LMFDB - Embedded newform 2592.2.p.g.2159.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2592,2,Mod(431,2592)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2592, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 3, 1]))

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

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

chi := DirichletCharacter("2592.431");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2592 = 2^{5} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2592.p (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:	$20.6972242039$
Analytic rank:	$0$
Dimension:	$48$
Relative dimension:	$24$ over $\Q(\zeta_{6})$
Twist minimal:	no (minimal twist has level 648)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			2159.20
Character	$\chi$	$=$	2592.2159
Dual form			2592.2.p.g.431.20

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.893655 - 1.54786i) q^{5} +(-2.90922 + 1.67964i) q^{7} +O(q^{10})$	$q+(0.893655 - 1.54786i) q^{5} +(-2.90922 + 1.67964i) q^{7} +(-3.13955 + 1.81262i) q^{11} +(1.48804 + 0.859122i) q^{13} -6.25325i q^{17} +4.57204 q^{19} +(0.308666 - 0.534624i) q^{23} +(0.902761 + 1.56363i) q^{25} +(2.83781 + 4.91523i) q^{29} +(-7.01048 - 4.04750i) q^{31} +6.00408i q^{35} +7.06318i q^{37} +(-5.99854 - 3.46326i) q^{41} +(-3.33310 - 5.77310i) q^{43} +(0.506189 + 0.876744i) q^{47} +(2.14239 - 3.71072i) q^{49} -10.5493 q^{53} +6.47944i q^{55} +(4.33478 + 2.50269i) q^{59} +(-12.3813 + 7.14832i) q^{61} +(2.65959 - 1.53552i) q^{65} +(-4.69155 + 8.12601i) q^{67} -9.70917 q^{71} -15.2438 q^{73} +(6.08911 - 10.5466i) q^{77} +(-13.3590 + 7.71280i) q^{79} +(5.60217 - 3.23441i) q^{83} +(-9.67913 - 5.58825i) q^{85} -1.45494i q^{89} -5.77206 q^{91} +(4.08583 - 7.07686i) q^{95} +(2.92965 + 5.07431i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q+O(q^{10}) $ 48 * q	$ 48 q - 24 q^{25} + 24 q^{49} + 48 q^{67}+O(q^{100}) $ 48 * q - 24 * q^25 + 24 * q^49 + 48 * q^67

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2592\mathbb{Z}\right)^\times$.

$n$	$325$	$1217$	$2431$
$\chi(n)$	$-1$	$e\left(\frac{5}{6}\right)$	$-1$

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
$5$	0.893655	−	1.54786i	0.399655	−	0.692222i	−0.594029	−	0.804444i	$-0.702463\pi$
$5$	0.893655	−	1.54786i	0.399655	−	0.692222i	0.993683	+	0.112222i	$0.0357967\pi$
$6$		0			0
$7$	−2.90922	+	1.67964i	−1.09958	+	0.634845i	−0.936111	−	0.351703i	$-0.885603\pi$
$7$	−2.90922	+	1.67964i	−1.09958	+	0.634845i	−0.163472	+	0.986548i	$0.552269\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−3.13955	+	1.81262i	−0.946611	+	0.546526i	−0.892027	−	0.451983i	$-0.850717\pi$
$11$	−3.13955	+	1.81262i	−0.946611	+	0.546526i	−0.0545848	+	0.998509i	$0.517384\pi$
$12$		0			0
$13$	1.48804	+	0.859122i	0.412709	+	0.238277i	0.691953	−	0.721943i	$-0.256751\pi$
$13$	1.48804	+	0.859122i	0.412709	+	0.238277i	−0.279244	+	0.960220i	$0.590084\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	6.25325i		−	1.51664i	−0.651885	−	0.758318i	$-0.726022\pi$
$17$		−	6.25325i		−	1.51664i	0.651885	−	0.758318i	$-0.273978\pi$
$18$		0			0
$19$	4.57204			1.04890			0.524449	−	0.851442i	$-0.324271\pi$
$19$	4.57204			1.04890			0.524449	+	0.851442i	$0.324271\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	0.308666	−	0.534624i	0.0643612	−	0.111477i	−0.832049	−	0.554702i	$-0.812832\pi$
$23$	0.308666	−	0.534624i	0.0643612	−	0.111477i	0.896410	+	0.443225i	$0.146166\pi$
$24$		0			0
$25$	0.902761	+	1.56363i	0.180552	+	0.312726i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	2.83781	+	4.91523i	0.526968	+	0.912735i	0.999506	+	0.0314246i	$0.0100044\pi$
$29$	2.83781	+	4.91523i	0.526968	+	0.912735i	−0.472539	+	0.881310i	$0.656662\pi$
$30$		0			0
$31$	−7.01048	−	4.04750i	−1.25912	−	0.726953i	−0.286216	−	0.958165i	$-0.592397\pi$
$31$	−7.01048	−	4.04750i	−1.25912	−	0.726953i	−0.972903	+	0.231212i	$0.925731\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$			6.00408i			1.01487i
$36$		0			0
$37$			7.06318i			1.16118i	0.814196	+	0.580590i	$0.197178\pi$
$37$			7.06318i			1.16118i	−0.814196	+	0.580590i	$0.802822\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−5.99854	−	3.46326i	−0.936814	−	0.540870i	−0.0478541	−	0.998854i	$-0.515238\pi$
$41$	−5.99854	−	3.46326i	−0.936814	−	0.540870i	−0.888960	+	0.457984i	$0.848572\pi$
$42$		0			0
$43$	−3.33310	−	5.77310i	−0.508294	−	0.880390i	−0.999954	−	0.00960315i	$-0.996943\pi$
$43$	−3.33310	−	5.77310i	−0.508294	−	0.880390i	0.491660	−	0.870787i	$-0.336390\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	0.506189	+	0.876744i	0.0738352	+	0.127886i	0.900579	−	0.434692i	$-0.143143\pi$
$47$	0.506189	+	0.876744i	0.0738352	+	0.127886i	−0.826744	+	0.562578i	$0.809809\pi$
$48$		0			0
$49$	2.14239	−	3.71072i	0.306055	−	0.530103i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−10.5493			−1.44906			−0.724529	−	0.689244i	$-0.757943\pi$
$53$	−10.5493			−1.44906			−0.724529	+	0.689244i	$0.757943\pi$
$54$		0			0
$55$			6.47944i			0.873687i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	4.33478	+	2.50269i	0.564341	+	0.325822i	0.754886	−	0.655856i	$-0.227692\pi$
$59$	4.33478	+	2.50269i	0.564341	+	0.325822i	−0.190545	+	0.981678i	$0.561026\pi$
$60$		0			0
$61$	−12.3813	+	7.14832i	−1.58526	+	0.915249i	−0.591185	+	0.806536i	$0.701340\pi$
$61$	−12.3813	+	7.14832i	−1.58526	+	0.915249i	−0.994073	+	0.108713i	$0.965327\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	2.65959	−	1.53552i	0.329882	−	0.190457i
$66$		0			0
$67$	−4.69155	+	8.12601i	−0.573165	+	0.992750i	0.423074	+	0.906095i	$0.360951\pi$
$67$	−4.69155	+	8.12601i	−0.573165	+	0.992750i	−0.996238	+	0.0866550i	$0.972382\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−9.70917			−1.15227			−0.576133	−	0.817356i	$-0.695439\pi$
$71$	−9.70917			−1.15227			−0.576133	+	0.817356i	$0.695439\pi$
$72$		0			0
$73$	−15.2438			−1.78415			−0.892075	−	0.451886i	$-0.850751\pi$
$73$	−15.2438			−1.78415			−0.892075	+	0.451886i	$0.850751\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	6.08911	−	10.5466i	0.693919	−	1.20190i
$78$		0			0
$79$	−13.3590	+	7.71280i	−1.50300	+	0.867758i	−0.503006	+	0.864283i	$0.667773\pi$
$79$	−13.3590	+	7.71280i	−1.50300	+	0.867758i	−0.999994	+	0.00347484i	$0.998894\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	5.60217	−	3.23441i	0.614918	−	0.355023i	−0.159970	−	0.987122i	$-0.551140\pi$
$83$	5.60217	−	3.23441i	0.614918	−	0.355023i	0.774888	+	0.632099i	$0.217806\pi$
$84$		0			0
$85$	−9.67913	−	5.58825i	−1.04985	−	0.606130i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	1.45494i		−	0.154223i	−0.997022	−	0.0771115i	$-0.975430\pi$
$89$		−	1.45494i		−	0.154223i	0.997022	−	0.0771115i	$-0.0245697\pi$
$90$		0			0
$91$	−5.77206			−0.605077
$92$		0			0
$93$		0			0
$94$		0			0
$95$	4.08583	−	7.07686i	0.419197	−	0.726071i
$96$		0			0
$97$	2.92965	+	5.07431i	0.297461	+	0.515218i	0.975554	−	0.219758i	$-0.0705267\pi$
$97$	2.92965	+	5.07431i	0.297461	+	0.515218i	−0.678093	+	0.734976i	$0.737193\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	1.43179	+	2.47993i	0.142468	+	0.246762i	0.928426	−	0.371518i	$-0.121163\pi$
$101$	1.43179	+	2.47993i	0.142468	+	0.246762i	−0.785957	+	0.618281i	$0.787829\pi$
$102$		0			0
$103$	4.19436	+	2.42161i	0.413283	+	0.238609i	0.692199	−	0.721707i	$-0.256642\pi$
$103$	4.19436	+	2.42161i	0.413283	+	0.238609i	−0.278917	+	0.960315i	$0.589975\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			10.8641i			1.05027i	0.851018	+	0.525137i	$0.175986\pi$
$107$			10.8641i			1.05027i	−0.851018	+	0.525137i	$0.824014\pi$
$108$		0			0
$109$			0.887009i			0.0849601i	0.999097	+	0.0424800i	$0.0135259\pi$
$109$			0.887009i			0.0849601i	−0.999097	+	0.0424800i	$0.986474\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	3.61952	+	2.08973i	0.340496	+	0.196585i	0.660491	−	0.750834i	$-0.270348\pi$
$113$	3.61952	+	2.08973i	0.340496	+	0.196585i	−0.319996	+	0.947419i	$0.603681\pi$
$114$		0			0
$115$	−0.551681	−	0.955540i	−0.0514445	−	0.0891045i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	10.5032	+	18.1921i	0.962828	+	1.66767i
$120$		0			0
$121$	1.07120	−	1.85538i	0.0973821	−	0.168671i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	12.1636			1.08794
$126$		0			0
$127$			11.0834i			0.983494i	0.870738	+	0.491747i	$0.163641\pi$
$127$			11.0834i			0.983494i	−0.870738	+	0.491747i	$0.836359\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	8.32098	+	4.80412i	0.727008	+	0.419738i	0.817327	−	0.576175i	$-0.195455\pi$
$131$	8.32098	+	4.80412i	0.727008	+	0.419738i	−0.0903187	+	0.995913i	$0.528789\pi$
$132$		0			0
$133$	−13.3011	+	7.67939i	−1.15335	+	0.665887i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	7.34340	−	4.23972i	0.627389	−	0.362223i	−0.152351	−	0.988326i	$-0.548684\pi$
$137$	7.34340	−	4.23972i	0.627389	−	0.362223i	0.779740	+	0.626103i	$0.215351\pi$
$138$		0			0
$139$	4.83770	−	8.37914i	0.410328	−	0.710710i	−0.584597	−	0.811324i	$-0.698747\pi$
$139$	4.83770	−	8.37914i	0.410328	−	0.710710i	0.994926	+	0.100614i	$0.0320807\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−6.22905			−0.520900
$144$		0			0
$145$	10.1441			0.842420
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−5.56943	+	9.64653i	−0.456265	+	0.790275i	−0.998760	−	0.0497846i	$-0.984147\pi$
$149$	−5.56943	+	9.64653i	−0.456265	+	0.790275i	0.542495	+	0.840059i	$0.317480\pi$
$150$		0			0
$151$	−4.32622	+	2.49775i	−0.352063	+	0.203264i	−0.665593	−	0.746314i	$-0.731822\pi$
$151$	−4.32622	+	2.49775i	−0.352063	+	0.203264i	0.313531	+	0.949578i	$0.398488\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−12.5299	+	7.23414i	−1.00643	+	0.581060i
$156$		0			0
$157$	−7.45814	−	4.30596i	−0.595225	−	0.343653i	0.171936	−	0.985108i	$-0.444998\pi$
$157$	−7.45814	−	4.30596i	−0.595225	−	0.343653i	−0.767161	+	0.641455i	$0.778331\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			2.07379i			0.163437i
$162$		0			0
$163$	2.04820			0.160427			0.0802137	−	0.996778i	$-0.474440\pi$
$163$	2.04820			0.160427			0.0802137	+	0.996778i	$0.474440\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	10.5032	−	18.1921i	0.812763	−	1.40775i	−0.0981605	−	0.995171i	$-0.531296\pi$
$167$	10.5032	−	18.1921i	0.812763	−	1.40775i	0.910923	−	0.412576i	$-0.135371\pi$
$168$		0			0
$169$	−5.02382	−	8.70151i	−0.386448	−	0.669347i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	3.40970	+	5.90577i	0.259234	+	0.449007i	0.966037	−	0.258404i	$-0.0831964\pi$
$173$	3.40970	+	5.90577i	0.259234	+	0.449007i	−0.706803	+	0.707411i	$0.749863\pi$
$174$		0			0
$175$	−5.25267	−	3.03263i	−0.397064	−	0.229245i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	22.5099i		−	1.68247i	−0.540671	−	0.841234i	$-0.681830\pi$
$179$		−	22.5099i		−	1.68247i	0.540671	−	0.841234i	$-0.318170\pi$
$180$		0			0
$181$			8.43198i			0.626744i	0.949630	+	0.313372i	$0.101459\pi$
$181$			8.43198i			0.626744i	−0.949630	+	0.313372i	$0.898541\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	10.9328	+	6.31205i	0.803794	+	0.464071i
$186$		0			0
$187$	11.3348	+	19.6324i	0.828881	+	1.43566i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−3.04029	−	5.26593i	−0.219987	−	0.381029i	0.734816	−	0.678266i	$-0.237268\pi$
$191$	−3.04029	−	5.26593i	−0.219987	−	0.381029i	−0.954804	+	0.297237i	$0.903935\pi$
$192$		0			0
$193$	0.500000	−	0.866025i	0.0359908	−	0.0623379i	−0.847469	−	0.530845i	$-0.821875\pi$
$193$	0.500000	−	0.866025i	0.0359908	−	0.0623379i	0.883460	+	0.468507i	$0.155208\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−22.4901			−1.60235			−0.801176	−	0.598429i	$-0.795792\pi$
$197$	−22.4901			−1.60235			−0.801176	+	0.598429i	$0.795792\pi$
$198$		0			0
$199$			9.83872i			0.697448i	0.937225	+	0.348724i	$0.113385\pi$
$199$			9.83872i			0.697448i	−0.937225	+	0.348724i	$0.886615\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−16.5116	−	9.53299i	−1.15889	−	0.669085i
$204$		0			0
$205$	−10.7212	+	6.18992i	−0.748804	+	0.432322i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−14.3542	+	8.28739i	−0.992899	+	0.573250i
$210$		0			0
$211$	6.11453	−	10.5907i	0.420941	−	0.729092i	−0.575090	−	0.818090i	$-0.695033\pi$
$211$	6.11453	−	10.5907i	0.420941	−	0.729092i	0.996032	+	0.0889981i	$0.0283665\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−11.9146			−0.812567
$216$		0			0
$217$	27.1934			1.84601
$218$		0			0
$219$		0			0
$220$		0			0
$221$	5.37230	−	9.30510i	0.361380	−	0.625928i
$222$		0			0
$223$	−5.59348	+	3.22940i	−0.374567	+	0.216256i	−0.675452	−	0.737404i	$-0.736051\pi$
$223$	−5.59348	+	3.22940i	−0.374567	+	0.216256i	0.300885	+	0.953661i	$0.402718\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	10.8063	−	6.23903i	0.717241	−	0.414099i	−0.0964955	−	0.995333i	$-0.530763\pi$
$227$	10.8063	−	6.23903i	0.717241	−	0.414099i	0.813736	+	0.581234i	$0.197430\pi$
$228$		0			0
$229$	−7.47846	−	4.31769i	−0.494190	−	0.285321i	0.232121	−	0.972687i	$-0.425434\pi$
$229$	−7.47846	−	4.31769i	−0.494190	−	0.285321i	−0.726311	+	0.687366i	$0.758767\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			6.26702i			0.410566i	0.978703	+	0.205283i	$0.0658115\pi$
$233$			6.26702i			0.410566i	−0.978703	+	0.205283i	$0.934189\pi$
$234$		0			0
$235$	1.80943			0.118034
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−10.5082	+	18.2008i	−0.679721	+	1.17731i	0.295344	+	0.955391i	$0.404566\pi$
$239$	−10.5082	+	18.2008i	−0.679721	+	1.17731i	−0.975065	+	0.221920i	$0.928768\pi$
$240$		0			0
$241$	−3.16928	−	5.48935i	−0.204151	−	0.353600i	0.745711	−	0.666270i	$-0.232110\pi$
$241$	−3.16928	−	5.48935i	−0.204151	−	0.353600i	−0.949862	+	0.312670i	$0.898777\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−3.82911	−	6.63221i	−0.244633	−	0.423716i
$246$		0			0
$247$	6.80339	+	3.92794i	0.432889	+	0.249929i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	8.50717i		−	0.536968i	−0.963284	−	0.268484i	$-0.913477\pi$
$251$		−	8.50717i		−	0.536968i	0.963284	−	0.268484i	$-0.0865226\pi$
$252$		0			0
$253$			2.23798i			0.140700i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−5.06622	−	2.92499i	−0.316022	−	0.182456i	0.333596	−	0.942716i	$-0.391738\pi$
$257$	−5.06622	−	2.92499i	−0.316022	−	0.182456i	−0.649618	+	0.760261i	$0.725071\pi$
$258$		0			0
$259$	−11.8636	−	20.5484i	−0.737169	−	1.27681i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−5.53748	−	9.59120i	−0.341456	−	0.591419i	0.643248	−	0.765658i	$-0.277587\pi$
$263$	−5.53748	−	9.59120i	−0.341456	−	0.591419i	−0.984703	+	0.174240i	$0.944253\pi$
$264$		0			0
$265$	−9.42744	+	16.3288i	−0.579123	+	1.00307i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−5.18889			−0.316372			−0.158186	−	0.987409i	$-0.550565\pi$
$269$	−5.18889			−0.316372			−0.158186	+	0.987409i	$0.550565\pi$
$270$		0			0
$271$		−	18.3273i		−	1.11330i	−0.830747	−	0.556651i	$-0.812086\pi$
$271$		−	18.3273i		−	1.11330i	0.830747	−	0.556651i	$-0.187914\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−5.66854	−	3.27273i	−0.341826	−	0.197353i
$276$		0			0
$277$	18.2153	−	10.5166i	1.09445	−	0.631882i	0.159694	−	0.987167i	$-0.448949\pi$
$277$	18.2153	−	10.5166i	1.09445	−	0.631882i	0.934758	+	0.355284i	$0.115616\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−14.9127	+	8.60983i	−0.889615	+	0.513619i	−0.873816	−	0.486256i	$-0.838362\pi$
$281$	−14.9127	+	8.60983i	−0.889615	+	0.513619i	−0.0157981	+	0.999875i	$0.505029\pi$
$282$		0			0
$283$	−4.02298	+	6.96801i	−0.239141	+	0.414205i	−0.960468	−	0.278390i	$-0.910199\pi$
$283$	−4.02298	+	6.96801i	−0.239141	+	0.414205i	0.721327	+	0.692595i	$0.243533\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	23.2681			1.37347
$288$		0			0
$289$	−22.1031			−1.30018
$290$		0			0
$291$		0			0
$292$		0			0
$293$	6.58772	−	11.4103i	0.384859	−	0.666595i	−0.606891	−	0.794785i	$-0.707584\pi$
$293$	6.58772	−	11.4103i	0.384859	−	0.666595i	0.991750	+	0.128190i	$0.0409168\pi$
$294$		0			0
$295$	7.74760	−	4.47308i	0.451083	−	0.260433i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	0.918615	−	0.530362i	0.0531249	−	0.0306717i
$300$		0			0
$301$	19.3935	+	11.1968i	1.11782	+	0.645375i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			25.5525i			1.46313i
$306$		0			0
$307$	−6.45699			−0.368520			−0.184260	−	0.982878i	$-0.558989\pi$
$307$	−6.45699			−0.368520			−0.184260	+	0.982878i	$0.558989\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−5.49720	+	9.52142i	−0.311717	+	0.539910i	−0.978734	−	0.205132i	$-0.934238\pi$
$311$	−5.49720	+	9.52142i	−0.311717	+	0.539910i	0.667017	+	0.745043i	$0.267571\pi$
$312$		0			0
$313$	−0.119404	−	0.206814i	−0.00674913	−	0.0116898i	0.862631	−	0.505834i	$-0.168815\pi$
$313$	−0.119404	−	0.206814i	−0.00674913	−	0.0116898i	−0.869380	+	0.494144i	$0.835482\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	3.03493	+	5.25666i	0.170459	+	0.295243i	0.938580	−	0.345061i	$-0.112142\pi$
$317$	3.03493	+	5.25666i	0.170459	+	0.295243i	−0.768121	+	0.640304i	$0.778808\pi$
$318$		0			0
$319$	−17.8189	−	10.2877i	−0.997667	−	0.576003i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	28.5901i		−	1.59080i
$324$		0			0
$325$			3.10233i			0.172086i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−2.94523	−	1.70043i	−0.162376	−	0.0937477i
$330$		0			0
$331$	10.6301	+	18.4119i	0.584284	+	1.01201i	0.994964	+	0.100231i	$0.0319581\pi$
$331$	10.6301	+	18.4119i	0.584284	+	1.01201i	−0.410680	+	0.911780i	$0.634709\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	8.38526	+	14.5237i	0.458136	+	0.793514i
$336$		0			0
$337$	−7.28561	+	12.6190i	−0.396873	+	0.687403i	−0.993338	−	0.115235i	$-0.963238\pi$
$337$	−7.28561	+	12.6190i	−0.396873	+	0.687403i	0.596466	+	0.802639i	$0.296571\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	29.3464			1.58920
$342$		0			0
$343$		−	9.12121i		−	0.492499i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−20.6647	−	11.9308i	−1.10934	−	0.640477i	−0.170680	−	0.985326i	$-0.554597\pi$
$347$	−20.6647	−	11.9308i	−1.10934	−	0.640477i	−0.938658	+	0.344850i	$0.887930\pi$
$348$		0			0
$349$	18.1752	−	10.4935i	0.972897	−	0.561702i	0.0727786	−	0.997348i	$-0.476813\pi$
$349$	18.1752	−	10.4935i	0.972897	−	0.561702i	0.900118	+	0.435646i	$0.143480\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	2.88786	−	1.66730i	0.153705	−	0.0887416i	−0.421175	−	0.906979i	$-0.638382\pi$
$353$	2.88786	−	1.66730i	0.153705	−	0.0887416i	0.574880	+	0.818238i	$0.305049\pi$
$354$		0			0
$355$	−8.67665	+	15.0284i	−0.460509	+	0.797625i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	24.4127			1.28845			0.644227	−	0.764834i	$-0.277179\pi$
$359$	24.4127			1.28845			0.644227	+	0.764834i	$0.277179\pi$
$360$		0			0
$361$	1.90356			0.100187
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−13.6227	+	23.5952i	−0.713044	+	1.23503i
$366$		0			0
$367$	−2.94984	+	1.70309i	−0.153980	+	0.0889007i	−0.575010	−	0.818146i	$-0.695002\pi$
$367$	−2.94984	+	1.70309i	−0.153980	+	0.0889007i	0.421030	+	0.907047i	$0.361669\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	30.6903	−	17.7190i	1.59336	−	0.919927i
$372$		0			0
$373$	23.8917	+	13.7939i	1.23706	+	0.714219i	0.968493	−	0.249041i	$-0.0801153\pi$
$373$	23.8917	+	13.7939i	1.23706	+	0.714219i	0.268571	+	0.963260i	$0.413449\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			9.75209i			0.502258i
$378$		0			0
$379$	−30.2051			−1.55153			−0.775766	−	0.631021i	$-0.782636\pi$
$379$	−30.2051			−1.55153			−0.775766	+	0.631021i	$0.782636\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	7.92015	−	13.7181i	0.404701	−	0.700962i	−0.589586	−	0.807706i	$-0.700709\pi$
$383$	7.92015	−	13.7181i	0.404701	−	0.700962i	0.994287	+	0.106743i	$0.0340423\pi$
$384$		0			0
$385$	−10.8831	−	18.8501i	−0.554656	−	0.960692i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	3.26398	+	5.65339i	0.165491	+	0.286638i	0.936829	−	0.349787i	$-0.113746\pi$
$389$	3.26398	+	5.65339i	0.165491	+	0.286638i	−0.771339	+	0.636425i	$0.780413\pi$
$390$		0			0
$391$	−3.34314	−	1.93016i	−0.169070	−	0.0976125i
$392$		0			0
$393$		0			0
$394$		0			0
$395$			27.5703i			1.38721i
$396$		0			0
$397$		−	16.9116i		−	0.848767i	−0.905483	−	0.424383i	$-0.860491\pi$
$397$		−	16.9116i		−	0.848767i	0.905483	−	0.424383i	$-0.139509\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	1.89780	+	1.09569i	0.0947715	+	0.0547163i	0.546637	−	0.837370i	$-0.315908\pi$
$401$	1.89780	+	1.09569i	0.0947715	+	0.0547163i	−0.451865	+	0.892086i	$0.649241\pi$
$402$		0			0
$403$	−6.95459	−	12.0457i	−0.346433	−	0.600039i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−12.8029	−	22.1752i	−0.634615	−	1.09919i
$408$		0			0
$409$	6.42658	−	11.1312i	0.317774	−	0.550401i	−0.662249	−	0.749284i	$-0.730398\pi$
$409$	6.42658	−	11.1312i	0.317774	−	0.550401i	0.980023	+	0.198883i	$0.0637313\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−16.8145			−0.827386
$414$		0			0
$415$		−	11.5618i		−	0.567546i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	18.2642	+	10.5448i	0.892262	+	0.515148i	0.874682	−	0.484697i	$-0.161070\pi$
$419$	18.2642	+	10.5448i	0.892262	+	0.515148i	0.0175806	+	0.999845i	$0.494404\pi$
$420$		0			0
$421$	6.71848	−	3.87892i	0.327439	−	0.189047i	−0.327265	−	0.944933i	$-0.606127\pi$
$421$	6.71848	−	3.87892i	0.327439	−	0.189047i	0.654703	+	0.755886i	$0.272794\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	9.77776	−	5.64519i	0.474291	−	0.273832i
$426$		0			0
$427$	24.0132	−	41.5921i	1.16208	−	2.01278i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−15.8966			−0.765712			−0.382856	−	0.923808i	$-0.625059\pi$
$431$	−15.8966			−0.765712			−0.382856	+	0.923808i	$0.625059\pi$
$432$		0			0
$433$	0.620460			0.0298174			0.0149087	−	0.999889i	$-0.495254\pi$
$433$	0.620460			0.0298174			0.0149087	+	0.999889i	$0.495254\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	1.41123	−	2.44432i	0.0675084	−	0.116928i
$438$		0			0
$439$	11.3367	−	6.54525i	0.541071	−	0.312388i	−0.204442	−	0.978879i	$-0.565538\pi$
$439$	11.3367	−	6.54525i	0.541071	−	0.312388i	0.745513	+	0.666491i	$0.232205\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−30.5417	+	17.6333i	−1.45108	+	0.837781i	−0.998543	−	0.0539647i	$-0.982814\pi$
$443$	−30.5417	+	17.6333i	−1.45108	+	0.837781i	−0.452537	+	0.891746i	$0.649481\pi$
$444$		0			0
$445$	−2.25203	−	1.30021i	−0.106757	−	0.0616359i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	2.94608i		−	0.139034i	−0.997581	−	0.0695171i	$-0.977854\pi$
$449$		−	2.94608i		−	0.139034i	0.997581	−	0.0695171i	$-0.0221459\pi$
$450$		0			0
$451$	25.1103			1.18240
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−5.15823	+	8.93432i	−0.241822	+	0.418847i
$456$		0			0
$457$	12.5837	+	21.7957i	0.588642	+	1.01956i	0.994411	+	0.105582i	$0.0336706\pi$
$457$	12.5837	+	21.7957i	0.588642	+	1.01956i	−0.405769	+	0.913976i	$0.632996\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−17.7689	−	30.7766i	−0.827579	−	1.43341i	−0.899932	−	0.436029i	$-0.856384\pi$
$461$	−17.7689	−	30.7766i	−0.827579	−	1.43341i	0.0723537	−	0.997379i	$-0.476949\pi$
$462$		0			0
$463$	26.0948	+	15.0658i	1.21273	+	0.700168i	0.963352	−	0.268239i	$-0.0864416\pi$
$463$	26.0948	+	15.0658i	1.21273	+	0.700168i	0.249375	+	0.968407i	$0.419775\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			8.04243i			0.372159i	0.982535	+	0.186080i	$0.0595782\pi$
$467$			8.04243i			0.372159i	−0.982535	+	0.186080i	$0.940422\pi$
$468$		0			0
$469$		−	31.5205i		−	1.45548i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	20.9289	+	12.0833i	0.962313	+	0.555592i
$474$		0			0
$475$	4.12746	+	7.14897i	0.189381	+	0.328017i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−18.9554	−	32.8317i	−0.866094	−	1.50012i	−0.865957	−	0.500118i	$-0.833290\pi$
$479$	−18.9554	−	32.8317i	−0.866094	−	1.50012i	−0.000136370	−	1.00000i	$-0.500043\pi$
$480$		0			0
$481$	−6.06813	+	10.5103i	−0.276683	+	0.479229i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	10.4724			0.475527
$486$		0			0
$487$			0.286027i			0.0129611i	0.999979	+	0.00648057i	$0.00206284\pi$
$487$			0.286027i			0.0129611i	−0.999979	+	0.00648057i	$0.997937\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	6.74349	+	3.89336i	0.304330	+	0.175705i	0.644386	−	0.764700i	$-0.277113\pi$
$491$	6.74349	+	3.89336i	0.304330	+	0.175705i	−0.340057	+	0.940405i	$0.610446\pi$
$492$		0			0
$493$	30.7361	−	17.7455i	1.38429	−	0.799218i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	28.2461	−	16.3079i	1.26701	−	0.731510i
$498$		0			0
$499$	−11.9068	+	20.6232i	−0.533022	+	0.923222i	0.466234	+	0.884662i	$0.345611\pi$
$499$	−11.9068	+	20.6232i	−0.533022	+	0.923222i	−0.999256	+	0.0385604i	$0.987723\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	11.8583			0.528735			0.264367	−	0.964422i	$-0.414837\pi$
$503$	11.8583			0.528735			0.264367	+	0.964422i	$0.414837\pi$
$504$		0			0
$505$	5.11810			0.227753
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−9.66297	+	16.7368i	−0.428304	+	0.741844i	−0.996723	−	0.0808956i	$-0.974222\pi$
$509$	−9.66297	+	16.7368i	−0.428304	+	0.741844i	0.568419	+	0.822739i	$0.307555\pi$
$510$		0			0
$511$	44.3476	−	25.6041i	1.96182	−	1.13266i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	7.49662	−	4.32818i	0.330341	−	0.190722i
$516$		0			0
$517$	−3.17841	−	1.83506i	−0.139786	−	0.0807058i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	41.6156i		−	1.82321i	−0.411063	−	0.911607i	$-0.634842\pi$
$521$		−	41.6156i		−	1.82321i	0.411063	−	0.911607i	$-0.365158\pi$
$522$		0			0
$523$	−6.05911			−0.264946			−0.132473	−	0.991187i	$-0.542292\pi$
$523$	−6.05911			−0.264946			−0.132473	+	0.991187i	$0.542292\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−25.3100	+	43.8383i	−1.10252	+	1.90962i
$528$		0			0
$529$	11.3095	+	19.5885i	0.491715	+	0.851676i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−5.95072	−	10.3069i	−0.257754	−	0.446443i
$534$		0			0
$535$	16.8161	+	9.70877i	0.727023	+	0.419747i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			15.5334i			0.669069i
$540$		0			0
$541$			17.8716i			0.768360i	0.923258	+	0.384180i	$0.125516\pi$
$541$			17.8716i			0.768360i	−0.923258	+	0.384180i	$0.874484\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	1.37296	+	0.792680i	0.0588113	+	0.0339547i
$546$		0			0
$547$	15.4706	+	26.7959i	0.661476	+	1.14571i	0.980228	+	0.197871i	$0.0634028\pi$
$547$	15.4706	+	26.7959i	0.661476	+	1.14571i	−0.318752	+	0.947838i	$0.603264\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	12.9746	+	22.4726i	0.552735	+	0.957366i
$552$		0			0
$553$	25.9095	−	44.8765i	1.10178	−	1.90834i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	25.4742			1.07938			0.539689	−	0.841864i	$-0.318542\pi$
$557$	25.4742			1.07938			0.539689	+	0.841864i	$0.318542\pi$
$558$		0			0
$559$		−	11.4542i		−	0.484460i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	22.0296	+	12.7188i	0.928437	+	0.536034i	0.886317	−	0.463079i	$-0.153255\pi$
$563$	22.0296	+	12.7188i	0.928437	+	0.536034i	0.0421203	+	0.999113i	$0.486589\pi$
$564$		0			0
$565$	6.46920	−	3.73499i	0.272161	−	0.157132i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−20.5771	+	11.8802i	−0.862638	+	0.498044i	−0.864895	−	0.501953i	$-0.832615\pi$
$569$	−20.5771	+	11.8802i	−0.862638	+	0.498044i	0.00225663	+	0.999997i	$0.499282\pi$
$570$		0			0
$571$	−0.636562	+	1.10256i	−0.0266393	+	0.0461406i	−0.879038	−	0.476752i	$-0.841814\pi$
$571$	−0.636562	+	1.10256i	−0.0266393	+	0.0461406i	0.852398	+	0.522893i	$0.175147\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	1.11461			0.0464823
$576$		0			0
$577$	7.62218			0.317315			0.158658	−	0.987334i	$-0.449283\pi$
$577$	7.62218			0.317315			0.158658	+	0.987334i	$0.449283\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−10.8653	+	18.8193i	−0.450769	+	0.780754i
$582$		0			0
$583$	33.1201	−	19.1219i	1.37170	−	0.791949i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−38.8377	+	22.4230i	−1.60300	+	0.925495i	−0.612122	+	0.790763i	$0.709684\pi$
$587$	−38.8377	+	22.4230i	−1.60300	+	0.925495i	−0.990882	+	0.134732i	$0.956983\pi$
$588$		0			0
$589$	−32.0522	−	18.5053i	−1.32069	−	0.762499i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	21.3237i		−	0.875658i	−0.899058	−	0.437829i	$-0.855748\pi$
$593$		−	21.3237i		−	0.875658i	0.899058	−	0.437829i	$-0.144252\pi$
$594$		0			0
$595$	37.5450			1.53919
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−0.992117	+	1.71840i	−0.0405368	+	0.0702118i	−0.885582	−	0.464483i	$-0.846240\pi$
$599$	−0.992117	+	1.71840i	−0.0405368	+	0.0702118i	0.845045	+	0.534695i	$0.179573\pi$
$600$		0			0
$601$	−4.78811	−	8.29324i	−0.195311	−	0.338288i	0.751692	−	0.659515i	$-0.229238\pi$
$601$	−4.78811	−	8.29324i	−0.195311	−	0.338288i	−0.947002	+	0.321227i	$0.895905\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−1.91457	−	3.31614i	−0.0778384	−	0.134820i
$606$		0			0
$607$	26.6428	+	15.3822i	1.08140	+	0.624345i	0.931273	−	0.364322i	$-0.118699\pi$
$607$	26.6428	+	15.3822i	1.08140	+	0.624345i	0.150125	+	0.988667i	$0.452033\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			1.73951i			0.0703731i
$612$		0			0
$613$		−	20.1482i		−	0.813777i	−0.913478	−	0.406889i	$-0.866614\pi$
$613$		−	20.1482i		−	0.813777i	0.913478	−	0.406889i	$-0.133386\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−24.4308	−	14.1051i	−0.983547	−	0.567851i	−0.0802077	−	0.996778i	$-0.525558\pi$
$617$	−24.4308	−	14.1051i	−0.983547	−	0.567851i	−0.903339	+	0.428927i	$0.858892\pi$
$618$		0			0
$619$	−12.0197	−	20.8187i	−0.483111	−	0.836773i	0.516701	−	0.856166i	$-0.327160\pi$
$619$	−12.0197	−	20.8187i	−0.483111	−	0.836773i	−0.999812	+	0.0193928i	$0.993827\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	2.44377	+	4.23274i	0.0979076	+	0.169581i
$624$		0			0
$625$	6.35624	−	11.0093i	0.254249	−	0.440373i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	44.1678			1.76109
$630$		0			0
$631$		−	28.1191i		−	1.11940i	−0.828694	−	0.559701i	$-0.810916\pi$
$631$		−	28.1191i		−	1.11940i	0.828694	−	0.559701i	$-0.189084\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	17.1555	+	9.90474i	0.680796	+	0.393058i
$636$		0			0
$637$	6.37592	−	3.68114i	0.252623	−	0.145852i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	8.98279	−	5.18622i	0.354799	−	0.204843i	−0.311998	−	0.950083i	$-0.600998\pi$
$641$	8.98279	−	5.18622i	0.354799	−	0.204843i	0.666797	+	0.745239i	$0.267665\pi$
$642$		0			0
$643$	−9.55211	+	16.5447i	−0.376698	+	0.652461i	−0.990580	−	0.136938i	$-0.956274\pi$
$643$	−9.55211	+	16.5447i	−0.376698	+	0.652461i	0.613881	+	0.789398i	$0.289607\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−3.61125			−0.141973			−0.0709864	−	0.997477i	$-0.522615\pi$
$647$	−3.61125			−0.141973			−0.0709864	+	0.997477i	$0.522615\pi$
$648$		0			0
$649$	−18.1457			−0.712282
$650$		0			0
$651$		0			0
$652$		0			0
$653$	1.18998	−	2.06111i	0.0465675	−	0.0806574i	−0.841802	−	0.539786i	$-0.818505\pi$
$653$	1.18998	−	2.06111i	0.0465675	−	0.0806574i	0.888370	+	0.459129i	$0.151838\pi$
$654$		0			0
$655$	14.8722	−	8.58646i	0.581104	−	0.335501i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	37.0796	−	21.4079i	1.44442	−	0.833934i	0.446276	−	0.894895i	$-0.352750\pi$
$659$	37.0796	−	21.4079i	1.44442	−	0.833934i	0.998140	+	0.0609613i	$0.0194166\pi$
$660$		0			0
$661$	30.7637	+	17.7614i	1.19657	+	0.690839i	0.959788	−	0.280724i	$-0.0905747\pi$
$661$	30.7637	+	17.7614i	1.19657	+	0.690839i	0.236780	+	0.971563i	$0.423908\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			27.4509i			1.06450i
$666$		0			0
$667$	3.50373			0.135665
$668$		0			0
$669$		0			0
$670$		0			0
$671$	25.9144	−	44.8851i	1.00042	−	1.73277i
$672$		0			0
$673$	8.55684	+	14.8209i	0.329842	+	0.571303i	0.982480	−	0.186367i	$-0.0596713\pi$
$673$	8.55684	+	14.8209i	0.329842	+	0.571303i	−0.652638	+	0.757669i	$0.726338\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−1.09679	−	1.89969i	−0.0421530	−	0.0730111i	0.844179	−	0.536061i	$-0.180088\pi$
$677$	−1.09679	−	1.89969i	−0.0421530	−	0.0730111i	−0.886332	+	0.463050i	$0.846755\pi$
$678$		0			0
$679$	−17.0460	−	9.84153i	−0.654167	−	0.377683i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			10.3775i			0.397083i	0.980092	+	0.198542i	$0.0636205\pi$
$683$			10.3775i			0.397083i	−0.980092	+	0.198542i	$0.936380\pi$
$684$		0			0
$685$		−	15.1554i		−	0.579057i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−15.6978	−	9.06314i	−0.598039	−	0.345278i
$690$		0			0
$691$	−19.4597	−	33.7053i	−0.740284	−	1.28221i	−0.952366	−	0.304957i	$-0.901358\pi$
$691$	−19.4597	−	33.7053i	−0.740284	−	1.28221i	0.212082	−	0.977252i	$-0.431975\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−8.64647	−	14.9761i	−0.327979	−	0.568077i
$696$		0			0
$697$	−21.6566	+	37.5103i	−0.820303	+	1.42081i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−26.9799			−1.01902			−0.509509	−	0.860465i	$-0.670173\pi$
$701$	−26.9799			−1.01902			−0.509509	+	0.860465i	$0.670173\pi$
$702$		0			0
$703$			32.2932i			1.21796i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−8.33079	−	4.80978i	−0.313312	−	0.180891i
$708$		0			0
$709$	−29.8798	+	17.2511i	−1.12216	+	0.647878i	−0.941951	−	0.335750i	$-0.891010\pi$
$709$	−29.8798	+	17.2511i	−1.12216	+	0.647878i	−0.180207	+	0.983629i	$0.557677\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−4.32779	+	2.49865i	−0.162077	+	0.0935751i
$714$		0			0
$715$	−5.56662	+	9.64168i	−0.208180	+	0.360578i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	34.7134			1.29459			0.647296	−	0.762239i	$-0.275900\pi$
$719$	34.7134			1.29459			0.647296	+	0.762239i	$0.275900\pi$
$720$		0			0
$721$	−16.2698			−0.605918
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−5.12373	+	8.87455i	−0.190290	+	0.329593i
$726$		0			0
$727$	−15.2326	+	8.79455i	−0.564946	+	0.326172i	−0.755128	−	0.655577i	$-0.772426\pi$
$727$	−15.2326	+	8.79455i	−0.564946	+	0.326172i	0.190182	+	0.981749i	$0.439092\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−36.1007	+	20.8427i	−1.33523	+	0.770896i
$732$		0			0
$733$	14.4793	+	8.35960i	0.534803	+	0.308769i	0.742970	−	0.669324i	$-0.233416\pi$
$733$	14.4793	+	8.35960i	0.534803	+	0.308769i	−0.208167	+	0.978093i	$0.566750\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	34.0161i		−	1.25300i
$738$		0			0
$739$	20.6373			0.759154			0.379577	−	0.925160i	$-0.376070\pi$
$739$	20.6373			0.759154			0.379577	+	0.925160i	$0.376070\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−10.6388	+	18.4270i	−0.390301	+	0.676021i	−0.992489	−	0.122333i	$-0.960962\pi$
$743$	−10.6388	+	18.4270i	−0.390301	+	0.676021i	0.602188	+	0.798354i	$0.294296\pi$
$744$		0			0
$745$	9.95429	+	17.2413i	0.364697	+	0.631674i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−18.2478	−	31.6061i	−0.666760	−	1.15486i
$750$		0			0
$751$	25.3009	+	14.6075i	0.923244	+	0.533035i	0.884668	−	0.466221i	$-0.154385\pi$
$751$	25.3009	+	14.6075i	0.923244	+	0.533035i	0.0385752	+	0.999256i	$0.487718\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			8.92849i			0.324941i
$756$		0			0
$757$		−	48.0411i		−	1.74608i	−0.487644	−	0.873042i	$-0.662144\pi$
$757$		−	48.0411i		−	1.74608i	0.487644	−	0.873042i	$-0.337856\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−31.4120	−	18.1357i	−1.13869	−	0.657420i	−0.192581	−	0.981281i	$-0.561686\pi$
$761$	−31.4120	−	18.1357i	−1.13869	−	0.657420i	−0.946105	+	0.323861i	$0.895019\pi$
$762$		0			0
$763$	−1.48986	−	2.58051i	−0.0539364	−	0.0934207i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	4.30023	+	7.44821i	0.155272	+	0.268939i
$768$		0			0
$769$	16.2620	−	28.1667i	0.586424	−	1.01572i	−0.408272	−	0.912860i	$-0.633868\pi$
$769$	16.2620	−	28.1667i	0.586424	−	1.01572i	0.994696	−	0.102857i	$-0.0327983\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	29.4139			1.05794			0.528972	−	0.848639i	$-0.322578\pi$
$773$	29.4139			1.05794			0.528972	+	0.848639i	$0.322578\pi$
$774$		0			0
$775$		−	14.6157i		−	0.525012i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−27.4256	−	15.8342i	−0.982623	−	0.567318i
$780$		0			0
$781$	30.4825	−	17.5991i	1.09075	−	0.629744i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−13.3300	+	7.69609i	−0.475769	+	0.274685i
$786$		0			0
$787$	−23.1068	+	40.0222i	−0.823670	+	1.42664i	0.0792616	+	0.996854i	$0.474744\pi$
$787$	−23.1068	+	40.0222i	−0.823670	+	1.42664i	−0.902932	+	0.429784i	$0.858590\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−14.0400			−0.499204
$792$		0			0
$793$	−24.5651			−0.872333
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−9.28549	+	16.0829i	−0.328909	+	0.569687i	−0.982296	−	0.187337i	$-0.940014\pi$
$797$	−9.28549	+	16.0829i	−0.328909	+	0.569687i	0.653387	+	0.757024i	$0.273348\pi$
$798$		0			0
$799$	5.48250	−	3.16532i	0.193957	−	0.111981i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	47.8587	−	27.6312i	1.68890	−	0.975086i
$804$		0			0
$805$	3.20993	+	1.85325i	0.113135	+	0.0653185i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	22.3374i		−	0.785340i	−0.919679	−	0.392670i	$-0.871551\pi$
$809$		−	22.3374i		−	0.785340i	0.919679	−	0.392670i	$-0.128449\pi$
$810$		0			0
$811$	−44.5840			−1.56556			−0.782778	−	0.622301i	$-0.786198\pi$
$811$	−44.5840			−1.56556			−0.782778	+	0.622301i	$0.786198\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	1.83039	−	3.17032i	0.0641156	−	0.111051i
$816$		0			0
$817$	−15.2391	−	26.3949i	−0.533148	−	0.923440i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	15.1993	+	26.3260i	0.530459	+	0.918783i	0.999368	+	0.0355362i	$0.0113139\pi$
$821$	15.1993	+	26.3260i	0.530459	+	0.918783i	−0.468909	+	0.883246i	$0.655353\pi$
$822$		0			0
$823$	−12.5306	−	7.23454i	−0.436789	−	0.252180i	0.265446	−	0.964126i	$-0.414481\pi$
$823$	−12.5306	−	7.23454i	−0.436789	−	0.252180i	−0.702235	+	0.711946i	$0.747814\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			52.0725i			1.81074i	0.424627	+	0.905368i	$0.360405\pi$
$827$			52.0725i			1.81074i	−0.424627	+	0.905368i	$0.639595\pi$
$828$		0			0
$829$			4.34421i			0.150881i	0.997150	+	0.0754404i	$0.0240363\pi$
$829$			4.34421i			0.150881i	−0.997150	+	0.0754404i	$0.975964\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−23.2041	−	13.3969i	−0.803973	−	0.464174i
$834$		0			0
$835$	−18.7725	−	32.5149i	−0.649649	−	1.12522i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−22.9240	−	39.7055i	−0.791424	−	1.37079i	−0.925086	−	0.379759i	$-0.876007\pi$
$839$	−22.9240	−	39.7055i	−0.791424	−	1.37079i	0.133662	−	0.991027i	$-0.457326\pi$
$840$		0			0
$841$	−1.60630	+	2.78219i	−0.0553896	+	0.0959376i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−17.9582			−0.617783
$846$		0			0
$847$			7.19695i			0.247290i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	3.77615	+	2.18016i	0.129445	+	0.0747349i
$852$		0			0
$853$	44.1912	−	25.5138i	1.51308	−	0.873577i	0.513196	−	0.858271i	$-0.328461\pi$
$853$	44.1912	−	25.5138i	1.51308	−	0.873577i	0.999883	−	0.0153054i	$-0.00487206\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	45.5385	−	26.2917i	1.55557	−	0.898107i	0.557894	−	0.829912i	$-0.311609\pi$
$857$	45.5385	−	26.2917i	1.55557	−	0.898107i	0.997672	−	0.0681949i	$-0.0217240\pi$
$858$		0			0
$859$	8.00975	−	13.8733i	0.273289	−	0.473351i	−0.696413	−	0.717641i	$-0.745222\pi$
$859$	8.00975	−	13.8733i	0.273289	−	0.473351i	0.969702	+	0.244291i	$0.0785550\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−25.3104			−0.861575			−0.430788	−	0.902453i	$-0.641764\pi$
$863$	−25.3104			−0.861575			−0.430788	+	0.902453i	$0.641764\pi$
$864$		0			0
$865$	12.1884			0.414417
$866$		0			0
$867$		0			0
$868$		0			0
$869$	27.9608	−	48.4295i	0.948505	−	1.64286i
$870$		0			0
$871$	−13.9625	+	8.06123i	−0.473100	+	0.273144i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−35.3866	+	20.4304i	−1.19628	+	0.690675i
$876$		0			0
$877$	2.73500	+	1.57905i	0.0923543	+	0.0533208i	0.545466	−	0.838133i	$-0.316353\pi$
$877$	2.73500	+	1.57905i	0.0923543	+	0.0533208i	−0.453112	+	0.891454i	$0.649686\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			17.1113i			0.576493i	0.957556	+	0.288246i	$0.0930722\pi$
$881$			17.1113i			0.576493i	−0.957556	+	0.288246i	$0.906928\pi$
$882$		0			0
$883$	19.0162			0.639945			0.319972	−	0.947427i	$-0.396326\pi$
$883$	19.0162			0.639945			0.319972	+	0.947427i	$0.396326\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−19.3348	+	33.4888i	−0.649199	+	1.12445i	0.334116	+	0.942532i	$0.391562\pi$
$887$	−19.3348	+	33.4888i	−0.649199	+	1.12445i	−0.983315	+	0.181913i	$0.941771\pi$
$888$		0			0
$889$	−18.6161	−	32.2441i	−0.624366	−	1.08143i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	2.31432	+	4.00851i	0.0774456	+	0.134140i
$894$		0			0
$895$	−34.8421	−	20.1161i	−1.16464	−	0.672406i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	45.9441i		−	1.53232i
$900$		0			0
$901$			65.9674i			2.19769i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	13.0515	+	7.53528i	0.433846	+	0.250481i
$906$		0			0
$907$	22.0826	+	38.2482i	0.733241	+	1.27001i	0.955491	+	0.295021i	$0.0953267\pi$
$907$	22.0826	+	38.2482i	0.733241	+	1.27001i	−0.222250	+	0.974990i	$0.571340\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	15.0699	+	26.1019i	0.499290	+	0.864795i	1.00000		0.000820108i	$-0.000261049\pi$
$911$	15.0699	+	26.1019i	0.499290	+	0.864795i	−0.500710	+	0.865615i	$0.666928\pi$
$912$		0			0
$913$	−11.7255	+	20.3092i	−0.388059	+	0.672137i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−32.2768			−1.06587
$918$		0			0
$919$		−	20.7713i		−	0.685183i	−0.939485	−	0.342591i	$-0.888695\pi$
$919$		−	20.7713i		−	0.685183i	0.939485	−	0.342591i	$-0.111305\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−14.4477	−	8.34136i	−0.475550	−	0.274559i
$924$		0			0
$925$	−11.0442	+	6.37637i	−0.363131	+	0.209654i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−24.6970	+	14.2588i	−0.810282	+	0.467816i	−0.847054	−	0.531507i	$-0.821626\pi$
$929$	−24.6970	+	14.2588i	−0.810282	+	0.467816i	0.0367720	+	0.999324i	$0.488292\pi$
$930$		0			0
$931$	9.79508	−	16.9656i	0.321021	−	0.556024i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	40.5175			1.32506
$936$		0			0
$937$	8.91394			0.291206			0.145603	−	0.989343i	$-0.453488\pi$
$937$	8.91394			0.291206			0.145603	+	0.989343i	$0.453488\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−6.27248	+	10.8643i	−0.204477	+	0.354165i	−0.949966	−	0.312353i	$-0.898883\pi$
$941$	−6.27248	+	10.8643i	−0.204477	+	0.354165i	0.745489	+	0.666518i	$0.232216\pi$
$942$		0			0
$943$	−3.70308	+	2.13798i	−0.120589	+	0.0696221i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−17.8198	+	10.2883i	−0.579066	+	0.334324i	−0.760762	−	0.649031i	$-0.775175\pi$
$947$	−17.8198	+	10.2883i	−0.579066	+	0.334324i	0.181696	+	0.983355i	$0.441841\pi$
$948$		0			0
$949$	−22.6834	−	13.0963i	−0.736335	−	0.425123i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			35.3391i			1.14475i	0.819993	+	0.572374i	$0.193977\pi$
$953$			35.3391i			1.14475i	−0.819993	+	0.572374i	$0.806023\pi$
$954$		0			0
$955$	−10.8679			−0.351676
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−14.2424	+	24.6686i	−0.459911	+	0.796589i
$960$		0			0
$961$	17.2645	+	29.9031i	0.556921	+	0.964615i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−0.893655	−	1.54786i	−0.0287678	−	0.0498272i
$966$		0			0
$967$	−17.4196	−	10.0572i	−0.560176	−	0.323418i	0.193040	−	0.981191i	$-0.438165\pi$
$967$	−17.4196	−	10.0572i	−0.560176	−	0.323418i	−0.753216	+	0.657773i	$0.771499\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			2.49582i			0.0800947i	0.999198	+	0.0400474i	$0.0127509\pi$
$971$			2.49582i			0.0800947i	−0.999198	+	0.0400474i	$0.987249\pi$
$972$		0			0
$973$			32.5024i			1.04198i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	53.8526	+	31.0918i	1.72290	+	0.994715i	0.912786	+	0.408439i	$0.133927\pi$
$977$	53.8526	+	31.0918i	1.72290	+	0.994715i	0.810111	+	0.586276i	$0.199407\pi$
$978$		0			0
$979$	2.63725	+	4.56785i	0.0842869	+	0.145989i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	11.9377	+	20.6767i	0.380754	+	0.659485i	0.991170	−	0.132596i	$-0.0423312\pi$
$983$	11.9377	+	20.6767i	0.380754	+	0.659485i	−0.610417	+	0.792081i	$0.708998\pi$
$984$		0			0
$985$	−20.0984	+	34.8114i	−0.640387	+	1.10918i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−4.11526			−0.130858
$990$		0			0
$991$			2.22210i			0.0705873i	0.999377	+	0.0352937i	$0.0112367\pi$
$991$			2.22210i			0.0705873i	−0.999377	+	0.0352937i	$0.988763\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	15.2289	+	8.79242i	0.482789	+	0.278738i
$996$		0			0
$997$	−32.3781	+	18.6935i	−1.02543	+	0.592030i	−0.915671	−	0.401929i	$-0.868340\pi$
$997$	−32.3781	+	18.6935i	−1.02543	+	0.592030i	−0.109755	+	0.993959i	$0.535007\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2592.2.p.g.2159.20		48
3.2	odd	2	inner	2592.2.p.g.2159.6		48
4.3	odd	2		648.2.l.g.539.17		48
8.3	odd	2	inner	2592.2.p.g.2159.5		48
8.5	even	2		648.2.l.g.539.10		48
9.2	odd	6	inner	2592.2.p.g.431.5		48
9.4	even	3		2592.2.f.c.1295.5		24
9.5	odd	6		2592.2.f.c.1295.19		24
9.7	even	3	inner	2592.2.p.g.431.19		48
12.11	even	2		648.2.l.g.539.8		48
24.5	odd	2		648.2.l.g.539.15		48
24.11	even	2	inner	2592.2.p.g.2159.19		48
36.7	odd	6		648.2.l.g.107.15		48
36.11	even	6		648.2.l.g.107.10		48
36.23	even	6		648.2.f.c.323.23	yes	24
36.31	odd	6		648.2.f.c.323.2	yes	24
72.5	odd	6		648.2.f.c.323.1	✓	24
72.11	even	6	inner	2592.2.p.g.431.20		48
72.13	even	6		648.2.f.c.323.24	yes	24
72.29	odd	6		648.2.l.g.107.17		48
72.43	odd	6	inner	2592.2.p.g.431.6		48
72.59	even	6		2592.2.f.c.1295.6		24
72.61	even	6		648.2.l.g.107.8		48
72.67	odd	6		2592.2.f.c.1295.20		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.2.f.c.323.1	✓	24	72.5	odd	6
648.2.f.c.323.2	yes	24	36.31	odd	6
648.2.f.c.323.23	yes	24	36.23	even	6
648.2.f.c.323.24	yes	24	72.13	even	6
648.2.l.g.107.8		48	72.61	even	6
648.2.l.g.107.10		48	36.11	even	6
648.2.l.g.107.15		48	36.7	odd	6
648.2.l.g.107.17		48	72.29	odd	6
648.2.l.g.539.8		48	12.11	even	2
648.2.l.g.539.10		48	8.5	even	2
648.2.l.g.539.15		48	24.5	odd	2
648.2.l.g.539.17		48	4.3	odd	2
2592.2.f.c.1295.5		24	9.4	even	3
2592.2.f.c.1295.6		24	72.59	even	6
2592.2.f.c.1295.19		24	9.5	odd	6
2592.2.f.c.1295.20		24	72.67	odd	6
2592.2.p.g.431.5		48	9.2	odd	6	inner
2592.2.p.g.431.6		48	72.43	odd	6	inner
2592.2.p.g.431.19		48	9.7	even	3	inner
2592.2.p.g.431.20		48	72.11	even	6	inner
2592.2.p.g.2159.5		48	8.3	odd	2	inner
2592.2.p.g.2159.6		48	3.2	odd	2	inner
2592.2.p.g.2159.19		48	24.11	even	2	inner
2592.2.p.g.2159.20		48	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2592 = 2^{5} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2592.p (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.893655 - 1.54786i) q^{5} +(-2.90922 + 1.67964i) q^{7} +O(q^{10})\)	\(q+(0.893655 - 1.54786i) q^{5} +(-2.90922 + 1.67964i) q^{7} +(-3.13955 + 1.81262i) q^{11} +(1.48804 + 0.859122i) q^{13} -6.25325i q^{17} +4.57204 q^{19} +(0.308666 - 0.534624i) q^{23} +(0.902761 + 1.56363i) q^{25} +(2.83781 + 4.91523i) q^{29} +(-7.01048 - 4.04750i) q^{31} +6.00408i q^{35} +7.06318i q^{37} +(-5.99854 - 3.46326i) q^{41} +(-3.33310 - 5.77310i) q^{43} +(0.506189 + 0.876744i) q^{47} +(2.14239 - 3.71072i) q^{49} -10.5493 q^{53} +6.47944i q^{55} +(4.33478 + 2.50269i) q^{59} +(-12.3813 + 7.14832i) q^{61} +(2.65959 - 1.53552i) q^{65} +(-4.69155 + 8.12601i) q^{67} -9.70917 q^{71} -15.2438 q^{73} +(6.08911 - 10.5466i) q^{77} +(-13.3590 + 7.71280i) q^{79} +(5.60217 - 3.23441i) q^{83} +(-9.67913 - 5.58825i) q^{85} -1.45494i q^{89} -5.77206 q^{91} +(4.08583 - 7.07686i) q^{95} +(2.92965 + 5.07431i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q+O(q^{10}) \) 48 * q	\( 48 q - 24 q^{25} + 24 q^{49} + 48 q^{67}+O(q^{100}) \) 48 * q - 24 * q^25 + 24 * q^49 + 48 * q^67

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