LMFDB - Embedded newform 6030.2.d.l.2411.1

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			2411.1
Character	$\chi$	$=$	6030.2411
Dual form			6030.2.d.l.2411.24

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -4.73932i q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -4.73932i q^{7} +1.00000 q^{8} +1.00000 q^{10} -5.12191 q^{11} -1.74819i q^{13} -4.73932i q^{14} +1.00000 q^{16} -3.24090i q^{17} -0.725860 q^{19} +1.00000 q^{20} -5.12191 q^{22} -4.25195i q^{23} +1.00000 q^{25} -1.74819i q^{26} -4.73932i q^{28} -2.63599i q^{29} +8.54983i q^{31} +1.00000 q^{32} -3.24090i q^{34} -4.73932i q^{35} -10.8157 q^{37} -0.725860 q^{38} +1.00000 q^{40} -1.43175 q^{41} +8.70086i q^{43} -5.12191 q^{44} -4.25195i q^{46} +5.76550i q^{47} -15.4612 q^{49} +1.00000 q^{50} -1.74819i q^{52} +6.68834 q^{53} -5.12191 q^{55} -4.73932i q^{56} -2.63599i q^{58} -4.44291i q^{59} -2.57925i q^{61} +8.54983i q^{62} +1.00000 q^{64} -1.74819i q^{65} +(5.92780 + 5.64457i) q^{67} -3.24090i q^{68} -4.73932i q^{70} -14.2941i q^{71} -10.1723 q^{73} -10.8157 q^{74} -0.725860 q^{76} +24.2744i q^{77} +3.03585i q^{79} +1.00000 q^{80} -1.43175 q^{82} -4.51804i q^{83} -3.24090i q^{85} +8.70086i q^{86} -5.12191 q^{88} +3.75518i q^{89} -8.28522 q^{91} -4.25195i q^{92} +5.76550i q^{94} -0.725860 q^{95} +10.8215i q^{97} -15.4612 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$		−	4.73932i		−	1.79129i	−0.444765	−	0.895647i	$-0.646713\pi$
$7$		−	4.73932i		−	1.79129i	0.444765	−	0.895647i	$-0.353287\pi$
$8$	1.00000			0.353553
$9$		0			0
$10$	1.00000			0.316228
$11$	−5.12191			−1.54431			−0.772157	−	0.635431i	$-0.780822\pi$
$11$	−5.12191			−1.54431			−0.772157	+	0.635431i	$0.780822\pi$
$12$		0			0
$13$		−	1.74819i		−	0.484860i	−0.970169	−	0.242430i	$-0.922056\pi$
$13$		−	1.74819i		−	0.484860i	0.970169	−	0.242430i	$-0.0779444\pi$
$14$		−	4.73932i		−	1.26664i
$15$		0			0
$16$	1.00000			0.250000
$17$		−	3.24090i		−	0.786034i	−0.919531	−	0.393017i	$-0.871431\pi$
$17$		−	3.24090i		−	0.786034i	0.919531	−	0.393017i	$-0.128569\pi$
$18$		0			0
$19$	−0.725860			−0.166524			−0.0832618	−	0.996528i	$-0.526534\pi$
$19$	−0.725860			−0.166524			−0.0832618	+	0.996528i	$0.526534\pi$
$20$	1.00000			0.223607
$21$		0			0
$22$	−5.12191			−1.09200
$23$		−	4.25195i		−	0.886594i	−0.896375	−	0.443297i	$-0.853809\pi$
$23$		−	4.25195i		−	0.886594i	0.896375	−	0.443297i	$-0.146191\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		−	1.74819i		−	0.342848i
$27$		0			0
$28$		−	4.73932i		−	0.895647i
$29$		−	2.63599i		−	0.489491i	−0.969587	−	0.244745i	$-0.921296\pi$
$29$		−	2.63599i		−	0.489491i	0.969587	−	0.244745i	$-0.0787043\pi$
$30$		0			0
$31$			8.54983i			1.53560i	0.640692	+	0.767798i	$0.278647\pi$
$31$			8.54983i			1.53560i	−0.640692	+	0.767798i	$0.721353\pi$
$32$	1.00000			0.176777
$33$		0			0
$34$		−	3.24090i		−	0.555810i
$35$		−	4.73932i		−	0.801091i
$36$		0			0
$37$	−10.8157			−1.77810			−0.889048	−	0.457814i	$-0.848633\pi$
$37$	−10.8157			−1.77810			−0.889048	+	0.457814i	$0.848633\pi$
$38$	−0.725860			−0.117750
$39$		0			0
$40$	1.00000			0.158114
$41$	−1.43175			−0.223602			−0.111801	−	0.993731i	$-0.535662\pi$
$41$	−1.43175			−0.223602			−0.111801	+	0.993731i	$0.535662\pi$
$42$		0			0
$43$			8.70086i			1.32687i	0.748235	+	0.663434i	$0.230902\pi$
$43$			8.70086i			1.32687i	−0.748235	+	0.663434i	$0.769098\pi$
$44$	−5.12191			−0.772157
$45$		0			0
$46$		−	4.25195i		−	0.626916i
$47$			5.76550i			0.840984i	0.907296	+	0.420492i	$0.138143\pi$
$47$			5.76550i			0.840984i	−0.907296	+	0.420492i	$0.861857\pi$
$48$		0			0
$49$	−15.4612			−2.20874
$50$	1.00000			0.141421
$51$		0			0
$52$		−	1.74819i		−	0.242430i
$53$	6.68834			0.918715			0.459357	−	0.888252i	$-0.348080\pi$
$53$	6.68834			0.918715			0.459357	+	0.888252i	$0.348080\pi$
$54$		0			0
$55$	−5.12191			−0.690639
$56$		−	4.73932i		−	0.633318i
$57$		0			0
$58$		−	2.63599i		−	0.346122i
$59$		−	4.44291i		−	0.578418i	−0.957266	−	0.289209i	$-0.906608\pi$
$59$		−	4.44291i		−	0.578418i	0.957266	−	0.289209i	$-0.0933923\pi$
$60$		0			0
$61$		−	2.57925i		−	0.330240i	−0.986274	−	0.165120i	$-0.947199\pi$
$61$		−	2.57925i		−	0.330240i	0.986274	−	0.165120i	$-0.0528011\pi$
$62$			8.54983i			1.08583i
$63$		0			0
$64$	1.00000			0.125000
$65$		−	1.74819i		−	0.216836i
$66$		0			0
$67$	5.92780	+	5.64457i	0.724196	+	0.689594i
$67$	5.92780	+	5.64457i	0.724196	+	0.689594i
$68$		−	3.24090i		−	0.393017i
$69$		0			0
$70$		−	4.73932i		−	0.566457i
$71$		−	14.2941i		−	1.69639i	−0.529682	−	0.848196i	$-0.677689\pi$
$71$		−	14.2941i		−	1.69639i	0.529682	−	0.848196i	$-0.322311\pi$
$72$		0			0
$73$	−10.1723			−1.19058			−0.595290	−	0.803511i	$-0.702963\pi$
$73$	−10.1723			−1.19058			−0.595290	+	0.803511i	$0.702963\pi$
$74$	−10.8157			−1.25730
$75$		0			0
$76$	−0.725860			−0.0832618
$77$			24.2744i			2.76632i
$78$		0			0
$79$			3.03585i			0.341560i	0.985309	+	0.170780i	$0.0546287\pi$
$79$			3.03585i			0.341560i	−0.985309	+	0.170780i	$0.945371\pi$
$80$	1.00000			0.111803
$81$		0			0
$82$	−1.43175			−0.158110
$83$		−	4.51804i		−	0.495920i	−0.968770	−	0.247960i	$-0.920240\pi$
$83$		−	4.51804i		−	0.495920i	0.968770	−	0.247960i	$-0.0797601\pi$
$84$		0			0
$85$		−	3.24090i		−	0.351525i
$86$			8.70086i			0.938237i
$87$		0			0
$88$	−5.12191			−0.545998
$89$			3.75518i			0.398048i	0.979995	+	0.199024i	$0.0637772\pi$
$89$			3.75518i			0.398048i	−0.979995	+	0.199024i	$0.936223\pi$
$90$		0			0
$91$	−8.28522			−0.868527
$92$		−	4.25195i		−	0.443297i
$93$		0			0
$94$			5.76550i			0.594665i
$95$	−0.725860			−0.0744716
$96$		0			0
$97$			10.8215i			1.09876i	0.835572	+	0.549381i	$0.185136\pi$
$97$			10.8215i			1.09876i	−0.835572	+	0.549381i	$0.814864\pi$
$98$	−15.4612			−1.56181
$99$		0			0
$100$	1.00000			0.100000
$101$	2.89331			0.287895			0.143948	−	0.989585i	$-0.454020\pi$
$101$	2.89331			0.287895			0.143948	+	0.989585i	$0.454020\pi$
$102$		0			0
$103$	−10.2758			−1.01250			−0.506252	−	0.862385i	$-0.668970\pi$
$103$	−10.2758			−1.01250			−0.506252	+	0.862385i	$0.668970\pi$
$104$		−	1.74819i		−	0.171424i
$105$		0			0
$106$	6.68834			0.649629
$107$			13.1871i			1.27485i	0.770513	+	0.637425i	$0.220000\pi$
$107$			13.1871i			1.27485i	−0.770513	+	0.637425i	$0.780000\pi$
$108$		0			0
$109$			0.261847i			0.0250804i	0.999921	+	0.0125402i	$0.00399177\pi$
$109$			0.261847i			0.0250804i	−0.999921	+	0.0125402i	$0.996008\pi$
$110$	−5.12191			−0.488355
$111$		0			0
$112$		−	4.73932i		−	0.447824i
$113$	−5.01843			−0.472094			−0.236047	−	0.971742i	$-0.575852\pi$
$113$	−5.01843			−0.472094			−0.236047	+	0.971742i	$0.575852\pi$
$114$		0			0
$115$		−	4.25195i		−	0.396497i
$116$		−	2.63599i		−	0.244745i
$117$		0			0
$118$		−	4.44291i		−	0.409003i
$119$	−15.3597			−1.40802
$120$		0			0
$121$	15.2340			1.38491
$122$		−	2.57925i		−	0.233515i
$123$		0			0
$124$			8.54983i			0.767798i
$125$	1.00000			0.0894427
$126$		0			0
$127$	−20.3949			−1.80976			−0.904879	−	0.425670i	$-0.860039\pi$
$127$	−20.3949			−1.80976			−0.904879	+	0.425670i	$0.860039\pi$
$128$	1.00000			0.0883883
$129$		0			0
$130$		−	1.74819i		−	0.153326i
$131$		−	16.1889i		−	1.41443i	−0.706996	−	0.707217i	$-0.749950\pi$
$131$		−	16.1889i		−	1.41443i	0.706996	−	0.707217i	$-0.250050\pi$
$132$		0			0
$133$			3.44008i			0.298293i
$134$	5.92780	+	5.64457i	0.512084	+	0.487617i
$135$		0			0
$136$		−	3.24090i		−	0.277905i
$137$	−16.2496			−1.38829			−0.694147	−	0.719833i	$-0.744218\pi$
$137$	−16.2496			−1.38829			−0.694147	+	0.719833i	$0.744218\pi$
$138$		0			0
$139$		−	20.0574i		−	1.70125i	−0.525776	−	0.850623i	$-0.676225\pi$
$139$		−	20.0574i		−	1.70125i	0.525776	−	0.850623i	$-0.323775\pi$
$140$		−	4.73932i		−	0.400546i
$141$		0			0
$142$		−	14.2941i		−	1.19953i
$143$			8.95406i			0.748776i
$144$		0			0
$145$		−	2.63599i		−	0.218907i
$146$	−10.1723			−0.841867
$147$		0			0
$148$	−10.8157			−0.889048
$149$		−	20.5053i		−	1.67986i	−0.542697	−	0.839928i	$-0.682597\pi$
$149$		−	20.5053i		−	1.67986i	0.542697	−	0.839928i	$-0.317403\pi$
$150$		0			0
$151$	−4.99057			−0.406127			−0.203063	−	0.979166i	$-0.565090\pi$
$151$	−4.99057			−0.406127			−0.203063	+	0.979166i	$0.565090\pi$
$152$	−0.725860			−0.0588750
$153$		0			0
$154$			24.2744i			1.95609i
$155$			8.54983i			0.686739i
$156$		0			0
$157$	7.62941			0.608893			0.304447	−	0.952529i	$-0.401528\pi$
$157$	7.62941			0.608893			0.304447	+	0.952529i	$0.401528\pi$
$158$			3.03585i			0.241519i
$159$		0			0
$160$	1.00000			0.0790569
$161$	−20.1514			−1.58815
$162$		0			0
$163$	−10.3590			−0.811377			−0.405689	−	0.914011i	$-0.632968\pi$
$163$	−10.3590			−0.811377			−0.405689	+	0.914011i	$0.632968\pi$
$164$	−1.43175			−0.111801
$165$		0			0
$166$		−	4.51804i		−	0.350668i
$167$			8.79713i			0.680743i	0.940291	+	0.340371i	$0.110553\pi$
$167$			8.79713i			0.680743i	−0.940291	+	0.340371i	$0.889447\pi$
$168$		0			0
$169$	9.94384			0.764911
$170$		−	3.24090i		−	0.248566i
$171$		0			0
$172$			8.70086i			0.663434i
$173$		−	1.33593i		−	0.101569i	−0.998710	−	0.0507844i	$-0.983828\pi$
$173$		−	1.33593i		−	0.101569i	0.998710	−	0.0507844i	$-0.0161721\pi$
$174$		0			0
$175$		−	4.73932i		−	0.358259i
$176$	−5.12191			−0.386079
$177$		0			0
$178$			3.75518i			0.281462i
$179$	13.4349			1.00417			0.502086	−	0.864818i	$-0.332566\pi$
$179$	13.4349			1.00417			0.502086	+	0.864818i	$0.332566\pi$
$180$		0			0
$181$	23.9891			1.78309			0.891547	−	0.452929i	$-0.149621\pi$
$181$	23.9891			1.78309			0.891547	+	0.452929i	$0.149621\pi$
$182$	−8.28522			−0.614141
$183$		0			0
$184$		−	4.25195i		−	0.313458i
$185$	−10.8157			−0.795189
$186$		0			0
$187$			16.5996i			1.21388i
$188$			5.76550i			0.420492i
$189$		0			0
$190$	−0.725860			−0.0526594
$191$	−4.75588			−0.344124			−0.172062	−	0.985086i	$-0.555043\pi$
$191$	−4.75588			−0.344124			−0.172062	+	0.985086i	$0.555043\pi$
$192$		0			0
$193$	5.23513			0.376833			0.188417	−	0.982089i	$-0.439664\pi$
$193$	5.23513			0.376833			0.188417	+	0.982089i	$0.439664\pi$
$194$			10.8215i			0.776941i
$195$		0			0
$196$	−15.4612			−1.10437
$197$	27.1637			1.93533			0.967666	−	0.252237i	$-0.0811662\pi$
$197$	27.1637			1.93533			0.967666	+	0.252237i	$0.0811662\pi$
$198$		0			0
$199$	−4.72867			−0.335207			−0.167603	−	0.985855i	$-0.553603\pi$
$199$	−4.72867			−0.335207			−0.167603	+	0.985855i	$0.553603\pi$
$200$	1.00000			0.0707107
$201$		0			0
$202$	2.89331			0.203573
$203$	−12.4928			−0.876822
$204$		0			0
$205$	−1.43175			−0.0999977
$206$	−10.2758			−0.715949
$207$		0			0
$208$		−	1.74819i		−	0.121215i
$209$	3.71779			0.257165
$210$		0			0
$211$	−21.9100			−1.50834			−0.754172	−	0.656677i	$-0.771961\pi$
$211$	−21.9100			−1.50834			−0.754172	+	0.656677i	$0.771961\pi$
$212$	6.68834			0.459357
$213$		0			0
$214$			13.1871i			0.901455i
$215$			8.70086i			0.593393i
$216$		0			0
$217$	40.5204			2.75070
$218$			0.261847i			0.0177345i
$219$		0			0
$220$	−5.12191			−0.345319
$221$	−5.66570			−0.381116
$222$		0			0
$223$	18.9417			1.26843			0.634215	−	0.773157i	$-0.281323\pi$
$223$	18.9417			1.26843			0.634215	+	0.773157i	$0.281323\pi$
$224$		−	4.73932i		−	0.316659i
$225$		0			0
$226$	−5.01843			−0.333821
$227$		−	25.5367i		−	1.69493i	−0.530850	−	0.847466i	$-0.678127\pi$
$227$		−	25.5367i		−	1.69493i	0.530850	−	0.847466i	$-0.321873\pi$
$228$		0			0
$229$			2.49414i			0.164818i	0.996599	+	0.0824088i	$0.0262613\pi$
$229$			2.49414i			0.164818i	−0.996599	+	0.0824088i	$0.973739\pi$
$230$		−	4.25195i		−	0.280366i
$231$		0			0
$232$		−	2.63599i		−	0.173061i
$233$	7.31040			0.478920			0.239460	−	0.970906i	$-0.423030\pi$
$233$	7.31040			0.478920			0.239460	+	0.970906i	$0.423030\pi$
$234$		0			0
$235$			5.76550i			0.376099i
$236$		−	4.44291i		−	0.289209i
$237$		0			0
$238$	−15.3597			−0.995619
$239$	1.61683			0.104584			0.0522921	−	0.998632i	$-0.483347\pi$
$239$	1.61683			0.104584			0.0522921	+	0.998632i	$0.483347\pi$
$240$		0			0
$241$	11.6519			0.750563			0.375282	−	0.926911i	$-0.377546\pi$
$241$	11.6519			0.750563			0.375282	+	0.926911i	$0.377546\pi$
$242$	15.2340			0.979278
$243$		0			0
$244$		−	2.57925i		−	0.165120i
$245$	−15.4612			−0.987777
$246$		0			0
$247$			1.26894i			0.0807406i
$248$			8.54983i			0.542915i
$249$		0			0
$250$	1.00000			0.0632456
$251$	−22.3366			−1.40988			−0.704938	−	0.709269i	$-0.749025\pi$
$251$	−22.3366			−1.40988			−0.704938	+	0.709269i	$0.749025\pi$
$252$		0			0
$253$			21.7781i			1.36918i
$254$	−20.3949			−1.27969
$255$		0			0
$256$	1.00000			0.0625000
$257$		−	14.9459i		−	0.932301i	−0.884706	−	0.466150i	$-0.845641\pi$
$257$		−	14.9459i		−	0.932301i	0.884706	−	0.466150i	$-0.154359\pi$
$258$		0			0
$259$			51.2592i			3.18509i
$260$		−	1.74819i		−	0.108418i
$261$		0			0
$262$		−	16.1889i		−	1.00016i
$263$		−	4.48000i		−	0.276248i	−0.990415	−	0.138124i	$-0.955893\pi$
$263$		−	4.48000i		−	0.276248i	0.990415	−	0.138124i	$-0.0441073\pi$
$264$		0			0
$265$	6.68834			0.410862
$266$			3.44008i			0.210925i
$267$		0			0
$268$	5.92780	+	5.64457i	0.362098	+	0.344797i
$269$		−	14.4866i		−	0.883264i	−0.897196	−	0.441632i	$-0.854400\pi$
$269$		−	14.4866i		−	0.883264i	0.897196	−	0.441632i	$-0.145600\pi$
$270$		0			0
$271$		−	5.83478i		−	0.354438i	−0.984171	−	0.177219i	$-0.943290\pi$
$271$		−	5.83478i		−	0.354438i	0.984171	−	0.177219i	$-0.0567101\pi$
$272$		−	3.24090i		−	0.196508i
$273$		0			0
$274$	−16.2496			−0.981673
$275$	−5.12191			−0.308863
$276$		0			0
$277$	−17.6693			−1.06164			−0.530822	−	0.847483i	$-0.678117\pi$
$277$	−17.6693			−1.06164			−0.530822	+	0.847483i	$0.678117\pi$
$278$		−	20.0574i		−	1.20296i
$279$		0			0
$280$		−	4.73932i		−	0.283229i
$281$	11.2237			0.669551			0.334775	−	0.942298i	$-0.391340\pi$
$281$	11.2237			0.669551			0.334775	+	0.942298i	$0.391340\pi$
$282$		0			0
$283$	10.2135			0.607130			0.303565	−	0.952811i	$-0.401823\pi$
$283$	10.2135			0.607130			0.303565	+	0.952811i	$0.401823\pi$
$284$		−	14.2941i		−	0.848196i
$285$		0			0
$286$			8.95406i			0.529465i
$287$			6.78552i			0.400536i
$288$		0			0
$289$	6.49657			0.382151
$290$		−	2.63599i		−	0.154790i
$291$		0			0
$292$	−10.1723			−0.595290
$293$		−	33.0445i		−	1.93048i	−0.261363	−	0.965241i	$-0.584172\pi$
$293$		−	33.0445i		−	1.93048i	0.261363	−	0.965241i	$-0.415828\pi$
$294$		0			0
$295$		−	4.44291i		−	0.258676i
$296$	−10.8157			−0.628652
$297$		0			0
$298$		−	20.5053i		−	1.18784i
$299$	−7.43321			−0.429874
$300$		0			0
$301$	41.2361			2.37681
$302$	−4.99057			−0.287175
$303$		0			0
$304$	−0.725860			−0.0416309
$305$		−	2.57925i		−	0.147688i
$306$		0			0
$307$	−20.3347			−1.16056			−0.580282	−	0.814415i	$-0.697058\pi$
$307$	−20.3347			−1.16056			−0.580282	+	0.814415i	$0.697058\pi$
$308$			24.2744i			1.38316i
$309$		0			0
$310$			8.54983i			0.485598i
$311$	9.20403			0.521912			0.260956	−	0.965351i	$-0.415962\pi$
$311$	9.20403			0.521912			0.260956	+	0.965351i	$0.415962\pi$
$312$		0			0
$313$		−	18.0289i		−	1.01905i	−0.860455	−	0.509527i	$-0.829820\pi$
$313$		−	18.0289i		−	1.01905i	0.860455	−	0.509527i	$-0.170180\pi$
$314$	7.62941			0.430553
$315$		0			0
$316$			3.03585i			0.170780i
$317$			28.2321i			1.58567i	0.609435	+	0.792836i	$0.291396\pi$
$317$			28.2321i			1.58567i	−0.609435	+	0.792836i	$0.708604\pi$
$318$		0			0
$319$			13.5013i			0.755927i
$320$	1.00000			0.0559017
$321$		0			0
$322$	−20.1514			−1.12299
$323$			2.35244i			0.130893i
$324$		0			0
$325$		−	1.74819i		−	0.0969720i
$326$	−10.3590			−0.573730
$327$		0			0
$328$	−1.43175			−0.0790551
$329$	27.3245			1.50645
$330$		0			0
$331$		−	10.5972i		−	0.582472i	−0.956651	−	0.291236i	$-0.905933\pi$
$331$		−	10.5972i		−	0.582472i	0.956651	−	0.291236i	$-0.0940665\pi$
$332$		−	4.51804i		−	0.247960i
$333$		0			0
$334$			8.79713i			0.481358i
$335$	5.92780	+	5.64457i	0.323870	+	0.308396i
$336$		0			0
$337$		−	24.5719i		−	1.33852i	−0.743030	−	0.669258i	$-0.766612\pi$
$337$		−	24.5719i		−	1.33852i	0.743030	−	0.669258i	$-0.233388\pi$
$338$	9.94384			0.540874
$339$		0			0
$340$		−	3.24090i		−	0.175762i
$341$		−	43.7915i		−	2.37144i
$342$		0			0
$343$			40.1001i			2.16520i
$344$			8.70086i			0.469119i
$345$		0			0
$346$		−	1.33593i		−	0.0718199i
$347$	−29.5983			−1.58892			−0.794460	−	0.607317i	$-0.792246\pi$
$347$	−29.5983			−1.58892			−0.794460	+	0.607317i	$0.792246\pi$
$348$		0			0
$349$	−24.3079			−1.30117			−0.650587	−	0.759432i	$-0.725477\pi$
$349$	−24.3079			−1.30117			−0.650587	+	0.759432i	$0.725477\pi$
$350$		−	4.73932i		−	0.253327i
$351$		0			0
$352$	−5.12191			−0.272999
$353$	26.9058			1.43205			0.716026	−	0.698074i	$-0.245959\pi$
$353$	26.9058			1.43205			0.716026	+	0.698074i	$0.245959\pi$
$354$		0			0
$355$		−	14.2941i		−	0.758650i
$356$			3.75518i			0.199024i
$357$		0			0
$358$	13.4349			0.710057
$359$			12.1558i			0.641560i	0.947154	+	0.320780i	$0.103945\pi$
$359$			12.1558i			0.641560i	−0.947154	+	0.320780i	$0.896055\pi$
$360$		0			0
$361$	−18.4731			−0.972270
$362$	23.9891			1.26084
$363$		0			0
$364$	−8.28522			−0.434263
$365$	−10.1723			−0.532443
$366$		0			0
$367$			26.7187i			1.39470i	0.716728	+	0.697352i	$0.245639\pi$
$367$			26.7187i			1.39470i	−0.716728	+	0.697352i	$0.754361\pi$
$368$		−	4.25195i		−	0.221648i
$369$		0			0
$370$	−10.8157			−0.562283
$371$		−	31.6982i		−	1.64569i
$372$		0			0
$373$		−	33.3618i		−	1.72741i	−0.503999	−	0.863705i	$-0.668138\pi$
$373$		−	33.3618i		−	1.72741i	0.503999	−	0.863705i	$-0.331862\pi$
$374$			16.5996i			0.858345i
$375$		0			0
$376$			5.76550i			0.297333i
$377$	−4.60820			−0.237334
$378$		0			0
$379$		−	11.1801i		−	0.574283i	−0.957888	−	0.287142i	$-0.907295\pi$
$379$		−	11.1801i		−	0.574283i	0.957888	−	0.287142i	$-0.0927050\pi$
$380$	−0.725860			−0.0372358
$381$		0			0
$382$	−4.75588			−0.243332
$383$	8.20272			0.419140			0.209570	−	0.977794i	$-0.432794\pi$
$383$	8.20272			0.419140			0.209570	+	0.977794i	$0.432794\pi$
$384$		0			0
$385$			24.2744i			1.23714i
$386$	5.23513			0.266461
$387$		0			0
$388$			10.8215i			0.549381i
$389$			15.7469i			0.798401i	0.916864	+	0.399201i	$0.130712\pi$
$389$			15.7469i			0.798401i	−0.916864	+	0.399201i	$0.869288\pi$
$390$		0			0
$391$	−13.7802			−0.696892
$392$	−15.4612			−0.780906
$393$		0			0
$394$	27.1637			1.36849
$395$			3.03585i			0.152750i
$396$		0			0
$397$	11.3144			0.567853			0.283927	−	0.958846i	$-0.408363\pi$
$397$	11.3144			0.567853			0.283927	+	0.958846i	$0.408363\pi$
$398$	−4.72867			−0.237027
$399$		0			0
$400$	1.00000			0.0500000
$401$	25.3643			1.26663			0.633317	−	0.773892i	$-0.281693\pi$
$401$	25.3643			1.26663			0.633317	+	0.773892i	$0.281693\pi$
$402$		0			0
$403$	14.9467			0.744548
$404$	2.89331			0.143948
$405$		0			0
$406$	−12.4928			−0.620007
$407$	55.3973			2.74594
$408$		0			0
$409$			7.38365i			0.365098i	0.983197	+	0.182549i	$0.0584348\pi$
$409$			7.38365i			0.365098i	−0.983197	+	0.182549i	$0.941565\pi$
$410$	−1.43175			−0.0707090
$411$		0			0
$412$	−10.2758			−0.506252
$413$	−21.0564			−1.03612
$414$		0			0
$415$		−	4.51804i		−	0.221782i
$416$		−	1.74819i		−	0.0857119i
$417$		0			0
$418$	3.71779			0.181843
$419$		−	8.10986i		−	0.396193i	−0.980183	−	0.198096i	$-0.936524\pi$
$419$		−	8.10986i		−	0.396193i	0.980183	−	0.198096i	$-0.0634759\pi$
$420$		0			0
$421$	−14.5262			−0.707962			−0.353981	−	0.935253i	$-0.615172\pi$
$421$	−14.5262			−0.707962			−0.353981	+	0.935253i	$0.615172\pi$
$422$	−21.9100			−1.06656
$423$		0			0
$424$	6.68834			0.324815
$425$		−	3.24090i		−	0.157207i
$426$		0			0
$427$	−12.2239			−0.591557
$428$			13.1871i			0.637425i
$429$		0			0
$430$			8.70086i			0.419593i
$431$		−	16.6118i		−	0.800163i	−0.916480	−	0.400081i	$-0.868982\pi$
$431$		−	16.6118i		−	0.800163i	0.916480	−	0.400081i	$-0.131018\pi$
$432$		0			0
$433$			29.9304i			1.43836i	0.694823	+	0.719180i	$0.255483\pi$
$433$			29.9304i			1.43836i	−0.694823	+	0.719180i	$0.744517\pi$
$434$	40.5204			1.94504
$435$		0			0
$436$			0.261847i			0.0125402i
$437$			3.08632i			0.147639i
$438$		0			0
$439$	−11.0670			−0.528199			−0.264099	−	0.964495i	$-0.585075\pi$
$439$	−11.0670			−0.528199			−0.264099	+	0.964495i	$0.585075\pi$
$440$	−5.12191			−0.244178
$441$		0			0
$442$	−5.66570			−0.269490
$443$	−27.8383			−1.32264			−0.661320	−	0.750104i	$-0.730003\pi$
$443$	−27.8383			−1.32264			−0.661320	+	0.750104i	$0.730003\pi$
$444$		0			0
$445$			3.75518i			0.178012i
$446$	18.9417			0.896916
$447$		0			0
$448$		−	4.73932i		−	0.223912i
$449$		−	4.37833i		−	0.206626i	−0.994649	−	0.103313i	$-0.967056\pi$
$449$		−	4.37833i		−	0.206626i	0.994649	−	0.103313i	$-0.0329443\pi$
$450$		0			0
$451$	7.33329			0.345311
$452$	−5.01843			−0.236047
$453$		0			0
$454$		−	25.5367i		−	1.19850i
$455$	−8.28522			−0.388417
$456$		0			0
$457$	−7.67344			−0.358948			−0.179474	−	0.983763i	$-0.557440\pi$
$457$	−7.67344			−0.358948			−0.179474	+	0.983763i	$0.557440\pi$
$458$			2.49414i			0.116544i
$459$		0			0
$460$		−	4.25195i		−	0.198248i
$461$			1.75815i			0.0818852i	0.999162	+	0.0409426i	$0.0130361\pi$
$461$			1.75815i			0.0818852i	−0.999162	+	0.0409426i	$0.986964\pi$
$462$		0			0
$463$		−	29.9782i		−	1.39320i	−0.717458	−	0.696602i	$-0.754694\pi$
$463$		−	29.9782i		−	1.39320i	0.717458	−	0.696602i	$-0.245306\pi$
$464$		−	2.63599i		−	0.122373i
$465$		0			0
$466$	7.31040			0.338648
$467$			15.3752i			0.711478i	0.934585	+	0.355739i	$0.115771\pi$
$467$			15.3752i			0.711478i	−0.934585	+	0.355739i	$0.884229\pi$
$468$		0			0
$469$	26.7514	−	28.0937i	1.23527	−	1.29725i
$470$			5.76550i			0.265942i
$471$		0			0
$472$		−	4.44291i		−	0.204502i
$473$		−	44.5650i		−	2.04910i
$474$		0			0
$475$	−0.725860			−0.0333047
$476$	−15.3597			−0.704009
$477$		0			0
$478$	1.61683			0.0739523
$479$		−	19.8201i		−	0.905603i	−0.891611	−	0.452801i	$-0.850425\pi$
$479$		−	19.8201i		−	0.905603i	0.891611	−	0.452801i	$-0.149575\pi$
$480$		0			0
$481$			18.9079i			0.862127i
$482$	11.6519			0.530728
$483$		0			0
$484$	15.2340			0.692454
$485$			10.8215i			0.491381i
$486$		0			0
$487$		−	21.1972i		−	0.960538i	−0.877121	−	0.480269i	$-0.840539\pi$
$487$		−	21.1972i		−	0.960538i	0.877121	−	0.480269i	$-0.159461\pi$
$488$		−	2.57925i		−	0.116757i
$489$		0			0
$490$	−15.4612			−0.698464
$491$		−	33.3322i		−	1.50426i	−0.659013	−	0.752131i	$-0.729026\pi$
$491$		−	33.3322i		−	1.50426i	0.659013	−	0.752131i	$-0.270974\pi$
$492$		0			0
$493$	−8.54297			−0.384756
$494$			1.26894i			0.0570922i
$495$		0			0
$496$			8.54983i			0.383899i
$497$	−67.7441			−3.03874
$498$		0			0
$499$		−	20.2880i		−	0.908215i	−0.890947	−	0.454108i	$-0.849958\pi$
$499$		−	20.2880i		−	0.908215i	0.890947	−	0.454108i	$-0.150042\pi$
$500$	1.00000			0.0447214
$501$		0			0
$502$	−22.3366			−0.996933
$503$	−5.83795			−0.260301			−0.130151	−	0.991494i	$-0.541546\pi$
$503$	−5.83795			−0.260301			−0.130151	+	0.991494i	$0.541546\pi$
$504$		0			0
$505$	2.89331			0.128751
$506$			21.7781i			0.968156i
$507$		0			0
$508$	−20.3949			−0.904879
$509$			14.9129i			0.661003i	0.943805	+	0.330501i	$0.107218\pi$
$509$			14.9129i			0.661003i	−0.943805	+	0.330501i	$0.892782\pi$
$510$		0			0
$511$			48.2099i			2.13268i
$512$	1.00000			0.0441942
$513$		0			0
$514$		−	14.9459i		−	0.659236i
$515$	−10.2758			−0.452806
$516$		0			0
$517$		−	29.5304i		−	1.29874i
$518$			51.2592i			2.25220i
$519$		0			0
$520$		−	1.74819i		−	0.0766631i
$521$	−2.82880			−0.123932			−0.0619660	−	0.998078i	$-0.519737\pi$
$521$	−2.82880			−0.123932			−0.0619660	+	0.998078i	$0.519737\pi$
$522$		0			0
$523$	−11.1033			−0.485513			−0.242756	−	0.970087i	$-0.578052\pi$
$523$	−11.1033			−0.485513			−0.242756	+	0.970087i	$0.578052\pi$
$524$		−	16.1889i		−	0.707217i
$525$		0			0
$526$		−	4.48000i		−	0.195337i
$527$	27.7092			1.20703
$528$		0			0
$529$	4.92089			0.213952
$530$	6.68834			0.290523
$531$		0			0
$532$			3.44008i			0.149146i
$533$			2.50296i			0.108415i
$534$		0			0
$535$			13.1871i			0.570130i
$536$	5.92780	+	5.64457i	0.256042	+	0.243808i
$537$		0			0
$538$		−	14.4866i		−	0.624562i
$539$	79.1907			3.41099
$540$		0			0
$541$			15.5871i			0.670142i	0.942193	+	0.335071i	$0.108760\pi$
$541$			15.5871i			0.670142i	−0.942193	+	0.335071i	$0.891240\pi$
$542$		−	5.83478i		−	0.250625i
$543$		0			0
$544$		−	3.24090i		−	0.138952i
$545$			0.261847i			0.0112163i
$546$		0			0
$547$			12.2195i			0.522467i	0.965276	+	0.261233i	$0.0841292\pi$
$547$			12.2195i			0.522467i	−0.965276	+	0.261233i	$0.915871\pi$
$548$	−16.2496			−0.694147
$549$		0			0
$550$	−5.12191			−0.218399
$551$			1.91336i			0.0815117i
$552$		0			0
$553$	14.3879			0.611835
$554$	−17.6693			−0.750695
$555$		0			0
$556$		−	20.0574i		−	0.850623i
$557$		−	9.66440i		−	0.409494i	−0.978815	−	0.204747i	$-0.934363\pi$
$557$		−	9.66440i		−	0.409494i	0.978815	−	0.204747i	$-0.0656371\pi$
$558$		0			0
$559$	15.2107			0.643345
$560$		−	4.73932i		−	0.200273i
$561$		0			0
$562$	11.2237			0.473444
$563$	−27.0699			−1.14086			−0.570431	−	0.821346i	$-0.693224\pi$
$563$	−27.0699			−1.14086			−0.570431	+	0.821346i	$0.693224\pi$
$564$		0			0
$565$	−5.01843			−0.211127
$566$	10.2135			0.429306
$567$		0			0
$568$		−	14.2941i		−	0.599765i
$569$		−	30.4424i		−	1.27621i	−0.769949	−	0.638106i	$-0.779718\pi$
$569$		−	30.4424i		−	1.27621i	0.769949	−	0.638106i	$-0.220282\pi$
$570$		0			0
$571$	−30.4476			−1.27419			−0.637097	−	0.770784i	$-0.719865\pi$
$571$	−30.4476			−1.27419			−0.637097	+	0.770784i	$0.719865\pi$
$572$			8.95406i			0.374388i
$573$		0			0
$574$			6.78552i			0.283222i
$575$		−	4.25195i		−	0.177319i
$576$		0			0
$577$		−	6.29146i		−	0.261917i	−0.991388	−	0.130959i	$-0.958195\pi$
$577$		−	6.29146i		−	0.261917i	0.991388	−	0.130959i	$-0.0418055\pi$
$578$	6.49657			0.270222
$579$		0			0
$580$		−	2.63599i		−	0.109453i
$581$	−21.4125			−0.888339
$582$		0			0
$583$	−34.2571			−1.41878
$584$	−10.1723			−0.420933
$585$		0			0
$586$		−	33.0445i		−	1.36506i
$587$	−32.4287			−1.33848			−0.669239	−	0.743048i	$-0.733380\pi$
$587$	−32.4287			−1.33848			−0.669239	+	0.743048i	$0.733380\pi$
$588$		0			0
$589$		−	6.20598i		−	0.255713i
$590$		−	4.44291i		−	0.182912i
$591$		0			0
$592$	−10.8157			−0.444524
$593$	14.7412			0.605349			0.302675	−	0.953094i	$-0.402120\pi$
$593$	14.7412			0.605349			0.302675	+	0.953094i	$0.402120\pi$
$594$		0			0
$595$	−15.3597			−0.629685
$596$		−	20.5053i		−	0.839928i
$597$		0			0
$598$	−7.43321			−0.303967
$599$	−7.10071			−0.290127			−0.145064	−	0.989422i	$-0.546339\pi$
$599$	−7.10071			−0.290127			−0.145064	+	0.989422i	$0.546339\pi$
$600$		0			0
$601$	29.2996			1.19516			0.597579	−	0.801810i	$-0.296130\pi$
$601$	29.2996			1.19516			0.597579	+	0.801810i	$0.296130\pi$
$602$	41.2361			1.68066
$603$		0			0
$604$	−4.99057			−0.203063
$605$	15.2340			0.619350
$606$		0			0
$607$	−2.78840			−0.113178			−0.0565889	−	0.998398i	$-0.518022\pi$
$607$	−2.78840			−0.113178			−0.0565889	+	0.998398i	$0.518022\pi$
$608$	−0.725860			−0.0294375
$609$		0			0
$610$		−	2.57925i		−	0.104431i
$611$	10.0792			0.407759
$612$		0			0
$613$	−1.77568			−0.0717191			−0.0358596	−	0.999357i	$-0.511417\pi$
$613$	−1.77568			−0.0717191			−0.0358596	+	0.999357i	$0.511417\pi$
$614$	−20.3347			−0.820643
$615$		0			0
$616$			24.2744i			0.978043i
$617$		−	26.8186i		−	1.07968i	−0.841769	−	0.539838i	$-0.818485\pi$
$617$		−	26.8186i		−	1.07968i	0.841769	−	0.539838i	$-0.181515\pi$
$618$		0			0
$619$	37.9583			1.52567			0.762837	−	0.646591i	$-0.223806\pi$
$619$	37.9583			1.52567			0.762837	+	0.646591i	$0.223806\pi$
$620$			8.54983i			0.343370i
$621$		0			0
$622$	9.20403			0.369048
$623$	17.7970			0.713021
$624$		0			0
$625$	1.00000			0.0400000
$626$		−	18.0289i		−	0.720580i
$627$		0			0
$628$	7.62941			0.304447
$629$			35.0527i			1.39764i
$630$		0			0
$631$			12.1189i			0.482444i	0.970470	+	0.241222i	$0.0775482\pi$
$631$			12.1189i			0.482444i	−0.970470	+	0.241222i	$0.922452\pi$
$632$			3.03585i			0.120760i
$633$		0			0
$634$			28.2321i			1.12124i
$635$	−20.3949			−0.809348
$636$		0			0
$637$			27.0290i			1.07093i
$638$			13.5013i			0.534521i
$639$		0			0
$640$	1.00000			0.0395285
$641$	−1.16583			−0.0460477			−0.0230238	−	0.999735i	$-0.507329\pi$
$641$	−1.16583			−0.0460477			−0.0230238	+	0.999735i	$0.507329\pi$
$642$		0			0
$643$	43.7597			1.72572			0.862858	−	0.505447i	$-0.168672\pi$
$643$	43.7597			1.72572			0.862858	+	0.505447i	$0.168672\pi$
$644$	−20.1514			−0.794075
$645$		0			0
$646$			2.35244i			0.0925555i
$647$	−26.2125			−1.03052			−0.515261	−	0.857033i	$-0.672305\pi$
$647$	−26.2125			−1.03052			−0.515261	+	0.857033i	$0.672305\pi$
$648$		0			0
$649$			22.7562i			0.893260i
$650$		−	1.74819i		−	0.0685695i
$651$		0			0
$652$	−10.3590			−0.405689
$653$	−8.27075			−0.323660			−0.161830	−	0.986819i	$-0.551740\pi$
$653$	−8.27075			−0.323660			−0.161830	+	0.986819i	$0.551740\pi$
$654$		0			0
$655$		−	16.1889i		−	0.632554i
$656$	−1.43175			−0.0559004
$657$		0			0
$658$	27.3245			1.06522
$659$		−	33.4488i		−	1.30298i	−0.758657	−	0.651490i	$-0.774144\pi$
$659$		−	33.4488i		−	1.30298i	0.758657	−	0.651490i	$-0.225856\pi$
$660$		0			0
$661$			17.5033i			0.680801i	0.940281	+	0.340401i	$0.110563\pi$
$661$			17.5033i			0.680801i	−0.940281	+	0.340401i	$0.889437\pi$
$662$		−	10.5972i		−	0.411870i
$663$		0			0
$664$		−	4.51804i		−	0.175334i
$665$			3.44008i			0.133401i
$666$		0			0
$667$	−11.2081			−0.433979
$668$			8.79713i			0.340371i
$669$		0			0
$670$	5.92780	+	5.64457i	0.229011	+	0.218069i
$671$			13.2107i			0.509994i
$672$		0			0
$673$			15.9887i			0.616321i	0.951334	+	0.308160i	$0.0997133\pi$
$673$			15.9887i			0.616321i	−0.951334	+	0.308160i	$0.900287\pi$
$674$		−	24.5719i		−	0.946473i
$675$		0			0
$676$	9.94384			0.382455
$677$	29.7097			1.14184			0.570919	−	0.821007i	$-0.306587\pi$
$677$	29.7097			1.14184			0.570919	+	0.821007i	$0.306587\pi$
$678$		0			0
$679$	51.2868			1.96820
$680$		−	3.24090i		−	0.124283i
$681$		0			0
$682$		−	43.7915i		−	1.67686i
$683$	−13.0329			−0.498689			−0.249345	−	0.968415i	$-0.580215\pi$
$683$	−13.0329			−0.498689			−0.249345	+	0.968415i	$0.580215\pi$
$684$		0			0
$685$	−16.2496			−0.620864
$686$			40.1001i			1.53103i
$687$		0			0
$688$			8.70086i			0.331717i
$689$		−	11.6925i		−	0.445448i
$690$		0			0
$691$	−28.1951			−1.07259			−0.536297	−	0.844030i	$-0.680177\pi$
$691$	−28.1951			−1.07259			−0.536297	+	0.844030i	$0.680177\pi$
$692$		−	1.33593i		−	0.0507844i
$693$		0			0
$694$	−29.5983			−1.12354
$695$		−	20.0574i		−	0.760820i
$696$		0			0
$697$			4.64015i			0.175758i
$698$	−24.3079			−0.920068
$699$		0			0
$700$		−	4.73932i		−	0.179129i
$701$	−27.1029			−1.02366			−0.511831	−	0.859086i	$-0.671033\pi$
$701$	−27.1029			−1.02366			−0.511831	+	0.859086i	$0.671033\pi$
$702$		0			0
$703$	7.85071			0.296095
$704$	−5.12191			−0.193039
$705$		0			0
$706$	26.9058			1.01261
$707$		−	13.7123i		−	0.515705i
$708$		0			0
$709$	43.0524			1.61687			0.808433	−	0.588589i	$-0.200316\pi$
$709$	43.0524			1.61687			0.808433	+	0.588589i	$0.200316\pi$
$710$		−	14.2941i		−	0.536446i
$711$		0			0
$712$			3.75518i			0.140731i
$713$	36.3535			1.36145
$714$		0			0
$715$			8.95406i			0.334863i
$716$	13.4349			0.502086
$717$		0			0
$718$			12.1558i			0.453651i
$719$			26.7373i			0.997134i	0.866851	+	0.498567i	$0.166140\pi$
$719$			26.7373i			0.997134i	−0.866851	+	0.498567i	$0.833860\pi$
$720$		0			0
$721$			48.7003i			1.81369i
$722$	−18.4731			−0.687499
$723$		0			0
$724$	23.9891			0.891547
$725$		−	2.63599i		−	0.0978981i
$726$		0			0
$727$			4.55293i			0.168859i	0.996429	+	0.0844295i	$0.0269068\pi$
$727$			4.55293i			0.168859i	−0.996429	+	0.0844295i	$0.973093\pi$
$728$	−8.28522			−0.307071
$729$		0			0
$730$	−10.1723			−0.376494
$731$	28.1986			1.04296
$732$		0			0
$733$			38.3148i			1.41519i	0.706618	+	0.707595i	$0.250220\pi$
$733$			38.3148i			1.41519i	−0.706618	+	0.707595i	$0.749780\pi$
$734$			26.7187i			0.986205i
$735$		0			0
$736$		−	4.25195i		−	0.156729i
$737$	−30.3617	−	28.9110i	−1.11839	−	1.06495i
$738$		0			0
$739$			16.0421i			0.590117i	0.955479	+	0.295059i	$0.0953392\pi$
$739$			16.0421i			0.590117i	−0.955479	+	0.295059i	$0.904661\pi$
$740$	−10.8157			−0.397594
$741$		0			0
$742$		−	31.6982i		−	1.16368i
$743$		−	5.60705i		−	0.205703i	−0.994697	−	0.102851i	$-0.967203\pi$
$743$		−	5.60705i		−	0.205703i	0.994697	−	0.102851i	$-0.0327966\pi$
$744$		0			0
$745$		−	20.5053i		−	0.751255i
$746$		−	33.3618i		−	1.22146i
$747$		0			0
$748$			16.5996i			0.606942i
$749$	62.4981			2.28363
$750$		0			0
$751$	−50.0435			−1.82611			−0.913057	−	0.407832i	$-0.866285\pi$
$751$	−50.0435			−1.82611			−0.913057	+	0.407832i	$0.866285\pi$
$752$			5.76550i			0.210246i
$753$		0			0
$754$	−4.60820			−0.167821
$755$	−4.99057			−0.181625
$756$		0			0
$757$		−	27.2689i		−	0.991104i	−0.868578	−	0.495552i	$-0.834966\pi$
$757$		−	27.2689i		−	0.991104i	0.868578	−	0.495552i	$-0.165034\pi$
$758$		−	11.1801i		−	0.406080i
$759$		0			0
$760$	−0.725860			−0.0263297
$761$		−	13.5242i		−	0.490251i	−0.969491	−	0.245125i	$-0.921171\pi$
$761$		−	13.5242i		−	0.490251i	0.969491	−	0.245125i	$-0.0788291\pi$
$762$		0			0
$763$	1.24098			0.0449264
$764$	−4.75588			−0.172062
$765$		0			0
$766$	8.20272			0.296376
$767$	−7.76704			−0.280452
$768$		0			0
$769$		−	40.8646i		−	1.47361i	−0.676103	−	0.736807i	$-0.736333\pi$
$769$		−	40.8646i		−	1.47361i	0.676103	−	0.736807i	$-0.263667\pi$
$770$			24.2744i			0.874788i
$771$		0			0
$772$	5.23513			0.188417
$773$		−	38.8760i		−	1.39827i	−0.714988	−	0.699137i	$-0.753568\pi$
$773$		−	38.8760i		−	1.39827i	0.714988	−	0.699137i	$-0.246432\pi$
$774$		0			0
$775$			8.54983i			0.307119i
$776$			10.8215i			0.388471i
$777$		0			0
$778$			15.7469i			0.564555i
$779$	1.03925			0.0372349
$780$		0			0
$781$			73.2129i			2.61976i
$782$	−13.7802			−0.492777
$783$		0			0
$784$	−15.4612			−0.552184
$785$	7.62941			0.272305
$786$		0			0
$787$			25.7666i			0.918482i	0.888312	+	0.459241i	$0.151879\pi$
$787$			25.7666i			0.918482i	−0.888312	+	0.459241i	$0.848121\pi$
$788$	27.1637			0.967666
$789$		0			0
$790$			3.03585i			0.108011i
$791$			23.7840i			0.845660i
$792$		0			0
$793$	−4.50902			−0.160120
$794$	11.3144			0.401533
$795$		0			0
$796$	−4.72867			−0.167603
$797$			47.8910i			1.69639i	0.529686	+	0.848194i	$0.322310\pi$
$797$			47.8910i			1.69639i	−0.529686	+	0.848194i	$0.677690\pi$
$798$		0			0
$799$	18.6854			0.661042
$800$	1.00000			0.0353553
$801$		0			0
$802$	25.3643			0.895646
$803$	52.1017			1.83863
$804$		0			0
$805$	−20.1514			−0.710242
$806$	14.9467			0.526475
$807$		0			0
$808$	2.89331			0.101786
$809$	37.3290			1.31242			0.656208	−	0.754580i	$-0.272159\pi$
$809$	37.3290			1.31242			0.656208	+	0.754580i	$0.272159\pi$
$810$		0			0
$811$		−	25.6651i		−	0.901222i	−0.892720	−	0.450611i	$-0.851206\pi$
$811$		−	25.6651i		−	0.901222i	0.892720	−	0.450611i	$-0.148794\pi$
$812$	−12.4928			−0.438411
$813$		0			0
$814$	55.3973			1.94167
$815$	−10.3590			−0.362859
$816$		0			0
$817$		−	6.31560i		−	0.220955i
$818$			7.38365i			0.258163i
$819$		0			0
$820$	−1.43175			−0.0499988
$821$		−	20.1599i		−	0.703586i	−0.936078	−	0.351793i	$-0.885572\pi$
$821$		−	20.1599i		−	0.703586i	0.936078	−	0.351793i	$-0.114428\pi$
$822$		0			0
$823$	54.9480			1.91537			0.957684	−	0.287822i	$-0.0929312\pi$
$823$	54.9480			1.91537			0.957684	+	0.287822i	$0.0929312\pi$
$824$	−10.2758			−0.357974
$825$		0			0
$826$	−21.0564			−0.732646
$827$			37.1501i			1.29184i	0.763407	+	0.645918i	$0.223525\pi$
$827$			37.1501i			1.29184i	−0.763407	+	0.645918i	$0.776475\pi$
$828$		0			0
$829$	22.6700			0.787360			0.393680	−	0.919248i	$-0.371202\pi$
$829$	22.6700			0.787360			0.393680	+	0.919248i	$0.371202\pi$
$830$		−	4.51804i		−	0.156824i
$831$		0			0
$832$		−	1.74819i		−	0.0606075i
$833$			50.1081i			1.73614i
$834$		0			0
$835$			8.79713i			0.304437i
$836$	3.71779			0.128582
$837$		0			0
$838$		−	8.10986i		−	0.280150i
$839$			7.15825i			0.247130i	0.992336	+	0.123565i	$0.0394328\pi$
$839$			7.15825i			0.247130i	−0.992336	+	0.123565i	$0.960567\pi$
$840$		0			0
$841$	22.0516			0.760399
$842$	−14.5262			−0.500605
$843$		0			0
$844$	−21.9100			−0.754172
$845$	9.94384			0.342079
$846$		0			0
$847$		−	72.1988i		−	2.48078i
$848$	6.68834			0.229679
$849$		0			0
$850$		−	3.24090i		−	0.111162i
$851$			45.9880i			1.57645i
$852$		0			0
$853$	7.27427			0.249066			0.124533	−	0.992215i	$-0.460257\pi$
$853$	7.27427			0.249066			0.124533	+	0.992215i	$0.460257\pi$
$854$	−12.2239			−0.418294
$855$		0			0
$856$			13.1871i			0.450727i
$857$	−16.3846			−0.559686			−0.279843	−	0.960046i	$-0.590282\pi$
$857$	−16.3846			−0.559686			−0.279843	+	0.960046i	$0.590282\pi$
$858$		0			0
$859$	0.680275			0.0232107			0.0116053	−	0.999933i	$-0.496306\pi$
$859$	0.680275			0.0232107			0.0116053	+	0.999933i	$0.496306\pi$
$860$			8.70086i			0.296697i
$861$		0			0
$862$		−	16.6118i		−	0.565801i
$863$			28.4253i			0.967609i	0.875176	+	0.483804i	$0.160745\pi$
$863$			28.4253i			0.967609i	−0.875176	+	0.483804i	$0.839255\pi$
$864$		0			0
$865$		−	1.33593i		−	0.0454229i
$866$			29.9304i			1.01707i
$867$		0			0
$868$	40.5204			1.37535
$869$		−	15.5494i		−	0.527476i
$870$		0			0
$871$	9.86777	−	10.3629i	0.334357	−	0.351133i
$872$			0.261847i			0.00886726i
$873$		0			0
$874$			3.08632i			0.104396i
$875$		−	4.73932i		−	0.160218i
$876$		0			0
$877$	17.7256			0.598551			0.299276	−	0.954167i	$-0.403255\pi$
$877$	17.7256			0.598551			0.299276	+	0.954167i	$0.403255\pi$
$878$	−11.0670			−0.373493
$879$		0			0
$880$	−5.12191			−0.172660
$881$			8.53896i			0.287685i	0.989601	+	0.143842i	$0.0459458\pi$
$881$			8.53896i			0.287685i	−0.989601	+	0.143842i	$0.954054\pi$
$882$		0			0
$883$			35.9425i			1.20956i	0.796392	+	0.604781i	$0.206739\pi$
$883$			35.9425i			1.20956i	−0.796392	+	0.604781i	$0.793261\pi$
$884$	−5.66570			−0.190558
$885$		0			0
$886$	−27.8383			−0.935247
$887$		−	0.838487i		−	0.0281536i	−0.999901	−	0.0140768i	$-0.995519\pi$
$887$		−	0.838487i		−	0.0281536i	0.999901	−	0.0140768i	$-0.00448094\pi$
$888$		0			0
$889$			96.6581i			3.24181i
$890$			3.75518i			0.125874i
$891$		0			0
$892$	18.9417			0.634215
$893$		−	4.18494i		−	0.140044i
$894$		0			0
$895$	13.4349			0.449079
$896$		−	4.73932i		−	0.158330i
$897$		0			0
$898$		−	4.37833i		−	0.146107i
$899$	22.5372			0.751659
$900$		0			0
$901$		−	21.6763i		−	0.722141i
$902$	7.33329			0.244172
$903$		0			0
$904$	−5.01843			−0.166911
$905$	23.9891			0.797424
$906$		0			0
$907$	56.4114			1.87311			0.936555	−	0.350521i	$-0.113996\pi$
$907$	56.4114			1.87311			0.936555	+	0.350521i	$0.113996\pi$
$908$		−	25.5367i		−	0.847466i
$909$		0			0
$910$	−8.28522			−0.274652
$911$			7.40421i			0.245313i	0.992449	+	0.122656i	$0.0391413\pi$
$911$			7.40421i			0.245313i	−0.992449	+	0.122656i	$0.960859\pi$
$912$		0			0
$913$			23.1410i			0.765856i
$914$	−7.67344			−0.253815
$915$		0			0
$916$			2.49414i			0.0824088i
$917$	−76.7246			−2.53367
$918$		0			0
$919$			8.50690i			0.280617i	0.990108	+	0.140308i	$0.0448094\pi$
$919$			8.50690i			0.280617i	−0.990108	+	0.140308i	$0.955191\pi$
$920$		−	4.25195i		−	0.140183i
$921$		0			0
$922$			1.75815i			0.0579016i
$923$	−24.9887			−0.822512
$924$		0			0
$925$	−10.8157			−0.355619
$926$		−	29.9782i		−	0.985144i
$927$		0			0
$928$		−	2.63599i		−	0.0865305i
$929$	38.6038			1.26655			0.633275	−	0.773927i	$-0.281710\pi$
$929$	38.6038			1.26655			0.633275	+	0.773927i	$0.281710\pi$
$930$		0			0
$931$	11.2226			0.367807
$932$	7.31040			0.239460
$933$		0			0
$934$			15.3752i			0.503091i
$935$			16.5996i			0.542865i
$936$		0			0
$937$		−	0.630761i		−	0.0206061i	−0.999947	−	0.0103030i	$-0.996720\pi$
$937$		−	0.630761i		−	0.0206061i	0.999947	−	0.0103030i	$-0.00327962\pi$
$938$	26.7514	−	28.0937i	0.873466	−	0.917293i
$939$		0			0
$940$			5.76550i			0.188050i
$941$	−23.7299			−0.773574			−0.386787	−	0.922169i	$-0.626415\pi$
$941$	−23.7299			−0.773574			−0.386787	+	0.922169i	$0.626415\pi$
$942$		0			0
$943$			6.08773i			0.198244i
$944$		−	4.44291i		−	0.144605i
$945$		0			0
$946$		−	44.5650i		−	1.44893i
$947$			18.8995i			0.614151i	0.951685	+	0.307076i	$0.0993505\pi$
$947$			18.8995i			0.614151i	−0.951685	+	0.307076i	$0.900650\pi$
$948$		0			0
$949$			17.7831i			0.577264i
$950$	−0.725860			−0.0235500
$951$		0			0
$952$	−15.3597			−0.497810
$953$		−	44.9448i		−	1.45591i	−0.685628	−	0.727953i	$-0.740472\pi$
$953$		−	44.9448i		−	1.45591i	0.685628	−	0.727953i	$-0.259528\pi$
$954$		0			0
$955$	−4.75588			−0.153897
$956$	1.61683			0.0522921
$957$		0			0
$958$		−	19.8201i		−	0.640358i
$959$			77.0119i			2.48684i
$960$		0			0
$961$	−42.0996			−1.35805
$962$			18.9079i			0.609616i
$963$		0			0
$964$	11.6519			0.375282
$965$	5.23513			0.168525
$966$		0			0
$967$	39.1824			1.26002			0.630011	−	0.776586i	$-0.283050\pi$
$967$	39.1824			1.26002			0.630011	+	0.776586i	$0.283050\pi$
$968$	15.2340			0.489639
$969$		0			0
$970$			10.8215i			0.347459i
$971$		−	2.14814i		−	0.0689371i	−0.999406	−	0.0344686i	$-0.989026\pi$
$971$		−	2.14814i		−	0.0689371i	0.999406	−	0.0344686i	$-0.0109739\pi$
$972$		0			0
$973$	−95.0584			−3.04743
$974$		−	21.1972i		−	0.679203i
$975$		0			0
$976$		−	2.57925i		−	0.0825599i
$977$			23.3470i			0.746936i	0.927643	+	0.373468i	$0.121831\pi$
$977$			23.3470i			0.746936i	−0.927643	+	0.373468i	$0.878169\pi$
$978$		0			0
$979$		−	19.2337i		−	0.614711i
$980$	−15.4612			−0.493889
$981$		0			0
$982$		−	33.3322i		−	1.06367i
$983$	58.9923			1.88156			0.940781	−	0.339016i	$-0.110094\pi$
$983$	58.9923			1.88156			0.940781	+	0.339016i	$0.110094\pi$
$984$		0			0
$985$	27.1637			0.865506
$986$	−8.54297			−0.272064
$987$		0			0
$988$			1.26894i			0.0403703i
$989$	36.9956			1.17639
$990$		0			0
$991$			32.8957i			1.04497i	0.852650	+	0.522483i	$0.174994\pi$
$991$			32.8957i			1.04497i	−0.852650	+	0.522483i	$0.825006\pi$
$992$			8.54983i			0.271457i
$993$		0			0
$994$	−67.7441			−2.14871
$995$	−4.72867			−0.149909
$996$		0			0
$997$	2.33677			0.0740062			0.0370031	−	0.999315i	$-0.488219\pi$
$997$	2.33677			0.0740062			0.0370031	+	0.999315i	$0.488219\pi$
$998$		−	20.2880i		−	0.642205i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6030.2.d.l.2411.1	yes	24
3.2	odd	2		6030.2.d.k.2411.1	✓	24
67.66	odd	2		6030.2.d.k.2411.24	yes	24
201.200	even	2	inner	6030.2.d.l.2411.24	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6030.2.d.k.2411.1	✓	24	3.2	odd	2
6030.2.d.k.2411.24	yes	24	67.66	odd	2
6030.2.d.l.2411.1	yes	24	1.1	even	1	trivial
6030.2.d.l.2411.24	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -4.73932i q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -4.73932i q^{7} +1.00000 q^{8} +1.00000 q^{10} -5.12191 q^{11} -1.74819i q^{13} -4.73932i q^{14} +1.00000 q^{16} -3.24090i q^{17} -0.725860 q^{19} +1.00000 q^{20} -5.12191 q^{22} -4.25195i q^{23} +1.00000 q^{25} -1.74819i q^{26} -4.73932i q^{28} -2.63599i q^{29} +8.54983i q^{31} +1.00000 q^{32} -3.24090i q^{34} -4.73932i q^{35} -10.8157 q^{37} -0.725860 q^{38} +1.00000 q^{40} -1.43175 q^{41} +8.70086i q^{43} -5.12191 q^{44} -4.25195i q^{46} +5.76550i q^{47} -15.4612 q^{49} +1.00000 q^{50} -1.74819i q^{52} +6.68834 q^{53} -5.12191 q^{55} -4.73932i q^{56} -2.63599i q^{58} -4.44291i q^{59} -2.57925i q^{61} +8.54983i q^{62} +1.00000 q^{64} -1.74819i q^{65} +(5.92780 + 5.64457i) q^{67} -3.24090i q^{68} -4.73932i q^{70} -14.2941i q^{71} -10.1723 q^{73} -10.8157 q^{74} -0.725860 q^{76} +24.2744i q^{77} +3.03585i q^{79} +1.00000 q^{80} -1.43175 q^{82} -4.51804i q^{83} -3.24090i q^{85} +8.70086i q^{86} -5.12191 q^{88} +3.75518i q^{89} -8.28522 q^{91} -4.25195i q^{92} +5.76550i q^{94} -0.725860 q^{95} +10.8215i q^{97} -15.4612 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 24 q^{2} + 24 q^{4} + 24 q^{5} + 24 q^{8}+O(q^{10}) \) 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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