LMFDB - Embedded newform 6030.2.d.j.2411.14

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:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 36x^{14} + 519x^{12} + 3876x^{10} + 16111x^{8} + 36772x^{6} + 41293x^{4} + 16036x^{2} + 4 $ x^16 + 36x^14 + 519x^12 + 3876x^10 + 16111x^8 + 36772x^6 + 41293x^4 + 16036*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{67}]$
Coefficient ring index:	$ 2^{9}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2411.14
Root			$-1.72684i$ of defining polynomial
Character	$\chi$	$=$	6030.2411
Dual form			6030.2.d.j.2411.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +4.32270i q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +4.32270i q^{7} +1.00000 q^{8} -1.00000 q^{10} +2.34280 q^{11} +1.88618i q^{13} +4.32270i q^{14} +1.00000 q^{16} +4.76961i q^{17} -7.45207 q^{19} -1.00000 q^{20} +2.34280 q^{22} -0.306025i q^{23} +1.00000 q^{25} +1.88618i q^{26} +4.32270i q^{28} -1.02344i q^{29} -3.24681i q^{31} +1.00000 q^{32} +4.76961i q^{34} -4.32270i q^{35} +4.46849 q^{37} -7.45207 q^{38} -1.00000 q^{40} -1.04283 q^{41} +7.21170i q^{43} +2.34280 q^{44} -0.306025i q^{46} +8.58757i q^{47} -11.6857 q^{49} +1.00000 q^{50} +1.88618i q^{52} -0.890987 q^{53} -2.34280 q^{55} +4.32270i q^{56} -1.02344i q^{58} -2.70252i q^{59} -8.48161i q^{61} -3.24681i q^{62} +1.00000 q^{64} -1.88618i q^{65} +(-7.20903 + 3.87683i) q^{67} +4.76961i q^{68} -4.32270i q^{70} -7.06714i q^{71} -8.74524 q^{73} +4.46849 q^{74} -7.45207 q^{76} +10.1272i q^{77} -10.0945i q^{79} -1.00000 q^{80} -1.04283 q^{82} +3.81814i q^{83} -4.76961i q^{85} +7.21170i q^{86} +2.34280 q^{88} +7.38611i q^{89} -8.15341 q^{91} -0.306025i q^{92} +8.58757i q^{94} +7.45207 q^{95} -11.2205i q^{97} -11.6857 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 16 q^{2} + 16 q^{4} - 16 q^{5} + 16 q^{8}+O(q^{10}) $ 16 * q + 16 * q^2 + 16 * q^4 - 16 * q^5 + 16 * q^8	$ 16 q + 16 q^{2} + 16 q^{4} - 16 q^{5} + 16 q^{8} - 16 q^{10} - 20 q^{11} + 16 q^{16} - 8 q^{19} - 16 q^{20} - 20 q^{22} + 16 q^{25} + 16 q^{32} + 32 q^{37} - 8 q^{38} - 16 q^{40} - 8 q^{41} - 20 q^{44} - 88 q^{49} + 16 q^{50} + 8 q^{53} + 20 q^{55} + 16 q^{64} + 4 q^{67} + 16 q^{73} + 32 q^{74} - 8 q^{76} - 16 q^{80} - 8 q^{82} - 20 q^{88} + 40 q^{91} + 8 q^{95} - 88 q^{98}+O(q^{100}) $ 16 * q + 16 * q^2 + 16 * q^4 - 16 * q^5 + 16 * q^8 - 16 * q^10 - 20 * q^11 + 16 * q^16 - 8 * q^19 - 16 * q^20 - 20 * q^22 + 16 * q^25 + 16 * q^32 + 32 * q^37 - 8 * q^38 - 16 * q^40 - 8 * q^41 - 20 * q^44 - 88 * q^49 + 16 * q^50 + 8 * q^53 + 20 * q^55 + 16 * q^64 + 4 * q^67 + 16 * q^73 + 32 * q^74 - 8 * q^76 - 16 * q^80 - 8 * q^82 - 20 * q^88 + 40 * q^91 + 8 * q^95 - 88 * 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
$7$			4.32270i			1.63383i	0.576760	+	0.816914i	$0.304317\pi$
$7$			4.32270i			1.63383i	−0.576760	+	0.816914i	$0.695683\pi$
$8$	1.00000			0.353553
$9$		0			0
$10$	−1.00000			−0.316228
$11$	2.34280			0.706380			0.353190	−	0.935552i	$-0.385097\pi$
$11$	2.34280			0.706380			0.353190	+	0.935552i	$0.385097\pi$
$12$		0			0
$13$			1.88618i			0.523133i	0.965185	+	0.261567i	$0.0842391\pi$
$13$			1.88618i			0.523133i	−0.965185	+	0.261567i	$0.915761\pi$
$14$			4.32270i			1.15529i
$15$		0			0
$16$	1.00000			0.250000
$17$			4.76961i			1.15680i	0.815754	+	0.578400i	$0.196323\pi$
$17$			4.76961i			1.15680i	−0.815754	+	0.578400i	$0.803677\pi$
$18$		0			0
$19$	−7.45207			−1.70962			−0.854811	−	0.518939i	$-0.826327\pi$
$19$	−7.45207			−1.70962			−0.854811	+	0.518939i	$0.826327\pi$
$20$	−1.00000			−0.223607
$21$		0			0
$22$	2.34280			0.499486
$23$		−	0.306025i		−	0.0638107i	−0.999491	−	0.0319053i	$-0.989842\pi$
$23$		−	0.306025i		−	0.0638107i	0.999491	−	0.0319053i	$-0.0101575\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$			1.88618i			0.369911i
$27$		0			0
$28$			4.32270i			0.816914i
$29$		−	1.02344i		−	0.190049i	−0.995475	−	0.0950243i	$-0.969707\pi$
$29$		−	1.02344i		−	0.190049i	0.995475	−	0.0950243i	$-0.0302929\pi$
$30$		0			0
$31$		−	3.24681i		−	0.583144i	−0.956549	−	0.291572i	$-0.905822\pi$
$31$		−	3.24681i		−	0.583144i	0.956549	−	0.291572i	$-0.0941784\pi$
$32$	1.00000			0.176777
$33$		0			0
$34$			4.76961i			0.817981i
$35$		−	4.32270i		−	0.730670i
$36$		0			0
$37$	4.46849			0.734616			0.367308	−	0.930099i	$-0.380280\pi$
$37$	4.46849			0.734616			0.367308	+	0.930099i	$0.380280\pi$
$38$	−7.45207			−1.20889
$39$		0			0
$40$	−1.00000			−0.158114
$41$	−1.04283			−0.162863			−0.0814317	−	0.996679i	$-0.525949\pi$
$41$	−1.04283			−0.162863			−0.0814317	+	0.996679i	$0.525949\pi$
$42$		0			0
$43$			7.21170i			1.09977i	0.835239	+	0.549887i	$0.185329\pi$
$43$			7.21170i			1.09977i	−0.835239	+	0.549887i	$0.814671\pi$
$44$	2.34280			0.353190
$45$		0			0
$46$		−	0.306025i		−	0.0451210i
$47$			8.58757i			1.25263i	0.779572	+	0.626313i	$0.215437\pi$
$47$			8.58757i			1.25263i	−0.779572	+	0.626313i	$0.784563\pi$
$48$		0			0
$49$	−11.6857			−1.66939
$50$	1.00000			0.141421
$51$		0			0
$52$			1.88618i			0.261567i
$53$	−0.890987			−0.122386			−0.0611932	−	0.998126i	$-0.519491\pi$
$53$	−0.890987			−0.122386			−0.0611932	+	0.998126i	$0.519491\pi$
$54$		0			0
$55$	−2.34280			−0.315903
$56$			4.32270i			0.577645i
$57$		0			0
$58$		−	1.02344i		−	0.134385i
$59$		−	2.70252i		−	0.351838i	−0.984405	−	0.175919i	$-0.943710\pi$
$59$		−	2.70252i		−	0.351838i	0.984405	−	0.175919i	$-0.0562897\pi$
$60$		0			0
$61$		−	8.48161i		−	1.08596i	−0.839746	−	0.542979i	$-0.817296\pi$
$61$		−	8.48161i		−	1.08596i	0.839746	−	0.542979i	$-0.182704\pi$
$62$		−	3.24681i		−	0.412345i
$63$		0			0
$64$	1.00000			0.125000
$65$		−	1.88618i		−	0.233952i
$66$		0			0
$67$	−7.20903	+	3.87683i	−0.880724	+	0.473631i
$67$	−7.20903	+	3.87683i	−0.880724	+	0.473631i
$68$			4.76961i			0.578400i
$69$		0			0
$70$		−	4.32270i		−	0.516662i
$71$		−	7.06714i		−	0.838716i	−0.907821	−	0.419358i	$-0.862255\pi$
$71$		−	7.06714i		−	0.838716i	0.907821	−	0.419358i	$-0.137745\pi$
$72$		0			0
$73$	−8.74524			−1.02355			−0.511776	−	0.859119i	$-0.671012\pi$
$73$	−8.74524			−1.02355			−0.511776	+	0.859119i	$0.671012\pi$
$74$	4.46849			0.519452
$75$		0			0
$76$	−7.45207			−0.854811
$77$			10.1272i			1.15410i
$78$		0			0
$79$		−	10.0945i		−	1.13572i	−0.823127	−	0.567858i	$-0.807772\pi$
$79$		−	10.0945i		−	1.13572i	0.823127	−	0.567858i	$-0.192228\pi$
$80$	−1.00000			−0.111803
$81$		0			0
$82$	−1.04283			−0.115162
$83$			3.81814i			0.419096i	0.977798	+	0.209548i	$0.0671992\pi$
$83$			3.81814i			0.419096i	−0.977798	+	0.209548i	$0.932801\pi$
$84$		0			0
$85$		−	4.76961i		−	0.517336i
$86$			7.21170i			0.777658i
$87$		0			0
$88$	2.34280			0.249743
$89$			7.38611i			0.782927i	0.920194	+	0.391463i	$0.128031\pi$
$89$			7.38611i			0.782927i	−0.920194	+	0.391463i	$0.871969\pi$
$90$		0			0
$91$	−8.15341			−0.854709
$92$		−	0.306025i		−	0.0319053i
$93$		0			0
$94$			8.58757i			0.885741i
$95$	7.45207			0.764566
$96$		0			0
$97$		−	11.2205i		−	1.13927i	−0.821898	−	0.569634i	$-0.807085\pi$
$97$		−	11.2205i		−	1.13927i	0.821898	−	0.569634i	$-0.192915\pi$
$98$	−11.6857			−1.18044
$99$		0			0
$100$	1.00000			0.100000
$101$	9.06607			0.902108			0.451054	−	0.892497i	$-0.351048\pi$
$101$	9.06607			0.902108			0.451054	+	0.892497i	$0.351048\pi$
$102$		0			0
$103$	−13.7126			−1.35114			−0.675572	−	0.737294i	$-0.736103\pi$
$103$	−13.7126			−1.35114			−0.675572	+	0.737294i	$0.736103\pi$
$104$			1.88618i			0.184956i
$105$		0			0
$106$	−0.890987			−0.0865403
$107$			4.47918i			0.433019i	0.976281	+	0.216509i	$0.0694672\pi$
$107$			4.47918i			0.433019i	−0.976281	+	0.216509i	$0.930533\pi$
$108$		0			0
$109$			8.94539i			0.856813i	0.903586	+	0.428406i	$0.140925\pi$
$109$			8.94539i			0.856813i	−0.903586	+	0.428406i	$0.859075\pi$
$110$	−2.34280			−0.223377
$111$		0			0
$112$			4.32270i			0.408457i
$113$	−17.2592			−1.62361			−0.811803	−	0.583931i	$-0.801514\pi$
$113$	−17.2592			−1.62361			−0.811803	+	0.583931i	$0.801514\pi$
$114$		0			0
$115$			0.306025i			0.0285370i
$116$		−	1.02344i		−	0.0950243i
$117$		0			0
$118$		−	2.70252i		−	0.248787i
$119$	−20.6176			−1.89001
$120$		0			0
$121$	−5.51130			−0.501027
$122$		−	8.48161i		−	0.767889i
$123$		0			0
$124$		−	3.24681i		−	0.291572i
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	11.5888			1.02834			0.514169	−	0.857689i	$-0.328100\pi$
$127$	11.5888			1.02834			0.514169	+	0.857689i	$0.328100\pi$
$128$	1.00000			0.0883883
$129$		0			0
$130$		−	1.88618i		−	0.165429i
$131$		−	16.2521i		−	1.41996i	−0.704223	−	0.709978i	$-0.748705\pi$
$131$		−	16.2521i		−	1.41996i	0.704223	−	0.709978i	$-0.251295\pi$
$132$		0			0
$133$		−	32.2131i		−	2.79323i
$134$	−7.20903	+	3.87683i	−0.622766	+	0.334907i
$135$		0			0
$136$			4.76961i			0.408990i
$137$	7.20634			0.615679			0.307840	−	0.951438i	$-0.400394\pi$
$137$	7.20634			0.615679			0.307840	+	0.951438i	$0.400394\pi$
$138$		0			0
$139$		−	4.80943i		−	0.407930i	−0.978978	−	0.203965i	$-0.934617\pi$
$139$		−	4.80943i		−	0.407930i	0.978978	−	0.203965i	$-0.0653829\pi$
$140$		−	4.32270i		−	0.365335i
$141$		0			0
$142$		−	7.06714i		−	0.593061i
$143$			4.41895i			0.369531i
$144$		0			0
$145$			1.02344i			0.0849924i
$146$	−8.74524			−0.723761
$147$		0			0
$148$	4.46849			0.367308
$149$			12.8615i			1.05365i	0.849973	+	0.526827i	$0.176618\pi$
$149$			12.8615i			1.05365i	−0.849973	+	0.526827i	$0.823382\pi$
$150$		0			0
$151$	−6.26170			−0.509570			−0.254785	−	0.966998i	$-0.582005\pi$
$151$	−6.26170			−0.509570			−0.254785	+	0.966998i	$0.582005\pi$
$152$	−7.45207			−0.604443
$153$		0			0
$154$			10.1272i			0.816074i
$155$			3.24681i			0.260790i
$156$		0			0
$157$	−10.8380			−0.864965			−0.432482	−	0.901642i	$-0.642362\pi$
$157$	−10.8380			−0.864965			−0.432482	+	0.901642i	$0.642362\pi$
$158$		−	10.0945i		−	0.803072i
$159$		0			0
$160$	−1.00000			−0.0790569
$161$	1.32286			0.104256
$162$		0			0
$163$	2.78197			0.217901			0.108950	−	0.994047i	$-0.465251\pi$
$163$	2.78197			0.217901			0.108950	+	0.994047i	$0.465251\pi$
$164$	−1.04283			−0.0814317
$165$		0			0
$166$			3.81814i			0.296345i
$167$			8.88154i			0.687274i	0.939103	+	0.343637i	$0.111659\pi$
$167$			8.88154i			0.687274i	−0.939103	+	0.343637i	$0.888341\pi$
$168$		0			0
$169$	9.44231			0.726332
$170$		−	4.76961i		−	0.365812i
$171$		0			0
$172$			7.21170i			0.549887i
$173$			15.4549i			1.17502i	0.809218	+	0.587509i	$0.199891\pi$
$173$			15.4549i			1.17502i	−0.809218	+	0.587509i	$0.800109\pi$
$174$		0			0
$175$			4.32270i			0.326765i
$176$	2.34280			0.176595
$177$		0			0
$178$			7.38611i			0.553613i
$179$	−4.25504			−0.318036			−0.159018	−	0.987276i	$-0.550833\pi$
$179$	−4.25504			−0.318036			−0.159018	+	0.987276i	$0.550833\pi$
$180$		0			0
$181$	14.3515			1.06674			0.533370	−	0.845882i	$-0.320926\pi$
$181$	14.3515			1.06674			0.533370	+	0.845882i	$0.320926\pi$
$182$	−8.15341			−0.604371
$183$		0			0
$184$		−	0.306025i		−	0.0225605i
$185$	−4.46849			−0.328530
$186$		0			0
$187$			11.1742i			0.817140i
$188$			8.58757i			0.626313i
$189$		0			0
$190$	7.45207			0.540630
$191$	1.08212			0.0782998			0.0391499	−	0.999233i	$-0.487535\pi$
$191$	1.08212			0.0782998			0.0391499	+	0.999233i	$0.487535\pi$
$192$		0			0
$193$	22.9265			1.65029			0.825143	−	0.564924i	$-0.191095\pi$
$193$	22.9265			1.65029			0.825143	+	0.564924i	$0.191095\pi$
$194$		−	11.2205i		−	0.805585i
$195$		0			0
$196$	−11.6857			−0.834696
$197$	−5.68166			−0.404801			−0.202401	−	0.979303i	$-0.564874\pi$
$197$	−5.68166			−0.404801			−0.202401	+	0.979303i	$0.564874\pi$
$198$		0			0
$199$	0.952599			0.0675280			0.0337640	−	0.999430i	$-0.489251\pi$
$199$	0.952599			0.0675280			0.0337640	+	0.999430i	$0.489251\pi$
$200$	1.00000			0.0707107
$201$		0			0
$202$	9.06607			0.637886
$203$	4.42404			0.310507
$204$		0			0
$205$	1.04283			0.0728347
$206$	−13.7126			−0.955404
$207$		0			0
$208$			1.88618i			0.130783i
$209$	−17.4587			−1.20764
$210$		0			0
$211$	−8.72943			−0.600959			−0.300480	−	0.953788i	$-0.597147\pi$
$211$	−8.72943			−0.600959			−0.300480	+	0.953788i	$0.597147\pi$
$212$	−0.890987			−0.0611932
$213$		0			0
$214$			4.47918i			0.306191i
$215$		−	7.21170i		−	0.491834i
$216$		0			0
$217$	14.0350			0.952757
$218$			8.94539i			0.605858i
$219$		0			0
$220$	−2.34280			−0.157951
$221$	−8.99635			−0.605160
$222$		0			0
$223$	8.04774			0.538917			0.269458	−	0.963012i	$-0.413155\pi$
$223$	8.04774			0.538917			0.269458	+	0.963012i	$0.413155\pi$
$224$			4.32270i			0.288823i
$225$		0			0
$226$	−17.2592			−1.14806
$227$			21.1569i			1.40423i	0.712064	+	0.702115i	$0.247761\pi$
$227$			21.1569i			1.40423i	−0.712064	+	0.702115i	$0.752239\pi$
$228$		0			0
$229$		−	2.10421i		−	0.139050i	−0.997580	−	0.0695250i	$-0.977852\pi$
$229$		−	2.10421i		−	0.139050i	0.997580	−	0.0695250i	$-0.0221484\pi$
$230$			0.306025i			0.0201787i
$231$		0			0
$232$		−	1.02344i		−	0.0671924i
$233$	−9.63658			−0.631313			−0.315657	−	0.948873i	$-0.602225\pi$
$233$	−9.63658			−0.631313			−0.315657	+	0.948873i	$0.602225\pi$
$234$		0			0
$235$		−	8.58757i		−	0.560192i
$236$		−	2.70252i		−	0.175919i
$237$		0			0
$238$	−20.6176			−1.33644
$239$	−1.59177			−0.102963			−0.0514814	−	0.998674i	$-0.516394\pi$
$239$	−1.59177			−0.102963			−0.0514814	+	0.998674i	$0.516394\pi$
$240$		0			0
$241$	−27.4018			−1.76511			−0.882554	−	0.470211i	$-0.844178\pi$
$241$	−27.4018			−1.76511			−0.882554	+	0.470211i	$0.844178\pi$
$242$	−5.51130			−0.354280
$243$		0			0
$244$		−	8.48161i		−	0.542979i
$245$	11.6857			0.746575
$246$		0			0
$247$		−	14.0560i		−	0.894360i
$248$		−	3.24681i		−	0.206173i
$249$		0			0
$250$	−1.00000			−0.0632456
$251$	−9.42979			−0.595203			−0.297601	−	0.954690i	$-0.596187\pi$
$251$	−9.42979			−0.595203			−0.297601	+	0.954690i	$0.596187\pi$
$252$		0			0
$253$		−	0.716955i		−	0.0450746i
$254$	11.5888			0.727145
$255$		0			0
$256$	1.00000			0.0625000
$257$			14.1197i			0.880761i	0.897811	+	0.440381i	$0.145156\pi$
$257$			14.1197i			0.880761i	−0.897811	+	0.440381i	$0.854844\pi$
$258$		0			0
$259$			19.3160i			1.20024i
$260$		−	1.88618i		−	0.116976i
$261$		0			0
$262$		−	16.2521i		−	1.00406i
$263$			14.8618i			0.916419i	0.888844	+	0.458210i	$0.151509\pi$
$263$			14.8618i			0.916419i	−0.888844	+	0.458210i	$0.848491\pi$
$264$		0			0
$265$	0.890987			0.0547329
$266$		−	32.2131i		−	1.97511i
$267$		0			0
$268$	−7.20903	+	3.87683i	−0.440362	+	0.236815i
$269$			3.47163i			0.211669i	0.994384	+	0.105834i	$0.0337513\pi$
$269$			3.47163i			0.211669i	−0.994384	+	0.105834i	$0.966249\pi$
$270$		0			0
$271$			17.2172i			1.04587i	0.852373	+	0.522935i	$0.175163\pi$
$271$			17.2172i			1.04587i	−0.852373	+	0.522935i	$0.824837\pi$
$272$			4.76961i			0.289200i
$273$		0			0
$274$	7.20634			0.435351
$275$	2.34280			0.141276
$276$		0			0
$277$	14.4541			0.868465			0.434232	−	0.900801i	$-0.357020\pi$
$277$	14.4541			0.868465			0.434232	+	0.900801i	$0.357020\pi$
$278$		−	4.80943i		−	0.288450i
$279$		0			0
$280$		−	4.32270i		−	0.258331i
$281$	7.19074			0.428964			0.214482	−	0.976728i	$-0.431194\pi$
$281$	7.19074			0.428964			0.214482	+	0.976728i	$0.431194\pi$
$282$		0			0
$283$	−11.4906			−0.683046			−0.341523	−	0.939873i	$-0.610943\pi$
$283$	−11.4906			−0.683046			−0.341523	+	0.939873i	$0.610943\pi$
$284$		−	7.06714i		−	0.419358i
$285$		0			0
$286$			4.41895i			0.261298i
$287$		−	4.50786i		−	0.266091i
$288$		0			0
$289$	−5.74913			−0.338184
$290$			1.02344i			0.0600987i
$291$		0			0
$292$	−8.74524			−0.511776
$293$			14.7402i			0.861134i	0.902559	+	0.430567i	$0.141686\pi$
$293$			14.7402i			0.861134i	−0.902559	+	0.430567i	$0.858314\pi$
$294$		0			0
$295$			2.70252i			0.157347i
$296$	4.46849			0.259726
$297$		0			0
$298$			12.8615i			0.745046i
$299$	0.577220			0.0333815
$300$		0			0
$301$	−31.1740			−1.79684
$302$	−6.26170			−0.360320
$303$		0			0
$304$	−7.45207			−0.427406
$305$			8.48161i			0.485656i
$306$		0			0
$307$	−22.7483			−1.29832			−0.649158	−	0.760653i	$-0.724879\pi$
$307$	−22.7483			−1.29832			−0.649158	+	0.760653i	$0.724879\pi$
$308$			10.1272i			0.577051i
$309$		0			0
$310$			3.24681i			0.184406i
$311$	−11.9661			−0.678538			−0.339269	−	0.940689i	$-0.610180\pi$
$311$	−11.9661			−0.678538			−0.339269	+	0.940689i	$0.610180\pi$
$312$		0			0
$313$		−	12.8024i		−	0.723632i	−0.932249	−	0.361816i	$-0.882157\pi$
$313$		−	12.8024i		−	0.723632i	0.932249	−	0.361816i	$-0.117843\pi$
$314$	−10.8380			−0.611622
$315$		0			0
$316$		−	10.0945i		−	0.567858i
$317$			16.1832i			0.908940i	0.890762	+	0.454470i	$0.150171\pi$
$317$			16.1832i			0.908940i	−0.890762	+	0.454470i	$0.849829\pi$
$318$		0			0
$319$		−	2.39772i		−	0.134247i
$320$	−1.00000			−0.0559017
$321$		0			0
$322$	1.32286			0.0737199
$323$		−	35.5434i		−	1.97769i
$324$		0			0
$325$			1.88618i			0.104627i
$326$	2.78197			0.154079
$327$		0			0
$328$	−1.04283			−0.0575809
$329$	−37.1215			−2.04658
$330$		0			0
$331$		−	2.54548i		−	0.139912i	−0.997550	−	0.0699561i	$-0.977714\pi$
$331$		−	2.54548i		−	0.139912i	0.997550	−	0.0699561i	$-0.0222859\pi$
$332$			3.81814i			0.209548i
$333$		0			0
$334$			8.88154i			0.485976i
$335$	7.20903	−	3.87683i	0.393872	−	0.211814i
$336$		0			0
$337$			16.2938i			0.887579i	0.896131	+	0.443789i	$0.146366\pi$
$337$			16.2938i			0.887579i	−0.896131	+	0.443789i	$0.853634\pi$
$338$	9.44231			0.513594
$339$		0			0
$340$		−	4.76961i		−	0.258668i
$341$		−	7.60662i		−	0.411922i
$342$		0			0
$343$		−	20.2551i		−	1.09367i
$344$			7.21170i			0.388829i
$345$		0			0
$346$			15.4549i			0.830863i
$347$	−15.5444			−0.834468			−0.417234	−	0.908799i	$-0.637000\pi$
$347$	−15.5444			−0.834468			−0.417234	+	0.908799i	$0.637000\pi$
$348$		0			0
$349$	−4.94933			−0.264931			−0.132466	−	0.991188i	$-0.542289\pi$
$349$	−4.94933			−0.264931			−0.132466	+	0.991188i	$0.542289\pi$
$350$			4.32270i			0.231058i
$351$		0			0
$352$	2.34280			0.124872
$353$	−4.05153			−0.215641			−0.107820	−	0.994170i	$-0.534387\pi$
$353$	−4.05153			−0.215641			−0.107820	+	0.994170i	$0.534387\pi$
$354$		0			0
$355$			7.06714i			0.375085i
$356$			7.38611i			0.391463i
$357$		0			0
$358$	−4.25504			−0.224886
$359$		−	0.936911i		−	0.0494482i	−0.999694	−	0.0247241i	$-0.992129\pi$
$359$		−	0.936911i		−	0.0494482i	0.999694	−	0.0247241i	$-0.00787074\pi$
$360$		0			0
$361$	36.5333			1.92281
$362$	14.3515			0.754298
$363$		0			0
$364$	−8.15341			−0.427355
$365$	8.74524			0.457747
$366$		0			0
$367$		−	19.7243i		−	1.02960i	−0.857310	−	0.514800i	$-0.827866\pi$
$367$		−	19.7243i		−	1.02960i	0.857310	−	0.514800i	$-0.172134\pi$
$368$		−	0.306025i		−	0.0159527i
$369$		0			0
$370$	−4.46849			−0.232306
$371$		−	3.85147i		−	0.199958i
$372$		0			0
$373$			25.9100i			1.34157i	0.741652	+	0.670785i	$0.234043\pi$
$373$			25.9100i			1.34157i	−0.741652	+	0.670785i	$0.765957\pi$
$374$			11.1742i			0.577805i
$375$		0			0
$376$			8.58757i			0.442870i
$377$	1.93040			0.0994208
$378$		0			0
$379$			27.1419i			1.39418i	0.716982	+	0.697092i	$0.245523\pi$
$379$			27.1419i			1.39418i	−0.716982	+	0.697092i	$0.754477\pi$
$380$	7.45207			0.382283
$381$		0			0
$382$	1.08212			0.0553663
$383$	19.0306			0.972420			0.486210	−	0.873842i	$-0.338379\pi$
$383$	19.0306			0.972420			0.486210	+	0.873842i	$0.338379\pi$
$384$		0			0
$385$		−	10.1272i		−	0.516131i
$386$	22.9265			1.16693
$387$		0			0
$388$		−	11.2205i		−	0.569634i
$389$			7.72549i			0.391698i	0.980634	+	0.195849i	$0.0627462\pi$
$389$			7.72549i			0.391698i	−0.980634	+	0.195849i	$0.937254\pi$
$390$		0			0
$391$	1.45962			0.0738162
$392$	−11.6857			−0.590219
$393$		0			0
$394$	−5.68166			−0.286238
$395$			10.0945i			0.507907i
$396$		0			0
$397$	−21.5728			−1.08271			−0.541354	−	0.840795i	$-0.682088\pi$
$397$	−21.5728			−1.08271			−0.541354	+	0.840795i	$0.682088\pi$
$398$	0.952599			0.0477495
$399$		0			0
$400$	1.00000			0.0500000
$401$	12.9657			0.647474			0.323737	−	0.946147i	$-0.395061\pi$
$401$	12.9657			0.647474			0.323737	+	0.946147i	$0.395061\pi$
$402$		0			0
$403$	6.12408			0.305062
$404$	9.06607			0.451054
$405$		0			0
$406$	4.42404			0.219561
$407$	10.4688			0.518918
$408$		0			0
$409$		−	14.1103i		−	0.697708i	−0.937177	−	0.348854i	$-0.886571\pi$
$409$		−	14.1103i		−	0.697708i	0.937177	−	0.348854i	$-0.113429\pi$
$410$	1.04283			0.0515019
$411$		0			0
$412$	−13.7126			−0.675572
$413$	11.6822			0.574842
$414$		0			0
$415$		−	3.81814i		−	0.187425i
$416$			1.88618i			0.0924778i
$417$		0			0
$418$	−17.4587			−0.853932
$419$		−	5.13468i		−	0.250846i	−0.992103	−	0.125423i	$-0.959971\pi$
$419$		−	5.13468i		−	0.250846i	0.992103	−	0.125423i	$-0.0400287\pi$
$420$		0			0
$421$	30.3752			1.48039			0.740197	−	0.672390i	$-0.234732\pi$
$421$	30.3752			1.48039			0.740197	+	0.672390i	$0.234732\pi$
$422$	−8.72943			−0.424942
$423$		0			0
$424$	−0.890987			−0.0432701
$425$			4.76961i			0.231360i
$426$		0			0
$427$	36.6635			1.77427
$428$			4.47918i			0.216509i
$429$		0			0
$430$		−	7.21170i		−	0.347779i
$431$		−	29.9389i		−	1.44210i	−0.692880	−	0.721052i	$-0.743659\pi$
$431$		−	29.9389i		−	1.44210i	0.692880	−	0.721052i	$-0.256341\pi$
$432$		0			0
$433$			2.16181i			0.103890i	0.998650	+	0.0519451i	$0.0165421\pi$
$433$			2.16181i			0.103890i	−0.998650	+	0.0519451i	$0.983458\pi$
$434$	14.0350			0.673701
$435$		0			0
$436$			8.94539i			0.428406i
$437$			2.28052i			0.109092i
$438$		0			0
$439$	−27.5160			−1.31327			−0.656635	−	0.754209i	$-0.728021\pi$
$439$	−27.5160			−1.31327			−0.656635	+	0.754209i	$0.728021\pi$
$440$	−2.34280			−0.111688
$441$		0			0
$442$	−8.99635			−0.427913
$443$	24.7781			1.17724			0.588621	−	0.808409i	$-0.299671\pi$
$443$	24.7781			1.17724			0.588621	+	0.808409i	$0.299671\pi$
$444$		0			0
$445$		−	7.38611i		−	0.350135i
$446$	8.04774			0.381072
$447$		0			0
$448$			4.32270i			0.204228i
$449$			8.62480i			0.407029i	0.979072	+	0.203515i	$0.0652365\pi$
$449$			8.62480i			0.407029i	−0.979072	+	0.203515i	$0.934764\pi$
$450$		0			0
$451$	−2.44315			−0.115043
$452$	−17.2592			−0.811803
$453$		0			0
$454$			21.1569i			0.992941i
$455$	8.15341			0.382238
$456$		0			0
$457$	12.6443			0.591475			0.295737	−	0.955269i	$-0.404435\pi$
$457$	12.6443			0.591475			0.295737	+	0.955269i	$0.404435\pi$
$458$		−	2.10421i		−	0.0983232i
$459$		0			0
$460$			0.306025i			0.0142685i
$461$		−	7.48976i		−	0.348833i	−0.984672	−	0.174416i	$-0.944196\pi$
$461$		−	7.48976i		−	0.348833i	0.984672	−	0.174416i	$-0.0558039\pi$
$462$		0			0
$463$		−	15.4778i		−	0.719316i	−0.933084	−	0.359658i	$-0.882893\pi$
$463$		−	15.4778i		−	0.719316i	0.933084	−	0.359658i	$-0.117107\pi$
$464$		−	1.02344i		−	0.0475122i
$465$		0			0
$466$	−9.63658			−0.446406
$467$			3.87833i			0.179468i	0.995966	+	0.0897339i	$0.0286017\pi$
$467$			3.87833i			0.179468i	−0.995966	+	0.0897339i	$0.971398\pi$
$468$		0			0
$469$	−16.7584	−	31.1625i	−0.773831	−	1.43895i
$470$		−	8.58757i		−	0.396115i
$471$		0			0
$472$		−	2.70252i		−	0.124394i
$473$			16.8956i			0.776858i
$474$		0			0
$475$	−7.45207			−0.341924
$476$	−20.6176			−0.945005
$477$		0			0
$478$	−1.59177			−0.0728057
$479$			12.3695i			0.565176i	0.959241	+	0.282588i	$0.0911929\pi$
$479$			12.3695i			0.565176i	−0.959241	+	0.282588i	$0.908807\pi$
$480$		0			0
$481$			8.42840i			0.384302i
$482$	−27.4018			−1.24812
$483$		0			0
$484$	−5.51130			−0.250514
$485$			11.2205i			0.509496i
$486$		0			0
$487$		−	40.7837i		−	1.84809i	−0.382288	−	0.924043i	$-0.624864\pi$
$487$		−	40.7837i		−	1.84809i	0.382288	−	0.924043i	$-0.375136\pi$
$488$		−	8.48161i		−	0.383944i
$489$		0			0
$490$	11.6857			0.527908
$491$			37.0202i			1.67070i	0.549719	+	0.835350i	$0.314735\pi$
$491$			37.0202i			1.67070i	−0.549719	+	0.835350i	$0.685265\pi$
$492$		0			0
$493$	4.88142			0.219848
$494$		−	14.0560i		−	0.632408i
$495$		0			0
$496$		−	3.24681i		−	0.145786i
$497$	30.5491			1.37032
$498$		0			0
$499$		−	23.9267i		−	1.07111i	−0.844501	−	0.535553i	$-0.820103\pi$
$499$		−	23.9267i		−	1.07111i	0.844501	−	0.535553i	$-0.179897\pi$
$500$	−1.00000			−0.0447214
$501$		0			0
$502$	−9.42979			−0.420872
$503$	20.8861			0.931266			0.465633	−	0.884978i	$-0.345827\pi$
$503$	20.8861			0.931266			0.465633	+	0.884978i	$0.345827\pi$
$504$		0			0
$505$	−9.06607			−0.403435
$506$		−	0.716955i		−	0.0318726i
$507$		0			0
$508$	11.5888			0.514169
$509$			18.1910i			0.806303i	0.915133	+	0.403152i	$0.132085\pi$
$509$			18.1910i			0.806303i	−0.915133	+	0.403152i	$0.867915\pi$
$510$		0			0
$511$		−	37.8031i		−	1.67231i
$512$	1.00000			0.0441942
$513$		0			0
$514$			14.1197i			0.622792i
$515$	13.7126			0.604250
$516$		0			0
$517$			20.1189i			0.884830i
$518$			19.3160i			0.848694i
$519$		0			0
$520$		−	1.88618i		−	0.0827146i
$521$	14.2382			0.623787			0.311893	−	0.950117i	$-0.399037\pi$
$521$	14.2382			0.623787			0.311893	+	0.950117i	$0.399037\pi$
$522$		0			0
$523$	−5.77717			−0.252618			−0.126309	−	0.991991i	$-0.540313\pi$
$523$	−5.77717			−0.252618			−0.126309	+	0.991991i	$0.540313\pi$
$524$		−	16.2521i		−	0.709978i
$525$		0			0
$526$			14.8618i			0.648006i
$527$	15.4860			0.674581
$528$		0			0
$529$	22.9063			0.995928
$530$	0.890987			0.0387020
$531$		0			0
$532$		−	32.2131i		−	1.39661i
$533$		−	1.96698i		−	0.0851993i
$534$		0			0
$535$		−	4.47918i		−	0.193652i
$536$	−7.20903	+	3.87683i	−0.311383	+	0.167454i
$537$		0			0
$538$			3.47163i			0.149672i
$539$	−27.3773			−1.17922
$540$		0			0
$541$			38.0356i			1.63528i	0.575730	+	0.817640i	$0.304718\pi$
$541$			38.0356i			1.63528i	−0.575730	+	0.817640i	$0.695282\pi$
$542$			17.2172i			0.739541i
$543$		0			0
$544$			4.76961i			0.204495i
$545$		−	8.94539i		−	0.383178i
$546$		0			0
$547$		−	29.8854i		−	1.27781i	−0.769287	−	0.638904i	$-0.779388\pi$
$547$		−	29.8854i		−	1.27781i	0.769287	−	0.638904i	$-0.220612\pi$
$548$	7.20634			0.307840
$549$		0			0
$550$	2.34280			0.0998972
$551$			7.62677i			0.324911i
$552$		0			0
$553$	43.6353			1.85556
$554$	14.4541			0.614097
$555$		0			0
$556$		−	4.80943i		−	0.203965i
$557$			30.2994i			1.28383i	0.766777	+	0.641913i	$0.221859\pi$
$557$			30.2994i			1.28383i	−0.766777	+	0.641913i	$0.778141\pi$
$558$		0			0
$559$	−13.6026			−0.575328
$560$		−	4.32270i		−	0.182667i
$561$		0			0
$562$	7.19074			0.303323
$563$	15.2563			0.642975			0.321488	−	0.946914i	$-0.395817\pi$
$563$	15.2563			0.642975			0.321488	+	0.946914i	$0.395817\pi$
$564$		0			0
$565$	17.2592			0.726099
$566$	−11.4906			−0.482986
$567$		0			0
$568$		−	7.06714i		−	0.296531i
$569$		−	15.6483i		−	0.656010i	−0.944676	−	0.328005i	$-0.893624\pi$
$569$		−	15.6483i		−	0.656010i	0.944676	−	0.328005i	$-0.106376\pi$
$570$		0			0
$571$	40.8813			1.71083			0.855415	−	0.517943i	$-0.173302\pi$
$571$	40.8813			1.71083			0.855415	+	0.517943i	$0.173302\pi$
$572$			4.41895i			0.184765i
$573$		0			0
$574$		−	4.50786i		−	0.188155i
$575$		−	0.306025i		−	0.0127621i
$576$		0			0
$577$			1.45710i			0.0606600i	0.999540	+	0.0303300i	$0.00965582\pi$
$577$			1.45710i			0.0606600i	−0.999540	+	0.0303300i	$0.990344\pi$
$578$	−5.74913			−0.239132
$579$		0			0
$580$			1.02344i			0.0424962i
$581$	−16.5047			−0.684730
$582$		0			0
$583$	−2.08740			−0.0864513
$584$	−8.74524			−0.361881
$585$		0			0
$586$			14.7402i			0.608914i
$587$	42.5020			1.75425			0.877123	−	0.480265i	$-0.159460\pi$
$587$	42.5020			1.75425			0.877123	+	0.480265i	$0.159460\pi$
$588$		0			0
$589$			24.1955i			0.996957i
$590$			2.70252i			0.111261i
$591$		0			0
$592$	4.46849			0.183654
$593$	23.5614			0.967549			0.483774	−	0.875193i	$-0.339266\pi$
$593$	23.5614			0.967549			0.483774	+	0.875193i	$0.339266\pi$
$594$		0			0
$595$	20.6176			0.845238
$596$			12.8615i			0.526827i
$597$		0			0
$598$	0.577220			0.0236043
$599$	25.1096			1.02595			0.512975	−	0.858404i	$-0.328543\pi$
$599$	25.1096			1.02595			0.512975	+	0.858404i	$0.328543\pi$
$600$		0			0
$601$	2.54590			0.103849			0.0519247	−	0.998651i	$-0.483464\pi$
$601$	2.54590			0.103849			0.0519247	+	0.998651i	$0.483464\pi$
$602$	−31.1740			−1.27056
$603$		0			0
$604$	−6.26170			−0.254785
$605$	5.51130			0.224066
$606$		0			0
$607$	−33.2700			−1.35039			−0.675195	−	0.737639i	$-0.735940\pi$
$607$	−33.2700			−1.35039			−0.675195	+	0.737639i	$0.735940\pi$
$608$	−7.45207			−0.302221
$609$		0			0
$610$			8.48161i			0.343410i
$611$	−16.1977			−0.655291
$612$		0			0
$613$	−8.84869			−0.357395			−0.178698	−	0.983904i	$-0.557188\pi$
$613$	−8.84869			−0.357395			−0.178698	+	0.983904i	$0.557188\pi$
$614$	−22.7483			−0.918049
$615$		0			0
$616$			10.1272i			0.408037i
$617$			16.0919i			0.647836i	0.946085	+	0.323918i	$0.105000\pi$
$617$			16.0919i			0.647836i	−0.946085	+	0.323918i	$0.895000\pi$
$618$		0			0
$619$	28.9494			1.16358			0.581788	−	0.813341i	$-0.302354\pi$
$619$	28.9494			1.16358			0.581788	+	0.813341i	$0.302354\pi$
$620$			3.24681i			0.130395i
$621$		0			0
$622$	−11.9661			−0.479799
$623$	−31.9280			−1.27917
$624$		0			0
$625$	1.00000			0.0400000
$626$		−	12.8024i		−	0.511685i
$627$		0			0
$628$	−10.8380			−0.432482
$629$			21.3129i			0.849803i
$630$		0			0
$631$			15.5328i			0.618351i	0.951005	+	0.309176i	$0.100053\pi$
$631$			15.5328i			0.618351i	−0.951005	+	0.309176i	$0.899947\pi$
$632$		−	10.0945i		−	0.401536i
$633$		0			0
$634$			16.1832i			0.642718i
$635$	−11.5888			−0.459887
$636$		0			0
$637$		−	22.0415i		−	0.873314i
$638$		−	2.39772i		−	0.0949267i
$639$		0			0
$640$	−1.00000			−0.0395285
$641$	47.7374			1.88551			0.942756	−	0.333482i	$-0.108224\pi$
$641$	47.7374			1.88551			0.942756	+	0.333482i	$0.108224\pi$
$642$		0			0
$643$	29.7601			1.17363			0.586813	−	0.809723i	$-0.300383\pi$
$643$	29.7601			1.17363			0.586813	+	0.809723i	$0.300383\pi$
$644$	1.32286			0.0521278
$645$		0			0
$646$		−	35.5434i		−	1.39844i
$647$	−14.4065			−0.566378			−0.283189	−	0.959064i	$-0.591392\pi$
$647$	−14.4065			−0.566378			−0.283189	+	0.959064i	$0.591392\pi$
$648$		0			0
$649$		−	6.33145i		−	0.248531i
$650$			1.88618i			0.0739822i
$651$		0			0
$652$	2.78197			0.108950
$653$	3.98162			0.155813			0.0779063	−	0.996961i	$-0.475176\pi$
$653$	3.98162			0.155813			0.0779063	+	0.996961i	$0.475176\pi$
$654$		0			0
$655$			16.2521i			0.635024i
$656$	−1.04283			−0.0407159
$657$		0			0
$658$	−37.1215			−1.44715
$659$		−	38.5891i		−	1.50322i	−0.659609	−	0.751609i	$-0.729278\pi$
$659$		−	38.5891i		−	1.50322i	0.659609	−	0.751609i	$-0.270722\pi$
$660$		0			0
$661$		−	0.211003i		−	0.00820708i	−0.999992	−	0.00410354i	$-0.998694\pi$
$661$		−	0.211003i		−	0.00820708i	0.999992	−	0.00410354i	$-0.00130620\pi$
$662$		−	2.54548i		−	0.0989329i
$663$		0			0
$664$			3.81814i			0.148173i
$665$			32.2131i			1.24917i
$666$		0			0
$667$	−0.313200			−0.0121271
$668$			8.88154i			0.343637i
$669$		0			0
$670$	7.20903	−	3.87683i	0.278509	−	0.149775i
$671$		−	19.8707i		−	0.767099i
$672$		0			0
$673$		−	26.2921i		−	1.01349i	−0.862097	−	0.506744i	$-0.830849\pi$
$673$		−	26.2921i		−	1.01349i	0.862097	−	0.506744i	$-0.169151\pi$
$674$			16.2938i			0.627613i
$675$		0			0
$676$	9.44231			0.363166
$677$	2.01759			0.0775422			0.0387711	−	0.999248i	$-0.487656\pi$
$677$	2.01759			0.0775422			0.0387711	+	0.999248i	$0.487656\pi$
$678$		0			0
$679$	48.5028			1.86137
$680$		−	4.76961i		−	0.182906i
$681$		0			0
$682$		−	7.60662i		−	0.291273i
$683$	−20.4030			−0.780701			−0.390350	−	0.920666i	$-0.627646\pi$
$683$	−20.4030			−0.780701			−0.390350	+	0.920666i	$0.627646\pi$
$684$		0			0
$685$	−7.20634			−0.275340
$686$		−	20.2551i		−	0.773342i
$687$		0			0
$688$			7.21170i			0.274943i
$689$		−	1.68056i		−	0.0640244i
$690$		0			0
$691$	−11.6919			−0.444781			−0.222390	−	0.974958i	$-0.571386\pi$
$691$	−11.6919			−0.444781			−0.222390	+	0.974958i	$0.571386\pi$
$692$			15.4549i			0.587509i
$693$		0			0
$694$	−15.5444			−0.590058
$695$			4.80943i			0.182432i
$696$		0			0
$697$		−	4.97391i		−	0.188400i
$698$	−4.94933			−0.187335
$699$		0			0
$700$			4.32270i			0.163383i
$701$	47.0578			1.77735			0.888673	−	0.458541i	$-0.151628\pi$
$701$	47.0578			1.77735			0.888673	+	0.458541i	$0.151628\pi$
$702$		0			0
$703$	−33.2995			−1.25592
$704$	2.34280			0.0882975
$705$		0			0
$706$	−4.05153			−0.152481
$707$			39.1899i			1.47389i
$708$		0			0
$709$	−20.8134			−0.781665			−0.390832	−	0.920462i	$-0.627813\pi$
$709$	−20.8134			−0.781665			−0.390832	+	0.920462i	$0.627813\pi$
$710$			7.06714i			0.265225i
$711$		0			0
$712$			7.38611i			0.276806i
$713$	−0.993606			−0.0372109
$714$		0			0
$715$		−	4.41895i		−	0.165259i
$716$	−4.25504			−0.159018
$717$		0			0
$718$		−	0.936911i		−	0.0349652i
$719$			20.2510i			0.755236i	0.925961	+	0.377618i	$0.123257\pi$
$719$			20.2510i			0.755236i	−0.925961	+	0.377618i	$0.876743\pi$
$720$		0			0
$721$		−	59.2756i		−	2.20754i
$722$	36.5333			1.35963
$723$		0			0
$724$	14.3515			0.533370
$725$		−	1.02344i		−	0.0380097i
$726$		0			0
$727$		−	45.2879i		−	1.67964i	−0.542868	−	0.839818i	$-0.682662\pi$
$727$		−	45.2879i		−	1.67964i	0.542868	−	0.839818i	$-0.317338\pi$
$728$	−8.15341			−0.302185
$729$		0			0
$730$	8.74524			0.323676
$731$	−34.3970			−1.27222
$732$		0			0
$733$			27.4413i			1.01357i	0.862073	+	0.506784i	$0.169166\pi$
$733$			27.4413i			1.01357i	−0.862073	+	0.506784i	$0.830834\pi$
$734$		−	19.7243i		−	0.728037i
$735$		0			0
$736$		−	0.306025i		−	0.0112802i
$737$	−16.8893	+	9.08264i	−0.622125	+	0.334563i
$738$		0			0
$739$			9.62548i			0.354079i	0.984204	+	0.177039i	$0.0566520\pi$
$739$			9.62548i			0.354079i	−0.984204	+	0.177039i	$0.943348\pi$
$740$	−4.46849			−0.164265
$741$		0			0
$742$		−	3.85147i		−	0.141392i
$743$			38.9980i			1.43070i	0.698767	+	0.715349i	$0.253732\pi$
$743$			38.9980i			1.43070i	−0.698767	+	0.715349i	$0.746268\pi$
$744$		0			0
$745$		−	12.8615i		−	0.471208i
$746$			25.9100i			0.948634i
$747$		0			0
$748$			11.1742i			0.408570i
$749$	−19.3622			−0.707478
$750$		0			0
$751$	31.3215			1.14294			0.571469	−	0.820624i	$-0.306374\pi$
$751$	31.3215			1.14294			0.571469	+	0.820624i	$0.306374\pi$
$752$			8.58757i			0.313157i
$753$		0			0
$754$	1.93040			0.0703011
$755$	6.26170			0.227887
$756$		0			0
$757$		−	46.9075i		−	1.70488i	−0.522825	−	0.852440i	$-0.675122\pi$
$757$		−	46.9075i		−	1.70488i	0.522825	−	0.852440i	$-0.324878\pi$
$758$			27.1419i			0.985837i
$759$		0			0
$760$	7.45207			0.270315
$761$			30.9907i			1.12341i	0.827337	+	0.561706i	$0.189855\pi$
$761$			30.9907i			1.12341i	−0.827337	+	0.561706i	$0.810145\pi$
$762$		0			0
$763$	−38.6682			−1.39988
$764$	1.08212			0.0391499
$765$		0			0
$766$	19.0306			0.687605
$767$	5.09745			0.184058
$768$		0			0
$769$		−	37.1288i		−	1.33890i	−0.742857	−	0.669450i	$-0.766530\pi$
$769$		−	37.1288i		−	1.33890i	0.742857	−	0.669450i	$-0.233470\pi$
$770$		−	10.1272i		−	0.364959i
$771$		0			0
$772$	22.9265			0.825143
$773$		−	37.3193i		−	1.34228i	−0.741329	−	0.671142i	$-0.765804\pi$
$773$		−	37.3193i		−	1.34228i	0.741329	−	0.671142i	$-0.234196\pi$
$774$		0			0
$775$		−	3.24681i		−	0.116629i
$776$		−	11.2205i		−	0.402792i
$777$		0			0
$778$			7.72549i			0.276972i
$779$	7.77128			0.278435
$780$		0			0
$781$		−	16.5569i		−	0.592452i
$782$	1.45962			0.0521959
$783$		0			0
$784$	−11.6857			−0.417348
$785$	10.8380			0.386824
$786$		0			0
$787$			18.6340i			0.664231i	0.943239	+	0.332115i	$0.107762\pi$
$787$			18.6340i			0.664231i	−0.943239	+	0.332115i	$0.892238\pi$
$788$	−5.68166			−0.202401
$789$		0			0
$790$			10.0945i			0.359145i
$791$		−	74.6062i		−	2.65269i
$792$		0			0
$793$	15.9979			0.568101
$794$	−21.5728			−0.765591
$795$		0			0
$796$	0.952599			0.0337640
$797$		−	32.2118i		−	1.14100i	−0.821297	−	0.570501i	$-0.806749\pi$
$797$		−	32.2118i		−	1.14100i	0.821297	−	0.570501i	$-0.193251\pi$
$798$		0			0
$799$	−40.9593			−1.44904
$800$	1.00000			0.0353553
$801$		0			0
$802$	12.9657			0.457833
$803$	−20.4883			−0.723017
$804$		0			0
$805$	−1.32286			−0.0466245
$806$	6.12408			0.215712
$807$		0			0
$808$	9.06607			0.318943
$809$	−46.0579			−1.61931			−0.809655	−	0.586906i	$-0.800346\pi$
$809$	−46.0579			−1.61931			−0.809655	+	0.586906i	$0.800346\pi$
$810$		0			0
$811$			31.5881i			1.10921i	0.832114	+	0.554605i	$0.187130\pi$
$811$			31.5881i			1.10921i	−0.832114	+	0.554605i	$0.812870\pi$
$812$	4.42404			0.155253
$813$		0			0
$814$	10.4688			0.366930
$815$	−2.78197			−0.0974482
$816$		0			0
$817$		−	53.7421i		−	1.88020i
$818$		−	14.1103i		−	0.493354i
$819$		0			0
$820$	1.04283			0.0364174
$821$			22.5266i			0.786185i	0.919499	+	0.393093i	$0.128595\pi$
$821$			22.5266i			0.786185i	−0.919499	+	0.393093i	$0.871405\pi$
$822$		0			0
$823$	−51.8439			−1.80716			−0.903582	−	0.428416i	$-0.859072\pi$
$823$	−51.8439			−1.80716			−0.903582	+	0.428416i	$0.859072\pi$
$824$	−13.7126			−0.477702
$825$		0			0
$826$	11.6822			0.406475
$827$		−	35.1577i		−	1.22255i	−0.791417	−	0.611276i	$-0.790657\pi$
$827$		−	35.1577i		−	1.22255i	0.791417	−	0.611276i	$-0.209343\pi$
$828$		0			0
$829$	−34.0245			−1.18172			−0.590859	−	0.806775i	$-0.701211\pi$
$829$	−34.0245			−1.18172			−0.590859	+	0.806775i	$0.701211\pi$
$830$		−	3.81814i		−	0.132530i
$831$		0			0
$832$			1.88618i			0.0653917i
$833$		−	55.7364i		−	1.93115i
$834$		0			0
$835$		−	8.88154i		−	0.307358i
$836$	−17.4587			−0.603821
$837$		0			0
$838$		−	5.13468i		−	0.177375i
$839$		−	26.0290i		−	0.898622i	−0.893375	−	0.449311i	$-0.851669\pi$
$839$		−	26.0290i		−	0.898622i	0.893375	−	0.449311i	$-0.148331\pi$
$840$		0			0
$841$	27.9526			0.963881
$842$	30.3752			1.04680
$843$		0			0
$844$	−8.72943			−0.300480
$845$	−9.44231			−0.324825
$846$		0			0
$847$		−	23.8237i		−	0.818592i
$848$	−0.890987			−0.0305966
$849$		0			0
$850$			4.76961i			0.163596i
$851$		−	1.36747i		−	0.0468763i
$852$		0			0
$853$	34.4188			1.17848			0.589239	−	0.807959i	$-0.299428\pi$
$853$	34.4188			1.17848			0.589239	+	0.807959i	$0.299428\pi$
$854$	36.6635			1.25460
$855$		0			0
$856$			4.47918i			0.153095i
$857$	44.0594			1.50504			0.752520	−	0.658569i	$-0.228838\pi$
$857$	44.0594			1.50504			0.752520	+	0.658569i	$0.228838\pi$
$858$		0			0
$859$	−16.7960			−0.573071			−0.286536	−	0.958070i	$-0.592504\pi$
$859$	−16.7960			−0.573071			−0.286536	+	0.958070i	$0.592504\pi$
$860$		−	7.21170i		−	0.245917i
$861$		0			0
$862$		−	29.9389i		−	1.01972i
$863$		−	3.60890i		−	0.122849i	−0.998112	−	0.0614243i	$-0.980436\pi$
$863$		−	3.60890i		−	0.122849i	0.998112	−	0.0614243i	$-0.0195643\pi$
$864$		0			0
$865$		−	15.4549i		−	0.525484i
$866$			2.16181i			0.0734614i
$867$		0			0
$868$	14.0350			0.476379
$869$		−	23.6493i		−	0.802247i
$870$		0			0
$871$	−7.31242	−	13.5976i	−0.247772	−	0.460736i
$872$			8.94539i			0.302929i
$873$		0			0
$874$			2.28052i			0.0771398i
$875$		−	4.32270i		−	0.146134i
$876$		0			0
$877$	−16.7003			−0.563929			−0.281964	−	0.959425i	$-0.590986\pi$
$877$	−16.7003			−0.563929			−0.281964	+	0.959425i	$0.590986\pi$
$878$	−27.5160			−0.928621
$879$		0			0
$880$	−2.34280			−0.0789757
$881$			17.3651i			0.585045i	0.956259	+	0.292522i	$0.0944946\pi$
$881$			17.3651i			0.585045i	−0.956259	+	0.292522i	$0.905505\pi$
$882$		0			0
$883$			46.1923i			1.55450i	0.629195	+	0.777248i	$0.283385\pi$
$883$			46.1923i			1.55450i	−0.629195	+	0.777248i	$0.716615\pi$
$884$	−8.99635			−0.302580
$885$		0			0
$886$	24.7781			0.832435
$887$		−	28.4787i		−	0.956220i	−0.878300	−	0.478110i	$-0.841322\pi$
$887$		−	28.4787i		−	0.956220i	0.878300	−	0.478110i	$-0.158678\pi$
$888$		0			0
$889$			50.0948i			1.68013i
$890$		−	7.38611i		−	0.247583i
$891$		0			0
$892$	8.04774			0.269458
$893$		−	63.9952i		−	2.14152i
$894$		0			0
$895$	4.25504			0.142230
$896$			4.32270i			0.144411i
$897$		0			0
$898$			8.62480i			0.287813i
$899$	−3.32293			−0.110826
$900$		0			0
$901$		−	4.24965i		−	0.141577i
$902$	−2.44315			−0.0813480
$903$		0			0
$904$	−17.2592			−0.574032
$905$	−14.3515			−0.477060
$906$		0			0
$907$	27.5658			0.915308			0.457654	−	0.889130i	$-0.348690\pi$
$907$	27.5658			0.915308			0.457654	+	0.889130i	$0.348690\pi$
$908$			21.1569i			0.702115i
$909$		0			0
$910$	8.15341			0.270283
$911$			10.8416i			0.359197i	0.983740	+	0.179599i	$0.0574799\pi$
$911$			10.8416i			0.359197i	−0.983740	+	0.179599i	$0.942520\pi$
$912$		0			0
$913$			8.94514i			0.296041i
$914$	12.6443			0.418236
$915$		0			0
$916$		−	2.10421i		−	0.0695250i
$917$	70.2532			2.31996
$918$		0			0
$919$			30.5690i			1.00838i	0.863593	+	0.504189i	$0.168209\pi$
$919$			30.5690i			1.00838i	−0.863593	+	0.504189i	$0.831791\pi$
$920$			0.306025i			0.0100894i
$921$		0			0
$922$		−	7.48976i		−	0.246662i
$923$	13.3299			0.438760
$924$		0			0
$925$	4.46849			0.146923
$926$		−	15.4778i		−	0.508633i
$927$		0			0
$928$		−	1.02344i		−	0.0335962i
$929$	55.1620			1.80981			0.904903	−	0.425618i	$-0.139943\pi$
$929$	55.1620			1.80981			0.904903	+	0.425618i	$0.139943\pi$
$930$		0			0
$931$	87.0830			2.85403
$932$	−9.63658			−0.315657
$933$		0			0
$934$			3.87833i			0.126903i
$935$		−	11.1742i		−	0.365436i
$936$		0			0
$937$		−	9.70567i		−	0.317070i	−0.987353	−	0.158535i	$-0.949323\pi$
$937$		−	9.70567i		−	0.317070i	0.987353	−	0.158535i	$-0.0506771\pi$
$938$	−16.7584	−	31.1625i	−0.547181	−	1.01749i
$939$		0			0
$940$		−	8.58757i		−	0.280096i
$941$	−11.8701			−0.386953			−0.193477	−	0.981105i	$-0.561976\pi$
$941$	−11.8701			−0.386953			−0.193477	+	0.981105i	$0.561976\pi$
$942$		0			0
$943$			0.319134i			0.0103924i
$944$		−	2.70252i		−	0.0879595i
$945$		0			0
$946$			16.8956i			0.549322i
$947$			45.9117i			1.49193i	0.665985	+	0.745965i	$0.268011\pi$
$947$			45.9117i			1.49193i	−0.665985	+	0.745965i	$0.731989\pi$
$948$		0			0
$949$		−	16.4951i		−	0.535455i
$950$	−7.45207			−0.241777
$951$		0			0
$952$	−20.6176			−0.668219
$953$		−	37.0264i		−	1.19940i	−0.800224	−	0.599701i	$-0.795286\pi$
$953$		−	37.0264i		−	1.19940i	0.800224	−	0.599701i	$-0.204714\pi$
$954$		0			0
$955$	−1.08212			−0.0350167
$956$	−1.59177			−0.0514814
$957$		0			0
$958$			12.3695i			0.399640i
$959$			31.1509i			1.00591i
$960$		0			0
$961$	20.4582			0.659942
$962$			8.42840i			0.271743i
$963$		0			0
$964$	−27.4018			−0.882554
$965$	−22.9265			−0.738030
$966$		0			0
$967$	−41.6323			−1.33880			−0.669402	−	0.742900i	$-0.733450\pi$
$967$	−41.6323			−1.33880			−0.669402	+	0.742900i	$0.733450\pi$
$968$	−5.51130			−0.177140
$969$		0			0
$970$			11.2205i			0.360268i
$971$		−	25.5130i		−	0.818753i	−0.912366	−	0.409376i	$-0.865746\pi$
$971$		−	25.5130i		−	0.818753i	0.912366	−	0.409376i	$-0.134254\pi$
$972$		0			0
$973$	20.7897			0.666488
$974$		−	40.7837i		−	1.30679i
$975$		0			0
$976$		−	8.48161i		−	0.271490i
$977$		−	39.3862i		−	1.26008i	−0.776564	−	0.630038i	$-0.783039\pi$
$977$		−	39.3862i		−	1.26008i	0.776564	−	0.630038i	$-0.216961\pi$
$978$		0			0
$979$			17.3042i			0.553044i
$980$	11.6857			0.373287
$981$		0			0
$982$			37.0202i			1.18136i
$983$	−20.4889			−0.653495			−0.326747	−	0.945112i	$-0.605953\pi$
$983$	−20.4889			−0.653495			−0.326747	+	0.945112i	$0.605953\pi$
$984$		0			0
$985$	5.68166			0.181033
$986$	4.88142			0.155456
$987$		0			0
$988$		−	14.0560i		−	0.447180i
$989$	2.20696			0.0701773
$990$		0			0
$991$		−	9.76728i		−	0.310268i	−0.987893	−	0.155134i	$-0.950419\pi$
$991$		−	9.76728i		−	0.310268i	0.987893	−	0.155134i	$-0.0495809\pi$
$992$		−	3.24681i		−	0.103086i
$993$		0			0
$994$	30.5491			0.968960
$995$	−0.952599			−0.0301994
$996$		0			0
$997$	−18.5379			−0.587100			−0.293550	−	0.955944i	$-0.594837\pi$
$997$	−18.5379			−0.587100			−0.293550	+	0.955944i	$0.594837\pi$
$998$		−	23.9267i		−	0.757387i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6030.2.d.j.2411.14	yes	16
3.2	odd	2		6030.2.d.i.2411.14	yes	16
67.66	odd	2		6030.2.d.i.2411.3	✓	16
201.200	even	2	inner	6030.2.d.j.2411.3	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6030.2.d.i.2411.3	✓	16	67.66	odd	2
6030.2.d.i.2411.14	yes	16	3.2	odd	2
6030.2.d.j.2411.3	yes	16	201.200	even	2	inner
6030.2.d.j.2411.14	yes	16	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +4.32270i q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +4.32270i q^{7} +1.00000 q^{8} -1.00000 q^{10} +2.34280 q^{11} +1.88618i q^{13} +4.32270i q^{14} +1.00000 q^{16} +4.76961i q^{17} -7.45207 q^{19} -1.00000 q^{20} +2.34280 q^{22} -0.306025i q^{23} +1.00000 q^{25} +1.88618i q^{26} +4.32270i q^{28} -1.02344i q^{29} -3.24681i q^{31} +1.00000 q^{32} +4.76961i q^{34} -4.32270i q^{35} +4.46849 q^{37} -7.45207 q^{38} -1.00000 q^{40} -1.04283 q^{41} +7.21170i q^{43} +2.34280 q^{44} -0.306025i q^{46} +8.58757i q^{47} -11.6857 q^{49} +1.00000 q^{50} +1.88618i q^{52} -0.890987 q^{53} -2.34280 q^{55} +4.32270i q^{56} -1.02344i q^{58} -2.70252i q^{59} -8.48161i q^{61} -3.24681i q^{62} +1.00000 q^{64} -1.88618i q^{65} +(-7.20903 + 3.87683i) q^{67} +4.76961i q^{68} -4.32270i q^{70} -7.06714i q^{71} -8.74524 q^{73} +4.46849 q^{74} -7.45207 q^{76} +10.1272i q^{77} -10.0945i q^{79} -1.00000 q^{80} -1.04283 q^{82} +3.81814i q^{83} -4.76961i q^{85} +7.21170i q^{86} +2.34280 q^{88} +7.38611i q^{89} -8.15341 q^{91} -0.306025i q^{92} +8.58757i q^{94} +7.45207 q^{95} -11.2205i q^{97} -11.6857 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{2} + 16 q^{4} - 16 q^{5} + 16 q^{8}+O(q^{10}) \) 16 * q + 16 * q^2 + 16 * q^4 - 16 * q^5 + 16 * q^8	\( 16 q + 16 q^{2} + 16 q^{4} - 16 q^{5} + 16 q^{8} - 16 q^{10} - 20 q^{11} + 16 q^{16} - 8 q^{19} - 16 q^{20} - 20 q^{22} + 16 q^{25} + 16 q^{32} + 32 q^{37} - 8 q^{38} - 16 q^{40} - 8 q^{41} - 20 q^{44} - 88 q^{49} + 16 q^{50} + 8 q^{53} + 20 q^{55} + 16 q^{64} + 4 q^{67} + 16 q^{73} + 32 q^{74} - 8 q^{76} - 16 q^{80} - 8 q^{82} - 20 q^{88} + 40 q^{91} + 8 q^{95} - 88 q^{98}+O(q^{100}) \) 16 * q + 16 * q^2 + 16 * q^4 - 16 * q^5 + 16 * q^8 - 16 * q^10 - 20 * q^11 + 16 * q^16 - 8 * q^19 - 16 * q^20 - 20 * q^22 + 16 * q^25 + 16 * q^32 + 32 * q^37 - 8 * q^38 - 16 * q^40 - 8 * q^41 - 20 * q^44 - 88 * q^49 + 16 * q^50 + 8 * q^53 + 20 * q^55 + 16 * q^64 + 4 * q^67 + 16 * q^73 + 32 * q^74 - 8 * q^76 - 16 * q^80 - 8 * q^82 - 20 * q^88 + 40 * q^91 + 8 * q^95 - 88 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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