LMFDB - Embedded newform 8001.2.a.o.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8001,2,Mod(1,8001)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8001, 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("8001.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8001 = 3^{2} \cdot 7 \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8001.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:	$63.8883066572$
Analytic rank:	$1$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 4 x^{12} - 10 x^{11} + 53 x^{10} + 19 x^{9} - 242 x^{8} + 61 x^{7} + 467 x^{6} - 211 x^{5} + \cdots - 8 $ x^13 - 4x^12 - 10x^11 + 53x^10 + 19x^9 - 242x^8 + 61x^7 + 467x^6 - 211x^5 - 372x^4 + 146x^3 + 116x^2 - 12x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 2667)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-0.305711$ of defining polynomial
Character	$\chi$	$=$	8001.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.305711 q^{2} -1.90654 q^{4} +0.276458 q^{5} +1.00000 q^{7} -1.19427 q^{8} +O(q^{10})$	\(q+0.305711 q^{2} -1.90654 q^{4} +0.276458 q^{5} +1.00000 q^{7} -1.19427 q^{8} +0.0845162 q^{10} +5.49757 q^{11} +5.85411 q^{13} +0.305711 q^{14} +3.44798 q^{16} -1.15122 q^{17} -6.10623 q^{19} -0.527078 q^{20} +1.68067 q^{22} -4.90674 q^{23} -4.92357 q^{25} +1.78966 q^{26} -1.90654 q^{28} -5.37546 q^{29} -9.23065 q^{31} +3.44263 q^{32} -0.351941 q^{34} +0.276458 q^{35} +3.98305 q^{37} -1.86674 q^{38} -0.330166 q^{40} -1.75449 q^{41} -10.6978 q^{43} -10.4813 q^{44} -1.50004 q^{46} +2.53420 q^{47} +1.00000 q^{49} -1.50519 q^{50} -11.1611 q^{52} +6.75216 q^{53} +1.51985 q^{55} -1.19427 q^{56} -1.64334 q^{58} -10.1143 q^{59} +6.36077 q^{61} -2.82191 q^{62} -5.84351 q^{64} +1.61841 q^{65} +3.74324 q^{67} +2.19485 q^{68} +0.0845162 q^{70} -10.2468 q^{71} +0.769235 q^{73} +1.21766 q^{74} +11.6418 q^{76} +5.49757 q^{77} -0.396745 q^{79} +0.953221 q^{80} -0.536368 q^{82} +2.27133 q^{83} -0.318264 q^{85} -3.27044 q^{86} -6.56559 q^{88} +6.61497 q^{89} +5.85411 q^{91} +9.35489 q^{92} +0.774732 q^{94} -1.68811 q^{95} +6.50324 q^{97} +0.305711 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - 4 q^{2} + 10 q^{4} - 12 q^{5} + 13 q^{7} - 9 q^{8}+O(q^{10}) $ 13 * q - 4 * q^2 + 10 * q^4 - 12 * q^5 + 13 * q^7 - 9 * q^8	$ 13 q - 4 q^{2} + 10 q^{4} - 12 q^{5} + 13 q^{7} - 9 q^{8} + 6 q^{10} - 3 q^{11} + 21 q^{13} - 4 q^{14} + 8 q^{16} - 17 q^{17} + 5 q^{19} - 29 q^{20} + q^{22} - 4 q^{23} + q^{25} - 22 q^{26} + 10 q^{28} - 21 q^{29} - 7 q^{31} - 12 q^{32} + 2 q^{34} - 12 q^{35} + 7 q^{37} + 9 q^{38} + 29 q^{40} - 21 q^{41} - 9 q^{43} + 2 q^{44} - 28 q^{46} - 23 q^{47} + 13 q^{49} - 15 q^{50} + 15 q^{52} - 31 q^{53} - 8 q^{55} - 9 q^{56} - 25 q^{58} - 28 q^{59} + 29 q^{61} + 3 q^{62} + 9 q^{64} - 30 q^{65} - 18 q^{67} - 34 q^{68} + 6 q^{70} - 10 q^{71} + 24 q^{73} + 19 q^{74} - 3 q^{77} - 28 q^{79} - 26 q^{80} + 18 q^{82} - 26 q^{83} + 20 q^{85} + 2 q^{86} - 17 q^{88} - 44 q^{89} + 21 q^{91} - 6 q^{92} - 9 q^{94} + 2 q^{95} + 17 q^{97} - 4 q^{98}+O(q^{100}) $ 13 * q - 4 * q^2 + 10 * q^4 - 12 * q^5 + 13 * q^7 - 9 * q^8 + 6 * q^10 - 3 * q^11 + 21 * q^13 - 4 * q^14 + 8 * q^16 - 17 * q^17 + 5 * q^19 - 29 * q^20 + q^22 - 4 * q^23 + q^25 - 22 * q^26 + 10 * q^28 - 21 * q^29 - 7 * q^31 - 12 * q^32 + 2 * q^34 - 12 * q^35 + 7 * q^37 + 9 * q^38 + 29 * q^40 - 21 * q^41 - 9 * q^43 + 2 * q^44 - 28 * q^46 - 23 * q^47 + 13 * q^49 - 15 * q^50 + 15 * q^52 - 31 * q^53 - 8 * q^55 - 9 * q^56 - 25 * q^58 - 28 * q^59 + 29 * q^61 + 3 * q^62 + 9 * q^64 - 30 * q^65 - 18 * q^67 - 34 * q^68 + 6 * q^70 - 10 * q^71 + 24 * q^73 + 19 * q^74 - 3 * q^77 - 28 * q^79 - 26 * q^80 + 18 * q^82 - 26 * q^83 + 20 * q^85 + 2 * q^86 - 17 * q^88 - 44 * q^89 + 21 * q^91 - 6 * q^92 - 9 * q^94 + 2 * q^95 + 17 * q^97 - 4 * 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 $

$2$	0.305711	0.216170	0.108085	−	0.994142i	$-0.465528\pi$
$2$	0.305711	0.216170	0.108085	+	0.994142i	$0.465528\pi$
$3$	0	0
$3$	0	0
$4$	−1.90654	−0.953270
$5$	0.276458	0.123636	0.0618178	−	0.998087i	$-0.480310\pi$
$5$	0.276458	0.123636	0.0618178	+	0.998087i	$0.480310\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.19427	−0.422239
$9$	0	0
$10$	0.0845162	0.0267264
$11$	5.49757	1.65758	0.828790	−	0.559560i	$-0.189030\pi$
$11$	5.49757	1.65758	0.828790	+	0.559560i	$0.189030\pi$
$12$	0	0
$13$	5.85411	1.62364	0.811819	−	0.583910i	$-0.198478\pi$
$13$	5.85411	1.62364	0.811819	+	0.583910i	$0.198478\pi$
$14$	0.305711	0.0817047
$15$	0	0
$16$	3.44798	0.861995
$17$	−1.15122	−0.279213	−0.139606	−	0.990207i	$-0.544584\pi$
$17$	−1.15122	−0.279213	−0.139606	+	0.990207i	$0.544584\pi$
$18$	0	0
$19$	−6.10623	−1.40087	−0.700433	−	0.713718i	$-0.747010\pi$
$19$	−6.10623	−1.40087	−0.700433	+	0.713718i	$0.747010\pi$
$20$	−0.527078	−0.117858
$21$	0	0
$22$	1.68067	0.358319
$23$	−4.90674	−1.02313	−0.511563	−	0.859246i	$-0.670933\pi$
$23$	−4.90674	−1.02313	−0.511563	+	0.859246i	$0.670933\pi$
$24$	0	0
$25$	−4.92357	−0.984714
$26$	1.78966	0.350982
$27$	0	0
$28$	−1.90654	−0.360302
$29$	−5.37546	−0.998198	−0.499099	−	0.866545i	$-0.666336\pi$
$29$	−5.37546	−0.998198	−0.499099	+	0.866545i	$0.666336\pi$
$30$	0	0
$31$	−9.23065	−1.65787	−0.828937	−	0.559342i	$-0.811054\pi$
$31$	−9.23065	−1.65787	−0.828937	+	0.559342i	$0.811054\pi$
$32$	3.44263	0.608577
$33$	0	0
$34$	−0.351941	−0.0603574
$35$	0.276458	0.0467299
$36$	0	0
$37$	3.98305	0.654810	0.327405	−	0.944884i	$-0.393826\pi$
$37$	3.98305	0.654810	0.327405	+	0.944884i	$0.393826\pi$
$38$	−1.86674	−0.302825
$39$	0	0
$40$	−0.330166	−0.0522038
$41$	−1.75449	−0.274006	−0.137003	−	0.990571i	$-0.543747\pi$
$41$	−1.75449	−0.274006	−0.137003	+	0.990571i	$0.543747\pi$
$42$	0	0
$43$	−10.6978	−1.63140	−0.815701	−	0.578474i	$-0.803649\pi$
$43$	−10.6978	−1.63140	−0.815701	+	0.578474i	$0.803649\pi$
$44$	−10.4813	−1.58012
$45$	0	0
$46$	−1.50004	−0.221169
$47$	2.53420	0.369651	0.184825	−	0.982771i	$-0.440828\pi$
$47$	2.53420	0.369651	0.184825	+	0.982771i	$0.440828\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−1.50519	−0.212866
$51$	0	0
$52$	−11.1611	−1.54777
$53$	6.75216	0.927481	0.463740	−	0.885971i	$-0.346507\pi$
$53$	6.75216	0.927481	0.463740	+	0.885971i	$0.346507\pi$
$54$	0	0
$55$	1.51985	0.204936
$56$	−1.19427	−0.159591
$57$	0	0
$58$	−1.64334	−0.215781
$59$	−10.1143	−1.31678	−0.658388	−	0.752679i	$-0.728761\pi$
$59$	−10.1143	−1.31678	−0.658388	+	0.752679i	$0.728761\pi$
$60$	0	0
$61$	6.36077	0.814413	0.407206	−	0.913336i	$-0.366503\pi$
$61$	6.36077	0.814413	0.407206	+	0.913336i	$0.366503\pi$
$62$	−2.82191	−0.358383
$63$	0	0
$64$	−5.84351	−0.730439
$65$	1.61841	0.200740
$66$	0	0
$67$	3.74324	0.457310	0.228655	−	0.973508i	$-0.426567\pi$
$67$	3.74324	0.457310	0.228655	+	0.973508i	$0.426567\pi$
$68$	2.19485	0.266165
$69$	0	0
$70$	0.0845162	0.0101016
$71$	−10.2468	−1.21607	−0.608036	−	0.793910i	$-0.708042\pi$
$71$	−10.2468	−1.21607	−0.608036	+	0.793910i	$0.708042\pi$
$72$	0	0
$73$	0.769235	0.0900321	0.0450161	−	0.998986i	$-0.485666\pi$
$73$	0.769235	0.0900321	0.0450161	+	0.998986i	$0.485666\pi$
$74$	1.21766	0.141550
$75$	0	0
$76$	11.6418	1.33540
$77$	5.49757	0.626506
$78$	0	0
$79$	−0.396745	−0.0446372	−0.0223186	−	0.999751i	$-0.507105\pi$
$79$	−0.396745	−0.0446372	−0.0223186	+	0.999751i	$0.507105\pi$
$80$	0.953221	0.106573
$81$	0	0
$82$	−0.536368	−0.0592319
$83$	2.27133	0.249311	0.124656	−	0.992200i	$-0.460217\pi$
$83$	2.27133	0.249311	0.124656	+	0.992200i	$0.460217\pi$
$84$	0	0
$85$	−0.318264	−0.0345206
$86$	−3.27044	−0.352661
$87$	0	0
$88$	−6.56559	−0.699895
$89$	6.61497	0.701185	0.350593	−	0.936528i	$-0.385980\pi$
$89$	6.61497	0.701185	0.350593	+	0.936528i	$0.385980\pi$
$90$	0	0
$91$	5.85411	0.613677
$92$	9.35489	0.975315
$93$	0	0
$94$	0.774732	0.0799075
$95$	−1.68811	−0.173197
$96$	0	0
$97$	6.50324	0.660304	0.330152	−	0.943928i	$-0.392900\pi$
$97$	6.50324	0.660304	0.330152	+	0.943928i	$0.392900\pi$
$98$	0.305711	0.0308815
$99$	0	0
$100$	9.38699	0.938699
$101$	−6.45399	−0.642196	−0.321098	−	0.947046i	$-0.604052\pi$
$101$	−6.45399	−0.642196	−0.321098	+	0.947046i	$0.604052\pi$
$102$	0	0
$103$	−7.17745	−0.707215	−0.353608	−	0.935394i	$-0.615045\pi$
$103$	−7.17745	−0.707215	−0.353608	+	0.935394i	$0.615045\pi$
$104$	−6.99140	−0.685563
$105$	0	0
$106$	2.06421	0.200494
$107$	−2.58615	−0.250012	−0.125006	−	0.992156i	$-0.539895\pi$
$107$	−2.58615	−0.250012	−0.125006	+	0.992156i	$0.539895\pi$
$108$	0	0
$109$	16.7781	1.60705	0.803524	−	0.595272i	$-0.202956\pi$
$109$	16.7781	1.60705	0.803524	+	0.595272i	$0.202956\pi$
$110$	0.464633	0.0443011
$111$	0	0
$112$	3.44798	0.325803
$113$	−10.3192	−0.970744	−0.485372	−	0.874308i	$-0.661316\pi$
$113$	−10.3192	−0.970744	−0.485372	+	0.874308i	$0.661316\pi$
$114$	0	0
$115$	−1.35651	−0.126495
$116$	10.2485	0.951553
$117$	0	0
$118$	−3.09207	−0.284648
$119$	−1.15122	−0.105532
$120$	0	0
$121$	19.2233	1.74757
$122$	1.94456	0.176052
$123$	0	0
$124$	17.5986	1.58040
$125$	−2.74345	−0.245381
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	−8.67168	−0.766476
$129$	0	0
$130$	0.494767	0.0433939
$131$	−11.9275	−1.04211	−0.521056	−	0.853522i	$-0.674462\pi$
$131$	−11.9275	−1.04211	−0.521056	+	0.853522i	$0.674462\pi$
$132$	0	0
$133$	−6.10623	−0.529477
$134$	1.14435	0.0988568
$135$	0	0
$136$	1.37487	0.117894
$137$	−8.19429	−0.700086	−0.350043	−	0.936734i	$-0.613833\pi$
$137$	−8.19429	−0.700086	−0.350043	+	0.936734i	$0.613833\pi$
$138$	0	0
$139$	−13.5478	−1.14911	−0.574553	−	0.818467i	$-0.694824\pi$
$139$	−13.5478	−1.14911	−0.574553	+	0.818467i	$0.694824\pi$
$140$	−0.527078	−0.0445462
$141$	0	0
$142$	−3.13256	−0.262878
$143$	32.1834	2.69131
$144$	0	0
$145$	−1.48609	−0.123413
$146$	0.235164	0.0194623
$147$	0	0
$148$	−7.59385	−0.624211
$149$	−4.98742	−0.408585	−0.204293	−	0.978910i	$-0.565489\pi$
$149$	−4.98742	−0.408585	−0.204293	+	0.978910i	$0.565489\pi$
$150$	0	0
$151$	4.17561	0.339806	0.169903	−	0.985461i	$-0.445655\pi$
$151$	4.17561	0.339806	0.169903	+	0.985461i	$0.445655\pi$
$152$	7.29250	0.591500
$153$	0	0
$154$	1.68067	0.135432
$155$	−2.55189	−0.204972
$156$	0	0
$157$	5.35019	0.426991	0.213496	−	0.976944i	$-0.431515\pi$
$157$	5.35019	0.426991	0.213496	+	0.976944i	$0.431515\pi$
$158$	−0.121289	−0.00964925
$159$	0	0
$160$	0.951742	0.0752418
$161$	−4.90674	−0.386705
$162$	0	0
$163$	−7.82336	−0.612773	−0.306386	−	0.951907i	$-0.599120\pi$
$163$	−7.82336	−0.612773	−0.306386	+	0.951907i	$0.599120\pi$
$164$	3.34501	0.261202
$165$	0	0
$166$	0.694371	0.0538937
$167$	−8.35675	−0.646665	−0.323332	−	0.946285i	$-0.604803\pi$
$167$	−8.35675	−0.646665	−0.323332	+	0.946285i	$0.604803\pi$
$168$	0	0
$169$	21.2706	1.63620
$170$	−0.0972969	−0.00746233
$171$	0	0
$172$	20.3958	1.55517
$173$	−7.00766	−0.532783	−0.266391	−	0.963865i	$-0.585831\pi$
$173$	−7.00766	−0.532783	−0.266391	+	0.963865i	$0.585831\pi$
$174$	0	0
$175$	−4.92357	−0.372187
$176$	18.9555	1.42883
$177$	0	0
$178$	2.02227	0.151575
$179$	−25.1904	−1.88282	−0.941410	−	0.337263i	$-0.890499\pi$
$179$	−25.1904	−1.88282	−0.941410	+	0.337263i	$0.890499\pi$
$180$	0	0
$181$	16.9017	1.25629	0.628147	−	0.778095i	$-0.283814\pi$
$181$	16.9017	1.25629	0.628147	+	0.778095i	$0.283814\pi$
$182$	1.78966	0.132659
$183$	0	0
$184$	5.85998	0.432003
$185$	1.10115	0.0809578
$186$	0	0
$187$	−6.32893	−0.462817
$188$	−4.83155	−0.352377
$189$	0	0
$190$	−0.516075	−0.0374400
$191$	21.8166	1.57859	0.789296	−	0.614013i	$-0.210446\pi$
$191$	21.8166	1.57859	0.789296	+	0.614013i	$0.210446\pi$
$192$	0	0
$193$	−17.6982	−1.27395	−0.636973	−	0.770886i	$-0.719814\pi$
$193$	−17.6982	−1.27395	−0.636973	+	0.770886i	$0.719814\pi$
$194$	1.98811	0.142738
$195$	0	0
$196$	−1.90654	−0.136181
$197$	−9.11313	−0.649284	−0.324642	−	0.945837i	$-0.605244\pi$
$197$	−9.11313	−0.649284	−0.324642	+	0.945837i	$0.605244\pi$
$198$	0	0
$199$	−0.0226194	−0.00160344	−0.000801722	−	1.00000i	$-0.500255\pi$
$199$	−0.0226194	−0.00160344	−0.000801722		1.00000i	$0.500255\pi$
$200$	5.88008	0.415785
$201$	0	0
$202$	−1.97306	−0.138824
$203$	−5.37546	−0.377283
$204$	0	0
$205$	−0.485043	−0.0338769
$206$	−2.19423	−0.152879
$207$	0	0
$208$	20.1848	1.39957
$209$	−33.5694	−2.32205
$210$	0	0
$211$	13.0645	0.899399	0.449700	−	0.893180i	$-0.351531\pi$
$211$	13.0645	0.899399	0.449700	+	0.893180i	$0.351531\pi$
$212$	−12.8733	−0.884140
$213$	0	0
$214$	−0.790614	−0.0540452
$215$	−2.95750	−0.201700
$216$	0	0
$217$	−9.23065	−0.626618
$218$	5.12924	0.347396
$219$	0	0
$220$	−2.89765	−0.195359
$221$	−6.73938	−0.453340
$222$	0	0
$223$	14.4547	0.967958	0.483979	−	0.875080i	$-0.339191\pi$
$223$	14.4547	0.967958	0.483979	+	0.875080i	$0.339191\pi$
$224$	3.44263	0.230020
$225$	0	0
$226$	−3.15468	−0.209846
$227$	−5.34653	−0.354862	−0.177431	−	0.984133i	$-0.556779\pi$
$227$	−5.34653	−0.354862	−0.177431	+	0.984133i	$0.556779\pi$
$228$	0	0
$229$	12.2399	0.808839	0.404419	−	0.914574i	$-0.367474\pi$
$229$	12.2399	0.808839	0.404419	+	0.914574i	$0.367474\pi$
$230$	−0.414698	−0.0273444
$231$	0	0
$232$	6.41976	0.421478
$233$	−20.6821	−1.35493	−0.677465	−	0.735555i	$-0.736921\pi$
$233$	−20.6821	−1.35493	−0.677465	+	0.735555i	$0.736921\pi$
$234$	0	0
$235$	0.700599	0.0457020
$236$	19.2834	1.25524
$237$	0	0
$238$	−0.351941	−0.0228130
$239$	−19.8538	−1.28424	−0.642119	−	0.766605i	$-0.721944\pi$
$239$	−19.8538	−1.28424	−0.642119	+	0.766605i	$0.721944\pi$
$240$	0	0
$241$	3.39756	0.218856	0.109428	−	0.993995i	$-0.465098\pi$
$241$	3.39756	0.218856	0.109428	+	0.993995i	$0.465098\pi$
$242$	5.87676	0.377773
$243$	0	0
$244$	−12.1271	−0.776356
$245$	0.276458	0.0176622
$246$	0	0
$247$	−35.7465	−2.27450
$248$	11.0239	0.700019
$249$	0	0
$250$	−0.838702	−0.0530442
$251$	−16.9646	−1.07080	−0.535398	−	0.844600i	$-0.679839\pi$
$251$	−16.9646	−1.07080	−0.535398	+	0.844600i	$0.679839\pi$
$252$	0	0
$253$	−26.9751	−1.69591
$254$	−0.305711	−0.0191820
$255$	0	0
$256$	9.03599	0.564749
$257$	9.88570	0.616653	0.308327	−	0.951281i	$-0.400231\pi$
$257$	9.88570	0.616653	0.308327	+	0.951281i	$0.400231\pi$
$258$	0	0
$259$	3.98305	0.247495
$260$	−3.08557	−0.191359
$261$	0	0
$262$	−3.64637	−0.225274
$263$	−15.4492	−0.952640	−0.476320	−	0.879272i	$-0.658030\pi$
$263$	−15.4492	−0.952640	−0.476320	+	0.879272i	$0.658030\pi$
$264$	0	0
$265$	1.86669	0.114670
$266$	−1.86674	−0.114457
$267$	0	0
$268$	−7.13664	−0.435940
$269$	20.0563	1.22285	0.611427	−	0.791301i	$-0.290596\pi$
$269$	20.0563	1.22285	0.611427	+	0.791301i	$0.290596\pi$
$270$	0	0
$271$	1.52170	0.0924367	0.0462183	−	0.998931i	$-0.485283\pi$
$271$	1.52170	0.0924367	0.0462183	+	0.998931i	$0.485283\pi$
$272$	−3.96939	−0.240680
$273$	0	0
$274$	−2.50509	−0.151338
$275$	−27.0677	−1.63224
$276$	0	0
$277$	−2.88589	−0.173396	−0.0866981	−	0.996235i	$-0.527632\pi$
$277$	−2.88589	−0.173396	−0.0866981	+	0.996235i	$0.527632\pi$
$278$	−4.14170	−0.248403
$279$	0	0
$280$	−0.330166	−0.0197312
$281$	1.40750	0.0839643	0.0419822	−	0.999118i	$-0.486633\pi$
$281$	1.40750	0.0839643	0.0419822	+	0.999118i	$0.486633\pi$
$282$	0	0
$283$	18.7659	1.11552	0.557759	−	0.830003i	$-0.311661\pi$
$283$	18.7659	1.11552	0.557759	+	0.830003i	$0.311661\pi$
$284$	19.5359	1.15924
$285$	0	0
$286$	9.83881	0.581781
$287$	−1.75449	−0.103564
$288$	0	0
$289$	−15.6747	−0.922040
$290$	−0.454313	−0.0266782
$291$	0	0
$292$	−1.46658	−0.0858250
$293$	−12.6713	−0.740266	−0.370133	−	0.928979i	$-0.620688\pi$
$293$	−12.6713	−0.740266	−0.370133	+	0.928979i	$0.620688\pi$
$294$	0	0
$295$	−2.79619	−0.162800
$296$	−4.75685	−0.276486
$297$	0	0
$298$	−1.52471	−0.0883240
$299$	−28.7246	−1.66118
$300$	0	0
$301$	−10.6978	−0.616612
$302$	1.27653	0.0734560
$303$	0	0
$304$	−21.0542	−1.20754
$305$	1.75848	0.100690
$306$	0	0
$307$	−8.37854	−0.478189	−0.239094	−	0.970996i	$-0.576851\pi$
$307$	−8.37854	−0.478189	−0.239094	+	0.970996i	$0.576851\pi$
$308$	−10.4813	−0.597230
$309$	0	0
$310$	−0.780139	−0.0443089
$311$	−26.6946	−1.51371	−0.756856	−	0.653581i	$-0.773266\pi$
$311$	−26.6946	−1.51371	−0.756856	+	0.653581i	$0.773266\pi$
$312$	0	0
$313$	−4.74932	−0.268447	−0.134224	−	0.990951i	$-0.542854\pi$
$313$	−4.74932	−0.268447	−0.134224	+	0.990951i	$0.542854\pi$
$314$	1.63561	0.0923028
$315$	0	0
$316$	0.756410	0.0425514
$317$	12.7833	0.717979	0.358990	−	0.933342i	$-0.383121\pi$
$317$	12.7833	0.717979	0.358990	+	0.933342i	$0.383121\pi$
$318$	0	0
$319$	−29.5520	−1.65459
$320$	−1.61548	−0.0903083
$321$	0	0
$322$	−1.50004	−0.0835941
$323$	7.02963	0.391139
$324$	0	0
$325$	−28.8231	−1.59882
$326$	−2.39169	−0.132463
$327$	0	0
$328$	2.09534	0.115696
$329$	2.53420	0.139715
$330$	0	0
$331$	30.0111	1.64956	0.824780	−	0.565454i	$-0.191299\pi$
$331$	30.0111	1.64956	0.824780	+	0.565454i	$0.191299\pi$
$332$	−4.33039	−0.237661
$333$	0	0
$334$	−2.55475	−0.139790
$335$	1.03485	0.0565398
$336$	0	0
$337$	−33.6687	−1.83405	−0.917026	−	0.398827i	$-0.869418\pi$
$337$	−33.6687	−1.83405	−0.917026	+	0.398827i	$0.869418\pi$
$338$	6.50265	0.353698
$339$	0	0
$340$	0.606784	0.0329075
$341$	−50.7462	−2.74806
$342$	0	0
$343$	1.00000	0.0539949
$344$	12.7761	0.688842
$345$	0	0
$346$	−2.14232	−0.115172
$347$	−10.1973	−0.547418	−0.273709	−	0.961813i	$-0.588251\pi$
$347$	−10.1973	−0.547418	−0.273709	+	0.961813i	$0.588251\pi$
$348$	0	0
$349$	−21.7938	−1.16660	−0.583299	−	0.812258i	$-0.698238\pi$
$349$	−21.7938	−1.16660	−0.583299	+	0.812258i	$0.698238\pi$
$350$	−1.50519	−0.0804558
$351$	0	0
$352$	18.9261	1.00876
$353$	−10.8660	−0.578340	−0.289170	−	0.957278i	$-0.593379\pi$
$353$	−10.8660	−0.578340	−0.289170	+	0.957278i	$0.593379\pi$
$354$	0	0
$355$	−2.83281	−0.150350
$356$	−12.6117	−0.668419
$357$	0	0
$358$	−7.70099	−0.407010
$359$	32.9974	1.74153	0.870767	−	0.491696i	$-0.163623\pi$
$359$	32.9974	1.74153	0.870767	+	0.491696i	$0.163623\pi$
$360$	0	0
$361$	18.2861	0.962424
$362$	5.16703	0.271573
$363$	0	0
$364$	−11.1611	−0.585000
$365$	0.212661	0.0111312
$366$	0	0
$367$	−11.0294	−0.575732	−0.287866	−	0.957671i	$-0.592946\pi$
$367$	−11.0294	−0.575732	−0.287866	+	0.957671i	$0.592946\pi$
$368$	−16.9183	−0.881929
$369$	0	0
$370$	0.336632	0.0175007
$371$	6.75216	0.350555
$372$	0	0
$373$	−33.0143	−1.70942	−0.854708	−	0.519109i	$-0.826264\pi$
$373$	−33.0143	−1.70942	−0.854708	+	0.519109i	$0.826264\pi$
$374$	−1.93482	−0.100047
$375$	0	0
$376$	−3.02652	−0.156081
$377$	−31.4685	−1.62071
$378$	0	0
$379$	−5.90961	−0.303556	−0.151778	−	0.988415i	$-0.548500\pi$
$379$	−5.90961	−0.303556	−0.151778	+	0.988415i	$0.548500\pi$
$380$	3.21846	0.165104
$381$	0	0
$382$	6.66956	0.341245
$383$	−24.2106	−1.23710	−0.618551	−	0.785745i	$-0.712280\pi$
$383$	−24.2106	−1.23710	−0.618551	+	0.785745i	$0.712280\pi$
$384$	0	0
$385$	1.51985	0.0774585
$386$	−5.41054	−0.275389
$387$	0	0
$388$	−12.3987	−0.629448
$389$	32.5969	1.65273	0.826363	−	0.563137i	$-0.190406\pi$
$389$	32.5969	1.65273	0.826363	+	0.563137i	$0.190406\pi$
$390$	0	0
$391$	5.64875	0.285669
$392$	−1.19427	−0.0603199
$393$	0	0
$394$	−2.78598	−0.140356
$395$	−0.109683	−0.00551876
$396$	0	0
$397$	12.9830	0.651597	0.325799	−	0.945439i	$-0.394367\pi$
$397$	12.9830	0.651597	0.325799	+	0.945439i	$0.394367\pi$
$398$	−0.00691498	−0.000346617 0
$399$	0	0
$400$	−16.9764	−0.848819
$401$	36.1973	1.80760	0.903802	−	0.427950i	$-0.140764\pi$
$401$	36.1973	1.80760	0.903802	+	0.427950i	$0.140764\pi$
$402$	0	0
$403$	−54.0372	−2.69179
$404$	12.3048	0.612186
$405$	0	0
$406$	−1.64334	−0.0815574
$407$	21.8971	1.08540
$408$	0	0
$409$	−9.96969	−0.492969	−0.246485	−	0.969147i	$-0.579276\pi$
$409$	−9.96969	−0.492969	−0.246485	+	0.969147i	$0.579276\pi$
$410$	−0.148283	−0.00732318
$411$	0	0
$412$	13.6841	0.674167
$413$	−10.1143	−0.497694
$414$	0	0
$415$	0.627928	0.0308237
$416$	20.1535	0.988108
$417$	0	0
$418$	−10.2625	−0.501957
$419$	−5.11772	−0.250017	−0.125008	−	0.992156i	$-0.539896\pi$
$419$	−5.11772	−0.250017	−0.125008	+	0.992156i	$0.539896\pi$
$420$	0	0
$421$	−32.5904	−1.58836	−0.794179	−	0.607684i	$-0.792099\pi$
$421$	−32.5904	−1.58836	−0.794179	+	0.607684i	$0.792099\pi$
$422$	3.99397	0.194423
$423$	0	0
$424$	−8.06392	−0.391619
$425$	5.66813	0.274945
$426$	0	0
$427$	6.36077	0.307819
$428$	4.93060	0.238329
$429$	0	0
$430$	−0.904139	−0.0436014
$431$	5.40245	0.260227	0.130113	−	0.991499i	$-0.458466\pi$
$431$	5.40245	0.260227	0.130113	+	0.991499i	$0.458466\pi$
$432$	0	0
$433$	−15.1606	−0.728572	−0.364286	−	0.931287i	$-0.618687\pi$
$433$	−15.1606	−0.728572	−0.364286	+	0.931287i	$0.618687\pi$
$434$	−2.82191	−0.135456
$435$	0	0
$436$	−31.9881	−1.53195
$437$	29.9617	1.43326
$438$	0	0
$439$	25.7263	1.22785	0.613925	−	0.789364i	$-0.289590\pi$
$439$	25.7263	1.22785	0.613925	+	0.789364i	$0.289590\pi$
$440$	−1.81511	−0.0865319
$441$	0	0
$442$	−2.06030	−0.0979986
$443$	−28.5560	−1.35674	−0.678368	−	0.734722i	$-0.737312\pi$
$443$	−28.5560	−1.35674	−0.678368	+	0.734722i	$0.737312\pi$
$444$	0	0
$445$	1.82876	0.0866915
$446$	4.41896	0.209244
$447$	0	0
$448$	−5.84351	−0.276080
$449$	33.0302	1.55879	0.779396	−	0.626532i	$-0.215526\pi$
$449$	33.0302	1.55879	0.779396	+	0.626532i	$0.215526\pi$
$450$	0	0
$451$	−9.64545	−0.454186
$452$	19.6739	0.925382
$453$	0	0
$454$	−1.63449	−0.0767106
$455$	1.61841	0.0758724
$456$	0	0
$457$	26.3102	1.23074	0.615370	−	0.788239i	$-0.289007\pi$
$457$	26.3102	1.23074	0.615370	+	0.788239i	$0.289007\pi$
$458$	3.74189	0.174847
$459$	0	0
$460$	2.58623	0.120584
$461$	−16.9634	−0.790064	−0.395032	−	0.918667i	$-0.629267\pi$
$461$	−16.9634	−0.790064	−0.395032	+	0.918667i	$0.629267\pi$
$462$	0	0
$463$	23.7165	1.10220	0.551099	−	0.834440i	$-0.314209\pi$
$463$	23.7165	1.10220	0.551099	+	0.834440i	$0.314209\pi$
$464$	−18.5345	−0.860441
$465$	0	0
$466$	−6.32275	−0.292896
$467$	34.2579	1.58526	0.792632	−	0.609700i	$-0.208710\pi$
$467$	34.2579	1.58526	0.792632	+	0.609700i	$0.208710\pi$
$468$	0	0
$469$	3.74324	0.172847
$470$	0.214181	0.00987942
$471$	0	0
$472$	12.0793	0.555994
$473$	−58.8120	−2.70418
$474$	0	0
$475$	30.0645	1.37945
$476$	2.19485	0.100601
$477$	0	0
$478$	−6.06953	−0.277614
$479$	−23.5026	−1.07386	−0.536932	−	0.843626i	$-0.680417\pi$
$479$	−23.5026	−1.07386	−0.536932	+	0.843626i	$0.680417\pi$
$480$	0	0
$481$	23.3172	1.06317
$482$	1.03867	0.0473102
$483$	0	0
$484$	−36.6499	−1.66591
$485$	1.79787	0.0816371
$486$	0	0
$487$	−0.487332	−0.0220831	−0.0110416	−	0.999939i	$-0.503515\pi$
$487$	−0.487332	−0.0220831	−0.0110416	+	0.999939i	$0.503515\pi$
$488$	−7.59649	−0.343877
$489$	0	0
$490$	0.0845162	0.00381805
$491$	−15.3844	−0.694290	−0.347145	−	0.937811i	$-0.612849\pi$
$491$	−15.3844	−0.694290	−0.347145	+	0.937811i	$0.612849\pi$
$492$	0	0
$493$	6.18835	0.278709
$494$	−10.9281	−0.491679
$495$	0	0
$496$	−31.8271	−1.42908
$497$	−10.2468	−0.459632
$498$	0	0
$499$	−3.97860	−0.178107	−0.0890533	−	0.996027i	$-0.528384\pi$
$499$	−3.97860	−0.178107	−0.0890533	+	0.996027i	$0.528384\pi$
$500$	5.23050	0.233915
$501$	0	0
$502$	−5.18627	−0.231474
$503$	39.1786	1.74689	0.873445	−	0.486924i	$-0.161881\pi$
$503$	39.1786	1.74689	0.873445	+	0.486924i	$0.161881\pi$
$504$	0	0
$505$	−1.78426	−0.0793983
$506$	−8.24659	−0.366606
$507$	0	0
$508$	1.90654	0.0845891
$509$	−28.7484	−1.27425	−0.637126	−	0.770760i	$-0.719877\pi$
$509$	−28.7484	−1.27425	−0.637126	+	0.770760i	$0.719877\pi$
$510$	0	0
$511$	0.769235	0.0340290
$512$	20.1058	0.888558
$513$	0	0
$514$	3.02217	0.133302
$515$	−1.98426	−0.0874370
$516$	0	0
$517$	13.9319	0.612726
$518$	1.21766	0.0535010
$519$	0	0
$520$	−1.93283	−0.0847600
$521$	−28.5996	−1.25297	−0.626487	−	0.779432i	$-0.715508\pi$
$521$	−28.5996	−1.25297	−0.626487	+	0.779432i	$0.715508\pi$
$522$	0	0
$523$	41.7036	1.82357	0.911785	−	0.410667i	$-0.134704\pi$
$523$	41.7036	1.82357	0.911785	+	0.410667i	$0.134704\pi$
$524$	22.7403	0.993415
$525$	0	0
$526$	−4.72300	−0.205933
$527$	10.6265	0.462899
$528$	0	0
$529$	1.07606	0.0467851
$530$	0.570667	0.0247882
$531$	0	0
$532$	11.6418	0.504735
$533$	−10.2710	−0.444886
$534$	0	0
$535$	−0.714961	−0.0309104
$536$	−4.47045	−0.193094
$537$	0	0
$538$	6.13142	0.264344
$539$	5.49757	0.236797
$540$	0	0
$541$	16.8941	0.726333	0.363166	−	0.931724i	$-0.381696\pi$
$541$	16.8941	0.726333	0.363166	+	0.931724i	$0.381696\pi$
$542$	0.465200	0.0199821
$543$	0	0
$544$	−3.96323	−0.169922
$545$	4.63843	0.198688
$546$	0	0
$547$	−12.9628	−0.554250	−0.277125	−	0.960834i	$-0.589382\pi$
$547$	−12.9628	−0.554250	−0.277125	+	0.960834i	$0.589382\pi$
$548$	15.6228	0.667371
$549$	0	0
$550$	−8.27488	−0.352842
$551$	32.8238	1.39834
$552$	0	0
$553$	−0.396745	−0.0168713
$554$	−0.882248	−0.0374831
$555$	0	0
$556$	25.8294	1.09541
$557$	32.8548	1.39210	0.696052	−	0.717992i	$-0.254938\pi$
$557$	32.8548	1.39210	0.696052	+	0.717992i	$0.254938\pi$
$558$	0	0
$559$	−62.6262	−2.64881
$560$	0.953221	0.0402809
$561$	0	0
$562$	0.430288	0.0181506
$563$	−37.3311	−1.57332	−0.786660	−	0.617386i	$-0.788192\pi$
$563$	−37.3311	−1.57332	−0.786660	+	0.617386i	$0.788192\pi$
$564$	0	0
$565$	−2.85281	−0.120019
$566$	5.73694	0.241142
$567$	0	0
$568$	12.2375	0.513473
$569$	20.1928	0.846526	0.423263	−	0.906007i	$-0.360885\pi$
$569$	20.1928	0.846526	0.423263	+	0.906007i	$0.360885\pi$
$570$	0	0
$571$	22.1763	0.928048	0.464024	−	0.885823i	$-0.346405\pi$
$571$	22.1763	0.928048	0.464024	+	0.885823i	$0.346405\pi$
$572$	−61.3589	−2.56554
$573$	0	0
$574$	−0.536368	−0.0223876
$575$	24.1587	1.00749
$576$	0	0
$577$	35.6411	1.48376	0.741878	−	0.670535i	$-0.233935\pi$
$577$	35.6411	1.48376	0.741878	+	0.670535i	$0.233935\pi$
$578$	−4.79192	−0.199318
$579$	0	0
$580$	2.83329	0.117646
$581$	2.27133	0.0942308
$582$	0	0
$583$	37.1205	1.53737
$584$	−0.918676	−0.0380151
$585$	0	0
$586$	−3.87376	−0.160023
$587$	−3.23639	−0.133580	−0.0667900	−	0.997767i	$-0.521276\pi$
$587$	−3.23639	−0.133580	−0.0667900	+	0.997767i	$0.521276\pi$
$588$	0	0
$589$	56.3645	2.32246
$590$	−0.854825	−0.0351926
$591$	0	0
$592$	13.7335	0.564443
$593$	−18.1133	−0.743825	−0.371912	−	0.928268i	$-0.621298\pi$
$593$	−18.1133	−0.743825	−0.371912	+	0.928268i	$0.621298\pi$
$594$	0	0
$595$	−0.318264	−0.0130476
$596$	9.50872	0.389492
$597$	0	0
$598$	−8.78141	−0.359099
$599$	22.2405	0.908721	0.454361	−	0.890818i	$-0.349868\pi$
$599$	22.2405	0.908721	0.454361	+	0.890818i	$0.349868\pi$
$600$	0	0
$601$	34.6167	1.41205	0.706023	−	0.708189i	$-0.250488\pi$
$601$	34.6167	1.41205	0.706023	+	0.708189i	$0.250488\pi$
$602$	−3.27044	−0.133293
$603$	0	0
$604$	−7.96097	−0.323927
$605$	5.31442	0.216062
$606$	0	0
$607$	−39.8677	−1.61818	−0.809089	−	0.587686i	$-0.800039\pi$
$607$	−39.8677	−1.61818	−0.809089	+	0.587686i	$0.800039\pi$
$608$	−21.0215	−0.852534
$609$	0	0
$610$	0.537588	0.0217663
$611$	14.8355	0.600179
$612$	0	0
$613$	−15.6926	−0.633818	−0.316909	−	0.948456i	$-0.602645\pi$
$613$	−15.6926	−0.633818	−0.316909	+	0.948456i	$0.602645\pi$
$614$	−2.56141	−0.103370
$615$	0	0
$616$	−6.56559	−0.264535
$617$	−26.1395	−1.05234	−0.526169	−	0.850380i	$-0.676372\pi$
$617$	−26.1395	−1.05234	−0.526169	+	0.850380i	$0.676372\pi$
$618$	0	0
$619$	4.72169	0.189781	0.0948904	−	0.995488i	$-0.469750\pi$
$619$	4.72169	0.189781	0.0948904	+	0.995488i	$0.469750\pi$
$620$	4.86527	0.195394
$621$	0	0
$622$	−8.16084	−0.327220
$623$	6.61497	0.265023
$624$	0	0
$625$	23.8594	0.954376
$626$	−1.45192	−0.0580303
$627$	0	0
$628$	−10.2003	−0.407038
$629$	−4.58538	−0.182831
$630$	0	0
$631$	−8.69806	−0.346264	−0.173132	−	0.984899i	$-0.555389\pi$
$631$	−8.69806	−0.346264	−0.173132	+	0.984899i	$0.555389\pi$
$632$	0.473821	0.0188476
$633$	0	0
$634$	3.90798	0.155206
$635$	−0.276458	−0.0109709
$636$	0	0
$637$	5.85411	0.231948
$638$	−9.03436	−0.357674
$639$	0	0
$640$	−2.39735	−0.0947637
$641$	0.289431	0.0114318	0.00571592	−	0.999984i	$-0.498181\pi$
$641$	0.289431	0.0114318	0.00571592	+	0.999984i	$0.498181\pi$
$642$	0	0
$643$	25.5065	1.00588	0.502939	−	0.864322i	$-0.332252\pi$
$643$	25.5065	1.00588	0.502939	+	0.864322i	$0.332252\pi$
$644$	9.35489	0.368634
$645$	0	0
$646$	2.14904	0.0845527
$647$	7.05434	0.277335	0.138667	−	0.990339i	$-0.455718\pi$
$647$	7.05434	0.277335	0.138667	+	0.990339i	$0.455718\pi$
$648$	0	0
$649$	−55.6043	−2.18266
$650$	−8.81154	−0.345617
$651$	0	0
$652$	14.9156	0.584138
$653$	−29.0397	−1.13641	−0.568206	−	0.822886i	$-0.692362\pi$
$653$	−29.0397	−1.13641	−0.568206	+	0.822886i	$0.692362\pi$
$654$	0	0
$655$	−3.29746	−0.128842
$656$	−6.04946	−0.236192
$657$	0	0
$658$	0.774732	0.0302022
$659$	−5.95542	−0.231990	−0.115995	−	0.993250i	$-0.537006\pi$
$659$	−5.95542	−0.231990	−0.115995	+	0.993250i	$0.537006\pi$
$660$	0	0
$661$	18.6354	0.724832	0.362416	−	0.932016i	$-0.381952\pi$
$661$	18.6354	0.724832	0.362416	+	0.932016i	$0.381952\pi$
$662$	9.17473	0.356586
$663$	0	0
$664$	−2.71259	−0.105269
$665$	−1.68811	−0.0654623
$666$	0	0
$667$	26.3760	1.02128
$668$	15.9325	0.616446
$669$	0	0
$670$	0.316364	0.0122222
$671$	34.9688	1.34995
$672$	0	0
$673$	21.1789	0.816387	0.408193	−	0.912896i	$-0.366159\pi$
$673$	21.1789	0.816387	0.408193	+	0.912896i	$0.366159\pi$
$674$	−10.2929	−0.396468
$675$	0	0
$676$	−40.5532	−1.55974
$677$	−21.4495	−0.824370	−0.412185	−	0.911100i	$-0.635234\pi$
$677$	−21.4495	−0.824370	−0.412185	+	0.911100i	$0.635234\pi$
$678$	0	0
$679$	6.50324	0.249571
$680$	0.380094	0.0145760
$681$	0	0
$682$	−15.5137	−0.594048
$683$	−42.3011	−1.61861	−0.809303	−	0.587391i	$-0.800155\pi$
$683$	−42.3011	−1.61861	−0.809303	+	0.587391i	$0.800155\pi$
$684$	0	0
$685$	−2.26538	−0.0865556
$686$	0.305711	0.0116721
$687$	0	0
$688$	−36.8859	−1.40626
$689$	39.5279	1.50589
$690$	0	0
$691$	−13.9263	−0.529782	−0.264891	−	0.964278i	$-0.585336\pi$
$691$	−13.9263	−0.529782	−0.264891	+	0.964278i	$0.585336\pi$
$692$	13.3604	0.507886
$693$	0	0
$694$	−3.11742	−0.118336
$695$	−3.74538	−0.142071
$696$	0	0
$697$	2.01981	0.0765059
$698$	−6.66261	−0.252184
$699$	0	0
$700$	9.38699	0.354795
$701$	34.0403	1.28568	0.642842	−	0.765999i	$-0.277755\pi$
$701$	34.0403	1.28568	0.642842	+	0.765999i	$0.277755\pi$
$702$	0	0
$703$	−24.3214	−0.917300
$704$	−32.1251	−1.21076
$705$	0	0
$706$	−3.32186	−0.125020
$707$	−6.45399	−0.242727
$708$	0	0
$709$	−34.3771	−1.29106	−0.645529	−	0.763736i	$-0.723363\pi$
$709$	−34.3771	−1.29106	−0.645529	+	0.763736i	$0.723363\pi$
$710$	−0.866020	−0.0325012
$711$	0	0
$712$	−7.90007	−0.296068
$713$	45.2924	1.69621
$714$	0	0
$715$	8.89734	0.332742
$716$	48.0266	1.79484
$717$	0	0
$718$	10.0877	0.376468
$719$	40.4073	1.50694	0.753469	−	0.657484i	$-0.228379\pi$
$719$	40.4073	1.50694	0.753469	+	0.657484i	$0.228379\pi$
$720$	0	0
$721$	−7.17745	−0.267302
$722$	5.59025	0.208047
$723$	0	0
$724$	−32.2238	−1.19759
$725$	26.4665	0.982940
$726$	0	0
$727$	3.23987	0.120160	0.0600800	−	0.998194i	$-0.480864\pi$
$727$	3.23987	0.120160	0.0600800	+	0.998194i	$0.480864\pi$
$728$	−6.99140	−0.259118
$729$	0	0
$730$	0.0650128	0.00240623
$731$	12.3156	0.455508
$732$	0	0
$733$	8.50412	0.314107	0.157053	−	0.987590i	$-0.449801\pi$
$733$	8.50412	0.314107	0.157053	+	0.987590i	$0.449801\pi$
$734$	−3.37182	−0.124456
$735$	0	0
$736$	−16.8921	−0.622650
$737$	20.5787	0.758027
$738$	0	0
$739$	−17.5860	−0.646913	−0.323456	−	0.946243i	$-0.604845\pi$
$739$	−17.5860	−0.646913	−0.323456	+	0.946243i	$0.604845\pi$
$740$	−2.09938	−0.0771747
$741$	0	0
$742$	2.06421	0.0757795
$743$	26.7552	0.981551	0.490776	−	0.871286i	$-0.336713\pi$
$743$	26.7552	0.981551	0.490776	+	0.871286i	$0.336713\pi$
$744$	0	0
$745$	−1.37881	−0.0505157
$746$	−10.0928	−0.369525
$747$	0	0
$748$	12.0664	0.441190
$749$	−2.58615	−0.0944958
$750$	0	0
$751$	−26.4154	−0.963910	−0.481955	−	0.876196i	$-0.660073\pi$
$751$	−26.4154	−0.963910	−0.481955	+	0.876196i	$0.660073\pi$
$752$	8.73786	0.318637
$753$	0	0
$754$	−9.62027	−0.350350
$755$	1.15438	0.0420122
$756$	0	0
$757$	−46.7851	−1.70043	−0.850217	−	0.526433i	$-0.823529\pi$
$757$	−46.7851	−1.70043	−0.850217	+	0.526433i	$0.823529\pi$
$758$	−1.80663	−0.0656199
$759$	0	0
$760$	2.01607	0.0731305
$761$	27.0165	0.979349	0.489674	−	0.871905i	$-0.337116\pi$
$761$	27.0165	0.979349	0.489674	+	0.871905i	$0.337116\pi$
$762$	0	0
$763$	16.7781	0.607407
$764$	−41.5942	−1.50482
$765$	0	0
$766$	−7.40143	−0.267425
$767$	−59.2105	−2.13797
$768$	0	0
$769$	8.57917	0.309373	0.154686	−	0.987964i	$-0.450563\pi$
$769$	8.57917	0.309373	0.154686	+	0.987964i	$0.450563\pi$
$770$	0.464633	0.0167442
$771$	0	0
$772$	33.7424	1.21441
$773$	−37.9118	−1.36359	−0.681796	−	0.731542i	$-0.738801\pi$
$773$	−37.9118	−1.36359	−0.681796	+	0.731542i	$0.738801\pi$
$774$	0	0
$775$	45.4478	1.63253
$776$	−7.76664	−0.278806
$777$	0	0
$778$	9.96522	0.357270
$779$	10.7133	0.383845
$780$	0	0
$781$	−56.3325	−2.01573
$782$	1.72688	0.0617532
$783$	0	0
$784$	3.44798	0.123142
$785$	1.47910	0.0527913
$786$	0	0
$787$	−40.7590	−1.45290	−0.726450	−	0.687219i	$-0.758831\pi$
$787$	−40.7590	−1.45290	−0.726450	+	0.687219i	$0.758831\pi$
$788$	17.3746	0.618943
$789$	0	0
$790$	−0.0335313	−0.00119299
$791$	−10.3192	−0.366907
$792$	0	0
$793$	37.2366	1.32231
$794$	3.96904	0.140856
$795$	0	0
$796$	0.0431247	0.00152852
$797$	5.07686	0.179831	0.0899157	−	0.995949i	$-0.471340\pi$
$797$	5.07686	0.179831	0.0899157	+	0.995949i	$0.471340\pi$
$798$	0	0
$799$	−2.91743	−0.103211
$800$	−16.9500	−0.599274
$801$	0	0
$802$	11.0659	0.390750
$803$	4.22892	0.149235
$804$	0	0
$805$	−1.35651	−0.0478105
$806$	−16.5198	−0.581884
$807$	0	0
$808$	7.70782	0.271160
$809$	−50.5158	−1.77604	−0.888021	−	0.459803i	$-0.847920\pi$
$809$	−50.5158	−1.77604	−0.888021	+	0.459803i	$0.847920\pi$
$810$	0	0
$811$	−16.8436	−0.591458	−0.295729	−	0.955272i	$-0.595563\pi$
$811$	−16.8436	−0.591458	−0.295729	+	0.955272i	$0.595563\pi$
$812$	10.2485	0.359653
$813$	0	0
$814$	6.69418	0.234631
$815$	−2.16283	−0.0757606
$816$	0	0
$817$	65.3234	2.28537
$818$	−3.04784	−0.106565
$819$	0	0
$820$	0.924755	0.0322938
$821$	15.9626	0.557100	0.278550	−	0.960422i	$-0.410146\pi$
$821$	15.9626	0.557100	0.278550	+	0.960422i	$0.410146\pi$
$822$	0	0
$823$	35.8972	1.25130	0.625650	−	0.780104i	$-0.284834\pi$
$823$	35.8972	1.25130	0.625650	+	0.780104i	$0.284834\pi$
$824$	8.57183	0.298614
$825$	0	0
$826$	−3.09207	−0.107587
$827$	20.2205	0.703137	0.351568	−	0.936162i	$-0.385648\pi$
$827$	20.2205	0.703137	0.351568	+	0.936162i	$0.385648\pi$
$828$	0	0
$829$	−30.6787	−1.06552	−0.532758	−	0.846268i	$-0.678844\pi$
$829$	−30.6787	−1.06552	−0.532758	+	0.846268i	$0.678844\pi$
$830$	0.191964	0.00666318
$831$	0	0
$832$	−34.2085	−1.18597
$833$	−1.15122	−0.0398875
$834$	0	0
$835$	−2.31029	−0.0799508
$836$	64.0015	2.21354
$837$	0	0
$838$	−1.56454	−0.0540462
$839$	−32.8827	−1.13524	−0.567618	−	0.823292i	$-0.692135\pi$
$839$	−32.8827	−1.13524	−0.567618	+	0.823292i	$0.692135\pi$
$840$	0	0
$841$	−0.104428	−0.00360097
$842$	−9.96323	−0.343356
$843$	0	0
$844$	−24.9081	−0.857371
$845$	5.88042	0.202293
$846$	0	0
$847$	19.2233	0.660519
$848$	23.2813	0.799484
$849$	0	0
$850$	1.73281	0.0594348
$851$	−19.5438	−0.669952
$852$	0	0
$853$	8.11977	0.278016	0.139008	−	0.990291i	$-0.455609\pi$
$853$	8.11977	0.278016	0.139008	+	0.990291i	$0.455609\pi$
$854$	1.94456	0.0665413
$855$	0	0
$856$	3.08856	0.105565
$857$	39.4412	1.34729	0.673644	−	0.739056i	$-0.264728\pi$
$857$	39.4412	1.34729	0.673644	+	0.739056i	$0.264728\pi$
$858$	0	0
$859$	50.5091	1.72335	0.861675	−	0.507461i	$-0.169416\pi$
$859$	50.5091	1.72335	0.861675	+	0.507461i	$0.169416\pi$
$860$	5.63859	0.192274
$861$	0	0
$862$	1.65159	0.0562533
$863$	−18.8260	−0.640844	−0.320422	−	0.947275i	$-0.603825\pi$
$863$	−18.8260	−0.640844	−0.320422	+	0.947275i	$0.603825\pi$
$864$	0	0
$865$	−1.93732	−0.0658710
$866$	−4.63476	−0.157496
$867$	0	0
$868$	17.5986	0.597336
$869$	−2.18113	−0.0739898
$870$	0	0
$871$	21.9133	0.742505
$872$	−20.0376	−0.678558
$873$	0	0
$874$	9.15961	0.309828
$875$	−2.74345	−0.0927455
$876$	0	0
$877$	51.8551	1.75102	0.875511	−	0.483197i	$-0.160525\pi$
$877$	51.8551	1.75102	0.875511	+	0.483197i	$0.160525\pi$
$878$	7.86482	0.265425
$879$	0	0
$880$	5.24040	0.176654
$881$	−18.0437	−0.607907	−0.303953	−	0.952687i	$-0.598307\pi$
$881$	−18.0437	−0.607907	−0.303953	+	0.952687i	$0.598307\pi$
$882$	0	0
$883$	−38.3129	−1.28933	−0.644665	−	0.764465i	$-0.723003\pi$
$883$	−38.3129	−1.28933	−0.644665	+	0.764465i	$0.723003\pi$
$884$	12.8489	0.432156
$885$	0	0
$886$	−8.72988	−0.293286
$887$	39.8828	1.33913	0.669567	−	0.742751i	$-0.266480\pi$
$887$	39.8828	1.33913	0.669567	+	0.742751i	$0.266480\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	0.559072	0.0187401
$891$	0	0
$892$	−27.5585	−0.922726
$893$	−15.4744	−0.517831
$894$	0	0
$895$	−6.96409	−0.232784
$896$	−8.67168	−0.289701
$897$	0	0
$898$	10.0977	0.336965
$899$	49.6190	1.65489
$900$	0	0
$901$	−7.77324	−0.258964
$902$	−2.94872	−0.0981816
$903$	0	0
$904$	12.3239	0.409886
$905$	4.67261	0.155323
$906$	0	0
$907$	15.9249	0.528779	0.264389	−	0.964416i	$-0.414830\pi$
$907$	15.9249	0.528779	0.264389	+	0.964416i	$0.414830\pi$
$908$	10.1934	0.338279
$909$	0	0
$910$	0.494767	0.0164014
$911$	−30.4445	−1.00867	−0.504335	−	0.863508i	$-0.668262\pi$
$911$	−30.4445	−1.00867	−0.504335	+	0.863508i	$0.668262\pi$
$912$	0	0
$913$	12.4868	0.413253
$914$	8.04332	0.266049
$915$	0	0
$916$	−23.3360	−0.771042
$917$	−11.9275	−0.393881
$918$	0	0
$919$	−32.5351	−1.07323	−0.536617	−	0.843826i	$-0.680298\pi$
$919$	−32.5351	−1.07323	−0.536617	+	0.843826i	$0.680298\pi$
$920$	1.62004	0.0534110
$921$	0	0
$922$	−5.18590	−0.170788
$923$	−59.9859	−1.97446
$924$	0	0
$925$	−19.6108	−0.644800
$926$	7.25038	0.238262
$927$	0	0
$928$	−18.5057	−0.607480
$929$	−24.8695	−0.815942	−0.407971	−	0.912995i	$-0.633764\pi$
$929$	−24.8695	−0.815942	−0.407971	+	0.912995i	$0.633764\pi$
$930$	0	0
$931$	−6.10623	−0.200124
$932$	39.4313	1.29161
$933$	0	0
$934$	10.4730	0.342687
$935$	−1.74968	−0.0572207
$936$	0	0
$937$	−49.0788	−1.60333	−0.801667	−	0.597770i	$-0.796053\pi$
$937$	−49.0788	−1.60333	−0.801667	+	0.597770i	$0.796053\pi$
$938$	1.14435	0.0373644
$939$	0	0
$940$	−1.33572	−0.0435664
$941$	−41.2182	−1.34367	−0.671837	−	0.740699i	$-0.734495\pi$
$941$	−41.2182	−1.34367	−0.671837	+	0.740699i	$0.734495\pi$
$942$	0	0
$943$	8.60884	0.280342
$944$	−34.8740	−1.13505
$945$	0	0
$946$	−17.9795	−0.584563
$947$	7.08226	0.230143	0.115071	−	0.993357i	$-0.463290\pi$
$947$	7.08226	0.230143	0.115071	+	0.993357i	$0.463290\pi$
$948$	0	0
$949$	4.50318	0.146180
$950$	9.19103	0.298197
$951$	0	0
$952$	1.37487	0.0445599
$953$	−11.4314	−0.370300	−0.185150	−	0.982710i	$-0.559277\pi$
$953$	−11.4314	−0.370300	−0.185150	+	0.982710i	$0.559277\pi$
$954$	0	0
$955$	6.03136	0.195170
$956$	37.8522	1.22423
$957$	0	0
$958$	−7.18501	−0.232137
$959$	−8.19429	−0.264608
$960$	0	0
$961$	54.2049	1.74855
$962$	7.12833	0.229827
$963$	0	0
$964$	−6.47759	−0.208629
$965$	−4.89281	−0.157505
$966$	0	0
$967$	45.9374	1.47725	0.738624	−	0.674117i	$-0.235476\pi$
$967$	45.9374	1.47725	0.738624	+	0.674117i	$0.235476\pi$
$968$	−22.9578	−0.737892
$969$	0	0
$970$	0.549629	0.0176475
$971$	−21.6154	−0.693671	−0.346835	−	0.937926i	$-0.612744\pi$
$971$	−21.6154	−0.693671	−0.346835	+	0.937926i	$0.612744\pi$
$972$	0	0
$973$	−13.5478	−0.434321
$974$	−0.148983	−0.00477371
$975$	0	0
$976$	21.9318	0.702020
$977$	−20.9557	−0.670432	−0.335216	−	0.942141i	$-0.608809\pi$
$977$	−20.9557	−0.670432	−0.335216	+	0.942141i	$0.608809\pi$
$978$	0	0
$979$	36.3662	1.16227
$980$	−0.527078	−0.0168369
$981$	0	0
$982$	−4.70319	−0.150085
$983$	41.0432	1.30908	0.654538	−	0.756029i	$-0.272863\pi$
$983$	41.0432	1.30908	0.654538	+	0.756029i	$0.272863\pi$
$984$	0	0
$985$	−2.51940	−0.0802746
$986$	1.89185	0.0602487
$987$	0	0
$988$	68.1522	2.16821
$989$	52.4914	1.66913
$990$	0	0
$991$	−57.2547	−1.81875	−0.909377	−	0.415973i	$-0.863441\pi$
$991$	−57.2547	−1.81875	−0.909377	+	0.415973i	$0.863441\pi$
$992$	−31.7777	−1.00894
$993$	0	0
$994$	−3.13256	−0.0993587
$995$	−0.00625330	−0.000198243 0
$996$	0	0
$997$	−6.92036	−0.219170	−0.109585	−	0.993977i	$-0.534952\pi$
$997$	−6.92036	−0.219170	−0.109585	+	0.993977i	$0.534952\pi$
$998$	−1.21630	−0.0385013
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.o.1.8		13
3.2	odd	2		2667.2.a.l.1.6	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.l.1.6	✓	13	3.2	odd	2
8001.2.a.o.1.8		13	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8001.a (trivial)

\(f(q)\)	\(=\)	\(q+0.305711 q^{2} -1.90654 q^{4} +0.276458 q^{5} +1.00000 q^{7} -1.19427 q^{8} +O(q^{10})\)	\(q+0.305711 q^{2} -1.90654 q^{4} +0.276458 q^{5} +1.00000 q^{7} -1.19427 q^{8} +0.0845162 q^{10} +5.49757 q^{11} +5.85411 q^{13} +0.305711 q^{14} +3.44798 q^{16} -1.15122 q^{17} -6.10623 q^{19} -0.527078 q^{20} +1.68067 q^{22} -4.90674 q^{23} -4.92357 q^{25} +1.78966 q^{26} -1.90654 q^{28} -5.37546 q^{29} -9.23065 q^{31} +3.44263 q^{32} -0.351941 q^{34} +0.276458 q^{35} +3.98305 q^{37} -1.86674 q^{38} -0.330166 q^{40} -1.75449 q^{41} -10.6978 q^{43} -10.4813 q^{44} -1.50004 q^{46} +2.53420 q^{47} +1.00000 q^{49} -1.50519 q^{50} -11.1611 q^{52} +6.75216 q^{53} +1.51985 q^{55} -1.19427 q^{56} -1.64334 q^{58} -10.1143 q^{59} +6.36077 q^{61} -2.82191 q^{62} -5.84351 q^{64} +1.61841 q^{65} +3.74324 q^{67} +2.19485 q^{68} +0.0845162 q^{70} -10.2468 q^{71} +0.769235 q^{73} +1.21766 q^{74} +11.6418 q^{76} +5.49757 q^{77} -0.396745 q^{79} +0.953221 q^{80} -0.536368 q^{82} +2.27133 q^{83} -0.318264 q^{85} -3.27044 q^{86} -6.56559 q^{88} +6.61497 q^{89} +5.85411 q^{91} +9.35489 q^{92} +0.774732 q^{94} -1.68811 q^{95} +6.50324 q^{97} +0.305711 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 4 q^{2} + 10 q^{4} - 12 q^{5} + 13 q^{7} - 9 q^{8}+O(q^{10}) \) 13 * q - 4 * q^2 + 10 * q^4 - 12 * q^5 + 13 * q^7 - 9 * q^8	\( 13 q - 4 q^{2} + 10 q^{4} - 12 q^{5} + 13 q^{7} - 9 q^{8} + 6 q^{10} - 3 q^{11} + 21 q^{13} - 4 q^{14} + 8 q^{16} - 17 q^{17} + 5 q^{19} - 29 q^{20} + q^{22} - 4 q^{23} + q^{25} - 22 q^{26} + 10 q^{28} - 21 q^{29} - 7 q^{31} - 12 q^{32} + 2 q^{34} - 12 q^{35} + 7 q^{37} + 9 q^{38} + 29 q^{40} - 21 q^{41} - 9 q^{43} + 2 q^{44} - 28 q^{46} - 23 q^{47} + 13 q^{49} - 15 q^{50} + 15 q^{52} - 31 q^{53} - 8 q^{55} - 9 q^{56} - 25 q^{58} - 28 q^{59} + 29 q^{61} + 3 q^{62} + 9 q^{64} - 30 q^{65} - 18 q^{67} - 34 q^{68} + 6 q^{70} - 10 q^{71} + 24 q^{73} + 19 q^{74} - 3 q^{77} - 28 q^{79} - 26 q^{80} + 18 q^{82} - 26 q^{83} + 20 q^{85} + 2 q^{86} - 17 q^{88} - 44 q^{89} + 21 q^{91} - 6 q^{92} - 9 q^{94} + 2 q^{95} + 17 q^{97} - 4 q^{98}+O(q^{100}) \) 13 * q - 4 * q^2 + 10 * q^4 - 12 * q^5 + 13 * q^7 - 9 * q^8 + 6 * q^10 - 3 * q^11 + 21 * q^13 - 4 * q^14 + 8 * q^16 - 17 * q^17 + 5 * q^19 - 29 * q^20 + q^22 - 4 * q^23 + q^25 - 22 * q^26 + 10 * q^28 - 21 * q^29 - 7 * q^31 - 12 * q^32 + 2 * q^34 - 12 * q^35 + 7 * q^37 + 9 * q^38 + 29 * q^40 - 21 * q^41 - 9 * q^43 + 2 * q^44 - 28 * q^46 - 23 * q^47 + 13 * q^49 - 15 * q^50 + 15 * q^52 - 31 * q^53 - 8 * q^55 - 9 * q^56 - 25 * q^58 - 28 * q^59 + 29 * q^61 + 3 * q^62 + 9 * q^64 - 30 * q^65 - 18 * q^67 - 34 * q^68 + 6 * q^70 - 10 * q^71 + 24 * q^73 + 19 * q^74 - 3 * q^77 - 28 * q^79 - 26 * q^80 + 18 * q^82 - 26 * q^83 + 20 * q^85 + 2 * q^86 - 17 * q^88 - 44 * q^89 + 21 * q^91 - 6 * q^92 - 9 * q^94 + 2 * q^95 + 17 * q^97 - 4 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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