LMFDB - Embedded newform 8020.2.a.c.1.1

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			1.1
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)

$f(q)$	$=$	$q-3.10189 q^{3} -1.00000 q^{5} +1.97262 q^{7} +6.62170 q^{9} +O(q^{10})$	$q-3.10189 q^{3} -1.00000 q^{5} +1.97262 q^{7} +6.62170 q^{9} +0.185891 q^{11} +5.99884 q^{13} +3.10189 q^{15} -1.15817 q^{17} +2.61335 q^{19} -6.11884 q^{21} -0.915400 q^{23} +1.00000 q^{25} -11.2341 q^{27} -5.92842 q^{29} +1.52735 q^{31} -0.576612 q^{33} -1.97262 q^{35} -4.46101 q^{37} -18.6077 q^{39} -9.05750 q^{41} +1.79219 q^{43} -6.62170 q^{45} -4.47790 q^{47} -3.10878 q^{49} +3.59252 q^{51} -12.0796 q^{53} -0.185891 q^{55} -8.10633 q^{57} -1.16916 q^{59} +9.27981 q^{61} +13.0621 q^{63} -5.99884 q^{65} +12.9087 q^{67} +2.83947 q^{69} -4.83239 q^{71} +3.01365 q^{73} -3.10189 q^{75} +0.366691 q^{77} +6.21687 q^{79} +14.9818 q^{81} -9.34234 q^{83} +1.15817 q^{85} +18.3893 q^{87} -17.4169 q^{89} +11.8334 q^{91} -4.73766 q^{93} -2.61335 q^{95} +4.82293 q^{97} +1.23091 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−3.10189	−1.79088	−0.895438	−	0.445187i	$-0.853137\pi$
$3$	−3.10189	−1.79088	−0.895438	+	0.445187i	$0.853137\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	1.97262	0.745579	0.372790	−	0.927916i	$-0.378401\pi$
$7$	1.97262	0.745579	0.372790	+	0.927916i	$0.378401\pi$
$8$	0	0
$9$	6.62170	2.20723
$10$	0	0
$11$	0.185891	0.0560482	0.0280241	−	0.999607i	$-0.491078\pi$
$11$	0.185891	0.0560482	0.0280241	+	0.999607i	$0.491078\pi$
$12$	0	0
$13$	5.99884	1.66378	0.831889	−	0.554942i	$-0.187259\pi$
$13$	5.99884	1.66378	0.831889	+	0.554942i	$0.187259\pi$
$14$	0	0
$15$	3.10189	0.800904
$16$	0	0
$17$	−1.15817	−0.280898	−0.140449	−	0.990088i	$-0.544855\pi$
$17$	−1.15817	−0.280898	−0.140449	+	0.990088i	$0.544855\pi$
$18$	0	0
$19$	2.61335	0.599545	0.299772	−	0.954011i	$-0.403089\pi$
$19$	2.61335	0.599545	0.299772	+	0.954011i	$0.403089\pi$
$20$	0	0
$21$	−6.11884	−1.33524
$22$	0	0
$23$	−0.915400	−0.190874	−0.0954370	−	0.995435i	$-0.530425\pi$
$23$	−0.915400	−0.190874	−0.0954370	+	0.995435i	$0.530425\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−11.2341	−2.16201
$28$	0	0
$29$	−5.92842	−1.10088	−0.550440	−	0.834875i	$-0.685540\pi$
$29$	−5.92842	−1.10088	−0.550440	+	0.834875i	$0.685540\pi$
$30$	0	0
$31$	1.52735	0.274320	0.137160	−	0.990549i	$-0.456203\pi$
$31$	1.52735	0.274320	0.137160	+	0.990549i	$0.456203\pi$
$32$	0	0
$33$	−0.576612	−0.100375
$34$	0	0
$35$	−1.97262	−0.333433
$36$	0	0
$37$	−4.46101	−0.733386	−0.366693	−	0.930342i	$-0.619510\pi$
$37$	−4.46101	−0.733386	−0.366693	+	0.930342i	$0.619510\pi$
$38$	0	0
$39$	−18.6077	−2.97962
$40$	0	0
$41$	−9.05750	−1.41454	−0.707272	−	0.706942i	$-0.750074\pi$
$41$	−9.05750	−1.41454	−0.707272	+	0.706942i	$0.750074\pi$
$42$	0	0
$43$	1.79219	0.273306	0.136653	−	0.990619i	$-0.456365\pi$
$43$	1.79219	0.273306	0.136653	+	0.990619i	$0.456365\pi$
$44$	0	0
$45$	−6.62170	−0.987105
$46$	0	0
$47$	−4.47790	−0.653169	−0.326585	−	0.945168i	$-0.605898\pi$
$47$	−4.47790	−0.653169	−0.326585	+	0.945168i	$0.605898\pi$
$48$	0	0
$49$	−3.10878	−0.444112
$50$	0	0
$51$	3.59252	0.503054
$52$	0	0
$53$	−12.0796	−1.65925	−0.829627	−	0.558318i	$-0.811447\pi$
$53$	−12.0796	−1.65925	−0.829627	+	0.558318i	$0.811447\pi$
$54$	0	0
$55$	−0.185891	−0.0250655
$56$	0	0
$57$	−8.10633	−1.07371
$58$	0	0
$59$	−1.16916	−0.152212	−0.0761058	−	0.997100i	$-0.524249\pi$
$59$	−1.16916	−0.152212	−0.0761058	+	0.997100i	$0.524249\pi$
$60$	0	0
$61$	9.27981	1.18816	0.594079	−	0.804407i	$-0.297517\pi$
$61$	9.27981	1.18816	0.594079	+	0.804407i	$0.297517\pi$
$62$	0	0
$63$	13.0621	1.64567
$64$	0	0
$65$	−5.99884	−0.744064
$66$	0	0
$67$	12.9087	1.57705	0.788527	−	0.615001i	$-0.210844\pi$
$67$	12.9087	1.57705	0.788527	+	0.615001i	$0.210844\pi$
$68$	0	0
$69$	2.83947	0.341832
$70$	0	0
$71$	−4.83239	−0.573499	−0.286750	−	0.958006i	$-0.592575\pi$
$71$	−4.83239	−0.573499	−0.286750	+	0.958006i	$0.592575\pi$
$72$	0	0
$73$	3.01365	0.352721	0.176360	−	0.984326i	$-0.443568\pi$
$73$	3.01365	0.352721	0.176360	+	0.984326i	$0.443568\pi$
$74$	0	0
$75$	−3.10189	−0.358175
$76$	0	0
$77$	0.366691	0.0417884
$78$	0	0
$79$	6.21687	0.699453	0.349727	−	0.936852i	$-0.386275\pi$
$79$	6.21687	0.699453	0.349727	+	0.936852i	$0.386275\pi$
$80$	0	0
$81$	14.9818	1.66465
$82$	0	0
$83$	−9.34234	−1.02545	−0.512727	−	0.858551i	$-0.671365\pi$
$83$	−9.34234	−1.02545	−0.512727	+	0.858551i	$0.671365\pi$
$84$	0	0
$85$	1.15817	0.125622
$86$	0	0
$87$	18.3893	1.97154
$88$	0	0
$89$	−17.4169	−1.84619	−0.923096	−	0.384569i	$-0.874350\pi$
$89$	−17.4169	−1.84619	−0.923096	+	0.384569i	$0.874350\pi$
$90$	0	0
$91$	11.8334	1.24048
$92$	0	0
$93$	−4.73766	−0.491272
$94$	0	0
$95$	−2.61335	−0.268125
$96$	0	0
$97$	4.82293	0.489694	0.244847	−	0.969562i	$-0.421262\pi$
$97$	4.82293	0.489694	0.244847	+	0.969562i	$0.421262\pi$
$98$	0	0
$99$	1.23091	0.123711
$100$	0	0
$101$	0.282599	0.0281197	0.0140598	−	0.999901i	$-0.495524\pi$
$101$	0.282599	0.0281197	0.0140598	+	0.999901i	$0.495524\pi$
$102$	0	0
$103$	−5.60175	−0.551957	−0.275979	−	0.961164i	$-0.589002\pi$
$103$	−5.60175	−0.551957	−0.275979	+	0.961164i	$0.589002\pi$
$104$	0	0
$105$	6.11884	0.597137
$106$	0	0
$107$	19.7182	1.90623	0.953116	−	0.302605i	$-0.0978562\pi$
$107$	19.7182	1.90623	0.953116	+	0.302605i	$0.0978562\pi$
$108$	0	0
$109$	1.92431	0.184315	0.0921576	−	0.995744i	$-0.470624\pi$
$109$	1.92431	0.184315	0.0921576	+	0.995744i	$0.470624\pi$
$110$	0	0
$111$	13.8376	1.31340
$112$	0	0
$113$	7.77312	0.731234	0.365617	−	0.930765i	$-0.380858\pi$
$113$	7.77312	0.731234	0.365617	+	0.930765i	$0.380858\pi$
$114$	0	0
$115$	0.915400	0.0853615
$116$	0	0
$117$	39.7225	3.67235
$118$	0	0
$119$	−2.28463	−0.209432
$120$	0	0
$121$	−10.9654	−0.996859
$122$	0	0
$123$	28.0953	2.53327
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	0.234435	0.0208028	0.0104014	−	0.999946i	$-0.496689\pi$
$127$	0.234435	0.0208028	0.0104014	+	0.999946i	$0.496689\pi$
$128$	0	0
$129$	−5.55916	−0.489457
$130$	0	0
$131$	4.81052	0.420297	0.210149	−	0.977669i	$-0.432605\pi$
$131$	4.81052	0.420297	0.210149	+	0.977669i	$0.432605\pi$
$132$	0	0
$133$	5.15515	0.447008
$134$	0	0
$135$	11.2341	0.966878
$136$	0	0
$137$	−15.7908	−1.34910	−0.674550	−	0.738229i	$-0.735663\pi$
$137$	−15.7908	−1.34910	−0.674550	+	0.738229i	$0.735663\pi$
$138$	0	0
$139$	−8.36842	−0.709800	−0.354900	−	0.934904i	$-0.615485\pi$
$139$	−8.36842	−0.709800	−0.354900	+	0.934904i	$0.615485\pi$
$140$	0	0
$141$	13.8899	1.16974
$142$	0	0
$143$	1.11513	0.0932517
$144$	0	0
$145$	5.92842	0.492328
$146$	0	0
$147$	9.64309	0.795348
$148$	0	0
$149$	−10.4799	−0.858546	−0.429273	−	0.903175i	$-0.641230\pi$
$149$	−10.4799	−0.858546	−0.429273	+	0.903175i	$0.641230\pi$
$150$	0	0
$151$	−14.4639	−1.17705	−0.588526	−	0.808478i	$-0.700292\pi$
$151$	−14.4639	−1.17705	−0.588526	+	0.808478i	$0.700292\pi$
$152$	0	0
$153$	−7.66908	−0.620009
$154$	0	0
$155$	−1.52735	−0.122679
$156$	0	0
$157$	−1.19408	−0.0952980	−0.0476490	−	0.998864i	$-0.515173\pi$
$157$	−1.19408	−0.0952980	−0.0476490	+	0.998864i	$0.515173\pi$
$158$	0	0
$159$	37.4694	2.97152
$160$	0	0
$161$	−1.80573	−0.142312
$162$	0	0
$163$	−0.455128	−0.0356484	−0.0178242	−	0.999841i	$-0.505674\pi$
$163$	−0.455128	−0.0356484	−0.0178242	+	0.999841i	$0.505674\pi$
$164$	0	0
$165$	0.576612	0.0448892
$166$	0	0
$167$	−1.62963	−0.126105	−0.0630524	−	0.998010i	$-0.520084\pi$
$167$	−1.62963	−0.126105	−0.0630524	+	0.998010i	$0.520084\pi$
$168$	0	0
$169$	22.9861	1.76816
$170$	0	0
$171$	17.3049	1.32334
$172$	0	0
$173$	−14.8042	−1.12554	−0.562771	−	0.826613i	$-0.690265\pi$
$173$	−14.8042	−1.12554	−0.562771	+	0.826613i	$0.690265\pi$
$174$	0	0
$175$	1.97262	0.149116
$176$	0	0
$177$	3.62660	0.272592
$178$	0	0
$179$	7.43781	0.555928	0.277964	−	0.960591i	$-0.410340\pi$
$179$	7.43781	0.555928	0.277964	+	0.960591i	$0.410340\pi$
$180$	0	0
$181$	−13.8852	−1.03208	−0.516038	−	0.856565i	$-0.672594\pi$
$181$	−13.8852	−1.03208	−0.516038	+	0.856565i	$0.672594\pi$
$182$	0	0
$183$	−28.7849	−2.12784
$184$	0	0
$185$	4.46101	0.327980
$186$	0	0
$187$	−0.215294	−0.0157438
$188$	0	0
$189$	−22.1606	−1.61195
$190$	0	0
$191$	11.1009	0.803231	0.401616	−	0.915808i	$-0.368449\pi$
$191$	11.1009	0.803231	0.401616	+	0.915808i	$0.368449\pi$
$192$	0	0
$193$	4.90003	0.352712	0.176356	−	0.984326i	$-0.443569\pi$
$193$	4.90003	0.352712	0.176356	+	0.984326i	$0.443569\pi$
$194$	0	0
$195$	18.6077	1.33253
$196$	0	0
$197$	9.43054	0.671898	0.335949	−	0.941880i	$-0.390943\pi$
$197$	9.43054	0.671898	0.335949	+	0.941880i	$0.390943\pi$
$198$	0	0
$199$	8.03587	0.569648	0.284824	−	0.958580i	$-0.408065\pi$
$199$	8.03587	0.569648	0.284824	+	0.958580i	$0.408065\pi$
$200$	0	0
$201$	−40.0414	−2.82431
$202$	0	0
$203$	−11.6945	−0.820793
$204$	0	0
$205$	9.05750	0.632603
$206$	0	0
$207$	−6.06150	−0.421304
$208$	0	0
$209$	0.485798	0.0336034
$210$	0	0
$211$	−19.5735	−1.34750	−0.673748	−	0.738961i	$-0.735317\pi$
$211$	−19.5735	−1.34750	−0.673748	+	0.738961i	$0.735317\pi$
$212$	0	0
$213$	14.9895	1.02707
$214$	0	0
$215$	−1.79219	−0.122226
$216$	0	0
$217$	3.01287	0.204527
$218$	0	0
$219$	−9.34799	−0.631679
$220$	0	0
$221$	−6.94770	−0.467353
$222$	0	0
$223$	−6.78366	−0.454268	−0.227134	−	0.973864i	$-0.572935\pi$
$223$	−6.78366	−0.454268	−0.227134	+	0.973864i	$0.572935\pi$
$224$	0	0
$225$	6.62170	0.441447
$226$	0	0
$227$	20.8382	1.38308	0.691541	−	0.722337i	$-0.256932\pi$
$227$	20.8382	1.38308	0.691541	+	0.722337i	$0.256932\pi$
$228$	0	0
$229$	−4.23931	−0.280142	−0.140071	−	0.990141i	$-0.544733\pi$
$229$	−4.23931	−0.280142	−0.140071	+	0.990141i	$0.544733\pi$
$230$	0	0
$231$	−1.13743	−0.0748377
$232$	0	0
$233$	0.201662	0.0132113	0.00660565	−	0.999978i	$-0.497897\pi$
$233$	0.201662	0.0132113	0.00660565	+	0.999978i	$0.497897\pi$
$234$	0	0
$235$	4.47790	0.292106
$236$	0	0
$237$	−19.2840	−1.25263
$238$	0	0
$239$	−18.8098	−1.21670	−0.608351	−	0.793668i	$-0.708169\pi$
$239$	−18.8098	−1.21670	−0.608351	+	0.793668i	$0.708169\pi$
$240$	0	0
$241$	−6.32279	−0.407286	−0.203643	−	0.979045i	$-0.565278\pi$
$241$	−6.32279	−0.407286	−0.203643	+	0.979045i	$0.565278\pi$
$242$	0	0
$243$	−12.7696	−0.819172
$244$	0	0
$245$	3.10878	0.198613
$246$	0	0
$247$	15.6771	0.997510
$248$	0	0
$249$	28.9789	1.83646
$250$	0	0
$251$	14.8817	0.939322	0.469661	−	0.882847i	$-0.344376\pi$
$251$	14.8817	0.939322	0.469661	+	0.882847i	$0.344376\pi$
$252$	0	0
$253$	−0.170164	−0.0106981
$254$	0	0
$255$	−3.59252	−0.224973
$256$	0	0
$257$	0.743600	0.0463845	0.0231923	−	0.999731i	$-0.492617\pi$
$257$	0.743600	0.0463845	0.0231923	+	0.999731i	$0.492617\pi$
$258$	0	0
$259$	−8.79987	−0.546798
$260$	0	0
$261$	−39.2562	−2.42990
$262$	0	0
$263$	7.29543	0.449855	0.224928	−	0.974375i	$-0.427785\pi$
$263$	7.29543	0.449855	0.224928	+	0.974375i	$0.427785\pi$
$264$	0	0
$265$	12.0796	0.742041
$266$	0	0
$267$	54.0254	3.30630
$268$	0	0
$269$	22.1052	1.34778	0.673889	−	0.738833i	$-0.264623\pi$
$269$	22.1052	1.34778	0.673889	+	0.738833i	$0.264623\pi$
$270$	0	0
$271$	−16.5100	−1.00291	−0.501456	−	0.865183i	$-0.667202\pi$
$271$	−16.5100	−1.00291	−0.501456	+	0.865183i	$0.667202\pi$
$272$	0	0
$273$	−36.7059	−2.22154
$274$	0	0
$275$	0.185891	0.0112096
$276$	0	0
$277$	0.738435	0.0443682	0.0221841	−	0.999754i	$-0.492938\pi$
$277$	0.738435	0.0443682	0.0221841	+	0.999754i	$0.492938\pi$
$278$	0	0
$279$	10.1136	0.605488
$280$	0	0
$281$	−4.15931	−0.248124	−0.124062	−	0.992274i	$-0.539592\pi$
$281$	−4.15931	−0.248124	−0.124062	+	0.992274i	$0.539592\pi$
$282$	0	0
$283$	4.32278	0.256963	0.128481	−	0.991712i	$-0.458990\pi$
$283$	4.32278	0.256963	0.128481	+	0.991712i	$0.458990\pi$
$284$	0	0
$285$	8.10633	0.480178
$286$	0	0
$287$	−17.8670	−1.05465
$288$	0	0
$289$	−15.6586	−0.921096
$290$	0	0
$291$	−14.9602	−0.876981
$292$	0	0
$293$	32.2955	1.88672	0.943361	−	0.331769i	$-0.107645\pi$
$293$	32.2955	1.88672	0.943361	+	0.331769i	$0.107645\pi$
$294$	0	0
$295$	1.16916	0.0680711
$296$	0	0
$297$	−2.08832	−0.121176
$298$	0	0
$299$	−5.49134	−0.317572
$300$	0	0
$301$	3.53530	0.203771
$302$	0	0
$303$	−0.876590	−0.0503588
$304$	0	0
$305$	−9.27981	−0.531361
$306$	0	0
$307$	1.40664	0.0802814	0.0401407	−	0.999194i	$-0.487219\pi$
$307$	1.40664	0.0802814	0.0401407	+	0.999194i	$0.487219\pi$
$308$	0	0
$309$	17.3760	0.988487
$310$	0	0
$311$	−16.6181	−0.942323	−0.471162	−	0.882047i	$-0.656165\pi$
$311$	−16.6181	−0.942323	−0.471162	+	0.882047i	$0.656165\pi$
$312$	0	0
$313$	−8.97109	−0.507076	−0.253538	−	0.967325i	$-0.581594\pi$
$313$	−8.97109	−0.507076	−0.253538	+	0.967325i	$0.581594\pi$
$314$	0	0
$315$	−13.0621	−0.735965
$316$	0	0
$317$	−27.1399	−1.52433	−0.762166	−	0.647382i	$-0.775864\pi$
$317$	−27.1399	−1.52433	−0.762166	+	0.647382i	$0.775864\pi$
$318$	0	0
$319$	−1.10204	−0.0617023
$320$	0	0
$321$	−61.1637	−3.41382
$322$	0	0
$323$	−3.02672	−0.168411
$324$	0	0
$325$	5.99884	0.332756
$326$	0	0
$327$	−5.96898	−0.330085
$328$	0	0
$329$	−8.83319	−0.486990
$330$	0	0
$331$	23.2415	1.27747	0.638734	−	0.769428i	$-0.279459\pi$
$331$	23.2415	1.27747	0.638734	+	0.769428i	$0.279459\pi$
$332$	0	0
$333$	−29.5395	−1.61876
$334$	0	0
$335$	−12.9087	−0.705280
$336$	0	0
$337$	14.1771	0.772279	0.386139	−	0.922440i	$-0.373808\pi$
$337$	14.1771	0.772279	0.386139	+	0.922440i	$0.373808\pi$
$338$	0	0
$339$	−24.1113	−1.30955
$340$	0	0
$341$	0.283920	0.0153751
$342$	0	0
$343$	−19.9408	−1.07670
$344$	0	0
$345$	−2.83947	−0.152872
$346$	0	0
$347$	5.11881	0.274792	0.137396	−	0.990516i	$-0.456127\pi$
$347$	5.11881	0.274792	0.137396	+	0.990516i	$0.456127\pi$
$348$	0	0
$349$	2.91019	0.155779	0.0778894	−	0.996962i	$-0.475182\pi$
$349$	2.91019	0.155779	0.0778894	+	0.996962i	$0.475182\pi$
$350$	0	0
$351$	−67.3916	−3.59710
$352$	0	0
$353$	15.4289	0.821198	0.410599	−	0.911816i	$-0.365320\pi$
$353$	15.4289	0.821198	0.410599	+	0.911816i	$0.365320\pi$
$354$	0	0
$355$	4.83239	0.256477
$356$	0	0
$357$	7.08667	0.375067
$358$	0	0
$359$	−20.9945	−1.10805	−0.554024	−	0.832501i	$-0.686908\pi$
$359$	−20.9945	−1.10805	−0.554024	+	0.832501i	$0.686908\pi$
$360$	0	0
$361$	−12.1704	−0.640546
$362$	0	0
$363$	34.0136	1.78525
$364$	0	0
$365$	−3.01365	−0.157742
$366$	0	0
$367$	9.41327	0.491369	0.245684	−	0.969350i	$-0.420987\pi$
$367$	9.41327	0.491369	0.245684	+	0.969350i	$0.420987\pi$
$368$	0	0
$369$	−59.9761	−3.12223
$370$	0	0
$371$	−23.8283	−1.23711
$372$	0	0
$373$	−10.6443	−0.551143	−0.275571	−	0.961281i	$-0.588867\pi$
$373$	−10.6443	−0.551143	−0.275571	+	0.961281i	$0.588867\pi$
$374$	0	0
$375$	3.10189	0.160181
$376$	0	0
$377$	−35.5636	−1.83162
$378$	0	0
$379$	−16.9872	−0.872573	−0.436286	−	0.899808i	$-0.643706\pi$
$379$	−16.9872	−0.872573	−0.436286	+	0.899808i	$0.643706\pi$
$380$	0	0
$381$	−0.727192	−0.0372552
$382$	0	0
$383$	5.60790	0.286550	0.143275	−	0.989683i	$-0.454237\pi$
$383$	5.60790	0.286550	0.143275	+	0.989683i	$0.454237\pi$
$384$	0	0
$385$	−0.366691	−0.0186883
$386$	0	0
$387$	11.8673	0.603251
$388$	0	0
$389$	13.2801	0.673326	0.336663	−	0.941625i	$-0.390702\pi$
$389$	13.2801	0.673326	0.336663	+	0.941625i	$0.390702\pi$
$390$	0	0
$391$	1.06019	0.0536162
$392$	0	0
$393$	−14.9217	−0.752700
$394$	0	0
$395$	−6.21687	−0.312805
$396$	0	0
$397$	−17.0017	−0.853292	−0.426646	−	0.904419i	$-0.640305\pi$
$397$	−17.0017	−0.853292	−0.426646	+	0.904419i	$0.640305\pi$
$398$	0	0
$399$	−15.9907	−0.800536
$400$	0	0
$401$	1.00000	0.0499376
$401$	1.00000	0.0499376
$402$	0	0
$403$	9.16231	0.456407
$404$	0	0
$405$	−14.9818	−0.744454
$406$	0	0
$407$	−0.829261	−0.0411049
$408$	0	0
$409$	2.95348	0.146040	0.0730202	−	0.997330i	$-0.476736\pi$
$409$	2.95348	0.146040	0.0730202	+	0.997330i	$0.476736\pi$
$410$	0	0
$411$	48.9813	2.41607
$412$	0	0
$413$	−2.30631	−0.113486
$414$	0	0
$415$	9.34234	0.458597
$416$	0	0
$417$	25.9579	1.27116
$418$	0	0
$419$	−11.5570	−0.564598	−0.282299	−	0.959326i	$-0.591097\pi$
$419$	−11.5570	−0.564598	−0.282299	+	0.959326i	$0.591097\pi$
$420$	0	0
$421$	−31.0211	−1.51187	−0.755937	−	0.654644i	$-0.772818\pi$
$421$	−31.0211	−1.51187	−0.755937	+	0.654644i	$0.772818\pi$
$422$	0	0
$423$	−29.6513	−1.44170
$424$	0	0
$425$	−1.15817	−0.0561797
$426$	0	0
$427$	18.3055	0.885866
$428$	0	0
$429$	−3.45900	−0.167002
$430$	0	0
$431$	−3.74633	−0.180454	−0.0902272	−	0.995921i	$-0.528759\pi$
$431$	−3.74633	−0.180454	−0.0902272	+	0.995921i	$0.528759\pi$
$432$	0	0
$433$	−32.0977	−1.54252	−0.771258	−	0.636523i	$-0.780372\pi$
$433$	−32.0977	−1.54252	−0.771258	+	0.636523i	$0.780372\pi$
$434$	0	0
$435$	−18.3893	−0.881699
$436$	0	0
$437$	−2.39226	−0.114437
$438$	0	0
$439$	−12.6297	−0.602782	−0.301391	−	0.953501i	$-0.597451\pi$
$439$	−12.6297	−0.602782	−0.301391	+	0.953501i	$0.597451\pi$
$440$	0	0
$441$	−20.5854	−0.980258
$442$	0	0
$443$	4.97080	0.236170	0.118085	−	0.993004i	$-0.462325\pi$
$443$	4.97080	0.236170	0.118085	+	0.993004i	$0.462325\pi$
$444$	0	0
$445$	17.4169	0.825642
$446$	0	0
$447$	32.5074	1.53755
$448$	0	0
$449$	13.8969	0.655833	0.327917	−	0.944707i	$-0.393653\pi$
$449$	13.8969	0.655833	0.327917	+	0.944707i	$0.393653\pi$
$450$	0	0
$451$	−1.68371	−0.0792826
$452$	0	0
$453$	44.8653	2.10795
$454$	0	0
$455$	−11.8334	−0.554759
$456$	0	0
$457$	−19.1624	−0.896379	−0.448190	−	0.893939i	$-0.647931\pi$
$457$	−19.1624	−0.896379	−0.448190	+	0.893939i	$0.647931\pi$
$458$	0	0
$459$	13.0111	0.607304
$460$	0	0
$461$	−34.8438	−1.62284	−0.811420	−	0.584464i	$-0.801305\pi$
$461$	−34.8438	−1.62284	−0.811420	+	0.584464i	$0.801305\pi$
$462$	0	0
$463$	22.7936	1.05931	0.529655	−	0.848213i	$-0.322321\pi$
$463$	22.7936	1.05931	0.529655	+	0.848213i	$0.322321\pi$
$464$	0	0
$465$	4.73766	0.219704
$466$	0	0
$467$	−15.7097	−0.726957	−0.363478	−	0.931603i	$-0.618411\pi$
$467$	−15.7097	−0.726957	−0.363478	+	0.931603i	$0.618411\pi$
$468$	0	0
$469$	25.4640	1.17582
$470$	0	0
$471$	3.70390	0.170667
$472$	0	0
$473$	0.333151	0.0153183
$474$	0	0
$475$	2.61335	0.119909
$476$	0	0
$477$	−79.9872	−3.66236
$478$	0	0
$479$	20.2742	0.926351	0.463176	−	0.886267i	$-0.346710\pi$
$479$	20.2742	0.926351	0.463176	+	0.886267i	$0.346710\pi$
$480$	0	0
$481$	−26.7609	−1.22019
$482$	0	0
$483$	5.60118	0.254863
$484$	0	0
$485$	−4.82293	−0.218998
$486$	0	0
$487$	−23.0295	−1.04357	−0.521784	−	0.853078i	$-0.674733\pi$
$487$	−23.0295	−1.04357	−0.521784	+	0.853078i	$0.674733\pi$
$488$	0	0
$489$	1.41176	0.0638419
$490$	0	0
$491$	−32.6056	−1.47147	−0.735735	−	0.677269i	$-0.763163\pi$
$491$	−32.6056	−1.47147	−0.735735	+	0.677269i	$0.763163\pi$
$492$	0	0
$493$	6.86614	0.309235
$494$	0	0
$495$	−1.23091	−0.0553254
$496$	0	0
$497$	−9.53246	−0.427589
$498$	0	0
$499$	−5.04163	−0.225694	−0.112847	−	0.993612i	$-0.535997\pi$
$499$	−5.04163	−0.225694	−0.112847	+	0.993612i	$0.535997\pi$
$500$	0	0
$501$	5.05494	0.225838
$502$	0	0
$503$	10.9995	0.490445	0.245223	−	0.969467i	$-0.421139\pi$
$503$	10.9995	0.490445	0.245223	+	0.969467i	$0.421139\pi$
$504$	0	0
$505$	−0.282599	−0.0125755
$506$	0	0
$507$	−71.3002	−3.16655
$508$	0	0
$509$	37.1759	1.64779	0.823897	−	0.566739i	$-0.191795\pi$
$509$	37.1759	1.64779	0.823897	+	0.566739i	$0.191795\pi$
$510$	0	0
$511$	5.94477	0.262981
$512$	0	0
$513$	−29.3587	−1.29622
$514$	0	0
$515$	5.60175	0.246843
$516$	0	0
$517$	−0.832401	−0.0366089
$518$	0	0
$519$	45.9209	2.01570
$520$	0	0
$521$	−19.8544	−0.869837	−0.434919	−	0.900470i	$-0.643223\pi$
$521$	−19.8544	−0.869837	−0.434919	+	0.900470i	$0.643223\pi$
$522$	0	0
$523$	−3.93838	−0.172213	−0.0861066	−	0.996286i	$-0.527443\pi$
$523$	−3.93838	−0.172213	−0.0861066	+	0.996286i	$0.527443\pi$
$524$	0	0
$525$	−6.11884	−0.267048
$526$	0	0
$527$	−1.76893	−0.0770559
$528$	0	0
$529$	−22.1620	−0.963567
$530$	0	0
$531$	−7.74183	−0.335967
$532$	0	0
$533$	−54.3345	−2.35349
$534$	0	0
$535$	−19.7182	−0.852493
$536$	0	0
$537$	−23.0712	−0.995598
$538$	0	0
$539$	−0.577894	−0.0248916
$540$	0	0
$541$	13.0238	0.559939	0.279969	−	0.960009i	$-0.409676\pi$
$541$	13.0238	0.559939	0.279969	+	0.960009i	$0.409676\pi$
$542$	0	0
$543$	43.0702	1.84832
$544$	0	0
$545$	−1.92431	−0.0824283
$546$	0	0
$547$	20.3242	0.869002	0.434501	−	0.900671i	$-0.356925\pi$
$547$	20.3242	0.869002	0.434501	+	0.900671i	$0.356925\pi$
$548$	0	0
$549$	61.4482	2.62254
$550$	0	0
$551$	−15.4931	−0.660026
$552$	0	0
$553$	12.2635	0.521498
$554$	0	0
$555$	−13.8376	−0.587372
$556$	0	0
$557$	−2.39829	−0.101619	−0.0508094	−	0.998708i	$-0.516180\pi$
$557$	−2.39829	−0.101619	−0.0508094	+	0.998708i	$0.516180\pi$
$558$	0	0
$559$	10.7510	0.454721
$560$	0	0
$561$	0.667817	0.0281953
$562$	0	0
$563$	−3.83710	−0.161715	−0.0808573	−	0.996726i	$-0.525766\pi$
$563$	−3.83710	−0.161715	−0.0808573	+	0.996726i	$0.525766\pi$
$564$	0	0
$565$	−7.77312	−0.327018
$566$	0	0
$567$	29.5534	1.24113
$568$	0	0
$569$	−15.1032	−0.633160	−0.316580	−	0.948566i	$-0.602535\pi$
$569$	−15.1032	−0.633160	−0.316580	+	0.948566i	$0.602535\pi$
$570$	0	0
$571$	−6.96447	−0.291454	−0.145727	−	0.989325i	$-0.546552\pi$
$571$	−6.96447	−0.291454	−0.145727	+	0.989325i	$0.546552\pi$
$572$	0	0
$573$	−34.4337	−1.43849
$574$	0	0
$575$	−0.915400	−0.0381748
$576$	0	0
$577$	23.3917	0.973811	0.486906	−	0.873455i	$-0.338126\pi$
$577$	23.3917	0.973811	0.486906	+	0.873455i	$0.338126\pi$
$578$	0	0
$579$	−15.1993	−0.631663
$580$	0	0
$581$	−18.4289	−0.764558
$582$	0	0
$583$	−2.24548	−0.0929982
$584$	0	0
$585$	−39.7225	−1.64232
$586$	0	0
$587$	5.99646	0.247501	0.123750	−	0.992313i	$-0.460508\pi$
$587$	5.99646	0.247501	0.123750	+	0.992313i	$0.460508\pi$
$588$	0	0
$589$	3.99150	0.164467
$590$	0	0
$591$	−29.2525	−1.20329
$592$	0	0
$593$	12.7066	0.521798	0.260899	−	0.965366i	$-0.415981\pi$
$593$	12.7066	0.521798	0.260899	+	0.965366i	$0.415981\pi$
$594$	0	0
$595$	2.28463	0.0936608
$596$	0	0
$597$	−24.9264	−1.02017
$598$	0	0
$599$	−39.7648	−1.62475	−0.812373	−	0.583139i	$-0.801824\pi$
$599$	−39.7648	−1.62475	−0.812373	+	0.583139i	$0.801824\pi$
$600$	0	0
$601$	39.7160	1.62005	0.810026	−	0.586394i	$-0.199453\pi$
$601$	39.7160	1.62005	0.810026	+	0.586394i	$0.199453\pi$
$602$	0	0
$603$	85.4778	3.48093
$604$	0	0
$605$	10.9654	0.445809
$606$	0	0
$607$	31.1461	1.26418	0.632091	−	0.774894i	$-0.282197\pi$
$607$	31.1461	1.26418	0.632091	+	0.774894i	$0.282197\pi$
$608$	0	0
$609$	36.2750	1.46994
$610$	0	0
$611$	−26.8622	−1.08673
$612$	0	0
$613$	33.5549	1.35527	0.677635	−	0.735398i	$-0.263005\pi$
$613$	33.5549	1.35527	0.677635	+	0.735398i	$0.263005\pi$
$614$	0	0
$615$	−28.0953	−1.13291
$616$	0	0
$617$	−15.6154	−0.628653	−0.314326	−	0.949315i	$-0.601779\pi$
$617$	−15.6154	−0.628653	−0.314326	+	0.949315i	$0.601779\pi$
$618$	0	0
$619$	30.4666	1.22456	0.612279	−	0.790642i	$-0.290253\pi$
$619$	30.4666	1.22456	0.612279	+	0.790642i	$0.290253\pi$
$620$	0	0
$621$	10.2837	0.412671
$622$	0	0
$623$	−34.3570	−1.37648
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−1.50689	−0.0601795
$628$	0	0
$629$	5.16663	0.206007
$630$	0	0
$631$	−27.8821	−1.10997	−0.554984	−	0.831861i	$-0.687275\pi$
$631$	−27.8821	−1.10997	−0.554984	+	0.831861i	$0.687275\pi$
$632$	0	0
$633$	60.7148	2.41320
$634$	0	0
$635$	−0.234435	−0.00930328
$636$	0	0
$637$	−18.6491	−0.738903
$638$	0	0
$639$	−31.9986	−1.26585
$640$	0	0
$641$	−28.9800	−1.14464	−0.572320	−	0.820031i	$-0.693956\pi$
$641$	−28.9800	−1.14464	−0.572320	+	0.820031i	$0.693956\pi$
$642$	0	0
$643$	−25.7800	−1.01667	−0.508333	−	0.861161i	$-0.669738\pi$
$643$	−25.7800	−1.01667	−0.508333	+	0.861161i	$0.669738\pi$
$644$	0	0
$645$	5.55916	0.218892
$646$	0	0
$647$	10.2704	0.403772	0.201886	−	0.979409i	$-0.435293\pi$
$647$	10.2704	0.403772	0.201886	+	0.979409i	$0.435293\pi$
$648$	0	0
$649$	−0.217336	−0.00853119
$650$	0	0
$651$	−9.34559	−0.366282
$652$	0	0
$653$	−36.2398	−1.41817	−0.709087	−	0.705121i	$-0.750892\pi$
$653$	−36.2398	−1.41817	−0.709087	+	0.705121i	$0.750892\pi$
$654$	0	0
$655$	−4.81052	−0.187963
$656$	0	0
$657$	19.9555	0.778537
$658$	0	0
$659$	19.2951	0.751630	0.375815	−	0.926695i	$-0.377363\pi$
$659$	19.2951	0.751630	0.375815	+	0.926695i	$0.377363\pi$
$660$	0	0
$661$	−24.5310	−0.954147	−0.477074	−	0.878863i	$-0.658302\pi$
$661$	−24.5310	−0.954147	−0.477074	+	0.878863i	$0.658302\pi$
$662$	0	0
$663$	21.5510	0.836970
$664$	0	0
$665$	−5.15515	−0.199908
$666$	0	0
$667$	5.42687	0.210129
$668$	0	0
$669$	21.0421	0.813536
$670$	0	0
$671$	1.72503	0.0665941
$672$	0	0
$673$	−27.7957	−1.07145	−0.535724	−	0.844393i	$-0.679961\pi$
$673$	−27.7957	−1.07145	−0.535724	+	0.844393i	$0.679961\pi$
$674$	0	0
$675$	−11.2341	−0.432401
$676$	0	0
$677$	−20.4172	−0.784697	−0.392349	−	0.919817i	$-0.628337\pi$
$677$	−20.4172	−0.784697	−0.392349	+	0.919817i	$0.628337\pi$
$678$	0	0
$679$	9.51379	0.365106
$680$	0	0
$681$	−64.6379	−2.47693
$682$	0	0
$683$	19.3296	0.739627	0.369813	−	0.929106i	$-0.379422\pi$
$683$	19.3296	0.739627	0.369813	+	0.929106i	$0.379422\pi$
$684$	0	0
$685$	15.7908	0.603336
$686$	0	0
$687$	13.1499	0.501699
$688$	0	0
$689$	−72.4633	−2.76063
$690$	0	0
$691$	−36.8036	−1.40007	−0.700037	−	0.714106i	$-0.746833\pi$
$691$	−36.8036	−1.40007	−0.700037	+	0.714106i	$0.746833\pi$
$692$	0	0
$693$	2.42812	0.0922367
$694$	0	0
$695$	8.36842	0.317432
$696$	0	0
$697$	10.4902	0.397343
$698$	0	0
$699$	−0.625532	−0.0236598
$700$	0	0
$701$	−19.1881	−0.724725	−0.362363	−	0.932037i	$-0.618030\pi$
$701$	−19.1881	−0.724725	−0.362363	+	0.932037i	$0.618030\pi$
$702$	0	0
$703$	−11.6582	−0.439698
$704$	0	0
$705$	−13.8899	−0.523126
$706$	0	0
$707$	0.557460	0.0209654
$708$	0	0
$709$	18.1262	0.680742	0.340371	−	0.940291i	$-0.389447\pi$
$709$	18.1262	0.680742	0.340371	+	0.940291i	$0.389447\pi$
$710$	0	0
$711$	41.1663	1.54386
$712$	0	0
$713$	−1.39813	−0.0523605
$714$	0	0
$715$	−1.11513	−0.0417034
$716$	0	0
$717$	58.3458	2.17896
$718$	0	0
$719$	20.6737	0.770997	0.385499	−	0.922708i	$-0.374029\pi$
$719$	20.6737	0.770997	0.385499	+	0.922708i	$0.374029\pi$
$720$	0	0
$721$	−11.0501	−0.411528
$722$	0	0
$723$	19.6126	0.729399
$724$	0	0
$725$	−5.92842	−0.220176
$726$	0	0
$727$	−13.5398	−0.502164	−0.251082	−	0.967966i	$-0.580786\pi$
$727$	−13.5398	−0.502164	−0.251082	+	0.967966i	$0.580786\pi$
$728$	0	0
$729$	−5.33556	−0.197613
$730$	0	0
$731$	−2.07566	−0.0767712
$732$	0	0
$733$	−14.4540	−0.533871	−0.266936	−	0.963714i	$-0.586011\pi$
$733$	−14.4540	−0.533871	−0.266936	+	0.963714i	$0.586011\pi$
$734$	0	0
$735$	−9.64309	−0.355691
$736$	0	0
$737$	2.39961	0.0883909
$738$	0	0
$739$	−5.56089	−0.204561	−0.102280	−	0.994756i	$-0.532614\pi$
$739$	−5.56089	−0.204561	−0.102280	+	0.994756i	$0.532614\pi$
$740$	0	0
$741$	−48.6286	−1.78642
$742$	0	0
$743$	46.8370	1.71828	0.859142	−	0.511737i	$-0.170998\pi$
$743$	46.8370	1.71828	0.859142	+	0.511737i	$0.170998\pi$
$744$	0	0
$745$	10.4799	0.383954
$746$	0	0
$747$	−61.8622	−2.26342
$748$	0	0
$749$	38.8965	1.42125
$750$	0	0
$751$	−44.7896	−1.63440	−0.817199	−	0.576356i	$-0.804474\pi$
$751$	−44.7896	−1.63440	−0.817199	+	0.576356i	$0.804474\pi$
$752$	0	0
$753$	−46.1612	−1.68221
$754$	0	0
$755$	14.4639	0.526394
$756$	0	0
$757$	31.9019	1.15949	0.579746	−	0.814797i	$-0.303152\pi$
$757$	31.9019	1.15949	0.579746	+	0.814797i	$0.303152\pi$
$758$	0	0
$759$	0.527830	0.0191590
$760$	0	0
$761$	2.52380	0.0914878	0.0457439	−	0.998953i	$-0.485434\pi$
$761$	2.52380	0.0914878	0.0457439	+	0.998953i	$0.485434\pi$
$762$	0	0
$763$	3.79592	0.137422
$764$	0	0
$765$	7.66908	0.277276
$766$	0	0
$767$	−7.01360	−0.253247
$768$	0	0
$769$	−44.9644	−1.62146	−0.810729	−	0.585422i	$-0.800929\pi$
$769$	−44.9644	−1.62146	−0.810729	+	0.585422i	$0.800929\pi$
$770$	0	0
$771$	−2.30656	−0.0830689
$772$	0	0
$773$	26.7131	0.960803	0.480402	−	0.877049i	$-0.340491\pi$
$773$	26.7131	0.960803	0.480402	+	0.877049i	$0.340491\pi$
$774$	0	0
$775$	1.52735	0.0548639
$776$	0	0
$777$	27.2962	0.979246
$778$	0	0
$779$	−23.6705	−0.848082
$780$	0	0
$781$	−0.898297	−0.0321436
$782$	0	0
$783$	66.6005	2.38011
$784$	0	0
$785$	1.19408	0.0426186
$786$	0	0
$787$	−4.04080	−0.144039	−0.0720195	−	0.997403i	$-0.522944\pi$
$787$	−4.04080	−0.144039	−0.0720195	+	0.997403i	$0.522944\pi$
$788$	0	0
$789$	−22.6296	−0.805635
$790$	0	0
$791$	15.3334	0.545193
$792$	0	0
$793$	55.6681	1.97683
$794$	0	0
$795$	−37.4694	−1.32890
$796$	0	0
$797$	51.1792	1.81286	0.906430	−	0.422356i	$-0.138797\pi$
$797$	51.1792	1.81286	0.906430	+	0.422356i	$0.138797\pi$
$798$	0	0
$799$	5.18619	0.183474
$800$	0	0
$801$	−115.330	−4.07498
$802$	0	0
$803$	0.560209	0.0197694
$804$	0	0
$805$	1.80573	0.0636437
$806$	0	0
$807$	−68.5678	−2.41370
$808$	0	0
$809$	−0.745105	−0.0261965	−0.0130982	−	0.999914i	$-0.504169\pi$
$809$	−0.745105	−0.0261965	−0.0130982	+	0.999914i	$0.504169\pi$
$810$	0	0
$811$	26.9104	0.944954	0.472477	−	0.881343i	$-0.343360\pi$
$811$	26.9104	0.944954	0.472477	+	0.881343i	$0.343360\pi$
$812$	0	0
$813$	51.2122	1.79609
$814$	0	0
$815$	0.455128	0.0159425
$816$	0	0
$817$	4.68362	0.163859
$818$	0	0
$819$	78.3573	2.73803
$820$	0	0
$821$	51.6654	1.80313	0.901567	−	0.432639i	$-0.142418\pi$
$821$	51.6654	1.80313	0.901567	+	0.432639i	$0.142418\pi$
$822$	0	0
$823$	−33.5710	−1.17021	−0.585106	−	0.810957i	$-0.698947\pi$
$823$	−33.5710	−1.17021	−0.585106	+	0.810957i	$0.698947\pi$
$824$	0	0
$825$	−0.576612	−0.0200751
$826$	0	0
$827$	34.2714	1.19173	0.595867	−	0.803083i	$-0.296809\pi$
$827$	34.2714	1.19173	0.595867	+	0.803083i	$0.296809\pi$
$828$	0	0
$829$	44.2439	1.53665	0.768327	−	0.640057i	$-0.221089\pi$
$829$	44.2439	1.53665	0.768327	+	0.640057i	$0.221089\pi$
$830$	0	0
$831$	−2.29054	−0.0794580
$832$	0	0
$833$	3.60051	0.124750
$834$	0	0
$835$	1.62963	0.0563958
$836$	0	0
$837$	−17.1584	−0.593081
$838$	0	0
$839$	−33.2835	−1.14908	−0.574538	−	0.818478i	$-0.694818\pi$
$839$	−33.2835	−1.14908	−0.574538	+	0.818478i	$0.694818\pi$
$840$	0	0
$841$	6.14613	0.211936
$842$	0	0
$843$	12.9017	0.444358
$844$	0	0
$845$	−22.9861	−0.790745
$846$	0	0
$847$	−21.6306	−0.743237
$848$	0	0
$849$	−13.4088	−0.460188
$850$	0	0
$851$	4.08361	0.139984
$852$	0	0
$853$	−1.89277	−0.0648071	−0.0324035	−	0.999475i	$-0.510316\pi$
$853$	−1.89277	−0.0648071	−0.0324035	+	0.999475i	$0.510316\pi$
$854$	0	0
$855$	−17.3049	−0.591814
$856$	0	0
$857$	10.6471	0.363700	0.181850	−	0.983326i	$-0.441792\pi$
$857$	10.6471	0.363700	0.181850	+	0.983326i	$0.441792\pi$
$858$	0	0
$859$	−10.5542	−0.360106	−0.180053	−	0.983657i	$-0.557627\pi$
$859$	−10.5542	−0.360106	−0.180053	+	0.983657i	$0.557627\pi$
$860$	0	0
$861$	55.4214	1.88875
$862$	0	0
$863$	−22.0399	−0.750247	−0.375124	−	0.926975i	$-0.622400\pi$
$863$	−22.0399	−0.750247	−0.375124	+	0.926975i	$0.622400\pi$
$864$	0	0
$865$	14.8042	0.503357
$866$	0	0
$867$	48.5713	1.64957
$868$	0	0
$869$	1.15566	0.0392031
$870$	0	0
$871$	77.4374	2.62387
$872$	0	0
$873$	31.9360	1.08087
$874$	0	0
$875$	−1.97262	−0.0666866
$876$	0	0
$877$	−57.3912	−1.93796	−0.968982	−	0.247133i	$-0.920512\pi$
$877$	−57.3912	−1.93796	−0.968982	+	0.247133i	$0.920512\pi$
$878$	0	0
$879$	−100.177	−3.37888
$880$	0	0
$881$	−18.6979	−0.629947	−0.314973	−	0.949100i	$-0.601996\pi$
$881$	−18.6979	−0.629947	−0.314973	+	0.949100i	$0.601996\pi$
$882$	0	0
$883$	−26.8895	−0.904903	−0.452451	−	0.891789i	$-0.649450\pi$
$883$	−26.8895	−0.904903	−0.452451	+	0.891789i	$0.649450\pi$
$884$	0	0
$885$	−3.62660	−0.121907
$886$	0	0
$887$	22.3004	0.748775	0.374387	−	0.927272i	$-0.377853\pi$
$887$	22.3004	0.748775	0.374387	+	0.927272i	$0.377853\pi$
$888$	0	0
$889$	0.462451	0.0155101
$890$	0	0
$891$	2.78499	0.0933005
$892$	0	0
$893$	−11.7023	−0.391604
$894$	0	0
$895$	−7.43781	−0.248619
$896$	0	0
$897$	17.0335	0.568732
$898$	0	0
$899$	−9.05475	−0.301993
$900$	0	0
$901$	13.9902	0.466082
$902$	0	0
$903$	−10.9661	−0.364929
$904$	0	0
$905$	13.8852	0.461559
$906$	0	0
$907$	−34.7322	−1.15326	−0.576631	−	0.817005i	$-0.695633\pi$
$907$	−34.7322	−1.15326	−0.576631	+	0.817005i	$0.695633\pi$
$908$	0	0
$909$	1.87129	0.0620667
$910$	0	0
$911$	44.5643	1.47648	0.738241	−	0.674537i	$-0.235657\pi$
$911$	44.5643	1.47648	0.738241	+	0.674537i	$0.235657\pi$
$912$	0	0
$913$	−1.73665	−0.0574749
$914$	0	0
$915$	28.7849	0.951601
$916$	0	0
$917$	9.48932	0.313365
$918$	0	0
$919$	−4.73926	−0.156334	−0.0781669	−	0.996940i	$-0.524907\pi$
$919$	−4.73926	−0.156334	−0.0781669	+	0.996940i	$0.524907\pi$
$920$	0	0
$921$	−4.36325	−0.143774
$922$	0	0
$923$	−28.9887	−0.954176
$924$	0	0
$925$	−4.46101	−0.146677
$926$	0	0
$927$	−37.0931	−1.21830
$928$	0	0
$929$	25.5550	0.838433	0.419216	−	0.907886i	$-0.362305\pi$
$929$	25.5550	0.838433	0.419216	+	0.907886i	$0.362305\pi$
$930$	0	0
$931$	−8.12435	−0.266265
$932$	0	0
$933$	51.5473	1.68758
$934$	0	0
$935$	0.215294	0.00704086
$936$	0	0
$937$	−4.24355	−0.138631	−0.0693153	−	0.997595i	$-0.522081\pi$
$937$	−4.24355	−0.138631	−0.0693153	+	0.997595i	$0.522081\pi$
$938$	0	0
$939$	27.8273	0.908110
$940$	0	0
$941$	−24.1928	−0.788664	−0.394332	−	0.918968i	$-0.629024\pi$
$941$	−24.1928	−0.788664	−0.394332	+	0.918968i	$0.629024\pi$
$942$	0	0
$943$	8.29123	0.270000
$944$	0	0
$945$	22.1606	0.720885
$946$	0	0
$947$	−0.205536	−0.00667902	−0.00333951	−	0.999994i	$-0.501063\pi$
$947$	−0.205536	−0.00667902	−0.00333951	+	0.999994i	$0.501063\pi$
$948$	0	0
$949$	18.0784	0.586849
$950$	0	0
$951$	84.1851	2.72989
$952$	0	0
$953$	3.60112	0.116652	0.0583259	−	0.998298i	$-0.481424\pi$
$953$	3.60112	0.116652	0.0583259	+	0.998298i	$0.481424\pi$
$954$	0	0
$955$	−11.1009	−0.359216
$956$	0	0
$957$	3.41840	0.110501
$958$	0	0
$959$	−31.1492	−1.00586
$960$	0	0
$961$	−28.6672	−0.924749
$962$	0	0
$963$	130.568	4.20750
$964$	0	0
$965$	−4.90003	−0.157737
$966$	0	0
$967$	−43.2378	−1.39044	−0.695218	−	0.718799i	$-0.744692\pi$
$967$	−43.2378	−1.39044	−0.695218	+	0.718799i	$0.744692\pi$
$968$	0	0
$969$	9.38854	0.301603
$970$	0	0
$971$	21.6240	0.693947	0.346974	−	0.937875i	$-0.387209\pi$
$971$	21.6240	0.693947	0.346974	+	0.937875i	$0.387209\pi$
$972$	0	0
$973$	−16.5077	−0.529212
$974$	0	0
$975$	−18.6077	−0.595924
$976$	0	0
$977$	31.0512	0.993415	0.496708	−	0.867918i	$-0.334542\pi$
$977$	31.0512	0.993415	0.496708	+	0.867918i	$0.334542\pi$
$978$	0	0
$979$	−3.23765	−0.103476
$980$	0	0
$981$	12.7422	0.406827
$982$	0	0
$983$	34.1386	1.08885	0.544427	−	0.838808i	$-0.316747\pi$
$983$	34.1386	1.08885	0.544427	+	0.838808i	$0.316747\pi$
$984$	0	0
$985$	−9.43054	−0.300482
$986$	0	0
$987$	27.3996	0.872137
$988$	0	0
$989$	−1.64057	−0.0521670
$990$	0	0
$991$	−45.6659	−1.45062	−0.725312	−	0.688420i	$-0.758305\pi$
$991$	−45.6659	−1.45062	−0.725312	+	0.688420i	$0.758305\pi$
$992$	0	0
$993$	−72.0924	−2.28778
$994$	0	0
$995$	−8.03587	−0.254754
$996$	0	0
$997$	−8.30234	−0.262938	−0.131469	−	0.991320i	$-0.541969\pi$
$997$	−8.30234	−0.262938	−0.131469	+	0.991320i	$0.541969\pi$
$998$	0	0
$999$	50.1155	1.58559

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8020.2.a.c.1.1	✓	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)

\(f(q)\)	\(=\)	\(q-3.10189 q^{3} -1.00000 q^{5} +1.97262 q^{7} +6.62170 q^{9} +O(q^{10})\)	\(q-3.10189 q^{3} -1.00000 q^{5} +1.97262 q^{7} +6.62170 q^{9} +0.185891 q^{11} +5.99884 q^{13} +3.10189 q^{15} -1.15817 q^{17} +2.61335 q^{19} -6.11884 q^{21} -0.915400 q^{23} +1.00000 q^{25} -11.2341 q^{27} -5.92842 q^{29} +1.52735 q^{31} -0.576612 q^{33} -1.97262 q^{35} -4.46101 q^{37} -18.6077 q^{39} -9.05750 q^{41} +1.79219 q^{43} -6.62170 q^{45} -4.47790 q^{47} -3.10878 q^{49} +3.59252 q^{51} -12.0796 q^{53} -0.185891 q^{55} -8.10633 q^{57} -1.16916 q^{59} +9.27981 q^{61} +13.0621 q^{63} -5.99884 q^{65} +12.9087 q^{67} +2.83947 q^{69} -4.83239 q^{71} +3.01365 q^{73} -3.10189 q^{75} +0.366691 q^{77} +6.21687 q^{79} +14.9818 q^{81} -9.34234 q^{83} +1.15817 q^{85} +18.3893 q^{87} -17.4169 q^{89} +11.8334 q^{91} -4.73766 q^{93} -2.61335 q^{95} +4.82293 q^{97} +1.23091 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 3 q^{3} - 28 q^{5} - 4 q^{7} + 17 q^{9}+O(q^{10}) \) 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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