LMFDB - Embedded newform 8020.2.a.f.1.3

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:	$37$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
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)

$f(q)$	$=$	$q-3.05263 q^{3} +1.00000 q^{5} +2.86058 q^{7} +6.31852 q^{9} +O(q^{10})$	$q-3.05263 q^{3} +1.00000 q^{5} +2.86058 q^{7} +6.31852 q^{9} -2.92493 q^{11} +2.59039 q^{13} -3.05263 q^{15} -3.00659 q^{17} -6.44674 q^{19} -8.73228 q^{21} -6.65623 q^{23} +1.00000 q^{25} -10.1302 q^{27} +0.745798 q^{29} -6.77167 q^{31} +8.92872 q^{33} +2.86058 q^{35} +3.42952 q^{37} -7.90748 q^{39} +10.4927 q^{41} -9.67354 q^{43} +6.31852 q^{45} +1.71217 q^{47} +1.18292 q^{49} +9.17798 q^{51} -1.95028 q^{53} -2.92493 q^{55} +19.6795 q^{57} +2.01945 q^{59} +4.02714 q^{61} +18.0746 q^{63} +2.59039 q^{65} +11.7041 q^{67} +20.3190 q^{69} +13.9308 q^{71} +1.82517 q^{73} -3.05263 q^{75} -8.36700 q^{77} +7.86911 q^{79} +11.9681 q^{81} -0.167275 q^{83} -3.00659 q^{85} -2.27664 q^{87} -3.06659 q^{89} +7.41001 q^{91} +20.6714 q^{93} -6.44674 q^{95} -2.34116 q^{97} -18.4812 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 37 q + 3 q^{3} + 37 q^{5} + 4 q^{7} + 50 q^{9}+O(q^{10}) $ 37 * q + 3 * q^3 + 37 * q^5 + 4 * q^7 + 50 * q^9	$ 37 q + 3 q^{3} + 37 q^{5} + 4 q^{7} + 50 q^{9} + 2 q^{11} + 27 q^{13} + 3 q^{15} + 36 q^{17} - 6 q^{19} + 20 q^{21} + 17 q^{23} + 37 q^{25} + 9 q^{27} + 29 q^{29} + 5 q^{31} + 36 q^{33} + 4 q^{35} + 35 q^{37} + 21 q^{39} + 24 q^{41} + 11 q^{43} + 50 q^{45} + 19 q^{47} + 57 q^{49} + 8 q^{51} + 65 q^{53} + 2 q^{55} + 62 q^{57} - 9 q^{59} + 13 q^{61} + 26 q^{63} + 27 q^{65} + 13 q^{67} + 20 q^{69} + 33 q^{71} + 67 q^{73} + 3 q^{75} + 62 q^{77} + 23 q^{79} + 97 q^{81} + 2 q^{83} + 36 q^{85} + 32 q^{87} + 34 q^{89} + q^{91} + 41 q^{93} - 6 q^{95} + 66 q^{97} - 7 q^{99}+O(q^{100}) $ 37 * q + 3 * q^3 + 37 * q^5 + 4 * q^7 + 50 * q^9 + 2 * q^11 + 27 * q^13 + 3 * q^15 + 36 * q^17 - 6 * q^19 + 20 * q^21 + 17 * q^23 + 37 * q^25 + 9 * q^27 + 29 * q^29 + 5 * q^31 + 36 * q^33 + 4 * q^35 + 35 * q^37 + 21 * q^39 + 24 * q^41 + 11 * q^43 + 50 * q^45 + 19 * q^47 + 57 * q^49 + 8 * q^51 + 65 * q^53 + 2 * q^55 + 62 * q^57 - 9 * q^59 + 13 * q^61 + 26 * q^63 + 27 * q^65 + 13 * q^67 + 20 * q^69 + 33 * q^71 + 67 * q^73 + 3 * q^75 + 62 * q^77 + 23 * q^79 + 97 * q^81 + 2 * q^83 + 36 * q^85 + 32 * q^87 + 34 * q^89 + q^91 + 41 * q^93 - 6 * q^95 + 66 * q^97 - 7 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−3.05263	−1.76243	−0.881217	−	0.472712i	$-0.843275\pi$
$3$	−3.05263	−1.76243	−0.881217	+	0.472712i	$0.843275\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	2.86058	1.08120	0.540599	−	0.841281i	$-0.318198\pi$
$7$	2.86058	1.08120	0.540599	+	0.841281i	$0.318198\pi$
$8$	0	0
$9$	6.31852	2.10617
$10$	0	0
$11$	−2.92493	−0.881900	−0.440950	−	0.897532i	$-0.645358\pi$
$11$	−2.92493	−0.881900	−0.440950	+	0.897532i	$0.645358\pi$
$12$	0	0
$13$	2.59039	0.718444	0.359222	−	0.933252i	$-0.383042\pi$
$13$	2.59039	0.718444	0.359222	+	0.933252i	$0.383042\pi$
$14$	0	0
$15$	−3.05263	−0.788184
$16$	0	0
$17$	−3.00659	−0.729204	−0.364602	−	0.931163i	$-0.618795\pi$
$17$	−3.00659	−0.729204	−0.364602	+	0.931163i	$0.618795\pi$
$18$	0	0
$19$	−6.44674	−1.47898	−0.739492	−	0.673165i	$-0.764934\pi$
$19$	−6.44674	−1.47898	−0.739492	+	0.673165i	$0.764934\pi$
$20$	0	0
$21$	−8.73228	−1.90554
$22$	0	0
$23$	−6.65623	−1.38792	−0.693960	−	0.720014i	$-0.744136\pi$
$23$	−6.65623	−1.38792	−0.693960	+	0.720014i	$0.744136\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−10.1302	−1.94956
$28$	0	0
$29$	0.745798	0.138491	0.0692456	−	0.997600i	$-0.477941\pi$
$29$	0.745798	0.138491	0.0692456	+	0.997600i	$0.477941\pi$
$30$	0	0
$31$	−6.77167	−1.21623	−0.608114	−	0.793850i	$-0.708074\pi$
$31$	−6.77167	−1.21623	−0.608114	+	0.793850i	$0.708074\pi$
$32$	0	0
$33$	8.92872	1.55429
$34$	0	0
$35$	2.86058	0.483526
$36$	0	0
$37$	3.42952	0.563810	0.281905	−	0.959442i	$-0.409034\pi$
$37$	3.42952	0.563810	0.281905	+	0.959442i	$0.409034\pi$
$38$	0	0
$39$	−7.90748	−1.26621
$40$	0	0
$41$	10.4927	1.63869	0.819345	−	0.573300i	$-0.194337\pi$
$41$	10.4927	1.63869	0.819345	+	0.573300i	$0.194337\pi$
$42$	0	0
$43$	−9.67354	−1.47520	−0.737600	−	0.675238i	$-0.764041\pi$
$43$	−9.67354	−1.47520	−0.737600	+	0.675238i	$0.764041\pi$
$44$	0	0
$45$	6.31852	0.941909
$46$	0	0
$47$	1.71217	0.249746	0.124873	−	0.992173i	$-0.460148\pi$
$47$	1.71217	0.249746	0.124873	+	0.992173i	$0.460148\pi$
$48$	0	0
$49$	1.18292	0.168988
$50$	0	0
$51$	9.17798	1.28517
$52$	0	0
$53$	−1.95028	−0.267892	−0.133946	−	0.990989i	$-0.542765\pi$
$53$	−1.95028	−0.267892	−0.133946	+	0.990989i	$0.542765\pi$
$54$	0	0
$55$	−2.92493	−0.394398
$56$	0	0
$57$	19.6795	2.60661
$58$	0	0
$59$	2.01945	0.262910	0.131455	−	0.991322i	$-0.458035\pi$
$59$	2.01945	0.262910	0.131455	+	0.991322i	$0.458035\pi$
$60$	0	0
$61$	4.02714	0.515623	0.257811	−	0.966195i	$-0.416999\pi$
$61$	4.02714	0.515623	0.257811	+	0.966195i	$0.416999\pi$
$62$	0	0
$63$	18.0746	2.27719
$64$	0	0
$65$	2.59039	0.321298
$66$	0	0
$67$	11.7041	1.42988	0.714940	−	0.699186i	$-0.246454\pi$
$67$	11.7041	1.42988	0.714940	+	0.699186i	$0.246454\pi$
$68$	0	0
$69$	20.3190	2.44612
$70$	0	0
$71$	13.9308	1.65328	0.826639	−	0.562732i	$-0.190250\pi$
$71$	13.9308	1.65328	0.826639	+	0.562732i	$0.190250\pi$
$72$	0	0
$73$	1.82517	0.213621	0.106810	−	0.994279i	$-0.465936\pi$
$73$	1.82517	0.213621	0.106810	+	0.994279i	$0.465936\pi$
$74$	0	0
$75$	−3.05263	−0.352487
$76$	0	0
$77$	−8.36700	−0.953508
$78$	0	0
$79$	7.86911	0.885344	0.442672	−	0.896684i	$-0.354031\pi$
$79$	7.86911	0.885344	0.442672	+	0.896684i	$0.354031\pi$
$80$	0	0
$81$	11.9681	1.32979
$82$	0	0
$83$	−0.167275	−0.0183608	−0.00918039	−	0.999958i	$-0.502922\pi$
$83$	−0.167275	−0.0183608	−0.00918039	+	0.999958i	$0.502922\pi$
$84$	0	0
$85$	−3.00659	−0.326110
$86$	0	0
$87$	−2.27664	−0.244082
$88$	0	0
$89$	−3.06659	−0.325058	−0.162529	−	0.986704i	$-0.551965\pi$
$89$	−3.06659	−0.325058	−0.162529	+	0.986704i	$0.551965\pi$
$90$	0	0
$91$	7.41001	0.776780
$92$	0	0
$93$	20.6714	2.14352
$94$	0	0
$95$	−6.44674	−0.661422
$96$	0	0
$97$	−2.34116	−0.237709	−0.118855	−	0.992912i	$-0.537922\pi$
$97$	−2.34116	−0.237709	−0.118855	+	0.992912i	$0.537922\pi$
$98$	0	0
$99$	−18.4812	−1.85743
$100$	0	0
$101$	−8.61102	−0.856829	−0.428414	−	0.903582i	$-0.640928\pi$
$101$	−8.61102	−0.856829	−0.428414	+	0.903582i	$0.640928\pi$
$102$	0	0
$103$	−9.06113	−0.892819	−0.446410	−	0.894829i	$-0.647298\pi$
$103$	−9.06113	−0.892819	−0.446410	+	0.894829i	$0.647298\pi$
$104$	0	0
$105$	−8.73228	−0.852183
$106$	0	0
$107$	6.10615	0.590304	0.295152	−	0.955450i	$-0.404630\pi$
$107$	6.10615	0.590304	0.295152	+	0.955450i	$0.404630\pi$
$108$	0	0
$109$	−7.55800	−0.723925	−0.361962	−	0.932193i	$-0.617893\pi$
$109$	−7.55800	−0.723925	−0.361962	+	0.932193i	$0.617893\pi$
$110$	0	0
$111$	−10.4690	−0.993677
$112$	0	0
$113$	−0.988484	−0.0929888	−0.0464944	−	0.998919i	$-0.514805\pi$
$113$	−0.988484	−0.0929888	−0.0464944	+	0.998919i	$0.514805\pi$
$114$	0	0
$115$	−6.65623	−0.620697
$116$	0	0
$117$	16.3674	1.51317
$118$	0	0
$119$	−8.60058	−0.788414
$120$	0	0
$121$	−2.44477	−0.222252
$122$	0	0
$123$	−32.0304	−2.88808
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	9.92527	0.880725	0.440363	−	0.897820i	$-0.354850\pi$
$127$	9.92527	0.880725	0.440363	+	0.897820i	$0.354850\pi$
$128$	0	0
$129$	29.5297	2.59994
$130$	0	0
$131$	−6.55529	−0.572738	−0.286369	−	0.958119i	$-0.592448\pi$
$131$	−6.55529	−0.572738	−0.286369	+	0.958119i	$0.592448\pi$
$132$	0	0
$133$	−18.4414	−1.59907
$134$	0	0
$135$	−10.1302	−0.871869
$136$	0	0
$137$	9.36789	0.800353	0.400176	−	0.916438i	$-0.368949\pi$
$137$	9.36789	0.800353	0.400176	+	0.916438i	$0.368949\pi$
$138$	0	0
$139$	−14.0226	−1.18939	−0.594693	−	0.803953i	$-0.702726\pi$
$139$	−14.0226	−1.18939	−0.594693	+	0.803953i	$0.702726\pi$
$140$	0	0
$141$	−5.22661	−0.440160
$142$	0	0
$143$	−7.57671	−0.633596
$144$	0	0
$145$	0.745798	0.0619352
$146$	0	0
$147$	−3.61101	−0.297831
$148$	0	0
$149$	1.31198	0.107481	0.0537407	−	0.998555i	$-0.482886\pi$
$149$	1.31198	0.107481	0.0537407	+	0.998555i	$0.482886\pi$
$150$	0	0
$151$	16.4541	1.33902	0.669508	−	0.742805i	$-0.266505\pi$
$151$	16.4541	1.33902	0.669508	+	0.742805i	$0.266505\pi$
$152$	0	0
$153$	−18.9972	−1.53583
$154$	0	0
$155$	−6.77167	−0.543914
$156$	0	0
$157$	7.49351	0.598048	0.299024	−	0.954246i	$-0.403339\pi$
$157$	7.49351	0.598048	0.299024	+	0.954246i	$0.403339\pi$
$158$	0	0
$159$	5.95348	0.472141
$160$	0	0
$161$	−19.0407	−1.50062
$162$	0	0
$163$	2.31609	0.181410	0.0907052	−	0.995878i	$-0.471088\pi$
$163$	2.31609	0.181410	0.0907052	+	0.995878i	$0.471088\pi$
$164$	0	0
$165$	8.92872	0.695100
$166$	0	0
$167$	25.3823	1.96414	0.982069	−	0.188523i	$-0.0603699\pi$
$167$	25.3823	1.96414	0.982069	+	0.188523i	$0.0603699\pi$
$168$	0	0
$169$	−6.28989	−0.483838
$170$	0	0
$171$	−40.7339	−3.11500
$172$	0	0
$173$	9.29523	0.706703	0.353352	−	0.935491i	$-0.385042\pi$
$173$	9.29523	0.706703	0.353352	+	0.935491i	$0.385042\pi$
$174$	0	0
$175$	2.86058	0.216240
$176$	0	0
$177$	−6.16462	−0.463361
$178$	0	0
$179$	1.75043	0.130833	0.0654166	−	0.997858i	$-0.479162\pi$
$179$	1.75043	0.130833	0.0654166	+	0.997858i	$0.479162\pi$
$180$	0	0
$181$	21.0794	1.56682	0.783411	−	0.621504i	$-0.213478\pi$
$181$	21.0794	1.56682	0.783411	+	0.621504i	$0.213478\pi$
$182$	0	0
$183$	−12.2934	−0.908751
$184$	0	0
$185$	3.42952	0.252143
$186$	0	0
$187$	8.79406	0.643086
$188$	0	0
$189$	−28.9782	−2.10786
$190$	0	0
$191$	13.4601	0.973940	0.486970	−	0.873419i	$-0.338102\pi$
$191$	13.4601	0.973940	0.486970	+	0.873419i	$0.338102\pi$
$192$	0	0
$193$	−7.30592	−0.525892	−0.262946	−	0.964811i	$-0.584694\pi$
$193$	−7.30592	−0.525892	−0.262946	+	0.964811i	$0.584694\pi$
$194$	0	0
$195$	−7.90748	−0.566267
$196$	0	0
$197$	−1.52017	−0.108307	−0.0541537	−	0.998533i	$-0.517246\pi$
$197$	−1.52017	−0.108307	−0.0541537	+	0.998533i	$0.517246\pi$
$198$	0	0
$199$	11.2747	0.799241	0.399620	−	0.916681i	$-0.369142\pi$
$199$	11.2747	0.799241	0.399620	+	0.916681i	$0.369142\pi$
$200$	0	0
$201$	−35.7282	−2.52007
$202$	0	0
$203$	2.13342	0.149736
$204$	0	0
$205$	10.4927	0.732845
$206$	0	0
$207$	−42.0575	−2.92320
$208$	0	0
$209$	18.8563	1.30432
$210$	0	0
$211$	−2.30900	−0.158958	−0.0794791	−	0.996837i	$-0.525326\pi$
$211$	−2.30900	−0.158958	−0.0794791	+	0.996837i	$0.525326\pi$
$212$	0	0
$213$	−42.5254	−2.91379
$214$	0	0
$215$	−9.67354	−0.659730
$216$	0	0
$217$	−19.3709	−1.31498
$218$	0	0
$219$	−5.57158	−0.376492
$220$	0	0
$221$	−7.78823	−0.523893
$222$	0	0
$223$	11.4994	0.770055	0.385028	−	0.922905i	$-0.374192\pi$
$223$	11.4994	0.770055	0.385028	+	0.922905i	$0.374192\pi$
$224$	0	0
$225$	6.31852	0.421235
$226$	0	0
$227$	21.0572	1.39762	0.698808	−	0.715309i	$-0.253714\pi$
$227$	21.0572	1.39762	0.698808	+	0.715309i	$0.253714\pi$
$228$	0	0
$229$	−26.8773	−1.77610	−0.888051	−	0.459744i	$-0.847941\pi$
$229$	−26.8773	−1.77610	−0.888051	+	0.459744i	$0.847941\pi$
$230$	0	0
$231$	25.5413	1.68050
$232$	0	0
$233$	27.7194	1.81596	0.907979	−	0.419015i	$-0.137624\pi$
$233$	27.7194	1.81596	0.907979	+	0.419015i	$0.137624\pi$
$234$	0	0
$235$	1.71217	0.111690
$236$	0	0
$237$	−24.0214	−1.56036
$238$	0	0
$239$	−1.76058	−0.113882	−0.0569411	−	0.998378i	$-0.518135\pi$
$239$	−1.76058	−0.113882	−0.0569411	+	0.998378i	$0.518135\pi$
$240$	0	0
$241$	−13.6638	−0.880165	−0.440083	−	0.897957i	$-0.645051\pi$
$241$	−13.6638	−0.880165	−0.440083	+	0.897957i	$0.645051\pi$
$242$	0	0
$243$	−6.14365	−0.394115
$244$	0	0
$245$	1.18292	0.0755739
$246$	0	0
$247$	−16.6996	−1.06257
$248$	0	0
$249$	0.510627	0.0323597
$250$	0	0
$251$	−17.1937	−1.08526	−0.542628	−	0.839973i	$-0.682571\pi$
$251$	−17.1937	−1.08526	−0.542628	+	0.839973i	$0.682571\pi$
$252$	0	0
$253$	19.4690	1.22401
$254$	0	0
$255$	9.17798	0.574748
$256$	0	0
$257$	26.6213	1.66059	0.830296	−	0.557323i	$-0.188171\pi$
$257$	26.6213	1.66059	0.830296	+	0.557323i	$0.188171\pi$
$258$	0	0
$259$	9.81042	0.609590
$260$	0	0
$261$	4.71234	0.291687
$262$	0	0
$263$	−6.55395	−0.404134	−0.202067	−	0.979372i	$-0.564766\pi$
$263$	−6.55395	−0.404134	−0.202067	+	0.979372i	$0.564766\pi$
$264$	0	0
$265$	−1.95028	−0.119805
$266$	0	0
$267$	9.36114	0.572892
$268$	0	0
$269$	−10.6974	−0.652235	−0.326117	−	0.945329i	$-0.605740\pi$
$269$	−10.6974	−0.652235	−0.326117	+	0.945329i	$0.605740\pi$
$270$	0	0
$271$	−0.957730	−0.0581780	−0.0290890	−	0.999577i	$-0.509261\pi$
$271$	−0.957730	−0.0581780	−0.0290890	+	0.999577i	$0.509261\pi$
$272$	0	0
$273$	−22.6200	−1.36902
$274$	0	0
$275$	−2.92493	−0.176380
$276$	0	0
$277$	−27.2598	−1.63788	−0.818941	−	0.573877i	$-0.805439\pi$
$277$	−27.2598	−1.63788	−0.818941	+	0.573877i	$0.805439\pi$
$278$	0	0
$279$	−42.7869	−2.56159
$280$	0	0
$281$	−5.96181	−0.355652	−0.177826	−	0.984062i	$-0.556906\pi$
$281$	−5.96181	−0.355652	−0.177826	+	0.984062i	$0.556906\pi$
$282$	0	0
$283$	−7.27319	−0.432346	−0.216173	−	0.976355i	$-0.569358\pi$
$283$	−7.27319	−0.432346	−0.216173	+	0.976355i	$0.569358\pi$
$284$	0	0
$285$	19.6795	1.16571
$286$	0	0
$287$	30.0153	1.77175
$288$	0	0
$289$	−7.96043	−0.468261
$290$	0	0
$291$	7.14670	0.418947
$292$	0	0
$293$	19.2335	1.12363	0.561815	−	0.827263i	$-0.310103\pi$
$293$	19.2335	1.12363	0.561815	+	0.827263i	$0.310103\pi$
$294$	0	0
$295$	2.01945	0.117577
$296$	0	0
$297$	29.6301	1.71932
$298$	0	0
$299$	−17.2422	−0.997143
$300$	0	0
$301$	−27.6719	−1.59498
$302$	0	0
$303$	26.2862	1.51010
$304$	0	0
$305$	4.02714	0.230593
$306$	0	0
$307$	−5.75387	−0.328391	−0.164195	−	0.986428i	$-0.552503\pi$
$307$	−5.75387	−0.328391	−0.164195	+	0.986428i	$0.552503\pi$
$308$	0	0
$309$	27.6602	1.57353
$310$	0	0
$311$	−16.3787	−0.928750	−0.464375	−	0.885639i	$-0.653721\pi$
$311$	−16.3787	−0.928750	−0.464375	+	0.885639i	$0.653721\pi$
$312$	0	0
$313$	16.3276	0.922889	0.461444	−	0.887169i	$-0.347331\pi$
$313$	16.3276	0.922889	0.461444	+	0.887169i	$0.347331\pi$
$314$	0	0
$315$	18.0746	1.01839
$316$	0	0
$317$	27.0092	1.51699	0.758494	−	0.651680i	$-0.225936\pi$
$317$	27.0092	1.51699	0.758494	+	0.651680i	$0.225936\pi$
$318$	0	0
$319$	−2.18141	−0.122135
$320$	0	0
$321$	−18.6398	−1.04037
$322$	0	0
$323$	19.3827	1.07848
$324$	0	0
$325$	2.59039	0.143689
$326$	0	0
$327$	23.0717	1.27587
$328$	0	0
$329$	4.89780	0.270024
$330$	0	0
$331$	−5.15765	−0.283490	−0.141745	−	0.989903i	$-0.545271\pi$
$331$	−5.15765	−0.283490	−0.141745	+	0.989903i	$0.545271\pi$
$332$	0	0
$333$	21.6695	1.18748
$334$	0	0
$335$	11.7041	0.639462
$336$	0	0
$337$	13.9862	0.761875	0.380937	−	0.924601i	$-0.375601\pi$
$337$	13.9862	0.761875	0.380937	+	0.924601i	$0.375601\pi$
$338$	0	0
$339$	3.01747	0.163887
$340$	0	0
$341$	19.8067	1.07259
$342$	0	0
$343$	−16.6402	−0.898488
$344$	0	0
$345$	20.3190	1.09394
$346$	0	0
$347$	11.5384	0.619416	0.309708	−	0.950832i	$-0.399769\pi$
$347$	11.5384	0.619416	0.309708	+	0.950832i	$0.399769\pi$
$348$	0	0
$349$	22.7370	1.21708	0.608542	−	0.793522i	$-0.291755\pi$
$349$	22.7370	1.21708	0.608542	+	0.793522i	$0.291755\pi$
$350$	0	0
$351$	−26.2411	−1.40065
$352$	0	0
$353$	−11.1764	−0.594860	−0.297430	−	0.954744i	$-0.596130\pi$
$353$	−11.1764	−0.594860	−0.297430	+	0.954744i	$0.596130\pi$
$354$	0	0
$355$	13.9308	0.739369
$356$	0	0
$357$	26.2544	1.38953
$358$	0	0
$359$	25.3425	1.33753	0.668764	−	0.743475i	$-0.266824\pi$
$359$	25.3425	1.33753	0.668764	+	0.743475i	$0.266824\pi$
$360$	0	0
$361$	22.5605	1.18739
$362$	0	0
$363$	7.46298	0.391705
$364$	0	0
$365$	1.82517	0.0955340
$366$	0	0
$367$	27.2597	1.42295	0.711473	−	0.702713i	$-0.248028\pi$
$367$	27.2597	1.42295	0.711473	+	0.702713i	$0.248028\pi$
$368$	0	0
$369$	66.2986	3.45137
$370$	0	0
$371$	−5.57893	−0.289644
$372$	0	0
$373$	−5.05538	−0.261758	−0.130879	−	0.991398i	$-0.541780\pi$
$373$	−5.05538	−0.261758	−0.130879	+	0.991398i	$0.541780\pi$
$374$	0	0
$375$	−3.05263	−0.157637
$376$	0	0
$377$	1.93191	0.0994982
$378$	0	0
$379$	−26.3595	−1.35400	−0.676999	−	0.735984i	$-0.736720\pi$
$379$	−26.3595	−1.35400	−0.676999	+	0.735984i	$0.736720\pi$
$380$	0	0
$381$	−30.2981	−1.55222
$382$	0	0
$383$	−13.9525	−0.712938	−0.356469	−	0.934307i	$-0.616020\pi$
$383$	−13.9525	−0.712938	−0.356469	+	0.934307i	$0.616020\pi$
$384$	0	0
$385$	−8.36700	−0.426422
$386$	0	0
$387$	−61.1224	−3.10703
$388$	0	0
$389$	22.9227	1.16223	0.581114	−	0.813822i	$-0.302617\pi$
$389$	22.9227	1.16223	0.581114	+	0.813822i	$0.302617\pi$
$390$	0	0
$391$	20.0125	1.01208
$392$	0	0
$393$	20.0108	1.00941
$394$	0	0
$395$	7.86911	0.395938
$396$	0	0
$397$	1.44746	0.0726460	0.0363230	−	0.999340i	$-0.488435\pi$
$397$	1.44746	0.0726460	0.0363230	+	0.999340i	$0.488435\pi$
$398$	0	0
$399$	56.2948	2.81826
$400$	0	0
$401$	1.00000	0.0499376
$401$	1.00000	0.0499376
$402$	0	0
$403$	−17.5413	−0.873792
$404$	0	0
$405$	11.9681	0.594702
$406$	0	0
$407$	−10.0311	−0.497224
$408$	0	0
$409$	11.9081	0.588815	0.294408	−	0.955680i	$-0.404878\pi$
$409$	11.9081	0.588815	0.294408	+	0.955680i	$0.404878\pi$
$410$	0	0
$411$	−28.5967	−1.41057
$412$	0	0
$413$	5.77679	0.284257
$414$	0	0
$415$	−0.167275	−0.00821119
$416$	0	0
$417$	42.8059	2.09621
$418$	0	0
$419$	5.67647	0.277314	0.138657	−	0.990340i	$-0.455721\pi$
$419$	5.67647	0.277314	0.138657	+	0.990340i	$0.455721\pi$
$420$	0	0
$421$	−26.7639	−1.30439	−0.652197	−	0.758049i	$-0.726153\pi$
$421$	−26.7639	−1.30439	−0.652197	+	0.758049i	$0.726153\pi$
$422$	0	0
$423$	10.8184	0.526008
$424$	0	0
$425$	−3.00659	−0.145841
$426$	0	0
$427$	11.5200	0.557490
$428$	0	0
$429$	23.1288	1.11667
$430$	0	0
$431$	−9.75239	−0.469756	−0.234878	−	0.972025i	$-0.575469\pi$
$431$	−9.75239	−0.469756	−0.234878	+	0.972025i	$0.575469\pi$
$432$	0	0
$433$	26.1114	1.25483	0.627416	−	0.778684i	$-0.284112\pi$
$433$	26.1114	1.25483	0.627416	+	0.778684i	$0.284112\pi$
$434$	0	0
$435$	−2.27664	−0.109157
$436$	0	0
$437$	42.9110	2.05271
$438$	0	0
$439$	−4.95151	−0.236323	−0.118161	−	0.992994i	$-0.537700\pi$
$439$	−4.95151	−0.236323	−0.118161	+	0.992994i	$0.537700\pi$
$440$	0	0
$441$	7.47430	0.355919
$442$	0	0
$443$	1.94427	0.0923751	0.0461876	−	0.998933i	$-0.485293\pi$
$443$	1.94427	0.0923751	0.0461876	+	0.998933i	$0.485293\pi$
$444$	0	0
$445$	−3.06659	−0.145370
$446$	0	0
$447$	−4.00498	−0.189429
$448$	0	0
$449$	26.0304	1.22845	0.614225	−	0.789131i	$-0.289469\pi$
$449$	26.0304	1.22845	0.614225	+	0.789131i	$0.289469\pi$
$450$	0	0
$451$	−30.6905	−1.44516
$452$	0	0
$453$	−50.2282	−2.35993
$454$	0	0
$455$	7.41001	0.347387
$456$	0	0
$457$	−10.3105	−0.482304	−0.241152	−	0.970487i	$-0.577525\pi$
$457$	−10.3105	−0.482304	−0.241152	+	0.970487i	$0.577525\pi$
$458$	0	0
$459$	30.4573	1.42163
$460$	0	0
$461$	12.3246	0.574015	0.287007	−	0.957928i	$-0.407340\pi$
$461$	12.3246	0.574015	0.287007	+	0.957928i	$0.407340\pi$
$462$	0	0
$463$	5.54613	0.257750	0.128875	−	0.991661i	$-0.458863\pi$
$463$	5.54613	0.257750	0.128875	+	0.991661i	$0.458863\pi$
$464$	0	0
$465$	20.6714	0.958612
$466$	0	0
$467$	17.4008	0.805211	0.402606	−	0.915374i	$-0.368105\pi$
$467$	17.4008	0.805211	0.402606	+	0.915374i	$0.368105\pi$
$468$	0	0
$469$	33.4804	1.54598
$470$	0	0
$471$	−22.8749	−1.05402
$472$	0	0
$473$	28.2944	1.30098
$474$	0	0
$475$	−6.44674	−0.295797
$476$	0	0
$477$	−12.3229	−0.564226
$478$	0	0
$479$	−25.8264	−1.18004	−0.590018	−	0.807390i	$-0.700879\pi$
$479$	−25.8264	−1.18004	−0.590018	+	0.807390i	$0.700879\pi$
$480$	0	0
$481$	8.88379	0.405066
$482$	0	0
$483$	58.1241	2.64474
$484$	0	0
$485$	−2.34116	−0.106307
$486$	0	0
$487$	2.72729	0.123585	0.0617926	−	0.998089i	$-0.480318\pi$
$487$	2.72729	0.123585	0.0617926	+	0.998089i	$0.480318\pi$
$488$	0	0
$489$	−7.07016	−0.319724
$490$	0	0
$491$	−17.0909	−0.771300	−0.385650	−	0.922645i	$-0.626023\pi$
$491$	−17.0909	−0.771300	−0.385650	+	0.922645i	$0.626023\pi$
$492$	0	0
$493$	−2.24231	−0.100988
$494$	0	0
$495$	−18.4812	−0.830670
$496$	0	0
$497$	39.8501	1.78752
$498$	0	0
$499$	22.3057	0.998542	0.499271	−	0.866446i	$-0.333601\pi$
$499$	22.3057	0.998542	0.499271	+	0.866446i	$0.333601\pi$
$500$	0	0
$501$	−77.4825	−3.46166
$502$	0	0
$503$	27.9017	1.24407	0.622037	−	0.782988i	$-0.286305\pi$
$503$	27.9017	1.24407	0.622037	+	0.782988i	$0.286305\pi$
$504$	0	0
$505$	−8.61102	−0.383186
$506$	0	0
$507$	19.2007	0.852732
$508$	0	0
$509$	−44.3081	−1.96392	−0.981961	−	0.189085i	$-0.939448\pi$
$509$	−44.3081	−1.96392	−0.981961	+	0.189085i	$0.939448\pi$
$510$	0	0
$511$	5.22106	0.230966
$512$	0	0
$513$	65.3068	2.88337
$514$	0	0
$515$	−9.06113	−0.399281
$516$	0	0
$517$	−5.00798	−0.220251
$518$	0	0
$519$	−28.3748	−1.24552
$520$	0	0
$521$	8.58798	0.376246	0.188123	−	0.982145i	$-0.439760\pi$
$521$	8.58798	0.376246	0.188123	+	0.982145i	$0.439760\pi$
$522$	0	0
$523$	−24.6429	−1.07756	−0.538780	−	0.842447i	$-0.681114\pi$
$523$	−24.6429	−1.07756	−0.538780	+	0.842447i	$0.681114\pi$
$524$	0	0
$525$	−8.73228	−0.381108
$526$	0	0
$527$	20.3596	0.886879
$528$	0	0
$529$	21.3054	0.926322
$530$	0	0
$531$	12.7599	0.553733
$532$	0	0
$533$	27.1803	1.17731
$534$	0	0
$535$	6.10615	0.263992
$536$	0	0
$537$	−5.34340	−0.230585
$538$	0	0
$539$	−3.45996	−0.149031
$540$	0	0
$541$	−12.9545	−0.556957	−0.278479	−	0.960442i	$-0.589830\pi$
$541$	−12.9545	−0.556957	−0.278479	+	0.960442i	$0.589830\pi$
$542$	0	0
$543$	−64.3476	−2.76142
$544$	0	0
$545$	−7.55800	−0.323749
$546$	0	0
$547$	−29.0492	−1.24205	−0.621027	−	0.783789i	$-0.713284\pi$
$547$	−29.0492	−1.24205	−0.621027	+	0.783789i	$0.713284\pi$
$548$	0	0
$549$	25.4456	1.08599
$550$	0	0
$551$	−4.80797	−0.204826
$552$	0	0
$553$	22.5102	0.957231
$554$	0	0
$555$	−10.4690	−0.444386
$556$	0	0
$557$	0.630134	0.0266996	0.0133498	−	0.999911i	$-0.495750\pi$
$557$	0.630134	0.0266996	0.0133498	+	0.999911i	$0.495750\pi$
$558$	0	0
$559$	−25.0582	−1.05985
$560$	0	0
$561$	−26.8450	−1.13340
$562$	0	0
$563$	39.5920	1.66861	0.834303	−	0.551307i	$-0.185871\pi$
$563$	39.5920	1.66861	0.834303	+	0.551307i	$0.185871\pi$
$564$	0	0
$565$	−0.988484	−0.0415858
$566$	0	0
$567$	34.2358	1.43777
$568$	0	0
$569$	11.5173	0.482832	0.241416	−	0.970422i	$-0.422388\pi$
$569$	11.5173	0.482832	0.241416	+	0.970422i	$0.422388\pi$
$570$	0	0
$571$	43.2715	1.81085	0.905427	−	0.424501i	$-0.139551\pi$
$571$	43.2715	1.81085	0.905427	+	0.424501i	$0.139551\pi$
$572$	0	0
$573$	−41.0887	−1.71651
$574$	0	0
$575$	−6.65623	−0.277584
$576$	0	0
$577$	3.29147	0.137026	0.0685129	−	0.997650i	$-0.478175\pi$
$577$	3.29147	0.137026	0.0685129	+	0.997650i	$0.478175\pi$
$578$	0	0
$579$	22.3022	0.926849
$580$	0	0
$581$	−0.478503	−0.0198516
$582$	0	0
$583$	5.70444	0.236254
$584$	0	0
$585$	16.3674	0.676709
$586$	0	0
$587$	4.08482	0.168599	0.0842994	−	0.996440i	$-0.473135\pi$
$587$	4.08482	0.168599	0.0842994	+	0.996440i	$0.473135\pi$
$588$	0	0
$589$	43.6552	1.79878
$590$	0	0
$591$	4.64050	0.190885
$592$	0	0
$593$	40.3043	1.65510	0.827549	−	0.561394i	$-0.189735\pi$
$593$	40.3043	1.65510	0.827549	+	0.561394i	$0.189735\pi$
$594$	0	0
$595$	−8.60058	−0.352590
$596$	0	0
$597$	−34.4174	−1.40861
$598$	0	0
$599$	21.4353	0.875825	0.437912	−	0.899018i	$-0.355718\pi$
$599$	21.4353	0.875825	0.437912	+	0.899018i	$0.355718\pi$
$600$	0	0
$601$	−10.9110	−0.445070	−0.222535	−	0.974925i	$-0.571433\pi$
$601$	−10.9110	−0.445070	−0.222535	+	0.974925i	$0.571433\pi$
$602$	0	0
$603$	73.9524	3.01158
$604$	0	0
$605$	−2.44477	−0.0993942
$606$	0	0
$607$	−30.5957	−1.24184	−0.620920	−	0.783874i	$-0.713241\pi$
$607$	−30.5957	−1.24184	−0.620920	+	0.783874i	$0.713241\pi$
$608$	0	0
$609$	−6.51252	−0.263901
$610$	0	0
$611$	4.43518	0.179428
$612$	0	0
$613$	−3.31651	−0.133953	−0.0669764	−	0.997755i	$-0.521335\pi$
$613$	−3.31651	−0.133953	−0.0669764	+	0.997755i	$0.521335\pi$
$614$	0	0
$615$	−32.0304	−1.29159
$616$	0	0
$617$	30.0839	1.21113	0.605567	−	0.795794i	$-0.292946\pi$
$617$	30.0839	1.21113	0.605567	+	0.795794i	$0.292946\pi$
$618$	0	0
$619$	−32.3276	−1.29935	−0.649677	−	0.760210i	$-0.725096\pi$
$619$	−32.3276	−1.29935	−0.649677	+	0.760210i	$0.725096\pi$
$620$	0	0
$621$	67.4289	2.70583
$622$	0	0
$623$	−8.77222	−0.351451
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−57.5612	−2.29877
$628$	0	0
$629$	−10.3112	−0.411133
$630$	0	0
$631$	23.2411	0.925214	0.462607	−	0.886563i	$-0.346914\pi$
$631$	23.2411	0.925214	0.462607	+	0.886563i	$0.346914\pi$
$632$	0	0
$633$	7.04852	0.280153
$634$	0	0
$635$	9.92527	0.393872
$636$	0	0
$637$	3.06422	0.121409
$638$	0	0
$639$	88.0219	3.48209
$640$	0	0
$641$	−21.1099	−0.833792	−0.416896	−	0.908954i	$-0.636882\pi$
$641$	−21.1099	−0.833792	−0.416896	+	0.908954i	$0.636882\pi$
$642$	0	0
$643$	9.98245	0.393669	0.196835	−	0.980437i	$-0.436934\pi$
$643$	9.98245	0.393669	0.196835	+	0.980437i	$0.436934\pi$
$644$	0	0
$645$	29.5297	1.16273
$646$	0	0
$647$	−36.3477	−1.42897	−0.714487	−	0.699648i	$-0.753340\pi$
$647$	−36.3477	−1.42897	−0.714487	+	0.699648i	$0.753340\pi$
$648$	0	0
$649$	−5.90675	−0.231860
$650$	0	0
$651$	59.1321	2.31757
$652$	0	0
$653$	8.82308	0.345274	0.172637	−	0.984986i	$-0.444771\pi$
$653$	8.82308	0.345274	0.172637	+	0.984986i	$0.444771\pi$
$654$	0	0
$655$	−6.55529	−0.256136
$656$	0	0
$657$	11.5324	0.449922
$658$	0	0
$659$	35.1418	1.36893	0.684465	−	0.729046i	$-0.260036\pi$
$659$	35.1418	1.36893	0.684465	+	0.729046i	$0.260036\pi$
$660$	0	0
$661$	4.67205	0.181722	0.0908608	−	0.995864i	$-0.471038\pi$
$661$	4.67205	0.181722	0.0908608	+	0.995864i	$0.471038\pi$
$662$	0	0
$663$	23.7745	0.923326
$664$	0	0
$665$	−18.4414	−0.715128
$666$	0	0
$667$	−4.96420	−0.192215
$668$	0	0
$669$	−35.1033	−1.35717
$670$	0	0
$671$	−11.7791	−0.454728
$672$	0	0
$673$	32.8059	1.26457	0.632287	−	0.774734i	$-0.282116\pi$
$673$	32.8059	1.26457	0.632287	+	0.774734i	$0.282116\pi$
$674$	0	0
$675$	−10.1302	−0.389912
$676$	0	0
$677$	40.4758	1.55561	0.777806	−	0.628504i	$-0.216332\pi$
$677$	40.4758	1.55561	0.777806	+	0.628504i	$0.216332\pi$
$678$	0	0
$679$	−6.69709	−0.257011
$680$	0	0
$681$	−64.2798	−2.46321
$682$	0	0
$683$	−1.92336	−0.0735953	−0.0367976	−	0.999323i	$-0.511716\pi$
$683$	−1.92336	−0.0735953	−0.0367976	+	0.999323i	$0.511716\pi$
$684$	0	0
$685$	9.36789	0.357929
$686$	0	0
$687$	82.0464	3.13026
$688$	0	0
$689$	−5.05198	−0.192465
$690$	0	0
$691$	−3.69412	−0.140531	−0.0702655	−	0.997528i	$-0.522385\pi$
$691$	−3.69412	−0.140531	−0.0702655	+	0.997528i	$0.522385\pi$
$692$	0	0
$693$	−52.8671	−2.00825
$694$	0	0
$695$	−14.0226	−0.531909
$696$	0	0
$697$	−31.5473	−1.19494
$698$	0	0
$699$	−84.6170	−3.20051
$700$	0	0
$701$	−17.9489	−0.677921	−0.338961	−	0.940801i	$-0.610075\pi$
$701$	−17.9489	−0.677921	−0.338961	+	0.940801i	$0.610075\pi$
$702$	0	0
$703$	−22.1092	−0.833866
$704$	0	0
$705$	−5.22661	−0.196846
$706$	0	0
$707$	−24.6325	−0.926402
$708$	0	0
$709$	52.1970	1.96030	0.980150	−	0.198255i	$-0.0635276\pi$
$709$	52.1970	1.96030	0.980150	+	0.198255i	$0.0635276\pi$
$710$	0	0
$711$	49.7211	1.86469
$712$	0	0
$713$	45.0738	1.68803
$714$	0	0
$715$	−7.57671	−0.283353
$716$	0	0
$717$	5.37438	0.200710
$718$	0	0
$719$	0.818480	0.0305242	0.0152621	−	0.999884i	$-0.495142\pi$
$719$	0.818480	0.0305242	0.0152621	+	0.999884i	$0.495142\pi$
$720$	0	0
$721$	−25.9201	−0.965314
$722$	0	0
$723$	41.7106	1.55123
$724$	0	0
$725$	0.745798	0.0276982
$726$	0	0
$727$	46.2500	1.71532	0.857659	−	0.514219i	$-0.171918\pi$
$727$	46.2500	1.71532	0.857659	+	0.514219i	$0.171918\pi$
$728$	0	0
$729$	−17.1502	−0.635191
$730$	0	0
$731$	29.0843	1.07572
$732$	0	0
$733$	44.6598	1.64955	0.824773	−	0.565464i	$-0.191303\pi$
$733$	44.6598	1.64955	0.824773	+	0.565464i	$0.191303\pi$
$734$	0	0
$735$	−3.61101	−0.133194
$736$	0	0
$737$	−34.2336	−1.26101
$738$	0	0
$739$	24.5393	0.902693	0.451346	−	0.892349i	$-0.350944\pi$
$739$	24.5393	0.902693	0.451346	+	0.892349i	$0.350944\pi$
$740$	0	0
$741$	50.9775	1.87271
$742$	0	0
$743$	21.0560	0.772470	0.386235	−	0.922400i	$-0.373775\pi$
$743$	21.0560	0.772470	0.386235	+	0.922400i	$0.373775\pi$
$744$	0	0
$745$	1.31198	0.0480672
$746$	0	0
$747$	−1.05693	−0.0386710
$748$	0	0
$749$	17.4671	0.638236
$750$	0	0
$751$	18.8230	0.686861	0.343431	−	0.939178i	$-0.388411\pi$
$751$	18.8230	0.686861	0.343431	+	0.939178i	$0.388411\pi$
$752$	0	0
$753$	52.4859	1.91269
$754$	0	0
$755$	16.4541	0.598826
$756$	0	0
$757$	−26.3678	−0.958355	−0.479178	−	0.877718i	$-0.659065\pi$
$757$	−26.3678	−0.958355	−0.479178	+	0.877718i	$0.659065\pi$
$758$	0	0
$759$	−59.4316	−2.15723
$760$	0	0
$761$	−1.44433	−0.0523568	−0.0261784	−	0.999657i	$-0.508334\pi$
$761$	−1.44433	−0.0523568	−0.0261784	+	0.999657i	$0.508334\pi$
$762$	0	0
$763$	−21.6203	−0.782706
$764$	0	0
$765$	−18.9972	−0.686845
$766$	0	0
$767$	5.23115	0.188886
$768$	0	0
$769$	9.70321	0.349907	0.174953	−	0.984577i	$-0.444023\pi$
$769$	9.70321	0.349907	0.174953	+	0.984577i	$0.444023\pi$
$770$	0	0
$771$	−81.2649	−2.92668
$772$	0	0
$773$	−9.27245	−0.333507	−0.166753	−	0.985999i	$-0.553328\pi$
$773$	−9.27245	−0.333507	−0.166753	+	0.985999i	$0.553328\pi$
$774$	0	0
$775$	−6.77167	−0.243246
$776$	0	0
$777$	−29.9475	−1.07436
$778$	0	0
$779$	−67.6440	−2.42360
$780$	0	0
$781$	−40.7466	−1.45803
$782$	0	0
$783$	−7.55508	−0.269997
$784$	0	0
$785$	7.49351	0.267455
$786$	0	0
$787$	−11.1181	−0.396318	−0.198159	−	0.980170i	$-0.563496\pi$
$787$	−11.1181	−0.396318	−0.198159	+	0.980170i	$0.563496\pi$
$788$	0	0
$789$	20.0067	0.712259
$790$	0	0
$791$	−2.82764	−0.100539
$792$	0	0
$793$	10.4319	0.370446
$794$	0	0
$795$	5.95348	0.211148
$796$	0	0
$797$	−43.8032	−1.55159	−0.775795	−	0.630985i	$-0.782651\pi$
$797$	−43.8032	−1.55159	−0.775795	+	0.630985i	$0.782651\pi$
$798$	0	0
$799$	−5.14779	−0.182116
$800$	0	0
$801$	−19.3763	−0.684628
$802$	0	0
$803$	−5.33851	−0.188392
$804$	0	0
$805$	−19.0407	−0.671096
$806$	0	0
$807$	32.6553	1.14952
$808$	0	0
$809$	−23.6397	−0.831126	−0.415563	−	0.909564i	$-0.636415\pi$
$809$	−23.6397	−0.831126	−0.415563	+	0.909564i	$0.636415\pi$
$810$	0	0
$811$	−20.4254	−0.717233	−0.358616	−	0.933485i	$-0.616751\pi$
$811$	−20.4254	−0.717233	−0.358616	+	0.933485i	$0.616751\pi$
$812$	0	0
$813$	2.92359	0.102535
$814$	0	0
$815$	2.31609	0.0811292
$816$	0	0
$817$	62.3628	2.18180
$818$	0	0
$819$	46.8203	1.63603
$820$	0	0
$821$	1.25415	0.0437703	0.0218851	−	0.999760i	$-0.493033\pi$
$821$	1.25415	0.0437703	0.0218851	+	0.999760i	$0.493033\pi$
$822$	0	0
$823$	7.39540	0.257787	0.128894	−	0.991658i	$-0.458857\pi$
$823$	7.39540	0.257787	0.128894	+	0.991658i	$0.458857\pi$
$824$	0	0
$825$	8.92872	0.310858
$826$	0	0
$827$	−39.9523	−1.38928	−0.694638	−	0.719359i	$-0.744435\pi$
$827$	−39.9523	−1.38928	−0.694638	+	0.719359i	$0.744435\pi$
$828$	0	0
$829$	−8.12106	−0.282056	−0.141028	−	0.990006i	$-0.545041\pi$
$829$	−8.12106	−0.282056	−0.141028	+	0.990006i	$0.545041\pi$
$830$	0	0
$831$	83.2139	2.88666
$832$	0	0
$833$	−3.55655	−0.123227
$834$	0	0
$835$	25.3823	0.878389
$836$	0	0
$837$	68.5984	2.37111
$838$	0	0
$839$	−22.1334	−0.764131	−0.382066	−	0.924135i	$-0.624787\pi$
$839$	−22.1334	−0.764131	−0.382066	+	0.924135i	$0.624787\pi$
$840$	0	0
$841$	−28.4438	−0.980820
$842$	0	0
$843$	18.1992	0.626813
$844$	0	0
$845$	−6.28989	−0.216379
$846$	0	0
$847$	−6.99347	−0.240299
$848$	0	0
$849$	22.2023	0.761981
$850$	0	0
$851$	−22.8277	−0.782523
$852$	0	0
$853$	−22.6945	−0.777045	−0.388522	−	0.921439i	$-0.627014\pi$
$853$	−22.6945	−0.777045	−0.388522	+	0.921439i	$0.627014\pi$
$854$	0	0
$855$	−40.7339	−1.39307
$856$	0	0
$857$	1.93153	0.0659796	0.0329898	−	0.999456i	$-0.489497\pi$
$857$	1.93153	0.0659796	0.0329898	+	0.999456i	$0.489497\pi$
$858$	0	0
$859$	19.6083	0.669028	0.334514	−	0.942391i	$-0.391428\pi$
$859$	19.6083	0.669028	0.334514	+	0.942391i	$0.391428\pi$
$860$	0	0
$861$	−91.6255	−3.12259
$862$	0	0
$863$	−32.8368	−1.11778	−0.558889	−	0.829242i	$-0.688772\pi$
$863$	−32.8368	−1.11778	−0.558889	+	0.829242i	$0.688772\pi$
$864$	0	0
$865$	9.29523	0.316047
$866$	0	0
$867$	24.3002	0.825279
$868$	0	0
$869$	−23.0166	−0.780785
$870$	0	0
$871$	30.3181	1.02729
$872$	0	0
$873$	−14.7927	−0.500657
$874$	0	0
$875$	2.86058	0.0967053
$876$	0	0
$877$	30.5559	1.03180	0.515899	−	0.856649i	$-0.327458\pi$
$877$	30.5559	1.03180	0.515899	+	0.856649i	$0.327458\pi$
$878$	0	0
$879$	−58.7125	−1.98032
$880$	0	0
$881$	15.9169	0.536254	0.268127	−	0.963384i	$-0.413595\pi$
$881$	15.9169	0.536254	0.268127	+	0.963384i	$0.413595\pi$
$882$	0	0
$883$	−3.41714	−0.114996	−0.0574980	−	0.998346i	$-0.518312\pi$
$883$	−3.41714	−0.114996	−0.0574980	+	0.998346i	$0.518312\pi$
$884$	0	0
$885$	−6.16462	−0.207221
$886$	0	0
$887$	−25.2393	−0.847453	−0.423726	−	0.905790i	$-0.639278\pi$
$887$	−25.2393	−0.847453	−0.423726	+	0.905790i	$0.639278\pi$
$888$	0	0
$889$	28.3920	0.952238
$890$	0	0
$891$	−35.0060	−1.17275
$892$	0	0
$893$	−11.0379	−0.369370
$894$	0	0
$895$	1.75043	0.0585104
$896$	0	0
$897$	52.6340	1.75740
$898$	0	0
$899$	−5.05030	−0.168437
$900$	0	0
$901$	5.86369	0.195348
$902$	0	0
$903$	84.4720	2.81105
$904$	0	0
$905$	21.0794	0.700704
$906$	0	0
$907$	8.13473	0.270109	0.135055	−	0.990838i	$-0.456879\pi$
$907$	8.13473	0.270109	0.135055	+	0.990838i	$0.456879\pi$
$908$	0	0
$909$	−54.4089	−1.80463
$910$	0	0
$911$	42.4244	1.40558	0.702791	−	0.711396i	$-0.251937\pi$
$911$	42.4244	1.40558	0.702791	+	0.711396i	$0.251937\pi$
$912$	0	0
$913$	0.489267	0.0161924
$914$	0	0
$915$	−12.2934	−0.406406
$916$	0	0
$917$	−18.7519	−0.619243
$918$	0	0
$919$	−1.43185	−0.0472323	−0.0236162	−	0.999721i	$-0.507518\pi$
$919$	−1.43185	−0.0472323	−0.0236162	+	0.999721i	$0.507518\pi$
$920$	0	0
$921$	17.5644	0.578767
$922$	0	0
$923$	36.0861	1.18779
$924$	0	0
$925$	3.42952	0.112762
$926$	0	0
$927$	−57.2529	−1.88043
$928$	0	0
$929$	60.8202	1.99544	0.997722	−	0.0674553i	$-0.0214880\pi$
$929$	60.8202	1.99544	0.997722	+	0.0674553i	$0.0214880\pi$
$930$	0	0
$931$	−7.62597	−0.249931
$932$	0	0
$933$	49.9980	1.63686
$934$	0	0
$935$	8.79406	0.287597
$936$	0	0
$937$	24.9296	0.814416	0.407208	−	0.913335i	$-0.366502\pi$
$937$	24.9296	0.814416	0.407208	+	0.913335i	$0.366502\pi$
$938$	0	0
$939$	−49.8419	−1.62653
$940$	0	0
$941$	−52.8068	−1.72145	−0.860726	−	0.509068i	$-0.829990\pi$
$941$	−52.8068	−1.72145	−0.860726	+	0.509068i	$0.829990\pi$
$942$	0	0
$943$	−69.8421	−2.27437
$944$	0	0
$945$	−28.9782	−0.942662
$946$	0	0
$947$	10.7313	0.348721	0.174360	−	0.984682i	$-0.444214\pi$
$947$	10.7313	0.348721	0.174360	+	0.984682i	$0.444214\pi$
$948$	0	0
$949$	4.72791	0.153474
$950$	0	0
$951$	−82.4490	−2.67359
$952$	0	0
$953$	19.6434	0.636313	0.318156	−	0.948038i	$-0.396936\pi$
$953$	19.6434	0.636313	0.318156	+	0.948038i	$0.396936\pi$
$954$	0	0
$955$	13.4601	0.435559
$956$	0	0
$957$	6.65902	0.215256
$958$	0	0
$959$	26.7976	0.865340
$960$	0	0
$961$	14.8555	0.479210
$962$	0	0
$963$	38.5819	1.24328
$964$	0	0
$965$	−7.30592	−0.235186
$966$	0	0
$967$	43.9618	1.41372	0.706858	−	0.707356i	$-0.250112\pi$
$967$	43.9618	1.41372	0.706858	+	0.707356i	$0.250112\pi$
$968$	0	0
$969$	−59.1681	−1.90075
$970$	0	0
$971$	27.6999	0.888931	0.444466	−	0.895796i	$-0.353394\pi$
$971$	27.6999	0.888931	0.444466	+	0.895796i	$0.353394\pi$
$972$	0	0
$973$	−40.1129	−1.28596
$974$	0	0
$975$	−7.90748	−0.253242
$976$	0	0
$977$	−52.4315	−1.67743	−0.838715	−	0.544570i	$-0.816693\pi$
$977$	−52.4315	−1.67743	−0.838715	+	0.544570i	$0.816693\pi$
$978$	0	0
$979$	8.96956	0.286668
$980$	0	0
$981$	−47.7554	−1.52471
$982$	0	0
$983$	−61.3074	−1.95540	−0.977702	−	0.209998i	$-0.932654\pi$
$983$	−61.3074	−1.95540	−0.977702	+	0.209998i	$0.932654\pi$
$984$	0	0
$985$	−1.52017	−0.0484366
$986$	0	0
$987$	−14.9511	−0.475900
$988$	0	0
$989$	64.3893	2.04746
$990$	0	0
$991$	29.0578	0.923050	0.461525	−	0.887127i	$-0.347302\pi$
$991$	29.0578	0.923050	0.461525	+	0.887127i	$0.347302\pi$
$992$	0	0
$993$	15.7444	0.499632
$994$	0	0
$995$	11.2747	0.357431
$996$	0	0
$997$	−37.3748	−1.18367	−0.591837	−	0.806058i	$-0.701597\pi$
$997$	−37.3748	−1.18367	−0.591837	+	0.806058i	$0.701597\pi$
$998$	0	0
$999$	−34.7417	−1.09918

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8020.2.a.f.1.3	✓	37

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8020.2.a.f.1.3	✓	37	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)

\(f(q)\)	\(=\)	\(q-3.05263 q^{3} +1.00000 q^{5} +2.86058 q^{7} +6.31852 q^{9} +O(q^{10})\)	\(q-3.05263 q^{3} +1.00000 q^{5} +2.86058 q^{7} +6.31852 q^{9} -2.92493 q^{11} +2.59039 q^{13} -3.05263 q^{15} -3.00659 q^{17} -6.44674 q^{19} -8.73228 q^{21} -6.65623 q^{23} +1.00000 q^{25} -10.1302 q^{27} +0.745798 q^{29} -6.77167 q^{31} +8.92872 q^{33} +2.86058 q^{35} +3.42952 q^{37} -7.90748 q^{39} +10.4927 q^{41} -9.67354 q^{43} +6.31852 q^{45} +1.71217 q^{47} +1.18292 q^{49} +9.17798 q^{51} -1.95028 q^{53} -2.92493 q^{55} +19.6795 q^{57} +2.01945 q^{59} +4.02714 q^{61} +18.0746 q^{63} +2.59039 q^{65} +11.7041 q^{67} +20.3190 q^{69} +13.9308 q^{71} +1.82517 q^{73} -3.05263 q^{75} -8.36700 q^{77} +7.86911 q^{79} +11.9681 q^{81} -0.167275 q^{83} -3.00659 q^{85} -2.27664 q^{87} -3.06659 q^{89} +7.41001 q^{91} +20.6714 q^{93} -6.44674 q^{95} -2.34116 q^{97} -18.4812 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 37 q + 3 q^{3} + 37 q^{5} + 4 q^{7} + 50 q^{9}+O(q^{10}) \) 37 * q + 3 * q^3 + 37 * q^5 + 4 * q^7 + 50 * q^9	\( 37 q + 3 q^{3} + 37 q^{5} + 4 q^{7} + 50 q^{9} + 2 q^{11} + 27 q^{13} + 3 q^{15} + 36 q^{17} - 6 q^{19} + 20 q^{21} + 17 q^{23} + 37 q^{25} + 9 q^{27} + 29 q^{29} + 5 q^{31} + 36 q^{33} + 4 q^{35} + 35 q^{37} + 21 q^{39} + 24 q^{41} + 11 q^{43} + 50 q^{45} + 19 q^{47} + 57 q^{49} + 8 q^{51} + 65 q^{53} + 2 q^{55} + 62 q^{57} - 9 q^{59} + 13 q^{61} + 26 q^{63} + 27 q^{65} + 13 q^{67} + 20 q^{69} + 33 q^{71} + 67 q^{73} + 3 q^{75} + 62 q^{77} + 23 q^{79} + 97 q^{81} + 2 q^{83} + 36 q^{85} + 32 q^{87} + 34 q^{89} + q^{91} + 41 q^{93} - 6 q^{95} + 66 q^{97} - 7 q^{99}+O(q^{100}) \) 37 * q + 3 * q^3 + 37 * q^5 + 4 * q^7 + 50 * q^9 + 2 * q^11 + 27 * q^13 + 3 * q^15 + 36 * q^17 - 6 * q^19 + 20 * q^21 + 17 * q^23 + 37 * q^25 + 9 * q^27 + 29 * q^29 + 5 * q^31 + 36 * q^33 + 4 * q^35 + 35 * q^37 + 21 * q^39 + 24 * q^41 + 11 * q^43 + 50 * q^45 + 19 * q^47 + 57 * q^49 + 8 * q^51 + 65 * q^53 + 2 * q^55 + 62 * q^57 - 9 * q^59 + 13 * q^61 + 26 * q^63 + 27 * q^65 + 13 * q^67 + 20 * q^69 + 33 * q^71 + 67 * q^73 + 3 * q^75 + 62 * q^77 + 23 * q^79 + 97 * q^81 + 2 * q^83 + 36 * q^85 + 32 * q^87 + 34 * q^89 + q^91 + 41 * q^93 - 6 * q^95 + 66 * q^97 - 7 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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