LMFDB - Embedded newform 4005.2.a.s.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4005, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4005.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$31.9800860095$
Analytic rank:	$1$
Dimension:	$10$
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} - x^{9} - 16x^{8} + 15x^{7} + 85x^{6} - 75x^{5} - 163x^{4} + 138x^{3} + 78x^{2} - 67x + 4 $ x^10 - x^9 - 16x^8 + 15x^7 + 85x^6 - 75x^5 - 163x^4 + 138x^3 + 78x^2 - 67x + 4
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			1.6
Root			$0.0651711$ 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)

$f(q)$	$=$	$q-0.0651711 q^{2} -1.99575 q^{4} -1.00000 q^{5} +4.79409 q^{7} +0.260408 q^{8} +O(q^{10})$	\(q-0.0651711 q^{2} -1.99575 q^{4} -1.00000 q^{5} +4.79409 q^{7} +0.260408 q^{8} +0.0651711 q^{10} +2.88226 q^{11} +0.934389 q^{13} -0.312436 q^{14} +3.97453 q^{16} -7.62856 q^{17} -8.57962 q^{19} +1.99575 q^{20} -0.187840 q^{22} -1.92666 q^{23} +1.00000 q^{25} -0.0608952 q^{26} -9.56781 q^{28} +1.72232 q^{29} -6.74445 q^{31} -0.779840 q^{32} +0.497162 q^{34} -4.79409 q^{35} +8.20230 q^{37} +0.559144 q^{38} -0.260408 q^{40} -4.25189 q^{41} -0.864185 q^{43} -5.75228 q^{44} +0.125563 q^{46} -0.169086 q^{47} +15.9833 q^{49} -0.0651711 q^{50} -1.86481 q^{52} -4.85706 q^{53} -2.88226 q^{55} +1.24842 q^{56} -0.112246 q^{58} -5.97438 q^{59} +10.6089 q^{61} +0.439543 q^{62} -7.89825 q^{64} -0.934389 q^{65} +4.63152 q^{67} +15.2247 q^{68} +0.312436 q^{70} -8.96578 q^{71} -10.6455 q^{73} -0.534553 q^{74} +17.1228 q^{76} +13.8178 q^{77} -2.14337 q^{79} -3.97453 q^{80} +0.277101 q^{82} -13.1053 q^{83} +7.62856 q^{85} +0.0563199 q^{86} +0.750562 q^{88} +1.00000 q^{89} +4.47954 q^{91} +3.84515 q^{92} +0.0110195 q^{94} +8.57962 q^{95} +5.75047 q^{97} -1.04165 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - q^{2} + 13 q^{4} - 10 q^{5} - q^{7}+O(q^{10}) $ 10 * q - q^2 + 13 * q^4 - 10 * q^5 - q^7	$ 10 q - q^{2} + 13 q^{4} - 10 q^{5} - q^{7} + q^{10} - 10 q^{11} + 5 q^{13} - 13 q^{14} + 19 q^{16} - 9 q^{17} - 6 q^{19} - 13 q^{20} + 3 q^{23} + 10 q^{25} - 14 q^{26} - 18 q^{28} - 38 q^{29} + 2 q^{31} - 16 q^{32} - 8 q^{34} + q^{35} + 9 q^{37} - 20 q^{38} - 36 q^{41} - 7 q^{43} - 16 q^{44} + 2 q^{46} + 23 q^{47} + 25 q^{49} - q^{50} + 13 q^{52} - 27 q^{53} + 10 q^{55} - 41 q^{56} - 32 q^{58} - 20 q^{59} + 30 q^{61} + 2 q^{62} - 2 q^{64} - 5 q^{65} - 5 q^{67} + 10 q^{68} + 13 q^{70} - 24 q^{71} - 19 q^{73} - 42 q^{74} - 30 q^{76} - 18 q^{77} + 12 q^{79} - 19 q^{80} + 29 q^{82} - 3 q^{83} + 9 q^{85} - 38 q^{86} - 16 q^{88} + 10 q^{89} - 6 q^{91} - 32 q^{92} + 17 q^{94} + 6 q^{95} + 3 q^{97} + 37 q^{98}+O(q^{100}) $ 10 * q - q^2 + 13 * q^4 - 10 * q^5 - q^7 + q^10 - 10 * q^11 + 5 * q^13 - 13 * q^14 + 19 * q^16 - 9 * q^17 - 6 * q^19 - 13 * q^20 + 3 * q^23 + 10 * q^25 - 14 * q^26 - 18 * q^28 - 38 * q^29 + 2 * q^31 - 16 * q^32 - 8 * q^34 + q^35 + 9 * q^37 - 20 * q^38 - 36 * q^41 - 7 * q^43 - 16 * q^44 + 2 * q^46 + 23 * q^47 + 25 * q^49 - q^50 + 13 * q^52 - 27 * q^53 + 10 * q^55 - 41 * q^56 - 32 * q^58 - 20 * q^59 + 30 * q^61 + 2 * q^62 - 2 * q^64 - 5 * q^65 - 5 * q^67 + 10 * q^68 + 13 * q^70 - 24 * q^71 - 19 * q^73 - 42 * q^74 - 30 * q^76 - 18 * q^77 + 12 * q^79 - 19 * q^80 + 29 * q^82 - 3 * q^83 + 9 * q^85 - 38 * q^86 - 16 * q^88 + 10 * q^89 - 6 * q^91 - 32 * q^92 + 17 * q^94 + 6 * q^95 + 3 * q^97 + 37 * 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 $

$2$	−0.0651711	−0.0460829	−0.0230415	−	0.999735i	$-0.507335\pi$
$2$	−0.0651711	−0.0460829	−0.0230415	+	0.999735i	$0.507335\pi$
$3$	0	0
$3$	0	0
$4$	−1.99575	−0.997876
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	4.79409	1.81199	0.905997	−	0.423284i	$-0.139123\pi$
$7$	4.79409	1.81199	0.905997	+	0.423284i	$0.139123\pi$
$8$	0.260408	0.0920680
$9$	0	0
$10$	0.0651711	0.0206089
$11$	2.88226	0.869034	0.434517	−	0.900664i	$-0.356919\pi$
$11$	2.88226	0.869034	0.434517	+	0.900664i	$0.356919\pi$
$12$	0	0
$13$	0.934389	0.259153	0.129576	−	0.991569i	$-0.458638\pi$
$13$	0.934389	0.259153	0.129576	+	0.991569i	$0.458638\pi$
$14$	−0.312436	−0.0835020
$15$	0	0
$16$	3.97453	0.993634
$17$	−7.62856	−1.85020	−0.925099	−	0.379726i	$-0.876018\pi$
$17$	−7.62856	−1.85020	−0.925099	+	0.379726i	$0.876018\pi$
$18$	0	0
$19$	−8.57962	−1.96830	−0.984150	−	0.177336i	$-0.943252\pi$
$19$	−8.57962	−1.96830	−0.984150	+	0.177336i	$0.943252\pi$
$20$	1.99575	0.446264
$21$	0	0
$22$	−0.187840	−0.0400476
$23$	−1.92666	−0.401737	−0.200869	−	0.979618i	$-0.564376\pi$
$23$	−1.92666	−0.401737	−0.200869	+	0.979618i	$0.564376\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−0.0608952	−0.0119425
$27$	0	0
$28$	−9.56781	−1.80815
$29$	1.72232	0.319827	0.159914	−	0.987131i	$-0.448878\pi$
$29$	1.72232	0.319827	0.159914	+	0.987131i	$0.448878\pi$
$30$	0	0
$31$	−6.74445	−1.21134	−0.605670	−	0.795716i	$-0.707095\pi$
$31$	−6.74445	−1.21134	−0.605670	+	0.795716i	$0.707095\pi$
$32$	−0.779840	−0.137858
$33$	0	0
$34$	0.497162	0.0852626
$35$	−4.79409	−0.810348
$36$	0	0
$37$	8.20230	1.34845	0.674225	−	0.738526i	$-0.264478\pi$
$37$	8.20230	1.34845	0.674225	+	0.738526i	$0.264478\pi$
$38$	0.559144	0.0907051
$39$	0	0
$40$	−0.260408	−0.0411741
$41$	−4.25189	−0.664034	−0.332017	−	0.943273i	$-0.607729\pi$
$41$	−4.25189	−0.664034	−0.332017	+	0.943273i	$0.607729\pi$
$42$	0	0
$43$	−0.864185	−0.131787	−0.0658935	−	0.997827i	$-0.520990\pi$
$43$	−0.864185	−0.131787	−0.0658935	+	0.997827i	$0.520990\pi$
$44$	−5.75228	−0.867188
$45$	0	0
$46$	0.125563	0.0185132
$47$	−0.169086	−0.0246638	−0.0123319	−	0.999924i	$-0.503925\pi$
$47$	−0.169086	−0.0246638	−0.0123319	+	0.999924i	$0.503925\pi$
$48$	0	0
$49$	15.9833	2.28332
$50$	−0.0651711	−0.00921658
$51$	0	0
$52$	−1.86481	−0.258603
$53$	−4.85706	−0.667168	−0.333584	−	0.942720i	$-0.608258\pi$
$53$	−4.85706	−0.667168	−0.333584	+	0.942720i	$0.608258\pi$
$54$	0	0
$55$	−2.88226	−0.388644
$56$	1.24842	0.166827
$57$	0	0
$58$	−0.112246	−0.0147386
$59$	−5.97438	−0.777799	−0.388899	−	0.921280i	$-0.627145\pi$
$59$	−5.97438	−0.777799	−0.388899	+	0.921280i	$0.627145\pi$
$60$	0	0
$61$	10.6089	1.35833	0.679166	−	0.733985i	$-0.262342\pi$
$61$	10.6089	1.35833	0.679166	+	0.733985i	$0.262342\pi$
$62$	0.439543	0.0558221
$63$	0	0
$64$	−7.89825	−0.987281
$65$	−0.934389	−0.115897
$66$	0	0
$67$	4.63152	0.565830	0.282915	−	0.959145i	$-0.408699\pi$
$67$	4.63152	0.565830	0.282915	+	0.959145i	$0.408699\pi$
$68$	15.2247	1.84627
$69$	0	0
$70$	0.312436	0.0373432
$71$	−8.96578	−1.06404	−0.532021	−	0.846731i	$-0.678567\pi$
$71$	−8.96578	−1.06404	−0.532021	+	0.846731i	$0.678567\pi$
$72$	0	0
$73$	−10.6455	−1.24596	−0.622980	−	0.782237i	$-0.714078\pi$
$73$	−10.6455	−1.24596	−0.622980	+	0.782237i	$0.714078\pi$
$74$	−0.534553	−0.0621405
$75$	0	0
$76$	17.1228	1.96412
$77$	13.8178	1.57468
$78$	0	0
$79$	−2.14337	−0.241148	−0.120574	−	0.992704i	$-0.538474\pi$
$79$	−2.14337	−0.241148	−0.120574	+	0.992704i	$0.538474\pi$
$80$	−3.97453	−0.444366
$81$	0	0
$82$	0.277101	0.0306006
$83$	−13.1053	−1.43849	−0.719247	−	0.694754i	$-0.755513\pi$
$83$	−13.1053	−1.43849	−0.719247	+	0.694754i	$0.755513\pi$
$84$	0	0
$85$	7.62856	0.827434
$86$	0.0563199	0.00607313
$87$	0	0
$88$	0.750562	0.0800102
$89$	1.00000	0.106000
$89$	1.00000	0.106000
$90$	0	0
$91$	4.47954	0.469584
$92$	3.84515	0.400884
$93$	0	0
$94$	0.0110195	0.00113658
$95$	8.57962	0.880251
$96$	0	0
$97$	5.75047	0.583872	0.291936	−	0.956438i	$-0.405701\pi$
$97$	5.75047	0.583872	0.291936	+	0.956438i	$0.405701\pi$
$98$	−1.04165	−0.105222
$99$	0	0
$100$	−1.99575	−0.199575
$101$	8.22080	0.818001	0.409000	−	0.912534i	$-0.365878\pi$
$101$	8.22080	0.818001	0.409000	+	0.912534i	$0.365878\pi$
$102$	0	0
$103$	6.15534	0.606504	0.303252	−	0.952910i	$-0.401928\pi$
$103$	6.15534	0.606504	0.303252	+	0.952910i	$0.401928\pi$
$104$	0.243322	0.0238597
$105$	0	0
$106$	0.316540	0.0307450
$107$	−10.9117	−1.05487	−0.527436	−	0.849595i	$-0.676847\pi$
$107$	−10.9117	−1.05487	−0.527436	+	0.849595i	$0.676847\pi$
$108$	0	0
$109$	3.46666	0.332046	0.166023	−	0.986122i	$-0.446907\pi$
$109$	3.46666	0.332046	0.166023	+	0.986122i	$0.446907\pi$
$110$	0.187840	0.0179098
$111$	0	0
$112$	19.0543	1.80046
$113$	−12.5195	−1.17774	−0.588870	−	0.808228i	$-0.700427\pi$
$113$	−12.5195	−1.17774	−0.588870	+	0.808228i	$0.700427\pi$
$114$	0	0
$115$	1.92666	0.179662
$116$	−3.43733	−0.319148
$117$	0	0
$118$	0.389357	0.0358432
$119$	−36.5720	−3.35255
$120$	0	0
$121$	−2.69259	−0.244780
$122$	−0.691394	−0.0625959
$123$	0	0
$124$	13.4603	1.20877
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−4.23393	−0.375701	−0.187850	−	0.982198i	$-0.560152\pi$
$127$	−4.23393	−0.375701	−0.187850	+	0.982198i	$0.560152\pi$
$128$	2.07442	0.183354
$129$	0	0
$130$	0.0608952	0.00534086
$131$	14.1416	1.23556	0.617778	−	0.786353i	$-0.288033\pi$
$131$	14.1416	1.23556	0.617778	+	0.786353i	$0.288033\pi$
$132$	0	0
$133$	−41.1315	−3.56655
$134$	−0.301841	−0.0260751
$135$	0	0
$136$	−1.98654	−0.170344
$137$	4.99715	0.426935	0.213468	−	0.976950i	$-0.431524\pi$
$137$	4.99715	0.426935	0.213468	+	0.976950i	$0.431524\pi$
$138$	0	0
$139$	14.4369	1.22452	0.612260	−	0.790656i	$-0.290260\pi$
$139$	14.4369	1.22452	0.612260	+	0.790656i	$0.290260\pi$
$140$	9.56781	0.808627
$141$	0	0
$142$	0.584310	0.0490342
$143$	2.69315	0.225213
$144$	0	0
$145$	−1.72232	−0.143031
$146$	0.693779	0.0574175
$147$	0	0
$148$	−16.3698	−1.34559
$149$	−21.1029	−1.72882	−0.864410	−	0.502787i	$-0.832308\pi$
$149$	−21.1029	−1.72882	−0.864410	+	0.502787i	$0.832308\pi$
$150$	0	0
$151$	8.51950	0.693307	0.346654	−	0.937993i	$-0.387318\pi$
$151$	8.51950	0.693307	0.346654	+	0.937993i	$0.387318\pi$
$152$	−2.23420	−0.181217
$153$	0	0
$154$	−0.900521	−0.0725660
$155$	6.74445	0.541727
$156$	0	0
$157$	−10.2488	−0.817941	−0.408971	−	0.912548i	$-0.634112\pi$
$157$	−10.2488	−0.817941	−0.408971	+	0.912548i	$0.634112\pi$
$158$	0.139686	0.0111128
$159$	0	0
$160$	0.779840	0.0616518
$161$	−9.23660	−0.727946
$162$	0	0
$163$	−2.61970	−0.205190	−0.102595	−	0.994723i	$-0.532715\pi$
$163$	−2.61970	−0.205190	−0.102595	+	0.994723i	$0.532715\pi$
$164$	8.48573	0.662624
$165$	0	0
$166$	0.854088	0.0662901
$167$	−21.7294	−1.68147	−0.840735	−	0.541447i	$-0.817877\pi$
$167$	−21.7294	−1.68147	−0.840735	+	0.541447i	$0.817877\pi$
$168$	0	0
$169$	−12.1269	−0.932840
$170$	−0.497162	−0.0381306
$171$	0	0
$172$	1.72470	0.131507
$173$	−5.11768	−0.389090	−0.194545	−	0.980894i	$-0.562323\pi$
$173$	−5.11768	−0.389090	−0.194545	+	0.980894i	$0.562323\pi$
$174$	0	0
$175$	4.79409	0.362399
$176$	11.4556	0.863501
$177$	0	0
$178$	−0.0651711	−0.00488478
$179$	24.9011	1.86119	0.930596	−	0.366047i	$-0.119289\pi$
$179$	24.9011	1.86119	0.930596	+	0.366047i	$0.119289\pi$
$180$	0	0
$181$	−24.1221	−1.79298	−0.896491	−	0.443062i	$-0.853892\pi$
$181$	−24.1221	−1.79298	−0.896491	+	0.443062i	$0.853892\pi$
$182$	−0.291937	−0.0216398
$183$	0	0
$184$	−0.501718	−0.0369872
$185$	−8.20230	−0.603045
$186$	0	0
$187$	−21.9875	−1.60788
$188$	0.337455	0.0246114
$189$	0	0
$190$	−0.559144	−0.0405645
$191$	10.5927	0.766459	0.383230	−	0.923653i	$-0.374812\pi$
$191$	10.5927	0.766459	0.383230	+	0.923653i	$0.374812\pi$
$192$	0	0
$193$	−1.90524	−0.137142	−0.0685711	−	0.997646i	$-0.521844\pi$
$193$	−1.90524	−0.137142	−0.0685711	+	0.997646i	$0.521844\pi$
$194$	−0.374765	−0.0269065
$195$	0	0
$196$	−31.8986	−2.27847
$197$	3.23704	0.230629	0.115315	−	0.993329i	$-0.463212\pi$
$197$	3.23704	0.230629	0.115315	+	0.993329i	$0.463212\pi$
$198$	0	0
$199$	−19.6336	−1.39179	−0.695895	−	0.718144i	$-0.744992\pi$
$199$	−19.6336	−1.39179	−0.695895	+	0.718144i	$0.744992\pi$
$200$	0.260408	0.0184136
$201$	0	0
$202$	−0.535759	−0.0376959
$203$	8.25696	0.579525
$204$	0	0
$205$	4.25189	0.296965
$206$	−0.401150	−0.0279495
$207$	0	0
$208$	3.71376	0.257503
$209$	−24.7287	−1.71052
$210$	0	0
$211$	−12.0943	−0.832605	−0.416303	−	0.909226i	$-0.636674\pi$
$211$	−12.0943	−0.832605	−0.416303	+	0.909226i	$0.636674\pi$
$212$	9.69348	0.665751
$213$	0	0
$214$	0.711127	0.0486116
$215$	0.864185	0.0589369
$216$	0	0
$217$	−32.3335	−2.19494
$218$	−0.225926	−0.0153017
$219$	0	0
$220$	5.75228	0.387818
$221$	−7.12805	−0.479484
$222$	0	0
$223$	−2.75761	−0.184663	−0.0923316	−	0.995728i	$-0.529432\pi$
$223$	−2.75761	−0.184663	−0.0923316	+	0.995728i	$0.529432\pi$
$224$	−3.73862	−0.249797
$225$	0	0
$226$	0.815912	0.0542737
$227$	24.5545	1.62974	0.814871	−	0.579642i	$-0.196808\pi$
$227$	24.5545	1.62974	0.814871	+	0.579642i	$0.196808\pi$
$228$	0	0
$229$	−14.0838	−0.930681	−0.465340	−	0.885132i	$-0.654068\pi$
$229$	−14.0838	−0.930681	−0.465340	+	0.885132i	$0.654068\pi$
$230$	−0.125563	−0.00827937
$231$	0	0
$232$	0.448506	0.0294459
$233$	−5.31368	−0.348111	−0.174055	−	0.984736i	$-0.555687\pi$
$233$	−5.31368	−0.348111	−0.174055	+	0.984736i	$0.555687\pi$
$234$	0	0
$235$	0.169086	0.0110300
$236$	11.9234	0.776147
$237$	0	0
$238$	2.38344	0.154495
$239$	−29.5068	−1.90863	−0.954317	−	0.298795i	$-0.903415\pi$
$239$	−29.5068	−1.90863	−0.954317	+	0.298795i	$0.903415\pi$
$240$	0	0
$241$	25.2659	1.62752	0.813761	−	0.581199i	$-0.197416\pi$
$241$	25.2659	1.62752	0.813761	+	0.581199i	$0.197416\pi$
$242$	0.175479	0.0112802
$243$	0	0
$244$	−21.1728	−1.35545
$245$	−15.9833	−1.02113
$246$	0	0
$247$	−8.01671	−0.510091
$248$	−1.75631	−0.111526
$249$	0	0
$250$	0.0651711	0.00412178
$251$	−9.64282	−0.608649	−0.304325	−	0.952568i	$-0.598431\pi$
$251$	−9.64282	−0.608649	−0.304325	+	0.952568i	$0.598431\pi$
$252$	0	0
$253$	−5.55315	−0.349123
$254$	0.275930	0.0173134
$255$	0	0
$256$	15.6613	0.978831
$257$	−20.3084	−1.26680	−0.633401	−	0.773824i	$-0.718342\pi$
$257$	−20.3084	−1.26680	−0.633401	+	0.773824i	$0.718342\pi$
$258$	0	0
$259$	39.3225	2.44338
$260$	1.86481	0.115651
$261$	0	0
$262$	−0.921622	−0.0569380
$263$	3.78362	0.233308	0.116654	−	0.993173i	$-0.462783\pi$
$263$	3.78362	0.233308	0.116654	+	0.993173i	$0.462783\pi$
$264$	0	0
$265$	4.85706	0.298367
$266$	2.68058	0.164357
$267$	0	0
$268$	−9.24337	−0.564629
$269$	−17.9804	−1.09628	−0.548141	−	0.836386i	$-0.684664\pi$
$269$	−17.9804	−1.09628	−0.548141	+	0.836386i	$0.684664\pi$
$270$	0	0
$271$	2.37495	0.144268	0.0721341	−	0.997395i	$-0.477019\pi$
$271$	2.37495	0.144268	0.0721341	+	0.997395i	$0.477019\pi$
$272$	−30.3200	−1.83842
$273$	0	0
$274$	−0.325670	−0.0196744
$275$	2.88226	0.173807
$276$	0	0
$277$	−30.8104	−1.85122	−0.925609	−	0.378481i	$-0.876446\pi$
$277$	−30.8104	−1.85122	−0.925609	+	0.378481i	$0.876446\pi$
$278$	−0.940868	−0.0564295
$279$	0	0
$280$	−1.24842	−0.0746071
$281$	10.4688	0.624519	0.312259	−	0.949997i	$-0.398914\pi$
$281$	10.4688	0.624519	0.312259	+	0.949997i	$0.398914\pi$
$282$	0	0
$283$	−11.0046	−0.654154	−0.327077	−	0.944998i	$-0.606064\pi$
$283$	−11.0046	−0.654154	−0.327077	+	0.944998i	$0.606064\pi$
$284$	17.8935	1.06178
$285$	0	0
$286$	−0.175516	−0.0103785
$287$	−20.3839	−1.20323
$288$	0	0
$289$	41.1950	2.42323
$290$	0.112246	0.00659129
$291$	0	0
$292$	21.2458	1.24331
$293$	−7.88428	−0.460605	−0.230302	−	0.973119i	$-0.573971\pi$
$293$	−7.88428	−0.460605	−0.230302	+	0.973119i	$0.573971\pi$
$294$	0	0
$295$	5.97438	0.347842
$296$	2.13594	0.124149
$297$	0	0
$298$	1.37530	0.0796691
$299$	−1.80026	−0.104111
$300$	0	0
$301$	−4.14298	−0.238797
$302$	−0.555225	−0.0319496
$303$	0	0
$304$	−34.1000	−1.95577
$305$	−10.6089	−0.607465
$306$	0	0
$307$	−20.4414	−1.16665	−0.583325	−	0.812239i	$-0.698249\pi$
$307$	−20.4414	−1.16665	−0.583325	+	0.812239i	$0.698249\pi$
$308$	−27.5769	−1.57134
$309$	0	0
$310$	−0.439543	−0.0249644
$311$	−9.20229	−0.521814	−0.260907	−	0.965364i	$-0.584022\pi$
$311$	−9.20229	−0.521814	−0.260907	+	0.965364i	$0.584022\pi$
$312$	0	0
$313$	−11.2361	−0.635100	−0.317550	−	0.948241i	$-0.602860\pi$
$313$	−11.2361	−0.635100	−0.317550	+	0.948241i	$0.602860\pi$
$314$	0.667924	0.0376931
$315$	0	0
$316$	4.27764	0.240636
$317$	−3.89254	−0.218627	−0.109313	−	0.994007i	$-0.534865\pi$
$317$	−3.89254	−0.218627	−0.109313	+	0.994007i	$0.534865\pi$
$318$	0	0
$319$	4.96418	0.277941
$320$	7.89825	0.441525
$321$	0	0
$322$	0.601959	0.0335459
$323$	65.4502	3.64175
$324$	0	0
$325$	0.934389	0.0518306
$326$	0.170728	0.00945578
$327$	0	0
$328$	−1.10723	−0.0611363
$329$	−0.810614	−0.0446906
$330$	0	0
$331$	−18.2985	−1.00578	−0.502889	−	0.864351i	$-0.667730\pi$
$331$	−18.2985	−1.00578	−0.502889	+	0.864351i	$0.667730\pi$
$332$	26.1550	1.43544
$333$	0	0
$334$	1.41613	0.0774870
$335$	−4.63152	−0.253047
$336$	0	0
$337$	−8.79510	−0.479100	−0.239550	−	0.970884i	$-0.577000\pi$
$337$	−8.79510	−0.479100	−0.239550	+	0.970884i	$0.577000\pi$
$338$	0.790324	0.0429880
$339$	0	0
$340$	−15.2247	−0.825677
$341$	−19.4393	−1.05269
$342$	0	0
$343$	43.0665	2.32537
$344$	−0.225040	−0.0121334
$345$	0	0
$346$	0.333525	0.0179304
$347$	19.2400	1.03286	0.516428	−	0.856331i	$-0.327261\pi$
$347$	19.2400	1.03286	0.516428	+	0.856331i	$0.327261\pi$
$348$	0	0
$349$	−15.5117	−0.830322	−0.415161	−	0.909748i	$-0.636275\pi$
$349$	−15.5117	−0.830322	−0.415161	+	0.909748i	$0.636275\pi$
$350$	−0.312436	−0.0167004
$351$	0	0
$352$	−2.24770	−0.119803
$353$	21.9875	1.17027	0.585137	−	0.810934i	$-0.301041\pi$
$353$	21.9875	1.17027	0.585137	+	0.810934i	$0.301041\pi$
$354$	0	0
$355$	8.96578	0.475854
$356$	−1.99575	−0.105775
$357$	0	0
$358$	−1.62283	−0.0857692
$359$	22.3791	1.18112	0.590562	−	0.806992i	$-0.298906\pi$
$359$	22.3791	1.18112	0.590562	+	0.806992i	$0.298906\pi$
$360$	0	0
$361$	54.6099	2.87421
$362$	1.57206	0.0826259
$363$	0	0
$364$	−8.94006	−0.468586
$365$	10.6455	0.557211
$366$	0	0
$367$	23.7285	1.23862	0.619310	−	0.785147i	$-0.287413\pi$
$367$	23.7285	1.23862	0.619310	+	0.785147i	$0.287413\pi$
$368$	−7.65760	−0.399180
$369$	0	0
$370$	0.534553	0.0277901
$371$	−23.2851	−1.20890
$372$	0	0
$373$	11.8280	0.612430	0.306215	−	0.951962i	$-0.400937\pi$
$373$	11.8280	0.612430	0.306215	+	0.951962i	$0.400937\pi$
$374$	1.43295	0.0740960
$375$	0	0
$376$	−0.0440314	−0.00227074
$377$	1.60932	0.0828842
$378$	0	0
$379$	−20.9881	−1.07809	−0.539044	−	0.842278i	$-0.681214\pi$
$379$	−20.9881	−1.07809	−0.539044	+	0.842278i	$0.681214\pi$
$380$	−17.1228	−0.878382
$381$	0	0
$382$	−0.690336	−0.0353207
$383$	−22.5341	−1.15144	−0.575718	−	0.817648i	$-0.695277\pi$
$383$	−22.5341	−1.15144	−0.575718	+	0.817648i	$0.695277\pi$
$384$	0	0
$385$	−13.8178	−0.704220
$386$	0.124167	0.00631991
$387$	0	0
$388$	−11.4765	−0.582632
$389$	−12.4354	−0.630500	−0.315250	−	0.949009i	$-0.602088\pi$
$389$	−12.4354	−0.630500	−0.315250	+	0.949009i	$0.602088\pi$
$390$	0	0
$391$	14.6977	0.743294
$392$	4.16216	0.210221
$393$	0	0
$394$	−0.210961	−0.0106281
$395$	2.14337	0.107845
$396$	0	0
$397$	13.7795	0.691575	0.345788	−	0.938313i	$-0.387612\pi$
$397$	13.7795	0.691575	0.345788	+	0.938313i	$0.387612\pi$
$398$	1.27954	0.0641377
$399$	0	0
$400$	3.97453	0.198727
$401$	29.6489	1.48060	0.740299	−	0.672278i	$-0.234684\pi$
$401$	29.6489	1.48060	0.740299	+	0.672278i	$0.234684\pi$
$402$	0	0
$403$	−6.30194	−0.313922
$404$	−16.4067	−0.816263
$405$	0	0
$406$	−0.538115	−0.0267062
$407$	23.6411	1.17185
$408$	0	0
$409$	3.36819	0.166546	0.0832731	−	0.996527i	$-0.473463\pi$
$409$	3.36819	0.166546	0.0832731	+	0.996527i	$0.473463\pi$
$410$	−0.277101	−0.0136850
$411$	0	0
$412$	−12.2845	−0.605216
$413$	−28.6417	−1.40937
$414$	0	0
$415$	13.1053	0.643314
$416$	−0.728674	−0.0357262
$417$	0	0
$418$	1.61160	0.0788257
$419$	−8.87280	−0.433465	−0.216732	−	0.976231i	$-0.569540\pi$
$419$	−8.87280	−0.433465	−0.216732	+	0.976231i	$0.569540\pi$
$420$	0	0
$421$	12.4510	0.606823	0.303412	−	0.952860i	$-0.401874\pi$
$421$	12.4510	0.606823	0.303412	+	0.952860i	$0.401874\pi$
$422$	0.788198	0.0383689
$423$	0	0
$424$	−1.26481	−0.0614248
$425$	−7.62856	−0.370040
$426$	0	0
$427$	50.8600	2.46129
$428$	21.7770	1.05263
$429$	0	0
$430$	−0.0563199	−0.00271599
$431$	−32.6548	−1.57293	−0.786463	−	0.617638i	$-0.788090\pi$
$431$	−32.6548	−1.57293	−0.786463	+	0.617638i	$0.788090\pi$
$432$	0	0
$433$	−8.52529	−0.409699	−0.204850	−	0.978793i	$-0.565671\pi$
$433$	−8.52529	−0.409699	−0.204850	+	0.978793i	$0.565671\pi$
$434$	2.10721	0.101149
$435$	0	0
$436$	−6.91860	−0.331341
$437$	16.5301	0.790740
$438$	0	0
$439$	−24.6206	−1.17508	−0.587539	−	0.809196i	$-0.699903\pi$
$439$	−24.6206	−1.17508	−0.587539	+	0.809196i	$0.699903\pi$
$440$	−0.750562	−0.0357816
$441$	0	0
$442$	0.464543	0.0220960
$443$	−0.621545	−0.0295305	−0.0147652	−	0.999891i	$-0.504700\pi$
$443$	−0.621545	−0.0295305	−0.0147652	+	0.999891i	$0.504700\pi$
$444$	0	0
$445$	−1.00000	−0.0474045
$446$	0.179716	0.00850982
$447$	0	0
$448$	−37.8649	−1.78895
$449$	33.1033	1.56224	0.781120	−	0.624381i	$-0.214649\pi$
$449$	33.1033	1.56224	0.781120	+	0.624381i	$0.214649\pi$
$450$	0	0
$451$	−12.2551	−0.577068
$452$	24.9859	1.17524
$453$	0	0
$454$	−1.60025	−0.0751033
$455$	−4.47954	−0.210004
$456$	0	0
$457$	3.71520	0.173790	0.0868948	−	0.996217i	$-0.472306\pi$
$457$	3.71520	0.173790	0.0868948	+	0.996217i	$0.472306\pi$
$458$	0.917854	0.0428885
$459$	0	0
$460$	−3.84515	−0.179281
$461$	24.6472	1.14794	0.573968	−	0.818878i	$-0.305403\pi$
$461$	24.6472	1.14794	0.573968	+	0.818878i	$0.305403\pi$
$462$	0	0
$463$	−9.84407	−0.457493	−0.228746	−	0.973486i	$-0.573463\pi$
$463$	−9.84407	−0.457493	−0.228746	+	0.973486i	$0.573463\pi$
$464$	6.84543	0.317791
$465$	0	0
$466$	0.346298	0.0160420
$467$	−7.03673	−0.325621	−0.162811	−	0.986657i	$-0.552056\pi$
$467$	−7.03673	−0.325621	−0.162811	+	0.986657i	$0.552056\pi$
$468$	0	0
$469$	22.2039	1.02528
$470$	−0.0110195	−0.000508294 0
$471$	0	0
$472$	−1.55578	−0.0716104
$473$	−2.49080	−0.114527
$474$	0	0
$475$	−8.57962	−0.393660
$476$	72.9886	3.34543
$477$	0	0
$478$	1.92299	0.0879555
$479$	3.99695	0.182625	0.0913127	−	0.995822i	$-0.470894\pi$
$479$	3.99695	0.182625	0.0913127	+	0.995822i	$0.470894\pi$
$480$	0	0
$481$	7.66414	0.349455
$482$	−1.64661	−0.0750010
$483$	0	0
$484$	5.37373	0.244261
$485$	−5.75047	−0.261115
$486$	0	0
$487$	42.5911	1.92999	0.964994	−	0.262270i	$-0.0844712\pi$
$487$	42.5911	1.92999	0.964994	+	0.262270i	$0.0844712\pi$
$488$	2.76264	0.125059
$489$	0	0
$490$	1.04165	0.0470568
$491$	−9.53084	−0.430121	−0.215060	−	0.976601i	$-0.568995\pi$
$491$	−9.53084	−0.430121	−0.215060	+	0.976601i	$0.568995\pi$
$492$	0	0
$493$	−13.1389	−0.591744
$494$	0.522458	0.0235065
$495$	0	0
$496$	−26.8061	−1.20363
$497$	−42.9827	−1.92804
$498$	0	0
$499$	9.38164	0.419980	0.209990	−	0.977704i	$-0.432657\pi$
$499$	9.38164	0.419980	0.209990	+	0.977704i	$0.432657\pi$
$500$	1.99575	0.0892528
$501$	0	0
$502$	0.628433	0.0280483
$503$	36.4882	1.62693	0.813464	−	0.581615i	$-0.197579\pi$
$503$	36.4882	1.62693	0.813464	+	0.581615i	$0.197579\pi$
$504$	0	0
$505$	−8.22080	−0.365821
$506$	0.361905	0.0160886
$507$	0	0
$508$	8.44988	0.374903
$509$	−11.0073	−0.487891	−0.243946	−	0.969789i	$-0.578442\pi$
$509$	−11.0073	−0.487891	−0.243946	+	0.969789i	$0.578442\pi$
$510$	0	0
$511$	−51.0354	−2.25767
$512$	−5.16950	−0.228462
$513$	0	0
$514$	1.32352	0.0583779
$515$	−6.15534	−0.271237
$516$	0	0
$517$	−0.487351	−0.0214337
$518$	−2.56269	−0.112598
$519$	0	0
$520$	−0.243322	−0.0106704
$521$	−16.8851	−0.739750	−0.369875	−	0.929081i	$-0.620600\pi$
$521$	−16.8851	−0.739750	−0.369875	+	0.929081i	$0.620600\pi$
$522$	0	0
$523$	2.99803	0.131095	0.0655473	−	0.997849i	$-0.479121\pi$
$523$	2.99803	0.131095	0.0655473	+	0.997849i	$0.479121\pi$
$524$	−28.2231	−1.23293
$525$	0	0
$526$	−0.246583	−0.0107515
$527$	51.4505	2.24122
$528$	0	0
$529$	−19.2880	−0.838607
$530$	−0.316540	−0.0137496
$531$	0	0
$532$	82.0882	3.55898
$533$	−3.97292	−0.172086
$534$	0	0
$535$	10.9117	0.471753
$536$	1.20608	0.0520949
$537$	0	0
$538$	1.17180	0.0505199
$539$	46.0679	1.98428
$540$	0	0
$541$	−8.58315	−0.369018	−0.184509	−	0.982831i	$-0.559070\pi$
$541$	−8.58315	−0.369018	−0.184509	+	0.982831i	$0.559070\pi$
$542$	−0.154778	−0.00664830
$543$	0	0
$544$	5.94906	0.255064
$545$	−3.46666	−0.148496
$546$	0	0
$547$	15.4189	0.659264	0.329632	−	0.944110i	$-0.393075\pi$
$547$	15.4189	0.659264	0.329632	+	0.944110i	$0.393075\pi$
$548$	−9.97308	−0.426029
$549$	0	0
$550$	−0.187840	−0.00800952
$551$	−14.7769	−0.629516
$552$	0	0
$553$	−10.2755	−0.436959
$554$	2.00795	0.0853095
$555$	0	0
$556$	−28.8125	−1.22192
$557$	37.7811	1.60084	0.800418	−	0.599443i	$-0.204611\pi$
$557$	37.7811	1.60084	0.800418	+	0.599443i	$0.204611\pi$
$558$	0	0
$559$	−0.807485	−0.0341530
$560$	−19.0543	−0.805189
$561$	0	0
$562$	−0.682266	−0.0287796
$563$	3.88503	0.163734	0.0818672	−	0.996643i	$-0.473912\pi$
$563$	3.88503	0.163734	0.0818672	+	0.996643i	$0.473912\pi$
$564$	0	0
$565$	12.5195	0.526701
$566$	0.717180	0.0301453
$567$	0	0
$568$	−2.33476	−0.0979642
$569$	−2.88059	−0.120761	−0.0603803	−	0.998175i	$-0.519231\pi$
$569$	−2.88059	−0.120761	−0.0603803	+	0.998175i	$0.519231\pi$
$570$	0	0
$571$	−39.0709	−1.63507	−0.817533	−	0.575882i	$-0.804659\pi$
$571$	−39.0709	−1.63507	−0.817533	+	0.575882i	$0.804659\pi$
$572$	−5.37487	−0.224734
$573$	0	0
$574$	1.32844	0.0554482
$575$	−1.92666	−0.0803475
$576$	0	0
$577$	32.6209	1.35802	0.679012	−	0.734127i	$-0.262408\pi$
$577$	32.6209	1.35802	0.679012	+	0.734127i	$0.262408\pi$
$578$	−2.68472	−0.111670
$579$	0	0
$580$	3.43733	0.142727
$581$	−62.8280	−2.60654
$582$	0	0
$583$	−13.9993	−0.579791
$584$	−2.77217	−0.114713
$585$	0	0
$586$	0.513827	0.0212260
$587$	10.9715	0.452843	0.226422	−	0.974029i	$-0.427297\pi$
$587$	10.9715	0.452843	0.226422	+	0.974029i	$0.427297\pi$
$588$	0	0
$589$	57.8649	2.38428
$590$	−0.389357	−0.0160296
$591$	0	0
$592$	32.6003	1.33986
$593$	−13.1231	−0.538902	−0.269451	−	0.963014i	$-0.586842\pi$
$593$	−13.1231	−0.538902	−0.269451	+	0.963014i	$0.586842\pi$
$594$	0	0
$595$	36.5720	1.49931
$596$	42.1163	1.72515
$597$	0	0
$598$	0.117325	0.00479776
$599$	37.4109	1.52857	0.764284	−	0.644880i	$-0.223093\pi$
$599$	37.4109	1.52857	0.764284	+	0.644880i	$0.223093\pi$
$600$	0	0
$601$	−26.3678	−1.07557	−0.537784	−	0.843083i	$-0.680738\pi$
$601$	−26.3678	−1.07557	−0.537784	+	0.843083i	$0.680738\pi$
$602$	0.270002	0.0110045
$603$	0	0
$604$	−17.0028	−0.691835
$605$	2.69259	0.109469
$606$	0	0
$607$	20.2431	0.821641	0.410820	−	0.911716i	$-0.365242\pi$
$607$	20.2431	0.821641	0.410820	+	0.911716i	$0.365242\pi$
$608$	6.69073	0.271345
$609$	0	0
$610$	0.691394	0.0279937
$611$	−0.157992	−0.00639169
$612$	0	0
$613$	21.9533	0.886684	0.443342	−	0.896353i	$-0.353793\pi$
$613$	21.9533	0.886684	0.443342	+	0.896353i	$0.353793\pi$
$614$	1.33219	0.0537627
$615$	0	0
$616$	3.59826	0.144978
$617$	5.03072	0.202529	0.101265	−	0.994860i	$-0.467711\pi$
$617$	5.03072	0.202529	0.101265	+	0.994860i	$0.467711\pi$
$618$	0	0
$619$	−6.54088	−0.262900	−0.131450	−	0.991323i	$-0.541963\pi$
$619$	−6.54088	−0.262900	−0.131450	+	0.991323i	$0.541963\pi$
$620$	−13.4603	−0.540577
$621$	0	0
$622$	0.599723	0.0240467
$623$	4.79409	0.192071
$624$	0	0
$625$	1.00000	0.0400000
$626$	0.732267	0.0292673
$627$	0	0
$628$	20.4540	0.816204
$629$	−62.5717	−2.49490
$630$	0	0
$631$	19.5524	0.778368	0.389184	−	0.921160i	$-0.372757\pi$
$631$	19.5524	0.778368	0.389184	+	0.921160i	$0.372757\pi$
$632$	−0.558151	−0.0222020
$633$	0	0
$634$	0.253681	0.0100750
$635$	4.23393	0.168018
$636$	0	0
$637$	14.9346	0.591730
$638$	−0.323521	−0.0128083
$639$	0	0
$640$	−2.07442	−0.0819985
$641$	17.4060	0.687497	0.343749	−	0.939062i	$-0.388303\pi$
$641$	17.4060	0.687497	0.343749	+	0.939062i	$0.388303\pi$
$642$	0	0
$643$	−34.2668	−1.35135	−0.675676	−	0.737199i	$-0.736148\pi$
$643$	−34.2668	−1.35135	−0.675676	+	0.737199i	$0.736148\pi$
$644$	18.4340	0.726400
$645$	0	0
$646$	−4.26546	−0.167822
$647$	35.9287	1.41250	0.706252	−	0.707961i	$-0.250385\pi$
$647$	35.9287	1.41250	0.706252	+	0.707961i	$0.250385\pi$
$648$	0	0
$649$	−17.2197	−0.675933
$650$	−0.0608952	−0.00238851
$651$	0	0
$652$	5.22827	0.204755
$653$	−18.4266	−0.721087	−0.360544	−	0.932742i	$-0.617409\pi$
$653$	−18.4266	−0.721087	−0.360544	+	0.932742i	$0.617409\pi$
$654$	0	0
$655$	−14.1416	−0.552557
$656$	−16.8993	−0.659807
$657$	0	0
$658$	0.0528286	0.00205947
$659$	43.9371	1.71155	0.855773	−	0.517351i	$-0.173082\pi$
$659$	43.9371	1.71155	0.855773	+	0.517351i	$0.173082\pi$
$660$	0	0
$661$	10.6403	0.413862	0.206931	−	0.978356i	$-0.433653\pi$
$661$	10.6403	0.413862	0.206931	+	0.978356i	$0.433653\pi$
$662$	1.19254	0.0463492
$663$	0	0
$664$	−3.41272	−0.132439
$665$	41.1315	1.59501
$666$	0	0
$667$	−3.31834	−0.128487
$668$	43.3665	1.67790
$669$	0	0
$670$	0.301841	0.0116611
$671$	30.5776	1.18044
$672$	0	0
$673$	22.3901	0.863074	0.431537	−	0.902095i	$-0.357971\pi$
$673$	22.3901	0.863074	0.431537	+	0.902095i	$0.357971\pi$
$674$	0.573186	0.0220783
$675$	0	0
$676$	24.2023	0.930859
$677$	−7.16665	−0.275437	−0.137718	−	0.990471i	$-0.543977\pi$
$677$	−7.16665	−0.275437	−0.137718	+	0.990471i	$0.543977\pi$
$678$	0	0
$679$	27.5683	1.05797
$680$	1.98654	0.0761802
$681$	0	0
$682$	1.26688	0.0485112
$683$	36.2531	1.38719	0.693593	−	0.720367i	$-0.256027\pi$
$683$	36.2531	1.38719	0.693593	+	0.720367i	$0.256027\pi$
$684$	0	0
$685$	−4.99715	−0.190931
$686$	−2.80669	−0.107160
$687$	0	0
$688$	−3.43473	−0.130948
$689$	−4.53838	−0.172899
$690$	0	0
$691$	−33.5789	−1.27740	−0.638701	−	0.769455i	$-0.720528\pi$
$691$	−33.5789	−1.27740	−0.638701	+	0.769455i	$0.720528\pi$
$692$	10.2136	0.388264
$693$	0	0
$694$	−1.25389	−0.0475970
$695$	−14.4369	−0.547622
$696$	0	0
$697$	32.4358	1.22859
$698$	1.01091	0.0382637
$699$	0	0
$700$	−9.56781	−0.361629
$701$	14.1707	0.535219	0.267610	−	0.963527i	$-0.413766\pi$
$701$	14.1707	0.535219	0.267610	+	0.963527i	$0.413766\pi$
$702$	0	0
$703$	−70.3726	−2.65415
$704$	−22.7648	−0.857980
$705$	0	0
$706$	−1.43295	−0.0539297
$707$	39.4112	1.48221
$708$	0	0
$709$	2.27036	0.0852653	0.0426326	−	0.999091i	$-0.486425\pi$
$709$	2.27036	0.0852653	0.0426326	+	0.999091i	$0.486425\pi$
$710$	−0.584310	−0.0219288
$711$	0	0
$712$	0.260408	0.00975919
$713$	12.9943	0.486640
$714$	0	0
$715$	−2.69315	−0.100718
$716$	−49.6964	−1.85724
$717$	0	0
$718$	−1.45847	−0.0544296
$719$	−16.3844	−0.611036	−0.305518	−	0.952186i	$-0.598830\pi$
$719$	−16.3844	−0.611036	−0.305518	+	0.952186i	$0.598830\pi$
$720$	0	0
$721$	29.5092	1.09898
$722$	−3.55899	−0.132452
$723$	0	0
$724$	48.1418	1.78917
$725$	1.72232	0.0639655
$726$	0	0
$727$	33.2005	1.23134	0.615669	−	0.788004i	$-0.288886\pi$
$727$	33.2005	1.23134	0.615669	+	0.788004i	$0.288886\pi$
$728$	1.16651	0.0432336
$729$	0	0
$730$	−0.693779	−0.0256779
$731$	6.59249	0.243832
$732$	0	0
$733$	42.7742	1.57990	0.789951	−	0.613170i	$-0.210106\pi$
$733$	42.7742	1.57990	0.789951	+	0.613170i	$0.210106\pi$
$734$	−1.54641	−0.0570792
$735$	0	0
$736$	1.50249	0.0553825
$737$	13.3492	0.491726
$738$	0	0
$739$	−50.2767	−1.84946	−0.924729	−	0.380626i	$-0.875708\pi$
$739$	−50.2767	−1.84946	−0.924729	+	0.380626i	$0.875708\pi$
$740$	16.3698	0.601764
$741$	0	0
$742$	1.51752	0.0557098
$743$	−50.3608	−1.84756	−0.923780	−	0.382925i	$-0.874917\pi$
$743$	−50.3608	−1.84756	−0.923780	+	0.382925i	$0.874917\pi$
$744$	0	0
$745$	21.1029	0.773152
$746$	−0.770843	−0.0282226
$747$	0	0
$748$	43.8816	1.60447
$749$	−52.3116	−1.91142
$750$	0	0
$751$	−35.6501	−1.30089	−0.650446	−	0.759553i	$-0.725418\pi$
$751$	−35.6501	−1.30089	−0.650446	+	0.759553i	$0.725418\pi$
$752$	−0.672040	−0.0245068
$753$	0	0
$754$	−0.104881	−0.00381955
$755$	−8.51950	−0.310057
$756$	0	0
$757$	−31.0690	−1.12922	−0.564611	−	0.825357i	$-0.690974\pi$
$757$	−31.0690	−1.12922	−0.564611	+	0.825357i	$0.690974\pi$
$758$	1.36782	0.0496814
$759$	0	0
$760$	2.23420	0.0810429
$761$	−20.4591	−0.741644	−0.370822	−	0.928704i	$-0.620924\pi$
$761$	−20.4591	−0.741644	−0.370822	+	0.928704i	$0.620924\pi$
$762$	0	0
$763$	16.6195	0.601666
$764$	−21.1404	−0.764832
$765$	0	0
$766$	1.46857	0.0530616
$767$	−5.58240	−0.201569
$768$	0	0
$769$	−43.3724	−1.56405	−0.782024	−	0.623248i	$-0.785813\pi$
$769$	−43.3724	−1.56405	−0.782024	+	0.623248i	$0.785813\pi$
$770$	0.900521	0.0324525
$771$	0	0
$772$	3.80239	0.136851
$773$	39.1552	1.40831	0.704157	−	0.710045i	$-0.251325\pi$
$773$	39.1552	1.40831	0.704157	+	0.710045i	$0.251325\pi$
$774$	0	0
$775$	−6.74445	−0.242268
$776$	1.49747	0.0537559
$777$	0	0
$778$	0.810429	0.0290553
$779$	36.4796	1.30702
$780$	0	0
$781$	−25.8417	−0.924689
$782$	−0.957864	−0.0342532
$783$	0	0
$784$	63.5260	2.26879
$785$	10.2488	0.365794
$786$	0	0
$787$	−26.3318	−0.938629	−0.469314	−	0.883031i	$-0.655499\pi$
$787$	−26.3318	−0.938629	−0.469314	+	0.883031i	$0.655499\pi$
$788$	−6.46033	−0.230140
$789$	0	0
$790$	−0.139686	−0.00496981
$791$	−60.0197	−2.13406
$792$	0	0
$793$	9.91285	0.352016
$794$	−0.898027	−0.0318698
$795$	0	0
$796$	39.1838	1.38883
$797$	−6.42562	−0.227607	−0.113803	−	0.993503i	$-0.536303\pi$
$797$	−6.42562	−0.227607	−0.113803	+	0.993503i	$0.536303\pi$
$798$	0	0
$799$	1.28989	0.0456329
$800$	−0.779840	−0.0275715
$801$	0	0
$802$	−1.93225	−0.0682302
$803$	−30.6831	−1.08278
$804$	0	0
$805$	9.23660	0.325547
$806$	0.410705	0.0144665
$807$	0	0
$808$	2.14076	0.0753117
$809$	−52.3420	−1.84025	−0.920124	−	0.391627i	$-0.871912\pi$
$809$	−52.3420	−1.84025	−0.920124	+	0.391627i	$0.871912\pi$
$810$	0	0
$811$	−8.33025	−0.292515	−0.146257	−	0.989247i	$-0.546723\pi$
$811$	−8.33025	−0.292515	−0.146257	+	0.989247i	$0.546723\pi$
$812$	−16.4789	−0.578295
$813$	0	0
$814$	−1.54072	−0.0540022
$815$	2.61970	0.0917640
$816$	0	0
$817$	7.41438	0.259396
$818$	−0.219509	−0.00767494
$819$	0	0
$820$	−8.48573	−0.296334
$821$	−29.8702	−1.04248	−0.521239	−	0.853411i	$-0.674530\pi$
$821$	−29.8702	−1.04248	−0.521239	+	0.853411i	$0.674530\pi$
$822$	0	0
$823$	−3.51777	−0.122622	−0.0613108	−	0.998119i	$-0.519528\pi$
$823$	−3.51777	−0.122622	−0.0613108	+	0.998119i	$0.519528\pi$
$824$	1.60290	0.0558396
$825$	0	0
$826$	1.86661	0.0649477
$827$	−51.1824	−1.77979	−0.889893	−	0.456168i	$-0.849221\pi$
$827$	−51.1824	−1.77979	−0.889893	+	0.456168i	$0.849221\pi$
$828$	0	0
$829$	−11.6152	−0.403412	−0.201706	−	0.979446i	$-0.564649\pi$
$829$	−11.6152	−0.403412	−0.201706	+	0.979446i	$0.564649\pi$
$830$	−0.854088	−0.0296458
$831$	0	0
$832$	−7.38004	−0.255857
$833$	−121.929	−4.22460
$834$	0	0
$835$	21.7294	0.751976
$836$	49.3524	1.70689
$837$	0	0
$838$	0.578250	0.0199753
$839$	17.3168	0.597843	0.298921	−	0.954278i	$-0.403373\pi$
$839$	17.3168	0.597843	0.298921	+	0.954278i	$0.403373\pi$
$840$	0	0
$841$	−26.0336	−0.897710
$842$	−0.811444	−0.0279642
$843$	0	0
$844$	24.1372	0.830837
$845$	12.1269	0.417179
$846$	0	0
$847$	−12.9085	−0.443541
$848$	−19.3045	−0.662920
$849$	0	0
$850$	0.497162	0.0170525
$851$	−15.8031	−0.541722
$852$	0	0
$853$	−40.2249	−1.37727	−0.688637	−	0.725107i	$-0.741790\pi$
$853$	−40.2249	−1.37727	−0.688637	+	0.725107i	$0.741790\pi$
$854$	−3.31460	−0.113423
$855$	0	0
$856$	−2.84149	−0.0971200
$857$	26.2666	0.897250	0.448625	−	0.893720i	$-0.351914\pi$
$857$	26.2666	0.897250	0.448625	+	0.893720i	$0.351914\pi$
$858$	0	0
$859$	−24.9036	−0.849701	−0.424851	−	0.905263i	$-0.639673\pi$
$859$	−24.9036	−0.849701	−0.424851	+	0.905263i	$0.639673\pi$
$860$	−1.72470	−0.0588118
$861$	0	0
$862$	2.12815	0.0724850
$863$	−20.4158	−0.694960	−0.347480	−	0.937687i	$-0.612963\pi$
$863$	−20.4158	−0.694960	−0.347480	+	0.937687i	$0.612963\pi$
$864$	0	0
$865$	5.11768	0.174006
$866$	0.555602	0.0188801
$867$	0	0
$868$	64.5296	2.19028
$869$	−6.17776	−0.209566
$870$	0	0
$871$	4.32764	0.146637
$872$	0.902745	0.0305708
$873$	0	0
$874$	−1.07728	−0.0364396
$875$	−4.79409	−0.162070
$876$	0	0
$877$	−25.1713	−0.849973	−0.424987	−	0.905200i	$-0.639721\pi$
$877$	−25.1713	−0.849973	−0.424987	+	0.905200i	$0.639721\pi$
$878$	1.60455	0.0541510
$879$	0	0
$880$	−11.4556	−0.386169
$881$	32.4462	1.09314	0.546570	−	0.837413i	$-0.315933\pi$
$881$	32.4462	1.09314	0.546570	+	0.837413i	$0.315933\pi$
$882$	0	0
$883$	5.23959	0.176326	0.0881632	−	0.996106i	$-0.471900\pi$
$883$	5.23959	0.176326	0.0881632	+	0.996106i	$0.471900\pi$
$884$	14.2258	0.478466
$885$	0	0
$886$	0.0405067	0.00136085
$887$	−52.8979	−1.77614	−0.888068	−	0.459711i	$-0.847953\pi$
$887$	−52.8979	−1.77614	−0.888068	+	0.459711i	$0.847953\pi$
$888$	0	0
$889$	−20.2978	−0.680767
$890$	0.0651711	0.00218454
$891$	0	0
$892$	5.50351	0.184271
$893$	1.45070	0.0485457
$894$	0	0
$895$	−24.9011	−0.832351
$896$	9.94493	0.332237
$897$	0	0
$898$	−2.15738	−0.0719926
$899$	−11.6161	−0.387419
$900$	0	0
$901$	37.0524	1.23439
$902$	0.798675	0.0265930
$903$	0	0
$904$	−3.26018	−0.108432
$905$	24.1221	0.801846
$906$	0	0
$907$	55.4646	1.84167	0.920836	−	0.389951i	$-0.127508\pi$
$907$	55.4646	1.84167	0.920836	+	0.389951i	$0.127508\pi$
$908$	−49.0048	−1.62628
$909$	0	0
$910$	0.291937	0.00967761
$911$	−24.8426	−0.823071	−0.411536	−	0.911394i	$-0.635007\pi$
$911$	−24.8426	−0.823071	−0.411536	+	0.911394i	$0.635007\pi$
$912$	0	0
$913$	−37.7729	−1.25010
$914$	−0.242123	−0.00800873
$915$	0	0
$916$	28.1077	0.928704
$917$	67.7959	2.23882
$918$	0	0
$919$	−22.6793	−0.748122	−0.374061	−	0.927404i	$-0.622035\pi$
$919$	−22.6793	−0.748122	−0.374061	+	0.927404i	$0.622035\pi$
$920$	0.501718	0.0165412
$921$	0	0
$922$	−1.60629	−0.0529003
$923$	−8.37753	−0.275750
$924$	0	0
$925$	8.20230	0.269690
$926$	0.641549	0.0210826
$927$	0	0
$928$	−1.34314	−0.0440906
$929$	21.3802	0.701460	0.350730	−	0.936477i	$-0.385933\pi$
$929$	21.3802	0.701460	0.350730	+	0.936477i	$0.385933\pi$
$930$	0	0
$931$	−137.130	−4.49426
$932$	10.6048	0.347371
$933$	0	0
$934$	0.458592	0.0150056
$935$	21.9875	0.719068
$936$	0	0
$937$	−5.96372	−0.194826	−0.0974132	−	0.995244i	$-0.531057\pi$
$937$	−5.96372	−0.194826	−0.0974132	+	0.995244i	$0.531057\pi$
$938$	−1.44705	−0.0472480
$939$	0	0
$940$	−0.337455	−0.0110066
$941$	−33.6345	−1.09645	−0.548226	−	0.836330i	$-0.684697\pi$
$941$	−33.6345	−1.09645	−0.548226	+	0.836330i	$0.684697\pi$
$942$	0	0
$943$	8.19197	0.266767
$944$	−23.7454	−0.772847
$945$	0	0
$946$	0.162328	0.00527775
$947$	−32.0303	−1.04085	−0.520423	−	0.853909i	$-0.674226\pi$
$947$	−32.0303	−1.04085	−0.520423	+	0.853909i	$0.674226\pi$
$948$	0	0
$949$	−9.94704	−0.322894
$950$	0.559144	0.0181410
$951$	0	0
$952$	−9.52362	−0.308662
$953$	41.3684	1.34005	0.670026	−	0.742337i	$-0.266283\pi$
$953$	41.3684	1.34005	0.670026	+	0.742337i	$0.266283\pi$
$954$	0	0
$955$	−10.5927	−0.342771
$956$	58.8882	1.90458
$957$	0	0
$958$	−0.260486	−0.00841592
$959$	23.9568	0.773605
$960$	0	0
$961$	14.4876	0.467343
$962$	−0.499480	−0.0161039
$963$	0	0
$964$	−50.4246	−1.62407
$965$	1.90524	0.0613318
$966$	0	0
$967$	−15.8416	−0.509432	−0.254716	−	0.967016i	$-0.581982\pi$
$967$	−15.8416	−0.509432	−0.254716	+	0.967016i	$0.581982\pi$
$968$	−0.701170	−0.0225364
$969$	0	0
$970$	0.374765	0.0120330
$971$	−31.6192	−1.01471	−0.507355	−	0.861737i	$-0.669377\pi$
$971$	−31.6192	−1.01471	−0.507355	+	0.861737i	$0.669377\pi$
$972$	0	0
$973$	69.2117	2.21882
$974$	−2.77571	−0.0889395
$975$	0	0
$976$	42.1655	1.34968
$977$	58.5271	1.87245	0.936223	−	0.351406i	$-0.114296\pi$
$977$	58.5271	1.87245	0.936223	+	0.351406i	$0.114296\pi$
$978$	0	0
$979$	2.88226	0.0921174
$980$	31.8986	1.01896
$981$	0	0
$982$	0.621135	0.0198212
$983$	55.9034	1.78304	0.891522	−	0.452978i	$-0.149639\pi$
$983$	55.9034	1.78304	0.891522	+	0.452978i	$0.149639\pi$
$984$	0	0
$985$	−3.23704	−0.103141
$986$	0.856273	0.0272693
$987$	0	0
$988$	15.9994	0.509008
$989$	1.66499	0.0529438
$990$	0	0
$991$	40.3583	1.28203	0.641013	−	0.767530i	$-0.278515\pi$
$991$	40.3583	1.28203	0.641013	+	0.767530i	$0.278515\pi$
$992$	5.25959	0.166992
$993$	0	0
$994$	2.80123	0.0888497
$995$	19.6336	0.622427
$996$	0	0
$997$	−55.4125	−1.75493	−0.877465	−	0.479640i	$-0.840767\pi$
$997$	−55.4125	−1.75493	−0.877465	+	0.479640i	$0.840767\pi$
$998$	−0.611412	−0.0193539
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4005.2.a.s.1.6		10
3.2	odd	2		1335.2.a.j.1.5	✓	10
15.14	odd	2		6675.2.a.z.1.6		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1335.2.a.j.1.5	✓	10	3.2	odd	2
4005.2.a.s.1.6		10	1.1	even	1	trivial
6675.2.a.z.1.6		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)

\(f(q)\)	\(=\)	\(q-0.0651711 q^{2} -1.99575 q^{4} -1.00000 q^{5} +4.79409 q^{7} +0.260408 q^{8} +O(q^{10})\)	\(q-0.0651711 q^{2} -1.99575 q^{4} -1.00000 q^{5} +4.79409 q^{7} +0.260408 q^{8} +0.0651711 q^{10} +2.88226 q^{11} +0.934389 q^{13} -0.312436 q^{14} +3.97453 q^{16} -7.62856 q^{17} -8.57962 q^{19} +1.99575 q^{20} -0.187840 q^{22} -1.92666 q^{23} +1.00000 q^{25} -0.0608952 q^{26} -9.56781 q^{28} +1.72232 q^{29} -6.74445 q^{31} -0.779840 q^{32} +0.497162 q^{34} -4.79409 q^{35} +8.20230 q^{37} +0.559144 q^{38} -0.260408 q^{40} -4.25189 q^{41} -0.864185 q^{43} -5.75228 q^{44} +0.125563 q^{46} -0.169086 q^{47} +15.9833 q^{49} -0.0651711 q^{50} -1.86481 q^{52} -4.85706 q^{53} -2.88226 q^{55} +1.24842 q^{56} -0.112246 q^{58} -5.97438 q^{59} +10.6089 q^{61} +0.439543 q^{62} -7.89825 q^{64} -0.934389 q^{65} +4.63152 q^{67} +15.2247 q^{68} +0.312436 q^{70} -8.96578 q^{71} -10.6455 q^{73} -0.534553 q^{74} +17.1228 q^{76} +13.8178 q^{77} -2.14337 q^{79} -3.97453 q^{80} +0.277101 q^{82} -13.1053 q^{83} +7.62856 q^{85} +0.0563199 q^{86} +0.750562 q^{88} +1.00000 q^{89} +4.47954 q^{91} +3.84515 q^{92} +0.0110195 q^{94} +8.57962 q^{95} +5.75047 q^{97} -1.04165 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - q^{2} + 13 q^{4} - 10 q^{5} - q^{7}+O(q^{10}) \) 10 * q - q^2 + 13 * q^4 - 10 * q^5 - q^7	\( 10 q - q^{2} + 13 q^{4} - 10 q^{5} - q^{7} + q^{10} - 10 q^{11} + 5 q^{13} - 13 q^{14} + 19 q^{16} - 9 q^{17} - 6 q^{19} - 13 q^{20} + 3 q^{23} + 10 q^{25} - 14 q^{26} - 18 q^{28} - 38 q^{29} + 2 q^{31} - 16 q^{32} - 8 q^{34} + q^{35} + 9 q^{37} - 20 q^{38} - 36 q^{41} - 7 q^{43} - 16 q^{44} + 2 q^{46} + 23 q^{47} + 25 q^{49} - q^{50} + 13 q^{52} - 27 q^{53} + 10 q^{55} - 41 q^{56} - 32 q^{58} - 20 q^{59} + 30 q^{61} + 2 q^{62} - 2 q^{64} - 5 q^{65} - 5 q^{67} + 10 q^{68} + 13 q^{70} - 24 q^{71} - 19 q^{73} - 42 q^{74} - 30 q^{76} - 18 q^{77} + 12 q^{79} - 19 q^{80} + 29 q^{82} - 3 q^{83} + 9 q^{85} - 38 q^{86} - 16 q^{88} + 10 q^{89} - 6 q^{91} - 32 q^{92} + 17 q^{94} + 6 q^{95} + 3 q^{97} + 37 q^{98}+O(q^{100}) \) 10 * q - q^2 + 13 * q^4 - 10 * q^5 - q^7 + q^10 - 10 * q^11 + 5 * q^13 - 13 * q^14 + 19 * q^16 - 9 * q^17 - 6 * q^19 - 13 * q^20 + 3 * q^23 + 10 * q^25 - 14 * q^26 - 18 * q^28 - 38 * q^29 + 2 * q^31 - 16 * q^32 - 8 * q^34 + q^35 + 9 * q^37 - 20 * q^38 - 36 * q^41 - 7 * q^43 - 16 * q^44 + 2 * q^46 + 23 * q^47 + 25 * q^49 - q^50 + 13 * q^52 - 27 * q^53 + 10 * q^55 - 41 * q^56 - 32 * q^58 - 20 * q^59 + 30 * q^61 + 2 * q^62 - 2 * q^64 - 5 * q^65 - 5 * q^67 + 10 * q^68 + 13 * q^70 - 24 * q^71 - 19 * q^73 - 42 * q^74 - 30 * q^76 - 18 * q^77 + 12 * q^79 - 19 * q^80 + 29 * q^82 - 3 * q^83 + 9 * q^85 - 38 * q^86 - 16 * q^88 + 10 * q^89 - 6 * q^91 - 32 * q^92 + 17 * q^94 + 6 * q^95 + 3 * q^97 + 37 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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