LMFDB - Embedded newform 4012.2.a.j.1.17

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			1.17
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)

$f(q)$	$=$	$q+2.29806 q^{3} +4.01491 q^{5} +4.07651 q^{7} +2.28106 q^{9} +O(q^{10})$	$q+2.29806 q^{3} +4.01491 q^{5} +4.07651 q^{7} +2.28106 q^{9} -1.78190 q^{11} -1.70458 q^{13} +9.22649 q^{15} +1.00000 q^{17} +5.98109 q^{19} +9.36805 q^{21} -7.29452 q^{23} +11.1195 q^{25} -1.65216 q^{27} -0.931389 q^{29} -3.34797 q^{31} -4.09490 q^{33} +16.3668 q^{35} -11.5216 q^{37} -3.91722 q^{39} +6.51764 q^{41} +1.48131 q^{43} +9.15827 q^{45} +8.42177 q^{47} +9.61792 q^{49} +2.29806 q^{51} -13.6213 q^{53} -7.15416 q^{55} +13.7449 q^{57} +1.00000 q^{59} +7.12292 q^{61} +9.29878 q^{63} -6.84373 q^{65} -10.1467 q^{67} -16.7632 q^{69} +1.51685 q^{71} +1.78218 q^{73} +25.5533 q^{75} -7.26392 q^{77} -6.46031 q^{79} -10.6399 q^{81} +15.5672 q^{83} +4.01491 q^{85} -2.14038 q^{87} +3.46879 q^{89} -6.94872 q^{91} -7.69383 q^{93} +24.0135 q^{95} +3.18902 q^{97} -4.06462 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	2.29806	1.32678	0.663392	−	0.748272i	$-0.269116\pi$
$3$	2.29806	1.32678	0.663392	+	0.748272i	$0.269116\pi$
$4$	0	0
$5$	4.01491	1.79552	0.897761	−	0.440482i	$-0.145193\pi$
$5$	4.01491	1.79552	0.897761	+	0.440482i	$0.145193\pi$
$6$	0	0
$7$	4.07651	1.54078	0.770388	−	0.637576i	$-0.220063\pi$
$7$	4.07651	1.54078	0.770388	+	0.637576i	$0.220063\pi$
$8$	0	0
$9$	2.28106	0.760355
$10$	0	0
$11$	−1.78190	−0.537263	−0.268631	−	0.963243i	$-0.586571\pi$
$11$	−1.78190	−0.537263	−0.268631	+	0.963243i	$0.586571\pi$
$12$	0	0
$13$	−1.70458	−0.472765	−0.236382	−	0.971660i	$-0.575962\pi$
$13$	−1.70458	−0.472765	−0.236382	+	0.971660i	$0.575962\pi$
$14$	0	0
$15$	9.22649	2.38227
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	5.98109	1.37216	0.686078	−	0.727528i	$-0.259331\pi$
$19$	5.98109	1.37216	0.686078	+	0.727528i	$0.259331\pi$
$20$	0	0
$21$	9.36805	2.04428
$22$	0	0
$23$	−7.29452	−1.52101	−0.760506	−	0.649330i	$-0.775049\pi$
$23$	−7.29452	−1.52101	−0.760506	+	0.649330i	$0.775049\pi$
$24$	0	0
$25$	11.1195	2.22390
$26$	0	0
$27$	−1.65216	−0.317957
$28$	0	0
$29$	−0.931389	−0.172955	−0.0864773	−	0.996254i	$-0.527561\pi$
$29$	−0.931389	−0.172955	−0.0864773	+	0.996254i	$0.527561\pi$
$30$	0	0
$31$	−3.34797	−0.601313	−0.300657	−	0.953732i	$-0.597206\pi$
$31$	−3.34797	−0.601313	−0.300657	+	0.953732i	$0.597206\pi$
$32$	0	0
$33$	−4.09490	−0.712831
$34$	0	0
$35$	16.3668	2.76650
$36$	0	0
$37$	−11.5216	−1.89413	−0.947067	−	0.321035i	$-0.895969\pi$
$37$	−11.5216	−1.89413	−0.947067	+	0.321035i	$0.895969\pi$
$38$	0	0
$39$	−3.91722	−0.627257
$40$	0	0
$41$	6.51764	1.01788	0.508942	−	0.860801i	$-0.330037\pi$
$41$	6.51764	1.01788	0.508942	+	0.860801i	$0.330037\pi$
$42$	0	0
$43$	1.48131	0.225897	0.112949	−	0.993601i	$-0.463970\pi$
$43$	1.48131	0.225897	0.112949	+	0.993601i	$0.463970\pi$
$44$	0	0
$45$	9.15827	1.36523
$46$	0	0
$47$	8.42177	1.22844	0.614220	−	0.789135i	$-0.289471\pi$
$47$	8.42177	1.22844	0.614220	+	0.789135i	$0.289471\pi$
$48$	0	0
$49$	9.61792	1.37399
$50$	0	0
$51$	2.29806	0.321792
$52$	0	0
$53$	−13.6213	−1.87102	−0.935512	−	0.353294i	$-0.885061\pi$
$53$	−13.6213	−1.87102	−0.935512	+	0.353294i	$0.885061\pi$
$54$	0	0
$55$	−7.15416	−0.964667
$56$	0	0
$57$	13.7449	1.82055
$58$	0	0
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	0	0
$61$	7.12292	0.911997	0.455998	−	0.889981i	$-0.349282\pi$
$61$	7.12292	0.911997	0.455998	+	0.889981i	$0.349282\pi$
$62$	0	0
$63$	9.29878	1.17154
$64$	0	0
$65$	−6.84373	−0.848860
$66$	0	0
$67$	−10.1467	−1.23961	−0.619807	−	0.784754i	$-0.712789\pi$
$67$	−10.1467	−1.23961	−0.619807	+	0.784754i	$0.712789\pi$
$68$	0	0
$69$	−16.7632	−2.01806
$70$	0	0
$71$	1.51685	0.180017	0.0900084	−	0.995941i	$-0.471311\pi$
$71$	1.51685	0.180017	0.0900084	+	0.995941i	$0.471311\pi$
$72$	0	0
$73$	1.78218	0.208589	0.104294	−	0.994546i	$-0.466742\pi$
$73$	1.78218	0.208589	0.104294	+	0.994546i	$0.466742\pi$
$74$	0	0
$75$	25.5533	2.95064
$76$	0	0
$77$	−7.26392	−0.827801
$78$	0	0
$79$	−6.46031	−0.726842	−0.363421	−	0.931625i	$-0.618391\pi$
$79$	−6.46031	−0.726842	−0.363421	+	0.931625i	$0.618391\pi$
$80$	0	0
$81$	−10.6399	−1.18222
$82$	0	0
$83$	15.5672	1.70872	0.854360	−	0.519682i	$-0.173950\pi$
$83$	15.5672	1.70872	0.854360	+	0.519682i	$0.173950\pi$
$84$	0	0
$85$	4.01491	0.435478
$86$	0	0
$87$	−2.14038	−0.229473
$88$	0	0
$89$	3.46879	0.367691	0.183846	−	0.982955i	$-0.441145\pi$
$89$	3.46879	0.367691	0.183846	+	0.982955i	$0.441145\pi$
$90$	0	0
$91$	−6.94872	−0.728424
$92$	0	0
$93$	−7.69383	−0.797813
$94$	0	0
$95$	24.0135	2.46374
$96$	0	0
$97$	3.18902	0.323796	0.161898	−	0.986807i	$-0.448238\pi$
$97$	3.18902	0.323796	0.161898	+	0.986807i	$0.448238\pi$
$98$	0	0
$99$	−4.06462	−0.408510
$100$	0	0
$101$	−1.38467	−0.137780	−0.0688901	−	0.997624i	$-0.521946\pi$
$101$	−1.38467	−0.137780	−0.0688901	+	0.997624i	$0.521946\pi$
$102$	0	0
$103$	−2.58504	−0.254711	−0.127356	−	0.991857i	$-0.540649\pi$
$103$	−2.58504	−0.254711	−0.127356	+	0.991857i	$0.540649\pi$
$104$	0	0
$105$	37.6119	3.67054
$106$	0	0
$107$	−2.63716	−0.254944	−0.127472	−	0.991842i	$-0.540686\pi$
$107$	−2.63716	−0.254944	−0.127472	+	0.991842i	$0.540686\pi$
$108$	0	0
$109$	2.59205	0.248274	0.124137	−	0.992265i	$-0.460384\pi$
$109$	2.59205	0.248274	0.124137	+	0.992265i	$0.460384\pi$
$110$	0	0
$111$	−26.4772	−2.51311
$112$	0	0
$113$	18.6718	1.75649	0.878247	−	0.478207i	$-0.158713\pi$
$113$	18.6718	1.75649	0.878247	+	0.478207i	$0.158713\pi$
$114$	0	0
$115$	−29.2869	−2.73101
$116$	0	0
$117$	−3.88825	−0.359469
$118$	0	0
$119$	4.07651	0.373693
$120$	0	0
$121$	−7.82484	−0.711349
$122$	0	0
$123$	14.9779	1.35051
$124$	0	0
$125$	24.5693	2.19754
$126$	0	0
$127$	11.4567	1.01662	0.508309	−	0.861175i	$-0.330271\pi$
$127$	11.4567	1.01662	0.508309	+	0.861175i	$0.330271\pi$
$128$	0	0
$129$	3.40413	0.299717
$130$	0	0
$131$	−13.7807	−1.20402	−0.602012	−	0.798487i	$-0.705634\pi$
$131$	−13.7807	−1.20402	−0.602012	+	0.798487i	$0.705634\pi$
$132$	0	0
$133$	24.3820	2.11418
$134$	0	0
$135$	−6.63326	−0.570900
$136$	0	0
$137$	−12.0364	−1.02834	−0.514168	−	0.857690i	$-0.671899\pi$
$137$	−12.0364	−1.02834	−0.514168	+	0.857690i	$0.671899\pi$
$138$	0	0
$139$	−13.1795	−1.11787	−0.558935	−	0.829211i	$-0.688790\pi$
$139$	−13.1795	−1.11787	−0.558935	+	0.829211i	$0.688790\pi$
$140$	0	0
$141$	19.3537	1.62987
$142$	0	0
$143$	3.03738	0.253999
$144$	0	0
$145$	−3.73944	−0.310544
$146$	0	0
$147$	22.1025	1.82298
$148$	0	0
$149$	5.57321	0.456575	0.228287	−	0.973594i	$-0.426687\pi$
$149$	5.57321	0.456575	0.228287	+	0.973594i	$0.426687\pi$
$150$	0	0
$151$	−19.5589	−1.59168	−0.795841	−	0.605506i	$-0.792971\pi$
$151$	−19.5589	−1.59168	−0.795841	+	0.605506i	$0.792971\pi$
$152$	0	0
$153$	2.28106	0.184413
$154$	0	0
$155$	−13.4418	−1.07967
$156$	0	0
$157$	−11.3964	−0.909529	−0.454765	−	0.890612i	$-0.650277\pi$
$157$	−11.3964	−0.909529	−0.454765	+	0.890612i	$0.650277\pi$
$158$	0	0
$159$	−31.3024	−2.48244
$160$	0	0
$161$	−29.7362	−2.34354
$162$	0	0
$163$	9.04240	0.708255	0.354128	−	0.935197i	$-0.384778\pi$
$163$	9.04240	0.708255	0.354128	+	0.935197i	$0.384778\pi$
$164$	0	0
$165$	−16.4407	−1.27990
$166$	0	0
$167$	−9.71944	−0.752113	−0.376056	−	0.926597i	$-0.622720\pi$
$167$	−9.71944	−0.752113	−0.376056	+	0.926597i	$0.622720\pi$
$168$	0	0
$169$	−10.0944	−0.776493
$170$	0	0
$171$	13.6432	1.04333
$172$	0	0
$173$	14.5518	1.10635	0.553175	−	0.833065i	$-0.313416\pi$
$173$	14.5518	1.10635	0.553175	+	0.833065i	$0.313416\pi$
$174$	0	0
$175$	45.3288	3.42653
$176$	0	0
$177$	2.29806	0.172733
$178$	0	0
$179$	−9.79342	−0.731995	−0.365997	−	0.930616i	$-0.619272\pi$
$179$	−9.79342	−0.731995	−0.365997	+	0.930616i	$0.619272\pi$
$180$	0	0
$181$	−24.9434	−1.85403	−0.927016	−	0.375023i	$-0.877635\pi$
$181$	−24.9434	−1.85403	−0.927016	+	0.375023i	$0.877635\pi$
$182$	0	0
$183$	16.3689	1.21002
$184$	0	0
$185$	−46.2581	−3.40096
$186$	0	0
$187$	−1.78190	−0.130305
$188$	0	0
$189$	−6.73502	−0.489901
$190$	0	0
$191$	17.5740	1.27161	0.635803	−	0.771851i	$-0.280669\pi$
$191$	17.5740	1.27161	0.635803	+	0.771851i	$0.280669\pi$
$192$	0	0
$193$	−24.1538	−1.73863	−0.869316	−	0.494257i	$-0.835440\pi$
$193$	−24.1538	−1.73863	−0.869316	+	0.494257i	$0.835440\pi$
$194$	0	0
$195$	−15.7273	−1.12625
$196$	0	0
$197$	−12.8788	−0.917575	−0.458787	−	0.888546i	$-0.651716\pi$
$197$	−12.8788	−0.917575	−0.458787	+	0.888546i	$0.651716\pi$
$198$	0	0
$199$	0.706416	0.0500765	0.0250383	−	0.999686i	$-0.492029\pi$
$199$	0.706416	0.0500765	0.0250383	+	0.999686i	$0.492029\pi$
$200$	0	0
$201$	−23.3176	−1.64470
$202$	0	0
$203$	−3.79681	−0.266484
$204$	0	0
$205$	26.1678	1.82764
$206$	0	0
$207$	−16.6393	−1.15651
$208$	0	0
$209$	−10.6577	−0.737208
$210$	0	0
$211$	10.4896	0.722134	0.361067	−	0.932540i	$-0.382413\pi$
$211$	10.4896	0.722134	0.361067	+	0.932540i	$0.382413\pi$
$212$	0	0
$213$	3.48580	0.238843
$214$	0	0
$215$	5.94732	0.405604
$216$	0	0
$217$	−13.6480	−0.926489
$218$	0	0
$219$	4.09556	0.276752
$220$	0	0
$221$	−1.70458	−0.114662
$222$	0	0
$223$	17.8641	1.19627	0.598133	−	0.801397i	$-0.295909\pi$
$223$	17.8641	1.19627	0.598133	+	0.801397i	$0.295909\pi$
$224$	0	0
$225$	25.3643	1.69095
$226$	0	0
$227$	−6.83358	−0.453561	−0.226780	−	0.973946i	$-0.572820\pi$
$227$	−6.83358	−0.453561	−0.226780	+	0.973946i	$0.572820\pi$
$228$	0	0
$229$	−20.4859	−1.35375	−0.676874	−	0.736099i	$-0.736666\pi$
$229$	−20.4859	−1.35375	−0.676874	+	0.736099i	$0.736666\pi$
$230$	0	0
$231$	−16.6929	−1.09831
$232$	0	0
$233$	12.9060	0.845498	0.422749	−	0.906247i	$-0.361065\pi$
$233$	12.9060	0.845498	0.422749	+	0.906247i	$0.361065\pi$
$234$	0	0
$235$	33.8126	2.20569
$236$	0	0
$237$	−14.8462	−0.964362
$238$	0	0
$239$	27.3721	1.77056	0.885278	−	0.465061i	$-0.153968\pi$
$239$	27.3721	1.77056	0.885278	+	0.465061i	$0.153968\pi$
$240$	0	0
$241$	10.3980	0.669794	0.334897	−	0.942255i	$-0.391298\pi$
$241$	10.3980	0.669794	0.334897	+	0.942255i	$0.391298\pi$
$242$	0	0
$243$	−19.4947	−1.25059
$244$	0	0
$245$	38.6151	2.46703
$246$	0	0
$247$	−10.1952	−0.648707
$248$	0	0
$249$	35.7743	2.26710
$250$	0	0
$251$	−17.5692	−1.10896	−0.554480	−	0.832197i	$-0.687083\pi$
$251$	−17.5692	−1.10896	−0.554480	+	0.832197i	$0.687083\pi$
$252$	0	0
$253$	12.9981	0.817183
$254$	0	0
$255$	9.22649	0.577785
$256$	0	0
$257$	21.7570	1.35717	0.678583	−	0.734523i	$-0.262594\pi$
$257$	21.7570	1.35717	0.678583	+	0.734523i	$0.262594\pi$
$258$	0	0
$259$	−46.9678	−2.91844
$260$	0	0
$261$	−2.12456	−0.131507
$262$	0	0
$263$	−15.3748	−0.948050	−0.474025	−	0.880511i	$-0.657199\pi$
$263$	−15.3748	−0.948050	−0.474025	+	0.880511i	$0.657199\pi$
$264$	0	0
$265$	−54.6882	−3.35947
$266$	0	0
$267$	7.97148	0.487847
$268$	0	0
$269$	−4.40959	−0.268857	−0.134429	−	0.990923i	$-0.542920\pi$
$269$	−4.40959	−0.268857	−0.134429	+	0.990923i	$0.542920\pi$
$270$	0	0
$271$	9.01438	0.547585	0.273792	−	0.961789i	$-0.411722\pi$
$271$	9.01438	0.547585	0.273792	+	0.961789i	$0.411722\pi$
$272$	0	0
$273$	−15.9686	−0.966461
$274$	0	0
$275$	−19.8138	−1.19482
$276$	0	0
$277$	22.9099	1.37652	0.688260	−	0.725464i	$-0.258375\pi$
$277$	22.9099	1.37652	0.688260	+	0.725464i	$0.258375\pi$
$278$	0	0
$279$	−7.63694	−0.457211
$280$	0	0
$281$	3.95651	0.236025	0.118013	−	0.993012i	$-0.462348\pi$
$281$	3.95651	0.236025	0.118013	+	0.993012i	$0.462348\pi$
$282$	0	0
$283$	−10.2427	−0.608863	−0.304431	−	0.952534i	$-0.598466\pi$
$283$	−10.2427	−0.608863	−0.304431	+	0.952534i	$0.598466\pi$
$284$	0	0
$285$	55.1845	3.26885
$286$	0	0
$287$	26.5692	1.56833
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	7.32856	0.429608
$292$	0	0
$293$	−10.2295	−0.597613	−0.298806	−	0.954314i	$-0.596588\pi$
$293$	−10.2295	−0.597613	−0.298806	+	0.954314i	$0.596588\pi$
$294$	0	0
$295$	4.01491	0.233757
$296$	0	0
$297$	2.94397	0.170827
$298$	0	0
$299$	12.4341	0.719081
$300$	0	0
$301$	6.03856	0.348057
$302$	0	0
$303$	−3.18206	−0.182805
$304$	0	0
$305$	28.5979	1.63751
$306$	0	0
$307$	23.1490	1.32118	0.660591	−	0.750746i	$-0.270306\pi$
$307$	23.1490	1.32118	0.660591	+	0.750746i	$0.270306\pi$
$308$	0	0
$309$	−5.94057	−0.337947
$310$	0	0
$311$	29.0126	1.64515	0.822575	−	0.568656i	$-0.192536\pi$
$311$	29.0126	1.64515	0.822575	+	0.568656i	$0.192536\pi$
$312$	0	0
$313$	19.5108	1.10282	0.551408	−	0.834236i	$-0.314091\pi$
$313$	19.5108	1.10282	0.551408	+	0.834236i	$0.314091\pi$
$314$	0	0
$315$	37.3338	2.10352
$316$	0	0
$317$	−26.3031	−1.47733	−0.738665	−	0.674073i	$-0.764543\pi$
$317$	−26.3031	−1.47733	−0.738665	+	0.674073i	$0.764543\pi$
$318$	0	0
$319$	1.65964	0.0929220
$320$	0	0
$321$	−6.06034	−0.338255
$322$	0	0
$323$	5.98109	0.332797
$324$	0	0
$325$	−18.9541	−1.05138
$326$	0	0
$327$	5.95669	0.329406
$328$	0	0
$329$	34.3314	1.89275
$330$	0	0
$331$	−4.07045	−0.223732	−0.111866	−	0.993723i	$-0.535683\pi$
$331$	−4.07045	−0.223732	−0.111866	+	0.993723i	$0.535683\pi$
$332$	0	0
$333$	−26.2814	−1.44021
$334$	0	0
$335$	−40.7380	−2.22575
$336$	0	0
$337$	−34.5000	−1.87933	−0.939667	−	0.342090i	$-0.888865\pi$
$337$	−34.5000	−1.87933	−0.939667	+	0.342090i	$0.888865\pi$
$338$	0	0
$339$	42.9088	2.33049
$340$	0	0
$341$	5.96574	0.323063
$342$	0	0
$343$	10.6720	0.576232
$344$	0	0
$345$	−67.3028	−3.62346
$346$	0	0
$347$	7.56983	0.406370	0.203185	−	0.979140i	$-0.434871\pi$
$347$	7.56983	0.406370	0.203185	+	0.979140i	$0.434871\pi$
$348$	0	0
$349$	−3.38077	−0.180968	−0.0904842	−	0.995898i	$-0.528841\pi$
$349$	−3.38077	−0.180968	−0.0904842	+	0.995898i	$0.528841\pi$
$350$	0	0
$351$	2.81623	0.150319
$352$	0	0
$353$	34.2014	1.82036	0.910180	−	0.414214i	$-0.135943\pi$
$353$	34.2014	1.82036	0.910180	+	0.414214i	$0.135943\pi$
$354$	0	0
$355$	6.09001	0.323224
$356$	0	0
$357$	9.36805	0.495810
$358$	0	0
$359$	−21.4836	−1.13386	−0.566931	−	0.823765i	$-0.691869\pi$
$359$	−21.4836	−1.13386	−0.566931	+	0.823765i	$0.691869\pi$
$360$	0	0
$361$	16.7734	0.882812
$362$	0	0
$363$	−17.9819	−0.943806
$364$	0	0
$365$	7.15530	0.374526
$366$	0	0
$367$	13.1799	0.687983	0.343991	−	0.938973i	$-0.388221\pi$
$367$	13.1799	0.687983	0.343991	+	0.938973i	$0.388221\pi$
$368$	0	0
$369$	14.8672	0.773954
$370$	0	0
$371$	−55.5272	−2.88283
$372$	0	0
$373$	2.67801	0.138662	0.0693312	−	0.997594i	$-0.477913\pi$
$373$	2.67801	0.138662	0.0693312	+	0.997594i	$0.477913\pi$
$374$	0	0
$375$	56.4616	2.91566
$376$	0	0
$377$	1.58762	0.0817668
$378$	0	0
$379$	12.9685	0.666148	0.333074	−	0.942901i	$-0.391914\pi$
$379$	12.9685	0.666148	0.333074	+	0.942901i	$0.391914\pi$
$380$	0	0
$381$	26.3281	1.34883
$382$	0	0
$383$	0.176613	0.00902450	0.00451225	−	0.999990i	$-0.498564\pi$
$383$	0.176613	0.00902450	0.00451225	+	0.999990i	$0.498564\pi$
$384$	0	0
$385$	−29.1640	−1.48633
$386$	0	0
$387$	3.37896	0.171762
$388$	0	0
$389$	−32.7560	−1.66080	−0.830398	−	0.557171i	$-0.811887\pi$
$389$	−32.7560	−1.66080	−0.830398	+	0.557171i	$0.811887\pi$
$390$	0	0
$391$	−7.29452	−0.368900
$392$	0	0
$393$	−31.6688	−1.59748
$394$	0	0
$395$	−25.9376	−1.30506
$396$	0	0
$397$	19.1210	0.959658	0.479829	−	0.877362i	$-0.340699\pi$
$397$	19.1210	0.959658	0.479829	+	0.877362i	$0.340699\pi$
$398$	0	0
$399$	56.0311	2.80506
$400$	0	0
$401$	27.4574	1.37116	0.685579	−	0.727998i	$-0.259549\pi$
$401$	27.4574	1.37116	0.685579	+	0.727998i	$0.259549\pi$
$402$	0	0
$403$	5.70688	0.284280
$404$	0	0
$405$	−42.7184	−2.12269
$406$	0	0
$407$	20.5303	1.01765
$408$	0	0
$409$	15.8144	0.781970	0.390985	−	0.920397i	$-0.372135\pi$
$409$	15.8144	0.781970	0.390985	+	0.920397i	$0.372135\pi$
$410$	0	0
$411$	−27.6602	−1.36438
$412$	0	0
$413$	4.07651	0.200592
$414$	0	0
$415$	62.5008	3.06804
$416$	0	0
$417$	−30.2872	−1.48317
$418$	0	0
$419$	3.43085	0.167608	0.0838039	−	0.996482i	$-0.473293\pi$
$419$	3.43085	0.167608	0.0838039	+	0.996482i	$0.473293\pi$
$420$	0	0
$421$	15.5705	0.758861	0.379431	−	0.925220i	$-0.376120\pi$
$421$	15.5705	0.758861	0.379431	+	0.925220i	$0.376120\pi$
$422$	0	0
$423$	19.2106	0.934051
$424$	0	0
$425$	11.1195	0.539375
$426$	0	0
$427$	29.0367	1.40518
$428$	0	0
$429$	6.98008	0.337001
$430$	0	0
$431$	−32.9016	−1.58482	−0.792408	−	0.609991i	$-0.791173\pi$
$431$	−32.9016	−1.58482	−0.792408	+	0.609991i	$0.791173\pi$
$432$	0	0
$433$	−23.3006	−1.11975	−0.559877	−	0.828576i	$-0.689152\pi$
$433$	−23.3006	−1.11975	−0.559877	+	0.828576i	$0.689152\pi$
$434$	0	0
$435$	−8.59345	−0.412024
$436$	0	0
$437$	−43.6292	−2.08707
$438$	0	0
$439$	22.7739	1.08694	0.543470	−	0.839429i	$-0.317110\pi$
$439$	22.7739	1.08694	0.543470	+	0.839429i	$0.317110\pi$
$440$	0	0
$441$	21.9391	1.04472
$442$	0	0
$443$	3.03806	0.144342	0.0721712	−	0.997392i	$-0.477007\pi$
$443$	3.03806	0.144342	0.0721712	+	0.997392i	$0.477007\pi$
$444$	0	0
$445$	13.9269	0.660198
$446$	0	0
$447$	12.8075	0.605776
$448$	0	0
$449$	16.8879	0.796990	0.398495	−	0.917171i	$-0.369533\pi$
$449$	16.8879	0.796990	0.398495	+	0.917171i	$0.369533\pi$
$450$	0	0
$451$	−11.6138	−0.546871
$452$	0	0
$453$	−44.9475	−2.11182
$454$	0	0
$455$	−27.8985	−1.30790
$456$	0	0
$457$	−11.1632	−0.522194	−0.261097	−	0.965313i	$-0.584084\pi$
$457$	−11.1632	−0.522194	−0.261097	+	0.965313i	$0.584084\pi$
$458$	0	0
$459$	−1.65216	−0.0771160
$460$	0	0
$461$	35.8536	1.66987	0.834934	−	0.550350i	$-0.185506\pi$
$461$	35.8536	1.66987	0.834934	+	0.550350i	$0.185506\pi$
$462$	0	0
$463$	−9.35812	−0.434909	−0.217454	−	0.976070i	$-0.569775\pi$
$463$	−9.35812	−0.434909	−0.217454	+	0.976070i	$0.569775\pi$
$464$	0	0
$465$	−30.8900	−1.43249
$466$	0	0
$467$	−19.6465	−0.909133	−0.454567	−	0.890713i	$-0.650206\pi$
$467$	−19.6465	−0.909133	−0.454567	+	0.890713i	$0.650206\pi$
$468$	0	0
$469$	−41.3630	−1.90997
$470$	0	0
$471$	−26.1895	−1.20675
$472$	0	0
$473$	−2.63954	−0.121366
$474$	0	0
$475$	66.5068	3.05154
$476$	0	0
$477$	−31.0710	−1.42264
$478$	0	0
$479$	−0.0576119	−0.00263235	−0.00131618	−	0.999999i	$-0.500419\pi$
$479$	−0.0576119	−0.00263235	−0.00131618	+	0.999999i	$0.500419\pi$
$480$	0	0
$481$	19.6394	0.895480
$482$	0	0
$483$	−68.3354	−3.10937
$484$	0	0
$485$	12.8036	0.581384
$486$	0	0
$487$	4.69169	0.212601	0.106300	−	0.994334i	$-0.466099\pi$
$487$	4.69169	0.212601	0.106300	+	0.994334i	$0.466099\pi$
$488$	0	0
$489$	20.7799	0.939702
$490$	0	0
$491$	11.2222	0.506452	0.253226	−	0.967407i	$-0.418508\pi$
$491$	11.2222	0.506452	0.253226	+	0.967407i	$0.418508\pi$
$492$	0	0
$493$	−0.931389	−0.0419476
$494$	0	0
$495$	−16.3191	−0.733489
$496$	0	0
$497$	6.18344	0.277365
$498$	0	0
$499$	25.1952	1.12789	0.563946	−	0.825812i	$-0.309283\pi$
$499$	25.1952	1.12789	0.563946	+	0.825812i	$0.309283\pi$
$500$	0	0
$501$	−22.3358	−0.997891
$502$	0	0
$503$	15.4976	0.691002	0.345501	−	0.938418i	$-0.387709\pi$
$503$	15.4976	0.691002	0.345501	+	0.938418i	$0.387709\pi$
$504$	0	0
$505$	−5.55934	−0.247387
$506$	0	0
$507$	−23.1975	−1.03024
$508$	0	0
$509$	−28.2460	−1.25198	−0.625991	−	0.779830i	$-0.715305\pi$
$509$	−28.2460	−1.25198	−0.625991	+	0.779830i	$0.715305\pi$
$510$	0	0
$511$	7.26508	0.321388
$512$	0	0
$513$	−9.88169	−0.436287
$514$	0	0
$515$	−10.3787	−0.457340
$516$	0	0
$517$	−15.0067	−0.659995
$518$	0	0
$519$	33.4408	1.46789
$520$	0	0
$521$	−9.69421	−0.424711	−0.212355	−	0.977192i	$-0.568113\pi$
$521$	−9.69421	−0.424711	−0.212355	+	0.977192i	$0.568113\pi$
$522$	0	0
$523$	17.7830	0.777595	0.388798	−	0.921323i	$-0.372891\pi$
$523$	17.7830	0.777595	0.388798	+	0.921323i	$0.372891\pi$
$524$	0	0
$525$	104.168	4.54627
$526$	0	0
$527$	−3.34797	−0.145840
$528$	0	0
$529$	30.2100	1.31348
$530$	0	0
$531$	2.28106	0.0989898
$532$	0	0
$533$	−11.1098	−0.481220
$534$	0	0
$535$	−10.5880	−0.457757
$536$	0	0
$537$	−22.5058	−0.971198
$538$	0	0
$539$	−17.1381	−0.738192
$540$	0	0
$541$	−39.6957	−1.70665	−0.853325	−	0.521379i	$-0.825418\pi$
$541$	−39.6957	−1.70665	−0.853325	+	0.521379i	$0.825418\pi$
$542$	0	0
$543$	−57.3214	−2.45990
$544$	0	0
$545$	10.4069	0.445781
$546$	0	0
$547$	39.3920	1.68428	0.842140	−	0.539259i	$-0.181296\pi$
$547$	39.3920	1.68428	0.842140	+	0.539259i	$0.181296\pi$
$548$	0	0
$549$	16.2478	0.693441
$550$	0	0
$551$	−5.57072	−0.237321
$552$	0	0
$553$	−26.3355	−1.11990
$554$	0	0
$555$	−106.304	−4.51234
$556$	0	0
$557$	25.0923	1.06320	0.531598	−	0.846997i	$-0.321592\pi$
$557$	25.0923	1.06320	0.531598	+	0.846997i	$0.321592\pi$
$558$	0	0
$559$	−2.52500	−0.106796
$560$	0	0
$561$	−4.09490	−0.172887
$562$	0	0
$563$	22.3329	0.941221	0.470611	−	0.882341i	$-0.344034\pi$
$563$	22.3329	0.941221	0.470611	+	0.882341i	$0.344034\pi$
$564$	0	0
$565$	74.9656	3.15383
$566$	0	0
$567$	−43.3738	−1.82153
$568$	0	0
$569$	−23.4545	−0.983265	−0.491633	−	0.870803i	$-0.663600\pi$
$569$	−23.4545	−0.983265	−0.491633	+	0.870803i	$0.663600\pi$
$570$	0	0
$571$	5.08932	0.212981	0.106491	−	0.994314i	$-0.466039\pi$
$571$	5.08932	0.212981	0.106491	+	0.994314i	$0.466039\pi$
$572$	0	0
$573$	40.3859	1.68715
$574$	0	0
$575$	−81.1115	−3.38258
$576$	0	0
$577$	−23.3296	−0.971223	−0.485612	−	0.874175i	$-0.661403\pi$
$577$	−23.3296	−0.971223	−0.485612	+	0.874175i	$0.661403\pi$
$578$	0	0
$579$	−55.5069	−2.30679
$580$	0	0
$581$	63.4597	2.63275
$582$	0	0
$583$	24.2717	1.00523
$584$	0	0
$585$	−15.6110	−0.645435
$586$	0	0
$587$	45.9103	1.89492	0.947461	−	0.319871i	$-0.103640\pi$
$587$	45.9103	1.89492	0.947461	+	0.319871i	$0.103640\pi$
$588$	0	0
$589$	−20.0245	−0.825096
$590$	0	0
$591$	−29.5961	−1.21742
$592$	0	0
$593$	−13.8706	−0.569596	−0.284798	−	0.958588i	$-0.591926\pi$
$593$	−13.8706	−0.569596	−0.284798	+	0.958588i	$0.591926\pi$
$594$	0	0
$595$	16.3668	0.670974
$596$	0	0
$597$	1.62338	0.0664407
$598$	0	0
$599$	−14.7158	−0.601270	−0.300635	−	0.953739i	$-0.597199\pi$
$599$	−14.7158	−0.601270	−0.300635	+	0.953739i	$0.597199\pi$
$600$	0	0
$601$	11.8810	0.484637	0.242318	−	0.970197i	$-0.422092\pi$
$601$	11.8810	0.484637	0.242318	+	0.970197i	$0.422092\pi$
$602$	0	0
$603$	−23.1452	−0.942546
$604$	0	0
$605$	−31.4160	−1.27724
$606$	0	0
$607$	−14.9106	−0.605203	−0.302602	−	0.953117i	$-0.597855\pi$
$607$	−14.9106	−0.605203	−0.302602	+	0.953117i	$0.597855\pi$
$608$	0	0
$609$	−8.72529	−0.353567
$610$	0	0
$611$	−14.3556	−0.580764
$612$	0	0
$613$	4.21272	0.170150	0.0850752	−	0.996375i	$-0.472887\pi$
$613$	4.21272	0.170150	0.0850752	+	0.996375i	$0.472887\pi$
$614$	0	0
$615$	60.1350	2.42488
$616$	0	0
$617$	−37.1964	−1.49747	−0.748736	−	0.662868i	$-0.769339\pi$
$617$	−37.1964	−1.49747	−0.748736	+	0.662868i	$0.769339\pi$
$618$	0	0
$619$	−3.97739	−0.159865	−0.0799325	−	0.996800i	$-0.525470\pi$
$619$	−3.97739	−0.159865	−0.0799325	+	0.996800i	$0.525470\pi$
$620$	0	0
$621$	12.0517	0.483617
$622$	0	0
$623$	14.1406	0.566530
$624$	0	0
$625$	43.0459	1.72184
$626$	0	0
$627$	−24.4920	−0.978115
$628$	0	0
$629$	−11.5216	−0.459395
$630$	0	0
$631$	−5.44626	−0.216812	−0.108406	−	0.994107i	$-0.534575\pi$
$631$	−5.44626	−0.216812	−0.108406	+	0.994107i	$0.534575\pi$
$632$	0	0
$633$	24.1057	0.958116
$634$	0	0
$635$	45.9976	1.82536
$636$	0	0
$637$	−16.3945	−0.649573
$638$	0	0
$639$	3.46003	0.136877
$640$	0	0
$641$	30.6020	1.20871	0.604354	−	0.796716i	$-0.293431\pi$
$641$	30.6020	1.20871	0.604354	+	0.796716i	$0.293431\pi$
$642$	0	0
$643$	3.97037	0.156576	0.0782880	−	0.996931i	$-0.475055\pi$
$643$	3.97037	0.156576	0.0782880	+	0.996931i	$0.475055\pi$
$644$	0	0
$645$	13.6673	0.538148
$646$	0	0
$647$	22.7945	0.896146	0.448073	−	0.893997i	$-0.352111\pi$
$647$	22.7945	0.896146	0.448073	+	0.893997i	$0.352111\pi$
$648$	0	0
$649$	−1.78190	−0.0699456
$650$	0	0
$651$	−31.3639	−1.22925
$652$	0	0
$653$	9.94888	0.389330	0.194665	−	0.980870i	$-0.437638\pi$
$653$	9.94888	0.389330	0.194665	+	0.980870i	$0.437638\pi$
$654$	0	0
$655$	−55.3282	−2.16185
$656$	0	0
$657$	4.06527	0.158601
$658$	0	0
$659$	−20.9897	−0.817644	−0.408822	−	0.912614i	$-0.634060\pi$
$659$	−20.9897	−0.817644	−0.408822	+	0.912614i	$0.634060\pi$
$660$	0	0
$661$	−29.4220	−1.14438	−0.572192	−	0.820120i	$-0.693907\pi$
$661$	−29.4220	−1.14438	−0.572192	+	0.820120i	$0.693907\pi$
$662$	0	0
$663$	−3.91722	−0.152132
$664$	0	0
$665$	97.8914	3.79606
$666$	0	0
$667$	6.79404	0.263066
$668$	0	0
$669$	41.0526	1.58719
$670$	0	0
$671$	−12.6923	−0.489982
$672$	0	0
$673$	−30.6723	−1.18233	−0.591165	−	0.806550i	$-0.701332\pi$
$673$	−30.6723	−1.18233	−0.591165	+	0.806550i	$0.701332\pi$
$674$	0	0
$675$	−18.3712	−0.707106
$676$	0	0
$677$	−4.06075	−0.156067	−0.0780336	−	0.996951i	$-0.524864\pi$
$677$	−4.06075	−0.156067	−0.0780336	+	0.996951i	$0.524864\pi$
$678$	0	0
$679$	13.0001	0.498897
$680$	0	0
$681$	−15.7040	−0.601777
$682$	0	0
$683$	19.4486	0.744179	0.372090	−	0.928197i	$-0.378641\pi$
$683$	19.4486	0.744179	0.372090	+	0.928197i	$0.378641\pi$
$684$	0	0
$685$	−48.3249	−1.84640
$686$	0	0
$687$	−47.0778	−1.79613
$688$	0	0
$689$	23.2185	0.884555
$690$	0	0
$691$	20.9819	0.798189	0.399095	−	0.916910i	$-0.369324\pi$
$691$	20.9819	0.798189	0.399095	+	0.916910i	$0.369324\pi$
$692$	0	0
$693$	−16.5695	−0.629422
$694$	0	0
$695$	−52.9145	−2.00716
$696$	0	0
$697$	6.51764	0.246873
$698$	0	0
$699$	29.6586	1.12179
$700$	0	0
$701$	23.2156	0.876842	0.438421	−	0.898770i	$-0.355538\pi$
$701$	23.2156	0.876842	0.438421	+	0.898770i	$0.355538\pi$
$702$	0	0
$703$	−68.9115	−2.59905
$704$	0	0
$705$	77.7033	2.92648
$706$	0	0
$707$	−5.64463	−0.212288
$708$	0	0
$709$	−13.9110	−0.522439	−0.261220	−	0.965279i	$-0.584125\pi$
$709$	−13.9110	−0.522439	−0.261220	+	0.965279i	$0.584125\pi$
$710$	0	0
$711$	−14.7364	−0.552658
$712$	0	0
$713$	24.4218	0.914605
$714$	0	0
$715$	12.1948	0.456061
$716$	0	0
$717$	62.9027	2.34915
$718$	0	0
$719$	−1.21703	−0.0453876	−0.0226938	−	0.999742i	$-0.507224\pi$
$719$	−1.21703	−0.0453876	−0.0226938	+	0.999742i	$0.507224\pi$
$720$	0	0
$721$	−10.5379	−0.392453
$722$	0	0
$723$	23.8952	0.888672
$724$	0	0
$725$	−10.3566	−0.384634
$726$	0	0
$727$	15.0917	0.559721	0.279861	−	0.960041i	$-0.409712\pi$
$727$	15.0917	0.559721	0.279861	+	0.960041i	$0.409712\pi$
$728$	0	0
$729$	−12.8801	−0.477042
$730$	0	0
$731$	1.48131	0.0547881
$732$	0	0
$733$	−26.6340	−0.983751	−0.491875	−	0.870666i	$-0.663688\pi$
$733$	−26.6340	−0.983751	−0.491875	+	0.870666i	$0.663688\pi$
$734$	0	0
$735$	88.7396	3.27321
$736$	0	0
$737$	18.0803	0.665998
$738$	0	0
$739$	43.7900	1.61084	0.805421	−	0.592703i	$-0.201939\pi$
$739$	43.7900	1.61084	0.805421	+	0.592703i	$0.201939\pi$
$740$	0	0
$741$	−23.4292	−0.860694
$742$	0	0
$743$	15.4703	0.567552	0.283776	−	0.958891i	$-0.408413\pi$
$743$	15.4703	0.567552	0.283776	+	0.958891i	$0.408413\pi$
$744$	0	0
$745$	22.3759	0.819791
$746$	0	0
$747$	35.5097	1.29923
$748$	0	0
$749$	−10.7504	−0.392811
$750$	0	0
$751$	13.5136	0.493117	0.246559	−	0.969128i	$-0.420700\pi$
$751$	13.5136	0.493117	0.246559	+	0.969128i	$0.420700\pi$
$752$	0	0
$753$	−40.3751	−1.47135
$754$	0	0
$755$	−78.5273	−2.85790
$756$	0	0
$757$	53.8763	1.95817	0.979085	−	0.203454i	$-0.0652167\pi$
$757$	53.8763	1.95817	0.979085	+	0.203454i	$0.0652167\pi$
$758$	0	0
$759$	29.8704	1.08423
$760$	0	0
$761$	−33.1510	−1.20172	−0.600861	−	0.799354i	$-0.705175\pi$
$761$	−33.1510	−1.20172	−0.600861	+	0.799354i	$0.705175\pi$
$762$	0	0
$763$	10.5665	0.382534
$764$	0	0
$765$	9.15827	0.331118
$766$	0	0
$767$	−1.70458	−0.0615487
$768$	0	0
$769$	−12.3659	−0.445926	−0.222963	−	0.974827i	$-0.571573\pi$
$769$	−12.3659	−0.445926	−0.222963	+	0.974827i	$0.571573\pi$
$770$	0	0
$771$	49.9989	1.80067
$772$	0	0
$773$	−53.4567	−1.92270	−0.961352	−	0.275324i	$-0.911215\pi$
$773$	−53.4567	−1.92270	−0.961352	+	0.275324i	$0.911215\pi$
$774$	0	0
$775$	−37.2278	−1.33726
$776$	0	0
$777$	−107.935	−3.87213
$778$	0	0
$779$	38.9826	1.39670
$780$	0	0
$781$	−2.70287	−0.0967163
$782$	0	0
$783$	1.53880	0.0549922
$784$	0	0
$785$	−45.7554	−1.63308
$786$	0	0
$787$	−43.7566	−1.55975	−0.779877	−	0.625933i	$-0.784718\pi$
$787$	−43.7566	−1.55975	−0.779877	+	0.625933i	$0.784718\pi$
$788$	0	0
$789$	−35.3321	−1.25786
$790$	0	0
$791$	76.1157	2.70636
$792$	0	0
$793$	−12.1416	−0.431160
$794$	0	0
$795$	−125.676	−4.45729
$796$	0	0
$797$	37.2229	1.31850	0.659251	−	0.751923i	$-0.270874\pi$
$797$	37.2229	1.31850	0.659251	+	0.751923i	$0.270874\pi$
$798$	0	0
$799$	8.42177	0.297941
$800$	0	0
$801$	7.91254	0.279576
$802$	0	0
$803$	−3.17567	−0.112067
$804$	0	0
$805$	−119.388	−4.20788
$806$	0	0
$807$	−10.1335	−0.356716
$808$	0	0
$809$	0.195713	0.00688090	0.00344045	−	0.999994i	$-0.498905\pi$
$809$	0.195713	0.00688090	0.00344045	+	0.999994i	$0.498905\pi$
$810$	0	0
$811$	15.9675	0.560696	0.280348	−	0.959898i	$-0.409550\pi$
$811$	15.9675	0.560696	0.280348	+	0.959898i	$0.409550\pi$
$812$	0	0
$813$	20.7156	0.726526
$814$	0	0
$815$	36.3044	1.27169
$816$	0	0
$817$	8.85983	0.309966
$818$	0	0
$819$	−15.8505	−0.553861
$820$	0	0
$821$	4.21337	0.147047	0.0735237	−	0.997293i	$-0.476576\pi$
$821$	4.21337	0.147047	0.0735237	+	0.997293i	$0.476576\pi$
$822$	0	0
$823$	−23.8731	−0.832163	−0.416082	−	0.909327i	$-0.636597\pi$
$823$	−23.8731	−0.832163	−0.416082	+	0.909327i	$0.636597\pi$
$824$	0	0
$825$	−45.5333	−1.58527
$826$	0	0
$827$	−17.2488	−0.599798	−0.299899	−	0.953971i	$-0.596953\pi$
$827$	−17.2488	−0.599798	−0.299899	+	0.953971i	$0.596953\pi$
$828$	0	0
$829$	−27.3331	−0.949319	−0.474660	−	0.880169i	$-0.657429\pi$
$829$	−27.3331	−0.949319	−0.474660	+	0.880169i	$0.657429\pi$
$830$	0	0
$831$	52.6481	1.82634
$832$	0	0
$833$	9.61792	0.333241
$834$	0	0
$835$	−39.0227	−1.35044
$836$	0	0
$837$	5.53137	0.191192
$838$	0	0
$839$	7.11422	0.245610	0.122805	−	0.992431i	$-0.460811\pi$
$839$	7.11422	0.245610	0.122805	+	0.992431i	$0.460811\pi$
$840$	0	0
$841$	−28.1325	−0.970087
$842$	0	0
$843$	9.09227	0.313155
$844$	0	0
$845$	−40.5282	−1.39421
$846$	0	0
$847$	−31.8980	−1.09603
$848$	0	0
$849$	−23.5382	−0.807829
$850$	0	0
$851$	84.0444	2.88100
$852$	0	0
$853$	12.5697	0.430380	0.215190	−	0.976572i	$-0.430963\pi$
$853$	12.5697	0.430380	0.215190	+	0.976572i	$0.430963\pi$
$854$	0	0
$855$	54.7764	1.87331
$856$	0	0
$857$	47.0208	1.60620	0.803100	−	0.595844i	$-0.203182\pi$
$857$	47.0208	1.60620	0.803100	+	0.595844i	$0.203182\pi$
$858$	0	0
$859$	−52.2762	−1.78364	−0.891821	−	0.452389i	$-0.850572\pi$
$859$	−52.2762	−1.78364	−0.891821	+	0.452389i	$0.850572\pi$
$860$	0	0
$861$	61.0576	2.08084
$862$	0	0
$863$	23.4546	0.798405	0.399202	−	0.916863i	$-0.369287\pi$
$863$	23.4546	0.798405	0.399202	+	0.916863i	$0.369287\pi$
$864$	0	0
$865$	58.4240	1.98648
$866$	0	0
$867$	2.29806	0.0780461
$868$	0	0
$869$	11.5116	0.390505
$870$	0	0
$871$	17.2958	0.586046
$872$	0	0
$873$	7.27437	0.246200
$874$	0	0
$875$	100.157	3.38592
$876$	0	0
$877$	−15.4597	−0.522039	−0.261019	−	0.965334i	$-0.584059\pi$
$877$	−15.4597	−0.522039	−0.261019	+	0.965334i	$0.584059\pi$
$878$	0	0
$879$	−23.5079	−0.792902
$880$	0	0
$881$	9.07731	0.305823	0.152911	−	0.988240i	$-0.451135\pi$
$881$	9.07731	0.305823	0.152911	+	0.988240i	$0.451135\pi$
$882$	0	0
$883$	−21.4868	−0.723089	−0.361544	−	0.932355i	$-0.617750\pi$
$883$	−21.4868	−0.723089	−0.361544	+	0.932355i	$0.617750\pi$
$884$	0	0
$885$	9.22649	0.310145
$886$	0	0
$887$	−17.8488	−0.599303	−0.299651	−	0.954049i	$-0.596870\pi$
$887$	−17.8488	−0.599303	−0.299651	+	0.954049i	$0.596870\pi$
$888$	0	0
$889$	46.7033	1.56638
$890$	0	0
$891$	18.9593	0.635160
$892$	0	0
$893$	50.3713	1.68561
$894$	0	0
$895$	−39.3197	−1.31431
$896$	0	0
$897$	28.5742	0.954065
$898$	0	0
$899$	3.11826	0.104000
$900$	0	0
$901$	−13.6213	−0.453790
$902$	0	0
$903$	13.8770	0.461796
$904$	0	0
$905$	−100.146	−3.32895
$906$	0	0
$907$	27.6005	0.916459	0.458230	−	0.888834i	$-0.348484\pi$
$907$	27.6005	0.916459	0.458230	+	0.888834i	$0.348484\pi$
$908$	0	0
$909$	−3.15853	−0.104762
$910$	0	0
$911$	9.47688	0.313983	0.156991	−	0.987600i	$-0.449820\pi$
$911$	9.47688	0.313983	0.156991	+	0.987600i	$0.449820\pi$
$912$	0	0
$913$	−27.7391	−0.918031
$914$	0	0
$915$	65.7196	2.17262
$916$	0	0
$917$	−56.1771	−1.85513
$918$	0	0
$919$	−28.6489	−0.945040	−0.472520	−	0.881320i	$-0.656656\pi$
$919$	−28.6489	−0.945040	−0.472520	+	0.881320i	$0.656656\pi$
$920$	0	0
$921$	53.1976	1.75292
$922$	0	0
$923$	−2.58559	−0.0851056
$924$	0	0
$925$	−128.114	−4.21237
$926$	0	0
$927$	−5.89664	−0.193671
$928$	0	0
$929$	47.0289	1.54297	0.771484	−	0.636249i	$-0.219515\pi$
$929$	47.0289	1.54297	0.771484	+	0.636249i	$0.219515\pi$
$930$	0	0
$931$	57.5256	1.88533
$932$	0	0
$933$	66.6725	2.18276
$934$	0	0
$935$	−7.15416	−0.233966
$936$	0	0
$937$	−47.8459	−1.56306	−0.781529	−	0.623869i	$-0.785560\pi$
$937$	−47.8459	−1.56306	−0.781529	+	0.623869i	$0.785560\pi$
$938$	0	0
$939$	44.8369	1.46320
$940$	0	0
$941$	0.439200	0.0143175	0.00715876	−	0.999974i	$-0.497721\pi$
$941$	0.439200	0.0143175	0.00715876	+	0.999974i	$0.497721\pi$
$942$	0	0
$943$	−47.5431	−1.54822
$944$	0	0
$945$	−27.0405	−0.879628
$946$	0	0
$947$	35.0470	1.13888	0.569438	−	0.822035i	$-0.307161\pi$
$947$	35.0470	1.13888	0.569438	+	0.822035i	$0.307161\pi$
$948$	0	0
$949$	−3.03787	−0.0986134
$950$	0	0
$951$	−60.4460	−1.96010
$952$	0	0
$953$	36.5882	1.18521	0.592604	−	0.805494i	$-0.298100\pi$
$953$	36.5882	1.18521	0.592604	+	0.805494i	$0.298100\pi$
$954$	0	0
$955$	70.5579	2.28320
$956$	0	0
$957$	3.81395	0.123287
$958$	0	0
$959$	−49.0663	−1.58443
$960$	0	0
$961$	−19.7911	−0.638422
$962$	0	0
$963$	−6.01553	−0.193848
$964$	0	0
$965$	−96.9755	−3.12175
$966$	0	0
$967$	2.24389	0.0721586	0.0360793	−	0.999349i	$-0.488513\pi$
$967$	2.24389	0.0721586	0.0360793	+	0.999349i	$0.488513\pi$
$968$	0	0
$969$	13.7449	0.441549
$970$	0	0
$971$	−3.21079	−0.103039	−0.0515195	−	0.998672i	$-0.516406\pi$
$971$	−3.21079	−0.103039	−0.0515195	+	0.998672i	$0.516406\pi$
$972$	0	0
$973$	−53.7263	−1.72239
$974$	0	0
$975$	−43.5575	−1.39496
$976$	0	0
$977$	−55.7462	−1.78348	−0.891740	−	0.452549i	$-0.850515\pi$
$977$	−55.7462	−1.78348	−0.891740	+	0.452549i	$0.850515\pi$
$978$	0	0
$979$	−6.18104	−0.197547
$980$	0	0
$981$	5.91264	0.188776
$982$	0	0
$983$	42.0609	1.34153	0.670767	−	0.741669i	$-0.265965\pi$
$983$	42.0609	1.34153	0.670767	+	0.741669i	$0.265965\pi$
$984$	0	0
$985$	−51.7071	−1.64753
$986$	0	0
$987$	78.8955	2.51127
$988$	0	0
$989$	−10.8054	−0.343593
$990$	0	0
$991$	−6.55930	−0.208363	−0.104182	−	0.994558i	$-0.533222\pi$
$991$	−6.55930	−0.208363	−0.104182	+	0.994558i	$0.533222\pi$
$992$	0	0
$993$	−9.35413	−0.296844
$994$	0	0
$995$	2.83620	0.0899135
$996$	0	0
$997$	−7.30631	−0.231393	−0.115697	−	0.993285i	$-0.536910\pi$
$997$	−7.30631	−0.231393	−0.115697	+	0.993285i	$0.536910\pi$
$998$	0	0
$999$	19.0354	0.602254

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.a.j.1.17	✓	21

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.a.j.1.17	✓	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)

\(f(q)\)	\(=\)	\(q+2.29806 q^{3} +4.01491 q^{5} +4.07651 q^{7} +2.28106 q^{9} +O(q^{10})\)	\(q+2.29806 q^{3} +4.01491 q^{5} +4.07651 q^{7} +2.28106 q^{9} -1.78190 q^{11} -1.70458 q^{13} +9.22649 q^{15} +1.00000 q^{17} +5.98109 q^{19} +9.36805 q^{21} -7.29452 q^{23} +11.1195 q^{25} -1.65216 q^{27} -0.931389 q^{29} -3.34797 q^{31} -4.09490 q^{33} +16.3668 q^{35} -11.5216 q^{37} -3.91722 q^{39} +6.51764 q^{41} +1.48131 q^{43} +9.15827 q^{45} +8.42177 q^{47} +9.61792 q^{49} +2.29806 q^{51} -13.6213 q^{53} -7.15416 q^{55} +13.7449 q^{57} +1.00000 q^{59} +7.12292 q^{61} +9.29878 q^{63} -6.84373 q^{65} -10.1467 q^{67} -16.7632 q^{69} +1.51685 q^{71} +1.78218 q^{73} +25.5533 q^{75} -7.26392 q^{77} -6.46031 q^{79} -10.6399 q^{81} +15.5672 q^{83} +4.01491 q^{85} -2.14038 q^{87} +3.46879 q^{89} -6.94872 q^{91} -7.69383 q^{93} +24.0135 q^{95} +3.18902 q^{97} -4.06462 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 21 q + 9 q^{3} + q^{5} + 11 q^{7} + 26 q^{9}+O(q^{10}) \) 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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