LMFDB - Embedded newform 8020.2.a.d.1.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8020,2,Mod(1,8020)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8020, 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("8020.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8020 = 2^{2} \cdot 5 \cdot 401 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8020.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:	$64.0400224211$
Analytic rank:	$1$
Dimension:	$29$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
Character	$\chi$	$=$	8020.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.358058 q^{3} +1.00000 q^{5} +1.78556 q^{7} -2.87179 q^{9} +O(q^{10})$	$q+0.358058 q^{3} +1.00000 q^{5} +1.78556 q^{7} -2.87179 q^{9} +3.64957 q^{11} -1.14680 q^{13} +0.358058 q^{15} +4.34377 q^{17} -6.49377 q^{19} +0.639335 q^{21} +4.28783 q^{23} +1.00000 q^{25} -2.10244 q^{27} -8.72153 q^{29} -10.1100 q^{31} +1.30676 q^{33} +1.78556 q^{35} -5.97550 q^{37} -0.410619 q^{39} +1.06863 q^{41} -8.89369 q^{43} -2.87179 q^{45} -2.51548 q^{47} -3.81177 q^{49} +1.55532 q^{51} -3.49802 q^{53} +3.64957 q^{55} -2.32515 q^{57} +1.19226 q^{59} -9.61645 q^{61} -5.12777 q^{63} -1.14680 q^{65} +15.8145 q^{67} +1.53529 q^{69} -10.7681 q^{71} +0.472248 q^{73} +0.358058 q^{75} +6.51654 q^{77} -15.3419 q^{79} +7.86259 q^{81} -4.75671 q^{83} +4.34377 q^{85} -3.12281 q^{87} +10.2927 q^{89} -2.04768 q^{91} -3.61997 q^{93} -6.49377 q^{95} +3.41846 q^{97} -10.4808 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 29 q - 3 q^{3} + 29 q^{5} - 8 q^{7} + 10 q^{9}+O(q^{10}) $ 29 * q - 3 * q^3 + 29 * q^5 - 8 * q^7 + 10 * q^9	$ 29 q - 3 q^{3} + 29 q^{5} - 8 q^{7} + 10 q^{9} + 2 q^{11} - 23 q^{13} - 3 q^{15} - 30 q^{17} - 6 q^{19} - 16 q^{21} - 21 q^{23} + 29 q^{25} - 15 q^{27} - 35 q^{29} - 7 q^{31} - 36 q^{33} - 8 q^{35} - 31 q^{37} - 11 q^{39} - 24 q^{41} - 17 q^{43} + 10 q^{45} - 17 q^{47} + q^{49} + 8 q^{51} - 57 q^{53} + 2 q^{55} - 46 q^{57} - 9 q^{59} - 27 q^{61} - 34 q^{63} - 23 q^{65} - 21 q^{67} - 28 q^{69} - 19 q^{71} - 81 q^{73} - 3 q^{75} - 66 q^{77} - 17 q^{79} - 39 q^{81} - 30 q^{83} - 30 q^{85} - 20 q^{87} - 38 q^{89} + q^{91} - 75 q^{93} - 6 q^{95} - 48 q^{97} - 7 q^{99}+O(q^{100}) $ 29 * q - 3 * q^3 + 29 * q^5 - 8 * q^7 + 10 * q^9 + 2 * q^11 - 23 * q^13 - 3 * q^15 - 30 * q^17 - 6 * q^19 - 16 * q^21 - 21 * q^23 + 29 * q^25 - 15 * q^27 - 35 * q^29 - 7 * q^31 - 36 * q^33 - 8 * q^35 - 31 * q^37 - 11 * q^39 - 24 * q^41 - 17 * q^43 + 10 * q^45 - 17 * q^47 + q^49 + 8 * q^51 - 57 * q^53 + 2 * q^55 - 46 * q^57 - 9 * q^59 - 27 * q^61 - 34 * q^63 - 23 * q^65 - 21 * q^67 - 28 * q^69 - 19 * q^71 - 81 * q^73 - 3 * q^75 - 66 * q^77 - 17 * q^79 - 39 * q^81 - 30 * q^83 - 30 * q^85 - 20 * q^87 - 38 * q^89 + q^91 - 75 * q^93 - 6 * q^95 - 48 * 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
$3$	0.358058	0.206725	0.103362	−	0.994644i	$-0.467040\pi$
$3$	0.358058	0.206725	0.103362	+	0.994644i	$0.467040\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.78556	0.674879	0.337440	−	0.941347i	$-0.390439\pi$
$7$	1.78556	0.674879	0.337440	+	0.941347i	$0.390439\pi$
$8$	0	0
$9$	−2.87179	−0.957265
$10$	0	0
$11$	3.64957	1.10039	0.550193	−	0.835037i	$-0.314554\pi$
$11$	3.64957	1.10039	0.550193	+	0.835037i	$0.314554\pi$
$12$	0	0
$13$	−1.14680	−0.318064	−0.159032	−	0.987273i	$-0.550837\pi$
$13$	−1.14680	−0.318064	−0.159032	+	0.987273i	$0.550837\pi$
$14$	0	0
$15$	0.358058	0.0924501
$16$	0	0
$17$	4.34377	1.05352	0.526759	−	0.850015i	$-0.323407\pi$
$17$	4.34377	1.05352	0.526759	+	0.850015i	$0.323407\pi$
$18$	0	0
$19$	−6.49377	−1.48977	−0.744887	−	0.667191i	$-0.767497\pi$
$19$	−6.49377	−1.48977	−0.744887	+	0.667191i	$0.767497\pi$
$20$	0	0
$21$	0.639335	0.139514
$22$	0	0
$23$	4.28783	0.894074	0.447037	−	0.894515i	$-0.352479\pi$
$23$	4.28783	0.894074	0.447037	+	0.894515i	$0.352479\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−2.10244	−0.404615
$28$	0	0
$29$	−8.72153	−1.61955	−0.809774	−	0.586742i	$-0.800410\pi$
$29$	−8.72153	−1.61955	−0.809774	+	0.586742i	$0.800410\pi$
$30$	0	0
$31$	−10.1100	−1.81581	−0.907906	−	0.419174i	$-0.862320\pi$
$31$	−10.1100	−1.81581	−0.907906	+	0.419174i	$0.862320\pi$
$32$	0	0
$33$	1.30676	0.227477
$34$	0	0
$35$	1.78556	0.301815
$36$	0	0
$37$	−5.97550	−0.982366	−0.491183	−	0.871056i	$-0.663435\pi$
$37$	−5.97550	−0.982366	−0.491183	+	0.871056i	$0.663435\pi$
$38$	0	0
$39$	−0.410619	−0.0657517
$40$	0	0
$41$	1.06863	0.166892	0.0834461	−	0.996512i	$-0.473407\pi$
$41$	1.06863	0.166892	0.0834461	+	0.996512i	$0.473407\pi$
$42$	0	0
$43$	−8.89369	−1.35628	−0.678138	−	0.734935i	$-0.737213\pi$
$43$	−8.89369	−1.35628	−0.678138	+	0.734935i	$0.737213\pi$
$44$	0	0
$45$	−2.87179	−0.428102
$46$	0	0
$47$	−2.51548	−0.366920	−0.183460	−	0.983027i	$-0.558730\pi$
$47$	−2.51548	−0.366920	−0.183460	+	0.983027i	$0.558730\pi$
$48$	0	0
$49$	−3.81177	−0.544538
$50$	0	0
$51$	1.55532	0.217788
$52$	0	0
$53$	−3.49802	−0.480489	−0.240245	−	0.970712i	$-0.577228\pi$
$53$	−3.49802	−0.480489	−0.240245	+	0.970712i	$0.577228\pi$
$54$	0	0
$55$	3.64957	0.492108
$56$	0	0
$57$	−2.32515	−0.307973
$58$	0	0
$59$	1.19226	0.155219	0.0776093	−	0.996984i	$-0.475271\pi$
$59$	1.19226	0.155219	0.0776093	+	0.996984i	$0.475271\pi$
$60$	0	0
$61$	−9.61645	−1.23126	−0.615630	−	0.788035i	$-0.711098\pi$
$61$	−9.61645	−1.23126	−0.615630	+	0.788035i	$0.711098\pi$
$62$	0	0
$63$	−5.12777	−0.646038
$64$	0	0
$65$	−1.14680	−0.142243
$66$	0	0
$67$	15.8145	1.93204	0.966022	−	0.258459i	$-0.0832147\pi$
$67$	15.8145	1.93204	0.966022	+	0.258459i	$0.0832147\pi$
$68$	0	0
$69$	1.53529	0.184827
$70$	0	0
$71$	−10.7681	−1.27794	−0.638968	−	0.769233i	$-0.720638\pi$
$71$	−10.7681	−1.27794	−0.638968	+	0.769233i	$0.720638\pi$
$72$	0	0
$73$	0.472248	0.0552725	0.0276363	−	0.999618i	$-0.491202\pi$
$73$	0.472248	0.0552725	0.0276363	+	0.999618i	$0.491202\pi$
$74$	0	0
$75$	0.358058	0.0413449
$76$	0	0
$77$	6.51654	0.742628
$78$	0	0
$79$	−15.3419	−1.72610	−0.863052	−	0.505115i	$-0.831450\pi$
$79$	−15.3419	−1.72610	−0.863052	+	0.505115i	$0.831450\pi$
$80$	0	0
$81$	7.86259	0.873621
$82$	0	0
$83$	−4.75671	−0.522116	−0.261058	−	0.965323i	$-0.584071\pi$
$83$	−4.75671	−0.522116	−0.261058	+	0.965323i	$0.584071\pi$
$84$	0	0
$85$	4.34377	0.471148
$86$	0	0
$87$	−3.12281	−0.334801
$88$	0	0
$89$	10.2927	1.09102	0.545510	−	0.838105i	$-0.316336\pi$
$89$	10.2927	1.09102	0.545510	+	0.838105i	$0.316336\pi$
$90$	0	0
$91$	−2.04768	−0.214655
$92$	0	0
$93$	−3.61997	−0.375373
$94$	0	0
$95$	−6.49377	−0.666247
$96$	0	0
$97$	3.41846	0.347092	0.173546	−	0.984826i	$-0.444477\pi$
$97$	3.41846	0.347092	0.173546	+	0.984826i	$0.444477\pi$
$98$	0	0
$99$	−10.4808	−1.05336
$100$	0	0
$101$	2.89799	0.288361	0.144180	−	0.989551i	$-0.453945\pi$
$101$	2.89799	0.288361	0.144180	+	0.989551i	$0.453945\pi$
$102$	0	0
$103$	−0.0338357	−0.00333393	−0.00166697	−	0.999999i	$-0.500531\pi$
$103$	−0.0338357	−0.00333393	−0.00166697	+	0.999999i	$0.500531\pi$
$104$	0	0
$105$	0.639335	0.0623927
$106$	0	0
$107$	13.4995	1.30505	0.652525	−	0.757767i	$-0.273710\pi$
$107$	13.4995	1.30505	0.652525	+	0.757767i	$0.273710\pi$
$108$	0	0
$109$	5.75590	0.551315	0.275658	−	0.961256i	$-0.411104\pi$
$109$	5.75590	0.551315	0.275658	+	0.961256i	$0.411104\pi$
$110$	0	0
$111$	−2.13957	−0.203079
$112$	0	0
$113$	−12.1271	−1.14082	−0.570409	−	0.821361i	$-0.693215\pi$
$113$	−12.1271	−1.14082	−0.570409	+	0.821361i	$0.693215\pi$
$114$	0	0
$115$	4.28783	0.399842
$116$	0	0
$117$	3.29337	0.304472
$118$	0	0
$119$	7.75607	0.710998
$120$	0	0
$121$	2.31936	0.210851
$122$	0	0
$123$	0.382632	0.0345008
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	6.43541	0.571050	0.285525	−	0.958371i	$-0.407832\pi$
$127$	6.43541	0.571050	0.285525	+	0.958371i	$0.407832\pi$
$128$	0	0
$129$	−3.18446	−0.280376
$130$	0	0
$131$	11.5295	1.00734	0.503669	−	0.863897i	$-0.331983\pi$
$131$	11.5295	1.00734	0.503669	+	0.863897i	$0.331983\pi$
$132$	0	0
$133$	−11.5950	−1.00542
$134$	0	0
$135$	−2.10244	−0.180949
$136$	0	0
$137$	11.3148	0.966689	0.483345	−	0.875430i	$-0.339422\pi$
$137$	11.3148	0.966689	0.483345	+	0.875430i	$0.339422\pi$
$138$	0	0
$139$	−20.4210	−1.73209	−0.866045	−	0.499967i	$-0.833346\pi$
$139$	−20.4210	−1.73209	−0.866045	+	0.499967i	$0.833346\pi$
$140$	0	0
$141$	−0.900687	−0.0758515
$142$	0	0
$143$	−4.18532	−0.349994
$144$	0	0
$145$	−8.72153	−0.724284
$146$	0	0
$147$	−1.36483	−0.112569
$148$	0	0
$149$	−7.03694	−0.576489	−0.288244	−	0.957557i	$-0.593072\pi$
$149$	−7.03694	−0.576489	−0.288244	+	0.957557i	$0.593072\pi$
$150$	0	0
$151$	−18.5918	−1.51298	−0.756488	−	0.654007i	$-0.773086\pi$
$151$	−18.5918	−1.51298	−0.756488	+	0.654007i	$0.773086\pi$
$152$	0	0
$153$	−12.4744	−1.00850
$154$	0	0
$155$	−10.1100	−0.812056
$156$	0	0
$157$	−2.87847	−0.229727	−0.114864	−	0.993381i	$-0.536643\pi$
$157$	−2.87847	−0.229727	−0.114864	+	0.993381i	$0.536643\pi$
$158$	0	0
$159$	−1.25249	−0.0993290
$160$	0	0
$161$	7.65619	0.603392
$162$	0	0
$163$	16.1121	1.26200	0.630998	−	0.775785i	$-0.282646\pi$
$163$	16.1121	1.26200	0.630998	+	0.775785i	$0.282646\pi$
$164$	0	0
$165$	1.30676	0.101731
$166$	0	0
$167$	6.81538	0.527390	0.263695	−	0.964606i	$-0.415059\pi$
$167$	6.81538	0.527390	0.263695	+	0.964606i	$0.415059\pi$
$168$	0	0
$169$	−11.6849	−0.898835
$170$	0	0
$171$	18.6488	1.42611
$172$	0	0
$173$	21.9520	1.66898	0.834491	−	0.551021i	$-0.185762\pi$
$173$	21.9520	1.66898	0.834491	+	0.551021i	$0.185762\pi$
$174$	0	0
$175$	1.78556	0.134976
$176$	0	0
$177$	0.426897	0.0320875
$178$	0	0
$179$	16.0237	1.19767	0.598835	−	0.800873i	$-0.295631\pi$
$179$	16.0237	1.19767	0.598835	+	0.800873i	$0.295631\pi$
$180$	0	0
$181$	−6.66018	−0.495048	−0.247524	−	0.968882i	$-0.579617\pi$
$181$	−6.66018	−0.495048	−0.247524	+	0.968882i	$0.579617\pi$
$182$	0	0
$183$	−3.44324	−0.254532
$184$	0	0
$185$	−5.97550	−0.439327
$186$	0	0
$187$	15.8529	1.15928
$188$	0	0
$189$	−3.75404	−0.273066
$190$	0	0
$191$	15.3483	1.11056	0.555280	−	0.831663i	$-0.312611\pi$
$191$	15.3483	1.11056	0.555280	+	0.831663i	$0.312611\pi$
$192$	0	0
$193$	−1.97031	−0.141826	−0.0709131	−	0.997482i	$-0.522591\pi$
$193$	−1.97031	−0.141826	−0.0709131	+	0.997482i	$0.522591\pi$
$194$	0	0
$195$	−0.410619	−0.0294051
$196$	0	0
$197$	−9.24337	−0.658563	−0.329282	−	0.944232i	$-0.606807\pi$
$197$	−9.24337	−0.658563	−0.329282	+	0.944232i	$0.606807\pi$
$198$	0	0
$199$	−19.1371	−1.35659	−0.678297	−	0.734788i	$-0.737282\pi$
$199$	−19.1371	−1.35659	−0.678297	+	0.734788i	$0.737282\pi$
$200$	0	0
$201$	5.66249	0.399401
$202$	0	0
$203$	−15.5728	−1.09300
$204$	0	0
$205$	1.06863	0.0746365
$206$	0	0
$207$	−12.3138	−0.855866
$208$	0	0
$209$	−23.6995	−1.63933
$210$	0	0
$211$	−6.32660	−0.435541	−0.217771	−	0.976000i	$-0.569878\pi$
$211$	−6.32660	−0.435541	−0.217771	+	0.976000i	$0.569878\pi$
$212$	0	0
$213$	−3.85559	−0.264181
$214$	0	0
$215$	−8.89369	−0.606545
$216$	0	0
$217$	−18.0521	−1.22545
$218$	0	0
$219$	0.169092	0.0114262
$220$	0	0
$221$	−4.98142	−0.335087
$222$	0	0
$223$	−21.3960	−1.43278	−0.716391	−	0.697699i	$-0.754208\pi$
$223$	−21.3960	−1.43278	−0.716391	+	0.697699i	$0.754208\pi$
$224$	0	0
$225$	−2.87179	−0.191453
$226$	0	0
$227$	−22.2238	−1.47505	−0.737524	−	0.675321i	$-0.764005\pi$
$227$	−22.2238	−1.47505	−0.737524	+	0.675321i	$0.764005\pi$
$228$	0	0
$229$	2.09728	0.138592	0.0692961	−	0.997596i	$-0.477925\pi$
$229$	2.09728	0.138592	0.0692961	+	0.997596i	$0.477925\pi$
$230$	0	0
$231$	2.33330	0.153520
$232$	0	0
$233$	−2.59191	−0.169802	−0.0849008	−	0.996389i	$-0.527057\pi$
$233$	−2.59191	−0.169802	−0.0849008	+	0.996389i	$0.527057\pi$
$234$	0	0
$235$	−2.51548	−0.164092
$236$	0	0
$237$	−5.49330	−0.356828
$238$	0	0
$239$	−7.52943	−0.487038	−0.243519	−	0.969896i	$-0.578302\pi$
$239$	−7.52943	−0.487038	−0.243519	+	0.969896i	$0.578302\pi$
$240$	0	0
$241$	−3.88696	−0.250381	−0.125190	−	0.992133i	$-0.539954\pi$
$241$	−3.88696	−0.250381	−0.125190	+	0.992133i	$0.539954\pi$
$242$	0	0
$243$	9.12258	0.585214
$244$	0	0
$245$	−3.81177	−0.243525
$246$	0	0
$247$	7.44704	0.473844
$248$	0	0
$249$	−1.70318	−0.107934
$250$	0	0
$251$	10.0536	0.634580	0.317290	−	0.948329i	$-0.397227\pi$
$251$	10.0536	0.634580	0.317290	+	0.948329i	$0.397227\pi$
$252$	0	0
$253$	15.6487	0.983828
$254$	0	0
$255$	1.55532	0.0973979
$256$	0	0
$257$	23.1228	1.44236	0.721179	−	0.692749i	$-0.243601\pi$
$257$	23.1228	1.44236	0.721179	+	0.692749i	$0.243601\pi$
$258$	0	0
$259$	−10.6696	−0.662978
$260$	0	0
$261$	25.0465	1.55034
$262$	0	0
$263$	−8.46856	−0.522194	−0.261097	−	0.965313i	$-0.584084\pi$
$263$	−8.46856	−0.522194	−0.261097	+	0.965313i	$0.584084\pi$
$264$	0	0
$265$	−3.49802	−0.214881
$266$	0	0
$267$	3.68537	0.225541
$268$	0	0
$269$	11.9107	0.726209	0.363105	−	0.931748i	$-0.381717\pi$
$269$	11.9107	0.726209	0.363105	+	0.931748i	$0.381717\pi$
$270$	0	0
$271$	−2.05016	−0.124539	−0.0622693	−	0.998059i	$-0.519834\pi$
$271$	−2.05016	−0.124539	−0.0622693	+	0.998059i	$0.519834\pi$
$272$	0	0
$273$	−0.733187	−0.0443745
$274$	0	0
$275$	3.64957	0.220077
$276$	0	0
$277$	16.6047	0.997678	0.498839	−	0.866695i	$-0.333760\pi$
$277$	16.6047	0.997678	0.498839	+	0.866695i	$0.333760\pi$
$278$	0	0
$279$	29.0339	1.73821
$280$	0	0
$281$	4.74619	0.283134	0.141567	−	0.989929i	$-0.454786\pi$
$281$	4.74619	0.283134	0.141567	+	0.989929i	$0.454786\pi$
$282$	0	0
$283$	−15.7667	−0.937234	−0.468617	−	0.883402i	$-0.655248\pi$
$283$	−15.7667	−0.937234	−0.468617	+	0.883402i	$0.655248\pi$
$284$	0	0
$285$	−2.32515	−0.137730
$286$	0	0
$287$	1.90811	0.112632
$288$	0	0
$289$	1.86832	0.109901
$290$	0	0
$291$	1.22401	0.0717526
$292$	0	0
$293$	23.0110	1.34432	0.672159	−	0.740407i	$-0.265367\pi$
$293$	23.0110	1.34432	0.672159	+	0.740407i	$0.265367\pi$
$294$	0	0
$295$	1.19226	0.0694159
$296$	0	0
$297$	−7.67301	−0.445233
$298$	0	0
$299$	−4.91727	−0.284373
$300$	0	0
$301$	−15.8802	−0.915322
$302$	0	0
$303$	1.03765	0.0596113
$304$	0	0
$305$	−9.61645	−0.550636
$306$	0	0
$307$	−28.0318	−1.59986	−0.799929	−	0.600095i	$-0.795129\pi$
$307$	−28.0318	−1.59986	−0.799929	+	0.600095i	$0.795129\pi$
$308$	0	0
$309$	−0.0121151	−0.000689206 0
$310$	0	0
$311$	−1.99073	−0.112884	−0.0564419	−	0.998406i	$-0.517976\pi$
$311$	−1.99073	−0.112884	−0.0564419	+	0.998406i	$0.517976\pi$
$312$	0	0
$313$	−22.8267	−1.29024	−0.645120	−	0.764082i	$-0.723192\pi$
$313$	−22.8267	−1.29024	−0.645120	+	0.764082i	$0.723192\pi$
$314$	0	0
$315$	−5.12777	−0.288917
$316$	0	0
$317$	−14.5019	−0.814506	−0.407253	−	0.913315i	$-0.633513\pi$
$317$	−14.5019	−0.814506	−0.407253	+	0.913315i	$0.633513\pi$
$318$	0	0
$319$	−31.8298	−1.78213
$320$	0	0
$321$	4.83362	0.269786
$322$	0	0
$323$	−28.2074	−1.56950
$324$	0	0
$325$	−1.14680	−0.0636128
$326$	0	0
$327$	2.06094	0.113971
$328$	0	0
$329$	−4.49155	−0.247627
$330$	0	0
$331$	13.6721	0.751485	0.375743	−	0.926724i	$-0.377388\pi$
$331$	13.6721	0.751485	0.375743	+	0.926724i	$0.377388\pi$
$332$	0	0
$333$	17.1604	0.940384
$334$	0	0
$335$	15.8145	0.864037
$336$	0	0
$337$	9.36685	0.510245	0.255123	−	0.966909i	$-0.417884\pi$
$337$	9.36685	0.510245	0.255123	+	0.966909i	$0.417884\pi$
$338$	0	0
$339$	−4.34219	−0.235835
$340$	0	0
$341$	−36.8972	−1.99810
$342$	0	0
$343$	−19.3051	−1.04238
$344$	0	0
$345$	1.53529	0.0826573
$346$	0	0
$347$	−0.663475	−0.0356172	−0.0178086	−	0.999841i	$-0.505669\pi$
$347$	−0.663475	−0.0356172	−0.0178086	+	0.999841i	$0.505669\pi$
$348$	0	0
$349$	19.6449	1.05157	0.525784	−	0.850618i	$-0.323772\pi$
$349$	19.6449	1.05157	0.525784	+	0.850618i	$0.323772\pi$
$350$	0	0
$351$	2.41107	0.128694
$352$	0	0
$353$	−2.57617	−0.137116	−0.0685578	−	0.997647i	$-0.521840\pi$
$353$	−2.57617	−0.137116	−0.0685578	+	0.997647i	$0.521840\pi$
$354$	0	0
$355$	−10.7681	−0.571510
$356$	0	0
$357$	2.77712	0.146981
$358$	0	0
$359$	4.69582	0.247836	0.123918	−	0.992292i	$-0.460454\pi$
$359$	4.69582	0.247836	0.123918	+	0.992292i	$0.460454\pi$
$360$	0	0
$361$	23.1691	1.21943
$362$	0	0
$363$	0.830465	0.0435881
$364$	0	0
$365$	0.472248	0.0247186
$366$	0	0
$367$	−16.8970	−0.882014	−0.441007	−	0.897504i	$-0.645379\pi$
$367$	−16.8970	−0.882014	−0.441007	+	0.897504i	$0.645379\pi$
$368$	0	0
$369$	−3.06889	−0.159760
$370$	0	0
$371$	−6.24593	−0.324272
$372$	0	0
$373$	6.53505	0.338372	0.169186	−	0.985584i	$-0.445886\pi$
$373$	6.53505	0.338372	0.169186	+	0.985584i	$0.445886\pi$
$374$	0	0
$375$	0.358058	0.0184900
$376$	0	0
$377$	10.0018	0.515120
$378$	0	0
$379$	14.8750	0.764075	0.382038	−	0.924147i	$-0.375222\pi$
$379$	14.8750	0.764075	0.382038	+	0.924147i	$0.375222\pi$
$380$	0	0
$381$	2.30425	0.118050
$382$	0	0
$383$	25.3746	1.29658	0.648291	−	0.761392i	$-0.275484\pi$
$383$	25.3746	1.29658	0.648291	+	0.761392i	$0.275484\pi$
$384$	0	0
$385$	6.51654	0.332113
$386$	0	0
$387$	25.5409	1.29831
$388$	0	0
$389$	−18.6928	−0.947762	−0.473881	−	0.880589i	$-0.657147\pi$
$389$	−18.6928	−0.947762	−0.473881	+	0.880589i	$0.657147\pi$
$390$	0	0
$391$	18.6253	0.941924
$392$	0	0
$393$	4.12823	0.208242
$394$	0	0
$395$	−15.3419	−0.771937
$396$	0	0
$397$	7.34109	0.368439	0.184219	−	0.982885i	$-0.441024\pi$
$397$	7.34109	0.368439	0.184219	+	0.982885i	$0.441024\pi$
$398$	0	0
$399$	−4.15169	−0.207845
$400$	0	0
$401$	−1.00000	−0.0499376
$401$	−1.00000	−0.0499376
$402$	0	0
$403$	11.5941	0.577545
$404$	0	0
$405$	7.86259	0.390695
$406$	0	0
$407$	−21.8080	−1.08098
$408$	0	0
$409$	−4.44620	−0.219850	−0.109925	−	0.993940i	$-0.535061\pi$
$409$	−4.44620	−0.219850	−0.109925	+	0.993940i	$0.535061\pi$
$410$	0	0
$411$	4.05136	0.199839
$412$	0	0
$413$	2.12885	0.104754
$414$	0	0
$415$	−4.75671	−0.233498
$416$	0	0
$417$	−7.31191	−0.358066
$418$	0	0
$419$	30.3182	1.48114	0.740570	−	0.671979i	$-0.234556\pi$
$419$	30.3182	1.48114	0.740570	+	0.671979i	$0.234556\pi$
$420$	0	0
$421$	3.26973	0.159357	0.0796785	−	0.996821i	$-0.474611\pi$
$421$	3.26973	0.159357	0.0796785	+	0.996821i	$0.474611\pi$
$422$	0	0
$423$	7.22394	0.351240
$424$	0	0
$425$	4.34377	0.210704
$426$	0	0
$427$	−17.1708	−0.830952
$428$	0	0
$429$	−1.49858	−0.0723523
$430$	0	0
$431$	13.3343	0.642290	0.321145	−	0.947030i	$-0.395932\pi$
$431$	13.3343	0.642290	0.321145	+	0.947030i	$0.395932\pi$
$432$	0	0
$433$	0.245919	0.0118181	0.00590906	−	0.999983i	$-0.498119\pi$
$433$	0.245919	0.0118181	0.00590906	+	0.999983i	$0.498119\pi$
$434$	0	0
$435$	−3.12281	−0.149727
$436$	0	0
$437$	−27.8442	−1.33197
$438$	0	0
$439$	23.9966	1.14529	0.572647	−	0.819802i	$-0.305917\pi$
$439$	23.9966	1.14529	0.572647	+	0.819802i	$0.305917\pi$
$440$	0	0
$441$	10.9466	0.521267
$442$	0	0
$443$	−18.4283	−0.875554	−0.437777	−	0.899084i	$-0.644234\pi$
$443$	−18.4283	−0.875554	−0.437777	+	0.899084i	$0.644234\pi$
$444$	0	0
$445$	10.2927	0.487919
$446$	0	0
$447$	−2.51963	−0.119175
$448$	0	0
$449$	−1.17519	−0.0554607	−0.0277303	−	0.999615i	$-0.508828\pi$
$449$	−1.17519	−0.0554607	−0.0277303	+	0.999615i	$0.508828\pi$
$450$	0	0
$451$	3.90005	0.183646
$452$	0	0
$453$	−6.65692	−0.312770
$454$	0	0
$455$	−2.04768	−0.0959966
$456$	0	0
$457$	−32.6542	−1.52750	−0.763749	−	0.645514i	$-0.776643\pi$
$457$	−32.6542	−1.52750	−0.763749	+	0.645514i	$0.776643\pi$
$458$	0	0
$459$	−9.13252	−0.426269
$460$	0	0
$461$	−16.6604	−0.775950	−0.387975	−	0.921670i	$-0.626825\pi$
$461$	−16.6604	−0.775950	−0.387975	+	0.921670i	$0.626825\pi$
$462$	0	0
$463$	−24.4751	−1.13745	−0.568727	−	0.822526i	$-0.692564\pi$
$463$	−24.4751	−1.13745	−0.568727	+	0.822526i	$0.692564\pi$
$464$	0	0
$465$	−3.61997	−0.167872
$466$	0	0
$467$	7.30031	0.337818	0.168909	−	0.985632i	$-0.445976\pi$
$467$	7.30031	0.337818	0.168909	+	0.985632i	$0.445976\pi$
$468$	0	0
$469$	28.2377	1.30390
$470$	0	0
$471$	−1.03066	−0.0474903
$472$	0	0
$473$	−32.4582	−1.49243
$474$	0	0
$475$	−6.49377	−0.297955
$476$	0	0
$477$	10.0456	0.459956
$478$	0	0
$479$	−35.2397	−1.61014	−0.805071	−	0.593178i	$-0.797873\pi$
$479$	−35.2397	−1.61014	−0.805071	+	0.593178i	$0.797873\pi$
$480$	0	0
$481$	6.85268	0.312455
$482$	0	0
$483$	2.74136	0.124736
$484$	0	0
$485$	3.41846	0.155224
$486$	0	0
$487$	42.2255	1.91342	0.956710	−	0.291042i	$-0.0940018\pi$
$487$	42.2255	1.91342	0.956710	+	0.291042i	$0.0940018\pi$
$488$	0	0
$489$	5.76905	0.260886
$490$	0	0
$491$	−1.24489	−0.0561810	−0.0280905	−	0.999605i	$-0.508943\pi$
$491$	−1.24489	−0.0561810	−0.0280905	+	0.999605i	$0.508943\pi$
$492$	0	0
$493$	−37.8843	−1.70622
$494$	0	0
$495$	−10.4808	−0.471078
$496$	0	0
$497$	−19.2271	−0.862453
$498$	0	0
$499$	31.6301	1.41596	0.707980	−	0.706233i	$-0.249607\pi$
$499$	31.6301	1.41596	0.707980	+	0.706233i	$0.249607\pi$
$500$	0	0
$501$	2.44030	0.109024
$502$	0	0
$503$	3.97352	0.177171	0.0885853	−	0.996069i	$-0.471765\pi$
$503$	3.97352	0.177171	0.0885853	+	0.996069i	$0.471765\pi$
$504$	0	0
$505$	2.89799	0.128959
$506$	0	0
$507$	−4.18385	−0.185811
$508$	0	0
$509$	−10.6995	−0.474245	−0.237123	−	0.971480i	$-0.576204\pi$
$509$	−10.6995	−0.474245	−0.237123	+	0.971480i	$0.576204\pi$
$510$	0	0
$511$	0.843229	0.0373023
$512$	0	0
$513$	13.6528	0.602785
$514$	0	0
$515$	−0.0338357	−0.00149098
$516$	0	0
$517$	−9.18042	−0.403754
$518$	0	0
$519$	7.86009	0.345020
$520$	0	0
$521$	−40.0684	−1.75543	−0.877715	−	0.479183i	$-0.840933\pi$
$521$	−40.0684	−1.75543	−0.877715	+	0.479183i	$0.840933\pi$
$522$	0	0
$523$	27.2924	1.19341	0.596707	−	0.802459i	$-0.296475\pi$
$523$	27.2924	1.19341	0.596707	+	0.802459i	$0.296475\pi$
$524$	0	0
$525$	0.639335	0.0279028
$526$	0	0
$527$	−43.9155	−1.91299
$528$	0	0
$529$	−4.61451	−0.200631
$530$	0	0
$531$	−3.42392	−0.148585
$532$	0	0
$533$	−1.22550	−0.0530825
$534$	0	0
$535$	13.4995	0.583636
$536$	0	0
$537$	5.73742	0.247588
$538$	0	0
$539$	−13.9113	−0.599202
$540$	0	0
$541$	−20.9555	−0.900949	−0.450474	−	0.892789i	$-0.648745\pi$
$541$	−20.9555	−0.900949	−0.450474	+	0.892789i	$0.648745\pi$
$542$	0	0
$543$	−2.38473	−0.102339
$544$	0	0
$545$	5.75590	0.246556
$546$	0	0
$547$	11.5928	0.495671	0.247836	−	0.968802i	$-0.420281\pi$
$547$	11.5928	0.495671	0.247836	+	0.968802i	$0.420281\pi$
$548$	0	0
$549$	27.6165	1.17864
$550$	0	0
$551$	56.6357	2.41276
$552$	0	0
$553$	−27.3940	−1.16491
$554$	0	0
$555$	−2.13957	−0.0908198
$556$	0	0
$557$	−21.7676	−0.922321	−0.461160	−	0.887317i	$-0.652567\pi$
$557$	−21.7676	−0.922321	−0.461160	+	0.887317i	$0.652567\pi$
$558$	0	0
$559$	10.1993	0.431383
$560$	0	0
$561$	5.67625	0.239651
$562$	0	0
$563$	18.2284	0.768236	0.384118	−	0.923284i	$-0.374506\pi$
$563$	18.2284	0.768236	0.384118	+	0.923284i	$0.374506\pi$
$564$	0	0
$565$	−12.1271	−0.510189
$566$	0	0
$567$	14.0391	0.589589
$568$	0	0
$569$	−26.8693	−1.12642	−0.563209	−	0.826315i	$-0.690433\pi$
$569$	−26.8693	−1.12642	−0.563209	+	0.826315i	$0.690433\pi$
$570$	0	0
$571$	−3.52936	−0.147699	−0.0738496	−	0.997269i	$-0.523528\pi$
$571$	−3.52936	−0.147699	−0.0738496	+	0.997269i	$0.523528\pi$
$572$	0	0
$573$	5.49556	0.229580
$574$	0	0
$575$	4.28783	0.178815
$576$	0	0
$577$	−0.788984	−0.0328458	−0.0164229	−	0.999865i	$-0.505228\pi$
$577$	−0.788984	−0.0328458	−0.0164229	+	0.999865i	$0.505228\pi$
$578$	0	0
$579$	−0.705486	−0.0293190
$580$	0	0
$581$	−8.49340	−0.352366
$582$	0	0
$583$	−12.7663	−0.528724
$584$	0	0
$585$	3.29337	0.136164
$586$	0	0
$587$	46.7191	1.92830	0.964152	−	0.265351i	$-0.0854880\pi$
$587$	46.7191	1.92830	0.964152	+	0.265351i	$0.0854880\pi$
$588$	0	0
$589$	65.6521	2.70515
$590$	0	0
$591$	−3.30966	−0.136141
$592$	0	0
$593$	−31.7724	−1.30474	−0.652368	−	0.757902i	$-0.726224\pi$
$593$	−31.7724	−1.30474	−0.652368	+	0.757902i	$0.726224\pi$
$594$	0	0
$595$	7.75607	0.317968
$596$	0	0
$597$	−6.85219	−0.280441
$598$	0	0
$599$	5.96704	0.243807	0.121903	−	0.992542i	$-0.461100\pi$
$599$	5.96704	0.243807	0.121903	+	0.992542i	$0.461100\pi$
$600$	0	0
$601$	−13.6400	−0.556386	−0.278193	−	0.960525i	$-0.589735\pi$
$601$	−13.6400	−0.556386	−0.278193	+	0.960525i	$0.589735\pi$
$602$	0	0
$603$	−45.4159	−1.84948
$604$	0	0
$605$	2.31936	0.0942954
$606$	0	0
$607$	−3.02295	−0.122698	−0.0613489	−	0.998116i	$-0.519540\pi$
$607$	−3.02295	−0.122698	−0.0613489	+	0.998116i	$0.519540\pi$
$608$	0	0
$609$	−5.57598	−0.225950
$610$	0	0
$611$	2.88474	0.116704
$612$	0	0
$613$	−25.6807	−1.03724	−0.518618	−	0.855006i	$-0.673553\pi$
$613$	−25.6807	−1.03724	−0.518618	+	0.855006i	$0.673553\pi$
$614$	0	0
$615$	0.382632	0.0154292
$616$	0	0
$617$	−26.5761	−1.06991	−0.534957	−	0.844879i	$-0.679672\pi$
$617$	−26.5761	−1.06991	−0.534957	+	0.844879i	$0.679672\pi$
$618$	0	0
$619$	−37.7473	−1.51719	−0.758595	−	0.651562i	$-0.774114\pi$
$619$	−37.7473	−1.51719	−0.758595	+	0.651562i	$0.774114\pi$
$620$	0	0
$621$	−9.01491	−0.361756
$622$	0	0
$623$	18.3782	0.736306
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−8.48578	−0.338890
$628$	0	0
$629$	−25.9562	−1.03494
$630$	0	0
$631$	13.0162	0.518168	0.259084	−	0.965855i	$-0.416579\pi$
$631$	13.0162	0.518168	0.259084	+	0.965855i	$0.416579\pi$
$632$	0	0
$633$	−2.26529	−0.0900371
$634$	0	0
$635$	6.43541	0.255381
$636$	0	0
$637$	4.37132	0.173198
$638$	0	0
$639$	30.9237	1.22332
$640$	0	0
$641$	−35.5238	−1.40311	−0.701553	−	0.712617i	$-0.747510\pi$
$641$	−35.5238	−1.40311	−0.701553	+	0.712617i	$0.747510\pi$
$642$	0	0
$643$	14.7738	0.582623	0.291312	−	0.956628i	$-0.405908\pi$
$643$	14.7738	0.582623	0.291312	+	0.956628i	$0.405908\pi$
$644$	0	0
$645$	−3.18446	−0.125388
$646$	0	0
$647$	−9.97767	−0.392263	−0.196131	−	0.980578i	$-0.562838\pi$
$647$	−9.97767	−0.392263	−0.196131	+	0.980578i	$0.562838\pi$
$648$	0	0
$649$	4.35123	0.170801
$650$	0	0
$651$	−6.46368	−0.253332
$652$	0	0
$653$	−19.1683	−0.750115	−0.375057	−	0.927002i	$-0.622377\pi$
$653$	−19.1683	−0.750115	−0.375057	+	0.927002i	$0.622377\pi$
$654$	0	0
$655$	11.5295	0.450495
$656$	0	0
$657$	−1.35620	−0.0529104
$658$	0	0
$659$	34.4830	1.34327	0.671633	−	0.740884i	$-0.265593\pi$
$659$	34.4830	1.34327	0.671633	+	0.740884i	$0.265593\pi$
$660$	0	0
$661$	−29.1413	−1.13347	−0.566734	−	0.823901i	$-0.691793\pi$
$661$	−29.1413	−1.13347	−0.566734	+	0.823901i	$0.691793\pi$
$662$	0	0
$663$	−1.78364	−0.0692707
$664$	0	0
$665$	−11.5950	−0.449636
$666$	0	0
$667$	−37.3965	−1.44800
$668$	0	0
$669$	−7.66101	−0.296192
$670$	0	0
$671$	−35.0959	−1.35486
$672$	0	0
$673$	20.7599	0.800237	0.400118	−	0.916463i	$-0.368969\pi$
$673$	20.7599	0.800237	0.400118	+	0.916463i	$0.368969\pi$
$674$	0	0
$675$	−2.10244	−0.0809230
$676$	0	0
$677$	36.0623	1.38599	0.692993	−	0.720944i	$-0.256292\pi$
$677$	36.0623	1.38599	0.692993	+	0.720944i	$0.256292\pi$
$678$	0	0
$679$	6.10388	0.234246
$680$	0	0
$681$	−7.95742	−0.304929
$682$	0	0
$683$	35.0285	1.34033	0.670164	−	0.742213i	$-0.266224\pi$
$683$	35.0285	1.34033	0.670164	+	0.742213i	$0.266224\pi$
$684$	0	0
$685$	11.3148	0.432317
$686$	0	0
$687$	0.750948	0.0286504
$688$	0	0
$689$	4.01151	0.152826
$690$	0	0
$691$	4.41228	0.167851	0.0839255	−	0.996472i	$-0.473254\pi$
$691$	4.41228	0.167851	0.0839255	+	0.996472i	$0.473254\pi$
$692$	0	0
$693$	−18.7142	−0.710892
$694$	0	0
$695$	−20.4210	−0.774614
$696$	0	0
$697$	4.64189	0.175824
$698$	0	0
$699$	−0.928053	−0.0351022
$700$	0	0
$701$	1.65820	0.0626295	0.0313148	−	0.999510i	$-0.490031\pi$
$701$	1.65820	0.0626295	0.0313148	+	0.999510i	$0.490031\pi$
$702$	0	0
$703$	38.8035	1.46350
$704$	0	0
$705$	−0.900687	−0.0339218
$706$	0	0
$707$	5.17454	0.194609
$708$	0	0
$709$	−3.95765	−0.148633	−0.0743163	−	0.997235i	$-0.523677\pi$
$709$	−3.95765	−0.148633	−0.0743163	+	0.997235i	$0.523677\pi$
$710$	0	0
$711$	44.0589	1.65234
$712$	0	0
$713$	−43.3500	−1.62347
$714$	0	0
$715$	−4.18532	−0.156522
$716$	0	0
$717$	−2.69597	−0.100683
$718$	0	0
$719$	26.7512	0.997652	0.498826	−	0.866702i	$-0.333765\pi$
$719$	26.7512	0.997652	0.498826	+	0.866702i	$0.333765\pi$
$720$	0	0
$721$	−0.0604158	−0.00225000
$722$	0	0
$723$	−1.39176	−0.0517599
$724$	0	0
$725$	−8.72153	−0.323910
$726$	0	0
$727$	50.1877	1.86136	0.930679	−	0.365836i	$-0.119217\pi$
$727$	50.1877	1.86136	0.930679	+	0.365836i	$0.119217\pi$
$728$	0	0
$729$	−20.3214	−0.752643
$730$	0	0
$731$	−38.6321	−1.42886
$732$	0	0
$733$	−18.3173	−0.676563	−0.338282	−	0.941045i	$-0.609846\pi$
$733$	−18.3173	−0.676563	−0.338282	+	0.941045i	$0.609846\pi$
$734$	0	0
$735$	−1.36483	−0.0503426
$736$	0	0
$737$	57.7160	2.12600
$738$	0	0
$739$	−45.4204	−1.67082	−0.835408	−	0.549631i	$-0.814768\pi$
$739$	−45.4204	−1.67082	−0.835408	+	0.549631i	$0.814768\pi$
$740$	0	0
$741$	2.66647	0.0979552
$742$	0	0
$743$	−5.93948	−0.217898	−0.108949	−	0.994047i	$-0.534749\pi$
$743$	−5.93948	−0.217898	−0.108949	+	0.994047i	$0.534749\pi$
$744$	0	0
$745$	−7.03694	−0.257814
$746$	0	0
$747$	13.6603	0.499804
$748$	0	0
$749$	24.1043	0.880751
$750$	0	0
$751$	−13.1331	−0.479232	−0.239616	−	0.970868i	$-0.577022\pi$
$751$	−13.1331	−0.479232	−0.239616	+	0.970868i	$0.577022\pi$
$752$	0	0
$753$	3.59978	0.131183
$754$	0	0
$755$	−18.5918	−0.676624
$756$	0	0
$757$	−21.0863	−0.766393	−0.383197	−	0.923667i	$-0.625177\pi$
$757$	−21.0863	−0.766393	−0.383197	+	0.923667i	$0.625177\pi$
$758$	0	0
$759$	5.60315	0.203381
$760$	0	0
$761$	43.0783	1.56159	0.780793	−	0.624789i	$-0.214815\pi$
$761$	43.0783	1.56159	0.780793	+	0.624789i	$0.214815\pi$
$762$	0	0
$763$	10.2775	0.372071
$764$	0	0
$765$	−12.4744	−0.451013
$766$	0	0
$767$	−1.36728	−0.0493695
$768$	0	0
$769$	17.7581	0.640374	0.320187	−	0.947354i	$-0.396254\pi$
$769$	17.7581	0.640374	0.320187	+	0.947354i	$0.396254\pi$
$770$	0	0
$771$	8.27928	0.298171
$772$	0	0
$773$	51.3409	1.84660	0.923302	−	0.384074i	$-0.125479\pi$
$773$	51.3409	1.84660	0.923302	+	0.384074i	$0.125479\pi$
$774$	0	0
$775$	−10.1100	−0.363162
$776$	0	0
$777$	−3.82034	−0.137054
$778$	0	0
$779$	−6.93945	−0.248632
$780$	0	0
$781$	−39.2989	−1.40622
$782$	0	0
$783$	18.3365	0.655294
$784$	0	0
$785$	−2.87847	−0.102737
$786$	0	0
$787$	20.4613	0.729368	0.364684	−	0.931131i	$-0.381177\pi$
$787$	20.4613	0.729368	0.364684	+	0.931131i	$0.381177\pi$
$788$	0	0
$789$	−3.03223	−0.107950
$790$	0	0
$791$	−21.6536	−0.769914
$792$	0	0
$793$	11.0281	0.391620
$794$	0	0
$795$	−1.25249	−0.0444213
$796$	0	0
$797$	−12.5377	−0.444109	−0.222054	−	0.975034i	$-0.571276\pi$
$797$	−12.5377	−0.444109	−0.222054	+	0.975034i	$0.571276\pi$
$798$	0	0
$799$	−10.9267	−0.386557
$800$	0	0
$801$	−29.5584	−1.04439
$802$	0	0
$803$	1.72350	0.0608211
$804$	0	0
$805$	7.65619	0.269845
$806$	0	0
$807$	4.26472	0.150125
$808$	0	0
$809$	23.7003	0.833259	0.416629	−	0.909076i	$-0.363211\pi$
$809$	23.7003	0.833259	0.416629	+	0.909076i	$0.363211\pi$
$810$	0	0
$811$	−18.6316	−0.654245	−0.327122	−	0.944982i	$-0.606079\pi$
$811$	−18.6316	−0.654245	−0.327122	+	0.944982i	$0.606079\pi$
$812$	0	0
$813$	−0.734077	−0.0257452
$814$	0	0
$815$	16.1121	0.564381
$816$	0	0
$817$	57.7536	2.02054
$818$	0	0
$819$	5.88051	0.205482
$820$	0	0
$821$	−38.5512	−1.34545	−0.672724	−	0.739894i	$-0.734876\pi$
$821$	−38.5512	−1.34545	−0.672724	+	0.739894i	$0.734876\pi$
$822$	0	0
$823$	−12.6849	−0.442168	−0.221084	−	0.975255i	$-0.570959\pi$
$823$	−12.6849	−0.442168	−0.221084	+	0.975255i	$0.570959\pi$
$824$	0	0
$825$	1.30676	0.0454954
$826$	0	0
$827$	−7.60397	−0.264416	−0.132208	−	0.991222i	$-0.542207\pi$
$827$	−7.60397	−0.264416	−0.132208	+	0.991222i	$0.542207\pi$
$828$	0	0
$829$	15.4699	0.537293	0.268646	−	0.963239i	$-0.413424\pi$
$829$	15.4699	0.537293	0.268646	+	0.963239i	$0.413424\pi$
$830$	0	0
$831$	5.94543	0.206245
$832$	0	0
$833$	−16.5574	−0.573681
$834$	0	0
$835$	6.81538	0.235856
$836$	0	0
$837$	21.2557	0.734705
$838$	0	0
$839$	−48.4724	−1.67345	−0.836726	−	0.547622i	$-0.815533\pi$
$839$	−48.4724	−1.67345	−0.836726	+	0.547622i	$0.815533\pi$
$840$	0	0
$841$	47.0652	1.62294
$842$	0	0
$843$	1.69941	0.0585308
$844$	0	0
$845$	−11.6849	−0.401971
$846$	0	0
$847$	4.14136	0.142299
$848$	0	0
$849$	−5.64539	−0.193749
$850$	0	0
$851$	−25.6219	−0.878308
$852$	0	0
$853$	−7.89573	−0.270345	−0.135172	−	0.990822i	$-0.543159\pi$
$853$	−7.89573	−0.270345	−0.135172	+	0.990822i	$0.543159\pi$
$854$	0	0
$855$	18.6488	0.637775
$856$	0	0
$857$	−25.8231	−0.882099	−0.441050	−	0.897483i	$-0.645394\pi$
$857$	−25.8231	−0.882099	−0.441050	+	0.897483i	$0.645394\pi$
$858$	0	0
$859$	−45.4717	−1.55147	−0.775737	−	0.631056i	$-0.782622\pi$
$859$	−45.4717	−1.55147	−0.775737	+	0.631056i	$0.782622\pi$
$860$	0	0
$861$	0.683213	0.0232838
$862$	0	0
$863$	33.1976	1.13006	0.565029	−	0.825071i	$-0.308865\pi$
$863$	33.1976	1.13006	0.565029	+	0.825071i	$0.308865\pi$
$864$	0	0
$865$	21.9520	0.746392
$866$	0	0
$867$	0.668965	0.0227193
$868$	0	0
$869$	−55.9915	−1.89938
$870$	0	0
$871$	−18.1360	−0.614514
$872$	0	0
$873$	−9.81713	−0.332259
$874$	0	0
$875$	1.78556	0.0603630
$876$	0	0
$877$	7.47679	0.252473	0.126237	−	0.992000i	$-0.459710\pi$
$877$	7.47679	0.252473	0.126237	+	0.992000i	$0.459710\pi$
$878$	0	0
$879$	8.23927	0.277904
$880$	0	0
$881$	38.6078	1.30073	0.650365	−	0.759622i	$-0.274616\pi$
$881$	38.6078	1.30073	0.650365	+	0.759622i	$0.274616\pi$
$882$	0	0
$883$	9.99669	0.336416	0.168208	−	0.985752i	$-0.446202\pi$
$883$	9.99669	0.336416	0.168208	+	0.985752i	$0.446202\pi$
$884$	0	0
$885$	0.426897	0.0143500
$886$	0	0
$887$	−46.7821	−1.57079	−0.785394	−	0.618996i	$-0.787540\pi$
$887$	−46.7821	−1.57079	−0.785394	+	0.618996i	$0.787540\pi$
$888$	0	0
$889$	11.4908	0.385390
$890$	0	0
$891$	28.6951	0.961321
$892$	0	0
$893$	16.3350	0.546628
$894$	0	0
$895$	16.0237	0.535614
$896$	0	0
$897$	−1.76067	−0.0587869
$898$	0	0
$899$	88.1748	2.94080
$900$	0	0
$901$	−15.1946	−0.506204
$902$	0	0
$903$	−5.68605	−0.189220
$904$	0	0
$905$	−6.66018	−0.221392
$906$	0	0
$907$	−11.3497	−0.376860	−0.188430	−	0.982087i	$-0.560340\pi$
$907$	−11.3497	−0.376860	−0.188430	+	0.982087i	$0.560340\pi$
$908$	0	0
$909$	−8.32243	−0.276038
$910$	0	0
$911$	−19.0887	−0.632438	−0.316219	−	0.948686i	$-0.602414\pi$
$911$	−19.0887	−0.632438	−0.316219	+	0.948686i	$0.602414\pi$
$912$	0	0
$913$	−17.3599	−0.574530
$914$	0	0
$915$	−3.44324	−0.113830
$916$	0	0
$917$	20.5867	0.679832
$918$	0	0
$919$	−46.8653	−1.54594	−0.772972	−	0.634440i	$-0.781231\pi$
$919$	−46.8653	−1.54594	−0.772972	+	0.634440i	$0.781231\pi$
$920$	0	0
$921$	−10.0370	−0.330730
$922$	0	0
$923$	12.3488	0.406466
$924$	0	0
$925$	−5.97550	−0.196473
$926$	0	0
$927$	0.0971692	0.00319145
$928$	0	0
$929$	−19.8167	−0.650166	−0.325083	−	0.945685i	$-0.605392\pi$
$929$	−19.8167	−0.650166	−0.325083	+	0.945685i	$0.605392\pi$
$930$	0	0
$931$	24.7527	0.811238
$932$	0	0
$933$	−0.712796	−0.0233359
$934$	0	0
$935$	15.8529	0.518445
$936$	0	0
$937$	−37.1264	−1.21287	−0.606433	−	0.795135i	$-0.707400\pi$
$937$	−37.1264	−1.21287	−0.606433	+	0.795135i	$0.707400\pi$
$938$	0	0
$939$	−8.17326	−0.266724
$940$	0	0
$941$	27.1483	0.885008	0.442504	−	0.896767i	$-0.354090\pi$
$941$	27.1483	0.885008	0.442504	+	0.896767i	$0.354090\pi$
$942$	0	0
$943$	4.58211	0.149214
$944$	0	0
$945$	−3.75404	−0.122119
$946$	0	0
$947$	−30.7958	−1.00073	−0.500364	−	0.865815i	$-0.666801\pi$
$947$	−30.7958	−1.00073	−0.500364	+	0.865815i	$0.666801\pi$
$948$	0	0
$949$	−0.541573	−0.0175802
$950$	0	0
$951$	−5.19251	−0.168379
$952$	0	0
$953$	−47.5060	−1.53887	−0.769435	−	0.638725i	$-0.779462\pi$
$953$	−47.5060	−1.53887	−0.769435	+	0.638725i	$0.779462\pi$
$954$	0	0
$955$	15.3483	0.496658
$956$	0	0
$957$	−11.3969	−0.368410
$958$	0	0
$959$	20.2033	0.652399
$960$	0	0
$961$	71.2124	2.29717
$962$	0	0
$963$	−38.7679	−1.24928
$964$	0	0
$965$	−1.97031	−0.0634266
$966$	0	0
$967$	50.3000	1.61754	0.808770	−	0.588126i	$-0.200134\pi$
$967$	50.3000	1.61754	0.808770	+	0.588126i	$0.200134\pi$
$968$	0	0
$969$	−10.0999	−0.324455
$970$	0	0
$971$	−24.0155	−0.770693	−0.385346	−	0.922772i	$-0.625918\pi$
$971$	−24.0155	−0.770693	−0.385346	+	0.922772i	$0.625918\pi$
$972$	0	0
$973$	−36.4630	−1.16895
$974$	0	0
$975$	−0.410619	−0.0131503
$976$	0	0
$977$	−47.8530	−1.53095	−0.765476	−	0.643465i	$-0.777496\pi$
$977$	−47.8530	−1.53095	−0.765476	+	0.643465i	$0.777496\pi$
$978$	0	0
$979$	37.5638	1.20054
$980$	0	0
$981$	−16.5298	−0.527755
$982$	0	0
$983$	60.3297	1.92422	0.962110	−	0.272663i	$-0.0879044\pi$
$983$	60.3297	1.92422	0.962110	+	0.272663i	$0.0879044\pi$
$984$	0	0
$985$	−9.24337	−0.294518
$986$	0	0
$987$	−1.60823	−0.0511906
$988$	0	0
$989$	−38.1346	−1.21261
$990$	0	0
$991$	24.1114	0.765923	0.382961	−	0.923764i	$-0.374904\pi$
$991$	24.1114	0.765923	0.382961	+	0.923764i	$0.374904\pi$
$992$	0	0
$993$	4.89539	0.155351
$994$	0	0
$995$	−19.1371	−0.606687
$996$	0	0
$997$	41.3815	1.31056	0.655282	−	0.755384i	$-0.272550\pi$
$997$	41.3815	1.31056	0.655282	+	0.755384i	$0.272550\pi$
$998$	0	0
$999$	12.5631	0.397480

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8020.2.a.d.1.18	✓	29

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8020.2.a.d.1.18	✓	29	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:	\( N \)	\(=\)	\( 8020 = 2^{2} \cdot 5 \cdot 401 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8020.a (trivial)

\(f(q)\)	\(=\)	\(q+0.358058 q^{3} +1.00000 q^{5} +1.78556 q^{7} -2.87179 q^{9} +O(q^{10})\)	\(q+0.358058 q^{3} +1.00000 q^{5} +1.78556 q^{7} -2.87179 q^{9} +3.64957 q^{11} -1.14680 q^{13} +0.358058 q^{15} +4.34377 q^{17} -6.49377 q^{19} +0.639335 q^{21} +4.28783 q^{23} +1.00000 q^{25} -2.10244 q^{27} -8.72153 q^{29} -10.1100 q^{31} +1.30676 q^{33} +1.78556 q^{35} -5.97550 q^{37} -0.410619 q^{39} +1.06863 q^{41} -8.89369 q^{43} -2.87179 q^{45} -2.51548 q^{47} -3.81177 q^{49} +1.55532 q^{51} -3.49802 q^{53} +3.64957 q^{55} -2.32515 q^{57} +1.19226 q^{59} -9.61645 q^{61} -5.12777 q^{63} -1.14680 q^{65} +15.8145 q^{67} +1.53529 q^{69} -10.7681 q^{71} +0.472248 q^{73} +0.358058 q^{75} +6.51654 q^{77} -15.3419 q^{79} +7.86259 q^{81} -4.75671 q^{83} +4.34377 q^{85} -3.12281 q^{87} +10.2927 q^{89} -2.04768 q^{91} -3.61997 q^{93} -6.49377 q^{95} +3.41846 q^{97} -10.4808 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 29 q - 3 q^{3} + 29 q^{5} - 8 q^{7} + 10 q^{9}+O(q^{10}) \) 29 * q - 3 * q^3 + 29 * q^5 - 8 * q^7 + 10 * q^9	\( 29 q - 3 q^{3} + 29 q^{5} - 8 q^{7} + 10 q^{9} + 2 q^{11} - 23 q^{13} - 3 q^{15} - 30 q^{17} - 6 q^{19} - 16 q^{21} - 21 q^{23} + 29 q^{25} - 15 q^{27} - 35 q^{29} - 7 q^{31} - 36 q^{33} - 8 q^{35} - 31 q^{37} - 11 q^{39} - 24 q^{41} - 17 q^{43} + 10 q^{45} - 17 q^{47} + q^{49} + 8 q^{51} - 57 q^{53} + 2 q^{55} - 46 q^{57} - 9 q^{59} - 27 q^{61} - 34 q^{63} - 23 q^{65} - 21 q^{67} - 28 q^{69} - 19 q^{71} - 81 q^{73} - 3 q^{75} - 66 q^{77} - 17 q^{79} - 39 q^{81} - 30 q^{83} - 30 q^{85} - 20 q^{87} - 38 q^{89} + q^{91} - 75 q^{93} - 6 q^{95} - 48 q^{97} - 7 q^{99}+O(q^{100}) \) 29 * q - 3 * q^3 + 29 * q^5 - 8 * q^7 + 10 * q^9 + 2 * q^11 - 23 * q^13 - 3 * q^15 - 30 * q^17 - 6 * q^19 - 16 * q^21 - 21 * q^23 + 29 * q^25 - 15 * q^27 - 35 * q^29 - 7 * q^31 - 36 * q^33 - 8 * q^35 - 31 * q^37 - 11 * q^39 - 24 * q^41 - 17 * q^43 + 10 * q^45 - 17 * q^47 + q^49 + 8 * q^51 - 57 * q^53 + 2 * q^55 - 46 * q^57 - 9 * q^59 - 27 * q^61 - 34 * q^63 - 23 * q^65 - 21 * q^67 - 28 * q^69 - 19 * q^71 - 81 * q^73 - 3 * q^75 - 66 * q^77 - 17 * q^79 - 39 * q^81 - 30 * q^83 - 30 * q^85 - 20 * q^87 - 38 * q^89 + q^91 - 75 * q^93 - 6 * q^95 - 48 * q^97 - 7 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more