LMFDB - Embedded newform 8024.2.a.ba.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8024,2,Mod(1,8024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8024, 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("8024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8024 = 2^{3} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8024.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:	$64.0719625819$
Analytic rank:	$0$
Dimension:	$30$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Character	$\chi$	$=$	8024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.64071 q^{3} -0.794586 q^{5} +0.767339 q^{7} +3.97338 q^{9} +O(q^{10})$	$q-2.64071 q^{3} -0.794586 q^{5} +0.767339 q^{7} +3.97338 q^{9} -2.79373 q^{11} -2.21305 q^{13} +2.09828 q^{15} -1.00000 q^{17} +5.83521 q^{19} -2.02632 q^{21} -3.67115 q^{23} -4.36863 q^{25} -2.57041 q^{27} -2.04867 q^{29} -4.02655 q^{31} +7.37745 q^{33} -0.609717 q^{35} +3.02611 q^{37} +5.84403 q^{39} +8.11469 q^{41} +8.81621 q^{43} -3.15719 q^{45} -10.5538 q^{47} -6.41119 q^{49} +2.64071 q^{51} -11.6488 q^{53} +2.21986 q^{55} -15.4091 q^{57} +1.00000 q^{59} -0.299414 q^{61} +3.04893 q^{63} +1.75846 q^{65} +5.19725 q^{67} +9.69445 q^{69} +6.38523 q^{71} -11.4523 q^{73} +11.5363 q^{75} -2.14374 q^{77} +7.13756 q^{79} -5.13241 q^{81} -7.37549 q^{83} +0.794586 q^{85} +5.40995 q^{87} -9.04725 q^{89} -1.69816 q^{91} +10.6330 q^{93} -4.63658 q^{95} +2.01803 q^{97} -11.1005 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q + 4 q^{3} + 2 q^{5} + 3 q^{7} + 34 q^{9}+O(q^{10}) $ 30 * q + 4 * q^3 + 2 * q^5 + 3 * q^7 + 34 * q^9	$ 30 q + 4 q^{3} + 2 q^{5} + 3 q^{7} + 34 q^{9} + 3 q^{11} + 9 q^{13} + 14 q^{15} - 30 q^{17} + 24 q^{19} + 7 q^{21} + 9 q^{23} + 40 q^{25} + 19 q^{27} + 9 q^{29} + 11 q^{31} - 14 q^{33} + 30 q^{35} - 13 q^{37} + 16 q^{39} - 13 q^{41} + 23 q^{43} + 12 q^{45} + 43 q^{47} + 35 q^{49} - 4 q^{51} - 4 q^{53} + 43 q^{55} + 3 q^{57} + 30 q^{59} + 43 q^{61} + 38 q^{63} + 3 q^{65} + 50 q^{67} + 34 q^{69} + 3 q^{71} - 16 q^{73} + 21 q^{75} + 18 q^{77} + 45 q^{79} + 6 q^{81} + 63 q^{83} - 2 q^{85} + 42 q^{87} + 6 q^{89} + 22 q^{91} - 2 q^{93} + 19 q^{95} - 28 q^{97} + 51 q^{99}+O(q^{100}) $ 30 * q + 4 * q^3 + 2 * q^5 + 3 * q^7 + 34 * q^9 + 3 * q^11 + 9 * q^13 + 14 * q^15 - 30 * q^17 + 24 * q^19 + 7 * q^21 + 9 * q^23 + 40 * q^25 + 19 * q^27 + 9 * q^29 + 11 * q^31 - 14 * q^33 + 30 * q^35 - 13 * q^37 + 16 * q^39 - 13 * q^41 + 23 * q^43 + 12 * q^45 + 43 * q^47 + 35 * q^49 - 4 * q^51 - 4 * q^53 + 43 * q^55 + 3 * q^57 + 30 * q^59 + 43 * q^61 + 38 * q^63 + 3 * q^65 + 50 * q^67 + 34 * q^69 + 3 * q^71 - 16 * q^73 + 21 * q^75 + 18 * q^77 + 45 * q^79 + 6 * q^81 + 63 * q^83 - 2 * q^85 + 42 * q^87 + 6 * q^89 + 22 * q^91 - 2 * q^93 + 19 * q^95 - 28 * q^97 + 51 * 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
$3$	−2.64071	−1.52462	−0.762309	−	0.647214i	$-0.775934\pi$
$3$	−2.64071	−1.52462	−0.762309	+	0.647214i	$0.775934\pi$
$4$	0	0
$5$	−0.794586	−0.355350	−0.177675	−	0.984089i	$-0.556858\pi$
$5$	−0.794586	−0.355350	−0.177675	+	0.984089i	$0.556858\pi$
$6$	0	0
$7$	0.767339	0.290027	0.145013	−	0.989430i	$-0.453677\pi$
$7$	0.767339	0.290027	0.145013	+	0.989430i	$0.453677\pi$
$8$	0	0
$9$	3.97338	1.32446
$10$	0	0
$11$	−2.79373	−0.842342	−0.421171	−	0.906981i	$-0.638381\pi$
$11$	−2.79373	−0.842342	−0.421171	+	0.906981i	$0.638381\pi$
$12$	0	0
$13$	−2.21305	−0.613789	−0.306895	−	0.951744i	$-0.599290\pi$
$13$	−2.21305	−0.613789	−0.306895	+	0.951744i	$0.599290\pi$
$14$	0	0
$15$	2.09828	0.541772
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	5.83521	1.33869	0.669344	−	0.742952i	$-0.266575\pi$
$19$	5.83521	1.33869	0.669344	+	0.742952i	$0.266575\pi$
$20$	0	0
$21$	−2.02632	−0.442180
$22$	0	0
$23$	−3.67115	−0.765487	−0.382743	−	0.923855i	$-0.625021\pi$
$23$	−3.67115	−0.765487	−0.382743	+	0.923855i	$0.625021\pi$
$24$	0	0
$25$	−4.36863	−0.873727
$26$	0	0
$27$	−2.57041	−0.494675
$28$	0	0
$29$	−2.04867	−0.380428	−0.190214	−	0.981743i	$-0.560918\pi$
$29$	−2.04867	−0.380428	−0.190214	+	0.981743i	$0.560918\pi$
$30$	0	0
$31$	−4.02655	−0.723190	−0.361595	−	0.932335i	$-0.617768\pi$
$31$	−4.02655	−0.723190	−0.361595	+	0.932335i	$0.617768\pi$
$32$	0	0
$33$	7.37745	1.28425
$34$	0	0
$35$	−0.609717	−0.103061
$36$	0	0
$37$	3.02611	0.497489	0.248744	−	0.968569i	$-0.419982\pi$
$37$	3.02611	0.497489	0.248744	+	0.968569i	$0.419982\pi$
$38$	0	0
$39$	5.84403	0.935794
$40$	0	0
$41$	8.11469	1.26730	0.633651	−	0.773619i	$-0.281556\pi$
$41$	8.11469	1.26730	0.633651	+	0.773619i	$0.281556\pi$
$42$	0	0
$43$	8.81621	1.34446	0.672230	−	0.740342i	$-0.265337\pi$
$43$	8.81621	1.34446	0.672230	+	0.740342i	$0.265337\pi$
$44$	0	0
$45$	−3.15719	−0.470646
$46$	0	0
$47$	−10.5538	−1.53943	−0.769716	−	0.638387i	$-0.779602\pi$
$47$	−10.5538	−1.53943	−0.769716	+	0.638387i	$0.779602\pi$
$48$	0	0
$49$	−6.41119	−0.915884
$50$	0	0
$51$	2.64071	0.369774
$52$	0	0
$53$	−11.6488	−1.60008	−0.800040	−	0.599947i	$-0.795188\pi$
$53$	−11.6488	−1.60008	−0.800040	+	0.599947i	$0.795188\pi$
$54$	0	0
$55$	2.21986	0.299326
$56$	0	0
$57$	−15.4091	−2.04099
$58$	0	0
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	0	0
$61$	−0.299414	−0.0383360	−0.0191680	−	0.999816i	$-0.506102\pi$
$61$	−0.299414	−0.0383360	−0.0191680	+	0.999816i	$0.506102\pi$
$62$	0	0
$63$	3.04893	0.384129
$64$	0	0
$65$	1.75846	0.218110
$66$	0	0
$67$	5.19725	0.634945	0.317473	−	0.948267i	$-0.397166\pi$
$67$	5.19725	0.634945	0.317473	+	0.948267i	$0.397166\pi$
$68$	0	0
$69$	9.69445	1.16707
$70$	0	0
$71$	6.38523	0.757788	0.378894	−	0.925440i	$-0.376305\pi$
$71$	6.38523	0.757788	0.378894	+	0.925440i	$0.376305\pi$
$72$	0	0
$73$	−11.4523	−1.34039	−0.670196	−	0.742184i	$-0.733790\pi$
$73$	−11.4523	−1.34039	−0.670196	+	0.742184i	$0.733790\pi$
$74$	0	0
$75$	11.5363	1.33210
$76$	0	0
$77$	−2.14374	−0.244302
$78$	0	0
$79$	7.13756	0.803038	0.401519	−	0.915851i	$-0.368482\pi$
$79$	7.13756	0.803038	0.401519	+	0.915851i	$0.368482\pi$
$80$	0	0
$81$	−5.13241	−0.570268
$82$	0	0
$83$	−7.37549	−0.809565	−0.404782	−	0.914413i	$-0.632653\pi$
$83$	−7.37549	−0.809565	−0.404782	+	0.914413i	$0.632653\pi$
$84$	0	0
$85$	0.794586	0.0861850
$86$	0	0
$87$	5.40995	0.580007
$88$	0	0
$89$	−9.04725	−0.959007	−0.479504	−	0.877540i	$-0.659183\pi$
$89$	−9.04725	−0.959007	−0.479504	+	0.877540i	$0.659183\pi$
$90$	0	0
$91$	−1.69816	−0.178015
$92$	0	0
$93$	10.6330	1.10259
$94$	0	0
$95$	−4.63658	−0.475703
$96$	0	0
$97$	2.01803	0.204900	0.102450	−	0.994738i	$-0.467332\pi$
$97$	2.01803	0.204900	0.102450	+	0.994738i	$0.467332\pi$
$98$	0	0
$99$	−11.1005	−1.11565
$100$	0	0
$101$	9.06414	0.901915	0.450958	−	0.892545i	$-0.351083\pi$
$101$	9.06414	0.901915	0.450958	+	0.892545i	$0.351083\pi$
$102$	0	0
$103$	11.2979	1.11321	0.556606	−	0.830777i	$-0.312103\pi$
$103$	11.2979	1.11321	0.556606	+	0.830777i	$0.312103\pi$
$104$	0	0
$105$	1.61009	0.157129
$106$	0	0
$107$	−6.90754	−0.667777	−0.333889	−	0.942613i	$-0.608361\pi$
$107$	−6.90754	−0.667777	−0.333889	+	0.942613i	$0.608361\pi$
$108$	0	0
$109$	2.86094	0.274028	0.137014	−	0.990569i	$-0.456249\pi$
$109$	2.86094	0.274028	0.137014	+	0.990569i	$0.456249\pi$
$110$	0	0
$111$	−7.99109	−0.758480
$112$	0	0
$113$	−11.7596	−1.10625	−0.553127	−	0.833097i	$-0.686566\pi$
$113$	−11.7596	−1.10625	−0.553127	+	0.833097i	$0.686566\pi$
$114$	0	0
$115$	2.91704	0.272016
$116$	0	0
$117$	−8.79327	−0.812938
$118$	0	0
$119$	−0.767339	−0.0703418
$120$	0	0
$121$	−3.19506	−0.290460
$122$	0	0
$123$	−21.4286	−1.93215
$124$	0	0
$125$	7.44419	0.665828
$126$	0	0
$127$	9.48374	0.841546	0.420773	−	0.907166i	$-0.361759\pi$
$127$	9.48374	0.841546	0.420773	+	0.907166i	$0.361759\pi$
$128$	0	0
$129$	−23.2811	−2.04979
$130$	0	0
$131$	−8.56173	−0.748042	−0.374021	−	0.927420i	$-0.622021\pi$
$131$	−8.56173	−0.748042	−0.374021	+	0.927420i	$0.622021\pi$
$132$	0	0
$133$	4.47758	0.388256
$134$	0	0
$135$	2.04241	0.175783
$136$	0	0
$137$	6.64486	0.567709	0.283854	−	0.958867i	$-0.408387\pi$
$137$	6.64486	0.567709	0.283854	+	0.958867i	$0.408387\pi$
$138$	0	0
$139$	−2.97006	−0.251917	−0.125958	−	0.992036i	$-0.540201\pi$
$139$	−2.97006	−0.251917	−0.125958	+	0.992036i	$0.540201\pi$
$140$	0	0
$141$	27.8696	2.34704
$142$	0	0
$143$	6.18267	0.517020
$144$	0	0
$145$	1.62784	0.135185
$146$	0	0
$147$	16.9301	1.39637
$148$	0	0
$149$	14.2669	1.16879	0.584397	−	0.811468i	$-0.301331\pi$
$149$	14.2669	1.16879	0.584397	+	0.811468i	$0.301331\pi$
$150$	0	0
$151$	−6.08701	−0.495354	−0.247677	−	0.968843i	$-0.579667\pi$
$151$	−6.08701	−0.495354	−0.247677	+	0.968843i	$0.579667\pi$
$152$	0	0
$153$	−3.97338	−0.321228
$154$	0	0
$155$	3.19944	0.256985
$156$	0	0
$157$	−4.14540	−0.330839	−0.165419	−	0.986223i	$-0.552898\pi$
$157$	−4.14540	−0.330839	−0.165419	+	0.986223i	$0.552898\pi$
$158$	0	0
$159$	30.7610	2.43951
$160$	0	0
$161$	−2.81701	−0.222012
$162$	0	0
$163$	1.22629	0.0960508	0.0480254	−	0.998846i	$-0.484707\pi$
$163$	1.22629	0.0960508	0.0480254	+	0.998846i	$0.484707\pi$
$164$	0	0
$165$	−5.86202	−0.456358
$166$	0	0
$167$	−11.3216	−0.876088	−0.438044	−	0.898953i	$-0.644329\pi$
$167$	−11.3216	−0.876088	−0.438044	+	0.898953i	$0.644329\pi$
$168$	0	0
$169$	−8.10242	−0.623263
$170$	0	0
$171$	23.1855	1.77304
$172$	0	0
$173$	−3.24692	−0.246859	−0.123429	−	0.992353i	$-0.539389\pi$
$173$	−3.24692	−0.246859	−0.123429	+	0.992353i	$0.539389\pi$
$174$	0	0
$175$	−3.35222	−0.253404
$176$	0	0
$177$	−2.64071	−0.198488
$178$	0	0
$179$	−3.54440	−0.264921	−0.132460	−	0.991188i	$-0.542288\pi$
$179$	−3.54440	−0.264921	−0.132460	+	0.991188i	$0.542288\pi$
$180$	0	0
$181$	−20.7698	−1.54381	−0.771905	−	0.635738i	$-0.780696\pi$
$181$	−20.7698	−1.54381	−0.771905	+	0.635738i	$0.780696\pi$
$182$	0	0
$183$	0.790667	0.0584478
$184$	0	0
$185$	−2.40450	−0.176783
$186$	0	0
$187$	2.79373	0.204298
$188$	0	0
$189$	−1.97237	−0.143469
$190$	0	0
$191$	−3.48457	−0.252135	−0.126067	−	0.992022i	$-0.540236\pi$
$191$	−3.48457	−0.252135	−0.126067	+	0.992022i	$0.540236\pi$
$192$	0	0
$193$	−14.2762	−1.02762	−0.513811	−	0.857903i	$-0.671767\pi$
$193$	−14.2762	−1.02762	−0.513811	+	0.857903i	$0.671767\pi$
$194$	0	0
$195$	−4.64359	−0.332534
$196$	0	0
$197$	−16.5381	−1.17829	−0.589144	−	0.808028i	$-0.700535\pi$
$197$	−16.5381	−1.17829	−0.589144	+	0.808028i	$0.700535\pi$
$198$	0	0
$199$	−15.0262	−1.06518	−0.532591	−	0.846373i	$-0.678782\pi$
$199$	−15.0262	−1.06518	−0.532591	+	0.846373i	$0.678782\pi$
$200$	0	0
$201$	−13.7245	−0.968048
$202$	0	0
$203$	−1.57202	−0.110334
$204$	0	0
$205$	−6.44782	−0.450335
$206$	0	0
$207$	−14.5868	−1.01386
$208$	0	0
$209$	−16.3020	−1.12763
$210$	0	0
$211$	8.14182	0.560506	0.280253	−	0.959926i	$-0.409582\pi$
$211$	8.14182	0.560506	0.280253	+	0.959926i	$0.409582\pi$
$212$	0	0
$213$	−16.8616	−1.15534
$214$	0	0
$215$	−7.00524	−0.477753
$216$	0	0
$217$	−3.08973	−0.209745
$218$	0	0
$219$	30.2423	2.04359
$220$	0	0
$221$	2.21305	0.148866
$222$	0	0
$223$	5.77633	0.386811	0.193406	−	0.981119i	$-0.438047\pi$
$223$	5.77633	0.386811	0.193406	+	0.981119i	$0.438047\pi$
$224$	0	0
$225$	−17.3582	−1.15721
$226$	0	0
$227$	−5.56820	−0.369574	−0.184787	−	0.982779i	$-0.559160\pi$
$227$	−5.56820	−0.369574	−0.184787	+	0.982779i	$0.559160\pi$
$228$	0	0
$229$	22.3194	1.47491	0.737454	−	0.675398i	$-0.236028\pi$
$229$	22.3194	1.47491	0.737454	+	0.675398i	$0.236028\pi$
$230$	0	0
$231$	5.66101	0.372467
$232$	0	0
$233$	5.57207	0.365039	0.182519	−	0.983202i	$-0.441575\pi$
$233$	5.57207	0.365039	0.182519	+	0.983202i	$0.441575\pi$
$234$	0	0
$235$	8.38591	0.547037
$236$	0	0
$237$	−18.8483	−1.22433
$238$	0	0
$239$	21.4874	1.38990	0.694951	−	0.719057i	$-0.255426\pi$
$239$	21.4874	1.38990	0.694951	+	0.719057i	$0.255426\pi$
$240$	0	0
$241$	−16.1968	−1.04333	−0.521665	−	0.853151i	$-0.674689\pi$
$241$	−16.1968	−1.04333	−0.521665	+	0.853151i	$0.674689\pi$
$242$	0	0
$243$	21.2645	1.36412
$244$	0	0
$245$	5.09424	0.325459
$246$	0	0
$247$	−12.9136	−0.821673
$248$	0	0
$249$	19.4766	1.23428
$250$	0	0
$251$	9.38664	0.592479	0.296240	−	0.955114i	$-0.404267\pi$
$251$	9.38664	0.592479	0.296240	+	0.955114i	$0.404267\pi$
$252$	0	0
$253$	10.2562	0.644802
$254$	0	0
$255$	−2.09828	−0.131399
$256$	0	0
$257$	−18.0213	−1.12414	−0.562070	−	0.827090i	$-0.689995\pi$
$257$	−18.0213	−1.12414	−0.562070	+	0.827090i	$0.689995\pi$
$258$	0	0
$259$	2.32205	0.144285
$260$	0	0
$261$	−8.14013	−0.503861
$262$	0	0
$263$	6.93909	0.427883	0.213941	−	0.976847i	$-0.431370\pi$
$263$	6.93909	0.427883	0.213941	+	0.976847i	$0.431370\pi$
$264$	0	0
$265$	9.25594	0.568588
$266$	0	0
$267$	23.8912	1.46212
$268$	0	0
$269$	−2.23835	−0.136474	−0.0682372	−	0.997669i	$-0.521737\pi$
$269$	−2.23835	−0.136474	−0.0682372	+	0.997669i	$0.521737\pi$
$270$	0	0
$271$	12.8082	0.778043	0.389022	−	0.921229i	$-0.372813\pi$
$271$	12.8082	0.778043	0.389022	+	0.921229i	$0.372813\pi$
$272$	0	0
$273$	4.48435	0.271405
$274$	0	0
$275$	12.2048	0.735977
$276$	0	0
$277$	−6.82901	−0.410315	−0.205158	−	0.978729i	$-0.565771\pi$
$277$	−6.82901	−0.410315	−0.205158	+	0.978729i	$0.565771\pi$
$278$	0	0
$279$	−15.9990	−0.957835
$280$	0	0
$281$	9.20103	0.548887	0.274444	−	0.961603i	$-0.411506\pi$
$281$	9.20103	0.548887	0.274444	+	0.961603i	$0.411506\pi$
$282$	0	0
$283$	27.9834	1.66344	0.831721	−	0.555193i	$-0.187356\pi$
$283$	27.9834	1.66344	0.831721	+	0.555193i	$0.187356\pi$
$284$	0	0
$285$	12.2439	0.725265
$286$	0	0
$287$	6.22671	0.367551
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−5.32903	−0.312394
$292$	0	0
$293$	26.6208	1.55520	0.777601	−	0.628758i	$-0.216437\pi$
$293$	26.6208	1.55520	0.777601	+	0.628758i	$0.216437\pi$
$294$	0	0
$295$	−0.794586	−0.0462626
$296$	0	0
$297$	7.18103	0.416686
$298$	0	0
$299$	8.12443	0.469848
$300$	0	0
$301$	6.76502	0.389929
$302$	0	0
$303$	−23.9358	−1.37508
$304$	0	0
$305$	0.237910	0.0136227
$306$	0	0
$307$	32.6644	1.86425	0.932127	−	0.362133i	$-0.117951\pi$
$307$	32.6644	1.86425	0.932127	+	0.362133i	$0.117951\pi$
$308$	0	0
$309$	−29.8344	−1.69722
$310$	0	0
$311$	1.68737	0.0956822	0.0478411	−	0.998855i	$-0.484766\pi$
$311$	1.68737	0.0956822	0.0478411	+	0.998855i	$0.484766\pi$
$312$	0	0
$313$	15.5306	0.877842	0.438921	−	0.898526i	$-0.355361\pi$
$313$	15.5306	0.877842	0.438921	+	0.898526i	$0.355361\pi$
$314$	0	0
$315$	−2.42263	−0.136500
$316$	0	0
$317$	−11.3696	−0.638581	−0.319290	−	0.947657i	$-0.603445\pi$
$317$	−11.3696	−0.638581	−0.319290	+	0.947657i	$0.603445\pi$
$318$	0	0
$319$	5.72343	0.320451
$320$	0	0
$321$	18.2408	1.01810
$322$	0	0
$323$	−5.83521	−0.324680
$324$	0	0
$325$	9.66800	0.536284
$326$	0	0
$327$	−7.55493	−0.417788
$328$	0	0
$329$	−8.09835	−0.446477
$330$	0	0
$331$	10.9127	0.599818	0.299909	−	0.953968i	$-0.403044\pi$
$331$	10.9127	0.599818	0.299909	+	0.953968i	$0.403044\pi$
$332$	0	0
$333$	12.0239	0.658903
$334$	0	0
$335$	−4.12966	−0.225628
$336$	0	0
$337$	−14.1607	−0.771385	−0.385692	−	0.922627i	$-0.626037\pi$
$337$	−14.1607	−0.771385	−0.385692	+	0.922627i	$0.626037\pi$
$338$	0	0
$339$	31.0539	1.68661
$340$	0	0
$341$	11.2491	0.609173
$342$	0	0
$343$	−10.2909	−0.555658
$344$	0	0
$345$	−7.70308	−0.414720
$346$	0	0
$347$	−3.20993	−0.172318	−0.0861589	−	0.996281i	$-0.527459\pi$
$347$	−3.20993	−0.172318	−0.0861589	+	0.996281i	$0.527459\pi$
$348$	0	0
$349$	34.8244	1.86411	0.932054	−	0.362318i	$-0.118015\pi$
$349$	34.8244	1.86411	0.932054	+	0.362318i	$0.118015\pi$
$350$	0	0
$351$	5.68844	0.303626
$352$	0	0
$353$	4.16358	0.221605	0.110803	−	0.993842i	$-0.464658\pi$
$353$	4.16358	0.221605	0.110803	+	0.993842i	$0.464658\pi$
$354$	0	0
$355$	−5.07362	−0.269280
$356$	0	0
$357$	2.02632	0.107244
$358$	0	0
$359$	10.9344	0.577094	0.288547	−	0.957466i	$-0.406828\pi$
$359$	10.9344	0.577094	0.288547	+	0.957466i	$0.406828\pi$
$360$	0	0
$361$	15.0497	0.792088
$362$	0	0
$363$	8.43724	0.442840
$364$	0	0
$365$	9.09985	0.476308
$366$	0	0
$367$	19.3149	1.00823	0.504116	−	0.863636i	$-0.331818\pi$
$367$	19.3149	1.00823	0.504116	+	0.863636i	$0.331818\pi$
$368$	0	0
$369$	32.2427	1.67849
$370$	0	0
$371$	−8.93854	−0.464066
$372$	0	0
$373$	−0.887139	−0.0459343	−0.0229672	−	0.999736i	$-0.507311\pi$
$373$	−0.887139	−0.0459343	−0.0229672	+	0.999736i	$0.507311\pi$
$374$	0	0
$375$	−19.6580	−1.01513
$376$	0	0
$377$	4.53380	0.233503
$378$	0	0
$379$	−5.25388	−0.269873	−0.134937	−	0.990854i	$-0.543083\pi$
$379$	−5.25388	−0.269873	−0.134937	+	0.990854i	$0.543083\pi$
$380$	0	0
$381$	−25.0439	−1.28304
$382$	0	0
$383$	14.9410	0.763451	0.381725	−	0.924276i	$-0.375330\pi$
$383$	14.9410	0.763451	0.381725	+	0.924276i	$0.375330\pi$
$384$	0	0
$385$	1.70339	0.0868126
$386$	0	0
$387$	35.0301	1.78068
$388$	0	0
$389$	−3.69605	−0.187397	−0.0936987	−	0.995601i	$-0.529869\pi$
$389$	−3.69605	−0.187397	−0.0936987	+	0.995601i	$0.529869\pi$
$390$	0	0
$391$	3.67115	0.185658
$392$	0	0
$393$	22.6091	1.14048
$394$	0	0
$395$	−5.67141	−0.285359
$396$	0	0
$397$	−0.0808047	−0.00405547	−0.00202774	−	0.999998i	$-0.500645\pi$
$397$	−0.0808047	−0.00405547	−0.00202774	+	0.999998i	$0.500645\pi$
$398$	0	0
$399$	−11.8240	−0.591941
$400$	0	0
$401$	−7.28750	−0.363920	−0.181960	−	0.983306i	$-0.558244\pi$
$401$	−7.28750	−0.363920	−0.181960	+	0.983306i	$0.558244\pi$
$402$	0	0
$403$	8.91096	0.443886
$404$	0	0
$405$	4.07814	0.202645
$406$	0	0
$407$	−8.45413	−0.419056
$408$	0	0
$409$	−13.7133	−0.678077	−0.339039	−	0.940772i	$-0.610102\pi$
$409$	−13.7133	−0.678077	−0.339039	+	0.940772i	$0.610102\pi$
$410$	0	0
$411$	−17.5472	−0.865539
$412$	0	0
$413$	0.767339	0.0377583
$414$	0	0
$415$	5.86046	0.287679
$416$	0	0
$417$	7.84307	0.384077
$418$	0	0
$419$	39.7538	1.94210	0.971051	−	0.238873i	$-0.0767779\pi$
$419$	39.7538	1.94210	0.971051	+	0.238873i	$0.0767779\pi$
$420$	0	0
$421$	26.2860	1.28110	0.640551	−	0.767915i	$-0.278706\pi$
$421$	26.2860	1.28110	0.640551	+	0.767915i	$0.278706\pi$
$422$	0	0
$423$	−41.9343	−2.03891
$424$	0	0
$425$	4.36863	0.211910
$426$	0	0
$427$	−0.229752	−0.0111185
$428$	0	0
$429$	−16.3267	−0.788258
$430$	0	0
$431$	−32.3975	−1.56053	−0.780267	−	0.625447i	$-0.784917\pi$
$431$	−32.3975	−1.56053	−0.780267	+	0.625447i	$0.784917\pi$
$432$	0	0
$433$	−16.6329	−0.799326	−0.399663	−	0.916662i	$-0.630873\pi$
$433$	−16.6329	−0.799326	−0.399663	+	0.916662i	$0.630873\pi$
$434$	0	0
$435$	−4.29867	−0.206105
$436$	0	0
$437$	−21.4219	−1.02475
$438$	0	0
$439$	27.3680	1.30620	0.653102	−	0.757270i	$-0.273467\pi$
$439$	27.3680	1.30620	0.653102	+	0.757270i	$0.273467\pi$
$440$	0	0
$441$	−25.4741	−1.21305
$442$	0	0
$443$	−14.7840	−0.702408	−0.351204	−	0.936299i	$-0.614228\pi$
$443$	−14.7840	−0.702408	−0.351204	+	0.936299i	$0.614228\pi$
$444$	0	0
$445$	7.18882	0.340783
$446$	0	0
$447$	−37.6749	−1.78196
$448$	0	0
$449$	31.7724	1.49943	0.749717	−	0.661759i	$-0.230190\pi$
$449$	31.7724	1.49943	0.749717	+	0.661759i	$0.230190\pi$
$450$	0	0
$451$	−22.6703	−1.06750
$452$	0	0
$453$	16.0741	0.755225
$454$	0	0
$455$	1.34933	0.0632577
$456$	0	0
$457$	3.49700	0.163583	0.0817915	−	0.996649i	$-0.473936\pi$
$457$	3.49700	0.163583	0.0817915	+	0.996649i	$0.473936\pi$
$458$	0	0
$459$	2.57041	0.119976
$460$	0	0
$461$	27.2992	1.27145	0.635726	−	0.771915i	$-0.280701\pi$
$461$	27.2992	1.27145	0.635726	+	0.771915i	$0.280701\pi$
$462$	0	0
$463$	7.73350	0.359406	0.179703	−	0.983721i	$-0.442486\pi$
$463$	7.73350	0.359406	0.179703	+	0.983721i	$0.442486\pi$
$464$	0	0
$465$	−8.44882	−0.391804
$466$	0	0
$467$	−21.9428	−1.01539	−0.507695	−	0.861537i	$-0.669502\pi$
$467$	−21.9428	−1.01539	−0.507695	+	0.861537i	$0.669502\pi$
$468$	0	0
$469$	3.98805	0.184151
$470$	0	0
$471$	10.9468	0.504402
$472$	0	0
$473$	−24.6301	−1.13250
$474$	0	0
$475$	−25.4919	−1.16965
$476$	0	0
$477$	−46.2849	−2.11924
$478$	0	0
$479$	−12.8569	−0.587448	−0.293724	−	0.955890i	$-0.594895\pi$
$479$	−12.8569	−0.587448	−0.293724	+	0.955890i	$0.594895\pi$
$480$	0	0
$481$	−6.69692	−0.305353
$482$	0	0
$483$	7.43893	0.338483
$484$	0	0
$485$	−1.60350	−0.0728110
$486$	0	0
$487$	20.9381	0.948795	0.474397	−	0.880311i	$-0.342666\pi$
$487$	20.9381	0.948795	0.474397	+	0.880311i	$0.342666\pi$
$488$	0	0
$489$	−3.23829	−0.146441
$490$	0	0
$491$	13.8210	0.623731	0.311866	−	0.950126i	$-0.399046\pi$
$491$	13.8210	0.623731	0.311866	+	0.950126i	$0.399046\pi$
$492$	0	0
$493$	2.04867	0.0922673
$494$	0	0
$495$	8.82034	0.396445
$496$	0	0
$497$	4.89964	0.219779
$498$	0	0
$499$	17.0448	0.763029	0.381515	−	0.924363i	$-0.375403\pi$
$499$	17.0448	0.763029	0.381515	+	0.924363i	$0.375403\pi$
$500$	0	0
$501$	29.8970	1.33570
$502$	0	0
$503$	−3.76243	−0.167758	−0.0838791	−	0.996476i	$-0.526731\pi$
$503$	−3.76243	−0.167758	−0.0838791	+	0.996476i	$0.526731\pi$
$504$	0	0
$505$	−7.20224	−0.320495
$506$	0	0
$507$	21.3962	0.950237
$508$	0	0
$509$	10.4596	0.463613	0.231807	−	0.972762i	$-0.425536\pi$
$509$	10.4596	0.463613	0.231807	+	0.972762i	$0.425536\pi$
$510$	0	0
$511$	−8.78781	−0.388750
$512$	0	0
$513$	−14.9989	−0.662216
$514$	0	0
$515$	−8.97713	−0.395579
$516$	0	0
$517$	29.4845	1.29673
$518$	0	0
$519$	8.57419	0.376365
$520$	0	0
$521$	0.595687	0.0260975	0.0130488	−	0.999915i	$-0.495846\pi$
$521$	0.595687	0.0260975	0.0130488	+	0.999915i	$0.495846\pi$
$522$	0	0
$523$	16.2309	0.709729	0.354864	−	0.934918i	$-0.384527\pi$
$523$	16.2309	0.709729	0.354864	+	0.934918i	$0.384527\pi$
$524$	0	0
$525$	8.85226	0.386344
$526$	0	0
$527$	4.02655	0.175399
$528$	0	0
$529$	−9.52268	−0.414030
$530$	0	0
$531$	3.97338	0.172430
$532$	0	0
$533$	−17.9582	−0.777856
$534$	0	0
$535$	5.48864	0.237294
$536$	0	0
$537$	9.35975	0.403903
$538$	0	0
$539$	17.9112	0.771488
$540$	0	0
$541$	42.7643	1.83858	0.919290	−	0.393581i	$-0.128764\pi$
$541$	42.7643	1.83858	0.919290	+	0.393581i	$0.128764\pi$
$542$	0	0
$543$	54.8472	2.35372
$544$	0	0
$545$	−2.27326	−0.0973759
$546$	0	0
$547$	−32.0073	−1.36853	−0.684267	−	0.729231i	$-0.739878\pi$
$547$	−32.0073	−1.36853	−0.684267	+	0.729231i	$0.739878\pi$
$548$	0	0
$549$	−1.18968	−0.0507745
$550$	0	0
$551$	−11.9544	−0.509275
$552$	0	0
$553$	5.47693	0.232903
$554$	0	0
$555$	6.34961	0.269526
$556$	0	0
$557$	0.428065	0.0181377	0.00906884	−	0.999959i	$-0.497113\pi$
$557$	0.428065	0.0181377	0.00906884	+	0.999959i	$0.497113\pi$
$558$	0	0
$559$	−19.5107	−0.825215
$560$	0	0
$561$	−7.37745	−0.311476
$562$	0	0
$563$	34.4678	1.45265	0.726323	−	0.687354i	$-0.241228\pi$
$563$	34.4678	1.45265	0.726323	+	0.687354i	$0.241228\pi$
$564$	0	0
$565$	9.34405	0.393107
$566$	0	0
$567$	−3.93830	−0.165393
$568$	0	0
$569$	41.0291	1.72003	0.860015	−	0.510269i	$-0.170454\pi$
$569$	41.0291	1.72003	0.860015	+	0.510269i	$0.170454\pi$
$570$	0	0
$571$	28.3342	1.18575	0.592875	−	0.805295i	$-0.297993\pi$
$571$	28.3342	1.18575	0.592875	+	0.805295i	$0.297993\pi$
$572$	0	0
$573$	9.20176	0.384409
$574$	0	0
$575$	16.0379	0.668826
$576$	0	0
$577$	7.38685	0.307518	0.153759	−	0.988108i	$-0.450862\pi$
$577$	7.38685	0.307518	0.153759	+	0.988108i	$0.450862\pi$
$578$	0	0
$579$	37.6993	1.56673
$580$	0	0
$581$	−5.65950	−0.234796
$582$	0	0
$583$	32.5435	1.34781
$584$	0	0
$585$	6.98701	0.288877
$586$	0	0
$587$	−4.12819	−0.170389	−0.0851944	−	0.996364i	$-0.527151\pi$
$587$	−4.12819	−0.170389	−0.0851944	+	0.996364i	$0.527151\pi$
$588$	0	0
$589$	−23.4958	−0.968127
$590$	0	0
$591$	43.6723	1.79644
$592$	0	0
$593$	44.1631	1.81356	0.906780	−	0.421603i	$-0.138533\pi$
$593$	44.1631	1.81356	0.906780	+	0.421603i	$0.138533\pi$
$594$	0	0
$595$	0.609717	0.0249960
$596$	0	0
$597$	39.6800	1.62399
$598$	0	0
$599$	16.1547	0.660064	0.330032	−	0.943970i	$-0.392940\pi$
$599$	16.1547	0.660064	0.330032	+	0.943970i	$0.392940\pi$
$600$	0	0
$601$	−17.8425	−0.727811	−0.363906	−	0.931436i	$-0.618557\pi$
$601$	−17.8425	−0.727811	−0.363906	+	0.931436i	$0.618557\pi$
$602$	0	0
$603$	20.6506	0.840958
$604$	0	0
$605$	2.53875	0.103215
$606$	0	0
$607$	−29.9137	−1.21416	−0.607080	−	0.794641i	$-0.707659\pi$
$607$	−29.9137	−1.21416	−0.607080	+	0.794641i	$0.707659\pi$
$608$	0	0
$609$	4.15126	0.168218
$610$	0	0
$611$	23.3561	0.944887
$612$	0	0
$613$	32.0782	1.29563	0.647813	−	0.761800i	$-0.275684\pi$
$613$	32.0782	1.29563	0.647813	+	0.761800i	$0.275684\pi$
$614$	0	0
$615$	17.0268	0.686589
$616$	0	0
$617$	−16.2483	−0.654134	−0.327067	−	0.945001i	$-0.606060\pi$
$617$	−16.2483	−0.654134	−0.327067	+	0.945001i	$0.606060\pi$
$618$	0	0
$619$	21.0111	0.844507	0.422253	−	0.906478i	$-0.361239\pi$
$619$	21.0111	0.844507	0.422253	+	0.906478i	$0.361239\pi$
$620$	0	0
$621$	9.43634	0.378667
$622$	0	0
$623$	−6.94231	−0.278138
$624$	0	0
$625$	15.9281	0.637125
$626$	0	0
$627$	43.0490	1.71921
$628$	0	0
$629$	−3.02611	−0.120659
$630$	0	0
$631$	19.4046	0.772485	0.386242	−	0.922397i	$-0.373773\pi$
$631$	19.4046	0.772485	0.386242	+	0.922397i	$0.373773\pi$
$632$	0	0
$633$	−21.5002	−0.854557
$634$	0	0
$635$	−7.53565	−0.299043
$636$	0	0
$637$	14.1883	0.562160
$638$	0	0
$639$	25.3709	1.00366
$640$	0	0
$641$	23.8736	0.942951	0.471475	−	0.881879i	$-0.343722\pi$
$641$	23.8736	0.942951	0.471475	+	0.881879i	$0.343722\pi$
$642$	0	0
$643$	0.240467	0.00948309	0.00474154	−	0.999989i	$-0.498491\pi$
$643$	0.240467	0.00948309	0.00474154	+	0.999989i	$0.498491\pi$
$644$	0	0
$645$	18.4988	0.728391
$646$	0	0
$647$	13.1949	0.518744	0.259372	−	0.965777i	$-0.416484\pi$
$647$	13.1949	0.518744	0.259372	+	0.965777i	$0.416484\pi$
$648$	0	0
$649$	−2.79373	−0.109664
$650$	0	0
$651$	8.15910	0.319780
$652$	0	0
$653$	11.5366	0.451461	0.225731	−	0.974190i	$-0.427523\pi$
$653$	11.5366	0.451461	0.225731	+	0.974190i	$0.427523\pi$
$654$	0	0
$655$	6.80303	0.265816
$656$	0	0
$657$	−45.5043	−1.77529
$658$	0	0
$659$	−29.1233	−1.13448	−0.567242	−	0.823551i	$-0.691989\pi$
$659$	−29.1233	−1.13448	−0.567242	+	0.823551i	$0.691989\pi$
$660$	0	0
$661$	−6.04510	−0.235127	−0.117564	−	0.993065i	$-0.537508\pi$
$661$	−6.04510	−0.235127	−0.117564	+	0.993065i	$0.537508\pi$
$662$	0	0
$663$	−5.84403	−0.226963
$664$	0	0
$665$	−3.55783	−0.137967
$666$	0	0
$667$	7.52096	0.291213
$668$	0	0
$669$	−15.2536	−0.589740
$670$	0	0
$671$	0.836483	0.0322921
$672$	0	0
$673$	5.22391	0.201367	0.100683	−	0.994919i	$-0.467897\pi$
$673$	5.22391	0.201367	0.100683	+	0.994919i	$0.467897\pi$
$674$	0	0
$675$	11.2292	0.432211
$676$	0	0
$677$	−40.2011	−1.54505	−0.772527	−	0.634982i	$-0.781008\pi$
$677$	−40.2011	−1.54505	−0.772527	+	0.634982i	$0.781008\pi$
$678$	0	0
$679$	1.54851	0.0594264
$680$	0	0
$681$	14.7040	0.563460
$682$	0	0
$683$	−20.4350	−0.781925	−0.390963	−	0.920407i	$-0.627858\pi$
$683$	−20.4350	−0.781925	−0.390963	+	0.920407i	$0.627858\pi$
$684$	0	0
$685$	−5.27991	−0.201735
$686$	0	0
$687$	−58.9392	−2.24867
$688$	0	0
$689$	25.7793	0.982112
$690$	0	0
$691$	13.4495	0.511643	0.255822	−	0.966724i	$-0.417654\pi$
$691$	13.4495	0.511643	0.255822	+	0.966724i	$0.417654\pi$
$692$	0	0
$693$	−8.51788	−0.323568
$694$	0	0
$695$	2.35996	0.0895186
$696$	0	0
$697$	−8.11469	−0.307366
$698$	0	0
$699$	−14.7143	−0.556544
$700$	0	0
$701$	30.9699	1.16972	0.584859	−	0.811135i	$-0.301150\pi$
$701$	30.9699	1.16972	0.584859	+	0.811135i	$0.301150\pi$
$702$	0	0
$703$	17.6580	0.665983
$704$	0	0
$705$	−22.1448	−0.834022
$706$	0	0
$707$	6.95527	0.261580
$708$	0	0
$709$	−36.5292	−1.37188	−0.685942	−	0.727656i	$-0.740610\pi$
$709$	−36.5292	−1.37188	−0.685942	+	0.727656i	$0.740610\pi$
$710$	0	0
$711$	28.3602	1.06359
$712$	0	0
$713$	14.7821	0.553593
$714$	0	0
$715$	−4.91266	−0.183723
$716$	0	0
$717$	−56.7420	−2.11907
$718$	0	0
$719$	−23.7592	−0.886070	−0.443035	−	0.896504i	$-0.646098\pi$
$719$	−23.7592	−0.886070	−0.443035	+	0.896504i	$0.646098\pi$
$720$	0	0
$721$	8.66929	0.322861
$722$	0	0
$723$	42.7712	1.59068
$724$	0	0
$725$	8.94988	0.332390
$726$	0	0
$727$	−31.3921	−1.16427	−0.582135	−	0.813092i	$-0.697782\pi$
$727$	−31.3921	−1.16427	−0.582135	+	0.813092i	$0.697782\pi$
$728$	0	0
$729$	−40.7561	−1.50949
$730$	0	0
$731$	−8.81621	−0.326079
$732$	0	0
$733$	24.9607	0.921946	0.460973	−	0.887414i	$-0.347501\pi$
$733$	24.9607	0.921946	0.460973	+	0.887414i	$0.347501\pi$
$734$	0	0
$735$	−13.4524	−0.496201
$736$	0	0
$737$	−14.5197	−0.534841
$738$	0	0
$739$	−6.97275	−0.256497	−0.128248	−	0.991742i	$-0.540935\pi$
$739$	−6.97275	−0.256497	−0.128248	+	0.991742i	$0.540935\pi$
$740$	0	0
$741$	34.1011	1.25274
$742$	0	0
$743$	52.0833	1.91075	0.955375	−	0.295397i	$-0.0954519\pi$
$743$	52.0833	1.91075	0.955375	+	0.295397i	$0.0954519\pi$
$744$	0	0
$745$	−11.3363	−0.415331
$746$	0	0
$747$	−29.3056	−1.07224
$748$	0	0
$749$	−5.30042	−0.193673
$750$	0	0
$751$	−13.2511	−0.483538	−0.241769	−	0.970334i	$-0.577728\pi$
$751$	−13.2511	−0.483538	−0.241769	+	0.970334i	$0.577728\pi$
$752$	0	0
$753$	−24.7874	−0.903304
$754$	0	0
$755$	4.83666	0.176024
$756$	0	0
$757$	46.8029	1.70108	0.850540	−	0.525911i	$-0.176276\pi$
$757$	46.8029	1.70108	0.850540	+	0.525911i	$0.176276\pi$
$758$	0	0
$759$	−27.0837	−0.983076
$760$	0	0
$761$	−33.5216	−1.21516	−0.607578	−	0.794260i	$-0.707859\pi$
$761$	−33.5216	−1.21516	−0.607578	+	0.794260i	$0.707859\pi$
$762$	0	0
$763$	2.19531	0.0794756
$764$	0	0
$765$	3.15719	0.114148
$766$	0	0
$767$	−2.21305	−0.0799085
$768$	0	0
$769$	−1.98505	−0.0715829	−0.0357914	−	0.999359i	$-0.511395\pi$
$769$	−1.98505	−0.0715829	−0.0357914	+	0.999359i	$0.511395\pi$
$770$	0	0
$771$	47.5892	1.71388
$772$	0	0
$773$	−9.18287	−0.330285	−0.165142	−	0.986270i	$-0.552808\pi$
$773$	−9.18287	−0.330285	−0.165142	+	0.986270i	$0.552808\pi$
$774$	0	0
$775$	17.5905	0.631870
$776$	0	0
$777$	−6.13187	−0.219980
$778$	0	0
$779$	47.3509	1.69652
$780$	0	0
$781$	−17.8386	−0.638316
$782$	0	0
$783$	5.26591	0.188188
$784$	0	0
$785$	3.29387	0.117563
$786$	0	0
$787$	4.96528	0.176993	0.0884965	−	0.996076i	$-0.471794\pi$
$787$	4.96528	0.176993	0.0884965	+	0.996076i	$0.471794\pi$
$788$	0	0
$789$	−18.3242	−0.652357
$790$	0	0
$791$	−9.02363	−0.320843
$792$	0	0
$793$	0.662618	0.0235302
$794$	0	0
$795$	−24.4423	−0.866879
$796$	0	0
$797$	6.73621	0.238609	0.119304	−	0.992858i	$-0.461934\pi$
$797$	6.73621	0.238609	0.119304	+	0.992858i	$0.461934\pi$
$798$	0	0
$799$	10.5538	0.373367
$800$	0	0
$801$	−35.9481	−1.27017
$802$	0	0
$803$	31.9947	1.12907
$804$	0	0
$805$	2.23836	0.0788918
$806$	0	0
$807$	5.91084	0.208071
$808$	0	0
$809$	−47.6772	−1.67624	−0.838121	−	0.545484i	$-0.816346\pi$
$809$	−47.6772	−1.67624	−0.838121	+	0.545484i	$0.816346\pi$
$810$	0	0
$811$	23.7110	0.832604	0.416302	−	0.909226i	$-0.363326\pi$
$811$	23.7110	0.832604	0.416302	+	0.909226i	$0.363326\pi$
$812$	0	0
$813$	−33.8228	−1.18622
$814$	0	0
$815$	−0.974397	−0.0341316
$816$	0	0
$817$	51.4445	1.79981
$818$	0	0
$819$	−6.74742	−0.235774
$820$	0	0
$821$	−17.6741	−0.616832	−0.308416	−	0.951252i	$-0.599799\pi$
$821$	−17.6741	−0.616832	−0.308416	+	0.951252i	$0.599799\pi$
$822$	0	0
$823$	−7.62285	−0.265716	−0.132858	−	0.991135i	$-0.542415\pi$
$823$	−7.62285	−0.265716	−0.132858	+	0.991135i	$0.542415\pi$
$824$	0	0
$825$	−32.2294	−1.12208
$826$	0	0
$827$	−1.88140	−0.0654225	−0.0327113	−	0.999465i	$-0.510414\pi$
$827$	−1.88140	−0.0654225	−0.0327113	+	0.999465i	$0.510414\pi$
$828$	0	0
$829$	−12.9940	−0.451300	−0.225650	−	0.974208i	$-0.572451\pi$
$829$	−12.9940	−0.451300	−0.225650	+	0.974208i	$0.572451\pi$
$830$	0	0
$831$	18.0335	0.625574
$832$	0	0
$833$	6.41119	0.222135
$834$	0	0
$835$	8.99595	0.311318
$836$	0	0
$837$	10.3499	0.357744
$838$	0	0
$839$	9.38723	0.324083	0.162042	−	0.986784i	$-0.448192\pi$
$839$	9.38723	0.324083	0.162042	+	0.986784i	$0.448192\pi$
$840$	0	0
$841$	−24.8030	−0.855275
$842$	0	0
$843$	−24.2973	−0.836843
$844$	0	0
$845$	6.43807	0.221476
$846$	0	0
$847$	−2.45169	−0.0842411
$848$	0	0
$849$	−73.8963	−2.53611
$850$	0	0
$851$	−11.1093	−0.380821
$852$	0	0
$853$	36.7853	1.25951	0.629753	−	0.776795i	$-0.283156\pi$
$853$	36.7853	1.25951	0.629753	+	0.776795i	$0.283156\pi$
$854$	0	0
$855$	−18.4229	−0.630049
$856$	0	0
$857$	17.0595	0.582741	0.291370	−	0.956610i	$-0.405889\pi$
$857$	17.0595	0.582741	0.291370	+	0.956610i	$0.405889\pi$
$858$	0	0
$859$	16.5119	0.563379	0.281689	−	0.959506i	$-0.409105\pi$
$859$	16.5119	0.563379	0.281689	+	0.959506i	$0.409105\pi$
$860$	0	0
$861$	−16.4430	−0.560375
$862$	0	0
$863$	14.3790	0.489468	0.244734	−	0.969590i	$-0.421299\pi$
$863$	14.3790	0.489468	0.244734	+	0.969590i	$0.421299\pi$
$864$	0	0
$865$	2.57996	0.0877212
$866$	0	0
$867$	−2.64071	−0.0896834
$868$	0	0
$869$	−19.9404	−0.676433
$870$	0	0
$871$	−11.5018	−0.389722
$872$	0	0
$873$	8.01838	0.271381
$874$	0	0
$875$	5.71221	0.193108
$876$	0	0
$877$	−38.3830	−1.29610	−0.648050	−	0.761598i	$-0.724415\pi$
$877$	−38.3830	−1.29610	−0.648050	+	0.761598i	$0.724415\pi$
$878$	0	0
$879$	−70.2978	−2.37109
$880$	0	0
$881$	33.1239	1.11597	0.557987	−	0.829850i	$-0.311574\pi$
$881$	33.1239	1.11597	0.557987	+	0.829850i	$0.311574\pi$
$882$	0	0
$883$	−50.0491	−1.68429	−0.842143	−	0.539254i	$-0.818706\pi$
$883$	−50.0491	−1.68429	−0.842143	+	0.539254i	$0.818706\pi$
$884$	0	0
$885$	2.09828	0.0705328
$886$	0	0
$887$	−11.6465	−0.391050	−0.195525	−	0.980699i	$-0.562641\pi$
$887$	−11.6465	−0.391050	−0.195525	+	0.980699i	$0.562641\pi$
$888$	0	0
$889$	7.27724	0.244071
$890$	0	0
$891$	14.3386	0.480361
$892$	0	0
$893$	−61.5837	−2.06082
$894$	0	0
$895$	2.81633	0.0941396
$896$	0	0
$897$	−21.4543	−0.716338
$898$	0	0
$899$	8.24907	0.275122
$900$	0	0
$901$	11.6488	0.388076
$902$	0	0
$903$	−17.8645	−0.594493
$904$	0	0
$905$	16.5034	0.548593
$906$	0	0
$907$	58.8883	1.95536	0.977678	−	0.210110i	$-0.0673823\pi$
$907$	58.8883	1.95536	0.977678	+	0.210110i	$0.0673823\pi$
$908$	0	0
$909$	36.0152	1.19455
$910$	0	0
$911$	6.57448	0.217822	0.108911	−	0.994051i	$-0.465264\pi$
$911$	6.57448	0.217822	0.108911	+	0.994051i	$0.465264\pi$
$912$	0	0
$913$	20.6051	0.681931
$914$	0	0
$915$	−0.628253	−0.0207694
$916$	0	0
$917$	−6.56975	−0.216952
$918$	0	0
$919$	−25.5164	−0.841708	−0.420854	−	0.907128i	$-0.638269\pi$
$919$	−25.5164	−0.841708	−0.420854	+	0.907128i	$0.638269\pi$
$920$	0	0
$921$	−86.2572	−2.84227
$922$	0	0
$923$	−14.1308	−0.465122
$924$	0	0
$925$	−13.2200	−0.434669
$926$	0	0
$927$	44.8907	1.47440
$928$	0	0
$929$	−7.26373	−0.238315	−0.119158	−	0.992875i	$-0.538019\pi$
$929$	−7.26373	−0.238315	−0.119158	+	0.992875i	$0.538019\pi$
$930$	0	0
$931$	−37.4106	−1.22608
$932$	0	0
$933$	−4.45588	−0.145879
$934$	0	0
$935$	−2.21986	−0.0725972
$936$	0	0
$937$	26.9341	0.879900	0.439950	−	0.898022i	$-0.354996\pi$
$937$	26.9341	0.879900	0.439950	+	0.898022i	$0.354996\pi$
$938$	0	0
$939$	−41.0119	−1.33837
$940$	0	0
$941$	20.0954	0.655091	0.327545	−	0.944835i	$-0.393779\pi$
$941$	20.0954	0.655091	0.327545	+	0.944835i	$0.393779\pi$
$942$	0	0
$943$	−29.7902	−0.970103
$944$	0	0
$945$	1.56722	0.0509817
$946$	0	0
$947$	−32.8432	−1.06726	−0.533630	−	0.845718i	$-0.679173\pi$
$947$	−32.8432	−1.06726	−0.533630	+	0.845718i	$0.679173\pi$
$948$	0	0
$949$	25.3445	0.822718
$950$	0	0
$951$	30.0239	0.973591
$952$	0	0
$953$	49.1462	1.59200	0.796001	−	0.605295i	$-0.206945\pi$
$953$	49.1462	1.59200	0.796001	+	0.605295i	$0.206945\pi$
$954$	0	0
$955$	2.76879	0.0895961
$956$	0	0
$957$	−15.1139	−0.488565
$958$	0	0
$959$	5.09886	0.164651
$960$	0	0
$961$	−14.7869	−0.476996
$962$	0	0
$963$	−27.4463	−0.884443
$964$	0	0
$965$	11.3437	0.365165
$966$	0	0
$967$	−5.12208	−0.164715	−0.0823576	−	0.996603i	$-0.526245\pi$
$967$	−5.12208	−0.164715	−0.0823576	+	0.996603i	$0.526245\pi$
$968$	0	0
$969$	15.4091	0.495012
$970$	0	0
$971$	5.84149	0.187462	0.0937311	−	0.995598i	$-0.470121\pi$
$971$	5.84149	0.187462	0.0937311	+	0.995598i	$0.470121\pi$
$972$	0	0
$973$	−2.27904	−0.0730626
$974$	0	0
$975$	−25.5304	−0.817628
$976$	0	0
$977$	−54.4507	−1.74203	−0.871017	−	0.491253i	$-0.836539\pi$
$977$	−54.4507	−1.74203	−0.871017	+	0.491253i	$0.836539\pi$
$978$	0	0
$979$	25.2756	0.807812
$980$	0	0
$981$	11.3676	0.362939
$982$	0	0
$983$	5.44456	0.173654	0.0868272	−	0.996223i	$-0.472327\pi$
$983$	5.44456	0.173654	0.0868272	+	0.996223i	$0.472327\pi$
$984$	0	0
$985$	13.1409	0.418704
$986$	0	0
$987$	21.3854	0.680706
$988$	0	0
$989$	−32.3656	−1.02917
$990$	0	0
$991$	2.07648	0.0659617	0.0329809	−	0.999456i	$-0.489500\pi$
$991$	2.07648	0.0659617	0.0329809	+	0.999456i	$0.489500\pi$
$992$	0	0
$993$	−28.8174	−0.914494
$994$	0	0
$995$	11.9396	0.378512
$996$	0	0
$997$	24.1571	0.765062	0.382531	−	0.923943i	$-0.375053\pi$
$997$	24.1571	0.765062	0.382531	+	0.923943i	$0.375053\pi$
$998$	0	0
$999$	−7.77833	−0.246095

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8024.2.a.ba.1.3	✓	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8024.2.a.ba.1.3	✓	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:	\( N \)	\(=\)	\( 8024 = 2^{3} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8024.a (trivial)

\(f(q)\)	\(=\)	\(q-2.64071 q^{3} -0.794586 q^{5} +0.767339 q^{7} +3.97338 q^{9} +O(q^{10})\)	\(q-2.64071 q^{3} -0.794586 q^{5} +0.767339 q^{7} +3.97338 q^{9} -2.79373 q^{11} -2.21305 q^{13} +2.09828 q^{15} -1.00000 q^{17} +5.83521 q^{19} -2.02632 q^{21} -3.67115 q^{23} -4.36863 q^{25} -2.57041 q^{27} -2.04867 q^{29} -4.02655 q^{31} +7.37745 q^{33} -0.609717 q^{35} +3.02611 q^{37} +5.84403 q^{39} +8.11469 q^{41} +8.81621 q^{43} -3.15719 q^{45} -10.5538 q^{47} -6.41119 q^{49} +2.64071 q^{51} -11.6488 q^{53} +2.21986 q^{55} -15.4091 q^{57} +1.00000 q^{59} -0.299414 q^{61} +3.04893 q^{63} +1.75846 q^{65} +5.19725 q^{67} +9.69445 q^{69} +6.38523 q^{71} -11.4523 q^{73} +11.5363 q^{75} -2.14374 q^{77} +7.13756 q^{79} -5.13241 q^{81} -7.37549 q^{83} +0.794586 q^{85} +5.40995 q^{87} -9.04725 q^{89} -1.69816 q^{91} +10.6330 q^{93} -4.63658 q^{95} +2.01803 q^{97} -11.1005 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 4 q^{3} + 2 q^{5} + 3 q^{7} + 34 q^{9}+O(q^{10}) \) 30 * q + 4 * q^3 + 2 * q^5 + 3 * q^7 + 34 * q^9	\( 30 q + 4 q^{3} + 2 q^{5} + 3 q^{7} + 34 q^{9} + 3 q^{11} + 9 q^{13} + 14 q^{15} - 30 q^{17} + 24 q^{19} + 7 q^{21} + 9 q^{23} + 40 q^{25} + 19 q^{27} + 9 q^{29} + 11 q^{31} - 14 q^{33} + 30 q^{35} - 13 q^{37} + 16 q^{39} - 13 q^{41} + 23 q^{43} + 12 q^{45} + 43 q^{47} + 35 q^{49} - 4 q^{51} - 4 q^{53} + 43 q^{55} + 3 q^{57} + 30 q^{59} + 43 q^{61} + 38 q^{63} + 3 q^{65} + 50 q^{67} + 34 q^{69} + 3 q^{71} - 16 q^{73} + 21 q^{75} + 18 q^{77} + 45 q^{79} + 6 q^{81} + 63 q^{83} - 2 q^{85} + 42 q^{87} + 6 q^{89} + 22 q^{91} - 2 q^{93} + 19 q^{95} - 28 q^{97} + 51 q^{99}+O(q^{100}) \) 30 * q + 4 * q^3 + 2 * q^5 + 3 * q^7 + 34 * q^9 + 3 * q^11 + 9 * q^13 + 14 * q^15 - 30 * q^17 + 24 * q^19 + 7 * q^21 + 9 * q^23 + 40 * q^25 + 19 * q^27 + 9 * q^29 + 11 * q^31 - 14 * q^33 + 30 * q^35 - 13 * q^37 + 16 * q^39 - 13 * q^41 + 23 * q^43 + 12 * q^45 + 43 * q^47 + 35 * q^49 - 4 * q^51 - 4 * q^53 + 43 * q^55 + 3 * q^57 + 30 * q^59 + 43 * q^61 + 38 * q^63 + 3 * q^65 + 50 * q^67 + 34 * q^69 + 3 * q^71 - 16 * q^73 + 21 * q^75 + 18 * q^77 + 45 * q^79 + 6 * q^81 + 63 * q^83 - 2 * q^85 + 42 * q^87 + 6 * q^89 + 22 * q^91 - 2 * q^93 + 19 * q^95 - 28 * q^97 + 51 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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