LMFDB - Embedded newform 4005.2.a.p.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4005,2,Mod(1,4005)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4005, 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("4005.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4005 = 3^{2} \cdot 5 \cdot 89 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4005.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:	$31.9800860095$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 11x^{6} + 9x^{5} + 34x^{4} - 19x^{3} - 27x^{2} + 11x - 1 $ x^8 - x^7 - 11x^6 + 9x^5 + 34x^4 - 19x^3 - 27x^2 + 11x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 445)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.51468$ of defining polynomial
Character	$\chi$	$=$	4005.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.51468 q^{2} +4.32363 q^{4} +1.00000 q^{5} -2.83060 q^{7} -5.84320 q^{8} +O(q^{10})$	\(q-2.51468 q^{2} +4.32363 q^{4} +1.00000 q^{5} -2.83060 q^{7} -5.84320 q^{8} -2.51468 q^{10} +1.32852 q^{11} -5.12107 q^{13} +7.11806 q^{14} +6.04654 q^{16} -4.61208 q^{17} +1.84782 q^{19} +4.32363 q^{20} -3.34081 q^{22} +2.29047 q^{23} +1.00000 q^{25} +12.8779 q^{26} -12.2385 q^{28} -0.167368 q^{29} +9.85168 q^{31} -3.51873 q^{32} +11.5979 q^{34} -2.83060 q^{35} -0.207451 q^{37} -4.64669 q^{38} -5.84320 q^{40} -5.20162 q^{41} +8.88527 q^{43} +5.74404 q^{44} -5.75981 q^{46} +5.91188 q^{47} +1.01229 q^{49} -2.51468 q^{50} -22.1416 q^{52} +3.11452 q^{53} +1.32852 q^{55} +16.5398 q^{56} +0.420877 q^{58} +7.20896 q^{59} +12.1210 q^{61} -24.7739 q^{62} -3.24458 q^{64} -5.12107 q^{65} +9.69323 q^{67} -19.9409 q^{68} +7.11806 q^{70} -0.991867 q^{71} -12.1815 q^{73} +0.521675 q^{74} +7.98932 q^{76} -3.76051 q^{77} -8.00154 q^{79} +6.04654 q^{80} +13.0804 q^{82} -14.9877 q^{83} -4.61208 q^{85} -22.3437 q^{86} -7.76282 q^{88} -1.00000 q^{89} +14.4957 q^{91} +9.90316 q^{92} -14.8665 q^{94} +1.84782 q^{95} +1.80568 q^{97} -2.54559 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - q^{2} + 7 q^{4} + 8 q^{5} - 6 q^{7} - 3 q^{8}+O(q^{10}) $ 8 * q - q^2 + 7 * q^4 + 8 * q^5 - 6 * q^7 - 3 * q^8	$ 8 q - q^{2} + 7 q^{4} + 8 q^{5} - 6 q^{7} - 3 q^{8} - q^{10} - 14 q^{11} - 7 q^{13} - 15 q^{14} + 9 q^{16} - 17 q^{17} + 17 q^{19} + 7 q^{20} + 2 q^{22} + q^{23} + 8 q^{25} - 3 q^{26} - 29 q^{28} - 10 q^{29} + q^{31} - 2 q^{32} - 16 q^{34} - 6 q^{35} - 11 q^{37} + 30 q^{38} - 3 q^{40} - 15 q^{41} - 5 q^{43} - 7 q^{44} - 12 q^{46} - 12 q^{47} + 4 q^{49} - q^{50} - 14 q^{52} + q^{53} - 14 q^{55} - 3 q^{56} - 37 q^{58} - 26 q^{59} + 13 q^{61} - 22 q^{62} - 15 q^{64} - 7 q^{65} - 25 q^{67} - 23 q^{68} - 15 q^{70} - 28 q^{71} - 17 q^{73} + 5 q^{74} + 8 q^{76} - 7 q^{79} + 9 q^{80} + 5 q^{82} - 44 q^{83} - 17 q^{85} + 13 q^{86} - 66 q^{88} - 8 q^{89} + 27 q^{91} - 15 q^{92} - 27 q^{94} + 17 q^{95} + q^{97} + 34 q^{98}+O(q^{100}) $ 8 * q - q^2 + 7 * q^4 + 8 * q^5 - 6 * q^7 - 3 * q^8 - q^10 - 14 * q^11 - 7 * q^13 - 15 * q^14 + 9 * q^16 - 17 * q^17 + 17 * q^19 + 7 * q^20 + 2 * q^22 + q^23 + 8 * q^25 - 3 * q^26 - 29 * q^28 - 10 * q^29 + q^31 - 2 * q^32 - 16 * q^34 - 6 * q^35 - 11 * q^37 + 30 * q^38 - 3 * q^40 - 15 * q^41 - 5 * q^43 - 7 * q^44 - 12 * q^46 - 12 * q^47 + 4 * q^49 - q^50 - 14 * q^52 + q^53 - 14 * q^55 - 3 * q^56 - 37 * q^58 - 26 * q^59 + 13 * q^61 - 22 * q^62 - 15 * q^64 - 7 * q^65 - 25 * q^67 - 23 * q^68 - 15 * q^70 - 28 * q^71 - 17 * q^73 + 5 * q^74 + 8 * q^76 - 7 * q^79 + 9 * q^80 + 5 * q^82 - 44 * q^83 - 17 * q^85 + 13 * q^86 - 66 * q^88 - 8 * q^89 + 27 * q^91 - 15 * q^92 - 27 * q^94 + 17 * q^95 + q^97 + 34 * 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 $

$2$	−2.51468	−1.77815	−0.889075	−	0.457762i	$-0.848651\pi$
$2$	−2.51468	−1.77815	−0.889075	+	0.457762i	$0.848651\pi$
$3$	0	0
$3$	0	0
$4$	4.32363	2.16182
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−2.83060	−1.06987	−0.534933	−	0.844895i	$-0.679663\pi$
$7$	−2.83060	−1.06987	−0.534933	+	0.844895i	$0.679663\pi$
$8$	−5.84320	−2.06588
$9$	0	0
$10$	−2.51468	−0.795213
$11$	1.32852	0.400564	0.200282	−	0.979738i	$-0.435814\pi$
$11$	1.32852	0.400564	0.200282	+	0.979738i	$0.435814\pi$
$12$	0	0
$13$	−5.12107	−1.42033	−0.710165	−	0.704036i	$-0.751379\pi$
$13$	−5.12107	−1.42033	−0.710165	+	0.704036i	$0.751379\pi$
$14$	7.11806	1.90238
$15$	0	0
$16$	6.04654	1.51164
$17$	−4.61208	−1.11859	−0.559297	−	0.828968i	$-0.688929\pi$
$17$	−4.61208	−1.11859	−0.559297	+	0.828968i	$0.688929\pi$
$18$	0	0
$19$	1.84782	0.423920	0.211960	−	0.977278i	$-0.432015\pi$
$19$	1.84782	0.423920	0.211960	+	0.977278i	$0.432015\pi$
$20$	4.32363	0.966794
$21$	0	0
$22$	−3.34081	−0.712263
$23$	2.29047	0.477596	0.238798	−	0.971069i	$-0.423247\pi$
$23$	2.29047	0.477596	0.238798	+	0.971069i	$0.423247\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	12.8779	2.52556
$27$	0	0
$28$	−12.2385	−2.31285
$29$	−0.167368	−0.0310794	−0.0155397	−	0.999879i	$-0.504947\pi$
$29$	−0.167368	−0.0310794	−0.0155397	+	0.999879i	$0.504947\pi$
$30$	0	0
$31$	9.85168	1.76941	0.884707	−	0.466148i	$-0.154359\pi$
$31$	9.85168	1.76941	0.884707	+	0.466148i	$0.154359\pi$
$32$	−3.51873	−0.622030
$33$	0	0
$34$	11.5979	1.98903
$35$	−2.83060	−0.478458
$36$	0	0
$37$	−0.207451	−0.0341048	−0.0170524	−	0.999855i	$-0.505428\pi$
$37$	−0.207451	−0.0341048	−0.0170524	+	0.999855i	$0.505428\pi$
$38$	−4.64669	−0.753793
$39$	0	0
$40$	−5.84320	−0.923892
$41$	−5.20162	−0.812357	−0.406178	−	0.913794i	$-0.633139\pi$
$41$	−5.20162	−0.812357	−0.406178	+	0.913794i	$0.633139\pi$
$42$	0	0
$43$	8.88527	1.35499	0.677496	−	0.735527i	$-0.263065\pi$
$43$	8.88527	1.35499	0.677496	+	0.735527i	$0.263065\pi$
$44$	5.74404	0.865946
$45$	0	0
$46$	−5.75981	−0.849238
$47$	5.91188	0.862336	0.431168	−	0.902272i	$-0.358102\pi$
$47$	5.91188	0.862336	0.431168	+	0.902272i	$0.358102\pi$
$48$	0	0
$49$	1.01229	0.144613
$50$	−2.51468	−0.355630
$51$	0	0
$52$	−22.1416	−3.07049
$53$	3.11452	0.427812	0.213906	−	0.976854i	$-0.431381\pi$
$53$	3.11452	0.427812	0.213906	+	0.976854i	$0.431381\pi$
$54$	0	0
$55$	1.32852	0.179138
$56$	16.5398	2.21022
$57$	0	0
$58$	0.420877	0.0552639
$59$	7.20896	0.938526	0.469263	−	0.883058i	$-0.344520\pi$
$59$	7.20896	0.938526	0.469263	+	0.883058i	$0.344520\pi$
$60$	0	0
$61$	12.1210	1.55193	0.775966	−	0.630775i	$-0.217263\pi$
$61$	12.1210	1.55193	0.775966	+	0.630775i	$0.217263\pi$
$62$	−24.7739	−3.14628
$63$	0	0
$64$	−3.24458	−0.405573
$65$	−5.12107	−0.635190
$66$	0	0
$67$	9.69323	1.18422	0.592108	−	0.805858i	$-0.298296\pi$
$67$	9.69323	1.18422	0.592108	+	0.805858i	$0.298296\pi$
$68$	−19.9409	−2.41819
$69$	0	0
$70$	7.11806	0.850771
$71$	−0.991867	−0.117713	−0.0588565	−	0.998266i	$-0.518745\pi$
$71$	−0.991867	−0.117713	−0.0588565	+	0.998266i	$0.518745\pi$
$72$	0	0
$73$	−12.1815	−1.42573	−0.712867	−	0.701300i	$-0.752603\pi$
$73$	−12.1815	−1.42573	−0.712867	+	0.701300i	$0.752603\pi$
$74$	0.521675	0.0606435
$75$	0	0
$76$	7.98932	0.916437
$77$	−3.76051	−0.428550
$78$	0	0
$79$	−8.00154	−0.900244	−0.450122	−	0.892967i	$-0.648619\pi$
$79$	−8.00154	−0.900244	−0.450122	+	0.892967i	$0.648619\pi$
$80$	6.04654	0.676024
$81$	0	0
$82$	13.0804	1.44449
$83$	−14.9877	−1.64511	−0.822557	−	0.568683i	$-0.807453\pi$
$83$	−14.9877	−1.64511	−0.822557	+	0.568683i	$0.807453\pi$
$84$	0	0
$85$	−4.61208	−0.500250
$86$	−22.3437	−2.40938
$87$	0	0
$88$	−7.76282	−0.827519
$89$	−1.00000	−0.106000
$89$	−1.00000	−0.106000
$90$	0	0
$91$	14.4957	1.51956
$92$	9.90316	1.03248
$93$	0	0
$94$	−14.8665	−1.53336
$95$	1.84782	0.189583
$96$	0	0
$97$	1.80568	0.183339	0.0916697	−	0.995789i	$-0.470780\pi$
$97$	1.80568	0.183339	0.0916697	+	0.995789i	$0.470780\pi$
$98$	−2.54559	−0.257143
$99$	0	0
$100$	4.32363	0.432363
$101$	−7.12949	−0.709411	−0.354705	−	0.934978i	$-0.615419\pi$
$101$	−7.12949	−0.709411	−0.354705	+	0.934978i	$0.615419\pi$
$102$	0	0
$103$	−4.87235	−0.480087	−0.240043	−	0.970762i	$-0.577162\pi$
$103$	−4.87235	−0.480087	−0.240043	+	0.970762i	$0.577162\pi$
$104$	29.9235	2.93424
$105$	0	0
$106$	−7.83202	−0.760713
$107$	15.4342	1.49208	0.746042	−	0.665899i	$-0.231952\pi$
$107$	15.4342	1.49208	0.746042	+	0.665899i	$0.231952\pi$
$108$	0	0
$109$	−16.8174	−1.61081	−0.805407	−	0.592722i	$-0.798053\pi$
$109$	−16.8174	−1.61081	−0.805407	+	0.592722i	$0.798053\pi$
$110$	−3.34081	−0.318534
$111$	0	0
$112$	−17.1153	−1.61725
$113$	−20.6237	−1.94011	−0.970056	−	0.242880i	$-0.921908\pi$
$113$	−20.6237	−1.94011	−0.970056	+	0.242880i	$0.921908\pi$
$114$	0	0
$115$	2.29047	0.213587
$116$	−0.723638	−0.0671881
$117$	0	0
$118$	−18.1282	−1.66884
$119$	13.0549	1.19674
$120$	0	0
$121$	−9.23503	−0.839548
$122$	−30.4804	−2.75957
$123$	0	0
$124$	42.5950	3.82515
$125$	1.00000	0.0894427
$126$	0	0
$127$	−12.4045	−1.10072	−0.550362	−	0.834926i	$-0.685510\pi$
$127$	−12.4045	−1.10072	−0.550362	+	0.834926i	$0.685510\pi$
$128$	15.1966	1.34320
$129$	0	0
$130$	12.8779	1.12946
$131$	−12.0122	−1.04951	−0.524755	−	0.851254i	$-0.675843\pi$
$131$	−12.0122	−1.04951	−0.524755	+	0.851254i	$0.675843\pi$
$132$	0	0
$133$	−5.23045	−0.453537
$134$	−24.3754	−2.10571
$135$	0	0
$136$	26.9493	2.31089
$137$	10.4285	0.890970	0.445485	−	0.895289i	$-0.353031\pi$
$137$	10.4285	0.890970	0.445485	+	0.895289i	$0.353031\pi$
$138$	0	0
$139$	−7.42653	−0.629910	−0.314955	−	0.949107i	$-0.601989\pi$
$139$	−7.42653	−0.629910	−0.314955	+	0.949107i	$0.601989\pi$
$140$	−12.2385	−1.03434
$141$	0	0
$142$	2.49423	0.209311
$143$	−6.80345	−0.568933
$144$	0	0
$145$	−0.167368	−0.0138992
$146$	30.6325	2.53517
$147$	0	0
$148$	−0.896944	−0.0737283
$149$	1.06192	0.0869958	0.0434979	−	0.999054i	$-0.486150\pi$
$149$	1.06192	0.0869958	0.0434979	+	0.999054i	$0.486150\pi$
$150$	0	0
$151$	−22.5617	−1.83605	−0.918024	−	0.396525i	$-0.870216\pi$
$151$	−22.5617	−1.83605	−0.918024	+	0.396525i	$0.870216\pi$
$152$	−10.7972	−0.875770
$153$	0	0
$154$	9.45649	0.762026
$155$	9.85168	0.791306
$156$	0	0
$157$	15.1708	1.21076	0.605379	−	0.795937i	$-0.293022\pi$
$157$	15.1708	1.21076	0.605379	+	0.795937i	$0.293022\pi$
$158$	20.1213	1.60077
$159$	0	0
$160$	−3.51873	−0.278180
$161$	−6.48340	−0.510964
$162$	0	0
$163$	−15.0509	−1.17888	−0.589439	−	0.807813i	$-0.700651\pi$
$163$	−15.0509	−1.17888	−0.589439	+	0.807813i	$0.700651\pi$
$164$	−22.4899	−1.75617
$165$	0	0
$166$	37.6893	2.92526
$167$	−14.3817	−1.11289	−0.556445	−	0.830885i	$-0.687835\pi$
$167$	−14.3817	−1.11289	−0.556445	+	0.830885i	$0.687835\pi$
$168$	0	0
$169$	13.2254	1.01733
$170$	11.5979	0.889520
$171$	0	0
$172$	38.4167	2.92924
$173$	21.2741	1.61744	0.808720	−	0.588194i	$-0.200161\pi$
$173$	21.2741	1.61744	0.808720	+	0.588194i	$0.200161\pi$
$174$	0	0
$175$	−2.83060	−0.213973
$176$	8.03296	0.605507
$177$	0	0
$178$	2.51468	0.188484
$179$	−14.7720	−1.10411	−0.552057	−	0.833807i	$-0.686157\pi$
$179$	−14.7720	−1.10411	−0.552057	+	0.833807i	$0.686157\pi$
$180$	0	0
$181$	1.86322	0.138492	0.0692461	−	0.997600i	$-0.477941\pi$
$181$	1.86322	0.138492	0.0692461	+	0.997600i	$0.477941\pi$
$182$	−36.4521	−2.70201
$183$	0	0
$184$	−13.3837	−0.986659
$185$	−0.207451	−0.0152521
$186$	0	0
$187$	−6.12724	−0.448068
$188$	25.5608	1.86421
$189$	0	0
$190$	−4.64669	−0.337107
$191$	−10.5662	−0.764547	−0.382273	−	0.924049i	$-0.624859\pi$
$191$	−10.5662	−0.764547	−0.382273	+	0.924049i	$0.624859\pi$
$192$	0	0
$193$	0.0385817	0.00277717	0.00138859	−	0.999999i	$-0.499558\pi$
$193$	0.0385817	0.00277717	0.00138859	+	0.999999i	$0.499558\pi$
$194$	−4.54072	−0.326005
$195$	0	0
$196$	4.37677	0.312626
$197$	−1.23776	−0.0881868	−0.0440934	−	0.999027i	$-0.514040\pi$
$197$	−1.23776	−0.0881868	−0.0440934	+	0.999027i	$0.514040\pi$
$198$	0	0
$199$	15.9428	1.13016	0.565078	−	0.825038i	$-0.308846\pi$
$199$	15.9428	1.13016	0.565078	+	0.825038i	$0.308846\pi$
$200$	−5.84320	−0.413177
$201$	0	0
$202$	17.9284	1.26144
$203$	0.473751	0.0332508
$204$	0	0
$205$	−5.20162	−0.363297
$206$	12.2524	0.853666
$207$	0	0
$208$	−30.9648	−2.14702
$209$	2.45487	0.169807
$210$	0	0
$211$	23.8439	1.64148	0.820742	−	0.571299i	$-0.193560\pi$
$211$	23.8439	1.64148	0.820742	+	0.571299i	$0.193560\pi$
$212$	13.4660	0.924851
$213$	0	0
$214$	−38.8122	−2.65315
$215$	8.88527	0.605971
$216$	0	0
$217$	−27.8861	−1.89303
$218$	42.2904	2.86427
$219$	0	0
$220$	5.74404	0.387263
$221$	23.6188	1.58877
$222$	0	0
$223$	−23.2041	−1.55386	−0.776929	−	0.629588i	$-0.783224\pi$
$223$	−23.2041	−1.55386	−0.776929	+	0.629588i	$0.783224\pi$
$224$	9.96012	0.665489
$225$	0	0
$226$	51.8620	3.44981
$227$	6.90337	0.458193	0.229096	−	0.973404i	$-0.426423\pi$
$227$	6.90337	0.458193	0.229096	+	0.973404i	$0.426423\pi$
$228$	0	0
$229$	−13.8134	−0.912816	−0.456408	−	0.889770i	$-0.650864\pi$
$229$	−13.8134	−0.912816	−0.456408	+	0.889770i	$0.650864\pi$
$230$	−5.75981	−0.379791
$231$	0	0
$232$	0.977965	0.0642066
$233$	1.20571	0.0789889	0.0394944	−	0.999220i	$-0.487425\pi$
$233$	1.20571	0.0789889	0.0394944	+	0.999220i	$0.487425\pi$
$234$	0	0
$235$	5.91188	0.385648
$236$	31.1689	2.02892
$237$	0	0
$238$	−32.8291	−2.12799
$239$	−27.0870	−1.75211	−0.876055	−	0.482211i	$-0.839834\pi$
$239$	−27.0870	−1.75211	−0.876055	+	0.482211i	$0.839834\pi$
$240$	0	0
$241$	7.66571	0.493792	0.246896	−	0.969042i	$-0.420589\pi$
$241$	7.66571	0.493792	0.246896	+	0.969042i	$0.420589\pi$
$242$	23.2232	1.49284
$243$	0	0
$244$	52.4067	3.35499
$245$	1.01229	0.0646728
$246$	0	0
$247$	−9.46284	−0.602106
$248$	−57.5654	−3.65540
$249$	0	0
$250$	−2.51468	−0.159043
$251$	−10.6839	−0.674363	−0.337182	−	0.941440i	$-0.609474\pi$
$251$	−10.6839	−0.674363	−0.337182	+	0.941440i	$0.609474\pi$
$252$	0	0
$253$	3.04294	0.191308
$254$	31.1935	1.95725
$255$	0	0
$256$	−31.7254	−1.98284
$257$	0.623084	0.0388669	0.0194335	−	0.999811i	$-0.493814\pi$
$257$	0.623084	0.0388669	0.0194335	+	0.999811i	$0.493814\pi$
$258$	0	0
$259$	0.587212	0.0364876
$260$	−22.1416	−1.37317
$261$	0	0
$262$	30.2068	1.86618
$263$	13.6749	0.843228	0.421614	−	0.906775i	$-0.361464\pi$
$263$	13.6749	0.843228	0.421614	+	0.906775i	$0.361464\pi$
$264$	0	0
$265$	3.11452	0.191323
$266$	13.1529	0.806457
$267$	0	0
$268$	41.9100	2.56006
$269$	−3.83259	−0.233677	−0.116838	−	0.993151i	$-0.537276\pi$
$269$	−3.83259	−0.233677	−0.116838	+	0.993151i	$0.537276\pi$
$270$	0	0
$271$	24.2650	1.47399	0.736995	−	0.675898i	$-0.236244\pi$
$271$	24.2650	1.47399	0.736995	+	0.675898i	$0.236244\pi$
$272$	−27.8871	−1.69091
$273$	0	0
$274$	−26.2245	−1.58428
$275$	1.32852	0.0801128
$276$	0	0
$277$	18.2047	1.09382	0.546908	−	0.837193i	$-0.315805\pi$
$277$	18.2047	1.09382	0.546908	+	0.837193i	$0.315805\pi$
$278$	18.6754	1.12007
$279$	0	0
$280$	16.5398	0.988440
$281$	4.11252	0.245332	0.122666	−	0.992448i	$-0.460856\pi$
$281$	4.11252	0.245332	0.122666	+	0.992448i	$0.460856\pi$
$282$	0	0
$283$	−6.51597	−0.387334	−0.193667	−	0.981067i	$-0.562038\pi$
$283$	−6.51597	−0.387334	−0.193667	+	0.981067i	$0.562038\pi$
$284$	−4.28847	−0.254474
$285$	0	0
$286$	17.1085	1.01165
$287$	14.7237	0.869113
$288$	0	0
$289$	4.27127	0.251251
$290$	0.420877	0.0247148
$291$	0	0
$292$	−52.6682	−3.08217
$293$	−18.8557	−1.10156	−0.550781	−	0.834650i	$-0.685670\pi$
$293$	−18.8557	−1.10156	−0.550781	+	0.834650i	$0.685670\pi$
$294$	0	0
$295$	7.20896	0.419722
$296$	1.21218	0.0704566
$297$	0	0
$298$	−2.67039	−0.154692
$299$	−11.7297	−0.678344
$300$	0	0
$301$	−25.1506	−1.44966
$302$	56.7356	3.26477
$303$	0	0
$304$	11.1729	0.640812
$305$	12.1210	0.694045
$306$	0	0
$307$	13.6486	0.778966	0.389483	−	0.921034i	$-0.372654\pi$
$307$	13.6486	0.778966	0.389483	+	0.921034i	$0.372654\pi$
$308$	−16.2591	−0.926446
$309$	0	0
$310$	−24.7739	−1.40706
$311$	−25.7543	−1.46040	−0.730198	−	0.683236i	$-0.760572\pi$
$311$	−25.7543	−1.46040	−0.730198	+	0.683236i	$0.760572\pi$
$312$	0	0
$313$	11.1065	0.627776	0.313888	−	0.949460i	$-0.398368\pi$
$313$	11.1065	0.627776	0.313888	+	0.949460i	$0.398368\pi$
$314$	−38.1496	−2.15291
$315$	0	0
$316$	−34.5957	−1.94616
$317$	−11.4835	−0.644977	−0.322488	−	0.946573i	$-0.604519\pi$
$317$	−11.4835	−0.644977	−0.322488	+	0.946573i	$0.604519\pi$
$318$	0	0
$319$	−0.222352	−0.0124493
$320$	−3.24458	−0.181378
$321$	0	0
$322$	16.3037	0.908570
$323$	−8.52231	−0.474194
$324$	0	0
$325$	−5.12107	−0.284066
$326$	37.8483	2.09622
$327$	0	0
$328$	30.3941	1.67824
$329$	−16.7342	−0.922584
$330$	0	0
$331$	15.6240	0.858773	0.429386	−	0.903121i	$-0.358730\pi$
$331$	15.6240	0.858773	0.429386	+	0.903121i	$0.358730\pi$
$332$	−64.8013	−3.55643
$333$	0	0
$334$	36.1654	1.97888
$335$	9.69323	0.529598
$336$	0	0
$337$	8.62551	0.469861	0.234931	−	0.972012i	$-0.424514\pi$
$337$	8.62551	0.469861	0.234931	+	0.972012i	$0.424514\pi$
$338$	−33.2576	−1.80897
$339$	0	0
$340$	−19.9409	−1.08145
$341$	13.0882	0.708763
$342$	0	0
$343$	16.9488	0.915150
$344$	−51.9185	−2.79926
$345$	0	0
$346$	−53.4976	−2.87605
$347$	−18.8508	−1.01197	−0.505983	−	0.862544i	$-0.668870\pi$
$347$	−18.8508	−1.01197	−0.505983	+	0.862544i	$0.668870\pi$
$348$	0	0
$349$	−20.2101	−1.08182	−0.540910	−	0.841080i	$-0.681920\pi$
$349$	−20.2101	−1.08182	−0.540910	+	0.841080i	$0.681920\pi$
$350$	7.11806	0.380476
$351$	0	0
$352$	−4.67471	−0.249163
$353$	25.4587	1.35503	0.677514	−	0.735510i	$-0.263057\pi$
$353$	25.4587	1.35503	0.677514	+	0.735510i	$0.263057\pi$
$354$	0	0
$355$	−0.991867	−0.0526428
$356$	−4.32363	−0.229152
$357$	0	0
$358$	37.1470	1.96328
$359$	12.7517	0.673011	0.336505	−	0.941682i	$-0.390755\pi$
$359$	12.7517	0.673011	0.336505	+	0.941682i	$0.390755\pi$
$360$	0	0
$361$	−15.5855	−0.820292
$362$	−4.68541	−0.246260
$363$	0	0
$364$	62.6741	3.28501
$365$	−12.1815	−0.637607
$366$	0	0
$367$	−27.0102	−1.40992	−0.704961	−	0.709246i	$-0.749036\pi$
$367$	−27.0102	−1.40992	−0.704961	+	0.709246i	$0.749036\pi$
$368$	13.8494	0.721951
$369$	0	0
$370$	0.521675	0.0271206
$371$	−8.81595	−0.457701
$372$	0	0
$373$	−27.1822	−1.40744	−0.703720	−	0.710478i	$-0.748479\pi$
$373$	−27.1822	−1.40744	−0.703720	+	0.710478i	$0.748479\pi$
$374$	15.4081	0.796733
$375$	0	0
$376$	−34.5443	−1.78149
$377$	0.857103	0.0441430
$378$	0	0
$379$	−7.72849	−0.396986	−0.198493	−	0.980102i	$-0.563605\pi$
$379$	−7.72849	−0.396986	−0.198493	+	0.980102i	$0.563605\pi$
$380$	7.98932	0.409843
$381$	0	0
$382$	26.5708	1.35948
$383$	−28.0352	−1.43253	−0.716267	−	0.697827i	$-0.754151\pi$
$383$	−28.0352	−1.43253	−0.716267	+	0.697827i	$0.754151\pi$
$384$	0	0
$385$	−3.76051	−0.191653
$386$	−0.0970208	−0.00493823
$387$	0	0
$388$	7.80711	0.396346
$389$	0.595763	0.0302064	0.0151032	−	0.999886i	$-0.495192\pi$
$389$	0.595763	0.0302064	0.0151032	+	0.999886i	$0.495192\pi$
$390$	0	0
$391$	−10.5638	−0.534236
$392$	−5.91501	−0.298753
$393$	0	0
$394$	3.11258	0.156809
$395$	−8.00154	−0.402601
$396$	0	0
$397$	−35.5629	−1.78485	−0.892426	−	0.451193i	$-0.850999\pi$
$397$	−35.5629	−1.78485	−0.892426	+	0.451193i	$0.850999\pi$
$398$	−40.0911	−2.00959
$399$	0	0
$400$	6.04654	0.302327
$401$	0.912908	0.0455884	0.0227942	−	0.999740i	$-0.492744\pi$
$401$	0.912908	0.0455884	0.0227942	+	0.999740i	$0.492744\pi$
$402$	0	0
$403$	−50.4511	−2.51315
$404$	−30.8253	−1.53362
$405$	0	0
$406$	−1.19134	−0.0591250
$407$	−0.275604	−0.0136612
$408$	0	0
$409$	−7.04632	−0.348418	−0.174209	−	0.984709i	$-0.555737\pi$
$409$	−7.04632	−0.348418	−0.174209	+	0.984709i	$0.555737\pi$
$410$	13.0804	0.645997
$411$	0	0
$412$	−21.0662	−1.03786
$413$	−20.4057	−1.00410
$414$	0	0
$415$	−14.9877	−0.735717
$416$	18.0197	0.883487
$417$	0	0
$418$	−6.17323	−0.301942
$419$	−27.1256	−1.32517	−0.662587	−	0.748985i	$-0.730541\pi$
$419$	−27.1256	−1.32517	−0.662587	+	0.748985i	$0.730541\pi$
$420$	0	0
$421$	0.264561	0.0128939	0.00644696	−	0.999979i	$-0.497948\pi$
$421$	0.264561	0.0128939	0.00644696	+	0.999979i	$0.497948\pi$
$422$	−59.9600	−2.91881
$423$	0	0
$424$	−18.1988	−0.883810
$425$	−4.61208	−0.223719
$426$	0	0
$427$	−34.3096	−1.66036
$428$	66.7320	3.22561
$429$	0	0
$430$	−22.3437	−1.07751
$431$	20.5851	0.991549	0.495774	−	0.868451i	$-0.334884\pi$
$431$	20.5851	0.991549	0.495774	+	0.868451i	$0.334884\pi$
$432$	0	0
$433$	−26.3333	−1.26550	−0.632749	−	0.774357i	$-0.718074\pi$
$433$	−26.3333	−1.26550	−0.632749	+	0.774357i	$0.718074\pi$
$434$	70.1248	3.36610
$435$	0	0
$436$	−72.7122	−3.48228
$437$	4.23239	0.202463
$438$	0	0
$439$	36.9387	1.76299	0.881494	−	0.472195i	$-0.156538\pi$
$439$	36.9387	1.76299	0.881494	+	0.472195i	$0.156538\pi$
$440$	−7.76282	−0.370078
$441$	0	0
$442$	−59.3938	−2.82507
$443$	6.92362	0.328951	0.164476	−	0.986381i	$-0.447407\pi$
$443$	6.92362	0.328951	0.164476	+	0.986381i	$0.447407\pi$
$444$	0	0
$445$	−1.00000	−0.0474045
$446$	58.3509	2.76299
$447$	0	0
$448$	9.18411	0.433909
$449$	13.7840	0.650509	0.325255	−	0.945626i	$-0.394550\pi$
$449$	13.7840	0.650509	0.325255	+	0.945626i	$0.394550\pi$
$450$	0	0
$451$	−6.91046	−0.325401
$452$	−89.1692	−4.19417
$453$	0	0
$454$	−17.3598	−0.814735
$455$	14.4957	0.679569
$456$	0	0
$457$	−32.0535	−1.49940	−0.749700	−	0.661778i	$-0.769802\pi$
$457$	−32.0535	−1.49940	−0.749700	+	0.661778i	$0.769802\pi$
$458$	34.7364	1.62312
$459$	0	0
$460$	9.90316	0.461737
$461$	27.4316	1.27762	0.638809	−	0.769365i	$-0.279427\pi$
$461$	27.4316	1.27762	0.638809	+	0.769365i	$0.279427\pi$
$462$	0	0
$463$	−11.4729	−0.533193	−0.266596	−	0.963808i	$-0.585899\pi$
$463$	−11.4729	−0.533193	−0.266596	+	0.963808i	$0.585899\pi$
$464$	−1.01200	−0.0469808
$465$	0	0
$466$	−3.03199	−0.140454
$467$	18.2580	0.844880	0.422440	−	0.906391i	$-0.361174\pi$
$467$	18.2580	0.844880	0.422440	+	0.906391i	$0.361174\pi$
$468$	0	0
$469$	−27.4376	−1.26695
$470$	−14.8665	−0.685741
$471$	0	0
$472$	−42.1234	−1.93889
$473$	11.8043	0.542761
$474$	0	0
$475$	1.84782	0.0847840
$476$	56.4448	2.58714
$477$	0	0
$478$	68.1151	3.11551
$479$	−42.7672	−1.95408	−0.977042	−	0.213048i	$-0.931661\pi$
$479$	−42.7672	−1.95408	−0.977042	+	0.213048i	$0.931661\pi$
$480$	0	0
$481$	1.06237	0.0484400
$482$	−19.2768	−0.878036
$483$	0	0
$484$	−39.9289	−1.81495
$485$	1.80568	0.0819918
$486$	0	0
$487$	−35.4303	−1.60550	−0.802751	−	0.596314i	$-0.796631\pi$
$487$	−35.4303	−1.60550	−0.802751	+	0.596314i	$0.796631\pi$
$488$	−70.8254	−3.20611
$489$	0	0
$490$	−2.54559	−0.114998
$491$	−10.3361	−0.466463	−0.233232	−	0.972421i	$-0.574930\pi$
$491$	−10.3361	−0.466463	−0.233232	+	0.972421i	$0.574930\pi$
$492$	0	0
$493$	0.771914	0.0347653
$494$	23.7960	1.07063
$495$	0	0
$496$	59.5686	2.67471
$497$	2.80758	0.125937
$498$	0	0
$499$	−29.1616	−1.30545	−0.652726	−	0.757594i	$-0.726375\pi$
$499$	−29.1616	−1.30545	−0.652726	+	0.757594i	$0.726375\pi$
$500$	4.32363	0.193359
$501$	0	0
$502$	26.8667	1.19912
$503$	−35.1802	−1.56861	−0.784303	−	0.620378i	$-0.786979\pi$
$503$	−35.1802	−1.56861	−0.784303	+	0.620378i	$0.786979\pi$
$504$	0	0
$505$	−7.12949	−0.317258
$506$	−7.65203	−0.340174
$507$	0	0
$508$	−53.6326	−2.37956
$509$	11.1436	0.493932	0.246966	−	0.969024i	$-0.420566\pi$
$509$	11.1436	0.493932	0.246966	+	0.969024i	$0.420566\pi$
$510$	0	0
$511$	34.4808	1.52534
$512$	49.3862	2.18258
$513$	0	0
$514$	−1.56686	−0.0691112
$515$	−4.87235	−0.214701
$516$	0	0
$517$	7.85405	0.345421
$518$	−1.47665	−0.0648804
$519$	0	0
$520$	29.9235	1.31223
$521$	27.8902	1.22189	0.610947	−	0.791672i	$-0.290789\pi$
$521$	27.8902	1.22189	0.610947	+	0.791672i	$0.290789\pi$
$522$	0	0
$523$	−19.0875	−0.834637	−0.417319	−	0.908760i	$-0.637030\pi$
$523$	−19.0875	−0.834637	−0.417319	+	0.908760i	$0.637030\pi$
$524$	−51.9363	−2.26885
$525$	0	0
$526$	−34.3879	−1.49939
$527$	−45.4367	−1.97925
$528$	0	0
$529$	−17.7537	−0.771902
$530$	−7.83202	−0.340201
$531$	0	0
$532$	−22.6145	−0.980465
$533$	26.6379	1.15381
$534$	0	0
$535$	15.4342	0.667280
$536$	−56.6395	−2.44646
$537$	0	0
$538$	9.63774	0.415513
$539$	1.34485	0.0579266
$540$	0	0
$541$	36.2025	1.55647	0.778233	−	0.627975i	$-0.216116\pi$
$541$	36.2025	1.55647	0.778233	+	0.627975i	$0.216116\pi$
$542$	−61.0187	−2.62098
$543$	0	0
$544$	16.2287	0.695799
$545$	−16.8174	−0.720378
$546$	0	0
$547$	10.3493	0.442504	0.221252	−	0.975217i	$-0.428986\pi$
$547$	10.3493	0.442504	0.221252	+	0.975217i	$0.428986\pi$
$548$	45.0892	1.92612
$549$	0	0
$550$	−3.34081	−0.142453
$551$	−0.309267	−0.0131752
$552$	0	0
$553$	22.6492	0.963140
$554$	−45.7791	−1.94497
$555$	0	0
$556$	−32.1096	−1.36175
$557$	27.7262	1.17479	0.587397	−	0.809299i	$-0.300153\pi$
$557$	27.7262	1.17479	0.587397	+	0.809299i	$0.300153\pi$
$558$	0	0
$559$	−45.5021	−1.92453
$560$	−17.1153	−0.723255
$561$	0	0
$562$	−10.3417	−0.436238
$563$	−37.3785	−1.57532	−0.787659	−	0.616112i	$-0.788707\pi$
$563$	−37.3785	−1.57532	−0.787659	+	0.616112i	$0.788707\pi$
$564$	0	0
$565$	−20.6237	−0.867645
$566$	16.3856	0.688739
$567$	0	0
$568$	5.79568	0.243181
$569$	13.8419	0.580285	0.290142	−	0.956983i	$-0.406297\pi$
$569$	13.8419	0.580285	0.290142	+	0.956983i	$0.406297\pi$
$570$	0	0
$571$	11.9908	0.501797	0.250899	−	0.968013i	$-0.419274\pi$
$571$	11.9908	0.501797	0.250899	+	0.968013i	$0.419274\pi$
$572$	−29.4156	−1.22993
$573$	0	0
$574$	−37.0255	−1.54541
$575$	2.29047	0.0955192
$576$	0	0
$577$	−11.9615	−0.497962	−0.248981	−	0.968508i	$-0.580096\pi$
$577$	−11.9615	−0.497962	−0.248981	+	0.968508i	$0.580096\pi$
$578$	−10.7409	−0.446763
$579$	0	0
$580$	−0.723638	−0.0300474
$581$	42.4241	1.76005
$582$	0	0
$583$	4.13770	0.171366
$584$	71.1788	2.94540
$585$	0	0
$586$	47.4161	1.95874
$587$	−5.14399	−0.212315	−0.106158	−	0.994349i	$-0.533855\pi$
$587$	−5.14399	−0.212315	−0.106158	+	0.994349i	$0.533855\pi$
$588$	0	0
$589$	18.2042	0.750090
$590$	−18.1282	−0.746328
$591$	0	0
$592$	−1.25436	−0.0515540
$593$	25.9237	1.06456	0.532279	−	0.846569i	$-0.321336\pi$
$593$	25.9237	1.06456	0.532279	+	0.846569i	$0.321336\pi$
$594$	0	0
$595$	13.0549	0.535201
$596$	4.59135	0.188069
$597$	0	0
$598$	29.4964	1.20620
$599$	−15.8637	−0.648175	−0.324087	−	0.946027i	$-0.605057\pi$
$599$	−15.8637	−0.648175	−0.324087	+	0.946027i	$0.605057\pi$
$600$	0	0
$601$	19.9231	0.812678	0.406339	−	0.913722i	$-0.366805\pi$
$601$	19.9231	0.812678	0.406339	+	0.913722i	$0.366805\pi$
$602$	63.2459	2.57771
$603$	0	0
$604$	−97.5487	−3.96920
$605$	−9.23503	−0.375457
$606$	0	0
$607$	9.07779	0.368456	0.184228	−	0.982884i	$-0.441021\pi$
$607$	9.07779	0.368456	0.184228	+	0.982884i	$0.441021\pi$
$608$	−6.50200	−0.263691
$609$	0	0
$610$	−30.4804	−1.23412
$611$	−30.2751	−1.22480
$612$	0	0
$613$	11.0839	0.447674	0.223837	−	0.974627i	$-0.428142\pi$
$613$	11.0839	0.447674	0.223837	+	0.974627i	$0.428142\pi$
$614$	−34.3219	−1.38512
$615$	0	0
$616$	21.9734	0.885334
$617$	15.0662	0.606542	0.303271	−	0.952904i	$-0.401921\pi$
$617$	15.0662	0.606542	0.303271	+	0.952904i	$0.401921\pi$
$618$	0	0
$619$	42.8816	1.72356	0.861778	−	0.507286i	$-0.169351\pi$
$619$	42.8816	1.72356	0.861778	+	0.507286i	$0.169351\pi$
$620$	42.5950	1.71066
$621$	0	0
$622$	64.7640	2.59680
$623$	2.83060	0.113406
$624$	0	0
$625$	1.00000	0.0400000
$626$	−27.9293	−1.11628
$627$	0	0
$628$	65.5928	2.61744
$629$	0.956782	0.0381494
$630$	0	0
$631$	−0.952046	−0.0379004	−0.0189502	−	0.999820i	$-0.506032\pi$
$631$	−0.952046	−0.0379004	−0.0189502	+	0.999820i	$0.506032\pi$
$632$	46.7546	1.85980
$633$	0	0
$634$	28.8773	1.14686
$635$	−12.4045	−0.492259
$636$	0	0
$637$	−5.18400	−0.205398
$638$	0.559144	0.0221367
$639$	0	0
$640$	15.1966	0.600697
$641$	−2.20464	−0.0870782	−0.0435391	−	0.999052i	$-0.513863\pi$
$641$	−2.20464	−0.0870782	−0.0435391	+	0.999052i	$0.513863\pi$
$642$	0	0
$643$	−12.6354	−0.498292	−0.249146	−	0.968466i	$-0.580150\pi$
$643$	−12.6354	−0.498292	−0.249146	+	0.968466i	$0.580150\pi$
$644$	−28.0319	−1.10461
$645$	0	0
$646$	21.4309	0.843188
$647$	−12.1752	−0.478656	−0.239328	−	0.970939i	$-0.576927\pi$
$647$	−12.1752	−0.478656	−0.239328	+	0.970939i	$0.576927\pi$
$648$	0	0
$649$	9.57725	0.375940
$650$	12.8779	0.505112
$651$	0	0
$652$	−65.0746	−2.54852
$653$	18.0294	0.705545	0.352773	−	0.935709i	$-0.385239\pi$
$653$	18.0294	0.705545	0.352773	+	0.935709i	$0.385239\pi$
$654$	0	0
$655$	−12.0122	−0.469355
$656$	−31.4518	−1.22799
$657$	0	0
$658$	42.0811	1.64049
$659$	−13.5430	−0.527562	−0.263781	−	0.964583i	$-0.584970\pi$
$659$	−13.5430	−0.527562	−0.263781	+	0.964583i	$0.584970\pi$
$660$	0	0
$661$	−12.2540	−0.476627	−0.238314	−	0.971188i	$-0.576595\pi$
$661$	−12.2540	−0.476627	−0.238314	+	0.971188i	$0.576595\pi$
$662$	−39.2894	−1.52703
$663$	0	0
$664$	87.5762	3.39861
$665$	−5.23045	−0.202828
$666$	0	0
$667$	−0.383351	−0.0148434
$668$	−62.1812	−2.40586
$669$	0	0
$670$	−24.3754	−0.941704
$671$	16.1030	0.621648
$672$	0	0
$673$	−2.00909	−0.0774448	−0.0387224	−	0.999250i	$-0.512329\pi$
$673$	−2.00909	−0.0774448	−0.0387224	+	0.999250i	$0.512329\pi$
$674$	−21.6904	−0.835484
$675$	0	0
$676$	57.1816	2.19929
$677$	−20.3448	−0.781913	−0.390956	−	0.920409i	$-0.627856\pi$
$677$	−20.3448	−0.781913	−0.390956	+	0.920409i	$0.627856\pi$
$678$	0	0
$679$	−5.11116	−0.196148
$680$	26.9493	1.03346
$681$	0	0
$682$	−32.9126	−1.26029
$683$	15.6710	0.599636	0.299818	−	0.953996i	$-0.403074\pi$
$683$	15.6710	0.599636	0.299818	+	0.953996i	$0.403074\pi$
$684$	0	0
$685$	10.4285	0.398454
$686$	−42.6209	−1.62727
$687$	0	0
$688$	53.7252	2.04825
$689$	−15.9497	−0.607634
$690$	0	0
$691$	−9.63284	−0.366450	−0.183225	−	0.983071i	$-0.558654\pi$
$691$	−9.63284	−0.366450	−0.183225	+	0.983071i	$0.558654\pi$
$692$	91.9814	3.49661
$693$	0	0
$694$	47.4039	1.79943
$695$	−7.42653	−0.281704
$696$	0	0
$697$	23.9903	0.908697
$698$	50.8220	1.92364
$699$	0	0
$700$	−12.2385	−0.462571
$701$	−13.8413	−0.522779	−0.261390	−	0.965233i	$-0.584181\pi$
$701$	−13.8413	−0.522779	−0.261390	+	0.965233i	$0.584181\pi$
$702$	0	0
$703$	−0.383334	−0.0144577
$704$	−4.31050	−0.162458
$705$	0	0
$706$	−64.0205	−2.40944
$707$	20.1807	0.758974
$708$	0	0
$709$	19.8323	0.744817	0.372409	−	0.928069i	$-0.378532\pi$
$709$	19.8323	0.744817	0.372409	+	0.928069i	$0.378532\pi$
$710$	2.49423	0.0936069
$711$	0	0
$712$	5.84320	0.218983
$713$	22.5650	0.845065
$714$	0	0
$715$	−6.80345	−0.254434
$716$	−63.8688	−2.38689
$717$	0	0
$718$	−32.0666	−1.19671
$719$	−16.3095	−0.608243	−0.304122	−	0.952633i	$-0.598363\pi$
$719$	−16.3095	−0.608243	−0.304122	+	0.952633i	$0.598363\pi$
$720$	0	0
$721$	13.7917	0.513628
$722$	39.1927	1.45860
$723$	0	0
$724$	8.05589	0.299395
$725$	−0.167368	−0.00621589
$726$	0	0
$727$	−2.99661	−0.111138	−0.0555691	−	0.998455i	$-0.517697\pi$
$727$	−2.99661	−0.111138	−0.0555691	+	0.998455i	$0.517697\pi$
$728$	−84.7013	−3.13924
$729$	0	0
$730$	30.6325	1.13376
$731$	−40.9796	−1.51568
$732$	0	0
$733$	46.6651	1.72362	0.861808	−	0.507235i	$-0.169333\pi$
$733$	46.6651	1.72362	0.861808	+	0.507235i	$0.169333\pi$
$734$	67.9222	2.50705
$735$	0	0
$736$	−8.05956	−0.297079
$737$	12.8777	0.474355
$738$	0	0
$739$	10.7663	0.396045	0.198022	−	0.980198i	$-0.436548\pi$
$739$	10.7663	0.396045	0.198022	+	0.980198i	$0.436548\pi$
$740$	−0.896944	−0.0329723
$741$	0	0
$742$	22.1693	0.813861
$743$	48.3152	1.77251	0.886257	−	0.463194i	$-0.153297\pi$
$743$	48.3152	1.77251	0.886257	+	0.463194i	$0.153297\pi$
$744$	0	0
$745$	1.06192	0.0389057
$746$	68.3546	2.50264
$747$	0	0
$748$	−26.4920	−0.968642
$749$	−43.6881	−1.59633
$750$	0	0
$751$	−20.6257	−0.752641	−0.376321	−	0.926490i	$-0.622811\pi$
$751$	−20.6257	−0.752641	−0.376321	+	0.926490i	$0.622811\pi$
$752$	35.7464	1.30354
$753$	0	0
$754$	−2.15534	−0.0784929
$755$	−22.5617	−0.821105
$756$	0	0
$757$	17.1099	0.621871	0.310936	−	0.950431i	$-0.399358\pi$
$757$	17.1099	0.621871	0.310936	+	0.950431i	$0.399358\pi$
$758$	19.4347	0.705901
$759$	0	0
$760$	−10.7972	−0.391656
$761$	5.86395	0.212568	0.106284	−	0.994336i	$-0.466105\pi$
$761$	5.86395	0.212568	0.106284	+	0.994336i	$0.466105\pi$
$762$	0	0
$763$	47.6033	1.72335
$764$	−45.6846	−1.65281
$765$	0	0
$766$	70.4997	2.54726
$767$	−36.9176	−1.33302
$768$	0	0
$769$	18.3365	0.661231	0.330615	−	0.943766i	$-0.392744\pi$
$769$	18.3365	0.661231	0.330615	+	0.943766i	$0.392744\pi$
$770$	9.45649	0.340788
$771$	0	0
$772$	0.166813	0.00600374
$773$	−0.361308	−0.0129954	−0.00649768	−	0.999979i	$-0.502068\pi$
$773$	−0.361308	−0.0129954	−0.00649768	+	0.999979i	$0.502068\pi$
$774$	0	0
$775$	9.85168	0.353883
$776$	−10.5510	−0.378758
$777$	0	0
$778$	−1.49816	−0.0537115
$779$	−9.61168	−0.344374
$780$	0	0
$781$	−1.31772	−0.0471516
$782$	26.5647	0.949952
$783$	0	0
$784$	6.12085	0.218602
$785$	15.1708	0.541467
$786$	0	0
$787$	24.4430	0.871300	0.435650	−	0.900116i	$-0.356519\pi$
$787$	24.4430	0.871300	0.435650	+	0.900116i	$0.356519\pi$
$788$	−5.35163	−0.190644
$789$	0	0
$790$	20.1213	0.715885
$791$	58.3774	2.07566
$792$	0	0
$793$	−62.0724	−2.20425
$794$	89.4296	3.17374
$795$	0	0
$796$	68.9309	2.44319
$797$	−8.59641	−0.304500	−0.152250	−	0.988342i	$-0.548652\pi$
$797$	−8.59641	−0.304500	−0.152250	+	0.988342i	$0.548652\pi$
$798$	0	0
$799$	−27.2661	−0.964604
$800$	−3.51873	−0.124406
$801$	0	0
$802$	−2.29567	−0.0810631
$803$	−16.1833	−0.571097
$804$	0	0
$805$	−6.48340	−0.228510
$806$	126.869	4.46876
$807$	0	0
$808$	41.6591	1.46556
$809$	−52.3619	−1.84094	−0.920472	−	0.390808i	$-0.872196\pi$
$809$	−52.3619	−1.84094	−0.920472	+	0.390808i	$0.872196\pi$
$810$	0	0
$811$	9.71912	0.341285	0.170642	−	0.985333i	$-0.445416\pi$
$811$	9.71912	0.341285	0.170642	+	0.985333i	$0.445416\pi$
$812$	2.04833	0.0718822
$813$	0	0
$814$	0.693056	0.0242916
$815$	−15.0509	−0.527210
$816$	0	0
$817$	16.4184	0.574408
$818$	17.7193	0.619540
$819$	0	0
$820$	−22.4899	−0.785382
$821$	−11.4338	−0.399044	−0.199522	−	0.979893i	$-0.563939\pi$
$821$	−11.4338	−0.399044	−0.199522	+	0.979893i	$0.563939\pi$
$822$	0	0
$823$	−14.3109	−0.498845	−0.249422	−	0.968395i	$-0.580241\pi$
$823$	−14.3109	−0.498845	−0.249422	+	0.968395i	$0.580241\pi$
$824$	28.4701	0.991804
$825$	0	0
$826$	51.3138	1.78544
$827$	−3.13835	−0.109131	−0.0545655	−	0.998510i	$-0.517377\pi$
$827$	−3.13835	−0.109131	−0.0545655	+	0.998510i	$0.517377\pi$
$828$	0	0
$829$	−45.3160	−1.57389	−0.786944	−	0.617024i	$-0.788338\pi$
$829$	−45.3160	−1.57389	−0.786944	+	0.617024i	$0.788338\pi$
$830$	37.6893	1.30821
$831$	0	0
$832$	16.6157	0.576047
$833$	−4.66876	−0.161763
$834$	0	0
$835$	−14.3817	−0.497699
$836$	10.6140	0.367092
$837$	0	0
$838$	68.2124	2.35636
$839$	43.2811	1.49423	0.747115	−	0.664695i	$-0.231439\pi$
$839$	43.2811	1.49423	0.747115	+	0.664695i	$0.231439\pi$
$840$	0	0
$841$	−28.9720	−0.999034
$842$	−0.665287	−0.0229273
$843$	0	0
$844$	103.092	3.54859
$845$	13.2254	0.454966
$846$	0	0
$847$	26.1407	0.898204
$848$	18.8321	0.646696
$849$	0	0
$850$	11.5979	0.397805
$851$	−0.475161	−0.0162883
$852$	0	0
$853$	51.6286	1.76773	0.883864	−	0.467744i	$-0.154933\pi$
$853$	51.6286	1.76773	0.883864	+	0.467744i	$0.154933\pi$
$854$	86.2779	2.95237
$855$	0	0
$856$	−90.1854	−3.08247
$857$	−29.9889	−1.02440	−0.512201	−	0.858866i	$-0.671170\pi$
$857$	−29.9889	−1.02440	−0.512201	+	0.858866i	$0.671170\pi$
$858$	0	0
$859$	4.50797	0.153810	0.0769049	−	0.997038i	$-0.475496\pi$
$859$	4.50797	0.153810	0.0769049	+	0.997038i	$0.475496\pi$
$860$	38.4167	1.31000
$861$	0	0
$862$	−51.7650	−1.76312
$863$	−29.1380	−0.991870	−0.495935	−	0.868360i	$-0.665175\pi$
$863$	−29.1380	−0.991870	−0.495935	+	0.868360i	$0.665175\pi$
$864$	0	0
$865$	21.2741	0.723341
$866$	66.2200	2.25025
$867$	0	0
$868$	−120.569	−4.09239
$869$	−10.6302	−0.360605
$870$	0	0
$871$	−49.6397	−1.68198
$872$	98.2674	3.32776
$873$	0	0
$874$	−10.6431	−0.360009
$875$	−2.83060	−0.0956917
$876$	0	0
$877$	−17.6137	−0.594772	−0.297386	−	0.954757i	$-0.596115\pi$
$877$	−17.6137	−0.594772	−0.297386	+	0.954757i	$0.596115\pi$
$878$	−92.8892	−3.13486
$879$	0	0
$880$	8.03296	0.270791
$881$	−27.6785	−0.932512	−0.466256	−	0.884650i	$-0.654398\pi$
$881$	−27.6785	−0.932512	−0.466256	+	0.884650i	$0.654398\pi$
$882$	0	0
$883$	23.7126	0.797994	0.398997	−	0.916952i	$-0.369358\pi$
$883$	23.7126	0.797994	0.398997	+	0.916952i	$0.369358\pi$
$884$	102.119	3.43463
$885$	0	0
$886$	−17.4107	−0.584925
$887$	28.0883	0.943112	0.471556	−	0.881836i	$-0.343692\pi$
$887$	28.0883	0.943112	0.471556	+	0.881836i	$0.343692\pi$
$888$	0	0
$889$	35.1122	1.17763
$890$	2.51468	0.0842924
$891$	0	0
$892$	−100.326	−3.35916
$893$	10.9241	0.365562
$894$	0	0
$895$	−14.7720	−0.493774
$896$	−43.0154	−1.43704
$897$	0	0
$898$	−34.6625	−1.15670
$899$	−1.64885	−0.0549924
$900$	0	0
$901$	−14.3644	−0.478547
$902$	17.3776	0.578612
$903$	0	0
$904$	120.508	4.00805
$905$	1.86322	0.0619356
$906$	0	0
$907$	−19.9123	−0.661178	−0.330589	−	0.943775i	$-0.607247\pi$
$907$	−19.9123	−0.661178	−0.330589	+	0.943775i	$0.607247\pi$
$908$	29.8476	0.990529
$909$	0	0
$910$	−36.4521	−1.20837
$911$	17.1810	0.569233	0.284617	−	0.958641i	$-0.408134\pi$
$911$	17.1810	0.569233	0.284617	+	0.958641i	$0.408134\pi$
$912$	0	0
$913$	−19.9115	−0.658973
$914$	80.6045	2.66616
$915$	0	0
$916$	−59.7242	−1.97334
$917$	34.0017	1.12283
$918$	0	0
$919$	3.26343	0.107651	0.0538253	−	0.998550i	$-0.482859\pi$
$919$	3.26343	0.107651	0.0538253	+	0.998550i	$0.482859\pi$
$920$	−13.3837	−0.441247
$921$	0	0
$922$	−68.9819	−2.27180
$923$	5.07942	0.167191
$924$	0	0
$925$	−0.207451	−0.00682096
$926$	28.8508	0.948097
$927$	0	0
$928$	0.588923	0.0193324
$929$	−52.8522	−1.73402	−0.867012	−	0.498288i	$-0.833962\pi$
$929$	−52.8522	−1.73402	−0.867012	+	0.498288i	$0.833962\pi$
$930$	0	0
$931$	1.87053	0.0613042
$932$	5.21306	0.170759
$933$	0	0
$934$	−45.9131	−1.50232
$935$	−6.12724	−0.200382
$936$	0	0
$937$	−32.9905	−1.07775	−0.538876	−	0.842385i	$-0.681151\pi$
$937$	−32.9905	−1.07775	−0.538876	+	0.842385i	$0.681151\pi$
$938$	68.9970	2.25283
$939$	0	0
$940$	25.5608	0.833701
$941$	8.91420	0.290595	0.145297	−	0.989388i	$-0.453586\pi$
$941$	8.91420	0.290595	0.145297	+	0.989388i	$0.453586\pi$
$942$	0	0
$943$	−11.9142	−0.387979
$944$	43.5893	1.41871
$945$	0	0
$946$	−29.6840	−0.965110
$947$	−18.7920	−0.610659	−0.305329	−	0.952247i	$-0.598767\pi$
$947$	−18.7920	−0.610659	−0.305329	+	0.952247i	$0.598767\pi$
$948$	0	0
$949$	62.3821	2.02501
$950$	−4.64669	−0.150759
$951$	0	0
$952$	−76.2827	−2.47234
$953$	13.2480	0.429145	0.214573	−	0.976708i	$-0.431164\pi$
$953$	13.2480	0.429145	0.214573	+	0.976708i	$0.431164\pi$
$954$	0	0
$955$	−10.5662	−0.341916
$956$	−117.114	−3.78774
$957$	0	0
$958$	107.546	3.47465
$959$	−29.5190	−0.953219
$960$	0	0
$961$	66.0555	2.13082
$962$	−2.67153	−0.0861337
$963$	0	0
$964$	33.1437	1.06749
$965$	0.0385817	0.00124199
$966$	0	0
$967$	23.7207	0.762808	0.381404	−	0.924409i	$-0.375441\pi$
$967$	23.7207	0.762808	0.381404	+	0.924409i	$0.375441\pi$
$968$	53.9622	1.73441
$969$	0	0
$970$	−4.54072	−0.145794
$971$	12.4868	0.400720	0.200360	−	0.979722i	$-0.435789\pi$
$971$	12.4868	0.400720	0.200360	+	0.979722i	$0.435789\pi$
$972$	0	0
$973$	21.0215	0.673919
$974$	89.0961	2.85482
$975$	0	0
$976$	73.2900	2.34596
$977$	2.08604	0.0667383	0.0333692	−	0.999443i	$-0.489376\pi$
$977$	2.08604	0.0667383	0.0333692	+	0.999443i	$0.489376\pi$
$978$	0	0
$979$	−1.32852	−0.0424597
$980$	4.37677	0.139811
$981$	0	0
$982$	25.9921	0.829442
$983$	32.8063	1.04636	0.523180	−	0.852222i	$-0.324746\pi$
$983$	32.8063	1.04636	0.523180	+	0.852222i	$0.324746\pi$
$984$	0	0
$985$	−1.23776	−0.0394383
$986$	−1.94112	−0.0618179
$987$	0	0
$988$	−40.9138	−1.30164
$989$	20.3515	0.647139
$990$	0	0
$991$	15.6986	0.498683	0.249341	−	0.968416i	$-0.419786\pi$
$991$	15.6986	0.498683	0.249341	+	0.968416i	$0.419786\pi$
$992$	−34.6654	−1.10063
$993$	0	0
$994$	−7.06017	−0.223935
$995$	15.9428	0.505421
$996$	0	0
$997$	−32.8993	−1.04193	−0.520966	−	0.853578i	$-0.674428\pi$
$997$	−32.8993	−1.04193	−0.520966	+	0.853578i	$0.674428\pi$
$998$	73.3322	2.32129
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4005.2.a.p.1.1		8
3.2	odd	2		445.2.a.g.1.8	✓	8
12.11	even	2		7120.2.a.bk.1.3		8
15.14	odd	2		2225.2.a.l.1.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
445.2.a.g.1.8	✓	8	3.2	odd	2
2225.2.a.l.1.1		8	15.14	odd	2
4005.2.a.p.1.1		8	1.1	even	1	trivial
7120.2.a.bk.1.3		8	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4005 = 3^{2} \cdot 5 \cdot 89 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4005.a (trivial)

\(f(q)\)	\(=\)	\(q-2.51468 q^{2} +4.32363 q^{4} +1.00000 q^{5} -2.83060 q^{7} -5.84320 q^{8} +O(q^{10})\)	\(q-2.51468 q^{2} +4.32363 q^{4} +1.00000 q^{5} -2.83060 q^{7} -5.84320 q^{8} -2.51468 q^{10} +1.32852 q^{11} -5.12107 q^{13} +7.11806 q^{14} +6.04654 q^{16} -4.61208 q^{17} +1.84782 q^{19} +4.32363 q^{20} -3.34081 q^{22} +2.29047 q^{23} +1.00000 q^{25} +12.8779 q^{26} -12.2385 q^{28} -0.167368 q^{29} +9.85168 q^{31} -3.51873 q^{32} +11.5979 q^{34} -2.83060 q^{35} -0.207451 q^{37} -4.64669 q^{38} -5.84320 q^{40} -5.20162 q^{41} +8.88527 q^{43} +5.74404 q^{44} -5.75981 q^{46} +5.91188 q^{47} +1.01229 q^{49} -2.51468 q^{50} -22.1416 q^{52} +3.11452 q^{53} +1.32852 q^{55} +16.5398 q^{56} +0.420877 q^{58} +7.20896 q^{59} +12.1210 q^{61} -24.7739 q^{62} -3.24458 q^{64} -5.12107 q^{65} +9.69323 q^{67} -19.9409 q^{68} +7.11806 q^{70} -0.991867 q^{71} -12.1815 q^{73} +0.521675 q^{74} +7.98932 q^{76} -3.76051 q^{77} -8.00154 q^{79} +6.04654 q^{80} +13.0804 q^{82} -14.9877 q^{83} -4.61208 q^{85} -22.3437 q^{86} -7.76282 q^{88} -1.00000 q^{89} +14.4957 q^{91} +9.90316 q^{92} -14.8665 q^{94} +1.84782 q^{95} +1.80568 q^{97} -2.54559 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - q^{2} + 7 q^{4} + 8 q^{5} - 6 q^{7} - 3 q^{8}+O(q^{10}) \) 8 * q - q^2 + 7 * q^4 + 8 * q^5 - 6 * q^7 - 3 * q^8	\( 8 q - q^{2} + 7 q^{4} + 8 q^{5} - 6 q^{7} - 3 q^{8} - q^{10} - 14 q^{11} - 7 q^{13} - 15 q^{14} + 9 q^{16} - 17 q^{17} + 17 q^{19} + 7 q^{20} + 2 q^{22} + q^{23} + 8 q^{25} - 3 q^{26} - 29 q^{28} - 10 q^{29} + q^{31} - 2 q^{32} - 16 q^{34} - 6 q^{35} - 11 q^{37} + 30 q^{38} - 3 q^{40} - 15 q^{41} - 5 q^{43} - 7 q^{44} - 12 q^{46} - 12 q^{47} + 4 q^{49} - q^{50} - 14 q^{52} + q^{53} - 14 q^{55} - 3 q^{56} - 37 q^{58} - 26 q^{59} + 13 q^{61} - 22 q^{62} - 15 q^{64} - 7 q^{65} - 25 q^{67} - 23 q^{68} - 15 q^{70} - 28 q^{71} - 17 q^{73} + 5 q^{74} + 8 q^{76} - 7 q^{79} + 9 q^{80} + 5 q^{82} - 44 q^{83} - 17 q^{85} + 13 q^{86} - 66 q^{88} - 8 q^{89} + 27 q^{91} - 15 q^{92} - 27 q^{94} + 17 q^{95} + q^{97} + 34 q^{98}+O(q^{100}) \) 8 * q - q^2 + 7 * q^4 + 8 * q^5 - 6 * q^7 - 3 * q^8 - q^10 - 14 * q^11 - 7 * q^13 - 15 * q^14 + 9 * q^16 - 17 * q^17 + 17 * q^19 + 7 * q^20 + 2 * q^22 + q^23 + 8 * q^25 - 3 * q^26 - 29 * q^28 - 10 * q^29 + q^31 - 2 * q^32 - 16 * q^34 - 6 * q^35 - 11 * q^37 + 30 * q^38 - 3 * q^40 - 15 * q^41 - 5 * q^43 - 7 * q^44 - 12 * q^46 - 12 * q^47 + 4 * q^49 - q^50 - 14 * q^52 + q^53 - 14 * q^55 - 3 * q^56 - 37 * q^58 - 26 * q^59 + 13 * q^61 - 22 * q^62 - 15 * q^64 - 7 * q^65 - 25 * q^67 - 23 * q^68 - 15 * q^70 - 28 * q^71 - 17 * q^73 + 5 * q^74 + 8 * q^76 - 7 * q^79 + 9 * q^80 + 5 * q^82 - 44 * q^83 - 17 * q^85 + 13 * q^86 - 66 * q^88 - 8 * q^89 + 27 * q^91 - 15 * q^92 - 27 * q^94 + 17 * q^95 + q^97 + 34 * q^98

Introduction

L-functions

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Database

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