LMFDB - Embedded newform 8032.2.a.i.1.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8032,2,Mod(1,8032)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8032, 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("8032.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8032 = 2^{5} \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8032.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:	$64.1358429035$
Analytic rank:	$0$
Dimension:	$30$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
Character	$\chi$	$=$	8032.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.07677 q^{3} -3.37253 q^{5} +5.03712 q^{7} -1.84058 q^{9} +O(q^{10})$	$q+1.07677 q^{3} -3.37253 q^{5} +5.03712 q^{7} -1.84058 q^{9} -0.757023 q^{11} -5.91567 q^{13} -3.63142 q^{15} -0.402177 q^{17} -1.31113 q^{19} +5.42379 q^{21} +4.16789 q^{23} +6.37394 q^{25} -5.21217 q^{27} +4.50326 q^{29} -4.73960 q^{31} -0.815136 q^{33} -16.9878 q^{35} +0.609418 q^{37} -6.36979 q^{39} +2.90904 q^{41} -3.27317 q^{43} +6.20739 q^{45} -7.08721 q^{47} +18.3725 q^{49} -0.433050 q^{51} +10.6039 q^{53} +2.55308 q^{55} -1.41178 q^{57} +1.30223 q^{59} -10.2255 q^{61} -9.27120 q^{63} +19.9507 q^{65} +3.47089 q^{67} +4.48784 q^{69} +5.95862 q^{71} +4.14982 q^{73} +6.86324 q^{75} -3.81321 q^{77} +13.9915 q^{79} -0.0905505 q^{81} +10.2343 q^{83} +1.35635 q^{85} +4.84895 q^{87} -4.81173 q^{89} -29.7979 q^{91} -5.10344 q^{93} +4.42183 q^{95} -2.06276 q^{97} +1.39336 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q + 3 q^{3} + 13 q^{7} + 35 q^{9}+O(q^{10}) $ 30 * q + 3 * q^3 + 13 * q^7 + 35 * q^9	$ 30 q + 3 q^{3} + 13 q^{7} + 35 q^{9} + 13 q^{11} - 7 q^{13} + 28 q^{15} - 9 q^{17} - 17 q^{19} - 6 q^{21} + 43 q^{23} + 34 q^{25} + 12 q^{27} - q^{29} + 39 q^{31} + 17 q^{35} - q^{37} + 48 q^{39} - 3 q^{41} - 19 q^{43} + 66 q^{47} + 25 q^{49} - 14 q^{51} + 3 q^{53} + 50 q^{55} - 14 q^{57} + 27 q^{59} + 15 q^{61} + 75 q^{63} - 6 q^{65} + 8 q^{67} + 18 q^{69} + 64 q^{71} - 15 q^{73} + 9 q^{75} + 71 q^{79} + 6 q^{81} + 60 q^{83} + 15 q^{85} + 64 q^{87} - 32 q^{89} - 26 q^{91} - 4 q^{93} + 72 q^{95} - 4 q^{97} + 13 q^{99}+O(q^{100}) $ 30 * q + 3 * q^3 + 13 * q^7 + 35 * q^9 + 13 * q^11 - 7 * q^13 + 28 * q^15 - 9 * q^17 - 17 * q^19 - 6 * q^21 + 43 * q^23 + 34 * q^25 + 12 * q^27 - q^29 + 39 * q^31 + 17 * q^35 - q^37 + 48 * q^39 - 3 * q^41 - 19 * q^43 + 66 * q^47 + 25 * q^49 - 14 * q^51 + 3 * q^53 + 50 * q^55 - 14 * q^57 + 27 * q^59 + 15 * q^61 + 75 * q^63 - 6 * q^65 + 8 * q^67 + 18 * q^69 + 64 * q^71 - 15 * q^73 + 9 * q^75 + 71 * q^79 + 6 * q^81 + 60 * q^83 + 15 * q^85 + 64 * q^87 - 32 * q^89 - 26 * q^91 - 4 * q^93 + 72 * q^95 - 4 * q^97 + 13 * 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
$3$	1.07677	0.621671	0.310835	−	0.950464i	$-0.399391\pi$
$3$	1.07677	0.621671	0.310835	+	0.950464i	$0.399391\pi$
$4$	0	0
$5$	−3.37253	−1.50824	−0.754120	−	0.656737i	$-0.771936\pi$
$5$	−3.37253	−1.50824	−0.754120	+	0.656737i	$0.771936\pi$
$6$	0	0
$7$	5.03712	1.90385	0.951926	−	0.306329i	$-0.0991008\pi$
$7$	5.03712	1.90385	0.951926	+	0.306329i	$0.0991008\pi$
$8$	0	0
$9$	−1.84058	−0.613525
$10$	0	0
$11$	−0.757023	−0.228251	−0.114126	−	0.993466i	$-0.536407\pi$
$11$	−0.757023	−0.228251	−0.114126	+	0.993466i	$0.536407\pi$
$12$	0	0
$13$	−5.91567	−1.64071	−0.820355	−	0.571854i	$-0.806224\pi$
$13$	−5.91567	−1.64071	−0.820355	+	0.571854i	$0.806224\pi$
$14$	0	0
$15$	−3.63142	−0.937629
$16$	0	0
$17$	−0.402177	−0.0975421	−0.0487711	−	0.998810i	$-0.515530\pi$
$17$	−0.402177	−0.0975421	−0.0487711	+	0.998810i	$0.515530\pi$
$18$	0	0
$19$	−1.31113	−0.300794	−0.150397	−	0.988626i	$-0.548055\pi$
$19$	−1.31113	−0.300794	−0.150397	+	0.988626i	$0.548055\pi$
$20$	0	0
$21$	5.42379	1.18357
$22$	0	0
$23$	4.16789	0.869065	0.434533	−	0.900656i	$-0.356914\pi$
$23$	4.16789	0.869065	0.434533	+	0.900656i	$0.356914\pi$
$24$	0	0
$25$	6.37394	1.27479
$26$	0	0
$27$	−5.21217	−1.00308
$28$	0	0
$29$	4.50326	0.836234	0.418117	−	0.908393i	$-0.362690\pi$
$29$	4.50326	0.836234	0.418117	+	0.908393i	$0.362690\pi$
$30$	0	0
$31$	−4.73960	−0.851258	−0.425629	−	0.904898i	$-0.639947\pi$
$31$	−4.73960	−0.851258	−0.425629	+	0.904898i	$0.639947\pi$
$32$	0	0
$33$	−0.815136	−0.141897
$34$	0	0
$35$	−16.9878	−2.87146
$36$	0	0
$37$	0.609418	0.100188	0.0500939	−	0.998745i	$-0.484048\pi$
$37$	0.609418	0.100188	0.0500939	+	0.998745i	$0.484048\pi$
$38$	0	0
$39$	−6.36979	−1.01998
$40$	0	0
$41$	2.90904	0.454315	0.227158	−	0.973858i	$-0.427057\pi$
$41$	2.90904	0.454315	0.227158	+	0.973858i	$0.427057\pi$
$42$	0	0
$43$	−3.27317	−0.499154	−0.249577	−	0.968355i	$-0.580291\pi$
$43$	−3.27317	−0.499154	−0.249577	+	0.968355i	$0.580291\pi$
$44$	0	0
$45$	6.20739	0.925344
$46$	0	0
$47$	−7.08721	−1.03378	−0.516888	−	0.856053i	$-0.672909\pi$
$47$	−7.08721	−1.03378	−0.516888	+	0.856053i	$0.672909\pi$
$48$	0	0
$49$	18.3725	2.62465
$50$	0	0
$51$	−0.433050	−0.0606391
$52$	0	0
$53$	10.6039	1.45655	0.728277	−	0.685282i	$-0.240321\pi$
$53$	10.6039	1.45655	0.728277	+	0.685282i	$0.240321\pi$
$54$	0	0
$55$	2.55308	0.344257
$56$	0	0
$57$	−1.41178	−0.186995
$58$	0	0
$59$	1.30223	0.169536	0.0847678	−	0.996401i	$-0.472985\pi$
$59$	1.30223	0.169536	0.0847678	+	0.996401i	$0.472985\pi$
$60$	0	0
$61$	−10.2255	−1.30924	−0.654621	−	0.755958i	$-0.727172\pi$
$61$	−10.2255	−1.30924	−0.654621	+	0.755958i	$0.727172\pi$
$62$	0	0
$63$	−9.27120	−1.16806
$64$	0	0
$65$	19.9507	2.47459
$66$	0	0
$67$	3.47089	0.424037	0.212019	−	0.977266i	$-0.431996\pi$
$67$	3.47089	0.424037	0.212019	+	0.977266i	$0.431996\pi$
$68$	0	0
$69$	4.48784	0.540273
$70$	0	0
$71$	5.95862	0.707159	0.353579	−	0.935405i	$-0.384964\pi$
$71$	5.95862	0.707159	0.353579	+	0.935405i	$0.384964\pi$
$72$	0	0
$73$	4.14982	0.485700	0.242850	−	0.970064i	$-0.421918\pi$
$73$	4.14982	0.485700	0.242850	+	0.970064i	$0.421918\pi$
$74$	0	0
$75$	6.86324	0.792498
$76$	0	0
$77$	−3.81321	−0.434556
$78$	0	0
$79$	13.9915	1.57417	0.787085	−	0.616844i	$-0.211589\pi$
$79$	13.9915	1.57417	0.787085	+	0.616844i	$0.211589\pi$
$80$	0	0
$81$	−0.0905505	−0.0100612
$82$	0	0
$83$	10.2343	1.12336	0.561679	−	0.827355i	$-0.310156\pi$
$83$	10.2343	1.12336	0.561679	+	0.827355i	$0.310156\pi$
$84$	0	0
$85$	1.35635	0.147117
$86$	0	0
$87$	4.84895	0.519862
$88$	0	0
$89$	−4.81173	−0.510043	−0.255021	−	0.966935i	$-0.582083\pi$
$89$	−4.81173	−0.510043	−0.255021	+	0.966935i	$0.582083\pi$
$90$	0	0
$91$	−29.7979	−3.12367
$92$	0	0
$93$	−5.10344	−0.529202
$94$	0	0
$95$	4.42183	0.453670
$96$	0	0
$97$	−2.06276	−0.209441	−0.104721	−	0.994502i	$-0.533395\pi$
$97$	−2.06276	−0.209441	−0.104721	+	0.994502i	$0.533395\pi$
$98$	0	0
$99$	1.39336	0.140038
$100$	0	0
$101$	0.262418	0.0261116	0.0130558	−	0.999915i	$-0.495844\pi$
$101$	0.262418	0.0261116	0.0130558	+	0.999915i	$0.495844\pi$
$102$	0	0
$103$	3.02231	0.297797	0.148898	−	0.988853i	$-0.452427\pi$
$103$	3.02231	0.297797	0.148898	+	0.988853i	$0.452427\pi$
$104$	0	0
$105$	−18.2919	−1.78511
$106$	0	0
$107$	6.31397	0.610395	0.305197	−	0.952289i	$-0.401278\pi$
$107$	6.31397	0.610395	0.305197	+	0.952289i	$0.401278\pi$
$108$	0	0
$109$	2.88471	0.276305	0.138153	−	0.990411i	$-0.455884\pi$
$109$	2.88471	0.276305	0.138153	+	0.990411i	$0.455884\pi$
$110$	0	0
$111$	0.656200	0.0622838
$112$	0	0
$113$	0.326348	0.0307003	0.0153501	−	0.999882i	$-0.495114\pi$
$113$	0.326348	0.0307003	0.0153501	+	0.999882i	$0.495114\pi$
$114$	0	0
$115$	−14.0563	−1.31076
$116$	0	0
$117$	10.8882	1.00662
$118$	0	0
$119$	−2.02581	−0.185706
$120$	0	0
$121$	−10.4269	−0.947901
$122$	0	0
$123$	3.13235	0.282435
$124$	0	0
$125$	−4.63364	−0.414446
$126$	0	0
$127$	19.6086	1.73999	0.869993	−	0.493064i	$-0.164123\pi$
$127$	19.6086	1.73999	0.869993	+	0.493064i	$0.164123\pi$
$128$	0	0
$129$	−3.52443	−0.310309
$130$	0	0
$131$	−8.82335	−0.770900	−0.385450	−	0.922729i	$-0.625954\pi$
$131$	−8.82335	−0.770900	−0.385450	+	0.922729i	$0.625954\pi$
$132$	0	0
$133$	−6.60433	−0.572668
$134$	0	0
$135$	17.5782	1.51289
$136$	0	0
$137$	4.92333	0.420628	0.210314	−	0.977634i	$-0.432551\pi$
$137$	4.92333	0.420628	0.210314	+	0.977634i	$0.432551\pi$
$138$	0	0
$139$	−8.72988	−0.740459	−0.370229	−	0.928940i	$-0.620721\pi$
$139$	−8.72988	−0.740459	−0.370229	+	0.928940i	$0.620721\pi$
$140$	0	0
$141$	−7.63126	−0.642668
$142$	0	0
$143$	4.47830	0.374494
$144$	0	0
$145$	−15.1874	−1.26124
$146$	0	0
$147$	19.7829	1.63167
$148$	0	0
$149$	23.4492	1.92103	0.960517	−	0.278221i	$-0.0897447\pi$
$149$	23.4492	1.92103	0.960517	+	0.278221i	$0.0897447\pi$
$150$	0	0
$151$	9.51376	0.774219	0.387109	−	0.922034i	$-0.373474\pi$
$151$	9.51376	0.774219	0.387109	+	0.922034i	$0.373474\pi$
$152$	0	0
$153$	0.740237	0.0598446
$154$	0	0
$155$	15.9844	1.28390
$156$	0	0
$157$	−6.76204	−0.539670	−0.269835	−	0.962907i	$-0.586969\pi$
$157$	−6.76204	−0.539670	−0.269835	+	0.962907i	$0.586969\pi$
$158$	0	0
$159$	11.4179	0.905498
$160$	0	0
$161$	20.9942	1.65457
$162$	0	0
$163$	−23.1117	−1.81025	−0.905126	−	0.425144i	$-0.860223\pi$
$163$	−23.1117	−1.81025	−0.905126	+	0.425144i	$0.860223\pi$
$164$	0	0
$165$	2.74907	0.214015
$166$	0	0
$167$	3.38510	0.261947	0.130974	−	0.991386i	$-0.458190\pi$
$167$	3.38510	0.261947	0.130974	+	0.991386i	$0.458190\pi$
$168$	0	0
$169$	21.9951	1.69193
$170$	0	0
$171$	2.41324	0.184545
$172$	0	0
$173$	23.1066	1.75677	0.878383	−	0.477958i	$-0.158623\pi$
$173$	23.1066	1.75677	0.878383	+	0.477958i	$0.158623\pi$
$174$	0	0
$175$	32.1063	2.42701
$176$	0	0
$177$	1.40219	0.105395
$178$	0	0
$179$	−6.38253	−0.477053	−0.238526	−	0.971136i	$-0.576664\pi$
$179$	−6.38253	−0.477053	−0.238526	+	0.971136i	$0.576664\pi$
$180$	0	0
$181$	10.4954	0.780119	0.390060	−	0.920790i	$-0.372454\pi$
$181$	10.4954	0.780119	0.390060	+	0.920790i	$0.372454\pi$
$182$	0	0
$183$	−11.0105	−0.813917
$184$	0	0
$185$	−2.05528	−0.151107
$186$	0	0
$187$	0.304457	0.0222641
$188$	0	0
$189$	−26.2543	−1.90972
$190$	0	0
$191$	6.85197	0.495791	0.247896	−	0.968787i	$-0.420261\pi$
$191$	6.85197	0.495791	0.247896	+	0.968787i	$0.420261\pi$
$192$	0	0
$193$	−7.59744	−0.546876	−0.273438	−	0.961890i	$-0.588161\pi$
$193$	−7.59744	−0.546876	−0.273438	+	0.961890i	$0.588161\pi$
$194$	0	0
$195$	21.4823	1.53838
$196$	0	0
$197$	−10.1345	−0.722053	−0.361026	−	0.932556i	$-0.617574\pi$
$197$	−10.1345	−0.722053	−0.361026	+	0.932556i	$0.617574\pi$
$198$	0	0
$199$	−0.534190	−0.0378677	−0.0189339	−	0.999821i	$-0.506027\pi$
$199$	−0.534190	−0.0378677	−0.0189339	+	0.999821i	$0.506027\pi$
$200$	0	0
$201$	3.73734	0.263611
$202$	0	0
$203$	22.6834	1.59207
$204$	0	0
$205$	−9.81081	−0.685217
$206$	0	0
$207$	−7.67132	−0.533194
$208$	0	0
$209$	0.992557	0.0686566
$210$	0	0
$211$	11.1227	0.765721	0.382860	−	0.923806i	$-0.374939\pi$
$211$	11.1227	0.765721	0.382860	+	0.923806i	$0.374939\pi$
$212$	0	0
$213$	6.41604	0.439620
$214$	0	0
$215$	11.0388	0.752843
$216$	0	0
$217$	−23.8739	−1.62067
$218$	0	0
$219$	4.46839	0.301946
$220$	0	0
$221$	2.37914	0.160038
$222$	0	0
$223$	−3.75907	−0.251726	−0.125863	−	0.992048i	$-0.540170\pi$
$223$	−3.75907	−0.251726	−0.125863	+	0.992048i	$0.540170\pi$
$224$	0	0
$225$	−11.7317	−0.782115
$226$	0	0
$227$	−8.64553	−0.573824	−0.286912	−	0.957957i	$-0.592629\pi$
$227$	−8.64553	−0.573824	−0.286912	+	0.957957i	$0.592629\pi$
$228$	0	0
$229$	23.3923	1.54580	0.772902	−	0.634526i	$-0.218804\pi$
$229$	23.3923	1.54580	0.772902	+	0.634526i	$0.218804\pi$
$230$	0	0
$231$	−4.10594	−0.270151
$232$	0	0
$233$	−10.9596	−0.717990	−0.358995	−	0.933339i	$-0.616881\pi$
$233$	−10.9596	−0.717990	−0.358995	+	0.933339i	$0.616881\pi$
$234$	0	0
$235$	23.9018	1.55918
$236$	0	0
$237$	15.0656	0.978616
$238$	0	0
$239$	16.5392	1.06983	0.534916	−	0.844905i	$-0.320343\pi$
$239$	16.5392	1.06983	0.534916	+	0.844905i	$0.320343\pi$
$240$	0	0
$241$	25.9029	1.66855	0.834277	−	0.551345i	$-0.185885\pi$
$241$	25.9029	1.66855	0.834277	+	0.551345i	$0.185885\pi$
$242$	0	0
$243$	15.5390	0.996827
$244$	0	0
$245$	−61.9619	−3.95860
$246$	0	0
$247$	7.75622	0.493517
$248$	0	0
$249$	11.0199	0.698359
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	−3.15519	−0.198365
$254$	0	0
$255$	1.46047	0.0914583
$256$	0	0
$257$	29.4066	1.83433	0.917167	−	0.398503i	$-0.130470\pi$
$257$	29.4066	1.83433	0.917167	+	0.398503i	$0.130470\pi$
$258$	0	0
$259$	3.06971	0.190743
$260$	0	0
$261$	−8.28859	−0.513051
$262$	0	0
$263$	19.7686	1.21899	0.609493	−	0.792791i	$-0.291373\pi$
$263$	19.7686	1.21899	0.609493	+	0.792791i	$0.291373\pi$
$264$	0	0
$265$	−35.7619	−2.19683
$266$	0	0
$267$	−5.18111	−0.317079
$268$	0	0
$269$	−27.0026	−1.64638	−0.823190	−	0.567766i	$-0.807808\pi$
$269$	−27.0026	−1.64638	−0.823190	+	0.567766i	$0.807808\pi$
$270$	0	0
$271$	−13.2890	−0.807249	−0.403624	−	0.914925i	$-0.632250\pi$
$271$	−13.2890	−0.807249	−0.403624	+	0.914925i	$0.632250\pi$
$272$	0	0
$273$	−32.0854	−1.94189
$274$	0	0
$275$	−4.82522	−0.290972
$276$	0	0
$277$	19.7914	1.18915	0.594574	−	0.804041i	$-0.297321\pi$
$277$	19.7914	1.18915	0.594574	+	0.804041i	$0.297321\pi$
$278$	0	0
$279$	8.72360	0.522268
$280$	0	0
$281$	20.5795	1.22767	0.613834	−	0.789435i	$-0.289626\pi$
$281$	20.5795	1.22767	0.613834	+	0.789435i	$0.289626\pi$
$282$	0	0
$283$	−20.6925	−1.23004	−0.615020	−	0.788511i	$-0.710852\pi$
$283$	−20.6925	−1.23004	−0.615020	+	0.788511i	$0.710852\pi$
$284$	0	0
$285$	4.76127	0.282033
$286$	0	0
$287$	14.6532	0.864949
$288$	0	0
$289$	−16.8383	−0.990486
$290$	0	0
$291$	−2.22110	−0.130203
$292$	0	0
$293$	−14.0104	−0.818493	−0.409247	−	0.912424i	$-0.634208\pi$
$293$	−14.0104	−0.818493	−0.409247	+	0.912424i	$0.634208\pi$
$294$	0	0
$295$	−4.39180	−0.255700
$296$	0	0
$297$	3.94573	0.228954
$298$	0	0
$299$	−24.6559	−1.42588
$300$	0	0
$301$	−16.4873	−0.950314
$302$	0	0
$303$	0.282563	0.0162328
$304$	0	0
$305$	34.4858	1.97465
$306$	0	0
$307$	−8.08209	−0.461269	−0.230635	−	0.973040i	$-0.574080\pi$
$307$	−8.08209	−0.461269	−0.230635	+	0.973040i	$0.574080\pi$
$308$	0	0
$309$	3.25431	0.185131
$310$	0	0
$311$	−29.8303	−1.69152	−0.845760	−	0.533564i	$-0.820852\pi$
$311$	−29.8303	−1.69152	−0.845760	+	0.533564i	$0.820852\pi$
$312$	0	0
$313$	−9.97460	−0.563798	−0.281899	−	0.959444i	$-0.590964\pi$
$313$	−9.97460	−0.563798	−0.281899	+	0.959444i	$0.590964\pi$
$314$	0	0
$315$	31.2674	1.76172
$316$	0	0
$317$	19.3799	1.08848	0.544241	−	0.838929i	$-0.316818\pi$
$317$	19.3799	1.08848	0.544241	+	0.838929i	$0.316818\pi$
$318$	0	0
$319$	−3.40907	−0.190871
$320$	0	0
$321$	6.79867	0.379464
$322$	0	0
$323$	0.527307	0.0293401
$324$	0	0
$325$	−37.7061	−2.09156
$326$	0	0
$327$	3.10616	0.171771
$328$	0	0
$329$	−35.6991	−1.96815
$330$	0	0
$331$	−11.4999	−0.632089	−0.316045	−	0.948744i	$-0.602355\pi$
$331$	−11.4999	−0.632089	−0.316045	+	0.948744i	$0.602355\pi$
$332$	0	0
$333$	−1.12168	−0.0614677
$334$	0	0
$335$	−11.7057	−0.639550
$336$	0	0
$337$	−6.09477	−0.332003	−0.166002	−	0.986125i	$-0.553086\pi$
$337$	−6.09477	−0.332003	−0.166002	+	0.986125i	$0.553086\pi$
$338$	0	0
$339$	0.351400	0.0190855
$340$	0	0
$341$	3.58799	0.194300
$342$	0	0
$343$	57.2849	3.09309
$344$	0	0
$345$	−15.1354	−0.814861
$346$	0	0
$347$	−8.88292	−0.476860	−0.238430	−	0.971160i	$-0.576633\pi$
$347$	−8.88292	−0.476860	−0.238430	+	0.971160i	$0.576633\pi$
$348$	0	0
$349$	6.75274	0.361466	0.180733	−	0.983532i	$-0.442153\pi$
$349$	6.75274	0.361466	0.180733	+	0.983532i	$0.442153\pi$
$350$	0	0
$351$	30.8334	1.64577
$352$	0	0
$353$	−18.5511	−0.987375	−0.493687	−	0.869639i	$-0.664351\pi$
$353$	−18.5511	−0.987375	−0.493687	+	0.869639i	$0.664351\pi$
$354$	0	0
$355$	−20.0956	−1.06656
$356$	0	0
$357$	−2.18132	−0.115448
$358$	0	0
$359$	−14.2524	−0.752211	−0.376106	−	0.926577i	$-0.622737\pi$
$359$	−14.2524	−0.752211	−0.376106	+	0.926577i	$0.622737\pi$
$360$	0	0
$361$	−17.2809	−0.909523
$362$	0	0
$363$	−11.2273	−0.589283
$364$	0	0
$365$	−13.9954	−0.732552
$366$	0	0
$367$	−12.8909	−0.672900	−0.336450	−	0.941701i	$-0.609226\pi$
$367$	−12.8909	−0.672900	−0.336450	+	0.941701i	$0.609226\pi$
$368$	0	0
$369$	−5.35431	−0.278734
$370$	0	0
$371$	53.4130	2.77306
$372$	0	0
$373$	20.3696	1.05470	0.527348	−	0.849649i	$-0.323186\pi$
$373$	20.3696	1.05470	0.527348	+	0.849649i	$0.323186\pi$
$374$	0	0
$375$	−4.98935	−0.257649
$376$	0	0
$377$	−26.6398	−1.37202
$378$	0	0
$379$	10.6904	0.549129	0.274565	−	0.961569i	$-0.411466\pi$
$379$	10.6904	0.549129	0.274565	+	0.961569i	$0.411466\pi$
$380$	0	0
$381$	21.1139	1.08170
$382$	0	0
$383$	35.2270	1.80001	0.900007	−	0.435875i	$-0.143561\pi$
$383$	35.2270	1.80001	0.900007	+	0.435875i	$0.143561\pi$
$384$	0	0
$385$	12.8602	0.655415
$386$	0	0
$387$	6.02452	0.306243
$388$	0	0
$389$	13.3639	0.677577	0.338788	−	0.940863i	$-0.389983\pi$
$389$	13.3639	0.677577	0.338788	+	0.940863i	$0.389983\pi$
$390$	0	0
$391$	−1.67623	−0.0847705
$392$	0	0
$393$	−9.50068	−0.479246
$394$	0	0
$395$	−47.1868	−2.37423
$396$	0	0
$397$	−10.1264	−0.508231	−0.254116	−	0.967174i	$-0.581784\pi$
$397$	−10.1264	−0.508231	−0.254116	+	0.967174i	$0.581784\pi$
$398$	0	0
$399$	−7.11131	−0.356011
$400$	0	0
$401$	−10.2017	−0.509449	−0.254725	−	0.967014i	$-0.581985\pi$
$401$	−10.2017	−0.509449	−0.254725	+	0.967014i	$0.581985\pi$
$402$	0	0
$403$	28.0379	1.39667
$404$	0	0
$405$	0.305384	0.0151747
$406$	0	0
$407$	−0.461344	−0.0228680
$408$	0	0
$409$	−15.4489	−0.763899	−0.381949	−	0.924183i	$-0.624747\pi$
$409$	−15.4489	−0.763899	−0.381949	+	0.924183i	$0.624747\pi$
$410$	0	0
$411$	5.30127	0.261492
$412$	0	0
$413$	6.55947	0.322770
$414$	0	0
$415$	−34.5154	−1.69429
$416$	0	0
$417$	−9.40003	−0.460321
$418$	0	0
$419$	−1.35657	−0.0662726	−0.0331363	−	0.999451i	$-0.510550\pi$
$419$	−1.35657	−0.0662726	−0.0331363	+	0.999451i	$0.510550\pi$
$420$	0	0
$421$	12.8027	0.623964	0.311982	−	0.950088i	$-0.399007\pi$
$421$	12.8027	0.623964	0.311982	+	0.950088i	$0.399007\pi$
$422$	0	0
$423$	13.0445	0.634247
$424$	0	0
$425$	−2.56345	−0.124346
$426$	0	0
$427$	−51.5070	−2.49260
$428$	0	0
$429$	4.82207	0.232812
$430$	0	0
$431$	−15.8635	−0.764116	−0.382058	−	0.924138i	$-0.624785\pi$
$431$	−15.8635	−0.764116	−0.382058	+	0.924138i	$0.624785\pi$
$432$	0	0
$433$	34.2384	1.64539	0.822697	−	0.568480i	$-0.192468\pi$
$433$	34.2384	1.64539	0.822697	+	0.568480i	$0.192468\pi$
$434$	0	0
$435$	−16.3532	−0.784077
$436$	0	0
$437$	−5.46466	−0.261410
$438$	0	0
$439$	−18.6928	−0.892157	−0.446078	−	0.894994i	$-0.647180\pi$
$439$	−18.6928	−0.892157	−0.446078	+	0.894994i	$0.647180\pi$
$440$	0	0
$441$	−33.8161	−1.61029
$442$	0	0
$443$	33.2679	1.58061	0.790303	−	0.612717i	$-0.209923\pi$
$443$	33.2679	1.58061	0.790303	+	0.612717i	$0.209923\pi$
$444$	0	0
$445$	16.2277	0.769267
$446$	0	0
$447$	25.2493	1.19425
$448$	0	0
$449$	0.684242	0.0322914	0.0161457	−	0.999870i	$-0.494860\pi$
$449$	0.684242	0.0322914	0.0161457	+	0.999870i	$0.494860\pi$
$450$	0	0
$451$	−2.20221	−0.103698
$452$	0	0
$453$	10.2441	0.481309
$454$	0	0
$455$	100.494	4.71124
$456$	0	0
$457$	25.2488	1.18109	0.590544	−	0.807005i	$-0.298913\pi$
$457$	25.2488	1.18109	0.590544	+	0.807005i	$0.298913\pi$
$458$	0	0
$459$	2.09621	0.0978427
$460$	0	0
$461$	37.6134	1.75183	0.875915	−	0.482466i	$-0.160259\pi$
$461$	37.6134	1.75183	0.875915	+	0.482466i	$0.160259\pi$
$462$	0	0
$463$	16.2676	0.756022	0.378011	−	0.925801i	$-0.376608\pi$
$463$	16.2676	0.756022	0.378011	+	0.925801i	$0.376608\pi$
$464$	0	0
$465$	17.2115	0.798164
$466$	0	0
$467$	36.8920	1.70716	0.853578	−	0.520965i	$-0.174428\pi$
$467$	36.8920	1.70716	0.853578	+	0.520965i	$0.174428\pi$
$468$	0	0
$469$	17.4833	0.807304
$470$	0	0
$471$	−7.28113	−0.335497
$472$	0	0
$473$	2.47786	0.113932
$474$	0	0
$475$	−8.35708	−0.383449
$476$	0	0
$477$	−19.5172	−0.893633
$478$	0	0
$479$	−26.9055	−1.22934	−0.614672	−	0.788783i	$-0.710712\pi$
$479$	−26.9055	−1.22934	−0.614672	+	0.788783i	$0.710712\pi$
$480$	0	0
$481$	−3.60512	−0.164379
$482$	0	0
$483$	22.6058	1.02860
$484$	0	0
$485$	6.95670	0.315888
$486$	0	0
$487$	37.3191	1.69109	0.845544	−	0.533905i	$-0.179276\pi$
$487$	37.3191	1.69109	0.845544	+	0.533905i	$0.179276\pi$
$488$	0	0
$489$	−24.8859	−1.12538
$490$	0	0
$491$	36.2808	1.63733	0.818665	−	0.574271i	$-0.194714\pi$
$491$	36.2808	1.63733	0.818665	+	0.574271i	$0.194714\pi$
$492$	0	0
$493$	−1.81110	−0.0815681
$494$	0	0
$495$	−4.69914	−0.211211
$496$	0	0
$497$	30.0143	1.34632
$498$	0	0
$499$	27.3565	1.22464	0.612322	−	0.790608i	$-0.290235\pi$
$499$	27.3565	1.22464	0.612322	+	0.790608i	$0.290235\pi$
$500$	0	0
$501$	3.64496	0.162845
$502$	0	0
$503$	9.89556	0.441221	0.220611	−	0.975362i	$-0.429195\pi$
$503$	9.89556	0.441221	0.220611	+	0.975362i	$0.429195\pi$
$504$	0	0
$505$	−0.885013	−0.0393825
$506$	0	0
$507$	23.6836	1.05182
$508$	0	0
$509$	40.3080	1.78662	0.893310	−	0.449440i	$-0.148377\pi$
$509$	40.3080	1.78662	0.893310	+	0.449440i	$0.148377\pi$
$510$	0	0
$511$	20.9032	0.924701
$512$	0	0
$513$	6.83384	0.301721
$514$	0	0
$515$	−10.1928	−0.449149
$516$	0	0
$517$	5.36518	0.235960
$518$	0	0
$519$	24.8804	1.09213
$520$	0	0
$521$	11.5647	0.506661	0.253330	−	0.967380i	$-0.418474\pi$
$521$	11.5647	0.506661	0.253330	+	0.967380i	$0.418474\pi$
$522$	0	0
$523$	8.26387	0.361354	0.180677	−	0.983543i	$-0.442171\pi$
$523$	8.26387	0.361354	0.180677	+	0.983543i	$0.442171\pi$
$524$	0	0
$525$	34.5709	1.50880
$526$	0	0
$527$	1.90616	0.0830335
$528$	0	0
$529$	−5.62868	−0.244725
$530$	0	0
$531$	−2.39685	−0.104014
$532$	0	0
$533$	−17.2089	−0.745400
$534$	0	0
$535$	−21.2940	−0.920621
$536$	0	0
$537$	−6.87249	−0.296570
$538$	0	0
$539$	−13.9084	−0.599079
$540$	0	0
$541$	0.828440	0.0356174	0.0178087	−	0.999841i	$-0.494331\pi$
$541$	0.828440	0.0356174	0.0178087	+	0.999841i	$0.494331\pi$
$542$	0	0
$543$	11.3011	0.484978
$544$	0	0
$545$	−9.72876	−0.416734
$546$	0	0
$547$	−13.0696	−0.558814	−0.279407	−	0.960173i	$-0.590138\pi$
$547$	−13.0696	−0.558814	−0.279407	+	0.960173i	$0.590138\pi$
$548$	0	0
$549$	18.8208	0.803253
$550$	0	0
$551$	−5.90437	−0.251534
$552$	0	0
$553$	70.4770	2.99699
$554$	0	0
$555$	−2.21305	−0.0939389
$556$	0	0
$557$	21.0805	0.893211	0.446605	−	0.894731i	$-0.352633\pi$
$557$	21.0805	0.893211	0.446605	+	0.894731i	$0.352633\pi$
$558$	0	0
$559$	19.3630	0.818967
$560$	0	0
$561$	0.327829	0.0138409
$562$	0	0
$563$	29.2459	1.23257	0.616285	−	0.787523i	$-0.288637\pi$
$563$	29.2459	1.23257	0.616285	+	0.787523i	$0.288637\pi$
$564$	0	0
$565$	−1.10062	−0.0463034
$566$	0	0
$567$	−0.456113	−0.0191550
$568$	0	0
$569$	3.63091	0.152216	0.0761078	−	0.997100i	$-0.475751\pi$
$569$	3.63091	0.152216	0.0761078	+	0.997100i	$0.475751\pi$
$570$	0	0
$571$	−31.9734	−1.33805	−0.669024	−	0.743241i	$-0.733287\pi$
$571$	−31.9734	−1.33805	−0.669024	+	0.743241i	$0.733287\pi$
$572$	0	0
$573$	7.37797	0.308219
$574$	0	0
$575$	26.5659	1.10787
$576$	0	0
$577$	−28.5282	−1.18765	−0.593823	−	0.804596i	$-0.702382\pi$
$577$	−28.5282	−1.18765	−0.593823	+	0.804596i	$0.702382\pi$
$578$	0	0
$579$	−8.18066	−0.339977
$580$	0	0
$581$	51.5513	2.13871
$582$	0	0
$583$	−8.02738	−0.332460
$584$	0	0
$585$	−36.7209	−1.51822
$586$	0	0
$587$	16.6603	0.687646	0.343823	−	0.939035i	$-0.388278\pi$
$587$	16.6603	0.687646	0.343823	+	0.939035i	$0.388278\pi$
$588$	0	0
$589$	6.21425	0.256054
$590$	0	0
$591$	−10.9125	−0.448879
$592$	0	0
$593$	−12.9127	−0.530259	−0.265130	−	0.964213i	$-0.585415\pi$
$593$	−12.9127	−0.530259	−0.265130	+	0.964213i	$0.585415\pi$
$594$	0	0
$595$	6.83210	0.280089
$596$	0	0
$597$	−0.575198	−0.0235413
$598$	0	0
$599$	−2.38552	−0.0974696	−0.0487348	−	0.998812i	$-0.515519\pi$
$599$	−2.38552	−0.0974696	−0.0487348	+	0.998812i	$0.515519\pi$
$600$	0	0
$601$	−5.45238	−0.222407	−0.111204	−	0.993798i	$-0.535471\pi$
$601$	−5.45238	−0.222407	−0.111204	+	0.993798i	$0.535471\pi$
$602$	0	0
$603$	−6.38844	−0.260158
$604$	0	0
$605$	35.1651	1.42966
$606$	0	0
$607$	−17.6882	−0.717942	−0.358971	−	0.933349i	$-0.616872\pi$
$607$	−17.6882	−0.717942	−0.358971	+	0.933349i	$0.616872\pi$
$608$	0	0
$609$	24.4247	0.989740
$610$	0	0
$611$	41.9255	1.69613
$612$	0	0
$613$	−45.9927	−1.85763	−0.928813	−	0.370549i	$-0.879170\pi$
$613$	−45.9927	−1.85763	−0.928813	+	0.370549i	$0.879170\pi$
$614$	0	0
$615$	−10.5639	−0.425979
$616$	0	0
$617$	−14.0145	−0.564201	−0.282101	−	0.959385i	$-0.591031\pi$
$617$	−14.0145	−0.564201	−0.282101	+	0.959385i	$0.591031\pi$
$618$	0	0
$619$	−14.1986	−0.570690	−0.285345	−	0.958425i	$-0.592108\pi$
$619$	−14.1986	−0.570690	−0.285345	+	0.958425i	$0.592108\pi$
$620$	0	0
$621$	−21.7237	−0.871744
$622$	0	0
$623$	−24.2373	−0.971046
$624$	0	0
$625$	−16.2426	−0.649704
$626$	0	0
$627$	1.06875	0.0426818
$628$	0	0
$629$	−0.245094	−0.00977253
$630$	0	0
$631$	31.9143	1.27049	0.635244	−	0.772311i	$-0.280900\pi$
$631$	31.9143	1.27049	0.635244	+	0.772311i	$0.280900\pi$
$632$	0	0
$633$	11.9766	0.476026
$634$	0	0
$635$	−66.1307	−2.62432
$636$	0	0
$637$	−108.686	−4.30629
$638$	0	0
$639$	−10.9673	−0.433860
$640$	0	0
$641$	8.75146	0.345662	0.172831	−	0.984951i	$-0.444709\pi$
$641$	8.75146	0.345662	0.172831	+	0.984951i	$0.444709\pi$
$642$	0	0
$643$	22.5093	0.887682	0.443841	−	0.896106i	$-0.353616\pi$
$643$	22.5093	0.887682	0.443841	+	0.896106i	$0.353616\pi$
$644$	0	0
$645$	11.8863	0.468021
$646$	0	0
$647$	−11.9176	−0.468529	−0.234265	−	0.972173i	$-0.575268\pi$
$647$	−11.9176	−0.468529	−0.234265	+	0.972173i	$0.575268\pi$
$648$	0	0
$649$	−0.985816	−0.0386967
$650$	0	0
$651$	−25.7066	−1.00752
$652$	0	0
$653$	−22.6529	−0.886476	−0.443238	−	0.896404i	$-0.646170\pi$
$653$	−22.6529	−0.886476	−0.443238	+	0.896404i	$0.646170\pi$
$654$	0	0
$655$	29.7570	1.16270
$656$	0	0
$657$	−7.63807	−0.297989
$658$	0	0
$659$	−48.1314	−1.87493	−0.937466	−	0.348076i	$-0.886835\pi$
$659$	−48.1314	−1.87493	−0.937466	+	0.348076i	$0.886835\pi$
$660$	0	0
$661$	16.9712	0.660105	0.330052	−	0.943963i	$-0.392934\pi$
$661$	16.9712	0.660105	0.330052	+	0.943963i	$0.392934\pi$
$662$	0	0
$663$	2.56178	0.0994912
$664$	0	0
$665$	22.2733	0.863720
$666$	0	0
$667$	18.7691	0.726742
$668$	0	0
$669$	−4.04763	−0.156491
$670$	0	0
$671$	7.74094	0.298836
$672$	0	0
$673$	−0.427078	−0.0164627	−0.00823133	−	0.999966i	$-0.502620\pi$
$673$	−0.427078	−0.0164627	−0.00823133	+	0.999966i	$0.502620\pi$
$674$	0	0
$675$	−33.2220	−1.27872
$676$	0	0
$677$	28.6019	1.09926	0.549629	−	0.835409i	$-0.314769\pi$
$677$	28.6019	1.09926	0.549629	+	0.835409i	$0.314769\pi$
$678$	0	0
$679$	−10.3903	−0.398745
$680$	0	0
$681$	−9.30921	−0.356730
$682$	0	0
$683$	−32.9411	−1.26045	−0.630227	−	0.776411i	$-0.717038\pi$
$683$	−32.9411	−1.26045	−0.630227	+	0.776411i	$0.717038\pi$
$684$	0	0
$685$	−16.6041	−0.634409
$686$	0	0
$687$	25.1880	0.960981
$688$	0	0
$689$	−62.7290	−2.38979
$690$	0	0
$691$	11.4295	0.434799	0.217400	−	0.976083i	$-0.430243\pi$
$691$	11.4295	0.434799	0.217400	+	0.976083i	$0.430243\pi$
$692$	0	0
$693$	7.01851	0.266611
$694$	0	0
$695$	29.4417	1.11679
$696$	0	0
$697$	−1.16995	−0.0443149
$698$	0	0
$699$	−11.8010	−0.446354
$700$	0	0
$701$	1.68525	0.0636510	0.0318255	−	0.999493i	$-0.489868\pi$
$701$	1.68525	0.0636510	0.0318255	+	0.999493i	$0.489868\pi$
$702$	0	0
$703$	−0.799028	−0.0301359
$704$	0	0
$705$	25.7366	0.969297
$706$	0	0
$707$	1.32183	0.0497126
$708$	0	0
$709$	−8.86001	−0.332745	−0.166372	−	0.986063i	$-0.553205\pi$
$709$	−8.86001	−0.332745	−0.166372	+	0.986063i	$0.553205\pi$
$710$	0	0
$711$	−25.7525	−0.965794
$712$	0	0
$713$	−19.7542	−0.739799
$714$	0	0
$715$	−15.1032	−0.564827
$716$	0	0
$717$	17.8088	0.665083
$718$	0	0
$719$	−14.6226	−0.545332	−0.272666	−	0.962109i	$-0.587905\pi$
$719$	−14.6226	−0.545332	−0.272666	+	0.962109i	$0.587905\pi$
$720$	0	0
$721$	15.2237	0.566961
$722$	0	0
$723$	27.8914	1.03729
$724$	0	0
$725$	28.7035	1.06602
$726$	0	0
$727$	−33.8957	−1.25712	−0.628561	−	0.777760i	$-0.716356\pi$
$727$	−33.8957	−1.25712	−0.628561	+	0.777760i	$0.716356\pi$
$728$	0	0
$729$	17.0035	0.629759
$730$	0	0
$731$	1.31639	0.0486885
$732$	0	0
$733$	18.2473	0.673978	0.336989	−	0.941509i	$-0.390591\pi$
$733$	18.2473	0.673978	0.336989	+	0.941509i	$0.390591\pi$
$734$	0	0
$735$	−66.7185	−2.46095
$736$	0	0
$737$	−2.62755	−0.0967869
$738$	0	0
$739$	16.2631	0.598247	0.299123	−	0.954214i	$-0.403306\pi$
$739$	16.2631	0.598247	0.299123	+	0.954214i	$0.403306\pi$
$740$	0	0
$741$	8.35163	0.306805
$742$	0	0
$743$	34.3062	1.25857	0.629287	−	0.777173i	$-0.283347\pi$
$743$	34.3062	1.25857	0.629287	+	0.777173i	$0.283347\pi$
$744$	0	0
$745$	−79.0831	−2.89738
$746$	0	0
$747$	−18.8370	−0.689209
$748$	0	0
$749$	31.8042	1.16210
$750$	0	0
$751$	12.3105	0.449218	0.224609	−	0.974449i	$-0.427890\pi$
$751$	12.3105	0.449218	0.224609	+	0.974449i	$0.427890\pi$
$752$	0	0
$753$	1.07677	0.0392395
$754$	0	0
$755$	−32.0854	−1.16771
$756$	0	0
$757$	−44.7805	−1.62757	−0.813787	−	0.581164i	$-0.802598\pi$
$757$	−44.7805	−1.62757	−0.813787	+	0.581164i	$0.802598\pi$
$758$	0	0
$759$	−3.39740	−0.123318
$760$	0	0
$761$	0.747027	0.0270797	0.0135399	−	0.999908i	$-0.495690\pi$
$761$	0.747027	0.0270797	0.0135399	+	0.999908i	$0.495690\pi$
$762$	0	0
$763$	14.5306	0.526044
$764$	0	0
$765$	−2.49647	−0.0902600
$766$	0	0
$767$	−7.70354	−0.278159
$768$	0	0
$769$	21.7945	0.785931	0.392966	−	0.919553i	$-0.371449\pi$
$769$	21.7945	0.785931	0.392966	+	0.919553i	$0.371449\pi$
$770$	0	0
$771$	31.6640	1.14035
$772$	0	0
$773$	−11.2792	−0.405685	−0.202842	−	0.979211i	$-0.565018\pi$
$773$	−11.2792	−0.405685	−0.202842	+	0.979211i	$0.565018\pi$
$774$	0	0
$775$	−30.2099	−1.08517
$776$	0	0
$777$	3.30536	0.118579
$778$	0	0
$779$	−3.81413	−0.136656
$780$	0	0
$781$	−4.51081	−0.161410
$782$	0	0
$783$	−23.4717	−0.838811
$784$	0	0
$785$	22.8052	0.813952
$786$	0	0
$787$	19.4722	0.694108	0.347054	−	0.937845i	$-0.387182\pi$
$787$	19.4722	0.694108	0.347054	+	0.937845i	$0.387182\pi$
$788$	0	0
$789$	21.2862	0.757808
$790$	0	0
$791$	1.64385	0.0584487
$792$	0	0
$793$	60.4906	2.14809
$794$	0	0
$795$	−38.5071	−1.36571
$796$	0	0
$797$	−13.6123	−0.482172	−0.241086	−	0.970504i	$-0.577504\pi$
$797$	−13.6123	−0.482172	−0.241086	+	0.970504i	$0.577504\pi$
$798$	0	0
$799$	2.85031	0.100837
$800$	0	0
$801$	8.85636	0.312924
$802$	0	0
$803$	−3.14151	−0.110862
$804$	0	0
$805$	−70.8034	−2.49549
$806$	0	0
$807$	−29.0755	−1.02351
$808$	0	0
$809$	27.2604	0.958425	0.479213	−	0.877699i	$-0.340922\pi$
$809$	27.2604	0.958425	0.479213	+	0.877699i	$0.340922\pi$
$810$	0	0
$811$	−42.8383	−1.50425	−0.752127	−	0.659018i	$-0.770972\pi$
$811$	−42.8383	−1.50425	−0.752127	+	0.659018i	$0.770972\pi$
$812$	0	0
$813$	−14.3091	−0.501843
$814$	0	0
$815$	77.9450	2.73029
$816$	0	0
$817$	4.29156	0.150143
$818$	0	0
$819$	54.8453	1.91645
$820$	0	0
$821$	40.9824	1.43030	0.715148	−	0.698973i	$-0.246359\pi$
$821$	40.9824	1.43030	0.715148	+	0.698973i	$0.246359\pi$
$822$	0	0
$823$	−16.2394	−0.566069	−0.283035	−	0.959110i	$-0.591341\pi$
$823$	−16.2394	−0.566069	−0.283035	+	0.959110i	$0.591341\pi$
$824$	0	0
$825$	−5.19563	−0.180889
$826$	0	0
$827$	−0.366511	−0.0127448	−0.00637241	−	0.999980i	$-0.502028\pi$
$827$	−0.366511	−0.0127448	−0.00637241	+	0.999980i	$0.502028\pi$
$828$	0	0
$829$	22.9086	0.795648	0.397824	−	0.917462i	$-0.369766\pi$
$829$	22.9086	0.795648	0.397824	+	0.917462i	$0.369766\pi$
$830$	0	0
$831$	21.3106	0.739258
$832$	0	0
$833$	−7.38901	−0.256014
$834$	0	0
$835$	−11.4163	−0.395079
$836$	0	0
$837$	24.7036	0.853881
$838$	0	0
$839$	24.7954	0.856031	0.428016	−	0.903771i	$-0.359213\pi$
$839$	24.7954	0.856031	0.428016	+	0.903771i	$0.359213\pi$
$840$	0	0
$841$	−8.72067	−0.300713
$842$	0	0
$843$	22.1593	0.763205
$844$	0	0
$845$	−74.1791	−2.55184
$846$	0	0
$847$	−52.5216	−1.80466
$848$	0	0
$849$	−22.2810	−0.764681
$850$	0	0
$851$	2.53999	0.0870697
$852$	0	0
$853$	−47.1337	−1.61383	−0.806914	−	0.590669i	$-0.798864\pi$
$853$	−47.1337	−1.61383	−0.806914	+	0.590669i	$0.798864\pi$
$854$	0	0
$855$	−8.13871	−0.278338
$856$	0	0
$857$	−11.8321	−0.404178	−0.202089	−	0.979367i	$-0.564773\pi$
$857$	−11.8321	−0.404178	−0.202089	+	0.979367i	$0.564773\pi$
$858$	0	0
$859$	−22.9727	−0.783817	−0.391909	−	0.920004i	$-0.628185\pi$
$859$	−22.9727	−0.783817	−0.391909	+	0.920004i	$0.628185\pi$
$860$	0	0
$861$	15.7780	0.537714
$862$	0	0
$863$	39.7533	1.35322	0.676609	−	0.736342i	$-0.263449\pi$
$863$	39.7533	1.35322	0.676609	+	0.736342i	$0.263449\pi$
$864$	0	0
$865$	−77.9278	−2.64962
$866$	0	0
$867$	−18.1308	−0.615756
$868$	0	0
$869$	−10.5919	−0.359306
$870$	0	0
$871$	−20.5326	−0.695722
$872$	0	0
$873$	3.79666	0.128497
$874$	0	0
$875$	−23.3402	−0.789043
$876$	0	0
$877$	−2.37640	−0.0802453	−0.0401226	−	0.999195i	$-0.512775\pi$
$877$	−2.37640	−0.0802453	−0.0401226	+	0.999195i	$0.512775\pi$
$878$	0	0
$879$	−15.0859	−0.508833
$880$	0	0
$881$	35.7782	1.20540	0.602699	−	0.797969i	$-0.294092\pi$
$881$	35.7782	1.20540	0.602699	+	0.797969i	$0.294092\pi$
$882$	0	0
$883$	46.2612	1.55681	0.778406	−	0.627761i	$-0.216028\pi$
$883$	46.2612	1.55681	0.778406	+	0.627761i	$0.216028\pi$
$884$	0	0
$885$	−4.72893	−0.158961
$886$	0	0
$887$	31.1304	1.04526	0.522628	−	0.852561i	$-0.324951\pi$
$887$	31.1304	1.04526	0.522628	+	0.852561i	$0.324951\pi$
$888$	0	0
$889$	98.7711	3.31268
$890$	0	0
$891$	0.0685488	0.00229647
$892$	0	0
$893$	9.29226	0.310954
$894$	0	0
$895$	21.5253	0.719510
$896$	0	0
$897$	−26.5486	−0.886431
$898$	0	0
$899$	−21.3437	−0.711851
$900$	0	0
$901$	−4.26463	−0.142075
$902$	0	0
$903$	−17.7530	−0.590783
$904$	0	0
$905$	−35.3961	−1.17661
$906$	0	0
$907$	−54.3704	−1.80534	−0.902669	−	0.430335i	$-0.858395\pi$
$907$	−54.3704	−1.80534	−0.902669	+	0.430335i	$0.858395\pi$
$908$	0	0
$909$	−0.483001	−0.0160201
$910$	0	0
$911$	33.1041	1.09679	0.548394	−	0.836220i	$-0.315240\pi$
$911$	33.1041	1.09679	0.548394	+	0.836220i	$0.315240\pi$
$912$	0	0
$913$	−7.74758	−0.256408
$914$	0	0
$915$	37.1331	1.22758
$916$	0	0
$917$	−44.4443	−1.46768
$918$	0	0
$919$	−51.6113	−1.70250	−0.851249	−	0.524762i	$-0.824154\pi$
$919$	−51.6113	−1.70250	−0.851249	+	0.524762i	$0.824154\pi$
$920$	0	0
$921$	−8.70252	−0.286758
$922$	0	0
$923$	−35.2492	−1.16024
$924$	0	0
$925$	3.88439	0.127718
$926$	0	0
$927$	−5.56278	−0.182706
$928$	0	0
$929$	30.4435	0.998818	0.499409	−	0.866366i	$-0.333551\pi$
$929$	30.4435	0.998818	0.499409	+	0.866366i	$0.333551\pi$
$930$	0	0
$931$	−24.0888	−0.789480
$932$	0	0
$933$	−32.1202	−1.05157
$934$	0	0
$935$	−1.02679	−0.0335796
$936$	0	0
$937$	−19.0953	−0.623818	−0.311909	−	0.950112i	$-0.600968\pi$
$937$	−19.0953	−0.623818	−0.311909	+	0.950112i	$0.600968\pi$
$938$	0	0
$939$	−10.7403	−0.350497
$940$	0	0
$941$	−33.7958	−1.10171	−0.550856	−	0.834600i	$-0.685699\pi$
$941$	−33.7958	−1.10171	−0.550856	+	0.834600i	$0.685699\pi$
$942$	0	0
$943$	12.1246	0.394830
$944$	0	0
$945$	88.5433	2.88031
$946$	0	0
$947$	23.1946	0.753723	0.376861	−	0.926270i	$-0.377003\pi$
$947$	23.1946	0.753723	0.376861	+	0.926270i	$0.377003\pi$
$948$	0	0
$949$	−24.5490	−0.796894
$950$	0	0
$951$	20.8676	0.676677
$952$	0	0
$953$	21.9648	0.711509	0.355755	−	0.934579i	$-0.384224\pi$
$953$	21.9648	0.711509	0.355755	+	0.934579i	$0.384224\pi$
$954$	0	0
$955$	−23.1085	−0.747772
$956$	0	0
$957$	−3.67077	−0.118659
$958$	0	0
$959$	24.7994	0.800814
$960$	0	0
$961$	−8.53616	−0.275360
$962$	0	0
$963$	−11.6213	−0.374493
$964$	0	0
$965$	25.6226	0.824820
$966$	0	0
$967$	−54.3458	−1.74764	−0.873822	−	0.486246i	$-0.838366\pi$
$967$	−54.3458	−1.74764	−0.873822	+	0.486246i	$0.838366\pi$
$968$	0	0
$969$	0.567786	0.0182399
$970$	0	0
$971$	41.5898	1.33468	0.667340	−	0.744753i	$-0.267433\pi$
$971$	41.5898	1.33468	0.667340	+	0.744753i	$0.267433\pi$
$972$	0	0
$973$	−43.9734	−1.40972
$974$	0	0
$975$	−40.6006	−1.30026
$976$	0	0
$977$	−35.8834	−1.14801	−0.574006	−	0.818851i	$-0.694611\pi$
$977$	−35.8834	−1.14801	−0.574006	+	0.818851i	$0.694611\pi$
$978$	0	0
$979$	3.64259	0.116418
$980$	0	0
$981$	−5.30953	−0.169520
$982$	0	0
$983$	−12.9093	−0.411743	−0.205871	−	0.978579i	$-0.566003\pi$
$983$	−12.9093	−0.411743	−0.205871	+	0.978579i	$0.566003\pi$
$984$	0	0
$985$	34.1789	1.08903
$986$	0	0
$987$	−38.4395	−1.22354
$988$	0	0
$989$	−13.6422	−0.433797
$990$	0	0
$991$	−57.6024	−1.82980	−0.914900	−	0.403681i	$-0.867731\pi$
$991$	−57.6024	−1.82980	−0.914900	+	0.403681i	$0.867731\pi$
$992$	0	0
$993$	−12.3826	−0.392951
$994$	0	0
$995$	1.80157	0.0571136
$996$	0	0
$997$	−51.5223	−1.63173	−0.815864	−	0.578243i	$-0.803738\pi$
$997$	−51.5223	−1.63173	−0.815864	+	0.578243i	$0.803738\pi$
$998$	0	0
$999$	−3.17639	−0.100496

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8032.2.a.i.1.19	yes	30
4.3	odd	2		8032.2.a.h.1.12	✓	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8032.2.a.h.1.12	✓	30	4.3	odd	2
8032.2.a.i.1.19	yes	30	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:	\( N \)	\(=\)	\( 8032 = 2^{5} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8032.a (trivial)

\(f(q)\)	\(=\)	\(q+1.07677 q^{3} -3.37253 q^{5} +5.03712 q^{7} -1.84058 q^{9} +O(q^{10})\)	\(q+1.07677 q^{3} -3.37253 q^{5} +5.03712 q^{7} -1.84058 q^{9} -0.757023 q^{11} -5.91567 q^{13} -3.63142 q^{15} -0.402177 q^{17} -1.31113 q^{19} +5.42379 q^{21} +4.16789 q^{23} +6.37394 q^{25} -5.21217 q^{27} +4.50326 q^{29} -4.73960 q^{31} -0.815136 q^{33} -16.9878 q^{35} +0.609418 q^{37} -6.36979 q^{39} +2.90904 q^{41} -3.27317 q^{43} +6.20739 q^{45} -7.08721 q^{47} +18.3725 q^{49} -0.433050 q^{51} +10.6039 q^{53} +2.55308 q^{55} -1.41178 q^{57} +1.30223 q^{59} -10.2255 q^{61} -9.27120 q^{63} +19.9507 q^{65} +3.47089 q^{67} +4.48784 q^{69} +5.95862 q^{71} +4.14982 q^{73} +6.86324 q^{75} -3.81321 q^{77} +13.9915 q^{79} -0.0905505 q^{81} +10.2343 q^{83} +1.35635 q^{85} +4.84895 q^{87} -4.81173 q^{89} -29.7979 q^{91} -5.10344 q^{93} +4.42183 q^{95} -2.06276 q^{97} +1.39336 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 3 q^{3} + 13 q^{7} + 35 q^{9}+O(q^{10}) \) 30 * q + 3 * q^3 + 13 * q^7 + 35 * q^9	\( 30 q + 3 q^{3} + 13 q^{7} + 35 q^{9} + 13 q^{11} - 7 q^{13} + 28 q^{15} - 9 q^{17} - 17 q^{19} - 6 q^{21} + 43 q^{23} + 34 q^{25} + 12 q^{27} - q^{29} + 39 q^{31} + 17 q^{35} - q^{37} + 48 q^{39} - 3 q^{41} - 19 q^{43} + 66 q^{47} + 25 q^{49} - 14 q^{51} + 3 q^{53} + 50 q^{55} - 14 q^{57} + 27 q^{59} + 15 q^{61} + 75 q^{63} - 6 q^{65} + 8 q^{67} + 18 q^{69} + 64 q^{71} - 15 q^{73} + 9 q^{75} + 71 q^{79} + 6 q^{81} + 60 q^{83} + 15 q^{85} + 64 q^{87} - 32 q^{89} - 26 q^{91} - 4 q^{93} + 72 q^{95} - 4 q^{97} + 13 q^{99}+O(q^{100}) \) 30 * q + 3 * q^3 + 13 * q^7 + 35 * q^9 + 13 * q^11 - 7 * q^13 + 28 * q^15 - 9 * q^17 - 17 * q^19 - 6 * q^21 + 43 * q^23 + 34 * q^25 + 12 * q^27 - q^29 + 39 * q^31 + 17 * q^35 - q^37 + 48 * q^39 - 3 * q^41 - 19 * q^43 + 66 * q^47 + 25 * q^49 - 14 * q^51 + 3 * q^53 + 50 * q^55 - 14 * q^57 + 27 * q^59 + 15 * q^61 + 75 * q^63 - 6 * q^65 + 8 * q^67 + 18 * q^69 + 64 * q^71 - 15 * q^73 + 9 * q^75 + 71 * q^79 + 6 * q^81 + 60 * q^83 + 15 * q^85 + 64 * q^87 - 32 * q^89 - 26 * q^91 - 4 * q^93 + 72 * q^95 - 4 * q^97 + 13 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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