LMFDB - Embedded newform 8020.2.a.c.1.24

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			1.24
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)

$f(q)$	$=$	$q+2.32475 q^{3} -1.00000 q^{5} -2.44792 q^{7} +2.40447 q^{9} +O(q^{10})$	$q+2.32475 q^{3} -1.00000 q^{5} -2.44792 q^{7} +2.40447 q^{9} -5.19969 q^{11} +2.45589 q^{13} -2.32475 q^{15} +3.30408 q^{17} +7.37799 q^{19} -5.69080 q^{21} -2.41239 q^{23} +1.00000 q^{25} -1.38446 q^{27} -0.883029 q^{29} -0.774173 q^{31} -12.0880 q^{33} +2.44792 q^{35} -3.11454 q^{37} +5.70933 q^{39} +11.0899 q^{41} -7.16054 q^{43} -2.40447 q^{45} +5.67529 q^{47} -1.00771 q^{49} +7.68117 q^{51} -10.4209 q^{53} +5.19969 q^{55} +17.1520 q^{57} -8.65032 q^{59} -11.8162 q^{61} -5.88594 q^{63} -2.45589 q^{65} +7.03182 q^{67} -5.60822 q^{69} -7.87312 q^{71} +8.30031 q^{73} +2.32475 q^{75} +12.7284 q^{77} -12.0439 q^{79} -10.4319 q^{81} -4.17482 q^{83} -3.30408 q^{85} -2.05282 q^{87} -9.39112 q^{89} -6.01181 q^{91} -1.79976 q^{93} -7.37799 q^{95} -16.7742 q^{97} -12.5025 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	2.32475	1.34220	0.671098	−	0.741369i	$-0.265823\pi$
$3$	2.32475	1.34220	0.671098	+	0.741369i	$0.265823\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−2.44792	−0.925225	−0.462612	−	0.886561i	$-0.653088\pi$
$7$	−2.44792	−0.925225	−0.462612	+	0.886561i	$0.653088\pi$
$8$	0	0
$9$	2.40447	0.801490
$10$	0	0
$11$	−5.19969	−1.56777	−0.783883	−	0.620909i	$-0.786764\pi$
$11$	−5.19969	−1.56777	−0.783883	+	0.620909i	$0.786764\pi$
$12$	0	0
$13$	2.45589	0.681141	0.340571	−	0.940219i	$-0.389380\pi$
$13$	2.45589	0.681141	0.340571	+	0.940219i	$0.389380\pi$
$14$	0	0
$15$	−2.32475	−0.600248
$16$	0	0
$17$	3.30408	0.801357	0.400679	−	0.916219i	$-0.368774\pi$
$17$	3.30408	0.801357	0.400679	+	0.916219i	$0.368774\pi$
$18$	0	0
$19$	7.37799	1.69263	0.846313	−	0.532685i	$-0.178817\pi$
$19$	7.37799	1.69263	0.846313	+	0.532685i	$0.178817\pi$
$20$	0	0
$21$	−5.69080	−1.24183
$22$	0	0
$23$	−2.41239	−0.503019	−0.251510	−	0.967855i	$-0.580927\pi$
$23$	−2.41239	−0.503019	−0.251510	+	0.967855i	$0.580927\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.38446	−0.266439
$28$	0	0
$29$	−0.883029	−0.163974	−0.0819872	−	0.996633i	$-0.526127\pi$
$29$	−0.883029	−0.163974	−0.0819872	+	0.996633i	$0.526127\pi$
$30$	0	0
$31$	−0.774173	−0.139046	−0.0695228	−	0.997580i	$-0.522148\pi$
$31$	−0.774173	−0.139046	−0.0695228	+	0.997580i	$0.522148\pi$
$32$	0	0
$33$	−12.0880	−2.10425
$34$	0	0
$35$	2.44792	0.413773
$36$	0	0
$37$	−3.11454	−0.512028	−0.256014	−	0.966673i	$-0.582409\pi$
$37$	−3.11454	−0.512028	−0.256014	+	0.966673i	$0.582409\pi$
$38$	0	0
$39$	5.70933	0.914225
$40$	0	0
$41$	11.0899	1.73196	0.865979	−	0.500081i	$-0.166696\pi$
$41$	11.0899	1.73196	0.865979	+	0.500081i	$0.166696\pi$
$42$	0	0
$43$	−7.16054	−1.09197	−0.545986	−	0.837794i	$-0.683845\pi$
$43$	−7.16054	−1.09197	−0.545986	+	0.837794i	$0.683845\pi$
$44$	0	0
$45$	−2.40447	−0.358437
$46$	0	0
$47$	5.67529	0.827826	0.413913	−	0.910316i	$-0.364162\pi$
$47$	5.67529	0.827826	0.413913	+	0.910316i	$0.364162\pi$
$48$	0	0
$49$	−1.00771	−0.143959
$50$	0	0
$51$	7.68117	1.07558
$52$	0	0
$53$	−10.4209	−1.43142	−0.715712	−	0.698395i	$-0.753898\pi$
$53$	−10.4209	−1.43142	−0.715712	+	0.698395i	$0.753898\pi$
$54$	0	0
$55$	5.19969	0.701126
$56$	0	0
$57$	17.1520	2.27184
$58$	0	0
$59$	−8.65032	−1.12618	−0.563088	−	0.826397i	$-0.690387\pi$
$59$	−8.65032	−1.12618	−0.563088	+	0.826397i	$0.690387\pi$
$60$	0	0
$61$	−11.8162	−1.51291	−0.756455	−	0.654046i	$-0.773070\pi$
$61$	−11.8162	−1.51291	−0.756455	+	0.654046i	$0.773070\pi$
$62$	0	0
$63$	−5.88594	−0.741559
$64$	0	0
$65$	−2.45589	−0.304616
$66$	0	0
$67$	7.03182	0.859073	0.429537	−	0.903049i	$-0.358677\pi$
$67$	7.03182	0.859073	0.429537	+	0.903049i	$0.358677\pi$
$68$	0	0
$69$	−5.60822	−0.675150
$70$	0	0
$71$	−7.87312	−0.934367	−0.467184	−	0.884160i	$-0.654731\pi$
$71$	−7.87312	−0.934367	−0.467184	+	0.884160i	$0.654731\pi$
$72$	0	0
$73$	8.30031	0.971478	0.485739	−	0.874104i	$-0.338551\pi$
$73$	8.30031	0.971478	0.485739	+	0.874104i	$0.338551\pi$
$74$	0	0
$75$	2.32475	0.268439
$76$	0	0
$77$	12.7284	1.45054
$78$	0	0
$79$	−12.0439	−1.35504	−0.677520	−	0.735504i	$-0.736945\pi$
$79$	−12.0439	−1.35504	−0.677520	+	0.735504i	$0.736945\pi$
$80$	0	0
$81$	−10.4319	−1.15910
$82$	0	0
$83$	−4.17482	−0.458246	−0.229123	−	0.973398i	$-0.573586\pi$
$83$	−4.17482	−0.458246	−0.229123	+	0.973398i	$0.573586\pi$
$84$	0	0
$85$	−3.30408	−0.358378
$86$	0	0
$87$	−2.05282	−0.220086
$88$	0	0
$89$	−9.39112	−0.995456	−0.497728	−	0.867333i	$-0.665832\pi$
$89$	−9.39112	−0.995456	−0.497728	+	0.867333i	$0.665832\pi$
$90$	0	0
$91$	−6.01181	−0.630209
$92$	0	0
$93$	−1.79976	−0.186626
$94$	0	0
$95$	−7.37799	−0.756966
$96$	0	0
$97$	−16.7742	−1.70316	−0.851579	−	0.524226i	$-0.824355\pi$
$97$	−16.7742	−1.70316	−0.851579	+	0.524226i	$0.824355\pi$
$98$	0	0
$99$	−12.5025	−1.25655
$100$	0	0
$101$	8.18129	0.814069	0.407034	−	0.913413i	$-0.366563\pi$
$101$	8.18129	0.814069	0.407034	+	0.913413i	$0.366563\pi$
$102$	0	0
$103$	3.34929	0.330015	0.165008	−	0.986292i	$-0.447235\pi$
$103$	3.34929	0.330015	0.165008	+	0.986292i	$0.447235\pi$
$104$	0	0
$105$	5.69080	0.555365
$106$	0	0
$107$	5.36392	0.518550	0.259275	−	0.965804i	$-0.416516\pi$
$107$	5.36392	0.518550	0.259275	+	0.965804i	$0.416516\pi$
$108$	0	0
$109$	−8.77781	−0.840761	−0.420381	−	0.907348i	$-0.638103\pi$
$109$	−8.77781	−0.840761	−0.420381	+	0.907348i	$0.638103\pi$
$110$	0	0
$111$	−7.24054	−0.687241
$112$	0	0
$113$	−6.88312	−0.647509	−0.323755	−	0.946141i	$-0.604945\pi$
$113$	−6.88312	−0.647509	−0.323755	+	0.946141i	$0.604945\pi$
$114$	0	0
$115$	2.41239	0.224957
$116$	0	0
$117$	5.90511	0.545928
$118$	0	0
$119$	−8.08811	−0.741436
$120$	0	0
$121$	16.0368	1.45789
$122$	0	0
$123$	25.7814	2.32463
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	19.2474	1.70793	0.853967	−	0.520327i	$-0.174190\pi$
$127$	19.2474	1.70793	0.853967	+	0.520327i	$0.174190\pi$
$128$	0	0
$129$	−16.6465	−1.46564
$130$	0	0
$131$	17.3834	1.51880	0.759399	−	0.650625i	$-0.225493\pi$
$131$	17.3834	1.51880	0.759399	+	0.650625i	$0.225493\pi$
$132$	0	0
$133$	−18.0607	−1.56606
$134$	0	0
$135$	1.38446	0.119155
$136$	0	0
$137$	−17.9231	−1.53128	−0.765639	−	0.643271i	$-0.777577\pi$
$137$	−17.9231	−1.53128	−0.765639	+	0.643271i	$0.777577\pi$
$138$	0	0
$139$	16.9382	1.43668	0.718340	−	0.695692i	$-0.244902\pi$
$139$	16.9382	1.43668	0.718340	+	0.695692i	$0.244902\pi$
$140$	0	0
$141$	13.1936	1.11111
$142$	0	0
$143$	−12.7699	−1.06787
$144$	0	0
$145$	0.883029	0.0733316
$146$	0	0
$147$	−2.34268	−0.193221
$148$	0	0
$149$	−14.9303	−1.22314	−0.611569	−	0.791191i	$-0.709461\pi$
$149$	−14.9303	−1.22314	−0.611569	+	0.791191i	$0.709461\pi$
$150$	0	0
$151$	5.95322	0.484466	0.242233	−	0.970218i	$-0.422120\pi$
$151$	5.95322	0.484466	0.242233	+	0.970218i	$0.422120\pi$
$152$	0	0
$153$	7.94457	0.642280
$154$	0	0
$155$	0.774173	0.0621831
$156$	0	0
$157$	−10.3852	−0.828832	−0.414416	−	0.910088i	$-0.636014\pi$
$157$	−10.3852	−0.828832	−0.414416	+	0.910088i	$0.636014\pi$
$158$	0	0
$159$	−24.2261	−1.92125
$160$	0	0
$161$	5.90534	0.465406
$162$	0	0
$163$	11.8877	0.931117	0.465558	−	0.885017i	$-0.345854\pi$
$163$	11.8877	0.931117	0.465558	+	0.885017i	$0.345854\pi$
$164$	0	0
$165$	12.0880	0.941049
$166$	0	0
$167$	−20.7939	−1.60908	−0.804539	−	0.593899i	$-0.797588\pi$
$167$	−20.7939	−1.60908	−0.804539	+	0.593899i	$0.797588\pi$
$168$	0	0
$169$	−6.96861	−0.536047
$170$	0	0
$171$	17.7402	1.35662
$172$	0	0
$173$	12.6244	0.959818	0.479909	−	0.877318i	$-0.340670\pi$
$173$	12.6244	0.959818	0.479909	+	0.877318i	$0.340670\pi$
$174$	0	0
$175$	−2.44792	−0.185045
$176$	0	0
$177$	−20.1098	−1.51155
$178$	0	0
$179$	−11.6539	−0.871051	−0.435526	−	0.900176i	$-0.643438\pi$
$179$	−11.6539	−0.871051	−0.435526	+	0.900176i	$0.643438\pi$
$180$	0	0
$181$	−13.4608	−1.00053	−0.500265	−	0.865872i	$-0.666764\pi$
$181$	−13.4608	−1.00053	−0.500265	+	0.865872i	$0.666764\pi$
$182$	0	0
$183$	−27.4697	−2.03062
$184$	0	0
$185$	3.11454	0.228986
$186$	0	0
$187$	−17.1802	−1.25634
$188$	0	0
$189$	3.38903	0.246516
$190$	0	0
$191$	−11.6463	−0.842694	−0.421347	−	0.906899i	$-0.638443\pi$
$191$	−11.6463	−0.842694	−0.421347	+	0.906899i	$0.638443\pi$
$192$	0	0
$193$	−5.38386	−0.387539	−0.193769	−	0.981047i	$-0.562071\pi$
$193$	−5.38386	−0.387539	−0.193769	+	0.981047i	$0.562071\pi$
$194$	0	0
$195$	−5.70933	−0.408854
$196$	0	0
$197$	−1.70954	−0.121799	−0.0608997	−	0.998144i	$-0.519397\pi$
$197$	−1.70954	−0.121799	−0.0608997	+	0.998144i	$0.519397\pi$
$198$	0	0
$199$	15.1288	1.07245	0.536226	−	0.844075i	$-0.319850\pi$
$199$	15.1288	1.07245	0.536226	+	0.844075i	$0.319850\pi$
$200$	0	0
$201$	16.3472	1.15304
$202$	0	0
$203$	2.16158	0.151713
$204$	0	0
$205$	−11.0899	−0.774555
$206$	0	0
$207$	−5.80053	−0.403165
$208$	0	0
$209$	−38.3633	−2.65364
$210$	0	0
$211$	−3.20461	−0.220614	−0.110307	−	0.993898i	$-0.535183\pi$
$211$	−3.20461	−0.220614	−0.110307	+	0.993898i	$0.535183\pi$
$212$	0	0
$213$	−18.3030	−1.25410
$214$	0	0
$215$	7.16054	0.488345
$216$	0	0
$217$	1.89511	0.128648
$218$	0	0
$219$	19.2962	1.30391
$220$	0	0
$221$	8.11446	0.545837
$222$	0	0
$223$	−17.0668	−1.14287	−0.571437	−	0.820646i	$-0.693614\pi$
$223$	−17.0668	−1.14287	−0.571437	+	0.820646i	$0.693614\pi$
$224$	0	0
$225$	2.40447	0.160298
$226$	0	0
$227$	5.16990	0.343138	0.171569	−	0.985172i	$-0.445116\pi$
$227$	5.16990	0.343138	0.171569	+	0.985172i	$0.445116\pi$
$228$	0	0
$229$	−16.5678	−1.09483	−0.547416	−	0.836861i	$-0.684388\pi$
$229$	−16.5678	−1.09483	−0.547416	+	0.836861i	$0.684388\pi$
$230$	0	0
$231$	29.5904	1.94690
$232$	0	0
$233$	−26.2444	−1.71933	−0.859663	−	0.510861i	$-0.829327\pi$
$233$	−26.2444	−1.71933	−0.859663	+	0.510861i	$0.829327\pi$
$234$	0	0
$235$	−5.67529	−0.370215
$236$	0	0
$237$	−27.9990	−1.81873
$238$	0	0
$239$	6.62179	0.428328	0.214164	−	0.976798i	$-0.431297\pi$
$239$	6.62179	0.428328	0.214164	+	0.976798i	$0.431297\pi$
$240$	0	0
$241$	−19.1285	−1.23217	−0.616086	−	0.787679i	$-0.711283\pi$
$241$	−19.1285	−1.23217	−0.616086	+	0.787679i	$0.711283\pi$
$242$	0	0
$243$	−20.0983	−1.28931
$244$	0	0
$245$	1.00771	0.0643803
$246$	0	0
$247$	18.1195	1.15292
$248$	0	0
$249$	−9.70541	−0.615056
$250$	0	0
$251$	27.4056	1.72983	0.864914	−	0.501920i	$-0.167373\pi$
$251$	27.4056	1.72983	0.864914	+	0.501920i	$0.167373\pi$
$252$	0	0
$253$	12.5437	0.788616
$254$	0	0
$255$	−7.68117	−0.481013
$256$	0	0
$257$	−13.1026	−0.817320	−0.408660	−	0.912687i	$-0.634004\pi$
$257$	−13.1026	−0.817320	−0.408660	+	0.912687i	$0.634004\pi$
$258$	0	0
$259$	7.62414	0.473741
$260$	0	0
$261$	−2.12322	−0.131424
$262$	0	0
$263$	−9.99373	−0.616240	−0.308120	−	0.951348i	$-0.599700\pi$
$263$	−9.99373	−0.616240	−0.308120	+	0.951348i	$0.599700\pi$
$264$	0	0
$265$	10.4209	0.640152
$266$	0	0
$267$	−21.8320	−1.33610
$268$	0	0
$269$	7.26689	0.443070	0.221535	−	0.975152i	$-0.428893\pi$
$269$	7.26689	0.443070	0.221535	+	0.975152i	$0.428893\pi$
$270$	0	0
$271$	17.4293	1.05876	0.529378	−	0.848386i	$-0.322425\pi$
$271$	17.4293	1.05876	0.529378	+	0.848386i	$0.322425\pi$
$272$	0	0
$273$	−13.9760	−0.845864
$274$	0	0
$275$	−5.19969	−0.313553
$276$	0	0
$277$	8.68281	0.521699	0.260850	−	0.965379i	$-0.415997\pi$
$277$	8.68281	0.521699	0.260850	+	0.965379i	$0.415997\pi$
$278$	0	0
$279$	−1.86148	−0.111444
$280$	0	0
$281$	2.22948	0.133000	0.0664999	−	0.997786i	$-0.478817\pi$
$281$	2.22948	0.133000	0.0664999	+	0.997786i	$0.478817\pi$
$282$	0	0
$283$	−3.16428	−0.188097	−0.0940484	−	0.995568i	$-0.529981\pi$
$283$	−3.16428	−0.188097	−0.0940484	+	0.995568i	$0.529981\pi$
$284$	0	0
$285$	−17.1520	−1.01600
$286$	0	0
$287$	−27.1472	−1.60245
$288$	0	0
$289$	−6.08305	−0.357826
$290$	0	0
$291$	−38.9958	−2.28597
$292$	0	0
$293$	−27.0843	−1.58228	−0.791141	−	0.611634i	$-0.790513\pi$
$293$	−27.0843	−1.58228	−0.791141	+	0.611634i	$0.790513\pi$
$294$	0	0
$295$	8.65032	0.503641
$296$	0	0
$297$	7.19875	0.417714
$298$	0	0
$299$	−5.92457	−0.342627
$300$	0	0
$301$	17.5284	1.01032
$302$	0	0
$303$	19.0195	1.09264
$304$	0	0
$305$	11.8162	0.676593
$306$	0	0
$307$	16.0289	0.914819	0.457410	−	0.889256i	$-0.348777\pi$
$307$	16.0289	0.914819	0.457410	+	0.889256i	$0.348777\pi$
$308$	0	0
$309$	7.78626	0.442945
$310$	0	0
$311$	−31.7170	−1.79851	−0.899253	−	0.437429i	$-0.855889\pi$
$311$	−31.7170	−1.79851	−0.899253	+	0.437429i	$0.855889\pi$
$312$	0	0
$313$	15.7872	0.892343	0.446172	−	0.894947i	$-0.352787\pi$
$313$	15.7872	0.892343	0.446172	+	0.894947i	$0.352787\pi$
$314$	0	0
$315$	5.88594	0.331635
$316$	0	0
$317$	−16.5970	−0.932179	−0.466090	−	0.884738i	$-0.654338\pi$
$317$	−16.5970	−0.932179	−0.466090	+	0.884738i	$0.654338\pi$
$318$	0	0
$319$	4.59148	0.257073
$320$	0	0
$321$	12.4698	0.695996
$322$	0	0
$323$	24.3775	1.35640
$324$	0	0
$325$	2.45589	0.136228
$326$	0	0
$327$	−20.4062	−1.12847
$328$	0	0
$329$	−13.8926	−0.765925
$330$	0	0
$331$	−31.4163	−1.72680	−0.863398	−	0.504523i	$-0.831668\pi$
$331$	−31.4163	−1.72680	−0.863398	+	0.504523i	$0.831668\pi$
$332$	0	0
$333$	−7.48883	−0.410385
$334$	0	0
$335$	−7.03182	−0.384189
$336$	0	0
$337$	−23.0214	−1.25405	−0.627026	−	0.778998i	$-0.715728\pi$
$337$	−23.0214	−1.25405	−0.627026	+	0.778998i	$0.715728\pi$
$338$	0	0
$339$	−16.0015	−0.869084
$340$	0	0
$341$	4.02546	0.217991
$342$	0	0
$343$	19.6022	1.05842
$344$	0	0
$345$	5.60822	0.301936
$346$	0	0
$347$	−4.17107	−0.223915	−0.111957	−	0.993713i	$-0.535712\pi$
$347$	−4.17107	−0.223915	−0.111957	+	0.993713i	$0.535712\pi$
$348$	0	0
$349$	14.7129	0.787564	0.393782	−	0.919204i	$-0.371166\pi$
$349$	14.7129	0.787564	0.393782	+	0.919204i	$0.371166\pi$
$350$	0	0
$351$	−3.40007	−0.181482
$352$	0	0
$353$	−7.59985	−0.404499	−0.202250	−	0.979334i	$-0.564825\pi$
$353$	−7.59985	−0.404499	−0.202250	+	0.979334i	$0.564825\pi$
$354$	0	0
$355$	7.87312	0.417862
$356$	0	0
$357$	−18.8029	−0.995152
$358$	0	0
$359$	−9.92150	−0.523637	−0.261818	−	0.965117i	$-0.584322\pi$
$359$	−9.92150	−0.523637	−0.261818	+	0.965117i	$0.584322\pi$
$360$	0	0
$361$	35.4347	1.86499
$362$	0	0
$363$	37.2815	1.95677
$364$	0	0
$365$	−8.30031	−0.434458
$366$	0	0
$367$	−11.2601	−0.587771	−0.293885	−	0.955841i	$-0.594948\pi$
$367$	−11.2601	−0.587771	−0.293885	+	0.955841i	$0.594948\pi$
$368$	0	0
$369$	26.6654	1.38815
$370$	0	0
$371$	25.5095	1.32439
$372$	0	0
$373$	−25.9527	−1.34378	−0.671890	−	0.740651i	$-0.734517\pi$
$373$	−25.9527	−1.34378	−0.671890	+	0.740651i	$0.734517\pi$
$374$	0	0
$375$	−2.32475	−0.120050
$376$	0	0
$377$	−2.16862	−0.111690
$378$	0	0
$379$	4.51228	0.231780	0.115890	−	0.993262i	$-0.463028\pi$
$379$	4.51228	0.231780	0.115890	+	0.993262i	$0.463028\pi$
$380$	0	0
$381$	44.7455	2.29238
$382$	0	0
$383$	−31.7652	−1.62313	−0.811564	−	0.584264i	$-0.801383\pi$
$383$	−31.7652	−1.62313	−0.811564	+	0.584264i	$0.801383\pi$
$384$	0	0
$385$	−12.7284	−0.648699
$386$	0	0
$387$	−17.2173	−0.875205
$388$	0	0
$389$	16.7692	0.850230	0.425115	−	0.905139i	$-0.360234\pi$
$389$	16.7692	0.850230	0.425115	+	0.905139i	$0.360234\pi$
$390$	0	0
$391$	−7.97075	−0.403098
$392$	0	0
$393$	40.4122	2.03853
$394$	0	0
$395$	12.0439	0.605993
$396$	0	0
$397$	35.5602	1.78472	0.892358	−	0.451327i	$-0.149049\pi$
$397$	35.5602	1.78472	0.892358	+	0.451327i	$0.149049\pi$
$398$	0	0
$399$	−41.9866	−2.10196
$400$	0	0
$401$	1.00000	0.0499376
$401$	1.00000	0.0499376
$402$	0	0
$403$	−1.90128	−0.0947097
$404$	0	0
$405$	10.4319	0.518367
$406$	0	0
$407$	16.1947	0.802739
$408$	0	0
$409$	15.8513	0.783799	0.391899	−	0.920008i	$-0.371818\pi$
$409$	15.8513	0.783799	0.391899	+	0.920008i	$0.371818\pi$
$410$	0	0
$411$	−41.6669	−2.05528
$412$	0	0
$413$	21.1752	1.04197
$414$	0	0
$415$	4.17482	0.204934
$416$	0	0
$417$	39.3771	1.92831
$418$	0	0
$419$	−22.5089	−1.09963	−0.549816	−	0.835286i	$-0.685302\pi$
$419$	−22.5089	−1.09963	−0.549816	+	0.835286i	$0.685302\pi$
$420$	0	0
$421$	−10.4152	−0.507608	−0.253804	−	0.967256i	$-0.581682\pi$
$421$	−10.4152	−0.507608	−0.253804	+	0.967256i	$0.581682\pi$
$422$	0	0
$423$	13.6461	0.663495
$424$	0	0
$425$	3.30408	0.160271
$426$	0	0
$427$	28.9250	1.39978
$428$	0	0
$429$	−29.6868	−1.43329
$430$	0	0
$431$	39.5349	1.90433	0.952163	−	0.305589i	$-0.0988534\pi$
$431$	39.5349	1.90433	0.952163	+	0.305589i	$0.0988534\pi$
$432$	0	0
$433$	35.2994	1.69638	0.848192	−	0.529690i	$-0.177692\pi$
$433$	35.2994	1.69638	0.848192	+	0.529690i	$0.177692\pi$
$434$	0	0
$435$	2.05282	0.0984253
$436$	0	0
$437$	−17.7986	−0.851424
$438$	0	0
$439$	25.8433	1.23343	0.616716	−	0.787186i	$-0.288463\pi$
$439$	25.8433	1.23343	0.616716	+	0.787186i	$0.288463\pi$
$440$	0	0
$441$	−2.42301	−0.115382
$442$	0	0
$443$	14.1589	0.672711	0.336355	−	0.941735i	$-0.390806\pi$
$443$	14.1589	0.672711	0.336355	+	0.941735i	$0.390806\pi$
$444$	0	0
$445$	9.39112	0.445182
$446$	0	0
$447$	−34.7093	−1.64169
$448$	0	0
$449$	29.7552	1.40424	0.702118	−	0.712060i	$-0.252238\pi$
$449$	29.7552	1.40424	0.702118	+	0.712060i	$0.252238\pi$
$450$	0	0
$451$	−57.6642	−2.71530
$452$	0	0
$453$	13.8398	0.650249
$454$	0	0
$455$	6.01181	0.281838
$456$	0	0
$457$	9.17522	0.429199	0.214599	−	0.976702i	$-0.431155\pi$
$457$	9.17522	0.429199	0.214599	+	0.976702i	$0.431155\pi$
$458$	0	0
$459$	−4.57436	−0.213513
$460$	0	0
$461$	−29.0912	−1.35491	−0.677457	−	0.735563i	$-0.736918\pi$
$461$	−29.0912	−1.35491	−0.677457	+	0.735563i	$0.736918\pi$
$462$	0	0
$463$	0.928084	0.0431317	0.0215659	−	0.999767i	$-0.493135\pi$
$463$	0.928084	0.0431317	0.0215659	+	0.999767i	$0.493135\pi$
$464$	0	0
$465$	1.79976	0.0834619
$466$	0	0
$467$	−32.1055	−1.48567	−0.742833	−	0.669477i	$-0.766518\pi$
$467$	−32.1055	−1.48567	−0.742833	+	0.669477i	$0.766518\pi$
$468$	0	0
$469$	−17.2133	−0.794836
$470$	0	0
$471$	−24.1431	−1.11245
$472$	0	0
$473$	37.2326	1.71196
$474$	0	0
$475$	7.37799	0.338525
$476$	0	0
$477$	−25.0568	−1.14727
$478$	0	0
$479$	9.83303	0.449283	0.224641	−	0.974442i	$-0.427879\pi$
$479$	9.83303	0.449283	0.224641	+	0.974442i	$0.427879\pi$
$480$	0	0
$481$	−7.64897	−0.348763
$482$	0	0
$483$	13.7284	0.624666
$484$	0	0
$485$	16.7742	0.761675
$486$	0	0
$487$	−3.55039	−0.160884	−0.0804419	−	0.996759i	$-0.525633\pi$
$487$	−3.55039	−0.160884	−0.0804419	+	0.996759i	$0.525633\pi$
$488$	0	0
$489$	27.6360	1.24974
$490$	0	0
$491$	−3.94669	−0.178112	−0.0890559	−	0.996027i	$-0.528385\pi$
$491$	−3.94669	−0.178112	−0.0890559	+	0.996027i	$0.528385\pi$
$492$	0	0
$493$	−2.91760	−0.131402
$494$	0	0
$495$	12.5025	0.561946
$496$	0	0
$497$	19.2727	0.864500
$498$	0	0
$499$	15.8569	0.709850	0.354925	−	0.934895i	$-0.384506\pi$
$499$	15.8569	0.709850	0.354925	+	0.934895i	$0.384506\pi$
$500$	0	0
$501$	−48.3406	−2.15970
$502$	0	0
$503$	31.2479	1.39327	0.696637	−	0.717424i	$-0.254679\pi$
$503$	31.2479	1.39327	0.696637	+	0.717424i	$0.254679\pi$
$504$	0	0
$505$	−8.18129	−0.364063
$506$	0	0
$507$	−16.2003	−0.719480
$508$	0	0
$509$	−12.7905	−0.566928	−0.283464	−	0.958983i	$-0.591484\pi$
$509$	−12.7905	−0.566928	−0.283464	+	0.958983i	$0.591484\pi$
$510$	0	0
$511$	−20.3185	−0.898836
$512$	0	0
$513$	−10.2145	−0.450982
$514$	0	0
$515$	−3.34929	−0.147587
$516$	0	0
$517$	−29.5097	−1.29784
$518$	0	0
$519$	29.3487	1.28826
$520$	0	0
$521$	−29.8043	−1.30575	−0.652875	−	0.757466i	$-0.726437\pi$
$521$	−29.8043	−1.30575	−0.652875	+	0.757466i	$0.726437\pi$
$522$	0	0
$523$	29.7129	1.29925	0.649627	−	0.760253i	$-0.274925\pi$
$523$	29.7129	1.29925	0.649627	+	0.760253i	$0.274925\pi$
$524$	0	0
$525$	−5.69080	−0.248367
$526$	0	0
$527$	−2.55793	−0.111425
$528$	0	0
$529$	−17.1804	−0.746972
$530$	0	0
$531$	−20.7994	−0.902619
$532$	0	0
$533$	27.2357	1.17971
$534$	0	0
$535$	−5.36392	−0.231903
$536$	0	0
$537$	−27.0924	−1.16912
$538$	0	0
$539$	5.23979	0.225694
$540$	0	0
$541$	22.7229	0.976934	0.488467	−	0.872582i	$-0.337556\pi$
$541$	22.7229	0.976934	0.488467	+	0.872582i	$0.337556\pi$
$542$	0	0
$543$	−31.2929	−1.34291
$544$	0	0
$545$	8.77781	0.376000
$546$	0	0
$547$	2.77036	0.118452	0.0592261	−	0.998245i	$-0.481137\pi$
$547$	2.77036	0.118452	0.0592261	+	0.998245i	$0.481137\pi$
$548$	0	0
$549$	−28.4117	−1.21258
$550$	0	0
$551$	−6.51498	−0.277547
$552$	0	0
$553$	29.4824	1.25372
$554$	0	0
$555$	7.24054	0.307344
$556$	0	0
$557$	5.25530	0.222674	0.111337	−	0.993783i	$-0.464487\pi$
$557$	5.25530	0.222674	0.111337	+	0.993783i	$0.464487\pi$
$558$	0	0
$559$	−17.5855	−0.743787
$560$	0	0
$561$	−39.9397	−1.68626
$562$	0	0
$563$	6.11309	0.257636	0.128818	−	0.991668i	$-0.458882\pi$
$563$	6.11309	0.257636	0.128818	+	0.991668i	$0.458882\pi$
$564$	0	0
$565$	6.88312	0.289575
$566$	0	0
$567$	25.5365	1.07243
$568$	0	0
$569$	−33.4724	−1.40324	−0.701618	−	0.712553i	$-0.747539\pi$
$569$	−33.4724	−1.40324	−0.701618	+	0.712553i	$0.747539\pi$
$570$	0	0
$571$	27.0336	1.13132	0.565660	−	0.824638i	$-0.308621\pi$
$571$	27.0336	1.13132	0.565660	+	0.824638i	$0.308621\pi$
$572$	0	0
$573$	−27.0747	−1.13106
$574$	0	0
$575$	−2.41239	−0.100604
$576$	0	0
$577$	6.86862	0.285944	0.142972	−	0.989727i	$-0.454334\pi$
$577$	6.86862	0.285944	0.142972	+	0.989727i	$0.454334\pi$
$578$	0	0
$579$	−12.5161	−0.520153
$580$	0	0
$581$	10.2196	0.423980
$582$	0	0
$583$	54.1856	2.24414
$584$	0	0
$585$	−5.90511	−0.244146
$586$	0	0
$587$	44.1853	1.82372	0.911860	−	0.410501i	$-0.134646\pi$
$587$	44.1853	1.82372	0.911860	+	0.410501i	$0.134646\pi$
$588$	0	0
$589$	−5.71184	−0.235352
$590$	0	0
$591$	−3.97425	−0.163479
$592$	0	0
$593$	19.2535	0.790645	0.395323	−	0.918542i	$-0.370633\pi$
$593$	19.2535	0.790645	0.395323	+	0.918542i	$0.370633\pi$
$594$	0	0
$595$	8.08811	0.331580
$596$	0	0
$597$	35.1707	1.43944
$598$	0	0
$599$	−24.8385	−1.01487	−0.507437	−	0.861689i	$-0.669407\pi$
$599$	−24.8385	−1.01487	−0.507437	+	0.861689i	$0.669407\pi$
$600$	0	0
$601$	25.8731	1.05539	0.527693	−	0.849435i	$-0.323057\pi$
$601$	25.8731	1.05539	0.527693	+	0.849435i	$0.323057\pi$
$602$	0	0
$603$	16.9078	0.688539
$604$	0	0
$605$	−16.0368	−0.651988
$606$	0	0
$607$	−13.5573	−0.550274	−0.275137	−	0.961405i	$-0.588723\pi$
$607$	−13.5573	−0.550274	−0.275137	+	0.961405i	$0.588723\pi$
$608$	0	0
$609$	5.02514	0.203629
$610$	0	0
$611$	13.9379	0.563866
$612$	0	0
$613$	0.467171	0.0188689	0.00943444	−	0.999955i	$-0.496997\pi$
$613$	0.467171	0.0188689	0.00943444	+	0.999955i	$0.496997\pi$
$614$	0	0
$615$	−25.7814	−1.03960
$616$	0	0
$617$	25.7074	1.03494	0.517470	−	0.855701i	$-0.326874\pi$
$617$	25.7074	1.03494	0.517470	+	0.855701i	$0.326874\pi$
$618$	0	0
$619$	−27.2402	−1.09488	−0.547438	−	0.836846i	$-0.684397\pi$
$619$	−27.2402	−1.09488	−0.547438	+	0.836846i	$0.684397\pi$
$620$	0	0
$621$	3.33986	0.134024
$622$	0	0
$623$	22.9887	0.921021
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−89.1851	−3.56171
$628$	0	0
$629$	−10.2907	−0.410317
$630$	0	0
$631$	25.4615	1.01361	0.506803	−	0.862062i	$-0.330827\pi$
$631$	25.4615	1.01361	0.506803	+	0.862062i	$0.330827\pi$
$632$	0	0
$633$	−7.44992	−0.296108
$634$	0	0
$635$	−19.2474	−0.763811
$636$	0	0
$637$	−2.47483	−0.0980562
$638$	0	0
$639$	−18.9307	−0.748887
$640$	0	0
$641$	43.0962	1.70220	0.851098	−	0.525007i	$-0.175937\pi$
$641$	43.0962	1.70220	0.851098	+	0.525007i	$0.175937\pi$
$642$	0	0
$643$	−18.3857	−0.725062	−0.362531	−	0.931972i	$-0.618087\pi$
$643$	−18.3857	−0.725062	−0.362531	+	0.931972i	$0.618087\pi$
$644$	0	0
$645$	16.6465	0.655454
$646$	0	0
$647$	−32.4412	−1.27540	−0.637698	−	0.770287i	$-0.720113\pi$
$647$	−32.4412	−1.27540	−0.637698	+	0.770287i	$0.720113\pi$
$648$	0	0
$649$	44.9790	1.76558
$650$	0	0
$651$	4.40566	0.172671
$652$	0	0
$653$	24.8831	0.973751	0.486875	−	0.873471i	$-0.338137\pi$
$653$	24.8831	0.973751	0.486875	+	0.873471i	$0.338137\pi$
$654$	0	0
$655$	−17.3834	−0.679227
$656$	0	0
$657$	19.9579	0.778630
$658$	0	0
$659$	38.2465	1.48987	0.744936	−	0.667136i	$-0.232480\pi$
$659$	38.2465	1.48987	0.744936	+	0.667136i	$0.232480\pi$
$660$	0	0
$661$	14.3456	0.557980	0.278990	−	0.960294i	$-0.410000\pi$
$661$	14.3456	0.557980	0.278990	+	0.960294i	$0.410000\pi$
$662$	0	0
$663$	18.8641	0.732621
$664$	0	0
$665$	18.0607	0.700364
$666$	0	0
$667$	2.13021	0.0824822
$668$	0	0
$669$	−39.6760	−1.53396
$670$	0	0
$671$	61.4406	2.37189
$672$	0	0
$673$	−8.22085	−0.316890	−0.158445	−	0.987368i	$-0.550648\pi$
$673$	−8.22085	−0.316890	−0.158445	+	0.987368i	$0.550648\pi$
$674$	0	0
$675$	−1.38446	−0.0532878
$676$	0	0
$677$	−9.37819	−0.360433	−0.180216	−	0.983627i	$-0.557680\pi$
$677$	−9.37819	−0.360433	−0.180216	+	0.983627i	$0.557680\pi$
$678$	0	0
$679$	41.0617	1.57580
$680$	0	0
$681$	12.0187	0.460559
$682$	0	0
$683$	−31.0163	−1.18681	−0.593403	−	0.804905i	$-0.702216\pi$
$683$	−31.0163	−1.18681	−0.593403	+	0.804905i	$0.702216\pi$
$684$	0	0
$685$	17.9231	0.684808
$686$	0	0
$687$	−38.5160	−1.46948
$688$	0	0
$689$	−25.5926	−0.975002
$690$	0	0
$691$	−22.6421	−0.861347	−0.430674	−	0.902508i	$-0.641724\pi$
$691$	−22.6421	−0.861347	−0.430674	+	0.902508i	$0.641724\pi$
$692$	0	0
$693$	30.6051	1.16259
$694$	0	0
$695$	−16.9382	−0.642503
$696$	0	0
$697$	36.6421	1.38792
$698$	0	0
$699$	−61.0117	−2.30767
$700$	0	0
$701$	−44.2958	−1.67303	−0.836515	−	0.547943i	$-0.815411\pi$
$701$	−44.2958	−1.67303	−0.836515	+	0.547943i	$0.815411\pi$
$702$	0	0
$703$	−22.9791	−0.866672
$704$	0	0
$705$	−13.1936	−0.496901
$706$	0	0
$707$	−20.0271	−0.753197
$708$	0	0
$709$	3.31491	0.124494	0.0622470	−	0.998061i	$-0.480173\pi$
$709$	3.31491	0.124494	0.0622470	+	0.998061i	$0.480173\pi$
$710$	0	0
$711$	−28.9591	−1.08605
$712$	0	0
$713$	1.86761	0.0699426
$714$	0	0
$715$	12.7699	0.477566
$716$	0	0
$717$	15.3940	0.574900
$718$	0	0
$719$	23.7708	0.886502	0.443251	−	0.896398i	$-0.353825\pi$
$719$	23.7708	0.886502	0.443251	+	0.896398i	$0.353825\pi$
$720$	0	0
$721$	−8.19877	−0.305338
$722$	0	0
$723$	−44.4689	−1.65382
$724$	0	0
$725$	−0.883029	−0.0327949
$726$	0	0
$727$	−21.9798	−0.815186	−0.407593	−	0.913164i	$-0.633632\pi$
$727$	−21.9798	−0.815186	−0.407593	+	0.913164i	$0.633632\pi$
$728$	0	0
$729$	−15.4277	−0.571397
$730$	0	0
$731$	−23.6590	−0.875060
$732$	0	0
$733$	37.4119	1.38184	0.690919	−	0.722932i	$-0.257206\pi$
$733$	37.4119	1.38184	0.690919	+	0.722932i	$0.257206\pi$
$734$	0	0
$735$	2.34268	0.0864110
$736$	0	0
$737$	−36.5633	−1.34683
$738$	0	0
$739$	33.9772	1.24987	0.624935	−	0.780677i	$-0.285125\pi$
$739$	33.9772	1.24987	0.624935	+	0.780677i	$0.285125\pi$
$740$	0	0
$741$	42.1234	1.54744
$742$	0	0
$743$	−2.62475	−0.0962926	−0.0481463	−	0.998840i	$-0.515331\pi$
$743$	−2.62475	−0.0962926	−0.0481463	+	0.998840i	$0.515331\pi$
$744$	0	0
$745$	14.9303	0.547004
$746$	0	0
$747$	−10.0382	−0.367280
$748$	0	0
$749$	−13.1304	−0.479775
$750$	0	0
$751$	−12.1603	−0.443737	−0.221869	−	0.975077i	$-0.571216\pi$
$751$	−12.1603	−0.443737	−0.221869	+	0.975077i	$0.571216\pi$
$752$	0	0
$753$	63.7113	2.32177
$754$	0	0
$755$	−5.95322	−0.216660
$756$	0	0
$757$	41.2286	1.49848	0.749240	−	0.662299i	$-0.230419\pi$
$757$	41.2286	1.49848	0.749240	+	0.662299i	$0.230419\pi$
$758$	0	0
$759$	29.1610	1.05848
$760$	0	0
$761$	43.2886	1.56921	0.784606	−	0.619995i	$-0.212865\pi$
$761$	43.2886	1.56921	0.784606	+	0.619995i	$0.212865\pi$
$762$	0	0
$763$	21.4873	0.777893
$764$	0	0
$765$	−7.94457	−0.287236
$766$	0	0
$767$	−21.2442	−0.767084
$768$	0	0
$769$	−33.0243	−1.19089	−0.595443	−	0.803397i	$-0.703023\pi$
$769$	−33.0243	−1.19089	−0.595443	+	0.803397i	$0.703023\pi$
$770$	0	0
$771$	−30.4604	−1.09700
$772$	0	0
$773$	26.7414	0.961820	0.480910	−	0.876770i	$-0.340306\pi$
$773$	26.7414	0.961820	0.480910	+	0.876770i	$0.340306\pi$
$774$	0	0
$775$	−0.774173	−0.0278091
$776$	0	0
$777$	17.7242	0.635853
$778$	0	0
$779$	81.8214	2.93156
$780$	0	0
$781$	40.9378	1.46487
$782$	0	0
$783$	1.22252	0.0436891
$784$	0	0
$785$	10.3852	0.370665
$786$	0	0
$787$	22.8241	0.813590	0.406795	−	0.913520i	$-0.366646\pi$
$787$	22.8241	0.813590	0.406795	+	0.913520i	$0.366646\pi$
$788$	0	0
$789$	−23.2329	−0.827115
$790$	0	0
$791$	16.8493	0.599092
$792$	0	0
$793$	−29.0193	−1.03050
$794$	0	0
$795$	24.2261	0.859210
$796$	0	0
$797$	−32.9312	−1.16648	−0.583241	−	0.812299i	$-0.698216\pi$
$797$	−32.9312	−1.16648	−0.583241	+	0.812299i	$0.698216\pi$
$798$	0	0
$799$	18.7516	0.663385
$800$	0	0
$801$	−22.5807	−0.797849
$802$	0	0
$803$	−43.1590	−1.52305
$804$	0	0
$805$	−5.90534	−0.208136
$806$	0	0
$807$	16.8937	0.594687
$808$	0	0
$809$	4.37099	0.153676	0.0768379	−	0.997044i	$-0.475518\pi$
$809$	4.37099	0.153676	0.0768379	+	0.997044i	$0.475518\pi$
$810$	0	0
$811$	−20.8882	−0.733483	−0.366742	−	0.930323i	$-0.619527\pi$
$811$	−20.8882	−0.733483	−0.366742	+	0.930323i	$0.619527\pi$
$812$	0	0
$813$	40.5188	1.42106
$814$	0	0
$815$	−11.8877	−0.416408
$816$	0	0
$817$	−52.8304	−1.84830
$818$	0	0
$819$	−14.4552	−0.505106
$820$	0	0
$821$	−41.9809	−1.46514	−0.732572	−	0.680689i	$-0.761680\pi$
$821$	−41.9809	−1.46514	−0.732572	+	0.680689i	$0.761680\pi$
$822$	0	0
$823$	−2.64112	−0.0920635	−0.0460318	−	0.998940i	$-0.514658\pi$
$823$	−2.64112	−0.0920635	−0.0460318	+	0.998940i	$0.514658\pi$
$824$	0	0
$825$	−12.0880	−0.420850
$826$	0	0
$827$	50.0018	1.73873	0.869366	−	0.494169i	$-0.164528\pi$
$827$	50.0018	1.73873	0.869366	+	0.494169i	$0.164528\pi$
$828$	0	0
$829$	−25.9460	−0.901143	−0.450572	−	0.892740i	$-0.648780\pi$
$829$	−25.9460	−0.901143	−0.450572	+	0.892740i	$0.648780\pi$
$830$	0	0
$831$	20.1854	0.700223
$832$	0	0
$833$	−3.32956	−0.115362
$834$	0	0
$835$	20.7939	0.719602
$836$	0	0
$837$	1.07181	0.0370471
$838$	0	0
$839$	−3.56947	−0.123232	−0.0616158	−	0.998100i	$-0.519625\pi$
$839$	−3.56947	−0.123232	−0.0616158	+	0.998100i	$0.519625\pi$
$840$	0	0
$841$	−28.2203	−0.973112
$842$	0	0
$843$	5.18299	0.178512
$844$	0	0
$845$	6.96861	0.239727
$846$	0	0
$847$	−39.2567	−1.34887
$848$	0	0
$849$	−7.35616	−0.252463
$850$	0	0
$851$	7.51350	0.257560
$852$	0	0
$853$	13.0398	0.446473	0.223236	−	0.974764i	$-0.428338\pi$
$853$	13.0398	0.446473	0.223236	+	0.974764i	$0.428338\pi$
$854$	0	0
$855$	−17.7402	−0.606701
$856$	0	0
$857$	23.0455	0.787218	0.393609	−	0.919278i	$-0.371226\pi$
$857$	23.0455	0.787218	0.393609	+	0.919278i	$0.371226\pi$
$858$	0	0
$859$	18.1903	0.620646	0.310323	−	0.950631i	$-0.399563\pi$
$859$	18.1903	0.620646	0.310323	+	0.950631i	$0.399563\pi$
$860$	0	0
$861$	−63.1106	−2.15080
$862$	0	0
$863$	−40.7654	−1.38767	−0.693835	−	0.720134i	$-0.744080\pi$
$863$	−40.7654	−1.38767	−0.693835	+	0.720134i	$0.744080\pi$
$864$	0	0
$865$	−12.6244	−0.429244
$866$	0	0
$867$	−14.1416	−0.480273
$868$	0	0
$869$	62.6244	2.12439
$870$	0	0
$871$	17.2694	0.585150
$872$	0	0
$873$	−40.3330	−1.36506
$874$	0	0
$875$	2.44792	0.0827546
$876$	0	0
$877$	20.6337	0.696752	0.348376	−	0.937355i	$-0.386733\pi$
$877$	20.6337	0.696752	0.348376	+	0.937355i	$0.386733\pi$
$878$	0	0
$879$	−62.9643	−2.12373
$880$	0	0
$881$	32.3915	1.09130	0.545648	−	0.838014i	$-0.316284\pi$
$881$	32.3915	1.09130	0.545648	+	0.838014i	$0.316284\pi$
$882$	0	0
$883$	23.2727	0.783189	0.391594	−	0.920138i	$-0.371924\pi$
$883$	23.2727	0.783189	0.391594	+	0.920138i	$0.371924\pi$
$884$	0	0
$885$	20.1098	0.675985
$886$	0	0
$887$	3.02081	0.101429	0.0507144	−	0.998713i	$-0.483850\pi$
$887$	3.02081	0.101429	0.0507144	+	0.998713i	$0.483850\pi$
$888$	0	0
$889$	−47.1161	−1.58022
$890$	0	0
$891$	54.2428	1.81720
$892$	0	0
$893$	41.8722	1.40120
$894$	0	0
$895$	11.6539	0.389546
$896$	0	0
$897$	−13.7732	−0.459873
$898$	0	0
$899$	0.683617	0.0227999
$900$	0	0
$901$	−34.4316	−1.14708
$902$	0	0
$903$	40.7492	1.35605
$904$	0	0
$905$	13.4608	0.447451
$906$	0	0
$907$	16.2038	0.538037	0.269019	−	0.963135i	$-0.413301\pi$
$907$	16.2038	0.538037	0.269019	+	0.963135i	$0.413301\pi$
$908$	0	0
$909$	19.6717	0.652468
$910$	0	0
$911$	14.8492	0.491976	0.245988	−	0.969273i	$-0.420888\pi$
$911$	14.8492	0.491976	0.245988	+	0.969273i	$0.420888\pi$
$912$	0	0
$913$	21.7078	0.718422
$914$	0	0
$915$	27.4697	0.908121
$916$	0	0
$917$	−42.5532	−1.40523
$918$	0	0
$919$	7.65288	0.252445	0.126223	−	0.992002i	$-0.459715\pi$
$919$	7.65288	0.252445	0.126223	+	0.992002i	$0.459715\pi$
$920$	0	0
$921$	37.2633	1.22787
$922$	0	0
$923$	−19.3355	−0.636436
$924$	0	0
$925$	−3.11454	−0.102406
$926$	0	0
$927$	8.05326	0.264504
$928$	0	0
$929$	42.8564	1.40607	0.703037	−	0.711153i	$-0.251827\pi$
$929$	42.8564	1.40607	0.703037	+	0.711153i	$0.251827\pi$
$930$	0	0
$931$	−7.43488	−0.243668
$932$	0	0
$933$	−73.7341	−2.41395
$934$	0	0
$935$	17.1802	0.561853
$936$	0	0
$937$	0.588174	0.0192148	0.00960740	−	0.999954i	$-0.496942\pi$
$937$	0.588174	0.0192148	0.00960740	+	0.999954i	$0.496942\pi$
$938$	0	0
$939$	36.7012	1.19770
$940$	0	0
$941$	53.3191	1.73815	0.869076	−	0.494678i	$-0.164714\pi$
$941$	53.3191	1.73815	0.869076	+	0.494678i	$0.164714\pi$
$942$	0	0
$943$	−26.7533	−0.871207
$944$	0	0
$945$	−3.38903	−0.110245
$946$	0	0
$947$	−11.5755	−0.376154	−0.188077	−	0.982154i	$-0.560225\pi$
$947$	−11.5755	−0.376154	−0.188077	+	0.982154i	$0.560225\pi$
$948$	0	0
$949$	20.3846	0.661714
$950$	0	0
$951$	−38.5839	−1.25117
$952$	0	0
$953$	−40.6032	−1.31527	−0.657634	−	0.753338i	$-0.728443\pi$
$953$	−40.6032	−1.31527	−0.657634	+	0.753338i	$0.728443\pi$
$954$	0	0
$955$	11.6463	0.376864
$956$	0	0
$957$	10.6740	0.345043
$958$	0	0
$959$	43.8743	1.41678
$960$	0	0
$961$	−30.4007	−0.980666
$962$	0	0
$963$	12.8974	0.415613
$964$	0	0
$965$	5.38386	0.173313
$966$	0	0
$967$	55.7241	1.79197	0.895983	−	0.444089i	$-0.146473\pi$
$967$	55.7241	1.79197	0.895983	+	0.444089i	$0.146473\pi$
$968$	0	0
$969$	56.6716	1.82055
$970$	0	0
$971$	19.8536	0.637132	0.318566	−	0.947901i	$-0.396799\pi$
$971$	19.8536	0.637132	0.318566	+	0.947901i	$0.396799\pi$
$972$	0	0
$973$	−41.4633	−1.32925
$974$	0	0
$975$	5.70933	0.182845
$976$	0	0
$977$	35.7654	1.14424	0.572119	−	0.820171i	$-0.306122\pi$
$977$	35.7654	1.14424	0.572119	+	0.820171i	$0.306122\pi$
$978$	0	0
$979$	48.8309	1.56064
$980$	0	0
$981$	−21.1060	−0.673862
$982$	0	0
$983$	28.1853	0.898972	0.449486	−	0.893287i	$-0.351607\pi$
$983$	28.1853	0.898972	0.449486	+	0.893287i	$0.351607\pi$
$984$	0	0
$985$	1.70954	0.0544703
$986$	0	0
$987$	−32.2969	−1.02802
$988$	0	0
$989$	17.2740	0.549283
$990$	0	0
$991$	−9.02095	−0.286560	−0.143280	−	0.989682i	$-0.545765\pi$
$991$	−9.02095	−0.286560	−0.143280	+	0.989682i	$0.545765\pi$
$992$	0	0
$993$	−73.0351	−2.31770
$994$	0	0
$995$	−15.1288	−0.479615
$996$	0	0
$997$	−34.0413	−1.07810	−0.539050	−	0.842274i	$-0.681217\pi$
$997$	−34.0413	−1.07810	−0.539050	+	0.842274i	$0.681217\pi$
$998$	0	0
$999$	4.31195	0.136424

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8020.2.a.c.1.24	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8020.2.a.c.1.24	✓	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)

\(f(q)\)	\(=\)	\(q+2.32475 q^{3} -1.00000 q^{5} -2.44792 q^{7} +2.40447 q^{9} +O(q^{10})\)	\(q+2.32475 q^{3} -1.00000 q^{5} -2.44792 q^{7} +2.40447 q^{9} -5.19969 q^{11} +2.45589 q^{13} -2.32475 q^{15} +3.30408 q^{17} +7.37799 q^{19} -5.69080 q^{21} -2.41239 q^{23} +1.00000 q^{25} -1.38446 q^{27} -0.883029 q^{29} -0.774173 q^{31} -12.0880 q^{33} +2.44792 q^{35} -3.11454 q^{37} +5.70933 q^{39} +11.0899 q^{41} -7.16054 q^{43} -2.40447 q^{45} +5.67529 q^{47} -1.00771 q^{49} +7.68117 q^{51} -10.4209 q^{53} +5.19969 q^{55} +17.1520 q^{57} -8.65032 q^{59} -11.8162 q^{61} -5.88594 q^{63} -2.45589 q^{65} +7.03182 q^{67} -5.60822 q^{69} -7.87312 q^{71} +8.30031 q^{73} +2.32475 q^{75} +12.7284 q^{77} -12.0439 q^{79} -10.4319 q^{81} -4.17482 q^{83} -3.30408 q^{85} -2.05282 q^{87} -9.39112 q^{89} -6.01181 q^{91} -1.79976 q^{93} -7.37799 q^{95} -16.7742 q^{97} -12.5025 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 3 q^{3} - 28 q^{5} - 4 q^{7} + 17 q^{9}+O(q^{10}) \) 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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