LMFDB - Embedded newform 8020.2.a.d.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:	$1$
Dimension:	$29$
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)

$f(q)$	$=$	$q-2.73282 q^{3} +1.00000 q^{5} +1.81499 q^{7} +4.46832 q^{9} +O(q^{10})$	$q-2.73282 q^{3} +1.00000 q^{5} +1.81499 q^{7} +4.46832 q^{9} -4.80451 q^{11} -3.54236 q^{13} -2.73282 q^{15} -1.11901 q^{17} +5.63113 q^{19} -4.96006 q^{21} +4.39969 q^{23} +1.00000 q^{25} -4.01266 q^{27} -0.434967 q^{29} -7.95658 q^{31} +13.1299 q^{33} +1.81499 q^{35} -3.14040 q^{37} +9.68064 q^{39} -1.77980 q^{41} +6.23083 q^{43} +4.46832 q^{45} +4.42433 q^{47} -3.70580 q^{49} +3.05806 q^{51} +4.03595 q^{53} -4.80451 q^{55} -15.3889 q^{57} +12.1462 q^{59} +11.2611 q^{61} +8.10998 q^{63} -3.54236 q^{65} -1.21318 q^{67} -12.0236 q^{69} -12.6854 q^{71} -6.23845 q^{73} -2.73282 q^{75} -8.72016 q^{77} +6.17709 q^{79} -2.43908 q^{81} -10.6394 q^{83} -1.11901 q^{85} +1.18869 q^{87} +14.2844 q^{89} -6.42936 q^{91} +21.7439 q^{93} +5.63113 q^{95} -18.3652 q^{97} -21.4681 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 29 q - 3 q^{3} + 29 q^{5} - 8 q^{7} + 10 q^{9}+O(q^{10}) $ 29 * q - 3 * q^3 + 29 * q^5 - 8 * q^7 + 10 * q^9	$ 29 q - 3 q^{3} + 29 q^{5} - 8 q^{7} + 10 q^{9} + 2 q^{11} - 23 q^{13} - 3 q^{15} - 30 q^{17} - 6 q^{19} - 16 q^{21} - 21 q^{23} + 29 q^{25} - 15 q^{27} - 35 q^{29} - 7 q^{31} - 36 q^{33} - 8 q^{35} - 31 q^{37} - 11 q^{39} - 24 q^{41} - 17 q^{43} + 10 q^{45} - 17 q^{47} + q^{49} + 8 q^{51} - 57 q^{53} + 2 q^{55} - 46 q^{57} - 9 q^{59} - 27 q^{61} - 34 q^{63} - 23 q^{65} - 21 q^{67} - 28 q^{69} - 19 q^{71} - 81 q^{73} - 3 q^{75} - 66 q^{77} - 17 q^{79} - 39 q^{81} - 30 q^{83} - 30 q^{85} - 20 q^{87} - 38 q^{89} + q^{91} - 75 q^{93} - 6 q^{95} - 48 q^{97} - 7 q^{99}+O(q^{100}) $ 29 * q - 3 * q^3 + 29 * q^5 - 8 * q^7 + 10 * q^9 + 2 * q^11 - 23 * q^13 - 3 * q^15 - 30 * q^17 - 6 * q^19 - 16 * q^21 - 21 * q^23 + 29 * q^25 - 15 * q^27 - 35 * q^29 - 7 * q^31 - 36 * q^33 - 8 * q^35 - 31 * q^37 - 11 * q^39 - 24 * q^41 - 17 * q^43 + 10 * q^45 - 17 * q^47 + q^49 + 8 * q^51 - 57 * q^53 + 2 * q^55 - 46 * q^57 - 9 * q^59 - 27 * q^61 - 34 * q^63 - 23 * q^65 - 21 * q^67 - 28 * q^69 - 19 * q^71 - 81 * q^73 - 3 * q^75 - 66 * q^77 - 17 * q^79 - 39 * q^81 - 30 * q^83 - 30 * q^85 - 20 * q^87 - 38 * q^89 + q^91 - 75 * q^93 - 6 * q^95 - 48 * 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
$3$	−2.73282	−1.57780	−0.788898	−	0.614524i	$-0.789348\pi$
$3$	−2.73282	−1.57780	−0.788898	+	0.614524i	$0.789348\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.81499	0.686003	0.343002	−	0.939335i	$-0.388556\pi$
$7$	1.81499	0.686003	0.343002	+	0.939335i	$0.388556\pi$
$8$	0	0
$9$	4.46832	1.48944
$10$	0	0
$11$	−4.80451	−1.44861	−0.724307	−	0.689478i	$-0.757840\pi$
$11$	−4.80451	−1.44861	−0.724307	+	0.689478i	$0.757840\pi$
$12$	0	0
$13$	−3.54236	−0.982473	−0.491237	−	0.871026i	$-0.663455\pi$
$13$	−3.54236	−0.982473	−0.491237	+	0.871026i	$0.663455\pi$
$14$	0	0
$15$	−2.73282	−0.705612
$16$	0	0
$17$	−1.11901	−0.271400	−0.135700	−	0.990750i	$-0.543328\pi$
$17$	−1.11901	−0.271400	−0.135700	+	0.990750i	$0.543328\pi$
$18$	0	0
$19$	5.63113	1.29187	0.645935	−	0.763392i	$-0.276468\pi$
$19$	5.63113	1.29187	0.645935	+	0.763392i	$0.276468\pi$
$20$	0	0
$21$	−4.96006	−1.08237
$22$	0	0
$23$	4.39969	0.917399	0.458699	−	0.888592i	$-0.348316\pi$
$23$	4.39969	0.917399	0.458699	+	0.888592i	$0.348316\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−4.01266	−0.772236
$28$	0	0
$29$	−0.434967	−0.0807714	−0.0403857	−	0.999184i	$-0.512859\pi$
$29$	−0.434967	−0.0807714	−0.0403857	+	0.999184i	$0.512859\pi$
$30$	0	0
$31$	−7.95658	−1.42904	−0.714522	−	0.699613i	$-0.753356\pi$
$31$	−7.95658	−1.42904	−0.714522	+	0.699613i	$0.753356\pi$
$32$	0	0
$33$	13.1299	2.28562
$34$	0	0
$35$	1.81499	0.306790
$36$	0	0
$37$	−3.14040	−0.516278	−0.258139	−	0.966108i	$-0.583109\pi$
$37$	−3.14040	−0.516278	−0.258139	+	0.966108i	$0.583109\pi$
$38$	0	0
$39$	9.68064	1.55014
$40$	0	0
$41$	−1.77980	−0.277959	−0.138979	−	0.990295i	$-0.544382\pi$
$41$	−1.77980	−0.277959	−0.138979	+	0.990295i	$0.544382\pi$
$42$	0	0
$43$	6.23083	0.950193	0.475097	−	0.879934i	$-0.342413\pi$
$43$	6.23083	0.950193	0.475097	+	0.879934i	$0.342413\pi$
$44$	0	0
$45$	4.46832	0.666098
$46$	0	0
$47$	4.42433	0.645355	0.322678	−	0.946509i	$-0.395417\pi$
$47$	4.42433	0.645355	0.322678	+	0.946509i	$0.395417\pi$
$48$	0	0
$49$	−3.70580	−0.529399
$50$	0	0
$51$	3.05806	0.428215
$52$	0	0
$53$	4.03595	0.554381	0.277190	−	0.960815i	$-0.410597\pi$
$53$	4.03595	0.554381	0.277190	+	0.960815i	$0.410597\pi$
$54$	0	0
$55$	−4.80451	−0.647840
$56$	0	0
$57$	−15.3889	−2.03831
$58$	0	0
$59$	12.1462	1.58131	0.790653	−	0.612265i	$-0.209741\pi$
$59$	12.1462	1.58131	0.790653	+	0.612265i	$0.209741\pi$
$60$	0	0
$61$	11.2611	1.44184	0.720920	−	0.693018i	$-0.243719\pi$
$61$	11.2611	1.44184	0.720920	+	0.693018i	$0.243719\pi$
$62$	0	0
$63$	8.10998	1.02176
$64$	0	0
$65$	−3.54236	−0.439375
$66$	0	0
$67$	−1.21318	−0.148214	−0.0741071	−	0.997250i	$-0.523611\pi$
$67$	−1.21318	−0.148214	−0.0741071	+	0.997250i	$0.523611\pi$
$68$	0	0
$69$	−12.0236	−1.44747
$70$	0	0
$71$	−12.6854	−1.50548	−0.752742	−	0.658316i	$-0.771269\pi$
$71$	−12.6854	−1.50548	−0.752742	+	0.658316i	$0.771269\pi$
$72$	0	0
$73$	−6.23845	−0.730156	−0.365078	−	0.930977i	$-0.618958\pi$
$73$	−6.23845	−0.730156	−0.365078	+	0.930977i	$0.618958\pi$
$74$	0	0
$75$	−2.73282	−0.315559
$76$	0	0
$77$	−8.72016	−0.993754
$78$	0	0
$79$	6.17709	0.694977	0.347489	−	0.937684i	$-0.387035\pi$
$79$	6.17709	0.694977	0.347489	+	0.937684i	$0.387035\pi$
$80$	0	0
$81$	−2.43908	−0.271008
$82$	0	0
$83$	−10.6394	−1.16783	−0.583915	−	0.811815i	$-0.698480\pi$
$83$	−10.6394	−1.16783	−0.583915	+	0.811815i	$0.698480\pi$
$84$	0	0
$85$	−1.11901	−0.121374
$86$	0	0
$87$	1.18869	0.127441
$88$	0	0
$89$	14.2844	1.51415	0.757074	−	0.653329i	$-0.226628\pi$
$89$	14.2844	1.51415	0.757074	+	0.653329i	$0.226628\pi$
$90$	0	0
$91$	−6.42936	−0.673980
$92$	0	0
$93$	21.7439	2.25474
$94$	0	0
$95$	5.63113	0.577742
$96$	0	0
$97$	−18.3652	−1.86471	−0.932353	−	0.361550i	$-0.882248\pi$
$97$	−18.3652	−1.86471	−0.932353	+	0.361550i	$0.882248\pi$
$98$	0	0
$99$	−21.4681	−2.15762
$100$	0	0
$101$	−13.8484	−1.37797	−0.688984	−	0.724776i	$-0.741943\pi$
$101$	−13.8484	−1.37797	−0.688984	+	0.724776i	$0.741943\pi$
$102$	0	0
$103$	−17.2217	−1.69690	−0.848451	−	0.529273i	$-0.822465\pi$
$103$	−17.2217	−1.69690	−0.848451	+	0.529273i	$0.822465\pi$
$104$	0	0
$105$	−4.96006	−0.484052
$106$	0	0
$107$	12.7061	1.22834	0.614172	−	0.789172i	$-0.289490\pi$
$107$	12.7061	1.22834	0.614172	+	0.789172i	$0.289490\pi$
$108$	0	0
$109$	3.46613	0.331995	0.165998	−	0.986126i	$-0.446916\pi$
$109$	3.46613	0.331995	0.165998	+	0.986126i	$0.446916\pi$
$110$	0	0
$111$	8.58215	0.814582
$112$	0	0
$113$	1.56052	0.146801	0.0734007	−	0.997303i	$-0.476615\pi$
$113$	1.56052	0.146801	0.0734007	+	0.997303i	$0.476615\pi$
$114$	0	0
$115$	4.39969	0.410273
$116$	0	0
$117$	−15.8284	−1.46333
$118$	0	0
$119$	−2.03100	−0.186182
$120$	0	0
$121$	12.0833	1.09848
$122$	0	0
$123$	4.86389	0.438562
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−12.2567	−1.08761	−0.543804	−	0.839212i	$-0.683017\pi$
$127$	−12.2567	−1.08761	−0.543804	+	0.839212i	$0.683017\pi$
$128$	0	0
$129$	−17.0278	−1.49921
$130$	0	0
$131$	11.0489	0.965349	0.482674	−	0.875800i	$-0.339665\pi$
$131$	11.0489	0.965349	0.482674	+	0.875800i	$0.339665\pi$
$132$	0	0
$133$	10.2205	0.886227
$134$	0	0
$135$	−4.01266	−0.345355
$136$	0	0
$137$	−3.33244	−0.284710	−0.142355	−	0.989816i	$-0.545467\pi$
$137$	−3.33244	−0.284710	−0.142355	+	0.989816i	$0.545467\pi$
$138$	0	0
$139$	14.9701	1.26975	0.634873	−	0.772617i	$-0.281052\pi$
$139$	14.9701	1.26975	0.634873	+	0.772617i	$0.281052\pi$
$140$	0	0
$141$	−12.0909	−1.01824
$142$	0	0
$143$	17.0193	1.42322
$144$	0	0
$145$	−0.434967	−0.0361221
$146$	0	0
$147$	10.1273	0.835284
$148$	0	0
$149$	−14.7557	−1.20883	−0.604416	−	0.796669i	$-0.706593\pi$
$149$	−14.7557	−1.20883	−0.604416	+	0.796669i	$0.706593\pi$
$150$	0	0
$151$	14.6059	1.18861	0.594306	−	0.804239i	$-0.297427\pi$
$151$	14.6059	1.18861	0.594306	+	0.804239i	$0.297427\pi$
$152$	0	0
$153$	−5.00011	−0.404235
$154$	0	0
$155$	−7.95658	−0.639088
$156$	0	0
$157$	2.46102	0.196411	0.0982053	−	0.995166i	$-0.468690\pi$
$157$	2.46102	0.196411	0.0982053	+	0.995166i	$0.468690\pi$
$158$	0	0
$159$	−11.0295	−0.874700
$160$	0	0
$161$	7.98541	0.629339
$162$	0	0
$163$	12.5899	0.986113	0.493057	−	0.869997i	$-0.335880\pi$
$163$	12.5899	0.986113	0.493057	+	0.869997i	$0.335880\pi$
$164$	0	0
$165$	13.1299	1.02216
$166$	0	0
$167$	5.09082	0.393940	0.196970	−	0.980410i	$-0.436890\pi$
$167$	5.09082	0.393940	0.196970	+	0.980410i	$0.436890\pi$
$168$	0	0
$169$	−0.451703	−0.0347464
$170$	0	0
$171$	25.1617	1.92416
$172$	0	0
$173$	−10.8805	−0.827226	−0.413613	−	0.910453i	$-0.635733\pi$
$173$	−10.8805	−0.827226	−0.413613	+	0.910453i	$0.635733\pi$
$174$	0	0
$175$	1.81499	0.137201
$176$	0	0
$177$	−33.1935	−2.49498
$178$	0	0
$179$	8.73037	0.652539	0.326269	−	0.945277i	$-0.394208\pi$
$179$	8.73037	0.652539	0.326269	+	0.945277i	$0.394208\pi$
$180$	0	0
$181$	14.8136	1.10109	0.550543	−	0.834807i	$-0.314421\pi$
$181$	14.8136	1.10109	0.550543	+	0.834807i	$0.314421\pi$
$182$	0	0
$183$	−30.7747	−2.27493
$184$	0	0
$185$	−3.14040	−0.230887
$186$	0	0
$187$	5.37631	0.393155
$188$	0	0
$189$	−7.28295	−0.529757
$190$	0	0
$191$	−1.15519	−0.0835864	−0.0417932	−	0.999126i	$-0.513307\pi$
$191$	−1.15519	−0.0835864	−0.0417932	+	0.999126i	$0.513307\pi$
$192$	0	0
$193$	3.28966	0.236795	0.118397	−	0.992966i	$-0.462224\pi$
$193$	3.28966	0.236795	0.118397	+	0.992966i	$0.462224\pi$
$194$	0	0
$195$	9.68064	0.693245
$196$	0	0
$197$	−12.6195	−0.899099	−0.449550	−	0.893255i	$-0.648416\pi$
$197$	−12.6195	−0.899099	−0.449550	+	0.893255i	$0.648416\pi$
$198$	0	0
$199$	3.91354	0.277424	0.138712	−	0.990333i	$-0.455704\pi$
$199$	3.91354	0.277424	0.138712	+	0.990333i	$0.455704\pi$
$200$	0	0
$201$	3.31542	0.233852
$202$	0	0
$203$	−0.789463	−0.0554095
$204$	0	0
$205$	−1.77980	−0.124307
$206$	0	0
$207$	19.6592	1.36641
$208$	0	0
$209$	−27.0548	−1.87142
$210$	0	0
$211$	−2.64230	−0.181904	−0.0909519	−	0.995855i	$-0.528991\pi$
$211$	−2.64230	−0.181904	−0.0909519	+	0.995855i	$0.528991\pi$
$212$	0	0
$213$	34.6670	2.37535
$214$	0	0
$215$	6.23083	0.424939
$216$	0	0
$217$	−14.4411	−0.980329
$218$	0	0
$219$	17.0486	1.15204
$220$	0	0
$221$	3.96394	0.266644
$222$	0	0
$223$	19.4178	1.30031	0.650155	−	0.759802i	$-0.274704\pi$
$223$	19.4178	1.30031	0.650155	+	0.759802i	$0.274704\pi$
$224$	0	0
$225$	4.46832	0.297888
$226$	0	0
$227$	−10.2364	−0.679411	−0.339706	−	0.940532i	$-0.610327\pi$
$227$	−10.2364	−0.679411	−0.339706	+	0.940532i	$0.610327\pi$
$228$	0	0
$229$	23.4962	1.55267	0.776337	−	0.630318i	$-0.217075\pi$
$229$	23.4962	1.55267	0.776337	+	0.630318i	$0.217075\pi$
$230$	0	0
$231$	23.8306	1.56794
$232$	0	0
$233$	−17.5267	−1.14821	−0.574106	−	0.818781i	$-0.694650\pi$
$233$	−17.5267	−1.14821	−0.574106	+	0.818781i	$0.694650\pi$
$234$	0	0
$235$	4.42433	0.288612
$236$	0	0
$237$	−16.8809	−1.09653
$238$	0	0
$239$	−25.0693	−1.62160	−0.810799	−	0.585324i	$-0.800967\pi$
$239$	−25.0693	−1.62160	−0.810799	+	0.585324i	$0.800967\pi$
$240$	0	0
$241$	−6.01394	−0.387392	−0.193696	−	0.981062i	$-0.562048\pi$
$241$	−6.01394	−0.387392	−0.193696	+	0.981062i	$0.562048\pi$
$242$	0	0
$243$	18.7035	1.19983
$244$	0	0
$245$	−3.70580	−0.236755
$246$	0	0
$247$	−19.9475	−1.26923
$248$	0	0
$249$	29.0757	1.84260
$250$	0	0
$251$	7.35943	0.464523	0.232262	−	0.972653i	$-0.425387\pi$
$251$	7.35943	0.464523	0.232262	+	0.972653i	$0.425387\pi$
$252$	0	0
$253$	−21.1384	−1.32896
$254$	0	0
$255$	3.05806	0.191503
$256$	0	0
$257$	15.9670	0.995994	0.497997	−	0.867179i	$-0.334069\pi$
$257$	15.9670	0.995994	0.497997	+	0.867179i	$0.334069\pi$
$258$	0	0
$259$	−5.69981	−0.354169
$260$	0	0
$261$	−1.94357	−0.120304
$262$	0	0
$263$	−18.9495	−1.16848	−0.584239	−	0.811582i	$-0.698607\pi$
$263$	−18.9495	−1.16848	−0.584239	+	0.811582i	$0.698607\pi$
$264$	0	0
$265$	4.03595	0.247927
$266$	0	0
$267$	−39.0368	−2.38902
$268$	0	0
$269$	−14.9509	−0.911570	−0.455785	−	0.890090i	$-0.650641\pi$
$269$	−14.9509	−0.911570	−0.455785	+	0.890090i	$0.650641\pi$
$270$	0	0
$271$	−22.9339	−1.39314	−0.696569	−	0.717490i	$-0.745291\pi$
$271$	−22.9339	−1.39314	−0.696569	+	0.717490i	$0.745291\pi$
$272$	0	0
$273$	17.5703	1.06340
$274$	0	0
$275$	−4.80451	−0.289723
$276$	0	0
$277$	−6.47342	−0.388950	−0.194475	−	0.980907i	$-0.562300\pi$
$277$	−6.47342	−0.388950	−0.194475	+	0.980907i	$0.562300\pi$
$278$	0	0
$279$	−35.5525	−2.12847
$280$	0	0
$281$	10.4083	0.620908	0.310454	−	0.950588i	$-0.399519\pi$
$281$	10.4083	0.620908	0.310454	+	0.950588i	$0.399519\pi$
$282$	0	0
$283$	7.31561	0.434868	0.217434	−	0.976075i	$-0.430231\pi$
$283$	7.31561	0.434868	0.217434	+	0.976075i	$0.430231\pi$
$284$	0	0
$285$	−15.3889	−0.911559
$286$	0	0
$287$	−3.23034	−0.190681
$288$	0	0
$289$	−15.7478	−0.926342
$290$	0	0
$291$	50.1889	2.94213
$292$	0	0
$293$	−23.6807	−1.38344	−0.691721	−	0.722165i	$-0.743147\pi$
$293$	−23.6807	−1.38344	−0.691721	+	0.722165i	$0.743147\pi$
$294$	0	0
$295$	12.1462	0.707181
$296$	0	0
$297$	19.2789	1.11867
$298$	0	0
$299$	−15.5853	−0.901320
$300$	0	0
$301$	11.3089	0.651836
$302$	0	0
$303$	37.8453	2.17415
$304$	0	0
$305$	11.2611	0.644811
$306$	0	0
$307$	13.3853	0.763940	0.381970	−	0.924175i	$-0.375246\pi$
$307$	13.3853	0.763940	0.381970	+	0.924175i	$0.375246\pi$
$308$	0	0
$309$	47.0638	2.67737
$310$	0	0
$311$	24.2388	1.37446	0.687229	−	0.726441i	$-0.258827\pi$
$311$	24.2388	1.37446	0.687229	+	0.726441i	$0.258827\pi$
$312$	0	0
$313$	−29.6565	−1.67628	−0.838141	−	0.545453i	$-0.816358\pi$
$313$	−29.6565	−1.67628	−0.838141	+	0.545453i	$0.816358\pi$
$314$	0	0
$315$	8.10998	0.456945
$316$	0	0
$317$	−8.59044	−0.482487	−0.241243	−	0.970465i	$-0.577555\pi$
$317$	−8.59044	−0.482487	−0.241243	+	0.970465i	$0.577555\pi$
$318$	0	0
$319$	2.08981	0.117007
$320$	0	0
$321$	−34.7235	−1.93808
$322$	0	0
$323$	−6.30131	−0.350614
$324$	0	0
$325$	−3.54236	−0.196495
$326$	0	0
$327$	−9.47232	−0.523821
$328$	0	0
$329$	8.03014	0.442716
$330$	0	0
$331$	25.8038	1.41830	0.709152	−	0.705056i	$-0.249078\pi$
$331$	25.8038	1.41830	0.709152	+	0.705056i	$0.249078\pi$
$332$	0	0
$333$	−14.0323	−0.768966
$334$	0	0
$335$	−1.21318	−0.0662834
$336$	0	0
$337$	−4.15121	−0.226131	−0.113065	−	0.993588i	$-0.536067\pi$
$337$	−4.15121	−0.226131	−0.113065	+	0.993588i	$0.536067\pi$
$338$	0	0
$339$	−4.26463	−0.231623
$340$	0	0
$341$	38.2275	2.07013
$342$	0	0
$343$	−19.4310	−1.04917
$344$	0	0
$345$	−12.0236	−0.647327
$346$	0	0
$347$	−2.41242	−0.129506	−0.0647529	−	0.997901i	$-0.520626\pi$
$347$	−2.41242	−0.129506	−0.0647529	+	0.997901i	$0.520626\pi$
$348$	0	0
$349$	−9.33395	−0.499635	−0.249818	−	0.968293i	$-0.580371\pi$
$349$	−9.33395	−0.499635	−0.249818	+	0.968293i	$0.580371\pi$
$350$	0	0
$351$	14.2143	0.758702
$352$	0	0
$353$	10.7243	0.570795	0.285398	−	0.958409i	$-0.407874\pi$
$353$	10.7243	0.570795	0.285398	+	0.958409i	$0.407874\pi$
$354$	0	0
$355$	−12.6854	−0.673273
$356$	0	0
$357$	5.55037	0.293757
$358$	0	0
$359$	−17.0940	−0.902187	−0.451093	−	0.892477i	$-0.648966\pi$
$359$	−17.0940	−0.902187	−0.451093	+	0.892477i	$0.648966\pi$
$360$	0	0
$361$	12.7096	0.668928
$362$	0	0
$363$	−33.0216	−1.73318
$364$	0	0
$365$	−6.23845	−0.326535
$366$	0	0
$367$	−22.7463	−1.18735	−0.593675	−	0.804705i	$-0.702323\pi$
$367$	−22.7463	−1.18735	−0.593675	+	0.804705i	$0.702323\pi$
$368$	0	0
$369$	−7.95274	−0.414003
$370$	0	0
$371$	7.32523	0.380307
$372$	0	0
$373$	−6.48116	−0.335582	−0.167791	−	0.985823i	$-0.553663\pi$
$373$	−6.48116	−0.335582	−0.167791	+	0.985823i	$0.553663\pi$
$374$	0	0
$375$	−2.73282	−0.141122
$376$	0	0
$377$	1.54081	0.0793558
$378$	0	0
$379$	30.8629	1.58532	0.792660	−	0.609665i	$-0.208696\pi$
$379$	30.8629	1.58532	0.792660	+	0.609665i	$0.208696\pi$
$380$	0	0
$381$	33.4955	1.71602
$382$	0	0
$383$	−27.6078	−1.41069	−0.705347	−	0.708862i	$-0.749209\pi$
$383$	−27.6078	−1.41069	−0.705347	+	0.708862i	$0.749209\pi$
$384$	0	0
$385$	−8.72016	−0.444420
$386$	0	0
$387$	27.8414	1.41526
$388$	0	0
$389$	−8.05140	−0.408222	−0.204111	−	0.978948i	$-0.565430\pi$
$389$	−8.05140	−0.408222	−0.204111	+	0.978948i	$0.565430\pi$
$390$	0	0
$391$	−4.92331	−0.248982
$392$	0	0
$393$	−30.1947	−1.52312
$394$	0	0
$395$	6.17709	0.310803
$396$	0	0
$397$	−30.4957	−1.53054	−0.765268	−	0.643712i	$-0.777393\pi$
$397$	−30.4957	−1.53054	−0.765268	+	0.643712i	$0.777393\pi$
$398$	0	0
$399$	−27.9307	−1.39829
$400$	0	0
$401$	−1.00000	−0.0499376
$401$	−1.00000	−0.0499376
$402$	0	0
$403$	28.1850	1.40400
$404$	0	0
$405$	−2.43908	−0.121199
$406$	0	0
$407$	15.0881	0.747888
$408$	0	0
$409$	−1.57844	−0.0780486	−0.0390243	−	0.999238i	$-0.512425\pi$
$409$	−1.57844	−0.0780486	−0.0390243	+	0.999238i	$0.512425\pi$
$410$	0	0
$411$	9.10697	0.449214
$412$	0	0
$413$	22.0454	1.08478
$414$	0	0
$415$	−10.6394	−0.522269
$416$	0	0
$417$	−40.9106	−2.00340
$418$	0	0
$419$	19.6145	0.958233	0.479117	−	0.877751i	$-0.340957\pi$
$419$	19.6145	0.958233	0.479117	+	0.877751i	$0.340957\pi$
$420$	0	0
$421$	−10.3836	−0.506065	−0.253033	−	0.967458i	$-0.581428\pi$
$421$	−10.3836	−0.506065	−0.253033	+	0.967458i	$0.581428\pi$
$422$	0	0
$423$	19.7693	0.961218
$424$	0	0
$425$	−1.11901	−0.0542801
$426$	0	0
$427$	20.4389	0.989108
$428$	0	0
$429$	−46.5107	−2.24556
$430$	0	0
$431$	−8.71293	−0.419687	−0.209844	−	0.977735i	$-0.567295\pi$
$431$	−8.71293	−0.419687	−0.209844	+	0.977735i	$0.567295\pi$
$432$	0	0
$433$	−13.6148	−0.654283	−0.327142	−	0.944975i	$-0.606085\pi$
$433$	−13.6148	−0.654283	−0.327142	+	0.944975i	$0.606085\pi$
$434$	0	0
$435$	1.18869	0.0569933
$436$	0	0
$437$	24.7752	1.18516
$438$	0	0
$439$	13.8634	0.661663	0.330832	−	0.943690i	$-0.392671\pi$
$439$	13.8634	0.661663	0.330832	+	0.943690i	$0.392671\pi$
$440$	0	0
$441$	−16.5587	−0.788509
$442$	0	0
$443$	−22.3874	−1.06366	−0.531828	−	0.846852i	$-0.678495\pi$
$443$	−22.3874	−1.06366	−0.531828	+	0.846852i	$0.678495\pi$
$444$	0	0
$445$	14.2844	0.677147
$446$	0	0
$447$	40.3246	1.90729
$448$	0	0
$449$	−39.7761	−1.87715	−0.938576	−	0.345072i	$-0.887854\pi$
$449$	−39.7761	−1.87715	−0.938576	+	0.345072i	$0.887854\pi$
$450$	0	0
$451$	8.55109	0.402655
$452$	0	0
$453$	−39.9153	−1.87539
$454$	0	0
$455$	−6.42936	−0.301413
$456$	0	0
$457$	33.0583	1.54640	0.773202	−	0.634160i	$-0.218654\pi$
$457$	33.0583	1.54640	0.773202	+	0.634160i	$0.218654\pi$
$458$	0	0
$459$	4.49022	0.209585
$460$	0	0
$461$	−34.6163	−1.61224	−0.806120	−	0.591752i	$-0.798436\pi$
$461$	−34.6163	−1.61224	−0.806120	+	0.591752i	$0.798436\pi$
$462$	0	0
$463$	−0.403312	−0.0187435	−0.00937175	−	0.999956i	$-0.502983\pi$
$463$	−0.403312	−0.0187435	−0.00937175	+	0.999956i	$0.502983\pi$
$464$	0	0
$465$	21.7439	1.00835
$466$	0	0
$467$	−31.1901	−1.44330	−0.721652	−	0.692256i	$-0.756617\pi$
$467$	−31.1901	−1.44330	−0.721652	+	0.692256i	$0.756617\pi$
$468$	0	0
$469$	−2.20192	−0.101675
$470$	0	0
$471$	−6.72553	−0.309896
$472$	0	0
$473$	−29.9361	−1.37646
$474$	0	0
$475$	5.63113	0.258374
$476$	0	0
$477$	18.0339	0.825717
$478$	0	0
$479$	−22.8797	−1.04540	−0.522701	−	0.852516i	$-0.675075\pi$
$479$	−22.8797	−1.04540	−0.522701	+	0.852516i	$0.675075\pi$
$480$	0	0
$481$	11.1244	0.507230
$482$	0	0
$483$	−21.8227	−0.992968
$484$	0	0
$485$	−18.3652	−0.833922
$486$	0	0
$487$	−7.91692	−0.358750	−0.179375	−	0.983781i	$-0.557408\pi$
$487$	−7.91692	−0.358750	−0.179375	+	0.983781i	$0.557408\pi$
$488$	0	0
$489$	−34.4058	−1.55589
$490$	0	0
$491$	−38.5150	−1.73816	−0.869078	−	0.494675i	$-0.835287\pi$
$491$	−38.5150	−1.73816	−0.869078	+	0.494675i	$0.835287\pi$
$492$	0	0
$493$	0.486734	0.0219214
$494$	0	0
$495$	−21.4681	−0.964919
$496$	0	0
$497$	−23.0240	−1.03277
$498$	0	0
$499$	6.91115	0.309385	0.154693	−	0.987963i	$-0.450561\pi$
$499$	6.91115	0.309385	0.154693	+	0.987963i	$0.450561\pi$
$500$	0	0
$501$	−13.9123	−0.621556
$502$	0	0
$503$	16.1202	0.718763	0.359381	−	0.933191i	$-0.382988\pi$
$503$	16.1202	0.718763	0.359381	+	0.933191i	$0.382988\pi$
$504$	0	0
$505$	−13.8484	−0.616246
$506$	0	0
$507$	1.23442	0.0548227
$508$	0	0
$509$	−12.1744	−0.539622	−0.269811	−	0.962913i	$-0.586961\pi$
$509$	−12.1744	−0.539622	−0.269811	+	0.962913i	$0.586961\pi$
$510$	0	0
$511$	−11.3228	−0.500889
$512$	0	0
$513$	−22.5958	−0.997629
$514$	0	0
$515$	−17.2217	−0.758878
$516$	0	0
$517$	−21.2568	−0.934871
$518$	0	0
$519$	29.7344	1.30519
$520$	0	0
$521$	20.1795	0.884081	0.442040	−	0.896995i	$-0.354255\pi$
$521$	20.1795	0.884081	0.442040	+	0.896995i	$0.354255\pi$
$522$	0	0
$523$	4.69194	0.205164	0.102582	−	0.994725i	$-0.467290\pi$
$523$	4.69194	0.205164	0.102582	+	0.994725i	$0.467290\pi$
$524$	0	0
$525$	−4.96006	−0.216475
$526$	0	0
$527$	8.90351	0.387843
$528$	0	0
$529$	−3.64273	−0.158380
$530$	0	0
$531$	54.2733	2.35526
$532$	0	0
$533$	6.30470	0.273087
$534$	0	0
$535$	12.7061	0.549332
$536$	0	0
$537$	−23.8586	−1.02957
$538$	0	0
$539$	17.8045	0.766895
$540$	0	0
$541$	−30.2969	−1.30257	−0.651283	−	0.758835i	$-0.725769\pi$
$541$	−30.2969	−1.30257	−0.651283	+	0.758835i	$0.725769\pi$
$542$	0	0
$543$	−40.4829	−1.73729
$544$	0	0
$545$	3.46613	0.148473
$546$	0	0
$547$	14.0848	0.602224	0.301112	−	0.953589i	$-0.402642\pi$
$547$	14.0848	0.602224	0.301112	+	0.953589i	$0.402642\pi$
$548$	0	0
$549$	50.3184	2.14754
$550$	0	0
$551$	−2.44936	−0.104346
$552$	0	0
$553$	11.2114	0.476757
$554$	0	0
$555$	8.58215	0.364292
$556$	0	0
$557$	−33.2163	−1.40742	−0.703711	−	0.710487i	$-0.748475\pi$
$557$	−33.2163	−1.40742	−0.703711	+	0.710487i	$0.748475\pi$
$558$	0	0
$559$	−22.0718	−0.933539
$560$	0	0
$561$	−14.6925	−0.620318
$562$	0	0
$563$	−23.2999	−0.981975	−0.490988	−	0.871167i	$-0.663364\pi$
$563$	−23.2999	−0.981975	−0.490988	+	0.871167i	$0.663364\pi$
$564$	0	0
$565$	1.56052	0.0656516
$566$	0	0
$567$	−4.42691	−0.185913
$568$	0	0
$569$	6.99712	0.293335	0.146667	−	0.989186i	$-0.453145\pi$
$569$	6.99712	0.293335	0.146667	+	0.989186i	$0.453145\pi$
$570$	0	0
$571$	−29.4331	−1.23174	−0.615868	−	0.787849i	$-0.711195\pi$
$571$	−29.4331	−1.23174	−0.615868	+	0.787849i	$0.711195\pi$
$572$	0	0
$573$	3.15692	0.131882
$574$	0	0
$575$	4.39969	0.183480
$576$	0	0
$577$	−20.8319	−0.867245	−0.433623	−	0.901095i	$-0.642765\pi$
$577$	−20.8319	−0.867245	−0.433623	+	0.901095i	$0.642765\pi$
$578$	0	0
$579$	−8.99005	−0.373614
$580$	0	0
$581$	−19.3105	−0.801135
$582$	0	0
$583$	−19.3908	−0.803084
$584$	0	0
$585$	−15.8284	−0.654423
$586$	0	0
$587$	−22.4405	−0.926218	−0.463109	−	0.886301i	$-0.653266\pi$
$587$	−22.4405	−0.926218	−0.463109	+	0.886301i	$0.653266\pi$
$588$	0	0
$589$	−44.8045	−1.84614
$590$	0	0
$591$	34.4868	1.41860
$592$	0	0
$593$	−37.7277	−1.54929	−0.774646	−	0.632395i	$-0.782072\pi$
$593$	−37.7277	−1.54929	−0.774646	+	0.632395i	$0.782072\pi$
$594$	0	0
$595$	−2.03100	−0.0832630
$596$	0	0
$597$	−10.6950	−0.437718
$598$	0	0
$599$	−11.6173	−0.474670	−0.237335	−	0.971428i	$-0.576274\pi$
$599$	−11.6173	−0.474670	−0.237335	+	0.971428i	$0.576274\pi$
$600$	0	0
$601$	19.9628	0.814300	0.407150	−	0.913361i	$-0.366523\pi$
$601$	19.9628	0.814300	0.407150	+	0.913361i	$0.366523\pi$
$602$	0	0
$603$	−5.42090	−0.220756
$604$	0	0
$605$	12.0833	0.491257
$606$	0	0
$607$	7.25296	0.294389	0.147194	−	0.989108i	$-0.452976\pi$
$607$	7.25296	0.294389	0.147194	+	0.989108i	$0.452976\pi$
$608$	0	0
$609$	2.15746	0.0874248
$610$	0	0
$611$	−15.6726	−0.634044
$612$	0	0
$613$	−32.8164	−1.32544	−0.662721	−	0.748866i	$-0.730599\pi$
$613$	−32.8164	−1.32544	−0.662721	+	0.748866i	$0.730599\pi$
$614$	0	0
$615$	4.86389	0.196131
$616$	0	0
$617$	−34.3146	−1.38145	−0.690727	−	0.723116i	$-0.742709\pi$
$617$	−34.3146	−1.38145	−0.690727	+	0.723116i	$0.742709\pi$
$618$	0	0
$619$	8.29147	0.333262	0.166631	−	0.986019i	$-0.446711\pi$
$619$	8.29147	0.333262	0.166631	+	0.986019i	$0.446711\pi$
$620$	0	0
$621$	−17.6545	−0.708449
$622$	0	0
$623$	25.9262	1.03871
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	73.9360	2.95272
$628$	0	0
$629$	3.51415	0.140118
$630$	0	0
$631$	−23.2309	−0.924806	−0.462403	−	0.886670i	$-0.653013\pi$
$631$	−23.2309	−0.924806	−0.462403	+	0.886670i	$0.653013\pi$
$632$	0	0
$633$	7.22095	0.287007
$634$	0	0
$635$	−12.2567	−0.486393
$636$	0	0
$637$	13.1273	0.520121
$638$	0	0
$639$	−56.6826	−2.24233
$640$	0	0
$641$	−17.2428	−0.681052	−0.340526	−	0.940235i	$-0.610605\pi$
$641$	−17.2428	−0.681052	−0.340526	+	0.940235i	$0.610605\pi$
$642$	0	0
$643$	4.19170	0.165304	0.0826522	−	0.996578i	$-0.473661\pi$
$643$	4.19170	0.165304	0.0826522	+	0.996578i	$0.473661\pi$
$644$	0	0
$645$	−17.0278	−0.670468
$646$	0	0
$647$	−33.7177	−1.32558	−0.662790	−	0.748805i	$-0.730628\pi$
$647$	−33.7177	−1.32558	−0.662790	+	0.748805i	$0.730628\pi$
$648$	0	0
$649$	−58.3567	−2.29070
$650$	0	0
$651$	39.4651	1.54676
$652$	0	0
$653$	33.1979	1.29914	0.649568	−	0.760304i	$-0.274950\pi$
$653$	33.1979	1.29914	0.649568	+	0.760304i	$0.274950\pi$
$654$	0	0
$655$	11.0489	0.431717
$656$	0	0
$657$	−27.8754	−1.08752
$658$	0	0
$659$	−27.0912	−1.05533	−0.527663	−	0.849454i	$-0.676931\pi$
$659$	−27.0912	−1.05533	−0.527663	+	0.849454i	$0.676931\pi$
$660$	0	0
$661$	24.6320	0.958075	0.479038	−	0.877794i	$-0.340986\pi$
$661$	24.6320	0.958075	0.479038	+	0.877794i	$0.340986\pi$
$662$	0	0
$663$	−10.8328	−0.420709
$664$	0	0
$665$	10.2205	0.396333
$666$	0	0
$667$	−1.91372	−0.0740996
$668$	0	0
$669$	−53.0653	−2.05162
$670$	0	0
$671$	−54.1042	−2.08867
$672$	0	0
$673$	−32.2578	−1.24345	−0.621724	−	0.783236i	$-0.713567\pi$
$673$	−32.2578	−1.24345	−0.621724	+	0.783236i	$0.713567\pi$
$674$	0	0
$675$	−4.01266	−0.154447
$676$	0	0
$677$	42.8731	1.64775	0.823874	−	0.566773i	$-0.191808\pi$
$677$	42.8731	1.64775	0.823874	+	0.566773i	$0.191808\pi$
$678$	0	0
$679$	−33.3328	−1.27919
$680$	0	0
$681$	27.9742	1.07197
$682$	0	0
$683$	8.17020	0.312624	0.156312	−	0.987708i	$-0.450039\pi$
$683$	8.17020	0.312624	0.156312	+	0.987708i	$0.450039\pi$
$684$	0	0
$685$	−3.33244	−0.127326
$686$	0	0
$687$	−64.2110	−2.44980
$688$	0	0
$689$	−14.2968	−0.544664
$690$	0	0
$691$	−25.4041	−0.966419	−0.483209	−	0.875505i	$-0.660529\pi$
$691$	−25.4041	−0.966419	−0.483209	+	0.875505i	$0.660529\pi$
$692$	0	0
$693$	−38.9645	−1.48014
$694$	0	0
$695$	14.9701	0.567848
$696$	0	0
$697$	1.99162	0.0754381
$698$	0	0
$699$	47.8973	1.81164
$700$	0	0
$701$	−22.3120	−0.842713	−0.421357	−	0.906895i	$-0.638446\pi$
$701$	−22.3120	−0.842713	−0.421357	+	0.906895i	$0.638446\pi$
$702$	0	0
$703$	−17.6840	−0.666965
$704$	0	0
$705$	−12.0909	−0.455370
$706$	0	0
$707$	−25.1348	−0.945291
$708$	0	0
$709$	44.7789	1.68171	0.840854	−	0.541263i	$-0.182053\pi$
$709$	44.7789	1.68171	0.840854	+	0.541263i	$0.182053\pi$
$710$	0	0
$711$	27.6012	1.03513
$712$	0	0
$713$	−35.0065	−1.31100
$714$	0	0
$715$	17.0193	0.636485
$716$	0	0
$717$	68.5100	2.55855
$718$	0	0
$719$	−48.5659	−1.81120	−0.905602	−	0.424130i	$-0.860580\pi$
$719$	−48.5659	−1.81120	−0.905602	+	0.424130i	$0.860580\pi$
$720$	0	0
$721$	−31.2573	−1.16408
$722$	0	0
$723$	16.4350	0.611225
$724$	0	0
$725$	−0.434967	−0.0161543
$726$	0	0
$727$	2.27850	0.0845051	0.0422525	−	0.999107i	$-0.486547\pi$
$727$	2.27850	0.0845051	0.0422525	+	0.999107i	$0.486547\pi$
$728$	0	0
$729$	−43.7962	−1.62208
$730$	0	0
$731$	−6.97238	−0.257883
$732$	0	0
$733$	14.1486	0.522590	0.261295	−	0.965259i	$-0.415850\pi$
$733$	14.1486	0.522590	0.261295	+	0.965259i	$0.415850\pi$
$734$	0	0
$735$	10.1273	0.373550
$736$	0	0
$737$	5.82876	0.214705
$738$	0	0
$739$	48.8043	1.79530	0.897648	−	0.440713i	$-0.145274\pi$
$739$	48.8043	1.79530	0.897648	+	0.440713i	$0.145274\pi$
$740$	0	0
$741$	54.5129	2.00258
$742$	0	0
$743$	−1.85765	−0.0681505	−0.0340753	−	0.999419i	$-0.510849\pi$
$743$	−1.85765	−0.0681505	−0.0340753	+	0.999419i	$0.510849\pi$
$744$	0	0
$745$	−14.7557	−0.540606
$746$	0	0
$747$	−47.5404	−1.73941
$748$	0	0
$749$	23.0615	0.842648
$750$	0	0
$751$	−29.6087	−1.08044	−0.540219	−	0.841524i	$-0.681659\pi$
$751$	−29.6087	−1.08044	−0.540219	+	0.841524i	$0.681659\pi$
$752$	0	0
$753$	−20.1120	−0.732923
$754$	0	0
$755$	14.6059	0.531563
$756$	0	0
$757$	39.2697	1.42728	0.713640	−	0.700513i	$-0.247045\pi$
$757$	39.2697	1.42728	0.713640	+	0.700513i	$0.247045\pi$
$758$	0	0
$759$	57.7674	2.09682
$760$	0	0
$761$	16.2420	0.588773	0.294386	−	0.955686i	$-0.404885\pi$
$761$	16.2420	0.588773	0.294386	+	0.955686i	$0.404885\pi$
$762$	0	0
$763$	6.29101	0.227750
$764$	0	0
$765$	−5.00011	−0.180779
$766$	0	0
$767$	−43.0263	−1.55359
$768$	0	0
$769$	50.7271	1.82927	0.914633	−	0.404284i	$-0.132479\pi$
$769$	50.7271	1.82927	0.914633	+	0.404284i	$0.132479\pi$
$770$	0	0
$771$	−43.6350	−1.57148
$772$	0	0
$773$	2.60716	0.0937732	0.0468866	−	0.998900i	$-0.485070\pi$
$773$	2.60716	0.0937732	0.0468866	+	0.998900i	$0.485070\pi$
$774$	0	0
$775$	−7.95658	−0.285809
$776$	0	0
$777$	15.5766	0.558806
$778$	0	0
$779$	−10.0223	−0.359087
$780$	0	0
$781$	60.9473	2.18087
$782$	0	0
$783$	1.74538	0.0623746
$784$	0	0
$785$	2.46102	0.0878375
$786$	0	0
$787$	−2.06384	−0.0735679	−0.0367839	−	0.999323i	$-0.511711\pi$
$787$	−2.06384	−0.0735679	−0.0367839	+	0.999323i	$0.511711\pi$
$788$	0	0
$789$	51.7857	1.84362
$790$	0	0
$791$	2.83234	0.100706
$792$	0	0
$793$	−39.8910	−1.41657
$794$	0	0
$795$	−11.0295	−0.391178
$796$	0	0
$797$	−26.8237	−0.950142	−0.475071	−	0.879947i	$-0.657578\pi$
$797$	−26.8237	−0.950142	−0.475071	+	0.879947i	$0.657578\pi$
$798$	0	0
$799$	−4.95089	−0.175150
$800$	0	0
$801$	63.8275	2.25523
$802$	0	0
$803$	29.9727	1.05771
$804$	0	0
$805$	7.98541	0.281449
$806$	0	0
$807$	40.8581	1.43827
$808$	0	0
$809$	38.2127	1.34349	0.671744	−	0.740783i	$-0.265546\pi$
$809$	38.2127	1.34349	0.671744	+	0.740783i	$0.265546\pi$
$810$	0	0
$811$	−31.1179	−1.09270	−0.546348	−	0.837558i	$-0.683982\pi$
$811$	−31.1179	−1.09270	−0.546348	+	0.837558i	$0.683982\pi$
$812$	0	0
$813$	62.6744	2.19809
$814$	0	0
$815$	12.5899	0.441003
$816$	0	0
$817$	35.0866	1.22753
$818$	0	0
$819$	−28.7284	−1.00385
$820$	0	0
$821$	46.2108	1.61277	0.806385	−	0.591391i	$-0.201421\pi$
$821$	46.2108	1.61277	0.806385	+	0.591391i	$0.201421\pi$
$822$	0	0
$823$	18.3427	0.639387	0.319693	−	0.947521i	$-0.396420\pi$
$823$	18.3427	0.639387	0.319693	+	0.947521i	$0.396420\pi$
$824$	0	0
$825$	13.1299	0.457124
$826$	0	0
$827$	31.5819	1.09821	0.549106	−	0.835753i	$-0.314969\pi$
$827$	31.5819	1.09821	0.549106	+	0.835753i	$0.314969\pi$
$828$	0	0
$829$	9.60273	0.333517	0.166758	−	0.985998i	$-0.446670\pi$
$829$	9.60273	0.333517	0.166758	+	0.985998i	$0.446670\pi$
$830$	0	0
$831$	17.6907	0.613684
$832$	0	0
$833$	4.14683	0.143679
$834$	0	0
$835$	5.09082	0.176175
$836$	0	0
$837$	31.9270	1.10356
$838$	0	0
$839$	−11.4527	−0.395390	−0.197695	−	0.980264i	$-0.563346\pi$
$839$	−11.4527	−0.395390	−0.197695	+	0.980264i	$0.563346\pi$
$840$	0	0
$841$	−28.8108	−0.993476
$842$	0	0
$843$	−28.4441	−0.979666
$844$	0	0
$845$	−0.451703	−0.0155391
$846$	0	0
$847$	21.9311	0.753563
$848$	0	0
$849$	−19.9923	−0.686133
$850$	0	0
$851$	−13.8168	−0.473633
$852$	0	0
$853$	−37.3547	−1.27900	−0.639500	−	0.768791i	$-0.720859\pi$
$853$	−37.3547	−1.27900	−0.639500	+	0.768791i	$0.720859\pi$
$854$	0	0
$855$	25.1617	0.860512
$856$	0	0
$857$	−41.0380	−1.40183	−0.700917	−	0.713243i	$-0.747225\pi$
$857$	−41.0380	−1.40183	−0.700917	+	0.713243i	$0.747225\pi$
$858$	0	0
$859$	−24.7367	−0.844004	−0.422002	−	0.906595i	$-0.638673\pi$
$859$	−24.7367	−0.844004	−0.422002	+	0.906595i	$0.638673\pi$
$860$	0	0
$861$	8.82793	0.300855
$862$	0	0
$863$	26.4969	0.901965	0.450983	−	0.892533i	$-0.351074\pi$
$863$	26.4969	0.901965	0.450983	+	0.892533i	$0.351074\pi$
$864$	0	0
$865$	−10.8805	−0.369947
$866$	0	0
$867$	43.0360	1.46158
$868$	0	0
$869$	−29.6779	−1.00675
$870$	0	0
$871$	4.29753	0.145616
$872$	0	0
$873$	−82.0617	−2.77737
$874$	0	0
$875$	1.81499	0.0613580
$876$	0	0
$877$	22.9337	0.774417	0.387209	−	0.921992i	$-0.373439\pi$
$877$	22.9337	0.774417	0.387209	+	0.921992i	$0.373439\pi$
$878$	0	0
$879$	64.7152	2.18279
$880$	0	0
$881$	54.9962	1.85287	0.926434	−	0.376457i	$-0.122858\pi$
$881$	54.9962	1.85287	0.926434	+	0.376457i	$0.122858\pi$
$882$	0	0
$883$	−26.2927	−0.884819	−0.442410	−	0.896813i	$-0.645876\pi$
$883$	−26.2927	−0.884819	−0.442410	+	0.896813i	$0.645876\pi$
$884$	0	0
$885$	−33.1935	−1.11579
$886$	0	0
$887$	57.9561	1.94598	0.972988	−	0.230855i	$-0.0741524\pi$
$887$	57.9561	1.94598	0.972988	+	0.230855i	$0.0741524\pi$
$888$	0	0
$889$	−22.2459	−0.746103
$890$	0	0
$891$	11.7186	0.392587
$892$	0	0
$893$	24.9140	0.833715
$894$	0	0
$895$	8.73037	0.291824
$896$	0	0
$897$	42.5918	1.42210
$898$	0	0
$899$	3.46085	0.115426
$900$	0	0
$901$	−4.51628	−0.150459
$902$	0	0
$903$	−30.9053	−1.02846
$904$	0	0
$905$	14.8136	0.492420
$906$	0	0
$907$	−52.8946	−1.75634	−0.878168	−	0.478352i	$-0.841234\pi$
$907$	−52.8946	−1.75634	−0.878168	+	0.478352i	$0.841234\pi$
$908$	0	0
$909$	−61.8792	−2.05240
$910$	0	0
$911$	34.0884	1.12940	0.564700	−	0.825296i	$-0.308992\pi$
$911$	34.0884	1.12940	0.564700	+	0.825296i	$0.308992\pi$
$912$	0	0
$913$	51.1173	1.69173
$914$	0	0
$915$	−30.7747	−1.01738
$916$	0	0
$917$	20.0537	0.662232
$918$	0	0
$919$	24.9788	0.823975	0.411988	−	0.911189i	$-0.364835\pi$
$919$	24.9788	0.823975	0.411988	+	0.911189i	$0.364835\pi$
$920$	0	0
$921$	−36.5797	−1.20534
$922$	0	0
$923$	44.9363	1.47910
$924$	0	0
$925$	−3.14040	−0.103256
$926$	0	0
$927$	−76.9520	−2.52743
$928$	0	0
$929$	51.0525	1.67498	0.837489	−	0.546454i	$-0.184023\pi$
$929$	51.0525	1.67498	0.837489	+	0.546454i	$0.184023\pi$
$930$	0	0
$931$	−20.8678	−0.683915
$932$	0	0
$933$	−66.2404	−2.16861
$934$	0	0
$935$	5.37631	0.175824
$936$	0	0
$937$	−25.9652	−0.848248	−0.424124	−	0.905604i	$-0.639418\pi$
$937$	−25.9652	−0.848248	−0.424124	+	0.905604i	$0.639418\pi$
$938$	0	0
$939$	81.0459	2.64483
$940$	0	0
$941$	−30.0342	−0.979088	−0.489544	−	0.871979i	$-0.662837\pi$
$941$	−30.0342	−0.979088	−0.489544	+	0.871979i	$0.662837\pi$
$942$	0	0
$943$	−7.83059	−0.254999
$944$	0	0
$945$	−7.28295	−0.236914
$946$	0	0
$947$	21.7181	0.705743	0.352871	−	0.935672i	$-0.385205\pi$
$947$	21.7181	0.705743	0.352871	+	0.935672i	$0.385205\pi$
$948$	0	0
$949$	22.0988	0.717358
$950$	0	0
$951$	23.4761	0.761266
$952$	0	0
$953$	−10.3334	−0.334730	−0.167365	−	0.985895i	$-0.553526\pi$
$953$	−10.3334	−0.334730	−0.167365	+	0.985895i	$0.553526\pi$
$954$	0	0
$955$	−1.15519	−0.0373810
$956$	0	0
$957$	−5.71107	−0.184613
$958$	0	0
$959$	−6.04836	−0.195312
$960$	0	0
$961$	32.3071	1.04216
$962$	0	0
$963$	56.7749	1.82955
$964$	0	0
$965$	3.28966	0.105898
$966$	0	0
$967$	59.6816	1.91923	0.959616	−	0.281314i	$-0.0907703\pi$
$967$	59.6816	1.91923	0.959616	+	0.281314i	$0.0907703\pi$
$968$	0	0
$969$	17.2204	0.553198
$970$	0	0
$971$	−52.4154	−1.68209	−0.841045	−	0.540965i	$-0.818059\pi$
$971$	−52.4154	−1.68209	−0.841045	+	0.540965i	$0.818059\pi$
$972$	0	0
$973$	27.1706	0.871050
$974$	0	0
$975$	9.68064	0.310028
$976$	0	0
$977$	−7.01118	−0.224308	−0.112154	−	0.993691i	$-0.535775\pi$
$977$	−7.01118	−0.224308	−0.112154	+	0.993691i	$0.535775\pi$
$978$	0	0
$979$	−68.6297	−2.19342
$980$	0	0
$981$	15.4878	0.494487
$982$	0	0
$983$	−21.4903	−0.685433	−0.342717	−	0.939439i	$-0.611347\pi$
$983$	−21.4903	−0.685433	−0.342717	+	0.939439i	$0.611347\pi$
$984$	0	0
$985$	−12.6195	−0.402089
$986$	0	0
$987$	−21.9449	−0.698515
$988$	0	0
$989$	27.4137	0.871706
$990$	0	0
$991$	49.9465	1.58660	0.793301	−	0.608830i	$-0.208361\pi$
$991$	49.9465	1.58660	0.793301	+	0.608830i	$0.208361\pi$
$992$	0	0
$993$	−70.5171	−2.23779
$994$	0	0
$995$	3.91354	0.124068
$996$	0	0
$997$	−27.9619	−0.885563	−0.442782	−	0.896630i	$-0.646008\pi$
$997$	−27.9619	−0.885563	−0.442782	+	0.896630i	$0.646008\pi$
$998$	0	0
$999$	12.6013	0.398689

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8020.2.a.d.1.3	✓	29

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8020.2.a.d.1.3	✓	29	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)

\(f(q)\)	\(=\)	\(q-2.73282 q^{3} +1.00000 q^{5} +1.81499 q^{7} +4.46832 q^{9} +O(q^{10})\)	\(q-2.73282 q^{3} +1.00000 q^{5} +1.81499 q^{7} +4.46832 q^{9} -4.80451 q^{11} -3.54236 q^{13} -2.73282 q^{15} -1.11901 q^{17} +5.63113 q^{19} -4.96006 q^{21} +4.39969 q^{23} +1.00000 q^{25} -4.01266 q^{27} -0.434967 q^{29} -7.95658 q^{31} +13.1299 q^{33} +1.81499 q^{35} -3.14040 q^{37} +9.68064 q^{39} -1.77980 q^{41} +6.23083 q^{43} +4.46832 q^{45} +4.42433 q^{47} -3.70580 q^{49} +3.05806 q^{51} +4.03595 q^{53} -4.80451 q^{55} -15.3889 q^{57} +12.1462 q^{59} +11.2611 q^{61} +8.10998 q^{63} -3.54236 q^{65} -1.21318 q^{67} -12.0236 q^{69} -12.6854 q^{71} -6.23845 q^{73} -2.73282 q^{75} -8.72016 q^{77} +6.17709 q^{79} -2.43908 q^{81} -10.6394 q^{83} -1.11901 q^{85} +1.18869 q^{87} +14.2844 q^{89} -6.42936 q^{91} +21.7439 q^{93} +5.63113 q^{95} -18.3652 q^{97} -21.4681 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 29 q - 3 q^{3} + 29 q^{5} - 8 q^{7} + 10 q^{9}+O(q^{10}) \) 29 * q - 3 * q^3 + 29 * q^5 - 8 * q^7 + 10 * q^9	\( 29 q - 3 q^{3} + 29 q^{5} - 8 q^{7} + 10 q^{9} + 2 q^{11} - 23 q^{13} - 3 q^{15} - 30 q^{17} - 6 q^{19} - 16 q^{21} - 21 q^{23} + 29 q^{25} - 15 q^{27} - 35 q^{29} - 7 q^{31} - 36 q^{33} - 8 q^{35} - 31 q^{37} - 11 q^{39} - 24 q^{41} - 17 q^{43} + 10 q^{45} - 17 q^{47} + q^{49} + 8 q^{51} - 57 q^{53} + 2 q^{55} - 46 q^{57} - 9 q^{59} - 27 q^{61} - 34 q^{63} - 23 q^{65} - 21 q^{67} - 28 q^{69} - 19 q^{71} - 81 q^{73} - 3 q^{75} - 66 q^{77} - 17 q^{79} - 39 q^{81} - 30 q^{83} - 30 q^{85} - 20 q^{87} - 38 q^{89} + q^{91} - 75 q^{93} - 6 q^{95} - 48 q^{97} - 7 q^{99}+O(q^{100}) \) 29 * q - 3 * q^3 + 29 * q^5 - 8 * q^7 + 10 * q^9 + 2 * q^11 - 23 * q^13 - 3 * q^15 - 30 * q^17 - 6 * q^19 - 16 * q^21 - 21 * q^23 + 29 * q^25 - 15 * q^27 - 35 * q^29 - 7 * q^31 - 36 * q^33 - 8 * q^35 - 31 * q^37 - 11 * q^39 - 24 * q^41 - 17 * q^43 + 10 * q^45 - 17 * q^47 + q^49 + 8 * q^51 - 57 * q^53 + 2 * q^55 - 46 * q^57 - 9 * q^59 - 27 * q^61 - 34 * q^63 - 23 * q^65 - 21 * q^67 - 28 * q^69 - 19 * q^71 - 81 * q^73 - 3 * q^75 - 66 * q^77 - 17 * q^79 - 39 * q^81 - 30 * q^83 - 30 * q^85 - 20 * q^87 - 38 * q^89 + q^91 - 75 * q^93 - 6 * q^95 - 48 * q^97 - 7 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

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Groups

Database

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