LMFDB - Embedded newform 6021.2.a.t.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6021,2,Mod(1,6021)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6021, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6021.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6021 = 3^{3} \cdot 223 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6021.a (trivial)

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:	yes
Analytic conductor:	$48.0779270570$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Character	$\chi$	$=$	6021.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.74618 q^{2} +1.04913 q^{4} -3.84450 q^{5} +0.0845192 q^{7} +1.66038 q^{8} +O(q^{10})$	\(q-1.74618 q^{2} +1.04913 q^{4} -3.84450 q^{5} +0.0845192 q^{7} +1.66038 q^{8} +6.71318 q^{10} -0.570057 q^{11} +1.00746 q^{13} -0.147585 q^{14} -4.99759 q^{16} +6.52034 q^{17} +0.290274 q^{19} -4.03339 q^{20} +0.995420 q^{22} -1.26750 q^{23} +9.78021 q^{25} -1.75920 q^{26} +0.0886717 q^{28} -8.61807 q^{29} +4.67153 q^{31} +5.40590 q^{32} -11.3857 q^{34} -0.324935 q^{35} +8.22063 q^{37} -0.506869 q^{38} -6.38336 q^{40} +1.13205 q^{41} +9.24965 q^{43} -0.598065 q^{44} +2.21328 q^{46} -7.64016 q^{47} -6.99286 q^{49} -17.0780 q^{50} +1.05696 q^{52} +5.51167 q^{53} +2.19159 q^{55} +0.140334 q^{56} +15.0487 q^{58} -0.577510 q^{59} -0.979782 q^{61} -8.15731 q^{62} +0.555527 q^{64} -3.87318 q^{65} +10.5036 q^{67} +6.84069 q^{68} +0.567393 q^{70} -11.6854 q^{71} +13.2639 q^{73} -14.3547 q^{74} +0.304535 q^{76} -0.0481808 q^{77} -11.1799 q^{79} +19.2132 q^{80} -1.97676 q^{82} -4.27142 q^{83} -25.0675 q^{85} -16.1515 q^{86} -0.946514 q^{88} -12.6061 q^{89} +0.0851497 q^{91} -1.32977 q^{92} +13.3411 q^{94} -1.11596 q^{95} +0.781592 q^{97} +12.2108 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q + 46 q^{4} + 16 q^{7}+O(q^{10}) $ 40 * q + 46 * q^4 + 16 * q^7	$ 40 q + 46 q^{4} + 16 q^{7} + 22 q^{10} + 14 q^{13} + 50 q^{16} + 64 q^{19} + 12 q^{22} + 40 q^{25} + 48 q^{28} + 54 q^{31} + 32 q^{34} + 24 q^{37} + 40 q^{40} + 24 q^{43} + 52 q^{46} + 64 q^{49} + 18 q^{52} + 36 q^{55} + 8 q^{58} + 58 q^{61} + 120 q^{64} + 52 q^{67} - 30 q^{70} + 50 q^{73} + 112 q^{76} + 60 q^{79} + 50 q^{82} + 38 q^{85} + 16 q^{88} + 118 q^{91} + 44 q^{94} + 38 q^{97}+O(q^{100}) $ 40 * q + 46 * q^4 + 16 * q^7 + 22 * q^10 + 14 * q^13 + 50 * q^16 + 64 * q^19 + 12 * q^22 + 40 * q^25 + 48 * q^28 + 54 * q^31 + 32 * q^34 + 24 * q^37 + 40 * q^40 + 24 * q^43 + 52 * q^46 + 64 * q^49 + 18 * q^52 + 36 * q^55 + 8 * q^58 + 58 * q^61 + 120 * q^64 + 52 * q^67 - 30 * q^70 + 50 * q^73 + 112 * q^76 + 60 * q^79 + 50 * q^82 + 38 * q^85 + 16 * q^88 + 118 * q^91 + 44 * q^94 + 38 * q^97

Display 100 coefficients 10 coefficients

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 $

$2$	−1.74618	−1.23473	−0.617366	−	0.786676i	$-0.711800\pi$
$2$	−1.74618	−1.23473	−0.617366	+	0.786676i	$0.711800\pi$
$3$	0	0
$3$	0	0
$4$	1.04913	0.524565
$5$	−3.84450	−1.71931	−0.859657	−	0.510871i	$-0.829323\pi$
$5$	−3.84450	−1.71931	−0.859657	+	0.510871i	$0.829323\pi$
$6$	0	0
$7$	0.0845192	0.0319453	0.0159726	−	0.999872i	$-0.494916\pi$
$7$	0.0845192	0.0319453	0.0159726	+	0.999872i	$0.494916\pi$
$8$	1.66038	0.587035
$9$	0	0
$10$	6.71318	2.12289
$11$	−0.570057	−0.171879	−0.0859394	−	0.996300i	$-0.527389\pi$
$11$	−0.570057	−0.171879	−0.0859394	+	0.996300i	$0.527389\pi$
$12$	0	0
$13$	1.00746	0.279419	0.139710	−	0.990193i	$-0.455383\pi$
$13$	1.00746	0.279419	0.139710	+	0.990193i	$0.455383\pi$
$14$	−0.147585	−0.0394439
$15$	0	0
$16$	−4.99759	−1.24940
$17$	6.52034	1.58141	0.790707	−	0.612195i	$-0.209713\pi$
$17$	6.52034	1.58141	0.790707	+	0.612195i	$0.209713\pi$
$18$	0	0
$19$	0.290274	0.0665933	0.0332967	−	0.999446i	$-0.489399\pi$
$19$	0.290274	0.0665933	0.0332967	+	0.999446i	$0.489399\pi$
$20$	−4.03339	−0.901893
$21$	0	0
$22$	0.995420	0.212224
$23$	−1.26750	−0.264292	−0.132146	−	0.991230i	$-0.542187\pi$
$23$	−1.26750	−0.264292	−0.132146	+	0.991230i	$0.542187\pi$
$24$	0	0
$25$	9.78021	1.95604
$26$	−1.75920	−0.345008
$27$	0	0
$28$	0.0886717	0.0167574
$29$	−8.61807	−1.60034	−0.800168	−	0.599776i	$-0.795256\pi$
$29$	−8.61807	−1.60034	−0.800168	+	0.599776i	$0.795256\pi$
$30$	0	0
$31$	4.67153	0.839031	0.419515	−	0.907748i	$-0.362200\pi$
$31$	4.67153	0.839031	0.419515	+	0.907748i	$0.362200\pi$
$32$	5.40590	0.955636
$33$	0	0
$34$	−11.3857	−1.95262
$35$	−0.324935	−0.0549240
$36$	0	0
$37$	8.22063	1.35146	0.675732	−	0.737147i	$-0.263828\pi$
$37$	8.22063	1.35146	0.675732	+	0.737147i	$0.263828\pi$
$38$	−0.506869	−0.0822250
$39$	0	0
$40$	−6.38336	−1.00930
$41$	1.13205	0.176797	0.0883983	−	0.996085i	$-0.471825\pi$
$41$	1.13205	0.176797	0.0883983	+	0.996085i	$0.471825\pi$
$42$	0	0
$43$	9.24965	1.41056	0.705279	−	0.708929i	$-0.250822\pi$
$43$	9.24965	1.41056	0.705279	+	0.708929i	$0.250822\pi$
$44$	−0.598065	−0.0901616
$45$	0	0
$46$	2.21328	0.326330
$47$	−7.64016	−1.11443	−0.557216	−	0.830367i	$-0.688131\pi$
$47$	−7.64016	−1.11443	−0.557216	+	0.830367i	$0.688131\pi$
$48$	0	0
$49$	−6.99286	−0.998979
$50$	−17.0780	−2.41519
$51$	0	0
$52$	1.05696	0.146574
$53$	5.51167	0.757086	0.378543	−	0.925584i	$-0.376425\pi$
$53$	5.51167	0.757086	0.378543	+	0.925584i	$0.376425\pi$
$54$	0	0
$55$	2.19159	0.295514
$56$	0.140334	0.0187530
$57$	0	0
$58$	15.0487	1.97599
$59$	−0.577510	−0.0751854	−0.0375927	−	0.999293i	$-0.511969\pi$
$59$	−0.577510	−0.0751854	−0.0375927	+	0.999293i	$0.511969\pi$
$60$	0	0
$61$	−0.979782	−0.125448	−0.0627241	−	0.998031i	$-0.519979\pi$
$61$	−0.979782	−0.125448	−0.0627241	+	0.998031i	$0.519979\pi$
$62$	−8.15731	−1.03598
$63$	0	0
$64$	0.555527	0.0694409
$65$	−3.87318	−0.480409
$66$	0	0
$67$	10.5036	1.28322	0.641611	−	0.767030i	$-0.278266\pi$
$67$	10.5036	1.28322	0.641611	+	0.767030i	$0.278266\pi$
$68$	6.84069	0.829555
$69$	0	0
$70$	0.567393	0.0678164
$71$	−11.6854	−1.38680	−0.693401	−	0.720552i	$-0.743889\pi$
$71$	−11.6854	−1.38680	−0.693401	+	0.720552i	$0.743889\pi$
$72$	0	0
$73$	13.2639	1.55243	0.776213	−	0.630471i	$-0.217138\pi$
$73$	13.2639	1.55243	0.776213	+	0.630471i	$0.217138\pi$
$74$	−14.3547	−1.66870
$75$	0	0
$76$	0.304535	0.0349326
$77$	−0.0481808	−0.00549071
$78$	0	0
$79$	−11.1799	−1.25784	−0.628921	−	0.777470i	$-0.716503\pi$
$79$	−11.1799	−1.25784	−0.628921	+	0.777470i	$0.716503\pi$
$80$	19.2132	2.14811
$81$	0	0
$82$	−1.97676	−0.218297
$83$	−4.27142	−0.468849	−0.234425	−	0.972134i	$-0.575321\pi$
$83$	−4.27142	−0.468849	−0.234425	+	0.972134i	$0.575321\pi$
$84$	0	0
$85$	−25.0675	−2.71895
$86$	−16.1515	−1.74166
$87$	0	0
$88$	−0.946514	−0.100899
$89$	−12.6061	−1.33625	−0.668124	−	0.744050i	$-0.732903\pi$
$89$	−12.6061	−1.33625	−0.668124	+	0.744050i	$0.732903\pi$
$90$	0	0
$91$	0.0851497	0.00892612
$92$	−1.32977	−0.138638
$93$	0	0
$94$	13.3411	1.37603
$95$	−1.11596	−0.114495
$96$	0	0
$97$	0.781592	0.0793586	0.0396793	−	0.999212i	$-0.487366\pi$
$97$	0.781592	0.0793586	0.0396793	+	0.999212i	$0.487366\pi$
$98$	12.2108	1.23347
$99$	0	0
$100$	10.2607	1.02607
$101$	−10.7403	−1.06870	−0.534350	−	0.845264i	$-0.679443\pi$
$101$	−10.7403	−1.06870	−0.534350	+	0.845264i	$0.679443\pi$
$102$	0	0
$103$	−1.86897	−0.184156	−0.0920778	−	0.995752i	$-0.529351\pi$
$103$	−1.86897	−0.184156	−0.0920778	+	0.995752i	$0.529351\pi$
$104$	1.67277	0.164029
$105$	0	0
$106$	−9.62434	−0.934799
$107$	5.02150	0.485447	0.242723	−	0.970096i	$-0.421959\pi$
$107$	5.02150	0.485447	0.242723	+	0.970096i	$0.421959\pi$
$108$	0	0
$109$	5.59482	0.535887	0.267944	−	0.963435i	$-0.413656\pi$
$109$	5.59482	0.535887	0.267944	+	0.963435i	$0.413656\pi$
$110$	−3.82690	−0.364880
$111$	0	0
$112$	−0.422392	−0.0399123
$113$	6.30339	0.592973	0.296486	−	0.955037i	$-0.404185\pi$
$113$	6.30339	0.592973	0.296486	+	0.955037i	$0.404185\pi$
$114$	0	0
$115$	4.87290	0.454401
$116$	−9.04148	−0.839481
$117$	0	0
$118$	1.00843	0.0928338
$119$	0.551094	0.0505187
$120$	0	0
$121$	−10.6750	−0.970458
$122$	1.71087	0.154895
$123$	0	0
$124$	4.90104	0.440127
$125$	−18.3776	−1.64374
$126$	0	0
$127$	−12.8217	−1.13774	−0.568870	−	0.822428i	$-0.692619\pi$
$127$	−12.8217	−1.13774	−0.568870	+	0.822428i	$0.692619\pi$
$128$	−11.7818	−1.04138
$129$	0	0
$130$	6.76326	0.593177
$131$	16.7216	1.46097	0.730486	−	0.682928i	$-0.239294\pi$
$131$	16.7216	1.46097	0.730486	+	0.682928i	$0.239294\pi$
$132$	0	0
$133$	0.0245337	0.00212734
$134$	−18.3412	−1.58444
$135$	0	0
$136$	10.8263	0.928345
$137$	3.78444	0.323327	0.161663	−	0.986846i	$-0.448314\pi$
$137$	3.78444	0.323327	0.161663	+	0.986846i	$0.448314\pi$
$138$	0	0
$139$	3.99601	0.338937	0.169469	−	0.985536i	$-0.445795\pi$
$139$	3.99601	0.338937	0.169469	+	0.985536i	$0.445795\pi$
$140$	−0.340899	−0.0288112
$141$	0	0
$142$	20.4048	1.71233
$143$	−0.574310	−0.0480262
$144$	0	0
$145$	33.1322	2.75148
$146$	−23.1612	−1.91683
$147$	0	0
$148$	8.62452	0.708931
$149$	−7.49430	−0.613957	−0.306978	−	0.951716i	$-0.599318\pi$
$149$	−7.49430	−0.613957	−0.306978	+	0.951716i	$0.599318\pi$
$150$	0	0
$151$	−4.83429	−0.393409	−0.196705	−	0.980463i	$-0.563024\pi$
$151$	−4.83429	−0.393409	−0.196705	+	0.980463i	$0.563024\pi$
$152$	0.481966	0.0390926
$153$	0	0
$154$	0.0841322	0.00677956
$155$	−17.9597	−1.44256
$156$	0	0
$157$	−22.9840	−1.83432	−0.917162	−	0.398514i	$-0.869526\pi$
$157$	−22.9840	−1.83432	−0.917162	+	0.398514i	$0.869526\pi$
$158$	19.5221	1.55310
$159$	0	0
$160$	−20.7830	−1.64304
$161$	−0.107128	−0.00844287
$162$	0	0
$163$	−10.0587	−0.787859	−0.393929	−	0.919141i	$-0.628885\pi$
$163$	−10.0587	−0.787859	−0.393929	+	0.919141i	$0.628885\pi$
$164$	1.18767	0.0927413
$165$	0	0
$166$	7.45865	0.578904
$167$	5.22914	0.404643	0.202322	−	0.979319i	$-0.435151\pi$
$167$	5.22914	0.404643	0.202322	+	0.979319i	$0.435151\pi$
$168$	0	0
$169$	−11.9850	−0.921925
$170$	43.7722	3.35717
$171$	0	0
$172$	9.70410	0.739930
$173$	6.37362	0.484578	0.242289	−	0.970204i	$-0.422102\pi$
$173$	6.37362	0.484578	0.242289	+	0.970204i	$0.422102\pi$
$174$	0	0
$175$	0.826616	0.0624863
$176$	2.84891	0.214745
$177$	0	0
$178$	22.0125	1.64991
$179$	6.87135	0.513589	0.256795	−	0.966466i	$-0.417334\pi$
$179$	6.87135	0.513589	0.256795	+	0.966466i	$0.417334\pi$
$180$	0	0
$181$	−16.3400	−1.21455	−0.607273	−	0.794493i	$-0.707737\pi$
$181$	−16.3400	−1.21455	−0.607273	+	0.794493i	$0.707737\pi$
$182$	−0.148686	−0.0110214
$183$	0	0
$184$	−2.10454	−0.155148
$185$	−31.6043	−2.32359
$186$	0	0
$187$	−3.71696	−0.271811
$188$	−8.01553	−0.584593
$189$	0	0
$190$	1.94866	0.141371
$191$	17.9185	1.29654	0.648270	−	0.761410i	$-0.275493\pi$
$191$	17.9185	1.29654	0.648270	+	0.761410i	$0.275493\pi$
$192$	0	0
$193$	2.11874	0.152510	0.0762551	−	0.997088i	$-0.475704\pi$
$193$	2.11874	0.152510	0.0762551	+	0.997088i	$0.475704\pi$
$194$	−1.36480	−0.0979867
$195$	0	0
$196$	−7.33642	−0.524030
$197$	−4.88867	−0.348303	−0.174152	−	0.984719i	$-0.555718\pi$
$197$	−4.88867	−0.348303	−0.174152	+	0.984719i	$0.555718\pi$
$198$	0	0
$199$	−10.9320	−0.774946	−0.387473	−	0.921881i	$-0.626652\pi$
$199$	−10.9320	−0.774946	−0.387473	+	0.921881i	$0.626652\pi$
$200$	16.2389	1.14826
$201$	0	0
$202$	18.7544	1.31956
$203$	−0.728393	−0.0511232
$204$	0	0
$205$	−4.35217	−0.303969
$206$	3.26356	0.227383
$207$	0	0
$208$	−5.03487	−0.349105
$209$	−0.165473	−0.0114460
$210$	0	0
$211$	−1.04208	−0.0717398	−0.0358699	−	0.999356i	$-0.511420\pi$
$211$	−1.04208	−0.0717398	−0.0358699	+	0.999356i	$0.511420\pi$
$212$	5.78246	0.397141
$213$	0	0
$214$	−8.76842	−0.599397
$215$	−35.5603	−2.42519
$216$	0	0
$217$	0.394834	0.0268031
$218$	−9.76955	−0.661677
$219$	0	0
$220$	2.29926	0.155016
$221$	6.56898	0.441877
$222$	0	0
$223$	1.00000	0.0669650
$223$	1.00000	0.0669650
$224$	0.456902	0.0305281
$225$	0	0
$226$	−11.0068	−0.732163
$227$	−1.92344	−0.127663	−0.0638316	−	0.997961i	$-0.520332\pi$
$227$	−1.92344	−0.127663	−0.0638316	+	0.997961i	$0.520332\pi$
$228$	0	0
$229$	4.74351	0.313460	0.156730	−	0.987641i	$-0.449905\pi$
$229$	4.74351	0.313460	0.156730	+	0.987641i	$0.449905\pi$
$230$	−8.50895	−0.561063
$231$	0	0
$232$	−14.3093	−0.939453
$233$	−4.47780	−0.293351	−0.146675	−	0.989185i	$-0.546857\pi$
$233$	−4.47780	−0.293351	−0.146675	+	0.989185i	$0.546857\pi$
$234$	0	0
$235$	29.3726	1.91606
$236$	−0.605883	−0.0394396
$237$	0	0
$238$	−0.962307	−0.0623771
$239$	15.5704	1.00717	0.503584	−	0.863946i	$-0.332014\pi$
$239$	15.5704	1.00717	0.503584	+	0.863946i	$0.332014\pi$
$240$	0	0
$241$	−20.0749	−1.29313	−0.646567	−	0.762857i	$-0.723796\pi$
$241$	−20.0749	−1.29313	−0.646567	+	0.762857i	$0.723796\pi$
$242$	18.6405	1.19826
$243$	0	0
$244$	−1.02792	−0.0658058
$245$	26.8841	1.71756
$246$	0	0
$247$	0.292439	0.0186074
$248$	7.75653	0.492540
$249$	0	0
$250$	32.0904	2.02958
$251$	12.4164	0.783716	0.391858	−	0.920026i	$-0.371833\pi$
$251$	12.4164	0.783716	0.391858	+	0.920026i	$0.371833\pi$
$252$	0	0
$253$	0.722547	0.0454261
$254$	22.3889	1.40480
$255$	0	0
$256$	19.4621	1.21638
$257$	21.9318	1.36807	0.684036	−	0.729449i	$-0.260223\pi$
$257$	21.9318	1.36807	0.684036	+	0.729449i	$0.260223\pi$
$258$	0	0
$259$	0.694802	0.0431729
$260$	−4.06348	−0.252006
$261$	0	0
$262$	−29.1988	−1.80391
$263$	19.5923	1.20811	0.604056	−	0.796942i	$-0.293550\pi$
$263$	19.5923	1.20811	0.604056	+	0.796942i	$0.293550\pi$
$264$	0	0
$265$	−21.1896	−1.30167
$266$	−0.0428402	−0.00262670
$267$	0	0
$268$	11.0197	0.673134
$269$	28.8226	1.75735	0.878674	−	0.477423i	$-0.158429\pi$
$269$	28.8226	1.75735	0.878674	+	0.477423i	$0.158429\pi$
$270$	0	0
$271$	20.4087	1.23974	0.619871	−	0.784704i	$-0.287185\pi$
$271$	20.4087	1.23974	0.619871	+	0.784704i	$0.287185\pi$
$272$	−32.5859	−1.97581
$273$	0	0
$274$	−6.60830	−0.399222
$275$	−5.57528	−0.336202
$276$	0	0
$277$	16.9083	1.01592	0.507960	−	0.861381i	$-0.330400\pi$
$277$	16.9083	1.01592	0.507960	+	0.861381i	$0.330400\pi$
$278$	−6.97774	−0.418497
$279$	0	0
$280$	−0.539516	−0.0322423
$281$	25.9751	1.54954	0.774771	−	0.632242i	$-0.217865\pi$
$281$	25.9751	1.54954	0.774771	+	0.632242i	$0.217865\pi$
$282$	0	0
$283$	14.7023	0.873962	0.436981	−	0.899471i	$-0.356048\pi$
$283$	14.7023	0.873962	0.436981	+	0.899471i	$0.356048\pi$
$284$	−12.2595	−0.727469
$285$	0	0
$286$	1.00285	0.0592995
$287$	0.0956800	0.00564781
$288$	0	0
$289$	25.5148	1.50087
$290$	−57.8547	−3.39734
$291$	0	0
$292$	13.9156	0.814349
$293$	−5.90617	−0.345042	−0.172521	−	0.985006i	$-0.555191\pi$
$293$	−5.90617	−0.345042	−0.172521	+	0.985006i	$0.555191\pi$
$294$	0	0
$295$	2.22024	0.129267
$296$	13.6494	0.793356
$297$	0	0
$298$	13.0864	0.758073
$299$	−1.27695	−0.0738481
$300$	0	0
$301$	0.781774	0.0450607
$302$	8.44153	0.485755
$303$	0	0
$304$	−1.45067	−0.0832015
$305$	3.76678	0.215685
$306$	0	0
$307$	19.6478	1.12136	0.560680	−	0.828033i	$-0.310540\pi$
$307$	19.6478	1.12136	0.560680	+	0.828033i	$0.310540\pi$
$308$	−0.0505480	−0.00288024
$309$	0	0
$310$	31.3608	1.78117
$311$	−27.1357	−1.53872	−0.769362	−	0.638813i	$-0.779426\pi$
$311$	−27.1357	−1.53872	−0.769362	+	0.638813i	$0.779426\pi$
$312$	0	0
$313$	30.0054	1.69600	0.848002	−	0.529993i	$-0.177805\pi$
$313$	30.0054	1.69600	0.848002	+	0.529993i	$0.177805\pi$
$314$	40.1341	2.26490
$315$	0	0
$316$	−11.7292	−0.659820
$317$	−7.54089	−0.423539	−0.211769	−	0.977320i	$-0.567923\pi$
$317$	−7.54089	−0.423539	−0.211769	+	0.977320i	$0.567923\pi$
$318$	0	0
$319$	4.91279	0.275064
$320$	−2.13573	−0.119391
$321$	0	0
$322$	0.187064	0.0104247
$323$	1.89268	0.105312
$324$	0	0
$325$	9.85317	0.546556
$326$	17.5643	0.972795
$327$	0	0
$328$	1.87964	0.103786
$329$	−0.645741	−0.0356008
$330$	0	0
$331$	14.4268	0.792970	0.396485	−	0.918041i	$-0.370230\pi$
$331$	14.4268	0.792970	0.396485	+	0.918041i	$0.370230\pi$
$332$	−4.48128	−0.245942
$333$	0	0
$334$	−9.13100	−0.499626
$335$	−40.3813	−2.20626
$336$	0	0
$337$	8.92666	0.486266	0.243133	−	0.969993i	$-0.421825\pi$
$337$	8.92666	0.486266	0.243133	+	0.969993i	$0.421825\pi$
$338$	20.9280	1.13833
$339$	0	0
$340$	−26.2990	−1.42627
$341$	−2.66304	−0.144212
$342$	0	0
$343$	−1.18267	−0.0638579
$344$	15.3580	0.828047
$345$	0	0
$346$	−11.1295	−0.598324
$347$	−8.04115	−0.431672	−0.215836	−	0.976430i	$-0.569248\pi$
$347$	−8.04115	−0.431672	−0.215836	+	0.976430i	$0.569248\pi$
$348$	0	0
$349$	0.522661	0.0279774	0.0139887	−	0.999902i	$-0.495547\pi$
$349$	0.522661	0.0279774	0.0139887	+	0.999902i	$0.495547\pi$
$350$	−1.44342	−0.0771539
$351$	0	0
$352$	−3.08167	−0.164254
$353$	7.15856	0.381011	0.190506	−	0.981686i	$-0.438987\pi$
$353$	7.15856	0.381011	0.190506	+	0.981686i	$0.438987\pi$
$354$	0	0
$355$	44.9246	2.38435
$356$	−13.2255	−0.700950
$357$	0	0
$358$	−11.9986	−0.634145
$359$	−1.65582	−0.0873909	−0.0436954	−	0.999045i	$-0.513913\pi$
$359$	−1.65582	−0.0873909	−0.0436954	+	0.999045i	$0.513913\pi$
$360$	0	0
$361$	−18.9157	−0.995565
$362$	28.5326	1.49964
$363$	0	0
$364$	0.0893332	0.00468233
$365$	−50.9932	−2.66911
$366$	0	0
$367$	27.4586	1.43333	0.716663	−	0.697419i	$-0.245669\pi$
$367$	27.4586	1.43333	0.716663	+	0.697419i	$0.245669\pi$
$368$	6.33443	0.330205
$369$	0	0
$370$	55.1866	2.86901
$371$	0.465842	0.0241853
$372$	0	0
$373$	−14.8378	−0.768274	−0.384137	−	0.923276i	$-0.625501\pi$
$373$	−14.8378	−0.768274	−0.384137	+	0.923276i	$0.625501\pi$
$374$	6.49047	0.335614
$375$	0	0
$376$	−12.6856	−0.654211
$377$	−8.68236	−0.447164
$378$	0	0
$379$	−11.8936	−0.610935	−0.305467	−	0.952203i	$-0.598813\pi$
$379$	−11.8936	−0.610935	−0.305467	+	0.952203i	$0.598813\pi$
$380$	−1.17079	−0.0600601
$381$	0	0
$382$	−31.2889	−1.60088
$383$	−34.4810	−1.76190	−0.880949	−	0.473211i	$-0.843095\pi$
$383$	−34.4810	−1.76190	−0.880949	+	0.473211i	$0.843095\pi$
$384$	0	0
$385$	0.185231	0.00944026
$386$	−3.69969	−0.188309
$387$	0	0
$388$	0.819992	0.0416288
$389$	14.0528	0.712503	0.356251	−	0.934390i	$-0.384055\pi$
$389$	14.0528	0.712503	0.356251	+	0.934390i	$0.384055\pi$
$390$	0	0
$391$	−8.26451	−0.417954
$392$	−11.6108	−0.586436
$393$	0	0
$394$	8.53648	0.430062
$395$	42.9813	2.16262
$396$	0	0
$397$	−4.84521	−0.243174	−0.121587	−	0.992581i	$-0.538798\pi$
$397$	−4.84521	−0.243174	−0.121587	+	0.992581i	$0.538798\pi$
$398$	19.0891	0.956851
$399$	0	0
$400$	−48.8775	−2.44387
$401$	0.959233	0.0479018	0.0239509	−	0.999713i	$-0.492375\pi$
$401$	0.959233	0.0479018	0.0239509	+	0.999713i	$0.492375\pi$
$402$	0	0
$403$	4.70637	0.234441
$404$	−11.2680	−0.560603
$405$	0	0
$406$	1.27190	0.0631234
$407$	−4.68623	−0.232288
$408$	0	0
$409$	10.2423	0.506448	0.253224	−	0.967408i	$-0.418509\pi$
$409$	10.2423	0.506448	0.253224	+	0.967408i	$0.418509\pi$
$410$	7.59966	0.375320
$411$	0	0
$412$	−1.96080	−0.0966016
$413$	−0.0488107	−0.00240182
$414$	0	0
$415$	16.4215	0.806099
$416$	5.44622	0.267023
$417$	0	0
$418$	0.288944	0.0141327
$419$	−25.4950	−1.24551	−0.622755	−	0.782417i	$-0.713987\pi$
$419$	−25.4950	−1.24551	−0.622755	+	0.782417i	$0.713987\pi$
$420$	0	0
$421$	30.7678	1.49953	0.749766	−	0.661704i	$-0.230166\pi$
$421$	30.7678	1.49953	0.749766	+	0.661704i	$0.230166\pi$
$422$	1.81966	0.0885795
$423$	0	0
$424$	9.15149	0.444436
$425$	63.7703	3.09331
$426$	0	0
$427$	−0.0828104	−0.00400748
$428$	5.26821	0.254649
$429$	0	0
$430$	62.0946	2.99447
$431$	−33.8864	−1.63225	−0.816126	−	0.577874i	$-0.803882\pi$
$431$	−33.8864	−1.63225	−0.816126	+	0.577874i	$0.803882\pi$
$432$	0	0
$433$	−30.4573	−1.46368	−0.731842	−	0.681474i	$-0.761339\pi$
$433$	−30.4573	−1.46368	−0.731842	+	0.681474i	$0.761339\pi$
$434$	−0.689449	−0.0330946
$435$	0	0
$436$	5.86970	0.281108
$437$	−0.367921	−0.0176001
$438$	0	0
$439$	32.6163	1.55669	0.778344	−	0.627837i	$-0.216060\pi$
$439$	32.6163	1.55669	0.778344	+	0.627837i	$0.216060\pi$
$440$	3.63888	0.173477
$441$	0	0
$442$	−11.4706	−0.545600
$443$	11.8235	0.561752	0.280876	−	0.959744i	$-0.409375\pi$
$443$	11.8235	0.561752	0.280876	+	0.959744i	$0.409375\pi$
$444$	0	0
$445$	48.4644	2.29743
$446$	−1.74618	−0.0826838
$447$	0	0
$448$	0.0469527	0.00221831
$449$	26.0531	1.22952	0.614762	−	0.788713i	$-0.289252\pi$
$449$	26.0531	1.22952	0.614762	+	0.788713i	$0.289252\pi$
$450$	0	0
$451$	−0.645333	−0.0303876
$452$	6.61308	0.311053
$453$	0	0
$454$	3.35866	0.157630
$455$	−0.327358	−0.0153468
$456$	0	0
$457$	−17.1061	−0.800187	−0.400094	−	0.916474i	$-0.631022\pi$
$457$	−17.1061	−0.800187	−0.400094	+	0.916474i	$0.631022\pi$
$458$	−8.28300	−0.387039
$459$	0	0
$460$	5.11231	0.238363
$461$	−15.4453	−0.719358	−0.359679	−	0.933076i	$-0.617114\pi$
$461$	−15.4453	−0.719358	−0.359679	+	0.933076i	$0.617114\pi$
$462$	0	0
$463$	15.1486	0.704015	0.352008	−	0.935997i	$-0.385499\pi$
$463$	15.1486	0.704015	0.352008	+	0.935997i	$0.385499\pi$
$464$	43.0696	1.99945
$465$	0	0
$466$	7.81903	0.362210
$467$	17.9449	0.830392	0.415196	−	0.909732i	$-0.363713\pi$
$467$	17.9449	0.830392	0.415196	+	0.909732i	$0.363713\pi$
$468$	0	0
$469$	0.887759	0.0409929
$470$	−51.2898	−2.36582
$471$	0	0
$472$	−0.958888	−0.0441364
$473$	−5.27283	−0.242445
$474$	0	0
$475$	2.83894	0.130259
$476$	0.578169	0.0265004
$477$	0	0
$478$	−27.1887	−1.24358
$479$	−11.5568	−0.528042	−0.264021	−	0.964517i	$-0.585049\pi$
$479$	−11.5568	−0.528042	−0.264021	+	0.964517i	$0.585049\pi$
$480$	0	0
$481$	8.28196	0.377625
$482$	35.0542	1.59668
$483$	0	0
$484$	−11.1995	−0.509069
$485$	−3.00483	−0.136442
$486$	0	0
$487$	−4.46550	−0.202351	−0.101176	−	0.994869i	$-0.532260\pi$
$487$	−4.46550	−0.202351	−0.101176	+	0.994869i	$0.532260\pi$
$488$	−1.62682	−0.0736425
$489$	0	0
$490$	−46.9443	−2.12073
$491$	9.92883	0.448082	0.224041	−	0.974580i	$-0.428075\pi$
$491$	9.92883	0.448082	0.224041	+	0.974580i	$0.428075\pi$
$492$	0	0
$493$	−56.1927	−2.53079
$494$	−0.510650	−0.0229752
$495$	0	0
$496$	−23.3464	−1.04828
$497$	−0.987641	−0.0443018
$498$	0	0
$499$	−23.1175	−1.03488	−0.517440	−	0.855720i	$-0.673115\pi$
$499$	−23.1175	−1.03488	−0.517440	+	0.855720i	$0.673115\pi$
$500$	−19.2805	−0.862248
$501$	0	0
$502$	−21.6812	−0.967679
$503$	34.8790	1.55518	0.777588	−	0.628774i	$-0.216443\pi$
$503$	34.8790	1.55518	0.777588	+	0.628774i	$0.216443\pi$
$504$	0	0
$505$	41.2911	1.83743
$506$	−1.26169	−0.0560891
$507$	0	0
$508$	−13.4516	−0.596819
$509$	17.3556	0.769275	0.384638	−	0.923068i	$-0.374326\pi$
$509$	17.3556	0.769275	0.384638	+	0.923068i	$0.374326\pi$
$510$	0	0
$511$	1.12106	0.0495927
$512$	−10.4206	−0.460530
$513$	0	0
$514$	−38.2969	−1.68920
$515$	7.18528	0.316621
$516$	0	0
$517$	4.35533	0.191547
$518$	−1.21325	−0.0533070
$519$	0	0
$520$	−6.43098	−0.282017
$521$	38.6525	1.69340	0.846698	−	0.532074i	$-0.178587\pi$
$521$	38.6525	1.69340	0.846698	+	0.532074i	$0.178587\pi$
$522$	0	0
$523$	−33.1427	−1.44923	−0.724614	−	0.689155i	$-0.757982\pi$
$523$	−33.1427	−1.44923	−0.724614	+	0.689155i	$0.757982\pi$
$524$	17.5431	0.766375
$525$	0	0
$526$	−34.2116	−1.49170
$527$	30.4599	1.32685
$528$	0	0
$529$	−21.3934	−0.930150
$530$	37.0008	1.60721
$531$	0	0
$532$	0.0257391	0.00111593
$533$	1.14049	0.0494003
$534$	0	0
$535$	−19.3052	−0.834636
$536$	17.4401	0.753296
$537$	0	0
$538$	−50.3294	−2.16985
$539$	3.98633	0.171703
$540$	0	0
$541$	2.85988	0.122956	0.0614779	−	0.998108i	$-0.480419\pi$
$541$	2.85988	0.122956	0.0614779	+	0.998108i	$0.480419\pi$
$542$	−35.6372	−1.53075
$543$	0	0
$544$	35.2483	1.51126
$545$	−21.5093	−0.921358
$546$	0	0
$547$	25.1470	1.07521	0.537604	−	0.843197i	$-0.319329\pi$
$547$	25.1470	1.07521	0.537604	+	0.843197i	$0.319329\pi$
$548$	3.97038	0.169606
$549$	0	0
$550$	9.73542	0.415120
$551$	−2.50160	−0.106572
$552$	0	0
$553$	−0.944920	−0.0401821
$554$	−29.5248	−1.25439
$555$	0	0
$556$	4.19234	0.177795
$557$	14.5967	0.618481	0.309240	−	0.950984i	$-0.399925\pi$
$557$	14.5967	0.618481	0.309240	+	0.950984i	$0.399925\pi$
$558$	0	0
$559$	9.31865	0.394137
$560$	1.62389	0.0686218
$561$	0	0
$562$	−45.3570	−1.91327
$563$	9.43881	0.397798	0.198899	−	0.980020i	$-0.436263\pi$
$563$	9.43881	0.397798	0.198899	+	0.980020i	$0.436263\pi$
$564$	0	0
$565$	−24.2334	−1.01951
$566$	−25.6728	−1.07911
$567$	0	0
$568$	−19.4023	−0.814101
$569$	−42.5557	−1.78403	−0.892014	−	0.452008i	$-0.850708\pi$
$569$	−42.5557	−1.78403	−0.892014	+	0.452008i	$0.850708\pi$
$570$	0	0
$571$	5.38826	0.225492	0.112746	−	0.993624i	$-0.464035\pi$
$571$	5.38826	0.225492	0.112746	+	0.993624i	$0.464035\pi$
$572$	−0.602526	−0.0251929
$573$	0	0
$574$	−0.167074	−0.00697354
$575$	−12.3964	−0.516966
$576$	0	0
$577$	−1.71672	−0.0714679	−0.0357340	−	0.999361i	$-0.511377\pi$
$577$	−1.71672	−0.0714679	−0.0357340	+	0.999361i	$0.511377\pi$
$578$	−44.5533	−1.85317
$579$	0	0
$580$	34.7600	1.44333
$581$	−0.361017	−0.0149775
$582$	0	0
$583$	−3.14197	−0.130127
$584$	22.0232	0.911328
$585$	0	0
$586$	10.3132	0.426035
$587$	43.7561	1.80601	0.903004	−	0.429633i	$-0.141357\pi$
$587$	43.7561	1.80601	0.903004	+	0.429633i	$0.141357\pi$
$588$	0	0
$589$	1.35602	0.0558739
$590$	−3.87693	−0.159611
$591$	0	0
$592$	−41.0833	−1.68851
$593$	21.7198	0.891924	0.445962	−	0.895052i	$-0.352862\pi$
$593$	21.7198	0.891924	0.445962	+	0.895052i	$0.352862\pi$
$594$	0	0
$595$	−2.11868	−0.0868575
$596$	−7.86250	−0.322061
$597$	0	0
$598$	2.22979	0.0911827
$599$	−23.2326	−0.949258	−0.474629	−	0.880186i	$-0.657418\pi$
$599$	−23.2326	−0.949258	−0.474629	+	0.880186i	$0.657418\pi$
$600$	0	0
$601$	−6.79367	−0.277120	−0.138560	−	0.990354i	$-0.544247\pi$
$601$	−6.79367	−0.277120	−0.138560	+	0.990354i	$0.544247\pi$
$602$	−1.36511	−0.0556379
$603$	0	0
$604$	−5.07181	−0.206369
$605$	41.0402	1.66852
$606$	0	0
$607$	30.6766	1.24513	0.622563	−	0.782570i	$-0.286091\pi$
$607$	30.6766	1.24513	0.622563	+	0.782570i	$0.286091\pi$
$608$	1.56919	0.0636390
$609$	0	0
$610$	−6.57745	−0.266313
$611$	−7.69716	−0.311394
$612$	0	0
$613$	25.4502	1.02792	0.513962	−	0.857813i	$-0.328177\pi$
$613$	25.4502	1.02792	0.513962	+	0.857813i	$0.328177\pi$
$614$	−34.3085	−1.38458
$615$	0	0
$616$	−0.0799987	−0.00322324
$617$	−14.7401	−0.593414	−0.296707	−	0.954969i	$-0.595889\pi$
$617$	−14.7401	−0.593414	−0.296707	+	0.954969i	$0.595889\pi$
$618$	0	0
$619$	−20.6717	−0.830865	−0.415433	−	0.909624i	$-0.636370\pi$
$619$	−20.6717	−0.830865	−0.415433	+	0.909624i	$0.636370\pi$
$620$	−18.8421	−0.756716
$621$	0	0
$622$	47.3837	1.89991
$623$	−1.06546	−0.0426868
$624$	0	0
$625$	21.7515	0.870060
$626$	−52.3947	−2.09411
$627$	0	0
$628$	−24.1132	−0.962223
$629$	53.6013	2.13722
$630$	0	0
$631$	34.0627	1.35602	0.678008	−	0.735055i	$-0.262844\pi$
$631$	34.0627	1.35602	0.678008	+	0.735055i	$0.262844\pi$
$632$	−18.5630	−0.738396
$633$	0	0
$634$	13.1677	0.522957
$635$	49.2930	1.95613
$636$	0	0
$637$	−7.04502	−0.279134
$638$	−8.57860	−0.339630
$639$	0	0
$640$	45.2953	1.79046
$641$	43.4684	1.71690	0.858449	−	0.512899i	$-0.171429\pi$
$641$	43.4684	1.71690	0.858449	+	0.512899i	$0.171429\pi$
$642$	0	0
$643$	−39.8565	−1.57179	−0.785894	−	0.618361i	$-0.787797\pi$
$643$	−39.8565	−1.57179	−0.785894	+	0.618361i	$0.787797\pi$
$644$	−0.112391	−0.00442884
$645$	0	0
$646$	−3.30495	−0.130032
$647$	−12.5169	−0.492091	−0.246046	−	0.969258i	$-0.579131\pi$
$647$	−12.5169	−0.492091	−0.246046	+	0.969258i	$0.579131\pi$
$648$	0	0
$649$	0.329214	0.0129228
$650$	−17.2054	−0.674850
$651$	0	0
$652$	−10.5529	−0.413283
$653$	2.75517	0.107818	0.0539091	−	0.998546i	$-0.482832\pi$
$653$	2.75517	0.107818	0.0539091	+	0.998546i	$0.482832\pi$
$654$	0	0
$655$	−64.2862	−2.51187
$656$	−5.65752	−0.220889
$657$	0	0
$658$	1.12758	0.0439575
$659$	−22.6099	−0.880756	−0.440378	−	0.897813i	$-0.645156\pi$
$659$	−22.6099	−0.880756	−0.440378	+	0.897813i	$0.645156\pi$
$660$	0	0
$661$	38.0331	1.47932	0.739659	−	0.672982i	$-0.234987\pi$
$661$	38.0331	1.47932	0.739659	+	0.672982i	$0.234987\pi$
$662$	−25.1918	−0.979106
$663$	0	0
$664$	−7.09220	−0.275231
$665$	−0.0943199	−0.00365757
$666$	0	0
$667$	10.9234	0.422955
$668$	5.48605	0.212262
$669$	0	0
$670$	70.5128	2.72415
$671$	0.558532	0.0215619
$672$	0	0
$673$	8.68091	0.334624	0.167312	−	0.985904i	$-0.446491\pi$
$673$	8.68091	0.334624	0.167312	+	0.985904i	$0.446491\pi$
$674$	−15.5875	−0.600409
$675$	0	0
$676$	−12.5739	−0.483610
$677$	45.7684	1.75902	0.879511	−	0.475879i	$-0.157870\pi$
$677$	45.7684	1.75902	0.879511	+	0.475879i	$0.157870\pi$
$678$	0	0
$679$	0.0660595	0.00253513
$680$	−41.6216	−1.59612
$681$	0	0
$682$	4.65013	0.178063
$683$	15.7397	0.602264	0.301132	−	0.953582i	$-0.402636\pi$
$683$	15.7397	0.602264	0.301132	+	0.953582i	$0.402636\pi$
$684$	0	0
$685$	−14.5493	−0.555900
$686$	2.06514	0.0788475
$687$	0	0
$688$	−46.2259	−1.76235
$689$	5.55278	0.211544
$690$	0	0
$691$	−12.4440	−0.473391	−0.236696	−	0.971584i	$-0.576064\pi$
$691$	−12.4440	−0.473391	−0.236696	+	0.971584i	$0.576064\pi$
$692$	6.68676	0.254193
$693$	0	0
$694$	14.0413	0.532999
$695$	−15.3627	−0.582740
$696$	0	0
$697$	7.38135	0.279588
$698$	−0.912658	−0.0345446
$699$	0	0
$700$	0.867228	0.0327782
$701$	−28.9536	−1.09356	−0.546782	−	0.837275i	$-0.684147\pi$
$701$	−28.9536	−1.09356	−0.546782	+	0.837275i	$0.684147\pi$
$702$	0	0
$703$	2.38623	0.0899985
$704$	−0.316682	−0.0119354
$705$	0	0
$706$	−12.5001	−0.470447
$707$	−0.907761	−0.0341399
$708$	0	0
$709$	2.16925	0.0814680	0.0407340	−	0.999170i	$-0.487030\pi$
$709$	2.16925	0.0814680	0.0407340	+	0.999170i	$0.487030\pi$
$710$	−78.4462	−2.94404
$711$	0	0
$712$	−20.9310	−0.784424
$713$	−5.92115	−0.221749
$714$	0	0
$715$	2.20794	0.0825721
$716$	7.20895	0.269411
$717$	0	0
$718$	2.89136	0.107904
$719$	−30.5570	−1.13958	−0.569791	−	0.821790i	$-0.692976\pi$
$719$	−30.5570	−1.13958	−0.569791	+	0.821790i	$0.692976\pi$
$720$	0	0
$721$	−0.157964	−0.00588290
$722$	33.0302	1.22926
$723$	0	0
$724$	−17.1428	−0.637109
$725$	−84.2866	−3.13032
$726$	0	0
$727$	30.1081	1.11665	0.558323	−	0.829624i	$-0.311445\pi$
$727$	30.1081	1.11665	0.558323	+	0.829624i	$0.311445\pi$
$728$	0.141381	0.00523994
$729$	0	0
$730$	89.0432	3.29564
$731$	60.3108	2.23068
$732$	0	0
$733$	26.4010	0.975142	0.487571	−	0.873083i	$-0.337883\pi$
$733$	26.4010	0.975142	0.487571	+	0.873083i	$0.337883\pi$
$734$	−47.9475	−1.76978
$735$	0	0
$736$	−6.85196	−0.252567
$737$	−5.98767	−0.220559
$738$	0	0
$739$	−7.71903	−0.283949	−0.141974	−	0.989870i	$-0.545345\pi$
$739$	−7.71903	−0.283949	−0.141974	+	0.989870i	$0.545345\pi$
$740$	−33.1570	−1.21888
$741$	0	0
$742$	−0.813442	−0.0298624
$743$	21.2662	0.780183	0.390091	−	0.920776i	$-0.372443\pi$
$743$	21.2662	0.780183	0.390091	+	0.920776i	$0.372443\pi$
$744$	0	0
$745$	28.8119	1.05559
$746$	25.9095	0.948613
$747$	0	0
$748$	−3.89958	−0.142583
$749$	0.424413	0.0155077
$750$	0	0
$751$	22.1755	0.809195	0.404597	−	0.914495i	$-0.367412\pi$
$751$	22.1755	0.809195	0.404597	+	0.914495i	$0.367412\pi$
$752$	38.1824	1.39237
$753$	0	0
$754$	15.1609	0.552128
$755$	18.5855	0.676394
$756$	0	0
$757$	−28.0123	−1.01812	−0.509062	−	0.860730i	$-0.670008\pi$
$757$	−28.0123	−1.01812	−0.509062	+	0.860730i	$0.670008\pi$
$758$	20.7684	0.754341
$759$	0	0
$760$	−1.85292	−0.0672125
$761$	−25.1891	−0.913104	−0.456552	−	0.889697i	$-0.650916\pi$
$761$	−25.1891	−0.913104	−0.456552	+	0.889697i	$0.650916\pi$
$762$	0	0
$763$	0.472870	0.0171191
$764$	18.7989	0.680120
$765$	0	0
$766$	60.2100	2.17547
$767$	−0.581818	−0.0210082
$768$	0	0
$769$	−15.7842	−0.569192	−0.284596	−	0.958648i	$-0.591859\pi$
$769$	−15.7842	−0.569192	−0.284596	+	0.958648i	$0.591859\pi$
$770$	−0.323446	−0.0116562
$771$	0	0
$772$	2.22284	0.0800016
$773$	23.0433	0.828809	0.414404	−	0.910093i	$-0.363990\pi$
$773$	23.0433	0.828809	0.414404	+	0.910093i	$0.363990\pi$
$774$	0	0
$775$	45.6885	1.64118
$776$	1.29774	0.0465863
$777$	0	0
$778$	−24.5386	−0.879751
$779$	0.328604	0.0117735
$780$	0	0
$781$	6.66135	0.238362
$782$	14.4313	0.516062
$783$	0	0
$784$	34.9474	1.24812
$785$	88.3622	3.15378
$786$	0	0
$787$	39.8122	1.41915	0.709576	−	0.704629i	$-0.248887\pi$
$787$	39.8122	1.41915	0.709576	+	0.704629i	$0.248887\pi$
$788$	−5.12886	−0.182708
$789$	0	0
$790$	−75.0529	−2.67026
$791$	0.532757	0.0189427
$792$	0	0
$793$	−0.987091	−0.0350526
$794$	8.46058	0.300255
$795$	0	0
$796$	−11.4690	−0.406510
$797$	4.60478	0.163110	0.0815549	−	0.996669i	$-0.474011\pi$
$797$	4.60478	0.163110	0.0815549	+	0.996669i	$0.474011\pi$
$798$	0	0
$799$	−49.8164	−1.76238
$800$	52.8708	1.86927
$801$	0	0
$802$	−1.67499	−0.0591459
$803$	−7.56120	−0.266829
$804$	0	0
$805$	0.411854	0.0145159
$806$	−8.21816	−0.289472
$807$	0	0
$808$	−17.8330	−0.627364
$809$	−0.521603	−0.0183386	−0.00916928	−	0.999958i	$-0.502919\pi$
$809$	−0.521603	−0.0183386	−0.00916928	+	0.999958i	$0.502919\pi$
$810$	0	0
$811$	22.4149	0.787092	0.393546	−	0.919305i	$-0.371248\pi$
$811$	22.4149	0.787092	0.393546	+	0.919305i	$0.371248\pi$
$812$	−0.764179	−0.0268174
$813$	0	0
$814$	8.18298	0.286813
$815$	38.6707	1.35458
$816$	0	0
$817$	2.68493	0.0939338
$818$	−17.8848	−0.625327
$819$	0	0
$820$	−4.56600	−0.159452
$821$	44.7330	1.56119	0.780595	−	0.625037i	$-0.214916\pi$
$821$	44.7330	1.56119	0.780595	+	0.625037i	$0.214916\pi$
$822$	0	0
$823$	−39.9676	−1.39318	−0.696592	−	0.717467i	$-0.745301\pi$
$823$	−39.9676	−1.39318	−0.696592	+	0.717467i	$0.745301\pi$
$824$	−3.10322	−0.108106
$825$	0	0
$826$	0.0852320	0.00296560
$827$	−12.9305	−0.449637	−0.224819	−	0.974401i	$-0.572179\pi$
$827$	−12.9305	−0.449637	−0.224819	+	0.974401i	$0.572179\pi$
$828$	0	0
$829$	51.1141	1.77527	0.887633	−	0.460552i	$-0.152348\pi$
$829$	51.1141	1.77527	0.887633	+	0.460552i	$0.152348\pi$
$830$	−28.6748	−0.995317
$831$	0	0
$832$	0.559671	0.0194031
$833$	−45.5958	−1.57980
$834$	0	0
$835$	−20.1035	−0.695709
$836$	−0.173602	−0.00600416
$837$	0	0
$838$	44.5187	1.53787
$839$	19.8940	0.686817	0.343409	−	0.939186i	$-0.388418\pi$
$839$	19.8940	0.686817	0.343409	+	0.939186i	$0.388418\pi$
$840$	0	0
$841$	45.2712	1.56107
$842$	−53.7260	−1.85152
$843$	0	0
$844$	−1.09328	−0.0376322
$845$	46.0765	1.58508
$846$	0	0
$847$	−0.902246	−0.0310015
$848$	−27.5450	−0.945900
$849$	0	0
$850$	−111.354	−3.81941
$851$	−10.4196	−0.357181
$852$	0	0
$853$	−29.4124	−1.00706	−0.503530	−	0.863978i	$-0.667966\pi$
$853$	−29.4124	−1.00706	−0.503530	+	0.863978i	$0.667966\pi$
$854$	0.144602	0.00494816
$855$	0	0
$856$	8.33762	0.284974
$857$	−1.43432	−0.0489954	−0.0244977	−	0.999700i	$-0.507799\pi$
$857$	−1.43432	−0.0489954	−0.0244977	+	0.999700i	$0.507799\pi$
$858$	0	0
$859$	31.2524	1.06632	0.533160	−	0.846015i	$-0.321005\pi$
$859$	31.2524	1.06632	0.533160	+	0.846015i	$0.321005\pi$
$860$	−37.3074	−1.27217
$861$	0	0
$862$	59.1717	2.01540
$863$	41.7037	1.41961	0.709805	−	0.704398i	$-0.248783\pi$
$863$	41.7037	1.41961	0.709805	+	0.704398i	$0.248783\pi$
$864$	0	0
$865$	−24.5034	−0.833141
$866$	53.1838	1.80726
$867$	0	0
$868$	0.414232	0.0140600
$869$	6.37320	0.216196
$870$	0	0
$871$	10.5820	0.358557
$872$	9.28956	0.314584
$873$	0	0
$874$	0.642455	0.0217314
$875$	−1.55326	−0.0525097
$876$	0	0
$877$	−42.8736	−1.44774	−0.723869	−	0.689937i	$-0.757638\pi$
$877$	−42.8736	−1.44774	−0.723869	+	0.689937i	$0.757638\pi$
$878$	−56.9537	−1.92210
$879$	0	0
$880$	−10.9526	−0.369214
$881$	23.9917	0.808302	0.404151	−	0.914692i	$-0.367567\pi$
$881$	23.9917	0.808302	0.404151	+	0.914692i	$0.367567\pi$
$882$	0	0
$883$	43.8222	1.47474	0.737368	−	0.675492i	$-0.236069\pi$
$883$	43.8222	1.47474	0.737368	+	0.675492i	$0.236069\pi$
$884$	6.89171	0.231793
$885$	0	0
$886$	−20.6459	−0.693614
$887$	−46.7431	−1.56948	−0.784740	−	0.619826i	$-0.787203\pi$
$887$	−46.7431	−1.56948	−0.784740	+	0.619826i	$0.787203\pi$
$888$	0	0
$889$	−1.08368	−0.0363454
$890$	−84.6273	−2.83671
$891$	0	0
$892$	1.04913	0.0351275
$893$	−2.21774	−0.0742138
$894$	0	0
$895$	−26.4169	−0.883021
$896$	−0.995792	−0.0332671
$897$	0	0
$898$	−45.4934	−1.51813
$899$	−40.2595	−1.34273
$900$	0	0
$901$	35.9379	1.19727
$902$	1.12687	0.0375205
$903$	0	0
$904$	10.4660	0.348096
$905$	62.8194	2.08819
$906$	0	0
$907$	−19.5895	−0.650458	−0.325229	−	0.945635i	$-0.605441\pi$
$907$	−19.5895	−0.650458	−0.325229	+	0.945635i	$0.605441\pi$
$908$	−2.01794	−0.0669677
$909$	0	0
$910$	0.571626	0.0189492
$911$	−25.5793	−0.847480	−0.423740	−	0.905784i	$-0.639283\pi$
$911$	−25.5793	−0.847480	−0.423740	+	0.905784i	$0.639283\pi$
$912$	0	0
$913$	2.43495	0.0805852
$914$	29.8702	0.988018
$915$	0	0
$916$	4.97656	0.164430
$917$	1.41330	0.0466711
$918$	0	0
$919$	44.6296	1.47220	0.736098	−	0.676875i	$-0.236666\pi$
$919$	44.6296	1.47220	0.736098	+	0.676875i	$0.236666\pi$
$920$	8.09089	0.266749
$921$	0	0
$922$	26.9702	0.888215
$923$	−11.7726	−0.387499
$924$	0	0
$925$	80.3995	2.64352
$926$	−26.4521	−0.869271
$927$	0	0
$928$	−46.5884	−1.52934
$929$	−8.60536	−0.282333	−0.141166	−	0.989986i	$-0.545085\pi$
$929$	−8.60536	−0.282333	−0.141166	+	0.989986i	$0.545085\pi$
$930$	0	0
$931$	−2.02984	−0.0665254
$932$	−4.69780	−0.153882
$933$	0	0
$934$	−31.3350	−1.02531
$935$	14.2899	0.467329
$936$	0	0
$937$	−30.7280	−1.00384	−0.501920	−	0.864914i	$-0.667373\pi$
$937$	−30.7280	−1.00384	−0.501920	+	0.864914i	$0.667373\pi$
$938$	−1.55018	−0.0506153
$939$	0	0
$940$	30.8157	1.00510
$941$	20.1753	0.657696	0.328848	−	0.944383i	$-0.393340\pi$
$941$	20.1753	0.657696	0.328848	+	0.944383i	$0.393340\pi$
$942$	0	0
$943$	−1.43487	−0.0467259
$944$	2.88615	0.0939363
$945$	0	0
$946$	9.20729	0.299355
$947$	41.5654	1.35069	0.675347	−	0.737500i	$-0.263994\pi$
$947$	41.5654	1.35069	0.675347	+	0.737500i	$0.263994\pi$
$948$	0	0
$949$	13.3629	0.433777
$950$	−4.95729	−0.160836
$951$	0	0
$952$	0.915028	0.0296562
$953$	−23.0719	−0.747372	−0.373686	−	0.927555i	$-0.621906\pi$
$953$	−23.0719	−0.747372	−0.373686	+	0.927555i	$0.621906\pi$
$954$	0	0
$955$	−68.8879	−2.22916
$956$	16.3354	0.528326
$957$	0	0
$958$	20.1802	0.651991
$959$	0.319858	0.0103288
$960$	0	0
$961$	−9.17684	−0.296027
$962$	−14.4618	−0.466266
$963$	0	0
$964$	−21.0611	−0.678334
$965$	−8.14551	−0.262213
$966$	0	0
$967$	4.42041	0.142151	0.0710754	−	0.997471i	$-0.477357\pi$
$967$	4.42041	0.142151	0.0710754	+	0.997471i	$0.477357\pi$
$968$	−17.7247	−0.569692
$969$	0	0
$970$	5.24697	0.168470
$971$	−20.5881	−0.660705	−0.330352	−	0.943858i	$-0.607168\pi$
$971$	−20.5881	−0.660705	−0.330352	+	0.943858i	$0.607168\pi$
$972$	0	0
$973$	0.337740	0.0108274
$974$	7.79755	0.249850
$975$	0	0
$976$	4.89654	0.156735
$977$	22.9847	0.735346	0.367673	−	0.929955i	$-0.380155\pi$
$977$	22.9847	0.735346	0.367673	+	0.929955i	$0.380155\pi$
$978$	0	0
$979$	7.18622	0.229673
$980$	28.2049	0.900973
$981$	0	0
$982$	−17.3375	−0.553262
$983$	46.7872	1.49228	0.746140	−	0.665789i	$-0.231905\pi$
$983$	46.7872	1.49228	0.746140	+	0.665789i	$0.231905\pi$
$984$	0	0
$985$	18.7945	0.598843
$986$	98.1224	3.12485
$987$	0	0
$988$	0.306807	0.00976082
$989$	−11.7239	−0.372799
$990$	0	0
$991$	29.0669	0.923341	0.461671	−	0.887051i	$-0.347250\pi$
$991$	29.0669	0.923341	0.461671	+	0.887051i	$0.347250\pi$
$992$	25.2538	0.801808
$993$	0	0
$994$	1.72460	0.0547009
$995$	42.0279	1.33238
$996$	0	0
$997$	−3.12320	−0.0989128	−0.0494564	−	0.998776i	$-0.515749\pi$
$997$	−3.12320	−0.0989128	−0.0494564	+	0.998776i	$0.515749\pi$
$998$	40.3672	1.27780
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6021.2.a.t.1.9	✓	40
3.2	odd	2	inner	6021.2.a.t.1.32	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6021.2.a.t.1.9	✓	40	1.1	even	1	trivial
6021.2.a.t.1.32	yes	40	3.2	odd	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:	\( N \)	\(=\)	\( 6021 = 3^{3} \cdot 223 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6021.a (trivial)

\(f(q)\)	\(=\)	\(q-1.74618 q^{2} +1.04913 q^{4} -3.84450 q^{5} +0.0845192 q^{7} +1.66038 q^{8} +O(q^{10})\)	\(q-1.74618 q^{2} +1.04913 q^{4} -3.84450 q^{5} +0.0845192 q^{7} +1.66038 q^{8} +6.71318 q^{10} -0.570057 q^{11} +1.00746 q^{13} -0.147585 q^{14} -4.99759 q^{16} +6.52034 q^{17} +0.290274 q^{19} -4.03339 q^{20} +0.995420 q^{22} -1.26750 q^{23} +9.78021 q^{25} -1.75920 q^{26} +0.0886717 q^{28} -8.61807 q^{29} +4.67153 q^{31} +5.40590 q^{32} -11.3857 q^{34} -0.324935 q^{35} +8.22063 q^{37} -0.506869 q^{38} -6.38336 q^{40} +1.13205 q^{41} +9.24965 q^{43} -0.598065 q^{44} +2.21328 q^{46} -7.64016 q^{47} -6.99286 q^{49} -17.0780 q^{50} +1.05696 q^{52} +5.51167 q^{53} +2.19159 q^{55} +0.140334 q^{56} +15.0487 q^{58} -0.577510 q^{59} -0.979782 q^{61} -8.15731 q^{62} +0.555527 q^{64} -3.87318 q^{65} +10.5036 q^{67} +6.84069 q^{68} +0.567393 q^{70} -11.6854 q^{71} +13.2639 q^{73} -14.3547 q^{74} +0.304535 q^{76} -0.0481808 q^{77} -11.1799 q^{79} +19.2132 q^{80} -1.97676 q^{82} -4.27142 q^{83} -25.0675 q^{85} -16.1515 q^{86} -0.946514 q^{88} -12.6061 q^{89} +0.0851497 q^{91} -1.32977 q^{92} +13.3411 q^{94} -1.11596 q^{95} +0.781592 q^{97} +12.2108 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 46 q^{4} + 16 q^{7}+O(q^{10}) \) 40 * q + 46 * q^4 + 16 * q^7	\( 40 q + 46 q^{4} + 16 q^{7} + 22 q^{10} + 14 q^{13} + 50 q^{16} + 64 q^{19} + 12 q^{22} + 40 q^{25} + 48 q^{28} + 54 q^{31} + 32 q^{34} + 24 q^{37} + 40 q^{40} + 24 q^{43} + 52 q^{46} + 64 q^{49} + 18 q^{52} + 36 q^{55} + 8 q^{58} + 58 q^{61} + 120 q^{64} + 52 q^{67} - 30 q^{70} + 50 q^{73} + 112 q^{76} + 60 q^{79} + 50 q^{82} + 38 q^{85} + 16 q^{88} + 118 q^{91} + 44 q^{94} + 38 q^{97}+O(q^{100}) \) 40 * q + 46 * q^4 + 16 * q^7 + 22 * q^10 + 14 * q^13 + 50 * q^16 + 64 * q^19 + 12 * q^22 + 40 * q^25 + 48 * q^28 + 54 * q^31 + 32 * q^34 + 24 * q^37 + 40 * q^40 + 24 * q^43 + 52 * q^46 + 64 * q^49 + 18 * q^52 + 36 * q^55 + 8 * q^58 + 58 * q^61 + 120 * q^64 + 52 * q^67 - 30 * q^70 + 50 * q^73 + 112 * q^76 + 60 * q^79 + 50 * q^82 + 38 * q^85 + 16 * q^88 + 118 * q^91 + 44 * q^94 + 38 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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