LMFDB - Embedded newform 8020.2.a.e.1.6

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			1.6
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)

$f(q)$	$=$	$q-2.89929 q^{3} -1.00000 q^{5} +2.68176 q^{7} +5.40586 q^{9} +O(q^{10})$	$q-2.89929 q^{3} -1.00000 q^{5} +2.68176 q^{7} +5.40586 q^{9} +5.20461 q^{11} -0.720281 q^{13} +2.89929 q^{15} -6.59547 q^{17} +5.39842 q^{19} -7.77518 q^{21} -2.07968 q^{23} +1.00000 q^{25} -6.97529 q^{27} +3.46977 q^{29} +6.64257 q^{31} -15.0897 q^{33} -2.68176 q^{35} +2.32741 q^{37} +2.08830 q^{39} +6.40479 q^{41} +4.43441 q^{43} -5.40586 q^{45} +3.76316 q^{47} +0.191812 q^{49} +19.1222 q^{51} +11.1129 q^{53} -5.20461 q^{55} -15.6516 q^{57} +11.8211 q^{59} -1.92175 q^{61} +14.4972 q^{63} +0.720281 q^{65} -9.76769 q^{67} +6.02958 q^{69} +11.6573 q^{71} +0.405425 q^{73} -2.89929 q^{75} +13.9575 q^{77} +7.64892 q^{79} +4.00578 q^{81} -7.18866 q^{83} +6.59547 q^{85} -10.0598 q^{87} -3.82899 q^{89} -1.93162 q^{91} -19.2587 q^{93} -5.39842 q^{95} +2.72830 q^{97} +28.1354 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.89929	−1.67390	−0.836952	−	0.547276i	$-0.815665\pi$
$3$	−2.89929	−1.67390	−0.836952	+	0.547276i	$0.815665\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	2.68176	1.01361	0.506804	−	0.862061i	$-0.330827\pi$
$7$	2.68176	1.01361	0.506804	+	0.862061i	$0.330827\pi$
$8$	0	0
$9$	5.40586	1.80195
$10$	0	0
$11$	5.20461	1.56925	0.784625	−	0.619971i	$-0.212856\pi$
$11$	5.20461	1.56925	0.784625	+	0.619971i	$0.212856\pi$
$12$	0	0
$13$	−0.720281	−0.199770	−0.0998850	−	0.994999i	$-0.531847\pi$
$13$	−0.720281	−0.199770	−0.0998850	+	0.994999i	$0.531847\pi$
$14$	0	0
$15$	2.89929	0.748593
$16$	0	0
$17$	−6.59547	−1.59964	−0.799819	−	0.600242i	$-0.795071\pi$
$17$	−6.59547	−1.59964	−0.799819	+	0.600242i	$0.795071\pi$
$18$	0	0
$19$	5.39842	1.23848	0.619242	−	0.785201i	$-0.287440\pi$
$19$	5.39842	1.23848	0.619242	+	0.785201i	$0.287440\pi$
$20$	0	0
$21$	−7.77518	−1.69668
$22$	0	0
$23$	−2.07968	−0.433643	−0.216821	−	0.976211i	$-0.569569\pi$
$23$	−2.07968	−0.433643	−0.216821	+	0.976211i	$0.569569\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−6.97529	−1.34240
$28$	0	0
$29$	3.46977	0.644319	0.322160	−	0.946685i	$-0.395591\pi$
$29$	3.46977	0.644319	0.322160	+	0.946685i	$0.395591\pi$
$30$	0	0
$31$	6.64257	1.19304	0.596520	−	0.802598i	$-0.296549\pi$
$31$	6.64257	1.19304	0.596520	+	0.802598i	$0.296549\pi$
$32$	0	0
$33$	−15.0897	−2.62677
$34$	0	0
$35$	−2.68176	−0.453299
$36$	0	0
$37$	2.32741	0.382625	0.191312	−	0.981529i	$-0.438726\pi$
$37$	2.32741	0.382625	0.191312	+	0.981529i	$0.438726\pi$
$38$	0	0
$39$	2.08830	0.334396
$40$	0	0
$41$	6.40479	1.00026	0.500130	−	0.865950i	$-0.333286\pi$
$41$	6.40479	1.00026	0.500130	+	0.865950i	$0.333286\pi$
$42$	0	0
$43$	4.43441	0.676242	0.338121	−	0.941103i	$-0.390209\pi$
$43$	4.43441	0.676242	0.338121	+	0.941103i	$0.390209\pi$
$44$	0	0
$45$	−5.40586	−0.805859
$46$	0	0
$47$	3.76316	0.548913	0.274457	−	0.961599i	$-0.411502\pi$
$47$	3.76316	0.548913	0.274457	+	0.961599i	$0.411502\pi$
$48$	0	0
$49$	0.191812	0.0274017
$50$	0	0
$51$	19.1222	2.67764
$52$	0	0
$53$	11.1129	1.52647	0.763235	−	0.646121i	$-0.223610\pi$
$53$	11.1129	1.52647	0.763235	+	0.646121i	$0.223610\pi$
$54$	0	0
$55$	−5.20461	−0.701790
$56$	0	0
$57$	−15.6516	−2.07310
$58$	0	0
$59$	11.8211	1.53897	0.769486	−	0.638664i	$-0.220513\pi$
$59$	11.8211	1.53897	0.769486	+	0.638664i	$0.220513\pi$
$60$	0	0
$61$	−1.92175	−0.246054	−0.123027	−	0.992403i	$-0.539260\pi$
$61$	−1.92175	−0.246054	−0.123027	+	0.992403i	$0.539260\pi$
$62$	0	0
$63$	14.4972	1.82648
$64$	0	0
$65$	0.720281	0.0893398
$66$	0	0
$67$	−9.76769	−1.19331	−0.596657	−	0.802497i	$-0.703505\pi$
$67$	−9.76769	−1.19331	−0.596657	+	0.802497i	$0.703505\pi$
$68$	0	0
$69$	6.02958	0.725876
$70$	0	0
$71$	11.6573	1.38346	0.691732	−	0.722155i	$-0.256848\pi$
$71$	11.6573	1.38346	0.691732	+	0.722155i	$0.256848\pi$
$72$	0	0
$73$	0.405425	0.0474514	0.0237257	−	0.999719i	$-0.492447\pi$
$73$	0.405425	0.0474514	0.0237257	+	0.999719i	$0.492447\pi$
$74$	0	0
$75$	−2.89929	−0.334781
$76$	0	0
$77$	13.9575	1.59060
$78$	0	0
$79$	7.64892	0.860571	0.430285	−	0.902693i	$-0.358413\pi$
$79$	7.64892	0.860571	0.430285	+	0.902693i	$0.358413\pi$
$80$	0	0
$81$	4.00578	0.445086
$82$	0	0
$83$	−7.18866	−0.789058	−0.394529	−	0.918883i	$-0.629092\pi$
$83$	−7.18866	−0.789058	−0.394529	+	0.918883i	$0.629092\pi$
$84$	0	0
$85$	6.59547	0.715380
$86$	0	0
$87$	−10.0598	−1.07853
$88$	0	0
$89$	−3.82899	−0.405872	−0.202936	−	0.979192i	$-0.565048\pi$
$89$	−3.82899	−0.405872	−0.202936	+	0.979192i	$0.565048\pi$
$90$	0	0
$91$	−1.93162	−0.202488
$92$	0	0
$93$	−19.2587	−1.99704
$94$	0	0
$95$	−5.39842	−0.553867
$96$	0	0
$97$	2.72830	0.277017	0.138508	−	0.990361i	$-0.455769\pi$
$97$	2.72830	0.277017	0.138508	+	0.990361i	$0.455769\pi$
$98$	0	0
$99$	28.1354	2.82772
$100$	0	0
$101$	−15.6653	−1.55876	−0.779378	−	0.626554i	$-0.784465\pi$
$101$	−15.6653	−1.55876	−0.779378	+	0.626554i	$0.784465\pi$
$102$	0	0
$103$	−3.53280	−0.348097	−0.174049	−	0.984737i	$-0.555685\pi$
$103$	−3.53280	−0.348097	−0.174049	+	0.984737i	$0.555685\pi$
$104$	0	0
$105$	7.77518	0.758780
$106$	0	0
$107$	−18.7041	−1.80819	−0.904095	−	0.427331i	$-0.859454\pi$
$107$	−18.7041	−1.80819	−0.904095	+	0.427331i	$0.859454\pi$
$108$	0	0
$109$	−17.2562	−1.65285	−0.826424	−	0.563048i	$-0.809629\pi$
$109$	−17.2562	−1.65285	−0.826424	+	0.563048i	$0.809629\pi$
$110$	0	0
$111$	−6.74784	−0.640477
$112$	0	0
$113$	11.1501	1.04891	0.524455	−	0.851438i	$-0.324269\pi$
$113$	11.1501	1.04891	0.524455	+	0.851438i	$0.324269\pi$
$114$	0	0
$115$	2.07968	0.193931
$116$	0	0
$117$	−3.89374	−0.359976
$118$	0	0
$119$	−17.6874	−1.62141
$120$	0	0
$121$	16.0880	1.46254
$122$	0	0
$123$	−18.5693	−1.67434
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−19.6702	−1.74545	−0.872726	−	0.488210i	$-0.837650\pi$
$127$	−19.6702	−1.74545	−0.872726	+	0.488210i	$0.837650\pi$
$128$	0	0
$129$	−12.8566	−1.13196
$130$	0	0
$131$	8.87200	0.775150	0.387575	−	0.921838i	$-0.373313\pi$
$131$	8.87200	0.775150	0.387575	+	0.921838i	$0.373313\pi$
$132$	0	0
$133$	14.4772	1.25534
$134$	0	0
$135$	6.97529	0.600337
$136$	0	0
$137$	18.3957	1.57165	0.785827	−	0.618446i	$-0.212237\pi$
$137$	18.3957	1.57165	0.785827	+	0.618446i	$0.212237\pi$
$138$	0	0
$139$	−14.7511	−1.25117	−0.625585	−	0.780156i	$-0.715140\pi$
$139$	−14.7511	−1.25117	−0.625585	+	0.780156i	$0.715140\pi$
$140$	0	0
$141$	−10.9105	−0.918828
$142$	0	0
$143$	−3.74878	−0.313489
$144$	0	0
$145$	−3.46977	−0.288148
$146$	0	0
$147$	−0.556117	−0.0458678
$148$	0	0
$149$	5.65888	0.463594	0.231797	−	0.972764i	$-0.425540\pi$
$149$	5.65888	0.463594	0.231797	+	0.972764i	$0.425540\pi$
$150$	0	0
$151$	−16.3044	−1.32683	−0.663415	−	0.748252i	$-0.730893\pi$
$151$	−16.3044	−1.32683	−0.663415	+	0.748252i	$0.730893\pi$
$152$	0	0
$153$	−35.6542	−2.88247
$154$	0	0
$155$	−6.64257	−0.533544
$156$	0	0
$157$	6.09538	0.486464	0.243232	−	0.969968i	$-0.421792\pi$
$157$	6.09538	0.486464	0.243232	+	0.969968i	$0.421792\pi$
$158$	0	0
$159$	−32.2194	−2.55517
$160$	0	0
$161$	−5.57719	−0.439544
$162$	0	0
$163$	4.37534	0.342703	0.171351	−	0.985210i	$-0.445187\pi$
$163$	4.37534	0.342703	0.171351	+	0.985210i	$0.445187\pi$
$164$	0	0
$165$	15.0897	1.17473
$166$	0	0
$167$	−7.97440	−0.617077	−0.308539	−	0.951212i	$-0.599840\pi$
$167$	−7.97440	−0.617077	−0.308539	+	0.951212i	$0.599840\pi$
$168$	0	0
$169$	−12.4812	−0.960092
$170$	0	0
$171$	29.1831	2.23169
$172$	0	0
$173$	7.36502	0.559952	0.279976	−	0.960007i	$-0.409673\pi$
$173$	7.36502	0.559952	0.279976	+	0.960007i	$0.409673\pi$
$174$	0	0
$175$	2.68176	0.202722
$176$	0	0
$177$	−34.2727	−2.57609
$178$	0	0
$179$	7.68172	0.574158	0.287079	−	0.957907i	$-0.407316\pi$
$179$	7.68172	0.574158	0.287079	+	0.957907i	$0.407316\pi$
$180$	0	0
$181$	7.16190	0.532340	0.266170	−	0.963926i	$-0.414242\pi$
$181$	7.16190	0.532340	0.266170	+	0.963926i	$0.414242\pi$
$182$	0	0
$183$	5.57169	0.411871
$184$	0	0
$185$	−2.32741	−0.171115
$186$	0	0
$187$	−34.3269	−2.51023
$188$	0	0
$189$	−18.7060	−1.36066
$190$	0	0
$191$	11.9736	0.866376	0.433188	−	0.901304i	$-0.357389\pi$
$191$	11.9736	0.866376	0.433188	+	0.901304i	$0.357389\pi$
$192$	0	0
$193$	18.3368	1.31991	0.659956	−	0.751304i	$-0.270575\pi$
$193$	18.3368	1.31991	0.659956	+	0.751304i	$0.270575\pi$
$194$	0	0
$195$	−2.08830	−0.149546
$196$	0	0
$197$	−6.45422	−0.459844	−0.229922	−	0.973209i	$-0.573847\pi$
$197$	−6.45422	−0.459844	−0.229922	+	0.973209i	$0.573847\pi$
$198$	0	0
$199$	6.20058	0.439547	0.219774	−	0.975551i	$-0.429468\pi$
$199$	6.20058	0.439547	0.219774	+	0.975551i	$0.429468\pi$
$200$	0	0
$201$	28.3193	1.99749
$202$	0	0
$203$	9.30506	0.653087
$204$	0	0
$205$	−6.40479	−0.447330
$206$	0	0
$207$	−11.2425	−0.781405
$208$	0	0
$209$	28.0967	1.94349
$210$	0	0
$211$	−14.0027	−0.963988	−0.481994	−	0.876174i	$-0.660087\pi$
$211$	−14.0027	−0.963988	−0.481994	+	0.876174i	$0.660087\pi$
$212$	0	0
$213$	−33.7978	−2.31579
$214$	0	0
$215$	−4.43441	−0.302425
$216$	0	0
$217$	17.8137	1.20928
$218$	0	0
$219$	−1.17544	−0.0794292
$220$	0	0
$221$	4.75059	0.319559
$222$	0	0
$223$	15.5540	1.04157	0.520785	−	0.853688i	$-0.325639\pi$
$223$	15.5540	1.04157	0.520785	+	0.853688i	$0.325639\pi$
$224$	0	0
$225$	5.40586	0.360391
$226$	0	0
$227$	−3.13017	−0.207757	−0.103878	−	0.994590i	$-0.533125\pi$
$227$	−3.13017	−0.207757	−0.103878	+	0.994590i	$0.533125\pi$
$228$	0	0
$229$	−27.3075	−1.80453	−0.902264	−	0.431183i	$-0.858096\pi$
$229$	−27.3075	−1.80453	−0.902264	+	0.431183i	$0.858096\pi$
$230$	0	0
$231$	−40.4668	−2.66252
$232$	0	0
$233$	22.0481	1.44442	0.722209	−	0.691675i	$-0.243127\pi$
$233$	22.0481	1.44442	0.722209	+	0.691675i	$0.243127\pi$
$234$	0	0
$235$	−3.76316	−0.245482
$236$	0	0
$237$	−22.1764	−1.44051
$238$	0	0
$239$	8.71399	0.563661	0.281831	−	0.959464i	$-0.409058\pi$
$239$	8.71399	0.563661	0.281831	+	0.959464i	$0.409058\pi$
$240$	0	0
$241$	9.58470	0.617405	0.308702	−	0.951159i	$-0.400105\pi$
$241$	9.58470	0.617405	0.308702	+	0.951159i	$0.400105\pi$
$242$	0	0
$243$	9.31198	0.597364
$244$	0	0
$245$	−0.191812	−0.0122544
$246$	0	0
$247$	−3.88838	−0.247412
$248$	0	0
$249$	20.8420	1.32081
$250$	0	0
$251$	−7.20543	−0.454803	−0.227401	−	0.973801i	$-0.573023\pi$
$251$	−7.20543	−0.454803	−0.227401	+	0.973801i	$0.573023\pi$
$252$	0	0
$253$	−10.8239	−0.680494
$254$	0	0
$255$	−19.1222	−1.19748
$256$	0	0
$257$	9.98343	0.622749	0.311375	−	0.950287i	$-0.399211\pi$
$257$	9.98343	0.622749	0.311375	+	0.950287i	$0.399211\pi$
$258$	0	0
$259$	6.24156	0.387831
$260$	0	0
$261$	18.7571	1.16103
$262$	0	0
$263$	9.82517	0.605846	0.302923	−	0.953015i	$-0.402038\pi$
$263$	9.82517	0.605846	0.302923	+	0.953015i	$0.402038\pi$
$264$	0	0
$265$	−11.1129	−0.682658
$266$	0	0
$267$	11.1013	0.679391
$268$	0	0
$269$	−20.8759	−1.27283	−0.636413	−	0.771348i	$-0.719583\pi$
$269$	−20.8759	−1.27283	−0.636413	+	0.771348i	$0.719583\pi$
$270$	0	0
$271$	−19.1095	−1.16082	−0.580408	−	0.814326i	$-0.697107\pi$
$271$	−19.1095	−1.16082	−0.580408	+	0.814326i	$0.697107\pi$
$272$	0	0
$273$	5.60031	0.338946
$274$	0	0
$275$	5.20461	0.313850
$276$	0	0
$277$	15.2212	0.914555	0.457278	−	0.889324i	$-0.348825\pi$
$277$	15.2212	0.914555	0.457278	+	0.889324i	$0.348825\pi$
$278$	0	0
$279$	35.9088	2.14981
$280$	0	0
$281$	−0.725066	−0.0432538	−0.0216269	−	0.999766i	$-0.506885\pi$
$281$	−0.725066	−0.0432538	−0.0216269	+	0.999766i	$0.506885\pi$
$282$	0	0
$283$	−13.3536	−0.793787	−0.396893	−	0.917865i	$-0.629912\pi$
$283$	−13.3536	−0.793787	−0.396893	+	0.917865i	$0.629912\pi$
$284$	0	0
$285$	15.6516	0.927119
$286$	0	0
$287$	17.1761	1.01387
$288$	0	0
$289$	26.5003	1.55884
$290$	0	0
$291$	−7.91012	−0.463700
$292$	0	0
$293$	19.8342	1.15873	0.579364	−	0.815069i	$-0.303301\pi$
$293$	19.8342	1.15873	0.579364	+	0.815069i	$0.303301\pi$
$294$	0	0
$295$	−11.8211	−0.688249
$296$	0	0
$297$	−36.3037	−2.10655
$298$	0	0
$299$	1.49795	0.0866288
$300$	0	0
$301$	11.8920	0.685444
$302$	0	0
$303$	45.4182	2.60921
$304$	0	0
$305$	1.92175	0.110039
$306$	0	0
$307$	18.9930	1.08399	0.541995	−	0.840382i	$-0.317669\pi$
$307$	18.9930	1.08399	0.541995	+	0.840382i	$0.317669\pi$
$308$	0	0
$309$	10.2426	0.582681
$310$	0	0
$311$	32.0890	1.81960	0.909799	−	0.415049i	$-0.136235\pi$
$311$	32.0890	1.81960	0.909799	+	0.415049i	$0.136235\pi$
$312$	0	0
$313$	−26.8963	−1.52027	−0.760135	−	0.649765i	$-0.774867\pi$
$313$	−26.8963	−1.52027	−0.760135	+	0.649765i	$0.774867\pi$
$314$	0	0
$315$	−14.4972	−0.816825
$316$	0	0
$317$	−16.4533	−0.924108	−0.462054	−	0.886852i	$-0.652887\pi$
$317$	−16.4533	−0.924108	−0.462054	+	0.886852i	$0.652887\pi$
$318$	0	0
$319$	18.0588	1.01110
$320$	0	0
$321$	54.2285	3.02674
$322$	0	0
$323$	−35.6052	−1.98112
$324$	0	0
$325$	−0.720281	−0.0399540
$326$	0	0
$327$	50.0308	2.76671
$328$	0	0
$329$	10.0919	0.556383
$330$	0	0
$331$	−10.3848	−0.570799	−0.285400	−	0.958409i	$-0.592126\pi$
$331$	−10.3848	−0.570799	−0.285400	+	0.958409i	$0.592126\pi$
$332$	0	0
$333$	12.5817	0.689472
$334$	0	0
$335$	9.76769	0.533666
$336$	0	0
$337$	−5.96173	−0.324756	−0.162378	−	0.986729i	$-0.551916\pi$
$337$	−5.96173	−0.324756	−0.162378	+	0.986729i	$0.551916\pi$
$338$	0	0
$339$	−32.3272	−1.75577
$340$	0	0
$341$	34.5720	1.87218
$342$	0	0
$343$	−18.2579	−0.985834
$344$	0	0
$345$	−6.02958	−0.324622
$346$	0	0
$347$	17.9959	0.966069	0.483034	−	0.875601i	$-0.339535\pi$
$347$	17.9959	0.966069	0.483034	+	0.875601i	$0.339535\pi$
$348$	0	0
$349$	−29.8638	−1.59857	−0.799286	−	0.600951i	$-0.794789\pi$
$349$	−29.8638	−1.59857	−0.799286	+	0.600951i	$0.794789\pi$
$350$	0	0
$351$	5.02417	0.268170
$352$	0	0
$353$	−22.7822	−1.21257	−0.606287	−	0.795246i	$-0.707342\pi$
$353$	−22.7822	−1.21257	−0.606287	+	0.795246i	$0.707342\pi$
$354$	0	0
$355$	−11.6573	−0.618704
$356$	0	0
$357$	51.2810	2.71408
$358$	0	0
$359$	−7.39873	−0.390490	−0.195245	−	0.980754i	$-0.562550\pi$
$359$	−7.39873	−0.390490	−0.195245	+	0.980754i	$0.562550\pi$
$360$	0	0
$361$	10.1430	0.533841
$362$	0	0
$363$	−46.6437	−2.44816
$364$	0	0
$365$	−0.405425	−0.0212209
$366$	0	0
$367$	−26.6979	−1.39362	−0.696811	−	0.717255i	$-0.745398\pi$
$367$	−26.6979	−1.39362	−0.696811	+	0.717255i	$0.745398\pi$
$368$	0	0
$369$	34.6234	1.80242
$370$	0	0
$371$	29.8020	1.54724
$372$	0	0
$373$	13.7382	0.711336	0.355668	−	0.934612i	$-0.384253\pi$
$373$	13.7382	0.711336	0.355668	+	0.934612i	$0.384253\pi$
$374$	0	0
$375$	2.89929	0.149719
$376$	0	0
$377$	−2.49921	−0.128716
$378$	0	0
$379$	9.81686	0.504258	0.252129	−	0.967694i	$-0.418869\pi$
$379$	9.81686	0.504258	0.252129	+	0.967694i	$0.418869\pi$
$380$	0	0
$381$	57.0297	2.92172
$382$	0	0
$383$	21.5724	1.10230	0.551149	−	0.834407i	$-0.314190\pi$
$383$	21.5724	1.10230	0.551149	+	0.834407i	$0.314190\pi$
$384$	0	0
$385$	−13.9575	−0.711340
$386$	0	0
$387$	23.9718	1.21856
$388$	0	0
$389$	19.5242	0.989914	0.494957	−	0.868917i	$-0.335184\pi$
$389$	19.5242	0.989914	0.494957	+	0.868917i	$0.335184\pi$
$390$	0	0
$391$	13.7165	0.693671
$392$	0	0
$393$	−25.7225	−1.29753
$394$	0	0
$395$	−7.64892	−0.384859
$396$	0	0
$397$	31.9060	1.60131	0.800657	−	0.599123i	$-0.204484\pi$
$397$	31.9060	1.60131	0.800657	+	0.599123i	$0.204484\pi$
$398$	0	0
$399$	−41.9737	−2.10131
$400$	0	0
$401$	−1.00000	−0.0499376
$401$	−1.00000	−0.0499376
$402$	0	0
$403$	−4.78452	−0.238334
$404$	0	0
$405$	−4.00578	−0.199049
$406$	0	0
$407$	12.1133	0.600434
$408$	0	0
$409$	−4.18557	−0.206963	−0.103482	−	0.994631i	$-0.532998\pi$
$409$	−4.18557	−0.206963	−0.103482	+	0.994631i	$0.532998\pi$
$410$	0	0
$411$	−53.3345	−2.63080
$412$	0	0
$413$	31.7012	1.55991
$414$	0	0
$415$	7.18866	0.352878
$416$	0	0
$417$	42.7676	2.09434
$418$	0	0
$419$	−35.1980	−1.71954	−0.859768	−	0.510685i	$-0.829392\pi$
$419$	−35.1980	−1.71954	−0.859768	+	0.510685i	$0.829392\pi$
$420$	0	0
$421$	27.4574	1.33819	0.669097	−	0.743175i	$-0.266681\pi$
$421$	27.4574	1.33819	0.669097	+	0.743175i	$0.266681\pi$
$422$	0	0
$423$	20.3431	0.989117
$424$	0	0
$425$	−6.59547	−0.319927
$426$	0	0
$427$	−5.15365	−0.249403
$428$	0	0
$429$	10.8688	0.524750
$430$	0	0
$431$	2.23280	0.107550	0.0537750	−	0.998553i	$-0.482875\pi$
$431$	2.23280	0.107550	0.0537750	+	0.998553i	$0.482875\pi$
$432$	0	0
$433$	1.53372	0.0737057	0.0368529	−	0.999321i	$-0.488267\pi$
$433$	1.53372	0.0737057	0.0368529	+	0.999321i	$0.488267\pi$
$434$	0	0
$435$	10.0598	0.482333
$436$	0	0
$437$	−11.2270	−0.537059
$438$	0	0
$439$	−1.71786	−0.0819889	−0.0409944	−	0.999159i	$-0.513053\pi$
$439$	−1.71786	−0.0819889	−0.0409944	+	0.999159i	$0.513053\pi$
$440$	0	0
$441$	1.03691	0.0493766
$442$	0	0
$443$	10.3446	0.491489	0.245744	−	0.969335i	$-0.420968\pi$
$443$	10.3446	0.491489	0.245744	+	0.969335i	$0.420968\pi$
$444$	0	0
$445$	3.82899	0.181512
$446$	0	0
$447$	−16.4067	−0.776011
$448$	0	0
$449$	15.1007	0.712647	0.356324	−	0.934363i	$-0.384030\pi$
$449$	15.1007	0.712647	0.356324	+	0.934363i	$0.384030\pi$
$450$	0	0
$451$	33.3344	1.56966
$452$	0	0
$453$	47.2710	2.22099
$454$	0	0
$455$	1.93162	0.0905556
$456$	0	0
$457$	28.3116	1.32436	0.662181	−	0.749344i	$-0.269631\pi$
$457$	28.3116	1.32436	0.662181	+	0.749344i	$0.269631\pi$
$458$	0	0
$459$	46.0054	2.14735
$460$	0	0
$461$	−26.9674	−1.25600	−0.627999	−	0.778214i	$-0.716126\pi$
$461$	−26.9674	−1.25600	−0.627999	+	0.778214i	$0.716126\pi$
$462$	0	0
$463$	35.9050	1.66865	0.834323	−	0.551275i	$-0.185859\pi$
$463$	35.9050	1.66865	0.834323	+	0.551275i	$0.185859\pi$
$464$	0	0
$465$	19.2587	0.893102
$466$	0	0
$467$	−20.6131	−0.953859	−0.476929	−	0.878942i	$-0.658250\pi$
$467$	−20.6131	−0.953859	−0.476929	+	0.878942i	$0.658250\pi$
$468$	0	0
$469$	−26.1946	−1.20955
$470$	0	0
$471$	−17.6722	−0.814294
$472$	0	0
$473$	23.0794	1.06119
$474$	0	0
$475$	5.39842	0.247697
$476$	0	0
$477$	60.0747	2.75063
$478$	0	0
$479$	4.60617	0.210461	0.105231	−	0.994448i	$-0.466442\pi$
$479$	4.60617	0.210461	0.105231	+	0.994448i	$0.466442\pi$
$480$	0	0
$481$	−1.67639	−0.0764369
$482$	0	0
$483$	16.1699	0.735754
$484$	0	0
$485$	−2.72830	−0.123886
$486$	0	0
$487$	−28.5954	−1.29578	−0.647890	−	0.761734i	$-0.724348\pi$
$487$	−28.5954	−1.29578	−0.647890	+	0.761734i	$0.724348\pi$
$488$	0	0
$489$	−12.6854	−0.573652
$490$	0	0
$491$	27.4839	1.24033	0.620166	−	0.784470i	$-0.287065\pi$
$491$	27.4839	1.24033	0.620166	+	0.784470i	$0.287065\pi$
$492$	0	0
$493$	−22.8848	−1.03068
$494$	0	0
$495$	−28.1354	−1.26459
$496$	0	0
$497$	31.2619	1.40229
$498$	0	0
$499$	19.7754	0.885268	0.442634	−	0.896702i	$-0.354044\pi$
$499$	19.7754	0.885268	0.442634	+	0.896702i	$0.354044\pi$
$500$	0	0
$501$	23.1201	1.03293
$502$	0	0
$503$	23.4588	1.04598	0.522988	−	0.852340i	$-0.324817\pi$
$503$	23.4588	1.04598	0.522988	+	0.852340i	$0.324817\pi$
$504$	0	0
$505$	15.6653	0.697097
$506$	0	0
$507$	36.1866	1.60710
$508$	0	0
$509$	−25.0457	−1.11013	−0.555066	−	0.831806i	$-0.687307\pi$
$509$	−25.0457	−1.11013	−0.555066	+	0.831806i	$0.687307\pi$
$510$	0	0
$511$	1.08725	0.0480972
$512$	0	0
$513$	−37.6556	−1.66253
$514$	0	0
$515$	3.53280	0.155674
$516$	0	0
$517$	19.5858	0.861382
$518$	0	0
$519$	−21.3533	−0.937307
$520$	0	0
$521$	32.4472	1.42154	0.710770	−	0.703425i	$-0.248347\pi$
$521$	32.4472	1.42154	0.710770	+	0.703425i	$0.248347\pi$
$522$	0	0
$523$	−23.9845	−1.04877	−0.524385	−	0.851482i	$-0.675705\pi$
$523$	−23.9845	−1.04877	−0.524385	+	0.851482i	$0.675705\pi$
$524$	0	0
$525$	−7.77518	−0.339337
$526$	0	0
$527$	−43.8109	−1.90843
$528$	0	0
$529$	−18.6749	−0.811954
$530$	0	0
$531$	63.9031	2.77316
$532$	0	0
$533$	−4.61325	−0.199822
$534$	0	0
$535$	18.7041	0.808648
$536$	0	0
$537$	−22.2715	−0.961086
$538$	0	0
$539$	0.998305	0.0430001
$540$	0	0
$541$	26.4893	1.13887	0.569433	−	0.822038i	$-0.307163\pi$
$541$	26.4893	1.13887	0.569433	+	0.822038i	$0.307163\pi$
$542$	0	0
$543$	−20.7644	−0.891086
$544$	0	0
$545$	17.2562	0.739176
$546$	0	0
$547$	0.947306	0.0405039	0.0202519	−	0.999795i	$-0.493553\pi$
$547$	0.947306	0.0405039	0.0202519	+	0.999795i	$0.493553\pi$
$548$	0	0
$549$	−10.3887	−0.443379
$550$	0	0
$551$	18.7313	0.797979
$552$	0	0
$553$	20.5125	0.872281
$554$	0	0
$555$	6.74784	0.286430
$556$	0	0
$557$	5.39085	0.228418	0.114209	−	0.993457i	$-0.463567\pi$
$557$	5.39085	0.228418	0.114209	+	0.993457i	$0.463567\pi$
$558$	0	0
$559$	−3.19402	−0.135093
$560$	0	0
$561$	99.5235	4.20188
$562$	0	0
$563$	−21.7057	−0.914788	−0.457394	−	0.889264i	$-0.651217\pi$
$563$	−21.7057	−0.914788	−0.457394	+	0.889264i	$0.651217\pi$
$564$	0	0
$565$	−11.1501	−0.469086
$566$	0	0
$567$	10.7425	0.451143
$568$	0	0
$569$	7.08425	0.296987	0.148494	−	0.988913i	$-0.452558\pi$
$569$	7.08425	0.296987	0.148494	+	0.988913i	$0.452558\pi$
$570$	0	0
$571$	9.87561	0.413281	0.206641	−	0.978417i	$-0.433747\pi$
$571$	9.87561	0.413281	0.206641	+	0.978417i	$0.433747\pi$
$572$	0	0
$573$	−34.7148	−1.45023
$574$	0	0
$575$	−2.07968	−0.0867286
$576$	0	0
$577$	30.0149	1.24954	0.624769	−	0.780810i	$-0.285193\pi$
$577$	30.0149	1.24954	0.624769	+	0.780810i	$0.285193\pi$
$578$	0	0
$579$	−53.1636	−2.20941
$580$	0	0
$581$	−19.2782	−0.799796
$582$	0	0
$583$	57.8382	2.39541
$584$	0	0
$585$	3.89374	0.160986
$586$	0	0
$587$	−30.0449	−1.24008	−0.620042	−	0.784568i	$-0.712885\pi$
$587$	−30.0449	−1.24008	−0.620042	+	0.784568i	$0.712885\pi$
$588$	0	0
$589$	35.8594	1.47756
$590$	0	0
$591$	18.7126	0.769735
$592$	0	0
$593$	25.6124	1.05178	0.525888	−	0.850554i	$-0.323733\pi$
$593$	25.6124	1.05178	0.525888	+	0.850554i	$0.323733\pi$
$594$	0	0
$595$	17.6874	0.725115
$596$	0	0
$597$	−17.9772	−0.735760
$598$	0	0
$599$	2.85506	0.116655	0.0583273	−	0.998298i	$-0.481423\pi$
$599$	2.85506	0.116655	0.0583273	+	0.998298i	$0.481423\pi$
$600$	0	0
$601$	19.8339	0.809041	0.404521	−	0.914529i	$-0.367438\pi$
$601$	19.8339	0.809041	0.404521	+	0.914529i	$0.367438\pi$
$602$	0	0
$603$	−52.8028	−2.15030
$604$	0	0
$605$	−16.0880	−0.654070
$606$	0	0
$607$	24.6666	1.00119	0.500594	−	0.865682i	$-0.333115\pi$
$607$	24.6666	1.00119	0.500594	+	0.865682i	$0.333115\pi$
$608$	0	0
$609$	−26.9780	−1.09321
$610$	0	0
$611$	−2.71053	−0.109656
$612$	0	0
$613$	−43.7380	−1.76656	−0.883281	−	0.468844i	$-0.844671\pi$
$613$	−43.7380	−1.76656	−0.883281	+	0.468844i	$0.844671\pi$
$614$	0	0
$615$	18.5693	0.748787
$616$	0	0
$617$	−13.7791	−0.554728	−0.277364	−	0.960765i	$-0.589461\pi$
$617$	−13.7791	−0.554728	−0.277364	+	0.960765i	$0.589461\pi$
$618$	0	0
$619$	10.9099	0.438506	0.219253	−	0.975668i	$-0.429638\pi$
$619$	10.9099	0.438506	0.219253	+	0.975668i	$0.429638\pi$
$620$	0	0
$621$	14.5064	0.582120
$622$	0	0
$623$	−10.2684	−0.411395
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−81.4604	−3.25321
$628$	0	0
$629$	−15.3504	−0.612061
$630$	0	0
$631$	−41.4208	−1.64894	−0.824468	−	0.565908i	$-0.808526\pi$
$631$	−41.4208	−1.64894	−0.824468	+	0.565908i	$0.808526\pi$
$632$	0	0
$633$	40.5980	1.61362
$634$	0	0
$635$	19.6702	0.780590
$636$	0	0
$637$	−0.138158	−0.00547403
$638$	0	0
$639$	63.0176	2.49294
$640$	0	0
$641$	23.0696	0.911193	0.455596	−	0.890186i	$-0.349426\pi$
$641$	23.0696	0.911193	0.455596	+	0.890186i	$0.349426\pi$
$642$	0	0
$643$	−1.22116	−0.0481577	−0.0240788	−	0.999710i	$-0.507665\pi$
$643$	−1.22116	−0.0481577	−0.0240788	+	0.999710i	$0.507665\pi$
$644$	0	0
$645$	12.8566	0.506230
$646$	0	0
$647$	−38.9778	−1.53237	−0.766187	−	0.642617i	$-0.777849\pi$
$647$	−38.9778	−1.53237	−0.766187	+	0.642617i	$0.777849\pi$
$648$	0	0
$649$	61.5241	2.41503
$650$	0	0
$651$	−51.6472	−2.02421
$652$	0	0
$653$	7.53108	0.294714	0.147357	−	0.989083i	$-0.452923\pi$
$653$	7.53108	0.294714	0.147357	+	0.989083i	$0.452923\pi$
$654$	0	0
$655$	−8.87200	−0.346658
$656$	0	0
$657$	2.19167	0.0855054
$658$	0	0
$659$	−10.9646	−0.427120	−0.213560	−	0.976930i	$-0.568506\pi$
$659$	−10.9646	−0.427120	−0.213560	+	0.976930i	$0.568506\pi$
$660$	0	0
$661$	4.56177	0.177432	0.0887161	−	0.996057i	$-0.471724\pi$
$661$	4.56177	0.177432	0.0887161	+	0.996057i	$0.471724\pi$
$662$	0	0
$663$	−13.7733	−0.534912
$664$	0	0
$665$	−14.4772	−0.561404
$666$	0	0
$667$	−7.21599	−0.279404
$668$	0	0
$669$	−45.0954	−1.74349
$670$	0	0
$671$	−10.0019	−0.386121
$672$	0	0
$673$	17.2875	0.666385	0.333193	−	0.942859i	$-0.391874\pi$
$673$	17.2875	0.666385	0.333193	+	0.942859i	$0.391874\pi$
$674$	0	0
$675$	−6.97529	−0.268479
$676$	0	0
$677$	−5.33959	−0.205217	−0.102608	−	0.994722i	$-0.532719\pi$
$677$	−5.33959	−0.205217	−0.102608	+	0.994722i	$0.532719\pi$
$678$	0	0
$679$	7.31663	0.280787
$680$	0	0
$681$	9.07526	0.347765
$682$	0	0
$683$	−50.9740	−1.95047	−0.975234	−	0.221176i	$-0.929010\pi$
$683$	−50.9740	−1.95047	−0.975234	+	0.221176i	$0.929010\pi$
$684$	0	0
$685$	−18.3957	−0.702865
$686$	0	0
$687$	79.1722	3.02061
$688$	0	0
$689$	−8.00439	−0.304943
$690$	0	0
$691$	−33.2831	−1.26615	−0.633075	−	0.774091i	$-0.718208\pi$
$691$	−33.2831	−1.26615	−0.633075	+	0.774091i	$0.718208\pi$
$692$	0	0
$693$	75.4523	2.86620
$694$	0	0
$695$	14.7511	0.559540
$696$	0	0
$697$	−42.2426	−1.60005
$698$	0	0
$699$	−63.9237	−2.41782
$700$	0	0
$701$	51.3949	1.94116	0.970579	−	0.240784i	$-0.0774044\pi$
$701$	51.3949	1.94116	0.970579	+	0.240784i	$0.0774044\pi$
$702$	0	0
$703$	12.5644	0.473874
$704$	0	0
$705$	10.9105	0.410912
$706$	0	0
$707$	−42.0105	−1.57997
$708$	0	0
$709$	25.6730	0.964170	0.482085	−	0.876125i	$-0.339880\pi$
$709$	25.6730	0.964170	0.482085	+	0.876125i	$0.339880\pi$
$710$	0	0
$711$	41.3490	1.55071
$712$	0	0
$713$	−13.8144	−0.517354
$714$	0	0
$715$	3.74878	0.140196
$716$	0	0
$717$	−25.2644	−0.943515
$718$	0	0
$719$	−1.12956	−0.0421255	−0.0210628	−	0.999778i	$-0.506705\pi$
$719$	−1.12956	−0.0421255	−0.0210628	+	0.999778i	$0.506705\pi$
$720$	0	0
$721$	−9.47411	−0.352834
$722$	0	0
$723$	−27.7888	−1.03348
$724$	0	0
$725$	3.46977	0.128864
$726$	0	0
$727$	1.62966	0.0604408	0.0302204	−	0.999543i	$-0.490379\pi$
$727$	1.62966	0.0604408	0.0302204	+	0.999543i	$0.490379\pi$
$728$	0	0
$729$	−39.0154	−1.44502
$730$	0	0
$731$	−29.2471	−1.08174
$732$	0	0
$733$	3.26676	0.120661	0.0603303	−	0.998178i	$-0.480785\pi$
$733$	3.26676	0.120661	0.0603303	+	0.998178i	$0.480785\pi$
$734$	0	0
$735$	0.556117	0.0205127
$736$	0	0
$737$	−50.8370	−1.87261
$738$	0	0
$739$	−11.3822	−0.418702	−0.209351	−	0.977841i	$-0.567135\pi$
$739$	−11.3822	−0.418702	−0.209351	+	0.977841i	$0.567135\pi$
$740$	0	0
$741$	11.2735	0.414143
$742$	0	0
$743$	38.9054	1.42730	0.713649	−	0.700503i	$-0.247041\pi$
$743$	38.9054	1.42730	0.713649	+	0.700503i	$0.247041\pi$
$744$	0	0
$745$	−5.65888	−0.207325
$746$	0	0
$747$	−38.8609	−1.42185
$748$	0	0
$749$	−50.1597	−1.83280
$750$	0	0
$751$	8.93160	0.325919	0.162959	−	0.986633i	$-0.447896\pi$
$751$	8.93160	0.325919	0.162959	+	0.986633i	$0.447896\pi$
$752$	0	0
$753$	20.8906	0.761296
$754$	0	0
$755$	16.3044	0.593376
$756$	0	0
$757$	33.0074	1.19968	0.599838	−	0.800122i	$-0.295232\pi$
$757$	33.0074	1.19968	0.599838	+	0.800122i	$0.295232\pi$
$758$	0	0
$759$	31.3816	1.13908
$760$	0	0
$761$	47.2147	1.71153	0.855766	−	0.517363i	$-0.173086\pi$
$761$	47.2147	1.71153	0.855766	+	0.517363i	$0.173086\pi$
$762$	0	0
$763$	−46.2770	−1.67534
$764$	0	0
$765$	35.6542	1.28908
$766$	0	0
$767$	−8.51448	−0.307440
$768$	0	0
$769$	−48.0340	−1.73215	−0.866075	−	0.499913i	$-0.833365\pi$
$769$	−48.0340	−1.73215	−0.866075	+	0.499913i	$0.833365\pi$
$770$	0	0
$771$	−28.9448	−1.04242
$772$	0	0
$773$	−49.2032	−1.76972	−0.884858	−	0.465860i	$-0.845745\pi$
$773$	−49.2032	−1.76972	−0.884858	+	0.465860i	$0.845745\pi$
$774$	0	0
$775$	6.64257	0.238608
$776$	0	0
$777$	−18.0961	−0.649193
$778$	0	0
$779$	34.5758	1.23881
$780$	0	0
$781$	60.6716	2.17100
$782$	0	0
$783$	−24.2026	−0.864931
$784$	0	0
$785$	−6.09538	−0.217553
$786$	0	0
$787$	−1.06319	−0.0378986	−0.0189493	−	0.999820i	$-0.506032\pi$
$787$	−1.06319	−0.0378986	−0.0189493	+	0.999820i	$0.506032\pi$
$788$	0	0
$789$	−28.4860	−1.01413
$790$	0	0
$791$	29.9017	1.06318
$792$	0	0
$793$	1.38420	0.0491542
$794$	0	0
$795$	32.2194	1.14270
$796$	0	0
$797$	0.552750	0.0195794	0.00978970	−	0.999952i	$-0.496884\pi$
$797$	0.552750	0.0195794	0.00978970	+	0.999952i	$0.496884\pi$
$798$	0	0
$799$	−24.8198	−0.878062
$800$	0	0
$801$	−20.6990	−0.731363
$802$	0	0
$803$	2.11008	0.0744632
$804$	0	0
$805$	5.57719	0.196570
$806$	0	0
$807$	60.5252	2.13059
$808$	0	0
$809$	−7.49546	−0.263526	−0.131763	−	0.991281i	$-0.542064\pi$
$809$	−7.49546	−0.263526	−0.131763	+	0.991281i	$0.542064\pi$
$810$	0	0
$811$	28.3653	0.996041	0.498020	−	0.867165i	$-0.334061\pi$
$811$	28.3653	0.996041	0.498020	+	0.867165i	$0.334061\pi$
$812$	0	0
$813$	55.4038	1.94310
$814$	0	0
$815$	−4.37534	−0.153261
$816$	0	0
$817$	23.9388	0.837514
$818$	0	0
$819$	−10.4421	−0.364875
$820$	0	0
$821$	19.6917	0.687244	0.343622	−	0.939108i	$-0.388346\pi$
$821$	19.6917	0.687244	0.343622	+	0.939108i	$0.388346\pi$
$822$	0	0
$823$	3.26072	0.113662	0.0568308	−	0.998384i	$-0.481900\pi$
$823$	3.26072	0.113662	0.0568308	+	0.998384i	$0.481900\pi$
$824$	0	0
$825$	−15.0897	−0.525355
$826$	0	0
$827$	36.8768	1.28233	0.641167	−	0.767402i	$-0.278451\pi$
$827$	36.8768	1.28233	0.641167	+	0.767402i	$0.278451\pi$
$828$	0	0
$829$	2.64177	0.0917524	0.0458762	−	0.998947i	$-0.485392\pi$
$829$	2.64177	0.0917524	0.0458762	+	0.998947i	$0.485392\pi$
$830$	0	0
$831$	−44.1307	−1.53088
$832$	0	0
$833$	−1.26509	−0.0438327
$834$	0	0
$835$	7.97440	0.275965
$836$	0	0
$837$	−46.3339	−1.60153
$838$	0	0
$839$	25.3267	0.874375	0.437187	−	0.899370i	$-0.355975\pi$
$839$	25.3267	0.874375	0.437187	+	0.899370i	$0.355975\pi$
$840$	0	0
$841$	−16.9607	−0.584853
$842$	0	0
$843$	2.10217	0.0724027
$844$	0	0
$845$	12.4812	0.429366
$846$	0	0
$847$	43.1440	1.48245
$848$	0	0
$849$	38.7158	1.32872
$850$	0	0
$851$	−4.84027	−0.165922
$852$	0	0
$853$	17.4598	0.597813	0.298907	−	0.954282i	$-0.403378\pi$
$853$	17.4598	0.597813	0.298907	+	0.954282i	$0.403378\pi$
$854$	0	0
$855$	−29.1831	−0.998042
$856$	0	0
$857$	−5.57716	−0.190512	−0.0952560	−	0.995453i	$-0.530367\pi$
$857$	−5.57716	−0.190512	−0.0952560	+	0.995453i	$0.530367\pi$
$858$	0	0
$859$	5.65389	0.192908	0.0964541	−	0.995337i	$-0.469250\pi$
$859$	5.65389	0.192908	0.0964541	+	0.995337i	$0.469250\pi$
$860$	0	0
$861$	−49.7984	−1.69712
$862$	0	0
$863$	45.0445	1.53333	0.766667	−	0.642045i	$-0.221914\pi$
$863$	45.0445	1.53333	0.766667	+	0.642045i	$0.221914\pi$
$864$	0	0
$865$	−7.36502	−0.250418
$866$	0	0
$867$	−76.8319	−2.60935
$868$	0	0
$869$	39.8097	1.35045
$870$	0	0
$871$	7.03548	0.238388
$872$	0	0
$873$	14.7488	0.499172
$874$	0	0
$875$	−2.68176	−0.0906599
$876$	0	0
$877$	−23.6000	−0.796915	−0.398458	−	0.917187i	$-0.630454\pi$
$877$	−23.6000	−0.796915	−0.398458	+	0.917187i	$0.630454\pi$
$878$	0	0
$879$	−57.5051	−1.93960
$880$	0	0
$881$	37.1629	1.25205	0.626025	−	0.779803i	$-0.284681\pi$
$881$	37.1629	1.25205	0.626025	+	0.779803i	$0.284681\pi$
$882$	0	0
$883$	50.9165	1.71348	0.856739	−	0.515751i	$-0.172487\pi$
$883$	50.9165	1.71348	0.856739	+	0.515751i	$0.172487\pi$
$884$	0	0
$885$	34.2727	1.15206
$886$	0	0
$887$	−36.5246	−1.22638	−0.613188	−	0.789937i	$-0.710113\pi$
$887$	−36.5246	−1.22638	−0.613188	+	0.789937i	$0.710113\pi$
$888$	0	0
$889$	−52.7508	−1.76920
$890$	0	0
$891$	20.8485	0.698451
$892$	0	0
$893$	20.3151	0.679820
$894$	0	0
$895$	−7.68172	−0.256771
$896$	0	0
$897$	−4.34299	−0.145008
$898$	0	0
$899$	23.0482	0.768699
$900$	0	0
$901$	−73.2947	−2.44180
$902$	0	0
$903$	−34.4784	−1.14737
$904$	0	0
$905$	−7.16190	−0.238070
$906$	0	0
$907$	41.7707	1.38697	0.693486	−	0.720470i	$-0.256074\pi$
$907$	41.7707	1.38697	0.693486	+	0.720470i	$0.256074\pi$
$908$	0	0
$909$	−84.6845	−2.80881
$910$	0	0
$911$	−28.7763	−0.953402	−0.476701	−	0.879065i	$-0.658168\pi$
$911$	−28.7763	−0.953402	−0.476701	+	0.879065i	$0.658168\pi$
$912$	0	0
$913$	−37.4142	−1.23823
$914$	0	0
$915$	−5.57169	−0.184194
$916$	0	0
$917$	23.7925	0.785699
$918$	0	0
$919$	−0.397684	−0.0131184	−0.00655919	−	0.999978i	$-0.502088\pi$
$919$	−0.397684	−0.0131184	−0.00655919	+	0.999978i	$0.502088\pi$
$920$	0	0
$921$	−55.0663	−1.81449
$922$	0	0
$923$	−8.39651	−0.276374
$924$	0	0
$925$	2.32741	0.0765249
$926$	0	0
$927$	−19.0978	−0.627256
$928$	0	0
$929$	25.6885	0.842812	0.421406	−	0.906872i	$-0.361537\pi$
$929$	25.6885	0.842812	0.421406	+	0.906872i	$0.361537\pi$
$930$	0	0
$931$	1.03548	0.0339365
$932$	0	0
$933$	−93.0351	−3.04583
$934$	0	0
$935$	34.3269	1.12261
$936$	0	0
$937$	27.5212	0.899079	0.449539	−	0.893260i	$-0.351588\pi$
$937$	27.5212	0.899079	0.449539	+	0.893260i	$0.351588\pi$
$938$	0	0
$939$	77.9802	2.54479
$940$	0	0
$941$	−47.5375	−1.54968	−0.774840	−	0.632158i	$-0.782169\pi$
$941$	−47.5375	−1.54968	−0.774840	+	0.632158i	$0.782169\pi$
$942$	0	0
$943$	−13.3199	−0.433756
$944$	0	0
$945$	18.7060	0.608507
$946$	0	0
$947$	−12.9450	−0.420656	−0.210328	−	0.977631i	$-0.567453\pi$
$947$	−12.9450	−0.420656	−0.210328	+	0.977631i	$0.567453\pi$
$948$	0	0
$949$	−0.292020	−0.00947937
$950$	0	0
$951$	47.7028	1.54687
$952$	0	0
$953$	−39.9767	−1.29497	−0.647487	−	0.762077i	$-0.724180\pi$
$953$	−39.9767	−1.29497	−0.647487	+	0.762077i	$0.724180\pi$
$954$	0	0
$955$	−11.9736	−0.387455
$956$	0	0
$957$	−52.3576	−1.69248
$958$	0	0
$959$	49.3329	1.59304
$960$	0	0
$961$	13.1237	0.423347
$962$	0	0
$963$	−101.112	−3.25828
$964$	0	0
$965$	−18.3368	−0.590283
$966$	0	0
$967$	45.8868	1.47562	0.737810	−	0.675008i	$-0.235860\pi$
$967$	45.8868	1.47562	0.737810	+	0.675008i	$0.235860\pi$
$968$	0	0
$969$	103.230	3.31621
$970$	0	0
$971$	17.6207	0.565474	0.282737	−	0.959197i	$-0.408758\pi$
$971$	17.6207	0.565474	0.282737	+	0.959197i	$0.408758\pi$
$972$	0	0
$973$	−39.5588	−1.26820
$974$	0	0
$975$	2.08830	0.0668791
$976$	0	0
$977$	22.2272	0.711110	0.355555	−	0.934655i	$-0.384292\pi$
$977$	22.2272	0.711110	0.355555	+	0.934655i	$0.384292\pi$
$978$	0	0
$979$	−19.9284	−0.636915
$980$	0	0
$981$	−93.2849	−2.97836
$982$	0	0
$983$	12.4228	0.396225	0.198112	−	0.980179i	$-0.436519\pi$
$983$	12.4228	0.396225	0.198112	+	0.980179i	$0.436519\pi$
$984$	0	0
$985$	6.45422	0.205649
$986$	0	0
$987$	−29.2592	−0.931332
$988$	0	0
$989$	−9.22215	−0.293247
$990$	0	0
$991$	−26.8414	−0.852644	−0.426322	−	0.904571i	$-0.640191\pi$
$991$	−26.8414	−0.852644	−0.426322	+	0.904571i	$0.640191\pi$
$992$	0	0
$993$	30.1085	0.955463
$994$	0	0
$995$	−6.20058	−0.196571
$996$	0	0
$997$	−32.8851	−1.04148	−0.520741	−	0.853715i	$-0.674344\pi$
$997$	−32.8851	−1.04148	−0.520741	+	0.853715i	$0.674344\pi$
$998$	0	0
$999$	−16.2344	−0.513634

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8020.2.a.e.1.6	✓	35

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8020.2.a.e.1.6	✓	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)

\(f(q)\)	\(=\)	\(q-2.89929 q^{3} -1.00000 q^{5} +2.68176 q^{7} +5.40586 q^{9} +O(q^{10})\)	\(q-2.89929 q^{3} -1.00000 q^{5} +2.68176 q^{7} +5.40586 q^{9} +5.20461 q^{11} -0.720281 q^{13} +2.89929 q^{15} -6.59547 q^{17} +5.39842 q^{19} -7.77518 q^{21} -2.07968 q^{23} +1.00000 q^{25} -6.97529 q^{27} +3.46977 q^{29} +6.64257 q^{31} -15.0897 q^{33} -2.68176 q^{35} +2.32741 q^{37} +2.08830 q^{39} +6.40479 q^{41} +4.43441 q^{43} -5.40586 q^{45} +3.76316 q^{47} +0.191812 q^{49} +19.1222 q^{51} +11.1129 q^{53} -5.20461 q^{55} -15.6516 q^{57} +11.8211 q^{59} -1.92175 q^{61} +14.4972 q^{63} +0.720281 q^{65} -9.76769 q^{67} +6.02958 q^{69} +11.6573 q^{71} +0.405425 q^{73} -2.89929 q^{75} +13.9575 q^{77} +7.64892 q^{79} +4.00578 q^{81} -7.18866 q^{83} +6.59547 q^{85} -10.0598 q^{87} -3.82899 q^{89} -1.93162 q^{91} -19.2587 q^{93} -5.39842 q^{95} +2.72830 q^{97} +28.1354 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 35 q - q^{3} - 35 q^{5} + 6 q^{7} + 52 q^{9}+O(q^{10}) \) 35 * q - q^3 - 35 * q^5 + 6 * q^7 + 52 * q^9	\( 35 q - q^{3} - 35 q^{5} + 6 q^{7} + 52 q^{9} - 2 q^{11} - q^{13} + q^{15} + 18 q^{17} + 2 q^{19} + 12 q^{21} + 13 q^{23} + 35 q^{25} - 7 q^{27} + 25 q^{29} + 13 q^{31} + 14 q^{33} - 6 q^{35} - 19 q^{37} - 3 q^{39} + 24 q^{41} - 5 q^{43} - 52 q^{45} + 19 q^{47} + 55 q^{49} + 41 q^{53} + 2 q^{55} + 14 q^{57} + 3 q^{59} + 13 q^{61} + 70 q^{63} + q^{65} - 17 q^{67} + 64 q^{69} + 17 q^{71} - 63 q^{73} - q^{75} + 54 q^{77} + 11 q^{79} + 107 q^{81} - 8 q^{83} - 18 q^{85} + 36 q^{87} + 38 q^{89} - 27 q^{91} + q^{93} - 2 q^{95} - 54 q^{97} - 51 q^{99}+O(q^{100}) \) 35 * q - q^3 - 35 * q^5 + 6 * q^7 + 52 * q^9 - 2 * q^11 - q^13 + q^15 + 18 * q^17 + 2 * q^19 + 12 * q^21 + 13 * q^23 + 35 * q^25 - 7 * q^27 + 25 * q^29 + 13 * q^31 + 14 * q^33 - 6 * q^35 - 19 * q^37 - 3 * q^39 + 24 * q^41 - 5 * q^43 - 52 * q^45 + 19 * q^47 + 55 * q^49 + 41 * q^53 + 2 * q^55 + 14 * q^57 + 3 * q^59 + 13 * q^61 + 70 * q^63 + q^65 - 17 * q^67 + 64 * q^69 + 17 * q^71 - 63 * q^73 - q^75 + 54 * q^77 + 11 * q^79 + 107 * q^81 - 8 * q^83 - 18 * q^85 + 36 * q^87 + 38 * q^89 - 27 * q^91 + q^93 - 2 * q^95 - 54 * q^97 - 51 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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