LMFDB - Embedded newform 4005.2.a.w.1.5

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:	$17$
Coefficient field:	$\mathbb{Q}[x]/(x^{17} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{17} - 5 x^{16} - 15 x^{15} + 105 x^{14} + 45 x^{13} - 849 x^{12} + 320 x^{11} + 3371 x^{10} + \cdots + 24 $ x^17 - 5x^16 - 15x^15 + 105x^14 + 45x^13 - 849x^12 + 320x^11 + 3371x^10 - 2456x^9 - 7002x^8 + 6279x^7 + 7299x^6 - 7119x^5 - 3066x^4 + 3184x^3 + 99x^2 - 231x + 24
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$1.89951$ 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)

$f(q)$	$=$	$q-1.89951 q^{2} +1.60815 q^{4} -1.00000 q^{5} +2.75204 q^{7} +0.744322 q^{8} +O(q^{10})$	\(q-1.89951 q^{2} +1.60815 q^{4} -1.00000 q^{5} +2.75204 q^{7} +0.744322 q^{8} +1.89951 q^{10} -3.41222 q^{11} +3.95657 q^{13} -5.22753 q^{14} -4.63015 q^{16} +4.75130 q^{17} +8.61014 q^{19} -1.60815 q^{20} +6.48156 q^{22} -0.0307956 q^{23} +1.00000 q^{25} -7.51555 q^{26} +4.42569 q^{28} +6.50199 q^{29} -4.13065 q^{31} +7.30639 q^{32} -9.02516 q^{34} -2.75204 q^{35} +9.66032 q^{37} -16.3551 q^{38} -0.744322 q^{40} -5.42123 q^{41} -2.68962 q^{43} -5.48737 q^{44} +0.0584967 q^{46} +6.91513 q^{47} +0.573715 q^{49} -1.89951 q^{50} +6.36276 q^{52} -6.20902 q^{53} +3.41222 q^{55} +2.04840 q^{56} -12.3506 q^{58} -7.93018 q^{59} +0.746311 q^{61} +7.84622 q^{62} -4.61828 q^{64} -3.95657 q^{65} -3.61139 q^{67} +7.64081 q^{68} +5.22753 q^{70} +10.8219 q^{71} +6.97045 q^{73} -18.3499 q^{74} +13.8464 q^{76} -9.39056 q^{77} +10.1580 q^{79} +4.63015 q^{80} +10.2977 q^{82} +13.2235 q^{83} -4.75130 q^{85} +5.10897 q^{86} -2.53979 q^{88} +1.00000 q^{89} +10.8886 q^{91} -0.0495240 q^{92} -13.1354 q^{94} -8.61014 q^{95} -14.3084 q^{97} -1.08978 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 17 q - 5 q^{2} + 21 q^{4} - 17 q^{5} + 12 q^{7} - 15 q^{8}+O(q^{10}) $ 17 * q - 5 * q^2 + 21 * q^4 - 17 * q^5 + 12 * q^7 - 15 * q^8	$ 17 q - 5 q^{2} + 21 q^{4} - 17 q^{5} + 12 q^{7} - 15 q^{8} + 5 q^{10} - 2 q^{11} + 8 q^{13} - 4 q^{14} + 33 q^{16} - 10 q^{17} + 32 q^{19} - 21 q^{20} + 8 q^{22} - 15 q^{23} + 17 q^{25} + 15 q^{26} + 24 q^{28} - q^{29} + 18 q^{31} - 25 q^{32} + 14 q^{34} - 12 q^{35} + 12 q^{37} - 22 q^{38} + 15 q^{40} + 7 q^{41} + 28 q^{43} + 14 q^{44} + 4 q^{46} - 26 q^{47} + 41 q^{49} - 5 q^{50} + 10 q^{52} - 12 q^{53} + 2 q^{55} - 13 q^{56} + 16 q^{58} + 23 q^{59} + 26 q^{61} - 10 q^{62} + 59 q^{64} - 8 q^{65} + 31 q^{67} + q^{68} + 4 q^{70} + 2 q^{71} + 33 q^{73} + 10 q^{74} + 66 q^{76} - 12 q^{77} + 33 q^{79} - 33 q^{80} + 30 q^{82} - 13 q^{83} + 10 q^{85} + 20 q^{86} + 12 q^{88} + 17 q^{89} + 40 q^{91} - 16 q^{92} + 38 q^{94} - 32 q^{95} + 45 q^{97} - 2 q^{98}+O(q^{100}) $ 17 * q - 5 * q^2 + 21 * q^4 - 17 * q^5 + 12 * q^7 - 15 * q^8 + 5 * q^10 - 2 * q^11 + 8 * q^13 - 4 * q^14 + 33 * q^16 - 10 * q^17 + 32 * q^19 - 21 * q^20 + 8 * q^22 - 15 * q^23 + 17 * q^25 + 15 * q^26 + 24 * q^28 - q^29 + 18 * q^31 - 25 * q^32 + 14 * q^34 - 12 * q^35 + 12 * q^37 - 22 * q^38 + 15 * q^40 + 7 * q^41 + 28 * q^43 + 14 * q^44 + 4 * q^46 - 26 * q^47 + 41 * q^49 - 5 * q^50 + 10 * q^52 - 12 * q^53 + 2 * q^55 - 13 * q^56 + 16 * q^58 + 23 * q^59 + 26 * q^61 - 10 * q^62 + 59 * q^64 - 8 * q^65 + 31 * q^67 + q^68 + 4 * q^70 + 2 * q^71 + 33 * q^73 + 10 * q^74 + 66 * q^76 - 12 * q^77 + 33 * q^79 - 33 * q^80 + 30 * q^82 - 13 * q^83 + 10 * q^85 + 20 * q^86 + 12 * q^88 + 17 * q^89 + 40 * q^91 - 16 * q^92 + 38 * q^94 - 32 * q^95 + 45 * q^97 - 2 * 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 $

$2$	−1.89951	−1.34316	−0.671579	−	0.740933i	$-0.734384\pi$
$2$	−1.89951	−1.34316	−0.671579	+	0.740933i	$0.734384\pi$
$3$	0	0
$3$	0	0
$4$	1.60815	0.804076
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	2.75204	1.04017	0.520086	−	0.854114i	$-0.325900\pi$
$7$	2.75204	1.04017	0.520086	+	0.854114i	$0.325900\pi$
$8$	0.744322	0.263158
$9$	0	0
$10$	1.89951	0.600679
$11$	−3.41222	−1.02882	−0.514412	−	0.857543i	$-0.671990\pi$
$11$	−3.41222	−1.02882	−0.514412	+	0.857543i	$0.671990\pi$
$12$	0	0
$13$	3.95657	1.09735	0.548677	−	0.836034i	$-0.315132\pi$
$13$	3.95657	1.09735	0.548677	+	0.836034i	$0.315132\pi$
$14$	−5.22753	−1.39712
$15$	0	0
$16$	−4.63015	−1.15754
$17$	4.75130	1.15236	0.576180	−	0.817323i	$-0.304543\pi$
$17$	4.75130	1.15236	0.576180	+	0.817323i	$0.304543\pi$
$18$	0	0
$19$	8.61014	1.97530	0.987651	−	0.156668i	$-0.0500754\pi$
$19$	8.61014	1.97530	0.987651	+	0.156668i	$0.0500754\pi$
$20$	−1.60815	−0.359594
$21$	0	0
$22$	6.48156	1.38187
$23$	−0.0307956	−0.00642133	−0.00321066	−	0.999995i	$-0.501022\pi$
$23$	−0.0307956	−0.00642133	−0.00321066	+	0.999995i	$0.501022\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−7.51555	−1.47392
$27$	0	0
$28$	4.42569	0.836377
$29$	6.50199	1.20739	0.603694	−	0.797216i	$-0.293695\pi$
$29$	6.50199	1.20739	0.603694	+	0.797216i	$0.293695\pi$
$30$	0	0
$31$	−4.13065	−0.741887	−0.370943	−	0.928656i	$-0.620966\pi$
$31$	−4.13065	−0.741887	−0.370943	+	0.928656i	$0.620966\pi$
$32$	7.30639	1.29160
$33$	0	0
$34$	−9.02516	−1.54780
$35$	−2.75204	−0.465179
$36$	0	0
$37$	9.66032	1.58815	0.794073	−	0.607822i	$-0.207957\pi$
$37$	9.66032	1.58815	0.794073	+	0.607822i	$0.207957\pi$
$38$	−16.3551	−2.65314
$39$	0	0
$40$	−0.744322	−0.117688
$41$	−5.42123	−0.846654	−0.423327	−	0.905977i	$-0.639138\pi$
$41$	−5.42123	−0.846654	−0.423327	+	0.905977i	$0.639138\pi$
$42$	0	0
$43$	−2.68962	−0.410163	−0.205082	−	0.978745i	$-0.565746\pi$
$43$	−2.68962	−0.410163	−0.205082	+	0.978745i	$0.565746\pi$
$44$	−5.48737	−0.827252
$45$	0	0
$46$	0.0584967	0.00862486
$47$	6.91513	1.00868	0.504338	−	0.863506i	$-0.331737\pi$
$47$	6.91513	1.00868	0.504338	+	0.863506i	$0.331737\pi$
$48$	0	0
$49$	0.573715	0.0819594
$50$	−1.89951	−0.268632
$51$	0	0
$52$	6.36276	0.882356
$53$	−6.20902	−0.852874	−0.426437	−	0.904517i	$-0.640231\pi$
$53$	−6.20902	−0.852874	−0.426437	+	0.904517i	$0.640231\pi$
$54$	0	0
$55$	3.41222	0.460104
$56$	2.04840	0.273729
$57$	0	0
$58$	−12.3506	−1.62171
$59$	−7.93018	−1.03242	−0.516211	−	0.856462i	$-0.672658\pi$
$59$	−7.93018	−1.03242	−0.516211	+	0.856462i	$0.672658\pi$
$60$	0	0
$61$	0.746311	0.0955553	0.0477777	−	0.998858i	$-0.484786\pi$
$61$	0.746311	0.0955553	0.0477777	+	0.998858i	$0.484786\pi$
$62$	7.84622	0.996471
$63$	0	0
$64$	−4.61828	−0.577286
$65$	−3.95657	−0.490752
$66$	0	0
$67$	−3.61139	−0.441202	−0.220601	−	0.975364i	$-0.570802\pi$
$67$	−3.61139	−0.441202	−0.220601	+	0.975364i	$0.570802\pi$
$68$	7.64081	0.926584
$69$	0	0
$70$	5.22753	0.624810
$71$	10.8219	1.28432	0.642159	−	0.766571i	$-0.278039\pi$
$71$	10.8219	1.28432	0.642159	+	0.766571i	$0.278039\pi$
$72$	0	0
$73$	6.97045	0.815829	0.407914	−	0.913020i	$-0.366256\pi$
$73$	6.97045	0.815829	0.407914	+	0.913020i	$0.366256\pi$
$74$	−18.3499	−2.13313
$75$	0	0
$76$	13.8464	1.58829
$77$	−9.39056	−1.07015
$78$	0	0
$79$	10.1580	1.14286	0.571432	−	0.820650i	$-0.306388\pi$
$79$	10.1580	1.14286	0.571432	+	0.820650i	$0.306388\pi$
$80$	4.63015	0.517667
$81$	0	0
$82$	10.2977	1.13719
$83$	13.2235	1.45147	0.725734	−	0.687976i	$-0.241500\pi$
$83$	13.2235	1.45147	0.725734	+	0.687976i	$0.241500\pi$
$84$	0	0
$85$	−4.75130	−0.515351
$86$	5.10897	0.550915
$87$	0	0
$88$	−2.53979	−0.270743
$89$	1.00000	0.106000
$89$	1.00000	0.106000
$90$	0	0
$91$	10.8886	1.14144
$92$	−0.0495240	−0.00516323
$93$	0	0
$94$	−13.1354	−1.35481
$95$	−8.61014	−0.883382
$96$	0	0
$97$	−14.3084	−1.45280	−0.726401	−	0.687271i	$-0.758809\pi$
$97$	−14.3084	−1.45280	−0.726401	+	0.687271i	$0.758809\pi$
$98$	−1.08978	−0.110084
$99$	0	0
$100$	1.60815	0.160815
$101$	3.39493	0.337808	0.168904	−	0.985632i	$-0.445977\pi$
$101$	3.39493	0.337808	0.168904	+	0.985632i	$0.445977\pi$
$102$	0	0
$103$	14.2785	1.40691	0.703454	−	0.710741i	$-0.251640\pi$
$103$	14.2785	1.40691	0.703454	+	0.710741i	$0.251640\pi$
$104$	2.94496	0.288777
$105$	0	0
$106$	11.7941	1.14554
$107$	−13.1543	−1.27168	−0.635839	−	0.771822i	$-0.719346\pi$
$107$	−13.1543	−1.27168	−0.635839	+	0.771822i	$0.719346\pi$
$108$	0	0
$109$	−7.74090	−0.741443	−0.370722	−	0.928744i	$-0.620890\pi$
$109$	−7.74090	−0.741443	−0.370722	+	0.928744i	$0.620890\pi$
$110$	−6.48156	−0.617992
$111$	0	0
$112$	−12.7424	−1.20404
$113$	−10.7748	−1.01361	−0.506804	−	0.862061i	$-0.669173\pi$
$113$	−10.7748	−1.01361	−0.506804	+	0.862061i	$0.669173\pi$
$114$	0	0
$115$	0.0307956	0.00287171
$116$	10.4562	0.970831
$117$	0	0
$118$	15.0635	1.38671
$119$	13.0758	1.19865
$120$	0	0
$121$	0.643250	0.0584773
$122$	−1.41763	−0.128346
$123$	0	0
$124$	−6.64271	−0.596533
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−5.55836	−0.493225	−0.246612	−	0.969114i	$-0.579317\pi$
$127$	−5.55836	−0.493225	−0.246612	+	0.969114i	$0.579317\pi$
$128$	−5.84029	−0.516214
$129$	0	0
$130$	7.51555	0.659157
$131$	−13.2634	−1.15883	−0.579415	−	0.815033i	$-0.696719\pi$
$131$	−13.2634	−1.15883	−0.579415	+	0.815033i	$0.696719\pi$
$132$	0	0
$133$	23.6954	2.05466
$134$	6.85989	0.592604
$135$	0	0
$136$	3.53650	0.303252
$137$	−21.0843	−1.80136	−0.900678	−	0.434487i	$-0.856930\pi$
$137$	−21.0843	−1.80136	−0.900678	+	0.434487i	$0.856930\pi$
$138$	0	0
$139$	−20.4772	−1.73685	−0.868426	−	0.495819i	$-0.834868\pi$
$139$	−20.4772	−1.73685	−0.868426	+	0.495819i	$0.834868\pi$
$140$	−4.42569	−0.374039
$141$	0	0
$142$	−20.5563	−1.72504
$143$	−13.5007	−1.12898
$144$	0	0
$145$	−6.50199	−0.539960
$146$	−13.2405	−1.09579
$147$	0	0
$148$	15.5353	1.27699
$149$	15.0054	1.22929	0.614645	−	0.788804i	$-0.289299\pi$
$149$	15.0054	1.22929	0.614645	+	0.788804i	$0.289299\pi$
$150$	0	0
$151$	21.9070	1.78276	0.891382	−	0.453252i	$-0.149736\pi$
$151$	21.9070	1.78276	0.891382	+	0.453252i	$0.149736\pi$
$152$	6.40872	0.519816
$153$	0	0
$154$	17.8375	1.43739
$155$	4.13065	0.331782
$156$	0	0
$157$	9.79206	0.781491	0.390746	−	0.920499i	$-0.372217\pi$
$157$	9.79206	0.781491	0.390746	+	0.920499i	$0.372217\pi$
$158$	−19.2952	−1.53505
$159$	0	0
$160$	−7.30639	−0.577621
$161$	−0.0847507	−0.00667929
$162$	0	0
$163$	−5.37190	−0.420760	−0.210380	−	0.977620i	$-0.567470\pi$
$163$	−5.37190	−0.420760	−0.210380	+	0.977620i	$0.567470\pi$
$164$	−8.71816	−0.680774
$165$	0	0
$166$	−25.1182	−1.94955
$167$	8.18049	0.633025	0.316513	−	0.948588i	$-0.397488\pi$
$167$	8.18049	0.633025	0.316513	+	0.948588i	$0.397488\pi$
$168$	0	0
$169$	2.65442	0.204186
$170$	9.02516	0.692198
$171$	0	0
$172$	−4.32532	−0.329802
$173$	−8.99311	−0.683734	−0.341867	−	0.939748i	$-0.611059\pi$
$173$	−8.99311	−0.683734	−0.341867	+	0.939748i	$0.611059\pi$
$174$	0	0
$175$	2.75204	0.208035
$176$	15.7991	1.19090
$177$	0	0
$178$	−1.89951	−0.142375
$179$	5.78367	0.432292	0.216146	−	0.976361i	$-0.430651\pi$
$179$	5.78367	0.432292	0.216146	+	0.976361i	$0.430651\pi$
$180$	0	0
$181$	−9.54965	−0.709820	−0.354910	−	0.934900i	$-0.615488\pi$
$181$	−9.54965	−0.709820	−0.354910	+	0.934900i	$0.615488\pi$
$182$	−20.6831	−1.53313
$183$	0	0
$184$	−0.0229219	−0.00168982
$185$	−9.66032	−0.710241
$186$	0	0
$187$	−16.2125	−1.18557
$188$	11.1206	0.811051
$189$	0	0
$190$	16.3551	1.18652
$191$	−5.35698	−0.387618	−0.193809	−	0.981039i	$-0.562084\pi$
$191$	−5.35698	−0.387618	−0.193809	+	0.981039i	$0.562084\pi$
$192$	0	0
$193$	−5.20843	−0.374911	−0.187455	−	0.982273i	$-0.560024\pi$
$193$	−5.20843	−0.374911	−0.187455	+	0.982273i	$0.560024\pi$
$194$	27.1791	1.95134
$195$	0	0
$196$	0.922621	0.0659015
$197$	1.09172	0.0777817	0.0388908	−	0.999243i	$-0.487618\pi$
$197$	1.09172	0.0777817	0.0388908	+	0.999243i	$0.487618\pi$
$198$	0	0
$199$	−7.49622	−0.531393	−0.265696	−	0.964057i	$-0.585602\pi$
$199$	−7.49622	−0.531393	−0.265696	+	0.964057i	$0.585602\pi$
$200$	0.744322	0.0526315
$201$	0	0
$202$	−6.44872	−0.453730
$203$	17.8937	1.25589
$204$	0	0
$205$	5.42123	0.378635
$206$	−27.1223	−1.88970
$207$	0	0
$208$	−18.3195	−1.27023
$209$	−29.3797	−2.03224
$210$	0	0
$211$	15.2590	1.05047	0.525237	−	0.850956i	$-0.323977\pi$
$211$	15.2590	1.05047	0.525237	+	0.850956i	$0.323977\pi$
$212$	−9.98503	−0.685775
$213$	0	0
$214$	24.9868	1.70807
$215$	2.68962	0.183431
$216$	0	0
$217$	−11.3677	−0.771690
$218$	14.7039	0.995876
$219$	0	0
$220$	5.48737	0.369958
$221$	18.7988	1.26455
$222$	0	0
$223$	−9.48971	−0.635478	−0.317739	−	0.948178i	$-0.602924\pi$
$223$	−9.48971	−0.635478	−0.317739	+	0.948178i	$0.602924\pi$
$224$	20.1075	1.34349
$225$	0	0
$226$	20.4669	1.36144
$227$	20.7935	1.38011	0.690055	−	0.723757i	$-0.257586\pi$
$227$	20.7935	1.38011	0.690055	+	0.723757i	$0.257586\pi$
$228$	0	0
$229$	−4.42775	−0.292594	−0.146297	−	0.989241i	$-0.546735\pi$
$229$	−4.42775	−0.292594	−0.146297	+	0.989241i	$0.546735\pi$
$230$	−0.0584967	−0.00385716
$231$	0	0
$232$	4.83957	0.317734
$233$	−0.0879849	−0.00576408	−0.00288204	−	0.999996i	$-0.500917\pi$
$233$	−0.0879849	−0.00576408	−0.00288204	+	0.999996i	$0.500917\pi$
$234$	0	0
$235$	−6.91513	−0.451093
$236$	−12.7529	−0.830145
$237$	0	0
$238$	−24.8376	−1.60998
$239$	6.39982	0.413970	0.206985	−	0.978344i	$-0.433635\pi$
$239$	6.39982	0.413970	0.206985	+	0.978344i	$0.433635\pi$
$240$	0	0
$241$	28.7036	1.84896	0.924481	−	0.381228i	$-0.124499\pi$
$241$	28.7036	1.84896	0.924481	+	0.381228i	$0.124499\pi$
$242$	−1.22186	−0.0785443
$243$	0	0
$244$	1.20018	0.0768337
$245$	−0.573715	−0.0366533
$246$	0	0
$247$	34.0666	2.16761
$248$	−3.07453	−0.195233
$249$	0	0
$250$	1.89951	0.120136
$251$	31.4866	1.98741	0.993707	−	0.112013i	$-0.0357298\pi$
$251$	31.4866	1.98741	0.993707	+	0.112013i	$0.0357298\pi$
$252$	0	0
$253$	0.105081	0.00660641
$254$	10.5582	0.662479
$255$	0	0
$256$	20.3303	1.27064
$257$	−8.96747	−0.559375	−0.279688	−	0.960091i	$-0.590231\pi$
$257$	−8.96747	−0.559375	−0.279688	+	0.960091i	$0.590231\pi$
$258$	0	0
$259$	26.5856	1.65195
$260$	−6.36276	−0.394601
$261$	0	0
$262$	25.1940	1.55649
$263$	5.19178	0.320139	0.160069	−	0.987106i	$-0.448828\pi$
$263$	5.19178	0.320139	0.160069	+	0.987106i	$0.448828\pi$
$264$	0	0
$265$	6.20902	0.381417
$266$	−45.0098	−2.75973
$267$	0	0
$268$	−5.80766	−0.354759
$269$	16.2429	0.990349	0.495175	−	0.868793i	$-0.335104\pi$
$269$	16.2429	0.990349	0.495175	+	0.868793i	$0.335104\pi$
$270$	0	0
$271$	−7.32797	−0.445142	−0.222571	−	0.974916i	$-0.571445\pi$
$271$	−7.32797	−0.445142	−0.222571	+	0.974916i	$0.571445\pi$
$272$	−21.9992	−1.33390
$273$	0	0
$274$	40.0500	2.41951
$275$	−3.41222	−0.205765
$276$	0	0
$277$	−28.7711	−1.72869	−0.864344	−	0.502901i	$-0.832266\pi$
$277$	−28.7711	−1.72869	−0.864344	+	0.502901i	$0.832266\pi$
$278$	38.8967	2.33287
$279$	0	0
$280$	−2.04840	−0.122416
$281$	−2.95688	−0.176393	−0.0881963	−	0.996103i	$-0.528110\pi$
$281$	−2.95688	−0.176393	−0.0881963	+	0.996103i	$0.528110\pi$
$282$	0	0
$283$	6.68149	0.397173	0.198587	−	0.980083i	$-0.436365\pi$
$283$	6.68149	0.397173	0.198587	+	0.980083i	$0.436365\pi$
$284$	17.4032	1.03269
$285$	0	0
$286$	25.6447	1.51640
$287$	−14.9194	−0.880667
$288$	0	0
$289$	5.57485	0.327932
$290$	12.3506	0.725253
$291$	0	0
$292$	11.2095	0.655988
$293$	−5.37223	−0.313849	−0.156925	−	0.987611i	$-0.550158\pi$
$293$	−5.37223	−0.313849	−0.156925	+	0.987611i	$0.550158\pi$
$294$	0	0
$295$	7.93018	0.461713
$296$	7.19039	0.417933
$297$	0	0
$298$	−28.5030	−1.65113
$299$	−0.121845	−0.00704647
$300$	0	0
$301$	−7.40194	−0.426641
$302$	−41.6126	−2.39454
$303$	0	0
$304$	−39.8663	−2.28649
$305$	−0.746311	−0.0427336
$306$	0	0
$307$	−27.8227	−1.58792	−0.793962	−	0.607967i	$-0.791985\pi$
$307$	−27.8227	−1.58792	−0.793962	+	0.607967i	$0.791985\pi$
$308$	−15.1014	−0.860485
$309$	0	0
$310$	−7.84622	−0.445636
$311$	14.5242	0.823590	0.411795	−	0.911277i	$-0.364902\pi$
$311$	14.5242	0.823590	0.411795	+	0.911277i	$0.364902\pi$
$312$	0	0
$313$	29.8704	1.68838	0.844189	−	0.536046i	$-0.180083\pi$
$313$	29.8704	1.68838	0.844189	+	0.536046i	$0.180083\pi$
$314$	−18.6001	−1.04967
$315$	0	0
$316$	16.3356	0.918949
$317$	27.7729	1.55988	0.779940	−	0.625854i	$-0.215250\pi$
$317$	27.7729	1.55988	0.779940	+	0.625854i	$0.215250\pi$
$318$	0	0
$319$	−22.1862	−1.24219
$320$	4.61828	0.258170
$321$	0	0
$322$	0.160985	0.00897135
$323$	40.9094	2.27626
$324$	0	0
$325$	3.95657	0.219471
$326$	10.2040	0.565147
$327$	0	0
$328$	−4.03514	−0.222804
$329$	19.0307	1.04920
$330$	0	0
$331$	3.25598	0.178965	0.0894823	−	0.995988i	$-0.471479\pi$
$331$	3.25598	0.178965	0.0894823	+	0.995988i	$0.471479\pi$
$332$	21.2654	1.16709
$333$	0	0
$334$	−15.5389	−0.850253
$335$	3.61139	0.197311
$336$	0	0
$337$	−0.139154	−0.00758020	−0.00379010	−	0.999993i	$-0.501206\pi$
$337$	−0.139154	−0.00758020	−0.00379010	+	0.999993i	$0.501206\pi$
$338$	−5.04210	−0.274254
$339$	0	0
$340$	−7.64081	−0.414381
$341$	14.0947	0.763270
$342$	0	0
$343$	−17.6854	−0.954921
$344$	−2.00194	−0.107938
$345$	0	0
$346$	17.0825	0.918363
$347$	11.1919	0.600813	0.300407	−	0.953811i	$-0.402878\pi$
$347$	11.1919	0.600813	0.300407	+	0.953811i	$0.402878\pi$
$348$	0	0
$349$	16.9140	0.905386	0.452693	−	0.891667i	$-0.350463\pi$
$349$	16.9140	0.905386	0.452693	+	0.891667i	$0.350463\pi$
$350$	−5.22753	−0.279423
$351$	0	0
$352$	−24.9310	−1.32883
$353$	−13.1929	−0.702185	−0.351093	−	0.936341i	$-0.614190\pi$
$353$	−13.1929	−0.702185	−0.351093	+	0.936341i	$0.614190\pi$
$354$	0	0
$355$	−10.8219	−0.574365
$356$	1.60815	0.0852318
$357$	0	0
$358$	−10.9862	−0.580636
$359$	13.7262	0.724440	0.362220	−	0.932093i	$-0.382019\pi$
$359$	13.7262	0.724440	0.362220	+	0.932093i	$0.382019\pi$
$360$	0	0
$361$	55.1346	2.90182
$362$	18.1397	0.953401
$363$	0	0
$364$	17.5105	0.917802
$365$	−6.97045	−0.364850
$366$	0	0
$367$	−28.8107	−1.50391	−0.751954	−	0.659216i	$-0.770888\pi$
$367$	−28.8107	−1.50391	−0.751954	+	0.659216i	$0.770888\pi$
$368$	0.142588	0.00743293
$369$	0	0
$370$	18.3499	0.953966
$371$	−17.0874	−0.887136
$372$	0	0
$373$	−2.50516	−0.129712	−0.0648561	−	0.997895i	$-0.520659\pi$
$373$	−2.50516	−0.129712	−0.0648561	+	0.997895i	$0.520659\pi$
$374$	30.7958	1.59241
$375$	0	0
$376$	5.14709	0.265441
$377$	25.7255	1.32493
$378$	0	0
$379$	−11.8616	−0.609292	−0.304646	−	0.952466i	$-0.598538\pi$
$379$	−11.8616	−0.609292	−0.304646	+	0.952466i	$0.598538\pi$
$380$	−13.8464	−0.710306
$381$	0	0
$382$	10.1757	0.520632
$383$	13.9435	0.712481	0.356240	−	0.934394i	$-0.384058\pi$
$383$	13.9435	0.712481	0.356240	+	0.934394i	$0.384058\pi$
$384$	0	0
$385$	9.39056	0.478587
$386$	9.89348	0.503565
$387$	0	0
$388$	−23.0101	−1.16816
$389$	16.7184	0.847659	0.423829	−	0.905742i	$-0.360686\pi$
$389$	16.7184	0.847659	0.423829	+	0.905742i	$0.360686\pi$
$390$	0	0
$391$	−0.146319	−0.00739968
$392$	0.427029	0.0215682
$393$	0	0
$394$	−2.07373	−0.104473
$395$	−10.1580	−0.511104
$396$	0	0
$397$	10.8302	0.543550	0.271775	−	0.962361i	$-0.412389\pi$
$397$	10.8302	0.543550	0.271775	+	0.962361i	$0.412389\pi$
$398$	14.2392	0.713745
$399$	0	0
$400$	−4.63015	−0.231508
$401$	30.9138	1.54376	0.771880	−	0.635768i	$-0.219317\pi$
$401$	30.9138	1.54376	0.771880	+	0.635768i	$0.219317\pi$
$402$	0	0
$403$	−16.3432	−0.814112
$404$	5.45956	0.271623
$405$	0	0
$406$	−33.9893	−1.68686
$407$	−32.9631	−1.63392
$408$	0	0
$409$	−23.6836	−1.17108	−0.585539	−	0.810644i	$-0.699117\pi$
$409$	−23.6836	−1.17108	−0.585539	+	0.810644i	$0.699117\pi$
$410$	−10.2977	−0.508567
$411$	0	0
$412$	22.9621	1.13126
$413$	−21.8242	−1.07390
$414$	0	0
$415$	−13.2235	−0.649116
$416$	28.9082	1.41734
$417$	0	0
$418$	55.8072	2.72962
$419$	8.17388	0.399320	0.199660	−	0.979865i	$-0.436016\pi$
$419$	8.17388	0.399320	0.199660	+	0.979865i	$0.436016\pi$
$420$	0	0
$421$	−5.49736	−0.267925	−0.133962	−	0.990986i	$-0.542770\pi$
$421$	−5.49736	−0.267925	−0.133962	+	0.990986i	$0.542770\pi$
$422$	−28.9847	−1.41095
$423$	0	0
$424$	−4.62151	−0.224440
$425$	4.75130	0.230472
$426$	0	0
$427$	2.05388	0.0993940
$428$	−21.1542	−1.02252
$429$	0	0
$430$	−5.10897	−0.246376
$431$	−30.9380	−1.49023	−0.745116	−	0.666935i	$-0.767606\pi$
$431$	−30.9380	−1.49023	−0.745116	+	0.666935i	$0.767606\pi$
$432$	0	0
$433$	17.2838	0.830604	0.415302	−	0.909684i	$-0.363676\pi$
$433$	17.2838	0.830604	0.415302	+	0.909684i	$0.363676\pi$
$434$	21.5931	1.03650
$435$	0	0
$436$	−12.4485	−0.596177
$437$	−0.265155	−0.0126841
$438$	0	0
$439$	17.3465	0.827905	0.413952	−	0.910299i	$-0.364148\pi$
$439$	17.3465	0.827905	0.413952	+	0.910299i	$0.364148\pi$
$440$	2.53979	0.121080
$441$	0	0
$442$	−35.7086	−1.69849
$443$	9.66578	0.459235	0.229617	−	0.973281i	$-0.426253\pi$
$443$	9.66578	0.459235	0.229617	+	0.973281i	$0.426253\pi$
$444$	0	0
$445$	−1.00000	−0.0474045
$446$	18.0258	0.853547
$447$	0	0
$448$	−12.7097	−0.600477
$449$	15.0389	0.709730	0.354865	−	0.934917i	$-0.384527\pi$
$449$	15.0389	0.709730	0.354865	+	0.934917i	$0.384527\pi$
$450$	0	0
$451$	18.4984	0.871058
$452$	−17.3275	−0.815018
$453$	0	0
$454$	−39.4974	−1.85371
$455$	−10.8886	−0.510466
$456$	0	0
$457$	−17.7563	−0.830604	−0.415302	−	0.909684i	$-0.636324\pi$
$457$	−17.7563	−0.830604	−0.415302	+	0.909684i	$0.636324\pi$
$458$	8.41057	0.393000
$459$	0	0
$460$	0.0495240	0.00230907
$461$	18.6101	0.866760	0.433380	−	0.901211i	$-0.357321\pi$
$461$	18.6101	0.866760	0.433380	+	0.901211i	$0.357321\pi$
$462$	0	0
$463$	−6.15953	−0.286258	−0.143129	−	0.989704i	$-0.545716\pi$
$463$	−6.15953	−0.286258	−0.143129	+	0.989704i	$0.545716\pi$
$464$	−30.1052	−1.39760
$465$	0	0
$466$	0.167128	0.00774208
$467$	16.1880	0.749093	0.374547	−	0.927208i	$-0.377798\pi$
$467$	16.1880	0.749093	0.374547	+	0.927208i	$0.377798\pi$
$468$	0	0
$469$	−9.93869	−0.458926
$470$	13.1354	0.605890
$471$	0	0
$472$	−5.90261	−0.271690
$473$	9.17758	0.421986
$474$	0	0
$475$	8.61014	0.395061
$476$	21.0278	0.963807
$477$	0	0
$478$	−12.1565	−0.556027
$479$	15.9586	0.729165	0.364583	−	0.931171i	$-0.381212\pi$
$479$	15.9586	0.729165	0.364583	+	0.931171i	$0.381212\pi$
$480$	0	0
$481$	38.2217	1.74276
$482$	−54.5229	−2.48345
$483$	0	0
$484$	1.03444	0.0470201
$485$	14.3084	0.649713
$486$	0	0
$487$	26.7296	1.21123	0.605617	−	0.795756i	$-0.292926\pi$
$487$	26.7296	1.21123	0.605617	+	0.795756i	$0.292926\pi$
$488$	0.555496	0.0251461
$489$	0	0
$490$	1.08978	0.0492313
$491$	−16.7794	−0.757244	−0.378622	−	0.925551i	$-0.623602\pi$
$491$	−16.7794	−0.757244	−0.378622	+	0.925551i	$0.623602\pi$
$492$	0	0
$493$	30.8929	1.39135
$494$	−64.7100	−2.91144
$495$	0	0
$496$	19.1255	0.858762
$497$	29.7822	1.33591
$498$	0	0
$499$	4.68037	0.209522	0.104761	−	0.994497i	$-0.466592\pi$
$499$	4.68037	0.209522	0.104761	+	0.994497i	$0.466592\pi$
$500$	−1.60815	−0.0719187
$501$	0	0
$502$	−59.8091	−2.66941
$503$	−2.09355	−0.0933468	−0.0466734	−	0.998910i	$-0.514862\pi$
$503$	−2.09355	−0.0933468	−0.0466734	+	0.998910i	$0.514862\pi$
$504$	0	0
$505$	−3.39493	−0.151073
$506$	−0.199604	−0.00887346
$507$	0	0
$508$	−8.93868	−0.396590
$509$	−15.9305	−0.706107	−0.353053	−	0.935603i	$-0.614857\pi$
$509$	−15.9305	−0.706107	−0.353053	+	0.935603i	$0.614857\pi$
$510$	0	0
$511$	19.1829	0.848603
$512$	−26.9371	−1.19046
$513$	0	0
$514$	17.0338	0.751330
$515$	−14.2785	−0.629188
$516$	0	0
$517$	−23.5960	−1.03775
$518$	−50.4996	−2.21883
$519$	0	0
$520$	−2.94496	−0.129145
$521$	−23.8189	−1.04352	−0.521762	−	0.853091i	$-0.674725\pi$
$521$	−23.8189	−1.04352	−0.521762	+	0.853091i	$0.674725\pi$
$522$	0	0
$523$	28.2698	1.23615	0.618076	−	0.786118i	$-0.287912\pi$
$523$	28.2698	1.23615	0.618076	+	0.786118i	$0.287912\pi$
$524$	−21.3296	−0.931787
$525$	0	0
$526$	−9.86185	−0.429997
$527$	−19.6260	−0.854920
$528$	0	0
$529$	−22.9991	−0.999959
$530$	−11.7941	−0.512303
$531$	0	0
$532$	38.1059	1.65210
$533$	−21.4495	−0.929080
$534$	0	0
$535$	13.1543	0.568712
$536$	−2.68804	−0.116106
$537$	0	0
$538$	−30.8537	−1.33020
$539$	−1.95764	−0.0843217
$540$	0	0
$541$	11.9149	0.512264	0.256132	−	0.966642i	$-0.417552\pi$
$541$	11.9149	0.512264	0.256132	+	0.966642i	$0.417552\pi$
$542$	13.9196	0.597897
$543$	0	0
$544$	34.7149	1.48839
$545$	7.74090	0.331584
$546$	0	0
$547$	25.4523	1.08826	0.544132	−	0.839000i	$-0.316859\pi$
$547$	25.4523	1.08826	0.544132	+	0.839000i	$0.316859\pi$
$548$	−33.9068	−1.44843
$549$	0	0
$550$	6.48156	0.276375
$551$	55.9830	2.38496
$552$	0	0
$553$	27.9552	1.18878
$554$	54.6511	2.32190
$555$	0	0
$556$	−32.9304	−1.39656
$557$	42.2055	1.78831	0.894153	−	0.447762i	$-0.147779\pi$
$557$	42.2055	1.78831	0.894153	+	0.447762i	$0.147779\pi$
$558$	0	0
$559$	−10.6417	−0.450094
$560$	12.7424	0.538463
$561$	0	0
$562$	5.61663	0.236923
$563$	−33.9250	−1.42977	−0.714884	−	0.699243i	$-0.753521\pi$
$563$	−33.9250	−1.42977	−0.714884	+	0.699243i	$0.753521\pi$
$564$	0	0
$565$	10.7748	0.453299
$566$	−12.6916	−0.533467
$567$	0	0
$568$	8.05495	0.337978
$569$	9.48928	0.397811	0.198906	−	0.980019i	$-0.436261\pi$
$569$	9.48928	0.397811	0.198906	+	0.980019i	$0.436261\pi$
$570$	0	0
$571$	30.5674	1.27921	0.639603	−	0.768705i	$-0.279099\pi$
$571$	30.5674	1.27921	0.639603	+	0.768705i	$0.279099\pi$
$572$	−21.7111	−0.907788
$573$	0	0
$574$	28.3397	1.18288
$575$	−0.0307956	−0.00128427
$576$	0	0
$577$	1.95662	0.0814551	0.0407276	−	0.999170i	$-0.487032\pi$
$577$	1.95662	0.0814551	0.0407276	+	0.999170i	$0.487032\pi$
$578$	−10.5895	−0.440465
$579$	0	0
$580$	−10.4562	−0.434169
$581$	36.3916	1.50978
$582$	0	0
$583$	21.1865	0.877456
$584$	5.18826	0.214692
$585$	0	0
$586$	10.2046	0.421549
$587$	−29.2878	−1.20884	−0.604418	−	0.796668i	$-0.706594\pi$
$587$	−29.2878	−1.20884	−0.604418	+	0.796668i	$0.706594\pi$
$588$	0	0
$589$	−35.5655	−1.46545
$590$	−15.0635	−0.620154
$591$	0	0
$592$	−44.7287	−1.83834
$593$	11.8724	0.487540	0.243770	−	0.969833i	$-0.421616\pi$
$593$	11.8724	0.487540	0.243770	+	0.969833i	$0.421616\pi$
$594$	0	0
$595$	−13.0758	−0.536054
$596$	24.1309	0.988442
$597$	0	0
$598$	0.231446	0.00946453
$599$	17.5343	0.716434	0.358217	−	0.933638i	$-0.383385\pi$
$599$	17.5343	0.716434	0.358217	+	0.933638i	$0.383385\pi$
$600$	0	0
$601$	−45.0554	−1.83785	−0.918923	−	0.394436i	$-0.870940\pi$
$601$	−45.0554	−1.83785	−0.918923	+	0.394436i	$0.870940\pi$
$602$	14.0601	0.573046
$603$	0	0
$604$	35.2297	1.43348
$605$	−0.643250	−0.0261518
$606$	0	0
$607$	−0.188420	−0.00764775	−0.00382388	−	0.999993i	$-0.501217\pi$
$607$	−0.188420	−0.00764775	−0.00382388	+	0.999993i	$0.501217\pi$
$608$	62.9091	2.55130
$609$	0	0
$610$	1.41763	0.0573981
$611$	27.3602	1.10687
$612$	0	0
$613$	−45.7708	−1.84866	−0.924332	−	0.381589i	$-0.875377\pi$
$613$	−45.7708	−1.84866	−0.924332	+	0.381589i	$0.875377\pi$
$614$	52.8495	2.13283
$615$	0	0
$616$	−6.98960	−0.281619
$617$	9.51782	0.383173	0.191587	−	0.981476i	$-0.438637\pi$
$617$	9.51782	0.383173	0.191587	+	0.981476i	$0.438637\pi$
$618$	0	0
$619$	42.0325	1.68943	0.844715	−	0.535217i	$-0.179770\pi$
$619$	42.0325	1.68943	0.844715	+	0.535217i	$0.179770\pi$
$620$	6.64271	0.266778
$621$	0	0
$622$	−27.5888	−1.10621
$623$	2.75204	0.110258
$624$	0	0
$625$	1.00000	0.0400000
$626$	−56.7393	−2.26776
$627$	0	0
$628$	15.7471	0.628378
$629$	45.8991	1.83012
$630$	0	0
$631$	1.20898	0.0481285	0.0240643	−	0.999710i	$-0.492339\pi$
$631$	1.20898	0.0481285	0.0240643	+	0.999710i	$0.492339\pi$
$632$	7.56082	0.300753
$633$	0	0
$634$	−52.7549	−2.09517
$635$	5.55836	0.220577
$636$	0	0
$637$	2.26994	0.0899384
$638$	42.1430	1.66846
$639$	0	0
$640$	5.84029	0.230858
$641$	33.0964	1.30723	0.653615	−	0.756827i	$-0.273252\pi$
$641$	33.0964	1.30723	0.653615	+	0.756827i	$0.273252\pi$
$642$	0	0
$643$	−15.2892	−0.602946	−0.301473	−	0.953475i	$-0.597478\pi$
$643$	−15.2892	−0.602946	−0.301473	+	0.953475i	$0.597478\pi$
$644$	−0.136292	−0.00537065
$645$	0	0
$646$	−77.7079	−3.05738
$647$	15.7702	0.619991	0.309995	−	0.950738i	$-0.399673\pi$
$647$	15.7702	0.619991	0.309995	+	0.950738i	$0.399673\pi$
$648$	0	0
$649$	27.0595	1.06218
$650$	−7.51555	−0.294784
$651$	0	0
$652$	−8.63883	−0.338323
$653$	−3.64118	−0.142490	−0.0712452	−	0.997459i	$-0.522697\pi$
$653$	−3.64118	−0.142490	−0.0712452	+	0.997459i	$0.522697\pi$
$654$	0	0
$655$	13.2634	0.518244
$656$	25.1011	0.980035
$657$	0	0
$658$	−36.1491	−1.40924
$659$	−13.8986	−0.541414	−0.270707	−	0.962662i	$-0.587258\pi$
$659$	−13.8986	−0.541414	−0.270707	+	0.962662i	$0.587258\pi$
$660$	0	0
$661$	−37.0308	−1.44033	−0.720166	−	0.693802i	$-0.755934\pi$
$661$	−37.0308	−1.44033	−0.720166	+	0.693802i	$0.755934\pi$
$662$	−6.18477	−0.240378
$663$	0	0
$664$	9.84254	0.381965
$665$	−23.6954	−0.918870
$666$	0	0
$667$	−0.200233	−0.00775304
$668$	13.1555	0.509000
$669$	0	0
$670$	−6.85989	−0.265021
$671$	−2.54658	−0.0983095
$672$	0	0
$673$	−0.454869	−0.0175339	−0.00876695	−	0.999962i	$-0.502791\pi$
$673$	−0.454869	−0.0175339	−0.00876695	+	0.999962i	$0.502791\pi$
$674$	0.264325	0.0101814
$675$	0	0
$676$	4.26870	0.164181
$677$	30.9187	1.18830	0.594150	−	0.804354i	$-0.297488\pi$
$677$	30.9187	1.18830	0.594150	+	0.804354i	$0.297488\pi$
$678$	0	0
$679$	−39.3774	−1.51117
$680$	−3.53650	−0.135619
$681$	0	0
$682$	−26.7730	−1.02519
$683$	45.0988	1.72566	0.862829	−	0.505496i	$-0.168690\pi$
$683$	45.0988	1.72566	0.862829	+	0.505496i	$0.168690\pi$
$684$	0	0
$685$	21.0843	0.805591
$686$	33.5936	1.28261
$687$	0	0
$688$	12.4534	0.474780
$689$	−24.5664	−0.935904
$690$	0	0
$691$	−15.2962	−0.581896	−0.290948	−	0.956739i	$-0.593971\pi$
$691$	−15.2962	−0.581896	−0.290948	+	0.956739i	$0.593971\pi$
$692$	−14.4623	−0.549773
$693$	0	0
$694$	−21.2592	−0.806988
$695$	20.4772	0.776744
$696$	0	0
$697$	−25.7579	−0.975650
$698$	−32.1284	−1.21608
$699$	0	0
$700$	4.42569	0.167275
$701$	−29.1321	−1.10030	−0.550152	−	0.835065i	$-0.685430\pi$
$701$	−29.1321	−1.10030	−0.550152	+	0.835065i	$0.685430\pi$
$702$	0	0
$703$	83.1767	3.13707
$704$	15.7586	0.593925
$705$	0	0
$706$	25.0600	0.943146
$707$	9.34298	0.351379
$708$	0	0
$709$	5.79196	0.217522	0.108761	−	0.994068i	$-0.465312\pi$
$709$	5.79196	0.217522	0.108761	+	0.994068i	$0.465312\pi$
$710$	20.5563	0.771463
$711$	0	0
$712$	0.744322	0.0278947
$713$	0.127206	0.00476390
$714$	0	0
$715$	13.5007	0.504897
$716$	9.30101	0.347595
$717$	0	0
$718$	−26.0731	−0.973038
$719$	−15.7750	−0.588309	−0.294154	−	0.955758i	$-0.595038\pi$
$719$	−15.7750	−0.588309	−0.294154	+	0.955758i	$0.595038\pi$
$720$	0	0
$721$	39.2951	1.46343
$722$	−104.729	−3.89761
$723$	0	0
$724$	−15.3573	−0.570749
$725$	6.50199	0.241478
$726$	0	0
$727$	−7.12821	−0.264371	−0.132185	−	0.991225i	$-0.542199\pi$
$727$	−7.12821	−0.264371	−0.132185	+	0.991225i	$0.542199\pi$
$728$	8.10464	0.300378
$729$	0	0
$730$	13.2405	0.490051
$731$	−12.7792	−0.472656
$732$	0	0
$733$	7.93931	0.293245	0.146623	−	0.989193i	$-0.453160\pi$
$733$	7.93931	0.293245	0.146623	+	0.989193i	$0.453160\pi$
$734$	54.7264	2.01999
$735$	0	0
$736$	−0.225005	−0.00829379
$737$	12.3229	0.453919
$738$	0	0
$739$	−43.4611	−1.59874	−0.799370	−	0.600839i	$-0.794833\pi$
$739$	−43.4611	−1.59874	−0.799370	+	0.600839i	$0.794833\pi$
$740$	−15.5353	−0.571087
$741$	0	0
$742$	32.4578	1.19156
$743$	−14.7561	−0.541347	−0.270674	−	0.962671i	$-0.587246\pi$
$743$	−14.7561	−0.541347	−0.270674	+	0.962671i	$0.587246\pi$
$744$	0	0
$745$	−15.0054	−0.549755
$746$	4.75858	0.174224
$747$	0	0
$748$	−26.0721	−0.953291
$749$	−36.2012	−1.32276
$750$	0	0
$751$	32.7889	1.19648	0.598241	−	0.801316i	$-0.295867\pi$
$751$	32.7889	1.19648	0.598241	+	0.801316i	$0.295867\pi$
$752$	−32.0181	−1.16758
$753$	0	0
$754$	−48.8660	−1.77959
$755$	−21.9070	−0.797277
$756$	0	0
$757$	0.271152	0.00985520	0.00492760	−	0.999988i	$-0.498431\pi$
$757$	0.271152	0.00985520	0.00492760	+	0.999988i	$0.498431\pi$
$758$	22.5314	0.818376
$759$	0	0
$760$	−6.40872	−0.232469
$761$	14.2436	0.516331	0.258165	−	0.966101i	$-0.416882\pi$
$761$	14.2436	0.516331	0.258165	+	0.966101i	$0.416882\pi$
$762$	0	0
$763$	−21.3032	−0.771229
$764$	−8.61484	−0.311674
$765$	0	0
$766$	−26.4859	−0.956975
$767$	−31.3763	−1.13293
$768$	0	0
$769$	−43.6194	−1.57296	−0.786478	−	0.617619i	$-0.788098\pi$
$769$	−43.6194	−1.57296	−0.786478	+	0.617619i	$0.788098\pi$
$770$	−17.8375	−0.642819
$771$	0	0
$772$	−8.37594	−0.301457
$773$	0.0896627	0.00322494	0.00161247	−	0.999999i	$-0.499487\pi$
$773$	0.0896627	0.00322494	0.00161247	+	0.999999i	$0.499487\pi$
$774$	0	0
$775$	−4.13065	−0.148377
$776$	−10.6501	−0.382316
$777$	0	0
$778$	−31.7569	−1.13854
$779$	−46.6776	−1.67240
$780$	0	0
$781$	−36.9266	−1.32134
$782$	0.277935	0.00993894
$783$	0	0
$784$	−2.65639	−0.0948711
$785$	−9.79206	−0.349494
$786$	0	0
$787$	1.32800	0.0473380	0.0236690	−	0.999720i	$-0.492465\pi$
$787$	1.32800	0.0473380	0.0236690	+	0.999720i	$0.492465\pi$
$788$	1.75565	0.0625423
$789$	0	0
$790$	19.2952	0.686494
$791$	−29.6527	−1.05433
$792$	0	0
$793$	2.95283	0.104858
$794$	−20.5720	−0.730074
$795$	0	0
$796$	−12.0550	−0.427280
$797$	−55.3943	−1.96217	−0.981083	−	0.193586i	$-0.937988\pi$
$797$	−55.3943	−1.96217	−0.981083	+	0.193586i	$0.937988\pi$
$798$	0	0
$799$	32.8559	1.16236
$800$	7.30639	0.258320
$801$	0	0
$802$	−58.7211	−2.07351
$803$	−23.7847	−0.839344
$804$	0	0
$805$	0.0847507	0.00298707
$806$	31.0441	1.09348
$807$	0	0
$808$	2.52692	0.0888969
$809$	−16.9376	−0.595496	−0.297748	−	0.954645i	$-0.596235\pi$
$809$	−16.9376	−0.595496	−0.297748	+	0.954645i	$0.596235\pi$
$810$	0	0
$811$	−38.8593	−1.36453	−0.682267	−	0.731103i	$-0.739006\pi$
$811$	−38.8593	−1.36453	−0.682267	+	0.731103i	$0.739006\pi$
$812$	28.7758	1.00983
$813$	0	0
$814$	62.6139	2.19462
$815$	5.37190	0.188170
$816$	0	0
$817$	−23.1580	−0.810197
$818$	44.9873	1.57294
$819$	0	0
$820$	8.71816	0.304451
$821$	−1.24933	−0.0436021	−0.0218010	−	0.999762i	$-0.506940\pi$
$821$	−1.24933	−0.0436021	−0.0218010	+	0.999762i	$0.506940\pi$
$822$	0	0
$823$	35.6194	1.24161	0.620806	−	0.783964i	$-0.286805\pi$
$823$	35.6194	1.24161	0.620806	+	0.783964i	$0.286805\pi$
$824$	10.6278	0.370238
$825$	0	0
$826$	41.4553	1.44241
$827$	27.2662	0.948140	0.474070	−	0.880487i	$-0.342784\pi$
$827$	27.2662	0.948140	0.474070	+	0.880487i	$0.342784\pi$
$828$	0	0
$829$	5.76554	0.200246	0.100123	−	0.994975i	$-0.468076\pi$
$829$	5.76554	0.200246	0.100123	+	0.994975i	$0.468076\pi$
$830$	25.1182	0.871866
$831$	0	0
$832$	−18.2725	−0.633487
$833$	2.72589	0.0944466
$834$	0	0
$835$	−8.18049	−0.283097
$836$	−47.2470	−1.63407
$837$	0	0
$838$	−15.5264	−0.536350
$839$	−33.2585	−1.14821	−0.574106	−	0.818781i	$-0.694650\pi$
$839$	−33.2585	−1.14821	−0.574106	+	0.818781i	$0.694650\pi$
$840$	0	0
$841$	13.2758	0.457787
$842$	10.4423	0.359866
$843$	0	0
$844$	24.5388	0.844660
$845$	−2.65442	−0.0913147
$846$	0	0
$847$	1.77025	0.0608265
$848$	28.7487	0.987234
$849$	0	0
$850$	−9.02516	−0.309560
$851$	−0.297495	−0.0101980
$852$	0	0
$853$	−5.88360	−0.201451	−0.100725	−	0.994914i	$-0.532116\pi$
$853$	−5.88360	−0.201451	−0.100725	+	0.994914i	$0.532116\pi$
$854$	−3.90137	−0.133502
$855$	0	0
$856$	−9.79107	−0.334652
$857$	12.9883	0.443673	0.221837	−	0.975084i	$-0.428795\pi$
$857$	12.9883	0.443673	0.221837	+	0.975084i	$0.428795\pi$
$858$	0	0
$859$	10.9313	0.372972	0.186486	−	0.982458i	$-0.440290\pi$
$859$	10.9313	0.372972	0.186486	+	0.982458i	$0.440290\pi$
$860$	4.32532	0.147492
$861$	0	0
$862$	58.7672	2.00162
$863$	47.1364	1.60454	0.802270	−	0.596961i	$-0.203625\pi$
$863$	47.1364	1.60454	0.802270	+	0.596961i	$0.203625\pi$
$864$	0	0
$865$	8.99311	0.305775
$866$	−32.8307	−1.11563
$867$	0	0
$868$	−18.2810	−0.620497
$869$	−34.6613	−1.17580
$870$	0	0
$871$	−14.2887	−0.484154
$872$	−5.76172	−0.195117
$873$	0	0
$874$	0.503665	0.0170367
$875$	−2.75204	−0.0930359
$876$	0	0
$877$	−23.9693	−0.809387	−0.404693	−	0.914452i	$-0.632622\pi$
$877$	−23.9693	−0.809387	−0.404693	+	0.914452i	$0.632622\pi$
$878$	−32.9500	−1.11201
$879$	0	0
$880$	−15.7991	−0.532588
$881$	−42.5357	−1.43306	−0.716531	−	0.697555i	$-0.754271\pi$
$881$	−42.5357	−1.43306	−0.716531	+	0.697555i	$0.754271\pi$
$882$	0	0
$883$	41.4069	1.39345	0.696726	−	0.717337i	$-0.254639\pi$
$883$	41.4069	1.39345	0.696726	+	0.717337i	$0.254639\pi$
$884$	30.2314	1.01679
$885$	0	0
$886$	−18.3603	−0.616825
$887$	−29.9187	−1.00457	−0.502286	−	0.864701i	$-0.667508\pi$
$887$	−29.9187	−1.00457	−0.502286	+	0.864701i	$0.667508\pi$
$888$	0	0
$889$	−15.2968	−0.513039
$890$	1.89951	0.0636718
$891$	0	0
$892$	−15.2609	−0.510972
$893$	59.5403	1.99244
$894$	0	0
$895$	−5.78367	−0.193327
$896$	−16.0727	−0.536951
$897$	0	0
$898$	−28.5666	−0.953281
$899$	−26.8574	−0.895745
$900$	0	0
$901$	−29.5009	−0.982817
$902$	−35.1380	−1.16997
$903$	0	0
$904$	−8.01993	−0.266739
$905$	9.54965	0.317441
$906$	0	0
$907$	−49.4971	−1.64352	−0.821762	−	0.569831i	$-0.807009\pi$
$907$	−49.4971	−1.64352	−0.821762	+	0.569831i	$0.807009\pi$
$908$	33.4390	1.10971
$909$	0	0
$910$	20.6831	0.685638
$911$	−12.7892	−0.423725	−0.211862	−	0.977299i	$-0.567953\pi$
$911$	−12.7892	−0.423725	−0.211862	+	0.977299i	$0.567953\pi$
$912$	0	0
$913$	−45.1215	−1.49330
$914$	33.7283	1.11563
$915$	0	0
$916$	−7.12049	−0.235268
$917$	−36.5014	−1.20538
$918$	0	0
$919$	−15.7765	−0.520420	−0.260210	−	0.965552i	$-0.583792\pi$
$919$	−15.7765	−0.520420	−0.260210	+	0.965552i	$0.583792\pi$
$920$	0.0229219	0.000755711 0
$921$	0	0
$922$	−35.3502	−1.16420
$923$	42.8174	1.40935
$924$	0	0
$925$	9.66032	0.317629
$926$	11.7001	0.384489
$927$	0	0
$928$	47.5061	1.55946
$929$	30.9474	1.01535	0.507675	−	0.861549i	$-0.330505\pi$
$929$	30.9474	1.01535	0.507675	+	0.861549i	$0.330505\pi$
$930$	0	0
$931$	4.93977	0.161895
$932$	−0.141493	−0.00463476
$933$	0	0
$934$	−30.7494	−1.00615
$935$	16.2125	0.530205
$936$	0	0
$937$	7.49125	0.244728	0.122364	−	0.992485i	$-0.460952\pi$
$937$	7.49125	0.244728	0.122364	+	0.992485i	$0.460952\pi$
$938$	18.8787	0.616410
$939$	0	0
$940$	−11.1206	−0.362713
$941$	40.5743	1.32269	0.661343	−	0.750084i	$-0.269987\pi$
$941$	40.5743	1.32269	0.661343	+	0.750084i	$0.269987\pi$
$942$	0	0
$943$	0.166950	0.00543665
$944$	36.7179	1.19507
$945$	0	0
$946$	−17.4329	−0.566794
$947$	−54.2814	−1.76391	−0.881954	−	0.471335i	$-0.843772\pi$
$947$	−54.2814	−1.76391	−0.881954	+	0.471335i	$0.843772\pi$
$948$	0	0
$949$	27.5790	0.895253
$950$	−16.3551	−0.530629
$951$	0	0
$952$	9.73258	0.315435
$953$	−48.1013	−1.55815	−0.779077	−	0.626929i	$-0.784312\pi$
$953$	−48.1013	−1.55815	−0.779077	+	0.626929i	$0.784312\pi$
$954$	0	0
$955$	5.35698	0.173348
$956$	10.2919	0.332863
$957$	0	0
$958$	−30.3135	−0.979385
$959$	−58.0249	−1.87372
$960$	0	0
$961$	−13.9377	−0.449604
$962$	−72.6026	−2.34080
$963$	0	0
$964$	46.1598	1.48671
$965$	5.20843	0.167665
$966$	0	0
$967$	46.9816	1.51082	0.755412	−	0.655250i	$-0.227437\pi$
$967$	46.9816	1.51082	0.755412	+	0.655250i	$0.227437\pi$
$968$	0.478785	0.0153887
$969$	0	0
$970$	−27.1791	−0.872668
$971$	35.4999	1.13925	0.569623	−	0.821906i	$-0.307089\pi$
$971$	35.4999	1.13925	0.569623	+	0.821906i	$0.307089\pi$
$972$	0	0
$973$	−56.3540	−1.80663
$974$	−50.7733	−1.62688
$975$	0	0
$976$	−3.45553	−0.110609
$977$	14.0708	0.450166	0.225083	−	0.974340i	$-0.427735\pi$
$977$	14.0708	0.450166	0.225083	+	0.974340i	$0.427735\pi$
$978$	0	0
$979$	−3.41222	−0.109055
$980$	−0.922621	−0.0294721
$981$	0	0
$982$	31.8727	1.01710
$983$	−24.7855	−0.790534	−0.395267	−	0.918566i	$-0.629348\pi$
$983$	−24.7855	−0.790534	−0.395267	+	0.918566i	$0.629348\pi$
$984$	0	0
$985$	−1.09172	−0.0347850
$986$	−58.6814	−1.86880
$987$	0	0
$988$	54.7842	1.74292
$989$	0.0828285	0.00263379
$990$	0	0
$991$	19.5347	0.620540	0.310270	−	0.950648i	$-0.399580\pi$
$991$	19.5347	0.620540	0.310270	+	0.950648i	$0.399580\pi$
$992$	−30.1801	−0.958221
$993$	0	0
$994$	−56.5716	−1.79434
$995$	7.49622	0.237646
$996$	0	0
$997$	44.6486	1.41404	0.707018	−	0.707195i	$-0.250040\pi$
$997$	44.6486	1.41404	0.707018	+	0.707195i	$0.250040\pi$
$998$	−8.89042	−0.281421
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4005.2.a.w.1.5	✓	17
3.2	odd	2		4005.2.a.x.1.13	yes	17

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4005.2.a.w.1.5	✓	17	1.1	even	1	trivial
4005.2.a.x.1.13	yes	17	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)

\(f(q)\)	\(=\)	\(q-1.89951 q^{2} +1.60815 q^{4} -1.00000 q^{5} +2.75204 q^{7} +0.744322 q^{8} +O(q^{10})\)	\(q-1.89951 q^{2} +1.60815 q^{4} -1.00000 q^{5} +2.75204 q^{7} +0.744322 q^{8} +1.89951 q^{10} -3.41222 q^{11} +3.95657 q^{13} -5.22753 q^{14} -4.63015 q^{16} +4.75130 q^{17} +8.61014 q^{19} -1.60815 q^{20} +6.48156 q^{22} -0.0307956 q^{23} +1.00000 q^{25} -7.51555 q^{26} +4.42569 q^{28} +6.50199 q^{29} -4.13065 q^{31} +7.30639 q^{32} -9.02516 q^{34} -2.75204 q^{35} +9.66032 q^{37} -16.3551 q^{38} -0.744322 q^{40} -5.42123 q^{41} -2.68962 q^{43} -5.48737 q^{44} +0.0584967 q^{46} +6.91513 q^{47} +0.573715 q^{49} -1.89951 q^{50} +6.36276 q^{52} -6.20902 q^{53} +3.41222 q^{55} +2.04840 q^{56} -12.3506 q^{58} -7.93018 q^{59} +0.746311 q^{61} +7.84622 q^{62} -4.61828 q^{64} -3.95657 q^{65} -3.61139 q^{67} +7.64081 q^{68} +5.22753 q^{70} +10.8219 q^{71} +6.97045 q^{73} -18.3499 q^{74} +13.8464 q^{76} -9.39056 q^{77} +10.1580 q^{79} +4.63015 q^{80} +10.2977 q^{82} +13.2235 q^{83} -4.75130 q^{85} +5.10897 q^{86} -2.53979 q^{88} +1.00000 q^{89} +10.8886 q^{91} -0.0495240 q^{92} -13.1354 q^{94} -8.61014 q^{95} -14.3084 q^{97} -1.08978 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 17 q - 5 q^{2} + 21 q^{4} - 17 q^{5} + 12 q^{7} - 15 q^{8}+O(q^{10}) \) 17 * q - 5 * q^2 + 21 * q^4 - 17 * q^5 + 12 * q^7 - 15 * q^8	\( 17 q - 5 q^{2} + 21 q^{4} - 17 q^{5} + 12 q^{7} - 15 q^{8} + 5 q^{10} - 2 q^{11} + 8 q^{13} - 4 q^{14} + 33 q^{16} - 10 q^{17} + 32 q^{19} - 21 q^{20} + 8 q^{22} - 15 q^{23} + 17 q^{25} + 15 q^{26} + 24 q^{28} - q^{29} + 18 q^{31} - 25 q^{32} + 14 q^{34} - 12 q^{35} + 12 q^{37} - 22 q^{38} + 15 q^{40} + 7 q^{41} + 28 q^{43} + 14 q^{44} + 4 q^{46} - 26 q^{47} + 41 q^{49} - 5 q^{50} + 10 q^{52} - 12 q^{53} + 2 q^{55} - 13 q^{56} + 16 q^{58} + 23 q^{59} + 26 q^{61} - 10 q^{62} + 59 q^{64} - 8 q^{65} + 31 q^{67} + q^{68} + 4 q^{70} + 2 q^{71} + 33 q^{73} + 10 q^{74} + 66 q^{76} - 12 q^{77} + 33 q^{79} - 33 q^{80} + 30 q^{82} - 13 q^{83} + 10 q^{85} + 20 q^{86} + 12 q^{88} + 17 q^{89} + 40 q^{91} - 16 q^{92} + 38 q^{94} - 32 q^{95} + 45 q^{97} - 2 q^{98}+O(q^{100}) \) 17 * q - 5 * q^2 + 21 * q^4 - 17 * q^5 + 12 * q^7 - 15 * q^8 + 5 * q^10 - 2 * q^11 + 8 * q^13 - 4 * q^14 + 33 * q^16 - 10 * q^17 + 32 * q^19 - 21 * q^20 + 8 * q^22 - 15 * q^23 + 17 * q^25 + 15 * q^26 + 24 * q^28 - q^29 + 18 * q^31 - 25 * q^32 + 14 * q^34 - 12 * q^35 + 12 * q^37 - 22 * q^38 + 15 * q^40 + 7 * q^41 + 28 * q^43 + 14 * q^44 + 4 * q^46 - 26 * q^47 + 41 * q^49 - 5 * q^50 + 10 * q^52 - 12 * q^53 + 2 * q^55 - 13 * q^56 + 16 * q^58 + 23 * q^59 + 26 * q^61 - 10 * q^62 + 59 * q^64 - 8 * q^65 + 31 * q^67 + q^68 + 4 * q^70 + 2 * q^71 + 33 * q^73 + 10 * q^74 + 66 * q^76 - 12 * q^77 + 33 * q^79 - 33 * q^80 + 30 * q^82 - 13 * q^83 + 10 * q^85 + 20 * q^86 + 12 * q^88 + 17 * q^89 + 40 * q^91 - 16 * q^92 + 38 * q^94 - 32 * q^95 + 45 * q^97 - 2 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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