LMFDB - Embedded newform 6030.2.d.k.2411.4

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			2411.4
Character	$\chi$	$=$	6030.2411
Dual form			6030.2.d.k.2411.21

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -3.30563i q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -3.30563i q^{7} -1.00000 q^{8} +1.00000 q^{10} -2.68928 q^{11} +4.10204i q^{13} +3.30563i q^{14} +1.00000 q^{16} +4.71660i q^{17} +2.81067 q^{19} -1.00000 q^{20} +2.68928 q^{22} -8.60320i q^{23} +1.00000 q^{25} -4.10204i q^{26} -3.30563i q^{28} -8.18213i q^{29} +5.37819i q^{31} -1.00000 q^{32} -4.71660i q^{34} +3.30563i q^{35} +6.45255 q^{37} -2.81067 q^{38} +1.00000 q^{40} -1.14877 q^{41} +11.9768i q^{43} -2.68928 q^{44} +8.60320i q^{46} -3.96365i q^{47} -3.92720 q^{49} -1.00000 q^{50} +4.10204i q^{52} -4.32739 q^{53} +2.68928 q^{55} +3.30563i q^{56} +8.18213i q^{58} +14.1803i q^{59} -4.25770i q^{61} -5.37819i q^{62} +1.00000 q^{64} -4.10204i q^{65} +(-4.90665 - 6.55170i) q^{67} +4.71660i q^{68} -3.30563i q^{70} -2.94339i q^{71} +4.89553 q^{73} -6.45255 q^{74} +2.81067 q^{76} +8.88978i q^{77} -14.9444i q^{79} -1.00000 q^{80} +1.14877 q^{82} +3.04264i q^{83} -4.71660i q^{85} -11.9768i q^{86} +2.68928 q^{88} +0.141891i q^{89} +13.5598 q^{91} -8.60320i q^{92} +3.96365i q^{94} -2.81067 q^{95} -15.8601i q^{97} +3.92720 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$		−	3.30563i		−	1.24941i	−0.780860	−	0.624706i	$-0.785219\pi$
$7$		−	3.30563i		−	1.24941i	0.780860	−	0.624706i	$-0.214781\pi$
$8$	−1.00000			−0.353553
$9$		0			0
$10$	1.00000			0.316228
$11$	−2.68928			−0.810850			−0.405425	−	0.914128i	$-0.632876\pi$
$11$	−2.68928			−0.810850			−0.405425	+	0.914128i	$0.632876\pi$
$12$		0			0
$13$			4.10204i			1.13770i	0.822441	+	0.568851i	$0.192612\pi$
$13$			4.10204i			1.13770i	−0.822441	+	0.568851i	$0.807388\pi$
$14$			3.30563i			0.883467i
$15$		0			0
$16$	1.00000			0.250000
$17$			4.71660i			1.14394i	0.820273	+	0.571972i	$0.193821\pi$
$17$			4.71660i			1.14394i	−0.820273	+	0.571972i	$0.806179\pi$
$18$		0			0
$19$	2.81067			0.644812			0.322406	−	0.946601i	$-0.395508\pi$
$19$	2.81067			0.644812			0.322406	+	0.946601i	$0.395508\pi$
$20$	−1.00000			−0.223607
$21$		0			0
$22$	2.68928			0.573357
$23$		−	8.60320i		−	1.79389i	−0.442140	−	0.896946i	$-0.645781\pi$
$23$		−	8.60320i		−	1.79389i	0.442140	−	0.896946i	$-0.354219\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		−	4.10204i		−	0.804477i
$27$		0			0
$28$		−	3.30563i		−	0.624706i
$29$		−	8.18213i		−	1.51938i	−0.650283	−	0.759692i	$-0.725350\pi$
$29$		−	8.18213i		−	1.51938i	0.650283	−	0.759692i	$-0.274650\pi$
$30$		0			0
$31$			5.37819i			0.965952i	0.875634	+	0.482976i	$0.160444\pi$
$31$			5.37819i			0.965952i	−0.875634	+	0.482976i	$0.839556\pi$
$32$	−1.00000			−0.176777
$33$		0			0
$34$		−	4.71660i		−	0.808891i
$35$			3.30563i			0.558754i
$36$		0			0
$37$	6.45255			1.06079			0.530396	−	0.847750i	$-0.322043\pi$
$37$	6.45255			1.06079			0.530396	+	0.847750i	$0.322043\pi$
$38$	−2.81067			−0.455951
$39$		0			0
$40$	1.00000			0.158114
$41$	−1.14877			−0.179408			−0.0897038	−	0.995968i	$-0.528592\pi$
$41$	−1.14877			−0.179408			−0.0897038	+	0.995968i	$0.528592\pi$
$42$		0			0
$43$			11.9768i			1.82645i	0.407458	+	0.913224i	$0.366415\pi$
$43$			11.9768i			1.82645i	−0.407458	+	0.913224i	$0.633585\pi$
$44$	−2.68928			−0.405425
$45$		0			0
$46$			8.60320i			1.26847i
$47$		−	3.96365i		−	0.578158i	−0.957305	−	0.289079i	$-0.906651\pi$
$47$		−	3.96365i		−	0.578158i	0.957305	−	0.289079i	$-0.0933490\pi$
$48$		0			0
$49$	−3.92720			−0.561029
$50$	−1.00000			−0.141421
$51$		0			0
$52$			4.10204i			0.568851i
$53$	−4.32739			−0.594412			−0.297206	−	0.954813i	$-0.596055\pi$
$53$	−4.32739			−0.594412			−0.297206	+	0.954813i	$0.596055\pi$
$54$		0			0
$55$	2.68928			0.362623
$56$			3.30563i			0.441734i
$57$		0			0
$58$			8.18213i			1.07437i
$59$			14.1803i			1.84611i	0.384665	+	0.923056i	$0.374317\pi$
$59$			14.1803i			1.84611i	−0.384665	+	0.923056i	$0.625683\pi$
$60$		0			0
$61$		−	4.25770i		−	0.545143i	−0.962136	−	0.272572i	$-0.912126\pi$
$61$		−	4.25770i		−	0.545143i	0.962136	−	0.272572i	$-0.0878741\pi$
$62$		−	5.37819i		−	0.683031i
$63$		0			0
$64$	1.00000			0.125000
$65$		−	4.10204i		−	0.508796i
$66$		0			0
$67$	−4.90665	−	6.55170i	−0.599442	−	0.800418i
$67$	−4.90665	−	6.55170i	−0.599442	−	0.800418i
$68$			4.71660i			0.571972i
$69$		0			0
$70$		−	3.30563i		−	0.395099i
$71$		−	2.94339i		−	0.349316i	−0.984629	−	0.174658i	$-0.944118\pi$
$71$		−	2.94339i		−	0.349316i	0.984629	−	0.174658i	$-0.0558820\pi$
$72$		0			0
$73$	4.89553			0.572979			0.286489	−	0.958083i	$-0.407512\pi$
$73$	4.89553			0.572979			0.286489	+	0.958083i	$0.407512\pi$
$74$	−6.45255			−0.750094
$75$		0			0
$76$	2.81067			0.322406
$77$			8.88978i			1.01308i
$78$		0			0
$79$		−	14.9444i		−	1.68138i	−0.541519	−	0.840689i	$-0.682150\pi$
$79$		−	14.9444i		−	1.68138i	0.541519	−	0.840689i	$-0.317850\pi$
$80$	−1.00000			−0.111803
$81$		0			0
$82$	1.14877			0.126860
$83$			3.04264i			0.333974i	0.985959	+	0.166987i	$0.0534037\pi$
$83$			3.04264i			0.333974i	−0.985959	+	0.166987i	$0.946596\pi$
$84$		0			0
$85$		−	4.71660i		−	0.511587i
$86$		−	11.9768i		−	1.29149i
$87$		0			0
$88$	2.68928			0.286679
$89$			0.141891i			0.0150404i	0.999972	+	0.00752021i	$0.00239378\pi$
$89$			0.141891i			0.0150404i	−0.999972	+	0.00752021i	$0.997606\pi$
$90$		0			0
$91$	13.5598			1.42146
$92$		−	8.60320i		−	0.896946i
$93$		0			0
$94$			3.96365i			0.408819i
$95$	−2.81067			−0.288369
$96$		0			0
$97$		−	15.8601i		−	1.61035i	−0.593035	−	0.805176i	$-0.702071\pi$
$97$		−	15.8601i		−	1.61035i	0.593035	−	0.805176i	$-0.297929\pi$
$98$	3.92720			0.396707
$99$		0			0
$100$	1.00000			0.100000
$101$	0.194963			0.0193995			0.00969977	−	0.999953i	$-0.496912\pi$
$101$	0.194963			0.0193995			0.00969977	+	0.999953i	$0.496912\pi$
$102$		0			0
$103$	18.4691			1.81981			0.909907	−	0.414812i	$-0.136153\pi$
$103$	18.4691			1.81981			0.909907	+	0.414812i	$0.136153\pi$
$104$		−	4.10204i		−	0.402238i
$105$		0			0
$106$	4.32739			0.420313
$107$			7.28319i			0.704093i	0.935983	+	0.352046i	$0.114514\pi$
$107$			7.28319i			0.704093i	−0.935983	+	0.352046i	$0.885486\pi$
$108$		0			0
$109$			8.39609i			0.804200i	0.915596	+	0.402100i	$0.131720\pi$
$109$			8.39609i			0.804200i	−0.915596	+	0.402100i	$0.868280\pi$
$110$	−2.68928			−0.256413
$111$		0			0
$112$		−	3.30563i		−	0.312353i
$113$	10.8843			1.02391			0.511954	−	0.859013i	$-0.328922\pi$
$113$	10.8843			1.02391			0.511954	+	0.859013i	$0.328922\pi$
$114$		0			0
$115$			8.60320i			0.802253i
$116$		−	8.18213i		−	0.759692i
$117$		0			0
$118$		−	14.1803i		−	1.30540i
$119$	15.5914			1.42926
$120$		0			0
$121$	−3.76775			−0.342523
$122$			4.25770i			0.385474i
$123$		0			0
$124$			5.37819i			0.482976i
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	21.3401			1.89363			0.946814	−	0.321781i	$-0.104282\pi$
$127$	21.3401			1.89363			0.946814	+	0.321781i	$0.104282\pi$
$128$	−1.00000			−0.0883883
$129$		0			0
$130$			4.10204i			0.359773i
$131$		−	16.4565i		−	1.43782i	−0.695106	−	0.718908i	$-0.744642\pi$
$131$		−	16.4565i		−	1.43782i	0.695106	−	0.718908i	$-0.255358\pi$
$132$		0			0
$133$		−	9.29104i		−	0.805635i
$134$	4.90665	+	6.55170i	0.423870	+	0.565981i
$135$		0			0
$136$		−	4.71660i		−	0.404445i
$137$	−15.2622			−1.30394			−0.651971	−	0.758244i	$-0.726058\pi$
$137$	−15.2622			−1.30394			−0.651971	+	0.758244i	$0.726058\pi$
$138$		0			0
$139$		−	20.0554i		−	1.70108i	−0.525912	−	0.850539i	$-0.676276\pi$
$139$		−	20.0554i		−	1.70108i	0.525912	−	0.850539i	$-0.323724\pi$
$140$			3.30563i			0.279377i
$141$		0			0
$142$			2.94339i			0.247004i
$143$		−	11.0316i		−	0.922505i
$144$		0			0
$145$			8.18213i			0.679489i
$146$	−4.89553			−0.405157
$147$		0			0
$148$	6.45255			0.530396
$149$		−	12.0920i		−	0.990618i	−0.868717	−	0.495309i	$-0.835055\pi$
$149$		−	12.0920i		−	0.990618i	0.868717	−	0.495309i	$-0.164945\pi$
$150$		0			0
$151$	−9.49449			−0.772651			−0.386325	−	0.922363i	$-0.626256\pi$
$151$	−9.49449			−0.772651			−0.386325	+	0.922363i	$0.626256\pi$
$152$	−2.81067			−0.227975
$153$		0			0
$154$		−	8.88978i		−	0.716359i
$155$		−	5.37819i		−	0.431987i
$156$		0			0
$157$	−11.0453			−0.881513			−0.440756	−	0.897627i	$-0.645290\pi$
$157$	−11.0453			−0.881513			−0.440756	+	0.897627i	$0.645290\pi$
$158$			14.9444i			1.18891i
$159$		0			0
$160$	1.00000			0.0790569
$161$	−28.4390			−2.24131
$162$		0			0
$163$	−20.5292			−1.60797			−0.803985	−	0.594650i	$-0.797291\pi$
$163$	−20.5292			−1.60797			−0.803985	+	0.594650i	$0.797291\pi$
$164$	−1.14877			−0.0897038
$165$		0			0
$166$		−	3.04264i		−	0.236155i
$167$			16.8432i			1.30337i	0.758491	+	0.651684i	$0.225937\pi$
$167$			16.8432i			1.30337i	−0.758491	+	0.651684i	$0.774063\pi$
$168$		0			0
$169$	−3.82676			−0.294366
$170$			4.71660i			0.361747i
$171$		0			0
$172$			11.9768i			0.913224i
$173$			1.30491i			0.0992108i	0.998769	+	0.0496054i	$0.0157964\pi$
$173$			1.30491i			0.0992108i	−0.998769	+	0.0496054i	$0.984204\pi$
$174$		0			0
$175$		−	3.30563i		−	0.249882i
$176$	−2.68928			−0.202712
$177$		0			0
$178$		−	0.141891i		−	0.0106352i
$179$	−13.3950			−1.00119			−0.500594	−	0.865682i	$-0.666885\pi$
$179$	−13.3950			−1.00119			−0.500594	+	0.865682i	$0.666885\pi$
$180$		0			0
$181$	−19.9900			−1.48585			−0.742924	−	0.669375i	$-0.766562\pi$
$181$	−19.9900			−1.48585			−0.742924	+	0.669375i	$0.766562\pi$
$182$	−13.5598			−1.00512
$183$		0			0
$184$			8.60320i			0.634237i
$185$	−6.45255			−0.474401
$186$		0			0
$187$		−	12.6843i		−	0.927567i
$188$		−	3.96365i		−	0.289079i
$189$		0			0
$190$	2.81067			0.203907
$191$	20.4225			1.47772			0.738862	−	0.673857i	$-0.235364\pi$
$191$	20.4225			1.47772			0.738862	+	0.673857i	$0.235364\pi$
$192$		0			0
$193$	0.661297			0.0476012			0.0238006	−	0.999717i	$-0.492423\pi$
$193$	0.661297			0.0476012			0.0238006	+	0.999717i	$0.492423\pi$
$194$			15.8601i			1.13869i
$195$		0			0
$196$	−3.92720			−0.280515
$197$	−2.96805			−0.211465			−0.105732	−	0.994395i	$-0.533719\pi$
$197$	−2.96805			−0.211465			−0.105732	+	0.994395i	$0.533719\pi$
$198$		0			0
$199$	13.4754			0.955248			0.477624	−	0.878564i	$-0.341498\pi$
$199$	13.4754			0.955248			0.477624	+	0.878564i	$0.341498\pi$
$200$	−1.00000			−0.0707107
$201$		0			0
$202$	−0.194963			−0.0137175
$203$	−27.0471			−1.89834
$204$		0			0
$205$	1.14877			0.0802335
$206$	−18.4691			−1.28680
$207$		0			0
$208$			4.10204i			0.284426i
$209$	−7.55869			−0.522846
$210$		0			0
$211$	−17.8794			−1.23087			−0.615436	−	0.788187i	$-0.711020\pi$
$211$	−17.8794			−1.23087			−0.615436	+	0.788187i	$0.711020\pi$
$212$	−4.32739			−0.297206
$213$		0			0
$214$		−	7.28319i		−	0.497869i
$215$		−	11.9768i		−	0.816812i
$216$		0			0
$217$	17.7783			1.20687
$218$		−	8.39609i		−	0.568655i
$219$		0			0
$220$	2.68928			0.181312
$221$	−19.3477			−1.30147
$222$		0			0
$223$	21.8794			1.46515			0.732577	−	0.680685i	$-0.238318\pi$
$223$	21.8794			1.46515			0.732577	+	0.680685i	$0.238318\pi$
$224$			3.30563i			0.220867i
$225$		0			0
$226$	−10.8843			−0.724012
$227$		−	1.01576i		−	0.0674183i	−0.999432	−	0.0337091i	$-0.989268\pi$
$227$		−	1.01576i		−	0.0674183i	0.999432	−	0.0337091i	$-0.0107320\pi$
$228$		0			0
$229$			2.40187i			0.158720i	0.996846	+	0.0793601i	$0.0252877\pi$
$229$			2.40187i			0.158720i	−0.996846	+	0.0793601i	$0.974712\pi$
$230$		−	8.60320i		−	0.567278i
$231$		0			0
$232$			8.18213i			0.537183i
$233$	−5.07539			−0.332500			−0.166250	−	0.986084i	$-0.553166\pi$
$233$	−5.07539			−0.332500			−0.166250	+	0.986084i	$0.553166\pi$
$234$		0			0
$235$			3.96365i			0.258560i
$236$			14.1803i			0.923056i
$237$		0			0
$238$	−15.5914			−1.01064
$239$	24.0427			1.55519			0.777597	−	0.628762i	$-0.216438\pi$
$239$	24.0427			1.55519			0.777597	+	0.628762i	$0.216438\pi$
$240$		0			0
$241$	−25.7150			−1.65645			−0.828224	−	0.560397i	$-0.810649\pi$
$241$	−25.7150			−1.65645			−0.828224	+	0.560397i	$0.810649\pi$
$242$	3.76775			0.242200
$243$		0			0
$244$		−	4.25770i		−	0.272572i
$245$	3.92720			0.250900
$246$		0			0
$247$			11.5295i			0.733604i
$248$		−	5.37819i		−	0.341515i
$249$		0			0
$250$	1.00000			0.0632456
$251$	−8.23937			−0.520064			−0.260032	−	0.965600i	$-0.583733\pi$
$251$	−8.23937			−0.520064			−0.260032	+	0.965600i	$0.583733\pi$
$252$		0			0
$253$			23.1365i			1.45458i
$254$	−21.3401			−1.33900
$255$		0			0
$256$	1.00000			0.0625000
$257$			1.95202i			0.121764i	0.998145	+	0.0608819i	$0.0193913\pi$
$257$			1.95202i			0.121764i	−0.998145	+	0.0608819i	$0.980609\pi$
$258$		0			0
$259$		−	21.3298i		−	1.32537i
$260$		−	4.10204i		−	0.254398i
$261$		0			0
$262$			16.4565i			1.01669i
$263$		−	16.5607i		−	1.02118i	−0.859826	−	0.510588i	$-0.829428\pi$
$263$		−	16.5607i		−	1.02118i	0.859826	−	0.510588i	$-0.170572\pi$
$264$		0			0
$265$	4.32739			0.265829
$266$			9.29104i			0.569670i
$267$		0			0
$268$	−4.90665	−	6.55170i	−0.299721	−	0.400209i
$269$		−	27.5029i		−	1.67688i	−0.544992	−	0.838441i	$-0.683467\pi$
$269$		−	27.5029i		−	1.67688i	0.544992	−	0.838441i	$-0.316533\pi$
$270$		0			0
$271$		−	11.7163i		−	0.711717i	−0.934540	−	0.355859i	$-0.884188\pi$
$271$		−	11.7163i		−	0.711717i	0.934540	−	0.355859i	$-0.115812\pi$
$272$			4.71660i			0.285986i
$273$		0			0
$274$	15.2622			0.922026
$275$	−2.68928			−0.162170
$276$		0			0
$277$	−24.0327			−1.44398			−0.721991	−	0.691902i	$-0.756773\pi$
$277$	−24.0327			−1.44398			−0.721991	+	0.691902i	$0.756773\pi$
$278$			20.0554i			1.20284i
$279$		0			0
$280$		−	3.30563i		−	0.197549i
$281$	−13.0652			−0.779405			−0.389702	−	0.920941i	$-0.627422\pi$
$281$	−13.0652			−0.779405			−0.389702	+	0.920941i	$0.627422\pi$
$282$		0			0
$283$	−10.0211			−0.595694			−0.297847	−	0.954614i	$-0.596269\pi$
$283$	−10.0211			−0.595694			−0.297847	+	0.954614i	$0.596269\pi$
$284$		−	2.94339i		−	0.174658i
$285$		0			0
$286$			11.0316i			0.652310i
$287$			3.79741i			0.224154i
$288$		0			0
$289$	−5.24634			−0.308608
$290$		−	8.18213i		−	0.480471i
$291$		0			0
$292$	4.89553			0.286489
$293$			1.73967i			0.101633i	0.998708	+	0.0508164i	$0.0161823\pi$
$293$			1.73967i			0.101633i	−0.998708	+	0.0508164i	$0.983818\pi$
$294$		0			0
$295$		−	14.1803i		−	0.825607i
$296$	−6.45255			−0.375047
$297$		0			0
$298$			12.0920i			0.700473i
$299$	35.2907			2.04091
$300$		0			0
$301$	39.5910			2.28199
$302$	9.49449			0.546346
$303$		0			0
$304$	2.81067			0.161203
$305$			4.25770i			0.243795i
$306$		0			0
$307$	27.9276			1.59392			0.796958	−	0.604035i	$-0.206441\pi$
$307$	27.9276			1.59392			0.796958	+	0.604035i	$0.206441\pi$
$308$			8.88978i			0.506542i
$309$		0			0
$310$			5.37819i			0.305461i
$311$	−9.38052			−0.531921			−0.265960	−	0.963984i	$-0.585689\pi$
$311$	−9.38052			−0.531921			−0.265960	+	0.963984i	$0.585689\pi$
$312$		0			0
$313$		−	19.4430i		−	1.09898i	−0.835499	−	0.549491i	$-0.814821\pi$
$313$		−	19.4430i		−	1.09898i	0.835499	−	0.549491i	$-0.185179\pi$
$314$	11.0453			0.623324
$315$		0			0
$316$		−	14.9444i		−	0.840689i
$317$		−	17.2994i		−	0.971629i	−0.874062	−	0.485815i	$-0.838523\pi$
$317$		−	17.2994i		−	0.971629i	0.874062	−	0.485815i	$-0.161477\pi$
$318$		0			0
$319$			22.0041i			1.23199i
$320$	−1.00000			−0.0559017
$321$		0			0
$322$	28.4390			1.58484
$323$			13.2568i			0.737629i
$324$		0			0
$325$			4.10204i			0.227540i
$326$	20.5292			1.13701
$327$		0			0
$328$	1.14877			0.0634302
$329$	−13.1024			−0.722357
$330$		0			0
$331$		−	28.5513i		−	1.56932i	−0.619924	−	0.784661i	$-0.712837\pi$
$331$		−	28.5513i		−	1.56932i	0.619924	−	0.784661i	$-0.287163\pi$
$332$			3.04264i			0.166987i
$333$		0			0
$334$		−	16.8432i		−	0.921620i
$335$	4.90665	+	6.55170i	0.268079	+	0.357958i
$336$		0			0
$337$		−	17.0521i		−	0.928888i	−0.885603	−	0.464444i	$-0.846254\pi$
$337$		−	17.0521i		−	0.928888i	0.885603	−	0.464444i	$-0.153746\pi$
$338$	3.82676			0.208148
$339$		0			0
$340$		−	4.71660i		−	0.255794i
$341$		−	14.4635i		−	0.783242i
$342$		0			0
$343$		−	10.1575i		−	0.548455i
$344$		−	11.9768i		−	0.645747i
$345$		0			0
$346$		−	1.30491i		−	0.0701526i
$347$	−0.646328			−0.0346967			−0.0173484	−	0.999850i	$-0.505522\pi$
$347$	−0.646328			−0.0346967			−0.0173484	+	0.999850i	$0.505522\pi$
$348$		0			0
$349$	−25.2033			−1.34910			−0.674552	−	0.738227i	$-0.735663\pi$
$349$	−25.2033			−1.34910			−0.674552	+	0.738227i	$0.735663\pi$
$350$			3.30563i			0.176693i
$351$		0			0
$352$	2.68928			0.143339
$353$	1.82608			0.0971922			0.0485961	−	0.998819i	$-0.484525\pi$
$353$	1.82608			0.0971922			0.0485961	+	0.998819i	$0.484525\pi$
$354$		0			0
$355$			2.94339i			0.156219i
$356$			0.141891i			0.00752021i
$357$		0			0
$358$	13.3950			0.707946
$359$			10.2897i			0.543070i	0.962429	+	0.271535i	$0.0875312\pi$
$359$			10.2897i			0.543070i	−0.962429	+	0.271535i	$0.912469\pi$
$360$		0			0
$361$	−11.1001			−0.584218
$362$	19.9900			1.05065
$363$		0			0
$364$	13.5598			0.710729
$365$	−4.89553			−0.256244
$366$		0			0
$367$			1.02445i			0.0534757i	0.999642	+	0.0267378i	$0.00851193\pi$
$367$			1.02445i			0.0534757i	−0.999642	+	0.0267378i	$0.991488\pi$
$368$		−	8.60320i		−	0.448473i
$369$		0			0
$370$	6.45255			0.335452
$371$			14.3048i			0.742666i
$372$		0			0
$373$			14.9557i			0.774377i	0.922000	+	0.387189i	$0.126554\pi$
$373$			14.9557i			0.774377i	−0.922000	+	0.387189i	$0.873446\pi$
$374$			12.6843i			0.655889i
$375$		0			0
$376$			3.96365i			0.204410i
$377$	33.5635			1.72861
$378$		0			0
$379$		−	10.1853i		−	0.523182i	−0.965179	−	0.261591i	$-0.915753\pi$
$379$		−	10.1853i		−	0.523182i	0.965179	−	0.261591i	$-0.0842471\pi$
$380$	−2.81067			−0.144184
$381$		0			0
$382$	−20.4225			−1.04491
$383$	−0.756650			−0.0386630			−0.0193315	−	0.999813i	$-0.506154\pi$
$383$	−0.756650			−0.0386630			−0.0193315	+	0.999813i	$0.506154\pi$
$384$		0			0
$385$		−	8.88978i		−	0.453065i
$386$	−0.661297			−0.0336591
$387$		0			0
$388$		−	15.8601i		−	0.805176i
$389$		−	10.5446i		−	0.534630i	−0.963609	−	0.267315i	$-0.913864\pi$
$389$		−	10.5446i		−	0.534630i	0.963609	−	0.267315i	$-0.0861364\pi$
$390$		0			0
$391$	40.5779			2.05211
$392$	3.92720			0.198354
$393$		0			0
$394$	2.96805			0.149528
$395$			14.9444i			0.751935i
$396$		0			0
$397$	15.1027			0.757985			0.378993	−	0.925400i	$-0.376271\pi$
$397$	15.1027			0.757985			0.378993	+	0.925400i	$0.376271\pi$
$398$	−13.4754			−0.675462
$399$		0			0
$400$	1.00000			0.0500000
$401$	−17.5526			−0.876536			−0.438268	−	0.898844i	$-0.644408\pi$
$401$	−17.5526			−0.876536			−0.438268	+	0.898844i	$0.644408\pi$
$402$		0			0
$403$	−22.0616			−1.09897
$404$	0.194963			0.00969977
$405$		0			0
$406$	27.0471			1.34233
$407$	−17.3527			−0.860144
$408$		0			0
$409$			6.37681i			0.315313i	0.987494	+	0.157656i	$0.0503939\pi$
$409$			6.37681i			0.315313i	−0.987494	+	0.157656i	$0.949606\pi$
$410$	−1.14877			−0.0567337
$411$		0			0
$412$	18.4691			0.909907
$413$	46.8747			2.30655
$414$		0			0
$415$		−	3.04264i		−	0.149358i
$416$		−	4.10204i		−	0.201119i
$417$		0			0
$418$	7.55869			0.369708
$419$		−	31.8808i		−	1.55748i	−0.627348	−	0.778739i	$-0.715860\pi$
$419$		−	31.8808i		−	1.55748i	0.627348	−	0.778739i	$-0.284140\pi$
$420$		0			0
$421$	−31.9854			−1.55888			−0.779438	−	0.626480i	$-0.784495\pi$
$421$	−31.9854			−1.55888			−0.779438	+	0.626480i	$0.784495\pi$
$422$	17.8794			0.870357
$423$		0			0
$424$	4.32739			0.210157
$425$			4.71660i			0.228789i
$426$		0			0
$427$	−14.0744			−0.681108
$428$			7.28319i			0.352046i
$429$		0			0
$430$			11.9768i			0.577574i
$431$		−	19.8635i		−	0.956789i	−0.878145	−	0.478394i	$-0.841219\pi$
$431$		−	19.8635i		−	0.956789i	0.878145	−	0.478394i	$-0.158781\pi$
$432$		0			0
$433$		−	22.7355i		−	1.09260i	−0.837590	−	0.546300i	$-0.816036\pi$
$433$		−	22.7355i		−	1.09260i	0.837590	−	0.546300i	$-0.183964\pi$
$434$	−17.7783			−0.853387
$435$		0			0
$436$			8.39609i			0.402100i
$437$		−	24.1808i		−	1.15672i
$438$		0			0
$439$	12.0596			0.575573			0.287787	−	0.957695i	$-0.407081\pi$
$439$	12.0596			0.575573			0.287787	+	0.957695i	$0.407081\pi$
$440$	−2.68928			−0.128207
$441$		0			0
$442$	19.3477			0.920277
$443$	−8.67567			−0.412193			−0.206097	−	0.978532i	$-0.566076\pi$
$443$	−8.67567			−0.412193			−0.206097	+	0.978532i	$0.566076\pi$
$444$		0			0
$445$		−	0.141891i		−	0.00672628i
$446$	−21.8794			−1.03602
$447$		0			0
$448$		−	3.30563i		−	0.156176i
$449$		−	15.4850i		−	0.730782i	−0.930854	−	0.365391i	$-0.880935\pi$
$449$		−	15.4850i		−	0.730782i	0.930854	−	0.365391i	$-0.119065\pi$
$450$		0			0
$451$	3.08937			0.145473
$452$	10.8843			0.511954
$453$		0			0
$454$			1.01576i			0.0476719i
$455$	−13.5598			−0.635695
$456$		0			0
$457$	−5.53703			−0.259012			−0.129506	−	0.991579i	$-0.541339\pi$
$457$	−5.53703			−0.259012			−0.129506	+	0.991579i	$0.541339\pi$
$458$		−	2.40187i		−	0.112232i
$459$		0			0
$460$			8.60320i			0.401126i
$461$		−	27.4327i		−	1.27767i	−0.769344	−	0.638835i	$-0.779417\pi$
$461$		−	27.4327i		−	1.27767i	0.769344	−	0.638835i	$-0.220583\pi$
$462$		0			0
$463$		−	41.2078i		−	1.91509i	−0.288289	−	0.957544i	$-0.593086\pi$
$463$		−	41.2078i		−	1.91509i	0.288289	−	0.957544i	$-0.406914\pi$
$464$		−	8.18213i		−	0.379846i
$465$		0			0
$466$	5.07539			0.235113
$467$			32.6360i			1.51021i	0.655603	+	0.755106i	$0.272415\pi$
$467$			32.6360i			1.51021i	−0.655603	+	0.755106i	$0.727585\pi$
$468$		0			0
$469$	−21.6575	+	16.2196i	−1.00005	+	0.748950i
$470$		−	3.96365i		−	0.182830i
$471$		0			0
$472$		−	14.1803i		−	0.652699i
$473$		−	32.2091i		−	1.48098i
$474$		0			0
$475$	2.81067			0.128962
$476$	15.5914			0.714628
$477$		0			0
$478$	−24.0427			−1.09969
$479$		−	8.53795i		−	0.390109i	−0.980792	−	0.195055i	$-0.937512\pi$
$479$		−	8.53795i		−	0.390109i	0.980792	−	0.195055i	$-0.0624884\pi$
$480$		0			0
$481$			26.4686i			1.20687i
$482$	25.7150			1.17129
$483$		0			0
$484$	−3.76775			−0.171261
$485$			15.8601i			0.720172i
$486$		0			0
$487$			33.5595i			1.52073i	0.649498	+	0.760363i	$0.274979\pi$
$487$			33.5595i			1.52073i	−0.649498	+	0.760363i	$0.725021\pi$
$488$			4.25770i			0.192737i
$489$		0			0
$490$	−3.92720			−0.177413
$491$		−	31.8313i		−	1.43653i	−0.695770	−	0.718264i	$-0.744937\pi$
$491$		−	31.8313i		−	1.43653i	0.695770	−	0.718264i	$-0.255063\pi$
$492$		0			0
$493$	38.5919			1.73809
$494$		−	11.5295i		−	0.518736i
$495$		0			0
$496$			5.37819i			0.241488i
$497$	−9.72976			−0.436439
$498$		0			0
$499$			23.2861i			1.04243i	0.853426	+	0.521213i	$0.174520\pi$
$499$			23.2861i			1.04243i	−0.853426	+	0.521213i	$0.825480\pi$
$500$	−1.00000			−0.0447214
$501$		0			0
$502$	8.23937			0.367741
$503$	−41.6904			−1.85889			−0.929443	−	0.368967i	$-0.879712\pi$
$503$	−41.6904			−1.85889			−0.929443	+	0.368967i	$0.879712\pi$
$504$		0			0
$505$	−0.194963			−0.00867574
$506$		−	23.1365i		−	1.02854i
$507$		0			0
$508$	21.3401			0.946814
$509$		−	31.5567i		−	1.39872i	−0.714767	−	0.699362i	$-0.753467\pi$
$509$		−	31.5567i		−	1.39872i	0.714767	−	0.699362i	$-0.246533\pi$
$510$		0			0
$511$		−	16.1828i		−	0.715886i
$512$	−1.00000			−0.0441942
$513$		0			0
$514$		−	1.95202i		−	0.0861000i
$515$	−18.4691			−0.813846
$516$		0			0
$517$			10.6594i			0.468799i
$518$			21.3298i			0.937176i
$519$		0			0
$520$			4.10204i			0.179886i
$521$	29.7132			1.30176			0.650880	−	0.759181i	$-0.274400\pi$
$521$	29.7132			1.30176			0.650880	+	0.759181i	$0.274400\pi$
$522$		0			0
$523$	−26.4738			−1.15762			−0.578809	−	0.815463i	$-0.696482\pi$
$523$	−26.4738			−1.15762			−0.578809	+	0.815463i	$0.696482\pi$
$524$		−	16.4565i		−	0.718908i
$525$		0			0
$526$			16.5607i			0.722080i
$527$	−25.3668			−1.10499
$528$		0			0
$529$	−51.0151			−2.21805
$530$	−4.32739			−0.187970
$531$		0			0
$532$		−	9.29104i		−	0.402818i
$533$		−	4.71230i		−	0.204112i
$534$		0			0
$535$		−	7.28319i		−	0.314880i
$536$	4.90665	+	6.55170i	0.211935	+	0.282990i
$537$		0			0
$538$			27.5029i			1.18574i
$539$	10.5614			0.454910
$540$		0			0
$541$		−	33.6640i		−	1.44733i	−0.690153	−	0.723663i	$-0.742457\pi$
$541$		−	33.6640i		−	1.44733i	0.690153	−	0.723663i	$-0.257543\pi$
$542$			11.7163i			0.503260i
$543$		0			0
$544$		−	4.71660i		−	0.202223i
$545$		−	8.39609i		−	0.359649i
$546$		0			0
$547$			42.0608i			1.79839i	0.437547	+	0.899196i	$0.355847\pi$
$547$			42.0608i			1.79839i	−0.437547	+	0.899196i	$0.644153\pi$
$548$	−15.2622			−0.651971
$549$		0			0
$550$	2.68928			0.114671
$551$		−	22.9973i		−	0.979717i
$552$		0			0
$553$	−49.4007			−2.10073
$554$	24.0327			1.02105
$555$		0			0
$556$		−	20.0554i		−	0.850539i
$557$			2.05555i			0.0870965i	0.999051	+	0.0435482i	$0.0138662\pi$
$557$			2.05555i			0.0870965i	−0.999051	+	0.0435482i	$0.986134\pi$
$558$		0			0
$559$	−49.1294			−2.07795
$560$			3.30563i			0.139688i
$561$		0			0
$562$	13.0652			0.551122
$563$	−22.9467			−0.967089			−0.483545	−	0.875320i	$-0.660651\pi$
$563$	−22.9467			−0.967089			−0.483545	+	0.875320i	$0.660651\pi$
$564$		0			0
$565$	−10.8843			−0.457905
$566$	10.0211			0.421220
$567$		0			0
$568$			2.94339i			0.123502i
$569$			9.93180i			0.416363i	0.978090	+	0.208181i	$0.0667544\pi$
$569$			9.93180i			0.416363i	−0.978090	+	0.208181i	$0.933246\pi$
$570$		0			0
$571$	16.8695			0.705965			0.352982	−	0.935630i	$-0.385168\pi$
$571$	16.8695			0.705965			0.352982	+	0.935630i	$0.385168\pi$
$572$		−	11.0316i		−	0.461253i
$573$		0			0
$574$		−	3.79741i		−	0.158501i
$575$		−	8.60320i		−	0.358778i
$576$		0			0
$577$			15.5688i			0.648139i	0.946033	+	0.324069i	$0.105051\pi$
$577$			15.5688i			0.648139i	−0.946033	+	0.324069i	$0.894949\pi$
$578$	5.24634			0.218219
$579$		0			0
$580$			8.18213i			0.339744i
$581$	10.0579			0.417270
$582$		0			0
$583$	11.6376			0.481979
$584$	−4.89553			−0.202579
$585$		0			0
$586$		−	1.73967i		−	0.0718652i
$587$	32.1199			1.32573			0.662864	−	0.748740i	$-0.269341\pi$
$587$	32.1199			1.32573			0.662864	+	0.748740i	$0.269341\pi$
$588$		0			0
$589$			15.1163i			0.622857i
$590$			14.1803i			0.583792i
$591$		0			0
$592$	6.45255			0.265198
$593$	3.49676			0.143595			0.0717973	−	0.997419i	$-0.477127\pi$
$593$	3.49676			0.143595			0.0717973	+	0.997419i	$0.477127\pi$
$594$		0			0
$595$	−15.5914			−0.639183
$596$		−	12.0920i		−	0.495309i
$597$		0			0
$598$	−35.2907			−1.44314
$599$	13.3690			0.546241			0.273120	−	0.961980i	$-0.411944\pi$
$599$	13.3690			0.546241			0.273120	+	0.961980i	$0.411944\pi$
$600$		0			0
$601$	32.0229			1.30624			0.653121	−	0.757253i	$-0.273459\pi$
$601$	32.0229			1.30624			0.653121	+	0.757253i	$0.273459\pi$
$602$	−39.5910			−1.61361
$603$		0			0
$604$	−9.49449			−0.386325
$605$	3.76775			0.153181
$606$		0			0
$607$	−18.4437			−0.748608			−0.374304	−	0.927306i	$-0.622118\pi$
$607$	−18.4437			−0.748608			−0.374304	+	0.927306i	$0.622118\pi$
$608$	−2.81067			−0.113988
$609$		0			0
$610$		−	4.25770i		−	0.172389i
$611$	16.2591			0.657771
$612$		0			0
$613$	34.1435			1.37904			0.689521	−	0.724266i	$-0.257821\pi$
$613$	34.1435			1.37904			0.689521	+	0.724266i	$0.257821\pi$
$614$	−27.9276			−1.12707
$615$		0			0
$616$		−	8.88978i		−	0.358180i
$617$			18.8310i			0.758108i	0.925375	+	0.379054i	$0.123751\pi$
$617$			18.8310i			0.758108i	−0.925375	+	0.379054i	$0.876249\pi$
$618$		0			0
$619$	25.0488			1.00680			0.503398	−	0.864055i	$-0.332083\pi$
$619$	25.0488			1.00680			0.503398	+	0.864055i	$0.332083\pi$
$620$		−	5.37819i		−	0.215993i
$621$		0			0
$622$	9.38052			0.376125
$623$	0.469039			0.0187917
$624$		0			0
$625$	1.00000			0.0400000
$626$			19.4430i			0.777098i
$627$		0			0
$628$	−11.0453			−0.440756
$629$			30.4341i			1.21349i
$630$		0			0
$631$		−	25.7832i		−	1.02641i	−0.858265	−	0.513206i	$-0.828458\pi$
$631$		−	25.7832i		−	1.02641i	0.858265	−	0.513206i	$-0.171542\pi$
$632$			14.9444i			0.594457i
$633$		0			0
$634$			17.2994i			0.687046i
$635$	−21.3401			−0.846856
$636$		0			0
$637$		−	16.1096i		−	0.638284i
$638$		−	22.0041i		−	0.871150i
$639$		0			0
$640$	1.00000			0.0395285
$641$	40.4566			1.59794			0.798971	−	0.601370i	$-0.205378\pi$
$641$	40.4566			1.59794			0.798971	+	0.601370i	$0.205378\pi$
$642$		0			0
$643$	18.8694			0.744136			0.372068	−	0.928206i	$-0.378649\pi$
$643$	18.8694			0.744136			0.372068	+	0.928206i	$0.378649\pi$
$644$	−28.4390			−1.12065
$645$		0			0
$646$		−	13.2568i		−	0.521582i
$647$	−13.0716			−0.513897			−0.256949	−	0.966425i	$-0.582717\pi$
$647$	−13.0716			−0.513897			−0.256949	+	0.966425i	$0.582717\pi$
$648$		0			0
$649$		−	38.1348i		−	1.49692i
$650$		−	4.10204i		−	0.160895i
$651$		0			0
$652$	−20.5292			−0.803985
$653$	7.13777			0.279323			0.139661	−	0.990199i	$-0.455399\pi$
$653$	7.13777			0.279323			0.139661	+	0.990199i	$0.455399\pi$
$654$		0			0
$655$			16.4565i			0.643011i
$656$	−1.14877			−0.0448519
$657$		0			0
$658$	13.1024			0.510784
$659$		−	0.609904i		−	0.0237585i	−0.999929	−	0.0118792i	$-0.996219\pi$
$659$		−	0.609904i		−	0.0237585i	0.999929	−	0.0118792i	$-0.00378137\pi$
$660$		0			0
$661$		−	10.3153i		−	0.401218i	−0.979671	−	0.200609i	$-0.935708\pi$
$661$		−	10.3153i		−	0.401218i	0.979671	−	0.200609i	$-0.0642921\pi$
$662$			28.5513i			1.10968i
$663$		0			0
$664$		−	3.04264i		−	0.118077i
$665$			9.29104i			0.360291i
$666$		0			0
$667$	−70.3925			−2.72561
$668$			16.8432i			0.651684i
$669$		0			0
$670$	−4.90665	−	6.55170i	−0.189560	−	0.253114i
$671$			11.4502i			0.442029i
$672$		0			0
$673$		−	28.6749i		−	1.10533i	−0.833402	−	0.552667i	$-0.813610\pi$
$673$		−	28.6749i		−	1.10533i	0.833402	−	0.552667i	$-0.186390\pi$
$674$			17.0521i			0.656823i
$675$		0			0
$676$	−3.82676			−0.147183
$677$	34.1394			1.31209			0.656043	−	0.754724i	$-0.272229\pi$
$677$	34.1394			1.31209			0.656043	+	0.754724i	$0.272229\pi$
$678$		0			0
$679$	−52.4278			−2.01199
$680$			4.71660i			0.180873i
$681$		0			0
$682$			14.4635i			0.553835i
$683$	42.3404			1.62011			0.810056	−	0.586353i	$-0.199437\pi$
$683$	42.3404			1.62011			0.810056	+	0.586353i	$0.199437\pi$
$684$		0			0
$685$	15.2622			0.583140
$686$			10.1575i			0.387817i
$687$		0			0
$688$			11.9768i			0.456612i
$689$		−	17.7511i		−	0.676264i
$690$		0			0
$691$	19.4005			0.738031			0.369015	−	0.929423i	$-0.379695\pi$
$691$	19.4005			0.738031			0.369015	+	0.929423i	$0.379695\pi$
$692$			1.30491i			0.0496054i
$693$		0			0
$694$	0.646328			0.0245343
$695$			20.0554i			0.760745i
$696$		0			0
$697$		−	5.41829i		−	0.205232i
$698$	25.2033			0.953960
$699$		0			0
$700$		−	3.30563i		−	0.124941i
$701$	−33.1831			−1.25331			−0.626654	−	0.779297i	$-0.715576\pi$
$701$	−33.1831			−1.25331			−0.626654	+	0.779297i	$0.715576\pi$
$702$		0			0
$703$	18.1360			0.684012
$704$	−2.68928			−0.101356
$705$		0			0
$706$	−1.82608			−0.0687253
$707$		−	0.644476i		−	0.0242380i
$708$		0			0
$709$	7.63754			0.286834			0.143417	−	0.989662i	$-0.454191\pi$
$709$	7.63754			0.286834			0.143417	+	0.989662i	$0.454191\pi$
$710$		−	2.94339i		−	0.110463i
$711$		0			0
$712$		−	0.141891i		−	0.00531759i
$713$	46.2697			1.73281
$714$		0			0
$715$			11.0316i			0.412557i
$716$	−13.3950			−0.500594
$717$		0			0
$718$		−	10.2897i		−	0.384008i
$719$			2.61194i			0.0974091i	0.998813	+	0.0487045i	$0.0155093\pi$
$719$			2.61194i			0.0974091i	−0.998813	+	0.0487045i	$0.984491\pi$
$720$		0			0
$721$		−	61.0521i		−	2.27370i
$722$	11.1001			0.413104
$723$		0			0
$724$	−19.9900			−0.742924
$725$		−	8.18213i		−	0.303877i
$726$		0			0
$727$		−	28.9746i		−	1.07461i	−0.843389	−	0.537304i	$-0.819443\pi$
$727$		−	28.9746i		−	1.07461i	0.843389	−	0.537304i	$-0.180557\pi$
$728$	−13.5598			−0.502561
$729$		0			0
$730$	4.89553			0.181192
$731$	−56.4899			−2.08935
$732$		0			0
$733$		−	32.1476i		−	1.18740i	−0.804687	−	0.593700i	$-0.797667\pi$
$733$		−	32.1476i		−	1.18740i	0.804687	−	0.593700i	$-0.202333\pi$
$734$		−	1.02445i		−	0.0378130i
$735$		0			0
$736$			8.60320i			0.317118i
$737$	13.1954	+	17.6194i	0.486058	+	0.649019i
$738$		0			0
$739$		−	13.0696i		−	0.480774i	−0.970677	−	0.240387i	$-0.922726\pi$
$739$		−	13.0696i		−	0.480774i	0.970677	−	0.240387i	$-0.0772743\pi$
$740$	−6.45255			−0.237200
$741$		0			0
$742$		−	14.3048i		−	0.525144i
$743$			42.0318i			1.54200i	0.636837	+	0.770999i	$0.280242\pi$
$743$			42.0318i			1.54200i	−0.636837	+	0.770999i	$0.719758\pi$
$744$		0			0
$745$			12.0920i			0.443018i
$746$		−	14.9557i		−	0.547568i
$747$		0			0
$748$		−	12.6843i		−	0.463783i
$749$	24.0755			0.879701
$750$		0			0
$751$	−13.8159			−0.504149			−0.252074	−	0.967708i	$-0.581113\pi$
$751$	−13.8159			−0.504149			−0.252074	+	0.967708i	$0.581113\pi$
$752$		−	3.96365i		−	0.144539i
$753$		0			0
$754$	−33.5635			−1.22231
$755$	9.49449			0.345540
$756$		0			0
$757$			45.2956i			1.64629i	0.567828	+	0.823147i	$0.307784\pi$
$757$			45.2956i			1.64629i	−0.567828	+	0.823147i	$0.692216\pi$
$758$			10.1853i			0.369945i
$759$		0			0
$760$	2.81067			0.101954
$761$			17.8606i			0.647448i	0.946152	+	0.323724i	$0.104935\pi$
$761$			17.8606i			0.647448i	−0.946152	+	0.323724i	$0.895065\pi$
$762$		0			0
$763$	27.7544			1.00478
$764$	20.4225			0.738862
$765$		0			0
$766$	0.756650			0.0273389
$767$	−58.1680			−2.10033
$768$		0			0
$769$		−	37.6392i		−	1.35730i	−0.734460	−	0.678652i	$-0.762564\pi$
$769$		−	37.6392i		−	1.35730i	0.734460	−	0.678652i	$-0.237436\pi$
$770$			8.88978i			0.320366i
$771$		0			0
$772$	0.661297			0.0238006
$773$		−	52.1708i		−	1.87645i	−0.346021	−	0.938227i	$-0.612467\pi$
$773$		−	52.1708i		−	1.87645i	0.346021	−	0.938227i	$-0.387533\pi$
$774$		0			0
$775$			5.37819i			0.193190i
$776$			15.8601i			0.569346i
$777$		0			0
$778$			10.5446i			0.378041i
$779$	−3.22881			−0.115684
$780$		0			0
$781$			7.91561i			0.283243i
$782$	−40.5779			−1.45106
$783$		0			0
$784$	−3.92720			−0.140257
$785$	11.0453			0.394224
$786$		0			0
$787$			48.4770i			1.72802i	0.503477	+	0.864008i	$0.332054\pi$
$787$			48.4770i			1.72802i	−0.503477	+	0.864008i	$0.667946\pi$
$788$	−2.96805			−0.105732
$789$		0			0
$790$		−	14.9444i		−	0.531698i
$791$		−	35.9794i		−	1.27928i
$792$		0			0
$793$	17.4653			0.620210
$794$	−15.1027			−0.535976
$795$		0			0
$796$	13.4754			0.477624
$797$			9.15705i			0.324359i	0.986761	+	0.162180i	$0.0518524\pi$
$797$			9.15705i			0.324359i	−0.986761	+	0.162180i	$0.948148\pi$
$798$		0			0
$799$	18.6950			0.661380
$800$	−1.00000			−0.0353553
$801$		0			0
$802$	17.5526			0.619804
$803$	−13.1655			−0.464599
$804$		0			0
$805$	28.4390			1.00234
$806$	22.0616			0.777086
$807$		0			0
$808$	−0.194963			−0.00685877
$809$	−38.5297			−1.35463			−0.677316	−	0.735692i	$-0.736857\pi$
$809$	−38.5297			−1.35463			−0.677316	+	0.735692i	$0.736857\pi$
$810$		0			0
$811$			18.2665i			0.641424i	0.947177	+	0.320712i	$0.103922\pi$
$811$			18.2665i			0.641424i	−0.947177	+	0.320712i	$0.896078\pi$
$812$	−27.0471			−0.949168
$813$		0			0
$814$	17.3527			0.608213
$815$	20.5292			0.719106
$816$		0			0
$817$			33.6629i			1.17772i
$818$		−	6.37681i		−	0.222960i
$819$		0			0
$820$	1.14877			0.0401168
$821$		−	3.25900i		−	0.113740i	−0.998382	−	0.0568699i	$-0.981888\pi$
$821$		−	3.25900i		−	0.113740i	0.998382	−	0.0568699i	$-0.0181120\pi$
$822$		0			0
$823$	2.71181			0.0945277			0.0472638	−	0.998882i	$-0.484950\pi$
$823$	2.71181			0.0945277			0.0472638	+	0.998882i	$0.484950\pi$
$824$	−18.4691			−0.643402
$825$		0			0
$826$	−46.8747			−1.63098
$827$			25.7808i			0.896487i	0.893912	+	0.448243i	$0.147950\pi$
$827$			25.7808i			0.896487i	−0.893912	+	0.448243i	$0.852050\pi$
$828$		0			0
$829$	−33.4154			−1.16057			−0.580283	−	0.814415i	$-0.697058\pi$
$829$	−33.4154			−1.16057			−0.580283	+	0.814415i	$0.697058\pi$
$830$			3.04264i			0.105612i
$831$		0			0
$832$			4.10204i			0.142213i
$833$		−	18.5231i		−	0.641786i
$834$		0			0
$835$		−	16.8432i		−	0.582884i
$836$	−7.55869			−0.261423
$837$		0			0
$838$			31.8808i			1.10130i
$839$			8.62308i			0.297702i	0.988860	+	0.148851i	$0.0475574\pi$
$839$			8.62308i			0.297702i	−0.988860	+	0.148851i	$0.952443\pi$
$840$		0			0
$841$	−37.9473			−1.30853
$842$	31.9854			1.10229
$843$		0			0
$844$	−17.8794			−0.615436
$845$	3.82676			0.131645
$846$		0			0
$847$			12.4548i			0.427952i
$848$	−4.32739			−0.148603
$849$		0			0
$850$		−	4.71660i		−	0.161778i
$851$		−	55.5126i		−	1.90295i
$852$		0			0
$853$	−25.0044			−0.856133			−0.428066	−	0.903747i	$-0.640805\pi$
$853$	−25.0044			−0.856133			−0.428066	+	0.903747i	$0.640805\pi$
$854$	14.0744			0.481616
$855$		0			0
$856$		−	7.28319i		−	0.248934i
$857$	23.6454			0.807712			0.403856	−	0.914823i	$-0.367670\pi$
$857$	23.6454			0.807712			0.403856	+	0.914823i	$0.367670\pi$
$858$		0			0
$859$	15.6198			0.532939			0.266470	−	0.963843i	$-0.414143\pi$
$859$	15.6198			0.532939			0.266470	+	0.963843i	$0.414143\pi$
$860$		−	11.9768i		−	0.408406i
$861$		0			0
$862$			19.8635i			0.676552i
$863$		−	18.9911i		−	0.646465i	−0.946320	−	0.323232i	$-0.895230\pi$
$863$		−	18.9911i		−	0.646465i	0.946320	−	0.323232i	$-0.104770\pi$
$864$		0			0
$865$		−	1.30491i		−	0.0443684i
$866$			22.7355i			0.772584i
$867$		0			0
$868$	17.7783			0.603436
$869$			40.1898i			1.36334i
$870$		0			0
$871$	26.8754	−	20.1273i	0.910637	−	0.681987i
$872$		−	8.39609i		−	0.284328i
$873$		0			0
$874$			24.1808i			0.817927i
$875$			3.30563i			0.111751i
$876$		0			0
$877$	4.06130			0.137140			0.0685702	−	0.997646i	$-0.478156\pi$
$877$	4.06130			0.137140			0.0685702	+	0.997646i	$0.478156\pi$
$878$	−12.0596			−0.406992
$879$		0			0
$880$	2.68928			0.0906558
$881$		−	42.4088i		−	1.42879i	−0.699744	−	0.714394i	$-0.746703\pi$
$881$		−	42.4088i		−	1.42879i	0.699744	−	0.714394i	$-0.253297\pi$
$882$		0			0
$883$		−	55.6595i		−	1.87309i	−0.350547	−	0.936545i	$-0.614004\pi$
$883$		−	55.6595i		−	1.87309i	0.350547	−	0.936545i	$-0.385996\pi$
$884$	−19.3477			−0.650734
$885$		0			0
$886$	8.67567			0.291465
$887$			27.3012i			0.916686i	0.888775	+	0.458343i	$0.151557\pi$
$887$			27.3012i			0.916686i	−0.888775	+	0.458343i	$0.848443\pi$
$888$		0			0
$889$		−	70.5425i		−	2.36592i
$890$			0.141891i			0.00475620i
$891$		0			0
$892$	21.8794			0.732577
$893$		−	11.1405i		−	0.372803i
$894$		0			0
$895$	13.3950			0.447744
$896$			3.30563i			0.110433i
$897$		0			0
$898$			15.4850i			0.516741i
$899$	44.0051			1.46765
$900$		0			0
$901$		−	20.4106i		−	0.679975i
$902$	−3.08937			−0.102865
$903$		0			0
$904$	−10.8843			−0.362006
$905$	19.9900			0.664492
$906$		0			0
$907$	14.2251			0.472335			0.236168	−	0.971712i	$-0.424109\pi$
$907$	14.2251			0.472335			0.236168	+	0.971712i	$0.424109\pi$
$908$		−	1.01576i		−	0.0337091i
$909$		0			0
$910$	13.5598			0.449505
$911$			56.5873i			1.87482i	0.348228	+	0.937410i	$0.386784\pi$
$911$			56.5873i			1.87482i	−0.348228	+	0.937410i	$0.613216\pi$
$912$		0			0
$913$		−	8.18253i		−	0.270802i
$914$	5.53703			0.183149
$915$		0			0
$916$			2.40187i			0.0793601i
$917$	−54.3993			−1.79642
$918$		0			0
$919$			22.9265i			0.756274i	0.925750	+	0.378137i	$0.123435\pi$
$919$			22.9265i			0.756274i	−0.925750	+	0.378137i	$0.876565\pi$
$920$		−	8.60320i		−	0.283639i
$921$		0			0
$922$			27.4327i			0.903449i
$923$	12.0739			0.397417
$924$		0			0
$925$	6.45255			0.212159
$926$			41.2078i			1.35417i
$927$		0			0
$928$			8.18213i			0.268592i
$929$	36.7607			1.20608			0.603040	−	0.797711i	$-0.293956\pi$
$929$	36.7607			1.20608			0.603040	+	0.797711i	$0.293956\pi$
$930$		0			0
$931$	−11.0381			−0.361758
$932$	−5.07539			−0.166250
$933$		0			0
$934$		−	32.6360i		−	1.06788i
$935$			12.6843i			0.414820i
$936$		0			0
$937$		−	31.5996i		−	1.03231i	−0.856494	−	0.516157i	$-0.827362\pi$
$937$		−	31.5996i		−	1.03231i	0.856494	−	0.516157i	$-0.172638\pi$
$938$	21.6575	−	16.2196i	0.707143	−	0.529588i
$939$		0			0
$940$			3.96365i			0.129280i
$941$	43.7421			1.42595			0.712975	−	0.701189i	$-0.247347\pi$
$941$	43.7421			1.42595			0.712975	+	0.701189i	$0.247347\pi$
$942$		0			0
$943$			9.88310i			0.321838i
$944$			14.1803i			0.461528i
$945$		0			0
$946$			32.2091i			1.04721i
$947$		−	27.7710i		−	0.902436i	−0.892414	−	0.451218i	$-0.850990\pi$
$947$		−	27.7710i		−	0.902436i	0.892414	−	0.451218i	$-0.149010\pi$
$948$		0			0
$949$			20.0817i			0.651879i
$950$	−2.81067			−0.0911902
$951$		0			0
$952$	−15.5914			−0.505319
$953$			28.7103i			0.930017i	0.885306	+	0.465009i	$0.153949\pi$
$953$			28.7103i			0.930017i	−0.885306	+	0.465009i	$0.846051\pi$
$954$		0			0
$955$	−20.4225			−0.660858
$956$	24.0427			0.777597
$957$		0			0
$958$			8.53795i			0.275849i
$959$			50.4514i			1.62916i
$960$		0			0
$961$	2.07506			0.0669374
$962$		−	26.4686i		−	0.853383i
$963$		0			0
$964$	−25.7150			−0.828224
$965$	−0.661297			−0.0212879
$966$		0			0
$967$	5.66724			0.182246			0.0911230	−	0.995840i	$-0.470954\pi$
$967$	5.66724			0.182246			0.0911230	+	0.995840i	$0.470954\pi$
$968$	3.76775			0.121100
$969$		0			0
$970$		−	15.8601i		−	0.509238i
$971$			35.8315i			1.14989i	0.818193	+	0.574943i	$0.194976\pi$
$971$			35.8315i			1.14989i	−0.818193	+	0.574943i	$0.805024\pi$
$972$		0			0
$973$	−66.2958			−2.12535
$974$		−	33.5595i		−	1.07532i
$975$		0			0
$976$		−	4.25770i		−	0.136286i
$977$			40.9859i			1.31125i	0.755085	+	0.655627i	$0.227596\pi$
$977$			40.9859i			1.31125i	−0.755085	+	0.655627i	$0.772404\pi$
$978$		0			0
$979$		−	0.381585i		−	0.0121955i
$980$	3.92720			0.125450
$981$		0			0
$982$			31.8313i			1.01578i
$983$	31.9710			1.01972			0.509858	−	0.860259i	$-0.329698\pi$
$983$	31.9710			1.01972			0.509858	+	0.860259i	$0.329698\pi$
$984$		0			0
$985$	2.96805			0.0945699
$986$	−38.5919			−1.22902
$987$		0			0
$988$			11.5295i			0.366802i
$989$	103.039			3.27645
$990$		0			0
$991$			1.92040i			0.0610036i	0.999535	+	0.0305018i	$0.00971053\pi$
$991$			1.92040i			0.0610036i	−0.999535	+	0.0305018i	$0.990289\pi$
$992$		−	5.37819i		−	0.170758i
$993$		0			0
$994$	9.72976			0.308609
$995$	−13.4754			−0.427200
$996$		0			0
$997$	−21.1336			−0.669307			−0.334653	−	0.942341i	$-0.608619\pi$
$997$	−21.1336			−0.669307			−0.334653	+	0.942341i	$0.608619\pi$
$998$		−	23.2861i		−	0.737107i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6030.2.d.k.2411.4	✓	24
3.2	odd	2		6030.2.d.l.2411.4	yes	24
67.66	odd	2		6030.2.d.l.2411.21	yes	24
201.200	even	2	inner	6030.2.d.k.2411.21	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6030.2.d.k.2411.4	✓	24	1.1	even	1	trivial
6030.2.d.k.2411.21	yes	24	201.200	even	2	inner
6030.2.d.l.2411.4	yes	24	3.2	odd	2
6030.2.d.l.2411.21	yes	24	67.66	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -3.30563i q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -3.30563i q^{7} -1.00000 q^{8} +1.00000 q^{10} -2.68928 q^{11} +4.10204i q^{13} +3.30563i q^{14} +1.00000 q^{16} +4.71660i q^{17} +2.81067 q^{19} -1.00000 q^{20} +2.68928 q^{22} -8.60320i q^{23} +1.00000 q^{25} -4.10204i q^{26} -3.30563i q^{28} -8.18213i q^{29} +5.37819i q^{31} -1.00000 q^{32} -4.71660i q^{34} +3.30563i q^{35} +6.45255 q^{37} -2.81067 q^{38} +1.00000 q^{40} -1.14877 q^{41} +11.9768i q^{43} -2.68928 q^{44} +8.60320i q^{46} -3.96365i q^{47} -3.92720 q^{49} -1.00000 q^{50} +4.10204i q^{52} -4.32739 q^{53} +2.68928 q^{55} +3.30563i q^{56} +8.18213i q^{58} +14.1803i q^{59} -4.25770i q^{61} -5.37819i q^{62} +1.00000 q^{64} -4.10204i q^{65} +(-4.90665 - 6.55170i) q^{67} +4.71660i q^{68} -3.30563i q^{70} -2.94339i q^{71} +4.89553 q^{73} -6.45255 q^{74} +2.81067 q^{76} +8.88978i q^{77} -14.9444i q^{79} -1.00000 q^{80} +1.14877 q^{82} +3.04264i q^{83} -4.71660i q^{85} -11.9768i q^{86} +2.68928 q^{88} +0.141891i q^{89} +13.5598 q^{91} -8.60320i q^{92} +3.96365i q^{94} -2.81067 q^{95} -15.8601i q^{97} +3.92720 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 24 q^{2} + 24 q^{4} - 24 q^{5} - 24 q^{8}+O(q^{10}) \) 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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