LMFDB - Embedded newform 6030.2.d.l.2411.6

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			2411.6
Character	$\chi$	$=$	6030.2411
Dual form			6030.2.d.l.2411.19

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -2.79591i q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -2.79591i q^{7} +1.00000 q^{8} +1.00000 q^{10} +6.41047 q^{11} +0.0604140i q^{13} -2.79591i q^{14} +1.00000 q^{16} -2.85617i q^{17} -6.91454 q^{19} +1.00000 q^{20} +6.41047 q^{22} -4.79630i q^{23} +1.00000 q^{25} +0.0604140i q^{26} -2.79591i q^{28} +8.56199i q^{29} -8.20722i q^{31} +1.00000 q^{32} -2.85617i q^{34} -2.79591i q^{35} +4.75004 q^{37} -6.91454 q^{38} +1.00000 q^{40} +6.22096 q^{41} -4.96367i q^{43} +6.41047 q^{44} -4.79630i q^{46} +4.52707i q^{47} -0.817085 q^{49} +1.00000 q^{50} +0.0604140i q^{52} +1.15009 q^{53} +6.41047 q^{55} -2.79591i q^{56} +8.56199i q^{58} -3.38038i q^{59} -3.47889i q^{61} -8.20722i q^{62} +1.00000 q^{64} +0.0604140i q^{65} +(-8.06126 - 1.41987i) q^{67} -2.85617i q^{68} -2.79591i q^{70} -12.0457i q^{71} -6.84440 q^{73} +4.75004 q^{74} -6.91454 q^{76} -17.9231i q^{77} +10.7379i q^{79} +1.00000 q^{80} +6.22096 q^{82} +7.74203i q^{83} -2.85617i q^{85} -4.96367i q^{86} +6.41047 q^{88} -9.32337i q^{89} +0.168912 q^{91} -4.79630i q^{92} +4.52707i q^{94} -6.91454 q^{95} -1.95056i q^{97} -0.817085 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$		−	2.79591i		−	1.05675i	−0.849010	−	0.528376i	$-0.822801\pi$
$7$		−	2.79591i		−	1.05675i	0.849010	−	0.528376i	$-0.177199\pi$
$8$	1.00000			0.353553
$9$		0			0
$10$	1.00000			0.316228
$11$	6.41047			1.93283			0.966414	−	0.256990i	$-0.0827307\pi$
$11$	6.41047			1.93283			0.966414	+	0.256990i	$0.0827307\pi$
$12$		0			0
$13$			0.0604140i			0.0167558i	0.999965	+	0.00837791i	$0.00266680\pi$
$13$			0.0604140i			0.0167558i	−0.999965	+	0.00837791i	$0.997333\pi$
$14$		−	2.79591i		−	0.747237i
$15$		0			0
$16$	1.00000			0.250000
$17$		−	2.85617i		−	0.692722i	−0.938101	−	0.346361i	$-0.887417\pi$
$17$		−	2.85617i		−	0.692722i	0.938101	−	0.346361i	$-0.112583\pi$
$18$		0			0
$19$	−6.91454			−1.58631			−0.793153	−	0.609023i	$-0.791562\pi$
$19$	−6.91454			−1.58631			−0.793153	+	0.609023i	$0.791562\pi$
$20$	1.00000			0.223607
$21$		0			0
$22$	6.41047			1.36672
$23$		−	4.79630i		−	1.00010i	−0.865997	−	0.500049i	$-0.833315\pi$
$23$		−	4.79630i		−	1.00010i	0.865997	−	0.500049i	$-0.166685\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$			0.0604140i			0.0118482i
$27$		0			0
$28$		−	2.79591i		−	0.528376i
$29$			8.56199i			1.58992i	0.606661	+	0.794961i	$0.292509\pi$
$29$			8.56199i			1.58992i	−0.606661	+	0.794961i	$0.707491\pi$
$30$		0			0
$31$		−	8.20722i		−	1.47406i	−0.675860	−	0.737030i	$-0.736227\pi$
$31$		−	8.20722i		−	1.47406i	0.675860	−	0.737030i	$-0.263773\pi$
$32$	1.00000			0.176777
$33$		0			0
$34$		−	2.85617i		−	0.489828i
$35$		−	2.79591i		−	0.472594i
$36$		0			0
$37$	4.75004			0.780901			0.390451	−	0.920624i	$-0.372319\pi$
$37$	4.75004			0.780901			0.390451	+	0.920624i	$0.372319\pi$
$38$	−6.91454			−1.12169
$39$		0			0
$40$	1.00000			0.158114
$41$	6.22096			0.971551			0.485776	−	0.874084i	$-0.338537\pi$
$41$	6.22096			0.971551			0.485776	+	0.874084i	$0.338537\pi$
$42$		0			0
$43$		−	4.96367i		−	0.756952i	−0.925611	−	0.378476i	$-0.876448\pi$
$43$		−	4.96367i		−	0.756952i	0.925611	−	0.378476i	$-0.123552\pi$
$44$	6.41047			0.966414
$45$		0			0
$46$		−	4.79630i		−	0.707176i
$47$			4.52707i			0.660340i	0.943921	+	0.330170i	$0.107106\pi$
$47$			4.52707i			0.660340i	−0.943921	+	0.330170i	$0.892894\pi$
$48$		0			0
$49$	−0.817085			−0.116726
$50$	1.00000			0.141421
$51$		0			0
$52$			0.0604140i			0.00837791i
$53$	1.15009			0.157977			0.0789885	−	0.996876i	$-0.474831\pi$
$53$	1.15009			0.157977			0.0789885	+	0.996876i	$0.474831\pi$
$54$		0			0
$55$	6.41047			0.864387
$56$		−	2.79591i		−	0.373619i
$57$		0			0
$58$			8.56199i			1.12424i
$59$		−	3.38038i		−	0.440088i	−0.975490	−	0.220044i	$-0.929380\pi$
$59$		−	3.38038i		−	0.440088i	0.975490	−	0.220044i	$-0.0706201\pi$
$60$		0			0
$61$		−	3.47889i		−	0.445426i	−0.974884	−	0.222713i	$-0.928509\pi$
$61$		−	3.47889i		−	0.445426i	0.974884	−	0.222713i	$-0.0714912\pi$
$62$		−	8.20722i		−	1.04232i
$63$		0			0
$64$	1.00000			0.125000
$65$			0.0604140i			0.00749343i
$66$		0			0
$67$	−8.06126	−	1.41987i	−0.984840	−	0.173465i
$67$	−8.06126	−	1.41987i	−0.984840	−	0.173465i
$68$		−	2.85617i		−	0.346361i
$69$		0			0
$70$		−	2.79591i		−	0.334175i
$71$		−	12.0457i		−	1.42956i	−0.699348	−	0.714781i	$-0.746526\pi$
$71$		−	12.0457i		−	1.42956i	0.699348	−	0.714781i	$-0.253474\pi$
$72$		0			0
$73$	−6.84440			−0.801077			−0.400538	−	0.916280i	$-0.631177\pi$
$73$	−6.84440			−0.801077			−0.400538	+	0.916280i	$0.631177\pi$
$74$	4.75004			0.552181
$75$		0			0
$76$	−6.91454			−0.793153
$77$		−	17.9231i		−	2.04252i
$78$		0			0
$79$			10.7379i			1.20811i	0.796943	+	0.604054i	$0.206449\pi$
$79$			10.7379i			1.20811i	−0.796943	+	0.604054i	$0.793551\pi$
$80$	1.00000			0.111803
$81$		0			0
$82$	6.22096			0.686990
$83$			7.74203i			0.849799i	0.905241	+	0.424899i	$0.139690\pi$
$83$			7.74203i			0.849799i	−0.905241	+	0.424899i	$0.860310\pi$
$84$		0			0
$85$		−	2.85617i		−	0.309795i
$86$		−	4.96367i		−	0.535246i
$87$		0			0
$88$	6.41047			0.683358
$89$		−	9.32337i		−	0.988276i	−0.869384	−	0.494138i	$-0.835484\pi$
$89$		−	9.32337i		−	0.988276i	0.869384	−	0.494138i	$-0.164516\pi$
$90$		0			0
$91$	0.168912			0.0177068
$92$		−	4.79630i		−	0.500049i
$93$		0			0
$94$			4.52707i			0.466931i
$95$	−6.91454			−0.709417
$96$		0			0
$97$		−	1.95056i		−	0.198050i	−0.995085	−	0.0990249i	$-0.968428\pi$
$97$		−	1.95056i		−	0.198050i	0.995085	−	0.0990249i	$-0.0315723\pi$
$98$	−0.817085			−0.0825381
$99$		0			0
$100$	1.00000			0.100000
$101$	2.15469			0.214400			0.107200	−	0.994238i	$-0.465812\pi$
$101$	2.15469			0.214400			0.107200	+	0.994238i	$0.465812\pi$
$102$		0			0
$103$	0.900096			0.0886891			0.0443446	−	0.999016i	$-0.485880\pi$
$103$	0.900096			0.0886891			0.0443446	+	0.999016i	$0.485880\pi$
$104$			0.0604140i			0.00592408i
$105$		0			0
$106$	1.15009			0.111707
$107$		−	2.02101i		−	0.195378i	−0.995217	−	0.0976892i	$-0.968855\pi$
$107$		−	2.02101i		−	0.195378i	0.995217	−	0.0976892i	$-0.0311451\pi$
$108$		0			0
$109$			18.8117i			1.80184i	0.433987	+	0.900919i	$0.357106\pi$
$109$			18.8117i			1.80184i	−0.433987	+	0.900919i	$0.642894\pi$
$110$	6.41047			0.611214
$111$		0			0
$112$		−	2.79591i		−	0.264188i
$113$	−1.33403			−0.125495			−0.0627477	−	0.998029i	$-0.519986\pi$
$113$	−1.33403			−0.125495			−0.0627477	+	0.998029i	$0.519986\pi$
$114$		0			0
$115$		−	4.79630i		−	0.447258i
$116$			8.56199i			0.794961i
$117$		0			0
$118$		−	3.38038i		−	0.311189i
$119$	−7.98557			−0.732036
$120$		0			0
$121$	30.0941			2.73582
$122$		−	3.47889i		−	0.314964i
$123$		0			0
$124$		−	8.20722i		−	0.737030i
$125$	1.00000			0.0894427
$126$		0			0
$127$	−18.0562			−1.60223			−0.801114	−	0.598512i	$-0.795759\pi$
$127$	−18.0562			−1.60223			−0.801114	+	0.598512i	$0.795759\pi$
$128$	1.00000			0.0883883
$129$		0			0
$130$			0.0604140i			0.00529866i
$131$			6.48608i			0.566692i	0.959018	+	0.283346i	$0.0914445\pi$
$131$			6.48608i			0.566692i	−0.959018	+	0.283346i	$0.908556\pi$
$132$		0			0
$133$			19.3324i			1.67633i
$134$	−8.06126	−	1.41987i	−0.696387	−	0.122658i
$135$		0			0
$136$		−	2.85617i		−	0.244914i
$137$	−9.09174			−0.776760			−0.388380	−	0.921499i	$-0.626965\pi$
$137$	−9.09174			−0.776760			−0.388380	+	0.921499i	$0.626965\pi$
$138$		0			0
$139$		−	4.15522i		−	0.352441i	−0.984351	−	0.176221i	$-0.943613\pi$
$139$		−	4.15522i		−	0.352441i	0.984351	−	0.176221i	$-0.0563872\pi$
$140$		−	2.79591i		−	0.236297i
$141$		0			0
$142$		−	12.0457i		−	1.01085i
$143$			0.387282i			0.0323861i
$144$		0			0
$145$			8.56199i			0.711035i
$146$	−6.84440			−0.566447
$147$		0			0
$148$	4.75004			0.390451
$149$			12.5719i			1.02993i	0.857210	+	0.514966i	$0.172196\pi$
$149$			12.5719i			1.02993i	−0.857210	+	0.514966i	$0.827804\pi$
$150$		0			0
$151$	−3.94978			−0.321428			−0.160714	−	0.987001i	$-0.551380\pi$
$151$	−3.94978			−0.321428			−0.160714	+	0.987001i	$0.551380\pi$
$152$	−6.91454			−0.560844
$153$		0			0
$154$		−	17.9231i		−	1.44428i
$155$		−	8.20722i		−	0.659220i
$156$		0			0
$157$	17.1422			1.36810			0.684049	−	0.729436i	$-0.260218\pi$
$157$	17.1422			1.36810			0.684049	+	0.729436i	$0.260218\pi$
$158$			10.7379i			0.854262i
$159$		0			0
$160$	1.00000			0.0790569
$161$	−13.4100			−1.05686
$162$		0			0
$163$	−12.3650			−0.968501			−0.484251	−	0.874929i	$-0.660908\pi$
$163$	−12.3650			−0.968501			−0.484251	+	0.874929i	$0.660908\pi$
$164$	6.22096			0.485776
$165$		0			0
$166$			7.74203i			0.600898i
$167$		−	9.33519i		−	0.722378i	−0.932493	−	0.361189i	$-0.882371\pi$
$167$		−	9.33519i		−	0.722378i	0.932493	−	0.361189i	$-0.117629\pi$
$168$		0			0
$169$	12.9964			0.999719
$170$		−	2.85617i		−	0.219058i
$171$		0			0
$172$		−	4.96367i		−	0.378476i
$173$		−	16.6031i		−	1.26231i	−0.775658	−	0.631154i	$-0.782582\pi$
$173$		−	16.6031i		−	1.26231i	0.775658	−	0.631154i	$-0.217418\pi$
$174$		0			0
$175$		−	2.79591i		−	0.211351i
$176$	6.41047			0.483207
$177$		0			0
$178$		−	9.32337i		−	0.698816i
$179$	2.90147			0.216866			0.108433	−	0.994104i	$-0.465417\pi$
$179$	2.90147			0.216866			0.108433	+	0.994104i	$0.465417\pi$
$180$		0			0
$181$	24.0476			1.78744			0.893722	−	0.448621i	$-0.148085\pi$
$181$	24.0476			1.78744			0.893722	+	0.448621i	$0.148085\pi$
$182$	0.168912			0.0125206
$183$		0			0
$184$		−	4.79630i		−	0.353588i
$185$	4.75004			0.349230
$186$		0			0
$187$		−	18.3094i		−	1.33891i
$188$			4.52707i			0.330170i
$189$		0			0
$190$	−6.91454			−0.501634
$191$	−15.9603			−1.15485			−0.577425	−	0.816444i	$-0.695942\pi$
$191$	−15.9603			−1.15485			−0.577425	+	0.816444i	$0.695942\pi$
$192$		0			0
$193$	−12.7068			−0.914656			−0.457328	−	0.889298i	$-0.651193\pi$
$193$	−12.7068			−0.914656			−0.457328	+	0.889298i	$0.651193\pi$
$194$		−	1.95056i		−	0.140042i
$195$		0			0
$196$	−0.817085			−0.0583632
$197$	−7.69464			−0.548221			−0.274110	−	0.961698i	$-0.588383\pi$
$197$	−7.69464			−0.548221			−0.274110	+	0.961698i	$0.588383\pi$
$198$		0			0
$199$	−5.58968			−0.396242			−0.198121	−	0.980178i	$-0.563484\pi$
$199$	−5.58968			−0.396242			−0.198121	+	0.980178i	$0.563484\pi$
$200$	1.00000			0.0707107
$201$		0			0
$202$	2.15469			0.151603
$203$	23.9385			1.68015
$204$		0			0
$205$	6.22096			0.434491
$206$	0.900096			0.0627127
$207$		0			0
$208$			0.0604140i			0.00418896i
$209$	−44.3255			−3.06606
$210$		0			0
$211$	11.4170			0.785977			0.392988	−	0.919543i	$-0.371441\pi$
$211$	11.4170			0.785977			0.392988	+	0.919543i	$0.371441\pi$
$212$	1.15009			0.0789885
$213$		0			0
$214$		−	2.02101i		−	0.138153i
$215$		−	4.96367i		−	0.338519i
$216$		0			0
$217$	−22.9466			−1.55772
$218$			18.8117i			1.27409i
$219$		0			0
$220$	6.41047			0.432194
$221$	0.172552			0.0116071
$222$		0			0
$223$	21.2012			1.41974			0.709870	−	0.704332i	$-0.248754\pi$
$223$	21.2012			1.41974			0.709870	+	0.704332i	$0.248754\pi$
$224$		−	2.79591i		−	0.186809i
$225$		0			0
$226$	−1.33403			−0.0887387
$227$		−	1.65845i		−	0.110075i	−0.998484	−	0.0550377i	$-0.982472\pi$
$227$		−	1.65845i		−	0.110075i	0.998484	−	0.0550377i	$-0.0175279\pi$
$228$		0			0
$229$		−	1.51630i		−	0.100200i	−0.998744	−	0.0500999i	$-0.984046\pi$
$229$		−	1.51630i		−	0.100200i	0.998744	−	0.0500999i	$-0.0159540\pi$
$230$		−	4.79630i		−	0.316259i
$231$		0			0
$232$			8.56199i			0.562122i
$233$	17.0922			1.11974			0.559872	−	0.828579i	$-0.310850\pi$
$233$	17.0922			1.11974			0.559872	+	0.828579i	$0.310850\pi$
$234$		0			0
$235$			4.52707i			0.295313i
$236$		−	3.38038i		−	0.220044i
$237$		0			0
$238$	−7.98557			−0.517628
$239$	16.2332			1.05004			0.525018	−	0.851091i	$-0.324059\pi$
$239$	16.2332			1.05004			0.525018	+	0.851091i	$0.324059\pi$
$240$		0			0
$241$	12.7140			0.818980			0.409490	−	0.912315i	$-0.365707\pi$
$241$	12.7140			0.818980			0.409490	+	0.912315i	$0.365707\pi$
$242$	30.0941			1.93452
$243$		0			0
$244$		−	3.47889i		−	0.222713i
$245$	−0.817085			−0.0522017
$246$		0			0
$247$		−	0.417735i		−	0.0265798i
$248$		−	8.20722i		−	0.521159i
$249$		0			0
$250$	1.00000			0.0632456
$251$	1.24797			0.0787713			0.0393856	−	0.999224i	$-0.487460\pi$
$251$	1.24797			0.0787713			0.0393856	+	0.999224i	$0.487460\pi$
$252$		0			0
$253$		−	30.7465i		−	1.93302i
$254$	−18.0562			−1.13295
$255$		0			0
$256$	1.00000			0.0625000
$257$			3.90208i			0.243405i	0.992567	+	0.121703i	$0.0388354\pi$
$257$			3.90208i			0.243405i	−0.992567	+	0.121703i	$0.961165\pi$
$258$		0			0
$259$		−	13.2807i		−	0.825220i
$260$			0.0604140i			0.00374672i
$261$		0			0
$262$			6.48608i			0.400712i
$263$		−	4.63041i		−	0.285523i	−0.989757	−	0.142762i	$-0.954402\pi$
$263$		−	4.63041i		−	0.285523i	0.989757	−	0.142762i	$-0.0455982\pi$
$264$		0			0
$265$	1.15009			0.0706495
$266$			19.3324i			1.18535i
$267$		0			0
$268$	−8.06126	−	1.41987i	−0.492420	−	0.0867324i
$269$		−	8.38696i		−	0.511362i	−0.966761	−	0.255681i	$-0.917700\pi$
$269$		−	8.38696i		−	0.511362i	0.966761	−	0.255681i	$-0.0822996\pi$
$270$		0			0
$271$			11.9480i			0.725790i	0.931830	+	0.362895i	$0.118212\pi$
$271$			11.9480i			0.725790i	−0.931830	+	0.362895i	$0.881788\pi$
$272$		−	2.85617i		−	0.173181i
$273$		0			0
$274$	−9.09174			−0.549252
$275$	6.41047			0.386566
$276$		0			0
$277$	0.782432			0.0470118			0.0235059	−	0.999724i	$-0.492517\pi$
$277$	0.782432			0.0470118			0.0235059	+	0.999724i	$0.492517\pi$
$278$		−	4.15522i		−	0.249213i
$279$		0			0
$280$		−	2.79591i		−	0.167087i
$281$	−7.18102			−0.428384			−0.214192	−	0.976792i	$-0.568712\pi$
$281$	−7.18102			−0.428384			−0.214192	+	0.976792i	$0.568712\pi$
$282$		0			0
$283$	32.4774			1.93058			0.965292	−	0.261175i	$-0.0841098\pi$
$283$	32.4774			1.93058			0.965292	+	0.261175i	$0.0841098\pi$
$284$		−	12.0457i		−	0.714781i
$285$		0			0
$286$			0.387282i			0.0229004i
$287$		−	17.3932i		−	1.02669i
$288$		0			0
$289$	8.84232			0.520136
$290$			8.56199i			0.502777i
$291$		0			0
$292$	−6.84440			−0.400538
$293$		−	21.0840i		−	1.23174i	−0.787848	−	0.615869i	$-0.788805\pi$
$293$		−	21.0840i		−	1.23174i	0.787848	−	0.615869i	$-0.211195\pi$
$294$		0			0
$295$		−	3.38038i		−	0.196813i
$296$	4.75004			0.276090
$297$		0			0
$298$			12.5719i			0.728272i
$299$	0.289764			0.0167575
$300$		0			0
$301$	−13.8779			−0.799912
$302$	−3.94978			−0.227284
$303$		0			0
$304$	−6.91454			−0.396576
$305$		−	3.47889i		−	0.199200i
$306$		0			0
$307$	−0.229221			−0.0130824			−0.00654118	−	0.999979i	$-0.502082\pi$
$307$	−0.229221			−0.0130824			−0.00654118	+	0.999979i	$0.502082\pi$
$308$		−	17.9231i		−	1.02126i
$309$		0			0
$310$		−	8.20722i		−	0.466139i
$311$	−23.1997			−1.31553			−0.657767	−	0.753221i	$-0.728499\pi$
$311$	−23.1997			−1.31553			−0.657767	+	0.753221i	$0.728499\pi$
$312$		0			0
$313$		−	21.2486i		−	1.20104i	−0.799609	−	0.600521i	$-0.794960\pi$
$313$		−	21.2486i		−	1.20104i	0.799609	−	0.600521i	$-0.205040\pi$
$314$	17.1422			0.967391
$315$		0			0
$316$			10.7379i			0.604054i
$317$		−	8.37838i		−	0.470577i	−0.971926	−	0.235288i	$-0.924397\pi$
$317$		−	8.37838i		−	0.470577i	0.971926	−	0.235288i	$-0.0756034\pi$
$318$		0			0
$319$			54.8864i			3.07305i
$320$	1.00000			0.0559017
$321$		0			0
$322$	−13.4100			−0.747311
$323$			19.7491i			1.09887i
$324$		0			0
$325$			0.0604140i			0.00335116i
$326$	−12.3650			−0.684834
$327$		0			0
$328$	6.22096			0.343495
$329$	12.6572			0.697817
$330$		0			0
$331$			23.8399i			1.31036i	0.755473	+	0.655180i	$0.227407\pi$
$331$			23.8399i			1.31036i	−0.755473	+	0.655180i	$0.772593\pi$
$332$			7.74203i			0.424899i
$333$		0			0
$334$		−	9.33519i		−	0.510799i
$335$	−8.06126	−	1.41987i	−0.440434	−	0.0775758i
$336$		0			0
$337$		−	24.2755i		−	1.32237i	−0.750222	−	0.661186i	$-0.770053\pi$
$337$		−	24.2755i		−	1.32237i	0.750222	−	0.661186i	$-0.229947\pi$
$338$	12.9964			0.706908
$339$		0			0
$340$		−	2.85617i		−	0.154897i
$341$		−	52.6121i		−	2.84911i
$342$		0			0
$343$		−	17.2868i		−	0.933402i
$344$		−	4.96367i		−	0.267623i
$345$		0			0
$346$		−	16.6031i		−	0.892586i
$347$	8.84916			0.475048			0.237524	−	0.971382i	$-0.423664\pi$
$347$	8.84916			0.475048			0.237524	+	0.971382i	$0.423664\pi$
$348$		0			0
$349$	−14.5659			−0.779694			−0.389847	−	0.920880i	$-0.627472\pi$
$349$	−14.5659			−0.779694			−0.389847	+	0.920880i	$0.627472\pi$
$350$		−	2.79591i		−	0.149447i
$351$		0			0
$352$	6.41047			0.341679
$353$	−2.22195			−0.118263			−0.0591313	−	0.998250i	$-0.518833\pi$
$353$	−2.22195			−0.118263			−0.0591313	+	0.998250i	$0.518833\pi$
$354$		0			0
$355$		−	12.0457i		−	0.639320i
$356$		−	9.32337i		−	0.494138i
$357$		0			0
$358$	2.90147			0.153347
$359$		−	2.23140i		−	0.117769i	−0.998265	−	0.0588843i	$-0.981246\pi$
$359$		−	2.23140i		−	0.117769i	0.998265	−	0.0588843i	$-0.0187543\pi$
$360$		0			0
$361$	28.8109			1.51636
$362$	24.0476			1.26391
$363$		0			0
$364$	0.168912			0.00885338
$365$	−6.84440			−0.358252
$366$		0			0
$367$		−	3.11629i		−	0.162669i	−0.996687	−	0.0813346i	$-0.974082\pi$
$367$		−	3.11629i		−	0.162669i	0.996687	−	0.0813346i	$-0.0259182\pi$
$368$		−	4.79630i		−	0.250025i
$369$		0			0
$370$	4.75004			0.246943
$371$		−	3.21554i		−	0.166943i
$372$		0			0
$373$			22.0978i			1.14418i	0.820191	+	0.572089i	$0.193867\pi$
$373$			22.0978i			1.14418i	−0.820191	+	0.572089i	$0.806133\pi$
$374$		−	18.3094i		−	0.946754i
$375$		0			0
$376$			4.52707i			0.233466i
$377$	−0.517264			−0.0266404
$378$		0			0
$379$			1.32443i			0.0680316i	0.999421	+	0.0340158i	$0.0108297\pi$
$379$			1.32443i			0.0680316i	−0.999421	+	0.0340158i	$0.989170\pi$
$380$	−6.91454			−0.354709
$381$		0			0
$382$	−15.9603			−0.816603
$383$	11.6994			0.597813			0.298906	−	0.954282i	$-0.403378\pi$
$383$	11.6994			0.597813			0.298906	+	0.954282i	$0.403378\pi$
$384$		0			0
$385$		−	17.9231i		−	0.913443i
$386$	−12.7068			−0.646760
$387$		0			0
$388$		−	1.95056i		−	0.0990249i
$389$			23.3510i			1.18394i	0.805959	+	0.591971i	$0.201650\pi$
$389$			23.3510i			1.18394i	−0.805959	+	0.591971i	$0.798350\pi$
$390$		0			0
$391$	−13.6990			−0.692790
$392$	−0.817085			−0.0412690
$393$		0			0
$394$	−7.69464			−0.387651
$395$			10.7379i			0.540283i
$396$		0			0
$397$	33.6777			1.69024			0.845118	−	0.534580i	$-0.179530\pi$
$397$	33.6777			1.69024			0.845118	+	0.534580i	$0.179530\pi$
$398$	−5.58968			−0.280185
$399$		0			0
$400$	1.00000			0.0500000
$401$	−14.8976			−0.743949			−0.371975	−	0.928243i	$-0.621319\pi$
$401$	−14.8976			−0.743949			−0.371975	+	0.928243i	$0.621319\pi$
$402$		0			0
$403$	0.495831			0.0246991
$404$	2.15469			0.107200
$405$		0			0
$406$	23.9385			1.18805
$407$	30.4500			1.50935
$408$		0			0
$409$			15.6501i			0.773846i	0.922112	+	0.386923i	$0.126462\pi$
$409$			15.6501i			0.773846i	−0.922112	+	0.386923i	$0.873538\pi$
$410$	6.22096			0.307231
$411$		0			0
$412$	0.900096			0.0443446
$413$	−9.45122			−0.465064
$414$		0			0
$415$			7.74203i			0.380041i
$416$			0.0604140i			0.00296204i
$417$		0			0
$418$	−44.3255			−2.16803
$419$		−	9.09061i		−	0.444106i	−0.975035	−	0.222053i	$-0.928724\pi$
$419$		−	9.09061i		−	0.444106i	0.975035	−	0.222053i	$-0.0712757\pi$
$420$		0			0
$421$	−32.9281			−1.60482			−0.802409	−	0.596774i	$-0.796449\pi$
$421$	−32.9281			−1.60482			−0.802409	+	0.596774i	$0.796449\pi$
$422$	11.4170			0.555770
$423$		0			0
$424$	1.15009			0.0558533
$425$		−	2.85617i		−	0.138544i
$426$		0			0
$427$	−9.72663			−0.470705
$428$		−	2.02101i		−	0.0976892i
$429$		0			0
$430$		−	4.96367i		−	0.239369i
$431$		−	8.65247i		−	0.416775i	−0.978046	−	0.208387i	$-0.933178\pi$
$431$		−	8.65247i		−	0.416775i	0.978046	−	0.208387i	$-0.0668215\pi$
$432$		0			0
$433$			31.8247i			1.52940i	0.644387	+	0.764700i	$0.277113\pi$
$433$			31.8247i			1.52940i	−0.644387	+	0.764700i	$0.722887\pi$
$434$	−22.9466			−1.10147
$435$		0			0
$436$			18.8117i			0.900919i
$437$			33.1643i			1.58646i
$438$		0			0
$439$	35.1326			1.67679			0.838395	−	0.545063i	$-0.183494\pi$
$439$	35.1326			1.67679			0.838395	+	0.545063i	$0.183494\pi$
$440$	6.41047			0.305607
$441$		0			0
$442$	0.172552			0.00820748
$443$	−14.0194			−0.666080			−0.333040	−	0.942913i	$-0.608074\pi$
$443$	−14.0194			−0.666080			−0.333040	+	0.942913i	$0.608074\pi$
$444$		0			0
$445$		−	9.32337i		−	0.441970i
$446$	21.2012			1.00391
$447$		0			0
$448$		−	2.79591i		−	0.132094i
$449$			35.1304i			1.65790i	0.559319	+	0.828952i	$0.311063\pi$
$449$			35.1304i			1.65790i	−0.559319	+	0.828952i	$0.688937\pi$
$450$		0			0
$451$	39.8793			1.87784
$452$	−1.33403			−0.0627477
$453$		0			0
$454$		−	1.65845i		−	0.0778350i
$455$	0.168912			0.00791870
$456$		0			0
$457$	−9.89004			−0.462636			−0.231318	−	0.972878i	$-0.574304\pi$
$457$	−9.89004			−0.462636			−0.231318	+	0.972878i	$0.574304\pi$
$458$		−	1.51630i		−	0.0708519i
$459$		0			0
$460$		−	4.79630i		−	0.223629i
$461$			13.8490i			0.645014i	0.946567	+	0.322507i	$0.104526\pi$
$461$			13.8490i			0.645014i	−0.946567	+	0.322507i	$0.895474\pi$
$462$		0			0
$463$		−	19.5604i		−	0.909047i	−0.890735	−	0.454524i	$-0.849810\pi$
$463$		−	19.5604i		−	0.909047i	0.890735	−	0.454524i	$-0.150190\pi$
$464$			8.56199i			0.397480i
$465$		0			0
$466$	17.0922			0.791779
$467$			24.4260i			1.13030i	0.824989	+	0.565149i	$0.191181\pi$
$467$			24.4260i			1.13030i	−0.824989	+	0.565149i	$0.808819\pi$
$468$		0			0
$469$	−3.96982	+	22.5385i	−0.183309	+	1.04073i
$470$			4.52707i			0.208818i
$471$		0			0
$472$		−	3.38038i		−	0.155595i
$473$		−	31.8194i		−	1.46306i
$474$		0			0
$475$	−6.91454			−0.317261
$476$	−7.98557			−0.366018
$477$		0			0
$478$	16.2332			0.742487
$479$		−	24.5764i		−	1.12292i	−0.827503	−	0.561462i	$-0.810239\pi$
$479$		−	24.5764i		−	1.12292i	0.827503	−	0.561462i	$-0.189761\pi$
$480$		0			0
$481$			0.286969i			0.0130846i
$482$	12.7140			0.579106
$483$		0			0
$484$	30.0941			1.36791
$485$		−	1.95056i		−	0.0885705i
$486$		0			0
$487$			30.8317i			1.39712i	0.715553	+	0.698559i	$0.246175\pi$
$487$			30.8317i			1.39712i	−0.715553	+	0.698559i	$0.753825\pi$
$488$		−	3.47889i		−	0.157482i
$489$		0			0
$490$	−0.817085			−0.0369122
$491$			27.6790i			1.24914i	0.780970	+	0.624568i	$0.214725\pi$
$491$			27.6790i			1.24914i	−0.780970	+	0.624568i	$0.785275\pi$
$492$		0			0
$493$	24.4545			1.10137
$494$		−	0.417735i		−	0.0187948i
$495$		0			0
$496$		−	8.20722i		−	0.368515i
$497$	−33.6787			−1.51069
$498$		0			0
$499$			27.3679i			1.22515i	0.790411	+	0.612577i	$0.209867\pi$
$499$			27.3679i			1.22515i	−0.790411	+	0.612577i	$0.790133\pi$
$500$	1.00000			0.0447214
$501$		0			0
$502$	1.24797			0.0556997
$503$	−42.1004			−1.87716			−0.938582	−	0.345056i	$-0.887860\pi$
$503$	−42.1004			−1.87716			−0.938582	+	0.345056i	$0.887860\pi$
$504$		0			0
$505$	2.15469			0.0958824
$506$		−	30.7465i		−	1.36685i
$507$		0			0
$508$	−18.0562			−0.801114
$509$			7.84075i			0.347535i	0.984787	+	0.173768i	$0.0555942\pi$
$509$			7.84075i			0.347535i	−0.984787	+	0.173768i	$0.944406\pi$
$510$		0			0
$511$			19.1363i			0.846540i
$512$	1.00000			0.0441942
$513$		0			0
$514$			3.90208i			0.172114i
$515$	0.900096			0.0396630
$516$		0			0
$517$			29.0206i			1.27632i
$518$		−	13.2807i		−	0.583518i
$519$		0			0
$520$			0.0604140i			0.00264933i
$521$	16.5544			0.725263			0.362632	−	0.931933i	$-0.381878\pi$
$521$	16.5544			0.725263			0.362632	+	0.931933i	$0.381878\pi$
$522$		0			0
$523$	−21.2612			−0.929689			−0.464845	−	0.885392i	$-0.653890\pi$
$523$	−21.2612			−0.929689			−0.464845	+	0.885392i	$0.653890\pi$
$524$			6.48608i			0.283346i
$525$		0			0
$526$		−	4.63041i		−	0.201895i
$527$	−23.4412			−1.02111
$528$		0			0
$529$	−0.00452664			−0.000196810
$530$	1.15009			0.0499567
$531$		0			0
$532$			19.3324i			0.838166i
$533$			0.375833i			0.0162791i
$534$		0			0
$535$		−	2.02101i		−	0.0873758i
$536$	−8.06126	−	1.41987i	−0.348194	−	0.0613290i
$537$		0			0
$538$		−	8.38696i		−	0.361587i
$539$	−5.23790			−0.225612
$540$		0			0
$541$			25.9118i			1.11404i	0.830500	+	0.557018i	$0.188055\pi$
$541$			25.9118i			1.11404i	−0.830500	+	0.557018i	$0.811945\pi$
$542$			11.9480i			0.513211i
$543$		0			0
$544$		−	2.85617i		−	0.122457i
$545$			18.8117i			0.805806i
$546$		0			0
$547$		−	4.88834i		−	0.209010i	−0.994524	−	0.104505i	$-0.966674\pi$
$547$		−	4.88834i		−	0.209010i	0.994524	−	0.104505i	$-0.0333259\pi$
$548$	−9.09174			−0.388380
$549$		0			0
$550$	6.41047			0.273343
$551$		−	59.2023i		−	2.52210i
$552$		0			0
$553$	30.0222			1.27667
$554$	0.782432			0.0332423
$555$		0			0
$556$		−	4.15522i		−	0.176221i
$557$			45.8626i			1.94326i	0.236507	+	0.971630i	$0.423997\pi$
$557$			45.8626i			1.94326i	−0.236507	+	0.971630i	$0.576003\pi$
$558$		0			0
$559$	0.299875			0.0126834
$560$		−	2.79591i		−	0.118149i
$561$		0			0
$562$	−7.18102			−0.302913
$563$	6.94381			0.292647			0.146323	−	0.989237i	$-0.453256\pi$
$563$	6.94381			0.292647			0.146323	+	0.989237i	$0.453256\pi$
$564$		0			0
$565$	−1.33403			−0.0561233
$566$	32.4774			1.36513
$567$		0			0
$568$		−	12.0457i		−	0.505427i
$569$		−	9.73949i		−	0.408301i	−0.978940	−	0.204150i	$-0.934557\pi$
$569$		−	9.73949i		−	0.408301i	0.978940	−	0.204150i	$-0.0654431\pi$
$570$		0			0
$571$	−31.3118			−1.31036			−0.655178	−	0.755474i	$-0.727406\pi$
$571$	−31.3118			−1.31036			−0.655178	+	0.755474i	$0.727406\pi$
$572$			0.387282i			0.0161931i
$573$		0			0
$574$		−	17.3932i		−	0.725979i
$575$		−	4.79630i		−	0.200020i
$576$		0			0
$577$			9.82670i			0.409091i	0.978857	+	0.204545i	$0.0655716\pi$
$577$			9.82670i			0.409091i	−0.978857	+	0.204545i	$0.934428\pi$
$578$	8.84232			0.367792
$579$		0			0
$580$			8.56199i			0.355517i
$581$	21.6460			0.898027
$582$		0			0
$583$	7.37261			0.305342
$584$	−6.84440			−0.283223
$585$		0			0
$586$		−	21.0840i		−	0.870970i
$587$	20.2864			0.837309			0.418655	−	0.908146i	$-0.362502\pi$
$587$	20.2864			0.837309			0.418655	+	0.908146i	$0.362502\pi$
$588$		0			0
$589$			56.7492i			2.33831i
$590$		−	3.38038i		−	0.139168i
$591$		0			0
$592$	4.75004			0.195225
$593$	−28.4755			−1.16935			−0.584675	−	0.811268i	$-0.698778\pi$
$593$	−28.4755			−1.16935			−0.584675	+	0.811268i	$0.698778\pi$
$594$		0			0
$595$	−7.98557			−0.327376
$596$			12.5719i			0.514966i
$597$		0			0
$598$	0.289764			0.0118493
$599$	−28.2745			−1.15527			−0.577633	−	0.816297i	$-0.696023\pi$
$599$	−28.2745			−1.15527			−0.577633	+	0.816297i	$0.696023\pi$
$600$		0			0
$601$	−11.0304			−0.449938			−0.224969	−	0.974366i	$-0.572228\pi$
$601$	−11.0304			−0.449938			−0.224969	+	0.974366i	$0.572228\pi$
$602$	−13.8779			−0.565623
$603$		0			0
$604$	−3.94978			−0.160714
$605$	30.0941			1.22350
$606$		0			0
$607$	−44.2590			−1.79642			−0.898208	−	0.439570i	$-0.855131\pi$
$607$	−44.2590			−1.79642			−0.898208	+	0.439570i	$0.855131\pi$
$608$	−6.91454			−0.280422
$609$		0			0
$610$		−	3.47889i		−	0.140856i
$611$	−0.273498			−0.0110645
$612$		0			0
$613$	1.27034			0.0513084			0.0256542	−	0.999671i	$-0.491833\pi$
$613$	1.27034			0.0513084			0.0256542	+	0.999671i	$0.491833\pi$
$614$	−0.229221			−0.00925063
$615$		0			0
$616$		−	17.9231i		−	0.722140i
$617$			22.7275i			0.914975i	0.889216	+	0.457488i	$0.151251\pi$
$617$			22.7275i			0.914975i	−0.889216	+	0.457488i	$0.848749\pi$
$618$		0			0
$619$	4.78372			0.192274			0.0961369	−	0.995368i	$-0.469351\pi$
$619$	4.78372			0.192274			0.0961369	+	0.995368i	$0.469351\pi$
$620$		−	8.20722i		−	0.329610i
$621$		0			0
$622$	−23.1997			−0.930224
$623$	−26.0673			−1.04436
$624$		0			0
$625$	1.00000			0.0400000
$626$		−	21.2486i		−	0.849265i
$627$		0			0
$628$	17.1422			0.684049
$629$		−	13.5669i		−	0.540948i
$630$		0			0
$631$			25.3963i			1.01101i	0.862823	+	0.505506i	$0.168694\pi$
$631$			25.3963i			1.01101i	−0.862823	+	0.505506i	$0.831306\pi$
$632$			10.7379i			0.427131i
$633$		0			0
$634$		−	8.37838i		−	0.332748i
$635$	−18.0562			−0.716538
$636$		0			0
$637$		−	0.0493634i		−	0.00195585i
$638$			54.8864i			2.17297i
$639$		0			0
$640$	1.00000			0.0395285
$641$	7.80763			0.308383			0.154191	−	0.988041i	$-0.450723\pi$
$641$	7.80763			0.308383			0.154191	+	0.988041i	$0.450723\pi$
$642$		0			0
$643$	−14.3054			−0.564152			−0.282076	−	0.959392i	$-0.591023\pi$
$643$	−14.3054			−0.564152			−0.282076	+	0.959392i	$0.591023\pi$
$644$	−13.4100			−0.528428
$645$		0			0
$646$			19.7491i			0.777017i
$647$	44.8162			1.76191			0.880954	−	0.473202i	$-0.156902\pi$
$647$	44.8162			1.76191			0.880954	+	0.473202i	$0.156902\pi$
$648$		0			0
$649$		−	21.6698i		−	0.850614i
$650$			0.0604140i			0.00236963i
$651$		0			0
$652$	−12.3650			−0.484251
$653$	7.55436			0.295625			0.147813	−	0.989015i	$-0.452777\pi$
$653$	7.55436			0.295625			0.147813	+	0.989015i	$0.452777\pi$
$654$		0			0
$655$			6.48608i			0.253432i
$656$	6.22096			0.242888
$657$		0			0
$658$	12.6572			0.493431
$659$		−	3.17339i		−	0.123618i	−0.998088	−	0.0618090i	$-0.980313\pi$
$659$		−	3.17339i		−	0.123618i	0.998088	−	0.0618090i	$-0.0196870\pi$
$660$		0			0
$661$			22.0214i			0.856532i	0.903653	+	0.428266i	$0.140875\pi$
$661$			22.0214i			0.856532i	−0.903653	+	0.428266i	$0.859125\pi$
$662$			23.8399i			0.926565i
$663$		0			0
$664$			7.74203i			0.300449i
$665$			19.3324i			0.749679i
$666$		0			0
$667$	41.0659			1.59008
$668$		−	9.33519i		−	0.361189i
$669$		0			0
$670$	−8.06126	−	1.41987i	−0.311434	−	0.0548544i
$671$		−	22.3013i		−	0.860931i
$672$		0			0
$673$			41.6895i			1.60701i	0.595296	+	0.803507i	$0.297035\pi$
$673$			41.6895i			1.60701i	−0.595296	+	0.803507i	$0.702965\pi$
$674$		−	24.2755i		−	0.935058i
$675$		0			0
$676$	12.9964			0.499860
$677$	43.0600			1.65493			0.827464	−	0.561518i	$-0.189783\pi$
$677$	43.0600			1.65493			0.827464	+	0.561518i	$0.189783\pi$
$678$		0			0
$679$	−5.45359			−0.209290
$680$		−	2.85617i		−	0.109529i
$681$		0			0
$682$		−	52.6121i		−	2.01462i
$683$	21.5831			0.825853			0.412926	−	0.910764i	$-0.364507\pi$
$683$	21.5831			0.825853			0.412926	+	0.910764i	$0.364507\pi$
$684$		0			0
$685$	−9.09174			−0.347378
$686$		−	17.2868i		−	0.660015i
$687$		0			0
$688$		−	4.96367i		−	0.189238i
$689$			0.0694815i			0.00264703i
$690$		0			0
$691$	23.6463			0.899547			0.449774	−	0.893143i	$-0.351505\pi$
$691$	23.6463			0.899547			0.449774	+	0.893143i	$0.351505\pi$
$692$		−	16.6031i		−	0.631154i
$693$		0			0
$694$	8.84916			0.335909
$695$		−	4.15522i		−	0.157616i
$696$		0			0
$697$		−	17.7681i		−	0.673015i
$698$	−14.5659			−0.551327
$699$		0			0
$700$		−	2.79591i		−	0.105675i
$701$	−38.0415			−1.43681			−0.718403	−	0.695627i	$-0.755127\pi$
$701$	−38.0415			−1.43681			−0.718403	+	0.695627i	$0.755127\pi$
$702$		0			0
$703$	−32.8443			−1.23875
$704$	6.41047			0.241604
$705$		0			0
$706$	−2.22195			−0.0836243
$707$		−	6.02431i		−	0.226567i
$708$		0			0
$709$	2.07580			0.0779583			0.0389792	−	0.999240i	$-0.487589\pi$
$709$	2.07580			0.0779583			0.0389792	+	0.999240i	$0.487589\pi$
$710$		−	12.0457i		−	0.452067i
$711$		0			0
$712$		−	9.32337i		−	0.349408i
$713$	−39.3643			−1.47421
$714$		0			0
$715$			0.387282i			0.0144835i
$716$	2.90147			0.108433
$717$		0			0
$718$		−	2.23140i		−	0.0832749i
$719$		−	46.7994i		−	1.74532i	−0.488327	−	0.872661i	$-0.662393\pi$
$719$		−	46.7994i		−	1.74532i	0.488327	−	0.872661i	$-0.337607\pi$
$720$		0			0
$721$		−	2.51658i		−	0.0937225i
$722$	28.8109			1.07223
$723$		0			0
$724$	24.0476			0.893722
$725$			8.56199i			0.317984i
$726$		0			0
$727$		−	22.8819i		−	0.848642i	−0.905512	−	0.424321i	$-0.860513\pi$
$727$		−	22.8819i		−	0.848642i	0.905512	−	0.424321i	$-0.139487\pi$
$728$	0.168912			0.00626029
$729$		0			0
$730$	−6.84440			−0.253323
$731$	−14.1771			−0.524358
$732$		0			0
$733$		−	0.385622i		−	0.0142433i	−0.999975	−	0.00712164i	$-0.997733\pi$
$733$		−	0.385622i		−	0.0142433i	0.999975	−	0.00712164i	$-0.00226691\pi$
$734$		−	3.11629i		−	0.115025i
$735$		0			0
$736$		−	4.79630i		−	0.176794i
$737$	−51.6765	−	9.10203i	−1.90353	−	0.335278i
$738$		0			0
$739$			28.4608i			1.04695i	0.852042	+	0.523473i	$0.175364\pi$
$739$			28.4608i			1.04695i	−0.852042	+	0.523473i	$0.824636\pi$
$740$	4.75004			0.174615
$741$		0			0
$742$		−	3.21554i		−	0.118046i
$743$			1.53158i			0.0561881i	0.999605	+	0.0280940i	$0.00894379\pi$
$743$			1.53158i			0.0561881i	−0.999605	+	0.0280940i	$0.991056\pi$
$744$		0			0
$745$			12.5719i			0.460600i
$746$			22.0978i			0.809057i
$747$		0			0
$748$		−	18.3094i		−	0.669456i
$749$	−5.65055			−0.206467
$750$		0			0
$751$	37.6841			1.37511			0.687556	−	0.726131i	$-0.258684\pi$
$751$	37.6841			1.37511			0.687556	+	0.726131i	$0.258684\pi$
$752$			4.52707i			0.165085i
$753$		0			0
$754$	−0.517264			−0.0188376
$755$	−3.94978			−0.143747
$756$		0			0
$757$		−	47.3302i		−	1.72025i	−0.510087	−	0.860123i	$-0.670387\pi$
$757$		−	47.3302i		−	1.72025i	0.510087	−	0.860123i	$-0.329613\pi$
$758$			1.32443i			0.0481056i
$759$		0			0
$760$	−6.91454			−0.250817
$761$		−	21.9961i		−	0.797359i	−0.917090	−	0.398680i	$-0.869469\pi$
$761$		−	21.9961i		−	0.797359i	0.917090	−	0.398680i	$-0.130531\pi$
$762$		0			0
$763$	52.5958			1.90410
$764$	−15.9603			−0.577425
$765$		0			0
$766$	11.6994			0.422718
$767$	0.204222			0.00737403
$768$		0			0
$769$			23.1525i			0.834901i	0.908700	+	0.417450i	$0.137076\pi$
$769$			23.1525i			0.834901i	−0.908700	+	0.417450i	$0.862924\pi$
$770$		−	17.9231i		−	0.645902i
$771$		0			0
$772$	−12.7068			−0.457328
$773$			19.4881i			0.700939i	0.936574	+	0.350469i	$0.113978\pi$
$773$			19.4881i			0.700939i	−0.936574	+	0.350469i	$0.886022\pi$
$774$		0			0
$775$		−	8.20722i		−	0.294812i
$776$		−	1.95056i		−	0.0700211i
$777$		0			0
$778$			23.3510i			0.837173i
$779$	−43.0151			−1.54118
$780$		0			0
$781$		−	77.2186i		−	2.76310i
$782$	−13.6990			−0.489877
$783$		0			0
$784$	−0.817085			−0.0291816
$785$	17.1422			0.611832
$786$		0			0
$787$		−	41.6933i		−	1.48621i	−0.669176	−	0.743104i	$-0.733353\pi$
$787$		−	41.6933i		−	1.48621i	0.669176	−	0.743104i	$-0.266647\pi$
$788$	−7.69464			−0.274110
$789$		0			0
$790$			10.7379i			0.382038i
$791$			3.72983i			0.132618i
$792$		0			0
$793$	0.210173			0.00746347
$794$	33.6777			1.19518
$795$		0			0
$796$	−5.58968			−0.198121
$797$			38.7306i			1.37191i	0.727644	+	0.685955i	$0.240615\pi$
$797$			38.7306i			1.37191i	−0.727644	+	0.685955i	$0.759385\pi$
$798$		0			0
$799$	12.9301			0.457432
$800$	1.00000			0.0353553
$801$		0			0
$802$	−14.8976			−0.526051
$803$	−43.8758			−1.54834
$804$		0			0
$805$	−13.4100			−0.472641
$806$	0.495831			0.0174649
$807$		0			0
$808$	2.15469			0.0758017
$809$	−14.0683			−0.494617			−0.247308	−	0.968937i	$-0.579546\pi$
$809$	−14.0683			−0.494617			−0.247308	+	0.968937i	$0.579546\pi$
$810$		0			0
$811$		−	26.0783i		−	0.915732i	−0.889021	−	0.457866i	$-0.848614\pi$
$811$		−	26.0783i		−	0.915732i	0.889021	−	0.457866i	$-0.151386\pi$
$812$	23.9385			0.840077
$813$		0			0
$814$	30.4500			1.06727
$815$	−12.3650			−0.433127
$816$		0			0
$817$			34.3215i			1.20076i
$818$			15.6501i			0.547192i
$819$		0			0
$820$	6.22096			0.217245
$821$		−	13.0059i		−	0.453910i	−0.973905	−	0.226955i	$-0.927123\pi$
$821$		−	13.0059i		−	0.453910i	0.973905	−	0.226955i	$-0.0728770\pi$
$822$		0			0
$823$	19.3886			0.675844			0.337922	−	0.941174i	$-0.390276\pi$
$823$	19.3886			0.675844			0.337922	+	0.941174i	$0.390276\pi$
$824$	0.900096			0.0313563
$825$		0			0
$826$	−9.45122			−0.328850
$827$			20.0361i			0.696724i	0.937360	+	0.348362i	$0.113262\pi$
$827$			20.0361i			0.696724i	−0.937360	+	0.348362i	$0.886738\pi$
$828$		0			0
$829$	24.9886			0.867889			0.433945	−	0.900940i	$-0.357121\pi$
$829$	24.9886			0.867889			0.433945	+	0.900940i	$0.357121\pi$
$830$			7.74203i			0.268730i
$831$		0			0
$832$			0.0604140i			0.00209448i
$833$			2.33373i			0.0808590i
$834$		0			0
$835$		−	9.33519i		−	0.323057i
$836$	−44.3255			−1.53303
$837$		0			0
$838$		−	9.09061i		−	0.314030i
$839$			33.0038i			1.13942i	0.821847	+	0.569708i	$0.192944\pi$
$839$			33.0038i			1.13942i	−0.821847	+	0.569708i	$0.807056\pi$
$840$		0			0
$841$	−44.3077			−1.52785
$842$	−32.9281			−1.13478
$843$		0			0
$844$	11.4170			0.392988
$845$	12.9964			0.447088
$846$		0			0
$847$		−	84.1402i		−	2.89109i
$848$	1.15009			0.0394942
$849$		0			0
$850$		−	2.85617i		−	0.0979657i
$851$		−	22.7826i		−	0.780978i
$852$		0			0
$853$	−6.21852			−0.212918			−0.106459	−	0.994317i	$-0.533951\pi$
$853$	−6.21852			−0.212918			−0.106459	+	0.994317i	$0.533951\pi$
$854$	−9.72663			−0.332839
$855$		0			0
$856$		−	2.02101i		−	0.0690767i
$857$	2.89844			0.0990088			0.0495044	−	0.998774i	$-0.484236\pi$
$857$	2.89844			0.0990088			0.0495044	+	0.998774i	$0.484236\pi$
$858$		0			0
$859$	−51.2605			−1.74899			−0.874493	−	0.485038i	$-0.838806\pi$
$859$	−51.2605			−1.74899			−0.874493	+	0.485038i	$0.838806\pi$
$860$		−	4.96367i		−	0.169260i
$861$		0			0
$862$		−	8.65247i		−	0.294704i
$863$		−	20.6040i		−	0.701370i	−0.936494	−	0.350685i	$-0.885949\pi$
$863$		−	20.6040i		−	0.701370i	0.936494	−	0.350685i	$-0.114051\pi$
$864$		0			0
$865$		−	16.6031i		−	0.564521i
$866$			31.8247i			1.08145i
$867$		0			0
$868$	−22.9466			−0.778859
$869$			68.8350i			2.33507i
$870$		0			0
$871$	0.0857800	−	0.487013i	0.00290654	−	0.0165018i
$872$			18.8117i			0.637046i
$873$		0			0
$874$			33.1643i			1.12180i
$875$		−	2.79591i		−	0.0945188i
$876$		0			0
$877$	−3.48332			−0.117623			−0.0588117	−	0.998269i	$-0.518731\pi$
$877$	−3.48332			−0.117623			−0.0588117	+	0.998269i	$0.518731\pi$
$878$	35.1326			1.18567
$879$		0			0
$880$	6.41047			0.216097
$881$		−	7.45634i		−	0.251210i	−0.992080	−	0.125605i	$-0.959913\pi$
$881$		−	7.45634i		−	0.251210i	0.992080	−	0.125605i	$-0.0400873\pi$
$882$		0			0
$883$			58.1214i			1.95594i	0.208743	+	0.977971i	$0.433063\pi$
$883$			58.1214i			1.95594i	−0.208743	+	0.977971i	$0.566937\pi$
$884$	0.172552			0.00580356
$885$		0			0
$886$	−14.0194			−0.470990
$887$		−	24.4383i		−	0.820556i	−0.911960	−	0.410278i	$-0.865432\pi$
$887$		−	24.4383i		−	0.820556i	0.911960	−	0.410278i	$-0.134568\pi$
$888$		0			0
$889$			50.4834i			1.69316i
$890$		−	9.32337i		−	0.312520i
$891$		0			0
$892$	21.2012			0.709870
$893$		−	31.3026i		−	1.04750i
$894$		0			0
$895$	2.90147			0.0969855
$896$		−	2.79591i		−	0.0934046i
$897$		0			0
$898$			35.1304i			1.17232i
$899$	70.2702			2.34364
$900$		0			0
$901$		−	3.28485i		−	0.109434i
$902$	39.8793			1.32783
$903$		0			0
$904$	−1.33403			−0.0443693
$905$	24.0476			0.799370
$906$		0			0
$907$	−35.0612			−1.16419			−0.582094	−	0.813121i	$-0.697767\pi$
$907$	−35.0612			−1.16419			−0.582094	+	0.813121i	$0.697767\pi$
$908$		−	1.65845i		−	0.0550377i
$909$		0			0
$910$	0.168912			0.00559937
$911$			4.04022i			0.133858i	0.997758	+	0.0669292i	$0.0213202\pi$
$911$			4.04022i			0.133858i	−0.997758	+	0.0669292i	$0.978680\pi$
$912$		0			0
$913$			49.6300i			1.64251i
$914$	−9.89004			−0.327133
$915$		0			0
$916$		−	1.51630i		−	0.0500999i
$917$	18.1345			0.598853
$918$		0			0
$919$		−	45.1688i		−	1.48998i	−0.667076	−	0.744990i	$-0.732454\pi$
$919$		−	45.1688i		−	1.48998i	0.667076	−	0.744990i	$-0.267546\pi$
$920$		−	4.79630i		−	0.158129i
$921$		0			0
$922$			13.8490i			0.456094i
$923$	0.727729			0.0239535
$924$		0			0
$925$	4.75004			0.156180
$926$		−	19.5604i		−	0.642793i
$927$		0			0
$928$			8.56199i			0.281061i
$929$	−43.3681			−1.42286			−0.711430	−	0.702757i	$-0.751952\pi$
$929$	−43.3681			−1.42286			−0.711430	+	0.702757i	$0.751952\pi$
$930$		0			0
$931$	5.64977			0.185164
$932$	17.0922			0.559872
$933$		0			0
$934$			24.4260i			0.799242i
$935$		−	18.3094i		−	0.598780i
$936$		0			0
$937$			50.4225i			1.64723i	0.567148	+	0.823616i	$0.308047\pi$
$937$			50.4225i			1.64723i	−0.567148	+	0.823616i	$0.691953\pi$
$938$	−3.96982	+	22.5385i	−0.129619	+	0.735909i
$939$		0			0
$940$			4.52707i			0.147657i
$941$	15.8245			0.515864			0.257932	−	0.966163i	$-0.416959\pi$
$941$	15.8245			0.515864			0.257932	+	0.966163i	$0.416959\pi$
$942$		0			0
$943$		−	29.8376i		−	0.971647i
$944$		−	3.38038i		−	0.110022i
$945$		0			0
$946$		−	31.8194i		−	1.03454i
$947$			33.1209i			1.07628i	0.842854	+	0.538142i	$0.180873\pi$
$947$			33.1209i			1.07628i	−0.842854	+	0.538142i	$0.819127\pi$
$948$		0			0
$949$		−	0.413498i		−	0.0134227i
$950$	−6.91454			−0.224337
$951$		0			0
$952$	−7.98557			−0.258814
$953$		−	31.2636i		−	1.01273i	−0.862320	−	0.506364i	$-0.830989\pi$
$953$		−	31.2636i		−	1.01273i	0.862320	−	0.506364i	$-0.169011\pi$
$954$		0			0
$955$	−15.9603			−0.516465
$956$	16.2332			0.525018
$957$		0			0
$958$		−	24.5764i		−	0.794026i
$959$			25.4196i			0.820843i
$960$		0			0
$961$	−36.3585			−1.17286
$962$			0.286969i			0.00925224i
$963$		0			0
$964$	12.7140			0.409490
$965$	−12.7068			−0.409047
$966$		0			0
$967$	−48.5562			−1.56146			−0.780730	−	0.624868i	$-0.785153\pi$
$967$	−48.5562			−1.56146			−0.780730	+	0.624868i	$0.785153\pi$
$968$	30.0941			0.967260
$969$		0			0
$970$		−	1.95056i		−	0.0626288i
$971$			53.2613i			1.70924i	0.519256	+	0.854619i	$0.326209\pi$
$971$			53.2613i			1.70924i	−0.519256	+	0.854619i	$0.673791\pi$
$972$		0			0
$973$	−11.6176			−0.372443
$974$			30.8317i			0.987912i
$975$		0			0
$976$		−	3.47889i		−	0.111356i
$977$		−	9.15802i		−	0.292991i	−0.989211	−	0.146496i	$-0.953201\pi$
$977$		−	9.15802i		−	0.292991i	0.989211	−	0.146496i	$-0.0467994\pi$
$978$		0			0
$979$		−	59.7672i		−	1.91017i
$980$	−0.817085			−0.0261008
$981$		0			0
$982$			27.6790i			0.883273i
$983$	−10.2854			−0.328053			−0.164027	−	0.986456i	$-0.552448\pi$
$983$	−10.2854			−0.328053			−0.164027	+	0.986456i	$0.552448\pi$
$984$		0			0
$985$	−7.69464			−0.245172
$986$	24.4545			0.778789
$987$		0			0
$988$		−	0.417735i		−	0.0132899i
$989$	−23.8073			−0.757027
$990$		0			0
$991$		−	16.6943i		−	0.530310i	−0.964206	−	0.265155i	$-0.914577\pi$
$991$		−	16.6943i		−	0.530310i	0.964206	−	0.265155i	$-0.0854232\pi$
$992$		−	8.20722i		−	0.260580i
$993$		0			0
$994$	−33.6787			−1.06822
$995$	−5.58968			−0.177205
$996$		0			0
$997$	5.36222			0.169823			0.0849116	−	0.996388i	$-0.472939\pi$
$997$	5.36222			0.169823			0.0849116	+	0.996388i	$0.472939\pi$
$998$			27.3679i			0.866314i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6030.2.d.l.2411.6	yes	24
3.2	odd	2		6030.2.d.k.2411.6	✓	24
67.66	odd	2		6030.2.d.k.2411.19	yes	24
201.200	even	2	inner	6030.2.d.l.2411.19	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6030.2.d.k.2411.6	✓	24	3.2	odd	2
6030.2.d.k.2411.19	yes	24	67.66	odd	2
6030.2.d.l.2411.6	yes	24	1.1	even	1	trivial
6030.2.d.l.2411.19	yes	24	201.200	even	2	inner

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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -2.79591i q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -2.79591i q^{7} +1.00000 q^{8} +1.00000 q^{10} +6.41047 q^{11} +0.0604140i q^{13} -2.79591i q^{14} +1.00000 q^{16} -2.85617i q^{17} -6.91454 q^{19} +1.00000 q^{20} +6.41047 q^{22} -4.79630i q^{23} +1.00000 q^{25} +0.0604140i q^{26} -2.79591i q^{28} +8.56199i q^{29} -8.20722i q^{31} +1.00000 q^{32} -2.85617i q^{34} -2.79591i q^{35} +4.75004 q^{37} -6.91454 q^{38} +1.00000 q^{40} +6.22096 q^{41} -4.96367i q^{43} +6.41047 q^{44} -4.79630i q^{46} +4.52707i q^{47} -0.817085 q^{49} +1.00000 q^{50} +0.0604140i q^{52} +1.15009 q^{53} +6.41047 q^{55} -2.79591i q^{56} +8.56199i q^{58} -3.38038i q^{59} -3.47889i q^{61} -8.20722i q^{62} +1.00000 q^{64} +0.0604140i q^{65} +(-8.06126 - 1.41987i) q^{67} -2.85617i q^{68} -2.79591i q^{70} -12.0457i q^{71} -6.84440 q^{73} +4.75004 q^{74} -6.91454 q^{76} -17.9231i q^{77} +10.7379i q^{79} +1.00000 q^{80} +6.22096 q^{82} +7.74203i q^{83} -2.85617i q^{85} -4.96367i q^{86} +6.41047 q^{88} -9.32337i q^{89} +0.168912 q^{91} -4.79630i q^{92} +4.52707i q^{94} -6.91454 q^{95} -1.95056i q^{97} -0.817085 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 24 q^{2} + 24 q^{4} + 24 q^{5} + 24 q^{8}+O(q^{10}) \) 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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