LMFDB - Embedded newform 4012.2.a.j.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4012,2,Mod(1,4012)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4012, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4012.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4012 = 2^{2} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4012.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:	$32.0359812909$
Analytic rank:	$0$
Dimension:	$21$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Character	$\chi$	$=$	4012.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.765447 q^{3} -3.22340 q^{5} -4.55619 q^{7} -2.41409 q^{9} +O(q^{10})$	$q+0.765447 q^{3} -3.22340 q^{5} -4.55619 q^{7} -2.41409 q^{9} -3.45003 q^{11} -1.86746 q^{13} -2.46734 q^{15} +1.00000 q^{17} -6.05262 q^{19} -3.48752 q^{21} +3.04103 q^{23} +5.39028 q^{25} -4.14420 q^{27} -4.74716 q^{29} +2.29554 q^{31} -2.64081 q^{33} +14.6864 q^{35} +0.243036 q^{37} -1.42944 q^{39} -7.57727 q^{41} -11.0811 q^{43} +7.78157 q^{45} -0.420648 q^{47} +13.7588 q^{49} +0.765447 q^{51} -11.6932 q^{53} +11.1208 q^{55} -4.63296 q^{57} +1.00000 q^{59} -12.8159 q^{61} +10.9990 q^{63} +6.01955 q^{65} +4.15051 q^{67} +2.32775 q^{69} +7.72149 q^{71} +8.51891 q^{73} +4.12598 q^{75} +15.7190 q^{77} +12.2708 q^{79} +4.07011 q^{81} -1.34361 q^{83} -3.22340 q^{85} -3.63370 q^{87} +4.23549 q^{89} +8.50848 q^{91} +1.75712 q^{93} +19.5100 q^{95} +12.5185 q^{97} +8.32868 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 21 q + 9 q^{3} + q^{5} + 11 q^{7} + 26 q^{9}+O(q^{10}) $ 21 * q + 9 * q^3 + q^5 + 11 * q^7 + 26 * q^9	$ 21 q + 9 q^{3} + q^{5} + 11 q^{7} + 26 q^{9} + 12 q^{11} - 4 q^{13} + 9 q^{15} + 21 q^{17} + 4 q^{19} + 8 q^{21} + 19 q^{23} + 26 q^{25} + 33 q^{27} - q^{29} + 13 q^{31} + 11 q^{33} + 15 q^{35} - 4 q^{37} + 18 q^{39} + 9 q^{41} + 7 q^{43} + 7 q^{45} + 33 q^{47} + 36 q^{49} + 9 q^{51} + 7 q^{53} + 12 q^{55} + 26 q^{57} + 21 q^{59} + 3 q^{61} + 51 q^{63} + q^{65} + 8 q^{69} + 55 q^{71} + 22 q^{73} + 14 q^{75} - 15 q^{77} + 28 q^{79} + 25 q^{81} + 54 q^{83} + q^{85} + 34 q^{87} + 30 q^{89} + 35 q^{91} - 5 q^{93} + 42 q^{95} + 9 q^{97} + 20 q^{99}+O(q^{100}) $ 21 * q + 9 * q^3 + q^5 + 11 * q^7 + 26 * q^9 + 12 * q^11 - 4 * q^13 + 9 * q^15 + 21 * q^17 + 4 * q^19 + 8 * q^21 + 19 * q^23 + 26 * q^25 + 33 * q^27 - q^29 + 13 * q^31 + 11 * q^33 + 15 * q^35 - 4 * q^37 + 18 * q^39 + 9 * q^41 + 7 * q^43 + 7 * q^45 + 33 * q^47 + 36 * q^49 + 9 * q^51 + 7 * q^53 + 12 * q^55 + 26 * q^57 + 21 * q^59 + 3 * q^61 + 51 * q^63 + q^65 + 8 * q^69 + 55 * q^71 + 22 * q^73 + 14 * q^75 - 15 * q^77 + 28 * q^79 + 25 * q^81 + 54 * q^83 + q^85 + 34 * q^87 + 30 * q^89 + 35 * q^91 - 5 * q^93 + 42 * q^95 + 9 * q^97 + 20 * q^99

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	0
$2$	0	0
$3$	0.765447	0.441931	0.220966	−	0.975282i	$-0.429079\pi$
$3$	0.765447	0.441931	0.220966	+	0.975282i	$0.429079\pi$
$4$	0	0
$5$	−3.22340	−1.44155	−0.720773	−	0.693171i	$-0.756213\pi$
$5$	−3.22340	−1.44155	−0.720773	+	0.693171i	$0.756213\pi$
$6$	0	0
$7$	−4.55619	−1.72208	−0.861039	−	0.508540i	$-0.830185\pi$
$7$	−4.55619	−1.72208	−0.861039	+	0.508540i	$0.830185\pi$
$8$	0	0
$9$	−2.41409	−0.804697
$10$	0	0
$11$	−3.45003	−1.04022	−0.520111	−	0.854099i	$-0.674109\pi$
$11$	−3.45003	−1.04022	−0.520111	+	0.854099i	$0.674109\pi$
$12$	0	0
$13$	−1.86746	−0.517939	−0.258970	−	0.965885i	$-0.583383\pi$
$13$	−1.86746	−0.517939	−0.258970	+	0.965885i	$0.583383\pi$
$14$	0	0
$15$	−2.46734	−0.637064
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	−6.05262	−1.38857	−0.694283	−	0.719702i	$-0.744278\pi$
$19$	−6.05262	−1.38857	−0.694283	+	0.719702i	$0.744278\pi$
$20$	0	0
$21$	−3.48752	−0.761039
$22$	0	0
$23$	3.04103	0.634098	0.317049	−	0.948409i	$-0.397308\pi$
$23$	3.04103	0.634098	0.317049	+	0.948409i	$0.397308\pi$
$24$	0	0
$25$	5.39028	1.07806
$26$	0	0
$27$	−4.14420	−0.797552
$28$	0	0
$29$	−4.74716	−0.881525	−0.440763	−	0.897624i	$-0.645292\pi$
$29$	−4.74716	−0.881525	−0.440763	+	0.897624i	$0.645292\pi$
$30$	0	0
$31$	2.29554	0.412291	0.206146	−	0.978521i	$-0.433908\pi$
$31$	2.29554	0.412291	0.206146	+	0.978521i	$0.433908\pi$
$32$	0	0
$33$	−2.64081	−0.459707
$34$	0	0
$35$	14.6864	2.48245
$36$	0	0
$37$	0.243036	0.0399549	0.0199774	−	0.999800i	$-0.493641\pi$
$37$	0.243036	0.0399549	0.0199774	+	0.999800i	$0.493641\pi$
$38$	0	0
$39$	−1.42944	−0.228893
$40$	0	0
$41$	−7.57727	−1.18337	−0.591686	−	0.806169i	$-0.701537\pi$
$41$	−7.57727	−1.18337	−0.591686	+	0.806169i	$0.701537\pi$
$42$	0	0
$43$	−11.0811	−1.68986	−0.844929	−	0.534878i	$-0.820357\pi$
$43$	−11.0811	−1.68986	−0.844929	+	0.534878i	$0.820357\pi$
$44$	0	0
$45$	7.78157	1.16001
$46$	0	0
$47$	−0.420648	−0.0613578	−0.0306789	−	0.999529i	$-0.509767\pi$
$47$	−0.420648	−0.0613578	−0.0306789	+	0.999529i	$0.509767\pi$
$48$	0	0
$49$	13.7588	1.96555
$50$	0	0
$51$	0.765447	0.107184
$52$	0	0
$53$	−11.6932	−1.60619	−0.803093	−	0.595854i	$-0.796814\pi$
$53$	−11.6932	−1.60619	−0.803093	+	0.595854i	$0.796814\pi$
$54$	0	0
$55$	11.1208	1.49953
$56$	0	0
$57$	−4.63296	−0.613651
$58$	0	0
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	0	0
$61$	−12.8159	−1.64091	−0.820454	−	0.571712i	$-0.806279\pi$
$61$	−12.8159	−1.64091	−0.820454	+	0.571712i	$0.806279\pi$
$62$	0	0
$63$	10.9990	1.38575
$64$	0	0
$65$	6.01955	0.746633
$66$	0	0
$67$	4.15051	0.507066	0.253533	−	0.967327i	$-0.418407\pi$
$67$	4.15051	0.507066	0.253533	+	0.967327i	$0.418407\pi$
$68$	0	0
$69$	2.32775	0.280228
$70$	0	0
$71$	7.72149	0.916373	0.458186	−	0.888856i	$-0.348499\pi$
$71$	7.72149	0.916373	0.458186	+	0.888856i	$0.348499\pi$
$72$	0	0
$73$	8.51891	0.997063	0.498532	−	0.866871i	$-0.333873\pi$
$73$	8.51891	0.997063	0.498532	+	0.866871i	$0.333873\pi$
$74$	0	0
$75$	4.12598	0.476427
$76$	0	0
$77$	15.7190	1.79134
$78$	0	0
$79$	12.2708	1.38057	0.690287	−	0.723535i	$-0.257484\pi$
$79$	12.2708	1.38057	0.690287	+	0.723535i	$0.257484\pi$
$80$	0	0
$81$	4.07011	0.452234
$82$	0	0
$83$	−1.34361	−0.147481	−0.0737404	−	0.997277i	$-0.523494\pi$
$83$	−1.34361	−0.147481	−0.0737404	+	0.997277i	$0.523494\pi$
$84$	0	0
$85$	−3.22340	−0.349626
$86$	0	0
$87$	−3.63370	−0.389573
$88$	0	0
$89$	4.23549	0.448961	0.224481	−	0.974479i	$-0.427931\pi$
$89$	4.23549	0.448961	0.224481	+	0.974479i	$0.427931\pi$
$90$	0	0
$91$	8.50848	0.891931
$92$	0	0
$93$	1.75712	0.182204
$94$	0	0
$95$	19.5100	2.00168
$96$	0	0
$97$	12.5185	1.27107	0.635533	−	0.772074i	$-0.280780\pi$
$97$	12.5185	1.27107	0.635533	+	0.772074i	$0.280780\pi$
$98$	0	0
$99$	8.32868	0.837064
$100$	0	0
$101$	−13.4781	−1.34112	−0.670559	−	0.741856i	$-0.733946\pi$
$101$	−13.4781	−1.34112	−0.670559	+	0.741856i	$0.733946\pi$
$102$	0	0
$103$	−13.8317	−1.36288	−0.681439	−	0.731875i	$-0.738646\pi$
$103$	−13.8317	−1.36288	−0.681439	+	0.731875i	$0.738646\pi$
$104$	0	0
$105$	11.2417	1.09707
$106$	0	0
$107$	−13.1613	−1.27235	−0.636177	−	0.771543i	$-0.719485\pi$
$107$	−13.1613	−1.27235	−0.636177	+	0.771543i	$0.719485\pi$
$108$	0	0
$109$	−6.37181	−0.610308	−0.305154	−	0.952303i	$-0.598708\pi$
$109$	−6.37181	−0.610308	−0.305154	+	0.952303i	$0.598708\pi$
$110$	0	0
$111$	0.186031	0.0176573
$112$	0	0
$113$	0.356332	0.0335209	0.0167605	−	0.999860i	$-0.494665\pi$
$113$	0.356332	0.0335209	0.0167605	+	0.999860i	$0.494665\pi$
$114$	0	0
$115$	−9.80243	−0.914082
$116$	0	0
$117$	4.50821	0.416784
$118$	0	0
$119$	−4.55619	−0.417665
$120$	0	0
$121$	0.902685	0.0820622
$122$	0	0
$123$	−5.80000	−0.522969
$124$	0	0
$125$	−1.25804	−0.112522
$126$	0	0
$127$	−7.78749	−0.691028	−0.345514	−	0.938414i	$-0.612295\pi$
$127$	−7.78749	−0.691028	−0.345514	+	0.938414i	$0.612295\pi$
$128$	0	0
$129$	−8.48203	−0.746801
$130$	0	0
$131$	17.6428	1.54146	0.770730	−	0.637161i	$-0.219892\pi$
$131$	17.6428	1.54146	0.770730	+	0.637161i	$0.219892\pi$
$132$	0	0
$133$	27.5769	2.39122
$134$	0	0
$135$	13.3584	1.14971
$136$	0	0
$137$	14.9946	1.28107	0.640537	−	0.767927i	$-0.278712\pi$
$137$	14.9946	1.28107	0.640537	+	0.767927i	$0.278712\pi$
$138$	0	0
$139$	16.0388	1.36039	0.680196	−	0.733030i	$-0.261895\pi$
$139$	16.0388	1.36039	0.680196	+	0.733030i	$0.261895\pi$
$140$	0	0
$141$	−0.321984	−0.0271159
$142$	0	0
$143$	6.44277	0.538772
$144$	0	0
$145$	15.3020	1.27076
$146$	0	0
$147$	10.5317	0.868637
$148$	0	0
$149$	−2.34188	−0.191854	−0.0959271	−	0.995388i	$-0.530582\pi$
$149$	−2.34188	−0.191854	−0.0959271	+	0.995388i	$0.530582\pi$
$150$	0	0
$151$	19.2082	1.56314	0.781569	−	0.623819i	$-0.214420\pi$
$151$	19.2082	1.56314	0.781569	+	0.623819i	$0.214420\pi$
$152$	0	0
$153$	−2.41409	−0.195168
$154$	0	0
$155$	−7.39944	−0.594337
$156$	0	0
$157$	−6.68261	−0.533331	−0.266665	−	0.963789i	$-0.585922\pi$
$157$	−6.68261	−0.533331	−0.266665	+	0.963789i	$0.585922\pi$
$158$	0	0
$159$	−8.95054	−0.709824
$160$	0	0
$161$	−13.8555	−1.09197
$162$	0	0
$163$	1.62247	0.127081	0.0635407	−	0.997979i	$-0.479761\pi$
$163$	1.62247	0.127081	0.0635407	+	0.997979i	$0.479761\pi$
$164$	0	0
$165$	8.51239	0.662688
$166$	0	0
$167$	−3.82850	−0.296258	−0.148129	−	0.988968i	$-0.547325\pi$
$167$	−3.82850	−0.296258	−0.148129	+	0.988968i	$0.547325\pi$
$168$	0	0
$169$	−9.51261	−0.731739
$170$	0	0
$171$	14.6116	1.11738
$172$	0	0
$173$	20.0963	1.52789	0.763945	−	0.645281i	$-0.223260\pi$
$173$	20.0963	1.52789	0.763945	+	0.645281i	$0.223260\pi$
$174$	0	0
$175$	−24.5591	−1.85650
$176$	0	0
$177$	0.765447	0.0575345
$178$	0	0
$179$	−3.88051	−0.290043	−0.145021	−	0.989429i	$-0.546325\pi$
$179$	−3.88051	−0.290043	−0.145021	+	0.989429i	$0.546325\pi$
$180$	0	0
$181$	−4.39936	−0.327002	−0.163501	−	0.986543i	$-0.552279\pi$
$181$	−4.39936	−0.327002	−0.163501	+	0.986543i	$0.552279\pi$
$182$	0	0
$183$	−9.80990	−0.725169
$184$	0	0
$185$	−0.783401	−0.0575968
$186$	0	0
$187$	−3.45003	−0.252291
$188$	0	0
$189$	18.8818	1.37345
$190$	0	0
$191$	−14.1205	−1.02173	−0.510863	−	0.859662i	$-0.670674\pi$
$191$	−14.1205	−1.02173	−0.510863	+	0.859662i	$0.670674\pi$
$192$	0	0
$193$	10.9859	0.790784	0.395392	−	0.918512i	$-0.370609\pi$
$193$	10.9859	0.790784	0.395392	+	0.918512i	$0.370609\pi$
$194$	0	0
$195$	4.60765	0.329960
$196$	0	0
$197$	−7.14114	−0.508785	−0.254392	−	0.967101i	$-0.581876\pi$
$197$	−7.14114	−0.508785	−0.254392	+	0.967101i	$0.581876\pi$
$198$	0	0
$199$	−14.9910	−1.06268	−0.531341	−	0.847158i	$-0.678312\pi$
$199$	−14.9910	−1.06268	−0.531341	+	0.847158i	$0.678312\pi$
$200$	0	0
$201$	3.17700	0.224088
$202$	0	0
$203$	21.6289	1.51805
$204$	0	0
$205$	24.4246	1.70588
$206$	0	0
$207$	−7.34131	−0.510257
$208$	0	0
$209$	20.8817	1.44442
$210$	0	0
$211$	−5.15539	−0.354911	−0.177456	−	0.984129i	$-0.556787\pi$
$211$	−5.15539	−0.354911	−0.177456	+	0.984129i	$0.556787\pi$
$212$	0	0
$213$	5.91040	0.404974
$214$	0	0
$215$	35.7189	2.43601
$216$	0	0
$217$	−10.4589	−0.709997
$218$	0	0
$219$	6.52078	0.440633
$220$	0	0
$221$	−1.86746	−0.125619
$222$	0	0
$223$	−25.0303	−1.67615	−0.838076	−	0.545554i	$-0.816319\pi$
$223$	−25.0303	−1.67615	−0.838076	+	0.545554i	$0.816319\pi$
$224$	0	0
$225$	−13.0126	−0.867509
$226$	0	0
$227$	11.2166	0.744475	0.372237	−	0.928138i	$-0.378591\pi$
$227$	11.2166	0.744475	0.372237	+	0.928138i	$0.378591\pi$
$228$	0	0
$229$	−10.0292	−0.662750	−0.331375	−	0.943499i	$-0.607513\pi$
$229$	−10.0292	−0.662750	−0.331375	+	0.943499i	$0.607513\pi$
$230$	0	0
$231$	12.0320	0.791650
$232$	0	0
$233$	−17.8907	−1.17206	−0.586030	−	0.810290i	$-0.699310\pi$
$233$	−17.8907	−1.17206	−0.586030	+	0.810290i	$0.699310\pi$
$234$	0	0
$235$	1.35592	0.0884502
$236$	0	0
$237$	9.39266	0.610119
$238$	0	0
$239$	−11.8913	−0.769186	−0.384593	−	0.923086i	$-0.625658\pi$
$239$	−11.8913	−0.769186	−0.384593	+	0.923086i	$0.625658\pi$
$240$	0	0
$241$	−7.92687	−0.510615	−0.255307	−	0.966860i	$-0.582177\pi$
$241$	−7.92687	−0.510615	−0.255307	+	0.966860i	$0.582177\pi$
$242$	0	0
$243$	15.5481	0.997408
$244$	0	0
$245$	−44.3502	−2.83343
$246$	0	0
$247$	11.3030	0.719193
$248$	0	0
$249$	−1.02847	−0.0651764
$250$	0	0
$251$	10.9373	0.690359	0.345179	−	0.938537i	$-0.387818\pi$
$251$	10.9373	0.690359	0.345179	+	0.938537i	$0.387818\pi$
$252$	0	0
$253$	−10.4916	−0.659603
$254$	0	0
$255$	−2.46734	−0.154511
$256$	0	0
$257$	−1.06187	−0.0662374	−0.0331187	−	0.999451i	$-0.510544\pi$
$257$	−1.06187	−0.0662374	−0.0331187	+	0.999451i	$0.510544\pi$
$258$	0	0
$259$	−1.10732	−0.0688054
$260$	0	0
$261$	11.4601	0.709361
$262$	0	0
$263$	−26.8374	−1.65486	−0.827432	−	0.561566i	$-0.810199\pi$
$263$	−26.8374	−1.65486	−0.827432	+	0.561566i	$0.810199\pi$
$264$	0	0
$265$	37.6919	2.31539
$266$	0	0
$267$	3.24205	0.198410
$268$	0	0
$269$	−19.3543	−1.18005	−0.590027	−	0.807384i	$-0.700883\pi$
$269$	−19.3543	−1.18005	−0.590027	+	0.807384i	$0.700883\pi$
$270$	0	0
$271$	0.405458	0.0246298	0.0123149	−	0.999924i	$-0.496080\pi$
$271$	0.405458	0.0246298	0.0123149	+	0.999924i	$0.496080\pi$
$272$	0	0
$273$	6.51279	0.394172
$274$	0	0
$275$	−18.5966	−1.12142
$276$	0	0
$277$	−19.1013	−1.14768	−0.573842	−	0.818966i	$-0.694548\pi$
$277$	−19.1013	−1.14768	−0.573842	+	0.818966i	$0.694548\pi$
$278$	0	0
$279$	−5.54164	−0.331770
$280$	0	0
$281$	−9.79228	−0.584158	−0.292079	−	0.956394i	$-0.594347\pi$
$281$	−9.79228	−0.584158	−0.292079	+	0.956394i	$0.594347\pi$
$282$	0	0
$283$	−25.6971	−1.52753	−0.763767	−	0.645492i	$-0.776652\pi$
$283$	−25.6971	−1.52753	−0.763767	+	0.645492i	$0.776652\pi$
$284$	0	0
$285$	14.9339	0.884606
$286$	0	0
$287$	34.5235	2.03786
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	9.58229	0.561724
$292$	0	0
$293$	−27.0663	−1.58123	−0.790616	−	0.612312i	$-0.790240\pi$
$293$	−27.0663	−1.58123	−0.790616	+	0.612312i	$0.790240\pi$
$294$	0	0
$295$	−3.22340	−0.187673
$296$	0	0
$297$	14.2976	0.829631
$298$	0	0
$299$	−5.67898	−0.328424
$300$	0	0
$301$	50.4878	2.91007
$302$	0	0
$303$	−10.3168	−0.592682
$304$	0	0
$305$	41.3107	2.36545
$306$	0	0
$307$	−24.9602	−1.42455	−0.712276	−	0.701900i	$-0.752336\pi$
$307$	−24.9602	−1.42455	−0.712276	+	0.701900i	$0.752336\pi$
$308$	0	0
$309$	−10.5874	−0.602298
$310$	0	0
$311$	18.7633	1.06397	0.531986	−	0.846753i	$-0.321446\pi$
$311$	18.7633	1.06397	0.531986	+	0.846753i	$0.321446\pi$
$312$	0	0
$313$	−21.7724	−1.23065	−0.615325	−	0.788274i	$-0.710975\pi$
$313$	−21.7724	−1.23065	−0.615325	+	0.788274i	$0.710975\pi$
$314$	0	0
$315$	−35.4543	−1.99762
$316$	0	0
$317$	−4.95601	−0.278357	−0.139179	−	0.990267i	$-0.544446\pi$
$317$	−4.95601	−0.278357	−0.139179	+	0.990267i	$0.544446\pi$
$318$	0	0
$319$	16.3778	0.916982
$320$	0	0
$321$	−10.0743	−0.562293
$322$	0	0
$323$	−6.05262	−0.336777
$324$	0	0
$325$	−10.0661	−0.558368
$326$	0	0
$327$	−4.87728	−0.269714
$328$	0	0
$329$	1.91655	0.105663
$330$	0	0
$331$	−0.0637060	−0.00350160	−0.00175080	−	0.999998i	$-0.500557\pi$
$331$	−0.0637060	−0.00350160	−0.00175080	+	0.999998i	$0.500557\pi$
$332$	0	0
$333$	−0.586711	−0.0321516
$334$	0	0
$335$	−13.3787	−0.730959
$336$	0	0
$337$	−7.22893	−0.393785	−0.196892	−	0.980425i	$-0.563085\pi$
$337$	−7.22893	−0.393785	−0.196892	+	0.980425i	$0.563085\pi$
$338$	0	0
$339$	0.272753	0.0148139
$340$	0	0
$341$	−7.91968	−0.428875
$342$	0	0
$343$	−30.7946	−1.66275
$344$	0	0
$345$	−7.50325	−0.403961
$346$	0	0
$347$	14.5748	0.782414	0.391207	−	0.920303i	$-0.372058\pi$
$347$	14.5748	0.782414	0.391207	+	0.920303i	$0.372058\pi$
$348$	0	0
$349$	0.138361	0.00740628	0.00370314	−	0.999993i	$-0.498821\pi$
$349$	0.138361	0.00740628	0.00370314	+	0.999993i	$0.498821\pi$
$350$	0	0
$351$	7.73911	0.413083
$352$	0	0
$353$	31.2012	1.66067	0.830336	−	0.557263i	$-0.188148\pi$
$353$	31.2012	1.66067	0.830336	+	0.557263i	$0.188148\pi$
$354$	0	0
$355$	−24.8894	−1.32099
$356$	0	0
$357$	−3.48752	−0.184579
$358$	0	0
$359$	−18.8614	−0.995466	−0.497733	−	0.867330i	$-0.665834\pi$
$359$	−18.8614	−0.995466	−0.497733	+	0.867330i	$0.665834\pi$
$360$	0	0
$361$	17.6342	0.928117
$362$	0	0
$363$	0.690957	0.0362659
$364$	0	0
$365$	−27.4598	−1.43731
$366$	0	0
$367$	−21.8750	−1.14187	−0.570933	−	0.820997i	$-0.693418\pi$
$367$	−21.8750	−1.14187	−0.570933	+	0.820997i	$0.693418\pi$
$368$	0	0
$369$	18.2922	0.952255
$370$	0	0
$371$	53.2765	2.76598
$372$	0	0
$373$	−16.8185	−0.870831	−0.435416	−	0.900230i	$-0.643399\pi$
$373$	−16.8185	−0.870831	−0.435416	+	0.900230i	$0.643399\pi$
$374$	0	0
$375$	−0.962963	−0.0497272
$376$	0	0
$377$	8.86511	0.456576
$378$	0	0
$379$	−33.4730	−1.71939	−0.859697	−	0.510804i	$-0.829348\pi$
$379$	−33.4730	−1.71939	−0.859697	+	0.510804i	$0.829348\pi$
$380$	0	0
$381$	−5.96091	−0.305387
$382$	0	0
$383$	37.5387	1.91814	0.959070	−	0.283169i	$-0.0913858\pi$
$383$	37.5387	1.91814	0.959070	+	0.283169i	$0.0913858\pi$
$384$	0	0
$385$	−50.6685	−2.58230
$386$	0	0
$387$	26.7509	1.35982
$388$	0	0
$389$	3.60262	0.182660	0.0913300	−	0.995821i	$-0.470888\pi$
$389$	3.60262	0.182660	0.0913300	+	0.995821i	$0.470888\pi$
$390$	0	0
$391$	3.04103	0.153791
$392$	0	0
$393$	13.5047	0.681220
$394$	0	0
$395$	−39.5537	−1.99016
$396$	0	0
$397$	−3.52016	−0.176672	−0.0883359	−	0.996091i	$-0.528155\pi$
$397$	−3.52016	−0.176672	−0.0883359	+	0.996091i	$0.528155\pi$
$398$	0	0
$399$	21.1086	1.05675
$400$	0	0
$401$	24.5531	1.22612	0.613060	−	0.790036i	$-0.289938\pi$
$401$	24.5531	1.22612	0.613060	+	0.790036i	$0.289938\pi$
$402$	0	0
$403$	−4.28682	−0.213542
$404$	0	0
$405$	−13.1196	−0.651916
$406$	0	0
$407$	−0.838481	−0.0415619
$408$	0	0
$409$	−16.6485	−0.823213	−0.411607	−	0.911362i	$-0.635032\pi$
$409$	−16.6485	−0.823213	−0.411607	+	0.911362i	$0.635032\pi$
$410$	0	0
$411$	11.4776	0.566147
$412$	0	0
$413$	−4.55619	−0.224195
$414$	0	0
$415$	4.33100	0.212600
$416$	0	0
$417$	12.2768	0.601199
$418$	0	0
$419$	27.3807	1.33763	0.668816	−	0.743428i	$-0.266801\pi$
$419$	27.3807	1.33763	0.668816	+	0.743428i	$0.266801\pi$
$420$	0	0
$421$	16.8407	0.820768	0.410384	−	0.911913i	$-0.365395\pi$
$421$	16.8407	0.820768	0.410384	+	0.911913i	$0.365395\pi$
$422$	0	0
$423$	1.01548	0.0493745
$424$	0	0
$425$	5.39028	0.261467
$426$	0	0
$427$	58.3917	2.82577
$428$	0	0
$429$	4.93160	0.238100
$430$	0	0
$431$	29.1760	1.40536	0.702680	−	0.711506i	$-0.251987\pi$
$431$	29.1760	1.40536	0.702680	+	0.711506i	$0.251987\pi$
$432$	0	0
$433$	−9.58529	−0.460640	−0.230320	−	0.973115i	$-0.573977\pi$
$433$	−9.58529	−0.460640	−0.230320	+	0.973115i	$0.573977\pi$
$434$	0	0
$435$	11.7129	0.561588
$436$	0	0
$437$	−18.4062	−0.880487
$438$	0	0
$439$	−37.5794	−1.79357	−0.896784	−	0.442468i	$-0.854103\pi$
$439$	−37.5794	−1.79357	−0.896784	+	0.442468i	$0.854103\pi$
$440$	0	0
$441$	−33.2151	−1.58167
$442$	0	0
$443$	26.9076	1.27842	0.639210	−	0.769032i	$-0.279261\pi$
$443$	26.9076	1.27842	0.639210	+	0.769032i	$0.279261\pi$
$444$	0	0
$445$	−13.6527	−0.647199
$446$	0	0
$447$	−1.79258	−0.0847863
$448$	0	0
$449$	−35.9929	−1.69861	−0.849306	−	0.527902i	$-0.822979\pi$
$449$	−35.9929	−1.69861	−0.849306	+	0.527902i	$0.822979\pi$
$450$	0	0
$451$	26.1418	1.23097
$452$	0	0
$453$	14.7028	0.690799
$454$	0	0
$455$	−27.4262	−1.28576
$456$	0	0
$457$	−18.3625	−0.858962	−0.429481	−	0.903076i	$-0.641303\pi$
$457$	−18.3625	−0.858962	−0.429481	+	0.903076i	$0.641303\pi$
$458$	0	0
$459$	−4.14420	−0.193435
$460$	0	0
$461$	−27.6529	−1.28792	−0.643962	−	0.765058i	$-0.722710\pi$
$461$	−27.6529	−1.28792	−0.643962	+	0.765058i	$0.722710\pi$
$462$	0	0
$463$	−7.88671	−0.366527	−0.183263	−	0.983064i	$-0.558666\pi$
$463$	−7.88671	−0.366527	−0.183263	+	0.983064i	$0.558666\pi$
$464$	0	0
$465$	−5.66388	−0.262656
$466$	0	0
$467$	−12.8534	−0.594785	−0.297393	−	0.954755i	$-0.596117\pi$
$467$	−12.8534	−0.594785	−0.297393	+	0.954755i	$0.596117\pi$
$468$	0	0
$469$	−18.9105	−0.873206
$470$	0	0
$471$	−5.11519	−0.235695
$472$	0	0
$473$	38.2302	1.75783
$474$	0	0
$475$	−32.6253	−1.49695
$476$	0	0
$477$	28.2285	1.29249
$478$	0	0
$479$	−22.6569	−1.03522	−0.517609	−	0.855617i	$-0.673178\pi$
$479$	−22.6569	−1.03522	−0.517609	+	0.855617i	$0.673178\pi$
$480$	0	0
$481$	−0.453859	−0.0206942
$482$	0	0
$483$	−10.6056	−0.482573
$484$	0	0
$485$	−40.3522	−1.83230
$486$	0	0
$487$	8.24739	0.373725	0.186862	−	0.982386i	$-0.440168\pi$
$487$	8.24739	0.373725	0.186862	+	0.982386i	$0.440168\pi$
$488$	0	0
$489$	1.24191	0.0561612
$490$	0	0
$491$	0.0300056	0.00135413	0.000677066	−	1.00000i	$-0.499784\pi$
$491$	0.0300056	0.00135413	0.000677066		1.00000i	$0.499784\pi$
$492$	0	0
$493$	−4.74716	−0.213801
$494$	0	0
$495$	−26.8466	−1.20667
$496$	0	0
$497$	−35.1806	−1.57806
$498$	0	0
$499$	26.3652	1.18027	0.590135	−	0.807305i	$-0.299075\pi$
$499$	26.3652	1.18027	0.590135	+	0.807305i	$0.299075\pi$
$500$	0	0
$501$	−2.93051	−0.130926
$502$	0	0
$503$	10.2253	0.455922	0.227961	−	0.973670i	$-0.426794\pi$
$503$	10.2253	0.455922	0.227961	+	0.973670i	$0.426794\pi$
$504$	0	0
$505$	43.4452	1.93328
$506$	0	0
$507$	−7.28140	−0.323378
$508$	0	0
$509$	27.8122	1.23275	0.616377	−	0.787451i	$-0.288600\pi$
$509$	27.8122	1.23275	0.616377	+	0.787451i	$0.288600\pi$
$510$	0	0
$511$	−38.8138	−1.71702
$512$	0	0
$513$	25.0833	1.10745
$514$	0	0
$515$	44.5851	1.96465
$516$	0	0
$517$	1.45125	0.0638258
$518$	0	0
$519$	15.3826	0.675223
$520$	0	0
$521$	−11.8890	−0.520869	−0.260434	−	0.965492i	$-0.583866\pi$
$521$	−11.8890	−0.520869	−0.260434	+	0.965492i	$0.583866\pi$
$522$	0	0
$523$	23.4292	1.02449	0.512243	−	0.858841i	$-0.328815\pi$
$523$	23.4292	1.02449	0.512243	+	0.858841i	$0.328815\pi$
$524$	0	0
$525$	−18.7987	−0.820444
$526$	0	0
$527$	2.29554	0.0999953
$528$	0	0
$529$	−13.7522	−0.597920
$530$	0	0
$531$	−2.41409	−0.104763
$532$	0	0
$533$	14.1502	0.612914
$534$	0	0
$535$	42.4242	1.83416
$536$	0	0
$537$	−2.97032	−0.128179
$538$	0	0
$539$	−47.4684	−2.04461
$540$	0	0
$541$	16.2778	0.699838	0.349919	−	0.936780i	$-0.386209\pi$
$541$	16.2778	0.699838	0.349919	+	0.936780i	$0.386209\pi$
$542$	0	0
$543$	−3.36748	−0.144512
$544$	0	0
$545$	20.5389	0.879788
$546$	0	0
$547$	3.30399	0.141269	0.0706343	−	0.997502i	$-0.477498\pi$
$547$	3.30399	0.141269	0.0706343	+	0.997502i	$0.477498\pi$
$548$	0	0
$549$	30.9388	1.32043
$550$	0	0
$551$	28.7328	1.22406
$552$	0	0
$553$	−55.9081	−2.37746
$554$	0	0
$555$	−0.599652	−0.0254538
$556$	0	0
$557$	−26.0064	−1.10193	−0.550963	−	0.834530i	$-0.685739\pi$
$557$	−26.0064	−1.10193	−0.550963	+	0.834530i	$0.685739\pi$
$558$	0	0
$559$	20.6935	0.875243
$560$	0	0
$561$	−2.64081	−0.111495
$562$	0	0
$563$	−9.14724	−0.385510	−0.192755	−	0.981247i	$-0.561742\pi$
$563$	−9.14724	−0.385510	−0.192755	+	0.981247i	$0.561742\pi$
$564$	0	0
$565$	−1.14860	−0.0483220
$566$	0	0
$567$	−18.5442	−0.778782
$568$	0	0
$569$	6.42076	0.269172	0.134586	−	0.990902i	$-0.457030\pi$
$569$	6.42076	0.269172	0.134586	+	0.990902i	$0.457030\pi$
$570$	0	0
$571$	−2.62147	−0.109705	−0.0548526	−	0.998494i	$-0.517469\pi$
$571$	−2.62147	−0.109705	−0.0548526	+	0.998494i	$0.517469\pi$
$572$	0	0
$573$	−10.8085	−0.451532
$574$	0	0
$575$	16.3920	0.683593
$576$	0	0
$577$	28.4310	1.18360	0.591799	−	0.806086i	$-0.298418\pi$
$577$	28.4310	1.18360	0.591799	+	0.806086i	$0.298418\pi$
$578$	0	0
$579$	8.40915	0.349472
$580$	0	0
$581$	6.12176	0.253973
$582$	0	0
$583$	40.3419	1.67079
$584$	0	0
$585$	−14.5317	−0.600813
$586$	0	0
$587$	31.5739	1.30319	0.651597	−	0.758566i	$-0.274099\pi$
$587$	31.5739	1.30319	0.651597	+	0.758566i	$0.274099\pi$
$588$	0	0
$589$	−13.8940	−0.572494
$590$	0	0
$591$	−5.46616	−0.224848
$592$	0	0
$593$	−14.3213	−0.588104	−0.294052	−	0.955789i	$-0.595004\pi$
$593$	−14.3213	−0.588104	−0.294052	+	0.955789i	$0.595004\pi$
$594$	0	0
$595$	14.6864	0.602084
$596$	0	0
$597$	−11.4748	−0.469632
$598$	0	0
$599$	23.2989	0.951968	0.475984	−	0.879454i	$-0.342092\pi$
$599$	23.2989	0.951968	0.475984	+	0.879454i	$0.342092\pi$
$600$	0	0
$601$	−23.3767	−0.953556	−0.476778	−	0.879024i	$-0.658195\pi$
$601$	−23.3767	−0.953556	−0.476778	+	0.879024i	$0.658195\pi$
$602$	0	0
$603$	−10.0197	−0.408034
$604$	0	0
$605$	−2.90971	−0.118297
$606$	0	0
$607$	30.8269	1.25123	0.625614	−	0.780133i	$-0.284849\pi$
$607$	30.8269	1.25123	0.625614	+	0.780133i	$0.284849\pi$
$608$	0	0
$609$	16.5558	0.670875
$610$	0	0
$611$	0.785542	0.0317796
$612$	0	0
$613$	−1.21175	−0.0489420	−0.0244710	−	0.999701i	$-0.507790\pi$
$613$	−1.21175	−0.0489420	−0.0244710	+	0.999701i	$0.507790\pi$
$614$	0	0
$615$	18.6957	0.753883
$616$	0	0
$617$	3.59785	0.144844	0.0724221	−	0.997374i	$-0.476927\pi$
$617$	3.59785	0.144844	0.0724221	+	0.997374i	$0.476927\pi$
$618$	0	0
$619$	21.9979	0.884170	0.442085	−	0.896973i	$-0.354239\pi$
$619$	21.9979	0.884170	0.442085	+	0.896973i	$0.354239\pi$
$620$	0	0
$621$	−12.6026	−0.505726
$622$	0	0
$623$	−19.2977	−0.773146
$624$	0	0
$625$	−22.8963	−0.915850
$626$	0	0
$627$	15.9838	0.638333
$628$	0	0
$629$	0.243036	0.00969048
$630$	0	0
$631$	−25.9723	−1.03394	−0.516971	−	0.856003i	$-0.672940\pi$
$631$	−25.9723	−1.03394	−0.516971	+	0.856003i	$0.672940\pi$
$632$	0	0
$633$	−3.94618	−0.156846
$634$	0	0
$635$	25.1022	0.996150
$636$	0	0
$637$	−25.6940	−1.01803
$638$	0	0
$639$	−18.6404	−0.737402
$640$	0	0
$641$	35.5952	1.40593	0.702963	−	0.711226i	$-0.251860\pi$
$641$	35.5952	1.40593	0.702963	+	0.711226i	$0.251860\pi$
$642$	0	0
$643$	29.8812	1.17840	0.589200	−	0.807987i	$-0.299443\pi$
$643$	29.8812	1.17840	0.589200	+	0.807987i	$0.299443\pi$
$644$	0	0
$645$	27.3409	1.07655
$646$	0	0
$647$	34.2877	1.34799	0.673995	−	0.738736i	$-0.264577\pi$
$647$	34.2877	1.34799	0.673995	+	0.738736i	$0.264577\pi$
$648$	0	0
$649$	−3.45003	−0.135425
$650$	0	0
$651$	−8.00575	−0.313770
$652$	0	0
$653$	−27.3908	−1.07189	−0.535943	−	0.844254i	$-0.680044\pi$
$653$	−27.3908	−1.07189	−0.535943	+	0.844254i	$0.680044\pi$
$654$	0	0
$655$	−56.8698	−2.22209
$656$	0	0
$657$	−20.5654	−0.802334
$658$	0	0
$659$	−10.2003	−0.397349	−0.198674	−	0.980066i	$-0.563664\pi$
$659$	−10.2003	−0.397349	−0.198674	+	0.980066i	$0.563664\pi$
$660$	0	0
$661$	15.1099	0.587706	0.293853	−	0.955851i	$-0.405062\pi$
$661$	15.1099	0.587706	0.293853	+	0.955851i	$0.405062\pi$
$662$	0	0
$663$	−1.42944	−0.0555148
$664$	0	0
$665$	−88.8912	−3.44705
$666$	0	0
$667$	−14.4362	−0.558973
$668$	0	0
$669$	−19.1594	−0.740744
$670$	0	0
$671$	44.2152	1.70691
$672$	0	0
$673$	16.7127	0.644227	0.322114	−	0.946701i	$-0.395607\pi$
$673$	16.7127	0.644227	0.322114	+	0.946701i	$0.395607\pi$
$674$	0	0
$675$	−22.3384	−0.859806
$676$	0	0
$677$	41.7210	1.60347	0.801735	−	0.597680i	$-0.203911\pi$
$677$	41.7210	1.60347	0.801735	+	0.597680i	$0.203911\pi$
$678$	0	0
$679$	−57.0369	−2.18887
$680$	0	0
$681$	8.58575	0.329006
$682$	0	0
$683$	−3.59184	−0.137438	−0.0687189	−	0.997636i	$-0.521891\pi$
$683$	−3.59184	−0.137438	−0.0687189	+	0.997636i	$0.521891\pi$
$684$	0	0
$685$	−48.3335	−1.84673
$686$	0	0
$687$	−7.67685	−0.292890
$688$	0	0
$689$	21.8366	0.831906
$690$	0	0
$691$	−23.7942	−0.905175	−0.452588	−	0.891720i	$-0.649499\pi$
$691$	−23.7942	−0.905175	−0.452588	+	0.891720i	$0.649499\pi$
$692$	0	0
$693$	−37.9470	−1.44149
$694$	0	0
$695$	−51.6994	−1.96107
$696$	0	0
$697$	−7.57727	−0.287010
$698$	0	0
$699$	−13.6944	−0.517969
$700$	0	0
$701$	−16.2411	−0.613418	−0.306709	−	0.951803i	$-0.599228\pi$
$701$	−16.2411	−0.613418	−0.306709	+	0.951803i	$0.599228\pi$
$702$	0	0
$703$	−1.47100	−0.0554800
$704$	0	0
$705$	1.03788	0.0390889
$706$	0	0
$707$	61.4086	2.30951
$708$	0	0
$709$	26.8873	1.00977	0.504886	−	0.863186i	$-0.331534\pi$
$709$	26.8873	1.00977	0.504886	+	0.863186i	$0.331534\pi$
$710$	0	0
$711$	−29.6229	−1.11094
$712$	0	0
$713$	6.98080	0.261433
$714$	0	0
$715$	−20.7676	−0.776665
$716$	0	0
$717$	−9.10218	−0.339927
$718$	0	0
$719$	30.4391	1.13519	0.567594	−	0.823309i	$-0.307874\pi$
$719$	30.4391	1.13519	0.567594	+	0.823309i	$0.307874\pi$
$720$	0	0
$721$	63.0198	2.34698
$722$	0	0
$723$	−6.06760	−0.225657
$724$	0	0
$725$	−25.5885	−0.950334
$726$	0	0
$727$	16.5806	0.614942	0.307471	−	0.951557i	$-0.400517\pi$
$727$	16.5806	0.614942	0.307471	+	0.951557i	$0.400517\pi$
$728$	0	0
$729$	−0.309104	−0.0114483
$730$	0	0
$731$	−11.0811	−0.409851
$732$	0	0
$733$	−22.0297	−0.813685	−0.406842	−	0.913498i	$-0.633370\pi$
$733$	−22.0297	−0.813685	−0.406842	+	0.913498i	$0.633370\pi$
$734$	0	0
$735$	−33.9477	−1.25218
$736$	0	0
$737$	−14.3194	−0.527461
$738$	0	0
$739$	2.85250	0.104931	0.0524654	−	0.998623i	$-0.483292\pi$
$739$	2.85250	0.104931	0.0524654	+	0.998623i	$0.483292\pi$
$740$	0	0
$741$	8.65185	0.317834
$742$	0	0
$743$	−19.3152	−0.708607	−0.354303	−	0.935131i	$-0.615282\pi$
$743$	−19.3152	−0.708607	−0.354303	+	0.935131i	$0.615282\pi$
$744$	0	0
$745$	7.54880	0.276567
$746$	0	0
$747$	3.24361	0.118677
$748$	0	0
$749$	59.9655	2.19109
$750$	0	0
$751$	20.8988	0.762607	0.381303	−	0.924450i	$-0.375475\pi$
$751$	20.8988	0.762607	0.381303	+	0.924450i	$0.375475\pi$
$752$	0	0
$753$	8.37195	0.305091
$754$	0	0
$755$	−61.9155	−2.25334
$756$	0	0
$757$	2.79910	0.101735	0.0508674	−	0.998705i	$-0.483801\pi$
$757$	2.79910	0.101735	0.0508674	+	0.998705i	$0.483801\pi$
$758$	0	0
$759$	−8.03078	−0.291499
$760$	0	0
$761$	−20.7212	−0.751144	−0.375572	−	0.926793i	$-0.622554\pi$
$761$	−20.7212	−0.751144	−0.375572	+	0.926793i	$0.622554\pi$
$762$	0	0
$763$	29.0311	1.05100
$764$	0	0
$765$	7.78157	0.281343
$766$	0	0
$767$	−1.86746	−0.0674299
$768$	0	0
$769$	−14.9520	−0.539183	−0.269592	−	0.962975i	$-0.586889\pi$
$769$	−14.9520	−0.539183	−0.269592	+	0.962975i	$0.586889\pi$
$770$	0	0
$771$	−0.812803	−0.0292724
$772$	0	0
$773$	38.9914	1.40242	0.701211	−	0.712954i	$-0.252643\pi$
$773$	38.9914	1.40242	0.701211	+	0.712954i	$0.252643\pi$
$774$	0	0
$775$	12.3736	0.444473
$776$	0	0
$777$	−0.847593	−0.0304072
$778$	0	0
$779$	45.8624	1.64319
$780$	0	0
$781$	−26.6394	−0.953231
$782$	0	0
$783$	19.6732	0.703062
$784$	0	0
$785$	21.5407	0.768821
$786$	0	0
$787$	14.3978	0.513226	0.256613	−	0.966514i	$-0.417393\pi$
$787$	14.3978	0.513226	0.256613	+	0.966514i	$0.417393\pi$
$788$	0	0
$789$	−20.5426	−0.731336
$790$	0	0
$791$	−1.62352	−0.0577256
$792$	0	0
$793$	23.9331	0.849891
$794$	0	0
$795$	28.8511	1.02324
$796$	0	0
$797$	7.13899	0.252876	0.126438	−	0.991975i	$-0.459645\pi$
$797$	7.13899	0.252876	0.126438	+	0.991975i	$0.459645\pi$
$798$	0	0
$799$	−0.420648	−0.0148815
$800$	0	0
$801$	−10.2249	−0.361278
$802$	0	0
$803$	−29.3905	−1.03717
$804$	0	0
$805$	44.6617	1.57412
$806$	0	0
$807$	−14.8147	−0.521502
$808$	0	0
$809$	22.2938	0.783808	0.391904	−	0.920006i	$-0.371817\pi$
$809$	22.2938	0.783808	0.391904	+	0.920006i	$0.371817\pi$
$810$	0	0
$811$	20.3801	0.715641	0.357820	−	0.933790i	$-0.383520\pi$
$811$	20.3801	0.715641	0.357820	+	0.933790i	$0.383520\pi$
$812$	0	0
$813$	0.310357	0.0108847
$814$	0	0
$815$	−5.22985	−0.183194
$816$	0	0
$817$	67.0699	2.34648
$818$	0	0
$819$	−20.5402	−0.717734
$820$	0	0
$821$	14.4396	0.503945	0.251972	−	0.967734i	$-0.418921\pi$
$821$	14.4396	0.503945	0.251972	+	0.967734i	$0.418921\pi$
$822$	0	0
$823$	−42.2431	−1.47250	−0.736251	−	0.676708i	$-0.763406\pi$
$823$	−42.2431	−1.47250	−0.736251	+	0.676708i	$0.763406\pi$
$824$	0	0
$825$	−14.2347	−0.495590
$826$	0	0
$827$	19.5571	0.680067	0.340034	−	0.940413i	$-0.389562\pi$
$827$	19.5571	0.680067	0.340034	+	0.940413i	$0.389562\pi$
$828$	0	0
$829$	−56.1273	−1.94938	−0.974691	−	0.223557i	$-0.928233\pi$
$829$	−56.1273	−1.94938	−0.974691	+	0.223557i	$0.928233\pi$
$830$	0	0
$831$	−14.6210	−0.507198
$832$	0	0
$833$	13.7588	0.476716
$834$	0	0
$835$	12.3408	0.427070
$836$	0	0
$837$	−9.51318	−0.328824
$838$	0	0
$839$	4.13583	0.142785	0.0713923	−	0.997448i	$-0.477256\pi$
$839$	4.13583	0.142785	0.0713923	+	0.997448i	$0.477256\pi$
$840$	0	0
$841$	−6.46448	−0.222913
$842$	0	0
$843$	−7.49547	−0.258158
$844$	0	0
$845$	30.6629	1.05484
$846$	0	0
$847$	−4.11280	−0.141317
$848$	0	0
$849$	−19.6698	−0.675065
$850$	0	0
$851$	0.739079	0.0253353
$852$	0	0
$853$	−37.4351	−1.28175	−0.640877	−	0.767644i	$-0.721429\pi$
$853$	−37.4351	−1.28175	−0.640877	+	0.767644i	$0.721429\pi$
$854$	0	0
$855$	−47.0989	−1.61075
$856$	0	0
$857$	40.8451	1.39524	0.697622	−	0.716466i	$-0.254242\pi$
$857$	40.8451	1.39524	0.697622	+	0.716466i	$0.254242\pi$
$858$	0	0
$859$	49.6713	1.69476	0.847382	−	0.530984i	$-0.178178\pi$
$859$	49.6713	1.69476	0.847382	+	0.530984i	$0.178178\pi$
$860$	0	0
$861$	26.4259	0.900592
$862$	0	0
$863$	4.27660	0.145577	0.0727885	−	0.997347i	$-0.476810\pi$
$863$	4.27660	0.145577	0.0727885	+	0.997347i	$0.476810\pi$
$864$	0	0
$865$	−64.7782	−2.20253
$866$	0	0
$867$	0.765447	0.0259959
$868$	0	0
$869$	−42.3347	−1.43610
$870$	0	0
$871$	−7.75090	−0.262629
$872$	0	0
$873$	−30.2209	−1.02282
$874$	0	0
$875$	5.73186	0.193772
$876$	0	0
$877$	17.0912	0.577128	0.288564	−	0.957461i	$-0.406822\pi$
$877$	17.0912	0.577128	0.288564	+	0.957461i	$0.406822\pi$
$878$	0	0
$879$	−20.7179	−0.698796
$880$	0	0
$881$	5.16000	0.173845	0.0869224	−	0.996215i	$-0.472297\pi$
$881$	5.16000	0.173845	0.0869224	+	0.996215i	$0.472297\pi$
$882$	0	0
$883$	50.2301	1.69038	0.845189	−	0.534467i	$-0.179488\pi$
$883$	50.2301	1.69038	0.845189	+	0.534467i	$0.179488\pi$
$884$	0	0
$885$	−2.46734	−0.0829387
$886$	0	0
$887$	32.0607	1.07649	0.538247	−	0.842787i	$-0.319087\pi$
$887$	32.0607	1.07649	0.538247	+	0.842787i	$0.319087\pi$
$888$	0	0
$889$	35.4813	1.19000
$890$	0	0
$891$	−14.0420	−0.470424
$892$	0	0
$893$	2.54602	0.0851994
$894$	0	0
$895$	12.5084	0.418110
$896$	0	0
$897$	−4.34696	−0.145141
$898$	0	0
$899$	−10.8973	−0.363445
$900$	0	0
$901$	−11.6932	−0.389557
$902$	0	0
$903$	38.6457	1.28605
$904$	0	0
$905$	14.1809	0.471388
$906$	0	0
$907$	3.63634	0.120743	0.0603713	−	0.998176i	$-0.480772\pi$
$907$	3.63634	0.120743	0.0603713	+	0.998176i	$0.480772\pi$
$908$	0	0
$909$	32.5373	1.07919
$910$	0	0
$911$	15.3288	0.507867	0.253933	−	0.967222i	$-0.418276\pi$
$911$	15.3288	0.507867	0.253933	+	0.967222i	$0.418276\pi$
$912$	0	0
$913$	4.63551	0.153413
$914$	0	0
$915$	31.6212	1.04536
$916$	0	0
$917$	−80.3840	−2.65451
$918$	0	0
$919$	10.1130	0.333598	0.166799	−	0.985991i	$-0.446657\pi$
$919$	10.1130	0.333598	0.166799	+	0.985991i	$0.446657\pi$
$920$	0	0
$921$	−19.1057	−0.629554
$922$	0	0
$923$	−14.4195	−0.474625
$924$	0	0
$925$	1.31003	0.0430736
$926$	0	0
$927$	33.3910	1.09670
$928$	0	0
$929$	9.06522	0.297420	0.148710	−	0.988881i	$-0.452488\pi$
$929$	9.06522	0.297420	0.148710	+	0.988881i	$0.452488\pi$
$930$	0	0
$931$	−83.2771	−2.72930
$932$	0	0
$933$	14.3624	0.470202
$934$	0	0
$935$	11.1208	0.363689
$936$	0	0
$937$	−2.24278	−0.0732685	−0.0366342	−	0.999329i	$-0.511664\pi$
$937$	−2.24278	−0.0732685	−0.0366342	+	0.999329i	$0.511664\pi$
$938$	0	0
$939$	−16.6656	−0.543862
$940$	0	0
$941$	−0.457837	−0.0149250	−0.00746252	−	0.999972i	$-0.502375\pi$
$941$	−0.457837	−0.0149250	−0.00746252	+	0.999972i	$0.502375\pi$
$942$	0	0
$943$	−23.0427	−0.750373
$944$	0	0
$945$	−60.8634	−1.97989
$946$	0	0
$947$	23.6496	0.768508	0.384254	−	0.923228i	$-0.374459\pi$
$947$	23.6496	0.768508	0.384254	+	0.923228i	$0.374459\pi$
$948$	0	0
$949$	−15.9087	−0.516418
$950$	0	0
$951$	−3.79357	−0.123015
$952$	0	0
$953$	−3.33804	−0.108130	−0.0540649	−	0.998537i	$-0.517218\pi$
$953$	−3.33804	−0.108130	−0.0540649	+	0.998537i	$0.517218\pi$
$954$	0	0
$955$	45.5160	1.47286
$956$	0	0
$957$	12.5364	0.405243
$958$	0	0
$959$	−68.3182	−2.20611
$960$	0	0
$961$	−25.7305	−0.830016
$962$	0	0
$963$	31.7727	1.02386
$964$	0	0
$965$	−35.4120	−1.13995
$966$	0	0
$967$	49.7503	1.59986	0.799931	−	0.600092i	$-0.204869\pi$
$967$	49.7503	1.59986	0.799931	+	0.600092i	$0.204869\pi$
$968$	0	0
$969$	−4.63296	−0.148832
$970$	0	0
$971$	−8.20494	−0.263309	−0.131655	−	0.991296i	$-0.542029\pi$
$971$	−8.20494	−0.263309	−0.131655	+	0.991296i	$0.542029\pi$
$972$	0	0
$973$	−73.0757	−2.34270
$974$	0	0
$975$	−7.70508	−0.246760
$976$	0	0
$977$	−28.1757	−0.901421	−0.450711	−	0.892670i	$-0.648829\pi$
$977$	−28.1757	−0.901421	−0.450711	+	0.892670i	$0.648829\pi$
$978$	0	0
$979$	−14.6126	−0.467020
$980$	0	0
$981$	15.3821	0.491113
$982$	0	0
$983$	−54.1253	−1.72633	−0.863164	−	0.504924i	$-0.831520\pi$
$983$	−54.1253	−1.72633	−0.863164	+	0.504924i	$0.831520\pi$
$984$	0	0
$985$	23.0187	0.733437
$986$	0	0
$987$	1.46702	0.0466957
$988$	0	0
$989$	−33.6980	−1.07154
$990$	0	0
$991$	−39.0352	−1.23999	−0.619997	−	0.784604i	$-0.712866\pi$
$991$	−39.0352	−1.23999	−0.619997	+	0.784604i	$0.712866\pi$
$992$	0	0
$993$	−0.0487636	−0.00154747
$994$	0	0
$995$	48.3218	1.53190
$996$	0	0
$997$	−34.2181	−1.08370	−0.541849	−	0.840476i	$-0.682276\pi$
$997$	−34.2181	−1.08370	−0.541849	+	0.840476i	$0.682276\pi$
$998$	0	0
$999$	−1.00719	−0.0318661

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.a.j.1.12	✓	21

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.a.j.1.12	✓	21	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 4012 = 2^{2} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4012.a (trivial)

\(f(q)\)	\(=\)	\(q+0.765447 q^{3} -3.22340 q^{5} -4.55619 q^{7} -2.41409 q^{9} +O(q^{10})\)	\(q+0.765447 q^{3} -3.22340 q^{5} -4.55619 q^{7} -2.41409 q^{9} -3.45003 q^{11} -1.86746 q^{13} -2.46734 q^{15} +1.00000 q^{17} -6.05262 q^{19} -3.48752 q^{21} +3.04103 q^{23} +5.39028 q^{25} -4.14420 q^{27} -4.74716 q^{29} +2.29554 q^{31} -2.64081 q^{33} +14.6864 q^{35} +0.243036 q^{37} -1.42944 q^{39} -7.57727 q^{41} -11.0811 q^{43} +7.78157 q^{45} -0.420648 q^{47} +13.7588 q^{49} +0.765447 q^{51} -11.6932 q^{53} +11.1208 q^{55} -4.63296 q^{57} +1.00000 q^{59} -12.8159 q^{61} +10.9990 q^{63} +6.01955 q^{65} +4.15051 q^{67} +2.32775 q^{69} +7.72149 q^{71} +8.51891 q^{73} +4.12598 q^{75} +15.7190 q^{77} +12.2708 q^{79} +4.07011 q^{81} -1.34361 q^{83} -3.22340 q^{85} -3.63370 q^{87} +4.23549 q^{89} +8.50848 q^{91} +1.75712 q^{93} +19.5100 q^{95} +12.5185 q^{97} +8.32868 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 21 q + 9 q^{3} + q^{5} + 11 q^{7} + 26 q^{9}+O(q^{10}) \) 21 * q + 9 * q^3 + q^5 + 11 * q^7 + 26 * q^9	\( 21 q + 9 q^{3} + q^{5} + 11 q^{7} + 26 q^{9} + 12 q^{11} - 4 q^{13} + 9 q^{15} + 21 q^{17} + 4 q^{19} + 8 q^{21} + 19 q^{23} + 26 q^{25} + 33 q^{27} - q^{29} + 13 q^{31} + 11 q^{33} + 15 q^{35} - 4 q^{37} + 18 q^{39} + 9 q^{41} + 7 q^{43} + 7 q^{45} + 33 q^{47} + 36 q^{49} + 9 q^{51} + 7 q^{53} + 12 q^{55} + 26 q^{57} + 21 q^{59} + 3 q^{61} + 51 q^{63} + q^{65} + 8 q^{69} + 55 q^{71} + 22 q^{73} + 14 q^{75} - 15 q^{77} + 28 q^{79} + 25 q^{81} + 54 q^{83} + q^{85} + 34 q^{87} + 30 q^{89} + 35 q^{91} - 5 q^{93} + 42 q^{95} + 9 q^{97} + 20 q^{99}+O(q^{100}) \) 21 * q + 9 * q^3 + q^5 + 11 * q^7 + 26 * q^9 + 12 * q^11 - 4 * q^13 + 9 * q^15 + 21 * q^17 + 4 * q^19 + 8 * q^21 + 19 * q^23 + 26 * q^25 + 33 * q^27 - q^29 + 13 * q^31 + 11 * q^33 + 15 * q^35 - 4 * q^37 + 18 * q^39 + 9 * q^41 + 7 * q^43 + 7 * q^45 + 33 * q^47 + 36 * q^49 + 9 * q^51 + 7 * q^53 + 12 * q^55 + 26 * q^57 + 21 * q^59 + 3 * q^61 + 51 * q^63 + q^65 + 8 * q^69 + 55 * q^71 + 22 * q^73 + 14 * q^75 - 15 * q^77 + 28 * q^79 + 25 * q^81 + 54 * q^83 + q^85 + 34 * q^87 + 30 * q^89 + 35 * q^91 - 5 * q^93 + 42 * q^95 + 9 * q^97 + 20 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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