LMFDB - Embedded newform 6300.2.ch.c.4301.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6300,2,Mod(1601,6300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6300, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("6300.1601");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6300 = 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6300.ch (of order $6$, degree $2$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$50.3057532734$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 2 x^{11} - 9 x^{10} + 58 x^{9} - 78 x^{8} - 298 x^{7} + 1341 x^{6} - 2086 x^{5} - 3822 x^{4} + \cdots + 117649 $ x^12 - 2x^11 - 9x^10 + 58x^9 - 78x^8 - 298x^7 + 1341x^6 - 2086x^5 - 3822x^4 + 19894x^3 - 21609x^2 - 33614*x + 117649
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	no (minimal twist has level 1260)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			4301.6
Root			$-2.64559 - 0.0290059i$ of defining polynomial
Character	$\chi$	$=$	6300.4301
Dual form			6300.2.ch.c.1601.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.64559 + 0.0290059i) q^{7} +O(q^{10})$	$q+(2.64559 + 0.0290059i) q^{7} +(1.55024 + 0.895031i) q^{11} +2.19816i q^{13} +(-0.468089 + 0.810754i) q^{17} +(0.366307 - 0.211487i) q^{19} +(3.53961 - 2.04359i) q^{23} +8.43097i q^{29} +(-9.20510 - 5.31456i) q^{31} +(4.18775 + 7.25340i) q^{37} -6.29118 q^{41} +9.56621 q^{43} +(1.52383 + 2.63936i) q^{47} +(6.99832 + 0.153476i) q^{49} +(5.75120 + 3.32045i) q^{53} +(-6.36526 + 11.0249i) q^{59} +(1.68487 - 0.972758i) q^{61} +(3.58177 - 6.20381i) q^{67} +4.91826i q^{71} +(3.99351 + 2.30565i) q^{73} +(4.07534 + 2.41285i) q^{77} +(-2.78022 - 4.81548i) q^{79} -15.1448 q^{83} +(2.15103 + 3.72570i) q^{89} +(-0.0637596 + 5.81543i) q^{91} +4.55329i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 2 q^{7}+O(q^{10}) $ 12 * q - 2 * q^7	$ 12 q - 2 q^{7} + 12 q^{11} - 6 q^{19} + 12 q^{23} - 6 q^{31} + 2 q^{37} - 8 q^{41} - 36 q^{43} + 16 q^{47} + 22 q^{49} - 12 q^{53} - 2 q^{67} - 6 q^{73} - 28 q^{77} + 14 q^{79} - 40 q^{83} - 20 q^{89} + 10 q^{91}+O(q^{100}) $ 12 * q - 2 * q^7 + 12 * q^11 - 6 * q^19 + 12 * q^23 - 6 * q^31 + 2 * q^37 - 8 * q^41 - 36 * q^43 + 16 * q^47 + 22 * q^49 - 12 * q^53 - 2 * q^67 - 6 * q^73 - 28 * q^77 + 14 * q^79 - 40 * q^83 - 20 * q^89 + 10 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/6300\mathbb{Z}\right)^\times$.

$n$	$2801$	$3151$	$3277$	$3601$
$\chi(n)$	$-1$	$1$	$1$	$e\left(\frac{1}{6}\right)$

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			0
$5$		0			0
$6$		0			0
$7$	2.64559	+	0.0290059i	0.999940	+	0.0109632i
$7$	2.64559	+	0.0290059i	0.999940	+	0.0109632i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.55024	+	0.895031i	0.467415	+	0.269862i	0.715157	−	0.698964i	$-0.246355\pi$
$11$	1.55024	+	0.895031i	0.467415	+	0.269862i	−0.247742	+	0.968826i	$0.579689\pi$
$12$		0			0
$13$			2.19816i			0.609659i	0.952407	+	0.304830i	$0.0985995\pi$
$13$			2.19816i			0.609659i	−0.952407	+	0.304830i	$0.901400\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−0.468089	+	0.810754i	−0.113528	+	0.196637i	−0.917190	−	0.398449i	$-0.869549\pi$
$17$	−0.468089	+	0.810754i	−0.113528	+	0.196637i	0.803662	+	0.595086i	$0.202882\pi$
$18$		0			0
$19$	0.366307	−	0.211487i	0.0840366	−	0.0485186i	−0.457393	−	0.889265i	$-0.651217\pi$
$19$	0.366307	−	0.211487i	0.0840366	−	0.0485186i	0.541429	+	0.840746i	$0.317883\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	3.53961	−	2.04359i	0.738059	−	0.426119i	−0.0833039	−	0.996524i	$-0.526547\pi$
$23$	3.53961	−	2.04359i	0.738059	−	0.426119i	0.821363	+	0.570405i	$0.193214\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$			8.43097i			1.56559i	0.622278	+	0.782796i	$0.286207\pi$
$29$			8.43097i			1.56559i	−0.622278	+	0.782796i	$0.713793\pi$
$30$		0			0
$31$	−9.20510	−	5.31456i	−1.65328	−	0.954524i	−0.975709	−	0.219070i	$-0.929698\pi$
$31$	−9.20510	−	5.31456i	−1.65328	−	0.954524i	−0.677575	−	0.735454i	$-0.736969\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	4.18775	+	7.25340i	0.688463	+	1.19245i	0.972335	+	0.233590i	$0.0750474\pi$
$37$	4.18775	+	7.25340i	0.688463	+	1.19245i	−0.283873	+	0.958862i	$0.591619\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−6.29118			−0.982518			−0.491259	−	0.871014i	$-0.663463\pi$
$41$	−6.29118			−0.982518			−0.491259	+	0.871014i	$0.663463\pi$
$42$		0			0
$43$	9.56621			1.45883			0.729417	−	0.684069i	$-0.239791\pi$
$43$	9.56621			1.45883			0.729417	+	0.684069i	$0.239791\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	1.52383	+	2.63936i	0.222274	+	0.384990i	0.955498	−	0.294997i	$-0.0953187\pi$
$47$	1.52383	+	2.63936i	0.222274	+	0.384990i	−0.733224	+	0.679987i	$0.761985\pi$
$48$		0			0
$49$	6.99832	+	0.153476i	0.999760	+	0.0219251i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	5.75120	+	3.32045i	0.789988	+	0.456100i	0.839958	−	0.542651i	$-0.182579\pi$
$53$	5.75120	+	3.32045i	0.789988	+	0.456100i	−0.0499706	+	0.998751i	$0.515913\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−6.36526	+	11.0249i	−0.828686	+	1.43533i	0.0703834	+	0.997520i	$0.477578\pi$
$59$	−6.36526	+	11.0249i	−0.828686	+	1.43533i	−0.899069	+	0.437806i	$0.855756\pi$
$60$		0			0
$61$	1.68487	−	0.972758i	0.215725	−	0.124549i	−0.388244	−	0.921557i	$-0.626918\pi$
$61$	1.68487	−	0.972758i	0.215725	−	0.124549i	0.603969	+	0.797008i	$0.293585\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	3.58177	−	6.20381i	0.437583	−	0.757916i	−0.559920	−	0.828547i	$-0.689168\pi$
$67$	3.58177	−	6.20381i	0.437583	−	0.757916i	0.997503	+	0.0706311i	$0.0225013\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			4.91826i			0.583690i	0.956466	+	0.291845i	$0.0942691\pi$
$71$			4.91826i			0.583690i	−0.956466	+	0.291845i	$0.905731\pi$
$72$		0			0
$73$	3.99351	+	2.30565i	0.467405	+	0.269856i	0.715153	−	0.698968i	$-0.246357\pi$
$73$	3.99351	+	2.30565i	0.467405	+	0.269856i	−0.247748	+	0.968825i	$0.579690\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	4.07534	+	2.41285i	0.464428	+	0.274970i
$78$		0			0
$79$	−2.78022	−	4.81548i	−0.312799	−	0.541784i	0.666168	−	0.745802i	$-0.267933\pi$
$79$	−2.78022	−	4.81548i	−0.312799	−	0.541784i	−0.978967	+	0.204018i	$0.934600\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−15.1448			−1.66236			−0.831178	−	0.556007i	$-0.812333\pi$
$83$	−15.1448			−1.66236			−0.831178	+	0.556007i	$0.812333\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	2.15103	+	3.72570i	0.228009	+	0.394924i	0.957218	−	0.289368i	$-0.0934450\pi$
$89$	2.15103	+	3.72570i	0.228009	+	0.394924i	−0.729209	+	0.684291i	$0.760112\pi$
$90$		0			0
$91$	−0.0637596	+	5.81543i	−0.00668382	+	0.609623i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$			4.55329i			0.462317i	0.972916	+	0.231158i	$0.0742516\pi$
$97$			4.55329i			0.462317i	−0.972916	+	0.231158i	$0.925748\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	4.40748	−	7.63398i	0.438561	−	0.759610i	−0.559018	−	0.829156i	$-0.688822\pi$
$101$	4.40748	−	7.63398i	0.438561	−	0.759610i	0.997579	+	0.0695459i	$0.0221550\pi$
$102$		0			0
$103$	−1.40106	+	0.808903i	−0.138051	+	0.0797036i	−0.567435	−	0.823418i	$-0.692064\pi$
$103$	−1.40106	+	0.808903i	−0.138051	+	0.0797036i	0.429384	+	0.903122i	$0.358731\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	5.58345	−	3.22361i	0.539773	−	0.311638i	−0.205214	−	0.978717i	$-0.565789\pi$
$107$	5.58345	−	3.22361i	0.539773	−	0.311638i	0.744987	+	0.667079i	$0.232456\pi$
$108$		0			0
$109$	1.39335	−	2.41335i	0.133458	−	0.231157i	−0.791549	−	0.611106i	$-0.790725\pi$
$109$	1.39335	−	2.41335i	0.133458	−	0.231157i	0.925008	+	0.379949i	$0.124058\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			7.71522i			0.725787i	0.931831	+	0.362894i	$0.118211\pi$
$113$			7.71522i			0.725787i	−0.931831	+	0.362894i	$0.881789\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−1.26189	+	2.13135i	−0.115677	+	0.195380i
$120$		0			0
$121$	−3.89784	−	6.75125i	−0.354349	−	0.613750i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	1.80732			0.160374			0.0801869	−	0.996780i	$-0.474448\pi$
$127$	1.80732			0.160374			0.0801869	+	0.996780i	$0.474448\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−5.53153	−	9.58089i	−0.483292	−	0.837086i	0.516524	−	0.856273i	$-0.327226\pi$
$131$	−5.53153	−	9.58089i	−0.483292	−	0.837086i	−0.999816	+	0.0191864i	$0.993892\pi$
$132$		0			0
$133$	0.975233	−	0.548885i	0.0845635	−	0.0475943i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−15.8746	−	9.16520i	−1.35626	−	0.783035i	−0.367140	−	0.930166i	$-0.619663\pi$
$137$	−15.8746	−	9.16520i	−1.35626	−	0.783035i	−0.989117	+	0.147130i	$0.952996\pi$
$138$		0			0
$139$			5.62390i			0.477013i	0.971141	+	0.238507i	$0.0766579\pi$
$139$			5.62390i			0.477013i	−0.971141	+	0.238507i	$0.923342\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−1.96742	+	3.40767i	−0.164524	+	0.284964i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−12.8179	+	7.40039i	−1.05008	+	0.606264i	−0.922672	−	0.385587i	$-0.873999\pi$
$149$	−12.8179	+	7.40039i	−1.05008	+	0.606264i	−0.127408	+	0.991850i	$0.540666\pi$
$150$		0			0
$151$	−3.87557	+	6.71269i	−0.315390	+	0.546271i	−0.979520	−	0.201346i	$-0.935468\pi$
$151$	−3.87557	+	6.71269i	−0.315390	+	0.546271i	0.664131	+	0.747616i	$0.268802\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−3.90780	−	2.25617i	−0.311876	−	0.180062i	0.335889	−	0.941901i	$-0.390963\pi$
$157$	−3.90780	−	2.25617i	−0.311876	−	0.180062i	−0.647766	+	0.761840i	$0.724296\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	9.42364	−	5.30385i	0.742687	−	0.418002i
$162$		0			0
$163$	10.1542	+	17.5876i	0.795340	+	1.37757i	0.922623	+	0.385703i	$0.126041\pi$
$163$	10.1542	+	17.5876i	0.795340	+	1.37757i	−0.127283	+	0.991866i	$0.540626\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	4.32784			0.334898			0.167449	−	0.985881i	$-0.446447\pi$
$167$	4.32784			0.334898			0.167449	+	0.985881i	$0.446447\pi$
$168$		0			0
$169$	8.16810			0.628316
$170$		0			0
$171$		0			0
$172$		0			0
$173$	12.6629	+	21.9327i	0.962740	+	1.66752i	0.715567	+	0.698544i	$0.246168\pi$
$173$	12.6629	+	21.9327i	0.962740	+	1.66752i	0.247173	+	0.968971i	$0.420498\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	14.9074	+	8.60678i	1.11423	+	0.643301i	0.939922	−	0.341390i	$-0.110898\pi$
$179$	14.9074	+	8.60678i	1.11423	+	0.643301i	0.174308	+	0.984691i	$0.444231\pi$
$180$		0			0
$181$			3.08246i			0.229118i	0.993416	+	0.114559i	$0.0365454\pi$
$181$			3.08246i			0.229118i	−0.993416	+	0.114559i	$0.963455\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−1.45130	+	0.837909i	−0.106130	+	0.0612740i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−3.82256	+	2.20696i	−0.276591	+	0.159690i	−0.631879	−	0.775067i	$-0.717716\pi$
$191$	−3.82256	+	2.20696i	−0.276591	+	0.159690i	0.355288	+	0.934757i	$0.384383\pi$
$192$		0			0
$193$	10.8104	−	18.7242i	0.778153	−	1.34780i	−0.154852	−	0.987938i	$-0.549490\pi$
$193$	10.8104	−	18.7242i	0.778153	−	1.34780i	0.933005	−	0.359863i	$-0.117177\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			2.65937i			0.189472i	0.995502	+	0.0947362i	$0.0302008\pi$
$197$			2.65937i			0.189472i	−0.995502	+	0.0947362i	$0.969799\pi$
$198$		0			0
$199$	10.1095	+	5.83671i	0.716641	+	0.413753i	0.813515	−	0.581544i	$-0.197551\pi$
$199$	10.1095	+	5.83671i	0.716641	+	0.413753i	−0.0968739	+	0.995297i	$0.530884\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−0.244548	+	22.3049i	−0.0171639	+	1.56550i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	0.757152			0.0523733
$210$		0			0
$211$	−5.93357			−0.408484			−0.204242	−	0.978920i	$-0.565473\pi$
$211$	−5.93357			−0.408484			−0.204242	+	0.978920i	$0.565473\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−24.1988	−	14.3272i	−1.64272	−	0.972592i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−1.78217	−	1.02893i	−0.119881	−	0.0692136i
$222$		0			0
$223$		−	16.2789i		−	1.09011i	−0.838399	−	0.545057i	$-0.816508\pi$
$223$		−	16.2789i		−	1.09011i	0.838399	−	0.545057i	$-0.183492\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	0.367610	−	0.636719i	0.0243991	−	0.0422605i	−0.853568	−	0.520981i	$-0.825566\pi$
$227$	0.367610	−	0.636719i	0.0243991	−	0.0422605i	0.877967	+	0.478721i	$0.158899\pi$
$228$		0			0
$229$	25.4384	−	14.6868i	1.68101	−	0.970534i	0.720025	−	0.693948i	$-0.244130\pi$
$229$	25.4384	−	14.6868i	1.68101	−	0.970534i	0.960989	−	0.276585i	$-0.0892029\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−21.1298	+	12.1993i	−1.38426	+	0.799203i	−0.992661	−	0.120932i	$-0.961412\pi$
$233$	−21.1298	+	12.1993i	−1.38426	+	0.799203i	−0.391600	+	0.920136i	$0.628078\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$			6.84320i			0.442650i	0.975200	+	0.221325i	$0.0710381\pi$
$239$			6.84320i			0.442650i	−0.975200	+	0.221325i	$0.928962\pi$
$240$		0			0
$241$	−0.0607315	−	0.0350634i	−0.00391206	−	0.00225863i	0.498043	−	0.867153i	$-0.334052\pi$
$241$	−0.0607315	−	0.0350634i	−0.00391206	−	0.00225863i	−0.501955	+	0.864894i	$0.667386\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	0.464883	+	0.805201i	0.0295798	+	0.0512337i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	14.5728			0.919824			0.459912	−	0.887965i	$-0.347881\pi$
$251$	14.5728			0.919824			0.459912	+	0.887965i	$0.347881\pi$
$252$		0			0
$253$	7.31632			0.459973
$254$		0			0
$255$		0			0
$256$		0			0
$257$	5.60343	+	9.70543i	0.349532	+	0.605408i	0.986166	−	0.165758i	$-0.0530071\pi$
$257$	5.60343	+	9.70543i	0.349532	+	0.605408i	−0.636634	+	0.771166i	$0.719674\pi$
$258$		0			0
$259$	10.8687	+	19.3110i	0.675348	+	1.19993i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−0.563331	−	0.325239i	−0.0347365	−	0.0200551i	0.482531	−	0.875879i	$-0.339717\pi$
$263$	−0.563331	−	0.325239i	−0.0347365	−	0.0200551i	−0.517268	+	0.855824i	$0.673051\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	2.26420	−	3.92171i	0.138051	−	0.239111i	−0.788708	−	0.614768i	$-0.789250\pi$
$269$	2.26420	−	3.92171i	0.138051	−	0.239111i	0.926759	+	0.375657i	$0.122583\pi$
$270$		0			0
$271$	6.11430	−	3.53009i	0.371417	−	0.214438i	−0.302660	−	0.953098i	$-0.597875\pi$
$271$	6.11430	−	3.53009i	0.371417	−	0.214438i	0.674077	+	0.738661i	$0.264541\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−8.66200	+	15.0030i	−0.520449	+	0.901444i	0.479268	+	0.877669i	$0.340902\pi$
$277$	−8.66200	+	15.0030i	−0.520449	+	0.901444i	−0.999717	+	0.0237759i	$0.992431\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	26.9791i		−	1.60944i	−0.593655	−	0.804720i	$-0.702316\pi$
$281$		−	26.9791i		−	1.60944i	0.593655	−	0.804720i	$-0.297684\pi$
$282$		0			0
$283$	−4.37436	−	2.52554i	−0.260029	−	0.150128i	0.364319	−	0.931274i	$-0.381302\pi$
$283$	−4.37436	−	2.52554i	−0.260029	−	0.150128i	−0.624348	+	0.781147i	$0.714635\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−16.6439	−	0.182482i	−0.982459	−	0.0107715i
$288$		0			0
$289$	8.06179	+	13.9634i	0.474223	+	0.821378i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	9.78083			0.571402			0.285701	−	0.958319i	$-0.407774\pi$
$293$	9.78083			0.571402			0.285701	+	0.958319i	$0.407774\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	4.49214	+	7.78062i	0.259787	+	0.449965i
$300$		0			0
$301$	25.3083	+	0.277477i	1.45875	+	0.0159935i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	26.9503i		−	1.53813i	−0.639168	−	0.769067i	$-0.720721\pi$
$307$		−	26.9503i		−	1.53813i	0.639168	−	0.769067i	$-0.279279\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	6.35136	−	11.0009i	0.360152	−	0.623802i	−0.627833	−	0.778348i	$-0.716058\pi$
$311$	6.35136	−	11.0009i	0.360152	−	0.623802i	0.987986	+	0.154546i	$0.0493913\pi$
$312$		0			0
$313$	−20.7569	+	11.9840i	−1.17325	+	0.677375i	−0.954443	−	0.298394i	$-0.903549\pi$
$313$	−20.7569	+	11.9840i	−1.17325	+	0.677375i	−0.218805	+	0.975769i	$0.570216\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	0.350824	−	0.202548i	0.0197042	−	0.0113762i	−0.490116	−	0.871657i	$-0.663045\pi$
$317$	0.350824	−	0.202548i	0.0197042	−	0.0113762i	0.509820	+	0.860281i	$0.329712\pi$
$318$		0			0
$319$	−7.54598	+	13.0700i	−0.422494	+	0.731781i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			0.395980i			0.0220329i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	3.95488	+	7.02686i	0.218040	+	0.387403i
$330$		0			0
$331$	−4.18929	−	7.25607i	−0.230264	−	0.398830i	0.727621	−	0.685979i	$-0.240626\pi$
$331$	−4.18929	−	7.25607i	−0.230264	−	0.398830i	−0.957886	+	0.287149i	$0.907292\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	13.8266			0.753181			0.376590	−	0.926380i	$-0.377096\pi$
$337$	13.8266			0.753181			0.376590	+	0.926380i	$0.377096\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−9.51340	−	16.4777i	−0.515180	−	0.892317i
$342$		0			0
$343$	18.5102	+	0.609027i	0.999459	+	0.0328843i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	10.7124	+	6.18483i	0.575074	+	0.332019i	0.759173	−	0.650889i	$-0.225603\pi$
$347$	10.7124	+	6.18483i	0.575074	+	0.332019i	−0.184099	+	0.982908i	$0.558937\pi$
$348$		0			0
$349$		−	6.67915i		−	0.357527i	−0.983892	−	0.178763i	$-0.942790\pi$
$349$		−	6.67915i		−	0.357527i	0.983892	−	0.178763i	$-0.0572096\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−2.85014	+	4.93658i	−0.151697	+	0.262748i	−0.931852	−	0.362840i	$-0.881807\pi$
$353$	−2.85014	+	4.93658i	−0.151697	+	0.262748i	0.780154	+	0.625587i	$0.215141\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−21.8381	+	12.6082i	−1.15257	+	0.665436i	−0.949512	−	0.313731i	$-0.898421\pi$
$359$	−21.8381	+	12.6082i	−1.15257	+	0.665436i	−0.203057	+	0.979167i	$0.565088\pi$
$360$		0			0
$361$	−9.41055	+	16.2995i	−0.495292	+	0.857871i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	24.4862	+	14.1371i	1.27817	+	0.737952i	0.976512	−	0.215465i	$-0.0691266\pi$
$367$	24.4862	+	14.1371i	1.27817	+	0.737952i	0.301658	+	0.953416i	$0.402460\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	15.1190	+	8.95139i	0.784940	+	0.464733i
$372$		0			0
$373$	8.22479	+	14.2458i	0.425863	+	0.737617i	0.996501	−	0.0835851i	$-0.0266371\pi$
$373$	8.22479	+	14.2458i	0.425863	+	0.737617i	−0.570637	+	0.821202i	$0.693304\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−18.5326			−0.954478
$378$		0			0
$379$	6.16869			0.316864			0.158432	−	0.987370i	$-0.449356\pi$
$379$	6.16869			0.316864			0.158432	+	0.987370i	$0.449356\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	5.06190	+	8.76747i	0.258651	+	0.447997i	0.965881	−	0.258987i	$-0.0833887\pi$
$383$	5.06190	+	8.76747i	0.258651	+	0.447997i	−0.707230	+	0.706984i	$0.750055\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	23.7572	+	13.7162i	1.20454	+	0.695441i	0.961561	−	0.274591i	$-0.0885425\pi$
$389$	23.7572	+	13.7162i	1.20454	+	0.695441i	0.242978	+	0.970032i	$0.421876\pi$
$390$		0			0
$391$			3.82634i			0.193506i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−21.4517	+	12.3851i	−1.07663	+	0.621592i	−0.929985	−	0.367598i	$-0.880180\pi$
$397$	−21.4517	+	12.3851i	−1.07663	+	0.621592i	−0.146643	+	0.989189i	$0.546847\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−20.8748	+	12.0521i	−1.04244	+	0.601851i	−0.920522	−	0.390690i	$-0.872236\pi$
$401$	−20.8748	+	12.0521i	−1.04244	+	0.601851i	−0.121914	+	0.992541i	$0.538903\pi$
$402$		0			0
$403$	11.6822	−	20.2343i	0.581934	−	1.00794i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			14.9927i			0.743160i
$408$		0			0
$409$	21.1709	+	12.2230i	1.04683	+	0.604388i	0.921761	−	0.387760i	$-0.126751\pi$
$409$	21.1709	+	12.2230i	1.04683	+	0.604388i	0.125071	+	0.992148i	$0.460084\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−17.1597	+	28.9829i	−0.844372	+	1.42615i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−6.05848			−0.295976			−0.147988	−	0.988989i	$-0.547280\pi$
$419$	−6.05848			−0.295976			−0.147988	+	0.988989i	$0.547280\pi$
$420$		0			0
$421$	−19.0817			−0.929986			−0.464993	−	0.885314i	$-0.653943\pi$
$421$	−19.0817			−0.929986			−0.464993	+	0.885314i	$0.653943\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	4.48568	−	2.52465i	0.217077	−	0.122176i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	29.4339	+	16.9937i	1.41778	+	0.818557i	0.996104	−	0.0881879i	$-0.0281076\pi$
$431$	29.4339	+	16.9937i	1.41778	+	0.818557i	0.421679	+	0.906745i	$0.361441\pi$
$432$		0			0
$433$			19.2496i			0.925075i	0.886600	+	0.462537i	$0.153061\pi$
$433$			19.2496i			0.925075i	−0.886600	+	0.462537i	$0.846939\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	0.864389	−	1.49717i	0.0413493	−	0.0716191i
$438$		0			0
$439$	29.0204	−	16.7549i	1.38507	−	0.799668i	0.392312	−	0.919832i	$-0.371675\pi$
$439$	29.0204	−	16.7549i	1.38507	−	0.799668i	0.992754	+	0.120164i	$0.0383421\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	2.40673	−	1.38952i	0.114347	−	0.0660183i	−0.441736	−	0.897145i	$-0.645637\pi$
$443$	2.40673	−	1.38952i	0.114347	−	0.0660183i	0.556083	+	0.831127i	$0.312304\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$			40.7948i			1.92523i	0.270881	+	0.962613i	$0.412685\pi$
$449$			40.7948i			1.92523i	−0.270881	+	0.962613i	$0.587315\pi$
$450$		0			0
$451$	−9.75284	−	5.63081i	−0.459243	−	0.265144i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	1.12014	+	1.94014i	0.0523979	+	0.0907558i	0.891035	−	0.453935i	$-0.149980\pi$
$457$	1.12014	+	1.94014i	0.0523979	+	0.0907558i	−0.838637	+	0.544691i	$0.816647\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−36.5014			−1.70004			−0.850020	−	0.526751i	$-0.823410\pi$
$461$	−36.5014			−1.70004			−0.850020	+	0.526751i	$0.823410\pi$
$462$		0			0
$463$	6.02692			0.280095			0.140047	−	0.990145i	$-0.455275\pi$
$463$	6.02692			0.280095			0.140047	+	0.990145i	$0.455275\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	13.6725	+	23.6814i	0.632687	+	1.09585i	0.987000	+	0.160720i	$0.0513815\pi$
$467$	13.6725	+	23.6814i	0.632687	+	1.09585i	−0.354313	+	0.935127i	$0.615285\pi$
$468$		0			0
$469$	9.65585	−	16.3089i	0.445866	−	0.753073i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	14.8299	+	8.56206i	0.681881	+	0.393684i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−18.4048	+	31.8781i	−0.840937	+	1.45655i	0.0481665	+	0.998839i	$0.484662\pi$
$479$	−18.4048	+	31.8781i	−0.840937	+	1.45655i	−0.889103	+	0.457706i	$0.848671\pi$
$480$		0			0
$481$	−15.9441	+	9.20534i	−0.726989	+	0.419728i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	15.8806	−	27.5060i	0.719619	−	1.24642i	−0.241532	−	0.970393i	$-0.577650\pi$
$487$	15.8806	−	27.5060i	0.719619	−	1.24642i	0.961151	−	0.276023i	$-0.0890167\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	0.651553i		−	0.0294042i	−0.999892	−	0.0147021i	$-0.995320\pi$
$491$		−	0.651553i		−	0.0294042i	0.999892	−	0.0147021i	$-0.00467999\pi$
$492$		0			0
$493$	−6.83545	−	3.94645i	−0.307853	−	0.177739i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−0.142659	+	13.0117i	−0.00639911	+	0.583655i
$498$		0			0
$499$	−10.9966	−	19.0467i	−0.492276	−	0.852647i	0.507684	−	0.861543i	$-0.330502\pi$
$499$	−10.9966	−	19.0467i	−0.492276	−	0.852647i	−0.999960	+	0.00889594i	$0.997168\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−26.7369			−1.19214			−0.596070	−	0.802933i	$-0.703272\pi$
$503$	−26.7369			−1.19214			−0.596070	+	0.802933i	$0.703272\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−7.01326	−	12.1473i	−0.310857	−	0.538421i	0.667691	−	0.744439i	$-0.267283\pi$
$509$	−7.01326	−	12.1473i	−0.310857	−	0.538421i	−0.978548	+	0.206018i	$0.933950\pi$
$510$		0			0
$511$	10.4983	+	6.21565i	0.464418	+	0.274964i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$			5.45551i			0.239933i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	15.0733	−	26.1077i	0.660372	−	1.14380i	−0.320146	−	0.947368i	$-0.603732\pi$
$521$	15.0733	−	26.1077i	0.660372	−	1.14380i	0.980518	−	0.196429i	$-0.0629345\pi$
$522$		0			0
$523$	−0.625635	+	0.361210i	−0.0273571	+	0.0157946i	−0.513616	−	0.858020i	$-0.671694\pi$
$523$	−0.625635	+	0.361210i	−0.0273571	+	0.0157946i	0.486259	+	0.873815i	$0.338361\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	8.61761	−	4.97538i	0.375389	−	0.216731i
$528$		0			0
$529$	−3.14745	+	5.45154i	−0.136846	+	0.237023i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	13.8290i		−	0.599001i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	10.7117	+	6.50164i	0.461386	+	0.280045i
$540$		0			0
$541$	−6.31658	−	10.9406i	−0.271571	−	0.470375i	0.697693	−	0.716397i	$-0.254210\pi$
$541$	−6.31658	−	10.9406i	−0.271571	−	0.470375i	−0.969264	+	0.246022i	$0.920877\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	10.0662			0.430401			0.215201	−	0.976570i	$-0.430959\pi$
$547$	10.0662			0.430401			0.215201	+	0.976570i	$0.430959\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	1.78304	+	3.08832i	0.0759603	+	0.131567i
$552$		0			0
$553$	−7.21565	−	12.8204i	−0.306841	−	0.545181i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−12.1440	−	7.01135i	−0.514559	−	0.297081i	0.220147	−	0.975467i	$-0.429346\pi$
$557$	−12.1440	−	7.01135i	−0.514559	−	0.297081i	−0.734706	+	0.678386i	$0.762680\pi$
$558$		0			0
$559$			21.0280i			0.889392i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	14.4528	−	25.0329i	0.609112	−	1.05501i	−0.382275	−	0.924048i	$-0.624859\pi$
$563$	14.4528	−	25.0329i	0.609112	−	1.05501i	0.991387	−	0.130964i	$-0.0418072\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−0.728101	+	0.420369i	−0.0305236	+	0.0176228i	−0.515184	−	0.857080i	$-0.672276\pi$
$569$	−0.728101	+	0.420369i	−0.0305236	+	0.0176228i	0.484661	+	0.874702i	$0.338943\pi$
$570$		0			0
$571$	22.6922	−	39.3040i	0.949638	−	1.64482i	0.203451	−	0.979085i	$-0.434784\pi$
$571$	22.6922	−	39.3040i	0.949638	−	1.64482i	0.746187	−	0.665736i	$-0.231882\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	17.2807	+	9.97704i	0.719406	+	0.415350i	0.814534	−	0.580116i	$-0.196993\pi$
$577$	17.2807	+	9.97704i	0.719406	+	0.415350i	−0.0951277	+	0.995465i	$0.530326\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−40.0669	−	0.439288i	−1.66226	−	0.0182247i
$582$		0			0
$583$	5.94382	+	10.2950i	0.246168	+	0.426375i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	24.0763			0.993733			0.496867	−	0.867827i	$-0.334484\pi$
$587$	24.0763			0.993733			0.496867	+	0.867827i	$0.334484\pi$
$588$		0			0
$589$	−4.49586			−0.185248
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−17.9389	−	31.0711i	−0.736663	−	1.27594i	−0.953990	−	0.299839i	$-0.903067\pi$
$593$	−17.9389	−	31.0711i	−0.736663	−	1.27594i	0.217327	−	0.976099i	$-0.430266\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−21.4343	−	12.3751i	−0.875783	−	0.505633i	−0.00651720	−	0.999979i	$-0.502075\pi$
$599$	−21.4343	−	12.3751i	−0.875783	−	0.505633i	−0.869266	+	0.494345i	$0.835408\pi$
$600$		0			0
$601$			35.0712i			1.43058i	0.698826	+	0.715291i	$0.253706\pi$
$601$			35.0712i			1.43058i	−0.698826	+	0.715291i	$0.746294\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−2.00722	+	1.15887i	−0.0814706	+	0.0470371i	−0.540182	−	0.841548i	$-0.681645\pi$
$607$	−2.00722	+	1.15887i	−0.0814706	+	0.0470371i	0.458711	+	0.888585i	$0.348311\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−5.80172	+	3.34963i	−0.234713	+	0.135511i
$612$		0			0
$613$	6.42396	−	11.1266i	0.259461	−	0.449400i	−0.706636	−	0.707577i	$-0.749788\pi$
$613$	6.42396	−	11.1266i	0.259461	−	0.449400i	0.966098	+	0.258177i	$0.0831217\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	29.7437i		−	1.19743i	−0.800960	−	0.598717i	$-0.795677\pi$
$617$		−	29.7437i		−	1.19743i	0.800960	−	0.598717i	$-0.204323\pi$
$618$		0			0
$619$	−37.6945	−	21.7629i	−1.51507	−	0.874726i	−0.999844	−	0.0176719i	$-0.994375\pi$
$619$	−37.6945	−	21.7629i	−1.51507	−	0.874726i	−0.515226	−	0.857054i	$-0.672292\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	5.58269	+	9.91908i	0.223666	+	0.397400i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−7.84097			−0.312640
$630$		0			0
$631$	4.21974			0.167985			0.0839925	−	0.996466i	$-0.473233\pi$
$631$	4.21974			0.167985			0.0839925	+	0.996466i	$0.473233\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−0.337364	+	15.3834i	−0.0133668	+	0.609513i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	38.7681	+	22.3828i	1.53125	+	0.884066i	0.999305	+	0.0372866i	$0.0118715\pi$
$641$	38.7681	+	22.3828i	1.53125	+	0.884066i	0.531943	+	0.846780i	$0.321462\pi$
$642$		0			0
$643$			27.0494i			1.06672i	0.845887	+	0.533362i	$0.179072\pi$
$643$			27.0494i			1.06672i	−0.845887	+	0.533362i	$0.820928\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−9.63234	+	16.6837i	−0.378686	+	0.655904i	−0.990871	−	0.134811i	$-0.956957\pi$
$647$	−9.63234	+	16.6837i	−0.378686	+	0.655904i	0.612185	+	0.790714i	$0.290291\pi$
$648$		0			0
$649$	−19.7354	+	11.3942i	−0.774680	+	0.447262i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	26.1038	−	15.0711i	1.02152	−	0.589776i	0.106978	−	0.994261i	$-0.465883\pi$
$653$	26.1038	−	15.0711i	1.02152	−	0.589776i	0.914544	+	0.404485i	$0.132549\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	21.2084i		−	0.826164i	−0.910694	−	0.413082i	$-0.864452\pi$
$659$		−	21.2084i		−	0.826164i	0.910694	−	0.413082i	$-0.135548\pi$
$660$		0			0
$661$	−37.3887	−	21.5864i	−1.45425	−	0.839614i	−0.455535	−	0.890218i	$-0.650552\pi$
$661$	−37.3887	−	21.5864i	−1.45425	−	0.839614i	−0.998719	+	0.0506043i	$0.983885\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	17.2295	+	29.8423i	0.667128	+	1.15550i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	3.48259			0.134444
$672$		0			0
$673$	31.7115			1.22239			0.611195	−	0.791480i	$-0.290689\pi$
$673$	31.7115			1.22239			0.611195	+	0.791480i	$0.290689\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−17.1064	−	29.6292i	−0.657453	−	1.13874i	−0.981273	−	0.192623i	$-0.938301\pi$
$677$	−17.1064	−	29.6292i	−0.657453	−	1.13874i	0.323820	−	0.946119i	$-0.395033\pi$
$678$		0			0
$679$	−0.132072	+	12.0462i	−0.00506848	+	0.462289i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	41.5412	+	23.9838i	1.58953	+	0.917715i	0.993384	+	0.114838i	$0.0366348\pi$
$683$	41.5412	+	23.9838i	1.58953	+	0.917715i	0.596144	+	0.802877i	$0.296699\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−7.29888	+	12.6420i	−0.278065	+	0.481623i
$690$		0			0
$691$	24.7315	−	14.2787i	0.940831	−	0.543189i	0.0506104	−	0.998718i	$-0.483883\pi$
$691$	24.7315	−	14.2787i	0.940831	−	0.543189i	0.890221	+	0.455529i	$0.150550\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	2.94484	−	5.10060i	0.111544	−	0.193199i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			25.7765i			0.973564i	0.873523	+	0.486782i	$0.161829\pi$
$701$			25.7765i			0.973564i	−0.873523	+	0.486782i	$0.838171\pi$
$702$		0			0
$703$	3.06801	+	1.77132i	0.115712	+	0.0668064i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	11.8818	−	20.0686i	0.446862	−	0.754756i
$708$		0			0
$709$	−21.7874	−	37.7369i	−0.818243	−	1.41724i	−0.906976	−	0.421183i	$-0.861615\pi$
$709$	−21.7874	−	37.7369i	−0.818243	−	1.41724i	0.0887325	−	0.996055i	$-0.471718\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−43.4432			−1.62696
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−0.203564	−	0.352584i	−0.00759166	−	0.0131491i	0.862205	−	0.506560i	$-0.169083\pi$
$719$	−0.203564	−	0.352584i	−0.00759166	−	0.0131491i	−0.869796	+	0.493411i	$0.835750\pi$
$720$		0			0
$721$	−3.73010	+	2.09939i	−0.138916	+	0.0781853i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$			17.8033i			0.660288i	0.943931	+	0.330144i	$0.107097\pi$
$727$			17.8033i			0.660288i	−0.943931	+	0.330144i	$0.892903\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−4.47784	+	7.75585i	−0.165619	+	0.286860i
$732$		0			0
$733$	3.00442	−	1.73460i	0.110971	−	0.0640690i	−0.443487	−	0.896281i	$-0.646259\pi$
$733$	3.00442	−	1.73460i	0.110971	−	0.0640690i	0.554458	+	0.832212i	$0.312926\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	11.1052	−	6.41159i	0.409066	−	0.236174i
$738$		0			0
$739$	12.4022	−	21.4812i	0.456222	−	0.790201i	−0.542535	−	0.840033i	$-0.682535\pi$
$739$	12.4022	−	21.4812i	0.456222	−	0.790201i	0.998758	+	0.0498326i	$0.0158688\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			29.8141i			1.09377i	0.837206	+	0.546887i	$0.184187\pi$
$743$			29.8141i			1.09377i	−0.837206	+	0.546887i	$0.815813\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	14.8650	−	8.36640i	0.543157	−	0.305702i
$750$		0			0
$751$	−16.9247	−	29.3145i	−0.617592	−	1.06970i	−0.989924	−	0.141601i	$-0.954775\pi$
$751$	−16.9247	−	29.3145i	−0.617592	−	1.06970i	0.372332	−	0.928100i	$-0.378558\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	1.19627			0.0434791			0.0217395	−	0.999764i	$-0.493080\pi$
$757$	1.19627			0.0434791			0.0217395	+	0.999764i	$0.493080\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−24.3373	−	42.1534i	−0.882226	−	1.52806i	−0.848861	−	0.528616i	$-0.822711\pi$
$761$	−24.3373	−	42.1534i	−0.882226	−	1.52806i	−0.0333649	−	0.999443i	$-0.510622\pi$
$762$		0			0
$763$	3.75623	−	6.34432i	0.135985	−	0.229680i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−24.2346	−	13.9918i	−0.875060	−	0.505216i
$768$		0			0
$769$			9.21143i			0.332173i	0.986111	+	0.166086i	$0.0531131\pi$
$769$			9.21143i			0.332173i	−0.986111	+	0.166086i	$0.946887\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	22.0186	−	38.1373i	0.791954	−	1.37170i	−0.132801	−	0.991143i	$-0.542397\pi$
$773$	22.0186	−	38.1373i	0.791954	−	1.37170i	0.924755	−	0.380562i	$-0.124269\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−2.30451	+	1.33051i	−0.0825675	+	0.0476703i
$780$		0			0
$781$	−4.40199	+	7.62448i	−0.157516	+	0.272825i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−36.4112	−	21.0220i	−1.29792	−	0.749354i	−0.317875	−	0.948133i	$-0.602969\pi$
$787$	−36.4112	−	21.0220i	−1.29792	−	0.749354i	−0.980044	+	0.198778i	$0.936303\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−0.223787	+	20.4113i	−0.00795695	+	0.725743i
$792$		0			0
$793$	2.13827	+	3.70360i	0.0759324	+	0.131519i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	33.6692			1.19263			0.596313	−	0.802752i	$-0.296632\pi$
$797$	33.6692			1.19263			0.596313	+	0.802752i	$0.296632\pi$
$798$		0			0
$799$	−2.85316			−0.100938
$800$		0			0
$801$		0			0
$802$		0			0
$803$	4.12726	+	7.14863i	0.145648	+	0.252270i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	8.22735	+	4.75006i	0.289258	+	0.167003i	0.637607	−	0.770362i	$-0.279924\pi$
$809$	8.22735	+	4.75006i	0.289258	+	0.167003i	−0.348349	+	0.937365i	$0.613258\pi$
$810$		0			0
$811$			14.8210i			0.520435i	0.965550	+	0.260217i	$0.0837942\pi$
$811$			14.8210i			0.520435i	−0.965550	+	0.260217i	$0.916206\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	3.50417	−	2.02313i	0.122595	−	0.0707805i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	15.6109	−	9.01294i	0.544823	−	0.314554i	−0.202208	−	0.979343i	$-0.564812\pi$
$821$	15.6109	−	9.01294i	0.544823	−	0.314554i	0.747031	+	0.664789i	$0.231478\pi$
$822$		0			0
$823$	12.9801	−	22.4821i	0.452456	−	0.783677i	−0.546082	−	0.837732i	$-0.683881\pi$
$823$	12.9801	−	22.4821i	0.452456	−	0.783677i	0.998538	+	0.0540548i	$0.0172146\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	24.4040i		−	0.848612i	−0.905519	−	0.424306i	$-0.860518\pi$
$827$		−	24.4040i		−	0.848612i	0.905519	−	0.424306i	$-0.139482\pi$
$828$		0			0
$829$	−8.99187	−	5.19146i	−0.312301	−	0.180307i	0.335655	−	0.941985i	$-0.391042\pi$
$829$	−8.99187	−	5.19146i	−0.312301	−	0.180307i	−0.647956	+	0.761678i	$0.724376\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−3.40027	+	5.60207i	−0.117812	+	0.194100i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	29.7021			1.02543			0.512715	−	0.858559i	$-0.328640\pi$
$839$	29.7021			1.02543			0.512715	+	0.858559i	$0.328640\pi$
$840$		0			0
$841$	−42.0813			−1.45108
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−10.1163	−	17.9741i	−0.347599	−	0.617598i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	29.6460	+	17.1161i	1.01625	+	0.586734i
$852$		0			0
$853$			47.4739i			1.62548i	0.582629	+	0.812738i	$0.302024\pi$
$853$			47.4739i			1.62548i	−0.582629	+	0.812738i	$0.697976\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−16.7597	+	29.0287i	−0.572501	+	0.991602i	0.423807	+	0.905753i	$0.360694\pi$
$857$	−16.7597	+	29.0287i	−0.572501	+	0.991602i	−0.996308	+	0.0858490i	$0.972640\pi$
$858$		0			0
$859$	−29.1235	+	16.8144i	−0.993680	+	0.573702i	−0.906372	−	0.422480i	$-0.861160\pi$
$859$	−29.1235	+	16.8144i	−0.993680	+	0.573702i	−0.0873078	+	0.996181i	$0.527826\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	17.4477	−	10.0735i	0.593928	−	0.342904i	−0.172721	−	0.984971i	$-0.555256\pi$
$863$	17.4477	−	10.0735i	0.593928	−	0.342904i	0.766649	+	0.642066i	$0.221923\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	9.95353i		−	0.337650i
$870$		0			0
$871$	13.6369	+	7.87330i	0.462070	+	0.266776i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−14.2444	−	24.6721i	−0.481001	−	0.833118i	0.518761	−	0.854919i	$-0.326393\pi$
$877$	−14.2444	−	24.6721i	−0.481001	−	0.833118i	−0.999762	+	0.0218011i	$0.993060\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−27.5545			−0.928336			−0.464168	−	0.885747i	$-0.653647\pi$
$881$	−27.5545			−0.928336			−0.464168	+	0.885747i	$0.653647\pi$
$882$		0			0
$883$	−48.1856			−1.62158			−0.810788	−	0.585340i	$-0.800961\pi$
$883$	−48.1856			−1.62158			−0.810788	+	0.585340i	$0.800961\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	23.9872	+	41.5470i	0.805410	+	1.39501i	0.916014	+	0.401147i	$0.131388\pi$
$887$	23.9872	+	41.5470i	0.805410	+	1.39501i	−0.110604	+	0.993865i	$0.535278\pi$
$888$		0			0
$889$	4.78143	+	0.0524230i	0.160364	+	0.00175821i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	1.11638	+	0.644543i	0.0373583	+	0.0215688i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	44.8069	−	77.6079i	1.49440	−	2.58837i
$900$		0			0
$901$	−5.38414	+	3.10854i	−0.179372	+	0.103560i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−11.3530	+	19.6639i	−0.376970	+	0.652931i	−0.990620	−	0.136647i	$-0.956367\pi$
$907$	−11.3530	+	19.6639i	−0.376970	+	0.652931i	0.613650	+	0.789578i	$0.289701\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			6.95948i			0.230578i	0.993332	+	0.115289i	$0.0367794\pi$
$911$			6.95948i			0.230578i	−0.993332	+	0.115289i	$0.963221\pi$
$912$		0			0
$913$	−23.4780	−	13.5551i	−0.777010	−	0.448607i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−14.3563	−	25.5076i	−0.474086	−	0.842335i
$918$		0			0
$919$	18.7082	+	32.4036i	0.617128	+	1.06890i	0.990007	+	0.141017i	$0.0450373\pi$
$919$	18.7082	+	32.4036i	0.617128	+	1.06890i	−0.372879	+	0.927880i	$0.621629\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−10.8111			−0.355852
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−17.1344	−	29.6776i	−0.562160	−	0.973689i	−0.997308	−	0.0733304i	$-0.976637\pi$
$929$	−17.1344	−	29.6776i	−0.562160	−	0.973689i	0.435148	−	0.900359i	$-0.356696\pi$
$930$		0			0
$931$	2.59599	−	1.42384i	0.0850802	−	0.0466644i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	10.0892i		−	0.329600i	−0.986327	−	0.164800i	$-0.947302\pi$
$937$		−	10.0892i		−	0.329600i	0.986327	−	0.164800i	$-0.0526979\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−3.02257	+	5.23524i	−0.0985328	+	0.170664i	−0.911078	−	0.412235i	$-0.864748\pi$
$941$	−3.02257	+	5.23524i	−0.0985328	+	0.170664i	0.812545	+	0.582899i	$0.198082\pi$
$942$		0			0
$943$	−22.2683	+	12.8566i	−0.725157	+	0.418669i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	18.5297	−	10.6982i	0.602136	−	0.347643i	−0.167746	−	0.985830i	$-0.553649\pi$
$947$	18.5297	−	10.6982i	0.602136	−	0.347643i	0.769881	+	0.638187i	$0.220315\pi$
$948$		0			0
$949$	−5.06819	+	8.77836i	−0.164520	+	0.284958i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	29.6828i		−	0.961519i	−0.876852	−	0.480760i	$-0.840361\pi$
$953$		−	29.6828i		−	0.961519i	0.876852	−	0.480760i	$-0.159639\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−41.7318	−	24.7078i	−1.34759	−	0.797857i
$960$		0			0
$961$	40.9892	+	70.9954i	1.32223	+	2.29017i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−41.7092			−1.34128			−0.670639	−	0.741783i	$-0.733980\pi$
$967$	−41.7092			−1.34128			−0.670639	+	0.741783i	$0.733980\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	10.7837	+	18.6779i	0.346065	+	0.599402i	0.985547	−	0.169405i	$-0.0541844\pi$
$971$	10.7837	+	18.6779i	0.346065	+	0.599402i	−0.639482	+	0.768806i	$0.720851\pi$
$972$		0			0
$973$	−0.163126	+	14.8786i	−0.00522960	+	0.476985i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−27.1798	−	15.6923i	−0.869558	−	0.502040i	−0.00235693	−	0.999997i	$-0.500750\pi$
$977$	−27.1798	−	15.6923i	−0.869558	−	0.502040i	−0.867201	+	0.497957i	$0.834084\pi$
$978$		0			0
$979$			7.70097i			0.246124i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	27.9768	−	48.4572i	0.892322	−	1.54555i	0.0552365	−	0.998473i	$-0.482409\pi$
$983$	27.9768	−	48.4572i	0.892322	−	1.54555i	0.837085	−	0.547073i	$-0.184258\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	33.8607	−	19.5495i	1.07671	−	0.621637i
$990$		0			0
$991$	31.1869	−	54.0173i	0.990685	−	1.71592i	0.377415	−	0.926044i	$-0.376813\pi$
$991$	31.1869	−	54.0173i	0.990685	−	1.71592i	0.613271	−	0.789873i	$-0.289854\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	25.1507	+	14.5208i	0.796532	+	0.459878i	0.842257	−	0.539076i	$-0.181227\pi$
$997$	25.1507	+	14.5208i	0.796532	+	0.459878i	−0.0457254	+	0.998954i	$0.514560\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6300.2.ch.c.4301.6		12
3.2	odd	2		6300.2.ch.b.4301.6		12
5.2	odd	4		6300.2.dd.c.4049.6		24
5.3	odd	4		6300.2.dd.c.4049.7		24
5.4	even	2		1260.2.cg.a.521.1	yes	12
7.5	odd	6		6300.2.ch.b.1601.6		12
15.2	even	4		6300.2.dd.b.4049.6		24
15.8	even	4		6300.2.dd.b.4049.7		24
15.14	odd	2		1260.2.cg.b.521.1	yes	12
21.5	even	6	inner	6300.2.ch.c.1601.6		12
35.4	even	6		8820.2.d.b.881.6		12
35.12	even	12		6300.2.dd.b.1349.7		24
35.19	odd	6		1260.2.cg.b.341.1	yes	12
35.24	odd	6		8820.2.d.a.881.6		12
35.33	even	12		6300.2.dd.b.1349.6		24
105.47	odd	12		6300.2.dd.c.1349.7		24
105.59	even	6		8820.2.d.b.881.7		12
105.68	odd	12		6300.2.dd.c.1349.6		24
105.74	odd	6		8820.2.d.a.881.7		12
105.89	even	6		1260.2.cg.a.341.1	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1260.2.cg.a.341.1	✓	12	105.89	even	6
1260.2.cg.a.521.1	yes	12	5.4	even	2
1260.2.cg.b.341.1	yes	12	35.19	odd	6
1260.2.cg.b.521.1	yes	12	15.14	odd	2
6300.2.ch.b.1601.6		12	7.5	odd	6
6300.2.ch.b.4301.6		12	3.2	odd	2
6300.2.ch.c.1601.6		12	21.5	even	6	inner
6300.2.ch.c.4301.6		12	1.1	even	1	trivial
6300.2.dd.b.1349.6		24	35.33	even	12
6300.2.dd.b.1349.7		24	35.12	even	12
6300.2.dd.b.4049.6		24	15.2	even	4
6300.2.dd.b.4049.7		24	15.8	even	4
6300.2.dd.c.1349.6		24	105.68	odd	12
6300.2.dd.c.1349.7		24	105.47	odd	12
6300.2.dd.c.4049.6		24	5.2	odd	4
6300.2.dd.c.4049.7		24	5.3	odd	4
8820.2.d.a.881.6		12	35.24	odd	6
8820.2.d.a.881.7		12	105.74	odd	6
8820.2.d.b.881.6		12	35.4	even	6
8820.2.d.b.881.7		12	105.59	even	6

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:	\( N \)	\(=\)	\( 6300 = 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6300.ch (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(2.64559 + 0.0290059i) q^{7} +O(q^{10})\)	\(q+(2.64559 + 0.0290059i) q^{7} +(1.55024 + 0.895031i) q^{11} +2.19816i q^{13} +(-0.468089 + 0.810754i) q^{17} +(0.366307 - 0.211487i) q^{19} +(3.53961 - 2.04359i) q^{23} +8.43097i q^{29} +(-9.20510 - 5.31456i) q^{31} +(4.18775 + 7.25340i) q^{37} -6.29118 q^{41} +9.56621 q^{43} +(1.52383 + 2.63936i) q^{47} +(6.99832 + 0.153476i) q^{49} +(5.75120 + 3.32045i) q^{53} +(-6.36526 + 11.0249i) q^{59} +(1.68487 - 0.972758i) q^{61} +(3.58177 - 6.20381i) q^{67} +4.91826i q^{71} +(3.99351 + 2.30565i) q^{73} +(4.07534 + 2.41285i) q^{77} +(-2.78022 - 4.81548i) q^{79} -15.1448 q^{83} +(2.15103 + 3.72570i) q^{89} +(-0.0637596 + 5.81543i) q^{91} +4.55329i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 2 q^{7}+O(q^{10}) \) 12 * q - 2 * q^7	\( 12 q - 2 q^{7} + 12 q^{11} - 6 q^{19} + 12 q^{23} - 6 q^{31} + 2 q^{37} - 8 q^{41} - 36 q^{43} + 16 q^{47} + 22 q^{49} - 12 q^{53} - 2 q^{67} - 6 q^{73} - 28 q^{77} + 14 q^{79} - 40 q^{83} - 20 q^{89} + 10 q^{91}+O(q^{100}) \) 12 * q - 2 * q^7 + 12 * q^11 - 6 * q^19 + 12 * q^23 - 6 * q^31 + 2 * q^37 - 8 * q^41 - 36 * q^43 + 16 * q^47 + 22 * q^49 - 12 * q^53 - 2 * q^67 - 6 * q^73 - 28 * q^77 + 14 * q^79 - 40 * q^83 - 20 * q^89 + 10 * q^91

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