LMFDB - Embedded newform 8020.2.a.e.1.2

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			1.2
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)

$f(q)$	$=$	$q-3.20562 q^{3} -1.00000 q^{5} -2.96453 q^{7} +7.27598 q^{9} +O(q^{10})$	$q-3.20562 q^{3} -1.00000 q^{5} -2.96453 q^{7} +7.27598 q^{9} +0.803393 q^{11} -0.822974 q^{13} +3.20562 q^{15} -4.32489 q^{17} -5.88345 q^{19} +9.50314 q^{21} -3.95100 q^{23} +1.00000 q^{25} -13.7072 q^{27} +8.32845 q^{29} -4.02664 q^{31} -2.57537 q^{33} +2.96453 q^{35} +2.20697 q^{37} +2.63814 q^{39} -0.284071 q^{41} +3.27752 q^{43} -7.27598 q^{45} -0.459673 q^{47} +1.78841 q^{49} +13.8639 q^{51} -12.2186 q^{53} -0.803393 q^{55} +18.8601 q^{57} -12.4434 q^{59} +0.769354 q^{61} -21.5698 q^{63} +0.822974 q^{65} -5.42443 q^{67} +12.6654 q^{69} +1.98779 q^{71} -1.80313 q^{73} -3.20562 q^{75} -2.38168 q^{77} +1.11823 q^{79} +22.1120 q^{81} -12.2376 q^{83} +4.32489 q^{85} -26.6978 q^{87} +14.7222 q^{89} +2.43973 q^{91} +12.9079 q^{93} +5.88345 q^{95} -14.0145 q^{97} +5.84547 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−3.20562	−1.85076	−0.925382	−	0.379036i	$-0.876256\pi$
$3$	−3.20562	−1.85076	−0.925382	+	0.379036i	$0.876256\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−2.96453	−1.12049	−0.560243	−	0.828328i	$-0.689292\pi$
$7$	−2.96453	−1.12049	−0.560243	+	0.828328i	$0.689292\pi$
$8$	0	0
$9$	7.27598	2.42533
$10$	0	0
$11$	0.803393	0.242232	0.121116	−	0.992638i	$-0.461353\pi$
$11$	0.803393	0.242232	0.121116	+	0.992638i	$0.461353\pi$
$12$	0	0
$13$	−0.822974	−0.228252	−0.114126	−	0.993466i	$-0.536407\pi$
$13$	−0.822974	−0.228252	−0.114126	+	0.993466i	$0.536407\pi$
$14$	0	0
$15$	3.20562	0.827687
$16$	0	0
$17$	−4.32489	−1.04894	−0.524470	−	0.851429i	$-0.675736\pi$
$17$	−4.32489	−1.04894	−0.524470	+	0.851429i	$0.675736\pi$
$18$	0	0
$19$	−5.88345	−1.34976	−0.674878	−	0.737930i	$-0.735804\pi$
$19$	−5.88345	−1.34976	−0.674878	+	0.737930i	$0.735804\pi$
$20$	0	0
$21$	9.50314	2.07375
$22$	0	0
$23$	−3.95100	−0.823840	−0.411920	−	0.911220i	$-0.635142\pi$
$23$	−3.95100	−0.823840	−0.411920	+	0.911220i	$0.635142\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−13.7072	−2.63795
$28$	0	0
$29$	8.32845	1.54655	0.773277	−	0.634068i	$-0.218616\pi$
$29$	8.32845	1.54655	0.773277	+	0.634068i	$0.218616\pi$
$30$	0	0
$31$	−4.02664	−0.723205	−0.361603	−	0.932332i	$-0.617770\pi$
$31$	−4.02664	−0.723205	−0.361603	+	0.932332i	$0.617770\pi$
$32$	0	0
$33$	−2.57537	−0.448314
$34$	0	0
$35$	2.96453	0.501096
$36$	0	0
$37$	2.20697	0.362823	0.181412	−	0.983407i	$-0.441933\pi$
$37$	2.20697	0.362823	0.181412	+	0.983407i	$0.441933\pi$
$38$	0	0
$39$	2.63814	0.422440
$40$	0	0
$41$	−0.284071	−0.0443644	−0.0221822	−	0.999754i	$-0.507061\pi$
$41$	−0.284071	−0.0443644	−0.0221822	+	0.999754i	$0.507061\pi$
$42$	0	0
$43$	3.27752	0.499817	0.249908	−	0.968269i	$-0.419600\pi$
$43$	3.27752	0.499817	0.249908	+	0.968269i	$0.419600\pi$
$44$	0	0
$45$	−7.27598	−1.08464
$46$	0	0
$47$	−0.459673	−0.0670502	−0.0335251	−	0.999438i	$-0.510673\pi$
$47$	−0.459673	−0.0670502	−0.0335251	+	0.999438i	$0.510673\pi$
$48$	0	0
$49$	1.78841	0.255488
$50$	0	0
$51$	13.8639	1.94134
$52$	0	0
$53$	−12.2186	−1.67836	−0.839180	−	0.543854i	$-0.816965\pi$
$53$	−12.2186	−1.67836	−0.839180	+	0.543854i	$0.816965\pi$
$54$	0	0
$55$	−0.803393	−0.108329
$56$	0	0
$57$	18.8601	2.49808
$58$	0	0
$59$	−12.4434	−1.61999	−0.809997	−	0.586434i	$-0.800532\pi$
$59$	−12.4434	−1.61999	−0.809997	+	0.586434i	$0.800532\pi$
$60$	0	0
$61$	0.769354	0.0985057	0.0492529	−	0.998786i	$-0.484316\pi$
$61$	0.769354	0.0985057	0.0492529	+	0.998786i	$0.484316\pi$
$62$	0	0
$63$	−21.5698	−2.71755
$64$	0	0
$65$	0.822974	0.102077
$66$	0	0
$67$	−5.42443	−0.662700	−0.331350	−	0.943508i	$-0.607504\pi$
$67$	−5.42443	−0.662700	−0.331350	+	0.943508i	$0.607504\pi$
$68$	0	0
$69$	12.6654	1.52473
$70$	0	0
$71$	1.98779	0.235907	0.117954	−	0.993019i	$-0.462367\pi$
$71$	1.98779	0.235907	0.117954	+	0.993019i	$0.462367\pi$
$72$	0	0
$73$	−1.80313	−0.211040	−0.105520	−	0.994417i	$-0.533651\pi$
$73$	−1.80313	−0.211040	−0.105520	+	0.994417i	$0.533651\pi$
$74$	0	0
$75$	−3.20562	−0.370153
$76$	0	0
$77$	−2.38168	−0.271417
$78$	0	0
$79$	1.11823	0.125811	0.0629055	−	0.998019i	$-0.479963\pi$
$79$	1.11823	0.125811	0.0629055	+	0.998019i	$0.479963\pi$
$80$	0	0
$81$	22.1120	2.45689
$82$	0	0
$83$	−12.2376	−1.34325	−0.671625	−	0.740891i	$-0.734403\pi$
$83$	−12.2376	−1.34325	−0.671625	+	0.740891i	$0.734403\pi$
$84$	0	0
$85$	4.32489	0.469100
$86$	0	0
$87$	−26.6978	−2.86231
$88$	0	0
$89$	14.7222	1.56055	0.780276	−	0.625435i	$-0.215079\pi$
$89$	14.7222	1.56055	0.780276	+	0.625435i	$0.215079\pi$
$90$	0	0
$91$	2.43973	0.255753
$92$	0	0
$93$	12.9079	1.33848
$94$	0	0
$95$	5.88345	0.603629
$96$	0	0
$97$	−14.0145	−1.42296	−0.711480	−	0.702707i	$-0.751975\pi$
$97$	−14.0145	−1.42296	−0.711480	+	0.702707i	$0.751975\pi$
$98$	0	0
$99$	5.84547	0.587492
$100$	0	0
$101$	−15.0876	−1.50128	−0.750638	−	0.660713i	$-0.770254\pi$
$101$	−15.0876	−1.50128	−0.750638	+	0.660713i	$0.770254\pi$
$102$	0	0
$103$	−0.338255	−0.0333293	−0.0166646	−	0.999861i	$-0.505305\pi$
$103$	−0.338255	−0.0333293	−0.0166646	+	0.999861i	$0.505305\pi$
$104$	0	0
$105$	−9.50314	−0.927411
$106$	0	0
$107$	0.0618977	0.00598388	0.00299194	−	0.999996i	$-0.499048\pi$
$107$	0.0618977	0.00598388	0.00299194	+	0.999996i	$0.499048\pi$
$108$	0	0
$109$	−4.67543	−0.447825	−0.223912	−	0.974609i	$-0.571883\pi$
$109$	−4.67543	−0.447825	−0.223912	+	0.974609i	$0.571883\pi$
$110$	0	0
$111$	−7.07470	−0.671500
$112$	0	0
$113$	−2.66331	−0.250543	−0.125271	−	0.992123i	$-0.539980\pi$
$113$	−2.66331	−0.250543	−0.125271	+	0.992123i	$0.539980\pi$
$114$	0	0
$115$	3.95100	0.368433
$116$	0	0
$117$	−5.98795	−0.553586
$118$	0	0
$119$	12.8212	1.17532
$120$	0	0
$121$	−10.3546	−0.941324
$122$	0	0
$123$	0.910623	0.0821081
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	7.52209	0.667477	0.333739	−	0.942666i	$-0.391690\pi$
$127$	7.52209	0.667477	0.333739	+	0.942666i	$0.391690\pi$
$128$	0	0
$129$	−10.5065	−0.925043
$130$	0	0
$131$	0.940352	0.0821590	0.0410795	−	0.999156i	$-0.486920\pi$
$131$	0.940352	0.0821590	0.0410795	+	0.999156i	$0.486920\pi$
$132$	0	0
$133$	17.4416	1.51238
$134$	0	0
$135$	13.7072	1.17973
$136$	0	0
$137$	−2.21303	−0.189072	−0.0945360	−	0.995521i	$-0.530137\pi$
$137$	−2.21303	−0.189072	−0.0945360	+	0.995521i	$0.530137\pi$
$138$	0	0
$139$	−20.7636	−1.76114	−0.880572	−	0.473912i	$-0.842841\pi$
$139$	−20.7636	−1.76114	−0.880572	+	0.473912i	$0.842841\pi$
$140$	0	0
$141$	1.47354	0.124094
$142$	0	0
$143$	−0.661171	−0.0552899
$144$	0	0
$145$	−8.32845	−0.691640
$146$	0	0
$147$	−5.73297	−0.472847
$148$	0	0
$149$	−4.25428	−0.348524	−0.174262	−	0.984699i	$-0.555754\pi$
$149$	−4.25428	−0.348524	−0.174262	+	0.984699i	$0.555754\pi$
$150$	0	0
$151$	−13.6616	−1.11176	−0.555882	−	0.831261i	$-0.687619\pi$
$151$	−13.6616	−1.11176	−0.555882	+	0.831261i	$0.687619\pi$
$152$	0	0
$153$	−31.4678	−2.54402
$154$	0	0
$155$	4.02664	0.323427
$156$	0	0
$157$	−13.8297	−1.10373	−0.551865	−	0.833933i	$-0.686084\pi$
$157$	−13.8297	−1.10373	−0.551865	+	0.833933i	$0.686084\pi$
$158$	0	0
$159$	39.1683	3.10625
$160$	0	0
$161$	11.7128	0.923101
$162$	0	0
$163$	14.9886	1.17400	0.586998	−	0.809589i	$-0.300310\pi$
$163$	14.9886	1.17400	0.586998	+	0.809589i	$0.300310\pi$
$164$	0	0
$165$	2.57537	0.200492
$166$	0	0
$167$	−19.0434	−1.47362	−0.736812	−	0.676098i	$-0.763670\pi$
$167$	−19.0434	−1.47362	−0.736812	+	0.676098i	$0.763670\pi$
$168$	0	0
$169$	−12.3227	−0.947901
$170$	0	0
$171$	−42.8079	−3.27360
$172$	0	0
$173$	−3.98745	−0.303160	−0.151580	−	0.988445i	$-0.548436\pi$
$173$	−3.98745	−0.303160	−0.151580	+	0.988445i	$0.548436\pi$
$174$	0	0
$175$	−2.96453	−0.224097
$176$	0	0
$177$	39.8888	2.99823
$178$	0	0
$179$	−3.35987	−0.251129	−0.125564	−	0.992085i	$-0.540074\pi$
$179$	−3.35987	−0.251129	−0.125564	+	0.992085i	$0.540074\pi$
$180$	0	0
$181$	1.04165	0.0774256	0.0387128	−	0.999250i	$-0.487674\pi$
$181$	1.04165	0.0774256	0.0387128	+	0.999250i	$0.487674\pi$
$182$	0	0
$183$	−2.46626	−0.182311
$184$	0	0
$185$	−2.20697	−0.162260
$186$	0	0
$187$	−3.47459	−0.254087
$188$	0	0
$189$	40.6353	2.95578
$190$	0	0
$191$	−24.5474	−1.77619	−0.888094	−	0.459663i	$-0.847970\pi$
$191$	−24.5474	−1.77619	−0.888094	+	0.459663i	$0.847970\pi$
$192$	0	0
$193$	−5.26079	−0.378680	−0.189340	−	0.981912i	$-0.560635\pi$
$193$	−5.26079	−0.378680	−0.189340	+	0.981912i	$0.560635\pi$
$194$	0	0
$195$	−2.63814	−0.188921
$196$	0	0
$197$	6.63208	0.472516	0.236258	−	0.971690i	$-0.424079\pi$
$197$	6.63208	0.472516	0.236258	+	0.971690i	$0.424079\pi$
$198$	0	0
$199$	−6.82204	−0.483602	−0.241801	−	0.970326i	$-0.577738\pi$
$199$	−6.82204	−0.483602	−0.241801	+	0.970326i	$0.577738\pi$
$200$	0	0
$201$	17.3887	1.22650
$202$	0	0
$203$	−24.6899	−1.73289
$204$	0	0
$205$	0.284071	0.0198404
$206$	0	0
$207$	−28.7474	−1.99808
$208$	0	0
$209$	−4.72672	−0.326954
$210$	0	0
$211$	2.07182	0.142630	0.0713149	−	0.997454i	$-0.477280\pi$
$211$	2.07182	0.142630	0.0713149	+	0.997454i	$0.477280\pi$
$212$	0	0
$213$	−6.37209	−0.436609
$214$	0	0
$215$	−3.27752	−0.223525
$216$	0	0
$217$	11.9371	0.810341
$218$	0	0
$219$	5.78013	0.390585
$220$	0	0
$221$	3.55927	0.239423
$222$	0	0
$223$	5.51003	0.368979	0.184490	−	0.982834i	$-0.440937\pi$
$223$	5.51003	0.368979	0.184490	+	0.982834i	$0.440937\pi$
$224$	0	0
$225$	7.27598	0.485066
$226$	0	0
$227$	−22.4830	−1.49225	−0.746126	−	0.665805i	$-0.768088\pi$
$227$	−22.4830	−1.49225	−0.746126	+	0.665805i	$0.768088\pi$
$228$	0	0
$229$	11.3527	0.750209	0.375105	−	0.926983i	$-0.377607\pi$
$229$	11.3527	0.750209	0.375105	+	0.926983i	$0.377607\pi$
$230$	0	0
$231$	7.63475	0.502330
$232$	0	0
$233$	−14.8322	−0.971691	−0.485845	−	0.874045i	$-0.661488\pi$
$233$	−14.8322	−0.971691	−0.485845	+	0.874045i	$0.661488\pi$
$234$	0	0
$235$	0.459673	0.0299857
$236$	0	0
$237$	−3.58463	−0.232847
$238$	0	0
$239$	9.07428	0.586966	0.293483	−	0.955964i	$-0.405186\pi$
$239$	9.07428	0.586966	0.293483	+	0.955964i	$0.405186\pi$
$240$	0	0
$241$	20.7649	1.33758	0.668792	−	0.743450i	$-0.266812\pi$
$241$	20.7649	1.33758	0.668792	+	0.743450i	$0.266812\pi$
$242$	0	0
$243$	−29.7611	−1.90918
$244$	0	0
$245$	−1.78841	−0.114258
$246$	0	0
$247$	4.84193	0.308084
$248$	0	0
$249$	39.2290	2.48604
$250$	0	0
$251$	26.7495	1.68841	0.844207	−	0.536017i	$-0.180072\pi$
$251$	26.7495	1.68841	0.844207	+	0.536017i	$0.180072\pi$
$252$	0	0
$253$	−3.17420	−0.199560
$254$	0	0
$255$	−13.8639	−0.868194
$256$	0	0
$257$	14.6443	0.913489	0.456745	−	0.889598i	$-0.349015\pi$
$257$	14.6443	0.913489	0.456745	+	0.889598i	$0.349015\pi$
$258$	0	0
$259$	−6.54261	−0.406538
$260$	0	0
$261$	60.5977	3.75090
$262$	0	0
$263$	12.2377	0.754611	0.377306	−	0.926089i	$-0.376851\pi$
$263$	12.2377	0.754611	0.377306	+	0.926089i	$0.376851\pi$
$264$	0	0
$265$	12.2186	0.750585
$266$	0	0
$267$	−47.1938	−2.88821
$268$	0	0
$269$	−5.05708	−0.308336	−0.154168	−	0.988045i	$-0.549270\pi$
$269$	−5.05708	−0.308336	−0.154168	+	0.988045i	$0.549270\pi$
$270$	0	0
$271$	21.2198	1.28901	0.644506	−	0.764600i	$-0.277063\pi$
$271$	21.2198	1.28901	0.644506	+	0.764600i	$0.277063\pi$
$272$	0	0
$273$	−7.82083	−0.473338
$274$	0	0
$275$	0.803393	0.0484464
$276$	0	0
$277$	−18.2944	−1.09921	−0.549603	−	0.835426i	$-0.685221\pi$
$277$	−18.2944	−1.09921	−0.549603	+	0.835426i	$0.685221\pi$
$278$	0	0
$279$	−29.2977	−1.75401
$280$	0	0
$281$	21.8980	1.30633	0.653163	−	0.757217i	$-0.273442\pi$
$281$	21.8980	1.30633	0.653163	+	0.757217i	$0.273442\pi$
$282$	0	0
$283$	16.6496	0.989717	0.494859	−	0.868974i	$-0.335220\pi$
$283$	16.6496	0.989717	0.494859	+	0.868974i	$0.335220\pi$
$284$	0	0
$285$	−18.8601	−1.11718
$286$	0	0
$287$	0.842136	0.0497097
$288$	0	0
$289$	1.70468	0.100275
$290$	0	0
$291$	44.9252	2.63356
$292$	0	0
$293$	−19.5003	−1.13922	−0.569609	−	0.821916i	$-0.692905\pi$
$293$	−19.5003	−1.13922	−0.569609	+	0.821916i	$0.692905\pi$
$294$	0	0
$295$	12.4434	0.724484
$296$	0	0
$297$	−11.0122	−0.638995
$298$	0	0
$299$	3.25157	0.188043
$300$	0	0
$301$	−9.71629	−0.560037
$302$	0	0
$303$	48.3652	2.77851
$304$	0	0
$305$	−0.769354	−0.0440531
$306$	0	0
$307$	18.1294	1.03470	0.517351	−	0.855773i	$-0.326918\pi$
$307$	18.1294	1.03470	0.517351	+	0.855773i	$0.326918\pi$
$308$	0	0
$309$	1.08432	0.0616846
$310$	0	0
$311$	−19.5612	−1.10921	−0.554606	−	0.832113i	$-0.687131\pi$
$311$	−19.5612	−1.10921	−0.554606	+	0.832113i	$0.687131\pi$
$312$	0	0
$313$	13.4653	0.761105	0.380552	−	0.924759i	$-0.375734\pi$
$313$	13.4653	0.761105	0.380552	+	0.924759i	$0.375734\pi$
$314$	0	0
$315$	21.5698	1.21532
$316$	0	0
$317$	4.84105	0.271900	0.135950	−	0.990716i	$-0.456591\pi$
$317$	4.84105	0.271900	0.135950	+	0.990716i	$0.456591\pi$
$318$	0	0
$319$	6.69101	0.374625
$320$	0	0
$321$	−0.198421	−0.0110748
$322$	0	0
$323$	25.4453	1.41581
$324$	0	0
$325$	−0.822974	−0.0456504
$326$	0	0
$327$	14.9876	0.828818
$328$	0	0
$329$	1.36271	0.0751287
$330$	0	0
$331$	9.88402	0.543275	0.271638	−	0.962400i	$-0.412435\pi$
$331$	9.88402	0.543275	0.271638	+	0.962400i	$0.412435\pi$
$332$	0	0
$333$	16.0579	0.879966
$334$	0	0
$335$	5.42443	0.296368
$336$	0	0
$337$	12.3527	0.672895	0.336448	−	0.941702i	$-0.390774\pi$
$337$	12.3527	0.672895	0.336448	+	0.941702i	$0.390774\pi$
$338$	0	0
$339$	8.53754	0.463696
$340$	0	0
$341$	−3.23497	−0.175183
$342$	0	0
$343$	15.4499	0.834215
$344$	0	0
$345$	−12.6654	−0.681882
$346$	0	0
$347$	7.31733	0.392815	0.196407	−	0.980522i	$-0.437072\pi$
$347$	7.31733	0.392815	0.196407	+	0.980522i	$0.437072\pi$
$348$	0	0
$349$	−2.61621	−0.140042	−0.0700212	−	0.997546i	$-0.522307\pi$
$349$	−2.61621	−0.140042	−0.0700212	+	0.997546i	$0.522307\pi$
$350$	0	0
$351$	11.2806	0.602116
$352$	0	0
$353$	19.2889	1.02664	0.513322	−	0.858196i	$-0.328415\pi$
$353$	19.2889	1.02664	0.513322	+	0.858196i	$0.328415\pi$
$354$	0	0
$355$	−1.98779	−0.105501
$356$	0	0
$357$	−41.1000	−2.17524
$358$	0	0
$359$	−30.6289	−1.61653	−0.808267	−	0.588816i	$-0.799594\pi$
$359$	−30.6289	−1.61653	−0.808267	+	0.588816i	$0.799594\pi$
$360$	0	0
$361$	15.6150	0.821840
$362$	0	0
$363$	33.1928	1.74217
$364$	0	0
$365$	1.80313	0.0943799
$366$	0	0
$367$	−1.73147	−0.0903819	−0.0451909	−	0.998978i	$-0.514390\pi$
$367$	−1.73147	−0.0903819	−0.0451909	+	0.998978i	$0.514390\pi$
$368$	0	0
$369$	−2.06690	−0.107598
$370$	0	0
$371$	36.2225	1.88058
$372$	0	0
$373$	−16.1688	−0.837190	−0.418595	−	0.908173i	$-0.637477\pi$
$373$	−16.1688	−0.837190	−0.418595	+	0.908173i	$0.637477\pi$
$374$	0	0
$375$	3.20562	0.165537
$376$	0	0
$377$	−6.85410	−0.353004
$378$	0	0
$379$	−10.9140	−0.560616	−0.280308	−	0.959910i	$-0.590437\pi$
$379$	−10.9140	−0.560616	−0.280308	+	0.959910i	$0.590437\pi$
$380$	0	0
$381$	−24.1129	−1.23534
$382$	0	0
$383$	−10.7505	−0.549323	−0.274662	−	0.961541i	$-0.588566\pi$
$383$	−10.7505	−0.549323	−0.274662	+	0.961541i	$0.588566\pi$
$384$	0	0
$385$	2.38168	0.121382
$386$	0	0
$387$	23.8472	1.21222
$388$	0	0
$389$	−1.02411	−0.0519246	−0.0259623	−	0.999663i	$-0.508265\pi$
$389$	−1.02411	−0.0519246	−0.0259623	+	0.999663i	$0.508265\pi$
$390$	0	0
$391$	17.0876	0.864159
$392$	0	0
$393$	−3.01441	−0.152057
$394$	0	0
$395$	−1.11823	−0.0562644
$396$	0	0
$397$	−31.7309	−1.59253	−0.796264	−	0.604949i	$-0.793193\pi$
$397$	−31.7309	−1.59253	−0.796264	+	0.604949i	$0.793193\pi$
$398$	0	0
$399$	−55.9112	−2.79906
$400$	0	0
$401$	−1.00000	−0.0499376
$401$	−1.00000	−0.0499376
$402$	0	0
$403$	3.31382	0.165073
$404$	0	0
$405$	−22.1120	−1.09875
$406$	0	0
$407$	1.77306	0.0878874
$408$	0	0
$409$	27.2004	1.34497	0.672486	−	0.740110i	$-0.265226\pi$
$409$	27.2004	1.34497	0.672486	+	0.740110i	$0.265226\pi$
$410$	0	0
$411$	7.09413	0.349928
$412$	0	0
$413$	36.8888	1.81518
$414$	0	0
$415$	12.2376	0.600719
$416$	0	0
$417$	66.5601	3.25946
$418$	0	0
$419$	−9.58624	−0.468319	−0.234159	−	0.972198i	$-0.575234\pi$
$419$	−9.58624	−0.468319	−0.234159	+	0.972198i	$0.575234\pi$
$420$	0	0
$421$	31.5373	1.53703	0.768516	−	0.639830i	$-0.220995\pi$
$421$	31.5373	1.53703	0.768516	+	0.639830i	$0.220995\pi$
$422$	0	0
$423$	−3.34457	−0.162619
$424$	0	0
$425$	−4.32489	−0.209788
$426$	0	0
$427$	−2.28077	−0.110374
$428$	0	0
$429$	2.11946	0.102329
$430$	0	0
$431$	5.91929	0.285122	0.142561	−	0.989786i	$-0.454466\pi$
$431$	5.91929	0.285122	0.142561	+	0.989786i	$0.454466\pi$
$432$	0	0
$433$	−10.4623	−0.502787	−0.251394	−	0.967885i	$-0.580889\pi$
$433$	−10.4623	−0.502787	−0.251394	+	0.967885i	$0.580889\pi$
$434$	0	0
$435$	26.6978	1.28006
$436$	0	0
$437$	23.2455	1.11198
$438$	0	0
$439$	−20.6945	−0.987694	−0.493847	−	0.869549i	$-0.664410\pi$
$439$	−20.6945	−0.987694	−0.493847	+	0.869549i	$0.664410\pi$
$440$	0	0
$441$	13.0125	0.619641
$442$	0	0
$443$	−17.9618	−0.853392	−0.426696	−	0.904395i	$-0.640323\pi$
$443$	−17.9618	−0.853392	−0.426696	+	0.904395i	$0.640323\pi$
$444$	0	0
$445$	−14.7222	−0.697900
$446$	0	0
$447$	13.6376	0.645037
$448$	0	0
$449$	10.7676	0.508156	0.254078	−	0.967184i	$-0.418228\pi$
$449$	10.7676	0.508156	0.254078	+	0.967184i	$0.418228\pi$
$450$	0	0
$451$	−0.228220	−0.0107465
$452$	0	0
$453$	43.7938	2.05761
$454$	0	0
$455$	−2.43973	−0.114376
$456$	0	0
$457$	35.0331	1.63878	0.819390	−	0.573237i	$-0.194313\pi$
$457$	35.0331	1.63878	0.819390	+	0.573237i	$0.194313\pi$
$458$	0	0
$459$	59.2820	2.76705
$460$	0	0
$461$	25.9092	1.20671	0.603356	−	0.797472i	$-0.293830\pi$
$461$	25.9092	1.20671	0.603356	+	0.797472i	$0.293830\pi$
$462$	0	0
$463$	22.0529	1.02488	0.512442	−	0.858722i	$-0.328741\pi$
$463$	22.0529	1.02488	0.512442	+	0.858722i	$0.328741\pi$
$464$	0	0
$465$	−12.9079	−0.598587
$466$	0	0
$467$	−4.63676	−0.214564	−0.107282	−	0.994229i	$-0.534215\pi$
$467$	−4.63676	−0.214564	−0.107282	+	0.994229i	$0.534215\pi$
$468$	0	0
$469$	16.0809	0.742545
$470$	0	0
$471$	44.3327	2.04275
$472$	0	0
$473$	2.63313	0.121072
$474$	0	0
$475$	−5.88345	−0.269951
$476$	0	0
$477$	−88.9027	−4.07057
$478$	0	0
$479$	20.5332	0.938187	0.469093	−	0.883149i	$-0.344581\pi$
$479$	20.5332	0.938187	0.469093	+	0.883149i	$0.344581\pi$
$480$	0	0
$481$	−1.81628	−0.0828151
$482$	0	0
$483$	−37.5469	−1.70844
$484$	0	0
$485$	14.0145	0.636367
$486$	0	0
$487$	27.1911	1.23215	0.616074	−	0.787688i	$-0.288722\pi$
$487$	27.1911	1.23215	0.616074	+	0.787688i	$0.288722\pi$
$488$	0	0
$489$	−48.0476	−2.17279
$490$	0	0
$491$	34.4193	1.55332	0.776661	−	0.629918i	$-0.216912\pi$
$491$	34.4193	1.55332	0.776661	+	0.629918i	$0.216912\pi$
$492$	0	0
$493$	−36.0196	−1.62224
$494$	0	0
$495$	−5.84547	−0.262734
$496$	0	0
$497$	−5.89285	−0.264331
$498$	0	0
$499$	−37.1641	−1.66369	−0.831846	−	0.555007i	$-0.812716\pi$
$499$	−37.1641	−1.66369	−0.831846	+	0.555007i	$0.812716\pi$
$500$	0	0
$501$	61.0459	2.72733
$502$	0	0
$503$	−11.7666	−0.524649	−0.262324	−	0.964980i	$-0.584489\pi$
$503$	−11.7666	−0.524649	−0.262324	+	0.964980i	$0.584489\pi$
$504$	0	0
$505$	15.0876	0.671391
$506$	0	0
$507$	39.5019	1.75434
$508$	0	0
$509$	25.0826	1.11177	0.555883	−	0.831261i	$-0.312380\pi$
$509$	25.0826	1.11177	0.555883	+	0.831261i	$0.312380\pi$
$510$	0	0
$511$	5.34541	0.236467
$512$	0	0
$513$	80.6454	3.56058
$514$	0	0
$515$	0.338255	0.0149053
$516$	0	0
$517$	−0.369298	−0.0162417
$518$	0	0
$519$	12.7822	0.561078
$520$	0	0
$521$	11.7432	0.514479	0.257240	−	0.966348i	$-0.417187\pi$
$521$	11.7432	0.514479	0.257240	+	0.966348i	$0.417187\pi$
$522$	0	0
$523$	−1.40510	−0.0614406	−0.0307203	−	0.999528i	$-0.509780\pi$
$523$	−1.40510	−0.0614406	−0.0307203	+	0.999528i	$0.509780\pi$
$524$	0	0
$525$	9.50314	0.414751
$526$	0	0
$527$	17.4148	0.758599
$528$	0	0
$529$	−7.38961	−0.321287
$530$	0	0
$531$	−90.5381	−3.92902
$532$	0	0
$533$	0.233783	0.0101263
$534$	0	0
$535$	−0.0618977	−0.00267607
$536$	0	0
$537$	10.7705	0.464780
$538$	0	0
$539$	1.43680	0.0618873
$540$	0	0
$541$	−26.9545	−1.15886	−0.579431	−	0.815021i	$-0.696725\pi$
$541$	−26.9545	−1.15886	−0.579431	+	0.815021i	$0.696725\pi$
$542$	0	0
$543$	−3.33915	−0.143297
$544$	0	0
$545$	4.67543	0.200273
$546$	0	0
$547$	−22.8459	−0.976818	−0.488409	−	0.872615i	$-0.662423\pi$
$547$	−22.8459	−0.976818	−0.488409	+	0.872615i	$0.662423\pi$
$548$	0	0
$549$	5.59781	0.238909
$550$	0	0
$551$	−49.0000	−2.08747
$552$	0	0
$553$	−3.31503	−0.140969
$554$	0	0
$555$	7.07470	0.300304
$556$	0	0
$557$	11.0065	0.466360	0.233180	−	0.972434i	$-0.425087\pi$
$557$	11.0065	0.466360	0.233180	+	0.972434i	$0.425087\pi$
$558$	0	0
$559$	−2.69731	−0.114084
$560$	0	0
$561$	11.1382	0.470255
$562$	0	0
$563$	33.7486	1.42234	0.711168	−	0.703022i	$-0.248167\pi$
$563$	33.7486	1.42234	0.711168	+	0.703022i	$0.248167\pi$
$564$	0	0
$565$	2.66331	0.112046
$566$	0	0
$567$	−65.5516	−2.75291
$568$	0	0
$569$	21.8206	0.914767	0.457384	−	0.889270i	$-0.348787\pi$
$569$	21.8206	0.914767	0.457384	+	0.889270i	$0.348787\pi$
$570$	0	0
$571$	−38.1536	−1.59668	−0.798339	−	0.602208i	$-0.794288\pi$
$571$	−38.1536	−1.59668	−0.798339	+	0.602208i	$0.794288\pi$
$572$	0	0
$573$	78.6896	3.28730
$574$	0	0
$575$	−3.95100	−0.164768
$576$	0	0
$577$	12.2491	0.509936	0.254968	−	0.966949i	$-0.417935\pi$
$577$	12.2491	0.509936	0.254968	+	0.966949i	$0.417935\pi$
$578$	0	0
$579$	16.8641	0.700847
$580$	0	0
$581$	36.2786	1.50509
$582$	0	0
$583$	−9.81637	−0.406553
$584$	0	0
$585$	5.98795	0.247571
$586$	0	0
$587$	−15.8888	−0.655799	−0.327899	−	0.944713i	$-0.606341\pi$
$587$	−15.8888	−0.655799	−0.327899	+	0.944713i	$0.606341\pi$
$588$	0	0
$589$	23.6905	0.976150
$590$	0	0
$591$	−21.2599	−0.874516
$592$	0	0
$593$	23.3069	0.957101	0.478550	−	0.878060i	$-0.341162\pi$
$593$	23.3069	0.957101	0.478550	+	0.878060i	$0.341162\pi$
$594$	0	0
$595$	−12.8212	−0.525620
$596$	0	0
$597$	21.8689	0.895033
$598$	0	0
$599$	−1.41860	−0.0579624	−0.0289812	−	0.999580i	$-0.509226\pi$
$599$	−1.41860	−0.0579624	−0.0289812	+	0.999580i	$0.509226\pi$
$600$	0	0
$601$	−23.5320	−0.959890	−0.479945	−	0.877299i	$-0.659343\pi$
$601$	−23.5320	−0.959890	−0.479945	+	0.877299i	$0.659343\pi$
$602$	0	0
$603$	−39.4681	−1.60726
$604$	0	0
$605$	10.3546	0.420973
$606$	0	0
$607$	−14.6341	−0.593978	−0.296989	−	0.954881i	$-0.595982\pi$
$607$	−14.6341	−0.593978	−0.296989	+	0.954881i	$0.595982\pi$
$608$	0	0
$609$	79.1464	3.20717
$610$	0	0
$611$	0.378299	0.0153043
$612$	0	0
$613$	−24.6911	−0.997263	−0.498632	−	0.866814i	$-0.666164\pi$
$613$	−24.6911	−0.997263	−0.498632	+	0.866814i	$0.666164\pi$
$614$	0	0
$615$	−0.910623	−0.0367199
$616$	0	0
$617$	−24.6394	−0.991945	−0.495973	−	0.868338i	$-0.665188\pi$
$617$	−24.6394	−0.991945	−0.495973	+	0.868338i	$0.665188\pi$
$618$	0	0
$619$	−10.7947	−0.433874	−0.216937	−	0.976186i	$-0.569607\pi$
$619$	−10.7947	−0.433874	−0.216937	+	0.976186i	$0.569607\pi$
$620$	0	0
$621$	54.1570	2.17325
$622$	0	0
$623$	−43.6444	−1.74858
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	15.1521	0.605115
$628$	0	0
$629$	−9.54489	−0.380580
$630$	0	0
$631$	39.3162	1.56515	0.782577	−	0.622554i	$-0.213905\pi$
$631$	39.3162	1.56515	0.782577	+	0.622554i	$0.213905\pi$
$632$	0	0
$633$	−6.64145	−0.263974
$634$	0	0
$635$	−7.52209	−0.298505
$636$	0	0
$637$	−1.47182	−0.0583156
$638$	0	0
$639$	14.4631	0.572152
$640$	0	0
$641$	−13.8861	−0.548468	−0.274234	−	0.961663i	$-0.588424\pi$
$641$	−13.8861	−0.548468	−0.274234	+	0.961663i	$0.588424\pi$
$642$	0	0
$643$	36.3122	1.43201	0.716006	−	0.698094i	$-0.245968\pi$
$643$	36.3122	1.43201	0.716006	+	0.698094i	$0.245968\pi$
$644$	0	0
$645$	10.5065	0.413692
$646$	0	0
$647$	31.6255	1.24333	0.621663	−	0.783285i	$-0.286457\pi$
$647$	31.6255	1.24333	0.621663	+	0.783285i	$0.286457\pi$
$648$	0	0
$649$	−9.99695	−0.392415
$650$	0	0
$651$	−38.2657	−1.49975
$652$	0	0
$653$	−30.0500	−1.17595	−0.587973	−	0.808881i	$-0.700074\pi$
$653$	−30.0500	−1.17595	−0.587973	+	0.808881i	$0.700074\pi$
$654$	0	0
$655$	−0.940352	−0.0367426
$656$	0	0
$657$	−13.1195	−0.511841
$658$	0	0
$659$	−42.7519	−1.66538	−0.832689	−	0.553741i	$-0.813200\pi$
$659$	−42.7519	−1.66538	−0.832689	+	0.553741i	$0.813200\pi$
$660$	0	0
$661$	−11.6754	−0.454120	−0.227060	−	0.973881i	$-0.572911\pi$
$661$	−11.6754	−0.454120	−0.227060	+	0.973881i	$0.572911\pi$
$662$	0	0
$663$	−11.4097	−0.443115
$664$	0	0
$665$	−17.4416	−0.676358
$666$	0	0
$667$	−32.9057	−1.27411
$668$	0	0
$669$	−17.6631	−0.682894
$670$	0	0
$671$	0.618094	0.0238612
$672$	0	0
$673$	23.8482	0.919280	0.459640	−	0.888105i	$-0.347978\pi$
$673$	23.8482	0.919280	0.459640	+	0.888105i	$0.347978\pi$
$674$	0	0
$675$	−13.7072	−0.527589
$676$	0	0
$677$	18.7995	0.722524	0.361262	−	0.932464i	$-0.382346\pi$
$677$	18.7995	0.722524	0.361262	+	0.932464i	$0.382346\pi$
$678$	0	0
$679$	41.5464	1.59440
$680$	0	0
$681$	72.0720	2.76181
$682$	0	0
$683$	20.7242	0.792988	0.396494	−	0.918037i	$-0.370227\pi$
$683$	20.7242	0.792988	0.396494	+	0.918037i	$0.370227\pi$
$684$	0	0
$685$	2.21303	0.0845555
$686$	0	0
$687$	−36.3925	−1.38846
$688$	0	0
$689$	10.0556	0.383089
$690$	0	0
$691$	−24.7988	−0.943389	−0.471695	−	0.881762i	$-0.656358\pi$
$691$	−24.7988	−0.943389	−0.471695	+	0.881762i	$0.656358\pi$
$692$	0	0
$693$	−17.3291	−0.658276
$694$	0	0
$695$	20.7636	0.787608
$696$	0	0
$697$	1.22858	0.0465356
$698$	0	0
$699$	47.5464	1.79837
$700$	0	0
$701$	19.6499	0.742167	0.371084	−	0.928599i	$-0.378986\pi$
$701$	19.6499	0.742167	0.371084	+	0.928599i	$0.378986\pi$
$702$	0	0
$703$	−12.9846	−0.489723
$704$	0	0
$705$	−1.47354	−0.0554966
$706$	0	0
$707$	44.7277	1.68216
$708$	0	0
$709$	13.0933	0.491730	0.245865	−	0.969304i	$-0.420928\pi$
$709$	13.0933	0.491730	0.245865	+	0.969304i	$0.420928\pi$
$710$	0	0
$711$	8.13625	0.305133
$712$	0	0
$713$	15.9092	0.595805
$714$	0	0
$715$	0.661171	0.0247264
$716$	0	0
$717$	−29.0887	−1.08634
$718$	0	0
$719$	−24.3250	−0.907170	−0.453585	−	0.891213i	$-0.649855\pi$
$719$	−24.3250	−0.907170	−0.453585	+	0.891213i	$0.649855\pi$
$720$	0	0
$721$	1.00277	0.0373450
$722$	0	0
$723$	−66.5643	−2.47555
$724$	0	0
$725$	8.32845	0.309311
$726$	0	0
$727$	−8.60441	−0.319120	−0.159560	−	0.987188i	$-0.551008\pi$
$727$	−8.60441	−0.319120	−0.159560	+	0.987188i	$0.551008\pi$
$728$	0	0
$729$	29.0667	1.07654
$730$	0	0
$731$	−14.1749	−0.524278
$732$	0	0
$733$	−20.5355	−0.758497	−0.379248	−	0.925295i	$-0.623817\pi$
$733$	−20.5355	−0.758497	−0.379248	+	0.925295i	$0.623817\pi$
$734$	0	0
$735$	5.73297	0.211464
$736$	0	0
$737$	−4.35795	−0.160527
$738$	0	0
$739$	4.72543	0.173828	0.0869138	−	0.996216i	$-0.472300\pi$
$739$	4.72543	0.173828	0.0869138	+	0.996216i	$0.472300\pi$
$740$	0	0
$741$	−15.5214	−0.570191
$742$	0	0
$743$	13.4118	0.492031	0.246016	−	0.969266i	$-0.420879\pi$
$743$	13.4118	0.492031	0.246016	+	0.969266i	$0.420879\pi$
$744$	0	0
$745$	4.25428	0.155865
$746$	0	0
$747$	−89.0405	−3.25782
$748$	0	0
$749$	−0.183497	−0.00670485
$750$	0	0
$751$	38.3470	1.39930	0.699651	−	0.714485i	$-0.253339\pi$
$751$	38.3470	1.39930	0.699651	+	0.714485i	$0.253339\pi$
$752$	0	0
$753$	−85.7487	−3.12486
$754$	0	0
$755$	13.6616	0.497196
$756$	0	0
$757$	29.5779	1.07503	0.537514	−	0.843255i	$-0.319364\pi$
$757$	29.5779	1.07503	0.537514	+	0.843255i	$0.319364\pi$
$758$	0	0
$759$	10.1753	0.369339
$760$	0	0
$761$	−13.6591	−0.495143	−0.247572	−	0.968870i	$-0.579633\pi$
$761$	−13.6591	−0.495143	−0.247572	+	0.968870i	$0.579633\pi$
$762$	0	0
$763$	13.8604	0.501781
$764$	0	0
$765$	31.4678	1.13772
$766$	0	0
$767$	10.2406	0.369767
$768$	0	0
$769$	−25.0231	−0.902356	−0.451178	−	0.892434i	$-0.648996\pi$
$769$	−25.0231	−0.902356	−0.451178	+	0.892434i	$0.648996\pi$
$770$	0	0
$771$	−46.9442	−1.69065
$772$	0	0
$773$	18.1089	0.651331	0.325666	−	0.945485i	$-0.394412\pi$
$773$	18.1089	0.651331	0.325666	+	0.945485i	$0.394412\pi$
$774$	0	0
$775$	−4.02664	−0.144641
$776$	0	0
$777$	20.9731	0.752406
$778$	0	0
$779$	1.67132	0.0598811
$780$	0	0
$781$	1.59698	0.0571443
$782$	0	0
$783$	−114.159	−4.07973
$784$	0	0
$785$	13.8297	0.493603
$786$	0	0
$787$	23.5546	0.839632	0.419816	−	0.907609i	$-0.362095\pi$
$787$	23.5546	0.839632	0.419816	+	0.907609i	$0.362095\pi$
$788$	0	0
$789$	−39.2295	−1.39661
$790$	0	0
$791$	7.89544	0.280730
$792$	0	0
$793$	−0.633159	−0.0224841
$794$	0	0
$795$	−39.1683	−1.38916
$796$	0	0
$797$	3.65847	0.129590	0.0647948	−	0.997899i	$-0.479361\pi$
$797$	3.65847	0.129590	0.0647948	+	0.997899i	$0.479361\pi$
$798$	0	0
$799$	1.98803	0.0703316
$800$	0	0
$801$	107.119	3.78485
$802$	0	0
$803$	−1.44862	−0.0511206
$804$	0	0
$805$	−11.7128	−0.412823
$806$	0	0
$807$	16.2111	0.570656
$808$	0	0
$809$	25.0116	0.879361	0.439681	−	0.898154i	$-0.355092\pi$
$809$	25.0116	0.879361	0.439681	+	0.898154i	$0.355092\pi$
$810$	0	0
$811$	−24.7281	−0.868321	−0.434160	−	0.900836i	$-0.642955\pi$
$811$	−24.7281	−0.868321	−0.434160	+	0.900836i	$0.642955\pi$
$812$	0	0
$813$	−68.0226	−2.38566
$814$	0	0
$815$	−14.9886	−0.525027
$816$	0	0
$817$	−19.2831	−0.674630
$818$	0	0
$819$	17.7514	0.620285
$820$	0	0
$821$	7.34552	0.256361	0.128180	−	0.991751i	$-0.459086\pi$
$821$	7.34552	0.256361	0.128180	+	0.991751i	$0.459086\pi$
$822$	0	0
$823$	17.4613	0.608664	0.304332	−	0.952566i	$-0.401567\pi$
$823$	17.4613	0.608664	0.304332	+	0.952566i	$0.401567\pi$
$824$	0	0
$825$	−2.57537	−0.0896629
$826$	0	0
$827$	−39.5733	−1.37610	−0.688049	−	0.725665i	$-0.741532\pi$
$827$	−39.5733	−1.37610	−0.688049	+	0.725665i	$0.741532\pi$
$828$	0	0
$829$	−54.8002	−1.90329	−0.951645	−	0.307199i	$-0.900608\pi$
$829$	−54.8002	−1.90329	−0.951645	+	0.307199i	$0.900608\pi$
$830$	0	0
$831$	58.6450	2.03437
$832$	0	0
$833$	−7.73469	−0.267991
$834$	0	0
$835$	19.0434	0.659025
$836$	0	0
$837$	55.1938	1.90778
$838$	0	0
$839$	−14.3105	−0.494052	−0.247026	−	0.969009i	$-0.579453\pi$
$839$	−14.3105	−0.494052	−0.247026	+	0.969009i	$0.579453\pi$
$840$	0	0
$841$	40.3630	1.39183
$842$	0	0
$843$	−70.1966	−2.41770
$844$	0	0
$845$	12.3227	0.423914
$846$	0	0
$847$	30.6964	1.05474
$848$	0	0
$849$	−53.3723	−1.83173
$850$	0	0
$851$	−8.71973	−0.298908
$852$	0	0
$853$	30.7688	1.05350	0.526751	−	0.850020i	$-0.323410\pi$
$853$	30.7688	1.05350	0.526751	+	0.850020i	$0.323410\pi$
$854$	0	0
$855$	42.8079	1.46400
$856$	0	0
$857$	−35.5127	−1.21309	−0.606545	−	0.795049i	$-0.707445\pi$
$857$	−35.5127	−1.21309	−0.606545	+	0.795049i	$0.707445\pi$
$858$	0	0
$859$	−26.4082	−0.901038	−0.450519	−	0.892767i	$-0.648761\pi$
$859$	−26.4082	−0.901038	−0.450519	+	0.892767i	$0.648761\pi$
$860$	0	0
$861$	−2.69956	−0.0920009
$862$	0	0
$863$	17.7126	0.602945	0.301472	−	0.953475i	$-0.402522\pi$
$863$	17.7126	0.602945	0.301472	+	0.953475i	$0.402522\pi$
$864$	0	0
$865$	3.98745	0.135577
$866$	0	0
$867$	−5.46454	−0.185586
$868$	0	0
$869$	0.898381	0.0304755
$870$	0	0
$871$	4.46417	0.151262
$872$	0	0
$873$	−101.969	−3.45114
$874$	0	0
$875$	2.96453	0.100219
$876$	0	0
$877$	−41.0391	−1.38579	−0.692896	−	0.721037i	$-0.743666\pi$
$877$	−41.0391	−1.38579	−0.692896	+	0.721037i	$0.743666\pi$
$878$	0	0
$879$	62.5104	2.10842
$880$	0	0
$881$	57.3981	1.93379	0.966895	−	0.255174i	$-0.0821327\pi$
$881$	57.3981	1.93379	0.966895	+	0.255174i	$0.0821327\pi$
$882$	0	0
$883$	−58.3052	−1.96213	−0.981063	−	0.193687i	$-0.937956\pi$
$883$	−58.3052	−1.96213	−0.981063	+	0.193687i	$0.937956\pi$
$884$	0	0
$885$	−39.8888	−1.34085
$886$	0	0
$887$	56.9076	1.91077	0.955385	−	0.295362i	$-0.0954402\pi$
$887$	56.9076	1.91077	0.955385	+	0.295362i	$0.0954402\pi$
$888$	0	0
$889$	−22.2994	−0.747899
$890$	0	0
$891$	17.7646	0.595137
$892$	0	0
$893$	2.70446	0.0905013
$894$	0	0
$895$	3.35987	0.112308
$896$	0	0
$897$	−10.4233	−0.348023
$898$	0	0
$899$	−33.5356	−1.11848
$900$	0	0
$901$	52.8443	1.76050
$902$	0	0
$903$	31.1467	1.03650
$904$	0	0
$905$	−1.04165	−0.0346258
$906$	0	0
$907$	17.1392	0.569097	0.284548	−	0.958662i	$-0.408156\pi$
$907$	17.1392	0.569097	0.284548	+	0.958662i	$0.408156\pi$
$908$	0	0
$909$	−109.777	−3.64109
$910$	0	0
$911$	5.39001	0.178579	0.0892895	−	0.996006i	$-0.471540\pi$
$911$	5.39001	0.178579	0.0892895	+	0.996006i	$0.471540\pi$
$912$	0	0
$913$	−9.83159	−0.325378
$914$	0	0
$915$	2.46626	0.0815319
$916$	0	0
$917$	−2.78770	−0.0920579
$918$	0	0
$919$	−23.7307	−0.782805	−0.391403	−	0.920220i	$-0.628010\pi$
$919$	−23.7307	−0.782805	−0.391403	+	0.920220i	$0.628010\pi$
$920$	0	0
$921$	−58.1160	−1.91499
$922$	0	0
$923$	−1.63590	−0.0538463
$924$	0	0
$925$	2.20697	0.0725647
$926$	0	0
$927$	−2.46114	−0.0808344
$928$	0	0
$929$	50.2519	1.64871	0.824356	−	0.566072i	$-0.191537\pi$
$929$	50.2519	1.64871	0.824356	+	0.566072i	$0.191537\pi$
$930$	0	0
$931$	−10.5220	−0.344846
$932$	0	0
$933$	62.7057	2.05289
$934$	0	0
$935$	3.47459	0.113631
$936$	0	0
$937$	33.1352	1.08248	0.541240	−	0.840868i	$-0.317955\pi$
$937$	33.1352	1.08248	0.541240	+	0.840868i	$0.317955\pi$
$938$	0	0
$939$	−43.1647	−1.40863
$940$	0	0
$941$	2.70070	0.0880404	0.0440202	−	0.999031i	$-0.485983\pi$
$941$	2.70070	0.0880404	0.0440202	+	0.999031i	$0.485983\pi$
$942$	0	0
$943$	1.12236	0.0365492
$944$	0	0
$945$	−40.6353	−1.32187
$946$	0	0
$947$	−14.1591	−0.460109	−0.230055	−	0.973178i	$-0.573890\pi$
$947$	−14.1591	−0.460109	−0.230055	+	0.973178i	$0.573890\pi$
$948$	0	0
$949$	1.48393	0.0481703
$950$	0	0
$951$	−15.5186	−0.503224
$952$	0	0
$953$	−28.6248	−0.927247	−0.463624	−	0.886032i	$-0.653451\pi$
$953$	−28.6248	−0.927247	−0.463624	+	0.886032i	$0.653451\pi$
$954$	0	0
$955$	24.5474	0.794335
$956$	0	0
$957$	−21.4488	−0.693342
$958$	0	0
$959$	6.56059	0.211852
$960$	0	0
$961$	−14.7862	−0.476974
$962$	0	0
$963$	0.450367	0.0145129
$964$	0	0
$965$	5.26079	0.169351
$966$	0	0
$967$	−1.51458	−0.0487057	−0.0243529	−	0.999703i	$-0.507753\pi$
$967$	−1.51458	−0.0487057	−0.0243529	+	0.999703i	$0.507753\pi$
$968$	0	0
$969$	−81.5678	−2.62034
$970$	0	0
$971$	54.2848	1.74208	0.871041	−	0.491211i	$-0.163445\pi$
$971$	54.2848	1.74208	0.871041	+	0.491211i	$0.163445\pi$
$972$	0	0
$973$	61.5542	1.97334
$974$	0	0
$975$	2.63814	0.0844881
$976$	0	0
$977$	−18.9218	−0.605362	−0.302681	−	0.953092i	$-0.597882\pi$
$977$	−18.9218	−0.605362	−0.302681	+	0.953092i	$0.597882\pi$
$978$	0	0
$979$	11.8277	0.378016
$980$	0	0
$981$	−34.0183	−1.08612
$982$	0	0
$983$	−41.5472	−1.32515	−0.662575	−	0.748995i	$-0.730536\pi$
$983$	−41.5472	−1.32515	−0.662575	+	0.748995i	$0.730536\pi$
$984$	0	0
$985$	−6.63208	−0.211316
$986$	0	0
$987$	−4.36833	−0.139046
$988$	0	0
$989$	−12.9495	−0.411769
$990$	0	0
$991$	−9.78225	−0.310744	−0.155372	−	0.987856i	$-0.549658\pi$
$991$	−9.78225	−0.310744	−0.155372	+	0.987856i	$0.549658\pi$
$992$	0	0
$993$	−31.6844	−1.00547
$994$	0	0
$995$	6.82204	0.216273
$996$	0	0
$997$	50.3865	1.59576	0.797878	−	0.602819i	$-0.205956\pi$
$997$	50.3865	1.59576	0.797878	+	0.602819i	$0.205956\pi$
$998$	0	0
$999$	−30.2513	−0.957108

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8020.2.a.e.1.2	✓	35

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8020.2.a.e.1.2	✓	35	1.1	even	1	trivial

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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)

\(f(q)\)	\(=\)	\(q-3.20562 q^{3} -1.00000 q^{5} -2.96453 q^{7} +7.27598 q^{9} +O(q^{10})\)	\(q-3.20562 q^{3} -1.00000 q^{5} -2.96453 q^{7} +7.27598 q^{9} +0.803393 q^{11} -0.822974 q^{13} +3.20562 q^{15} -4.32489 q^{17} -5.88345 q^{19} +9.50314 q^{21} -3.95100 q^{23} +1.00000 q^{25} -13.7072 q^{27} +8.32845 q^{29} -4.02664 q^{31} -2.57537 q^{33} +2.96453 q^{35} +2.20697 q^{37} +2.63814 q^{39} -0.284071 q^{41} +3.27752 q^{43} -7.27598 q^{45} -0.459673 q^{47} +1.78841 q^{49} +13.8639 q^{51} -12.2186 q^{53} -0.803393 q^{55} +18.8601 q^{57} -12.4434 q^{59} +0.769354 q^{61} -21.5698 q^{63} +0.822974 q^{65} -5.42443 q^{67} +12.6654 q^{69} +1.98779 q^{71} -1.80313 q^{73} -3.20562 q^{75} -2.38168 q^{77} +1.11823 q^{79} +22.1120 q^{81} -12.2376 q^{83} +4.32489 q^{85} -26.6978 q^{87} +14.7222 q^{89} +2.43973 q^{91} +12.9079 q^{93} +5.88345 q^{95} -14.0145 q^{97} +5.84547 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 35 q - q^{3} - 35 q^{5} + 6 q^{7} + 52 q^{9}+O(q^{10}) \) 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

Varieties

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Database

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