LMFDB - Embedded newform 2736.2.dc.d.449.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2736,2,Mod(449,2736)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2736, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 3, 5]))

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

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

chi := DirichletCharacter("2736.449");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2736 = 2^{4} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2736.dc (of order $6$, degree $2$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$21.8470699930$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 10x^{14} + 46x^{12} + 126x^{10} + 315x^{8} + 1134x^{6} + 3726x^{4} + 7290x^{2} + 6561 $ x^16 + 10x^14 + 46x^12 + 126x^10 + 315x^8 + 1134x^6 + 3726x^4 + 7290*x^2 + 6561
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{7} $
Twist minimal:	no (minimal twist has level 684)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			449.4
Root			$-0.654543 - 1.60361i$ of defining polynomial
Character	$\chi$	$=$	2736.449
Dual form			2736.2.dc.d.1889.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.406956 + 0.234956i) q^{5} +3.49289 q^{7} +O(q^{10})$	$q+(-0.406956 + 0.234956i) q^{5} +3.49289 q^{7} +1.34919i q^{11} +(4.30672 + 2.48649i) q^{13} +(-6.31655 + 3.64686i) q^{17} +(-1.88959 - 3.92803i) q^{19} +(6.97653 + 4.02790i) q^{23} +(-2.38959 + 4.13889i) q^{25} +(-4.74117 + 8.21194i) q^{29} -3.95998i q^{31} +(-1.42145 + 0.820674i) q^{35} -0.581007i q^{37} +(-0.761473 - 1.31891i) q^{41} +(4.63603 + 8.02985i) q^{43} +(1.06693 + 0.615993i) q^{47} +5.20026 q^{49} +(5.09568 - 8.82598i) q^{53} +(-0.316999 - 0.549059i) q^{55} +(-5.09568 - 8.82598i) q^{59} +(-3.59618 + 6.22877i) q^{61} -2.33686 q^{65} +(-9.23616 - 5.33250i) q^{67} +(3.67423 + 6.36396i) q^{71} +(-0.703424 - 1.21837i) q^{73} +4.71255i q^{77} +(1.56422 - 0.903104i) q^{79} +5.53518i q^{83} +(1.71370 - 2.96822i) q^{85} +(-7.73800 + 13.4026i) q^{89} +(15.0429 + 8.68501i) q^{91} +(1.69189 + 1.15456i) q^{95} +(9.66561 - 5.58044i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q + 12 q^{19} + 4 q^{25} + 4 q^{43} - 20 q^{55} + 12 q^{61} - 36 q^{67} + 44 q^{73} + 12 q^{79} + 56 q^{85} + 60 q^{91}+O(q^{100}) $ 16 * q + 12 * q^19 + 4 * q^25 + 4 * q^43 - 20 * q^55 + 12 * q^61 - 36 * q^67 + 44 * q^73 + 12 * q^79 + 56 * q^85 + 60 * q^91

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−0.406956	+	0.234956i	−0.181996	+	0.105075i	−0.588230	−	0.808694i	$-0.700175\pi$
$5$	−0.406956	+	0.234956i	−0.181996	+	0.105075i	0.406234	+	0.913769i	$0.366842\pi$
$6$		0			0
$7$	3.49289			1.32019			0.660094	−	0.751183i	$-0.270517\pi$
$7$	3.49289			1.32019			0.660094	+	0.751183i	$0.270517\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			1.34919i			0.406795i	0.979096	+	0.203397i	$0.0651983\pi$
$11$			1.34919i			0.406795i	−0.979096	+	0.203397i	$0.934802\pi$
$12$		0			0
$13$	4.30672	+	2.48649i	1.19447	+	0.689627i	0.959317	−	0.282332i	$-0.0911078\pi$
$13$	4.30672	+	2.48649i	1.19447	+	0.689627i	0.235152	+	0.971959i	$0.424441\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−6.31655	+	3.64686i	−1.53199	+	0.884494i	−0.532719	+	0.846292i	$0.678830\pi$
$17$	−6.31655	+	3.64686i	−1.53199	+	0.884494i	−0.999270	+	0.0382017i	$0.987837\pi$
$18$		0			0
$19$	−1.88959	−	3.92803i	−0.433502	−	0.901153i
$19$	−1.88959	−	3.92803i	−0.433502	−	0.901153i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	6.97653	+	4.02790i	1.45471	+	0.839875i	0.998743	−	0.0501233i	$-0.0159614\pi$
$23$	6.97653	+	4.02790i	1.45471	+	0.839875i	0.455964	+	0.889998i	$0.349295\pi$
$24$		0			0
$25$	−2.38959	+	4.13889i	−0.477918	+	0.827779i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−4.74117	+	8.21194i	−0.880413	+	1.52492i	−0.0295296	+	0.999564i	$0.509401\pi$
$29$	−4.74117	+	8.21194i	−0.880413	+	1.52492i	−0.850883	+	0.525355i	$0.823932\pi$
$30$		0			0
$31$		−	3.95998i		−	0.711234i	−0.934632	−	0.355617i	$-0.884271\pi$
$31$		−	3.95998i		−	0.711234i	0.934632	−	0.355617i	$-0.115729\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−1.42145	+	0.820674i	−0.240269	+	0.138719i
$36$		0			0
$37$		−	0.581007i		−	0.0955169i	−0.998859	−	0.0477584i	$-0.984792\pi$
$37$		−	0.581007i		−	0.0955169i	0.998859	−	0.0477584i	$-0.0152078\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−0.761473	−	1.31891i	−0.118922	−	0.205979i	0.800419	−	0.599441i	$-0.204611\pi$
$41$	−0.761473	−	1.31891i	−0.118922	−	0.205979i	−0.919341	+	0.393462i	$0.871277\pi$
$42$		0			0
$43$	4.63603	+	8.02985i	0.706989	+	1.22454i	0.965969	+	0.258658i	$0.0832802\pi$
$43$	4.63603	+	8.02985i	0.706989	+	1.22454i	−0.258980	+	0.965883i	$0.583386\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	1.06693	+	0.615993i	0.155628	+	0.0898519i	0.575792	−	0.817597i	$-0.304694\pi$
$47$	1.06693	+	0.615993i	0.155628	+	0.0898519i	−0.420164	+	0.907448i	$0.638027\pi$
$48$		0			0
$49$	5.20026			0.742894
$50$		0			0
$51$		0			0
$52$		0			0
$53$	5.09568	−	8.82598i	0.699946	−	1.21234i	−0.268539	−	0.963269i	$-0.586541\pi$
$53$	5.09568	−	8.82598i	0.699946	−	1.21234i	0.968485	−	0.249073i	$-0.0801260\pi$
$54$		0			0
$55$	−0.316999	−	0.549059i	−0.0427442	−	0.0740350i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−5.09568	−	8.82598i	−0.663402	−	1.14905i	−0.979716	−	0.200391i	$-0.935779\pi$
$59$	−5.09568	−	8.82598i	−0.663402	−	1.14905i	0.316314	−	0.948654i	$-0.397555\pi$
$60$		0			0
$61$	−3.59618	+	6.22877i	−0.460444	+	0.797512i	−0.998983	−	0.0450883i	$-0.985643\pi$
$61$	−3.59618	+	6.22877i	−0.460444	+	0.797512i	0.538539	+	0.842601i	$0.318976\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−2.33686			−0.289852
$66$		0			0
$67$	−9.23616	−	5.33250i	−1.12838	−	0.651469i	−0.184851	−	0.982767i	$-0.559180\pi$
$67$	−9.23616	−	5.33250i	−1.12838	−	0.651469i	−0.943526	+	0.331298i	$0.892514\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	3.67423	+	6.36396i	0.436051	+	0.755263i	0.997381	−	0.0723293i	$-0.0230432\pi$
$71$	3.67423	+	6.36396i	0.436051	+	0.755263i	−0.561329	+	0.827592i	$0.689710\pi$
$72$		0			0
$73$	−0.703424	−	1.21837i	−0.0823296	−	0.142599i	0.821921	−	0.569602i	$-0.192903\pi$
$73$	−0.703424	−	1.21837i	−0.0823296	−	0.142599i	−0.904250	+	0.427003i	$0.859569\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			4.71255i			0.537045i
$78$		0			0
$79$	1.56422	−	0.903104i	0.175989	−	0.101607i	−0.409418	−	0.912347i	$-0.634268\pi$
$79$	1.56422	−	0.903104i	0.175989	−	0.101607i	0.585407	+	0.810740i	$0.300935\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			5.53518i			0.607565i	0.952741	+	0.303782i	$0.0982496\pi$
$83$			5.53518i			0.607565i	−0.952741	+	0.303782i	$0.901750\pi$
$84$		0			0
$85$	1.71370	−	2.96822i	0.185877	−	0.321949i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−7.73800	+	13.4026i	−0.820226	+	1.42067i	0.0852870	+	0.996356i	$0.472819\pi$
$89$	−7.73800	+	13.4026i	−0.820226	+	1.42067i	−0.905513	+	0.424317i	$0.860514\pi$
$90$		0			0
$91$	15.0429	+	8.68501i	1.57692	+	0.910437i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	1.69189	+	1.15456i	0.173585	+	0.118456i
$96$		0			0
$97$	9.66561	−	5.58044i	0.981394	−	0.566608i	0.0787031	−	0.996898i	$-0.474922\pi$
$97$	9.66561	−	5.58044i	0.981394	−	0.566608i	0.902691	+	0.430290i	$0.141589\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−5.75566	−	3.32303i	−0.572710	−	0.330654i	0.185521	−	0.982640i	$-0.440603\pi$
$101$	−5.75566	−	3.32303i	−0.572710	−	0.330654i	−0.758231	+	0.651986i	$0.773936\pi$
$102$		0			0
$103$		−	4.89881i		−	0.482694i	−0.970439	−	0.241347i	$-0.922411\pi$
$103$		−	4.89881i		−	0.482694i	0.970439	−	0.241347i	$-0.0775893\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−5.28463			−0.510885			−0.255442	−	0.966824i	$-0.582221\pi$
$107$	−5.28463			−0.510885			−0.255442	+	0.966824i	$0.582221\pi$
$108$		0			0
$109$	−4.19011	+	2.41916i	−0.401340	+	0.231714i	−0.687062	−	0.726599i	$-0.741100\pi$
$109$	−4.19011	+	2.41916i	−0.401340	+	0.231714i	0.285722	+	0.958313i	$0.407767\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	20.7955			1.95627			0.978137	−	0.207960i	$-0.0666825\pi$
$113$	20.7955			1.95627			0.978137	+	0.207960i	$0.0666825\pi$
$114$		0			0
$115$	−3.78552			−0.353001
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−22.0630	+	12.7381i	−2.02251	+	1.16770i
$120$		0			0
$121$	9.17970			0.834518
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	4.59535i		−	0.411021i
$126$		0			0
$127$	12.6103	+	7.28054i	1.11898	+	0.646044i	0.941141	−	0.338016i	$-0.109756\pi$
$127$	12.6103	+	7.28054i	1.11898	+	0.646044i	0.177840	+	0.984059i	$0.443089\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−1.06693	+	0.615993i	−0.0932183	+	0.0538196i	−0.545884	−	0.837860i	$-0.683806\pi$
$131$	−1.06693	+	0.615993i	−0.0932183	+	0.0538196i	0.452666	+	0.891680i	$0.350473\pi$
$132$		0			0
$133$	−6.60013	−	13.7202i	−0.572304	−	1.18969i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	13.1566	+	7.59595i	1.12404	+	0.648966i	0.942430	−	0.334405i	$-0.108535\pi$
$137$	13.1566	+	7.59595i	1.12404	+	0.648966i	0.181612	+	0.983370i	$0.441869\pi$
$138$		0			0
$139$	−4.34657	+	7.52848i	−0.368671	+	0.638558i	−0.989358	−	0.145501i	$-0.953521\pi$
$139$	−4.34657	+	7.52848i	−0.368671	+	0.638558i	0.620687	+	0.784059i	$0.286854\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−3.35473	+	5.81056i	−0.280537	+	0.485904i
$144$		0			0
$145$		−	4.45586i		−	0.370039i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−12.1911	+	7.03855i	−0.998737	+	0.576621i	−0.907874	−	0.419243i	$-0.862296\pi$
$149$	−12.1911	+	7.03855i	−0.998737	+	0.576621i	−0.0908623	+	0.995863i	$0.528962\pi$
$150$		0			0
$151$			14.4128i			1.17289i	0.809987	+	0.586447i	$0.199474\pi$
$151$			14.4128i			1.17289i	−0.809987	+	0.586447i	$0.800526\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	0.930421	+	1.61154i	0.0747332	+	0.129442i
$156$		0			0
$157$	2.27918	+	3.94766i	0.181899	+	0.315058i	0.942527	−	0.334130i	$-0.108442\pi$
$157$	2.27918	+	3.94766i	0.181899	+	0.315058i	−0.760628	+	0.649187i	$0.775109\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	24.3682	+	14.0690i	1.92048	+	1.10879i
$162$		0			0
$163$	20.3066			1.59053			0.795267	−	0.606259i	$-0.207331\pi$
$163$	20.3066			1.59053			0.795267	+	0.606259i	$0.207331\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−7.75542	+	13.4328i	−0.600133	+	1.03946i	0.392668	+	0.919680i	$0.371552\pi$
$167$	−7.75542	+	13.4328i	−0.600133	+	1.03946i	−0.992800	+	0.119780i	$0.961781\pi$
$168$		0			0
$169$	5.86522	+	10.1589i	0.451171	+	0.781451i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−0.813911	−	1.40974i	−0.0618805	−	0.107180i	0.833425	−	0.552632i	$-0.186376\pi$
$173$	−0.813911	−	1.40974i	−0.0618805	−	0.107180i	−0.895306	+	0.445452i	$0.853043\pi$
$174$		0			0
$175$	−8.34657	+	14.4567i	−0.630942	+	1.09282i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	10.8023			0.807401			0.403700	−	0.914891i	$-0.367724\pi$
$179$	10.8023			0.807401			0.403700	+	0.914891i	$0.367724\pi$
$180$		0			0
$181$	3.19328	+	1.84364i	0.237355	+	0.137037i	0.613960	−	0.789337i	$-0.289575\pi$
$181$	3.19328	+	1.84364i	0.237355	+	0.137037i	−0.376606	+	0.926374i	$0.622909\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	0.136511	+	0.236444i	0.0100365	+	0.0173837i
$186$		0			0
$187$	−4.92029	−	8.52220i	−0.359808	−	0.623205i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	18.2996i		−	1.32411i	−0.749453	−	0.662057i	$-0.769684\pi$
$191$		−	18.2996i		−	1.32411i	0.749453	−	0.662057i	$-0.230316\pi$
$192$		0			0
$193$	−6.17194	+	3.56337i	−0.444266	+	0.256497i	−0.705406	−	0.708804i	$-0.749235\pi$
$193$	−6.17194	+	3.56337i	−0.444266	+	0.256497i	0.261139	+	0.965301i	$0.415902\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	5.88677i		−	0.419415i	−0.977764	−	0.209708i	$-0.932749\pi$
$197$		−	5.88677i		−	0.419415i	0.977764	−	0.209708i	$-0.0672512\pi$
$198$		0			0
$199$	−1.22602	+	2.12353i	−0.0869103	+	0.150533i	−0.906204	−	0.422842i	$-0.861033\pi$
$199$	−1.22602	+	2.12353i	−0.0869103	+	0.150533i	0.819293	+	0.573375i	$0.194366\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−16.5604	+	28.6834i	−1.16231	+	2.01318i
$204$		0			0
$205$	0.619772	+	0.357825i	0.0432867	+	0.0249916i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	5.29965	−	2.54941i	0.366584	−	0.176346i
$210$		0			0
$211$	−13.1777	+	7.60813i	−0.907188	+	0.523765i	−0.879525	−	0.475852i	$-0.842140\pi$
$211$	−13.1777	+	7.60813i	−0.907188	+	0.523765i	−0.0276624	+	0.999617i	$0.508806\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−3.77332	−	2.17853i	−0.257338	−	0.148574i
$216$		0			0
$217$		−	13.8318i		−	0.938961i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−36.2715			−2.43988
$222$		0			0
$223$	21.1808	−	12.2288i	1.41837	−	0.818898i	0.422217	−	0.906495i	$-0.361252\pi$
$223$	21.1808	−	12.2288i	1.41837	−	0.818898i	0.996156	+	0.0875966i	$0.0279186\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	2.84290			0.188690			0.0943449	−	0.995540i	$-0.469924\pi$
$227$	2.84290			0.188690			0.0943449	+	0.995540i	$0.469924\pi$
$228$		0			0
$229$	6.64189			0.438909			0.219454	−	0.975623i	$-0.429572\pi$
$229$	6.64189			0.438909			0.219454	+	0.975623i	$0.429572\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−3.14836	+	1.81771i	−0.206256	+	0.119082i	−0.599570	−	0.800322i	$-0.704662\pi$
$233$	−3.14836	+	1.81771i	−0.206256	+	0.119082i	0.393314	+	0.919404i	$0.371328\pi$
$234$		0			0
$235$	−0.578925			−0.0377649
$236$		0			0
$237$		0			0
$238$		0			0
$239$			0.762075i			0.0492945i	0.999696	+	0.0246473i	$0.00784626\pi$
$239$			0.762075i			0.0492945i	−0.999696	+	0.0246473i	$0.992154\pi$
$240$		0			0
$241$	6.97549	+	4.02730i	0.449331	+	0.259421i	0.707548	−	0.706666i	$-0.249801\pi$
$241$	6.97549	+	4.02730i	0.449331	+	0.259421i	−0.258217	+	0.966087i	$0.583135\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−2.11627	+	1.22183i	−0.135204	+	0.0780599i
$246$		0			0
$247$	1.62906	−	21.6154i	0.103654	−	1.37535i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−21.2665	−	12.2782i	−1.34233	−	0.774995i	−0.355181	−	0.934797i	$-0.615581\pi$
$251$	−21.2665	−	12.2782i	−1.34233	−	0.774995i	−0.987149	+	0.159803i	$0.948914\pi$
$252$		0			0
$253$	−5.43438	+	9.41263i	−0.341657	+	0.591767i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−9.33081	+	16.1614i	−0.582040	+	1.00812i	0.413197	+	0.910641i	$0.364412\pi$
$257$	−9.33081	+	16.1614i	−0.582040	+	1.00812i	−0.995237	+	0.0974812i	$0.968921\pi$
$258$		0			0
$259$		−	2.02939i		−	0.126100i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	18.9821	−	10.9593i	1.17049	−	0.675780i	0.216691	−	0.976240i	$-0.430473\pi$
$263$	18.9821	−	10.9593i	1.17049	−	0.675780i	0.953794	+	0.300460i	$0.0971401\pi$
$264$		0			0
$265$			4.78904i			0.294189i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−6.97653	−	12.0837i	−0.425366	−	0.736756i	0.571088	−	0.820889i	$-0.306521\pi$
$269$	−6.97653	−	12.0837i	−0.425366	−	0.736756i	−0.996455	+	0.0841325i	$0.973188\pi$
$270$		0			0
$271$	1.00000	+	1.73205i	0.0607457	+	0.105215i	0.894799	−	0.446469i	$-0.147319\pi$
$271$	1.00000	+	1.73205i	0.0607457	+	0.105215i	−0.834053	+	0.551684i	$0.813985\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−5.58414	−	3.22400i	−0.336736	−	0.194415i
$276$		0			0
$277$	−8.35811			−0.502190			−0.251095	−	0.967962i	$-0.580791\pi$
$277$	−8.35811			−0.502190			−0.251095	+	0.967962i	$0.580791\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−4.87768	+	8.44839i	−0.290978	+	0.503988i	−0.974041	−	0.226371i	$-0.927314\pi$
$281$	−4.87768	+	8.44839i	−0.290978	+	0.503988i	0.683063	+	0.730359i	$0.260647\pi$
$282$		0			0
$283$	4.41318	+	7.64385i	0.262336	+	0.454380i	0.966862	−	0.255298i	$-0.0821737\pi$
$283$	4.41318	+	7.64385i	0.262336	+	0.454380i	−0.704526	+	0.709678i	$0.748840\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−2.65974	−	4.60681i	−0.157000	−	0.271931i
$288$		0			0
$289$	18.0992	−	31.3488i	1.06466	−	1.84404i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	19.3358			1.12961			0.564805	−	0.825224i	$-0.308951\pi$
$293$	19.3358			1.12961			0.564805	+	0.825224i	$0.308951\pi$
$294$		0			0
$295$	4.14743	+	2.39452i	0.241473	+	0.139414i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	20.0306	+	34.6941i	1.15840	+	2.00641i
$300$		0			0
$301$	16.1931	+	28.0473i	0.933357	+	1.61662i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	3.37978i		−	0.193525i
$306$		0			0
$307$	8.04900	−	4.64709i	0.459381	−	0.265224i	−0.252403	−	0.967622i	$-0.581221\pi$
$307$	8.04900	−	4.64709i	0.459381	−	0.265224i	0.711784	+	0.702399i	$0.247888\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	18.7734i		−	1.06454i	−0.846573	−	0.532272i	$-0.821338\pi$
$311$		−	18.7734i		−	1.06454i	0.846573	−	0.532272i	$-0.178662\pi$
$312$		0			0
$313$	14.5891	−	25.2690i	0.824623	−	1.42829i	−0.0775842	−	0.996986i	$-0.524721\pi$
$313$	14.5891	−	25.2690i	0.824623	−	1.42829i	0.902207	−	0.431303i	$-0.141946\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	7.43254	−	12.8735i	0.417453	−	0.723050i	−0.578229	−	0.815874i	$-0.696256\pi$
$317$	7.43254	−	12.8735i	0.417453	−	0.723050i	0.995683	+	0.0928242i	$0.0295894\pi$
$318$		0			0
$319$	−11.0794	−	6.39671i	−0.620329	−	0.358147i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	26.2607	+	17.9205i	1.46118	+	0.997126i
$324$		0			0
$325$	−20.5826	+	11.8834i	−1.14172	+	0.659171i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	3.72667	+	2.15160i	0.205458	+	0.118621i
$330$		0			0
$331$		−	22.5415i		−	1.23899i	−0.784999	−	0.619497i	$-0.787336\pi$
$331$		−	22.5415i		−	1.23899i	0.784999	−	0.619497i	$-0.212664\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	5.01161			0.273814
$336$		0			0
$337$	−8.17511	+	4.71990i	−0.445326	+	0.257109i	−0.705854	−	0.708357i	$-0.749437\pi$
$337$	−8.17511	+	4.71990i	−0.445326	+	0.257109i	0.260528	+	0.965466i	$0.416103\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	5.34275			0.289326
$342$		0			0
$343$	−6.28630			−0.339428
$344$		0			0
$345$		0			0
$346$		0			0
$347$	24.8418	−	14.3424i	1.33358	−	0.769942i	0.347733	−	0.937594i	$-0.386952\pi$
$347$	24.8418	−	14.3424i	1.33358	−	0.769942i	0.985846	+	0.167651i	$0.0536183\pi$
$348$		0			0
$349$	−33.1576			−1.77488			−0.887442	−	0.460919i	$-0.847520\pi$
$349$	−33.1576			−1.77488			−0.887442	+	0.460919i	$0.847520\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			0.993688i			0.0528887i	0.999650	+	0.0264443i	$0.00841848\pi$
$353$			0.993688i			0.0528887i	−0.999650	+	0.0264443i	$0.991582\pi$
$354$		0			0
$355$	−2.99050	−	1.72657i	−0.158719	−	0.0916366i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	15.8513	−	9.15177i	0.836601	−	0.483012i	−0.0195066	−	0.999810i	$-0.506210\pi$
$359$	15.8513	−	9.15177i	0.836601	−	0.483012i	0.856107	+	0.516798i	$0.172876\pi$
$360$		0			0
$361$	−11.8589	+	14.8448i	−0.624152	+	0.781303i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	0.572525	+	0.330547i	0.0299673	+	0.0173016i
$366$		0			0
$367$	−10.0460	+	17.4003i	−0.524400	+	0.908287i	0.475197	+	0.879879i	$0.342377\pi$
$367$	−10.0460	+	17.4003i	−0.524400	+	0.908287i	−0.999596	+	0.0284071i	$0.990957\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	17.7986	−	30.8282i	0.924060	−	1.60052i
$372$		0			0
$373$			32.6611i			1.69113i	0.533873	+	0.845565i	$0.320736\pi$
$373$			32.6611i			1.69113i	−0.533873	+	0.845565i	$0.679264\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−40.8377	+	23.5777i	−2.10325	+	1.21431i
$378$		0			0
$379$		−	16.9346i		−	0.869874i	−0.900461	−	0.434937i	$-0.856771\pi$
$379$		−	16.9346i		−	0.869874i	0.900461	−	0.434937i	$-0.143229\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−0.912997	−	1.58136i	−0.0466520	−	0.0808036i	0.841756	−	0.539857i	$-0.181522\pi$
$383$	−0.912997	−	1.58136i	−0.0466520	−	0.0808036i	−0.888408	+	0.459054i	$0.848189\pi$
$384$		0			0
$385$	−1.10724	−	1.91780i	−0.0564303	−	0.0977401i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	0.336928	+	0.194525i	0.0170829	+	0.00986283i	0.508517	−	0.861052i	$-0.330194\pi$
$389$	0.336928	+	0.194525i	0.0170829	+	0.00986283i	−0.491434	+	0.870915i	$0.663527\pi$
$390$		0			0
$391$	−58.7568			−2.97146
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−0.424379	+	0.735047i	−0.0213528	+	0.0369842i
$396$		0			0
$397$	10.4415	+	18.0852i	0.524044	+	0.907670i	0.999608	+	0.0279895i	$0.00891051\pi$
$397$	10.4415	+	18.0852i	0.524044	+	0.907670i	−0.475564	+	0.879681i	$0.657756\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−6.11018	−	10.5831i	−0.305128	−	0.528497i	0.672162	−	0.740404i	$-0.265366\pi$
$401$	−6.11018	−	10.5831i	−0.305128	−	0.528497i	−0.977290	+	0.211907i	$0.932033\pi$
$402$		0			0
$403$	9.84644	−	17.0545i	0.490486	−	0.849547i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	0.783886			0.0388558
$408$		0			0
$409$	16.4233	+	9.48201i	0.812081	+	0.468855i	0.847678	−	0.530511i	$-0.178000\pi$
$409$	16.4233	+	9.48201i	0.812081	+	0.468855i	−0.0355969	+	0.999366i	$0.511333\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−17.7986	−	30.8282i	−0.875814	−	1.51695i
$414$		0			0
$415$	−1.30052	−	2.25257i	−0.0638401	−	0.110574i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	15.3264i		−	0.748745i	−0.927278	−	0.374373i	$-0.877858\pi$
$419$		−	15.3264i		−	0.748745i	0.927278	−	0.374373i	$-0.122142\pi$
$420$		0			0
$421$	10.6624	−	6.15596i	0.519656	−	0.300023i	−0.217138	−	0.976141i	$-0.569672\pi$
$421$	10.6624	−	6.15596i	0.519656	−	0.300023i	0.736794	+	0.676118i	$0.236339\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	34.8580i		−	1.69086i
$426$		0			0
$427$	−12.5611	+	21.7564i	−0.607872	+	1.05287i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	2.69475	−	4.66745i	0.129802	−	0.224823i	−0.793798	−	0.608182i	$-0.791899\pi$
$431$	2.69475	−	4.66745i	0.129802	−	0.224823i	0.923600	+	0.383358i	$0.125233\pi$
$432$		0			0
$433$	−28.5304	−	16.4721i	−1.37108	−	0.791596i	−0.380020	−	0.924978i	$-0.624083\pi$
$433$	−28.5304	−	16.4721i	−1.37108	−	0.791596i	−0.991065	+	0.133382i	$0.957416\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	2.63894	−	35.0151i	0.126238	−	1.67500i
$438$		0			0
$439$	30.6658	−	17.7049i	1.46360	−	0.845009i	0.464424	−	0.885613i	$-0.346261\pi$
$439$	30.6658	−	17.7049i	1.46360	−	0.845009i	0.999175	+	0.0406035i	$0.0129281\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−20.4202	−	11.7896i	−0.970191	−	0.560140i	−0.0708967	−	0.997484i	$-0.522586\pi$
$443$	−20.4202	−	11.7896i	−0.970191	−	0.560140i	−0.899295	+	0.437344i	$0.855919\pi$
$444$		0			0
$445$		−	7.27236i		−	0.344743i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	27.9061			1.31697			0.658485	−	0.752594i	$-0.271197\pi$
$449$	27.9061			1.31697			0.658485	+	0.752594i	$0.271197\pi$
$450$		0			0
$451$	1.77946	−	1.02737i	0.0837913	−	0.0483769i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−8.16238			−0.382658
$456$		0			0
$457$	1.38023			0.0645644			0.0322822	−	0.999479i	$-0.489722\pi$
$457$	1.38023			0.0645644			0.0322822	+	0.999479i	$0.489722\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−8.29648	+	4.78998i	−0.386406	+	0.223091i	−0.680602	−	0.732654i	$-0.738281\pi$
$461$	−8.29648	+	4.78998i	−0.386406	+	0.223091i	0.294196	+	0.955745i	$0.404948\pi$
$462$		0			0
$463$	−17.3114			−0.804527			−0.402263	−	0.915524i	$-0.631776\pi$
$463$	−17.3114			−0.804527			−0.402263	+	0.915524i	$0.631776\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	14.7691i		−	0.683432i	−0.939803	−	0.341716i	$-0.888992\pi$
$467$		−	14.7691i		−	0.683432i	0.939803	−	0.341716i	$-0.111008\pi$
$468$		0			0
$469$	−32.2609	−	18.6258i	−1.48967	−	0.860061i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−10.8338	+	6.25487i	−0.498137	+	0.287599i
$474$		0			0
$475$	20.7731	+	1.56558i	0.953134	+	0.0718336i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−27.8909	−	16.1028i	−1.27437	−	0.735757i	−0.298562	−	0.954390i	$-0.596507\pi$
$479$	−27.8909	−	16.1028i	−1.27437	−	0.735757i	−0.975807	+	0.218633i	$0.929840\pi$
$480$		0			0
$481$	1.44466	−	2.50223i	0.0658710	−	0.114092i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−2.62232	+	4.54198i	−0.119073	+	0.206241i
$486$		0			0
$487$		−	5.60690i		−	0.254073i	−0.991898	−	0.127037i	$-0.959453\pi$
$487$		−	5.60690i		−	0.254073i	0.991898	−	0.127037i	$-0.0405466\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	3.87723	−	2.23852i	0.174977	−	0.101023i	−0.409954	−	0.912106i	$-0.634455\pi$
$491$	3.87723	−	2.23852i	0.174977	−	0.101023i	0.584930	+	0.811083i	$0.301122\pi$
$492$		0			0
$493$		−	69.1615i		−	3.11488i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	12.8337	+	22.2286i	0.575669	+	0.997089i
$498$		0			0
$499$	6.05013	+	10.4791i	0.270841	+	0.469111i	0.969077	−	0.246757i	$-0.0793649\pi$
$499$	6.05013	+	10.4791i	0.270841	+	0.469111i	−0.698236	+	0.715867i	$0.746032\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−14.4265	−	8.32914i	−0.643246	−	0.371378i	0.142618	−	0.989778i	$-0.454448\pi$
$503$	−14.4265	−	8.32914i	−0.643246	−	0.371378i	−0.785864	+	0.618400i	$0.787781\pi$
$504$		0			0
$505$	3.12306			0.138975
$506$		0			0
$507$		0			0
$508$		0			0
$509$	11.5662	−	20.0332i	0.512661	−	0.887956i	−0.487231	−	0.873273i	$-0.661993\pi$
$509$	11.5662	−	20.0332i	0.512661	−	0.887956i	0.999892	−	0.0146825i	$-0.00467374\pi$
$510$		0			0
$511$	−2.45698	−	4.25562i	−0.108690	−	0.188257i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	1.15101	+	1.99360i	0.0507193	+	0.0878485i
$516$		0			0
$517$	−0.831089	+	1.43949i	−0.0365513	+	0.0633087i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	28.3189			1.24067			0.620336	−	0.784336i	$-0.286996\pi$
$521$	28.3189			1.24067			0.620336	+	0.784336i	$0.286996\pi$
$522$		0			0
$523$	−21.2457	−	12.2662i	−0.929008	−	0.536363i	−0.0425103	−	0.999096i	$-0.513536\pi$
$523$	−21.2457	−	12.2662i	−0.929008	−	0.536363i	−0.886498	+	0.462733i	$0.846869\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	14.4415	+	25.0134i	0.629082	+	1.08960i
$528$		0			0
$529$	20.9480	+	36.2829i	0.910781	+	1.57752i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	7.57357i		−	0.328048i
$534$		0			0
$535$	2.15061	−	1.24166i	0.0929790	−	0.0536815i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			7.01611i			0.302205i
$540$		0			0
$541$	−8.97233	+	15.5405i	−0.385750	+	0.668139i	−0.991873	−	0.127232i	$-0.959391\pi$
$541$	−8.97233	+	15.5405i	−0.385750	+	0.668139i	0.606123	+	0.795371i	$0.292724\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	1.13679	−	1.96898i	0.0486949	−	0.0843420i
$546$		0			0
$547$	−9.51521	−	5.49361i	−0.406841	−	0.234890i	0.282591	−	0.959241i	$-0.408806\pi$
$547$	−9.51521	−	5.49361i	−0.406841	−	0.234890i	−0.689432	+	0.724351i	$0.742140\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	41.2156	+	3.10625i	1.75585	+	0.132330i
$552$		0			0
$553$	5.46365	−	3.15444i	0.232338	−	0.134141i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−3.00262	−	1.73357i	−0.127225	−	0.0734535i	0.435037	−	0.900413i	$-0.356735\pi$
$557$	−3.00262	−	1.73357i	−0.127225	−	0.0734535i	−0.562262	+	0.826959i	$0.690069\pi$
$558$		0			0
$559$			46.1097i			1.95023i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	1.55779			0.0656532			0.0328266	−	0.999461i	$-0.489549\pi$
$563$	1.55779			0.0656532			0.0328266	+	0.999461i	$0.489549\pi$
$564$		0			0
$565$	−8.46284	+	4.88602i	−0.356034	+	0.205556i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	18.3886			0.770890			0.385445	−	0.922731i	$-0.374048\pi$
$569$	18.3886			0.770890			0.385445	+	0.922731i	$0.374048\pi$
$570$		0			0
$571$	−28.4660			−1.19126			−0.595632	−	0.803257i	$-0.703098\pi$
$571$	−28.4660			−1.19126			−0.595632	+	0.803257i	$0.703098\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−33.3421	+	19.2501i	−1.39046	+	0.802783i
$576$		0			0
$577$	−18.6213			−0.775216			−0.387608	−	0.921824i	$-0.626699\pi$
$577$	−18.6213			−0.775216			−0.387608	+	0.921824i	$0.626699\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			19.3337i			0.802099i
$582$		0			0
$583$	11.9079	+	6.87502i	0.493174	+	0.284734i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−5.35787	+	3.09337i	−0.221143	+	0.127677i	−0.606479	−	0.795099i	$-0.707419\pi$
$587$	−5.35787	+	3.09337i	−0.221143	+	0.127677i	0.385336	+	0.922776i	$0.374085\pi$
$588$		0			0
$589$	−15.5549	+	7.48275i	−0.640930	+	0.308321i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	16.5420	+	9.55051i	0.679297	+	0.392192i	0.799590	−	0.600546i	$-0.205050\pi$
$593$	16.5420	+	9.55051i	0.679297	+	0.392192i	−0.120293	+	0.992738i	$0.538383\pi$
$594$		0			0
$595$	5.98577	−	10.3677i	0.245393	−	0.425033i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	21.9789	−	38.0686i	0.898035	−	1.55544i	0.0680310	−	0.997683i	$-0.478328\pi$
$599$	21.9789	−	38.0686i	0.898035	−	1.55544i	0.830003	−	0.557758i	$-0.188338\pi$
$600$		0			0
$601$		−	25.3002i		−	1.03202i	−0.856583	−	0.516009i	$-0.827417\pi$
$601$		−	25.3002i		−	1.03202i	0.856583	−	0.516009i	$-0.172583\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−3.73573	+	2.15682i	−0.151879	+	0.0876874i
$606$		0			0
$607$		−	12.2880i		−	0.498754i	−0.968406	−	0.249377i	$-0.919774\pi$
$607$		−	12.2880i		−	0.498754i	0.968406	−	0.249377i	$-0.0802259\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	3.06332	+	5.30582i	0.123929	+	0.214651i
$612$		0			0
$613$	5.52042	+	9.56165i	0.222968	+	0.386192i	0.955708	−	0.294317i	$-0.0950922\pi$
$613$	5.52042	+	9.56165i	0.222968	+	0.386192i	−0.732740	+	0.680509i	$0.761759\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	27.4315	+	15.8376i	1.10435	+	0.637598i	0.937361	−	0.348360i	$-0.113261\pi$
$617$	27.4315	+	15.8376i	1.10435	+	0.637598i	0.166991	+	0.985958i	$0.446595\pi$
$618$		0			0
$619$	34.0244			1.36755			0.683777	−	0.729691i	$-0.260336\pi$
$619$	34.0244			1.36755			0.683777	+	0.729691i	$0.260336\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−27.0280	+	46.8138i	−1.08285	+	1.87556i
$624$		0			0
$625$	−10.8683	−	18.8244i	−0.434730	−	0.752975i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	2.11885	+	3.66996i	0.0844841	+	0.146331i
$630$		0			0
$631$	−13.9530	+	24.1674i	−0.555462	+	0.962088i	0.442406	+	0.896815i	$0.354125\pi$
$631$	−13.9530	+	24.1674i	−0.555462	+	0.962088i	−0.997867	+	0.0652727i	$0.979208\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−6.84243			−0.271533
$636$		0			0
$637$	22.3961	+	12.9304i	0.887364	+	0.512320i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	0.0350138	+	0.0606458i	0.00138296	+	0.00239536i	0.866716	−	0.498802i	$-0.166226\pi$
$641$	0.0350138	+	0.0606458i	0.00138296	+	0.00239536i	−0.865333	+	0.501197i	$0.832893\pi$
$642$		0			0
$643$	16.0562	+	27.8101i	0.633194	+	1.09672i	0.986895	+	0.161366i	$0.0515900\pi$
$643$	16.0562	+	27.8101i	0.633194	+	1.09672i	−0.353700	+	0.935359i	$0.615077\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	5.51784i		−	0.216929i	−0.994100	−	0.108464i	$-0.965407\pi$
$647$		−	5.51784i		−	0.216929i	0.994100	−	0.108464i	$-0.0345933\pi$
$648$		0			0
$649$	11.9079	−	6.87502i	0.467426	−	0.269868i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	24.9898i		−	0.977928i	−0.872304	−	0.488964i	$-0.837375\pi$
$653$		−	24.9898i		−	0.977928i	0.872304	−	0.488964i	$-0.162625\pi$
$654$		0			0
$655$	0.289463	−	0.501364i	0.0113102	−	0.0195899i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−1.43887	+	2.49220i	−0.0560506	+	0.0970824i	−0.892689	−	0.450673i	$-0.851184\pi$
$659$	−1.43887	+	2.49220i	−0.0560506	+	0.0970824i	0.836639	+	0.547755i	$0.184517\pi$
$660$		0			0
$661$	−28.9510	−	16.7149i	−1.12606	−	0.650133i	−0.183121	−	0.983090i	$-0.558620\pi$
$661$	−28.9510	−	16.7149i	−1.12606	−	0.650133i	−0.942942	+	0.332958i	$0.891953\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	5.90960	+	4.03276i	0.229164	+	0.156384i
$666$		0			0
$667$	−66.1538	+	38.1939i	−2.56148	+	1.47887i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−8.40377	−	4.85192i	−0.324424	−	0.187306i
$672$		0			0
$673$			1.66815i			0.0643026i	0.999483	+	0.0321513i	$0.0102358\pi$
$673$			1.66815i			0.0643026i	−0.999483	+	0.0321513i	$0.989764\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−4.30258			−0.165361			−0.0826807	−	0.996576i	$-0.526348\pi$
$677$	−4.30258			−0.165361			−0.0826807	+	0.996576i	$0.526348\pi$
$678$		0			0
$679$	33.7609	−	19.4918i	1.29562	−	0.748029i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	0.412745			0.0157933			0.00789663	−	0.999969i	$-0.497486\pi$
$683$	0.412745			0.0157933			0.00789663	+	0.999969i	$0.497486\pi$
$684$		0			0
$685$	−7.13885			−0.272762
$686$		0			0
$687$		0			0
$688$		0			0
$689$	43.8914	−	25.3407i	1.67213	−	0.965403i
$690$		0			0
$691$	12.9573			0.492920			0.246460	−	0.969153i	$-0.420733\pi$
$691$	12.9573			0.492920			0.246460	+	0.969153i	$0.420733\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	4.08501i		−	0.154953i
$696$		0			0
$697$	9.61977	+	5.55398i	0.364375	+	0.210372i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−16.1208	+	9.30735i	−0.608874	+	0.351534i	−0.772525	−	0.634985i	$-0.781006\pi$
$701$	−16.1208	+	9.30735i	−0.608874	+	0.351534i	0.163651	+	0.986518i	$0.447673\pi$
$702$		0			0
$703$	−2.28221	+	1.09787i	−0.0860753	+	0.0414068i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−20.1039	−	11.6070i	−0.756084	−	0.436525i
$708$		0			0
$709$	17.7822	−	30.7997i	0.667825	−	1.15671i	−0.310686	−	0.950513i	$-0.600559\pi$
$709$	17.7822	−	30.7997i	0.667825	−	1.15671i	0.978511	−	0.206194i	$-0.0661079\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	15.9504	−	27.6269i	0.597347	−	1.03464i
$714$		0			0
$715$		−	3.15286i		−	0.117910i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−10.6858	+	6.16944i	−0.398512	+	0.230081i	−0.685842	−	0.727751i	$-0.740566\pi$
$719$	−10.6858	+	6.16944i	−0.398512	+	0.230081i	0.287330	+	0.957832i	$0.407232\pi$
$720$		0			0
$721$		−	17.1110i		−	0.637247i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−22.6589	−	39.2464i	−0.841530	−	1.45757i
$726$		0			0
$727$	−5.42944	−	9.40407i	−0.201367	−	0.348778i	0.747602	−	0.664147i	$-0.231205\pi$
$727$	−5.42944	−	9.40407i	−0.201367	−	0.348778i	−0.948969	+	0.315369i	$0.897872\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−58.5675	−	33.8140i	−2.16620	−	1.25065i
$732$		0			0
$733$	−28.5581			−1.05482			−0.527409	−	0.849612i	$-0.676836\pi$
$733$	−28.5581			−1.05482			−0.527409	+	0.849612i	$0.676836\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	7.19453	−	12.4613i	0.265014	−	0.459018i
$738$		0			0
$739$	−18.0090	−	31.1925i	−0.662472	−	1.14744i	−0.979964	−	0.199175i	$-0.936174\pi$
$739$	−18.0090	−	31.1925i	−0.662472	−	1.14744i	0.317492	−	0.948261i	$-0.397159\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−7.65055	−	13.2511i	−0.280671	−	0.486137i	0.690879	−	0.722971i	$-0.257224\pi$
$743$	−7.65055	−	13.2511i	−0.280671	−	0.486137i	−0.971550	+	0.236833i	$0.923890\pi$
$744$		0			0
$745$	3.30750	−	5.72876i	0.121177	−	0.209885i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−18.4586			−0.674463
$750$		0			0
$751$	−1.48161	−	0.855410i	−0.0540648	−	0.0312143i	0.472724	−	0.881211i	$-0.343271\pi$
$751$	−1.48161	−	0.855410i	−0.0540648	−	0.0312143i	−0.526789	+	0.849996i	$0.676604\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−3.38637	−	5.86536i	−0.123242	−	0.213462i
$756$		0			0
$757$	6.74519	+	11.6830i	0.245158	+	0.424626i	0.962176	−	0.272429i	$-0.0878268\pi$
$757$	6.74519	+	11.6830i	0.245158	+	0.424626i	−0.717018	+	0.697055i	$0.754493\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			32.3374i			1.17223i	0.810228	+	0.586115i	$0.199343\pi$
$761$			32.3374i			1.17223i	−0.810228	+	0.586115i	$0.800657\pi$
$762$		0			0
$763$	−14.6356	+	8.44986i	−0.529844	+	0.305906i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	50.6814i		−	1.83000i
$768$		0			0
$769$	8.87233	−	15.3673i	0.319945	−	0.554160i	−0.660532	−	0.750798i	$-0.729669\pi$
$769$	8.87233	−	15.3673i	0.319945	−	0.554160i	0.980476	+	0.196638i	$0.0630024\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	22.4209	−	38.8341i	0.806424	−	1.39677i	−0.108902	−	0.994052i	$-0.534734\pi$
$773$	22.4209	−	38.8341i	0.806424	−	1.39677i	0.915326	−	0.402714i	$-0.131933\pi$
$774$		0			0
$775$	16.3899	+	9.46274i	0.588744	+	0.339912i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−3.74185	+	5.48330i	−0.134066	+	0.196459i
$780$		0			0
$781$	−8.58616	+	4.95722i	−0.307237	+	0.177383i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−1.85505	−	1.07102i	−0.0662097	−	0.0382262i
$786$		0			0
$787$			2.31647i			0.0825731i	0.999147	+	0.0412866i	$0.0131457\pi$
$787$			2.31647i			0.0825731i	−0.999147	+	0.0412866i	$0.986854\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	72.6363			2.58265
$792$		0			0
$793$	−30.9755	+	17.8837i	−1.09997	+	0.635069i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	47.2067			1.67215			0.836074	−	0.548616i	$-0.184845\pi$
$797$	47.2067			1.67215			0.836074	+	0.548616i	$0.184845\pi$
$798$		0			0
$799$	−8.98577			−0.317894
$800$		0			0
$801$		0			0
$802$		0			0
$803$	1.64380	−	0.949050i	0.0580085	−	0.0334912i
$804$		0			0
$805$	−13.2224			−0.466028
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	26.6610i		−	0.937352i	−0.883370	−	0.468676i	$-0.844731\pi$
$809$		−	26.6610i		−	0.937352i	0.883370	−	0.468676i	$-0.155269\pi$
$810$		0			0
$811$	−11.9478	−	6.89808i	−0.419545	−	0.242224i	0.275338	−	0.961348i	$-0.411210\pi$
$811$	−11.9478	−	6.89808i	−0.419545	−	0.242224i	−0.694883	+	0.719123i	$0.744544\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−8.26388	+	4.77115i	−0.289471	+	0.167126i
$816$		0			0
$817$	22.7813	−	33.3836i	0.797017	−	1.16795i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−41.3464	−	23.8713i	−1.44300	−	0.833115i	−0.444949	−	0.895556i	$-0.646778\pi$
$821$	−41.3464	−	23.8713i	−1.44300	−	0.833115i	−0.998049	+	0.0624403i	$0.980112\pi$
$822$		0			0
$823$	19.3097	−	33.4455i	0.673096	−	1.16584i	−0.303926	−	0.952696i	$-0.598298\pi$
$823$	19.3097	−	33.4455i	0.673096	−	1.16584i	0.977022	−	0.213140i	$-0.0683691\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	21.4671	−	37.1821i	0.746484	−	1.29295i	−0.203014	−	0.979176i	$-0.565074\pi$
$827$	21.4671	−	37.1821i	0.746484	−	1.29295i	0.949498	−	0.313772i	$-0.101593\pi$
$828$		0			0
$829$		−	12.0605i		−	0.418877i	−0.977822	−	0.209439i	$-0.932836\pi$
$829$		−	12.0605i		−	0.418877i	0.977822	−	0.209439i	$-0.0671636\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−32.8477	+	18.9646i	−1.13811	+	0.657085i
$834$		0			0
$835$		−	7.28873i		−	0.252237i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	16.4589	+	28.5076i	0.568223	+	0.984191i	0.996742	+	0.0806582i	$0.0257022\pi$
$839$	16.4589	+	28.5076i	0.568223	+	0.984191i	−0.428519	+	0.903533i	$0.640964\pi$
$840$		0			0
$841$	−30.4573	−	52.7536i	−1.05025	−	1.81909i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−4.77377	−	2.75614i	−0.164223	−	0.0948140i
$846$		0			0
$847$	32.0636			1.10172
$848$		0			0
$849$		0			0
$850$		0			0
$851$	2.34024	−	4.05341i	0.0802223	−	0.138949i
$852$		0			0
$853$	18.9818	+	32.8775i	0.649925	+	1.12570i	0.983140	+	0.182853i	$0.0585332\pi$
$853$	18.9818	+	32.8775i	0.649925	+	1.12570i	−0.333215	+	0.942851i	$0.608133\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	17.3194	+	29.9981i	0.591620	+	1.02472i	0.994014	+	0.109250i	$0.0348448\pi$
$857$	17.3194	+	29.9981i	0.591620	+	1.02472i	−0.402394	+	0.915466i	$0.631822\pi$
$858$		0			0
$859$	−10.7044	+	18.5406i	−0.365230	+	0.632597i	−0.988813	−	0.149160i	$-0.952343\pi$
$859$	−10.7044	+	18.5406i	−0.365230	+	0.632597i	0.623583	+	0.781757i	$0.285676\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	15.2030			0.517515			0.258758	−	0.965942i	$-0.416687\pi$
$863$	15.2030			0.517515			0.258758	+	0.965942i	$0.416687\pi$
$864$		0			0
$865$	0.662451	+	0.382466i	0.0225240	+	0.0130043i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	1.21846	+	2.11043i	0.0413333	+	0.0715913i
$870$		0			0
$871$	−26.5184	−	45.9312i	−0.898541	−	1.55632i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		−	16.0511i		−	0.542625i
$876$		0			0
$877$	−47.6506	+	27.5111i	−1.60905	+	0.928983i	−0.619462	+	0.785027i	$0.712649\pi$
$877$	−47.6506	+	27.5111i	−1.60905	+	0.928983i	−0.989584	+	0.143957i	$0.954017\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	34.6052i		−	1.16588i	−0.812516	−	0.582939i	$-0.801903\pi$
$881$		−	34.6052i		−	1.16588i	0.812516	−	0.582939i	$-0.198097\pi$
$882$		0			0
$883$	−17.1634	+	29.7279i	−0.577596	+	1.00042i	0.418159	+	0.908374i	$0.362676\pi$
$883$	−17.1634	+	29.7279i	−0.577596	+	1.00042i	−0.995754	+	0.0920509i	$0.970658\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	24.0978	−	41.7386i	0.809124	−	1.40144i	−0.104347	−	0.994541i	$-0.533275\pi$
$887$	24.0978	−	41.7386i	0.809124	−	1.40144i	0.913471	−	0.406903i	$-0.133391\pi$
$888$		0			0
$889$	44.0463	+	25.4301i	1.47726	+	0.852899i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	0.403577	−	5.35492i	0.0135052	−	0.179196i
$894$		0			0
$895$	−4.39605	+	2.53806i	−0.146944	+	0.0848380i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	32.5191	+	18.7749i	1.08457	+	0.626179i
$900$		0			0
$901$			74.3330i			2.47639i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−1.73270			−0.0575968
$906$		0			0
$907$	14.7241	−	8.50097i	0.488906	−	0.282270i	−0.235214	−	0.971943i	$-0.575579\pi$
$907$	14.7241	−	8.50097i	0.488906	−	0.282270i	0.724121	+	0.689673i	$0.242246\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−28.0110			−0.928045			−0.464023	−	0.885823i	$-0.653594\pi$
$911$	−28.0110			−0.928045			−0.464023	+	0.885823i	$0.653594\pi$
$912$		0			0
$913$	−7.46798			−0.247154
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−3.72667	+	2.15160i	−0.123066	+	0.0710519i
$918$		0			0
$919$	−46.1408			−1.52205			−0.761023	−	0.648725i	$-0.775303\pi$
$919$	−46.1408			−1.52205			−0.761023	+	0.648725i	$0.775303\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			36.5437i			1.20285i
$924$		0			0
$925$	2.40472	+	1.38837i	0.0790669	+	0.0456493i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−41.6526	+	24.0482i	−1.36658	+	0.788994i	−0.990489	−	0.137589i	$-0.956065\pi$
$929$	−41.6526	+	24.0482i	−1.36658	+	0.788994i	−0.376089	+	0.926584i	$0.622731\pi$
$930$		0			0
$931$	−9.82636	−	20.4268i	−0.322046	−	0.669461i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	4.00468	+	2.31210i	0.130967	+	0.0756139i
$936$		0			0
$937$	−1.17194	+	2.02986i	−0.0382856	+	0.0663127i	−0.884533	−	0.466477i	$-0.845523\pi$
$937$	−1.17194	+	2.02986i	−0.0382856	+	0.0663127i	0.846248	+	0.532790i	$0.178856\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	5.24721	−	9.08843i	0.171054	−	0.296274i	−0.767735	−	0.640768i	$-0.778616\pi$
$941$	5.24721	−	9.08843i	0.171054	−	0.296274i	0.938789	+	0.344494i	$0.111949\pi$
$942$		0			0
$943$		−	12.2686i		−	0.399519i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	15.6357	−	9.02729i	0.508093	−	0.293348i	−0.223957	−	0.974599i	$-0.571897\pi$
$947$	15.6357	−	9.02729i	0.508093	−	0.293348i	0.732049	+	0.681252i	$0.238564\pi$
$948$		0			0
$949$		−	6.99622i		−	0.227107i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	11.1592	+	19.3283i	0.361482	+	0.626106i	0.988205	−	0.153136i	$-0.0489374\pi$
$953$	11.1592	+	19.3283i	0.361482	+	0.626106i	−0.626723	+	0.779242i	$0.715604\pi$
$954$		0			0
$955$	4.29961	+	7.44714i	0.139132	+	0.240984i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	45.9544	+	26.5318i	1.48394	+	0.856756i
$960$		0			0
$961$	15.3185			0.494147
$962$		0			0
$963$		0			0
$964$		0			0
$965$	1.67447	−	2.90027i	0.0539031	−	0.0933629i
$966$		0			0
$967$	10.6439	+	18.4358i	0.342286	+	0.592856i	0.984857	−	0.173370i	$-0.0554656\pi$
$967$	10.6439	+	18.4358i	0.342286	+	0.592856i	−0.642571	+	0.766226i	$0.722132\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−22.1329	−	38.3352i	−0.710277	−	1.23024i	−0.964753	−	0.263157i	$-0.915236\pi$
$971$	−22.1329	−	38.3352i	−0.710277	−	1.23024i	0.254476	−	0.967079i	$-0.418097\pi$
$972$		0			0
$973$	−15.1821	+	26.2961i	−0.486715	+	0.843016i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−56.8291			−1.81812			−0.909062	−	0.416660i	$-0.863201\pi$
$977$	−56.8291			−1.81812			−0.909062	+	0.416660i	$0.863201\pi$
$978$		0			0
$979$	−18.0826	−	10.4400i	−0.577923	−	0.333664i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	22.7404	+	39.3875i	0.725306	+	1.25627i	0.958848	+	0.283920i	$0.0916351\pi$
$983$	22.7404	+	39.3875i	0.725306	+	1.25627i	−0.233542	+	0.972347i	$0.575032\pi$
$984$		0			0
$985$	1.38313	+	2.39566i	0.0440703	+	0.0763319i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			74.6939i			2.37513i
$990$		0			0
$991$	8.64805	−	4.99295i	0.274714	−	0.158606i	−0.356314	−	0.934366i	$-0.615967\pi$
$991$	8.64805	−	4.99295i	0.274714	−	0.158606i	0.631028	+	0.775760i	$0.282633\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	1.15224i		−	0.0365285i
$996$		0			0
$997$	11.2792	−	19.5361i	0.357215	−	0.618715i	−0.630279	−	0.776369i	$-0.717060\pi$
$997$	11.2792	−	19.5361i	0.357215	−	0.618715i	0.987494	+	0.157654i	$0.0503929\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2736.2.dc.d.449.4		16
3.2	odd	2	inner	2736.2.dc.d.449.5		16
4.3	odd	2		684.2.bk.a.449.4	✓	16
12.11	even	2		684.2.bk.a.449.5	yes	16
19.8	odd	6	inner	2736.2.dc.d.1889.5		16
57.8	even	6	inner	2736.2.dc.d.1889.4		16
76.27	even	6		684.2.bk.a.521.5	yes	16
228.179	odd	6		684.2.bk.a.521.4	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
684.2.bk.a.449.4	✓	16	4.3	odd	2
684.2.bk.a.449.5	yes	16	12.11	even	2
684.2.bk.a.521.4	yes	16	228.179	odd	6
684.2.bk.a.521.5	yes	16	76.27	even	6
2736.2.dc.d.449.4		16	1.1	even	1	trivial
2736.2.dc.d.449.5		16	3.2	odd	2	inner
2736.2.dc.d.1889.4		16	57.8	even	6	inner
2736.2.dc.d.1889.5		16	19.8	odd	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.dc (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.406956 + 0.234956i) q^{5} +3.49289 q^{7} +O(q^{10})\)	\(q+(-0.406956 + 0.234956i) q^{5} +3.49289 q^{7} +1.34919i q^{11} +(4.30672 + 2.48649i) q^{13} +(-6.31655 + 3.64686i) q^{17} +(-1.88959 - 3.92803i) q^{19} +(6.97653 + 4.02790i) q^{23} +(-2.38959 + 4.13889i) q^{25} +(-4.74117 + 8.21194i) q^{29} -3.95998i q^{31} +(-1.42145 + 0.820674i) q^{35} -0.581007i q^{37} +(-0.761473 - 1.31891i) q^{41} +(4.63603 + 8.02985i) q^{43} +(1.06693 + 0.615993i) q^{47} +5.20026 q^{49} +(5.09568 - 8.82598i) q^{53} +(-0.316999 - 0.549059i) q^{55} +(-5.09568 - 8.82598i) q^{59} +(-3.59618 + 6.22877i) q^{61} -2.33686 q^{65} +(-9.23616 - 5.33250i) q^{67} +(3.67423 + 6.36396i) q^{71} +(-0.703424 - 1.21837i) q^{73} +4.71255i q^{77} +(1.56422 - 0.903104i) q^{79} +5.53518i q^{83} +(1.71370 - 2.96822i) q^{85} +(-7.73800 + 13.4026i) q^{89} +(15.0429 + 8.68501i) q^{91} +(1.69189 + 1.15456i) q^{95} +(9.66561 - 5.58044i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q + 12 q^{19} + 4 q^{25} + 4 q^{43} - 20 q^{55} + 12 q^{61} - 36 q^{67} + 44 q^{73} + 12 q^{79} + 56 q^{85} + 60 q^{91}+O(q^{100}) \) 16 * q + 12 * q^19 + 4 * q^25 + 4 * q^43 - 20 * q^55 + 12 * q^61 - 36 * q^67 + 44 * q^73 + 12 * q^79 + 56 * q^85 + 60 * q^91

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