LMFDB - Embedded newform 2736.2.cg.a.2591.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2736,2,Mod(1151,2736)]

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, 3, 4]))

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

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

chi := DirichletCharacter("2736.1151");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2736 = 2^{4} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2736.cg (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:	$24$
Relative dimension:	$12$ over $\Q(\zeta_{6})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			2591.3
Character	$\chi$	$=$	2736.2591
Dual form			2736.2.cg.a.1151.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.15944 + 1.24676i) q^{5} +0.872468i q^{7} +O(q^{10})$	$q+(-2.15944 + 1.24676i) q^{5} +0.872468i q^{7} +2.41030 q^{11} +(-1.98130 + 3.43171i) q^{13} +(0.303025 - 0.174952i) q^{17} +(-1.35622 - 4.14254i) q^{19} +(-3.88372 + 6.72680i) q^{23} +(0.608801 - 1.05447i) q^{25} +(-7.96395 - 4.59799i) q^{29} -2.43850i q^{31} +(-1.08776 - 1.88405i) q^{35} +6.18021 q^{37} +(-1.31650 + 0.760080i) q^{41} +(5.24748 - 3.02963i) q^{43} +(-4.05152 + 7.01744i) q^{47} +6.23880 q^{49} +(-3.41943 - 1.97421i) q^{53} +(-5.20491 + 3.00506i) q^{55} +(1.55163 + 2.68750i) q^{59} +(6.53184 - 11.3135i) q^{61} -9.88080i q^{65} +(5.04454 + 2.91247i) q^{67} +(-4.65667 - 8.06559i) q^{71} +(-1.01576 - 1.75935i) q^{73} +2.10291i q^{77} +(-10.8347 + 6.25541i) q^{79} -6.29106 q^{83} +(-0.436244 + 0.755598i) q^{85} +(-14.0641 - 8.11993i) q^{89} +(-2.99406 - 1.72862i) q^{91} +(8.09342 + 7.25472i) q^{95} +(-8.90950 - 15.4317i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q+O(q^{10}) $ 24 * q	$ 24 q + 12 q^{13} - 12 q^{19} + 8 q^{25} + 16 q^{37} - 12 q^{43} + 16 q^{49} + 12 q^{55} - 60 q^{67} + 8 q^{73} + 12 q^{79} + 16 q^{85} - 12 q^{91} - 32 q^{97}+O(q^{100}) $ 24 * q + 12 * q^13 - 12 * q^19 + 8 * q^25 + 16 * q^37 - 12 * q^43 + 16 * q^49 + 12 * q^55 - 60 * q^67 + 8 * q^73 + 12 * q^79 + 16 * q^85 - 12 * q^91 - 32 * 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
$5$	−2.15944	+	1.24676i	−0.965733	+	0.557566i	−0.897933	−	0.440133i	$-0.854931\pi$
$5$	−2.15944	+	1.24676i	−0.965733	+	0.557566i	−0.0678003	+	0.997699i	$0.521598\pi$
$6$		0			0
$7$			0.872468i			0.329762i	0.986313	+	0.164881i	$0.0527240\pi$
$7$			0.872468i			0.329762i	−0.986313	+	0.164881i	$0.947276\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.41030			0.726733			0.363367	−	0.931646i	$-0.381627\pi$
$11$	2.41030			0.726733			0.363367	+	0.931646i	$0.381627\pi$
$12$		0			0
$13$	−1.98130	+	3.43171i	−0.549514	+	0.951786i	0.448794	+	0.893635i	$0.351854\pi$
$13$	−1.98130	+	3.43171i	−0.549514	+	0.951786i	−0.998308	+	0.0581510i	$0.981480\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	0.303025	−	0.174952i	0.0734945	−	0.0424321i	−0.462802	−	0.886461i	$-0.653156\pi$
$17$	0.303025	−	0.174952i	0.0734945	−	0.0424321i	0.536297	+	0.844029i	$0.319823\pi$
$18$		0			0
$19$	−1.35622	−	4.14254i	−0.311138	−	0.950365i
$19$	−1.35622	−	4.14254i	−0.311138	−	0.950365i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−3.88372	+	6.72680i	−0.809812	+	1.40264i	0.103182	+	0.994663i	$0.467098\pi$
$23$	−3.88372	+	6.72680i	−0.809812	+	1.40264i	−0.912994	+	0.407973i	$0.866236\pi$
$24$		0			0
$25$	0.608801	−	1.05447i	0.121760	−	0.210895i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−7.96395	−	4.59799i	−1.47887	−	0.853825i	−0.479154	−	0.877731i	$-0.659057\pi$
$29$	−7.96395	−	4.59799i	−1.47887	−	0.853825i	−0.999714	+	0.0239054i	$0.992390\pi$
$30$		0			0
$31$		−	2.43850i		−	0.437967i	−0.975729	−	0.218983i	$-0.929726\pi$
$31$		−	2.43850i		−	0.437967i	0.975729	−	0.218983i	$-0.0702741\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−1.08776	−	1.88405i	−0.183864	−	0.318462i
$36$		0			0
$37$	6.18021			1.01602			0.508010	−	0.861351i	$-0.330381\pi$
$37$	6.18021			1.01602			0.508010	+	0.861351i	$0.330381\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−1.31650	+	0.760080i	−0.205602	+	0.118705i	−0.599266	−	0.800550i	$-0.704541\pi$
$41$	−1.31650	+	0.760080i	−0.205602	+	0.118705i	0.393664	+	0.919255i	$0.371207\pi$
$42$		0			0
$43$	5.24748	−	3.02963i	0.800233	−	0.462015i	−0.0433195	−	0.999061i	$-0.513793\pi$
$43$	5.24748	−	3.02963i	0.800233	−	0.462015i	0.843553	+	0.537046i	$0.180460\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−4.05152	+	7.01744i	−0.590975	+	1.02360i	0.403126	+	0.915145i	$0.367924\pi$
$47$	−4.05152	+	7.01744i	−0.590975	+	1.02360i	−0.994101	+	0.108455i	$0.965410\pi$
$48$		0			0
$49$	6.23880			0.891257
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−3.41943	−	1.97421i	−0.469695	−	0.271179i	0.246417	−	0.969164i	$-0.420747\pi$
$53$	−3.41943	−	1.97421i	−0.469695	−	0.271179i	−0.716112	+	0.697985i	$0.754080\pi$
$54$		0			0
$55$	−5.20491	+	3.00506i	−0.701830	+	0.405202i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	1.55163	+	2.68750i	0.202005	+	0.349883i	0.949174	−	0.314751i	$-0.101921\pi$
$59$	1.55163	+	2.68750i	0.202005	+	0.349883i	−0.747170	+	0.664634i	$0.768588\pi$
$60$		0			0
$61$	6.53184	−	11.3135i	0.836316	−	1.44854i	−0.0566386	−	0.998395i	$-0.518038\pi$
$61$	6.53184	−	11.3135i	0.836316	−	1.44854i	0.892954	−	0.450147i	$-0.148628\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	9.88080i		−	1.22556i
$66$		0			0
$67$	5.04454	+	2.91247i	0.616289	+	0.355815i	0.775423	−	0.631442i	$-0.217537\pi$
$67$	5.04454	+	2.91247i	0.616289	+	0.355815i	−0.159134	+	0.987257i	$0.550870\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−4.65667	−	8.06559i	−0.552645	−	0.957209i	−0.998083	−	0.0618964i	$-0.980285\pi$
$71$	−4.65667	−	8.06559i	−0.552645	−	0.957209i	0.445437	−	0.895313i	$-0.353048\pi$
$72$		0			0
$73$	−1.01576	−	1.75935i	−0.118886	−	0.205916i	0.800441	−	0.599412i	$-0.204599\pi$
$73$	−1.01576	−	1.75935i	−0.118886	−	0.205916i	−0.919326	+	0.393496i	$0.871266\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			2.10291i			0.239649i
$78$		0			0
$79$	−10.8347	+	6.25541i	−1.21900	+	0.703788i	−0.964703	−	0.263339i	$-0.915176\pi$
$79$	−10.8347	+	6.25541i	−1.21900	+	0.703788i	−0.254294	+	0.967127i	$0.581843\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−6.29106			−0.690533			−0.345267	−	0.938505i	$-0.612212\pi$
$83$	−6.29106			−0.690533			−0.345267	+	0.938505i	$0.612212\pi$
$84$		0			0
$85$	−0.436244	+	0.755598i	−0.0473174	+	0.0819561i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−14.0641	−	8.11993i	−1.49080	−	0.860711i	−0.490851	−	0.871243i	$-0.663314\pi$
$89$	−14.0641	−	8.11993i	−1.49080	−	0.860711i	−0.999945	+	0.0105322i	$0.996647\pi$
$90$		0			0
$91$	−2.99406	−	1.72862i	−0.313863	−	0.181209i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	8.09342	+	7.25472i	0.830368	+	0.744318i
$96$		0			0
$97$	−8.90950	−	15.4317i	−0.904622	−	1.56685i	−0.821423	−	0.570319i	$-0.806820\pi$
$97$	−8.90950	−	15.4317i	−0.904622	−	1.56685i	−0.0831993	−	0.996533i	$-0.526514\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−0.725701	−	0.418984i	−0.0722099	−	0.0416904i	0.463460	−	0.886118i	$-0.346608\pi$
$101$	−0.725701	−	0.418984i	−0.0722099	−	0.0416904i	−0.535670	+	0.844427i	$0.679941\pi$
$102$		0			0
$103$			5.56741i			0.548573i	0.961648	+	0.274287i	$0.0884417\pi$
$103$			5.56741i			0.548573i	−0.961648	+	0.274287i	$0.911558\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	2.97716			0.287813			0.143907	−	0.989591i	$-0.454033\pi$
$107$	2.97716			0.287813			0.143907	+	0.989591i	$0.454033\pi$
$108$		0			0
$109$	−7.33514	−	12.7048i	−0.702579	−	1.21690i	−0.967558	−	0.252649i	$-0.918698\pi$
$109$	−7.33514	−	12.7048i	−0.702579	−	1.21690i	0.264979	−	0.964254i	$-0.414635\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	1.07542i		−	0.101167i	−0.998720	−	0.0505835i	$-0.983892\pi$
$113$		−	1.07542i		−	0.101167i	0.998720	−	0.0505835i	$-0.0161081\pi$
$114$		0			0
$115$		−	19.3682i		−	1.80610i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	0.152640	+	0.264380i	0.0139925	+	0.0242357i
$120$		0			0
$121$	−5.19045			−0.471859
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	9.43145i		−	0.843575i
$126$		0			0
$127$	−8.85067	−	5.10994i	−0.785370	−	0.453434i	0.0529600	−	0.998597i	$-0.483134\pi$
$127$	−8.85067	−	5.10994i	−0.785370	−	0.453434i	−0.838330	+	0.545163i	$0.816468\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−3.78592	−	6.55741i	−0.330777	−	0.572923i	0.651887	−	0.758316i	$-0.273978\pi$
$131$	−3.78592	−	6.55741i	−0.330777	−	0.572923i	−0.982665	+	0.185393i	$0.940644\pi$
$132$		0			0
$133$	3.61424	−	1.18326i	0.313394	−	0.102602i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	1.81417	+	1.04741i	0.154995	+	0.0894862i	0.575491	−	0.817808i	$-0.304811\pi$
$137$	1.81417	+	1.04741i	0.154995	+	0.0894862i	−0.420497	+	0.907294i	$0.638144\pi$
$138$		0			0
$139$	−17.2674	−	9.96934i	−1.46460	−	0.845588i	−0.465383	−	0.885109i	$-0.654083\pi$
$139$	−17.2674	−	9.96934i	−1.46460	−	0.845588i	−0.999219	+	0.0395212i	$0.987417\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−4.77553	+	8.27147i	−0.399350	+	0.691695i
$144$		0			0
$145$	22.9303			1.90426
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−6.14614	+	3.54847i	−0.503511	+	0.290702i	−0.730162	−	0.683274i	$-0.760556\pi$
$149$	−6.14614	+	3.54847i	−0.503511	+	0.290702i	0.226651	+	0.973976i	$0.427222\pi$
$150$		0			0
$151$			5.25075i			0.427300i	0.976910	+	0.213650i	$0.0685352\pi$
$151$			5.25075i			0.427300i	−0.976910	+	0.213650i	$0.931465\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	3.04021	+	5.26580i	0.244196	+	0.422959i
$156$		0			0
$157$	−0.877101	−	1.51918i	−0.0700003	−	0.121244i	0.828901	−	0.559396i	$-0.188967\pi$
$157$	−0.877101	−	1.51918i	−0.0700003	−	0.121244i	−0.898901	+	0.438151i	$0.855633\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−5.86892	−	3.38842i	−0.462536	−	0.267045i
$162$		0			0
$163$			15.6318i			1.22438i	0.790712	+	0.612188i	$0.209710\pi$
$163$			15.6318i			1.22438i	−0.790712	+	0.612188i	$0.790290\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−1.96384	+	3.40147i	−0.151967	+	0.263214i	−0.931950	−	0.362586i	$-0.881894\pi$
$167$	−1.96384	+	3.40147i	−0.151967	+	0.263214i	0.779984	+	0.625800i	$0.215227\pi$
$168$		0			0
$169$	−1.35111	−	2.34019i	−0.103932	−	0.180015i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	6.93385	−	4.00326i	0.527171	−	0.304362i	−0.212693	−	0.977119i	$-0.568223\pi$
$173$	6.93385	−	4.00326i	0.527171	−	0.304362i	0.739864	+	0.672757i	$0.234890\pi$
$174$		0			0
$175$	0.919995	+	0.531160i	0.0695451	+	0.0401519i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−20.8431			−1.55789			−0.778945	−	0.627092i	$-0.784245\pi$
$179$	−20.8431			−1.55789			−0.778945	+	0.627092i	$0.784245\pi$
$180$		0			0
$181$	2.38314	−	4.12772i	0.177137	−	0.306811i	−0.763762	−	0.645499i	$-0.776650\pi$
$181$	2.38314	−	4.12772i	0.177137	−	0.306811i	0.940899	+	0.338688i	$0.109983\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−13.3458	+	7.70521i	−0.981204	+	0.566498i
$186$		0			0
$187$	0.730383	−	0.421687i	0.0534109	−	0.0308368i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−11.9420			−0.864094			−0.432047	−	0.901851i	$-0.642209\pi$
$191$	−11.9420			−0.864094			−0.432047	+	0.901851i	$0.642209\pi$
$192$		0			0
$193$	0.108240	+	0.187477i	0.00779129	+	0.0134949i	0.869895	−	0.493237i	$-0.164187\pi$
$193$	0.108240	+	0.187477i	0.00779129	+	0.0134949i	−0.862103	+	0.506732i	$0.830853\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	24.3961i		−	1.73815i	−0.494681	−	0.869075i	$-0.664715\pi$
$197$		−	24.3961i		−	1.73815i	0.494681	−	0.869075i	$-0.335285\pi$
$198$		0			0
$199$	2.63892	+	1.52358i	0.187068	+	0.108004i	0.590609	−	0.806958i	$-0.298887\pi$
$199$	2.63892	+	1.52358i	0.187068	+	0.108004i	−0.403541	+	0.914962i	$0.632221\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	4.01160	−	6.94830i	0.281559	−	0.487675i
$204$		0			0
$205$	1.89527	−	3.28270i	0.132371	−	0.229274i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−3.26890	−	9.98478i	−0.226115	−	0.690662i
$210$		0			0
$211$	16.3816	−	9.45790i	1.12775	−	0.651109i	0.184385	−	0.982854i	$-0.440971\pi$
$211$	16.3816	−	9.45790i	1.12775	−	0.651109i	0.943369	+	0.331745i	$0.107637\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−7.55443	+	13.0847i	−0.515208	+	0.892366i
$216$		0			0
$217$	2.12751			0.144425
$218$		0			0
$219$		0			0
$220$		0			0
$221$			1.38653i			0.0932680i
$222$		0			0
$223$	−15.5577	+	8.98227i	−1.04182	+	0.601497i	−0.920349	−	0.391097i	$-0.872096\pi$
$223$	−15.5577	+	8.98227i	−1.04182	+	0.601497i	−0.121474	+	0.992595i	$0.538762\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	4.86062			0.322611			0.161305	−	0.986905i	$-0.448430\pi$
$227$	4.86062			0.322611			0.161305	+	0.986905i	$0.448430\pi$
$228$		0			0
$229$	13.8282			0.913791			0.456896	−	0.889520i	$-0.348961\pi$
$229$	13.8282			0.913791			0.456896	+	0.889520i	$0.348961\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	2.77985	−	1.60495i	0.182114	−	0.105144i	−0.406172	−	0.913797i	$-0.633136\pi$
$233$	2.77985	−	1.60495i	0.182114	−	0.105144i	0.588286	+	0.808653i	$0.299803\pi$
$234$		0			0
$235$		−	20.2050i		−	1.31803i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	24.5476			1.58785			0.793926	−	0.608014i	$-0.208033\pi$
$239$	24.5476			1.58785			0.793926	+	0.608014i	$0.208033\pi$
$240$		0			0
$241$	0.0130168	−	0.0225458i	0.000838488	−	0.00145230i	−0.865606	−	0.500726i	$-0.833066\pi$
$241$	0.0130168	−	0.0225458i	0.000838488	−	0.00145230i	0.866444	+	0.499274i	$0.166400\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−13.4723	+	7.77826i	−0.860716	+	0.496935i
$246$		0			0
$247$	16.9031	+	3.55347i	1.07552	+	0.226102i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−5.69579	+	9.86539i	−0.359515	+	0.622698i	−0.987880	−	0.155221i	$-0.950391\pi$
$251$	−5.69579	+	9.86539i	−0.359515	+	0.622698i	0.628365	+	0.777919i	$0.283724\pi$
$252$		0			0
$253$	−9.36094	+	16.2136i	−0.588517	+	1.01934i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−2.41804	−	1.39606i	−0.150833	−	0.0870837i	0.422684	−	0.906277i	$-0.361088\pi$
$257$	−2.41804	−	1.39606i	−0.150833	−	0.0870837i	−0.573517	+	0.819194i	$0.694421\pi$
$258$		0			0
$259$			5.39203i			0.335045i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	13.5051	+	23.3915i	0.832759	+	1.44238i	0.895842	+	0.444373i	$0.146574\pi$
$263$	13.5051	+	23.3915i	0.832759	+	1.44238i	−0.0630823	+	0.998008i	$0.520093\pi$
$264$		0			0
$265$	9.84544			0.604800
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−5.17984	+	2.99058i	−0.315820	+	0.182339i	−0.649528	−	0.760338i	$-0.725034\pi$
$269$	−5.17984	+	2.99058i	−0.315820	+	0.182339i	0.333708	+	0.942677i	$0.391700\pi$
$270$		0			0
$271$	6.27136	−	3.62077i	0.380958	−	0.219946i	−0.297277	−	0.954791i	$-0.596078\pi$
$271$	6.27136	−	3.62077i	0.380958	−	0.219946i	0.678235	+	0.734845i	$0.262745\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.46739	−	2.54160i	0.0884872	−	0.153264i
$276$		0			0
$277$	−2.27144			−0.136478			−0.0682388	−	0.997669i	$-0.521738\pi$
$277$	−2.27144			−0.136478			−0.0682388	+	0.997669i	$0.521738\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	10.3360	+	5.96752i	0.616597	+	0.355992i	0.775543	−	0.631295i	$-0.217476\pi$
$281$	10.3360	+	5.96752i	0.616597	+	0.355992i	−0.158946	+	0.987287i	$0.550810\pi$
$282$		0			0
$283$	2.08849	−	1.20579i	0.124148	−	0.0716769i	−0.436640	−	0.899636i	$-0.643832\pi$
$283$	2.08849	−	1.20579i	0.124148	−	0.0716769i	0.560788	+	0.827959i	$0.310498\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−0.663145	−	1.14860i	−0.0391442	−	0.0677998i
$288$		0			0
$289$	−8.43878	+	14.6164i	−0.496399	+	0.859788i
$290$		0			0
$291$		0			0
$292$		0			0
$293$			21.5726i			1.26028i	0.776480	+	0.630142i	$0.217003\pi$
$293$			21.5726i			1.26028i	−0.776480	+	0.630142i	$0.782997\pi$
$294$		0			0
$295$	−6.70131	−	3.86900i	−0.390165	−	0.225262i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−15.3896	−	26.6557i	−0.890006	−	1.54154i
$300$		0			0
$301$	2.64326	+	4.57826i	0.152355	+	0.263887i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			32.5744i			1.86521i
$306$		0			0
$307$	24.3229	−	14.0428i	1.38818	−	0.801466i	0.395069	−	0.918651i	$-0.370721\pi$
$307$	24.3229	−	14.0428i	1.38818	−	0.801466i	0.993110	+	0.117186i	$0.0373873\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	17.3069			0.981382			0.490691	−	0.871334i	$-0.336744\pi$
$311$	17.3069			0.981382			0.490691	+	0.871334i	$0.336744\pi$
$312$		0			0
$313$	2.82124	−	4.88653i	0.159466	−	0.276203i	−0.775210	−	0.631703i	$-0.782356\pi$
$313$	2.82124	−	4.88653i	0.159466	−	0.276203i	0.934676	+	0.355500i	$0.115689\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−8.65525	−	4.99711i	−0.486127	−	0.280666i	0.236839	−	0.971549i	$-0.423889\pi$
$317$	−8.65525	−	4.99711i	−0.486127	−	0.280666i	−0.722966	+	0.690883i	$0.757222\pi$
$318$		0			0
$319$	−19.1955	−	11.0825i	−1.07474	−	0.620503i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−1.13571	−	1.01802i	−0.0631929	−	0.0566443i
$324$		0			0
$325$	2.41244	+	4.17846i	0.133818	+	0.231779i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−6.12250	−	3.53483i	−0.337544	−	0.194881i
$330$		0			0
$331$			29.8452i			1.64044i	0.572047	+	0.820221i	$0.306149\pi$
$331$			29.8452i			1.64044i	−0.572047	+	0.820221i	$0.693851\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−14.5246			−0.793561
$336$		0			0
$337$	9.01263	+	15.6103i	0.490949	+	0.850349i	0.999946	−	0.0104196i	$-0.00331673\pi$
$337$	9.01263	+	15.6103i	0.490949	+	0.850349i	−0.508997	+	0.860769i	$0.669983\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	5.87751i		−	0.318285i
$342$		0			0
$343$			11.5504i			0.623665i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−6.35411	−	11.0056i	−0.341106	−	0.590813i	0.643532	−	0.765419i	$-0.277468\pi$
$347$	−6.35411	−	11.0056i	−0.341106	−	0.590813i	−0.984638	+	0.174606i	$0.944135\pi$
$348$		0			0
$349$	−12.0832			−0.646799			−0.323399	−	0.946263i	$-0.604826\pi$
$349$	−12.0832			−0.646799			−0.323399	+	0.946263i	$0.604826\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	2.13272i		−	0.113513i	−0.998388	−	0.0567566i	$-0.981924\pi$
$353$		−	2.13272i		−	0.113513i	0.998388	−	0.0567566i	$-0.0180759\pi$
$354$		0			0
$355$	20.1116	+	11.6115i	1.06742	+	0.616273i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	15.1267	+	26.2002i	0.798355	+	1.38279i	0.920687	+	0.390302i	$0.127629\pi$
$359$	15.1267	+	26.2002i	0.798355	+	1.38279i	−0.122332	+	0.992489i	$0.539037\pi$
$360$		0			0
$361$	−15.3213	+	11.2364i	−0.806386	+	0.591390i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	4.38696	+	2.53281i	0.229624	+	0.132573i
$366$		0			0
$367$	−31.9344	−	18.4373i	−1.66696	−	0.962421i	−0.969262	−	0.246031i	$-0.920874\pi$
$367$	−31.9344	−	18.4373i	−1.66696	−	0.962421i	−0.697700	−	0.716390i	$-0.745793\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	1.72244	−	2.98335i	0.0894245	−	0.154888i
$372$		0			0
$373$	−20.0837			−1.03990			−0.519948	−	0.854198i	$-0.674049\pi$
$373$	−20.0837			−1.03990			−0.519948	+	0.854198i	$0.674049\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	31.5580	−	18.2200i	1.62532	−	0.938378i
$378$		0			0
$379$			14.8800i			0.764332i	0.924094	+	0.382166i	$0.124822\pi$
$379$			14.8800i			0.764332i	−0.924094	+	0.382166i	$0.875178\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−0.105222	−	0.182250i	−0.00537660	−	0.00931254i	0.863325	−	0.504649i	$-0.168378\pi$
$383$	−0.105222	−	0.182250i	−0.00537660	−	0.00931254i	−0.868701	+	0.495337i	$0.835045\pi$
$384$		0			0
$385$	−2.62182	−	4.54112i	−0.133620	−	0.231437i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	3.88933	+	2.24551i	0.197197	+	0.113852i	0.595347	−	0.803468i	$-0.297014\pi$
$389$	3.88933	+	2.24551i	0.197197	+	0.113852i	−0.398150	+	0.917320i	$0.630348\pi$
$390$		0			0
$391$			2.71786i			0.137448i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	15.5979	−	27.0164i	0.784817	−	1.35934i
$396$		0			0
$397$	13.0390	+	22.5842i	0.654407	+	1.13347i	0.982042	+	0.188662i	$0.0604152\pi$
$397$	13.0390	+	22.5842i	0.654407	+	1.13347i	−0.327635	+	0.944805i	$0.606251\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−18.2353	+	10.5282i	−0.910629	+	0.525752i	−0.880633	−	0.473798i	$-0.842883\pi$
$401$	−18.2353	+	10.5282i	−0.910629	+	0.525752i	−0.0299955	+	0.999550i	$0.509549\pi$
$402$		0			0
$403$	8.36822	+	4.83140i	0.416851	+	0.240669i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	14.8962			0.738375
$408$		0			0
$409$	6.13874	−	10.6326i	0.303541	−	0.525749i	−0.673394	−	0.739283i	$-0.735164\pi$
$409$	6.13874	−	10.6326i	0.303541	−	0.525749i	0.976935	+	0.213535i	$0.0684977\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−2.34476	+	1.35375i	−0.115378	+	0.0666135i
$414$		0			0
$415$	13.5852	−	7.84342i	0.666871	−	0.385018i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−25.9467			−1.26758			−0.633791	−	0.773505i	$-0.718502\pi$
$419$	−25.9467			−1.26758			−0.633791	+	0.773505i	$0.718502\pi$
$420$		0			0
$421$	−15.8963	−	27.5332i	−0.774737	−	1.34188i	−0.934942	−	0.354800i	$-0.884549\pi$
$421$	−15.8963	−	27.5332i	−0.774737	−	1.34188i	0.160205	−	0.987084i	$-0.448784\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	0.426043i		−	0.0206661i
$426$		0			0
$427$	9.87065	+	5.69882i	0.477674	+	0.275785i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	13.1686	−	22.8087i	0.634308	−	1.09865i	−0.352353	−	0.935867i	$-0.614618\pi$
$431$	13.1686	−	22.8087i	0.634308	−	1.09865i	0.986661	−	0.162787i	$-0.0520483\pi$
$432$		0			0
$433$	1.38362	−	2.39650i	0.0664925	−	0.115168i	−0.830863	−	0.556478i	$-0.812152\pi$
$433$	1.38362	−	2.39650i	0.0664925	−	0.115168i	0.897355	+	0.441309i	$0.145486\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	33.1333	+	6.96546i	1.58498	+	0.333203i
$438$		0			0
$439$	−0.903581	+	0.521683i	−0.0431256	+	0.0248986i	−0.521408	−	0.853308i	$-0.674593\pi$
$439$	−0.903581	+	0.521683i	−0.0431256	+	0.0248986i	0.478282	+	0.878206i	$0.341260\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	20.5933	−	35.6686i	0.978416	−	1.69467i	0.310247	−	0.950656i	$-0.399588\pi$
$443$	20.5933	−	35.6686i	0.978416	−	1.69467i	0.668169	−	0.744010i	$-0.267079\pi$
$444$		0			0
$445$	40.4943			1.91961
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	7.61751i		−	0.359492i	−0.983713	−	0.179746i	$-0.942472\pi$
$449$		−	7.61751i		−	0.359492i	0.983713	−	0.179746i	$-0.0575277\pi$
$450$		0			0
$451$	−3.17315	+	1.83202i	−0.149418	+	0.0862665i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	8.62068			0.404144
$456$		0			0
$457$	−27.2754			−1.27589			−0.637945	−	0.770082i	$-0.720215\pi$
$457$	−27.2754			−1.27589			−0.637945	+	0.770082i	$0.720215\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	9.16049	−	5.28881i	0.426647	−	0.246324i	−0.271270	−	0.962503i	$-0.587444\pi$
$461$	9.16049	−	5.28881i	0.426647	−	0.246324i	0.697917	+	0.716179i	$0.254110\pi$
$462$		0			0
$463$		−	33.5895i		−	1.56104i	−0.625134	−	0.780518i	$-0.714956\pi$
$463$		−	33.5895i		−	1.56104i	0.625134	−	0.780518i	$-0.285044\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−12.7016			−0.587758			−0.293879	−	0.955843i	$-0.594946\pi$
$467$	−12.7016			−0.587758			−0.293879	+	0.955843i	$0.594946\pi$
$468$		0			0
$469$	−2.54104	+	4.40121i	−0.117334	+	0.203229i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	12.6480	−	7.30233i	0.581556	−	0.335762i
$474$		0			0
$475$	−5.19387	−	1.09189i	−0.238311	−	0.0500991i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−10.5935	+	18.3485i	−0.484029	+	0.838363i	−0.999832	−	0.0183448i	$-0.994160\pi$
$479$	−10.5935	+	18.3485i	−0.484029	+	0.838363i	0.515803	+	0.856707i	$0.327494\pi$
$480$		0			0
$481$	−12.2448	+	21.2087i	−0.558317	+	0.967034i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	38.4791	+	22.2159i	1.74725	+	1.00877i
$486$		0			0
$487$			32.5829i			1.47647i	0.674541	+	0.738237i	$0.264341\pi$
$487$			32.5829i			1.47647i	−0.674541	+	0.738237i	$0.735659\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	3.36736	+	5.83244i	0.151967	+	0.263214i	0.931950	−	0.362586i	$-0.118106\pi$
$491$	3.36736	+	5.83244i	0.151967	+	0.263214i	−0.779984	+	0.625800i	$0.784773\pi$
$492$		0			0
$493$	−3.21771			−0.144918
$494$		0			0
$495$		0			0
$496$		0			0
$497$	7.03697	−	4.06280i	0.315651	−	0.182241i
$498$		0			0
$499$	14.1798	−	8.18673i	0.634777	−	0.366489i	−0.147823	−	0.989014i	$-0.547227\pi$
$499$	14.1798	−	8.18673i	0.634777	−	0.366489i	0.782600	+	0.622525i	$0.213893\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	12.2751	−	21.2611i	0.547320	−	0.947986i	−0.451137	−	0.892455i	$-0.648981\pi$
$503$	12.2751	−	21.2611i	0.547320	−	0.947986i	0.998457	−	0.0555311i	$-0.0176852\pi$
$504$		0			0
$505$	2.08948			0.0929807
$506$		0			0
$507$		0			0
$508$		0			0
$509$	0.111741	+	0.0645136i	0.00495282	+	0.00285951i	0.502474	−	0.864592i	$-0.332423\pi$
$509$	0.111741	+	0.0645136i	0.00495282	+	0.00285951i	−0.497522	+	0.867452i	$0.665756\pi$
$510$		0			0
$511$	1.53498	−	0.886218i	0.0679033	−	0.0392040i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−6.94120	−	12.0225i	−0.305866	−	0.529775i
$516$		0			0
$517$	−9.76539	+	16.9142i	−0.429482	+	0.743884i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	2.37366i		−	0.103992i	−0.998647	−	0.0519960i	$-0.983442\pi$
$521$		−	2.37366i		−	0.103992i	0.998647	−	0.0519960i	$-0.0165583\pi$
$522$		0			0
$523$	−14.9732	−	8.64479i	−0.654733	−	0.378010i	0.135534	−	0.990773i	$-0.456725\pi$
$523$	−14.9732	−	8.64479i	−0.654733	−	0.378010i	−0.790267	+	0.612762i	$0.790058\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−0.426619	−	0.738927i	−0.0185838	−	0.0321881i
$528$		0			0
$529$	−18.6666	−	32.3315i	−0.811591	−	1.40572i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	6.02379i		−	0.260919i
$534$		0			0
$535$	−6.42902	+	3.71180i	−0.277951	+	0.160475i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	15.0374			0.647706
$540$		0			0
$541$	−1.88044	+	3.25701i	−0.0808463	+	0.140030i	−0.903614	−	0.428348i	$-0.859096\pi$
$541$	−1.88044	+	3.25701i	−0.0808463	+	0.140030i	0.822767	+	0.568378i	$0.192429\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	31.6797	+	18.2903i	1.35701	+	0.783469i
$546$		0			0
$547$	14.2655	+	8.23618i	0.609948	+	0.352154i	0.772945	−	0.634473i	$-0.218783\pi$
$547$	14.2655	+	8.23618i	0.609948	+	0.352154i	−0.162997	+	0.986627i	$0.552116\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−8.24650	+	39.2269i	−0.351313	+	1.67112i
$552$		0			0
$553$	−5.45765	−	9.45292i	−0.232083	−	0.401979i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	24.8362	+	14.3392i	1.05234	+	0.607570i	0.923304	−	0.384069i	$-0.125478\pi$
$557$	24.8362	+	14.3392i	1.05234	+	0.607570i	0.129038	+	0.991640i	$0.458811\pi$
$558$		0			0
$559$			24.0105i			1.01553i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−29.9715			−1.26315			−0.631574	−	0.775316i	$-0.717591\pi$
$563$	−29.9715			−1.26315			−0.631574	+	0.775316i	$0.717591\pi$
$564$		0			0
$565$	1.34079	+	2.32231i	0.0564073	+	0.0977002i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			41.8354i			1.75383i	0.480645	+	0.876915i	$0.340403\pi$
$569$			41.8354i			1.75383i	−0.480645	+	0.876915i	$0.659597\pi$
$570$		0			0
$571$			12.8770i			0.538886i	0.963016	+	0.269443i	$0.0868396\pi$
$571$			12.8770i			0.538886i	−0.963016	+	0.269443i	$0.913160\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	4.72883	+	8.19057i	0.197206	+	0.341570i
$576$		0			0
$577$	−15.3277			−0.638101			−0.319050	−	0.947738i	$-0.603364\pi$
$577$	−15.3277			−0.638101			−0.319050	+	0.947738i	$0.603364\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	5.48875i		−	0.227712i
$582$		0			0
$583$	−8.24187	−	4.75844i	−0.341343	−	0.197075i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−16.4588	−	28.5074i	−0.679326	−	1.17663i	−0.975184	−	0.221395i	$-0.928939\pi$
$587$	−16.4588	−	28.5074i	−0.679326	−	1.17663i	0.295858	−	0.955232i	$-0.404394\pi$
$588$		0			0
$589$	−10.1016	+	3.30714i	−0.416228	+	0.136268i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	6.43447	+	3.71494i	0.264232	+	0.152554i	0.626264	−	0.779611i	$-0.284583\pi$
$593$	6.43447	+	3.71494i	0.264232	+	0.152554i	−0.362032	+	0.932166i	$0.617917\pi$
$594$		0			0
$595$	−0.659235	−	0.380609i	−0.0270260	−	0.0156035i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−14.4459	+	25.0210i	−0.590242	+	1.02233i	0.403958	+	0.914778i	$0.367634\pi$
$599$	−14.4459	+	25.0210i	−0.590242	+	1.02233i	−0.994200	+	0.107551i	$0.965699\pi$
$600$		0			0
$601$	−5.79064			−0.236205			−0.118103	−	0.993001i	$-0.537681\pi$
$601$	−5.79064			−0.236205			−0.118103	+	0.993001i	$0.537681\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	11.2085	−	6.47122i	0.455690	−	0.263093i
$606$		0			0
$607$			13.2721i			0.538698i	0.963043	+	0.269349i	$0.0868084\pi$
$607$			13.2721i			0.538698i	−0.963043	+	0.269349i	$0.913192\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−16.0546	−	27.8073i	−0.649499	−	1.12496i
$612$		0			0
$613$	−11.8422	−	20.5113i	−0.478302	−	0.828443i	0.521388	−	0.853319i	$-0.325414\pi$
$613$	−11.8422	−	20.5113i	−0.478302	−	0.828443i	−0.999691	+	0.0248760i	$0.992081\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	16.2958	+	9.40837i	0.656044	+	0.378767i	0.790768	−	0.612116i	$-0.209682\pi$
$617$	16.2958	+	9.40837i	0.656044	+	0.378767i	−0.134724	+	0.990883i	$0.543015\pi$
$618$		0			0
$619$			41.7559i			1.67831i	0.543892	+	0.839155i	$0.316950\pi$
$619$			41.7559i			1.67831i	−0.543892	+	0.839155i	$0.683050\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	7.08439	−	12.2705i	0.283830	−	0.491608i
$624$		0			0
$625$	14.8027	+	25.6391i	0.592109	+	1.02556i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	1.87276	−	1.08124i	0.0746718	−	0.0431118i
$630$		0			0
$631$	−2.42656	−	1.40098i	−0.0965999	−	0.0557720i	0.450922	−	0.892563i	$-0.351095\pi$
$631$	−2.42656	−	1.40098i	−0.0965999	−	0.0557720i	−0.547522	+	0.836791i	$0.684429\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	25.4834			1.01128
$636$		0			0
$637$	−12.3609	+	21.4098i	−0.489758	+	0.848286i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	0.697803	−	0.402877i	0.0275615	−	0.0159127i	−0.486156	−	0.873872i	$-0.661601\pi$
$641$	0.697803	−	0.402877i	0.0275615	−	0.0159127i	0.513717	+	0.857959i	$0.328268\pi$
$642$		0			0
$643$	−28.1911	+	16.2761i	−1.11175	+	0.641869i	−0.939282	−	0.343147i	$-0.888507\pi$
$643$	−28.1911	+	16.2761i	−1.11175	+	0.641869i	−0.172467	+	0.985015i	$0.555174\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	19.3718			0.761582			0.380791	−	0.924661i	$-0.375652\pi$
$647$	19.3718			0.761582			0.380791	+	0.924661i	$0.375652\pi$
$648$		0			0
$649$	3.73989	+	6.47768i	0.146804	+	0.254271i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			24.9996i			0.978310i	0.872197	+	0.489155i	$0.162695\pi$
$653$			24.9996i			0.978310i	−0.872197	+	0.489155i	$0.837305\pi$
$654$		0			0
$655$	16.3510	+	9.44024i	0.638885	+	0.368861i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	15.1955	−	26.3194i	0.591932	−	1.02526i	−0.402040	−	0.915622i	$-0.631699\pi$
$659$	15.1955	−	26.3194i	0.591932	−	1.02526i	0.993972	−	0.109635i	$-0.0349681\pi$
$660$		0			0
$661$	−25.1929	+	43.6354i	−0.979890	+	1.69722i	−0.317141	+	0.948378i	$0.602723\pi$
$661$	−25.1929	+	43.6354i	−0.979890	+	1.69722i	−0.662749	+	0.748842i	$0.730610\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−6.32951	+	7.06126i	−0.245448	+	0.273824i
$666$		0			0
$667$	61.8596	−	35.7146i	2.39521	−	1.38288i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	15.7437	−	27.2689i	0.607779	−	1.05270i
$672$		0			0
$673$	15.3999			0.593621			0.296811	−	0.954936i	$-0.404077\pi$
$673$	15.3999			0.593621			0.296811	+	0.954936i	$0.404077\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			24.1461i			0.928011i	0.885832	+	0.464006i	$0.153588\pi$
$677$			24.1461i			0.928011i	−0.885832	+	0.464006i	$0.846412\pi$
$678$		0			0
$679$	13.4637	−	7.77326i	0.516688	−	0.298310i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	6.74618			0.258136			0.129068	−	0.991636i	$-0.458802\pi$
$683$	6.74618			0.258136			0.129068	+	0.991636i	$0.458802\pi$
$684$		0			0
$685$	−5.22346			−0.199578
$686$		0			0
$687$		0			0
$688$		0			0
$689$	13.5499	−	7.82301i	0.516209	−	0.298033i
$690$		0			0
$691$		−	16.8381i		−	0.640553i	−0.947324	−	0.320276i	$-0.896224\pi$
$691$		−	16.8381i		−	0.640553i	0.947324	−	0.320276i	$-0.103776\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	49.7173			1.88589
$696$		0			0
$697$	−0.265955	+	0.460647i	−0.0100738	+	0.0174482i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−26.6147	+	15.3660i	−1.00522	+	0.580365i	−0.909789	−	0.415071i	$-0.863757\pi$
$701$	−26.6147	+	15.3660i	−1.00522	+	0.580365i	−0.0954327	+	0.995436i	$0.530423\pi$
$702$		0			0
$703$	−8.38172	−	25.6018i	−0.316123	−	0.965589i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	0.365550	−	0.633151i	0.0137479	−	0.0238121i
$708$		0			0
$709$	−0.147095	+	0.254777i	−0.00552428	+	0.00956834i	−0.868774	−	0.495208i	$-0.835092\pi$
$709$	−0.147095	+	0.254777i	−0.00552428	+	0.00956834i	0.863250	+	0.504777i	$0.168425\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	16.4033	+	9.47044i	0.614308	+	0.354671i
$714$		0			0
$715$		−	23.8157i		−	0.890657i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	25.4185	+	44.0261i	0.947948	+	1.64189i	0.749738	+	0.661735i	$0.230180\pi$
$719$	25.4185	+	44.0261i	0.947948	+	1.64189i	0.198211	+	0.980159i	$0.436487\pi$
$720$		0			0
$721$	−4.85739			−0.180899
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−9.69692	+	5.59852i	−0.360135	+	0.207924i
$726$		0			0
$727$	−25.2657	+	14.5871i	−0.937052	+	0.541007i	−0.889035	−	0.457839i	$-0.848624\pi$
$727$	−25.2657	+	14.5871i	−0.937052	+	0.541007i	−0.0480173	+	0.998847i	$0.515290\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	1.06008	−	1.83611i	0.0392085	−	0.0679111i
$732$		0			0
$733$	0.798680			0.0294999			0.0147500	−	0.999891i	$-0.495305\pi$
$733$	0.798680			0.0294999			0.0147500	+	0.999891i	$0.495305\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	12.1589	+	7.01993i	0.447878	+	0.258582i
$738$		0			0
$739$	13.3843	−	7.72742i	0.492349	−	0.284258i	−0.233199	−	0.972429i	$-0.574919\pi$
$739$	13.3843	−	7.72742i	0.492349	−	0.284258i	0.725549	+	0.688171i	$0.241586\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−8.37017	−	14.4976i	−0.307072	−	0.531864i	0.670649	−	0.741775i	$-0.266016\pi$
$743$	−8.37017	−	14.4976i	−0.307072	−	0.531864i	−0.977720	+	0.209911i	$0.932682\pi$
$744$		0			0
$745$	8.84816	−	15.3255i	0.324172	−	0.561482i
$746$		0			0
$747$		0			0
$748$		0			0
$749$			2.59748i			0.0949099i
$750$		0			0
$751$	−4.27226	−	2.46659i	−0.155897	−	0.0900071i	0.420022	−	0.907514i	$-0.362022\pi$
$751$	−4.27226	−	2.46659i	−0.155897	−	0.0900071i	−0.575919	+	0.817507i	$0.695356\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−6.54640	−	11.3387i	−0.238248	−	0.412657i
$756$		0			0
$757$	24.3296	+	42.1402i	0.884276	+	1.53161i	0.846542	+	0.532323i	$0.178681\pi$
$757$	24.3296	+	42.1402i	0.884276	+	1.53161i	0.0377340	+	0.999288i	$0.487986\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	40.7399i		−	1.47682i	−0.674351	−	0.738411i	$-0.735576\pi$
$761$		−	40.7399i		−	1.47682i	0.674351	−	0.738411i	$-0.264424\pi$
$762$		0			0
$763$	11.0846	−	6.39968i	0.401288	−	0.231684i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−12.2970			−0.444018
$768$		0			0
$769$	−9.00912	+	15.6043i	−0.324877	+	0.562704i	−0.981487	−	0.191526i	$-0.938656\pi$
$769$	−9.00912	+	15.6043i	−0.324877	+	0.562704i	0.656610	+	0.754230i	$0.271990\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−40.4933	−	23.3788i	−1.45644	−	0.840878i	−0.457610	−	0.889153i	$-0.651294\pi$
$773$	−40.4933	−	23.3788i	−1.45644	−	0.840878i	−0.998834	+	0.0482746i	$0.984628\pi$
$774$		0			0
$775$	−2.57133	−	1.48456i	−0.0923650	−	0.0533269i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	4.93412	+	4.42281i	0.176783	+	0.158464i
$780$		0			0
$781$	−11.2240	−	19.4405i	−0.401626	−	0.695636i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	3.78810	+	2.18706i	0.135203	+	0.0780596i
$786$		0			0
$787$			17.9843i			0.641070i	0.947237	+	0.320535i	$0.103863\pi$
$787$			17.9843i			0.641070i	−0.947237	+	0.320535i	$0.896137\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	0.938269			0.0333610
$792$		0			0
$793$	25.8831	+	44.8308i	0.919135	+	1.59199i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	47.4462i		−	1.68063i	−0.542099	−	0.840315i	$-0.682370\pi$
$797$		−	47.4462i		−	1.68063i	0.542099	−	0.840315i	$-0.317630\pi$
$798$		0			0
$799$			2.83529i			0.100305i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−2.44829	−	4.24056i	−0.0863982	−	0.149646i
$804$		0			0
$805$	16.8982			0.595582
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	2.25995i		−	0.0794558i	−0.999211	−	0.0397279i	$-0.987351\pi$
$809$		−	2.25995i		−	0.0794558i	0.999211	−	0.0397279i	$-0.0126491\pi$
$810$		0			0
$811$	−14.1260	−	8.15565i	−0.496031	−	0.286383i	0.231042	−	0.972944i	$-0.425786\pi$
$811$	−14.1260	−	8.15565i	−0.496031	−	0.286383i	−0.727073	+	0.686560i	$0.759120\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−19.4890	−	33.7560i	−0.682671	−	1.18242i
$816$		0			0
$817$	−19.6671	−	17.6291i	−0.688066	−	0.616763i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	17.5399	+	10.1267i	0.612148	+	0.353424i	0.773806	−	0.633423i	$-0.218351\pi$
$821$	17.5399	+	10.1267i	0.612148	+	0.353424i	−0.161658	+	0.986847i	$0.551684\pi$
$822$		0			0
$823$	23.6500	+	13.6544i	0.824389	+	0.475961i	0.851928	−	0.523660i	$-0.175434\pi$
$823$	23.6500	+	13.6544i	0.824389	+	0.475961i	−0.0275388	+	0.999621i	$0.508767\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−18.8405	+	32.6326i	−0.655147	+	1.13475i	0.326710	+	0.945125i	$0.394060\pi$
$827$	−18.8405	+	32.6326i	−0.655147	+	1.13475i	−0.981857	+	0.189623i	$0.939273\pi$
$828$		0			0
$829$	−1.47580			−0.0512568			−0.0256284	−	0.999672i	$-0.508159\pi$
$829$	−1.47580			−0.0512568			−0.0256284	+	0.999672i	$0.508159\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	1.89051	−	1.09149i	0.0655025	−	0.0378179i
$834$		0			0
$835$		−	9.79373i		−	0.338926i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−5.93971	−	10.2879i	−0.205062	−	0.355177i	0.745091	−	0.666963i	$-0.232406\pi$
$839$	−5.93971	−	10.2879i	−0.205062	−	0.355177i	−0.950152	+	0.311786i	$0.899073\pi$
$840$		0			0
$841$	27.7830	+	48.1216i	0.958035	+	1.65937i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	5.83530	+	3.36901i	0.200740	+	0.115898i
$846$		0			0
$847$		−	4.52850i		−	0.155601i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−24.0022	+	41.5730i	−0.822785	+	1.42511i
$852$		0			0
$853$	−26.0432	−	45.1081i	−0.891701	−	1.54447i	−0.837835	−	0.545923i	$-0.816179\pi$
$853$	−26.0432	−	45.1081i	−0.891701	−	1.54447i	−0.0538658	−	0.998548i	$-0.517154\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−22.2925	+	12.8706i	−0.761499	+	0.439652i	−0.829834	−	0.558011i	$-0.811565\pi$
$857$	−22.2925	+	12.8706i	−0.761499	+	0.439652i	0.0683348	+	0.997662i	$0.478231\pi$
$858$		0			0
$859$	34.9616	+	20.1851i	1.19287	+	0.688706i	0.958957	−	0.283551i	$-0.0915125\pi$
$859$	34.9616	+	20.1851i	1.19287	+	0.688706i	0.233916	+	0.972257i	$0.424846\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−8.58624			−0.292279			−0.146139	−	0.989264i	$-0.546685\pi$
$863$	−8.58624			−0.292279			−0.146139	+	0.989264i	$0.546685\pi$
$864$		0			0
$865$	−9.98218	+	17.2896i	−0.339404	+	0.587865i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−26.1149	+	15.0774i	−0.885886	+	0.511466i
$870$		0			0
$871$	−19.9895	+	11.5410i	−0.677319	+	0.391051i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	8.22865			0.278179
$876$		0			0
$877$	15.1777	+	26.2885i	0.512514	+	0.887700i	0.999895	+	0.0145104i	$0.00461895\pi$
$877$	15.1777	+	26.2885i	0.512514	+	0.887700i	−0.487381	+	0.873189i	$0.662048\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			23.7828i			0.801264i	0.916239	+	0.400632i	$0.131209\pi$
$881$			23.7828i			0.801264i	−0.916239	+	0.400632i	$0.868791\pi$
$882$		0			0
$883$	17.2046	+	9.93310i	0.578982	+	0.334275i	0.760729	−	0.649070i	$-0.224842\pi$
$883$	17.2046	+	9.93310i	0.578982	+	0.334275i	−0.181747	+	0.983345i	$0.558175\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	23.1517	−	40.0999i	0.777358	−	1.34642i	−0.156101	−	0.987741i	$-0.549893\pi$
$887$	23.1517	−	40.0999i	0.777358	−	1.34642i	0.933459	−	0.358683i	$-0.116774\pi$
$888$		0			0
$889$	4.45826	−	7.72193i	0.149525	−	0.258985i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	34.5648	+	7.26641i	1.15667	+	0.243161i
$894$		0			0
$895$	45.0096	−	25.9863i	1.50451	−	0.868627i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−11.2122	+	19.4201i	−0.373947	+	0.647696i
$900$		0			0
$901$	−1.38157			−0.0460267
$902$		0			0
$903$		0			0
$904$		0			0
$905$			11.8848i			0.395063i
$906$		0			0
$907$	−27.3210	+	15.7738i	−0.907178	+	0.523759i	−0.879522	−	0.475858i	$-0.842138\pi$
$907$	−27.3210	+	15.7738i	−0.907178	+	0.523759i	−0.0276559	+	0.999618i	$0.508804\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−27.7826			−0.920480			−0.460240	−	0.887795i	$-0.652237\pi$
$911$	−27.7826			−0.920480			−0.460240	+	0.887795i	$0.652237\pi$
$912$		0			0
$913$	−15.1633			−0.501834
$914$		0			0
$915$		0			0
$916$		0			0
$917$	5.72113	−	3.30310i	0.188928	−	0.109078i
$918$		0			0
$919$			40.8311i			1.34690i	0.739235	+	0.673448i	$0.235187\pi$
$919$			40.8311i			1.34690i	−0.739235	+	0.673448i	$0.764813\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	36.9051			1.21475
$924$		0			0
$925$	3.76252	−	6.51687i	0.123711	−	0.214273i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	4.27054	−	2.46560i	0.140112	−	0.0808936i	−0.428305	−	0.903634i	$-0.640889\pi$
$929$	4.27054	−	2.46560i	0.140112	−	0.0808936i	0.568417	+	0.822740i	$0.307556\pi$
$930$		0			0
$931$	−8.46119	−	25.8445i	−0.277304	−	0.847019i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−1.05148	+	1.82122i	−0.0343871	+	0.0595602i
$936$		0			0
$937$	2.91420	−	5.04754i	0.0952028	−	0.164896i	−0.814490	−	0.580177i	$-0.802983\pi$
$937$	2.91420	−	5.04754i	0.0952028	−	0.164896i	0.909693	+	0.415281i	$0.136317\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	11.9364	+	6.89149i	0.389116	+	0.224656i	0.681777	−	0.731560i	$-0.261207\pi$
$941$	11.9364	+	6.89149i	0.389116	+	0.224656i	−0.292661	+	0.956216i	$0.594541\pi$
$942$		0			0
$943$		−	11.8078i		−	0.384513i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	3.20691	+	5.55454i	0.104211	+	0.180498i	0.913415	−	0.407028i	$-0.133435\pi$
$947$	3.20691	+	5.55454i	0.104211	+	0.180498i	−0.809205	+	0.587527i	$0.800102\pi$
$948$		0			0
$949$	8.05011			0.261318
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−40.4432	+	23.3499i	−1.31008	+	0.756377i	−0.982110	−	0.188310i	$-0.939699\pi$
$953$	−40.4432	+	23.3499i	−1.31008	+	0.756377i	−0.327974	+	0.944687i	$0.606366\pi$
$954$		0			0
$955$	25.7881	−	14.8888i	0.834484	−	0.481790i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−0.913832	+	1.58280i	−0.0295092	+	0.0511114i
$960$		0			0
$961$	25.0537			0.808185
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−0.467477	−	0.269898i	−0.0150486	−	0.00868832i
$966$		0			0
$967$	−19.6384	+	11.3382i	−0.631528	+	0.364613i	−0.781343	−	0.624101i	$-0.785465\pi$
$967$	−19.6384	+	11.3382i	−0.631528	+	0.364613i	0.149816	+	0.988714i	$0.452132\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−11.1706	−	19.3480i	−0.358481	−	0.620908i	0.629226	−	0.777222i	$-0.283372\pi$
$971$	−11.1706	−	19.3480i	−0.358481	−	0.620908i	−0.987707	+	0.156315i	$0.950039\pi$
$972$		0			0
$973$	8.69793	−	15.0653i	0.278843	−	0.482970i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	54.3329i		−	1.73826i	−0.494581	−	0.869131i	$-0.664679\pi$
$977$		−	54.3329i		−	1.73826i	0.494581	−	0.869131i	$-0.335321\pi$
$978$		0			0
$979$	−33.8988	−	19.5715i	−1.08341	−	0.625508i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	18.6715	+	32.3399i	0.595528	+	1.03148i	0.993472	+	0.114075i	$0.0363903\pi$
$983$	18.6715	+	32.3399i	0.595528	+	1.03148i	−0.397945	+	0.917409i	$0.630276\pi$
$984$		0			0
$985$	30.4160	+	52.6820i	0.969133	+	1.67859i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			47.0650i			1.49658i
$990$		0			0
$991$	−6.22238	+	3.59249i	−0.197660	+	0.114119i	−0.595564	−	0.803308i	$-0.703071\pi$
$991$	−6.22238	+	3.59249i	−0.197660	+	0.114119i	0.397903	+	0.917427i	$0.369738\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−7.59815			−0.240877
$996$		0			0
$997$	30.8737	−	53.4748i	0.977779	−	1.69356i	0.307339	−	0.951600i	$-0.400562\pi$
$997$	30.8737	−	53.4748i	0.977779	−	1.69356i	0.670441	−	0.741963i	$-0.266105\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2736.2.cg.a.2591.3	yes	24
3.2	odd	2	inner	2736.2.cg.a.2591.10	yes	24
4.3	odd	2		2736.2.cg.b.2591.3	yes	24
12.11	even	2		2736.2.cg.b.2591.10	yes	24
19.11	even	3		2736.2.cg.b.1151.10	yes	24
57.11	odd	6		2736.2.cg.b.1151.3	yes	24
76.11	odd	6	inner	2736.2.cg.a.1151.10	yes	24
228.11	even	6	inner	2736.2.cg.a.1151.3	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2736.2.cg.a.1151.3	✓	24	228.11	even	6	inner
2736.2.cg.a.1151.10	yes	24	76.11	odd	6	inner
2736.2.cg.a.2591.3	yes	24	1.1	even	1	trivial
2736.2.cg.a.2591.10	yes	24	3.2	odd	2	inner
2736.2.cg.b.1151.3	yes	24	57.11	odd	6
2736.2.cg.b.1151.10	yes	24	19.11	even	3
2736.2.cg.b.2591.3	yes	24	4.3	odd	2
2736.2.cg.b.2591.10	yes	24	12.11	even	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.cg (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.15944 + 1.24676i) q^{5} +0.872468i q^{7} +O(q^{10})\)	\(q+(-2.15944 + 1.24676i) q^{5} +0.872468i q^{7} +2.41030 q^{11} +(-1.98130 + 3.43171i) q^{13} +(0.303025 - 0.174952i) q^{17} +(-1.35622 - 4.14254i) q^{19} +(-3.88372 + 6.72680i) q^{23} +(0.608801 - 1.05447i) q^{25} +(-7.96395 - 4.59799i) q^{29} -2.43850i q^{31} +(-1.08776 - 1.88405i) q^{35} +6.18021 q^{37} +(-1.31650 + 0.760080i) q^{41} +(5.24748 - 3.02963i) q^{43} +(-4.05152 + 7.01744i) q^{47} +6.23880 q^{49} +(-3.41943 - 1.97421i) q^{53} +(-5.20491 + 3.00506i) q^{55} +(1.55163 + 2.68750i) q^{59} +(6.53184 - 11.3135i) q^{61} -9.88080i q^{65} +(5.04454 + 2.91247i) q^{67} +(-4.65667 - 8.06559i) q^{71} +(-1.01576 - 1.75935i) q^{73} +2.10291i q^{77} +(-10.8347 + 6.25541i) q^{79} -6.29106 q^{83} +(-0.436244 + 0.755598i) q^{85} +(-14.0641 - 8.11993i) q^{89} +(-2.99406 - 1.72862i) q^{91} +(8.09342 + 7.25472i) q^{95} +(-8.90950 - 15.4317i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 24 * q	\( 24 q + 12 q^{13} - 12 q^{19} + 8 q^{25} + 16 q^{37} - 12 q^{43} + 16 q^{49} + 12 q^{55} - 60 q^{67} + 8 q^{73} + 12 q^{79} + 16 q^{85} - 12 q^{91} - 32 q^{97}+O(q^{100}) \) 24 * q + 12 * q^13 - 12 * q^19 + 8 * q^25 + 16 * q^37 - 12 * q^43 + 16 * q^49 + 12 * q^55 - 60 * q^67 + 8 * q^73 + 12 * q^79 + 16 * q^85 - 12 * q^91 - 32 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more