LMFDB - Embedded newform 8020.2.a.e.1.4

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			1.4
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)

$f(q)$	$=$	$q-2.93370 q^{3} -1.00000 q^{5} +4.42731 q^{7} +5.60662 q^{9} +O(q^{10})$	$q-2.93370 q^{3} -1.00000 q^{5} +4.42731 q^{7} +5.60662 q^{9} -4.88604 q^{11} +5.14097 q^{13} +2.93370 q^{15} +5.88727 q^{17} +1.35267 q^{19} -12.9884 q^{21} +6.77123 q^{23} +1.00000 q^{25} -7.64704 q^{27} +10.1828 q^{29} -7.69699 q^{31} +14.3342 q^{33} -4.42731 q^{35} +10.4292 q^{37} -15.0821 q^{39} +1.04474 q^{41} +1.57740 q^{43} -5.60662 q^{45} +10.9624 q^{47} +12.6011 q^{49} -17.2715 q^{51} +8.65866 q^{53} +4.88604 q^{55} -3.96832 q^{57} +0.162182 q^{59} -8.71052 q^{61} +24.8222 q^{63} -5.14097 q^{65} -9.19604 q^{67} -19.8648 q^{69} +14.3054 q^{71} -7.41197 q^{73} -2.93370 q^{75} -21.6320 q^{77} +0.446476 q^{79} +5.61430 q^{81} -2.85233 q^{83} -5.88727 q^{85} -29.8733 q^{87} +13.3205 q^{89} +22.7607 q^{91} +22.5807 q^{93} -1.35267 q^{95} +2.63422 q^{97} -27.3942 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.93370	−1.69377	−0.846887	−	0.531772i	$-0.821526\pi$
$3$	−2.93370	−1.69377	−0.846887	+	0.531772i	$0.821526\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	4.42731	1.67337	0.836683	−	0.547687i	$-0.184492\pi$
$7$	4.42731	1.67337	0.836683	+	0.547687i	$0.184492\pi$
$8$	0	0
$9$	5.60662	1.86887
$10$	0	0
$11$	−4.88604	−1.47320	−0.736599	−	0.676330i	$-0.763569\pi$
$11$	−4.88604	−1.47320	−0.736599	+	0.676330i	$0.763569\pi$
$12$	0	0
$13$	5.14097	1.42585	0.712925	−	0.701240i	$-0.247370\pi$
$13$	5.14097	1.42585	0.712925	+	0.701240i	$0.247370\pi$
$14$	0	0
$15$	2.93370	0.757479
$16$	0	0
$17$	5.88727	1.42787	0.713937	−	0.700210i	$-0.246910\pi$
$17$	5.88727	1.42787	0.713937	+	0.700210i	$0.246910\pi$
$18$	0	0
$19$	1.35267	0.310323	0.155162	−	0.987889i	$-0.450410\pi$
$19$	1.35267	0.310323	0.155162	+	0.987889i	$0.450410\pi$
$20$	0	0
$21$	−12.9884	−2.83431
$22$	0	0
$23$	6.77123	1.41190	0.705950	−	0.708262i	$-0.250520\pi$
$23$	6.77123	1.41190	0.705950	+	0.708262i	$0.250520\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−7.64704	−1.47167
$28$	0	0
$29$	10.1828	1.89090	0.945449	−	0.325770i	$-0.105624\pi$
$29$	10.1828	1.89090	0.945449	+	0.325770i	$0.105624\pi$
$30$	0	0
$31$	−7.69699	−1.38242	−0.691210	−	0.722654i	$-0.742922\pi$
$31$	−7.69699	−1.38242	−0.691210	+	0.722654i	$0.742922\pi$
$32$	0	0
$33$	14.3342	2.49527
$34$	0	0
$35$	−4.42731	−0.748352
$36$	0	0
$37$	10.4292	1.71455	0.857274	−	0.514860i	$-0.172156\pi$
$37$	10.4292	1.71455	0.857274	+	0.514860i	$0.172156\pi$
$38$	0	0
$39$	−15.0821	−2.41507
$40$	0	0
$41$	1.04474	0.163161	0.0815803	−	0.996667i	$-0.474003\pi$
$41$	1.04474	0.163161	0.0815803	+	0.996667i	$0.474003\pi$
$42$	0	0
$43$	1.57740	0.240551	0.120275	−	0.992741i	$-0.461622\pi$
$43$	1.57740	0.240551	0.120275	+	0.992741i	$0.461622\pi$
$44$	0	0
$45$	−5.60662	−0.835785
$46$	0	0
$47$	10.9624	1.59903	0.799515	−	0.600647i	$-0.205090\pi$
$47$	10.9624	1.59903	0.799515	+	0.600647i	$0.205090\pi$
$48$	0	0
$49$	12.6011	1.80016
$50$	0	0
$51$	−17.2715	−2.41850
$52$	0	0
$53$	8.65866	1.18936	0.594679	−	0.803963i	$-0.297279\pi$
$53$	8.65866	1.18936	0.594679	+	0.803963i	$0.297279\pi$
$54$	0	0
$55$	4.88604	0.658834
$56$	0	0
$57$	−3.96832	−0.525617
$58$	0	0
$59$	0.162182	0.0211143	0.0105572	−	0.999944i	$-0.496639\pi$
$59$	0.162182	0.0211143	0.0105572	+	0.999944i	$0.496639\pi$
$60$	0	0
$61$	−8.71052	−1.11527	−0.557634	−	0.830087i	$-0.688291\pi$
$61$	−8.71052	−1.11527	−0.557634	+	0.830087i	$0.688291\pi$
$62$	0	0
$63$	24.8222	3.12731
$64$	0	0
$65$	−5.14097	−0.637659
$66$	0	0
$67$	−9.19604	−1.12348	−0.561738	−	0.827315i	$-0.689867\pi$
$67$	−9.19604	−1.12348	−0.561738	+	0.827315i	$0.689867\pi$
$68$	0	0
$69$	−19.8648	−2.39144
$70$	0	0
$71$	14.3054	1.69774	0.848872	−	0.528599i	$-0.177282\pi$
$71$	14.3054	1.69774	0.848872	+	0.528599i	$0.177282\pi$
$72$	0	0
$73$	−7.41197	−0.867505	−0.433753	−	0.901032i	$-0.642811\pi$
$73$	−7.41197	−0.867505	−0.433753	+	0.901032i	$0.642811\pi$
$74$	0	0
$75$	−2.93370	−0.338755
$76$	0	0
$77$	−21.6320	−2.46520
$78$	0	0
$79$	0.446476	0.0502325	0.0251162	−	0.999685i	$-0.492004\pi$
$79$	0.446476	0.0502325	0.0251162	+	0.999685i	$0.492004\pi$
$80$	0	0
$81$	5.61430	0.623811
$82$	0	0
$83$	−2.85233	−0.313084	−0.156542	−	0.987671i	$-0.550035\pi$
$83$	−2.85233	−0.313084	−0.156542	+	0.987671i	$0.550035\pi$
$84$	0	0
$85$	−5.88727	−0.638564
$86$	0	0
$87$	−29.8733	−3.20275
$88$	0	0
$89$	13.3205	1.41197	0.705984	−	0.708228i	$-0.250505\pi$
$89$	13.3205	1.41197	0.705984	+	0.708228i	$0.250505\pi$
$90$	0	0
$91$	22.7607	2.38597
$92$	0	0
$93$	22.5807	2.34151
$94$	0	0
$95$	−1.35267	−0.138781
$96$	0	0
$97$	2.63422	0.267465	0.133732	−	0.991017i	$-0.457304\pi$
$97$	2.63422	0.267465	0.133732	+	0.991017i	$0.457304\pi$
$98$	0	0
$99$	−27.3942	−2.75322
$100$	0	0
$101$	10.7324	1.06791	0.533957	−	0.845512i	$-0.320704\pi$
$101$	10.7324	1.06791	0.533957	+	0.845512i	$0.320704\pi$
$102$	0	0
$103$	10.3791	1.02268	0.511342	−	0.859377i	$-0.329149\pi$
$103$	10.3791	1.02268	0.511342	+	0.859377i	$0.329149\pi$
$104$	0	0
$105$	12.9884	1.26754
$106$	0	0
$107$	−2.22637	−0.215231	−0.107615	−	0.994193i	$-0.534322\pi$
$107$	−2.22637	−0.215231	−0.107615	+	0.994193i	$0.534322\pi$
$108$	0	0
$109$	−10.7347	−1.02820	−0.514100	−	0.857730i	$-0.671874\pi$
$109$	−10.7347	−1.02820	−0.514100	+	0.857730i	$0.671874\pi$
$110$	0	0
$111$	−30.5961	−2.90406
$112$	0	0
$113$	11.8136	1.11133	0.555663	−	0.831408i	$-0.312464\pi$
$113$	11.8136	1.11133	0.555663	+	0.831408i	$0.312464\pi$
$114$	0	0
$115$	−6.77123	−0.631421
$116$	0	0
$117$	28.8235	2.66473
$118$	0	0
$119$	26.0648	2.38936
$120$	0	0
$121$	12.8734	1.17031
$122$	0	0
$123$	−3.06495	−0.276357
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	5.76361	0.511437	0.255719	−	0.966751i	$-0.417688\pi$
$127$	5.76361	0.511437	0.255719	+	0.966751i	$0.417688\pi$
$128$	0	0
$129$	−4.62761	−0.407439
$130$	0	0
$131$	−20.9937	−1.83423	−0.917114	−	0.398624i	$-0.869488\pi$
$131$	−20.9937	−1.83423	−0.917114	+	0.398624i	$0.869488\pi$
$132$	0	0
$133$	5.98868	0.519284
$134$	0	0
$135$	7.64704	0.658152
$136$	0	0
$137$	−16.6704	−1.42425	−0.712123	−	0.702055i	$-0.752266\pi$
$137$	−16.6704	−1.42425	−0.712123	+	0.702055i	$0.752266\pi$
$138$	0	0
$139$	11.2975	0.958243	0.479122	−	0.877749i	$-0.340955\pi$
$139$	11.2975	0.958243	0.479122	+	0.877749i	$0.340955\pi$
$140$	0	0
$141$	−32.1604	−2.70840
$142$	0	0
$143$	−25.1190	−2.10056
$144$	0	0
$145$	−10.1828	−0.845635
$146$	0	0
$147$	−36.9679	−3.04906
$148$	0	0
$149$	−21.4555	−1.75770	−0.878850	−	0.477099i	$-0.841688\pi$
$149$	−21.4555	−1.75770	−0.878850	+	0.477099i	$0.841688\pi$
$150$	0	0
$151$	−21.0536	−1.71331	−0.856657	−	0.515886i	$-0.827463\pi$
$151$	−21.0536	−1.71331	−0.856657	+	0.515886i	$0.827463\pi$
$152$	0	0
$153$	33.0077	2.66851
$154$	0	0
$155$	7.69699	0.618237
$156$	0	0
$157$	−7.04653	−0.562374	−0.281187	−	0.959653i	$-0.590728\pi$
$157$	−7.04653	−0.562374	−0.281187	+	0.959653i	$0.590728\pi$
$158$	0	0
$159$	−25.4020	−2.01451
$160$	0	0
$161$	29.9784	2.36263
$162$	0	0
$163$	−6.14528	−0.481335	−0.240668	−	0.970608i	$-0.577366\pi$
$163$	−6.14528	−0.481335	−0.240668	+	0.970608i	$0.577366\pi$
$164$	0	0
$165$	−14.3342	−1.11592
$166$	0	0
$167$	−15.4110	−1.19254	−0.596270	−	0.802784i	$-0.703351\pi$
$167$	−15.4110	−1.19254	−0.596270	+	0.802784i	$0.703351\pi$
$168$	0	0
$169$	13.4296	1.03305
$170$	0	0
$171$	7.58388	0.579954
$172$	0	0
$173$	−17.1572	−1.30444	−0.652218	−	0.758031i	$-0.726161\pi$
$173$	−17.1572	−1.30444	−0.652218	+	0.758031i	$0.726161\pi$
$174$	0	0
$175$	4.42731	0.334673
$176$	0	0
$177$	−0.475795	−0.0357629
$178$	0	0
$179$	4.27305	0.319383	0.159692	−	0.987167i	$-0.448950\pi$
$179$	4.27305	0.319383	0.159692	+	0.987167i	$0.448950\pi$
$180$	0	0
$181$	−1.00665	−0.0748239	−0.0374119	−	0.999300i	$-0.511911\pi$
$181$	−1.00665	−0.0748239	−0.0374119	+	0.999300i	$0.511911\pi$
$182$	0	0
$183$	25.5541	1.88901
$184$	0	0
$185$	−10.4292	−0.766769
$186$	0	0
$187$	−28.7655	−2.10354
$188$	0	0
$189$	−33.8558	−2.46265
$190$	0	0
$191$	−3.59706	−0.260274	−0.130137	−	0.991496i	$-0.541542\pi$
$191$	−3.59706	−0.260274	−0.130137	+	0.991496i	$0.541542\pi$
$192$	0	0
$193$	−10.7610	−0.774592	−0.387296	−	0.921956i	$-0.626591\pi$
$193$	−10.7610	−0.774592	−0.387296	+	0.921956i	$0.626591\pi$
$194$	0	0
$195$	15.0821	1.08005
$196$	0	0
$197$	15.2035	1.08320	0.541601	−	0.840636i	$-0.317818\pi$
$197$	15.2035	1.08320	0.541601	+	0.840636i	$0.317818\pi$
$198$	0	0
$199$	4.14885	0.294104	0.147052	−	0.989129i	$-0.453022\pi$
$199$	4.14885	0.294104	0.147052	+	0.989129i	$0.453022\pi$
$200$	0	0
$201$	26.9785	1.90291
$202$	0	0
$203$	45.0824	3.16417
$204$	0	0
$205$	−1.04474	−0.0729677
$206$	0	0
$207$	37.9637	2.63866
$208$	0	0
$209$	−6.60919	−0.457167
$210$	0	0
$211$	−9.02318	−0.621182	−0.310591	−	0.950544i	$-0.600527\pi$
$211$	−9.02318	−0.621182	−0.310591	+	0.950544i	$0.600527\pi$
$212$	0	0
$213$	−41.9679	−2.87560
$214$	0	0
$215$	−1.57740	−0.107578
$216$	0	0
$217$	−34.0770	−2.31330
$218$	0	0
$219$	21.7445	1.46936
$220$	0	0
$221$	30.2663	2.03593
$222$	0	0
$223$	−4.60813	−0.308583	−0.154291	−	0.988025i	$-0.549309\pi$
$223$	−4.60813	−0.308583	−0.154291	+	0.988025i	$0.549309\pi$
$224$	0	0
$225$	5.60662	0.373774
$226$	0	0
$227$	−6.67409	−0.442975	−0.221487	−	0.975163i	$-0.571091\pi$
$227$	−6.67409	−0.442975	−0.221487	+	0.975163i	$0.571091\pi$
$228$	0	0
$229$	−12.3547	−0.816421	−0.408211	−	0.912888i	$-0.633847\pi$
$229$	−12.3547	−0.816421	−0.408211	+	0.912888i	$0.633847\pi$
$230$	0	0
$231$	63.4620	4.17549
$232$	0	0
$233$	−25.8047	−1.69052	−0.845262	−	0.534352i	$-0.820556\pi$
$233$	−25.8047	−1.69052	−0.845262	+	0.534352i	$0.820556\pi$
$234$	0	0
$235$	−10.9624	−0.715108
$236$	0	0
$237$	−1.30983	−0.0850825
$238$	0	0
$239$	4.78171	0.309303	0.154652	−	0.987969i	$-0.450575\pi$
$239$	4.78171	0.309303	0.154652	+	0.987969i	$0.450575\pi$
$240$	0	0
$241$	−12.1372	−0.781829	−0.390914	−	0.920427i	$-0.627841\pi$
$241$	−12.1372	−0.781829	−0.390914	+	0.920427i	$0.627841\pi$
$242$	0	0
$243$	6.47043	0.415078
$244$	0	0
$245$	−12.6011	−0.805054
$246$	0	0
$247$	6.95402	0.442474
$248$	0	0
$249$	8.36789	0.530294
$250$	0	0
$251$	2.54915	0.160901	0.0804506	−	0.996759i	$-0.474364\pi$
$251$	2.54915	0.160901	0.0804506	+	0.996759i	$0.474364\pi$
$252$	0	0
$253$	−33.0845	−2.08001
$254$	0	0
$255$	17.2715	1.08158
$256$	0	0
$257$	−5.64962	−0.352413	−0.176207	−	0.984353i	$-0.556383\pi$
$257$	−5.64962	−0.352413	−0.176207	+	0.984353i	$0.556383\pi$
$258$	0	0
$259$	46.1733	2.86907
$260$	0	0
$261$	57.0910	3.53385
$262$	0	0
$263$	19.2578	1.18749	0.593744	−	0.804654i	$-0.297649\pi$
$263$	19.2578	1.18749	0.593744	+	0.804654i	$0.297649\pi$
$264$	0	0
$265$	−8.65866	−0.531897
$266$	0	0
$267$	−39.0783	−2.39155
$268$	0	0
$269$	12.0330	0.733666	0.366833	−	0.930287i	$-0.380442\pi$
$269$	12.0330	0.733666	0.366833	+	0.930287i	$0.380442\pi$
$270$	0	0
$271$	1.09425	0.0664708	0.0332354	−	0.999448i	$-0.489419\pi$
$271$	1.09425	0.0664708	0.0332354	+	0.999448i	$0.489419\pi$
$272$	0	0
$273$	−66.7731	−4.04129
$274$	0	0
$275$	−4.88604	−0.294640
$276$	0	0
$277$	14.2767	0.857805	0.428903	−	0.903351i	$-0.358900\pi$
$277$	14.2767	0.857805	0.428903	+	0.903351i	$0.358900\pi$
$278$	0	0
$279$	−43.1541	−2.58357
$280$	0	0
$281$	15.2441	0.909384	0.454692	−	0.890649i	$-0.349749\pi$
$281$	15.2441	0.909384	0.454692	+	0.890649i	$0.349749\pi$
$282$	0	0
$283$	−27.0367	−1.60716	−0.803581	−	0.595195i	$-0.797075\pi$
$283$	−27.0367	−1.60716	−0.803581	+	0.595195i	$0.797075\pi$
$284$	0	0
$285$	3.96832	0.235063
$286$	0	0
$287$	4.62538	0.273028
$288$	0	0
$289$	17.6600	1.03882
$290$	0	0
$291$	−7.72802	−0.453025
$292$	0	0
$293$	−20.9831	−1.22585	−0.612923	−	0.790143i	$-0.710006\pi$
$293$	−20.9831	−1.22585	−0.612923	+	0.790143i	$0.710006\pi$
$294$	0	0
$295$	−0.162182	−0.00944262
$296$	0	0
$297$	37.3638	2.16807
$298$	0	0
$299$	34.8107	2.01316
$300$	0	0
$301$	6.98363	0.402529
$302$	0	0
$303$	−31.4857	−1.80881
$304$	0	0
$305$	8.71052	0.498763
$306$	0	0
$307$	4.05050	0.231174	0.115587	−	0.993297i	$-0.463125\pi$
$307$	4.05050	0.231174	0.115587	+	0.993297i	$0.463125\pi$
$308$	0	0
$309$	−30.4492	−1.73220
$310$	0	0
$311$	−2.03759	−0.115541	−0.0577707	−	0.998330i	$-0.518399\pi$
$311$	−2.03759	−0.115541	−0.0577707	+	0.998330i	$0.518399\pi$
$312$	0	0
$313$	32.6599	1.84605	0.923024	−	0.384742i	$-0.125710\pi$
$313$	32.6599	1.84605	0.923024	+	0.384742i	$0.125710\pi$
$314$	0	0
$315$	−24.8222	−1.39857
$316$	0	0
$317$	−1.99713	−0.112170	−0.0560850	−	0.998426i	$-0.517862\pi$
$317$	−1.99713	−0.112170	−0.0560850	+	0.998426i	$0.517862\pi$
$318$	0	0
$319$	−49.7536	−2.78567
$320$	0	0
$321$	6.53150	0.364553
$322$	0	0
$323$	7.96352	0.443102
$324$	0	0
$325$	5.14097	0.285170
$326$	0	0
$327$	31.4925	1.74154
$328$	0	0
$329$	48.5339	2.67576
$330$	0	0
$331$	12.5464	0.689611	0.344805	−	0.938674i	$-0.387945\pi$
$331$	12.5464	0.689611	0.344805	+	0.938674i	$0.387945\pi$
$332$	0	0
$333$	58.4725	3.20427
$334$	0	0
$335$	9.19604	0.502433
$336$	0	0
$337$	30.6680	1.67059	0.835295	−	0.549801i	$-0.185297\pi$
$337$	30.6680	1.67059	0.835295	+	0.549801i	$0.185297\pi$
$338$	0	0
$339$	−34.6575	−1.88234
$340$	0	0
$341$	37.6078	2.03658
$342$	0	0
$343$	24.7978	1.33895
$344$	0	0
$345$	19.8648	1.06948
$346$	0	0
$347$	28.9200	1.55251	0.776254	−	0.630420i	$-0.217117\pi$
$347$	28.9200	1.55251	0.776254	+	0.630420i	$0.217117\pi$
$348$	0	0
$349$	18.6106	0.996202	0.498101	−	0.867119i	$-0.334031\pi$
$349$	18.6106	0.996202	0.498101	+	0.867119i	$0.334031\pi$
$350$	0	0
$351$	−39.3132	−2.09839
$352$	0	0
$353$	−5.27534	−0.280778	−0.140389	−	0.990096i	$-0.544835\pi$
$353$	−5.27534	−0.280778	−0.140389	+	0.990096i	$0.544835\pi$
$354$	0	0
$355$	−14.3054	−0.759254
$356$	0	0
$357$	−76.4664	−4.04703
$358$	0	0
$359$	13.4709	0.710968	0.355484	−	0.934682i	$-0.384316\pi$
$359$	13.4709	0.710968	0.355484	+	0.934682i	$0.384316\pi$
$360$	0	0
$361$	−17.1703	−0.903700
$362$	0	0
$363$	−37.7668	−1.98225
$364$	0	0
$365$	7.41197	0.387960
$366$	0	0
$367$	−11.8944	−0.620883	−0.310441	−	0.950593i	$-0.600477\pi$
$367$	−11.8944	−0.620883	−0.310441	+	0.950593i	$0.600477\pi$
$368$	0	0
$369$	5.85744	0.304926
$370$	0	0
$371$	38.3346	1.99023
$372$	0	0
$373$	−12.0340	−0.623098	−0.311549	−	0.950230i	$-0.600848\pi$
$373$	−12.0340	−0.623098	−0.311549	+	0.950230i	$0.600848\pi$
$374$	0	0
$375$	2.93370	0.151496
$376$	0	0
$377$	52.3495	2.69614
$378$	0	0
$379$	0.251051	0.0128956	0.00644781	−	0.999979i	$-0.497948\pi$
$379$	0.251051	0.0128956	0.00644781	+	0.999979i	$0.497948\pi$
$380$	0	0
$381$	−16.9087	−0.866260
$382$	0	0
$383$	28.6593	1.46442	0.732212	−	0.681077i	$-0.238488\pi$
$383$	28.6593	1.46442	0.732212	+	0.681077i	$0.238488\pi$
$384$	0	0
$385$	21.6320	1.10247
$386$	0	0
$387$	8.84386	0.449558
$388$	0	0
$389$	6.09274	0.308914	0.154457	−	0.988000i	$-0.450637\pi$
$389$	6.09274	0.308914	0.154457	+	0.988000i	$0.450637\pi$
$390$	0	0
$391$	39.8641	2.01601
$392$	0	0
$393$	61.5893	3.10677
$394$	0	0
$395$	−0.446476	−0.0224647
$396$	0	0
$397$	−4.79140	−0.240473	−0.120237	−	0.992745i	$-0.538365\pi$
$397$	−4.79140	−0.240473	−0.120237	+	0.992745i	$0.538365\pi$
$398$	0	0
$399$	−17.5690	−0.879550
$400$	0	0
$401$	−1.00000	−0.0499376
$401$	−1.00000	−0.0499376
$402$	0	0
$403$	−39.5700	−1.97112
$404$	0	0
$405$	−5.61430	−0.278977
$406$	0	0
$407$	−50.9575	−2.52587
$408$	0	0
$409$	−29.7951	−1.47327	−0.736635	−	0.676290i	$-0.763587\pi$
$409$	−29.7951	−1.47327	−0.736635	+	0.676290i	$0.763587\pi$
$410$	0	0
$411$	48.9059	2.41235
$412$	0	0
$413$	0.718032	0.0353320
$414$	0	0
$415$	2.85233	0.140015
$416$	0	0
$417$	−33.1436	−1.62305
$418$	0	0
$419$	−24.2813	−1.18622	−0.593108	−	0.805123i	$-0.702099\pi$
$419$	−24.2813	−1.18622	−0.593108	+	0.805123i	$0.702099\pi$
$420$	0	0
$421$	1.68344	0.0820459	0.0410230	−	0.999158i	$-0.486938\pi$
$421$	1.68344	0.0820459	0.0410230	+	0.999158i	$0.486938\pi$
$422$	0	0
$423$	61.4619	2.98838
$424$	0	0
$425$	5.88727	0.285575
$426$	0	0
$427$	−38.5642	−1.86625
$428$	0	0
$429$	73.6918	3.55787
$430$	0	0
$431$	0.159481	0.00768193	0.00384097	−	0.999993i	$-0.498777\pi$
$431$	0.159481	0.00768193	0.00384097	+	0.999993i	$0.498777\pi$
$432$	0	0
$433$	10.4425	0.501837	0.250918	−	0.968008i	$-0.419267\pi$
$433$	10.4425	0.501837	0.250918	+	0.968008i	$0.419267\pi$
$434$	0	0
$435$	29.8733	1.43232
$436$	0	0
$437$	9.15922	0.438145
$438$	0	0
$439$	−2.87787	−0.137353	−0.0686767	−	0.997639i	$-0.521878\pi$
$439$	−2.87787	−0.137353	−0.0686767	+	0.997639i	$0.521878\pi$
$440$	0	0
$441$	70.6495	3.36426
$442$	0	0
$443$	−3.67968	−0.174827	−0.0874135	−	0.996172i	$-0.527860\pi$
$443$	−3.67968	−0.174827	−0.0874135	+	0.996172i	$0.527860\pi$
$444$	0	0
$445$	−13.3205	−0.631451
$446$	0	0
$447$	62.9439	2.97715
$448$	0	0
$449$	−21.2261	−1.00172	−0.500861	−	0.865527i	$-0.666983\pi$
$449$	−21.2261	−1.00172	−0.500861	+	0.865527i	$0.666983\pi$
$450$	0	0
$451$	−5.10464	−0.240368
$452$	0	0
$453$	61.7649	2.90197
$454$	0	0
$455$	−22.7607	−1.06704
$456$	0	0
$457$	−12.6640	−0.592395	−0.296197	−	0.955127i	$-0.595719\pi$
$457$	−12.6640	−0.592395	−0.296197	+	0.955127i	$0.595719\pi$
$458$	0	0
$459$	−45.0202	−2.10136
$460$	0	0
$461$	28.9241	1.34713	0.673564	−	0.739129i	$-0.264763\pi$
$461$	28.9241	1.34713	0.673564	+	0.739129i	$0.264763\pi$
$462$	0	0
$463$	−2.67541	−0.124337	−0.0621685	−	0.998066i	$-0.519802\pi$
$463$	−2.67541	−0.124337	−0.0621685	+	0.998066i	$0.519802\pi$
$464$	0	0
$465$	−22.5807	−1.04715
$466$	0	0
$467$	2.32837	0.107744	0.0538721	−	0.998548i	$-0.482844\pi$
$467$	2.32837	0.107744	0.0538721	+	0.998548i	$0.482844\pi$
$468$	0	0
$469$	−40.7137	−1.87999
$470$	0	0
$471$	20.6724	0.952536
$472$	0	0
$473$	−7.70723	−0.354379
$474$	0	0
$475$	1.35267	0.0620646
$476$	0	0
$477$	48.5458	2.22276
$478$	0	0
$479$	−25.8388	−1.18061	−0.590303	−	0.807182i	$-0.700992\pi$
$479$	−25.8388	−1.18061	−0.590303	+	0.807182i	$0.700992\pi$
$480$	0	0
$481$	53.6162	2.44469
$482$	0	0
$483$	−87.9476	−4.00176
$484$	0	0
$485$	−2.63422	−0.119614
$486$	0	0
$487$	−37.6336	−1.70534	−0.852672	−	0.522447i	$-0.825019\pi$
$487$	−37.6336	−1.70534	−0.852672	+	0.522447i	$0.825019\pi$
$488$	0	0
$489$	18.0284	0.815274
$490$	0	0
$491$	−40.7881	−1.84074	−0.920369	−	0.391050i	$-0.872112\pi$
$491$	−40.7881	−1.84074	−0.920369	+	0.391050i	$0.872112\pi$
$492$	0	0
$493$	59.9489	2.69996
$494$	0	0
$495$	27.3942	1.23128
$496$	0	0
$497$	63.3347	2.84095
$498$	0	0
$499$	31.5598	1.41281	0.706405	−	0.707807i	$-0.250316\pi$
$499$	31.5598	1.41281	0.706405	+	0.707807i	$0.250316\pi$
$500$	0	0
$501$	45.2113	2.01989
$502$	0	0
$503$	21.5833	0.962351	0.481176	−	0.876624i	$-0.340210\pi$
$503$	21.5833	0.962351	0.481176	+	0.876624i	$0.340210\pi$
$504$	0	0
$505$	−10.7324	−0.477586
$506$	0	0
$507$	−39.3985	−1.74975
$508$	0	0
$509$	32.1942	1.42698	0.713491	−	0.700664i	$-0.247113\pi$
$509$	32.1942	1.42698	0.713491	+	0.700664i	$0.247113\pi$
$510$	0	0
$511$	−32.8151	−1.45165
$512$	0	0
$513$	−10.3439	−0.456694
$514$	0	0
$515$	−10.3791	−0.457358
$516$	0	0
$517$	−53.5627	−2.35569
$518$	0	0
$519$	50.3341	2.20942
$520$	0	0
$521$	7.36896	0.322840	0.161420	−	0.986886i	$-0.448393\pi$
$521$	7.36896	0.322840	0.161420	+	0.986886i	$0.448393\pi$
$522$	0	0
$523$	2.78301	0.121693	0.0608463	−	0.998147i	$-0.480620\pi$
$523$	2.78301	0.121693	0.0608463	+	0.998147i	$0.480620\pi$
$524$	0	0
$525$	−12.9884	−0.566861
$526$	0	0
$527$	−45.3143	−1.97392
$528$	0	0
$529$	22.8496	0.993461
$530$	0	0
$531$	0.909295	0.0394600
$532$	0	0
$533$	5.37097	0.232643
$534$	0	0
$535$	2.22637	0.0962542
$536$	0	0
$537$	−12.5359	−0.540963
$538$	0	0
$539$	−61.5695	−2.65199
$540$	0	0
$541$	−23.7209	−1.01984	−0.509921	−	0.860221i	$-0.670325\pi$
$541$	−23.7209	−1.01984	−0.509921	+	0.860221i	$0.670325\pi$
$542$	0	0
$543$	2.95322	0.126735
$544$	0	0
$545$	10.7347	0.459825
$546$	0	0
$547$	30.2236	1.29227	0.646135	−	0.763223i	$-0.276384\pi$
$547$	30.2236	1.29227	0.646135	+	0.763223i	$0.276384\pi$
$548$	0	0
$549$	−48.8365	−2.08429
$550$	0	0
$551$	13.7739	0.586789
$552$	0	0
$553$	1.97669	0.0840574
$554$	0	0
$555$	30.5961	1.29873
$556$	0	0
$557$	42.5626	1.80343	0.901717	−	0.432326i	$-0.142307\pi$
$557$	42.5626	1.80343	0.901717	+	0.432326i	$0.142307\pi$
$558$	0	0
$559$	8.10935	0.342989
$560$	0	0
$561$	84.3894	3.56292
$562$	0	0
$563$	−32.9489	−1.38863	−0.694316	−	0.719670i	$-0.744293\pi$
$563$	−32.9489	−1.38863	−0.694316	+	0.719670i	$0.744293\pi$
$564$	0	0
$565$	−11.8136	−0.497000
$566$	0	0
$567$	24.8563	1.04386
$568$	0	0
$569$	−33.7109	−1.41323	−0.706616	−	0.707597i	$-0.749779\pi$
$569$	−33.7109	−1.41323	−0.706616	+	0.707597i	$0.749779\pi$
$570$	0	0
$571$	14.9994	0.627704	0.313852	−	0.949472i	$-0.398380\pi$
$571$	14.9994	0.627704	0.313852	+	0.949472i	$0.398380\pi$
$572$	0	0
$573$	10.5527	0.440846
$574$	0	0
$575$	6.77123	0.282380
$576$	0	0
$577$	20.5560	0.855759	0.427880	−	0.903836i	$-0.359261\pi$
$577$	20.5560	0.855759	0.427880	+	0.903836i	$0.359261\pi$
$578$	0	0
$579$	31.5695	1.31198
$580$	0	0
$581$	−12.6282	−0.523904
$582$	0	0
$583$	−42.3066	−1.75216
$584$	0	0
$585$	−28.8235	−1.19170
$586$	0	0
$587$	17.8249	0.735711	0.367855	−	0.929883i	$-0.380092\pi$
$587$	17.8249	0.735711	0.367855	+	0.929883i	$0.380092\pi$
$588$	0	0
$589$	−10.4115	−0.428997
$590$	0	0
$591$	−44.6025	−1.83470
$592$	0	0
$593$	34.5016	1.41681	0.708405	−	0.705806i	$-0.249415\pi$
$593$	34.5016	1.41681	0.708405	+	0.705806i	$0.249415\pi$
$594$	0	0
$595$	−26.0648	−1.06855
$596$	0	0
$597$	−12.1715	−0.498146
$598$	0	0
$599$	3.01662	0.123256	0.0616278	−	0.998099i	$-0.480371\pi$
$599$	3.01662	0.123256	0.0616278	+	0.998099i	$0.480371\pi$
$600$	0	0
$601$	15.8695	0.647332	0.323666	−	0.946171i	$-0.395085\pi$
$601$	15.8695	0.647332	0.323666	+	0.946171i	$0.395085\pi$
$602$	0	0
$603$	−51.5587	−2.09963
$604$	0	0
$605$	−12.8734	−0.523380
$606$	0	0
$607$	10.4673	0.424856	0.212428	−	0.977177i	$-0.431863\pi$
$607$	10.4673	0.424856	0.212428	+	0.977177i	$0.431863\pi$
$608$	0	0
$609$	−132.258	−5.35938
$610$	0	0
$611$	56.3574	2.27998
$612$	0	0
$613$	−8.51113	−0.343761	−0.171881	−	0.985118i	$-0.554984\pi$
$613$	−8.51113	−0.343761	−0.171881	+	0.985118i	$0.554984\pi$
$614$	0	0
$615$	3.06495	0.123591
$616$	0	0
$617$	−23.9567	−0.964461	−0.482231	−	0.876044i	$-0.660173\pi$
$617$	−23.9567	−0.964461	−0.482231	+	0.876044i	$0.660173\pi$
$618$	0	0
$619$	−31.6536	−1.27227	−0.636134	−	0.771579i	$-0.719467\pi$
$619$	−31.6536	−1.27227	−0.636134	+	0.771579i	$0.719467\pi$
$620$	0	0
$621$	−51.7799	−2.07786
$622$	0	0
$623$	58.9739	2.36274
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	19.3894	0.774338
$628$	0	0
$629$	61.3995	2.44816
$630$	0	0
$631$	3.29459	0.131156	0.0655778	−	0.997847i	$-0.479111\pi$
$631$	3.29459	0.131156	0.0655778	+	0.997847i	$0.479111\pi$
$632$	0	0
$633$	26.4713	1.05214
$634$	0	0
$635$	−5.76361	−0.228722
$636$	0	0
$637$	64.7819	2.56675
$638$	0	0
$639$	80.2051	3.17287
$640$	0	0
$641$	24.2278	0.956939	0.478470	−	0.878104i	$-0.341192\pi$
$641$	24.2278	0.956939	0.478470	+	0.878104i	$0.341192\pi$
$642$	0	0
$643$	2.82809	0.111529	0.0557644	−	0.998444i	$-0.482240\pi$
$643$	2.82809	0.111529	0.0557644	+	0.998444i	$0.482240\pi$
$644$	0	0
$645$	4.62761	0.182212
$646$	0	0
$647$	4.94948	0.194584	0.0972921	−	0.995256i	$-0.468982\pi$
$647$	4.94948	0.194584	0.0972921	+	0.995256i	$0.468982\pi$
$648$	0	0
$649$	−0.792430	−0.0311056
$650$	0	0
$651$	99.9717	3.91820
$652$	0	0
$653$	−24.3673	−0.953565	−0.476783	−	0.879021i	$-0.658197\pi$
$653$	−24.3673	−0.953565	−0.476783	+	0.879021i	$0.658197\pi$
$654$	0	0
$655$	20.9937	0.820292
$656$	0	0
$657$	−41.5561	−1.62126
$658$	0	0
$659$	35.1649	1.36983	0.684914	−	0.728624i	$-0.259840\pi$
$659$	35.1649	1.36983	0.684914	+	0.728624i	$0.259840\pi$
$660$	0	0
$661$	4.14797	0.161337	0.0806686	−	0.996741i	$-0.474294\pi$
$661$	4.14797	0.161337	0.0806686	+	0.996741i	$0.474294\pi$
$662$	0	0
$663$	−88.7924	−3.44841
$664$	0	0
$665$	−5.98868	−0.232231
$666$	0	0
$667$	68.9501	2.66976
$668$	0	0
$669$	13.5189	0.522670
$670$	0	0
$671$	42.5600	1.64301
$672$	0	0
$673$	−0.620296	−0.0239107	−0.0119553	−	0.999929i	$-0.503806\pi$
$673$	−0.620296	−0.0239107	−0.0119553	+	0.999929i	$0.503806\pi$
$674$	0	0
$675$	−7.64704	−0.294335
$676$	0	0
$677$	37.5961	1.44494	0.722468	−	0.691405i	$-0.243008\pi$
$677$	37.5961	1.44494	0.722468	+	0.691405i	$0.243008\pi$
$678$	0	0
$679$	11.6625	0.447566
$680$	0	0
$681$	19.5798	0.750299
$682$	0	0
$683$	2.22494	0.0851348	0.0425674	−	0.999094i	$-0.486446\pi$
$683$	2.22494	0.0851348	0.0425674	+	0.999094i	$0.486446\pi$
$684$	0	0
$685$	16.6704	0.636942
$686$	0	0
$687$	36.2450	1.38283
$688$	0	0
$689$	44.5140	1.69585
$690$	0	0
$691$	−27.0079	−1.02743	−0.513715	−	0.857961i	$-0.671731\pi$
$691$	−27.0079	−1.02743	−0.513715	+	0.857961i	$0.671731\pi$
$692$	0	0
$693$	−121.283	−4.60714
$694$	0	0
$695$	−11.2975	−0.428539
$696$	0	0
$697$	6.15066	0.232973
$698$	0	0
$699$	75.7034	2.86337
$700$	0	0
$701$	37.4675	1.41513	0.707564	−	0.706649i	$-0.249794\pi$
$701$	37.4675	1.41513	0.707564	+	0.706649i	$0.249794\pi$
$702$	0	0
$703$	14.1072	0.532064
$704$	0	0
$705$	32.1604	1.21123
$706$	0	0
$707$	47.5157	1.78701
$708$	0	0
$709$	−2.75023	−0.103287	−0.0516436	−	0.998666i	$-0.516446\pi$
$709$	−2.75023	−0.103287	−0.0516436	+	0.998666i	$0.516446\pi$
$710$	0	0
$711$	2.50322	0.0938781
$712$	0	0
$713$	−52.1181	−1.95184
$714$	0	0
$715$	25.1190	0.939398
$716$	0	0
$717$	−14.0281	−0.523890
$718$	0	0
$719$	24.1067	0.899028	0.449514	−	0.893273i	$-0.351597\pi$
$719$	24.1067	0.899028	0.449514	+	0.893273i	$0.351597\pi$
$720$	0	0
$721$	45.9516	1.71133
$722$	0	0
$723$	35.6071	1.32424
$724$	0	0
$725$	10.1828	0.378180
$726$	0	0
$727$	−24.3431	−0.902836	−0.451418	−	0.892313i	$-0.649082\pi$
$727$	−24.3431	−0.902836	−0.451418	+	0.892313i	$0.649082\pi$
$728$	0	0
$729$	−35.8252	−1.32686
$730$	0	0
$731$	9.28656	0.343476
$732$	0	0
$733$	−48.0804	−1.77589	−0.887945	−	0.459950i	$-0.847867\pi$
$733$	−48.0804	−1.77589	−0.887945	+	0.459950i	$0.847867\pi$
$734$	0	0
$735$	36.9679	1.36358
$736$	0	0
$737$	44.9323	1.65510
$738$	0	0
$739$	−38.8478	−1.42904	−0.714519	−	0.699616i	$-0.753354\pi$
$739$	−38.8478	−1.42904	−0.714519	+	0.699616i	$0.753354\pi$
$740$	0	0
$741$	−20.4010	−0.749451
$742$	0	0
$743$	−32.1391	−1.17907	−0.589534	−	0.807744i	$-0.700689\pi$
$743$	−32.1391	−1.17907	−0.589534	+	0.807744i	$0.700689\pi$
$744$	0	0
$745$	21.4555	0.786067
$746$	0	0
$747$	−15.9919	−0.585114
$748$	0	0
$749$	−9.85682	−0.360160
$750$	0	0
$751$	−26.6146	−0.971182	−0.485591	−	0.874186i	$-0.661395\pi$
$751$	−26.6146	−0.971182	−0.485591	+	0.874186i	$0.661395\pi$
$752$	0	0
$753$	−7.47846	−0.272530
$754$	0	0
$755$	21.0536	0.766218
$756$	0	0
$757$	−20.7601	−0.754540	−0.377270	−	0.926103i	$-0.623137\pi$
$757$	−20.7601	−0.754540	−0.377270	+	0.926103i	$0.623137\pi$
$758$	0	0
$759$	97.0603	3.52306
$760$	0	0
$761$	−20.0227	−0.725821	−0.362910	−	0.931824i	$-0.618217\pi$
$761$	−20.0227	−0.725821	−0.362910	+	0.931824i	$0.618217\pi$
$762$	0	0
$763$	−47.5260	−1.72056
$764$	0	0
$765$	−33.0077	−1.19340
$766$	0	0
$767$	0.833775	0.0301059
$768$	0	0
$769$	31.4005	1.13233	0.566165	−	0.824292i	$-0.308427\pi$
$769$	31.4005	1.13233	0.566165	+	0.824292i	$0.308427\pi$
$770$	0	0
$771$	16.5743	0.596909
$772$	0	0
$773$	9.57211	0.344285	0.172142	−	0.985072i	$-0.444931\pi$
$773$	9.57211	0.344285	0.172142	+	0.985072i	$0.444931\pi$
$774$	0	0
$775$	−7.69699	−0.276484
$776$	0	0
$777$	−135.459	−4.85955
$778$	0	0
$779$	1.41318	0.0506325
$780$	0	0
$781$	−69.8970	−2.50111
$782$	0	0
$783$	−77.8683	−2.78278
$784$	0	0
$785$	7.04653	0.251501
$786$	0	0
$787$	−42.3209	−1.50858	−0.754289	−	0.656543i	$-0.772018\pi$
$787$	−42.3209	−1.50858	−0.754289	+	0.656543i	$0.772018\pi$
$788$	0	0
$789$	−56.4968	−2.01134
$790$	0	0
$791$	52.3023	1.85966
$792$	0	0
$793$	−44.7806	−1.59020
$794$	0	0
$795$	25.4020	0.900914
$796$	0	0
$797$	−12.7077	−0.450131	−0.225065	−	0.974344i	$-0.572260\pi$
$797$	−12.7077	−0.450131	−0.225065	+	0.974344i	$0.572260\pi$
$798$	0	0
$799$	64.5386	2.28321
$800$	0	0
$801$	74.6828	2.63879
$802$	0	0
$803$	36.2152	1.27801
$804$	0	0
$805$	−29.9784	−1.05660
$806$	0	0
$807$	−35.3013	−1.24266
$808$	0	0
$809$	14.6710	0.515804	0.257902	−	0.966171i	$-0.416969\pi$
$809$	14.6710	0.515804	0.257902	+	0.966171i	$0.416969\pi$
$810$	0	0
$811$	−14.9622	−0.525393	−0.262697	−	0.964878i	$-0.584612\pi$
$811$	−14.9622	−0.525393	−0.262697	+	0.964878i	$0.584612\pi$
$812$	0	0
$813$	−3.21020	−0.112586
$814$	0	0
$815$	6.14528	0.215260
$816$	0	0
$817$	2.13369	0.0746484
$818$	0	0
$819$	127.610	4.45907
$820$	0	0
$821$	−36.7024	−1.28092	−0.640461	−	0.767991i	$-0.721257\pi$
$821$	−36.7024	−1.28092	−0.640461	+	0.767991i	$0.721257\pi$
$822$	0	0
$823$	−27.6216	−0.962827	−0.481414	−	0.876494i	$-0.659876\pi$
$823$	−27.6216	−0.962827	−0.481414	+	0.876494i	$0.659876\pi$
$824$	0	0
$825$	14.3342	0.499053
$826$	0	0
$827$	−28.8439	−1.00300	−0.501500	−	0.865157i	$-0.667218\pi$
$827$	−28.8439	−1.00300	−0.501500	+	0.865157i	$0.667218\pi$
$828$	0	0
$829$	−56.3289	−1.95638	−0.978192	−	0.207703i	$-0.933401\pi$
$829$	−56.3289	−1.95638	−0.978192	+	0.207703i	$0.933401\pi$
$830$	0	0
$831$	−41.8837	−1.45293
$832$	0	0
$833$	74.1861	2.57039
$834$	0	0
$835$	15.4110	0.533320
$836$	0	0
$837$	58.8592	2.03447
$838$	0	0
$839$	−16.4945	−0.569454	−0.284727	−	0.958609i	$-0.591903\pi$
$839$	−16.4945	−0.569454	−0.284727	+	0.958609i	$0.591903\pi$
$840$	0	0
$841$	74.6894	2.57549
$842$	0	0
$843$	−44.7215	−1.54029
$844$	0	0
$845$	−13.4296	−0.461993
$846$	0	0
$847$	56.9947	1.95836
$848$	0	0
$849$	79.3175	2.72217
$850$	0	0
$851$	70.6185	2.42077
$852$	0	0
$853$	−45.6164	−1.56188	−0.780938	−	0.624608i	$-0.785259\pi$
$853$	−45.6164	−1.56188	−0.780938	+	0.624608i	$0.785259\pi$
$854$	0	0
$855$	−7.58388	−0.259363
$856$	0	0
$857$	36.5871	1.24979	0.624897	−	0.780708i	$-0.285141\pi$
$857$	36.5871	1.24979	0.624897	+	0.780708i	$0.285141\pi$
$858$	0	0
$859$	−54.8589	−1.87176	−0.935881	−	0.352316i	$-0.885394\pi$
$859$	−54.8589	−1.87176	−0.935881	+	0.352316i	$0.885394\pi$
$860$	0	0
$861$	−13.5695	−0.462447
$862$	0	0
$863$	−22.9671	−0.781809	−0.390904	−	0.920431i	$-0.627838\pi$
$863$	−22.9671	−0.781809	−0.390904	+	0.920431i	$0.627838\pi$
$864$	0	0
$865$	17.1572	0.583362
$866$	0	0
$867$	−51.8092	−1.75953
$868$	0	0
$869$	−2.18150	−0.0740024
$870$	0	0
$871$	−47.2766	−1.60191
$872$	0	0
$873$	14.7691	0.499857
$874$	0	0
$875$	−4.42731	−0.149670
$876$	0	0
$877$	−20.5914	−0.695323	−0.347662	−	0.937620i	$-0.613024\pi$
$877$	−20.5914	−0.695323	−0.347662	+	0.937620i	$0.613024\pi$
$878$	0	0
$879$	61.5582	2.07631
$880$	0	0
$881$	−34.2076	−1.15248	−0.576242	−	0.817279i	$-0.695482\pi$
$881$	−34.2076	−1.15248	−0.576242	+	0.817279i	$0.695482\pi$
$882$	0	0
$883$	36.2765	1.22080	0.610401	−	0.792093i	$-0.291008\pi$
$883$	36.2765	1.22080	0.610401	+	0.792093i	$0.291008\pi$
$884$	0	0
$885$	0.475795	0.0159937
$886$	0	0
$887$	41.2884	1.38633	0.693164	−	0.720780i	$-0.256216\pi$
$887$	41.2884	1.38633	0.693164	+	0.720780i	$0.256216\pi$
$888$	0	0
$889$	25.5173	0.855822
$890$	0	0
$891$	−27.4317	−0.918997
$892$	0	0
$893$	14.8285	0.496216
$894$	0	0
$895$	−4.27305	−0.142832
$896$	0	0
$897$	−102.124	−3.40983
$898$	0	0
$899$	−78.3769	−2.61401
$900$	0	0
$901$	50.9759	1.69825
$902$	0	0
$903$	−20.4879	−0.681794
$904$	0	0
$905$	1.00665	0.0334623
$906$	0	0
$907$	58.6359	1.94697	0.973487	−	0.228741i	$-0.0734610\pi$
$907$	58.6359	1.94697	0.973487	+	0.228741i	$0.0734610\pi$
$908$	0	0
$909$	60.1725	1.99579
$910$	0	0
$911$	−57.1275	−1.89272	−0.946359	−	0.323116i	$-0.895270\pi$
$911$	−57.1275	−1.89272	−0.946359	+	0.323116i	$0.895270\pi$
$912$	0	0
$913$	13.9366	0.461235
$914$	0	0
$915$	−25.5541	−0.844792
$916$	0	0
$917$	−92.9457	−3.06934
$918$	0	0
$919$	−26.9643	−0.889471	−0.444736	−	0.895662i	$-0.646702\pi$
$919$	−26.9643	−0.889471	−0.444736	+	0.895662i	$0.646702\pi$
$920$	0	0
$921$	−11.8830	−0.391557
$922$	0	0
$923$	73.5439	2.42073
$924$	0	0
$925$	10.4292	0.342910
$926$	0	0
$927$	58.1917	1.91127
$928$	0	0
$929$	1.44721	0.0474816	0.0237408	−	0.999718i	$-0.492442\pi$
$929$	1.44721	0.0474816	0.0237408	+	0.999718i	$0.492442\pi$
$930$	0	0
$931$	17.0451	0.558630
$932$	0	0
$933$	5.97770	0.195701
$934$	0	0
$935$	28.7655	0.940732
$936$	0	0
$937$	20.4139	0.666894	0.333447	−	0.942769i	$-0.391788\pi$
$937$	20.4139	0.666894	0.333447	+	0.942769i	$0.391788\pi$
$938$	0	0
$939$	−95.8145	−3.12679
$940$	0	0
$941$	−6.96075	−0.226914	−0.113457	−	0.993543i	$-0.536192\pi$
$941$	−6.96075	−0.226914	−0.113457	+	0.993543i	$0.536192\pi$
$942$	0	0
$943$	7.07416	0.230366
$944$	0	0
$945$	33.8558	1.10133
$946$	0	0
$947$	45.0373	1.46351	0.731757	−	0.681565i	$-0.238700\pi$
$947$	45.0373	1.46351	0.731757	+	0.681565i	$0.238700\pi$
$948$	0	0
$949$	−38.1047	−1.23693
$950$	0	0
$951$	5.85898	0.189991
$952$	0	0
$953$	−3.52998	−0.114347	−0.0571736	−	0.998364i	$-0.518209\pi$
$953$	−3.52998	−0.114347	−0.0571736	+	0.998364i	$0.518209\pi$
$954$	0	0
$955$	3.59706	0.116398
$956$	0	0
$957$	145.962	4.71829
$958$	0	0
$959$	−73.8049	−2.38328
$960$	0	0
$961$	28.2436	0.911084
$962$	0	0
$963$	−12.4824	−0.402239
$964$	0	0
$965$	10.7610	0.346408
$966$	0	0
$967$	−27.0754	−0.870685	−0.435342	−	0.900265i	$-0.643373\pi$
$967$	−27.0754	−0.870685	−0.435342	+	0.900265i	$0.643373\pi$
$968$	0	0
$969$	−23.3626	−0.750515
$970$	0	0
$971$	−37.9587	−1.21815	−0.609076	−	0.793112i	$-0.708459\pi$
$971$	−37.9587	−1.21815	−0.609076	+	0.793112i	$0.708459\pi$
$972$	0	0
$973$	50.0176	1.60349
$974$	0	0
$975$	−15.0821	−0.483014
$976$	0	0
$977$	−31.8142	−1.01783	−0.508913	−	0.860818i	$-0.669953\pi$
$977$	−31.8142	−1.01783	−0.508913	+	0.860818i	$0.669953\pi$
$978$	0	0
$979$	−65.0844	−2.08011
$980$	0	0
$981$	−60.1855	−1.92158
$982$	0	0
$983$	2.41258	0.0769495	0.0384747	−	0.999260i	$-0.487750\pi$
$983$	2.41258	0.0769495	0.0384747	+	0.999260i	$0.487750\pi$
$984$	0	0
$985$	−15.2035	−0.484423
$986$	0	0
$987$	−142.384	−4.53214
$988$	0	0
$989$	10.6809	0.339633
$990$	0	0
$991$	−25.6808	−0.815778	−0.407889	−	0.913031i	$-0.633735\pi$
$991$	−25.6808	−0.815778	−0.407889	+	0.913031i	$0.633735\pi$
$992$	0	0
$993$	−36.8073	−1.16805
$994$	0	0
$995$	−4.14885	−0.131527
$996$	0	0
$997$	−36.0894	−1.14296	−0.571481	−	0.820615i	$-0.693631\pi$
$997$	−36.0894	−1.14296	−0.571481	+	0.820615i	$0.693631\pi$
$998$	0	0
$999$	−79.7524	−2.52326

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8020.2.a.e.1.4	✓	35

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8020.2.a.e.1.4	✓	35	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)

\(f(q)\)	\(=\)	\(q-2.93370 q^{3} -1.00000 q^{5} +4.42731 q^{7} +5.60662 q^{9} +O(q^{10})\)	\(q-2.93370 q^{3} -1.00000 q^{5} +4.42731 q^{7} +5.60662 q^{9} -4.88604 q^{11} +5.14097 q^{13} +2.93370 q^{15} +5.88727 q^{17} +1.35267 q^{19} -12.9884 q^{21} +6.77123 q^{23} +1.00000 q^{25} -7.64704 q^{27} +10.1828 q^{29} -7.69699 q^{31} +14.3342 q^{33} -4.42731 q^{35} +10.4292 q^{37} -15.0821 q^{39} +1.04474 q^{41} +1.57740 q^{43} -5.60662 q^{45} +10.9624 q^{47} +12.6011 q^{49} -17.2715 q^{51} +8.65866 q^{53} +4.88604 q^{55} -3.96832 q^{57} +0.162182 q^{59} -8.71052 q^{61} +24.8222 q^{63} -5.14097 q^{65} -9.19604 q^{67} -19.8648 q^{69} +14.3054 q^{71} -7.41197 q^{73} -2.93370 q^{75} -21.6320 q^{77} +0.446476 q^{79} +5.61430 q^{81} -2.85233 q^{83} -5.88727 q^{85} -29.8733 q^{87} +13.3205 q^{89} +22.7607 q^{91} +22.5807 q^{93} -1.35267 q^{95} +2.63422 q^{97} -27.3942 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 35 q - q^{3} - 35 q^{5} + 6 q^{7} + 52 q^{9}+O(q^{10}) \) 35 * q - q^3 - 35 * q^5 + 6 * q^7 + 52 * q^9	\( 35 q - q^{3} - 35 q^{5} + 6 q^{7} + 52 q^{9} - 2 q^{11} - q^{13} + q^{15} + 18 q^{17} + 2 q^{19} + 12 q^{21} + 13 q^{23} + 35 q^{25} - 7 q^{27} + 25 q^{29} + 13 q^{31} + 14 q^{33} - 6 q^{35} - 19 q^{37} - 3 q^{39} + 24 q^{41} - 5 q^{43} - 52 q^{45} + 19 q^{47} + 55 q^{49} + 41 q^{53} + 2 q^{55} + 14 q^{57} + 3 q^{59} + 13 q^{61} + 70 q^{63} + q^{65} - 17 q^{67} + 64 q^{69} + 17 q^{71} - 63 q^{73} - q^{75} + 54 q^{77} + 11 q^{79} + 107 q^{81} - 8 q^{83} - 18 q^{85} + 36 q^{87} + 38 q^{89} - 27 q^{91} + q^{93} - 2 q^{95} - 54 q^{97} - 51 q^{99}+O(q^{100}) \) 35 * q - q^3 - 35 * q^5 + 6 * q^7 + 52 * q^9 - 2 * q^11 - q^13 + q^15 + 18 * q^17 + 2 * q^19 + 12 * q^21 + 13 * q^23 + 35 * q^25 - 7 * q^27 + 25 * q^29 + 13 * q^31 + 14 * q^33 - 6 * q^35 - 19 * q^37 - 3 * q^39 + 24 * q^41 - 5 * q^43 - 52 * q^45 + 19 * q^47 + 55 * q^49 + 41 * q^53 + 2 * q^55 + 14 * q^57 + 3 * q^59 + 13 * q^61 + 70 * q^63 + q^65 - 17 * q^67 + 64 * q^69 + 17 * q^71 - 63 * q^73 - q^75 + 54 * q^77 + 11 * q^79 + 107 * q^81 - 8 * q^83 - 18 * q^85 + 36 * q^87 + 38 * q^89 - 27 * q^91 + q^93 - 2 * q^95 - 54 * q^97 - 51 * q^99

Introduction

L-functions

Modular forms

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Database

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