LMFDB - Embedded newform 1080.4.q.b.721.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1080,4,Mod(361,1080)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1080.361"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1080, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1080 = 2^{3} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1080.q (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,40,0,-34] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$63.7220628062$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 6 x^{15} + 16 x^{14} + 14 x^{13} - 284 x^{12} + 764 x^{11} + 19770 x^{10} + 55106 x^{9} + \cdots + 193472540143 $ x^16 - 6x^15 + 16x^14 + 14x^13 - 284x^12 + 764x^11 + 19770x^10 + 55106x^9 + 266991x^8 + 335444x^7 - 4573302x^6 + 2211506x^5 + 251845642x^4 + 1438820210x^3 + 12022873630x^2 + 75245140836*x + 193472540143
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{14} $
Twist minimal:	no (minimal twist has level 360)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			721.3
Root			$5.28147 + 2.94432i$ of defining polynomial
Character	$\chi$	$=$	1080.721
Dual form			1080.4.q.b.361.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.50000 + 4.33013i) q^{5} +(-5.91307 + 10.2417i) q^{7} +(5.28382 - 9.15185i) q^{11} +(10.0854 + 17.4685i) q^{13} +95.1376 q^{17} -8.49782 q^{19} +(-31.7906 - 55.0629i) q^{23} +(-12.5000 + 21.6506i) q^{25} +(90.4720 - 156.702i) q^{29} +(67.8404 + 117.503i) q^{31} -59.1307 q^{35} -82.5485 q^{37} +(59.9638 + 103.860i) q^{41} +(-11.9830 + 20.7552i) q^{43} +(24.8195 - 42.9886i) q^{47} +(101.571 + 175.927i) q^{49} +70.1397 q^{53} +52.8382 q^{55} +(214.018 + 370.691i) q^{59} +(-359.534 + 622.730i) q^{61} +(-50.4272 + 87.3425i) q^{65} +(-130.985 - 226.872i) q^{67} +77.8019 q^{71} +626.739 q^{73} +(62.4872 + 108.231i) q^{77} +(-186.943 + 323.796i) q^{79} +(-457.533 + 792.471i) q^{83} +(237.844 + 411.958i) q^{85} -905.545 q^{89} -238.544 q^{91} +(-21.2446 - 36.7967i) q^{95} +(-366.564 + 634.907i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 40 q^{5} - 34 q^{7} + 64 q^{11} - 48 q^{13} - 212 q^{17} + 456 q^{19} + 166 q^{23} - 200 q^{25} + 110 q^{29} - 160 q^{31} - 340 q^{35} + 104 q^{37} + 280 q^{41} + 136 q^{43} + 594 q^{47} - 1094 q^{49}+ \cdots + 182 q^{97}+O(q^{100}) $ 16 * q + 40 * q^5 - 34 * q^7 + 64 * q^11 - 48 * q^13 - 212 * q^17 + 456 * q^19 + 166 * q^23 - 200 * q^25 + 110 * q^29 - 160 * q^31 - 340 * q^35 + 104 * q^37 + 280 * q^41 + 136 * q^43 + 594 * q^47 - 1094 * q^49 - 752 * q^53 + 640 * q^55 + 24 * q^59 - 950 * q^61 + 240 * q^65 - 426 * q^67 + 136 * q^71 + 308 * q^73 - 696 * q^77 - 764 * q^79 + 1362 * q^83 - 530 * q^85 + 172 * q^89 - 1224 * q^91 + 1140 * q^95 + 182 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1080\mathbb{Z}\right)^\times$.

$n$	$217$	$271$	$541$	$1001$
$\chi(n)$	$1$	$1$	$1$	$e\left(\frac{2}{3}\right)$

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
$5$	2.50000	+	4.33013i	0.223607	+	0.387298i
$5$	2.50000	+	4.33013i	0.223607	+	0.387298i
$6$		0			0
$7$	−5.91307	+	10.2417i	−0.319276	+	0.553002i	−0.980337	−	0.197330i	$-0.936773\pi$
$7$	−5.91307	+	10.2417i	−0.319276	+	0.553002i	0.661061	+	0.750332i	$0.270106\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	5.28382	−	9.15185i	0.144830	−	0.250853i	−0.784479	−	0.620155i	$-0.787070\pi$
$11$	5.28382	−	9.15185i	0.144830	−	0.250853i	0.929310	+	0.369302i	$0.120403\pi$
$12$		0			0
$13$	10.0854	+	17.4685i	0.215169	+	0.372684i	0.953325	−	0.301946i	$-0.0976364\pi$
$13$	10.0854	+	17.4685i	0.215169	+	0.372684i	−0.738156	+	0.674630i	$0.764303\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	95.1376			1.35731			0.678655	−	0.734457i	$-0.262563\pi$
$17$	95.1376			1.35731			0.678655	+	0.734457i	$0.262563\pi$
$18$		0			0
$19$	−8.49782			−0.102607			−0.0513035	−	0.998683i	$-0.516338\pi$
$19$	−8.49782			−0.102607			−0.0513035	+	0.998683i	$0.516338\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−31.7906	−	55.0629i	−0.288208	−	0.499191i	0.685174	−	0.728380i	$-0.259726\pi$
$23$	−31.7906	−	55.0629i	−0.288208	−	0.499191i	−0.973382	+	0.229188i	$0.926393\pi$
$24$		0			0
$25$	−12.5000	+	21.6506i	−0.100000	+	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	90.4720	−	156.702i	0.579318	−	1.00341i	−0.416239	−	0.909255i	$-0.636652\pi$
$29$	90.4720	−	156.702i	0.579318	−	1.00341i	0.995558	−	0.0941536i	$-0.0300145\pi$
$30$		0			0
$31$	67.8404	+	117.503i	0.393049	+	0.680780i	0.992850	−	0.119369i	$-0.0380871\pi$
$31$	67.8404	+	117.503i	0.393049	+	0.680780i	−0.599801	+	0.800149i	$0.704754\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−59.1307			−0.285569
$36$		0			0
$37$	−82.5485			−0.366781			−0.183390	−	0.983040i	$-0.558707\pi$
$37$	−82.5485			−0.366781			−0.183390	+	0.983040i	$0.558707\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	59.9638	+	103.860i	0.228409	+	0.395616i	0.957337	−	0.288974i	$-0.0933143\pi$
$41$	59.9638	+	103.860i	0.228409	+	0.395616i	−0.728928	+	0.684591i	$0.759981\pi$
$42$		0			0
$43$	−11.9830	+	20.7552i	−0.0424976	+	0.0736080i	−0.886492	−	0.462744i	$-0.846865\pi$
$43$	−11.9830	+	20.7552i	−0.0424976	+	0.0736080i	0.843994	+	0.536352i	$0.180198\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	24.8195	−	42.9886i	0.0770275	−	0.133415i	−0.824939	−	0.565222i	$-0.808790\pi$
$47$	24.8195	−	42.9886i	0.0770275	−	0.133415i	0.901966	+	0.431807i	$0.142124\pi$
$48$		0			0
$49$	101.571	+	175.927i	0.296126	+	0.512905i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	70.1397			0.181782			0.0908909	−	0.995861i	$-0.471029\pi$
$53$	70.1397			0.181782			0.0908909	+	0.995861i	$0.471029\pi$
$54$		0			0
$55$	52.8382			0.129540
$56$		0			0
$57$		0			0
$58$		0			0
$59$	214.018	+	370.691i	0.472251	+	0.817963i	0.999496	−	0.0317502i	$-0.0101081\pi$
$59$	214.018	+	370.691i	0.472251	+	0.817963i	−0.527244	+	0.849714i	$0.676775\pi$
$60$		0			0
$61$	−359.534	+	622.730i	−0.754648	+	1.30709i	0.190901	+	0.981609i	$0.438859\pi$
$61$	−359.534	+	622.730i	−0.754648	+	1.30709i	−0.945549	+	0.325480i	$0.894474\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−50.4272	+	87.3425i	−0.0962266	+	0.166669i
$66$		0			0
$67$	−130.985	−	226.872i	−0.238840	−	0.413684i	0.721541	−	0.692371i	$-0.243434\pi$
$67$	−130.985	−	226.872i	−0.238840	−	0.413684i	−0.960382	+	0.278687i	$0.910101\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	77.8019			0.130048			0.0650238	−	0.997884i	$-0.479288\pi$
$71$	77.8019			0.130048			0.0650238	+	0.997884i	$0.479288\pi$
$72$		0			0
$73$	626.739			1.00485			0.502426	−	0.864620i	$-0.332441\pi$
$73$	626.739			1.00485			0.502426	+	0.864620i	$0.332441\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	62.4872	+	108.231i	0.0924815	+	0.160183i
$78$		0			0
$79$	−186.943	+	323.796i	−0.266238	+	0.461137i	−0.967887	−	0.251385i	$-0.919114\pi$
$79$	−186.943	+	323.796i	−0.266238	+	0.461137i	0.701649	+	0.712522i	$0.252447\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−457.533	+	792.471i	−0.605070	+	1.04801i	0.386970	+	0.922092i	$0.373522\pi$
$83$	−457.533	+	792.471i	−0.605070	+	1.04801i	−0.992040	+	0.125920i	$0.959812\pi$
$84$		0			0
$85$	237.844	+	411.958i	0.303504	+	0.525684i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−905.545			−1.07851			−0.539256	−	0.842142i	$-0.681294\pi$
$89$	−905.545			−1.07851			−0.539256	+	0.842142i	$0.681294\pi$
$90$		0			0
$91$	−238.544			−0.274793
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−21.2446	−	36.7967i	−0.0229436	−	0.0397396i
$96$		0			0
$97$	−366.564	+	634.907i	−0.383700	+	0.664588i	−0.991588	−	0.129435i	$-0.958684\pi$
$97$	−366.564	+	634.907i	−0.383700	+	0.664588i	0.607888	+	0.794023i	$0.292017\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	722.727	−	1251.80i	0.712020	−	1.23325i	−0.252078	−	0.967707i	$-0.581114\pi$
$101$	722.727	−	1251.80i	0.712020	−	1.23325i	0.964098	−	0.265548i	$-0.0855529\pi$
$102$		0			0
$103$	−121.055	−	209.673i	−0.115804	−	0.200579i	0.802297	−	0.596926i	$-0.203611\pi$
$103$	−121.055	−	209.673i	−0.115804	−	0.200579i	−0.918101	+	0.396346i	$0.870278\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1094.76			−0.989108			−0.494554	−	0.869147i	$-0.664669\pi$
$107$	−1094.76			−0.989108			−0.494554	+	0.869147i	$0.664669\pi$
$108$		0			0
$109$	1759.83			1.54643			0.773214	−	0.634145i	$-0.218648\pi$
$109$	1759.83			1.54643			0.773214	+	0.634145i	$0.218648\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	214.669	+	371.818i	0.178711	+	0.309537i	0.941439	−	0.337182i	$-0.109474\pi$
$113$	214.669	+	371.818i	0.178711	+	0.309537i	−0.762728	+	0.646719i	$0.776141\pi$
$114$		0			0
$115$	158.953	−	275.314i	0.128891	−	0.223245i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−562.555	+	974.374i	−0.433356	+	0.750595i
$120$		0			0
$121$	609.662	+	1055.97i	0.458048	+	0.793363i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−125.000			−0.0894427
$126$		0			0
$127$	−271.242			−0.189518			−0.0947592	−	0.995500i	$-0.530208\pi$
$127$	−271.242			−0.189518			−0.0947592	+	0.995500i	$0.530208\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	1404.04	+	2431.87i	0.936424	+	1.62193i	0.772075	+	0.635532i	$0.219219\pi$
$131$	1404.04	+	2431.87i	0.936424	+	1.62193i	0.164349	+	0.986402i	$0.447448\pi$
$132$		0			0
$133$	50.2482	−	87.0325i	0.0327599	−	0.0567419i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−1094.02	+	1894.89i	−0.682250	+	1.18169i	0.292043	+	0.956405i	$0.405665\pi$
$137$	−1094.02	+	1894.89i	−0.682250	+	1.18169i	−0.974293	+	0.225286i	$0.927668\pi$
$138$		0			0
$139$	349.941	+	606.116i	0.213537	+	0.369857i	0.952819	−	0.303539i	$-0.0981683\pi$
$139$	349.941	+	606.116i	0.213537	+	0.369857i	−0.739282	+	0.673396i	$0.764835\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	213.159			0.124652
$144$		0			0
$145$	904.720			0.518158
$146$		0			0
$147$		0			0
$148$		0			0
$149$	111.212	+	192.626i	0.0611468	+	0.105909i	0.894978	−	0.446110i	$-0.147191\pi$
$149$	111.212	+	192.626i	0.0611468	+	0.105909i	−0.833831	+	0.552019i	$0.813858\pi$
$150$		0			0
$151$	1198.98	−	2076.69i	0.646169	−	1.11920i	−0.337861	−	0.941196i	$-0.609703\pi$
$151$	1198.98	−	2076.69i	0.646169	−	1.11920i	0.984030	−	0.178002i	$-0.0569634\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−339.202	+	587.516i	−0.175777	+	0.304454i
$156$		0			0
$157$	1165.79	+	2019.21i	0.592612	+	1.02643i	0.993879	+	0.110473i	$0.0352367\pi$
$157$	1165.79	+	2019.21i	0.592612	+	1.02643i	−0.401267	+	0.915961i	$0.631430\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	751.919			0.368072
$162$		0			0
$163$	75.4086			0.0362359			0.0181180	−	0.999836i	$-0.494233\pi$
$163$	75.4086			0.0362359			0.0181180	+	0.999836i	$0.494233\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	1162.13	+	2012.88i	0.538495	+	0.932701i	0.998985	+	0.0450362i	$0.0143403\pi$
$167$	1162.13	+	2012.88i	0.538495	+	0.932701i	−0.460490	+	0.887665i	$0.652326\pi$
$168$		0			0
$169$	895.068	−	1550.30i	0.407404	−	0.705645i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−917.804	+	1589.68i	−0.403349	+	0.698621i	−0.994128	−	0.108213i	$-0.965487\pi$
$173$	−917.804	+	1589.68i	−0.403349	+	0.698621i	0.590779	+	0.806834i	$0.298821\pi$
$174$		0			0
$175$	−147.827	−	256.043i	−0.0638551	−	0.110600i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−4347.88			−1.81551			−0.907754	−	0.419502i	$-0.862205\pi$
$179$	−4347.88			−1.81551			−0.907754	+	0.419502i	$0.862205\pi$
$180$		0			0
$181$	2194.28			0.901103			0.450551	−	0.892751i	$-0.351227\pi$
$181$	2194.28			0.901103			0.450551	+	0.892751i	$0.351227\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−206.371	−	357.445i	−0.0820147	−	0.142054i
$186$		0			0
$187$	502.690	−	870.685i	0.196579	−	0.340486i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−1271.94	+	2203.07i	−0.481857	+	0.834601i	−0.999783	−	0.0208244i	$-0.993371\pi$
$191$	−1271.94	+	2203.07i	−0.481857	+	0.834601i	0.517926	+	0.855425i	$0.326704\pi$
$192$		0			0
$193$	212.642	+	368.306i	0.0793071	+	0.137364i	0.902951	−	0.429743i	$-0.141396\pi$
$193$	212.642	+	368.306i	0.0793071	+	0.137364i	−0.823644	+	0.567107i	$0.808063\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−625.217			−0.226116			−0.113058	−	0.993588i	$-0.536065\pi$
$197$	−625.217			−0.226116			−0.113058	+	0.993588i	$0.536065\pi$
$198$		0			0
$199$	3705.77			1.32007			0.660037	−	0.751233i	$-0.270541\pi$
$199$	3705.77			1.32007			0.660037	+	0.751233i	$0.270541\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	1069.93	+	1853.18i	0.369924	+	0.640728i
$204$		0			0
$205$	−299.819	+	519.302i	−0.102148	+	0.176925i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−44.9010	+	77.7708i	−0.0148606	+	0.0257393i
$210$		0			0
$211$	108.659	+	188.203i	0.0354521	+	0.0614049i	0.883207	−	0.468983i	$-0.155380\pi$
$211$	108.659	+	188.203i	0.0354521	+	0.0614049i	−0.847755	+	0.530388i	$0.822046\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−119.830			−0.0380110
$216$		0			0
$217$	−1604.58			−0.501963
$218$		0			0
$219$		0			0
$220$		0			0
$221$	959.506	+	1661.91i	0.292051	+	0.505848i
$222$		0			0
$223$	−2588.06	+	4482.65i	−0.777172	+	1.34610i	0.156394	+	0.987695i	$0.450013\pi$
$223$	−2588.06	+	4482.65i	−0.777172	+	1.34610i	−0.933566	+	0.358406i	$0.883320\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−722.731	+	1251.81i	−0.211319	+	0.366015i	−0.952127	−	0.305701i	$-0.901109\pi$
$227$	−722.731	+	1251.81i	−0.211319	+	0.366015i	0.740809	+	0.671716i	$0.234442\pi$
$228$		0			0
$229$	2949.65	+	5108.95i	0.851173	+	1.47428i	0.880150	+	0.474695i	$0.157442\pi$
$229$	2949.65	+	5108.95i	0.851173	+	1.47428i	−0.0289770	+	0.999580i	$0.509225\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	4962.50			1.39530			0.697648	−	0.716441i	$-0.254230\pi$
$233$	4962.50			1.39530			0.697648	+	0.716441i	$0.254230\pi$
$234$		0			0
$235$	248.195			0.0688954
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−2785.66	−	4824.91i	−0.753931	−	1.30585i	−0.945904	−	0.324446i	$-0.894822\pi$
$239$	−2785.66	−	4824.91i	−0.753931	−	1.30585i	0.191973	−	0.981400i	$-0.438511\pi$
$240$		0			0
$241$	−2711.29	+	4696.09i	−0.724687	+	1.25519i	0.234416	+	0.972136i	$0.424682\pi$
$241$	−2711.29	+	4696.09i	−0.724687	+	1.25519i	−0.959103	+	0.283058i	$0.908651\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−507.856	+	879.633i	−0.132432	+	0.229378i
$246$		0			0
$247$	−85.7043	−	148.444i	−0.0220779	−	0.0382400i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−6523.01			−1.64035			−0.820176	−	0.572111i	$-0.806125\pi$
$251$	−6523.01			−1.64035			−0.820176	+	0.572111i	$0.806125\pi$
$252$		0			0
$253$	−671.903			−0.166965
$254$		0			0
$255$		0			0
$256$		0			0
$257$	2104.56	+	3645.21i	0.510813	+	0.884755i	0.999921	+	0.0125315i	$0.00398899\pi$
$257$	2104.56	+	3645.21i	0.510813	+	0.884755i	−0.489108	+	0.872223i	$0.662678\pi$
$258$		0			0
$259$	488.115	−	845.440i	0.117104	−	0.202830i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	3441.10	−	5960.16i	0.806796	−	1.39741i	−0.108275	−	0.994121i	$-0.534533\pi$
$263$	3441.10	−	5960.16i	0.806796	−	1.39741i	0.915072	−	0.403291i	$-0.132134\pi$
$264$		0			0
$265$	175.349	+	303.714i	0.0406476	+	0.0704038i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	1959.61			0.444162			0.222081	−	0.975028i	$-0.428715\pi$
$269$	1959.61			0.444162			0.222081	+	0.975028i	$0.428715\pi$
$270$		0			0
$271$	−2727.70			−0.611424			−0.305712	−	0.952124i	$-0.598894\pi$
$271$	−2727.70			−0.611424			−0.305712	+	0.952124i	$0.598894\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	132.096	+	228.796i	0.0289660	+	0.0501707i
$276$		0			0
$277$	1287.18	−	2229.47i	0.279204	−	0.483595i	−0.691984	−	0.721913i	$-0.743263\pi$
$277$	1287.18	−	2229.47i	0.279204	−	0.483595i	0.971187	+	0.238319i	$0.0765963\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	2200.93	−	3812.12i	0.467247	−	0.809296i	−0.532053	−	0.846711i	$-0.678579\pi$
$281$	2200.93	−	3812.12i	0.467247	−	0.809296i	0.999300	+	0.0374156i	$0.0119125\pi$
$282$		0			0
$283$	−2241.80	−	3882.92i	−0.470888	−	0.815602i	0.528557	−	0.848898i	$-0.322733\pi$
$283$	−2241.80	−	3882.92i	−0.470888	−	0.815602i	−0.999446	+	0.0332954i	$0.989400\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−1418.28			−0.291702
$288$		0			0
$289$	4138.17			0.842290
$290$		0			0
$291$		0			0
$292$		0			0
$293$	2304.41	+	3991.36i	0.459472	+	0.795829i	0.998933	−	0.0461815i	$-0.0147053\pi$
$293$	2304.41	+	3991.36i	0.459472	+	0.795829i	−0.539461	+	0.842011i	$0.681372\pi$
$294$		0			0
$295$	−1070.09	+	1853.45i	−0.211197	+	0.365804i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	641.244	−	1110.67i	0.124027	−	0.214821i
$300$		0			0
$301$	−141.713	−	245.454i	−0.0271369	−	0.0470025i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−3595.34			−0.674978
$306$		0			0
$307$	−3962.95			−0.736735			−0.368368	−	0.929680i	$-0.620083\pi$
$307$	−3962.95			−0.736735			−0.368368	+	0.929680i	$0.620083\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−794.224	−	1375.64i	−0.144811	−	0.250821i	0.784491	−	0.620140i	$-0.212924\pi$
$311$	−794.224	−	1375.64i	−0.144811	−	0.250821i	−0.929303	+	0.369319i	$0.879591\pi$
$312$		0			0
$313$	721.627	−	1249.90i	0.130316	−	0.225713i	−0.793483	−	0.608593i	$-0.791734\pi$
$313$	721.627	−	1249.90i	0.130316	−	0.225713i	0.923798	+	0.382880i	$0.125068\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	1831.70	−	3172.59i	0.324538	−	0.562116i	−0.656881	−	0.753994i	$-0.728125\pi$
$317$	1831.70	−	3172.59i	0.324538	−	0.562116i	0.981419	+	0.191879i	$0.0614580\pi$
$318$		0			0
$319$	−956.076	−	1655.97i	−0.167806	−	0.290648i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−808.463			−0.139270
$324$		0			0
$325$	−504.272			−0.0860677
$326$		0			0
$327$		0			0
$328$		0			0
$329$	293.518	+	508.389i	0.0491860	+	0.0851926i
$330$		0			0
$331$	2292.45	−	3970.64i	0.380678	−	0.659354i	−0.610481	−	0.792031i	$-0.709024\pi$
$331$	2292.45	−	3970.64i	0.380678	−	0.659354i	0.991159	+	0.132677i	$0.0423572\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	654.923	−	1134.36i	0.106813	−	0.185005i
$336$		0			0
$337$	1479.45	+	2562.49i	0.239142	+	0.414206i	0.960468	−	0.278389i	$-0.0898005\pi$
$337$	1479.45	+	2562.49i	0.239142	+	0.414206i	−0.721326	+	0.692595i	$0.756467\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	1433.83			0.227701
$342$		0			0
$343$	−6458.76			−1.01673
$344$		0			0
$345$		0			0
$346$		0			0
$347$	1591.87	+	2757.20i	0.246271	+	0.426553i	0.962488	−	0.271324i	$-0.0874614\pi$
$347$	1591.87	+	2757.20i	0.246271	+	0.426553i	−0.716217	+	0.697877i	$0.754128\pi$
$348$		0			0
$349$	2715.50	−	4703.38i	0.416496	−	0.721392i	−0.579088	−	0.815265i	$-0.696591\pi$
$349$	2715.50	−	4703.38i	0.416496	−	0.721392i	0.995584	+	0.0938725i	$0.0299246\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	6296.17	−	10905.3i	0.949324	−	1.64428i	0.202470	−	0.979288i	$-0.435103\pi$
$353$	6296.17	−	10905.3i	0.949324	−	1.64428i	0.746854	−	0.664988i	$-0.231564\pi$
$354$		0			0
$355$	194.505	+	336.892i	0.0290795	+	0.0503672i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−4529.29			−0.665868			−0.332934	−	0.942950i	$-0.608039\pi$
$359$	−4529.29			−0.665868			−0.332934	+	0.942950i	$0.608039\pi$
$360$		0			0
$361$	−6786.79			−0.989472
$362$		0			0
$363$		0			0
$364$		0			0
$365$	1566.85	+	2713.86i	0.224692	+	0.389178i
$366$		0			0
$367$	−3984.83	+	6901.93i	−0.566775	+	0.981683i	0.430107	+	0.902778i	$0.358476\pi$
$367$	−3984.83	+	6901.93i	−0.566775	+	0.981683i	−0.996882	+	0.0789054i	$0.974857\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−414.741	+	718.353i	−0.0580385	+	0.100526i
$372$		0			0
$373$	−2315.26	−	4010.15i	−0.321393	−	0.556669i	0.659383	−	0.751807i	$-0.270818\pi$
$373$	−2315.26	−	4010.15i	−0.321393	−	0.556669i	−0.980776	+	0.195138i	$0.937484\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	3649.80			0.498606
$378$		0			0
$379$	5684.71			0.770459			0.385229	−	0.922821i	$-0.374122\pi$
$379$	5684.71			0.770459			0.385229	+	0.922821i	$0.374122\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	45.7892	+	79.3092i	0.00610893	+	0.0105810i	0.869064	−	0.494700i	$-0.164722\pi$
$383$	45.7892	+	79.3092i	0.00610893	+	0.0105810i	−0.862955	+	0.505281i	$0.831389\pi$
$384$		0			0
$385$	−312.436	+	541.155i	−0.0413590	+	0.0716359i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	5545.59	−	9605.24i	0.722808	−	1.25194i	−0.237062	−	0.971495i	$-0.576184\pi$
$389$	5545.59	−	9605.24i	0.722808	−	1.25194i	0.959870	−	0.280446i	$-0.0904823\pi$
$390$		0			0
$391$	−3024.48	−	5238.55i	−0.391188	−	0.677557i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−1869.43			−0.238130
$396$		0			0
$397$	−9625.41			−1.21684			−0.608420	−	0.793615i	$-0.708196\pi$
$397$	−9625.41			−1.21684			−0.608420	+	0.793615i	$0.708196\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	1767.56	+	3061.50i	0.220119	+	0.381257i	0.954844	−	0.297108i	$-0.0960222\pi$
$401$	1767.56	+	3061.50i	0.220119	+	0.381257i	−0.734725	+	0.678365i	$0.762689\pi$
$402$		0			0
$403$	−1368.40	+	2370.14i	−0.169144	+	0.292966i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−436.172	+	755.471i	−0.0531209	+	0.0920082i
$408$		0			0
$409$	5166.68	+	8948.96i	0.624636	+	1.08190i	0.988611	+	0.150493i	$0.0480859\pi$
$409$	5166.68	+	8948.96i	0.624636	+	1.08190i	−0.363975	+	0.931409i	$0.618581\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−5062.02			−0.603114
$414$		0			0
$415$	−4575.33			−0.541191
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−1340.11	−	2321.15i	−0.156250	−	0.270634i	0.777263	−	0.629176i	$-0.216607\pi$
$419$	−1340.11	−	2321.15i	−0.156250	−	0.270634i	−0.933514	+	0.358542i	$0.883274\pi$
$420$		0			0
$421$	6031.60	−	10447.0i	0.698248	−	1.20940i	−0.270826	−	0.962628i	$-0.587297\pi$
$421$	6031.60	−	10447.0i	0.698248	−	1.20940i	0.969074	−	0.246772i	$-0.0793698\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−1189.22	+	2059.79i	−0.135731	+	0.235093i
$426$		0			0
$427$	−4251.89	−	7364.49i	−0.481882	−	0.834644i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	719.508			0.0804118			0.0402059	−	0.999191i	$-0.487199\pi$
$431$	719.508			0.0804118			0.0402059	+	0.999191i	$0.487199\pi$
$432$		0			0
$433$	3689.01			0.409429			0.204714	−	0.978822i	$-0.434373\pi$
$433$	3689.01			0.409429			0.204714	+	0.978822i	$0.434373\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	270.151	+	467.915i	0.0295722	+	0.0512206i
$438$		0			0
$439$	1128.49	−	1954.61i	0.122688	−	0.212502i	−0.798139	−	0.602474i	$-0.794182\pi$
$439$	1128.49	−	1954.61i	0.122688	−	0.212502i	0.920827	+	0.389972i	$0.127515\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	3072.90	−	5322.42i	0.329566	−	0.570825i	−0.652860	−	0.757479i	$-0.726431\pi$
$443$	3072.90	−	5322.42i	0.329566	−	0.570825i	0.982426	+	0.186654i	$0.0597642\pi$
$444$		0			0
$445$	−2263.86	−	3921.12i	−0.241163	−	0.417706i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−4288.40			−0.450740			−0.225370	−	0.974273i	$-0.572359\pi$
$449$	−4288.40			−0.450740			−0.225370	+	0.974273i	$0.572359\pi$
$450$		0			0
$451$	1267.35			0.132322
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−596.359	−	1032.92i	−0.0614456	−	0.106427i
$456$		0			0
$457$	−6463.45	+	11195.0i	−0.661592	+	1.14591i	0.318605	+	0.947888i	$0.396786\pi$
$457$	−6463.45	+	11195.0i	−0.661592	+	1.14591i	−0.980197	+	0.198024i	$0.936548\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	7265.08	−	12583.5i	0.733988	−	1.27130i	−0.221178	−	0.975233i	$-0.570990\pi$
$461$	7265.08	−	12583.5i	0.733988	−	1.27130i	0.955166	−	0.296071i	$-0.0956764\pi$
$462$		0			0
$463$	765.864	+	1326.51i	0.0768741	+	0.133150i	0.901900	−	0.431946i	$-0.142173\pi$
$463$	765.864	+	1326.51i	0.0768741	+	0.133150i	−0.825026	+	0.565095i	$0.808839\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−16271.7			−1.61234			−0.806171	−	0.591683i	$-0.798464\pi$
$467$	−16271.7			−1.61234			−0.806171	+	0.591683i	$0.798464\pi$
$468$		0			0
$469$	3098.08			0.305024
$470$		0			0
$471$		0			0
$472$		0			0
$473$	126.633	+	219.334i	0.0123099	+	0.0213213i
$474$		0			0
$475$	106.223	−	183.983i	0.0102607	−	0.0177721i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	9326.56	−	16154.1i	0.889648	−	1.54092i	0.0493566	−	0.998781i	$-0.484283\pi$
$479$	9326.56	−	16154.1i	0.889648	−	1.54092i	0.840292	−	0.542135i	$-0.182384\pi$
$480$		0			0
$481$	−832.538	−	1442.00i	−0.0789199	−	0.136693i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−3665.64			−0.343192
$486$		0			0
$487$	−6328.11			−0.588817			−0.294409	−	0.955680i	$-0.595123\pi$
$487$	−6328.11			−0.588817			−0.294409	+	0.955680i	$0.595123\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−2178.77	−	3773.74i	−0.200258	−	0.346857i	0.748354	−	0.663300i	$-0.230845\pi$
$491$	−2178.77	−	3773.74i	−0.200258	−	0.346857i	−0.948611	+	0.316443i	$0.897511\pi$
$492$		0			0
$493$	8607.29	−	14908.3i	0.786314	−	1.36194i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−460.048	+	796.826i	−0.0415211	+	0.0719166i
$498$		0			0
$499$	−6847.63	−	11860.4i	−0.614312	−	1.06402i	−0.990505	−	0.137479i	$-0.956100\pi$
$499$	−6847.63	−	11860.4i	−0.614312	−	1.06402i	0.376192	−	0.926542i	$-0.377233\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−6429.71			−0.569953			−0.284977	−	0.958534i	$-0.591986\pi$
$503$	−6429.71			−0.569953			−0.284977	+	0.958534i	$0.591986\pi$
$504$		0			0
$505$	7227.27			0.636850
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−1599.64	−	2770.65i	−0.139298	−	0.241271i	0.787933	−	0.615761i	$-0.211151\pi$
$509$	−1599.64	−	2770.65i	−0.139298	−	0.241271i	−0.927231	+	0.374490i	$0.877818\pi$
$510$		0			0
$511$	−3705.95	+	6418.89i	−0.320825	+	0.555685i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	605.273	−	1048.36i	0.0517893	−	0.0897017i
$516$		0			0
$517$	−262.283	−	454.288i	−0.0223118	−	0.0386452i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	11378.1			0.956783			0.478391	−	0.878147i	$-0.341220\pi$
$521$	11378.1			0.956783			0.478391	+	0.878147i	$0.341220\pi$
$522$		0			0
$523$	15286.8			1.27810			0.639051	−	0.769164i	$-0.279327\pi$
$523$	15286.8			1.27810			0.639051	+	0.769164i	$0.279327\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	6454.18	+	11179.0i	0.533489	+	0.924029i
$528$		0			0
$529$	4062.22	−	7035.97i	0.333872	−	0.578283i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−1209.52	+	2094.96i	−0.0982932	+	0.170249i
$534$		0			0
$535$	−2736.90	−	4740.45i	−0.221171	−	0.383080i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	2146.74			0.171552
$540$		0			0
$541$	−8233.95			−0.654354			−0.327177	−	0.944963i	$-0.606097\pi$
$541$	−8233.95			−0.654354			−0.327177	+	0.944963i	$0.606097\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	4399.56	+	7620.27i	0.345792	+	0.598929i
$546$		0			0
$547$	5101.54	−	8836.12i	0.398768	−	0.690686i	−0.594806	−	0.803869i	$-0.702771\pi$
$547$	5101.54	−	8836.12i	0.398768	−	0.690686i	0.993574	+	0.113183i	$0.0361046\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−768.815	+	1331.63i	−0.0594422	+	0.102957i
$552$		0			0
$553$	−2210.82	−	3829.25i	−0.170006	−	0.294460i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	47.2572			0.00359489			0.00179744	−	0.999998i	$-0.499428\pi$
$557$	47.2572			0.00359489			0.00179744	+	0.999998i	$0.499428\pi$
$558$		0			0
$559$	−483.417			−0.0365767
$560$		0			0
$561$		0			0
$562$		0			0
$563$	2850.76	+	4937.66i	0.213402	+	0.369623i	0.952777	−	0.303671i	$-0.0982124\pi$
$563$	2850.76	+	4937.66i	0.213402	+	0.369623i	−0.739375	+	0.673294i	$0.764879\pi$
$564$		0			0
$565$	−1073.35	+	1859.09i	−0.0799221	+	0.138429i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	5681.99	−	9841.50i	0.418632	−	0.725091i	−0.577170	−	0.816624i	$-0.695843\pi$
$569$	5681.99	−	9841.50i	0.418632	−	0.725091i	0.995802	+	0.0915323i	$0.0291765\pi$
$570$		0			0
$571$	−5894.71	−	10209.9i	−0.432025	−	0.748289i	0.565023	−	0.825075i	$-0.308867\pi$
$571$	−5894.71	−	10209.9i	−0.432025	−	0.748289i	−0.997048	+	0.0767867i	$0.975534\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	1589.53			0.115283
$576$		0			0
$577$	−24947.7			−1.79997			−0.899987	−	0.435918i	$-0.856424\pi$
$577$	−24947.7			−1.79997			−0.899987	+	0.435918i	$0.856424\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−5410.85	−	9371.87i	−0.386368	−	0.669209i
$582$		0			0
$583$	370.606	−	641.908i	0.0263275	−	0.0456006i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−10734.6	+	18592.9i	−0.754794	+	1.30734i	0.190683	+	0.981652i	$0.438930\pi$
$587$	−10734.6	+	18592.9i	−0.754794	+	1.30734i	−0.945477	+	0.325690i	$0.894403\pi$
$588$		0			0
$589$	−576.496	−	998.521i	−0.0403296	−	0.0698528i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−12062.0			−0.835290			−0.417645	−	0.908610i	$-0.637144\pi$
$593$	−12062.0			−0.835290			−0.417645	+	0.908610i	$0.637144\pi$
$594$		0			0
$595$	−5625.55			−0.387605
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−717.294	−	1242.39i	−0.0489279	−	0.0847457i	0.840524	−	0.541774i	$-0.182247\pi$
$599$	−717.294	−	1242.39i	−0.0489279	−	0.0847457i	−0.889452	+	0.457028i	$0.848914\pi$
$600$		0			0
$601$	−6603.99	+	11438.4i	−0.448224	+	0.776346i	−0.998271	−	0.0587878i	$-0.981276\pi$
$601$	−6603.99	+	11438.4i	−0.448224	+	0.776346i	0.550047	+	0.835134i	$0.314610\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−3048.31	+	5279.83i	−0.204845	+	0.354803i
$606$		0			0
$607$	−14532.8	−	25171.6i	−0.971777	−	1.68317i	−0.690184	−	0.723634i	$-0.742470\pi$
$607$	−14532.8	−	25171.6i	−0.971777	−	1.68317i	−0.281594	−	0.959534i	$-0.590863\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	1001.26			0.0662957
$612$		0			0
$613$	12478.5			0.822186			0.411093	−	0.911593i	$-0.365147\pi$
$613$	12478.5			0.822186			0.411093	+	0.911593i	$0.365147\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	4363.83	+	7558.38i	0.284735	+	0.493175i	0.972545	−	0.232716i	$-0.0747612\pi$
$617$	4363.83	+	7558.38i	0.284735	+	0.493175i	−0.687810	+	0.725891i	$0.741428\pi$
$618$		0			0
$619$	11067.4	−	19169.4i	0.718639	−	1.24472i	−0.242900	−	0.970051i	$-0.578099\pi$
$619$	11067.4	−	19169.4i	0.718639	−	1.24472i	0.961539	−	0.274668i	$-0.0885680\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	5354.55	−	9274.35i	0.344343	−	0.596419i
$624$		0			0
$625$	−312.500	−	541.266i	−0.0200000	−	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−7853.47			−0.497835
$630$		0			0
$631$	22905.4			1.44509			0.722543	−	0.691326i	$-0.242973\pi$
$631$	22905.4			1.44509			0.722543	+	0.691326i	$0.242973\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−678.105	−	1174.51i	−0.0423776	−	0.0734002i
$636$		0			0
$637$	−2048.78	+	3548.60i	−0.127434	+	0.220723i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	5677.78	−	9834.20i	0.349858	−	0.605971i	−0.636366	−	0.771387i	$-0.719563\pi$
$641$	5677.78	−	9834.20i	0.349858	−	0.605971i	0.986224	+	0.165416i	$0.0528967\pi$
$642$		0			0
$643$	−1307.23	−	2264.19i	−0.0801745	−	0.138866i	0.823150	−	0.567824i	$-0.192214\pi$
$643$	−1307.23	−	2264.19i	−0.0801745	−	0.138866i	−0.903325	+	0.428957i	$0.858881\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−1495.04			−0.0908439			−0.0454220	−	0.998968i	$-0.514463\pi$
$647$	−1495.04			−0.0908439			−0.0454220	+	0.998968i	$0.514463\pi$
$648$		0			0
$649$	4523.34			0.273585
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−12457.7	−	21577.3i	−0.746565	−	1.29309i	−0.949460	−	0.313888i	$-0.898368\pi$
$653$	−12457.7	−	21577.3i	−0.746565	−	1.29309i	0.202895	−	0.979201i	$-0.434965\pi$
$654$		0			0
$655$	−7020.20	+	12159.3i	−0.418781	+	0.725351i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	11250.4	−	19486.3i	0.665028	−	1.15186i	−0.314249	−	0.949341i	$-0.601753\pi$
$659$	11250.4	−	19486.3i	0.665028	−	1.15186i	0.979278	−	0.202523i	$-0.0649139\pi$
$660$		0			0
$661$	7031.71	+	12179.3i	0.413770	+	0.716671i	0.995298	−	0.0968554i	$-0.0308784\pi$
$661$	7031.71	+	12179.3i	0.413770	+	0.716671i	−0.581528	+	0.813526i	$0.697545\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	502.482			0.0293014
$666$		0			0
$667$	−11504.6			−0.667857
$668$		0			0
$669$		0			0
$670$		0			0
$671$	3799.42	+	6580.79i	0.218592	+	0.378612i
$672$		0			0
$673$	2518.43	−	4362.05i	0.144247	−	0.249843i	−0.784845	−	0.619693i	$-0.787257\pi$
$673$	2518.43	−	4362.05i	0.144247	−	0.249843i	0.929092	+	0.369849i	$0.120591\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	929.496	−	1609.93i	0.0527673	−	0.0913956i	−0.838435	−	0.545001i	$-0.816529\pi$
$677$	929.496	−	1609.93i	0.0527673	−	0.0913956i	0.891203	+	0.453606i	$0.149863\pi$
$678$		0			0
$679$	−4335.03	−	7508.50i	−0.245012	−	0.424374i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	10695.1			0.599176			0.299588	−	0.954069i	$-0.403151\pi$
$683$	10695.1			0.599176			0.299588	+	0.954069i	$0.403151\pi$
$684$		0			0
$685$	−10940.2			−0.610223
$686$		0			0
$687$		0			0
$688$		0			0
$689$	707.391	+	1225.24i	0.0391138	+	0.0677472i
$690$		0			0
$691$	−4189.77	+	7256.90i	−0.230661	+	0.399516i	−0.958003	−	0.286759i	$-0.907422\pi$
$691$	−4189.77	+	7256.90i	−0.230661	+	0.399516i	0.727342	+	0.686275i	$0.240755\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−1749.71	+	3030.58i	−0.0954966	+	0.165405i
$696$		0			0
$697$	5704.82	+	9881.03i	0.310022	+	0.536974i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−25278.4			−1.36199			−0.680994	−	0.732289i	$-0.738452\pi$
$701$	−25278.4			−1.36199			−0.680994	+	0.732289i	$0.738452\pi$
$702$		0			0
$703$	701.482			0.0376343
$704$		0			0
$705$		0			0
$706$		0			0
$707$	8547.07	+	14804.0i	0.454661	+	0.787497i
$708$		0			0
$709$	−3208.62	+	5557.50i	−0.169961	+	0.294381i	−0.938406	−	0.345535i	$-0.887697\pi$
$709$	−3208.62	+	5557.50i	−0.169961	+	0.294381i	0.768445	+	0.639916i	$0.221031\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	4313.37	−	7470.98i	0.226560	−	0.392413i
$714$		0			0
$715$	532.897	+	923.005i	0.0278730	+	0.0482775i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	26029.4			1.35011			0.675057	−	0.737765i	$-0.264119\pi$
$719$	26029.4			1.35011			0.675057	+	0.737765i	$0.264119\pi$
$720$		0			0
$721$	2863.22			0.147894
$722$		0			0
$723$		0			0
$724$		0			0
$725$	2261.80	+	3917.55i	0.115864	+	0.200682i
$726$		0			0
$727$	−10621.8	+	18397.5i	−0.541873	+	0.938551i	0.456924	+	0.889506i	$0.348951\pi$
$727$	−10621.8	+	18397.5i	−0.541873	+	0.938551i	−0.998797	+	0.0490451i	$0.984382\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−1140.04	+	1974.60i	−0.0576824	+	0.0999089i
$732$		0			0
$733$	6960.53	+	12056.0i	0.350741	+	0.607501i	0.986379	−	0.164486i	$-0.0525964\pi$
$733$	6960.53	+	12056.0i	0.350741	+	0.607501i	−0.635638	+	0.771987i	$0.719263\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−2768.40			−0.138365
$738$		0			0
$739$	−11206.6			−0.557838			−0.278919	−	0.960315i	$-0.589976\pi$
$739$	−11206.6			−0.557838			−0.278919	+	0.960315i	$0.589976\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−7510.94	−	13009.3i	−0.370861	−	0.642350i	0.618837	−	0.785519i	$-0.287604\pi$
$743$	−7510.94	−	13009.3i	−0.370861	−	0.642350i	−0.989698	+	0.143169i	$0.954271\pi$
$744$		0			0
$745$	−556.062	+	963.128i	−0.0273457	+	0.0473641i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	6473.39	−	11212.2i	0.315798	−	0.546978i
$750$		0			0
$751$	5236.36	+	9069.65i	0.254431	+	0.440687i	0.964741	−	0.263202i	$-0.0847785\pi$
$751$	5236.36	+	9069.65i	0.254431	+	0.440687i	−0.710310	+	0.703889i	$0.751445\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	11989.8			0.577952
$756$		0			0
$757$	−11485.8			−0.551463			−0.275731	−	0.961235i	$-0.588920\pi$
$757$	−11485.8			−0.551463			−0.275731	+	0.961235i	$0.588920\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	6794.91	+	11769.1i	0.323673	+	0.560619i	0.981243	−	0.192775i	$-0.0617488\pi$
$761$	6794.91	+	11769.1i	0.323673	+	0.560619i	−0.657570	+	0.753394i	$0.728415\pi$
$762$		0			0
$763$	−10406.0	+	18023.7i	−0.493737	+	0.855178i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−4316.94	+	7477.17i	−0.203228	+	0.352001i
$768$		0			0
$769$	12406.9	+	21489.3i	0.581798	+	1.00770i	0.995266	+	0.0971852i	$0.0309839\pi$
$769$	12406.9	+	21489.3i	0.581798	+	1.00770i	−0.413468	+	0.910519i	$0.635683\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−19071.6			−0.887397			−0.443699	−	0.896176i	$-0.646334\pi$
$773$	−19071.6			−0.887397			−0.443699	+	0.896176i	$0.646334\pi$
$774$		0			0
$775$	−3392.02			−0.157219
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−509.562	−	882.587i	−0.0234364	−	0.0405930i
$780$		0			0
$781$	411.091	−	712.031i	0.0188348	−	0.0326229i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−5828.95	+	10096.0i	−0.265024	+	0.459036i
$786$		0			0
$787$	21719.8	+	37619.7i	0.983768	+	1.70394i	0.647287	+	0.762247i	$0.275904\pi$
$787$	21719.8	+	37619.7i	0.983768	+	1.70394i	0.336482	+	0.941690i	$0.390763\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−5077.41			−0.228233
$792$		0			0
$793$	−14504.2			−0.649508
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−320.709	−	555.484i	−0.0142536	−	0.0246879i	0.858811	−	0.512293i	$-0.171204\pi$
$797$	−320.709	−	555.484i	−0.0142536	−	0.0246879i	−0.873064	+	0.487605i	$0.837871\pi$
$798$		0			0
$799$	2361.26	−	4089.83i	0.104550	−	0.181086i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	3311.58	−	5735.82i	0.145533	−	0.252071i
$804$		0			0
$805$	1879.80	+	3255.90i	0.0823033	+	0.142553i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	19703.7			0.856297			0.428149	−	0.903708i	$-0.359166\pi$
$809$	19703.7			0.856297			0.428149	+	0.903708i	$0.359166\pi$
$810$		0			0
$811$	44804.2			1.93994			0.969969	−	0.243230i	$-0.0782071\pi$
$811$	44804.2			1.93994			0.969969	+	0.243230i	$0.0782071\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	188.522	+	326.529i	0.00810260	+	0.0140341i
$816$		0			0
$817$	101.830	−	176.374i	0.00436055	−	0.00755270i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	5624.46	−	9741.86i	0.239093	−	0.414121i	−0.721361	−	0.692559i	$-0.756483\pi$
$821$	5624.46	−	9741.86i	0.239093	−	0.414121i	0.960454	+	0.278438i	$0.0898167\pi$
$822$		0			0
$823$	19807.4	+	34307.4i	0.838934	+	1.45308i	0.890787	+	0.454421i	$0.150154\pi$
$823$	19807.4	+	34307.4i	0.838934	+	1.45308i	−0.0518529	+	0.998655i	$0.516513\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−29446.4			−1.23815			−0.619077	−	0.785331i	$-0.712493\pi$
$827$	−29446.4			−1.23815			−0.619077	+	0.785331i	$0.712493\pi$
$828$		0			0
$829$	−12267.1			−0.513938			−0.256969	−	0.966420i	$-0.582724\pi$
$829$	−12267.1			−0.513938			−0.256969	+	0.966420i	$0.582724\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	9663.25	+	16737.2i	0.401935	+	0.696172i
$834$		0			0
$835$	−5810.67	+	10064.4i	−0.240822	+	0.417117i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	8147.47	−	14111.8i	0.335258	−	0.580685i	−0.648276	−	0.761405i	$-0.724510\pi$
$839$	8147.47	−	14111.8i	0.335258	−	0.580685i	0.983534	+	0.180721i	$0.0578430\pi$
$840$		0			0
$841$	−4175.87	−	7232.81i	−0.171219	−	0.296560i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	8950.68			0.364394
$846$		0			0
$847$	−14419.9			−0.584975
$848$		0			0
$849$		0			0
$850$		0			0
$851$	2624.26	+	4545.36i	0.105709	+	0.183094i
$852$		0			0
$853$	22554.3	−	39065.2i	0.905327	−	1.56807i	0.0848497	−	0.996394i	$-0.472959\pi$
$853$	22554.3	−	39065.2i	0.905327	−	1.56807i	0.820477	−	0.571679i	$-0.193708\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−22328.2	+	38673.5i	−0.889984	+	1.54150i	−0.0500915	+	0.998745i	$0.515951\pi$
$857$	−22328.2	+	38673.5i	−0.889984	+	1.54150i	−0.839892	+	0.542753i	$0.817382\pi$
$858$		0			0
$859$	589.701	+	1021.39i	0.0234230	+	0.0405698i	0.877499	−	0.479578i	$-0.159210\pi$
$859$	589.701	+	1021.39i	0.0234230	+	0.0405698i	−0.854076	+	0.520148i	$0.825877\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	4489.17			0.177072			0.0885359	−	0.996073i	$-0.471781\pi$
$863$	4489.17			0.177072			0.0885359	+	0.996073i	$0.471781\pi$
$864$		0			0
$865$	−9178.04			−0.360766
$866$		0			0
$867$		0			0
$868$		0			0
$869$	1975.55	+	3421.76i	0.0771185	+	0.133573i
$870$		0			0
$871$	2642.08	−	4576.21i	0.102782	−	0.178024i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	739.133	−	1280.22i	0.0285569	−	0.0494620i
$876$		0			0
$877$	−23466.7	−	40645.5i	−0.903550	−	1.56500i	−0.822851	−	0.568257i	$-0.807618\pi$
$877$	−23466.7	−	40645.5i	−0.903550	−	1.56500i	−0.0806990	−	0.996739i	$-0.525715\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−11123.2			−0.425369			−0.212684	−	0.977121i	$-0.568221\pi$
$881$	−11123.2			−0.425369			−0.212684	+	0.977121i	$0.568221\pi$
$882$		0			0
$883$	44352.9			1.69037			0.845183	−	0.534477i	$-0.179491\pi$
$883$	44352.9			1.69037			0.845183	+	0.534477i	$0.179491\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	20058.7	+	34742.6i	0.759306	+	1.31516i	0.943205	+	0.332211i	$0.107795\pi$
$887$	20058.7	+	34742.6i	0.759306	+	1.31516i	−0.183899	+	0.982945i	$0.558872\pi$
$888$		0			0
$889$	1603.87	−	2777.99i	0.0605086	−	0.104804i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−210.911	+	365.309i	−0.00790356	+	0.0136894i
$894$		0			0
$895$	−10869.7	−	18826.9i	−0.405960	−	0.703143i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	24550.6			0.910801
$900$		0			0
$901$	6672.93			0.246734
$902$		0			0
$903$		0			0
$904$		0			0
$905$	5485.70	+	9501.51i	0.201493	+	0.348996i
$906$		0			0
$907$	−13295.3	+	23028.1i	−0.486728	+	0.843038i	−0.999884	−	0.0152576i	$-0.995143\pi$
$907$	−13295.3	+	23028.1i	−0.486728	+	0.843038i	0.513155	+	0.858296i	$0.328476\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−7942.08	+	13756.1i	−0.288840	+	0.500285i	−0.973533	−	0.228546i	$-0.926603\pi$
$911$	−7942.08	+	13756.1i	−0.288840	+	0.500285i	0.684693	+	0.728831i	$0.259936\pi$
$912$		0			0
$913$	4835.05	+	8374.55i	0.175265	+	0.303568i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−33208.7			−1.19591
$918$		0			0
$919$	33673.9			1.20871			0.604353	−	0.796717i	$-0.293432\pi$
$919$	33673.9			1.20871			0.604353	+	0.796717i	$0.293432\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	784.667	+	1359.08i	0.0279823	+	0.0484667i
$924$		0			0
$925$	1031.86	−	1787.23i	0.0366781	−	0.0635283i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−26355.1	+	45648.4i	−0.930768	+	1.61214i	−0.148757	+	0.988874i	$0.547527\pi$
$929$	−26355.1	+	45648.4i	−0.930768	+	1.61214i	−0.782011	+	0.623264i	$0.785806\pi$
$930$		0			0
$931$	−863.135	−	1494.99i	−0.0303846	−	0.0526277i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	5026.90			0.175826
$936$		0			0
$937$	−20591.1			−0.717912			−0.358956	−	0.933355i	$-0.616867\pi$
$937$	−20591.1			−0.717912			−0.358956	+	0.933355i	$0.616867\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−4192.91	−	7262.33i	−0.145255	−	0.251589i	0.784213	−	0.620492i	$-0.213067\pi$
$941$	−4192.91	−	7262.33i	−0.145255	−	0.251589i	−0.929468	+	0.368903i	$0.879734\pi$
$942$		0			0
$943$	3812.57	−	6603.56i	0.131659	−	0.228040i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−4020.09	+	6962.99i	−0.137946	+	0.238930i	−0.926719	−	0.375755i	$-0.877384\pi$
$947$	−4020.09	+	6962.99i	−0.137946	+	0.238930i	0.788773	+	0.614685i	$0.210717\pi$
$948$		0			0
$949$	6320.94	+	10948.2i	0.216213	+	0.374493i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	14235.6			0.483877			0.241939	−	0.970292i	$-0.422217\pi$
$953$	14235.6			0.483877			0.241939	+	0.970292i	$0.422217\pi$
$954$		0			0
$955$	−12719.4			−0.430986
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−12938.0	−	22409.3i	−0.435651	−	0.754570i
$960$		0			0
$961$	5690.85	−	9856.84i	0.191026	−	0.330866i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−1063.21	+	1841.53i	−0.0354672	+	0.0614311i
$966$		0			0
$967$	26243.2	+	45454.5i	0.872724	+	1.51160i	0.859167	+	0.511695i	$0.170982\pi$
$967$	26243.2	+	45454.5i	0.872724	+	1.51160i	0.0135567	+	0.999908i	$0.495685\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−5728.28			−0.189320			−0.0946598	−	0.995510i	$-0.530176\pi$
$971$	−5728.28			−0.189320			−0.0946598	+	0.995510i	$0.530176\pi$
$972$		0			0
$973$	−8276.90			−0.272708
$974$		0			0
$975$		0			0
$976$		0			0
$977$	14740.3	+	25530.9i	0.482685	+	0.836035i	0.999802	−	0.0198796i	$-0.00632830\pi$
$977$	14740.3	+	25530.9i	0.482685	+	0.836035i	−0.517117	+	0.855914i	$0.672995\pi$
$978$		0			0
$979$	−4784.74	+	8287.41i	−0.156201	+	0.270548i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−20420.9	+	35370.0i	−0.662589	+	1.14764i	0.317344	+	0.948311i	$0.397209\pi$
$983$	−20420.9	+	35370.0i	−0.662589	+	1.14764i	−0.979933	+	0.199328i	$0.936124\pi$
$984$		0			0
$985$	−1563.04	−	2707.27i	−0.0505611	−	0.0875744i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	1523.79			0.0489926
$990$		0			0
$991$	53919.4			1.72836			0.864181	−	0.503181i	$-0.167837\pi$
$991$	53919.4			1.72836			0.864181	+	0.503181i	$0.167837\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	9264.41	+	16046.4i	0.295178	+	0.511262i
$996$		0			0
$997$	11727.7	−	20313.0i	0.372538	−	0.645254i	−0.617418	−	0.786636i	$-0.711821\pi$
$997$	11727.7	−	20313.0i	0.372538	−	0.645254i	0.989955	+	0.141382i	$0.0451544\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1080.4.q.b.721.3		16
3.2	odd	2		360.4.q.b.241.7	yes	16
9.4	even	3	inner	1080.4.q.b.361.3		16
9.5	odd	6		360.4.q.b.121.7	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
360.4.q.b.121.7	✓	16	9.5	odd	6
360.4.q.b.241.7	yes	16	3.2	odd	2
1080.4.q.b.361.3		16	9.4	even	3	inner
1080.4.q.b.721.3		16	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1080 = 2^{3} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1080.q (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(2.50000 + 4.33013i) q^{5} +(-5.91307 + 10.2417i) q^{7} +(5.28382 - 9.15185i) q^{11} +(10.0854 + 17.4685i) q^{13} +95.1376 q^{17} -8.49782 q^{19} +(-31.7906 - 55.0629i) q^{23} +(-12.5000 + 21.6506i) q^{25} +(90.4720 - 156.702i) q^{29} +(67.8404 + 117.503i) q^{31} -59.1307 q^{35} -82.5485 q^{37} +(59.9638 + 103.860i) q^{41} +(-11.9830 + 20.7552i) q^{43} +(24.8195 - 42.9886i) q^{47} +(101.571 + 175.927i) q^{49} +70.1397 q^{53} +52.8382 q^{55} +(214.018 + 370.691i) q^{59} +(-359.534 + 622.730i) q^{61} +(-50.4272 + 87.3425i) q^{65} +(-130.985 - 226.872i) q^{67} +77.8019 q^{71} +626.739 q^{73} +(62.4872 + 108.231i) q^{77} +(-186.943 + 323.796i) q^{79} +(-457.533 + 792.471i) q^{83} +(237.844 + 411.958i) q^{85} -905.545 q^{89} -238.544 q^{91} +(-21.2446 - 36.7967i) q^{95} +(-366.564 + 634.907i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 40 q^{5} - 34 q^{7} + 64 q^{11} - 48 q^{13} - 212 q^{17} + 456 q^{19} + 166 q^{23} - 200 q^{25} + 110 q^{29} - 160 q^{31} - 340 q^{35} + 104 q^{37} + 280 q^{41} + 136 q^{43} + 594 q^{47} - 1094 q^{49}+ \cdots + 182 q^{97}+O(q^{100}) \) 16 * q + 40 * q^5 - 34 * q^7 + 64 * q^11 - 48 * q^13 - 212 * q^17 + 456 * q^19 + 166 * q^23 - 200 * q^25 + 110 * q^29 - 160 * q^31 - 340 * q^35 + 104 * q^37 + 280 * q^41 + 136 * q^43 + 594 * q^47 - 1094 * q^49 - 752 * q^53 + 640 * q^55 + 24 * q^59 - 950 * q^61 + 240 * q^65 - 426 * q^67 + 136 * q^71 + 308 * q^73 - 696 * q^77 - 764 * q^79 + 1362 * q^83 - 530 * q^85 + 172 * q^89 - 1224 * q^91 + 1140 * q^95 + 182 * q^97

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