LMFDB - Embedded newform 8020.2.a.c.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8020,2,Mod(1,8020)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8020, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8020.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8020 = 2^{2} \cdot 5 \cdot 401 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8020.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$64.0400224211$
Analytic rank:	$1$
Dimension:	$28$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Character	$\chi$	$=$	8020.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.58112 q^{3} -1.00000 q^{5} +3.33363 q^{7} -0.500055 q^{9} +O(q^{10})$	$q-1.58112 q^{3} -1.00000 q^{5} +3.33363 q^{7} -0.500055 q^{9} -4.48018 q^{11} +5.14806 q^{13} +1.58112 q^{15} -2.08313 q^{17} -1.21732 q^{19} -5.27087 q^{21} -6.31215 q^{23} +1.00000 q^{25} +5.53401 q^{27} -2.70237 q^{29} +9.30315 q^{31} +7.08371 q^{33} -3.33363 q^{35} -2.68867 q^{37} -8.13971 q^{39} +10.6715 q^{41} +9.65978 q^{43} +0.500055 q^{45} -12.9089 q^{47} +4.11307 q^{49} +3.29369 q^{51} -1.27197 q^{53} +4.48018 q^{55} +1.92473 q^{57} +6.58994 q^{59} -13.6691 q^{61} -1.66700 q^{63} -5.14806 q^{65} -3.07957 q^{67} +9.98027 q^{69} +6.24891 q^{71} -5.77379 q^{73} -1.58112 q^{75} -14.9353 q^{77} -4.70756 q^{79} -7.24978 q^{81} +8.49193 q^{83} +2.08313 q^{85} +4.27278 q^{87} +4.55626 q^{89} +17.1617 q^{91} -14.7094 q^{93} +1.21732 q^{95} -14.3702 q^{97} +2.24034 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 3 q^{3} - 28 q^{5} - 4 q^{7} + 17 q^{9}+O(q^{10}) $ 28 * q + 3 * q^3 - 28 * q^5 - 4 * q^7 + 17 * q^9	$ 28 q + 3 q^{3} - 28 q^{5} - 4 q^{7} + 17 q^{9} + 2 q^{11} + 3 q^{13} - 3 q^{15} - 10 q^{17} - 2 q^{19} - 12 q^{21} - 23 q^{23} + 28 q^{25} + 9 q^{27} - 37 q^{29} - 11 q^{31} + 2 q^{33} + 4 q^{35} - 3 q^{37} - 19 q^{39} - 30 q^{41} + 13 q^{43} - 17 q^{45} - 15 q^{47} + 12 q^{49} - 8 q^{51} - 35 q^{53} - 2 q^{55} - 22 q^{57} - q^{59} - 33 q^{61} - 20 q^{63} - 3 q^{65} + 19 q^{67} - 8 q^{69} - 31 q^{71} + 31 q^{73} + 3 q^{75} - 42 q^{77} - 29 q^{79} - 36 q^{81} + 14 q^{83} + 10 q^{85} - 32 q^{87} - 32 q^{89} - 7 q^{91} - 11 q^{93} + 2 q^{95} + 2 q^{97} - 39 q^{99}+O(q^{100}) $ 28 * q + 3 * q^3 - 28 * q^5 - 4 * q^7 + 17 * q^9 + 2 * q^11 + 3 * q^13 - 3 * q^15 - 10 * q^17 - 2 * q^19 - 12 * q^21 - 23 * q^23 + 28 * q^25 + 9 * q^27 - 37 * q^29 - 11 * q^31 + 2 * q^33 + 4 * q^35 - 3 * q^37 - 19 * q^39 - 30 * q^41 + 13 * q^43 - 17 * q^45 - 15 * q^47 + 12 * q^49 - 8 * q^51 - 35 * q^53 - 2 * q^55 - 22 * q^57 - q^59 - 33 * q^61 - 20 * q^63 - 3 * q^65 + 19 * q^67 - 8 * q^69 - 31 * q^71 + 31 * q^73 + 3 * q^75 - 42 * q^77 - 29 * q^79 - 36 * q^81 + 14 * q^83 + 10 * q^85 - 32 * q^87 - 32 * q^89 - 7 * q^91 - 11 * q^93 + 2 * q^95 + 2 * q^97 - 39 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.58112	−0.912861	−0.456430	−	0.889759i	$-0.650872\pi$
$3$	−1.58112	−0.912861	−0.456430	+	0.889759i	$0.650872\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	3.33363	1.25999	0.629996	−	0.776598i	$-0.283056\pi$
$7$	3.33363	1.25999	0.629996	+	0.776598i	$0.283056\pi$
$8$	0	0
$9$	−0.500055	−0.166685
$10$	0	0
$11$	−4.48018	−1.35083	−0.675413	−	0.737440i	$-0.736035\pi$
$11$	−4.48018	−1.35083	−0.675413	+	0.737440i	$0.736035\pi$
$12$	0	0
$13$	5.14806	1.42782	0.713908	−	0.700240i	$-0.246923\pi$
$13$	5.14806	1.42782	0.713908	+	0.700240i	$0.246923\pi$
$14$	0	0
$15$	1.58112	0.408244
$16$	0	0
$17$	−2.08313	−0.505234	−0.252617	−	0.967566i	$-0.581291\pi$
$17$	−2.08313	−0.505234	−0.252617	+	0.967566i	$0.581291\pi$
$18$	0	0
$19$	−1.21732	−0.279272	−0.139636	−	0.990203i	$-0.544593\pi$
$19$	−1.21732	−0.279272	−0.139636	+	0.990203i	$0.544593\pi$
$20$	0	0
$21$	−5.27087	−1.15020
$22$	0	0
$23$	−6.31215	−1.31617	−0.658087	−	0.752942i	$-0.728634\pi$
$23$	−6.31215	−1.31617	−0.658087	+	0.752942i	$0.728634\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.53401	1.06502
$28$	0	0
$29$	−2.70237	−0.501818	−0.250909	−	0.968011i	$-0.580730\pi$
$29$	−2.70237	−0.501818	−0.250909	+	0.968011i	$0.580730\pi$
$30$	0	0
$31$	9.30315	1.67089	0.835447	−	0.549570i	$-0.185209\pi$
$31$	9.30315	1.67089	0.835447	+	0.549570i	$0.185209\pi$
$32$	0	0
$33$	7.08371	1.23312
$34$	0	0
$35$	−3.33363	−0.563486
$36$	0	0
$37$	−2.68867	−0.442015	−0.221008	−	0.975272i	$-0.570935\pi$
$37$	−2.68867	−0.442015	−0.221008	+	0.975272i	$0.570935\pi$
$38$	0	0
$39$	−8.13971	−1.30340
$40$	0	0
$41$	10.6715	1.66661	0.833303	−	0.552817i	$-0.186447\pi$
$41$	10.6715	1.66661	0.833303	+	0.552817i	$0.186447\pi$
$42$	0	0
$43$	9.65978	1.47310	0.736552	−	0.676381i	$-0.236453\pi$
$43$	9.65978	1.47310	0.736552	+	0.676381i	$0.236453\pi$
$44$	0	0
$45$	0.500055	0.0745439
$46$	0	0
$47$	−12.9089	−1.88295	−0.941476	−	0.337080i	$-0.890561\pi$
$47$	−12.9089	−1.88295	−0.941476	+	0.337080i	$0.890561\pi$
$48$	0	0
$49$	4.11307	0.587581
$50$	0	0
$51$	3.29369	0.461209
$52$	0	0
$53$	−1.27197	−0.174718	−0.0873591	−	0.996177i	$-0.527843\pi$
$53$	−1.27197	−0.174718	−0.0873591	+	0.996177i	$0.527843\pi$
$54$	0	0
$55$	4.48018	0.604108
$56$	0	0
$57$	1.92473	0.254936
$58$	0	0
$59$	6.58994	0.857937	0.428969	−	0.903319i	$-0.358877\pi$
$59$	6.58994	0.857937	0.428969	+	0.903319i	$0.358877\pi$
$60$	0	0
$61$	−13.6691	−1.75015	−0.875074	−	0.483989i	$-0.839188\pi$
$61$	−13.6691	−1.75015	−0.875074	+	0.483989i	$0.839188\pi$
$62$	0	0
$63$	−1.66700	−0.210022
$64$	0	0
$65$	−5.14806	−0.638539
$66$	0	0
$67$	−3.07957	−0.376229	−0.188115	−	0.982147i	$-0.560238\pi$
$67$	−3.07957	−0.376229	−0.188115	+	0.982147i	$0.560238\pi$
$68$	0	0
$69$	9.98027	1.20148
$70$	0	0
$71$	6.24891	0.741609	0.370804	−	0.928711i	$-0.379082\pi$
$71$	6.24891	0.741609	0.370804	+	0.928711i	$0.379082\pi$
$72$	0	0
$73$	−5.77379	−0.675771	−0.337886	−	0.941187i	$-0.609712\pi$
$73$	−5.77379	−0.675771	−0.337886	+	0.941187i	$0.609712\pi$
$74$	0	0
$75$	−1.58112	−0.182572
$76$	0	0
$77$	−14.9353	−1.70203
$78$	0	0
$79$	−4.70756	−0.529642	−0.264821	−	0.964298i	$-0.585313\pi$
$79$	−4.70756	−0.529642	−0.264821	+	0.964298i	$0.585313\pi$
$80$	0	0
$81$	−7.24978	−0.805531
$82$	0	0
$83$	8.49193	0.932111	0.466055	−	0.884756i	$-0.345675\pi$
$83$	8.49193	0.932111	0.466055	+	0.884756i	$0.345675\pi$
$84$	0	0
$85$	2.08313	0.225948
$86$	0	0
$87$	4.27278	0.458090
$88$	0	0
$89$	4.55626	0.482962	0.241481	−	0.970406i	$-0.422367\pi$
$89$	4.55626	0.482962	0.241481	+	0.970406i	$0.422367\pi$
$90$	0	0
$91$	17.1617	1.79904
$92$	0	0
$93$	−14.7094	−1.52529
$94$	0	0
$95$	1.21732	0.124894
$96$	0	0
$97$	−14.3702	−1.45907	−0.729534	−	0.683944i	$-0.760263\pi$
$97$	−14.3702	−1.45907	−0.729534	+	0.683944i	$0.760263\pi$
$98$	0	0
$99$	2.24034	0.225163
$100$	0	0
$101$	−12.7935	−1.27300	−0.636498	−	0.771278i	$-0.719618\pi$
$101$	−12.7935	−1.27300	−0.636498	+	0.771278i	$0.719618\pi$
$102$	0	0
$103$	−15.8202	−1.55881	−0.779406	−	0.626519i	$-0.784479\pi$
$103$	−15.8202	−1.55881	−0.779406	+	0.626519i	$0.784479\pi$
$104$	0	0
$105$	5.27087	0.514384
$106$	0	0
$107$	−5.45121	−0.526988	−0.263494	−	0.964661i	$-0.584875\pi$
$107$	−5.45121	−0.526988	−0.263494	+	0.964661i	$0.584875\pi$
$108$	0	0
$109$	16.6376	1.59359	0.796795	−	0.604250i	$-0.206527\pi$
$109$	16.6376	1.59359	0.796795	+	0.604250i	$0.206527\pi$
$110$	0	0
$111$	4.25112	0.403498
$112$	0	0
$113$	−3.45935	−0.325429	−0.162714	−	0.986673i	$-0.552025\pi$
$113$	−3.45935	−0.325429	−0.162714	+	0.986673i	$0.552025\pi$
$114$	0	0
$115$	6.31215	0.588611
$116$	0	0
$117$	−2.57432	−0.237996
$118$	0	0
$119$	−6.94439	−0.636591
$120$	0	0
$121$	9.07204	0.824731
$122$	0	0
$123$	−16.8729	−1.52138
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	17.0434	1.51235	0.756177	−	0.654367i	$-0.227065\pi$
$127$	17.0434	1.51235	0.756177	+	0.654367i	$0.227065\pi$
$128$	0	0
$129$	−15.2733	−1.34474
$130$	0	0
$131$	−6.81219	−0.595184	−0.297592	−	0.954693i	$-0.596183\pi$
$131$	−6.81219	−0.595184	−0.297592	+	0.954693i	$0.596183\pi$
$132$	0	0
$133$	−4.05808	−0.351880
$134$	0	0
$135$	−5.53401	−0.476292
$136$	0	0
$137$	7.62695	0.651614	0.325807	−	0.945436i	$-0.394364\pi$
$137$	7.62695	0.651614	0.325807	+	0.945436i	$0.394364\pi$
$138$	0	0
$139$	19.2383	1.63177	0.815885	−	0.578214i	$-0.196250\pi$
$139$	19.2383	1.63177	0.815885	+	0.578214i	$0.196250\pi$
$140$	0	0
$141$	20.4105	1.71887
$142$	0	0
$143$	−23.0643	−1.92873
$144$	0	0
$145$	2.70237	0.224420
$146$	0	0
$147$	−6.50326	−0.536380
$148$	0	0
$149$	5.59856	0.458652	0.229326	−	0.973350i	$-0.426348\pi$
$149$	5.59856	0.458652	0.229326	+	0.973350i	$0.426348\pi$
$150$	0	0
$151$	−10.5085	−0.855173	−0.427586	−	0.903975i	$-0.640636\pi$
$151$	−10.5085	−0.855173	−0.427586	+	0.903975i	$0.640636\pi$
$152$	0	0
$153$	1.04168	0.0842150
$154$	0	0
$155$	−9.30315	−0.747247
$156$	0	0
$157$	11.5928	0.925208	0.462604	−	0.886565i	$-0.346915\pi$
$157$	11.5928	0.925208	0.462604	+	0.886565i	$0.346915\pi$
$158$	0	0
$159$	2.01113	0.159493
$160$	0	0
$161$	−21.0424	−1.65837
$162$	0	0
$163$	19.5693	1.53278	0.766392	−	0.642374i	$-0.222050\pi$
$163$	19.5693	1.53278	0.766392	+	0.642374i	$0.222050\pi$
$164$	0	0
$165$	−7.08371	−0.551466
$166$	0	0
$167$	8.84859	0.684724	0.342362	−	0.939568i	$-0.388773\pi$
$167$	8.84859	0.684724	0.342362	+	0.939568i	$0.388773\pi$
$168$	0	0
$169$	13.5026	1.03866
$170$	0	0
$171$	0.608726	0.0465504
$172$	0	0
$173$	6.19390	0.470914	0.235457	−	0.971885i	$-0.424341\pi$
$173$	6.19390	0.470914	0.235457	+	0.971885i	$0.424341\pi$
$174$	0	0
$175$	3.33363	0.251998
$176$	0	0
$177$	−10.4195	−0.783178
$178$	0	0
$179$	6.66383	0.498078	0.249039	−	0.968493i	$-0.419885\pi$
$179$	6.66383	0.498078	0.249039	+	0.968493i	$0.419885\pi$
$180$	0	0
$181$	−9.78563	−0.727360	−0.363680	−	0.931524i	$-0.618480\pi$
$181$	−9.78563	−0.727360	−0.363680	+	0.931524i	$0.618480\pi$
$182$	0	0
$183$	21.6125	1.59764
$184$	0	0
$185$	2.68867	0.197675
$186$	0	0
$187$	9.33282	0.682484
$188$	0	0
$189$	18.4483	1.34192
$190$	0	0
$191$	−26.1568	−1.89264	−0.946322	−	0.323227i	$-0.895232\pi$
$191$	−26.1568	−1.89264	−0.946322	+	0.323227i	$0.895232\pi$
$192$	0	0
$193$	−15.5525	−1.11949	−0.559745	−	0.828665i	$-0.689101\pi$
$193$	−15.5525	−1.11949	−0.559745	+	0.828665i	$0.689101\pi$
$194$	0	0
$195$	8.13971	0.582897
$196$	0	0
$197$	6.10796	0.435174	0.217587	−	0.976041i	$-0.430181\pi$
$197$	6.10796	0.435174	0.217587	+	0.976041i	$0.430181\pi$
$198$	0	0
$199$	−12.9613	−0.918806	−0.459403	−	0.888228i	$-0.651937\pi$
$199$	−12.9613	−0.918806	−0.459403	+	0.888228i	$0.651937\pi$
$200$	0	0
$201$	4.86917	0.343445
$202$	0	0
$203$	−9.00871	−0.632287
$204$	0	0
$205$	−10.6715	−0.745329
$206$	0	0
$207$	3.15642	0.219387
$208$	0	0
$209$	5.45380	0.377247
$210$	0	0
$211$	12.2788	0.845310	0.422655	−	0.906291i	$-0.361098\pi$
$211$	12.2788	0.845310	0.422655	+	0.906291i	$0.361098\pi$
$212$	0	0
$213$	−9.88028	−0.676986
$214$	0	0
$215$	−9.65978	−0.658792
$216$	0	0
$217$	31.0132	2.10531
$218$	0	0
$219$	9.12907	0.616885
$220$	0	0
$221$	−10.7241	−0.721382
$222$	0	0
$223$	−6.97483	−0.467069	−0.233534	−	0.972349i	$-0.575029\pi$
$223$	−6.97483	−0.467069	−0.233534	+	0.972349i	$0.575029\pi$
$224$	0	0
$225$	−0.500055	−0.0333370
$226$	0	0
$227$	−17.1680	−1.13948	−0.569739	−	0.821825i	$-0.692956\pi$
$227$	−17.1680	−1.13948	−0.569739	+	0.821825i	$0.692956\pi$
$228$	0	0
$229$	15.6548	1.03450	0.517250	−	0.855834i	$-0.326956\pi$
$229$	15.6548	1.03450	0.517250	+	0.855834i	$0.326956\pi$
$230$	0	0
$231$	23.6145	1.55372
$232$	0	0
$233$	11.4766	0.751859	0.375930	−	0.926648i	$-0.377323\pi$
$233$	11.4766	0.751859	0.375930	+	0.926648i	$0.377323\pi$
$234$	0	0
$235$	12.9089	0.842082
$236$	0	0
$237$	7.44322	0.483489
$238$	0	0
$239$	−20.2157	−1.30764	−0.653821	−	0.756649i	$-0.726835\pi$
$239$	−20.2157	−1.30764	−0.653821	+	0.756649i	$0.726835\pi$
$240$	0	0
$241$	−15.2223	−0.980557	−0.490279	−	0.871566i	$-0.663105\pi$
$241$	−15.2223	−0.980557	−0.490279	+	0.871566i	$0.663105\pi$
$242$	0	0
$243$	−5.13926	−0.329684
$244$	0	0
$245$	−4.11307	−0.262774
$246$	0	0
$247$	−6.26682	−0.398749
$248$	0	0
$249$	−13.4268	−0.850887
$250$	0	0
$251$	−13.0250	−0.822130	−0.411065	−	0.911606i	$-0.634843\pi$
$251$	−13.0250	−0.822130	−0.411065	+	0.911606i	$0.634843\pi$
$252$	0	0
$253$	28.2796	1.77792
$254$	0	0
$255$	−3.29369	−0.206259
$256$	0	0
$257$	15.4276	0.962348	0.481174	−	0.876625i	$-0.340211\pi$
$257$	15.4276	0.962348	0.481174	+	0.876625i	$0.340211\pi$
$258$	0	0
$259$	−8.96304	−0.556936
$260$	0	0
$261$	1.35134	0.0836457
$262$	0	0
$263$	−10.0190	−0.617797	−0.308898	−	0.951095i	$-0.599960\pi$
$263$	−10.0190	−0.617797	−0.308898	+	0.951095i	$0.599960\pi$
$264$	0	0
$265$	1.27197	0.0781363
$266$	0	0
$267$	−7.20399	−0.440877
$268$	0	0
$269$	6.18894	0.377346	0.188673	−	0.982040i	$-0.439581\pi$
$269$	6.18894	0.377346	0.188673	+	0.982040i	$0.439581\pi$
$270$	0	0
$271$	5.19686	0.315687	0.157844	−	0.987464i	$-0.449546\pi$
$271$	5.19686	0.315687	0.157844	+	0.987464i	$0.449546\pi$
$272$	0	0
$273$	−27.1348	−1.64227
$274$	0	0
$275$	−4.48018	−0.270165
$276$	0	0
$277$	−16.1319	−0.969271	−0.484636	−	0.874716i	$-0.661048\pi$
$277$	−16.1319	−0.969271	−0.484636	+	0.874716i	$0.661048\pi$
$278$	0	0
$279$	−4.65209	−0.278513
$280$	0	0
$281$	−13.4448	−0.802052	−0.401026	−	0.916067i	$-0.631346\pi$
$281$	−13.4448	−0.802052	−0.401026	+	0.916067i	$0.631346\pi$
$282$	0	0
$283$	−6.22533	−0.370058	−0.185029	−	0.982733i	$-0.559238\pi$
$283$	−6.22533	−0.370058	−0.185029	+	0.982733i	$0.559238\pi$
$284$	0	0
$285$	−1.92473	−0.114011
$286$	0	0
$287$	35.5747	2.09991
$288$	0	0
$289$	−12.6606	−0.744738
$290$	0	0
$291$	22.7210	1.33193
$292$	0	0
$293$	−30.7314	−1.79535	−0.897673	−	0.440662i	$-0.854744\pi$
$293$	−30.7314	−1.79535	−0.897673	+	0.440662i	$0.854744\pi$
$294$	0	0
$295$	−6.58994	−0.383681
$296$	0	0
$297$	−24.7934	−1.43866
$298$	0	0
$299$	−32.4953	−1.87925
$300$	0	0
$301$	32.2021	1.85610
$302$	0	0
$303$	20.2280	1.16207
$304$	0	0
$305$	13.6691	0.782690
$306$	0	0
$307$	19.2133	1.09656	0.548281	−	0.836294i	$-0.315282\pi$
$307$	19.2133	1.09656	0.548281	+	0.836294i	$0.315282\pi$
$308$	0	0
$309$	25.0137	1.42298
$310$	0	0
$311$	−24.7589	−1.40395	−0.701975	−	0.712202i	$-0.747698\pi$
$311$	−24.7589	−1.40395	−0.701975	+	0.712202i	$0.747698\pi$
$312$	0	0
$313$	−13.3884	−0.756757	−0.378378	−	0.925651i	$-0.623518\pi$
$313$	−13.3884	−0.756757	−0.378378	+	0.925651i	$0.623518\pi$
$314$	0	0
$315$	1.66700	0.0939247
$316$	0	0
$317$	−3.06024	−0.171880	−0.0859401	−	0.996300i	$-0.527389\pi$
$317$	−3.06024	−0.171880	−0.0859401	+	0.996300i	$0.527389\pi$
$318$	0	0
$319$	12.1071	0.677869
$320$	0	0
$321$	8.61902	0.481067
$322$	0	0
$323$	2.53583	0.141098
$324$	0	0
$325$	5.14806	0.285563
$326$	0	0
$327$	−26.3060	−1.45473
$328$	0	0
$329$	−43.0333	−2.37251
$330$	0	0
$331$	−5.46302	−0.300275	−0.150137	−	0.988665i	$-0.547972\pi$
$331$	−5.46302	−0.300275	−0.150137	+	0.988665i	$0.547972\pi$
$332$	0	0
$333$	1.34449	0.0736774
$334$	0	0
$335$	3.07957	0.168255
$336$	0	0
$337$	11.2431	0.612450	0.306225	−	0.951959i	$-0.400934\pi$
$337$	11.2431	0.612450	0.306225	+	0.951959i	$0.400934\pi$
$338$	0	0
$339$	5.46966	0.297071
$340$	0	0
$341$	−41.6798	−2.25709
$342$	0	0
$343$	−9.62396	−0.519645
$344$	0	0
$345$	−9.98027	−0.537320
$346$	0	0
$347$	7.80699	0.419101	0.209551	−	0.977798i	$-0.432800\pi$
$347$	7.80699	0.419101	0.209551	+	0.977798i	$0.432800\pi$
$348$	0	0
$349$	−18.6022	−0.995752	−0.497876	−	0.867248i	$-0.665886\pi$
$349$	−18.6022	−0.995752	−0.497876	+	0.867248i	$0.665886\pi$
$350$	0	0
$351$	28.4894	1.52065
$352$	0	0
$353$	−30.5010	−1.62340	−0.811702	−	0.584072i	$-0.801458\pi$
$353$	−30.5010	−1.62340	−0.811702	+	0.584072i	$0.801458\pi$
$354$	0	0
$355$	−6.24891	−0.331658
$356$	0	0
$357$	10.9799	0.581119
$358$	0	0
$359$	−1.03237	−0.0544865	−0.0272432	−	0.999629i	$-0.508673\pi$
$359$	−1.03237	−0.0544865	−0.0272432	+	0.999629i	$0.508673\pi$
$360$	0	0
$361$	−17.5181	−0.922007
$362$	0	0
$363$	−14.3440	−0.752865
$364$	0	0
$365$	5.77379	0.302214
$366$	0	0
$367$	−3.60726	−0.188298	−0.0941488	−	0.995558i	$-0.530013\pi$
$367$	−3.60726	−0.188298	−0.0941488	+	0.995558i	$0.530013\pi$
$368$	0	0
$369$	−5.33633	−0.277798
$370$	0	0
$371$	−4.24026	−0.220144
$372$	0	0
$373$	−26.8687	−1.39121	−0.695604	−	0.718425i	$-0.744863\pi$
$373$	−26.8687	−1.39121	−0.695604	+	0.718425i	$0.744863\pi$
$374$	0	0
$375$	1.58112	0.0816488
$376$	0	0
$377$	−13.9120	−0.716504
$378$	0	0
$379$	15.3718	0.789597	0.394799	−	0.918768i	$-0.370814\pi$
$379$	15.3718	0.789597	0.394799	+	0.918768i	$0.370814\pi$
$380$	0	0
$381$	−26.9476	−1.38057
$382$	0	0
$383$	2.18488	0.111642	0.0558210	−	0.998441i	$-0.482222\pi$
$383$	2.18488	0.111642	0.0558210	+	0.998441i	$0.482222\pi$
$384$	0	0
$385$	14.9353	0.761171
$386$	0	0
$387$	−4.83043	−0.245544
$388$	0	0
$389$	−18.6007	−0.943092	−0.471546	−	0.881841i	$-0.656304\pi$
$389$	−18.6007	−0.943092	−0.471546	+	0.881841i	$0.656304\pi$
$390$	0	0
$391$	13.1491	0.664976
$392$	0	0
$393$	10.7709	0.543320
$394$	0	0
$395$	4.70756	0.236863
$396$	0	0
$397$	−16.0609	−0.806076	−0.403038	−	0.915183i	$-0.632046\pi$
$397$	−16.0609	−0.806076	−0.403038	+	0.915183i	$0.632046\pi$
$398$	0	0
$399$	6.41632	0.321218
$400$	0	0
$401$	1.00000	0.0499376
$401$	1.00000	0.0499376
$402$	0	0
$403$	47.8932	2.38573
$404$	0	0
$405$	7.24978	0.360244
$406$	0	0
$407$	12.0458	0.597086
$408$	0	0
$409$	−7.21244	−0.356632	−0.178316	−	0.983973i	$-0.557065\pi$
$409$	−7.21244	−0.356632	−0.178316	+	0.983973i	$0.557065\pi$
$410$	0	0
$411$	−12.0591	−0.594833
$412$	0	0
$413$	21.9684	1.08099
$414$	0	0
$415$	−8.49193	−0.416853
$416$	0	0
$417$	−30.4181	−1.48958
$418$	0	0
$419$	−31.3528	−1.53168	−0.765841	−	0.643030i	$-0.777677\pi$
$419$	−31.3528	−1.53168	−0.765841	+	0.643030i	$0.777677\pi$
$420$	0	0
$421$	−26.4448	−1.28884	−0.644420	−	0.764671i	$-0.722901\pi$
$421$	−26.4448	−1.28884	−0.644420	+	0.764671i	$0.722901\pi$
$422$	0	0
$423$	6.45515	0.313860
$424$	0	0
$425$	−2.08313	−0.101047
$426$	0	0
$427$	−45.5677	−2.20517
$428$	0	0
$429$	36.4674	1.76066
$430$	0	0
$431$	3.52695	0.169887	0.0849436	−	0.996386i	$-0.472929\pi$
$431$	3.52695	0.169887	0.0849436	+	0.996386i	$0.472929\pi$
$432$	0	0
$433$	−35.8244	−1.72161	−0.860805	−	0.508934i	$-0.830040\pi$
$433$	−35.8244	−1.72161	−0.860805	+	0.508934i	$0.830040\pi$
$434$	0	0
$435$	−4.27278	−0.204864
$436$	0	0
$437$	7.68389	0.367570
$438$	0	0
$439$	31.6303	1.50963	0.754816	−	0.655937i	$-0.227726\pi$
$439$	31.6303	1.50963	0.754816	+	0.655937i	$0.227726\pi$
$440$	0	0
$441$	−2.05676	−0.0979410
$442$	0	0
$443$	33.1780	1.57634	0.788168	−	0.615460i	$-0.211030\pi$
$443$	33.1780	1.57634	0.788168	+	0.615460i	$0.211030\pi$
$444$	0	0
$445$	−4.55626	−0.215987
$446$	0	0
$447$	−8.85201	−0.418686
$448$	0	0
$449$	16.8498	0.795190	0.397595	−	0.917561i	$-0.369845\pi$
$449$	16.8498	0.795190	0.397595	+	0.917561i	$0.369845\pi$
$450$	0	0
$451$	−47.8102	−2.25129
$452$	0	0
$453$	16.6153	0.780654
$454$	0	0
$455$	−17.1617	−0.804554
$456$	0	0
$457$	−3.51590	−0.164467	−0.0822334	−	0.996613i	$-0.526205\pi$
$457$	−3.51590	−0.164467	−0.0822334	+	0.996613i	$0.526205\pi$
$458$	0	0
$459$	−11.5281	−0.538085
$460$	0	0
$461$	−30.0753	−1.40074	−0.700372	−	0.713778i	$-0.746983\pi$
$461$	−30.0753	−1.40074	−0.700372	+	0.713778i	$0.746983\pi$
$462$	0	0
$463$	3.00741	0.139766	0.0698831	−	0.997555i	$-0.477737\pi$
$463$	3.00741	0.139766	0.0698831	+	0.997555i	$0.477737\pi$
$464$	0	0
$465$	14.7094	0.682132
$466$	0	0
$467$	−3.30093	−0.152749	−0.0763745	−	0.997079i	$-0.524334\pi$
$467$	−3.30093	−0.152749	−0.0763745	+	0.997079i	$0.524334\pi$
$468$	0	0
$469$	−10.2661	−0.474046
$470$	0	0
$471$	−18.3297	−0.844586
$472$	0	0
$473$	−43.2776	−1.98991
$474$	0	0
$475$	−1.21732	−0.0558543
$476$	0	0
$477$	0.636054	0.0291229
$478$	0	0
$479$	−5.91575	−0.270298	−0.135149	−	0.990825i	$-0.543151\pi$
$479$	−5.91575	−0.270298	−0.135149	+	0.990825i	$0.543151\pi$
$480$	0	0
$481$	−13.8415	−0.631117
$482$	0	0
$483$	33.2705	1.51386
$484$	0	0
$485$	14.3702	0.652515
$486$	0	0
$487$	−18.8555	−0.854423	−0.427212	−	0.904152i	$-0.640504\pi$
$487$	−18.8555	−0.854423	−0.427212	+	0.904152i	$0.640504\pi$
$488$	0	0
$489$	−30.9414	−1.39922
$490$	0	0
$491$	28.4685	1.28476	0.642382	−	0.766385i	$-0.277946\pi$
$491$	28.4685	1.28476	0.642382	+	0.766385i	$0.277946\pi$
$492$	0	0
$493$	5.62941	0.253536
$494$	0	0
$495$	−2.24034	−0.100696
$496$	0	0
$497$	20.8315	0.934421
$498$	0	0
$499$	−39.5742	−1.77159	−0.885793	−	0.464082i	$-0.846384\pi$
$499$	−39.5742	−1.77159	−0.885793	+	0.464082i	$0.846384\pi$
$500$	0	0
$501$	−13.9907	−0.625058
$502$	0	0
$503$	23.0644	1.02839	0.514195	−	0.857673i	$-0.328091\pi$
$503$	23.0644	1.02839	0.514195	+	0.857673i	$0.328091\pi$
$504$	0	0
$505$	12.7935	0.569301
$506$	0	0
$507$	−21.3492	−0.948151
$508$	0	0
$509$	36.0771	1.59909	0.799544	−	0.600607i	$-0.205075\pi$
$509$	36.0771	1.59909	0.799544	+	0.600607i	$0.205075\pi$
$510$	0	0
$511$	−19.2477	−0.851467
$512$	0	0
$513$	−6.73665	−0.297430
$514$	0	0
$515$	15.8202	0.697122
$516$	0	0
$517$	57.8341	2.54354
$518$	0	0
$519$	−9.79331	−0.429879
$520$	0	0
$521$	−14.7342	−0.645517	−0.322759	−	0.946481i	$-0.604610\pi$
$521$	−14.7342	−0.645517	−0.322759	+	0.946481i	$0.604610\pi$
$522$	0	0
$523$	13.3205	0.582464	0.291232	−	0.956652i	$-0.405935\pi$
$523$	13.3205	0.582464	0.291232	+	0.956652i	$0.405935\pi$
$524$	0	0
$525$	−5.27087	−0.230040
$526$	0	0
$527$	−19.3797	−0.844193
$528$	0	0
$529$	16.8432	0.732314
$530$	0	0
$531$	−3.29534	−0.143005
$532$	0	0
$533$	54.9375	2.37961
$534$	0	0
$535$	5.45121	0.235676
$536$	0	0
$537$	−10.5363	−0.454676
$538$	0	0
$539$	−18.4273	−0.793720
$540$	0	0
$541$	−19.4187	−0.834877	−0.417438	−	0.908705i	$-0.637072\pi$
$541$	−19.4187	−0.834877	−0.417438	+	0.908705i	$0.637072\pi$
$542$	0	0
$543$	15.4723	0.663979
$544$	0	0
$545$	−16.6376	−0.712675
$546$	0	0
$547$	0.537700	0.0229904	0.0114952	−	0.999934i	$-0.496341\pi$
$547$	0.537700	0.0229904	0.0114952	+	0.999934i	$0.496341\pi$
$548$	0	0
$549$	6.83531	0.291724
$550$	0	0
$551$	3.28965	0.140144
$552$	0	0
$553$	−15.6932	−0.667344
$554$	0	0
$555$	−4.25112	−0.180450
$556$	0	0
$557$	−20.1125	−0.852193	−0.426097	−	0.904678i	$-0.640112\pi$
$557$	−20.1125	−0.852193	−0.426097	+	0.904678i	$0.640112\pi$
$558$	0	0
$559$	49.7292	2.10332
$560$	0	0
$561$	−14.7563	−0.623013
$562$	0	0
$563$	0.0804882	0.00339217	0.00169609	−	0.999999i	$-0.499460\pi$
$563$	0.0804882	0.00339217	0.00169609	+	0.999999i	$0.499460\pi$
$564$	0	0
$565$	3.45935	0.145536
$566$	0	0
$567$	−24.1681	−1.01496
$568$	0	0
$569$	11.9728	0.501925	0.250963	−	0.967997i	$-0.419253\pi$
$569$	11.9728	0.501925	0.250963	+	0.967997i	$0.419253\pi$
$570$	0	0
$571$	−32.6082	−1.36461	−0.682305	−	0.731068i	$-0.739022\pi$
$571$	−32.6082	−1.36461	−0.682305	+	0.731068i	$0.739022\pi$
$572$	0	0
$573$	41.3572	1.72772
$574$	0	0
$575$	−6.31215	−0.263235
$576$	0	0
$577$	14.4477	0.601465	0.300732	−	0.953709i	$-0.402769\pi$
$577$	14.4477	0.601465	0.300732	+	0.953709i	$0.402769\pi$
$578$	0	0
$579$	24.5903	1.02194
$580$	0	0
$581$	28.3089	1.17445
$582$	0	0
$583$	5.69865	0.236014
$584$	0	0
$585$	2.57432	0.106435
$586$	0	0
$587$	−14.5432	−0.600261	−0.300131	−	0.953898i	$-0.597030\pi$
$587$	−14.5432	−0.600261	−0.300131	+	0.953898i	$0.597030\pi$
$588$	0	0
$589$	−11.3249	−0.466634
$590$	0	0
$591$	−9.65743	−0.397253
$592$	0	0
$593$	38.7817	1.59257	0.796286	−	0.604921i	$-0.206795\pi$
$593$	38.7817	1.59257	0.796286	+	0.604921i	$0.206795\pi$
$594$	0	0
$595$	6.94439	0.284692
$596$	0	0
$597$	20.4935	0.838742
$598$	0	0
$599$	0.763380	0.0311908	0.0155954	−	0.999878i	$-0.495036\pi$
$599$	0.763380	0.0311908	0.0155954	+	0.999878i	$0.495036\pi$
$600$	0	0
$601$	−8.15875	−0.332802	−0.166401	−	0.986058i	$-0.553215\pi$
$601$	−8.15875	−0.332802	−0.166401	+	0.986058i	$0.553215\pi$
$602$	0	0
$603$	1.53996	0.0627119
$604$	0	0
$605$	−9.07204	−0.368831
$606$	0	0
$607$	−13.6422	−0.553720	−0.276860	−	0.960910i	$-0.589294\pi$
$607$	−13.6422	−0.553720	−0.276860	+	0.960910i	$0.589294\pi$
$608$	0	0
$609$	14.2439	0.577190
$610$	0	0
$611$	−66.4557	−2.68851
$612$	0	0
$613$	−14.0134	−0.565998	−0.282999	−	0.959120i	$-0.591329\pi$
$613$	−14.0134	−0.565998	−0.282999	+	0.959120i	$0.591329\pi$
$614$	0	0
$615$	16.8729	0.680381
$616$	0	0
$617$	−34.4494	−1.38688	−0.693440	−	0.720515i	$-0.743906\pi$
$617$	−34.4494	−1.38688	−0.693440	+	0.720515i	$0.743906\pi$
$618$	0	0
$619$	28.3073	1.13777	0.568883	−	0.822419i	$-0.307376\pi$
$619$	28.3073	1.13777	0.568883	+	0.822419i	$0.307376\pi$
$620$	0	0
$621$	−34.9315	−1.40175
$622$	0	0
$623$	15.1889	0.608529
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−8.62312	−0.344374
$628$	0	0
$629$	5.60087	0.223321
$630$	0	0
$631$	18.9140	0.752956	0.376478	−	0.926425i	$-0.377135\pi$
$631$	18.9140	0.752956	0.376478	+	0.926425i	$0.377135\pi$
$632$	0	0
$633$	−19.4143	−0.771650
$634$	0	0
$635$	−17.0434	−0.676346
$636$	0	0
$637$	21.1743	0.838958
$638$	0	0
$639$	−3.12480	−0.123615
$640$	0	0
$641$	−36.1534	−1.42797	−0.713987	−	0.700159i	$-0.753113\pi$
$641$	−36.1534	−1.42797	−0.713987	+	0.700159i	$0.753113\pi$
$642$	0	0
$643$	10.9574	0.432119	0.216060	−	0.976380i	$-0.430679\pi$
$643$	10.9574	0.432119	0.216060	+	0.976380i	$0.430679\pi$
$644$	0	0
$645$	15.2733	0.601385
$646$	0	0
$647$	21.9336	0.862298	0.431149	−	0.902281i	$-0.358108\pi$
$647$	21.9336	0.862298	0.431149	+	0.902281i	$0.358108\pi$
$648$	0	0
$649$	−29.5242	−1.15892
$650$	0	0
$651$	−49.0357	−1.92186
$652$	0	0
$653$	26.8361	1.05018	0.525088	−	0.851048i	$-0.324032\pi$
$653$	26.8361	1.05018	0.525088	+	0.851048i	$0.324032\pi$
$654$	0	0
$655$	6.81219	0.266174
$656$	0	0
$657$	2.88722	0.112641
$658$	0	0
$659$	−37.8865	−1.47585	−0.737925	−	0.674883i	$-0.764194\pi$
$659$	−37.8865	−1.47585	−0.737925	+	0.674883i	$0.764194\pi$
$660$	0	0
$661$	−13.4541	−0.523302	−0.261651	−	0.965163i	$-0.584267\pi$
$661$	−13.4541	−0.523302	−0.261651	+	0.965163i	$0.584267\pi$
$662$	0	0
$663$	16.9561	0.658521
$664$	0	0
$665$	4.05808	0.157366
$666$	0	0
$667$	17.0578	0.660480
$668$	0	0
$669$	11.0280	0.426369
$670$	0	0
$671$	61.2401	2.36415
$672$	0	0
$673$	−21.0749	−0.812378	−0.406189	−	0.913789i	$-0.633143\pi$
$673$	−21.0749	−0.812378	−0.406189	+	0.913789i	$0.633143\pi$
$674$	0	0
$675$	5.53401	0.213004
$676$	0	0
$677$	6.60271	0.253763	0.126881	−	0.991918i	$-0.459503\pi$
$677$	6.60271	0.253763	0.126881	+	0.991918i	$0.459503\pi$
$678$	0	0
$679$	−47.9047	−1.83842
$680$	0	0
$681$	27.1447	1.04019
$682$	0	0
$683$	−10.9345	−0.418395	−0.209198	−	0.977873i	$-0.567085\pi$
$683$	−10.9345	−0.418395	−0.209198	+	0.977873i	$0.567085\pi$
$684$	0	0
$685$	−7.62695	−0.291411
$686$	0	0
$687$	−24.7522	−0.944355
$688$	0	0
$689$	−6.54817	−0.249465
$690$	0	0
$691$	45.5552	1.73300	0.866501	−	0.499176i	$-0.166364\pi$
$691$	45.5552	1.73300	0.866501	+	0.499176i	$0.166364\pi$
$692$	0	0
$693$	7.46846	0.283703
$694$	0	0
$695$	−19.2383	−0.729750
$696$	0	0
$697$	−22.2301	−0.842026
$698$	0	0
$699$	−18.1460	−0.686343
$700$	0	0
$701$	−18.6798	−0.705527	−0.352764	−	0.935712i	$-0.614758\pi$
$701$	−18.6798	−0.705527	−0.352764	+	0.935712i	$0.614758\pi$
$702$	0	0
$703$	3.27297	0.123442
$704$	0	0
$705$	−20.4105	−0.768703
$706$	0	0
$707$	−42.6486	−1.60397
$708$	0	0
$709$	−24.1207	−0.905871	−0.452935	−	0.891543i	$-0.649623\pi$
$709$	−24.1207	−0.905871	−0.452935	+	0.891543i	$0.649623\pi$
$710$	0	0
$711$	2.35404	0.0882834
$712$	0	0
$713$	−58.7229	−2.19919
$714$	0	0
$715$	23.0643	0.862555
$716$	0	0
$717$	31.9634	1.19370
$718$	0	0
$719$	−8.94887	−0.333736	−0.166868	−	0.985979i	$-0.553365\pi$
$719$	−8.94887	−0.333736	−0.166868	+	0.985979i	$0.553365\pi$
$720$	0	0
$721$	−52.7387	−1.96409
$722$	0	0
$723$	24.0684	0.895112
$724$	0	0
$725$	−2.70237	−0.100364
$726$	0	0
$727$	−41.9938	−1.55746	−0.778732	−	0.627357i	$-0.784137\pi$
$727$	−41.9938	−1.55746	−0.778732	+	0.627357i	$0.784137\pi$
$728$	0	0
$729$	29.8751	1.10649
$730$	0	0
$731$	−20.1226	−0.744262
$732$	0	0
$733$	−25.8416	−0.954480	−0.477240	−	0.878773i	$-0.658363\pi$
$733$	−25.8416	−0.954480	−0.477240	+	0.878773i	$0.658363\pi$
$734$	0	0
$735$	6.50326	0.239876
$736$	0	0
$737$	13.7970	0.508221
$738$	0	0
$739$	35.4614	1.30447	0.652235	−	0.758017i	$-0.273832\pi$
$739$	35.4614	1.30447	0.652235	+	0.758017i	$0.273832\pi$
$740$	0	0
$741$	9.90861	0.364002
$742$	0	0
$743$	18.0214	0.661140	0.330570	−	0.943781i	$-0.392759\pi$
$743$	18.0214	0.661140	0.330570	+	0.943781i	$0.392759\pi$
$744$	0	0
$745$	−5.59856	−0.205115
$746$	0	0
$747$	−4.24644	−0.155369
$748$	0	0
$749$	−18.1723	−0.664001
$750$	0	0
$751$	11.1189	0.405735	0.202868	−	0.979206i	$-0.434974\pi$
$751$	11.1189	0.405735	0.202868	+	0.979206i	$0.434974\pi$
$752$	0	0
$753$	20.5941	0.750490
$754$	0	0
$755$	10.5085	0.382445
$756$	0	0
$757$	15.7093	0.570964	0.285482	−	0.958384i	$-0.407846\pi$
$757$	15.7093	0.570964	0.285482	+	0.958384i	$0.407846\pi$
$758$	0	0
$759$	−44.7135	−1.62300
$760$	0	0
$761$	−38.3518	−1.39025	−0.695126	−	0.718888i	$-0.744651\pi$
$761$	−38.3518	−1.39025	−0.695126	+	0.718888i	$0.744651\pi$
$762$	0	0
$763$	55.4634	2.00791
$764$	0	0
$765$	−1.04168	−0.0376621
$766$	0	0
$767$	33.9254	1.22498
$768$	0	0
$769$	−2.57147	−0.0927297	−0.0463648	−	0.998925i	$-0.514764\pi$
$769$	−2.57147	−0.0927297	−0.0463648	+	0.998925i	$0.514764\pi$
$770$	0	0
$771$	−24.3929	−0.878490
$772$	0	0
$773$	5.36452	0.192948	0.0964741	−	0.995335i	$-0.469244\pi$
$773$	5.36452	0.192948	0.0964741	+	0.995335i	$0.469244\pi$
$774$	0	0
$775$	9.30315	0.334179
$776$	0	0
$777$	14.1716	0.508405
$778$	0	0
$779$	−12.9906	−0.465436
$780$	0	0
$781$	−27.9962	−1.00178
$782$	0	0
$783$	−14.9550	−0.534447
$784$	0	0
$785$	−11.5928	−0.413766
$786$	0	0
$787$	−52.1708	−1.85969	−0.929845	−	0.367952i	$-0.880059\pi$
$787$	−52.1708	−1.85969	−0.929845	+	0.367952i	$0.880059\pi$
$788$	0	0
$789$	15.8412	0.563963
$790$	0	0
$791$	−11.5322	−0.410038
$792$	0	0
$793$	−70.3694	−2.49889
$794$	0	0
$795$	−2.01113	−0.0713276
$796$	0	0
$797$	−22.8205	−0.808345	−0.404173	−	0.914683i	$-0.632440\pi$
$797$	−22.8205	−0.808345	−0.404173	+	0.914683i	$0.632440\pi$
$798$	0	0
$799$	26.8909	0.951332
$800$	0	0
$801$	−2.27838	−0.0805026
$802$	0	0
$803$	25.8677	0.912850
$804$	0	0
$805$	21.0424	0.741645
$806$	0	0
$807$	−9.78546	−0.344465
$808$	0	0
$809$	31.5301	1.10854	0.554269	−	0.832338i	$-0.312998\pi$
$809$	31.5301	1.10854	0.554269	+	0.832338i	$0.312998\pi$
$810$	0	0
$811$	−33.1959	−1.16567	−0.582833	−	0.812592i	$-0.698056\pi$
$811$	−33.1959	−1.16567	−0.582833	+	0.812592i	$0.698056\pi$
$812$	0	0
$813$	−8.21687	−0.288178
$814$	0	0
$815$	−19.5693	−0.685482
$816$	0	0
$817$	−11.7590	−0.411396
$818$	0	0
$819$	−8.58181	−0.299873
$820$	0	0
$821$	−48.2413	−1.68363	−0.841816	−	0.539764i	$-0.818513\pi$
$821$	−48.2413	−1.68363	−0.841816	+	0.539764i	$0.818513\pi$
$822$	0	0
$823$	8.88606	0.309749	0.154874	−	0.987934i	$-0.450503\pi$
$823$	8.88606	0.309749	0.154874	+	0.987934i	$0.450503\pi$
$824$	0	0
$825$	7.08371	0.246623
$826$	0	0
$827$	−50.7133	−1.76347	−0.881737	−	0.471742i	$-0.843625\pi$
$827$	−50.7133	−1.76347	−0.881737	+	0.471742i	$0.843625\pi$
$828$	0	0
$829$	13.3560	0.463874	0.231937	−	0.972731i	$-0.425494\pi$
$829$	13.3560	0.463874	0.231937	+	0.972731i	$0.425494\pi$
$830$	0	0
$831$	25.5065	0.884810
$832$	0	0
$833$	−8.56807	−0.296866
$834$	0	0
$835$	−8.84859	−0.306218
$836$	0	0
$837$	51.4837	1.77954
$838$	0	0
$839$	−12.9792	−0.448090	−0.224045	−	0.974579i	$-0.571926\pi$
$839$	−12.9792	−0.448090	−0.224045	+	0.974579i	$0.571926\pi$
$840$	0	0
$841$	−21.6972	−0.748178
$842$	0	0
$843$	21.2579	0.732162
$844$	0	0
$845$	−13.5026	−0.464502
$846$	0	0
$847$	30.2428	1.03916
$848$	0	0
$849$	9.84301	0.337811
$850$	0	0
$851$	16.9713	0.581769
$852$	0	0
$853$	9.60993	0.329038	0.164519	−	0.986374i	$-0.447393\pi$
$853$	9.60993	0.329038	0.164519	+	0.986374i	$0.447393\pi$
$854$	0	0
$855$	−0.608726	−0.0208180
$856$	0	0
$857$	−8.27086	−0.282527	−0.141264	−	0.989972i	$-0.545117\pi$
$857$	−8.27086	−0.282527	−0.141264	+	0.989972i	$0.545117\pi$
$858$	0	0
$859$	−10.8254	−0.369357	−0.184678	−	0.982799i	$-0.559124\pi$
$859$	−10.8254	−0.369357	−0.184678	+	0.982799i	$0.559124\pi$
$860$	0	0
$861$	−56.2480	−1.91693
$862$	0	0
$863$	−10.3887	−0.353636	−0.176818	−	0.984244i	$-0.556580\pi$
$863$	−10.3887	−0.353636	−0.176818	+	0.984244i	$0.556580\pi$
$864$	0	0
$865$	−6.19390	−0.210599
$866$	0	0
$867$	20.0179	0.679842
$868$	0	0
$869$	21.0907	0.715454
$870$	0	0
$871$	−15.8538	−0.537186
$872$	0	0
$873$	7.18587	0.243205
$874$	0	0
$875$	−3.33363	−0.112697
$876$	0	0
$877$	14.1104	0.476473	0.238236	−	0.971207i	$-0.423431\pi$
$877$	14.1104	0.476473	0.238236	+	0.971207i	$0.423431\pi$
$878$	0	0
$879$	48.5900	1.63890
$880$	0	0
$881$	−16.1368	−0.543664	−0.271832	−	0.962345i	$-0.587629\pi$
$881$	−16.1368	−0.543664	−0.271832	+	0.962345i	$0.587629\pi$
$882$	0	0
$883$	−10.2405	−0.344621	−0.172310	−	0.985043i	$-0.555123\pi$
$883$	−10.2405	−0.344621	−0.172310	+	0.985043i	$0.555123\pi$
$884$	0	0
$885$	10.4195	0.350248
$886$	0	0
$887$	2.55618	0.0858283	0.0429141	−	0.999079i	$-0.486336\pi$
$887$	2.55618	0.0858283	0.0429141	+	0.999079i	$0.486336\pi$
$888$	0	0
$889$	56.8162	1.90556
$890$	0	0
$891$	32.4803	1.08813
$892$	0	0
$893$	15.7142	0.525855
$894$	0	0
$895$	−6.66383	−0.222747
$896$	0	0
$897$	51.3791	1.71550
$898$	0	0
$899$	−25.1406	−0.838486
$900$	0	0
$901$	2.64968	0.0882736
$902$	0	0
$903$	−50.9154	−1.69436
$904$	0	0
$905$	9.78563	0.325285
$906$	0	0
$907$	52.6337	1.74767	0.873837	−	0.486218i	$-0.161624\pi$
$907$	52.6337	1.74767	0.873837	+	0.486218i	$0.161624\pi$
$908$	0	0
$909$	6.39744	0.212190
$910$	0	0
$911$	−43.1084	−1.42825	−0.714123	−	0.700020i	$-0.753174\pi$
$911$	−43.1084	−1.42825	−0.714123	+	0.700020i	$0.753174\pi$
$912$	0	0
$913$	−38.0454	−1.25912
$914$	0	0
$915$	−21.6125	−0.714487
$916$	0	0
$917$	−22.7093	−0.749927
$918$	0	0
$919$	28.6885	0.946346	0.473173	−	0.880969i	$-0.343108\pi$
$919$	28.6885	0.946346	0.473173	+	0.880969i	$0.343108\pi$
$920$	0	0
$921$	−30.3786	−1.00101
$922$	0	0
$923$	32.1698	1.05888
$924$	0	0
$925$	−2.68867	−0.0884031
$926$	0	0
$927$	7.91098	0.259831
$928$	0	0
$929$	−58.1269	−1.90708	−0.953540	−	0.301266i	$-0.902591\pi$
$929$	−58.1269	−1.90708	−0.953540	+	0.301266i	$0.902591\pi$
$930$	0	0
$931$	−5.00691	−0.164095
$932$	0	0
$933$	39.1468	1.28161
$934$	0	0
$935$	−9.33282	−0.305216
$936$	0	0
$937$	−16.1199	−0.526613	−0.263307	−	0.964712i	$-0.584813\pi$
$937$	−16.1199	−0.526613	−0.263307	+	0.964712i	$0.584813\pi$
$938$	0	0
$939$	21.1687	0.690814
$940$	0	0
$941$	0.469047	0.0152905	0.00764524	−	0.999971i	$-0.497566\pi$
$941$	0.469047	0.0152905	0.00764524	+	0.999971i	$0.497566\pi$
$942$	0	0
$943$	−67.3600	−2.19354
$944$	0	0
$945$	−18.4483	−0.600124
$946$	0	0
$947$	46.1472	1.49958	0.749791	−	0.661674i	$-0.230154\pi$
$947$	46.1472	1.49958	0.749791	+	0.661674i	$0.230154\pi$
$948$	0	0
$949$	−29.7239	−0.964877
$950$	0	0
$951$	4.83861	0.156903
$952$	0	0
$953$	17.3350	0.561535	0.280768	−	0.959776i	$-0.409411\pi$
$953$	17.3350	0.561535	0.280768	+	0.959776i	$0.409411\pi$
$954$	0	0
$955$	26.1568	0.846416
$956$	0	0
$957$	−19.1428	−0.618800
$958$	0	0
$959$	25.4254	0.821029
$960$	0	0
$961$	55.5486	1.79189
$962$	0	0
$963$	2.72591	0.0878411
$964$	0	0
$965$	15.5525	0.500651
$966$	0	0
$967$	50.1909	1.61403	0.807015	−	0.590531i	$-0.201082\pi$
$967$	50.1909	1.61403	0.807015	+	0.590531i	$0.201082\pi$
$968$	0	0
$969$	−4.00946	−0.128802
$970$	0	0
$971$	0.641284	0.0205798	0.0102899	−	0.999947i	$-0.496725\pi$
$971$	0.641284	0.0205798	0.0102899	+	0.999947i	$0.496725\pi$
$972$	0	0
$973$	64.1333	2.05602
$974$	0	0
$975$	−8.13971	−0.260679
$976$	0	0
$977$	−34.9365	−1.11772	−0.558858	−	0.829263i	$-0.688760\pi$
$977$	−34.9365	−1.11772	−0.558858	+	0.829263i	$0.688760\pi$
$978$	0	0
$979$	−20.4129	−0.652398
$980$	0	0
$981$	−8.31971	−0.265628
$982$	0	0
$983$	12.1249	0.386725	0.193363	−	0.981127i	$-0.438061\pi$
$983$	12.1249	0.386725	0.193363	+	0.981127i	$0.438061\pi$
$984$	0	0
$985$	−6.10796	−0.194616
$986$	0	0
$987$	68.0409	2.16577
$988$	0	0
$989$	−60.9740	−1.93886
$990$	0	0
$991$	16.6528	0.528992	0.264496	−	0.964387i	$-0.414794\pi$
$991$	16.6528	0.528992	0.264496	+	0.964387i	$0.414794\pi$
$992$	0	0
$993$	8.63769	0.274109
$994$	0	0
$995$	12.9613	0.410902
$996$	0	0
$997$	40.9160	1.29582	0.647911	−	0.761716i	$-0.275643\pi$
$997$	40.9160	1.29582	0.647911	+	0.761716i	$0.275643\pi$
$998$	0	0
$999$	−14.8792	−0.470756

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8020.2.a.c.1.7	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8020.2.a.c.1.7	✓	28	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8020 = 2^{2} \cdot 5 \cdot 401 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8020.a (trivial)

\(f(q)\)	\(=\)	\(q-1.58112 q^{3} -1.00000 q^{5} +3.33363 q^{7} -0.500055 q^{9} +O(q^{10})\)	\(q-1.58112 q^{3} -1.00000 q^{5} +3.33363 q^{7} -0.500055 q^{9} -4.48018 q^{11} +5.14806 q^{13} +1.58112 q^{15} -2.08313 q^{17} -1.21732 q^{19} -5.27087 q^{21} -6.31215 q^{23} +1.00000 q^{25} +5.53401 q^{27} -2.70237 q^{29} +9.30315 q^{31} +7.08371 q^{33} -3.33363 q^{35} -2.68867 q^{37} -8.13971 q^{39} +10.6715 q^{41} +9.65978 q^{43} +0.500055 q^{45} -12.9089 q^{47} +4.11307 q^{49} +3.29369 q^{51} -1.27197 q^{53} +4.48018 q^{55} +1.92473 q^{57} +6.58994 q^{59} -13.6691 q^{61} -1.66700 q^{63} -5.14806 q^{65} -3.07957 q^{67} +9.98027 q^{69} +6.24891 q^{71} -5.77379 q^{73} -1.58112 q^{75} -14.9353 q^{77} -4.70756 q^{79} -7.24978 q^{81} +8.49193 q^{83} +2.08313 q^{85} +4.27278 q^{87} +4.55626 q^{89} +17.1617 q^{91} -14.7094 q^{93} +1.21732 q^{95} -14.3702 q^{97} +2.24034 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 3 q^{3} - 28 q^{5} - 4 q^{7} + 17 q^{9}+O(q^{10}) \) 28 * q + 3 * q^3 - 28 * q^5 - 4 * q^7 + 17 * q^9	\( 28 q + 3 q^{3} - 28 q^{5} - 4 q^{7} + 17 q^{9} + 2 q^{11} + 3 q^{13} - 3 q^{15} - 10 q^{17} - 2 q^{19} - 12 q^{21} - 23 q^{23} + 28 q^{25} + 9 q^{27} - 37 q^{29} - 11 q^{31} + 2 q^{33} + 4 q^{35} - 3 q^{37} - 19 q^{39} - 30 q^{41} + 13 q^{43} - 17 q^{45} - 15 q^{47} + 12 q^{49} - 8 q^{51} - 35 q^{53} - 2 q^{55} - 22 q^{57} - q^{59} - 33 q^{61} - 20 q^{63} - 3 q^{65} + 19 q^{67} - 8 q^{69} - 31 q^{71} + 31 q^{73} + 3 q^{75} - 42 q^{77} - 29 q^{79} - 36 q^{81} + 14 q^{83} + 10 q^{85} - 32 q^{87} - 32 q^{89} - 7 q^{91} - 11 q^{93} + 2 q^{95} + 2 q^{97} - 39 q^{99}+O(q^{100}) \) 28 * q + 3 * q^3 - 28 * q^5 - 4 * q^7 + 17 * q^9 + 2 * q^11 + 3 * q^13 - 3 * q^15 - 10 * q^17 - 2 * q^19 - 12 * q^21 - 23 * q^23 + 28 * q^25 + 9 * q^27 - 37 * q^29 - 11 * q^31 + 2 * q^33 + 4 * q^35 - 3 * q^37 - 19 * q^39 - 30 * q^41 + 13 * q^43 - 17 * q^45 - 15 * q^47 + 12 * q^49 - 8 * q^51 - 35 * q^53 - 2 * q^55 - 22 * q^57 - q^59 - 33 * q^61 - 20 * q^63 - 3 * q^65 + 19 * q^67 - 8 * q^69 - 31 * q^71 + 31 * q^73 + 3 * q^75 - 42 * q^77 - 29 * q^79 - 36 * q^81 + 14 * q^83 + 10 * q^85 - 32 * q^87 - 32 * q^89 - 7 * q^91 - 11 * q^93 + 2 * q^95 + 2 * q^97 - 39 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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