LMFDB - Embedded newform 4005.2.a.w.1.15

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			1.15
Root			$-1.59605$ 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)

$f(q)$	$=$	$q+1.59605 q^{2} +0.547372 q^{4} -1.00000 q^{5} +1.65189 q^{7} -2.31847 q^{8} +O(q^{10})$	\(q+1.59605 q^{2} +0.547372 q^{4} -1.00000 q^{5} +1.65189 q^{7} -2.31847 q^{8} -1.59605 q^{10} +0.312701 q^{11} -4.26577 q^{13} +2.63650 q^{14} -4.79513 q^{16} -3.29407 q^{17} +2.57815 q^{19} -0.547372 q^{20} +0.499086 q^{22} +5.06311 q^{23} +1.00000 q^{25} -6.80838 q^{26} +0.904200 q^{28} +5.30438 q^{29} +6.85629 q^{31} -3.01633 q^{32} -5.25750 q^{34} -1.65189 q^{35} +9.86431 q^{37} +4.11485 q^{38} +2.31847 q^{40} +4.30603 q^{41} +1.15263 q^{43} +0.171164 q^{44} +8.08097 q^{46} +8.26666 q^{47} -4.27125 q^{49} +1.59605 q^{50} -2.33496 q^{52} -6.28163 q^{53} -0.312701 q^{55} -3.82986 q^{56} +8.46605 q^{58} -5.37742 q^{59} +11.7981 q^{61} +10.9430 q^{62} +4.77605 q^{64} +4.26577 q^{65} +0.735092 q^{67} -1.80308 q^{68} -2.63650 q^{70} -3.08726 q^{71} +4.86570 q^{73} +15.7439 q^{74} +1.41120 q^{76} +0.516549 q^{77} +10.4024 q^{79} +4.79513 q^{80} +6.87263 q^{82} -6.74042 q^{83} +3.29407 q^{85} +1.83965 q^{86} -0.724987 q^{88} +1.00000 q^{89} -7.04660 q^{91} +2.77140 q^{92} +13.1940 q^{94} -2.57815 q^{95} +4.83568 q^{97} -6.81712 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.59605	1.12858	0.564288	−	0.825578i	$-0.309151\pi$
$2$	1.59605	1.12858	0.564288	+	0.825578i	$0.309151\pi$
$3$	0	0
$3$	0	0
$4$	0.547372	0.273686
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	1.65189	0.624357	0.312179	−	0.950023i	$-0.398941\pi$
$7$	1.65189	0.624357	0.312179	+	0.950023i	$0.398941\pi$
$8$	−2.31847	−0.819701
$9$	0	0
$10$	−1.59605	−0.504715
$11$	0.312701	0.0942830	0.0471415	−	0.998888i	$-0.484989\pi$
$11$	0.312701	0.0942830	0.0471415	+	0.998888i	$0.484989\pi$
$12$	0	0
$13$	−4.26577	−1.18311	−0.591556	−	0.806264i	$-0.701486\pi$
$13$	−4.26577	−1.18311	−0.591556	+	0.806264i	$0.701486\pi$
$14$	2.63650	0.704635
$15$	0	0
$16$	−4.79513	−1.19878
$17$	−3.29407	−0.798929	−0.399465	−	0.916749i	$-0.630804\pi$
$17$	−3.29407	−0.798929	−0.399465	+	0.916749i	$0.630804\pi$
$18$	0	0
$19$	2.57815	0.591467	0.295734	−	0.955270i	$-0.404436\pi$
$19$	2.57815	0.591467	0.295734	+	0.955270i	$0.404436\pi$
$20$	−0.547372	−0.122396
$21$	0	0
$22$	0.499086	0.106406
$23$	5.06311	1.05573	0.527866	−	0.849328i	$-0.322992\pi$
$23$	5.06311	1.05573	0.527866	+	0.849328i	$0.322992\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−6.80838	−1.33523
$27$	0	0
$28$	0.904200	0.170878
$29$	5.30438	0.984999	0.492499	−	0.870313i	$-0.336083\pi$
$29$	5.30438	0.984999	0.492499	+	0.870313i	$0.336083\pi$
$30$	0	0
$31$	6.85629	1.23143	0.615713	−	0.787971i	$-0.288868\pi$
$31$	6.85629	1.23143	0.615713	+	0.787971i	$0.288868\pi$
$32$	−3.01633	−0.533216
$33$	0	0
$34$	−5.25750	−0.901653
$35$	−1.65189	−0.279221
$36$	0	0
$37$	9.86431	1.62168	0.810841	−	0.585266i	$-0.199010\pi$
$37$	9.86431	1.62168	0.810841	+	0.585266i	$0.199010\pi$
$38$	4.11485	0.667516
$39$	0	0
$40$	2.31847	0.366582
$41$	4.30603	0.672488	0.336244	−	0.941775i	$-0.390843\pi$
$41$	4.30603	0.672488	0.336244	+	0.941775i	$0.390843\pi$
$42$	0	0
$43$	1.15263	0.175774	0.0878868	−	0.996130i	$-0.471989\pi$
$43$	1.15263	0.175774	0.0878868	+	0.996130i	$0.471989\pi$
$44$	0.171164	0.0258039
$45$	0	0
$46$	8.08097	1.19147
$47$	8.26666	1.20582	0.602908	−	0.797810i	$-0.294008\pi$
$47$	8.26666	1.20582	0.602908	+	0.797810i	$0.294008\pi$
$48$	0	0
$49$	−4.27125	−0.610178
$50$	1.59605	0.225715
$51$	0	0
$52$	−2.33496	−0.323801
$53$	−6.28163	−0.862848	−0.431424	−	0.902149i	$-0.641989\pi$
$53$	−6.28163	−0.862848	−0.431424	+	0.902149i	$0.641989\pi$
$54$	0	0
$55$	−0.312701	−0.0421646
$56$	−3.82986	−0.511786
$57$	0	0
$58$	8.46605	1.11165
$59$	−5.37742	−0.700081	−0.350040	−	0.936735i	$-0.613832\pi$
$59$	−5.37742	−0.700081	−0.350040	+	0.936735i	$0.613832\pi$
$60$	0	0
$61$	11.7981	1.51059	0.755297	−	0.655383i	$-0.227493\pi$
$61$	11.7981	1.51059	0.755297	+	0.655383i	$0.227493\pi$
$62$	10.9430	1.38976
$63$	0	0
$64$	4.77605	0.597006
$65$	4.26577	0.529104
$66$	0	0
$67$	0.735092	0.0898058	0.0449029	−	0.998991i	$-0.485702\pi$
$67$	0.735092	0.0898058	0.0449029	+	0.998991i	$0.485702\pi$
$68$	−1.80308	−0.218656
$69$	0	0
$70$	−2.63650	−0.315122
$71$	−3.08726	−0.366390	−0.183195	−	0.983077i	$-0.558644\pi$
$71$	−3.08726	−0.366390	−0.183195	+	0.983077i	$0.558644\pi$
$72$	0	0
$73$	4.86570	0.569487	0.284743	−	0.958604i	$-0.408092\pi$
$73$	4.86570	0.569487	0.284743	+	0.958604i	$0.408092\pi$
$74$	15.7439	1.83019
$75$	0	0
$76$	1.41120	0.161876
$77$	0.516549	0.0588663
$78$	0	0
$79$	10.4024	1.17036	0.585182	−	0.810902i	$-0.301023\pi$
$79$	10.4024	1.17036	0.585182	+	0.810902i	$0.301023\pi$
$80$	4.79513	0.536112
$81$	0	0
$82$	6.87263	0.758955
$83$	−6.74042	−0.739858	−0.369929	−	0.929060i	$-0.620618\pi$
$83$	−6.74042	−0.739858	−0.369929	+	0.929060i	$0.620618\pi$
$84$	0	0
$85$	3.29407	0.357292
$86$	1.83965	0.198374
$87$	0	0
$88$	−0.724987	−0.0772839
$89$	1.00000	0.106000
$89$	1.00000	0.106000
$90$	0	0
$91$	−7.04660	−0.738685
$92$	2.77140	0.288939
$93$	0	0
$94$	13.1940	1.36086
$95$	−2.57815	−0.264512
$96$	0	0
$97$	4.83568	0.490989	0.245495	−	0.969398i	$-0.421050\pi$
$97$	4.83568	0.490989	0.245495	+	0.969398i	$0.421050\pi$
$98$	−6.81712	−0.688633
$99$	0	0
$100$	0.547372	0.0547372
$101$	17.9368	1.78478	0.892389	−	0.451268i	$-0.149028\pi$
$101$	17.9368	1.78478	0.892389	+	0.451268i	$0.149028\pi$
$102$	0	0
$103$	−12.1543	−1.19759	−0.598797	−	0.800901i	$-0.704354\pi$
$103$	−12.1543	−1.19759	−0.598797	+	0.800901i	$0.704354\pi$
$104$	9.89004	0.969799
$105$	0	0
$106$	−10.0258	−0.973790
$107$	−2.47111	−0.238891	−0.119446	−	0.992841i	$-0.538112\pi$
$107$	−2.47111	−0.238891	−0.119446	+	0.992841i	$0.538112\pi$
$108$	0	0
$109$	10.6997	1.02485	0.512425	−	0.858732i	$-0.328747\pi$
$109$	10.6997	1.02485	0.512425	+	0.858732i	$0.328747\pi$
$110$	−0.499086	−0.0475860
$111$	0	0
$112$	−7.92104	−0.748468
$113$	−17.1698	−1.61520	−0.807599	−	0.589732i	$-0.799233\pi$
$113$	−17.1698	−1.61520	−0.807599	+	0.589732i	$0.799233\pi$
$114$	0	0
$115$	−5.06311	−0.472138
$116$	2.90347	0.269580
$117$	0	0
$118$	−8.58263	−0.790095
$119$	−5.44145	−0.498817
$120$	0	0
$121$	−10.9022	−0.991111
$122$	18.8304	1.70482
$123$	0	0
$124$	3.75294	0.337024
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−0.530525	−0.0470765	−0.0235382	−	0.999723i	$-0.507493\pi$
$127$	−0.530525	−0.0470765	−0.0235382	+	0.999723i	$0.507493\pi$
$128$	13.6555	1.20698
$129$	0	0
$130$	6.80838	0.597134
$131$	3.01167	0.263131	0.131565	−	0.991308i	$-0.458000\pi$
$131$	3.01167	0.263131	0.131565	+	0.991308i	$0.458000\pi$
$132$	0	0
$133$	4.25882	0.369287
$134$	1.17324	0.101353
$135$	0	0
$136$	7.63719	0.654883
$137$	11.0835	0.946925	0.473463	−	0.880814i	$-0.343004\pi$
$137$	11.0835	0.946925	0.473463	+	0.880814i	$0.343004\pi$
$138$	0	0
$139$	5.13255	0.435337	0.217669	−	0.976023i	$-0.430155\pi$
$139$	5.13255	0.435337	0.217669	+	0.976023i	$0.430155\pi$
$140$	−0.904200	−0.0764188
$141$	0	0
$142$	−4.92742	−0.413500
$143$	−1.33391	−0.111547
$144$	0	0
$145$	−5.30438	−0.440505
$146$	7.76589	0.642710
$147$	0	0
$148$	5.39944	0.443831
$149$	1.19416	0.0978294	0.0489147	−	0.998803i	$-0.484424\pi$
$149$	1.19416	0.0978294	0.0489147	+	0.998803i	$0.484424\pi$
$150$	0	0
$151$	−1.29569	−0.105442	−0.0527208	−	0.998609i	$-0.516789\pi$
$151$	−1.29569	−0.105442	−0.0527208	+	0.998609i	$0.516789\pi$
$152$	−5.97734	−0.484826
$153$	0	0
$154$	0.824438	0.0664351
$155$	−6.85629	−0.550710
$156$	0	0
$157$	19.1427	1.52775	0.763876	−	0.645363i	$-0.223294\pi$
$157$	19.1427	1.52775	0.763876	+	0.645363i	$0.223294\pi$
$158$	16.6028	1.32084
$159$	0	0
$160$	3.01633	0.238462
$161$	8.36373	0.659154
$162$	0	0
$163$	−21.1277	−1.65485	−0.827424	−	0.561578i	$-0.810195\pi$
$163$	−21.1277	−1.65485	−0.827424	+	0.561578i	$0.810195\pi$
$164$	2.35700	0.184050
$165$	0	0
$166$	−10.7580	−0.834986
$167$	−11.5765	−0.895813	−0.447906	−	0.894080i	$-0.647830\pi$
$167$	−11.5765	−0.895813	−0.447906	+	0.894080i	$0.647830\pi$
$168$	0	0
$169$	5.19681	0.399754
$170$	5.25750	0.403232
$171$	0	0
$172$	0.630914	0.0481068
$173$	16.3647	1.24419	0.622093	−	0.782943i	$-0.286282\pi$
$173$	16.3647	1.24419	0.622093	+	0.782943i	$0.286282\pi$
$174$	0	0
$175$	1.65189	0.124871
$176$	−1.49944	−0.113025
$177$	0	0
$178$	1.59605	0.119629
$179$	14.5100	1.08452	0.542262	−	0.840209i	$-0.317568\pi$
$179$	14.5100	1.08452	0.542262	+	0.840209i	$0.317568\pi$
$180$	0	0
$181$	−20.6306	−1.53346	−0.766731	−	0.641968i	$-0.778118\pi$
$181$	−20.6306	−1.53346	−0.766731	+	0.641968i	$0.778118\pi$
$182$	−11.2467	−0.833662
$183$	0	0
$184$	−11.7387	−0.865385
$185$	−9.86431	−0.725238
$186$	0	0
$187$	−1.03006	−0.0753254
$188$	4.52494	0.330015
$189$	0	0
$190$	−4.11485	−0.298522
$191$	−19.7013	−1.42554	−0.712768	−	0.701400i	$-0.752559\pi$
$191$	−19.7013	−1.42554	−0.712768	+	0.701400i	$0.752559\pi$
$192$	0	0
$193$	18.5596	1.33595	0.667975	−	0.744184i	$-0.267161\pi$
$193$	18.5596	1.33595	0.667975	+	0.744184i	$0.267161\pi$
$194$	7.71799	0.554119
$195$	0	0
$196$	−2.33796	−0.166997
$197$	−13.1125	−0.934229	−0.467115	−	0.884197i	$-0.654706\pi$
$197$	−13.1125	−0.934229	−0.467115	+	0.884197i	$0.654706\pi$
$198$	0	0
$199$	22.3651	1.58542	0.792711	−	0.609598i	$-0.208669\pi$
$199$	22.3651	1.58542	0.792711	+	0.609598i	$0.208669\pi$
$200$	−2.31847	−0.163940
$201$	0	0
$202$	28.6280	2.01426
$203$	8.76227	0.614991
$204$	0	0
$205$	−4.30603	−0.300746
$206$	−19.3988	−1.35158
$207$	0	0
$208$	20.4549	1.41829
$209$	0.806189	0.0557653
$210$	0	0
$211$	−17.5062	−1.20518	−0.602589	−	0.798051i	$-0.705864\pi$
$211$	−17.5062	−1.20518	−0.602589	+	0.798051i	$0.705864\pi$
$212$	−3.43838	−0.236149
$213$	0	0
$214$	−3.94401	−0.269607
$215$	−1.15263	−0.0786084
$216$	0	0
$217$	11.3259	0.768850
$218$	17.0773	1.15662
$219$	0	0
$220$	−0.171164	−0.0115399
$221$	14.0517	0.945223
$222$	0	0
$223$	22.4098	1.50067	0.750336	−	0.661057i	$-0.229892\pi$
$223$	22.4098	1.50067	0.750336	+	0.661057i	$0.229892\pi$
$224$	−4.98265	−0.332917
$225$	0	0
$226$	−27.4038	−1.82288
$227$	−27.4458	−1.82164	−0.910821	−	0.412801i	$-0.864551\pi$
$227$	−27.4458	−1.82164	−0.910821	+	0.412801i	$0.864551\pi$
$228$	0	0
$229$	26.7472	1.76750	0.883751	−	0.467958i	$-0.155010\pi$
$229$	26.7472	1.76750	0.883751	+	0.467958i	$0.155010\pi$
$230$	−8.08097	−0.532844
$231$	0	0
$232$	−12.2980	−0.807405
$233$	−3.78149	−0.247734	−0.123867	−	0.992299i	$-0.539530\pi$
$233$	−3.78149	−0.247734	−0.123867	+	0.992299i	$0.539530\pi$
$234$	0	0
$235$	−8.26666	−0.539258
$236$	−2.94345	−0.191602
$237$	0	0
$238$	−8.68483	−0.562954
$239$	20.3619	1.31710	0.658550	−	0.752537i	$-0.271170\pi$
$239$	20.3619	1.31710	0.658550	+	0.752537i	$0.271170\pi$
$240$	0	0
$241$	−2.45826	−0.158350	−0.0791752	−	0.996861i	$-0.525229\pi$
$241$	−2.45826	−0.158350	−0.0791752	+	0.996861i	$0.525229\pi$
$242$	−17.4005	−1.11854
$243$	0	0
$244$	6.45795	0.413428
$245$	4.27125	0.272880
$246$	0	0
$247$	−10.9978	−0.699772
$248$	−15.8961	−1.00940
$249$	0	0
$250$	−1.59605	−0.100943
$251$	20.7248	1.30814	0.654070	−	0.756434i	$-0.273060\pi$
$251$	20.7248	1.30814	0.654070	+	0.756434i	$0.273060\pi$
$252$	0	0
$253$	1.58324	0.0995376
$254$	−0.846743	−0.0531294
$255$	0	0
$256$	12.2427	0.765168
$257$	−11.2294	−0.700473	−0.350237	−	0.936661i	$-0.613899\pi$
$257$	−11.2294	−0.700473	−0.350237	+	0.936661i	$0.613899\pi$
$258$	0	0
$259$	16.2948	1.01251
$260$	2.33496	0.144808
$261$	0	0
$262$	4.80677	0.296963
$263$	−2.14224	−0.132096	−0.0660480	−	0.997816i	$-0.521039\pi$
$263$	−2.14224	−0.132096	−0.0660480	+	0.997816i	$0.521039\pi$
$264$	0	0
$265$	6.28163	0.385877
$266$	6.79729	0.416768
$267$	0	0
$268$	0.402369	0.0245786
$269$	−12.2972	−0.749774	−0.374887	−	0.927070i	$-0.622319\pi$
$269$	−12.2972	−0.749774	−0.374887	+	0.927070i	$0.622319\pi$
$270$	0	0
$271$	6.67702	0.405600	0.202800	−	0.979220i	$-0.434996\pi$
$271$	6.67702	0.405600	0.202800	+	0.979220i	$0.434996\pi$
$272$	15.7955	0.957742
$273$	0	0
$274$	17.6898	1.06868
$275$	0.312701	0.0188566
$276$	0	0
$277$	9.65816	0.580303	0.290151	−	0.956981i	$-0.406294\pi$
$277$	9.65816	0.580303	0.290151	+	0.956981i	$0.406294\pi$
$278$	8.19180	0.491312
$279$	0	0
$280$	3.82986	0.228878
$281$	−19.0745	−1.13789	−0.568944	−	0.822376i	$-0.692648\pi$
$281$	−19.0745	−1.13789	−0.568944	+	0.822376i	$0.692648\pi$
$282$	0	0
$283$	3.01850	0.179431	0.0897156	−	0.995967i	$-0.471404\pi$
$283$	3.01850	0.179431	0.0897156	+	0.995967i	$0.471404\pi$
$284$	−1.68988	−0.100276
$285$	0	0
$286$	−2.12899	−0.125890
$287$	7.11310	0.419873
$288$	0	0
$289$	−6.14910	−0.361712
$290$	−8.46605	−0.497144
$291$	0	0
$292$	2.66334	0.155860
$293$	22.7079	1.32661	0.663305	−	0.748349i	$-0.269153\pi$
$293$	22.7079	1.32661	0.663305	+	0.748349i	$0.269153\pi$
$294$	0	0
$295$	5.37742	0.313086
$296$	−22.8701	−1.32930
$297$	0	0
$298$	1.90594	0.110408
$299$	−21.5981	−1.24905
$300$	0	0
$301$	1.90401	0.109746
$302$	−2.06798	−0.118999
$303$	0	0
$304$	−12.3625	−0.709040
$305$	−11.7981	−0.675558
$306$	0	0
$307$	22.4280	1.28003	0.640016	−	0.768362i	$-0.278928\pi$
$307$	22.4280	1.28003	0.640016	+	0.768362i	$0.278928\pi$
$308$	0.282744	0.0161109
$309$	0	0
$310$	−10.9430	−0.621519
$311$	−2.80828	−0.159243	−0.0796213	−	0.996825i	$-0.525371\pi$
$311$	−2.80828	−0.159243	−0.0796213	+	0.996825i	$0.525371\pi$
$312$	0	0
$313$	24.0478	1.35926	0.679631	−	0.733554i	$-0.262140\pi$
$313$	24.0478	1.35926	0.679631	+	0.733554i	$0.262140\pi$
$314$	30.5527	1.72419
$315$	0	0
$316$	5.69399	0.320312
$317$	0.567561	0.0318774	0.0159387	−	0.999873i	$-0.494926\pi$
$317$	0.567561	0.0318774	0.0159387	+	0.999873i	$0.494926\pi$
$318$	0	0
$319$	1.65869	0.0928686
$320$	−4.77605	−0.266989
$321$	0	0
$322$	13.3489	0.743906
$323$	−8.49259	−0.472540
$324$	0	0
$325$	−4.26577	−0.236622
$326$	−33.7208	−1.86762
$327$	0	0
$328$	−9.98337	−0.551239
$329$	13.6557	0.752860
$330$	0	0
$331$	21.8342	1.20012	0.600059	−	0.799956i	$-0.295144\pi$
$331$	21.8342	1.20012	0.600059	+	0.799956i	$0.295144\pi$
$332$	−3.68952	−0.202489
$333$	0	0
$334$	−18.4766	−1.01099
$335$	−0.735092	−0.0401624
$336$	0	0
$337$	26.8567	1.46298	0.731490	−	0.681852i	$-0.238825\pi$
$337$	26.8567	1.46298	0.731490	+	0.681852i	$0.238825\pi$
$338$	8.29436	0.451153
$339$	0	0
$340$	1.80308	0.0977858
$341$	2.14397	0.116102
$342$	0	0
$343$	−18.6189	−1.00533
$344$	−2.67232	−0.144082
$345$	0	0
$346$	26.1189	1.40416
$347$	−19.1660	−1.02889	−0.514443	−	0.857525i	$-0.672001\pi$
$347$	−19.1660	−1.02889	−0.514443	+	0.857525i	$0.672001\pi$
$348$	0	0
$349$	−16.8496	−0.901937	−0.450968	−	0.892540i	$-0.648921\pi$
$349$	−16.8496	−0.901937	−0.450968	+	0.892540i	$0.648921\pi$
$350$	2.63650	0.140927
$351$	0	0
$352$	−0.943209	−0.0502732
$353$	−20.7985	−1.10699	−0.553497	−	0.832851i	$-0.686707\pi$
$353$	−20.7985	−1.10699	−0.553497	+	0.832851i	$0.686707\pi$
$354$	0	0
$355$	3.08726	0.163855
$356$	0.547372	0.0290106
$357$	0	0
$358$	23.1586	1.22397
$359$	−24.1169	−1.27284	−0.636419	−	0.771343i	$-0.719585\pi$
$359$	−24.1169	−1.27284	−0.636419	+	0.771343i	$0.719585\pi$
$360$	0	0
$361$	−12.3532	−0.650167
$362$	−32.9275	−1.73063
$363$	0	0
$364$	−3.85711	−0.202167
$365$	−4.86570	−0.254682
$366$	0	0
$367$	1.78328	0.0930864	0.0465432	−	0.998916i	$-0.485179\pi$
$367$	1.78328	0.0930864	0.0465432	+	0.998916i	$0.485179\pi$
$368$	−24.2783	−1.26559
$369$	0	0
$370$	−15.7439	−0.818487
$371$	−10.3766	−0.538725
$372$	0	0
$373$	4.80838	0.248968	0.124484	−	0.992222i	$-0.460272\pi$
$373$	4.80838	0.248968	0.124484	+	0.992222i	$0.460272\pi$
$374$	−1.64403	−0.0850105
$375$	0	0
$376$	−19.1660	−0.988410
$377$	−22.6273	−1.16536
$378$	0	0
$379$	14.9770	0.769319	0.384659	−	0.923059i	$-0.374319\pi$
$379$	14.9770	0.769319	0.384659	+	0.923059i	$0.374319\pi$
$380$	−1.41120	−0.0723932
$381$	0	0
$382$	−31.4442	−1.60883
$383$	−16.5874	−0.847579	−0.423789	−	0.905761i	$-0.639300\pi$
$383$	−16.5874	−0.847579	−0.423789	+	0.905761i	$0.639300\pi$
$384$	0	0
$385$	−0.516549	−0.0263258
$386$	29.6220	1.50772
$387$	0	0
$388$	2.64692	0.134377
$389$	−4.03093	−0.204376	−0.102188	−	0.994765i	$-0.532584\pi$
$389$	−4.03093	−0.204376	−0.102188	+	0.994765i	$0.532584\pi$
$390$	0	0
$391$	−16.6782	−0.843455
$392$	9.90274	0.500164
$393$	0	0
$394$	−20.9282	−1.05435
$395$	−10.4024	−0.523402
$396$	0	0
$397$	3.69116	0.185254	0.0926270	−	0.995701i	$-0.470474\pi$
$397$	3.69116	0.185254	0.0926270	+	0.995701i	$0.470474\pi$
$398$	35.6958	1.78927
$399$	0	0
$400$	−4.79513	−0.239756
$401$	20.7374	1.03558	0.517788	−	0.855509i	$-0.326756\pi$
$401$	20.7374	1.03558	0.517788	+	0.855509i	$0.326756\pi$
$402$	0	0
$403$	−29.2474	−1.45691
$404$	9.81809	0.488468
$405$	0	0
$406$	13.9850	0.694065
$407$	3.08458	0.152897
$408$	0	0
$409$	3.59295	0.177660	0.0888300	−	0.996047i	$-0.471687\pi$
$409$	3.59295	0.177660	0.0888300	+	0.996047i	$0.471687\pi$
$410$	−6.87263	−0.339415
$411$	0	0
$412$	−6.65289	−0.327765
$413$	−8.88293	−0.437101
$414$	0	0
$415$	6.74042	0.330874
$416$	12.8670	0.630854
$417$	0	0
$418$	1.28672	0.0629354
$419$	17.8945	0.874204	0.437102	−	0.899412i	$-0.356005\pi$
$419$	17.8945	0.874204	0.437102	+	0.899412i	$0.356005\pi$
$420$	0	0
$421$	−15.0678	−0.734362	−0.367181	−	0.930150i	$-0.619677\pi$
$421$	−15.0678	−0.734362	−0.367181	+	0.930150i	$0.619677\pi$
$422$	−27.9408	−1.36014
$423$	0	0
$424$	14.5637	0.707278
$425$	−3.29407	−0.159786
$426$	0	0
$427$	19.4892	0.943150
$428$	−1.35262	−0.0653811
$429$	0	0
$430$	−1.83965	−0.0887156
$431$	−21.4719	−1.03426	−0.517132	−	0.855905i	$-0.673001\pi$
$431$	−21.4719	−1.03426	−0.517132	+	0.855905i	$0.673001\pi$
$432$	0	0
$433$	−2.12381	−0.102064	−0.0510319	−	0.998697i	$-0.516251\pi$
$433$	−2.12381	−0.102064	−0.0510319	+	0.998697i	$0.516251\pi$
$434$	18.0766	0.867706
$435$	0	0
$436$	5.85674	0.280487
$437$	13.0534	0.624431
$438$	0	0
$439$	−38.5896	−1.84178	−0.920891	−	0.389819i	$-0.872538\pi$
$439$	−38.5896	−1.84178	−0.920891	+	0.389819i	$0.872538\pi$
$440$	0.724987	0.0345624
$441$	0	0
$442$	22.4273	1.06676
$443$	18.5685	0.882217	0.441108	−	0.897454i	$-0.354585\pi$
$443$	18.5685	0.882217	0.441108	+	0.897454i	$0.354585\pi$
$444$	0	0
$445$	−1.00000	−0.0474045
$446$	35.7671	1.69362
$447$	0	0
$448$	7.88953	0.372745
$449$	−36.0323	−1.70047	−0.850236	−	0.526402i	$-0.823541\pi$
$449$	−36.0323	−1.70047	−0.850236	+	0.526402i	$0.823541\pi$
$450$	0	0
$451$	1.34650	0.0634042
$452$	−9.39826	−0.442057
$453$	0	0
$454$	−43.8049	−2.05586
$455$	7.04660	0.330350
$456$	0	0
$457$	10.4574	0.489175	0.244588	−	0.969627i	$-0.421347\pi$
$457$	10.4574	0.489175	0.244588	+	0.969627i	$0.421347\pi$
$458$	42.6898	1.99476
$459$	0	0
$460$	−2.77140	−0.129217
$461$	−0.185494	−0.00863930	−0.00431965	−	0.999991i	$-0.501375\pi$
$461$	−0.185494	−0.00863930	−0.00431965	+	0.999991i	$0.501375\pi$
$462$	0	0
$463$	−7.19924	−0.334577	−0.167288	−	0.985908i	$-0.553501\pi$
$463$	−7.19924	−0.334577	−0.167288	+	0.985908i	$0.553501\pi$
$464$	−25.4352	−1.18080
$465$	0	0
$466$	−6.03544	−0.279587
$467$	−18.1602	−0.840354	−0.420177	−	0.907442i	$-0.638032\pi$
$467$	−18.1602	−0.840354	−0.420177	+	0.907442i	$0.638032\pi$
$468$	0	0
$469$	1.21429	0.0560709
$470$	−13.1940	−0.608594
$471$	0	0
$472$	12.4674	0.573857
$473$	0.360427	0.0165725
$474$	0	0
$475$	2.57815	0.118293
$476$	−2.97850	−0.136519
$477$	0	0
$478$	32.4985	1.48645
$479$	14.5331	0.664035	0.332018	−	0.943273i	$-0.392271\pi$
$479$	14.5331	0.664035	0.332018	+	0.943273i	$0.392271\pi$
$480$	0	0
$481$	−42.0789	−1.91863
$482$	−3.92350	−0.178710
$483$	0	0
$484$	−5.96756	−0.271253
$485$	−4.83568	−0.219577
$486$	0	0
$487$	22.5099	1.02002	0.510011	−	0.860168i	$-0.329641\pi$
$487$	22.5099	1.02002	0.510011	+	0.860168i	$0.329641\pi$
$488$	−27.3535	−1.23824
$489$	0	0
$490$	6.81712	0.307966
$491$	14.0158	0.632524	0.316262	−	0.948672i	$-0.397572\pi$
$491$	14.0158	0.632524	0.316262	+	0.948672i	$0.397572\pi$
$492$	0	0
$493$	−17.4730	−0.786944
$494$	−17.5530	−0.789746
$495$	0	0
$496$	−32.8768	−1.47621
$497$	−5.09983	−0.228758
$498$	0	0
$499$	21.7705	0.974583	0.487292	−	0.873239i	$-0.337985\pi$
$499$	21.7705	0.974583	0.487292	+	0.873239i	$0.337985\pi$
$500$	−0.547372	−0.0244792
$501$	0	0
$502$	33.0779	1.47634
$503$	−21.0760	−0.939731	−0.469865	−	0.882738i	$-0.655698\pi$
$503$	−21.0760	−0.939731	−0.469865	+	0.882738i	$0.655698\pi$
$504$	0	0
$505$	−17.9368	−0.798177
$506$	2.52693	0.112336
$507$	0	0
$508$	−0.290394	−0.0128842
$509$	13.1967	0.584934	0.292467	−	0.956276i	$-0.405524\pi$
$509$	13.1967	0.584934	0.292467	+	0.956276i	$0.405524\pi$
$510$	0	0
$511$	8.03761	0.355563
$512$	−7.77101	−0.343433
$513$	0	0
$514$	−17.9227	−0.790538
$515$	12.1543	0.535581
$516$	0	0
$517$	2.58500	0.113688
$518$	26.0073	1.14269
$519$	0	0
$520$	−9.89004	−0.433707
$521$	31.6444	1.38637	0.693183	−	0.720762i	$-0.256208\pi$
$521$	31.6444	1.38637	0.693183	+	0.720762i	$0.256208\pi$
$522$	0	0
$523$	19.5164	0.853393	0.426696	−	0.904395i	$-0.359677\pi$
$523$	19.5164	0.853393	0.426696	+	0.904395i	$0.359677\pi$
$524$	1.64850	0.0720151
$525$	0	0
$526$	−3.41911	−0.149080
$527$	−22.5851	−0.983822
$528$	0	0
$529$	2.63511	0.114570
$530$	10.0258	0.435492
$531$	0	0
$532$	2.33116	0.101069
$533$	−18.3685	−0.795629
$534$	0	0
$535$	2.47111	0.106835
$536$	−1.70429	−0.0736139
$537$	0	0
$538$	−19.6269	−0.846178
$539$	−1.33562	−0.0575294
$540$	0	0
$541$	−16.8616	−0.724938	−0.362469	−	0.931996i	$-0.618066\pi$
$541$	−16.8616	−0.724938	−0.362469	+	0.931996i	$0.618066\pi$
$542$	10.6569	0.457751
$543$	0	0
$544$	9.93599	0.426002
$545$	−10.6997	−0.458327
$546$	0	0
$547$	−3.59443	−0.153687	−0.0768433	−	0.997043i	$-0.524484\pi$
$547$	−3.59443	−0.153687	−0.0768433	+	0.997043i	$0.524484\pi$
$548$	6.06678	0.259160
$549$	0	0
$550$	0.499086	0.0212811
$551$	13.6755	0.582594
$552$	0	0
$553$	17.1837	0.730725
$554$	15.4149	0.654916
$555$	0	0
$556$	2.80941	0.119146
$557$	0.0100663	0.000426524 0	0.000213262	−	1.00000i	$-0.499932\pi$
$557$	0.0100663	0.000426524 0	0.000213262		1.00000i	$0.499932\pi$
$558$	0	0
$559$	−4.91684	−0.207960
$560$	7.92104	0.334725
$561$	0	0
$562$	−30.4438	−1.28419
$563$	−37.8978	−1.59720	−0.798601	−	0.601861i	$-0.794426\pi$
$563$	−37.8978	−1.59720	−0.798601	+	0.601861i	$0.794426\pi$
$564$	0	0
$565$	17.1698	0.722339
$566$	4.81768	0.202502
$567$	0	0
$568$	7.15771	0.300331
$569$	−23.1594	−0.970891	−0.485446	−	0.874267i	$-0.661343\pi$
$569$	−23.1594	−0.970891	−0.485446	+	0.874267i	$0.661343\pi$
$570$	0	0
$571$	−24.5298	−1.02654	−0.513271	−	0.858227i	$-0.671566\pi$
$571$	−24.5298	−1.02654	−0.513271	+	0.858227i	$0.671566\pi$
$572$	−0.730146	−0.0305289
$573$	0	0
$574$	11.3528	0.473859
$575$	5.06311	0.211146
$576$	0	0
$577$	−11.7052	−0.487292	−0.243646	−	0.969864i	$-0.578343\pi$
$577$	−11.7052	−0.487292	−0.243646	+	0.969864i	$0.578343\pi$
$578$	−9.81427	−0.408220
$579$	0	0
$580$	−2.90347	−0.120560
$581$	−11.1345	−0.461935
$582$	0	0
$583$	−1.96427	−0.0813519
$584$	−11.2810	−0.466809
$585$	0	0
$586$	36.2429	1.49718
$587$	−6.12814	−0.252936	−0.126468	−	0.991971i	$-0.540364\pi$
$587$	−6.12814	−0.252936	−0.126468	+	0.991971i	$0.540364\pi$
$588$	0	0
$589$	17.6765	0.728348
$590$	8.58263	0.353341
$591$	0	0
$592$	−47.3006	−1.94404
$593$	−18.5032	−0.759837	−0.379918	−	0.925020i	$-0.624048\pi$
$593$	−18.5032	−0.759837	−0.379918	+	0.925020i	$0.624048\pi$
$594$	0	0
$595$	5.44145	0.223078
$596$	0.653649	0.0267745
$597$	0	0
$598$	−34.4716	−1.40965
$599$	45.8000	1.87134	0.935668	−	0.352882i	$-0.114798\pi$
$599$	45.8000	1.87134	0.935668	+	0.352882i	$0.114798\pi$
$600$	0	0
$601$	−35.5468	−1.44998	−0.724991	−	0.688758i	$-0.758156\pi$
$601$	−35.5468	−1.44998	−0.724991	+	0.688758i	$0.758156\pi$
$602$	3.03890	0.123856
$603$	0	0
$604$	−0.709224	−0.0288579
$605$	10.9022	0.443238
$606$	0	0
$607$	−32.3881	−1.31459	−0.657297	−	0.753632i	$-0.728300\pi$
$607$	−32.3881	−1.31459	−0.657297	+	0.753632i	$0.728300\pi$
$608$	−7.77653	−0.315380
$609$	0	0
$610$	−18.8304	−0.762419
$611$	−35.2637	−1.42662
$612$	0	0
$613$	9.42107	0.380513	0.190257	−	0.981734i	$-0.439068\pi$
$613$	9.42107	0.380513	0.190257	+	0.981734i	$0.439068\pi$
$614$	35.7961	1.44461
$615$	0	0
$616$	−1.19760	−0.0482528
$617$	−32.5296	−1.30959	−0.654796	−	0.755805i	$-0.727246\pi$
$617$	−32.5296	−1.30959	−0.654796	+	0.755805i	$0.727246\pi$
$618$	0	0
$619$	3.60286	0.144811	0.0724056	−	0.997375i	$-0.476932\pi$
$619$	3.60286	0.144811	0.0724056	+	0.997375i	$0.476932\pi$
$620$	−3.75294	−0.150722
$621$	0	0
$622$	−4.48214	−0.179718
$623$	1.65189	0.0661817
$624$	0	0
$625$	1.00000	0.0400000
$626$	38.3815	1.53403
$627$	0	0
$628$	10.4782	0.418124
$629$	−32.4937	−1.29561
$630$	0	0
$631$	6.77736	0.269802	0.134901	−	0.990859i	$-0.456928\pi$
$631$	6.77736	0.269802	0.134901	+	0.990859i	$0.456928\pi$
$632$	−24.1176	−0.959348
$633$	0	0
$634$	0.905855	0.0359761
$635$	0.530525	0.0210532
$636$	0	0
$637$	18.2202	0.721909
$638$	2.64734	0.104809
$639$	0	0
$640$	−13.6555	−0.539780
$641$	−12.3607	−0.488220	−0.244110	−	0.969748i	$-0.578496\pi$
$641$	−12.3607	−0.488220	−0.244110	+	0.969748i	$0.578496\pi$
$642$	0	0
$643$	−13.7782	−0.543360	−0.271680	−	0.962388i	$-0.587579\pi$
$643$	−13.7782	−0.543360	−0.271680	+	0.962388i	$0.587579\pi$
$644$	4.57807	0.180401
$645$	0	0
$646$	−13.5546	−0.533298
$647$	−15.0963	−0.593499	−0.296749	−	0.954955i	$-0.595903\pi$
$647$	−15.0963	−0.593499	−0.296749	+	0.954955i	$0.595903\pi$
$648$	0	0
$649$	−1.68153	−0.0660057
$650$	−6.80838	−0.267047
$651$	0	0
$652$	−11.5647	−0.452908
$653$	8.93166	0.349523	0.174761	−	0.984611i	$-0.444085\pi$
$653$	8.93166	0.349523	0.174761	+	0.984611i	$0.444085\pi$
$654$	0	0
$655$	−3.01167	−0.117676
$656$	−20.6479	−0.806167
$657$	0	0
$658$	21.7951	0.849661
$659$	−23.2632	−0.906205	−0.453103	−	0.891458i	$-0.649683\pi$
$659$	−23.2632	−0.906205	−0.453103	+	0.891458i	$0.649683\pi$
$660$	0	0
$661$	32.4474	1.26206	0.631028	−	0.775760i	$-0.282633\pi$
$661$	32.4474	1.26206	0.631028	+	0.775760i	$0.282633\pi$
$662$	34.8485	1.35443
$663$	0	0
$664$	15.6274	0.606462
$665$	−4.25882	−0.165150
$666$	0	0
$667$	26.8567	1.03989
$668$	−6.33662	−0.245171
$669$	0	0
$670$	−1.17324	−0.0453263
$671$	3.68928	0.142423
$672$	0	0
$673$	9.30803	0.358798	0.179399	−	0.983776i	$-0.442585\pi$
$673$	9.30803	0.358798	0.179399	+	0.983776i	$0.442585\pi$
$674$	42.8647	1.65109
$675$	0	0
$676$	2.84458	0.109407
$677$	13.4553	0.517127	0.258564	−	0.965994i	$-0.416751\pi$
$677$	13.4553	0.517127	0.258564	+	0.965994i	$0.416751\pi$
$678$	0	0
$679$	7.98803	0.306553
$680$	−7.63719	−0.292873
$681$	0	0
$682$	3.42188	0.131031
$683$	5.46981	0.209296	0.104648	−	0.994509i	$-0.466628\pi$
$683$	5.46981	0.209296	0.104648	+	0.994509i	$0.466628\pi$
$684$	0	0
$685$	−11.0835	−0.423478
$686$	−29.7167	−1.13459
$687$	0	0
$688$	−5.52698	−0.210714
$689$	26.7960	1.02085
$690$	0	0
$691$	43.5511	1.65676	0.828382	−	0.560164i	$-0.189262\pi$
$691$	43.5511	1.65676	0.828382	+	0.560164i	$0.189262\pi$
$692$	8.95758	0.340516
$693$	0	0
$694$	−30.5899	−1.16118
$695$	−5.13255	−0.194689
$696$	0	0
$697$	−14.1843	−0.537270
$698$	−26.8927	−1.01790
$699$	0	0
$700$	0.904200	0.0341755
$701$	−2.98694	−0.112815	−0.0564075	−	0.998408i	$-0.517965\pi$
$701$	−2.98694	−0.112815	−0.0564075	+	0.998408i	$0.517965\pi$
$702$	0	0
$703$	25.4316	0.959172
$704$	1.49348	0.0562875
$705$	0	0
$706$	−33.1954	−1.24933
$707$	29.6297	1.11434
$708$	0	0
$709$	19.1866	0.720568	0.360284	−	0.932843i	$-0.382680\pi$
$709$	19.1866	0.720568	0.360284	+	0.932843i	$0.382680\pi$
$710$	4.92742	0.184923
$711$	0	0
$712$	−2.31847	−0.0868882
$713$	34.7142	1.30006
$714$	0	0
$715$	1.33391	0.0498855
$716$	7.94234	0.296819
$717$	0	0
$718$	−38.4917	−1.43650
$719$	−43.3314	−1.61599	−0.807994	−	0.589190i	$-0.799447\pi$
$719$	−43.3314	−1.61599	−0.807994	+	0.589190i	$0.799447\pi$
$720$	0	0
$721$	−20.0775	−0.747727
$722$	−19.7163	−0.733763
$723$	0	0
$724$	−11.2926	−0.419687
$725$	5.30438	0.197000
$726$	0	0
$727$	38.6380	1.43300	0.716502	−	0.697585i	$-0.245742\pi$
$727$	38.6380	1.43300	0.716502	+	0.697585i	$0.245742\pi$
$728$	16.3373	0.605501
$729$	0	0
$730$	−7.76589	−0.287428
$731$	−3.79683	−0.140431
$732$	0	0
$733$	26.6371	0.983865	0.491932	−	0.870633i	$-0.336291\pi$
$733$	26.6371	0.983865	0.491932	+	0.870633i	$0.336291\pi$
$734$	2.84620	0.105055
$735$	0	0
$736$	−15.2720	−0.562933
$737$	0.229864	0.00846716
$738$	0	0
$739$	16.8083	0.618304	0.309152	−	0.951013i	$-0.399955\pi$
$739$	16.8083	0.618304	0.309152	+	0.951013i	$0.399955\pi$
$740$	−5.39944	−0.198487
$741$	0	0
$742$	−16.5615	−0.607993
$743$	−30.2622	−1.11021	−0.555107	−	0.831779i	$-0.687323\pi$
$743$	−30.2622	−1.11021	−0.555107	+	0.831779i	$0.687323\pi$
$744$	0	0
$745$	−1.19416	−0.0437506
$746$	7.67440	0.280980
$747$	0	0
$748$	−0.563826	−0.0206155
$749$	−4.08201	−0.149153
$750$	0	0
$751$	−23.5893	−0.860787	−0.430394	−	0.902641i	$-0.641625\pi$
$751$	−23.5893	−0.860787	−0.430394	+	0.902641i	$0.641625\pi$
$752$	−39.6397	−1.44551
$753$	0	0
$754$	−36.1142	−1.31520
$755$	1.29569	0.0471550
$756$	0	0
$757$	−45.5701	−1.65627	−0.828137	−	0.560526i	$-0.810599\pi$
$757$	−45.5701	−1.65627	−0.828137	+	0.560526i	$0.810599\pi$
$758$	23.9041	0.868235
$759$	0	0
$760$	5.97734	0.216821
$761$	34.3822	1.24636	0.623178	−	0.782080i	$-0.285841\pi$
$761$	34.3822	1.24636	0.623178	+	0.782080i	$0.285841\pi$
$762$	0	0
$763$	17.6748	0.639872
$764$	−10.7839	−0.390149
$765$	0	0
$766$	−26.4744	−0.956558
$767$	22.9389	0.828274
$768$	0	0
$769$	−5.77880	−0.208389	−0.104194	−	0.994557i	$-0.533226\pi$
$769$	−5.77880	−0.208389	−0.104194	+	0.994557i	$0.533226\pi$
$770$	−0.824438	−0.0297107
$771$	0	0
$772$	10.1590	0.365631
$773$	7.43120	0.267282	0.133641	−	0.991030i	$-0.457333\pi$
$773$	7.43120	0.267282	0.133641	+	0.991030i	$0.457333\pi$
$774$	0	0
$775$	6.85629	0.246285
$776$	−11.2114	−0.402465
$777$	0	0
$778$	−6.43356	−0.230654
$779$	11.1016	0.397755
$780$	0	0
$781$	−0.965390	−0.0345444
$782$	−26.6193	−0.951904
$783$	0	0
$784$	20.4812	0.731470
$785$	−19.1427	−0.683232
$786$	0	0
$787$	−6.45412	−0.230064	−0.115032	−	0.993362i	$-0.536697\pi$
$787$	−6.45412	−0.230064	−0.115032	+	0.993362i	$0.536697\pi$
$788$	−7.17743	−0.255685
$789$	0	0
$790$	−16.6028	−0.590700
$791$	−28.3627	−1.00846
$792$	0	0
$793$	−50.3280	−1.78720
$794$	5.89127	0.209073
$795$	0	0
$796$	12.2420	0.433907
$797$	−18.2999	−0.648216	−0.324108	−	0.946020i	$-0.605064\pi$
$797$	−18.2999	−0.648216	−0.324108	+	0.946020i	$0.605064\pi$
$798$	0	0
$799$	−27.2310	−0.963362
$800$	−3.01633	−0.106643
$801$	0	0
$802$	33.0979	1.16873
$803$	1.52151	0.0536929
$804$	0	0
$805$	−8.36373	−0.294783
$806$	−46.6802	−1.64424
$807$	0	0
$808$	−41.5858	−1.46298
$809$	48.0281	1.68858	0.844288	−	0.535889i	$-0.180023\pi$
$809$	48.0281	1.68858	0.844288	+	0.535889i	$0.180023\pi$
$810$	0	0
$811$	20.2349	0.710543	0.355272	−	0.934763i	$-0.384388\pi$
$811$	20.2349	0.710543	0.355272	+	0.934763i	$0.384388\pi$
$812$	4.79622	0.168314
$813$	0	0
$814$	4.92314	0.172556
$815$	21.1277	0.740071
$816$	0	0
$817$	2.97164	0.103964
$818$	5.73452	0.200503
$819$	0	0
$820$	−2.35700	−0.0823099
$821$	−18.6121	−0.649566	−0.324783	−	0.945788i	$-0.605291\pi$
$821$	−18.6121	−0.649566	−0.324783	+	0.945788i	$0.605291\pi$
$822$	0	0
$823$	−39.6408	−1.38179	−0.690896	−	0.722954i	$-0.742784\pi$
$823$	−39.6408	−1.38179	−0.690896	+	0.722954i	$0.742784\pi$
$824$	28.1792	0.981670
$825$	0	0
$826$	−14.1776	−0.493302
$827$	−46.0444	−1.60112	−0.800561	−	0.599251i	$-0.795465\pi$
$827$	−46.0444	−1.60112	−0.800561	+	0.599251i	$0.795465\pi$
$828$	0	0
$829$	−22.8796	−0.794641	−0.397320	−	0.917680i	$-0.630060\pi$
$829$	−22.8796	−0.794641	−0.397320	+	0.917680i	$0.630060\pi$
$830$	10.7580	0.373417
$831$	0	0
$832$	−20.3735	−0.706326
$833$	14.0698	0.487489
$834$	0	0
$835$	11.5765	0.400620
$836$	0.441285	0.0152622
$837$	0	0
$838$	28.5605	0.986607
$839$	51.4718	1.77700	0.888502	−	0.458872i	$-0.151746\pi$
$839$	51.4718	1.77700	0.888502	+	0.458872i	$0.151746\pi$
$840$	0	0
$841$	−0.863551	−0.0297776
$842$	−24.0490	−0.828784
$843$	0	0
$844$	−9.58242	−0.329840
$845$	−5.19681	−0.178776
$846$	0	0
$847$	−18.0093	−0.618807
$848$	30.1212	1.03437
$849$	0	0
$850$	−5.25750	−0.180331
$851$	49.9441	1.71206
$852$	0	0
$853$	−52.3073	−1.79097	−0.895485	−	0.445093i	$-0.853171\pi$
$853$	−52.3073	−1.79097	−0.895485	+	0.445093i	$0.853171\pi$
$854$	31.1058	1.06442
$855$	0	0
$856$	5.72918	0.195819
$857$	44.2539	1.51169	0.755843	−	0.654753i	$-0.227227\pi$
$857$	44.2539	1.51169	0.755843	+	0.654753i	$0.227227\pi$
$858$	0	0
$859$	19.9864	0.681928	0.340964	−	0.940076i	$-0.389247\pi$
$859$	19.9864	0.681928	0.340964	+	0.940076i	$0.389247\pi$
$860$	−0.630914	−0.0215140
$861$	0	0
$862$	−34.2702	−1.16725
$863$	43.4641	1.47953	0.739767	−	0.672863i	$-0.234936\pi$
$863$	43.4641	1.47953	0.739767	+	0.672863i	$0.234936\pi$
$864$	0	0
$865$	−16.3647	−0.556417
$866$	−3.38970	−0.115187
$867$	0	0
$868$	6.19946	0.210423
$869$	3.25285	0.110345
$870$	0	0
$871$	−3.13573	−0.106250
$872$	−24.8070	−0.840071
$873$	0	0
$874$	20.8339	0.704718
$875$	−1.65189	−0.0558442
$876$	0	0
$877$	−0.561222	−0.0189511	−0.00947555	−	0.999955i	$-0.503016\pi$
$877$	−0.561222	−0.0189511	−0.00947555	+	0.999955i	$0.503016\pi$
$878$	−61.5909	−2.07859
$879$	0	0
$880$	1.49944	0.0505462
$881$	38.9725	1.31302	0.656509	−	0.754318i	$-0.272032\pi$
$881$	38.9725	1.31302	0.656509	+	0.754318i	$0.272032\pi$
$882$	0	0
$883$	45.0122	1.51478	0.757390	−	0.652963i	$-0.226474\pi$
$883$	45.0122	1.51478	0.757390	+	0.652963i	$0.226474\pi$
$884$	7.69153	0.258694
$885$	0	0
$886$	29.6362	0.995649
$887$	50.6113	1.69936	0.849681	−	0.527297i	$-0.176794\pi$
$887$	50.6113	1.69936	0.849681	+	0.527297i	$0.176794\pi$
$888$	0	0
$889$	−0.876371	−0.0293925
$890$	−1.59605	−0.0534997
$891$	0	0
$892$	12.2665	0.410712
$893$	21.3127	0.713201
$894$	0	0
$895$	−14.5100	−0.485014
$896$	22.5574	0.753589
$897$	0	0
$898$	−57.5094	−1.91911
$899$	36.3684	1.21295
$900$	0	0
$901$	20.6921	0.689354
$902$	2.14908	0.0715565
$903$	0	0
$904$	39.8076	1.32398
$905$	20.6306	0.685785
$906$	0	0
$907$	−18.0299	−0.598672	−0.299336	−	0.954148i	$-0.596765\pi$
$907$	−18.0299	−0.598672	−0.299336	+	0.954148i	$0.596765\pi$
$908$	−15.0231	−0.498558
$909$	0	0
$910$	11.2467	0.372825
$911$	−12.8735	−0.426517	−0.213258	−	0.976996i	$-0.568408\pi$
$911$	−12.8735	−0.426517	−0.213258	+	0.976996i	$0.568408\pi$
$912$	0	0
$913$	−2.10774	−0.0697560
$914$	16.6905	0.552072
$915$	0	0
$916$	14.6406	0.483740
$917$	4.97496	0.164288
$918$	0	0
$919$	7.98015	0.263241	0.131621	−	0.991300i	$-0.457982\pi$
$919$	7.98015	0.263241	0.131621	+	0.991300i	$0.457982\pi$
$920$	11.7387	0.387012
$921$	0	0
$922$	−0.296057	−0.00975011
$923$	13.1695	0.433481
$924$	0	0
$925$	9.86431	0.324336
$926$	−11.4903	−0.377596
$927$	0	0
$928$	−15.9997	−0.525217
$929$	−59.0076	−1.93598	−0.967989	−	0.250994i	$-0.919243\pi$
$929$	−59.0076	−1.93598	−0.967989	+	0.250994i	$0.919243\pi$
$930$	0	0
$931$	−11.0119	−0.360900
$932$	−2.06988	−0.0678012
$933$	0	0
$934$	−28.9846	−0.948404
$935$	1.03006	0.0336866
$936$	0	0
$937$	−1.28011	−0.0418194	−0.0209097	−	0.999781i	$-0.506656\pi$
$937$	−1.28011	−0.0418194	−0.0209097	+	0.999781i	$0.506656\pi$
$938$	1.93807	0.0632803
$939$	0	0
$940$	−4.52494	−0.147587
$941$	−17.8453	−0.581739	−0.290870	−	0.956763i	$-0.593945\pi$
$941$	−17.8453	−0.581739	−0.290870	+	0.956763i	$0.593945\pi$
$942$	0	0
$943$	21.8019	0.709967
$944$	25.7854	0.839244
$945$	0	0
$946$	0.575260	0.0187033
$947$	37.6144	1.22230	0.611152	−	0.791513i	$-0.290707\pi$
$947$	37.6144	1.22230	0.611152	+	0.791513i	$0.290707\pi$
$948$	0	0
$949$	−20.7560	−0.673767
$950$	4.11485	0.133503
$951$	0	0
$952$	12.6158	0.408881
$953$	31.4840	1.01987	0.509933	−	0.860214i	$-0.329670\pi$
$953$	31.4840	1.01987	0.509933	+	0.860214i	$0.329670\pi$
$954$	0	0
$955$	19.7013	0.637519
$956$	11.1455	0.360471
$957$	0	0
$958$	23.1956	0.749415
$959$	18.3087	0.591220
$960$	0	0
$961$	16.0087	0.516410
$962$	−67.1599	−2.16532
$963$	0	0
$964$	−1.34558	−0.0433382
$965$	−18.5596	−0.597455
$966$	0	0
$967$	17.5286	0.563681	0.281841	−	0.959461i	$-0.409055\pi$
$967$	17.5286	0.563681	0.281841	+	0.959461i	$0.409055\pi$
$968$	25.2764	0.812415
$969$	0	0
$970$	−7.71799	−0.247810
$971$	37.3269	1.19788	0.598939	−	0.800795i	$-0.295589\pi$
$971$	37.3269	1.19788	0.598939	+	0.800795i	$0.295589\pi$
$972$	0	0
$973$	8.47843	0.271806
$974$	35.9269	1.15117
$975$	0	0
$976$	−56.5734	−1.81087
$977$	−11.3776	−0.364001	−0.182000	−	0.983298i	$-0.558257\pi$
$977$	−11.3776	−0.364001	−0.182000	+	0.983298i	$0.558257\pi$
$978$	0	0
$979$	0.312701	0.00999398
$980$	2.33796	0.0746834
$981$	0	0
$982$	22.3699	0.713852
$983$	17.9011	0.570958	0.285479	−	0.958385i	$-0.407847\pi$
$983$	17.9011	0.570958	0.285479	+	0.958385i	$0.407847\pi$
$984$	0	0
$985$	13.1125	0.417800
$986$	−27.8878	−0.888127
$987$	0	0
$988$	−6.01987	−0.191518
$989$	5.83587	0.185570
$990$	0	0
$991$	−41.1988	−1.30872	−0.654362	−	0.756181i	$-0.727063\pi$
$991$	−41.1988	−1.30872	−0.654362	+	0.756181i	$0.727063\pi$
$992$	−20.6808	−0.656616
$993$	0	0
$994$	−8.13957	−0.258171
$995$	−22.3651	−0.709022
$996$	0	0
$997$	−38.0200	−1.20411	−0.602053	−	0.798456i	$-0.705650\pi$
$997$	−38.0200	−1.20411	−0.602053	+	0.798456i	$0.705650\pi$
$998$	34.7468	1.09989
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4005.2.a.w.1.15	✓	17
3.2	odd	2		4005.2.a.x.1.3	yes	17

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4005.2.a.w.1.15	✓	17	1.1	even	1	trivial
4005.2.a.x.1.3	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)

\(f(q)\)	\(=\)	\(q+1.59605 q^{2} +0.547372 q^{4} -1.00000 q^{5} +1.65189 q^{7} -2.31847 q^{8} +O(q^{10})\)	\(q+1.59605 q^{2} +0.547372 q^{4} -1.00000 q^{5} +1.65189 q^{7} -2.31847 q^{8} -1.59605 q^{10} +0.312701 q^{11} -4.26577 q^{13} +2.63650 q^{14} -4.79513 q^{16} -3.29407 q^{17} +2.57815 q^{19} -0.547372 q^{20} +0.499086 q^{22} +5.06311 q^{23} +1.00000 q^{25} -6.80838 q^{26} +0.904200 q^{28} +5.30438 q^{29} +6.85629 q^{31} -3.01633 q^{32} -5.25750 q^{34} -1.65189 q^{35} +9.86431 q^{37} +4.11485 q^{38} +2.31847 q^{40} +4.30603 q^{41} +1.15263 q^{43} +0.171164 q^{44} +8.08097 q^{46} +8.26666 q^{47} -4.27125 q^{49} +1.59605 q^{50} -2.33496 q^{52} -6.28163 q^{53} -0.312701 q^{55} -3.82986 q^{56} +8.46605 q^{58} -5.37742 q^{59} +11.7981 q^{61} +10.9430 q^{62} +4.77605 q^{64} +4.26577 q^{65} +0.735092 q^{67} -1.80308 q^{68} -2.63650 q^{70} -3.08726 q^{71} +4.86570 q^{73} +15.7439 q^{74} +1.41120 q^{76} +0.516549 q^{77} +10.4024 q^{79} +4.79513 q^{80} +6.87263 q^{82} -6.74042 q^{83} +3.29407 q^{85} +1.83965 q^{86} -0.724987 q^{88} +1.00000 q^{89} -7.04660 q^{91} +2.77140 q^{92} +13.1940 q^{94} -2.57815 q^{95} +4.83568 q^{97} -6.81712 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 17 q - 5 q^{2} + 21 q^{4} - 17 q^{5} + 12 q^{7} - 15 q^{8}+O(q^{10}) \) 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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