LMFDB - Embedded newform 6030.2.d.k.2411.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6030,2,Mod(2411,6030)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6030, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6030.2411");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6030 = 2 \cdot 3^{2} \cdot 5 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6030.d (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:	$48.1497924188$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2411.15
Character	$\chi$	$=$	6030.2411
Dual form			6030.2.d.k.2411.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.26508i q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.26508i q^{7} -1.00000 q^{8} +1.00000 q^{10} +1.70850 q^{11} +4.06756i q^{13} -1.26508i q^{14} +1.00000 q^{16} +3.79637i q^{17} -5.81136 q^{19} -1.00000 q^{20} -1.70850 q^{22} +3.51697i q^{23} +1.00000 q^{25} -4.06756i q^{26} +1.26508i q^{28} +1.09253i q^{29} +9.21868i q^{31} -1.00000 q^{32} -3.79637i q^{34} -1.26508i q^{35} +4.05609 q^{37} +5.81136 q^{38} +1.00000 q^{40} -6.43163 q^{41} +3.64024i q^{43} +1.70850 q^{44} -3.51697i q^{46} -9.46841i q^{47} +5.39958 q^{49} -1.00000 q^{50} +4.06756i q^{52} -8.27249 q^{53} -1.70850 q^{55} -1.26508i q^{56} -1.09253i q^{58} -11.9595i q^{59} +1.02086i q^{61} -9.21868i q^{62} +1.00000 q^{64} -4.06756i q^{65} +(6.24461 + 5.29196i) q^{67} +3.79637i q^{68} +1.26508i q^{70} -6.95627i q^{71} -13.1318 q^{73} -4.05609 q^{74} -5.81136 q^{76} +2.16138i q^{77} +8.81630i q^{79} -1.00000 q^{80} +6.43163 q^{82} -6.03823i q^{83} -3.79637i q^{85} -3.64024i q^{86} -1.70850 q^{88} -2.09365i q^{89} -5.14578 q^{91} +3.51697i q^{92} +9.46841i q^{94} +5.81136 q^{95} +7.10610i q^{97} -5.39958 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 24 q^{2} + 24 q^{4} - 24 q^{5} - 24 q^{8}+O(q^{10}) $ 24 * q - 24 * q^2 + 24 * q^4 - 24 * q^5 - 24 * q^8	$ 24 q - 24 q^{2} + 24 q^{4} - 24 q^{5} - 24 q^{8} + 24 q^{10} - 12 q^{11} + 24 q^{16} + 4 q^{19} - 24 q^{20} + 12 q^{22} + 24 q^{25} - 24 q^{32} - 16 q^{37} - 4 q^{38} + 24 q^{40} - 8 q^{41} - 12 q^{44} - 20 q^{49} - 24 q^{50} - 24 q^{53} + 12 q^{55} + 24 q^{64} - 32 q^{67} - 4 q^{73} + 16 q^{74} + 4 q^{76} - 24 q^{80} + 8 q^{82} + 12 q^{88} - 4 q^{95} + 20 q^{98}+O(q^{100}) $ 24 * q - 24 * q^2 + 24 * q^4 - 24 * q^5 - 24 * q^8 + 24 * q^10 - 12 * q^11 + 24 * q^16 + 4 * q^19 - 24 * q^20 + 12 * q^22 + 24 * q^25 - 24 * q^32 - 16 * q^37 - 4 * q^38 + 24 * q^40 - 8 * q^41 - 12 * q^44 - 20 * q^49 - 24 * q^50 - 24 * q^53 + 12 * q^55 + 24 * q^64 - 32 * q^67 - 4 * q^73 + 16 * q^74 + 4 * q^76 - 24 * q^80 + 8 * q^82 + 12 * q^88 - 4 * q^95 + 20 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/6030\mathbb{Z}\right)^\times$.

$n$	$1207$	$3151$	$4691$
$\chi(n)$	$1$	$-1$	$-1$

Coefficient data

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

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

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

$2$	−1.00000			−0.707107
$2$	−1.00000			−0.707107
$3$		0			0
$3$		0			0
$4$	1.00000			0.500000
$5$	−1.00000			−0.447214
$5$	−1.00000			−0.447214
$6$		0			0
$7$			1.26508i			0.478154i	0.971001	+	0.239077i	$0.0768448\pi$
$7$			1.26508i			0.478154i	−0.971001	+	0.239077i	$0.923155\pi$
$8$	−1.00000			−0.353553
$9$		0			0
$10$	1.00000			0.316228
$11$	1.70850			0.515133			0.257566	−	0.966261i	$-0.417079\pi$
$11$	1.70850			0.515133			0.257566	+	0.966261i	$0.417079\pi$
$12$		0			0
$13$			4.06756i			1.12814i	0.825727	+	0.564069i	$0.190765\pi$
$13$			4.06756i			1.12814i	−0.825727	+	0.564069i	$0.809235\pi$
$14$		−	1.26508i		−	0.338106i
$15$		0			0
$16$	1.00000			0.250000
$17$			3.79637i			0.920755i	0.887723	+	0.460378i	$0.152286\pi$
$17$			3.79637i			0.920755i	−0.887723	+	0.460378i	$0.847714\pi$
$18$		0			0
$19$	−5.81136			−1.33322			−0.666609	−	0.745408i	$-0.732255\pi$
$19$	−5.81136			−1.33322			−0.666609	+	0.745408i	$0.732255\pi$
$20$	−1.00000			−0.223607
$21$		0			0
$22$	−1.70850			−0.364254
$23$			3.51697i			0.733338i	0.930351	+	0.366669i	$0.119502\pi$
$23$			3.51697i			0.733338i	−0.930351	+	0.366669i	$0.880498\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		−	4.06756i		−	0.797715i
$27$		0			0
$28$			1.26508i			0.239077i
$29$			1.09253i			0.202878i	0.994842	+	0.101439i	$0.0323447\pi$
$29$			1.09253i			0.202878i	−0.994842	+	0.101439i	$0.967655\pi$
$30$		0			0
$31$			9.21868i			1.65572i	0.560932	+	0.827862i	$0.310443\pi$
$31$			9.21868i			1.65572i	−0.560932	+	0.827862i	$0.689557\pi$
$32$	−1.00000			−0.176777
$33$		0			0
$34$		−	3.79637i		−	0.651072i
$35$		−	1.26508i		−	0.213837i
$36$		0			0
$37$	4.05609			0.666817			0.333409	−	0.942782i	$-0.391801\pi$
$37$	4.05609			0.666817			0.333409	+	0.942782i	$0.391801\pi$
$38$	5.81136			0.942728
$39$		0			0
$40$	1.00000			0.158114
$41$	−6.43163			−1.00445			−0.502226	−	0.864736i	$-0.667486\pi$
$41$	−6.43163			−1.00445			−0.502226	+	0.864736i	$0.667486\pi$
$42$		0			0
$43$			3.64024i			0.555132i	0.960707	+	0.277566i	$0.0895276\pi$
$43$			3.64024i			0.555132i	−0.960707	+	0.277566i	$0.910472\pi$
$44$	1.70850			0.257566
$45$		0			0
$46$		−	3.51697i		−	0.518548i
$47$		−	9.46841i		−	1.38111i	−0.723280	−	0.690555i	$-0.757366\pi$
$47$		−	9.46841i		−	1.38111i	0.723280	−	0.690555i	$-0.242634\pi$
$48$		0			0
$49$	5.39958			0.771369
$50$	−1.00000			−0.141421
$51$		0			0
$52$			4.06756i			0.564069i
$53$	−8.27249			−1.13631			−0.568157	−	0.822920i	$-0.692343\pi$
$53$	−8.27249			−1.13631			−0.568157	+	0.822920i	$0.692343\pi$
$54$		0			0
$55$	−1.70850			−0.230374
$56$		−	1.26508i		−	0.169053i
$57$		0			0
$58$		−	1.09253i		−	0.143456i
$59$		−	11.9595i		−	1.55700i	−0.627645	−	0.778499i	$-0.715981\pi$
$59$		−	11.9595i		−	1.55700i	0.627645	−	0.778499i	$-0.284019\pi$
$60$		0			0
$61$			1.02086i			0.130707i	0.997862	+	0.0653537i	$0.0208176\pi$
$61$			1.02086i			0.130707i	−0.997862	+	0.0653537i	$0.979182\pi$
$62$		−	9.21868i		−	1.17077i
$63$		0			0
$64$	1.00000			0.125000
$65$		−	4.06756i		−	0.504519i
$66$		0			0
$67$	6.24461	+	5.29196i	0.762901	+	0.646516i
$67$	6.24461	+	5.29196i	0.762901	+	0.646516i
$68$			3.79637i			0.460378i
$69$		0			0
$70$			1.26508i			0.151206i
$71$		−	6.95627i		−	0.825557i	−0.910831	−	0.412778i	$-0.864558\pi$
$71$		−	6.95627i		−	0.825557i	0.910831	−	0.412778i	$-0.135442\pi$
$72$		0			0
$73$	−13.1318			−1.53697			−0.768483	−	0.639870i	$-0.778988\pi$
$73$	−13.1318			−1.53697			−0.768483	+	0.639870i	$0.778988\pi$
$74$	−4.05609			−0.471511
$75$		0			0
$76$	−5.81136			−0.666609
$77$			2.16138i			0.246313i
$78$		0			0
$79$			8.81630i			0.991911i	0.868348	+	0.495955i	$0.165182\pi$
$79$			8.81630i			0.991911i	−0.868348	+	0.495955i	$0.834818\pi$
$80$	−1.00000			−0.111803
$81$		0			0
$82$	6.43163			0.710255
$83$		−	6.03823i		−	0.662782i	−0.943494	−	0.331391i	$-0.892482\pi$
$83$		−	6.03823i		−	0.662782i	0.943494	−	0.331391i	$-0.107518\pi$
$84$		0			0
$85$		−	3.79637i		−	0.411774i
$86$		−	3.64024i		−	0.392537i
$87$		0			0
$88$	−1.70850			−0.182127
$89$		−	2.09365i		−	0.221926i	−0.993825	−	0.110963i	$-0.964606\pi$
$89$		−	2.09365i		−	0.221926i	0.993825	−	0.110963i	$-0.0353935\pi$
$90$		0			0
$91$	−5.14578			−0.539424
$92$			3.51697i			0.366669i
$93$		0			0
$94$			9.46841i			0.976592i
$95$	5.81136			0.596233
$96$		0			0
$97$			7.10610i			0.721515i	0.932660	+	0.360758i	$0.117482\pi$
$97$			7.10610i			0.721515i	−0.932660	+	0.360758i	$0.882518\pi$
$98$	−5.39958			−0.545440
$99$		0			0
$100$	1.00000			0.100000
$101$	3.09307			0.307772			0.153886	−	0.988089i	$-0.450821\pi$
$101$	3.09307			0.307772			0.153886	+	0.988089i	$0.450821\pi$
$102$		0			0
$103$	−8.04736			−0.792930			−0.396465	−	0.918050i	$-0.629763\pi$
$103$	−8.04736			−0.792930			−0.396465	+	0.918050i	$0.629763\pi$
$104$		−	4.06756i		−	0.398857i
$105$		0			0
$106$	8.27249			0.803495
$107$			8.44792i			0.816692i	0.912827	+	0.408346i	$0.133894\pi$
$107$			8.44792i			0.816692i	−0.912827	+	0.408346i	$0.866106\pi$
$108$		0			0
$109$		−	12.3763i		−	1.18544i	−0.805410	−	0.592718i	$-0.798055\pi$
$109$		−	12.3763i		−	1.18544i	0.805410	−	0.592718i	$-0.201945\pi$
$110$	1.70850			0.162899
$111$		0			0
$112$			1.26508i			0.119538i
$113$	6.79299			0.639031			0.319516	−	0.947581i	$-0.396480\pi$
$113$	6.79299			0.639031			0.319516	+	0.947581i	$0.396480\pi$
$114$		0			0
$115$		−	3.51697i		−	0.327959i
$116$			1.09253i			0.101439i
$117$		0			0
$118$			11.9595i			1.10096i
$119$	−4.80270			−0.440263
$120$		0			0
$121$	−8.08102			−0.734639
$122$		−	1.02086i		−	0.0924242i
$123$		0			0
$124$			9.21868i			0.827862i
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	11.9143			1.05723			0.528614	−	0.848863i	$-0.322712\pi$
$127$	11.9143			1.05723			0.528614	+	0.848863i	$0.322712\pi$
$128$	−1.00000			−0.0883883
$129$		0			0
$130$			4.06756i			0.356749i
$131$		−	15.7949i		−	1.38001i	−0.723807	−	0.690003i	$-0.757609\pi$
$131$		−	15.7949i		−	1.38001i	0.723807	−	0.690003i	$-0.242391\pi$
$132$		0			0
$133$		−	7.35182i		−	0.637484i
$134$	−6.24461	−	5.29196i	−0.539452	−	0.457156i
$135$		0			0
$136$		−	3.79637i		−	0.325536i
$137$	7.69589			0.657504			0.328752	−	0.944416i	$-0.393372\pi$
$137$	7.69589			0.657504			0.328752	+	0.944416i	$0.393372\pi$
$138$		0			0
$139$		−	4.89338i		−	0.415051i	−0.978230	−	0.207525i	$-0.933459\pi$
$139$		−	4.89338i		−	0.415051i	0.978230	−	0.207525i	$-0.0665410\pi$
$140$		−	1.26508i		−	0.106918i
$141$		0			0
$142$			6.95627i			0.583757i
$143$			6.94944i			0.581141i
$144$		0			0
$145$		−	1.09253i		−	0.0907298i
$146$	13.1318			1.08680
$147$		0			0
$148$	4.05609			0.333409
$149$			5.34707i			0.438049i	0.975719	+	0.219025i	$0.0702875\pi$
$149$			5.34707i			0.438049i	−0.975719	+	0.219025i	$0.929712\pi$
$150$		0			0
$151$	−0.609563			−0.0496056			−0.0248028	−	0.999692i	$-0.507896\pi$
$151$	−0.609563			−0.0496056			−0.0248028	+	0.999692i	$0.507896\pi$
$152$	5.81136			0.471364
$153$		0			0
$154$		−	2.16138i		−	0.174169i
$155$		−	9.21868i		−	0.740462i
$156$		0			0
$157$	−9.00059			−0.718325			−0.359163	−	0.933275i	$-0.616938\pi$
$157$	−9.00059			−0.718325			−0.359163	+	0.933275i	$0.616938\pi$
$158$		−	8.81630i		−	0.701387i
$159$		0			0
$160$	1.00000			0.0790569
$161$	−4.44923			−0.350648
$162$		0			0
$163$	11.7829			0.922911			0.461455	−	0.887163i	$-0.347327\pi$
$163$	11.7829			0.922911			0.461455	+	0.887163i	$0.347327\pi$
$164$	−6.43163			−0.502226
$165$		0			0
$166$			6.03823i			0.468658i
$167$			10.7252i			0.829944i	0.909834	+	0.414972i	$0.136209\pi$
$167$			10.7252i			0.829944i	−0.909834	+	0.414972i	$0.863791\pi$
$168$		0			0
$169$	−3.54506			−0.272697
$170$			3.79637i			0.291168i
$171$		0			0
$172$			3.64024i			0.277566i
$173$		−	9.92373i		−	0.754487i	−0.926114	−	0.377243i	$-0.876872\pi$
$173$		−	9.92373i		−	0.754487i	0.926114	−	0.377243i	$-0.123128\pi$
$174$		0			0
$175$			1.26508i			0.0956308i
$176$	1.70850			0.128783
$177$		0			0
$178$			2.09365i			0.156925i
$179$	−1.07332			−0.0802236			−0.0401118	−	0.999195i	$-0.512771\pi$
$179$	−1.07332			−0.0802236			−0.0401118	+	0.999195i	$0.512771\pi$
$180$		0			0
$181$	−21.5659			−1.60298			−0.801489	−	0.598010i	$-0.795958\pi$
$181$	−21.5659			−1.60298			−0.801489	+	0.598010i	$0.795958\pi$
$182$	5.14578			0.381430
$183$		0			0
$184$		−	3.51697i		−	0.259274i
$185$	−4.05609			−0.298210
$186$		0			0
$187$			6.48611i			0.474311i
$188$		−	9.46841i		−	0.690555i
$189$		0			0
$190$	−5.81136			−0.421601
$191$	−3.98820			−0.288576			−0.144288	−	0.989536i	$-0.546089\pi$
$191$	−3.98820			−0.288576			−0.144288	+	0.989536i	$0.546089\pi$
$192$		0			0
$193$	12.7661			0.918923			0.459462	−	0.888198i	$-0.348042\pi$
$193$	12.7661			0.918923			0.459462	+	0.888198i	$0.348042\pi$
$194$		−	7.10610i		−	0.510188i
$195$		0			0
$196$	5.39958			0.385684
$197$	15.5044			1.10464			0.552321	−	0.833631i	$-0.313742\pi$
$197$	15.5044			1.10464			0.552321	+	0.833631i	$0.313742\pi$
$198$		0			0
$199$	−18.7350			−1.32809			−0.664045	−	0.747693i	$-0.731162\pi$
$199$	−18.7350			−1.32809			−0.664045	+	0.747693i	$0.731162\pi$
$200$	−1.00000			−0.0707107
$201$		0			0
$202$	−3.09307			−0.217628
$203$	−1.38214			−0.0970069
$204$		0			0
$205$	6.43163			0.449205
$206$	8.04736			0.560686
$207$		0			0
$208$			4.06756i			0.282035i
$209$	−9.92872			−0.686784
$210$		0			0
$211$	9.93702			0.684093			0.342046	−	0.939683i	$-0.388880\pi$
$211$	9.93702			0.684093			0.342046	+	0.939683i	$0.388880\pi$
$212$	−8.27249			−0.568157
$213$		0			0
$214$		−	8.44792i		−	0.577488i
$215$		−	3.64024i		−	0.248262i
$216$		0			0
$217$	−11.6623			−0.791691
$218$			12.3763i			0.838230i
$219$		0			0
$220$	−1.70850			−0.115187
$221$	−15.4420			−1.03874
$222$		0			0
$223$	−11.0452			−0.739645			−0.369822	−	0.929103i	$-0.620581\pi$
$223$	−11.0452			−0.739645			−0.369822	+	0.929103i	$0.620581\pi$
$224$		−	1.26508i		−	0.0845265i
$225$		0			0
$226$	−6.79299			−0.451863
$227$			3.94267i			0.261684i	0.991403	+	0.130842i	$0.0417681\pi$
$227$			3.94267i			0.261684i	−0.991403	+	0.130842i	$0.958232\pi$
$228$		0			0
$229$			10.1938i			0.673623i	0.941572	+	0.336812i	$0.109349\pi$
$229$			10.1938i			0.673623i	−0.941572	+	0.336812i	$0.890651\pi$
$230$			3.51697i			0.231902i
$231$		0			0
$232$		−	1.09253i		−	0.0717282i
$233$	−2.12468			−0.139192			−0.0695962	−	0.997575i	$-0.522171\pi$
$233$	−2.12468			−0.139192			−0.0695962	+	0.997575i	$0.522171\pi$
$234$		0			0
$235$			9.46841i			0.617651i
$236$		−	11.9595i		−	0.778499i
$237$		0			0
$238$	4.80270			0.311313
$239$	−9.57621			−0.619434			−0.309717	−	0.950829i	$-0.600234\pi$
$239$	−9.57621			−0.619434			−0.309717	+	0.950829i	$0.600234\pi$
$240$		0			0
$241$	−2.29675			−0.147946			−0.0739732	−	0.997260i	$-0.523568\pi$
$241$	−2.29675			−0.147946			−0.0739732	+	0.997260i	$0.523568\pi$
$242$	8.08102			0.519468
$243$		0			0
$244$			1.02086i			0.0653537i
$245$	−5.39958			−0.344967
$246$		0			0
$247$		−	23.6381i		−	1.50406i
$248$		−	9.21868i		−	0.585387i
$249$		0			0
$250$	1.00000			0.0632456
$251$	−15.0952			−0.952801			−0.476401	−	0.879228i	$-0.658059\pi$
$251$	−15.0952			−0.952801			−0.476401	+	0.879228i	$0.658059\pi$
$252$		0			0
$253$			6.00874i			0.377766i
$254$	−11.9143			−0.747573
$255$		0			0
$256$	1.00000			0.0625000
$257$		−	11.1062i		−	0.692785i	−0.938090	−	0.346392i	$-0.887407\pi$
$257$		−	11.1062i		−	0.692785i	0.938090	−	0.346392i	$-0.112593\pi$
$258$		0			0
$259$			5.13127i			0.318841i
$260$		−	4.06756i		−	0.252260i
$261$		0			0
$262$			15.7949i			0.975811i
$263$			0.485188i			0.0299180i	0.999888	+	0.0149590i	$0.00476177\pi$
$263$			0.485188i			0.0299180i	−0.999888	+	0.0149590i	$0.995238\pi$
$264$		0			0
$265$	8.27249			0.508175
$266$			7.35182i			0.450769i
$267$		0			0
$268$	6.24461	+	5.29196i	0.381450	+	0.323258i
$269$			9.69432i			0.591073i	0.955331	+	0.295536i	$0.0954984\pi$
$269$			9.69432i			0.591073i	−0.955331	+	0.295536i	$0.904502\pi$
$270$		0			0
$271$		−	10.8863i		−	0.661297i	−0.943754	−	0.330649i	$-0.892732\pi$
$271$		−	10.8863i		−	0.661297i	0.943754	−	0.330649i	$-0.107268\pi$
$272$			3.79637i			0.230189i
$273$		0			0
$274$	−7.69589			−0.464926
$275$	1.70850			0.103027
$276$		0			0
$277$	−22.8780			−1.37461			−0.687304	−	0.726370i	$-0.741206\pi$
$277$	−22.8780			−1.37461			−0.687304	+	0.726370i	$0.741206\pi$
$278$			4.89338i			0.293485i
$279$		0			0
$280$			1.26508i			0.0756028i
$281$	−18.8633			−1.12529			−0.562644	−	0.826699i	$-0.690216\pi$
$281$	−18.8633			−1.12529			−0.562644	+	0.826699i	$0.690216\pi$
$282$		0			0
$283$	6.98390			0.415150			0.207575	−	0.978219i	$-0.433443\pi$
$283$	6.98390			0.415150			0.207575	+	0.978219i	$0.433443\pi$
$284$		−	6.95627i		−	0.412778i
$285$		0			0
$286$		−	6.94944i		−	0.410929i
$287$		−	8.13651i		−	0.480283i
$288$		0			0
$289$	2.58756			0.152209
$290$			1.09253i			0.0641557i
$291$		0			0
$292$	−13.1318			−0.768483
$293$		−	3.10398i		−	0.181337i	−0.995881	−	0.0906684i	$-0.971100\pi$
$293$		−	3.10398i		−	0.181337i	0.995881	−	0.0906684i	$-0.0289003\pi$
$294$		0			0
$295$			11.9595i			0.696311i
$296$	−4.05609			−0.235756
$297$		0			0
$298$		−	5.34707i		−	0.309748i
$299$	−14.3055			−0.827307
$300$		0			0
$301$	−4.60518			−0.265438
$302$	0.609563			0.0350764
$303$		0			0
$304$	−5.81136			−0.333305
$305$		−	1.02086i		−	0.0584542i
$306$		0			0
$307$	10.2684			0.586050			0.293025	−	0.956105i	$-0.405338\pi$
$307$	10.2684			0.586050			0.293025	+	0.956105i	$0.405338\pi$
$308$			2.16138i			0.123156i
$309$		0			0
$310$			9.21868i			0.523586i
$311$	−32.0671			−1.81836			−0.909178	−	0.416407i	$-0.863289\pi$
$311$	−32.0671			−1.81836			−0.909178	+	0.416407i	$0.863289\pi$
$312$		0			0
$313$		−	18.9782i		−	1.07271i	−0.843993	−	0.536354i	$-0.819801\pi$
$313$		−	18.9782i		−	1.07271i	0.843993	−	0.536354i	$-0.180199\pi$
$314$	9.00059			0.507933
$315$		0			0
$316$			8.81630i			0.495955i
$317$		−	9.63852i		−	0.541353i	−0.962670	−	0.270677i	$-0.912753\pi$
$317$		−	9.63852i		−	0.541353i	0.962670	−	0.270677i	$-0.0872474\pi$
$318$		0			0
$319$			1.86659i			0.104509i
$320$	−1.00000			−0.0559017
$321$		0			0
$322$	4.44923			0.247946
$323$		−	22.0621i		−	1.22757i
$324$		0			0
$325$			4.06756i			0.225628i
$326$	−11.7829			−0.652596
$327$		0			0
$328$	6.43163			0.355128
$329$	11.9783			0.660383
$330$		0			0
$331$		−	13.3559i		−	0.734108i	−0.930200	−	0.367054i	$-0.880366\pi$
$331$		−	13.3559i		−	0.734108i	0.930200	−	0.367054i	$-0.119634\pi$
$332$		−	6.03823i		−	0.331391i
$333$		0			0
$334$		−	10.7252i		−	0.586859i
$335$	−6.24461	−	5.29196i	−0.341180	−	0.289131i
$336$		0			0
$337$		−	11.1170i		−	0.605583i	−0.953057	−	0.302792i	$-0.902081\pi$
$337$		−	11.1170i		−	0.605583i	0.953057	−	0.302792i	$-0.0979186\pi$
$338$	3.54506			0.192826
$339$		0			0
$340$		−	3.79637i		−	0.205887i
$341$			15.7501i			0.852917i
$342$		0			0
$343$			15.6864i			0.846987i
$344$		−	3.64024i		−	0.196269i
$345$		0			0
$346$			9.92373i			0.533503i
$347$	−4.07565			−0.218793			−0.109396	−	0.993998i	$-0.534892\pi$
$347$	−4.07565			−0.218793			−0.109396	+	0.993998i	$0.534892\pi$
$348$		0			0
$349$	10.4925			0.561650			0.280825	−	0.959759i	$-0.409392\pi$
$349$	10.4925			0.561650			0.280825	+	0.959759i	$0.409392\pi$
$350$		−	1.26508i		−	0.0676212i
$351$		0			0
$352$	−1.70850			−0.0910634
$353$	−29.1312			−1.55050			−0.775248	−	0.631657i	$-0.782375\pi$
$353$	−29.1312			−1.55050			−0.775248	+	0.631657i	$0.782375\pi$
$354$		0			0
$355$			6.95627i			0.369200i
$356$		−	2.09365i		−	0.110963i
$357$		0			0
$358$	1.07332			0.0567266
$359$			6.16562i			0.325409i	0.986675	+	0.162705i	$0.0520217\pi$
$359$			6.16562i			0.325409i	−0.986675	+	0.162705i	$0.947978\pi$
$360$		0			0
$361$	14.7719			0.777470
$362$	21.5659			1.13348
$363$		0			0
$364$	−5.14578			−0.269712
$365$	13.1318			0.687352
$366$		0			0
$367$			4.32675i			0.225855i	0.993603	+	0.112927i	$0.0360227\pi$
$367$			4.32675i			0.225855i	−0.993603	+	0.112927i	$0.963977\pi$
$368$			3.51697i			0.183334i
$369$		0			0
$370$	4.05609			0.210866
$371$		−	10.4653i		−	0.543333i
$372$		0			0
$373$			26.1670i			1.35488i	0.735580	+	0.677438i	$0.236910\pi$
$373$			26.1670i			1.35488i	−0.735580	+	0.677438i	$0.763090\pi$
$374$		−	6.48611i		−	0.335389i
$375$		0			0
$376$			9.46841i			0.488296i
$377$	−4.44394			−0.228875
$378$		0			0
$379$		−	24.9543i		−	1.28182i	−0.767617	−	0.640909i	$-0.778558\pi$
$379$		−	24.9543i		−	1.28182i	0.767617	−	0.640909i	$-0.221442\pi$
$380$	5.81136			0.298117
$381$		0			0
$382$	3.98820			0.204054
$383$	34.3314			1.75425			0.877127	−	0.480259i	$-0.159457\pi$
$383$	34.3314			1.75425			0.877127	+	0.480259i	$0.159457\pi$
$384$		0			0
$385$		−	2.16138i		−	0.110154i
$386$	−12.7661			−0.649777
$387$		0			0
$388$			7.10610i			0.360758i
$389$			21.6678i			1.09860i	0.835626	+	0.549299i	$0.185105\pi$
$389$			21.6678i			1.09860i	−0.835626	+	0.549299i	$0.814895\pi$
$390$		0			0
$391$	−13.3517			−0.675225
$392$	−5.39958			−0.272720
$393$		0			0
$394$	−15.5044			−0.781100
$395$		−	8.81630i		−	0.443596i
$396$		0			0
$397$	13.4815			0.676618			0.338309	−	0.941035i	$-0.390145\pi$
$397$	13.4815			0.676618			0.338309	+	0.941035i	$0.390145\pi$
$398$	18.7350			0.939101
$399$		0			0
$400$	1.00000			0.0500000
$401$	−4.93469			−0.246427			−0.123213	−	0.992380i	$-0.539320\pi$
$401$	−4.93469			−0.246427			−0.123213	+	0.992380i	$0.539320\pi$
$402$		0			0
$403$	−37.4976			−1.86789
$404$	3.09307			0.153886
$405$		0			0
$406$	1.38214			0.0685943
$407$	6.92984			0.343499
$408$		0			0
$409$		−	15.4700i		−	0.764940i	−0.923968	−	0.382470i	$-0.875074\pi$
$409$		−	15.4700i		−	0.764940i	0.923968	−	0.382470i	$-0.124926\pi$
$410$	−6.43163			−0.317636
$411$		0			0
$412$	−8.04736			−0.396465
$413$	15.1297			0.744485
$414$		0			0
$415$			6.03823i			0.296405i
$416$		−	4.06756i		−	0.199429i
$417$		0			0
$418$	9.92872			0.485630
$419$			15.7784i			0.770823i	0.922745	+	0.385412i	$0.125941\pi$
$419$			15.7784i			0.770823i	−0.922745	+	0.385412i	$0.874059\pi$
$420$		0			0
$421$	−26.8544			−1.30880			−0.654402	−	0.756147i	$-0.727080\pi$
$421$	−26.8544			−1.30880			−0.654402	+	0.756147i	$0.727080\pi$
$422$	−9.93702			−0.483726
$423$		0			0
$424$	8.27249			0.401747
$425$			3.79637i			0.184151i
$426$		0			0
$427$	−1.29146			−0.0624983
$428$			8.44792i			0.408346i
$429$		0			0
$430$			3.64024i			0.175548i
$431$		−	12.6248i		−	0.608114i	−0.952654	−	0.304057i	$-0.901659\pi$
$431$		−	12.6248i		−	0.608114i	0.952654	−	0.304057i	$-0.0983413\pi$
$432$		0			0
$433$		−	17.6647i		−	0.848914i	−0.905448	−	0.424457i	$-0.860465\pi$
$433$		−	17.6647i		−	0.848914i	0.905448	−	0.424457i	$-0.139535\pi$
$434$	11.6623			0.559810
$435$		0			0
$436$		−	12.3763i		−	0.592718i
$437$		−	20.4384i		−	0.977699i
$438$		0			0
$439$	−9.53770			−0.455209			−0.227605	−	0.973754i	$-0.573089\pi$
$439$	−9.53770			−0.455209			−0.227605	+	0.973754i	$0.573089\pi$
$440$	1.70850			0.0814496
$441$		0			0
$442$	15.4420			0.734500
$443$	−22.8020			−1.08336			−0.541678	−	0.840586i	$-0.682211\pi$
$443$	−22.8020			−1.08336			−0.541678	+	0.840586i	$0.682211\pi$
$444$		0			0
$445$			2.09365i			0.0992483i
$446$	11.0452			0.523008
$447$		0			0
$448$			1.26508i			0.0597692i
$449$			10.0894i			0.476149i	0.971247	+	0.238075i	$0.0765163\pi$
$449$			10.0894i			0.476149i	−0.971247	+	0.238075i	$0.923484\pi$
$450$		0			0
$451$	−10.9885			−0.517426
$452$	6.79299			0.319516
$453$		0			0
$454$		−	3.94267i		−	0.185039i
$455$	5.14578			0.241238
$456$		0			0
$457$	7.13178			0.333611			0.166805	−	0.985990i	$-0.446655\pi$
$457$	7.13178			0.333611			0.166805	+	0.985990i	$0.446655\pi$
$458$		−	10.1938i		−	0.476324i
$459$		0			0
$460$		−	3.51697i		−	0.163979i
$461$			9.70607i			0.452057i	0.974121	+	0.226028i	$0.0725742\pi$
$461$			9.70607i			0.452057i	−0.974121	+	0.226028i	$0.927426\pi$
$462$		0			0
$463$		−	24.3701i		−	1.13258i	−0.824207	−	0.566288i	$-0.808379\pi$
$463$		−	24.3701i		−	1.13258i	0.824207	−	0.566288i	$-0.191621\pi$
$464$			1.09253i			0.0507195i
$465$		0			0
$466$	2.12468			0.0984240
$467$		−	5.58652i		−	0.258513i	−0.991611	−	0.129257i	$-0.958741\pi$
$467$		−	5.58652i		−	0.258513i	0.991611	−	0.129257i	$-0.0412591\pi$
$468$		0			0
$469$	−6.69473	+	7.89991i	−0.309134	+	0.364784i
$470$		−	9.46841i		−	0.436745i
$471$		0			0
$472$			11.9595i			0.550482i
$473$			6.21936i			0.285966i
$474$		0			0
$475$	−5.81136			−0.266644
$476$	−4.80270			−0.220131
$477$		0			0
$478$	9.57621			0.438006
$479$			32.3635i			1.47872i	0.673308	+	0.739362i	$0.264873\pi$
$479$			32.3635i			1.47872i	−0.673308	+	0.739362i	$0.735127\pi$
$480$		0			0
$481$			16.4984i			0.752263i
$482$	2.29675			0.104614
$483$		0			0
$484$	−8.08102			−0.367319
$485$		−	7.10610i		−	0.322671i
$486$		0			0
$487$		−	9.18614i		−	0.416264i	−0.978101	−	0.208132i	$-0.933262\pi$
$487$		−	9.18614i		−	0.416264i	0.978101	−	0.208132i	$-0.0667383\pi$
$488$		−	1.02086i		−	0.0462121i
$489$		0			0
$490$	5.39958			0.243928
$491$		−	10.2189i		−	0.461173i	−0.973052	−	0.230587i	$-0.925935\pi$
$491$		−	10.2189i		−	0.461173i	0.973052	−	0.230587i	$-0.0740645\pi$
$492$		0			0
$493$	−4.14766			−0.186801
$494$			23.6381i			1.06353i
$495$		0			0
$496$			9.21868i			0.413931i
$497$	8.80021			0.394743
$498$		0			0
$499$			24.8528i			1.11257i	0.830993	+	0.556283i	$0.187773\pi$
$499$			24.8528i			1.11257i	−0.830993	+	0.556283i	$0.812227\pi$
$500$	−1.00000			−0.0447214
$501$		0			0
$502$	15.0952			0.673732
$503$	12.0829			0.538750			0.269375	−	0.963035i	$-0.413183\pi$
$503$	12.0829			0.538750			0.269375	+	0.963035i	$0.413183\pi$
$504$		0			0
$505$	−3.09307			−0.137640
$506$		−	6.00874i		−	0.267121i
$507$		0			0
$508$	11.9143			0.528614
$509$		−	7.47552i		−	0.331347i	−0.986181	−	0.165673i	$-0.947020\pi$
$509$		−	7.47552i		−	0.331347i	0.986181	−	0.165673i	$-0.0529797\pi$
$510$		0			0
$511$		−	16.6128i		−	0.734907i
$512$	−1.00000			−0.0441942
$513$		0			0
$514$			11.1062i			0.489873i
$515$	8.04736			0.354609
$516$		0			0
$517$		−	16.1768i		−	0.711455i
$518$		−	5.13127i		−	0.225455i
$519$		0			0
$520$			4.06756i			0.178374i
$521$	−20.6670			−0.905437			−0.452718	−	0.891654i	$-0.649546\pi$
$521$	−20.6670			−0.905437			−0.452718	+	0.891654i	$0.649546\pi$
$522$		0			0
$523$	−23.9815			−1.04864			−0.524319	−	0.851522i	$-0.675680\pi$
$523$	−23.9815			−1.04864			−0.524319	+	0.851522i	$0.675680\pi$
$524$		−	15.7949i		−	0.690003i
$525$		0			0
$526$		−	0.485188i		−	0.0211552i
$527$	−34.9975			−1.52452
$528$		0			0
$529$	10.6310			0.462215
$530$	−8.27249			−0.359334
$531$		0			0
$532$		−	7.35182i		−	0.318742i
$533$		−	26.1611i		−	1.13316i
$534$		0			0
$535$		−	8.44792i		−	0.365236i
$536$	−6.24461	−	5.29196i	−0.269726	−	0.228578i
$537$		0			0
$538$		−	9.69432i		−	0.417952i
$539$	9.22519			0.397357
$540$		0			0
$541$			11.6002i			0.498733i	0.968409	+	0.249366i	$0.0802223\pi$
$541$			11.6002i			0.498733i	−0.968409	+	0.249366i	$0.919778\pi$
$542$			10.8863i			0.467608i
$543$		0			0
$544$		−	3.79637i		−	0.162768i
$545$			12.3763i			0.530143i
$546$		0			0
$547$			31.3525i			1.34054i	0.742119	+	0.670268i	$0.233821\pi$
$547$			31.3525i			1.34054i	−0.742119	+	0.670268i	$0.766179\pi$
$548$	7.69589			0.328752
$549$		0			0
$550$	−1.70850			−0.0728507
$551$		−	6.34910i		−	0.270481i
$552$		0			0
$553$	−11.1533			−0.474286
$554$	22.8780			0.971995
$555$		0			0
$556$		−	4.89338i		−	0.207525i
$557$			16.4778i			0.698186i	0.937088	+	0.349093i	$0.113510\pi$
$557$			16.4778i			0.698186i	−0.937088	+	0.349093i	$0.886490\pi$
$558$		0			0
$559$	−14.8069			−0.626265
$560$		−	1.26508i		−	0.0534592i
$561$		0			0
$562$	18.8633			0.795699
$563$	−0.0136229			−0.000574135			−0.000287068	−	1.00000i	$-0.500091\pi$
$563$	−0.0136229			−0.000574135			−0.000287068		1.00000i	$0.500091\pi$
$564$		0			0
$565$	−6.79299			−0.285783
$566$	−6.98390			−0.293555
$567$		0			0
$568$			6.95627i			0.291878i
$569$			37.8719i			1.58767i	0.608132	+	0.793836i	$0.291919\pi$
$569$			37.8719i			1.58767i	−0.608132	+	0.793836i	$0.708081\pi$
$570$		0			0
$571$	10.9431			0.457954			0.228977	−	0.973432i	$-0.426462\pi$
$571$	10.9431			0.457954			0.228977	+	0.973432i	$0.426462\pi$
$572$			6.94944i			0.290570i
$573$		0			0
$574$			8.13651i			0.339611i
$575$			3.51697i			0.146668i
$576$		0			0
$577$			15.6339i			0.650846i	0.945569	+	0.325423i	$0.105507\pi$
$577$			15.6339i			0.650846i	−0.945569	+	0.325423i	$0.894493\pi$
$578$	−2.58756			−0.107628
$579$		0			0
$580$		−	1.09253i		−	0.0453649i
$581$	7.63882			0.316912
$582$		0			0
$583$	−14.1336			−0.585352
$584$	13.1318			0.543400
$585$		0			0
$586$			3.10398i			0.128224i
$587$	2.99108			0.123455			0.0617275	−	0.998093i	$-0.480339\pi$
$587$	2.99108			0.123455			0.0617275	+	0.998093i	$0.480339\pi$
$588$		0			0
$589$		−	53.5731i		−	2.20744i
$590$		−	11.9595i		−	0.492366i
$591$		0			0
$592$	4.05609			0.166704
$593$	−38.0749			−1.56355			−0.781774	−	0.623562i	$-0.785685\pi$
$593$	−38.0749			−1.56355			−0.781774	+	0.623562i	$0.785685\pi$
$594$		0			0
$595$	4.80270			0.196892
$596$			5.34707i			0.219025i
$597$		0			0
$598$	14.3055			0.584994
$599$	−19.5979			−0.800747			−0.400373	−	0.916352i	$-0.631120\pi$
$599$	−19.5979			−0.800747			−0.400373	+	0.916352i	$0.631120\pi$
$600$		0			0
$601$	−46.0861			−1.87989			−0.939946	−	0.341323i	$-0.889125\pi$
$601$	−46.0861			−1.87989			−0.939946	+	0.341323i	$0.889125\pi$
$602$	4.60518			0.187693
$603$		0			0
$604$	−0.609563			−0.0248028
$605$	8.08102			0.328540
$606$		0			0
$607$	28.3088			1.14902			0.574510	−	0.818498i	$-0.305193\pi$
$607$	28.3088			1.14902			0.574510	+	0.818498i	$0.305193\pi$
$608$	5.81136			0.235682
$609$		0			0
$610$			1.02086i			0.0413333i
$611$	38.5134			1.55808
$612$		0			0
$613$	11.4876			0.463979			0.231990	−	0.972718i	$-0.425476\pi$
$613$	11.4876			0.463979			0.231990	+	0.972718i	$0.425476\pi$
$614$	−10.2684			−0.414400
$615$		0			0
$616$		−	2.16138i		−	0.0870847i
$617$		−	6.32986i		−	0.254831i	−0.991849	−	0.127415i	$-0.959332\pi$
$617$		−	6.32986i		−	0.254831i	0.991849	−	0.127415i	$-0.0406681\pi$
$618$		0			0
$619$	−45.7725			−1.83975			−0.919876	−	0.392209i	$-0.871711\pi$
$619$	−45.7725			−1.83975			−0.919876	+	0.392209i	$0.871711\pi$
$620$		−	9.21868i		−	0.370231i
$621$		0			0
$622$	32.0671			1.28577
$623$	2.64862			0.106115
$624$		0			0
$625$	1.00000			0.0400000
$626$			18.9782i			0.758520i
$627$		0			0
$628$	−9.00059			−0.359163
$629$			15.3984i			0.613976i
$630$		0			0
$631$			47.4104i			1.88738i	0.330835	+	0.943689i	$0.392670\pi$
$631$			47.4104i			1.88738i	−0.330835	+	0.943689i	$0.607330\pi$
$632$		−	8.81630i		−	0.350693i
$633$		0			0
$634$			9.63852i			0.382795i
$635$	−11.9143			−0.472806
$636$		0			0
$637$			21.9631i			0.870211i
$638$		−	1.86659i		−	0.0738991i
$639$		0			0
$640$	1.00000			0.0395285
$641$	−39.0918			−1.54403			−0.772017	−	0.635602i	$-0.780752\pi$
$641$	−39.0918			−1.54403			−0.772017	+	0.635602i	$0.780752\pi$
$642$		0			0
$643$	35.3851			1.39545			0.697726	−	0.716364i	$-0.254195\pi$
$643$	35.3851			1.39545			0.697726	+	0.716364i	$0.254195\pi$
$644$	−4.44923			−0.175324
$645$		0			0
$646$			22.0621i			0.868022i
$647$	−38.7395			−1.52301			−0.761503	−	0.648162i	$-0.775538\pi$
$647$	−38.7395			−1.52301			−0.761503	+	0.648162i	$0.775538\pi$
$648$		0			0
$649$		−	20.4329i		−	0.802061i
$650$		−	4.06756i		−	0.159543i
$651$		0			0
$652$	11.7829			0.461455
$653$	−30.1454			−1.17968			−0.589840	−	0.807520i	$-0.700809\pi$
$653$	−30.1454			−1.17968			−0.589840	+	0.807520i	$0.700809\pi$
$654$		0			0
$655$			15.7949i			0.617157i
$656$	−6.43163			−0.251113
$657$		0			0
$658$	−11.9783			−0.466961
$659$		−	41.4008i		−	1.61275i	−0.591407	−	0.806374i	$-0.701427\pi$
$659$		−	41.4008i		−	1.61275i	0.591407	−	0.806374i	$-0.298573\pi$
$660$		0			0
$661$		−	39.0504i		−	1.51888i	−0.650576	−	0.759441i	$-0.725472\pi$
$661$		−	39.0504i		−	1.51888i	0.650576	−	0.759441i	$-0.274528\pi$
$662$			13.3559i			0.519093i
$663$		0			0
$664$			6.03823i			0.234329i
$665$			7.35182i			0.285091i
$666$		0			0
$667$	−3.84240			−0.148778
$668$			10.7252i			0.414972i
$669$		0			0
$670$	6.24461	+	5.29196i	0.241250	+	0.204446i
$671$			1.74414i			0.0673317i
$672$		0			0
$673$		−	30.7327i		−	1.18466i	−0.805696	−	0.592329i	$-0.798208\pi$
$673$		−	30.7327i		−	1.18466i	0.805696	−	0.592329i	$-0.201792\pi$
$674$			11.1170i			0.428212i
$675$		0			0
$676$	−3.54506			−0.136349
$677$	25.4061			0.976437			0.488219	−	0.872721i	$-0.337647\pi$
$677$	25.4061			0.976437			0.488219	+	0.872721i	$0.337647\pi$
$678$		0			0
$679$	−8.98976			−0.344995
$680$			3.79637i			0.145584i
$681$		0			0
$682$		−	15.7501i		−	0.603104i
$683$	−15.1050			−0.577978			−0.288989	−	0.957332i	$-0.593319\pi$
$683$	−15.1050			−0.577978			−0.288989	+	0.957332i	$0.593319\pi$
$684$		0			0
$685$	−7.69589			−0.294045
$686$		−	15.6864i		−	0.598910i
$687$		0			0
$688$			3.64024i			0.138783i
$689$		−	33.6489i		−	1.28192i
$690$		0			0
$691$	36.0912			1.37297			0.686487	−	0.727142i	$-0.259152\pi$
$691$	36.0912			1.37297			0.686487	+	0.727142i	$0.259152\pi$
$692$		−	9.92373i		−	0.377243i
$693$		0			0
$694$	4.07565			0.154710
$695$			4.89338i			0.185616i
$696$		0			0
$697$		−	24.4169i		−	0.924855i
$698$	−10.4925			−0.397147
$699$		0			0
$700$			1.26508i			0.0478154i
$701$	−7.98260			−0.301499			−0.150749	−	0.988572i	$-0.548169\pi$
$701$	−7.98260			−0.301499			−0.150749	+	0.988572i	$0.548169\pi$
$702$		0			0
$703$	−23.5714			−0.889013
$704$	1.70850			0.0643916
$705$		0			0
$706$	29.1312			1.09637
$707$			3.91298i			0.147163i
$708$		0			0
$709$	−32.2953			−1.21288			−0.606438	−	0.795131i	$-0.707402\pi$
$709$	−32.2953			−1.21288			−0.606438	+	0.795131i	$0.707402\pi$
$710$		−	6.95627i		−	0.261064i
$711$		0			0
$712$			2.09365i			0.0784627i
$713$	−32.4218			−1.21421
$714$		0			0
$715$		−	6.94944i		−	0.259894i
$716$	−1.07332			−0.0401118
$717$		0			0
$718$		−	6.16562i		−	0.230099i
$719$			27.4921i			1.02528i	0.858603	+	0.512641i	$0.171333\pi$
$719$			27.4921i			1.02528i	−0.858603	+	0.512641i	$0.828667\pi$
$720$		0			0
$721$		−	10.1805i		−	0.379142i
$722$	−14.7719			−0.549755
$723$		0			0
$724$	−21.5659			−0.801489
$725$			1.09253i			0.0405756i
$726$		0			0
$727$			6.85116i			0.254095i	0.991897	+	0.127048i	$0.0405501\pi$
$727$			6.85116i			0.254095i	−0.991897	+	0.127048i	$0.959450\pi$
$728$	5.14578			0.190715
$729$		0			0
$730$	−13.1318			−0.486032
$731$	−13.8197			−0.511140
$732$		0			0
$733$		−	8.20320i		−	0.302992i	−0.988458	−	0.151496i	$-0.951591\pi$
$733$		−	8.20320i		−	0.302992i	0.988458	−	0.151496i	$-0.0484091\pi$
$734$		−	4.32675i		−	0.159703i
$735$		0			0
$736$		−	3.51697i		−	0.129637i
$737$	10.6689	+	9.04132i	0.392995	+	0.333041i
$738$		0			0
$739$		−	42.6045i		−	1.56723i	−0.621245	−	0.783616i	$-0.713373\pi$
$739$		−	42.6045i		−	1.56723i	0.621245	−	0.783616i	$-0.286627\pi$
$740$	−4.05609			−0.149105
$741$		0			0
$742$			10.4653i			0.384194i
$743$			33.5303i			1.23011i	0.788485	+	0.615054i	$0.210866\pi$
$743$			33.5303i			1.23011i	−0.788485	+	0.615054i	$0.789134\pi$
$744$		0			0
$745$		−	5.34707i		−	0.195902i
$746$		−	26.1670i		−	0.958041i
$747$		0			0
$748$			6.48611i			0.237156i
$749$	−10.6873			−0.390504
$750$		0			0
$751$	−33.0593			−1.20635			−0.603176	−	0.797608i	$-0.706098\pi$
$751$	−33.0593			−1.20635			−0.603176	+	0.797608i	$0.706098\pi$
$752$		−	9.46841i		−	0.345277i
$753$		0			0
$754$	4.44394			0.161839
$755$	0.609563			0.0221843
$756$		0			0
$757$			14.7918i			0.537618i	0.963193	+	0.268809i	$0.0866301\pi$
$757$			14.7918i			0.537618i	−0.963193	+	0.268809i	$0.913370\pi$
$758$			24.9543i			0.906382i
$759$		0			0
$760$	−5.81136			−0.210800
$761$		−	27.3606i		−	0.991819i	−0.868374	−	0.495910i	$-0.834835\pi$
$761$		−	27.3606i		−	0.991819i	0.868374	−	0.495910i	$-0.165165\pi$
$762$		0			0
$763$	15.6570			0.566821
$764$	−3.98820			−0.144288
$765$		0			0
$766$	−34.3314			−1.24044
$767$	48.6461			1.75651
$768$		0			0
$769$		−	49.6864i		−	1.79174i	−0.444319	−	0.895869i	$-0.646554\pi$
$769$		−	49.6864i		−	1.79174i	0.444319	−	0.895869i	$-0.353446\pi$
$770$			2.16138i			0.0778909i
$771$		0			0
$772$	12.7661			0.459462
$773$		−	21.7154i		−	0.781050i	−0.920592	−	0.390525i	$-0.872293\pi$
$773$		−	21.7154i		−	0.781050i	0.920592	−	0.390525i	$-0.127707\pi$
$774$		0			0
$775$			9.21868i			0.331145i
$776$		−	7.10610i		−	0.255094i
$777$		0			0
$778$		−	21.6678i		−	0.776827i
$779$	37.3766			1.33915
$780$		0			0
$781$		−	11.8848i		−	0.425271i
$782$	13.3517			0.477456
$783$		0			0
$784$	5.39958			0.192842
$785$	9.00059			0.321245
$786$		0			0
$787$			53.4700i			1.90600i	0.302972	+	0.953000i	$0.402021\pi$
$787$			53.4700i			1.90600i	−0.302972	+	0.953000i	$0.597979\pi$
$788$	15.5044			0.552321
$789$		0			0
$790$			8.81630i			0.313670i
$791$			8.59366i			0.305555i
$792$		0			0
$793$	−4.15240			−0.147456
$794$	−13.4815			−0.478441
$795$		0			0
$796$	−18.7350			−0.664045
$797$		−	44.0273i		−	1.55953i	−0.626075	−	0.779763i	$-0.715340\pi$
$797$		−	44.0273i		−	1.55953i	0.626075	−	0.779763i	$-0.284660\pi$
$798$		0			0
$799$	35.9456			1.27166
$800$	−1.00000			−0.0353553
$801$		0			0
$802$	4.93469			0.174250
$803$	−22.4358			−0.791741
$804$		0			0
$805$	4.44923			0.156815
$806$	37.4976			1.32080
$807$		0			0
$808$	−3.09307			−0.108814
$809$	−33.9434			−1.19339			−0.596693	−	0.802469i	$-0.703519\pi$
$809$	−33.9434			−1.19339			−0.596693	+	0.802469i	$0.703519\pi$
$810$		0			0
$811$		−	24.2809i		−	0.852617i	−0.904578	−	0.426309i	$-0.859814\pi$
$811$		−	24.2809i		−	0.852617i	0.904578	−	0.426309i	$-0.140186\pi$
$812$	−1.38214			−0.0485035
$813$		0			0
$814$	−6.92984			−0.242891
$815$	−11.7829			−0.412738
$816$		0			0
$817$		−	21.1548i		−	0.740111i
$818$			15.4700i			0.540894i
$819$		0			0
$820$	6.43163			0.224602
$821$			16.0973i			0.561801i	0.959737	+	0.280901i	$0.0906331\pi$
$821$			16.0973i			0.561801i	−0.959737	+	0.280901i	$0.909367\pi$
$822$		0			0
$823$	−43.7870			−1.52632			−0.763159	−	0.646211i	$-0.776353\pi$
$823$	−43.7870			−1.52632			−0.763159	+	0.646211i	$0.776353\pi$
$824$	8.04736			0.280343
$825$		0			0
$826$	−15.1297			−0.526431
$827$			5.13161i			0.178444i	0.996012	+	0.0892218i	$0.0284380\pi$
$827$			5.13161i			0.178444i	−0.996012	+	0.0892218i	$0.971562\pi$
$828$		0			0
$829$	−19.9427			−0.692640			−0.346320	−	0.938116i	$-0.612569\pi$
$829$	−19.9427			−0.692640			−0.346320	+	0.938116i	$0.612569\pi$
$830$		−	6.03823i		−	0.209590i
$831$		0			0
$832$			4.06756i			0.141017i
$833$			20.4988i			0.710242i
$834$		0			0
$835$		−	10.7252i		−	0.371162i
$836$	−9.92872			−0.343392
$837$		0			0
$838$		−	15.7784i		−	0.545054i
$839$			7.33000i			0.253060i	0.991963	+	0.126530i	$0.0403840\pi$
$839$			7.33000i			0.253060i	−0.991963	+	0.126530i	$0.959616\pi$
$840$		0			0
$841$	27.8064			0.958841
$842$	26.8544			0.925464
$843$		0			0
$844$	9.93702			0.342046
$845$	3.54506			0.121954
$846$		0			0
$847$		−	10.2231i		−	0.351270i
$848$	−8.27249			−0.284078
$849$		0			0
$850$		−	3.79637i		−	0.130214i
$851$			14.2651i			0.489003i
$852$		0			0
$853$	12.9801			0.444430			0.222215	−	0.974998i	$-0.428671\pi$
$853$	12.9801			0.444430			0.222215	+	0.974998i	$0.428671\pi$
$854$	1.29146			0.0441930
$855$		0			0
$856$		−	8.44792i		−	0.288744i
$857$	44.4137			1.51714			0.758572	−	0.651589i	$-0.225897\pi$
$857$	44.4137			1.51714			0.758572	+	0.651589i	$0.225897\pi$
$858$		0			0
$859$	42.8706			1.46273			0.731364	−	0.681988i	$-0.238884\pi$
$859$	42.8706			1.46273			0.731364	+	0.681988i	$0.238884\pi$
$860$		−	3.64024i		−	0.124131i
$861$		0			0
$862$			12.6248i			0.430001i
$863$			5.40636i			0.184035i	0.995757	+	0.0920174i	$0.0293315\pi$
$863$			5.40636i			0.184035i	−0.995757	+	0.0920174i	$0.970668\pi$
$864$		0			0
$865$			9.92373i			0.337417i
$866$			17.6647i			0.600273i
$867$		0			0
$868$	−11.6623			−0.395845
$869$			15.0627i			0.510966i
$870$		0			0
$871$	−21.5254	+	25.4003i	−0.729359	+	0.860658i
$872$			12.3763i			0.419115i
$873$		0			0
$874$			20.4384i			0.691338i
$875$		−	1.26508i		−	0.0427674i
$876$		0			0
$877$	20.6710			0.698012			0.349006	−	0.937121i	$-0.386519\pi$
$877$	20.6710			0.698012			0.349006	+	0.937121i	$0.386519\pi$
$878$	9.53770			0.321882
$879$		0			0
$880$	−1.70850			−0.0575936
$881$		−	30.0675i		−	1.01300i	−0.862240	−	0.506500i	$-0.830939\pi$
$881$		−	30.0675i		−	1.01300i	0.862240	−	0.506500i	$-0.169061\pi$
$882$		0			0
$883$			9.39649i			0.316217i	0.987422	+	0.158109i	$0.0505396\pi$
$883$			9.39649i			0.316217i	−0.987422	+	0.158109i	$0.949460\pi$
$884$	−15.4420			−0.519370
$885$		0			0
$886$	22.8020			0.766048
$887$			32.9459i			1.10622i	0.833110	+	0.553108i	$0.186558\pi$
$887$			32.9459i			1.10622i	−0.833110	+	0.553108i	$0.813442\pi$
$888$		0			0
$889$			15.0726i			0.505517i
$890$		−	2.09365i		−	0.0701792i
$891$		0			0
$892$	−11.0452			−0.369822
$893$			55.0244i			1.84132i
$894$		0			0
$895$	1.07332			0.0358771
$896$		−	1.26508i		−	0.0422632i
$897$		0			0
$898$		−	10.0894i		−	0.336688i
$899$	−10.0717			−0.335910
$900$		0			0
$901$		−	31.4054i		−	1.04627i
$902$	10.9885			0.365876
$903$		0			0
$904$	−6.79299			−0.225932
$905$	21.5659			0.716873
$906$		0			0
$907$	−6.36804			−0.211447			−0.105724	−	0.994396i	$-0.533716\pi$
$907$	−6.36804			−0.211447			−0.105724	+	0.994396i	$0.533716\pi$
$908$			3.94267i			0.130842i
$909$		0			0
$910$	−5.14578			−0.170581
$911$			27.3561i			0.906347i	0.891422	+	0.453174i	$0.149708\pi$
$911$			27.3561i			0.906347i	−0.891422	+	0.453174i	$0.850292\pi$
$912$		0			0
$913$		−	10.3163i		−	0.341421i
$914$	−7.13178			−0.235898
$915$		0			0
$916$			10.1938i			0.336812i
$917$	19.9817			0.659855
$918$		0			0
$919$		−	10.3178i		−	0.340352i	−0.985414	−	0.170176i	$-0.945566\pi$
$919$		−	10.3178i		−	0.340352i	0.985414	−	0.170176i	$-0.0544336\pi$
$920$			3.51697i			0.115951i
$921$		0			0
$922$		−	9.70607i		−	0.319652i
$923$	28.2950			0.931343
$924$		0			0
$925$	4.05609			0.133363
$926$			24.3701i			0.800852i
$927$		0			0
$928$		−	1.09253i		−	0.0358641i
$929$	−28.0609			−0.920648			−0.460324	−	0.887751i	$-0.652267\pi$
$929$	−28.0609			−0.920648			−0.460324	+	0.887751i	$0.652267\pi$
$930$		0			0
$931$	−31.3789			−1.02840
$932$	−2.12468			−0.0695962
$933$		0			0
$934$			5.58652i			0.182797i
$935$		−	6.48611i		−	0.212118i
$936$		0			0
$937$			46.3777i			1.51509i	0.652780	+	0.757547i	$0.273603\pi$
$937$			46.3777i			1.51509i	−0.652780	+	0.757547i	$0.726397\pi$
$938$	6.69473	−	7.89991i	0.218591	−	0.257941i
$939$		0			0
$940$			9.46841i			0.308826i
$941$	36.7900			1.19932			0.599660	−	0.800255i	$-0.295302\pi$
$941$	36.7900			1.19932			0.599660	+	0.800255i	$0.295302\pi$
$942$		0			0
$943$		−	22.6198i		−	0.736603i
$944$		−	11.9595i		−	0.389250i
$945$		0			0
$946$		−	6.21936i		−	0.202209i
$947$		−	21.2258i		−	0.689745i	−0.938649	−	0.344873i	$-0.887922\pi$
$947$		−	21.2258i		−	0.689745i	0.938649	−	0.344873i	$-0.112078\pi$
$948$		0			0
$949$		−	53.4146i		−	1.73391i
$950$	5.81136			0.188546
$951$		0			0
$952$	4.80270			0.155656
$953$			8.59541i			0.278433i	0.990262	+	0.139216i	$0.0444583\pi$
$953$			8.59541i			0.278433i	−0.990262	+	0.139216i	$0.955542\pi$
$954$		0			0
$955$	3.98820			0.129055
$956$	−9.57621			−0.309717
$957$		0			0
$958$		−	32.3635i		−	1.04562i
$959$			9.73589i			0.314388i
$960$		0			0
$961$	−53.9841			−1.74142
$962$		−	16.4984i		−	0.531930i
$963$		0			0
$964$	−2.29675			−0.0739732
$965$	−12.7661			−0.410955
$966$		0			0
$967$	49.4056			1.58878			0.794389	−	0.607410i	$-0.207791\pi$
$967$	49.4056			1.58878			0.794389	+	0.607410i	$0.207791\pi$
$968$	8.08102			0.259734
$969$		0			0
$970$			7.10610i			0.228163i
$971$		−	34.7931i		−	1.11656i	−0.829652	−	0.558281i	$-0.811461\pi$
$971$		−	34.7931i		−	1.11656i	0.829652	−	0.558281i	$-0.188539\pi$
$972$		0			0
$973$	6.19050			0.198458
$974$			9.18614i			0.294343i
$975$		0			0
$976$			1.02086i			0.0326769i
$977$		−	4.34279i		−	0.138938i	−0.997584	−	0.0694690i	$-0.977869\pi$
$977$		−	4.34279i		−	0.138938i	0.997584	−	0.0694690i	$-0.0221305\pi$
$978$		0			0
$979$		−	3.57700i		−	0.114321i
$980$	−5.39958			−0.172483
$981$		0			0
$982$			10.2189i			0.326099i
$983$	2.97043			0.0947420			0.0473710	−	0.998877i	$-0.484916\pi$
$983$	2.97043			0.0947420			0.0473710	+	0.998877i	$0.484916\pi$
$984$		0			0
$985$	−15.5044			−0.494011
$986$	4.14766			0.132088
$987$		0			0
$988$		−	23.6381i		−	0.752028i
$989$	−12.8026			−0.407099
$990$		0			0
$991$		−	1.08501i		−	0.0344666i	−0.999851	−	0.0172333i	$-0.994514\pi$
$991$		−	1.08501i		−	0.0344666i	0.999851	−	0.0172333i	$-0.00548580\pi$
$992$		−	9.21868i		−	0.292693i
$993$		0			0
$994$	−8.80021			−0.279126
$995$	18.7350			0.593940
$996$		0			0
$997$	−16.9064			−0.535432			−0.267716	−	0.963498i	$-0.586269\pi$
$997$	−16.9064			−0.535432			−0.267716	+	0.963498i	$0.586269\pi$
$998$		−	24.8528i		−	0.786703i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6030.2.d.k.2411.15	yes	24
3.2	odd	2		6030.2.d.l.2411.15	yes	24
67.66	odd	2		6030.2.d.l.2411.10	yes	24
201.200	even	2	inner	6030.2.d.k.2411.10	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6030.2.d.k.2411.10	✓	24	201.200	even	2	inner
6030.2.d.k.2411.15	yes	24	1.1	even	1	trivial
6030.2.d.l.2411.10	yes	24	67.66	odd	2
6030.2.d.l.2411.15	yes	24	3.2	odd	2

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:	\( N \)	\(=\)	\( 6030 = 2 \cdot 3^{2} \cdot 5 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6030.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.26508i q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.26508i q^{7} -1.00000 q^{8} +1.00000 q^{10} +1.70850 q^{11} +4.06756i q^{13} -1.26508i q^{14} +1.00000 q^{16} +3.79637i q^{17} -5.81136 q^{19} -1.00000 q^{20} -1.70850 q^{22} +3.51697i q^{23} +1.00000 q^{25} -4.06756i q^{26} +1.26508i q^{28} +1.09253i q^{29} +9.21868i q^{31} -1.00000 q^{32} -3.79637i q^{34} -1.26508i q^{35} +4.05609 q^{37} +5.81136 q^{38} +1.00000 q^{40} -6.43163 q^{41} +3.64024i q^{43} +1.70850 q^{44} -3.51697i q^{46} -9.46841i q^{47} +5.39958 q^{49} -1.00000 q^{50} +4.06756i q^{52} -8.27249 q^{53} -1.70850 q^{55} -1.26508i q^{56} -1.09253i q^{58} -11.9595i q^{59} +1.02086i q^{61} -9.21868i q^{62} +1.00000 q^{64} -4.06756i q^{65} +(6.24461 + 5.29196i) q^{67} +3.79637i q^{68} +1.26508i q^{70} -6.95627i q^{71} -13.1318 q^{73} -4.05609 q^{74} -5.81136 q^{76} +2.16138i q^{77} +8.81630i q^{79} -1.00000 q^{80} +6.43163 q^{82} -6.03823i q^{83} -3.79637i q^{85} -3.64024i q^{86} -1.70850 q^{88} -2.09365i q^{89} -5.14578 q^{91} +3.51697i q^{92} +9.46841i q^{94} +5.81136 q^{95} +7.10610i q^{97} -5.39958 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 24 q^{2} + 24 q^{4} - 24 q^{5} - 24 q^{8}+O(q^{10}) \) 24 * q - 24 * q^2 + 24 * q^4 - 24 * q^5 - 24 * q^8	\( 24 q - 24 q^{2} + 24 q^{4} - 24 q^{5} - 24 q^{8} + 24 q^{10} - 12 q^{11} + 24 q^{16} + 4 q^{19} - 24 q^{20} + 12 q^{22} + 24 q^{25} - 24 q^{32} - 16 q^{37} - 4 q^{38} + 24 q^{40} - 8 q^{41} - 12 q^{44} - 20 q^{49} - 24 q^{50} - 24 q^{53} + 12 q^{55} + 24 q^{64} - 32 q^{67} - 4 q^{73} + 16 q^{74} + 4 q^{76} - 24 q^{80} + 8 q^{82} + 12 q^{88} - 4 q^{95} + 20 q^{98}+O(q^{100}) \) 24 * q - 24 * q^2 + 24 * q^4 - 24 * q^5 - 24 * q^8 + 24 * q^10 - 12 * q^11 + 24 * q^16 + 4 * q^19 - 24 * q^20 + 12 * q^22 + 24 * q^25 - 24 * q^32 - 16 * q^37 - 4 * q^38 + 24 * q^40 - 8 * q^41 - 12 * q^44 - 20 * q^49 - 24 * q^50 - 24 * q^53 + 12 * q^55 + 24 * q^64 - 32 * q^67 - 4 * q^73 + 16 * q^74 + 4 * q^76 - 24 * q^80 + 8 * q^82 + 12 * q^88 - 4 * q^95 + 20 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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