LMFDB - Embedded newform 4320.2.cc.a.1871.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4320,2,Mod(1871,4320)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4320, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 3, 1, 0]))

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

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

chi := DirichletCharacter("4320.1871");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4320 = 2^{5} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4320.cc (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:	$34.4953736732$
Analytic rank:	$0$
Dimension:	$48$
Relative dimension:	$24$ over $\Q(\zeta_{6})$
Twist minimal:	no (minimal twist has level 360)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			1871.18
Character	$\chi$	$=$	4320.1871
Dual form			4320.2.cc.a.3311.18

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 - 0.866025i) q^{5} +(2.20775 + 1.27465i) q^{7} +O(q^{10})$	$q+(-0.500000 - 0.866025i) q^{5} +(2.20775 + 1.27465i) q^{7} +(-4.02755 - 2.32531i) q^{11} +(3.23898 - 1.87003i) q^{13} +3.38689i q^{17} +5.12923 q^{19} +(4.16526 + 7.21444i) q^{23} +(-0.500000 + 0.866025i) q^{25} +(-2.01987 + 3.49852i) q^{29} +(-4.88169 + 2.81845i) q^{31} -2.54929i q^{35} -7.40207i q^{37} +(5.23156 - 3.02044i) q^{41} +(-5.26273 + 9.11531i) q^{43} +(0.621010 - 1.07562i) q^{47} +(-0.250553 - 0.433970i) q^{49} -2.05427 q^{53} +4.65062i q^{55} +(-0.351174 + 0.202750i) q^{59} +(-4.57621 - 2.64207i) q^{61} +(-3.23898 - 1.87003i) q^{65} +(6.17580 + 10.6968i) q^{67} +5.43205 q^{71} +13.3990 q^{73} +(-5.92789 - 10.2674i) q^{77} +(-5.40268 - 3.11924i) q^{79} +(12.0774 + 6.97289i) q^{83} +(2.93313 - 1.69345i) q^{85} +2.72618i q^{89} +9.53450 q^{91} +(-2.56462 - 4.44204i) q^{95} +(1.31601 - 2.27939i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 24 q^{5}+O(q^{10}) $ 48 * q - 24 * q^5	$ 48 q - 24 q^{5} - 24 q^{25} - 12 q^{41} - 12 q^{47} + 24 q^{49} - 36 q^{59} - 12 q^{61} - 60 q^{83}+O(q^{100}) $ 48 * q - 24 * q^5 - 24 * q^25 - 12 * q^41 - 12 * q^47 + 24 * q^49 - 36 * q^59 - 12 * q^61 - 60 * q^83

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4320\mathbb{Z}\right)^\times$.

$n$	$2081$	$2431$	$3457$	$3781$
$\chi(n)$	$e\left(\frac{1}{6}\right)$	$-1$	$1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−0.500000	−	0.866025i	−0.223607	−	0.387298i
$5$	−0.500000	−	0.866025i	−0.223607	−	0.387298i
$6$		0			0
$7$	2.20775	+	1.27465i	0.834452	+	0.481771i	0.855375	−	0.518010i	$-0.173327\pi$
$7$	2.20775	+	1.27465i	0.834452	+	0.481771i	−0.0209226	+	0.999781i	$0.506660\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−4.02755	−	2.32531i	−1.21435	−	0.701107i	−0.250649	−	0.968078i	$-0.580644\pi$
$11$	−4.02755	−	2.32531i	−1.21435	−	0.701107i	−0.963705	+	0.266971i	$0.913977\pi$
$12$		0			0
$13$	3.23898	−	1.87003i	0.898332	−	0.518653i	0.0216737	−	0.999765i	$-0.493101\pi$
$13$	3.23898	−	1.87003i	0.898332	−	0.518653i	0.876659	+	0.481113i	$0.159767\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			3.38689i			0.821442i	0.911761	+	0.410721i	$0.134723\pi$
$17$			3.38689i			0.821442i	−0.911761	+	0.410721i	$0.865277\pi$
$18$		0			0
$19$	5.12923			1.17673			0.588363	−	0.808597i	$-0.299773\pi$
$19$	5.12923			1.17673			0.588363	+	0.808597i	$0.299773\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	4.16526	+	7.21444i	0.868517	+	1.50432i	0.863512	+	0.504328i	$0.168260\pi$
$23$	4.16526	+	7.21444i	0.868517	+	1.50432i	0.00500444	+	0.999987i	$0.498407\pi$
$24$		0			0
$25$	−0.500000	+	0.866025i	−0.100000	+	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−2.01987	+	3.49852i	−0.375081	+	0.649659i	−0.990339	−	0.138666i	$-0.955718\pi$
$29$	−2.01987	+	3.49852i	−0.375081	+	0.649659i	0.615258	+	0.788326i	$0.289052\pi$
$30$		0			0
$31$	−4.88169	+	2.81845i	−0.876778	+	0.506208i	−0.869595	−	0.493766i	$-0.835620\pi$
$31$	−4.88169	+	2.81845i	−0.876778	+	0.506208i	−0.00718342	+	0.999974i	$0.502287\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		−	2.54929i		−	0.430909i
$36$		0			0
$37$		−	7.40207i		−	1.21689i	−0.793595	−	0.608446i	$-0.791793\pi$
$37$		−	7.40207i		−	1.21689i	0.793595	−	0.608446i	$-0.208207\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	5.23156	−	3.02044i	0.817032	−	0.471714i	−0.0323597	−	0.999476i	$-0.510302\pi$
$41$	5.23156	−	3.02044i	0.817032	−	0.471714i	0.849392	+	0.527762i	$0.176969\pi$
$42$		0			0
$43$	−5.26273	+	9.11531i	−0.802558	+	1.39007i	0.115369	+	0.993323i	$0.463195\pi$
$43$	−5.26273	+	9.11531i	−0.802558	+	1.39007i	−0.917927	+	0.396749i	$0.870138\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	0.621010	−	1.07562i	0.0905837	−	0.156896i	−0.817173	−	0.576392i	$-0.804460\pi$
$47$	0.621010	−	1.07562i	0.0905837	−	0.156896i	0.907757	+	0.419497i	$0.137793\pi$
$48$		0			0
$49$	−0.250553	−	0.433970i	−0.0357933	−	0.0619958i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−2.05427			−0.282176			−0.141088	−	0.989997i	$-0.545060\pi$
$53$	−2.05427			−0.282176			−0.141088	+	0.989997i	$0.545060\pi$
$54$		0			0
$55$			4.65062i			0.627089i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−0.351174	+	0.202750i	−0.0457190	+	0.0263959i	−0.522685	−	0.852526i	$-0.675070\pi$
$59$	−0.351174	+	0.202750i	−0.0457190	+	0.0263959i	0.476966	+	0.878922i	$0.341736\pi$
$60$		0			0
$61$	−4.57621	−	2.64207i	−0.585923	−	0.338283i	0.177561	−	0.984110i	$-0.443179\pi$
$61$	−4.57621	−	2.64207i	−0.585923	−	0.338283i	−0.763484	+	0.645827i	$0.776513\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−3.23898	−	1.87003i	−0.401746	−	0.231948i
$66$		0			0
$67$	6.17580	+	10.6968i	0.754494	+	1.30682i	0.945625	+	0.325258i	$0.105451\pi$
$67$	6.17580	+	10.6968i	0.754494	+	1.30682i	−0.191131	+	0.981565i	$0.561216\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	5.43205			0.644666			0.322333	−	0.946626i	$-0.395533\pi$
$71$	5.43205			0.644666			0.322333	+	0.946626i	$0.395533\pi$
$72$		0			0
$73$	13.3990			1.56823			0.784115	−	0.620616i	$-0.213117\pi$
$73$	13.3990			1.56823			0.784115	+	0.620616i	$0.213117\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−5.92789	−	10.2674i	−0.675546	−	1.17008i
$78$		0			0
$79$	−5.40268	−	3.11924i	−0.607849	−	0.350942i	0.164274	−	0.986415i	$-0.447472\pi$
$79$	−5.40268	−	3.11924i	−0.607849	−	0.350942i	−0.772123	+	0.635473i	$0.780805\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	12.0774	+	6.97289i	1.32567	+	0.765374i	0.984626	−	0.174675i	$-0.0558875\pi$
$83$	12.0774	+	6.97289i	1.32567	+	0.765374i	0.341040	+	0.940049i	$0.389221\pi$
$84$		0			0
$85$	2.93313	−	1.69345i	0.318143	−	0.183680i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			2.72618i			0.288975i	0.989507	+	0.144487i	$0.0461533\pi$
$89$			2.72618i			0.288975i	−0.989507	+	0.144487i	$0.953847\pi$
$90$		0			0
$91$	9.53450			0.999487
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−2.56462	−	4.44204i	−0.263124	−	0.455744i
$96$		0			0
$97$	1.31601	−	2.27939i	0.133620	−	0.231437i	−0.791449	−	0.611235i	$-0.790673\pi$
$97$	1.31601	−	2.27939i	0.133620	−	0.231437i	0.925070	+	0.379798i	$0.124006\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	7.87081	−	13.6326i	0.783175	−	1.35650i	−0.146909	−	0.989150i	$-0.546932\pi$
$101$	7.87081	−	13.6326i	0.783175	−	1.35650i	0.930083	−	0.367348i	$-0.119734\pi$
$102$		0			0
$103$	8.19174	−	4.72950i	0.807156	−	0.466012i	−0.0388112	−	0.999247i	$-0.512357\pi$
$103$	8.19174	−	4.72950i	0.807156	−	0.466012i	0.845967	+	0.533235i	$0.179024\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			9.14609i			0.884186i	0.896969	+	0.442093i	$0.145764\pi$
$107$			9.14609i			0.884186i	−0.896969	+	0.442093i	$0.854236\pi$
$108$		0			0
$109$		−	2.24057i		−	0.214608i	−0.994226	−	0.107304i	$-0.965778\pi$
$109$		−	2.24057i		−	0.214608i	0.994226	−	0.107304i	$-0.0342218\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	5.37513	−	3.10333i	0.505650	−	0.291937i	−0.225394	−	0.974268i	$-0.572367\pi$
$113$	5.37513	−	3.10333i	0.505650	−	0.291937i	0.731044	+	0.682331i	$0.239034\pi$
$114$		0			0
$115$	4.16526	−	7.21444i	0.388413	−	0.672750i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−4.31709	+	7.47741i	−0.395747	+	0.685453i
$120$		0			0
$121$	5.31412	+	9.20433i	0.483102	+	0.836758i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$			12.8277i			1.13827i	0.822243	+	0.569137i	$0.192723\pi$
$127$			12.8277i			1.13827i	−0.822243	+	0.569137i	$0.807277\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	8.33011	−	4.80939i	0.727805	−	0.420199i	−0.0898134	−	0.995959i	$-0.528627\pi$
$131$	8.33011	−	4.80939i	0.727805	−	0.420199i	0.817619	+	0.575760i	$0.195294\pi$
$132$		0			0
$133$	11.3241	+	6.53796i	0.981922	+	0.566913i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	8.47690	+	4.89414i	0.724231	+	0.418135i	0.816308	−	0.577617i	$-0.196017\pi$
$137$	8.47690	+	4.89414i	0.724231	+	0.418135i	−0.0920770	+	0.995752i	$0.529351\pi$
$138$		0			0
$139$	−3.13540	−	5.43067i	−0.265941	−	0.460624i	0.701869	−	0.712306i	$-0.252349\pi$
$139$	−3.13540	−	5.43067i	−0.265941	−	0.460624i	−0.967810	+	0.251683i	$0.919016\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−17.3936			−1.45452
$144$		0			0
$145$	4.03975			0.335483
$146$		0			0
$147$		0			0
$148$		0			0
$149$	2.41284	+	4.17916i	0.197667	+	0.342370i	0.947772	−	0.318950i	$-0.103330\pi$
$149$	2.41284	+	4.17916i	0.197667	+	0.342370i	−0.750104	+	0.661319i	$0.769997\pi$
$150$		0			0
$151$	−10.6615	−	6.15545i	−0.867624	−	0.500923i	−0.00106627	−	0.999999i	$-0.500339\pi$
$151$	−10.6615	−	6.15545i	−0.867624	−	0.500923i	−0.866558	+	0.499076i	$0.833673\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	4.88169	+	2.81845i	0.392107	+	0.226383i
$156$		0			0
$157$	−6.14993	+	3.55066i	−0.490818	+	0.283374i	−0.724914	−	0.688840i	$-0.758120\pi$
$157$	−6.14993	+	3.55066i	−0.490818	+	0.283374i	0.234096	+	0.972213i	$0.424787\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			21.2369i			1.67371i
$162$		0			0
$163$	4.88666			0.382753			0.191377	−	0.981517i	$-0.438705\pi$
$163$	4.88666			0.382753			0.191377	+	0.981517i	$0.438705\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	7.41709	+	12.8468i	0.573951	+	0.994113i	0.996155	+	0.0876117i	$0.0279235\pi$
$167$	7.41709	+	12.8468i	0.573951	+	0.994113i	−0.422203	+	0.906501i	$0.638743\pi$
$168$		0			0
$169$	0.494011	−	0.855652i	0.0380008	−	0.0658194i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	3.22057	−	5.57819i	0.244855	−	0.424102i	−0.717236	−	0.696831i	$-0.754593\pi$
$173$	3.22057	−	5.57819i	0.244855	−	0.424102i	0.962091	+	0.272729i	$0.0879262\pi$
$174$		0			0
$175$	−2.20775	+	1.27465i	−0.166890	+	0.0963542i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			8.76690i			0.655269i	0.944805	+	0.327634i	$0.106251\pi$
$179$			8.76690i			0.655269i	−0.944805	+	0.327634i	$0.893749\pi$
$180$		0			0
$181$			15.3572i			1.14149i	0.821126	+	0.570747i	$0.193346\pi$
$181$			15.3572i			1.14149i	−0.821126	+	0.570747i	$0.806654\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−6.41038	+	3.70104i	−0.471301	+	0.272106i
$186$		0			0
$187$	7.87557	−	13.6409i	0.575918	−	0.997520i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	1.49605	−	2.59124i	0.108251	−	0.187496i	−0.806811	−	0.590810i	$-0.798808\pi$
$191$	1.49605	−	2.59124i	0.108251	−	0.187496i	0.915062	+	0.403314i	$0.132142\pi$
$192$		0			0
$193$	1.27611	+	2.21028i	0.0918562	+	0.159100i	0.908292	−	0.418336i	$-0.137387\pi$
$193$	1.27611	+	2.21028i	0.0918562	+	0.159100i	−0.816436	+	0.577436i	$0.804053\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−10.5167			−0.749283			−0.374642	−	0.927170i	$-0.622234\pi$
$197$	−10.5167			−0.749283			−0.374642	+	0.927170i	$0.622234\pi$
$198$		0			0
$199$		−	12.4162i		−	0.880162i	−0.897958	−	0.440081i	$-0.854950\pi$
$199$		−	12.4162i		−	0.880162i	0.897958	−	0.440081i	$-0.145050\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−8.91876	+	5.14925i	−0.625974	+	0.361406i
$204$		0			0
$205$	−5.23156	−	3.02044i	−0.365388	−	0.210957i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−20.6583	−	11.9270i	−1.42896	−	0.825011i
$210$		0			0
$211$	7.79890	+	13.5081i	0.536899	+	0.929936i	0.999069	+	0.0431444i	$0.0137376\pi$
$211$	7.79890	+	13.5081i	0.536899	+	0.929936i	−0.462170	+	0.886791i	$0.652929\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	10.5255			0.717830
$216$		0			0
$217$	−14.3701			−0.975506
$218$		0			0
$219$		0			0
$220$		0			0
$221$	6.33358	+	10.9701i	0.426043	+	0.737928i
$222$		0			0
$223$	−1.43709	−	0.829706i	−0.0962348	−	0.0555612i	0.451110	−	0.892468i	$-0.351028\pi$
$223$	−1.43709	−	0.829706i	−0.0962348	−	0.0555612i	−0.547345	+	0.836907i	$0.684361\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−8.51386	−	4.91548i	−0.565085	−	0.326252i	0.190099	−	0.981765i	$-0.439119\pi$
$227$	−8.51386	−	4.91548i	−0.565085	−	0.326252i	−0.755184	+	0.655513i	$0.772452\pi$
$228$		0			0
$229$	9.39914	−	5.42659i	0.621112	−	0.358599i	−0.156190	−	0.987727i	$-0.549921\pi$
$229$	9.39914	−	5.42659i	0.621112	−	0.358599i	0.777302	+	0.629128i	$0.216588\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			26.3031i			1.72318i	0.507609	+	0.861588i	$0.330530\pi$
$233$			26.3031i			1.72318i	−0.507609	+	0.861588i	$0.669470\pi$
$234$		0			0
$235$	−1.24202			−0.0810205
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−3.17976	−	5.50750i	−0.205681	−	0.356251i	0.744668	−	0.667435i	$-0.232608\pi$
$239$	−3.17976	−	5.50750i	−0.205681	−	0.356251i	−0.950350	+	0.311184i	$0.899274\pi$
$240$		0			0
$241$	5.71542	−	9.89940i	0.368163	−	0.637677i	−0.621116	−	0.783719i	$-0.713320\pi$
$241$	5.71542	−	9.89940i	0.368163	−	0.637677i	0.989278	+	0.146042i	$0.0466536\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−0.250553	+	0.433970i	−0.0160072	+	0.0277253i
$246$		0			0
$247$	16.6135	−	9.59181i	1.05709	−	0.610312i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			17.3175i			1.09307i	0.837436	+	0.546535i	$0.184054\pi$
$251$			17.3175i			1.09307i	−0.837436	+	0.546535i	$0.815946\pi$
$252$		0			0
$253$		−	38.7421i		−	2.43569i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	5.77481	−	3.33409i	0.360223	−	0.207975i	−0.308956	−	0.951076i	$-0.599980\pi$
$257$	5.77481	−	3.33409i	0.360223	−	0.207975i	0.669178	+	0.743102i	$0.266646\pi$
$258$		0			0
$259$	9.43502	−	16.3419i	0.586264	−	1.01544i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	12.8435	−	22.2455i	0.791962	−	1.37172i	−0.132788	−	0.991144i	$-0.542393\pi$
$263$	12.8435	−	22.2455i	0.791962	−	1.37172i	0.924750	−	0.380575i	$-0.124274\pi$
$264$		0			0
$265$	1.02714	+	1.77905i	0.0630965	+	0.109286i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	17.0236			1.03794			0.518972	−	0.854791i	$-0.326315\pi$
$269$	17.0236			1.03794			0.518972	+	0.854791i	$0.326315\pi$
$270$		0			0
$271$			31.1779i			1.89392i	0.321350	+	0.946960i	$0.395863\pi$
$271$			31.1779i			1.89392i	−0.321350	+	0.946960i	$0.604137\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	4.02755	−	2.32531i	0.242871	−	0.140221i
$276$		0			0
$277$	9.16573	+	5.29183i	0.550715	+	0.317956i	0.749410	−	0.662106i	$-0.230337\pi$
$277$	9.16573	+	5.29183i	0.550715	+	0.317956i	−0.198695	+	0.980061i	$0.563670\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	9.33995	+	5.39242i	0.557175	+	0.321685i	0.752011	−	0.659151i	$-0.229084\pi$
$281$	9.33995	+	5.39242i	0.557175	+	0.321685i	−0.194836	+	0.980836i	$0.562418\pi$
$282$		0			0
$283$	−6.51583	−	11.2858i	−0.387326	−	0.670868i	0.604763	−	0.796406i	$-0.293268\pi$
$283$	−6.51583	−	11.2858i	−0.387326	−	0.670868i	−0.992089	+	0.125537i	$0.959935\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	15.4000			0.909032
$288$		0			0
$289$	5.52898			0.325234
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−8.77812	−	15.2041i	−0.512823	−	0.888236i	−0.999889	−	0.0148707i	$-0.995266\pi$
$293$	−8.77812	−	15.2041i	−0.512823	−	0.888236i	0.487066	−	0.873365i	$-0.338067\pi$
$294$		0			0
$295$	0.351174	+	0.202750i	0.0204461	+	0.0118046i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	26.9824	+	15.5783i	1.56043	+	0.900917i
$300$		0			0
$301$	−23.2376	+	13.4162i	−1.33939	+	0.773299i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			5.28415i			0.302569i
$306$		0			0
$307$	14.8826			0.849397			0.424699	−	0.905335i	$-0.360380\pi$
$307$	14.8826			0.849397			0.424699	+	0.905335i	$0.360380\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	3.87344	+	6.70899i	0.219642	+	0.380432i	0.954699	−	0.297574i	$-0.0961776\pi$
$311$	3.87344	+	6.70899i	0.219642	+	0.380432i	−0.735056	+	0.678006i	$0.762844\pi$
$312$		0			0
$313$	9.95594	−	17.2442i	0.562743	−	0.974700i	−0.434513	−	0.900666i	$-0.643079\pi$
$313$	9.95594	−	17.2442i	0.562743	−	0.974700i	0.997256	−	0.0740338i	$-0.0235873\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−17.6886	+	30.6376i	−0.993493	+	1.72078i	−0.398109	+	0.917338i	$0.630334\pi$
$317$	−17.6886	+	30.6376i	−0.993493	+	1.72078i	−0.595383	+	0.803442i	$0.703000\pi$
$318$		0			0
$319$	16.2703	−	9.39366i	0.910962	−	0.525944i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			17.3721i			0.966612i
$324$		0			0
$325$			3.74006i			0.207461i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	2.74207	−	1.58314i	0.151175	−	0.0872812i
$330$		0			0
$331$	−0.0768415	+	0.133093i	−0.00422359	+	0.00731547i	−0.868129	−	0.496338i	$-0.834678\pi$
$331$	−0.0768415	+	0.133093i	−0.00422359	+	0.00731547i	0.863906	+	0.503653i	$0.168011\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	6.17580	−	10.6968i	0.337420	−	0.584429i
$336$		0			0
$337$	9.38668	+	16.2582i	0.511325	+	0.885641i	0.999914	+	0.0131265i	$0.00417842\pi$
$337$	9.38668	+	16.2582i	0.511325	+	0.885641i	−0.488589	+	0.872514i	$0.662488\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	26.2150			1.41962
$342$		0			0
$343$		−	19.1225i		−	1.03252i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	21.2257	−	12.2547i	1.13945	−	0.657864i	0.193159	−	0.981168i	$-0.438127\pi$
$347$	21.2257	−	12.2547i	1.13945	−	0.657864i	0.946295	+	0.323303i	$0.104793\pi$
$348$		0			0
$349$	20.5838	+	11.8841i	1.10183	+	0.636140i	0.936700	−	0.350133i	$-0.113863\pi$
$349$	20.5838	+	11.8841i	1.10183	+	0.636140i	0.165126	+	0.986272i	$0.447197\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	0.676798	+	0.390750i	0.0360223	+	0.0207975i	0.517903	−	0.855439i	$-0.326713\pi$
$353$	0.676798	+	0.390750i	0.0360223	+	0.0207975i	−0.481881	+	0.876237i	$0.660046\pi$
$354$		0			0
$355$	−2.71603	−	4.70430i	−0.144152	−	0.249678i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−10.9728			−0.579123			−0.289561	−	0.957159i	$-0.593509\pi$
$359$	−10.9728			−0.579123			−0.289561	+	0.957159i	$0.593509\pi$
$360$		0			0
$361$	7.30901			0.384685
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−6.69948	−	11.6038i	−0.350667	−	0.607373i
$366$		0			0
$367$	−7.62051	−	4.39970i	−0.397787	−	0.229663i	0.287741	−	0.957708i	$-0.407096\pi$
$367$	−7.62051	−	4.39970i	−0.397787	−	0.229663i	−0.685529	+	0.728045i	$0.740429\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−4.53533	−	2.61847i	−0.235462	−	0.135944i
$372$		0			0
$373$	−15.1314	+	8.73611i	−0.783473	+	0.452338i	−0.837660	−	0.546192i	$-0.816077\pi$
$373$	−15.1314	+	8.73611i	−0.783473	+	0.452338i	0.0541866	+	0.998531i	$0.482743\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			15.1089i			0.778147i
$378$		0			0
$379$	25.7713			1.32378			0.661892	−	0.749600i	$-0.269754\pi$
$379$	25.7713			1.32378			0.661892	+	0.749600i	$0.269754\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−18.0206	−	31.2126i	−0.920809	−	1.59489i	−0.798166	−	0.602437i	$-0.794196\pi$
$383$	−18.0206	−	31.2126i	−0.920809	−	1.59489i	−0.122643	−	0.992451i	$-0.539137\pi$
$384$		0			0
$385$	−5.92789	+	10.2674i	−0.302113	+	0.523276i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−9.31753	+	16.1384i	−0.472418	+	0.818251i	−0.999502	−	0.0315617i	$-0.989952\pi$
$389$	−9.31753	+	16.1384i	−0.472418	+	0.818251i	0.527084	+	0.849813i	$0.323285\pi$
$390$		0			0
$391$	−24.4345	+	14.1073i	−1.23571	+	0.713436i
$392$		0			0
$393$		0			0
$394$		0			0
$395$			6.23847i			0.313892i
$396$		0			0
$397$		−	27.9154i		−	1.40103i	−0.713635	−	0.700517i	$-0.752953\pi$
$397$		−	27.9154i		−	1.40103i	0.713635	−	0.700517i	$-0.247047\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	12.3568	−	7.13421i	0.617070	−	0.356265i	−0.158657	−	0.987334i	$-0.550717\pi$
$401$	12.3568	−	7.13421i	0.617070	−	0.356265i	0.775727	+	0.631068i	$0.217383\pi$
$402$		0			0
$403$	−10.5412	+	18.2578i	−0.525092	+	0.909486i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−17.2121	+	29.8122i	−0.853172	+	1.47774i
$408$		0			0
$409$	2.20480	+	3.81883i	0.109020	+	0.188829i	0.915374	−	0.402605i	$-0.131895\pi$
$409$	2.20480	+	3.81883i	0.109020	+	0.188829i	−0.806353	+	0.591434i	$0.798562\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−1.03374			−0.0508670
$414$		0			0
$415$		−	13.9458i		−	0.684571i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	17.8653	−	10.3145i	0.872777	−	0.503898i	0.00450661	−	0.999990i	$-0.498565\pi$
$419$	17.8653	−	10.3145i	0.872777	−	0.503898i	0.868270	+	0.496092i	$0.165232\pi$
$420$		0			0
$421$	3.57346	+	2.06314i	0.174160	+	0.100551i	0.584546	−	0.811361i	$-0.301273\pi$
$421$	3.57346	+	2.06314i	0.174160	+	0.100551i	−0.410386	+	0.911912i	$0.634606\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−2.93313	−	1.69345i	−0.142278	−	0.0821442i
$426$		0			0
$427$	−6.73542	−	11.6661i	−0.325950	−	0.564562i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	26.0944			1.25692			0.628462	−	0.777841i	$-0.283685\pi$
$431$	26.0944			1.25692			0.628462	+	0.777841i	$0.283685\pi$
$432$		0			0
$433$	0.448125			0.0215355			0.0107677	−	0.999942i	$-0.496572\pi$
$433$	0.448125			0.0215355			0.0107677	+	0.999942i	$0.496572\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	21.3646	+	37.0045i	1.02201	+	1.77017i
$438$		0			0
$439$	−29.0457	−	16.7695i	−1.38627	−	0.800366i	−0.393381	−	0.919376i	$-0.628695\pi$
$439$	−29.0457	−	16.7695i	−1.38627	−	0.800366i	−0.992893	+	0.119010i	$0.962028\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−32.1997	−	18.5905i	−1.52986	−	0.883262i	−0.999367	−	0.0355701i	$-0.988675\pi$
$443$	−32.1997	−	18.5905i	−1.52986	−	0.883262i	−0.530488	−	0.847692i	$-0.677991\pi$
$444$		0			0
$445$	2.36094	−	1.36309i	0.111919	−	0.0646167i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			26.7099i			1.26052i	0.776384	+	0.630260i	$0.217052\pi$
$449$			26.7099i			1.26052i	−0.776384	+	0.630260i	$0.782948\pi$
$450$		0			0
$451$	−28.0938			−1.32289
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−4.76725	−	8.25712i	−0.223492	−	0.387100i
$456$		0			0
$457$	20.4459	−	35.4134i	0.956421	−	1.65657i	0.225338	−	0.974281i	$-0.427651\pi$
$457$	20.4459	−	35.4134i	0.956421	−	1.65657i	0.731083	−	0.682289i	$-0.239015\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−5.52803	+	9.57483i	−0.257466	+	0.445944i	−0.965562	−	0.260171i	$-0.916221\pi$
$461$	−5.52803	+	9.57483i	−0.257466	+	0.445944i	0.708096	+	0.706116i	$0.249554\pi$
$462$		0			0
$463$	19.4701	−	11.2411i	0.904851	−	0.522416i	0.0260804	−	0.999660i	$-0.491697\pi$
$463$	19.4701	−	11.2411i	0.904851	−	0.522416i	0.878771	+	0.477244i	$0.158364\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			20.5034i			0.948784i	0.880314	+	0.474392i	$0.157332\pi$
$467$			20.5034i			0.948784i	−0.880314	+	0.474392i	$0.842668\pi$
$468$		0			0
$469$			31.4879i			1.45397i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	42.3918	−	24.4749i	1.94918	−	1.12536i
$474$		0			0
$475$	−2.56462	+	4.44204i	−0.117673	+	0.203815i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−0.870737	+	1.50816i	−0.0397850	+	0.0689096i	−0.885232	−	0.465149i	$-0.846001\pi$
$479$	−0.870737	+	1.50816i	−0.0397850	+	0.0689096i	0.845447	+	0.534059i	$0.179334\pi$
$480$		0			0
$481$	−13.8421	−	23.9752i	−0.631145	−	1.09317i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−2.63202			−0.119514
$486$		0			0
$487$		−	0.442827i		−	0.0200664i	−0.999950	−	0.0100332i	$-0.996806\pi$
$487$		−	0.442827i		−	0.0200664i	0.999950	−	0.0100332i	$-0.00319372\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−25.3759	+	14.6508i	−1.14520	+	0.661181i	−0.947713	−	0.319124i	$-0.896611\pi$
$491$	−25.3759	+	14.6508i	−1.14520	+	0.661181i	−0.197487	+	0.980306i	$0.563278\pi$
$492$		0			0
$493$	−11.8491	−	6.84109i	−0.533657	−	0.308107i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	11.9926	+	6.92395i	0.537943	+	0.310582i
$498$		0			0
$499$	−14.0808	−	24.3886i	−0.630341	−	1.09178i	−0.987482	−	0.157733i	$-0.949582\pi$
$499$	−14.0808	−	24.3886i	−0.630341	−	1.09178i	0.357141	−	0.934051i	$-0.383752\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−15.0167			−0.669560			−0.334780	−	0.942296i	$-0.608662\pi$
$503$	−15.0167			−0.669560			−0.334780	+	0.942296i	$0.608662\pi$
$504$		0			0
$505$	−15.7416			−0.700493
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−9.57636	−	16.5867i	−0.424465	−	0.735194i	0.571906	−	0.820319i	$-0.306204\pi$
$509$	−9.57636	−	16.5867i	−0.424465	−	0.735194i	−0.996370	+	0.0851250i	$0.972871\pi$
$510$		0			0
$511$	29.5816	+	17.0789i	1.30861	+	0.755528i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−8.19174	−	4.72950i	−0.360971	−	0.208407i
$516$		0			0
$517$	−5.00231	+	2.88808i	−0.220001	+	0.127018i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	36.1713i		−	1.58469i	−0.610072	−	0.792346i	$-0.708859\pi$
$521$		−	36.1713i		−	1.58469i	0.610072	−	0.792346i	$-0.291141\pi$
$522$		0			0
$523$	−12.2116			−0.533975			−0.266988	−	0.963700i	$-0.586028\pi$
$523$	−12.2116			−0.533975			−0.266988	+	0.963700i	$0.586028\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−9.54577	−	16.5338i	−0.415820	−	0.720222i
$528$		0			0
$529$	−23.1988	+	40.1815i	−1.00864	+	1.74702i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	11.2966	−	19.5663i	0.489311	−	0.847512i
$534$		0			0
$535$	7.92074	−	4.57304i	0.342444	−	0.197710i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			2.33045i			0.100380i
$540$		0			0
$541$			34.9053i			1.50070i	0.661042	+	0.750349i	$0.270114\pi$
$541$			34.9053i			1.50070i	−0.661042	+	0.750349i	$0.729886\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−1.94039	+	1.12029i	−0.0831174	+	0.0479878i
$546$		0			0
$547$	−11.0482	+	19.1361i	−0.472388	+	0.818200i	−0.999501	−	0.0315955i	$-0.989941\pi$
$547$	−11.0482	+	19.1361i	−0.472388	+	0.818200i	0.527113	+	0.849795i	$0.323275\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−10.3604	+	17.9447i	−0.441368	+	0.764471i
$552$		0			0
$553$	−7.95185	−	13.7730i	−0.338147	−	0.585688i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−11.3188			−0.479591			−0.239796	−	0.970823i	$-0.577080\pi$
$557$	−11.3188			−0.479591			−0.239796	+	0.970823i	$0.577080\pi$
$558$		0			0
$559$			39.3658i			1.66500i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−21.3524	+	12.3278i	−0.899896	+	0.519555i	−0.877167	−	0.480186i	$-0.840569\pi$
$563$	−21.3524	+	12.3278i	−0.899896	+	0.519555i	−0.0227299	+	0.999742i	$0.507236\pi$
$564$		0			0
$565$	−5.37513	−	3.10333i	−0.226134	−	0.130558i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−10.4402	−	6.02768i	−0.437678	−	0.252693i	0.264934	−	0.964266i	$-0.414650\pi$
$569$	−10.4402	−	6.02768i	−0.437678	−	0.252693i	−0.702612	+	0.711573i	$0.747983\pi$
$570$		0			0
$571$	4.73898	+	8.20815i	0.198320	+	0.343501i	0.947984	−	0.318318i	$-0.103118\pi$
$571$	4.73898	+	8.20815i	0.198320	+	0.343501i	−0.749664	+	0.661819i	$0.769785\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−8.33052			−0.347407
$576$		0			0
$577$	7.47489			0.311184			0.155592	−	0.987821i	$-0.450272\pi$
$577$	7.47489			0.311184			0.155592	+	0.987821i	$0.450272\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	17.7759	+	30.7888i	0.737470	+	1.27734i
$582$		0			0
$583$	8.27369	+	4.77682i	0.342661	+	0.197836i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−29.4957	−	17.0293i	−1.21742	−	0.702876i	−0.253052	−	0.967453i	$-0.581434\pi$
$587$	−29.4957	−	17.0293i	−1.21742	−	0.702876i	−0.964365	+	0.264577i	$0.914768\pi$
$588$		0			0
$589$	−25.0393	+	14.4565i	−1.03173	+	0.595668i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	5.46511i		−	0.224425i	−0.993684	−	0.112213i	$-0.964206\pi$
$593$		−	5.46511i		−	0.224425i	0.993684	−	0.112213i	$-0.0357937\pi$
$594$		0			0
$595$	8.63417			0.353967
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−8.10483	−	14.0380i	−0.331154	−	0.573576i	0.651584	−	0.758576i	$-0.274105\pi$
$599$	−8.10483	−	14.0380i	−0.331154	−	0.573576i	−0.982738	+	0.185000i	$0.940771\pi$
$600$		0			0
$601$	−4.73112	+	8.19455i	−0.192987	+	0.334263i	−0.946239	−	0.323469i	$-0.895151\pi$
$601$	−4.73112	+	8.19455i	−0.192987	+	0.334263i	0.753252	+	0.657732i	$0.228484\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	5.31412	−	9.20433i	0.216050	−	0.374209i
$606$		0			0
$607$	13.9365	−	8.04627i	0.565667	−	0.326588i	−0.189750	−	0.981832i	$-0.560768\pi$
$607$	13.9365	−	8.04627i	0.565667	−	0.326588i	0.755417	+	0.655244i	$0.227434\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	4.64523i		−	0.187926i
$612$		0			0
$613$		−	9.72482i		−	0.392782i	−0.980526	−	0.196391i	$-0.937078\pi$
$613$		−	9.72482i		−	0.392782i	0.980526	−	0.196391i	$-0.0629221\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−26.6764	+	15.4016i	−1.07395	+	0.620046i	−0.929258	−	0.369431i	$-0.879553\pi$
$617$	−26.6764	+	15.4016i	−1.07395	+	0.620046i	−0.144692	+	0.989477i	$0.546219\pi$
$618$		0			0
$619$	−15.0957	+	26.1466i	−0.606749	+	1.05092i	0.385024	+	0.922907i	$0.374193\pi$
$619$	−15.0957	+	26.1466i	−0.606749	+	1.05092i	−0.991772	+	0.128013i	$0.959140\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−3.47492	+	6.01873i	−0.139220	+	0.241135i
$624$		0			0
$625$	−0.500000	−	0.866025i	−0.0200000	−	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	25.0700			0.999606
$630$		0			0
$631$		−	28.3960i		−	1.13043i	−0.824944	−	0.565214i	$-0.808794\pi$
$631$		−	28.3960i		−	1.13043i	0.824944	−	0.565214i	$-0.191206\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	11.1091	−	6.41385i	0.440852	−	0.254526i
$636$		0			0
$637$	−1.62307	−	0.937082i	−0.0643085	−	0.0371285i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−3.53067	−	2.03844i	−0.139453	−	0.0805134i	0.428650	−	0.903471i	$-0.358989\pi$
$641$	−3.53067	−	2.03844i	−0.139453	−	0.0805134i	−0.568103	+	0.822957i	$0.692323\pi$
$642$		0			0
$643$	11.1471	+	19.3074i	0.439600	+	0.761410i	0.997659	−	0.0683917i	$-0.0217868\pi$
$643$	11.1471	+	19.3074i	0.439600	+	0.761410i	−0.558058	+	0.829802i	$0.688453\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−11.9935			−0.471513			−0.235757	−	0.971812i	$-0.575757\pi$
$647$	−11.9935			−0.471513			−0.235757	+	0.971812i	$0.575757\pi$
$648$		0			0
$649$	1.88583			0.0740253
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−22.7175	−	39.3478i	−0.889004	−	1.53980i	−0.841055	−	0.540950i	$-0.818065\pi$
$653$	−22.7175	−	39.3478i	−0.889004	−	1.53980i	−0.0479492	−	0.998850i	$-0.515269\pi$
$654$		0			0
$655$	−8.33011	−	4.80939i	−0.325484	−	0.187919i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−11.4821	−	6.62918i	−0.447278	−	0.258236i	0.259402	−	0.965769i	$-0.416475\pi$
$659$	−11.4821	−	6.62918i	−0.447278	−	0.258236i	−0.706680	+	0.707533i	$0.749808\pi$
$660$		0			0
$661$	−38.9627	+	22.4951i	−1.51547	+	0.874959i	−0.515638	+	0.856807i	$0.672445\pi$
$661$	−38.9627	+	22.4951i	−1.51547	+	0.874959i	−0.999835	+	0.0181521i	$0.994222\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	13.0759i		−	0.507062i
$666$		0			0
$667$	−33.6532			−1.30306
$668$		0			0
$669$		0			0
$670$		0			0
$671$	12.2873	+	21.2822i	0.474345	+	0.821590i
$672$		0			0
$673$	8.61513	−	14.9218i	0.332089	−	0.575195i	−0.650832	−	0.759221i	$-0.725580\pi$
$673$	8.61513	−	14.9218i	0.332089	−	0.575195i	0.982921	+	0.184027i	$0.0589133\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	14.7680	−	25.5790i	0.567581	−	0.983079i	−0.429223	−	0.903198i	$-0.641213\pi$
$677$	14.7680	−	25.5790i	0.567581	−	0.983079i	0.996804	−	0.0798808i	$-0.0254540\pi$
$678$		0			0
$679$	5.81084	−	3.35489i	0.223000	−	0.128749i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	31.4320i		−	1.20271i	−0.798982	−	0.601355i	$-0.794628\pi$
$683$		−	31.4320i		−	1.20271i	0.798982	−	0.601355i	$-0.205372\pi$
$684$		0			0
$685$		−	9.78829i		−	0.373991i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−6.65376	+	3.84155i	−0.253488	+	0.146351i
$690$		0			0
$691$	−0.540149	+	0.935565i	−0.0205482	+	0.0355906i	−0.876117	−	0.482099i	$-0.839874\pi$
$691$	−0.540149	+	0.935565i	−0.0205482	+	0.0355906i	0.855568	+	0.517690i	$0.173208\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−3.13540	+	5.43067i	−0.118932	+	0.205997i
$696$		0			0
$697$	10.2299	+	17.7187i	0.387485	+	0.671144i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−31.7591			−1.19952			−0.599762	−	0.800179i	$-0.704738\pi$
$701$	−31.7591			−1.19952			−0.599762	+	0.800179i	$0.704738\pi$
$702$		0			0
$703$		−	37.9669i		−	1.43195i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	34.7536	−	20.0650i	1.30704	−	0.754622i
$708$		0			0
$709$	−5.26182	−	3.03791i	−0.197612	−	0.114091i	0.397929	−	0.917416i	$-0.369729\pi$
$709$	−5.26182	−	3.03791i	−0.197612	−	0.114091i	−0.595541	+	0.803325i	$0.703062\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−40.6671	−	23.4791i	−1.52299	−	0.879301i
$714$		0			0
$715$	8.69679	+	15.0633i	0.325241	+	0.563335i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	3.28970			0.122685			0.0613425	−	0.998117i	$-0.480462\pi$
$719$	3.28970			0.122685			0.0613425	+	0.998117i	$0.480462\pi$
$720$		0			0
$721$	24.1138			0.898044
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−2.01987	−	3.49852i	−0.0750162	−	0.129932i
$726$		0			0
$727$	0.500514	+	0.288972i	0.0185630	+	0.0107174i	0.509253	−	0.860617i	$-0.329922\pi$
$727$	0.500514	+	0.288972i	0.0185630	+	0.0107174i	−0.490690	+	0.871334i	$0.663255\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−30.8726	−	17.8243i	−1.14186	−	0.659255i
$732$		0			0
$733$	42.8431	−	24.7354i	1.58244	−	0.913625i	0.587943	−	0.808903i	$-0.299938\pi$
$733$	42.8431	−	24.7354i	1.58244	−	0.913625i	0.994502	−	0.104722i	$-0.0333953\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	57.4426i		−	2.11593i
$738$		0			0
$739$	−5.51433			−0.202848			−0.101424	−	0.994843i	$-0.532340\pi$
$739$	−5.51433			−0.202848			−0.101424	+	0.994843i	$0.532340\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−7.32352	−	12.6847i	−0.268674	−	0.465357i	0.699846	−	0.714294i	$-0.253252\pi$
$743$	−7.32352	−	12.6847i	−0.268674	−	0.465357i	−0.968520	+	0.248937i	$0.919919\pi$
$744$		0			0
$745$	2.41284	−	4.17916i	0.0883995	−	0.153112i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−11.6580	+	20.1923i	−0.425975	+	0.737810i
$750$		0			0
$751$	−37.8937	+	21.8780i	−1.38276	+	0.798338i	−0.992486	−	0.122360i	$-0.960954\pi$
$751$	−37.8937	+	21.8780i	−1.38276	+	0.798338i	−0.390276	+	0.920698i	$0.627620\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			12.3109i			0.448039i
$756$		0			0
$757$			16.8688i			0.613108i	0.951853	+	0.306554i	$0.0991759\pi$
$757$			16.8688i			0.613108i	−0.951853	+	0.306554i	$0.900824\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−0.899605	+	0.519387i	−0.0326107	+	0.0188278i	−0.516217	−	0.856458i	$-0.672660\pi$
$761$	−0.899605	+	0.519387i	−0.0326107	+	0.0188278i	0.483606	+	0.875286i	$0.339327\pi$
$762$		0			0
$763$	2.85594	−	4.94663i	0.103392	−	0.179080i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−0.758298	+	1.31341i	−0.0273806	+	0.0474245i
$768$		0			0
$769$	13.8495	+	23.9881i	0.499427	+	0.865032i	1.00000		0.000662003i	$-0.000210722\pi$
$769$	13.8495	+	23.9881i	0.499427	+	0.865032i	−0.500573	+	0.865694i	$0.666877\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−8.84895			−0.318275			−0.159137	−	0.987256i	$-0.550871\pi$
$773$	−8.84895			−0.318275			−0.159137	+	0.987256i	$0.550871\pi$
$774$		0			0
$775$		−	5.63690i		−	0.202483i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	26.8339	−	15.4925i	0.961423	−	0.555078i
$780$		0			0
$781$	−21.8779	−	12.6312i	−0.782852	−	0.451980i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	6.14993	+	3.55066i	0.219500	+	0.126729i
$786$		0			0
$787$	−10.0300	−	17.3725i	−0.357531	−	0.619263i	0.630016	−	0.776582i	$-0.283048\pi$
$787$	−10.0300	−	17.3725i	−0.357531	−	0.619263i	−0.987548	+	0.157319i	$0.949715\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	15.8226			0.562588
$792$		0			0
$793$	−19.7630			−0.701805
$794$		0			0
$795$		0			0
$796$		0			0
$797$	8.24761	+	14.2853i	0.292145	+	0.506011i	0.974317	−	0.225182i	$-0.0722975\pi$
$797$	8.24761	+	14.2853i	0.292145	+	0.506011i	−0.682171	+	0.731192i	$0.738964\pi$
$798$		0			0
$799$	3.64301	+	2.10329i	0.128881	+	0.0744092i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−53.9650	−	31.1567i	−1.90438	−	1.09950i
$804$		0			0
$805$	18.3917	−	10.6185i	0.648223	−	0.374252i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	15.2109i		−	0.534787i	−0.963587	−	0.267394i	$-0.913838\pi$
$809$		−	15.2109i		−	0.534787i	0.963587	−	0.267394i	$-0.0861624\pi$
$810$		0			0
$811$	9.18448			0.322511			0.161255	−	0.986913i	$-0.448446\pi$
$811$	9.18448			0.322511			0.161255	+	0.986913i	$0.448446\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−2.44333	−	4.23198i	−0.0855862	−	0.148240i
$816$		0			0
$817$	−26.9937	+	46.7545i	−0.944391	+	1.63573i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−1.37455	+	2.38079i	−0.0479720	+	0.0830900i	−0.889014	−	0.457879i	$-0.848609\pi$
$821$	−1.37455	+	2.38079i	−0.0479720	+	0.0830900i	0.841042	+	0.540969i	$0.181943\pi$
$822$		0			0
$823$	1.97316	−	1.13920i	0.0687799	−	0.0397101i	−0.465216	−	0.885197i	$-0.654023\pi$
$823$	1.97316	−	1.13920i	0.0687799	−	0.0397101i	0.533996	+	0.845487i	$0.320690\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			8.53141i			0.296666i	0.988937	+	0.148333i	$0.0473908\pi$
$827$			8.53141i			0.296666i	−0.988937	+	0.148333i	$0.952609\pi$
$828$		0			0
$829$		−	28.8554i		−	1.00219i	−0.865393	−	0.501094i	$-0.832931\pi$
$829$		−	28.8554i		−	1.00219i	0.865393	−	0.501094i	$-0.167069\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	1.46981	−	0.848595i	0.0509259	−	0.0294021i
$834$		0			0
$835$	7.41709	−	12.8468i	0.256679	−	0.444581i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	16.6368	−	28.8158i	0.574367	−	0.994833i	−0.421743	−	0.906716i	$-0.638581\pi$
$839$	16.6368	−	28.8158i	0.574367	−	0.994833i	0.996110	−	0.0881178i	$-0.0280852\pi$
$840$		0			0
$841$	6.34023	+	10.9816i	0.218628	+	0.378676i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−0.988022			−0.0339890
$846$		0			0
$847$			27.0945i			0.930979i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	53.4018	−	30.8316i	1.83059	−	1.05689i
$852$		0			0
$853$	42.4777	+	24.5245i	1.45441	+	0.839704i	0.998727	−	0.0504383i	$-0.0160618\pi$
$853$	42.4777	+	24.5245i	1.45441	+	0.839704i	0.455683	+	0.890142i	$0.349395\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−11.4294	−	6.59876i	−0.390420	−	0.225409i	0.291922	−	0.956442i	$-0.405705\pi$
$857$	−11.4294	−	6.59876i	−0.390420	−	0.225409i	−0.682342	+	0.731033i	$0.739039\pi$
$858$		0			0
$859$	6.61059	+	11.4499i	0.225550	+	0.390665i	0.956484	−	0.291783i	$-0.0942486\pi$
$859$	6.61059	+	11.4499i	0.225550	+	0.390665i	−0.730934	+	0.682448i	$0.760915\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	8.26851			0.281463			0.140732	−	0.990048i	$-0.455055\pi$
$863$	8.26851			0.281463			0.140732	+	0.990048i	$0.455055\pi$
$864$		0			0
$865$	−6.44113			−0.219005
$866$		0			0
$867$		0			0
$868$		0			0
$869$	14.5064	+	25.1258i	0.492095	+	0.852334i
$870$		0			0
$871$	40.0066	+	23.0978i	1.35557	+	0.782641i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	2.20775	+	1.27465i	0.0746357	+	0.0430909i
$876$		0			0
$877$	4.51357	−	2.60591i	0.152412	−	0.0879953i	−0.421854	−	0.906664i	$-0.638621\pi$
$877$	4.51357	−	2.60591i	0.152412	−	0.0879953i	0.574267	+	0.818668i	$0.305287\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	17.9255i		−	0.603925i	−0.953320	−	0.301963i	$-0.902358\pi$
$881$		−	17.9255i		−	0.603925i	0.953320	−	0.301963i	$-0.0976418\pi$
$882$		0			0
$883$	−20.5203			−0.690564			−0.345282	−	0.938499i	$-0.612217\pi$
$883$	−20.5203			−0.690564			−0.345282	+	0.938499i	$0.612217\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	19.1974	+	33.2509i	0.644587	+	1.11646i	0.984397	+	0.175963i	$0.0563039\pi$
$887$	19.1974	+	33.2509i	0.644587	+	1.11646i	−0.339810	+	0.940494i	$0.610363\pi$
$888$		0			0
$889$	−16.3508	+	28.3204i	−0.548388	+	0.949835i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	3.18531	−	5.51711i	0.106592	−	0.184623i
$894$		0			0
$895$	7.59236	−	4.38345i	0.253784	−	0.146523i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	22.7716i		−	0.759476i
$900$		0			0
$901$		−	6.95760i		−	0.231791i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	13.2997	−	7.67861i	0.442099	−	0.255246i
$906$		0			0
$907$	1.90633	−	3.30185i	0.0632985	−	0.109636i	−0.832639	−	0.553815i	$-0.813171\pi$
$907$	1.90633	−	3.30185i	0.0632985	−	0.109636i	0.895938	+	0.444179i	$0.146505\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	4.06826	−	7.04644i	0.134788	−	0.233459i	−0.790729	−	0.612167i	$-0.790298\pi$
$911$	4.06826	−	7.04644i	0.134788	−	0.233459i	0.925516	+	0.378708i	$0.123631\pi$
$912$		0			0
$913$	−32.4282	−	56.1673i	−1.07322	−	1.85887i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	24.5211			0.809758
$918$		0			0
$919$			25.3748i			0.837038i	0.908208	+	0.418519i	$0.137451\pi$
$919$			25.3748i			0.837038i	−0.908208	+	0.418519i	$0.862549\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	17.5943	−	10.1581i	0.579125	−	0.334358i
$924$		0			0
$925$	6.41038	+	3.70104i	0.210772	+	0.121689i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−35.7951	−	20.6663i	−1.17440	−	0.678040i	−0.219688	−	0.975570i	$-0.570504\pi$
$929$	−35.7951	−	20.6663i	−1.17440	−	0.678040i	−0.954713	+	0.297530i	$0.903837\pi$
$930$		0			0
$931$	−1.28514	−	2.22593i	−0.0421189	−	0.0729520i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−15.7511			−0.515117
$936$		0			0
$937$	−48.8089			−1.59452			−0.797258	−	0.603639i	$-0.793717\pi$
$937$	−48.8089			−1.59452			−0.797258	+	0.603639i	$0.793717\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−7.53979	−	13.0593i	−0.245790	−	0.425721i	0.716563	−	0.697522i	$-0.245714\pi$
$941$	−7.53979	−	13.0593i	−0.245790	−	0.425721i	−0.962353	+	0.271801i	$0.912381\pi$
$942$		0			0
$943$	43.5816	+	25.1619i	1.41921	+	0.819383i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−20.3781	−	11.7653i	−0.662198	−	0.382320i	0.130916	−	0.991394i	$-0.458208\pi$
$947$	−20.3781	−	11.7653i	−0.662198	−	0.382320i	−0.793114	+	0.609073i	$0.791542\pi$
$948$		0			0
$949$	43.3990	−	25.0564i	1.40879	−	0.813366i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			45.3356i			1.46856i	0.678845	+	0.734282i	$0.262481\pi$
$953$			45.3356i			1.46856i	−0.678845	+	0.734282i	$0.737519\pi$
$954$		0			0
$955$	−2.99211			−0.0968224
$956$		0			0
$957$		0			0
$958$		0			0
$959$	12.4766	+	21.6101i	0.402891	+	0.697827i
$960$		0			0
$961$	0.387293	−	0.670812i	0.0124933	−	0.0216391i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	1.27611	−	2.21028i	0.0410793	−	0.0711515i
$966$		0			0
$967$	−50.3479	+	29.0684i	−1.61908	+	0.934776i	−0.631921	+	0.775032i	$0.717733\pi$
$967$	−50.3479	+	29.0684i	−1.61908	+	0.934776i	−0.987159	+	0.159744i	$0.948933\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	15.8715i		−	0.509340i	−0.967028	−	0.254670i	$-0.918033\pi$
$971$		−	15.8715i		−	0.509340i	0.967028	−	0.254670i	$-0.0819669\pi$
$972$		0			0
$973$		−	15.9861i		−	0.512491i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	5.73553	−	3.31141i	0.183496	−	0.105941i	−0.405438	−	0.914122i	$-0.632881\pi$
$977$	5.73553	−	3.31141i	0.183496	−	0.105941i	0.588934	+	0.808181i	$0.299548\pi$
$978$		0			0
$979$	6.33921	−	10.9798i	0.202602	−	0.350917i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−7.73291	+	13.3938i	−0.246642	+	0.427196i	−0.962592	−	0.270956i	$-0.912660\pi$
$983$	−7.73291	+	13.3938i	−0.246642	+	0.427196i	0.715950	+	0.698151i	$0.245994\pi$
$984$		0			0
$985$	5.25835	+	9.10772i	0.167545	+	0.290196i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−87.6825			−2.78814
$990$		0			0
$991$		−	7.14184i		−	0.226868i	−0.993546	−	0.113434i	$-0.963815\pi$
$991$		−	7.14184i		−	0.226868i	0.993546	−	0.113434i	$-0.0361850\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−10.7528	+	6.20811i	−0.340885	+	0.196810i
$996$		0			0
$997$	14.6617	+	8.46495i	0.464341	+	0.268088i	0.713868	−	0.700280i	$-0.246942\pi$
$997$	14.6617	+	8.46495i	0.464341	+	0.268088i	−0.249527	+	0.968368i	$0.580275\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4320.2.cc.a.1871.18		48
3.2	odd	2		1440.2.cc.b.911.18		48
4.3	odd	2		1080.2.bm.a.251.9		48
8.3	odd	2		4320.2.cc.b.1871.7		48
8.5	even	2		1080.2.bm.b.251.16		48
9.4	even	3		1440.2.cc.a.1391.18		48
9.5	odd	6		4320.2.cc.b.3311.7		48
12.11	even	2		360.2.bm.b.11.16	yes	48
24.5	odd	2		360.2.bm.a.11.9	✓	48
24.11	even	2		1440.2.cc.a.911.18		48
36.23	even	6		1080.2.bm.b.611.16		48
36.31	odd	6		360.2.bm.a.131.9	yes	48
72.5	odd	6		1080.2.bm.a.611.9		48
72.13	even	6		360.2.bm.b.131.16	yes	48
72.59	even	6	inner	4320.2.cc.a.3311.18		48
72.67	odd	6		1440.2.cc.b.1391.18		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
360.2.bm.a.11.9	✓	48	24.5	odd	2
360.2.bm.a.131.9	yes	48	36.31	odd	6
360.2.bm.b.11.16	yes	48	12.11	even	2
360.2.bm.b.131.16	yes	48	72.13	even	6
1080.2.bm.a.251.9		48	4.3	odd	2
1080.2.bm.a.611.9		48	72.5	odd	6
1080.2.bm.b.251.16		48	8.5	even	2
1080.2.bm.b.611.16		48	36.23	even	6
1440.2.cc.a.911.18		48	24.11	even	2
1440.2.cc.a.1391.18		48	9.4	even	3
1440.2.cc.b.911.18		48	3.2	odd	2
1440.2.cc.b.1391.18		48	72.67	odd	6
4320.2.cc.a.1871.18		48	1.1	even	1	trivial
4320.2.cc.a.3311.18		48	72.59	even	6	inner
4320.2.cc.b.1871.7		48	8.3	odd	2
4320.2.cc.b.3311.7		48	9.5	odd	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:	\( N \)	\(=\)	\( 4320 = 2^{5} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4320.cc (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 - 0.866025i) q^{5} +(2.20775 + 1.27465i) q^{7} +O(q^{10})\)	\(q+(-0.500000 - 0.866025i) q^{5} +(2.20775 + 1.27465i) q^{7} +(-4.02755 - 2.32531i) q^{11} +(3.23898 - 1.87003i) q^{13} +3.38689i q^{17} +5.12923 q^{19} +(4.16526 + 7.21444i) q^{23} +(-0.500000 + 0.866025i) q^{25} +(-2.01987 + 3.49852i) q^{29} +(-4.88169 + 2.81845i) q^{31} -2.54929i q^{35} -7.40207i q^{37} +(5.23156 - 3.02044i) q^{41} +(-5.26273 + 9.11531i) q^{43} +(0.621010 - 1.07562i) q^{47} +(-0.250553 - 0.433970i) q^{49} -2.05427 q^{53} +4.65062i q^{55} +(-0.351174 + 0.202750i) q^{59} +(-4.57621 - 2.64207i) q^{61} +(-3.23898 - 1.87003i) q^{65} +(6.17580 + 10.6968i) q^{67} +5.43205 q^{71} +13.3990 q^{73} +(-5.92789 - 10.2674i) q^{77} +(-5.40268 - 3.11924i) q^{79} +(12.0774 + 6.97289i) q^{83} +(2.93313 - 1.69345i) q^{85} +2.72618i q^{89} +9.53450 q^{91} +(-2.56462 - 4.44204i) q^{95} +(1.31601 - 2.27939i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 24 q^{5}+O(q^{10}) \) 48 * q - 24 * q^5	\( 48 q - 24 q^{5} - 24 q^{25} - 12 q^{41} - 12 q^{47} + 24 q^{49} - 36 q^{59} - 12 q^{61} - 60 q^{83}+O(q^{100}) \) 48 * q - 24 * q^5 - 24 * q^25 - 12 * q^41 - 12 * q^47 + 24 * q^49 - 36 * q^59 - 12 * q^61 - 60 * q^83

Introduction

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