LMFDB - Embedded newform 8001.2.a.h.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8001,2,Mod(1,8001)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8001, 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("8001.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8001 = 3^{2} \cdot 7 \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8001.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:	$63.8883066572$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2667)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	8001.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.56155 q^{2} +4.56155 q^{4} -3.56155 q^{5} -1.00000 q^{7} -6.56155 q^{8} +O(q^{10})$	\(q-2.56155 q^{2} +4.56155 q^{4} -3.56155 q^{5} -1.00000 q^{7} -6.56155 q^{8} +9.12311 q^{10} -3.12311 q^{11} -3.56155 q^{13} +2.56155 q^{14} +7.68466 q^{16} -4.00000 q^{17} -4.00000 q^{19} -16.2462 q^{20} +8.00000 q^{22} +0.438447 q^{23} +7.68466 q^{25} +9.12311 q^{26} -4.56155 q^{28} -2.43845 q^{29} +2.43845 q^{31} -6.56155 q^{32} +10.2462 q^{34} +3.56155 q^{35} -0.438447 q^{37} +10.2462 q^{38} +23.3693 q^{40} -7.12311 q^{41} +7.12311 q^{43} -14.2462 q^{44} -1.12311 q^{46} +10.0000 q^{47} +1.00000 q^{49} -19.6847 q^{50} -16.2462 q^{52} +8.68466 q^{53} +11.1231 q^{55} +6.56155 q^{56} +6.24621 q^{58} +8.68466 q^{59} -4.43845 q^{61} -6.24621 q^{62} +1.43845 q^{64} +12.6847 q^{65} +7.12311 q^{67} -18.2462 q^{68} -9.12311 q^{70} +12.0000 q^{71} -4.43845 q^{73} +1.12311 q^{74} -18.2462 q^{76} +3.12311 q^{77} +6.24621 q^{79} -27.3693 q^{80} +18.2462 q^{82} +4.68466 q^{83} +14.2462 q^{85} -18.2462 q^{86} +20.4924 q^{88} -5.31534 q^{89} +3.56155 q^{91} +2.00000 q^{92} -25.6155 q^{94} +14.2462 q^{95} -17.1231 q^{97} -2.56155 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} + 5 q^{4} - 3 q^{5} - 2 q^{7} - 9 q^{8}+O(q^{10}) $ 2 * q - q^2 + 5 * q^4 - 3 * q^5 - 2 * q^7 - 9 * q^8	$ 2 q - q^{2} + 5 q^{4} - 3 q^{5} - 2 q^{7} - 9 q^{8} + 10 q^{10} + 2 q^{11} - 3 q^{13} + q^{14} + 3 q^{16} - 8 q^{17} - 8 q^{19} - 16 q^{20} + 16 q^{22} + 5 q^{23} + 3 q^{25} + 10 q^{26} - 5 q^{28} - 9 q^{29} + 9 q^{31} - 9 q^{32} + 4 q^{34} + 3 q^{35} - 5 q^{37} + 4 q^{38} + 22 q^{40} - 6 q^{41} + 6 q^{43} - 12 q^{44} + 6 q^{46} + 20 q^{47} + 2 q^{49} - 27 q^{50} - 16 q^{52} + 5 q^{53} + 14 q^{55} + 9 q^{56} - 4 q^{58} + 5 q^{59} - 13 q^{61} + 4 q^{62} + 7 q^{64} + 13 q^{65} + 6 q^{67} - 20 q^{68} - 10 q^{70} + 24 q^{71} - 13 q^{73} - 6 q^{74} - 20 q^{76} - 2 q^{77} - 4 q^{79} - 30 q^{80} + 20 q^{82} - 3 q^{83} + 12 q^{85} - 20 q^{86} + 8 q^{88} - 23 q^{89} + 3 q^{91} + 4 q^{92} - 10 q^{94} + 12 q^{95} - 26 q^{97} - q^{98}+O(q^{100}) $ 2 * q - q^2 + 5 * q^4 - 3 * q^5 - 2 * q^7 - 9 * q^8 + 10 * q^10 + 2 * q^11 - 3 * q^13 + q^14 + 3 * q^16 - 8 * q^17 - 8 * q^19 - 16 * q^20 + 16 * q^22 + 5 * q^23 + 3 * q^25 + 10 * q^26 - 5 * q^28 - 9 * q^29 + 9 * q^31 - 9 * q^32 + 4 * q^34 + 3 * q^35 - 5 * q^37 + 4 * q^38 + 22 * q^40 - 6 * q^41 + 6 * q^43 - 12 * q^44 + 6 * q^46 + 20 * q^47 + 2 * q^49 - 27 * q^50 - 16 * q^52 + 5 * q^53 + 14 * q^55 + 9 * q^56 - 4 * q^58 + 5 * q^59 - 13 * q^61 + 4 * q^62 + 7 * q^64 + 13 * q^65 + 6 * q^67 - 20 * q^68 - 10 * q^70 + 24 * q^71 - 13 * q^73 - 6 * q^74 - 20 * q^76 - 2 * q^77 - 4 * q^79 - 30 * q^80 + 20 * q^82 - 3 * q^83 + 12 * q^85 - 20 * q^86 + 8 * q^88 - 23 * q^89 + 3 * q^91 + 4 * q^92 - 10 * q^94 + 12 * q^95 - 26 * q^97 - q^98

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 $

$2$	−2.56155	−1.81129	−0.905646	−	0.424035i	$-0.860613\pi$
$2$	−2.56155	−1.81129	−0.905646	+	0.424035i	$0.860613\pi$
$3$	0	0
$3$	0	0
$4$	4.56155	2.28078
$5$	−3.56155	−1.59277	−0.796387	−	0.604787i	$-0.793258\pi$
$5$	−3.56155	−1.59277	−0.796387	+	0.604787i	$0.793258\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−6.56155	−2.31986
$9$	0	0
$10$	9.12311	2.88498
$11$	−3.12311	−0.941652	−0.470826	−	0.882226i	$-0.656044\pi$
$11$	−3.12311	−0.941652	−0.470826	+	0.882226i	$0.656044\pi$
$12$	0	0
$13$	−3.56155	−0.987797	−0.493899	−	0.869520i	$-0.664429\pi$
$13$	−3.56155	−0.987797	−0.493899	+	0.869520i	$0.664429\pi$
$14$	2.56155	0.684604
$15$	0	0
$16$	7.68466	1.92116
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	0	0
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	−16.2462	−3.63276
$21$	0	0
$22$	8.00000	1.70561
$23$	0.438447	0.0914226	0.0457113	−	0.998955i	$-0.485445\pi$
$23$	0.438447	0.0914226	0.0457113	+	0.998955i	$0.485445\pi$
$24$	0	0
$25$	7.68466	1.53693
$26$	9.12311	1.78919
$27$	0	0
$28$	−4.56155	−0.862052
$29$	−2.43845	−0.452808	−0.226404	−	0.974033i	$-0.572697\pi$
$29$	−2.43845	−0.452808	−0.226404	+	0.974033i	$0.572697\pi$
$30$	0	0
$31$	2.43845	0.437958	0.218979	−	0.975730i	$-0.429727\pi$
$31$	2.43845	0.437958	0.218979	+	0.975730i	$0.429727\pi$
$32$	−6.56155	−1.15993
$33$	0	0
$34$	10.2462	1.75721
$35$	3.56155	0.602012
$36$	0	0
$37$	−0.438447	−0.0720803	−0.0360401	−	0.999350i	$-0.511474\pi$
$37$	−0.438447	−0.0720803	−0.0360401	+	0.999350i	$0.511474\pi$
$38$	10.2462	1.66215
$39$	0	0
$40$	23.3693	3.69501
$41$	−7.12311	−1.11244	−0.556221	−	0.831034i	$-0.687749\pi$
$41$	−7.12311	−1.11244	−0.556221	+	0.831034i	$0.687749\pi$
$42$	0	0
$43$	7.12311	1.08626	0.543132	−	0.839648i	$-0.317238\pi$
$43$	7.12311	1.08626	0.543132	+	0.839648i	$0.317238\pi$
$44$	−14.2462	−2.14770
$45$	0	0
$46$	−1.12311	−0.165593
$47$	10.0000	1.45865	0.729325	−	0.684167i	$-0.239834\pi$
$47$	10.0000	1.45865	0.729325	+	0.684167i	$0.239834\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−19.6847	−2.78383
$51$	0	0
$52$	−16.2462	−2.25294
$53$	8.68466	1.19293	0.596465	−	0.802639i	$-0.296572\pi$
$53$	8.68466	1.19293	0.596465	+	0.802639i	$0.296572\pi$
$54$	0	0
$55$	11.1231	1.49984
$56$	6.56155	0.876824
$57$	0	0
$58$	6.24621	0.820168
$59$	8.68466	1.13065	0.565323	−	0.824870i	$-0.308751\pi$
$59$	8.68466	1.13065	0.565323	+	0.824870i	$0.308751\pi$
$60$	0	0
$61$	−4.43845	−0.568285	−0.284142	−	0.958782i	$-0.591709\pi$
$61$	−4.43845	−0.568285	−0.284142	+	0.958782i	$0.591709\pi$
$62$	−6.24621	−0.793270
$63$	0	0
$64$	1.43845	0.179806
$65$	12.6847	1.57334
$66$	0	0
$67$	7.12311	0.870226	0.435113	−	0.900376i	$-0.356708\pi$
$67$	7.12311	0.870226	0.435113	+	0.900376i	$0.356708\pi$
$68$	−18.2462	−2.21268
$69$	0	0
$70$	−9.12311	−1.09042
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	0	0
$73$	−4.43845	−0.519481	−0.259740	−	0.965678i	$-0.583637\pi$
$73$	−4.43845	−0.519481	−0.259740	+	0.965678i	$0.583637\pi$
$74$	1.12311	0.130558
$75$	0	0
$76$	−18.2462	−2.09298
$77$	3.12311	0.355911
$78$	0	0
$79$	6.24621	0.702754	0.351377	−	0.936234i	$-0.385714\pi$
$79$	6.24621	0.702754	0.351377	+	0.936234i	$0.385714\pi$
$80$	−27.3693	−3.05998
$81$	0	0
$82$	18.2462	2.01496
$83$	4.68466	0.514208	0.257104	−	0.966384i	$-0.417232\pi$
$83$	4.68466	0.514208	0.257104	+	0.966384i	$0.417232\pi$
$84$	0	0
$85$	14.2462	1.54522
$86$	−18.2462	−1.96754
$87$	0	0
$88$	20.4924	2.18450
$89$	−5.31534	−0.563425	−0.281713	−	0.959499i	$-0.590902\pi$
$89$	−5.31534	−0.563425	−0.281713	+	0.959499i	$0.590902\pi$
$90$	0	0
$91$	3.56155	0.373352
$92$	2.00000	0.208514
$93$	0	0
$94$	−25.6155	−2.64204
$95$	14.2462	1.46163
$96$	0	0
$97$	−17.1231	−1.73859	−0.869294	−	0.494295i	$-0.835426\pi$
$97$	−17.1231	−1.73859	−0.869294	+	0.494295i	$0.835426\pi$
$98$	−2.56155	−0.258756
$99$	0	0
$100$	35.0540	3.50540
$101$	−1.31534	−0.130881	−0.0654407	−	0.997856i	$-0.520845\pi$
$101$	−1.31534	−0.130881	−0.0654407	+	0.997856i	$0.520845\pi$
$102$	0	0
$103$	14.2462	1.40372	0.701860	−	0.712314i	$-0.252353\pi$
$103$	14.2462	1.40372	0.701860	+	0.712314i	$0.252353\pi$
$104$	23.3693	2.29155
$105$	0	0
$106$	−22.2462	−2.16074
$107$	−4.00000	−0.386695	−0.193347	−	0.981130i	$-0.561934\pi$
$107$	−4.00000	−0.386695	−0.193347	+	0.981130i	$0.561934\pi$
$108$	0	0
$109$	8.24621	0.789844	0.394922	−	0.918715i	$-0.370772\pi$
$109$	8.24621	0.789844	0.394922	+	0.918715i	$0.370772\pi$
$110$	−28.4924	−2.71665
$111$	0	0
$112$	−7.68466	−0.726132
$113$	−19.3693	−1.82211	−0.911056	−	0.412283i	$-0.864732\pi$
$113$	−19.3693	−1.82211	−0.911056	+	0.412283i	$0.864732\pi$
$114$	0	0
$115$	−1.56155	−0.145616
$116$	−11.1231	−1.03275
$117$	0	0
$118$	−22.2462	−2.04793
$119$	4.00000	0.366679
$120$	0	0
$121$	−1.24621	−0.113292
$122$	11.3693	1.02933
$123$	0	0
$124$	11.1231	0.998884
$125$	−9.56155	−0.855211
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	9.43845	0.834249
$129$	0	0
$130$	−32.4924	−2.84977
$131$	18.0000	1.57267	0.786334	−	0.617802i	$-0.211977\pi$
$131$	18.0000	1.57267	0.786334	+	0.617802i	$0.211977\pi$
$132$	0	0
$133$	4.00000	0.346844
$134$	−18.2462	−1.57623
$135$	0	0
$136$	26.2462	2.25059
$137$	−6.43845	−0.550074	−0.275037	−	0.961434i	$-0.588690\pi$
$137$	−6.43845	−0.550074	−0.275037	+	0.961434i	$0.588690\pi$
$138$	0	0
$139$	−15.1231	−1.28273	−0.641363	−	0.767238i	$-0.721631\pi$
$139$	−15.1231	−1.28273	−0.641363	+	0.767238i	$0.721631\pi$
$140$	16.2462	1.37306
$141$	0	0
$142$	−30.7386	−2.57953
$143$	11.1231	0.930161
$144$	0	0
$145$	8.68466	0.721222
$146$	11.3693	0.940931
$147$	0	0
$148$	−2.00000	−0.164399
$149$	4.24621	0.347863	0.173932	−	0.984758i	$-0.444353\pi$
$149$	4.24621	0.347863	0.173932	+	0.984758i	$0.444353\pi$
$150$	0	0
$151$	−13.3693	−1.08798	−0.543990	−	0.839092i	$-0.683087\pi$
$151$	−13.3693	−1.08798	−0.543990	+	0.839092i	$0.683087\pi$
$152$	26.2462	2.12885
$153$	0	0
$154$	−8.00000	−0.644658
$155$	−8.68466	−0.697569
$156$	0	0
$157$	14.0000	1.11732	0.558661	−	0.829396i	$-0.311315\pi$
$157$	14.0000	1.11732	0.558661	+	0.829396i	$0.311315\pi$
$158$	−16.0000	−1.27289
$159$	0	0
$160$	23.3693	1.84751
$161$	−0.438447	−0.0345545
$162$	0	0
$163$	17.5616	1.37553	0.687763	−	0.725935i	$-0.258593\pi$
$163$	17.5616	1.37553	0.687763	+	0.725935i	$0.258593\pi$
$164$	−32.4924	−2.53723
$165$	0	0
$166$	−12.0000	−0.931381
$167$	12.4924	0.966693	0.483346	−	0.875429i	$-0.339421\pi$
$167$	12.4924	0.966693	0.483346	+	0.875429i	$0.339421\pi$
$168$	0	0
$169$	−0.315342	−0.0242570
$170$	−36.4924	−2.79884
$171$	0	0
$172$	32.4924	2.47752
$173$	−20.2462	−1.53929	−0.769645	−	0.638471i	$-0.779567\pi$
$173$	−20.2462	−1.53929	−0.769645	+	0.638471i	$0.779567\pi$
$174$	0	0
$175$	−7.68466	−0.580906
$176$	−24.0000	−1.80907
$177$	0	0
$178$	13.6155	1.02053
$179$	18.2462	1.36379	0.681893	−	0.731452i	$-0.261157\pi$
$179$	18.2462	1.36379	0.681893	+	0.731452i	$0.261157\pi$
$180$	0	0
$181$	17.1231	1.27275	0.636375	−	0.771380i	$-0.280433\pi$
$181$	17.1231	1.27275	0.636375	+	0.771380i	$0.280433\pi$
$182$	−9.12311	−0.676250
$183$	0	0
$184$	−2.87689	−0.212087
$185$	1.56155	0.114808
$186$	0	0
$187$	12.4924	0.913536
$188$	45.6155	3.32685
$189$	0	0
$190$	−36.4924	−2.64744
$191$	18.2462	1.32025	0.660125	−	0.751156i	$-0.270503\pi$
$191$	18.2462	1.32025	0.660125	+	0.751156i	$0.270503\pi$
$192$	0	0
$193$	3.75379	0.270204	0.135102	−	0.990832i	$-0.456864\pi$
$193$	3.75379	0.270204	0.135102	+	0.990832i	$0.456864\pi$
$194$	43.8617	3.14909
$195$	0	0
$196$	4.56155	0.325825
$197$	−9.12311	−0.649994	−0.324997	−	0.945715i	$-0.605363\pi$
$197$	−9.12311	−0.649994	−0.324997	+	0.945715i	$0.605363\pi$
$198$	0	0
$199$	−1.56155	−0.110696	−0.0553478	−	0.998467i	$-0.517627\pi$
$199$	−1.56155	−0.110696	−0.0553478	+	0.998467i	$0.517627\pi$
$200$	−50.4233	−3.56547
$201$	0	0
$202$	3.36932	0.237064
$203$	2.43845	0.171145
$204$	0	0
$205$	25.3693	1.77187
$206$	−36.4924	−2.54255
$207$	0	0
$208$	−27.3693	−1.89772
$209$	12.4924	0.864119
$210$	0	0
$211$	−4.68466	−0.322505	−0.161253	−	0.986913i	$-0.551553\pi$
$211$	−4.68466	−0.322505	−0.161253	+	0.986913i	$0.551553\pi$
$212$	39.6155	2.72081
$213$	0	0
$214$	10.2462	0.700417
$215$	−25.3693	−1.73017
$216$	0	0
$217$	−2.43845	−0.165533
$218$	−21.1231	−1.43064
$219$	0	0
$220$	50.7386	3.42080
$221$	14.2462	0.958304
$222$	0	0
$223$	−17.3693	−1.16314	−0.581568	−	0.813498i	$-0.697560\pi$
$223$	−17.3693	−1.16314	−0.581568	+	0.813498i	$0.697560\pi$
$224$	6.56155	0.438412
$225$	0	0
$226$	49.6155	3.30038
$227$	−16.2462	−1.07830	−0.539149	−	0.842210i	$-0.681254\pi$
$227$	−16.2462	−1.07830	−0.539149	+	0.842210i	$0.681254\pi$
$228$	0	0
$229$	−7.36932	−0.486978	−0.243489	−	0.969904i	$-0.578292\pi$
$229$	−7.36932	−0.486978	−0.243489	+	0.969904i	$0.578292\pi$
$230$	4.00000	0.263752
$231$	0	0
$232$	16.0000	1.05045
$233$	14.9309	0.978154	0.489077	−	0.872241i	$-0.337334\pi$
$233$	14.9309	0.978154	0.489077	+	0.872241i	$0.337334\pi$
$234$	0	0
$235$	−35.6155	−2.32330
$236$	39.6155	2.57875
$237$	0	0
$238$	−10.2462	−0.664163
$239$	−17.3153	−1.12004	−0.560018	−	0.828480i	$-0.689206\pi$
$239$	−17.3153	−1.12004	−0.560018	+	0.828480i	$0.689206\pi$
$240$	0	0
$241$	−29.1231	−1.87598	−0.937992	−	0.346657i	$-0.887317\pi$
$241$	−29.1231	−1.87598	−0.937992	+	0.346657i	$0.887317\pi$
$242$	3.19224	0.205205
$243$	0	0
$244$	−20.2462	−1.29613
$245$	−3.56155	−0.227539
$246$	0	0
$247$	14.2462	0.906465
$248$	−16.0000	−1.01600
$249$	0	0
$250$	24.4924	1.54904
$251$	−10.0000	−0.631194	−0.315597	−	0.948893i	$-0.602205\pi$
$251$	−10.0000	−0.631194	−0.315597	+	0.948893i	$0.602205\pi$
$252$	0	0
$253$	−1.36932	−0.0860882
$254$	−2.56155	−0.160726
$255$	0	0
$256$	−27.0540	−1.69087
$257$	9.80776	0.611792	0.305896	−	0.952065i	$-0.401044\pi$
$257$	9.80776	0.611792	0.305896	+	0.952065i	$0.401044\pi$
$258$	0	0
$259$	0.438447	0.0272438
$260$	57.8617	3.58843
$261$	0	0
$262$	−46.1080	−2.84856
$263$	−8.00000	−0.493301	−0.246651	−	0.969104i	$-0.579330\pi$
$263$	−8.00000	−0.493301	−0.246651	+	0.969104i	$0.579330\pi$
$264$	0	0
$265$	−30.9309	−1.90007
$266$	−10.2462	−0.628236
$267$	0	0
$268$	32.4924	1.98479
$269$	−4.00000	−0.243884	−0.121942	−	0.992537i	$-0.538912\pi$
$269$	−4.00000	−0.243884	−0.121942	+	0.992537i	$0.538912\pi$
$270$	0	0
$271$	−20.6847	−1.25650	−0.628252	−	0.778010i	$-0.716229\pi$
$271$	−20.6847	−1.25650	−0.628252	+	0.778010i	$0.716229\pi$
$272$	−30.7386	−1.86380
$273$	0	0
$274$	16.4924	0.996344
$275$	−24.0000	−1.44725
$276$	0	0
$277$	11.3693	0.683116	0.341558	−	0.939861i	$-0.389045\pi$
$277$	11.3693	0.683116	0.341558	+	0.939861i	$0.389045\pi$
$278$	38.7386	2.32339
$279$	0	0
$280$	−23.3693	−1.39658
$281$	9.36932	0.558927	0.279463	−	0.960156i	$-0.409843\pi$
$281$	9.36932	0.558927	0.279463	+	0.960156i	$0.409843\pi$
$282$	0	0
$283$	−12.0000	−0.713326	−0.356663	−	0.934233i	$-0.616086\pi$
$283$	−12.0000	−0.713326	−0.356663	+	0.934233i	$0.616086\pi$
$284$	54.7386	3.24814
$285$	0	0
$286$	−28.4924	−1.68479
$287$	7.12311	0.420464
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	−22.2462	−1.30634
$291$	0	0
$292$	−20.2462	−1.18482
$293$	16.2462	0.949114	0.474557	−	0.880225i	$-0.342608\pi$
$293$	16.2462	0.949114	0.474557	+	0.880225i	$0.342608\pi$
$294$	0	0
$295$	−30.9309	−1.80086
$296$	2.87689	0.167216
$297$	0	0
$298$	−10.8769	−0.630082
$299$	−1.56155	−0.0903069
$300$	0	0
$301$	−7.12311	−0.410569
$302$	34.2462	1.97065
$303$	0	0
$304$	−30.7386	−1.76298
$305$	15.8078	0.905150
$306$	0	0
$307$	0.876894	0.0500470	0.0250235	−	0.999687i	$-0.492034\pi$
$307$	0.876894	0.0500470	0.0250235	+	0.999687i	$0.492034\pi$
$308$	14.2462	0.811753
$309$	0	0
$310$	22.2462	1.26350
$311$	2.43845	0.138272	0.0691358	−	0.997607i	$-0.477976\pi$
$311$	2.43845	0.138272	0.0691358	+	0.997607i	$0.477976\pi$
$312$	0	0
$313$	12.2462	0.692197	0.346098	−	0.938198i	$-0.387506\pi$
$313$	12.2462	0.692197	0.346098	+	0.938198i	$0.387506\pi$
$314$	−35.8617	−2.02380
$315$	0	0
$316$	28.4924	1.60282
$317$	−15.1231	−0.849398	−0.424699	−	0.905335i	$-0.639620\pi$
$317$	−15.1231	−0.849398	−0.424699	+	0.905335i	$0.639620\pi$
$318$	0	0
$319$	7.61553	0.426388
$320$	−5.12311	−0.286390
$321$	0	0
$322$	1.12311	0.0625882
$323$	16.0000	0.890264
$324$	0	0
$325$	−27.3693	−1.51818
$326$	−44.9848	−2.49148
$327$	0	0
$328$	46.7386	2.58071
$329$	−10.0000	−0.551318
$330$	0	0
$331$	−18.7386	−1.02997	−0.514984	−	0.857200i	$-0.672202\pi$
$331$	−18.7386	−1.02997	−0.514984	+	0.857200i	$0.672202\pi$
$332$	21.3693	1.17279
$333$	0	0
$334$	−32.0000	−1.75096
$335$	−25.3693	−1.38607
$336$	0	0
$337$	31.3693	1.70880	0.854398	−	0.519619i	$-0.173926\pi$
$337$	31.3693	1.70880	0.854398	+	0.519619i	$0.173926\pi$
$338$	0.807764	0.0439366
$339$	0	0
$340$	64.9848	3.52430
$341$	−7.61553	−0.412404
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−46.7386	−2.51998
$345$	0	0
$346$	51.8617	2.78810
$347$	16.9309	0.908897	0.454448	−	0.890773i	$-0.349836\pi$
$347$	16.9309	0.908897	0.454448	+	0.890773i	$0.349836\pi$
$348$	0	0
$349$	−28.2462	−1.51199	−0.755993	−	0.654580i	$-0.772845\pi$
$349$	−28.2462	−1.51199	−0.755993	+	0.654580i	$0.772845\pi$
$350$	19.6847	1.05219
$351$	0	0
$352$	20.4924	1.09225
$353$	2.24621	0.119554	0.0597769	−	0.998212i	$-0.480961\pi$
$353$	2.24621	0.119554	0.0597769	+	0.998212i	$0.480961\pi$
$354$	0	0
$355$	−42.7386	−2.26833
$356$	−24.2462	−1.28505
$357$	0	0
$358$	−46.7386	−2.47021
$359$	2.87689	0.151837	0.0759183	−	0.997114i	$-0.475811\pi$
$359$	2.87689	0.151837	0.0759183	+	0.997114i	$0.475811\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	−43.8617	−2.30532
$363$	0	0
$364$	16.2462	0.851533
$365$	15.8078	0.827416
$366$	0	0
$367$	22.9309	1.19698	0.598491	−	0.801130i	$-0.295767\pi$
$367$	22.9309	1.19698	0.598491	+	0.801130i	$0.295767\pi$
$368$	3.36932	0.175638
$369$	0	0
$370$	−4.00000	−0.207950
$371$	−8.68466	−0.450885
$372$	0	0
$373$	14.0000	0.724893	0.362446	−	0.932005i	$-0.381942\pi$
$373$	14.0000	0.724893	0.362446	+	0.932005i	$0.381942\pi$
$374$	−32.0000	−1.65468
$375$	0	0
$376$	−65.6155	−3.38386
$377$	8.68466	0.447283
$378$	0	0
$379$	16.8769	0.866908	0.433454	−	0.901176i	$-0.357295\pi$
$379$	16.8769	0.866908	0.433454	+	0.901176i	$0.357295\pi$
$380$	64.9848	3.33365
$381$	0	0
$382$	−46.7386	−2.39136
$383$	27.3693	1.39851	0.699253	−	0.714874i	$-0.253516\pi$
$383$	27.3693	1.39851	0.699253	+	0.714874i	$0.253516\pi$
$384$	0	0
$385$	−11.1231	−0.566886
$386$	−9.61553	−0.489417
$387$	0	0
$388$	−78.1080	−3.96533
$389$	9.12311	0.462560	0.231280	−	0.972887i	$-0.425709\pi$
$389$	9.12311	0.462560	0.231280	+	0.972887i	$0.425709\pi$
$390$	0	0
$391$	−1.75379	−0.0886929
$392$	−6.56155	−0.331408
$393$	0	0
$394$	23.3693	1.17733
$395$	−22.2462	−1.11933
$396$	0	0
$397$	10.1922	0.511534	0.255767	−	0.966739i	$-0.417672\pi$
$397$	10.1922	0.511534	0.255767	+	0.966739i	$0.417672\pi$
$398$	4.00000	0.200502
$399$	0	0
$400$	59.0540	2.95270
$401$	−5.56155	−0.277731	−0.138865	−	0.990311i	$-0.544346\pi$
$401$	−5.56155	−0.277731	−0.138865	+	0.990311i	$0.544346\pi$
$402$	0	0
$403$	−8.68466	−0.432614
$404$	−6.00000	−0.298511
$405$	0	0
$406$	−6.24621	−0.309994
$407$	1.36932	0.0678745
$408$	0	0
$409$	8.24621	0.407749	0.203874	−	0.978997i	$-0.434647\pi$
$409$	8.24621	0.407749	0.203874	+	0.978997i	$0.434647\pi$
$410$	−64.9848	−3.20937
$411$	0	0
$412$	64.9848	3.20157
$413$	−8.68466	−0.427344
$414$	0	0
$415$	−16.6847	−0.819018
$416$	23.3693	1.14578
$417$	0	0
$418$	−32.0000	−1.56517
$419$	−14.0000	−0.683945	−0.341972	−	0.939710i	$-0.611095\pi$
$419$	−14.0000	−0.683945	−0.341972	+	0.939710i	$0.611095\pi$
$420$	0	0
$421$	5.12311	0.249685	0.124842	−	0.992177i	$-0.460157\pi$
$421$	5.12311	0.249685	0.124842	+	0.992177i	$0.460157\pi$
$422$	12.0000	0.584151
$423$	0	0
$424$	−56.9848	−2.76743
$425$	−30.7386	−1.49104
$426$	0	0
$427$	4.43845	0.214792
$428$	−18.2462	−0.881964
$429$	0	0
$430$	64.9848	3.13385
$431$	−25.3693	−1.22200	−0.610998	−	0.791632i	$-0.709232\pi$
$431$	−25.3693	−1.22200	−0.610998	+	0.791632i	$0.709232\pi$
$432$	0	0
$433$	−7.75379	−0.372623	−0.186312	−	0.982491i	$-0.559653\pi$
$433$	−7.75379	−0.372623	−0.186312	+	0.982491i	$0.559653\pi$
$434$	6.24621	0.299828
$435$	0	0
$436$	37.6155	1.80146
$437$	−1.75379	−0.0838951
$438$	0	0
$439$	36.1080	1.72334	0.861669	−	0.507470i	$-0.169419\pi$
$439$	36.1080	1.72334	0.861669	+	0.507470i	$0.169419\pi$
$440$	−72.9848	−3.47942
$441$	0	0
$442$	−36.4924	−1.73577
$443$	22.7386	1.08035	0.540173	−	0.841554i	$-0.318359\pi$
$443$	22.7386	1.08035	0.540173	+	0.841554i	$0.318359\pi$
$444$	0	0
$445$	18.9309	0.897409
$446$	44.4924	2.10678
$447$	0	0
$448$	−1.43845	−0.0679602
$449$	−6.00000	−0.283158	−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000	−0.283158	−0.141579	+	0.989927i	$0.545218\pi$
$450$	0	0
$451$	22.2462	1.04753
$452$	−88.3542	−4.15583
$453$	0	0
$454$	41.6155	1.95311
$455$	−12.6847	−0.594666
$456$	0	0
$457$	−13.8078	−0.645900	−0.322950	−	0.946416i	$-0.604675\pi$
$457$	−13.8078	−0.645900	−0.322950	+	0.946416i	$0.604675\pi$
$458$	18.8769	0.882059
$459$	0	0
$460$	−7.12311	−0.332117
$461$	−14.0000	−0.652045	−0.326023	−	0.945362i	$-0.605709\pi$
$461$	−14.0000	−0.652045	−0.326023	+	0.945362i	$0.605709\pi$
$462$	0	0
$463$	−3.80776	−0.176962	−0.0884809	−	0.996078i	$-0.528201\pi$
$463$	−3.80776	−0.176962	−0.0884809	+	0.996078i	$0.528201\pi$
$464$	−18.7386	−0.869919
$465$	0	0
$466$	−38.2462	−1.77172
$467$	34.5464	1.59862	0.799308	−	0.600921i	$-0.205199\pi$
$467$	34.5464	1.59862	0.799308	+	0.600921i	$0.205199\pi$
$468$	0	0
$469$	−7.12311	−0.328914
$470$	91.2311	4.20817
$471$	0	0
$472$	−56.9848	−2.62294
$473$	−22.2462	−1.02288
$474$	0	0
$475$	−30.7386	−1.41039
$476$	18.2462	0.836314
$477$	0	0
$478$	44.3542	2.02871
$479$	−10.4924	−0.479411	−0.239706	−	0.970846i	$-0.577051\pi$
$479$	−10.4924	−0.479411	−0.239706	+	0.970846i	$0.577051\pi$
$480$	0	0
$481$	1.56155	0.0712007
$482$	74.6004	3.39795
$483$	0	0
$484$	−5.68466	−0.258394
$485$	60.9848	2.76918
$486$	0	0
$487$	26.7386	1.21164	0.605822	−	0.795601i	$-0.292845\pi$
$487$	26.7386	1.21164	0.605822	+	0.795601i	$0.292845\pi$
$488$	29.1231	1.31834
$489$	0	0
$490$	9.12311	0.412140
$491$	39.8617	1.79894	0.899468	−	0.436988i	$-0.143955\pi$
$491$	39.8617	1.79894	0.899468	+	0.436988i	$0.143955\pi$
$492$	0	0
$493$	9.75379	0.439289
$494$	−36.4924	−1.64187
$495$	0	0
$496$	18.7386	0.841389
$497$	−12.0000	−0.538274
$498$	0	0
$499$	24.0000	1.07439	0.537194	−	0.843459i	$-0.319484\pi$
$499$	24.0000	1.07439	0.537194	+	0.843459i	$0.319484\pi$
$500$	−43.6155	−1.95055
$501$	0	0
$502$	25.6155	1.14328
$503$	10.8769	0.484977	0.242488	−	0.970154i	$-0.422036\pi$
$503$	10.8769	0.484977	0.242488	+	0.970154i	$0.422036\pi$
$504$	0	0
$505$	4.68466	0.208465
$506$	3.50758	0.155931
$507$	0	0
$508$	4.56155	0.202386
$509$	−34.7386	−1.53976	−0.769881	−	0.638187i	$-0.779685\pi$
$509$	−34.7386	−1.53976	−0.769881	+	0.638187i	$0.779685\pi$
$510$	0	0
$511$	4.43845	0.196345
$512$	50.4233	2.22842
$513$	0	0
$514$	−25.1231	−1.10813
$515$	−50.7386	−2.23581
$516$	0	0
$517$	−31.2311	−1.37354
$518$	−1.12311	−0.0493464
$519$	0	0
$520$	−83.2311	−3.64992
$521$	38.7386	1.69717	0.848585	−	0.529059i	$-0.177455\pi$
$521$	38.7386	1.69717	0.848585	+	0.529059i	$0.177455\pi$
$522$	0	0
$523$	35.4233	1.54895	0.774476	−	0.632603i	$-0.218014\pi$
$523$	35.4233	1.54895	0.774476	+	0.632603i	$0.218014\pi$
$524$	82.1080	3.58690
$525$	0	0
$526$	20.4924	0.893512
$527$	−9.75379	−0.424882
$528$	0	0
$529$	−22.8078	−0.991642
$530$	79.2311	3.44158
$531$	0	0
$532$	18.2462	0.791074
$533$	25.3693	1.09887
$534$	0	0
$535$	14.2462	0.615917
$536$	−46.7386	−2.01880
$537$	0	0
$538$	10.2462	0.441746
$539$	−3.12311	−0.134522
$540$	0	0
$541$	1.61553	0.0694570	0.0347285	−	0.999397i	$-0.488943\pi$
$541$	1.61553	0.0694570	0.0347285	+	0.999397i	$0.488943\pi$
$542$	52.9848	2.27589
$543$	0	0
$544$	26.2462	1.12530
$545$	−29.3693	−1.25804
$546$	0	0
$547$	−8.49242	−0.363110	−0.181555	−	0.983381i	$-0.558113\pi$
$547$	−8.49242	−0.363110	−0.181555	+	0.983381i	$0.558113\pi$
$548$	−29.3693	−1.25460
$549$	0	0
$550$	61.4773	2.62140
$551$	9.75379	0.415525
$552$	0	0
$553$	−6.24621	−0.265616
$554$	−29.1231	−1.23732
$555$	0	0
$556$	−68.9848	−2.92561
$557$	−22.4924	−0.953035	−0.476517	−	0.879165i	$-0.658101\pi$
$557$	−22.4924	−0.953035	−0.476517	+	0.879165i	$0.658101\pi$
$558$	0	0
$559$	−25.3693	−1.07301
$560$	27.3693	1.15656
$561$	0	0
$562$	−24.0000	−1.01238
$563$	20.0000	0.842900	0.421450	−	0.906852i	$-0.361521\pi$
$563$	20.0000	0.842900	0.421450	+	0.906852i	$0.361521\pi$
$564$	0	0
$565$	68.9848	2.90221
$566$	30.7386	1.29204
$567$	0	0
$568$	−78.7386	−3.30380
$569$	−44.3542	−1.85942	−0.929712	−	0.368288i	$-0.879944\pi$
$569$	−44.3542	−1.85942	−0.929712	+	0.368288i	$0.879944\pi$
$570$	0	0
$571$	−23.6155	−0.988279	−0.494140	−	0.869383i	$-0.664517\pi$
$571$	−23.6155	−0.988279	−0.494140	+	0.869383i	$0.664517\pi$
$572$	50.7386	2.12149
$573$	0	0
$574$	−18.2462	−0.761582
$575$	3.36932	0.140510
$576$	0	0
$577$	26.6847	1.11090	0.555448	−	0.831551i	$-0.312547\pi$
$577$	26.6847	1.11090	0.555448	+	0.831551i	$0.312547\pi$
$578$	2.56155	0.106547
$579$	0	0
$580$	39.6155	1.64495
$581$	−4.68466	−0.194352
$582$	0	0
$583$	−27.1231	−1.12332
$584$	29.1231	1.20512
$585$	0	0
$586$	−41.6155	−1.71912
$587$	8.24621	0.340358	0.170179	−	0.985413i	$-0.445565\pi$
$587$	8.24621	0.340358	0.170179	+	0.985413i	$0.445565\pi$
$588$	0	0
$589$	−9.75379	−0.401898
$590$	79.2311	3.26189
$591$	0	0
$592$	−3.36932	−0.138478
$593$	18.3002	0.751499	0.375749	−	0.926721i	$-0.377385\pi$
$593$	18.3002	0.751499	0.375749	+	0.926721i	$0.377385\pi$
$594$	0	0
$595$	−14.2462	−0.584038
$596$	19.3693	0.793398
$597$	0	0
$598$	4.00000	0.163572
$599$	−43.1771	−1.76417	−0.882084	−	0.471092i	$-0.843860\pi$
$599$	−43.1771	−1.76417	−0.882084	+	0.471092i	$0.843860\pi$
$600$	0	0
$601$	−28.2462	−1.15219	−0.576093	−	0.817384i	$-0.695424\pi$
$601$	−28.2462	−1.15219	−0.576093	+	0.817384i	$0.695424\pi$
$602$	18.2462	0.743660
$603$	0	0
$604$	−60.9848	−2.48144
$605$	4.43845	0.180449
$606$	0	0
$607$	6.43845	0.261329	0.130664	−	0.991427i	$-0.458289\pi$
$607$	6.43845	0.261329	0.130664	+	0.991427i	$0.458289\pi$
$608$	26.2462	1.06442
$609$	0	0
$610$	−40.4924	−1.63949
$611$	−35.6155	−1.44085
$612$	0	0
$613$	−3.36932	−0.136085	−0.0680427	−	0.997682i	$-0.521675\pi$
$613$	−3.36932	−0.136085	−0.0680427	+	0.997682i	$0.521675\pi$
$614$	−2.24621	−0.0906497
$615$	0	0
$616$	−20.4924	−0.825663
$617$	−2.93087	−0.117992	−0.0589962	−	0.998258i	$-0.518790\pi$
$617$	−2.93087	−0.117992	−0.0589962	+	0.998258i	$0.518790\pi$
$618$	0	0
$619$	−40.1080	−1.61207	−0.806037	−	0.591865i	$-0.798392\pi$
$619$	−40.1080	−1.61207	−0.806037	+	0.591865i	$0.798392\pi$
$620$	−39.6155	−1.59100
$621$	0	0
$622$	−6.24621	−0.250450
$623$	5.31534	0.212955
$624$	0	0
$625$	−4.36932	−0.174773
$626$	−31.3693	−1.25377
$627$	0	0
$628$	63.8617	2.54836
$629$	1.75379	0.0699281
$630$	0	0
$631$	19.1231	0.761279	0.380639	−	0.924724i	$-0.375704\pi$
$631$	19.1231	0.761279	0.380639	+	0.924724i	$0.375704\pi$
$632$	−40.9848	−1.63029
$633$	0	0
$634$	38.7386	1.53851
$635$	−3.56155	−0.141336
$636$	0	0
$637$	−3.56155	−0.141114
$638$	−19.5076	−0.772312
$639$	0	0
$640$	−33.6155	−1.32877
$641$	47.1231	1.86125	0.930625	−	0.365973i	$-0.119264\pi$
$641$	47.1231	1.86125	0.930625	+	0.365973i	$0.119264\pi$
$642$	0	0
$643$	−5.75379	−0.226907	−0.113454	−	0.993543i	$-0.536191\pi$
$643$	−5.75379	−0.226907	−0.113454	+	0.993543i	$0.536191\pi$
$644$	−2.00000	−0.0788110
$645$	0	0
$646$	−40.9848	−1.61253
$647$	−34.5464	−1.35816	−0.679080	−	0.734065i	$-0.737621\pi$
$647$	−34.5464	−1.35816	−0.679080	+	0.734065i	$0.737621\pi$
$648$	0	0
$649$	−27.1231	−1.06468
$650$	70.1080	2.74986
$651$	0	0
$652$	80.1080	3.13727
$653$	−12.2462	−0.479231	−0.239616	−	0.970868i	$-0.577021\pi$
$653$	−12.2462	−0.479231	−0.239616	+	0.970868i	$0.577021\pi$
$654$	0	0
$655$	−64.1080	−2.50490
$656$	−54.7386	−2.13718
$657$	0	0
$658$	25.6155	0.998597
$659$	4.82292	0.187874	0.0939371	−	0.995578i	$-0.470055\pi$
$659$	4.82292	0.187874	0.0939371	+	0.995578i	$0.470055\pi$
$660$	0	0
$661$	−12.4384	−0.483800	−0.241900	−	0.970301i	$-0.577771\pi$
$661$	−12.4384	−0.483800	−0.241900	+	0.970301i	$0.577771\pi$
$662$	48.0000	1.86557
$663$	0	0
$664$	−30.7386	−1.19289
$665$	−14.2462	−0.552444
$666$	0	0
$667$	−1.06913	−0.0413969
$668$	56.9848	2.20481
$669$	0	0
$670$	64.9848	2.51058
$671$	13.8617	0.535127
$672$	0	0
$673$	4.43845	0.171090	0.0855448	−	0.996334i	$-0.472737\pi$
$673$	4.43845	0.171090	0.0855448	+	0.996334i	$0.472737\pi$
$674$	−80.3542	−3.09513
$675$	0	0
$676$	−1.43845	−0.0553249
$677$	−44.9848	−1.72891	−0.864454	−	0.502712i	$-0.832336\pi$
$677$	−44.9848	−1.72891	−0.864454	+	0.502712i	$0.832336\pi$
$678$	0	0
$679$	17.1231	0.657124
$680$	−93.4773	−3.58469
$681$	0	0
$682$	19.5076	0.746984
$683$	49.6155	1.89849	0.949243	−	0.314545i	$-0.101852\pi$
$683$	49.6155	1.89849	0.949243	+	0.314545i	$0.101852\pi$
$684$	0	0
$685$	22.9309	0.876143
$686$	2.56155	0.0978005
$687$	0	0
$688$	54.7386	2.08689
$689$	−30.9309	−1.17837
$690$	0	0
$691$	42.2462	1.60712	0.803561	−	0.595223i	$-0.202936\pi$
$691$	42.2462	1.60712	0.803561	+	0.595223i	$0.202936\pi$
$692$	−92.3542	−3.51078
$693$	0	0
$694$	−43.3693	−1.64628
$695$	53.8617	2.04309
$696$	0	0
$697$	28.4924	1.07923
$698$	72.3542	2.73865
$699$	0	0
$700$	−35.0540	−1.32492
$701$	34.0540	1.28620	0.643100	−	0.765782i	$-0.277648\pi$
$701$	34.0540	1.28620	0.643100	+	0.765782i	$0.277648\pi$
$702$	0	0
$703$	1.75379	0.0661454
$704$	−4.49242	−0.169315
$705$	0	0
$706$	−5.75379	−0.216547
$707$	1.31534	0.0494685
$708$	0	0
$709$	15.7538	0.591646	0.295823	−	0.955243i	$-0.404406\pi$
$709$	15.7538	0.591646	0.295823	+	0.955243i	$0.404406\pi$
$710$	109.477	4.10861
$711$	0	0
$712$	34.8769	1.30707
$713$	1.06913	0.0400392
$714$	0	0
$715$	−39.6155	−1.48154
$716$	83.2311	3.11049
$717$	0	0
$718$	−7.36932	−0.275020
$719$	−44.3542	−1.65413	−0.827066	−	0.562105i	$-0.809991\pi$
$719$	−44.3542	−1.65413	−0.827066	+	0.562105i	$0.809991\pi$
$720$	0	0
$721$	−14.2462	−0.530557
$722$	7.68466	0.285993
$723$	0	0
$724$	78.1080	2.90286
$725$	−18.7386	−0.695935
$726$	0	0
$727$	−17.3693	−0.644192	−0.322096	−	0.946707i	$-0.604387\pi$
$727$	−17.3693	−0.644192	−0.322096	+	0.946707i	$0.604387\pi$
$728$	−23.3693	−0.866125
$729$	0	0
$730$	−40.4924	−1.49869
$731$	−28.4924	−1.05383
$732$	0	0
$733$	45.4233	1.67775	0.838874	−	0.544326i	$-0.183215\pi$
$733$	45.4233	1.67775	0.838874	+	0.544326i	$0.183215\pi$
$734$	−58.7386	−2.16808
$735$	0	0
$736$	−2.87689	−0.106044
$737$	−22.2462	−0.819450
$738$	0	0
$739$	−39.8078	−1.46435	−0.732176	−	0.681115i	$-0.761495\pi$
$739$	−39.8078	−1.46435	−0.732176	+	0.681115i	$0.761495\pi$
$740$	7.12311	0.261851
$741$	0	0
$742$	22.2462	0.816684
$743$	−19.1771	−0.703539	−0.351769	−	0.936087i	$-0.614420\pi$
$743$	−19.1771	−0.703539	−0.351769	+	0.936087i	$0.614420\pi$
$744$	0	0
$745$	−15.1231	−0.554068
$746$	−35.8617	−1.31299
$747$	0	0
$748$	56.9848	2.08357
$749$	4.00000	0.146157
$750$	0	0
$751$	−8.00000	−0.291924	−0.145962	−	0.989290i	$-0.546628\pi$
$751$	−8.00000	−0.291924	−0.145962	+	0.989290i	$0.546628\pi$
$752$	76.8466	2.80231
$753$	0	0
$754$	−22.2462	−0.810159
$755$	47.6155	1.73291
$756$	0	0
$757$	27.5616	1.00174	0.500871	−	0.865522i	$-0.333013\pi$
$757$	27.5616	1.00174	0.500871	+	0.865522i	$0.333013\pi$
$758$	−43.2311	−1.57022
$759$	0	0
$760$	−93.4773	−3.39078
$761$	−13.4233	−0.486594	−0.243297	−	0.969952i	$-0.578229\pi$
$761$	−13.4233	−0.486594	−0.243297	+	0.969952i	$0.578229\pi$
$762$	0	0
$763$	−8.24621	−0.298533
$764$	83.2311	3.01119
$765$	0	0
$766$	−70.1080	−2.53310
$767$	−30.9309	−1.11685
$768$	0	0
$769$	10.4924	0.378366	0.189183	−	0.981942i	$-0.439416\pi$
$769$	10.4924	0.378366	0.189183	+	0.981942i	$0.439416\pi$
$770$	28.4924	1.02680
$771$	0	0
$772$	17.1231	0.616274
$773$	−52.1080	−1.87419	−0.937096	−	0.349071i	$-0.886497\pi$
$773$	−52.1080	−1.87419	−0.937096	+	0.349071i	$0.886497\pi$
$774$	0	0
$775$	18.7386	0.673112
$776$	112.354	4.03328
$777$	0	0
$778$	−23.3693	−0.837831
$779$	28.4924	1.02085
$780$	0	0
$781$	−37.4773	−1.34104
$782$	4.49242	0.160649
$783$	0	0
$784$	7.68466	0.274452
$785$	−49.8617	−1.77964
$786$	0	0
$787$	42.0540	1.49906	0.749531	−	0.661969i	$-0.230279\pi$
$787$	42.0540	1.49906	0.749531	+	0.661969i	$0.230279\pi$
$788$	−41.6155	−1.48249
$789$	0	0
$790$	56.9848	2.02743
$791$	19.3693	0.688694
$792$	0	0
$793$	15.8078	0.561350
$794$	−26.1080	−0.926536
$795$	0	0
$796$	−7.12311	−0.252472
$797$	−40.1080	−1.42070	−0.710348	−	0.703850i	$-0.751463\pi$
$797$	−40.1080	−1.42070	−0.710348	+	0.703850i	$0.751463\pi$
$798$	0	0
$799$	−40.0000	−1.41510
$800$	−50.4233	−1.78273
$801$	0	0
$802$	14.2462	0.503051
$803$	13.8617	0.489170
$804$	0	0
$805$	1.56155	0.0550375
$806$	22.2462	0.783589
$807$	0	0
$808$	8.63068	0.303626
$809$	−18.1080	−0.636642	−0.318321	−	0.947983i	$-0.603119\pi$
$809$	−18.1080	−0.636642	−0.318321	+	0.947983i	$0.603119\pi$
$810$	0	0
$811$	6.05398	0.212584	0.106292	−	0.994335i	$-0.466102\pi$
$811$	6.05398	0.212584	0.106292	+	0.994335i	$0.466102\pi$
$812$	11.1231	0.390344
$813$	0	0
$814$	−3.50758	−0.122941
$815$	−62.5464	−2.19090
$816$	0	0
$817$	−28.4924	−0.996824
$818$	−21.1231	−0.738552
$819$	0	0
$820$	115.723	4.04124
$821$	11.4233	0.398676	0.199338	−	0.979931i	$-0.436121\pi$
$821$	11.4233	0.398676	0.199338	+	0.979931i	$0.436121\pi$
$822$	0	0
$823$	−40.9848	−1.42864	−0.714321	−	0.699818i	$-0.753264\pi$
$823$	−40.9848	−1.42864	−0.714321	+	0.699818i	$0.753264\pi$
$824$	−93.4773	−3.25643
$825$	0	0
$826$	22.2462	0.774045
$827$	14.3002	0.497266	0.248633	−	0.968598i	$-0.420019\pi$
$827$	14.3002	0.497266	0.248633	+	0.968598i	$0.420019\pi$
$828$	0	0
$829$	10.4924	0.364417	0.182208	−	0.983260i	$-0.441675\pi$
$829$	10.4924	0.364417	0.182208	+	0.983260i	$0.441675\pi$
$830$	42.7386	1.48348
$831$	0	0
$832$	−5.12311	−0.177612
$833$	−4.00000	−0.138592
$834$	0	0
$835$	−44.4924	−1.53972
$836$	56.9848	1.97086
$837$	0	0
$838$	35.8617	1.23882
$839$	−46.5464	−1.60696	−0.803480	−	0.595332i	$-0.797021\pi$
$839$	−46.5464	−1.60696	−0.803480	+	0.595332i	$0.797021\pi$
$840$	0	0
$841$	−23.0540	−0.794965
$842$	−13.1231	−0.452252
$843$	0	0
$844$	−21.3693	−0.735562
$845$	1.12311	0.0386360
$846$	0	0
$847$	1.24621	0.0428203
$848$	66.7386	2.29181
$849$	0	0
$850$	78.7386	2.70071
$851$	−0.192236	−0.00658976
$852$	0	0
$853$	−45.2311	−1.54868	−0.774341	−	0.632769i	$-0.781918\pi$
$853$	−45.2311	−1.54868	−0.774341	+	0.632769i	$0.781918\pi$
$854$	−11.3693	−0.389050
$855$	0	0
$856$	26.2462	0.897077
$857$	8.05398	0.275119	0.137559	−	0.990494i	$-0.456074\pi$
$857$	8.05398	0.275119	0.137559	+	0.990494i	$0.456074\pi$
$858$	0	0
$859$	44.0000	1.50126	0.750630	−	0.660722i	$-0.229750\pi$
$859$	44.0000	1.50126	0.750630	+	0.660722i	$0.229750\pi$
$860$	−115.723	−3.94614
$861$	0	0
$862$	64.9848	2.21339
$863$	−2.68466	−0.0913868	−0.0456934	−	0.998956i	$-0.514550\pi$
$863$	−2.68466	−0.0913868	−0.0456934	+	0.998956i	$0.514550\pi$
$864$	0	0
$865$	72.1080	2.45174
$866$	19.8617	0.674929
$867$	0	0
$868$	−11.1231	−0.377543
$869$	−19.5076	−0.661749
$870$	0	0
$871$	−25.3693	−0.859607
$872$	−54.1080	−1.83233
$873$	0	0
$874$	4.49242	0.151958
$875$	9.56155	0.323239
$876$	0	0
$877$	−33.2311	−1.12213	−0.561067	−	0.827771i	$-0.689609\pi$
$877$	−33.2311	−1.12213	−0.561067	+	0.827771i	$0.689609\pi$
$878$	−92.4924	−3.12147
$879$	0	0
$880$	85.4773	2.88144
$881$	−41.4233	−1.39559	−0.697793	−	0.716299i	$-0.745835\pi$
$881$	−41.4233	−1.39559	−0.697793	+	0.716299i	$0.745835\pi$
$882$	0	0
$883$	45.6695	1.53690	0.768451	−	0.639909i	$-0.221028\pi$
$883$	45.6695	1.53690	0.768451	+	0.639909i	$0.221028\pi$
$884$	64.9848	2.18568
$885$	0	0
$886$	−58.2462	−1.95682
$887$	−17.0691	−0.573125	−0.286563	−	0.958062i	$-0.592513\pi$
$887$	−17.0691	−0.573125	−0.286563	+	0.958062i	$0.592513\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	−48.4924	−1.62547
$891$	0	0
$892$	−79.2311	−2.65285
$893$	−40.0000	−1.33855
$894$	0	0
$895$	−64.9848	−2.17220
$896$	−9.43845	−0.315316
$897$	0	0
$898$	15.3693	0.512881
$899$	−5.94602	−0.198311
$900$	0	0
$901$	−34.7386	−1.15731
$902$	−56.9848	−1.89739
$903$	0	0
$904$	127.093	4.22704
$905$	−60.9848	−2.02720
$906$	0	0
$907$	19.3153	0.641355	0.320678	−	0.947188i	$-0.396089\pi$
$907$	19.3153	0.641355	0.320678	+	0.947188i	$0.396089\pi$
$908$	−74.1080	−2.45936
$909$	0	0
$910$	32.4924	1.07711
$911$	−21.7538	−0.720735	−0.360368	−	0.932810i	$-0.617349\pi$
$911$	−21.7538	−0.720735	−0.360368	+	0.932810i	$0.617349\pi$
$912$	0	0
$913$	−14.6307	−0.484205
$914$	35.3693	1.16991
$915$	0	0
$916$	−33.6155	−1.11069
$917$	−18.0000	−0.594412
$918$	0	0
$919$	5.94602	0.196141	0.0980706	−	0.995179i	$-0.468733\pi$
$919$	5.94602	0.196141	0.0980706	+	0.995179i	$0.468733\pi$
$920$	10.2462	0.337808
$921$	0	0
$922$	35.8617	1.18104
$923$	−42.7386	−1.40676
$924$	0	0
$925$	−3.36932	−0.110782
$926$	9.75379	0.320529
$927$	0	0
$928$	16.0000	0.525226
$929$	2.49242	0.0817737	0.0408869	−	0.999164i	$-0.486982\pi$
$929$	2.49242	0.0817737	0.0408869	+	0.999164i	$0.486982\pi$
$930$	0	0
$931$	−4.00000	−0.131095
$932$	68.1080	2.23095
$933$	0	0
$934$	−88.4924	−2.89556
$935$	−44.4924	−1.45506
$936$	0	0
$937$	−7.86174	−0.256832	−0.128416	−	0.991720i	$-0.540989\pi$
$937$	−7.86174	−0.256832	−0.128416	+	0.991720i	$0.540989\pi$
$938$	18.2462	0.595760
$939$	0	0
$940$	−162.462	−5.29893
$941$	−9.36932	−0.305431	−0.152716	−	0.988270i	$-0.548802\pi$
$941$	−9.36932	−0.305431	−0.152716	+	0.988270i	$0.548802\pi$
$942$	0	0
$943$	−3.12311	−0.101702
$944$	66.7386	2.17216
$945$	0	0
$946$	56.9848	1.85274
$947$	−5.61553	−0.182480	−0.0912401	−	0.995829i	$-0.529083\pi$
$947$	−5.61553	−0.182480	−0.0912401	+	0.995829i	$0.529083\pi$
$948$	0	0
$949$	15.8078	0.513142
$950$	78.7386	2.55462
$951$	0	0
$952$	−26.2462	−0.850645
$953$	22.9848	0.744552	0.372276	−	0.928122i	$-0.378577\pi$
$953$	22.9848	0.744552	0.372276	+	0.928122i	$0.378577\pi$
$954$	0	0
$955$	−64.9848	−2.10286
$956$	−78.9848	−2.55455
$957$	0	0
$958$	26.8769	0.868353
$959$	6.43845	0.207908
$960$	0	0
$961$	−25.0540	−0.808193
$962$	−4.00000	−0.128965
$963$	0	0
$964$	−132.847	−4.27870
$965$	−13.3693	−0.430374
$966$	0	0
$967$	−3.61553	−0.116268	−0.0581338	−	0.998309i	$-0.518515\pi$
$967$	−3.61553	−0.116268	−0.0581338	+	0.998309i	$0.518515\pi$
$968$	8.17708	0.262821
$969$	0	0
$970$	−156.216	−5.01579
$971$	21.2311	0.681337	0.340669	−	0.940183i	$-0.389347\pi$
$971$	21.2311	0.681337	0.340669	+	0.940183i	$0.389347\pi$
$972$	0	0
$973$	15.1231	0.484825
$974$	−68.4924	−2.19464
$975$	0	0
$976$	−34.1080	−1.09177
$977$	−18.3845	−0.588171	−0.294086	−	0.955779i	$-0.595015\pi$
$977$	−18.3845	−0.588171	−0.294086	+	0.955779i	$0.595015\pi$
$978$	0	0
$979$	16.6004	0.530550
$980$	−16.2462	−0.518966
$981$	0	0
$982$	−102.108	−3.25840
$983$	2.49242	0.0794959	0.0397480	−	0.999210i	$-0.487344\pi$
$983$	2.49242	0.0794959	0.0397480	+	0.999210i	$0.487344\pi$
$984$	0	0
$985$	32.4924	1.03529
$986$	−24.9848	−0.795680
$987$	0	0
$988$	64.9848	2.06744
$989$	3.12311	0.0993090
$990$	0	0
$991$	−32.9848	−1.04780	−0.523899	−	0.851780i	$-0.675523\pi$
$991$	−32.9848	−1.04780	−0.523899	+	0.851780i	$0.675523\pi$
$992$	−16.0000	−0.508001
$993$	0	0
$994$	30.7386	0.974970
$995$	5.56155	0.176313
$996$	0	0
$997$	−14.0000	−0.443384	−0.221692	−	0.975117i	$-0.571158\pi$
$997$	−14.0000	−0.443384	−0.221692	+	0.975117i	$0.571158\pi$
$998$	−61.4773	−1.94603
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.h.1.1		2
3.2	odd	2		2667.2.a.i.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.i.1.2	✓	2	3.2	odd	2
8001.2.a.h.1.1		2	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:	\( N \)	\(=\)	\( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8001.a (trivial)

\(f(q)\)	\(=\)	\(q-2.56155 q^{2} +4.56155 q^{4} -3.56155 q^{5} -1.00000 q^{7} -6.56155 q^{8} +O(q^{10})\)	\(q-2.56155 q^{2} +4.56155 q^{4} -3.56155 q^{5} -1.00000 q^{7} -6.56155 q^{8} +9.12311 q^{10} -3.12311 q^{11} -3.56155 q^{13} +2.56155 q^{14} +7.68466 q^{16} -4.00000 q^{17} -4.00000 q^{19} -16.2462 q^{20} +8.00000 q^{22} +0.438447 q^{23} +7.68466 q^{25} +9.12311 q^{26} -4.56155 q^{28} -2.43845 q^{29} +2.43845 q^{31} -6.56155 q^{32} +10.2462 q^{34} +3.56155 q^{35} -0.438447 q^{37} +10.2462 q^{38} +23.3693 q^{40} -7.12311 q^{41} +7.12311 q^{43} -14.2462 q^{44} -1.12311 q^{46} +10.0000 q^{47} +1.00000 q^{49} -19.6847 q^{50} -16.2462 q^{52} +8.68466 q^{53} +11.1231 q^{55} +6.56155 q^{56} +6.24621 q^{58} +8.68466 q^{59} -4.43845 q^{61} -6.24621 q^{62} +1.43845 q^{64} +12.6847 q^{65} +7.12311 q^{67} -18.2462 q^{68} -9.12311 q^{70} +12.0000 q^{71} -4.43845 q^{73} +1.12311 q^{74} -18.2462 q^{76} +3.12311 q^{77} +6.24621 q^{79} -27.3693 q^{80} +18.2462 q^{82} +4.68466 q^{83} +14.2462 q^{85} -18.2462 q^{86} +20.4924 q^{88} -5.31534 q^{89} +3.56155 q^{91} +2.00000 q^{92} -25.6155 q^{94} +14.2462 q^{95} -17.1231 q^{97} -2.56155 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} + 5 q^{4} - 3 q^{5} - 2 q^{7} - 9 q^{8}+O(q^{10}) \) 2 * q - q^2 + 5 * q^4 - 3 * q^5 - 2 * q^7 - 9 * q^8	\( 2 q - q^{2} + 5 q^{4} - 3 q^{5} - 2 q^{7} - 9 q^{8} + 10 q^{10} + 2 q^{11} - 3 q^{13} + q^{14} + 3 q^{16} - 8 q^{17} - 8 q^{19} - 16 q^{20} + 16 q^{22} + 5 q^{23} + 3 q^{25} + 10 q^{26} - 5 q^{28} - 9 q^{29} + 9 q^{31} - 9 q^{32} + 4 q^{34} + 3 q^{35} - 5 q^{37} + 4 q^{38} + 22 q^{40} - 6 q^{41} + 6 q^{43} - 12 q^{44} + 6 q^{46} + 20 q^{47} + 2 q^{49} - 27 q^{50} - 16 q^{52} + 5 q^{53} + 14 q^{55} + 9 q^{56} - 4 q^{58} + 5 q^{59} - 13 q^{61} + 4 q^{62} + 7 q^{64} + 13 q^{65} + 6 q^{67} - 20 q^{68} - 10 q^{70} + 24 q^{71} - 13 q^{73} - 6 q^{74} - 20 q^{76} - 2 q^{77} - 4 q^{79} - 30 q^{80} + 20 q^{82} - 3 q^{83} + 12 q^{85} - 20 q^{86} + 8 q^{88} - 23 q^{89} + 3 q^{91} + 4 q^{92} - 10 q^{94} + 12 q^{95} - 26 q^{97} - q^{98}+O(q^{100}) \) 2 * q - q^2 + 5 * q^4 - 3 * q^5 - 2 * q^7 - 9 * q^8 + 10 * q^10 + 2 * q^11 - 3 * q^13 + q^14 + 3 * q^16 - 8 * q^17 - 8 * q^19 - 16 * q^20 + 16 * q^22 + 5 * q^23 + 3 * q^25 + 10 * q^26 - 5 * q^28 - 9 * q^29 + 9 * q^31 - 9 * q^32 + 4 * q^34 + 3 * q^35 - 5 * q^37 + 4 * q^38 + 22 * q^40 - 6 * q^41 + 6 * q^43 - 12 * q^44 + 6 * q^46 + 20 * q^47 + 2 * q^49 - 27 * q^50 - 16 * q^52 + 5 * q^53 + 14 * q^55 + 9 * q^56 - 4 * q^58 + 5 * q^59 - 13 * q^61 + 4 * q^62 + 7 * q^64 + 13 * q^65 + 6 * q^67 - 20 * q^68 - 10 * q^70 + 24 * q^71 - 13 * q^73 - 6 * q^74 - 20 * q^76 - 2 * q^77 - 4 * q^79 - 30 * q^80 + 20 * q^82 - 3 * q^83 + 12 * q^85 - 20 * q^86 + 8 * q^88 - 23 * q^89 + 3 * q^91 + 4 * q^92 - 10 * q^94 + 12 * q^95 - 26 * q^97 - q^98

Introduction

L-functions

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Database

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