LMFDB - Embedded newform 6021.2.a.r.1.15

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:	$35$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
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)

$f(q)$	$=$	$q-0.544885 q^{2} -1.70310 q^{4} -3.52229 q^{5} -1.92545 q^{7} +2.01776 q^{8} +O(q^{10})$	\(q-0.544885 q^{2} -1.70310 q^{4} -3.52229 q^{5} -1.92545 q^{7} +2.01776 q^{8} +1.91924 q^{10} +4.90691 q^{11} +4.97668 q^{13} +1.04915 q^{14} +2.30675 q^{16} +5.35154 q^{17} +2.61074 q^{19} +5.99881 q^{20} -2.67370 q^{22} +5.79004 q^{23} +7.40654 q^{25} -2.71172 q^{26} +3.27923 q^{28} -7.33849 q^{29} +0.867380 q^{31} -5.29244 q^{32} -2.91597 q^{34} +6.78198 q^{35} +0.758117 q^{37} -1.42255 q^{38} -7.10715 q^{40} -7.41653 q^{41} +0.212950 q^{43} -8.35695 q^{44} -3.15491 q^{46} +1.46021 q^{47} -3.29266 q^{49} -4.03571 q^{50} -8.47578 q^{52} +8.24239 q^{53} -17.2836 q^{55} -3.88510 q^{56} +3.99863 q^{58} +8.59529 q^{59} +6.52396 q^{61} -0.472622 q^{62} -1.72973 q^{64} -17.5293 q^{65} -12.0983 q^{67} -9.11420 q^{68} -3.69540 q^{70} +11.1107 q^{71} +1.81284 q^{73} -0.413087 q^{74} -4.44635 q^{76} -9.44799 q^{77} -4.17924 q^{79} -8.12505 q^{80} +4.04116 q^{82} -2.07210 q^{83} -18.8497 q^{85} -0.116033 q^{86} +9.90098 q^{88} +1.15979 q^{89} -9.58233 q^{91} -9.86102 q^{92} -0.795645 q^{94} -9.19577 q^{95} +7.00327 q^{97} +1.79412 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 35 q + 4 q^{2} + 36 q^{4} + 10 q^{5} - 2 q^{7} + 15 q^{8}+O(q^{10}) $ 35 * q + 4 * q^2 + 36 * q^4 + 10 * q^5 - 2 * q^7 + 15 * q^8	$ 35 q + 4 q^{2} + 36 q^{4} + 10 q^{5} - 2 q^{7} + 15 q^{8} - 7 q^{10} + 34 q^{11} + 2 q^{13} + 18 q^{14} + 42 q^{16} + 20 q^{17} - 14 q^{19} + 27 q^{20} + 8 q^{22} + 8 q^{23} + 27 q^{25} + 28 q^{26} - 4 q^{28} + 23 q^{29} + 12 q^{31} + 29 q^{32} - 21 q^{34} + 41 q^{35} - 8 q^{37} + 18 q^{38} + 16 q^{40} + 50 q^{41} + 2 q^{43} + 83 q^{44} - 5 q^{46} + 21 q^{47} + 43 q^{49} + 39 q^{50} + 6 q^{52} + 37 q^{53} + 20 q^{55} + 33 q^{56} - 32 q^{58} + 81 q^{59} - 6 q^{61} + 26 q^{62} - q^{64} + 29 q^{65} + 12 q^{67} + 55 q^{68} + 50 q^{70} + 43 q^{71} - 20 q^{73} + 48 q^{74} - 15 q^{76} + 29 q^{77} + 28 q^{79} + 88 q^{80} - 6 q^{82} + 64 q^{83} - 67 q^{85} + 41 q^{86} + 10 q^{88} + 50 q^{89} + 2 q^{91} + 32 q^{92} + 15 q^{94} + 25 q^{95} + 9 q^{97} + 74 q^{98}+O(q^{100}) $ 35 * q + 4 * q^2 + 36 * q^4 + 10 * q^5 - 2 * q^7 + 15 * q^8 - 7 * q^10 + 34 * q^11 + 2 * q^13 + 18 * q^14 + 42 * q^16 + 20 * q^17 - 14 * q^19 + 27 * q^20 + 8 * q^22 + 8 * q^23 + 27 * q^25 + 28 * q^26 - 4 * q^28 + 23 * q^29 + 12 * q^31 + 29 * q^32 - 21 * q^34 + 41 * q^35 - 8 * q^37 + 18 * q^38 + 16 * q^40 + 50 * q^41 + 2 * q^43 + 83 * q^44 - 5 * q^46 + 21 * q^47 + 43 * q^49 + 39 * q^50 + 6 * q^52 + 37 * q^53 + 20 * q^55 + 33 * q^56 - 32 * q^58 + 81 * q^59 - 6 * q^61 + 26 * q^62 - q^64 + 29 * q^65 + 12 * q^67 + 55 * q^68 + 50 * q^70 + 43 * q^71 - 20 * q^73 + 48 * q^74 - 15 * q^76 + 29 * q^77 + 28 * q^79 + 88 * q^80 - 6 * q^82 + 64 * q^83 - 67 * q^85 + 41 * q^86 + 10 * q^88 + 50 * q^89 + 2 * q^91 + 32 * q^92 + 15 * q^94 + 25 * q^95 + 9 * q^97 + 74 * q^98

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 $

$2$	−0.544885	−0.385292	−0.192646	−	0.981268i	$-0.561707\pi$
$2$	−0.544885	−0.385292	−0.192646	+	0.981268i	$0.561707\pi$
$3$	0	0
$3$	0	0
$4$	−1.70310	−0.851550
$5$	−3.52229	−1.57522	−0.787608	−	0.616176i	$-0.788681\pi$
$5$	−3.52229	−1.57522	−0.787608	+	0.616176i	$0.788681\pi$
$6$	0	0
$7$	−1.92545	−0.727750	−0.363875	−	0.931448i	$-0.618547\pi$
$7$	−1.92545	−0.727750	−0.363875	+	0.931448i	$0.618547\pi$
$8$	2.01776	0.713387
$9$	0	0
$10$	1.91924	0.606918
$11$	4.90691	1.47949	0.739744	−	0.672888i	$-0.234947\pi$
$11$	4.90691	1.47949	0.739744	+	0.672888i	$0.234947\pi$
$12$	0	0
$13$	4.97668	1.38028	0.690141	−	0.723675i	$-0.257548\pi$
$13$	4.97668	1.38028	0.690141	+	0.723675i	$0.257548\pi$
$14$	1.04915	0.280396
$15$	0	0
$16$	2.30675	0.576688
$17$	5.35154	1.29794	0.648969	−	0.760815i	$-0.275200\pi$
$17$	5.35154	1.29794	0.648969	+	0.760815i	$0.275200\pi$
$18$	0	0
$19$	2.61074	0.598944	0.299472	−	0.954105i	$-0.403189\pi$
$19$	2.61074	0.598944	0.299472	+	0.954105i	$0.403189\pi$
$20$	5.99881	1.34138
$21$	0	0
$22$	−2.67370	−0.570035
$23$	5.79004	1.20731	0.603654	−	0.797247i	$-0.293711\pi$
$23$	5.79004	1.20731	0.603654	+	0.797247i	$0.293711\pi$
$24$	0	0
$25$	7.40654	1.48131
$26$	−2.71172	−0.531812
$27$	0	0
$28$	3.27923	0.619716
$29$	−7.33849	−1.36272	−0.681361	−	0.731947i	$-0.738612\pi$
$29$	−7.33849	−1.36272	−0.681361	+	0.731947i	$0.738612\pi$
$30$	0	0
$31$	0.867380	0.155786	0.0778930	−	0.996962i	$-0.475181\pi$
$31$	0.867380	0.155786	0.0778930	+	0.996962i	$0.475181\pi$
$32$	−5.29244	−0.935581
$33$	0	0
$34$	−2.91597	−0.500085
$35$	6.78198	1.14636
$36$	0	0
$37$	0.758117	0.124634	0.0623168	−	0.998056i	$-0.480151\pi$
$37$	0.758117	0.124634	0.0623168	+	0.998056i	$0.480151\pi$
$38$	−1.42255	−0.230768
$39$	0	0
$40$	−7.10715	−1.12374
$41$	−7.41653	−1.15827	−0.579133	−	0.815233i	$-0.696609\pi$
$41$	−7.41653	−1.15827	−0.579133	+	0.815233i	$0.696609\pi$
$42$	0	0
$43$	0.212950	0.0324745	0.0162373	−	0.999868i	$-0.494831\pi$
$43$	0.212950	0.0324745	0.0162373	+	0.999868i	$0.494831\pi$
$44$	−8.35695	−1.25986
$45$	0	0
$46$	−3.15491	−0.465166
$47$	1.46021	0.212993	0.106497	−	0.994313i	$-0.466037\pi$
$47$	1.46021	0.212993	0.106497	+	0.994313i	$0.466037\pi$
$48$	0	0
$49$	−3.29266	−0.470379
$50$	−4.03571	−0.570736
$51$	0	0
$52$	−8.47578	−1.17538
$53$	8.24239	1.13218	0.566090	−	0.824344i	$-0.308456\pi$
$53$	8.24239	1.13218	0.566090	+	0.824344i	$0.308456\pi$
$54$	0	0
$55$	−17.2836	−2.33051
$56$	−3.88510	−0.519168
$57$	0	0
$58$	3.99863	0.525046
$59$	8.59529	1.11901	0.559505	−	0.828827i	$-0.310991\pi$
$59$	8.59529	1.11901	0.559505	+	0.828827i	$0.310991\pi$
$60$	0	0
$61$	6.52396	0.835308	0.417654	−	0.908606i	$-0.362852\pi$
$61$	6.52396	0.835308	0.417654	+	0.908606i	$0.362852\pi$
$62$	−0.472622	−0.0600231
$63$	0	0
$64$	−1.72973	−0.216216
$65$	−17.5293	−2.17424
$66$	0	0
$67$	−12.0983	−1.47804	−0.739020	−	0.673683i	$-0.764711\pi$
$67$	−12.0983	−1.47804	−0.739020	+	0.673683i	$0.764711\pi$
$68$	−9.11420	−1.10526
$69$	0	0
$70$	−3.69540	−0.441685
$71$	11.1107	1.31859	0.659296	−	0.751883i	$-0.270854\pi$
$71$	11.1107	1.31859	0.659296	+	0.751883i	$0.270854\pi$
$72$	0	0
$73$	1.81284	0.212177	0.106089	−	0.994357i	$-0.466167\pi$
$73$	1.81284	0.212177	0.106089	+	0.994357i	$0.466167\pi$
$74$	−0.413087	−0.0480203
$75$	0	0
$76$	−4.44635	−0.510031
$77$	−9.44799	−1.07670
$78$	0	0
$79$	−4.17924	−0.470202	−0.235101	−	0.971971i	$-0.575542\pi$
$79$	−4.17924	−0.470202	−0.235101	+	0.971971i	$0.575542\pi$
$80$	−8.12505	−0.908408
$81$	0	0
$82$	4.04116	0.446271
$83$	−2.07210	−0.227442	−0.113721	−	0.993513i	$-0.536277\pi$
$83$	−2.07210	−0.227442	−0.113721	+	0.993513i	$0.536277\pi$
$84$	0	0
$85$	−18.8497	−2.04453
$86$	−0.116033	−0.0125122
$87$	0	0
$88$	9.90098	1.05545
$89$	1.15979	0.122937	0.0614686	−	0.998109i	$-0.480422\pi$
$89$	1.15979	0.122937	0.0614686	+	0.998109i	$0.480422\pi$
$90$	0	0
$91$	−9.58233	−1.00450
$92$	−9.86102	−1.02808
$93$	0	0
$94$	−0.795645	−0.0820645
$95$	−9.19577	−0.943466
$96$	0	0
$97$	7.00327	0.711074	0.355537	−	0.934662i	$-0.384298\pi$
$97$	7.00327	0.711074	0.355537	+	0.934662i	$0.384298\pi$
$98$	1.79412	0.181233
$99$	0	0
$100$	−12.6141	−1.26141
$101$	17.0983	1.70135	0.850673	−	0.525696i	$-0.176195\pi$
$101$	17.0983	1.70135	0.850673	+	0.525696i	$0.176195\pi$
$102$	0	0
$103$	−14.3823	−1.41713	−0.708565	−	0.705646i	$-0.750657\pi$
$103$	−14.3823	−1.41713	−0.708565	+	0.705646i	$0.750657\pi$
$104$	10.0418	0.984676
$105$	0	0
$106$	−4.49116	−0.436220
$107$	3.09972	0.299662	0.149831	−	0.988712i	$-0.452127\pi$
$107$	3.09972	0.299662	0.149831	+	0.988712i	$0.452127\pi$
$108$	0	0
$109$	−6.35621	−0.608815	−0.304407	−	0.952542i	$-0.598458\pi$
$109$	−6.35621	−0.608815	−0.304407	+	0.952542i	$0.598458\pi$
$110$	9.41755	0.897929
$111$	0	0
$112$	−4.44152	−0.419685
$113$	10.0384	0.944329	0.472165	−	0.881510i	$-0.343473\pi$
$113$	10.0384	0.944329	0.472165	+	0.881510i	$0.343473\pi$
$114$	0	0
$115$	−20.3942	−1.90177
$116$	12.4982	1.16043
$117$	0	0
$118$	−4.68344	−0.431146
$119$	−10.3041	−0.944575
$120$	0	0
$121$	13.0777	1.18889
$122$	−3.55481	−0.321837
$123$	0	0
$124$	−1.47723	−0.132660
$125$	−8.47652	−0.758163
$126$	0	0
$127$	−0.180841	−0.0160470	−0.00802350	−	0.999968i	$-0.502554\pi$
$127$	−0.180841	−0.0160470	−0.00802350	+	0.999968i	$0.502554\pi$
$128$	11.5274	1.01889
$129$	0	0
$130$	9.55146	0.837719
$131$	7.70412	0.673112	0.336556	−	0.941663i	$-0.390738\pi$
$131$	7.70412	0.673112	0.336556	+	0.941663i	$0.390738\pi$
$132$	0	0
$133$	−5.02683	−0.435882
$134$	6.59218	0.569477
$135$	0	0
$136$	10.7981	0.925933
$137$	11.3485	0.969565	0.484783	−	0.874635i	$-0.338899\pi$
$137$	11.3485	0.969565	0.484783	+	0.874635i	$0.338899\pi$
$138$	0	0
$139$	−6.58796	−0.558783	−0.279392	−	0.960177i	$-0.590133\pi$
$139$	−6.58796	−0.558783	−0.279392	+	0.960177i	$0.590133\pi$
$140$	−11.5504	−0.976187
$141$	0	0
$142$	−6.05403	−0.508043
$143$	24.4201	2.04211
$144$	0	0
$145$	25.8483	2.14658
$146$	−0.987791	−0.0817501
$147$	0	0
$148$	−1.29115	−0.106132
$149$	−13.9884	−1.14597	−0.572986	−	0.819565i	$-0.694215\pi$
$149$	−13.9884	−1.14597	−0.572986	+	0.819565i	$0.694215\pi$
$150$	0	0
$151$	9.84193	0.800925	0.400462	−	0.916313i	$-0.368850\pi$
$151$	9.84193	0.800925	0.400462	+	0.916313i	$0.368850\pi$
$152$	5.26785	0.427279
$153$	0	0
$154$	5.14807	0.414843
$155$	−3.05516	−0.245397
$156$	0	0
$157$	5.83842	0.465957	0.232979	−	0.972482i	$-0.425153\pi$
$157$	5.83842	0.465957	0.232979	+	0.972482i	$0.425153\pi$
$158$	2.27721	0.181165
$159$	0	0
$160$	18.6415	1.47374
$161$	−11.1484	−0.878618
$162$	0	0
$163$	5.32383	0.416994	0.208497	−	0.978023i	$-0.433143\pi$
$163$	5.32383	0.416994	0.208497	+	0.978023i	$0.433143\pi$
$164$	12.6311	0.986322
$165$	0	0
$166$	1.12905	0.0876316
$167$	−12.2569	−0.948469	−0.474234	−	0.880399i	$-0.657275\pi$
$167$	−12.2569	−0.948469	−0.474234	+	0.880399i	$0.657275\pi$
$168$	0	0
$169$	11.7673	0.905179
$170$	10.2709	0.787743
$171$	0	0
$172$	−0.362675	−0.0276537
$173$	−12.3207	−0.936723	−0.468361	−	0.883537i	$-0.655155\pi$
$173$	−12.3207	−0.936723	−0.468361	+	0.883537i	$0.655155\pi$
$174$	0	0
$175$	−14.2609	−1.07802
$176$	11.3190	0.853202
$177$	0	0
$178$	−0.631951	−0.0473667
$179$	10.1420	0.758052	0.379026	−	0.925386i	$-0.376259\pi$
$179$	10.1420	0.758052	0.379026	+	0.925386i	$0.376259\pi$
$180$	0	0
$181$	−21.5002	−1.59810	−0.799050	−	0.601265i	$-0.794664\pi$
$181$	−21.5002	−1.59810	−0.799050	+	0.601265i	$0.794664\pi$
$182$	5.22127	0.387026
$183$	0	0
$184$	11.6829	0.861278
$185$	−2.67031	−0.196325
$186$	0	0
$187$	26.2595	1.92028
$188$	−2.48688	−0.181374
$189$	0	0
$190$	5.01064	0.363510
$191$	10.8711	0.786605	0.393303	−	0.919409i	$-0.371332\pi$
$191$	10.8711	0.786605	0.393303	+	0.919409i	$0.371332\pi$
$192$	0	0
$193$	1.43694	0.103433	0.0517165	−	0.998662i	$-0.483531\pi$
$193$	1.43694	0.103433	0.0517165	+	0.998662i	$0.483531\pi$
$194$	−3.81598	−0.273971
$195$	0	0
$196$	5.60772	0.400552
$197$	−6.34500	−0.452063	−0.226031	−	0.974120i	$-0.572575\pi$
$197$	−6.34500	−0.452063	−0.226031	+	0.974120i	$0.572575\pi$
$198$	0	0
$199$	−11.6315	−0.824535	−0.412267	−	0.911063i	$-0.635263\pi$
$199$	−11.6315	−0.824535	−0.412267	+	0.911063i	$0.635263\pi$
$200$	14.9446	1.05675
$201$	0	0
$202$	−9.31662	−0.655515
$203$	14.1299	0.991722
$204$	0	0
$205$	26.1232	1.82452
$206$	7.83670	0.546009
$207$	0	0
$208$	11.4800	0.795992
$209$	12.8106	0.886131
$210$	0	0
$211$	−20.5550	−1.41506	−0.707531	−	0.706682i	$-0.750191\pi$
$211$	−20.5550	−1.41506	−0.707531	+	0.706682i	$0.750191\pi$
$212$	−14.0376	−0.964108
$213$	0	0
$214$	−1.68899	−0.115457
$215$	−0.750071	−0.0511544
$216$	0	0
$217$	−1.67009	−0.113373
$218$	3.46341	0.234571
$219$	0	0
$220$	29.4356	1.98455
$221$	26.6329	1.79152
$222$	0	0
$223$	1.00000	0.0669650
$223$	1.00000	0.0669650
$224$	10.1903	0.680869
$225$	0	0
$226$	−5.46975	−0.363843
$227$	12.9497	0.859504	0.429752	−	0.902947i	$-0.358601\pi$
$227$	12.9497	0.859504	0.429752	+	0.902947i	$0.358601\pi$
$228$	0	0
$229$	−10.9906	−0.726277	−0.363139	−	0.931735i	$-0.618295\pi$
$229$	−10.9906	−0.726277	−0.363139	+	0.931735i	$0.618295\pi$
$230$	11.1125	0.732737
$231$	0	0
$232$	−14.8073	−0.972149
$233$	29.8228	1.95376	0.976878	−	0.213799i	$-0.0685837\pi$
$233$	29.8228	1.95376	0.976878	+	0.213799i	$0.0685837\pi$
$234$	0	0
$235$	−5.14327	−0.335510
$236$	−14.6386	−0.952894
$237$	0	0
$238$	5.61455	0.363937
$239$	−29.3658	−1.89951	−0.949757	−	0.312987i	$-0.898670\pi$
$239$	−29.3658	−1.89951	−0.949757	+	0.312987i	$0.898670\pi$
$240$	0	0
$241$	−14.0603	−0.905703	−0.452852	−	0.891586i	$-0.649593\pi$
$241$	−14.0603	−0.905703	−0.452852	+	0.891586i	$0.649593\pi$
$242$	−7.12587	−0.458068
$243$	0	0
$244$	−11.1110	−0.711306
$245$	11.5977	0.740949
$246$	0	0
$247$	12.9928	0.826712
$248$	1.75017	0.111136
$249$	0	0
$250$	4.61873	0.292114
$251$	3.11928	0.196887	0.0984437	−	0.995143i	$-0.468614\pi$
$251$	3.11928	0.196887	0.0984437	+	0.995143i	$0.468614\pi$
$252$	0	0
$253$	28.4112	1.78620
$254$	0.0985374	0.00618278
$255$	0	0
$256$	−2.82165	−0.176353
$257$	11.3922	0.710625	0.355313	−	0.934748i	$-0.384374\pi$
$257$	11.3922	0.710625	0.355313	+	0.934748i	$0.384374\pi$
$258$	0	0
$259$	−1.45971	−0.0907022
$260$	29.8542	1.85148
$261$	0	0
$262$	−4.19786	−0.259345
$263$	−22.4755	−1.38590	−0.692948	−	0.720988i	$-0.743688\pi$
$263$	−22.4755	−1.38590	−0.692948	+	0.720988i	$0.743688\pi$
$264$	0	0
$265$	−29.0321	−1.78343
$266$	2.73905	0.167942
$267$	0	0
$268$	20.6046	1.25863
$269$	−26.0241	−1.58672	−0.793360	−	0.608753i	$-0.791670\pi$
$269$	−26.0241	−1.58672	−0.793360	+	0.608753i	$0.791670\pi$
$270$	0	0
$271$	16.1094	0.978579	0.489289	−	0.872122i	$-0.337256\pi$
$271$	16.1094	0.978579	0.489289	+	0.872122i	$0.337256\pi$
$272$	12.3447	0.748505
$273$	0	0
$274$	−6.18361	−0.373566
$275$	36.3432	2.19158
$276$	0	0
$277$	16.4605	0.989013	0.494506	−	0.869174i	$-0.335349\pi$
$277$	16.4605	0.989013	0.494506	+	0.869174i	$0.335349\pi$
$278$	3.58968	0.215295
$279$	0	0
$280$	13.6844	0.817802
$281$	15.2791	0.911476	0.455738	−	0.890114i	$-0.349375\pi$
$281$	15.2791	0.911476	0.455738	+	0.890114i	$0.349375\pi$
$282$	0	0
$283$	5.52998	0.328723	0.164361	−	0.986400i	$-0.447444\pi$
$283$	5.52998	0.328723	0.164361	+	0.986400i	$0.447444\pi$
$284$	−18.9226	−1.12285
$285$	0	0
$286$	−13.3062	−0.786809
$287$	14.2801	0.842929
$288$	0	0
$289$	11.6390	0.684644
$290$	−14.0844	−0.827062
$291$	0	0
$292$	−3.08745	−0.180679
$293$	33.3531	1.94851	0.974254	−	0.225454i	$-0.0723867\pi$
$293$	33.3531	1.94851	0.974254	+	0.225454i	$0.0723867\pi$
$294$	0	0
$295$	−30.2751	−1.76268
$296$	1.52970	0.0889121
$297$	0	0
$298$	7.62205	0.441534
$299$	28.8152	1.66642
$300$	0	0
$301$	−0.410023	−0.0236333
$302$	−5.36272	−0.308590
$303$	0	0
$304$	6.02232	0.345404
$305$	−22.9793	−1.31579
$306$	0	0
$307$	33.7603	1.92680	0.963401	−	0.268064i	$-0.0863839\pi$
$307$	33.7603	1.92680	0.963401	+	0.268064i	$0.0863839\pi$
$308$	16.0909	0.916862
$309$	0	0
$310$	1.66471	0.0945494
$311$	5.91329	0.335312	0.167656	−	0.985846i	$-0.446380\pi$
$311$	5.91329	0.335312	0.167656	+	0.985846i	$0.446380\pi$
$312$	0	0
$313$	−1.05584	−0.0596795	−0.0298398	−	0.999555i	$-0.509500\pi$
$313$	−1.05584	−0.0596795	−0.0298398	+	0.999555i	$0.509500\pi$
$314$	−3.18127	−0.179530
$315$	0	0
$316$	7.11767	0.400400
$317$	−6.94408	−0.390018	−0.195009	−	0.980801i	$-0.562474\pi$
$317$	−6.94408	−0.390018	−0.195009	+	0.980801i	$0.562474\pi$
$318$	0	0
$319$	−36.0093	−2.01613
$320$	6.09260	0.340587
$321$	0	0
$322$	6.07461	0.338525
$323$	13.9715	0.777392
$324$	0	0
$325$	36.8599	2.04462
$326$	−2.90088	−0.160665
$327$	0	0
$328$	−14.9648	−0.826293
$329$	−2.81155	−0.155006
$330$	0	0
$331$	−13.9349	−0.765932	−0.382966	−	0.923762i	$-0.625097\pi$
$331$	−13.9349	−0.765932	−0.382966	+	0.923762i	$0.625097\pi$
$332$	3.52899	0.193678
$333$	0	0
$334$	6.67861	0.365437
$335$	42.6137	2.32823
$336$	0	0
$337$	−6.88545	−0.375074	−0.187537	−	0.982258i	$-0.560050\pi$
$337$	−6.88545	−0.375074	−0.187537	+	0.982258i	$0.560050\pi$
$338$	−6.41184	−0.348758
$339$	0	0
$340$	32.1029	1.74102
$341$	4.25615	0.230484
$342$	0	0
$343$	19.8180	1.07007
$344$	0.429682	0.0231669
$345$	0	0
$346$	6.71335	0.360912
$347$	−4.39169	−0.235758	−0.117879	−	0.993028i	$-0.537610\pi$
$347$	−4.39169	−0.235758	−0.117879	+	0.993028i	$0.537610\pi$
$348$	0	0
$349$	−2.25332	−0.120617	−0.0603087	−	0.998180i	$-0.519209\pi$
$349$	−2.25332	−0.120617	−0.0603087	+	0.998180i	$0.519209\pi$
$350$	7.77055	0.415353
$351$	0	0
$352$	−25.9695	−1.38418
$353$	−5.65239	−0.300846	−0.150423	−	0.988622i	$-0.548064\pi$
$353$	−5.65239	−0.300846	−0.150423	+	0.988622i	$0.548064\pi$
$354$	0	0
$355$	−39.1350	−2.07707
$356$	−1.97523	−0.104687
$357$	0	0
$358$	−5.52625	−0.292071
$359$	−19.7561	−1.04269	−0.521343	−	0.853347i	$-0.674569\pi$
$359$	−19.7561	−1.04269	−0.521343	+	0.853347i	$0.674569\pi$
$360$	0	0
$361$	−12.1841	−0.641266
$362$	11.7152	0.615735
$363$	0	0
$364$	16.3197	0.855383
$365$	−6.38536	−0.334225
$366$	0	0
$367$	−10.4855	−0.547338	−0.273669	−	0.961824i	$-0.588237\pi$
$367$	−10.4855	−0.547338	−0.273669	+	0.961824i	$0.588237\pi$
$368$	13.3562	0.696239
$369$	0	0
$370$	1.45501	0.0756424
$371$	−15.8703	−0.823944
$372$	0	0
$373$	−12.3617	−0.640065	−0.320032	−	0.947407i	$-0.603694\pi$
$373$	−12.3617	−0.640065	−0.320032	+	0.947407i	$0.603694\pi$
$374$	−14.3084	−0.739870
$375$	0	0
$376$	2.94635	0.151947
$377$	−36.5213	−1.88094
$378$	0	0
$379$	20.3373	1.04466	0.522328	−	0.852745i	$-0.325064\pi$
$379$	20.3373	1.04466	0.522328	+	0.852745i	$0.325064\pi$
$380$	15.6613	0.803409
$381$	0	0
$382$	−5.92350	−0.303073
$383$	−17.5143	−0.894940	−0.447470	−	0.894299i	$-0.647675\pi$
$383$	−17.5143	−0.894940	−0.447470	+	0.894299i	$0.647675\pi$
$384$	0	0
$385$	33.2786	1.69603
$386$	−0.782966	−0.0398519
$387$	0	0
$388$	−11.9273	−0.605515
$389$	22.1559	1.12335	0.561673	−	0.827359i	$-0.310158\pi$
$389$	22.1559	1.12335	0.561673	+	0.827359i	$0.310158\pi$
$390$	0	0
$391$	30.9856	1.56701
$392$	−6.64380	−0.335563
$393$	0	0
$394$	3.45730	0.174176
$395$	14.7205	0.740669
$396$	0	0
$397$	0.692372	0.0347491	0.0173746	−	0.999849i	$-0.494469\pi$
$397$	0.692372	0.0347491	0.0173746	+	0.999849i	$0.494469\pi$
$398$	6.33783	0.317687
$399$	0	0
$400$	17.0850	0.854251
$401$	37.7823	1.88676	0.943380	−	0.331714i	$-0.107627\pi$
$401$	37.7823	1.88676	0.943380	+	0.331714i	$0.107627\pi$
$402$	0	0
$403$	4.31667	0.215029
$404$	−29.1201	−1.44878
$405$	0	0
$406$	−7.69916	−0.382103
$407$	3.72001	0.184394
$408$	0	0
$409$	−36.9457	−1.82685	−0.913423	−	0.407011i	$-0.866571\pi$
$409$	−36.9457	−1.82685	−0.913423	+	0.407011i	$0.866571\pi$
$410$	−14.2341	−0.702973
$411$	0	0
$412$	24.4945	1.20676
$413$	−16.5498	−0.814361
$414$	0	0
$415$	7.29853	0.358271
$416$	−26.3388	−1.29137
$417$	0	0
$418$	−6.98033	−0.341419
$419$	23.6740	1.15655	0.578275	−	0.815842i	$-0.303726\pi$
$419$	23.6740	1.15655	0.578275	+	0.815842i	$0.303726\pi$
$420$	0	0
$421$	−1.30547	−0.0636246	−0.0318123	−	0.999494i	$-0.510128\pi$
$421$	−1.30547	−0.0636246	−0.0318123	+	0.999494i	$0.510128\pi$
$422$	11.2001	0.545212
$423$	0	0
$424$	16.6312	0.807683
$425$	39.6364	1.92265
$426$	0	0
$427$	−12.5615	−0.607896
$428$	−5.27914	−0.255177
$429$	0	0
$430$	0.408703	0.0197094
$431$	16.1331	0.777106	0.388553	−	0.921426i	$-0.372975\pi$
$431$	16.1331	0.777106	0.388553	+	0.921426i	$0.372975\pi$
$432$	0	0
$433$	3.45048	0.165819	0.0829097	−	0.996557i	$-0.473579\pi$
$433$	3.45048	0.165819	0.0829097	+	0.996557i	$0.473579\pi$
$434$	0.910009	0.0436818
$435$	0	0
$436$	10.8253	0.518436
$437$	15.1163	0.723109
$438$	0	0
$439$	−18.7468	−0.894734	−0.447367	−	0.894350i	$-0.647638\pi$
$439$	−18.7468	−0.894734	−0.447367	+	0.894350i	$0.647638\pi$
$440$	−34.8741	−1.66256
$441$	0	0
$442$	−14.5119	−0.690259
$443$	−26.0021	−1.23540	−0.617699	−	0.786414i	$-0.711935\pi$
$443$	−26.0021	−1.23540	−0.617699	+	0.786414i	$0.711935\pi$
$444$	0	0
$445$	−4.08511	−0.193653
$446$	−0.544885	−0.0258011
$447$	0	0
$448$	3.33050	0.157351
$449$	−36.6882	−1.73142	−0.865712	−	0.500543i	$-0.833134\pi$
$449$	−36.6882	−1.73142	−0.865712	+	0.500543i	$0.833134\pi$
$450$	0	0
$451$	−36.3922	−1.71364
$452$	−17.0963	−0.804144
$453$	0	0
$454$	−7.05612	−0.331160
$455$	33.7518	1.58231
$456$	0	0
$457$	9.57404	0.447855	0.223927	−	0.974606i	$-0.428112\pi$
$457$	9.57404	0.447855	0.223927	+	0.974606i	$0.428112\pi$
$458$	5.98860	0.279829
$459$	0	0
$460$	34.7334	1.61945
$461$	−34.8481	−1.62304	−0.811519	−	0.584326i	$-0.801359\pi$
$461$	−34.8481	−1.62304	−0.811519	+	0.584326i	$0.801359\pi$
$462$	0	0
$463$	38.9400	1.80970	0.904848	−	0.425734i	$-0.139984\pi$
$463$	38.9400	1.80970	0.904848	+	0.425734i	$0.139984\pi$
$464$	−16.9281	−0.785865
$465$	0	0
$466$	−16.2500	−0.752766
$467$	−27.1146	−1.25472	−0.627358	−	0.778731i	$-0.715864\pi$
$467$	−27.1146	−1.25472	−0.627358	+	0.778731i	$0.715864\pi$
$468$	0	0
$469$	23.2946	1.07564
$470$	2.80249	0.129269
$471$	0	0
$472$	17.3433	0.798288
$473$	1.04492	0.0480457
$474$	0	0
$475$	19.3365	0.887220
$476$	17.5489	0.804353
$477$	0	0
$478$	16.0010	0.731868
$479$	32.0424	1.46406	0.732028	−	0.681275i	$-0.238574\pi$
$479$	32.0424	1.46406	0.732028	+	0.681275i	$0.238574\pi$
$480$	0	0
$481$	3.77290	0.172030
$482$	7.66125	0.348960
$483$	0	0
$484$	−22.2727	−1.01240
$485$	−24.6676	−1.12010
$486$	0	0
$487$	−6.75388	−0.306048	−0.153024	−	0.988222i	$-0.548901\pi$
$487$	−6.75388	−0.306048	−0.153024	+	0.988222i	$0.548901\pi$
$488$	13.1638	0.595898
$489$	0	0
$490$	−6.31941	−0.285482
$491$	2.70889	0.122250	0.0611252	−	0.998130i	$-0.480531\pi$
$491$	2.70889	0.122250	0.0611252	+	0.998130i	$0.480531\pi$
$492$	0	0
$493$	−39.2722	−1.76873
$494$	−7.07958	−0.318525
$495$	0	0
$496$	2.00083	0.0898399
$497$	−21.3930	−0.959606
$498$	0	0
$499$	35.0818	1.57048	0.785239	−	0.619193i	$-0.212540\pi$
$499$	35.0818	1.57048	0.785239	+	0.619193i	$0.212540\pi$
$500$	14.4364	0.645614
$501$	0	0
$502$	−1.69965	−0.0758592
$503$	29.5647	1.31822	0.659111	−	0.752045i	$-0.270933\pi$
$503$	29.5647	1.31822	0.659111	+	0.752045i	$0.270933\pi$
$504$	0	0
$505$	−60.2252	−2.67999
$506$	−15.4808	−0.688207
$507$	0	0
$508$	0.307990	0.0136648
$509$	40.8499	1.81064	0.905320	−	0.424731i	$-0.139631\pi$
$509$	40.8499	1.81064	0.905320	+	0.424731i	$0.139631\pi$
$510$	0	0
$511$	−3.49053	−0.154412
$512$	−21.5173	−0.950939
$513$	0	0
$514$	−6.20744	−0.273798
$515$	50.6586	2.23229
$516$	0	0
$517$	7.16510	0.315121
$518$	0.795376	0.0349468
$519$	0	0
$520$	−35.3700	−1.55108
$521$	−20.9938	−0.919754	−0.459877	−	0.887983i	$-0.652106\pi$
$521$	−20.9938	−0.919754	−0.459877	+	0.887983i	$0.652106\pi$
$522$	0	0
$523$	−10.8329	−0.473689	−0.236845	−	0.971548i	$-0.576113\pi$
$523$	−10.8329	−0.473689	−0.236845	+	0.971548i	$0.576113\pi$
$524$	−13.1209	−0.573189
$525$	0	0
$526$	12.2465	0.533975
$527$	4.64182	0.202201
$528$	0	0
$529$	10.5246	0.457590
$530$	15.8192	0.687141
$531$	0	0
$532$	8.56120	0.371175
$533$	−36.9097	−1.59874
$534$	0	0
$535$	−10.9181	−0.472032
$536$	−24.4115	−1.05442
$537$	0	0
$538$	14.1802	0.611351
$539$	−16.1568	−0.695921
$540$	0	0
$541$	35.6712	1.53362	0.766812	−	0.641872i	$-0.221842\pi$
$541$	35.6712	1.53362	0.766812	+	0.641872i	$0.221842\pi$
$542$	−8.77780	−0.377039
$543$	0	0
$544$	−28.3227	−1.21433
$545$	22.3884	0.959015
$546$	0	0
$547$	−20.3380	−0.869590	−0.434795	−	0.900530i	$-0.643179\pi$
$547$	−20.3380	−0.869590	−0.434795	+	0.900530i	$0.643179\pi$
$548$	−19.3276	−0.825633
$549$	0	0
$550$	−19.8029	−0.844397
$551$	−19.1589	−0.816195
$552$	0	0
$553$	8.04691	0.342189
$554$	−8.96906	−0.381059
$555$	0	0
$556$	11.2200	0.475832
$557$	12.7000	0.538116	0.269058	−	0.963124i	$-0.413288\pi$
$557$	12.7000	0.538116	0.269058	+	0.963124i	$0.413288\pi$
$558$	0	0
$559$	1.05978	0.0448240
$560$	15.6443	0.661094
$561$	0	0
$562$	−8.32537	−0.351184
$563$	−44.7684	−1.88676	−0.943381	−	0.331710i	$-0.892374\pi$
$563$	−44.7684	−1.88676	−0.943381	+	0.331710i	$0.892374\pi$
$564$	0	0
$565$	−35.3580	−1.48752
$566$	−3.01320	−0.126654
$567$	0	0
$568$	22.4187	0.940667
$569$	33.7867	1.41641	0.708207	−	0.706005i	$-0.249504\pi$
$569$	33.7867	1.41641	0.708207	+	0.706005i	$0.249504\pi$
$570$	0	0
$571$	10.4596	0.437723	0.218861	−	0.975756i	$-0.429766\pi$
$571$	10.4596	0.437723	0.218861	+	0.975756i	$0.429766\pi$
$572$	−41.5899	−1.73896
$573$	0	0
$574$	−7.78103	−0.324774
$575$	42.8842	1.78839
$576$	0	0
$577$	17.5236	0.729519	0.364759	−	0.931102i	$-0.381151\pi$
$577$	17.5236	0.729519	0.364759	+	0.931102i	$0.381151\pi$
$578$	−6.34189	−0.263788
$579$	0	0
$580$	−44.0222	−1.82792
$581$	3.98971	0.165521
$582$	0	0
$583$	40.4447	1.67505
$584$	3.65789	0.151364
$585$	0	0
$586$	−18.1736	−0.750744
$587$	−40.1825	−1.65851	−0.829255	−	0.558871i	$-0.811235\pi$
$587$	−40.1825	−1.65851	−0.829255	+	0.558871i	$0.811235\pi$
$588$	0	0
$589$	2.26450	0.0933071
$590$	16.4965	0.679148
$591$	0	0
$592$	1.74879	0.0718747
$593$	−4.57177	−0.187740	−0.0938700	−	0.995584i	$-0.529924\pi$
$593$	−4.57177	−0.187740	−0.0938700	+	0.995584i	$0.529924\pi$
$594$	0	0
$595$	36.2940	1.48791
$596$	23.8236	0.975852
$597$	0	0
$598$	−15.7010	−0.642060
$599$	26.8550	1.09726	0.548632	−	0.836064i	$-0.315149\pi$
$599$	26.8550	1.09726	0.548632	+	0.836064i	$0.315149\pi$
$600$	0	0
$601$	43.1286	1.75925	0.879627	−	0.475665i	$-0.157792\pi$
$601$	43.1286	1.75925	0.879627	+	0.475665i	$0.157792\pi$
$602$	0.223416	0.00910574
$603$	0	0
$604$	−16.7618	−0.682027
$605$	−46.0636	−1.87275
$606$	0	0
$607$	−23.1297	−0.938804	−0.469402	−	0.882985i	$-0.655530\pi$
$607$	−23.1297	−0.938804	−0.469402	+	0.882985i	$0.655530\pi$
$608$	−13.8172	−0.560360
$609$	0	0
$610$	12.5211	0.506964
$611$	7.26698	0.293990
$612$	0	0
$613$	−29.1544	−1.17754	−0.588768	−	0.808302i	$-0.700387\pi$
$613$	−29.1544	−1.17754	−0.588768	+	0.808302i	$0.700387\pi$
$614$	−18.3955	−0.742381
$615$	0	0
$616$	−19.0638	−0.768103
$617$	12.8386	0.516864	0.258432	−	0.966029i	$-0.416794\pi$
$617$	12.8386	0.516864	0.258432	+	0.966029i	$0.416794\pi$
$618$	0	0
$619$	0.201907	0.00811531	0.00405765	−	0.999992i	$-0.498708\pi$
$619$	0.201907	0.00811531	0.00405765	+	0.999992i	$0.498708\pi$
$620$	5.20325	0.208968
$621$	0	0
$622$	−3.22206	−0.129193
$623$	−2.23311	−0.0894676
$624$	0	0
$625$	−7.17591	−0.287036
$626$	0.575311	0.0229941
$627$	0	0
$628$	−9.94342	−0.396786
$629$	4.05709	0.161767
$630$	0	0
$631$	−45.7536	−1.82142	−0.910711	−	0.413044i	$-0.864466\pi$
$631$	−45.7536	−1.82142	−0.910711	+	0.413044i	$0.864466\pi$
$632$	−8.43273	−0.335436
$633$	0	0
$634$	3.78373	0.150271
$635$	0.636973	0.0252775
$636$	0	0
$637$	−16.3865	−0.649256
$638$	19.6209	0.776800
$639$	0	0
$640$	−40.6028	−1.60497
$641$	21.8315	0.862293	0.431146	−	0.902282i	$-0.358109\pi$
$641$	21.8315	0.862293	0.431146	+	0.902282i	$0.358109\pi$
$642$	0	0
$643$	11.0631	0.436285	0.218142	−	0.975917i	$-0.430000\pi$
$643$	11.0631	0.436285	0.218142	+	0.975917i	$0.430000\pi$
$644$	18.9869	0.748187
$645$	0	0
$646$	−7.61284	−0.299523
$647$	−14.4091	−0.566482	−0.283241	−	0.959049i	$-0.591410\pi$
$647$	−14.4091	−0.566482	−0.283241	+	0.959049i	$0.591410\pi$
$648$	0	0
$649$	42.1763	1.65556
$650$	−20.0844	−0.787777
$651$	0	0
$652$	−9.06701	−0.355092
$653$	17.5839	0.688110	0.344055	−	0.938950i	$-0.388199\pi$
$653$	17.5839	0.688110	0.344055	+	0.938950i	$0.388199\pi$
$654$	0	0
$655$	−27.1362	−1.06030
$656$	−17.1081	−0.667958
$657$	0	0
$658$	1.53197	0.0597225
$659$	−18.8503	−0.734305	−0.367153	−	0.930161i	$-0.619667\pi$
$659$	−18.8503	−0.734305	−0.367153	+	0.930161i	$0.619667\pi$
$660$	0	0
$661$	−42.2750	−1.64431	−0.822154	−	0.569266i	$-0.807228\pi$
$661$	−42.2750	−1.64431	−0.822154	+	0.569266i	$0.807228\pi$
$662$	7.59293	0.295108
$663$	0	0
$664$	−4.18100	−0.162254
$665$	17.7060	0.686608
$666$	0	0
$667$	−42.4901	−1.64523
$668$	20.8748	0.807669
$669$	0	0
$670$	−23.2196	−0.897050
$671$	32.0125	1.23583
$672$	0	0
$673$	−6.75497	−0.260385	−0.130193	−	0.991489i	$-0.541560\pi$
$673$	−6.75497	−0.260385	−0.130193	+	0.991489i	$0.541560\pi$
$674$	3.75178	0.144513
$675$	0	0
$676$	−20.0409	−0.770805
$677$	32.8325	1.26185	0.630927	−	0.775842i	$-0.282675\pi$
$677$	32.8325	1.26185	0.630927	+	0.775842i	$0.282675\pi$
$678$	0	0
$679$	−13.4844	−0.517485
$680$	−38.0342	−1.45855
$681$	0	0
$682$	−2.31911	−0.0888035
$683$	28.9759	1.10873	0.554366	−	0.832273i	$-0.312961\pi$
$683$	28.9759	1.10873	0.554366	+	0.832273i	$0.312961\pi$
$684$	0	0
$685$	−39.9726	−1.52728
$686$	−10.7985	−0.412289
$687$	0	0
$688$	0.491222	0.0187277
$689$	41.0197	1.56273
$690$	0	0
$691$	29.8526	1.13565	0.567823	−	0.823151i	$-0.307786\pi$
$691$	29.8526	1.13565	0.567823	+	0.823151i	$0.307786\pi$
$692$	20.9833	0.797666
$693$	0	0
$694$	2.39297	0.0908358
$695$	23.2047	0.880205
$696$	0	0
$697$	−39.6898	−1.50336
$698$	1.22780	0.0464729
$699$	0	0
$700$	24.2877	0.917990
$701$	0.208112	0.00786029	0.00393014	−	0.999992i	$-0.498749\pi$
$701$	0.208112	0.00786029	0.00393014	+	0.999992i	$0.498749\pi$
$702$	0	0
$703$	1.97924	0.0746486
$704$	−8.48761	−0.319889
$705$	0	0
$706$	3.07990	0.115914
$707$	−32.9219	−1.23815
$708$	0	0
$709$	−42.6951	−1.60345	−0.801724	−	0.597694i	$-0.796084\pi$
$709$	−42.6951	−1.60345	−0.801724	+	0.597694i	$0.796084\pi$
$710$	21.3241	0.800278
$711$	0	0
$712$	2.34018	0.0877019
$713$	5.02217	0.188082
$714$	0	0
$715$	−86.0147	−3.21677
$716$	−17.2729	−0.645519
$717$	0	0
$718$	10.7648	0.401738
$719$	33.8284	1.26159	0.630794	−	0.775950i	$-0.282729\pi$
$719$	33.8284	1.26159	0.630794	+	0.775950i	$0.282729\pi$
$720$	0	0
$721$	27.6923	1.03132
$722$	6.63891	0.247075
$723$	0	0
$724$	36.6170	1.36086
$725$	−54.3528	−2.01861
$726$	0	0
$727$	34.7217	1.28776	0.643878	−	0.765128i	$-0.277324\pi$
$727$	34.7217	1.28776	0.643878	+	0.765128i	$0.277324\pi$
$728$	−19.3349	−0.716598
$729$	0	0
$730$	3.47929	0.128774
$731$	1.13961	0.0421499
$732$	0	0
$733$	32.2249	1.19025	0.595127	−	0.803631i	$-0.297102\pi$
$733$	32.2249	1.19025	0.595127	+	0.803631i	$0.297102\pi$
$734$	5.71339	0.210885
$735$	0	0
$736$	−30.6435	−1.12953
$737$	−59.3652	−2.18674
$738$	0	0
$739$	9.80728	0.360767	0.180383	−	0.983596i	$-0.442266\pi$
$739$	9.80728	0.360767	0.180383	+	0.983596i	$0.442266\pi$
$740$	4.54780	0.167181
$741$	0	0
$742$	8.64749	0.317459
$743$	38.5113	1.41284	0.706421	−	0.707792i	$-0.250309\pi$
$743$	38.5113	1.41284	0.706421	+	0.707792i	$0.250309\pi$
$744$	0	0
$745$	49.2711	1.80515
$746$	6.73571	0.246612
$747$	0	0
$748$	−44.7226	−1.63522
$749$	−5.96835	−0.218079
$750$	0	0
$751$	13.1961	0.481532	0.240766	−	0.970583i	$-0.422601\pi$
$751$	13.1961	0.481532	0.240766	+	0.970583i	$0.422601\pi$
$752$	3.36833	0.122830
$753$	0	0
$754$	19.8999	0.724712
$755$	−34.6661	−1.26163
$756$	0	0
$757$	−31.1027	−1.13045	−0.565223	−	0.824938i	$-0.691210\pi$
$757$	−31.1027	−1.13045	−0.565223	+	0.824938i	$0.691210\pi$
$758$	−11.0815	−0.402498
$759$	0	0
$760$	−18.5549	−0.673057
$761$	18.7117	0.678298	0.339149	−	0.940733i	$-0.389861\pi$
$761$	18.7117	0.678298	0.339149	+	0.940733i	$0.389861\pi$
$762$	0	0
$763$	12.2385	0.443065
$764$	−18.5146	−0.669834
$765$	0	0
$766$	9.54330	0.344813
$767$	42.7760	1.54455
$768$	0	0
$769$	48.0627	1.73319	0.866593	−	0.499015i	$-0.166305\pi$
$769$	48.0627	1.73319	0.866593	+	0.499015i	$0.166305\pi$
$770$	−18.1330	−0.653468
$771$	0	0
$772$	−2.44725	−0.0880784
$773$	−4.75150	−0.170900	−0.0854498	−	0.996342i	$-0.527233\pi$
$773$	−4.75150	−0.170900	−0.0854498	+	0.996342i	$0.527233\pi$
$774$	0	0
$775$	6.42428	0.230767
$776$	14.1309	0.507271
$777$	0	0
$778$	−12.0724	−0.432816
$779$	−19.3626	−0.693737
$780$	0	0
$781$	54.5190	1.95084
$782$	−16.8836	−0.603757
$783$	0	0
$784$	−7.59533	−0.271262
$785$	−20.5646	−0.733983
$786$	0	0
$787$	49.1006	1.75025	0.875124	−	0.483900i	$-0.160780\pi$
$787$	49.1006	1.75025	0.875124	+	0.483900i	$0.160780\pi$
$788$	10.8062	0.384954
$789$	0	0
$790$	−8.02099	−0.285374
$791$	−19.3283	−0.687236
$792$	0	0
$793$	32.4677	1.15296
$794$	−0.377263	−0.0133886
$795$	0	0
$796$	19.8096	0.702133
$797$	7.49733	0.265569	0.132784	−	0.991145i	$-0.457608\pi$
$797$	7.49733	0.265569	0.132784	+	0.991145i	$0.457608\pi$
$798$	0	0
$799$	7.81435	0.276452
$800$	−39.1987	−1.38588
$801$	0	0
$802$	−20.5870	−0.726954
$803$	8.89545	0.313913
$804$	0	0
$805$	39.2680	1.38401
$806$	−2.35209	−0.0828488
$807$	0	0
$808$	34.5004	1.21372
$809$	4.36227	0.153369	0.0766847	−	0.997055i	$-0.475567\pi$
$809$	4.36227	0.153369	0.0766847	+	0.997055i	$0.475567\pi$
$810$	0	0
$811$	48.1691	1.69144	0.845722	−	0.533623i	$-0.179170\pi$
$811$	48.1691	1.69144	0.845722	+	0.533623i	$0.179170\pi$
$812$	−24.0646	−0.844501
$813$	0	0
$814$	−2.02698	−0.0710455
$815$	−18.7521	−0.656857
$816$	0	0
$817$	0.555955	0.0194504
$818$	20.1312	0.703869
$819$	0	0
$820$	−44.4904	−1.55367
$821$	−17.1215	−0.597545	−0.298773	−	0.954324i	$-0.596577\pi$
$821$	−17.1215	−0.597545	−0.298773	+	0.954324i	$0.596577\pi$
$822$	0	0
$823$	−13.1405	−0.458050	−0.229025	−	0.973421i	$-0.573554\pi$
$823$	−13.1405	−0.458050	−0.229025	+	0.973421i	$0.573554\pi$
$824$	−29.0201	−1.01096
$825$	0	0
$826$	9.01772	0.313767
$827$	30.8965	1.07438	0.537189	−	0.843462i	$-0.319486\pi$
$827$	30.8965	1.07438	0.537189	+	0.843462i	$0.319486\pi$
$828$	0	0
$829$	22.1990	0.771004	0.385502	−	0.922707i	$-0.374028\pi$
$829$	22.1990	0.771004	0.385502	+	0.922707i	$0.374028\pi$
$830$	−3.97686	−0.138039
$831$	0	0
$832$	−8.60829	−0.298439
$833$	−17.6208	−0.610523
$834$	0	0
$835$	43.1724	1.49404
$836$	−21.8178	−0.754584
$837$	0	0
$838$	−12.8996	−0.445609
$839$	7.02019	0.242364	0.121182	−	0.992630i	$-0.461332\pi$
$839$	7.02019	0.242364	0.121182	+	0.992630i	$0.461332\pi$
$840$	0	0
$841$	24.8534	0.857014
$842$	0.711330	0.0245140
$843$	0	0
$844$	35.0072	1.20500
$845$	−41.4480	−1.42585
$846$	0	0
$847$	−25.1805	−0.865212
$848$	19.0131	0.652914
$849$	0	0
$850$	−21.5973	−0.740780
$851$	4.38953	0.150471
$852$	0	0
$853$	46.5993	1.59553	0.797765	−	0.602969i	$-0.206016\pi$
$853$	46.5993	1.59553	0.797765	+	0.602969i	$0.206016\pi$
$854$	6.84460	0.234217
$855$	0	0
$856$	6.25451	0.213775
$857$	29.7179	1.01514	0.507572	−	0.861610i	$-0.330543\pi$
$857$	29.7179	1.01514	0.507572	+	0.861610i	$0.330543\pi$
$858$	0	0
$859$	7.80790	0.266402	0.133201	−	0.991089i	$-0.457474\pi$
$859$	7.80790	0.266402	0.133201	+	0.991089i	$0.457474\pi$
$860$	1.27745	0.0435605
$861$	0	0
$862$	−8.79071	−0.299413
$863$	19.5273	0.664718	0.332359	−	0.943153i	$-0.392155\pi$
$863$	19.5273	0.664718	0.332359	+	0.943153i	$0.392155\pi$
$864$	0	0
$865$	43.3970	1.47554
$866$	−1.88011	−0.0638889
$867$	0	0
$868$	2.84434	0.0965431
$869$	−20.5072	−0.695658
$870$	0	0
$871$	−60.2093	−2.04011
$872$	−12.8253	−0.434321
$873$	0	0
$874$	−8.23663	−0.278608
$875$	16.3211	0.551753
$876$	0	0
$877$	35.1264	1.18614	0.593068	−	0.805153i	$-0.297917\pi$
$877$	35.1264	1.18614	0.593068	+	0.805153i	$0.297917\pi$
$878$	10.2148	0.344734
$879$	0	0
$880$	−39.8688	−1.34398
$881$	−36.5729	−1.23217	−0.616086	−	0.787679i	$-0.711283\pi$
$881$	−36.5729	−1.23217	−0.616086	+	0.787679i	$0.711283\pi$
$882$	0	0
$883$	−17.1226	−0.576221	−0.288110	−	0.957597i	$-0.593027\pi$
$883$	−17.1226	−0.576221	−0.288110	+	0.957597i	$0.593027\pi$
$884$	−45.3585	−1.52557
$885$	0	0
$886$	14.1682	0.475989
$887$	−17.9234	−0.601809	−0.300904	−	0.953654i	$-0.597289\pi$
$887$	−17.9234	−0.601809	−0.300904	+	0.953654i	$0.597289\pi$
$888$	0	0
$889$	0.348199	0.0116782
$890$	2.22592	0.0746129
$891$	0	0
$892$	−1.70310	−0.0570240
$893$	3.81221	0.127571
$894$	0	0
$895$	−35.7232	−1.19410
$896$	−22.1954	−0.741495
$897$	0	0
$898$	19.9909	0.667104
$899$	−6.36526	−0.212293
$900$	0	0
$901$	44.1095	1.46950
$902$	19.8296	0.660253
$903$	0	0
$904$	20.2550	0.673673
$905$	75.7301	2.51735
$906$	0	0
$907$	20.7787	0.689947	0.344973	−	0.938612i	$-0.387888\pi$
$907$	20.7787	0.689947	0.344973	+	0.938612i	$0.387888\pi$
$908$	−22.0547	−0.731911
$909$	0	0
$910$	−18.3908	−0.609650
$911$	−26.1713	−0.867094	−0.433547	−	0.901131i	$-0.642738\pi$
$911$	−26.1713	−0.867094	−0.433547	+	0.901131i	$0.642738\pi$
$912$	0	0
$913$	−10.1676	−0.336498
$914$	−5.21675	−0.172555
$915$	0	0
$916$	18.7180	0.618462
$917$	−14.8339	−0.489858
$918$	0	0
$919$	−25.1458	−0.829484	−0.414742	−	0.909939i	$-0.636128\pi$
$919$	−25.1458	−0.829484	−0.414742	+	0.909939i	$0.636128\pi$
$920$	−41.1507	−1.35670
$921$	0	0
$922$	18.9882	0.625343
$923$	55.2942	1.82003
$924$	0	0
$925$	5.61502	0.184621
$926$	−21.2178	−0.697262
$927$	0	0
$928$	38.8385	1.27494
$929$	−3.99061	−0.130928	−0.0654638	−	0.997855i	$-0.520853\pi$
$929$	−3.99061	−0.130928	−0.0654638	+	0.997855i	$0.520853\pi$
$930$	0	0
$931$	−8.59625	−0.281731
$932$	−50.7912	−1.66372
$933$	0	0
$934$	14.7744	0.483432
$935$	−92.4936	−3.02486
$936$	0	0
$937$	30.9907	1.01242	0.506212	−	0.862409i	$-0.331045\pi$
$937$	30.9907	1.01242	0.506212	+	0.862409i	$0.331045\pi$
$938$	−12.6929	−0.414437
$939$	0	0
$940$	8.75951	0.285704
$941$	5.24153	0.170869	0.0854345	−	0.996344i	$-0.472772\pi$
$941$	5.24153	0.170869	0.0854345	+	0.996344i	$0.472772\pi$
$942$	0	0
$943$	−42.9420	−1.39838
$944$	19.8272	0.645320
$945$	0	0
$946$	−0.569364	−0.0185116
$947$	6.98091	0.226849	0.113425	−	0.993547i	$-0.463818\pi$
$947$	6.98091	0.226849	0.113425	+	0.993547i	$0.463818\pi$
$948$	0	0
$949$	9.02193	0.292864
$950$	−10.5362	−0.341839
$951$	0	0
$952$	−20.7912	−0.673848
$953$	45.1304	1.46192	0.730958	−	0.682423i	$-0.239074\pi$
$953$	45.1304	1.46192	0.730958	+	0.682423i	$0.239074\pi$
$954$	0	0
$955$	−38.2912	−1.23907
$956$	50.0129	1.61753
$957$	0	0
$958$	−17.4594	−0.564089
$959$	−21.8509	−0.705602
$960$	0	0
$961$	−30.2477	−0.975731
$962$	−2.05580	−0.0662816
$963$	0	0
$964$	23.9461	0.771252
$965$	−5.06131	−0.162929
$966$	0	0
$967$	−55.3089	−1.77861	−0.889306	−	0.457312i	$-0.848812\pi$
$967$	−55.3089	−1.77861	−0.889306	+	0.457312i	$0.848812\pi$
$968$	26.3878	0.848136
$969$	0	0
$970$	13.4410	0.431564
$971$	11.3627	0.364647	0.182323	−	0.983239i	$-0.441638\pi$
$971$	11.3627	0.364647	0.182323	+	0.983239i	$0.441638\pi$
$972$	0	0
$973$	12.6848	0.406655
$974$	3.68009	0.117918
$975$	0	0
$976$	15.0491	0.481712
$977$	−58.6995	−1.87796	−0.938982	−	0.343965i	$-0.888230\pi$
$977$	−58.6995	−1.87796	−0.938982	+	0.343965i	$0.888230\pi$
$978$	0	0
$979$	5.69097	0.181884
$980$	−19.7520	−0.630955
$981$	0	0
$982$	−1.47603	−0.0471021
$983$	−15.5307	−0.495353	−0.247676	−	0.968843i	$-0.579667\pi$
$983$	−15.5307	−0.495353	−0.247676	+	0.968843i	$0.579667\pi$
$984$	0	0
$985$	22.3489	0.712097
$986$	21.3988	0.681478
$987$	0	0
$988$	−22.1280	−0.703986
$989$	1.23299	0.0392067
$990$	0	0
$991$	2.73743	0.0869572	0.0434786	−	0.999054i	$-0.486156\pi$
$991$	2.73743	0.0869572	0.0434786	+	0.999054i	$0.486156\pi$
$992$	−4.59056	−0.145750
$993$	0	0
$994$	11.6567	0.369729
$995$	40.9695	1.29882
$996$	0	0
$997$	52.8817	1.67478	0.837389	−	0.546607i	$-0.184081\pi$
$997$	52.8817	1.67478	0.837389	+	0.546607i	$0.184081\pi$
$998$	−19.1156	−0.605093
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6021.2.a.r.1.15	yes	35
3.2	odd	2		6021.2.a.q.1.21	✓	35

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6021.2.a.q.1.21	✓	35	3.2	odd	2
6021.2.a.r.1.15	yes	35	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-0.544885 q^{2} -1.70310 q^{4} -3.52229 q^{5} -1.92545 q^{7} +2.01776 q^{8} +O(q^{10})\)	\(q-0.544885 q^{2} -1.70310 q^{4} -3.52229 q^{5} -1.92545 q^{7} +2.01776 q^{8} +1.91924 q^{10} +4.90691 q^{11} +4.97668 q^{13} +1.04915 q^{14} +2.30675 q^{16} +5.35154 q^{17} +2.61074 q^{19} +5.99881 q^{20} -2.67370 q^{22} +5.79004 q^{23} +7.40654 q^{25} -2.71172 q^{26} +3.27923 q^{28} -7.33849 q^{29} +0.867380 q^{31} -5.29244 q^{32} -2.91597 q^{34} +6.78198 q^{35} +0.758117 q^{37} -1.42255 q^{38} -7.10715 q^{40} -7.41653 q^{41} +0.212950 q^{43} -8.35695 q^{44} -3.15491 q^{46} +1.46021 q^{47} -3.29266 q^{49} -4.03571 q^{50} -8.47578 q^{52} +8.24239 q^{53} -17.2836 q^{55} -3.88510 q^{56} +3.99863 q^{58} +8.59529 q^{59} +6.52396 q^{61} -0.472622 q^{62} -1.72973 q^{64} -17.5293 q^{65} -12.0983 q^{67} -9.11420 q^{68} -3.69540 q^{70} +11.1107 q^{71} +1.81284 q^{73} -0.413087 q^{74} -4.44635 q^{76} -9.44799 q^{77} -4.17924 q^{79} -8.12505 q^{80} +4.04116 q^{82} -2.07210 q^{83} -18.8497 q^{85} -0.116033 q^{86} +9.90098 q^{88} +1.15979 q^{89} -9.58233 q^{91} -9.86102 q^{92} -0.795645 q^{94} -9.19577 q^{95} +7.00327 q^{97} +1.79412 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 35 q + 4 q^{2} + 36 q^{4} + 10 q^{5} - 2 q^{7} + 15 q^{8}+O(q^{10}) \) 35 * q + 4 * q^2 + 36 * q^4 + 10 * q^5 - 2 * q^7 + 15 * q^8	\( 35 q + 4 q^{2} + 36 q^{4} + 10 q^{5} - 2 q^{7} + 15 q^{8} - 7 q^{10} + 34 q^{11} + 2 q^{13} + 18 q^{14} + 42 q^{16} + 20 q^{17} - 14 q^{19} + 27 q^{20} + 8 q^{22} + 8 q^{23} + 27 q^{25} + 28 q^{26} - 4 q^{28} + 23 q^{29} + 12 q^{31} + 29 q^{32} - 21 q^{34} + 41 q^{35} - 8 q^{37} + 18 q^{38} + 16 q^{40} + 50 q^{41} + 2 q^{43} + 83 q^{44} - 5 q^{46} + 21 q^{47} + 43 q^{49} + 39 q^{50} + 6 q^{52} + 37 q^{53} + 20 q^{55} + 33 q^{56} - 32 q^{58} + 81 q^{59} - 6 q^{61} + 26 q^{62} - q^{64} + 29 q^{65} + 12 q^{67} + 55 q^{68} + 50 q^{70} + 43 q^{71} - 20 q^{73} + 48 q^{74} - 15 q^{76} + 29 q^{77} + 28 q^{79} + 88 q^{80} - 6 q^{82} + 64 q^{83} - 67 q^{85} + 41 q^{86} + 10 q^{88} + 50 q^{89} + 2 q^{91} + 32 q^{92} + 15 q^{94} + 25 q^{95} + 9 q^{97} + 74 q^{98}+O(q^{100}) \) 35 * q + 4 * q^2 + 36 * q^4 + 10 * q^5 - 2 * q^7 + 15 * q^8 - 7 * q^10 + 34 * q^11 + 2 * q^13 + 18 * q^14 + 42 * q^16 + 20 * q^17 - 14 * q^19 + 27 * q^20 + 8 * q^22 + 8 * q^23 + 27 * q^25 + 28 * q^26 - 4 * q^28 + 23 * q^29 + 12 * q^31 + 29 * q^32 - 21 * q^34 + 41 * q^35 - 8 * q^37 + 18 * q^38 + 16 * q^40 + 50 * q^41 + 2 * q^43 + 83 * q^44 - 5 * q^46 + 21 * q^47 + 43 * q^49 + 39 * q^50 + 6 * q^52 + 37 * q^53 + 20 * q^55 + 33 * q^56 - 32 * q^58 + 81 * q^59 - 6 * q^61 + 26 * q^62 - q^64 + 29 * q^65 + 12 * q^67 + 55 * q^68 + 50 * q^70 + 43 * q^71 - 20 * q^73 + 48 * q^74 - 15 * q^76 + 29 * q^77 + 28 * q^79 + 88 * q^80 - 6 * q^82 + 64 * q^83 - 67 * q^85 + 41 * q^86 + 10 * q^88 + 50 * q^89 + 2 * q^91 + 32 * q^92 + 15 * q^94 + 25 * q^95 + 9 * q^97 + 74 * q^98

Introduction

L-functions

Modular forms

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Database

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