LMFDB - Embedded newform 4005.2.a.u.1.11

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:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} - 13 x^{10} + 41 x^{9} + 58 x^{8} - 202 x^{7} - 95 x^{6} + 432 x^{5} + 4 x^{4} + \cdots - 27 $ x^12 - 3x^11 - 13x^10 + 41x^9 + 58x^8 - 202x^7 - 95x^6 + 432x^5 + 4x^4 - 368x^3 + 94x^2 + 77*x - 27
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$-1.85231$ 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)

$f(q)$	$=$	$q+1.85231 q^{2} +1.43107 q^{4} +1.00000 q^{5} -0.0119316 q^{7} -1.05384 q^{8} +O(q^{10})$	\(q+1.85231 q^{2} +1.43107 q^{4} +1.00000 q^{5} -0.0119316 q^{7} -1.05384 q^{8} +1.85231 q^{10} -3.66920 q^{11} +0.322749 q^{13} -0.0221011 q^{14} -4.81418 q^{16} +3.56432 q^{17} -3.88003 q^{19} +1.43107 q^{20} -6.79651 q^{22} -3.76817 q^{23} +1.00000 q^{25} +0.597833 q^{26} -0.0170749 q^{28} -10.2839 q^{29} -10.3612 q^{31} -6.80969 q^{32} +6.60224 q^{34} -0.0119316 q^{35} +2.45297 q^{37} -7.18704 q^{38} -1.05384 q^{40} +6.31765 q^{41} +3.04722 q^{43} -5.25088 q^{44} -6.97984 q^{46} +9.55117 q^{47} -6.99986 q^{49} +1.85231 q^{50} +0.461876 q^{52} +1.41730 q^{53} -3.66920 q^{55} +0.0125740 q^{56} -19.0490 q^{58} -6.60375 q^{59} -6.03944 q^{61} -19.1922 q^{62} -2.98533 q^{64} +0.322749 q^{65} +4.11863 q^{67} +5.10078 q^{68} -0.0221011 q^{70} -3.56519 q^{71} -5.44804 q^{73} +4.54367 q^{74} -5.55259 q^{76} +0.0437794 q^{77} -8.71250 q^{79} -4.81418 q^{80} +11.7023 q^{82} +0.742084 q^{83} +3.56432 q^{85} +5.64440 q^{86} +3.86676 q^{88} +1.00000 q^{89} -0.00385091 q^{91} -5.39251 q^{92} +17.6918 q^{94} -3.88003 q^{95} +18.4917 q^{97} -12.9659 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 3 q^{2} + 11 q^{4} + 12 q^{5} - 8 q^{7} - 9 q^{8}+O(q^{10}) $ 12 * q - 3 * q^2 + 11 * q^4 + 12 * q^5 - 8 * q^7 - 9 * q^8	$ 12 q - 3 q^{2} + 11 q^{4} + 12 q^{5} - 8 q^{7} - 9 q^{8} - 3 q^{10} - 12 q^{13} + 4 q^{14} + q^{16} - 24 q^{19} + 11 q^{20} - 16 q^{22} - 24 q^{23} + 12 q^{25} - q^{26} - 44 q^{28} + 8 q^{29} - 12 q^{31} - 31 q^{32} - 18 q^{34} - 8 q^{35} - 10 q^{37} + 2 q^{38} - 9 q^{40} - 10 q^{41} - 42 q^{43} + 42 q^{44} - 24 q^{46} - 22 q^{47} - 4 q^{49} - 3 q^{50} - 30 q^{52} + 8 q^{53} + 27 q^{56} - 12 q^{58} + 4 q^{59} - 52 q^{61} - 14 q^{62} + 7 q^{64} - 12 q^{65} - 40 q^{67} + 23 q^{68} + 4 q^{70} + 2 q^{71} - 8 q^{73} + 26 q^{74} - 46 q^{76} - 12 q^{77} - 26 q^{79} + q^{80} - 26 q^{82} - 14 q^{83} + 32 q^{86} - 60 q^{88} + 12 q^{89} - 24 q^{91} - 38 q^{92} - 26 q^{94} - 24 q^{95} - 6 q^{97} + 30 q^{98}+O(q^{100}) $ 12 * q - 3 * q^2 + 11 * q^4 + 12 * q^5 - 8 * q^7 - 9 * q^8 - 3 * q^10 - 12 * q^13 + 4 * q^14 + q^16 - 24 * q^19 + 11 * q^20 - 16 * q^22 - 24 * q^23 + 12 * q^25 - q^26 - 44 * q^28 + 8 * q^29 - 12 * q^31 - 31 * q^32 - 18 * q^34 - 8 * q^35 - 10 * q^37 + 2 * q^38 - 9 * q^40 - 10 * q^41 - 42 * q^43 + 42 * q^44 - 24 * q^46 - 22 * q^47 - 4 * q^49 - 3 * q^50 - 30 * q^52 + 8 * q^53 + 27 * q^56 - 12 * q^58 + 4 * q^59 - 52 * q^61 - 14 * q^62 + 7 * q^64 - 12 * q^65 - 40 * q^67 + 23 * q^68 + 4 * q^70 + 2 * q^71 - 8 * q^73 + 26 * q^74 - 46 * q^76 - 12 * q^77 - 26 * q^79 + q^80 - 26 * q^82 - 14 * q^83 + 32 * q^86 - 60 * q^88 + 12 * q^89 - 24 * q^91 - 38 * q^92 - 26 * q^94 - 24 * q^95 - 6 * q^97 + 30 * 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 $

$2$	1.85231	1.30978	0.654892	−	0.755723i	$-0.272714\pi$
$2$	1.85231	1.30978	0.654892	+	0.755723i	$0.272714\pi$
$3$	0	0
$3$	0	0
$4$	1.43107	0.715534
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−0.0119316	−0.00450972	−0.00225486	−	0.999997i	$-0.500718\pi$
$7$	−0.0119316	−0.00450972	−0.00225486	+	0.999997i	$0.500718\pi$
$8$	−1.05384	−0.372589
$9$	0	0
$10$	1.85231	0.585753
$11$	−3.66920	−1.10631	−0.553153	−	0.833080i	$-0.686576\pi$
$11$	−3.66920	−1.10631	−0.553153	+	0.833080i	$0.686576\pi$
$12$	0	0
$13$	0.322749	0.0895145	0.0447573	−	0.998998i	$-0.485749\pi$
$13$	0.322749	0.0895145	0.0447573	+	0.998998i	$0.485749\pi$
$14$	−0.0221011	−0.00590676
$15$	0	0
$16$	−4.81418	−1.20355
$17$	3.56432	0.864474	0.432237	−	0.901760i	$-0.357724\pi$
$17$	3.56432	0.864474	0.432237	+	0.901760i	$0.357724\pi$
$18$	0	0
$19$	−3.88003	−0.890141	−0.445070	−	0.895496i	$-0.646821\pi$
$19$	−3.88003	−0.890141	−0.445070	+	0.895496i	$0.646821\pi$
$20$	1.43107	0.319996
$21$	0	0
$22$	−6.79651	−1.44902
$23$	−3.76817	−0.785718	−0.392859	−	0.919599i	$-0.628514\pi$
$23$	−3.76817	−0.785718	−0.392859	+	0.919599i	$0.628514\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0.597833	0.117245
$27$	0	0
$28$	−0.0170749	−0.00322686
$29$	−10.2839	−1.90967	−0.954836	−	0.297133i	$-0.903970\pi$
$29$	−10.2839	−1.90967	−0.954836	+	0.297133i	$0.903970\pi$
$30$	0	0
$31$	−10.3612	−1.86092	−0.930462	−	0.366389i	$-0.880594\pi$
$31$	−10.3612	−1.86092	−0.930462	+	0.366389i	$0.880594\pi$
$32$	−6.80969	−1.20379
$33$	0	0
$34$	6.60224	1.13227
$35$	−0.0119316	−0.00201681
$36$	0	0
$37$	2.45297	0.403265	0.201633	−	0.979461i	$-0.435375\pi$
$37$	2.45297	0.403265	0.201633	+	0.979461i	$0.435375\pi$
$38$	−7.18704	−1.16589
$39$	0	0
$40$	−1.05384	−0.166627
$41$	6.31765	0.986651	0.493325	−	0.869845i	$-0.335781\pi$
$41$	6.31765	0.986651	0.493325	+	0.869845i	$0.335781\pi$
$42$	0	0
$43$	3.04722	0.464696	0.232348	−	0.972633i	$-0.425359\pi$
$43$	3.04722	0.464696	0.232348	+	0.972633i	$0.425359\pi$
$44$	−5.25088	−0.791599
$45$	0	0
$46$	−6.97984	−1.02912
$47$	9.55117	1.39318	0.696591	−	0.717469i	$-0.254699\pi$
$47$	9.55117	1.39318	0.696591	+	0.717469i	$0.254699\pi$
$48$	0	0
$49$	−6.99986	−0.999980
$50$	1.85231	0.261957
$51$	0	0
$52$	0.461876	0.0640507
$53$	1.41730	0.194682	0.0973409	−	0.995251i	$-0.468966\pi$
$53$	1.41730	0.194682	0.0973409	+	0.995251i	$0.468966\pi$
$54$	0	0
$55$	−3.66920	−0.494755
$56$	0.0125740	0.00168027
$57$	0	0
$58$	−19.0490	−2.50126
$59$	−6.60375	−0.859735	−0.429867	−	0.902892i	$-0.641440\pi$
$59$	−6.60375	−0.859735	−0.429867	+	0.902892i	$0.641440\pi$
$60$	0	0
$61$	−6.03944	−0.773270	−0.386635	−	0.922233i	$-0.626363\pi$
$61$	−6.03944	−0.773270	−0.386635	+	0.922233i	$0.626363\pi$
$62$	−19.1922	−2.43741
$63$	0	0
$64$	−2.98533	−0.373166
$65$	0.322749	0.0400321
$66$	0	0
$67$	4.11863	0.503171	0.251586	−	0.967835i	$-0.419048\pi$
$67$	4.11863	0.503171	0.251586	+	0.967835i	$0.419048\pi$
$68$	5.10078	0.618561
$69$	0	0
$70$	−0.0221011	−0.00264158
$71$	−3.56519	−0.423110	−0.211555	−	0.977366i	$-0.567853\pi$
$71$	−3.56519	−0.423110	−0.211555	+	0.977366i	$0.567853\pi$
$72$	0	0
$73$	−5.44804	−0.637645	−0.318822	−	0.947814i	$-0.603287\pi$
$73$	−5.44804	−0.637645	−0.318822	+	0.947814i	$0.603287\pi$
$74$	4.54367	0.528190
$75$	0	0
$76$	−5.55259	−0.636926
$77$	0.0437794	0.00498913
$78$	0	0
$79$	−8.71250	−0.980232	−0.490116	−	0.871657i	$-0.663046\pi$
$79$	−8.71250	−0.980232	−0.490116	+	0.871657i	$0.663046\pi$
$80$	−4.81418	−0.538242
$81$	0	0
$82$	11.7023	1.29230
$83$	0.742084	0.0814543	0.0407271	−	0.999170i	$-0.487033\pi$
$83$	0.742084	0.0814543	0.0407271	+	0.999170i	$0.487033\pi$
$84$	0	0
$85$	3.56432	0.386605
$86$	5.64440	0.608651
$87$	0	0
$88$	3.86676	0.412198
$89$	1.00000	0.106000
$89$	1.00000	0.106000
$90$	0	0
$91$	−0.00385091	−0.000403685 0
$92$	−5.39251	−0.562208
$93$	0	0
$94$	17.6918	1.82477
$95$	−3.88003	−0.398083
$96$	0	0
$97$	18.4917	1.87754	0.938771	−	0.344541i	$-0.111965\pi$
$97$	18.4917	1.87754	0.938771	+	0.344541i	$0.111965\pi$
$98$	−12.9659	−1.30976
$99$	0	0
$100$	1.43107	0.143107
$101$	6.77781	0.674417	0.337209	−	0.941430i	$-0.390517\pi$
$101$	6.77781	0.674417	0.337209	+	0.941430i	$0.390517\pi$
$102$	0	0
$103$	4.01640	0.395748	0.197874	−	0.980228i	$-0.436596\pi$
$103$	4.01640	0.395748	0.197874	+	0.980228i	$0.436596\pi$
$104$	−0.340126	−0.0333521
$105$	0	0
$106$	2.62529	0.254991
$107$	−8.04766	−0.777997	−0.388998	−	0.921238i	$-0.627179\pi$
$107$	−8.04766	−0.777997	−0.388998	+	0.921238i	$0.627179\pi$
$108$	0	0
$109$	−2.53532	−0.242840	−0.121420	−	0.992601i	$-0.538745\pi$
$109$	−2.53532	−0.242840	−0.121420	+	0.992601i	$0.538745\pi$
$110$	−6.79651	−0.648022
$111$	0	0
$112$	0.0574408	0.00542765
$113$	−8.53206	−0.802629	−0.401314	−	0.915940i	$-0.631447\pi$
$113$	−8.53206	−0.802629	−0.401314	+	0.915940i	$0.631447\pi$
$114$	0	0
$115$	−3.76817	−0.351384
$116$	−14.7170	−1.36644
$117$	0	0
$118$	−12.2322	−1.12607
$119$	−0.0425280	−0.00389853
$120$	0	0
$121$	2.46304	0.223913
$122$	−11.1869	−1.01282
$123$	0	0
$124$	−14.8276	−1.33155
$125$	1.00000	0.0894427
$126$	0	0
$127$	−10.2098	−0.905971	−0.452985	−	0.891518i	$-0.649641\pi$
$127$	−10.2098	−0.905971	−0.452985	+	0.891518i	$0.649641\pi$
$128$	8.08962	0.715028
$129$	0	0
$130$	0.597833	0.0524334
$131$	−17.1831	−1.50130	−0.750648	−	0.660702i	$-0.770259\pi$
$131$	−17.1831	−1.50130	−0.750648	+	0.660702i	$0.770259\pi$
$132$	0	0
$133$	0.0462950	0.00401428
$134$	7.62900	0.659046
$135$	0	0
$136$	−3.75623	−0.322094
$137$	2.62533	0.224297	0.112149	−	0.993691i	$-0.464227\pi$
$137$	2.62533	0.224297	0.112149	+	0.993691i	$0.464227\pi$
$138$	0	0
$139$	−0.352604	−0.0299075	−0.0149537	−	0.999888i	$-0.504760\pi$
$139$	−0.352604	−0.0299075	−0.0149537	+	0.999888i	$0.504760\pi$
$140$	−0.0170749	−0.00144309
$141$	0	0
$142$	−6.60385	−0.554182
$143$	−1.18423	−0.0990305
$144$	0	0
$145$	−10.2839	−0.854031
$146$	−10.0915	−0.835177
$147$	0	0
$148$	3.51036	0.288550
$149$	−9.51669	−0.779638	−0.389819	−	0.920892i	$-0.627462\pi$
$149$	−9.51669	−0.779638	−0.389819	+	0.920892i	$0.627462\pi$
$150$	0	0
$151$	1.88742	0.153596	0.0767981	−	0.997047i	$-0.475530\pi$
$151$	1.88742	0.153596	0.0767981	+	0.997047i	$0.475530\pi$
$152$	4.08894	0.331657
$153$	0	0
$154$	0.0810932	0.00653468
$155$	−10.3612	−0.832230
$156$	0	0
$157$	15.1002	1.20513	0.602565	−	0.798070i	$-0.294145\pi$
$157$	15.1002	1.20513	0.602565	+	0.798070i	$0.294145\pi$
$158$	−16.1383	−1.28389
$159$	0	0
$160$	−6.80969	−0.538353
$161$	0.0449603	0.00354337
$162$	0	0
$163$	−14.5732	−1.14146	−0.570732	−	0.821136i	$-0.693340\pi$
$163$	−14.5732	−1.14146	−0.570732	+	0.821136i	$0.693340\pi$
$164$	9.04098	0.705982
$165$	0	0
$166$	1.37457	0.106688
$167$	10.7749	0.833789	0.416895	−	0.908955i	$-0.363118\pi$
$167$	10.7749	0.833789	0.416895	+	0.908955i	$0.363118\pi$
$168$	0	0
$169$	−12.8958	−0.991987
$170$	6.60224	0.506368
$171$	0	0
$172$	4.36077	0.332506
$173$	7.44256	0.565848	0.282924	−	0.959142i	$-0.408696\pi$
$173$	7.44256	0.565848	0.282924	+	0.959142i	$0.408696\pi$
$174$	0	0
$175$	−0.0119316	−0.000901944 0
$176$	17.6642	1.33149
$177$	0	0
$178$	1.85231	0.138837
$179$	24.7379	1.84900	0.924498	−	0.381187i	$-0.124484\pi$
$179$	24.7379	1.84900	0.924498	+	0.381187i	$0.124484\pi$
$180$	0	0
$181$	−7.80139	−0.579873	−0.289937	−	0.957046i	$-0.593634\pi$
$181$	−7.80139	−0.579873	−0.289937	+	0.957046i	$0.593634\pi$
$182$	−0.00713310	−0.000528740 0
$183$	0	0
$184$	3.97105	0.292750
$185$	2.45297	0.180346
$186$	0	0
$187$	−13.0782	−0.956373
$188$	13.6684	0.996868
$189$	0	0
$190$	−7.18704	−0.521403
$191$	12.6385	0.914487	0.457243	−	0.889342i	$-0.348837\pi$
$191$	12.6385	0.914487	0.457243	+	0.889342i	$0.348837\pi$
$192$	0	0
$193$	−6.65090	−0.478742	−0.239371	−	0.970928i	$-0.576941\pi$
$193$	−6.65090	−0.478742	−0.239371	+	0.970928i	$0.576941\pi$
$194$	34.2524	2.45918
$195$	0	0
$196$	−10.0173	−0.715519
$197$	−10.1804	−0.725322	−0.362661	−	0.931921i	$-0.618132\pi$
$197$	−10.1804	−0.725322	−0.362661	+	0.931921i	$0.618132\pi$
$198$	0	0
$199$	0.563244	0.0399273	0.0199637	−	0.999801i	$-0.493645\pi$
$199$	0.563244	0.0399273	0.0199637	+	0.999801i	$0.493645\pi$
$200$	−1.05384	−0.0745178
$201$	0	0
$202$	12.5546	0.883341
$203$	0.122703	0.00861208
$204$	0	0
$205$	6.31765	0.441244
$206$	7.43963	0.518344
$207$	0	0
$208$	−1.55377	−0.107735
$209$	14.2366	0.984768
$210$	0	0
$211$	−26.6416	−1.83408	−0.917042	−	0.398790i	$-0.869430\pi$
$211$	−26.6416	−1.83408	−0.917042	+	0.398790i	$0.869430\pi$
$212$	2.02826	0.139301
$213$	0	0
$214$	−14.9068	−1.01901
$215$	3.04722	0.207818
$216$	0	0
$217$	0.123625	0.00839224
$218$	−4.69621	−0.318068
$219$	0	0
$220$	−5.25088	−0.354014
$221$	1.15038	0.0773830
$222$	0	0
$223$	14.8896	0.997082	0.498541	−	0.866866i	$-0.333869\pi$
$223$	14.8896	0.997082	0.498541	+	0.866866i	$0.333869\pi$
$224$	0.0812505	0.00542878
$225$	0	0
$226$	−15.8041	−1.05127
$227$	−20.6871	−1.37305	−0.686525	−	0.727106i	$-0.740865\pi$
$227$	−20.6871	−1.37305	−0.686525	+	0.727106i	$0.740865\pi$
$228$	0	0
$229$	27.5653	1.82157	0.910783	−	0.412885i	$-0.135479\pi$
$229$	27.5653	1.82157	0.910783	+	0.412885i	$0.135479\pi$
$230$	−6.97984	−0.460237
$231$	0	0
$232$	10.8376	0.711523
$233$	10.5269	0.689639	0.344819	−	0.938669i	$-0.387940\pi$
$233$	10.5269	0.689639	0.344819	+	0.938669i	$0.387940\pi$
$234$	0	0
$235$	9.55117	0.623050
$236$	−9.45041	−0.615169
$237$	0	0
$238$	−0.0787752	−0.00510624
$239$	−16.0957	−1.04115	−0.520573	−	0.853817i	$-0.674282\pi$
$239$	−16.0957	−1.04115	−0.520573	+	0.853817i	$0.674282\pi$
$240$	0	0
$241$	−12.0131	−0.773831	−0.386915	−	0.922115i	$-0.626459\pi$
$241$	−12.0131	−0.773831	−0.386915	+	0.922115i	$0.626459\pi$
$242$	4.56233	0.293277
$243$	0	0
$244$	−8.64284	−0.553301
$245$	−6.99986	−0.447205
$246$	0	0
$247$	−1.25228	−0.0796806
$248$	10.9190	0.693360
$249$	0	0
$250$	1.85231	0.117151
$251$	21.3373	1.34680	0.673400	−	0.739279i	$-0.264833\pi$
$251$	21.3373	1.34680	0.673400	+	0.739279i	$0.264833\pi$
$252$	0	0
$253$	13.8262	0.869245
$254$	−18.9117	−1.18663
$255$	0	0
$256$	20.9552	1.30970
$257$	24.8546	1.55039	0.775193	−	0.631724i	$-0.217652\pi$
$257$	24.8546	1.55039	0.775193	+	0.631724i	$0.217652\pi$
$258$	0	0
$259$	−0.0292678	−0.00181861
$260$	0.461876	0.0286443
$261$	0	0
$262$	−31.8285	−1.96637
$263$	−18.1771	−1.12085	−0.560423	−	0.828207i	$-0.689361\pi$
$263$	−18.1771	−1.12085	−0.560423	+	0.828207i	$0.689361\pi$
$264$	0	0
$265$	1.41730	0.0870643
$266$	0.0857529	0.00525785
$267$	0	0
$268$	5.89404	0.360036
$269$	13.6548	0.832546	0.416273	−	0.909240i	$-0.363336\pi$
$269$	13.6548	0.832546	0.416273	+	0.909240i	$0.363336\pi$
$270$	0	0
$271$	−12.5024	−0.759465	−0.379733	−	0.925096i	$-0.623984\pi$
$271$	−12.5024	−0.759465	−0.379733	+	0.925096i	$0.623984\pi$
$272$	−17.1593	−1.04043
$273$	0	0
$274$	4.86294	0.293781
$275$	−3.66920	−0.221261
$276$	0	0
$277$	13.4857	0.810279	0.405140	−	0.914255i	$-0.367223\pi$
$277$	13.4857	0.810279	0.405140	+	0.914255i	$0.367223\pi$
$278$	−0.653133	−0.0391723
$279$	0	0
$280$	0.0125740	0.000751440 0
$281$	16.9242	1.00962	0.504808	−	0.863232i	$-0.331563\pi$
$281$	16.9242	1.00962	0.504808	+	0.863232i	$0.331563\pi$
$282$	0	0
$283$	−3.11691	−0.185281	−0.0926404	−	0.995700i	$-0.529531\pi$
$283$	−3.11691	−0.185281	−0.0926404	+	0.995700i	$0.529531\pi$
$284$	−5.10202	−0.302749
$285$	0	0
$286$	−2.19357	−0.129709
$287$	−0.0753796	−0.00444952
$288$	0	0
$289$	−4.29564	−0.252684
$290$	−19.0490	−1.11860
$291$	0	0
$292$	−7.79652	−0.456257
$293$	−10.2842	−0.600807	−0.300403	−	0.953812i	$-0.597121\pi$
$293$	−10.2842	−0.600807	−0.300403	+	0.953812i	$0.597121\pi$
$294$	0	0
$295$	−6.60375	−0.384485
$296$	−2.58504	−0.150252
$297$	0	0
$298$	−17.6279	−1.02116
$299$	−1.21617	−0.0703332
$300$	0	0
$301$	−0.0363581	−0.00209565
$302$	3.49610	0.201178
$303$	0	0
$304$	18.6792	1.07132
$305$	−6.03944	−0.345817
$306$	0	0
$307$	−7.84612	−0.447802	−0.223901	−	0.974612i	$-0.571879\pi$
$307$	−7.84612	−0.447802	−0.223901	+	0.974612i	$0.571879\pi$
$308$	0.0626513	0.00356989
$309$	0	0
$310$	−19.1922	−1.09004
$311$	14.9499	0.847733	0.423866	−	0.905725i	$-0.360673\pi$
$311$	14.9499	0.847733	0.423866	+	0.905725i	$0.360673\pi$
$312$	0	0
$313$	−7.33302	−0.414487	−0.207243	−	0.978289i	$-0.566449\pi$
$313$	−7.33302	−0.414487	−0.207243	+	0.978289i	$0.566449\pi$
$314$	27.9704	1.57846
$315$	0	0
$316$	−12.4682	−0.701389
$317$	19.1401	1.07501	0.537506	−	0.843260i	$-0.319366\pi$
$317$	19.1401	1.07501	0.537506	+	0.843260i	$0.319366\pi$
$318$	0	0
$319$	37.7337	2.11268
$320$	−2.98533	−0.166885
$321$	0	0
$322$	0.0832806	0.00464104
$323$	−13.8297	−0.769504
$324$	0	0
$325$	0.322749	0.0179029
$326$	−26.9942	−1.49507
$327$	0	0
$328$	−6.65780	−0.367615
$329$	−0.113961	−0.00628285
$330$	0	0
$331$	−6.80478	−0.374025	−0.187012	−	0.982358i	$-0.559880\pi$
$331$	−6.80478	−0.374025	−0.187012	+	0.982358i	$0.559880\pi$
$332$	1.06197	0.0582833
$333$	0	0
$334$	19.9586	1.09208
$335$	4.11863	0.225025
$336$	0	0
$337$	−15.6219	−0.850980	−0.425490	−	0.904963i	$-0.639898\pi$
$337$	−15.6219	−0.850980	−0.425490	+	0.904963i	$0.639898\pi$
$338$	−23.8871	−1.29929
$339$	0	0
$340$	5.10078	0.276629
$341$	38.0173	2.05875
$342$	0	0
$343$	0.167041	0.00901934
$344$	−3.21128	−0.173141
$345$	0	0
$346$	13.7860	0.741138
$347$	−7.73328	−0.415144	−0.207572	−	0.978220i	$-0.566556\pi$
$347$	−7.73328	−0.415144	−0.207572	+	0.978220i	$0.566556\pi$
$348$	0	0
$349$	6.17602	0.330595	0.165297	−	0.986244i	$-0.447142\pi$
$349$	6.17602	0.330595	0.165297	+	0.986244i	$0.447142\pi$
$350$	−0.0221011	−0.00118135
$351$	0	0
$352$	24.9861	1.33177
$353$	36.2350	1.92860	0.964298	−	0.264818i	$-0.0853118\pi$
$353$	36.2350	1.92860	0.964298	+	0.264818i	$0.0853118\pi$
$354$	0	0
$355$	−3.56519	−0.189220
$356$	1.43107	0.0758464
$357$	0	0
$358$	45.8223	2.42179
$359$	24.0464	1.26912	0.634559	−	0.772874i	$-0.281182\pi$
$359$	24.0464	1.26912	0.634559	+	0.772874i	$0.281182\pi$
$360$	0	0
$361$	−3.94533	−0.207649
$362$	−14.4506	−0.759508
$363$	0	0
$364$	−0.00551092	−0.000288850 0
$365$	−5.44804	−0.285164
$366$	0	0
$367$	−30.7616	−1.60574	−0.802872	−	0.596151i	$-0.796696\pi$
$367$	−30.7616	−1.60574	−0.802872	+	0.596151i	$0.796696\pi$
$368$	18.1407	0.945647
$369$	0	0
$370$	4.54367	0.236214
$371$	−0.0169107	−0.000877960 0
$372$	0	0
$373$	−13.7232	−0.710563	−0.355281	−	0.934759i	$-0.615615\pi$
$373$	−13.7232	−0.710563	−0.355281	+	0.934759i	$0.615615\pi$
$374$	−24.2249	−1.25264
$375$	0	0
$376$	−10.0654	−0.519084
$377$	−3.31912	−0.170943
$378$	0	0
$379$	9.04980	0.464857	0.232429	−	0.972613i	$-0.425333\pi$
$379$	9.04980	0.464857	0.232429	+	0.972613i	$0.425333\pi$
$380$	−5.55259	−0.284842
$381$	0	0
$382$	23.4104	1.19778
$383$	3.99333	0.204050	0.102025	−	0.994782i	$-0.467468\pi$
$383$	3.99333	0.204050	0.102025	+	0.994782i	$0.467468\pi$
$384$	0	0
$385$	0.0437794	0.00223121
$386$	−12.3196	−0.627049
$387$	0	0
$388$	26.4628	1.34345
$389$	5.06796	0.256956	0.128478	−	0.991712i	$-0.458991\pi$
$389$	5.06796	0.256956	0.128478	+	0.991712i	$0.458991\pi$
$390$	0	0
$391$	−13.4310	−0.679233
$392$	7.37674	0.372582
$393$	0	0
$394$	−18.8573	−0.950015
$395$	−8.71250	−0.438373
$396$	0	0
$397$	4.22665	0.212129	0.106065	−	0.994359i	$-0.466175\pi$
$397$	4.22665	0.212129	0.106065	+	0.994359i	$0.466175\pi$
$398$	1.04331	0.0522962
$399$	0	0
$400$	−4.81418	−0.240709
$401$	3.22234	0.160916	0.0804579	−	0.996758i	$-0.474362\pi$
$401$	3.22234	0.160916	0.0804579	+	0.996758i	$0.474362\pi$
$402$	0	0
$403$	−3.34406	−0.166580
$404$	9.69951	0.482569
$405$	0	0
$406$	0.227285	0.0112800
$407$	−9.00043	−0.446135
$408$	0	0
$409$	−8.45061	−0.417856	−0.208928	−	0.977931i	$-0.566997\pi$
$409$	−8.45061	−0.417856	−0.208928	+	0.977931i	$0.566997\pi$
$410$	11.7023	0.577934
$411$	0	0
$412$	5.74774	0.283171
$413$	0.0787932	0.00387716
$414$	0	0
$415$	0.742084	0.0364275
$416$	−2.19782	−0.107757
$417$	0	0
$418$	26.3707	1.28983
$419$	−5.95559	−0.290950	−0.145475	−	0.989362i	$-0.546471\pi$
$419$	−5.95559	−0.290950	−0.145475	+	0.989362i	$0.546471\pi$
$420$	0	0
$421$	−19.6873	−0.959499	−0.479750	−	0.877406i	$-0.659272\pi$
$421$	−19.6873	−0.959499	−0.479750	+	0.877406i	$0.659272\pi$
$422$	−49.3486	−2.40225
$423$	0	0
$424$	−1.49361	−0.0725363
$425$	3.56432	0.172895
$426$	0	0
$427$	0.0720601	0.00348723
$428$	−11.5167	−0.556683
$429$	0	0
$430$	5.64440	0.272197
$431$	15.4350	0.743475	0.371738	−	0.928338i	$-0.378762\pi$
$431$	15.4350	0.743475	0.371738	+	0.928338i	$0.378762\pi$
$432$	0	0
$433$	−14.4969	−0.696676	−0.348338	−	0.937369i	$-0.613254\pi$
$433$	−14.4969	−0.696676	−0.348338	+	0.937369i	$0.613254\pi$
$434$	0.228993	0.0109920
$435$	0	0
$436$	−3.62822	−0.173760
$437$	14.6206	0.699400
$438$	0	0
$439$	−24.8628	−1.18663	−0.593317	−	0.804969i	$-0.702182\pi$
$439$	−24.8628	−1.18663	−0.593317	+	0.804969i	$0.702182\pi$
$440$	3.86676	0.184340
$441$	0	0
$442$	2.13087	0.101355
$443$	−25.5276	−1.21285	−0.606427	−	0.795139i	$-0.707398\pi$
$443$	−25.5276	−1.21285	−0.606427	+	0.795139i	$0.707398\pi$
$444$	0	0
$445$	1.00000	0.0474045
$446$	27.5802	1.30596
$447$	0	0
$448$	0.0356197	0.00168287
$449$	2.12082	0.100088	0.0500438	−	0.998747i	$-0.484064\pi$
$449$	2.12082	0.100088	0.0500438	+	0.998747i	$0.484064\pi$
$450$	0	0
$451$	−23.1807	−1.09154
$452$	−12.2100	−0.574308
$453$	0	0
$454$	−38.3190	−1.79840
$455$	−0.00385091	−0.000180534 0
$456$	0	0
$457$	22.8314	1.06801	0.534005	−	0.845482i	$-0.320687\pi$
$457$	22.8314	1.06801	0.534005	+	0.845482i	$0.320687\pi$
$458$	51.0596	2.38586
$459$	0	0
$460$	−5.39251	−0.251427
$461$	9.05032	0.421515	0.210758	−	0.977538i	$-0.432407\pi$
$461$	9.05032	0.421515	0.210758	+	0.977538i	$0.432407\pi$
$462$	0	0
$463$	11.8045	0.548604	0.274302	−	0.961644i	$-0.411553\pi$
$463$	11.8045	0.548604	0.274302	+	0.961644i	$0.411553\pi$
$464$	49.5086	2.29838
$465$	0	0
$466$	19.4991	0.903277
$467$	10.6314	0.491961	0.245981	−	0.969275i	$-0.420890\pi$
$467$	10.6314	0.491961	0.245981	+	0.969275i	$0.420890\pi$
$468$	0	0
$469$	−0.0491419	−0.00226916
$470$	17.6918	0.816060
$471$	0	0
$472$	6.95930	0.320328
$473$	−11.1808	−0.514096
$474$	0	0
$475$	−3.88003	−0.178028
$476$	−0.0608604	−0.00278953
$477$	0	0
$478$	−29.8143	−1.36368
$479$	27.8637	1.27312	0.636562	−	0.771225i	$-0.280356\pi$
$479$	27.8637	1.27312	0.636562	+	0.771225i	$0.280356\pi$
$480$	0	0
$481$	0.791693	0.0360981
$482$	−22.2520	−1.01355
$483$	0	0
$484$	3.52478	0.160217
$485$	18.4917	0.839663
$486$	0	0
$487$	−33.7998	−1.53161	−0.765807	−	0.643070i	$-0.777660\pi$
$487$	−33.7998	−1.53161	−0.765807	+	0.643070i	$0.777660\pi$
$488$	6.36461	0.288112
$489$	0	0
$490$	−12.9659	−0.585741
$491$	−33.6353	−1.51794	−0.758970	−	0.651125i	$-0.774297\pi$
$491$	−33.6353	−1.51794	−0.758970	+	0.651125i	$0.774297\pi$
$492$	0	0
$493$	−36.6551	−1.65086
$494$	−2.31961	−0.104364
$495$	0	0
$496$	49.8806	2.23971
$497$	0.0425383	0.00190811
$498$	0	0
$499$	1.00546	0.0450106	0.0225053	−	0.999747i	$-0.492836\pi$
$499$	1.00546	0.0450106	0.0225053	+	0.999747i	$0.492836\pi$
$500$	1.43107	0.0639993
$501$	0	0
$502$	39.5234	1.76402
$503$	28.3188	1.26268	0.631338	−	0.775508i	$-0.282506\pi$
$503$	28.3188	1.26268	0.631338	+	0.775508i	$0.282506\pi$
$504$	0	0
$505$	6.77781	0.301609
$506$	25.6104	1.13852
$507$	0	0
$508$	−14.6109	−0.648253
$509$	−4.10770	−0.182071	−0.0910353	−	0.995848i	$-0.529018\pi$
$509$	−4.10770	−0.182071	−0.0910353	+	0.995848i	$0.529018\pi$
$510$	0	0
$511$	0.0650038	0.00287560
$512$	22.6363	1.00039
$513$	0	0
$514$	46.0385	2.03067
$515$	4.01640	0.176984
$516$	0	0
$517$	−35.0452	−1.54128
$518$	−0.0542132	−0.00238199
$519$	0	0
$520$	−0.340126	−0.0149155
$521$	−19.7156	−0.863754	−0.431877	−	0.901932i	$-0.642149\pi$
$521$	−19.7156	−0.863754	−0.431877	+	0.901932i	$0.642149\pi$
$522$	0	0
$523$	34.6109	1.51343	0.756715	−	0.653744i	$-0.226803\pi$
$523$	34.6109	1.51343	0.756715	+	0.653744i	$0.226803\pi$
$524$	−24.5902	−1.07423
$525$	0	0
$526$	−33.6696	−1.46807
$527$	−36.9306	−1.60872
$528$	0	0
$529$	−8.80088	−0.382647
$530$	2.62529	0.114035
$531$	0	0
$532$	0.0662513	0.00287236
$533$	2.03902	0.0883196
$534$	0	0
$535$	−8.04766	−0.347931
$536$	−4.34039	−0.187476
$537$	0	0
$538$	25.2929	1.09046
$539$	25.6839	1.10628
$540$	0	0
$541$	−43.1373	−1.85462	−0.927308	−	0.374299i	$-0.877883\pi$
$541$	−43.1373	−1.85462	−0.927308	+	0.374299i	$0.877883\pi$
$542$	−23.1583	−0.994736
$543$	0	0
$544$	−24.2719	−1.04065
$545$	−2.53532	−0.108601
$546$	0	0
$547$	34.9943	1.49625	0.748123	−	0.663560i	$-0.230955\pi$
$547$	34.9943	1.49625	0.748123	+	0.663560i	$0.230955\pi$
$548$	3.75703	0.160492
$549$	0	0
$550$	−6.79651	−0.289804
$551$	39.9019	1.69988
$552$	0	0
$553$	0.103954	0.00442057
$554$	24.9798	1.06129
$555$	0	0
$556$	−0.504600	−0.0213998
$557$	−17.1210	−0.725440	−0.362720	−	0.931898i	$-0.618152\pi$
$557$	−17.1210	−0.725440	−0.362720	+	0.931898i	$0.618152\pi$
$558$	0	0
$559$	0.983487	0.0415971
$560$	0.0574408	0.00242732
$561$	0	0
$562$	31.3490	1.32238
$563$	44.8279	1.88927	0.944636	−	0.328121i	$-0.106415\pi$
$563$	44.8279	1.88927	0.944636	+	0.328121i	$0.106415\pi$
$564$	0	0
$565$	−8.53206	−0.358946
$566$	−5.77349	−0.242678
$567$	0	0
$568$	3.75714	0.157646
$569$	23.0662	0.966986	0.483493	−	0.875348i	$-0.339368\pi$
$569$	23.0662	0.966986	0.483493	+	0.875348i	$0.339368\pi$
$570$	0	0
$571$	−1.86574	−0.0780790	−0.0390395	−	0.999238i	$-0.512430\pi$
$571$	−1.86574	−0.0780790	−0.0390395	+	0.999238i	$0.512430\pi$
$572$	−1.69472	−0.0708597
$573$	0	0
$574$	−0.139627	−0.00582791
$575$	−3.76817	−0.157144
$576$	0	0
$577$	−14.4331	−0.600857	−0.300428	−	0.953804i	$-0.597130\pi$
$577$	−14.4331	−0.600857	−0.300428	+	0.953804i	$0.597130\pi$
$578$	−7.95687	−0.330962
$579$	0	0
$580$	−14.7170	−0.611088
$581$	−0.00885424	−0.000367336 0
$582$	0	0
$583$	−5.20038	−0.215378
$584$	5.74137	0.237580
$585$	0	0
$586$	−19.0495	−0.786927
$587$	−40.4742	−1.67055	−0.835275	−	0.549833i	$-0.814691\pi$
$587$	−40.4742	−1.67055	−0.835275	+	0.549833i	$0.814691\pi$
$588$	0	0
$589$	40.2018	1.65648
$590$	−12.2322	−0.503592
$591$	0	0
$592$	−11.8090	−0.485348
$593$	−24.6659	−1.01291	−0.506453	−	0.862268i	$-0.669044\pi$
$593$	−24.6659	−1.01291	−0.506453	+	0.862268i	$0.669044\pi$
$594$	0	0
$595$	−0.0425280	−0.00174348
$596$	−13.6190	−0.557857
$597$	0	0
$598$	−2.25274	−0.0921213
$599$	10.8767	0.444409	0.222205	−	0.975000i	$-0.428675\pi$
$599$	10.8767	0.444409	0.222205	+	0.975000i	$0.428675\pi$
$600$	0	0
$601$	−22.8536	−0.932218	−0.466109	−	0.884727i	$-0.654345\pi$
$601$	−22.8536	−0.932218	−0.466109	+	0.884727i	$0.654345\pi$
$602$	−0.0673467	−0.00274485
$603$	0	0
$604$	2.70103	0.109903
$605$	2.46304	0.100137
$606$	0	0
$607$	21.5052	0.872870	0.436435	−	0.899736i	$-0.356241\pi$
$607$	21.5052	0.872870	0.436435	+	0.899736i	$0.356241\pi$
$608$	26.4218	1.07155
$609$	0	0
$610$	−11.1869	−0.452946
$611$	3.08263	0.124710
$612$	0	0
$613$	−13.5395	−0.546856	−0.273428	−	0.961892i	$-0.588158\pi$
$613$	−13.5395	−0.546856	−0.273428	+	0.961892i	$0.588158\pi$
$614$	−14.5335	−0.586523
$615$	0	0
$616$	−0.0461366	−0.00185889
$617$	24.7239	0.995346	0.497673	−	0.867365i	$-0.334188\pi$
$617$	24.7239	0.995346	0.497673	+	0.867365i	$0.334188\pi$
$618$	0	0
$619$	36.8924	1.48283	0.741416	−	0.671046i	$-0.234155\pi$
$619$	36.8924	1.48283	0.741416	+	0.671046i	$0.234155\pi$
$620$	−14.8276	−0.595489
$621$	0	0
$622$	27.6920	1.11035
$623$	−0.0119316	−0.000478029 0
$624$	0	0
$625$	1.00000	0.0400000
$626$	−13.5831	−0.542888
$627$	0	0
$628$	21.6095	0.862311
$629$	8.74316	0.348612
$630$	0	0
$631$	−0.729265	−0.0290316	−0.0145158	−	0.999895i	$-0.504621\pi$
$631$	−0.729265	−0.0290316	−0.0145158	+	0.999895i	$0.504621\pi$
$632$	9.18159	0.365224
$633$	0	0
$634$	35.4534	1.40803
$635$	−10.2098	−0.405162
$636$	0	0
$637$	−2.25920	−0.0895127
$638$	69.8947	2.76716
$639$	0	0
$640$	8.08962	0.319770
$641$	0.557453	0.0220181	0.0110090	−	0.999939i	$-0.496496\pi$
$641$	0.557453	0.0220181	0.0110090	+	0.999939i	$0.496496\pi$
$642$	0	0
$643$	4.92726	0.194312	0.0971560	−	0.995269i	$-0.469025\pi$
$643$	4.92726	0.194312	0.0971560	+	0.995269i	$0.469025\pi$
$644$	0.0643412	0.00253540
$645$	0	0
$646$	−25.6169	−1.00788
$647$	19.3074	0.759051	0.379525	−	0.925181i	$-0.376087\pi$
$647$	19.3074	0.759051	0.379525	+	0.925181i	$0.376087\pi$
$648$	0	0
$649$	24.2305	0.951130
$650$	0.597833	0.0234489
$651$	0	0
$652$	−20.8553	−0.816756
$653$	−33.4001	−1.30705	−0.653523	−	0.756906i	$-0.726710\pi$
$653$	−33.4001	−1.30705	−0.653523	+	0.756906i	$0.726710\pi$
$654$	0	0
$655$	−17.1831	−0.671400
$656$	−30.4143	−1.18748
$657$	0	0
$658$	−0.211091	−0.00822918
$659$	−18.0245	−0.702136	−0.351068	−	0.936350i	$-0.614181\pi$
$659$	−18.0245	−0.702136	−0.351068	+	0.936350i	$0.614181\pi$
$660$	0	0
$661$	−16.3335	−0.635298	−0.317649	−	0.948208i	$-0.602893\pi$
$661$	−16.3335	−0.635298	−0.317649	+	0.948208i	$0.602893\pi$
$662$	−12.6046	−0.489892
$663$	0	0
$664$	−0.782039	−0.0303490
$665$	0.0462950	0.00179524
$666$	0	0
$667$	38.7515	1.50046
$668$	15.4197	0.596605
$669$	0	0
$670$	7.62900	0.294734
$671$	22.1599	0.855474
$672$	0	0
$673$	30.4912	1.17535	0.587675	−	0.809097i	$-0.300043\pi$
$673$	30.4912	1.17535	0.587675	+	0.809097i	$0.300043\pi$
$674$	−28.9367	−1.11460
$675$	0	0
$676$	−18.4548	−0.709800
$677$	32.5468	1.25088	0.625438	−	0.780274i	$-0.284920\pi$
$677$	32.5468	1.25088	0.625438	+	0.780274i	$0.284920\pi$
$678$	0	0
$679$	−0.220635	−0.00846719
$680$	−3.75623	−0.144045
$681$	0	0
$682$	70.4199	2.69652
$683$	−31.6799	−1.21220	−0.606099	−	0.795389i	$-0.707267\pi$
$683$	−31.6799	−1.21220	−0.606099	+	0.795389i	$0.707267\pi$
$684$	0	0
$685$	2.62533	0.100309
$686$	0.309412	0.0118134
$687$	0	0
$688$	−14.6698	−0.559283
$689$	0.457434	0.0174268
$690$	0	0
$691$	−48.0401	−1.82753	−0.913767	−	0.406240i	$-0.866840\pi$
$691$	−48.0401	−1.82753	−0.913767	+	0.406240i	$0.866840\pi$
$692$	10.6508	0.404883
$693$	0	0
$694$	−14.3245	−0.543749
$695$	−0.352604	−0.0133750
$696$	0	0
$697$	22.5181	0.852934
$698$	11.4399	0.433008
$699$	0	0
$700$	−0.0170749	−0.000645371 0
$701$	−14.7813	−0.558283	−0.279141	−	0.960250i	$-0.590050\pi$
$701$	−14.7813	−0.558283	−0.279141	+	0.960250i	$0.590050\pi$
$702$	0	0
$703$	−9.51760	−0.358963
$704$	10.9538	0.412836
$705$	0	0
$706$	67.1187	2.52605
$707$	−0.0808701	−0.00304143
$708$	0	0
$709$	45.4456	1.70675	0.853373	−	0.521300i	$-0.174553\pi$
$709$	45.4456	1.70675	0.853373	+	0.521300i	$0.174553\pi$
$710$	−6.60385	−0.247838
$711$	0	0
$712$	−1.05384	−0.0394944
$713$	39.0427	1.46216
$714$	0	0
$715$	−1.18423	−0.0442878
$716$	35.4016	1.32302
$717$	0	0
$718$	44.5414	1.66227
$719$	47.2191	1.76098	0.880488	−	0.474069i	$-0.157215\pi$
$719$	47.2191	1.76098	0.880488	+	0.474069i	$0.157215\pi$
$720$	0	0
$721$	−0.0479220	−0.00178471
$722$	−7.30800	−0.271975
$723$	0	0
$724$	−11.1643	−0.414919
$725$	−10.2839	−0.381934
$726$	0	0
$727$	25.2211	0.935400	0.467700	−	0.883887i	$-0.345083\pi$
$727$	25.2211	0.935400	0.467700	+	0.883887i	$0.345083\pi$
$728$	0.00405825	0.000150409 0
$729$	0	0
$730$	−10.0915	−0.373503
$731$	10.8612	0.401718
$732$	0	0
$733$	−34.9698	−1.29164	−0.645820	−	0.763490i	$-0.723484\pi$
$733$	−34.9698	−1.29164	−0.645820	+	0.763490i	$0.723484\pi$
$734$	−56.9802	−2.10318
$735$	0	0
$736$	25.6601	0.945843
$737$	−15.1121	−0.556661
$738$	0	0
$739$	−36.0257	−1.32523	−0.662614	−	0.748961i	$-0.730553\pi$
$739$	−36.0257	−1.32523	−0.662614	+	0.748961i	$0.730553\pi$
$740$	3.51036	0.129043
$741$	0	0
$742$	−0.0313239	−0.00114994
$743$	−27.5573	−1.01098	−0.505489	−	0.862833i	$-0.668688\pi$
$743$	−27.5573	−1.01098	−0.505489	+	0.862833i	$0.668688\pi$
$744$	0	0
$745$	−9.51669	−0.348665
$746$	−25.4198	−0.930683
$747$	0	0
$748$	−18.7158	−0.684317
$749$	0.0960214	0.00350855
$750$	0	0
$751$	18.4115	0.671844	0.335922	−	0.941890i	$-0.390952\pi$
$751$	18.4115	0.671844	0.335922	+	0.941890i	$0.390952\pi$
$752$	−45.9811	−1.67676
$753$	0	0
$754$	−6.14806	−0.223899
$755$	1.88742	0.0686903
$756$	0	0
$757$	39.8528	1.44847	0.724237	−	0.689551i	$-0.242192\pi$
$757$	39.8528	1.44847	0.724237	+	0.689551i	$0.242192\pi$
$758$	16.7631	0.608862
$759$	0	0
$760$	4.08894	0.148321
$761$	−40.0441	−1.45160	−0.725799	−	0.687907i	$-0.758530\pi$
$761$	−40.0441	−1.45160	−0.725799	+	0.687907i	$0.758530\pi$
$762$	0	0
$763$	0.0302504	0.00109514
$764$	18.0865	0.654346
$765$	0	0
$766$	7.39690	0.267261
$767$	−2.13136	−0.0769588
$768$	0	0
$769$	−28.5201	−1.02846	−0.514231	−	0.857651i	$-0.671923\pi$
$769$	−28.5201	−1.02846	−0.514231	+	0.857651i	$0.671923\pi$
$770$	0.0810932	0.00292240
$771$	0	0
$772$	−9.51789	−0.342556
$773$	44.9955	1.61838	0.809188	−	0.587550i	$-0.199908\pi$
$773$	44.9955	1.61838	0.809188	+	0.587550i	$0.199908\pi$
$774$	0	0
$775$	−10.3612	−0.372185
$776$	−19.4873	−0.699552
$777$	0	0
$778$	9.38746	0.336557
$779$	−24.5127	−0.878258
$780$	0	0
$781$	13.0814	0.468089
$782$	−24.8784	−0.889648
$783$	0	0
$784$	33.6986	1.20352
$785$	15.1002	0.538951
$786$	0	0
$787$	−22.0793	−0.787042	−0.393521	−	0.919316i	$-0.628743\pi$
$787$	−22.0793	−0.787042	−0.393521	+	0.919316i	$0.628743\pi$
$788$	−14.5688	−0.518993
$789$	0	0
$790$	−16.1383	−0.574174
$791$	0.101801	0.00361963
$792$	0	0
$793$	−1.94922	−0.0692189
$794$	7.82908	0.277843
$795$	0	0
$796$	0.806040	0.0285693
$797$	−12.3347	−0.436919	−0.218459	−	0.975846i	$-0.570103\pi$
$797$	−12.3347	−0.436919	−0.218459	+	0.975846i	$0.570103\pi$
$798$	0	0
$799$	34.0434	1.20437
$800$	−6.80969	−0.240759
$801$	0	0
$802$	5.96878	0.210765
$803$	19.9900	0.705430
$804$	0	0
$805$	0.0449603	0.00158464
$806$	−6.19426	−0.218183
$807$	0	0
$808$	−7.14274	−0.251281
$809$	−1.42979	−0.0502688	−0.0251344	−	0.999684i	$-0.508001\pi$
$809$	−1.42979	−0.0502688	−0.0251344	+	0.999684i	$0.508001\pi$
$810$	0	0
$811$	−33.3512	−1.17112	−0.585559	−	0.810630i	$-0.699125\pi$
$811$	−33.3512	−1.17112	−0.585559	+	0.810630i	$0.699125\pi$
$812$	0.175597	0.00616224
$813$	0	0
$814$	−16.6716	−0.584340
$815$	−14.5732	−0.510478
$816$	0	0
$817$	−11.8233	−0.413645
$818$	−15.6532	−0.547300
$819$	0	0
$820$	9.04098	0.315725
$821$	−3.74345	−0.130647	−0.0653236	−	0.997864i	$-0.520808\pi$
$821$	−3.74345	−0.130647	−0.0653236	+	0.997864i	$0.520808\pi$
$822$	0	0
$823$	21.0164	0.732586	0.366293	−	0.930500i	$-0.380627\pi$
$823$	21.0164	0.732586	0.366293	+	0.930500i	$0.380627\pi$
$824$	−4.23265	−0.147451
$825$	0	0
$826$	0.145950	0.00507824
$827$	0.0438441	0.00152461	0.000762304	−	1.00000i	$-0.499757\pi$
$827$	0.0438441	0.00152461	0.000762304		1.00000i	$0.499757\pi$
$828$	0	0
$829$	−25.9804	−0.902336	−0.451168	−	0.892439i	$-0.648993\pi$
$829$	−25.9804	−0.902336	−0.451168	+	0.892439i	$0.648993\pi$
$830$	1.37457	0.0477121
$831$	0	0
$832$	−0.963512	−0.0334038
$833$	−24.9497	−0.864457
$834$	0	0
$835$	10.7749	0.372882
$836$	20.3736	0.704635
$837$	0	0
$838$	−11.0316	−0.381081
$839$	−57.8376	−1.99678	−0.998388	−	0.0567595i	$-0.981923\pi$
$839$	−57.8376	−1.99678	−0.998388	+	0.0567595i	$0.981923\pi$
$840$	0	0
$841$	76.7586	2.64685
$842$	−36.4670	−1.25674
$843$	0	0
$844$	−38.1260	−1.31235
$845$	−12.8958	−0.443630
$846$	0	0
$847$	−0.0293880	−0.00100978
$848$	−6.82316	−0.234308
$849$	0	0
$850$	6.60224	0.226455
$851$	−9.24320	−0.316853
$852$	0	0
$853$	25.8224	0.884143	0.442071	−	0.896980i	$-0.354244\pi$
$853$	25.8224	0.884143	0.442071	+	0.896980i	$0.354244\pi$
$854$	0.133478	0.00456752
$855$	0	0
$856$	8.48096	0.289873
$857$	27.2503	0.930854	0.465427	−	0.885086i	$-0.345901\pi$
$857$	27.2503	0.930854	0.465427	+	0.885086i	$0.345901\pi$
$858$	0	0
$859$	−7.97809	−0.272209	−0.136104	−	0.990694i	$-0.543458\pi$
$859$	−7.97809	−0.272209	−0.136104	+	0.990694i	$0.543458\pi$
$860$	4.36077	0.148701
$861$	0	0
$862$	28.5904	0.973792
$863$	33.7674	1.14946	0.574728	−	0.818345i	$-0.305108\pi$
$863$	33.7674	1.14946	0.574728	+	0.818345i	$0.305108\pi$
$864$	0	0
$865$	7.44256	0.253055
$866$	−26.8528	−0.912494
$867$	0	0
$868$	0.176916	0.00600493
$869$	31.9679	1.08444
$870$	0	0
$871$	1.32929	0.0450411
$872$	2.67183	0.0904794
$873$	0	0
$874$	27.0820	0.916063
$875$	−0.0119316	−0.000403361 0
$876$	0	0
$877$	−15.3654	−0.518854	−0.259427	−	0.965763i	$-0.583534\pi$
$877$	−15.3654	−0.518854	−0.259427	+	0.965763i	$0.583534\pi$
$878$	−46.0536	−1.55423
$879$	0	0
$880$	17.6642	0.595460
$881$	4.35977	0.146884	0.0734421	−	0.997299i	$-0.476602\pi$
$881$	4.35977	0.146884	0.0734421	+	0.997299i	$0.476602\pi$
$882$	0	0
$883$	−7.77738	−0.261730	−0.130865	−	0.991400i	$-0.541775\pi$
$883$	−7.77738	−0.261730	−0.130865	+	0.991400i	$0.541775\pi$
$884$	1.64627	0.0553702
$885$	0	0
$886$	−47.2852	−1.58858
$887$	20.9414	0.703142	0.351571	−	0.936161i	$-0.385648\pi$
$887$	20.9414	0.703142	0.351571	+	0.936161i	$0.385648\pi$
$888$	0	0
$889$	0.121819	0.00408567
$890$	1.85231	0.0620897
$891$	0	0
$892$	21.3080	0.713446
$893$	−37.0589	−1.24013
$894$	0	0
$895$	24.7379	0.826896
$896$	−0.0965220	−0.00322457
$897$	0	0
$898$	3.92842	0.131093
$899$	106.553	3.55375
$900$	0	0
$901$	5.05173	0.168297
$902$	−42.9380	−1.42968
$903$	0	0
$904$	8.99144	0.299051
$905$	−7.80139	−0.259327
$906$	0	0
$907$	−4.47525	−0.148598	−0.0742991	−	0.997236i	$-0.523672\pi$
$907$	−4.47525	−0.148598	−0.0742991	+	0.997236i	$0.523672\pi$
$908$	−29.6046	−0.982464
$909$	0	0
$910$	−0.00713310	−0.000236460 0
$911$	4.98197	0.165060	0.0825300	−	0.996589i	$-0.473700\pi$
$911$	4.98197	0.165060	0.0825300	+	0.996589i	$0.473700\pi$
$912$	0	0
$913$	−2.72286	−0.0901134
$914$	42.2910	1.39886
$915$	0	0
$916$	39.4478	1.30339
$917$	0.205022	0.00677042
$918$	0	0
$919$	−4.88172	−0.161033	−0.0805165	−	0.996753i	$-0.525657\pi$
$919$	−4.88172	−0.161033	−0.0805165	+	0.996753i	$0.525657\pi$
$920$	3.97105	0.130922
$921$	0	0
$922$	16.7640	0.552094
$923$	−1.15066	−0.0378745
$924$	0	0
$925$	2.45297	0.0806531
$926$	21.8657	0.718552
$927$	0	0
$928$	70.0302	2.29885
$929$	−15.4262	−0.506116	−0.253058	−	0.967451i	$-0.581436\pi$
$929$	−15.4262	−0.506116	−0.253058	+	0.967451i	$0.581436\pi$
$930$	0	0
$931$	27.1597	0.890123
$932$	15.0647	0.493460
$933$	0	0
$934$	19.6926	0.644363
$935$	−13.0782	−0.427703
$936$	0	0
$937$	44.0224	1.43815	0.719075	−	0.694932i	$-0.244566\pi$
$937$	44.0224	1.43815	0.719075	+	0.694932i	$0.244566\pi$
$938$	−0.0910262	−0.00297211
$939$	0	0
$940$	13.6684	0.445813
$941$	−28.5460	−0.930574	−0.465287	−	0.885160i	$-0.654049\pi$
$941$	−28.5460	−0.930574	−0.465287	+	0.885160i	$0.654049\pi$
$942$	0	0
$943$	−23.8060	−0.775229
$944$	31.7916	1.03473
$945$	0	0
$946$	−20.7104	−0.673355
$947$	33.7592	1.09703	0.548513	−	0.836142i	$-0.315194\pi$
$947$	33.7592	1.09703	0.548513	+	0.836142i	$0.315194\pi$
$948$	0	0
$949$	−1.75835	−0.0570785
$950$	−7.18704	−0.233178
$951$	0	0
$952$	0.0448178	0.00145255
$953$	57.3474	1.85767	0.928833	−	0.370499i	$-0.120813\pi$
$953$	57.3474	1.85767	0.928833	+	0.370499i	$0.120813\pi$
$954$	0	0
$955$	12.6385	0.408971
$956$	−23.0341	−0.744975
$957$	0	0
$958$	51.6123	1.66752
$959$	−0.0313244	−0.00101152
$960$	0	0
$961$	76.3542	2.46304
$962$	1.46646	0.0472807
$963$	0	0
$964$	−17.1915	−0.553702
$965$	−6.65090	−0.214100
$966$	0	0
$967$	−33.7107	−1.08406	−0.542031	−	0.840359i	$-0.682344\pi$
$967$	−33.7107	−1.08406	−0.542031	+	0.840359i	$0.682344\pi$
$968$	−2.59565	−0.0834275
$969$	0	0
$970$	34.2524	1.09978
$971$	19.3779	0.621867	0.310933	−	0.950432i	$-0.399358\pi$
$971$	19.3779	0.621867	0.310933	+	0.950432i	$0.399358\pi$
$972$	0	0
$973$	0.00420713	0.000134874 0
$974$	−62.6078	−2.00608
$975$	0	0
$976$	29.0749	0.930666
$977$	−48.1262	−1.53969	−0.769847	−	0.638229i	$-0.779667\pi$
$977$	−48.1262	−1.53969	−0.769847	+	0.638229i	$0.779667\pi$
$978$	0	0
$979$	−3.66920	−0.117268
$980$	−10.0173	−0.319990
$981$	0	0
$982$	−62.3032	−1.98817
$983$	−22.6235	−0.721577	−0.360789	−	0.932648i	$-0.617492\pi$
$983$	−22.6235	−0.721577	−0.360789	+	0.932648i	$0.617492\pi$
$984$	0	0
$985$	−10.1804	−0.324374
$986$	−67.8968	−2.16227
$987$	0	0
$988$	−1.79209	−0.0570141
$989$	−11.4824	−0.365120
$990$	0	0
$991$	−18.5390	−0.588911	−0.294455	−	0.955665i	$-0.595138\pi$
$991$	−18.5390	−0.588911	−0.294455	+	0.955665i	$0.595138\pi$
$992$	70.5565	2.24017
$993$	0	0
$994$	0.0787944	0.00249921
$995$	0.563244	0.0178560
$996$	0	0
$997$	1.44225	0.0456765	0.0228383	−	0.999739i	$-0.492730\pi$
$997$	1.44225	0.0456765	0.0228383	+	0.999739i	$0.492730\pi$
$998$	1.86243	0.0589542
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4005.2.a.u.1.11	✓	12
3.2	odd	2		4005.2.a.v.1.2	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4005.2.a.u.1.11	✓	12	1.1	even	1	trivial
4005.2.a.v.1.2	yes	12	3.2	odd	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)

\(f(q)\)	\(=\)	\(q+1.85231 q^{2} +1.43107 q^{4} +1.00000 q^{5} -0.0119316 q^{7} -1.05384 q^{8} +O(q^{10})\)	\(q+1.85231 q^{2} +1.43107 q^{4} +1.00000 q^{5} -0.0119316 q^{7} -1.05384 q^{8} +1.85231 q^{10} -3.66920 q^{11} +0.322749 q^{13} -0.0221011 q^{14} -4.81418 q^{16} +3.56432 q^{17} -3.88003 q^{19} +1.43107 q^{20} -6.79651 q^{22} -3.76817 q^{23} +1.00000 q^{25} +0.597833 q^{26} -0.0170749 q^{28} -10.2839 q^{29} -10.3612 q^{31} -6.80969 q^{32} +6.60224 q^{34} -0.0119316 q^{35} +2.45297 q^{37} -7.18704 q^{38} -1.05384 q^{40} +6.31765 q^{41} +3.04722 q^{43} -5.25088 q^{44} -6.97984 q^{46} +9.55117 q^{47} -6.99986 q^{49} +1.85231 q^{50} +0.461876 q^{52} +1.41730 q^{53} -3.66920 q^{55} +0.0125740 q^{56} -19.0490 q^{58} -6.60375 q^{59} -6.03944 q^{61} -19.1922 q^{62} -2.98533 q^{64} +0.322749 q^{65} +4.11863 q^{67} +5.10078 q^{68} -0.0221011 q^{70} -3.56519 q^{71} -5.44804 q^{73} +4.54367 q^{74} -5.55259 q^{76} +0.0437794 q^{77} -8.71250 q^{79} -4.81418 q^{80} +11.7023 q^{82} +0.742084 q^{83} +3.56432 q^{85} +5.64440 q^{86} +3.86676 q^{88} +1.00000 q^{89} -0.00385091 q^{91} -5.39251 q^{92} +17.6918 q^{94} -3.88003 q^{95} +18.4917 q^{97} -12.9659 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 3 q^{2} + 11 q^{4} + 12 q^{5} - 8 q^{7} - 9 q^{8}+O(q^{10}) \) 12 * q - 3 * q^2 + 11 * q^4 + 12 * q^5 - 8 * q^7 - 9 * q^8	\( 12 q - 3 q^{2} + 11 q^{4} + 12 q^{5} - 8 q^{7} - 9 q^{8} - 3 q^{10} - 12 q^{13} + 4 q^{14} + q^{16} - 24 q^{19} + 11 q^{20} - 16 q^{22} - 24 q^{23} + 12 q^{25} - q^{26} - 44 q^{28} + 8 q^{29} - 12 q^{31} - 31 q^{32} - 18 q^{34} - 8 q^{35} - 10 q^{37} + 2 q^{38} - 9 q^{40} - 10 q^{41} - 42 q^{43} + 42 q^{44} - 24 q^{46} - 22 q^{47} - 4 q^{49} - 3 q^{50} - 30 q^{52} + 8 q^{53} + 27 q^{56} - 12 q^{58} + 4 q^{59} - 52 q^{61} - 14 q^{62} + 7 q^{64} - 12 q^{65} - 40 q^{67} + 23 q^{68} + 4 q^{70} + 2 q^{71} - 8 q^{73} + 26 q^{74} - 46 q^{76} - 12 q^{77} - 26 q^{79} + q^{80} - 26 q^{82} - 14 q^{83} + 32 q^{86} - 60 q^{88} + 12 q^{89} - 24 q^{91} - 38 q^{92} - 26 q^{94} - 24 q^{95} - 6 q^{97} + 30 q^{98}+O(q^{100}) \) 12 * q - 3 * q^2 + 11 * q^4 + 12 * q^5 - 8 * q^7 - 9 * q^8 - 3 * q^10 - 12 * q^13 + 4 * q^14 + q^16 - 24 * q^19 + 11 * q^20 - 16 * q^22 - 24 * q^23 + 12 * q^25 - q^26 - 44 * q^28 + 8 * q^29 - 12 * q^31 - 31 * q^32 - 18 * q^34 - 8 * q^35 - 10 * q^37 + 2 * q^38 - 9 * q^40 - 10 * q^41 - 42 * q^43 + 42 * q^44 - 24 * q^46 - 22 * q^47 - 4 * q^49 - 3 * q^50 - 30 * q^52 + 8 * q^53 + 27 * q^56 - 12 * q^58 + 4 * q^59 - 52 * q^61 - 14 * q^62 + 7 * q^64 - 12 * q^65 - 40 * q^67 + 23 * q^68 + 4 * q^70 + 2 * q^71 - 8 * q^73 + 26 * q^74 - 46 * q^76 - 12 * q^77 - 26 * q^79 + q^80 - 26 * q^82 - 14 * q^83 + 32 * q^86 - 60 * q^88 + 12 * q^89 - 24 * q^91 - 38 * q^92 - 26 * q^94 - 24 * q^95 - 6 * q^97 + 30 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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