LMFDB - Embedded newform 567.4.c.c.566.30

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [567,4,Mod(566,567)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(567, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("567.566");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 567 = 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	567.c (of order $2$, degree $1$, 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:	$33.4540829733$
Analytic rank:	$0$
Dimension:	$44$
Twist minimal:	no (minimal twist has level 63)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			566.30
Character	$\chi$	$=$	567.566
Dual form			567.4.c.c.566.29

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.23715i q^{2} +2.99518 q^{4} -5.51516 q^{5} +(4.78094 + 17.8925i) q^{7} +24.5978i q^{8} +O(q^{10})$	\(q+2.23715i q^{2} +2.99518 q^{4} -5.51516 q^{5} +(4.78094 + 17.8925i) q^{7} +24.5978i q^{8} -12.3382i q^{10} -34.9229i q^{11} +26.7350i q^{13} +(-40.0282 + 10.6957i) q^{14} -31.0675 q^{16} -35.9093 q^{17} +78.0198i q^{19} -16.5189 q^{20} +78.1276 q^{22} +57.8833i q^{23} -94.5830 q^{25} -59.8102 q^{26} +(14.3198 + 53.5913i) q^{28} -221.779i q^{29} +145.494i q^{31} +127.280i q^{32} -80.3345i q^{34} +(-26.3676 - 98.6801i) q^{35} +290.956 q^{37} -174.542 q^{38} -135.661i q^{40} -108.121 q^{41} -348.122 q^{43} -104.600i q^{44} -129.493 q^{46} +251.090 q^{47} +(-297.285 + 171.086i) q^{49} -211.596i q^{50} +80.0761i q^{52} -160.173i q^{53} +192.605i q^{55} +(-440.117 + 117.601i) q^{56} +496.151 q^{58} -734.849 q^{59} +671.285i q^{61} -325.492 q^{62} -533.284 q^{64} -147.448i q^{65} -688.727 q^{67} -107.555 q^{68} +(220.762 - 58.9883i) q^{70} +577.396i q^{71} -37.4543i q^{73} +650.911i q^{74} +233.683i q^{76} +(624.859 - 166.964i) q^{77} +856.000 q^{79} +171.342 q^{80} -241.883i q^{82} -1232.26 q^{83} +198.046 q^{85} -778.800i q^{86} +859.027 q^{88} -1505.64 q^{89} +(-478.358 + 127.819i) q^{91} +173.371i q^{92} +561.724i q^{94} -430.291i q^{95} -1157.41i q^{97} +(-382.745 - 665.071i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 44 q - 156 q^{4} - 10 q^{7}+O(q^{10}) $ 44 * q - 156 * q^4 - 10 * q^7	$ 44 q - 156 q^{4} - 10 q^{7} + 484 q^{16} + 68 q^{22} + 704 q^{25} + 300 q^{28} + 328 q^{37} + 340 q^{43} + 968 q^{46} + 158 q^{49} + 1076 q^{58} - 808 q^{64} + 1180 q^{67} - 768 q^{70} + 604 q^{79} + 1224 q^{85} - 2588 q^{88} + 210 q^{91}+O(q^{100}) $ 44 * q - 156 * q^4 - 10 * q^7 + 484 * q^16 + 68 * q^22 + 704 * q^25 + 300 * q^28 + 328 * q^37 + 340 * q^43 + 968 * q^46 + 158 * q^49 + 1076 * q^58 - 808 * q^64 + 1180 * q^67 - 768 * q^70 + 604 * q^79 + 1224 * q^85 - 2588 * q^88 + 210 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/567\mathbb{Z}\right)^\times$.

$n$	$325$	$407$
$\chi(n)$	$-1$	$-1$

Coefficient data

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

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

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

$2$			2.23715i			0.790951i	0.918477	+	0.395475i	$0.129420\pi$
$2$			2.23715i			0.790951i	−0.918477	+	0.395475i	$0.870580\pi$
$3$		0			0
$3$		0			0
$4$	2.99518			0.374397
$5$	−5.51516			−0.493291			−0.246645	−	0.969106i	$-0.579328\pi$
$5$	−5.51516			−0.493291			−0.246645	+	0.969106i	$0.579328\pi$
$6$		0			0
$7$	4.78094	+	17.8925i	0.258147	+	0.966106i
$7$	4.78094	+	17.8925i	0.258147	+	0.966106i
$8$			24.5978i			1.08708i
$9$		0			0
$10$		−	12.3382i		−	0.390169i
$11$		−	34.9229i		−	0.957241i	−0.878022	−	0.478620i	$-0.841137\pi$
$11$		−	34.9229i		−	0.957241i	0.878022	−	0.478620i	$-0.158863\pi$
$12$		0			0
$13$			26.7350i			0.570382i	0.958471	+	0.285191i	$0.0920571\pi$
$13$			26.7350i			0.570382i	−0.958471	+	0.285191i	$0.907943\pi$
$14$	−40.0282	+	10.6957i	−0.764142	+	0.204181i
$15$		0			0
$16$	−31.0675			−0.485430
$17$	−35.9093			−0.512311			−0.256156	−	0.966636i	$-0.582456\pi$
$17$	−35.9093			−0.512311			−0.256156	+	0.966636i	$0.582456\pi$
$18$		0			0
$19$			78.0198i			0.942051i	0.882120	+	0.471025i	$0.156116\pi$
$19$			78.0198i			0.942051i	−0.882120	+	0.471025i	$0.843884\pi$
$20$	−16.5189			−0.184686
$21$		0			0
$22$	78.1276			0.757130
$23$			57.8833i			0.524761i	0.964965	+	0.262380i	$0.0845075\pi$
$23$			57.8833i			0.524761i	−0.964965	+	0.262380i	$0.915492\pi$
$24$		0			0
$25$	−94.5830			−0.756664
$26$	−59.8102			−0.451144
$27$		0			0
$28$	14.3198	+	53.5913i	0.0966493	+	0.361707i
$29$		−	221.779i		−	1.42011i	−0.704145	−	0.710056i	$-0.748670\pi$
$29$		−	221.779i		−	1.42011i	0.704145	−	0.710056i	$-0.251330\pi$
$30$		0			0
$31$			145.494i			0.842953i	0.906839	+	0.421476i	$0.138488\pi$
$31$			145.494i			0.842953i	−0.906839	+	0.421476i	$0.861512\pi$
$32$			127.280i			0.703129i
$33$		0			0
$34$		−	80.3345i		−	0.405213i
$35$	−26.3676	−	98.6801i	−0.127341	−	0.476571i
$36$		0			0
$37$	290.956			1.29278			0.646390	−	0.763008i	$-0.276278\pi$
$37$	290.956			1.29278			0.646390	+	0.763008i	$0.276278\pi$
$38$	−174.542			−0.745116
$39$		0			0
$40$		−	135.661i		−	0.536246i
$41$	−108.121			−0.411846			−0.205923	−	0.978568i	$-0.566020\pi$
$41$	−108.121			−0.411846			−0.205923	+	0.978568i	$0.566020\pi$
$42$		0			0
$43$	−348.122			−1.23461			−0.617303	−	0.786725i	$-0.711775\pi$
$43$	−348.122			−1.23461			−0.617303	+	0.786725i	$0.711775\pi$
$44$		−	104.600i		−	0.358388i
$45$		0			0
$46$	−129.493			−0.415060
$47$	251.090			0.779259			0.389630	−	0.920972i	$-0.372603\pi$
$47$	251.090			0.779259			0.389630	+	0.920972i	$0.372603\pi$
$48$		0			0
$49$	−297.285	+	171.086i	−0.866721	+	0.498794i
$50$		−	211.596i		−	0.598484i
$51$		0			0
$52$			80.0761i			0.213549i
$53$		−	160.173i		−	0.415122i	−0.978222	−	0.207561i	$-0.933448\pi$
$53$		−	160.173i		−	0.415122i	0.978222	−	0.207561i	$-0.0665525\pi$
$54$		0			0
$55$			192.605i			0.472198i
$56$	−440.117	+	117.601i	−1.05023	+	0.280626i
$57$		0			0
$58$	496.151			1.12324
$59$	−734.849			−1.62151			−0.810756	−	0.585384i	$-0.800944\pi$
$59$	−734.849			−1.62151			−0.810756	+	0.585384i	$0.800944\pi$
$60$		0			0
$61$			671.285i			1.40900i	0.709702	+	0.704502i	$0.248830\pi$
$61$			671.285i			1.40900i	−0.709702	+	0.704502i	$0.751170\pi$
$62$	−325.492			−0.666734
$63$		0			0
$64$	−533.284			−1.04157
$65$		−	147.448i		−	0.281364i
$66$		0			0
$67$	−688.727			−1.25584			−0.627921	−	0.778277i	$-0.716094\pi$
$67$	−688.727			−1.25584			−0.627921	+	0.778277i	$0.716094\pi$
$68$	−107.555			−0.191808
$69$		0			0
$70$	220.762	−	58.9883i	0.376944	−	0.100721i
$71$			577.396i			0.965130i	0.875860	+	0.482565i	$0.160295\pi$
$71$			577.396i			0.965130i	−0.875860	+	0.482565i	$0.839705\pi$
$72$		0			0
$73$		−	37.4543i		−	0.0600506i	−0.999549	−	0.0300253i	$-0.990441\pi$
$73$		−	37.4543i		−	0.0600506i	0.999549	−	0.0300253i	$-0.00955879\pi$
$74$			650.911i			1.02252i
$75$		0			0
$76$			233.683i			0.352701i
$77$	624.859	−	166.964i	0.924796	−	0.247108i
$78$		0			0
$79$	856.000			1.21908			0.609541	−	0.792755i	$-0.291354\pi$
$79$	856.000			1.21908			0.609541	+	0.792755i	$0.291354\pi$
$80$	171.342			0.239458
$81$		0			0
$82$		−	241.883i		−	0.325750i
$83$	−1232.26			−1.62961			−0.814806	−	0.579733i	$-0.803157\pi$
$83$	−1232.26			−1.62961			−0.814806	+	0.579733i	$0.803157\pi$
$84$		0			0
$85$	198.046			0.252718
$86$		−	778.800i		−	0.976513i
$87$		0			0
$88$	859.027			1.04060
$89$	−1505.64			−1.79323			−0.896615	−	0.442811i	$-0.853981\pi$
$89$	−1505.64			−1.79323			−0.896615	+	0.442811i	$0.853981\pi$
$90$		0			0
$91$	−478.358	+	127.819i	−0.551050	+	0.147242i
$92$			173.371i			0.196469i
$93$		0			0
$94$			561.724i			0.616356i
$95$		−	430.291i		−	0.464705i
$96$		0			0
$97$		−	1157.41i		−	1.21152i	−0.795648	−	0.605759i	$-0.792870\pi$
$97$		−	1157.41i		−	1.21152i	0.795648	−	0.605759i	$-0.207130\pi$
$98$	−382.745	−	665.071i	−0.394521	−	0.685533i
$99$		0			0
$100$	−283.293			−0.283293
$101$	155.567			0.153262			0.0766311	−	0.997060i	$-0.475584\pi$
$101$	155.567			0.153262			0.0766311	+	0.997060i	$0.475584\pi$
$102$		0			0
$103$		−	1894.55i		−	1.81239i	−0.422864	−	0.906193i	$-0.638975\pi$
$103$		−	1894.55i		−	1.81239i	0.422864	−	0.906193i	$-0.361025\pi$
$104$	−657.624			−0.620051
$105$		0			0
$106$	358.330			0.328341
$107$			507.467i			0.458492i	0.973369	+	0.229246i	$0.0736260\pi$
$107$			507.467i			0.458492i	−0.973369	+	0.229246i	$0.926374\pi$
$108$		0			0
$109$	298.426			0.262239			0.131119	−	0.991367i	$-0.458143\pi$
$109$	298.426			0.262239			0.131119	+	0.991367i	$0.458143\pi$
$110$	−430.886			−0.373485
$111$		0			0
$112$	−148.532	−	555.877i	−0.125312	−	0.468977i
$113$		−	249.616i		−	0.207804i	−0.994588	−	0.103902i	$-0.966867\pi$
$113$		−	249.616i		−	0.207804i	0.994588	−	0.103902i	$-0.0331328\pi$
$114$		0			0
$115$		−	319.235i		−	0.258860i
$116$		−	664.266i		−	0.531686i
$117$		0			0
$118$		−	1643.97i		−	1.28254i
$119$	−171.680	−	642.509i	−0.132251	−	0.494947i
$120$		0			0
$121$	111.392			0.0836905
$122$	−1501.76			−1.11445
$123$		0			0
$124$			435.781i			0.315599i
$125$	1211.03			0.866546
$126$		0			0
$127$	1942.24			1.35706			0.678528	−	0.734574i	$-0.262618\pi$
$127$	1942.24			1.35706			0.678528	+	0.734574i	$0.262618\pi$
$128$		−	174.795i		−	0.120702i
$129$		0			0
$130$	329.863			0.222545
$131$	−26.5284			−0.0176931			−0.00884655	−	0.999961i	$-0.502816\pi$
$131$	−26.5284			−0.0176931			−0.00884655	+	0.999961i	$0.502816\pi$
$132$		0			0
$133$	−1395.97	+	373.008i	−0.910121	+	0.243187i
$134$		−	1540.78i		−	0.993309i
$135$		0			0
$136$		−	883.291i		−	0.556924i
$137$		−	266.674i		−	0.166303i	−0.996537	−	0.0831515i	$-0.973501\pi$
$137$		−	266.674i		−	0.166303i	0.996537	−	0.0831515i	$-0.0264985\pi$
$138$		0			0
$139$			513.619i			0.313414i	0.987645	+	0.156707i	$0.0500879\pi$
$139$			513.619i			0.313414i	−0.987645	+	0.156707i	$0.949912\pi$
$140$	−78.9757	−	295.564i	−0.0476762	−	0.178427i
$141$		0			0
$142$	−1291.72			−0.763371
$143$	933.665			0.545993
$144$		0			0
$145$			1223.14i			0.700528i
$146$	83.7908			0.0474971
$147$		0			0
$148$	871.463			0.484013
$149$			2119.54i			1.16537i	0.812699	+	0.582683i	$0.197997\pi$
$149$			2119.54i			1.16537i	−0.812699	+	0.582683i	$0.802003\pi$
$150$		0			0
$151$	−2283.70			−1.23076			−0.615380	−	0.788230i	$-0.710998\pi$
$151$	−2283.70			−1.23076			−0.615380	+	0.788230i	$0.710998\pi$
$152$	−1919.12			−1.02408
$153$		0			0
$154$	373.524	+	1397.90i	0.195451	+	0.731468i
$155$		−	802.423i		−	0.415821i
$156$		0			0
$157$			3671.03i			1.86611i	0.359730	+	0.933057i	$0.382869\pi$
$157$			3671.03i			1.86611i	−0.359730	+	0.933057i	$0.617131\pi$
$158$			1915.00i			0.964234i
$159$		0			0
$160$		−	701.969i		−	0.346847i
$161$	−1035.68	+	276.737i	−0.506974	+	0.135465i
$162$		0			0
$163$	169.826			0.0816063			0.0408032	−	0.999167i	$-0.487008\pi$
$163$	169.826			0.0816063			0.0408032	+	0.999167i	$0.487008\pi$
$164$	−323.842			−0.154194
$165$		0			0
$166$		−	2756.74i		−	1.28894i
$167$	436.626			0.202318			0.101159	−	0.994870i	$-0.467745\pi$
$167$	436.626			0.202318			0.101159	+	0.994870i	$0.467745\pi$
$168$		0			0
$169$	1482.24			0.674664
$170$			443.057i			0.199888i
$171$		0			0
$172$	−1042.69			−0.462233
$173$	1655.46			0.727529			0.363765	−	0.931491i	$-0.381491\pi$
$173$	1655.46			0.727529			0.363765	+	0.931491i	$0.381491\pi$
$174$		0			0
$175$	−452.196	−	1692.33i	−0.195330	−	0.731018i
$176$			1084.97i			0.464673i
$177$		0			0
$178$		−	3368.34i		−	1.41836i
$179$			4545.28i			1.89793i	0.315376	+	0.948967i	$0.397869\pi$
$179$			4545.28i			1.89793i	−0.315376	+	0.948967i	$0.602131\pi$
$180$		0			0
$181$			69.4780i			0.0285318i	0.999898	+	0.0142659i	$0.00454113\pi$
$181$			69.4780i			0.0285318i	−0.999898	+	0.0142659i	$0.995459\pi$
$182$	−285.949	−	1070.16i	−0.116461	−	0.435853i
$183$		0			0
$184$	−1423.80			−0.570457
$185$	−1604.67			−0.637716
$186$		0			0
$187$			1254.06i			0.490405i
$188$	752.057			0.291752
$189$		0			0
$190$	962.625			0.367559
$191$			1404.79i			0.532184i	0.963948	+	0.266092i	$0.0857325\pi$
$191$			1404.79i			0.532184i	−0.963948	+	0.266092i	$0.914267\pi$
$192$		0			0
$193$	2312.08			0.862316			0.431158	−	0.902277i	$-0.358105\pi$
$193$	2312.08			0.862316			0.431158	+	0.902277i	$0.358105\pi$
$194$	2589.30			0.958251
$195$		0			0
$196$	−890.421	+	512.433i	−0.324498	+	0.186747i
$197$		−	1556.72i		−	0.563005i	−0.959561	−	0.281502i	$-0.909167\pi$
$197$		−	1556.72i		−	0.563005i	0.959561	−	0.281502i	$-0.0908327\pi$
$198$		0			0
$199$		−	1187.87i		−	0.423146i	−0.977362	−	0.211573i	$-0.932141\pi$
$199$		−	1187.87i		−	0.423146i	0.977362	−	0.211573i	$-0.0678585\pi$
$200$		−	2326.54i		−	0.822555i
$201$		0			0
$202$			348.026i			0.121223i
$203$	3968.18	−	1060.31i	1.37198	−	0.366597i
$204$		0			0
$205$	596.305			0.203160
$206$	4238.39			1.43351
$207$		0			0
$208$		−	830.592i		−	0.276881i
$209$	2724.68			0.901769
$210$		0			0
$211$	245.992			0.0802596			0.0401298	−	0.999194i	$-0.487223\pi$
$211$	245.992			0.0802596			0.0401298	+	0.999194i	$0.487223\pi$
$212$		−	479.746i		−	0.155420i
$213$		0			0
$214$	−1135.28			−0.362645
$215$	1919.95			0.609020
$216$		0			0
$217$	−2603.26	+	695.599i	−0.814381	+	0.217605i
$218$			667.623i			0.207418i
$219$		0			0
$220$			576.886i			0.176789i
$221$		−	960.038i		−	0.292213i
$222$		0			0
$223$			1098.57i			0.329892i	0.986303	+	0.164946i	$0.0527450\pi$
$223$			1098.57i			0.329892i	−0.986303	+	0.164946i	$0.947255\pi$
$224$	−2277.36	+	608.518i	−0.679297	+	0.181510i
$225$		0			0
$226$	558.427			0.164363
$227$	4303.91			1.25842			0.629208	−	0.777237i	$-0.283379\pi$
$227$	4303.91			1.25842			0.629208	+	0.777237i	$0.283379\pi$
$228$		0			0
$229$			2859.63i			0.825194i	0.910914	+	0.412597i	$0.135378\pi$
$229$			2859.63i			0.825194i	−0.910914	+	0.412597i	$0.864622\pi$
$230$	714.176			0.204745
$231$		0			0
$232$	5455.27			1.54378
$233$		−	2256.65i		−	0.634497i	−0.948342	−	0.317249i	$-0.897241\pi$
$233$		−	2256.65i		−	0.634497i	0.948342	−	0.317249i	$-0.102759\pi$
$234$		0			0
$235$	−1384.80			−0.384401
$236$	−2201.00			−0.607089
$237$		0			0
$238$	1437.39	−	384.074i	0.391479	−	0.104604i
$239$		−	1578.58i		−	0.427239i	−0.976917	−	0.213620i	$-0.931475\pi$
$239$		−	1578.58i		−	0.427239i	0.976917	−	0.213620i	$-0.0685253\pi$
$240$		0			0
$241$			1441.42i			0.385269i	0.981271	+	0.192634i	$0.0617031\pi$
$241$			1441.42i			0.385269i	−0.981271	+	0.192634i	$0.938297\pi$
$242$			249.200i			0.0661951i
$243$		0			0
$244$			2010.62i			0.527527i
$245$	1639.57	−	943.567i	0.427545	−	0.246050i
$246$		0			0
$247$	−2085.86			−0.537329
$248$	−3578.84			−0.916357
$249$		0			0
$250$			2709.26i			0.685395i
$251$	2935.90			0.738296			0.369148	−	0.929371i	$-0.379650\pi$
$251$	2935.90			0.738296			0.369148	+	0.929371i	$0.379650\pi$
$252$		0			0
$253$	2021.45			0.502322
$254$			4345.08i			1.07337i
$255$		0			0
$256$	−3875.23			−0.946101
$257$	7276.08			1.76603			0.883015	−	0.469346i	$-0.155510\pi$
$257$	7276.08			1.76603			0.883015	+	0.469346i	$0.155510\pi$
$258$		0			0
$259$	1391.04	+	5205.93i	0.333727	+	1.24896i
$260$		−	441.632i		−	0.105342i
$261$		0			0
$262$		−	59.3479i		−	0.0139944i
$263$			6732.69i			1.57854i	0.614047	+	0.789269i	$0.289540\pi$
$263$			6732.69i			1.57854i	−0.614047	+	0.789269i	$0.710460\pi$
$264$		0			0
$265$			883.379i			0.204776i
$266$	−834.473	−	3122.99i	−0.192349	−	0.719861i
$267$		0			0
$268$	−2062.86			−0.470183
$269$	7501.34			1.70024			0.850121	−	0.526588i	$-0.176529\pi$
$269$	7501.34			1.70024			0.850121	+	0.526588i	$0.176529\pi$
$270$		0			0
$271$			4621.90i			1.03602i	0.855376	+	0.518008i	$0.173326\pi$
$271$			4621.90i			1.03602i	−0.855376	+	0.518008i	$0.826674\pi$
$272$	1115.61			0.248691
$273$		0			0
$274$	596.589			0.131537
$275$			3303.11i			0.724310i
$276$		0			0
$277$	−1964.73			−0.426171			−0.213085	−	0.977034i	$-0.568351\pi$
$277$	−1964.73			−0.426171			−0.213085	+	0.977034i	$0.568351\pi$
$278$	−1149.04			−0.247895
$279$		0			0
$280$	2427.32	−	648.586i	0.518071	−	0.138430i
$281$			1965.57i			0.417281i	0.977992	+	0.208641i	$0.0669039\pi$
$281$			1965.57i			0.417281i	−0.977992	+	0.208641i	$0.933096\pi$
$282$		0			0
$283$			5336.18i			1.12086i	0.828202	+	0.560429i	$0.189364\pi$
$283$			5336.18i			1.12086i	−0.828202	+	0.560429i	$0.810636\pi$
$284$			1729.40i			0.361342i
$285$		0			0
$286$			2088.75i			0.431854i
$287$	−516.921	−	1934.56i	−0.106317	−	0.397887i
$288$		0			0
$289$	−3623.52			−0.737537
$290$	−2736.35			−0.554083
$291$		0			0
$292$		−	112.182i		−	0.0224828i
$293$	−2064.99			−0.411734			−0.205867	−	0.978580i	$-0.566001\pi$
$293$	−2064.99			−0.411734			−0.205867	+	0.978580i	$0.566001\pi$
$294$		0			0
$295$	4052.81			0.799877
$296$			7156.88i			1.40535i
$297$		0			0
$298$	−4741.73			−0.921748
$299$	−1547.51			−0.299314
$300$		0			0
$301$	−1664.35	−	6228.78i	−0.318709	−	1.19276i
$302$		−	5108.97i		−	0.973471i
$303$		0			0
$304$		−	2423.88i		−	0.457300i
$305$		−	3702.24i		−	0.695049i
$306$		0			0
$307$			395.602i			0.0735446i	0.999324	+	0.0367723i	$0.0117076\pi$
$307$			395.602i			0.0735446i	−0.999324	+	0.0367723i	$0.988292\pi$
$308$	1871.56	−	500.087i	0.346241	−	0.0925166i
$309$		0			0
$310$	1795.14			0.328894
$311$	6310.26			1.15055			0.575277	−	0.817959i	$-0.304894\pi$
$311$	6310.26			1.15055			0.575277	+	0.817959i	$0.304894\pi$
$312$		0			0
$313$		−	10147.5i		−	1.83249i	−0.400620	−	0.916244i	$-0.631205\pi$
$313$		−	10147.5i		−	1.83249i	0.400620	−	0.916244i	$-0.368795\pi$
$314$	−8212.62			−1.47600
$315$		0			0
$316$	2563.87			0.456421
$317$			1774.85i			0.314466i	0.987562	+	0.157233i	$0.0502573\pi$
$317$			1774.85i			0.314466i	−0.987562	+	0.157233i	$0.949743\pi$
$318$		0			0
$319$	−7745.15			−1.35939
$320$	2941.15			0.513797
$321$		0			0
$322$	−619.100	−	2316.96i	−0.107146	−	0.400992i
$323$		−	2801.64i		−	0.482623i
$324$		0			0
$325$		−	2528.68i		−	0.431588i
$326$			379.927i			0.0645466i
$327$		0			0
$328$		−	2659.55i		−	0.447710i
$329$	1200.44	+	4492.63i	0.201163	+	0.752847i
$330$		0			0
$331$	4795.45			0.796319			0.398160	−	0.917316i	$-0.369649\pi$
$331$	4795.45			0.796319			0.398160	+	0.917316i	$0.369649\pi$
$332$	−3690.83			−0.610122
$333$		0			0
$334$			976.796i			0.160024i
$335$	3798.44			0.619495
$336$		0			0
$337$	7389.81			1.19451			0.597253	−	0.802053i	$-0.296259\pi$
$337$	7389.81			1.19451			0.597253	+	0.802053i	$0.296259\pi$
$338$			3315.98i			0.533626i
$339$		0			0
$340$	593.181			0.0946170
$341$	5081.08			0.806908
$342$		0			0
$343$	−4482.47	−	4501.23i	−0.705628	−	0.708582i
$344$		−	8563.04i		−	1.34212i
$345$		0			0
$346$			3703.51i			0.575440i
$347$		−	1807.43i		−	0.279619i	−0.990178	−	0.139810i	$-0.955351\pi$
$347$		−	1807.43i		−	0.279619i	0.990178	−	0.139810i	$-0.0446491\pi$
$348$		0			0
$349$			5882.51i			0.902245i	0.892462	+	0.451122i	$0.148976\pi$
$349$			5882.51i			0.902245i	−0.892462	+	0.451122i	$0.851024\pi$
$350$	3785.99	−	1011.63i	0.578199	−	0.154497i
$351$		0			0
$352$	4444.98			0.673064
$353$	−2393.09			−0.360825			−0.180413	−	0.983591i	$-0.557743\pi$
$353$	−2393.09			−0.360825			−0.180413	+	0.983591i	$0.557743\pi$
$354$		0			0
$355$		−	3184.43i		−	0.476090i
$356$	−4509.65			−0.671380
$357$		0			0
$358$	−10168.5			−1.50117
$359$			10728.4i			1.57723i	0.614887	+	0.788615i	$0.289202\pi$
$359$			10728.4i			1.57723i	−0.614887	+	0.788615i	$0.710798\pi$
$360$		0			0
$361$	771.915			0.112540
$362$	−155.432			−0.0225672
$363$		0			0
$364$	−1432.76	+	382.839i	−0.206311	+	0.0551270i
$365$			206.566i			0.0296224i
$366$		0			0
$367$		−	5476.89i		−	0.778995i	−0.921027	−	0.389498i	$-0.872649\pi$
$367$		−	5476.89i		−	0.778995i	0.921027	−	0.389498i	$-0.127351\pi$
$368$		−	1798.29i		−	0.254735i
$369$		0			0
$370$		−	3589.87i		−	0.504402i
$371$	2865.90	−	765.777i	0.401051	−	0.107162i
$372$		0			0
$373$	12104.8			1.68032			0.840162	−	0.542336i	$-0.182460\pi$
$373$	12104.8			1.68032			0.840162	+	0.542336i	$0.182460\pi$
$374$	−2805.51			−0.387886
$375$		0			0
$376$			6176.26i			0.847117i
$377$	5929.26			0.810007
$378$		0			0
$379$	2.33509			0.000316479			0.000158240	−	1.00000i	$-0.499950\pi$
$379$	2.33509			0.000316479			0.000158240		1.00000i	$0.499950\pi$
$380$		−	1288.80i		−	0.173984i
$381$		0			0
$382$	−3142.73			−0.420931
$383$	−8197.13			−1.09361			−0.546807	−	0.837259i	$-0.684157\pi$
$383$	−8197.13			−1.09361			−0.546807	+	0.837259i	$0.684157\pi$
$384$		0			0
$385$	−3446.19	+	920.834i	−0.456193	+	0.121896i
$386$			5172.45i			0.682049i
$387$		0			0
$388$		−	3466.65i		−	0.453588i
$389$		−	5181.87i		−	0.675402i	−0.941253	−	0.337701i	$-0.890351\pi$
$389$		−	5181.87i		−	0.675402i	0.941253	−	0.337701i	$-0.109649\pi$
$390$		0			0
$391$		−	2078.55i		−	0.268841i
$392$	−4208.35	−	7312.57i	−0.542229	−	0.942195i
$393$		0			0
$394$	3482.62			0.445309
$395$	−4720.97			−0.601362
$396$		0			0
$397$		−	6757.41i		−	0.854269i	−0.904188	−	0.427135i	$-0.859523\pi$
$397$		−	6757.41i		−	0.854269i	0.904188	−	0.427135i	$-0.140477\pi$
$398$	2657.44			0.334687
$399$		0			0
$400$	2938.46			0.367308
$401$			8857.10i			1.10300i	0.834175	+	0.551499i	$0.185944\pi$
$401$			8857.10i			1.10300i	−0.834175	+	0.551499i	$0.814056\pi$
$402$		0			0
$403$	−3889.79			−0.480805
$404$	465.950			0.0573809
$405$		0			0
$406$	2372.07	+	8877.40i	0.289960	+	1.08517i
$407$		−	10161.0i		−	1.23750i
$408$		0			0
$409$		−	11044.2i		−	1.33521i	−0.744514	−	0.667607i	$-0.767319\pi$
$409$		−	11044.2i		−	1.33521i	0.744514	−	0.667607i	$-0.232681\pi$
$410$			1334.02i			0.160689i
$411$		0			0
$412$		−	5674.52i		−	0.678552i
$413$	−3513.27	−	13148.3i	−0.418588	−	1.56655i
$414$		0			0
$415$	6796.10			0.803873
$416$	−3402.84			−0.401052
$417$		0			0
$418$			6095.50i			0.713255i
$419$	9654.55			1.12567			0.562835	−	0.826569i	$-0.309711\pi$
$419$	9654.55			1.12567			0.562835	+	0.826569i	$0.309711\pi$
$420$		0			0
$421$	−2622.35			−0.303576			−0.151788	−	0.988413i	$-0.548503\pi$
$421$	−2622.35			−0.303576			−0.151788	+	0.988413i	$0.548503\pi$
$422$			550.320i			0.0634814i
$423$		0			0
$424$	3939.91			0.451271
$425$	3396.41			0.387648
$426$		0			0
$427$	−12011.0	+	3209.38i	−1.36125	+	0.363730i
$428$			1519.95i			0.171658i
$429$		0			0
$430$			4295.20i			0.481705i
$431$			2142.38i			0.239431i	0.992808	+	0.119716i	$0.0381983\pi$
$431$			2142.38i			0.239431i	−0.992808	+	0.119716i	$0.961802\pi$
$432$		0			0
$433$			8576.98i			0.951925i	0.879466	+	0.475962i	$0.157900\pi$
$433$			8576.98i			0.951925i	−0.879466	+	0.475962i	$0.842100\pi$
$434$	−1556.16	−	5823.87i	−0.172115	−	0.644136i
$435$		0			0
$436$	893.838			0.0981814
$437$	−4516.04			−0.494351
$438$		0			0
$439$		−	7399.33i		−	0.804444i	−0.915542	−	0.402222i	$-0.868238\pi$
$439$		−	7399.33i		−	0.804444i	0.915542	−	0.402222i	$-0.131762\pi$
$440$	−4737.67			−0.513317
$441$		0			0
$442$	2147.75			0.231126
$443$			14178.8i			1.52067i	0.649533	+	0.760334i	$0.274965\pi$
$443$			14178.8i			1.52067i	−0.649533	+	0.760334i	$0.725035\pi$
$444$		0			0
$445$	8303.84			0.884584
$446$	−2457.67			−0.260928
$447$		0			0
$448$	−2549.60	−	9541.80i	−0.268878	−	1.00627i
$449$		−	6800.58i		−	0.714787i	−0.933954	−	0.357393i	$-0.883666\pi$
$449$		−	6800.58i		−	0.714787i	0.933954	−	0.357393i	$-0.116334\pi$
$450$		0			0
$451$			3775.90i			0.394236i
$452$		−	747.642i		−	0.0778012i
$453$		0			0
$454$			9628.47i			0.995345i
$455$	2638.22	−	704.940i	0.271828	−	0.0726332i
$456$		0			0
$457$	−3771.82			−0.386079			−0.193040	−	0.981191i	$-0.561835\pi$
$457$	−3771.82			−0.386079			−0.193040	+	0.981191i	$0.561835\pi$
$458$	−6397.41			−0.652688
$459$		0			0
$460$		−	956.166i		−	0.0969162i
$461$	−15614.2			−1.57750			−0.788750	−	0.614714i	$-0.789272\pi$
$461$	−15614.2			−1.57750			−0.788750	+	0.614714i	$0.789272\pi$
$462$		0			0
$463$	8111.33			0.814180			0.407090	−	0.913388i	$-0.366544\pi$
$463$	8111.33			0.814180			0.407090	+	0.913388i	$0.366544\pi$
$464$			6890.11i			0.689365i
$465$		0			0
$466$	5048.45			0.501856
$467$	−6607.80			−0.654759			−0.327380	−	0.944893i	$-0.606166\pi$
$467$	−6607.80			−0.654759			−0.327380	+	0.944893i	$0.606166\pi$
$468$		0			0
$469$	−3292.76	−	12323.1i	−0.324191	−	1.21328i
$470$		−	3098.00i		−	0.304042i
$471$		0			0
$472$		−	18075.7i		−	1.76271i
$473$			12157.4i			1.18182i
$474$		0			0
$475$		−	7379.35i		−	0.712816i
$476$	−514.213	−	1924.43i	−0.0495145	−	0.185307i
$477$		0			0
$478$	3531.53			0.337925
$479$	8276.26			0.789462			0.394731	−	0.918797i	$-0.370838\pi$
$479$	8276.26			0.789462			0.394731	+	0.918797i	$0.370838\pi$
$480$		0			0
$481$			7778.72i			0.737378i
$482$	−3224.66			−0.304728
$483$		0			0
$484$	333.639			0.0313335
$485$			6383.30i			0.597630i
$486$		0			0
$487$	229.243			0.0213306			0.0106653	−	0.999943i	$-0.496605\pi$
$487$	229.243			0.0213306			0.0106653	+	0.999943i	$0.496605\pi$
$488$	−16512.2			−1.53170
$489$		0			0
$490$	2110.90	+	3667.97i	0.194614	+	0.338167i
$491$		−	20696.4i		−	1.90228i	−0.308766	−	0.951138i	$-0.599916\pi$
$491$		−	20696.4i		−	1.90228i	0.308766	−	0.951138i	$-0.400084\pi$
$492$		0			0
$493$			7963.92i			0.727540i
$494$		−	4666.38i		−	0.425001i
$495$		0			0
$496$		−	4520.14i		−	0.409195i
$497$	−10331.1	+	2760.49i	−0.932418	+	0.249145i
$498$		0			0
$499$	−13104.7			−1.17565			−0.587824	−	0.808989i	$-0.700015\pi$
$499$	−13104.7			−1.17565			−0.587824	+	0.808989i	$0.700015\pi$
$500$	3627.26			0.324432
$501$		0			0
$502$			6568.03i			0.583956i
$503$	7415.97			0.657379			0.328690	−	0.944438i	$-0.393393\pi$
$503$	7415.97			0.657379			0.328690	+	0.944438i	$0.393393\pi$
$504$		0			0
$505$	−857.975			−0.0756028
$506$			4522.28i			0.397312i
$507$		0			0
$508$	5817.36			0.508078
$509$	−21073.8			−1.83513			−0.917565	−	0.397587i	$-0.869848\pi$
$509$	−21073.8			−1.83513			−0.917565	+	0.397587i	$0.869848\pi$
$510$		0			0
$511$	670.153	−	179.067i	0.0580153	−	0.0155019i
$512$		−	10067.8i		−	0.869021i
$513$		0			0
$514$			16277.7i			1.39684i
$515$			10448.8i			0.894033i
$516$		0			0
$517$		−	8768.77i		−	0.745939i
$518$	−11646.4	+	3111.97i	−0.987867	+	0.263961i
$519$		0			0
$520$	3626.90			0.305865
$521$	−19991.7			−1.68110			−0.840548	−	0.541736i	$-0.817767\pi$
$521$	−19991.7			−1.68110			−0.840548	+	0.541736i	$0.817767\pi$
$522$		0			0
$523$		−	3604.15i		−	0.301336i	−0.988584	−	0.150668i	$-0.951858\pi$
$523$		−	3604.15i		−	0.301336i	0.988584	−	0.150668i	$-0.0481424\pi$
$524$	−79.4571			−0.00662424
$525$		0			0
$526$	−15062.0			−1.24855
$527$		−	5224.60i		−	0.431854i
$528$		0			0
$529$	8816.53			0.724626
$530$	−1976.25			−0.161967
$531$		0			0
$532$	−4181.18	+	1117.22i	−0.340746	+	0.0910485i
$533$		−	2890.63i		−	0.234910i
$534$		0			0
$535$		−	2798.76i		−	0.226170i
$536$		−	16941.2i		−	1.36520i
$537$		0			0
$538$			16781.6i			1.34481i
$539$	5974.83	+	10382.1i	0.477466	+	0.829660i
$540$		0			0
$541$	−13211.0			−1.04988			−0.524939	−	0.851140i	$-0.675912\pi$
$541$	−13211.0			−1.04988			−0.524939	+	0.851140i	$0.675912\pi$
$542$	−10339.9			−0.819438
$543$		0			0
$544$		−	4570.54i		−	0.360221i
$545$	−1645.87			−0.129360
$546$		0			0
$547$	17717.1			1.38488			0.692439	−	0.721477i	$-0.256536\pi$
$547$	17717.1			1.38488			0.692439	+	0.721477i	$0.256536\pi$
$548$		−	798.735i		−	0.0622633i
$549$		0			0
$550$	−7389.55			−0.572893
$551$	17303.1			1.33782
$552$		0			0
$553$	4092.48	+	15316.0i	0.314702	+	1.17776i
$554$		−	4395.39i		−	0.337080i
$555$		0			0
$556$			1538.38i			0.117341i
$557$			13013.3i			0.989933i	0.868912	+	0.494966i	$0.164820\pi$
$557$			13013.3i			0.989933i	−0.868912	+	0.494966i	$0.835180\pi$
$558$		0			0
$559$		−	9307.05i		−	0.704198i
$560$	819.177	+	3065.75i	0.0618153	+	0.231342i
$561$		0			0
$562$	−4397.27			−0.330049
$563$	15733.9			1.17781			0.588904	−	0.808203i	$-0.299560\pi$
$563$	15733.9			1.17781			0.588904	+	0.808203i	$0.299560\pi$
$564$		0			0
$565$			1376.67i			0.102508i
$566$	−11937.8			−0.886544
$567$		0			0
$568$	−14202.7			−1.04917
$569$			6230.19i			0.459022i	0.973306	+	0.229511i	$0.0737126\pi$
$569$			6230.19i			0.459022i	−0.973306	+	0.229511i	$0.926287\pi$
$570$		0			0
$571$	−2364.39			−0.173286			−0.0866432	−	0.996239i	$-0.527614\pi$
$571$	−2364.39			−0.173286			−0.0866432	+	0.996239i	$0.527614\pi$
$572$	2796.49			0.204418
$573$		0			0
$574$	4327.90	−	1156.43i	0.314709	−	0.0840913i
$575$		−	5474.78i		−	0.397068i
$576$		0			0
$577$		−	4194.43i		−	0.302628i	−0.988486	−	0.151314i	$-0.951650\pi$
$577$		−	4194.43i		−	0.302628i	0.988486	−	0.151314i	$-0.0483505\pi$
$578$		−	8106.34i		−	0.583355i
$579$		0			0
$580$			3663.53i			0.262276i
$581$	−5891.35	−	22048.2i	−0.420679	−	1.57438i
$582$		0			0
$583$	−5593.70			−0.397371
$584$	921.295			0.0652799
$585$		0			0
$586$		−	4619.69i		−	0.325662i
$587$	−13406.7			−0.942683			−0.471341	−	0.881951i	$-0.656230\pi$
$587$	−13406.7			−0.942683			−0.471341	+	0.881951i	$0.656230\pi$
$588$		0			0
$589$	−11351.4			−0.794104
$590$			9066.73i			0.632663i
$591$		0			0
$592$	−9039.28			−0.627554
$593$	4672.62			0.323577			0.161789	−	0.986825i	$-0.448274\pi$
$593$	4672.62			0.323577			0.161789	+	0.986825i	$0.448274\pi$
$594$		0			0
$595$	946.845	+	3543.54i	0.0652384	+	0.244153i
$596$			6348.40i			0.436310i
$597$		0			0
$598$		−	3462.01i		−	0.236743i
$599$		−	6479.85i		−	0.442002i	−0.975274	−	0.221001i	$-0.929068\pi$
$599$		−	6479.85i		−	0.442002i	0.975274	−	0.221001i	$-0.0709324\pi$
$600$		0			0
$601$		−	2590.70i		−	0.175835i	−0.996128	−	0.0879174i	$-0.971979\pi$
$601$		−	2590.70i		−	0.175835i	0.996128	−	0.0879174i	$-0.0280212\pi$
$602$	13934.7	−	3723.39i	0.943415	−	0.252083i
$603$		0			0
$604$	−6840.08			−0.460793
$605$	−614.345			−0.0412837
$606$		0			0
$607$			2158.83i			0.144356i	0.997392	+	0.0721781i	$0.0229950\pi$
$607$			2158.83i			0.144356i	−0.997392	+	0.0721781i	$0.977005\pi$
$608$	−9930.35			−0.662383
$609$		0			0
$610$	8282.46			0.549749
$611$			6712.89i			0.444476i
$612$		0			0
$613$	6333.20			0.417285			0.208642	−	0.977992i	$-0.433096\pi$
$613$	6333.20			0.417285			0.208642	+	0.977992i	$0.433096\pi$
$614$	−885.019			−0.0581702
$615$		0			0
$616$	4106.96	+	15370.2i	0.268627	+	1.00533i
$617$		−	2081.75i		−	0.135831i	−0.997691	−	0.0679157i	$-0.978365\pi$
$617$		−	2081.75i		−	0.135831i	0.997691	−	0.0679157i	$-0.0216349\pi$
$618$		0			0
$619$			23537.5i			1.52836i	0.645004	+	0.764179i	$0.276856\pi$
$619$			23537.5i			1.52836i	−0.645004	+	0.764179i	$0.723144\pi$
$620$		−	2403.40i		−	0.155682i
$621$		0			0
$622$			14117.0i			0.910031i
$623$	−7198.37	−	26939.7i	−0.462916	−	1.73245i
$624$		0			0
$625$	5143.83			0.329205
$626$	22701.4			1.44941
$627$		0			0
$628$			10995.4i			0.698667i
$629$	−10448.0			−0.662306
$630$		0			0
$631$	13465.2			0.849510			0.424755	−	0.905308i	$-0.360360\pi$
$631$	13465.2			0.849510			0.424755	+	0.905308i	$0.360360\pi$
$632$			21055.7i			1.32524i
$633$		0			0
$634$	−3970.60			−0.248727
$635$	−10711.8			−0.669423
$636$		0			0
$637$	−4574.00	−	7947.93i	−0.284503	−	0.494362i
$638$		−	17327.0i		−	1.07521i
$639$		0			0
$640$			964.023i			0.0595412i
$641$		−	3115.35i		−	0.191964i	−0.995383	−	0.0959819i	$-0.969401\pi$
$641$		−	3115.35i		−	0.191964i	0.995383	−	0.0959819i	$-0.0305991\pi$
$642$		0			0
$643$		−	159.935i		−	0.00980905i	−0.999988	−	0.00490452i	$-0.998439\pi$
$643$		−	159.935i		−	0.00980905i	0.999988	−	0.00490452i	$-0.00156116\pi$
$644$	−3102.04	+	828.874i	−0.189810	+	0.0507177i
$645$		0			0
$646$	6267.68			0.381731
$647$	−11705.0			−0.711239			−0.355619	−	0.934631i	$-0.615730\pi$
$647$	−11705.0			−0.711239			−0.355619	+	0.934631i	$0.615730\pi$
$648$		0			0
$649$			25663.1i			1.55218i
$650$	5657.03			0.341365
$651$		0			0
$652$	508.660			0.0305532
$653$			26881.5i			1.61096i	0.592624	+	0.805479i	$0.298092\pi$
$653$			26881.5i			1.61096i	−0.592624	+	0.805479i	$0.701908\pi$
$654$		0			0
$655$	146.308			0.00872784
$656$	3359.06			0.199923
$657$		0			0
$658$	−10050.7	+	2685.57i	−0.595465	+	0.159110i
$659$		−	8300.96i		−	0.490682i	−0.969437	−	0.245341i	$-0.921100\pi$
$659$		−	8300.96i		−	0.490682i	0.969437	−	0.245341i	$-0.0789000\pi$
$660$		0			0
$661$		−	1441.47i		−	0.0848209i	−0.999100	−	0.0424105i	$-0.986496\pi$
$661$		−	1441.47i		−	0.0848209i	0.999100	−	0.0424105i	$-0.0135037\pi$
$662$			10728.1i			0.629849i
$663$		0			0
$664$		−	30310.9i		−	1.77152i
$665$	7699.00	−	2057.20i	0.448954	−	0.119962i
$666$		0			0
$667$	12837.3			0.745219
$668$	1307.77			0.0757473
$669$		0			0
$670$			8497.66i			0.489990i
$671$	23443.2			1.34876
$672$		0			0
$673$	−16024.3			−0.917818			−0.458909	−	0.888483i	$-0.651760\pi$
$673$	−16024.3			−0.917818			−0.458909	+	0.888483i	$0.651760\pi$
$674$			16532.1i			0.944796i
$675$		0			0
$676$	4439.56			0.252592
$677$	27510.4			1.56176			0.780880	−	0.624681i	$-0.214771\pi$
$677$	27510.4			1.56176			0.780880	+	0.624681i	$0.214771\pi$
$678$		0			0
$679$	20709.0	−	5533.51i	1.17045	−	0.312749i
$680$			4871.49i			0.274725i
$681$		0			0
$682$			11367.1i			0.638225i
$683$		−	2208.68i		−	0.123738i	−0.998084	−	0.0618688i	$-0.980294\pi$
$683$		−	2208.68i		−	0.123738i	0.998084	−	0.0618688i	$-0.0197060\pi$
$684$		0			0
$685$			1470.75i			0.0820357i
$686$	10069.9	−	10027.9i	0.560453	−	0.558117i
$687$		0			0
$688$	10815.3			0.599315
$689$	4282.23			0.236778
$690$		0			0
$691$		−	23568.0i		−	1.29749i	−0.761005	−	0.648746i	$-0.775294\pi$
$691$		−	23568.0i		−	1.29749i	0.761005	−	0.648746i	$-0.224706\pi$
$692$	4958.40			0.272385
$693$		0			0
$694$	4043.48			0.221165
$695$		−	2832.69i		−	0.154604i
$696$		0			0
$697$	3882.56			0.210994
$698$	−13160.0			−0.713631
$699$		0			0
$700$	−1354.41	−	5068.82i	−0.0731311	−	0.273691i
$701$		−	18383.9i		−	0.990516i	−0.868746	−	0.495258i	$-0.835074\pi$
$701$		−	18383.9i		−	0.990516i	0.868746	−	0.495258i	$-0.164926\pi$
$702$		0			0
$703$			22700.3i			1.21786i
$704$			18623.8i			0.997034i
$705$		0			0
$706$		−	5353.69i		−	0.285395i
$707$	743.756	+	2783.48i	0.0395641	+	0.148067i
$708$		0			0
$709$	−23216.7			−1.22979			−0.614895	−	0.788609i	$-0.710802\pi$
$709$	−23216.7			−1.22979			−0.614895	+	0.788609i	$0.710802\pi$
$710$	7124.03			0.376564
$711$		0			0
$712$		−	37035.5i		−	1.94939i
$713$	−8421.68			−0.442348
$714$		0			0
$715$	−5149.31			−0.269333
$716$			13613.9i			0.710580i
$717$		0			0
$718$	−24001.1			−1.24751
$719$	24589.3			1.27542			0.637709	−	0.770278i	$-0.279882\pi$
$719$	24589.3			1.27542			0.637709	+	0.770278i	$0.279882\pi$
$720$		0			0
$721$	33898.3	−	9057.74i	1.75096	−	0.467861i
$722$			1726.89i			0.0890139i
$723$		0			0
$724$			208.099i			0.0106822i
$725$			20976.5i			1.07455i
$726$		0			0
$727$			16419.1i			0.837620i	0.908074	+	0.418810i	$0.137553\pi$
$727$			16419.1i			0.837620i	−0.908074	+	0.418810i	$0.862447\pi$
$728$	−3144.06	−	11766.6i	−0.160064	−	0.599035i
$729$		0			0
$730$	−462.119			−0.0234299
$731$	12500.8			0.632503
$732$		0			0
$733$		−	1910.37i		−	0.0962635i	−0.998841	−	0.0481318i	$-0.984673\pi$
$733$		−	1910.37i		−	0.0962635i	0.998841	−	0.0481318i	$-0.0153267\pi$
$734$	12252.6			0.616147
$735$		0			0
$736$	−7367.38			−0.368974
$737$			24052.3i			1.20214i
$738$		0			0
$739$	10538.9			0.524601			0.262300	−	0.964986i	$-0.415519\pi$
$739$	10538.9			0.524601			0.262300	+	0.964986i	$0.415519\pi$
$740$	−4806.26			−0.238759
$741$		0			0
$742$	1713.16	+	6411.44i	0.0847600	+	0.317212i
$743$			10720.1i			0.529319i	0.964342	+	0.264660i	$0.0852596\pi$
$743$			10720.1i			0.529319i	−0.964342	+	0.264660i	$0.914740\pi$
$744$		0			0
$745$		−	11689.6i		−	0.574864i
$746$			27080.1i			1.32905i
$747$		0			0
$748$			3756.12i			0.183606i
$749$	−9079.86	+	2426.17i	−0.442952	+	0.118358i
$750$		0			0
$751$	−33949.0			−1.64956			−0.824779	−	0.565455i	$-0.808700\pi$
$751$	−33949.0			−1.64956			−0.824779	+	0.565455i	$0.808700\pi$
$752$	−7800.73			−0.378276
$753$		0			0
$754$			13264.6i			0.640676i
$755$	12595.0			0.607123
$756$		0			0
$757$	−4201.14			−0.201708			−0.100854	−	0.994901i	$-0.532158\pi$
$757$	−4201.14			−0.201708			−0.100854	+	0.994901i	$0.532158\pi$
$758$			5.22395i			0.000250320i
$759$		0			0
$760$	10584.2			0.505171
$761$	−3689.22			−0.175735			−0.0878674	−	0.996132i	$-0.528005\pi$
$761$	−3689.22			−0.175735			−0.0878674	+	0.996132i	$0.528005\pi$
$762$		0			0
$763$	1426.76	+	5339.59i	0.0676960	+	0.253350i
$764$			4207.60i			0.199248i
$765$		0			0
$766$		−	18338.2i		−	0.864994i
$767$		−	19646.2i		−	0.924882i
$768$		0			0
$769$		−	2918.13i		−	0.136841i	−0.997657	−	0.0684204i	$-0.978204\pi$
$769$		−	2918.13i		−	0.136841i	0.997657	−	0.0684204i	$-0.0217959\pi$
$770$	−2060.04	−	7709.64i	−0.0964139	−	0.360826i
$771$		0			0
$772$	6925.07			0.322848
$773$	−6902.85			−0.321188			−0.160594	−	0.987021i	$-0.551341\pi$
$773$	−6902.85			−0.321188			−0.160594	+	0.987021i	$0.551341\pi$
$774$		0			0
$775$		−	13761.3i		−	0.637832i
$776$	28469.8			1.31702
$777$		0			0
$778$	11592.6			0.534210
$779$		−	8435.59i		−	0.387980i
$780$		0			0
$781$	20164.3			0.923862
$782$	4650.02			0.212640
$783$		0			0
$784$	9235.92	−	5315.23i	0.420732	−	0.242130i
$785$		−	20246.3i		−	0.920536i
$786$		0			0
$787$		−	33704.8i		−	1.52662i	−0.646035	−	0.763308i	$-0.723574\pi$
$787$		−	33704.8i		−	1.52662i	0.646035	−	0.763308i	$-0.276426\pi$
$788$		−	4662.66i		−	0.210787i
$789$		0			0
$790$		−	10561.5i		−	0.475648i
$791$	4466.25	−	1193.40i	0.200761	−	0.0536439i
$792$		0			0
$793$	−17946.8			−0.803671
$794$	15117.3			0.675685
$795$		0			0
$796$		−	3557.88i		−	0.158424i
$797$	−5142.80			−0.228566			−0.114283	−	0.993448i	$-0.536457\pi$
$797$	−5142.80			−0.228566			−0.114283	+	0.993448i	$0.536457\pi$
$798$		0			0
$799$	−9016.46			−0.399223
$800$		−	12038.5i		−	0.532033i
$801$		0			0
$802$	−19814.6			−0.872418
$803$	−1308.01			−0.0574829
$804$		0			0
$805$	5711.93	−	1526.25i	0.250086	−	0.0668237i
$806$		−	8702.04i		−	0.380293i
$807$		0			0
$808$			3826.60i			0.166608i
$809$		−	4644.84i		−	0.201859i	−0.994894	−	0.100930i	$-0.967818\pi$
$809$		−	4644.84i		−	0.201859i	0.994894	−	0.100930i	$-0.0321817\pi$
$810$		0			0
$811$		−	8124.92i		−	0.351794i	−0.984409	−	0.175897i	$-0.943717\pi$
$811$		−	8124.92i		−	0.351794i	0.984409	−	0.175897i	$-0.0562825\pi$
$812$	11885.4	−	3175.82i	0.513665	−	0.137253i
$813$		0			0
$814$	22731.7			0.978802
$815$	−936.619			−0.0402556
$816$		0			0
$817$		−	27160.4i		−	1.16306i
$818$	24707.6			1.05609
$819$		0			0
$820$	1786.04			0.0760624
$821$			2602.19i			0.110618i	0.998469	+	0.0553088i	$0.0176143\pi$
$821$			2602.19i			0.110618i	−0.998469	+	0.0553088i	$0.982386\pi$
$822$		0			0
$823$	19000.6			0.804761			0.402381	−	0.915472i	$-0.368183\pi$
$823$	19000.6			0.804761			0.402381	+	0.915472i	$0.368183\pi$
$824$	46601.9			1.97021
$825$		0			0
$826$	29414.7	−	7859.70i	1.23907	−	0.331082i
$827$			20997.3i			0.882886i	0.897289	+	0.441443i	$0.145533\pi$
$827$			20997.3i			0.882886i	−0.897289	+	0.441443i	$0.854467\pi$
$828$		0			0
$829$			5441.05i			0.227956i	0.993483	+	0.113978i	$0.0363594\pi$
$829$			5441.05i			0.227956i	−0.993483	+	0.113978i	$0.963641\pi$
$830$			15203.9i			0.635824i
$831$		0			0
$832$		−	14257.4i		−	0.594093i
$833$	10675.3	−	6143.59i	0.444031	−	0.255538i
$834$		0			0
$835$	−2408.06			−0.0998016
$836$	8160.88			0.337620
$837$		0			0
$838$			21598.6i			0.890350i
$839$	30883.3			1.27081			0.635406	−	0.772179i	$-0.280833\pi$
$839$	30883.3			1.27081			0.635406	+	0.772179i	$0.280833\pi$
$840$		0			0
$841$	−24796.8			−1.01672
$842$		−	5866.59i		−	0.240114i
$843$		0			0
$844$	736.789			0.0300490
$845$	−8174.77			−0.332805
$846$		0			0
$847$	532.559	+	1993.09i	0.0216044	+	0.0808539i
$848$			4976.18i			0.201513i
$849$		0			0
$850$			7598.28i			0.306610i
$851$			16841.5i			0.678400i
$852$		0			0
$853$			8069.66i			0.323916i	0.986798	+	0.161958i	$0.0517808\pi$
$853$			8069.66i			0.323916i	−0.986798	+	0.161958i	$0.948219\pi$
$854$	−7179.85	−	26870.4i	−0.287692	−	1.07668i
$855$		0			0
$856$	−12482.6			−0.498418
$857$	−28497.2			−1.13588			−0.567938	−	0.823072i	$-0.692259\pi$
$857$	−28497.2			−1.13588			−0.567938	+	0.823072i	$0.692259\pi$
$858$		0			0
$859$			36415.2i			1.44642i	0.690631	+	0.723208i	$0.257333\pi$
$859$			36415.2i			1.44642i	−0.690631	+	0.723208i	$0.742667\pi$
$860$	5750.57			0.228015
$861$		0			0
$862$	−4792.82			−0.189378
$863$		−	784.791i		−	0.0309555i	−0.999880	−	0.0154778i	$-0.995073\pi$
$863$		−	784.791i		−	0.0309555i	0.999880	−	0.0154778i	$-0.00492692\pi$
$864$		0			0
$865$	−9130.14			−0.358883
$866$	−19188.0			−0.752926
$867$		0			0
$868$	−7797.22	+	2083.44i	−0.304902	+	0.0814707i
$869$		−	29894.0i		−	1.16695i
$870$		0			0
$871$		−	18413.2i		−	0.716310i
$872$			7340.63i			0.285075i
$873$		0			0
$874$		−	10103.0i		−	0.391008i
$875$	5789.89	+	21668.5i	0.223696	+	0.837175i
$876$		0			0
$877$	−15357.4			−0.591312			−0.295656	−	0.955294i	$-0.595538\pi$
$877$	−15357.4			−0.591312			−0.295656	+	0.955294i	$0.595538\pi$
$878$	16553.4			0.636276
$879$		0			0
$880$		−	5983.77i		−	0.229219i
$881$	−40911.2			−1.56451			−0.782254	−	0.622959i	$-0.785930\pi$
$881$	−40911.2			−1.56451			−0.782254	+	0.622959i	$0.785930\pi$
$882$		0			0
$883$	−33904.8			−1.29217			−0.646086	−	0.763265i	$-0.723595\pi$
$883$	−33904.8			−1.29217			−0.646086	+	0.763265i	$0.723595\pi$
$884$		−	2875.48i		−	0.109404i
$885$		0			0
$886$	−31720.1			−1.20277
$887$	17282.5			0.654216			0.327108	−	0.944987i	$-0.393926\pi$
$887$	17282.5			0.654216			0.327108	+	0.944987i	$0.393926\pi$
$888$		0			0
$889$	9285.75	+	34751.6i	0.350320	+	1.31106i
$890$			18576.9i			0.699662i
$891$		0			0
$892$			3290.42i			0.123511i
$893$			19590.0i			0.734102i
$894$		0			0
$895$		−	25067.9i		−	0.936233i
$896$	3127.53	−	835.686i	0.116611	−	0.0311588i
$897$		0			0
$898$	15213.9			0.565361
$899$	32267.5			1.19709
$900$		0			0
$901$			5751.71i			0.212672i
$902$	−8447.25			−0.311821
$903$		0			0
$904$	6140.00			0.225900
$905$		−	383.182i		−	0.0140745i
$906$		0			0
$907$	46397.0			1.69855			0.849276	−	0.527949i	$-0.177039\pi$
$907$	46397.0			1.69855			0.849276	+	0.527949i	$0.177039\pi$
$908$	12891.0			0.471147
$909$		0			0
$910$	1577.05	+	5902.08i	0.0574493	+	0.215002i
$911$		−	7236.49i		−	0.263179i	−0.991304	−	0.131589i	$-0.957992\pi$
$911$		−	7236.49i		−	0.263179i	0.991304	−	0.131589i	$-0.0420080\pi$
$912$		0			0
$913$			43034.0i			1.55993i
$914$		−	8438.12i		−	0.305370i
$915$		0			0
$916$			8565.09i			0.308950i
$917$	−126.831	−	474.660i	−0.00456741	−	0.0170934i
$918$		0			0
$919$	4016.19			0.144159			0.0720793	−	0.997399i	$-0.477037\pi$
$919$	4016.19			0.144159			0.0720793	+	0.997399i	$0.477037\pi$
$920$	7852.49			0.281401
$921$		0			0
$922$		−	34931.4i		−	1.24773i
$923$	−15436.7			−0.550493
$924$		0			0
$925$	−27519.5			−0.978200
$926$			18146.2i			0.643976i
$927$		0			0
$928$	28228.0			0.998522
$929$	−13191.2			−0.465865			−0.232932	−	0.972493i	$-0.574832\pi$
$929$	−13191.2			−0.465865			−0.232932	+	0.972493i	$0.574832\pi$
$930$		0			0
$931$	−13348.1	−	23194.1i	−0.469889	−	0.816495i
$932$		−	6759.05i		−	0.237554i
$933$		0			0
$934$		−	14782.6i		−	0.517882i
$935$		−	6916.33i		−	0.241912i
$936$		0			0
$937$		−	35573.0i		−	1.24025i	−0.784502	−	0.620127i	$-0.787081\pi$
$937$		−	35573.0i		−	1.24025i	0.784502	−	0.620127i	$-0.212919\pi$
$938$	27568.5	−	7366.40i	0.959642	−	0.256419i
$939$		0			0
$940$	−4147.71			−0.143919
$941$	−27185.4			−0.941786			−0.470893	−	0.882190i	$-0.656068\pi$
$941$	−27185.4			−0.941786			−0.470893	+	0.882190i	$0.656068\pi$
$942$		0			0
$943$		−	6258.41i		−	0.216121i
$944$	22829.9			0.787131
$945$		0			0
$946$	−27197.9			−0.934758
$947$			55034.1i			1.88846i	0.329291	+	0.944229i	$0.393190\pi$
$947$			55034.1i			1.88846i	−0.329291	+	0.944229i	$0.606810\pi$
$948$		0			0
$949$	1001.34			0.0342518
$950$	16508.7			0.563803
$951$		0			0
$952$	15804.3	−	4222.96i	0.538047	−	0.143768i
$953$		−	29456.0i		−	1.00123i	−0.865670	−	0.500615i	$-0.833107\pi$
$953$		−	29456.0i		−	1.00123i	0.865670	−	0.500615i	$-0.166893\pi$
$954$		0			0
$955$		−	7747.65i		−	0.262521i
$956$		−	4728.14i		−	0.159957i
$957$		0			0
$958$			18515.2i			0.624425i
$959$	4771.47	−	1274.95i	0.160666	−	0.0429305i
$960$		0			0
$961$	8622.44			0.289431
$962$	−17402.1			−0.583230
$963$		0			0
$964$			4317.29i			0.144243i
$965$	−12751.5			−0.425372
$966$		0			0
$967$	27773.4			0.923612			0.461806	−	0.886981i	$-0.347202\pi$
$967$	27773.4			0.923612			0.461806	+	0.886981i	$0.347202\pi$
$968$			2740.00i			0.0909783i
$969$		0			0
$970$	−14280.4			−0.472696
$971$	−13906.7			−0.459615			−0.229807	−	0.973236i	$-0.573810\pi$
$971$	−13906.7			−0.459615			−0.229807	+	0.973236i	$0.573810\pi$
$972$		0			0
$973$	−9189.95	+	2455.58i	−0.302792	+	0.0809069i
$974$			512.851i			0.0168715i
$975$		0			0
$976$		−	20855.2i		−	0.683973i
$977$		−	8361.53i		−	0.273807i	−0.990584	−	0.136903i	$-0.956285\pi$
$977$		−	8361.53i		−	0.273807i	0.990584	−	0.136903i	$-0.0437150\pi$
$978$		0			0
$979$			52581.3i			1.71655i
$980$	4910.81	−	2826.15i	0.160072	−	0.0921205i
$981$		0			0
$982$	46301.0			1.50461
$983$	−9693.02			−0.314506			−0.157253	−	0.987558i	$-0.550264\pi$
$983$	−9693.02			−0.314506			−0.157253	+	0.987558i	$0.550264\pi$
$984$		0			0
$985$			8585.57i			0.277725i
$986$	−17816.5			−0.575448
$987$		0			0
$988$	−6247.52			−0.201174
$989$		−	20150.4i		−	0.647873i
$990$		0			0
$991$	31419.3			1.00713			0.503566	−	0.863957i	$-0.332021\pi$
$991$	31419.3			1.00713			0.503566	+	0.863957i	$0.332021\pi$
$992$	−18518.5			−0.592704
$993$		0			0
$994$	−6175.63	−	23112.1i	−0.197062	−	0.737497i
$995$			6551.30i			0.208734i
$996$		0			0
$997$		−	13150.1i		−	0.417721i	−0.977945	−	0.208860i	$-0.933025\pi$
$997$		−	13150.1i		−	0.417721i	0.977945	−	0.208860i	$-0.0669754\pi$
$998$		−	29317.2i		−	0.929880i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	567.4.c.c.566.30		44
3.2	odd	2	inner	567.4.c.c.566.15		44
7.6	odd	2	inner	567.4.c.c.566.16		44
9.2	odd	6		189.4.o.a.125.7		44
9.4	even	3		189.4.o.a.62.8		44
9.5	odd	6		63.4.o.a.20.15	✓	44
9.7	even	3		63.4.o.a.41.16	yes	44
21.20	even	2	inner	567.4.c.c.566.29		44
63.13	odd	6		189.4.o.a.62.7		44
63.20	even	6		189.4.o.a.125.8		44
63.34	odd	6		63.4.o.a.41.15	yes	44
63.41	even	6		63.4.o.a.20.16	yes	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.4.o.a.20.15	✓	44	9.5	odd	6
63.4.o.a.20.16	yes	44	63.41	even	6
63.4.o.a.41.15	yes	44	63.34	odd	6
63.4.o.a.41.16	yes	44	9.7	even	3
189.4.o.a.62.7		44	63.13	odd	6
189.4.o.a.62.8		44	9.4	even	3
189.4.o.a.125.7		44	9.2	odd	6
189.4.o.a.125.8		44	63.20	even	6
567.4.c.c.566.15		44	3.2	odd	2	inner
567.4.c.c.566.16		44	7.6	odd	2	inner
567.4.c.c.566.29		44	21.20	even	2	inner
567.4.c.c.566.30		44	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 567 = 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	567.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.23715i q^{2} +2.99518 q^{4} -5.51516 q^{5} +(4.78094 + 17.8925i) q^{7} +24.5978i q^{8} +O(q^{10})\)	\(q+2.23715i q^{2} +2.99518 q^{4} -5.51516 q^{5} +(4.78094 + 17.8925i) q^{7} +24.5978i q^{8} -12.3382i q^{10} -34.9229i q^{11} +26.7350i q^{13} +(-40.0282 + 10.6957i) q^{14} -31.0675 q^{16} -35.9093 q^{17} +78.0198i q^{19} -16.5189 q^{20} +78.1276 q^{22} +57.8833i q^{23} -94.5830 q^{25} -59.8102 q^{26} +(14.3198 + 53.5913i) q^{28} -221.779i q^{29} +145.494i q^{31} +127.280i q^{32} -80.3345i q^{34} +(-26.3676 - 98.6801i) q^{35} +290.956 q^{37} -174.542 q^{38} -135.661i q^{40} -108.121 q^{41} -348.122 q^{43} -104.600i q^{44} -129.493 q^{46} +251.090 q^{47} +(-297.285 + 171.086i) q^{49} -211.596i q^{50} +80.0761i q^{52} -160.173i q^{53} +192.605i q^{55} +(-440.117 + 117.601i) q^{56} +496.151 q^{58} -734.849 q^{59} +671.285i q^{61} -325.492 q^{62} -533.284 q^{64} -147.448i q^{65} -688.727 q^{67} -107.555 q^{68} +(220.762 - 58.9883i) q^{70} +577.396i q^{71} -37.4543i q^{73} +650.911i q^{74} +233.683i q^{76} +(624.859 - 166.964i) q^{77} +856.000 q^{79} +171.342 q^{80} -241.883i q^{82} -1232.26 q^{83} +198.046 q^{85} -778.800i q^{86} +859.027 q^{88} -1505.64 q^{89} +(-478.358 + 127.819i) q^{91} +173.371i q^{92} +561.724i q^{94} -430.291i q^{95} -1157.41i q^{97} +(-382.745 - 665.071i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 156 q^{4} - 10 q^{7}+O(q^{10}) \) 44 * q - 156 * q^4 - 10 * q^7	\( 44 q - 156 q^{4} - 10 q^{7} + 484 q^{16} + 68 q^{22} + 704 q^{25} + 300 q^{28} + 328 q^{37} + 340 q^{43} + 968 q^{46} + 158 q^{49} + 1076 q^{58} - 808 q^{64} + 1180 q^{67} - 768 q^{70} + 604 q^{79} + 1224 q^{85} - 2588 q^{88} + 210 q^{91}+O(q^{100}) \) 44 * q - 156 * q^4 - 10 * q^7 + 484 * q^16 + 68 * q^22 + 704 * q^25 + 300 * q^28 + 328 * q^37 + 340 * q^43 + 968 * q^46 + 158 * q^49 + 1076 * q^58 - 808 * q^64 + 1180 * q^67 - 768 * q^70 + 604 * q^79 + 1224 * q^85 - 2588 * q^88 + 210 * q^91

Introduction

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