LMFDB - Embedded newform 8024.2.a.bb.1.7

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:	$32$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
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)

$f(q)$	$=$	$q-2.29360 q^{3} -3.85032 q^{5} +1.95004 q^{7} +2.26061 q^{9} +O(q^{10})$	$q-2.29360 q^{3} -3.85032 q^{5} +1.95004 q^{7} +2.26061 q^{9} -0.260097 q^{11} -6.61698 q^{13} +8.83111 q^{15} +1.00000 q^{17} +8.40744 q^{19} -4.47262 q^{21} +4.64992 q^{23} +9.82500 q^{25} +1.69587 q^{27} -5.50817 q^{29} -0.591068 q^{31} +0.596559 q^{33} -7.50829 q^{35} -4.47569 q^{37} +15.1767 q^{39} +1.62366 q^{41} -4.97374 q^{43} -8.70407 q^{45} -5.88682 q^{47} -3.19734 q^{49} -2.29360 q^{51} -7.20595 q^{53} +1.00146 q^{55} -19.2833 q^{57} +1.00000 q^{59} -7.00124 q^{61} +4.40828 q^{63} +25.4775 q^{65} +0.409412 q^{67} -10.6651 q^{69} -5.14411 q^{71} +11.1641 q^{73} -22.5346 q^{75} -0.507200 q^{77} +3.88052 q^{79} -10.6715 q^{81} +15.9045 q^{83} -3.85032 q^{85} +12.6336 q^{87} +0.185461 q^{89} -12.9034 q^{91} +1.35567 q^{93} -32.3714 q^{95} -2.56631 q^{97} -0.587977 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 8 q^{5} - 3 q^{7} + 40 q^{9}+O(q^{10}) $ 32 * q + 8 * q^5 - 3 * q^7 + 40 * q^9	$ 32 q + 8 q^{5} - 3 q^{7} + 40 q^{9} + 3 q^{11} + 13 q^{13} + 4 q^{15} + 32 q^{17} + 14 q^{19} - 7 q^{21} + 7 q^{23} + 38 q^{25} + 9 q^{27} + 17 q^{29} + 15 q^{31} + 18 q^{33} + 6 q^{35} + 21 q^{37} + 16 q^{39} + 49 q^{41} - 7 q^{43} + 14 q^{45} - 25 q^{47} + 37 q^{49} + 12 q^{53} + 15 q^{55} + 45 q^{57} + 32 q^{59} + 5 q^{61} - 12 q^{63} + 39 q^{65} + 12 q^{69} - 13 q^{71} + 70 q^{73} - 47 q^{75} - 10 q^{77} - q^{79} + 84 q^{81} - 17 q^{83} + 8 q^{85} + 20 q^{87} + 42 q^{89} + 36 q^{91} + 2 q^{93} - q^{95} + 58 q^{97} + 7 q^{99}+O(q^{100}) $ 32 * q + 8 * q^5 - 3 * q^7 + 40 * q^9 + 3 * q^11 + 13 * q^13 + 4 * q^15 + 32 * q^17 + 14 * q^19 - 7 * q^21 + 7 * q^23 + 38 * q^25 + 9 * q^27 + 17 * q^29 + 15 * q^31 + 18 * q^33 + 6 * q^35 + 21 * q^37 + 16 * q^39 + 49 * q^41 - 7 * q^43 + 14 * q^45 - 25 * q^47 + 37 * q^49 + 12 * q^53 + 15 * q^55 + 45 * q^57 + 32 * q^59 + 5 * q^61 - 12 * q^63 + 39 * q^65 + 12 * q^69 - 13 * q^71 + 70 * q^73 - 47 * q^75 - 10 * q^77 - q^79 + 84 * q^81 - 17 * q^83 + 8 * q^85 + 20 * q^87 + 42 * q^89 + 36 * q^91 + 2 * q^93 - q^95 + 58 * q^97 + 7 * 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
$3$	−2.29360	−1.32421	−0.662106	−	0.749410i	$-0.730337\pi$
$3$	−2.29360	−1.32421	−0.662106	+	0.749410i	$0.730337\pi$
$4$	0	0
$5$	−3.85032	−1.72192	−0.860959	−	0.508675i	$-0.830136\pi$
$5$	−3.85032	−1.72192	−0.860959	+	0.508675i	$0.830136\pi$
$6$	0	0
$7$	1.95004	0.737047	0.368523	−	0.929619i	$-0.379863\pi$
$7$	1.95004	0.737047	0.368523	+	0.929619i	$0.379863\pi$
$8$	0	0
$9$	2.26061	0.753536
$10$	0	0
$11$	−0.260097	−0.0784222	−0.0392111	−	0.999231i	$-0.512484\pi$
$11$	−0.260097	−0.0784222	−0.0392111	+	0.999231i	$0.512484\pi$
$12$	0	0
$13$	−6.61698	−1.83522	−0.917610	−	0.397483i	$-0.869884\pi$
$13$	−6.61698	−1.83522	−0.917610	+	0.397483i	$0.869884\pi$
$14$	0	0
$15$	8.83111	2.28018
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	8.40744	1.92880	0.964399	−	0.264451i	$-0.0851907\pi$
$19$	8.40744	1.92880	0.964399	+	0.264451i	$0.0851907\pi$
$20$	0	0
$21$	−4.47262	−0.976005
$22$	0	0
$23$	4.64992	0.969575	0.484787	−	0.874632i	$-0.338897\pi$
$23$	4.64992	0.969575	0.484787	+	0.874632i	$0.338897\pi$
$24$	0	0
$25$	9.82500	1.96500
$26$	0	0
$27$	1.69587	0.326371
$28$	0	0
$29$	−5.50817	−1.02284	−0.511421	−	0.859330i	$-0.670881\pi$
$29$	−5.50817	−1.02284	−0.511421	+	0.859330i	$0.670881\pi$
$30$	0	0
$31$	−0.591068	−0.106159	−0.0530795	−	0.998590i	$-0.516904\pi$
$31$	−0.591068	−0.106159	−0.0530795	+	0.998590i	$0.516904\pi$
$32$	0	0
$33$	0.596559	0.103848
$34$	0	0
$35$	−7.50829	−1.26913
$36$	0	0
$37$	−4.47569	−0.735800	−0.367900	−	0.929865i	$-0.619923\pi$
$37$	−4.47569	−0.735800	−0.367900	+	0.929865i	$0.619923\pi$
$38$	0	0
$39$	15.1767	2.43022
$40$	0	0
$41$	1.62366	0.253574	0.126787	−	0.991930i	$-0.459534\pi$
$41$	1.62366	0.253574	0.126787	+	0.991930i	$0.459534\pi$
$42$	0	0
$43$	−4.97374	−0.758488	−0.379244	−	0.925297i	$-0.623816\pi$
$43$	−4.97374	−0.758488	−0.379244	+	0.925297i	$0.623816\pi$
$44$	0	0
$45$	−8.70407	−1.29753
$46$	0	0
$47$	−5.88682	−0.858681	−0.429340	−	0.903143i	$-0.641254\pi$
$47$	−5.88682	−0.858681	−0.429340	+	0.903143i	$0.641254\pi$
$48$	0	0
$49$	−3.19734	−0.456762
$50$	0	0
$51$	−2.29360	−0.321168
$52$	0	0
$53$	−7.20595	−0.989813	−0.494906	−	0.868946i	$-0.664798\pi$
$53$	−7.20595	−0.989813	−0.494906	+	0.868946i	$0.664798\pi$
$54$	0	0
$55$	1.00146	0.135037
$56$	0	0
$57$	−19.2833	−2.55414
$58$	0	0
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	0	0
$61$	−7.00124	−0.896417	−0.448208	−	0.893929i	$-0.647938\pi$
$61$	−7.00124	−0.896417	−0.448208	+	0.893929i	$0.647938\pi$
$62$	0	0
$63$	4.40828	0.555391
$64$	0	0
$65$	25.4775	3.16010
$66$	0	0
$67$	0.409412	0.0500177	0.0250088	−	0.999687i	$-0.492039\pi$
$67$	0.409412	0.0500177	0.0250088	+	0.999687i	$0.492039\pi$
$68$	0	0
$69$	−10.6651	−1.28392
$70$	0	0
$71$	−5.14411	−0.610494	−0.305247	−	0.952273i	$-0.598739\pi$
$71$	−5.14411	−0.610494	−0.305247	+	0.952273i	$0.598739\pi$
$72$	0	0
$73$	11.1641	1.30666	0.653329	−	0.757074i	$-0.273372\pi$
$73$	11.1641	1.30666	0.653329	+	0.757074i	$0.273372\pi$
$74$	0	0
$75$	−22.5346	−2.60207
$76$	0	0
$77$	−0.507200	−0.0578008
$78$	0	0
$79$	3.88052	0.436593	0.218296	−	0.975883i	$-0.429950\pi$
$79$	3.88052	0.436593	0.218296	+	0.975883i	$0.429950\pi$
$80$	0	0
$81$	−10.6715	−1.18572
$82$	0	0
$83$	15.9045	1.74575	0.872874	−	0.487946i	$-0.162254\pi$
$83$	15.9045	1.74575	0.872874	+	0.487946i	$0.162254\pi$
$84$	0	0
$85$	−3.85032	−0.417626
$86$	0	0
$87$	12.6336	1.35446
$88$	0	0
$89$	0.185461	0.0196588	0.00982940	−	0.999952i	$-0.496871\pi$
$89$	0.185461	0.0196588	0.00982940	+	0.999952i	$0.496871\pi$
$90$	0	0
$91$	−12.9034	−1.35264
$92$	0	0
$93$	1.35567	0.140577
$94$	0	0
$95$	−32.3714	−3.32123
$96$	0	0
$97$	−2.56631	−0.260569	−0.130284	−	0.991477i	$-0.541589\pi$
$97$	−2.56631	−0.260569	−0.130284	+	0.991477i	$0.541589\pi$
$98$	0	0
$99$	−0.587977	−0.0590939
$100$	0	0
$101$	−15.6644	−1.55867	−0.779334	−	0.626608i	$-0.784443\pi$
$101$	−15.6644	−1.55867	−0.779334	+	0.626608i	$0.784443\pi$
$102$	0	0
$103$	−16.2375	−1.59993	−0.799966	−	0.600045i	$-0.795149\pi$
$103$	−16.2375	−1.59993	−0.799966	+	0.600045i	$0.795149\pi$
$104$	0	0
$105$	17.2210	1.68060
$106$	0	0
$107$	−8.96795	−0.866964	−0.433482	−	0.901162i	$-0.642715\pi$
$107$	−8.96795	−0.866964	−0.433482	+	0.901162i	$0.642715\pi$
$108$	0	0
$109$	0.127842	0.0122451	0.00612254	−	0.999981i	$-0.498051\pi$
$109$	0.127842	0.0122451	0.00612254	+	0.999981i	$0.498051\pi$
$110$	0	0
$111$	10.2655	0.974354
$112$	0	0
$113$	1.79385	0.168751	0.0843756	−	0.996434i	$-0.473110\pi$
$113$	1.79385	0.168751	0.0843756	+	0.996434i	$0.473110\pi$
$114$	0	0
$115$	−17.9037	−1.66953
$116$	0	0
$117$	−14.9584	−1.38290
$118$	0	0
$119$	1.95004	0.178760
$120$	0	0
$121$	−10.9323	−0.993850
$122$	0	0
$123$	−3.72404	−0.335785
$124$	0	0
$125$	−18.5778	−1.66165
$126$	0	0
$127$	3.39475	0.301235	0.150618	−	0.988592i	$-0.451874\pi$
$127$	3.39475	0.301235	0.150618	+	0.988592i	$0.451874\pi$
$128$	0	0
$129$	11.4078	1.00440
$130$	0	0
$131$	8.81741	0.770381	0.385190	−	0.922837i	$-0.374136\pi$
$131$	8.81741	0.770381	0.385190	+	0.922837i	$0.374136\pi$
$132$	0	0
$133$	16.3949	1.42161
$134$	0	0
$135$	−6.52966	−0.561983
$136$	0	0
$137$	−18.9254	−1.61691	−0.808454	−	0.588559i	$-0.799695\pi$
$137$	−18.9254	−1.61691	−0.808454	+	0.588559i	$0.799695\pi$
$138$	0	0
$139$	5.31334	0.450672	0.225336	−	0.974281i	$-0.427652\pi$
$139$	5.31334	0.450672	0.225336	+	0.974281i	$0.427652\pi$
$140$	0	0
$141$	13.5020	1.13708
$142$	0	0
$143$	1.72106	0.143922
$144$	0	0
$145$	21.2083	1.76125
$146$	0	0
$147$	7.33342	0.604850
$148$	0	0
$149$	12.4869	1.02297	0.511485	−	0.859292i	$-0.329096\pi$
$149$	12.4869	1.02297	0.511485	+	0.859292i	$0.329096\pi$
$150$	0	0
$151$	5.96944	0.485786	0.242893	−	0.970053i	$-0.421904\pi$
$151$	5.96944	0.485786	0.242893	+	0.970053i	$0.421904\pi$
$152$	0	0
$153$	2.26061	0.182759
$154$	0	0
$155$	2.27580	0.182797
$156$	0	0
$157$	−1.96555	−0.156868	−0.0784338	−	0.996919i	$-0.524992\pi$
$157$	−1.96555	−0.156868	−0.0784338	+	0.996919i	$0.524992\pi$
$158$	0	0
$159$	16.5276	1.31072
$160$	0	0
$161$	9.06754	0.714622
$162$	0	0
$163$	−12.1228	−0.949532	−0.474766	−	0.880112i	$-0.657467\pi$
$163$	−12.1228	−0.949532	−0.474766	+	0.880112i	$0.657467\pi$
$164$	0	0
$165$	−2.29694	−0.178817
$166$	0	0
$167$	9.96493	0.771109	0.385555	−	0.922685i	$-0.374010\pi$
$167$	9.96493	0.771109	0.385555	+	0.922685i	$0.374010\pi$
$168$	0	0
$169$	30.7844	2.36803
$170$	0	0
$171$	19.0059	1.45342
$172$	0	0
$173$	−7.68809	−0.584514	−0.292257	−	0.956340i	$-0.594406\pi$
$173$	−7.68809	−0.584514	−0.292257	+	0.956340i	$0.594406\pi$
$174$	0	0
$175$	19.1592	1.44830
$176$	0	0
$177$	−2.29360	−0.172398
$178$	0	0
$179$	−20.3510	−1.52111	−0.760554	−	0.649274i	$-0.775073\pi$
$179$	−20.3510	−1.52111	−0.760554	+	0.649274i	$0.775073\pi$
$180$	0	0
$181$	−6.17649	−0.459095	−0.229547	−	0.973297i	$-0.573725\pi$
$181$	−6.17649	−0.459095	−0.229547	+	0.973297i	$0.573725\pi$
$182$	0	0
$183$	16.0581	1.18705
$184$	0	0
$185$	17.2329	1.26699
$186$	0	0
$187$	−0.260097	−0.0190202
$188$	0	0
$189$	3.30702	0.240550
$190$	0	0
$191$	−15.0983	−1.09248	−0.546239	−	0.837629i	$-0.683941\pi$
$191$	−15.0983	−1.09248	−0.546239	+	0.837629i	$0.683941\pi$
$192$	0	0
$193$	26.3393	1.89594	0.947971	−	0.318357i	$-0.103131\pi$
$193$	26.3393	1.89594	0.947971	+	0.318357i	$0.103131\pi$
$194$	0	0
$195$	−58.4352	−4.18463
$196$	0	0
$197$	−18.9958	−1.35339	−0.676697	−	0.736262i	$-0.736589\pi$
$197$	−18.9958	−1.35339	−0.676697	+	0.736262i	$0.736589\pi$
$198$	0	0
$199$	−10.1216	−0.717500	−0.358750	−	0.933434i	$-0.616797\pi$
$199$	−10.1216	−0.717500	−0.358750	+	0.933434i	$0.616797\pi$
$200$	0	0
$201$	−0.939029	−0.0662340
$202$	0	0
$203$	−10.7412	−0.753882
$204$	0	0
$205$	−6.25163	−0.436633
$206$	0	0
$207$	10.5116	0.730609
$208$	0	0
$209$	−2.18675	−0.151261
$210$	0	0
$211$	24.7587	1.70446	0.852228	−	0.523170i	$-0.175251\pi$
$211$	24.7587	1.70446	0.852228	+	0.523170i	$0.175251\pi$
$212$	0	0
$213$	11.7985	0.808423
$214$	0	0
$215$	19.1505	1.30605
$216$	0	0
$217$	−1.15261	−0.0782441
$218$	0	0
$219$	−25.6060	−1.73029
$220$	0	0
$221$	−6.61698	−0.445106
$222$	0	0
$223$	−4.97866	−0.333396	−0.166698	−	0.986008i	$-0.553310\pi$
$223$	−4.97866	−0.333396	−0.166698	+	0.986008i	$0.553310\pi$
$224$	0	0
$225$	22.2105	1.48070
$226$	0	0
$227$	−6.77457	−0.449644	−0.224822	−	0.974400i	$-0.572180\pi$
$227$	−6.77457	−0.449644	−0.224822	+	0.974400i	$0.572180\pi$
$228$	0	0
$229$	27.8607	1.84108	0.920542	−	0.390643i	$-0.127747\pi$
$229$	27.8607	1.84108	0.920542	+	0.390643i	$0.127747\pi$
$230$	0	0
$231$	1.16331	0.0765405
$232$	0	0
$233$	−13.8677	−0.908501	−0.454250	−	0.890874i	$-0.650093\pi$
$233$	−13.8677	−0.908501	−0.454250	+	0.890874i	$0.650093\pi$
$234$	0	0
$235$	22.6662	1.47858
$236$	0	0
$237$	−8.90037	−0.578141
$238$	0	0
$239$	6.29824	0.407399	0.203700	−	0.979033i	$-0.434703\pi$
$239$	6.29824	0.407399	0.203700	+	0.979033i	$0.434703\pi$
$240$	0	0
$241$	27.0164	1.74028	0.870139	−	0.492807i	$-0.164029\pi$
$241$	27.0164	1.74028	0.870139	+	0.492807i	$0.164029\pi$
$242$	0	0
$243$	19.3885	1.24377
$244$	0	0
$245$	12.3108	0.786507
$246$	0	0
$247$	−55.6318	−3.53977
$248$	0	0
$249$	−36.4786	−2.31174
$250$	0	0
$251$	1.16130	0.0733008	0.0366504	−	0.999328i	$-0.488331\pi$
$251$	1.16130	0.0733008	0.0366504	+	0.999328i	$0.488331\pi$
$252$	0	0
$253$	−1.20943	−0.0760362
$254$	0	0
$255$	8.83111	0.553026
$256$	0	0
$257$	24.3730	1.52035	0.760173	−	0.649721i	$-0.225114\pi$
$257$	24.3730	1.52035	0.760173	+	0.649721i	$0.225114\pi$
$258$	0	0
$259$	−8.72779	−0.542319
$260$	0	0
$261$	−12.4518	−0.770748
$262$	0	0
$263$	−20.0599	−1.23695	−0.618473	−	0.785806i	$-0.712248\pi$
$263$	−20.0599	−1.23695	−0.618473	+	0.785806i	$0.712248\pi$
$264$	0	0
$265$	27.7452	1.70438
$266$	0	0
$267$	−0.425373	−0.0260324
$268$	0	0
$269$	6.71853	0.409636	0.204818	−	0.978800i	$-0.434340\pi$
$269$	6.71853	0.409636	0.204818	+	0.978800i	$0.434340\pi$
$270$	0	0
$271$	−11.3525	−0.689615	−0.344808	−	0.938673i	$-0.612056\pi$
$271$	−11.3525	−0.689615	−0.344808	+	0.938673i	$0.612056\pi$
$272$	0	0
$273$	29.5952	1.79118
$274$	0	0
$275$	−2.55545	−0.154100
$276$	0	0
$277$	5.06536	0.304348	0.152174	−	0.988354i	$-0.451373\pi$
$277$	5.06536	0.304348	0.152174	+	0.988354i	$0.451373\pi$
$278$	0	0
$279$	−1.33617	−0.0799946
$280$	0	0
$281$	−19.1832	−1.14437	−0.572186	−	0.820124i	$-0.693905\pi$
$281$	−19.1832	−1.14437	−0.572186	+	0.820124i	$0.693905\pi$
$282$	0	0
$283$	9.97166	0.592753	0.296377	−	0.955071i	$-0.404222\pi$
$283$	9.97166	0.592753	0.296377	+	0.955071i	$0.404222\pi$
$284$	0	0
$285$	74.2470	4.39801
$286$	0	0
$287$	3.16621	0.186896
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	5.88608	0.345048
$292$	0	0
$293$	1.78176	0.104092	0.0520458	−	0.998645i	$-0.483426\pi$
$293$	1.78176	0.104092	0.0520458	+	0.998645i	$0.483426\pi$
$294$	0	0
$295$	−3.85032	−0.224175
$296$	0	0
$297$	−0.441091	−0.0255947
$298$	0	0
$299$	−30.7684	−1.77938
$300$	0	0
$301$	−9.69900	−0.559041
$302$	0	0
$303$	35.9279	2.06401
$304$	0	0
$305$	26.9570	1.54356
$306$	0	0
$307$	−3.50234	−0.199889	−0.0999447	−	0.994993i	$-0.531867\pi$
$307$	−3.50234	−0.199889	−0.0999447	+	0.994993i	$0.531867\pi$
$308$	0	0
$309$	37.2425	2.11865
$310$	0	0
$311$	−19.6373	−1.11353	−0.556764	−	0.830671i	$-0.687957\pi$
$311$	−19.6373	−1.11353	−0.556764	+	0.830671i	$0.687957\pi$
$312$	0	0
$313$	−8.02754	−0.453743	−0.226872	−	0.973925i	$-0.572850\pi$
$313$	−8.02754	−0.453743	−0.226872	+	0.973925i	$0.572850\pi$
$314$	0	0
$315$	−16.9733	−0.956337
$316$	0	0
$317$	−11.5414	−0.648229	−0.324114	−	0.946018i	$-0.605066\pi$
$317$	−11.5414	−0.648229	−0.324114	+	0.946018i	$0.605066\pi$
$318$	0	0
$319$	1.43266	0.0802135
$320$	0	0
$321$	20.5689	1.14804
$322$	0	0
$323$	8.40744	0.467802
$324$	0	0
$325$	−65.0118	−3.60620
$326$	0	0
$327$	−0.293219	−0.0162151
$328$	0	0
$329$	−11.4795	−0.632888
$330$	0	0
$331$	32.8968	1.80817	0.904086	−	0.427351i	$-0.140553\pi$
$331$	32.8968	1.80817	0.904086	+	0.427351i	$0.140553\pi$
$332$	0	0
$333$	−10.1178	−0.554451
$334$	0	0
$335$	−1.57637	−0.0861263
$336$	0	0
$337$	−14.1553	−0.771091	−0.385545	−	0.922689i	$-0.625987\pi$
$337$	−14.1553	−0.771091	−0.385545	+	0.922689i	$0.625987\pi$
$338$	0	0
$339$	−4.11437	−0.223462
$340$	0	0
$341$	0.153735	0.00832522
$342$	0	0
$343$	−19.8852	−1.07370
$344$	0	0
$345$	41.0639	2.21081
$346$	0	0
$347$	5.36911	0.288229	0.144114	−	0.989561i	$-0.453967\pi$
$347$	5.36911	0.288229	0.144114	+	0.989561i	$0.453967\pi$
$348$	0	0
$349$	12.3788	0.662621	0.331310	−	0.943522i	$-0.392509\pi$
$349$	12.3788	0.662621	0.331310	+	0.943522i	$0.392509\pi$
$350$	0	0
$351$	−11.2215	−0.598962
$352$	0	0
$353$	0.0646768	0.00344240	0.00172120	−	0.999999i	$-0.499452\pi$
$353$	0.0646768	0.00344240	0.00172120	+	0.999999i	$0.499452\pi$
$354$	0	0
$355$	19.8065	1.05122
$356$	0	0
$357$	−4.47262	−0.236716
$358$	0	0
$359$	5.19421	0.274140	0.137070	−	0.990561i	$-0.456231\pi$
$359$	5.19421	0.274140	0.137070	+	0.990561i	$0.456231\pi$
$360$	0	0
$361$	51.6850	2.72026
$362$	0	0
$363$	25.0745	1.31607
$364$	0	0
$365$	−42.9853	−2.24996
$366$	0	0
$367$	33.4048	1.74372	0.871858	−	0.489759i	$-0.162915\pi$
$367$	33.4048	1.74372	0.871858	+	0.489759i	$0.162915\pi$
$368$	0	0
$369$	3.67046	0.191077
$370$	0	0
$371$	−14.0519	−0.729538
$372$	0	0
$373$	−1.02906	−0.0532829	−0.0266414	−	0.999645i	$-0.508481\pi$
$373$	−1.02906	−0.0532829	−0.0266414	+	0.999645i	$0.508481\pi$
$374$	0	0
$375$	42.6101	2.20037
$376$	0	0
$377$	36.4475	1.87714
$378$	0	0
$379$	−7.21680	−0.370702	−0.185351	−	0.982672i	$-0.559342\pi$
$379$	−7.21680	−0.370702	−0.185351	+	0.982672i	$0.559342\pi$
$380$	0	0
$381$	−7.78620	−0.398899
$382$	0	0
$383$	7.42244	0.379269	0.189634	−	0.981855i	$-0.439270\pi$
$383$	7.42244	0.379269	0.189634	+	0.981855i	$0.439270\pi$
$384$	0	0
$385$	1.95288	0.0995282
$386$	0	0
$387$	−11.2437	−0.571548
$388$	0	0
$389$	13.2389	0.671237	0.335619	−	0.941998i	$-0.391055\pi$
$389$	13.2389	0.671237	0.335619	+	0.941998i	$0.391055\pi$
$390$	0	0
$391$	4.64992	0.235156
$392$	0	0
$393$	−20.2236	−1.02015
$394$	0	0
$395$	−14.9413	−0.751777
$396$	0	0
$397$	31.0992	1.56082	0.780411	−	0.625267i	$-0.215010\pi$
$397$	31.0992	1.56082	0.780411	+	0.625267i	$0.215010\pi$
$398$	0	0
$399$	−37.6033	−1.88252
$400$	0	0
$401$	−12.8083	−0.639616	−0.319808	−	0.947482i	$-0.603618\pi$
$401$	−12.8083	−0.639616	−0.319808	+	0.947482i	$0.603618\pi$
$402$	0	0
$403$	3.91108	0.194825
$404$	0	0
$405$	41.0886	2.04171
$406$	0	0
$407$	1.16411	0.0577030
$408$	0	0
$409$	−1.09805	−0.0542952	−0.0271476	−	0.999631i	$-0.508642\pi$
$409$	−1.09805	−0.0542952	−0.0271476	+	0.999631i	$0.508642\pi$
$410$	0	0
$411$	43.4074	2.14113
$412$	0	0
$413$	1.95004	0.0959553
$414$	0	0
$415$	−61.2375	−3.00603
$416$	0	0
$417$	−12.1867	−0.596785
$418$	0	0
$419$	11.7055	0.571850	0.285925	−	0.958252i	$-0.407699\pi$
$419$	11.7055	0.571850	0.285925	+	0.958252i	$0.407699\pi$
$420$	0	0
$421$	−5.43800	−0.265032	−0.132516	−	0.991181i	$-0.542306\pi$
$421$	−5.43800	−0.265032	−0.132516	+	0.991181i	$0.542306\pi$
$422$	0	0
$423$	−13.3078	−0.647047
$424$	0	0
$425$	9.82500	0.476582
$426$	0	0
$427$	−13.6527	−0.660701
$428$	0	0
$429$	−3.94742	−0.190583
$430$	0	0
$431$	36.1719	1.74234	0.871170	−	0.490981i	$-0.163362\pi$
$431$	36.1719	1.74234	0.871170	+	0.490981i	$0.163362\pi$
$432$	0	0
$433$	−38.6135	−1.85565	−0.927824	−	0.373018i	$-0.878323\pi$
$433$	−38.6135	−1.85565	−0.927824	+	0.373018i	$0.878323\pi$
$434$	0	0
$435$	−48.6433	−2.33227
$436$	0	0
$437$	39.0939	1.87011
$438$	0	0
$439$	18.5418	0.884950	0.442475	−	0.896781i	$-0.354101\pi$
$439$	18.5418	0.884950	0.442475	+	0.896781i	$0.354101\pi$
$440$	0	0
$441$	−7.22792	−0.344187
$442$	0	0
$443$	−8.04134	−0.382056	−0.191028	−	0.981585i	$-0.561182\pi$
$443$	−8.04134	−0.382056	−0.191028	+	0.981585i	$0.561182\pi$
$444$	0	0
$445$	−0.714084	−0.0338508
$446$	0	0
$447$	−28.6401	−1.35463
$448$	0	0
$449$	2.30453	0.108758	0.0543788	−	0.998520i	$-0.482682\pi$
$449$	2.30453	0.108758	0.0543788	+	0.998520i	$0.482682\pi$
$450$	0	0
$451$	−0.422310	−0.0198858
$452$	0	0
$453$	−13.6915	−0.643284
$454$	0	0
$455$	49.6822	2.32914
$456$	0	0
$457$	−37.0585	−1.73352	−0.866762	−	0.498722i	$-0.833803\pi$
$457$	−37.0585	−1.73352	−0.866762	+	0.498722i	$0.833803\pi$
$458$	0	0
$459$	1.69587	0.0791565
$460$	0	0
$461$	21.1302	0.984131	0.492065	−	0.870558i	$-0.336242\pi$
$461$	21.1302	0.984131	0.492065	+	0.870558i	$0.336242\pi$
$462$	0	0
$463$	29.8393	1.38675	0.693375	−	0.720577i	$-0.256123\pi$
$463$	29.8393	1.38675	0.693375	+	0.720577i	$0.256123\pi$
$464$	0	0
$465$	−5.21979	−0.242062
$466$	0	0
$467$	−27.5813	−1.27631	−0.638155	−	0.769908i	$-0.720302\pi$
$467$	−27.5813	−1.27631	−0.638155	+	0.769908i	$0.720302\pi$
$468$	0	0
$469$	0.798371	0.0368654
$470$	0	0
$471$	4.50818	0.207726
$472$	0	0
$473$	1.29365	0.0594823
$474$	0	0
$475$	82.6030	3.79009
$476$	0	0
$477$	−16.2898	−0.745859
$478$	0	0
$479$	−6.78983	−0.310235	−0.155118	−	0.987896i	$-0.549576\pi$
$479$	−6.78983	−0.310235	−0.155118	+	0.987896i	$0.549576\pi$
$480$	0	0
$481$	29.6156	1.35035
$482$	0	0
$483$	−20.7973	−0.946310
$484$	0	0
$485$	9.88111	0.448678
$486$	0	0
$487$	12.3040	0.557549	0.278774	−	0.960357i	$-0.410072\pi$
$487$	12.3040	0.557549	0.278774	+	0.960357i	$0.410072\pi$
$488$	0	0
$489$	27.8049	1.25738
$490$	0	0
$491$	16.9322	0.764139	0.382069	−	0.924134i	$-0.375212\pi$
$491$	16.9322	0.764139	0.382069	+	0.924134i	$0.375212\pi$
$492$	0	0
$493$	−5.50817	−0.248076
$494$	0	0
$495$	2.26390	0.101755
$496$	0	0
$497$	−10.0312	−0.449962
$498$	0	0
$499$	3.43692	0.153858	0.0769289	−	0.997037i	$-0.475489\pi$
$499$	3.43692	0.153858	0.0769289	+	0.997037i	$0.475489\pi$
$500$	0	0
$501$	−22.8556	−1.02111
$502$	0	0
$503$	24.5798	1.09596	0.547979	−	0.836492i	$-0.315397\pi$
$503$	24.5798	1.09596	0.547979	+	0.836492i	$0.315397\pi$
$504$	0	0
$505$	60.3131	2.68390
$506$	0	0
$507$	−70.6071	−3.13577
$508$	0	0
$509$	24.6353	1.09194	0.545971	−	0.837804i	$-0.316161\pi$
$509$	24.6353	1.09194	0.545971	+	0.837804i	$0.316161\pi$
$510$	0	0
$511$	21.7704	0.963067
$512$	0	0
$513$	14.2579	0.629503
$514$	0	0
$515$	62.5198	2.75495
$516$	0	0
$517$	1.53114	0.0673396
$518$	0	0
$519$	17.6334	0.774021
$520$	0	0
$521$	21.8378	0.956732	0.478366	−	0.878161i	$-0.341229\pi$
$521$	21.8378	0.956732	0.478366	+	0.878161i	$0.341229\pi$
$522$	0	0
$523$	7.64106	0.334120	0.167060	−	0.985947i	$-0.446573\pi$
$523$	7.64106	0.334120	0.167060	+	0.985947i	$0.446573\pi$
$524$	0	0
$525$	−43.9435	−1.91785
$526$	0	0
$527$	−0.591068	−0.0257473
$528$	0	0
$529$	−1.37826	−0.0599244
$530$	0	0
$531$	2.26061	0.0981020
$532$	0	0
$533$	−10.7437	−0.465363
$534$	0	0
$535$	34.5295	1.49284
$536$	0	0
$537$	46.6772	2.01427
$538$	0	0
$539$	0.831618	0.0358203
$540$	0	0
$541$	−24.0887	−1.03566	−0.517828	−	0.855485i	$-0.673259\pi$
$541$	−24.0887	−1.03566	−0.517828	+	0.855485i	$0.673259\pi$
$542$	0	0
$543$	14.1664	0.607939
$544$	0	0
$545$	−0.492234	−0.0210850
$546$	0	0
$547$	38.9186	1.66404	0.832020	−	0.554746i	$-0.187185\pi$
$547$	38.9186	1.66404	0.832020	+	0.554746i	$0.187185\pi$
$548$	0	0
$549$	−15.8271	−0.675482
$550$	0	0
$551$	−46.3096	−1.97286
$552$	0	0
$553$	7.56718	0.321789
$554$	0	0
$555$	−39.5253	−1.67776
$556$	0	0
$557$	−38.0677	−1.61298	−0.806489	−	0.591249i	$-0.798635\pi$
$557$	−38.0677	−1.61298	−0.806489	+	0.591249i	$0.798635\pi$
$558$	0	0
$559$	32.9111	1.39199
$560$	0	0
$561$	0.596559	0.0251867
$562$	0	0
$563$	14.7230	0.620499	0.310250	−	0.950655i	$-0.399587\pi$
$563$	14.7230	0.620499	0.310250	+	0.950655i	$0.399587\pi$
$564$	0	0
$565$	−6.90690	−0.290575
$566$	0	0
$567$	−20.8098	−0.873931
$568$	0	0
$569$	−23.5561	−0.987522	−0.493761	−	0.869598i	$-0.664378\pi$
$569$	−23.5561	−0.987522	−0.493761	+	0.869598i	$0.664378\pi$
$570$	0	0
$571$	−34.4138	−1.44017	−0.720087	−	0.693884i	$-0.755898\pi$
$571$	−34.4138	−1.44017	−0.720087	+	0.693884i	$0.755898\pi$
$572$	0	0
$573$	34.6296	1.44667
$574$	0	0
$575$	45.6854	1.90521
$576$	0	0
$577$	9.25182	0.385158	0.192579	−	0.981281i	$-0.438315\pi$
$577$	9.25182	0.385158	0.192579	+	0.981281i	$0.438315\pi$
$578$	0	0
$579$	−60.4118	−2.51063
$580$	0	0
$581$	31.0145	1.28670
$582$	0	0
$583$	1.87424	0.0776233
$584$	0	0
$585$	57.5946	2.38125
$586$	0	0
$587$	−36.3366	−1.49977	−0.749886	−	0.661567i	$-0.769892\pi$
$587$	−36.3366	−1.49977	−0.749886	+	0.661567i	$0.769892\pi$
$588$	0	0
$589$	−4.96937	−0.204759
$590$	0	0
$591$	43.5688	1.79218
$592$	0	0
$593$	9.83445	0.403852	0.201926	−	0.979401i	$-0.435280\pi$
$593$	9.83445	0.403852	0.201926	+	0.979401i	$0.435280\pi$
$594$	0	0
$595$	−7.50829	−0.307810
$596$	0	0
$597$	23.2149	0.950122
$598$	0	0
$599$	19.7603	0.807385	0.403693	−	0.914895i	$-0.367727\pi$
$599$	19.7603	0.807385	0.403693	+	0.914895i	$0.367727\pi$
$600$	0	0
$601$	23.1726	0.945231	0.472616	−	0.881269i	$-0.343310\pi$
$601$	23.1726	0.945231	0.472616	+	0.881269i	$0.343310\pi$
$602$	0	0
$603$	0.925521	0.0376901
$604$	0	0
$605$	42.0931	1.71133
$606$	0	0
$607$	30.8745	1.25316	0.626578	−	0.779359i	$-0.284455\pi$
$607$	30.8745	1.25316	0.626578	+	0.779359i	$0.284455\pi$
$608$	0	0
$609$	24.6360	0.998300
$610$	0	0
$611$	38.9530	1.57587
$612$	0	0
$613$	−18.4018	−0.743241	−0.371621	−	0.928385i	$-0.621198\pi$
$613$	−18.4018	−0.743241	−0.371621	+	0.928385i	$0.621198\pi$
$614$	0	0
$615$	14.3387	0.578194
$616$	0	0
$617$	36.3637	1.46395	0.731974	−	0.681332i	$-0.238599\pi$
$617$	36.3637	1.46395	0.731974	+	0.681332i	$0.238599\pi$
$618$	0	0
$619$	−13.4833	−0.541938	−0.270969	−	0.962588i	$-0.587344\pi$
$619$	−13.4833	−0.541938	−0.270969	+	0.962588i	$0.587344\pi$
$620$	0	0
$621$	7.88567	0.316441
$622$	0	0
$623$	0.361656	0.0144895
$624$	0	0
$625$	22.4056	0.896223
$626$	0	0
$627$	5.01553	0.200301
$628$	0	0
$629$	−4.47569	−0.178458
$630$	0	0
$631$	39.7930	1.58413	0.792067	−	0.610434i	$-0.209005\pi$
$631$	39.7930	1.58413	0.792067	+	0.610434i	$0.209005\pi$
$632$	0	0
$633$	−56.7865	−2.25706
$634$	0	0
$635$	−13.0709	−0.518702
$636$	0	0
$637$	21.1567	0.838259
$638$	0	0
$639$	−11.6288	−0.460029
$640$	0	0
$641$	−11.3134	−0.446851	−0.223426	−	0.974721i	$-0.571724\pi$
$641$	−11.3134	−0.446851	−0.223426	+	0.974721i	$0.571724\pi$
$642$	0	0
$643$	34.6411	1.36611	0.683056	−	0.730366i	$-0.260651\pi$
$643$	34.6411	1.36611	0.683056	+	0.730366i	$0.260651\pi$
$644$	0	0
$645$	−43.9236	−1.72949
$646$	0	0
$647$	−23.7955	−0.935497	−0.467749	−	0.883862i	$-0.654935\pi$
$647$	−23.7955	−0.935497	−0.467749	+	0.883862i	$0.654935\pi$
$648$	0	0
$649$	−0.260097	−0.0102097
$650$	0	0
$651$	2.64362	0.103612
$652$	0	0
$653$	40.4352	1.58235	0.791176	−	0.611588i	$-0.209469\pi$
$653$	40.4352	1.58235	0.791176	+	0.611588i	$0.209469\pi$
$654$	0	0
$655$	−33.9499	−1.32653
$656$	0	0
$657$	25.2376	0.984613
$658$	0	0
$659$	6.03058	0.234918	0.117459	−	0.993078i	$-0.462525\pi$
$659$	6.03058	0.234918	0.117459	+	0.993078i	$0.462525\pi$
$660$	0	0
$661$	25.5325	0.993098	0.496549	−	0.868009i	$-0.334600\pi$
$661$	25.5325	0.993098	0.496549	+	0.868009i	$0.334600\pi$
$662$	0	0
$663$	15.1767	0.589414
$664$	0	0
$665$	−63.1255	−2.44790
$666$	0	0
$667$	−25.6126	−0.991722
$668$	0	0
$669$	11.4191	0.441486
$670$	0	0
$671$	1.82100	0.0702990
$672$	0	0
$673$	−44.8813	−1.73005	−0.865023	−	0.501732i	$-0.832696\pi$
$673$	−44.8813	−1.73005	−0.865023	+	0.501732i	$0.832696\pi$
$674$	0	0
$675$	16.6619	0.641318
$676$	0	0
$677$	−27.8331	−1.06971	−0.534857	−	0.844943i	$-0.679635\pi$
$677$	−27.8331	−1.06971	−0.534857	+	0.844943i	$0.679635\pi$
$678$	0	0
$679$	−5.00440	−0.192051
$680$	0	0
$681$	15.5382	0.595424
$682$	0	0
$683$	35.1828	1.34623	0.673117	−	0.739536i	$-0.264955\pi$
$683$	35.1828	1.34623	0.673117	+	0.739536i	$0.264955\pi$
$684$	0	0
$685$	72.8690	2.78418
$686$	0	0
$687$	−63.9013	−2.43798
$688$	0	0
$689$	47.6816	1.81652
$690$	0	0
$691$	33.2520	1.26497	0.632483	−	0.774575i	$-0.282036\pi$
$691$	33.2520	1.26497	0.632483	+	0.774575i	$0.282036\pi$
$692$	0	0
$693$	−1.14658	−0.0435550
$694$	0	0
$695$	−20.4581	−0.776019
$696$	0	0
$697$	1.62366	0.0615006
$698$	0	0
$699$	31.8069	1.20305
$700$	0	0
$701$	15.6218	0.590028	0.295014	−	0.955493i	$-0.404676\pi$
$701$	15.6218	0.590028	0.295014	+	0.955493i	$0.404676\pi$
$702$	0	0
$703$	−37.6291	−1.41921
$704$	0	0
$705$	−51.9871	−1.95795
$706$	0	0
$707$	−30.5463	−1.14881
$708$	0	0
$709$	41.1738	1.54631	0.773157	−	0.634215i	$-0.218677\pi$
$709$	41.1738	1.54631	0.773157	+	0.634215i	$0.218677\pi$
$710$	0	0
$711$	8.77234	0.328988
$712$	0	0
$713$	−2.74842	−0.102929
$714$	0	0
$715$	−6.62662	−0.247822
$716$	0	0
$717$	−14.4456	−0.539483
$718$	0	0
$719$	−38.7275	−1.44429	−0.722147	−	0.691740i	$-0.756844\pi$
$719$	−38.7275	−1.44429	−0.722147	+	0.691740i	$0.756844\pi$
$720$	0	0
$721$	−31.6639	−1.17922
$722$	0	0
$723$	−61.9648	−2.30449
$724$	0	0
$725$	−54.1178	−2.00988
$726$	0	0
$727$	9.71024	0.360133	0.180066	−	0.983654i	$-0.442369\pi$
$727$	9.71024	0.360133	0.180066	+	0.983654i	$0.442369\pi$
$728$	0	0
$729$	−12.4551	−0.461298
$730$	0	0
$731$	−4.97374	−0.183960
$732$	0	0
$733$	21.6918	0.801203	0.400602	−	0.916252i	$-0.368801\pi$
$733$	21.6918	0.801203	0.400602	+	0.916252i	$0.368801\pi$
$734$	0	0
$735$	−28.2360	−1.04150
$736$	0	0
$737$	−0.106487	−0.00392250
$738$	0	0
$739$	−23.3619	−0.859380	−0.429690	−	0.902976i	$-0.641377\pi$
$739$	−23.3619	−0.859380	−0.429690	+	0.902976i	$0.641377\pi$
$740$	0	0
$741$	127.597	4.68740
$742$	0	0
$743$	−16.1369	−0.592005	−0.296002	−	0.955187i	$-0.595654\pi$
$743$	−16.1369	−0.592005	−0.296002	+	0.955187i	$0.595654\pi$
$744$	0	0
$745$	−48.0788	−1.76147
$746$	0	0
$747$	35.9539	1.31548
$748$	0	0
$749$	−17.4879	−0.638993
$750$	0	0
$751$	46.2034	1.68599	0.842994	−	0.537924i	$-0.180791\pi$
$751$	46.2034	1.68599	0.842994	+	0.537924i	$0.180791\pi$
$752$	0	0
$753$	−2.66357	−0.0970657
$754$	0	0
$755$	−22.9843	−0.836484
$756$	0	0
$757$	−37.5117	−1.36339	−0.681693	−	0.731639i	$-0.738756\pi$
$757$	−37.5117	−1.36339	−0.681693	+	0.731639i	$0.738756\pi$
$758$	0	0
$759$	2.77395	0.100688
$760$	0	0
$761$	25.6212	0.928768	0.464384	−	0.885634i	$-0.346276\pi$
$761$	25.6212	0.928768	0.464384	+	0.885634i	$0.346276\pi$
$762$	0	0
$763$	0.249298	0.00902519
$764$	0	0
$765$	−8.70407	−0.314696
$766$	0	0
$767$	−6.61698	−0.238925
$768$	0	0
$769$	−30.5427	−1.10140	−0.550700	−	0.834703i	$-0.685639\pi$
$769$	−30.5427	−1.10140	−0.550700	+	0.834703i	$0.685639\pi$
$770$	0	0
$771$	−55.9019	−2.01326
$772$	0	0
$773$	28.3352	1.01915	0.509574	−	0.860427i	$-0.329803\pi$
$773$	28.3352	1.01915	0.509574	+	0.860427i	$0.329803\pi$
$774$	0	0
$775$	−5.80724	−0.208602
$776$	0	0
$777$	20.0181	0.718144
$778$	0	0
$779$	13.6508	0.489092
$780$	0	0
$781$	1.33797	0.0478763
$782$	0	0
$783$	−9.34116	−0.333826
$784$	0	0
$785$	7.56799	0.270113
$786$	0	0
$787$	−37.8685	−1.34987	−0.674933	−	0.737879i	$-0.735828\pi$
$787$	−37.8685	−1.34987	−0.674933	+	0.737879i	$0.735828\pi$
$788$	0	0
$789$	46.0094	1.63798
$790$	0	0
$791$	3.49808	0.124377
$792$	0	0
$793$	46.3270	1.64512
$794$	0	0
$795$	−63.6365	−2.25695
$796$	0	0
$797$	−46.4179	−1.64421	−0.822104	−	0.569337i	$-0.807200\pi$
$797$	−46.4179	−1.64421	−0.822104	+	0.569337i	$0.807200\pi$
$798$	0	0
$799$	−5.88682	−0.208261
$800$	0	0
$801$	0.419254	0.0148136
$802$	0	0
$803$	−2.90374	−0.102471
$804$	0	0
$805$	−34.9130	−1.23052
$806$	0	0
$807$	−15.4096	−0.542444
$808$	0	0
$809$	32.2967	1.13549	0.567746	−	0.823204i	$-0.307816\pi$
$809$	32.2967	1.13549	0.567746	+	0.823204i	$0.307816\pi$
$810$	0	0
$811$	35.4570	1.24506	0.622531	−	0.782595i	$-0.286104\pi$
$811$	35.4570	1.24506	0.622531	+	0.782595i	$0.286104\pi$
$812$	0	0
$813$	26.0381	0.913197
$814$	0	0
$815$	46.6767	1.63502
$816$	0	0
$817$	−41.8164	−1.46297
$818$	0	0
$819$	−29.1695	−1.01926
$820$	0	0
$821$	45.9048	1.60209	0.801044	−	0.598606i	$-0.204278\pi$
$821$	45.9048	1.60209	0.801044	+	0.598606i	$0.204278\pi$
$822$	0	0
$823$	−14.8501	−0.517644	−0.258822	−	0.965925i	$-0.583334\pi$
$823$	−14.8501	−0.517644	−0.258822	+	0.965925i	$0.583334\pi$
$824$	0	0
$825$	5.86119	0.204060
$826$	0	0
$827$	−2.59014	−0.0900681	−0.0450341	−	0.998985i	$-0.514340\pi$
$827$	−2.59014	−0.0900681	−0.0450341	+	0.998985i	$0.514340\pi$
$828$	0	0
$829$	−20.4292	−0.709536	−0.354768	−	0.934954i	$-0.615440\pi$
$829$	−20.4292	−0.709536	−0.354768	+	0.934954i	$0.615440\pi$
$830$	0	0
$831$	−11.6179	−0.403021
$832$	0	0
$833$	−3.19734	−0.110781
$834$	0	0
$835$	−38.3682	−1.32779
$836$	0	0
$837$	−1.00238	−0.0346472
$838$	0	0
$839$	−28.6167	−0.987957	−0.493978	−	0.869474i	$-0.664458\pi$
$839$	−28.6167	−0.987957	−0.493978	+	0.869474i	$0.664458\pi$
$840$	0	0
$841$	1.33998	0.0462063
$842$	0	0
$843$	43.9986	1.51539
$844$	0	0
$845$	−118.530	−4.07755
$846$	0	0
$847$	−21.3185	−0.732514
$848$	0	0
$849$	−22.8710	−0.784931
$850$	0	0
$851$	−20.8116	−0.713413
$852$	0	0
$853$	36.5299	1.25076	0.625380	−	0.780320i	$-0.284944\pi$
$853$	36.5299	1.25076	0.625380	+	0.780320i	$0.284944\pi$
$854$	0	0
$855$	−73.1789	−2.50267
$856$	0	0
$857$	28.6603	0.979017	0.489509	−	0.871998i	$-0.337176\pi$
$857$	28.6603	0.979017	0.489509	+	0.871998i	$0.337176\pi$
$858$	0	0
$859$	18.7600	0.640083	0.320042	−	0.947403i	$-0.396303\pi$
$859$	18.7600	0.640083	0.320042	+	0.947403i	$0.396303\pi$
$860$	0	0
$861$	−7.26203	−0.247489
$862$	0	0
$863$	−2.44782	−0.0833247	−0.0416624	−	0.999132i	$-0.513265\pi$
$863$	−2.44782	−0.0833247	−0.0416624	+	0.999132i	$0.513265\pi$
$864$	0	0
$865$	29.6016	1.00649
$866$	0	0
$867$	−2.29360	−0.0778948
$868$	0	0
$869$	−1.00931	−0.0342386
$870$	0	0
$871$	−2.70907	−0.0917934
$872$	0	0
$873$	−5.80141	−0.196348
$874$	0	0
$875$	−36.2275	−1.22471
$876$	0	0
$877$	−4.00693	−0.135304	−0.0676522	−	0.997709i	$-0.521551\pi$
$877$	−4.00693	−0.135304	−0.0676522	+	0.997709i	$0.521551\pi$
$878$	0	0
$879$	−4.08665	−0.137839
$880$	0	0
$881$	−34.5092	−1.16264	−0.581322	−	0.813674i	$-0.697464\pi$
$881$	−34.5092	−1.16264	−0.581322	+	0.813674i	$0.697464\pi$
$882$	0	0
$883$	−30.7321	−1.03422	−0.517109	−	0.855920i	$-0.672992\pi$
$883$	−30.7321	−1.03422	−0.517109	+	0.855920i	$0.672992\pi$
$884$	0	0
$885$	8.83111	0.296854
$886$	0	0
$887$	−1.38309	−0.0464397	−0.0232199	−	0.999730i	$-0.507392\pi$
$887$	−1.38309	−0.0464397	−0.0232199	+	0.999730i	$0.507392\pi$
$888$	0	0
$889$	6.61990	0.222024
$890$	0	0
$891$	2.77562	0.0929867
$892$	0	0
$893$	−49.4931	−1.65622
$894$	0	0
$895$	78.3581	2.61922
$896$	0	0
$897$	70.5704	2.35628
$898$	0	0
$899$	3.25571	0.108584
$900$	0	0
$901$	−7.20595	−0.240065
$902$	0	0
$903$	22.2456	0.740288
$904$	0	0
$905$	23.7815	0.790523
$906$	0	0
$907$	−15.6759	−0.520511	−0.260255	−	0.965540i	$-0.583807\pi$
$907$	−15.6759	−0.520511	−0.260255	+	0.965540i	$0.583807\pi$
$908$	0	0
$909$	−35.4111	−1.17451
$910$	0	0
$911$	8.97013	0.297194	0.148597	−	0.988898i	$-0.452524\pi$
$911$	8.97013	0.297194	0.148597	+	0.988898i	$0.452524\pi$
$912$	0	0
$913$	−4.13672	−0.136905
$914$	0	0
$915$	−61.8287	−2.04399
$916$	0	0
$917$	17.1943	0.567807
$918$	0	0
$919$	−49.9663	−1.64824	−0.824118	−	0.566419i	$-0.808328\pi$
$919$	−49.9663	−1.64824	−0.824118	+	0.566419i	$0.808328\pi$
$920$	0	0
$921$	8.03298	0.264696
$922$	0	0
$923$	34.0385	1.12039
$924$	0	0
$925$	−43.9737	−1.44585
$926$	0	0
$927$	−36.7067	−1.20561
$928$	0	0
$929$	9.00217	0.295352	0.147676	−	0.989036i	$-0.452821\pi$
$929$	9.00217	0.295352	0.147676	+	0.989036i	$0.452821\pi$
$930$	0	0
$931$	−26.8814	−0.881003
$932$	0	0
$933$	45.0401	1.47455
$934$	0	0
$935$	1.00146	0.0327512
$936$	0	0
$937$	42.0365	1.37327	0.686637	−	0.727001i	$-0.259086\pi$
$937$	42.0365	1.37327	0.686637	+	0.727001i	$0.259086\pi$
$938$	0	0
$939$	18.4120	0.600852
$940$	0	0
$941$	5.09027	0.165938	0.0829690	−	0.996552i	$-0.473560\pi$
$941$	5.09027	0.165938	0.0829690	+	0.996552i	$0.473560\pi$
$942$	0	0
$943$	7.54990	0.245859
$944$	0	0
$945$	−12.7331	−0.414208
$946$	0	0
$947$	−21.1894	−0.688563	−0.344281	−	0.938867i	$-0.611877\pi$
$947$	−21.1894	−0.688563	−0.344281	+	0.938867i	$0.611877\pi$
$948$	0	0
$949$	−73.8725	−2.39800
$950$	0	0
$951$	26.4713	0.858392
$952$	0	0
$953$	9.33545	0.302405	0.151202	−	0.988503i	$-0.451685\pi$
$953$	9.33545	0.302405	0.151202	+	0.988503i	$0.451685\pi$
$954$	0	0
$955$	58.1335	1.88116
$956$	0	0
$957$	−3.28595	−0.106220
$958$	0	0
$959$	−36.9054	−1.19174
$960$	0	0
$961$	−30.6506	−0.988730
$962$	0	0
$963$	−20.2730	−0.653289
$964$	0	0
$965$	−101.415	−3.26466
$966$	0	0
$967$	26.7362	0.859780	0.429890	−	0.902881i	$-0.358552\pi$
$967$	26.7362	0.859780	0.429890	+	0.902881i	$0.358552\pi$
$968$	0	0
$969$	−19.2833	−0.619469
$970$	0	0
$971$	−32.3015	−1.03660	−0.518302	−	0.855197i	$-0.673436\pi$
$971$	−32.3015	−1.03660	−0.518302	+	0.855197i	$0.673436\pi$
$972$	0	0
$973$	10.3612	0.332166
$974$	0	0
$975$	149.111	4.77538
$976$	0	0
$977$	−41.1956	−1.31796	−0.658982	−	0.752159i	$-0.729012\pi$
$977$	−41.1956	−1.31796	−0.658982	+	0.752159i	$0.729012\pi$
$978$	0	0
$979$	−0.0482378	−0.00154169
$980$	0	0
$981$	0.289001	0.00922710
$982$	0	0
$983$	26.3249	0.839633	0.419817	−	0.907609i	$-0.362094\pi$
$983$	26.3249	0.839633	0.419817	+	0.907609i	$0.362094\pi$
$984$	0	0
$985$	73.1399	2.33043
$986$	0	0
$987$	26.3295	0.838077
$988$	0	0
$989$	−23.1275	−0.735411
$990$	0	0
$991$	23.1908	0.736681	0.368341	−	0.929691i	$-0.379926\pi$
$991$	23.1908	0.736681	0.368341	+	0.929691i	$0.379926\pi$
$992$	0	0
$993$	−75.4521	−2.39440
$994$	0	0
$995$	38.9714	1.23548
$996$	0	0
$997$	27.1387	0.859492	0.429746	−	0.902950i	$-0.358603\pi$
$997$	27.1387	0.859492	0.429746	+	0.902950i	$0.358603\pi$
$998$	0	0
$999$	−7.59021	−0.240143

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8024.2.a.bb.1.7	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8024.2.a.bb.1.7	✓	32	1.1	even	1	trivial

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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)

\(f(q)\)	\(=\)	\(q-2.29360 q^{3} -3.85032 q^{5} +1.95004 q^{7} +2.26061 q^{9} +O(q^{10})\)	\(q-2.29360 q^{3} -3.85032 q^{5} +1.95004 q^{7} +2.26061 q^{9} -0.260097 q^{11} -6.61698 q^{13} +8.83111 q^{15} +1.00000 q^{17} +8.40744 q^{19} -4.47262 q^{21} +4.64992 q^{23} +9.82500 q^{25} +1.69587 q^{27} -5.50817 q^{29} -0.591068 q^{31} +0.596559 q^{33} -7.50829 q^{35} -4.47569 q^{37} +15.1767 q^{39} +1.62366 q^{41} -4.97374 q^{43} -8.70407 q^{45} -5.88682 q^{47} -3.19734 q^{49} -2.29360 q^{51} -7.20595 q^{53} +1.00146 q^{55} -19.2833 q^{57} +1.00000 q^{59} -7.00124 q^{61} +4.40828 q^{63} +25.4775 q^{65} +0.409412 q^{67} -10.6651 q^{69} -5.14411 q^{71} +11.1641 q^{73} -22.5346 q^{75} -0.507200 q^{77} +3.88052 q^{79} -10.6715 q^{81} +15.9045 q^{83} -3.85032 q^{85} +12.6336 q^{87} +0.185461 q^{89} -12.9034 q^{91} +1.35567 q^{93} -32.3714 q^{95} -2.56631 q^{97} -0.587977 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 8 q^{5} - 3 q^{7} + 40 q^{9}+O(q^{10}) \) 32 * q + 8 * q^5 - 3 * q^7 + 40 * q^9	\( 32 q + 8 q^{5} - 3 q^{7} + 40 q^{9} + 3 q^{11} + 13 q^{13} + 4 q^{15} + 32 q^{17} + 14 q^{19} - 7 q^{21} + 7 q^{23} + 38 q^{25} + 9 q^{27} + 17 q^{29} + 15 q^{31} + 18 q^{33} + 6 q^{35} + 21 q^{37} + 16 q^{39} + 49 q^{41} - 7 q^{43} + 14 q^{45} - 25 q^{47} + 37 q^{49} + 12 q^{53} + 15 q^{55} + 45 q^{57} + 32 q^{59} + 5 q^{61} - 12 q^{63} + 39 q^{65} + 12 q^{69} - 13 q^{71} + 70 q^{73} - 47 q^{75} - 10 q^{77} - q^{79} + 84 q^{81} - 17 q^{83} + 8 q^{85} + 20 q^{87} + 42 q^{89} + 36 q^{91} + 2 q^{93} - q^{95} + 58 q^{97} + 7 q^{99}+O(q^{100}) \) 32 * q + 8 * q^5 - 3 * q^7 + 40 * q^9 + 3 * q^11 + 13 * q^13 + 4 * q^15 + 32 * q^17 + 14 * q^19 - 7 * q^21 + 7 * q^23 + 38 * q^25 + 9 * q^27 + 17 * q^29 + 15 * q^31 + 18 * q^33 + 6 * q^35 + 21 * q^37 + 16 * q^39 + 49 * q^41 - 7 * q^43 + 14 * q^45 - 25 * q^47 + 37 * q^49 + 12 * q^53 + 15 * q^55 + 45 * q^57 + 32 * q^59 + 5 * q^61 - 12 * q^63 + 39 * q^65 + 12 * q^69 - 13 * q^71 + 70 * q^73 - 47 * q^75 - 10 * q^77 - q^79 + 84 * q^81 - 17 * q^83 + 8 * q^85 + 20 * q^87 + 42 * q^89 + 36 * q^91 + 2 * q^93 - q^95 + 58 * q^97 + 7 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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