LMFDB - Embedded newform 8020.2.a.f.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("8020.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$64.0400224211$
Analytic rank:	$0$
Dimension:	$37$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Character	$\chi$	$=$	8020.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.32921 q^{3} +1.00000 q^{5} -3.52319 q^{7} +8.08363 q^{9} +O(q^{10})$	$q-3.32921 q^{3} +1.00000 q^{5} -3.52319 q^{7} +8.08363 q^{9} -5.52528 q^{11} +1.73778 q^{13} -3.32921 q^{15} -0.387977 q^{17} -5.59581 q^{19} +11.7294 q^{21} +7.11555 q^{23} +1.00000 q^{25} -16.9245 q^{27} -6.31070 q^{29} +0.522093 q^{31} +18.3948 q^{33} -3.52319 q^{35} -6.36068 q^{37} -5.78542 q^{39} -0.378927 q^{41} +7.24950 q^{43} +8.08363 q^{45} -10.5319 q^{47} +5.41286 q^{49} +1.29166 q^{51} +2.65036 q^{53} -5.52528 q^{55} +18.6296 q^{57} +2.27360 q^{59} -12.4098 q^{61} -28.4802 q^{63} +1.73778 q^{65} -2.90341 q^{67} -23.6892 q^{69} -2.08172 q^{71} -5.92974 q^{73} -3.32921 q^{75} +19.4666 q^{77} -8.08975 q^{79} +32.0942 q^{81} +4.29939 q^{83} -0.387977 q^{85} +21.0096 q^{87} -10.2803 q^{89} -6.12252 q^{91} -1.73816 q^{93} -5.59581 q^{95} -3.85465 q^{97} -44.6643 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 37 q + 3 q^{3} + 37 q^{5} + 4 q^{7} + 50 q^{9}+O(q^{10}) $ 37 * q + 3 * q^3 + 37 * q^5 + 4 * q^7 + 50 * q^9	$ 37 q + 3 q^{3} + 37 q^{5} + 4 q^{7} + 50 q^{9} + 2 q^{11} + 27 q^{13} + 3 q^{15} + 36 q^{17} - 6 q^{19} + 20 q^{21} + 17 q^{23} + 37 q^{25} + 9 q^{27} + 29 q^{29} + 5 q^{31} + 36 q^{33} + 4 q^{35} + 35 q^{37} + 21 q^{39} + 24 q^{41} + 11 q^{43} + 50 q^{45} + 19 q^{47} + 57 q^{49} + 8 q^{51} + 65 q^{53} + 2 q^{55} + 62 q^{57} - 9 q^{59} + 13 q^{61} + 26 q^{63} + 27 q^{65} + 13 q^{67} + 20 q^{69} + 33 q^{71} + 67 q^{73} + 3 q^{75} + 62 q^{77} + 23 q^{79} + 97 q^{81} + 2 q^{83} + 36 q^{85} + 32 q^{87} + 34 q^{89} + q^{91} + 41 q^{93} - 6 q^{95} + 66 q^{97} - 7 q^{99}+O(q^{100}) $ 37 * q + 3 * q^3 + 37 * q^5 + 4 * q^7 + 50 * q^9 + 2 * q^11 + 27 * q^13 + 3 * q^15 + 36 * q^17 - 6 * q^19 + 20 * q^21 + 17 * q^23 + 37 * q^25 + 9 * q^27 + 29 * q^29 + 5 * q^31 + 36 * q^33 + 4 * q^35 + 35 * q^37 + 21 * q^39 + 24 * q^41 + 11 * q^43 + 50 * q^45 + 19 * q^47 + 57 * q^49 + 8 * q^51 + 65 * q^53 + 2 * q^55 + 62 * q^57 - 9 * q^59 + 13 * q^61 + 26 * q^63 + 27 * q^65 + 13 * q^67 + 20 * q^69 + 33 * q^71 + 67 * q^73 + 3 * q^75 + 62 * q^77 + 23 * q^79 + 97 * q^81 + 2 * q^83 + 36 * q^85 + 32 * q^87 + 34 * q^89 + q^91 + 41 * q^93 - 6 * q^95 + 66 * q^97 - 7 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−3.32921	−1.92212	−0.961060	−	0.276341i	$-0.910878\pi$
$3$	−3.32921	−1.92212	−0.961060	+	0.276341i	$0.910878\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−3.52319	−1.33164	−0.665820	−	0.746112i	$-0.731918\pi$
$7$	−3.52319	−1.33164	−0.665820	+	0.746112i	$0.731918\pi$
$8$	0	0
$9$	8.08363	2.69454
$10$	0	0
$11$	−5.52528	−1.66593	−0.832967	−	0.553323i	$-0.813360\pi$
$11$	−5.52528	−1.66593	−0.832967	+	0.553323i	$0.813360\pi$
$12$	0	0
$13$	1.73778	0.481973	0.240986	−	0.970529i	$-0.422529\pi$
$13$	1.73778	0.481973	0.240986	+	0.970529i	$0.422529\pi$
$14$	0	0
$15$	−3.32921	−0.859598
$16$	0	0
$17$	−0.387977	−0.0940983	−0.0470491	−	0.998893i	$-0.514982\pi$
$17$	−0.387977	−0.0940983	−0.0470491	+	0.998893i	$0.514982\pi$
$18$	0	0
$19$	−5.59581	−1.28377	−0.641884	−	0.766802i	$-0.721847\pi$
$19$	−5.59581	−1.28377	−0.641884	+	0.766802i	$0.721847\pi$
$20$	0	0
$21$	11.7294	2.55957
$22$	0	0
$23$	7.11555	1.48370	0.741848	−	0.670569i	$-0.233950\pi$
$23$	7.11555	1.48370	0.741848	+	0.670569i	$0.233950\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−16.9245	−3.25712
$28$	0	0
$29$	−6.31070	−1.17187	−0.585934	−	0.810359i	$-0.699272\pi$
$29$	−6.31070	−1.17187	−0.585934	+	0.810359i	$0.699272\pi$
$30$	0	0
$31$	0.522093	0.0937706	0.0468853	−	0.998900i	$-0.485070\pi$
$31$	0.522093	0.0937706	0.0468853	+	0.998900i	$0.485070\pi$
$32$	0	0
$33$	18.3948	3.20212
$34$	0	0
$35$	−3.52319	−0.595528
$36$	0	0
$37$	−6.36068	−1.04569	−0.522845	−	0.852428i	$-0.675129\pi$
$37$	−6.36068	−1.04569	−0.522845	+	0.852428i	$0.675129\pi$
$38$	0	0
$39$	−5.78542	−0.926409
$40$	0	0
$41$	−0.378927	−0.0591784	−0.0295892	−	0.999562i	$-0.509420\pi$
$41$	−0.378927	−0.0591784	−0.0295892	+	0.999562i	$0.509420\pi$
$42$	0	0
$43$	7.24950	1.10554	0.552770	−	0.833334i	$-0.313571\pi$
$43$	7.24950	1.10554	0.552770	+	0.833334i	$0.313571\pi$
$44$	0	0
$45$	8.08363	1.20504
$46$	0	0
$47$	−10.5319	−1.53623	−0.768117	−	0.640309i	$-0.778806\pi$
$47$	−10.5319	−1.53623	−0.768117	+	0.640309i	$0.778806\pi$
$48$	0	0
$49$	5.41286	0.773266
$50$	0	0
$51$	1.29166	0.180868
$52$	0	0
$53$	2.65036	0.364055	0.182028	−	0.983293i	$-0.441734\pi$
$53$	2.65036	0.364055	0.182028	+	0.983293i	$0.441734\pi$
$54$	0	0
$55$	−5.52528	−0.745028
$56$	0	0
$57$	18.6296	2.46755
$58$	0	0
$59$	2.27360	0.295997	0.147998	−	0.988988i	$-0.452717\pi$
$59$	2.27360	0.295997	0.147998	+	0.988988i	$0.452717\pi$
$60$	0	0
$61$	−12.4098	−1.58891	−0.794456	−	0.607322i	$-0.792244\pi$
$61$	−12.4098	−1.58891	−0.794456	+	0.607322i	$0.792244\pi$
$62$	0	0
$63$	−28.4802	−3.58816
$64$	0	0
$65$	1.73778	0.215545
$66$	0	0
$67$	−2.90341	−0.354708	−0.177354	−	0.984147i	$-0.556754\pi$
$67$	−2.90341	−0.354708	−0.177354	+	0.984147i	$0.556754\pi$
$68$	0	0
$69$	−23.6892	−2.85184
$70$	0	0
$71$	−2.08172	−0.247055	−0.123527	−	0.992341i	$-0.539421\pi$
$71$	−2.08172	−0.247055	−0.123527	+	0.992341i	$0.539421\pi$
$72$	0	0
$73$	−5.92974	−0.694024	−0.347012	−	0.937861i	$-0.612804\pi$
$73$	−5.92974	−0.694024	−0.347012	+	0.937861i	$0.612804\pi$
$74$	0	0
$75$	−3.32921	−0.384424
$76$	0	0
$77$	19.4666	2.21842
$78$	0	0
$79$	−8.08975	−0.910168	−0.455084	−	0.890448i	$-0.650391\pi$
$79$	−8.08975	−0.910168	−0.455084	+	0.890448i	$0.650391\pi$
$80$	0	0
$81$	32.0942	3.56602
$82$	0	0
$83$	4.29939	0.471919	0.235960	−	0.971763i	$-0.424177\pi$
$83$	4.29939	0.471919	0.235960	+	0.971763i	$0.424177\pi$
$84$	0	0
$85$	−0.387977	−0.0420820
$86$	0	0
$87$	21.0096	2.25247
$88$	0	0
$89$	−10.2803	−1.08971	−0.544854	−	0.838531i	$-0.683415\pi$
$89$	−10.2803	−1.08971	−0.544854	+	0.838531i	$0.683415\pi$
$90$	0	0
$91$	−6.12252	−0.641814
$92$	0	0
$93$	−1.73816	−0.180238
$94$	0	0
$95$	−5.59581	−0.574118
$96$	0	0
$97$	−3.85465	−0.391380	−0.195690	−	0.980666i	$-0.562695\pi$
$97$	−3.85465	−0.391380	−0.195690	+	0.980666i	$0.562695\pi$
$98$	0	0
$99$	−44.6643	−4.48893
$100$	0	0
$101$	−7.96394	−0.792442	−0.396221	−	0.918155i	$-0.629679\pi$
$101$	−7.96394	−0.792442	−0.396221	+	0.918155i	$0.629679\pi$
$102$	0	0
$103$	9.73166	0.958889	0.479444	−	0.877572i	$-0.340838\pi$
$103$	9.73166	0.958889	0.479444	+	0.877572i	$0.340838\pi$
$104$	0	0
$105$	11.7294	1.14468
$106$	0	0
$107$	−11.6679	−1.12798	−0.563990	−	0.825782i	$-0.690734\pi$
$107$	−11.6679	−1.12798	−0.563990	+	0.825782i	$0.690734\pi$
$108$	0	0
$109$	7.42609	0.711290	0.355645	−	0.934621i	$-0.384261\pi$
$109$	7.42609	0.711290	0.355645	+	0.934621i	$0.384261\pi$
$110$	0	0
$111$	21.1760	2.00994
$112$	0	0
$113$	11.2159	1.05510	0.527550	−	0.849524i	$-0.323111\pi$
$113$	11.2159	1.05510	0.527550	+	0.849524i	$0.323111\pi$
$114$	0	0
$115$	7.11555	0.663529
$116$	0	0
$117$	14.0475	1.29870
$118$	0	0
$119$	1.36692	0.125305
$120$	0	0
$121$	19.5287	1.77534
$122$	0	0
$123$	1.26153	0.113748
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	20.0569	1.77976	0.889879	−	0.456197i	$-0.150789\pi$
$127$	20.0569	1.77976	0.889879	+	0.456197i	$0.150789\pi$
$128$	0	0
$129$	−24.1351	−2.12498
$130$	0	0
$131$	−13.7178	−1.19853	−0.599264	−	0.800552i	$-0.704540\pi$
$131$	−13.7178	−1.19853	−0.599264	+	0.800552i	$0.704540\pi$
$132$	0	0
$133$	19.7151	1.70952
$134$	0	0
$135$	−16.9245	−1.45663
$136$	0	0
$137$	−10.6918	−0.913466	−0.456733	−	0.889604i	$-0.650981\pi$
$137$	−10.6918	−0.913466	−0.456733	+	0.889604i	$0.650981\pi$
$138$	0	0
$139$	−10.0365	−0.851283	−0.425641	−	0.904892i	$-0.639951\pi$
$139$	−10.0365	−0.851283	−0.425641	+	0.904892i	$0.639951\pi$
$140$	0	0
$141$	35.0629	2.95283
$142$	0	0
$143$	−9.60170	−0.802935
$144$	0	0
$145$	−6.31070	−0.524075
$146$	0	0
$147$	−18.0205	−1.48631
$148$	0	0
$149$	0.350067	0.0286786	0.0143393	−	0.999897i	$-0.495435\pi$
$149$	0.350067	0.0286786	0.0143393	+	0.999897i	$0.495435\pi$
$150$	0	0
$151$	−12.2194	−0.994397	−0.497198	−	0.867637i	$-0.665638\pi$
$151$	−12.2194	−0.994397	−0.497198	+	0.867637i	$0.665638\pi$
$152$	0	0
$153$	−3.13626	−0.253552
$154$	0	0
$155$	0.522093	0.0419355
$156$	0	0
$157$	−3.78816	−0.302328	−0.151164	−	0.988509i	$-0.548302\pi$
$157$	−3.78816	−0.302328	−0.151164	+	0.988509i	$0.548302\pi$
$158$	0	0
$159$	−8.82361	−0.699758
$160$	0	0
$161$	−25.0694	−1.97575
$162$	0	0
$163$	−12.3411	−0.966632	−0.483316	−	0.875446i	$-0.660568\pi$
$163$	−12.3411	−0.966632	−0.483316	+	0.875446i	$0.660568\pi$
$164$	0	0
$165$	18.3948	1.43203
$166$	0	0
$167$	3.32443	0.257252	0.128626	−	0.991693i	$-0.458943\pi$
$167$	3.32443	0.257252	0.128626	+	0.991693i	$0.458943\pi$
$168$	0	0
$169$	−9.98013	−0.767702
$170$	0	0
$171$	−45.2345	−3.45917
$172$	0	0
$173$	−15.2332	−1.15816	−0.579081	−	0.815270i	$-0.696588\pi$
$173$	−15.2332	−1.15816	−0.579081	+	0.815270i	$0.696588\pi$
$174$	0	0
$175$	−3.52319	−0.266328
$176$	0	0
$177$	−7.56927	−0.568942
$178$	0	0
$179$	−12.6343	−0.944329	−0.472164	−	0.881511i	$-0.656527\pi$
$179$	−12.6343	−0.944329	−0.472164	+	0.881511i	$0.656527\pi$
$180$	0	0
$181$	−5.70222	−0.423842	−0.211921	−	0.977287i	$-0.567972\pi$
$181$	−5.70222	−0.423842	−0.211921	+	0.977287i	$0.567972\pi$
$182$	0	0
$183$	41.3148	3.05408
$184$	0	0
$185$	−6.36068	−0.467647
$186$	0	0
$187$	2.14368	0.156761
$188$	0	0
$189$	59.6281	4.33731
$190$	0	0
$191$	16.7428	1.21147	0.605733	−	0.795668i	$-0.292880\pi$
$191$	16.7428	1.21147	0.605733	+	0.795668i	$0.292880\pi$
$192$	0	0
$193$	16.8560	1.21332	0.606661	−	0.794961i	$-0.292509\pi$
$193$	16.8560	1.21332	0.606661	+	0.794961i	$0.292509\pi$
$194$	0	0
$195$	−5.78542	−0.414303
$196$	0	0
$197$	−25.1703	−1.79331	−0.896655	−	0.442730i	$-0.854010\pi$
$197$	−25.1703	−1.79331	−0.896655	+	0.442730i	$0.854010\pi$
$198$	0	0
$199$	8.56757	0.607339	0.303669	−	0.952777i	$-0.401788\pi$
$199$	8.56757	0.607339	0.303669	+	0.952777i	$0.401788\pi$
$200$	0	0
$201$	9.66605	0.681791
$202$	0	0
$203$	22.2338	1.56051
$204$	0	0
$205$	−0.378927	−0.0264654
$206$	0	0
$207$	57.5195	3.99788
$208$	0	0
$209$	30.9184	2.13867
$210$	0	0
$211$	−25.0578	−1.72505	−0.862526	−	0.506012i	$-0.831119\pi$
$211$	−25.0578	−1.72505	−0.862526	+	0.506012i	$0.831119\pi$
$212$	0	0
$213$	6.93048	0.474868
$214$	0	0
$215$	7.24950	0.494412
$216$	0	0
$217$	−1.83943	−0.124869
$218$	0	0
$219$	19.7414	1.33400
$220$	0	0
$221$	−0.674218	−0.0453528
$222$	0	0
$223$	−24.6344	−1.64964	−0.824821	−	0.565394i	$-0.808724\pi$
$223$	−24.6344	−1.64964	−0.824821	+	0.565394i	$0.808724\pi$
$224$	0	0
$225$	8.08363	0.538909
$226$	0	0
$227$	−7.99145	−0.530411	−0.265206	−	0.964192i	$-0.585440\pi$
$227$	−7.99145	−0.530411	−0.265206	+	0.964192i	$0.585440\pi$
$228$	0	0
$229$	−7.67952	−0.507477	−0.253738	−	0.967273i	$-0.581660\pi$
$229$	−7.67952	−0.507477	−0.253738	+	0.967273i	$0.581660\pi$
$230$	0	0
$231$	−64.8084	−4.26408
$232$	0	0
$233$	−15.9288	−1.04353	−0.521764	−	0.853090i	$-0.674726\pi$
$233$	−15.9288	−1.04353	−0.521764	+	0.853090i	$0.674726\pi$
$234$	0	0
$235$	−10.5319	−0.687025
$236$	0	0
$237$	26.9325	1.74945
$238$	0	0
$239$	7.51600	0.486170	0.243085	−	0.970005i	$-0.421841\pi$
$239$	7.51600	0.486170	0.243085	+	0.970005i	$0.421841\pi$
$240$	0	0
$241$	−13.7076	−0.882981	−0.441491	−	0.897266i	$-0.645550\pi$
$241$	−13.7076	−0.882981	−0.441491	+	0.897266i	$0.645550\pi$
$242$	0	0
$243$	−56.0749	−3.59721
$244$	0	0
$245$	5.41286	0.345815
$246$	0	0
$247$	−9.72427	−0.618741
$248$	0	0
$249$	−14.3136	−0.907085
$250$	0	0
$251$	17.9123	1.13061	0.565307	−	0.824881i	$-0.308758\pi$
$251$	17.9123	1.13061	0.565307	+	0.824881i	$0.308758\pi$
$252$	0	0
$253$	−39.3154	−2.47174
$254$	0	0
$255$	1.29166	0.0808867
$256$	0	0
$257$	30.3400	1.89255	0.946277	−	0.323357i	$-0.104811\pi$
$257$	30.3400	1.89255	0.946277	+	0.323357i	$0.104811\pi$
$258$	0	0
$259$	22.4099	1.39248
$260$	0	0
$261$	−51.0134	−3.15765
$262$	0	0
$263$	−9.83186	−0.606258	−0.303129	−	0.952950i	$-0.598031\pi$
$263$	−9.83186	−0.606258	−0.303129	+	0.952950i	$0.598031\pi$
$264$	0	0
$265$	2.65036	0.162811
$266$	0	0
$267$	34.2252	2.09455
$268$	0	0
$269$	24.8472	1.51496	0.757479	−	0.652860i	$-0.226431\pi$
$269$	24.8472	1.51496	0.757479	+	0.652860i	$0.226431\pi$
$270$	0	0
$271$	1.65273	0.100396	0.0501981	−	0.998739i	$-0.484015\pi$
$271$	1.65273	0.100396	0.0501981	+	0.998739i	$0.484015\pi$
$272$	0	0
$273$	20.3831	1.23364
$274$	0	0
$275$	−5.52528	−0.333187
$276$	0	0
$277$	32.5275	1.95439	0.977194	−	0.212348i	$-0.0681109\pi$
$277$	32.5275	1.95439	0.977194	+	0.212348i	$0.0681109\pi$
$278$	0	0
$279$	4.22040	0.252669
$280$	0	0
$281$	24.1653	1.44158	0.720790	−	0.693153i	$-0.243779\pi$
$281$	24.1653	1.44158	0.720790	+	0.693153i	$0.243779\pi$
$282$	0	0
$283$	8.51886	0.506394	0.253197	−	0.967415i	$-0.418518\pi$
$283$	8.51886	0.506394	0.253197	+	0.967415i	$0.418518\pi$
$284$	0	0
$285$	18.6296	1.10352
$286$	0	0
$287$	1.33503	0.0788044
$288$	0	0
$289$	−16.8495	−0.991146
$290$	0	0
$291$	12.8329	0.752279
$292$	0	0
$293$	16.3056	0.952583	0.476291	−	0.879288i	$-0.341981\pi$
$293$	16.3056	0.952583	0.476291	+	0.879288i	$0.341981\pi$
$294$	0	0
$295$	2.27360	0.132374
$296$	0	0
$297$	93.5124	5.42614
$298$	0	0
$299$	12.3652	0.715100
$300$	0	0
$301$	−25.5414	−1.47218
$302$	0	0
$303$	26.5136	1.52317
$304$	0	0
$305$	−12.4098	−0.710583
$306$	0	0
$307$	30.8430	1.76030	0.880152	−	0.474691i	$-0.157440\pi$
$307$	30.8430	1.76030	0.880152	+	0.474691i	$0.157440\pi$
$308$	0	0
$309$	−32.3987	−1.84310
$310$	0	0
$311$	−15.3602	−0.870996	−0.435498	−	0.900190i	$-0.643428\pi$
$311$	−15.3602	−0.870996	−0.435498	+	0.900190i	$0.643428\pi$
$312$	0	0
$313$	−9.36416	−0.529293	−0.264647	−	0.964345i	$-0.585255\pi$
$313$	−9.36416	−0.529293	−0.264647	+	0.964345i	$0.585255\pi$
$314$	0	0
$315$	−28.4802	−1.60468
$316$	0	0
$317$	10.9725	0.616276	0.308138	−	0.951342i	$-0.400294\pi$
$317$	10.9725	0.616276	0.308138	+	0.951342i	$0.400294\pi$
$318$	0	0
$319$	34.8684	1.95225
$320$	0	0
$321$	38.8449	2.16811
$322$	0	0
$323$	2.17105	0.120800
$324$	0	0
$325$	1.73778	0.0963945
$326$	0	0
$327$	−24.7230	−1.36719
$328$	0	0
$329$	37.1059	2.04571
$330$	0	0
$331$	−12.5549	−0.690079	−0.345039	−	0.938588i	$-0.612134\pi$
$331$	−12.5549	−0.690079	−0.345039	+	0.938588i	$0.612134\pi$
$332$	0	0
$333$	−51.4174	−2.81766
$334$	0	0
$335$	−2.90341	−0.158630
$336$	0	0
$337$	−1.75915	−0.0958273	−0.0479136	−	0.998851i	$-0.515257\pi$
$337$	−1.75915	−0.0958273	−0.0479136	+	0.998851i	$0.515257\pi$
$338$	0	0
$339$	−37.3399	−2.02803
$340$	0	0
$341$	−2.88471	−0.156216
$342$	0	0
$343$	5.59180	0.301929
$344$	0	0
$345$	−23.6892	−1.27538
$346$	0	0
$347$	22.9952	1.23445	0.617224	−	0.786788i	$-0.288257\pi$
$347$	22.9952	1.23445	0.617224	+	0.786788i	$0.288257\pi$
$348$	0	0
$349$	10.9881	0.588182	0.294091	−	0.955777i	$-0.404983\pi$
$349$	10.9881	0.588182	0.294091	+	0.955777i	$0.404983\pi$
$350$	0	0
$351$	−29.4110	−1.56984
$352$	0	0
$353$	−9.29570	−0.494760	−0.247380	−	0.968919i	$-0.579570\pi$
$353$	−9.29570	−0.494760	−0.247380	+	0.968919i	$0.579570\pi$
$354$	0	0
$355$	−2.08172	−0.110486
$356$	0	0
$357$	−4.55075	−0.240851
$358$	0	0
$359$	6.92789	0.365640	0.182820	−	0.983146i	$-0.441477\pi$
$359$	6.92789	0.365640	0.182820	+	0.983146i	$0.441477\pi$
$360$	0	0
$361$	12.3131	0.648058
$362$	0	0
$363$	−65.0151	−3.41241
$364$	0	0
$365$	−5.92974	−0.310377
$366$	0	0
$367$	−22.8469	−1.19260	−0.596299	−	0.802762i	$-0.703363\pi$
$367$	−22.8469	−1.19260	−0.596299	+	0.802762i	$0.703363\pi$
$368$	0	0
$369$	−3.06310	−0.159459
$370$	0	0
$371$	−9.33773	−0.484791
$372$	0	0
$373$	34.2536	1.77358	0.886791	−	0.462170i	$-0.152929\pi$
$373$	34.2536	1.77358	0.886791	+	0.462170i	$0.152929\pi$
$374$	0	0
$375$	−3.32921	−0.171920
$376$	0	0
$377$	−10.9666	−0.564808
$378$	0	0
$379$	5.97208	0.306765	0.153383	−	0.988167i	$-0.450983\pi$
$379$	5.97208	0.306765	0.153383	+	0.988167i	$0.450983\pi$
$380$	0	0
$381$	−66.7735	−3.42091
$382$	0	0
$383$	0.609099	0.0311235	0.0155617	−	0.999879i	$-0.495046\pi$
$383$	0.609099	0.0311235	0.0155617	+	0.999879i	$0.495046\pi$
$384$	0	0
$385$	19.4666	0.992110
$386$	0	0
$387$	58.6023	2.97892
$388$	0	0
$389$	−25.5434	−1.29510	−0.647550	−	0.762023i	$-0.724206\pi$
$389$	−25.5434	−1.29510	−0.647550	+	0.762023i	$0.724206\pi$
$390$	0	0
$391$	−2.76067	−0.139613
$392$	0	0
$393$	45.6693	2.30371
$394$	0	0
$395$	−8.08975	−0.407040
$396$	0	0
$397$	7.07445	0.355057	0.177528	−	0.984116i	$-0.443190\pi$
$397$	7.07445	0.355057	0.177528	+	0.984116i	$0.443190\pi$
$398$	0	0
$399$	−65.6357	−3.28589
$400$	0	0
$401$	1.00000	0.0499376
$401$	1.00000	0.0499376
$402$	0	0
$403$	0.907280	0.0451949
$404$	0	0
$405$	32.0942	1.59477
$406$	0	0
$407$	35.1445	1.74205
$408$	0	0
$409$	−23.3593	−1.15504	−0.577520	−	0.816376i	$-0.695980\pi$
$409$	−23.3593	−1.15504	−0.577520	+	0.816376i	$0.695980\pi$
$410$	0	0
$411$	35.5954	1.75579
$412$	0	0
$413$	−8.01031	−0.394161
$414$	0	0
$415$	4.29939	0.211049
$416$	0	0
$417$	33.4135	1.63627
$418$	0	0
$419$	25.1281	1.22759	0.613794	−	0.789466i	$-0.289642\pi$
$419$	25.1281	1.22759	0.613794	+	0.789466i	$0.289642\pi$
$420$	0	0
$421$	−11.6301	−0.566816	−0.283408	−	0.958999i	$-0.591465\pi$
$421$	−11.6301	−0.566816	−0.283408	+	0.958999i	$0.591465\pi$
$422$	0	0
$423$	−85.1359	−4.13945
$424$	0	0
$425$	−0.387977	−0.0188197
$426$	0	0
$427$	43.7221	2.11586
$428$	0	0
$429$	31.9661	1.54334
$430$	0	0
$431$	−21.2487	−1.02352	−0.511758	−	0.859130i	$-0.671005\pi$
$431$	−21.2487	−1.02352	−0.511758	+	0.859130i	$0.671005\pi$
$432$	0	0
$433$	18.0804	0.868886	0.434443	−	0.900699i	$-0.356945\pi$
$433$	18.0804	0.868886	0.434443	+	0.900699i	$0.356945\pi$
$434$	0	0
$435$	21.0096	1.00733
$436$	0	0
$437$	−39.8173	−1.90472
$438$	0	0
$439$	21.9207	1.04622	0.523109	−	0.852266i	$-0.324772\pi$
$439$	21.9207	1.04622	0.523109	+	0.852266i	$0.324772\pi$
$440$	0	0
$441$	43.7556	2.08360
$442$	0	0
$443$	25.3202	1.20300	0.601499	−	0.798873i	$-0.294570\pi$
$443$	25.3202	1.20300	0.601499	+	0.798873i	$0.294570\pi$
$444$	0	0
$445$	−10.2803	−0.487332
$446$	0	0
$447$	−1.16545	−0.0551238
$448$	0	0
$449$	−7.82913	−0.369479	−0.184740	−	0.982787i	$-0.559144\pi$
$449$	−7.82913	−0.369479	−0.184740	+	0.982787i	$0.559144\pi$
$450$	0	0
$451$	2.09368	0.0985874
$452$	0	0
$453$	40.6808	1.91135
$454$	0	0
$455$	−6.12252	−0.287028
$456$	0	0
$457$	8.68277	0.406163	0.203081	−	0.979162i	$-0.434904\pi$
$457$	8.68277	0.406163	0.203081	+	0.979162i	$0.434904\pi$
$458$	0	0
$459$	6.56631	0.306489
$460$	0	0
$461$	14.3552	0.668589	0.334294	−	0.942469i	$-0.391502\pi$
$461$	14.3552	0.668589	0.334294	+	0.942469i	$0.391502\pi$
$462$	0	0
$463$	35.4161	1.64593	0.822963	−	0.568094i	$-0.192319\pi$
$463$	35.4161	1.64593	0.822963	+	0.568094i	$0.192319\pi$
$464$	0	0
$465$	−1.73816	−0.0806050
$466$	0	0
$467$	−15.3096	−0.708446	−0.354223	−	0.935161i	$-0.615255\pi$
$467$	−15.3096	−0.708446	−0.354223	+	0.935161i	$0.615255\pi$
$468$	0	0
$469$	10.2293	0.472343
$470$	0	0
$471$	12.6116	0.581111
$472$	0	0
$473$	−40.0555	−1.84176
$474$	0	0
$475$	−5.59581	−0.256753
$476$	0	0
$477$	21.4246	0.980963
$478$	0	0
$479$	14.4266	0.659169	0.329585	−	0.944126i	$-0.393091\pi$
$479$	14.4266	0.659169	0.329585	+	0.944126i	$0.393091\pi$
$480$	0	0
$481$	−11.0534	−0.503994
$482$	0	0
$483$	83.4614	3.79762
$484$	0	0
$485$	−3.85465	−0.175030
$486$	0	0
$487$	−31.3549	−1.42082	−0.710412	−	0.703786i	$-0.751491\pi$
$487$	−31.3549	−1.42082	−0.710412	+	0.703786i	$0.751491\pi$
$488$	0	0
$489$	41.0862	1.85798
$490$	0	0
$491$	29.3116	1.32282	0.661408	−	0.750026i	$-0.269959\pi$
$491$	29.3116	1.32282	0.661408	+	0.750026i	$0.269959\pi$
$492$	0	0
$493$	2.44841	0.110271
$494$	0	0
$495$	−44.6643	−2.00751
$496$	0	0
$497$	7.33429	0.328988
$498$	0	0
$499$	30.7006	1.37435	0.687173	−	0.726494i	$-0.258852\pi$
$499$	30.7006	1.37435	0.687173	+	0.726494i	$0.258852\pi$
$500$	0	0
$501$	−11.0677	−0.494470
$502$	0	0
$503$	14.3924	0.641726	0.320863	−	0.947126i	$-0.396027\pi$
$503$	14.3924	0.641726	0.320863	+	0.947126i	$0.396027\pi$
$504$	0	0
$505$	−7.96394	−0.354391
$506$	0	0
$507$	33.2259	1.47562
$508$	0	0
$509$	−25.7988	−1.14351	−0.571757	−	0.820423i	$-0.693738\pi$
$509$	−25.7988	−1.14351	−0.571757	+	0.820423i	$0.693738\pi$
$510$	0	0
$511$	20.8916	0.924190
$512$	0	0
$513$	94.7061	4.18138
$514$	0	0
$515$	9.73166	0.428828
$516$	0	0
$517$	58.1916	2.55927
$518$	0	0
$519$	50.7146	2.22612
$520$	0	0
$521$	−29.1557	−1.27733	−0.638667	−	0.769483i	$-0.720514\pi$
$521$	−29.1557	−1.27733	−0.638667	+	0.769483i	$0.720514\pi$
$522$	0	0
$523$	16.3934	0.716834	0.358417	−	0.933562i	$-0.383317\pi$
$523$	16.3934	0.716834	0.358417	+	0.933562i	$0.383317\pi$
$524$	0	0
$525$	11.7294	0.511914
$526$	0	0
$527$	−0.202560	−0.00882365
$528$	0	0
$529$	27.6311	1.20135
$530$	0	0
$531$	18.3789	0.797577
$532$	0	0
$533$	−0.658490	−0.0285224
$534$	0	0
$535$	−11.6679	−0.504448
$536$	0	0
$537$	42.0621	1.81511
$538$	0	0
$539$	−29.9075	−1.28821
$540$	0	0
$541$	−13.3310	−0.573144	−0.286572	−	0.958059i	$-0.592516\pi$
$541$	−13.3310	−0.573144	−0.286572	+	0.958059i	$0.592516\pi$
$542$	0	0
$543$	18.9839	0.814676
$544$	0	0
$545$	7.42609	0.318099
$546$	0	0
$547$	6.68959	0.286026	0.143013	−	0.989721i	$-0.454321\pi$
$547$	6.68959	0.286026	0.143013	+	0.989721i	$0.454321\pi$
$548$	0	0
$549$	−100.316	−4.28139
$550$	0	0
$551$	35.3135	1.50440
$552$	0	0
$553$	28.5017	1.21202
$554$	0	0
$555$	21.1760	0.898873
$556$	0	0
$557$	39.2134	1.66152	0.830762	−	0.556628i	$-0.187905\pi$
$557$	39.2134	1.66152	0.830762	+	0.556628i	$0.187905\pi$
$558$	0	0
$559$	12.5980	0.532840
$560$	0	0
$561$	−7.13676	−0.301314
$562$	0	0
$563$	−6.57461	−0.277087	−0.138543	−	0.990356i	$-0.544242\pi$
$563$	−6.57461	−0.277087	−0.138543	+	0.990356i	$0.544242\pi$
$564$	0	0
$565$	11.2159	0.471855
$566$	0	0
$567$	−113.074	−4.74866
$568$	0	0
$569$	15.6404	0.655678	0.327839	−	0.944734i	$-0.393680\pi$
$569$	15.6404	0.655678	0.327839	+	0.944734i	$0.393680\pi$
$570$	0	0
$571$	35.0018	1.46478	0.732391	−	0.680885i	$-0.238405\pi$
$571$	35.0018	1.46478	0.732391	+	0.680885i	$0.238405\pi$
$572$	0	0
$573$	−55.7402	−2.32858
$574$	0	0
$575$	7.11555	0.296739
$576$	0	0
$577$	14.7384	0.613569	0.306784	−	0.951779i	$-0.400747\pi$
$577$	14.7384	0.613569	0.306784	+	0.951779i	$0.400747\pi$
$578$	0	0
$579$	−56.1172	−2.33215
$580$	0	0
$581$	−15.1476	−0.628426
$582$	0	0
$583$	−14.6440	−0.606492
$584$	0	0
$585$	14.0475	0.580795
$586$	0	0
$587$	−47.1194	−1.94483	−0.972413	−	0.233267i	$-0.925059\pi$
$587$	−47.1194	−1.94483	−0.972413	+	0.233267i	$0.925059\pi$
$588$	0	0
$589$	−2.92153	−0.120380
$590$	0	0
$591$	83.7972	3.44696
$592$	0	0
$593$	19.2906	0.792171	0.396086	−	0.918214i	$-0.370368\pi$
$593$	19.2906	0.792171	0.396086	+	0.918214i	$0.370368\pi$
$594$	0	0
$595$	1.36692	0.0560381
$596$	0	0
$597$	−28.5232	−1.16738
$598$	0	0
$599$	18.0078	0.735779	0.367889	−	0.929870i	$-0.380081\pi$
$599$	18.0078	0.735779	0.367889	+	0.929870i	$0.380081\pi$
$600$	0	0
$601$	33.2601	1.35671	0.678354	−	0.734735i	$-0.262694\pi$
$601$	33.2601	1.35671	0.678354	+	0.734735i	$0.262694\pi$
$602$	0	0
$603$	−23.4701	−0.955775
$604$	0	0
$605$	19.5287	0.793955
$606$	0	0
$607$	41.6848	1.69193	0.845966	−	0.533236i	$-0.179024\pi$
$607$	41.6848	1.69193	0.845966	+	0.533236i	$0.179024\pi$
$608$	0	0
$609$	−74.0209	−2.99948
$610$	0	0
$611$	−18.3021	−0.740423
$612$	0	0
$613$	−12.9945	−0.524844	−0.262422	−	0.964953i	$-0.584521\pi$
$613$	−12.9945	−0.524844	−0.262422	+	0.964953i	$0.584521\pi$
$614$	0	0
$615$	1.26153	0.0508697
$616$	0	0
$617$	−20.4959	−0.825132	−0.412566	−	0.910928i	$-0.635367\pi$
$617$	−20.4959	−0.825132	−0.412566	+	0.910928i	$0.635367\pi$
$618$	0	0
$619$	−0.209415	−0.00841712	−0.00420856	−	0.999991i	$-0.501340\pi$
$619$	−0.209415	−0.00841712	−0.00420856	+	0.999991i	$0.501340\pi$
$620$	0	0
$621$	−120.427	−4.83257
$622$	0	0
$623$	36.2194	1.45110
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−102.934	−4.11078
$628$	0	0
$629$	2.46780	0.0983976
$630$	0	0
$631$	−16.8125	−0.669296	−0.334648	−	0.942343i	$-0.608617\pi$
$631$	−16.8125	−0.669296	−0.334648	+	0.942343i	$0.608617\pi$
$632$	0	0
$633$	83.4228	3.31576
$634$	0	0
$635$	20.0569	0.795932
$636$	0	0
$637$	9.40634	0.372693
$638$	0	0
$639$	−16.8278	−0.665699
$640$	0	0
$641$	44.1851	1.74521	0.872603	−	0.488430i	$-0.162430\pi$
$641$	44.1851	1.74521	0.872603	+	0.488430i	$0.162430\pi$
$642$	0	0
$643$	1.10631	0.0436287	0.0218143	−	0.999762i	$-0.493056\pi$
$643$	1.10631	0.0436287	0.0218143	+	0.999762i	$0.493056\pi$
$644$	0	0
$645$	−24.1351	−0.950319
$646$	0	0
$647$	−23.6633	−0.930300	−0.465150	−	0.885232i	$-0.654000\pi$
$647$	−23.6633	−0.930300	−0.465150	+	0.885232i	$0.654000\pi$
$648$	0	0
$649$	−12.5622	−0.493111
$650$	0	0
$651$	6.12385	0.240013
$652$	0	0
$653$	36.0014	1.40884	0.704421	−	0.709782i	$-0.251207\pi$
$653$	36.0014	1.40884	0.704421	+	0.709782i	$0.251207\pi$
$654$	0	0
$655$	−13.7178	−0.535998
$656$	0	0
$657$	−47.9339	−1.87008
$658$	0	0
$659$	−32.6542	−1.27203	−0.636013	−	0.771678i	$-0.719418\pi$
$659$	−32.6542	−1.27203	−0.636013	+	0.771678i	$0.719418\pi$
$660$	0	0
$661$	−16.8672	−0.656060	−0.328030	−	0.944667i	$-0.606385\pi$
$661$	−16.8672	−0.656060	−0.328030	+	0.944667i	$0.606385\pi$
$662$	0	0
$663$	2.24461	0.0871735
$664$	0	0
$665$	19.7151	0.764519
$666$	0	0
$667$	−44.9041	−1.73869
$668$	0	0
$669$	82.0131	3.17081
$670$	0	0
$671$	68.5676	2.64702
$672$	0	0
$673$	−10.0636	−0.387924	−0.193962	−	0.981009i	$-0.562134\pi$
$673$	−10.0636	−0.387924	−0.193962	+	0.981009i	$0.562134\pi$
$674$	0	0
$675$	−16.9245	−0.651423
$676$	0	0
$677$	4.33195	0.166490	0.0832451	−	0.996529i	$-0.473472\pi$
$677$	4.33195	0.166490	0.0832451	+	0.996529i	$0.473472\pi$
$678$	0	0
$679$	13.5806	0.521177
$680$	0	0
$681$	26.6052	1.01951
$682$	0	0
$683$	−3.37360	−0.129087	−0.0645437	−	0.997915i	$-0.520559\pi$
$683$	−3.37360	−0.129087	−0.0645437	+	0.997915i	$0.520559\pi$
$684$	0	0
$685$	−10.6918	−0.408515
$686$	0	0
$687$	25.5667	0.975431
$688$	0	0
$689$	4.60574	0.175465
$690$	0	0
$691$	32.5945	1.23995	0.619976	−	0.784621i	$-0.287142\pi$
$691$	32.5945	1.23995	0.619976	+	0.784621i	$0.287142\pi$
$692$	0	0
$693$	157.361	5.97764
$694$	0	0
$695$	−10.0365	−0.380705
$696$	0	0
$697$	0.147015	0.00556859
$698$	0	0
$699$	53.0302	2.00579
$700$	0	0
$701$	−18.5703	−0.701391	−0.350696	−	0.936489i	$-0.614055\pi$
$701$	−18.5703	−0.701391	−0.350696	+	0.936489i	$0.614055\pi$
$702$	0	0
$703$	35.5932	1.34242
$704$	0	0
$705$	35.0629	1.32054
$706$	0	0
$707$	28.0585	1.05525
$708$	0	0
$709$	−41.6997	−1.56607	−0.783033	−	0.621980i	$-0.786329\pi$
$709$	−41.6997	−1.56607	−0.783033	+	0.621980i	$0.786329\pi$
$710$	0	0
$711$	−65.3946	−2.45249
$712$	0	0
$713$	3.71498	0.139127
$714$	0	0
$715$	−9.60170	−0.359083
$716$	0	0
$717$	−25.0223	−0.934477
$718$	0	0
$719$	36.4863	1.36071	0.680356	−	0.732882i	$-0.261825\pi$
$719$	36.4863	1.36071	0.680356	+	0.732882i	$0.261825\pi$
$720$	0	0
$721$	−34.2865	−1.27689
$722$	0	0
$723$	45.6353	1.69720
$724$	0	0
$725$	−6.31070	−0.234373
$726$	0	0
$727$	16.6843	0.618787	0.309393	−	0.950934i	$-0.399874\pi$
$727$	16.6843	0.618787	0.309393	+	0.950934i	$0.399874\pi$
$728$	0	0
$729$	90.4024	3.34824
$730$	0	0
$731$	−2.81264	−0.104029
$732$	0	0
$733$	11.7386	0.433574	0.216787	−	0.976219i	$-0.430442\pi$
$733$	11.7386	0.433574	0.216787	+	0.976219i	$0.430442\pi$
$734$	0	0
$735$	−18.0205	−0.664697
$736$	0	0
$737$	16.0421	0.590920
$738$	0	0
$739$	−9.97439	−0.366914	−0.183457	−	0.983028i	$-0.558729\pi$
$739$	−9.97439	−0.366914	−0.183457	+	0.983028i	$0.558729\pi$
$740$	0	0
$741$	32.3741	1.18929
$742$	0	0
$743$	−27.5671	−1.01134	−0.505670	−	0.862727i	$-0.668755\pi$
$743$	−27.5671	−1.01134	−0.505670	+	0.862727i	$0.668755\pi$
$744$	0	0
$745$	0.350067	0.0128255
$746$	0	0
$747$	34.7547	1.27161
$748$	0	0
$749$	41.1083	1.50206
$750$	0	0
$751$	−7.82038	−0.285370	−0.142685	−	0.989768i	$-0.545574\pi$
$751$	−7.82038	−0.285370	−0.142685	+	0.989768i	$0.545574\pi$
$752$	0	0
$753$	−59.6338	−2.17318
$754$	0	0
$755$	−12.2194	−0.444708
$756$	0	0
$757$	−13.4393	−0.488459	−0.244229	−	0.969717i	$-0.578535\pi$
$757$	−13.4393	−0.488459	−0.244229	+	0.969717i	$0.578535\pi$
$758$	0	0
$759$	130.889	4.75098
$760$	0	0
$761$	−20.6221	−0.747552	−0.373776	−	0.927519i	$-0.621937\pi$
$761$	−20.6221	−0.747552	−0.373776	+	0.927519i	$0.621937\pi$
$762$	0	0
$763$	−26.1635	−0.947183
$764$	0	0
$765$	−3.13626	−0.113392
$766$	0	0
$767$	3.95100	0.142662
$768$	0	0
$769$	5.16314	0.186188	0.0930938	−	0.995657i	$-0.470324\pi$
$769$	5.16314	0.186188	0.0930938	+	0.995657i	$0.470324\pi$
$770$	0	0
$771$	−101.008	−3.63772
$772$	0	0
$773$	−18.7570	−0.674642	−0.337321	−	0.941390i	$-0.609521\pi$
$773$	−18.7570	−0.674642	−0.337321	+	0.941390i	$0.609521\pi$
$774$	0	0
$775$	0.522093	0.0187541
$776$	0	0
$777$	−74.6072	−2.67652
$778$	0	0
$779$	2.12040	0.0759713
$780$	0	0
$781$	11.5021	0.411577
$782$	0	0
$783$	106.805	3.81691
$784$	0	0
$785$	−3.78816	−0.135205
$786$	0	0
$787$	54.5548	1.94467	0.972334	−	0.233595i	$-0.0750491\pi$
$787$	54.5548	1.94467	0.972334	+	0.233595i	$0.0750491\pi$
$788$	0	0
$789$	32.7323	1.16530
$790$	0	0
$791$	−39.5156	−1.40501
$792$	0	0
$793$	−21.5655	−0.765812
$794$	0	0
$795$	−8.82361	−0.312941
$796$	0	0
$797$	−8.46737	−0.299930	−0.149965	−	0.988691i	$-0.547916\pi$
$797$	−8.46737	−0.299930	−0.149965	+	0.988691i	$0.547916\pi$
$798$	0	0
$799$	4.08613	0.144557
$800$	0	0
$801$	−83.1020	−2.93626
$802$	0	0
$803$	32.7635	1.15620
$804$	0	0
$805$	−25.0694	−0.883581
$806$	0	0
$807$	−82.7214	−2.91193
$808$	0	0
$809$	22.8394	0.802990	0.401495	−	0.915861i	$-0.368491\pi$
$809$	22.8394	0.802990	0.401495	+	0.915861i	$0.368491\pi$
$810$	0	0
$811$	48.6554	1.70852	0.854261	−	0.519845i	$-0.174010\pi$
$811$	48.6554	1.70852	0.854261	+	0.519845i	$0.174010\pi$
$812$	0	0
$813$	−5.50228	−0.192973
$814$	0	0
$815$	−12.3411	−0.432291
$816$	0	0
$817$	−40.5669	−1.41925
$818$	0	0
$819$	−49.4922	−1.72940
$820$	0	0
$821$	47.3604	1.65289	0.826444	−	0.563018i	$-0.190360\pi$
$821$	47.3604	1.65289	0.826444	+	0.563018i	$0.190360\pi$
$822$	0	0
$823$	23.9653	0.835378	0.417689	−	0.908590i	$-0.362840\pi$
$823$	23.9653	0.835378	0.417689	+	0.908590i	$0.362840\pi$
$824$	0	0
$825$	18.3948	0.640425
$826$	0	0
$827$	−9.80582	−0.340982	−0.170491	−	0.985359i	$-0.554535\pi$
$827$	−9.80582	−0.340982	−0.170491	+	0.985359i	$0.554535\pi$
$828$	0	0
$829$	−46.5094	−1.61534	−0.807669	−	0.589636i	$-0.799271\pi$
$829$	−46.5094	−1.61534	−0.807669	+	0.589636i	$0.799271\pi$
$830$	0	0
$831$	−108.291	−3.75657
$832$	0	0
$833$	−2.10006	−0.0727629
$834$	0	0
$835$	3.32443	0.115047
$836$	0	0
$837$	−8.83614	−0.305422
$838$	0	0
$839$	−44.6851	−1.54270	−0.771350	−	0.636411i	$-0.780418\pi$
$839$	−44.6851	−1.54270	−0.771350	+	0.636411i	$0.780418\pi$
$840$	0	0
$841$	10.8249	0.373273
$842$	0	0
$843$	−80.4513	−2.77089
$844$	0	0
$845$	−9.98013	−0.343327
$846$	0	0
$847$	−68.8033	−2.36411
$848$	0	0
$849$	−28.3611	−0.973349
$850$	0	0
$851$	−45.2598	−1.55148
$852$	0	0
$853$	−32.9669	−1.12877	−0.564383	−	0.825513i	$-0.690886\pi$
$853$	−32.9669	−1.12877	−0.564383	+	0.825513i	$0.690886\pi$
$854$	0	0
$855$	−45.2345	−1.54699
$856$	0	0
$857$	−19.4828	−0.665521	−0.332761	−	0.943011i	$-0.607980\pi$
$857$	−19.4828	−0.665521	−0.332761	+	0.943011i	$0.607980\pi$
$858$	0	0
$859$	6.73928	0.229941	0.114971	−	0.993369i	$-0.463323\pi$
$859$	6.73928	0.229941	0.114971	+	0.993369i	$0.463323\pi$
$860$	0	0
$861$	−4.44460	−0.151471
$862$	0	0
$863$	33.8087	1.15086	0.575431	−	0.817851i	$-0.304834\pi$
$863$	33.8087	1.15086	0.575431	+	0.817851i	$0.304834\pi$
$864$	0	0
$865$	−15.2332	−0.517945
$866$	0	0
$867$	56.0954	1.90510
$868$	0	0
$869$	44.6981	1.51628
$870$	0	0
$871$	−5.04547	−0.170959
$872$	0	0
$873$	−31.1595	−1.05459
$874$	0	0
$875$	−3.52319	−0.119106
$876$	0	0
$877$	−23.6520	−0.798671	−0.399336	−	0.916805i	$-0.630759\pi$
$877$	−23.6520	−0.798671	−0.399336	+	0.916805i	$0.630759\pi$
$878$	0	0
$879$	−54.2847	−1.83098
$880$	0	0
$881$	−46.5868	−1.56955	−0.784775	−	0.619781i	$-0.787222\pi$
$881$	−46.5868	−1.56955	−0.784775	+	0.619781i	$0.787222\pi$
$882$	0	0
$883$	34.2188	1.15155	0.575777	−	0.817607i	$-0.304700\pi$
$883$	34.2188	1.15155	0.575777	+	0.817607i	$0.304700\pi$
$884$	0	0
$885$	−7.56927	−0.254438
$886$	0	0
$887$	20.1754	0.677423	0.338712	−	0.940890i	$-0.390009\pi$
$887$	20.1754	0.677423	0.338712	+	0.940890i	$0.390009\pi$
$888$	0	0
$889$	−70.6641	−2.37000
$890$	0	0
$891$	−177.329	−5.94076
$892$	0	0
$893$	58.9345	1.97217
$894$	0	0
$895$	−12.6343	−0.422317
$896$	0	0
$897$	−41.1665	−1.37451
$898$	0	0
$899$	−3.29477	−0.109887
$900$	0	0
$901$	−1.02828	−0.0342570
$902$	0	0
$903$	85.0326	2.82971
$904$	0	0
$905$	−5.70222	−0.189548
$906$	0	0
$907$	2.96954	0.0986018	0.0493009	−	0.998784i	$-0.484301\pi$
$907$	2.96954	0.0986018	0.0493009	+	0.998784i	$0.484301\pi$
$908$	0	0
$909$	−64.3776	−2.13527
$910$	0	0
$911$	29.0975	0.964044	0.482022	−	0.876159i	$-0.339902\pi$
$911$	29.0975	0.964044	0.482022	+	0.876159i	$0.339902\pi$
$912$	0	0
$913$	−23.7553	−0.786186
$914$	0	0
$915$	41.3148	1.36583
$916$	0	0
$917$	48.3303	1.59601
$918$	0	0
$919$	38.9984	1.28644	0.643220	−	0.765682i	$-0.277598\pi$
$919$	38.9984	1.28644	0.643220	+	0.765682i	$0.277598\pi$
$920$	0	0
$921$	−102.683	−3.38352
$922$	0	0
$923$	−3.61756	−0.119074
$924$	0	0
$925$	−6.36068	−0.209138
$926$	0	0
$927$	78.6671	2.58377
$928$	0	0
$929$	34.2944	1.12516	0.562581	−	0.826742i	$-0.309808\pi$
$929$	34.2944	1.12516	0.562581	+	0.826742i	$0.309808\pi$
$930$	0	0
$931$	−30.2893	−0.992693
$932$	0	0
$933$	51.1372	1.67416
$934$	0	0
$935$	2.14368	0.0701059
$936$	0	0
$937$	43.9481	1.43572	0.717861	−	0.696187i	$-0.245122\pi$
$937$	43.9481	1.43572	0.717861	+	0.696187i	$0.245122\pi$
$938$	0	0
$939$	31.1752	1.01737
$940$	0	0
$941$	−8.11656	−0.264592	−0.132296	−	0.991210i	$-0.542235\pi$
$941$	−8.11656	−0.264592	−0.132296	+	0.991210i	$0.542235\pi$
$942$	0	0
$943$	−2.69627	−0.0878027
$944$	0	0
$945$	59.6281	1.93970
$946$	0	0
$947$	−8.65365	−0.281206	−0.140603	−	0.990066i	$-0.544904\pi$
$947$	−8.65365	−0.281206	−0.140603	+	0.990066i	$0.544904\pi$
$948$	0	0
$949$	−10.3046	−0.334501
$950$	0	0
$951$	−36.5297	−1.18456
$952$	0	0
$953$	9.73648	0.315395	0.157698	−	0.987487i	$-0.449593\pi$
$953$	9.73648	0.315395	0.157698	+	0.987487i	$0.449593\pi$
$954$	0	0
$955$	16.7428	0.541784
$956$	0	0
$957$	−116.084	−3.75247
$958$	0	0
$959$	37.6694	1.21641
$960$	0	0
$961$	−30.7274	−0.991207
$962$	0	0
$963$	−94.3191	−3.03939
$964$	0	0
$965$	16.8560	0.542614
$966$	0	0
$967$	−27.3305	−0.878888	−0.439444	−	0.898270i	$-0.644825\pi$
$967$	−27.3305	−0.878888	−0.439444	+	0.898270i	$0.644825\pi$
$968$	0	0
$969$	−7.22787	−0.232193
$970$	0	0
$971$	20.6894	0.663953	0.331977	−	0.943288i	$-0.392284\pi$
$971$	20.6894	0.663953	0.331977	+	0.943288i	$0.392284\pi$
$972$	0	0
$973$	35.3604	1.13360
$974$	0	0
$975$	−5.78542	−0.185282
$976$	0	0
$977$	13.8460	0.442972	0.221486	−	0.975164i	$-0.428909\pi$
$977$	13.8460	0.442972	0.221486	+	0.975164i	$0.428909\pi$
$978$	0	0
$979$	56.8014	1.81538
$980$	0	0
$981$	60.0298	1.91660
$982$	0	0
$983$	−13.8718	−0.442440	−0.221220	−	0.975224i	$-0.571004\pi$
$983$	−13.8718	−0.442440	−0.221220	+	0.975224i	$0.571004\pi$
$984$	0	0
$985$	−25.1703	−0.801993
$986$	0	0
$987$	−123.533	−3.93210
$988$	0	0
$989$	51.5842	1.64028
$990$	0	0
$991$	36.0743	1.14594	0.572968	−	0.819578i	$-0.305792\pi$
$991$	36.0743	1.14594	0.572968	+	0.819578i	$0.305792\pi$
$992$	0	0
$993$	41.7978	1.32641
$994$	0	0
$995$	8.56757	0.271610
$996$	0	0
$997$	15.5095	0.491191	0.245596	−	0.969372i	$-0.421017\pi$
$997$	15.5095	0.491191	0.245596	+	0.969372i	$0.421017\pi$
$998$	0	0
$999$	107.651	3.40593

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8020.2.a.f.1.2	✓	37

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8020.2.a.f.1.2	✓	37	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-3.32921 q^{3} +1.00000 q^{5} -3.52319 q^{7} +8.08363 q^{9} +O(q^{10})\)	\(q-3.32921 q^{3} +1.00000 q^{5} -3.52319 q^{7} +8.08363 q^{9} -5.52528 q^{11} +1.73778 q^{13} -3.32921 q^{15} -0.387977 q^{17} -5.59581 q^{19} +11.7294 q^{21} +7.11555 q^{23} +1.00000 q^{25} -16.9245 q^{27} -6.31070 q^{29} +0.522093 q^{31} +18.3948 q^{33} -3.52319 q^{35} -6.36068 q^{37} -5.78542 q^{39} -0.378927 q^{41} +7.24950 q^{43} +8.08363 q^{45} -10.5319 q^{47} +5.41286 q^{49} +1.29166 q^{51} +2.65036 q^{53} -5.52528 q^{55} +18.6296 q^{57} +2.27360 q^{59} -12.4098 q^{61} -28.4802 q^{63} +1.73778 q^{65} -2.90341 q^{67} -23.6892 q^{69} -2.08172 q^{71} -5.92974 q^{73} -3.32921 q^{75} +19.4666 q^{77} -8.08975 q^{79} +32.0942 q^{81} +4.29939 q^{83} -0.387977 q^{85} +21.0096 q^{87} -10.2803 q^{89} -6.12252 q^{91} -1.73816 q^{93} -5.59581 q^{95} -3.85465 q^{97} -44.6643 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 37 q + 3 q^{3} + 37 q^{5} + 4 q^{7} + 50 q^{9}+O(q^{10}) \) 37 * q + 3 * q^3 + 37 * q^5 + 4 * q^7 + 50 * q^9	\( 37 q + 3 q^{3} + 37 q^{5} + 4 q^{7} + 50 q^{9} + 2 q^{11} + 27 q^{13} + 3 q^{15} + 36 q^{17} - 6 q^{19} + 20 q^{21} + 17 q^{23} + 37 q^{25} + 9 q^{27} + 29 q^{29} + 5 q^{31} + 36 q^{33} + 4 q^{35} + 35 q^{37} + 21 q^{39} + 24 q^{41} + 11 q^{43} + 50 q^{45} + 19 q^{47} + 57 q^{49} + 8 q^{51} + 65 q^{53} + 2 q^{55} + 62 q^{57} - 9 q^{59} + 13 q^{61} + 26 q^{63} + 27 q^{65} + 13 q^{67} + 20 q^{69} + 33 q^{71} + 67 q^{73} + 3 q^{75} + 62 q^{77} + 23 q^{79} + 97 q^{81} + 2 q^{83} + 36 q^{85} + 32 q^{87} + 34 q^{89} + q^{91} + 41 q^{93} - 6 q^{95} + 66 q^{97} - 7 q^{99}+O(q^{100}) \) 37 * q + 3 * q^3 + 37 * q^5 + 4 * q^7 + 50 * q^9 + 2 * q^11 + 27 * q^13 + 3 * q^15 + 36 * q^17 - 6 * q^19 + 20 * q^21 + 17 * q^23 + 37 * q^25 + 9 * q^27 + 29 * q^29 + 5 * q^31 + 36 * q^33 + 4 * q^35 + 35 * q^37 + 21 * q^39 + 24 * q^41 + 11 * q^43 + 50 * q^45 + 19 * q^47 + 57 * q^49 + 8 * q^51 + 65 * q^53 + 2 * q^55 + 62 * q^57 - 9 * q^59 + 13 * q^61 + 26 * q^63 + 27 * q^65 + 13 * q^67 + 20 * q^69 + 33 * q^71 + 67 * q^73 + 3 * q^75 + 62 * q^77 + 23 * q^79 + 97 * q^81 + 2 * q^83 + 36 * q^85 + 32 * q^87 + 34 * q^89 + q^91 + 41 * q^93 - 6 * q^95 + 66 * q^97 - 7 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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