LMFDB - Embedded newform 4005.2.a.t.1.3

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:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 19x^{8} - x^{7} + 128x^{6} + 14x^{5} - 358x^{4} - 59x^{3} + 344x^{2} + 71x - 21 $ x^10 - 19x^8 - x^7 + 128x^6 + 14x^5 - 358x^4 - 59x^3 + 344x^2 + 71*x - 21
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1335)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.20255$ 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)

$f(q)$	$=$	$q-2.20255 q^{2} +2.85124 q^{4} +1.00000 q^{5} +4.87392 q^{7} -1.87489 q^{8} +O(q^{10})$	\(q-2.20255 q^{2} +2.85124 q^{4} +1.00000 q^{5} +4.87392 q^{7} -1.87489 q^{8} -2.20255 q^{10} +5.33481 q^{11} +2.01458 q^{13} -10.7351 q^{14} -1.57292 q^{16} -7.54803 q^{17} +3.88335 q^{19} +2.85124 q^{20} -11.7502 q^{22} +5.37621 q^{23} +1.00000 q^{25} -4.43722 q^{26} +13.8967 q^{28} +0.337682 q^{29} -7.70484 q^{31} +7.21423 q^{32} +16.6249 q^{34} +4.87392 q^{35} +3.90579 q^{37} -8.55328 q^{38} -1.87489 q^{40} -6.47297 q^{41} -1.00684 q^{43} +15.2108 q^{44} -11.8414 q^{46} +7.38596 q^{47} +16.7551 q^{49} -2.20255 q^{50} +5.74404 q^{52} +7.81944 q^{53} +5.33481 q^{55} -9.13808 q^{56} -0.743761 q^{58} +1.89012 q^{59} +9.00854 q^{61} +16.9703 q^{62} -12.7439 q^{64} +2.01458 q^{65} -14.1431 q^{67} -21.5212 q^{68} -10.7351 q^{70} +3.18252 q^{71} +4.54222 q^{73} -8.60271 q^{74} +11.0723 q^{76} +26.0014 q^{77} +0.958885 q^{79} -1.57292 q^{80} +14.2571 q^{82} +11.5468 q^{83} -7.54803 q^{85} +2.21763 q^{86} -10.0022 q^{88} +1.00000 q^{89} +9.81889 q^{91} +15.3288 q^{92} -16.2680 q^{94} +3.88335 q^{95} +0.264046 q^{97} -36.9039 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 18 q^{4} + 10 q^{5} + 9 q^{7} - 3 q^{8}+O(q^{10}) $ 10 * q + 18 * q^4 + 10 * q^5 + 9 * q^7 - 3 * q^8	$ 10 q + 18 q^{4} + 10 q^{5} + 9 q^{7} - 3 q^{8} - 4 q^{11} + 15 q^{13} + q^{14} + 22 q^{16} - 11 q^{17} + 14 q^{19} + 18 q^{20} + 10 q^{22} + 8 q^{23} + 10 q^{25} + 14 q^{26} + 36 q^{28} - 5 q^{29} - 8 q^{31} - q^{32} + 2 q^{34} + 9 q^{35} + 23 q^{37} - 8 q^{38} - 3 q^{40} - 13 q^{41} + 25 q^{43} + 8 q^{44} - 14 q^{46} + q^{47} + 21 q^{49} + 51 q^{52} - 9 q^{53} - 4 q^{55} - 15 q^{56} - 8 q^{58} + 15 q^{59} + 8 q^{61} + 8 q^{62} + 9 q^{64} + 15 q^{65} + 52 q^{67} + 28 q^{68} + q^{70} + 22 q^{71} + 34 q^{73} + 18 q^{74} + 14 q^{76} - 4 q^{77} - 3 q^{79} + 22 q^{80} + 17 q^{82} + 10 q^{83} - 11 q^{85} + 6 q^{86} + 4 q^{88} + 10 q^{89} + 18 q^{91} + 14 q^{92} - 43 q^{94} + 14 q^{95} + 34 q^{97} + 38 q^{98}+O(q^{100}) $ 10 * q + 18 * q^4 + 10 * q^5 + 9 * q^7 - 3 * q^8 - 4 * q^11 + 15 * q^13 + q^14 + 22 * q^16 - 11 * q^17 + 14 * q^19 + 18 * q^20 + 10 * q^22 + 8 * q^23 + 10 * q^25 + 14 * q^26 + 36 * q^28 - 5 * q^29 - 8 * q^31 - q^32 + 2 * q^34 + 9 * q^35 + 23 * q^37 - 8 * q^38 - 3 * q^40 - 13 * q^41 + 25 * q^43 + 8 * q^44 - 14 * q^46 + q^47 + 21 * q^49 + 51 * q^52 - 9 * q^53 - 4 * q^55 - 15 * q^56 - 8 * q^58 + 15 * q^59 + 8 * q^61 + 8 * q^62 + 9 * q^64 + 15 * q^65 + 52 * q^67 + 28 * q^68 + q^70 + 22 * q^71 + 34 * q^73 + 18 * q^74 + 14 * q^76 - 4 * q^77 - 3 * q^79 + 22 * q^80 + 17 * q^82 + 10 * q^83 - 11 * q^85 + 6 * q^86 + 4 * q^88 + 10 * q^89 + 18 * q^91 + 14 * q^92 - 43 * q^94 + 14 * q^95 + 34 * q^97 + 38 * 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 $

$2$	−2.20255	−1.55744	−0.778720	−	0.627372i	$-0.784131\pi$
$2$	−2.20255	−1.55744	−0.778720	+	0.627372i	$0.784131\pi$
$3$	0	0
$3$	0	0
$4$	2.85124	1.42562
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	4.87392	1.84217	0.921084	−	0.389365i	$-0.127305\pi$
$7$	4.87392	1.84217	0.921084	+	0.389365i	$0.127305\pi$
$8$	−1.87489	−0.662875
$9$	0	0
$10$	−2.20255	−0.696508
$11$	5.33481	1.60851	0.804253	−	0.594287i	$-0.202566\pi$
$11$	5.33481	1.60851	0.804253	+	0.594287i	$0.202566\pi$
$12$	0	0
$13$	2.01458	0.558744	0.279372	−	0.960183i	$-0.409874\pi$
$13$	2.01458	0.558744	0.279372	+	0.960183i	$0.409874\pi$
$14$	−10.7351	−2.86906
$15$	0	0
$16$	−1.57292	−0.393230
$17$	−7.54803	−1.83067	−0.915333	−	0.402697i	$-0.868073\pi$
$17$	−7.54803	−1.83067	−0.915333	+	0.402697i	$0.868073\pi$
$18$	0	0
$19$	3.88335	0.890901	0.445451	−	0.895306i	$-0.353043\pi$
$19$	3.88335	0.890901	0.445451	+	0.895306i	$0.353043\pi$
$20$	2.85124	0.637556
$21$	0	0
$22$	−11.7502	−2.50515
$23$	5.37621	1.12102	0.560508	−	0.828149i	$-0.310606\pi$
$23$	5.37621	1.12102	0.560508	+	0.828149i	$0.310606\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−4.43722	−0.870210
$27$	0	0
$28$	13.8967	2.62623
$29$	0.337682	0.0627059	0.0313529	−	0.999508i	$-0.490018\pi$
$29$	0.337682	0.0627059	0.0313529	+	0.999508i	$0.490018\pi$
$30$	0	0
$31$	−7.70484	−1.38383	−0.691915	−	0.721979i	$-0.743233\pi$
$31$	−7.70484	−1.38383	−0.691915	+	0.721979i	$0.743233\pi$
$32$	7.21423	1.27531
$33$	0	0
$34$	16.6249	2.85115
$35$	4.87392	0.823842
$36$	0	0
$37$	3.90579	0.642108	0.321054	−	0.947061i	$-0.395963\pi$
$37$	3.90579	0.642108	0.321054	+	0.947061i	$0.395963\pi$
$38$	−8.55328	−1.38753
$39$	0	0
$40$	−1.87489	−0.296447
$41$	−6.47297	−1.01091	−0.505454	−	0.862853i	$-0.668675\pi$
$41$	−6.47297	−1.01091	−0.505454	+	0.862853i	$0.668675\pi$
$42$	0	0
$43$	−1.00684	−0.153542	−0.0767711	−	0.997049i	$-0.524461\pi$
$43$	−1.00684	−0.153542	−0.0767711	+	0.997049i	$0.524461\pi$
$44$	15.2108	2.29312
$45$	0	0
$46$	−11.8414	−1.74592
$47$	7.38596	1.07735	0.538676	−	0.842513i	$-0.318925\pi$
$47$	7.38596	1.07735	0.538676	+	0.842513i	$0.318925\pi$
$48$	0	0
$49$	16.7551	2.39358
$50$	−2.20255	−0.311488
$51$	0	0
$52$	5.74404	0.796555
$53$	7.81944	1.07408	0.537041	−	0.843556i	$-0.319542\pi$
$53$	7.81944	1.07408	0.537041	+	0.843556i	$0.319542\pi$
$54$	0	0
$55$	5.33481	0.719346
$56$	−9.13808	−1.22113
$57$	0	0
$58$	−0.743761	−0.0976606
$59$	1.89012	0.246073	0.123036	−	0.992402i	$-0.460737\pi$
$59$	1.89012	0.246073	0.123036	+	0.992402i	$0.460737\pi$
$60$	0	0
$61$	9.00854	1.15343	0.576713	−	0.816947i	$-0.304335\pi$
$61$	9.00854	1.15343	0.576713	+	0.816947i	$0.304335\pi$
$62$	16.9703	2.15523
$63$	0	0
$64$	−12.7439	−1.59298
$65$	2.01458	0.249878
$66$	0	0
$67$	−14.1431	−1.72785	−0.863925	−	0.503620i	$-0.832001\pi$
$67$	−14.1431	−1.72785	−0.863925	+	0.503620i	$0.832001\pi$
$68$	−21.5212	−2.60983
$69$	0	0
$70$	−10.7351	−1.28308
$71$	3.18252	0.377695	0.188848	−	0.982006i	$-0.439525\pi$
$71$	3.18252	0.377695	0.188848	+	0.982006i	$0.439525\pi$
$72$	0	0
$73$	4.54222	0.531626	0.265813	−	0.964025i	$-0.414360\pi$
$73$	4.54222	0.531626	0.265813	+	0.964025i	$0.414360\pi$
$74$	−8.60271	−1.00005
$75$	0	0
$76$	11.0723	1.27009
$77$	26.0014	2.96314
$78$	0	0
$79$	0.958885	0.107883	0.0539415	−	0.998544i	$-0.482822\pi$
$79$	0.958885	0.107883	0.0539415	+	0.998544i	$0.482822\pi$
$80$	−1.57292	−0.175858
$81$	0	0
$82$	14.2571	1.57443
$83$	11.5468	1.26742	0.633712	−	0.773569i	$-0.281530\pi$
$83$	11.5468	1.26742	0.633712	+	0.773569i	$0.281530\pi$
$84$	0	0
$85$	−7.54803	−0.818699
$86$	2.21763	0.239133
$87$	0	0
$88$	−10.0022	−1.06624
$89$	1.00000	0.106000
$89$	1.00000	0.106000
$90$	0	0
$91$	9.81889	1.02930
$92$	15.3288	1.59814
$93$	0	0
$94$	−16.2680	−1.67791
$95$	3.88335	0.398423
$96$	0	0
$97$	0.264046	0.0268098	0.0134049	−	0.999910i	$-0.495733\pi$
$97$	0.264046	0.0268098	0.0134049	+	0.999910i	$0.495733\pi$
$98$	−36.9039	−3.72786
$99$	0	0
$100$	2.85124	0.285124
$101$	4.93728	0.491278	0.245639	−	0.969361i	$-0.421002\pi$
$101$	4.93728	0.491278	0.245639	+	0.969361i	$0.421002\pi$
$102$	0	0
$103$	−17.7283	−1.74682	−0.873409	−	0.486987i	$-0.838096\pi$
$103$	−17.7283	−1.74682	−0.873409	+	0.486987i	$0.838096\pi$
$104$	−3.77712	−0.370377
$105$	0	0
$106$	−17.2227	−1.67282
$107$	−7.78971	−0.753060	−0.376530	−	0.926404i	$-0.622883\pi$
$107$	−7.78971	−0.753060	−0.376530	+	0.926404i	$0.622883\pi$
$108$	0	0
$109$	−17.0663	−1.63465	−0.817327	−	0.576174i	$-0.804545\pi$
$109$	−17.0663	−1.63465	−0.817327	+	0.576174i	$0.804545\pi$
$110$	−11.7502	−1.12034
$111$	0	0
$112$	−7.66628	−0.724396
$113$	−9.16704	−0.862363	−0.431182	−	0.902265i	$-0.641903\pi$
$113$	−9.16704	−0.862363	−0.431182	+	0.902265i	$0.641903\pi$
$114$	0	0
$115$	5.37621	0.501334
$116$	0.962810	0.0893947
$117$	0	0
$118$	−4.16309	−0.383243
$119$	−36.7885	−3.37239
$120$	0	0
$121$	17.4602	1.58729
$122$	−19.8418	−1.79639
$123$	0	0
$124$	−21.9683	−1.97281
$125$	1.00000	0.0894427
$126$	0	0
$127$	19.7101	1.74899	0.874493	−	0.485037i	$-0.161194\pi$
$127$	19.7101	1.74899	0.874493	+	0.485037i	$0.161194\pi$
$128$	13.6406	1.20567
$129$	0	0
$130$	−4.43722	−0.389170
$131$	−17.3273	−1.51390	−0.756948	−	0.653475i	$-0.773310\pi$
$131$	−17.3273	−1.51390	−0.756948	+	0.653475i	$0.773310\pi$
$132$	0	0
$133$	18.9271	1.64119
$134$	31.1508	2.69102
$135$	0	0
$136$	14.1518	1.21350
$137$	13.0881	1.11820	0.559098	−	0.829102i	$-0.311148\pi$
$137$	13.0881	1.11820	0.559098	+	0.829102i	$0.311148\pi$
$138$	0	0
$139$	−0.760203	−0.0644796	−0.0322398	−	0.999480i	$-0.510264\pi$
$139$	−0.760203	−0.0644796	−0.0322398	+	0.999480i	$0.510264\pi$
$140$	13.8967	1.17448
$141$	0	0
$142$	−7.00966	−0.588237
$143$	10.7474	0.898742
$144$	0	0
$145$	0.337682	0.0280429
$146$	−10.0045	−0.827976
$147$	0	0
$148$	11.1363	0.915402
$149$	7.39004	0.605416	0.302708	−	0.953083i	$-0.402109\pi$
$149$	7.39004	0.605416	0.302708	+	0.953083i	$0.402109\pi$
$150$	0	0
$151$	−6.39367	−0.520309	−0.260155	−	0.965567i	$-0.583774\pi$
$151$	−6.39367	−0.520309	−0.260155	+	0.965567i	$0.583774\pi$
$152$	−7.28087	−0.590556
$153$	0	0
$154$	−57.2695	−4.61491
$155$	−7.70484	−0.618868
$156$	0	0
$157$	18.3410	1.46377	0.731886	−	0.681427i	$-0.238640\pi$
$157$	18.3410	1.46377	0.731886	+	0.681427i	$0.238640\pi$
$158$	−2.11200	−0.168021
$159$	0	0
$160$	7.21423	0.570335
$161$	26.2032	2.06510
$162$	0	0
$163$	−17.6947	−1.38595	−0.692977	−	0.720960i	$-0.743701\pi$
$163$	−17.6947	−1.38595	−0.692977	+	0.720960i	$0.743701\pi$
$164$	−18.4560	−1.44117
$165$	0	0
$166$	−25.4324	−1.97394
$167$	20.9480	1.62100	0.810501	−	0.585737i	$-0.199195\pi$
$167$	20.9480	1.62100	0.810501	+	0.585737i	$0.199195\pi$
$168$	0	0
$169$	−8.94147	−0.687805
$170$	16.6249	1.27507
$171$	0	0
$172$	−2.87075	−0.218893
$173$	−20.8654	−1.58637	−0.793184	−	0.608982i	$-0.791578\pi$
$173$	−20.8654	−1.58637	−0.793184	+	0.608982i	$0.791578\pi$
$174$	0	0
$175$	4.87392	0.368433
$176$	−8.39123	−0.632513
$177$	0	0
$178$	−2.20255	−0.165088
$179$	−6.84436	−0.511571	−0.255786	−	0.966734i	$-0.582334\pi$
$179$	−6.84436	−0.511571	−0.255786	+	0.966734i	$0.582334\pi$
$180$	0	0
$181$	−11.9988	−0.891861	−0.445931	−	0.895068i	$-0.647127\pi$
$181$	−11.9988	−0.891861	−0.445931	+	0.895068i	$0.647127\pi$
$182$	−21.6266	−1.60307
$183$	0	0
$184$	−10.0798	−0.743094
$185$	3.90579	0.287160
$186$	0	0
$187$	−40.2673	−2.94464
$188$	21.0591	1.53589
$189$	0	0
$190$	−8.55328	−0.620520
$191$	−3.78850	−0.274126	−0.137063	−	0.990562i	$-0.543766\pi$
$191$	−3.78850	−0.274126	−0.137063	+	0.990562i	$0.543766\pi$
$192$	0	0
$193$	12.8138	0.922361	0.461180	−	0.887306i	$-0.347426\pi$
$193$	12.8138	0.922361	0.461180	+	0.887306i	$0.347426\pi$
$194$	−0.581576	−0.0417547
$195$	0	0
$196$	47.7727	3.41233
$197$	8.23040	0.586392	0.293196	−	0.956052i	$-0.405281\pi$
$197$	8.23040	0.586392	0.293196	+	0.956052i	$0.405281\pi$
$198$	0	0
$199$	11.0179	0.781040	0.390520	−	0.920594i	$-0.372295\pi$
$199$	11.0179	0.781040	0.390520	+	0.920594i	$0.372295\pi$
$200$	−1.87489	−0.132575
$201$	0	0
$202$	−10.8746	−0.765135
$203$	1.64583	0.115515
$204$	0	0
$205$	−6.47297	−0.452092
$206$	39.0474	2.72056
$207$	0	0
$208$	−3.16877	−0.219715
$209$	20.7169	1.43302
$210$	0	0
$211$	−24.3196	−1.67423	−0.837114	−	0.547029i	$-0.815759\pi$
$211$	−24.3196	−1.67423	−0.837114	+	0.547029i	$0.815759\pi$
$212$	22.2951	1.53123
$213$	0	0
$214$	17.1572	1.17285
$215$	−1.00684	−0.0686662
$216$	0	0
$217$	−37.5527	−2.54925
$218$	37.5894	2.54588
$219$	0	0
$220$	15.2108	1.02551
$221$	−15.2061	−1.02287
$222$	0	0
$223$	21.6282	1.44833	0.724164	−	0.689628i	$-0.242226\pi$
$223$	21.6282	1.44833	0.724164	+	0.689628i	$0.242226\pi$
$224$	35.1616	2.34933
$225$	0	0
$226$	20.1909	1.34308
$227$	1.07413	0.0712924	0.0356462	−	0.999364i	$-0.488651\pi$
$227$	1.07413	0.0712924	0.0356462	+	0.999364i	$0.488651\pi$
$228$	0	0
$229$	8.48589	0.560763	0.280382	−	0.959889i	$-0.409539\pi$
$229$	8.48589	0.560763	0.280382	+	0.959889i	$0.409539\pi$
$230$	−11.8414	−0.780797
$231$	0	0
$232$	−0.633117	−0.0415662
$233$	0.303699	0.0198960	0.00994800	−	0.999951i	$-0.496833\pi$
$233$	0.303699	0.0198960	0.00994800	+	0.999951i	$0.496833\pi$
$234$	0	0
$235$	7.38596	0.481807
$236$	5.38918	0.350806
$237$	0	0
$238$	81.0285	5.25230
$239$	−8.46825	−0.547766	−0.273883	−	0.961763i	$-0.588308\pi$
$239$	−8.46825	−0.547766	−0.273883	+	0.961763i	$0.588308\pi$
$240$	0	0
$241$	−21.1992	−1.36556	−0.682780	−	0.730624i	$-0.739229\pi$
$241$	−21.1992	−1.36556	−0.682780	+	0.730624i	$0.739229\pi$
$242$	−38.4570	−2.47211
$243$	0	0
$244$	25.6855	1.64435
$245$	16.7551	1.07044
$246$	0	0
$247$	7.82331	0.497786
$248$	14.4458	0.917307
$249$	0	0
$250$	−2.20255	−0.139302
$251$	−28.4202	−1.79387	−0.896934	−	0.442165i	$-0.854211\pi$
$251$	−28.4202	−1.79387	−0.896934	+	0.442165i	$0.854211\pi$
$252$	0	0
$253$	28.6810	1.80316
$254$	−43.4125	−2.72394
$255$	0	0
$256$	−4.55638	−0.284774
$257$	1.99398	0.124381	0.0621907	−	0.998064i	$-0.480191\pi$
$257$	1.99398	0.124381	0.0621907	+	0.998064i	$0.480191\pi$
$258$	0	0
$259$	19.0365	1.18287
$260$	5.74404	0.356230
$261$	0	0
$262$	38.1644	2.35780
$263$	−6.55632	−0.404280	−0.202140	−	0.979357i	$-0.564790\pi$
$263$	−6.55632	−0.404280	−0.202140	+	0.979357i	$0.564790\pi$
$264$	0	0
$265$	7.81944	0.480344
$266$	−41.6880	−2.55605
$267$	0	0
$268$	−40.3252	−2.46326
$269$	−21.5473	−1.31376	−0.656882	−	0.753993i	$-0.728125\pi$
$269$	−21.5473	−1.31376	−0.656882	+	0.753993i	$0.728125\pi$
$270$	0	0
$271$	−2.69418	−0.163660	−0.0818300	−	0.996646i	$-0.526076\pi$
$271$	−2.69418	−0.163660	−0.0818300	+	0.996646i	$0.526076\pi$
$272$	11.8725	0.719873
$273$	0	0
$274$	−28.8273	−1.74152
$275$	5.33481	0.321701
$276$	0	0
$277$	−18.0899	−1.08692	−0.543459	−	0.839436i	$-0.682886\pi$
$277$	−18.0899	−1.08692	−0.543459	+	0.839436i	$0.682886\pi$
$278$	1.67439	0.100423
$279$	0	0
$280$	−9.13808	−0.546105
$281$	−14.8056	−0.883227	−0.441614	−	0.897205i	$-0.645594\pi$
$281$	−14.8056	−0.883227	−0.441614	+	0.897205i	$0.645594\pi$
$282$	0	0
$283$	−3.55868	−0.211541	−0.105771	−	0.994391i	$-0.533731\pi$
$283$	−3.55868	−0.211541	−0.105771	+	0.994391i	$0.533731\pi$
$284$	9.07411	0.538449
$285$	0	0
$286$	−23.6717	−1.39974
$287$	−31.5487	−1.86226
$288$	0	0
$289$	39.9728	2.35134
$290$	−0.743761	−0.0436752
$291$	0	0
$292$	12.9509	0.757896
$293$	23.9535	1.39938	0.699688	−	0.714448i	$-0.253322\pi$
$293$	23.9535	1.39938	0.699688	+	0.714448i	$0.253322\pi$
$294$	0	0
$295$	1.89012	0.110047
$296$	−7.32295	−0.425638
$297$	0	0
$298$	−16.2770	−0.942899
$299$	10.8308	0.626361
$300$	0	0
$301$	−4.90727	−0.282850
$302$	14.0824	0.810350
$303$	0	0
$304$	−6.10820	−0.350329
$305$	9.00854	0.515828
$306$	0	0
$307$	7.15829	0.408545	0.204273	−	0.978914i	$-0.434517\pi$
$307$	7.15829	0.408545	0.204273	+	0.978914i	$0.434517\pi$
$308$	74.1362	4.22430
$309$	0	0
$310$	16.9703	0.963849
$311$	−3.06998	−0.174083	−0.0870413	−	0.996205i	$-0.527741\pi$
$311$	−3.06998	−0.174083	−0.0870413	+	0.996205i	$0.527741\pi$
$312$	0	0
$313$	−10.6760	−0.603443	−0.301721	−	0.953396i	$-0.597561\pi$
$313$	−10.6760	−0.603443	−0.301721	+	0.953396i	$0.597561\pi$
$314$	−40.3970	−2.27974
$315$	0	0
$316$	2.73401	0.153800
$317$	1.18689	0.0666626	0.0333313	−	0.999444i	$-0.489388\pi$
$317$	1.18689	0.0666626	0.0333313	+	0.999444i	$0.489388\pi$
$318$	0	0
$319$	1.80147	0.100863
$320$	−12.7439	−0.712404
$321$	0	0
$322$	−57.7139	−3.21627
$323$	−29.3116	−1.63094
$324$	0	0
$325$	2.01458	0.111749
$326$	38.9735	2.15854
$327$	0	0
$328$	12.1361	0.670106
$329$	35.9985	1.98466
$330$	0	0
$331$	6.25664	0.343896	0.171948	−	0.985106i	$-0.444994\pi$
$331$	6.25664	0.343896	0.171948	+	0.985106i	$0.444994\pi$
$332$	32.9226	1.80686
$333$	0	0
$334$	−46.1390	−2.52461
$335$	−14.1431	−0.772718
$336$	0	0
$337$	24.2538	1.32119	0.660596	−	0.750742i	$-0.270304\pi$
$337$	24.2538	1.32119	0.660596	+	0.750742i	$0.270304\pi$
$338$	19.6941	1.07122
$339$	0	0
$340$	−21.5212	−1.16715
$341$	−41.1038	−2.22590
$342$	0	0
$343$	47.5454	2.56721
$344$	1.88773	0.101779
$345$	0	0
$346$	45.9572	2.47067
$347$	0.561391	0.0301371	0.0150685	−	0.999886i	$-0.495203\pi$
$347$	0.561391	0.0301371	0.0150685	+	0.999886i	$0.495203\pi$
$348$	0	0
$349$	−14.7322	−0.788597	−0.394299	−	0.918982i	$-0.629012\pi$
$349$	−14.7322	−0.788597	−0.394299	+	0.918982i	$0.629012\pi$
$350$	−10.7351	−0.573813
$351$	0	0
$352$	38.4865	2.05134
$353$	1.32139	0.0703304	0.0351652	−	0.999382i	$-0.488804\pi$
$353$	1.32139	0.0703304	0.0351652	+	0.999382i	$0.488804\pi$
$354$	0	0
$355$	3.18252	0.168910
$356$	2.85124	0.151115
$357$	0	0
$358$	15.0751	0.796741
$359$	−11.4983	−0.606858	−0.303429	−	0.952854i	$-0.598132\pi$
$359$	−11.4983	−0.606858	−0.303429	+	0.952854i	$0.598132\pi$
$360$	0	0
$361$	−3.91960	−0.206295
$362$	26.4279	1.38902
$363$	0	0
$364$	27.9960	1.46739
$365$	4.54222	0.237750
$366$	0	0
$367$	−6.14767	−0.320906	−0.160453	−	0.987043i	$-0.551295\pi$
$367$	−6.14767	−0.320906	−0.160453	+	0.987043i	$0.551295\pi$
$368$	−8.45635	−0.440818
$369$	0	0
$370$	−8.60271	−0.447234
$371$	38.1113	1.97864
$372$	0	0
$373$	1.53643	0.0795532	0.0397766	−	0.999209i	$-0.487335\pi$
$373$	1.53643	0.0795532	0.0397766	+	0.999209i	$0.487335\pi$
$374$	88.6909	4.58610
$375$	0	0
$376$	−13.8479	−0.714150
$377$	0.680286	0.0350365
$378$	0	0
$379$	−5.22168	−0.268219	−0.134110	−	0.990966i	$-0.542817\pi$
$379$	−5.22168	−0.268219	−0.134110	+	0.990966i	$0.542817\pi$
$380$	11.0723	0.567999
$381$	0	0
$382$	8.34436	0.426935
$383$	−0.219937	−0.0112383	−0.00561913	−	0.999984i	$-0.501789\pi$
$383$	−0.219937	−0.0112383	−0.00561913	+	0.999984i	$0.501789\pi$
$384$	0	0
$385$	26.0014	1.32515
$386$	−28.2232	−1.43652
$387$	0	0
$388$	0.752859	0.0382206
$389$	20.2844	1.02846	0.514229	−	0.857653i	$-0.328078\pi$
$389$	20.2844	1.02846	0.514229	+	0.857653i	$0.328078\pi$
$390$	0	0
$391$	−40.5798	−2.05221
$392$	−31.4140	−1.58665
$393$	0	0
$394$	−18.1279	−0.913270
$395$	0.958885	0.0482468
$396$	0	0
$397$	−4.43462	−0.222567	−0.111284	−	0.993789i	$-0.535496\pi$
$397$	−4.43462	−0.222567	−0.111284	+	0.993789i	$0.535496\pi$
$398$	−24.2676	−1.21642
$399$	0	0
$400$	−1.57292	−0.0786460
$401$	5.85929	0.292599	0.146299	−	0.989240i	$-0.453264\pi$
$401$	5.85929	0.292599	0.146299	+	0.989240i	$0.453264\pi$
$402$	0	0
$403$	−15.5220	−0.773206
$404$	14.0774	0.700374
$405$	0	0
$406$	−3.62503	−0.179907
$407$	20.8367	1.03284
$408$	0	0
$409$	26.3672	1.30378	0.651888	−	0.758316i	$-0.273977\pi$
$409$	26.3672	1.30378	0.651888	+	0.758316i	$0.273977\pi$
$410$	14.2571	0.704106
$411$	0	0
$412$	−50.5475	−2.49030
$413$	9.21229	0.453307
$414$	0	0
$415$	11.5468	0.566810
$416$	14.5336	0.712570
$417$	0	0
$418$	−45.6301	−2.23184
$419$	13.5003	0.659533	0.329767	−	0.944062i	$-0.393030\pi$
$419$	13.5003	0.659533	0.329767	+	0.944062i	$0.393030\pi$
$420$	0	0
$421$	30.7122	1.49682	0.748410	−	0.663237i	$-0.230818\pi$
$421$	30.7122	1.49682	0.748410	+	0.663237i	$0.230818\pi$
$422$	53.5651	2.60751
$423$	0	0
$424$	−14.6606	−0.711983
$425$	−7.54803	−0.366133
$426$	0	0
$427$	43.9069	2.12480
$428$	−22.2103	−1.07358
$429$	0	0
$430$	2.21763	0.106943
$431$	−29.1856	−1.40582	−0.702911	−	0.711278i	$-0.748117\pi$
$431$	−29.1856	−1.40582	−0.702911	+	0.711278i	$0.748117\pi$
$432$	0	0
$433$	9.29686	0.446779	0.223389	−	0.974729i	$-0.428288\pi$
$433$	9.29686	0.446779	0.223389	+	0.974729i	$0.428288\pi$
$434$	82.7119	3.97030
$435$	0	0
$436$	−48.6601	−2.33039
$437$	20.8777	0.998715
$438$	0	0
$439$	34.4501	1.64421	0.822106	−	0.569334i	$-0.192799\pi$
$439$	34.4501	1.64421	0.822106	+	0.569334i	$0.192799\pi$
$440$	−10.0022	−0.476836
$441$	0	0
$442$	33.4922	1.59306
$443$	−15.9583	−0.758200	−0.379100	−	0.925356i	$-0.623766\pi$
$443$	−15.9583	−0.758200	−0.379100	+	0.925356i	$0.623766\pi$
$444$	0	0
$445$	1.00000	0.0474045
$446$	−47.6371	−2.25568
$447$	0	0
$448$	−62.1126	−2.93454
$449$	−11.2338	−0.530156	−0.265078	−	0.964227i	$-0.585398\pi$
$449$	−11.2338	−0.530156	−0.265078	+	0.964227i	$0.585398\pi$
$450$	0	0
$451$	−34.5321	−1.62605
$452$	−26.1374	−1.22940
$453$	0	0
$454$	−2.36582	−0.111034
$455$	9.81889	0.460317
$456$	0	0
$457$	27.0994	1.26766	0.633828	−	0.773474i	$-0.281483\pi$
$457$	27.0994	1.26766	0.633828	+	0.773474i	$0.281483\pi$
$458$	−18.6906	−0.873355
$459$	0	0
$460$	15.3288	0.714711
$461$	7.89559	0.367734	0.183867	−	0.982951i	$-0.441138\pi$
$461$	7.89559	0.367734	0.183867	+	0.982951i	$0.441138\pi$
$462$	0	0
$463$	18.2915	0.850076	0.425038	−	0.905175i	$-0.360261\pi$
$463$	18.2915	0.850076	0.425038	+	0.905175i	$0.360261\pi$
$464$	−0.531146	−0.0246578
$465$	0	0
$466$	−0.668914	−0.0309868
$467$	−0.146968	−0.00680086	−0.00340043	−	0.999994i	$-0.501082\pi$
$467$	−0.146968	−0.00680086	−0.00340043	+	0.999994i	$0.501082\pi$
$468$	0	0
$469$	−68.9321	−3.18299
$470$	−16.2680	−0.750385
$471$	0	0
$472$	−3.54378	−0.163115
$473$	−5.37132	−0.246973
$474$	0	0
$475$	3.88335	0.178180
$476$	−104.893	−4.80775
$477$	0	0
$478$	18.6518	0.853112
$479$	21.6915	0.991110	0.495555	−	0.868577i	$-0.334965\pi$
$479$	21.6915	0.991110	0.495555	+	0.868577i	$0.334965\pi$
$480$	0	0
$481$	7.86853	0.358774
$482$	46.6923	2.12678
$483$	0	0
$484$	49.7831	2.26287
$485$	0.264046	0.0119897
$486$	0	0
$487$	32.5003	1.47273	0.736365	−	0.676585i	$-0.236541\pi$
$487$	32.5003	1.47273	0.736365	+	0.676585i	$0.236541\pi$
$488$	−16.8901	−0.764577
$489$	0	0
$490$	−36.9039	−1.66715
$491$	−26.6246	−1.20155	−0.600775	−	0.799418i	$-0.705141\pi$
$491$	−26.6246	−1.20155	−0.600775	+	0.799418i	$0.705141\pi$
$492$	0	0
$493$	−2.54883	−0.114794
$494$	−17.2313	−0.775271
$495$	0	0
$496$	12.1191	0.544164
$497$	15.5113	0.695777
$498$	0	0
$499$	17.8124	0.797393	0.398697	−	0.917083i	$-0.369463\pi$
$499$	17.8124	0.797393	0.398697	+	0.917083i	$0.369463\pi$
$500$	2.85124	0.127511
$501$	0	0
$502$	62.5970	2.79384
$503$	8.35268	0.372428	0.186214	−	0.982509i	$-0.440378\pi$
$503$	8.35268	0.372428	0.186214	+	0.982509i	$0.440378\pi$
$504$	0	0
$505$	4.93728	0.219706
$506$	−63.1715	−2.80832
$507$	0	0
$508$	56.1981	2.49339
$509$	36.0463	1.59772	0.798861	−	0.601515i	$-0.205436\pi$
$509$	36.0463	1.59772	0.798861	+	0.601515i	$0.205436\pi$
$510$	0	0
$511$	22.1384	0.979344
$512$	−17.2455	−0.762152
$513$	0	0
$514$	−4.39185	−0.193716
$515$	−17.7283	−0.781201
$516$	0	0
$517$	39.4027	1.73293
$518$	−41.9289	−1.84225
$519$	0	0
$520$	−3.77712	−0.165638
$521$	−15.7937	−0.691933	−0.345967	−	0.938247i	$-0.612449\pi$
$521$	−15.7937	−0.691933	−0.345967	+	0.938247i	$0.612449\pi$
$522$	0	0
$523$	18.9535	0.828777	0.414388	−	0.910100i	$-0.363995\pi$
$523$	18.9535	0.828777	0.414388	+	0.910100i	$0.363995\pi$
$524$	−49.4043	−2.15824
$525$	0	0
$526$	14.4406	0.629641
$527$	58.1564	2.53333
$528$	0	0
$529$	5.90360	0.256678
$530$	−17.2227	−0.748108
$531$	0	0
$532$	53.9657	2.33971
$533$	−13.0403	−0.564839
$534$	0	0
$535$	−7.78971	−0.336779
$536$	26.5168	1.14535
$537$	0	0
$538$	47.4591	2.04611
$539$	89.3851	3.85009
$540$	0	0
$541$	37.9051	1.62967	0.814833	−	0.579696i	$-0.196829\pi$
$541$	37.9051	1.62967	0.814833	+	0.579696i	$0.196829\pi$
$542$	5.93408	0.254890
$543$	0	0
$544$	−54.4532	−2.33466
$545$	−17.0663	−0.731040
$546$	0	0
$547$	−37.8684	−1.61914	−0.809568	−	0.587026i	$-0.800299\pi$
$547$	−37.8684	−1.61914	−0.809568	+	0.587026i	$0.800299\pi$
$548$	37.3174	1.59412
$549$	0	0
$550$	−11.7502	−0.501030
$551$	1.31134	0.0558648
$552$	0	0
$553$	4.67353	0.198739
$554$	39.8440	1.69281
$555$	0	0
$556$	−2.16752	−0.0919233
$557$	−41.4441	−1.75604	−0.878020	−	0.478623i	$-0.841136\pi$
$557$	−41.4441	−1.75604	−0.878020	+	0.478623i	$0.841136\pi$
$558$	0	0
$559$	−2.02837	−0.0857907
$560$	−7.66628	−0.323960
$561$	0	0
$562$	32.6101	1.37557
$563$	8.53452	0.359687	0.179844	−	0.983695i	$-0.442441\pi$
$563$	8.53452	0.359687	0.179844	+	0.983695i	$0.442441\pi$
$564$	0	0
$565$	−9.16704	−0.385660
$566$	7.83817	0.329463
$567$	0	0
$568$	−5.96688	−0.250365
$569$	−33.1696	−1.39054	−0.695271	−	0.718748i	$-0.744715\pi$
$569$	−33.1696	−1.39054	−0.695271	+	0.718748i	$0.744715\pi$
$570$	0	0
$571$	−43.5988	−1.82455	−0.912277	−	0.409574i	$-0.865677\pi$
$571$	−43.5988	−1.82455	−0.912277	+	0.409574i	$0.865677\pi$
$572$	30.6434	1.28126
$573$	0	0
$574$	69.4877	2.90036
$575$	5.37621	0.224203
$576$	0	0
$577$	5.86750	0.244267	0.122134	−	0.992514i	$-0.461026\pi$
$577$	5.86750	0.244267	0.122134	+	0.992514i	$0.461026\pi$
$578$	−88.0421	−3.66207
$579$	0	0
$580$	0.962810	0.0399785
$581$	56.2781	2.33481
$582$	0	0
$583$	41.7152	1.72767
$584$	−8.51618	−0.352402
$585$	0	0
$586$	−52.7588	−2.17944
$587$	25.1589	1.03842	0.519209	−	0.854647i	$-0.326226\pi$
$587$	25.1589	1.03842	0.519209	+	0.854647i	$0.326226\pi$
$588$	0	0
$589$	−29.9206	−1.23286
$590$	−4.16309	−0.171392
$591$	0	0
$592$	−6.14350	−0.252496
$593$	−38.9975	−1.60143	−0.800717	−	0.599043i	$-0.795548\pi$
$593$	−38.9975	−1.60143	−0.800717	+	0.599043i	$0.795548\pi$
$594$	0	0
$595$	−36.7885	−1.50818
$596$	21.0708	0.863092
$597$	0	0
$598$	−23.8554	−0.975520
$599$	−26.5564	−1.08506	−0.542532	−	0.840035i	$-0.682534\pi$
$599$	−26.5564	−1.08506	−0.542532	+	0.840035i	$0.682534\pi$
$600$	0	0
$601$	8.52501	0.347742	0.173871	−	0.984768i	$-0.444372\pi$
$601$	8.52501	0.347742	0.173871	+	0.984768i	$0.444372\pi$
$602$	10.8085	0.440522
$603$	0	0
$604$	−18.2299	−0.741763
$605$	17.4602	0.709858
$606$	0	0
$607$	8.67859	0.352253	0.176127	−	0.984368i	$-0.443643\pi$
$607$	8.67859	0.352253	0.176127	+	0.984368i	$0.443643\pi$
$608$	28.0154	1.13617
$609$	0	0
$610$	−19.8418	−0.803370
$611$	14.8796	0.601964
$612$	0	0
$613$	−2.92446	−0.118118	−0.0590590	−	0.998254i	$-0.518810\pi$
$613$	−2.92446	−0.118118	−0.0590590	+	0.998254i	$0.518810\pi$
$614$	−15.7665	−0.636285
$615$	0	0
$616$	−48.7499	−1.96419
$617$	−22.6961	−0.913711	−0.456855	−	0.889541i	$-0.651024\pi$
$617$	−22.6961	−0.913711	−0.456855	+	0.889541i	$0.651024\pi$
$618$	0	0
$619$	32.4818	1.30555	0.652777	−	0.757550i	$-0.273604\pi$
$619$	32.4818	1.30555	0.652777	+	0.757550i	$0.273604\pi$
$620$	−21.9683	−0.882269
$621$	0	0
$622$	6.76179	0.271123
$623$	4.87392	0.195269
$624$	0	0
$625$	1.00000	0.0400000
$626$	23.5144	0.939826
$627$	0	0
$628$	52.2946	2.08678
$629$	−29.4811	−1.17549
$630$	0	0
$631$	−12.5334	−0.498948	−0.249474	−	0.968381i	$-0.580258\pi$
$631$	−12.5334	−0.498948	−0.249474	+	0.968381i	$0.580258\pi$
$632$	−1.79781	−0.0715130
$633$	0	0
$634$	−2.61420	−0.103823
$635$	19.7101	0.782171
$636$	0	0
$637$	33.7544	1.33740
$638$	−3.96782	−0.157088
$639$	0	0
$640$	13.6406	0.539192
$641$	−32.0217	−1.26478	−0.632390	−	0.774650i	$-0.717926\pi$
$641$	−32.0217	−1.26478	−0.632390	+	0.774650i	$0.717926\pi$
$642$	0	0
$643$	1.83559	0.0723888	0.0361944	−	0.999345i	$-0.488476\pi$
$643$	1.83559	0.0723888	0.0361944	+	0.999345i	$0.488476\pi$
$644$	74.7115	2.94405
$645$	0	0
$646$	64.5604	2.54010
$647$	−0.111665	−0.00438999	−0.00219499	−	0.999998i	$-0.500699\pi$
$647$	−0.111665	−0.00438999	−0.00219499	+	0.999998i	$0.500699\pi$
$648$	0	0
$649$	10.0834	0.395809
$650$	−4.43722	−0.174042
$651$	0	0
$652$	−50.4517	−1.97584
$653$	−2.40942	−0.0942880	−0.0471440	−	0.998888i	$-0.515012\pi$
$653$	−2.40942	−0.0942880	−0.0471440	+	0.998888i	$0.515012\pi$
$654$	0	0
$655$	−17.3273	−0.677035
$656$	10.1815	0.397520
$657$	0	0
$658$	−79.2887	−3.09099
$659$	1.60633	0.0625739	0.0312870	−	0.999510i	$-0.490039\pi$
$659$	1.60633	0.0625739	0.0312870	+	0.999510i	$0.490039\pi$
$660$	0	0
$661$	−23.4794	−0.913242	−0.456621	−	0.889661i	$-0.650940\pi$
$661$	−23.4794	−0.913242	−0.456621	+	0.889661i	$0.650940\pi$
$662$	−13.7806	−0.535598
$663$	0	0
$664$	−21.6490	−0.840145
$665$	18.9271	0.733962
$666$	0	0
$667$	1.81545	0.0702943
$668$	59.7276	2.31093
$669$	0	0
$670$	31.1508	1.20346
$671$	48.0589	1.85529
$672$	0	0
$673$	−45.5574	−1.75611	−0.878054	−	0.478561i	$-0.841158\pi$
$673$	−45.5574	−1.75611	−0.878054	+	0.478561i	$0.841158\pi$
$674$	−53.4204	−2.05768
$675$	0	0
$676$	−25.4943	−0.980548
$677$	23.3799	0.898561	0.449281	−	0.893391i	$-0.351680\pi$
$677$	23.3799	0.898561	0.449281	+	0.893391i	$0.351680\pi$
$678$	0	0
$679$	1.28694	0.0493882
$680$	14.1518	0.542695
$681$	0	0
$682$	90.5334	3.46670
$683$	18.6861	0.715005	0.357503	−	0.933912i	$-0.383628\pi$
$683$	18.6861	0.715005	0.357503	+	0.933912i	$0.383628\pi$
$684$	0	0
$685$	13.0881	0.500072
$686$	−104.721	−3.99827
$687$	0	0
$688$	1.58368	0.0603774
$689$	15.7529	0.600137
$690$	0	0
$691$	−20.2360	−0.769812	−0.384906	−	0.922956i	$-0.625766\pi$
$691$	−20.2360	−0.769812	−0.384906	+	0.922956i	$0.625766\pi$
$692$	−59.4923	−2.26156
$693$	0	0
$694$	−1.23649	−0.0469367
$695$	−0.760203	−0.0288362
$696$	0	0
$697$	48.8582	1.85064
$698$	32.4485	1.22819
$699$	0	0
$700$	13.8967	0.525246
$701$	−41.8548	−1.58083	−0.790417	−	0.612569i	$-0.790136\pi$
$701$	−41.8548	−1.58083	−0.790417	+	0.612569i	$0.790136\pi$
$702$	0	0
$703$	15.1676	0.572055
$704$	−67.9862	−2.56232
$705$	0	0
$706$	−2.91043	−0.109535
$707$	24.0639	0.905015
$708$	0	0
$709$	−41.0332	−1.54103	−0.770517	−	0.637419i	$-0.780002\pi$
$709$	−41.0332	−1.54103	−0.770517	+	0.637419i	$0.780002\pi$
$710$	−7.00966	−0.263068
$711$	0	0
$712$	−1.87489	−0.0702646
$713$	−41.4228	−1.55130
$714$	0	0
$715$	10.7474	0.401930
$716$	−19.5149	−0.729305
$717$	0	0
$718$	25.3257	0.945145
$719$	30.1087	1.12287	0.561433	−	0.827522i	$-0.310250\pi$
$719$	30.1087	1.12287	0.561433	+	0.827522i	$0.310250\pi$
$720$	0	0
$721$	−86.4061	−3.21793
$722$	8.63313	0.321292
$723$	0	0
$724$	−34.2113	−1.27145
$725$	0.337682	0.0125412
$726$	0	0
$727$	−4.41714	−0.163823	−0.0819114	−	0.996640i	$-0.526102\pi$
$727$	−4.41714	−0.163823	−0.0819114	+	0.996640i	$0.526102\pi$
$728$	−18.4094	−0.682297
$729$	0	0
$730$	−10.0045	−0.370282
$731$	7.59969	0.281085
$732$	0	0
$733$	27.4633	1.01438	0.507189	−	0.861835i	$-0.330684\pi$
$733$	27.4633	1.01438	0.507189	+	0.861835i	$0.330684\pi$
$734$	13.5406	0.499791
$735$	0	0
$736$	38.7852	1.42964
$737$	−75.4505	−2.77926
$738$	0	0
$739$	3.77208	0.138758	0.0693791	−	0.997590i	$-0.477898\pi$
$739$	3.77208	0.138758	0.0693791	+	0.997590i	$0.477898\pi$
$740$	11.1363	0.409380
$741$	0	0
$742$	−83.9421	−3.08161
$743$	−11.1633	−0.409540	−0.204770	−	0.978810i	$-0.565645\pi$
$743$	−11.1633	−0.409540	−0.204770	+	0.978810i	$0.565645\pi$
$744$	0	0
$745$	7.39004	0.270750
$746$	−3.38406	−0.123899
$747$	0	0
$748$	−114.812	−4.19793
$749$	−37.9664	−1.38726
$750$	0	0
$751$	16.3361	0.596112	0.298056	−	0.954548i	$-0.403662\pi$
$751$	16.3361	0.596112	0.298056	+	0.954548i	$0.403662\pi$
$752$	−11.6175	−0.423648
$753$	0	0
$754$	−1.49837	−0.0545673
$755$	−6.39367	−0.232689
$756$	0	0
$757$	−15.4936	−0.563123	−0.281562	−	0.959543i	$-0.590852\pi$
$757$	−15.4936	−0.563123	−0.281562	+	0.959543i	$0.590852\pi$
$758$	11.5010	0.417736
$759$	0	0
$760$	−7.28087	−0.264105
$761$	33.2479	1.20524	0.602618	−	0.798030i	$-0.294124\pi$
$761$	33.2479	1.20524	0.602618	+	0.798030i	$0.294124\pi$
$762$	0	0
$763$	−83.1797	−3.01131
$764$	−10.8019	−0.390799
$765$	0	0
$766$	0.484423	0.0175029
$767$	3.80780	0.137492
$768$	0	0
$769$	−34.3864	−1.24001	−0.620003	−	0.784600i	$-0.712869\pi$
$769$	−34.3864	−1.24001	−0.620003	+	0.784600i	$0.712869\pi$
$770$	−57.2695	−2.06385
$771$	0	0
$772$	36.5353	1.31493
$773$	4.48568	0.161339	0.0806693	−	0.996741i	$-0.474294\pi$
$773$	4.48568	0.161339	0.0806693	+	0.996741i	$0.474294\pi$
$774$	0	0
$775$	−7.70484	−0.276766
$776$	−0.495059	−0.0177716
$777$	0	0
$778$	−44.6774	−1.60176
$779$	−25.1368	−0.900620
$780$	0	0
$781$	16.9781	0.607525
$782$	89.3791	3.19619
$783$	0	0
$784$	−26.3544	−0.941228
$785$	18.3410	0.654619
$786$	0	0
$787$	27.1311	0.967118	0.483559	−	0.875312i	$-0.339344\pi$
$787$	27.1311	0.967118	0.483559	+	0.875312i	$0.339344\pi$
$788$	23.4668	0.835971
$789$	0	0
$790$	−2.11200	−0.0751414
$791$	−44.6794	−1.58862
$792$	0	0
$793$	18.1484	0.644469
$794$	9.76748	0.346635
$795$	0	0
$796$	31.4147	1.11347
$797$	−15.5540	−0.550951	−0.275476	−	0.961308i	$-0.588835\pi$
$797$	−15.5540	−0.550951	−0.275476	+	0.961308i	$0.588835\pi$
$798$	0	0
$799$	−55.7494	−1.97227
$800$	7.21423	0.255062
$801$	0	0
$802$	−12.9054	−0.455705
$803$	24.2319	0.855124
$804$	0	0
$805$	26.2032	0.923541
$806$	34.1880	1.20422
$807$	0	0
$808$	−9.25688	−0.325656
$809$	15.7952	0.555330	0.277665	−	0.960678i	$-0.410439\pi$
$809$	15.7952	0.555330	0.277665	+	0.960678i	$0.410439\pi$
$810$	0	0
$811$	−43.9045	−1.54170	−0.770848	−	0.637019i	$-0.780167\pi$
$811$	−43.9045	−1.54170	−0.770848	+	0.637019i	$0.780167\pi$
$812$	4.69266	0.164680
$813$	0	0
$814$	−45.8938	−1.60858
$815$	−17.6947	−0.619817
$816$	0	0
$817$	−3.90992	−0.136791
$818$	−58.0752	−2.03055
$819$	0	0
$820$	−18.4560	−0.644511
$821$	34.0762	1.18927	0.594634	−	0.803996i	$-0.297297\pi$
$821$	34.0762	1.18927	0.594634	+	0.803996i	$0.297297\pi$
$822$	0	0
$823$	9.99487	0.348399	0.174200	−	0.984710i	$-0.444266\pi$
$823$	9.99487	0.348399	0.174200	+	0.984710i	$0.444266\pi$
$824$	33.2386	1.15792
$825$	0	0
$826$	−20.2905	−0.705998
$827$	−1.72028	−0.0598200	−0.0299100	−	0.999553i	$-0.509522\pi$
$827$	−1.72028	−0.0598200	−0.0299100	+	0.999553i	$0.509522\pi$
$828$	0	0
$829$	15.6999	0.545280	0.272640	−	0.962116i	$-0.412103\pi$
$829$	15.6999	0.545280	0.272640	+	0.962116i	$0.412103\pi$
$830$	−25.4324	−0.882772
$831$	0	0
$832$	−25.6736	−0.890070
$833$	−126.468	−4.38185
$834$	0	0
$835$	20.9480	0.724934
$836$	59.0689	2.04294
$837$	0	0
$838$	−29.7351	−1.02718
$839$	17.9809	0.620770	0.310385	−	0.950611i	$-0.399542\pi$
$839$	17.9809	0.620770	0.310385	+	0.950611i	$0.399542\pi$
$840$	0	0
$841$	−28.8860	−0.996068
$842$	−67.6452	−2.33121
$843$	0	0
$844$	−69.3409	−2.38681
$845$	−8.94147	−0.307596
$846$	0	0
$847$	85.0995	2.92405
$848$	−12.2994	−0.422362
$849$	0	0
$850$	16.6249	0.570231
$851$	20.9984	0.719814
$852$	0	0
$853$	28.0025	0.958786	0.479393	−	0.877600i	$-0.340857\pi$
$853$	28.0025	0.958786	0.479393	+	0.877600i	$0.340857\pi$
$854$	−96.7072	−3.30925
$855$	0	0
$856$	14.6049	0.499185
$857$	−37.2474	−1.27235	−0.636173	−	0.771547i	$-0.719483\pi$
$857$	−37.2474	−1.27235	−0.636173	+	0.771547i	$0.719483\pi$
$858$	0	0
$859$	17.3878	0.593266	0.296633	−	0.954992i	$-0.404136\pi$
$859$	17.3878	0.593266	0.296633	+	0.954992i	$0.404136\pi$
$860$	−2.87075	−0.0978917
$861$	0	0
$862$	64.2828	2.18948
$863$	−20.5136	−0.698290	−0.349145	−	0.937069i	$-0.613528\pi$
$863$	−20.5136	−0.698290	−0.349145	+	0.937069i	$0.613528\pi$
$864$	0	0
$865$	−20.8654	−0.709446
$866$	−20.4768	−0.695831
$867$	0	0
$868$	−107.072	−3.63425
$869$	5.11547	0.173530
$870$	0	0
$871$	−28.4923	−0.965425
$872$	31.9975	1.08357
$873$	0	0
$874$	−45.9842	−1.55544
$875$	4.87392	0.164768
$876$	0	0
$877$	0.924927	0.0312326	0.0156163	−	0.999878i	$-0.495029\pi$
$877$	0.924927	0.0312326	0.0156163	+	0.999878i	$0.495029\pi$
$878$	−75.8781	−2.56076
$879$	0	0
$880$	−8.39123	−0.282868
$881$	22.7220	0.765524	0.382762	−	0.923847i	$-0.374973\pi$
$881$	22.7220	0.765524	0.382762	+	0.923847i	$0.374973\pi$
$882$	0	0
$883$	−20.0534	−0.674852	−0.337426	−	0.941352i	$-0.609556\pi$
$883$	−20.0534	−0.674852	−0.337426	+	0.941352i	$0.609556\pi$
$884$	−43.3562	−1.45823
$885$	0	0
$886$	35.1489	1.18085
$887$	35.5012	1.19201	0.596007	−	0.802979i	$-0.296753\pi$
$887$	35.5012	1.19201	0.596007	+	0.802979i	$0.296753\pi$
$888$	0	0
$889$	96.0653	3.22193
$890$	−2.20255	−0.0738297
$891$	0	0
$892$	61.6670	2.06476
$893$	28.6822	0.959815
$894$	0	0
$895$	−6.84436	−0.228782
$896$	66.4831	2.22105
$897$	0	0
$898$	24.7430	0.825686
$899$	−2.60178	−0.0867743
$900$	0	0
$901$	−59.0214	−1.96629
$902$	76.0587	2.53248
$903$	0	0
$904$	17.1872	0.571639
$905$	−11.9988	−0.398852
$906$	0	0
$907$	20.8339	0.691780	0.345890	−	0.938275i	$-0.387577\pi$
$907$	20.8339	0.691780	0.345890	+	0.938275i	$0.387577\pi$
$908$	3.06259	0.101636
$909$	0	0
$910$	−21.6266	−0.716916
$911$	31.3773	1.03958	0.519788	−	0.854296i	$-0.326011\pi$
$911$	31.3773	1.03958	0.519788	+	0.854296i	$0.326011\pi$
$912$	0	0
$913$	61.5999	2.03866
$914$	−59.6878	−1.97430
$915$	0	0
$916$	24.1953	0.799435
$917$	−84.4520	−2.78885
$918$	0	0
$919$	3.01788	0.0995507	0.0497753	−	0.998760i	$-0.484149\pi$
$919$	3.01788	0.0995507	0.0497753	+	0.998760i	$0.484149\pi$
$920$	−10.0798	−0.332322
$921$	0	0
$922$	−17.3904	−0.572724
$923$	6.41143	0.211035
$924$	0	0
$925$	3.90579	0.128422
$926$	−40.2879	−1.32394
$927$	0	0
$928$	2.43611	0.0799693
$929$	30.5542	1.00245	0.501225	−	0.865317i	$-0.332883\pi$
$929$	30.5542	1.00245	0.501225	+	0.865317i	$0.332883\pi$
$930$	0	0
$931$	65.0657	2.13244
$932$	0.865919	0.0283641
$933$	0	0
$934$	0.323704	0.0105919
$935$	−40.2673	−1.31688
$936$	0	0
$937$	−22.4780	−0.734326	−0.367163	−	0.930157i	$-0.619671\pi$
$937$	−22.4780	−0.734326	−0.367163	+	0.930157i	$0.619671\pi$
$938$	151.827	4.95731
$939$	0	0
$940$	21.0591	0.686873
$941$	60.8171	1.98258	0.991290	−	0.131699i	$-0.0420433\pi$
$941$	60.8171	1.98258	0.991290	+	0.131699i	$0.0420433\pi$
$942$	0	0
$943$	−34.8000	−1.13325
$944$	−2.97301	−0.0967632
$945$	0	0
$946$	11.8306	0.384646
$947$	−36.7249	−1.19340	−0.596700	−	0.802464i	$-0.703522\pi$
$947$	−36.7249	−1.19340	−0.596700	+	0.802464i	$0.703522\pi$
$948$	0	0
$949$	9.15066	0.297043
$950$	−8.55328	−0.277505
$951$	0	0
$952$	68.9745	2.23548
$953$	−27.5716	−0.893131	−0.446565	−	0.894751i	$-0.647353\pi$
$953$	−27.5716	−0.893131	−0.446565	+	0.894751i	$0.647353\pi$
$954$	0	0
$955$	−3.78850	−0.122593
$956$	−24.1450	−0.780905
$957$	0	0
$958$	−47.7767	−1.54359
$959$	63.7905	2.05990
$960$	0	0
$961$	28.3645	0.914985
$962$	−17.3309	−0.558769
$963$	0	0
$964$	−60.4439	−1.94677
$965$	12.8138	0.412492
$966$	0	0
$967$	−33.1465	−1.06592	−0.532960	−	0.846141i	$-0.678920\pi$
$967$	−33.1465	−1.06592	−0.532960	+	0.846141i	$0.678920\pi$
$968$	−32.7360	−1.05218
$969$	0	0
$970$	−0.581576	−0.0186733
$971$	−15.0224	−0.482091	−0.241045	−	0.970514i	$-0.577490\pi$
$971$	−15.0224	−0.482091	−0.241045	+	0.970514i	$0.577490\pi$
$972$	0	0
$973$	−3.70517	−0.118782
$974$	−71.5836	−2.29369
$975$	0	0
$976$	−14.1697	−0.453562
$977$	−2.06728	−0.0661383	−0.0330691	−	0.999453i	$-0.510528\pi$
$977$	−2.06728	−0.0661383	−0.0330691	+	0.999453i	$0.510528\pi$
$978$	0	0
$979$	5.33481	0.170501
$980$	47.7727	1.52604
$981$	0	0
$982$	58.6420	1.87134
$983$	−54.6938	−1.74446	−0.872230	−	0.489096i	$-0.837327\pi$
$983$	−54.6938	−1.74446	−0.872230	+	0.489096i	$0.837327\pi$
$984$	0	0
$985$	8.23040	0.262243
$986$	5.61393	0.178784
$987$	0	0
$988$	22.3061	0.709652
$989$	−5.41300	−0.172123
$990$	0	0
$991$	−40.6412	−1.29101	−0.645505	−	0.763756i	$-0.723353\pi$
$991$	−40.6412	−1.29101	−0.645505	+	0.763756i	$0.723353\pi$
$992$	−55.5845	−1.76481
$993$	0	0
$994$	−34.1645	−1.08363
$995$	11.0179	0.349292
$996$	0	0
$997$	−50.8749	−1.61122	−0.805612	−	0.592444i	$-0.798163\pi$
$997$	−50.8749	−1.61122	−0.805612	+	0.592444i	$0.798163\pi$
$998$	−39.2328	−1.24189
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4005.2.a.t.1.3		10
3.2	odd	2		1335.2.a.i.1.8	✓	10
15.14	odd	2		6675.2.a.ba.1.3		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1335.2.a.i.1.8	✓	10	3.2	odd	2
4005.2.a.t.1.3		10	1.1	even	1	trivial
6675.2.a.ba.1.3		10	15.14	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)

\(f(q)\)	\(=\)	\(q-2.20255 q^{2} +2.85124 q^{4} +1.00000 q^{5} +4.87392 q^{7} -1.87489 q^{8} +O(q^{10})\)	\(q-2.20255 q^{2} +2.85124 q^{4} +1.00000 q^{5} +4.87392 q^{7} -1.87489 q^{8} -2.20255 q^{10} +5.33481 q^{11} +2.01458 q^{13} -10.7351 q^{14} -1.57292 q^{16} -7.54803 q^{17} +3.88335 q^{19} +2.85124 q^{20} -11.7502 q^{22} +5.37621 q^{23} +1.00000 q^{25} -4.43722 q^{26} +13.8967 q^{28} +0.337682 q^{29} -7.70484 q^{31} +7.21423 q^{32} +16.6249 q^{34} +4.87392 q^{35} +3.90579 q^{37} -8.55328 q^{38} -1.87489 q^{40} -6.47297 q^{41} -1.00684 q^{43} +15.2108 q^{44} -11.8414 q^{46} +7.38596 q^{47} +16.7551 q^{49} -2.20255 q^{50} +5.74404 q^{52} +7.81944 q^{53} +5.33481 q^{55} -9.13808 q^{56} -0.743761 q^{58} +1.89012 q^{59} +9.00854 q^{61} +16.9703 q^{62} -12.7439 q^{64} +2.01458 q^{65} -14.1431 q^{67} -21.5212 q^{68} -10.7351 q^{70} +3.18252 q^{71} +4.54222 q^{73} -8.60271 q^{74} +11.0723 q^{76} +26.0014 q^{77} +0.958885 q^{79} -1.57292 q^{80} +14.2571 q^{82} +11.5468 q^{83} -7.54803 q^{85} +2.21763 q^{86} -10.0022 q^{88} +1.00000 q^{89} +9.81889 q^{91} +15.3288 q^{92} -16.2680 q^{94} +3.88335 q^{95} +0.264046 q^{97} -36.9039 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 18 q^{4} + 10 q^{5} + 9 q^{7} - 3 q^{8}+O(q^{10}) \) 10 * q + 18 * q^4 + 10 * q^5 + 9 * q^7 - 3 * q^8	\( 10 q + 18 q^{4} + 10 q^{5} + 9 q^{7} - 3 q^{8} - 4 q^{11} + 15 q^{13} + q^{14} + 22 q^{16} - 11 q^{17} + 14 q^{19} + 18 q^{20} + 10 q^{22} + 8 q^{23} + 10 q^{25} + 14 q^{26} + 36 q^{28} - 5 q^{29} - 8 q^{31} - q^{32} + 2 q^{34} + 9 q^{35} + 23 q^{37} - 8 q^{38} - 3 q^{40} - 13 q^{41} + 25 q^{43} + 8 q^{44} - 14 q^{46} + q^{47} + 21 q^{49} + 51 q^{52} - 9 q^{53} - 4 q^{55} - 15 q^{56} - 8 q^{58} + 15 q^{59} + 8 q^{61} + 8 q^{62} + 9 q^{64} + 15 q^{65} + 52 q^{67} + 28 q^{68} + q^{70} + 22 q^{71} + 34 q^{73} + 18 q^{74} + 14 q^{76} - 4 q^{77} - 3 q^{79} + 22 q^{80} + 17 q^{82} + 10 q^{83} - 11 q^{85} + 6 q^{86} + 4 q^{88} + 10 q^{89} + 18 q^{91} + 14 q^{92} - 43 q^{94} + 14 q^{95} + 34 q^{97} + 38 q^{98}+O(q^{100}) \) 10 * q + 18 * q^4 + 10 * q^5 + 9 * q^7 - 3 * q^8 - 4 * q^11 + 15 * q^13 + q^14 + 22 * q^16 - 11 * q^17 + 14 * q^19 + 18 * q^20 + 10 * q^22 + 8 * q^23 + 10 * q^25 + 14 * q^26 + 36 * q^28 - 5 * q^29 - 8 * q^31 - q^32 + 2 * q^34 + 9 * q^35 + 23 * q^37 - 8 * q^38 - 3 * q^40 - 13 * q^41 + 25 * q^43 + 8 * q^44 - 14 * q^46 + q^47 + 21 * q^49 + 51 * q^52 - 9 * q^53 - 4 * q^55 - 15 * q^56 - 8 * q^58 + 15 * q^59 + 8 * q^61 + 8 * q^62 + 9 * q^64 + 15 * q^65 + 52 * q^67 + 28 * q^68 + q^70 + 22 * q^71 + 34 * q^73 + 18 * q^74 + 14 * q^76 - 4 * q^77 - 3 * q^79 + 22 * q^80 + 17 * q^82 + 10 * q^83 - 11 * q^85 + 6 * q^86 + 4 * q^88 + 10 * q^89 + 18 * q^91 + 14 * q^92 - 43 * q^94 + 14 * q^95 + 34 * q^97 + 38 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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