LMFDB - Embedded newform 540.4.i.b.361.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [540,4,Mod(181,540)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(540, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4, 0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("540.181");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 540 = 2^{2} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	540.i (of order $3$, 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:	$31.8610314031$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	8.0.192296282256.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} + 36x^{6} - 94x^{5} + 331x^{4} - 510x^{3} + 708x^{2} - 468x + 117 $ x^8 - 4x^7 + 36x^6 - 94x^5 + 331x^4 - 510x^3 + 708x^2 - 468*x + 117
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{6} $
Twist minimal:	no (minimal twist has level 180)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			361.1
Root			$0.500000 - 1.47446i$ of defining polynomial
Character	$\chi$	$=$	540.361
Dual form			540.4.i.b.181.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.50000 + 4.33013i) q^{5} +(-8.95174 - 15.5049i) q^{7} +(19.7807 + 34.2612i) q^{11} +(11.7420 - 20.3377i) q^{13} +16.2456 q^{17} -28.3443 q^{19} +(-52.7655 + 91.3926i) q^{23} +(-12.5000 - 21.6506i) q^{25} +(-49.9221 - 86.4676i) q^{29} +(160.949 - 278.773i) q^{31} +89.5174 q^{35} -205.760 q^{37} +(163.822 - 283.748i) q^{41} +(-187.531 - 324.812i) q^{43} +(44.4668 + 77.0187i) q^{47} +(11.2328 - 19.4557i) q^{49} -85.0314 q^{53} -197.807 q^{55} +(398.403 - 690.054i) q^{59} +(158.968 + 275.341i) q^{61} +(58.7098 + 101.688i) q^{65} +(262.745 - 455.088i) q^{67} +760.854 q^{71} -101.076 q^{73} +(354.143 - 613.394i) q^{77} +(112.817 + 195.404i) q^{79} +(-251.526 - 435.655i) q^{83} +(-40.6139 + 70.3453i) q^{85} -868.775 q^{89} -420.444 q^{91} +(70.8608 - 122.734i) q^{95} +(-874.721 - 1515.06i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 20 q^{5} + 13 q^{7} + 57 q^{11} - 14 q^{13} - 6 q^{17} - 62 q^{19} - 69 q^{23} - 100 q^{25} - 69 q^{29} + 58 q^{31} - 130 q^{35} - 776 q^{37} + 396 q^{41} - 371 q^{43} + 129 q^{47} - 111 q^{49} - 2712 q^{53}+ \cdots + 1495 q^{97}+O(q^{100}) $ 8 * q - 20 * q^5 + 13 * q^7 + 57 * q^11 - 14 * q^13 - 6 * q^17 - 62 * q^19 - 69 * q^23 - 100 * q^25 - 69 * q^29 + 58 * q^31 - 130 * q^35 - 776 * q^37 + 396 * q^41 - 371 * q^43 + 129 * q^47 - 111 * q^49 - 2712 * q^53 - 570 * q^55 + 15 * q^59 + 1441 * q^61 - 70 * q^65 - 368 * q^67 - 336 * q^71 - 1910 * q^73 + 342 * q^77 + 1408 * q^79 - 789 * q^83 + 15 * q^85 - 3234 * q^89 - 2812 * q^91 + 155 * q^95 + 1495 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/540\mathbb{Z}\right)^\times$.

$n$	$217$	$271$	$461$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{3}\right)$

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
$7$	−8.95174	−	15.5049i	−0.483348	−	0.837184i	0.516469	−	0.856306i	$-0.327246\pi$
$7$	−8.95174	−	15.5049i	−0.483348	−	0.837184i	−0.999817	+	0.0191220i	$0.993913\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	19.7807	+	34.2612i	0.542191	+	0.939103i	0.998778	+	0.0494239i	$0.0157385\pi$
$11$	19.7807	+	34.2612i	0.542191	+	0.939103i	−0.456587	+	0.889679i	$0.650928\pi$
$12$		0			0
$13$	11.7420	−	20.3377i	0.250510	−	0.433896i	−0.713156	−	0.701005i	$-0.752735\pi$
$13$	11.7420	−	20.3377i	0.250510	−	0.433896i	0.963666	+	0.267109i	$0.0860683\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	16.2456			0.231772			0.115886	−	0.993263i	$-0.463029\pi$
$17$	16.2456			0.231772			0.115886	+	0.993263i	$0.463029\pi$
$18$		0			0
$19$	−28.3443			−0.342244			−0.171122	−	0.985250i	$-0.554739\pi$
$19$	−28.3443			−0.342244			−0.171122	+	0.985250i	$0.554739\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−52.7655	+	91.3926i	−0.478364	+	0.828551i	−0.999692	−	0.0248052i	$-0.992103\pi$
$23$	−52.7655	+	91.3926i	−0.478364	+	0.828551i	0.521328	+	0.853356i	$0.325437\pi$
$24$		0			0
$25$	−12.5000	−	21.6506i	−0.100000	−	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−49.9221	−	86.4676i	−0.319666	−	0.553677i	0.660753	−	0.750604i	$-0.270237\pi$
$29$	−49.9221	−	86.4676i	−0.319666	−	0.553677i	−0.980418	+	0.196927i	$0.936904\pi$
$30$		0			0
$31$	160.949	−	278.773i	0.932496	−	1.61513i	0.153457	−	0.988155i	$-0.450959\pi$
$31$	160.949	−	278.773i	0.932496	−	1.61513i	0.779039	−	0.626975i	$-0.215707\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	89.5174			0.432320
$36$		0			0
$37$	−205.760			−0.914237			−0.457119	−	0.889406i	$-0.651118\pi$
$37$	−205.760			−0.914237			−0.457119	+	0.889406i	$0.651118\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	163.822	−	283.748i	0.624017	−	1.08083i	−0.364714	−	0.931120i	$-0.618833\pi$
$41$	163.822	−	283.748i	0.624017	−	1.08083i	0.988730	−	0.149709i	$-0.0478336\pi$
$42$		0			0
$43$	−187.531	−	324.812i	−0.665073	−	1.15194i	−0.979265	−	0.202581i	$-0.935067\pi$
$43$	−187.531	−	324.812i	−0.665073	−	1.15194i	0.314192	−	0.949359i	$-0.398266\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	44.4668	+	77.0187i	0.138003	+	0.239028i	0.926741	−	0.375702i	$-0.122598\pi$
$47$	44.4668	+	77.0187i	0.138003	+	0.239028i	−0.788738	+	0.614730i	$0.789265\pi$
$48$		0			0
$49$	11.2328	−	19.4557i	0.0327486	−	0.0567222i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−85.0314			−0.220377			−0.110188	−	0.993911i	$-0.535145\pi$
$53$	−85.0314			−0.220377			−0.110188	+	0.993911i	$0.535145\pi$
$54$		0			0
$55$	−197.807			−0.484951
$56$		0			0
$57$		0			0
$58$		0			0
$59$	398.403	−	690.054i	0.879113	−	1.52267i	0.0267963	−	0.999641i	$-0.491469\pi$
$59$	398.403	−	690.054i	0.879113	−	1.52267i	0.852316	−	0.523027i	$-0.175197\pi$
$60$		0			0
$61$	158.968	+	275.341i	0.333669	+	0.577931i	0.983228	−	0.182380i	$-0.0583800\pi$
$61$	158.968	+	275.341i	0.333669	+	0.577931i	−0.649559	+	0.760311i	$0.725047\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	58.7098	+	101.688i	0.112032	+	0.194044i
$66$		0			0
$67$	262.745	−	455.088i	0.479096	−	0.829818i	−0.520617	−	0.853790i	$-0.674298\pi$
$67$	262.745	−	455.088i	0.479096	−	0.829818i	0.999713	+	0.0239721i	$0.00763129\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	760.854			1.27179			0.635893	−	0.771778i	$-0.280632\pi$
$71$	760.854			1.27179			0.635893	+	0.771778i	$0.280632\pi$
$72$		0			0
$73$	−101.076			−0.162056			−0.0810279	−	0.996712i	$-0.525820\pi$
$73$	−101.076			−0.162056			−0.0810279	+	0.996712i	$0.525820\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	354.143	−	613.394i	0.524135	−	0.907828i
$78$		0			0
$79$	112.817	+	195.404i	0.160669	+	0.278287i	0.935109	−	0.354361i	$-0.115301\pi$
$79$	112.817	+	195.404i	0.160669	+	0.278287i	−0.774440	+	0.632648i	$0.781968\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−251.526	−	435.655i	−0.332633	−	0.576137i	0.650394	−	0.759597i	$-0.274604\pi$
$83$	−251.526	−	435.655i	−0.332633	−	0.576137i	−0.983027	+	0.183460i	$0.941270\pi$
$84$		0			0
$85$	−40.6139	+	70.3453i	−0.0518258	+	0.0897650i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−868.775			−1.03472			−0.517359	−	0.855768i	$-0.673085\pi$
$89$	−868.775			−1.03472			−0.517359	+	0.855768i	$0.673085\pi$
$90$		0			0
$91$	−420.444			−0.484335
$92$		0			0
$93$		0			0
$94$		0			0
$95$	70.8608	−	122.734i	0.0765280	−	0.132550i
$96$		0			0
$97$	−874.721	−	1515.06i	−0.915613	−	1.58589i	−0.806001	−	0.591914i	$-0.798372\pi$
$97$	−874.721	−	1515.06i	−0.915613	−	1.58589i	−0.109613	−	0.993974i	$-0.534961\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−647.078	−	1120.77i	−0.637492	−	1.10417i	−0.985981	−	0.166855i	$-0.946639\pi$
$101$	−647.078	−	1120.77i	−0.637492	−	1.10417i	0.348490	−	0.937313i	$-0.386695\pi$
$102$		0			0
$103$	−95.9034	+	166.110i	−0.0917441	+	0.158905i	−0.908245	−	0.418439i	$-0.862578\pi$
$103$	−95.9034	+	166.110i	−0.0917441	+	0.158905i	0.816501	+	0.577344i	$0.195911\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	1417.70			1.28089			0.640443	−	0.768006i	$-0.278751\pi$
$107$	1417.70			1.28089			0.640443	+	0.768006i	$0.278751\pi$
$108$		0			0
$109$	129.747			0.114014			0.0570071	−	0.998374i	$-0.481844\pi$
$109$	129.747			0.114014			0.0570071	+	0.998374i	$0.481844\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	415.427	−	719.541i	0.345842	−	0.599015i	−0.639665	−	0.768654i	$-0.720927\pi$
$113$	415.427	−	719.541i	0.345842	−	0.599015i	0.985506	+	0.169639i	$0.0542601\pi$
$114$		0			0
$115$	−263.828	−	456.963i	−0.213931	−	0.370539i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−145.426	−	251.885i	−0.112027	−	0.194036i
$120$		0			0
$121$	−117.052	+	202.740i	−0.0879429	+	0.152322i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	125.000			0.0894427
$126$		0			0
$127$	−2385.67			−1.66688			−0.833441	−	0.552608i	$-0.813633\pi$
$127$	−2385.67			−1.66688			−0.833441	+	0.552608i	$0.813633\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−452.158	+	783.161i	−0.301567	+	0.522329i	−0.976491	−	0.215558i	$-0.930843\pi$
$131$	−452.158	+	783.161i	−0.301567	+	0.522329i	0.674924	+	0.737887i	$0.264176\pi$
$132$		0			0
$133$	253.731	+	439.475i	0.165423	+	0.286521i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	404.182	+	700.064i	0.252056	+	0.436573i	0.964092	−	0.265570i	$-0.0855601\pi$
$137$	404.182	+	700.064i	0.252056	+	0.436573i	−0.712036	+	0.702143i	$0.752227\pi$
$138$		0			0
$139$	1432.36	−	2480.92i	0.874037	−	1.51388i	0.0162523	−	0.999868i	$-0.494827\pi$
$139$	1432.36	−	2480.92i	0.874037	−	1.51388i	0.857785	−	0.514009i	$-0.171840\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	929.056			0.543298
$144$		0			0
$145$	499.221			0.285918
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−339.705	+	588.387i	−0.186777	+	0.323507i	−0.944174	−	0.329448i	$-0.893137\pi$
$149$	−339.705	+	588.387i	−0.186777	+	0.323507i	0.757397	+	0.652955i	$0.226471\pi$
$150$		0			0
$151$	1193.11	+	2066.53i	0.643008	+	1.11372i	0.984758	+	0.173932i	$0.0556472\pi$
$151$	1193.11	+	2066.53i	0.643008	+	1.11372i	−0.341750	+	0.939791i	$0.611019\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	804.747	+	1393.86i	0.417025	+	0.722308i
$156$		0			0
$157$	1349.17	−	2336.84i	0.685833	−	1.18790i	−0.287341	−	0.957828i	$-0.592771\pi$
$157$	1349.17	−	2336.84i	0.685833	−	1.18790i	0.973174	−	0.230069i	$-0.0738953\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	1889.37			0.924866
$162$		0			0
$163$	1500.47			0.721017			0.360509	−	0.932756i	$-0.382603\pi$
$163$	1500.47			0.721017			0.360509	+	0.932756i	$0.382603\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−1714.72	+	2969.98i	−0.794543	+	1.37619i	0.128586	+	0.991698i	$0.458956\pi$
$167$	−1714.72	+	2969.98i	−0.794543	+	1.37619i	−0.923129	+	0.384491i	$0.874377\pi$
$168$		0			0
$169$	822.753	+	1425.05i	0.374489	+	0.648634i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1003.97	−	1738.93i	−0.441217	−	0.764211i	0.556563	−	0.830806i	$-0.312120\pi$
$173$	−1003.97	−	1738.93i	−0.441217	−	0.764211i	−0.997780	+	0.0665946i	$0.978787\pi$
$174$		0			0
$175$	−223.793	+	387.622i	−0.0966697	+	0.167437i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−3836.95			−1.60216			−0.801082	−	0.598555i	$-0.795742\pi$
$179$	−3836.95			−1.60216			−0.801082	+	0.598555i	$0.795742\pi$
$180$		0			0
$181$	−1822.10			−0.748263			−0.374131	−	0.927376i	$-0.622059\pi$
$181$	−1822.10			−0.748263			−0.374131	+	0.927376i	$0.622059\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	514.401	−	890.968i	0.204430	−	0.354082i
$186$		0			0
$187$	321.348	+	556.592i	0.125665	+	0.217658i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	2097.98	+	3633.82i	0.794790	+	1.37662i	0.922973	+	0.384866i	$0.125752\pi$
$191$	2097.98	+	3633.82i	0.794790	+	1.37662i	−0.128183	+	0.991751i	$0.540914\pi$
$192$		0			0
$193$	905.179	−	1567.82i	0.337597	−	0.584735i	−0.646383	−	0.763013i	$-0.723719\pi$
$193$	905.179	−	1567.82i	0.337597	−	0.584735i	0.983980	+	0.178278i	$0.0570526\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−3827.21			−1.38415			−0.692074	−	0.721826i	$-0.743303\pi$
$197$	−3827.21			−1.38415			−0.692074	+	0.721826i	$0.743303\pi$
$198$		0			0
$199$	−477.853			−0.170222			−0.0851108	−	0.996371i	$-0.527124\pi$
$199$	−477.853			−0.170222			−0.0851108	+	0.996371i	$0.527124\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−893.779	+	1548.07i	−0.309020	+	0.535238i
$204$		0			0
$205$	819.109	+	1418.74i	0.279069	+	0.483361i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−560.670	−	971.109i	−0.185562	−	0.321402i
$210$		0			0
$211$	108.133	−	187.291i	0.0352803	−	0.0611073i	−0.847846	−	0.530242i	$-0.822101\pi$
$211$	108.133	−	187.291i	0.0352803	−	0.0611073i	0.883126	+	0.469135i	$0.155434\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	1875.31			0.594860
$216$		0			0
$217$	−5763.11			−1.80288
$218$		0			0
$219$		0			0
$220$		0			0
$221$	190.755	−	330.397i	0.0580613	−	0.100565i
$222$		0			0
$223$	2690.38	+	4659.88i	0.807899	+	1.39932i	0.914317	+	0.405000i	$0.132729\pi$
$223$	2690.38	+	4659.88i	0.807899	+	1.39932i	−0.106418	+	0.994321i	$0.533938\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−277.295	−	480.289i	−0.0810780	−	0.140431i	0.822635	−	0.568570i	$-0.192503\pi$
$227$	−277.295	−	480.289i	−0.0810780	−	0.140431i	−0.903713	+	0.428138i	$0.859170\pi$
$228$		0			0
$229$	−1705.33	+	2953.72i	−0.492103	+	0.852348i	−0.999959	−	0.00909459i	$-0.997105\pi$
$229$	−1705.33	+	2953.72i	−0.492103	+	0.852348i	0.507855	+	0.861442i	$0.330438\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−2640.60			−0.742452			−0.371226	−	0.928543i	$-0.621062\pi$
$233$	−2640.60			−0.742452			−0.371226	+	0.928543i	$0.621062\pi$
$234$		0			0
$235$	−444.668			−0.123434
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−91.4754	+	158.440i	−0.0247575	+	0.0428813i	−0.878139	−	0.478406i	$-0.841215\pi$
$239$	−91.4754	+	158.440i	−0.0247575	+	0.0428813i	0.853381	+	0.521287i	$0.174548\pi$
$240$		0			0
$241$	3716.89	+	6437.83i	0.993467	+	1.72074i	0.595562	+	0.803309i	$0.296929\pi$
$241$	3716.89	+	6437.83i	0.993467	+	1.72074i	0.397905	+	0.917427i	$0.369737\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	56.1638	+	97.2786i	0.0146456	+	0.0253669i
$246$		0			0
$247$	−332.818	+	576.457i	−0.0857355	+	0.148498i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	2170.36			0.545785			0.272893	−	0.962045i	$-0.412020\pi$
$251$	2170.36			0.545785			0.272893	+	0.962045i	$0.412020\pi$
$252$		0			0
$253$	−4174.96			−1.03746
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−39.1832	+	67.8672i	−0.00951043	+	0.0164725i	−0.870741	−	0.491741i	$-0.836361\pi$
$257$	−39.1832	+	67.8672i	−0.00951043	+	0.0164725i	0.861231	+	0.508214i	$0.169694\pi$
$258$		0			0
$259$	1841.91	+	3190.28i	0.441895	+	0.765385i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	608.404	+	1053.79i	0.142646	+	0.247070i	0.928492	−	0.371352i	$-0.121106\pi$
$263$	608.404	+	1053.79i	0.142646	+	0.247070i	−0.785846	+	0.618422i	$0.787772\pi$
$264$		0			0
$265$	212.579	−	368.197i	0.0492777	−	0.0853515i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−1249.44			−0.283196			−0.141598	−	0.989924i	$-0.545224\pi$
$269$	−1249.44			−0.283196			−0.141598	+	0.989924i	$0.545224\pi$
$270$		0			0
$271$	7675.60			1.72051			0.860257	−	0.509860i	$-0.170303\pi$
$271$	7675.60			1.72051			0.860257	+	0.509860i	$0.170303\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	494.517	−	856.529i	0.108438	−	0.187821i
$276$		0			0
$277$	−365.873	−	633.711i	−0.0793617	−	0.137459i	0.823613	−	0.567152i	$-0.191955\pi$
$277$	−365.873	−	633.711i	−0.0793617	−	0.137459i	−0.902975	+	0.429694i	$0.858622\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	564.975	+	978.566i	0.119942	+	0.207745i	0.919744	−	0.392518i	$-0.128396\pi$
$281$	564.975	+	978.566i	0.119942	+	0.207745i	−0.799803	+	0.600263i	$0.795063\pi$
$282$		0			0
$283$	1225.63	−	2122.85i	0.257442	−	0.445902i	−0.708114	−	0.706098i	$-0.750454\pi$
$283$	1225.63	−	2122.85i	0.257442	−	0.445902i	0.965556	+	0.260196i	$0.0837871\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−5865.96			−1.20647
$288$		0			0
$289$	−4649.08			−0.946282
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−3122.93	+	5409.07i	−0.622674	+	1.07850i	0.366312	+	0.930492i	$0.380620\pi$
$293$	−3122.93	+	5409.07i	−0.622674	+	1.07850i	−0.988986	+	0.148011i	$0.952713\pi$
$294$		0			0
$295$	1992.01	+	3450.27i	0.393151	+	0.680958i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	1239.14	+	2146.26i	0.239670	+	0.415121i
$300$		0			0
$301$	−3357.45	+	5815.27i	−0.642924	+	1.11358i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−1589.68			−0.298442
$306$		0			0
$307$	10638.4			1.97774			0.988870	−	0.148782i	$-0.0475354\pi$
$307$	10638.4			1.97774			0.988870	+	0.148782i	$0.0475354\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−18.6347	+	32.2762i	−0.00339767	+	0.00588493i	−0.867719	−	0.497055i	$-0.834415\pi$
$311$	−18.6347	+	32.2762i	−0.00339767	+	0.00588493i	0.864322	+	0.502940i	$0.167748\pi$
$312$		0			0
$313$	−473.786	−	820.622i	−0.0855590	−	0.148193i	0.820070	−	0.572263i	$-0.193934\pi$
$313$	−473.786	−	820.622i	−0.0855590	−	0.148193i	−0.905629	+	0.424070i	$0.860601\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−5288.32	−	9159.63i	−0.936976	−	1.62289i	−0.771072	−	0.636748i	$-0.780279\pi$
$317$	−5288.32	−	9159.63i	−0.936976	−	1.62289i	−0.165904	−	0.986142i	$-0.553054\pi$
$318$		0			0
$319$	1974.99	−	3420.78i	0.346640	−	0.600398i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−460.469			−0.0793225
$324$		0			0
$325$	−587.098			−0.100204
$326$		0			0
$327$		0			0
$328$		0			0
$329$	796.110	−	1378.90i	0.133407	−	0.231068i
$330$		0			0
$331$	4359.68	+	7551.19i	0.723957	+	1.25393i	0.959402	+	0.282042i	$0.0910118\pi$
$331$	4359.68	+	7551.19i	0.723957	+	1.25393i	−0.235445	+	0.971888i	$0.575655\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	1313.73	+	2275.44i	0.214258	+	0.371106i
$336$		0			0
$337$	3886.81	−	6732.15i	0.628273	−	1.08820i	−0.359625	−	0.933097i	$-0.617095\pi$
$337$	3886.81	−	6732.15i	0.628273	−	1.08820i	0.987898	−	0.155104i	$-0.0495713\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	12734.8			2.02236
$342$		0			0
$343$	−6543.10			−1.03001
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−3544.97	+	6140.07i	−0.548427	+	0.949903i	0.449956	+	0.893051i	$0.351440\pi$
$347$	−3544.97	+	6140.07i	−0.548427	+	0.949903i	−0.998383	+	0.0568520i	$0.981894\pi$
$348$		0			0
$349$	−6042.73	−	10466.3i	−0.926819	−	1.60530i	−0.788609	−	0.614894i	$-0.789199\pi$
$349$	−6042.73	−	10466.3i	−0.926819	−	1.60530i	−0.138209	−	0.990403i	$-0.544135\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	4488.47	+	7774.27i	0.676763	+	1.17219i	0.975950	+	0.217994i	$0.0699512\pi$
$353$	4488.47	+	7774.27i	0.676763	+	1.17219i	−0.299187	+	0.954194i	$0.596715\pi$
$354$		0			0
$355$	−1902.13	+	3294.59i	−0.284380	+	0.492560i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	2038.36			0.299667			0.149834	−	0.988711i	$-0.452126\pi$
$359$	2038.36			0.299667			0.149834	+	0.988711i	$0.452126\pi$
$360$		0			0
$361$	−6055.60			−0.882869
$362$		0			0
$363$		0			0
$364$		0			0
$365$	252.691	−	437.673i	0.0362368	−	0.0627640i
$366$		0			0
$367$	−3523.91	−	6103.59i	−0.501217	−	0.868133i	−0.999999	−	0.00140579i	$-0.999553\pi$
$367$	−3523.91	−	6103.59i	−0.501217	−	0.868133i	0.498782	−	0.866727i	$-0.333781\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	761.179	+	1318.40i	0.106519	+	0.184496i
$372$		0			0
$373$	2142.75	−	3711.35i	0.297446	−	0.515191i	−0.678105	−	0.734965i	$-0.737199\pi$
$373$	2142.75	−	3711.35i	0.297446	−	0.515191i	0.975551	+	0.219774i	$0.0705319\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−2344.73			−0.320318
$378$		0			0
$379$	−1366.86			−0.185253			−0.0926267	−	0.995701i	$-0.529526\pi$
$379$	−1366.86			−0.185253			−0.0926267	+	0.995701i	$0.529526\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	6227.39	−	10786.2i	0.830822	−	1.43903i	−0.0665644	−	0.997782i	$-0.521204\pi$
$383$	6227.39	−	10786.2i	0.830822	−	1.43903i	0.897387	−	0.441245i	$-0.145463\pi$
$384$		0			0
$385$	1770.72	+	3066.97i	0.234400	+	0.405993i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	218.281	+	378.074i	0.0284506	+	0.0492779i	0.879900	−	0.475159i	$-0.157609\pi$
$389$	218.281	+	378.074i	0.0284506	+	0.0492779i	−0.851450	+	0.524437i	$0.824276\pi$
$390$		0			0
$391$	−857.206	+	1484.72i	−0.110871	+	0.192035i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−1128.17			−0.143707
$396$		0			0
$397$	−7117.74			−0.899822			−0.449911	−	0.893073i	$-0.648544\pi$
$397$	−7117.74			−0.899822			−0.449911	+	0.893073i	$0.648544\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	3884.74	−	6728.56i	0.483777	−	0.837926i	−0.516050	−	0.856559i	$-0.672598\pi$
$401$	3884.74	−	6728.56i	0.483777	−	0.837926i	0.999826	+	0.0186329i	$0.00593139\pi$
$402$		0			0
$403$	−3779.72	−	6546.67i	−0.467200	−	0.809213i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−4070.08	−	7049.59i	−0.495691	−	0.858563i
$408$		0			0
$409$	6416.30	−	11113.4i	0.775711	−	1.34357i	−0.158683	−	0.987330i	$-0.550725\pi$
$409$	6416.30	−	11113.4i	0.775711	−	1.34357i	0.934394	−	0.356241i	$-0.115942\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−14265.6			−1.69967
$414$		0			0
$415$	2515.26			0.297516
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−7451.60	+	12906.6i	−0.868818	+	1.50484i	−0.00561181	+	0.999984i	$0.501786\pi$
$419$	−7451.60	+	12906.6i	−0.868818	+	1.50484i	−0.863206	+	0.504852i	$0.831547\pi$
$420$		0			0
$421$	609.386	+	1055.49i	0.0705456	+	0.122188i	0.899141	−	0.437660i	$-0.144193\pi$
$421$	609.386	+	1055.49i	0.0705456	+	0.122188i	−0.828595	+	0.559849i	$0.810859\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−203.069	−	351.727i	−0.0231772	−	0.0401441i
$426$		0			0
$427$	2846.08	−	4929.56i	0.322556	−	0.558684i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	9989.48			1.11642			0.558209	−	0.829700i	$-0.311489\pi$
$431$	9989.48			1.11642			0.558209	+	0.829700i	$0.311489\pi$
$432$		0			0
$433$	−6010.34			−0.667063			−0.333532	−	0.942739i	$-0.608240\pi$
$433$	−6010.34			−0.667063			−0.333532	+	0.942739i	$0.608240\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	1495.60	−	2590.46i	0.163717	−	0.283566i
$438$		0			0
$439$	−5117.33	−	8863.48i	−0.556348	−	0.963624i	−0.997797	−	0.0663373i	$-0.978869\pi$
$439$	−5117.33	−	8863.48i	−0.556348	−	0.963624i	0.441449	−	0.897286i	$-0.354465\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	2479.28	+	4294.24i	0.265901	+	0.460554i	0.967799	−	0.251723i	$-0.0809973\pi$
$443$	2479.28	+	4294.24i	0.265901	+	0.460554i	−0.701898	+	0.712277i	$0.747664\pi$
$444$		0			0
$445$	2171.94	−	3761.91i	0.231370	−	0.400745i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	7715.87			0.810989			0.405495	−	0.914097i	$-0.367099\pi$
$449$	7715.87			0.810989			0.405495	+	0.914097i	$0.367099\pi$
$450$		0			0
$451$	12962.0			1.35335
$452$		0			0
$453$		0			0
$454$		0			0
$455$	1051.11	−	1820.57i	0.108301	−	0.187582i
$456$		0			0
$457$	245.867	+	425.854i	0.0251667	+	0.0435900i	0.878334	−	0.478047i	$-0.158655\pi$
$457$	245.867	+	425.854i	0.0251667	+	0.0435900i	−0.853168	+	0.521637i	$0.825322\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	1606.74	+	2782.96i	0.162329	+	0.281162i	0.935703	−	0.352788i	$-0.114766\pi$
$461$	1606.74	+	2782.96i	0.162329	+	0.281162i	−0.773375	+	0.633949i	$0.781433\pi$
$462$		0			0
$463$	−1662.35	+	2879.28i	−0.166860	+	0.289009i	−0.937314	−	0.348486i	$-0.886696\pi$
$463$	−1662.35	+	2879.28i	−0.166860	+	0.289009i	0.770455	+	0.637495i	$0.220029\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−11322.6			−1.12194			−0.560972	−	0.827835i	$-0.689572\pi$
$467$	−11322.6			−1.12194			−0.560972	+	0.827835i	$0.689572\pi$
$468$		0			0
$469$	−9408.10			−0.926281
$470$		0			0
$471$		0			0
$472$		0			0
$473$	7418.97	−	12850.0i	0.721194	−	1.24914i
$474$		0			0
$475$	354.304	+	613.672i	0.0342244	+	0.0592783i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−3591.16	−	6220.07i	−0.342556	−	0.593325i	0.642351	−	0.766411i	$-0.277959\pi$
$479$	−3591.16	−	6220.07i	−0.342556	−	0.593325i	−0.984907	+	0.173086i	$0.944626\pi$
$480$		0			0
$481$	−2416.03	+	4184.68i	−0.229026	+	0.396684i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	8747.21			0.818949
$486$		0			0
$487$	−242.899			−0.0226012			−0.0113006	−	0.999936i	$-0.503597\pi$
$487$	−242.899			−0.0226012			−0.0113006	+	0.999936i	$0.503597\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	1745.68	−	3023.60i	0.160451	−	0.277909i	−0.774580	−	0.632476i	$-0.782039\pi$
$491$	1745.68	−	3023.60i	0.160451	−	0.277909i	0.935030	+	0.354567i	$0.115372\pi$
$492$		0			0
$493$	−811.012	−	1404.71i	−0.0740896	−	0.128327i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−6810.97	−	11796.9i	−0.614715	−	1.06472i
$498$		0			0
$499$	357.109	−	618.532i	0.0320369	−	0.0554895i	−0.849562	−	0.527488i	$-0.823134\pi$
$499$	357.109	−	618.532i	0.0320369	−	0.0554895i	0.881599	+	0.471999i	$0.156467\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	18948.7			1.67968			0.839842	−	0.542832i	$-0.182648\pi$
$503$	18948.7			1.67968			0.839842	+	0.542832i	$0.182648\pi$
$504$		0			0
$505$	6470.78			0.570190
$506$		0			0
$507$		0			0
$508$		0			0
$509$	5920.63	−	10254.8i	0.515574	−	0.893001i	−0.484262	−	0.874923i	$-0.660912\pi$
$509$	5920.63	−	10254.8i	0.515574	−	0.893001i	0.999837	−	0.0180779i	$-0.00575468\pi$
$510$		0			0
$511$	904.808	+	1567.17i	0.0783294	+	0.135671i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−479.517	−	830.548i	−0.0410292	−	0.0710647i
$516$		0			0
$517$	−1759.17	+	3046.97i	−0.149648	+	0.259198i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	21836.5			1.83623			0.918114	−	0.396317i	$-0.129712\pi$
$521$	21836.5			1.83623			0.918114	+	0.396317i	$0.129712\pi$
$522$		0			0
$523$	6595.18			0.551409			0.275705	−	0.961242i	$-0.411089\pi$
$523$	6595.18			0.551409			0.275705	+	0.961242i	$0.411089\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	2614.71	−	4528.82i	0.216127	−	0.374342i
$528$		0			0
$529$	515.095	+	892.171i	0.0423354	+	0.0733271i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−3847.18	−	6663.51i	−0.312645	−	0.541517i
$534$		0			0
$535$	−3544.26	+	6138.84i	−0.286415	+	0.496085i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	888.768			0.0710240
$540$		0			0
$541$	−7877.56			−0.626031			−0.313015	−	0.949748i	$-0.601339\pi$
$541$	−7877.56			−0.626031			−0.313015	+	0.949748i	$0.601339\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−324.368	+	561.823i	−0.0254943	+	0.0441575i
$546$		0			0
$547$	−7710.52	−	13355.0i	−0.602702	−	1.04391i	−0.992410	−	0.122972i	$-0.960758\pi$
$547$	−7710.52	−	13355.0i	−0.602702	−	1.04391i	0.389708	−	0.920938i	$-0.372576\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	1415.01	+	2450.86i	0.109403	+	0.189492i
$552$		0			0
$553$	2019.81	−	3498.41i	0.155318	−	0.269019i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−17146.1			−1.30431			−0.652156	−	0.758085i	$-0.726135\pi$
$557$	−17146.1			−1.30431			−0.652156	+	0.758085i	$0.726135\pi$
$558$		0			0
$559$	−8807.90			−0.666430
$560$		0			0
$561$		0			0
$562$		0			0
$563$	6665.25	−	11544.5i	0.498946	−	0.864200i	−0.501053	−	0.865416i	$-0.667054\pi$
$563$	6665.25	−	11544.5i	0.498946	−	0.864200i	0.999999	+	0.00121656i	$0.000387245\pi$
$564$		0			0
$565$	2077.14	+	3597.71i	0.154665	+	0.267888i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−9291.05	−	16092.6i	−0.684536	−	1.18565i	−0.973582	−	0.228336i	$-0.926671\pi$
$569$	−9291.05	−	16092.6i	−0.684536	−	1.18565i	0.289046	−	0.957315i	$-0.406662\pi$
$570$		0			0
$571$	−2786.46	+	4826.29i	−0.204220	+	0.353719i	−0.949884	−	0.312603i	$-0.898799\pi$
$571$	−2786.46	+	4826.29i	−0.204220	+	0.353719i	0.745664	+	0.666322i	$0.232132\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	2638.28			0.191346
$576$		0			0
$577$	−11047.6			−0.797083			−0.398542	−	0.917150i	$-0.630484\pi$
$577$	−11047.6			−0.797083			−0.398542	+	0.917150i	$0.630484\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−4503.19	+	7799.75i	−0.321555	+	0.556950i
$582$		0			0
$583$	−1681.98	−	2913.28i	−0.119486	−	0.206956i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	10.1276	+	17.5414i	0.000712110	+	0.00123341i	0.866381	−	0.499383i	$-0.166440\pi$
$587$	10.1276	+	17.5414i	0.000712110	+	0.00123341i	−0.865669	+	0.500617i	$0.833107\pi$
$588$		0			0
$589$	−4562.00	+	7901.62i	−0.319141	+	0.552768i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−21868.4			−1.51438			−0.757190	−	0.653194i	$-0.773428\pi$
$593$	−21868.4			−1.51438			−0.757190	+	0.653194i	$0.773428\pi$
$594$		0			0
$595$	1454.26			0.100200
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−4679.83	+	8105.70i	−0.319220	+	0.552905i	−0.980325	−	0.197388i	$-0.936754\pi$
$599$	−4679.83	+	8105.70i	−0.319220	+	0.552905i	0.661106	+	0.750293i	$0.270087\pi$
$600$		0			0
$601$	−503.648	−	872.344i	−0.0341834	−	0.0592074i	0.848428	−	0.529311i	$-0.177550\pi$
$601$	−503.648	−	872.344i	−0.0341834	−	0.0592074i	−0.882611	+	0.470104i	$0.844216\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−585.260	−	1013.70i	−0.0393293	−	0.0681203i
$606$		0			0
$607$	3487.80	−	6041.05i	0.233222	−	0.403952i	−0.725533	−	0.688188i	$-0.758407\pi$
$607$	3487.80	−	6041.05i	0.233222	−	0.403952i	0.958754	+	0.284236i	$0.0917399\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	2088.51			0.138285
$612$		0			0
$613$	25416.8			1.67468			0.837338	−	0.546686i	$-0.184111\pi$
$613$	25416.8			1.67468			0.837338	+	0.546686i	$0.184111\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−4925.42	+	8531.08i	−0.321378	+	0.556643i	−0.980773	−	0.195154i	$-0.937479\pi$
$617$	−4925.42	+	8531.08i	−0.321378	+	0.556643i	0.659395	+	0.751797i	$0.270813\pi$
$618$		0			0
$619$	−8508.50	−	14737.2i	−0.552481	−	0.956925i	−0.998095	−	0.0616997i	$-0.980348\pi$
$619$	−8508.50	−	14737.2i	−0.552481	−	0.956925i	0.445614	−	0.895225i	$-0.352985\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	7777.05	+	13470.2i	0.500130	+	0.866250i
$624$		0			0
$625$	−312.500	+	541.266i	−0.0200000	+	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−3342.69			−0.211895
$630$		0			0
$631$	6738.60			0.425134			0.212567	−	0.977147i	$-0.431818\pi$
$631$	6738.60			0.425134			0.212567	+	0.977147i	$0.431818\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	5964.18	−	10330.3i	0.372726	−	0.645581i
$636$		0			0
$637$	−263.789	−	456.896i	−0.0164077	−	0.0284190i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−3145.07	−	5447.43i	−0.193796	−	0.335664i	0.752709	−	0.658353i	$-0.228747\pi$
$641$	−3145.07	−	5447.43i	−0.193796	−	0.335664i	−0.946505	+	0.322689i	$0.895413\pi$
$642$		0			0
$643$	−4441.83	+	7693.48i	−0.272424	+	0.471853i	−0.969482	−	0.245162i	$-0.921159\pi$
$643$	−4441.83	+	7693.48i	−0.272424	+	0.471853i	0.697058	+	0.717015i	$0.254492\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	13436.4			0.816444			0.408222	−	0.912883i	$-0.366149\pi$
$647$	13436.4			0.816444			0.408222	+	0.912883i	$0.366149\pi$
$648$		0			0
$649$	31522.8			1.90659
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−8583.02	+	14866.2i	−0.514363	+	0.890904i	0.485498	+	0.874238i	$0.338638\pi$
$653$	−8583.02	+	14866.2i	−0.514363	+	0.890904i	−0.999861	+	0.0166657i	$0.994695\pi$
$654$		0			0
$655$	−2260.79	−	3915.80i	−0.134865	−	0.233593i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	10209.4	+	17683.2i	0.603493	+	1.04528i	0.992288	+	0.123956i	$0.0395580\pi$
$659$	10209.4	+	17683.2i	0.603493	+	1.04528i	−0.388795	+	0.921324i	$0.627109\pi$
$660$		0			0
$661$	−2622.01	+	4541.46i	−0.154288	+	0.267235i	−0.932800	−	0.360395i	$-0.882642\pi$
$661$	−2622.01	+	4541.46i	−0.154288	+	0.267235i	0.778511	+	0.627630i	$0.215975\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−2537.31			−0.147959
$666$		0			0
$667$	10536.7			0.611666
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−6289.00	+	10892.9i	−0.361825	+	0.626699i
$672$		0			0
$673$	−7477.24	−	12951.0i	−0.428271	−	0.741787i	0.568449	−	0.822719i	$-0.307544\pi$
$673$	−7477.24	−	12951.0i	−0.428271	−	0.741787i	−0.996720	+	0.0809318i	$0.974210\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	13525.7	+	23427.2i	0.767850	+	1.32996i	0.938726	+	0.344663i	$0.112007\pi$
$677$	13525.7	+	23427.2i	0.767850	+	1.32996i	−0.170876	+	0.985293i	$0.554660\pi$
$678$		0			0
$679$	−15660.6	+	27124.9i	−0.885121	+	1.53307i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−26054.5			−1.45966			−0.729829	−	0.683630i	$-0.760400\pi$
$683$	−26054.5			−1.45966			−0.729829	+	0.683630i	$0.760400\pi$
$684$		0			0
$685$	−4041.82			−0.225445
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−998.435	+	1729.34i	−0.0552066	+	0.0956207i
$690$		0			0
$691$	3147.80	+	5452.15i	0.173297	+	0.300159i	0.939570	−	0.342356i	$-0.111225\pi$
$691$	3147.80	+	5452.15i	0.173297	+	0.300159i	−0.766274	+	0.642514i	$0.777891\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	7161.80	+	12404.6i	0.390881	+	0.677026i
$696$		0			0
$697$	2661.38	−	4609.64i	0.144630	−	0.250506i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−4014.82			−0.216316			−0.108158	−	0.994134i	$-0.534495\pi$
$701$	−4014.82			−0.216316			−0.108158	+	0.994134i	$0.534495\pi$
$702$		0			0
$703$	5832.13			0.312892
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−11584.9	+	20065.7i	−0.616261	+	1.06740i
$708$		0			0
$709$	11299.4	+	19571.1i	0.598528	+	1.03668i	0.993039	+	0.117790i	$0.0375809\pi$
$709$	11299.4	+	19571.1i	0.598528	+	1.03668i	−0.394510	+	0.918892i	$0.629086\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	16985.2	+	29419.2i	0.892145	+	1.54524i
$714$		0			0
$715$	−2322.64	+	4022.93i	−0.121485	+	0.210418i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−6579.83			−0.341288			−0.170644	−	0.985333i	$-0.554585\pi$
$719$	−6579.83			−0.341288			−0.170644	+	0.985333i	$0.554585\pi$
$720$		0			0
$721$	3434.01			0.177378
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−1248.05	+	2161.69i	−0.0639331	+	0.110735i
$726$		0			0
$727$	−16608.6	−	28767.0i	−0.847290	−	1.46755i	−0.883618	−	0.468209i	$-0.844900\pi$
$727$	−16608.6	−	28767.0i	−0.847290	−	1.46755i	0.0363284	−	0.999340i	$-0.488434\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−3046.54	−	5276.76i	−0.154145	−	0.266988i
$732$		0			0
$733$	−8025.62	+	13900.8i	−0.404411	+	0.700460i	−0.994253	−	0.107059i	$-0.965857\pi$
$733$	−8025.62	+	13900.8i	−0.404411	+	0.700460i	0.589842	+	0.807519i	$0.299190\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	20789.1			1.03905
$738$		0			0
$739$	14855.1			0.739452			0.369726	−	0.929141i	$-0.379452\pi$
$739$	14855.1			0.739452			0.369726	+	0.929141i	$0.379452\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	11747.5	−	20347.2i	0.580044	−	1.00467i	−0.415430	−	0.909625i	$-0.636369\pi$
$743$	11747.5	−	20347.2i	0.580044	−	1.00467i	0.995473	−	0.0950399i	$-0.0302979\pi$
$744$		0			0
$745$	−1698.53	−	2941.93i	−0.0835291	−	0.144677i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−12690.9	−	21981.3i	−0.619114	−	1.07234i
$750$		0			0
$751$	−4150.84	+	7189.47i	−0.201686	+	0.349331i	−0.949072	−	0.315060i	$-0.897975\pi$
$751$	−4150.84	+	7189.47i	−0.201686	+	0.349331i	0.747386	+	0.664390i	$0.231309\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−11931.1			−0.575124
$756$		0			0
$757$	7018.98			0.337000			0.168500	−	0.985702i	$-0.446108\pi$
$757$	7018.98			0.337000			0.168500	+	0.985702i	$0.446108\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−15904.2	+	27546.8i	−0.757589	+	1.31218i	0.186488	+	0.982457i	$0.440290\pi$
$761$	−15904.2	+	27546.8i	−0.757589	+	1.31218i	−0.944077	+	0.329725i	$0.893044\pi$
$762$		0			0
$763$	−1161.46	−	2011.72i	−0.0551086	−	0.0954509i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−9356.06	−	16205.2i	−0.440453	−	0.762888i
$768$		0			0
$769$	10966.5	−	18994.5i	0.514253	−	0.890712i	−0.485610	−	0.874175i	$-0.661403\pi$
$769$	10966.5	−	18994.5i	0.514253	−	0.890712i	0.999863	−	0.0165366i	$-0.00526402\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−25629.2			−1.19252			−0.596260	−	0.802791i	$-0.703347\pi$
$773$	−25629.2			−1.19252			−0.596260	+	0.802791i	$0.703347\pi$
$774$		0			0
$775$	−8047.47			−0.372998
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−4643.42	+	8042.63i	−0.213566	+	0.369907i
$780$		0			0
$781$	15050.2	+	26067.7i	0.689551	+	1.19434i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	6745.87	+	11684.2i	0.306714	+	0.531244i
$786$		0			0
$787$	−14894.6	+	25798.3i	−0.674633	+	1.16850i	0.301943	+	0.953326i	$0.402365\pi$
$787$	−14894.6	+	25798.3i	−0.674633	+	1.16850i	−0.976576	+	0.215173i	$0.930968\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−14875.2			−0.668648
$792$		0			0
$793$	7466.39			0.334350
$794$		0			0
$795$		0			0
$796$		0			0
$797$	6557.71	−	11358.3i	0.291450	−	0.504807i	−0.682703	−	0.730696i	$-0.739196\pi$
$797$	6557.71	−	11358.3i	0.291450	−	0.504807i	0.974153	+	0.225890i	$0.0725289\pi$
$798$		0			0
$799$	722.388	+	1251.21i	0.0319853	+	0.0554001i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−1999.36	−	3462.99i	−0.0878653	−	0.152187i
$804$		0			0
$805$	−4723.43	+	8181.23i	−0.206806	+	0.358199i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−9743.58			−0.423444			−0.211722	−	0.977330i	$-0.567907\pi$
$809$	−9743.58			−0.423444			−0.211722	+	0.977330i	$0.567907\pi$
$810$		0			0
$811$	13466.5			0.583075			0.291538	−	0.956559i	$-0.405833\pi$
$811$	13466.5			0.583075			0.291538	+	0.956559i	$0.405833\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−3751.17	+	6497.22i	−0.161224	+	0.279249i
$816$		0			0
$817$	5315.42	+	9206.58i	0.227617	+	0.394244i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−12564.1	−	21761.6i	−0.534092	−	0.925074i	−0.999207	−	0.0398240i	$-0.987320\pi$
$821$	−12564.1	−	21761.6i	−0.534092	−	0.925074i	0.465115	−	0.885250i	$-0.346013\pi$
$822$		0			0
$823$	−18516.2	+	32071.0i	−0.784245	+	1.35835i	0.145205	+	0.989402i	$0.453616\pi$
$823$	−18516.2	+	32071.0i	−0.784245	+	1.35835i	−0.929449	+	0.368950i	$0.879717\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	33611.4			1.41328			0.706640	−	0.707574i	$-0.250210\pi$
$827$	33611.4			1.41328			0.706640	+	0.707574i	$0.250210\pi$
$828$		0			0
$829$	17321.9			0.725710			0.362855	−	0.931846i	$-0.381802\pi$
$829$	17321.9			0.725710			0.362855	+	0.931846i	$0.381802\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	182.483	−	316.069i	0.00759021	−	0.0131466i
$834$		0			0
$835$	−8573.58	−	14849.9i	−0.355331	−	0.615451i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−6935.33	−	12012.4i	−0.285381	−	0.494294i	0.687321	−	0.726354i	$-0.258787\pi$
$839$	−6935.33	−	12012.4i	−0.285381	−	0.494294i	−0.972701	+	0.232060i	$0.925453\pi$
$840$		0			0
$841$	7210.07	−	12488.2i	0.295628	−	0.512043i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−8227.53			−0.334953
$846$		0			0
$847$	4191.27			0.170028
$848$		0			0
$849$		0			0
$850$		0			0
$851$	10857.0	−	18805.0i	0.437338	−	0.757492i
$852$		0			0
$853$	9118.48	+	15793.7i	0.366015	+	0.633957i	0.988938	−	0.148326i	$-0.0473885\pi$
$853$	9118.48	+	15793.7i	0.366015	+	0.633957i	−0.622923	+	0.782283i	$0.714055\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	18734.5	+	32449.1i	0.746741	+	1.29339i	0.949377	+	0.314140i	$0.101716\pi$
$857$	18734.5	+	32449.1i	0.746741	+	1.29339i	−0.202635	+	0.979254i	$0.564951\pi$
$858$		0			0
$859$	−4242.53	+	7348.29i	−0.168514	+	0.291875i	−0.937898	−	0.346912i	$-0.887230\pi$
$859$	−4242.53	+	7348.29i	−0.168514	+	0.291875i	0.769384	+	0.638787i	$0.220563\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−20829.5			−0.821603			−0.410801	−	0.911725i	$-0.634751\pi$
$863$	−20829.5			−0.821603			−0.410801	+	0.911725i	$0.634751\pi$
$864$		0			0
$865$	10039.7			0.394637
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−4463.19	+	7730.46i	−0.174227	+	0.301770i
$870$		0			0
$871$	−6170.28	−	10687.2i	−0.240037	−	0.415756i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−1118.97	−	1938.11i	−0.0432320	−	0.0748800i
$876$		0			0
$877$	−20795.6	+	36019.0i	−0.800704	+	1.38686i	0.118450	+	0.992960i	$0.462207\pi$
$877$	−20795.6	+	36019.0i	−0.800704	+	1.38686i	−0.919154	+	0.393899i	$0.871126\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	47222.3			1.80586			0.902928	−	0.429792i	$-0.141413\pi$
$881$	47222.3			1.80586			0.902928	+	0.429792i	$0.141413\pi$
$882$		0			0
$883$	−3201.36			−0.122010			−0.0610048	−	0.998137i	$-0.519430\pi$
$883$	−3201.36			−0.122010			−0.0610048	+	0.998137i	$0.519430\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−14557.9	+	25215.1i	−0.551080	+	0.954498i	0.447117	+	0.894475i	$0.352451\pi$
$887$	−14557.9	+	25215.1i	−0.551080	+	0.954498i	−0.998197	+	0.0600228i	$0.980883\pi$
$888$		0			0
$889$	21355.9	+	36989.5i	0.805685	+	1.39549i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−1260.38	−	2183.04i	−0.0472307	−	0.0818060i
$894$		0			0
$895$	9592.38	−	16614.5i	0.358255	−	0.620515i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−32139.7			−1.19235
$900$		0			0
$901$	−1381.38			−0.0510772
$902$		0			0
$903$		0			0
$904$		0			0
$905$	4555.25	−	7889.92i	0.167317	−	0.289801i
$906$		0			0
$907$	19627.7	+	33996.2i	0.718552	+	1.24457i	0.961573	+	0.274548i	$0.0885281\pi$
$907$	19627.7	+	33996.2i	0.718552	+	1.24457i	−0.243022	+	0.970021i	$0.578139\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−18333.4	−	31754.4i	−0.666754	−	1.15485i	−0.978807	−	0.204787i	$-0.934350\pi$
$911$	−18333.4	−	31754.4i	−0.666754	−	1.15485i	0.312052	−	0.950065i	$-0.398984\pi$
$912$		0			0
$913$	9950.71	−	17235.1i	0.360701	−	0.624753i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	16190.4			0.583047
$918$		0			0
$919$	−11786.5			−0.423070			−0.211535	−	0.977370i	$-0.567846\pi$
$919$	−11786.5			−0.423070			−0.211535	+	0.977370i	$0.567846\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	8933.91	−	15474.0i	0.318595	−	0.551823i
$924$		0			0
$925$	2572.00	+	4454.84i	0.0914237	+	0.158350i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	20004.6	+	34649.0i	0.706490	+	1.22368i	0.966151	+	0.257977i	$0.0830560\pi$
$929$	20004.6	+	34649.0i	0.706490	+	1.22368i	−0.259661	+	0.965700i	$0.583611\pi$
$930$		0			0
$931$	−318.385	+	551.459i	−0.0112080	+	0.0194128i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−3213.48			−0.112398
$936$		0			0
$937$	−14272.7			−0.497620			−0.248810	−	0.968552i	$-0.580039\pi$
$937$	−14272.7			−0.497620			−0.248810	+	0.968552i	$0.580039\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−13110.8	+	22708.6i	−0.454197	+	0.786693i	−0.998642	−	0.0521041i	$-0.983407\pi$
$941$	−13110.8	+	22708.6i	−0.454197	+	0.786693i	0.544444	+	0.838797i	$0.316741\pi$
$942$		0			0
$943$	17288.3	+	29944.2i	0.597014	+	1.03406i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−20835.9	−	36088.9i	−0.714971	−	1.23837i	−0.962971	−	0.269605i	$-0.913107\pi$
$947$	−20835.9	−	36088.9i	−0.714971	−	1.23837i	0.248000	−	0.968760i	$-0.420227\pi$
$948$		0			0
$949$	−1186.83	+	2055.65i	−0.0405966	+	0.0703155i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	24374.3			0.828500			0.414250	−	0.910163i	$-0.364044\pi$
$953$	24374.3			0.828500			0.414250	+	0.910163i	$0.364044\pi$
$954$		0			0
$955$	−20979.8			−0.710882
$956$		0			0
$957$		0			0
$958$		0			0
$959$	7236.27	−	12533.6i	0.243661	−	0.422034i
$960$		0			0
$961$	−36914.0	−	63936.8i	−1.23910	−	2.14618i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	4525.89	+	7839.08i	0.150978	+	0.261501i
$966$		0			0
$967$	9335.19	−	16169.0i	0.310444	−	0.537705i	−0.668015	−	0.744148i	$-0.732856\pi$
$967$	9335.19	−	16169.0i	0.310444	−	0.537705i	0.978459	+	0.206443i	$0.0661890\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−139.756			−0.00461893			−0.00230946	−	0.999997i	$-0.500735\pi$
$971$	−139.756			−0.00461893			−0.00230946	+	0.999997i	$0.500735\pi$
$972$		0			0
$973$	−51288.4			−1.68986
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−5752.25	+	9963.18i	−0.188363	+	0.326254i	−0.944705	−	0.327923i	$-0.893651\pi$
$977$	−5752.25	+	9963.18i	−0.188363	+	0.326254i	0.756342	+	0.654177i	$0.226985\pi$
$978$		0			0
$979$	−17185.0	−	29765.3i	−0.561016	−	0.971708i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	707.470	+	1225.37i	0.0229550	+	0.0397593i	0.877275	−	0.479989i	$-0.159359\pi$
$983$	707.470	+	1225.37i	0.0229550	+	0.0397593i	−0.854320	+	0.519748i	$0.826026\pi$
$984$		0			0
$985$	9568.02	−	16572.3i	0.309505	−	0.536078i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	39580.6			1.27259
$990$		0			0
$991$	12991.4			0.416434			0.208217	−	0.978083i	$-0.433234\pi$
$991$	12991.4			0.416434			0.208217	+	0.978083i	$0.433234\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	1194.63	−	2069.16i	0.0380627	−	0.0659266i
$996$		0			0
$997$	−10575.6	−	18317.4i	−0.335940	−	0.581865i	0.647725	−	0.761874i	$-0.275721\pi$
$997$	−10575.6	−	18317.4i	−0.335940	−	0.581865i	−0.983665	+	0.180009i	$0.942387\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	540.4.i.b.361.1		8
3.2	odd	2		180.4.i.b.121.3	yes	8
9.2	odd	6		180.4.i.b.61.3	✓	8
9.4	even	3		1620.4.a.h.1.4		4
9.5	odd	6		1620.4.a.g.1.4		4
9.7	even	3	inner	540.4.i.b.181.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
180.4.i.b.61.3	✓	8	9.2	odd	6
180.4.i.b.121.3	yes	8	3.2	odd	2
540.4.i.b.181.1		8	9.7	even	3	inner
540.4.i.b.361.1		8	1.1	even	1	trivial
1620.4.a.g.1.4		4	9.5	odd	6
1620.4.a.h.1.4		4	9.4	even	3

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:	\( N \)	\(=\)	\( 540 = 2^{2} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	540.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.50000 + 4.33013i) q^{5} +(-8.95174 - 15.5049i) q^{7} +(19.7807 + 34.2612i) q^{11} +(11.7420 - 20.3377i) q^{13} +16.2456 q^{17} -28.3443 q^{19} +(-52.7655 + 91.3926i) q^{23} +(-12.5000 - 21.6506i) q^{25} +(-49.9221 - 86.4676i) q^{29} +(160.949 - 278.773i) q^{31} +89.5174 q^{35} -205.760 q^{37} +(163.822 - 283.748i) q^{41} +(-187.531 - 324.812i) q^{43} +(44.4668 + 77.0187i) q^{47} +(11.2328 - 19.4557i) q^{49} -85.0314 q^{53} -197.807 q^{55} +(398.403 - 690.054i) q^{59} +(158.968 + 275.341i) q^{61} +(58.7098 + 101.688i) q^{65} +(262.745 - 455.088i) q^{67} +760.854 q^{71} -101.076 q^{73} +(354.143 - 613.394i) q^{77} +(112.817 + 195.404i) q^{79} +(-251.526 - 435.655i) q^{83} +(-40.6139 + 70.3453i) q^{85} -868.775 q^{89} -420.444 q^{91} +(70.8608 - 122.734i) q^{95} +(-874.721 - 1515.06i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 20 q^{5} + 13 q^{7} + 57 q^{11} - 14 q^{13} - 6 q^{17} - 62 q^{19} - 69 q^{23} - 100 q^{25} - 69 q^{29} + 58 q^{31} - 130 q^{35} - 776 q^{37} + 396 q^{41} - 371 q^{43} + 129 q^{47} - 111 q^{49} - 2712 q^{53}+ \cdots + 1495 q^{97}+O(q^{100}) \) 8 * q - 20 * q^5 + 13 * q^7 + 57 * q^11 - 14 * q^13 - 6 * q^17 - 62 * q^19 - 69 * q^23 - 100 * q^25 - 69 * q^29 + 58 * q^31 - 130 * q^35 - 776 * q^37 + 396 * q^41 - 371 * q^43 + 129 * q^47 - 111 * q^49 - 2712 * q^53 - 570 * q^55 + 15 * q^59 + 1441 * q^61 - 70 * q^65 - 368 * q^67 - 336 * q^71 - 1910 * q^73 + 342 * q^77 + 1408 * q^79 - 789 * q^83 + 15 * q^85 - 3234 * q^89 - 2812 * q^91 + 155 * q^95 + 1495 * q^97

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