LMFDB - Embedded newform 2592.2.p.g.431.19

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			431.19
Character	$\chi$	$=$	2592.431
Dual form			2592.2.p.g.2159.19

$q$-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 $\left(\mathbb{Z}/2592\mathbb{Z}\right)^\times$.

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

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	0.993683	−	0.112222i	$-0.0357967\pi$
$5$	0.893655	+	1.54786i	0.399655	+	0.692222i	−0.594029	+	0.804444i	$0.702463\pi$
$6$		0			0
$7$	2.90922	+	1.67964i	1.09958	+	0.634845i	0.936111	−	0.351703i	$-0.114397\pi$
$7$	2.90922	+	1.67964i	1.09958	+	0.634845i	0.163472	+	0.986548i	$0.447731\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	3.13955	+	1.81262i	0.946611	+	0.546526i	0.892027	−	0.451983i	$-0.149283\pi$
$11$	3.13955	+	1.81262i	0.946611	+	0.546526i	0.0545848	+	0.998509i	$0.482616\pi$
$12$		0			0
$13$	−1.48804	+	0.859122i	−0.412709	+	0.238277i	−0.691953	−	0.721943i	$-0.743249\pi$
$13$	−1.48804	+	0.859122i	−0.412709	+	0.238277i	0.279244	+	0.960220i	$0.409916\pi$
$14$		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	0.896410	−	0.443225i	$-0.146166\pi$
$23$	0.308666	+	0.534624i	0.0643612	+	0.111477i	−0.832049	+	0.554702i	$0.812832\pi$
$24$		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	−0.472539	−	0.881310i	$-0.656662\pi$
$29$	2.83781	−	4.91523i	0.526968	−	0.912735i	0.999506	−	0.0314246i	$-0.0100044\pi$
$30$		0			0
$31$	7.01048	−	4.04750i	1.25912	−	0.726953i	0.286216	−	0.958165i	$-0.407603\pi$
$31$	7.01048	−	4.04750i	1.25912	−	0.726953i	0.972903	+	0.231212i	$0.0742692\pi$
$32$		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	$-0.484762\pi$
$41$	5.99854	−	3.46326i	0.936814	−	0.540870i	0.888960	+	0.457984i	$0.151428\pi$
$42$		0			0
$43$	−3.33310	+	5.77310i	−0.508294	+	0.880390i	0.491660	+	0.870787i	$0.336390\pi$
$43$	−3.33310	+	5.77310i	−0.508294	+	0.880390i	−0.999954	+	0.00960315i	$0.996943\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	0.506189	−	0.876744i	0.0738352	−	0.127886i	−0.826744	−	0.562578i	$-0.809809\pi$
$47$	0.506189	−	0.876744i	0.0738352	−	0.127886i	0.900579	+	0.434692i	$0.143143\pi$
$48$		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	$-0.772308\pi$
$59$	−4.33478	+	2.50269i	−0.564341	+	0.325822i	0.190545	+	0.981678i	$0.438974\pi$
$60$		0			0
$61$	12.3813	+	7.14832i	1.58526	+	0.915249i	0.994073	+	0.108713i	$0.0346729\pi$
$61$	12.3813	+	7.14832i	1.58526	+	0.915249i	0.591185	+	0.806536i	$0.298660\pi$
$62$		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	−0.996238	−	0.0866550i	$-0.972382\pi$
$67$	−4.69155	−	8.12601i	−0.573165	−	0.992750i	0.423074	−	0.906095i	$-0.360951\pi$
$68$		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	0.999994	+	0.00347484i	$0.00110608\pi$
$79$	13.3590	+	7.71280i	1.50300	+	0.867758i	0.503006	+	0.864283i	$0.332227\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−5.60217	−	3.23441i	−0.614918	−	0.355023i	0.159970	−	0.987122i	$-0.448860\pi$
$83$	−5.60217	−	3.23441i	−0.614918	−	0.355023i	−0.774888	+	0.632099i	$0.782194\pi$
$84$		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	−0.678093	−	0.734976i	$-0.737193\pi$
$97$	2.92965	−	5.07431i	0.297461	−	0.515218i	0.975554	+	0.219758i	$0.0705267\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	1.43179	−	2.47993i	0.142468	−	0.246762i	−0.785957	−	0.618281i	$-0.787829\pi$
$101$	1.43179	−	2.47993i	0.142468	−	0.246762i	0.928426	+	0.371518i	$0.121163\pi$
$102$		0			0
$103$	−4.19436	+	2.42161i	−0.413283	+	0.238609i	−0.692199	−	0.721707i	$-0.743358\pi$
$103$	−4.19436	+	2.42161i	−0.413283	+	0.238609i	0.278917	+	0.960315i	$0.410025\pi$
$104$		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	$-0.729652\pi$
$113$	−3.61952	+	2.08973i	−0.340496	+	0.196585i	0.319996	+	0.947419i	$0.396319\pi$
$114$		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	$-0.804545\pi$
$131$	−8.32098	+	4.80412i	−0.727008	+	0.419738i	0.0903187	+	0.995913i	$0.471211\pi$
$132$		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	$-0.451316\pi$
$137$	−7.34340	−	4.23972i	−0.627389	−	0.362223i	−0.779740	+	0.626103i	$0.784649\pi$
$138$		0			0
$139$	4.83770	+	8.37914i	0.410328	+	0.710710i	0.994926	−	0.100614i	$-0.0320807\pi$
$139$	4.83770	+	8.37914i	0.410328	+	0.710710i	−0.584597	+	0.811324i	$0.698747\pi$
$140$		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	0.542495	−	0.840059i	$-0.317480\pi$
$149$	−5.56943	−	9.64653i	−0.456265	−	0.790275i	−0.998760	+	0.0497846i	$0.984147\pi$
$150$		0			0
$151$	4.32622	+	2.49775i	0.352063	+	0.203264i	0.665593	−	0.746314i	$-0.268178\pi$
$151$	4.32622	+	2.49775i	0.352063	+	0.203264i	−0.313531	+	0.949578i	$0.601512\pi$
$152$		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	$-0.555002\pi$
$157$	7.45814	−	4.30596i	0.595225	−	0.343653i	0.767161	+	0.641455i	$0.221669\pi$
$158$		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	0.910923	+	0.412576i	$0.135371\pi$
$167$	10.5032	+	18.1921i	0.812763	+	1.40775i	−0.0981605	+	0.995171i	$0.531296\pi$
$168$		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	−0.706803	−	0.707411i	$-0.749863\pi$
$173$	3.40970	−	5.90577i	0.259234	−	0.449007i	0.966037	+	0.258404i	$0.0831964\pi$
$174$		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	−0.954804	−	0.297237i	$-0.903935\pi$
$191$	−3.04029	+	5.26593i	−0.219987	+	0.381029i	0.734816	+	0.678266i	$0.237268\pi$
$192$		0			0
$193$	0.500000	+	0.866025i	0.0359908	+	0.0623379i	0.883460	−	0.468507i	$-0.155208\pi$
$193$	0.500000	+	0.866025i	0.0359908	+	0.0623379i	−0.847469	+	0.530845i	$0.821875\pi$
$194$		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	0.996032	−	0.0889981i	$-0.0283665\pi$
$211$	6.11453	+	10.5907i	0.420941	+	0.729092i	−0.575090	+	0.818090i	$0.695033\pi$
$212$		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	$-0.263949\pi$
$223$	5.59348	+	3.22940i	0.374567	+	0.216256i	−0.300885	+	0.953661i	$0.597282\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−10.8063	−	6.23903i	−0.717241	−	0.414099i	0.0964955	−	0.995333i	$-0.469237\pi$
$227$	−10.8063	−	6.23903i	−0.717241	−	0.414099i	−0.813736	+	0.581234i	$0.802570\pi$
$228$		0			0
$229$	7.47846	−	4.31769i	0.494190	−	0.285321i	−0.232121	−	0.972687i	$-0.574566\pi$
$229$	7.47846	−	4.31769i	0.494190	−	0.285321i	0.726311	+	0.687366i	$0.241233\pi$
$230$		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	−0.975065	−	0.221920i	$-0.928768\pi$
$239$	−10.5082	−	18.2008i	−0.679721	−	1.17731i	0.295344	−	0.955391i	$-0.404566\pi$
$240$		0			0
$241$	−3.16928	+	5.48935i	−0.204151	+	0.353600i	−0.949862	−	0.312670i	$-0.898777\pi$
$241$	−3.16928	+	5.48935i	−0.204151	+	0.353600i	0.745711	+	0.666270i	$0.232110\pi$
$242$		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	$-0.608262\pi$
$257$	5.06622	−	2.92499i	0.316022	−	0.182456i	0.649618	+	0.760261i	$0.274929\pi$
$258$		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	−0.984703	−	0.174240i	$-0.944253\pi$
$263$	−5.53748	+	9.59120i	−0.341456	+	0.591419i	0.643248	+	0.765658i	$0.277587\pi$
$264$		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	$-0.551051\pi$
$277$	−18.2153	−	10.5166i	−1.09445	−	0.631882i	−0.934758	+	0.355284i	$0.884384\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	14.9127	+	8.60983i	0.889615	+	0.513619i	0.873816	−	0.486256i	$-0.161638\pi$
$281$	14.9127	+	8.60983i	0.889615	+	0.513619i	0.0157981	+	0.999875i	$0.494971\pi$
$282$		0			0
$283$	−4.02298	−	6.96801i	−0.239141	−	0.414205i	0.721327	−	0.692595i	$-0.243533\pi$
$283$	−4.02298	−	6.96801i	−0.239141	−	0.414205i	−0.960468	+	0.278390i	$0.910199\pi$
$284$		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	0.991750	−	0.128190i	$-0.0409168\pi$
$293$	6.58772	+	11.4103i	0.384859	+	0.666595i	−0.606891	+	0.794785i	$0.707584\pi$
$294$		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	0.667017	−	0.745043i	$-0.267571\pi$
$311$	−5.49720	−	9.52142i	−0.311717	−	0.539910i	−0.978734	+	0.205132i	$0.934238\pi$
$312$		0			0
$313$	−0.119404	+	0.206814i	−0.00674913	+	0.0116898i	−0.869380	−	0.494144i	$-0.835482\pi$
$313$	−0.119404	+	0.206814i	−0.00674913	+	0.0116898i	0.862631	+	0.505834i	$0.168815\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	3.03493	−	5.25666i	0.170459	−	0.295243i	−0.768121	−	0.640304i	$-0.778808\pi$
$317$	3.03493	−	5.25666i	0.170459	−	0.295243i	0.938580	+	0.345061i	$0.112142\pi$
$318$		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	−0.410680	−	0.911780i	$-0.634709\pi$
$331$	10.6301	−	18.4119i	0.584284	−	1.01201i	0.994964	−	0.100231i	$-0.0319581\pi$
$332$		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	0.596466	−	0.802639i	$-0.296571\pi$
$337$	−7.28561	−	12.6190i	−0.396873	−	0.687403i	−0.993338	+	0.115235i	$0.963238\pi$
$338$		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	$-0.445403\pi$
$347$	20.6647	−	11.9308i	1.10934	−	0.640477i	0.938658	+	0.344850i	$0.112070\pi$
$348$		0			0
$349$	−18.1752	−	10.4935i	−0.972897	−	0.561702i	−0.0727786	−	0.997348i	$-0.523187\pi$
$349$	−18.1752	−	10.4935i	−0.972897	−	0.561702i	−0.900118	+	0.435646i	$0.856520\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−2.88786	−	1.66730i	−0.153705	−	0.0887416i	0.421175	−	0.906979i	$-0.361618\pi$
$353$	−2.88786	−	1.66730i	−0.153705	−	0.0887416i	−0.574880	+	0.818238i	$0.694951\pi$
$354$		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	$-0.304998\pi$
$367$	2.94984	+	1.70309i	0.153980	+	0.0889007i	−0.421030	+	0.907047i	$0.638331\pi$
$368$		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	$-0.919885\pi$
$373$	−23.8917	+	13.7939i	−1.23706	+	0.714219i	−0.268571	+	0.963260i	$0.586551\pi$
$374$		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	0.994287	−	0.106743i	$-0.0340423\pi$
$383$	7.92015	+	13.7181i	0.404701	+	0.700962i	−0.589586	+	0.807706i	$0.700709\pi$
$384$		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	−0.771339	−	0.636425i	$-0.780413\pi$
$389$	3.26398	−	5.65339i	0.165491	−	0.286638i	0.936829	+	0.349787i	$0.113746\pi$
$390$		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	$-0.684092\pi$
$401$	−1.89780	+	1.09569i	−0.0947715	+	0.0547163i	0.451865	+	0.892086i	$0.350759\pi$
$402$		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	0.980023	−	0.198883i	$-0.0637313\pi$
$409$	6.42658	+	11.1312i	0.317774	+	0.550401i	−0.662249	+	0.749284i	$0.730398\pi$
$410$		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	$-0.838930\pi$
$419$	−18.2642	+	10.5448i	−0.892262	+	0.515148i	−0.0175806	+	0.999845i	$0.505596\pi$
$420$		0			0
$421$	−6.71848	−	3.87892i	−0.327439	−	0.189047i	0.327265	−	0.944933i	$-0.393873\pi$
$421$	−6.71848	−	3.87892i	−0.327439	−	0.189047i	−0.654703	+	0.755886i	$0.727206\pi$
$422$		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	$-0.434462\pi$
$439$	−11.3367	−	6.54525i	−0.541071	−	0.312388i	−0.745513	+	0.666491i	$0.767795\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	30.5417	+	17.6333i	1.45108	+	0.837781i	0.998543	−	0.0539647i	$-0.0171858\pi$
$443$	30.5417	+	17.6333i	1.45108	+	0.837781i	0.452537	+	0.891746i	$0.350519\pi$
$444$		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	−0.405769	−	0.913976i	$-0.632996\pi$
$457$	12.5837	−	21.7957i	0.588642	−	1.01956i	0.994411	−	0.105582i	$-0.0336706\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−17.7689	+	30.7766i	−0.827579	+	1.43341i	0.0723537	+	0.997379i	$0.476949\pi$
$461$	−17.7689	+	30.7766i	−0.827579	+	1.43341i	−0.899932	+	0.436029i	$0.856384\pi$
$462$		0			0
$463$	−26.0948	+	15.0658i	−1.21273	+	0.700168i	−0.963352	−	0.268239i	$-0.913558\pi$
$463$	−26.0948	+	15.0658i	−1.21273	+	0.700168i	−0.249375	+	0.968407i	$0.580225\pi$
$464$		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	−0.000136370		1.00000i	$0.500043\pi$
$479$	−18.9554	+	32.8317i	−0.866094	+	1.50012i	−0.865957	+	0.500118i	$0.833290\pi$
$480$		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	$-0.722887\pi$
$491$	−6.74349	+	3.89336i	−0.304330	+	0.175705i	0.340057	+	0.940405i	$0.389554\pi$
$492$		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	−0.999256	−	0.0385604i	$-0.987723\pi$
$499$	−11.9068	−	20.6232i	−0.533022	−	0.923222i	0.466234	−	0.884662i	$-0.345611\pi$
$500$		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	0.568419	−	0.822739i	$-0.307555\pi$
$509$	−9.66297	−	16.7368i	−0.428304	−	0.741844i	−0.996723	+	0.0808956i	$0.974222\pi$
$510$		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	−0.318752	−	0.947838i	$-0.603264\pi$
$547$	15.4706	−	26.7959i	0.661476	−	1.14571i	0.980228	−	0.197871i	$-0.0634028\pi$
$548$		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	$-0.846745\pi$
$563$	−22.0296	+	12.7188i	−0.928437	+	0.536034i	−0.0421203	+	0.999113i	$0.513411\pi$
$564$		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	$-0.167385\pi$
$569$	20.5771	+	11.8802i	0.862638	+	0.498044i	−0.00225663	+	0.999997i	$0.500718\pi$
$570$		0			0
$571$	−0.636562	−	1.10256i	−0.0266393	−	0.0461406i	0.852398	−	0.522893i	$-0.175147\pi$
$571$	−0.636562	−	1.10256i	−0.0266393	−	0.0461406i	−0.879038	+	0.476752i	$0.841814\pi$
$572$		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	0.990882	+	0.134732i	$0.0430173\pi$
$587$	38.8377	+	22.4230i	1.60300	+	0.925495i	0.612122	+	0.790763i	$0.290316\pi$
$588$		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	0.845045	−	0.534695i	$-0.179573\pi$
$599$	−0.992117	−	1.71840i	−0.0405368	−	0.0702118i	−0.885582	+	0.464483i	$0.846240\pi$
$600$		0			0
$601$	−4.78811	+	8.29324i	−0.195311	+	0.338288i	−0.947002	−	0.321227i	$-0.895905\pi$
$601$	−4.78811	+	8.29324i	−0.195311	+	0.338288i	0.751692	+	0.659515i	$0.229238\pi$
$602$		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	$-0.881301\pi$
$607$	−26.6428	+	15.3822i	−1.08140	+	0.624345i	−0.150125	+	0.988667i	$0.547967\pi$
$608$		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	$-0.474442\pi$
$617$	24.4308	−	14.1051i	0.983547	−	0.567851i	0.903339	+	0.428927i	$0.141108\pi$
$618$		0			0
$619$	−12.0197	+	20.8187i	−0.483111	+	0.836773i	−0.999812	−	0.0193928i	$-0.993827\pi$
$619$	−12.0197	+	20.8187i	−0.483111	+	0.836773i	0.516701	+	0.856166i	$0.327160\pi$
$620$		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	$-0.399002\pi$
$641$	−8.98279	−	5.18622i	−0.354799	−	0.204843i	−0.666797	+	0.745239i	$0.732335\pi$
$642$		0			0
$643$	−9.55211	−	16.5447i	−0.376698	−	0.652461i	0.613881	−	0.789398i	$-0.289607\pi$
$643$	−9.55211	−	16.5447i	−0.376698	−	0.652461i	−0.990580	+	0.136938i	$0.956274\pi$
$644$		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	0.888370	−	0.459129i	$-0.151838\pi$
$653$	1.18998	+	2.06111i	0.0465675	+	0.0806574i	−0.841802	+	0.539786i	$0.818505\pi$
$654$		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	$-0.647250\pi$
$659$	−37.0796	−	21.4079i	−1.44442	−	0.833934i	−0.998140	+	0.0609613i	$0.980583\pi$
$660$		0			0
$661$	−30.7637	+	17.7614i	−1.19657	+	0.690839i	−0.959788	−	0.280724i	$-0.909425\pi$
$661$	−30.7637	+	17.7614i	−1.19657	+	0.690839i	−0.236780	+	0.971563i	$0.576092\pi$
$662$		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	−0.652638	−	0.757669i	$-0.726338\pi$
$673$	8.55684	−	14.8209i	0.329842	−	0.571303i	0.982480	+	0.186367i	$0.0596713\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−1.09679	+	1.89969i	−0.0421530	+	0.0730111i	−0.886332	−	0.463050i	$-0.846755\pi$
$677$	−1.09679	+	1.89969i	−0.0421530	+	0.0730111i	0.844179	+	0.536061i	$0.180088\pi$
$678$		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	0.212082	+	0.977252i	$0.431975\pi$
$691$	−19.4597	+	33.7053i	−0.740284	+	1.28221i	−0.952366	+	0.304957i	$0.901358\pi$
$692$		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	$-0.108990\pi$
$709$	29.8798	+	17.2511i	1.12216	+	0.647878i	0.180207	+	0.983629i	$0.442323\pi$
$710$		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	$-0.227574\pi$
$727$	15.2326	+	8.79455i	0.564946	+	0.326172i	−0.190182	+	0.981749i	$0.560908\pi$
$728$		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	$-0.766584\pi$
$733$	−14.4793	+	8.35960i	−0.534803	+	0.308769i	0.208167	+	0.978093i	$0.433250\pi$
$734$		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	0.602188	−	0.798354i	$-0.294296\pi$
$743$	−10.6388	−	18.4270i	−0.390301	−	0.676021i	−0.992489	+	0.122333i	$0.960962\pi$
$744$		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	$-0.845615\pi$
$751$	−25.3009	+	14.6075i	−0.923244	+	0.533035i	−0.0385752	+	0.999256i	$0.512282\pi$
$752$		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	$-0.438314\pi$
$761$	31.4120	−	18.1357i	1.13869	−	0.657420i	0.946105	+	0.323861i	$0.104981\pi$
$762$		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	0.994696	+	0.102857i	$0.0327983\pi$
$769$	16.2620	+	28.1667i	0.586424	+	1.01572i	−0.408272	+	0.912860i	$0.633868\pi$
$770$		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	−0.902932	−	0.429784i	$-0.858590\pi$
$787$	−23.1068	−	40.0222i	−0.823670	−	1.42664i	0.0792616	−	0.996854i	$-0.474744\pi$
$788$		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	0.653387	−	0.757024i	$-0.273348\pi$
$797$	−9.28549	−	16.0829i	−0.328909	−	0.569687i	−0.982296	+	0.187337i	$0.940014\pi$
$798$		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	−0.468909	−	0.883246i	$-0.655353\pi$
$821$	15.1993	−	26.3260i	0.530459	−	0.918783i	0.999368	−	0.0355362i	$-0.0113139\pi$
$822$		0			0
$823$	12.5306	−	7.23454i	0.436789	−	0.252180i	−0.265446	−	0.964126i	$-0.585519\pi$
$823$	12.5306	−	7.23454i	0.436789	−	0.252180i	0.702235	+	0.711946i	$0.252186\pi$
$824$		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	0.133662	+	0.991027i	$0.457326\pi$
$839$	−22.9240	+	39.7055i	−0.791424	+	1.37079i	−0.925086	+	0.379759i	$0.876007\pi$
$840$		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	−0.999883	−	0.0153054i	$-0.995128\pi$
$853$	−44.1912	−	25.5138i	−1.51308	−	0.873577i	−0.513196	−	0.858271i	$-0.671539\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−45.5385	−	26.2917i	−1.55557	−	0.898107i	−0.997672	−	0.0681949i	$-0.978276\pi$
$857$	−45.5385	−	26.2917i	−1.55557	−	0.898107i	−0.557894	−	0.829912i	$-0.688391\pi$
$858$		0			0
$859$	8.00975	+	13.8733i	0.273289	+	0.473351i	0.969702	−	0.244291i	$-0.0785550\pi$
$859$	8.00975	+	13.8733i	0.273289	+	0.473351i	−0.696413	+	0.717641i	$0.745222\pi$
$860$		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	$-0.683647\pi$
$877$	−2.73500	+	1.57905i	−0.0923543	+	0.0533208i	0.453112	+	0.891454i	$0.350314\pi$
$878$		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	−0.983315	−	0.181913i	$-0.941771\pi$
$887$	−19.3348	−	33.4888i	−0.649199	−	1.12445i	0.334116	−	0.942532i	$-0.391562\pi$
$888$		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	−0.222250	−	0.974990i	$-0.571340\pi$
$907$	22.0826	−	38.2482i	0.733241	−	1.27001i	0.955491	−	0.295021i	$-0.0953267\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	15.0699	−	26.1019i	0.499290	−	0.864795i	−0.500710	−	0.865615i	$-0.666928\pi$
$911$	15.0699	−	26.1019i	0.499290	−	0.864795i	1.00000		0.000820108i	$0.000261049\pi$
$912$		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	$-0.178374\pi$
$929$	24.6970	+	14.2588i	0.810282	+	0.467816i	−0.0367720	+	0.999324i	$0.511708\pi$
$930$		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	0.745489	−	0.666518i	$-0.232216\pi$
$941$	−6.27248	−	10.8643i	−0.204477	−	0.354165i	−0.949966	+	0.312353i	$0.898883\pi$
$942$		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	$-0.224825\pi$
$947$	17.8198	+	10.2883i	0.579066	+	0.334324i	−0.181696	+	0.983355i	$0.558159\pi$
$948$		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	$-0.561835\pi$
$967$	17.4196	−	10.0572i	0.560176	−	0.323418i	0.753216	+	0.657773i	$0.228501\pi$
$968$		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	−0.810111	+	0.586276i	$0.800593\pi$
$977$	−53.8526	+	31.0918i	−1.72290	+	0.994715i	−0.912786	+	0.408439i	$0.866073\pi$
$978$		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	−0.610417	−	0.792081i	$-0.708998\pi$
$983$	11.9377	−	20.6767i	0.380754	−	0.659485i	0.991170	+	0.132596i	$0.0423312\pi$
$984$		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	$-0.131660\pi$
$997$	32.3781	+	18.6935i	1.02543	+	0.592030i	0.109755	+	0.993959i	$0.464993\pi$
$998$		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.431.19		48
3.2	odd	2	inner	2592.2.p.g.431.5		48
4.3	odd	2		648.2.l.g.107.15		48
8.3	odd	2	inner	2592.2.p.g.431.6		48
8.5	even	2		648.2.l.g.107.8		48
9.2	odd	6		2592.2.f.c.1295.19		24
9.4	even	3	inner	2592.2.p.g.2159.20		48
9.5	odd	6	inner	2592.2.p.g.2159.6		48
9.7	even	3		2592.2.f.c.1295.5		24
12.11	even	2		648.2.l.g.107.10		48
24.5	odd	2		648.2.l.g.107.17		48
24.11	even	2	inner	2592.2.p.g.431.20		48
36.7	odd	6		648.2.f.c.323.2	yes	24
36.11	even	6		648.2.f.c.323.23	yes	24
36.23	even	6		648.2.l.g.539.8		48
36.31	odd	6		648.2.l.g.539.17		48
72.5	odd	6		648.2.l.g.539.15		48
72.11	even	6		2592.2.f.c.1295.6		24
72.13	even	6		648.2.l.g.539.10		48
72.29	odd	6		648.2.f.c.323.1	✓	24
72.43	odd	6		2592.2.f.c.1295.20		24
72.59	even	6	inner	2592.2.p.g.2159.19		48
72.61	even	6		648.2.f.c.323.24	yes	24
72.67	odd	6	inner	2592.2.p.g.2159.5		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.2.f.c.323.1	✓	24	72.29	odd	6
648.2.f.c.323.2	yes	24	36.7	odd	6
648.2.f.c.323.23	yes	24	36.11	even	6
648.2.f.c.323.24	yes	24	72.61	even	6
648.2.l.g.107.8		48	8.5	even	2
648.2.l.g.107.10		48	12.11	even	2
648.2.l.g.107.15		48	4.3	odd	2
648.2.l.g.107.17		48	24.5	odd	2
648.2.l.g.539.8		48	36.23	even	6
648.2.l.g.539.10		48	72.13	even	6
648.2.l.g.539.15		48	72.5	odd	6
648.2.l.g.539.17		48	36.31	odd	6
2592.2.f.c.1295.5		24	9.7	even	3
2592.2.f.c.1295.6		24	72.11	even	6
2592.2.f.c.1295.19		24	9.2	odd	6
2592.2.f.c.1295.20		24	72.43	odd	6
2592.2.p.g.431.5		48	3.2	odd	2	inner
2592.2.p.g.431.6		48	8.3	odd	2	inner
2592.2.p.g.431.19		48	1.1	even	1	trivial
2592.2.p.g.431.20		48	24.11	even	2	inner
2592.2.p.g.2159.5		48	72.67	odd	6	inner
2592.2.p.g.2159.6		48	9.5	odd	6	inner
2592.2.p.g.2159.19		48	72.59	even	6	inner
2592.2.p.g.2159.20		48	9.4	even	3	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:	\( N \)	\(=\)	\( 2592 = 2^{5} \cdot 3^{4} \)
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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