LMFDB - Embedded newform 2736.2.dc.c.1889.3

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} + 16x^{14} + 174x^{12} + 1012x^{10} + 4243x^{8} + 9708x^{6} + 15858x^{4} + 12150x^{2} + 6561 $ x^16 + 16x^14 + 174x^12 + 1012x^10 + 4243x^8 + 9708x^6 + 15858x^4 + 12150*x^2 + 6561
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 171)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			1889.3
Root			$1.38883 - 2.40552i$ of defining polynomial
Character	$\chi$	$=$	2736.1889
Dual form			2736.2.dc.c.449.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.00395 - 1.15698i) q^{5} +0.442911 q^{7} +O(q^{10})$	$q+(-2.00395 - 1.15698i) q^{5} +0.442911 q^{7} +4.35455i q^{11} +(1.90192 - 1.09807i) q^{13} +(-1.76720 - 1.02029i) q^{17} +(2.76570 + 3.36911i) q^{19} +(0.0969149 - 0.0559538i) q^{23} +(0.177216 + 0.306946i) q^{25} +(-2.77765 - 4.81103i) q^{29} -2.50698i q^{31} +(-0.887572 - 0.512440i) q^{35} -3.93598i q^{37} +(-2.99745 + 5.19173i) q^{41} +(-1.54424 + 2.67470i) q^{43} +(-7.99929 + 4.61839i) q^{47} -6.80383 q^{49} +(3.77115 + 6.53182i) q^{53} +(5.03813 - 8.72631i) q^{55} +(-0.993498 + 1.72079i) q^{59} +(2.98105 + 5.16332i) q^{61} -5.08180 q^{65} +(3.09793 - 1.78859i) q^{67} +(-3.86806 + 6.69968i) q^{71} +(-4.78179 + 8.28230i) q^{73} +1.92868i q^{77} +(5.17111 + 2.98554i) q^{79} +2.81116i q^{83} +(2.36092 + 4.08923i) q^{85} +(5.87202 + 10.1706i) q^{89} +(0.842379 - 0.486348i) q^{91} +(-1.64431 - 9.95140i) q^{95} +(10.7441 + 6.20313i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 8 q^{7}+O(q^{10}) $ 16 * q - 8 * q^7	$ 16 q - 8 q^{7} - 24 q^{13} + 12 q^{19} + 20 q^{25} + 4 q^{55} - 44 q^{61} + 24 q^{67} - 20 q^{73} + 48 q^{79} - 56 q^{85} + 24 q^{91} + 48 q^{97}+O(q^{100}) $ 16 * q - 8 * q^7 - 24 * q^13 + 12 * q^19 + 20 * q^25 + 4 * q^55 - 44 * q^61 + 24 * q^67 - 20 * q^73 + 48 * q^79 - 56 * q^85 + 24 * q^91 + 48 * q^97

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−2.00395	−	1.15698i	−0.896195	−	0.517418i	−0.0202309	−	0.999795i	$-0.506440\pi$
$5$	−2.00395	−	1.15698i	−0.896195	−	0.517418i	−0.875964	+	0.482377i	$0.839773\pi$
$6$		0			0
$7$	0.442911			0.167405			0.0837023	−	0.996491i	$-0.473326\pi$
$7$	0.442911			0.167405			0.0837023	+	0.996491i	$0.473326\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			4.35455i			1.31295i	0.754350	+	0.656473i	$0.227952\pi$
$11$			4.35455i			1.31295i	−0.754350	+	0.656473i	$0.772048\pi$
$12$		0			0
$13$	1.90192	−	1.09807i	0.527496	−	0.304550i	−0.212500	−	0.977161i	$-0.568161\pi$
$13$	1.90192	−	1.09807i	0.527496	−	0.304550i	0.739996	+	0.672611i	$0.234827\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.76720	−	1.02029i	−0.428608	−	0.247457i	0.270145	−	0.962820i	$-0.412928\pi$
$17$	−1.76720	−	1.02029i	−0.428608	−	0.247457i	−0.698754	+	0.715362i	$0.746262\pi$
$18$		0			0
$19$	2.76570	+	3.36911i	0.634494	+	0.772928i
$19$	2.76570	+	3.36911i	0.634494	+	0.772928i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	0.0969149	−	0.0559538i	0.0202082	−	0.0116672i	−0.489862	−	0.871800i	$-0.662953\pi$
$23$	0.0969149	−	0.0559538i	0.0202082	−	0.0116672i	0.510070	+	0.860133i	$0.329619\pi$
$24$		0			0
$25$	0.177216	+	0.306946i	0.0354431	+	0.0613893i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−2.77765	−	4.81103i	−0.515797	−	0.893387i	−0.999832	−	0.0183379i	$-0.994163\pi$
$29$	−2.77765	−	4.81103i	−0.515797	−	0.893387i	0.484035	−	0.875049i	$-0.339171\pi$
$30$		0			0
$31$		−	2.50698i		−	0.450267i	−0.974328	−	0.225134i	$-0.927718\pi$
$31$		−	2.50698i		−	0.450267i	0.974328	−	0.225134i	$-0.0722819\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−0.887572	−	0.512440i	−0.150027	−	0.0866181i
$36$		0			0
$37$		−	3.93598i		−	0.647071i	−0.946216	−	0.323536i	$-0.895128\pi$
$37$		−	3.93598i		−	0.647071i	0.946216	−	0.323536i	$-0.104872\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−2.99745	+	5.19173i	−0.468123	+	0.810813i	−0.999336	−	0.0364253i	$-0.988403\pi$
$41$	−2.99745	+	5.19173i	−0.468123	+	0.810813i	0.531213	+	0.847238i	$0.321736\pi$
$42$		0			0
$43$	−1.54424	+	2.67470i	−0.235494	+	0.407888i	−0.959416	−	0.281994i	$-0.909004\pi$
$43$	−1.54424	+	2.67470i	−0.235494	+	0.407888i	0.723922	+	0.689882i	$0.242338\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−7.99929	+	4.61839i	−1.16682	+	0.673661i	−0.952928	−	0.303197i	$-0.901946\pi$
$47$	−7.99929	+	4.61839i	−1.16682	+	0.673661i	−0.213888	+	0.976858i	$0.568613\pi$
$48$		0			0
$49$	−6.80383			−0.971976
$50$		0			0
$51$		0			0
$52$		0			0
$53$	3.77115	+	6.53182i	0.518007	+	0.897215i	0.999781	+	0.0209193i	$0.00665930\pi$
$53$	3.77115	+	6.53182i	0.518007	+	0.897215i	−0.481774	+	0.876296i	$0.660007\pi$
$54$		0			0
$55$	5.03813	−	8.72631i	0.679342	−	1.17665i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−0.993498	+	1.72079i	−0.129342	+	0.224028i	−0.923422	−	0.383786i	$-0.874620\pi$
$59$	−0.993498	+	1.72079i	−0.129342	+	0.224028i	0.794080	+	0.607814i	$0.207953\pi$
$60$		0			0
$61$	2.98105	+	5.16332i	0.381684	+	0.661096i	0.991303	−	0.131599i	$-0.0420110\pi$
$61$	2.98105	+	5.16332i	0.381684	+	0.661096i	−0.609619	+	0.792694i	$0.708678\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−5.08180			−0.630319
$66$		0			0
$67$	3.09793	−	1.78859i	0.378472	−	0.218511i	−0.298681	−	0.954353i	$-0.596547\pi$
$67$	3.09793	−	1.78859i	0.378472	−	0.218511i	0.677153	+	0.735842i	$0.263213\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−3.86806	+	6.69968i	−0.459055	+	0.795106i	−0.998911	−	0.0466509i	$-0.985145\pi$
$71$	−3.86806	+	6.69968i	−0.459055	+	0.795106i	0.539856	+	0.841757i	$0.318479\pi$
$72$		0			0
$73$	−4.78179	+	8.28230i	−0.559666	+	0.969370i	0.437858	+	0.899044i	$0.355737\pi$
$73$	−4.78179	+	8.28230i	−0.559666	+	0.969370i	−0.997524	+	0.0703259i	$0.977596\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			1.92868i			0.219793i
$78$		0			0
$79$	5.17111	+	2.98554i	0.581795	+	0.335900i	0.761847	−	0.647758i	$-0.224293\pi$
$79$	5.17111	+	2.98554i	0.581795	+	0.335900i	−0.180051	+	0.983657i	$0.557626\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			2.81116i			0.308565i	0.988027	+	0.154282i	$0.0493066\pi$
$83$			2.81116i			0.308565i	−0.988027	+	0.154282i	$0.950693\pi$
$84$		0			0
$85$	2.36092	+	4.08923i	0.256078	+	0.443540i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	5.87202	+	10.1706i	0.622432	+	1.07808i	0.989031	+	0.147706i	$0.0471888\pi$
$89$	5.87202	+	10.1706i	0.622432	+	1.07808i	−0.366599	+	0.930379i	$0.619478\pi$
$90$		0			0
$91$	0.842379	−	0.486348i	0.0883053	−	0.0509831i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−1.64431	−	9.95140i	−0.168703	−	1.02099i
$96$		0			0
$97$	10.7441	+	6.20313i	1.09090	+	0.629832i	0.933816	−	0.357753i	$-0.116457\pi$
$97$	10.7441	+	6.20313i	1.09090	+	0.629832i	0.157085	+	0.987585i	$0.449790\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	1.31021	−	0.756450i	0.130371	−	0.0752696i	−0.433396	−	0.901203i	$-0.642685\pi$
$101$	1.31021	−	0.756450i	0.130371	−	0.0752696i	0.563767	+	0.825934i	$0.309352\pi$
$102$		0			0
$103$			14.2916i			1.40819i	0.710106	+	0.704095i	$0.248647\pi$
$103$			14.2916i			1.40819i	−0.710106	+	0.704095i	$0.751353\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	4.20173			0.406197			0.203098	−	0.979158i	$-0.434899\pi$
$107$	4.20173			0.406197			0.203098	+	0.979158i	$0.434899\pi$
$108$		0			0
$109$	7.49326	+	4.32623i	0.717724	+	0.414378i	0.813914	−	0.580985i	$-0.197333\pi$
$109$	7.49326	+	4.32623i	0.717724	+	0.414378i	−0.0961906	+	0.995363i	$0.530666\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−10.9168			−1.02696			−0.513482	−	0.858100i	$-0.671645\pi$
$113$	−10.9168			−1.02696			−0.513482	+	0.858100i	$0.671645\pi$
$114$		0			0
$115$	−0.258950			−0.0241472
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−0.782711	−	0.451898i	−0.0717510	−	0.0414254i
$120$		0			0
$121$	−7.96209			−0.723826
$122$		0			0
$123$		0			0
$124$		0			0
$125$			10.7497i			0.961481i
$126$		0			0
$127$	−4.90866	+	2.83402i	−0.435573	+	0.251478i	−0.701718	−	0.712455i	$-0.747583\pi$
$127$	−4.90866	+	2.83402i	−0.435573	+	0.251478i	0.266145	+	0.963933i	$0.414250\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−8.66663	−	5.00368i	−0.757207	−	0.437173i	0.0710853	−	0.997470i	$-0.477354\pi$
$131$	−8.66663	−	5.00368i	−0.757207	−	0.437173i	−0.828292	+	0.560297i	$0.810687\pi$
$132$		0			0
$133$	1.22496	+	1.49222i	0.106217	+	0.129392i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−4.06189	+	2.34514i	−0.347031	+	0.200358i	−0.663377	−	0.748285i	$-0.730877\pi$
$137$	−4.06189	+	2.34514i	−0.347031	+	0.200358i	0.316346	+	0.948644i	$0.397544\pi$
$138$		0			0
$139$	7.89581	+	13.6759i	0.669714	+	1.15998i	0.977984	+	0.208680i	$0.0669166\pi$
$139$	7.89581	+	13.6759i	0.669714	+	1.15998i	−0.308270	+	0.951299i	$0.599750\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	4.78160	+	8.28198i	0.399858	+	0.692574i
$144$		0			0
$145$			12.8548i			1.06753i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	3.31416	+	1.91343i	0.271507	+	0.156754i	0.629572	−	0.776942i	$-0.283230\pi$
$149$	3.31416	+	1.91343i	0.271507	+	0.156754i	−0.358065	+	0.933696i	$0.616564\pi$
$150$		0			0
$151$			18.5228i			1.50736i	0.657239	+	0.753682i	$0.271724\pi$
$151$			18.5228i			1.50736i	−0.657239	+	0.753682i	$0.728276\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−2.90053	+	5.02387i	−0.232977	+	0.403527i
$156$		0			0
$157$	12.1897	−	21.1131i	0.972840	−	1.68501i	0.285954	−	0.958243i	$-0.407690\pi$
$157$	12.1897	−	21.1131i	0.972840	−	1.68501i	0.686886	−	0.726765i	$-0.258977\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	0.0429246	−	0.0247826i	0.00338294	−	0.00195314i
$162$		0			0
$163$	−18.4185			−1.44265			−0.721324	−	0.692598i	$-0.756466\pi$
$163$	−18.4185			−1.44265			−0.721324	+	0.692598i	$0.756466\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−4.78160	−	8.28198i	−0.370012	−	0.640879i	0.619555	−	0.784953i	$-0.287313\pi$
$167$	−4.78160	−	8.28198i	−0.370012	−	0.640879i	−0.989567	+	0.144074i	$0.953980\pi$
$168$		0			0
$169$	−4.08848	+	7.08145i	−0.314498	+	0.544727i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	9.56321	−	16.5640i	0.727077	−	1.25933i	−0.231036	−	0.972945i	$-0.574212\pi$
$173$	9.56321	−	16.5640i	0.727077	−	1.25933i	0.958113	−	0.286390i	$-0.0924551\pi$
$174$		0			0
$175$	0.0784907	+	0.135950i	0.00593334	+	0.0102768i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	9.84289			0.735692			0.367846	−	0.929887i	$-0.380095\pi$
$179$	9.84289			0.735692			0.367846	+	0.929887i	$0.380095\pi$
$180$		0			0
$181$	19.6805	−	11.3626i	1.46284	−	0.844572i	0.463700	−	0.885992i	$-0.346521\pi$
$181$	19.6805	−	11.3626i	1.46284	−	0.844572i	0.999142	+	0.0414197i	$0.0131881\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−4.55386	+	7.88752i	−0.334806	+	0.579902i
$186$		0			0
$187$	4.44291	−	7.69535i	0.324898	−	0.562740i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			21.9342i			1.58710i	0.608502	+	0.793552i	$0.291771\pi$
$191$			21.9342i			1.58710i	−0.608502	+	0.793552i	$0.708229\pi$
$192$		0			0
$193$	9.23065	+	5.32932i	0.664437	+	0.383613i	0.793965	−	0.607963i	$-0.208013\pi$
$193$	9.23065	+	5.32932i	0.664437	+	0.383613i	−0.129529	+	0.991576i	$0.541346\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			12.4637i			0.888003i	0.896026	+	0.444002i	$0.146442\pi$
$197$			12.4637i			0.888003i	−0.896026	+	0.444002i	$0.853558\pi$
$198$		0			0
$199$	1.64907	+	2.85627i	0.116899	+	0.202476i	0.918537	−	0.395334i	$-0.129371\pi$
$199$	1.64907	+	2.85627i	0.116899	+	0.202476i	−0.801638	+	0.597810i	$0.796038\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−1.23025	−	2.13086i	−0.0863468	−	0.149557i
$204$		0			0
$205$	12.0135	−	6.93599i	0.839059	−	0.484431i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−14.6710	+	12.0434i	−1.01481	+	0.833056i
$210$		0			0
$211$	−7.10782	−	4.10370i	−0.489322	−	0.282510i	0.234971	−	0.972002i	$-0.424500\pi$
$211$	−7.10782	−	4.10370i	−0.489322	−	0.282510i	−0.724293	+	0.689492i	$0.757834\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	6.18916	−	3.57332i	0.422098	−	0.243698i
$216$		0			0
$217$		−	1.11037i		−	0.0753768i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−4.48141			−0.301452
$222$		0			0
$223$	−22.3452	−	12.9010i	−1.49635	−	0.863916i	−0.496355	−	0.868120i	$-0.665328\pi$
$223$	−22.3452	−	12.9010i	−1.49635	−	0.863916i	−0.999991	+	0.00420411i	$0.998662\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−20.4461			−1.35705			−0.678527	−	0.734576i	$-0.737381\pi$
$227$	−20.4461			−1.35705			−0.678527	+	0.734576i	$0.737381\pi$
$228$		0			0
$229$	−16.8292			−1.11211			−0.556053	−	0.831147i	$-0.687685\pi$
$229$	−16.8292			−1.11211			−0.556053	+	0.831147i	$0.687685\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−14.7589	−	8.52104i	−0.966886	−	0.558232i	−0.0686005	−	0.997644i	$-0.521853\pi$
$233$	−14.7589	−	8.52104i	−0.966886	−	0.558232i	−0.898285	+	0.439412i	$0.855187\pi$
$234$		0			0
$235$	21.3736			1.39426
$236$		0			0
$237$		0			0
$238$		0			0
$239$			0.873308i			0.0564896i	0.999601	+	0.0282448i	$0.00899180\pi$
$239$			0.873308i			0.0564896i	−0.999601	+	0.0282448i	$0.991008\pi$
$240$		0			0
$241$	−11.8703	+	6.85330i	−0.764631	+	0.441460i	−0.830956	−	0.556338i	$-0.812206\pi$
$241$	−11.8703	+	6.85330i	−0.764631	+	0.441460i	0.0663249	+	0.997798i	$0.478873\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	13.6345	+	7.87191i	0.871079	+	0.502918i
$246$		0			0
$247$	8.95964	+	3.37084i	0.570089	+	0.214481i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−13.7744	+	7.95265i	−0.869432	+	0.501967i	−0.867159	−	0.498031i	$-0.834057\pi$
$251$	−13.7744	+	7.95265i	−0.869432	+	0.501967i	−0.00227233	+	0.999997i	$0.500723\pi$
$252$		0			0
$253$	0.243654	+	0.422021i	0.0153184	+	0.0265322i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−8.34197	−	14.4487i	−0.520358	−	0.901286i	−0.999720	−	0.0236687i	$-0.992465\pi$
$257$	−8.34197	−	14.4487i	−0.520358	−	0.901286i	0.479362	−	0.877617i	$-0.340868\pi$
$258$		0			0
$259$		−	1.74329i		−	0.108323i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	7.01480	+	4.05000i	0.432551	+	0.249733i	0.700433	−	0.713718i	$-0.252990\pi$
$263$	7.01480	+	4.05000i	0.432551	+	0.249733i	−0.267882	+	0.963452i	$0.586324\pi$
$264$		0			0
$265$		−	17.4526i		−	1.07211i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−4.70165	+	8.14349i	−0.286664	+	0.496517i	−0.973011	−	0.230757i	$-0.925880\pi$
$269$	−4.70165	+	8.14349i	−0.286664	+	0.496517i	0.686347	+	0.727274i	$0.259213\pi$
$270$		0			0
$271$	4.04408	−	7.00456i	0.245661	−	0.425497i	−0.716657	−	0.697426i	$-0.754328\pi$
$271$	4.04408	−	7.00456i	0.245661	−	0.425497i	0.962317	+	0.271930i	$0.0876618\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1.33661	+	0.771694i	−0.0806008	+	0.0465349i
$276$		0			0
$277$	−20.6025			−1.23788			−0.618941	−	0.785438i	$-0.712438\pi$
$277$	−20.6025			−1.23788			−0.618941	+	0.785438i	$0.712438\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−7.75309	−	13.4287i	−0.462510	−	0.801091i	0.536575	−	0.843853i	$-0.319718\pi$
$281$	−7.75309	−	13.4287i	−0.462510	−	0.801091i	−0.999085	+	0.0427613i	$0.986385\pi$
$282$		0			0
$283$	13.4307	−	23.2627i	0.798372	−	1.38282i	−0.122303	−	0.992493i	$-0.539028\pi$
$283$	13.4307	−	23.2627i	0.798372	−	1.38282i	0.920676	−	0.390329i	$-0.127639\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−1.32760	+	2.29948i	−0.0783659	+	0.135734i
$288$		0			0
$289$	−6.41801	−	11.1163i	−0.377530	−	0.653901i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	4.25367			0.248502			0.124251	−	0.992251i	$-0.460347\pi$
$293$	4.25367			0.248502			0.124251	+	0.992251i	$0.460347\pi$
$294$		0			0
$295$	3.98184	−	2.29892i	0.231832	−	0.133848i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	0.122883	−	0.212839i	0.00710648	−	0.0123088i
$300$		0			0
$301$	−0.683960	+	1.18465i	−0.0394228	+	0.0682823i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	13.7961i		−	0.789961i
$306$		0			0
$307$	−13.9950	−	8.08003i	−0.798738	−	0.461152i	0.0442918	−	0.999019i	$-0.485897\pi$
$307$	−13.9950	−	8.08003i	−0.798738	−	0.461152i	−0.843030	+	0.537867i	$0.819230\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	29.4966i		−	1.67260i	−0.548272	−	0.836300i	$-0.684714\pi$
$311$		−	29.4966i		−	1.67260i	0.548272	−	0.836300i	$-0.315286\pi$
$312$		0			0
$313$	11.8233	+	20.4785i	0.668291	+	1.15751i	0.978382	+	0.206806i	$0.0663070\pi$
$313$	11.8233	+	20.4785i	0.668291	+	1.15751i	−0.310091	+	0.950707i	$0.600360\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	7.33946	+	12.7123i	0.412225	+	0.713995i	0.995133	−	0.0985437i	$-0.0314184\pi$
$317$	7.33946	+	12.7123i	0.412225	+	0.713995i	−0.582908	+	0.812538i	$0.698085\pi$
$318$		0			0
$319$	20.9499	−	12.0954i	1.17297	−	0.677214i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−1.45005	−	8.77571i	−0.0806829	−	0.488293i
$324$		0			0
$325$	0.674098	+	0.389191i	0.0373922	+	0.0215884i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−3.54297	+	2.04553i	−0.195330	+	0.112774i
$330$		0			0
$331$			27.0043i			1.48429i	0.670238	+	0.742146i	$0.266192\pi$
$331$			27.0043i			1.48429i	−0.670238	+	0.742146i	$0.733808\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−8.27747			−0.452246
$336$		0			0
$337$	6.27218	+	3.62125i	0.341668	+	0.197262i	0.661009	−	0.750378i	$-0.270128\pi$
$337$	6.27218	+	3.62125i	0.341668	+	0.197262i	−0.319342	+	0.947640i	$0.603462\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	10.9168			0.591177
$342$		0			0
$343$	−6.11386			−0.330118
$344$		0			0
$345$		0			0
$346$		0			0
$347$	12.1501	+	7.01489i	0.652254	+	0.376579i	0.789319	−	0.613983i	$-0.210434\pi$
$347$	12.1501	+	7.01489i	0.652254	+	0.376579i	−0.137065	+	0.990562i	$0.543767\pi$
$348$		0			0
$349$	−24.7540			−1.32505			−0.662526	−	0.749039i	$-0.730516\pi$
$349$	−24.7540			−1.32505			−0.662526	+	0.749039i	$0.730516\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			4.21215i			0.224190i	0.993697	+	0.112095i	$0.0357561\pi$
$353$			4.21215i			0.224190i	−0.993697	+	0.112095i	$0.964244\pi$
$354$		0			0
$355$	15.5028	−	8.95056i	0.822805	−	0.475047i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	2.82959	+	1.63366i	0.149340	+	0.0862214i	0.572808	−	0.819690i	$-0.305854\pi$
$359$	2.82959	+	1.63366i	0.149340	+	0.0862214i	−0.423468	+	0.905911i	$0.639187\pi$
$360$		0			0
$361$	−3.70186	+	18.6359i	−0.194835	+	0.980836i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	19.1649	−	11.0649i	1.00314	−	0.579163i
$366$		0			0
$367$	1.26245	+	2.18663i	0.0658994	+	0.114141i	0.897093	−	0.441843i	$-0.145675\pi$
$367$	1.26245	+	2.18663i	0.0658994	+	0.114141i	−0.831193	+	0.555984i	$0.812342\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	1.67028	+	2.89301i	0.0867168	+	0.150198i
$372$		0			0
$373$			1.46572i			0.0758922i	0.999280	+	0.0379461i	$0.0120815\pi$
$373$			1.46572i			0.0758922i	−0.999280	+	0.0379461i	$0.987918\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−10.5657	−	6.10012i	−0.544162	−	0.314172i
$378$		0			0
$379$			10.0737i			0.517450i	0.965951	+	0.258725i	$0.0833023\pi$
$379$			10.0737i			0.517450i	−0.965951	+	0.258725i	$0.916698\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−0.807614	+	1.39883i	−0.0412671	+	0.0714768i	−0.885921	−	0.463836i	$-0.846473\pi$
$383$	−0.807614	+	1.39883i	−0.0412671	+	0.0714768i	0.844654	+	0.535312i	$0.179806\pi$
$384$		0			0
$385$	2.23144	−	3.86497i	0.113725	−	0.196977i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	20.9026	−	12.0681i	1.05981	−	0.611879i	0.134427	−	0.990924i	$-0.457081\pi$
$389$	20.9026	−	12.0681i	1.05981	−	0.611879i	0.925378	+	0.379045i	$0.123747\pi$
$390$		0			0
$391$	−0.228357			−0.0115485
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−6.90844	−	11.9658i	−0.347601	−	0.602063i
$396$		0			0
$397$	−1.61727	+	2.80119i	−0.0811682	+	0.140588i	−0.903752	−	0.428056i	$-0.859198\pi$
$397$	−1.61727	+	2.80119i	−0.0811682	+	0.140588i	0.822584	+	0.568644i	$0.192532\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	16.5950	−	28.7433i	0.828713	−	1.43537i	−0.0703352	−	0.997523i	$-0.522407\pi$
$401$	16.5950	−	28.7433i	0.828713	−	1.43537i	0.899048	−	0.437850i	$-0.144260\pi$
$402$		0			0
$403$	−2.75285	−	4.76807i	−0.137129	−	0.237514i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	17.1394			0.849570
$408$		0			0
$409$	−16.9680	+	9.79650i	−0.839016	+	0.484406i	−0.856929	−	0.515434i	$-0.827631\pi$
$409$	−16.9680	+	9.79650i	−0.839016	+	0.484406i	0.0179139	+	0.999840i	$0.494298\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−0.440031	+	0.762156i	−0.0216525	+	0.0375032i
$414$		0			0
$415$	3.25246	−	5.63343i	0.159657	−	0.276534i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			12.0352i			0.587956i	0.955812	+	0.293978i	$0.0949793\pi$
$419$			12.0352i			0.587956i	−0.955812	+	0.293978i	$0.905021\pi$
$420$		0			0
$421$	−13.1141	−	7.57142i	−0.639142	−	0.369009i	0.145142	−	0.989411i	$-0.453636\pi$
$421$	−13.1141	−	7.57142i	−0.639142	−	0.369009i	−0.784284	+	0.620402i	$0.786969\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	0.723247i		−	0.0350826i
$426$		0			0
$427$	1.32034	+	2.28689i	0.0638956	+	0.110670i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	7.08531	+	12.2721i	0.341287	+	0.591127i	0.984672	−	0.174416i	$-0.0558037\pi$
$431$	7.08531	+	12.2721i	0.341287	+	0.591127i	−0.643385	+	0.765543i	$0.722470\pi$
$432$		0			0
$433$	26.4317	−	15.2604i	1.27023	−	0.733366i	0.295197	−	0.955436i	$-0.404615\pi$
$433$	26.4317	−	15.2604i	1.27023	−	0.733366i	0.975031	+	0.222070i	$0.0712814\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	0.456552	+	0.171766i	0.0218398	+	0.00821669i
$438$		0			0
$439$	4.25778	+	2.45823i	0.203213	+	0.117325i	0.598153	−	0.801382i	$-0.295901\pi$
$439$	4.25778	+	2.45823i	0.203213	+	0.117325i	−0.394940	+	0.918707i	$0.629235\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	24.3646	−	14.0669i	1.15760	−	0.668338i	0.206869	−	0.978369i	$-0.433673\pi$
$443$	24.3646	−	14.0669i	1.15760	−	0.668338i	0.950727	+	0.310030i	$0.100339\pi$
$444$		0			0
$445$		−	27.1753i		−	1.28823i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−7.56032			−0.356794			−0.178397	−	0.983959i	$-0.557091\pi$
$449$	−7.56032			−0.356794			−0.178397	+	0.983959i	$0.557091\pi$
$450$		0			0
$451$	−22.6077	−	13.0525i	−1.06455	−	0.614620i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−2.25078			−0.105518
$456$		0			0
$457$	20.8749			0.976486			0.488243	−	0.872708i	$-0.337638\pi$
$457$	20.8749			0.976486			0.488243	+	0.872708i	$0.337638\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	15.3048	+	8.83625i	0.712817	+	0.411545i	0.812103	−	0.583514i	$-0.198323\pi$
$461$	15.3048	+	8.83625i	0.712817	+	0.411545i	−0.0992862	+	0.995059i	$0.531656\pi$
$462$		0			0
$463$	−24.3998			−1.13395			−0.566977	−	0.823733i	$-0.691887\pi$
$463$	−24.3998			−1.13395			−0.566977	+	0.823733i	$0.691887\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			5.00332i			0.231526i	0.993277	+	0.115763i	$0.0369313\pi$
$467$			5.00332i			0.231526i	−0.993277	+	0.115763i	$0.963069\pi$
$468$		0			0
$469$	1.37211	−	0.792186i	0.0633580	−	0.0365797i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−11.6471	−	6.72447i	−0.535535	−	0.309191i
$474$		0			0
$475$	−0.544013	+	1.44598i	−0.0249610	+	0.0663461i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	11.8134	−	6.82045i	0.539766	−	0.311634i	−0.205218	−	0.978716i	$-0.565790\pi$
$479$	11.8134	−	6.82045i	0.539766	−	0.311634i	0.744984	+	0.667082i	$0.232457\pi$
$480$		0			0
$481$	−4.32199	−	7.48590i	−0.197066	−	0.341328i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−14.3538	−	24.8616i	−0.651774	−	1.12890i
$486$		0			0
$487$		−	21.4543i		−	0.972185i	−0.873907	−	0.486092i	$-0.838422\pi$
$487$		−	21.4543i		−	0.972185i	0.873907	−	0.486092i	$-0.161578\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	35.7044	+	20.6140i	1.61132	+	0.930295i	0.989065	+	0.147479i	$0.0471158\pi$
$491$	35.7044	+	20.6140i	1.61132	+	0.930295i	0.622253	+	0.782816i	$0.286218\pi$
$492$		0			0
$493$			11.3361i			0.510551i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−1.71321	+	2.96736i	−0.0768479	+	0.133104i
$498$		0			0
$499$	−18.3512	+	31.7851i	−0.821511	+	1.42290i	0.0830465	+	0.996546i	$0.473535\pi$
$499$	−18.3512	+	31.7851i	−0.821511	+	1.42290i	−0.904557	+	0.426352i	$0.859798\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−18.1080	+	10.4547i	−0.807396	+	0.466151i	−0.846051	−	0.533102i	$-0.821026\pi$
$503$	−18.1080	+	10.4547i	−0.807396	+	0.466151i	0.0386545	+	0.999253i	$0.487693\pi$
$504$		0			0
$505$	−3.50080			−0.155783
$506$		0			0
$507$		0			0
$508$		0			0
$509$	13.3608	+	23.1415i	0.592205	+	1.02573i	0.993935	+	0.109971i	$0.0350759\pi$
$509$	13.3608	+	23.1415i	0.592205	+	1.02573i	−0.401729	+	0.915758i	$0.631591\pi$
$510$		0			0
$511$	−2.11791	+	3.66832i	−0.0936906	+	0.162277i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	16.5351	−	28.6396i	0.728623	−	1.26201i
$516$		0			0
$517$	−20.1110	−	34.8333i	−0.884481	−	1.53197i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	3.78016			0.165612			0.0828059	−	0.996566i	$-0.473612\pi$
$521$	3.78016			0.165612			0.0828059	+	0.996566i	$0.473612\pi$
$522$		0			0
$523$	−10.9565	+	6.32572i	−0.479093	+	0.276604i	−0.720038	−	0.693934i	$-0.755876\pi$
$523$	−10.9565	+	6.32572i	−0.479093	+	0.276604i	0.240945	+	0.970539i	$0.422543\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−2.55785	+	4.43033i	−0.111422	+	0.192988i
$528$		0			0
$529$	−11.4937	+	19.9077i	−0.499728	+	0.865554i
$530$		0			0
$531$		0			0
$532$		0			0
$533$			13.1657i			0.570268i
$534$		0			0
$535$	−8.42007	−	4.86133i	−0.364031	−	0.210174i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	29.6276i		−	1.27615i
$540$		0			0
$541$	−6.87282	−	11.9041i	−0.295485	−	0.511796i	0.679612	−	0.733571i	$-0.262148\pi$
$541$	−6.87282	−	11.9041i	−0.295485	−	0.511796i	−0.975098	+	0.221776i	$0.928815\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−10.0107	−	17.3391i	−0.428813	−	0.742727i
$546$		0			0
$547$	−10.3538	+	5.97776i	−0.442696	+	0.255591i	−0.704741	−	0.709465i	$-0.748937\pi$
$547$	−10.3538	+	5.97776i	−0.442696	+	0.255591i	0.262044	+	0.965056i	$0.415603\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	8.52679	−	22.6641i	0.363253	−	0.965522i
$552$		0			0
$553$	2.29034	+	1.32233i	0.0973952	+	0.0562311i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	37.5342	−	21.6704i	1.59037	−	0.918203i	0.597132	−	0.802143i	$-0.296307\pi$
$557$	37.5342	−	21.6704i	1.59037	−	0.918203i	0.993242	−	0.116060i	$-0.0370266\pi$
$558$		0			0
$559$			6.78274i			0.286879i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−3.49961			−0.147491			−0.0737455	−	0.997277i	$-0.523495\pi$
$563$	−3.49961			−0.147491			−0.0737455	+	0.997277i	$0.523495\pi$
$564$		0			0
$565$	21.8767	+	12.6305i	0.920360	+	0.531370i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−30.6088			−1.28319			−0.641594	−	0.767045i	$-0.721726\pi$
$569$	−30.6088			−1.28319			−0.641594	+	0.767045i	$0.721726\pi$
$570$		0			0
$571$	1.03663			0.0433816			0.0216908	−	0.999765i	$-0.493095\pi$
$571$	1.03663			0.0433816			0.0216908	+	0.999765i	$0.493095\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	0.0343497	+	0.0198318i	0.00143248	+	0.000827043i
$576$		0			0
$577$	−33.8235			−1.40809			−0.704045	−	0.710155i	$-0.748625\pi$
$577$	−33.8235			−1.40809			−0.704045	+	0.710155i	$0.748625\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			1.24509i			0.0516552i
$582$		0			0
$583$	−28.4431	+	16.4217i	−1.17799	+	0.680115i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	25.5448	+	14.7483i	1.05435	+	0.608728i	0.923863	−	0.382722i	$-0.125013\pi$
$587$	25.5448	+	14.7483i	1.05435	+	0.608728i	0.130484	+	0.991450i	$0.458347\pi$
$588$		0			0
$589$	8.44631	−	6.93355i	0.348024	−	0.285692i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−0.793776	+	0.458287i	−0.0325965	+	0.0188196i	−0.516210	−	0.856462i	$-0.672657\pi$
$593$	−0.793776	+	0.458287i	−0.0325965	+	0.0188196i	0.483613	+	0.875282i	$0.339324\pi$
$594$		0			0
$595$	1.04568	+	1.81116i	0.0428686	+	0.0742505i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−6.45145	−	11.1742i	−0.263599	−	0.456567i	0.703597	−	0.710600i	$-0.251576\pi$
$599$	−6.45145	−	11.1742i	−0.263599	−	0.456567i	−0.967196	+	0.254033i	$0.918243\pi$
$600$		0			0
$601$			2.13260i			0.0869908i	0.999054	+	0.0434954i	$0.0138494\pi$
$601$			2.13260i			0.0869908i	−0.999054	+	0.0434954i	$0.986151\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	15.9556	+	9.21200i	0.648689	+	0.374521i
$606$		0			0
$607$		−	26.8985i		−	1.09178i	−0.837858	−	0.545888i	$-0.816192\pi$
$607$		−	26.8985i		−	1.09178i	0.837858	−	0.545888i	$-0.183808\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−10.1426	+	17.5676i	−0.410327	+	0.710708i
$612$		0			0
$613$	5.19926	−	9.00538i	0.209996	−	0.363724i	−0.741717	−	0.670713i	$-0.765988\pi$
$613$	5.19926	−	9.00538i	0.209996	−	0.363724i	0.951713	+	0.306989i	$0.0993216\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	32.3209	−	18.6605i	1.30119	−	0.751243i	0.320583	−	0.947220i	$-0.396121\pi$
$617$	32.3209	−	18.6605i	1.30119	−	0.751243i	0.980609	+	0.195977i	$0.0627878\pi$
$618$		0			0
$619$	−1.38013			−0.0554721			−0.0277361	−	0.999615i	$-0.508830\pi$
$619$	−1.38013			−0.0554721			−0.0277361	+	0.999615i	$0.508830\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	2.60078	+	4.50468i	0.104198	+	0.180476i
$624$		0			0
$625$	13.3233	−	23.0766i	0.532931	−	0.923063i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−4.01585	+	6.95566i	−0.160122	+	0.277340i
$630$		0			0
$631$	−6.59458	−	11.4222i	−0.262526	−	0.454709i	0.704386	−	0.709817i	$-0.251222\pi$
$631$	−6.59458	−	11.4222i	−0.262526	−	0.454709i	−0.966913	+	0.255108i	$0.917889\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	13.1156			0.520478
$636$		0			0
$637$	−12.9403	+	7.47109i	−0.512714	+	0.296015i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−7.14519	+	12.3758i	−0.282218	+	0.488816i	−0.971931	−	0.235267i	$-0.924404\pi$
$641$	−7.14519	+	12.3758i	−0.282218	+	0.488816i	0.689713	+	0.724083i	$0.257737\pi$
$642$		0			0
$643$	4.69241	−	8.12750i	0.185051	−	0.320517i	−0.758543	−	0.651623i	$-0.774088\pi$
$643$	4.69241	−	8.12750i	0.185051	−	0.320517i	0.943594	+	0.331106i	$0.107422\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	31.2028i		−	1.22671i	−0.789807	−	0.613355i	$-0.789819\pi$
$647$		−	31.2028i		−	1.22671i	0.789807	−	0.613355i	$-0.210181\pi$
$648$		0			0
$649$	−7.49326	−	4.32623i	−0.294136	−	0.169819i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			7.50143i			0.293554i	0.989170	+	0.146777i	$0.0468899\pi$
$653$			7.50143i			0.293554i	−0.989170	+	0.146777i	$0.953110\pi$
$654$		0			0
$655$	11.5783	+	20.0543i	0.452403	+	0.783585i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−17.4317	−	30.1925i	−0.679041	−	1.17613i	−0.975270	−	0.221017i	$-0.929062\pi$
$659$	−17.4317	−	30.1925i	−0.679041	−	1.17613i	0.296229	−	0.955117i	$-0.404271\pi$
$660$		0			0
$661$	13.1329	−	7.58227i	0.510810	−	0.294916i	−0.222357	−	0.974965i	$-0.571375\pi$
$661$	13.1329	−	7.58227i	0.510810	−	0.294916i	0.733166	+	0.680049i	$0.238042\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−0.728285	−	4.40758i	−0.0282417	−	0.170919i
$666$		0			0
$667$	−0.538392	−	0.310841i	−0.0208466	−	0.0120358i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−22.4839	+	12.9811i	−0.867983	+	0.501130i
$672$		0			0
$673$			37.6595i			1.45167i	0.687870	+	0.725834i	$0.258546\pi$
$673$			37.6595i			1.45167i	−0.687870	+	0.725834i	$0.741454\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	26.3032			1.01091			0.505457	−	0.862852i	$-0.331324\pi$
$677$	26.3032			1.01091			0.505457	+	0.862852i	$0.331324\pi$
$678$		0			0
$679$	4.75869	+	2.74743i	0.182622	+	0.105437i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−17.0314			−0.651690			−0.325845	−	0.945423i	$-0.605649\pi$
$683$	−17.0314			−0.651690			−0.325845	+	0.945423i	$0.605649\pi$
$684$		0			0
$685$	10.8531			0.414676
$686$		0			0
$687$		0			0
$688$		0			0
$689$	14.3448	+	8.28198i	0.546494	+	0.315518i
$690$		0			0
$691$	−25.8728			−0.984247			−0.492124	−	0.870525i	$-0.663779\pi$
$691$	−25.8728			−0.984247			−0.492124	+	0.870525i	$0.663779\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	36.5412i		−	1.38609i
$696$		0			0
$697$	10.5942	−	6.11655i	0.401283	−	0.231681i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	16.8947	+	9.75417i	0.638105	+	0.368410i	0.783884	−	0.620907i	$-0.213236\pi$
$701$	16.8947	+	9.75417i	0.638105	+	0.368410i	−0.145779	+	0.989317i	$0.546569\pi$
$702$		0			0
$703$	13.2608	−	10.8857i	0.500139	−	0.410563i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	0.580306	−	0.335040i	0.0218247	−	0.0126005i
$708$		0			0
$709$	9.39305	+	16.2692i	0.352763	+	0.611004i	0.986733	−	0.162354i	$-0.0519087\pi$
$709$	9.39305	+	16.2692i	0.352763	+	0.611004i	−0.633969	+	0.773358i	$0.718575\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−0.140275	−	0.242964i	−0.00525335	−	0.00909907i
$714$		0			0
$715$		−	22.1289i		−	0.827575i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	8.42192	+	4.86240i	0.314085	+	0.181337i	0.648753	−	0.760999i	$-0.275291\pi$
$719$	8.42192	+	4.86240i	0.314085	+	0.181337i	−0.334668	+	0.942336i	$0.608624\pi$
$720$		0			0
$721$			6.32989i			0.235737i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	0.984487	−	1.70518i	0.0365629	−	0.0633288i
$726$		0			0
$727$	−8.51333	+	14.7455i	−0.315742	+	0.546881i	−0.979595	−	0.200982i	$-0.935587\pi$
$727$	−8.51333	+	14.7455i	−0.315742	+	0.546881i	0.663853	+	0.747863i	$0.268920\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	5.45795	−	3.15115i	0.201870	−	0.116550i
$732$		0			0
$733$	−26.2345			−0.968995			−0.484497	−	0.874793i	$-0.660997\pi$
$733$	−26.2345			−0.968995			−0.484497	+	0.874793i	$0.660997\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	7.78850	+	13.4901i	0.286893	+	0.496913i
$738$		0			0
$739$	20.0102	−	34.6587i	0.736088	−	1.27494i	−0.218157	−	0.975914i	$-0.570004\pi$
$739$	20.0102	−	34.6587i	0.736088	−	1.27494i	0.954244	−	0.299028i	$-0.0966622\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	6.38094	−	11.0521i	0.234094	−	0.405463i	−0.724915	−	0.688838i	$-0.758121\pi$
$743$	6.38094	−	11.0521i	0.234094	−	0.405463i	0.959009	+	0.283376i	$0.0914543\pi$
$744$		0			0
$745$	−4.42761	−	7.66885i	−0.162215	−	0.280965i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	1.86099			0.0679992
$750$		0			0
$751$	22.1942	−	12.8138i	0.809877	−	0.467583i	−0.0370362	−	0.999314i	$-0.511792\pi$
$751$	22.1942	−	12.8138i	0.809877	−	0.467583i	0.846913	+	0.531731i	$0.178458\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	21.4306	−	37.1188i	0.779938	−	1.35089i
$756$		0			0
$757$	6.13282	−	10.6224i	0.222901	−	0.386076i	−0.732787	−	0.680459i	$-0.761781\pi$
$757$	6.13282	−	10.6224i	0.222901	−	0.386076i	0.955688	+	0.294383i	$0.0951140\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	25.1024i		−	0.909962i	−0.890501	−	0.454981i	$-0.849646\pi$
$761$		−	25.1024i		−	0.909962i	0.890501	−	0.454981i	$-0.150354\pi$
$762$		0			0
$763$	3.31884	+	1.91614i	0.120150	+	0.0693687i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			4.36372i			0.157565i
$768$		0			0
$769$	3.18965	+	5.52464i	0.115022	+	0.199224i	0.917788	−	0.397070i	$-0.129973\pi$
$769$	3.18965	+	5.52464i	0.115022	+	0.199224i	−0.802767	+	0.596293i	$0.796640\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	3.50637	+	6.07322i	0.126115	+	0.218438i	0.922168	−	0.386789i	$-0.126416\pi$
$773$	3.50637	+	6.07322i	0.126115	+	0.218438i	−0.796053	+	0.605227i	$0.793082\pi$
$774$		0			0
$775$	0.769510	−	0.444277i	0.0276416	−	0.0159589i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−25.7816	+	4.26001i	−0.923721	+	0.152631i
$780$		0			0
$781$	−29.1741	−	16.8437i	−1.04393	−	0.602714i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−48.8550	+	28.2064i	−1.74371	+	1.00673i
$786$		0			0
$787$		−	37.2645i		−	1.32834i	−0.747583	−	0.664168i	$-0.768786\pi$
$787$		−	37.2645i		−	1.32834i	0.747583	−	0.664168i	$-0.231214\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−4.83516			−0.171918
$792$		0			0
$793$	11.3394	+	6.54680i	0.402674	+	0.232484i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−4.90386			−0.173704			−0.0868518	−	0.996221i	$-0.527681\pi$
$797$	−4.90386			−0.173704			−0.0868518	+	0.996221i	$0.527681\pi$
$798$		0			0
$799$	18.8484			0.666809
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−36.0657	−	20.8225i	−1.27273	−	0.734811i
$804$		0			0
$805$	−0.114692			−0.00404236
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	37.0689i		−	1.30327i	−0.758531	−	0.651637i	$-0.774083\pi$
$809$		−	37.0689i		−	1.30327i	0.758531	−	0.651637i	$-0.225917\pi$
$810$		0			0
$811$	13.2641	−	7.65801i	0.465764	−	0.268909i	−0.248701	−	0.968580i	$-0.580004\pi$
$811$	13.2641	−	7.65801i	0.465764	−	0.268909i	0.714465	+	0.699671i	$0.246670\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	36.9098	+	21.3099i	1.29289	+	0.746452i
$816$		0			0
$817$	−13.2823	+	2.19469i	−0.464688	+	0.0767825i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	39.4866	−	22.7976i	1.37809	−	0.795642i	0.386163	−	0.922430i	$-0.373800\pi$
$821$	39.4866	−	22.7976i	1.37809	−	0.795642i	0.991930	+	0.126788i	$0.0404668\pi$
$822$		0			0
$823$	−10.1367	−	17.5573i	−0.353343	−	0.612008i	0.633490	−	0.773751i	$-0.281622\pi$
$823$	−10.1367	−	17.5573i	−0.353343	−	0.612008i	−0.986833	+	0.161743i	$0.948289\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	7.03132	+	12.1786i	0.244503	+	0.423492i	0.961992	−	0.273078i	$-0.0880419\pi$
$827$	7.03132	+	12.1786i	0.244503	+	0.423492i	−0.717489	+	0.696570i	$0.754709\pi$
$828$		0			0
$829$		−	55.5594i		−	1.92966i	−0.262879	−	0.964829i	$-0.584672\pi$
$829$		−	55.5594i		−	1.92966i	0.262879	−	0.964829i	$-0.415328\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	12.0237	+	6.94189i	0.416597	+	0.240522i
$834$		0			0
$835$			22.1289i			0.765803i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	7.23353	−	12.5288i	0.249729	−	0.432544i	−0.713721	−	0.700430i	$-0.752992\pi$
$839$	7.23353	−	12.5288i	0.249729	−	0.432544i	0.963451	+	0.267886i	$0.0863250\pi$
$840$		0			0
$841$	−0.930701	+	1.61202i	−0.0320931	+	0.0555870i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	16.3862	−	9.46060i	0.563704	−	0.325454i
$846$		0			0
$847$	−3.52650			−0.121172
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−0.220233	−	0.381455i	−0.00754950	−	0.0130761i
$852$		0			0
$853$	5.25689	−	9.10519i	0.179992	−	0.311756i	−0.761885	−	0.647712i	$-0.775726\pi$
$853$	5.25689	−	9.10519i	0.179992	−	0.311756i	0.941878	+	0.335956i	$0.109059\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	3.44649	−	5.96950i	0.117730	−	0.203914i	−0.801138	−	0.598480i	$-0.795772\pi$
$857$	3.44649	−	5.96950i	0.117730	−	0.203914i	0.918868	+	0.394566i	$0.129105\pi$
$858$		0			0
$859$	1.14881	+	1.98980i	0.0391970	+	0.0678912i	0.884958	−	0.465670i	$-0.154187\pi$
$859$	1.14881	+	1.98980i	0.0391970	+	0.0678912i	−0.845761	+	0.533561i	$0.820853\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	40.8213			1.38957			0.694787	−	0.719216i	$-0.255499\pi$
$863$	40.8213			1.38957			0.694787	+	0.719216i	$0.255499\pi$
$864$		0			0
$865$	−38.3284	+	22.1289i	−1.30321	+	0.752406i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−13.0007	+	22.5179i	−0.441018	+	0.763866i
$870$		0			0
$871$	3.92800	−	6.80349i	0.133095	−	0.230527i
$872$		0			0
$873$		0			0
$874$		0			0
$875$			4.76115i			0.160956i
$876$		0			0
$877$	27.6606	+	15.9699i	0.934032	+	0.539264i	0.888085	−	0.459680i	$-0.152036\pi$
$877$	27.6606	+	15.9699i	0.934032	+	0.539264i	0.0459478	+	0.998944i	$0.485369\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	33.3540i		−	1.12372i	−0.827231	−	0.561862i	$-0.810085\pi$
$881$		−	33.3540i		−	1.12372i	0.827231	−	0.561862i	$-0.189915\pi$
$882$		0			0
$883$	15.8620	+	27.4739i	0.533800	+	0.924569i	0.999220	+	0.0394792i	$0.0125699\pi$
$883$	15.8620	+	27.4739i	0.533800	+	0.924569i	−0.465420	+	0.885090i	$0.654097\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−0.202841	−	0.351330i	−0.00681072	−	0.0117965i	0.862600	−	0.505887i	$-0.168835\pi$
$887$	−0.202841	−	0.351330i	−0.00681072	−	0.0117965i	−0.869411	+	0.494090i	$0.835501\pi$
$888$		0			0
$889$	−2.17410	+	1.25522i	−0.0729169	+	0.0420986i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−37.6835	−	14.1775i	−1.26103	−	0.474430i
$894$		0			0
$895$	−19.7247	−	11.3880i	−0.659323	−	0.380660i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−12.0612	+	6.96353i	−0.402263	+	0.232247i
$900$		0			0
$901$		−	15.3907i		−	0.512738i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−52.5851			−1.74799
$906$		0			0
$907$	22.6077	+	13.0525i	0.750675	+	0.433402i	0.825938	−	0.563761i	$-0.190646\pi$
$907$	22.6077	+	13.0525i	0.750675	+	0.433402i	−0.0752627	+	0.997164i	$0.523980\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−40.6992			−1.34842			−0.674212	−	0.738538i	$-0.735517\pi$
$911$	−40.6992			−1.34842			−0.674212	+	0.738538i	$0.735517\pi$
$912$		0			0
$913$	−12.2413			−0.405129
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−3.83854	−	2.21618i	−0.126760	−	0.0731848i
$918$		0			0
$919$	−50.1092			−1.65295			−0.826475	−	0.562973i	$-0.809658\pi$
$919$	−50.1092			−1.65295			−0.826475	+	0.562973i	$0.809658\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			16.9896i			0.559221i
$924$		0			0
$925$	1.20814	−	0.697517i	0.0397233	−	0.0229342i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−8.94942	−	5.16695i	−0.293621	−	0.169522i	0.345953	−	0.938252i	$-0.387556\pi$
$929$	−8.94942	−	5.16695i	−0.293621	−	0.169522i	−0.639574	+	0.768730i	$0.720889\pi$
$930$		0			0
$931$	−18.8173	−	22.9229i	−0.616713	−	0.751267i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−17.8068	+	10.2807i	−0.582343	+	0.336216i
$936$		0			0
$937$	−13.2882	−	23.0159i	−0.434107	−	0.751896i	0.563115	−	0.826378i	$-0.309603\pi$
$937$	−13.2882	−	23.0159i	−0.434107	−	0.751896i	−0.997222	+	0.0744827i	$0.976269\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	6.53979	+	11.3272i	0.213191	+	0.369258i	0.952711	−	0.303876i	$-0.0982810\pi$
$941$	6.53979	+	11.3272i	0.213191	+	0.369258i	−0.739520	+	0.673134i	$0.764948\pi$
$942$		0			0
$943$			0.670875i			0.0218467i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	45.3231	+	26.1673i	1.47280	+	0.850324i	0.999532	−	0.0305930i	$-0.00973957\pi$
$947$	45.3231	+	26.1673i	1.47280	+	0.850324i	0.473272	+	0.880917i	$0.343073\pi$
$948$		0			0
$949$			21.0030i			0.681785i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−15.9642	+	27.6508i	−0.517132	+	0.895699i	0.482670	+	0.875802i	$0.339667\pi$
$953$	−15.9642	+	27.6508i	−0.517132	+	0.895699i	−0.999802	+	0.0198965i	$0.993666\pi$
$954$		0			0
$955$	25.3775	−	43.9551i	0.821196	−	1.42235i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−1.79906	+	1.03869i	−0.0580946	+	0.0335409i
$960$		0			0
$961$	24.7150			0.797259
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−12.3318	−	21.3594i	−0.396976	−	0.687583i
$966$		0			0
$967$	−16.6781	+	28.8873i	−0.536331	+	0.928952i	0.462767	+	0.886480i	$0.346857\pi$
$967$	−16.6781	+	28.8873i	−0.536331	+	0.928952i	−0.999098	+	0.0424724i	$0.986477\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−3.86806	+	6.69968i	−0.124132	+	0.215003i	−0.921393	−	0.388631i	$-0.872948\pi$
$971$	−3.86806	+	6.69968i	−0.124132	+	0.215003i	0.797261	+	0.603634i	$0.206281\pi$
$972$		0			0
$973$	3.49714	+	6.05722i	0.112113	+	0.194186i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	32.2249			1.03097			0.515483	−	0.856900i	$-0.327613\pi$
$977$	32.2249			1.03097			0.515483	+	0.856900i	$0.327613\pi$
$978$		0			0
$979$	−44.2885	+	25.5700i	−1.41547	+	0.817220i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	4.33406	−	7.50682i	0.138235	−	0.239430i	−0.788594	−	0.614915i	$-0.789190\pi$
$983$	4.33406	−	7.50682i	0.138235	−	0.239430i	0.926829	+	0.375485i	$0.122524\pi$
$984$		0			0
$985$	14.4203	−	24.9767i	0.459469	−	0.795824i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			0.345625i			0.0109902i
$990$		0			0
$991$	22.3953	+	12.9299i	0.711411	+	0.410733i	0.811583	−	0.584237i	$-0.198606\pi$
$991$	22.3953	+	12.9299i	0.711411	+	0.410733i	−0.100172	+	0.994970i	$0.531939\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	7.63178i		−	0.241944i
$996$		0			0
$997$	−4.41801	−	7.65222i	−0.139920	−	0.242348i	0.787546	−	0.616255i	$-0.211351\pi$
$997$	−4.41801	−	7.65222i	−0.139920	−	0.242348i	−0.927466	+	0.373907i	$0.878018\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2736.2.dc.c.1889.3		16
3.2	odd	2	inner	2736.2.dc.c.1889.6		16
4.3	odd	2		171.2.m.a.8.1	✓	16
12.11	even	2		171.2.m.a.8.8	yes	16
19.12	odd	6	inner	2736.2.dc.c.449.6		16
57.50	even	6	inner	2736.2.dc.c.449.3		16
76.31	even	6		171.2.m.a.107.8	yes	16
228.107	odd	6		171.2.m.a.107.1	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
171.2.m.a.8.1	✓	16	4.3	odd	2
171.2.m.a.8.8	yes	16	12.11	even	2
171.2.m.a.107.1	yes	16	228.107	odd	6
171.2.m.a.107.8	yes	16	76.31	even	6
2736.2.dc.c.449.3		16	57.50	even	6	inner
2736.2.dc.c.449.6		16	19.12	odd	6	inner
2736.2.dc.c.1889.3		16	1.1	even	1	trivial
2736.2.dc.c.1889.6		16	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.dc (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.00395 - 1.15698i) q^{5} +0.442911 q^{7} +O(q^{10})\)	\(q+(-2.00395 - 1.15698i) q^{5} +0.442911 q^{7} +4.35455i q^{11} +(1.90192 - 1.09807i) q^{13} +(-1.76720 - 1.02029i) q^{17} +(2.76570 + 3.36911i) q^{19} +(0.0969149 - 0.0559538i) q^{23} +(0.177216 + 0.306946i) q^{25} +(-2.77765 - 4.81103i) q^{29} -2.50698i q^{31} +(-0.887572 - 0.512440i) q^{35} -3.93598i q^{37} +(-2.99745 + 5.19173i) q^{41} +(-1.54424 + 2.67470i) q^{43} +(-7.99929 + 4.61839i) q^{47} -6.80383 q^{49} +(3.77115 + 6.53182i) q^{53} +(5.03813 - 8.72631i) q^{55} +(-0.993498 + 1.72079i) q^{59} +(2.98105 + 5.16332i) q^{61} -5.08180 q^{65} +(3.09793 - 1.78859i) q^{67} +(-3.86806 + 6.69968i) q^{71} +(-4.78179 + 8.28230i) q^{73} +1.92868i q^{77} +(5.17111 + 2.98554i) q^{79} +2.81116i q^{83} +(2.36092 + 4.08923i) q^{85} +(5.87202 + 10.1706i) q^{89} +(0.842379 - 0.486348i) q^{91} +(-1.64431 - 9.95140i) q^{95} +(10.7441 + 6.20313i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 8 q^{7}+O(q^{10}) \) 16 * q - 8 * q^7	\( 16 q - 8 q^{7} - 24 q^{13} + 12 q^{19} + 20 q^{25} + 4 q^{55} - 44 q^{61} + 24 q^{67} - 20 q^{73} + 48 q^{79} - 56 q^{85} + 24 q^{91} + 48 q^{97}+O(q^{100}) \) 16 * q - 8 * q^7 - 24 * q^13 + 12 * q^19 + 20 * q^25 + 4 * q^55 - 44 * q^61 + 24 * q^67 - 20 * q^73 + 48 * q^79 - 56 * q^85 + 24 * q^91 + 48 * q^97

Introduction

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