LMFDB - Embedded newform 8020.2.a.e.1.1

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:	$0$
Dimension:	$35$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
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)

$f(q)$	$=$	$q-3.41155 q^{3} -1.00000 q^{5} +0.311908 q^{7} +8.63864 q^{9} +O(q^{10})$	$q-3.41155 q^{3} -1.00000 q^{5} +0.311908 q^{7} +8.63864 q^{9} -4.73347 q^{11} -4.87677 q^{13} +3.41155 q^{15} +5.19674 q^{17} +5.28429 q^{19} -1.06409 q^{21} -7.51907 q^{23} +1.00000 q^{25} -19.2365 q^{27} +1.85359 q^{29} +4.70742 q^{31} +16.1484 q^{33} -0.311908 q^{35} -3.55202 q^{37} +16.6373 q^{39} +9.83063 q^{41} -8.90007 q^{43} -8.63864 q^{45} -2.69834 q^{47} -6.90271 q^{49} -17.7289 q^{51} +4.15416 q^{53} +4.73347 q^{55} -18.0276 q^{57} +1.04264 q^{59} -0.751334 q^{61} +2.69446 q^{63} +4.87677 q^{65} -10.6699 q^{67} +25.6516 q^{69} -11.2670 q^{71} +11.8856 q^{73} -3.41155 q^{75} -1.47641 q^{77} +1.76323 q^{79} +39.7102 q^{81} +17.7359 q^{83} -5.19674 q^{85} -6.32362 q^{87} -18.6716 q^{89} -1.52110 q^{91} -16.0596 q^{93} -5.28429 q^{95} -10.7804 q^{97} -40.8907 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 35 q - q^{3} - 35 q^{5} + 6 q^{7} + 52 q^{9}+O(q^{10}) $ 35 * q - q^3 - 35 * q^5 + 6 * q^7 + 52 * q^9	$ 35 q - q^{3} - 35 q^{5} + 6 q^{7} + 52 q^{9} - 2 q^{11} - q^{13} + q^{15} + 18 q^{17} + 2 q^{19} + 12 q^{21} + 13 q^{23} + 35 q^{25} - 7 q^{27} + 25 q^{29} + 13 q^{31} + 14 q^{33} - 6 q^{35} - 19 q^{37} - 3 q^{39} + 24 q^{41} - 5 q^{43} - 52 q^{45} + 19 q^{47} + 55 q^{49} + 41 q^{53} + 2 q^{55} + 14 q^{57} + 3 q^{59} + 13 q^{61} + 70 q^{63} + q^{65} - 17 q^{67} + 64 q^{69} + 17 q^{71} - 63 q^{73} - q^{75} + 54 q^{77} + 11 q^{79} + 107 q^{81} - 8 q^{83} - 18 q^{85} + 36 q^{87} + 38 q^{89} - 27 q^{91} + q^{93} - 2 q^{95} - 54 q^{97} - 51 q^{99}+O(q^{100}) $ 35 * q - q^3 - 35 * q^5 + 6 * q^7 + 52 * q^9 - 2 * q^11 - q^13 + q^15 + 18 * q^17 + 2 * q^19 + 12 * q^21 + 13 * q^23 + 35 * q^25 - 7 * q^27 + 25 * q^29 + 13 * q^31 + 14 * q^33 - 6 * q^35 - 19 * q^37 - 3 * q^39 + 24 * q^41 - 5 * q^43 - 52 * q^45 + 19 * q^47 + 55 * q^49 + 41 * q^53 + 2 * q^55 + 14 * q^57 + 3 * q^59 + 13 * q^61 + 70 * q^63 + q^65 - 17 * q^67 + 64 * q^69 + 17 * q^71 - 63 * q^73 - q^75 + 54 * q^77 + 11 * q^79 + 107 * q^81 - 8 * q^83 - 18 * q^85 + 36 * q^87 + 38 * q^89 - 27 * q^91 + q^93 - 2 * q^95 - 54 * 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
$3$	−3.41155	−1.96966	−0.984828	−	0.173532i	$-0.944482\pi$
$3$	−3.41155	−1.96966	−0.984828	+	0.173532i	$0.944482\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	0.311908	0.117890	0.0589450	−	0.998261i	$-0.481226\pi$
$7$	0.311908	0.117890	0.0589450	+	0.998261i	$0.481226\pi$
$8$	0	0
$9$	8.63864	2.87955
$10$	0	0
$11$	−4.73347	−1.42719	−0.713597	−	0.700556i	$-0.752935\pi$
$11$	−4.73347	−1.42719	−0.713597	+	0.700556i	$0.752935\pi$
$12$	0	0
$13$	−4.87677	−1.35257	−0.676286	−	0.736639i	$-0.736412\pi$
$13$	−4.87677	−1.35257	−0.676286	+	0.736639i	$0.736412\pi$
$14$	0	0
$15$	3.41155	0.880857
$16$	0	0
$17$	5.19674	1.26039	0.630197	−	0.776435i	$-0.282974\pi$
$17$	5.19674	1.26039	0.630197	+	0.776435i	$0.282974\pi$
$18$	0	0
$19$	5.28429	1.21230	0.606150	−	0.795350i	$-0.292713\pi$
$19$	5.28429	1.21230	0.606150	+	0.795350i	$0.292713\pi$
$20$	0	0
$21$	−1.06409	−0.232203
$22$	0	0
$23$	−7.51907	−1.56783	−0.783917	−	0.620866i	$-0.786781\pi$
$23$	−7.51907	−1.56783	−0.783917	+	0.620866i	$0.786781\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−19.2365	−3.70206
$28$	0	0
$29$	1.85359	0.344204	0.172102	−	0.985079i	$-0.444944\pi$
$29$	1.85359	0.344204	0.172102	+	0.985079i	$0.444944\pi$
$30$	0	0
$31$	4.70742	0.845477	0.422738	−	0.906252i	$-0.361069\pi$
$31$	4.70742	0.845477	0.422738	+	0.906252i	$0.361069\pi$
$32$	0	0
$33$	16.1484	2.81108
$34$	0	0
$35$	−0.311908	−0.0527220
$36$	0	0
$37$	−3.55202	−0.583948	−0.291974	−	0.956426i	$-0.594312\pi$
$37$	−3.55202	−0.583948	−0.291974	+	0.956426i	$0.594312\pi$
$38$	0	0
$39$	16.6373	2.66410
$40$	0	0
$41$	9.83063	1.53529	0.767644	−	0.640877i	$-0.221429\pi$
$41$	9.83063	1.53529	0.767644	+	0.640877i	$0.221429\pi$
$42$	0	0
$43$	−8.90007	−1.35725	−0.678624	−	0.734486i	$-0.737423\pi$
$43$	−8.90007	−1.35725	−0.678624	+	0.734486i	$0.737423\pi$
$44$	0	0
$45$	−8.63864	−1.28777
$46$	0	0
$47$	−2.69834	−0.393594	−0.196797	−	0.980444i	$-0.563054\pi$
$47$	−2.69834	−0.393594	−0.196797	+	0.980444i	$0.563054\pi$
$48$	0	0
$49$	−6.90271	−0.986102
$50$	0	0
$51$	−17.7289	−2.48254
$52$	0	0
$53$	4.15416	0.570618	0.285309	−	0.958436i	$-0.407904\pi$
$53$	4.15416	0.570618	0.285309	+	0.958436i	$0.407904\pi$
$54$	0	0
$55$	4.73347	0.638261
$56$	0	0
$57$	−18.0276	−2.38781
$58$	0	0
$59$	1.04264	0.135740	0.0678699	−	0.997694i	$-0.478380\pi$
$59$	1.04264	0.135740	0.0678699	+	0.997694i	$0.478380\pi$
$60$	0	0
$61$	−0.751334	−0.0961985	−0.0480993	−	0.998843i	$-0.515316\pi$
$61$	−0.751334	−0.0961985	−0.0480993	+	0.998843i	$0.515316\pi$
$62$	0	0
$63$	2.69446	0.339470
$64$	0	0
$65$	4.87677	0.604889
$66$	0	0
$67$	−10.6699	−1.30354	−0.651768	−	0.758418i	$-0.725973\pi$
$67$	−10.6699	−1.30354	−0.651768	+	0.758418i	$0.725973\pi$
$68$	0	0
$69$	25.6516	3.08809
$70$	0	0
$71$	−11.2670	−1.33715	−0.668575	−	0.743645i	$-0.733095\pi$
$71$	−11.2670	−1.33715	−0.668575	+	0.743645i	$0.733095\pi$
$72$	0	0
$73$	11.8856	1.39110	0.695550	−	0.718478i	$-0.255161\pi$
$73$	11.8856	1.39110	0.695550	+	0.718478i	$0.255161\pi$
$74$	0	0
$75$	−3.41155	−0.393931
$76$	0	0
$77$	−1.47641	−0.168252
$78$	0	0
$79$	1.76323	0.198379	0.0991897	−	0.995069i	$-0.468375\pi$
$79$	1.76323	0.198379	0.0991897	+	0.995069i	$0.468375\pi$
$80$	0	0
$81$	39.7102	4.41224
$82$	0	0
$83$	17.7359	1.94677	0.973383	−	0.229186i	$-0.0736064\pi$
$83$	17.7359	1.94677	0.973383	+	0.229186i	$0.0736064\pi$
$84$	0	0
$85$	−5.19674	−0.563665
$86$	0	0
$87$	−6.32362	−0.677963
$88$	0	0
$89$	−18.6716	−1.97919	−0.989593	−	0.143892i	$-0.954038\pi$
$89$	−18.6716	−1.97919	−0.989593	+	0.143892i	$0.954038\pi$
$90$	0	0
$91$	−1.52110	−0.159455
$92$	0	0
$93$	−16.0596	−1.66530
$94$	0	0
$95$	−5.28429	−0.542157
$96$	0	0
$97$	−10.7804	−1.09458	−0.547292	−	0.836942i	$-0.684341\pi$
$97$	−10.7804	−1.09458	−0.547292	+	0.836942i	$0.684341\pi$
$98$	0	0
$99$	−40.8907	−4.10967
$100$	0	0
$101$	−12.8221	−1.27585	−0.637924	−	0.770099i	$-0.720207\pi$
$101$	−12.8221	−1.27585	−0.637924	+	0.770099i	$0.720207\pi$
$102$	0	0
$103$	5.42910	0.534945	0.267473	−	0.963565i	$-0.413811\pi$
$103$	5.42910	0.534945	0.267473	+	0.963565i	$0.413811\pi$
$104$	0	0
$105$	1.06409	0.103844
$106$	0	0
$107$	13.7236	1.32671	0.663357	−	0.748303i	$-0.269131\pi$
$107$	13.7236	1.32671	0.663357	+	0.748303i	$0.269131\pi$
$108$	0	0
$109$	15.5586	1.49025	0.745124	−	0.666926i	$-0.232390\pi$
$109$	15.5586	1.49025	0.745124	+	0.666926i	$0.232390\pi$
$110$	0	0
$111$	12.1179	1.15018
$112$	0	0
$113$	−6.81005	−0.640636	−0.320318	−	0.947310i	$-0.603790\pi$
$113$	−6.81005	−0.640636	−0.320318	+	0.947310i	$0.603790\pi$
$114$	0	0
$115$	7.51907	0.701157
$116$	0	0
$117$	−42.1287	−3.89480
$118$	0	0
$119$	1.62090	0.148588
$120$	0	0
$121$	11.4057	1.03688
$122$	0	0
$123$	−33.5377	−3.02399
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−14.1749	−1.25782	−0.628908	−	0.777480i	$-0.716498\pi$
$127$	−14.1749	−1.25782	−0.628908	+	0.777480i	$0.716498\pi$
$128$	0	0
$129$	30.3630	2.67331
$130$	0	0
$131$	−15.9757	−1.39580	−0.697902	−	0.716193i	$-0.745883\pi$
$131$	−15.9757	−1.39580	−0.697902	+	0.716193i	$0.745883\pi$
$132$	0	0
$133$	1.64821	0.142918
$134$	0	0
$135$	19.2365	1.65561
$136$	0	0
$137$	2.20823	0.188661	0.0943307	−	0.995541i	$-0.469929\pi$
$137$	2.20823	0.188661	0.0943307	+	0.995541i	$0.469929\pi$
$138$	0	0
$139$	−10.7834	−0.914640	−0.457320	−	0.889302i	$-0.651191\pi$
$139$	−10.7834	−0.914640	−0.457320	+	0.889302i	$0.651191\pi$
$140$	0	0
$141$	9.20552	0.775244
$142$	0	0
$143$	23.0840	1.93038
$144$	0	0
$145$	−1.85359	−0.153933
$146$	0	0
$147$	23.5489	1.94228
$148$	0	0
$149$	−6.40792	−0.524957	−0.262479	−	0.964938i	$-0.584540\pi$
$149$	−6.40792	−0.524957	−0.262479	+	0.964938i	$0.584540\pi$
$150$	0	0
$151$	14.4505	1.17596	0.587981	−	0.808875i	$-0.299923\pi$
$151$	14.4505	1.17596	0.587981	+	0.808875i	$0.299923\pi$
$152$	0	0
$153$	44.8927	3.62936
$154$	0	0
$155$	−4.70742	−0.378109
$156$	0	0
$157$	−20.4931	−1.63553	−0.817765	−	0.575553i	$-0.804787\pi$
$157$	−20.4931	−1.63553	−0.817765	+	0.575553i	$0.804787\pi$
$158$	0	0
$159$	−14.1721	−1.12392
$160$	0	0
$161$	−2.34526	−0.184832
$162$	0	0
$163$	−16.5857	−1.29909	−0.649546	−	0.760322i	$-0.725041\pi$
$163$	−16.5857	−1.29909	−0.649546	+	0.760322i	$0.725041\pi$
$164$	0	0
$165$	−16.1484	−1.25715
$166$	0	0
$167$	−3.34838	−0.259106	−0.129553	−	0.991573i	$-0.541354\pi$
$167$	−3.34838	−0.259106	−0.129553	+	0.991573i	$0.541354\pi$
$168$	0	0
$169$	10.7829	0.829452
$170$	0	0
$171$	45.6491	3.49088
$172$	0	0
$173$	−14.7337	−1.12018	−0.560091	−	0.828431i	$-0.689234\pi$
$173$	−14.7337	−1.12018	−0.560091	+	0.828431i	$0.689234\pi$
$174$	0	0
$175$	0.311908	0.0235780
$176$	0	0
$177$	−3.55700	−0.267361
$178$	0	0
$179$	4.80761	0.359338	0.179669	−	0.983727i	$-0.442497\pi$
$179$	4.80761	0.359338	0.179669	+	0.983727i	$0.442497\pi$
$180$	0	0
$181$	−11.6321	−0.864606	−0.432303	−	0.901728i	$-0.642299\pi$
$181$	−11.6321	−0.864606	−0.432303	+	0.901728i	$0.642299\pi$
$182$	0	0
$183$	2.56321	0.189478
$184$	0	0
$185$	3.55202	0.261150
$186$	0	0
$187$	−24.5986	−1.79883
$188$	0	0
$189$	−6.00001	−0.436436
$190$	0	0
$191$	−5.43944	−0.393584	−0.196792	−	0.980445i	$-0.563052\pi$
$191$	−5.43944	−0.393584	−0.196792	+	0.980445i	$0.563052\pi$
$192$	0	0
$193$	−5.66031	−0.407438	−0.203719	−	0.979029i	$-0.565303\pi$
$193$	−5.66031	−0.407438	−0.203719	+	0.979029i	$0.565303\pi$
$194$	0	0
$195$	−16.6373	−1.19142
$196$	0	0
$197$	14.0788	1.00308	0.501538	−	0.865135i	$-0.332768\pi$
$197$	14.0788	1.00308	0.501538	+	0.865135i	$0.332768\pi$
$198$	0	0
$199$	−23.6994	−1.68001	−0.840004	−	0.542581i	$-0.817447\pi$
$199$	−23.6994	−1.68001	−0.840004	+	0.542581i	$0.817447\pi$
$200$	0	0
$201$	36.4009	2.56752
$202$	0	0
$203$	0.578150	0.0405782
$204$	0	0
$205$	−9.83063	−0.686601
$206$	0	0
$207$	−64.9545	−4.51465
$208$	0	0
$209$	−25.0130	−1.73019
$210$	0	0
$211$	16.2113	1.11604	0.558018	−	0.829829i	$-0.311562\pi$
$211$	16.2113	1.11604	0.558018	+	0.829829i	$0.311562\pi$
$212$	0	0
$213$	38.4380	2.63373
$214$	0	0
$215$	8.90007	0.606980
$216$	0	0
$217$	1.46828	0.0996733
$218$	0	0
$219$	−40.5481	−2.73999
$220$	0	0
$221$	−25.3433	−1.70477
$222$	0	0
$223$	11.6982	0.783372	0.391686	−	0.920099i	$-0.371892\pi$
$223$	11.6982	0.783372	0.391686	+	0.920099i	$0.371892\pi$
$224$	0	0
$225$	8.63864	0.575909
$226$	0	0
$227$	−4.18102	−0.277504	−0.138752	−	0.990327i	$-0.544309\pi$
$227$	−4.18102	−0.277504	−0.138752	+	0.990327i	$0.544309\pi$
$228$	0	0
$229$	12.1594	0.803516	0.401758	−	0.915746i	$-0.368399\pi$
$229$	12.1594	0.803516	0.401758	+	0.915746i	$0.368399\pi$
$230$	0	0
$231$	5.03682	0.331399
$232$	0	0
$233$	10.6342	0.696667	0.348333	−	0.937371i	$-0.386748\pi$
$233$	10.6342	0.696667	0.348333	+	0.937371i	$0.386748\pi$
$234$	0	0
$235$	2.69834	0.176020
$236$	0	0
$237$	−6.01535	−0.390739
$238$	0	0
$239$	20.8285	1.34728	0.673640	−	0.739059i	$-0.264730\pi$
$239$	20.8285	1.34728	0.673640	+	0.739059i	$0.264730\pi$
$240$	0	0
$241$	−22.8053	−1.46902	−0.734508	−	0.678600i	$-0.762587\pi$
$241$	−22.8053	−1.46902	−0.734508	+	0.678600i	$0.762587\pi$
$242$	0	0
$243$	−77.7637	−4.98854
$244$	0	0
$245$	6.90271	0.440998
$246$	0	0
$247$	−25.7703	−1.63972
$248$	0	0
$249$	−60.5067	−3.83446
$250$	0	0
$251$	−15.2831	−0.964662	−0.482331	−	0.875989i	$-0.660210\pi$
$251$	−15.2831	−0.964662	−0.482331	+	0.875989i	$0.660210\pi$
$252$	0	0
$253$	35.5913	2.23760
$254$	0	0
$255$	17.7289	1.11023
$256$	0	0
$257$	−8.83881	−0.551350	−0.275675	−	0.961251i	$-0.588901\pi$
$257$	−8.83881	−0.551350	−0.275675	+	0.961251i	$0.588901\pi$
$258$	0	0
$259$	−1.10790	−0.0688417
$260$	0	0
$261$	16.0125	0.991151
$262$	0	0
$263$	−11.0608	−0.682035	−0.341018	−	0.940057i	$-0.610772\pi$
$263$	−11.0608	−0.682035	−0.341018	+	0.940057i	$0.610772\pi$
$264$	0	0
$265$	−4.15416	−0.255188
$266$	0	0
$267$	63.6990	3.89832
$268$	0	0
$269$	0.950123	0.0579300	0.0289650	−	0.999580i	$-0.490779\pi$
$269$	0.950123	0.0579300	0.0289650	+	0.999580i	$0.490779\pi$
$270$	0	0
$271$	−1.54506	−0.0938556	−0.0469278	−	0.998898i	$-0.514943\pi$
$271$	−1.54506	−0.0938556	−0.0469278	+	0.998898i	$0.514943\pi$
$272$	0	0
$273$	5.18931	0.314071
$274$	0	0
$275$	−4.73347	−0.285439
$276$	0	0
$277$	−2.25449	−0.135459	−0.0677297	−	0.997704i	$-0.521576\pi$
$277$	−2.25449	−0.135459	−0.0677297	+	0.997704i	$0.521576\pi$
$278$	0	0
$279$	40.6657	2.43459
$280$	0	0
$281$	32.6279	1.94642	0.973209	−	0.229922i	$-0.0738472\pi$
$281$	32.6279	1.94642	0.973209	+	0.229922i	$0.0738472\pi$
$282$	0	0
$283$	−29.7600	−1.76905	−0.884524	−	0.466494i	$-0.845517\pi$
$283$	−29.7600	−1.76905	−0.884524	+	0.466494i	$0.845517\pi$
$284$	0	0
$285$	18.0276	1.06786
$286$	0	0
$287$	3.06625	0.180995
$288$	0	0
$289$	10.0061	0.588592
$290$	0	0
$291$	36.7778	2.15595
$292$	0	0
$293$	27.3003	1.59490	0.797452	−	0.603383i	$-0.206181\pi$
$293$	27.3003	1.59490	0.797452	+	0.603383i	$0.206181\pi$
$294$	0	0
$295$	−1.04264	−0.0607046
$296$	0	0
$297$	91.0553	5.28356
$298$	0	0
$299$	36.6688	2.12061
$300$	0	0
$301$	−2.77600	−0.160006
$302$	0	0
$303$	43.7433	2.51298
$304$	0	0
$305$	0.751334	0.0430213
$306$	0	0
$307$	11.7917	0.672989	0.336494	−	0.941685i	$-0.390759\pi$
$307$	11.7917	0.672989	0.336494	+	0.941685i	$0.390759\pi$
$308$	0	0
$309$	−18.5216	−1.05366
$310$	0	0
$311$	23.8677	1.35341	0.676707	−	0.736252i	$-0.263406\pi$
$311$	23.8677	1.35341	0.676707	+	0.736252i	$0.263406\pi$
$312$	0	0
$313$	19.5820	1.10684	0.553420	−	0.832902i	$-0.313322\pi$
$313$	19.5820	1.10684	0.553420	+	0.832902i	$0.313322\pi$
$314$	0	0
$315$	−2.69446	−0.151816
$316$	0	0
$317$	20.3451	1.14269	0.571347	−	0.820709i	$-0.306421\pi$
$317$	20.3451	1.14269	0.571347	+	0.820709i	$0.306421\pi$
$318$	0	0
$319$	−8.77393	−0.491246
$320$	0	0
$321$	−46.8188	−2.61317
$322$	0	0
$323$	27.4611	1.52798
$324$	0	0
$325$	−4.87677	−0.270514
$326$	0	0
$327$	−53.0790	−2.93528
$328$	0	0
$329$	−0.841634	−0.0464008
$330$	0	0
$331$	1.27137	0.0698808	0.0349404	−	0.999389i	$-0.488876\pi$
$331$	1.27137	0.0698808	0.0349404	+	0.999389i	$0.488876\pi$
$332$	0	0
$333$	−30.6846	−1.68151
$334$	0	0
$335$	10.6699	0.582959
$336$	0	0
$337$	−18.8172	−1.02504	−0.512518	−	0.858677i	$-0.671287\pi$
$337$	−18.8172	−1.02504	−0.512518	+	0.858677i	$0.671287\pi$
$338$	0	0
$339$	23.2328	1.26183
$340$	0	0
$341$	−22.2824	−1.20666
$342$	0	0
$343$	−4.33637	−0.234142
$344$	0	0
$345$	−25.6516	−1.38104
$346$	0	0
$347$	−18.3009	−0.982444	−0.491222	−	0.871034i	$-0.663450\pi$
$347$	−18.3009	−0.982444	−0.491222	+	0.871034i	$0.663450\pi$
$348$	0	0
$349$	28.8632	1.54501	0.772505	−	0.635009i	$-0.219004\pi$
$349$	28.8632	1.54501	0.772505	+	0.635009i	$0.219004\pi$
$350$	0	0
$351$	93.8119	5.00731
$352$	0	0
$353$	23.7625	1.26475	0.632376	−	0.774662i	$-0.282080\pi$
$353$	23.7625	1.26475	0.632376	+	0.774662i	$0.282080\pi$
$354$	0	0
$355$	11.2670	0.597992
$356$	0	0
$357$	−5.52978	−0.292667
$358$	0	0
$359$	23.2845	1.22891	0.614455	−	0.788952i	$-0.289376\pi$
$359$	23.2845	1.22891	0.614455	+	0.788952i	$0.289376\pi$
$360$	0	0
$361$	8.92376	0.469672
$362$	0	0
$363$	−38.9111	−2.04230
$364$	0	0
$365$	−11.8856	−0.622119
$366$	0	0
$367$	15.8171	0.825646	0.412823	−	0.910811i	$-0.364543\pi$
$367$	15.8171	0.825646	0.412823	+	0.910811i	$0.364543\pi$
$368$	0	0
$369$	84.9233	4.42093
$370$	0	0
$371$	1.29571	0.0672702
$372$	0	0
$373$	27.7692	1.43783	0.718916	−	0.695097i	$-0.244638\pi$
$373$	27.7692	1.43783	0.718916	+	0.695097i	$0.244638\pi$
$374$	0	0
$375$	3.41155	0.176171
$376$	0	0
$377$	−9.03955	−0.465560
$378$	0	0
$379$	1.51071	0.0776000	0.0388000	−	0.999247i	$-0.487646\pi$
$379$	1.51071	0.0776000	0.0388000	+	0.999247i	$0.487646\pi$
$380$	0	0
$381$	48.3582	2.47747
$382$	0	0
$383$	27.5455	1.40751	0.703754	−	0.710444i	$-0.251506\pi$
$383$	27.5455	1.40751	0.703754	+	0.710444i	$0.251506\pi$
$384$	0	0
$385$	1.47641	0.0752446
$386$	0	0
$387$	−76.8845	−3.90826
$388$	0	0
$389$	2.08372	0.105649	0.0528244	−	0.998604i	$-0.483178\pi$
$389$	2.08372	0.105649	0.0528244	+	0.998604i	$0.483178\pi$
$390$	0	0
$391$	−39.0746	−1.97609
$392$	0	0
$393$	54.5018	2.74925
$394$	0	0
$395$	−1.76323	−0.0887179
$396$	0	0
$397$	15.6843	0.787173	0.393587	−	0.919288i	$-0.371234\pi$
$397$	15.6843	0.787173	0.393587	+	0.919288i	$0.371234\pi$
$398$	0	0
$399$	−5.62295	−0.281500
$400$	0	0
$401$	−1.00000	−0.0499376
$401$	−1.00000	−0.0499376
$402$	0	0
$403$	−22.9570	−1.14357
$404$	0	0
$405$	−39.7102	−1.97322
$406$	0	0
$407$	16.8134	0.833408
$408$	0	0
$409$	−21.5826	−1.06719	−0.533596	−	0.845740i	$-0.679160\pi$
$409$	−21.5826	−1.06719	−0.533596	+	0.845740i	$0.679160\pi$
$410$	0	0
$411$	−7.53346	−0.371598
$412$	0	0
$413$	0.325206	0.0160024
$414$	0	0
$415$	−17.7359	−0.870620
$416$	0	0
$417$	36.7882	1.80153
$418$	0	0
$419$	−3.95843	−0.193382	−0.0966909	−	0.995314i	$-0.530826\pi$
$419$	−3.95843	−0.193382	−0.0966909	+	0.995314i	$0.530826\pi$
$420$	0	0
$421$	15.2096	0.741269	0.370635	−	0.928779i	$-0.379140\pi$
$421$	15.2096	0.741269	0.370635	+	0.928779i	$0.379140\pi$
$422$	0	0
$423$	−23.3100	−1.13337
$424$	0	0
$425$	5.19674	0.252079
$426$	0	0
$427$	−0.234347	−0.0113408
$428$	0	0
$429$	−78.7522	−3.80219
$430$	0	0
$431$	11.1348	0.536343	0.268172	−	0.963371i	$-0.413581\pi$
$431$	11.1348	0.536343	0.268172	+	0.963371i	$0.413581\pi$
$432$	0	0
$433$	−5.90258	−0.283660	−0.141830	−	0.989891i	$-0.545299\pi$
$433$	−5.90258	−0.283660	−0.141830	+	0.989891i	$0.545299\pi$
$434$	0	0
$435$	6.32362	0.303194
$436$	0	0
$437$	−39.7330	−1.90069
$438$	0	0
$439$	4.22636	0.201713	0.100857	−	0.994901i	$-0.467842\pi$
$439$	4.22636	0.201713	0.100857	+	0.994901i	$0.467842\pi$
$440$	0	0
$441$	−59.6301	−2.83953
$442$	0	0
$443$	16.9870	0.807077	0.403538	−	0.914963i	$-0.367780\pi$
$443$	16.9870	0.807077	0.403538	+	0.914963i	$0.367780\pi$
$444$	0	0
$445$	18.6716	0.885119
$446$	0	0
$447$	21.8609	1.03399
$448$	0	0
$449$	−21.6565	−1.02203	−0.511016	−	0.859571i	$-0.670731\pi$
$449$	−21.6565	−1.02203	−0.511016	+	0.859571i	$0.670731\pi$
$450$	0	0
$451$	−46.5330	−2.19115
$452$	0	0
$453$	−49.2984	−2.31624
$454$	0	0
$455$	1.52110	0.0713104
$456$	0	0
$457$	−14.1961	−0.664063	−0.332032	−	0.943268i	$-0.607734\pi$
$457$	−14.1961	−0.664063	−0.332032	+	0.943268i	$0.607734\pi$
$458$	0	0
$459$	−99.9669	−4.66606
$460$	0	0
$461$	36.7167	1.71007	0.855033	−	0.518574i	$-0.173537\pi$
$461$	36.7167	1.71007	0.855033	+	0.518574i	$0.173537\pi$
$462$	0	0
$463$	6.44794	0.299661	0.149831	−	0.988712i	$-0.452127\pi$
$463$	6.44794	0.299661	0.149831	+	0.988712i	$0.452127\pi$
$464$	0	0
$465$	16.0596	0.744744
$466$	0	0
$467$	−17.6633	−0.817358	−0.408679	−	0.912678i	$-0.634010\pi$
$467$	−17.6633	−0.817358	−0.408679	+	0.912678i	$0.634010\pi$
$468$	0	0
$469$	−3.32803	−0.153674
$470$	0	0
$471$	69.9132	3.22143
$472$	0	0
$473$	42.1282	1.93706
$474$	0	0
$475$	5.28429	0.242460
$476$	0	0
$477$	35.8863	1.64312
$478$	0	0
$479$	9.94573	0.454432	0.227216	−	0.973844i	$-0.427038\pi$
$479$	9.94573	0.454432	0.227216	+	0.973844i	$0.427038\pi$
$480$	0	0
$481$	17.3224	0.789832
$482$	0	0
$483$	8.00095	0.364056
$484$	0	0
$485$	10.7804	0.489513
$486$	0	0
$487$	−4.01032	−0.181725	−0.0908624	−	0.995863i	$-0.528962\pi$
$487$	−4.01032	−0.181725	−0.0908624	+	0.995863i	$0.528962\pi$
$488$	0	0
$489$	56.5829	2.55877
$490$	0	0
$491$	−15.1764	−0.684901	−0.342450	−	0.939536i	$-0.611257\pi$
$491$	−15.1764	−0.684901	−0.342450	+	0.939536i	$0.611257\pi$
$492$	0	0
$493$	9.63264	0.433832
$494$	0	0
$495$	40.8907	1.83790
$496$	0	0
$497$	−3.51427	−0.157637
$498$	0	0
$499$	−3.90140	−0.174651	−0.0873253	−	0.996180i	$-0.527832\pi$
$499$	−3.90140	−0.174651	−0.0873253	+	0.996180i	$0.527832\pi$
$500$	0	0
$501$	11.4232	0.510349
$502$	0	0
$503$	2.67524	0.119283	0.0596415	−	0.998220i	$-0.481004\pi$
$503$	2.67524	0.119283	0.0596415	+	0.998220i	$0.481004\pi$
$504$	0	0
$505$	12.8221	0.570577
$506$	0	0
$507$	−36.7863	−1.63374
$508$	0	0
$509$	26.4985	1.17453	0.587263	−	0.809396i	$-0.300206\pi$
$509$	26.4985	1.17453	0.587263	+	0.809396i	$0.300206\pi$
$510$	0	0
$511$	3.70720	0.163997
$512$	0	0
$513$	−101.651	−4.48801
$514$	0	0
$515$	−5.42910	−0.239235
$516$	0	0
$517$	12.7725	0.561735
$518$	0	0
$519$	50.2646	2.20637
$520$	0	0
$521$	15.4949	0.678845	0.339422	−	0.940634i	$-0.389768\pi$
$521$	15.4949	0.678845	0.339422	+	0.940634i	$0.389768\pi$
$522$	0	0
$523$	−8.72507	−0.381521	−0.190760	−	0.981637i	$-0.561095\pi$
$523$	−8.72507	−0.381521	−0.190760	+	0.981637i	$0.561095\pi$
$524$	0	0
$525$	−1.06409	−0.0464406
$526$	0	0
$527$	24.4632	1.06563
$528$	0	0
$529$	33.5364	1.45810
$530$	0	0
$531$	9.00696	0.390869
$532$	0	0
$533$	−47.9417	−2.07659
$534$	0	0
$535$	−13.7236	−0.593325
$536$	0	0
$537$	−16.4014	−0.707772
$538$	0	0
$539$	32.6738	1.40736
$540$	0	0
$541$	12.9113	0.555099	0.277549	−	0.960711i	$-0.410478\pi$
$541$	12.9113	0.555099	0.277549	+	0.960711i	$0.410478\pi$
$542$	0	0
$543$	39.6834	1.70298
$544$	0	0
$545$	−15.5586	−0.666459
$546$	0	0
$547$	−6.99189	−0.298952	−0.149476	−	0.988765i	$-0.547759\pi$
$547$	−6.99189	−0.298952	−0.149476	+	0.988765i	$0.547759\pi$
$548$	0	0
$549$	−6.49051	−0.277008
$550$	0	0
$551$	9.79493	0.417278
$552$	0	0
$553$	0.549967	0.0233870
$554$	0	0
$555$	−12.1179	−0.514375
$556$	0	0
$557$	19.4728	0.825089	0.412544	−	0.910938i	$-0.364640\pi$
$557$	19.4728	0.825089	0.412544	+	0.910938i	$0.364640\pi$
$558$	0	0
$559$	43.4036	1.83578
$560$	0	0
$561$	83.9192	3.54307
$562$	0	0
$563$	33.1829	1.39849	0.699245	−	0.714882i	$-0.253519\pi$
$563$	33.1829	1.39849	0.699245	+	0.714882i	$0.253519\pi$
$564$	0	0
$565$	6.81005	0.286501
$566$	0	0
$567$	12.3859	0.520160
$568$	0	0
$569$	18.3338	0.768593	0.384296	−	0.923210i	$-0.374444\pi$
$569$	18.3338	0.768593	0.384296	+	0.923210i	$0.374444\pi$
$570$	0	0
$571$	−35.8641	−1.50087	−0.750433	−	0.660947i	$-0.770155\pi$
$571$	−35.8641	−1.50087	−0.750433	+	0.660947i	$0.770155\pi$
$572$	0	0
$573$	18.5569	0.775226
$574$	0	0
$575$	−7.51907	−0.313567
$576$	0	0
$577$	12.3705	0.514991	0.257495	−	0.966280i	$-0.417103\pi$
$577$	12.3705	0.514991	0.257495	+	0.966280i	$0.417103\pi$
$578$	0	0
$579$	19.3104	0.802512
$580$	0	0
$581$	5.53196	0.229504
$582$	0	0
$583$	−19.6636	−0.814382
$584$	0	0
$585$	42.1287	1.74181
$586$	0	0
$587$	−21.2216	−0.875908	−0.437954	−	0.898998i	$-0.644297\pi$
$587$	−21.2216	−0.875908	−0.437954	+	0.898998i	$0.644297\pi$
$588$	0	0
$589$	24.8754	1.02497
$590$	0	0
$591$	−48.0306	−1.97572
$592$	0	0
$593$	−46.6732	−1.91664	−0.958319	−	0.285700i	$-0.907774\pi$
$593$	−46.6732	−1.91664	−0.958319	+	0.285700i	$0.907774\pi$
$594$	0	0
$595$	−1.62090	−0.0664505
$596$	0	0
$597$	80.8517	3.30904
$598$	0	0
$599$	10.3008	0.420881	0.210440	−	0.977607i	$-0.432510\pi$
$599$	10.3008	0.420881	0.210440	+	0.977607i	$0.432510\pi$
$600$	0	0
$601$	9.99305	0.407625	0.203813	−	0.979010i	$-0.434667\pi$
$601$	9.99305	0.407625	0.203813	+	0.979010i	$0.434667\pi$
$602$	0	0
$603$	−92.1735	−3.75360
$604$	0	0
$605$	−11.4057	−0.463708
$606$	0	0
$607$	12.4057	0.503530	0.251765	−	0.967788i	$-0.418989\pi$
$607$	12.4057	0.503530	0.251765	+	0.967788i	$0.418989\pi$
$608$	0	0
$609$	−1.97239	−0.0799251
$610$	0	0
$611$	13.1592	0.532364
$612$	0	0
$613$	42.3291	1.70966	0.854828	−	0.518911i	$-0.173662\pi$
$613$	42.3291	1.70966	0.854828	+	0.518911i	$0.173662\pi$
$614$	0	0
$615$	33.5377	1.35237
$616$	0	0
$617$	39.8578	1.60461	0.802307	−	0.596912i	$-0.203606\pi$
$617$	39.8578	1.60461	0.802307	+	0.596912i	$0.203606\pi$
$618$	0	0
$619$	42.1834	1.69549	0.847747	−	0.530401i	$-0.177959\pi$
$619$	42.1834	1.69549	0.847747	+	0.530401i	$0.177959\pi$
$620$	0	0
$621$	144.640	5.80422
$622$	0	0
$623$	−5.82382	−0.233327
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	85.3331	3.40788
$628$	0	0
$629$	−18.4589	−0.736005
$630$	0	0
$631$	5.98050	0.238080	0.119040	−	0.992889i	$-0.462018\pi$
$631$	5.98050	0.238080	0.119040	+	0.992889i	$0.462018\pi$
$632$	0	0
$633$	−55.3058	−2.19821
$634$	0	0
$635$	14.1749	0.562513
$636$	0	0
$637$	33.6629	1.33377
$638$	0	0
$639$	−97.3318	−3.85039
$640$	0	0
$641$	17.2720	0.682203	0.341101	−	0.940027i	$-0.389200\pi$
$641$	17.2720	0.682203	0.341101	+	0.940027i	$0.389200\pi$
$642$	0	0
$643$	−8.41920	−0.332021	−0.166010	−	0.986124i	$-0.553089\pi$
$643$	−8.41920	−0.332021	−0.166010	+	0.986124i	$0.553089\pi$
$644$	0	0
$645$	−30.3630	−1.19554
$646$	0	0
$647$	9.10698	0.358032	0.179016	−	0.983846i	$-0.442709\pi$
$647$	9.10698	0.358032	0.179016	+	0.983846i	$0.442709\pi$
$648$	0	0
$649$	−4.93529	−0.193727
$650$	0	0
$651$	−5.00910	−0.196322
$652$	0	0
$653$	−47.6648	−1.86527	−0.932634	−	0.360823i	$-0.882496\pi$
$653$	−47.6648	−1.86527	−0.932634	+	0.360823i	$0.882496\pi$
$654$	0	0
$655$	15.9757	0.624222
$656$	0	0
$657$	102.675	4.00574
$658$	0	0
$659$	−9.51371	−0.370602	−0.185301	−	0.982682i	$-0.559326\pi$
$659$	−9.51371	−0.370602	−0.185301	+	0.982682i	$0.559326\pi$
$660$	0	0
$661$	7.58890	0.295174	0.147587	−	0.989049i	$-0.452849\pi$
$661$	7.58890	0.295174	0.147587	+	0.989049i	$0.452849\pi$
$662$	0	0
$663$	86.4598	3.35782
$664$	0	0
$665$	−1.64821	−0.0639149
$666$	0	0
$667$	−13.9373	−0.539654
$668$	0	0
$669$	−39.9091	−1.54297
$670$	0	0
$671$	3.55642	0.137294
$672$	0	0
$673$	−35.7465	−1.37793	−0.688964	−	0.724796i	$-0.741934\pi$
$673$	−35.7465	−1.37793	−0.688964	+	0.724796i	$0.741934\pi$
$674$	0	0
$675$	−19.2365	−0.740412
$676$	0	0
$677$	20.3137	0.780719	0.390360	−	0.920662i	$-0.372351\pi$
$677$	20.3137	0.780719	0.390360	+	0.920662i	$0.372351\pi$
$678$	0	0
$679$	−3.36249	−0.129041
$680$	0	0
$681$	14.2637	0.546587
$682$	0	0
$683$	4.28470	0.163950	0.0819748	−	0.996634i	$-0.473877\pi$
$683$	4.28470	0.163950	0.0819748	+	0.996634i	$0.473877\pi$
$684$	0	0
$685$	−2.20823	−0.0843720
$686$	0	0
$687$	−41.4823	−1.58265
$688$	0	0
$689$	−20.2589	−0.771802
$690$	0	0
$691$	15.0391	0.572115	0.286058	−	0.958212i	$-0.407655\pi$
$691$	15.0391	0.572115	0.286058	+	0.958212i	$0.407655\pi$
$692$	0	0
$693$	−12.7541	−0.484490
$694$	0	0
$695$	10.7834	0.409040
$696$	0	0
$697$	51.0872	1.93507
$698$	0	0
$699$	−36.2789	−1.37219
$700$	0	0
$701$	43.7329	1.65177	0.825885	−	0.563839i	$-0.190676\pi$
$701$	43.7329	1.65177	0.825885	+	0.563839i	$0.190676\pi$
$702$	0	0
$703$	−18.7699	−0.707921
$704$	0	0
$705$	−9.20552	−0.346700
$706$	0	0
$707$	−3.99932	−0.150410
$708$	0	0
$709$	21.5729	0.810188	0.405094	−	0.914275i	$-0.367239\pi$
$709$	21.5729	0.810188	0.405094	+	0.914275i	$0.367239\pi$
$710$	0	0
$711$	15.2319	0.571243
$712$	0	0
$713$	−35.3954	−1.32557
$714$	0	0
$715$	−23.0840	−0.863294
$716$	0	0
$717$	−71.0572	−2.65368
$718$	0	0
$719$	30.1888	1.12585	0.562927	−	0.826507i	$-0.309675\pi$
$719$	30.1888	1.12585	0.562927	+	0.826507i	$0.309675\pi$
$720$	0	0
$721$	1.69338	0.0630648
$722$	0	0
$723$	77.8012	2.89346
$724$	0	0
$725$	1.85359	0.0688407
$726$	0	0
$727$	−27.5553	−1.02197	−0.510986	−	0.859589i	$-0.670719\pi$
$727$	−27.5553	−1.02197	−0.510986	+	0.859589i	$0.670719\pi$
$728$	0	0
$729$	146.164	5.41347
$730$	0	0
$731$	−46.2513	−1.71067
$732$	0	0
$733$	25.3241	0.935367	0.467684	−	0.883896i	$-0.345089\pi$
$733$	25.3241	0.935367	0.467684	+	0.883896i	$0.345089\pi$
$734$	0	0
$735$	−23.5489	−0.868615
$736$	0	0
$737$	50.5057	1.86040
$738$	0	0
$739$	21.3644	0.785904	0.392952	−	0.919559i	$-0.371454\pi$
$739$	21.3644	0.785904	0.392952	+	0.919559i	$0.371454\pi$
$740$	0	0
$741$	87.9165	3.22969
$742$	0	0
$743$	−27.2428	−0.999443	−0.499721	−	0.866186i	$-0.666564\pi$
$743$	−27.2428	−0.999443	−0.499721	+	0.866186i	$0.666564\pi$
$744$	0	0
$745$	6.40792	0.234768
$746$	0	0
$747$	153.214	5.60580
$748$	0	0
$749$	4.28051	0.156407
$750$	0	0
$751$	0.116497	0.00425105	0.00212553	−	0.999998i	$-0.499323\pi$
$751$	0.116497	0.00425105	0.00212553	+	0.999998i	$0.499323\pi$
$752$	0	0
$753$	52.1391	1.90005
$754$	0	0
$755$	−14.4505	−0.525906
$756$	0	0
$757$	33.4710	1.21652	0.608262	−	0.793736i	$-0.291867\pi$
$757$	33.4710	1.21652	0.608262	+	0.793736i	$0.291867\pi$
$758$	0	0
$759$	−121.421	−4.40731
$760$	0	0
$761$	2.33066	0.0844863	0.0422432	−	0.999107i	$-0.486550\pi$
$761$	2.33066	0.0844863	0.0422432	+	0.999107i	$0.486550\pi$
$762$	0	0
$763$	4.85286	0.175685
$764$	0	0
$765$	−44.8927	−1.62310
$766$	0	0
$767$	−5.08470	−0.183598
$768$	0	0
$769$	1.18991	0.0429092	0.0214546	−	0.999770i	$-0.493170\pi$
$769$	1.18991	0.0429092	0.0214546	+	0.999770i	$0.493170\pi$
$770$	0	0
$771$	30.1540	1.08597
$772$	0	0
$773$	−53.6880	−1.93102	−0.965512	−	0.260359i	$-0.916159\pi$
$773$	−53.6880	−1.93102	−0.965512	+	0.260359i	$0.916159\pi$
$774$	0	0
$775$	4.70742	0.169095
$776$	0	0
$777$	3.77966	0.135595
$778$	0	0
$779$	51.9480	1.86123
$780$	0	0
$781$	53.3321	1.90837
$782$	0	0
$783$	−35.6566	−1.27426
$784$	0	0
$785$	20.4931	0.731431
$786$	0	0
$787$	−6.94470	−0.247552	−0.123776	−	0.992310i	$-0.539500\pi$
$787$	−6.94470	−0.247552	−0.123776	+	0.992310i	$0.539500\pi$
$788$	0	0
$789$	37.7343	1.34338
$790$	0	0
$791$	−2.12411	−0.0755246
$792$	0	0
$793$	3.66408	0.130115
$794$	0	0
$795$	14.1721	0.502633
$796$	0	0
$797$	−41.2305	−1.46046	−0.730230	−	0.683201i	$-0.760587\pi$
$797$	−41.2305	−1.46046	−0.730230	+	0.683201i	$0.760587\pi$
$798$	0	0
$799$	−14.0226	−0.496083
$800$	0	0
$801$	−161.297	−5.69916
$802$	0	0
$803$	−56.2599	−1.98537
$804$	0	0
$805$	2.34526	0.0826594
$806$	0	0
$807$	−3.24139	−0.114102
$808$	0	0
$809$	2.38406	0.0838192	0.0419096	−	0.999121i	$-0.486656\pi$
$809$	2.38406	0.0838192	0.0419096	+	0.999121i	$0.486656\pi$
$810$	0	0
$811$	13.9715	0.490607	0.245304	−	0.969446i	$-0.421112\pi$
$811$	13.9715	0.490607	0.245304	+	0.969446i	$0.421112\pi$
$812$	0	0
$813$	5.27104	0.184863
$814$	0	0
$815$	16.5857	0.580972
$816$	0	0
$817$	−47.0306	−1.64539
$818$	0	0
$819$	−13.1403	−0.459158
$820$	0	0
$821$	−17.7535	−0.619602	−0.309801	−	0.950801i	$-0.600262\pi$
$821$	−17.7535	−0.619602	−0.309801	+	0.950801i	$0.600262\pi$
$822$	0	0
$823$	−8.61322	−0.300238	−0.150119	−	0.988668i	$-0.547966\pi$
$823$	−8.61322	−0.300238	−0.150119	+	0.988668i	$0.547966\pi$
$824$	0	0
$825$	16.1484	0.562217
$826$	0	0
$827$	−33.2954	−1.15779	−0.578897	−	0.815401i	$-0.696517\pi$
$827$	−33.2954	−1.15779	−0.578897	+	0.815401i	$0.696517\pi$
$828$	0	0
$829$	−20.0542	−0.696513	−0.348256	−	0.937399i	$-0.613226\pi$
$829$	−20.0542	−0.696513	−0.348256	+	0.937399i	$0.613226\pi$
$830$	0	0
$831$	7.69130	0.266808
$832$	0	0
$833$	−35.8716	−1.24288
$834$	0	0
$835$	3.34838	0.115876
$836$	0	0
$837$	−90.5541	−3.13001
$838$	0	0
$839$	44.6913	1.54291	0.771457	−	0.636281i	$-0.219528\pi$
$839$	44.6913	1.54291	0.771457	+	0.636281i	$0.219528\pi$
$840$	0	0
$841$	−25.5642	−0.881524
$842$	0	0
$843$	−111.312	−3.83377
$844$	0	0
$845$	−10.7829	−0.370942
$846$	0	0
$847$	3.55753	0.122238
$848$	0	0
$849$	101.528	3.48442
$850$	0	0
$851$	26.7079	0.915534
$852$	0	0
$853$	−18.6541	−0.638704	−0.319352	−	0.947636i	$-0.603465\pi$
$853$	−18.6541	−0.638704	−0.319352	+	0.947636i	$0.603465\pi$
$854$	0	0
$855$	−45.6491	−1.56117
$856$	0	0
$857$	−22.6705	−0.774411	−0.387206	−	0.921993i	$-0.626560\pi$
$857$	−22.6705	−0.774411	−0.387206	+	0.921993i	$0.626560\pi$
$858$	0	0
$859$	−0.728185	−0.0248454	−0.0124227	−	0.999923i	$-0.503954\pi$
$859$	−0.728185	−0.0248454	−0.0124227	+	0.999923i	$0.503954\pi$
$860$	0	0
$861$	−10.4607	−0.356498
$862$	0	0
$863$	19.5851	0.666683	0.333342	−	0.942806i	$-0.391824\pi$
$863$	19.5851	0.666683	0.333342	+	0.942806i	$0.391824\pi$
$864$	0	0
$865$	14.7337	0.500960
$866$	0	0
$867$	−34.1362	−1.15933
$868$	0	0
$869$	−8.34621	−0.283126
$870$	0	0
$871$	52.0347	1.76313
$872$	0	0
$873$	−93.1280	−3.15191
$874$	0	0
$875$	−0.311908	−0.0105444
$876$	0	0
$877$	−4.55789	−0.153909	−0.0769546	−	0.997035i	$-0.524520\pi$
$877$	−4.55789	−0.153909	−0.0769546	+	0.997035i	$0.524520\pi$
$878$	0	0
$879$	−93.1364	−3.14141
$880$	0	0
$881$	−6.14567	−0.207053	−0.103526	−	0.994627i	$-0.533013\pi$
$881$	−6.14567	−0.207053	−0.103526	+	0.994627i	$0.533013\pi$
$882$	0	0
$883$	4.70133	0.158212	0.0791062	−	0.996866i	$-0.474793\pi$
$883$	4.70133	0.158212	0.0791062	+	0.996866i	$0.474793\pi$
$884$	0	0
$885$	3.55700	0.119567
$886$	0	0
$887$	43.6125	1.46436	0.732182	−	0.681109i	$-0.238502\pi$
$887$	43.6125	1.46436	0.732182	+	0.681109i	$0.238502\pi$
$888$	0	0
$889$	−4.42125	−0.148284
$890$	0	0
$891$	−187.967	−6.29713
$892$	0	0
$893$	−14.2588	−0.477154
$894$	0	0
$895$	−4.80761	−0.160701
$896$	0	0
$897$	−125.097	−4.17687
$898$	0	0
$899$	8.72564	0.291016
$900$	0	0
$901$	21.5881	0.719203
$902$	0	0
$903$	9.47046	0.315157
$904$	0	0
$905$	11.6321	0.386664
$906$	0	0
$907$	−11.8651	−0.393974	−0.196987	−	0.980406i	$-0.563116\pi$
$907$	−11.8651	−0.393974	−0.196987	+	0.980406i	$0.563116\pi$
$908$	0	0
$909$	−110.766	−3.67387
$910$	0	0
$911$	−15.2101	−0.503935	−0.251967	−	0.967736i	$-0.581078\pi$
$911$	−15.2101	−0.503935	−0.251967	+	0.967736i	$0.581078\pi$
$912$	0	0
$913$	−83.9522	−2.77841
$914$	0	0
$915$	−2.56321	−0.0847371
$916$	0	0
$917$	−4.98295	−0.164551
$918$	0	0
$919$	−27.0026	−0.890734	−0.445367	−	0.895348i	$-0.646927\pi$
$919$	−27.0026	−0.890734	−0.445367	+	0.895348i	$0.646927\pi$
$920$	0	0
$921$	−40.2280	−1.32556
$922$	0	0
$923$	54.9467	1.80859
$924$	0	0
$925$	−3.55202	−0.116790
$926$	0	0
$927$	46.9001	1.54040
$928$	0	0
$929$	−32.1384	−1.05443	−0.527214	−	0.849733i	$-0.676763\pi$
$929$	−32.1384	−1.05443	−0.527214	+	0.849733i	$0.676763\pi$
$930$	0	0
$931$	−36.4760	−1.19545
$932$	0	0
$933$	−81.4258	−2.66576
$934$	0	0
$935$	24.5986	0.804460
$936$	0	0
$937$	−31.9356	−1.04329	−0.521645	−	0.853163i	$-0.674681\pi$
$937$	−31.9356	−1.04329	−0.521645	+	0.853163i	$0.674681\pi$
$938$	0	0
$939$	−66.8049	−2.18010
$940$	0	0
$941$	−11.2217	−0.365818	−0.182909	−	0.983130i	$-0.558551\pi$
$941$	−11.2217	−0.365818	−0.182909	+	0.983130i	$0.558551\pi$
$942$	0	0
$943$	−73.9172	−2.40708
$944$	0	0
$945$	6.00001	0.195180
$946$	0	0
$947$	56.9486	1.85058	0.925290	−	0.379260i	$-0.123821\pi$
$947$	56.9486	1.85058	0.925290	+	0.379260i	$0.123821\pi$
$948$	0	0
$949$	−57.9632	−1.88156
$950$	0	0
$951$	−69.4082	−2.25071
$952$	0	0
$953$	34.8251	1.12810	0.564049	−	0.825742i	$-0.309243\pi$
$953$	34.8251	1.12810	0.564049	+	0.825742i	$0.309243\pi$
$954$	0	0
$955$	5.43944	0.176016
$956$	0	0
$957$	29.9326	0.967585
$958$	0	0
$959$	0.688763	0.0222413
$960$	0	0
$961$	−8.84024	−0.285169
$962$	0	0
$963$	118.554	3.82034
$964$	0	0
$965$	5.66031	0.182212
$966$	0	0
$967$	−9.51393	−0.305947	−0.152974	−	0.988230i	$-0.548885\pi$
$967$	−9.51393	−0.305947	−0.152974	+	0.988230i	$0.548885\pi$
$968$	0	0
$969$	−93.6847	−3.00959
$970$	0	0
$971$	−11.0502	−0.354617	−0.177308	−	0.984155i	$-0.556739\pi$
$971$	−11.0502	−0.354617	−0.177308	+	0.984155i	$0.556739\pi$
$972$	0	0
$973$	−3.36344	−0.107827
$974$	0	0
$975$	16.6373	0.532821
$976$	0	0
$977$	−36.4089	−1.16482	−0.582412	−	0.812894i	$-0.697891\pi$
$977$	−36.4089	−1.16482	−0.582412	+	0.812894i	$0.697891\pi$
$978$	0	0
$979$	88.3815	2.82468
$980$	0	0
$981$	134.406	4.29124
$982$	0	0
$983$	−17.4863	−0.557727	−0.278863	−	0.960331i	$-0.589958\pi$
$983$	−17.4863	−0.557727	−0.278863	+	0.960331i	$0.589958\pi$
$984$	0	0
$985$	−14.0788	−0.448589
$986$	0	0
$987$	2.87127	0.0913936
$988$	0	0
$989$	66.9202	2.12794
$990$	0	0
$991$	38.4049	1.21997	0.609985	−	0.792413i	$-0.291175\pi$
$991$	38.4049	1.21997	0.609985	+	0.792413i	$0.291175\pi$
$992$	0	0
$993$	−4.33734	−0.137641
$994$	0	0
$995$	23.6994	0.751322
$996$	0	0
$997$	4.31525	0.136665	0.0683327	−	0.997663i	$-0.478232\pi$
$997$	4.31525	0.136665	0.0683327	+	0.997663i	$0.478232\pi$
$998$	0	0
$999$	68.3283	2.16181

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8020.2.a.e.1.1	✓	35

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8020.2.a.e.1.1	✓	35	1.1	even	1	trivial

Overview	Random
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8020 = 2^{2} \cdot 5 \cdot 401 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8020.a (trivial)

\(f(q)\)	\(=\)	\(q-3.41155 q^{3} -1.00000 q^{5} +0.311908 q^{7} +8.63864 q^{9} +O(q^{10})\)	\(q-3.41155 q^{3} -1.00000 q^{5} +0.311908 q^{7} +8.63864 q^{9} -4.73347 q^{11} -4.87677 q^{13} +3.41155 q^{15} +5.19674 q^{17} +5.28429 q^{19} -1.06409 q^{21} -7.51907 q^{23} +1.00000 q^{25} -19.2365 q^{27} +1.85359 q^{29} +4.70742 q^{31} +16.1484 q^{33} -0.311908 q^{35} -3.55202 q^{37} +16.6373 q^{39} +9.83063 q^{41} -8.90007 q^{43} -8.63864 q^{45} -2.69834 q^{47} -6.90271 q^{49} -17.7289 q^{51} +4.15416 q^{53} +4.73347 q^{55} -18.0276 q^{57} +1.04264 q^{59} -0.751334 q^{61} +2.69446 q^{63} +4.87677 q^{65} -10.6699 q^{67} +25.6516 q^{69} -11.2670 q^{71} +11.8856 q^{73} -3.41155 q^{75} -1.47641 q^{77} +1.76323 q^{79} +39.7102 q^{81} +17.7359 q^{83} -5.19674 q^{85} -6.32362 q^{87} -18.6716 q^{89} -1.52110 q^{91} -16.0596 q^{93} -5.28429 q^{95} -10.7804 q^{97} -40.8907 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 35 q - q^{3} - 35 q^{5} + 6 q^{7} + 52 q^{9}+O(q^{10}) \) 35 * q - q^3 - 35 * q^5 + 6 * q^7 + 52 * q^9	\( 35 q - q^{3} - 35 q^{5} + 6 q^{7} + 52 q^{9} - 2 q^{11} - q^{13} + q^{15} + 18 q^{17} + 2 q^{19} + 12 q^{21} + 13 q^{23} + 35 q^{25} - 7 q^{27} + 25 q^{29} + 13 q^{31} + 14 q^{33} - 6 q^{35} - 19 q^{37} - 3 q^{39} + 24 q^{41} - 5 q^{43} - 52 q^{45} + 19 q^{47} + 55 q^{49} + 41 q^{53} + 2 q^{55} + 14 q^{57} + 3 q^{59} + 13 q^{61} + 70 q^{63} + q^{65} - 17 q^{67} + 64 q^{69} + 17 q^{71} - 63 q^{73} - q^{75} + 54 q^{77} + 11 q^{79} + 107 q^{81} - 8 q^{83} - 18 q^{85} + 36 q^{87} + 38 q^{89} - 27 q^{91} + q^{93} - 2 q^{95} - 54 q^{97} - 51 q^{99}+O(q^{100}) \) 35 * q - q^3 - 35 * q^5 + 6 * q^7 + 52 * q^9 - 2 * q^11 - q^13 + q^15 + 18 * q^17 + 2 * q^19 + 12 * q^21 + 13 * q^23 + 35 * q^25 - 7 * q^27 + 25 * q^29 + 13 * q^31 + 14 * q^33 - 6 * q^35 - 19 * q^37 - 3 * q^39 + 24 * q^41 - 5 * q^43 - 52 * q^45 + 19 * q^47 + 55 * q^49 + 41 * q^53 + 2 * q^55 + 14 * q^57 + 3 * q^59 + 13 * q^61 + 70 * q^63 + q^65 - 17 * q^67 + 64 * q^69 + 17 * q^71 - 63 * q^73 - q^75 + 54 * q^77 + 11 * q^79 + 107 * q^81 - 8 * q^83 - 18 * q^85 + 36 * q^87 + 38 * q^89 - 27 * q^91 + q^93 - 2 * q^95 - 54 * q^97 - 51 * q^99

Introduction

L-functions

Modular forms

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