LMFDB - Embedded newform 4005.2.a.l.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:	$4$
Coefficient field:	4.4.725.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 3x^{2} + x + 1 $ x^4 - x^3 - 3*x^2 + x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 445)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.35567$ 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)

$f(q)$	$=$	$q-0.162147 q^{2} -1.97371 q^{4} +1.00000 q^{5} -0.475281 q^{7} +0.644326 q^{8} +O(q^{10})$	\(q-0.162147 q^{2} -1.97371 q^{4} +1.00000 q^{5} -0.475281 q^{7} +0.644326 q^{8} -0.162147 q^{10} +5.19353 q^{11} -1.30490 q^{13} +0.0770654 q^{14} +3.84294 q^{16} +7.40330 q^{17} +6.83604 q^{19} -1.97371 q^{20} -0.842116 q^{22} -2.69195 q^{23} +1.00000 q^{25} +0.211586 q^{26} +0.938065 q^{28} +0.00689789 q^{29} -8.54230 q^{31} -1.91177 q^{32} -1.20042 q^{34} -0.475281 q^{35} -4.56119 q^{37} -1.10845 q^{38} +0.644326 q^{40} -3.41020 q^{41} +9.87858 q^{43} -10.2505 q^{44} +0.436493 q^{46} +4.62493 q^{47} -6.77411 q^{49} -0.162147 q^{50} +2.57549 q^{52} -2.86526 q^{53} +5.19353 q^{55} -0.306236 q^{56} -0.00111847 q^{58} +1.13767 q^{59} +7.30684 q^{61} +1.38511 q^{62} -7.37589 q^{64} -1.30490 q^{65} -1.79531 q^{67} -14.6120 q^{68} +0.0770654 q^{70} -2.91492 q^{71} -2.26369 q^{73} +0.739584 q^{74} -13.4924 q^{76} -2.46838 q^{77} -10.5104 q^{79} +3.84294 q^{80} +0.552955 q^{82} -6.35650 q^{83} +7.40330 q^{85} -1.60178 q^{86} +3.34632 q^{88} +1.00000 q^{89} +0.620194 q^{91} +5.31313 q^{92} -0.749920 q^{94} +6.83604 q^{95} +18.3113 q^{97} +1.09840 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{2} + 3 q^{4} + 4 q^{5} + 2 q^{7} + 9 q^{8}+O(q^{10}) $ 4 * q - q^2 + 3 * q^4 + 4 * q^5 + 2 * q^7 + 9 * q^8	$ 4 q - q^{2} + 3 q^{4} + 4 q^{5} + 2 q^{7} + 9 q^{8} - q^{10} + 14 q^{11} - 5 q^{13} + 5 q^{14} - 3 q^{16} + 3 q^{17} - q^{19} + 3 q^{20} + 2 q^{22} + 3 q^{23} + 4 q^{25} + 9 q^{26} + 5 q^{28} + 10 q^{29} - 11 q^{31} + 2 q^{32} - 8 q^{34} + 2 q^{35} + 3 q^{37} - 2 q^{38} + 9 q^{40} + 3 q^{41} + 9 q^{43} + 9 q^{44} - 4 q^{46} + 24 q^{47} - q^{50} + 8 q^{52} - 3 q^{53} + 14 q^{55} + 13 q^{56} + 19 q^{58} + 22 q^{59} - 3 q^{61} + 24 q^{62} - 11 q^{64} - 5 q^{65} - 9 q^{67} - 31 q^{68} + 5 q^{70} - 16 q^{71} + 3 q^{73} - 31 q^{74} - 24 q^{76} + 4 q^{77} - 27 q^{79} - 3 q^{80} - 15 q^{82} - 6 q^{83} + 3 q^{85} - 15 q^{86} + 30 q^{88} + 4 q^{89} - 29 q^{91} + 17 q^{92} + 13 q^{94} - q^{95} + 41 q^{97} - 22 q^{98}+O(q^{100}) $ 4 * q - q^2 + 3 * q^4 + 4 * q^5 + 2 * q^7 + 9 * q^8 - q^10 + 14 * q^11 - 5 * q^13 + 5 * q^14 - 3 * q^16 + 3 * q^17 - q^19 + 3 * q^20 + 2 * q^22 + 3 * q^23 + 4 * q^25 + 9 * q^26 + 5 * q^28 + 10 * q^29 - 11 * q^31 + 2 * q^32 - 8 * q^34 + 2 * q^35 + 3 * q^37 - 2 * q^38 + 9 * q^40 + 3 * q^41 + 9 * q^43 + 9 * q^44 - 4 * q^46 + 24 * q^47 - q^50 + 8 * q^52 - 3 * q^53 + 14 * q^55 + 13 * q^56 + 19 * q^58 + 22 * q^59 - 3 * q^61 + 24 * q^62 - 11 * q^64 - 5 * q^65 - 9 * q^67 - 31 * q^68 + 5 * q^70 - 16 * q^71 + 3 * q^73 - 31 * q^74 - 24 * q^76 + 4 * q^77 - 27 * q^79 - 3 * q^80 - 15 * q^82 - 6 * q^83 + 3 * q^85 - 15 * q^86 + 30 * q^88 + 4 * q^89 - 29 * q^91 + 17 * q^92 + 13 * q^94 - q^95 + 41 * q^97 - 22 * 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 $

$2$	−0.162147	−0.114655	−0.0573277	−	0.998355i	$-0.518258\pi$
$2$	−0.162147	−0.114655	−0.0573277	+	0.998355i	$0.518258\pi$
$3$	0	0
$3$	0	0
$4$	−1.97371	−0.986854
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−0.475281	−0.179639	−0.0898196	−	0.995958i	$-0.528629\pi$
$7$	−0.475281	−0.179639	−0.0898196	+	0.995958i	$0.528629\pi$
$8$	0.644326	0.227804
$9$	0	0
$10$	−0.162147	−0.0512754
$11$	5.19353	1.56591	0.782954	−	0.622080i	$-0.213712\pi$
$11$	5.19353	1.56591	0.782954	+	0.622080i	$0.213712\pi$
$12$	0	0
$13$	−1.30490	−0.361914	−0.180957	−	0.983491i	$-0.557920\pi$
$13$	−1.30490	−0.361914	−0.180957	+	0.983491i	$0.557920\pi$
$14$	0.0770654	0.0205966
$15$	0	0
$16$	3.84294	0.960735
$17$	7.40330	1.79556	0.897782	−	0.440439i	$-0.145177\pi$
$17$	7.40330	1.79556	0.897782	+	0.440439i	$0.145177\pi$
$18$	0	0
$19$	6.83604	1.56830	0.784148	−	0.620574i	$-0.213100\pi$
$19$	6.83604	1.56830	0.784148	+	0.620574i	$0.213100\pi$
$20$	−1.97371	−0.441335
$21$	0	0
$22$	−0.842116	−0.179540
$23$	−2.69195	−0.561311	−0.280656	−	0.959809i	$-0.590552\pi$
$23$	−2.69195	−0.561311	−0.280656	+	0.959809i	$0.590552\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0.211586	0.0414954
$27$	0	0
$28$	0.938065	0.177278
$29$	0.00689789	0.00128091	0.000640453	−	1.00000i	$-0.499796\pi$
$29$	0.00689789	0.00128091	0.000640453		1.00000i	$0.499796\pi$
$30$	0	0
$31$	−8.54230	−1.53424	−0.767121	−	0.641502i	$-0.778312\pi$
$31$	−8.54230	−1.53424	−0.767121	+	0.641502i	$0.778312\pi$
$32$	−1.91177	−0.337957
$33$	0	0
$34$	−1.20042	−0.205871
$35$	−0.475281	−0.0803371
$36$	0	0
$37$	−4.56119	−0.749855	−0.374927	−	0.927054i	$-0.622332\pi$
$37$	−4.56119	−0.749855	−0.374927	+	0.927054i	$0.622332\pi$
$38$	−1.10845	−0.179814
$39$	0	0
$40$	0.644326	0.101877
$41$	−3.41020	−0.532584	−0.266292	−	0.963892i	$-0.585799\pi$
$41$	−3.41020	−0.532584	−0.266292	+	0.963892i	$0.585799\pi$
$42$	0	0
$43$	9.87858	1.50647	0.753235	−	0.657752i	$-0.228492\pi$
$43$	9.87858	1.50647	0.753235	+	0.657752i	$0.228492\pi$
$44$	−10.2505	−1.54532
$45$	0	0
$46$	0.436493	0.0643574
$47$	4.62493	0.674616	0.337308	−	0.941394i	$-0.390484\pi$
$47$	4.62493	0.674616	0.337308	+	0.941394i	$0.390484\pi$
$48$	0	0
$49$	−6.77411	−0.967730
$50$	−0.162147	−0.0229311
$51$	0	0
$52$	2.57549	0.357157
$53$	−2.86526	−0.393574	−0.196787	−	0.980446i	$-0.563051\pi$
$53$	−2.86526	−0.393574	−0.196787	+	0.980446i	$0.563051\pi$
$54$	0	0
$55$	5.19353	0.700295
$56$	−0.306236	−0.0409224
$57$	0	0
$58$	−0.00111847	−0.000146863 0
$59$	1.13767	0.148111	0.0740557	−	0.997254i	$-0.476406\pi$
$59$	1.13767	0.148111	0.0740557	+	0.997254i	$0.476406\pi$
$60$	0	0
$61$	7.30684	0.935546	0.467773	−	0.883849i	$-0.345057\pi$
$61$	7.30684	0.935546	0.467773	+	0.883849i	$0.345057\pi$
$62$	1.38511	0.175909
$63$	0	0
$64$	−7.37589	−0.921987
$65$	−1.30490	−0.161853
$66$	0	0
$67$	−1.79531	−0.219332	−0.109666	−	0.993968i	$-0.534978\pi$
$67$	−1.79531	−0.219332	−0.109666	+	0.993968i	$0.534978\pi$
$68$	−14.6120	−1.77196
$69$	0	0
$70$	0.0770654	0.00921108
$71$	−2.91492	−0.345937	−0.172969	−	0.984927i	$-0.555336\pi$
$71$	−2.91492	−0.345937	−0.172969	+	0.984927i	$0.555336\pi$
$72$	0	0
$73$	−2.26369	−0.264945	−0.132473	−	0.991187i	$-0.542292\pi$
$73$	−2.26369	−0.264945	−0.132473	+	0.991187i	$0.542292\pi$
$74$	0.739584	0.0859749
$75$	0	0
$76$	−13.4924	−1.54768
$77$	−2.46838	−0.281298
$78$	0	0
$79$	−10.5104	−1.18251	−0.591257	−	0.806483i	$-0.701368\pi$
$79$	−10.5104	−1.18251	−0.591257	+	0.806483i	$0.701368\pi$
$80$	3.84294	0.429654
$81$	0	0
$82$	0.552955	0.0610636
$83$	−6.35650	−0.697716	−0.348858	−	0.937176i	$-0.613431\pi$
$83$	−6.35650	−0.697716	−0.348858	+	0.937176i	$0.613431\pi$
$84$	0	0
$85$	7.40330	0.803001
$86$	−1.60178	−0.172725
$87$	0	0
$88$	3.34632	0.356719
$89$	1.00000	0.106000
$89$	1.00000	0.106000
$90$	0	0
$91$	0.620194	0.0650140
$92$	5.31313	0.553932
$93$	0	0
$94$	−0.749920	−0.0773483
$95$	6.83604	0.701363
$96$	0	0
$97$	18.3113	1.85923	0.929617	−	0.368528i	$-0.120138\pi$
$97$	18.3113	1.85923	0.929617	+	0.368528i	$0.120138\pi$
$98$	1.09840	0.110955
$99$	0	0
$100$	−1.97371	−0.197371
$101$	14.3545	1.42832	0.714162	−	0.699981i	$-0.246808\pi$
$101$	14.3545	1.42832	0.714162	+	0.699981i	$0.246808\pi$
$102$	0	0
$103$	5.43964	0.535983	0.267992	−	0.963421i	$-0.413640\pi$
$103$	5.43964	0.535983	0.267992	+	0.963421i	$0.413640\pi$
$104$	−0.840781	−0.0824454
$105$	0	0
$106$	0.464594	0.0451254
$107$	7.27486	0.703287	0.351643	−	0.936134i	$-0.385623\pi$
$107$	7.27486	0.703287	0.351643	+	0.936134i	$0.385623\pi$
$108$	0	0
$109$	3.92293	0.375749	0.187874	−	0.982193i	$-0.439840\pi$
$109$	3.92293	0.375749	0.187874	+	0.982193i	$0.439840\pi$
$110$	−0.842116	−0.0802926
$111$	0	0
$112$	−1.82648	−0.172586
$113$	15.2119	1.43102	0.715509	−	0.698603i	$-0.246195\pi$
$113$	15.2119	1.43102	0.715509	+	0.698603i	$0.246195\pi$
$114$	0	0
$115$	−2.69195	−0.251026
$116$	−0.0136144	−0.00126407
$117$	0	0
$118$	−0.184469	−0.0169818
$119$	−3.51865	−0.322554
$120$	0	0
$121$	15.9727	1.45207
$122$	−1.18478	−0.107265
$123$	0	0
$124$	16.8600	1.51407
$125$	1.00000	0.0894427
$126$	0	0
$127$	14.3082	1.26965	0.634823	−	0.772658i	$-0.281073\pi$
$127$	14.3082	1.26965	0.634823	+	0.772658i	$0.281073\pi$
$128$	5.01953	0.443668
$129$	0	0
$130$	0.211586	0.0185573
$131$	5.16529	0.451294	0.225647	−	0.974209i	$-0.427550\pi$
$131$	5.16529	0.451294	0.225647	+	0.974209i	$0.427550\pi$
$132$	0	0
$133$	−3.24904	−0.281727
$134$	0.291105	0.0251476
$135$	0	0
$136$	4.77014	0.409036
$137$	10.1033	0.863181	0.431590	−	0.902070i	$-0.357953\pi$
$137$	10.1033	0.863181	0.431590	+	0.902070i	$0.357953\pi$
$138$	0	0
$139$	−21.0062	−1.78172	−0.890862	−	0.454275i	$-0.849898\pi$
$139$	−21.0062	−1.78172	−0.890862	+	0.454275i	$0.849898\pi$
$140$	0.938065	0.0792810
$141$	0	0
$142$	0.472646	0.0396636
$143$	−6.77704	−0.566724
$144$	0	0
$145$	0.00689789	0.000572838 0
$146$	0.367052	0.0303774
$147$	0	0
$148$	9.00245	0.739997
$149$	4.55954	0.373532	0.186766	−	0.982404i	$-0.440199\pi$
$149$	4.55954	0.373532	0.186766	+	0.982404i	$0.440199\pi$
$150$	0	0
$151$	3.64071	0.296276	0.148138	−	0.988967i	$-0.452672\pi$
$151$	3.64071	0.296276	0.148138	+	0.988967i	$0.452672\pi$
$152$	4.40464	0.357263
$153$	0	0
$154$	0.400241	0.0322524
$155$	−8.54230	−0.686134
$156$	0	0
$157$	17.3667	1.38601	0.693006	−	0.720932i	$-0.256286\pi$
$157$	17.3667	1.38601	0.693006	+	0.720932i	$0.256286\pi$
$158$	1.70423	0.135582
$159$	0	0
$160$	−1.91177	−0.151139
$161$	1.27943	0.100834
$162$	0	0
$163$	−9.06918	−0.710353	−0.355177	−	0.934799i	$-0.615579\pi$
$163$	−9.06918	−0.710353	−0.355177	+	0.934799i	$0.615579\pi$
$164$	6.73074	0.525583
$165$	0	0
$166$	1.03069	0.0799970
$167$	−4.29059	−0.332016	−0.166008	−	0.986124i	$-0.553088\pi$
$167$	−4.29059	−0.332016	−0.166008	+	0.986124i	$0.553088\pi$
$168$	0	0
$169$	−11.2972	−0.869018
$170$	−1.20042	−0.0920684
$171$	0	0
$172$	−19.4974	−1.48667
$173$	−21.1969	−1.61157	−0.805787	−	0.592206i	$-0.798257\pi$
$173$	−21.1969	−1.61157	−0.805787	+	0.592206i	$0.798257\pi$
$174$	0	0
$175$	−0.475281	−0.0359278
$176$	19.9584	1.50442
$177$	0	0
$178$	−0.162147	−0.0121534
$179$	−0.468991	−0.0350541	−0.0175270	−	0.999846i	$-0.505579\pi$
$179$	−0.468991	−0.0350541	−0.0175270	+	0.999846i	$0.505579\pi$
$180$	0	0
$181$	−22.2732	−1.65555	−0.827776	−	0.561058i	$-0.810394\pi$
$181$	−22.2732	−1.65555	−0.827776	+	0.561058i	$0.810394\pi$
$182$	−0.100563	−0.00745420
$183$	0	0
$184$	−1.73450	−0.127869
$185$	−4.56119	−0.335345
$186$	0	0
$187$	38.4493	2.81169
$188$	−9.12827	−0.665747
$189$	0	0
$190$	−1.10845	−0.0804151
$191$	21.3072	1.54173	0.770867	−	0.636996i	$-0.219823\pi$
$191$	21.3072	1.54173	0.770867	+	0.636996i	$0.219823\pi$
$192$	0	0
$193$	−1.33822	−0.0963275	−0.0481637	−	0.998839i	$-0.515337\pi$
$193$	−1.33822	−0.0963275	−0.0481637	+	0.998839i	$0.515337\pi$
$194$	−2.96913	−0.213171
$195$	0	0
$196$	13.3701	0.955008
$197$	9.65959	0.688217	0.344109	−	0.938930i	$-0.388181\pi$
$197$	9.65959	0.688217	0.344109	+	0.938930i	$0.388181\pi$
$198$	0	0
$199$	9.85738	0.698771	0.349386	−	0.936979i	$-0.386390\pi$
$199$	9.85738	0.698771	0.349386	+	0.936979i	$0.386390\pi$
$200$	0.644326	0.0455607
$201$	0	0
$202$	−2.32754	−0.163765
$203$	−0.00327843	−0.000230101 0
$204$	0	0
$205$	−3.41020	−0.238179
$206$	−0.882022	−0.0614534
$207$	0	0
$208$	−5.01466	−0.347704
$209$	35.5032	2.45581
$210$	0	0
$211$	−12.2510	−0.843396	−0.421698	−	0.906736i	$-0.638566\pi$
$211$	−12.2510	−0.843396	−0.421698	+	0.906736i	$0.638566\pi$
$212$	5.65519	0.388400
$213$	0	0
$214$	−1.17960	−0.0806356
$215$	9.87858	0.673714
$216$	0	0
$217$	4.05999	0.275610
$218$	−0.636093	−0.0430816
$219$	0	0
$220$	−10.2505	−0.691089
$221$	−9.66057	−0.649841
$222$	0	0
$223$	−22.4242	−1.50163	−0.750817	−	0.660511i	$-0.770340\pi$
$223$	−22.4242	−1.50163	−0.750817	+	0.660511i	$0.770340\pi$
$224$	0.908629	0.0607103
$225$	0	0
$226$	−2.46657	−0.164074
$227$	23.1665	1.53762	0.768809	−	0.639479i	$-0.220850\pi$
$227$	23.1665	1.53762	0.768809	+	0.639479i	$0.220850\pi$
$228$	0	0
$229$	4.15788	0.274761	0.137380	−	0.990518i	$-0.456132\pi$
$229$	4.15788	0.274761	0.137380	+	0.990518i	$0.456132\pi$
$230$	0.436493	0.0287815
$231$	0	0
$232$	0.00444449	0.000291795 0
$233$	11.7721	0.771215	0.385607	−	0.922663i	$-0.373992\pi$
$233$	11.7721	0.771215	0.385607	+	0.922663i	$0.373992\pi$
$234$	0	0
$235$	4.62493	0.301697
$236$	−2.24542	−0.146164
$237$	0	0
$238$	0.570539	0.0369825
$239$	20.3834	1.31849	0.659246	−	0.751927i	$-0.270875\pi$
$239$	20.3834	1.31849	0.659246	+	0.751927i	$0.270875\pi$
$240$	0	0
$241$	−11.3468	−0.730908	−0.365454	−	0.930829i	$-0.619086\pi$
$241$	−11.3468	−0.730908	−0.365454	+	0.930829i	$0.619086\pi$
$242$	−2.58993	−0.166487
$243$	0	0
$244$	−14.4216	−0.923247
$245$	−6.77411	−0.432782
$246$	0	0
$247$	−8.92036	−0.567589
$248$	−5.50403	−0.349506
$249$	0	0
$250$	−0.162147	−0.0102551
$251$	16.0015	1.01000	0.505002	−	0.863118i	$-0.331492\pi$
$251$	16.0015	1.01000	0.505002	+	0.863118i	$0.331492\pi$
$252$	0	0
$253$	−13.9807	−0.878962
$254$	−2.32003	−0.145572
$255$	0	0
$256$	13.9379	0.871118
$257$	−9.71532	−0.606025	−0.303012	−	0.952987i	$-0.597992\pi$
$257$	−9.71532	−0.606025	−0.303012	+	0.952987i	$0.597992\pi$
$258$	0	0
$259$	2.16784	0.134703
$260$	2.57549	0.159725
$261$	0	0
$262$	−0.837538	−0.0517432
$263$	−23.1738	−1.42896	−0.714479	−	0.699657i	$-0.753336\pi$
$263$	−23.1738	−1.42896	−0.714479	+	0.699657i	$0.753336\pi$
$264$	0	0
$265$	−2.86526	−0.176012
$266$	0.526823	0.0323016
$267$	0	0
$268$	3.54342	0.216449
$269$	4.92771	0.300448	0.150224	−	0.988652i	$-0.452001\pi$
$269$	4.92771	0.300448	0.150224	+	0.988652i	$0.452001\pi$
$270$	0	0
$271$	16.5952	1.00809	0.504044	−	0.863678i	$-0.331845\pi$
$271$	16.5952	1.00809	0.504044	+	0.863678i	$0.331845\pi$
$272$	28.4505	1.72506
$273$	0	0
$274$	−1.63822	−0.0989684
$275$	5.19353	0.313181
$276$	0	0
$277$	−12.9860	−0.780256	−0.390128	−	0.920761i	$-0.627569\pi$
$277$	−12.9860	−0.780256	−0.390128	+	0.920761i	$0.627569\pi$
$278$	3.40610	0.204284
$279$	0	0
$280$	−0.306236	−0.0183011
$281$	−29.4497	−1.75682	−0.878412	−	0.477904i	$-0.841397\pi$
$281$	−29.4497	−1.75682	−0.878412	+	0.477904i	$0.841397\pi$
$282$	0	0
$283$	−14.8999	−0.885708	−0.442854	−	0.896594i	$-0.646034\pi$
$283$	−14.8999	−0.885708	−0.442854	+	0.896594i	$0.646034\pi$
$284$	5.75320	0.341390
$285$	0	0
$286$	1.09888	0.0649780
$287$	1.62080	0.0956729
$288$	0	0
$289$	37.8089	2.22405
$290$	−0.00111847	−6.56790e−5 0
$291$	0	0
$292$	4.46787	0.261462
$293$	2.70575	0.158072	0.0790358	−	0.996872i	$-0.474816\pi$
$293$	2.70575	0.158072	0.0790358	+	0.996872i	$0.474816\pi$
$294$	0	0
$295$	1.13767	0.0662374
$296$	−2.93889	−0.170820
$297$	0	0
$298$	−0.739316	−0.0428274
$299$	3.51273	0.203147
$300$	0	0
$301$	−4.69510	−0.270621
$302$	−0.590330	−0.0339697
$303$	0	0
$304$	26.2705	1.50672
$305$	7.30684	0.418389
$306$	0	0
$307$	−15.4273	−0.880481	−0.440241	−	0.897880i	$-0.645107\pi$
$307$	−15.4273	−0.880481	−0.440241	+	0.897880i	$0.645107\pi$
$308$	4.87187	0.277600
$309$	0	0
$310$	1.38511	0.0786690
$311$	9.28894	0.526728	0.263364	−	0.964697i	$-0.415168\pi$
$311$	9.28894	0.526728	0.263364	+	0.964697i	$0.415168\pi$
$312$	0	0
$313$	−5.56675	−0.314651	−0.157326	−	0.987547i	$-0.550287\pi$
$313$	−5.56675	−0.314651	−0.157326	+	0.987547i	$0.550287\pi$
$314$	−2.81596	−0.158914
$315$	0	0
$316$	20.7445	1.16697
$317$	−30.9404	−1.73778	−0.868892	−	0.495002i	$-0.835167\pi$
$317$	−30.9404	−1.73778	−0.868892	+	0.495002i	$0.835167\pi$
$318$	0	0
$319$	0.0358244	0.00200578
$320$	−7.37589	−0.412325
$321$	0	0
$322$	−0.207457	−0.0115611
$323$	50.6093	2.81598
$324$	0	0
$325$	−1.30490	−0.0723829
$326$	1.47054	0.0814458
$327$	0	0
$328$	−2.19728	−0.121325
$329$	−2.19814	−0.121187
$330$	0	0
$331$	0.0734460	0.00403696	0.00201848	−	0.999998i	$-0.499357\pi$
$331$	0.0734460	0.00403696	0.00201848	+	0.999998i	$0.499357\pi$
$332$	12.5459	0.688544
$333$	0	0
$334$	0.695708	0.0380674
$335$	−1.79531	−0.0980884
$336$	0	0
$337$	16.9619	0.923971	0.461986	−	0.886887i	$-0.347137\pi$
$337$	16.9619	0.923971	0.461986	+	0.886887i	$0.347137\pi$
$338$	1.83182	0.0996376
$339$	0	0
$340$	−14.6120	−0.792445
$341$	−44.3647	−2.40248
$342$	0	0
$343$	6.54657	0.353481
$344$	6.36503	0.343179
$345$	0	0
$346$	3.43702	0.184776
$347$	21.2221	1.13926	0.569630	−	0.821901i	$-0.307087\pi$
$347$	21.2221	1.13926	0.569630	+	0.821901i	$0.307087\pi$
$348$	0	0
$349$	7.48298	0.400555	0.200277	−	0.979739i	$-0.435816\pi$
$349$	7.48298	0.400555	0.200277	+	0.979739i	$0.435816\pi$
$350$	0.0770654	0.00411932
$351$	0	0
$352$	−9.92885	−0.529209
$353$	−2.09475	−0.111492	−0.0557461	−	0.998445i	$-0.517754\pi$
$353$	−2.09475	−0.111492	−0.0557461	+	0.998445i	$0.517754\pi$
$354$	0	0
$355$	−2.91492	−0.154708
$356$	−1.97371	−0.104606
$357$	0	0
$358$	0.0760456	0.00401914
$359$	1.57842	0.0833059	0.0416529	−	0.999132i	$-0.486738\pi$
$359$	1.57842	0.0833059	0.0416529	+	0.999132i	$0.486738\pi$
$360$	0	0
$361$	27.7315	1.45955
$362$	3.61153	0.189818
$363$	0	0
$364$	−1.22408	−0.0641593
$365$	−2.26369	−0.118487
$366$	0	0
$367$	34.6898	1.81079	0.905397	−	0.424567i	$-0.139574\pi$
$367$	34.6898	1.81079	0.905397	+	0.424567i	$0.139574\pi$
$368$	−10.3450	−0.539272
$369$	0	0
$370$	0.739584	0.0384491
$371$	1.36180	0.0707013
$372$	0	0
$373$	28.3861	1.46977	0.734887	−	0.678189i	$-0.237235\pi$
$373$	28.3861	1.46977	0.734887	+	0.678189i	$0.237235\pi$
$374$	−6.23444	−0.322375
$375$	0	0
$376$	2.97996	0.153680
$377$	−0.00900106	−0.000463578 0
$378$	0	0
$379$	−15.3669	−0.789344	−0.394672	−	0.918822i	$-0.629142\pi$
$379$	−15.3669	−0.789344	−0.394672	+	0.918822i	$0.629142\pi$
$380$	−13.4924	−0.692143
$381$	0	0
$382$	−3.45490	−0.176768
$383$	23.8489	1.21862	0.609311	−	0.792932i	$-0.291446\pi$
$383$	23.8489	1.21862	0.609311	+	0.792932i	$0.291446\pi$
$384$	0	0
$385$	−2.46838	−0.125800
$386$	0.216989	0.0110445
$387$	0	0
$388$	−36.1412	−1.83479
$389$	−11.7669	−0.596603	−0.298302	−	0.954472i	$-0.596420\pi$
$389$	−11.7669	−0.596603	−0.298302	+	0.954472i	$0.596420\pi$
$390$	0	0
$391$	−19.9294	−1.00787
$392$	−4.36473	−0.220452
$393$	0	0
$394$	−1.56628	−0.0789078
$395$	−10.5104	−0.528836
$396$	0	0
$397$	−11.2778	−0.566019	−0.283009	−	0.959117i	$-0.591333\pi$
$397$	−11.2778	−0.566019	−0.283009	+	0.959117i	$0.591333\pi$
$398$	−1.59835	−0.0801179
$399$	0	0
$400$	3.84294	0.192147
$401$	−12.9916	−0.648770	−0.324385	−	0.945925i	$-0.605157\pi$
$401$	−12.9916	−0.648770	−0.324385	+	0.945925i	$0.605157\pi$
$402$	0	0
$403$	11.1469	0.555264
$404$	−28.3315	−1.40955
$405$	0	0
$406$	0.000531589 0	2.63823e−5 0
$407$	−23.6887	−1.17420
$408$	0	0
$409$	−14.2198	−0.703124	−0.351562	−	0.936165i	$-0.614349\pi$
$409$	−14.2198	−0.703124	−0.351562	+	0.936165i	$0.614349\pi$
$410$	0.552955	0.0273085
$411$	0	0
$412$	−10.7363	−0.528937
$413$	−0.540710	−0.0266066
$414$	0	0
$415$	−6.35650	−0.312028
$416$	2.49467	0.122311
$417$	0	0
$418$	−5.75674	−0.281571
$419$	32.8278	1.60374	0.801871	−	0.597497i	$-0.203838\pi$
$419$	32.8278	1.60374	0.801871	+	0.597497i	$0.203838\pi$
$420$	0	0
$421$	35.9329	1.75126	0.875630	−	0.482982i	$-0.160447\pi$
$421$	35.9329	1.75126	0.875630	+	0.482982i	$0.160447\pi$
$422$	1.98647	0.0966999
$423$	0	0
$424$	−1.84616	−0.0896576
$425$	7.40330	0.359113
$426$	0	0
$427$	−3.47280	−0.168061
$428$	−14.3584	−0.694042
$429$	0	0
$430$	−1.60178	−0.0772449
$431$	24.7729	1.19327	0.596634	−	0.802513i	$-0.296504\pi$
$431$	24.7729	1.19327	0.596634	+	0.802513i	$0.296504\pi$
$432$	0	0
$433$	28.1739	1.35395	0.676976	−	0.736005i	$-0.263290\pi$
$433$	28.1739	1.35395	0.676976	+	0.736005i	$0.263290\pi$
$434$	−0.658316	−0.0316002
$435$	0	0
$436$	−7.74273	−0.370809
$437$	−18.4023	−0.880302
$438$	0	0
$439$	−0.128231	−0.00612014	−0.00306007	−	0.999995i	$-0.500974\pi$
$439$	−0.128231	−0.00612014	−0.00306007	+	0.999995i	$0.500974\pi$
$440$	3.34632	0.159530
$441$	0	0
$442$	1.56644	0.0745077
$443$	−3.96327	−0.188301	−0.0941504	−	0.995558i	$-0.530013\pi$
$443$	−3.96327	−0.188301	−0.0941504	+	0.995558i	$0.530013\pi$
$444$	0	0
$445$	1.00000	0.0474045
$446$	3.63602	0.172170
$447$	0	0
$448$	3.50562	0.165625
$449$	5.39593	0.254650	0.127325	−	0.991861i	$-0.459361\pi$
$449$	5.39593	0.254650	0.127325	+	0.991861i	$0.459361\pi$
$450$	0	0
$451$	−17.7110	−0.833977
$452$	−30.0239	−1.41221
$453$	0	0
$454$	−3.75639	−0.176296
$455$	0.620194	0.0290751
$456$	0	0
$457$	0.665823	0.0311459	0.0155729	−	0.999879i	$-0.495043\pi$
$457$	0.665823	0.0311459	0.0155729	+	0.999879i	$0.495043\pi$
$458$	−0.674189	−0.0315028
$459$	0	0
$460$	5.31313	0.247726
$461$	−42.6998	−1.98873	−0.994365	−	0.106010i	$-0.966192\pi$
$461$	−42.6998	−1.98873	−0.994365	+	0.106010i	$0.966192\pi$
$462$	0	0
$463$	7.05789	0.328008	0.164004	−	0.986460i	$-0.447559\pi$
$463$	7.05789	0.328008	0.164004	+	0.986460i	$0.447559\pi$
$464$	0.0265082	0.00123061
$465$	0	0
$466$	−1.90881	−0.0884239
$467$	30.6233	1.41708	0.708540	−	0.705671i	$-0.249354\pi$
$467$	30.6233	1.41708	0.708540	+	0.705671i	$0.249354\pi$
$468$	0	0
$469$	0.853277	0.0394007
$470$	−0.749920	−0.0345912
$471$	0	0
$472$	0.733027	0.0337403
$473$	51.3047	2.35899
$474$	0	0
$475$	6.83604	0.313659
$476$	6.94478	0.318314
$477$	0	0
$478$	−3.30511	−0.151172
$479$	−24.7778	−1.13213	−0.566063	−	0.824362i	$-0.691534\pi$
$479$	−24.7778	−1.13213	−0.566063	+	0.824362i	$0.691534\pi$
$480$	0	0
$481$	5.95190	0.271383
$482$	1.83984	0.0838026
$483$	0	0
$484$	−31.5255	−1.43298
$485$	18.3113	0.831474
$486$	0	0
$487$	41.0479	1.86006	0.930029	−	0.367486i	$-0.119782\pi$
$487$	41.0479	1.86006	0.930029	+	0.367486i	$0.119782\pi$
$488$	4.70799	0.213121
$489$	0	0
$490$	1.09840	0.0496208
$491$	−32.1154	−1.44935	−0.724674	−	0.689092i	$-0.758010\pi$
$491$	−32.1154	−1.44935	−0.724674	+	0.689092i	$0.758010\pi$
$492$	0	0
$493$	0.0510671	0.00229995
$494$	1.44641	0.0650771
$495$	0	0
$496$	−32.8276	−1.47400
$497$	1.38540	0.0621439
$498$	0	0
$499$	−29.5346	−1.32215	−0.661074	−	0.750321i	$-0.729899\pi$
$499$	−29.5346	−1.32215	−0.661074	+	0.750321i	$0.729899\pi$
$500$	−1.97371	−0.0882669
$501$	0	0
$502$	−2.59459	−0.115802
$503$	−17.9883	−0.802058	−0.401029	−	0.916065i	$-0.631347\pi$
$503$	−17.9883	−0.802058	−0.401029	+	0.916065i	$0.631347\pi$
$504$	0	0
$505$	14.3545	0.638766
$506$	2.26694	0.100778
$507$	0	0
$508$	−28.2402	−1.25295
$509$	−28.2936	−1.25409	−0.627045	−	0.778983i	$-0.715736\pi$
$509$	−28.2936	−1.25409	−0.627045	+	0.778983i	$0.715736\pi$
$510$	0	0
$511$	1.07589	0.0475946
$512$	−12.2990	−0.543546
$513$	0	0
$514$	1.57531	0.0694840
$515$	5.43964	0.239699
$516$	0	0
$517$	24.0197	1.05639
$518$	−0.351510	−0.0154445
$519$	0	0
$520$	−0.840781	−0.0368707
$521$	13.2784	0.581739	0.290870	−	0.956763i	$-0.406055\pi$
$521$	13.2784	0.581739	0.290870	+	0.956763i	$0.406055\pi$
$522$	0	0
$523$	−23.7559	−1.03877	−0.519386	−	0.854540i	$-0.673839\pi$
$523$	−23.7559	−1.03877	−0.519386	+	0.854540i	$0.673839\pi$
$524$	−10.1948	−0.445361
$525$	0	0
$526$	3.75757	0.163838
$527$	−63.2413	−2.75483
$528$	0	0
$529$	−15.7534	−0.684930
$530$	0.464594	0.0201807
$531$	0	0
$532$	6.41265	0.278024
$533$	4.44997	0.192750
$534$	0	0
$535$	7.27486	0.314519
$536$	−1.15677	−0.0499647
$537$	0	0
$538$	−0.799014	−0.0344479
$539$	−35.1815	−1.51538
$540$	0	0
$541$	−12.9184	−0.555404	−0.277702	−	0.960667i	$-0.589573\pi$
$541$	−12.9184	−0.555404	−0.277702	+	0.960667i	$0.589573\pi$
$542$	−2.69087	−0.115583
$543$	0	0
$544$	−14.1534	−0.606824
$545$	3.92293	0.168040
$546$	0	0
$547$	39.0481	1.66958	0.834788	−	0.550571i	$-0.185590\pi$
$547$	39.0481	1.66958	0.834788	+	0.550571i	$0.185590\pi$
$548$	−19.9409	−0.851834
$549$	0	0
$550$	−0.842116	−0.0359079
$551$	0.0471543	0.00200884
$552$	0	0
$553$	4.99540	0.212426
$554$	2.10565	0.0894605
$555$	0	0
$556$	41.4601	1.75830
$557$	−5.51252	−0.233573	−0.116786	−	0.993157i	$-0.537259\pi$
$557$	−5.51252	−0.233573	−0.116786	+	0.993157i	$0.537259\pi$
$558$	0	0
$559$	−12.8906	−0.545213
$560$	−1.82648	−0.0771827
$561$	0	0
$562$	4.77519	0.201429
$563$	37.2064	1.56806	0.784031	−	0.620722i	$-0.213160\pi$
$563$	37.2064	1.56806	0.784031	+	0.620722i	$0.213160\pi$
$564$	0	0
$565$	15.2119	0.639971
$566$	2.41598	0.101551
$567$	0	0
$568$	−1.87816	−0.0788057
$569$	27.0351	1.13337	0.566685	−	0.823935i	$-0.308226\pi$
$569$	27.0351	1.13337	0.566685	+	0.823935i	$0.308226\pi$
$570$	0	0
$571$	32.8494	1.37471	0.687353	−	0.726324i	$-0.258773\pi$
$571$	32.8494	1.37471	0.687353	+	0.726324i	$0.258773\pi$
$572$	13.3759	0.559274
$573$	0	0
$574$	−0.262809	−0.0109694
$575$	−2.69195	−0.112262
$576$	0	0
$577$	35.2109	1.46585	0.732925	−	0.680309i	$-0.238155\pi$
$577$	35.2109	1.46585	0.732925	+	0.680309i	$0.238155\pi$
$578$	−6.13061	−0.255000
$579$	0	0
$580$	−0.0136144	−0.000565308 0
$581$	3.02112	0.125337
$582$	0	0
$583$	−14.8808	−0.616301
$584$	−1.45856	−0.0603555
$585$	0	0
$586$	−0.438730	−0.0181238
$587$	−40.2094	−1.65962	−0.829811	−	0.558045i	$-0.811552\pi$
$587$	−40.2094	−1.65962	−0.829811	+	0.558045i	$0.811552\pi$
$588$	0	0
$589$	−58.3956	−2.40615
$590$	−0.184469	−0.00759448
$591$	0	0
$592$	−17.5284	−0.720412
$593$	−29.5937	−1.21527	−0.607633	−	0.794218i	$-0.707881\pi$
$593$	−29.5937	−1.21527	−0.607633	+	0.794218i	$0.707881\pi$
$594$	0	0
$595$	−3.51865	−0.144250
$596$	−8.99920	−0.368621
$597$	0	0
$598$	−0.569580	−0.0232919
$599$	8.74518	0.357318	0.178659	−	0.983911i	$-0.442824\pi$
$599$	8.74518	0.357318	0.178659	+	0.983911i	$0.442824\pi$
$600$	0	0
$601$	3.42482	0.139701	0.0698507	−	0.997557i	$-0.477748\pi$
$601$	3.42482	0.139701	0.0698507	+	0.997557i	$0.477748\pi$
$602$	0.761297	0.0310282
$603$	0	0
$604$	−7.18569	−0.292382
$605$	15.9727	0.649384
$606$	0	0
$607$	−20.5908	−0.835756	−0.417878	−	0.908503i	$-0.637226\pi$
$607$	−20.5908	−0.835756	−0.417878	+	0.908503i	$0.637226\pi$
$608$	−13.0690	−0.530017
$609$	0	0
$610$	−1.18478	−0.0479705
$611$	−6.03508	−0.244153
$612$	0	0
$613$	−26.4547	−1.06850	−0.534248	−	0.845328i	$-0.679405\pi$
$613$	−26.4547	−1.06850	−0.534248	+	0.845328i	$0.679405\pi$
$614$	2.50149	0.100952
$615$	0	0
$616$	−1.59044	−0.0640808
$617$	43.8302	1.76454	0.882268	−	0.470747i	$-0.156015\pi$
$617$	43.8302	1.76454	0.882268	+	0.470747i	$0.156015\pi$
$618$	0	0
$619$	−0.0432321	−0.00173765	−0.000868823	−	1.00000i	$-0.500277\pi$
$619$	−0.0432321	−0.00173765	−0.000868823		1.00000i	$0.500277\pi$
$620$	16.8600	0.677115
$621$	0	0
$622$	−1.50618	−0.0603922
$623$	−0.475281	−0.0190417
$624$	0	0
$625$	1.00000	0.0400000
$626$	0.902633	0.0360765
$627$	0	0
$628$	−34.2768	−1.36779
$629$	−33.7679	−1.34641
$630$	0	0
$631$	−29.9042	−1.19047	−0.595233	−	0.803553i	$-0.702940\pi$
$631$	−29.9042	−1.19047	−0.595233	+	0.803553i	$0.702940\pi$
$632$	−6.77213	−0.269381
$633$	0	0
$634$	5.01689	0.199246
$635$	14.3082	0.567803
$636$	0	0
$637$	8.83954	0.350235
$638$	−0.00580882	−0.000229973 0
$639$	0	0
$640$	5.01953	0.198414
$641$	16.3131	0.644330	0.322165	−	0.946684i	$-0.395589\pi$
$641$	16.3131	0.644330	0.322165	+	0.946684i	$0.395589\pi$
$642$	0	0
$643$	17.2134	0.678832	0.339416	−	0.940636i	$-0.389771\pi$
$643$	17.2134	0.678832	0.339416	+	0.940636i	$0.389771\pi$
$644$	−2.52523	−0.0995080
$645$	0	0
$646$	−8.20616	−0.322867
$647$	−28.3178	−1.11329	−0.556645	−	0.830751i	$-0.687912\pi$
$647$	−28.3178	−1.11329	−0.556645	+	0.830751i	$0.687912\pi$
$648$	0	0
$649$	5.90850	0.231929
$650$	0.211586	0.00829909
$651$	0	0
$652$	17.8999	0.701015
$653$	−27.2771	−1.06743	−0.533717	−	0.845663i	$-0.679205\pi$
$653$	−27.2771	−1.06743	−0.533717	+	0.845663i	$0.679205\pi$
$654$	0	0
$655$	5.16529	0.201825
$656$	−13.1052	−0.511672
$657$	0	0
$658$	0.356422	0.0138948
$659$	36.6207	1.42654	0.713270	−	0.700889i	$-0.247213\pi$
$659$	36.6207	1.42654	0.713270	+	0.700889i	$0.247213\pi$
$660$	0	0
$661$	−39.6430	−1.54193	−0.770967	−	0.636875i	$-0.780227\pi$
$661$	−39.6430	−1.54193	−0.770967	+	0.636875i	$0.780227\pi$
$662$	−0.0119091	−0.000462859 0
$663$	0	0
$664$	−4.09566	−0.158942
$665$	−3.24904	−0.125992
$666$	0	0
$667$	−0.0185688	−0.000718987 0
$668$	8.46838	0.327652
$669$	0	0
$670$	0.291105	0.0112464
$671$	37.9483	1.46498
$672$	0	0
$673$	7.70607	0.297047	0.148524	−	0.988909i	$-0.452548\pi$
$673$	7.70607	0.297047	0.148524	+	0.988909i	$0.452548\pi$
$674$	−2.75032	−0.105938
$675$	0	0
$676$	22.2974	0.857594
$677$	16.9048	0.649706	0.324853	−	0.945765i	$-0.394685\pi$
$677$	16.9048	0.649706	0.324853	+	0.945765i	$0.394685\pi$
$678$	0	0
$679$	−8.70302	−0.333991
$680$	4.77014	0.182926
$681$	0	0
$682$	7.19361	0.275458
$683$	7.11167	0.272120	0.136060	−	0.990701i	$-0.456556\pi$
$683$	7.11167	0.272120	0.136060	+	0.990701i	$0.456556\pi$
$684$	0	0
$685$	10.1033	0.386026
$686$	−1.06151	−0.0405285
$687$	0	0
$688$	37.9628	1.44732
$689$	3.73888	0.142440
$690$	0	0
$691$	18.0632	0.687156	0.343578	−	0.939124i	$-0.388361\pi$
$691$	18.0632	0.687156	0.343578	+	0.939124i	$0.388361\pi$
$692$	41.8366	1.59039
$693$	0	0
$694$	−3.44110	−0.130622
$695$	−21.0062	−0.796811
$696$	0	0
$697$	−25.2468	−0.956289
$698$	−1.21334	−0.0459258
$699$	0	0
$700$	0.938065	0.0354555
$701$	−14.5894	−0.551034	−0.275517	−	0.961296i	$-0.588849\pi$
$701$	−14.5894	−0.551034	−0.275517	+	0.961296i	$0.588849\pi$
$702$	0	0
$703$	−31.1805	−1.17599
$704$	−38.3069	−1.44375
$705$	0	0
$706$	0.339658	0.0127832
$707$	−6.82240	−0.256583
$708$	0	0
$709$	19.9577	0.749527	0.374763	−	0.927120i	$-0.377724\pi$
$709$	19.9577	0.749527	0.374763	+	0.927120i	$0.377724\pi$
$710$	0.472646	0.0177381
$711$	0	0
$712$	0.644326	0.0241471
$713$	22.9955	0.861188
$714$	0	0
$715$	−6.77704	−0.253447
$716$	0.925652	0.0345932
$717$	0	0
$718$	−0.255937	−0.00955147
$719$	1.62038	0.0604298	0.0302149	−	0.999543i	$-0.490381\pi$
$719$	1.62038	0.0604298	0.0302149	+	0.999543i	$0.490381\pi$
$720$	0	0
$721$	−2.58535	−0.0962836
$722$	−4.49658	−0.167345
$723$	0	0
$724$	43.9608	1.63379
$725$	0.00689789	0.000256181 0
$726$	0	0
$727$	5.61596	0.208284	0.104142	−	0.994562i	$-0.466790\pi$
$727$	5.61596	0.208284	0.104142	+	0.994562i	$0.466790\pi$
$728$	0.399607	0.0148104
$729$	0	0
$730$	0.367052	0.0135852
$731$	73.1342	2.70496
$732$	0	0
$733$	48.1350	1.77791	0.888953	−	0.457998i	$-0.151433\pi$
$733$	48.1350	1.77791	0.888953	+	0.457998i	$0.151433\pi$
$734$	−5.62485	−0.207617
$735$	0	0
$736$	5.14641	0.189699
$737$	−9.32400	−0.343454
$738$	0	0
$739$	−8.98910	−0.330669	−0.165335	−	0.986238i	$-0.552870\pi$
$739$	−8.98910	−0.330669	−0.165335	+	0.986238i	$0.552870\pi$
$740$	9.00245	0.330937
$741$	0	0
$742$	−0.220813	−0.00810629
$743$	−2.20072	−0.0807365	−0.0403683	−	0.999185i	$-0.512853\pi$
$743$	−2.20072	−0.0807365	−0.0403683	+	0.999185i	$0.512853\pi$
$744$	0	0
$745$	4.55954	0.167049
$746$	−4.60272	−0.168518
$747$	0	0
$748$	−75.8876	−2.77473
$749$	−3.45760	−0.126338
$750$	0	0
$751$	−22.7671	−0.830781	−0.415391	−	0.909643i	$-0.636355\pi$
$751$	−22.7671	−0.830781	−0.415391	+	0.909643i	$0.636355\pi$
$752$	17.7733	0.648127
$753$	0	0
$754$	0.00145950	5.31517e−5 0
$755$	3.64071	0.132499
$756$	0	0
$757$	−30.2022	−1.09772	−0.548859	−	0.835915i	$-0.684937\pi$
$757$	−30.2022	−1.09772	−0.548859	+	0.835915i	$0.684937\pi$
$758$	2.49170	0.0905026
$759$	0	0
$760$	4.40464	0.159773
$761$	−17.4802	−0.633656	−0.316828	−	0.948483i	$-0.602618\pi$
$761$	−17.4802	−0.633656	−0.316828	+	0.948483i	$0.602618\pi$
$762$	0	0
$763$	−1.86449	−0.0674992
$764$	−42.0542	−1.52147
$765$	0	0
$766$	−3.86703	−0.139722
$767$	−1.48454	−0.0536036
$768$	0	0
$769$	−28.0588	−1.01183	−0.505913	−	0.862585i	$-0.668844\pi$
$769$	−28.0588	−1.01183	−0.505913	+	0.862585i	$0.668844\pi$
$770$	0.400241	0.0144237
$771$	0	0
$772$	2.64126	0.0950612
$773$	2.19223	0.0788489	0.0394245	−	0.999223i	$-0.487448\pi$
$773$	2.19223	0.0788489	0.0394245	+	0.999223i	$0.487448\pi$
$774$	0	0
$775$	−8.54230	−0.306849
$776$	11.7985	0.423540
$777$	0	0
$778$	1.90796	0.0684038
$779$	−23.3123	−0.835249
$780$	0	0
$781$	−15.1387	−0.541706
$782$	3.23149	0.115558
$783$	0	0
$784$	−26.0325	−0.929732
$785$	17.3667	0.619843
$786$	0	0
$787$	−36.7899	−1.31142	−0.655709	−	0.755013i	$-0.727630\pi$
$787$	−36.7899	−1.31142	−0.655709	+	0.755013i	$0.727630\pi$
$788$	−19.0652	−0.679170
$789$	0	0
$790$	1.70423	0.0606339
$791$	−7.22994	−0.257067
$792$	0	0
$793$	−9.53471	−0.338587
$794$	1.82867	0.0648971
$795$	0	0
$796$	−19.4556	−0.689585
$797$	42.4563	1.50388	0.751941	−	0.659231i	$-0.229118\pi$
$797$	42.4563	1.50388	0.751941	+	0.659231i	$0.229118\pi$
$798$	0	0
$799$	34.2398	1.21132
$800$	−1.91177	−0.0675914
$801$	0	0
$802$	2.10655	0.0743850
$803$	−11.7566	−0.414880
$804$	0	0
$805$	1.27943	0.0450941
$806$	−1.80743	−0.0636641
$807$	0	0
$808$	9.24896	0.325377
$809$	−28.4853	−1.00149	−0.500744	−	0.865595i	$-0.666940\pi$
$809$	−28.4853	−1.00149	−0.500744	+	0.865595i	$0.666940\pi$
$810$	0	0
$811$	−17.7783	−0.624282	−0.312141	−	0.950036i	$-0.601046\pi$
$811$	−17.7783	−0.624282	−0.312141	+	0.950036i	$0.601046\pi$
$812$	0.00647067	0.000227076 0
$813$	0	0
$814$	3.84105	0.134629
$815$	−9.06918	−0.317680
$816$	0	0
$817$	67.5304	2.36259
$818$	2.30570	0.0806169
$819$	0	0
$820$	6.73074	0.235048
$821$	−31.1566	−1.08737	−0.543687	−	0.839288i	$-0.682972\pi$
$821$	−31.1566	−1.08737	−0.543687	+	0.839288i	$0.682972\pi$
$822$	0	0
$823$	20.4644	0.713346	0.356673	−	0.934229i	$-0.383911\pi$
$823$	20.4644	0.713346	0.356673	+	0.934229i	$0.383911\pi$
$824$	3.50490	0.122099
$825$	0	0
$826$	0.0876747	0.00305059
$827$	16.0701	0.558812	0.279406	−	0.960173i	$-0.409862\pi$
$827$	16.0701	0.558812	0.279406	+	0.960173i	$0.409862\pi$
$828$	0	0
$829$	−34.1508	−1.18611	−0.593053	−	0.805163i	$-0.702078\pi$
$829$	−34.1508	−1.18611	−0.593053	+	0.805163i	$0.702078\pi$
$830$	1.03069	0.0357757
$831$	0	0
$832$	9.62481	0.333680
$833$	−50.1508	−1.73762
$834$	0	0
$835$	−4.29059	−0.148482
$836$	−70.0729	−2.42352
$837$	0	0
$838$	−5.32293	−0.183878
$839$	2.87004	0.0990846	0.0495423	−	0.998772i	$-0.484224\pi$
$839$	2.87004	0.0990846	0.0495423	+	0.998772i	$0.484224\pi$
$840$	0	0
$841$	−29.0000	−0.999998
$842$	−5.82641	−0.200791
$843$	0	0
$844$	24.1800	0.832309
$845$	−11.2972	−0.388637
$846$	0	0
$847$	−7.59153	−0.260848
$848$	−11.0110	−0.378121
$849$	0	0
$850$	−1.20042	−0.0411742
$851$	12.2785	0.420902
$852$	0	0
$853$	22.6448	0.775344	0.387672	−	0.921797i	$-0.373279\pi$
$853$	22.6448	0.775344	0.387672	+	0.921797i	$0.373279\pi$
$854$	0.563105	0.0192691
$855$	0	0
$856$	4.68738	0.160211
$857$	−22.5002	−0.768594	−0.384297	−	0.923210i	$-0.625556\pi$
$857$	−22.5002	−0.768594	−0.384297	+	0.923210i	$0.625556\pi$
$858$	0	0
$859$	−38.7537	−1.32226	−0.661129	−	0.750272i	$-0.729923\pi$
$859$	−38.7537	−1.32226	−0.661129	+	0.750272i	$0.729923\pi$
$860$	−19.4974	−0.664857
$861$	0	0
$862$	−4.01686	−0.136815
$863$	−29.5580	−1.00617	−0.503083	−	0.864238i	$-0.667801\pi$
$863$	−29.5580	−1.00617	−0.503083	+	0.864238i	$0.667801\pi$
$864$	0	0
$865$	−21.1969	−0.720718
$866$	−4.56832	−0.155238
$867$	0	0
$868$	−8.01324	−0.271987
$869$	−54.5861	−1.85171
$870$	0	0
$871$	2.34270	0.0793795
$872$	2.52765	0.0855969
$873$	0	0
$874$	2.98388	0.100931
$875$	−0.475281	−0.0160674
$876$	0	0
$877$	−0.306586	−0.0103527	−0.00517633	−	0.999987i	$-0.501648\pi$
$877$	−0.306586	−0.0103527	−0.00517633	+	0.999987i	$0.501648\pi$
$878$	0.0207923	0.000701707 0
$879$	0	0
$880$	19.9584	0.672798
$881$	−4.86851	−0.164024	−0.0820121	−	0.996631i	$-0.526135\pi$
$881$	−4.86851	−0.164024	−0.0820121	+	0.996631i	$0.526135\pi$
$882$	0	0
$883$	54.1155	1.82113	0.910565	−	0.413366i	$-0.135647\pi$
$883$	54.1155	1.82113	0.910565	+	0.413366i	$0.135647\pi$
$884$	19.0672	0.641298
$885$	0	0
$886$	0.642634	0.0215897
$887$	42.0954	1.41343	0.706713	−	0.707500i	$-0.250177\pi$
$887$	42.0954	1.41343	0.706713	+	0.707500i	$0.250177\pi$
$888$	0	0
$889$	−6.80040	−0.228078
$890$	−0.162147	−0.00543519
$891$	0	0
$892$	44.2588	1.48189
$893$	31.6162	1.05800
$894$	0	0
$895$	−0.468991	−0.0156767
$896$	−2.38568	−0.0797001
$897$	0	0
$898$	−0.874935	−0.0291970
$899$	−0.0589238	−0.00196522
$900$	0	0
$901$	−21.2124	−0.706688
$902$	2.87178	0.0956200
$903$	0	0
$904$	9.80144	0.325991
$905$	−22.2732	−0.740386
$906$	0	0
$907$	−3.59534	−0.119381	−0.0596907	−	0.998217i	$-0.519011\pi$
$907$	−3.59534	−0.119381	−0.0596907	+	0.998217i	$0.519011\pi$
$908$	−45.7240	−1.51740
$909$	0	0
$910$	−0.100563	−0.00333362
$911$	−51.3001	−1.69965	−0.849823	−	0.527068i	$-0.823291\pi$
$911$	−51.3001	−1.69965	−0.849823	+	0.527068i	$0.823291\pi$
$912$	0	0
$913$	−33.0127	−1.09256
$914$	−0.107961	−0.00357104
$915$	0	0
$916$	−8.20645	−0.271149
$917$	−2.45496	−0.0810700
$918$	0	0
$919$	−24.6752	−0.813959	−0.406979	−	0.913437i	$-0.633418\pi$
$919$	−24.6752	−0.813959	−0.406979	+	0.913437i	$0.633418\pi$
$920$	−1.73450	−0.0571846
$921$	0	0
$922$	6.92366	0.228019
$923$	3.80368	0.125200
$924$	0	0
$925$	−4.56119	−0.149971
$926$	−1.14442	−0.0376079
$927$	0	0
$928$	−0.0131872	−0.000432891 0
$929$	6.23375	0.204523	0.102261	−	0.994758i	$-0.467392\pi$
$929$	6.23375	0.204523	0.102261	+	0.994758i	$0.467392\pi$
$930$	0	0
$931$	−46.3081	−1.51769
$932$	−23.2347	−0.761076
$933$	0	0
$934$	−4.96549	−0.162476
$935$	38.4493	1.25743
$936$	0	0
$937$	−10.7283	−0.350479	−0.175240	−	0.984526i	$-0.556070\pi$
$937$	−10.7283	−0.350479	−0.175240	+	0.984526i	$0.556070\pi$
$938$	−0.138356	−0.00451750
$939$	0	0
$940$	−9.12827	−0.297731
$941$	−11.6152	−0.378644	−0.189322	−	0.981915i	$-0.560629\pi$
$941$	−11.6152	−0.378644	−0.189322	+	0.981915i	$0.560629\pi$
$942$	0	0
$943$	9.18011	0.298945
$944$	4.37198	0.142296
$945$	0	0
$946$	−8.31891	−0.270471
$947$	−9.67739	−0.314473	−0.157236	−	0.987561i	$-0.550258\pi$
$947$	−9.67739	−0.314473	−0.157236	+	0.987561i	$0.550258\pi$
$948$	0	0
$949$	2.95390	0.0958875
$950$	−1.10845	−0.0359627
$951$	0	0
$952$	−2.26715	−0.0734789
$953$	−18.4382	−0.597273	−0.298636	−	0.954367i	$-0.596532\pi$
$953$	−18.4382	−0.597273	−0.298636	+	0.954367i	$0.596532\pi$
$954$	0	0
$955$	21.3072	0.689485
$956$	−40.2309	−1.30116
$957$	0	0
$958$	4.01765	0.129804
$959$	−4.80189	−0.155061
$960$	0	0
$961$	41.9709	1.35390
$962$	−0.965083	−0.0311155
$963$	0	0
$964$	22.3952	0.721300
$965$	−1.33822	−0.0430790
$966$	0	0
$967$	−45.7382	−1.47084	−0.735420	−	0.677611i	$-0.763015\pi$
$967$	−45.7382	−1.47084	−0.735420	+	0.677611i	$0.763015\pi$
$968$	10.2916	0.330786
$969$	0	0
$970$	−2.96913	−0.0953330
$971$	18.8604	0.605259	0.302629	−	0.953108i	$-0.402136\pi$
$971$	18.8604	0.605259	0.302629	+	0.953108i	$0.402136\pi$
$972$	0	0
$973$	9.98384	0.320067
$974$	−6.65580	−0.213266
$975$	0	0
$976$	28.0798	0.898812
$977$	5.89342	0.188547	0.0942736	−	0.995546i	$-0.469947\pi$
$977$	5.89342	0.188547	0.0942736	+	0.995546i	$0.469947\pi$
$978$	0	0
$979$	5.19353	0.165986
$980$	13.3701	0.427093
$981$	0	0
$982$	5.20742	0.166175
$983$	−2.88660	−0.0920683	−0.0460341	−	0.998940i	$-0.514658\pi$
$983$	−2.88660	−0.0920683	−0.0460341	+	0.998940i	$0.514658\pi$
$984$	0	0
$985$	9.65959	0.307780
$986$	−0.00828040	−0.000263702 0
$987$	0	0
$988$	17.6062	0.560127
$989$	−26.5927	−0.845599
$990$	0	0
$991$	15.8725	0.504206	0.252103	−	0.967700i	$-0.418878\pi$
$991$	15.8725	0.504206	0.252103	+	0.967700i	$0.418878\pi$
$992$	16.3309	0.518508
$993$	0	0
$994$	−0.224639	−0.00712513
$995$	9.85738	0.312500
$996$	0	0
$997$	−28.4316	−0.900437	−0.450218	−	0.892918i	$-0.648654\pi$
$997$	−28.4316	−0.900437	−0.450218	+	0.892918i	$0.648654\pi$
$998$	4.78895	0.151591
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4005.2.a.l.1.3		4
3.2	odd	2		445.2.a.d.1.2	✓	4
12.11	even	2		7120.2.a.bc.1.1		4
15.14	odd	2		2225.2.a.i.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
445.2.a.d.1.2	✓	4	3.2	odd	2
2225.2.a.i.1.3		4	15.14	odd	2
4005.2.a.l.1.3		4	1.1	even	1	trivial
7120.2.a.bc.1.1		4	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)

\(f(q)\)	\(=\)	\(q-0.162147 q^{2} -1.97371 q^{4} +1.00000 q^{5} -0.475281 q^{7} +0.644326 q^{8} +O(q^{10})\)	\(q-0.162147 q^{2} -1.97371 q^{4} +1.00000 q^{5} -0.475281 q^{7} +0.644326 q^{8} -0.162147 q^{10} +5.19353 q^{11} -1.30490 q^{13} +0.0770654 q^{14} +3.84294 q^{16} +7.40330 q^{17} +6.83604 q^{19} -1.97371 q^{20} -0.842116 q^{22} -2.69195 q^{23} +1.00000 q^{25} +0.211586 q^{26} +0.938065 q^{28} +0.00689789 q^{29} -8.54230 q^{31} -1.91177 q^{32} -1.20042 q^{34} -0.475281 q^{35} -4.56119 q^{37} -1.10845 q^{38} +0.644326 q^{40} -3.41020 q^{41} +9.87858 q^{43} -10.2505 q^{44} +0.436493 q^{46} +4.62493 q^{47} -6.77411 q^{49} -0.162147 q^{50} +2.57549 q^{52} -2.86526 q^{53} +5.19353 q^{55} -0.306236 q^{56} -0.00111847 q^{58} +1.13767 q^{59} +7.30684 q^{61} +1.38511 q^{62} -7.37589 q^{64} -1.30490 q^{65} -1.79531 q^{67} -14.6120 q^{68} +0.0770654 q^{70} -2.91492 q^{71} -2.26369 q^{73} +0.739584 q^{74} -13.4924 q^{76} -2.46838 q^{77} -10.5104 q^{79} +3.84294 q^{80} +0.552955 q^{82} -6.35650 q^{83} +7.40330 q^{85} -1.60178 q^{86} +3.34632 q^{88} +1.00000 q^{89} +0.620194 q^{91} +5.31313 q^{92} -0.749920 q^{94} +6.83604 q^{95} +18.3113 q^{97} +1.09840 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{2} + 3 q^{4} + 4 q^{5} + 2 q^{7} + 9 q^{8}+O(q^{10}) \) 4 * q - q^2 + 3 * q^4 + 4 * q^5 + 2 * q^7 + 9 * q^8	\( 4 q - q^{2} + 3 q^{4} + 4 q^{5} + 2 q^{7} + 9 q^{8} - q^{10} + 14 q^{11} - 5 q^{13} + 5 q^{14} - 3 q^{16} + 3 q^{17} - q^{19} + 3 q^{20} + 2 q^{22} + 3 q^{23} + 4 q^{25} + 9 q^{26} + 5 q^{28} + 10 q^{29} - 11 q^{31} + 2 q^{32} - 8 q^{34} + 2 q^{35} + 3 q^{37} - 2 q^{38} + 9 q^{40} + 3 q^{41} + 9 q^{43} + 9 q^{44} - 4 q^{46} + 24 q^{47} - q^{50} + 8 q^{52} - 3 q^{53} + 14 q^{55} + 13 q^{56} + 19 q^{58} + 22 q^{59} - 3 q^{61} + 24 q^{62} - 11 q^{64} - 5 q^{65} - 9 q^{67} - 31 q^{68} + 5 q^{70} - 16 q^{71} + 3 q^{73} - 31 q^{74} - 24 q^{76} + 4 q^{77} - 27 q^{79} - 3 q^{80} - 15 q^{82} - 6 q^{83} + 3 q^{85} - 15 q^{86} + 30 q^{88} + 4 q^{89} - 29 q^{91} + 17 q^{92} + 13 q^{94} - q^{95} + 41 q^{97} - 22 q^{98}+O(q^{100}) \) 4 * q - q^2 + 3 * q^4 + 4 * q^5 + 2 * q^7 + 9 * q^8 - q^10 + 14 * q^11 - 5 * q^13 + 5 * q^14 - 3 * q^16 + 3 * q^17 - q^19 + 3 * q^20 + 2 * q^22 + 3 * q^23 + 4 * q^25 + 9 * q^26 + 5 * q^28 + 10 * q^29 - 11 * q^31 + 2 * q^32 - 8 * q^34 + 2 * q^35 + 3 * q^37 - 2 * q^38 + 9 * q^40 + 3 * q^41 + 9 * q^43 + 9 * q^44 - 4 * q^46 + 24 * q^47 - q^50 + 8 * q^52 - 3 * q^53 + 14 * q^55 + 13 * q^56 + 19 * q^58 + 22 * q^59 - 3 * q^61 + 24 * q^62 - 11 * q^64 - 5 * q^65 - 9 * q^67 - 31 * q^68 + 5 * q^70 - 16 * q^71 + 3 * q^73 - 31 * q^74 - 24 * q^76 + 4 * q^77 - 27 * q^79 - 3 * q^80 - 15 * q^82 - 6 * q^83 + 3 * q^85 - 15 * q^86 + 30 * q^88 + 4 * q^89 - 29 * q^91 + 17 * q^92 + 13 * q^94 - q^95 + 41 * q^97 - 22 * q^98

Introduction

L-functions

Modular forms

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