LMFDB - Embedded newform 8001.2.a.v.1.7

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:	$0$
Dimension:	$19$
Coefficient field:	$\mathbb{Q}[x]/(x^{19} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{19} - 4 x^{18} - 22 x^{17} + 101 x^{16} + 178 x^{15} - 1035 x^{14} - 583 x^{13} + 5572 x^{12} + \cdots + 210 $ x^19 - 4x^18 - 22x^17 + 101x^16 + 178x^15 - 1035x^14 - 583x^13 + 5572x^12 + 21x^11 - 17032x^10 + 4985x^9 + 29792x^8 - 13249x^7 - 28600x^6 + 14000x^5 + 13725x^4 - 5723x^3 - 2913x^2 + 608x + 210
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 2667)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$1.35771$ 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)

$f(q)$	$=$	$q-1.35771 q^{2} -0.156613 q^{4} -4.06757 q^{5} +1.00000 q^{7} +2.92806 q^{8} +O(q^{10})$	\(q-1.35771 q^{2} -0.156613 q^{4} -4.06757 q^{5} +1.00000 q^{7} +2.92806 q^{8} +5.52260 q^{10} -4.14107 q^{11} +3.33246 q^{13} -1.35771 q^{14} -3.66225 q^{16} +1.50369 q^{17} +5.47517 q^{19} +0.637032 q^{20} +5.62239 q^{22} +6.86481 q^{23} +11.5451 q^{25} -4.52453 q^{26} -0.156613 q^{28} +9.24608 q^{29} -7.77701 q^{31} -0.883841 q^{32} -2.04158 q^{34} -4.06757 q^{35} +8.24505 q^{37} -7.43372 q^{38} -11.9101 q^{40} +4.27213 q^{41} -4.47938 q^{43} +0.648543 q^{44} -9.32046 q^{46} +8.84174 q^{47} +1.00000 q^{49} -15.6750 q^{50} -0.521905 q^{52} -4.29491 q^{53} +16.8441 q^{55} +2.92806 q^{56} -12.5535 q^{58} -3.63171 q^{59} -3.59520 q^{61} +10.5590 q^{62} +8.52450 q^{64} -13.5550 q^{65} -0.939878 q^{67} -0.235496 q^{68} +5.52260 q^{70} -4.09929 q^{71} +6.53569 q^{73} -11.1944 q^{74} -0.857480 q^{76} -4.14107 q^{77} -7.66941 q^{79} +14.8964 q^{80} -5.80033 q^{82} +0.904225 q^{83} -6.11635 q^{85} +6.08172 q^{86} -12.1253 q^{88} +0.889515 q^{89} +3.33246 q^{91} -1.07512 q^{92} -12.0046 q^{94} -22.2706 q^{95} -10.8632 q^{97} -1.35771 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q - 4 q^{2} + 22 q^{4} - 5 q^{5} + 19 q^{7} - 9 q^{8}+O(q^{10}) $ 19 * q - 4 * q^2 + 22 * q^4 - 5 * q^5 + 19 * q^7 - 9 * q^8	$ 19 q - 4 q^{2} + 22 q^{4} - 5 q^{5} + 19 q^{7} - 9 q^{8} + 9 q^{11} + 24 q^{13} - 4 q^{14} + 20 q^{16} - 17 q^{17} + 23 q^{19} - 5 q^{20} - 3 q^{22} + 17 q^{23} + 38 q^{25} - 28 q^{26} + 22 q^{28} - 2 q^{29} + 16 q^{31} - 17 q^{32} + 29 q^{34} - 5 q^{35} + 56 q^{37} - 2 q^{38} - 13 q^{40} + 7 q^{41} + 19 q^{43} + 29 q^{44} + 10 q^{46} - 25 q^{47} + 19 q^{49} + 9 q^{50} + 16 q^{52} - 18 q^{53} + 10 q^{55} - 9 q^{56} + 31 q^{58} - 11 q^{59} + 26 q^{61} - 26 q^{62} + 45 q^{64} - 27 q^{65} + 24 q^{67} - 14 q^{68} + 32 q^{71} + 51 q^{73} + 12 q^{76} + 9 q^{77} + 30 q^{79} + 30 q^{80} - 52 q^{82} - q^{83} + 44 q^{85} + 24 q^{86} - 30 q^{88} - 5 q^{89} + 24 q^{91} + 88 q^{92} + 7 q^{94} + 24 q^{95} + 5 q^{97} - 4 q^{98}+O(q^{100}) $ 19 * q - 4 * q^2 + 22 * q^4 - 5 * q^5 + 19 * q^7 - 9 * q^8 + 9 * q^11 + 24 * q^13 - 4 * q^14 + 20 * q^16 - 17 * q^17 + 23 * q^19 - 5 * q^20 - 3 * q^22 + 17 * q^23 + 38 * q^25 - 28 * q^26 + 22 * q^28 - 2 * q^29 + 16 * q^31 - 17 * q^32 + 29 * q^34 - 5 * q^35 + 56 * q^37 - 2 * q^38 - 13 * q^40 + 7 * q^41 + 19 * q^43 + 29 * q^44 + 10 * q^46 - 25 * q^47 + 19 * q^49 + 9 * q^50 + 16 * q^52 - 18 * q^53 + 10 * q^55 - 9 * q^56 + 31 * q^58 - 11 * q^59 + 26 * q^61 - 26 * q^62 + 45 * q^64 - 27 * q^65 + 24 * q^67 - 14 * q^68 + 32 * q^71 + 51 * q^73 + 12 * q^76 + 9 * q^77 + 30 * q^79 + 30 * q^80 - 52 * q^82 - q^83 + 44 * q^85 + 24 * q^86 - 30 * q^88 - 5 * q^89 + 24 * q^91 + 88 * q^92 + 7 * q^94 + 24 * q^95 + 5 * q^97 - 4 * 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 $

$2$	−1.35771	−0.960049	−0.480024	−	0.877255i	$-0.659372\pi$
$2$	−1.35771	−0.960049	−0.480024	+	0.877255i	$0.659372\pi$
$3$	0	0
$3$	0	0
$4$	−0.156613	−0.0783063
$5$	−4.06757	−1.81907	−0.909536	−	0.415625i	$-0.863563\pi$
$5$	−4.06757	−1.81907	−0.909536	+	0.415625i	$0.863563\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	2.92806	1.03523
$9$	0	0
$10$	5.52260	1.74640
$11$	−4.14107	−1.24858	−0.624290	−	0.781193i	$-0.714612\pi$
$11$	−4.14107	−1.24858	−0.624290	+	0.781193i	$0.714612\pi$
$12$	0	0
$13$	3.33246	0.924258	0.462129	−	0.886813i	$-0.347086\pi$
$13$	3.33246	0.924258	0.462129	+	0.886813i	$0.347086\pi$
$14$	−1.35771	−0.362864
$15$	0	0
$16$	−3.66225	−0.915562
$17$	1.50369	0.364698	0.182349	−	0.983234i	$-0.441630\pi$
$17$	1.50369	0.364698	0.182349	+	0.983234i	$0.441630\pi$
$18$	0	0
$19$	5.47517	1.25609	0.628045	−	0.778177i	$-0.283855\pi$
$19$	5.47517	1.25609	0.628045	+	0.778177i	$0.283855\pi$
$20$	0.637032	0.142445
$21$	0	0
$22$	5.62239	1.19870
$23$	6.86481	1.43141	0.715706	−	0.698401i	$-0.246105\pi$
$23$	6.86481	1.43141	0.715706	+	0.698401i	$0.246105\pi$
$24$	0	0
$25$	11.5451	2.30902
$26$	−4.52453	−0.887333
$27$	0	0
$28$	−0.156613	−0.0295970
$29$	9.24608	1.71695	0.858477	−	0.512852i	$-0.171411\pi$
$29$	9.24608	1.71695	0.858477	+	0.512852i	$0.171411\pi$
$30$	0	0
$31$	−7.77701	−1.39679	−0.698396	−	0.715711i	$-0.746103\pi$
$31$	−7.77701	−1.39679	−0.698396	+	0.715711i	$0.746103\pi$
$32$	−0.883841	−0.156243
$33$	0	0
$34$	−2.04158	−0.350128
$35$	−4.06757	−0.687545
$36$	0	0
$37$	8.24505	1.35548	0.677739	−	0.735303i	$-0.262960\pi$
$37$	8.24505	1.35548	0.677739	+	0.735303i	$0.262960\pi$
$38$	−7.43372	−1.20591
$39$	0	0
$40$	−11.9101	−1.88315
$41$	4.27213	0.667194	0.333597	−	0.942716i	$-0.391737\pi$
$41$	4.27213	0.667194	0.333597	+	0.942716i	$0.391737\pi$
$42$	0	0
$43$	−4.47938	−0.683099	−0.341550	−	0.939864i	$-0.610952\pi$
$43$	−4.47938	−0.683099	−0.341550	+	0.939864i	$0.610952\pi$
$44$	0.648543	0.0977716
$45$	0	0
$46$	−9.32046	−1.37423
$47$	8.84174	1.28970	0.644850	−	0.764309i	$-0.276920\pi$
$47$	8.84174	1.28970	0.644850	+	0.764309i	$0.276920\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−15.6750	−2.21677
$51$	0	0
$52$	−0.521905	−0.0723752
$53$	−4.29491	−0.589952	−0.294976	−	0.955505i	$-0.595312\pi$
$53$	−4.29491	−0.589952	−0.294976	+	0.955505i	$0.595312\pi$
$54$	0	0
$55$	16.8441	2.27126
$56$	2.92806	0.391279
$57$	0	0
$58$	−12.5535	−1.64836
$59$	−3.63171	−0.472808	−0.236404	−	0.971655i	$-0.575969\pi$
$59$	−3.63171	−0.472808	−0.236404	+	0.971655i	$0.575969\pi$
$60$	0	0
$61$	−3.59520	−0.460318	−0.230159	−	0.973153i	$-0.573925\pi$
$61$	−3.59520	−0.460318	−0.230159	+	0.973153i	$0.573925\pi$
$62$	10.5590	1.34099
$63$	0	0
$64$	8.52450	1.06556
$65$	−13.5550	−1.68129
$66$	0	0
$67$	−0.939878	−0.114824	−0.0574122	−	0.998351i	$-0.518285\pi$
$67$	−0.939878	−0.114824	−0.0574122	+	0.998351i	$0.518285\pi$
$68$	−0.235496	−0.0285581
$69$	0	0
$70$	5.52260	0.660076
$71$	−4.09929	−0.486496	−0.243248	−	0.969964i	$-0.578213\pi$
$71$	−4.09929	−0.486496	−0.243248	+	0.969964i	$0.578213\pi$
$72$	0	0
$73$	6.53569	0.764945	0.382472	−	0.923967i	$-0.375073\pi$
$73$	6.53569	0.764945	0.382472	+	0.923967i	$0.375073\pi$
$74$	−11.1944	−1.30132
$75$	0	0
$76$	−0.857480	−0.0983597
$77$	−4.14107	−0.471919
$78$	0	0
$79$	−7.66941	−0.862876	−0.431438	−	0.902143i	$-0.641994\pi$
$79$	−7.66941	−0.862876	−0.431438	+	0.902143i	$0.641994\pi$
$80$	14.8964	1.66547
$81$	0	0
$82$	−5.80033	−0.640539
$83$	0.904225	0.0992515	0.0496258	−	0.998768i	$-0.484197\pi$
$83$	0.904225	0.0992515	0.0496258	+	0.998768i	$0.484197\pi$
$84$	0	0
$85$	−6.11635	−0.663412
$86$	6.08172	0.655808
$87$	0	0
$88$	−12.1253	−1.29256
$89$	0.889515	0.0942884	0.0471442	−	0.998888i	$-0.484988\pi$
$89$	0.889515	0.0942884	0.0471442	+	0.998888i	$0.484988\pi$
$90$	0	0
$91$	3.33246	0.349337
$92$	−1.07512	−0.112089
$93$	0	0
$94$	−12.0046	−1.23818
$95$	−22.2706	−2.28492
$96$	0	0
$97$	−10.8632	−1.10299	−0.551497	−	0.834177i	$-0.685943\pi$
$97$	−10.8632	−1.10299	−0.551497	+	0.834177i	$0.685943\pi$
$98$	−1.35771	−0.137150
$99$	0	0
$100$	−1.80811	−0.180811
$101$	−7.33839	−0.730197	−0.365098	−	0.930969i	$-0.618965\pi$
$101$	−7.33839	−0.730197	−0.365098	+	0.930969i	$0.618965\pi$
$102$	0	0
$103$	4.53321	0.446670	0.223335	−	0.974742i	$-0.428306\pi$
$103$	4.53321	0.446670	0.223335	+	0.974742i	$0.428306\pi$
$104$	9.75766	0.956817
$105$	0	0
$106$	5.83126	0.566382
$107$	3.67395	0.355175	0.177587	−	0.984105i	$-0.443171\pi$
$107$	3.67395	0.355175	0.177587	+	0.984105i	$0.443171\pi$
$108$	0	0
$109$	14.0170	1.34259	0.671293	−	0.741192i	$-0.265739\pi$
$109$	14.0170	1.34259	0.671293	+	0.741192i	$0.265739\pi$
$110$	−22.8695	−2.18052
$111$	0	0
$112$	−3.66225	−0.346050
$113$	−10.5328	−0.990845	−0.495423	−	0.868652i	$-0.664987\pi$
$113$	−10.5328	−0.990845	−0.495423	+	0.868652i	$0.664987\pi$
$114$	0	0
$115$	−27.9231	−2.60384
$116$	−1.44805	−0.134448
$117$	0	0
$118$	4.93082	0.453919
$119$	1.50369	0.137843
$120$	0	0
$121$	6.14846	0.558951
$122$	4.88126	0.441928
$123$	0	0
$124$	1.21798	0.109378
$125$	−26.6227	−2.38121
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	−9.80615	−0.866749
$129$	0	0
$130$	18.4038	1.61412
$131$	13.5622	1.18493	0.592466	−	0.805595i	$-0.298154\pi$
$131$	13.5622	1.18493	0.592466	+	0.805595i	$0.298154\pi$
$132$	0	0
$133$	5.47517	0.474758
$134$	1.27609	0.110237
$135$	0	0
$136$	4.40289	0.377545
$137$	21.5286	1.83931	0.919656	−	0.392724i	$-0.128467\pi$
$137$	21.5286	1.83931	0.919656	+	0.392724i	$0.128467\pi$
$138$	0	0
$139$	6.20276	0.526111	0.263056	−	0.964781i	$-0.415270\pi$
$139$	6.20276	0.526111	0.263056	+	0.964781i	$0.415270\pi$
$140$	0.637032	0.0538390
$141$	0	0
$142$	5.56566	0.467060
$143$	−13.8000	−1.15401
$144$	0	0
$145$	−37.6091	−3.12326
$146$	−8.87360	−0.734384
$147$	0	0
$148$	−1.29128	−0.106142
$149$	−20.7051	−1.69622	−0.848112	−	0.529817i	$-0.822261\pi$
$149$	−20.7051	−1.69622	−0.848112	+	0.529817i	$0.822261\pi$
$150$	0	0
$151$	4.24264	0.345261	0.172631	−	0.984987i	$-0.444773\pi$
$151$	4.24264	0.345261	0.172631	+	0.984987i	$0.444773\pi$
$152$	16.0316	1.30034
$153$	0	0
$154$	5.62239	0.453065
$155$	31.6335	2.54087
$156$	0	0
$157$	−7.13470	−0.569411	−0.284705	−	0.958615i	$-0.591896\pi$
$157$	−7.13470	−0.569411	−0.284705	+	0.958615i	$0.591896\pi$
$158$	10.4129	0.828403
$159$	0	0
$160$	3.59509	0.284216
$161$	6.86481	0.541023
$162$	0	0
$163$	24.3942	1.91070	0.955349	−	0.295479i	$-0.0954793\pi$
$163$	24.3942	1.91070	0.955349	+	0.295479i	$0.0954793\pi$
$164$	−0.669069	−0.0522455
$165$	0	0
$166$	−1.22768	−0.0952863
$167$	15.1000	1.16847	0.584237	−	0.811583i	$-0.301394\pi$
$167$	15.1000	1.16847	0.584237	+	0.811583i	$0.301394\pi$
$168$	0	0
$169$	−1.89470	−0.145746
$170$	8.30426	0.636908
$171$	0	0
$172$	0.701527	0.0534909
$173$	3.35769	0.255281	0.127640	−	0.991821i	$-0.459260\pi$
$173$	3.35769	0.255281	0.127640	+	0.991821i	$0.459260\pi$
$174$	0	0
$175$	11.5451	0.872729
$176$	15.1656	1.14315
$177$	0	0
$178$	−1.20771	−0.0905215
$179$	2.72845	0.203934	0.101967	−	0.994788i	$-0.467486\pi$
$179$	2.72845	0.203934	0.101967	+	0.994788i	$0.467486\pi$
$180$	0	0
$181$	−15.5927	−1.15899	−0.579497	−	0.814974i	$-0.696751\pi$
$181$	−15.5927	−1.15899	−0.579497	+	0.814974i	$0.696751\pi$
$182$	−4.52453	−0.335380
$183$	0	0
$184$	20.1006	1.48184
$185$	−33.5373	−2.46571
$186$	0	0
$187$	−6.22688	−0.455354
$188$	−1.38473	−0.100992
$189$	0	0
$190$	30.2372	2.19363
$191$	6.09888	0.441299	0.220650	−	0.975353i	$-0.429182\pi$
$191$	6.09888	0.441299	0.220650	+	0.975353i	$0.429182\pi$
$192$	0	0
$193$	−11.7723	−0.847390	−0.423695	−	0.905805i	$-0.639267\pi$
$193$	−11.7723	−0.847390	−0.423695	+	0.905805i	$0.639267\pi$
$194$	14.7492	1.05893
$195$	0	0
$196$	−0.156613	−0.0111866
$197$	−9.06253	−0.645679	−0.322839	−	0.946454i	$-0.604637\pi$
$197$	−9.06253	−0.645679	−0.322839	+	0.946454i	$0.604637\pi$
$198$	0	0
$199$	−14.7682	−1.04689	−0.523446	−	0.852059i	$-0.675354\pi$
$199$	−14.7682	−1.04689	−0.523446	+	0.852059i	$0.675354\pi$
$200$	33.8048	2.39036
$201$	0	0
$202$	9.96343	0.701024
$203$	9.24608	0.648948
$204$	0	0
$205$	−17.3772	−1.21367
$206$	−6.15480	−0.428825
$207$	0	0
$208$	−12.2043	−0.846216
$209$	−22.6731	−1.56833
$210$	0	0
$211$	−8.40152	−0.578385	−0.289192	−	0.957271i	$-0.593387\pi$
$211$	−8.40152	−0.578385	−0.289192	+	0.957271i	$0.593387\pi$
$212$	0.672637	0.0461969
$213$	0	0
$214$	−4.98818	−0.340985
$215$	18.2202	1.24261
$216$	0	0
$217$	−7.77701	−0.527938
$218$	−19.0311	−1.28895
$219$	0	0
$220$	−2.63799	−0.177854
$221$	5.01098	0.337075
$222$	0	0
$223$	20.1977	1.35254	0.676268	−	0.736656i	$-0.263596\pi$
$223$	20.1977	1.35254	0.676268	+	0.736656i	$0.263596\pi$
$224$	−0.883841	−0.0590541
$225$	0	0
$226$	14.3006	0.951260
$227$	6.19838	0.411401	0.205700	−	0.978615i	$-0.434053\pi$
$227$	6.19838	0.411401	0.205700	+	0.978615i	$0.434053\pi$
$228$	0	0
$229$	−19.1653	−1.26648	−0.633240	−	0.773955i	$-0.718275\pi$
$229$	−19.1653	−1.26648	−0.633240	+	0.773955i	$0.718275\pi$
$230$	37.9116	2.49982
$231$	0	0
$232$	27.0731	1.77744
$233$	28.1199	1.84220	0.921098	−	0.389331i	$-0.127294\pi$
$233$	28.1199	1.84220	0.921098	+	0.389331i	$0.127294\pi$
$234$	0	0
$235$	−35.9644	−2.34606
$236$	0.568771	0.0370238
$237$	0	0
$238$	−2.04158	−0.132336
$239$	1.16086	0.0750897	0.0375448	−	0.999295i	$-0.488046\pi$
$239$	1.16086	0.0750897	0.0375448	+	0.999295i	$0.488046\pi$
$240$	0	0
$241$	−3.36538	−0.216783	−0.108391	−	0.994108i	$-0.534570\pi$
$241$	−3.36538	−0.216783	−0.108391	+	0.994108i	$0.534570\pi$
$242$	−8.34785	−0.536620
$243$	0	0
$244$	0.563054	0.0360458
$245$	−4.06757	−0.259867
$246$	0	0
$247$	18.2458	1.16095
$248$	−22.7716	−1.44600
$249$	0	0
$250$	36.1460	2.28607
$251$	−27.6043	−1.74237	−0.871185	−	0.490955i	$-0.836648\pi$
$251$	−27.6043	−1.74237	−0.871185	+	0.490955i	$0.836648\pi$
$252$	0	0
$253$	−28.4277	−1.78723
$254$	−1.35771	−0.0851906
$255$	0	0
$256$	−3.73505	−0.233441
$257$	26.7692	1.66981	0.834907	−	0.550391i	$-0.185521\pi$
$257$	26.7692	1.66981	0.834907	+	0.550391i	$0.185521\pi$
$258$	0	0
$259$	8.24505	0.512322
$260$	2.12288	0.131656
$261$	0	0
$262$	−18.4136	−1.13759
$263$	−8.17676	−0.504201	−0.252100	−	0.967701i	$-0.581121\pi$
$263$	−8.17676	−0.504201	−0.252100	+	0.967701i	$0.581121\pi$
$264$	0	0
$265$	17.4698	1.07316
$266$	−7.43372	−0.455790
$267$	0	0
$268$	0.147197	0.00899147
$269$	−10.2343	−0.623994	−0.311997	−	0.950083i	$-0.600998\pi$
$269$	−10.2343	−0.623994	−0.311997	+	0.950083i	$0.600998\pi$
$270$	0	0
$271$	15.6853	0.952814	0.476407	−	0.879225i	$-0.341939\pi$
$271$	15.6853	0.952814	0.476407	+	0.879225i	$0.341939\pi$
$272$	−5.50688	−0.333904
$273$	0	0
$274$	−29.2297	−1.76583
$275$	−47.8091	−2.88300
$276$	0	0
$277$	15.1536	0.910489	0.455244	−	0.890366i	$-0.349552\pi$
$277$	15.1536	0.910489	0.455244	+	0.890366i	$0.349552\pi$
$278$	−8.42157	−0.505092
$279$	0	0
$280$	−11.9101	−0.711764
$281$	−19.6547	−1.17250	−0.586250	−	0.810130i	$-0.699396\pi$
$281$	−19.6547	−1.17250	−0.586250	+	0.810130i	$0.699396\pi$
$282$	0	0
$283$	−29.3642	−1.74552	−0.872761	−	0.488148i	$-0.837672\pi$
$283$	−29.3642	−1.74552	−0.872761	+	0.488148i	$0.837672\pi$
$284$	0.642000	0.0380957
$285$	0	0
$286$	18.7364	1.10791
$287$	4.27213	0.252176
$288$	0	0
$289$	−14.7389	−0.866995
$290$	51.0624	2.99849
$291$	0	0
$292$	−1.02357	−0.0599000
$293$	−4.40237	−0.257189	−0.128594	−	0.991697i	$-0.541047\pi$
$293$	−4.40237	−0.257189	−0.128594	+	0.991697i	$0.541047\pi$
$294$	0	0
$295$	14.7722	0.860072
$296$	24.1420	1.40323
$297$	0	0
$298$	28.1115	1.62846
$299$	22.8767	1.32300
$300$	0	0
$301$	−4.47938	−0.258187
$302$	−5.76030	−0.331468
$303$	0	0
$304$	−20.0514	−1.15003
$305$	14.6237	0.837352
$306$	0	0
$307$	33.2964	1.90032	0.950162	−	0.311756i	$-0.100917\pi$
$307$	33.2964	1.90032	0.950162	+	0.311756i	$0.100917\pi$
$308$	0.648543	0.0369542
$309$	0	0
$310$	−42.9493	−2.43936
$311$	−30.8539	−1.74956	−0.874782	−	0.484517i	$-0.838995\pi$
$311$	−30.8539	−1.74956	−0.874782	+	0.484517i	$0.838995\pi$
$312$	0	0
$313$	−9.70430	−0.548519	−0.274260	−	0.961656i	$-0.588433\pi$
$313$	−9.70430	−0.548519	−0.274260	+	0.961656i	$0.588433\pi$
$314$	9.68688	0.546662
$315$	0	0
$316$	1.20113	0.0675686
$317$	12.8962	0.724322	0.362161	−	0.932116i	$-0.382039\pi$
$317$	12.8962	0.724322	0.362161	+	0.932116i	$0.382039\pi$
$318$	0	0
$319$	−38.2887	−2.14375
$320$	−34.6740	−1.93833
$321$	0	0
$322$	−9.32046	−0.519409
$323$	8.23295	0.458094
$324$	0	0
$325$	38.4736	2.13413
$326$	−33.1203	−1.83436
$327$	0	0
$328$	12.5091	0.690697
$329$	8.84174	0.487461
$330$	0	0
$331$	−33.3548	−1.83335	−0.916674	−	0.399636i	$-0.869137\pi$
$331$	−33.3548	−1.83335	−0.916674	+	0.399636i	$0.869137\pi$
$332$	−0.141613	−0.00777202
$333$	0	0
$334$	−20.5015	−1.12179
$335$	3.82302	0.208874
$336$	0	0
$337$	31.6811	1.72578	0.862891	−	0.505390i	$-0.168651\pi$
$337$	31.6811	1.72578	0.862891	+	0.505390i	$0.168651\pi$
$338$	2.57246	0.139924
$339$	0	0
$340$	0.957898	0.0519493
$341$	32.2052	1.74401
$342$	0	0
$343$	1.00000	0.0539949
$344$	−13.1159	−0.707162
$345$	0	0
$346$	−4.55879	−0.245082
$347$	32.2561	1.73160	0.865800	−	0.500390i	$-0.166810\pi$
$347$	32.2561	1.73160	0.865800	+	0.500390i	$0.166810\pi$
$348$	0	0
$349$	21.2214	1.13596	0.567978	−	0.823044i	$-0.307726\pi$
$349$	21.2214	1.13596	0.567978	+	0.823044i	$0.307726\pi$
$350$	−15.6750	−0.837862
$351$	0	0
$352$	3.66005	0.195081
$353$	−21.6154	−1.15047	−0.575235	−	0.817988i	$-0.695089\pi$
$353$	−21.6154	−1.15047	−0.575235	+	0.817988i	$0.695089\pi$
$354$	0	0
$355$	16.6741	0.884971
$356$	−0.139309	−0.00738337
$357$	0	0
$358$	−3.70445	−0.195786
$359$	9.14005	0.482393	0.241197	−	0.970476i	$-0.422460\pi$
$359$	9.14005	0.482393	0.241197	+	0.970476i	$0.422460\pi$
$360$	0	0
$361$	10.9775	0.577763
$362$	21.1704	1.11269
$363$	0	0
$364$	−0.521905	−0.0273553
$365$	−26.5844	−1.39149
$366$	0	0
$367$	17.4322	0.909954	0.454977	−	0.890503i	$-0.349647\pi$
$367$	17.4322	0.909954	0.454977	+	0.890503i	$0.349647\pi$
$368$	−25.1406	−1.31055
$369$	0	0
$370$	45.5341	2.36720
$371$	−4.29491	−0.222981
$372$	0	0
$373$	23.2475	1.20371	0.601855	−	0.798606i	$-0.294429\pi$
$373$	23.2475	1.20371	0.601855	+	0.798606i	$0.294429\pi$
$374$	8.45432	0.437162
$375$	0	0
$376$	25.8892	1.33513
$377$	30.8122	1.58691
$378$	0	0
$379$	−19.8749	−1.02090	−0.510452	−	0.859906i	$-0.670522\pi$
$379$	−19.8749	−1.02090	−0.510452	+	0.859906i	$0.670522\pi$
$380$	3.48786	0.178923
$381$	0	0
$382$	−8.28053	−0.423669
$383$	−29.4149	−1.50303	−0.751514	−	0.659717i	$-0.770676\pi$
$383$	−29.4149	−1.50303	−0.751514	+	0.659717i	$0.770676\pi$
$384$	0	0
$385$	16.8441	0.858454
$386$	15.9834	0.813536
$387$	0	0
$388$	1.70132	0.0863713
$389$	7.56699	0.383662	0.191831	−	0.981428i	$-0.438557\pi$
$389$	7.56699	0.383662	0.191831	+	0.981428i	$0.438557\pi$
$390$	0	0
$391$	10.3225	0.522033
$392$	2.92806	0.147890
$393$	0	0
$394$	12.3043	0.619883
$395$	31.1958	1.56963
$396$	0	0
$397$	12.8178	0.643305	0.321653	−	0.946858i	$-0.395762\pi$
$397$	12.8178	0.643305	0.321653	+	0.946858i	$0.395762\pi$
$398$	20.0510	1.00507
$399$	0	0
$400$	−42.2811	−2.11405
$401$	−33.4344	−1.66963	−0.834816	−	0.550528i	$-0.814426\pi$
$401$	−33.4344	−1.66963	−0.834816	+	0.550528i	$0.814426\pi$
$402$	0	0
$403$	−25.9166	−1.29100
$404$	1.14928	0.0571790
$405$	0	0
$406$	−12.5535	−0.623021
$407$	−34.1433	−1.69242
$408$	0	0
$409$	−17.1785	−0.849422	−0.424711	−	0.905329i	$-0.639624\pi$
$409$	−17.1785	−0.849422	−0.424711	+	0.905329i	$0.639624\pi$
$410$	23.5932	1.16519
$411$	0	0
$412$	−0.709957	−0.0349771
$413$	−3.63171	−0.178705
$414$	0	0
$415$	−3.67800	−0.180546
$416$	−2.94537	−0.144408
$417$	0	0
$418$	30.7835	1.50567
$419$	29.5133	1.44182	0.720910	−	0.693029i	$-0.243724\pi$
$419$	29.5133	1.44182	0.720910	+	0.693029i	$0.243724\pi$
$420$	0	0
$421$	−20.9064	−1.01892	−0.509458	−	0.860495i	$-0.670154\pi$
$421$	−20.9064	−1.01892	−0.509458	+	0.860495i	$0.670154\pi$
$422$	11.4069	0.555278
$423$	0	0
$424$	−12.5758	−0.610734
$425$	17.3603	0.842096
$426$	0	0
$427$	−3.59520	−0.173984
$428$	−0.575387	−0.0278124
$429$	0	0
$430$	−24.7378	−1.19296
$431$	21.1316	1.01787	0.508937	−	0.860804i	$-0.330039\pi$
$431$	21.1316	1.01787	0.508937	+	0.860804i	$0.330039\pi$
$432$	0	0
$433$	4.15711	0.199778	0.0998890	−	0.994999i	$-0.468151\pi$
$433$	4.15711	0.199778	0.0998890	+	0.994999i	$0.468151\pi$
$434$	10.5590	0.506846
$435$	0	0
$436$	−2.19524	−0.105133
$437$	37.5860	1.79798
$438$	0	0
$439$	−6.72225	−0.320835	−0.160418	−	0.987049i	$-0.551284\pi$
$439$	−6.72225	−0.320835	−0.160418	+	0.987049i	$0.551284\pi$
$440$	49.3205	2.35126
$441$	0	0
$442$	−6.80348	−0.323609
$443$	22.5903	1.07330	0.536649	−	0.843805i	$-0.319690\pi$
$443$	22.5903	1.07330	0.536649	+	0.843805i	$0.319690\pi$
$444$	0	0
$445$	−3.61816	−0.171517
$446$	−27.4227	−1.29850
$447$	0	0
$448$	8.52450	0.402745
$449$	40.0006	1.88775	0.943873	−	0.330308i	$-0.107153\pi$
$449$	40.0006	1.88775	0.943873	+	0.330308i	$0.107153\pi$
$450$	0	0
$451$	−17.6912	−0.833045
$452$	1.64957	0.0775894
$453$	0	0
$454$	−8.41563	−0.394965
$455$	−13.5550	−0.635469
$456$	0	0
$457$	5.14695	0.240764	0.120382	−	0.992728i	$-0.461588\pi$
$457$	5.14695	0.240764	0.120382	+	0.992728i	$0.461588\pi$
$458$	26.0210	1.21588
$459$	0	0
$460$	4.37311	0.203897
$461$	1.65601	0.0771279	0.0385639	−	0.999256i	$-0.487722\pi$
$461$	1.65601	0.0771279	0.0385639	+	0.999256i	$0.487722\pi$
$462$	0	0
$463$	29.8032	1.38507	0.692537	−	0.721383i	$-0.256493\pi$
$463$	29.8032	1.38507	0.692537	+	0.721383i	$0.256493\pi$
$464$	−33.8614	−1.57198
$465$	0	0
$466$	−38.1788	−1.76860
$467$	12.4102	0.574277	0.287139	−	0.957889i	$-0.407296\pi$
$467$	12.4102	0.574277	0.287139	+	0.957889i	$0.407296\pi$
$468$	0	0
$469$	−0.939878	−0.0433995
$470$	48.8294	2.25233
$471$	0	0
$472$	−10.6339	−0.489464
$473$	18.5494	0.852904
$474$	0	0
$475$	63.2115	2.90034
$476$	−0.235496	−0.0107940
$477$	0	0
$478$	−1.57611	−0.0720898
$479$	23.2909	1.06419	0.532095	−	0.846685i	$-0.321405\pi$
$479$	23.2909	1.06419	0.532095	+	0.846685i	$0.321405\pi$
$480$	0	0
$481$	27.4763	1.25281
$482$	4.56922	0.208122
$483$	0	0
$484$	−0.962926	−0.0437694
$485$	44.1869	2.00642
$486$	0	0
$487$	−11.8607	−0.537460	−0.268730	−	0.963216i	$-0.586604\pi$
$487$	−11.8607	−0.537460	−0.268730	+	0.963216i	$0.586604\pi$
$488$	−10.5270	−0.476534
$489$	0	0
$490$	5.52260	0.249485
$491$	34.9281	1.57628	0.788142	−	0.615493i	$-0.211043\pi$
$491$	34.9281	1.57628	0.788142	+	0.615493i	$0.211043\pi$
$492$	0	0
$493$	13.9032	0.626170
$494$	−24.7726	−1.11457
$495$	0	0
$496$	28.4813	1.27885
$497$	−4.09929	−0.183878
$498$	0	0
$499$	13.9524	0.624597	0.312299	−	0.949984i	$-0.398901\pi$
$499$	13.9524	0.624597	0.312299	+	0.949984i	$0.398901\pi$
$500$	4.16945	0.186463
$501$	0	0
$502$	37.4788	1.67276
$503$	33.5301	1.49504	0.747518	−	0.664242i	$-0.231246\pi$
$503$	33.5301	1.49504	0.747518	+	0.664242i	$0.231246\pi$
$504$	0	0
$505$	29.8494	1.32828
$506$	38.5967	1.71583
$507$	0	0
$508$	−0.156613	−0.00694856
$509$	−8.96936	−0.397560	−0.198780	−	0.980044i	$-0.563698\pi$
$509$	−8.96936	−0.397560	−0.198780	+	0.980044i	$0.563698\pi$
$510$	0	0
$511$	6.53569	0.289122
$512$	24.6834	1.09086
$513$	0	0
$514$	−36.3449	−1.60310
$515$	−18.4391	−0.812525
$516$	0	0
$517$	−36.6143	−1.61029
$518$	−11.1944	−0.491855
$519$	0	0
$520$	−39.6899	−1.74052
$521$	−3.45790	−0.151493	−0.0757467	−	0.997127i	$-0.524134\pi$
$521$	−3.45790	−0.151493	−0.0757467	+	0.997127i	$0.524134\pi$
$522$	0	0
$523$	7.35603	0.321657	0.160828	−	0.986982i	$-0.448583\pi$
$523$	7.35603	0.321657	0.160828	+	0.986982i	$0.448583\pi$
$524$	−2.12401	−0.0927877
$525$	0	0
$526$	11.1017	0.484057
$527$	−11.6942	−0.509407
$528$	0	0
$529$	24.1257	1.04894
$530$	−23.7191	−1.03029
$531$	0	0
$532$	−0.857480	−0.0371765
$533$	14.2367	0.616660
$534$	0	0
$535$	−14.9441	−0.646088
$536$	−2.75202	−0.118869
$537$	0	0
$538$	13.8952	0.599065
$539$	−4.14107	−0.178369
$540$	0	0
$541$	13.1595	0.565769	0.282884	−	0.959154i	$-0.408709\pi$
$541$	13.1595	0.565769	0.282884	+	0.959154i	$0.408709\pi$
$542$	−21.2961	−0.914748
$543$	0	0
$544$	−1.32902	−0.0569813
$545$	−57.0151	−2.44226
$546$	0	0
$547$	−41.0280	−1.75423	−0.877116	−	0.480279i	$-0.840535\pi$
$547$	−41.0280	−1.75423	−0.877116	+	0.480279i	$0.840535\pi$
$548$	−3.37165	−0.144030
$549$	0	0
$550$	64.9111	2.76782
$551$	50.6239	2.15665
$552$	0	0
$553$	−7.66941	−0.326136
$554$	−20.5742	−0.874114
$555$	0	0
$556$	−0.971430	−0.0411978
$557$	−22.0235	−0.933165	−0.466582	−	0.884478i	$-0.654515\pi$
$557$	−22.0235	−0.933165	−0.466582	+	0.884478i	$0.654515\pi$
$558$	0	0
$559$	−14.9274	−0.631360
$560$	14.8964	0.629490
$561$	0	0
$562$	26.6854	1.12566
$563$	−25.8895	−1.09111	−0.545556	−	0.838074i	$-0.683682\pi$
$563$	−25.8895	−1.09111	−0.545556	+	0.838074i	$0.683682\pi$
$564$	0	0
$565$	42.8430	1.80242
$566$	39.8682	1.67579
$567$	0	0
$568$	−12.0030	−0.503634
$569$	−16.3568	−0.685713	−0.342856	−	0.939388i	$-0.611394\pi$
$569$	−16.3568	−0.685713	−0.342856	+	0.939388i	$0.611394\pi$
$570$	0	0
$571$	−43.7952	−1.83277	−0.916385	−	0.400298i	$-0.868907\pi$
$571$	−43.7952	−1.83277	−0.916385	+	0.400298i	$0.868907\pi$
$572$	2.16125	0.0903662
$573$	0	0
$574$	−5.80033	−0.242101
$575$	79.2551	3.30516
$576$	0	0
$577$	16.4323	0.684087	0.342043	−	0.939684i	$-0.388881\pi$
$577$	16.4323	0.684087	0.342043	+	0.939684i	$0.388881\pi$
$578$	20.0112	0.832358
$579$	0	0
$580$	5.89005	0.244571
$581$	0.904225	0.0375136
$582$	0	0
$583$	17.7855	0.736601
$584$	19.1369	0.791891
$585$	0	0
$586$	5.97715	0.246914
$587$	18.2183	0.751951	0.375976	−	0.926630i	$-0.377308\pi$
$587$	18.2183	0.751951	0.375976	+	0.926630i	$0.377308\pi$
$588$	0	0
$589$	−42.5805	−1.75450
$590$	−20.0565	−0.825711
$591$	0	0
$592$	−30.1954	−1.24102
$593$	−0.539852	−0.0221691	−0.0110845	−	0.999939i	$-0.503528\pi$
$593$	−0.539852	−0.0221691	−0.0110845	+	0.999939i	$0.503528\pi$
$594$	0	0
$595$	−6.11635	−0.250746
$596$	3.24267	0.132825
$597$	0	0
$598$	−31.0601	−1.27014
$599$	28.9266	1.18191	0.590954	−	0.806705i	$-0.298752\pi$
$599$	28.9266	1.18191	0.590954	+	0.806705i	$0.298752\pi$
$600$	0	0
$601$	6.27630	0.256016	0.128008	−	0.991773i	$-0.459142\pi$
$601$	6.27630	0.256016	0.128008	+	0.991773i	$0.459142\pi$
$602$	6.08172	0.247872
$603$	0	0
$604$	−0.664451	−0.0270361
$605$	−25.0093	−1.01677
$606$	0	0
$607$	−11.9901	−0.486664	−0.243332	−	0.969943i	$-0.578240\pi$
$607$	−11.9901	−0.486664	−0.243332	+	0.969943i	$0.578240\pi$
$608$	−4.83918	−0.196255
$609$	0	0
$610$	−19.8548	−0.803899
$611$	29.4648	1.19202
$612$	0	0
$613$	−18.0885	−0.730588	−0.365294	−	0.930892i	$-0.619031\pi$
$613$	−18.0885	−0.730588	−0.365294	+	0.930892i	$0.619031\pi$
$614$	−45.2069	−1.82440
$615$	0	0
$616$	−12.1253	−0.488543
$617$	33.8316	1.36201	0.681004	−	0.732280i	$-0.261544\pi$
$617$	33.8316	1.36201	0.681004	+	0.732280i	$0.261544\pi$
$618$	0	0
$619$	17.0117	0.683759	0.341879	−	0.939744i	$-0.388937\pi$
$619$	17.0117	0.683759	0.341879	+	0.939744i	$0.388937\pi$
$620$	−4.95421	−0.198966
$621$	0	0
$622$	41.8908	1.67967
$623$	0.889515	0.0356377
$624$	0	0
$625$	50.5641	2.02256
$626$	13.1757	0.526605
$627$	0	0
$628$	1.11738	0.0445884
$629$	12.3980	0.494340
$630$	0	0
$631$	40.9850	1.63159	0.815794	−	0.578342i	$-0.196300\pi$
$631$	40.9850	1.63159	0.815794	+	0.578342i	$0.196300\pi$
$632$	−22.4565	−0.893272
$633$	0	0
$634$	−17.5093	−0.695384
$635$	−4.06757	−0.161417
$636$	0	0
$637$	3.33246	0.132037
$638$	51.9851	2.05811
$639$	0	0
$640$	39.8872	1.57668
$641$	−18.3751	−0.725773	−0.362887	−	0.931833i	$-0.618209\pi$
$641$	−18.3751	−0.725773	−0.362887	+	0.931833i	$0.618209\pi$
$642$	0	0
$643$	−40.6254	−1.60211	−0.801054	−	0.598592i	$-0.795727\pi$
$643$	−40.6254	−1.60211	−0.801054	+	0.598592i	$0.795727\pi$
$644$	−1.07512	−0.0423655
$645$	0	0
$646$	−11.1780	−0.439792
$647$	17.3245	0.681095	0.340548	−	0.940227i	$-0.389388\pi$
$647$	17.3245	0.681095	0.340548	+	0.940227i	$0.389388\pi$
$648$	0	0
$649$	15.0392	0.590339
$650$	−52.2362	−2.04887
$651$	0	0
$652$	−3.82043	−0.149620
$653$	−3.87508	−0.151644	−0.0758219	−	0.997121i	$-0.524158\pi$
$653$	−3.87508	−0.151644	−0.0758219	+	0.997121i	$0.524158\pi$
$654$	0	0
$655$	−55.1651	−2.15548
$656$	−15.6456	−0.610858
$657$	0	0
$658$	−12.0046	−0.467986
$659$	−23.7917	−0.926795	−0.463397	−	0.886151i	$-0.653370\pi$
$659$	−23.7917	−0.926795	−0.463397	+	0.886151i	$0.653370\pi$
$660$	0	0
$661$	27.3539	1.06394	0.531972	−	0.846762i	$-0.321451\pi$
$661$	27.3539	1.06394	0.531972	+	0.846762i	$0.321451\pi$
$662$	45.2863	1.76010
$663$	0	0
$664$	2.64763	0.102748
$665$	−22.2706	−0.863618
$666$	0	0
$667$	63.4726	2.45767
$668$	−2.36485	−0.0914988
$669$	0	0
$670$	−5.19057	−0.200529
$671$	14.8880	0.574744
$672$	0	0
$673$	26.3576	1.01601	0.508006	−	0.861354i	$-0.330383\pi$
$673$	26.3576	1.01601	0.508006	+	0.861354i	$0.330383\pi$
$674$	−43.0139	−1.65684
$675$	0	0
$676$	0.296734	0.0114128
$677$	−39.4511	−1.51623	−0.758115	−	0.652121i	$-0.773879\pi$
$677$	−39.4511	−1.51623	−0.758115	+	0.652121i	$0.773879\pi$
$678$	0	0
$679$	−10.8632	−0.416892
$680$	−17.9091	−0.686782
$681$	0	0
$682$	−43.7254	−1.67433
$683$	8.64860	0.330930	0.165465	−	0.986216i	$-0.447088\pi$
$683$	8.64860	0.330930	0.165465	+	0.986216i	$0.447088\pi$
$684$	0	0
$685$	−87.5691	−3.34584
$686$	−1.35771	−0.0518378
$687$	0	0
$688$	16.4046	0.625419
$689$	−14.3126	−0.545268
$690$	0	0
$691$	−9.10919	−0.346530	−0.173265	−	0.984875i	$-0.555432\pi$
$691$	−9.10919	−0.346530	−0.173265	+	0.984875i	$0.555432\pi$
$692$	−0.525857	−0.0199901
$693$	0	0
$694$	−43.7946	−1.66242
$695$	−25.2301	−0.957034
$696$	0	0
$697$	6.42395	0.243324
$698$	−28.8126	−1.09057
$699$	0	0
$700$	−1.80811	−0.0683401
$701$	−9.44526	−0.356743	−0.178371	−	0.983963i	$-0.557083\pi$
$701$	−9.44526	−0.356743	−0.178371	+	0.983963i	$0.557083\pi$
$702$	0	0
$703$	45.1431	1.70260
$704$	−35.3005	−1.33044
$705$	0	0
$706$	29.3475	1.10451
$707$	−7.33839	−0.275988
$708$	0	0
$709$	−4.58727	−0.172279	−0.0861393	−	0.996283i	$-0.527453\pi$
$709$	−4.58727	−0.172279	−0.0861393	+	0.996283i	$0.527453\pi$
$710$	−22.6387	−0.849616
$711$	0	0
$712$	2.60456	0.0976099
$713$	−53.3878	−1.99939
$714$	0	0
$715$	56.1323	2.09923
$716$	−0.427309	−0.0159693
$717$	0	0
$718$	−12.4096	−0.463121
$719$	2.72744	0.101716	0.0508581	−	0.998706i	$-0.483804\pi$
$719$	2.72744	0.101716	0.0508581	+	0.998706i	$0.483804\pi$
$720$	0	0
$721$	4.53321	0.168825
$722$	−14.9043	−0.554681
$723$	0	0
$724$	2.44201	0.0907565
$725$	106.747	3.96449
$726$	0	0
$727$	18.5557	0.688194	0.344097	−	0.938934i	$-0.388185\pi$
$727$	18.5557	0.688194	0.344097	+	0.938934i	$0.388185\pi$
$728$	9.75766	0.361643
$729$	0	0
$730$	36.0940	1.33590
$731$	−6.73559	−0.249125
$732$	0	0
$733$	28.8879	1.06700	0.533499	−	0.845801i	$-0.320877\pi$
$733$	28.8879	1.06700	0.533499	+	0.845801i	$0.320877\pi$
$734$	−23.6680	−0.873600
$735$	0	0
$736$	−6.06741	−0.223648
$737$	3.89210	0.143367
$738$	0	0
$739$	36.6965	1.34990	0.674951	−	0.737863i	$-0.264165\pi$
$739$	36.6965	1.34990	0.674951	+	0.737863i	$0.264165\pi$
$740$	5.25236	0.193081
$741$	0	0
$742$	5.83126	0.214072
$743$	−23.9221	−0.877618	−0.438809	−	0.898580i	$-0.644600\pi$
$743$	−23.9221	−0.877618	−0.438809	+	0.898580i	$0.644600\pi$
$744$	0	0
$745$	84.2192	3.08555
$746$	−31.5634	−1.15562
$747$	0	0
$748$	0.975207	0.0356571
$749$	3.67395	0.134243
$750$	0	0
$751$	−35.6390	−1.30049	−0.650243	−	0.759727i	$-0.725333\pi$
$751$	−35.6390	−1.30049	−0.650243	+	0.759727i	$0.725333\pi$
$752$	−32.3806	−1.18080
$753$	0	0
$754$	−41.8342	−1.52351
$755$	−17.2572	−0.628055
$756$	0	0
$757$	−28.0965	−1.02118	−0.510592	−	0.859823i	$-0.670574\pi$
$757$	−28.0965	−1.02118	−0.510592	+	0.859823i	$0.670574\pi$
$758$	26.9844	0.980118
$759$	0	0
$760$	−65.2098	−2.36541
$761$	−22.6232	−0.820091	−0.410045	−	0.912065i	$-0.634487\pi$
$761$	−22.6232	−0.820091	−0.410045	+	0.912065i	$0.634487\pi$
$762$	0	0
$763$	14.0170	0.507450
$764$	−0.955161	−0.0345565
$765$	0	0
$766$	39.9370	1.44298
$767$	−12.1025	−0.436997
$768$	0	0
$769$	15.2047	0.548295	0.274148	−	0.961688i	$-0.411604\pi$
$769$	15.2047	0.548295	0.274148	+	0.961688i	$0.411604\pi$
$770$	−22.8695	−0.824158
$771$	0	0
$772$	1.84369	0.0663559
$773$	−7.25076	−0.260792	−0.130396	−	0.991462i	$-0.541625\pi$
$773$	−7.25076	−0.260792	−0.130396	+	0.991462i	$0.541625\pi$
$774$	0	0
$775$	−89.7865	−3.22523
$776$	−31.8082	−1.14185
$777$	0	0
$778$	−10.2738	−0.368334
$779$	23.3906	0.838056
$780$	0	0
$781$	16.9754	0.607429
$782$	−14.0151	−0.501177
$783$	0	0
$784$	−3.66225	−0.130795
$785$	29.0209	1.03580
$786$	0	0
$787$	−14.1413	−0.504083	−0.252041	−	0.967716i	$-0.581102\pi$
$787$	−14.1413	−0.504083	−0.252041	+	0.967716i	$0.581102\pi$
$788$	1.41931	0.0505607
$789$	0	0
$790$	−42.3550	−1.50692
$791$	−10.5328	−0.374504
$792$	0	0
$793$	−11.9809	−0.425453
$794$	−17.4029	−0.617604
$795$	0	0
$796$	2.31289	0.0819782
$797$	−46.8079	−1.65802	−0.829010	−	0.559234i	$-0.811095\pi$
$797$	−46.8079	−1.65802	−0.829010	+	0.559234i	$0.811095\pi$
$798$	0	0
$799$	13.2952	0.470351
$800$	−10.2040	−0.360768
$801$	0	0
$802$	45.3943	1.60293
$803$	−27.0648	−0.955094
$804$	0	0
$805$	−27.9231	−0.984160
$806$	35.1873	1.23942
$807$	0	0
$808$	−21.4873	−0.755919
$809$	−23.7428	−0.834753	−0.417377	−	0.908734i	$-0.637050\pi$
$809$	−23.7428	−0.834753	−0.417377	+	0.908734i	$0.637050\pi$
$810$	0	0
$811$	17.4329	0.612150	0.306075	−	0.952007i	$-0.400984\pi$
$811$	17.4329	0.612150	0.306075	+	0.952007i	$0.400984\pi$
$812$	−1.44805	−0.0508167
$813$	0	0
$814$	46.3569	1.62481
$815$	−99.2249	−3.47570
$816$	0	0
$817$	−24.5254	−0.858034
$818$	23.3235	0.815486
$819$	0	0
$820$	2.72148	0.0950383
$821$	0.499582	0.0174355	0.00871777	−	0.999962i	$-0.497225\pi$
$821$	0.499582	0.0174355	0.00871777	+	0.999962i	$0.497225\pi$
$822$	0	0
$823$	50.5433	1.76183	0.880915	−	0.473274i	$-0.156928\pi$
$823$	50.5433	1.76183	0.880915	+	0.473274i	$0.156928\pi$
$824$	13.2735	0.462405
$825$	0	0
$826$	4.93082	0.171565
$827$	5.11034	0.177704	0.0888520	−	0.996045i	$-0.471680\pi$
$827$	5.11034	0.177704	0.0888520	+	0.996045i	$0.471680\pi$
$828$	0	0
$829$	−50.7911	−1.76405	−0.882025	−	0.471203i	$-0.843820\pi$
$829$	−50.7911	−1.76405	−0.882025	+	0.471203i	$0.843820\pi$
$830$	4.99367	0.173333
$831$	0	0
$832$	28.4076	0.984855
$833$	1.50369	0.0520997
$834$	0	0
$835$	−61.4203	−2.12554
$836$	3.55089	0.122810
$837$	0	0
$838$	−40.0706	−1.38422
$839$	12.1037	0.417865	0.208932	−	0.977930i	$-0.433001\pi$
$839$	12.1037	0.417865	0.208932	+	0.977930i	$0.433001\pi$
$840$	0	0
$841$	56.4900	1.94793
$842$	28.3849	0.978210
$843$	0	0
$844$	1.31578	0.0452911
$845$	7.70683	0.265123
$846$	0	0
$847$	6.14846	0.211264
$848$	15.7290	0.540137
$849$	0	0
$850$	−23.5703	−0.808453
$851$	56.6007	1.94025
$852$	0	0
$853$	31.5795	1.08126	0.540631	−	0.841260i	$-0.318185\pi$
$853$	31.5795	1.08126	0.540631	+	0.841260i	$0.318185\pi$
$854$	4.88126	0.167033
$855$	0	0
$856$	10.7576	0.367686
$857$	36.7530	1.25546	0.627729	−	0.778432i	$-0.283985\pi$
$857$	36.7530	1.25546	0.627729	+	0.778432i	$0.283985\pi$
$858$	0	0
$859$	26.9999	0.921225	0.460613	−	0.887601i	$-0.347630\pi$
$859$	26.9999	0.921225	0.460613	+	0.887601i	$0.347630\pi$
$860$	−2.85351	−0.0973038
$861$	0	0
$862$	−28.6907	−0.977209
$863$	−10.8508	−0.369365	−0.184682	−	0.982798i	$-0.559126\pi$
$863$	−10.8508	−0.369365	−0.184682	+	0.982798i	$0.559126\pi$
$864$	0	0
$865$	−13.6576	−0.464374
$866$	−5.64417	−0.191797
$867$	0	0
$868$	1.21798	0.0413408
$869$	31.7596	1.07737
$870$	0	0
$871$	−3.13211	−0.106127
$872$	41.0427	1.38988
$873$	0	0
$874$	−51.0311	−1.72615
$875$	−26.6227	−0.900011
$876$	0	0
$877$	−12.3835	−0.418162	−0.209081	−	0.977898i	$-0.567047\pi$
$877$	−12.3835	−0.418162	−0.209081	+	0.977898i	$0.567047\pi$
$878$	9.12689	0.308018
$879$	0	0
$880$	−61.6872	−2.07948
$881$	−32.1116	−1.08187	−0.540933	−	0.841066i	$-0.681929\pi$
$881$	−32.1116	−1.08187	−0.540933	+	0.841066i	$0.681929\pi$
$882$	0	0
$883$	13.1303	0.441870	0.220935	−	0.975289i	$-0.429089\pi$
$883$	13.1303	0.441870	0.220935	+	0.975289i	$0.429089\pi$
$884$	−0.784783	−0.0263951
$885$	0	0
$886$	−30.6712	−1.03042
$887$	−16.1209	−0.541289	−0.270644	−	0.962679i	$-0.587237\pi$
$887$	−16.1209	−0.541289	−0.270644	+	0.962679i	$0.587237\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	4.91243	0.164665
$891$	0	0
$892$	−3.16321	−0.105912
$893$	48.4101	1.61998
$894$	0	0
$895$	−11.0982	−0.370970
$896$	−9.80615	−0.327600
$897$	0	0
$898$	−54.3094	−1.81233
$899$	−71.9069	−2.39823
$900$	0	0
$901$	−6.45821	−0.215154
$902$	24.0196	0.799764
$903$	0	0
$904$	−30.8408	−1.02575
$905$	63.4242	2.10829
$906$	0	0
$907$	−7.43444	−0.246857	−0.123428	−	0.992353i	$-0.539389\pi$
$907$	−7.43444	−0.246857	−0.123428	+	0.992353i	$0.539389\pi$
$908$	−0.970744	−0.0322153
$909$	0	0
$910$	18.4038	0.610081
$911$	18.7356	0.620740	0.310370	−	0.950616i	$-0.399547\pi$
$911$	18.7356	0.620740	0.310370	+	0.950616i	$0.399547\pi$
$912$	0	0
$913$	−3.74446	−0.123923
$914$	−6.98809	−0.231145
$915$	0	0
$916$	3.00153	0.0991733
$917$	13.5622	0.447863
$918$	0	0
$919$	59.5412	1.96408	0.982041	−	0.188666i	$-0.0604165\pi$
$919$	59.5412	1.96408	0.982041	+	0.188666i	$0.0604165\pi$
$920$	−81.7606	−2.69557
$921$	0	0
$922$	−2.24838	−0.0740465
$923$	−13.6607	−0.449648
$924$	0	0
$925$	95.1900	3.12983
$926$	−40.4643	−1.32974
$927$	0	0
$928$	−8.17207	−0.268261
$929$	−8.96931	−0.294274	−0.147137	−	0.989116i	$-0.547006\pi$
$929$	−8.96931	−0.294274	−0.147137	+	0.989116i	$0.547006\pi$
$930$	0	0
$931$	5.47517	0.179441
$932$	−4.40393	−0.144255
$933$	0	0
$934$	−16.8496	−0.551334
$935$	25.3283	0.828322
$936$	0	0
$937$	19.6061	0.640503	0.320252	−	0.947333i	$-0.396233\pi$
$937$	19.6061	0.640503	0.320252	+	0.947333i	$0.396233\pi$
$938$	1.27609	0.0416657
$939$	0	0
$940$	5.63247	0.183711
$941$	−31.9202	−1.04057	−0.520285	−	0.853993i	$-0.674174\pi$
$941$	−31.9202	−1.04057	−0.520285	+	0.853993i	$0.674174\pi$
$942$	0	0
$943$	29.3274	0.955030
$944$	13.3002	0.432885
$945$	0	0
$946$	−25.1848	−0.818829
$947$	1.95493	0.0635265	0.0317633	−	0.999495i	$-0.489888\pi$
$947$	1.95493	0.0635265	0.0317633	+	0.999495i	$0.489888\pi$
$948$	0	0
$949$	21.7799	0.707007
$950$	−85.8231	−2.78447
$951$	0	0
$952$	4.40289	0.142699
$953$	3.28363	0.106367	0.0531836	−	0.998585i	$-0.483063\pi$
$953$	3.28363	0.106367	0.0531836	+	0.998585i	$0.483063\pi$
$954$	0	0
$955$	−24.8076	−0.802755
$956$	−0.181805	−0.00587999
$957$	0	0
$958$	−31.6224	−1.02167
$959$	21.5286	0.695195
$960$	0	0
$961$	29.4819	0.951030
$962$	−37.3050	−1.20276
$963$	0	0
$964$	0.527060	0.0169755
$965$	47.8847	1.54146
$966$	0	0
$967$	−58.3821	−1.87744	−0.938720	−	0.344680i	$-0.887987\pi$
$967$	−58.3821	−1.87744	−0.938720	+	0.344680i	$0.887987\pi$
$968$	18.0031	0.578641
$969$	0	0
$970$	−59.9932	−1.92627
$971$	31.4550	1.00944	0.504719	−	0.863284i	$-0.331596\pi$
$971$	31.4550	1.00944	0.504719	+	0.863284i	$0.331596\pi$
$972$	0	0
$973$	6.20276	0.198851
$974$	16.1034	0.515988
$975$	0	0
$976$	13.1665	0.421450
$977$	13.3824	0.428141	0.214070	−	0.976818i	$-0.431328\pi$
$977$	13.3824	0.428141	0.214070	+	0.976818i	$0.431328\pi$
$978$	0	0
$979$	−3.68354	−0.117727
$980$	0.637032	0.0203492
$981$	0	0
$982$	−47.4224	−1.51331
$983$	59.8159	1.90783	0.953915	−	0.300077i	$-0.0970124\pi$
$983$	59.8159	1.90783	0.953915	+	0.300077i	$0.0970124\pi$
$984$	0	0
$985$	36.8625	1.17454
$986$	−18.8766	−0.601153
$987$	0	0
$988$	−2.85752	−0.0909098
$989$	−30.7501	−0.977797
$990$	0	0
$991$	23.0477	0.732134	0.366067	−	0.930588i	$-0.380704\pi$
$991$	23.0477	0.732134	0.366067	+	0.930588i	$0.380704\pi$
$992$	6.87365	0.218238
$993$	0	0
$994$	5.56566	0.176532
$995$	60.0708	1.90437
$996$	0	0
$997$	−11.0638	−0.350395	−0.175197	−	0.984533i	$-0.556056\pi$
$997$	−11.0638	−0.350395	−0.175197	+	0.984533i	$0.556056\pi$
$998$	−18.9434	−0.599644
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.v.1.7		19
3.2	odd	2		2667.2.a.q.1.13	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.q.1.13	✓	19	3.2	odd	2
8001.2.a.v.1.7		19	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)

\(f(q)\)	\(=\)	\(q-1.35771 q^{2} -0.156613 q^{4} -4.06757 q^{5} +1.00000 q^{7} +2.92806 q^{8} +O(q^{10})\)	\(q-1.35771 q^{2} -0.156613 q^{4} -4.06757 q^{5} +1.00000 q^{7} +2.92806 q^{8} +5.52260 q^{10} -4.14107 q^{11} +3.33246 q^{13} -1.35771 q^{14} -3.66225 q^{16} +1.50369 q^{17} +5.47517 q^{19} +0.637032 q^{20} +5.62239 q^{22} +6.86481 q^{23} +11.5451 q^{25} -4.52453 q^{26} -0.156613 q^{28} +9.24608 q^{29} -7.77701 q^{31} -0.883841 q^{32} -2.04158 q^{34} -4.06757 q^{35} +8.24505 q^{37} -7.43372 q^{38} -11.9101 q^{40} +4.27213 q^{41} -4.47938 q^{43} +0.648543 q^{44} -9.32046 q^{46} +8.84174 q^{47} +1.00000 q^{49} -15.6750 q^{50} -0.521905 q^{52} -4.29491 q^{53} +16.8441 q^{55} +2.92806 q^{56} -12.5535 q^{58} -3.63171 q^{59} -3.59520 q^{61} +10.5590 q^{62} +8.52450 q^{64} -13.5550 q^{65} -0.939878 q^{67} -0.235496 q^{68} +5.52260 q^{70} -4.09929 q^{71} +6.53569 q^{73} -11.1944 q^{74} -0.857480 q^{76} -4.14107 q^{77} -7.66941 q^{79} +14.8964 q^{80} -5.80033 q^{82} +0.904225 q^{83} -6.11635 q^{85} +6.08172 q^{86} -12.1253 q^{88} +0.889515 q^{89} +3.33246 q^{91} -1.07512 q^{92} -12.0046 q^{94} -22.2706 q^{95} -10.8632 q^{97} -1.35771 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q - 4 q^{2} + 22 q^{4} - 5 q^{5} + 19 q^{7} - 9 q^{8}+O(q^{10}) \) 19 * q - 4 * q^2 + 22 * q^4 - 5 * q^5 + 19 * q^7 - 9 * q^8	\( 19 q - 4 q^{2} + 22 q^{4} - 5 q^{5} + 19 q^{7} - 9 q^{8} + 9 q^{11} + 24 q^{13} - 4 q^{14} + 20 q^{16} - 17 q^{17} + 23 q^{19} - 5 q^{20} - 3 q^{22} + 17 q^{23} + 38 q^{25} - 28 q^{26} + 22 q^{28} - 2 q^{29} + 16 q^{31} - 17 q^{32} + 29 q^{34} - 5 q^{35} + 56 q^{37} - 2 q^{38} - 13 q^{40} + 7 q^{41} + 19 q^{43} + 29 q^{44} + 10 q^{46} - 25 q^{47} + 19 q^{49} + 9 q^{50} + 16 q^{52} - 18 q^{53} + 10 q^{55} - 9 q^{56} + 31 q^{58} - 11 q^{59} + 26 q^{61} - 26 q^{62} + 45 q^{64} - 27 q^{65} + 24 q^{67} - 14 q^{68} + 32 q^{71} + 51 q^{73} + 12 q^{76} + 9 q^{77} + 30 q^{79} + 30 q^{80} - 52 q^{82} - q^{83} + 44 q^{85} + 24 q^{86} - 30 q^{88} - 5 q^{89} + 24 q^{91} + 88 q^{92} + 7 q^{94} + 24 q^{95} + 5 q^{97} - 4 q^{98}+O(q^{100}) \) 19 * q - 4 * q^2 + 22 * q^4 - 5 * q^5 + 19 * q^7 - 9 * q^8 + 9 * q^11 + 24 * q^13 - 4 * q^14 + 20 * q^16 - 17 * q^17 + 23 * q^19 - 5 * q^20 - 3 * q^22 + 17 * q^23 + 38 * q^25 - 28 * q^26 + 22 * q^28 - 2 * q^29 + 16 * q^31 - 17 * q^32 + 29 * q^34 - 5 * q^35 + 56 * q^37 - 2 * q^38 - 13 * q^40 + 7 * q^41 + 19 * q^43 + 29 * q^44 + 10 * q^46 - 25 * q^47 + 19 * q^49 + 9 * q^50 + 16 * q^52 - 18 * q^53 + 10 * q^55 - 9 * q^56 + 31 * q^58 - 11 * q^59 + 26 * q^61 - 26 * q^62 + 45 * q^64 - 27 * q^65 + 24 * q^67 - 14 * q^68 + 32 * q^71 + 51 * q^73 + 12 * q^76 + 9 * q^77 + 30 * q^79 + 30 * q^80 - 52 * q^82 - q^83 + 44 * q^85 + 24 * q^86 - 30 * q^88 - 5 * q^89 + 24 * q^91 + 88 * q^92 + 7 * q^94 + 24 * q^95 + 5 * q^97 - 4 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more