LMFDB - Embedded newform 8000.2.a.g.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8000,2,Mod(1,8000)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8000, 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("8000.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8000 = 2^{6} \cdot 5^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8000.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:	$63.8803216170$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1000)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	8000.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.618034 q^{3} +0.236068 q^{7} -2.61803 q^{9} +O(q^{10})$	$q+0.618034 q^{3} +0.236068 q^{7} -2.61803 q^{9} -1.76393 q^{11} +4.61803 q^{13} +2.23607 q^{17} -7.09017 q^{19} +0.145898 q^{21} -0.763932 q^{23} -3.47214 q^{27} +5.76393 q^{29} -0.854102 q^{31} -1.09017 q^{33} +3.23607 q^{37} +2.85410 q^{39} -3.00000 q^{41} +10.7082 q^{43} -9.32624 q^{47} -6.94427 q^{49} +1.38197 q^{51} +3.14590 q^{53} -4.38197 q^{57} -5.61803 q^{59} -7.56231 q^{61} -0.618034 q^{63} +0.381966 q^{67} -0.472136 q^{69} +4.23607 q^{71} -12.8541 q^{73} -0.416408 q^{77} +6.23607 q^{79} +5.70820 q^{81} +10.9443 q^{83} +3.56231 q^{87} -8.47214 q^{89} +1.09017 q^{91} -0.527864 q^{93} +8.56231 q^{97} +4.61803 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} - 4 q^{7} - 3 q^{9}+O(q^{10}) $ 2 * q - q^3 - 4 * q^7 - 3 * q^9	$ 2 q - q^{3} - 4 q^{7} - 3 q^{9} - 8 q^{11} + 7 q^{13} - 3 q^{19} + 7 q^{21} - 6 q^{23} + 2 q^{27} + 16 q^{29} + 5 q^{31} + 9 q^{33} + 2 q^{37} - q^{39} - 6 q^{41} + 8 q^{43} - 3 q^{47} + 4 q^{49} + 5 q^{51} + 13 q^{53} - 11 q^{57} - 9 q^{59} + 5 q^{61} + q^{63} + 3 q^{67} + 8 q^{69} + 4 q^{71} - 19 q^{73} + 26 q^{77} + 8 q^{79} - 2 q^{81} + 4 q^{83} - 13 q^{87} - 8 q^{89} - 9 q^{91} - 10 q^{93} - 3 q^{97} + 7 q^{99}+O(q^{100}) $ 2 * q - q^3 - 4 * q^7 - 3 * q^9 - 8 * q^11 + 7 * q^13 - 3 * q^19 + 7 * q^21 - 6 * q^23 + 2 * q^27 + 16 * q^29 + 5 * q^31 + 9 * q^33 + 2 * q^37 - q^39 - 6 * q^41 + 8 * q^43 - 3 * q^47 + 4 * q^49 + 5 * q^51 + 13 * q^53 - 11 * q^57 - 9 * q^59 + 5 * q^61 + q^63 + 3 * q^67 + 8 * q^69 + 4 * q^71 - 19 * q^73 + 26 * q^77 + 8 * q^79 - 2 * q^81 + 4 * q^83 - 13 * q^87 - 8 * q^89 - 9 * q^91 - 10 * q^93 - 3 * 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
$3$	0.618034	0.356822	0.178411	−	0.983956i	$-0.442904\pi$
$3$	0.618034	0.356822	0.178411	+	0.983956i	$0.442904\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.236068	0.0892253	0.0446127	−	0.999004i	$-0.485795\pi$
$7$	0.236068	0.0892253	0.0446127	+	0.999004i	$0.485795\pi$
$8$	0	0
$9$	−2.61803	−0.872678
$10$	0	0
$11$	−1.76393	−0.531846	−0.265923	−	0.963994i	$-0.585677\pi$
$11$	−1.76393	−0.531846	−0.265923	+	0.963994i	$0.585677\pi$
$12$	0	0
$13$	4.61803	1.28081	0.640406	−	0.768036i	$-0.278766\pi$
$13$	4.61803	1.28081	0.640406	+	0.768036i	$0.278766\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.23607	0.542326	0.271163	−	0.962533i	$-0.412592\pi$
$17$	2.23607	0.542326	0.271163	+	0.962533i	$0.412592\pi$
$18$	0	0
$19$	−7.09017	−1.62660	−0.813298	−	0.581847i	$-0.802330\pi$
$19$	−7.09017	−1.62660	−0.813298	+	0.581847i	$0.802330\pi$
$20$	0	0
$21$	0.145898	0.0318376
$22$	0	0
$23$	−0.763932	−0.159291	−0.0796454	−	0.996823i	$-0.525379\pi$
$23$	−0.763932	−0.159291	−0.0796454	+	0.996823i	$0.525379\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−3.47214	−0.668213
$28$	0	0
$29$	5.76393	1.07034	0.535168	−	0.844746i	$-0.320248\pi$
$29$	5.76393	1.07034	0.535168	+	0.844746i	$0.320248\pi$
$30$	0	0
$31$	−0.854102	−0.153401	−0.0767006	−	0.997054i	$-0.524439\pi$
$31$	−0.854102	−0.153401	−0.0767006	+	0.997054i	$0.524439\pi$
$32$	0	0
$33$	−1.09017	−0.189774
$34$	0	0
$35$	0	0
$36$	0	0
$37$	3.23607	0.532006	0.266003	−	0.963972i	$-0.414297\pi$
$37$	3.23607	0.532006	0.266003	+	0.963972i	$0.414297\pi$
$38$	0	0
$39$	2.85410	0.457022
$40$	0	0
$41$	−3.00000	−0.468521	−0.234261	−	0.972174i	$-0.575267\pi$
$41$	−3.00000	−0.468521	−0.234261	+	0.972174i	$0.575267\pi$
$42$	0	0
$43$	10.7082	1.63299	0.816493	−	0.577355i	$-0.195915\pi$
$43$	10.7082	1.63299	0.816493	+	0.577355i	$0.195915\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−9.32624	−1.36037	−0.680186	−	0.733040i	$-0.738101\pi$
$47$	−9.32624	−1.36037	−0.680186	+	0.733040i	$0.738101\pi$
$48$	0	0
$49$	−6.94427	−0.992039
$50$	0	0
$51$	1.38197	0.193514
$52$	0	0
$53$	3.14590	0.432122	0.216061	−	0.976380i	$-0.430679\pi$
$53$	3.14590	0.432122	0.216061	+	0.976380i	$0.430679\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−4.38197	−0.580406
$58$	0	0
$59$	−5.61803	−0.731406	−0.365703	−	0.930732i	$-0.619171\pi$
$59$	−5.61803	−0.731406	−0.365703	+	0.930732i	$0.619171\pi$
$60$	0	0
$61$	−7.56231	−0.968254	−0.484127	−	0.874998i	$-0.660863\pi$
$61$	−7.56231	−0.968254	−0.484127	+	0.874998i	$0.660863\pi$
$62$	0	0
$63$	−0.618034	−0.0778650
$64$	0	0
$65$	0	0
$66$	0	0
$67$	0.381966	0.0466646	0.0233323	−	0.999728i	$-0.492572\pi$
$67$	0.381966	0.0466646	0.0233323	+	0.999728i	$0.492572\pi$
$68$	0	0
$69$	−0.472136	−0.0568385
$70$	0	0
$71$	4.23607	0.502729	0.251364	−	0.967893i	$-0.419121\pi$
$71$	4.23607	0.502729	0.251364	+	0.967893i	$0.419121\pi$
$72$	0	0
$73$	−12.8541	−1.50446	−0.752229	−	0.658901i	$-0.771021\pi$
$73$	−12.8541	−1.50446	−0.752229	+	0.658901i	$0.771021\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−0.416408	−0.0474541
$78$	0	0
$79$	6.23607	0.701612	0.350806	−	0.936448i	$-0.385908\pi$
$79$	6.23607	0.701612	0.350806	+	0.936448i	$0.385908\pi$
$80$	0	0
$81$	5.70820	0.634245
$82$	0	0
$83$	10.9443	1.20129	0.600645	−	0.799516i	$-0.294911\pi$
$83$	10.9443	1.20129	0.600645	+	0.799516i	$0.294911\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	3.56231	0.381919
$88$	0	0
$89$	−8.47214	−0.898045	−0.449022	−	0.893521i	$-0.648228\pi$
$89$	−8.47214	−0.898045	−0.449022	+	0.893521i	$0.648228\pi$
$90$	0	0
$91$	1.09017	0.114281
$92$	0	0
$93$	−0.527864	−0.0547370
$94$	0	0
$95$	0	0
$96$	0	0
$97$	8.56231	0.869370	0.434685	−	0.900582i	$-0.356860\pi$
$97$	8.56231	0.869370	0.434685	+	0.900582i	$0.356860\pi$
$98$	0	0
$99$	4.61803	0.464130
$100$	0	0
$101$	13.4721	1.34053	0.670264	−	0.742123i	$-0.266181\pi$
$101$	13.4721	1.34053	0.670264	+	0.742123i	$0.266181\pi$
$102$	0	0
$103$	−13.4721	−1.32745	−0.663724	−	0.747977i	$-0.731025\pi$
$103$	−13.4721	−1.32745	−0.663724	+	0.747977i	$0.731025\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−19.2705	−1.86295	−0.931475	−	0.363805i	$-0.881478\pi$
$107$	−19.2705	−1.86295	−0.931475	+	0.363805i	$0.881478\pi$
$108$	0	0
$109$	−8.14590	−0.780236	−0.390118	−	0.920765i	$-0.627566\pi$
$109$	−8.14590	−0.780236	−0.390118	+	0.920765i	$0.627566\pi$
$110$	0	0
$111$	2.00000	0.189832
$112$	0	0
$113$	−9.00000	−0.846649	−0.423324	−	0.905978i	$-0.639137\pi$
$113$	−9.00000	−0.846649	−0.423324	+	0.905978i	$0.639137\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−12.0902	−1.11774
$118$	0	0
$119$	0.527864	0.0483892
$120$	0	0
$121$	−7.88854	−0.717140
$122$	0	0
$123$	−1.85410	−0.167179
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−15.7082	−1.39388	−0.696939	−	0.717131i	$-0.745455\pi$
$127$	−15.7082	−1.39388	−0.696939	+	0.717131i	$0.745455\pi$
$128$	0	0
$129$	6.61803	0.582685
$130$	0	0
$131$	−2.32624	−0.203244	−0.101622	−	0.994823i	$-0.532403\pi$
$131$	−2.32624	−0.203244	−0.101622	+	0.994823i	$0.532403\pi$
$132$	0	0
$133$	−1.67376	−0.145134
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−9.56231	−0.816963	−0.408481	−	0.912767i	$-0.633942\pi$
$137$	−9.56231	−0.816963	−0.408481	+	0.912767i	$0.633942\pi$
$138$	0	0
$139$	16.8885	1.43247	0.716234	−	0.697860i	$-0.245864\pi$
$139$	16.8885	1.43247	0.716234	+	0.697860i	$0.245864\pi$
$140$	0	0
$141$	−5.76393	−0.485411
$142$	0	0
$143$	−8.14590	−0.681194
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−4.29180	−0.353981
$148$	0	0
$149$	−7.47214	−0.612141	−0.306071	−	0.952009i	$-0.599014\pi$
$149$	−7.47214	−0.612141	−0.306071	+	0.952009i	$0.599014\pi$
$150$	0	0
$151$	−20.0902	−1.63491	−0.817457	−	0.575989i	$-0.804617\pi$
$151$	−20.0902	−1.63491	−0.817457	+	0.575989i	$0.804617\pi$
$152$	0	0
$153$	−5.85410	−0.473276
$154$	0	0
$155$	0	0
$156$	0	0
$157$	16.0344	1.27969	0.639844	−	0.768505i	$-0.278999\pi$
$157$	16.0344	1.27969	0.639844	+	0.768505i	$0.278999\pi$
$158$	0	0
$159$	1.94427	0.154191
$160$	0	0
$161$	−0.180340	−0.0142128
$162$	0	0
$163$	−21.6180	−1.69326	−0.846628	−	0.532186i	$-0.821371\pi$
$163$	−21.6180	−1.69326	−0.846628	+	0.532186i	$0.821371\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−15.2361	−1.17900	−0.589501	−	0.807768i	$-0.700676\pi$
$167$	−15.2361	−1.17900	−0.589501	+	0.807768i	$0.700676\pi$
$168$	0	0
$169$	8.32624	0.640480
$170$	0	0
$171$	18.5623	1.41950
$172$	0	0
$173$	20.4164	1.55223	0.776115	−	0.630591i	$-0.217187\pi$
$173$	20.4164	1.55223	0.776115	+	0.630591i	$0.217187\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−3.47214	−0.260982
$178$	0	0
$179$	15.4721	1.15644	0.578221	−	0.815880i	$-0.303747\pi$
$179$	15.4721	1.15644	0.578221	+	0.815880i	$0.303747\pi$
$180$	0	0
$181$	0.763932	0.0567826	0.0283913	−	0.999597i	$-0.490962\pi$
$181$	0.763932	0.0567826	0.0283913	+	0.999597i	$0.490962\pi$
$182$	0	0
$183$	−4.67376	−0.345494
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−3.94427	−0.288434
$188$	0	0
$189$	−0.819660	−0.0596215
$190$	0	0
$191$	20.0000	1.44715	0.723575	−	0.690246i	$-0.242498\pi$
$191$	20.0000	1.44715	0.723575	+	0.690246i	$0.242498\pi$
$192$	0	0
$193$	19.9787	1.43810	0.719050	−	0.694959i	$-0.244577\pi$
$193$	19.9787	1.43810	0.719050	+	0.694959i	$0.244577\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−17.4721	−1.24484	−0.622419	−	0.782684i	$-0.713850\pi$
$197$	−17.4721	−1.24484	−0.622419	+	0.782684i	$0.713850\pi$
$198$	0	0
$199$	−12.6525	−0.896910	−0.448455	−	0.893805i	$-0.648026\pi$
$199$	−12.6525	−0.896910	−0.448455	+	0.893805i	$0.648026\pi$
$200$	0	0
$201$	0.236068	0.0166510
$202$	0	0
$203$	1.36068	0.0955010
$204$	0	0
$205$	0	0
$206$	0	0
$207$	2.00000	0.139010
$208$	0	0
$209$	12.5066	0.865098
$210$	0	0
$211$	−16.6525	−1.14640	−0.573202	−	0.819414i	$-0.694299\pi$
$211$	−16.6525	−1.14640	−0.573202	+	0.819414i	$0.694299\pi$
$212$	0	0
$213$	2.61803	0.179385
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−0.201626	−0.0136873
$218$	0	0
$219$	−7.94427	−0.536824
$220$	0	0
$221$	10.3262	0.694618
$222$	0	0
$223$	3.79837	0.254358	0.127179	−	0.991880i	$-0.459408\pi$
$223$	3.79837	0.254358	0.127179	+	0.991880i	$0.459408\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−21.7984	−1.44681	−0.723404	−	0.690425i	$-0.757424\pi$
$227$	−21.7984	−1.44681	−0.723404	+	0.690425i	$0.757424\pi$
$228$	0	0
$229$	−5.90983	−0.390533	−0.195266	−	0.980750i	$-0.562557\pi$
$229$	−5.90983	−0.390533	−0.195266	+	0.980750i	$0.562557\pi$
$230$	0	0
$231$	−0.257354	−0.0169327
$232$	0	0
$233$	11.0000	0.720634	0.360317	−	0.932830i	$-0.382669\pi$
$233$	11.0000	0.720634	0.360317	+	0.932830i	$0.382669\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	3.85410	0.250351
$238$	0	0
$239$	−25.5066	−1.64988	−0.824942	−	0.565218i	$-0.808792\pi$
$239$	−25.5066	−1.64988	−0.824942	+	0.565218i	$0.808792\pi$
$240$	0	0
$241$	15.7639	1.01544	0.507722	−	0.861521i	$-0.330488\pi$
$241$	15.7639	1.01544	0.507722	+	0.861521i	$0.330488\pi$
$242$	0	0
$243$	13.9443	0.894525
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−32.7426	−2.08336
$248$	0	0
$249$	6.76393	0.428647
$250$	0	0
$251$	−12.0344	−0.759607	−0.379804	−	0.925067i	$-0.624008\pi$
$251$	−12.0344	−0.759607	−0.379804	+	0.925067i	$0.624008\pi$
$252$	0	0
$253$	1.34752	0.0847181
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1.58359	0.0987818	0.0493909	−	0.998780i	$-0.484272\pi$
$257$	1.58359	0.0987818	0.0493909	+	0.998780i	$0.484272\pi$
$258$	0	0
$259$	0.763932	0.0474684
$260$	0	0
$261$	−15.0902	−0.934058
$262$	0	0
$263$	10.8541	0.669293	0.334646	−	0.942344i	$-0.391383\pi$
$263$	10.8541	0.669293	0.334646	+	0.942344i	$0.391383\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−5.23607	−0.320442
$268$	0	0
$269$	−27.7426	−1.69150	−0.845749	−	0.533580i	$-0.820846\pi$
$269$	−27.7426	−1.69150	−0.845749	+	0.533580i	$0.820846\pi$
$270$	0	0
$271$	−17.0902	−1.03815	−0.519077	−	0.854728i	$-0.673724\pi$
$271$	−17.0902	−1.03815	−0.519077	+	0.854728i	$0.673724\pi$
$272$	0	0
$273$	0.673762	0.0407779
$274$	0	0
$275$	0	0
$276$	0	0
$277$	14.2361	0.855362	0.427681	−	0.903930i	$-0.359331\pi$
$277$	14.2361	0.855362	0.427681	+	0.903930i	$0.359331\pi$
$278$	0	0
$279$	2.23607	0.133870
$280$	0	0
$281$	−12.0000	−0.715860	−0.357930	−	0.933748i	$-0.616517\pi$
$281$	−12.0000	−0.715860	−0.357930	+	0.933748i	$0.616517\pi$
$282$	0	0
$283$	−6.65248	−0.395449	−0.197724	−	0.980258i	$-0.563355\pi$
$283$	−6.65248	−0.395449	−0.197724	+	0.980258i	$0.563355\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−0.708204	−0.0418040
$288$	0	0
$289$	−12.0000	−0.705882
$290$	0	0
$291$	5.29180	0.310211
$292$	0	0
$293$	−23.0000	−1.34367	−0.671837	−	0.740699i	$-0.734495\pi$
$293$	−23.0000	−1.34367	−0.671837	+	0.740699i	$0.734495\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	6.12461	0.355386
$298$	0	0
$299$	−3.52786	−0.204022
$300$	0	0
$301$	2.52786	0.145704
$302$	0	0
$303$	8.32624	0.478330
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−20.6180	−1.17673	−0.588367	−	0.808594i	$-0.700229\pi$
$307$	−20.6180	−1.17673	−0.588367	+	0.808594i	$0.700229\pi$
$308$	0	0
$309$	−8.32624	−0.473663
$310$	0	0
$311$	15.7984	0.895844	0.447922	−	0.894073i	$-0.352164\pi$
$311$	15.7984	0.895844	0.447922	+	0.894073i	$0.352164\pi$
$312$	0	0
$313$	−15.4164	−0.871387	−0.435693	−	0.900095i	$-0.643497\pi$
$313$	−15.4164	−0.871387	−0.435693	+	0.900095i	$0.643497\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−10.9443	−0.614692	−0.307346	−	0.951598i	$-0.599441\pi$
$317$	−10.9443	−0.614692	−0.307346	+	0.951598i	$0.599441\pi$
$318$	0	0
$319$	−10.1672	−0.569253
$320$	0	0
$321$	−11.9098	−0.664742
$322$	0	0
$323$	−15.8541	−0.882146
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−5.03444	−0.278405
$328$	0	0
$329$	−2.20163	−0.121380
$330$	0	0
$331$	5.03444	0.276718	0.138359	−	0.990382i	$-0.455817\pi$
$331$	5.03444	0.276718	0.138359	+	0.990382i	$0.455817\pi$
$332$	0	0
$333$	−8.47214	−0.464270
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−8.88854	−0.484190	−0.242095	−	0.970253i	$-0.577835\pi$
$337$	−8.88854	−0.484190	−0.242095	+	0.970253i	$0.577835\pi$
$338$	0	0
$339$	−5.56231	−0.302103
$340$	0	0
$341$	1.50658	0.0815858
$342$	0	0
$343$	−3.29180	−0.177740
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−1.29180	−0.0693472	−0.0346736	−	0.999399i	$-0.511039\pi$
$347$	−1.29180	−0.0693472	−0.0346736	+	0.999399i	$0.511039\pi$
$348$	0	0
$349$	30.8885	1.65343	0.826713	−	0.562624i	$-0.190208\pi$
$349$	30.8885	1.65343	0.826713	+	0.562624i	$0.190208\pi$
$350$	0	0
$351$	−16.0344	−0.855855
$352$	0	0
$353$	30.0000	1.59674	0.798369	−	0.602168i	$-0.205696\pi$
$353$	30.0000	1.59674	0.798369	+	0.602168i	$0.205696\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0.326238	0.0172663
$358$	0	0
$359$	10.4377	0.550880	0.275440	−	0.961318i	$-0.411176\pi$
$359$	10.4377	0.550880	0.275440	+	0.961318i	$0.411176\pi$
$360$	0	0
$361$	31.2705	1.64582
$362$	0	0
$363$	−4.87539	−0.255892
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−17.5623	−0.916745	−0.458372	−	0.888760i	$-0.651567\pi$
$367$	−17.5623	−0.916745	−0.458372	+	0.888760i	$0.651567\pi$
$368$	0	0
$369$	7.85410	0.408868
$370$	0	0
$371$	0.742646	0.0385562
$372$	0	0
$373$	−34.9230	−1.80824	−0.904122	−	0.427275i	$-0.859474\pi$
$373$	−34.9230	−1.80824	−0.904122	+	0.427275i	$0.859474\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	26.6180	1.37090
$378$	0	0
$379$	4.67376	0.240075	0.120038	−	0.992769i	$-0.461699\pi$
$379$	4.67376	0.240075	0.120038	+	0.992769i	$0.461699\pi$
$380$	0	0
$381$	−9.70820	−0.497366
$382$	0	0
$383$	10.7082	0.547164	0.273582	−	0.961849i	$-0.411792\pi$
$383$	10.7082	0.547164	0.273582	+	0.961849i	$0.411792\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−28.0344	−1.42507
$388$	0	0
$389$	32.8885	1.66752	0.833758	−	0.552131i	$-0.186185\pi$
$389$	32.8885	1.66752	0.833758	+	0.552131i	$0.186185\pi$
$390$	0	0
$391$	−1.70820	−0.0863876
$392$	0	0
$393$	−1.43769	−0.0725221
$394$	0	0
$395$	0	0
$396$	0	0
$397$	9.94427	0.499089	0.249544	−	0.968363i	$-0.419719\pi$
$397$	9.94427	0.499089	0.249544	+	0.968363i	$0.419719\pi$
$398$	0	0
$399$	−1.03444	−0.0517869
$400$	0	0
$401$	−32.9443	−1.64516	−0.822579	−	0.568651i	$-0.807466\pi$
$401$	−32.9443	−1.64516	−0.822579	+	0.568651i	$0.807466\pi$
$402$	0	0
$403$	−3.94427	−0.196478
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−5.70820	−0.282945
$408$	0	0
$409$	11.2361	0.555588	0.277794	−	0.960641i	$-0.410397\pi$
$409$	11.2361	0.555588	0.277794	+	0.960641i	$0.410397\pi$
$410$	0	0
$411$	−5.90983	−0.291510
$412$	0	0
$413$	−1.32624	−0.0652599
$414$	0	0
$415$	0	0
$416$	0	0
$417$	10.4377	0.511136
$418$	0	0
$419$	−36.4164	−1.77906	−0.889529	−	0.456879i	$-0.848967\pi$
$419$	−36.4164	−1.77906	−0.889529	+	0.456879i	$0.848967\pi$
$420$	0	0
$421$	−1.61803	−0.0788582	−0.0394291	−	0.999222i	$-0.512554\pi$
$421$	−1.61803	−0.0788582	−0.0394291	+	0.999222i	$0.512554\pi$
$422$	0	0
$423$	24.4164	1.18717
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−1.78522	−0.0863928
$428$	0	0
$429$	−5.03444	−0.243065
$430$	0	0
$431$	37.7426	1.81800	0.909000	−	0.416797i	$-0.136847\pi$
$431$	37.7426	1.81800	0.909000	+	0.416797i	$0.136847\pi$
$432$	0	0
$433$	6.18034	0.297008	0.148504	−	0.988912i	$-0.452554\pi$
$433$	6.18034	0.297008	0.148504	+	0.988912i	$0.452554\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	5.41641	0.259102
$438$	0	0
$439$	2.41641	0.115329	0.0576644	−	0.998336i	$-0.481635\pi$
$439$	2.41641	0.115329	0.0576644	+	0.998336i	$0.481635\pi$
$440$	0	0
$441$	18.1803	0.865730
$442$	0	0
$443$	−15.4721	−0.735103	−0.367552	−	0.930003i	$-0.619804\pi$
$443$	−15.4721	−0.735103	−0.367552	+	0.930003i	$0.619804\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−4.61803	−0.218426
$448$	0	0
$449$	−16.5623	−0.781624	−0.390812	−	0.920471i	$-0.627806\pi$
$449$	−16.5623	−0.781624	−0.390812	+	0.920471i	$0.627806\pi$
$450$	0	0
$451$	5.29180	0.249181
$452$	0	0
$453$	−12.4164	−0.583374
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−31.8885	−1.49168	−0.745842	−	0.666123i	$-0.767952\pi$
$457$	−31.8885	−1.49168	−0.745842	+	0.666123i	$0.767952\pi$
$458$	0	0
$459$	−7.76393	−0.362389
$460$	0	0
$461$	17.2705	0.804368	0.402184	−	0.915559i	$-0.368251\pi$
$461$	17.2705	0.804368	0.402184	+	0.915559i	$0.368251\pi$
$462$	0	0
$463$	13.3607	0.620923	0.310462	−	0.950586i	$-0.399516\pi$
$463$	13.3607	0.620923	0.310462	+	0.950586i	$0.399516\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−14.1803	−0.656188	−0.328094	−	0.944645i	$-0.606406\pi$
$467$	−14.1803	−0.656188	−0.328094	+	0.944645i	$0.606406\pi$
$468$	0	0
$469$	0.0901699	0.00416366
$470$	0	0
$471$	9.90983	0.456621
$472$	0	0
$473$	−18.8885	−0.868496
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−8.23607	−0.377104
$478$	0	0
$479$	28.9787	1.32407	0.662036	−	0.749472i	$-0.269693\pi$
$479$	28.9787	1.32407	0.662036	+	0.749472i	$0.269693\pi$
$480$	0	0
$481$	14.9443	0.681400
$482$	0	0
$483$	−0.111456	−0.00507143
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−42.6525	−1.93277	−0.966384	−	0.257103i	$-0.917232\pi$
$487$	−42.6525	−1.93277	−0.966384	+	0.257103i	$0.917232\pi$
$488$	0	0
$489$	−13.3607	−0.604191
$490$	0	0
$491$	−8.14590	−0.367619	−0.183810	−	0.982962i	$-0.558843\pi$
$491$	−8.14590	−0.367619	−0.183810	+	0.982962i	$0.558843\pi$
$492$	0	0
$493$	12.8885	0.580471
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.00000	0.0448561
$498$	0	0
$499$	6.00000	0.268597	0.134298	−	0.990941i	$-0.457122\pi$
$499$	6.00000	0.268597	0.134298	+	0.990941i	$0.457122\pi$
$500$	0	0
$501$	−9.41641	−0.420694
$502$	0	0
$503$	31.5623	1.40729	0.703647	−	0.710550i	$-0.251554\pi$
$503$	31.5623	1.40729	0.703647	+	0.710550i	$0.251554\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	5.14590	0.228537
$508$	0	0
$509$	20.4721	0.907411	0.453706	−	0.891152i	$-0.350102\pi$
$509$	20.4721	0.907411	0.453706	+	0.891152i	$0.350102\pi$
$510$	0	0
$511$	−3.03444	−0.134236
$512$	0	0
$513$	24.6180	1.08691
$514$	0	0
$515$	0	0
$516$	0	0
$517$	16.4508	0.723508
$518$	0	0
$519$	12.6180	0.553870
$520$	0	0
$521$	2.03444	0.0891305	0.0445653	−	0.999006i	$-0.485810\pi$
$521$	2.03444	0.0891305	0.0445653	+	0.999006i	$0.485810\pi$
$522$	0	0
$523$	−1.05573	−0.0461638	−0.0230819	−	0.999734i	$-0.507348\pi$
$523$	−1.05573	−0.0461638	−0.0230819	+	0.999734i	$0.507348\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.90983	−0.0831935
$528$	0	0
$529$	−22.4164	−0.974626
$530$	0	0
$531$	14.7082	0.638282
$532$	0	0
$533$	−13.8541	−0.600088
$534$	0	0
$535$	0	0
$536$	0	0
$537$	9.56231	0.412644
$538$	0	0
$539$	12.2492	0.527611
$540$	0	0
$541$	10.2918	0.442479	0.221239	−	0.975220i	$-0.428990\pi$
$541$	10.2918	0.442479	0.221239	+	0.975220i	$0.428990\pi$
$542$	0	0
$543$	0.472136	0.0202613
$544$	0	0
$545$	0	0
$546$	0	0
$547$	1.72949	0.0739477	0.0369738	−	0.999316i	$-0.488228\pi$
$547$	1.72949	0.0739477	0.0369738	+	0.999316i	$0.488228\pi$
$548$	0	0
$549$	19.7984	0.844974
$550$	0	0
$551$	−40.8673	−1.74100
$552$	0	0
$553$	1.47214	0.0626016
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−4.36068	−0.184768	−0.0923840	−	0.995723i	$-0.529449\pi$
$557$	−4.36068	−0.184768	−0.0923840	+	0.995723i	$0.529449\pi$
$558$	0	0
$559$	49.4508	2.09155
$560$	0	0
$561$	−2.43769	−0.102920
$562$	0	0
$563$	28.0000	1.18006	0.590030	−	0.807382i	$-0.299116\pi$
$563$	28.0000	1.18006	0.590030	+	0.807382i	$0.299116\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1.34752	0.0565907
$568$	0	0
$569$	0.944272	0.0395859	0.0197930	−	0.999804i	$-0.493699\pi$
$569$	0.944272	0.0395859	0.0197930	+	0.999804i	$0.493699\pi$
$570$	0	0
$571$	35.6525	1.49201	0.746005	−	0.665941i	$-0.231970\pi$
$571$	35.6525	1.49201	0.746005	+	0.665941i	$0.231970\pi$
$572$	0	0
$573$	12.3607	0.516375
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−19.5066	−0.812069	−0.406035	−	0.913858i	$-0.633089\pi$
$577$	−19.5066	−0.812069	−0.406035	+	0.913858i	$0.633089\pi$
$578$	0	0
$579$	12.3475	0.513146
$580$	0	0
$581$	2.58359	0.107185
$582$	0	0
$583$	−5.54915	−0.229822
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−35.4164	−1.46179	−0.730896	−	0.682489i	$-0.760897\pi$
$587$	−35.4164	−1.46179	−0.730896	+	0.682489i	$0.760897\pi$
$588$	0	0
$589$	6.05573	0.249522
$590$	0	0
$591$	−10.7984	−0.444186
$592$	0	0
$593$	22.4164	0.920532	0.460266	−	0.887781i	$-0.347754\pi$
$593$	22.4164	0.920532	0.460266	+	0.887781i	$0.347754\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−7.81966	−0.320037
$598$	0	0
$599$	−2.03444	−0.0831250	−0.0415625	−	0.999136i	$-0.513234\pi$
$599$	−2.03444	−0.0831250	−0.0415625	+	0.999136i	$0.513234\pi$
$600$	0	0
$601$	−22.5279	−0.918931	−0.459465	−	0.888196i	$-0.651959\pi$
$601$	−22.5279	−0.918931	−0.459465	+	0.888196i	$0.651959\pi$
$602$	0	0
$603$	−1.00000	−0.0407231
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−12.7639	−0.518072	−0.259036	−	0.965868i	$-0.583405\pi$
$607$	−12.7639	−0.518072	−0.259036	+	0.965868i	$0.583405\pi$
$608$	0	0
$609$	0.840946	0.0340769
$610$	0	0
$611$	−43.0689	−1.74238
$612$	0	0
$613$	−2.88854	−0.116667	−0.0583336	−	0.998297i	$-0.518579\pi$
$613$	−2.88854	−0.116667	−0.0583336	+	0.998297i	$0.518579\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−6.81966	−0.274549	−0.137275	−	0.990533i	$-0.543834\pi$
$617$	−6.81966	−0.274549	−0.137275	+	0.990533i	$0.543834\pi$
$618$	0	0
$619$	−35.0689	−1.40954	−0.704769	−	0.709437i	$-0.748949\pi$
$619$	−35.0689	−1.40954	−0.704769	+	0.709437i	$0.748949\pi$
$620$	0	0
$621$	2.65248	0.106440
$622$	0	0
$623$	−2.00000	−0.0801283
$624$	0	0
$625$	0	0
$626$	0	0
$627$	7.72949	0.308686
$628$	0	0
$629$	7.23607	0.288521
$630$	0	0
$631$	4.47214	0.178033	0.0890165	−	0.996030i	$-0.471628\pi$
$631$	4.47214	0.178033	0.0890165	+	0.996030i	$0.471628\pi$
$632$	0	0
$633$	−10.2918	−0.409062
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−32.0689	−1.27062
$638$	0	0
$639$	−11.0902	−0.438720
$640$	0	0
$641$	−26.8541	−1.06067	−0.530337	−	0.847787i	$-0.677934\pi$
$641$	−26.8541	−1.06067	−0.530337	+	0.847787i	$0.677934\pi$
$642$	0	0
$643$	−28.9098	−1.14009	−0.570046	−	0.821613i	$-0.693075\pi$
$643$	−28.9098	−1.14009	−0.570046	+	0.821613i	$0.693075\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	38.5279	1.51469	0.757343	−	0.653017i	$-0.226497\pi$
$647$	38.5279	1.51469	0.757343	+	0.653017i	$0.226497\pi$
$648$	0	0
$649$	9.90983	0.388995
$650$	0	0
$651$	−0.124612	−0.00488392
$652$	0	0
$653$	23.5967	0.923412	0.461706	−	0.887033i	$-0.347237\pi$
$653$	23.5967	0.923412	0.461706	+	0.887033i	$0.347237\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	33.6525	1.31291
$658$	0	0
$659$	23.7984	0.927053	0.463527	−	0.886083i	$-0.346584\pi$
$659$	23.7984	0.927053	0.463527	+	0.886083i	$0.346584\pi$
$660$	0	0
$661$	−28.4721	−1.10744	−0.553719	−	0.832704i	$-0.686792\pi$
$661$	−28.4721	−1.10744	−0.553719	+	0.832704i	$0.686792\pi$
$662$	0	0
$663$	6.38197	0.247855
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−4.40325	−0.170495
$668$	0	0
$669$	2.34752	0.0907605
$670$	0	0
$671$	13.3394	0.514962
$672$	0	0
$673$	36.6869	1.41418	0.707088	−	0.707125i	$-0.250008\pi$
$673$	36.6869	1.41418	0.707088	+	0.707125i	$0.250008\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	10.9443	0.420623	0.210311	−	0.977634i	$-0.432552\pi$
$677$	10.9443	0.420623	0.210311	+	0.977634i	$0.432552\pi$
$678$	0	0
$679$	2.02129	0.0775698
$680$	0	0
$681$	−13.4721	−0.516253
$682$	0	0
$683$	−4.05573	−0.155188	−0.0775941	−	0.996985i	$-0.524724\pi$
$683$	−4.05573	−0.155188	−0.0775941	+	0.996985i	$0.524724\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−3.65248	−0.139351
$688$	0	0
$689$	14.5279	0.553468
$690$	0	0
$691$	−13.3475	−0.507764	−0.253882	−	0.967235i	$-0.581707\pi$
$691$	−13.3475	−0.507764	−0.253882	+	0.967235i	$0.581707\pi$
$692$	0	0
$693$	1.09017	0.0414121
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−6.70820	−0.254091
$698$	0	0
$699$	6.79837	0.257138
$700$	0	0
$701$	19.5066	0.736753	0.368377	−	0.929677i	$-0.379914\pi$
$701$	19.5066	0.736753	0.368377	+	0.929677i	$0.379914\pi$
$702$	0	0
$703$	−22.9443	−0.865360
$704$	0	0
$705$	0	0
$706$	0	0
$707$	3.18034	0.119609
$708$	0	0
$709$	−12.9787	−0.487426	−0.243713	−	0.969847i	$-0.578365\pi$
$709$	−12.9787	−0.487426	−0.243713	+	0.969847i	$0.578365\pi$
$710$	0	0
$711$	−16.3262	−0.612282
$712$	0	0
$713$	0.652476	0.0244354
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−15.7639	−0.588715
$718$	0	0
$719$	−5.79837	−0.216243	−0.108121	−	0.994138i	$-0.534484\pi$
$719$	−5.79837	−0.216243	−0.108121	+	0.994138i	$0.534484\pi$
$720$	0	0
$721$	−3.18034	−0.118442
$722$	0	0
$723$	9.74265	0.362333
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−8.00000	−0.296704	−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000	−0.296704	−0.148352	+	0.988935i	$0.547397\pi$
$728$	0	0
$729$	−8.50658	−0.315058
$730$	0	0
$731$	23.9443	0.885611
$732$	0	0
$733$	25.5066	0.942107	0.471053	−	0.882105i	$-0.343874\pi$
$733$	25.5066	0.942107	0.471053	+	0.882105i	$0.343874\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−0.673762	−0.0248183
$738$	0	0
$739$	−17.4721	−0.642723	−0.321361	−	0.946957i	$-0.604140\pi$
$739$	−17.4721	−0.642723	−0.321361	+	0.946957i	$0.604140\pi$
$740$	0	0
$741$	−20.2361	−0.743391
$742$	0	0
$743$	−6.50658	−0.238703	−0.119352	−	0.992852i	$-0.538082\pi$
$743$	−6.50658	−0.238703	−0.119352	+	0.992852i	$0.538082\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−28.6525	−1.04834
$748$	0	0
$749$	−4.54915	−0.166222
$750$	0	0
$751$	11.6525	0.425205	0.212602	−	0.977139i	$-0.431806\pi$
$751$	11.6525	0.425205	0.212602	+	0.977139i	$0.431806\pi$
$752$	0	0
$753$	−7.43769	−0.271045
$754$	0	0
$755$	0	0
$756$	0	0
$757$	7.47214	0.271579	0.135790	−	0.990738i	$-0.456643\pi$
$757$	7.47214	0.271579	0.135790	+	0.990738i	$0.456643\pi$
$758$	0	0
$759$	0.832816	0.0302293
$760$	0	0
$761$	37.7771	1.36942	0.684709	−	0.728816i	$-0.259929\pi$
$761$	37.7771	1.36942	0.684709	+	0.728816i	$0.259929\pi$
$762$	0	0
$763$	−1.92299	−0.0696168
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−25.9443	−0.936793
$768$	0	0
$769$	37.7214	1.36027	0.680134	−	0.733088i	$-0.261922\pi$
$769$	37.7214	1.36027	0.680134	+	0.733088i	$0.261922\pi$
$770$	0	0
$771$	0.978714	0.0352475
$772$	0	0
$773$	−32.7639	−1.17844	−0.589218	−	0.807974i	$-0.700564\pi$
$773$	−32.7639	−1.17844	−0.589218	+	0.807974i	$0.700564\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0.472136	0.0169378
$778$	0	0
$779$	21.2705	0.762095
$780$	0	0
$781$	−7.47214	−0.267374
$782$	0	0
$783$	−20.0132	−0.715212
$784$	0	0
$785$	0	0
$786$	0	0
$787$	29.8328	1.06343	0.531713	−	0.846925i	$-0.321549\pi$
$787$	29.8328	1.06343	0.531713	+	0.846925i	$0.321549\pi$
$788$	0	0
$789$	6.70820	0.238818
$790$	0	0
$791$	−2.12461	−0.0755425
$792$	0	0
$793$	−34.9230	−1.24015
$794$	0	0
$795$	0	0
$796$	0	0
$797$	20.4377	0.723940	0.361970	−	0.932190i	$-0.382104\pi$
$797$	20.4377	0.723940	0.361970	+	0.932190i	$0.382104\pi$
$798$	0	0
$799$	−20.8541	−0.737765
$800$	0	0
$801$	22.1803	0.783704
$802$	0	0
$803$	22.6738	0.800140
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−17.1459	−0.603564
$808$	0	0
$809$	8.65248	0.304205	0.152102	−	0.988365i	$-0.451396\pi$
$809$	8.65248	0.304205	0.152102	+	0.988365i	$0.451396\pi$
$810$	0	0
$811$	−22.1803	−0.778857	−0.389429	−	0.921057i	$-0.627327\pi$
$811$	−22.1803	−0.778857	−0.389429	+	0.921057i	$0.627327\pi$
$812$	0	0
$813$	−10.5623	−0.370436
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−75.9230	−2.65621
$818$	0	0
$819$	−2.85410	−0.0997304
$820$	0	0
$821$	39.1033	1.36472	0.682358	−	0.731018i	$-0.260955\pi$
$821$	39.1033	1.36472	0.682358	+	0.731018i	$0.260955\pi$
$822$	0	0
$823$	−48.1246	−1.67752	−0.838759	−	0.544502i	$-0.816719\pi$
$823$	−48.1246	−1.67752	−0.838759	+	0.544502i	$0.816719\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−0.618034	−0.0214911	−0.0107456	−	0.999942i	$-0.503420\pi$
$827$	−0.618034	−0.0214911	−0.0107456	+	0.999942i	$0.503420\pi$
$828$	0	0
$829$	−1.90983	−0.0663311	−0.0331656	−	0.999450i	$-0.510559\pi$
$829$	−1.90983	−0.0663311	−0.0331656	+	0.999450i	$0.510559\pi$
$830$	0	0
$831$	8.79837	0.305212
$832$	0	0
$833$	−15.5279	−0.538009
$834$	0	0
$835$	0	0
$836$	0	0
$837$	2.96556	0.102505
$838$	0	0
$839$	−37.0902	−1.28049	−0.640247	−	0.768169i	$-0.721168\pi$
$839$	−37.0902	−1.28049	−0.640247	+	0.768169i	$0.721168\pi$
$840$	0	0
$841$	4.22291	0.145618
$842$	0	0
$843$	−7.41641	−0.255435
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−1.86223	−0.0639871
$848$	0	0
$849$	−4.11146	−0.141105
$850$	0	0
$851$	−2.47214	−0.0847437
$852$	0	0
$853$	−18.5279	−0.634382	−0.317191	−	0.948362i	$-0.602740\pi$
$853$	−18.5279	−0.634382	−0.317191	+	0.948362i	$0.602740\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	49.4853	1.69039	0.845193	−	0.534462i	$-0.179486\pi$
$857$	49.4853	1.69039	0.845193	+	0.534462i	$0.179486\pi$
$858$	0	0
$859$	22.1803	0.756783	0.378392	−	0.925646i	$-0.376477\pi$
$859$	22.1803	0.756783	0.378392	+	0.925646i	$0.376477\pi$
$860$	0	0
$861$	−0.437694	−0.0149166
$862$	0	0
$863$	−26.2918	−0.894983	−0.447492	−	0.894288i	$-0.647683\pi$
$863$	−26.2918	−0.894983	−0.447492	+	0.894288i	$0.647683\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−7.41641	−0.251874
$868$	0	0
$869$	−11.0000	−0.373149
$870$	0	0
$871$	1.76393	0.0597686
$872$	0	0
$873$	−22.4164	−0.758680
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−5.50658	−0.185944	−0.0929720	−	0.995669i	$-0.529637\pi$
$877$	−5.50658	−0.185944	−0.0929720	+	0.995669i	$0.529637\pi$
$878$	0	0
$879$	−14.2148	−0.479453
$880$	0	0
$881$	16.7082	0.562914	0.281457	−	0.959574i	$-0.409182\pi$
$881$	16.7082	0.562914	0.281457	+	0.959574i	$0.409182\pi$
$882$	0	0
$883$	−40.1246	−1.35030	−0.675150	−	0.737680i	$-0.735921\pi$
$883$	−40.1246	−1.35030	−0.675150	+	0.737680i	$0.735921\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	15.4164	0.517632	0.258816	−	0.965927i	$-0.416668\pi$
$887$	15.4164	0.517632	0.258816	+	0.965927i	$0.416668\pi$
$888$	0	0
$889$	−3.70820	−0.124369
$890$	0	0
$891$	−10.0689	−0.337320
$892$	0	0
$893$	66.1246	2.21278
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−2.18034	−0.0727994
$898$	0	0
$899$	−4.92299	−0.164191
$900$	0	0
$901$	7.03444	0.234351
$902$	0	0
$903$	1.56231	0.0519903
$904$	0	0
$905$	0	0
$906$	0	0
$907$	35.0902	1.16515	0.582575	−	0.812777i	$-0.302045\pi$
$907$	35.0902	1.16515	0.582575	+	0.812777i	$0.302045\pi$
$908$	0	0
$909$	−35.2705	−1.16985
$910$	0	0
$911$	−19.7082	−0.652962	−0.326481	−	0.945204i	$-0.605863\pi$
$911$	−19.7082	−0.652962	−0.326481	+	0.945204i	$0.605863\pi$
$912$	0	0
$913$	−19.3050	−0.638901
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−0.549150	−0.0181345
$918$	0	0
$919$	−44.5410	−1.46927	−0.734636	−	0.678461i	$-0.762647\pi$
$919$	−44.5410	−1.46927	−0.734636	+	0.678461i	$0.762647\pi$
$920$	0	0
$921$	−12.7426	−0.419884
$922$	0	0
$923$	19.5623	0.643901
$924$	0	0
$925$	0	0
$926$	0	0
$927$	35.2705	1.15844
$928$	0	0
$929$	30.4853	1.00019	0.500095	−	0.865971i	$-0.333298\pi$
$929$	30.4853	1.00019	0.500095	+	0.865971i	$0.333298\pi$
$930$	0	0
$931$	49.2361	1.61365
$932$	0	0
$933$	9.76393	0.319657
$934$	0	0
$935$	0	0
$936$	0	0
$937$	15.4508	0.504757	0.252379	−	0.967629i	$-0.418787\pi$
$937$	15.4508	0.504757	0.252379	+	0.967629i	$0.418787\pi$
$938$	0	0
$939$	−9.52786	−0.310930
$940$	0	0
$941$	−23.7984	−0.775805	−0.387902	−	0.921700i	$-0.626800\pi$
$941$	−23.7984	−0.775805	−0.387902	+	0.921700i	$0.626800\pi$
$942$	0	0
$943$	2.29180	0.0746311
$944$	0	0
$945$	0	0
$946$	0	0
$947$	7.11146	0.231091	0.115546	−	0.993302i	$-0.463138\pi$
$947$	7.11146	0.231091	0.115546	+	0.993302i	$0.463138\pi$
$948$	0	0
$949$	−59.3607	−1.92693
$950$	0	0
$951$	−6.76393	−0.219336
$952$	0	0
$953$	−54.0476	−1.75077	−0.875387	−	0.483423i	$-0.839393\pi$
$953$	−54.0476	−1.75077	−0.875387	+	0.483423i	$0.839393\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−6.28367	−0.203122
$958$	0	0
$959$	−2.25735	−0.0728938
$960$	0	0
$961$	−30.2705	−0.976468
$962$	0	0
$963$	50.4508	1.62576
$964$	0	0
$965$	0	0
$966$	0	0
$967$	45.9787	1.47858	0.739288	−	0.673390i	$-0.235162\pi$
$967$	45.9787	1.47858	0.739288	+	0.673390i	$0.235162\pi$
$968$	0	0
$969$	−9.79837	−0.314769
$970$	0	0
$971$	0.0557281	0.00178840	0.000894200	−	1.00000i	$-0.499715\pi$
$971$	0.0557281	0.00178840	0.000894200		1.00000i	$0.499715\pi$
$972$	0	0
$973$	3.98684	0.127812
$974$	0	0
$975$	0	0
$976$	0	0
$977$	7.72949	0.247288	0.123644	−	0.992327i	$-0.460542\pi$
$977$	7.72949	0.247288	0.123644	+	0.992327i	$0.460542\pi$
$978$	0	0
$979$	14.9443	0.477621
$980$	0	0
$981$	21.3262	0.680894
$982$	0	0
$983$	−0.798374	−0.0254642	−0.0127321	−	0.999919i	$-0.504053\pi$
$983$	−0.798374	−0.0254642	−0.0127321	+	0.999919i	$0.504053\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−1.36068	−0.0433109
$988$	0	0
$989$	−8.18034	−0.260120
$990$	0	0
$991$	19.3607	0.615012	0.307506	−	0.951546i	$-0.400506\pi$
$991$	19.3607	0.615012	0.307506	+	0.951546i	$0.400506\pi$
$992$	0	0
$993$	3.11146	0.0987391
$994$	0	0
$995$	0	0
$996$	0	0
$997$	21.2705	0.673644	0.336822	−	0.941568i	$-0.390648\pi$
$997$	21.2705	0.673644	0.336822	+	0.941568i	$0.390648\pi$
$998$	0	0
$999$	−11.2361	−0.355493

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8000.2.a.g.1.2		2
4.3	odd	2		8000.2.a.r.1.1		2
5.4	even	2		8000.2.a.q.1.1		2
8.3	odd	2		1000.2.a.b.1.2	✓	2
8.5	even	2		2000.2.a.g.1.1		2
20.19	odd	2		8000.2.a.h.1.2		2
24.11	even	2		9000.2.a.o.1.1		2
40.3	even	4		1000.2.c.a.249.3		4
40.13	odd	4		2000.2.c.h.1249.2		4
40.19	odd	2		1000.2.a.c.1.1	yes	2
40.27	even	4		1000.2.c.a.249.2		4
40.29	even	2		2000.2.a.f.1.2		2
40.37	odd	4		2000.2.c.h.1249.3		4
120.59	even	2		9000.2.a.c.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1000.2.a.b.1.2	✓	2	8.3	odd	2
1000.2.a.c.1.1	yes	2	40.19	odd	2
1000.2.c.a.249.2		4	40.27	even	4
1000.2.c.a.249.3		4	40.3	even	4
2000.2.a.f.1.2		2	40.29	even	2
2000.2.a.g.1.1		2	8.5	even	2
2000.2.c.h.1249.2		4	40.13	odd	4
2000.2.c.h.1249.3		4	40.37	odd	4
8000.2.a.g.1.2		2	1.1	even	1	trivial
8000.2.a.h.1.2		2	20.19	odd	2
8000.2.a.q.1.1		2	5.4	even	2
8000.2.a.r.1.1		2	4.3	odd	2
9000.2.a.c.1.2		2	120.59	even	2
9000.2.a.o.1.1		2	24.11	even	2

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:	\( N \)	\(=\)	\( 8000 = 2^{6} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8000.a (trivial)

\(f(q)\)	\(=\)	\(q+0.618034 q^{3} +0.236068 q^{7} -2.61803 q^{9} +O(q^{10})\)	\(q+0.618034 q^{3} +0.236068 q^{7} -2.61803 q^{9} -1.76393 q^{11} +4.61803 q^{13} +2.23607 q^{17} -7.09017 q^{19} +0.145898 q^{21} -0.763932 q^{23} -3.47214 q^{27} +5.76393 q^{29} -0.854102 q^{31} -1.09017 q^{33} +3.23607 q^{37} +2.85410 q^{39} -3.00000 q^{41} +10.7082 q^{43} -9.32624 q^{47} -6.94427 q^{49} +1.38197 q^{51} +3.14590 q^{53} -4.38197 q^{57} -5.61803 q^{59} -7.56231 q^{61} -0.618034 q^{63} +0.381966 q^{67} -0.472136 q^{69} +4.23607 q^{71} -12.8541 q^{73} -0.416408 q^{77} +6.23607 q^{79} +5.70820 q^{81} +10.9443 q^{83} +3.56231 q^{87} -8.47214 q^{89} +1.09017 q^{91} -0.527864 q^{93} +8.56231 q^{97} +4.61803 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} - 4 q^{7} - 3 q^{9}+O(q^{10}) \) 2 * q - q^3 - 4 * q^7 - 3 * q^9	\( 2 q - q^{3} - 4 q^{7} - 3 q^{9} - 8 q^{11} + 7 q^{13} - 3 q^{19} + 7 q^{21} - 6 q^{23} + 2 q^{27} + 16 q^{29} + 5 q^{31} + 9 q^{33} + 2 q^{37} - q^{39} - 6 q^{41} + 8 q^{43} - 3 q^{47} + 4 q^{49} + 5 q^{51} + 13 q^{53} - 11 q^{57} - 9 q^{59} + 5 q^{61} + q^{63} + 3 q^{67} + 8 q^{69} + 4 q^{71} - 19 q^{73} + 26 q^{77} + 8 q^{79} - 2 q^{81} + 4 q^{83} - 13 q^{87} - 8 q^{89} - 9 q^{91} - 10 q^{93} - 3 q^{97} + 7 q^{99}+O(q^{100}) \) 2 * q - q^3 - 4 * q^7 - 3 * q^9 - 8 * q^11 + 7 * q^13 - 3 * q^19 + 7 * q^21 - 6 * q^23 + 2 * q^27 + 16 * q^29 + 5 * q^31 + 9 * q^33 + 2 * q^37 - q^39 - 6 * q^41 + 8 * q^43 - 3 * q^47 + 4 * q^49 + 5 * q^51 + 13 * q^53 - 11 * q^57 - 9 * q^59 + 5 * q^61 + q^63 + 3 * q^67 + 8 * q^69 + 4 * q^71 - 19 * q^73 + 26 * q^77 + 8 * q^79 - 2 * q^81 + 4 * q^83 - 13 * q^87 - 8 * q^89 - 9 * q^91 - 10 * q^93 - 3 * q^97 + 7 * q^99

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