LMFDB - Embedded newform 4004.2.a.g.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4004,2,Mod(1,4004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4004, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4004.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:	$31.9721009693$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.246302029.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 9x^{4} + 14x^{3} + 15x^{2} - 13x - 10 $ x^6 - 2x^5 - 9x^4 + 14x^3 + 15x^2 - 13*x - 10
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-0.820857$ of defining polynomial
Character	$\chi$	$=$	4004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.820857 q^{3} +0.0215726 q^{5} +1.00000 q^{7} -2.32619 q^{9} +O(q^{10})$	$q+0.820857 q^{3} +0.0215726 q^{5} +1.00000 q^{7} -2.32619 q^{9} -1.00000 q^{11} -1.00000 q^{13} +0.0177081 q^{15} +0.645197 q^{17} +2.66244 q^{19} +0.820857 q^{21} +0.816993 q^{23} -4.99953 q^{25} -4.37205 q^{27} -2.98229 q^{29} -2.72028 q^{31} -0.820857 q^{33} +0.0215726 q^{35} -3.34011 q^{37} -0.820857 q^{39} +3.93482 q^{41} -4.28305 q^{43} -0.0501821 q^{45} -7.90272 q^{47} +1.00000 q^{49} +0.529615 q^{51} +7.61883 q^{53} -0.0215726 q^{55} +2.18548 q^{57} -8.01678 q^{59} +7.80516 q^{61} -2.32619 q^{63} -0.0215726 q^{65} -2.10012 q^{67} +0.670635 q^{69} -10.1318 q^{71} -0.993429 q^{73} -4.10391 q^{75} -1.00000 q^{77} -7.44053 q^{79} +3.38975 q^{81} +11.6149 q^{83} +0.0139186 q^{85} -2.44804 q^{87} -4.20682 q^{89} -1.00000 q^{91} -2.23296 q^{93} +0.0574359 q^{95} -11.5872 q^{97} +2.32619 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{3} - 3 q^{5} + 6 q^{7} + 4 q^{9}+O(q^{10}) $ 6 * q - 2 * q^3 - 3 * q^5 + 6 * q^7 + 4 * q^9	$ 6 q - 2 q^{3} - 3 q^{5} + 6 q^{7} + 4 q^{9} - 6 q^{11} - 6 q^{13} + 4 q^{15} - q^{17} - 12 q^{19} - 2 q^{21} + 5 q^{23} - 3 q^{25} - 8 q^{27} - 14 q^{29} - 4 q^{31} + 2 q^{33} - 3 q^{35} - 3 q^{37} + 2 q^{39} + 6 q^{41} - 14 q^{43} - 20 q^{45} + 2 q^{47} + 6 q^{49} - 5 q^{51} - 3 q^{53} + 3 q^{55} - 22 q^{57} + 2 q^{59} - 26 q^{61} + 4 q^{63} + 3 q^{65} + 9 q^{67} - 11 q^{69} + 3 q^{71} - 7 q^{73} - 6 q^{75} - 6 q^{77} + 6 q^{81} - 15 q^{83} + q^{85} - 23 q^{87} - q^{89} - 6 q^{91} + 8 q^{93} + 12 q^{95} - 16 q^{97} - 4 q^{99}+O(q^{100}) $ 6 * q - 2 * q^3 - 3 * q^5 + 6 * q^7 + 4 * q^9 - 6 * q^11 - 6 * q^13 + 4 * q^15 - q^17 - 12 * q^19 - 2 * q^21 + 5 * q^23 - 3 * q^25 - 8 * q^27 - 14 * q^29 - 4 * q^31 + 2 * q^33 - 3 * q^35 - 3 * q^37 + 2 * q^39 + 6 * q^41 - 14 * q^43 - 20 * q^45 + 2 * q^47 + 6 * q^49 - 5 * q^51 - 3 * q^53 + 3 * q^55 - 22 * q^57 + 2 * q^59 - 26 * q^61 + 4 * q^63 + 3 * q^65 + 9 * q^67 - 11 * q^69 + 3 * q^71 - 7 * q^73 - 6 * q^75 - 6 * q^77 + 6 * q^81 - 15 * q^83 + q^85 - 23 * q^87 - q^89 - 6 * q^91 + 8 * q^93 + 12 * q^95 - 16 * q^97 - 4 * 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
$3$	0.820857	0.473922	0.236961	−	0.971519i	$-0.423849\pi$
$3$	0.820857	0.473922	0.236961	+	0.971519i	$0.423849\pi$
$4$	0	0
$5$	0.0215726	0.00964758	0.00482379	−	0.999988i	$-0.498465\pi$
$5$	0.0215726	0.00964758	0.00482379	+	0.999988i	$0.498465\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−2.32619	−0.775398
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0.0177081	0.00457220
$16$	0	0
$17$	0.645197	0.156483	0.0782417	−	0.996934i	$-0.475069\pi$
$17$	0.645197	0.156483	0.0782417	+	0.996934i	$0.475069\pi$
$18$	0	0
$19$	2.66244	0.610806	0.305403	−	0.952223i	$-0.401209\pi$
$19$	2.66244	0.610806	0.305403	+	0.952223i	$0.401209\pi$
$20$	0	0
$21$	0.820857	0.179126
$22$	0	0
$23$	0.816993	0.170355	0.0851774	−	0.996366i	$-0.472854\pi$
$23$	0.816993	0.170355	0.0851774	+	0.996366i	$0.472854\pi$
$24$	0	0
$25$	−4.99953	−0.999907
$26$	0	0
$27$	−4.37205	−0.841400
$28$	0	0
$29$	−2.98229	−0.553798	−0.276899	−	0.960899i	$-0.589307\pi$
$29$	−2.98229	−0.553798	−0.276899	+	0.960899i	$0.589307\pi$
$30$	0	0
$31$	−2.72028	−0.488576	−0.244288	−	0.969703i	$-0.578554\pi$
$31$	−2.72028	−0.488576	−0.244288	+	0.969703i	$0.578554\pi$
$32$	0	0
$33$	−0.820857	−0.142893
$34$	0	0
$35$	0.0215726	0.00364644
$36$	0	0
$37$	−3.34011	−0.549111	−0.274555	−	0.961571i	$-0.588531\pi$
$37$	−3.34011	−0.549111	−0.274555	+	0.961571i	$0.588531\pi$
$38$	0	0
$39$	−0.820857	−0.131442
$40$	0	0
$41$	3.93482	0.614515	0.307258	−	0.951626i	$-0.400589\pi$
$41$	3.93482	0.614515	0.307258	+	0.951626i	$0.400589\pi$
$42$	0	0
$43$	−4.28305	−0.653159	−0.326579	−	0.945170i	$-0.605896\pi$
$43$	−4.28305	−0.653159	−0.326579	+	0.945170i	$0.605896\pi$
$44$	0	0
$45$	−0.0501821	−0.00748071
$46$	0	0
$47$	−7.90272	−1.15273	−0.576365	−	0.817192i	$-0.695530\pi$
$47$	−7.90272	−1.15273	−0.576365	+	0.817192i	$0.695530\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0.529615	0.0741610
$52$	0	0
$53$	7.61883	1.04653	0.523263	−	0.852171i	$-0.324714\pi$
$53$	7.61883	1.04653	0.523263	+	0.852171i	$0.324714\pi$
$54$	0	0
$55$	−0.0215726	−0.00290885
$56$	0	0
$57$	2.18548	0.289474
$58$	0	0
$59$	−8.01678	−1.04370	−0.521848	−	0.853039i	$-0.674757\pi$
$59$	−8.01678	−1.04370	−0.521848	+	0.853039i	$0.674757\pi$
$60$	0	0
$61$	7.80516	0.999348	0.499674	−	0.866213i	$-0.333453\pi$
$61$	7.80516	0.999348	0.499674	+	0.866213i	$0.333453\pi$
$62$	0	0
$63$	−2.32619	−0.293073
$64$	0	0
$65$	−0.0215726	−0.00267576
$66$	0	0
$67$	−2.10012	−0.256570	−0.128285	−	0.991737i	$-0.540947\pi$
$67$	−2.10012	−0.256570	−0.128285	+	0.991737i	$0.540947\pi$
$68$	0	0
$69$	0.670635	0.0807349
$70$	0	0
$71$	−10.1318	−1.20243	−0.601213	−	0.799089i	$-0.705316\pi$
$71$	−10.1318	−1.20243	−0.601213	+	0.799089i	$0.705316\pi$
$72$	0	0
$73$	−0.993429	−0.116272	−0.0581360	−	0.998309i	$-0.518516\pi$
$73$	−0.993429	−0.116272	−0.0581360	+	0.998309i	$0.518516\pi$
$74$	0	0
$75$	−4.10391	−0.473878
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	−7.44053	−0.837126	−0.418563	−	0.908188i	$-0.637466\pi$
$79$	−7.44053	−0.837126	−0.418563	+	0.908188i	$0.637466\pi$
$80$	0	0
$81$	3.38975	0.376639
$82$	0	0
$83$	11.6149	1.27490	0.637449	−	0.770492i	$-0.279989\pi$
$83$	11.6149	1.27490	0.637449	+	0.770492i	$0.279989\pi$
$84$	0	0
$85$	0.0139186	0.00150969
$86$	0	0
$87$	−2.44804	−0.262457
$88$	0	0
$89$	−4.20682	−0.445922	−0.222961	−	0.974827i	$-0.571572\pi$
$89$	−4.20682	−0.445922	−0.222961	+	0.974827i	$0.571572\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	−2.23296	−0.231547
$94$	0	0
$95$	0.0574359	0.00589280
$96$	0	0
$97$	−11.5872	−1.17650	−0.588251	−	0.808678i	$-0.700183\pi$
$97$	−11.5872	−1.17650	−0.588251	+	0.808678i	$0.700183\pi$
$98$	0	0
$99$	2.32619	0.233791
$100$	0	0
$101$	−12.4432	−1.23814	−0.619071	−	0.785335i	$-0.712491\pi$
$101$	−12.4432	−1.23814	−0.619071	+	0.785335i	$0.712491\pi$
$102$	0	0
$103$	6.04539	0.595670	0.297835	−	0.954617i	$-0.403736\pi$
$103$	6.04539	0.595670	0.297835	+	0.954617i	$0.403736\pi$
$104$	0	0
$105$	0.0177081	0.00172813
$106$	0	0
$107$	−15.5909	−1.50723	−0.753615	−	0.657316i	$-0.771692\pi$
$107$	−15.5909	−1.50723	−0.753615	+	0.657316i	$0.771692\pi$
$108$	0	0
$109$	−6.64744	−0.636709	−0.318355	−	0.947972i	$-0.603130\pi$
$109$	−6.64744	−0.636709	−0.318355	+	0.947972i	$0.603130\pi$
$110$	0	0
$111$	−2.74176	−0.260236
$112$	0	0
$113$	−11.3465	−1.06738	−0.533692	−	0.845679i	$-0.679196\pi$
$113$	−11.3465	−1.06738	−0.533692	+	0.845679i	$0.679196\pi$
$114$	0	0
$115$	0.0176247	0.00164351
$116$	0	0
$117$	2.32619	0.215057
$118$	0	0
$119$	0.645197	0.0591452
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	3.22992	0.291232
$124$	0	0
$125$	−0.215716	−0.0192943
$126$	0	0
$127$	−6.37885	−0.566032	−0.283016	−	0.959115i	$-0.591335\pi$
$127$	−6.37885	−0.566032	−0.283016	+	0.959115i	$0.591335\pi$
$128$	0	0
$129$	−3.51577	−0.309546
$130$	0	0
$131$	10.7521	0.939411	0.469706	−	0.882823i	$-0.344360\pi$
$131$	10.7521	0.939411	0.469706	+	0.882823i	$0.344360\pi$
$132$	0	0
$133$	2.66244	0.230863
$134$	0	0
$135$	−0.0943166	−0.00811748
$136$	0	0
$137$	−0.0284544	−0.00243102	−0.00121551	−	0.999999i	$-0.500387\pi$
$137$	−0.0284544	−0.00243102	−0.00121551	+	0.999999i	$0.500387\pi$
$138$	0	0
$139$	−10.6978	−0.907379	−0.453689	−	0.891160i	$-0.649892\pi$
$139$	−10.6978	−0.907379	−0.453689	+	0.891160i	$0.649892\pi$
$140$	0	0
$141$	−6.48701	−0.546305
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	−0.0643359	−0.00534281
$146$	0	0
$147$	0.820857	0.0677032
$148$	0	0
$149$	−13.7261	−1.12448	−0.562241	−	0.826973i	$-0.690061\pi$
$149$	−13.7261	−1.12448	−0.562241	+	0.826973i	$0.690061\pi$
$150$	0	0
$151$	−8.70079	−0.708060	−0.354030	−	0.935234i	$-0.615189\pi$
$151$	−8.70079	−0.708060	−0.354030	+	0.935234i	$0.615189\pi$
$152$	0	0
$153$	−1.50085	−0.121337
$154$	0	0
$155$	−0.0586835	−0.00471358
$156$	0	0
$157$	3.41108	0.272234	0.136117	−	0.990693i	$-0.456538\pi$
$157$	3.41108	0.272234	0.136117	+	0.990693i	$0.456538\pi$
$158$	0	0
$159$	6.25397	0.495972
$160$	0	0
$161$	0.816993	0.0643881
$162$	0	0
$163$	−4.07515	−0.319191	−0.159595	−	0.987183i	$-0.551019\pi$
$163$	−4.07515	−0.319191	−0.159595	+	0.987183i	$0.551019\pi$
$164$	0	0
$165$	−0.0177081	−0.00137857
$166$	0	0
$167$	20.9512	1.62125	0.810626	−	0.585565i	$-0.199127\pi$
$167$	20.9512	1.62125	0.810626	+	0.585565i	$0.199127\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−6.19335	−0.473617
$172$	0	0
$173$	−13.0407	−0.991469	−0.495734	−	0.868474i	$-0.665101\pi$
$173$	−13.0407	−0.991469	−0.495734	+	0.868474i	$0.665101\pi$
$174$	0	0
$175$	−4.99953	−0.377929
$176$	0	0
$177$	−6.58063	−0.494631
$178$	0	0
$179$	22.8547	1.70824	0.854121	−	0.520074i	$-0.174096\pi$
$179$	22.8547	1.70824	0.854121	+	0.520074i	$0.174096\pi$
$180$	0	0
$181$	−6.37901	−0.474148	−0.237074	−	0.971492i	$-0.576188\pi$
$181$	−6.37901	−0.474148	−0.237074	+	0.971492i	$0.576188\pi$
$182$	0	0
$183$	6.40692	0.473613
$184$	0	0
$185$	−0.0720550	−0.00529759
$186$	0	0
$187$	−0.645197	−0.0471815
$188$	0	0
$189$	−4.37205	−0.318019
$190$	0	0
$191$	24.5396	1.77562	0.887811	−	0.460209i	$-0.152226\pi$
$191$	24.5396	1.77562	0.887811	+	0.460209i	$0.152226\pi$
$192$	0	0
$193$	17.7800	1.27983	0.639914	−	0.768446i	$-0.278970\pi$
$193$	17.7800	1.27983	0.639914	+	0.768446i	$0.278970\pi$
$194$	0	0
$195$	−0.0177081	−0.00126810
$196$	0	0
$197$	12.7055	0.905230	0.452615	−	0.891706i	$-0.350491\pi$
$197$	12.7055	0.905230	0.452615	+	0.891706i	$0.350491\pi$
$198$	0	0
$199$	17.2777	1.22479	0.612393	−	0.790554i	$-0.290207\pi$
$199$	17.2777	1.22479	0.612393	+	0.790554i	$0.290207\pi$
$200$	0	0
$201$	−1.72390	−0.121594
$202$	0	0
$203$	−2.98229	−0.209316
$204$	0	0
$205$	0.0848844	0.00592858
$206$	0	0
$207$	−1.90048	−0.132093
$208$	0	0
$209$	−2.66244	−0.184165
$210$	0	0
$211$	−23.5859	−1.62372	−0.811860	−	0.583852i	$-0.801545\pi$
$211$	−23.5859	−1.62372	−0.811860	+	0.583852i	$0.801545\pi$
$212$	0	0
$213$	−8.31678	−0.569856
$214$	0	0
$215$	−0.0923967	−0.00630140
$216$	0	0
$217$	−2.72028	−0.184664
$218$	0	0
$219$	−0.815463	−0.0551039
$220$	0	0
$221$	−0.645197	−0.0434007
$222$	0	0
$223$	−28.0114	−1.87578	−0.937889	−	0.346934i	$-0.887223\pi$
$223$	−28.0114	−1.87578	−0.937889	+	0.346934i	$0.887223\pi$
$224$	0	0
$225$	11.6299	0.775326
$226$	0	0
$227$	−3.35587	−0.222737	−0.111368	−	0.993779i	$-0.535523\pi$
$227$	−3.35587	−0.222737	−0.111368	+	0.993779i	$0.535523\pi$
$228$	0	0
$229$	−10.9705	−0.724954	−0.362477	−	0.931993i	$-0.618069\pi$
$229$	−10.9705	−0.724954	−0.362477	+	0.931993i	$0.618069\pi$
$230$	0	0
$231$	−0.820857	−0.0540085
$232$	0	0
$233$	2.32843	0.152541	0.0762704	−	0.997087i	$-0.475699\pi$
$233$	2.32843	0.152541	0.0762704	+	0.997087i	$0.475699\pi$
$234$	0	0
$235$	−0.170483	−0.0111211
$236$	0	0
$237$	−6.10762	−0.396732
$238$	0	0
$239$	−0.882251	−0.0570681	−0.0285340	−	0.999593i	$-0.509084\pi$
$239$	−0.882251	−0.0570681	−0.0285340	+	0.999593i	$0.509084\pi$
$240$	0	0
$241$	18.3280	1.18061	0.590306	−	0.807180i	$-0.299007\pi$
$241$	18.3280	1.18061	0.590306	+	0.807180i	$0.299007\pi$
$242$	0	0
$243$	15.8986	1.01990
$244$	0	0
$245$	0.0215726	0.00137823
$246$	0	0
$247$	−2.66244	−0.169407
$248$	0	0
$249$	9.53416	0.604203
$250$	0	0
$251$	−5.00641	−0.316002	−0.158001	−	0.987439i	$-0.550505\pi$
$251$	−5.00641	−0.316002	−0.158001	+	0.987439i	$0.550505\pi$
$252$	0	0
$253$	−0.816993	−0.0513639
$254$	0	0
$255$	0.0114252	0.000715474 0
$256$	0	0
$257$	2.69870	0.168341	0.0841703	−	0.996451i	$-0.473176\pi$
$257$	2.69870	0.168341	0.0841703	+	0.996451i	$0.473176\pi$
$258$	0	0
$259$	−3.34011	−0.207544
$260$	0	0
$261$	6.93739	0.429413
$262$	0	0
$263$	−7.53000	−0.464320	−0.232160	−	0.972678i	$-0.574579\pi$
$263$	−7.53000	−0.464320	−0.232160	+	0.972678i	$0.574579\pi$
$264$	0	0
$265$	0.164358	0.0100964
$266$	0	0
$267$	−3.45320	−0.211332
$268$	0	0
$269$	13.9094	0.848068	0.424034	−	0.905646i	$-0.360614\pi$
$269$	13.9094	0.848068	0.424034	+	0.905646i	$0.360614\pi$
$270$	0	0
$271$	−13.6792	−0.830953	−0.415477	−	0.909604i	$-0.636385\pi$
$271$	−13.6792	−0.830953	−0.415477	+	0.909604i	$0.636385\pi$
$272$	0	0
$273$	−0.820857	−0.0496806
$274$	0	0
$275$	4.99953	0.301483
$276$	0	0
$277$	13.3927	0.804692	0.402346	−	0.915488i	$-0.368195\pi$
$277$	13.3927	0.804692	0.402346	+	0.915488i	$0.368195\pi$
$278$	0	0
$279$	6.32789	0.378841
$280$	0	0
$281$	−16.8126	−1.00295	−0.501477	−	0.865171i	$-0.667210\pi$
$281$	−16.8126	−1.00295	−0.501477	+	0.865171i	$0.667210\pi$
$282$	0	0
$283$	3.53664	0.210231	0.105116	−	0.994460i	$-0.466479\pi$
$283$	3.53664	0.210231	0.105116	+	0.994460i	$0.466479\pi$
$284$	0	0
$285$	0.0471467	0.00279273
$286$	0	0
$287$	3.93482	0.232265
$288$	0	0
$289$	−16.5837	−0.975513
$290$	0	0
$291$	−9.51144	−0.557571
$292$	0	0
$293$	−6.67382	−0.389889	−0.194944	−	0.980814i	$-0.562453\pi$
$293$	−6.67382	−0.389889	−0.194944	+	0.980814i	$0.562453\pi$
$294$	0	0
$295$	−0.172943	−0.0100691
$296$	0	0
$297$	4.37205	0.253692
$298$	0	0
$299$	−0.816993	−0.0472479
$300$	0	0
$301$	−4.28305	−0.246871
$302$	0	0
$303$	−10.2141	−0.586783
$304$	0	0
$305$	0.168378	0.00964129
$306$	0	0
$307$	20.8927	1.19241	0.596204	−	0.802833i	$-0.296675\pi$
$307$	20.8927	1.19241	0.596204	+	0.802833i	$0.296675\pi$
$308$	0	0
$309$	4.96240	0.282301
$310$	0	0
$311$	−19.9530	−1.13143	−0.565715	−	0.824601i	$-0.691400\pi$
$311$	−19.9530	−1.13143	−0.565715	+	0.824601i	$0.691400\pi$
$312$	0	0
$313$	−14.3525	−0.811254	−0.405627	−	0.914039i	$-0.632947\pi$
$313$	−14.3525	−0.811254	−0.405627	+	0.914039i	$0.632947\pi$
$314$	0	0
$315$	−0.0501821	−0.00282744
$316$	0	0
$317$	−6.57787	−0.369450	−0.184725	−	0.982790i	$-0.559139\pi$
$317$	−6.57787	−0.369450	−0.184725	+	0.982790i	$0.559139\pi$
$318$	0	0
$319$	2.98229	0.166976
$320$	0	0
$321$	−12.7979	−0.714310
$322$	0	0
$323$	1.71780	0.0955809
$324$	0	0
$325$	4.99953	0.277324
$326$	0	0
$327$	−5.45660	−0.301751
$328$	0	0
$329$	−7.90272	−0.435691
$330$	0	0
$331$	7.12108	0.391410	0.195705	−	0.980663i	$-0.437300\pi$
$331$	7.12108	0.391410	0.195705	+	0.980663i	$0.437300\pi$
$332$	0	0
$333$	7.76974	0.425779
$334$	0	0
$335$	−0.0453050	−0.00247528
$336$	0	0
$337$	−34.7433	−1.89259	−0.946294	−	0.323308i	$-0.895205\pi$
$337$	−34.7433	−1.89259	−0.946294	+	0.323308i	$0.895205\pi$
$338$	0	0
$339$	−9.31382	−0.505857
$340$	0	0
$341$	2.72028	0.147311
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0.0144674	0.000778897 0
$346$	0	0
$347$	−0.630125	−0.0338269	−0.0169134	−	0.999857i	$-0.505384\pi$
$347$	−0.630125	−0.0338269	−0.0169134	+	0.999857i	$0.505384\pi$
$348$	0	0
$349$	−16.7994	−0.899249	−0.449625	−	0.893218i	$-0.648442\pi$
$349$	−16.7994	−0.899249	−0.449625	+	0.893218i	$0.648442\pi$
$350$	0	0
$351$	4.37205	0.233363
$352$	0	0
$353$	−19.0547	−1.01418	−0.507088	−	0.861894i	$-0.669278\pi$
$353$	−19.0547	−1.01418	−0.507088	+	0.861894i	$0.669278\pi$
$354$	0	0
$355$	−0.218570	−0.0116005
$356$	0	0
$357$	0.529615	0.0280302
$358$	0	0
$359$	−2.55213	−0.134696	−0.0673481	−	0.997730i	$-0.521454\pi$
$359$	−2.55213	−0.134696	−0.0673481	+	0.997730i	$0.521454\pi$
$360$	0	0
$361$	−11.9114	−0.626916
$362$	0	0
$363$	0.820857	0.0430838
$364$	0	0
$365$	−0.0214309	−0.00112174
$366$	0	0
$367$	−13.9572	−0.728558	−0.364279	−	0.931290i	$-0.618685\pi$
$367$	−13.9572	−0.728558	−0.364279	+	0.931290i	$0.618685\pi$
$368$	0	0
$369$	−9.15314	−0.476494
$370$	0	0
$371$	7.61883	0.395550
$372$	0	0
$373$	−25.8896	−1.34051	−0.670256	−	0.742130i	$-0.733816\pi$
$373$	−25.8896	−1.34051	−0.670256	+	0.742130i	$0.733816\pi$
$374$	0	0
$375$	−0.177072	−0.00914398
$376$	0	0
$377$	2.98229	0.153596
$378$	0	0
$379$	14.9202	0.766398	0.383199	−	0.923666i	$-0.374822\pi$
$379$	14.9202	0.766398	0.383199	+	0.923666i	$0.374822\pi$
$380$	0	0
$381$	−5.23613	−0.268255
$382$	0	0
$383$	33.0594	1.68926	0.844629	−	0.535352i	$-0.179821\pi$
$383$	33.0594	1.68926	0.844629	+	0.535352i	$0.179821\pi$
$384$	0	0
$385$	−0.0215726	−0.00109944
$386$	0	0
$387$	9.96320	0.506458
$388$	0	0
$389$	33.7504	1.71121	0.855606	−	0.517628i	$-0.173185\pi$
$389$	33.7504	1.71121	0.855606	+	0.517628i	$0.173185\pi$
$390$	0	0
$391$	0.527122	0.0266577
$392$	0	0
$393$	8.82590	0.445208
$394$	0	0
$395$	−0.160512	−0.00807623
$396$	0	0
$397$	−15.3011	−0.767942	−0.383971	−	0.923345i	$-0.625444\pi$
$397$	−15.3011	−0.767942	−0.383971	+	0.923345i	$0.625444\pi$
$398$	0	0
$399$	2.18548	0.109411
$400$	0	0
$401$	32.4884	1.62240	0.811198	−	0.584772i	$-0.198816\pi$
$401$	32.4884	1.62240	0.811198	+	0.584772i	$0.198816\pi$
$402$	0	0
$403$	2.72028	0.135507
$404$	0	0
$405$	0.0731259	0.00363366
$406$	0	0
$407$	3.34011	0.165563
$408$	0	0
$409$	4.59934	0.227423	0.113711	−	0.993514i	$-0.463726\pi$
$409$	4.59934	0.227423	0.113711	+	0.993514i	$0.463726\pi$
$410$	0	0
$411$	−0.0233570	−0.00115212
$412$	0	0
$413$	−8.01678	−0.394480
$414$	0	0
$415$	0.250564	0.0122997
$416$	0	0
$417$	−8.78140	−0.430027
$418$	0	0
$419$	−12.1421	−0.593180	−0.296590	−	0.955005i	$-0.595849\pi$
$419$	−12.1421	−0.593180	−0.296590	+	0.955005i	$0.595849\pi$
$420$	0	0
$421$	1.40042	0.0682525	0.0341263	−	0.999418i	$-0.489135\pi$
$421$	1.40042	0.0682525	0.0341263	+	0.999418i	$0.489135\pi$
$422$	0	0
$423$	18.3833	0.893825
$424$	0	0
$425$	−3.22569	−0.156469
$426$	0	0
$427$	7.80516	0.377718
$428$	0	0
$429$	0.820857	0.0396314
$430$	0	0
$431$	31.7948	1.53150	0.765751	−	0.643138i	$-0.222368\pi$
$431$	31.7948	1.53150	0.765751	+	0.643138i	$0.222368\pi$
$432$	0	0
$433$	31.3293	1.50559	0.752795	−	0.658255i	$-0.228705\pi$
$433$	31.3293	1.50559	0.752795	+	0.658255i	$0.228705\pi$
$434$	0	0
$435$	−0.0528106	−0.00253208
$436$	0	0
$437$	2.17519	0.104054
$438$	0	0
$439$	12.1209	0.578500	0.289250	−	0.957254i	$-0.406594\pi$
$439$	12.1209	0.578500	0.289250	+	0.957254i	$0.406594\pi$
$440$	0	0
$441$	−2.32619	−0.110771
$442$	0	0
$443$	−9.39855	−0.446539	−0.223269	−	0.974757i	$-0.571673\pi$
$443$	−9.39855	−0.446539	−0.223269	+	0.974757i	$0.571673\pi$
$444$	0	0
$445$	−0.0907522	−0.00430207
$446$	0	0
$447$	−11.2671	−0.532917
$448$	0	0
$449$	−20.3794	−0.961762	−0.480881	−	0.876786i	$-0.659683\pi$
$449$	−20.3794	−0.961762	−0.480881	+	0.876786i	$0.659683\pi$
$450$	0	0
$451$	−3.93482	−0.185283
$452$	0	0
$453$	−7.14211	−0.335566
$454$	0	0
$455$	−0.0215726	−0.00101134
$456$	0	0
$457$	24.6565	1.15338	0.576691	−	0.816963i	$-0.304344\pi$
$457$	24.6565	1.15338	0.576691	+	0.816963i	$0.304344\pi$
$458$	0	0
$459$	−2.82083	−0.131665
$460$	0	0
$461$	−2.47858	−0.115439	−0.0577195	−	0.998333i	$-0.518383\pi$
$461$	−2.47858	−0.115439	−0.0577195	+	0.998333i	$0.518383\pi$
$462$	0	0
$463$	−9.10213	−0.423012	−0.211506	−	0.977377i	$-0.567837\pi$
$463$	−9.10213	−0.423012	−0.211506	+	0.977377i	$0.567837\pi$
$464$	0	0
$465$	−0.0481708	−0.00223387
$466$	0	0
$467$	−29.5856	−1.36906	−0.684529	−	0.728986i	$-0.739992\pi$
$467$	−29.5856	−1.36906	−0.684529	+	0.728986i	$0.739992\pi$
$468$	0	0
$469$	−2.10012	−0.0969743
$470$	0	0
$471$	2.80001	0.129018
$472$	0	0
$473$	4.28305	0.196935
$474$	0	0
$475$	−13.3110	−0.610749
$476$	0	0
$477$	−17.7229	−0.811474
$478$	0	0
$479$	26.1391	1.19433	0.597164	−	0.802119i	$-0.296294\pi$
$479$	26.1391	1.19433	0.597164	+	0.802119i	$0.296294\pi$
$480$	0	0
$481$	3.34011	0.152296
$482$	0	0
$483$	0.670635	0.0305149
$484$	0	0
$485$	−0.249967	−0.0113504
$486$	0	0
$487$	27.1510	1.23033	0.615165	−	0.788399i	$-0.289089\pi$
$487$	27.1510	1.23033	0.615165	+	0.788399i	$0.289089\pi$
$488$	0	0
$489$	−3.34512	−0.151272
$490$	0	0
$491$	10.6721	0.481625	0.240812	−	0.970572i	$-0.422586\pi$
$491$	10.6721	0.481625	0.240812	+	0.970572i	$0.422586\pi$
$492$	0	0
$493$	−1.92417	−0.0866601
$494$	0	0
$495$	0.0501821	0.00225552
$496$	0	0
$497$	−10.1318	−0.454474
$498$	0	0
$499$	41.2614	1.84711	0.923556	−	0.383463i	$-0.125269\pi$
$499$	41.2614	1.84711	0.923556	+	0.383463i	$0.125269\pi$
$500$	0	0
$501$	17.1979	0.768347
$502$	0	0
$503$	−12.8541	−0.573137	−0.286568	−	0.958060i	$-0.592515\pi$
$503$	−12.8541	−0.573137	−0.286568	+	0.958060i	$0.592515\pi$
$504$	0	0
$505$	−0.268432	−0.0119451
$506$	0	0
$507$	0.820857	0.0364556
$508$	0	0
$509$	40.8706	1.81156	0.905780	−	0.423749i	$-0.139286\pi$
$509$	40.8706	1.81156	0.905780	+	0.423749i	$0.139286\pi$
$510$	0	0
$511$	−0.993429	−0.0439467
$512$	0	0
$513$	−11.6403	−0.513932
$514$	0	0
$515$	0.130415	0.00574677
$516$	0	0
$517$	7.90272	0.347561
$518$	0	0
$519$	−10.7046	−0.469879
$520$	0	0
$521$	−9.19615	−0.402891	−0.201445	−	0.979500i	$-0.564564\pi$
$521$	−9.19615	−0.402891	−0.201445	+	0.979500i	$0.564564\pi$
$522$	0	0
$523$	4.74818	0.207624	0.103812	−	0.994597i	$-0.466896\pi$
$523$	4.74818	0.207624	0.103812	+	0.994597i	$0.466896\pi$
$524$	0	0
$525$	−4.10391	−0.179109
$526$	0	0
$527$	−1.75512	−0.0764540
$528$	0	0
$529$	−22.3325	−0.970979
$530$	0	0
$531$	18.6486	0.809279
$532$	0	0
$533$	−3.93482	−0.170436
$534$	0	0
$535$	−0.336337	−0.0145411
$536$	0	0
$537$	18.7605	0.809574
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−39.3236	−1.69065	−0.845326	−	0.534251i	$-0.820594\pi$
$541$	−39.3236	−1.69065	−0.845326	+	0.534251i	$0.820594\pi$
$542$	0	0
$543$	−5.23626	−0.224709
$544$	0	0
$545$	−0.143403	−0.00614270
$546$	0	0
$547$	−11.2109	−0.479344	−0.239672	−	0.970854i	$-0.577040\pi$
$547$	−11.2109	−0.479344	−0.239672	+	0.970854i	$0.577040\pi$
$548$	0	0
$549$	−18.1563	−0.774892
$550$	0	0
$551$	−7.94017	−0.338263
$552$	0	0
$553$	−7.44053	−0.316404
$554$	0	0
$555$	−0.0591469	−0.00251065
$556$	0	0
$557$	−16.8196	−0.712670	−0.356335	−	0.934358i	$-0.615974\pi$
$557$	−16.8196	−0.712670	−0.356335	+	0.934358i	$0.615974\pi$
$558$	0	0
$559$	4.28305	0.181154
$560$	0	0
$561$	−0.529615	−0.0223604
$562$	0	0
$563$	−4.45420	−0.187722	−0.0938612	−	0.995585i	$-0.529921\pi$
$563$	−4.45420	−0.187722	−0.0938612	+	0.995585i	$0.529921\pi$
$564$	0	0
$565$	−0.244773	−0.0102977
$566$	0	0
$567$	3.38975	0.142356
$568$	0	0
$569$	8.26415	0.346451	0.173226	−	0.984882i	$-0.444581\pi$
$569$	8.26415	0.346451	0.173226	+	0.984882i	$0.444581\pi$
$570$	0	0
$571$	13.1936	0.552133	0.276067	−	0.961138i	$-0.410969\pi$
$571$	13.1936	0.552133	0.276067	+	0.961138i	$0.410969\pi$
$572$	0	0
$573$	20.1435	0.841506
$574$	0	0
$575$	−4.08458	−0.170339
$576$	0	0
$577$	−23.0206	−0.958358	−0.479179	−	0.877717i	$-0.659066\pi$
$577$	−23.0206	−0.958358	−0.479179	+	0.877717i	$0.659066\pi$
$578$	0	0
$579$	14.5948	0.606539
$580$	0	0
$581$	11.6149	0.481866
$582$	0	0
$583$	−7.61883	−0.315540
$584$	0	0
$585$	0.0501821	0.00207478
$586$	0	0
$587$	18.9507	0.782177	0.391089	−	0.920353i	$-0.372099\pi$
$587$	18.9507	0.782177	0.391089	+	0.920353i	$0.372099\pi$
$588$	0	0
$589$	−7.24257	−0.298425
$590$	0	0
$591$	10.4294	0.429008
$592$	0	0
$593$	5.52708	0.226970	0.113485	−	0.993540i	$-0.463799\pi$
$593$	5.52708	0.226970	0.113485	+	0.993540i	$0.463799\pi$
$594$	0	0
$595$	0.0139186	0.000570608 0
$596$	0	0
$597$	14.1825	0.580453
$598$	0	0
$599$	23.2467	0.949835	0.474918	−	0.880030i	$-0.342478\pi$
$599$	23.2467	0.949835	0.474918	+	0.880030i	$0.342478\pi$
$600$	0	0
$601$	−14.2865	−0.582757	−0.291378	−	0.956608i	$-0.594114\pi$
$601$	−14.2865	−0.582757	−0.291378	+	0.956608i	$0.594114\pi$
$602$	0	0
$603$	4.88527	0.198944
$604$	0	0
$605$	0.0215726	0.000877053 0
$606$	0	0
$607$	8.09285	0.328479	0.164239	−	0.986421i	$-0.447483\pi$
$607$	8.09285	0.328479	0.164239	+	0.986421i	$0.447483\pi$
$608$	0	0
$609$	−2.44804	−0.0991994
$610$	0	0
$611$	7.90272	0.319710
$612$	0	0
$613$	14.9787	0.604985	0.302492	−	0.953152i	$-0.402181\pi$
$613$	14.9787	0.604985	0.302492	+	0.953152i	$0.402181\pi$
$614$	0	0
$615$	0.0696780	0.00280969
$616$	0	0
$617$	19.9617	0.803626	0.401813	−	0.915722i	$-0.368380\pi$
$617$	19.9617	0.803626	0.401813	+	0.915722i	$0.368380\pi$
$618$	0	0
$619$	−44.8442	−1.80244	−0.901221	−	0.433360i	$-0.857328\pi$
$619$	−44.8442	−1.80244	−0.901221	+	0.433360i	$0.857328\pi$
$620$	0	0
$621$	−3.57193	−0.143337
$622$	0	0
$623$	−4.20682	−0.168543
$624$	0	0
$625$	24.9930	0.999721
$626$	0	0
$627$	−2.18548	−0.0872798
$628$	0	0
$629$	−2.15503	−0.0859267
$630$	0	0
$631$	37.8045	1.50497	0.752487	−	0.658607i	$-0.228854\pi$
$631$	37.8045	1.50497	0.752487	+	0.658607i	$0.228854\pi$
$632$	0	0
$633$	−19.3607	−0.769517
$634$	0	0
$635$	−0.137609	−0.00546083
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	23.5686	0.932358
$640$	0	0
$641$	−19.0388	−0.751987	−0.375994	−	0.926622i	$-0.622699\pi$
$641$	−19.0388	−0.751987	−0.375994	+	0.926622i	$0.622699\pi$
$642$	0	0
$643$	39.0027	1.53812	0.769058	−	0.639179i	$-0.220726\pi$
$643$	39.0027	1.53812	0.769058	+	0.639179i	$0.220726\pi$
$644$	0	0
$645$	−0.0758445	−0.00298637
$646$	0	0
$647$	−41.1462	−1.61762	−0.808812	−	0.588067i	$-0.799889\pi$
$647$	−41.1462	−1.61762	−0.808812	+	0.588067i	$0.799889\pi$
$648$	0	0
$649$	8.01678	0.314686
$650$	0	0
$651$	−2.23296	−0.0875166
$652$	0	0
$653$	−28.5409	−1.11689	−0.558446	−	0.829541i	$-0.688602\pi$
$653$	−28.5409	−1.11689	−0.558446	+	0.829541i	$0.688602\pi$
$654$	0	0
$655$	0.231950	0.00906304
$656$	0	0
$657$	2.31091	0.0901571
$658$	0	0
$659$	9.81452	0.382319	0.191160	−	0.981559i	$-0.438775\pi$
$659$	9.81452	0.382319	0.191160	+	0.981559i	$0.438775\pi$
$660$	0	0
$661$	33.1173	1.28811	0.644057	−	0.764978i	$-0.277250\pi$
$661$	33.1173	1.28811	0.644057	+	0.764978i	$0.277250\pi$
$662$	0	0
$663$	−0.529615	−0.0205685
$664$	0	0
$665$	0.0574359	0.00222727
$666$	0	0
$667$	−2.43651	−0.0943421
$668$	0	0
$669$	−22.9933	−0.888973
$670$	0	0
$671$	−7.80516	−0.301315
$672$	0	0
$673$	0.315820	0.0121740	0.00608699	−	0.999981i	$-0.498062\pi$
$673$	0.315820	0.0121740	0.00608699	+	0.999981i	$0.498062\pi$
$674$	0	0
$675$	21.8582	0.841322
$676$	0	0
$677$	3.55335	0.136567	0.0682833	−	0.997666i	$-0.478248\pi$
$677$	3.55335	0.136567	0.0682833	+	0.997666i	$0.478248\pi$
$678$	0	0
$679$	−11.5872	−0.444676
$680$	0	0
$681$	−2.75469	−0.105560
$682$	0	0
$683$	28.0851	1.07464	0.537322	−	0.843377i	$-0.319436\pi$
$683$	28.0851	1.07464	0.537322	+	0.843377i	$0.319436\pi$
$684$	0	0
$685$	−0.000613837 0	−2.34535e−5 0
$686$	0	0
$687$	−9.00525	−0.343572
$688$	0	0
$689$	−7.61883	−0.290254
$690$	0	0
$691$	−31.7383	−1.20738	−0.603692	−	0.797218i	$-0.706304\pi$
$691$	−31.7383	−1.20738	−0.603692	+	0.797218i	$0.706304\pi$
$692$	0	0
$693$	2.32619	0.0883648
$694$	0	0
$695$	−0.230781	−0.00875401
$696$	0	0
$697$	2.53873	0.0961614
$698$	0	0
$699$	1.91131	0.0722925
$700$	0	0
$701$	−12.4905	−0.471759	−0.235880	−	0.971782i	$-0.575797\pi$
$701$	−12.4905	−0.471759	−0.235880	+	0.971782i	$0.575797\pi$
$702$	0	0
$703$	−8.89285	−0.335400
$704$	0	0
$705$	−0.139942	−0.00527052
$706$	0	0
$707$	−12.4432	−0.467974
$708$	0	0
$709$	9.59380	0.360303	0.180151	−	0.983639i	$-0.442341\pi$
$709$	9.59380	0.360303	0.180151	+	0.983639i	$0.442341\pi$
$710$	0	0
$711$	17.3081	0.649105
$712$	0	0
$713$	−2.22245	−0.0832313
$714$	0	0
$715$	0.0215726	0.000806771 0
$716$	0	0
$717$	−0.724203	−0.0270458
$718$	0	0
$719$	45.1738	1.68470	0.842349	−	0.538932i	$-0.181172\pi$
$719$	45.1738	1.68470	0.842349	+	0.538932i	$0.181172\pi$
$720$	0	0
$721$	6.04539	0.225142
$722$	0	0
$723$	15.0447	0.559518
$724$	0	0
$725$	14.9101	0.553746
$726$	0	0
$727$	5.63694	0.209063	0.104531	−	0.994522i	$-0.466666\pi$
$727$	5.63694	0.209063	0.104531	+	0.994522i	$0.466666\pi$
$728$	0	0
$729$	2.88126	0.106713
$730$	0	0
$731$	−2.76341	−0.102208
$732$	0	0
$733$	31.7510	1.17275	0.586376	−	0.810039i	$-0.300554\pi$
$733$	31.7510	1.17275	0.586376	+	0.810039i	$0.300554\pi$
$734$	0	0
$735$	0.0177081	0.000653172 0
$736$	0	0
$737$	2.10012	0.0773588
$738$	0	0
$739$	15.3188	0.563513	0.281756	−	0.959486i	$-0.409083\pi$
$739$	15.3188	0.563513	0.281756	+	0.959486i	$0.409083\pi$
$740$	0	0
$741$	−2.18548	−0.0802858
$742$	0	0
$743$	30.5683	1.12144	0.560720	−	0.828005i	$-0.310524\pi$
$743$	30.5683	1.12144	0.560720	+	0.828005i	$0.310524\pi$
$744$	0	0
$745$	−0.296107	−0.0108485
$746$	0	0
$747$	−27.0185	−0.988554
$748$	0	0
$749$	−15.5909	−0.569679
$750$	0	0
$751$	28.1462	1.02707	0.513535	−	0.858069i	$-0.328336\pi$
$751$	28.1462	1.02707	0.513535	+	0.858069i	$0.328336\pi$
$752$	0	0
$753$	−4.10955	−0.149760
$754$	0	0
$755$	−0.187699	−0.00683107
$756$	0	0
$757$	9.56146	0.347517	0.173759	−	0.984788i	$-0.444409\pi$
$757$	9.56146	0.347517	0.173759	+	0.984788i	$0.444409\pi$
$758$	0	0
$759$	−0.670635	−0.0243425
$760$	0	0
$761$	11.3564	0.411668	0.205834	−	0.978587i	$-0.434009\pi$
$761$	11.3564	0.411668	0.205834	+	0.978587i	$0.434009\pi$
$762$	0	0
$763$	−6.64744	−0.240653
$764$	0	0
$765$	−0.0323774	−0.00117061
$766$	0	0
$767$	8.01678	0.289469
$768$	0	0
$769$	−32.0187	−1.15462	−0.577312	−	0.816523i	$-0.695898\pi$
$769$	−32.0187	−1.15462	−0.577312	+	0.816523i	$0.695898\pi$
$770$	0	0
$771$	2.21525	0.0797803
$772$	0	0
$773$	−7.95378	−0.286078	−0.143039	−	0.989717i	$-0.545687\pi$
$773$	−7.95378	−0.286078	−0.143039	+	0.989717i	$0.545687\pi$
$774$	0	0
$775$	13.6001	0.488531
$776$	0	0
$777$	−2.74176	−0.0983599
$778$	0	0
$779$	10.4762	0.375349
$780$	0	0
$781$	10.1318	0.362545
$782$	0	0
$783$	13.0387	0.465966
$784$	0	0
$785$	0.0735859	0.00262640
$786$	0	0
$787$	−34.8755	−1.24318	−0.621589	−	0.783344i	$-0.713513\pi$
$787$	−34.8755	−1.24318	−0.621589	+	0.783344i	$0.713513\pi$
$788$	0	0
$789$	−6.18105	−0.220051
$790$	0	0
$791$	−11.3465	−0.403433
$792$	0	0
$793$	−7.80516	−0.277169
$794$	0	0
$795$	0.134915	0.00478493
$796$	0	0
$797$	−5.44514	−0.192877	−0.0964384	−	0.995339i	$-0.530745\pi$
$797$	−5.44514	−0.192877	−0.0964384	+	0.995339i	$0.530745\pi$
$798$	0	0
$799$	−5.09882	−0.180383
$800$	0	0
$801$	9.78588	0.345767
$802$	0	0
$803$	0.993429	0.0350573
$804$	0	0
$805$	0.0176247	0.000621189 0
$806$	0	0
$807$	11.4176	0.401918
$808$	0	0
$809$	45.3990	1.59614	0.798072	−	0.602562i	$-0.205853\pi$
$809$	45.3990	1.59614	0.798072	+	0.602562i	$0.205853\pi$
$810$	0	0
$811$	−36.7005	−1.28873	−0.644365	−	0.764718i	$-0.722878\pi$
$811$	−36.7005	−1.28873	−0.644365	+	0.764718i	$0.722878\pi$
$812$	0	0
$813$	−11.2287	−0.393807
$814$	0	0
$815$	−0.0879118	−0.00307942
$816$	0	0
$817$	−11.4034	−0.398953
$818$	0	0
$819$	2.32619	0.0812838
$820$	0	0
$821$	−16.1474	−0.563547	−0.281774	−	0.959481i	$-0.590923\pi$
$821$	−16.1474	−0.563547	−0.281774	+	0.959481i	$0.590923\pi$
$822$	0	0
$823$	16.2633	0.566904	0.283452	−	0.958986i	$-0.408520\pi$
$823$	16.2633	0.566904	0.283452	+	0.958986i	$0.408520\pi$
$824$	0	0
$825$	4.10391	0.142880
$826$	0	0
$827$	26.5903	0.924635	0.462317	−	0.886715i	$-0.347018\pi$
$827$	26.5903	0.924635	0.462317	+	0.886715i	$0.347018\pi$
$828$	0	0
$829$	11.3096	0.392797	0.196399	−	0.980524i	$-0.437075\pi$
$829$	11.3096	0.392797	0.196399	+	0.980524i	$0.437075\pi$
$830$	0	0
$831$	10.9935	0.381361
$832$	0	0
$833$	0.645197	0.0223548
$834$	0	0
$835$	0.451972	0.0156412
$836$	0	0
$837$	11.8932	0.411088
$838$	0	0
$839$	19.5770	0.675873	0.337937	−	0.941169i	$-0.390271\pi$
$839$	19.5770	0.675873	0.337937	+	0.941169i	$0.390271\pi$
$840$	0	0
$841$	−20.1059	−0.693308
$842$	0	0
$843$	−13.8007	−0.475322
$844$	0	0
$845$	0.0215726	0.000742121 0
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	2.90308	0.0996333
$850$	0	0
$851$	−2.72885	−0.0935437
$852$	0	0
$853$	−26.6749	−0.913330	−0.456665	−	0.889639i	$-0.650956\pi$
$853$	−26.6749	−0.913330	−0.456665	+	0.889639i	$0.650956\pi$
$854$	0	0
$855$	−0.133607	−0.00456926
$856$	0	0
$857$	7.97688	0.272485	0.136242	−	0.990676i	$-0.456497\pi$
$857$	7.97688	0.272485	0.136242	+	0.990676i	$0.456497\pi$
$858$	0	0
$859$	−18.4442	−0.629308	−0.314654	−	0.949206i	$-0.601888\pi$
$859$	−18.4442	−0.629308	−0.314654	+	0.949206i	$0.601888\pi$
$860$	0	0
$861$	3.22992	0.110076
$862$	0	0
$863$	35.6907	1.21493	0.607463	−	0.794348i	$-0.292187\pi$
$863$	35.6907	1.21493	0.607463	+	0.794348i	$0.292187\pi$
$864$	0	0
$865$	−0.281323	−0.00956527
$866$	0	0
$867$	−13.6129	−0.462317
$868$	0	0
$869$	7.44053	0.252403
$870$	0	0
$871$	2.10012	0.0711597
$872$	0	0
$873$	26.9541	0.912257
$874$	0	0
$875$	−0.215716	−0.00729254
$876$	0	0
$877$	−17.1830	−0.580229	−0.290114	−	0.956992i	$-0.593693\pi$
$877$	−17.1830	−0.580229	−0.290114	+	0.956992i	$0.593693\pi$
$878$	0	0
$879$	−5.47825	−0.184777
$880$	0	0
$881$	−2.22513	−0.0749664	−0.0374832	−	0.999297i	$-0.511934\pi$
$881$	−2.22513	−0.0749664	−0.0374832	+	0.999297i	$0.511934\pi$
$882$	0	0
$883$	−20.0094	−0.673371	−0.336686	−	0.941617i	$-0.609306\pi$
$883$	−20.0094	−0.673371	−0.336686	+	0.941617i	$0.609306\pi$
$884$	0	0
$885$	−0.141962	−0.00477199
$886$	0	0
$887$	−8.52875	−0.286367	−0.143184	−	0.989696i	$-0.545734\pi$
$887$	−8.52875	−0.286367	−0.143184	+	0.989696i	$0.545734\pi$
$888$	0	0
$889$	−6.37885	−0.213940
$890$	0	0
$891$	−3.38975	−0.113561
$892$	0	0
$893$	−21.0405	−0.704095
$894$	0	0
$895$	0.493037	0.0164804
$896$	0	0
$897$	−0.670635	−0.0223918
$898$	0	0
$899$	8.11266	0.270572
$900$	0	0
$901$	4.91565	0.163764
$902$	0	0
$903$	−3.51577	−0.116998
$904$	0	0
$905$	−0.137612	−0.00457438
$906$	0	0
$907$	58.5910	1.94548	0.972742	−	0.231891i	$-0.0744913\pi$
$907$	58.5910	1.94548	0.972742	+	0.231891i	$0.0744913\pi$
$908$	0	0
$909$	28.9452	0.960052
$910$	0	0
$911$	−37.0677	−1.22811	−0.614054	−	0.789264i	$-0.710462\pi$
$911$	−37.0677	−1.22811	−0.614054	+	0.789264i	$0.710462\pi$
$912$	0	0
$913$	−11.6149	−0.384396
$914$	0	0
$915$	0.138214	0.00456922
$916$	0	0
$917$	10.7521	0.355064
$918$	0	0
$919$	20.7879	0.685730	0.342865	−	0.939385i	$-0.388603\pi$
$919$	20.7879	0.685730	0.342865	+	0.939385i	$0.388603\pi$
$920$	0	0
$921$	17.1499	0.565109
$922$	0	0
$923$	10.1318	0.333493
$924$	0	0
$925$	16.6990	0.549060
$926$	0	0
$927$	−14.0627	−0.461881
$928$	0	0
$929$	24.4209	0.801224	0.400612	−	0.916248i	$-0.368798\pi$
$929$	24.4209	0.801224	0.400612	+	0.916248i	$0.368798\pi$
$930$	0	0
$931$	2.66244	0.0872580
$932$	0	0
$933$	−16.3785	−0.536210
$934$	0	0
$935$	−0.0139186	−0.000455187 0
$936$	0	0
$937$	−51.4147	−1.67964	−0.839822	−	0.542861i	$-0.817341\pi$
$937$	−51.4147	−1.67964	−0.839822	+	0.542861i	$0.817341\pi$
$938$	0	0
$939$	−11.7814	−0.384471
$940$	0	0
$941$	47.6793	1.55430	0.777151	−	0.629315i	$-0.216664\pi$
$941$	47.6793	1.55430	0.777151	+	0.629315i	$0.216664\pi$
$942$	0	0
$943$	3.21472	0.104686
$944$	0	0
$945$	−0.0943166	−0.00306812
$946$	0	0
$947$	6.59325	0.214252	0.107126	−	0.994245i	$-0.465835\pi$
$947$	6.59325	0.214252	0.107126	+	0.994245i	$0.465835\pi$
$948$	0	0
$949$	0.993429	0.0322481
$950$	0	0
$951$	−5.39949	−0.175090
$952$	0	0
$953$	−19.9277	−0.645521	−0.322761	−	0.946481i	$-0.604611\pi$
$953$	−19.9277	−0.645521	−0.322761	+	0.946481i	$0.604611\pi$
$954$	0	0
$955$	0.529383	0.0171304
$956$	0	0
$957$	2.44804	0.0791338
$958$	0	0
$959$	−0.0284544	−0.000918841 0
$960$	0	0
$961$	−23.6001	−0.761293
$962$	0	0
$963$	36.2675	1.16870
$964$	0	0
$965$	0.383560	0.0123472
$966$	0	0
$967$	−47.6725	−1.53304	−0.766522	−	0.642218i	$-0.778014\pi$
$967$	−47.6725	−1.53304	−0.766522	+	0.642218i	$0.778014\pi$
$968$	0	0
$969$	1.41007	0.0452979
$970$	0	0
$971$	−18.1601	−0.582784	−0.291392	−	0.956604i	$-0.594118\pi$
$971$	−18.1601	−0.582784	−0.291392	+	0.956604i	$0.594118\pi$
$972$	0	0
$973$	−10.6978	−0.342957
$974$	0	0
$975$	4.10391	0.131430
$976$	0	0
$977$	6.83119	0.218549	0.109274	−	0.994012i	$-0.465147\pi$
$977$	6.83119	0.218549	0.109274	+	0.994012i	$0.465147\pi$
$978$	0	0
$979$	4.20682	0.134451
$980$	0	0
$981$	15.4632	0.493703
$982$	0	0
$983$	25.5051	0.813487	0.406743	−	0.913542i	$-0.366664\pi$
$983$	25.5051	0.813487	0.406743	+	0.913542i	$0.366664\pi$
$984$	0	0
$985$	0.274091	0.00873327
$986$	0	0
$987$	−6.48701	−0.206484
$988$	0	0
$989$	−3.49922	−0.111269
$990$	0	0
$991$	15.6773	0.498005	0.249003	−	0.968503i	$-0.419897\pi$
$991$	15.6773	0.498005	0.249003	+	0.968503i	$0.419897\pi$
$992$	0	0
$993$	5.84539	0.185498
$994$	0	0
$995$	0.372726	0.0118162
$996$	0	0
$997$	−0.967011	−0.0306255	−0.0153128	−	0.999883i	$-0.504874\pi$
$997$	−0.967011	−0.0306255	−0.0153128	+	0.999883i	$0.504874\pi$
$998$	0	0
$999$	14.6031	0.462022

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4004.2.a.g.1.5	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4004.2.a.g.1.5	✓	6	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:	\( N \)	\(=\)	\( 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4004.a (trivial)

\(f(q)\)	\(=\)	\(q+0.820857 q^{3} +0.0215726 q^{5} +1.00000 q^{7} -2.32619 q^{9} +O(q^{10})\)	\(q+0.820857 q^{3} +0.0215726 q^{5} +1.00000 q^{7} -2.32619 q^{9} -1.00000 q^{11} -1.00000 q^{13} +0.0177081 q^{15} +0.645197 q^{17} +2.66244 q^{19} +0.820857 q^{21} +0.816993 q^{23} -4.99953 q^{25} -4.37205 q^{27} -2.98229 q^{29} -2.72028 q^{31} -0.820857 q^{33} +0.0215726 q^{35} -3.34011 q^{37} -0.820857 q^{39} +3.93482 q^{41} -4.28305 q^{43} -0.0501821 q^{45} -7.90272 q^{47} +1.00000 q^{49} +0.529615 q^{51} +7.61883 q^{53} -0.0215726 q^{55} +2.18548 q^{57} -8.01678 q^{59} +7.80516 q^{61} -2.32619 q^{63} -0.0215726 q^{65} -2.10012 q^{67} +0.670635 q^{69} -10.1318 q^{71} -0.993429 q^{73} -4.10391 q^{75} -1.00000 q^{77} -7.44053 q^{79} +3.38975 q^{81} +11.6149 q^{83} +0.0139186 q^{85} -2.44804 q^{87} -4.20682 q^{89} -1.00000 q^{91} -2.23296 q^{93} +0.0574359 q^{95} -11.5872 q^{97} +2.32619 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{3} - 3 q^{5} + 6 q^{7} + 4 q^{9}+O(q^{10}) \) 6 * q - 2 * q^3 - 3 * q^5 + 6 * q^7 + 4 * q^9	\( 6 q - 2 q^{3} - 3 q^{5} + 6 q^{7} + 4 q^{9} - 6 q^{11} - 6 q^{13} + 4 q^{15} - q^{17} - 12 q^{19} - 2 q^{21} + 5 q^{23} - 3 q^{25} - 8 q^{27} - 14 q^{29} - 4 q^{31} + 2 q^{33} - 3 q^{35} - 3 q^{37} + 2 q^{39} + 6 q^{41} - 14 q^{43} - 20 q^{45} + 2 q^{47} + 6 q^{49} - 5 q^{51} - 3 q^{53} + 3 q^{55} - 22 q^{57} + 2 q^{59} - 26 q^{61} + 4 q^{63} + 3 q^{65} + 9 q^{67} - 11 q^{69} + 3 q^{71} - 7 q^{73} - 6 q^{75} - 6 q^{77} + 6 q^{81} - 15 q^{83} + q^{85} - 23 q^{87} - q^{89} - 6 q^{91} + 8 q^{93} + 12 q^{95} - 16 q^{97} - 4 q^{99}+O(q^{100}) \) 6 * q - 2 * q^3 - 3 * q^5 + 6 * q^7 + 4 * q^9 - 6 * q^11 - 6 * q^13 + 4 * q^15 - q^17 - 12 * q^19 - 2 * q^21 + 5 * q^23 - 3 * q^25 - 8 * q^27 - 14 * q^29 - 4 * q^31 + 2 * q^33 - 3 * q^35 - 3 * q^37 + 2 * q^39 + 6 * q^41 - 14 * q^43 - 20 * q^45 + 2 * q^47 + 6 * q^49 - 5 * q^51 - 3 * q^53 + 3 * q^55 - 22 * q^57 + 2 * q^59 - 26 * q^61 + 4 * q^63 + 3 * q^65 + 9 * q^67 - 11 * q^69 + 3 * q^71 - 7 * q^73 - 6 * q^75 - 6 * q^77 + 6 * q^81 - 15 * q^83 + q^85 - 23 * q^87 - q^89 - 6 * q^91 + 8 * q^93 + 12 * q^95 - 16 * q^97 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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