LMFDB - Embedded newform 2736.2.bm.p.1855.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2736,2,Mod(559,2736)]

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([3, 0, 0, 1]))

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

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

chi := DirichletCharacter("2736.559");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2736 = 2^{4} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2736.bm (of order $6$, 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:	$21.8470699930$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	8.0.897122304.10
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 8x^{6} + 51x^{4} - 104x^{2} + 169 $ x^8 - 8x^6 + 51x^4 - 104*x^2 + 169
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			1855.4
Root			$-1.30421 + 0.752986i$ of defining polynomial
Character	$\chi$	$=$	2736.1855
Dual form			2736.2.bm.p.559.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.05719 + 3.56317i) q^{5} -1.00000i q^{7} +O(q^{10})$	$q+(2.05719 + 3.56317i) q^{5} -1.00000i q^{7} -4.11439i q^{11} +(-1.50000 - 0.866025i) q^{13} +(1.50597 + 2.60842i) q^{17} +(1.73205 + 4.00000i) q^{19} +(0.954747 + 0.551224i) q^{23} +(-5.96410 + 10.3301i) q^{25} +(4.51791 + 2.60842i) q^{29} -4.26795 q^{31} +(3.56317 - 2.05719i) q^{35} +4.26795i q^{37} +(5.59808 - 3.23205i) q^{43} +(5.21684 + 3.01194i) q^{47} +6.00000 q^{49} +(1.65367 + 0.954747i) q^{53} +(14.6603 - 8.46410i) q^{55} +(-0.954747 - 1.65367i) q^{59} +(-1.96410 + 3.40192i) q^{61} -7.12633i q^{65} +(-5.59808 + 9.69615i) q^{67} +(4.51791 + 7.82526i) q^{71} +(2.69615 + 4.66987i) q^{73} -4.11439 q^{77} +(7.33013 + 12.6962i) q^{79} -11.2407i q^{83} +(-6.19615 + 10.7321i) q^{85} +(-10.6895 - 6.17158i) q^{89} +(-0.866025 + 1.50000i) q^{91} +(-10.6895 + 14.4004i) q^{95} +(-5.19615 + 3.00000i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 12 q^{13} - 20 q^{25} - 48 q^{31} + 24 q^{43} + 48 q^{49} + 48 q^{55} + 12 q^{61} - 24 q^{67} - 20 q^{73} + 24 q^{79} - 8 q^{85}+O(q^{100}) $ 8 * q - 12 * q^13 - 20 * q^25 - 48 * q^31 + 24 * q^43 + 48 * q^49 + 48 * q^55 + 12 * q^61 - 24 * q^67 - 20 * q^73 + 24 * q^79 - 8 * q^85

Display 100 coefficients 10 coefficients

Character values

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

$n$	$1009$	$1217$	$1711$	$2053$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$1$	$-1$	$1$

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
$5$	2.05719	+	3.56317i	0.920006	+	1.59350i	0.799403	+	0.600795i	$0.205149\pi$
$5$	2.05719	+	3.56317i	0.920006	+	1.59350i	0.120603	+	0.992701i	$0.461517\pi$
$6$		0			0
$7$		−	1.00000i		−	0.377964i	−0.981981	−	0.188982i	$-0.939481\pi$
$7$		−	1.00000i		−	0.377964i	0.981981	−	0.188982i	$-0.0605189\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	4.11439i		−	1.24054i	−0.784390	−	0.620268i	$-0.787024\pi$
$11$		−	4.11439i		−	1.24054i	0.784390	−	0.620268i	$-0.212976\pi$
$12$		0			0
$13$	−1.50000	−	0.866025i	−0.416025	−	0.240192i	0.277350	−	0.960769i	$-0.410544\pi$
$13$	−1.50000	−	0.866025i	−0.416025	−	0.240192i	−0.693375	+	0.720577i	$0.743877\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.50597	+	2.60842i	0.365252	+	0.632634i	0.988817	−	0.149137i	$-0.0476496\pi$
$17$	1.50597	+	2.60842i	0.365252	+	0.632634i	−0.623565	+	0.781772i	$0.714316\pi$
$18$		0			0
$19$	1.73205	+	4.00000i	0.397360	+	0.917663i
$19$	1.73205	+	4.00000i	0.397360	+	0.917663i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	0.954747	+	0.551224i	0.199079	+	0.114938i	0.596226	−	0.802817i	$-0.296666\pi$
$23$	0.954747	+	0.551224i	0.199079	+	0.114938i	−0.397147	+	0.917755i	$0.630000\pi$
$24$		0			0
$25$	−5.96410	+	10.3301i	−1.19282	+	2.06603i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	4.51791	+	2.60842i	0.838955	+	0.484371i	0.856909	−	0.515468i	$-0.172382\pi$
$29$	4.51791	+	2.60842i	0.838955	+	0.484371i	−0.0179536	+	0.999839i	$0.505715\pi$
$30$		0			0
$31$	−4.26795			−0.766546			−0.383273	−	0.923635i	$-0.625203\pi$
$31$	−4.26795			−0.766546			−0.383273	+	0.923635i	$0.625203\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	3.56317	−	2.05719i	0.602285	−	0.347729i
$36$		0			0
$37$			4.26795i			0.701647i	0.936442	+	0.350823i	$0.114098\pi$
$37$			4.26795i			0.701647i	−0.936442	+	0.350823i	$0.885902\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$41$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$42$		0			0
$43$	5.59808	−	3.23205i	0.853699	−	0.492883i	−0.00819845	−	0.999966i	$-0.502610\pi$
$43$	5.59808	−	3.23205i	0.853699	−	0.492883i	0.861897	+	0.507083i	$0.169276\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	5.21684	+	3.01194i	0.760954	+	0.439337i	0.829638	−	0.558302i	$-0.188547\pi$
$47$	5.21684	+	3.01194i	0.760954	+	0.439337i	−0.0686842	+	0.997638i	$0.521880\pi$
$48$		0			0
$49$	6.00000			0.857143
$50$		0			0
$51$		0			0
$52$		0			0
$53$	1.65367	+	0.954747i	0.227149	+	0.131145i	0.609256	−	0.792973i	$-0.291468\pi$
$53$	1.65367	+	0.954747i	0.227149	+	0.131145i	−0.382107	+	0.924118i	$0.624801\pi$
$54$		0			0
$55$	14.6603	−	8.46410i	1.97679	−	1.14130i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−0.954747	−	1.65367i	−0.124298	−	0.215290i	0.797161	−	0.603767i	$-0.206334\pi$
$59$	−0.954747	−	1.65367i	−0.124298	−	0.215290i	−0.921458	+	0.388478i	$0.873001\pi$
$60$		0			0
$61$	−1.96410	+	3.40192i	−0.251477	+	0.435572i	−0.963933	−	0.266146i	$-0.914250\pi$
$61$	−1.96410	+	3.40192i	−0.251477	+	0.435572i	0.712455	+	0.701717i	$0.247583\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	7.12633i		−	0.883913i
$66$		0			0
$67$	−5.59808	+	9.69615i	−0.683914	+	1.18457i	0.289863	+	0.957068i	$0.406390\pi$
$67$	−5.59808	+	9.69615i	−0.683914	+	1.18457i	−0.973777	+	0.227505i	$0.926943\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	4.51791	+	7.82526i	0.536178	+	0.928687i	0.999105	+	0.0422909i	$0.0134656\pi$
$71$	4.51791	+	7.82526i	0.536178	+	0.928687i	−0.462928	+	0.886396i	$0.653201\pi$
$72$		0			0
$73$	2.69615	+	4.66987i	0.315561	+	0.546567i	0.979557	−	0.201169i	$-0.0644741\pi$
$73$	2.69615	+	4.66987i	0.315561	+	0.546567i	−0.663996	+	0.747736i	$0.731141\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−4.11439			−0.468878
$78$		0			0
$79$	7.33013	+	12.6962i	0.824704	+	1.42843i	0.902146	+	0.431432i	$0.141991\pi$
$79$	7.33013	+	12.6962i	0.824704	+	1.42843i	−0.0774418	+	0.996997i	$0.524675\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	11.2407i		−	1.23383i	−0.787030	−	0.616915i	$-0.788382\pi$
$83$		−	11.2407i		−	1.23383i	0.787030	−	0.616915i	$-0.211618\pi$
$84$		0			0
$85$	−6.19615	+	10.7321i	−0.672067	+	1.16405i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−10.6895	−	6.17158i	−1.13308	−	0.654187i	−0.188376	−	0.982097i	$-0.560322\pi$
$89$	−10.6895	−	6.17158i	−1.13308	−	0.654187i	−0.944709	+	0.327910i	$0.893656\pi$
$90$		0			0
$91$	−0.866025	+	1.50000i	−0.0907841	+	0.157243i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−10.6895	+	14.4004i	−1.09672	+	1.47745i
$96$		0			0
$97$	−5.19615	+	3.00000i	−0.527589	+	0.304604i	−0.740034	−	0.672569i	$-0.765191\pi$
$97$	−5.19615	+	3.00000i	−0.527589	+	0.304604i	0.212445	+	0.977173i	$0.431857\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	5.62036	−	9.73475i	0.559247	−	0.968644i	−0.438313	−	0.898823i	$-0.644424\pi$
$101$	5.62036	−	9.73475i	0.559247	−	0.968644i	0.997560	−	0.0698213i	$-0.0222429\pi$
$102$		0			0
$103$	14.6603			1.44452			0.722259	−	0.691623i	$-0.243104\pi$
$103$	14.6603			1.44452			0.722259	+	0.691623i	$0.243104\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−19.4695			−1.88219			−0.941094	−	0.338145i	$-0.890200\pi$
$107$	−19.4695			−1.88219			−0.941094	+	0.338145i	$0.890200\pi$
$108$		0			0
$109$	−11.1962	+	6.46410i	−1.07240	+	0.619149i	−0.928835	−	0.370493i	$-0.879189\pi$
$109$	−11.1962	+	6.46410i	−1.07240	+	0.619149i	−0.143562	+	0.989641i	$0.545856\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			1.90949i			0.179630i	0.995958	+	0.0898151i	$0.0286276\pi$
$113$			1.90949i			0.179630i	−0.995958	+	0.0898151i	$0.971372\pi$
$114$		0			0
$115$			4.53590i			0.422975i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	2.60842	−	1.50597i	0.239113	−	0.138052i
$120$		0			0
$121$	−5.92820			−0.538928
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−28.5053			−2.54959
$126$		0			0
$127$	6.92820	−	12.0000i	0.614779	−	1.06483i	−0.375645	−	0.926764i	$-0.622579\pi$
$127$	6.92820	−	12.0000i	0.614779	−	1.06483i	0.990423	−	0.138064i	$-0.0440880\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−2.60842	+	1.50597i	−0.227899	+	0.131577i	−0.609602	−	0.792707i	$-0.708671\pi$
$131$	−2.60842	+	1.50597i	−0.227899	+	0.131577i	0.381704	+	0.924285i	$0.375337\pi$
$132$		0			0
$133$	4.00000	−	1.73205i	0.346844	−	0.150188i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	0.403524	−	0.698924i	0.0344754	−	0.0597131i	−0.848273	−	0.529559i	$-0.822357\pi$
$137$	0.403524	−	0.698924i	0.0344754	−	0.0597131i	0.882748	+	0.469846i	$0.155691\pi$
$138$		0			0
$139$	−9.86603	−	5.69615i	−0.836825	−	0.483141i	0.0193585	−	0.999813i	$-0.493838\pi$
$139$	−9.86603	−	5.69615i	−0.836825	−	0.483141i	−0.856184	+	0.516671i	$0.827171\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−3.56317	+	6.17158i	−0.297967	+	0.516094i
$144$		0			0
$145$			21.4641i			1.78250i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−3.15964	−	5.47266i	−0.258848	−	0.448338i	0.707086	−	0.707128i	$-0.250010\pi$
$149$	−3.15964	−	5.47266i	−0.258848	−	0.448338i	−0.965934	+	0.258790i	$0.916676\pi$
$150$		0			0
$151$	−5.07180			−0.412737			−0.206368	−	0.978474i	$-0.566165\pi$
$151$	−5.07180			−0.412737			−0.206368	+	0.978474i	$0.566165\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−8.78000	−	15.2074i	−0.705227	−	1.22149i
$156$		0			0
$157$	6.69615	+	11.5981i	0.534411	+	0.925627i	0.999192	+	0.0402013i	$0.0127999\pi$
$157$	6.69615	+	11.5981i	0.534411	+	0.925627i	−0.464780	+	0.885426i	$0.653867\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	0.551224	−	0.954747i	0.0434425	−	0.0752446i
$162$		0			0
$163$			5.00000i			0.391630i	0.980641	+	0.195815i	$0.0627352\pi$
$163$			5.00000i			0.391630i	−0.980641	+	0.195815i	$0.937265\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−11.3884	+	19.7253i	−0.881263	+	1.52639i	−0.0313242	+	0.999509i	$0.509972\pi$
$167$	−11.3884	+	19.7253i	−0.881263	+	1.52639i	−0.849938	+	0.526882i	$0.823361\pi$
$168$		0			0
$169$	−5.00000	−	8.66025i	−0.384615	−	0.666173i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	16.8611	−	9.73475i	1.28192	−	0.740119i	0.304724	−	0.952441i	$-0.401436\pi$
$173$	16.8611	−	9.73475i	1.28192	−	0.740119i	0.977200	+	0.212321i	$0.0681023\pi$
$174$		0			0
$175$	10.3301	+	5.96410i	0.780884	+	0.450844i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	22.7768			1.70242			0.851211	−	0.524824i	$-0.175869\pi$
$179$	22.7768			1.70242			0.851211	+	0.524824i	$0.175869\pi$
$180$		0			0
$181$	19.3923	+	11.1962i	1.44142	+	0.832203i	0.997945	−	0.0640837i	$-0.0204125\pi$
$181$	19.3923	+	11.1962i	1.44142	+	0.832203i	0.443474	+	0.896287i	$0.353746\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−15.2074	+	8.78000i	−1.11807	+	0.645519i
$186$		0			0
$187$	10.7321	−	6.19615i	0.784805	−	0.453108i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	6.31928i		−	0.457247i	−0.973515	−	0.228624i	$-0.926577\pi$
$191$		−	6.31928i		−	0.457247i	0.973515	−	0.228624i	$-0.0734225\pi$
$192$		0			0
$193$	−11.8923	+	6.86603i	−0.856027	+	0.494227i	−0.862680	−	0.505751i	$-0.831216\pi$
$193$	−11.8923	+	6.86603i	−0.856027	+	0.494227i	0.00665294	+	0.999978i	$0.497882\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−1.10245			−0.0785461			−0.0392731	−	0.999229i	$-0.512504\pi$
$197$	−1.10245			−0.0785461			−0.0392731	+	0.999229i	$0.512504\pi$
$198$		0			0
$199$	−0.401924	−	0.232051i	−0.0284916	−	0.0164496i	0.485687	−	0.874133i	$-0.338570\pi$
$199$	−0.401924	−	0.232051i	−0.0284916	−	0.0164496i	−0.514178	+	0.857683i	$0.671903\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	2.60842	−	4.51791i	0.183075	−	0.317095i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	16.4576	−	7.12633i	1.13839	−	0.492939i
$210$		0			0
$211$	7.33013	+	12.6962i	0.504627	+	0.874039i	0.999986	+	0.00535078i	$0.00170321\pi$
$211$	7.33013	+	12.6962i	0.504627	+	0.874039i	−0.495359	+	0.868688i	$0.664963\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	23.0327	+	13.2979i	1.57081	+	0.906910i
$216$		0			0
$217$			4.26795i			0.289727i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	5.21684i		−	0.350922i
$222$		0			0
$223$	−6.86603	−	11.8923i	−0.459783	−	0.796368i	0.539166	−	0.842199i	$-0.318740\pi$
$223$	−6.86603	−	11.8923i	−0.459783	−	0.796368i	−0.998949	+	0.0458318i	$0.985406\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−3.30734			−0.219516			−0.109758	−	0.993958i	$-0.535008\pi$
$227$	−3.30734			−0.219516			−0.109758	+	0.993958i	$0.535008\pi$
$228$		0			0
$229$	−26.3205			−1.73931			−0.869654	−	0.493662i	$-0.835658\pi$
$229$	−26.3205			−1.73931			−0.869654	+	0.493662i	$0.835658\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	7.52986	+	13.0421i	0.493297	+	0.854416i	0.999970	−	0.00772244i	$-0.00245815\pi$
$233$	7.52986	+	13.0421i	0.493297	+	0.854416i	−0.506673	+	0.862138i	$0.669125\pi$
$234$		0			0
$235$			24.7846i			1.61677i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	24.3909i		−	1.57772i	−0.614574	−	0.788859i	$-0.710672\pi$
$239$		−	24.3909i		−	1.57772i	0.614574	−	0.788859i	$-0.289328\pi$
$240$		0			0
$241$	9.69615	+	5.59808i	0.624584	+	0.360604i	0.778652	−	0.627457i	$-0.215904\pi$
$241$	9.69615	+	5.59808i	0.624584	+	0.360604i	−0.154068	+	0.988060i	$0.549237\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	12.3432	+	21.3790i	0.788576	+	1.36585i
$246$		0			0
$247$	0.866025	−	7.50000i	0.0551039	−	0.477214i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−14.2527	−	8.22878i	−0.899620	−	0.519396i	−0.0225432	−	0.999746i	$-0.507176\pi$
$251$	−14.2527	−	8.22878i	−0.899620	−	0.519396i	−0.877077	+	0.480350i	$0.840510\pi$
$252$		0			0
$253$	2.26795	−	3.92820i	0.142585	−	0.246964i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	23.0327	+	13.2979i	1.43674	+	0.829501i	0.997622	−	0.0689288i	$-0.0219581\pi$
$257$	23.0327	+	13.2979i	1.43674	+	0.829501i	0.439117	+	0.898430i	$0.355291\pi$
$258$		0			0
$259$	4.26795			0.265197
$260$		0			0
$261$		0			0
$262$		0			0
$263$	22.0779	−	12.7467i	1.36138	−	0.785995i	0.371575	−	0.928403i	$-0.378818\pi$
$263$	22.0779	−	12.7467i	1.36138	−	0.785995i	0.989808	+	0.142408i	$0.0454846\pi$
$264$		0			0
$265$			7.85641i			0.482615i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−13.9968	+	8.08108i	−0.853402	+	0.492712i	−0.861797	−	0.507253i	$-0.830661\pi$
$269$	−13.9968	+	8.08108i	−0.853402	+	0.492712i	0.00839497	+	0.999965i	$0.497328\pi$
$270$		0			0
$271$	1.39230	−	0.803848i	0.0845765	−	0.0488303i	−0.457115	−	0.889407i	$-0.651117\pi$
$271$	1.39230	−	0.803848i	0.0845765	−	0.0488303i	0.541692	+	0.840577i	$0.317784\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	42.5022	+	24.5386i	2.56298	+	1.47974i
$276$		0			0
$277$	−21.4641			−1.28965			−0.644826	−	0.764329i	$-0.723070\pi$
$277$	−21.4641			−1.28965			−0.644826	+	0.764329i	$0.723070\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	27.5506	+	15.9063i	1.64353	+	0.948892i	0.979565	+	0.201129i	$0.0644610\pi$
$281$	27.5506	+	15.9063i	1.64353	+	0.948892i	0.663965	+	0.747764i	$0.268872\pi$
$282$		0			0
$283$	−9.58846	+	5.53590i	−0.569975	+	0.329075i	−0.757139	−	0.653254i	$-0.773404\pi$
$283$	−9.58846	+	5.53590i	−0.569975	+	0.329075i	0.187165	+	0.982329i	$0.440070\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	3.96410	−	6.86603i	0.233182	−	0.403884i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	14.2527i		−	0.832650i	−0.909216	−	0.416325i	$-0.863318\pi$
$293$		−	14.2527i		−	0.832650i	0.909216	−	0.416325i	$-0.136682\pi$
$294$		0			0
$295$	3.92820	−	6.80385i	0.228709	−	0.396135i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−0.954747	−	1.65367i	−0.0552145	−	0.0956343i
$300$		0			0
$301$	−3.23205	−	5.59808i	−0.186292	−	0.322668i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−16.1622			−0.925443
$306$		0			0
$307$	−10.2679	−	17.7846i	−0.586023	−	1.01502i	−0.994747	−	0.102363i	$-0.967360\pi$
$307$	−10.2679	−	17.7846i	−0.586023	−	1.01502i	0.408724	−	0.912658i	$-0.365974\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	27.4029i		−	1.55387i	−0.629578	−	0.776937i	$-0.716772\pi$
$311$		−	27.4029i		−	1.55387i	0.629578	−	0.776937i	$-0.283228\pi$
$312$		0			0
$313$	8.19615	−	14.1962i	0.463274	−	0.802414i	−0.535848	−	0.844315i	$-0.680008\pi$
$313$	8.19615	−	14.1962i	0.463274	−	0.802414i	0.999122	+	0.0419006i	$0.0133413\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	15.2074	+	8.78000i	0.854133	+	0.493134i	0.862043	−	0.506835i	$-0.169185\pi$
$317$	15.2074	+	8.78000i	0.854133	+	0.493134i	−0.00790998	+	0.999969i	$0.502518\pi$
$318$		0			0
$319$	10.7321	−	18.5885i	0.600879	−	1.04075i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−7.82526	+	10.5418i	−0.435409	+	0.586561i
$324$		0			0
$325$	17.8923	−	10.3301i	0.992487	−	0.573012i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	3.01194	−	5.21684i	0.166054	−	0.287614i
$330$		0			0
$331$	−25.9808			−1.42803			−0.714016	−	0.700129i	$-0.753126\pi$
$331$	−25.9808			−1.42803			−0.714016	+	0.700129i	$0.753126\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−46.0653			−2.51682
$336$		0			0
$337$	26.0885	−	15.0622i	1.42113	−	0.820489i	0.424733	−	0.905319i	$-0.360368\pi$
$337$	26.0885	−	15.0622i	1.42113	−	0.820489i	0.996395	+	0.0848294i	$0.0270345\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			17.5600i			0.950928i
$342$		0			0
$343$		−	13.0000i		−	0.701934i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	5.47266	−	3.15964i	0.293788	−	0.169618i	−0.345861	−	0.938286i	$-0.612413\pi$
$347$	5.47266	−	3.15964i	0.293788	−	0.169618i	0.639649	+	0.768667i	$0.279080\pi$
$348$		0			0
$349$	23.0000			1.23116			0.615581	−	0.788074i	$-0.288921\pi$
$349$	23.0000			1.23116			0.615581	+	0.788074i	$0.288921\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−16.7530			−0.891670			−0.445835	−	0.895115i	$-0.647093\pi$
$353$	−16.7530			−0.891670			−0.445835	+	0.895115i	$0.647093\pi$
$354$		0			0
$355$	−18.5885	+	32.1962i	−0.986573	+	1.70879i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	10.9453	−	6.31928i	0.577672	−	0.333519i	−0.182536	−	0.983199i	$-0.558431\pi$
$359$	10.9453	−	6.31928i	0.577672	−	0.333519i	0.760208	+	0.649680i	$0.225097\pi$
$360$		0			0
$361$	−13.0000	+	13.8564i	−0.684211	+	0.729285i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−11.0930	+	19.2137i	−0.580635	+	1.00569i
$366$		0			0
$367$	21.9904	+	12.6962i	1.14789	+	0.662734i	0.948372	−	0.317161i	$-0.102729\pi$
$367$	21.9904	+	12.6962i	1.14789	+	0.662734i	0.199517	+	0.979894i	$0.436063\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	0.954747	−	1.65367i	0.0495680	−	0.0858543i
$372$		0			0
$373$		−	20.5359i		−	1.06331i	−0.846961	−	0.531654i	$-0.821571\pi$
$373$		−	20.5359i		−	1.06331i	0.846961	−	0.531654i	$-0.178429\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−4.51791	−	7.82526i	−0.232684	−	0.403021i
$378$		0			0
$379$	−21.5885			−1.10892			−0.554462	−	0.832209i	$-0.687076\pi$
$379$	−21.5885			−1.10892			−0.554462	+	0.832209i	$0.687076\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−17.8158	−	30.8579i	−0.910346	−	1.57677i	−0.813575	−	0.581460i	$-0.802482\pi$
$383$	−17.8158	−	30.8579i	−0.910346	−	1.57677i	−0.0967711	−	0.995307i	$-0.530851\pi$
$384$		0			0
$385$	−8.46410	−	14.6603i	−0.431371	−	0.747156i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	17.0088	−	29.4601i	0.862380	−	1.49369i	−0.00724629	−	0.999974i	$-0.502307\pi$
$389$	17.0088	−	29.4601i	0.862380	−	1.49369i	0.869626	−	0.493711i	$-0.164360\pi$
$390$		0			0
$391$			3.32051i			0.167925i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−30.1590	+	52.2369i	−1.51746	+	2.62832i
$396$		0			0
$397$	12.8923	+	22.3301i	0.647046	+	1.12072i	0.983825	+	0.179133i	$0.0573293\pi$
$397$	12.8923	+	22.3301i	0.647046	+	1.12072i	−0.336779	+	0.941584i	$0.609337\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	13.9968	−	8.08108i	0.698969	−	0.403550i	−0.107994	−	0.994152i	$-0.534443\pi$
$401$	13.9968	−	8.08108i	0.698969	−	0.403550i	0.806963	+	0.590602i	$0.201110\pi$
$402$		0			0
$403$	6.40192	+	3.69615i	0.318903	+	0.184118i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	17.5600			0.870417
$408$		0			0
$409$	−31.9808	−	18.4641i	−1.58135	−	0.912991i	−0.994663	−	0.103175i	$-0.967100\pi$
$409$	−31.9808	−	18.4641i	−1.58135	−	0.912991i	−0.586684	−	0.809816i	$-0.699567\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−1.65367	+	0.954747i	−0.0813718	+	0.0469801i
$414$		0			0
$415$	40.0526	−	23.1244i	1.96610	−	1.13513i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			16.7530i			0.818436i	0.912437	+	0.409218i	$0.134198\pi$
$419$			16.7530i			0.818436i	−0.912437	+	0.409218i	$0.865802\pi$
$420$		0			0
$421$	−9.00000	+	5.19615i	−0.438633	+	0.253245i	−0.703018	−	0.711172i	$-0.748165\pi$
$421$	−9.00000	+	5.19615i	−0.438633	+	0.253245i	0.264385	+	0.964417i	$0.414831\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−35.9271			−1.74272
$426$		0			0
$427$	3.40192	+	1.96410i	0.164631	+	0.0950495i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	5.21684	−	9.03583i	0.251286	−	0.435240i	−0.712594	−	0.701577i	$-0.752480\pi$
$431$	5.21684	−	9.03583i	0.251286	−	0.435240i	0.963880	+	0.266336i	$0.0858131\pi$
$432$		0			0
$433$	−19.5000	−	11.2583i	−0.937110	−	0.541041i	−0.0480569	−	0.998845i	$-0.515303\pi$
$433$	−19.5000	−	11.2583i	−0.937110	−	0.541041i	−0.889053	+	0.457804i	$0.848636\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−0.551224	+	4.77374i	−0.0263686	+	0.228359i
$438$		0			0
$439$	−1.79423	−	3.10770i	−0.0856339	−	0.148322i	0.820027	−	0.572324i	$-0.193958\pi$
$439$	−1.79423	−	3.10770i	−0.0856339	−	0.148322i	−0.905661	+	0.424002i	$0.860625\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	2.60842	+	1.50597i	0.123930	+	0.0715509i	0.560683	−	0.828030i	$-0.310539\pi$
$443$	2.60842	+	1.50597i	0.123930	+	0.0715509i	−0.436754	+	0.899581i	$0.643872\pi$
$444$		0			0
$445$		−	50.7846i		−	2.40742i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			12.8548i			0.606656i	0.952886	+	0.303328i	$0.0980978\pi$
$449$			12.8548i			0.606656i	−0.952886	+	0.303328i	$0.901902\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−7.12633			−0.334088
$456$		0			0
$457$	17.3923			0.813578			0.406789	−	0.913522i	$-0.366648\pi$
$457$	17.3923			0.813578			0.406789	+	0.913522i	$0.366648\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−4.37021	−	7.56943i	−0.203541	−	0.352544i	0.746126	−	0.665805i	$-0.231912\pi$
$461$	−4.37021	−	7.56943i	−0.203541	−	0.352544i	−0.949667	+	0.313261i	$0.898578\pi$
$462$		0			0
$463$			20.3205i			0.944374i	0.881498	+	0.472187i	$0.156535\pi$
$463$			20.3205i			0.944374i	−0.881498	+	0.472187i	$0.843465\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	9.84287i		−	0.455474i	−0.973723	−	0.227737i	$-0.926867\pi$
$467$		−	9.84287i		−	0.455474i	0.973723	−	0.227737i	$-0.0731326\pi$
$468$		0			0
$469$	9.69615	+	5.59808i	0.447727	+	0.258495i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−13.2979	−	23.0327i	−0.611439	−	1.05904i
$474$		0			0
$475$	−51.6506	−	5.96410i	−2.36989	−	0.273652i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−27.2948	−	15.7586i	−1.24713	−	0.720030i	−0.276593	−	0.960987i	$-0.589206\pi$
$479$	−27.2948	−	15.7586i	−1.24713	−	0.720030i	−0.970536	+	0.240957i	$0.922539\pi$
$480$		0			0
$481$	3.69615	−	6.40192i	0.168530	−	0.291903i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−21.3790	−	12.3432i	−0.970770	−	0.560474i
$486$		0			0
$487$	4.39230			0.199034			0.0995172	−	0.995036i	$-0.468270\pi$
$487$	4.39230			0.199034			0.0995172	+	0.995036i	$0.468270\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	4.00627	−	2.31302i	0.180800	−	0.104385i	−0.406868	−	0.913487i	$-0.633379\pi$
$491$	4.00627	−	2.31302i	0.180800	−	0.104385i	0.587669	+	0.809102i	$0.300046\pi$
$492$		0			0
$493$			15.7128i			0.707670i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	7.82526	−	4.51791i	0.351011	−	0.202656i
$498$		0			0
$499$	−30.3109	+	17.5000i	−1.35690	+	0.783408i	−0.989205	−	0.146538i	$-0.953187\pi$
$499$	−30.3109	+	17.5000i	−1.35690	+	0.783408i	−0.367697	+	0.929946i	$0.619854\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	27.2948	+	15.7586i	1.21701	+	0.702643i	0.964278	−	0.264893i	$-0.0853365\pi$
$503$	27.2948	+	15.7586i	1.21701	+	0.702643i	0.252735	+	0.967535i	$0.418670\pi$
$504$		0			0
$505$	46.2487			2.05804
$506$		0			0
$507$		0			0
$508$		0			0
$509$	12.3432	+	7.12633i	0.547101	+	0.315869i	0.747952	−	0.663753i	$-0.231037\pi$
$509$	12.3432	+	7.12633i	0.547101	+	0.315869i	−0.200851	+	0.979622i	$0.564371\pi$
$510$		0			0
$511$	4.66987	−	2.69615i	0.206583	−	0.119271i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	30.1590	+	52.2369i	1.32896	+	2.30183i
$516$		0			0
$517$	12.3923	−	21.4641i	0.545013	−	0.943990i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	35.6317i		−	1.56105i	−0.625124	−	0.780526i	$-0.714951\pi$
$521$		−	35.6317i		−	1.56105i	0.625124	−	0.780526i	$-0.285049\pi$
$522$		0			0
$523$	0.866025	−	1.50000i	0.0378686	−	0.0655904i	−0.846470	−	0.532437i	$-0.821276\pi$
$523$	0.866025	−	1.50000i	0.0378686	−	0.0655904i	0.884339	+	0.466846i	$0.154610\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−6.42741	−	11.1326i	−0.279982	−	0.484944i
$528$		0			0
$529$	−10.8923	−	18.8660i	−0.473578	−	0.820262i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−40.0526	−	69.3731i	−1.73162	−	2.99926i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	24.6863i		−	1.06332i
$540$		0			0
$541$	7.89230	−	13.6699i	0.339317	−	0.587714i	−0.644988	−	0.764193i	$-0.723138\pi$
$541$	7.89230	−	13.6699i	0.339317	−	0.587714i	0.984304	+	0.176479i	$0.0564708\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−46.0653	−	26.5958i	−1.97322	−	1.13924i
$546$		0			0
$547$	−1.79423	+	3.10770i	−0.0767157	+	0.132875i	−0.901831	−	0.432089i	$-0.857777\pi$
$547$	−1.79423	+	3.10770i	−0.0767157	+	0.132875i	0.825115	+	0.564964i	$0.191110\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−2.60842	+	22.5896i	−0.111122	+	0.962348i
$552$		0			0
$553$	12.6962	−	7.33013i	0.539895	−	0.311709i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	16.0540	−	27.8064i	0.680231	−	1.17820i	−0.294679	−	0.955596i	$-0.595213\pi$
$557$	16.0540	−	27.8064i	0.680231	−	1.17820i	0.974910	−	0.222599i	$-0.0714540\pi$
$558$		0			0
$559$	−11.1962			−0.473547
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−1.39785			−0.0589122			−0.0294561	−	0.999566i	$-0.509378\pi$
$563$	−1.39785			−0.0589122			−0.0294561	+	0.999566i	$0.509378\pi$
$564$		0			0
$565$	−6.80385	+	3.92820i	−0.286240	+	0.165261i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			10.4337i			0.437402i	0.975792	+	0.218701i	$0.0701820\pi$
$569$			10.4337i			0.437402i	−0.975792	+	0.218701i	$0.929818\pi$
$570$		0			0
$571$		−	17.7846i		−	0.744263i	−0.928180	−	0.372131i	$-0.878627\pi$
$571$		−	17.7846i		−	0.744263i	0.928180	−	0.372131i	$-0.121373\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−11.3884	+	6.57511i	−0.474930	+	0.274201i
$576$		0			0
$577$	23.3205			0.970845			0.485423	−	0.874280i	$-0.338666\pi$
$577$	23.3205			0.970845			0.485423	+	0.874280i	$0.338666\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−11.2407			−0.466344
$582$		0			0
$583$	3.92820	−	6.80385i	0.162690	−	0.281787i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−38.4959	+	22.2256i	−1.58890	+	0.917350i	−0.595407	+	0.803425i	$0.703009\pi$
$587$	−38.4959	+	22.2256i	−1.58890	+	0.917350i	−0.993489	+	0.113925i	$0.963658\pi$
$588$		0			0
$589$	−7.39230	−	17.0718i	−0.304595	−	0.703431i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	0.551224	−	0.954747i	0.0226360	−	0.0392068i	−0.854485	−	0.519475i	$-0.826127\pi$
$593$	0.551224	−	0.954747i	0.0226360	−	0.0392068i	0.877122	+	0.480268i	$0.159461\pi$
$594$		0			0
$595$	10.7321	+	6.19615i	0.439971	+	0.254017i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−10.6895	+	18.5148i	−0.436761	+	0.756492i	−0.997438	−	0.0715426i	$-0.977208\pi$
$599$	−10.6895	+	18.5148i	−0.436761	+	0.756492i	0.560676	+	0.828035i	$0.310541\pi$
$600$		0			0
$601$		−	19.7321i		−	0.804887i	−0.915445	−	0.402444i	$-0.868161\pi$
$601$		−	19.7321i		−	0.804887i	0.915445	−	0.402444i	$-0.131839\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−12.1955	−	21.1232i	−0.495816	−	0.858779i
$606$		0			0
$607$	−21.3397			−0.866154			−0.433077	−	0.901357i	$-0.642572\pi$
$607$	−21.3397			−0.866154			−0.433077	+	0.901357i	$0.642572\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−5.21684	−	9.03583i	−0.211051	−	0.365550i
$612$		0			0
$613$	3.46410	+	6.00000i	0.139914	+	0.242338i	0.927464	−	0.373913i	$-0.121984\pi$
$613$	3.46410	+	6.00000i	0.139914	+	0.242338i	−0.787550	+	0.616251i	$0.788651\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	2.05719	−	3.56317i	0.0828195	−	0.143448i	−0.821640	−	0.570006i	$-0.806941\pi$
$617$	2.05719	−	3.56317i	0.0828195	−	0.143448i	0.904460	+	0.426558i	$0.140274\pi$
$618$		0			0
$619$			5.39230i			0.216735i	0.994111	+	0.108368i	$0.0345623\pi$
$619$			5.39230i			0.216735i	−0.994111	+	0.108368i	$0.965438\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−6.17158	+	10.6895i	−0.247259	+	0.428266i
$624$		0			0
$625$	−28.8205	−	49.9186i	−1.15282	−	1.99674i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−11.1326	+	6.42741i	−0.443886	+	0.256278i
$630$		0			0
$631$	−1.91858	−	1.10770i	−0.0763776	−	0.0440966i	0.461325	−	0.887231i	$-0.347374\pi$
$631$	−1.91858	−	1.10770i	−0.0763776	−	0.0440966i	−0.537702	+	0.843135i	$0.680708\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	57.0107			2.26240
$636$		0			0
$637$	−9.00000	−	5.19615i	−0.356593	−	0.205879i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−3.30734	+	1.90949i	−0.130632	+	0.0754205i	−0.563892	−	0.825849i	$-0.690697\pi$
$641$	−3.30734	+	1.90949i	−0.130632	+	0.0754205i	0.433260	+	0.901269i	$0.357363\pi$
$642$		0			0
$643$	−30.7750	+	17.7679i	−1.21365	+	0.700700i	−0.963552	−	0.267522i	$-0.913795\pi$
$643$	−30.7750	+	17.7679i	−1.21365	+	0.700700i	−0.250095	+	0.968221i	$0.580462\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	25.4934i		−	1.00225i	−0.865375	−	0.501124i	$-0.832920\pi$
$647$		−	25.4934i		−	1.00225i	0.865375	−	0.501124i	$-0.167080\pi$
$648$		0			0
$649$	−6.80385	+	3.92820i	−0.267074	+	0.154195i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	25.4934			0.997633			0.498817	−	0.866708i	$-0.333768\pi$
$653$	25.4934			0.997633			0.498817	+	0.866708i	$0.333768\pi$
$654$		0			0
$655$	−10.7321	−	6.19615i	−0.419336	−	0.242104i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	5.47266	−	9.47893i	0.213185	−	0.369247i	−0.739525	−	0.673129i	$-0.764950\pi$
$659$	5.47266	−	9.47893i	0.213185	−	0.369247i	0.952709	+	0.303883i	$0.0982831\pi$
$660$		0			0
$661$	−20.7846	−	12.0000i	−0.808428	−	0.466746i	0.0379819	−	0.999278i	$-0.487907\pi$
$661$	−20.7846	−	12.0000i	−0.808428	−	0.466746i	−0.846410	+	0.532533i	$0.821240\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	14.4004	+	10.6895i	0.558422	+	0.414521i
$666$		0			0
$667$	2.87564	+	4.98076i	0.111345	+	0.192856i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	13.9968	+	8.08108i	0.540342	+	0.311967i
$672$		0			0
$673$			24.3731i			0.939513i	0.882796	+	0.469756i	$0.155658\pi$
$673$			24.3731i			0.939513i	−0.882796	+	0.469756i	$0.844342\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			20.8673i			0.801997i	0.916079	+	0.400999i	$0.131337\pi$
$677$			20.8673i			0.801997i	−0.916079	+	0.400999i	$0.868663\pi$
$678$		0			0
$679$	3.00000	+	5.19615i	0.115129	+	0.199410i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−49.8843			−1.90877			−0.954385	−	0.298578i	$-0.903488\pi$
$683$	−49.8843			−1.90877			−0.954385	+	0.298578i	$0.903488\pi$
$684$		0			0
$685$	3.32051			0.126870
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−1.65367	−	2.86424i	−0.0629999	−	0.109119i
$690$		0			0
$691$			36.2487i			1.37897i	0.724302	+	0.689483i	$0.242162\pi$
$691$			36.2487i			1.37897i	−0.724302	+	0.689483i	$0.757838\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	46.8724i		−	1.77797i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	17.4123	+	30.1590i	0.657654	+	1.13909i	0.981221	+	0.192885i	$0.0617843\pi$
$701$	17.4123	+	30.1590i	0.657654	+	1.13909i	−0.323568	+	0.946205i	$0.604882\pi$
$702$		0			0
$703$	−17.0718	+	7.39230i	−0.643875	+	0.278806i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−9.73475	−	5.62036i	−0.366113	−	0.211375i
$708$		0			0
$709$	19.0885	−	33.0622i	0.716882	−	1.24168i	−0.245347	−	0.969435i	$-0.578902\pi$
$709$	19.0885	−	33.0622i	0.716882	−	1.24168i	0.962229	−	0.272241i	$-0.0877647\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−4.07481	−	2.35259i	−0.152603	−	0.0881054i
$714$		0			0
$715$	−29.3205			−1.09652
$716$		0			0
$717$		0			0
$718$		0			0
$719$	17.1169	−	9.88245i	0.638353	−	0.368553i	−0.145627	−	0.989340i	$-0.546520\pi$
$719$	17.1169	−	9.88245i	0.638353	−	0.368553i	0.783980	+	0.620786i	$0.213187\pi$
$720$		0			0
$721$		−	14.6603i		−	0.545976i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−53.8906	+	31.1137i	−2.00145	+	1.15554i
$726$		0			0
$727$	28.7942	−	16.6244i	1.06792	−	0.616563i	0.140307	−	0.990108i	$-0.455191\pi$
$727$	28.7942	−	16.6244i	1.06792	−	0.616563i	0.927612	+	0.373545i	$0.121858\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	16.8611	+	9.73475i	0.623630	+	0.360053i
$732$		0			0
$733$	15.6077			0.576483			0.288242	−	0.957558i	$-0.406929\pi$
$733$	15.6077			0.576483			0.288242	+	0.957558i	$0.406929\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	39.8938	+	23.0327i	1.46951	+	0.848419i
$738$		0			0
$739$	22.6699	−	13.0885i	0.833925	−	0.481467i	−0.0212698	−	0.999774i	$-0.506771\pi$
$739$	22.6699	−	13.0885i	0.833925	−	0.481467i	0.855195	+	0.518307i	$0.173438\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−3.56317	−	6.17158i	−0.130720	−	0.226413i	0.793234	−	0.608916i	$-0.208396\pi$
$743$	−3.56317	−	6.17158i	−0.130720	−	0.226413i	−0.923954	+	0.382503i	$0.875062\pi$
$744$		0			0
$745$	13.0000	−	22.5167i	0.476283	−	0.824947i
$746$		0			0
$747$		0			0
$748$		0			0
$749$			19.4695i			0.711400i
$750$		0			0
$751$	−6.06218	+	10.5000i	−0.221212	+	0.383150i	−0.955176	−	0.296038i	$-0.904335\pi$
$751$	−6.06218	+	10.5000i	−0.221212	+	0.383150i	0.733964	+	0.679188i	$0.237668\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−10.4337	−	18.0717i	−0.379720	−	0.657695i
$756$		0			0
$757$	−1.37564	−	2.38269i	−0.0499986	−	0.0866002i	0.839943	−	0.542675i	$-0.182588\pi$
$757$	−1.37564	−	2.38269i	−0.0499986	−	0.0866002i	−0.889942	+	0.456075i	$0.849255\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	45.4745			1.64845			0.824225	−	0.566262i	$-0.191611\pi$
$761$	45.4745			1.64845			0.824225	+	0.566262i	$0.191611\pi$
$762$		0			0
$763$	6.46410	+	11.1962i	0.234016	+	0.405328i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			3.30734i			0.119421i
$768$		0			0
$769$	−7.03590	+	12.1865i	−0.253721	+	0.439458i	−0.964547	−	0.263910i	$-0.914988\pi$
$769$	−7.03590	+	12.1865i	−0.253721	+	0.439458i	0.710826	+	0.703368i	$0.248321\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	33.7222	+	19.4695i	1.21290	+	0.700269i	0.963390	−	0.268103i	$-0.0863966\pi$
$773$	33.7222	+	19.4695i	1.21290	+	0.700269i	0.249512	+	0.968372i	$0.419730\pi$
$774$		0			0
$775$	25.4545	−	44.0885i	0.914352	−	1.58370i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	32.1962	−	18.5885i	1.15207	−	0.665147i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−27.5506	+	47.7190i	−0.983322	+	1.70316i
$786$		0			0
$787$	45.5885			1.62505			0.812526	−	0.582924i	$-0.198092\pi$
$787$	45.5885			1.62505			0.812526	+	0.582924i	$0.198092\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	1.90949			0.0678938
$792$		0			0
$793$	5.89230	−	3.40192i	0.209242	−	0.120806i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	15.6505i		−	0.554370i	−0.960817	−	0.277185i	$-0.910599\pi$
$797$		−	15.6505i		−	0.554370i	0.960817	−	0.277185i	$-0.0894014\pi$
$798$		0			0
$799$			18.1436i			0.641874i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	19.2137	−	11.0930i	0.678036	−	0.391464i
$804$		0			0
$805$	4.53590			0.159869
$806$		0			0
$807$		0			0
$808$		0			0
$809$	43.0534			1.51368			0.756838	−	0.653602i	$-0.226743\pi$
$809$	43.0534			1.51368			0.756838	+	0.653602i	$0.226743\pi$
$810$		0			0
$811$	18.9282	−	32.7846i	0.664659	−	1.15122i	−0.314719	−	0.949185i	$-0.601910\pi$
$811$	18.9282	−	32.7846i	0.664659	−	1.15122i	0.979378	−	0.202038i	$-0.0647565\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−17.8158	+	10.2860i	−0.624061	+	0.360302i
$816$		0			0
$817$	22.6244	+	16.7942i	0.791526	+	0.587556i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−3.71087	+	6.42741i	−0.129510	+	0.224318i	−0.923487	−	0.383630i	$-0.874674\pi$
$821$	−3.71087	+	6.42741i	−0.129510	+	0.224318i	0.793977	+	0.607948i	$0.208007\pi$
$822$		0			0
$823$	−36.7128	−	21.1962i	−1.27973	−	0.738851i	−0.302930	−	0.953013i	$-0.597965\pi$
$823$	−36.7128	−	21.1962i	−1.27973	−	0.738851i	−0.976798	+	0.214161i	$0.931298\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	20.6801	−	35.8189i	0.719117	−	1.24555i	−0.242234	−	0.970218i	$-0.577880\pi$
$827$	20.6801	−	35.8189i	0.719117	−	1.24555i	0.961350	−	0.275329i	$-0.0887866\pi$
$828$		0			0
$829$			4.26795i			0.148232i	0.997250	+	0.0741160i	$0.0236135\pi$
$829$			4.26795i			0.148232i	−0.997250	+	0.0741160i	$0.976386\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	9.03583	+	15.6505i	0.313073	+	0.542258i
$834$		0			0
$835$	−93.7128			−3.24307
$836$		0			0
$837$		0			0
$838$		0			0
$839$	0.255824	+	0.443100i	0.00883202	+	0.0152975i	0.870408	−	0.492332i	$-0.163855\pi$
$839$	0.255824	+	0.443100i	0.00883202	+	0.0152975i	−0.861576	+	0.507629i	$0.830522\pi$
$840$		0			0
$841$	−0.892305	−	1.54552i	−0.0307691	−	0.0532937i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	20.5719	−	35.6317i	0.707697	−	1.22577i
$846$		0			0
$847$			5.92820i			0.203695i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−2.35259	+	4.07481i	−0.0806459	+	0.139683i
$852$		0			0
$853$	−16.8923	−	29.2583i	−0.578382	−	1.00179i	−0.995665	−	0.0930099i	$-0.970351\pi$
$853$	−16.8923	−	29.2583i	−0.578382	−	1.00179i	0.417284	−	0.908776i	$-0.362982\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−5.72848	+	3.30734i	−0.195681	+	0.112977i	−0.594639	−	0.803992i	$-0.702705\pi$
$857$	−5.72848	+	3.30734i	−0.195681	+	0.112977i	0.398958	+	0.916969i	$0.369372\pi$
$858$		0			0
$859$	9.86603	+	5.69615i	0.336624	+	0.194350i	0.658778	−	0.752337i	$-0.271074\pi$
$859$	9.86603	+	5.69615i	0.336624	+	0.194350i	−0.322154	+	0.946687i	$0.604407\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	24.6863			0.840333			0.420166	−	0.907447i	$-0.361972\pi$
$863$	24.6863			0.840333			0.420166	+	0.907447i	$0.361972\pi$
$864$		0			0
$865$	69.3731	+	40.0526i	2.35876	+	1.36183i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	52.2369	−	30.1590i	1.77202	−	1.02307i
$870$		0			0
$871$	16.7942	−	9.69615i	0.569051	−	0.328542i
$872$		0			0
$873$		0			0
$874$		0			0
$875$			28.5053i			0.963656i
$876$		0			0
$877$	−21.1077	+	12.1865i	−0.712756	+	0.411510i	−0.812081	−	0.583545i	$-0.801665\pi$
$877$	−21.1077	+	12.1865i	−0.712756	+	0.411510i	0.0993245	+	0.995055i	$0.468332\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−28.8007			−0.970321			−0.485161	−	0.874425i	$-0.661239\pi$
$881$	−28.8007			−0.970321			−0.485161	+	0.874425i	$0.661239\pi$
$882$		0			0
$883$	−31.6699	−	18.2846i	−1.06578	−	0.615326i	−0.138752	−	0.990327i	$-0.544309\pi$
$883$	−31.6699	−	18.2846i	−1.06578	−	0.615326i	−0.927024	+	0.375001i	$0.877642\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	0.954747	−	1.65367i	0.0320573	−	0.0555248i	−0.849552	−	0.527505i	$-0.823127\pi$
$887$	0.954747	−	1.65367i	0.0320573	−	0.0555248i	0.881609	+	0.471981i	$0.156461\pi$
$888$		0			0
$889$	−12.0000	−	6.92820i	−0.402467	−	0.232364i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−3.01194	+	26.0842i	−0.100791	+	0.872874i
$894$		0			0
$895$	46.8564	+	81.1577i	1.56624	+	2.71280i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−19.2822	−	11.1326i	−0.643098	−	0.371293i
$900$		0			0
$901$			5.75129i			0.191603i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			92.1307i			3.06253i
$906$		0			0
$907$	−24.5885	−	42.5885i	−0.816446	−	1.41413i	−0.908285	−	0.418353i	$-0.862608\pi$
$907$	−24.5885	−	42.5885i	−0.816446	−	1.41413i	0.0918384	−	0.995774i	$-0.470726\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	24.6863			0.817895			0.408947	−	0.912558i	$-0.365896\pi$
$911$	24.6863			0.817895			0.408947	+	0.912558i	$0.365896\pi$
$912$		0			0
$913$	−46.2487			−1.53061
$914$		0			0
$915$		0			0
$916$		0			0
$917$	1.50597	+	2.60842i	0.0497315	+	0.0861376i
$918$		0			0
$919$			8.17691i			0.269732i	0.990864	+	0.134866i	$0.0430603\pi$
$919$			8.17691i			0.269732i	−0.990864	+	0.134866i	$0.956940\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	15.6505i		−	0.515143i
$924$		0			0
$925$	−44.0885	−	25.4545i	−1.44962	−	0.836938i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−17.4123	−	30.1590i	−0.571279	−	0.989485i	−0.996435	−	0.0843645i	$-0.973114\pi$
$929$	−17.4123	−	30.1590i	−0.571279	−	0.989485i	0.425156	−	0.905120i	$-0.360219\pi$
$930$		0			0
$931$	10.3923	+	24.0000i	0.340594	+	0.786568i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	44.1558	+	25.4934i	1.44405	+	0.833723i
$936$		0			0
$937$	−10.6962	+	18.5263i	−0.349428	+	0.605227i	−0.986148	−	0.165868i	$-0.946957\pi$
$937$	−10.6962	+	18.5263i	−0.349428	+	0.605227i	0.636720	+	0.771095i	$0.280291\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−15.6505	−	9.03583i	−0.510192	−	0.294560i	0.222721	−	0.974882i	$-0.428506\pi$
$941$	−15.6505	−	9.03583i	−0.510192	−	0.294560i	−0.732913	+	0.680323i	$0.761840\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	31.1137	−	17.9635i	1.01106	−	0.583736i	0.0995588	−	0.995032i	$-0.468257\pi$
$947$	31.1137	−	17.9635i	1.01106	−	0.583736i	0.911502	+	0.411295i	$0.134924\pi$
$948$		0			0
$949$		−	9.33975i		−	0.303181i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	39.8938	−	23.0327i	1.29229	−	0.746101i	0.313226	−	0.949678i	$-0.398590\pi$
$953$	39.8938	−	23.0327i	1.29229	−	0.746101i	0.979059	+	0.203577i	$0.0652568\pi$
$954$		0			0
$955$	22.5167	−	13.0000i	0.728622	−	0.420670i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−0.698924	−	0.403524i	−0.0225694	−	0.0130305i
$960$		0			0
$961$	−12.7846			−0.412407
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−48.9296	−	28.2495i	−1.57510	−	0.909384i
$966$		0			0
$967$	46.9186	−	27.0885i	1.50880	−	0.871106i	0.508853	−	0.860854i	$-0.330070\pi$
$967$	46.9186	−	27.0885i	1.50880	−	0.871106i	0.999947	−	0.0102523i	$-0.00326348\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−12.3432	−	21.3790i	−0.396111	−	0.686085i	0.597131	−	0.802144i	$-0.296307\pi$
$971$	−12.3432	−	21.3790i	−0.396111	−	0.686085i	−0.993242	+	0.116059i	$0.962974\pi$
$972$		0			0
$973$	−5.69615	+	9.86603i	−0.182610	+	0.316290i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	45.5537i		−	1.45739i	−0.684838	−	0.728696i	$-0.740127\pi$
$977$		−	45.5537i		−	1.45739i	0.684838	−	0.728696i	$-0.259873\pi$
$978$		0			0
$979$	−25.3923	+	43.9808i	−0.811542	+	1.40563i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	10.6895	+	18.5148i	0.340942	+	0.590529i	0.984608	−	0.174778i	$-0.0559207\pi$
$983$	10.6895	+	18.5148i	0.340942	+	0.590529i	−0.643666	+	0.765307i	$0.722587\pi$
$984$		0			0
$985$	−2.26795	−	3.92820i	−0.0722629	−	0.125163i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	7.12633			0.226604
$990$		0			0
$991$	6.99038	+	12.1077i	0.222057	+	0.384614i	0.955432	−	0.295210i	$-0.0953896\pi$
$991$	6.99038	+	12.1077i	0.222057	+	0.384614i	−0.733376	+	0.679824i	$0.762056\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	1.90949i		−	0.0605351i
$996$		0			0
$997$	−13.1603	+	22.7942i	−0.416789	+	0.721900i	−0.995614	−	0.0935513i	$-0.970178\pi$
$997$	−13.1603	+	22.7942i	−0.416789	+	0.721900i	0.578825	+	0.815452i	$0.303511\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2736.2.bm.p.1855.4	yes	8
3.2	odd	2	inner	2736.2.bm.p.1855.1	yes	8
4.3	odd	2		2736.2.bm.q.1855.4	yes	8
12.11	even	2		2736.2.bm.q.1855.1	yes	8
19.8	odd	6		2736.2.bm.q.559.4	yes	8
57.8	even	6		2736.2.bm.q.559.1	yes	8
76.27	even	6	inner	2736.2.bm.p.559.4	yes	8
228.179	odd	6	inner	2736.2.bm.p.559.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2736.2.bm.p.559.1	✓	8	228.179	odd	6	inner
2736.2.bm.p.559.4	yes	8	76.27	even	6	inner
2736.2.bm.p.1855.1	yes	8	3.2	odd	2	inner
2736.2.bm.p.1855.4	yes	8	1.1	even	1	trivial
2736.2.bm.q.559.1	yes	8	57.8	even	6
2736.2.bm.q.559.4	yes	8	19.8	odd	6
2736.2.bm.q.1855.1	yes	8	12.11	even	2
2736.2.bm.q.1855.4	yes	8	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2736.bm (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(2.05719 + 3.56317i) q^{5} -1.00000i q^{7} +O(q^{10})\)	\(q+(2.05719 + 3.56317i) q^{5} -1.00000i q^{7} -4.11439i q^{11} +(-1.50000 - 0.866025i) q^{13} +(1.50597 + 2.60842i) q^{17} +(1.73205 + 4.00000i) q^{19} +(0.954747 + 0.551224i) q^{23} +(-5.96410 + 10.3301i) q^{25} +(4.51791 + 2.60842i) q^{29} -4.26795 q^{31} +(3.56317 - 2.05719i) q^{35} +4.26795i q^{37} +(5.59808 - 3.23205i) q^{43} +(5.21684 + 3.01194i) q^{47} +6.00000 q^{49} +(1.65367 + 0.954747i) q^{53} +(14.6603 - 8.46410i) q^{55} +(-0.954747 - 1.65367i) q^{59} +(-1.96410 + 3.40192i) q^{61} -7.12633i q^{65} +(-5.59808 + 9.69615i) q^{67} +(4.51791 + 7.82526i) q^{71} +(2.69615 + 4.66987i) q^{73} -4.11439 q^{77} +(7.33013 + 12.6962i) q^{79} -11.2407i q^{83} +(-6.19615 + 10.7321i) q^{85} +(-10.6895 - 6.17158i) q^{89} +(-0.866025 + 1.50000i) q^{91} +(-10.6895 + 14.4004i) q^{95} +(-5.19615 + 3.00000i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 12 q^{13} - 20 q^{25} - 48 q^{31} + 24 q^{43} + 48 q^{49} + 48 q^{55} + 12 q^{61} - 24 q^{67} - 20 q^{73} + 24 q^{79} - 8 q^{85}+O(q^{100}) \) 8 * q - 12 * q^13 - 20 * q^25 - 48 * q^31 + 24 * q^43 + 48 * q^49 + 48 * q^55 + 12 * q^61 - 24 * q^67 - 20 * q^73 + 24 * q^79 - 8 * q^85

Introduction

L-functions

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