LMFDB - Embedded newform 8001.2.a.m.1.9

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:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 2 x^{10} - 15 x^{9} + 25 x^{8} + 88 x^{7} - 112 x^{6} - 247 x^{5} + 215 x^{4} + 313 x^{3} + \cdots + 57 $ x^11 - 2x^10 - 15x^9 + 25x^8 + 88x^7 - 112x^6 - 247x^5 + 215x^4 + 313x^3 - 170x^2 - 121x + 57
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
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			1.9
Root			$2.14674$ 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)

$f(q)$	$=$	$q+2.14674 q^{2} +2.60851 q^{4} -3.76087 q^{5} -1.00000 q^{7} +1.30632 q^{8} +O(q^{10})$	\(q+2.14674 q^{2} +2.60851 q^{4} -3.76087 q^{5} -1.00000 q^{7} +1.30632 q^{8} -8.07362 q^{10} +4.50622 q^{11} -6.10521 q^{13} -2.14674 q^{14} -2.41269 q^{16} +5.55094 q^{17} +8.42678 q^{19} -9.81027 q^{20} +9.67370 q^{22} +0.794388 q^{23} +9.14413 q^{25} -13.1063 q^{26} -2.60851 q^{28} -1.41175 q^{29} -8.89305 q^{31} -7.79207 q^{32} +11.9164 q^{34} +3.76087 q^{35} +6.87494 q^{37} +18.0901 q^{38} -4.91289 q^{40} +5.52728 q^{41} -8.12045 q^{43} +11.7545 q^{44} +1.70535 q^{46} -5.95533 q^{47} +1.00000 q^{49} +19.6301 q^{50} -15.9255 q^{52} +11.7025 q^{53} -16.9473 q^{55} -1.30632 q^{56} -3.03067 q^{58} -5.35689 q^{59} -4.45521 q^{61} -19.0911 q^{62} -11.9022 q^{64} +22.9609 q^{65} -13.9075 q^{67} +14.4797 q^{68} +8.07362 q^{70} -8.38797 q^{71} -12.5654 q^{73} +14.7587 q^{74} +21.9813 q^{76} -4.50622 q^{77} -9.28601 q^{79} +9.07382 q^{80} +11.8657 q^{82} -12.3129 q^{83} -20.8764 q^{85} -17.4325 q^{86} +5.88656 q^{88} -0.471303 q^{89} +6.10521 q^{91} +2.07217 q^{92} -12.7846 q^{94} -31.6920 q^{95} -1.95818 q^{97} +2.14674 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q + 2 q^{2} + 12 q^{4} - q^{5} - 11 q^{7} + 15 q^{8}+O(q^{10}) $ 11 * q + 2 * q^2 + 12 * q^4 - q^5 - 11 * q^7 + 15 * q^8	$ 11 q + 2 q^{2} + 12 q^{4} - q^{5} - 11 q^{7} + 15 q^{8} - 12 q^{10} + 7 q^{11} - 24 q^{13} - 2 q^{14} - 6 q^{16} + 15 q^{17} - 19 q^{19} - 3 q^{20} - 3 q^{22} + 11 q^{23} + 10 q^{25} - 10 q^{26} - 12 q^{28} + 10 q^{29} - 20 q^{31} + 27 q^{32} - 9 q^{34} + q^{35} - 22 q^{37} - 8 q^{38} - 29 q^{40} - 9 q^{41} - 17 q^{43} + 9 q^{44} - 18 q^{46} + 7 q^{47} + 11 q^{49} + 47 q^{50} - 66 q^{52} + 28 q^{53} - 24 q^{55} - 15 q^{56} - 39 q^{58} - 35 q^{59} - 6 q^{61} - 18 q^{62} + 11 q^{64} + 43 q^{65} - 22 q^{67} + 12 q^{68} + 12 q^{70} + 22 q^{71} - 29 q^{73} - 14 q^{74} + 10 q^{76} - 7 q^{77} - 20 q^{79} - 66 q^{80} - 24 q^{82} - 17 q^{83} - 50 q^{85} + 12 q^{86} + 2 q^{88} + q^{89} + 24 q^{91} + 22 q^{92} + q^{94} - 10 q^{95} - 45 q^{97} + 2 q^{98}+O(q^{100}) $ 11 * q + 2 * q^2 + 12 * q^4 - q^5 - 11 * q^7 + 15 * q^8 - 12 * q^10 + 7 * q^11 - 24 * q^13 - 2 * q^14 - 6 * q^16 + 15 * q^17 - 19 * q^19 - 3 * q^20 - 3 * q^22 + 11 * q^23 + 10 * q^25 - 10 * q^26 - 12 * q^28 + 10 * q^29 - 20 * q^31 + 27 * q^32 - 9 * q^34 + q^35 - 22 * q^37 - 8 * q^38 - 29 * q^40 - 9 * q^41 - 17 * q^43 + 9 * q^44 - 18 * q^46 + 7 * q^47 + 11 * q^49 + 47 * q^50 - 66 * q^52 + 28 * q^53 - 24 * q^55 - 15 * q^56 - 39 * q^58 - 35 * q^59 - 6 * q^61 - 18 * q^62 + 11 * q^64 + 43 * q^65 - 22 * q^67 + 12 * q^68 + 12 * q^70 + 22 * q^71 - 29 * q^73 - 14 * q^74 + 10 * q^76 - 7 * q^77 - 20 * q^79 - 66 * q^80 - 24 * q^82 - 17 * q^83 - 50 * q^85 + 12 * q^86 + 2 * q^88 + q^89 + 24 * q^91 + 22 * q^92 + q^94 - 10 * q^95 - 45 * q^97 + 2 * 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 $

$2$	2.14674	1.51798	0.758989	−	0.651104i	$-0.225694\pi$
$2$	2.14674	1.51798	0.758989	+	0.651104i	$0.225694\pi$
$3$	0	0
$3$	0	0
$4$	2.60851	1.30426
$5$	−3.76087	−1.68191	−0.840956	−	0.541104i	$-0.818007\pi$
$5$	−3.76087	−1.68191	−0.840956	+	0.541104i	$0.818007\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.30632	0.461853
$9$	0	0
$10$	−8.07362	−2.55310
$11$	4.50622	1.35868	0.679338	−	0.733826i	$-0.262267\pi$
$11$	4.50622	1.35868	0.679338	+	0.733826i	$0.262267\pi$
$12$	0	0
$13$	−6.10521	−1.69328	−0.846640	−	0.532167i	$-0.821378\pi$
$13$	−6.10521	−1.69328	−0.846640	+	0.532167i	$0.821378\pi$
$14$	−2.14674	−0.573742
$15$	0	0
$16$	−2.41269	−0.603173
$17$	5.55094	1.34630	0.673150	−	0.739506i	$-0.264941\pi$
$17$	5.55094	1.34630	0.673150	+	0.739506i	$0.264941\pi$
$18$	0	0
$19$	8.42678	1.93324	0.966618	−	0.256223i	$-0.0824781\pi$
$19$	8.42678	1.93324	0.966618	+	0.256223i	$0.0824781\pi$
$20$	−9.81027	−2.19364
$21$	0	0
$22$	9.67370	2.06244
$23$	0.794388	0.165641	0.0828207	−	0.996564i	$-0.473607\pi$
$23$	0.794388	0.165641	0.0828207	+	0.996564i	$0.473607\pi$
$24$	0	0
$25$	9.14413	1.82883
$26$	−13.1063	−2.57036
$27$	0	0
$28$	−2.60851	−0.492962
$29$	−1.41175	−0.262156	−0.131078	−	0.991372i	$-0.541844\pi$
$29$	−1.41175	−0.262156	−0.131078	+	0.991372i	$0.541844\pi$
$30$	0	0
$31$	−8.89305	−1.59724	−0.798620	−	0.601836i	$-0.794436\pi$
$31$	−8.89305	−1.59724	−0.798620	+	0.601836i	$0.794436\pi$
$32$	−7.79207	−1.37746
$33$	0	0
$34$	11.9164	2.04365
$35$	3.76087	0.635703
$36$	0	0
$37$	6.87494	1.13023	0.565117	−	0.825011i	$-0.308831\pi$
$37$	6.87494	1.13023	0.565117	+	0.825011i	$0.308831\pi$
$38$	18.0901	2.93461
$39$	0	0
$40$	−4.91289	−0.776796
$41$	5.52728	0.863216	0.431608	−	0.902061i	$-0.357946\pi$
$41$	5.52728	0.863216	0.431608	+	0.902061i	$0.357946\pi$
$42$	0	0
$43$	−8.12045	−1.23836	−0.619179	−	0.785250i	$-0.712534\pi$
$43$	−8.12045	−1.23836	−0.619179	+	0.785250i	$0.712534\pi$
$44$	11.7545	1.77206
$45$	0	0
$46$	1.70535	0.251440
$47$	−5.95533	−0.868675	−0.434337	−	0.900750i	$-0.643017\pi$
$47$	−5.95533	−0.868675	−0.434337	+	0.900750i	$0.643017\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	19.6301	2.77612
$51$	0	0
$52$	−15.9255	−2.20847
$53$	11.7025	1.60746	0.803731	−	0.594993i	$-0.202845\pi$
$53$	11.7025	1.60746	0.803731	+	0.594993i	$0.202845\pi$
$54$	0	0
$55$	−16.9473	−2.28517
$56$	−1.30632	−0.174564
$57$	0	0
$58$	−3.03067	−0.397947
$59$	−5.35689	−0.697408	−0.348704	−	0.937233i	$-0.613378\pi$
$59$	−5.35689	−0.697408	−0.348704	+	0.937233i	$0.613378\pi$
$60$	0	0
$61$	−4.45521	−0.570431	−0.285215	−	0.958463i	$-0.592065\pi$
$61$	−4.45521	−0.570431	−0.285215	+	0.958463i	$0.592065\pi$
$62$	−19.0911	−2.42457
$63$	0	0
$64$	−11.9022	−1.48777
$65$	22.9609	2.84795
$66$	0	0
$67$	−13.9075	−1.69908	−0.849538	−	0.527527i	$-0.823119\pi$
$67$	−13.9075	−1.69908	−0.849538	+	0.527527i	$0.823119\pi$
$68$	14.4797	1.75592
$69$	0	0
$70$	8.07362	0.964983
$71$	−8.38797	−0.995469	−0.497734	−	0.867329i	$-0.665835\pi$
$71$	−8.38797	−0.995469	−0.497734	+	0.867329i	$0.665835\pi$
$72$	0	0
$73$	−12.5654	−1.47067	−0.735337	−	0.677701i	$-0.762976\pi$
$73$	−12.5654	−1.47067	−0.735337	+	0.677701i	$0.762976\pi$
$74$	14.7587	1.71567
$75$	0	0
$76$	21.9813	2.52143
$77$	−4.50622	−0.513531
$78$	0	0
$79$	−9.28601	−1.04476	−0.522379	−	0.852714i	$-0.674955\pi$
$79$	−9.28601	−1.04476	−0.522379	+	0.852714i	$0.674955\pi$
$80$	9.07382	1.01448
$81$	0	0
$82$	11.8657	1.31034
$83$	−12.3129	−1.35151	−0.675757	−	0.737124i	$-0.736183\pi$
$83$	−12.3129	−1.35151	−0.675757	+	0.737124i	$0.736183\pi$
$84$	0	0
$85$	−20.8764	−2.26436
$86$	−17.4325	−1.87980
$87$	0	0
$88$	5.88656	0.627509
$89$	−0.471303	−0.0499580	−0.0249790	−	0.999688i	$-0.507952\pi$
$89$	−0.471303	−0.0499580	−0.0249790	+	0.999688i	$0.507952\pi$
$90$	0	0
$91$	6.10521	0.639999
$92$	2.07217	0.216039
$93$	0	0
$94$	−12.7846	−1.31863
$95$	−31.6920	−3.25153
$96$	0	0
$97$	−1.95818	−0.198823	−0.0994115	−	0.995046i	$-0.531696\pi$
$97$	−1.95818	−0.198823	−0.0994115	+	0.995046i	$0.531696\pi$
$98$	2.14674	0.216854
$99$	0	0
$100$	23.8526	2.38526
$101$	−12.2060	−1.21454	−0.607270	−	0.794496i	$-0.707735\pi$
$101$	−12.2060	−1.21454	−0.607270	+	0.794496i	$0.707735\pi$
$102$	0	0
$103$	−13.1584	−1.29654	−0.648269	−	0.761412i	$-0.724507\pi$
$103$	−13.1584	−1.29654	−0.648269	+	0.761412i	$0.724507\pi$
$104$	−7.97534	−0.782047
$105$	0	0
$106$	25.1223	2.44009
$107$	−0.278639	−0.0269371	−0.0134685	−	0.999909i	$-0.504287\pi$
$107$	−0.278639	−0.0269371	−0.0134685	+	0.999909i	$0.504287\pi$
$108$	0	0
$109$	−0.226693	−0.0217133	−0.0108566	−	0.999941i	$-0.503456\pi$
$109$	−0.226693	−0.0217133	−0.0108566	+	0.999941i	$0.503456\pi$
$110$	−36.3815	−3.46884
$111$	0	0
$112$	2.41269	0.227978
$113$	13.1880	1.24062	0.620309	−	0.784357i	$-0.287007\pi$
$113$	13.1880	1.24062	0.620309	+	0.784357i	$0.287007\pi$
$114$	0	0
$115$	−2.98759	−0.278594
$116$	−3.68258	−0.341919
$117$	0	0
$118$	−11.4999	−1.05865
$119$	−5.55094	−0.508854
$120$	0	0
$121$	9.30600	0.846000
$122$	−9.56419	−0.865901
$123$	0	0
$124$	−23.1976	−2.08321
$125$	−15.5855	−1.39401
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	−9.96683	−0.880952
$129$	0	0
$130$	49.2911	4.32312
$131$	11.1870	0.977413	0.488706	−	0.872448i	$-0.337469\pi$
$131$	11.1870	0.977413	0.488706	+	0.872448i	$0.337469\pi$
$132$	0	0
$133$	−8.42678	−0.730694
$134$	−29.8559	−2.57916
$135$	0	0
$136$	7.25129	0.621793
$137$	−1.25995	−0.107644	−0.0538222	−	0.998551i	$-0.517140\pi$
$137$	−1.25995	−0.107644	−0.0538222	+	0.998551i	$0.517140\pi$
$138$	0	0
$139$	−14.5977	−1.23816	−0.619081	−	0.785327i	$-0.712495\pi$
$139$	−14.5977	−1.23816	−0.619081	+	0.785327i	$0.712495\pi$
$140$	9.81027	0.829119
$141$	0	0
$142$	−18.0068	−1.51110
$143$	−27.5114	−2.30062
$144$	0	0
$145$	5.30942	0.440923
$146$	−26.9748	−2.23245
$147$	0	0
$148$	17.9334	1.47411
$149$	−5.24399	−0.429605	−0.214802	−	0.976658i	$-0.568911\pi$
$149$	−5.24399	−0.429605	−0.214802	+	0.976658i	$0.568911\pi$
$150$	0	0
$151$	−14.7689	−1.20188	−0.600939	−	0.799295i	$-0.705206\pi$
$151$	−14.7689	−1.20188	−0.600939	+	0.799295i	$0.705206\pi$
$152$	11.0081	0.892871
$153$	0	0
$154$	−9.67370	−0.779529
$155$	33.4456	2.68642
$156$	0	0
$157$	−3.82422	−0.305206	−0.152603	−	0.988288i	$-0.548766\pi$
$157$	−3.82422	−0.305206	−0.152603	+	0.988288i	$0.548766\pi$
$158$	−19.9347	−1.58592
$159$	0	0
$160$	29.3049	2.31676
$161$	−0.794388	−0.0626066
$162$	0	0
$163$	13.6169	1.06656	0.533279	−	0.845939i	$-0.320960\pi$
$163$	13.6169	1.06656	0.533279	+	0.845939i	$0.320960\pi$
$164$	14.4180	1.12585
$165$	0	0
$166$	−26.4326	−2.05157
$167$	7.00092	0.541748	0.270874	−	0.962615i	$-0.412687\pi$
$167$	7.00092	0.541748	0.270874	+	0.962615i	$0.412687\pi$
$168$	0	0
$169$	24.2735	1.86719
$170$	−44.8162	−3.43725
$171$	0	0
$172$	−21.1823	−1.61513
$173$	5.51449	0.419259	0.209629	−	0.977781i	$-0.432774\pi$
$173$	5.51449	0.419259	0.209629	+	0.977781i	$0.432774\pi$
$174$	0	0
$175$	−9.14413	−0.691232
$176$	−10.8721	−0.819516
$177$	0	0
$178$	−1.01177	−0.0758351
$179$	−5.30592	−0.396583	−0.198291	−	0.980143i	$-0.563539\pi$
$179$	−5.30592	−0.396583	−0.198291	+	0.980143i	$0.563539\pi$
$180$	0	0
$181$	−10.0430	−0.746491	−0.373245	−	0.927733i	$-0.621755\pi$
$181$	−10.0430	−0.746491	−0.373245	+	0.927733i	$0.621755\pi$
$182$	13.1063	0.971505
$183$	0	0
$184$	1.03772	0.0765020
$185$	−25.8558	−1.90095
$186$	0	0
$187$	25.0137	1.82919
$188$	−15.5346	−1.13297
$189$	0	0
$190$	−68.0346	−4.93575
$191$	5.95234	0.430696	0.215348	−	0.976537i	$-0.430911\pi$
$191$	5.95234	0.430696	0.215348	+	0.976537i	$0.430911\pi$
$192$	0	0
$193$	4.28776	0.308640	0.154320	−	0.988021i	$-0.450681\pi$
$193$	4.28776	0.308640	0.154320	+	0.988021i	$0.450681\pi$
$194$	−4.20371	−0.301809
$195$	0	0
$196$	2.60851	0.186322
$197$	23.8910	1.70216	0.851081	−	0.525035i	$-0.175948\pi$
$197$	23.8910	1.70216	0.851081	+	0.525035i	$0.175948\pi$
$198$	0	0
$199$	3.85492	0.273268	0.136634	−	0.990622i	$-0.456372\pi$
$199$	3.85492	0.273268	0.136634	+	0.990622i	$0.456372\pi$
$200$	11.9451	0.844650
$201$	0	0
$202$	−26.2031	−1.84364
$203$	1.41175	0.0990857
$204$	0	0
$205$	−20.7874	−1.45185
$206$	−28.2478	−1.96811
$207$	0	0
$208$	14.7300	1.02134
$209$	37.9729	2.62664
$210$	0	0
$211$	1.58146	0.108872	0.0544360	−	0.998517i	$-0.482664\pi$
$211$	1.58146	0.108872	0.0544360	+	0.998517i	$0.482664\pi$
$212$	30.5261	2.09654
$213$	0	0
$214$	−0.598167	−0.0408899
$215$	30.5399	2.08281
$216$	0	0
$217$	8.89305	0.603700
$218$	−0.486653	−0.0329603
$219$	0	0
$220$	−44.2072	−2.98045
$221$	−33.8896	−2.27966
$222$	0	0
$223$	18.9301	1.26765	0.633827	−	0.773475i	$-0.281483\pi$
$223$	18.9301	1.26765	0.633827	+	0.773475i	$0.281483\pi$
$224$	7.79207	0.520629
$225$	0	0
$226$	28.3112	1.88323
$227$	−19.2889	−1.28025	−0.640123	−	0.768272i	$-0.721117\pi$
$227$	−19.2889	−1.28025	−0.640123	+	0.768272i	$0.721117\pi$
$228$	0	0
$229$	26.0675	1.72259	0.861295	−	0.508104i	$-0.169654\pi$
$229$	26.0675	1.72259	0.861295	+	0.508104i	$0.169654\pi$
$230$	−6.41359	−0.422900
$231$	0	0
$232$	−1.84420	−0.121078
$233$	12.5555	0.822535	0.411268	−	0.911515i	$-0.365086\pi$
$233$	12.5555	0.822535	0.411268	+	0.911515i	$0.365086\pi$
$234$	0	0
$235$	22.3972	1.46103
$236$	−13.9735	−0.909598
$237$	0	0
$238$	−11.9164	−0.772429
$239$	−6.37351	−0.412268	−0.206134	−	0.978524i	$-0.566088\pi$
$239$	−6.37351	−0.412268	−0.206134	+	0.978524i	$0.566088\pi$
$240$	0	0
$241$	−16.7581	−1.07948	−0.539742	−	0.841830i	$-0.681478\pi$
$241$	−16.7581	−1.07948	−0.539742	+	0.841830i	$0.681478\pi$
$242$	19.9776	1.28421
$243$	0	0
$244$	−11.6215	−0.743988
$245$	−3.76087	−0.240273
$246$	0	0
$247$	−51.4472	−3.27351
$248$	−11.6172	−0.737690
$249$	0	0
$250$	−33.4582	−2.11608
$251$	−21.2238	−1.33963	−0.669816	−	0.742527i	$-0.733627\pi$
$251$	−21.2238	−1.33963	−0.669816	+	0.742527i	$0.733627\pi$
$252$	0	0
$253$	3.57969	0.225053
$254$	2.14674	0.134699
$255$	0	0
$256$	2.40814	0.150509
$257$	−14.2228	−0.887192	−0.443596	−	0.896227i	$-0.646297\pi$
$257$	−14.2228	−0.887192	−0.443596	+	0.896227i	$0.646297\pi$
$258$	0	0
$259$	−6.87494	−0.427188
$260$	59.8937	3.71445
$261$	0	0
$262$	24.0156	1.48369
$263$	7.67303	0.473139	0.236570	−	0.971615i	$-0.423977\pi$
$263$	7.67303	0.473139	0.236570	+	0.971615i	$0.423977\pi$
$264$	0	0
$265$	−44.0115	−2.70361
$266$	−18.0901	−1.10918
$267$	0	0
$268$	−36.2780	−2.21603
$269$	−20.4166	−1.24482	−0.622410	−	0.782691i	$-0.713846\pi$
$269$	−20.4166	−1.24482	−0.622410	+	0.782691i	$0.713846\pi$
$270$	0	0
$271$	−22.4100	−1.36131	−0.680655	−	0.732604i	$-0.738305\pi$
$271$	−22.4100	−1.36131	−0.680655	+	0.732604i	$0.738305\pi$
$272$	−13.3927	−0.812052
$273$	0	0
$274$	−2.70478	−0.163402
$275$	41.2055	2.48478
$276$	0	0
$277$	10.1355	0.608984	0.304492	−	0.952515i	$-0.401513\pi$
$277$	10.1355	0.608984	0.304492	+	0.952515i	$0.401513\pi$
$278$	−31.3376	−1.87950
$279$	0	0
$280$	4.91289	0.293601
$281$	−2.48350	−0.148153	−0.0740767	−	0.997253i	$-0.523601\pi$
$281$	−2.48350	−0.148153	−0.0740767	+	0.997253i	$0.523601\pi$
$282$	0	0
$283$	9.19677	0.546691	0.273346	−	0.961916i	$-0.411870\pi$
$283$	9.19677	0.546691	0.273346	+	0.961916i	$0.411870\pi$
$284$	−21.8801	−1.29835
$285$	0	0
$286$	−59.0599	−3.49229
$287$	−5.52728	−0.326265
$288$	0	0
$289$	13.8129	0.812526
$290$	11.3980	0.669312
$291$	0	0
$292$	−32.7771	−1.91814
$293$	10.2716	0.600074	0.300037	−	0.953928i	$-0.403001\pi$
$293$	10.2716	0.600074	0.300037	+	0.953928i	$0.403001\pi$
$294$	0	0
$295$	20.1466	1.17298
$296$	8.98086	0.522002
$297$	0	0
$298$	−11.2575	−0.652130
$299$	−4.84990	−0.280477
$300$	0	0
$301$	8.12045	0.468055
$302$	−31.7051	−1.82442
$303$	0	0
$304$	−20.3312	−1.16608
$305$	16.7555	0.959414
$306$	0	0
$307$	5.17808	0.295529	0.147764	−	0.989023i	$-0.452792\pi$
$307$	5.17808	0.295529	0.147764	+	0.989023i	$0.452792\pi$
$308$	−11.7545	−0.669776
$309$	0	0
$310$	71.7992	4.07792
$311$	20.7814	1.17840	0.589202	−	0.807986i	$-0.299442\pi$
$311$	20.7814	1.17840	0.589202	+	0.807986i	$0.299442\pi$
$312$	0	0
$313$	−27.8210	−1.57254	−0.786268	−	0.617885i	$-0.787990\pi$
$313$	−27.8210	−1.57254	−0.786268	+	0.617885i	$0.787990\pi$
$314$	−8.20962	−0.463295
$315$	0	0
$316$	−24.2227	−1.36263
$317$	−12.9728	−0.728626	−0.364313	−	0.931277i	$-0.618696\pi$
$317$	−12.9728	−0.728626	−0.364313	+	0.931277i	$0.618696\pi$
$318$	0	0
$319$	−6.36167	−0.356185
$320$	44.7626	2.50231
$321$	0	0
$322$	−1.70535	−0.0950354
$323$	46.7765	2.60272
$324$	0	0
$325$	−55.8268	−3.09671
$326$	29.2320	1.61901
$327$	0	0
$328$	7.22039	0.398679
$329$	5.95533	0.328328
$330$	0	0
$331$	−15.6703	−0.861319	−0.430660	−	0.902514i	$-0.641719\pi$
$331$	−15.6703	−0.861319	−0.430660	+	0.902514i	$0.641719\pi$
$332$	−32.1183	−1.76272
$333$	0	0
$334$	15.0292	0.822361
$335$	52.3044	2.85770
$336$	0	0
$337$	−9.53314	−0.519303	−0.259652	−	0.965702i	$-0.583608\pi$
$337$	−9.53314	−0.519303	−0.259652	+	0.965702i	$0.583608\pi$
$338$	52.1091	2.83436
$339$	0	0
$340$	−54.4562	−2.95330
$341$	−40.0740	−2.17013
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−10.6079	−0.571939
$345$	0	0
$346$	11.8382	0.636425
$347$	−18.1615	−0.974963	−0.487482	−	0.873133i	$-0.662084\pi$
$347$	−18.1615	−0.974963	−0.487482	+	0.873133i	$0.662084\pi$
$348$	0	0
$349$	−31.4424	−1.68307	−0.841537	−	0.540199i	$-0.818349\pi$
$349$	−31.4424	−1.68307	−0.841537	+	0.540199i	$0.818349\pi$
$350$	−19.6301	−1.04927
$351$	0	0
$352$	−35.1128	−1.87152
$353$	36.3666	1.93560	0.967801	−	0.251718i	$-0.0809956\pi$
$353$	36.3666	1.93560	0.967801	+	0.251718i	$0.0809956\pi$
$354$	0	0
$355$	31.5461	1.67429
$356$	−1.22940	−0.0651580
$357$	0	0
$358$	−11.3904	−0.602004
$359$	−22.7443	−1.20040	−0.600199	−	0.799851i	$-0.704912\pi$
$359$	−22.7443	−1.20040	−0.600199	+	0.799851i	$0.704912\pi$
$360$	0	0
$361$	52.0106	2.73740
$362$	−21.5598	−1.13316
$363$	0	0
$364$	15.9255	0.834723
$365$	47.2570	2.47354
$366$	0	0
$367$	1.90500	0.0994401	0.0497201	−	0.998763i	$-0.484167\pi$
$367$	1.90500	0.0994401	0.0497201	+	0.998763i	$0.484167\pi$
$368$	−1.91661	−0.0999104
$369$	0	0
$370$	−55.5057	−2.88560
$371$	−11.7025	−0.607563
$372$	0	0
$373$	−19.0960	−0.988755	−0.494378	−	0.869247i	$-0.664604\pi$
$373$	−19.0960	−0.988755	−0.494378	+	0.869247i	$0.664604\pi$
$374$	53.6981	2.77666
$375$	0	0
$376$	−7.77956	−0.401200
$377$	8.61905	0.443903
$378$	0	0
$379$	25.1080	1.28971	0.644855	−	0.764305i	$-0.276918\pi$
$379$	25.1080	1.28971	0.644855	+	0.764305i	$0.276918\pi$
$380$	−82.6690	−4.24083
$381$	0	0
$382$	12.7782	0.653787
$383$	7.26619	0.371285	0.185642	−	0.982617i	$-0.440563\pi$
$383$	7.26619	0.371285	0.185642	+	0.982617i	$0.440563\pi$
$384$	0	0
$385$	16.9473	0.863714
$386$	9.20473	0.468509
$387$	0	0
$388$	−5.10793	−0.259316
$389$	25.9914	1.31782	0.658908	−	0.752223i	$-0.271019\pi$
$389$	25.9914	1.31782	0.658908	+	0.752223i	$0.271019\pi$
$390$	0	0
$391$	4.40960	0.223003
$392$	1.30632	0.0659790
$393$	0	0
$394$	51.2878	2.58384
$395$	34.9235	1.75719
$396$	0	0
$397$	−0.788194	−0.0395583	−0.0197792	−	0.999804i	$-0.506296\pi$
$397$	−0.788194	−0.0395583	−0.0197792	+	0.999804i	$0.506296\pi$
$398$	8.27553	0.414815
$399$	0	0
$400$	−22.0620	−1.10310
$401$	−20.4089	−1.01917	−0.509585	−	0.860420i	$-0.670201\pi$
$401$	−20.4089	−1.01917	−0.509585	+	0.860420i	$0.670201\pi$
$402$	0	0
$403$	54.2939	2.70457
$404$	−31.8394	−1.58407
$405$	0	0
$406$	3.03067	0.150410
$407$	30.9800	1.53562
$408$	0	0
$409$	1.06363	0.0525930	0.0262965	−	0.999654i	$-0.491629\pi$
$409$	1.06363	0.0525930	0.0262965	+	0.999654i	$0.491629\pi$
$410$	−44.6252	−2.20388
$411$	0	0
$412$	−34.3239	−1.69102
$413$	5.35689	0.263595
$414$	0	0
$415$	46.3071	2.27313
$416$	47.5722	2.33242
$417$	0	0
$418$	81.5181	3.98718
$419$	−9.44301	−0.461321	−0.230661	−	0.973034i	$-0.574089\pi$
$419$	−9.44301	−0.461321	−0.230661	+	0.973034i	$0.574089\pi$
$420$	0	0
$421$	−33.3356	−1.62468	−0.812340	−	0.583184i	$-0.801807\pi$
$421$	−33.3356	−1.62468	−0.812340	+	0.583184i	$0.801807\pi$
$422$	3.39498	0.165265
$423$	0	0
$424$	15.2872	0.742411
$425$	50.7585	2.46215
$426$	0	0
$427$	4.45521	0.215603
$428$	−0.726834	−0.0351328
$429$	0	0
$430$	65.5615	3.16165
$431$	−9.79350	−0.471736	−0.235868	−	0.971785i	$-0.575793\pi$
$431$	−9.79350	−0.471736	−0.235868	+	0.971785i	$0.575793\pi$
$432$	0	0
$433$	−2.28324	−0.109725	−0.0548626	−	0.998494i	$-0.517472\pi$
$433$	−2.28324	−0.109725	−0.0548626	+	0.998494i	$0.517472\pi$
$434$	19.0911	0.916403
$435$	0	0
$436$	−0.591332	−0.0283197
$437$	6.69413	0.320224
$438$	0	0
$439$	15.3687	0.733507	0.366754	−	0.930318i	$-0.380469\pi$
$439$	15.3687	0.733507	0.366754	+	0.930318i	$0.380469\pi$
$440$	−22.1386	−1.05541
$441$	0	0
$442$	−72.7524	−3.46048
$443$	−14.6632	−0.696669	−0.348335	−	0.937370i	$-0.613253\pi$
$443$	−14.6632	−0.696669	−0.348335	+	0.937370i	$0.613253\pi$
$444$	0	0
$445$	1.77251	0.0840249
$446$	40.6381	1.92427
$447$	0	0
$448$	11.9022	0.562326
$449$	−32.0243	−1.51132	−0.755661	−	0.654963i	$-0.772684\pi$
$449$	−32.0243	−1.51132	−0.755661	+	0.654963i	$0.772684\pi$
$450$	0	0
$451$	24.9071	1.17283
$452$	34.4009	1.61808
$453$	0	0
$454$	−41.4082	−1.94339
$455$	−22.9609	−1.07642
$456$	0	0
$457$	−9.86869	−0.461638	−0.230819	−	0.972997i	$-0.574140\pi$
$457$	−9.86869	−0.461638	−0.230819	+	0.972997i	$0.574140\pi$
$458$	55.9603	2.61485
$459$	0	0
$460$	−7.79316	−0.363358
$461$	26.6811	1.24266	0.621330	−	0.783549i	$-0.286593\pi$
$461$	26.6811	1.24266	0.621330	+	0.783549i	$0.286593\pi$
$462$	0	0
$463$	40.1469	1.86579	0.932893	−	0.360155i	$-0.117276\pi$
$463$	40.1469	1.86579	0.932893	+	0.360155i	$0.117276\pi$
$464$	3.40613	0.158125
$465$	0	0
$466$	26.9534	1.24859
$467$	−16.8989	−0.781987	−0.390994	−	0.920393i	$-0.627869\pi$
$467$	−16.8989	−0.781987	−0.390994	+	0.920393i	$0.627869\pi$
$468$	0	0
$469$	13.9075	0.642191
$470$	48.0811	2.21782
$471$	0	0
$472$	−6.99780	−0.322100
$473$	−36.5925	−1.68253
$474$	0	0
$475$	77.0556	3.53555
$476$	−14.4797	−0.663676
$477$	0	0
$478$	−13.6823	−0.625814
$479$	−20.1551	−0.920912	−0.460456	−	0.887683i	$-0.652314\pi$
$479$	−20.1551	−0.920912	−0.460456	+	0.887683i	$0.652314\pi$
$480$	0	0
$481$	−41.9729	−1.91380
$482$	−35.9754	−1.63863
$483$	0	0
$484$	24.2748	1.10340
$485$	7.36445	0.334403
$486$	0	0
$487$	−18.3826	−0.832993	−0.416497	−	0.909137i	$-0.636742\pi$
$487$	−18.3826	−0.832993	−0.416497	+	0.909137i	$0.636742\pi$
$488$	−5.81992	−0.263455
$489$	0	0
$490$	−8.07362	−0.364729
$491$	−9.75952	−0.440441	−0.220221	−	0.975450i	$-0.570678\pi$
$491$	−9.75952	−0.440441	−0.220221	+	0.975450i	$0.570678\pi$
$492$	0	0
$493$	−7.83656	−0.352941
$494$	−110.444	−4.96911
$495$	0	0
$496$	21.4562	0.963412
$497$	8.38797	0.376252
$498$	0	0
$499$	32.5664	1.45787	0.728936	−	0.684582i	$-0.240015\pi$
$499$	32.5664	1.45787	0.728936	+	0.684582i	$0.240015\pi$
$500$	−40.6551	−1.81815
$501$	0	0
$502$	−45.5620	−2.03353
$503$	−18.0394	−0.804338	−0.402169	−	0.915565i	$-0.631744\pi$
$503$	−18.0394	−0.804338	−0.402169	+	0.915565i	$0.631744\pi$
$504$	0	0
$505$	45.9051	2.04275
$506$	7.68467	0.341625
$507$	0	0
$508$	2.60851	0.115734
$509$	−5.63526	−0.249779	−0.124889	−	0.992171i	$-0.539858\pi$
$509$	−5.63526	−0.249779	−0.124889	+	0.992171i	$0.539858\pi$
$510$	0	0
$511$	12.5654	0.555863
$512$	25.1033	1.10942
$513$	0	0
$514$	−30.5326	−1.34674
$515$	49.4871	2.18066
$516$	0	0
$517$	−26.8360	−1.18025
$518$	−14.7587	−0.648462
$519$	0	0
$520$	29.9942	1.31533
$521$	−7.56432	−0.331399	−0.165699	−	0.986176i	$-0.552988\pi$
$521$	−7.56432	−0.331399	−0.165699	+	0.986176i	$0.552988\pi$
$522$	0	0
$523$	5.67584	0.248187	0.124094	−	0.992271i	$-0.460398\pi$
$523$	5.67584	0.248187	0.124094	+	0.992271i	$0.460398\pi$
$524$	29.1814	1.27480
$525$	0	0
$526$	16.4720	0.718214
$527$	−49.3648	−2.15036
$528$	0	0
$529$	−22.3689	−0.972563
$530$	−94.4815	−4.10402
$531$	0	0
$532$	−21.9813	−0.953012
$533$	−33.7452	−1.46167
$534$	0	0
$535$	1.04793	0.0453058
$536$	−18.1677	−0.784724
$537$	0	0
$538$	−43.8291	−1.88961
$539$	4.50622	0.194097
$540$	0	0
$541$	14.5587	0.625926	0.312963	−	0.949765i	$-0.398678\pi$
$541$	14.5587	0.625926	0.312963	+	0.949765i	$0.398678\pi$
$542$	−48.1085	−2.06644
$543$	0	0
$544$	−43.2533	−1.85447
$545$	0.852564	0.0365198
$546$	0	0
$547$	−11.4754	−0.490651	−0.245326	−	0.969441i	$-0.578895\pi$
$547$	−11.4754	−0.490651	−0.245326	+	0.969441i	$0.578895\pi$
$548$	−3.28658	−0.140396
$549$	0	0
$550$	88.4576	3.77184
$551$	−11.8965	−0.506809
$552$	0	0
$553$	9.28601	0.394881
$554$	21.7583	0.924423
$555$	0	0
$556$	−38.0783	−1.61488
$557$	−29.8779	−1.26597	−0.632984	−	0.774165i	$-0.718170\pi$
$557$	−29.8779	−1.26597	−0.632984	+	0.774165i	$0.718170\pi$
$558$	0	0
$559$	49.5770	2.09688
$560$	−9.07382	−0.383439
$561$	0	0
$562$	−5.33145	−0.224894
$563$	13.8570	0.584004	0.292002	−	0.956418i	$-0.405679\pi$
$563$	13.8570	0.584004	0.292002	+	0.956418i	$0.405679\pi$
$564$	0	0
$565$	−49.5982	−2.08661
$566$	19.7431	0.829865
$567$	0	0
$568$	−10.9574	−0.459761
$569$	25.5878	1.07270	0.536349	−	0.843996i	$-0.319803\pi$
$569$	25.5878	1.07270	0.536349	+	0.843996i	$0.319803\pi$
$570$	0	0
$571$	−24.0863	−1.00798	−0.503991	−	0.863709i	$-0.668136\pi$
$571$	−24.0863	−1.00798	−0.503991	+	0.863709i	$0.668136\pi$
$572$	−71.7638	−3.00059
$573$	0	0
$574$	−11.8657	−0.495263
$575$	7.26399	0.302929
$576$	0	0
$577$	−3.92715	−0.163489	−0.0817447	−	0.996653i	$-0.526049\pi$
$577$	−3.92715	−0.163489	−0.0817447	+	0.996653i	$0.526049\pi$
$578$	29.6528	1.23340
$579$	0	0
$580$	13.8497	0.575077
$581$	12.3129	0.510825
$582$	0	0
$583$	52.7340	2.18402
$584$	−16.4145	−0.679236
$585$	0	0
$586$	22.0505	0.910898
$587$	3.58758	0.148075	0.0740376	−	0.997255i	$-0.476412\pi$
$587$	3.58758	0.148075	0.0740376	+	0.997255i	$0.476412\pi$
$588$	0	0
$589$	−74.9398	−3.08784
$590$	43.2495	1.78055
$591$	0	0
$592$	−16.5871	−0.681726
$593$	12.3769	0.508258	0.254129	−	0.967170i	$-0.418211\pi$
$593$	12.3769	0.508258	0.254129	+	0.967170i	$0.418211\pi$
$594$	0	0
$595$	20.8764	0.855847
$596$	−13.6790	−0.560314
$597$	0	0
$598$	−10.4115	−0.425758
$599$	−23.5335	−0.961553	−0.480777	−	0.876843i	$-0.659645\pi$
$599$	−23.5335	−0.961553	−0.480777	+	0.876843i	$0.659645\pi$
$600$	0	0
$601$	−1.20245	−0.0490492	−0.0245246	−	0.999699i	$-0.507807\pi$
$601$	−1.20245	−0.0490492	−0.0245246	+	0.999699i	$0.507807\pi$
$602$	17.4325	0.710497
$603$	0	0
$604$	−38.5249	−1.56756
$605$	−34.9987	−1.42290
$606$	0	0
$607$	14.4517	0.586577	0.293289	−	0.956024i	$-0.405250\pi$
$607$	14.4517	0.586577	0.293289	+	0.956024i	$0.405250\pi$
$608$	−65.6620	−2.66295
$609$	0	0
$610$	35.9697	1.45637
$611$	36.3585	1.47091
$612$	0	0
$613$	27.4455	1.10851	0.554256	−	0.832346i	$-0.313003\pi$
$613$	27.4455	1.10851	0.554256	+	0.832346i	$0.313003\pi$
$614$	11.1160	0.448606
$615$	0	0
$616$	−5.88656	−0.237176
$617$	−3.30266	−0.132960	−0.0664801	−	0.997788i	$-0.521177\pi$
$617$	−3.30266	−0.132960	−0.0664801	+	0.997788i	$0.521177\pi$
$618$	0	0
$619$	1.72491	0.0693300	0.0346650	−	0.999399i	$-0.488964\pi$
$619$	1.72491	0.0693300	0.0346650	+	0.999399i	$0.488964\pi$
$620$	87.2433	3.50377
$621$	0	0
$622$	44.6123	1.78879
$623$	0.471303	0.0188823
$624$	0	0
$625$	12.8945	0.515781
$626$	−59.7246	−2.38707
$627$	0	0
$628$	−9.97551	−0.398066
$629$	38.1624	1.52163
$630$	0	0
$631$	−5.11689	−0.203700	−0.101850	−	0.994800i	$-0.532476\pi$
$631$	−5.11689	−0.203700	−0.101850	+	0.994800i	$0.532476\pi$
$632$	−12.1305	−0.482525
$633$	0	0
$634$	−27.8493	−1.10604
$635$	−3.76087	−0.149246
$636$	0	0
$637$	−6.10521	−0.241897
$638$	−13.6569	−0.540681
$639$	0	0
$640$	37.4840	1.48168
$641$	8.17501	0.322893	0.161447	−	0.986881i	$-0.448384\pi$
$641$	8.17501	0.322893	0.161447	+	0.986881i	$0.448384\pi$
$642$	0	0
$643$	−5.81134	−0.229177	−0.114588	−	0.993413i	$-0.536555\pi$
$643$	−5.81134	−0.229177	−0.114588	+	0.993413i	$0.536555\pi$
$644$	−2.07217	−0.0816550
$645$	0	0
$646$	100.417	3.95086
$647$	−2.09263	−0.0822699	−0.0411349	−	0.999154i	$-0.513097\pi$
$647$	−2.09263	−0.0822699	−0.0411349	+	0.999154i	$0.513097\pi$
$648$	0	0
$649$	−24.1393	−0.947551
$650$	−119.846	−4.70074
$651$	0	0
$652$	35.5198	1.39106
$653$	−6.62367	−0.259204	−0.129602	−	0.991566i	$-0.541370\pi$
$653$	−6.62367	−0.259204	−0.129602	+	0.991566i	$0.541370\pi$
$654$	0	0
$655$	−42.0728	−1.64392
$656$	−13.3356	−0.520668
$657$	0	0
$658$	12.7846	0.498395
$659$	−2.89048	−0.112597	−0.0562985	−	0.998414i	$-0.517930\pi$
$659$	−2.89048	−0.112597	−0.0562985	+	0.998414i	$0.517930\pi$
$660$	0	0
$661$	17.1370	0.666553	0.333277	−	0.942829i	$-0.391846\pi$
$661$	17.1370	0.666553	0.333277	+	0.942829i	$0.391846\pi$
$662$	−33.6402	−1.30746
$663$	0	0
$664$	−16.0845	−0.624201
$665$	31.6920	1.22896
$666$	0	0
$667$	−1.12148	−0.0434239
$668$	18.2620	0.706578
$669$	0	0
$670$	112.284	4.33792
$671$	−20.0761	−0.775031
$672$	0	0
$673$	−14.3209	−0.552029	−0.276015	−	0.961153i	$-0.589014\pi$
$673$	−14.3209	−0.552029	−0.276015	+	0.961153i	$0.589014\pi$
$674$	−20.4652	−0.788291
$675$	0	0
$676$	63.3178	2.43530
$677$	36.7484	1.41236	0.706178	−	0.708034i	$-0.250418\pi$
$677$	36.7484	1.41236	0.706178	+	0.708034i	$0.250418\pi$
$678$	0	0
$679$	1.95818	0.0751480
$680$	−27.2712	−1.04580
$681$	0	0
$682$	−86.0287	−3.29421
$683$	−12.9102	−0.493995	−0.246998	−	0.969016i	$-0.579444\pi$
$683$	−12.9102	−0.493995	−0.246998	+	0.969016i	$0.579444\pi$
$684$	0	0
$685$	4.73849	0.181048
$686$	−2.14674	−0.0819631
$687$	0	0
$688$	19.5921	0.746943
$689$	−71.4461	−2.72188
$690$	0	0
$691$	−31.6118	−1.20257	−0.601285	−	0.799034i	$-0.705345\pi$
$691$	−31.6118	−1.20257	−0.601285	+	0.799034i	$0.705345\pi$
$692$	14.3846	0.546820
$693$	0	0
$694$	−38.9882	−1.47997
$695$	54.9001	2.08248
$696$	0	0
$697$	30.6816	1.16215
$698$	−67.4989	−2.55487
$699$	0	0
$700$	−23.8526	−0.901543
$701$	−17.0083	−0.642393	−0.321197	−	0.947013i	$-0.604085\pi$
$701$	−17.0083	−0.642393	−0.321197	+	0.947013i	$0.604085\pi$
$702$	0	0
$703$	57.9336	2.18501
$704$	−53.6339	−2.02140
$705$	0	0
$706$	78.0699	2.93820
$707$	12.2060	0.459053
$708$	0	0
$709$	28.6723	1.07681	0.538406	−	0.842686i	$-0.319027\pi$
$709$	28.6723	1.07681	0.538406	+	0.842686i	$0.319027\pi$
$710$	67.7213	2.54154
$711$	0	0
$712$	−0.615671	−0.0230733
$713$	−7.06454	−0.264569
$714$	0	0
$715$	103.467	3.86944
$716$	−13.8405	−0.517245
$717$	0	0
$718$	−48.8262	−1.82218
$719$	−0.701383	−0.0261572	−0.0130786	−	0.999914i	$-0.504163\pi$
$719$	−0.701383	−0.0261572	−0.0130786	+	0.999914i	$0.504163\pi$
$720$	0	0
$721$	13.1584	0.490045
$722$	111.653	4.15531
$723$	0	0
$724$	−26.1973	−0.973615
$725$	−12.9093	−0.479438
$726$	0	0
$727$	22.9444	0.850962	0.425481	−	0.904967i	$-0.360105\pi$
$727$	22.9444	0.850962	0.425481	+	0.904967i	$0.360105\pi$
$728$	7.97534	0.295586
$729$	0	0
$730$	101.449	3.75478
$731$	−45.0761	−1.66720
$732$	0	0
$733$	−53.4312	−1.97353	−0.986764	−	0.162164i	$-0.948153\pi$
$733$	−53.4312	−1.97353	−0.986764	+	0.162164i	$0.948153\pi$
$734$	4.08955	0.150948
$735$	0	0
$736$	−6.18993	−0.228164
$737$	−62.6704	−2.30849
$738$	0	0
$739$	21.0934	0.775935	0.387967	−	0.921673i	$-0.373177\pi$
$739$	21.0934	0.775935	0.387967	+	0.921673i	$0.373177\pi$
$740$	−67.4450	−2.47933
$741$	0	0
$742$	−25.1223	−0.922267
$743$	−34.5024	−1.26577	−0.632885	−	0.774246i	$-0.718129\pi$
$743$	−34.5024	−1.26577	−0.632885	+	0.774246i	$0.718129\pi$
$744$	0	0
$745$	19.7220	0.722557
$746$	−40.9943	−1.50091
$747$	0	0
$748$	65.2486	2.38573
$749$	0.278639	0.0101813
$750$	0	0
$751$	26.6024	0.970737	0.485368	−	0.874310i	$-0.338686\pi$
$751$	26.6024	0.970737	0.485368	+	0.874310i	$0.338686\pi$
$752$	14.3684	0.523961
$753$	0	0
$754$	18.5029	0.673835
$755$	55.5440	2.02145
$756$	0	0
$757$	41.1608	1.49602	0.748008	−	0.663690i	$-0.231010\pi$
$757$	41.1608	1.49602	0.748008	+	0.663690i	$0.231010\pi$
$758$	53.9004	1.95775
$759$	0	0
$760$	−41.3998	−1.50173
$761$	−26.5892	−0.963859	−0.481929	−	0.876210i	$-0.660064\pi$
$761$	−26.5892	−0.963859	−0.481929	+	0.876210i	$0.660064\pi$
$762$	0	0
$763$	0.226693	0.00820685
$764$	15.5268	0.561738
$765$	0	0
$766$	15.5986	0.563602
$767$	32.7049	1.18091
$768$	0	0
$769$	−24.1014	−0.869120	−0.434560	−	0.900643i	$-0.643096\pi$
$769$	−24.1014	−0.869120	−0.434560	+	0.900643i	$0.643096\pi$
$770$	36.3815	1.31110
$771$	0	0
$772$	11.1847	0.402545
$773$	29.2512	1.05209	0.526046	−	0.850456i	$-0.323674\pi$
$773$	29.2512	1.05209	0.526046	+	0.850456i	$0.323674\pi$
$774$	0	0
$775$	−81.3193	−2.92107
$776$	−2.55800	−0.0918270
$777$	0	0
$778$	55.7969	2.00042
$779$	46.5772	1.66880
$780$	0	0
$781$	−37.7980	−1.35252
$782$	9.46629	0.338514
$783$	0	0
$784$	−2.41269	−0.0861675
$785$	14.3824	0.513329
$786$	0	0
$787$	−29.5342	−1.05278	−0.526389	−	0.850244i	$-0.676455\pi$
$787$	−29.5342	−1.05278	−0.526389	+	0.850244i	$0.676455\pi$
$788$	62.3199	2.22005
$789$	0	0
$790$	74.9717	2.66737
$791$	−13.1880	−0.468910
$792$	0	0
$793$	27.2000	0.965899
$794$	−1.69205	−0.0600487
$795$	0	0
$796$	10.0556	0.356412
$797$	6.48449	0.229692	0.114846	−	0.993383i	$-0.463362\pi$
$797$	6.48449	0.229692	0.114846	+	0.993383i	$0.463362\pi$
$798$	0	0
$799$	−33.0577	−1.16950
$800$	−71.2517	−2.51913
$801$	0	0
$802$	−43.8126	−1.54708
$803$	−56.6227	−1.99817
$804$	0	0
$805$	2.98759	0.105299
$806$	116.555	4.10548
$807$	0	0
$808$	−15.9449	−0.560939
$809$	32.0931	1.12833	0.564166	−	0.825662i	$-0.309198\pi$
$809$	32.0931	1.12833	0.564166	+	0.825662i	$0.309198\pi$
$810$	0	0
$811$	−15.0253	−0.527611	−0.263806	−	0.964576i	$-0.584978\pi$
$811$	−15.0253	−0.527611	−0.263806	+	0.964576i	$0.584978\pi$
$812$	3.68258	0.129233
$813$	0	0
$814$	66.5061	2.33104
$815$	−51.2114	−1.79386
$816$	0	0
$817$	−68.4292	−2.39404
$818$	2.28334	0.0798350
$819$	0	0
$820$	−54.2241	−1.89359
$821$	18.9953	0.662940	0.331470	−	0.943466i	$-0.392455\pi$
$821$	18.9953	0.662940	0.331470	+	0.943466i	$0.392455\pi$
$822$	0	0
$823$	−50.7523	−1.76911	−0.884557	−	0.466432i	$-0.845539\pi$
$823$	−50.7523	−1.76911	−0.884557	+	0.466432i	$0.845539\pi$
$824$	−17.1891	−0.598810
$825$	0	0
$826$	11.4999	0.400132
$827$	−23.8580	−0.829625	−0.414812	−	0.909907i	$-0.636153\pi$
$827$	−23.8580	−0.829625	−0.414812	+	0.909907i	$0.636153\pi$
$828$	0	0
$829$	−8.43067	−0.292809	−0.146405	−	0.989225i	$-0.546770\pi$
$829$	−8.43067	−0.292809	−0.146405	+	0.989225i	$0.546770\pi$
$830$	99.4096	3.45056
$831$	0	0
$832$	72.6653	2.51922
$833$	5.55094	0.192329
$834$	0	0
$835$	−26.3296	−0.911172
$836$	99.0527	3.42581
$837$	0	0
$838$	−20.2717	−0.700275
$839$	−18.2023	−0.628413	−0.314206	−	0.949355i	$-0.601738\pi$
$839$	−18.2023	−0.628413	−0.314206	+	0.949355i	$0.601738\pi$
$840$	0	0
$841$	−27.0070	−0.931274
$842$	−71.5631	−2.46623
$843$	0	0
$844$	4.12525	0.141997
$845$	−91.2896	−3.14046
$846$	0	0
$847$	−9.30600	−0.319758
$848$	−28.2345	−0.969577
$849$	0	0
$850$	108.966	3.73749
$851$	5.46137	0.187213
$852$	0	0
$853$	16.0465	0.549423	0.274711	−	0.961527i	$-0.411418\pi$
$853$	16.0465	0.549423	0.274711	+	0.961527i	$0.411418\pi$
$854$	9.56419	0.327280
$855$	0	0
$856$	−0.363991	−0.0124410
$857$	27.6010	0.942831	0.471416	−	0.881911i	$-0.343743\pi$
$857$	27.6010	0.942831	0.471416	+	0.881911i	$0.343743\pi$
$858$	0	0
$859$	25.4853	0.869548	0.434774	−	0.900540i	$-0.356828\pi$
$859$	25.4853	0.869548	0.434774	+	0.900540i	$0.356828\pi$
$860$	79.6638	2.71651
$861$	0	0
$862$	−21.0241	−0.716085
$863$	−1.32591	−0.0451347	−0.0225673	−	0.999745i	$-0.507184\pi$
$863$	−1.32591	−0.0451347	−0.0225673	+	0.999745i	$0.507184\pi$
$864$	0	0
$865$	−20.7393	−0.705156
$866$	−4.90152	−0.166561
$867$	0	0
$868$	23.1976	0.787379
$869$	−41.8448	−1.41949
$870$	0	0
$871$	84.9084	2.87701
$872$	−0.296134	−0.0100284
$873$	0	0
$874$	14.3706	0.486093
$875$	15.5855	0.526888
$876$	0	0
$877$	−2.51486	−0.0849209	−0.0424604	−	0.999098i	$-0.513520\pi$
$877$	−2.51486	−0.0849209	−0.0424604	+	0.999098i	$0.513520\pi$
$878$	32.9926	1.11345
$879$	0	0
$880$	40.8886	1.37835
$881$	−2.78025	−0.0936689	−0.0468344	−	0.998903i	$-0.514913\pi$
$881$	−2.78025	−0.0936689	−0.0468344	+	0.998903i	$0.514913\pi$
$882$	0	0
$883$	−35.9660	−1.21035	−0.605175	−	0.796092i	$-0.706897\pi$
$883$	−35.9660	−1.21035	−0.605175	+	0.796092i	$0.706897\pi$
$884$	−88.4015	−2.97326
$885$	0	0
$886$	−31.4781	−1.05753
$887$	−26.5645	−0.891948	−0.445974	−	0.895046i	$-0.647143\pi$
$887$	−26.5645	−0.891948	−0.445974	+	0.895046i	$0.647143\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	3.80512	0.127548
$891$	0	0
$892$	49.3794	1.65335
$893$	−50.1843	−1.67935
$894$	0	0
$895$	19.9549	0.667017
$896$	9.96683	0.332969
$897$	0	0
$898$	−68.7481	−2.29415
$899$	12.5548	0.418726
$900$	0	0
$901$	64.9598	2.16413
$902$	53.4692	1.78033
$903$	0	0
$904$	17.2277	0.572984
$905$	37.7704	1.25553
$906$	0	0
$907$	16.1989	0.537876	0.268938	−	0.963157i	$-0.413327\pi$
$907$	16.1989	0.537876	0.268938	+	0.963157i	$0.413327\pi$
$908$	−50.3152	−1.66977
$909$	0	0
$910$	−49.2911	−1.63399
$911$	15.6502	0.518515	0.259257	−	0.965808i	$-0.416522\pi$
$911$	15.6502	0.518515	0.259257	+	0.965808i	$0.416522\pi$
$912$	0	0
$913$	−55.4845	−1.83627
$914$	−21.1856	−0.700756
$915$	0	0
$916$	67.9974	2.24670
$917$	−11.1870	−0.369427
$918$	0	0
$919$	−7.42148	−0.244812	−0.122406	−	0.992480i	$-0.539061\pi$
$919$	−7.42148	−0.244812	−0.122406	+	0.992480i	$0.539061\pi$
$920$	−3.90274	−0.128670
$921$	0	0
$922$	57.2774	1.88633
$923$	51.2103	1.68561
$924$	0	0
$925$	62.8654	2.06700
$926$	86.1852	2.83222
$927$	0	0
$928$	11.0005	0.361108
$929$	−44.5635	−1.46208	−0.731040	−	0.682335i	$-0.760965\pi$
$929$	−44.5635	−1.46208	−0.731040	+	0.682335i	$0.760965\pi$
$930$	0	0
$931$	8.42678	0.276176
$932$	32.7510	1.07280
$933$	0	0
$934$	−36.2776	−1.18704
$935$	−94.0734	−3.07653
$936$	0	0
$937$	39.8727	1.30259	0.651293	−	0.758827i	$-0.274227\pi$
$937$	39.8727	1.30259	0.651293	+	0.758827i	$0.274227\pi$
$938$	29.8559	0.974831
$939$	0	0
$940$	58.4234	1.90556
$941$	33.4788	1.09138	0.545690	−	0.837987i	$-0.316268\pi$
$941$	33.4788	1.09138	0.545690	+	0.837987i	$0.316268\pi$
$942$	0	0
$943$	4.39081	0.142984
$944$	12.9245	0.420657
$945$	0	0
$946$	−78.5548	−2.55404
$947$	56.6022	1.83933	0.919663	−	0.392709i	$-0.128462\pi$
$947$	56.6022	1.83933	0.919663	+	0.392709i	$0.128462\pi$
$948$	0	0
$949$	76.7146	2.49026
$950$	165.419	5.36689
$951$	0	0
$952$	−7.25129	−0.235016
$953$	7.03236	0.227801	0.113900	−	0.993492i	$-0.463666\pi$
$953$	7.03236	0.227801	0.113900	+	0.993492i	$0.463666\pi$
$954$	0	0
$955$	−22.3860	−0.724393
$956$	−16.6254	−0.537703
$957$	0	0
$958$	−43.2679	−1.39792
$959$	1.25995	0.0406858
$960$	0	0
$961$	48.0864	1.55117
$962$	−90.1051	−2.90511
$963$	0	0
$964$	−43.7137	−1.40792
$965$	−16.1257	−0.519105
$966$	0	0
$967$	−50.3067	−1.61775	−0.808877	−	0.587978i	$-0.799924\pi$
$967$	−50.3067	−1.61775	−0.808877	+	0.587978i	$0.799924\pi$
$968$	12.1566	0.390728
$969$	0	0
$970$	15.8096	0.507616
$971$	31.7314	1.01831	0.509154	−	0.860675i	$-0.329958\pi$
$971$	31.7314	1.01831	0.509154	+	0.860675i	$0.329958\pi$
$972$	0	0
$973$	14.5977	0.467981
$974$	−39.4626	−1.26446
$975$	0	0
$976$	10.7490	0.344068
$977$	28.5006	0.911816	0.455908	−	0.890027i	$-0.349315\pi$
$977$	28.5006	0.911816	0.455908	+	0.890027i	$0.349315\pi$
$978$	0	0
$979$	−2.12379	−0.0678767
$980$	−9.81027	−0.313378
$981$	0	0
$982$	−20.9512	−0.668580
$983$	51.6542	1.64751	0.823756	−	0.566944i	$-0.191874\pi$
$983$	51.6542	1.64751	0.823756	+	0.566944i	$0.191874\pi$
$984$	0	0
$985$	−89.8508	−2.86288
$986$	−16.8231	−0.535756
$987$	0	0
$988$	−134.201	−4.26949
$989$	−6.45079	−0.205123
$990$	0	0
$991$	−27.4474	−0.871894	−0.435947	−	0.899972i	$-0.643587\pi$
$991$	−27.4474	−0.871894	−0.435947	+	0.899972i	$0.643587\pi$
$992$	69.2953	2.20013
$993$	0	0
$994$	18.0068	0.571142
$995$	−14.4979	−0.459613
$996$	0	0
$997$	40.0473	1.26831	0.634156	−	0.773205i	$-0.281348\pi$
$997$	40.0473	1.26831	0.634156	+	0.773205i	$0.281348\pi$
$998$	69.9117	2.21302
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.m.1.9		11
3.2	odd	2		2667.2.a.k.1.3	✓	11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.k.1.3	✓	11	3.2	odd	2
8001.2.a.m.1.9		11	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)

\(f(q)\)	\(=\)	\(q+2.14674 q^{2} +2.60851 q^{4} -3.76087 q^{5} -1.00000 q^{7} +1.30632 q^{8} +O(q^{10})\)	\(q+2.14674 q^{2} +2.60851 q^{4} -3.76087 q^{5} -1.00000 q^{7} +1.30632 q^{8} -8.07362 q^{10} +4.50622 q^{11} -6.10521 q^{13} -2.14674 q^{14} -2.41269 q^{16} +5.55094 q^{17} +8.42678 q^{19} -9.81027 q^{20} +9.67370 q^{22} +0.794388 q^{23} +9.14413 q^{25} -13.1063 q^{26} -2.60851 q^{28} -1.41175 q^{29} -8.89305 q^{31} -7.79207 q^{32} +11.9164 q^{34} +3.76087 q^{35} +6.87494 q^{37} +18.0901 q^{38} -4.91289 q^{40} +5.52728 q^{41} -8.12045 q^{43} +11.7545 q^{44} +1.70535 q^{46} -5.95533 q^{47} +1.00000 q^{49} +19.6301 q^{50} -15.9255 q^{52} +11.7025 q^{53} -16.9473 q^{55} -1.30632 q^{56} -3.03067 q^{58} -5.35689 q^{59} -4.45521 q^{61} -19.0911 q^{62} -11.9022 q^{64} +22.9609 q^{65} -13.9075 q^{67} +14.4797 q^{68} +8.07362 q^{70} -8.38797 q^{71} -12.5654 q^{73} +14.7587 q^{74} +21.9813 q^{76} -4.50622 q^{77} -9.28601 q^{79} +9.07382 q^{80} +11.8657 q^{82} -12.3129 q^{83} -20.8764 q^{85} -17.4325 q^{86} +5.88656 q^{88} -0.471303 q^{89} +6.10521 q^{91} +2.07217 q^{92} -12.7846 q^{94} -31.6920 q^{95} -1.95818 q^{97} +2.14674 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + 2 q^{2} + 12 q^{4} - q^{5} - 11 q^{7} + 15 q^{8}+O(q^{10}) \) 11 * q + 2 * q^2 + 12 * q^4 - q^5 - 11 * q^7 + 15 * q^8	\( 11 q + 2 q^{2} + 12 q^{4} - q^{5} - 11 q^{7} + 15 q^{8} - 12 q^{10} + 7 q^{11} - 24 q^{13} - 2 q^{14} - 6 q^{16} + 15 q^{17} - 19 q^{19} - 3 q^{20} - 3 q^{22} + 11 q^{23} + 10 q^{25} - 10 q^{26} - 12 q^{28} + 10 q^{29} - 20 q^{31} + 27 q^{32} - 9 q^{34} + q^{35} - 22 q^{37} - 8 q^{38} - 29 q^{40} - 9 q^{41} - 17 q^{43} + 9 q^{44} - 18 q^{46} + 7 q^{47} + 11 q^{49} + 47 q^{50} - 66 q^{52} + 28 q^{53} - 24 q^{55} - 15 q^{56} - 39 q^{58} - 35 q^{59} - 6 q^{61} - 18 q^{62} + 11 q^{64} + 43 q^{65} - 22 q^{67} + 12 q^{68} + 12 q^{70} + 22 q^{71} - 29 q^{73} - 14 q^{74} + 10 q^{76} - 7 q^{77} - 20 q^{79} - 66 q^{80} - 24 q^{82} - 17 q^{83} - 50 q^{85} + 12 q^{86} + 2 q^{88} + q^{89} + 24 q^{91} + 22 q^{92} + q^{94} - 10 q^{95} - 45 q^{97} + 2 q^{98}+O(q^{100}) \) 11 * q + 2 * q^2 + 12 * q^4 - q^5 - 11 * q^7 + 15 * q^8 - 12 * q^10 + 7 * q^11 - 24 * q^13 - 2 * q^14 - 6 * q^16 + 15 * q^17 - 19 * q^19 - 3 * q^20 - 3 * q^22 + 11 * q^23 + 10 * q^25 - 10 * q^26 - 12 * q^28 + 10 * q^29 - 20 * q^31 + 27 * q^32 - 9 * q^34 + q^35 - 22 * q^37 - 8 * q^38 - 29 * q^40 - 9 * q^41 - 17 * q^43 + 9 * q^44 - 18 * q^46 + 7 * q^47 + 11 * q^49 + 47 * q^50 - 66 * q^52 + 28 * q^53 - 24 * q^55 - 15 * q^56 - 39 * q^58 - 35 * q^59 - 6 * q^61 - 18 * q^62 + 11 * q^64 + 43 * q^65 - 22 * q^67 + 12 * q^68 + 12 * q^70 + 22 * q^71 - 29 * q^73 - 14 * q^74 + 10 * q^76 - 7 * q^77 - 20 * q^79 - 66 * q^80 - 24 * q^82 - 17 * q^83 - 50 * q^85 + 12 * q^86 + 2 * q^88 + q^89 + 24 * q^91 + 22 * q^92 + q^94 - 10 * q^95 - 45 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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