LMFDB - Embedded newform 8001.2.a.q.1.3

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:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 22 x^{13} + 186 x^{11} - 763 x^{9} - 7 x^{8} + 1588 x^{7} + 64 x^{6} - 1625 x^{5} - 185 x^{4} + \cdots - 13 $ x^15 - 22x^13 + 186x^11 - 763x^9 - 7x^8 + 1588x^7 + 64x^6 - 1625x^5 - 185x^4 + 726x^3 + 145x^2 - 83*x - 13
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 889)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.07483$ 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)

$f(q)$	$=$	$q-2.07483 q^{2} +2.30492 q^{4} -1.11193 q^{5} -1.00000 q^{7} -0.632654 q^{8} +O(q^{10})$	\(q-2.07483 q^{2} +2.30492 q^{4} -1.11193 q^{5} -1.00000 q^{7} -0.632654 q^{8} +2.30706 q^{10} -4.88004 q^{11} -0.190950 q^{13} +2.07483 q^{14} -3.29719 q^{16} -0.273165 q^{17} +1.55515 q^{19} -2.56290 q^{20} +10.1252 q^{22} -2.18830 q^{23} -3.76362 q^{25} +0.396188 q^{26} -2.30492 q^{28} +2.52648 q^{29} +0.702773 q^{31} +8.10641 q^{32} +0.566772 q^{34} +1.11193 q^{35} +8.99238 q^{37} -3.22667 q^{38} +0.703465 q^{40} +0.921889 q^{41} -8.65681 q^{43} -11.2481 q^{44} +4.54035 q^{46} +7.57880 q^{47} +1.00000 q^{49} +7.80887 q^{50} -0.440124 q^{52} +3.51283 q^{53} +5.42624 q^{55} +0.632654 q^{56} -5.24202 q^{58} +3.45670 q^{59} +8.93237 q^{61} -1.45814 q^{62} -10.2250 q^{64} +0.212322 q^{65} +3.52334 q^{67} -0.629624 q^{68} -2.30706 q^{70} +13.9899 q^{71} -8.11920 q^{73} -18.6577 q^{74} +3.58449 q^{76} +4.88004 q^{77} -12.2109 q^{79} +3.66623 q^{80} -1.91276 q^{82} -9.50363 q^{83} +0.303740 q^{85} +17.9614 q^{86} +3.08737 q^{88} +5.13628 q^{89} +0.190950 q^{91} -5.04386 q^{92} -15.7247 q^{94} -1.72921 q^{95} +11.6865 q^{97} -2.07483 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + 14 q^{4} - 7 q^{5} - 15 q^{7}+O(q^{10}) $ 15 * q + 14 * q^4 - 7 * q^5 - 15 * q^7	$ 15 q + 14 q^{4} - 7 q^{5} - 15 q^{7} + 10 q^{10} - 14 q^{11} + 6 q^{13} + 20 q^{16} - 10 q^{17} + 13 q^{19} - 8 q^{20} - 11 q^{22} - 15 q^{23} - 22 q^{26} - 14 q^{28} - 16 q^{29} + 22 q^{31} - 15 q^{34} + 7 q^{35} - 14 q^{37} + 6 q^{38} + 22 q^{40} - 19 q^{41} - q^{43} - 25 q^{44} - 28 q^{46} - 49 q^{47} + 15 q^{49} - 24 q^{50} - 17 q^{52} + 28 q^{53} + 39 q^{55} - 10 q^{58} - 43 q^{59} + 27 q^{61} - 14 q^{62} + 18 q^{64} + 8 q^{65} + 3 q^{67} - 13 q^{68} - 10 q^{70} - 55 q^{71} - 3 q^{73} + 12 q^{74} - 20 q^{76} + 14 q^{77} + 18 q^{79} - 29 q^{80} + 14 q^{82} - 17 q^{83} + 7 q^{85} - 4 q^{86} - 114 q^{88} - 36 q^{89} - 6 q^{91} - 45 q^{92} - 15 q^{94} - 59 q^{95} - 2 q^{97}+O(q^{100}) $ 15 * q + 14 * q^4 - 7 * q^5 - 15 * q^7 + 10 * q^10 - 14 * q^11 + 6 * q^13 + 20 * q^16 - 10 * q^17 + 13 * q^19 - 8 * q^20 - 11 * q^22 - 15 * q^23 - 22 * q^26 - 14 * q^28 - 16 * q^29 + 22 * q^31 - 15 * q^34 + 7 * q^35 - 14 * q^37 + 6 * q^38 + 22 * q^40 - 19 * q^41 - q^43 - 25 * q^44 - 28 * q^46 - 49 * q^47 + 15 * q^49 - 24 * q^50 - 17 * q^52 + 28 * q^53 + 39 * q^55 - 10 * q^58 - 43 * q^59 + 27 * q^61 - 14 * q^62 + 18 * q^64 + 8 * q^65 + 3 * q^67 - 13 * q^68 - 10 * q^70 - 55 * q^71 - 3 * q^73 + 12 * q^74 - 20 * q^76 + 14 * q^77 + 18 * q^79 - 29 * q^80 + 14 * q^82 - 17 * q^83 + 7 * q^85 - 4 * q^86 - 114 * q^88 - 36 * q^89 - 6 * q^91 - 45 * q^92 - 15 * q^94 - 59 * q^95 - 2 * q^97

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 $

$2$	−2.07483	−1.46713	−0.733563	−	0.679621i	$-0.762144\pi$
$2$	−2.07483	−1.46713	−0.733563	+	0.679621i	$0.762144\pi$
$3$	0	0
$3$	0	0
$4$	2.30492	1.15246
$5$	−1.11193	−0.497269	−0.248634	−	0.968597i	$-0.579982\pi$
$5$	−1.11193	−0.497269	−0.248634	+	0.968597i	$0.579982\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−0.632654	−0.223677
$9$	0	0
$10$	2.30706	0.729556
$11$	−4.88004	−1.47139	−0.735693	−	0.677315i	$-0.763143\pi$
$11$	−4.88004	−1.47139	−0.735693	+	0.677315i	$0.763143\pi$
$12$	0	0
$13$	−0.190950	−0.0529599	−0.0264800	−	0.999649i	$-0.508430\pi$
$13$	−0.190950	−0.0529599	−0.0264800	+	0.999649i	$0.508430\pi$
$14$	2.07483	0.554522
$15$	0	0
$16$	−3.29719	−0.824297
$17$	−0.273165	−0.0662523	−0.0331262	−	0.999451i	$-0.510546\pi$
$17$	−0.273165	−0.0662523	−0.0331262	+	0.999451i	$0.510546\pi$
$18$	0	0
$19$	1.55515	0.356776	0.178388	−	0.983960i	$-0.442912\pi$
$19$	1.55515	0.356776	0.178388	+	0.983960i	$0.442912\pi$
$20$	−2.56290	−0.573082
$21$	0	0
$22$	10.1252	2.15871
$23$	−2.18830	−0.456292	−0.228146	−	0.973627i	$-0.573266\pi$
$23$	−2.18830	−0.456292	−0.228146	+	0.973627i	$0.573266\pi$
$24$	0	0
$25$	−3.76362	−0.752724
$26$	0.396188	0.0776989
$27$	0	0
$28$	−2.30492	−0.435589
$29$	2.52648	0.469156	0.234578	−	0.972097i	$-0.424629\pi$
$29$	2.52648	0.469156	0.234578	+	0.972097i	$0.424629\pi$
$30$	0	0
$31$	0.702773	0.126222	0.0631109	−	0.998007i	$-0.479898\pi$
$31$	0.702773	0.126222	0.0631109	+	0.998007i	$0.479898\pi$
$32$	8.10641	1.43302
$33$	0	0
$34$	0.566772	0.0972005
$35$	1.11193	0.187950
$36$	0	0
$37$	8.99238	1.47834	0.739169	−	0.673520i	$-0.235218\pi$
$37$	8.99238	1.47834	0.739169	+	0.673520i	$0.235218\pi$
$38$	−3.22667	−0.523435
$39$	0	0
$40$	0.703465	0.111228
$41$	0.921889	0.143975	0.0719874	−	0.997406i	$-0.477066\pi$
$41$	0.921889	0.143975	0.0719874	+	0.997406i	$0.477066\pi$
$42$	0	0
$43$	−8.65681	−1.32015	−0.660076	−	0.751199i	$-0.729476\pi$
$43$	−8.65681	−1.32015	−0.660076	+	0.751199i	$0.729476\pi$
$44$	−11.2481	−1.69571
$45$	0	0
$46$	4.54035	0.669438
$47$	7.57880	1.10548	0.552741	−	0.833353i	$-0.313582\pi$
$47$	7.57880	1.10548	0.552741	+	0.833353i	$0.313582\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	7.80887	1.10434
$51$	0	0
$52$	−0.440124	−0.0610342
$53$	3.51283	0.482524	0.241262	−	0.970460i	$-0.422439\pi$
$53$	3.51283	0.482524	0.241262	+	0.970460i	$0.422439\pi$
$54$	0	0
$55$	5.42624	0.731674
$56$	0.632654	0.0845419
$57$	0	0
$58$	−5.24202	−0.688311
$59$	3.45670	0.450024	0.225012	−	0.974356i	$-0.427758\pi$
$59$	3.45670	0.450024	0.225012	+	0.974356i	$0.427758\pi$
$60$	0	0
$61$	8.93237	1.14367	0.571837	−	0.820368i	$-0.306231\pi$
$61$	8.93237	1.14367	0.571837	+	0.820368i	$0.306231\pi$
$62$	−1.45814	−0.185183
$63$	0	0
$64$	−10.2250	−1.27813
$65$	0.212322	0.0263353
$66$	0	0
$67$	3.52334	0.430444	0.215222	−	0.976565i	$-0.430952\pi$
$67$	3.52334	0.430444	0.215222	+	0.976565i	$0.430952\pi$
$68$	−0.629624	−0.0763531
$69$	0	0
$70$	−2.30706	−0.275746
$71$	13.9899	1.66029	0.830146	−	0.557545i	$-0.188257\pi$
$71$	13.9899	1.66029	0.830146	+	0.557545i	$0.188257\pi$
$72$	0	0
$73$	−8.11920	−0.950281	−0.475140	−	0.879910i	$-0.657603\pi$
$73$	−8.11920	−0.950281	−0.475140	+	0.879910i	$0.657603\pi$
$74$	−18.6577	−2.16891
$75$	0	0
$76$	3.58449	0.411169
$77$	4.88004	0.556132
$78$	0	0
$79$	−12.2109	−1.37383	−0.686916	−	0.726737i	$-0.741036\pi$
$79$	−12.2109	−1.37383	−0.686916	+	0.726737i	$0.741036\pi$
$80$	3.66623	0.409897
$81$	0	0
$82$	−1.91276	−0.211229
$83$	−9.50363	−1.04316	−0.521580	−	0.853203i	$-0.674657\pi$
$83$	−9.50363	−1.04316	−0.521580	+	0.853203i	$0.674657\pi$
$84$	0	0
$85$	0.303740	0.0329452
$86$	17.9614	1.93683
$87$	0	0
$88$	3.08737	0.329115
$89$	5.13628	0.544444	0.272222	−	0.962234i	$-0.412241\pi$
$89$	5.13628	0.544444	0.272222	+	0.962234i	$0.412241\pi$
$90$	0	0
$91$	0.190950	0.0200170
$92$	−5.04386	−0.525858
$93$	0	0
$94$	−15.7247	−1.62188
$95$	−1.72921	−0.177413
$96$	0	0
$97$	11.6865	1.18658	0.593290	−	0.804989i	$-0.297829\pi$
$97$	11.6865	1.18658	0.593290	+	0.804989i	$0.297829\pi$
$98$	−2.07483	−0.209589
$99$	0	0
$100$	−8.67483	−0.867483
$101$	−8.92138	−0.887711	−0.443855	−	0.896098i	$-0.646390\pi$
$101$	−8.92138	−0.887711	−0.443855	+	0.896098i	$0.646390\pi$
$102$	0	0
$103$	16.8803	1.66327	0.831633	−	0.555326i	$-0.187406\pi$
$103$	16.8803	1.66327	0.831633	+	0.555326i	$0.187406\pi$
$104$	0.120805	0.0118459
$105$	0	0
$106$	−7.28852	−0.707924
$107$	5.24332	0.506890	0.253445	−	0.967350i	$-0.418436\pi$
$107$	5.24332	0.506890	0.253445	+	0.967350i	$0.418436\pi$
$108$	0	0
$109$	4.99347	0.478288	0.239144	−	0.970984i	$-0.423133\pi$
$109$	4.99347	0.478288	0.239144	+	0.970984i	$0.423133\pi$
$110$	−11.2585	−1.07346
$111$	0	0
$112$	3.29719	0.311555
$113$	−7.31880	−0.688494	−0.344247	−	0.938879i	$-0.611866\pi$
$113$	−7.31880	−0.688494	−0.344247	+	0.938879i	$0.611866\pi$
$114$	0	0
$115$	2.43323	0.226900
$116$	5.82334	0.540683
$117$	0	0
$118$	−7.17206	−0.660242
$119$	0.273165	0.0250410
$120$	0	0
$121$	12.8147	1.16498
$122$	−18.5332	−1.67791
$123$	0	0
$124$	1.61984	0.145466
$125$	9.74450	0.871575
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	5.00241	0.442155
$129$	0	0
$130$	−0.440533	−0.0386373
$131$	−14.4436	−1.26194	−0.630971	−	0.775806i	$-0.717343\pi$
$131$	−14.4436	−1.26194	−0.630971	+	0.775806i	$0.717343\pi$
$132$	0	0
$133$	−1.55515	−0.134848
$134$	−7.31033	−0.631516
$135$	0	0
$136$	0.172819	0.0148191
$137$	−4.82562	−0.412280	−0.206140	−	0.978522i	$-0.566090\pi$
$137$	−4.82562	−0.412280	−0.206140	+	0.978522i	$0.566090\pi$
$138$	0	0
$139$	1.10975	0.0941279	0.0470639	−	0.998892i	$-0.485014\pi$
$139$	1.10975	0.0941279	0.0470639	+	0.998892i	$0.485014\pi$
$140$	2.56290	0.216605
$141$	0	0
$142$	−29.0266	−2.43586
$143$	0.931842	0.0779245
$144$	0	0
$145$	−2.80926	−0.233297
$146$	16.8460	1.39418
$147$	0	0
$148$	20.7267	1.70372
$149$	−2.93387	−0.240352	−0.120176	−	0.992753i	$-0.538346\pi$
$149$	−2.93387	−0.240352	−0.120176	+	0.992753i	$0.538346\pi$
$150$	0	0
$151$	13.9892	1.13842	0.569211	−	0.822191i	$-0.307249\pi$
$151$	13.9892	1.13842	0.569211	+	0.822191i	$0.307249\pi$
$152$	−0.983871	−0.0798025
$153$	0	0
$154$	−10.1252	−0.815915
$155$	−0.781433	−0.0627662
$156$	0	0
$157$	−4.10258	−0.327421	−0.163711	−	0.986508i	$-0.552346\pi$
$157$	−4.10258	−0.327421	−0.163711	+	0.986508i	$0.552346\pi$
$158$	25.3355	2.01559
$159$	0	0
$160$	−9.01374	−0.712599
$161$	2.18830	0.172462
$162$	0	0
$163$	−20.2773	−1.58824	−0.794119	−	0.607763i	$-0.792067\pi$
$163$	−20.2773	−1.58824	−0.794119	+	0.607763i	$0.792067\pi$
$164$	2.12488	0.165925
$165$	0	0
$166$	19.7184	1.53045
$167$	−20.4264	−1.58065	−0.790323	−	0.612691i	$-0.790087\pi$
$167$	−20.4264	−1.58065	−0.790323	+	0.612691i	$0.790087\pi$
$168$	0	0
$169$	−12.9635	−0.997195
$170$	−0.630209	−0.0483348
$171$	0	0
$172$	−19.9532	−1.52142
$173$	10.9250	0.830614	0.415307	−	0.909681i	$-0.363674\pi$
$173$	10.9250	0.830614	0.415307	+	0.909681i	$0.363674\pi$
$174$	0	0
$175$	3.76362	0.284503
$176$	16.0904	1.21286
$177$	0	0
$178$	−10.6569	−0.798768
$179$	−8.35743	−0.624664	−0.312332	−	0.949973i	$-0.601110\pi$
$179$	−8.35743	−0.624664	−0.312332	+	0.949973i	$0.601110\pi$
$180$	0	0
$181$	22.1099	1.64341	0.821707	−	0.569910i	$-0.193022\pi$
$181$	22.1099	1.64341	0.821707	+	0.569910i	$0.193022\pi$
$182$	−0.396188	−0.0293674
$183$	0	0
$184$	1.38444	0.102062
$185$	−9.99887	−0.735131
$186$	0	0
$187$	1.33306	0.0974827
$188$	17.4685	1.27402
$189$	0	0
$190$	3.58782	0.260288
$191$	1.18740	0.0859176	0.0429588	−	0.999077i	$-0.486322\pi$
$191$	1.18740	0.0859176	0.0429588	+	0.999077i	$0.486322\pi$
$192$	0	0
$193$	−20.7415	−1.49301	−0.746504	−	0.665381i	$-0.768269\pi$
$193$	−20.7415	−1.49301	−0.746504	+	0.665381i	$0.768269\pi$
$194$	−24.2474	−1.74086
$195$	0	0
$196$	2.30492	0.164637
$197$	10.9290	0.778656	0.389328	−	0.921099i	$-0.372707\pi$
$197$	10.9290	0.778656	0.389328	+	0.921099i	$0.372707\pi$
$198$	0	0
$199$	12.4856	0.885078	0.442539	−	0.896749i	$-0.354078\pi$
$199$	12.4856	0.885078	0.442539	+	0.896749i	$0.354078\pi$
$200$	2.38107	0.168367
$201$	0	0
$202$	18.5104	1.30238
$203$	−2.52648	−0.177324
$204$	0	0
$205$	−1.02507	−0.0715942
$206$	−35.0238	−2.44022
$207$	0	0
$208$	0.629597	0.0436547
$209$	−7.58918	−0.524955
$210$	0	0
$211$	5.40922	0.372386	0.186193	−	0.982513i	$-0.440385\pi$
$211$	5.40922	0.372386	0.186193	+	0.982513i	$0.440385\pi$
$212$	8.09679	0.556090
$213$	0	0
$214$	−10.8790	−0.743672
$215$	9.62574	0.656470
$216$	0	0
$217$	−0.702773	−0.0477074
$218$	−10.3606	−0.701709
$219$	0	0
$220$	12.5070	0.843225
$221$	0.0521609	0.00350872
$222$	0	0
$223$	−4.29677	−0.287733	−0.143867	−	0.989597i	$-0.545954\pi$
$223$	−4.29677	−0.287733	−0.143867	+	0.989597i	$0.545954\pi$
$224$	−8.10641	−0.541632
$225$	0	0
$226$	15.1853	1.01011
$227$	−13.5885	−0.901898	−0.450949	−	0.892550i	$-0.648914\pi$
$227$	−13.5885	−0.901898	−0.450949	+	0.892550i	$0.648914\pi$
$228$	0	0
$229$	6.28386	0.415249	0.207624	−	0.978209i	$-0.433427\pi$
$229$	6.28386	0.415249	0.207624	+	0.978209i	$0.433427\pi$
$230$	−5.04854	−0.332891
$231$	0	0
$232$	−1.59839	−0.104939
$233$	13.4057	0.878236	0.439118	−	0.898429i	$-0.355291\pi$
$233$	13.4057	0.878236	0.439118	+	0.898429i	$0.355291\pi$
$234$	0	0
$235$	−8.42708	−0.549722
$236$	7.96741	0.518634
$237$	0	0
$238$	−0.566772	−0.0367383
$239$	4.70806	0.304539	0.152270	−	0.988339i	$-0.451342\pi$
$239$	4.70806	0.304539	0.152270	+	0.988339i	$0.451342\pi$
$240$	0	0
$241$	−16.9491	−1.09179	−0.545893	−	0.837855i	$-0.683810\pi$
$241$	−16.9491	−1.09179	−0.545893	+	0.837855i	$0.683810\pi$
$242$	−26.5884	−1.70917
$243$	0	0
$244$	20.5884	1.31804
$245$	−1.11193	−0.0710384
$246$	0	0
$247$	−0.296955	−0.0188948
$248$	−0.444612	−0.0282329
$249$	0	0
$250$	−20.2182	−1.27871
$251$	−25.0271	−1.57970	−0.789849	−	0.613301i	$-0.789841\pi$
$251$	−25.0271	−1.57970	−0.789849	+	0.613301i	$0.789841\pi$
$252$	0	0
$253$	10.6790	0.671382
$254$	−2.07483	−0.130186
$255$	0	0
$256$	10.0709	0.629434
$257$	15.6182	0.974238	0.487119	−	0.873336i	$-0.338048\pi$
$257$	15.6182	0.974238	0.487119	+	0.873336i	$0.338048\pi$
$258$	0	0
$259$	−8.99238	−0.558759
$260$	0.489385	0.0303504
$261$	0	0
$262$	29.9680	1.85143
$263$	−10.7904	−0.665362	−0.332681	−	0.943039i	$-0.607953\pi$
$263$	−10.7904	−0.665362	−0.332681	+	0.943039i	$0.607953\pi$
$264$	0	0
$265$	−3.90601	−0.239944
$266$	3.22667	0.197840
$267$	0	0
$268$	8.12101	0.496070
$269$	−6.64295	−0.405028	−0.202514	−	0.979279i	$-0.564911\pi$
$269$	−6.64295	−0.405028	−0.202514	+	0.979279i	$0.564911\pi$
$270$	0	0
$271$	6.05639	0.367899	0.183950	−	0.982936i	$-0.441112\pi$
$271$	6.05639	0.367899	0.183950	+	0.982936i	$0.441112\pi$
$272$	0.900677	0.0546116
$273$	0	0
$274$	10.0123	0.604867
$275$	18.3666	1.10755
$276$	0	0
$277$	16.3385	0.981687	0.490843	−	0.871248i	$-0.336689\pi$
$277$	16.3385	0.981687	0.490843	+	0.871248i	$0.336689\pi$
$278$	−2.30255	−0.138098
$279$	0	0
$280$	−0.703465	−0.0420401
$281$	−2.81844	−0.168134	−0.0840670	−	0.996460i	$-0.526791\pi$
$281$	−2.81844	−0.168134	−0.0840670	+	0.996460i	$0.526791\pi$
$282$	0	0
$283$	9.50868	0.565232	0.282616	−	0.959233i	$-0.408798\pi$
$283$	9.50868	0.565232	0.282616	+	0.959233i	$0.408798\pi$
$284$	32.2455	1.91342
$285$	0	0
$286$	−1.93341	−0.114325
$287$	−0.921889	−0.0544174
$288$	0	0
$289$	−16.9254	−0.995611
$290$	5.82874	0.342276
$291$	0	0
$292$	−18.7141	−1.09516
$293$	−19.0133	−1.11077	−0.555385	−	0.831593i	$-0.687429\pi$
$293$	−19.0133	−1.11077	−0.555385	+	0.831593i	$0.687429\pi$
$294$	0	0
$295$	−3.84360	−0.223783
$296$	−5.68906	−0.330670
$297$	0	0
$298$	6.08728	0.352627
$299$	0.417856	0.0241652
$300$	0	0
$301$	8.65681	0.498970
$302$	−29.0252	−1.67021
$303$	0	0
$304$	−5.12762	−0.294089
$305$	−9.93215	−0.568713
$306$	0	0
$307$	−5.35407	−0.305573	−0.152787	−	0.988259i	$-0.548825\pi$
$307$	−5.35407	−0.305573	−0.152787	+	0.988259i	$0.548825\pi$
$308$	11.2481	0.640919
$309$	0	0
$310$	1.62134	0.0920859
$311$	20.7112	1.17442	0.587212	−	0.809433i	$-0.300225\pi$
$311$	20.7112	1.17442	0.587212	+	0.809433i	$0.300225\pi$
$312$	0	0
$313$	10.8427	0.612866	0.306433	−	0.951892i	$-0.400865\pi$
$313$	10.8427	0.612866	0.306433	+	0.951892i	$0.400865\pi$
$314$	8.51215	0.480369
$315$	0	0
$316$	−28.1451	−1.58329
$317$	2.06625	0.116052	0.0580260	−	0.998315i	$-0.481519\pi$
$317$	2.06625	0.116052	0.0580260	+	0.998315i	$0.481519\pi$
$318$	0	0
$319$	−12.3293	−0.690310
$320$	11.3695	0.635575
$321$	0	0
$322$	−4.54035	−0.253024
$323$	−0.424813	−0.0236372
$324$	0	0
$325$	0.718662	0.0398642
$326$	42.0719	2.33014
$327$	0	0
$328$	−0.583237	−0.0322039
$329$	−7.57880	−0.417833
$330$	0	0
$331$	10.4765	0.575840	0.287920	−	0.957654i	$-0.407036\pi$
$331$	10.4765	0.575840	0.287920	+	0.957654i	$0.407036\pi$
$332$	−21.9051	−1.20220
$333$	0	0
$334$	42.3814	2.31901
$335$	−3.91770	−0.214047
$336$	0	0
$337$	−10.2807	−0.560024	−0.280012	−	0.959997i	$-0.590338\pi$
$337$	−10.2807	−0.560024	−0.280012	+	0.959997i	$0.590338\pi$
$338$	26.8971	1.46301
$339$	0	0
$340$	0.700096	0.0379680
$341$	−3.42956	−0.185721
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	5.47676	0.295287
$345$	0	0
$346$	−22.6675	−1.21861
$347$	4.54733	0.244113	0.122057	−	0.992523i	$-0.461051\pi$
$347$	4.54733	0.244113	0.122057	+	0.992523i	$0.461051\pi$
$348$	0	0
$349$	36.0154	1.92786	0.963930	−	0.266157i	$-0.0857539\pi$
$349$	36.0154	1.92786	0.963930	+	0.266157i	$0.0857539\pi$
$350$	−7.80887	−0.417401
$351$	0	0
$352$	−39.5596	−2.10853
$353$	26.2971	1.39966	0.699828	−	0.714312i	$-0.253260\pi$
$353$	26.2971	1.39966	0.699828	+	0.714312i	$0.253260\pi$
$354$	0	0
$355$	−15.5557	−0.825612
$356$	11.8387	0.627450
$357$	0	0
$358$	17.3402	0.916460
$359$	−13.5542	−0.715362	−0.357681	−	0.933844i	$-0.616432\pi$
$359$	−13.5542	−0.715362	−0.357681	+	0.933844i	$0.616432\pi$
$360$	0	0
$361$	−16.5815	−0.872711
$362$	−45.8742	−2.41110
$363$	0	0
$364$	0.440124	0.0230688
$365$	9.02796	0.472545
$366$	0	0
$367$	−25.4349	−1.32769	−0.663847	−	0.747869i	$-0.731077\pi$
$367$	−25.4349	−1.32769	−0.663847	+	0.747869i	$0.731077\pi$
$368$	7.21524	0.376120
$369$	0	0
$370$	20.7460	1.07853
$371$	−3.51283	−0.182377
$372$	0	0
$373$	4.77570	0.247277	0.123638	−	0.992327i	$-0.460544\pi$
$373$	4.77570	0.247277	0.123638	+	0.992327i	$0.460544\pi$
$374$	−2.76587	−0.143019
$375$	0	0
$376$	−4.79476	−0.247271
$377$	−0.482431	−0.0248465
$378$	0	0
$379$	−16.3319	−0.838915	−0.419458	−	0.907775i	$-0.637780\pi$
$379$	−16.3319	−0.838915	−0.419458	+	0.907775i	$0.637780\pi$
$380$	−3.98569	−0.204462
$381$	0	0
$382$	−2.46366	−0.126052
$383$	15.5784	0.796018	0.398009	−	0.917382i	$-0.369701\pi$
$383$	15.5784	0.796018	0.398009	+	0.917382i	$0.369701\pi$
$384$	0	0
$385$	−5.42624	−0.276547
$386$	43.0351	2.19043
$387$	0	0
$388$	26.9363	1.36749
$389$	−26.8543	−1.36157	−0.680785	−	0.732484i	$-0.738361\pi$
$389$	−26.8543	−1.36157	−0.680785	+	0.732484i	$0.738361\pi$
$390$	0	0
$391$	0.597768	0.0302304
$392$	−0.632654	−0.0319538
$393$	0	0
$394$	−22.6757	−1.14239
$395$	13.5776	0.683164
$396$	0	0
$397$	−0.986077	−0.0494898	−0.0247449	−	0.999694i	$-0.507877\pi$
$397$	−0.986077	−0.0494898	−0.0247449	+	0.999694i	$0.507877\pi$
$398$	−25.9054	−1.29852
$399$	0	0
$400$	12.4094	0.620468
$401$	−4.54152	−0.226793	−0.113396	−	0.993550i	$-0.536173\pi$
$401$	−4.54152	−0.226793	−0.113396	+	0.993550i	$0.536173\pi$
$402$	0	0
$403$	−0.134194	−0.00668470
$404$	−20.5631	−1.02305
$405$	0	0
$406$	5.24202	0.260157
$407$	−43.8831	−2.17521
$408$	0	0
$409$	−21.6850	−1.07226	−0.536128	−	0.844137i	$-0.680114\pi$
$409$	−21.6850	−1.07226	−0.536128	+	0.844137i	$0.680114\pi$
$410$	2.12685	0.105038
$411$	0	0
$412$	38.9077	1.91685
$413$	−3.45670	−0.170093
$414$	0	0
$415$	10.5673	0.518731
$416$	−1.54792	−0.0758929
$417$	0	0
$418$	15.7463	0.770175
$419$	−3.65606	−0.178610	−0.0893052	−	0.996004i	$-0.528465\pi$
$419$	−3.65606	−0.178610	−0.0893052	+	0.996004i	$0.528465\pi$
$420$	0	0
$421$	−13.4669	−0.656336	−0.328168	−	0.944619i	$-0.606431\pi$
$421$	−13.4669	−0.656336	−0.328168	+	0.944619i	$0.606431\pi$
$422$	−11.2232	−0.546337
$423$	0	0
$424$	−2.22241	−0.107930
$425$	1.02809	0.0498697
$426$	0	0
$427$	−8.93237	−0.432268
$428$	12.0854	0.584171
$429$	0	0
$430$	−19.9718	−0.963125
$431$	−19.4996	−0.939265	−0.469632	−	0.882862i	$-0.655614\pi$
$431$	−19.4996	−0.939265	−0.469632	+	0.882862i	$0.655614\pi$
$432$	0	0
$433$	13.4941	0.648484	0.324242	−	0.945974i	$-0.394891\pi$
$433$	13.4941	0.648484	0.324242	+	0.945974i	$0.394891\pi$
$434$	1.45814	0.0699927
$435$	0	0
$436$	11.5095	0.551207
$437$	−3.40313	−0.162794
$438$	0	0
$439$	−16.5073	−0.787853	−0.393926	−	0.919142i	$-0.628883\pi$
$439$	−16.5073	−0.787853	−0.393926	+	0.919142i	$0.628883\pi$
$440$	−3.43293	−0.163659
$441$	0	0
$442$	−0.108225	−0.00514773
$443$	−23.3269	−1.10829	−0.554147	−	0.832419i	$-0.686956\pi$
$443$	−23.3269	−1.10829	−0.554147	+	0.832419i	$0.686956\pi$
$444$	0	0
$445$	−5.71117	−0.270735
$446$	8.91507	0.422141
$447$	0	0
$448$	10.2250	0.483088
$449$	−0.322351	−0.0152127	−0.00760634	−	0.999971i	$-0.502421\pi$
$449$	−0.322351	−0.0152127	−0.00760634	+	0.999971i	$0.502421\pi$
$450$	0	0
$451$	−4.49885	−0.211843
$452$	−16.8692	−0.793462
$453$	0	0
$454$	28.1937	1.32320
$455$	−0.212322	−0.00995382
$456$	0	0
$457$	−2.36275	−0.110525	−0.0552623	−	0.998472i	$-0.517600\pi$
$457$	−2.36275	−0.110525	−0.0552623	+	0.998472i	$0.517600\pi$
$458$	−13.0379	−0.609223
$459$	0	0
$460$	5.60840	0.261493
$461$	−30.6316	−1.42666	−0.713329	−	0.700829i	$-0.752813\pi$
$461$	−30.6316	−1.42666	−0.713329	+	0.700829i	$0.752813\pi$
$462$	0	0
$463$	23.7930	1.10576	0.552878	−	0.833262i	$-0.313530\pi$
$463$	23.7930	1.10576	0.552878	+	0.833262i	$0.313530\pi$
$464$	−8.33029	−0.386724
$465$	0	0
$466$	−27.8145	−1.28848
$467$	21.9185	1.01427	0.507134	−	0.861867i	$-0.330705\pi$
$467$	21.9185	1.01427	0.507134	+	0.861867i	$0.330705\pi$
$468$	0	0
$469$	−3.52334	−0.162693
$470$	17.4848	0.806511
$471$	0	0
$472$	−2.18689	−0.100660
$473$	42.2455	1.94245
$474$	0	0
$475$	−5.85299	−0.268553
$476$	0.629624	0.0288588
$477$	0	0
$478$	−9.76843	−0.446797
$479$	33.4689	1.52923	0.764617	−	0.644485i	$-0.222928\pi$
$479$	33.4689	1.52923	0.764617	+	0.644485i	$0.222928\pi$
$480$	0	0
$481$	−1.71709	−0.0782927
$482$	35.1665	1.60179
$483$	0	0
$484$	29.5369	1.34259
$485$	−12.9945	−0.590049
$486$	0	0
$487$	−38.7743	−1.75703	−0.878517	−	0.477712i	$-0.841466\pi$
$487$	−38.7743	−1.75703	−0.878517	+	0.477712i	$0.841466\pi$
$488$	−5.65110	−0.255813
$489$	0	0
$490$	2.30706	0.104222
$491$	−25.9164	−1.16959	−0.584795	−	0.811181i	$-0.698825\pi$
$491$	−25.9164	−1.16959	−0.584795	+	0.811181i	$0.698825\pi$
$492$	0	0
$493$	−0.690147	−0.0310827
$494$	0.616132	0.0277211
$495$	0	0
$496$	−2.31718	−0.104044
$497$	−13.9899	−0.627532
$498$	0	0
$499$	32.1750	1.44035	0.720175	−	0.693792i	$-0.244061\pi$
$499$	32.1750	1.44035	0.720175	+	0.693792i	$0.244061\pi$
$500$	22.4603	1.00445
$501$	0	0
$502$	51.9270	2.31762
$503$	−26.7576	−1.19306	−0.596530	−	0.802590i	$-0.703454\pi$
$503$	−26.7576	−1.19306	−0.596530	+	0.802590i	$0.703454\pi$
$504$	0	0
$505$	9.91993	0.441431
$506$	−22.1571	−0.985002
$507$	0	0
$508$	2.30492	0.102264
$509$	23.6043	1.04624	0.523120	−	0.852259i	$-0.324768\pi$
$509$	23.6043	1.04624	0.523120	+	0.852259i	$0.324768\pi$
$510$	0	0
$511$	8.11920	0.359172
$512$	−30.9003	−1.36561
$513$	0	0
$514$	−32.4051	−1.42933
$515$	−18.7697	−0.827090
$516$	0	0
$517$	−36.9848	−1.62659
$518$	18.6577	0.819770
$519$	0	0
$520$	−0.134326	−0.00589061
$521$	−17.3487	−0.760059	−0.380030	−	0.924974i	$-0.624086\pi$
$521$	−17.3487	−0.760059	−0.380030	+	0.924974i	$0.624086\pi$
$522$	0	0
$523$	30.0204	1.31270	0.656350	−	0.754457i	$-0.272100\pi$
$523$	30.0204	1.31270	0.656350	+	0.754457i	$0.272100\pi$
$524$	−33.2913	−1.45434
$525$	0	0
$526$	22.3881	0.976170
$527$	−0.191973	−0.00836249
$528$	0	0
$529$	−18.2113	−0.791797
$530$	8.10431	0.352029
$531$	0	0
$532$	−3.58449	−0.155407
$533$	−0.176035	−0.00762490
$534$	0	0
$535$	−5.83018	−0.252061
$536$	−2.22905	−0.0962805
$537$	0	0
$538$	13.7830	0.594227
$539$	−4.88004	−0.210198
$540$	0	0
$541$	32.3908	1.39259	0.696295	−	0.717755i	$-0.254830\pi$
$541$	32.3908	1.39259	0.696295	+	0.717755i	$0.254830\pi$
$542$	−12.5660	−0.539755
$543$	0	0
$544$	−2.21439	−0.0949412
$545$	−5.55238	−0.237838
$546$	0	0
$547$	30.6502	1.31051	0.655253	−	0.755409i	$-0.272562\pi$
$547$	30.6502	1.31051	0.655253	+	0.755409i	$0.272562\pi$
$548$	−11.1227	−0.475136
$549$	0	0
$550$	−38.1075	−1.62491
$551$	3.92906	0.167383
$552$	0	0
$553$	12.2109	0.519260
$554$	−33.8996	−1.44026
$555$	0	0
$556$	2.55789	0.108479
$557$	16.5560	0.701502	0.350751	−	0.936469i	$-0.385926\pi$
$557$	16.5560	0.701502	0.350751	+	0.936469i	$0.385926\pi$
$558$	0	0
$559$	1.65302	0.0699151
$560$	−3.66623	−0.154927
$561$	0	0
$562$	5.84778	0.246674
$563$	−3.28926	−0.138626	−0.0693130	−	0.997595i	$-0.522081\pi$
$563$	−3.28926	−0.138626	−0.0693130	+	0.997595i	$0.522081\pi$
$564$	0	0
$565$	8.13797	0.342367
$566$	−19.7289	−0.829267
$567$	0	0
$568$	−8.85075	−0.371369
$569$	27.3323	1.14583	0.572914	−	0.819616i	$-0.305813\pi$
$569$	27.3323	1.14583	0.572914	+	0.819616i	$0.305813\pi$
$570$	0	0
$571$	−17.4246	−0.729196	−0.364598	−	0.931165i	$-0.618794\pi$
$571$	−17.4246	−0.729196	−0.364598	+	0.931165i	$0.618794\pi$
$572$	2.14782	0.0898048
$573$	0	0
$574$	1.91276	0.0798372
$575$	8.23593	0.343462
$576$	0	0
$577$	−2.17562	−0.0905724	−0.0452862	−	0.998974i	$-0.514420\pi$
$577$	−2.17562	−0.0905724	−0.0452862	+	0.998974i	$0.514420\pi$
$578$	35.1173	1.46069
$579$	0	0
$580$	−6.47512	−0.268865
$581$	9.50363	0.394277
$582$	0	0
$583$	−17.1427	−0.709979
$584$	5.13664	0.212556
$585$	0	0
$586$	39.4494	1.62964
$587$	4.58747	0.189345	0.0946726	−	0.995508i	$-0.469820\pi$
$587$	4.58747	0.189345	0.0946726	+	0.995508i	$0.469820\pi$
$588$	0	0
$589$	1.09292	0.0450329
$590$	7.97481	0.328318
$591$	0	0
$592$	−29.6496	−1.21859
$593$	22.8809	0.939605	0.469803	−	0.882771i	$-0.344325\pi$
$593$	22.8809	0.939605	0.469803	+	0.882771i	$0.344325\pi$
$594$	0	0
$595$	−0.303740	−0.0124521
$596$	−6.76233	−0.276996
$597$	0	0
$598$	−0.866979	−0.0354534
$599$	−19.7022	−0.805008	−0.402504	−	0.915418i	$-0.631860\pi$
$599$	−19.7022	−0.805008	−0.402504	+	0.915418i	$0.631860\pi$
$600$	0	0
$601$	−1.09291	−0.0445807	−0.0222903	−	0.999752i	$-0.507096\pi$
$601$	−1.09291	−0.0445807	−0.0222903	+	0.999752i	$0.507096\pi$
$602$	−17.9614	−0.732052
$603$	0	0
$604$	32.2439	1.31199
$605$	−14.2491	−0.579307
$606$	0	0
$607$	17.4518	0.708345	0.354173	−	0.935180i	$-0.384763\pi$
$607$	17.4518	0.708345	0.354173	+	0.935180i	$0.384763\pi$
$608$	12.6067	0.511268
$609$	0	0
$610$	20.6075	0.834374
$611$	−1.44717	−0.0585463
$612$	0	0
$613$	−36.8071	−1.48662	−0.743312	−	0.668945i	$-0.766746\pi$
$613$	−36.8071	−1.48662	−0.743312	+	0.668945i	$0.766746\pi$
$614$	11.1088	0.448314
$615$	0	0
$616$	−3.08737	−0.124394
$617$	18.7716	0.755716	0.377858	−	0.925864i	$-0.376661\pi$
$617$	18.7716	0.755716	0.377858	+	0.925864i	$0.376661\pi$
$618$	0	0
$619$	10.9189	0.438868	0.219434	−	0.975627i	$-0.429579\pi$
$619$	10.9189	0.438868	0.219434	+	0.975627i	$0.429579\pi$
$620$	−1.80114	−0.0723355
$621$	0	0
$622$	−42.9722	−1.72303
$623$	−5.13628	−0.205781
$624$	0	0
$625$	7.98291	0.319317
$626$	−22.4968	−0.899151
$627$	0	0
$628$	−9.45611	−0.377340
$629$	−2.45641	−0.0979433
$630$	0	0
$631$	34.3955	1.36926	0.684632	−	0.728888i	$-0.259963\pi$
$631$	34.3955	1.36926	0.684632	+	0.728888i	$0.259963\pi$
$632$	7.72527	0.307295
$633$	0	0
$634$	−4.28711	−0.170263
$635$	−1.11193	−0.0441255
$636$	0	0
$637$	−0.190950	−0.00756571
$638$	25.5812	1.01277
$639$	0	0
$640$	−5.56232	−0.219870
$641$	−19.2398	−0.759925	−0.379962	−	0.925002i	$-0.624063\pi$
$641$	−19.2398	−0.759925	−0.379962	+	0.925002i	$0.624063\pi$
$642$	0	0
$643$	−27.3501	−1.07858	−0.539291	−	0.842119i	$-0.681308\pi$
$643$	−27.3501	−1.07858	−0.539291	+	0.842119i	$0.681308\pi$
$644$	5.04386	0.198756
$645$	0	0
$646$	0.881414	0.0346788
$647$	−16.2328	−0.638178	−0.319089	−	0.947725i	$-0.603377\pi$
$647$	−16.2328	−0.638178	−0.319089	+	0.947725i	$0.603377\pi$
$648$	0	0
$649$	−16.8688	−0.662159
$650$	−1.49110	−0.0584858
$651$	0	0
$652$	−46.7374	−1.83038
$653$	6.78390	0.265474	0.132737	−	0.991151i	$-0.457623\pi$
$653$	6.78390	0.265474	0.132737	+	0.991151i	$0.457623\pi$
$654$	0	0
$655$	16.0602	0.627525
$656$	−3.03964	−0.118678
$657$	0	0
$658$	15.7247	0.613014
$659$	−2.72500	−0.106151	−0.0530756	−	0.998590i	$-0.516902\pi$
$659$	−2.72500	−0.106151	−0.0530756	+	0.998590i	$0.516902\pi$
$660$	0	0
$661$	−22.2032	−0.863604	−0.431802	−	0.901968i	$-0.642122\pi$
$661$	−22.2032	−0.863604	−0.431802	+	0.901968i	$0.642122\pi$
$662$	−21.7369	−0.844830
$663$	0	0
$664$	6.01251	0.233331
$665$	1.72921	0.0670560
$666$	0	0
$667$	−5.52870	−0.214072
$668$	−47.0813	−1.82163
$669$	0	0
$670$	8.12855	0.314033
$671$	−43.5903	−1.68278
$672$	0	0
$673$	−18.8141	−0.725229	−0.362615	−	0.931939i	$-0.618116\pi$
$673$	−18.8141	−0.725229	−0.362615	+	0.931939i	$0.618116\pi$
$674$	21.3306	0.821625
$675$	0	0
$676$	−29.8799	−1.14923
$677$	5.75451	0.221164	0.110582	−	0.993867i	$-0.464729\pi$
$677$	5.75451	0.221164	0.110582	+	0.993867i	$0.464729\pi$
$678$	0	0
$679$	−11.6865	−0.448485
$680$	−0.192162	−0.00736909
$681$	0	0
$682$	7.11575	0.272476
$683$	−38.0212	−1.45484	−0.727420	−	0.686193i	$-0.759281\pi$
$683$	−38.0212	−1.45484	−0.727420	+	0.686193i	$0.759281\pi$
$684$	0	0
$685$	5.36574	0.205014
$686$	2.07483	0.0792174
$687$	0	0
$688$	28.5431	1.08820
$689$	−0.670774	−0.0255545
$690$	0	0
$691$	−37.4394	−1.42426	−0.712131	−	0.702047i	$-0.752270\pi$
$691$	−37.4394	−1.42426	−0.712131	+	0.702047i	$0.752270\pi$
$692$	25.1813	0.957248
$693$	0	0
$694$	−9.43493	−0.358145
$695$	−1.23396	−0.0468069
$696$	0	0
$697$	−0.251828	−0.00953867
$698$	−74.7258	−2.82841
$699$	0	0
$700$	8.67483	0.327878
$701$	10.9478	0.413494	0.206747	−	0.978394i	$-0.433712\pi$
$701$	10.9478	0.413494	0.206747	+	0.978394i	$0.433712\pi$
$702$	0	0
$703$	13.9845	0.527435
$704$	49.8986	1.88062
$705$	0	0
$706$	−54.5621	−2.05347
$707$	8.92138	0.335523
$708$	0	0
$709$	−0.472856	−0.0177585	−0.00887925	−	0.999961i	$-0.502826\pi$
$709$	−0.472856	−0.0177585	−0.00887925	+	0.999961i	$0.502826\pi$
$710$	32.2755	1.21128
$711$	0	0
$712$	−3.24949	−0.121780
$713$	−1.53788	−0.0575941
$714$	0	0
$715$	−1.03614	−0.0387494
$716$	−19.2632	−0.719899
$717$	0	0
$718$	28.1226	1.04953
$719$	11.4440	0.426788	0.213394	−	0.976966i	$-0.431548\pi$
$719$	11.4440	0.426788	0.213394	+	0.976966i	$0.431548\pi$
$720$	0	0
$721$	−16.8803	−0.628655
$722$	34.4038	1.28038
$723$	0	0
$724$	50.9614	1.89397
$725$	−9.50871	−0.353145
$726$	0	0
$727$	−2.02782	−0.0752077	−0.0376038	−	0.999293i	$-0.511972\pi$
$727$	−2.02782	−0.0752077	−0.0376038	+	0.999293i	$0.511972\pi$
$728$	−0.120805	−0.00447734
$729$	0	0
$730$	−18.7315	−0.693283
$731$	2.36474	0.0874631
$732$	0	0
$733$	−37.1334	−1.37155	−0.685777	−	0.727811i	$-0.740538\pi$
$733$	−37.1334	−1.37155	−0.685777	+	0.727811i	$0.740538\pi$
$734$	52.7732	1.94789
$735$	0	0
$736$	−17.7393	−0.653878
$737$	−17.1940	−0.633350
$738$	0	0
$739$	−22.4807	−0.826966	−0.413483	−	0.910512i	$-0.635688\pi$
$739$	−22.4807	−0.826966	−0.413483	+	0.910512i	$0.635688\pi$
$740$	−23.0466	−0.847209
$741$	0	0
$742$	7.28852	0.267570
$743$	−41.7849	−1.53294	−0.766469	−	0.642281i	$-0.777988\pi$
$743$	−41.7849	−1.53294	−0.766469	+	0.642281i	$0.777988\pi$
$744$	0	0
$745$	3.26225	0.119520
$746$	−9.90877	−0.362786
$747$	0	0
$748$	3.07259	0.112345
$749$	−5.24332	−0.191587
$750$	0	0
$751$	−36.8889	−1.34610	−0.673048	−	0.739599i	$-0.735015\pi$
$751$	−36.8889	−1.34610	−0.673048	+	0.739599i	$0.735015\pi$
$752$	−24.9887	−0.911246
$753$	0	0
$754$	1.00096	0.0364529
$755$	−15.5549	−0.566102
$756$	0	0
$757$	53.8923	1.95875	0.979374	−	0.202055i	$-0.0647620\pi$
$757$	53.8923	1.95875	0.979374	+	0.202055i	$0.0647620\pi$
$758$	33.8860	1.23079
$759$	0	0
$760$	1.09399	0.0396833
$761$	−25.3486	−0.918884	−0.459442	−	0.888208i	$-0.651951\pi$
$761$	−25.3486	−0.918884	−0.459442	+	0.888208i	$0.651951\pi$
$762$	0	0
$763$	−4.99347	−0.180776
$764$	2.73687	0.0990165
$765$	0	0
$766$	−32.3225	−1.16786
$767$	−0.660056	−0.0238332
$768$	0	0
$769$	−51.6702	−1.86328	−0.931638	−	0.363388i	$-0.881620\pi$
$769$	−51.6702	−1.86328	−0.931638	+	0.363388i	$0.881620\pi$
$770$	11.2585	0.405729
$771$	0	0
$772$	−47.8075	−1.72063
$773$	−16.2029	−0.582776	−0.291388	−	0.956605i	$-0.594117\pi$
$773$	−16.2029	−0.582776	−0.291388	+	0.956605i	$0.594117\pi$
$774$	0	0
$775$	−2.64497	−0.0950101
$776$	−7.39348	−0.265411
$777$	0	0
$778$	55.7182	1.99759
$779$	1.43367	0.0513667
$780$	0	0
$781$	−68.2711	−2.44293
$782$	−1.24027	−0.0443519
$783$	0	0
$784$	−3.29719	−0.117757
$785$	4.56177	0.162816
$786$	0	0
$787$	24.6797	0.879735	0.439867	−	0.898063i	$-0.355025\pi$
$787$	24.6797	0.879735	0.439867	+	0.898063i	$0.355025\pi$
$788$	25.1904	0.897369
$789$	0	0
$790$	−28.1713	−1.00229
$791$	7.31880	0.260226
$792$	0	0
$793$	−1.70563	−0.0605689
$794$	2.04594	0.0726078
$795$	0	0
$796$	28.7782	1.02002
$797$	−41.1052	−1.45602	−0.728011	−	0.685565i	$-0.759555\pi$
$797$	−41.1052	−1.45602	−0.728011	+	0.685565i	$0.759555\pi$
$798$	0	0
$799$	−2.07027	−0.0732408
$800$	−30.5094	−1.07867
$801$	0	0
$802$	9.42288	0.332734
$803$	39.6220	1.39823
$804$	0	0
$805$	−2.43323	−0.0857601
$806$	0.278431	0.00980730
$807$	0	0
$808$	5.64415	0.198560
$809$	40.3870	1.41993	0.709966	−	0.704236i	$-0.248710\pi$
$809$	40.3870	1.41993	0.709966	+	0.704236i	$0.248710\pi$
$810$	0	0
$811$	−41.7206	−1.46501	−0.732504	−	0.680763i	$-0.761648\pi$
$811$	−41.7206	−1.46501	−0.732504	+	0.680763i	$0.761648\pi$
$812$	−5.82334	−0.204359
$813$	0	0
$814$	91.0500	3.19130
$815$	22.5468	0.789781
$816$	0	0
$817$	−13.4626	−0.470998
$818$	44.9928	1.57313
$819$	0	0
$820$	−2.36271	−0.0825094
$821$	−3.27664	−0.114356	−0.0571778	−	0.998364i	$-0.518210\pi$
$821$	−3.27664	−0.114356	−0.0571778	+	0.998364i	$0.518210\pi$
$822$	0	0
$823$	−45.8404	−1.59789	−0.798947	−	0.601401i	$-0.794609\pi$
$823$	−45.8404	−1.59789	−0.798947	+	0.601401i	$0.794609\pi$
$824$	−10.6794	−0.372034
$825$	0	0
$826$	7.17206	0.249548
$827$	4.64256	0.161437	0.0807187	−	0.996737i	$-0.474278\pi$
$827$	4.64256	0.161437	0.0807187	+	0.996737i	$0.474278\pi$
$828$	0	0
$829$	−45.5084	−1.58057	−0.790287	−	0.612737i	$-0.790068\pi$
$829$	−45.5084	−1.58057	−0.790287	+	0.612737i	$0.790068\pi$
$830$	−21.9254	−0.761043
$831$	0	0
$832$	1.95247	0.0676897
$833$	−0.273165	−0.00946462
$834$	0	0
$835$	22.7127	0.786006
$836$	−17.4924	−0.604989
$837$	0	0
$838$	7.58571	0.262044
$839$	4.60975	0.159146	0.0795731	−	0.996829i	$-0.474644\pi$
$839$	4.60975	0.159146	0.0795731	+	0.996829i	$0.474644\pi$
$840$	0	0
$841$	−22.6169	−0.779893
$842$	27.9415	0.962928
$843$	0	0
$844$	12.4678	0.429160
$845$	14.4145	0.495874
$846$	0	0
$847$	−12.8147	−0.440320
$848$	−11.5825	−0.397743
$849$	0	0
$850$	−2.13311	−0.0731651
$851$	−19.6780	−0.674554
$852$	0	0
$853$	−8.75593	−0.299797	−0.149899	−	0.988701i	$-0.547895\pi$
$853$	−8.75593	−0.299797	−0.149899	+	0.988701i	$0.547895\pi$
$854$	18.5332	0.634191
$855$	0	0
$856$	−3.31720	−0.113380
$857$	11.6570	0.398197	0.199098	−	0.979980i	$-0.436199\pi$
$857$	11.6570	0.398197	0.199098	+	0.979980i	$0.436199\pi$
$858$	0	0
$859$	−12.4680	−0.425403	−0.212701	−	0.977117i	$-0.568226\pi$
$859$	−12.4680	−0.425403	−0.212701	+	0.977117i	$0.568226\pi$
$860$	22.1865	0.756555
$861$	0	0
$862$	40.4584	1.37802
$863$	48.3414	1.64556	0.822781	−	0.568359i	$-0.192422\pi$
$863$	48.3414	1.64556	0.822781	+	0.568359i	$0.192422\pi$
$864$	0	0
$865$	−12.1478	−0.413038
$866$	−27.9979	−0.951408
$867$	0	0
$868$	−1.61984	−0.0549808
$869$	59.5896	2.02144
$870$	0	0
$871$	−0.672781	−0.0227963
$872$	−3.15914	−0.106982
$873$	0	0
$874$	7.06092	0.238839
$875$	−9.74450	−0.329424
$876$	0	0
$877$	−14.1671	−0.478389	−0.239195	−	0.970972i	$-0.576883\pi$
$877$	−14.1671	−0.478389	−0.239195	+	0.970972i	$0.576883\pi$
$878$	34.2499	1.15588
$879$	0	0
$880$	−17.8913	−0.603117
$881$	−9.46264	−0.318804	−0.159402	−	0.987214i	$-0.550957\pi$
$881$	−9.46264	−0.318804	−0.159402	+	0.987214i	$0.550957\pi$
$882$	0	0
$883$	−3.07548	−0.103498	−0.0517490	−	0.998660i	$-0.516480\pi$
$883$	−3.07548	−0.103498	−0.0517490	+	0.998660i	$0.516480\pi$
$884$	0.120227	0.00404366
$885$	0	0
$886$	48.3994	1.62601
$887$	23.5275	0.789975	0.394988	−	0.918686i	$-0.370749\pi$
$887$	23.5275	0.789975	0.394988	+	0.918686i	$0.370749\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	11.8497	0.397203
$891$	0	0
$892$	−9.90371	−0.331601
$893$	11.7862	0.394409
$894$	0	0
$895$	9.29285	0.310626
$896$	−5.00241	−0.167119
$897$	0	0
$898$	0.668824	0.0223189
$899$	1.77554	0.0592177
$900$	0	0
$901$	−0.959583	−0.0319684
$902$	9.33435	0.310800
$903$	0	0
$904$	4.63026	0.154000
$905$	−24.5846	−0.817218
$906$	0	0
$907$	−34.7093	−1.15250	−0.576251	−	0.817273i	$-0.695485\pi$
$907$	−34.7093	−1.15250	−0.576251	+	0.817273i	$0.695485\pi$
$908$	−31.3203	−1.03940
$909$	0	0
$910$	0.440533	0.0146035
$911$	12.6282	0.418390	0.209195	−	0.977874i	$-0.432916\pi$
$911$	12.6282	0.418390	0.209195	+	0.977874i	$0.432916\pi$
$912$	0	0
$913$	46.3781	1.53489
$914$	4.90230	0.162154
$915$	0	0
$916$	14.4838	0.478558
$917$	14.4436	0.476970
$918$	0	0
$919$	−56.8401	−1.87498	−0.937490	−	0.348012i	$-0.886857\pi$
$919$	−56.8401	−1.87498	−0.937490	+	0.348012i	$0.886857\pi$
$920$	−1.53939	−0.0507523
$921$	0	0
$922$	63.5555	2.09309
$923$	−2.67136	−0.0879290
$924$	0	0
$925$	−33.8439	−1.11278
$926$	−49.3665	−1.62228
$927$	0	0
$928$	20.4807	0.672312
$929$	10.2646	0.336771	0.168385	−	0.985721i	$-0.446145\pi$
$929$	10.2646	0.336771	0.168385	+	0.985721i	$0.446145\pi$
$930$	0	0
$931$	1.55515	0.0509679
$932$	30.8990	1.01213
$933$	0	0
$934$	−45.4772	−1.48806
$935$	−1.48226	−0.0484751
$936$	0	0
$937$	−47.2431	−1.54336	−0.771682	−	0.636009i	$-0.780584\pi$
$937$	−47.2431	−1.54336	−0.771682	+	0.636009i	$0.780584\pi$
$938$	7.31033	0.238691
$939$	0	0
$940$	−19.4237	−0.633532
$941$	−17.0143	−0.554650	−0.277325	−	0.960776i	$-0.589448\pi$
$941$	−17.0143	−0.554650	−0.277325	+	0.960776i	$0.589448\pi$
$942$	0	0
$943$	−2.01737	−0.0656946
$944$	−11.3974	−0.370953
$945$	0	0
$946$	−87.6523	−2.84982
$947$	−18.0648	−0.587026	−0.293513	−	0.955955i	$-0.594824\pi$
$947$	−18.0648	−0.587026	−0.293513	+	0.955955i	$0.594824\pi$
$948$	0	0
$949$	1.55036	0.0503268
$950$	12.1439	0.394002
$951$	0	0
$952$	−0.172819	−0.00560110
$953$	43.1274	1.39703	0.698517	−	0.715594i	$-0.253844\pi$
$953$	43.1274	1.39703	0.698517	+	0.715594i	$0.253844\pi$
$954$	0	0
$955$	−1.32031	−0.0427242
$956$	10.8517	0.350969
$957$	0	0
$958$	−69.4423	−2.24358
$959$	4.82562	0.155827
$960$	0	0
$961$	−30.5061	−0.984068
$962$	3.56268	0.114865
$963$	0	0
$964$	−39.0663	−1.25824
$965$	23.0631	0.742426
$966$	0	0
$967$	−24.9390	−0.801983	−0.400991	−	0.916082i	$-0.631334\pi$
$967$	−24.9390	−0.801983	−0.400991	+	0.916082i	$0.631334\pi$
$968$	−8.10730	−0.260578
$969$	0	0
$970$	26.9614	0.865677
$971$	−59.4545	−1.90798	−0.953992	−	0.299831i	$-0.903070\pi$
$971$	−59.4545	−1.90798	−0.953992	+	0.299831i	$0.903070\pi$
$972$	0	0
$973$	−1.10975	−0.0355770
$974$	80.4502	2.57779
$975$	0	0
$976$	−29.4517	−0.942726
$977$	21.5671	0.689991	0.344996	−	0.938604i	$-0.387880\pi$
$977$	21.5671	0.689991	0.344996	+	0.938604i	$0.387880\pi$
$978$	0	0
$979$	−25.0652	−0.801088
$980$	−2.56290	−0.0818689
$981$	0	0
$982$	53.7721	1.71594
$983$	−17.6792	−0.563880	−0.281940	−	0.959432i	$-0.590978\pi$
$983$	−17.6792	−0.563880	−0.281940	+	0.959432i	$0.590978\pi$
$984$	0	0
$985$	−12.1522	−0.387201
$986$	1.43194	0.0456022
$987$	0	0
$988$	−0.684458	−0.0217755
$989$	18.9437	0.602375
$990$	0	0
$991$	40.2801	1.27954	0.639770	−	0.768566i	$-0.279030\pi$
$991$	40.2801	1.27954	0.639770	+	0.768566i	$0.279030\pi$
$992$	5.69697	0.180879
$993$	0	0
$994$	29.0266	0.920668
$995$	−13.8830	−0.440122
$996$	0	0
$997$	10.4111	0.329721	0.164861	−	0.986317i	$-0.447283\pi$
$997$	10.4111	0.329721	0.164861	+	0.986317i	$0.447283\pi$
$998$	−66.7576	−2.11318
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.q.1.3		15
3.2	odd	2		889.2.a.b.1.13	✓	15
21.20	even	2		6223.2.a.j.1.13		15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
889.2.a.b.1.13	✓	15	3.2	odd	2
6223.2.a.j.1.13		15	21.20	even	2
8001.2.a.q.1.3		15	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)

\(f(q)\)	\(=\)	\(q-2.07483 q^{2} +2.30492 q^{4} -1.11193 q^{5} -1.00000 q^{7} -0.632654 q^{8} +O(q^{10})\)	\(q-2.07483 q^{2} +2.30492 q^{4} -1.11193 q^{5} -1.00000 q^{7} -0.632654 q^{8} +2.30706 q^{10} -4.88004 q^{11} -0.190950 q^{13} +2.07483 q^{14} -3.29719 q^{16} -0.273165 q^{17} +1.55515 q^{19} -2.56290 q^{20} +10.1252 q^{22} -2.18830 q^{23} -3.76362 q^{25} +0.396188 q^{26} -2.30492 q^{28} +2.52648 q^{29} +0.702773 q^{31} +8.10641 q^{32} +0.566772 q^{34} +1.11193 q^{35} +8.99238 q^{37} -3.22667 q^{38} +0.703465 q^{40} +0.921889 q^{41} -8.65681 q^{43} -11.2481 q^{44} +4.54035 q^{46} +7.57880 q^{47} +1.00000 q^{49} +7.80887 q^{50} -0.440124 q^{52} +3.51283 q^{53} +5.42624 q^{55} +0.632654 q^{56} -5.24202 q^{58} +3.45670 q^{59} +8.93237 q^{61} -1.45814 q^{62} -10.2250 q^{64} +0.212322 q^{65} +3.52334 q^{67} -0.629624 q^{68} -2.30706 q^{70} +13.9899 q^{71} -8.11920 q^{73} -18.6577 q^{74} +3.58449 q^{76} +4.88004 q^{77} -12.2109 q^{79} +3.66623 q^{80} -1.91276 q^{82} -9.50363 q^{83} +0.303740 q^{85} +17.9614 q^{86} +3.08737 q^{88} +5.13628 q^{89} +0.190950 q^{91} -5.04386 q^{92} -15.7247 q^{94} -1.72921 q^{95} +11.6865 q^{97} -2.07483 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 14 q^{4} - 7 q^{5} - 15 q^{7}+O(q^{10}) \) 15 * q + 14 * q^4 - 7 * q^5 - 15 * q^7	\( 15 q + 14 q^{4} - 7 q^{5} - 15 q^{7} + 10 q^{10} - 14 q^{11} + 6 q^{13} + 20 q^{16} - 10 q^{17} + 13 q^{19} - 8 q^{20} - 11 q^{22} - 15 q^{23} - 22 q^{26} - 14 q^{28} - 16 q^{29} + 22 q^{31} - 15 q^{34} + 7 q^{35} - 14 q^{37} + 6 q^{38} + 22 q^{40} - 19 q^{41} - q^{43} - 25 q^{44} - 28 q^{46} - 49 q^{47} + 15 q^{49} - 24 q^{50} - 17 q^{52} + 28 q^{53} + 39 q^{55} - 10 q^{58} - 43 q^{59} + 27 q^{61} - 14 q^{62} + 18 q^{64} + 8 q^{65} + 3 q^{67} - 13 q^{68} - 10 q^{70} - 55 q^{71} - 3 q^{73} + 12 q^{74} - 20 q^{76} + 14 q^{77} + 18 q^{79} - 29 q^{80} + 14 q^{82} - 17 q^{83} + 7 q^{85} - 4 q^{86} - 114 q^{88} - 36 q^{89} - 6 q^{91} - 45 q^{92} - 15 q^{94} - 59 q^{95} - 2 q^{97}+O(q^{100}) \) 15 * q + 14 * q^4 - 7 * q^5 - 15 * q^7 + 10 * q^10 - 14 * q^11 + 6 * q^13 + 20 * q^16 - 10 * q^17 + 13 * q^19 - 8 * q^20 - 11 * q^22 - 15 * q^23 - 22 * q^26 - 14 * q^28 - 16 * q^29 + 22 * q^31 - 15 * q^34 + 7 * q^35 - 14 * q^37 + 6 * q^38 + 22 * q^40 - 19 * q^41 - q^43 - 25 * q^44 - 28 * q^46 - 49 * q^47 + 15 * q^49 - 24 * q^50 - 17 * q^52 + 28 * q^53 + 39 * q^55 - 10 * q^58 - 43 * q^59 + 27 * q^61 - 14 * q^62 + 18 * q^64 + 8 * q^65 + 3 * q^67 - 13 * q^68 - 10 * q^70 - 55 * q^71 - 3 * q^73 + 12 * q^74 - 20 * q^76 + 14 * q^77 + 18 * q^79 - 29 * q^80 + 14 * q^82 - 17 * q^83 + 7 * q^85 - 4 * q^86 - 114 * q^88 - 36 * q^89 - 6 * q^91 - 45 * q^92 - 15 * q^94 - 59 * q^95 - 2 * q^97

Introduction

L-functions

Modular forms

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Database

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