LMFDB - Embedded newform 2736.2.cg.b.1151.7

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			1151.7
Character	$\chi$	$=$	2736.1151
Dual form			2736.2.cg.b.2591.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.420258 + 0.242636i) q^{5} -1.92619i q^{7} +O(q^{10})$	$q+(0.420258 + 0.242636i) q^{5} -1.92619i q^{7} -0.610994 q^{11} +(-0.889419 - 1.54052i) q^{13} +(-3.89960 - 2.25144i) q^{17} +(1.30176 + 4.15998i) q^{19} +(-0.121114 - 0.209776i) q^{23} +(-2.38226 - 4.12619i) q^{25} +(-5.18126 + 2.99140i) q^{29} +0.423054i q^{31} +(0.467363 - 0.809497i) q^{35} -1.98567 q^{37} +(-5.42417 - 3.13165i) q^{41} +(-3.24844 - 1.87549i) q^{43} +(2.88383 + 4.99494i) q^{47} +3.28979 q^{49} +(-2.85356 + 1.64750i) q^{53} +(-0.256775 - 0.148249i) q^{55} +(-2.16651 + 3.75250i) q^{59} +(-1.59275 - 2.75873i) q^{61} -0.863220i q^{65} +(-4.13814 + 2.38916i) q^{67} +(-7.60211 + 13.1672i) q^{71} +(3.34087 - 5.78656i) q^{73} +1.17689i q^{77} +(-13.3701 - 7.71920i) q^{79} +8.99578 q^{83} +(-1.09256 - 1.89237i) q^{85} +(-8.14682 + 4.70357i) q^{89} +(-2.96733 + 1.71319i) q^{91} +(-0.462288 + 2.06412i) q^{95} +(3.18910 - 5.52368i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times$.

$n$	$1009$	$1217$	$1711$	$2053$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$-1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	0.420258	+	0.242636i	0.187945	+	0.108510i	0.591020	−	0.806657i	$-0.298725\pi$
$5$	0.420258	+	0.242636i	0.187945	+	0.108510i	−0.403075	+	0.915167i	$0.632059\pi$
$6$		0			0
$7$		−	1.92619i		−	0.728032i	−0.931393	−	0.364016i	$-0.881405\pi$
$7$		−	1.92619i		−	0.728032i	0.931393	−	0.364016i	$-0.118595\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−0.610994			−0.184222			−0.0921109	−	0.995749i	$-0.529361\pi$
$11$	−0.610994			−0.184222			−0.0921109	+	0.995749i	$0.529361\pi$
$12$		0			0
$13$	−0.889419	−	1.54052i	−0.246680	−	0.427263i	0.715922	−	0.698180i	$-0.246006\pi$
$13$	−0.889419	−	1.54052i	−0.246680	−	0.427263i	−0.962603	+	0.270917i	$0.912673\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−3.89960	−	2.25144i	−0.945792	−	0.546053i	−0.0540208	−	0.998540i	$-0.517204\pi$
$17$	−3.89960	−	2.25144i	−0.945792	−	0.546053i	−0.891771	+	0.452487i	$0.850537\pi$
$18$		0			0
$19$	1.30176	+	4.15998i	0.298643	+	0.954365i
$19$	1.30176	+	4.15998i	0.298643	+	0.954365i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−0.121114	−	0.209776i	−0.0252540	−	0.0437413i	0.853122	−	0.521711i	$-0.174706\pi$
$23$	−0.121114	−	0.209776i	−0.0252540	−	0.0437413i	−0.878376	+	0.477970i	$0.841373\pi$
$24$		0			0
$25$	−2.38226	−	4.12619i	−0.476451	−	0.825237i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−5.18126	+	2.99140i	−0.962136	+	0.555490i	−0.896830	−	0.442376i	$-0.854136\pi$
$29$	−5.18126	+	2.99140i	−0.962136	+	0.555490i	−0.0653063	+	0.997865i	$0.520802\pi$
$30$		0			0
$31$			0.423054i			0.0759827i	0.999278	+	0.0379913i	$0.0120959\pi$
$31$			0.423054i			0.0759827i	−0.999278	+	0.0379913i	$0.987904\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	0.467363	−	0.809497i	0.0789988	−	0.136830i
$36$		0			0
$37$	−1.98567			−0.326443			−0.163221	−	0.986589i	$-0.552188\pi$
$37$	−1.98567			−0.326443			−0.163221	+	0.986589i	$0.552188\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−5.42417	−	3.13165i	−0.847114	−	0.489081i	0.0125623	−	0.999921i	$-0.496001\pi$
$41$	−5.42417	−	3.13165i	−0.847114	−	0.489081i	−0.859676	+	0.510840i	$0.829335\pi$
$42$		0			0
$43$	−3.24844	−	1.87549i	−0.495382	−	0.286009i	0.231422	−	0.972853i	$-0.425662\pi$
$43$	−3.24844	−	1.87549i	−0.495382	−	0.286009i	−0.726805	+	0.686844i	$0.758995\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	2.88383	+	4.99494i	0.420649	+	0.728586i	0.996003	−	0.0893185i	$-0.0284689\pi$
$47$	2.88383	+	4.99494i	0.420649	+	0.728586i	−0.575354	+	0.817905i	$0.695136\pi$
$48$		0			0
$49$	3.28979			0.469970
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−2.85356	+	1.64750i	−0.391966	+	0.226302i	−0.683012	−	0.730407i	$-0.739330\pi$
$53$	−2.85356	+	1.64750i	−0.391966	+	0.226302i	0.291045	+	0.956709i	$0.405997\pi$
$54$		0			0
$55$	−0.256775	−	0.148249i	−0.0346236	−	0.0199899i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−2.16651	+	3.75250i	−0.282056	+	0.488534i	−0.971891	−	0.235432i	$-0.924350\pi$
$59$	−2.16651	+	3.75250i	−0.282056	+	0.488534i	0.689835	+	0.723966i	$0.257683\pi$
$60$		0			0
$61$	−1.59275	−	2.75873i	−0.203931	−	0.353219i	0.745861	−	0.666102i	$-0.232038\pi$
$61$	−1.59275	−	2.75873i	−0.203931	−	0.353219i	−0.949792	+	0.312883i	$0.898705\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	0.863220i		−	0.107069i
$66$		0			0
$67$	−4.13814	+	2.38916i	−0.505554	+	0.291882i	−0.731004	−	0.682373i	$-0.760948\pi$
$67$	−4.13814	+	2.38916i	−0.505554	+	0.291882i	0.225450	+	0.974255i	$0.427615\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−7.60211	+	13.1672i	−0.902205	+	1.56266i	−0.0775785	+	0.996986i	$0.524719\pi$
$71$	−7.60211	+	13.1672i	−0.902205	+	1.56266i	−0.824626	+	0.565678i	$0.808614\pi$
$72$		0			0
$73$	3.34087	−	5.78656i	0.391019	−	0.677265i	−0.601565	−	0.798824i	$-0.705456\pi$
$73$	3.34087	−	5.78656i	0.391019	−	0.677265i	0.992584	+	0.121558i	$0.0387892\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			1.17689i			0.134119i
$78$		0			0
$79$	−13.3701	−	7.71920i	−1.50425	−	0.868478i	−0.999988	−	0.00492653i	$-0.998432\pi$
$79$	−13.3701	−	7.71920i	−1.50425	−	0.868478i	−0.504260	−	0.863552i	$-0.668235\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	8.99578			0.987415			0.493707	−	0.869628i	$-0.335641\pi$
$83$	8.99578			0.987415			0.493707	+	0.869628i	$0.335641\pi$
$84$		0			0
$85$	−1.09256	−	1.89237i	−0.118505	−	0.205256i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−8.14682	+	4.70357i	−0.863562	+	0.498578i	−0.865203	−	0.501421i	$-0.832811\pi$
$89$	−8.14682	+	4.70357i	−0.863562	+	0.498578i	0.00164176	+	0.999999i	$0.499477\pi$
$90$		0			0
$91$	−2.96733	+	1.71319i	−0.311061	+	0.179591i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−0.462288	+	2.06412i	−0.0474298	+	0.211774i
$96$		0			0
$97$	3.18910	−	5.52368i	0.323804	−	0.560845i	−0.657466	−	0.753484i	$-0.728372\pi$
$97$	3.18910	−	5.52368i	0.323804	−	0.560845i	0.981270	+	0.192640i	$0.0617049\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−4.61192	+	2.66269i	−0.458903	+	0.264948i	−0.711583	−	0.702602i	$-0.752021\pi$
$101$	−4.61192	+	2.66269i	−0.458903	+	0.264948i	0.252680	+	0.967550i	$0.418688\pi$
$102$		0			0
$103$			3.57269i			0.352027i	0.984388	+	0.176014i	$0.0563203\pi$
$103$			3.57269i			0.352027i	−0.984388	+	0.176014i	$0.943680\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	4.12772			0.399042			0.199521	−	0.979894i	$-0.436061\pi$
$107$	4.12772			0.399042			0.199521	+	0.979894i	$0.436061\pi$
$108$		0			0
$109$	5.62369	−	9.74051i	0.538652	−	0.932972i	−0.460325	−	0.887750i	$-0.652267\pi$
$109$	5.62369	−	9.74051i	0.538652	−	0.932972i	0.998977	−	0.0452218i	$-0.0143995\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	6.15441i		−	0.578958i	−0.957184	−	0.289479i	$-0.906518\pi$
$113$		−	6.15441i		−	0.578958i	0.957184	−	0.289479i	$-0.0934820\pi$
$114$		0			0
$115$		−	0.117547i		−	0.0109613i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−4.33669	+	7.51138i	−0.397544	+	0.688567i
$120$		0			0
$121$	−10.6267			−0.966062
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	4.73845i		−	0.423819i
$126$		0			0
$127$	−0.925602	+	0.534397i	−0.0821339	+	0.0474200i	−0.540504	−	0.841341i	$-0.681767\pi$
$127$	−0.925602	+	0.534397i	−0.0821339	+	0.0474200i	0.458371	+	0.888761i	$0.348433\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−1.48949	+	2.57987i	−0.130137	+	0.225404i	−0.923729	−	0.383046i	$-0.874875\pi$
$131$	−1.48949	+	2.57987i	−0.130137	+	0.225404i	0.793592	+	0.608450i	$0.208208\pi$
$132$		0			0
$133$	8.01292	−	2.50743i	0.694808	−	0.217422i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−15.9996	+	9.23737i	−1.36694	+	0.789202i	−0.990536	−	0.137254i	$-0.956172\pi$
$137$	−15.9996	+	9.23737i	−1.36694	+	0.789202i	−0.376402	+	0.926456i	$0.622839\pi$
$138$		0			0
$139$	7.89522	−	4.55831i	0.669664	−	0.386631i	−0.126285	−	0.991994i	$-0.540305\pi$
$139$	7.89522	−	4.55831i	0.669664	−	0.386631i	0.795949	+	0.605363i	$0.206972\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	0.543430	+	0.941248i	0.0454439	+	0.0787111i
$144$		0			0
$145$	−2.90329			−0.241105
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−12.3348	−	7.12152i	−1.01051	−	0.583418i	−0.0991686	−	0.995071i	$-0.531618\pi$
$149$	−12.3348	−	7.12152i	−1.01051	−	0.583418i	−0.911341	+	0.411653i	$0.864952\pi$
$150$		0			0
$151$		−	9.84107i		−	0.800855i	−0.916329	−	0.400427i	$-0.868862\pi$
$151$		−	9.84107i		−	0.800855i	0.916329	−	0.400427i	$-0.131138\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−0.102648	+	0.177792i	−0.00824489	+	0.0142806i
$156$		0			0
$157$	−3.26706	+	5.65872i	−0.260740	+	0.451615i	−0.966439	−	0.256897i	$-0.917300\pi$
$157$	−3.26706	+	5.65872i	−0.260740	+	0.451615i	0.705699	+	0.708512i	$0.250633\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−0.404068	+	0.233289i	−0.0318450	+	0.0183857i
$162$		0			0
$163$			15.0135i			1.17595i	0.808879	+	0.587975i	$0.200075\pi$
$163$			15.0135i			1.17595i	−0.808879	+	0.587975i	$0.799925\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−3.15701	−	5.46810i	−0.244297	−	0.423134i	0.717637	−	0.696417i	$-0.245224\pi$
$167$	−3.15701	−	5.46810i	−0.244297	−	0.423134i	−0.961934	+	0.273283i	$0.911890\pi$
$168$		0			0
$169$	4.91787	−	8.51800i	0.378298	−	0.655231i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−11.1182	−	6.41908i	−0.845299	−	0.488033i	0.0137631	−	0.999905i	$-0.495619\pi$
$173$	−11.1182	−	6.41908i	−0.845299	−	0.488033i	−0.859062	+	0.511872i	$0.828952\pi$
$174$		0			0
$175$	−7.94782	+	4.58868i	−0.600799	+	0.346872i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	11.4674			0.857115			0.428558	−	0.903514i	$-0.359022\pi$
$179$	11.4674			0.857115			0.428558	+	0.903514i	$0.359022\pi$
$180$		0			0
$181$	−6.87538	−	11.9085i	−0.511043	−	0.885152i	−0.999918	−	0.0127983i	$-0.995926\pi$
$181$	−6.87538	−	11.9085i	−0.511043	−	0.885152i	0.488875	−	0.872354i	$-0.337407\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−0.834495	−	0.481796i	−0.0613533	−	0.0354223i
$186$		0			0
$187$	2.38263	+	1.37561i	0.174235	+	0.100595i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−17.4411			−1.26199			−0.630996	−	0.775786i	$-0.717353\pi$
$191$	−17.4411			−1.26199			−0.630996	+	0.775786i	$0.717353\pi$
$192$		0			0
$193$	3.48115	−	6.02954i	0.250579	−	0.434016i	−0.713106	−	0.701056i	$-0.752712\pi$
$193$	3.48115	−	6.02954i	0.250579	−	0.434016i	0.963685	+	0.267040i	$0.0860457\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			9.58982i			0.683246i	0.939837	+	0.341623i	$0.110977\pi$
$197$			9.58982i			0.683246i	−0.939837	+	0.341623i	$0.889023\pi$
$198$		0			0
$199$	3.32171	−	1.91779i	0.235470	−	0.135948i	−0.377623	−	0.925959i	$-0.623259\pi$
$199$	3.32171	−	1.91779i	0.235470	−	0.135948i	0.613093	+	0.790011i	$0.289925\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	5.76201	+	9.98010i	0.404414	+	0.700466i
$204$		0			0
$205$	−1.51970	−	2.63220i	−0.106141	−	0.183841i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−0.795365	−	2.54172i	−0.0550165	−	0.175815i
$210$		0			0
$211$	−9.19164	−	5.30679i	−0.632778	−	0.365335i	0.149049	−	0.988830i	$-0.452379\pi$
$211$	−9.19164	−	5.30679i	−0.632778	−	0.365335i	−0.781827	+	0.623495i	$0.785712\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−0.910122	−	1.57638i	−0.0620698	−	0.107508i
$216$		0			0
$217$	0.814882			0.0553178
$218$		0			0
$219$		0			0
$220$		0			0
$221$			8.00988i			0.538803i
$222$		0			0
$223$	−6.06628	−	3.50237i	−0.406228	−	0.234536i	0.282940	−	0.959138i	$-0.408690\pi$
$223$	−6.06628	−	3.50237i	−0.406228	−	0.234536i	−0.689168	+	0.724602i	$0.742024\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	11.1861			0.742450			0.371225	−	0.928543i	$-0.378938\pi$
$227$	11.1861			0.742450			0.371225	+	0.928543i	$0.378938\pi$
$228$		0			0
$229$	−10.5605			−0.697859			−0.348930	−	0.937149i	$-0.613455\pi$
$229$	−10.5605			−0.697859			−0.348930	+	0.937149i	$0.613455\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	5.49728	+	3.17386i	0.360139	+	0.207926i	0.669142	−	0.743135i	$-0.266662\pi$
$233$	5.49728	+	3.17386i	0.360139	+	0.207926i	−0.309003	+	0.951061i	$0.599995\pi$
$234$		0			0
$235$			2.79888i			0.182579i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	2.69586			0.174381			0.0871904	−	0.996192i	$-0.472211\pi$
$239$	2.69586			0.174381			0.0871904	+	0.996192i	$0.472211\pi$
$240$		0			0
$241$	0.667812	+	1.15668i	0.0430175	+	0.0745086i	0.886733	−	0.462283i	$-0.152970\pi$
$241$	0.667812	+	1.15668i	0.0430175	+	0.0745086i	−0.843715	+	0.536791i	$0.819636\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	1.38256	+	0.798222i	0.0883285	+	0.0509965i
$246$		0			0
$247$	5.25072	−	5.70534i	0.334095	−	0.363022i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−5.08912	−	8.81462i	−0.321223	−	0.556374i	0.659518	−	0.751689i	$-0.270760\pi$
$251$	−5.08912	−	8.81462i	−0.321223	−	0.556374i	−0.980741	+	0.195315i	$0.937427\pi$
$252$		0			0
$253$	0.0740000	+	0.128172i	0.00465234	+	0.00805809i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−13.9876	+	8.07575i	−0.872523	+	0.503751i	−0.868186	−	0.496239i	$-0.834714\pi$
$257$	−13.9876	+	8.07575i	−0.872523	+	0.503751i	−0.00433712	+	0.999991i	$0.501381\pi$
$258$		0			0
$259$			3.82479i			0.237661i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	0.492663	−	0.853317i	0.0303789	−	0.0526178i	−0.850436	−	0.526078i	$-0.823662\pi$
$263$	0.492663	−	0.853317i	0.0303789	−	0.0526178i	0.880815	+	0.473460i	$0.156995\pi$
$264$		0			0
$265$	−1.59897			−0.0982242
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−15.8778	−	9.16706i	−0.968087	−	0.558926i	−0.0694348	−	0.997586i	$-0.522120\pi$
$269$	−15.8778	−	9.16706i	−0.968087	−	0.558926i	−0.898653	+	0.438661i	$0.855453\pi$
$270$		0			0
$271$	−7.22963	−	4.17403i	−0.439169	−	0.253554i	0.264076	−	0.964502i	$-0.414933\pi$
$271$	−7.22963	−	4.17403i	−0.439169	−	0.253554i	−0.703245	+	0.710948i	$0.748266\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.45554	+	2.52108i	0.0877726	+	0.152027i
$276$		0			0
$277$	22.1188			1.32899			0.664494	−	0.747293i	$-0.268647\pi$
$277$	22.1188			1.32899			0.664494	+	0.747293i	$0.268647\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	9.79959	−	5.65780i	0.584595	−	0.337516i	−0.178363	−	0.983965i	$-0.557080\pi$
$281$	9.79959	−	5.65780i	0.584595	−	0.337516i	0.762957	+	0.646449i	$0.223747\pi$
$282$		0			0
$283$	9.12442	+	5.26799i	0.542390	+	0.313149i	0.746047	−	0.665893i	$-0.231949\pi$
$283$	9.12442	+	5.26799i	0.542390	+	0.313149i	−0.203657	+	0.979042i	$0.565283\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−6.03215	+	10.4480i	−0.356067	+	0.616726i
$288$		0			0
$289$	1.63792	+	2.83697i	0.0963484	+	0.166880i
$290$		0			0
$291$		0			0
$292$		0			0
$293$			6.28078i			0.366927i	0.983027	+	0.183464i	$0.0587309\pi$
$293$			6.28078i			0.366927i	−0.983027	+	0.183464i	$0.941269\pi$
$294$		0			0
$295$	−1.82099	+	1.05135i	−0.106022	+	0.0612118i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−0.215442	+	0.373157i	−0.0124593	+	0.0215802i
$300$		0			0
$301$	−3.61255	+	6.25711i	−0.208224	+	0.360654i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	1.54584i		−	0.0885143i
$306$		0			0
$307$	12.0020	+	6.92933i	0.684988	+	0.395478i	0.801732	−	0.597684i	$-0.203912\pi$
$307$	12.0020	+	6.92933i	0.684988	+	0.395478i	−0.116744	+	0.993162i	$0.537246\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−8.69554			−0.493079			−0.246539	−	0.969133i	$-0.579293\pi$
$311$	−8.69554			−0.493079			−0.246539	+	0.969133i	$0.579293\pi$
$312$		0			0
$313$	−0.278745	−	0.482801i	−0.0157556	−	0.0272895i	0.858040	−	0.513583i	$-0.171682\pi$
$313$	−0.278745	−	0.482801i	−0.0157556	−	0.0272895i	−0.873796	+	0.486293i	$0.838349\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	24.9649	−	14.4135i	1.40217	−	0.809542i	0.407553	−	0.913182i	$-0.366382\pi$
$317$	24.9649	−	14.4135i	1.40217	−	0.809542i	0.994615	+	0.103639i	$0.0330488\pi$
$318$		0			0
$319$	3.16572	−	1.82773i	0.177246	−	0.102333i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	4.28960	−	19.1531i	0.238680	−	1.06571i
$324$		0			0
$325$	−4.23765	+	7.33982i	−0.235062	+	0.407140i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	9.62120	−	5.55480i	0.530434	−	0.306246i
$330$		0			0
$331$			6.62797i			0.364306i	0.983270	+	0.182153i	$0.0583067\pi$
$331$			6.62797i			0.364306i	−0.983270	+	0.182153i	$0.941693\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−2.31878			−0.126689
$336$		0			0
$337$	−5.92757	+	10.2668i	−0.322895	+	0.559271i	−0.981084	−	0.193582i	$-0.937989\pi$
$337$	−5.92757	+	10.2668i	−0.322895	+	0.559271i	0.658189	+	0.752853i	$0.271323\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	0.258483i		−	0.0139977i
$342$		0			0
$343$		−	19.8201i		−	1.07018i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	13.2261	−	22.9084i	0.710017	−	1.22979i	−0.254833	−	0.966985i	$-0.582021\pi$
$347$	13.2261	−	22.9084i	0.710017	−	1.22979i	0.964850	−	0.262800i	$-0.0846460\pi$
$348$		0			0
$349$	14.9308			0.799226			0.399613	−	0.916684i	$-0.369145\pi$
$349$	14.9308			0.799226			0.399613	+	0.916684i	$0.369145\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	19.4562i		−	1.03555i	−0.855517	−	0.517775i	$-0.826760\pi$
$353$		−	19.4562i		−	1.03555i	0.855517	−	0.517775i	$-0.173240\pi$
$354$		0			0
$355$	−6.38970	+	3.68909i	−0.339130	+	0.195797i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−10.7512	+	18.6215i	−0.567424	+	0.982808i	0.429395	+	0.903117i	$0.358727\pi$
$359$	−10.7512	+	18.6215i	−0.567424	+	0.982808i	−0.996820	+	0.0796910i	$0.974607\pi$
$360$		0			0
$361$	−15.6109	+	10.8306i	−0.821625	+	0.570029i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	2.80806	−	1.62123i	0.146980	−	0.0848592i
$366$		0			0
$367$	8.60649	−	4.96896i	0.449255	−	0.259378i	−0.258260	−	0.966075i	$-0.583149\pi$
$367$	8.60649	−	4.96896i	0.449255	−	0.259378i	0.707516	+	0.706698i	$0.249816\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	3.17340	+	5.49650i	0.164755	+	0.285364i
$372$		0			0
$373$	−18.4225			−0.953881			−0.476941	−	0.878936i	$-0.658254\pi$
$373$	−18.4225			−0.953881			−0.476941	+	0.878936i	$0.658254\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	9.21662	+	5.32122i	0.474680	+	0.274057i
$378$		0			0
$379$			20.7734i			1.06706i	0.845783	+	0.533528i	$0.179134\pi$
$379$			20.7734i			1.06706i	−0.845783	+	0.533528i	$0.820866\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−13.3400	+	23.1055i	−0.681642	+	1.18064i	0.292838	+	0.956162i	$0.405400\pi$
$383$	−13.3400	+	23.1055i	−0.681642	+	1.18064i	−0.974480	+	0.224476i	$0.927933\pi$
$384$		0			0
$385$	−0.285556	+	0.494598i	−0.0145533	+	0.0252071i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	5.27614	−	3.04618i	0.267511	−	0.154447i	−0.360245	−	0.932858i	$-0.617307\pi$
$389$	5.27614	−	3.04618i	0.267511	−	0.154447i	0.627756	+	0.778410i	$0.283974\pi$
$390$		0			0
$391$			1.09072i			0.0551602i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−3.74591	−	6.48811i	−0.188477	−	0.326452i
$396$		0			0
$397$	−5.32784	+	9.22810i	−0.267397	+	0.463145i	−0.968189	−	0.250221i	$-0.919497\pi$
$397$	−5.32784	+	9.22810i	−0.267397	+	0.463145i	0.700792	+	0.713366i	$0.252830\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	22.9567	+	13.2540i	1.14640	+	0.661875i	0.948008	−	0.318248i	$-0.103094\pi$
$401$	22.9567	+	13.2540i	1.14640	+	0.661875i	0.198393	+	0.980123i	$0.436428\pi$
$402$		0			0
$403$	0.651722	−	0.376272i	0.0324646	−	0.0187434i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	1.21324			0.0601378
$408$		0			0
$409$	2.19512	+	3.80206i	0.108542	+	0.188000i	0.915180	−	0.403046i	$-0.132049\pi$
$409$	2.19512	+	3.80206i	0.108542	+	0.188000i	−0.806638	+	0.591046i	$0.798715\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	7.22804	+	4.17311i	0.355669	+	0.205345i
$414$		0			0
$415$	3.78055	+	2.18270i	0.185580	+	0.107145i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	5.42844			0.265197			0.132598	−	0.991170i	$-0.457668\pi$
$419$	5.42844			0.265197			0.132598	+	0.991170i	$0.457668\pi$
$420$		0			0
$421$	2.57871	−	4.46646i	0.125679	−	0.217682i	−0.796319	−	0.604876i	$-0.793223\pi$
$421$	2.57871	−	4.46646i	0.125679	−	0.217682i	0.921998	+	0.387195i	$0.126556\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			21.4540i			1.04067i
$426$		0			0
$427$	−5.31383	+	3.06794i	−0.257154	+	0.148468i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	9.18928	+	15.9163i	0.442632	+	0.766661i	0.997884	−	0.0650209i	$-0.0207114\pi$
$431$	9.18928	+	15.9163i	0.442632	+	0.766661i	−0.555252	+	0.831682i	$0.687378\pi$
$432$		0			0
$433$	11.1096	+	19.2424i	0.533894	+	0.924732i	0.999216	+	0.0395901i	$0.0126052\pi$
$433$	11.1096	+	19.2424i	0.533894	+	0.924732i	−0.465322	+	0.885141i	$0.654061\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	0.715002	−	0.776909i	0.0342032	−	0.0371646i
$438$		0			0
$439$	−25.6789	−	14.8257i	−1.22558	−	0.707592i	−0.259482	−	0.965748i	$-0.583552\pi$
$439$	−25.6789	−	14.8257i	−1.22558	−	0.707592i	−0.966103	+	0.258156i	$0.916885\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−11.3830	−	19.7160i	−0.540824	−	0.936734i	−0.998857	−	0.0477989i	$-0.984779\pi$
$443$	−11.3830	−	19.7160i	−0.540824	−	0.936734i	0.458033	−	0.888935i	$-0.348554\pi$
$444$		0			0
$445$	−4.56502			−0.216403
$446$		0			0
$447$		0			0
$448$		0			0
$449$			4.46867i			0.210889i	0.994425	+	0.105445i	$0.0336266\pi$
$449$			4.46867i			0.210889i	−0.994425	+	0.105445i	$0.966373\pi$
$450$		0			0
$451$	3.31414	+	1.91342i	0.156057	+	0.0900994i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−1.66273			−0.0779499
$456$		0			0
$457$	5.85292			0.273788			0.136894	−	0.990586i	$-0.456288\pi$
$457$	5.85292			0.273788			0.136894	+	0.990586i	$0.456288\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	13.0016	+	7.50646i	0.605544	+	0.349611i	0.771219	−	0.636569i	$-0.219647\pi$
$461$	13.0016	+	7.50646i	0.605544	+	0.349611i	−0.165676	+	0.986180i	$0.552980\pi$
$462$		0			0
$463$		−	1.11830i		−	0.0519719i	−0.999662	−	0.0259860i	$-0.991727\pi$
$463$		−	1.11830i		−	0.0519719i	0.999662	−	0.0259860i	$-0.00827252\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	41.5205			1.92134			0.960671	−	0.277691i	$-0.0895690\pi$
$467$	41.5205			1.92134			0.960671	+	0.277691i	$0.0895690\pi$
$468$		0			0
$469$	4.60197	+	7.97085i	0.212499	+	0.368060i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	1.98478	+	1.14591i	0.0912602	+	0.0526891i
$474$		0			0
$475$	14.0637	−	15.2814i	0.645289	−	0.701160i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−7.83998	−	13.5792i	−0.358218	−	0.620451i	0.629445	−	0.777045i	$-0.283282\pi$
$479$	−7.83998	−	13.5792i	−0.358218	−	0.620451i	−0.987663	+	0.156593i	$0.949949\pi$
$480$		0			0
$481$	1.76610	+	3.05897i	0.0805270	+	0.139477i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	2.68049	−	1.54758i	0.121715	−	0.0702720i
$486$		0			0
$487$		−	2.04224i		−	0.0925427i	−0.998929	−	0.0462714i	$-0.985266\pi$
$487$		−	2.04224i		−	0.0925427i	0.998929	−	0.0462714i	$-0.0147339\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	13.2695	−	22.9835i	0.598846	−	1.03723i	−0.394145	−	0.919048i	$-0.628959\pi$
$491$	13.2695	−	22.9835i	0.598846	−	1.03723i	0.992992	−	0.118184i	$-0.0377073\pi$
$492$		0			0
$493$	26.9398			1.21331
$494$		0			0
$495$		0			0
$496$		0			0
$497$	25.3626	+	14.6431i	1.13767	+	0.656834i
$498$		0			0
$499$	29.9939	+	17.3170i	1.34271	+	0.775214i	0.987204	−	0.159460i	$-0.0509752\pi$
$499$	29.9939	+	17.3170i	1.34271	+	0.775214i	0.355506	+	0.934674i	$0.384309\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−15.9791	−	27.6765i	−0.712471	−	1.23404i	−0.963927	−	0.266168i	$-0.914243\pi$
$503$	−15.9791	−	27.6765i	−0.712471	−	1.23404i	0.251456	−	0.967869i	$-0.419091\pi$
$504$		0			0
$505$	−2.58426			−0.114998
$506$		0			0
$507$		0			0
$508$		0			0
$509$	19.7589	−	11.4078i	0.875799	−	0.505643i	0.00652823	−	0.999979i	$-0.497922\pi$
$509$	19.7589	−	11.4078i	0.875799	−	0.505643i	0.869271	+	0.494336i	$0.164589\pi$
$510$		0			0
$511$	−11.1460	−	6.43516i	−0.493071	−	0.284675i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−0.866863	+	1.50145i	−0.0381985	+	0.0661618i
$516$		0			0
$517$	−1.76200	−	3.05188i	−0.0774928	−	0.134221i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	5.09569i		−	0.223246i	−0.993751	−	0.111623i	$-0.964395\pi$
$521$		−	5.09569i		−	0.223246i	0.993751	−	0.111623i	$-0.0356050\pi$
$522$		0			0
$523$	21.1712	−	12.2232i	0.925752	−	0.534483i	0.0402866	−	0.999188i	$-0.487173\pi$
$523$	21.1712	−	12.2232i	0.925752	−	0.534483i	0.885466	+	0.464705i	$0.153840\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	0.952478	−	1.64974i	0.0414906	−	0.0718638i
$528$		0			0
$529$	11.4707	−	19.8678i	0.498724	−	0.863816i
$530$		0			0
$531$		0			0
$532$		0			0
$533$			11.1414i			0.482587i
$534$		0			0
$535$	1.73471	+	1.00153i	0.0749980	+	0.0433001i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−2.01004			−0.0865787
$540$		0			0
$541$	15.0409	+	26.0516i	0.646659	+	1.12005i	0.983916	+	0.178633i	$0.0571675\pi$
$541$	15.0409	+	26.0516i	0.646659	+	1.12005i	−0.337257	+	0.941413i	$0.609499\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	4.72680	−	2.72902i	0.202474	−	0.116898i
$546$		0			0
$547$	17.2592	−	9.96461i	0.737951	−	0.426056i	−0.0833729	−	0.996518i	$-0.526569\pi$
$547$	17.2592	−	9.96461i	0.737951	−	0.426056i	0.821324	+	0.570462i	$0.193236\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−19.1889	−	17.6599i	−0.817475	−	0.752336i
$552$		0			0
$553$	−14.8687	+	25.7533i	−0.632280	+	1.09514i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	8.34354	−	4.81715i	0.353527	−	0.204109i	−0.312710	−	0.949849i	$-0.601237\pi$
$557$	8.34354	−	4.81715i	0.353527	−	0.204109i	0.666238	+	0.745739i	$0.267904\pi$
$558$		0			0
$559$			6.67237i			0.282211i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−1.81914			−0.0766677			−0.0383339	−	0.999265i	$-0.512205\pi$
$563$	−1.81914			−0.0766677			−0.0383339	+	0.999265i	$0.512205\pi$
$564$		0			0
$565$	1.49328	−	2.58644i	0.0628228	−	0.108812i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	16.0388i		−	0.672382i	−0.941794	−	0.336191i	$-0.890861\pi$
$569$		−	16.0388i		−	0.672382i	0.941794	−	0.336191i	$-0.109139\pi$
$570$		0			0
$571$			3.71146i			0.155320i	0.996980	+	0.0776598i	$0.0247448\pi$
$571$			3.71146i			0.155320i	−0.996980	+	0.0776598i	$0.975255\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−0.577049	+	0.999478i	−0.0240646	+	0.0416811i
$576$		0			0
$577$	−46.9587			−1.95492			−0.977459	−	0.211126i	$-0.932287\pi$
$577$	−46.9587			−1.95492			−0.977459	+	0.211126i	$0.932287\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	17.3276i		−	0.718869i
$582$		0			0
$583$	1.74351	−	1.00661i	0.0722087	−	0.0416897i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	22.0431	−	38.1798i	0.909816	−	1.57585i	0.0954964	−	0.995430i	$-0.469556\pi$
$587$	22.0431	−	38.1798i	0.909816	−	1.57585i	0.814319	−	0.580417i	$-0.197110\pi$
$588$		0			0
$589$	−1.75989	+	0.550712i	−0.0725152	+	0.0226917i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	10.9625	−	6.32922i	0.450177	−	0.259910i	−0.257728	−	0.966218i	$-0.582974\pi$
$593$	10.9625	−	6.32922i	0.450177	−	0.259910i	0.707905	+	0.706308i	$0.249641\pi$
$594$		0			0
$595$	−3.64506	+	2.10448i	−0.149433	+	0.0862752i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−24.2828	−	42.0590i	−0.992168	−	1.71849i	−0.604260	−	0.796787i	$-0.706531\pi$
$599$	−24.2828	−	42.0590i	−0.992168	−	1.71849i	−0.387908	−	0.921698i	$-0.626802\pi$
$600$		0			0
$601$	25.2147			1.02853			0.514264	−	0.857632i	$-0.328065\pi$
$601$	25.2147			1.02853			0.514264	+	0.857632i	$0.328065\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−4.46595	−	2.57842i	−0.181567	−	0.104828i
$606$		0			0
$607$			34.9811i			1.41984i	0.704283	+	0.709919i	$0.251269\pi$
$607$			34.9811i			1.41984i	−0.704283	+	0.709919i	$0.748731\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	5.12986	−	8.88518i	0.207532	−	0.359456i
$612$		0			0
$613$	−13.9896	+	24.2307i	−0.565035	+	0.978670i	0.432011	+	0.901868i	$0.357804\pi$
$613$	−13.9896	+	24.2307i	−0.565035	+	0.978670i	−0.997046	+	0.0768017i	$0.975529\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	16.0668	−	9.27615i	0.646824	−	0.373444i	−0.140415	−	0.990093i	$-0.544844\pi$
$617$	16.0668	−	9.27615i	0.646824	−	0.373444i	0.787238	+	0.616649i	$0.211510\pi$
$618$		0			0
$619$		−	3.22585i		−	0.129658i	−0.997896	−	0.0648290i	$-0.979350\pi$
$619$		−	3.22585i		−	0.129658i	0.997896	−	0.0648290i	$-0.0206502\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	9.05998	+	15.6923i	0.362980	+	0.628700i
$624$		0			0
$625$	−10.7616	+	18.6396i	−0.430462	+	0.745583i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	7.74333	+	4.47062i	0.308747	+	0.178255i
$630$		0			0
$631$	30.6237	−	17.6806i	1.21911	−	0.703852i	0.254381	−	0.967104i	$-0.418128\pi$
$631$	30.6237	−	17.6806i	1.21911	−	0.703852i	0.964727	+	0.263252i	$0.0847949\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−0.518656			−0.0205822
$636$		0			0
$637$	−2.92600	−	5.06798i	−0.115932	−	0.200801i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−1.15028	−	0.664116i	−0.0454334	−	0.0262310i	0.477111	−	0.878843i	$-0.341684\pi$
$641$	−1.15028	−	0.664116i	−0.0454334	−	0.0262310i	−0.522545	+	0.852612i	$0.675017\pi$
$642$		0			0
$643$	23.3711	+	13.4933i	0.921666	+	0.532124i	0.884166	−	0.467173i	$-0.154727\pi$
$643$	23.3711	+	13.4933i	0.921666	+	0.532124i	0.0374997	+	0.999297i	$0.488061\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−27.9228			−1.09776			−0.548878	−	0.835902i	$-0.684945\pi$
$647$	−27.9228			−1.09776			−0.548878	+	0.835902i	$0.684945\pi$
$648$		0			0
$649$	1.32373	−	2.29276i	0.0519608	−	0.0899987i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	1.51311i		−	0.0592127i	−0.999562	−	0.0296064i	$-0.990575\pi$
$653$		−	1.51311i		−	0.0592127i	0.999562	−	0.0296064i	$-0.00942538\pi$
$654$		0			0
$655$	−1.25194	+	0.722808i	−0.0489174	+	0.0282424i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	14.1199	+	24.4564i	0.550033	+	0.952686i	0.998271	+	0.0587714i	$0.0187183\pi$
$659$	14.1199	+	24.4564i	0.550033	+	0.952686i	−0.448238	+	0.893914i	$0.647948\pi$
$660$		0			0
$661$	−4.30326	−	7.45347i	−0.167377	−	0.289906i	0.770120	−	0.637900i	$-0.220197\pi$
$661$	−4.30326	−	7.45347i	−0.167377	−	0.289906i	−0.937497	+	0.347993i	$0.886863\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	3.97589	+	0.890455i	0.154178	+	0.0345304i
$666$		0			0
$667$	1.25505	+	0.724602i	0.0485956	+	0.0280567i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	0.973162	+	1.68557i	0.0375685	+	0.0650706i
$672$		0			0
$673$	25.3199			0.976009			0.488005	−	0.872841i	$-0.337725\pi$
$673$	25.3199			0.976009			0.488005	+	0.872841i	$0.337725\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			18.7436i			0.720376i	0.932880	+	0.360188i	$0.117287\pi$
$677$			18.7436i			0.720376i	−0.932880	+	0.360188i	$0.882713\pi$
$678$		0			0
$679$	−10.6397	−	6.14281i	−0.408313	−	0.235739i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−7.71791			−0.295318			−0.147659	−	0.989038i	$-0.547174\pi$
$683$	−7.71791			−0.295318			−0.147659	+	0.989038i	$0.547174\pi$
$684$		0			0
$685$	−8.96528			−0.342546
$686$		0			0
$687$		0			0
$688$		0			0
$689$	5.07602	+	2.93064i	0.193381	+	0.111648i
$690$		0			0
$691$		−	5.59423i		−	0.212814i	−0.994323	−	0.106407i	$-0.966065\pi$
$691$		−	5.59423i		−	0.212814i	0.994323	−	0.106407i	$-0.0339347\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	4.42404			0.167813
$696$		0			0
$697$	14.1014	+	24.4244i	0.534129	+	0.925138i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	26.9559	+	15.5630i	1.01811	+	0.587807i	0.913556	−	0.406713i	$-0.133325\pi$
$701$	26.9559	+	15.5630i	1.01811	+	0.587807i	0.104555	+	0.994519i	$0.466658\pi$
$702$		0			0
$703$	−2.58486	−	8.26036i	−0.0974899	−	0.311545i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	5.12886	+	8.88344i	0.192891	+	0.334096i
$708$		0			0
$709$	10.5873	+	18.3377i	0.397614	+	0.688688i	0.993431	−	0.114432i	$-0.0365049\pi$
$709$	10.5873	+	18.3377i	0.397614	+	0.688688i	−0.595817	+	0.803120i	$0.703172\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	0.0887463	−	0.0512377i	0.00332358	−	0.00191887i
$714$		0			0
$715$			0.527423i			0.0197245i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	16.0882	−	27.8656i	0.599988	−	1.03921i	−0.392834	−	0.919609i	$-0.628505\pi$
$719$	16.0882	−	27.8656i	0.599988	−	1.03921i	0.992822	−	0.119600i	$-0.0381613\pi$
$720$		0			0
$721$	6.88168			0.256287
$722$		0			0
$723$		0			0
$724$		0			0
$725$	24.6862	+	14.2526i	0.916822	+	0.529327i
$726$		0			0
$727$	−8.00241	−	4.62020i	−0.296793	−	0.171354i	0.344208	−	0.938893i	$-0.388147\pi$
$727$	−8.00241	−	4.62020i	−0.296793	−	0.171354i	−0.641001	+	0.767540i	$0.721481\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	8.44507	+	14.6273i	0.312352	+	0.541010i
$732$		0			0
$733$	−19.4170			−0.717184			−0.358592	−	0.933494i	$-0.616743\pi$
$733$	−19.4170			−0.717184			−0.358592	+	0.933494i	$0.616743\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	2.52838	−	1.45976i	0.0931341	−	0.0537710i
$738$		0			0
$739$	−5.33473	−	3.08001i	−0.196241	−	0.113300i	0.398660	−	0.917099i	$-0.369475\pi$
$739$	−5.33473	−	3.08001i	−0.196241	−	0.113300i	−0.594901	+	0.803799i	$0.702809\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−16.0767	+	27.8457i	−0.589798	+	1.02156i	0.404461	+	0.914555i	$0.367459\pi$
$743$	−16.0767	+	27.8457i	−0.589798	+	1.02156i	−0.994259	+	0.107004i	$0.965874\pi$
$744$		0			0
$745$	−3.45588	−	5.98575i	−0.126614	−	0.219301i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	7.95078i		−	0.290515i
$750$		0			0
$751$	19.9837	−	11.5376i	0.729216	−	0.421013i	−0.0889190	−	0.996039i	$-0.528341\pi$
$751$	19.9837	−	11.5376i	0.729216	−	0.421013i	0.818135	+	0.575026i	$0.195008\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	2.38780	−	4.13579i	0.0869009	−	0.150517i
$756$		0			0
$757$	−11.0252	+	19.0963i	−0.400719	+	0.694066i	−0.993813	−	0.111067i	$-0.964573\pi$
$757$	−11.0252	+	19.0963i	−0.400719	+	0.694066i	0.593093	+	0.805134i	$0.297906\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			7.20390i			0.261141i	0.991439	+	0.130571i	$0.0416809\pi$
$761$			7.20390i			0.261141i	−0.991439	+	0.130571i	$0.958319\pi$
$762$		0			0
$763$	−18.7621	−	10.8323i	−0.679233	−	0.392156i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	7.70774			0.278310
$768$		0			0
$769$	5.60217	+	9.70323i	0.202019	+	0.349908i	0.949179	−	0.314737i	$-0.101916\pi$
$769$	5.60217	+	9.70323i	0.202019	+	0.349908i	−0.747160	+	0.664645i	$0.768583\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	24.0909	−	13.9089i	0.866491	−	0.500269i	0.000310458	−	1.00000i	$-0.499901\pi$
$773$	24.0909	−	13.9089i	0.866491	−	0.500269i	0.866181	+	0.499731i	$0.166568\pi$
$774$		0			0
$775$	1.74560	−	1.00782i	0.0627037	−	0.0362020i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	5.96665	−	26.6411i	0.213777	−	0.954516i
$780$		0			0
$781$	4.64485	−	8.04511i	0.166206	−	0.287877i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−2.74602	+	1.58542i	−0.0980096	+	0.0565859i
$786$		0			0
$787$			49.1575i			1.75228i	0.482060	+	0.876138i	$0.339889\pi$
$787$			49.1575i			1.75228i	−0.482060	+	0.876138i	$0.660111\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−11.8546			−0.421500
$792$		0			0
$793$	−2.83325	+	4.90733i	−0.100612	+	0.174264i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			5.13639i			0.181940i	0.995854	+	0.0909701i	$0.0289968\pi$
$797$			5.13639i			0.181940i	−0.995854	+	0.0909701i	$0.971003\pi$
$798$		0			0
$799$		−	25.9710i		−	0.918788i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−2.04125	+	3.53555i	−0.0720343	+	0.124767i
$804$		0			0
$805$	−0.226417			−0.00798015
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	26.9399i		−	0.947156i	−0.880752	−	0.473578i	$-0.842962\pi$
$809$		−	26.9399i		−	0.947156i	0.880752	−	0.473578i	$-0.157038\pi$
$810$		0			0
$811$	−23.4757	+	13.5537i	−0.824345	+	0.475936i	−0.851912	−	0.523684i	$-0.824557\pi$
$811$	−23.4757	+	13.5537i	−0.824345	+	0.475936i	0.0275678	+	0.999620i	$0.491224\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−3.64282	+	6.30955i	−0.127602	+	0.221014i
$816$		0			0
$817$	3.57332	−	15.9549i	0.125015	−	0.558190i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	1.06111	−	0.612630i	0.0370329	−	0.0213809i	−0.481369	−	0.876518i	$-0.659860\pi$
$821$	1.06111	−	0.612630i	0.0370329	−	0.0213809i	0.518402	+	0.855137i	$0.326527\pi$
$822$		0			0
$823$	7.98619	−	4.61083i	0.278381	−	0.160723i	−0.354309	−	0.935128i	$-0.615284\pi$
$823$	7.98619	−	4.61083i	0.278381	−	0.160723i	0.632690	+	0.774405i	$0.281951\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−14.1285	−	24.4713i	−0.491297	−	0.850951i	0.508653	−	0.860972i	$-0.330144\pi$
$827$	−14.1285	−	24.4713i	−0.491297	−	0.850951i	−0.999950	+	0.0100207i	$0.996810\pi$
$828$		0			0
$829$	−4.20848			−0.146167			−0.0730833	−	0.997326i	$-0.523284\pi$
$829$	−4.20848			−0.146167			−0.0730833	+	0.997326i	$0.523284\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−12.8289	−	7.40675i	−0.444494	−	0.256629i
$834$		0			0
$835$		−	3.06402i		−	0.106035i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−20.7781	+	35.9888i	−0.717341	+	1.24247i	0.244709	+	0.969597i	$0.421308\pi$
$839$	−20.7781	+	35.9888i	−0.717341	+	1.24247i	−0.962050	+	0.272874i	$0.912026\pi$
$840$		0			0
$841$	3.39698	−	5.88374i	0.117137	−	0.202888i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	4.13355	−	2.38650i	0.142198	−	0.0820983i
$846$		0			0
$847$			20.4690i			0.703324i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	0.240493	+	0.416546i	0.00824399	+	0.0142790i
$852$		0			0
$853$	8.60095	−	14.8973i	0.294491	−	0.510073i	−0.680376	−	0.732864i	$-0.738183\pi$
$853$	8.60095	−	14.8973i	0.294491	−	0.510073i	0.974866	+	0.222791i	$0.0715166\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	28.5845	+	16.5032i	0.976426	+	0.563740i	0.901189	−	0.433426i	$-0.142695\pi$
$857$	28.5845	+	16.5032i	0.976426	+	0.563740i	0.0752371	+	0.997166i	$0.476029\pi$
$858$		0			0
$859$	31.1602	−	17.9904i	1.06317	−	0.613824i	0.136865	−	0.990590i	$-0.456297\pi$
$859$	31.1602	−	17.9904i	1.06317	−	0.613824i	0.926308	+	0.376766i	$0.122964\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	30.3018			1.03149			0.515743	−	0.856744i	$-0.327516\pi$
$863$	30.3018			1.03149			0.515743	+	0.856744i	$0.327516\pi$
$864$		0			0
$865$	−3.11500	−	5.39534i	−0.105913	−	0.183447i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	8.16903	+	4.71639i	0.277115	+	0.159993i
$870$		0			0
$871$	7.36108	+	4.24992i	0.249421	+	0.144003i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−9.12715			−0.308554
$876$		0			0
$877$	−22.6100	+	39.1616i	−0.763485	+	1.32239i	0.177559	+	0.984110i	$0.443180\pi$
$877$	−22.6100	+	39.1616i	−0.763485	+	1.32239i	−0.941044	+	0.338284i	$0.890153\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	5.91740i		−	0.199362i	−0.995019	−	0.0996812i	$-0.968218\pi$
$881$		−	5.91740i		−	0.199362i	0.995019	−	0.0996812i	$-0.0317823\pi$
$882$		0			0
$883$	−38.5219	+	22.2406i	−1.29637	+	0.748457i	−0.979774	−	0.200105i	$-0.935872\pi$
$883$	−38.5219	+	22.2406i	−1.29637	+	0.748457i	−0.316591	+	0.948562i	$0.602538\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−2.69968	−	4.67598i	−0.0906463	−	0.157004i	0.817137	−	0.576444i	$-0.195560\pi$
$887$	−2.69968	−	4.67598i	−0.0906463	−	0.157004i	−0.907783	+	0.419440i	$0.862227\pi$
$888$		0			0
$889$	1.02935	+	1.78289i	0.0345233	+	0.0597961i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−17.0248	+	18.4988i	−0.569713	+	0.619040i
$894$		0			0
$895$	4.81928	+	2.78241i	0.161091	+	0.0930057i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−1.26552	−	2.19195i	−0.0422076	−	0.0731057i
$900$		0			0
$901$	14.8370			0.494291
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	6.67286i		−	0.221813i
$906$		0			0
$907$	−0.958413	−	0.553340i	−0.0318236	−	0.0183734i	0.484004	−	0.875066i	$-0.339182\pi$
$907$	−0.958413	−	0.553340i	−0.0318236	−	0.0183734i	−0.515827	+	0.856693i	$0.672515\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	11.4474			0.379268			0.189634	−	0.981855i	$-0.439270\pi$
$911$	11.4474			0.379268			0.189634	+	0.981855i	$0.439270\pi$
$912$		0			0
$913$	−5.49637			−0.181903
$914$		0			0
$915$		0			0
$916$		0			0
$917$	4.96933	+	2.86904i	0.164102	+	0.0947441i
$918$		0			0
$919$		−	28.6358i		−	0.944607i	−0.881436	−	0.472303i	$-0.843423\pi$
$919$		−	28.6358i		−	0.944607i	0.881436	−	0.472303i	$-0.156577\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	27.0458			0.890225
$924$		0			0
$925$	4.73038	+	8.19326i	0.155534	+	0.269393i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−33.0773	−	19.0972i	−1.08523	−	0.626558i	−0.152928	−	0.988237i	$-0.548870\pi$
$929$	−33.0773	−	19.0972i	−1.08523	−	0.626558i	−0.932302	+	0.361680i	$0.882203\pi$
$930$		0			0
$931$	4.28250	+	13.6855i	0.140353	+	0.448523i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	0.667547	+	1.15623i	0.0218311	+	0.0378126i
$936$		0			0
$937$	−20.0248	−	34.6840i	−0.654182	−	1.13308i	−0.982098	−	0.188370i	$-0.939680\pi$
$937$	−20.0248	−	34.6840i	−0.654182	−	1.13308i	0.327916	−	0.944707i	$-0.393654\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−32.0454	+	18.5014i	−1.04465	+	0.603128i	−0.921147	−	0.389216i	$-0.872746\pi$
$941$	−32.0454	+	18.5014i	−1.04465	+	0.603128i	−0.123502	+	0.992344i	$0.539413\pi$
$942$		0			0
$943$			1.51715i			0.0494051i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	25.1399	−	43.5436i	0.816938	−	1.41498i	−0.0909907	−	0.995852i	$-0.529003\pi$
$947$	25.1399	−	43.5436i	0.816938	−	1.41498i	0.907928	−	0.419126i	$-0.137663\pi$
$948$		0			0
$949$	−11.8857			−0.385827
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−3.70820	−	2.14093i	−0.120121	−	0.0693516i	0.438736	−	0.898616i	$-0.355426\pi$
$953$	−3.70820	−	2.14093i	−0.120121	−	0.0693516i	−0.558856	+	0.829264i	$0.688760\pi$
$954$		0			0
$955$	−7.32975	−	4.23183i	−0.237185	−	0.136939i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	17.7929	+	30.8183i	0.574564	+	0.995174i
$960$		0			0
$961$	30.8210			0.994227
$962$		0			0
$963$		0			0
$964$		0			0
$965$	2.92597	−	1.68931i	0.0941902	−	0.0543807i
$966$		0			0
$967$	−6.59550	−	3.80791i	−0.212097	−	0.122454i	0.390189	−	0.920735i	$-0.372410\pi$
$967$	−6.59550	−	3.80791i	−0.212097	−	0.122454i	−0.602286	+	0.798281i	$0.705743\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−22.8677	+	39.6080i	−0.733858	+	1.27108i	0.221364	+	0.975191i	$0.428949\pi$
$971$	−22.8677	+	39.6080i	−0.733858	+	1.27108i	−0.955222	+	0.295888i	$0.904384\pi$
$972$		0			0
$973$	−8.78017	−	15.2077i	−0.281479	−	0.487537i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	26.3130i		−	0.841827i	−0.907101	−	0.420913i	$-0.861710\pi$
$977$		−	26.3130i		−	0.841827i	0.907101	−	0.420913i	$-0.138290\pi$
$978$		0			0
$979$	4.97766	−	2.87386i	0.159087	−	0.0918488i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−6.15030	+	10.6526i	−0.196164	+	0.339766i	−0.947281	−	0.320403i	$-0.896182\pi$
$983$	−6.15030	+	10.6526i	−0.196164	+	0.339766i	0.751117	+	0.660169i	$0.229515\pi$
$984$		0			0
$985$	−2.32684	+	4.03020i	−0.0741392	+	0.128413i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			0.908591i			0.0288915i
$990$		0			0
$991$	−14.9492	−	8.63095i	−0.474879	−	0.274171i	0.243401	−	0.969926i	$-0.421737\pi$
$991$	−14.9492	−	8.63095i	−0.474879	−	0.274171i	−0.718280	+	0.695754i	$0.755070\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	1.86130			0.0590072
$996$		0			0
$997$	−3.95927	−	6.85765i	−0.125391	−	0.217184i	0.796495	−	0.604646i	$-0.206685\pi$
$997$	−3.95927	−	6.85765i	−0.125391	−	0.217184i	−0.921886	+	0.387462i	$0.873352\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2736.2.cg.b.1151.7	yes	24
3.2	odd	2	inner	2736.2.cg.b.1151.6	yes	24
4.3	odd	2		2736.2.cg.a.1151.7	yes	24
12.11	even	2		2736.2.cg.a.1151.6	✓	24
19.7	even	3		2736.2.cg.a.2591.6	yes	24
57.26	odd	6		2736.2.cg.a.2591.7	yes	24
76.7	odd	6	inner	2736.2.cg.b.2591.6	yes	24
228.83	even	6	inner	2736.2.cg.b.2591.7	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2736.2.cg.a.1151.6	✓	24	12.11	even	2
2736.2.cg.a.1151.7	yes	24	4.3	odd	2
2736.2.cg.a.2591.6	yes	24	19.7	even	3
2736.2.cg.a.2591.7	yes	24	57.26	odd	6
2736.2.cg.b.1151.6	yes	24	3.2	odd	2	inner
2736.2.cg.b.1151.7	yes	24	1.1	even	1	trivial
2736.2.cg.b.2591.6	yes	24	76.7	odd	6	inner
2736.2.cg.b.2591.7	yes	24	228.83	even	6	inner

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)

\(f(q)\)	\(=\)	\(q+(0.420258 + 0.242636i) q^{5} -1.92619i q^{7} +O(q^{10})\)	\(q+(0.420258 + 0.242636i) q^{5} -1.92619i q^{7} -0.610994 q^{11} +(-0.889419 - 1.54052i) q^{13} +(-3.89960 - 2.25144i) q^{17} +(1.30176 + 4.15998i) q^{19} +(-0.121114 - 0.209776i) q^{23} +(-2.38226 - 4.12619i) q^{25} +(-5.18126 + 2.99140i) q^{29} +0.423054i q^{31} +(0.467363 - 0.809497i) q^{35} -1.98567 q^{37} +(-5.42417 - 3.13165i) q^{41} +(-3.24844 - 1.87549i) q^{43} +(2.88383 + 4.99494i) q^{47} +3.28979 q^{49} +(-2.85356 + 1.64750i) q^{53} +(-0.256775 - 0.148249i) q^{55} +(-2.16651 + 3.75250i) q^{59} +(-1.59275 - 2.75873i) q^{61} -0.863220i q^{65} +(-4.13814 + 2.38916i) q^{67} +(-7.60211 + 13.1672i) q^{71} +(3.34087 - 5.78656i) q^{73} +1.17689i q^{77} +(-13.3701 - 7.71920i) q^{79} +8.99578 q^{83} +(-1.09256 - 1.89237i) q^{85} +(-8.14682 + 4.70357i) q^{89} +(-2.96733 + 1.71319i) q^{91} +(-0.462288 + 2.06412i) q^{95} +(3.18910 - 5.52368i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 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