LMFDB - Embedded newform 4023.2.a.e.1.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4023,2,Mod(1,4023)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4023, 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("4023.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4023 = 3^{3} \cdot 149 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4023.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:	$32.1238167332$
Analytic rank:	$1$
Dimension:	$25$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
Character	$\chi$	$=$	4023.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.30329 q^{2} -0.301424 q^{4} +3.35120 q^{5} -0.705123 q^{7} -2.99943 q^{8} +O(q^{10})$	\(q+1.30329 q^{2} -0.301424 q^{4} +3.35120 q^{5} -0.705123 q^{7} -2.99943 q^{8} +4.36761 q^{10} -0.652483 q^{11} -6.43179 q^{13} -0.918983 q^{14} -3.30630 q^{16} +2.51909 q^{17} -3.54973 q^{19} -1.01013 q^{20} -0.850378 q^{22} -0.127282 q^{23} +6.23057 q^{25} -8.38251 q^{26} +0.212541 q^{28} +3.29408 q^{29} +0.205572 q^{31} +1.68979 q^{32} +3.28312 q^{34} -2.36301 q^{35} -11.2955 q^{37} -4.62634 q^{38} -10.0517 q^{40} -0.626938 q^{41} -1.83475 q^{43} +0.196674 q^{44} -0.165886 q^{46} -2.95179 q^{47} -6.50280 q^{49} +8.12026 q^{50} +1.93869 q^{52} -6.27772 q^{53} -2.18660 q^{55} +2.11497 q^{56} +4.29316 q^{58} +5.90664 q^{59} -10.5574 q^{61} +0.267921 q^{62} +8.81488 q^{64} -21.5542 q^{65} -11.6990 q^{67} -0.759314 q^{68} -3.07970 q^{70} +6.73009 q^{71} -3.36112 q^{73} -14.7214 q^{74} +1.06997 q^{76} +0.460081 q^{77} +6.77049 q^{79} -11.0801 q^{80} -0.817085 q^{82} -13.5176 q^{83} +8.44199 q^{85} -2.39122 q^{86} +1.95708 q^{88} +13.7643 q^{89} +4.53520 q^{91} +0.0383658 q^{92} -3.84706 q^{94} -11.8959 q^{95} -12.9506 q^{97} -8.47506 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 25 q - 7 q^{2} + 27 q^{4} - 12 q^{5} - 2 q^{7} - 21 q^{8}+O(q^{10}) $ 25 * q - 7 * q^2 + 27 * q^4 - 12 * q^5 - 2 * q^7 - 21 * q^8	$ 25 q - 7 q^{2} + 27 q^{4} - 12 q^{5} - 2 q^{7} - 21 q^{8} - 4 q^{10} - 12 q^{11} - 10 q^{14} + 35 q^{16} - 26 q^{17} - 30 q^{20} + 8 q^{22} - 26 q^{23} + 27 q^{25} - 16 q^{26} + 4 q^{28} - 20 q^{29} - 6 q^{31} - 49 q^{32} - 14 q^{34} - 16 q^{35} - 2 q^{37} - 27 q^{38} + 2 q^{40} - 35 q^{41} + 4 q^{43} - 22 q^{44} + 6 q^{46} - 38 q^{47} + 19 q^{49} - 22 q^{50} + 4 q^{52} - 36 q^{53} + 10 q^{55} - 79 q^{56} - 22 q^{58} - 15 q^{59} + 10 q^{61} + 14 q^{62} + 41 q^{64} - 80 q^{65} - 6 q^{67} - 33 q^{68} + 8 q^{70} - 26 q^{71} - 6 q^{73} - 75 q^{74} - 10 q^{76} - 47 q^{77} - 6 q^{79} - 66 q^{80} + 12 q^{82} - 40 q^{83} - 12 q^{85} - 4 q^{86} + 12 q^{88} - 30 q^{89} - 158 q^{92} + 18 q^{94} - 22 q^{95} - 20 q^{97} - 35 q^{98}+O(q^{100}) $ 25 * q - 7 * q^2 + 27 * q^4 - 12 * q^5 - 2 * q^7 - 21 * q^8 - 4 * q^10 - 12 * q^11 - 10 * q^14 + 35 * q^16 - 26 * q^17 - 30 * q^20 + 8 * q^22 - 26 * q^23 + 27 * q^25 - 16 * q^26 + 4 * q^28 - 20 * q^29 - 6 * q^31 - 49 * q^32 - 14 * q^34 - 16 * q^35 - 2 * q^37 - 27 * q^38 + 2 * q^40 - 35 * q^41 + 4 * q^43 - 22 * q^44 + 6 * q^46 - 38 * q^47 + 19 * q^49 - 22 * q^50 + 4 * q^52 - 36 * q^53 + 10 * q^55 - 79 * q^56 - 22 * q^58 - 15 * q^59 + 10 * q^61 + 14 * q^62 + 41 * q^64 - 80 * q^65 - 6 * q^67 - 33 * q^68 + 8 * q^70 - 26 * q^71 - 6 * q^73 - 75 * q^74 - 10 * q^76 - 47 * q^77 - 6 * q^79 - 66 * q^80 + 12 * q^82 - 40 * q^83 - 12 * q^85 - 4 * q^86 + 12 * q^88 - 30 * q^89 - 158 * q^92 + 18 * q^94 - 22 * q^95 - 20 * q^97 - 35 * 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 $

$2$	1.30329	0.921568	0.460784	−	0.887512i	$-0.347568\pi$
$2$	1.30329	0.921568	0.460784	+	0.887512i	$0.347568\pi$
$3$	0	0
$3$	0	0
$4$	−0.301424	−0.150712
$5$	3.35120	1.49870	0.749352	−	0.662172i	$-0.230365\pi$
$5$	3.35120	1.49870	0.749352	+	0.662172i	$0.230365\pi$
$6$	0	0
$7$	−0.705123	−0.266512	−0.133256	−	0.991082i	$-0.542543\pi$
$7$	−0.705123	−0.266512	−0.133256	+	0.991082i	$0.542543\pi$
$8$	−2.99943	−1.06046
$9$	0	0
$10$	4.36761	1.38116
$11$	−0.652483	−0.196731	−0.0983656	−	0.995150i	$-0.531361\pi$
$11$	−0.652483	−0.196731	−0.0983656	+	0.995150i	$0.531361\pi$
$12$	0	0
$13$	−6.43179	−1.78386	−0.891929	−	0.452176i	$-0.850648\pi$
$13$	−6.43179	−1.78386	−0.891929	+	0.452176i	$0.850648\pi$
$14$	−0.918983	−0.245609
$15$	0	0
$16$	−3.30630	−0.826574
$17$	2.51909	0.610970	0.305485	−	0.952197i	$-0.401181\pi$
$17$	2.51909	0.610970	0.305485	+	0.952197i	$0.401181\pi$
$18$	0	0
$19$	−3.54973	−0.814364	−0.407182	−	0.913347i	$-0.633489\pi$
$19$	−3.54973	−0.814364	−0.407182	+	0.913347i	$0.633489\pi$
$20$	−1.01013	−0.225872
$21$	0	0
$22$	−0.850378	−0.181301
$23$	−0.127282	−0.0265401	−0.0132701	−	0.999912i	$-0.504224\pi$
$23$	−0.127282	−0.0265401	−0.0132701	+	0.999912i	$0.504224\pi$
$24$	0	0
$25$	6.23057	1.24611
$26$	−8.38251	−1.64395
$27$	0	0
$28$	0.212541	0.0401664
$29$	3.29408	0.611695	0.305848	−	0.952080i	$-0.401060\pi$
$29$	3.29408	0.611695	0.305848	+	0.952080i	$0.401060\pi$
$30$	0	0
$31$	0.205572	0.0369219	0.0184609	−	0.999830i	$-0.494123\pi$
$31$	0.205572	0.0369219	0.0184609	+	0.999830i	$0.494123\pi$
$32$	1.68979	0.298715
$33$	0	0
$34$	3.28312	0.563050
$35$	−2.36301	−0.399422
$36$	0	0
$37$	−11.2955	−1.85698	−0.928488	−	0.371363i	$-0.878890\pi$
$37$	−11.2955	−1.85698	−0.928488	+	0.371363i	$0.878890\pi$
$38$	−4.62634	−0.750492
$39$	0	0
$40$	−10.0517	−1.58931
$41$	−0.626938	−0.0979113	−0.0489556	−	0.998801i	$-0.515589\pi$
$41$	−0.626938	−0.0979113	−0.0489556	+	0.998801i	$0.515589\pi$
$42$	0	0
$43$	−1.83475	−0.279797	−0.139898	−	0.990166i	$-0.544678\pi$
$43$	−1.83475	−0.279797	−0.139898	+	0.990166i	$0.544678\pi$
$44$	0.196674	0.0296497
$45$	0	0
$46$	−0.165886	−0.0244585
$47$	−2.95179	−0.430563	−0.215282	−	0.976552i	$-0.569067\pi$
$47$	−2.95179	−0.430563	−0.215282	+	0.976552i	$0.569067\pi$
$48$	0	0
$49$	−6.50280	−0.928972
$50$	8.12026	1.14838
$51$	0	0
$52$	1.93869	0.268848
$53$	−6.27772	−0.862311	−0.431155	−	0.902278i	$-0.641894\pi$
$53$	−6.27772	−0.862311	−0.431155	+	0.902278i	$0.641894\pi$
$54$	0	0
$55$	−2.18660	−0.294842
$56$	2.11497	0.282625
$57$	0	0
$58$	4.29316	0.563719
$59$	5.90664	0.768978	0.384489	−	0.923129i	$-0.374378\pi$
$59$	5.90664	0.768978	0.384489	+	0.923129i	$0.374378\pi$
$60$	0	0
$61$	−10.5574	−1.35174	−0.675868	−	0.737023i	$-0.736231\pi$
$61$	−10.5574	−1.35174	−0.675868	+	0.737023i	$0.736231\pi$
$62$	0.267921	0.0340260
$63$	0	0
$64$	8.81488	1.10186
$65$	−21.5542	−2.67347
$66$	0	0
$67$	−11.6990	−1.42927	−0.714633	−	0.699500i	$-0.753406\pi$
$67$	−11.6990	−1.42927	−0.714633	+	0.699500i	$0.753406\pi$
$68$	−0.759314	−0.0920803
$69$	0	0
$70$	−3.07970	−0.368095
$71$	6.73009	0.798714	0.399357	−	0.916795i	$-0.369233\pi$
$71$	6.73009	0.798714	0.399357	+	0.916795i	$0.369233\pi$
$72$	0	0
$73$	−3.36112	−0.393390	−0.196695	−	0.980465i	$-0.563021\pi$
$73$	−3.36112	−0.393390	−0.196695	+	0.980465i	$0.563021\pi$
$74$	−14.7214	−1.71133
$75$	0	0
$76$	1.06997	0.122734
$77$	0.460081	0.0524311
$78$	0	0
$79$	6.77049	0.761740	0.380870	−	0.924629i	$-0.375625\pi$
$79$	6.77049	0.761740	0.380870	+	0.924629i	$0.375625\pi$
$80$	−11.0801	−1.23879
$81$	0	0
$82$	−0.817085	−0.0902319
$83$	−13.5176	−1.48375	−0.741873	−	0.670540i	$-0.766062\pi$
$83$	−13.5176	−1.48375	−0.741873	+	0.670540i	$0.766062\pi$
$84$	0	0
$85$	8.44199	0.915663
$86$	−2.39122	−0.257852
$87$	0	0
$88$	1.95708	0.208625
$89$	13.7643	1.45902	0.729508	−	0.683972i	$-0.239749\pi$
$89$	13.7643	1.45902	0.729508	+	0.683972i	$0.239749\pi$
$90$	0	0
$91$	4.53520	0.475419
$92$	0.0383658	0.00399991
$93$	0	0
$94$	−3.84706	−0.396794
$95$	−11.8959	−1.22049
$96$	0	0
$97$	−12.9506	−1.31493	−0.657466	−	0.753484i	$-0.728372\pi$
$97$	−12.9506	−1.31493	−0.657466	+	0.753484i	$0.728372\pi$
$98$	−8.47506	−0.856111
$99$	0	0
$100$	−1.87804	−0.187804
$101$	11.8320	1.17733	0.588664	−	0.808378i	$-0.299654\pi$
$101$	11.8320	1.17733	0.588664	+	0.808378i	$0.299654\pi$
$102$	0	0
$103$	−12.7153	−1.25287	−0.626436	−	0.779473i	$-0.715487\pi$
$103$	−12.7153	−1.25287	−0.626436	+	0.779473i	$0.715487\pi$
$104$	19.2917	1.89171
$105$	0	0
$106$	−8.18171	−0.794678
$107$	−4.13934	−0.400165	−0.200083	−	0.979779i	$-0.564121\pi$
$107$	−4.13934	−0.400165	−0.200083	+	0.979779i	$0.564121\pi$
$108$	0	0
$109$	−0.484488	−0.0464055	−0.0232027	−	0.999731i	$-0.507386\pi$
$109$	−0.484488	−0.0464055	−0.0232027	+	0.999731i	$0.507386\pi$
$110$	−2.84979	−0.271717
$111$	0	0
$112$	2.33135	0.220292
$113$	10.9260	1.02783	0.513917	−	0.857840i	$-0.328194\pi$
$113$	10.9260	1.02783	0.513917	+	0.857840i	$0.328194\pi$
$114$	0	0
$115$	−0.426548	−0.0397758
$116$	−0.992913	−0.0921897
$117$	0	0
$118$	7.69809	0.708666
$119$	−1.77627	−0.162830
$120$	0	0
$121$	−10.5743	−0.961297
$122$	−13.7594	−1.24572
$123$	0	0
$124$	−0.0619643	−0.00556456
$125$	4.12388	0.368851
$126$	0	0
$127$	−0.398201	−0.0353346	−0.0176673	−	0.999844i	$-0.505624\pi$
$127$	−0.398201	−0.0353346	−0.0176673	+	0.999844i	$0.505624\pi$
$128$	8.10882	0.716725
$129$	0	0
$130$	−28.0915	−2.46379
$131$	14.9425	1.30554	0.652768	−	0.757558i	$-0.273608\pi$
$131$	14.9425	1.30554	0.652768	+	0.757558i	$0.273608\pi$
$132$	0	0
$133$	2.50300	0.217037
$134$	−15.2473	−1.31717
$135$	0	0
$136$	−7.55585	−0.647909
$137$	−10.2013	−0.871556	−0.435778	−	0.900054i	$-0.643527\pi$
$137$	−10.2013	−0.871556	−0.435778	+	0.900054i	$0.643527\pi$
$138$	0	0
$139$	17.1455	1.45426	0.727130	−	0.686499i	$-0.240854\pi$
$139$	17.1455	1.45426	0.727130	+	0.686499i	$0.240854\pi$
$140$	0.712268	0.0601976
$141$	0	0
$142$	8.77128	0.736070
$143$	4.19663	0.350940
$144$	0	0
$145$	11.0391	0.916750
$146$	−4.38053	−0.362535
$147$	0	0
$148$	3.40474	0.279868
$149$	−1.00000	−0.0819232
$149$	−1.00000	−0.0819232
$150$	0	0
$151$	10.4261	0.848466	0.424233	−	0.905553i	$-0.360544\pi$
$151$	10.4261	0.848466	0.424233	+	0.905553i	$0.360544\pi$
$152$	10.6472	0.863600
$153$	0	0
$154$	0.599621	0.0483189
$155$	0.688914	0.0553349
$156$	0	0
$157$	−1.86635	−0.148951	−0.0744754	−	0.997223i	$-0.523728\pi$
$157$	−1.86635	−0.148951	−0.0744754	+	0.997223i	$0.523728\pi$
$158$	8.82395	0.701996
$159$	0	0
$160$	5.66282	0.447685
$161$	0.0897495	0.00707325
$162$	0	0
$163$	11.5300	0.903101	0.451550	−	0.892246i	$-0.350871\pi$
$163$	11.5300	0.903101	0.451550	+	0.892246i	$0.350871\pi$
$164$	0.188974	0.0147564
$165$	0	0
$166$	−17.6174	−1.36737
$167$	−5.64730	−0.437001	−0.218501	−	0.975837i	$-0.570117\pi$
$167$	−5.64730	−0.437001	−0.218501	+	0.975837i	$0.570117\pi$
$168$	0	0
$169$	28.3679	2.18215
$170$	11.0024	0.843846
$171$	0	0
$172$	0.553037	0.0421687
$173$	3.33769	0.253760	0.126880	−	0.991918i	$-0.459504\pi$
$173$	3.33769	0.253760	0.126880	+	0.991918i	$0.459504\pi$
$174$	0	0
$175$	−4.39332	−0.332104
$176$	2.15730	0.162613
$177$	0	0
$178$	17.9390	1.34458
$179$	−5.46719	−0.408637	−0.204318	−	0.978904i	$-0.565498\pi$
$179$	−5.46719	−0.408637	−0.204318	+	0.978904i	$0.565498\pi$
$180$	0	0
$181$	−17.3752	−1.29149	−0.645745	−	0.763553i	$-0.723453\pi$
$181$	−17.3752	−1.29149	−0.645745	+	0.763553i	$0.723453\pi$
$182$	5.91071	0.438131
$183$	0	0
$184$	0.381774	0.0281447
$185$	−37.8537	−2.78306
$186$	0	0
$187$	−1.64367	−0.120197
$188$	0.889740	0.0648910
$189$	0	0
$190$	−15.5038	−1.12477
$191$	6.90474	0.499610	0.249805	−	0.968296i	$-0.419634\pi$
$191$	6.90474	0.499610	0.249805	+	0.968296i	$0.419634\pi$
$192$	0	0
$193$	−1.58177	−0.113858	−0.0569290	−	0.998378i	$-0.518131\pi$
$193$	−1.58177	−0.113858	−0.0569290	+	0.998378i	$0.518131\pi$
$194$	−16.8784	−1.21180
$195$	0	0
$196$	1.96010	0.140007
$197$	−16.5769	−1.18106	−0.590529	−	0.807016i	$-0.701081\pi$
$197$	−16.5769	−1.18106	−0.590529	+	0.807016i	$0.701081\pi$
$198$	0	0
$199$	−12.7548	−0.904163	−0.452081	−	0.891977i	$-0.649318\pi$
$199$	−12.7548	−0.904163	−0.452081	+	0.891977i	$0.649318\pi$
$200$	−18.6882	−1.32145
$201$	0	0
$202$	15.4206	1.08499
$203$	−2.32273	−0.163024
$204$	0	0
$205$	−2.10100	−0.146740
$206$	−16.5717	−1.15461
$207$	0	0
$208$	21.2654	1.47449
$209$	2.31614	0.160211
$210$	0	0
$211$	13.3152	0.916657	0.458329	−	0.888783i	$-0.348448\pi$
$211$	13.3152	0.916657	0.458329	+	0.888783i	$0.348448\pi$
$212$	1.89225	0.129960
$213$	0	0
$214$	−5.39478	−0.368780
$215$	−6.14862	−0.419333
$216$	0	0
$217$	−0.144954	−0.00984010
$218$	−0.631430	−0.0427658
$219$	0	0
$220$	0.659094	0.0444361
$221$	−16.2023	−1.08988
$222$	0	0
$223$	19.0431	1.27522	0.637611	−	0.770359i	$-0.279923\pi$
$223$	19.0431	1.27522	0.637611	+	0.770359i	$0.279923\pi$
$224$	−1.19151	−0.0796110
$225$	0	0
$226$	14.2398	0.947219
$227$	1.33186	0.0883987	0.0441994	−	0.999023i	$-0.485926\pi$
$227$	1.33186	0.0883987	0.0441994	+	0.999023i	$0.485926\pi$
$228$	0	0
$229$	19.5516	1.29201	0.646004	−	0.763334i	$-0.276439\pi$
$229$	19.5516	1.29201	0.646004	+	0.763334i	$0.276439\pi$
$230$	−0.555918	−0.0366561
$231$	0	0
$232$	−9.88037	−0.648678
$233$	−17.2060	−1.12720	−0.563600	−	0.826048i	$-0.690584\pi$
$233$	−17.2060	−1.12720	−0.563600	+	0.826048i	$0.690584\pi$
$234$	0	0
$235$	−9.89206	−0.645287
$236$	−1.78040	−0.115894
$237$	0	0
$238$	−2.31500	−0.150059
$239$	−18.2924	−1.18324	−0.591620	−	0.806217i	$-0.701511\pi$
$239$	−18.2924	−1.18324	−0.591620	+	0.806217i	$0.701511\pi$
$240$	0	0
$241$	−14.7763	−0.951826	−0.475913	−	0.879492i	$-0.657882\pi$
$241$	−14.7763	−0.951826	−0.475913	+	0.879492i	$0.657882\pi$
$242$	−13.7814	−0.885901
$243$	0	0
$244$	3.18225	0.203723
$245$	−21.7922	−1.39225
$246$	0	0
$247$	22.8311	1.45271
$248$	−0.616600	−0.0391541
$249$	0	0
$250$	5.37463	0.339922
$251$	−10.3208	−0.651441	−0.325721	−	0.945466i	$-0.605607\pi$
$251$	−10.3208	−0.651441	−0.325721	+	0.945466i	$0.605607\pi$
$252$	0	0
$253$	0.0830494	0.00522127
$254$	−0.518974	−0.0325633
$255$	0	0
$256$	−7.06159	−0.441350
$257$	−11.6385	−0.725988	−0.362994	−	0.931792i	$-0.618245\pi$
$257$	−11.6385	−0.725988	−0.362994	+	0.931792i	$0.618245\pi$
$258$	0	0
$259$	7.96475	0.494905
$260$	6.49695	0.402924
$261$	0	0
$262$	19.4745	1.20314
$263$	−23.4981	−1.44895	−0.724477	−	0.689299i	$-0.757919\pi$
$263$	−23.4981	−1.44895	−0.724477	+	0.689299i	$0.757919\pi$
$264$	0	0
$265$	−21.0379	−1.29235
$266$	3.26214	0.200015
$267$	0	0
$268$	3.52637	0.215407
$269$	18.3321	1.11773	0.558865	−	0.829258i	$-0.311237\pi$
$269$	18.3321	1.11773	0.558865	+	0.829258i	$0.311237\pi$
$270$	0	0
$271$	2.21863	0.134772	0.0673862	−	0.997727i	$-0.478534\pi$
$271$	2.21863	0.134772	0.0673862	+	0.997727i	$0.478534\pi$
$272$	−8.32887	−0.505012
$273$	0	0
$274$	−13.2953	−0.803198
$275$	−4.06534	−0.245149
$276$	0	0
$277$	3.84616	0.231093	0.115547	−	0.993302i	$-0.463138\pi$
$277$	3.84616	0.231093	0.115547	+	0.993302i	$0.463138\pi$
$278$	22.3456	1.34020
$279$	0	0
$280$	7.08770	0.423571
$281$	−4.16702	−0.248583	−0.124292	−	0.992246i	$-0.539666\pi$
$281$	−4.16702	−0.248583	−0.124292	+	0.992246i	$0.539666\pi$
$282$	0	0
$283$	27.4799	1.63351	0.816754	−	0.576986i	$-0.195771\pi$
$283$	27.4799	1.63351	0.816754	+	0.576986i	$0.195771\pi$
$284$	−2.02861	−0.120376
$285$	0	0
$286$	5.46945	0.323415
$287$	0.442069	0.0260945
$288$	0	0
$289$	−10.6542	−0.626716
$290$	14.3872	0.844848
$291$	0	0
$292$	1.01312	0.0592884
$293$	22.1245	1.29253	0.646264	−	0.763114i	$-0.276330\pi$
$293$	22.1245	1.29253	0.646264	+	0.763114i	$0.276330\pi$
$294$	0	0
$295$	19.7943	1.15247
$296$	33.8802	1.96925
$297$	0	0
$298$	−1.30329	−0.0754978
$299$	0.818651	0.0473438
$300$	0	0
$301$	1.29373	0.0745691
$302$	13.5883	0.781919
$303$	0	0
$304$	11.7365	0.673132
$305$	−35.3800	−2.02585
$306$	0	0
$307$	−4.85798	−0.277260	−0.138630	−	0.990344i	$-0.544270\pi$
$307$	−4.85798	−0.277260	−0.138630	+	0.990344i	$0.544270\pi$
$308$	−0.138679	−0.00790199
$309$	0	0
$310$	0.897858	0.0509949
$311$	−9.70430	−0.550280	−0.275140	−	0.961404i	$-0.588724\pi$
$311$	−9.70430	−0.550280	−0.275140	+	0.961404i	$0.588724\pi$
$312$	0	0
$313$	−9.12759	−0.515922	−0.257961	−	0.966155i	$-0.583051\pi$
$313$	−9.12759	−0.515922	−0.257961	+	0.966155i	$0.583051\pi$
$314$	−2.43240	−0.137268
$315$	0	0
$316$	−2.04079	−0.114803
$317$	25.8843	1.45381	0.726905	−	0.686738i	$-0.240958\pi$
$317$	25.8843	1.45381	0.726905	+	0.686738i	$0.240958\pi$
$318$	0	0
$319$	−2.14933	−0.120339
$320$	29.5405	1.65136
$321$	0	0
$322$	0.116970	0.00651849
$323$	−8.94210	−0.497552
$324$	0	0
$325$	−40.0737	−2.22289
$326$	15.0270	0.832269
$327$	0	0
$328$	1.88046	0.103831
$329$	2.08138	0.114750
$330$	0	0
$331$	−5.83824	−0.320899	−0.160449	−	0.987044i	$-0.551294\pi$
$331$	−5.83824	−0.320899	−0.160449	+	0.987044i	$0.551294\pi$
$332$	4.07451	0.223618
$333$	0	0
$334$	−7.36010	−0.402727
$335$	−39.2059	−2.14205
$336$	0	0
$337$	6.89032	0.375340	0.187670	−	0.982232i	$-0.439907\pi$
$337$	6.89032	0.375340	0.187670	+	0.982232i	$0.439907\pi$
$338$	36.9717	2.01100
$339$	0	0
$340$	−2.54461	−0.138001
$341$	−0.134132	−0.00726368
$342$	0	0
$343$	9.52114	0.514093
$344$	5.50321	0.296713
$345$	0	0
$346$	4.35000	0.233857
$347$	−10.2062	−0.547897	−0.273949	−	0.961744i	$-0.588330\pi$
$347$	−10.2062	−0.547897	−0.273949	+	0.961744i	$0.588330\pi$
$348$	0	0
$349$	−22.6599	−1.21296	−0.606478	−	0.795100i	$-0.707418\pi$
$349$	−22.6599	−1.21296	−0.606478	+	0.795100i	$0.707418\pi$
$350$	−5.72579	−0.306056
$351$	0	0
$352$	−1.10256	−0.0587665
$353$	−16.5745	−0.882172	−0.441086	−	0.897465i	$-0.645407\pi$
$353$	−16.5745	−0.882172	−0.441086	+	0.897465i	$0.645407\pi$
$354$	0	0
$355$	22.5539	1.19704
$356$	−4.14889	−0.219891
$357$	0	0
$358$	−7.12536	−0.376587
$359$	23.9611	1.26462	0.632310	−	0.774715i	$-0.282107\pi$
$359$	23.9611	1.26462	0.632310	+	0.774715i	$0.282107\pi$
$360$	0	0
$361$	−6.39942	−0.336811
$362$	−22.6450	−1.19020
$363$	0	0
$364$	−1.36702	−0.0716512
$365$	−11.2638	−0.589575
$366$	0	0
$367$	18.4219	0.961617	0.480808	−	0.876826i	$-0.340343\pi$
$367$	18.4219	0.961617	0.480808	+	0.876826i	$0.340343\pi$
$368$	0.420832	0.0219374
$369$	0	0
$370$	−49.3345	−2.56478
$371$	4.42656	0.229816
$372$	0	0
$373$	−13.9669	−0.723176	−0.361588	−	0.932338i	$-0.617765\pi$
$373$	−13.9669	−0.723176	−0.361588	+	0.932338i	$0.617765\pi$
$374$	−2.14218	−0.110769
$375$	0	0
$376$	8.85371	0.456595
$377$	−21.1868	−1.09118
$378$	0	0
$379$	14.6382	0.751914	0.375957	−	0.926637i	$-0.377314\pi$
$379$	14.6382	0.751914	0.375957	+	0.926637i	$0.377314\pi$
$380$	3.58569	0.183942
$381$	0	0
$382$	8.99891	0.460424
$383$	24.7759	1.26599	0.632994	−	0.774157i	$-0.281826\pi$
$383$	24.7759	1.26599	0.632994	+	0.774157i	$0.281826\pi$
$384$	0	0
$385$	1.54183	0.0785787
$386$	−2.06151	−0.104928
$387$	0	0
$388$	3.90361	0.198176
$389$	18.7545	0.950889	0.475444	−	0.879746i	$-0.342287\pi$
$389$	18.7545	0.950889	0.475444	+	0.879746i	$0.342287\pi$
$390$	0	0
$391$	−0.320635	−0.0162152
$392$	19.5047	0.985137
$393$	0	0
$394$	−21.6046	−1.08843
$395$	22.6893	1.14162
$396$	0	0
$397$	−12.0855	−0.606552	−0.303276	−	0.952903i	$-0.598080\pi$
$397$	−12.0855	−0.606552	−0.303276	+	0.952903i	$0.598080\pi$
$398$	−16.6232	−0.833248
$399$	0	0
$400$	−20.6001	−1.03001
$401$	−19.1972	−0.958663	−0.479331	−	0.877634i	$-0.659121\pi$
$401$	−19.1972	−0.958663	−0.479331	+	0.877634i	$0.659121\pi$
$402$	0	0
$403$	−1.32220	−0.0658633
$404$	−3.56645	−0.177437
$405$	0	0
$406$	−3.02720	−0.150238
$407$	7.37015	0.365325
$408$	0	0
$409$	14.3058	0.707378	0.353689	−	0.935363i	$-0.384927\pi$
$409$	14.3058	0.707378	0.353689	+	0.935363i	$0.384927\pi$
$410$	−2.73822	−0.135231
$411$	0	0
$412$	3.83268	0.188823
$413$	−4.16491	−0.204942
$414$	0	0
$415$	−45.3001	−2.22370
$416$	−10.8684	−0.532865
$417$	0	0
$418$	3.01861	0.147645
$419$	35.9064	1.75414	0.877072	−	0.480358i	$-0.159493\pi$
$419$	35.9064	1.75414	0.877072	+	0.480358i	$0.159493\pi$
$420$	0	0
$421$	−17.8353	−0.869241	−0.434621	−	0.900614i	$-0.643118\pi$
$421$	−17.8353	−0.869241	−0.434621	+	0.900614i	$0.643118\pi$
$422$	17.3536	0.844762
$423$	0	0
$424$	18.8296	0.914445
$425$	15.6954	0.761337
$426$	0	0
$427$	7.44427	0.360253
$428$	1.24770	0.0603096
$429$	0	0
$430$	−8.01347	−0.386444
$431$	−19.4070	−0.934801	−0.467401	−	0.884046i	$-0.654809\pi$
$431$	−19.4070	−0.934801	−0.467401	+	0.884046i	$0.654809\pi$
$432$	0	0
$433$	12.2858	0.590417	0.295208	−	0.955433i	$-0.404611\pi$
$433$	12.2858	0.590417	0.295208	+	0.955433i	$0.404611\pi$
$434$	−0.188917	−0.00906833
$435$	0	0
$436$	0.146036	0.00699385
$437$	0.451817	0.0216133
$438$	0	0
$439$	10.7560	0.513356	0.256678	−	0.966497i	$-0.417372\pi$
$439$	10.7560	0.513356	0.256678	+	0.966497i	$0.417372\pi$
$440$	6.55857	0.312668
$441$	0	0
$442$	−21.1163	−1.00440
$443$	9.46616	0.449751	0.224875	−	0.974388i	$-0.427802\pi$
$443$	9.46616	0.449751	0.224875	+	0.974388i	$0.427802\pi$
$444$	0	0
$445$	46.1271	2.18663
$446$	24.8188	1.17520
$447$	0	0
$448$	−6.21558	−0.293659
$449$	−7.50879	−0.354362	−0.177181	−	0.984178i	$-0.556698\pi$
$449$	−7.50879	−0.354362	−0.177181	+	0.984178i	$0.556698\pi$
$450$	0	0
$451$	0.409067	0.0192622
$452$	−3.29336	−0.154907
$453$	0	0
$454$	1.73581	0.0814655
$455$	15.1984	0.712512
$456$	0	0
$457$	8.12472	0.380059	0.190029	−	0.981778i	$-0.439142\pi$
$457$	8.12472	0.380059	0.190029	+	0.981778i	$0.439142\pi$
$458$	25.4815	1.19067
$459$	0	0
$460$	0.128572	0.00599468
$461$	28.7028	1.33682	0.668411	−	0.743792i	$-0.266975\pi$
$461$	28.7028	1.33682	0.668411	+	0.743792i	$0.266975\pi$
$462$	0	0
$463$	29.8856	1.38890	0.694450	−	0.719541i	$-0.255648\pi$
$463$	29.8856	1.38890	0.694450	+	0.719541i	$0.255648\pi$
$464$	−10.8912	−0.505611
$465$	0	0
$466$	−22.4244	−1.03879
$467$	−31.9678	−1.47929	−0.739646	−	0.672997i	$-0.765007\pi$
$467$	−31.9678	−1.47929	−0.739646	+	0.672997i	$0.765007\pi$
$468$	0	0
$469$	8.24927	0.380916
$470$	−12.8923	−0.594676
$471$	0	0
$472$	−17.7166	−0.815471
$473$	1.19714	0.0550448
$474$	0	0
$475$	−22.1168	−1.01479
$476$	0.535410	0.0245405
$477$	0	0
$478$	−23.8404	−1.09044
$479$	−39.9173	−1.82387	−0.911935	−	0.410334i	$-0.865412\pi$
$479$	−39.9173	−1.82387	−0.911935	+	0.410334i	$0.865412\pi$
$480$	0	0
$481$	72.6505	3.31258
$482$	−19.2579	−0.877172
$483$	0	0
$484$	3.18733	0.144879
$485$	−43.4000	−1.97069
$486$	0	0
$487$	−9.71286	−0.440131	−0.220066	−	0.975485i	$-0.570627\pi$
$487$	−9.71286	−0.440131	−0.220066	+	0.975485i	$0.570627\pi$
$488$	31.6662	1.43346
$489$	0	0
$490$	−28.4017	−1.28306
$491$	33.7385	1.52260	0.761298	−	0.648403i	$-0.224563\pi$
$491$	33.7385	1.52260	0.761298	+	0.648403i	$0.224563\pi$
$492$	0	0
$493$	8.29809	0.373727
$494$	29.7557	1.33877
$495$	0	0
$496$	−0.679683	−0.0305187
$497$	−4.74554	−0.212867
$498$	0	0
$499$	15.4423	0.691292	0.345646	−	0.938365i	$-0.387660\pi$
$499$	15.4423	0.691292	0.345646	+	0.938365i	$0.387660\pi$
$500$	−1.24304	−0.0555902
$501$	0	0
$502$	−13.4510	−0.600348
$503$	8.40426	0.374727	0.187364	−	0.982291i	$-0.440006\pi$
$503$	8.40426	0.374727	0.187364	+	0.982291i	$0.440006\pi$
$504$	0	0
$505$	39.6515	1.76447
$506$	0.108238	0.00481176
$507$	0	0
$508$	0.120027	0.00532535
$509$	25.2182	1.11778	0.558889	−	0.829243i	$-0.311228\pi$
$509$	25.2182	1.11778	0.558889	+	0.829243i	$0.311228\pi$
$510$	0	0
$511$	2.37001	0.104843
$512$	−25.4210	−1.12346
$513$	0	0
$514$	−15.1683	−0.669047
$515$	−42.6114	−1.87768
$516$	0	0
$517$	1.92600	0.0847052
$518$	10.3804	0.456089
$519$	0	0
$520$	64.6505	2.83511
$521$	−34.3918	−1.50673	−0.753366	−	0.657601i	$-0.771571\pi$
$521$	−34.3918	−1.50673	−0.753366	+	0.657601i	$0.771571\pi$
$522$	0	0
$523$	4.25582	0.186094	0.0930471	−	0.995662i	$-0.470339\pi$
$523$	4.25582	0.186094	0.0930471	+	0.995662i	$0.470339\pi$
$524$	−4.50403	−0.196760
$525$	0	0
$526$	−30.6249	−1.33531
$527$	0.517855	0.0225581
$528$	0	0
$529$	−22.9838	−0.999296
$530$	−27.4186	−1.19099
$531$	0	0
$532$	−0.754462	−0.0327101
$533$	4.03233	0.174660
$534$	0	0
$535$	−13.8718	−0.599729
$536$	35.0905	1.51568
$537$	0	0
$538$	23.8922	1.03007
$539$	4.24297	0.182758
$540$	0	0
$541$	−7.25204	−0.311790	−0.155895	−	0.987774i	$-0.549826\pi$
$541$	−7.25204	−0.311790	−0.155895	+	0.987774i	$0.549826\pi$
$542$	2.89153	0.124202
$543$	0	0
$544$	4.25673	0.182506
$545$	−1.62362	−0.0695481
$546$	0	0
$547$	−39.1657	−1.67460	−0.837302	−	0.546741i	$-0.815868\pi$
$547$	−39.1657	−1.67460	−0.837302	+	0.546741i	$0.815868\pi$
$548$	3.07491	0.131354
$549$	0	0
$550$	−5.29834	−0.225922
$551$	−11.6931	−0.498142
$552$	0	0
$553$	−4.77403	−0.203013
$554$	5.01268	0.212968
$555$	0	0
$556$	−5.16805	−0.219174
$557$	44.2977	1.87695	0.938477	−	0.345341i	$-0.112237\pi$
$557$	44.2977	1.87695	0.938477	+	0.345341i	$0.112237\pi$
$558$	0	0
$559$	11.8007	0.499118
$560$	7.81282	0.330152
$561$	0	0
$562$	−5.43085	−0.229087
$563$	42.5960	1.79521	0.897604	−	0.440803i	$-0.145306\pi$
$563$	42.5960	1.79521	0.897604	+	0.440803i	$0.145306\pi$
$564$	0	0
$565$	36.6153	1.54042
$566$	35.8143	1.50539
$567$	0	0
$568$	−20.1864	−0.847004
$569$	10.7105	0.449007	0.224504	−	0.974473i	$-0.427924\pi$
$569$	10.7105	0.449007	0.224504	+	0.974473i	$0.427924\pi$
$570$	0	0
$571$	26.1810	1.09564	0.547821	−	0.836596i	$-0.315457\pi$
$571$	26.1810	1.09564	0.547821	+	0.836596i	$0.315457\pi$
$572$	−1.26496	−0.0528908
$573$	0	0
$574$	0.576146	0.0240479
$575$	−0.793039	−0.0330720
$576$	0	0
$577$	−18.2356	−0.759158	−0.379579	−	0.925159i	$-0.623931\pi$
$577$	−18.2356	−0.759158	−0.379579	+	0.925159i	$0.623931\pi$
$578$	−13.8855	−0.577562
$579$	0	0
$580$	−3.32745	−0.138165
$581$	9.53156	0.395436
$582$	0	0
$583$	4.09610	0.169643
$584$	10.0815	0.417174
$585$	0	0
$586$	28.8348	1.19115
$587$	−39.8800	−1.64603	−0.823013	−	0.568023i	$-0.807708\pi$
$587$	−39.8800	−1.64603	−0.823013	+	0.568023i	$0.807708\pi$
$588$	0	0
$589$	−0.729726	−0.0300678
$590$	25.7979	1.06208
$591$	0	0
$592$	37.3464	1.53493
$593$	−46.9265	−1.92704	−0.963520	−	0.267635i	$-0.913758\pi$
$593$	−46.9265	−1.92704	−0.963520	+	0.267635i	$0.913758\pi$
$594$	0	0
$595$	−5.95265	−0.244035
$596$	0.301424	0.0123468
$597$	0	0
$598$	1.06694	0.0436306
$599$	−13.8523	−0.565988	−0.282994	−	0.959122i	$-0.591328\pi$
$599$	−13.8523	−0.565988	−0.282994	+	0.959122i	$0.591328\pi$
$600$	0	0
$601$	22.1301	0.902707	0.451354	−	0.892345i	$-0.350941\pi$
$601$	22.1301	0.902707	0.451354	+	0.892345i	$0.350941\pi$
$602$	1.68611	0.0687205
$603$	0	0
$604$	−3.14268	−0.127874
$605$	−35.4365	−1.44070
$606$	0	0
$607$	−40.5391	−1.64543	−0.822716	−	0.568452i	$-0.807542\pi$
$607$	−40.5391	−1.64543	−0.822716	+	0.568452i	$0.807542\pi$
$608$	−5.99829	−0.243263
$609$	0	0
$610$	−46.1105	−1.86696
$611$	18.9853	0.768064
$612$	0	0
$613$	7.41345	0.299426	0.149713	−	0.988729i	$-0.452165\pi$
$613$	7.41345	0.299426	0.149713	+	0.988729i	$0.452165\pi$
$614$	−6.33138	−0.255514
$615$	0	0
$616$	−1.37998	−0.0556011
$617$	−2.19650	−0.0884279	−0.0442139	−	0.999022i	$-0.514078\pi$
$617$	−2.19650	−0.0884279	−0.0442139	+	0.999022i	$0.514078\pi$
$618$	0	0
$619$	−20.6803	−0.831212	−0.415606	−	0.909545i	$-0.636430\pi$
$619$	−20.6803	−0.831212	−0.415606	+	0.909545i	$0.636430\pi$
$620$	−0.207655	−0.00833963
$621$	0	0
$622$	−12.6476	−0.507121
$623$	−9.70556	−0.388845
$624$	0	0
$625$	−17.3329	−0.693315
$626$	−11.8959	−0.475457
$627$	0	0
$628$	0.562561	0.0224486
$629$	−28.4545	−1.13456
$630$	0	0
$631$	−12.4661	−0.496268	−0.248134	−	0.968726i	$-0.579817\pi$
$631$	−12.4661	−0.496268	−0.248134	+	0.968726i	$0.579817\pi$
$632$	−20.3076	−0.807795
$633$	0	0
$634$	33.7349	1.33979
$635$	−1.33445	−0.0529562
$636$	0	0
$637$	41.8246	1.65715
$638$	−2.80121	−0.110901
$639$	0	0
$640$	27.1743	1.07416
$641$	2.39590	0.0946325	0.0473163	−	0.998880i	$-0.484933\pi$
$641$	2.39590	0.0946325	0.0473163	+	0.998880i	$0.484933\pi$
$642$	0	0
$643$	2.82329	0.111340	0.0556699	−	0.998449i	$-0.482271\pi$
$643$	2.82329	0.111340	0.0556699	+	0.998449i	$0.482271\pi$
$644$	−0.0270526	−0.00106602
$645$	0	0
$646$	−11.6542	−0.458528
$647$	19.8146	0.778991	0.389495	−	0.921028i	$-0.372649\pi$
$647$	19.8146	0.778991	0.389495	+	0.921028i	$0.372649\pi$
$648$	0	0
$649$	−3.85398	−0.151282
$650$	−52.2278	−2.04854
$651$	0	0
$652$	−3.47542	−0.136108
$653$	26.7544	1.04698	0.523490	−	0.852032i	$-0.324630\pi$
$653$	26.7544	1.04698	0.523490	+	0.852032i	$0.324630\pi$
$654$	0	0
$655$	50.0755	1.95661
$656$	2.07284	0.0809309
$657$	0	0
$658$	2.71265	0.105750
$659$	0.320343	0.0124788	0.00623940	−	0.999981i	$-0.498014\pi$
$659$	0.320343	0.0124788	0.00623940	+	0.999981i	$0.498014\pi$
$660$	0	0
$661$	−9.54786	−0.371369	−0.185684	−	0.982609i	$-0.559450\pi$
$661$	−9.54786	−0.371369	−0.185684	+	0.982609i	$0.559450\pi$
$662$	−7.60895	−0.295730
$663$	0	0
$664$	40.5450	1.57345
$665$	8.38806	0.325275
$666$	0	0
$667$	−0.419277	−0.0162345
$668$	1.70223	0.0658613
$669$	0	0
$670$	−51.0968	−1.97404
$671$	6.88852	0.265929
$672$	0	0
$673$	−14.4951	−0.558746	−0.279373	−	0.960183i	$-0.590127\pi$
$673$	−14.4951	−0.558746	−0.279373	+	0.960183i	$0.590127\pi$
$674$	8.98011	0.345901
$675$	0	0
$676$	−8.55075	−0.328875
$677$	46.1038	1.77191	0.885957	−	0.463768i	$-0.153503\pi$
$677$	46.1038	1.77191	0.885957	+	0.463768i	$0.153503\pi$
$678$	0	0
$679$	9.13176	0.350445
$680$	−25.3212	−0.971023
$681$	0	0
$682$	−0.174814	−0.00669398
$683$	−11.5155	−0.440628	−0.220314	−	0.975429i	$-0.570708\pi$
$683$	−11.5155	−0.440628	−0.220314	+	0.975429i	$0.570708\pi$
$684$	0	0
$685$	−34.1866	−1.30620
$686$	12.4089	0.473772
$687$	0	0
$688$	6.06623	0.231273
$689$	40.3769	1.53824
$690$	0	0
$691$	40.7486	1.55015	0.775074	−	0.631870i	$-0.217713\pi$
$691$	40.7486	1.55015	0.775074	+	0.631870i	$0.217713\pi$
$692$	−1.00606	−0.0382446
$693$	0	0
$694$	−13.3017	−0.504925
$695$	57.4580	2.17951
$696$	0	0
$697$	−1.57931	−0.0598208
$698$	−29.5325	−1.11782
$699$	0	0
$700$	1.32425	0.0500519
$701$	−10.3889	−0.392383	−0.196191	−	0.980566i	$-0.562857\pi$
$701$	−10.3889	−0.392383	−0.196191	+	0.980566i	$0.562857\pi$
$702$	0	0
$703$	40.0961	1.51225
$704$	−5.75156	−0.216770
$705$	0	0
$706$	−21.6015	−0.812982
$707$	−8.34302	−0.313772
$708$	0	0
$709$	24.5283	0.921180	0.460590	−	0.887613i	$-0.347638\pi$
$709$	24.5283	0.921180	0.460590	+	0.887613i	$0.347638\pi$
$710$	29.3944	1.10315
$711$	0	0
$712$	−41.2852	−1.54723
$713$	−0.0261656	−0.000979911 0
$714$	0	0
$715$	14.0638	0.525955
$716$	1.64794	0.0615864
$717$	0	0
$718$	31.2284	1.16543
$719$	−21.4872	−0.801339	−0.400670	−	0.916223i	$-0.631222\pi$
$719$	−21.4872	−0.801339	−0.400670	+	0.916223i	$0.631222\pi$
$720$	0	0
$721$	8.96583	0.333905
$722$	−8.34033	−0.310395
$723$	0	0
$724$	5.23730	0.194643
$725$	20.5240	0.762242
$726$	0	0
$727$	−10.3314	−0.383172	−0.191586	−	0.981476i	$-0.561363\pi$
$727$	−10.3314	−0.383172	−0.191586	+	0.981476i	$0.561363\pi$
$728$	−13.6030	−0.504162
$729$	0	0
$730$	−14.6801	−0.543333
$731$	−4.62191	−0.170947
$732$	0	0
$733$	15.0659	0.556471	0.278236	−	0.960513i	$-0.410250\pi$
$733$	15.0659	0.556471	0.278236	+	0.960513i	$0.410250\pi$
$734$	24.0092	0.886196
$735$	0	0
$736$	−0.215079	−0.00792793
$737$	7.63343	0.281181
$738$	0	0
$739$	35.1923	1.29457	0.647285	−	0.762248i	$-0.275904\pi$
$739$	35.1923	1.29457	0.647285	+	0.762248i	$0.275904\pi$
$740$	11.4100	0.419439
$741$	0	0
$742$	5.76912	0.211791
$743$	34.9355	1.28166	0.640829	−	0.767683i	$-0.278591\pi$
$743$	34.9355	1.28166	0.640829	+	0.767683i	$0.278591\pi$
$744$	0	0
$745$	−3.35120	−0.122779
$746$	−18.2029	−0.666456
$747$	0	0
$748$	0.495439	0.0181151
$749$	2.91875	0.106649
$750$	0	0
$751$	−35.1445	−1.28244	−0.641221	−	0.767356i	$-0.721572\pi$
$751$	−35.1445	−1.28244	−0.641221	+	0.767356i	$0.721572\pi$
$752$	9.75951	0.355893
$753$	0	0
$754$	−27.6127	−1.00559
$755$	34.9401	1.27160
$756$	0	0
$757$	31.8030	1.15590	0.577950	−	0.816072i	$-0.303853\pi$
$757$	31.8030	1.15590	0.577950	+	0.816072i	$0.303853\pi$
$758$	19.0779	0.692940
$759$	0	0
$760$	35.6809	1.29428
$761$	−15.9005	−0.576392	−0.288196	−	0.957571i	$-0.593055\pi$
$761$	−15.9005	−0.576392	−0.288196	+	0.957571i	$0.593055\pi$
$762$	0	0
$763$	0.341624	0.0123676
$764$	−2.08125	−0.0752970
$765$	0	0
$766$	32.2902	1.16669
$767$	−37.9902	−1.37175
$768$	0	0
$769$	−12.2802	−0.442836	−0.221418	−	0.975179i	$-0.571069\pi$
$769$	−12.2802	−0.442836	−0.221418	+	0.975179i	$0.571069\pi$
$770$	2.00945	0.0724157
$771$	0	0
$772$	0.476781	0.0171597
$773$	−14.7943	−0.532113	−0.266057	−	0.963957i	$-0.585721\pi$
$773$	−14.7943	−0.532113	−0.266057	+	0.963957i	$0.585721\pi$
$774$	0	0
$775$	1.28083	0.0460088
$776$	38.8444	1.39443
$777$	0	0
$778$	24.4426	0.876309
$779$	2.22546	0.0797354
$780$	0	0
$781$	−4.39127	−0.157132
$782$	−0.417882	−0.0149434
$783$	0	0
$784$	21.5002	0.767864
$785$	−6.25451	−0.223233
$786$	0	0
$787$	−17.6730	−0.629975	−0.314987	−	0.949096i	$-0.602000\pi$
$787$	−17.6730	−0.629975	−0.314987	+	0.949096i	$0.602000\pi$
$788$	4.99668	0.177999
$789$	0	0
$790$	29.5708	1.05208
$791$	−7.70419	−0.273929
$792$	0	0
$793$	67.9029	2.41130
$794$	−15.7509	−0.558979
$795$	0	0
$796$	3.84459	0.136268
$797$	21.9523	0.777591	0.388795	−	0.921324i	$-0.372891\pi$
$797$	21.9523	0.777591	0.388795	+	0.921324i	$0.372891\pi$
$798$	0	0
$799$	−7.43584	−0.263061
$800$	10.5283	0.372233
$801$	0	0
$802$	−25.0196	−0.883473
$803$	2.19308	0.0773920
$804$	0	0
$805$	0.300769	0.0106007
$806$	−1.72321	−0.0606976
$807$	0	0
$808$	−35.4893	−1.24851
$809$	−18.5949	−0.653762	−0.326881	−	0.945065i	$-0.605998\pi$
$809$	−18.5949	−0.653762	−0.326881	+	0.945065i	$0.605998\pi$
$810$	0	0
$811$	−14.1200	−0.495820	−0.247910	−	0.968783i	$-0.579744\pi$
$811$	−14.1200	−0.495820	−0.247910	+	0.968783i	$0.579744\pi$
$812$	0.700126	0.0245696
$813$	0	0
$814$	9.60548	0.336672
$815$	38.6394	1.35348
$816$	0	0
$817$	6.51287	0.227856
$818$	18.6447	0.651897
$819$	0	0
$820$	0.633290	0.0221154
$821$	55.9261	1.95183	0.975916	−	0.218145i	$-0.0700005\pi$
$821$	55.9261	1.95183	0.975916	+	0.218145i	$0.0700005\pi$
$822$	0	0
$823$	41.1911	1.43583	0.717916	−	0.696130i	$-0.245096\pi$
$823$	41.1911	1.43583	0.717916	+	0.696130i	$0.245096\pi$
$824$	38.1386	1.32862
$825$	0	0
$826$	−5.42810	−0.188868
$827$	32.5191	1.13080	0.565400	−	0.824817i	$-0.308722\pi$
$827$	32.5191	1.13080	0.565400	+	0.824817i	$0.308722\pi$
$828$	0	0
$829$	−0.000993012 0	−3.44887e−5 0	−1.72444e−5	−	1.00000i	$-0.500005\pi$
$829$	−0.000993012 0	−3.44887e−5 0	−1.72444e−5		1.00000i	$0.500005\pi$
$830$	−59.0394	−2.04929
$831$	0	0
$832$	−56.6955	−1.96556
$833$	−16.3812	−0.567573
$834$	0	0
$835$	−18.9253	−0.654936
$836$	−0.698139	−0.0241456
$837$	0	0
$838$	46.7967	1.61656
$839$	12.6260	0.435898	0.217949	−	0.975960i	$-0.430063\pi$
$839$	12.6260	0.435898	0.217949	+	0.975960i	$0.430063\pi$
$840$	0	0
$841$	−18.1490	−0.625829
$842$	−23.2447	−0.801065
$843$	0	0
$844$	−4.01352	−0.138151
$845$	95.0666	3.27039
$846$	0	0
$847$	7.45616	0.256197
$848$	20.7560	0.712764
$849$	0	0
$850$	20.4557	0.701625
$851$	1.43772	0.0492844
$852$	0	0
$853$	−54.7093	−1.87321	−0.936605	−	0.350388i	$-0.886050\pi$
$853$	−54.7093	−1.87321	−0.936605	+	0.350388i	$0.886050\pi$
$854$	9.70207	0.331998
$855$	0	0
$856$	12.4157	0.424359
$857$	−17.4880	−0.597380	−0.298690	−	0.954350i	$-0.596550\pi$
$857$	−17.4880	−0.597380	−0.298690	+	0.954350i	$0.596550\pi$
$858$	0	0
$859$	1.37851	0.0470342	0.0235171	−	0.999723i	$-0.492514\pi$
$859$	1.37851	0.0470342	0.0235171	+	0.999723i	$0.492514\pi$
$860$	1.85334	0.0631984
$861$	0	0
$862$	−25.2930	−0.861483
$863$	9.93558	0.338211	0.169105	−	0.985598i	$-0.445912\pi$
$863$	9.93558	0.338211	0.169105	+	0.985598i	$0.445912\pi$
$864$	0	0
$865$	11.1853	0.380311
$866$	16.0120	0.544109
$867$	0	0
$868$	0.0436925	0.00148302
$869$	−4.41763	−0.149858
$870$	0	0
$871$	75.2458	2.54961
$872$	1.45319	0.0492112
$873$	0	0
$874$	0.588850	0.0199182
$875$	−2.90785	−0.0983032
$876$	0	0
$877$	−6.92363	−0.233794	−0.116897	−	0.993144i	$-0.537295\pi$
$877$	−6.92363	−0.233794	−0.116897	+	0.993144i	$0.537295\pi$
$878$	14.0182	0.473093
$879$	0	0
$880$	7.22956	0.243709
$881$	−47.9623	−1.61589	−0.807946	−	0.589257i	$-0.799421\pi$
$881$	−47.9623	−1.61589	−0.807946	+	0.589257i	$0.799421\pi$
$882$	0	0
$883$	−3.56056	−0.119822	−0.0599111	−	0.998204i	$-0.519082\pi$
$883$	−3.56056	−0.119822	−0.0599111	+	0.998204i	$0.519082\pi$
$884$	4.88374	0.164258
$885$	0	0
$886$	12.3372	0.414476
$887$	21.9127	0.735756	0.367878	−	0.929874i	$-0.380084\pi$
$887$	21.9127	0.735756	0.367878	+	0.929874i	$0.380084\pi$
$888$	0	0
$889$	0.280781	0.00941709
$890$	60.1172	2.01513
$891$	0	0
$892$	−5.74004	−0.192191
$893$	10.4781	0.350635
$894$	0	0
$895$	−18.3217	−0.612426
$896$	−5.71772	−0.191015
$897$	0	0
$898$	−9.78616	−0.326568
$899$	0.677171	0.0225849
$900$	0	0
$901$	−15.8141	−0.526845
$902$	0.533134	0.0177514
$903$	0	0
$904$	−32.7719	−1.08998
$905$	−58.2279	−1.93556
$906$	0	0
$907$	44.8868	1.49044	0.745221	−	0.666818i	$-0.232344\pi$
$907$	44.8868	1.49044	0.745221	+	0.666818i	$0.232344\pi$
$908$	−0.401454	−0.0133227
$909$	0	0
$910$	19.8080	0.656628
$911$	36.5515	1.21100	0.605502	−	0.795844i	$-0.292972\pi$
$911$	36.5515	1.21100	0.605502	+	0.795844i	$0.292972\pi$
$912$	0	0
$913$	8.81999	0.291899
$914$	10.5889	0.350250
$915$	0	0
$916$	−5.89332	−0.194721
$917$	−10.5363	−0.347940
$918$	0	0
$919$	−40.4324	−1.33374	−0.666871	−	0.745174i	$-0.732367\pi$
$919$	−40.4324	−1.33374	−0.666871	+	0.745174i	$0.732367\pi$
$920$	1.27940	0.0421806
$921$	0	0
$922$	37.4082	1.23197
$923$	−43.2865	−1.42479
$924$	0	0
$925$	−70.3776	−2.31400
$926$	38.9497	1.27997
$927$	0	0
$928$	5.56629	0.182722
$929$	−17.1188	−0.561648	−0.280824	−	0.959759i	$-0.590608\pi$
$929$	−17.1188	−0.561648	−0.280824	+	0.959759i	$0.590608\pi$
$930$	0	0
$931$	23.0832	0.756521
$932$	5.18628	0.169882
$933$	0	0
$934$	−41.6634	−1.36327
$935$	−5.50826	−0.180139
$936$	0	0
$937$	−27.7824	−0.907613	−0.453806	−	0.891100i	$-0.649934\pi$
$937$	−27.7824	−0.907613	−0.453806	+	0.891100i	$0.649934\pi$
$938$	10.7512	0.351040
$939$	0	0
$940$	2.98170	0.0972524
$941$	34.3258	1.11899	0.559495	−	0.828834i	$-0.310995\pi$
$941$	34.3258	1.11899	0.559495	+	0.828834i	$0.310995\pi$
$942$	0	0
$943$	0.0797979	0.00259858
$944$	−19.5291	−0.635618
$945$	0	0
$946$	1.56023	0.0507275
$947$	−13.0412	−0.423781	−0.211890	−	0.977293i	$-0.567962\pi$
$947$	−13.0412	−0.423781	−0.211890	+	0.977293i	$0.567962\pi$
$948$	0	0
$949$	21.6180	0.701751
$950$	−28.8247	−0.935198
$951$	0	0
$952$	5.32780	0.172675
$953$	−32.1984	−1.04301	−0.521504	−	0.853249i	$-0.674629\pi$
$953$	−32.1984	−1.04301	−0.521504	+	0.853249i	$0.674629\pi$
$954$	0	0
$955$	23.1392	0.748767
$956$	5.51377	0.178328
$957$	0	0
$958$	−52.0241	−1.68082
$959$	7.19317	0.232280
$960$	0	0
$961$	−30.9577	−0.998637
$962$	94.6850	3.05277
$963$	0	0
$964$	4.45393	0.143451
$965$	−5.30082	−0.170639
$966$	0	0
$967$	32.4205	1.04257	0.521287	−	0.853382i	$-0.325452\pi$
$967$	32.4205	1.04257	0.521287	+	0.853382i	$0.325452\pi$
$968$	31.7168	1.01942
$969$	0	0
$970$	−56.5630	−1.81613
$971$	−7.25760	−0.232907	−0.116454	−	0.993196i	$-0.537153\pi$
$971$	−7.25760	−0.232907	−0.116454	+	0.993196i	$0.537153\pi$
$972$	0	0
$973$	−12.0897	−0.387577
$974$	−12.6587	−0.405611
$975$	0	0
$976$	34.9059	1.11731
$977$	40.7477	1.30363	0.651817	−	0.758376i	$-0.274007\pi$
$977$	40.7477	1.30363	0.651817	+	0.758376i	$0.274007\pi$
$978$	0	0
$979$	−8.98100	−0.287034
$980$	6.56869	0.209829
$981$	0	0
$982$	43.9711	1.40318
$983$	−17.8165	−0.568258	−0.284129	−	0.958786i	$-0.591704\pi$
$983$	−17.8165	−0.568258	−0.284129	+	0.958786i	$0.591704\pi$
$984$	0	0
$985$	−55.5527	−1.77006
$986$	10.8149	0.344415
$987$	0	0
$988$	−6.88183	−0.218940
$989$	0.233531	0.00742585
$990$	0	0
$991$	−50.1846	−1.59417	−0.797084	−	0.603869i	$-0.793625\pi$
$991$	−50.1846	−1.59417	−0.797084	+	0.603869i	$0.793625\pi$
$992$	0.347373	0.0110291
$993$	0	0
$994$	−6.18484	−0.196171
$995$	−42.7439	−1.35507
$996$	0	0
$997$	−55.2432	−1.74957	−0.874785	−	0.484511i	$-0.838998\pi$
$997$	−55.2432	−1.74957	−0.874785	+	0.484511i	$0.838998\pi$
$998$	20.1258	0.637073
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4023.2.a.e.1.19	✓	25
3.2	odd	2		4023.2.a.f.1.7	yes	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4023.2.a.e.1.19	✓	25	1.1	even	1	trivial
4023.2.a.f.1.7	yes	25	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4023 = 3^{3} \cdot 149 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4023.a (trivial)

\(f(q)\)	\(=\)	\(q+1.30329 q^{2} -0.301424 q^{4} +3.35120 q^{5} -0.705123 q^{7} -2.99943 q^{8} +O(q^{10})\)	\(q+1.30329 q^{2} -0.301424 q^{4} +3.35120 q^{5} -0.705123 q^{7} -2.99943 q^{8} +4.36761 q^{10} -0.652483 q^{11} -6.43179 q^{13} -0.918983 q^{14} -3.30630 q^{16} +2.51909 q^{17} -3.54973 q^{19} -1.01013 q^{20} -0.850378 q^{22} -0.127282 q^{23} +6.23057 q^{25} -8.38251 q^{26} +0.212541 q^{28} +3.29408 q^{29} +0.205572 q^{31} +1.68979 q^{32} +3.28312 q^{34} -2.36301 q^{35} -11.2955 q^{37} -4.62634 q^{38} -10.0517 q^{40} -0.626938 q^{41} -1.83475 q^{43} +0.196674 q^{44} -0.165886 q^{46} -2.95179 q^{47} -6.50280 q^{49} +8.12026 q^{50} +1.93869 q^{52} -6.27772 q^{53} -2.18660 q^{55} +2.11497 q^{56} +4.29316 q^{58} +5.90664 q^{59} -10.5574 q^{61} +0.267921 q^{62} +8.81488 q^{64} -21.5542 q^{65} -11.6990 q^{67} -0.759314 q^{68} -3.07970 q^{70} +6.73009 q^{71} -3.36112 q^{73} -14.7214 q^{74} +1.06997 q^{76} +0.460081 q^{77} +6.77049 q^{79} -11.0801 q^{80} -0.817085 q^{82} -13.5176 q^{83} +8.44199 q^{85} -2.39122 q^{86} +1.95708 q^{88} +13.7643 q^{89} +4.53520 q^{91} +0.0383658 q^{92} -3.84706 q^{94} -11.8959 q^{95} -12.9506 q^{97} -8.47506 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q - 7 q^{2} + 27 q^{4} - 12 q^{5} - 2 q^{7} - 21 q^{8}+O(q^{10}) \) 25 * q - 7 * q^2 + 27 * q^4 - 12 * q^5 - 2 * q^7 - 21 * q^8	\( 25 q - 7 q^{2} + 27 q^{4} - 12 q^{5} - 2 q^{7} - 21 q^{8} - 4 q^{10} - 12 q^{11} - 10 q^{14} + 35 q^{16} - 26 q^{17} - 30 q^{20} + 8 q^{22} - 26 q^{23} + 27 q^{25} - 16 q^{26} + 4 q^{28} - 20 q^{29} - 6 q^{31} - 49 q^{32} - 14 q^{34} - 16 q^{35} - 2 q^{37} - 27 q^{38} + 2 q^{40} - 35 q^{41} + 4 q^{43} - 22 q^{44} + 6 q^{46} - 38 q^{47} + 19 q^{49} - 22 q^{50} + 4 q^{52} - 36 q^{53} + 10 q^{55} - 79 q^{56} - 22 q^{58} - 15 q^{59} + 10 q^{61} + 14 q^{62} + 41 q^{64} - 80 q^{65} - 6 q^{67} - 33 q^{68} + 8 q^{70} - 26 q^{71} - 6 q^{73} - 75 q^{74} - 10 q^{76} - 47 q^{77} - 6 q^{79} - 66 q^{80} + 12 q^{82} - 40 q^{83} - 12 q^{85} - 4 q^{86} + 12 q^{88} - 30 q^{89} - 158 q^{92} + 18 q^{94} - 22 q^{95} - 20 q^{97} - 35 q^{98}+O(q^{100}) \) 25 * q - 7 * q^2 + 27 * q^4 - 12 * q^5 - 2 * q^7 - 21 * q^8 - 4 * q^10 - 12 * q^11 - 10 * q^14 + 35 * q^16 - 26 * q^17 - 30 * q^20 + 8 * q^22 - 26 * q^23 + 27 * q^25 - 16 * q^26 + 4 * q^28 - 20 * q^29 - 6 * q^31 - 49 * q^32 - 14 * q^34 - 16 * q^35 - 2 * q^37 - 27 * q^38 + 2 * q^40 - 35 * q^41 + 4 * q^43 - 22 * q^44 + 6 * q^46 - 38 * q^47 + 19 * q^49 - 22 * q^50 + 4 * q^52 - 36 * q^53 + 10 * q^55 - 79 * q^56 - 22 * q^58 - 15 * q^59 + 10 * q^61 + 14 * q^62 + 41 * q^64 - 80 * q^65 - 6 * q^67 - 33 * q^68 + 8 * q^70 - 26 * q^71 - 6 * q^73 - 75 * q^74 - 10 * q^76 - 47 * q^77 - 6 * q^79 - 66 * q^80 + 12 * q^82 - 40 * q^83 - 12 * q^85 - 4 * q^86 + 12 * q^88 - 30 * q^89 - 158 * q^92 + 18 * q^94 - 22 * q^95 - 20 * q^97 - 35 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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