LMFDB - Embedded newform 2736.2.cg.b.2591.6

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			2591.6
Character	$\chi$	$=$	2736.2591
Dual form			2736.2.cg.b.1151.6

$q$-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 $\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times$.

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

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
$5$	−0.420258	+	0.242636i	−0.187945	+	0.108510i	−0.591020	−	0.806657i	$-0.701275\pi$
$5$	−0.420258	+	0.242636i	−0.187945	+	0.108510i	0.403075	+	0.915167i	$0.367941\pi$
$6$		0			0
$7$			1.92619i			0.728032i	0.931393	+	0.364016i	$0.118595\pi$
$7$			1.92619i			0.728032i	−0.931393	+	0.364016i	$0.881405\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	0.610994			0.184222			0.0921109	−	0.995749i	$-0.470639\pi$
$11$	0.610994			0.184222			0.0921109	+	0.995749i	$0.470639\pi$
$12$		0			0
$13$	−0.889419	+	1.54052i	−0.246680	+	0.427263i	−0.962603	−	0.270917i	$-0.912673\pi$
$13$	−0.889419	+	1.54052i	−0.246680	+	0.427263i	0.715922	+	0.698180i	$0.246006\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	3.89960	−	2.25144i	0.945792	−	0.546053i	0.0540208	−	0.998540i	$-0.482796\pi$
$17$	3.89960	−	2.25144i	0.945792	−	0.546053i	0.891771	+	0.452487i	$0.149463\pi$
$18$		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	$-0.825294\pi$
$23$	0.121114	−	0.209776i	0.0252540	−	0.0437413i	0.878376	+	0.477970i	$0.158627\pi$
$24$		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	$-0.145864\pi$
$29$	5.18126	+	2.99140i	0.962136	+	0.555490i	0.0653063	+	0.997865i	$0.479198\pi$
$30$		0			0
$31$		−	0.423054i		−	0.0759827i	−0.999278	−	0.0379913i	$-0.987904\pi$
$31$		−	0.423054i		−	0.0759827i	0.999278	−	0.0379913i	$-0.0120959\pi$
$32$		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	$-0.503999\pi$
$41$	5.42417	−	3.13165i	0.847114	−	0.489081i	0.859676	+	0.510840i	$0.170665\pi$
$42$		0			0
$43$	−3.24844	+	1.87549i	−0.495382	+	0.286009i	−0.726805	−	0.686844i	$-0.758995\pi$
$43$	−3.24844	+	1.87549i	−0.495382	+	0.286009i	0.231422	+	0.972853i	$0.425662\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−2.88383	+	4.99494i	−0.420649	+	0.728586i	−0.996003	−	0.0893185i	$-0.971531\pi$
$47$	−2.88383	+	4.99494i	−0.420649	+	0.728586i	0.575354	+	0.817905i	$0.304864\pi$
$48$		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	$-0.260670\pi$
$53$	2.85356	+	1.64750i	0.391966	+	0.226302i	−0.291045	+	0.956709i	$0.594003\pi$
$54$		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	$-0.0756504\pi$
$59$	2.16651	+	3.75250i	0.282056	+	0.488534i	−0.689835	+	0.723966i	$0.742317\pi$
$60$		0			0
$61$	−1.59275	+	2.75873i	−0.203931	+	0.353219i	−0.949792	−	0.312883i	$-0.898705\pi$
$61$	−1.59275	+	2.75873i	−0.203931	+	0.353219i	0.745861	+	0.666102i	$0.232038\pi$
$62$		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	0.225450	−	0.974255i	$-0.427615\pi$
$67$	−4.13814	−	2.38916i	−0.505554	−	0.291882i	−0.731004	+	0.682373i	$0.760948\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	7.60211	+	13.1672i	0.902205	+	1.56266i	0.824626	+	0.565678i	$0.191386\pi$
$71$	7.60211	+	13.1672i	0.902205	+	1.56266i	0.0775785	+	0.996986i	$0.475281\pi$
$72$		0			0
$73$	3.34087	+	5.78656i	0.391019	+	0.677265i	0.992584	−	0.121558i	$-0.0387892\pi$
$73$	3.34087	+	5.78656i	0.391019	+	0.677265i	−0.601565	+	0.798824i	$0.705456\pi$
$74$		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	−0.504260	+	0.863552i	$0.668235\pi$
$79$	−13.3701	+	7.71920i	−1.50425	+	0.868478i	−0.999988	+	0.00492653i	$0.998432\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−8.99578			−0.987415			−0.493707	−	0.869628i	$-0.664359\pi$
$83$	−8.99578			−0.987415			−0.493707	+	0.869628i	$0.664359\pi$
$84$		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	$-0.167189\pi$
$89$	8.14682	+	4.70357i	0.863562	+	0.498578i	−0.00164176	+	0.999999i	$0.500523\pi$
$90$		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	0.981270	−	0.192640i	$-0.0617049\pi$
$97$	3.18910	+	5.52368i	0.323804	+	0.560845i	−0.657466	+	0.753484i	$0.728372\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	4.61192	+	2.66269i	0.458903	+	0.264948i	0.711583	−	0.702602i	$-0.247979\pi$
$101$	4.61192	+	2.66269i	0.458903	+	0.264948i	−0.252680	+	0.967550i	$0.581312\pi$
$102$		0			0
$103$		−	3.57269i		−	0.352027i	−0.984388	−	0.176014i	$-0.943680\pi$
$103$		−	3.57269i		−	0.352027i	0.984388	−	0.176014i	$-0.0563203\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−4.12772			−0.399042			−0.199521	−	0.979894i	$-0.563939\pi$
$107$	−4.12772			−0.399042			−0.199521	+	0.979894i	$0.563939\pi$
$108$		0			0
$109$	5.62369	+	9.74051i	0.538652	+	0.932972i	0.998977	+	0.0452218i	$0.0143995\pi$
$109$	5.62369	+	9.74051i	0.538652	+	0.932972i	−0.460325	+	0.887750i	$0.652267\pi$
$110$		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	0.458371	−	0.888761i	$-0.348433\pi$
$127$	−0.925602	−	0.534397i	−0.0821339	−	0.0474200i	−0.540504	+	0.841341i	$0.681767\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	1.48949	+	2.57987i	0.130137	+	0.225404i	0.923729	−	0.383046i	$-0.125125\pi$
$131$	1.48949	+	2.57987i	0.130137	+	0.225404i	−0.793592	+	0.608450i	$0.791792\pi$
$132$		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	$-0.0438277\pi$
$137$	15.9996	+	9.23737i	1.36694	+	0.789202i	0.376402	+	0.926456i	$0.377161\pi$
$138$		0			0
$139$	7.89522	+	4.55831i	0.669664	+	0.386631i	0.795949	−	0.605363i	$-0.206972\pi$
$139$	7.89522	+	4.55831i	0.669664	+	0.386631i	−0.126285	+	0.991994i	$0.540305\pi$
$140$		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	$-0.468382\pi$
$149$	12.3348	−	7.12152i	1.01051	−	0.583418i	0.911341	+	0.411653i	$0.135048\pi$
$150$		0			0
$151$			9.84107i			0.800855i	0.916329	+	0.400427i	$0.131138\pi$
$151$			9.84107i			0.800855i	−0.916329	+	0.400427i	$0.868862\pi$
$152$		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	0.705699	−	0.708512i	$-0.250633\pi$
$157$	−3.26706	−	5.65872i	−0.260740	−	0.451615i	−0.966439	+	0.256897i	$0.917300\pi$
$158$		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	$-0.799925\pi$
$163$		−	15.0135i		−	1.17595i	0.808879	−	0.587975i	$-0.200075\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	3.15701	−	5.46810i	0.244297	−	0.423134i	−0.717637	−	0.696417i	$-0.754776\pi$
$167$	3.15701	−	5.46810i	0.244297	−	0.423134i	0.961934	+	0.273283i	$0.0881096\pi$
$168$		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	$-0.504381\pi$
$173$	11.1182	−	6.41908i	0.845299	−	0.488033i	0.859062	+	0.511872i	$0.171048\pi$
$174$		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	$-0.640978\pi$
$179$	−11.4674			−0.857115			−0.428558	+	0.903514i	$0.640978\pi$
$180$		0			0
$181$	−6.87538	+	11.9085i	−0.511043	+	0.885152i	0.488875	+	0.872354i	$0.337407\pi$
$181$	−6.87538	+	11.9085i	−0.511043	+	0.885152i	−0.999918	+	0.0127983i	$0.995926\pi$
$182$		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	$-0.282647\pi$
$191$	17.4411			1.26199			0.630996	+	0.775786i	$0.282647\pi$
$192$		0			0
$193$	3.48115	+	6.02954i	0.250579	+	0.434016i	0.963685	−	0.267040i	$-0.0860457\pi$
$193$	3.48115	+	6.02954i	0.250579	+	0.434016i	−0.713106	+	0.701056i	$0.752712\pi$
$194$		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	0.613093	−	0.790011i	$-0.289925\pi$
$199$	3.32171	+	1.91779i	0.235470	+	0.135948i	−0.377623	+	0.925959i	$0.623259\pi$
$200$		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	−0.781827	−	0.623495i	$-0.785712\pi$
$211$	−9.19164	+	5.30679i	−0.632778	+	0.365335i	0.149049	+	0.988830i	$0.452379\pi$
$212$		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	−0.689168	−	0.724602i	$-0.742024\pi$
$223$	−6.06628	+	3.50237i	−0.406228	+	0.234536i	0.282940	+	0.959138i	$0.408690\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−11.1861			−0.742450			−0.371225	−	0.928543i	$-0.621062\pi$
$227$	−11.1861			−0.742450			−0.371225	+	0.928543i	$0.621062\pi$
$228$		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	$-0.733338\pi$
$233$	−5.49728	+	3.17386i	−0.360139	+	0.207926i	0.309003	+	0.951061i	$0.400005\pi$
$234$		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	$-0.527789\pi$
$239$	−2.69586			−0.174381			−0.0871904	+	0.996192i	$0.527789\pi$
$240$		0			0
$241$	0.667812	−	1.15668i	0.0430175	−	0.0745086i	−0.843715	−	0.536791i	$-0.819636\pi$
$241$	0.667812	−	1.15668i	0.0430175	−	0.0745086i	0.886733	+	0.462283i	$0.152970\pi$
$242$		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	$-0.729240\pi$
$251$	5.08912	−	8.81462i	0.321223	−	0.556374i	0.980741	+	0.195315i	$0.0625729\pi$
$252$		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	$-0.165286\pi$
$257$	13.9876	+	8.07575i	0.872523	+	0.503751i	0.00433712	+	0.999991i	$0.498619\pi$
$258$		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	$-0.176338\pi$
$263$	−0.492663	−	0.853317i	−0.0303789	−	0.0526178i	−0.880815	+	0.473460i	$0.843005\pi$
$264$		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	$-0.477880\pi$
$269$	15.8778	−	9.16706i	0.968087	−	0.558926i	0.898653	+	0.438661i	$0.144547\pi$
$270$		0			0
$271$	−7.22963	+	4.17403i	−0.439169	+	0.253554i	−0.703245	−	0.710948i	$-0.748266\pi$
$271$	−7.22963	+	4.17403i	−0.439169	+	0.253554i	0.264076	+	0.964502i	$0.414933\pi$
$272$		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	$-0.442920\pi$
$281$	−9.79959	−	5.65780i	−0.584595	−	0.337516i	−0.762957	+	0.646449i	$0.776253\pi$
$282$		0			0
$283$	9.12442	−	5.26799i	0.542390	−	0.313149i	−0.203657	−	0.979042i	$-0.565283\pi$
$283$	9.12442	−	5.26799i	0.542390	−	0.313149i	0.746047	+	0.665893i	$0.231949\pi$
$284$		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	−0.116744	−	0.993162i	$-0.537246\pi$
$307$	12.0020	−	6.92933i	0.684988	−	0.395478i	0.801732	+	0.597684i	$0.203912\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	8.69554			0.493079			0.246539	−	0.969133i	$-0.420707\pi$
$311$	8.69554			0.493079			0.246539	+	0.969133i	$0.420707\pi$
$312$		0			0
$313$	−0.278745	+	0.482801i	−0.0157556	+	0.0272895i	−0.873796	−	0.486293i	$-0.838349\pi$
$313$	−0.278745	+	0.482801i	−0.0157556	+	0.0272895i	0.858040	+	0.513583i	$0.171682\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−24.9649	−	14.4135i	−1.40217	−	0.809542i	−0.407553	−	0.913182i	$-0.633618\pi$
$317$	−24.9649	−	14.4135i	−1.40217	−	0.809542i	−0.994615	+	0.103639i	$0.966951\pi$
$318$		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	$-0.941693\pi$
$331$		−	6.62797i		−	0.364306i	0.983270	−	0.182153i	$-0.0583067\pi$
$332$		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	0.658189	−	0.752853i	$-0.271323\pi$
$337$	−5.92757	−	10.2668i	−0.322895	−	0.559271i	−0.981084	+	0.193582i	$0.937989\pi$
$338$		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	−0.964850	−	0.262800i	$-0.915354\pi$
$347$	−13.2261	−	22.9084i	−0.710017	−	1.22979i	0.254833	−	0.966985i	$-0.417979\pi$
$348$		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	0.996820	+	0.0796910i	$0.0253934\pi$
$359$	10.7512	+	18.6215i	0.567424	+	0.982808i	−0.429395	+	0.903117i	$0.641273\pi$
$360$		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	0.707516	−	0.706698i	$-0.249816\pi$
$367$	8.60649	+	4.96896i	0.449255	+	0.259378i	−0.258260	+	0.966075i	$0.583149\pi$
$368$		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	$-0.820866\pi$
$379$		−	20.7734i		−	1.06706i	0.845783	−	0.533528i	$-0.179134\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	13.3400	+	23.1055i	0.681642	+	1.18064i	0.974480	+	0.224476i	$0.0720670\pi$
$383$	13.3400	+	23.1055i	0.681642	+	1.18064i	−0.292838	+	0.956162i	$0.594600\pi$
$384$		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	$-0.382693\pi$
$389$	−5.27614	−	3.04618i	−0.267511	−	0.154447i	−0.627756	+	0.778410i	$0.716026\pi$
$390$		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	0.700792	−	0.713366i	$-0.252830\pi$
$397$	−5.32784	−	9.22810i	−0.267397	−	0.463145i	−0.968189	+	0.250221i	$0.919497\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−22.9567	+	13.2540i	−1.14640	+	0.661875i	−0.948008	−	0.318248i	$-0.896906\pi$
$401$	−22.9567	+	13.2540i	−1.14640	+	0.661875i	−0.198393	+	0.980123i	$0.563572\pi$
$402$		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	−0.806638	−	0.591046i	$-0.798715\pi$
$409$	2.19512	−	3.80206i	0.108542	−	0.188000i	0.915180	+	0.403046i	$0.132049\pi$
$410$		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	$-0.542332\pi$
$419$	−5.42844			−0.265197			−0.132598	+	0.991170i	$0.542332\pi$
$420$		0			0
$421$	2.57871	+	4.46646i	0.125679	+	0.217682i	0.921998	−	0.387195i	$-0.126556\pi$
$421$	2.57871	+	4.46646i	0.125679	+	0.217682i	−0.796319	+	0.604876i	$0.793223\pi$
$422$		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	$-0.979289\pi$
$431$	−9.18928	+	15.9163i	−0.442632	+	0.766661i	0.555252	+	0.831682i	$0.312622\pi$
$432$		0			0
$433$	11.1096	−	19.2424i	0.533894	−	0.924732i	−0.465322	−	0.885141i	$-0.654061\pi$
$433$	11.1096	−	19.2424i	0.533894	−	0.924732i	0.999216	−	0.0395901i	$-0.0126052\pi$
$434$		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	−0.966103	−	0.258156i	$-0.916885\pi$
$439$	−25.6789	+	14.8257i	−1.22558	+	0.707592i	−0.259482	+	0.965748i	$0.583552\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	11.3830	−	19.7160i	0.540824	−	0.936734i	−0.458033	−	0.888935i	$-0.651446\pi$
$443$	11.3830	−	19.7160i	0.540824	−	0.936734i	0.998857	−	0.0477989i	$-0.0152207\pi$
$444$		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	$-0.780353\pi$
$461$	−13.0016	+	7.50646i	−0.605544	+	0.349611i	0.165676	+	0.986180i	$0.447020\pi$
$462$		0			0
$463$			1.11830i			0.0519719i	0.999662	+	0.0259860i	$0.00827252\pi$
$463$			1.11830i			0.0519719i	−0.999662	+	0.0259860i	$0.991727\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−41.5205			−1.92134			−0.960671	−	0.277691i	$-0.910431\pi$
$467$	−41.5205			−1.92134			−0.960671	+	0.277691i	$0.910431\pi$
$468$		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	$-0.716718\pi$
$479$	7.83998	−	13.5792i	0.358218	−	0.620451i	0.987663	+	0.156593i	$0.0500512\pi$
$480$		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	$0.0147339\pi$
$487$			2.04224i			0.0925427i	−0.998929	+	0.0462714i	$0.985266\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−13.2695	−	22.9835i	−0.598846	−	1.03723i	−0.992992	−	0.118184i	$-0.962293\pi$
$491$	−13.2695	−	22.9835i	−0.598846	−	1.03723i	0.394145	−	0.919048i	$-0.371041\pi$
$492$		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	0.355506	−	0.934674i	$-0.384309\pi$
$499$	29.9939	−	17.3170i	1.34271	−	0.775214i	0.987204	+	0.159460i	$0.0509752\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	15.9791	−	27.6765i	0.712471	−	1.23404i	−0.251456	−	0.967869i	$-0.580909\pi$
$503$	15.9791	−	27.6765i	0.712471	−	1.23404i	0.963927	−	0.266168i	$-0.0857574\pi$
$504$		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	$-0.502078\pi$
$509$	−19.7589	−	11.4078i	−0.875799	−	0.505643i	−0.869271	+	0.494336i	$0.835411\pi$
$510$		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	0.885466	−	0.464705i	$-0.153840\pi$
$523$	21.1712	+	12.2232i	0.925752	+	0.534483i	0.0402866	+	0.999188i	$0.487173\pi$
$524$		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	−0.337257	−	0.941413i	$-0.609499\pi$
$541$	15.0409	−	26.0516i	0.646659	−	1.12005i	0.983916	−	0.178633i	$-0.0571675\pi$
$542$		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	0.821324	−	0.570462i	$-0.193236\pi$
$547$	17.2592	+	9.96461i	0.737951	+	0.426056i	−0.0833729	+	0.996518i	$0.526569\pi$
$548$		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	$-0.398763\pi$
$557$	−8.34354	−	4.81715i	−0.353527	−	0.204109i	−0.666238	+	0.745739i	$0.732096\pi$
$558$		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	$-0.487795\pi$
$563$	1.81914			0.0766677			0.0383339	+	0.999265i	$0.487795\pi$
$564$		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	$-0.975255\pi$
$571$		−	3.71146i		−	0.155320i	0.996980	−	0.0776598i	$-0.0247448\pi$
$572$		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	−0.814319	−	0.580417i	$-0.802890\pi$
$587$	−22.0431	−	38.1798i	−0.909816	−	1.57585i	−0.0954964	−	0.995430i	$-0.530444\pi$
$588$		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	$-0.417026\pi$
$593$	−10.9625	−	6.32922i	−0.450177	−	0.259910i	−0.707905	+	0.706308i	$0.750359\pi$
$594$		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	0.387908	−	0.921698i	$-0.373198\pi$
$599$	24.2828	−	42.0590i	0.992168	−	1.71849i	0.604260	−	0.796787i	$-0.293469\pi$
$600$		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	$-0.748731\pi$
$607$		−	34.9811i		−	1.41984i	0.704283	−	0.709919i	$-0.251269\pi$
$608$		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	−0.997046	−	0.0768017i	$-0.975529\pi$
$613$	−13.9896	−	24.2307i	−0.565035	−	0.978670i	0.432011	−	0.901868i	$-0.357804\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−16.0668	−	9.27615i	−0.646824	−	0.373444i	0.140415	−	0.990093i	$-0.455156\pi$
$617$	−16.0668	−	9.27615i	−0.646824	−	0.373444i	−0.787238	+	0.616649i	$0.788490\pi$
$618$		0			0
$619$			3.22585i			0.129658i	0.997896	+	0.0648290i	$0.0206502\pi$
$619$			3.22585i			0.129658i	−0.997896	+	0.0648290i	$0.979350\pi$
$620$		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	0.964727	−	0.263252i	$-0.0847949\pi$
$631$	30.6237	+	17.6806i	1.21911	+	0.703852i	0.254381	+	0.967104i	$0.418128\pi$
$632$		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	$-0.658316\pi$
$641$	1.15028	−	0.664116i	0.0454334	−	0.0262310i	0.522545	+	0.852612i	$0.324983\pi$
$642$		0			0
$643$	23.3711	−	13.4933i	0.921666	−	0.532124i	0.0374997	−	0.999297i	$-0.488061\pi$
$643$	23.3711	−	13.4933i	0.921666	−	0.532124i	0.884166	+	0.467173i	$0.154727\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	27.9228			1.09776			0.548878	−	0.835902i	$-0.315055\pi$
$647$	27.9228			1.09776			0.548878	+	0.835902i	$0.315055\pi$
$648$		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	0.448238	+	0.893914i	$0.352052\pi$
$659$	−14.1199	+	24.4564i	−0.550033	+	0.952686i	−0.998271	+	0.0587714i	$0.981282\pi$
$660$		0			0
$661$	−4.30326	+	7.45347i	−0.167377	+	0.289906i	−0.937497	−	0.347993i	$-0.886863\pi$
$661$	−4.30326	+	7.45347i	−0.167377	+	0.289906i	0.770120	+	0.637900i	$0.220197\pi$
$662$		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	$-0.452826\pi$
$683$	7.71791			0.295318			0.147659	+	0.989038i	$0.452826\pi$
$684$		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	$0.0339347\pi$
$691$			5.59423i			0.212814i	−0.994323	+	0.106407i	$0.966065\pi$
$692$		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	$-0.866675\pi$
$701$	−26.9559	+	15.5630i	−1.01811	+	0.587807i	−0.104555	+	0.994519i	$0.533342\pi$
$702$		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	−0.595817	−	0.803120i	$-0.703172\pi$
$709$	10.5873	−	18.3377i	0.397614	−	0.688688i	0.993431	+	0.114432i	$0.0365049\pi$
$710$		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	−0.992822	−	0.119600i	$-0.961839\pi$
$719$	−16.0882	−	27.8656i	−0.599988	−	1.03921i	0.392834	−	0.919609i	$-0.371495\pi$
$720$		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	−0.641001	−	0.767540i	$-0.721481\pi$
$727$	−8.00241	+	4.62020i	−0.296793	+	0.171354i	0.344208	+	0.938893i	$0.388147\pi$
$728$		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	−0.594901	−	0.803799i	$-0.702809\pi$
$739$	−5.33473	+	3.08001i	−0.196241	+	0.113300i	0.398660	+	0.917099i	$0.369475\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	16.0767	+	27.8457i	0.589798	+	1.02156i	0.994259	+	0.107004i	$0.0341258\pi$
$743$	16.0767	+	27.8457i	0.589798	+	1.02156i	−0.404461	+	0.914555i	$0.632541\pi$
$744$		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	0.818135	−	0.575026i	$-0.195008\pi$
$751$	19.9837	+	11.5376i	0.729216	+	0.421013i	−0.0889190	+	0.996039i	$0.528341\pi$
$752$		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	0.593093	−	0.805134i	$-0.297906\pi$
$757$	−11.0252	−	19.0963i	−0.400719	−	0.694066i	−0.993813	+	0.111067i	$0.964573\pi$
$758$		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	−0.747160	−	0.664645i	$-0.768583\pi$
$769$	5.60217	−	9.70323i	0.202019	−	0.349908i	0.949179	+	0.314737i	$0.101916\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−24.0909	−	13.9089i	−0.866491	−	0.500269i	−0.000310458	−	1.00000i	$-0.500099\pi$
$773$	−24.0909	−	13.9089i	−0.866491	−	0.500269i	−0.866181	+	0.499731i	$0.833432\pi$
$774$		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	$-0.660111\pi$
$787$		−	49.1575i		−	1.75228i	0.482060	−	0.876138i	$-0.339889\pi$
$788$		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	0.0275678	−	0.999620i	$-0.491224\pi$
$811$	−23.4757	−	13.5537i	−0.824345	−	0.475936i	−0.851912	+	0.523684i	$0.824557\pi$
$812$		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	$-0.340140\pi$
$821$	−1.06111	−	0.612630i	−0.0370329	−	0.0213809i	−0.518402	+	0.855137i	$0.673473\pi$
$822$		0			0
$823$	7.98619	+	4.61083i	0.278381	+	0.160723i	0.632690	−	0.774405i	$-0.281951\pi$
$823$	7.98619	+	4.61083i	0.278381	+	0.160723i	−0.354309	+	0.935128i	$0.615284\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	14.1285	−	24.4713i	0.491297	−	0.850951i	−0.508653	−	0.860972i	$-0.669856\pi$
$827$	14.1285	−	24.4713i	0.491297	−	0.850951i	0.999950	+	0.0100207i	$0.00318974\pi$
$828$		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	0.962050	+	0.272874i	$0.0879743\pi$
$839$	20.7781	+	35.9888i	0.717341	+	1.24247i	−0.244709	+	0.969597i	$0.578692\pi$
$840$		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	0.974866	−	0.222791i	$-0.0715166\pi$
$853$	8.60095	+	14.8973i	0.294491	+	0.510073i	−0.680376	+	0.732864i	$0.738183\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−28.5845	+	16.5032i	−0.976426	+	0.563740i	−0.901189	−	0.433426i	$-0.857305\pi$
$857$	−28.5845	+	16.5032i	−0.976426	+	0.563740i	−0.0752371	+	0.997166i	$0.523971\pi$
$858$		0			0
$859$	31.1602	+	17.9904i	1.06317	+	0.613824i	0.926308	−	0.376766i	$-0.122964\pi$
$859$	31.1602	+	17.9904i	1.06317	+	0.613824i	0.136865	+	0.990590i	$0.456297\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−30.3018			−1.03149			−0.515743	−	0.856744i	$-0.672484\pi$
$863$	−30.3018			−1.03149			−0.515743	+	0.856744i	$0.672484\pi$
$864$		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	−0.941044	−	0.338284i	$-0.890153\pi$
$877$	−22.6100	−	39.1616i	−0.763485	−	1.32239i	0.177559	−	0.984110i	$-0.443180\pi$
$878$		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	−0.316591	−	0.948562i	$-0.602538\pi$
$883$	−38.5219	−	22.2406i	−1.29637	−	0.748457i	−0.979774	+	0.200105i	$0.935872\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	2.69968	−	4.67598i	0.0906463	−	0.157004i	−0.817137	−	0.576444i	$-0.804440\pi$
$887$	2.69968	−	4.67598i	0.0906463	−	0.157004i	0.907783	+	0.419440i	$0.137773\pi$
$888$		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	−0.515827	−	0.856693i	$-0.672515\pi$
$907$	−0.958413	+	0.553340i	−0.0318236	+	0.0183734i	0.484004	+	0.875066i	$0.339182\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−11.4474			−0.379268			−0.189634	−	0.981855i	$-0.560730\pi$
$911$	−11.4474			−0.379268			−0.189634	+	0.981855i	$0.560730\pi$
$912$		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	$0.156577\pi$
$919$			28.6358i			0.944607i	−0.881436	+	0.472303i	$0.843423\pi$
$920$		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	$-0.451130\pi$
$929$	33.0773	−	19.0972i	1.08523	−	0.626558i	0.932302	+	0.361680i	$0.117797\pi$
$930$		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	0.327916	+	0.944707i	$0.393654\pi$
$937$	−20.0248	+	34.6840i	−0.654182	+	1.13308i	−0.982098	+	0.188370i	$0.939680\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	32.0454	+	18.5014i	1.04465	+	0.603128i	0.921147	−	0.389216i	$-0.127254\pi$
$941$	32.0454	+	18.5014i	1.04465	+	0.603128i	0.123502	+	0.992344i	$0.460587\pi$
$942$		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	−0.907928	−	0.419126i	$-0.862337\pi$
$947$	−25.1399	−	43.5436i	−0.816938	−	1.41498i	0.0909907	−	0.995852i	$-0.470997\pi$
$948$		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	$-0.644574\pi$
$953$	3.70820	−	2.14093i	0.120121	−	0.0693516i	0.558856	+	0.829264i	$0.311240\pi$
$954$		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	−0.602286	−	0.798281i	$-0.705743\pi$
$967$	−6.59550	+	3.80791i	−0.212097	+	0.122454i	0.390189	+	0.920735i	$0.372410\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	22.8677	+	39.6080i	0.733858	+	1.27108i	0.955222	+	0.295888i	$0.0956157\pi$
$971$	22.8677	+	39.6080i	0.733858	+	1.27108i	−0.221364	+	0.975191i	$0.571051\pi$
$972$		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	$-0.103818\pi$
$983$	6.15030	+	10.6526i	0.196164	+	0.339766i	−0.751117	+	0.660169i	$0.770485\pi$
$984$		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	−0.718280	−	0.695754i	$-0.755070\pi$
$991$	−14.9492	+	8.63095i	−0.474879	+	0.274171i	0.243401	+	0.969926i	$0.421737\pi$
$992$		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	−0.921886	−	0.387462i	$-0.873352\pi$
$997$	−3.95927	+	6.85765i	−0.125391	+	0.217184i	0.796495	+	0.604646i	$0.206685\pi$
$998$		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.2591.6	yes	24
3.2	odd	2	inner	2736.2.cg.b.2591.7	yes	24
4.3	odd	2		2736.2.cg.a.2591.6	yes	24
12.11	even	2		2736.2.cg.a.2591.7	yes	24
19.11	even	3		2736.2.cg.a.1151.7	yes	24
57.11	odd	6		2736.2.cg.a.1151.6	✓	24
76.11	odd	6	inner	2736.2.cg.b.1151.7	yes	24
228.11	even	6	inner	2736.2.cg.b.1151.6	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2736.2.cg.a.1151.6	✓	24	57.11	odd	6
2736.2.cg.a.1151.7	yes	24	19.11	even	3
2736.2.cg.a.2591.6	yes	24	4.3	odd	2
2736.2.cg.a.2591.7	yes	24	12.11	even	2
2736.2.cg.b.1151.6	yes	24	228.11	even	6	inner
2736.2.cg.b.1151.7	yes	24	76.11	odd	6	inner
2736.2.cg.b.2591.6	yes	24	1.1	even	1	trivial
2736.2.cg.b.2591.7	yes	24	3.2	odd	2	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

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