6.00000	+	10.3923i	−0.650791	+	1.12720i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−4.22474	+	7.31747i	−0.447822	+	0.775651i	−0.998244	−	0.0592361i	$-0.981134\pi$
$89$	−4.22474	+	7.31747i	−0.447822	+	0.775651i	0.550422	+	0.834887i	$0.314467\pi$
$90$		0			0
$91$	−0.724745	+	1.25529i	−0.0759739	+	0.131591i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	10.2247	−	3.07483i	1.04904	−	0.315471i
$96$		0			0
$97$	9.34847	+	16.1920i	0.949193	+	1.64405i	0.747129	+	0.664679i	$0.231431\pi$
$97$	9.34847	+	16.1920i	0.949193	+	1.64405i	0.202064	+	0.979372i	$0.435235\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	0.550510	−	0.953512i	0.0547778	−	0.0948780i	−0.837336	−	0.546688i	$-0.815888\pi$
$101$	0.550510	−	0.953512i	0.0547778	−	0.0948780i	0.892114	+	0.451810i	$0.149222\pi$
$102$		0			0
$103$	−8.55051			−0.842507			−0.421253	−	0.906943i	$-0.638410\pi$
$103$	−8.55051			−0.842507			−0.421253	+	0.906943i	$0.638410\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−15.7980			−1.52725			−0.763623	−	0.645662i	$-0.776581\pi$
$107$	−15.7980			−1.52725			−0.763623	+	0.645662i	$0.776581\pi$
$108$		0			0
$109$	−4.00000	−	6.92820i	−0.383131	−	0.663602i	0.608377	−	0.793648i	$-0.291821\pi$
$109$	−4.00000	−	6.92820i	−0.383131	−	0.663602i	−0.991508	+	0.130046i	$0.958487\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−6.24745			−0.587711			−0.293855	−	0.955850i	$-0.594938\pi$
$113$	−6.24745			−0.587711			−0.293855	+	0.955850i	$0.594938\pi$
$114$		0			0
$115$	−8.69694			−0.810994
$116$		0			0
$117$		0			0
$118$		0			0
$119$	3.55051	+	6.14966i	0.325475	+	0.563739i
$120$		0			0
$121$	−5.00000			−0.454545
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−9.79796			−0.876356
$126$		0			0
$127$	5.34847	−	9.26382i	0.474600	−	0.822031i	−0.524977	−	0.851116i	$-0.675926\pi$
$127$	5.34847	−	9.26382i	0.474600	−	0.822031i	0.999577	+	0.0290853i	$0.00925943\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−7.89898	−	13.6814i	−0.690137	−	1.19535i	−0.971793	−	0.235837i	$-0.924217\pi$
$131$	−7.89898	−	13.6814i	−0.690137	−	1.19535i	0.281656	−	0.959516i	$-0.409116\pi$
$132$		0			0
$133$	1.44949	−	6.14966i	0.125687	−	0.533244i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−11.4495	+	19.8311i	−0.978196	+	1.69429i	−0.309238	+	0.950985i	$0.600074\pi$
$137$	−11.4495	+	19.8311i	−0.978196	+	1.69429i	−0.668958	+	0.743300i	$0.733259\pi$
$138$		0			0
$139$	−2.27526	+	3.94086i	−0.192985	+	0.334259i	−0.946238	−	0.323471i	$-0.895150\pi$
$139$	−2.27526	+	3.94086i	−0.192985	+	0.334259i	0.753253	+	0.657730i	$0.228483\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	1.22474	−	2.12132i	0.102418	−	0.177394i
$144$		0			0
$145$	−12.0000			−0.996546
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−7.77526	−	13.4671i	−0.636974	−	1.10327i	−0.986093	−	0.166193i	$-0.946853\pi$
$149$	−7.77526	−	13.4671i	−0.636974	−	1.10327i	0.349120	−	0.937078i	$-0.386481\pi$
$150$		0			0
$151$	−22.6969			−1.84705			−0.923525	−	0.383537i	$-0.874706\pi$
$151$	−22.6969			−1.84705			−0.923525	+	0.383537i	$0.874706\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−5.57321	−	9.65309i	−0.447651	−	0.775355i
$156$		0			0
$157$	2.94949	+	5.10867i	0.235395	+	0.407716i	0.959387	−	0.282092i	$-0.0910283\pi$
$157$	2.94949	+	5.10867i	0.235395	+	0.407716i	−0.723992	+	0.689808i	$0.757695\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−2.57321	+	4.45694i	−0.202798	+	0.351256i
$162$		0			0
$163$	24.1464			1.89129			0.945647	−	0.325195i	$-0.105430\pi$
$163$	24.1464			1.89129			0.945647	+	0.325195i	$0.105430\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−9.67423	+	16.7563i	−0.748615	+	1.29664i	0.199872	+	0.979822i	$0.435947\pi$
$167$	−9.67423	+	16.7563i	−0.748615	+	1.29664i	−0.948487	+	0.316817i	$0.897386\pi$
$168$		0			0
$169$	6.00000	+	10.3923i	0.461538	+	0.799408i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	3.55051	+	6.14966i	0.269940	+	0.467550i	0.968846	−	0.247663i	$-0.0796627\pi$
$173$	3.55051	+	6.14966i	0.269940	+	0.467550i	−0.698906	+	0.715214i	$0.746329\pi$
$174$		0			0
$175$	0.724745	−	1.25529i	0.0547856	−	0.0948914i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−8.44949			−0.631545			−0.315772	−	0.948835i	$-0.602264\pi$
$179$	−8.44949			−0.631545			−0.315772	+	0.948835i	$0.602264\pi$
$180$		0			0
$181$	9.34847	−	16.1920i	0.694866	−	1.20354i	−0.275360	−	0.961341i	$-0.588797\pi$
$181$	9.34847	−	16.1920i	0.694866	−	1.20354i	0.970226	−	0.242202i	$-0.0778698\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	13.2247	+	22.9059i	0.972303	+	1.68408i
$186$		0			0
$187$	−6.00000	−	10.3923i	−0.438763	−	0.759961i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−6.24745			−0.452050			−0.226025	−	0.974122i	$-0.572573\pi$
$191$	−6.24745			−0.452050			−0.226025	+	0.974122i	$0.572573\pi$
$192$		0			0
$193$	1.05051	+	1.81954i	0.0756174	+	0.130973i	0.901355	−	0.433082i	$-0.142574\pi$
$193$	1.05051	+	1.81954i	0.0756174	+	0.130973i	−0.825737	+	0.564055i	$0.809241\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	23.1464			1.64911			0.824557	−	0.565778i	$-0.191424\pi$
$197$	23.1464			1.64911			0.824557	+	0.565778i	$0.191424\pi$
$198$		0			0
$199$	6.17423	−	10.6941i	0.437680	−	0.758084i	−0.559830	−	0.828607i	$-0.689134\pi$
$199$	6.17423	−	10.6941i	0.437680	−	0.758084i	0.997510	+	0.0705235i	$0.0224670\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−3.55051	+	6.14966i	−0.249197	+	0.431622i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−2.44949	+	10.3923i	−0.169435	+	0.718851i
$210$		0			0
$211$	0.174235	+	0.301783i	0.0119948	+	0.0207756i	0.871961	−	0.489576i	$-0.162849\pi$
$211$	0.174235	+	0.301783i	0.0119948	+	0.0207756i	−0.859966	+	0.510352i	$0.829515\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	1.77526	−	3.07483i	0.121071	−	0.209702i
$216$		0			0
$217$	−6.59592			−0.447760
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−4.89898			−0.329541
$222$		0			0
$223$	−3.62372	−	6.27647i	−0.242663	−	0.420304i	0.718809	−	0.695207i	$-0.244687\pi$
$223$	−3.62372	−	6.27647i	−0.242663	−	0.420304i	−0.961472	+	0.274903i	$0.911354\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	10.6515			0.706967			0.353483	−	0.935441i	$-0.384997\pi$
$227$	10.6515			0.706967			0.353483	+	0.935441i	$0.384997\pi$
$228$		0			0
$229$	1.20204			0.0794331			0.0397166	−	0.999211i	$-0.487355\pi$
$229$	1.20204			0.0794331			0.0397166	+	0.999211i	$0.487355\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−6.55051	−	11.3458i	−0.429138	−	0.743289i	0.567659	−	0.823264i	$-0.307849\pi$
$233$	−6.55051	−	11.3458i	−0.429138	−	0.743289i	−0.996797	+	0.0799748i	$0.974516\pi$
$234$		0			0
$235$	26.6969			1.74152
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−23.1464			−1.49722			−0.748609	−	0.663012i	$-0.769278\pi$
$239$	−23.1464			−1.49722			−0.748609	+	0.663012i	$0.769278\pi$
$240$		0			0
$241$	11.9495	−	20.6971i	0.769734	−	1.33322i	−0.167973	−	0.985792i	$-0.553722\pi$
$241$	11.9495	−	20.6971i	0.769734	−	1.33322i	0.937707	−	0.347427i	$-0.112945\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	6.00000	+	10.3923i	0.383326	+	0.663940i
$246$		0			0
$247$	3.17423	+	2.98735i	0.201972	+	0.190080i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	6.55051	−	11.3458i	0.413465	−	0.716142i	−0.581801	−	0.813331i	$-0.697652\pi$
$251$	6.55051	−	11.3458i	0.413465	−	0.716142i	0.995266	+	0.0971893i	$0.0309852\pi$
$252$		0			0
$253$	4.34847	−	7.53177i	0.273386	−	0.473518i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	10.7753	−	18.6633i	0.672142	−	1.16418i	−0.305154	−	0.952303i	$-0.598708\pi$
$257$	10.7753	−	18.6633i	0.672142	−	1.16418i	0.977296	−	0.211881i	$-0.0679589\pi$
$258$		0			0
$259$	15.6515			0.972539
$260$		0			0
$261$		0			0
$262$		0			0
$263$	12.7980	+	22.1667i	0.789156	+	1.36686i	0.926485	+	0.376333i	$0.122815\pi$
$263$	12.7980	+	22.1667i	0.789156	+	1.36686i	−0.137329	+	0.990526i	$0.543852\pi$
$264$		0			0
$265$	−6.00000			−0.368577
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−11.5732	−	20.0454i	−0.705631	−	1.22219i	−0.966463	−	0.256805i	$-0.917330\pi$
$269$	−11.5732	−	20.0454i	−0.705631	−	1.22219i	0.260832	−	0.965384i	$-0.416003\pi$
$270$		0			0
$271$	−2.00000	−	3.46410i	−0.121491	−	0.210429i	0.798865	−	0.601511i	$-0.205434\pi$
$271$	−2.00000	−	3.46410i	−0.121491	−	0.210429i	−0.920356	+	0.391082i	$0.872101\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1.22474	+	2.12132i	−0.0738549	+	0.127920i
$276$		0			0
$277$	−1.30306			−0.0782934			−0.0391467	−	0.999233i	$-0.512464\pi$
$277$	−1.30306			−0.0782934			−0.0391467	+	0.999233i	$0.512464\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−0.123724	+	0.214297i	−0.00738078	+	0.0127839i	−0.869692	−	0.493594i	$-0.835683\pi$
$281$	−0.123724	+	0.214297i	−0.00738078	+	0.0127839i	0.862311	+	0.506378i	$0.169016\pi$
$282$		0			0
$283$	−2.00000	−	3.46410i	−0.118888	−	0.205919i	0.800439	−	0.599414i	$-0.204600\pi$
$283$	−2.00000	−	3.46410i	−0.118888	−	0.205919i	−0.919327	+	0.393494i	$0.871266\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−3.50000	+	6.06218i	−0.205882	+	0.356599i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−15.7980			−0.922927			−0.461463	−	0.887159i	$-0.652675\pi$
$293$	−15.7980			−0.922927			−0.461463	+	0.887159i	$0.652675\pi$
$294$		0			0
$295$	−9.00000	+	15.5885i	−0.524000	+	0.907595i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−1.77526	−	3.07483i	−0.102666	−	0.177822i
$300$		0			0
$301$	−1.05051	−	1.81954i	−0.0605504	−	0.104876i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−29.1464			−1.66892
$306$		0			0
$307$	1.00000	+	1.73205i	0.0570730	+	0.0988534i	0.893150	−	0.449758i	$-0.148490\pi$
$307$	1.00000	+	1.73205i	0.0570730	+	0.0988534i	−0.836077	+	0.548612i	$0.815157\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	5.14643			0.291827			0.145914	−	0.989297i	$-0.453388\pi$
$311$	5.14643			0.291827			0.145914	+	0.989297i	$0.453388\pi$
$312$		0			0
$313$	2.00000	−	3.46410i	0.113047	−	0.195803i	−0.803951	−	0.594696i	$-0.797272\pi$
$313$	2.00000	−	3.46410i	0.113047	−	0.195803i	0.916997	+	0.398894i	$0.130606\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	2.57321	−	4.45694i	0.144526	−	0.250327i	−0.784670	−	0.619914i	$-0.787168\pi$
$317$	2.57321	−	4.45694i	0.144526	−	0.250327i	0.929196	+	0.369587i	$0.120501\pi$
$318$		0			0
$319$	6.00000	−	10.3923i	0.335936	−	0.581857i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	20.4495	−	6.14966i	1.13784	−	0.342176i
$324$		0			0
$325$	0.500000	+	0.866025i	0.0277350	+	0.0480384i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	7.89898	−	13.6814i	0.435485	−	0.754282i
$330$		0			0
$331$	−33.0454			−1.81634			−0.908170	−	0.418602i	$-0.862520\pi$
$331$	−33.0454			−1.81634			−0.908170	+	0.418602i	$0.862520\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	23.1464			1.26462
$336$		0			0
$337$	−10.3990	−	18.0116i	−0.566469	−	0.981152i	−0.996911	−	0.0785344i	$-0.974976\pi$
$337$	−10.3990	−	18.0116i	−0.566469	−	0.981152i	0.430443	−	0.902618i	$-0.358357\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	11.1464			0.603613
$342$		0			0
$343$	17.2474			0.931275
$344$		0			0
$345$		0			0
$346$		0			0
$347$	14.0227	+	24.2880i	0.752778	+	1.30385i	0.946471	+	0.322788i	$0.104620\pi$
$347$	14.0227	+	24.2880i	0.752778	+	1.30385i	−0.193693	+	0.981062i	$0.562047\pi$
$348$		0			0
$349$	36.5959			1.95893			0.979467	−	0.201603i	$-0.0646151\pi$
$349$	36.5959			1.95893			0.979467	+	0.201603i	$0.0646151\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−19.3485			−1.02982			−0.514908	−	0.857246i	$-0.672174\pi$
$353$	−19.3485			−1.02982			−0.514908	+	0.857246i	$0.672174\pi$
$354$		0			0
$355$	−7.34847	+	12.7279i	−0.390016	+	0.675528i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−7.10102	−	12.2993i	−0.374778	−	0.649134i	0.615516	−	0.788124i	$-0.288948\pi$
$359$	−7.10102	−	12.2993i	−0.374778	−	0.649134i	−0.990294	+	0.138991i	$0.955614\pi$
$360$		0			0
$361$	−17.0000	−	8.48528i	−0.894737	−	0.446594i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	2.57321	−	4.45694i	0.134688	−	0.233287i
$366$		0			0
$367$	0.724745	−	1.25529i	0.0378314	−	0.0655259i	−0.846490	−	0.532405i	$-0.821288\pi$
$367$	0.724745	−	1.25529i	0.0378314	−	0.0655259i	0.884321	+	0.466879i	$0.154622\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−1.77526	+	3.07483i	−0.0921667	+	0.159637i
$372$		0			0
$373$	−18.6969			−0.968091			−0.484045	−	0.875043i	$-0.660833\pi$
$373$	−18.6969			−0.968091			−0.484045	+	0.875043i	$0.660833\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−2.44949	−	4.24264i	−0.126155	−	0.218507i
$378$		0			0
$379$	−10.7526			−0.552321			−0.276161	−	0.961111i	$-0.589062\pi$
$379$	−10.7526			−0.552321			−0.276161	+	0.961111i	$0.589062\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	11.0227	+	19.0919i	0.563234	+	0.975550i	0.997212	+	0.0746261i	$0.0237763\pi$
$383$	11.0227	+	19.0919i	0.563234	+	0.975550i	−0.433978	+	0.900924i	$0.642890\pi$
$384$		0			0
$385$	4.34847	+	7.53177i	0.221619	+	0.383855i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	19.4722	−	33.7268i	0.987279	−	1.71002i	0.355946	−	0.934506i	$-0.384159\pi$
$389$	19.4722	−	33.7268i	0.987279	−	1.71002i	0.631333	−	0.775512i	$-0.282508\pi$
$390$		0			0
$391$	−17.3939			−0.879646
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−2.87628	+	4.98186i	−0.144721	+	0.250664i
$396$		0			0
$397$	4.60102	+	7.96920i	0.230919	+	0.399963i	0.958079	−	0.286505i	$-0.0924936\pi$
$397$	4.60102	+	7.96920i	0.230919	+	0.399963i	−0.727160	+	0.686468i	$0.759160\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	10.2247	+	17.7098i	0.510599	+	0.884384i	0.999925	+	0.0122827i	$0.00390980\pi$
$401$	10.2247	+	17.7098i	0.510599	+	0.884384i	−0.489325	+	0.872101i	$0.662757\pi$
$402$		0			0
$403$	2.27526	−	3.94086i	0.113339	−	0.196308i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−26.4495			−1.31105
$408$		0			0
$409$	−12.6969	+	21.9917i	−0.627823	+	1.08742i	0.360164	+	0.932889i	$0.382721\pi$
$409$	−12.6969	+	21.9917i	−0.627823	+	1.08742i	−0.987988	+	0.154533i	$0.950613\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	5.32577	+	9.22450i	0.262064	+	0.453908i
$414$		0			0
$415$	8.69694	+	15.0635i	0.426916	+	0.739440i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	24.2474			1.18457			0.592283	−	0.805730i	$-0.298227\pi$
$419$	24.2474			1.18457			0.592283	+	0.805730i	$0.298227\pi$
$420$		0			0
$421$	8.00000	+	13.8564i	0.389896	+	0.675320i	0.992435	−	0.122769i	$-0.0391776\pi$
$421$	8.00000	+	13.8564i	0.389896	+	0.675320i	−0.602539	+	0.798089i	$0.705844\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	4.89898			0.237635
$426$		0			0
$427$	−8.62372	+	14.9367i	−0.417331	+	0.722839i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−6.55051	+	11.3458i	−0.315527	+	0.546509i	−0.979549	−	0.201204i	$-0.935515\pi$
$431$	−6.55051	+	11.3458i	−0.315527	+	0.546509i	0.664022	+	0.747713i	$0.268848\pi$
$432$		0			0
$433$	4.60102	−	7.96920i	0.221111	−	0.382975i	−0.734035	−	0.679112i	$-0.762365\pi$
$433$	4.60102	−	7.96920i	0.221111	−	0.382975i	0.955146	+	0.296137i	$0.0956984\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	11.2702	+	10.6066i	0.539125	+	0.507383i
$438$		0			0
$439$	7.52270	+	13.0297i	0.359039	+	0.621874i	0.987801	−	0.155724i	$-0.0497711\pi$
$439$	7.52270	+	13.0297i	0.359039	+	0.621874i	−0.628761	+	0.777598i	$0.716438\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	3.55051	−	6.14966i	0.168690	−	0.292179i	−0.769270	−	0.638924i	$-0.779380\pi$
$443$	3.55051	−	6.14966i	0.168690	−	0.292179i	0.937959	+	0.346745i	$0.112713\pi$
$444$		0			0
$445$	20.6969			0.981129
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−19.1010			−0.901433			−0.450716	−	0.892667i	$-0.648831\pi$
$449$	−19.1010			−0.901433			−0.450716	+	0.892667i	$0.648831\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$	3.55051			0.166450
$456$		0			0
$457$	−22.7980			−1.06644			−0.533222	−	0.845975i	$-0.679019\pi$
$457$	−22.7980			−1.06644			−0.533222	+	0.845975i	$0.679019\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	2.02270	+	3.50343i	0.0942067	+	0.163171i	0.909277	−	0.416191i	$-0.136635\pi$
$461$	2.02270	+	3.50343i	0.0942067	+	0.163171i	−0.815071	+	0.579362i	$0.803302\pi$
$462$		0			0
$463$	14.3485			0.666830			0.333415	−	0.942780i	$-0.391799\pi$
$463$	14.3485			0.666830			0.333415	+	0.942780i	$0.391799\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$		0			0
$469$	6.84847	−	11.8619i	0.316233	−	0.547731i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	1.77526	+	3.07483i	0.0816263	+	0.141381i
$474$		0			0
$475$	−3.17423	−	2.98735i	−0.145644	−	0.137069i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	17.6969	−	30.6520i	0.808594	−	1.40053i	−0.105244	−	0.994446i	$-0.533562\pi$
$479$	17.6969	−	30.6520i	0.808594	−	1.40053i	0.913838	−	0.406079i	$-0.133104\pi$
$480$		0			0
$481$	−5.39898	+	9.35131i	−0.246172	+	0.426383i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	22.8990	−	39.6622i	1.03979	−	1.80097i
$486$		0			0
$487$	6.69694			0.303467			0.151734	−	0.988421i	$-0.451514\pi$
$487$	6.69694			0.303467			0.151734	+	0.988421i	$0.451514\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−12.7980	−	22.1667i	−0.577564	−	1.00037i	−0.995758	−	0.0920122i	$-0.970670\pi$
$491$	−12.7980	−	22.1667i	−0.577564	−	1.00037i	0.418194	−	0.908358i	$-0.362663\pi$
$492$		0			0
$493$	−24.0000			−1.08091
$494$		0			0
$495$		0			0
$496$		0			0
$497$	4.34847	+	7.53177i	0.195056	+	0.337846i
$498$		0			0
$499$	16.5227	+	28.6182i	0.739658	+	1.28112i	0.952649	+	0.304071i	$0.0983460\pi$
$499$	16.5227	+	28.6182i	0.739658	+	1.28112i	−0.212992	+	0.977054i	$0.568321\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	8.44949	−	14.6349i	0.376744	−	0.652540i	−0.613842	−	0.789429i	$-0.710377\pi$
$503$	8.44949	−	14.6349i	0.376744	−	0.652540i	0.990586	+	0.136889i	$0.0437103\pi$
$504$		0			0
$505$	−2.69694			−0.120012
$506$		0			0
$507$		0			0
$508$		0			0
$509$	7.89898	−	13.6814i	0.350116	−	0.606419i	−0.636153	−	0.771563i	$-0.719475\pi$
$509$	7.89898	−	13.6814i	0.350116	−	0.606419i	0.986270	+	0.165144i	$0.0528088\pi$
$510$		0			0
$511$	−1.52270	−	2.63740i	−0.0673605	−	0.116672i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	10.4722	+	18.1384i	0.461460	+	0.799272i
$516$		0			0
$517$	−13.3485	+	23.1202i	−0.587065	+	1.01683i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	44.9444			1.96905			0.984525	−	0.175246i	$-0.0560721\pi$
$521$	44.9444			1.96905			0.984525	+	0.175246i	$0.0560721\pi$
$522$		0			0
$523$	8.07321	−	13.9832i	0.353017	−	0.611443i	−0.633760	−	0.773530i	$-0.718489\pi$
$523$	8.07321	−	13.9832i	0.353017	−	0.611443i	0.986777	+	0.162087i	$0.0518224\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−11.1464	−	19.3062i	−0.485546	−	0.840990i
$528$		0			0
$529$	5.19694	+	9.00136i	0.225954	+	0.391364i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	19.3485	+	33.5125i	0.836507	+	1.44887i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−12.0000			−0.516877
$540$		0			0
$541$	−10.9495	+	18.9651i	−0.470755	+	0.815372i	−0.999441	−	0.0334458i	$-0.989352\pi$
$541$	−10.9495	+	18.9651i	−0.470755	+	0.815372i	0.528685	+	0.848818i	$0.322685\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−9.79796	+	16.9706i	−0.419698	+	0.726939i
$546$		0			0
$547$	0.724745	−	1.25529i	0.0309879	−	0.0536725i	−0.850116	−	0.526596i	$-0.823468\pi$
$547$	0.724745	−	1.25529i	0.0309879	−	0.0536725i	0.881103	+	0.472924i	$0.156801\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	15.5505	+	14.6349i	0.662474	+	0.623470i
$552$		0			0
$553$	1.70204	+	2.94802i	0.0723781	+	0.125363i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	11.4495	−	19.8311i	0.485130	−	0.840271i	−0.514724	−	0.857356i	$-0.672105\pi$
$557$	11.4495	−	19.8311i	0.485130	−	0.840271i	0.999854	+	0.0170856i	$0.00543878\pi$
$558$		0			0
$559$	1.44949			0.0613069
$560$		0			0
$561$		0			0
$562$		0			0
$563$	37.5959			1.58448			0.792240	−	0.610210i	$-0.208915\pi$
$563$	37.5959			1.58448			0.792240	+	0.610210i	$0.208915\pi$
$564$		0			0
$565$	7.65153	+	13.2528i	0.321902	+	0.557551i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	33.7980			1.41688			0.708442	−	0.705769i	$-0.249398\pi$
$569$	33.7980			1.41688			0.708442	+	0.705769i	$0.249398\pi$
$570$		0			0
$571$	15.4495			0.646541			0.323271	−	0.946307i	$-0.395218\pi$
$571$	15.4495			0.646541			0.323271	+	0.946307i	$0.395218\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	1.77526	+	3.07483i	0.0740333	+	0.128229i
$576$		0			0
$577$	37.3939			1.55673			0.778364	−	0.627814i	$-0.216050\pi$
$577$	37.3939			1.55673			0.778364	+	0.627814i	$0.216050\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	10.2929			0.427020
$582$		0			0
$583$	3.00000	−	5.19615i	0.124247	−	0.215203i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−13.2247	−	22.9059i	−0.545844	−	0.945429i	−0.998553	−	0.0537709i	$-0.982876\pi$
$587$	−13.2247	−	22.9059i	−0.545844	−	0.945429i	0.452710	−	0.891658i	$-0.350457\pi$
$588$		0			0
$589$	−4.55051	+	19.3062i	−0.187501	+	0.795497i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−17.5732	+	30.4377i	−0.721645	+	1.24993i	0.238695	+	0.971095i	$0.423281\pi$
$593$	−17.5732	+	30.4377i	−0.721645	+	1.24993i	−0.960340	+	0.278832i	$0.910053\pi$
$594$		0			0
$595$	8.69694	−	15.0635i	0.356540	−	0.617545i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	9.67423	−	16.7563i	0.395279	−	0.684642i	−0.597858	−	0.801602i	$-0.703981\pi$
$599$	9.67423	−	16.7563i	0.395279	−	0.684642i	0.993137	+	0.116959i	$0.0373147\pi$
$600$		0			0
$601$	14.7980			0.603621			0.301811	−	0.953368i	$-0.402409\pi$
$601$	14.7980			0.603621			0.301811	+	0.953368i	$0.402409\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	6.12372	+	10.6066i	0.248965	+	0.431220i
$606$		0			0
$607$	−22.1464			−0.898896			−0.449448	−	0.893307i	$-0.648379\pi$
$607$	−22.1464			−0.898896			−0.449448	+	0.893307i	$0.648379\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	5.44949	+	9.43879i	0.220463	+	0.381853i
$612$		0			0
$613$	−17.3485	−	30.0484i	−0.700698	−	1.21364i	−0.968222	−	0.250093i	$-0.919539\pi$
$613$	−17.3485	−	30.0484i	−0.700698	−	1.21364i	0.267524	−	0.963551i	$-0.413795\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−14.0227	+	24.2880i	−0.564533	+	0.977799i	0.432560	+	0.901605i	$0.357610\pi$
$617$	−14.0227	+	24.2880i	−0.564533	+	0.977799i	−0.997093	+	0.0761944i	$0.975723\pi$
$618$		0			0
$619$	−40.1464			−1.61362			−0.806811	−	0.590810i	$-0.798808\pi$
$619$	−40.1464			−1.61362			−0.806811	+	0.590810i	$0.798808\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	6.12372	−	10.6066i	0.245342	−	0.424945i
$624$		0			0
$625$	14.5000	+	25.1147i	0.580000	+	1.00459i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	26.4495	+	45.8119i	1.05461	+	1.82664i
$630$		0			0
$631$	2.92679	−	5.06934i	0.116514	−	0.201807i	−0.801870	−	0.597498i	$-0.796162\pi$
$631$	2.92679	−	5.06934i	0.116514	−	0.201807i	0.918384	+	0.395691i	$0.129495\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−26.2020			−1.03980
$636$		0			0
$637$	−2.44949	+	4.24264i	−0.0970523	+	0.168100i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−11.6969	−	20.2597i	−0.462001	−	0.800210i	0.537059	−	0.843544i	$-0.319535\pi$
$641$	−11.6969	−	20.2597i	−0.462001	−	0.800210i	−0.999061	+	0.0433348i	$0.986202\pi$
$642$		0			0
$643$	−18.8712	−	32.6858i	−0.744206	−	1.28900i	−0.950565	−	0.310527i	$-0.899495\pi$
$643$	−18.8712	−	32.6858i	−0.744206	−	1.28900i	0.206358	−	0.978477i	$-0.433839\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	4.89898			0.192599			0.0962994	−	0.995352i	$-0.469299\pi$
$647$	4.89898			0.192599			0.0962994	+	0.995352i	$0.469299\pi$
$648$		0			0
$649$	−9.00000	−	15.5885i	−0.353281	−	0.611900i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−22.8990			−0.896106			−0.448053	−	0.894007i	$-0.647882\pi$
$653$	−22.8990			−0.896106			−0.448053	+	0.894007i	$0.647882\pi$
$654$		0			0
$655$	−19.3485	+	33.5125i	−0.756007	+	1.30944i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−16.7753	+	29.0556i	−0.653471	+	1.13185i	0.328804	+	0.944398i	$0.393355\pi$
$659$	−16.7753	+	29.0556i	−0.653471	+	1.13185i	−0.982275	+	0.187447i	$0.939979\pi$
$660$		0			0
$661$	17.0000	−	29.4449i	0.661223	−	1.14527i	−0.319071	−	0.947731i	$-0.603371\pi$
$661$	17.0000	−	29.4449i	0.661223	−	1.14527i	0.980294	−	0.197542i	$-0.0632958\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−14.8207	+	4.45694i	−0.574721	+	0.172833i
$666$		0			0
$667$	−8.69694	−	15.0635i	−0.336747	−	0.583263i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	14.5732	−	25.2415i	0.562593	−	0.974439i
$672$		0			0
$673$	5.00000			0.192736			0.0963679	−	0.995346i	$-0.469277\pi$
$673$	5.00000			0.192736			0.0963679	+	0.995346i	$0.469277\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	1.59592			0.0613361			0.0306681	−	0.999530i	$-0.490237\pi$
$677$	1.59592			0.0613361			0.0306681	+	0.999530i	$0.490237\pi$
$678$		0			0
$679$	−13.5505	−	23.4702i	−0.520021	−	0.900703i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	11.1464			0.426506			0.213253	−	0.976997i	$-0.431594\pi$
$683$	11.1464			0.426506			0.213253	+	0.976997i	$0.431594\pi$
$684$		0			0
$685$	56.0908			2.14312
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−1.22474	−	2.12132i	−0.0466591	−	0.0808159i
$690$		0			0
$691$	21.3939			0.813861			0.406931	−	0.913459i	$-0.366599\pi$
$691$	21.3939			0.813861			0.406931	+	0.913459i	$0.366599\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	11.1464			0.422808
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	21.9217	+	37.9695i	0.827971	+	1.43409i	0.899628	+	0.436658i	$0.143838\pi$
$701$	21.9217	+	37.9695i	0.827971	+	1.43409i	−0.0716572	+	0.997429i	$0.522829\pi$
$702$		0			0
$703$	10.7980	−	45.8119i	0.407253	−	1.72783i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−0.797959	+	1.38211i	−0.0300103	+	0.0519794i
$708$		0			0
$709$	7.84847	−	13.5939i	0.294755	−	0.510531i	−0.680173	−	0.733052i	$-0.738095\pi$
$709$	7.84847	−	13.5939i	0.294755	−	0.510531i	0.974928	+	0.222521i	$0.0714285\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	8.07832	−	13.9921i	0.302535	−	0.524007i
$714$		0			0
$715$	−6.00000			−0.224387
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−0.674235	−	1.16781i	−0.0251447	−	0.0435519i	0.853179	−	0.521618i	$-0.174671\pi$
$719$	−0.674235	−	1.16781i	−0.0251447	−	0.0435519i	−0.878324	+	0.478066i	$0.841338\pi$
$720$		0			0
$721$	12.3939			0.461572
$722$		0			0
$723$		0			0
$724$		0			0
$725$	2.44949	+	4.24264i	0.0909718	+	0.157568i
$726$		0			0
$727$	−1.17423	−	2.03383i	−0.0435500	−	0.0754307i	0.843429	−	0.537241i	$-0.180533\pi$
$727$	−1.17423	−	2.03383i	−0.0435500	−	0.0754307i	−0.886979	+	0.461810i	$0.847200\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	3.55051	−	6.14966i	0.131320	−	0.227454i
$732$		0			0
$733$	−1.30306			−0.0481297			−0.0240648	−	0.999710i	$-0.507661\pi$
$733$	−1.30306			−0.0481297			−0.0240648	+	0.999710i	$0.507661\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−11.5732	+	20.0454i	−0.426305	+	0.738382i
$738$		0			0
$739$	−4.72474	−	8.18350i	−0.173803	−	0.301035i	0.765944	−	0.642908i	$-0.222272\pi$
$739$	−4.72474	−	8.18350i	−0.173803	−	0.301035i	−0.939746	+	0.341873i	$0.888939\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	9.67423	+	16.7563i	0.354913	+	0.614728i	0.987103	−	0.160086i	$-0.0511771\pi$
$743$	9.67423	+	16.7563i	0.354913	+	0.614728i	−0.632190	+	0.774814i	$0.717844\pi$
$744$		0			0
$745$	−19.0454	+	32.9876i	−0.697770	+	1.20857i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	22.8990			0.836711
$750$		0			0
$751$	−13.7247	+	23.7720i	−0.500823	+	0.867451i	0.499176	+	0.866500i	$0.333636\pi$
$751$	−13.7247	+	23.7720i	−0.500823	+	0.867451i	−1.00000		0.000950641i	$0.999697\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	27.7980	+	48.1475i	1.01167	+	1.75227i
$756$		0			0
$757$	−15.0505	−	26.0682i	−0.547020	−	0.947467i	−0.998477	−	0.0551736i	$-0.982429\pi$
$757$	−15.0505	−	26.0682i	−0.547020	−	0.947467i	0.451457	−	0.892293i	$-0.350905\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	28.0454			1.01665			0.508323	−	0.861167i	$-0.330266\pi$
$761$	28.0454			1.01665			0.508323	+	0.861167i	$0.330266\pi$
$762$		0			0
$763$	5.79796	+	10.0424i	0.209900	+	0.363558i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−7.34847			−0.265338
$768$		0			0
$769$	6.50000	−	11.2583i	0.234396	−	0.405986i	−0.724701	−	0.689063i	$-0.758022\pi$
$769$	6.50000	−	11.2583i	0.234396	−	0.405986i	0.959097	+	0.283078i	$0.0913554\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	12.7980	−	22.1667i	0.460311	−	0.797281i	−0.538666	−	0.842520i	$-0.681071\pi$
$773$	12.7980	−	22.1667i	0.460311	−	0.797281i	0.998976	+	0.0452383i	$0.0144047\pi$
$774$		0			0
$775$	−2.27526	+	3.94086i	−0.0817296	+	0.141560i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	−7.34847	−	12.7279i	−0.262949	−	0.455441i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	7.22474	−	12.5136i	0.257862	−	0.446630i
$786$		0			0
$787$	7.85357			0.279950			0.139975	−	0.990155i	$-0.455298\pi$
$787$	7.85357			0.279950			0.139975	+	0.990155i	$0.455298\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	9.05561			0.321981
$792$		0			0
$793$	−5.94949	−	10.3048i	−0.211273	−	0.365935i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−28.2929			−1.00218			−0.501092	−	0.865394i	$-0.667068\pi$
$797$	−28.2929			−1.00218			−0.501092	+	0.865394i	$0.667068\pi$
$798$		0			0
$799$	53.3939			1.88894
$800$		0			0
$801$		0			0
$802$		0			0
$803$	2.57321	+	4.45694i	0.0908068	+	0.157282i
$804$		0			0
$805$	12.6061			0.444307
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−45.5505			−1.60147			−0.800735	−	0.599018i	$-0.795558\pi$
$809$	−45.5505			−1.60147			−0.800735	+	0.599018i	$0.795558\pi$
$810$		0			0
$811$	26.0454	−	45.1120i	0.914578	−	1.58410i	0.107060	−	0.994253i	$-0.465856\pi$
$811$	26.0454	−	45.1120i	0.914578	−	1.58410i	0.807518	−	0.589843i	$-0.200810\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−29.5732	−	51.2223i	−1.03590	−	1.79424i
$816$		0			0
$817$	−6.05051	+	1.81954i	−0.211681	+	0.0636575i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−1.34847	+	2.33562i	−0.0470619	+	0.0815136i	−0.888597	−	0.458689i	$-0.848319\pi$
$821$	−1.34847	+	2.33562i	−0.0470619	+	0.0815136i	0.841535	+	0.540203i	$0.181652\pi$
$822$		0			0
$823$	2.65153	−	4.59259i	0.0924266	−	0.160087i	−0.816105	−	0.577903i	$-0.803871\pi$
$823$	2.65153	−	4.59259i	0.0924266	−	0.160087i	0.908532	+	0.417816i	$0.137204\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	2.44949	−	4.24264i	0.0851771	−	0.147531i	−0.820290	−	0.571948i	$-0.806188\pi$
$827$	2.44949	−	4.24264i	0.0851771	−	0.147531i	0.905467	+	0.424417i	$0.139521\pi$
$828$		0			0
$829$	13.6969			0.475714			0.237857	−	0.971300i	$-0.423555\pi$
$829$	13.6969			0.475714			0.237857	+	0.971300i	$0.423555\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	12.0000	+	20.7846i	0.415775	+	0.720144i
$834$		0			0
$835$	47.3939			1.64013
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−17.5732	−	30.4377i	−0.606695	−	1.05083i	−0.991781	−	0.127946i	$-0.959162\pi$
$839$	−17.5732	−	30.4377i	−0.606695	−	1.05083i	0.385086	−	0.922880i	$-0.374172\pi$
$840$		0			0
$841$	2.50000	+	4.33013i	0.0862069	+	0.149315i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	14.6969	−	25.4558i	0.505590	−	0.875708i
$846$		0			0
$847$	7.24745			0.249025
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−19.1691	+	33.2019i	−0.657109	+	1.13815i
$852$		0			0
$853$	−15.2980	−	26.4968i	−0.523792	−	0.907235i	−0.999616	−	0.0276942i	$-0.991184\pi$
$853$	−15.2980	−	26.4968i	−0.523792	−	0.907235i	0.475824	−	0.879540i	$-0.342150\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	16.5959	+	28.7450i	0.566906	+	0.981910i	0.996870	+	0.0790635i	$0.0251930\pi$
$857$	16.5959	+	28.7450i	0.566906	+	0.981910i	−0.429964	+	0.902846i	$0.641474\pi$
$858$		0			0
$859$	13.2753	−	22.9934i	0.452946	−	0.784525i	−0.545622	−	0.838032i	$-0.683706\pi$
$859$	13.2753	−	22.9934i	0.452946	−	0.784525i	0.998568	+	0.0535064i	$0.0170398\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−13.1010			−0.445964			−0.222982	−	0.974823i	$-0.571579\pi$
$863$	−13.1010			−0.445964			−0.222982	+	0.974823i	$0.571579\pi$
$864$		0			0
$865$	8.69694	−	15.0635i	0.295705	−	0.512176i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−2.87628	−	4.98186i	−0.0975710	−	0.168998i
$870$		0			0
$871$	4.72474	+	8.18350i	0.160092	+	0.277287i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	14.2020			0.480117
$876$		0			0
$877$	5.39898	+	9.35131i	0.182311	+	0.315771i	0.942667	−	0.333735i	$-0.108309\pi$
$877$	5.39898	+	9.35131i	0.182311	+	0.315771i	−0.760356	+	0.649506i	$0.774976\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−2.44949			−0.0825254			−0.0412627	−	0.999148i	$-0.513138\pi$
$881$	−2.44949			−0.0825254			−0.0412627	+	0.999148i	$0.513138\pi$
$882$		0			0
$883$	−22.1742	+	38.4069i	−0.746222	+	1.29250i	0.203399	+	0.979096i	$0.434801\pi$
$883$	−22.1742	+	38.4069i	−0.746222	+	1.29250i	−0.949621	+	0.313399i	$0.898532\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−21.9217	+	37.9695i	−0.736058	+	1.27489i	0.218199	+	0.975904i	$0.429982\pi$
$887$	−21.9217	+	37.9695i	−0.736058	+	1.27489i	−0.954258	+	0.298986i	$0.903352\pi$
$888$		0			0
$889$	−7.75255	+	13.4278i	−0.260012	+	0.450354i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−34.5959	−	32.5590i	−1.15771	−	1.08955i
$894$		0			0
$895$	10.3485	+	17.9241i	0.345911	+	0.599136i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	11.1464	−	19.3062i	0.371754	−	0.643897i
$900$		0			0
$901$	−12.0000			−0.399778
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−45.7980			−1.52238
$906$		0			0
$907$	−20.0000	−	34.6410i	−0.664089	−	1.15024i	−0.979531	−	0.201291i	$-0.935486\pi$
$907$	−20.0000	−	34.6410i	−0.664089	−	1.15024i	0.315442	−	0.948945i	$-0.397847\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−41.3939			−1.37144			−0.685720	−	0.727865i	$-0.740513\pi$
$911$	−41.3939			−1.37144			−0.685720	+	0.727865i	$0.740513\pi$
$912$		0			0
$913$	−17.3939			−0.575653
$914$		0			0
$915$		0			0
$916$		0			0
$917$	11.4495	+	19.8311i	0.378095	+	0.654881i
$918$		0			0
$919$	14.3485			0.473312			0.236656	−	0.971593i	$-0.423949\pi$
$919$	14.3485			0.473312			0.236656	+	0.971593i	$0.423949\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−6.00000			−0.197492
$924$		0			0
$925$	5.39898	−	9.35131i	0.177517	−	0.307469i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−17.2702	−	29.9128i	−0.566615	−	0.981407i	−0.996897	−	0.0787120i	$-0.974919\pi$
$929$	−17.2702	−	29.9128i	−0.566615	−	0.981407i	0.430282	−	0.902694i	$-0.358414\pi$
$930$		0			0
$931$	4.89898	−	20.7846i	0.160558	−	0.681188i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−14.6969	+	25.4558i	−0.480641	+	0.832495i
$936$		0			0
$937$	15.7474	−	27.2754i	0.514447	−	0.891048i	−0.485413	−	0.874285i	$-0.661331\pi$
$937$	15.7474	−	27.2754i	0.514447	−	0.891048i	0.999859	−	0.0167628i	$-0.00533600\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	27.7980	−	48.1475i	0.906188	−	1.56956i	0.0868724	−	0.996219i	$-0.472313\pi$
$941$	27.7980	−	48.1475i	0.906188	−	1.56956i	0.819315	−	0.573343i	$-0.194354\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1.89898	+	3.28913i	0.0617085	+	0.106882i	0.895229	−	0.445606i	$-0.147012\pi$
$947$	1.89898	+	3.28913i	0.0617085	+	0.106882i	−0.833521	+	0.552488i	$0.813678\pi$
$948$		0			0
$949$	2.10102			0.0682020
$950$		0			0
$951$		0			0
$952$		0			0
$953$	5.32577	+	9.22450i	0.172518	+	0.298811i	0.939300	−	0.343098i	$-0.111476\pi$
$953$	5.32577	+	9.22450i	0.172518	+	0.298811i	−0.766781	+	0.641909i	$0.778143\pi$
$954$		0			0
$955$	7.65153	+	13.2528i	0.247598	+	0.428852i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	16.5959	−	28.7450i	0.535910	−	0.928224i
$960$		0			0
$961$	−10.2929			−0.332028
$962$		0			0
$963$		0			0
$964$		0			0
$965$	2.57321	−	4.45694i	0.0828347	−	0.143474i
$966$		0			0
$967$	12.9722	+	22.4685i	0.417158	+	0.722538i	0.995652	−	0.0931480i	$-0.0296930\pi$
$967$	12.9722	+	22.4685i	0.417158	+	0.722538i	−0.578495	+	0.815686i	$0.696360\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	11.4495	+	19.8311i	0.367432	+	0.636410i	0.989163	−	0.146820i	$-0.0469039\pi$
$971$	11.4495	+	19.8311i	0.367432	+	0.636410i	−0.621732	+	0.783230i	$0.713571\pi$
$972$		0			0
$973$	3.29796	−	5.71223i	0.105728	−	0.183126i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−1.59592			−0.0510579			−0.0255290	−	0.999674i	$-0.508127\pi$
$977$	−1.59592			−0.0510579			−0.0255290	+	0.999674i	$0.508127\pi$
$978$		0			0
$979$	−10.3485	+	17.9241i	−0.330739	+	0.572856i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	2.57321	+	4.45694i	0.0820728	+	0.142154i	0.904140	−	0.427236i	$-0.140513\pi$
$983$	2.57321	+	4.45694i	0.0820728	+	0.142154i	−0.822067	+	0.569390i	$0.807179\pi$
$984$		0			0
$985$	−28.3485	−	49.1010i	−0.903257	−	1.56449i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	5.14643			0.163647
$990$		0			0
$991$	−7.97219	−	13.8082i	−0.253245	−	0.438633i	0.711172	−	0.703018i	$-0.248165\pi$
$991$	−7.97219	−	13.8082i	−0.253245	−	0.438633i	−0.964417	+	0.264384i	$0.914831\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−30.2474			−0.958909
$996$		0			0
$997$	−12.8485	+	22.2542i	−0.406915	+	0.704798i	−0.994542	−	0.104334i	$-0.966729\pi$
$997$	−12.8485	+	22.2542i	−0.406915	+	0.704798i	0.587627	+	0.809132i	$0.300062\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2736.2.s.t.577.1		4
3.2	odd	2		912.2.q.i.577.2		4
4.3	odd	2		684.2.k.f.577.1		4
12.11	even	2		228.2.i.a.121.2	yes	4
19.11	even	3	inner	2736.2.s.t.1873.1		4
57.11	odd	6		912.2.q.i.49.2		4
76.11	odd	6		684.2.k.f.505.1		4
228.11	even	6		228.2.i.a.49.2	✓	4
228.83	even	6		4332.2.a.l.1.1		2
228.107	odd	6		4332.2.a.g.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
228.2.i.a.49.2	✓	4	228.11	even	6
228.2.i.a.121.2	yes	4	12.11	even	2
684.2.k.f.505.1		4	76.11	odd	6
684.2.k.f.577.1		4	4.3	odd	2
912.2.q.i.49.2		4	57.11	odd	6
912.2.q.i.577.2		4	3.2	odd	2
2736.2.s.t.577.1		4	1.1	even	1	trivial
2736.2.s.t.1873.1		4	19.11	even	3	inner
4332.2.a.g.1.1		2	228.107	odd	6
4332.2.a.l.1.1		2	228.83	even	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:	\( N \)	\(=\)	\( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2736.s (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.22474 - 2.12132i) q^{5} -1.44949 q^{7} +O(q^{10})\)	\(q+(-1.22474 - 2.12132i) q^{5} -1.44949 q^{7} +2.44949 q^{11} +(0.500000 - 0.866025i) q^{13} +(-2.44949 - 4.24264i) q^{17} +(-1.00000 + 4.24264i) q^{19} +(1.77526 - 3.07483i) q^{23} +(-0.500000 + 0.866025i) q^{25} +(2.44949 - 4.24264i) q^{29} +4.55051 q^{31} +(1.77526 + 3.07483i) q^{35} -10.7980 q^{37} +(0.724745 + 1.25529i) q^{43} +(-5.44949 + 9.43879i) q^{47} -4.89898 q^{49} +(1.22474 - 2.12132i) q^{53} +(-3.00000 - 5.19615i) q^{55} +(-3.67423 - 6.36396i) q^{59} +(5.94949 - 10.3048i) q^{61} -2.44949 q^{65} +(-4.72474 + 8.18350i) q^{67} +(-3.00000 - 5.19615i) q^{71} +(1.05051 + 1.81954i) q^{73} -3.55051 q^{77} +(-1.17423 - 2.03383i) q^{79} -7.10102 q^{83} +(-6.00000 + 10.3923i) q^{85} +(-4.22474 + 7.31747i) q^{89} +(-0.724745 + 1.25529i) q^{91} +(10.2247 - 3.07483i) q^{95} +(9.34847 + 16.1920i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{7}+O(q^{10}) \) 4 * q + 4 * q^7	\( 4 q + 4 q^{7} + 2 q^{13} - 4 q^{19} + 12 q^{23} - 2 q^{25} + 28 q^{31} + 12 q^{35} - 4 q^{37} - 2 q^{43} - 12 q^{47} - 12 q^{55} + 14 q^{61} - 14 q^{67} - 12 q^{71} + 14 q^{73} - 24 q^{77} + 10 q^{79} - 48 q^{83} - 24 q^{85} - 12 q^{89} + 2 q^{91} + 36 q^{95} + 8 q^{97}+O(q^{100}) \) 4 * q + 4 * q^7 + 2 * q^13 - 4 * q^19 + 12 * q^23 - 2 * q^25 + 28 * q^31 + 12 * q^35 - 4 * q^37 - 2 * q^43 - 12 * q^47 - 12 * q^55 + 14 * q^61 - 14 * q^67 - 12 * q^71 + 14 * q^73 - 24 * q^77 + 10 * q^79 - 48 * q^83 - 24 * q^85 - 12 * q^89 + 2 * q^91 + 36 * q^95 + 8 * q^97

Introduction

L-functions

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