LMFDB - Embedded newform 8020.2.a.c.1.8

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:	$28$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
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)

$f(q)$	$=$	$q-1.53415 q^{3} -1.00000 q^{5} +0.759833 q^{7} -0.646394 q^{9} +O(q^{10})$	$q-1.53415 q^{3} -1.00000 q^{5} +0.759833 q^{7} -0.646394 q^{9} -2.75985 q^{11} -4.86468 q^{13} +1.53415 q^{15} +0.830882 q^{17} -3.43525 q^{19} -1.16570 q^{21} -1.12980 q^{23} +1.00000 q^{25} +5.59410 q^{27} +9.75363 q^{29} +8.03408 q^{31} +4.23402 q^{33} -0.759833 q^{35} +5.57488 q^{37} +7.46314 q^{39} -4.04451 q^{41} +6.17253 q^{43} +0.646394 q^{45} +3.41376 q^{47} -6.42265 q^{49} -1.27469 q^{51} +5.80300 q^{53} +2.75985 q^{55} +5.27017 q^{57} +0.686320 q^{59} +7.65711 q^{61} -0.491151 q^{63} +4.86468 q^{65} -0.747223 q^{67} +1.73328 q^{69} -11.8382 q^{71} -2.72697 q^{73} -1.53415 q^{75} -2.09703 q^{77} +14.7572 q^{79} -6.64299 q^{81} -10.0970 q^{83} -0.830882 q^{85} -14.9635 q^{87} +0.776847 q^{89} -3.69635 q^{91} -12.3255 q^{93} +3.43525 q^{95} -1.20228 q^{97} +1.78395 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 3 q^{3} - 28 q^{5} - 4 q^{7} + 17 q^{9}+O(q^{10}) $ 28 * q + 3 * q^3 - 28 * q^5 - 4 * q^7 + 17 * q^9	$ 28 q + 3 q^{3} - 28 q^{5} - 4 q^{7} + 17 q^{9} + 2 q^{11} + 3 q^{13} - 3 q^{15} - 10 q^{17} - 2 q^{19} - 12 q^{21} - 23 q^{23} + 28 q^{25} + 9 q^{27} - 37 q^{29} - 11 q^{31} + 2 q^{33} + 4 q^{35} - 3 q^{37} - 19 q^{39} - 30 q^{41} + 13 q^{43} - 17 q^{45} - 15 q^{47} + 12 q^{49} - 8 q^{51} - 35 q^{53} - 2 q^{55} - 22 q^{57} - q^{59} - 33 q^{61} - 20 q^{63} - 3 q^{65} + 19 q^{67} - 8 q^{69} - 31 q^{71} + 31 q^{73} + 3 q^{75} - 42 q^{77} - 29 q^{79} - 36 q^{81} + 14 q^{83} + 10 q^{85} - 32 q^{87} - 32 q^{89} - 7 q^{91} - 11 q^{93} + 2 q^{95} + 2 q^{97} - 39 q^{99}+O(q^{100}) $ 28 * q + 3 * q^3 - 28 * q^5 - 4 * q^7 + 17 * q^9 + 2 * q^11 + 3 * q^13 - 3 * q^15 - 10 * q^17 - 2 * q^19 - 12 * q^21 - 23 * q^23 + 28 * q^25 + 9 * q^27 - 37 * q^29 - 11 * q^31 + 2 * q^33 + 4 * q^35 - 3 * q^37 - 19 * q^39 - 30 * q^41 + 13 * q^43 - 17 * q^45 - 15 * q^47 + 12 * q^49 - 8 * q^51 - 35 * q^53 - 2 * q^55 - 22 * q^57 - q^59 - 33 * q^61 - 20 * q^63 - 3 * q^65 + 19 * q^67 - 8 * q^69 - 31 * q^71 + 31 * q^73 + 3 * q^75 - 42 * q^77 - 29 * q^79 - 36 * q^81 + 14 * q^83 + 10 * q^85 - 32 * q^87 - 32 * q^89 - 7 * q^91 - 11 * q^93 + 2 * q^95 + 2 * q^97 - 39 * 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
$3$	−1.53415	−0.885740	−0.442870	−	0.896586i	$-0.646040\pi$
$3$	−1.53415	−0.885740	−0.442870	+	0.896586i	$0.646040\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	0.759833	0.287190	0.143595	−	0.989637i	$-0.454134\pi$
$7$	0.759833	0.287190	0.143595	+	0.989637i	$0.454134\pi$
$8$	0	0
$9$	−0.646394	−0.215465
$10$	0	0
$11$	−2.75985	−0.832127	−0.416064	−	0.909335i	$-0.636591\pi$
$11$	−2.75985	−0.832127	−0.416064	+	0.909335i	$0.636591\pi$
$12$	0	0
$13$	−4.86468	−1.34922	−0.674610	−	0.738174i	$-0.735688\pi$
$13$	−4.86468	−1.34922	−0.674610	+	0.738174i	$0.735688\pi$
$14$	0	0
$15$	1.53415	0.396115
$16$	0	0
$17$	0.830882	0.201518	0.100759	−	0.994911i	$-0.467873\pi$
$17$	0.830882	0.201518	0.100759	+	0.994911i	$0.467873\pi$
$18$	0	0
$19$	−3.43525	−0.788099	−0.394050	−	0.919089i	$-0.628926\pi$
$19$	−3.43525	−0.788099	−0.394050	+	0.919089i	$0.628926\pi$
$20$	0	0
$21$	−1.16570	−0.254376
$22$	0	0
$23$	−1.12980	−0.235579	−0.117790	−	0.993039i	$-0.537581\pi$
$23$	−1.12980	−0.235579	−0.117790	+	0.993039i	$0.537581\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.59410	1.07659
$28$	0	0
$29$	9.75363	1.81120	0.905602	−	0.424130i	$-0.139420\pi$
$29$	9.75363	1.81120	0.905602	+	0.424130i	$0.139420\pi$
$30$	0	0
$31$	8.03408	1.44296	0.721482	−	0.692434i	$-0.243461\pi$
$31$	8.03408	1.44296	0.721482	+	0.692434i	$0.243461\pi$
$32$	0	0
$33$	4.23402	0.737049
$34$	0	0
$35$	−0.759833	−0.128435
$36$	0	0
$37$	5.57488	0.916505	0.458252	−	0.888822i	$-0.348476\pi$
$37$	5.57488	0.916505	0.458252	+	0.888822i	$0.348476\pi$
$38$	0	0
$39$	7.46314	1.19506
$40$	0	0
$41$	−4.04451	−0.631646	−0.315823	−	0.948818i	$-0.602281\pi$
$41$	−4.04451	−0.631646	−0.315823	+	0.948818i	$0.602281\pi$
$42$	0	0
$43$	6.17253	0.941302	0.470651	−	0.882320i	$-0.344019\pi$
$43$	6.17253	0.941302	0.470651	+	0.882320i	$0.344019\pi$
$44$	0	0
$45$	0.646394	0.0963587
$46$	0	0
$47$	3.41376	0.497948	0.248974	−	0.968510i	$-0.419907\pi$
$47$	3.41376	0.497948	0.248974	+	0.968510i	$0.419907\pi$
$48$	0	0
$49$	−6.42265	−0.917522
$50$	0	0
$51$	−1.27469	−0.178493
$52$	0	0
$53$	5.80300	0.797103	0.398552	−	0.917146i	$-0.369513\pi$
$53$	5.80300	0.797103	0.398552	+	0.917146i	$0.369513\pi$
$54$	0	0
$55$	2.75985	0.372139
$56$	0	0
$57$	5.27017	0.698051
$58$	0	0
$59$	0.686320	0.0893513	0.0446756	−	0.999002i	$-0.485775\pi$
$59$	0.686320	0.0893513	0.0446756	+	0.999002i	$0.485775\pi$
$60$	0	0
$61$	7.65711	0.980393	0.490196	−	0.871612i	$-0.336925\pi$
$61$	7.65711	0.980393	0.490196	+	0.871612i	$0.336925\pi$
$62$	0	0
$63$	−0.491151	−0.0618793
$64$	0	0
$65$	4.86468	0.603390
$66$	0	0
$67$	−0.747223	−0.0912879	−0.0456439	−	0.998958i	$-0.514534\pi$
$67$	−0.747223	−0.0912879	−0.0456439	+	0.998958i	$0.514534\pi$
$68$	0	0
$69$	1.73328	0.208662
$70$	0	0
$71$	−11.8382	−1.40493	−0.702465	−	0.711718i	$-0.747917\pi$
$71$	−11.8382	−1.40493	−0.702465	+	0.711718i	$0.747917\pi$
$72$	0	0
$73$	−2.72697	−0.319168	−0.159584	−	0.987184i	$-0.551015\pi$
$73$	−2.72697	−0.319168	−0.159584	+	0.987184i	$0.551015\pi$
$74$	0	0
$75$	−1.53415	−0.177148
$76$	0	0
$77$	−2.09703	−0.238979
$78$	0	0
$79$	14.7572	1.66032	0.830159	−	0.557526i	$-0.188249\pi$
$79$	14.7572	1.66032	0.830159	+	0.557526i	$0.188249\pi$
$80$	0	0
$81$	−6.64299	−0.738111
$82$	0	0
$83$	−10.0970	−1.10829	−0.554145	−	0.832420i	$-0.686955\pi$
$83$	−10.0970	−1.10829	−0.554145	+	0.832420i	$0.686955\pi$
$84$	0	0
$85$	−0.830882	−0.0901218
$86$	0	0
$87$	−14.9635	−1.60426
$88$	0	0
$89$	0.776847	0.0823456	0.0411728	−	0.999152i	$-0.486891\pi$
$89$	0.776847	0.0823456	0.0411728	+	0.999152i	$0.486891\pi$
$90$	0	0
$91$	−3.69635	−0.387483
$92$	0	0
$93$	−12.3255	−1.27809
$94$	0	0
$95$	3.43525	0.352449
$96$	0	0
$97$	−1.20228	−0.122073	−0.0610365	−	0.998136i	$-0.519441\pi$
$97$	−1.20228	−0.122073	−0.0610365	+	0.998136i	$0.519441\pi$
$98$	0	0
$99$	1.78395	0.179294
$100$	0	0
$101$	6.46240	0.643032	0.321516	−	0.946904i	$-0.395808\pi$
$101$	6.46240	0.643032	0.321516	+	0.946904i	$0.395808\pi$
$102$	0	0
$103$	−3.29037	−0.324209	−0.162105	−	0.986774i	$-0.551828\pi$
$103$	−3.29037	−0.324209	−0.162105	+	0.986774i	$0.551828\pi$
$104$	0	0
$105$	1.16570	0.113760
$106$	0	0
$107$	−5.44880	−0.526755	−0.263378	−	0.964693i	$-0.584837\pi$
$107$	−5.44880	−0.526755	−0.263378	+	0.964693i	$0.584837\pi$
$108$	0	0
$109$	−0.521292	−0.0499307	−0.0249653	−	0.999688i	$-0.507948\pi$
$109$	−0.521292	−0.0499307	−0.0249653	+	0.999688i	$0.507948\pi$
$110$	0	0
$111$	−8.55268	−0.811785
$112$	0	0
$113$	−7.94408	−0.747316	−0.373658	−	0.927567i	$-0.621897\pi$
$113$	−7.94408	−0.747316	−0.373658	+	0.927567i	$0.621897\pi$
$114$	0	0
$115$	1.12980	0.105354
$116$	0	0
$117$	3.14450	0.290709
$118$	0	0
$119$	0.631332	0.0578741
$120$	0	0
$121$	−3.38321	−0.307564
$122$	0	0
$123$	6.20487	0.559474
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−3.24379	−0.287840	−0.143920	−	0.989589i	$-0.545971\pi$
$127$	−3.24379	−0.287840	−0.143920	+	0.989589i	$0.545971\pi$
$128$	0	0
$129$	−9.46956	−0.833749
$130$	0	0
$131$	−9.24112	−0.807400	−0.403700	−	0.914891i	$-0.632276\pi$
$131$	−9.24112	−0.807400	−0.403700	+	0.914891i	$0.632276\pi$
$132$	0	0
$133$	−2.61021	−0.226334
$134$	0	0
$135$	−5.59410	−0.481464
$136$	0	0
$137$	−15.6925	−1.34070	−0.670349	−	0.742046i	$-0.733856\pi$
$137$	−15.6925	−1.34070	−0.670349	+	0.742046i	$0.733856\pi$
$138$	0	0
$139$	4.81186	0.408137	0.204069	−	0.978957i	$-0.434583\pi$
$139$	4.81186	0.408137	0.204069	+	0.978957i	$0.434583\pi$
$140$	0	0
$141$	−5.23721	−0.441053
$142$	0	0
$143$	13.4258	1.12272
$144$	0	0
$145$	−9.75363	−0.809995
$146$	0	0
$147$	9.85329	0.812686
$148$	0	0
$149$	−5.22558	−0.428096	−0.214048	−	0.976823i	$-0.568665\pi$
$149$	−5.22558	−0.428096	−0.214048	+	0.976823i	$0.568665\pi$
$150$	0	0
$151$	−13.9772	−1.13745	−0.568723	−	0.822529i	$-0.692562\pi$
$151$	−13.9772	−1.13745	−0.568723	+	0.822529i	$0.692562\pi$
$152$	0	0
$153$	−0.537077	−0.0434201
$154$	0	0
$155$	−8.03408	−0.645313
$156$	0	0
$157$	13.2320	1.05603	0.528013	−	0.849236i	$-0.322937\pi$
$157$	13.2320	1.05603	0.528013	+	0.849236i	$0.322937\pi$
$158$	0	0
$159$	−8.90265	−0.706026
$160$	0	0
$161$	−0.858459	−0.0676560
$162$	0	0
$163$	−7.04005	−0.551419	−0.275710	−	0.961241i	$-0.588913\pi$
$163$	−7.04005	−0.551419	−0.275710	+	0.961241i	$0.588913\pi$
$164$	0	0
$165$	−4.23402	−0.329618
$166$	0	0
$167$	13.0422	1.00924	0.504619	−	0.863342i	$-0.331633\pi$
$167$	13.0422	1.00924	0.504619	+	0.863342i	$0.331633\pi$
$168$	0	0
$169$	10.6652	0.820396
$170$	0	0
$171$	2.22052	0.169807
$172$	0	0
$173$	16.8374	1.28012	0.640060	−	0.768325i	$-0.278909\pi$
$173$	16.8374	1.28012	0.640060	+	0.768325i	$0.278909\pi$
$174$	0	0
$175$	0.759833	0.0574380
$176$	0	0
$177$	−1.05292	−0.0791420
$178$	0	0
$179$	4.08801	0.305552	0.152776	−	0.988261i	$-0.451179\pi$
$179$	4.08801	0.305552	0.152776	+	0.988261i	$0.451179\pi$
$180$	0	0
$181$	−20.0459	−1.49000	−0.745000	−	0.667064i	$-0.767551\pi$
$181$	−20.0459	−1.49000	−0.745000	+	0.667064i	$0.767551\pi$
$182$	0	0
$183$	−11.7471	−0.868373
$184$	0	0
$185$	−5.57488	−0.409873
$186$	0	0
$187$	−2.29311	−0.167689
$188$	0	0
$189$	4.25059	0.309185
$190$	0	0
$191$	11.6572	0.843487	0.421743	−	0.906715i	$-0.361418\pi$
$191$	11.6572	0.843487	0.421743	+	0.906715i	$0.361418\pi$
$192$	0	0
$193$	−17.4242	−1.25422	−0.627109	−	0.778931i	$-0.715762\pi$
$193$	−17.4242	−1.25422	−0.627109	+	0.778931i	$0.715762\pi$
$194$	0	0
$195$	−7.46314	−0.534447
$196$	0	0
$197$	−3.97131	−0.282944	−0.141472	−	0.989942i	$-0.545184\pi$
$197$	−3.97131	−0.282944	−0.141472	+	0.989942i	$0.545184\pi$
$198$	0	0
$199$	−13.8110	−0.979035	−0.489517	−	0.871994i	$-0.662827\pi$
$199$	−13.8110	−0.979035	−0.489517	+	0.871994i	$0.662827\pi$
$200$	0	0
$201$	1.14635	0.0808573
$202$	0	0
$203$	7.41113	0.520159
$204$	0	0
$205$	4.04451	0.282481
$206$	0	0
$207$	0.730295	0.0507590
$208$	0	0
$209$	9.48078	0.655799
$210$	0	0
$211$	−3.28172	−0.225923	−0.112962	−	0.993599i	$-0.536034\pi$
$211$	−3.28172	−0.225923	−0.112962	+	0.993599i	$0.536034\pi$
$212$	0	0
$213$	18.1615	1.24440
$214$	0	0
$215$	−6.17253	−0.420963
$216$	0	0
$217$	6.10456	0.414405
$218$	0	0
$219$	4.18358	0.282700
$220$	0	0
$221$	−4.04198	−0.271893
$222$	0	0
$223$	18.7777	1.25745	0.628725	−	0.777628i	$-0.283577\pi$
$223$	18.7777	1.25745	0.628725	+	0.777628i	$0.283577\pi$
$224$	0	0
$225$	−0.646394	−0.0430929
$226$	0	0
$227$	11.9728	0.794662	0.397331	−	0.917675i	$-0.369936\pi$
$227$	11.9728	0.794662	0.397331	+	0.917675i	$0.369936\pi$
$228$	0	0
$229$	−23.4706	−1.55098	−0.775491	−	0.631358i	$-0.782498\pi$
$229$	−23.4706	−1.55098	−0.775491	+	0.631358i	$0.782498\pi$
$230$	0	0
$231$	3.21715	0.211673
$232$	0	0
$233$	−8.02403	−0.525672	−0.262836	−	0.964841i	$-0.584658\pi$
$233$	−8.02403	−0.525672	−0.262836	+	0.964841i	$0.584658\pi$
$234$	0	0
$235$	−3.41376	−0.222689
$236$	0	0
$237$	−22.6398	−1.47061
$238$	0	0
$239$	12.8059	0.828344	0.414172	−	0.910199i	$-0.364071\pi$
$239$	12.8059	0.828344	0.414172	+	0.910199i	$0.364071\pi$
$240$	0	0
$241$	−22.4734	−1.44764	−0.723818	−	0.689991i	$-0.757614\pi$
$241$	−22.4734	−1.44764	−0.723818	+	0.689991i	$0.757614\pi$
$242$	0	0
$243$	−6.59098	−0.422812
$244$	0	0
$245$	6.42265	0.410328
$246$	0	0
$247$	16.7114	1.06332
$248$	0	0
$249$	15.4903	0.981656
$250$	0	0
$251$	−5.19371	−0.327824	−0.163912	−	0.986475i	$-0.552411\pi$
$251$	−5.19371	−0.327824	−0.163912	+	0.986475i	$0.552411\pi$
$252$	0	0
$253$	3.11808	0.196032
$254$	0	0
$255$	1.27469	0.0798245
$256$	0	0
$257$	19.3741	1.20852	0.604262	−	0.796785i	$-0.293468\pi$
$257$	19.3741	1.20852	0.604262	+	0.796785i	$0.293468\pi$
$258$	0	0
$259$	4.23598	0.263211
$260$	0	0
$261$	−6.30468	−0.390250
$262$	0	0
$263$	−10.7948	−0.665635	−0.332817	−	0.942991i	$-0.607999\pi$
$263$	−10.7948	−0.665635	−0.332817	+	0.942991i	$0.607999\pi$
$264$	0	0
$265$	−5.80300	−0.356475
$266$	0	0
$267$	−1.19180	−0.0729368
$268$	0	0
$269$	3.85567	0.235084	0.117542	−	0.993068i	$-0.462498\pi$
$269$	3.85567	0.235084	0.117542	+	0.993068i	$0.462498\pi$
$270$	0	0
$271$	28.3032	1.71930	0.859650	−	0.510884i	$-0.170682\pi$
$271$	28.3032	1.71930	0.859650	+	0.510884i	$0.170682\pi$
$272$	0	0
$273$	5.67074	0.343209
$274$	0	0
$275$	−2.75985	−0.166425
$276$	0	0
$277$	31.7035	1.90488	0.952441	−	0.304724i	$-0.0985641\pi$
$277$	31.7035	1.90488	0.952441	+	0.304724i	$0.0985641\pi$
$278$	0	0
$279$	−5.19318	−0.310907
$280$	0	0
$281$	−19.1110	−1.14007	−0.570034	−	0.821621i	$-0.693070\pi$
$281$	−19.1110	−1.14007	−0.570034	+	0.821621i	$0.693070\pi$
$282$	0	0
$283$	28.9062	1.71829	0.859147	−	0.511729i	$-0.170995\pi$
$283$	28.9062	1.71829	0.859147	+	0.511729i	$0.170995\pi$
$284$	0	0
$285$	−5.27017	−0.312178
$286$	0	0
$287$	−3.07315	−0.181402
$288$	0	0
$289$	−16.3096	−0.959390
$290$	0	0
$291$	1.84447	0.108125
$292$	0	0
$293$	23.4457	1.36971	0.684857	−	0.728677i	$-0.259865\pi$
$293$	23.4457	1.36971	0.684857	+	0.728677i	$0.259865\pi$
$294$	0	0
$295$	−0.686320	−0.0399591
$296$	0	0
$297$	−15.4389	−0.895856
$298$	0	0
$299$	5.49612	0.317849
$300$	0	0
$301$	4.69009	0.270332
$302$	0	0
$303$	−9.91427	−0.569560
$304$	0	0
$305$	−7.65711	−0.438445
$306$	0	0
$307$	13.3216	0.760305	0.380153	−	0.924924i	$-0.375871\pi$
$307$	13.3216	0.760305	0.380153	+	0.924924i	$0.375871\pi$
$308$	0	0
$309$	5.04790	0.287165
$310$	0	0
$311$	−22.8500	−1.29570	−0.647851	−	0.761767i	$-0.724332\pi$
$311$	−22.8500	−1.29570	−0.647851	+	0.761767i	$0.724332\pi$
$312$	0	0
$313$	−14.0076	−0.791759	−0.395879	−	0.918303i	$-0.629560\pi$
$313$	−14.0076	−0.791759	−0.395879	+	0.918303i	$0.629560\pi$
$314$	0	0
$315$	0.491151	0.0276732
$316$	0	0
$317$	10.7526	0.603924	0.301962	−	0.953320i	$-0.402358\pi$
$317$	10.7526	0.603924	0.301962	+	0.953320i	$0.402358\pi$
$318$	0	0
$319$	−26.9186	−1.50715
$320$	0	0
$321$	8.35926	0.466568
$322$	0	0
$323$	−2.85428	−0.158817
$324$	0	0
$325$	−4.86468	−0.269844
$326$	0	0
$327$	0.799738	0.0442256
$328$	0	0
$329$	2.59389	0.143006
$330$	0	0
$331$	15.8279	0.869979	0.434989	−	0.900436i	$-0.356752\pi$
$331$	15.8279	0.869979	0.434989	+	0.900436i	$0.356752\pi$
$332$	0	0
$333$	−3.60357	−0.197474
$334$	0	0
$335$	0.747223	0.0408252
$336$	0	0
$337$	−35.6496	−1.94196	−0.970979	−	0.239164i	$-0.923127\pi$
$337$	−35.6496	−1.94196	−0.970979	+	0.239164i	$0.923127\pi$
$338$	0	0
$339$	12.1874	0.661928
$340$	0	0
$341$	−22.1729	−1.20073
$342$	0	0
$343$	−10.1990	−0.550693
$344$	0	0
$345$	−1.73328	−0.0933165
$346$	0	0
$347$	−27.9865	−1.50239	−0.751196	−	0.660079i	$-0.770523\pi$
$347$	−27.9865	−1.50239	−0.751196	+	0.660079i	$0.770523\pi$
$348$	0	0
$349$	31.2666	1.67366	0.836831	−	0.547461i	$-0.184405\pi$
$349$	31.2666	1.67366	0.836831	+	0.547461i	$0.184405\pi$
$350$	0	0
$351$	−27.2135	−1.45255
$352$	0	0
$353$	13.7901	0.733973	0.366986	−	0.930226i	$-0.380390\pi$
$353$	13.7901	0.733973	0.366986	+	0.930226i	$0.380390\pi$
$354$	0	0
$355$	11.8382	0.628304
$356$	0	0
$357$	−0.968555	−0.0512614
$358$	0	0
$359$	−10.5760	−0.558180	−0.279090	−	0.960265i	$-0.590033\pi$
$359$	−10.5760	−0.558180	−0.279090	+	0.960265i	$0.590033\pi$
$360$	0	0
$361$	−7.19909	−0.378900
$362$	0	0
$363$	5.19033	0.272422
$364$	0	0
$365$	2.72697	0.142736
$366$	0	0
$367$	6.01748	0.314110	0.157055	−	0.987590i	$-0.449800\pi$
$367$	6.01748	0.314110	0.157055	+	0.987590i	$0.449800\pi$
$368$	0	0
$369$	2.61434	0.136097
$370$	0	0
$371$	4.40931	0.228920
$372$	0	0
$373$	−11.1189	−0.575715	−0.287858	−	0.957673i	$-0.592943\pi$
$373$	−11.1189	−0.575715	−0.287858	+	0.957673i	$0.592943\pi$
$374$	0	0
$375$	1.53415	0.0792230
$376$	0	0
$377$	−47.4483	−2.44371
$378$	0	0
$379$	−27.3547	−1.40512	−0.702559	−	0.711626i	$-0.747959\pi$
$379$	−27.3547	−1.40512	−0.702559	+	0.711626i	$0.747959\pi$
$380$	0	0
$381$	4.97646	0.254952
$382$	0	0
$383$	−35.7722	−1.82788	−0.913938	−	0.405855i	$-0.866974\pi$
$383$	−35.7722	−1.82788	−0.913938	+	0.405855i	$0.866974\pi$
$384$	0	0
$385$	2.09703	0.106874
$386$	0	0
$387$	−3.98988	−0.202817
$388$	0	0
$389$	−30.2658	−1.53454	−0.767269	−	0.641325i	$-0.778385\pi$
$389$	−30.2658	−1.53454	−0.767269	+	0.641325i	$0.778385\pi$
$390$	0	0
$391$	−0.938729	−0.0474736
$392$	0	0
$393$	14.1772	0.715147
$394$	0	0
$395$	−14.7572	−0.742517
$396$	0	0
$397$	−13.6559	−0.685370	−0.342685	−	0.939450i	$-0.611336\pi$
$397$	−13.6559	−0.685370	−0.342685	+	0.939450i	$0.611336\pi$
$398$	0	0
$399$	4.00445	0.200473
$400$	0	0
$401$	1.00000	0.0499376
$401$	1.00000	0.0499376
$402$	0	0
$403$	−39.0833	−1.94688
$404$	0	0
$405$	6.64299	0.330093
$406$	0	0
$407$	−15.3859	−0.762649
$408$	0	0
$409$	3.69437	0.182675	0.0913373	−	0.995820i	$-0.470886\pi$
$409$	3.69437	0.182675	0.0913373	+	0.995820i	$0.470886\pi$
$410$	0	0
$411$	24.0746	1.18751
$412$	0	0
$413$	0.521489	0.0256608
$414$	0	0
$415$	10.0970	0.495642
$416$	0	0
$417$	−7.38211	−0.361503
$418$	0	0
$419$	8.55113	0.417750	0.208875	−	0.977942i	$-0.433020\pi$
$419$	8.55113	0.417750	0.208875	+	0.977942i	$0.433020\pi$
$420$	0	0
$421$	−16.6874	−0.813294	−0.406647	−	0.913585i	$-0.633302\pi$
$421$	−16.6874	−0.813294	−0.406647	+	0.913585i	$0.633302\pi$
$422$	0	0
$423$	−2.20663	−0.107290
$424$	0	0
$425$	0.830882	0.0403037
$426$	0	0
$427$	5.81813	0.281559
$428$	0	0
$429$	−20.5972	−0.994441
$430$	0	0
$431$	−18.8109	−0.906089	−0.453045	−	0.891488i	$-0.649662\pi$
$431$	−18.8109	−0.906089	−0.453045	+	0.891488i	$0.649662\pi$
$432$	0	0
$433$	−15.7460	−0.756703	−0.378352	−	0.925662i	$-0.623509\pi$
$433$	−15.7460	−0.756703	−0.378352	+	0.925662i	$0.623509\pi$
$434$	0	0
$435$	14.9635	0.717445
$436$	0	0
$437$	3.88114	0.185660
$438$	0	0
$439$	−0.595803	−0.0284361	−0.0142181	−	0.999899i	$-0.504526\pi$
$439$	−0.595803	−0.0284361	−0.0142181	+	0.999899i	$0.504526\pi$
$440$	0	0
$441$	4.15156	0.197693
$442$	0	0
$443$	−3.15079	−0.149698	−0.0748492	−	0.997195i	$-0.523848\pi$
$443$	−3.15079	−0.149698	−0.0748492	+	0.997195i	$0.523848\pi$
$444$	0	0
$445$	−0.776847	−0.0368261
$446$	0	0
$447$	8.01681	0.379182
$448$	0	0
$449$	28.8574	1.36187	0.680933	−	0.732346i	$-0.261575\pi$
$449$	28.8574	1.36187	0.680933	+	0.732346i	$0.261575\pi$
$450$	0	0
$451$	11.1623	0.525610
$452$	0	0
$453$	21.4430	1.00748
$454$	0	0
$455$	3.69635	0.173288
$456$	0	0
$457$	0.990165	0.0463179	0.0231590	−	0.999732i	$-0.492628\pi$
$457$	0.990165	0.0463179	0.0231590	+	0.999732i	$0.492628\pi$
$458$	0	0
$459$	4.64804	0.216952
$460$	0	0
$461$	0.967691	0.0450699	0.0225349	−	0.999746i	$-0.492826\pi$
$461$	0.967691	0.0450699	0.0225349	+	0.999746i	$0.492826\pi$
$462$	0	0
$463$	20.3287	0.944755	0.472378	−	0.881396i	$-0.343396\pi$
$463$	20.3287	0.944755	0.472378	+	0.881396i	$0.343396\pi$
$464$	0	0
$465$	12.3255	0.571579
$466$	0	0
$467$	−38.8600	−1.79823	−0.899114	−	0.437714i	$-0.855788\pi$
$467$	−38.8600	−1.79823	−0.899114	+	0.437714i	$0.855788\pi$
$468$	0	0
$469$	−0.567765	−0.0262170
$470$	0	0
$471$	−20.2998	−0.935365
$472$	0	0
$473$	−17.0353	−0.783283
$474$	0	0
$475$	−3.43525	−0.157620
$476$	0	0
$477$	−3.75102	−0.171747
$478$	0	0
$479$	−1.66519	−0.0760844	−0.0380422	−	0.999276i	$-0.512112\pi$
$479$	−1.66519	−0.0760844	−0.0380422	+	0.999276i	$0.512112\pi$
$480$	0	0
$481$	−27.1200	−1.23657
$482$	0	0
$483$	1.31700	0.0599257
$484$	0	0
$485$	1.20228	0.0545927
$486$	0	0
$487$	4.83758	0.219212	0.109606	−	0.993975i	$-0.465041\pi$
$487$	4.83758	0.219212	0.109606	+	0.993975i	$0.465041\pi$
$488$	0	0
$489$	10.8005	0.488414
$490$	0	0
$491$	31.0655	1.40197	0.700984	−	0.713177i	$-0.252745\pi$
$491$	31.0655	1.40197	0.700984	+	0.713177i	$0.252745\pi$
$492$	0	0
$493$	8.10411	0.364991
$494$	0	0
$495$	−1.78395	−0.0801827
$496$	0	0
$497$	−8.99502	−0.403482
$498$	0	0
$499$	22.5844	1.01102	0.505509	−	0.862821i	$-0.331305\pi$
$499$	22.5844	1.01102	0.505509	+	0.862821i	$0.331305\pi$
$500$	0	0
$501$	−20.0087	−0.893923
$502$	0	0
$503$	20.1548	0.898660	0.449330	−	0.893366i	$-0.351663\pi$
$503$	20.1548	0.898660	0.449330	+	0.893366i	$0.351663\pi$
$504$	0	0
$505$	−6.46240	−0.287573
$506$	0	0
$507$	−16.3619	−0.726658
$508$	0	0
$509$	−28.5080	−1.26359	−0.631797	−	0.775134i	$-0.717682\pi$
$509$	−28.5080	−1.26359	−0.631797	+	0.775134i	$0.717682\pi$
$510$	0	0
$511$	−2.07204	−0.0916618
$512$	0	0
$513$	−19.2171	−0.848456
$514$	0	0
$515$	3.29037	0.144991
$516$	0	0
$517$	−9.42148	−0.414356
$518$	0	0
$519$	−25.8310	−1.13385
$520$	0	0
$521$	−11.3868	−0.498866	−0.249433	−	0.968392i	$-0.580244\pi$
$521$	−11.3868	−0.498866	−0.249433	+	0.968392i	$0.580244\pi$
$522$	0	0
$523$	−13.3107	−0.582036	−0.291018	−	0.956718i	$-0.593994\pi$
$523$	−13.3107	−0.582036	−0.291018	+	0.956718i	$0.593994\pi$
$524$	0	0
$525$	−1.16570	−0.0508751
$526$	0	0
$527$	6.67537	0.290784
$528$	0	0
$529$	−21.7236	−0.944502
$530$	0	0
$531$	−0.443633	−0.0192520
$532$	0	0
$533$	19.6753	0.852230
$534$	0	0
$535$	5.44880	0.235572
$536$	0	0
$537$	−6.27160	−0.270640
$538$	0	0
$539$	17.7256	0.763495
$540$	0	0
$541$	2.85131	0.122587	0.0612936	−	0.998120i	$-0.480477\pi$
$541$	2.85131	0.122587	0.0612936	+	0.998120i	$0.480477\pi$
$542$	0	0
$543$	30.7534	1.31975
$544$	0	0
$545$	0.521292	0.0223297
$546$	0	0
$547$	25.4604	1.08861	0.544305	−	0.838887i	$-0.316793\pi$
$547$	25.4604	1.08861	0.544305	+	0.838887i	$0.316793\pi$
$548$	0	0
$549$	−4.94951	−0.211240
$550$	0	0
$551$	−33.5061	−1.42741
$552$	0	0
$553$	11.2130	0.476827
$554$	0	0
$555$	8.55268	0.363041
$556$	0	0
$557$	−8.97044	−0.380090	−0.190045	−	0.981775i	$-0.560863\pi$
$557$	−8.97044	−0.380090	−0.190045	+	0.981775i	$0.560863\pi$
$558$	0	0
$559$	−30.0274	−1.27002
$560$	0	0
$561$	3.51797	0.148529
$562$	0	0
$563$	27.9367	1.17739	0.588695	−	0.808355i	$-0.299642\pi$
$563$	27.9367	1.17739	0.588695	+	0.808355i	$0.299642\pi$
$564$	0	0
$565$	7.94408	0.334210
$566$	0	0
$567$	−5.04757	−0.211978
$568$	0	0
$569$	−41.2323	−1.72855	−0.864274	−	0.503021i	$-0.832222\pi$
$569$	−41.2323	−1.72855	−0.864274	+	0.503021i	$0.832222\pi$
$570$	0	0
$571$	39.1954	1.64028	0.820139	−	0.572164i	$-0.193896\pi$
$571$	39.1954	1.64028	0.820139	+	0.572164i	$0.193896\pi$
$572$	0	0
$573$	−17.8839	−0.747110
$574$	0	0
$575$	−1.12980	−0.0471159
$576$	0	0
$577$	38.7606	1.61363	0.806813	−	0.590807i	$-0.201190\pi$
$577$	38.7606	1.61363	0.806813	+	0.590807i	$0.201190\pi$
$578$	0	0
$579$	26.7312	1.11091
$580$	0	0
$581$	−7.67203	−0.318290
$582$	0	0
$583$	−16.0154	−0.663291
$584$	0	0
$585$	−3.14450	−0.130009
$586$	0	0
$587$	−22.5665	−0.931418	−0.465709	−	0.884938i	$-0.654201\pi$
$587$	−22.5665	−0.931418	−0.465709	+	0.884938i	$0.654201\pi$
$588$	0	0
$589$	−27.5990	−1.13720
$590$	0	0
$591$	6.09257	0.250615
$592$	0	0
$593$	47.7257	1.95986	0.979929	−	0.199347i	$-0.0638820\pi$
$593$	47.7257	1.95986	0.979929	+	0.199347i	$0.0638820\pi$
$594$	0	0
$595$	−0.631332	−0.0258821
$596$	0	0
$597$	21.1881	0.867170
$598$	0	0
$599$	28.1861	1.15165	0.575826	−	0.817572i	$-0.304681\pi$
$599$	28.1861	1.15165	0.575826	+	0.817572i	$0.304681\pi$
$600$	0	0
$601$	−45.3474	−1.84976	−0.924880	−	0.380259i	$-0.875835\pi$
$601$	−45.3474	−1.84976	−0.924880	+	0.380259i	$0.875835\pi$
$602$	0	0
$603$	0.483000	0.0196693
$604$	0	0
$605$	3.38321	0.137547
$606$	0	0
$607$	28.2277	1.14573	0.572864	−	0.819651i	$-0.305832\pi$
$607$	28.2277	1.14573	0.572864	+	0.819651i	$0.305832\pi$
$608$	0	0
$609$	−11.3698	−0.460726
$610$	0	0
$611$	−16.6069	−0.671842
$612$	0	0
$613$	−25.9000	−1.04609	−0.523046	−	0.852304i	$-0.675204\pi$
$613$	−25.9000	−1.04609	−0.523046	+	0.852304i	$0.675204\pi$
$614$	0	0
$615$	−6.20487	−0.250204
$616$	0	0
$617$	0.401050	0.0161457	0.00807283	−	0.999967i	$-0.497430\pi$
$617$	0.401050	0.0161457	0.00807283	+	0.999967i	$0.497430\pi$
$618$	0	0
$619$	−25.8802	−1.04021	−0.520107	−	0.854101i	$-0.674108\pi$
$619$	−25.8802	−1.04021	−0.520107	+	0.854101i	$0.674108\pi$
$620$	0	0
$621$	−6.32021	−0.253621
$622$	0	0
$623$	0.590274	0.0236488
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−14.5449	−0.580867
$628$	0	0
$629$	4.63207	0.184693
$630$	0	0
$631$	14.7018	0.585271	0.292636	−	0.956224i	$-0.405468\pi$
$631$	14.7018	0.585271	0.292636	+	0.956224i	$0.405468\pi$
$632$	0	0
$633$	5.03465	0.200109
$634$	0	0
$635$	3.24379	0.128726
$636$	0	0
$637$	31.2442	1.23794
$638$	0	0
$639$	7.65211	0.302713
$640$	0	0
$641$	−34.8309	−1.37574	−0.687868	−	0.725836i	$-0.741453\pi$
$641$	−34.8309	−1.37574	−0.687868	+	0.725836i	$0.741453\pi$
$642$	0	0
$643$	−2.39935	−0.0946213	−0.0473107	−	0.998880i	$-0.515065\pi$
$643$	−2.39935	−0.0946213	−0.0473107	+	0.998880i	$0.515065\pi$
$644$	0	0
$645$	9.46956	0.372864
$646$	0	0
$647$	−44.7501	−1.75931	−0.879653	−	0.475616i	$-0.842225\pi$
$647$	−44.7501	−1.75931	−0.879653	+	0.475616i	$0.842225\pi$
$648$	0	0
$649$	−1.89414	−0.0743517
$650$	0	0
$651$	−9.36529	−0.367055
$652$	0	0
$653$	−13.7635	−0.538606	−0.269303	−	0.963056i	$-0.586793\pi$
$653$	−13.7635	−0.538606	−0.269303	+	0.963056i	$0.586793\pi$
$654$	0	0
$655$	9.24112	0.361080
$656$	0	0
$657$	1.76270	0.0687694
$658$	0	0
$659$	−13.1460	−0.512096	−0.256048	−	0.966664i	$-0.582421\pi$
$659$	−13.1460	−0.512096	−0.256048	+	0.966664i	$0.582421\pi$
$660$	0	0
$661$	−38.0880	−1.48145	−0.740726	−	0.671808i	$-0.765518\pi$
$661$	−38.0880	−1.48145	−0.740726	+	0.671808i	$0.765518\pi$
$662$	0	0
$663$	6.20099	0.240826
$664$	0	0
$665$	2.61021	0.101220
$666$	0	0
$667$	−11.0196	−0.426682
$668$	0	0
$669$	−28.8078	−1.11377
$670$	0	0
$671$	−21.1325	−0.815812
$672$	0	0
$673$	−5.93599	−0.228816	−0.114408	−	0.993434i	$-0.536497\pi$
$673$	−5.93599	−0.228816	−0.114408	+	0.993434i	$0.536497\pi$
$674$	0	0
$675$	5.59410	0.215317
$676$	0	0
$677$	−3.80771	−0.146342	−0.0731712	−	0.997319i	$-0.523312\pi$
$677$	−3.80771	−0.146342	−0.0731712	+	0.997319i	$0.523312\pi$
$678$	0	0
$679$	−0.913531	−0.0350581
$680$	0	0
$681$	−18.3680	−0.703864
$682$	0	0
$683$	8.79800	0.336646	0.168323	−	0.985732i	$-0.446165\pi$
$683$	8.79800	0.336646	0.168323	+	0.985732i	$0.446165\pi$
$684$	0	0
$685$	15.6925	0.599579
$686$	0	0
$687$	36.0074	1.37377
$688$	0	0
$689$	−28.2297	−1.07547
$690$	0	0
$691$	13.0002	0.494551	0.247276	−	0.968945i	$-0.420465\pi$
$691$	13.0002	0.494551	0.247276	+	0.968945i	$0.420465\pi$
$692$	0	0
$693$	1.35551	0.0514914
$694$	0	0
$695$	−4.81186	−0.182524
$696$	0	0
$697$	−3.36051	−0.127288
$698$	0	0
$699$	12.3100	0.465609
$700$	0	0
$701$	−17.2602	−0.651910	−0.325955	−	0.945385i	$-0.605686\pi$
$701$	−17.2602	−0.651910	−0.325955	+	0.945385i	$0.605686\pi$
$702$	0	0
$703$	−19.1511	−0.722297
$704$	0	0
$705$	5.23721	0.197245
$706$	0	0
$707$	4.91034	0.184672
$708$	0	0
$709$	−11.0965	−0.416739	−0.208369	−	0.978050i	$-0.566816\pi$
$709$	−11.0965	−0.416739	−0.208369	+	0.978050i	$0.566816\pi$
$710$	0	0
$711$	−9.53898	−0.357740
$712$	0	0
$713$	−9.07690	−0.339932
$714$	0	0
$715$	−13.4258	−0.502097
$716$	0	0
$717$	−19.6461	−0.733697
$718$	0	0
$719$	−27.0873	−1.01019	−0.505093	−	0.863065i	$-0.668542\pi$
$719$	−27.0873	−1.01019	−0.505093	+	0.863065i	$0.668542\pi$
$720$	0	0
$721$	−2.50013	−0.0931097
$722$	0	0
$723$	34.4774	1.28223
$724$	0	0
$725$	9.75363	0.362241
$726$	0	0
$727$	36.4438	1.35163	0.675813	−	0.737074i	$-0.263793\pi$
$727$	36.4438	1.35163	0.675813	+	0.737074i	$0.263793\pi$
$728$	0	0
$729$	30.0405	1.11261
$730$	0	0
$731$	5.12864	0.189690
$732$	0	0
$733$	−3.67189	−0.135624	−0.0678122	−	0.997698i	$-0.521602\pi$
$733$	−3.67189	−0.135624	−0.0678122	+	0.997698i	$0.521602\pi$
$734$	0	0
$735$	−9.85329	−0.363444
$736$	0	0
$737$	2.06223	0.0759631
$738$	0	0
$739$	26.8919	0.989233	0.494617	−	0.869111i	$-0.335308\pi$
$739$	26.8919	0.989233	0.494617	+	0.869111i	$0.335308\pi$
$740$	0	0
$741$	−25.6377	−0.941825
$742$	0	0
$743$	−31.6345	−1.16056	−0.580279	−	0.814418i	$-0.697056\pi$
$743$	−31.6345	−1.16056	−0.580279	+	0.814418i	$0.697056\pi$
$744$	0	0
$745$	5.22558	0.191450
$746$	0	0
$747$	6.52663	0.238797
$748$	0	0
$749$	−4.14018	−0.151279
$750$	0	0
$751$	−25.7965	−0.941329	−0.470664	−	0.882312i	$-0.655986\pi$
$751$	−25.7965	−0.941329	−0.470664	+	0.882312i	$0.655986\pi$
$752$	0	0
$753$	7.96791	0.290367
$754$	0	0
$755$	13.9772	0.508682
$756$	0	0
$757$	−8.21800	−0.298688	−0.149344	−	0.988785i	$-0.547716\pi$
$757$	−8.21800	−0.298688	−0.149344	+	0.988785i	$0.547716\pi$
$758$	0	0
$759$	−4.78359	−0.173633
$760$	0	0
$761$	−30.7715	−1.11547	−0.557733	−	0.830020i	$-0.688329\pi$
$761$	−30.7715	−1.11547	−0.557733	+	0.830020i	$0.688329\pi$
$762$	0	0
$763$	−0.396095	−0.0143396
$764$	0	0
$765$	0.537077	0.0194180
$766$	0	0
$767$	−3.33873	−0.120555
$768$	0	0
$769$	−2.79781	−0.100892	−0.0504458	−	0.998727i	$-0.516064\pi$
$769$	−2.79781	−0.100892	−0.0504458	+	0.998727i	$0.516064\pi$
$770$	0	0
$771$	−29.7228	−1.07044
$772$	0	0
$773$	18.1152	0.651559	0.325779	−	0.945446i	$-0.394373\pi$
$773$	18.1152	0.651559	0.325779	+	0.945446i	$0.394373\pi$
$774$	0	0
$775$	8.03408	0.288593
$776$	0	0
$777$	−6.49861	−0.233136
$778$	0	0
$779$	13.8939	0.497800
$780$	0	0
$781$	32.6716	1.16908
$782$	0	0
$783$	54.5628	1.94992
$784$	0	0
$785$	−13.2320	−0.472269
$786$	0	0
$787$	10.7805	0.384282	0.192141	−	0.981367i	$-0.438457\pi$
$787$	10.7805	0.384282	0.192141	+	0.981367i	$0.438457\pi$
$788$	0	0
$789$	16.5608	0.589580
$790$	0	0
$791$	−6.03618	−0.214622
$792$	0	0
$793$	−37.2494	−1.32277
$794$	0	0
$795$	8.90265	0.315744
$796$	0	0
$797$	−0.891987	−0.0315958	−0.0157979	−	0.999875i	$-0.505029\pi$
$797$	−0.891987	−0.0315958	−0.0157979	+	0.999875i	$0.505029\pi$
$798$	0	0
$799$	2.83643	0.100346
$800$	0	0
$801$	−0.502149	−0.0177426
$802$	0	0
$803$	7.52605	0.265588
$804$	0	0
$805$	0.858459	0.0302567
$806$	0	0
$807$	−5.91517	−0.208224
$808$	0	0
$809$	−53.9355	−1.89627	−0.948135	−	0.317867i	$-0.897033\pi$
$809$	−53.9355	−1.89627	−0.948135	+	0.317867i	$0.897033\pi$
$810$	0	0
$811$	20.2727	0.711872	0.355936	−	0.934510i	$-0.384162\pi$
$811$	20.2727	0.711872	0.355936	+	0.934510i	$0.384162\pi$
$812$	0	0
$813$	−43.4213	−1.52285
$814$	0	0
$815$	7.04005	0.246602
$816$	0	0
$817$	−21.2041	−0.741839
$818$	0	0
$819$	2.38930	0.0834888
$820$	0	0
$821$	11.7045	0.408490	0.204245	−	0.978920i	$-0.434526\pi$
$821$	11.7045	0.408490	0.204245	+	0.978920i	$0.434526\pi$
$822$	0	0
$823$	−6.24375	−0.217643	−0.108822	−	0.994061i	$-0.534708\pi$
$823$	−6.24375	−0.217643	−0.108822	+	0.994061i	$0.534708\pi$
$824$	0	0
$825$	4.23402	0.147410
$826$	0	0
$827$	−20.8295	−0.724314	−0.362157	−	0.932117i	$-0.617960\pi$
$827$	−20.8295	−0.724314	−0.362157	+	0.932117i	$0.617960\pi$
$828$	0	0
$829$	−31.3402	−1.08849	−0.544245	−	0.838926i	$-0.683184\pi$
$829$	−31.3402	−1.08849	−0.544245	+	0.838926i	$0.683184\pi$
$830$	0	0
$831$	−48.6379	−1.68723
$832$	0	0
$833$	−5.33647	−0.184898
$834$	0	0
$835$	−13.0422	−0.451345
$836$	0	0
$837$	44.9435	1.55347
$838$	0	0
$839$	46.5853	1.60830	0.804151	−	0.594425i	$-0.202620\pi$
$839$	46.5853	1.60830	0.804151	+	0.594425i	$0.202620\pi$
$840$	0	0
$841$	66.1332	2.28046
$842$	0	0
$843$	29.3191	1.00980
$844$	0	0
$845$	−10.6652	−0.366892
$846$	0	0
$847$	−2.57067	−0.0883293
$848$	0	0
$849$	−44.3463	−1.52196
$850$	0	0
$851$	−6.29849	−0.215910
$852$	0	0
$853$	25.7874	0.882945	0.441472	−	0.897275i	$-0.354456\pi$
$853$	25.7874	0.882945	0.441472	+	0.897275i	$0.354456\pi$
$854$	0	0
$855$	−2.22052	−0.0759402
$856$	0	0
$857$	−31.3117	−1.06959	−0.534793	−	0.844983i	$-0.679610\pi$
$857$	−31.3117	−1.06959	−0.534793	+	0.844983i	$0.679610\pi$
$858$	0	0
$859$	−2.00145	−0.0682887	−0.0341443	−	0.999417i	$-0.510871\pi$
$859$	−2.00145	−0.0682887	−0.0341443	+	0.999417i	$0.510871\pi$
$860$	0	0
$861$	4.71467	0.160675
$862$	0	0
$863$	−28.2136	−0.960402	−0.480201	−	0.877158i	$-0.659436\pi$
$863$	−28.2136	−0.960402	−0.480201	+	0.877158i	$0.659436\pi$
$864$	0	0
$865$	−16.8374	−0.572487
$866$	0	0
$867$	25.0214	0.849770
$868$	0	0
$869$	−40.7278	−1.38160
$870$	0	0
$871$	3.63501	0.123167
$872$	0	0
$873$	0.777145	0.0263024
$874$	0	0
$875$	−0.759833	−0.0256871
$876$	0	0
$877$	53.4420	1.80461	0.902303	−	0.431102i	$-0.141875\pi$
$877$	53.4420	1.80461	0.902303	+	0.431102i	$0.141875\pi$
$878$	0	0
$879$	−35.9692	−1.21321
$880$	0	0
$881$	−32.6836	−1.10114	−0.550569	−	0.834790i	$-0.685589\pi$
$881$	−32.6836	−1.10114	−0.550569	+	0.834790i	$0.685589\pi$
$882$	0	0
$883$	53.8651	1.81271	0.906353	−	0.422521i	$-0.138855\pi$
$883$	53.8651	1.81271	0.906353	+	0.422521i	$0.138855\pi$
$884$	0	0
$885$	1.05292	0.0353934
$886$	0	0
$887$	−21.7792	−0.731275	−0.365638	−	0.930757i	$-0.619149\pi$
$887$	−21.7792	−0.731275	−0.365638	+	0.930757i	$0.619149\pi$
$888$	0	0
$889$	−2.46474	−0.0826648
$890$	0	0
$891$	18.3337	0.614202
$892$	0	0
$893$	−11.7271	−0.392432
$894$	0	0
$895$	−4.08801	−0.136647
$896$	0	0
$897$	−8.43185	−0.281531
$898$	0	0
$899$	78.3614	2.61350
$900$	0	0
$901$	4.82160	0.160631
$902$	0	0
$903$	−7.19529	−0.239444
$904$	0	0
$905$	20.0459	0.666348
$906$	0	0
$907$	−33.0195	−1.09639	−0.548197	−	0.836349i	$-0.684686\pi$
$907$	−33.0195	−1.09639	−0.548197	+	0.836349i	$0.684686\pi$
$908$	0	0
$909$	−4.17725	−0.138551
$910$	0	0
$911$	−0.911206	−0.0301896	−0.0150948	−	0.999886i	$-0.504805\pi$
$911$	−0.911206	−0.0301896	−0.0150948	+	0.999886i	$0.504805\pi$
$912$	0	0
$913$	27.8662	0.922238
$914$	0	0
$915$	11.7471	0.388348
$916$	0	0
$917$	−7.02171	−0.231877
$918$	0	0
$919$	8.51531	0.280894	0.140447	−	0.990088i	$-0.455146\pi$
$919$	8.51531	0.280894	0.140447	+	0.990088i	$0.455146\pi$
$920$	0	0
$921$	−20.4373	−0.673433
$922$	0	0
$923$	57.5889	1.89556
$924$	0	0
$925$	5.57488	0.183301
$926$	0	0
$927$	2.12687	0.0698556
$928$	0	0
$929$	29.6015	0.971195	0.485598	−	0.874182i	$-0.338602\pi$
$929$	29.6015	0.971195	0.485598	+	0.874182i	$0.338602\pi$
$930$	0	0
$931$	22.0634	0.723098
$932$	0	0
$933$	35.0552	1.14766
$934$	0	0
$935$	2.29311	0.0749928
$936$	0	0
$937$	−4.12820	−0.134863	−0.0674313	−	0.997724i	$-0.521480\pi$
$937$	−4.12820	−0.134863	−0.0674313	+	0.997724i	$0.521480\pi$
$938$	0	0
$939$	21.4898	0.701292
$940$	0	0
$941$	8.36517	0.272697	0.136348	−	0.990661i	$-0.456463\pi$
$941$	8.36517	0.272697	0.136348	+	0.990661i	$0.456463\pi$
$942$	0	0
$943$	4.56948	0.148803
$944$	0	0
$945$	−4.25059	−0.138272
$946$	0	0
$947$	49.6566	1.61362	0.806811	−	0.590810i	$-0.201192\pi$
$947$	49.6566	1.61362	0.806811	+	0.590810i	$0.201192\pi$
$948$	0	0
$949$	13.2659	0.430628
$950$	0	0
$951$	−16.4960	−0.534920
$952$	0	0
$953$	−46.3342	−1.50091	−0.750457	−	0.660920i	$-0.770166\pi$
$953$	−46.3342	−1.50091	−0.750457	+	0.660920i	$0.770166\pi$
$954$	0	0
$955$	−11.6572	−0.377219
$956$	0	0
$957$	41.2971	1.33494
$958$	0	0
$959$	−11.9237	−0.385035
$960$	0	0
$961$	33.5464	1.08214
$962$	0	0
$963$	3.52207	0.113497
$964$	0	0
$965$	17.4242	0.560904
$966$	0	0
$967$	−12.3455	−0.397005	−0.198503	−	0.980100i	$-0.563608\pi$
$967$	−12.3455	−0.397005	−0.198503	+	0.980100i	$0.563608\pi$
$968$	0	0
$969$	4.37889	0.140670
$970$	0	0
$971$	−44.2040	−1.41857	−0.709287	−	0.704920i	$-0.750983\pi$
$971$	−44.2040	−1.41857	−0.709287	+	0.704920i	$0.750983\pi$
$972$	0	0
$973$	3.65621	0.117213
$974$	0	0
$975$	7.46314	0.239012
$976$	0	0
$977$	−47.6278	−1.52375	−0.761875	−	0.647724i	$-0.775721\pi$
$977$	−47.6278	−1.52375	−0.761875	+	0.647724i	$0.775721\pi$
$978$	0	0
$979$	−2.14399	−0.0685221
$980$	0	0
$981$	0.336960	0.0107583
$982$	0	0
$983$	49.1756	1.56846	0.784229	−	0.620471i	$-0.213059\pi$
$983$	49.1756	1.56846	0.784229	+	0.620471i	$0.213059\pi$
$984$	0	0
$985$	3.97131	0.126536
$986$	0	0
$987$	−3.97941	−0.126666
$988$	0	0
$989$	−6.97372	−0.221751
$990$	0	0
$991$	−11.9666	−0.380131	−0.190065	−	0.981771i	$-0.560870\pi$
$991$	−11.9666	−0.380131	−0.190065	+	0.981771i	$0.560870\pi$
$992$	0	0
$993$	−24.2823	−0.770575
$994$	0	0
$995$	13.8110	0.437838
$996$	0	0
$997$	−25.2431	−0.799456	−0.399728	−	0.916634i	$-0.630895\pi$
$997$	−25.2431	−0.799456	−0.399728	+	0.916634i	$0.630895\pi$
$998$	0	0
$999$	31.1865	0.986696

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8020.2.a.c.1.8	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8020.2.a.c.1.8	✓	28	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)

\(f(q)\)	\(=\)	\(q-1.53415 q^{3} -1.00000 q^{5} +0.759833 q^{7} -0.646394 q^{9} +O(q^{10})\)	\(q-1.53415 q^{3} -1.00000 q^{5} +0.759833 q^{7} -0.646394 q^{9} -2.75985 q^{11} -4.86468 q^{13} +1.53415 q^{15} +0.830882 q^{17} -3.43525 q^{19} -1.16570 q^{21} -1.12980 q^{23} +1.00000 q^{25} +5.59410 q^{27} +9.75363 q^{29} +8.03408 q^{31} +4.23402 q^{33} -0.759833 q^{35} +5.57488 q^{37} +7.46314 q^{39} -4.04451 q^{41} +6.17253 q^{43} +0.646394 q^{45} +3.41376 q^{47} -6.42265 q^{49} -1.27469 q^{51} +5.80300 q^{53} +2.75985 q^{55} +5.27017 q^{57} +0.686320 q^{59} +7.65711 q^{61} -0.491151 q^{63} +4.86468 q^{65} -0.747223 q^{67} +1.73328 q^{69} -11.8382 q^{71} -2.72697 q^{73} -1.53415 q^{75} -2.09703 q^{77} +14.7572 q^{79} -6.64299 q^{81} -10.0970 q^{83} -0.830882 q^{85} -14.9635 q^{87} +0.776847 q^{89} -3.69635 q^{91} -12.3255 q^{93} +3.43525 q^{95} -1.20228 q^{97} +1.78395 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 3 q^{3} - 28 q^{5} - 4 q^{7} + 17 q^{9}+O(q^{10}) \) 28 * q + 3 * q^3 - 28 * q^5 - 4 * q^7 + 17 * q^9	\( 28 q + 3 q^{3} - 28 q^{5} - 4 q^{7} + 17 q^{9} + 2 q^{11} + 3 q^{13} - 3 q^{15} - 10 q^{17} - 2 q^{19} - 12 q^{21} - 23 q^{23} + 28 q^{25} + 9 q^{27} - 37 q^{29} - 11 q^{31} + 2 q^{33} + 4 q^{35} - 3 q^{37} - 19 q^{39} - 30 q^{41} + 13 q^{43} - 17 q^{45} - 15 q^{47} + 12 q^{49} - 8 q^{51} - 35 q^{53} - 2 q^{55} - 22 q^{57} - q^{59} - 33 q^{61} - 20 q^{63} - 3 q^{65} + 19 q^{67} - 8 q^{69} - 31 q^{71} + 31 q^{73} + 3 q^{75} - 42 q^{77} - 29 q^{79} - 36 q^{81} + 14 q^{83} + 10 q^{85} - 32 q^{87} - 32 q^{89} - 7 q^{91} - 11 q^{93} + 2 q^{95} + 2 q^{97} - 39 q^{99}+O(q^{100}) \) 28 * q + 3 * q^3 - 28 * q^5 - 4 * q^7 + 17 * q^9 + 2 * q^11 + 3 * q^13 - 3 * q^15 - 10 * q^17 - 2 * q^19 - 12 * q^21 - 23 * q^23 + 28 * q^25 + 9 * q^27 - 37 * q^29 - 11 * q^31 + 2 * q^33 + 4 * q^35 - 3 * q^37 - 19 * q^39 - 30 * q^41 + 13 * q^43 - 17 * q^45 - 15 * q^47 + 12 * q^49 - 8 * q^51 - 35 * q^53 - 2 * q^55 - 22 * q^57 - q^59 - 33 * q^61 - 20 * q^63 - 3 * q^65 + 19 * q^67 - 8 * q^69 - 31 * q^71 + 31 * q^73 + 3 * q^75 - 42 * q^77 - 29 * q^79 - 36 * q^81 + 14 * q^83 + 10 * q^85 - 32 * q^87 - 32 * q^89 - 7 * q^91 - 11 * q^93 + 2 * q^95 + 2 * q^97 - 39 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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