LMFDB - Embedded newform 1503.2.a.e.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1503,2,Mod(1,1503)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1503, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1503.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1503 = 3^{2} \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1503.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:	$12.0015154238$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 8x^{6} + 28x^{5} + 9x^{4} - 64x^{3} + 17x^{2} + 23x + 1 $ x^8 - 3x^7 - 8x^6 + 28x^5 + 9x^4 - 64x^3 + 17x^2 + 23*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 501)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-2.45154$ of defining polynomial
Character	$\chi$	$=$	1503.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.45154 q^{2} +4.01004 q^{4} -1.48532 q^{5} -5.10210 q^{7} +4.92768 q^{8} +O(q^{10})$	\(q+2.45154 q^{2} +4.01004 q^{4} -1.48532 q^{5} -5.10210 q^{7} +4.92768 q^{8} -3.64131 q^{10} -4.83831 q^{11} -2.69442 q^{13} -12.5080 q^{14} +4.06032 q^{16} -5.76005 q^{17} +6.48887 q^{19} -5.95618 q^{20} -11.8613 q^{22} +2.39680 q^{23} -2.79383 q^{25} -6.60546 q^{26} -20.4596 q^{28} -1.46082 q^{29} +7.19438 q^{31} +0.0986604 q^{32} -14.1210 q^{34} +7.57824 q^{35} +3.46817 q^{37} +15.9077 q^{38} -7.31917 q^{40} +8.25442 q^{41} -0.556466 q^{43} -19.4018 q^{44} +5.87585 q^{46} -8.12240 q^{47} +19.0314 q^{49} -6.84918 q^{50} -10.8047 q^{52} -1.56013 q^{53} +7.18642 q^{55} -25.1415 q^{56} -3.58127 q^{58} -10.5071 q^{59} -0.565519 q^{61} +17.6373 q^{62} -7.87876 q^{64} +4.00206 q^{65} +8.66646 q^{67} -23.0980 q^{68} +18.5783 q^{70} -3.64683 q^{71} -1.86009 q^{73} +8.50235 q^{74} +26.0206 q^{76} +24.6855 q^{77} -9.58543 q^{79} -6.03086 q^{80} +20.2360 q^{82} -4.32336 q^{83} +8.55551 q^{85} -1.36420 q^{86} -23.8416 q^{88} +2.94180 q^{89} +13.7472 q^{91} +9.61127 q^{92} -19.9124 q^{94} -9.63803 q^{95} -18.4986 q^{97} +46.6562 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 3 q^{2} + 9 q^{4} - 7 q^{5} - 4 q^{7} - 3 q^{8}+O(q^{10}) $ 8 * q - 3 * q^2 + 9 * q^4 - 7 * q^5 - 4 * q^7 - 3 * q^8	$ 8 q - 3 q^{2} + 9 q^{4} - 7 q^{5} - 4 q^{7} - 3 q^{8} - q^{10} - 13 q^{11} - 5 q^{14} + 7 q^{16} - 11 q^{17} + 12 q^{19} - 9 q^{20} - 17 q^{22} - 7 q^{23} - 5 q^{25} - 3 q^{26} - 27 q^{28} - q^{29} - 2 q^{31} - 4 q^{32} - 14 q^{34} + 4 q^{35} - 9 q^{37} - 22 q^{40} - 4 q^{41} + 2 q^{43} - 3 q^{44} - 5 q^{46} - 17 q^{47} - 2 q^{49} + 4 q^{50} - 36 q^{52} - 9 q^{53} + 7 q^{55} - 9 q^{56} - 29 q^{58} - 29 q^{59} - 12 q^{61} + 34 q^{62} - 5 q^{64} - 8 q^{65} - 26 q^{68} + 5 q^{70} - 13 q^{71} - 20 q^{73} + 17 q^{74} + 30 q^{76} + 22 q^{77} + 8 q^{79} + 34 q^{80} + 15 q^{82} - 33 q^{83} - 31 q^{85} - 11 q^{86} - 44 q^{88} - 4 q^{89} + q^{91} + 33 q^{92} - 2 q^{94} - 3 q^{95} - 31 q^{97} + 57 q^{98}+O(q^{100}) $ 8 * q - 3 * q^2 + 9 * q^4 - 7 * q^5 - 4 * q^7 - 3 * q^8 - q^10 - 13 * q^11 - 5 * q^14 + 7 * q^16 - 11 * q^17 + 12 * q^19 - 9 * q^20 - 17 * q^22 - 7 * q^23 - 5 * q^25 - 3 * q^26 - 27 * q^28 - q^29 - 2 * q^31 - 4 * q^32 - 14 * q^34 + 4 * q^35 - 9 * q^37 - 22 * q^40 - 4 * q^41 + 2 * q^43 - 3 * q^44 - 5 * q^46 - 17 * q^47 - 2 * q^49 + 4 * q^50 - 36 * q^52 - 9 * q^53 + 7 * q^55 - 9 * q^56 - 29 * q^58 - 29 * q^59 - 12 * q^61 + 34 * q^62 - 5 * q^64 - 8 * q^65 - 26 * q^68 + 5 * q^70 - 13 * q^71 - 20 * q^73 + 17 * q^74 + 30 * q^76 + 22 * q^77 + 8 * q^79 + 34 * q^80 + 15 * q^82 - 33 * q^83 - 31 * q^85 - 11 * q^86 - 44 * q^88 - 4 * q^89 + q^91 + 33 * q^92 - 2 * q^94 - 3 * q^95 - 31 * q^97 + 57 * 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 $

$2$	2.45154	1.73350	0.866749	−	0.498744i	$-0.166205\pi$
$2$	2.45154	1.73350	0.866749	+	0.498744i	$0.166205\pi$
$3$	0	0
$3$	0	0
$4$	4.01004	2.00502
$5$	−1.48532	−0.664254	−0.332127	−	0.943235i	$-0.607766\pi$
$5$	−1.48532	−0.664254	−0.332127	+	0.943235i	$0.607766\pi$
$6$	0	0
$7$	−5.10210	−1.92841	−0.964206	−	0.265153i	$-0.914577\pi$
$7$	−5.10210	−1.92841	−0.964206	+	0.265153i	$0.914577\pi$
$8$	4.92768	1.74220
$9$	0	0
$10$	−3.64131	−1.15148
$11$	−4.83831	−1.45880	−0.729402	−	0.684085i	$-0.760202\pi$
$11$	−4.83831	−1.45880	−0.729402	+	0.684085i	$0.760202\pi$
$12$	0	0
$13$	−2.69442	−0.747297	−0.373648	−	0.927570i	$-0.621893\pi$
$13$	−2.69442	−0.747297	−0.373648	+	0.927570i	$0.621893\pi$
$14$	−12.5080	−3.34290
$15$	0	0
$16$	4.06032	1.01508
$17$	−5.76005	−1.39702	−0.698509	−	0.715602i	$-0.746153\pi$
$17$	−5.76005	−1.39702	−0.698509	+	0.715602i	$0.746153\pi$
$18$	0	0
$19$	6.48887	1.48865	0.744324	−	0.667819i	$-0.232772\pi$
$19$	6.48887	1.48865	0.744324	+	0.667819i	$0.232772\pi$
$20$	−5.95618	−1.33184
$21$	0	0
$22$	−11.8613	−2.52883
$23$	2.39680	0.499768	0.249884	−	0.968276i	$-0.419608\pi$
$23$	2.39680	0.499768	0.249884	+	0.968276i	$0.419608\pi$
$24$	0	0
$25$	−2.79383	−0.558766
$26$	−6.60546	−1.29544
$27$	0	0
$28$	−20.4596	−3.86650
$29$	−1.46082	−0.271268	−0.135634	−	0.990759i	$-0.543307\pi$
$29$	−1.46082	−0.271268	−0.135634	+	0.990759i	$0.543307\pi$
$30$	0	0
$31$	7.19438	1.29215	0.646075	−	0.763274i	$-0.276409\pi$
$31$	7.19438	1.29215	0.646075	+	0.763274i	$0.276409\pi$
$32$	0.0986604	0.0174409
$33$	0	0
$34$	−14.1210	−2.42173
$35$	7.57824	1.28096
$36$	0	0
$37$	3.46817	0.570164	0.285082	−	0.958503i	$-0.407979\pi$
$37$	3.46817	0.570164	0.285082	+	0.958503i	$0.407979\pi$
$38$	15.9077	2.58057
$39$	0	0
$40$	−7.31917	−1.15726
$41$	8.25442	1.28912	0.644562	−	0.764552i	$-0.277040\pi$
$41$	8.25442	1.28912	0.644562	+	0.764552i	$0.277040\pi$
$42$	0	0
$43$	−0.556466	−0.0848602	−0.0424301	−	0.999099i	$-0.513510\pi$
$43$	−0.556466	−0.0848602	−0.0424301	+	0.999099i	$0.513510\pi$
$44$	−19.4018	−2.92493
$45$	0	0
$46$	5.87585	0.866347
$47$	−8.12240	−1.18477	−0.592387	−	0.805654i	$-0.701814\pi$
$47$	−8.12240	−1.18477	−0.592387	+	0.805654i	$0.701814\pi$
$48$	0	0
$49$	19.0314	2.71877
$50$	−6.84918	−0.968620
$51$	0	0
$52$	−10.8047	−1.49834
$53$	−1.56013	−0.214300	−0.107150	−	0.994243i	$-0.534173\pi$
$53$	−1.56013	−0.214300	−0.107150	+	0.994243i	$0.534173\pi$
$54$	0	0
$55$	7.18642	0.969017
$56$	−25.1415	−3.35968
$57$	0	0
$58$	−3.58127	−0.470243
$59$	−10.5071	−1.36791	−0.683953	−	0.729526i	$-0.739741\pi$
$59$	−10.5071	−1.36791	−0.683953	+	0.729526i	$0.739741\pi$
$60$	0	0
$61$	−0.565519	−0.0724073	−0.0362037	−	0.999344i	$-0.511527\pi$
$61$	−0.565519	−0.0724073	−0.0362037	+	0.999344i	$0.511527\pi$
$62$	17.6373	2.23994
$63$	0	0
$64$	−7.87876	−0.984845
$65$	4.00206	0.496395
$66$	0	0
$67$	8.66646	1.05878	0.529388	−	0.848380i	$-0.322421\pi$
$67$	8.66646	1.05878	0.529388	+	0.848380i	$0.322421\pi$
$68$	−23.0980	−2.80104
$69$	0	0
$70$	18.5783	2.22054
$71$	−3.64683	−0.432799	−0.216399	−	0.976305i	$-0.569431\pi$
$71$	−3.64683	−0.432799	−0.216399	+	0.976305i	$0.569431\pi$
$72$	0	0
$73$	−1.86009	−0.217707	−0.108853	−	0.994058i	$-0.534718\pi$
$73$	−1.86009	−0.217707	−0.108853	+	0.994058i	$0.534718\pi$
$74$	8.50235	0.988378
$75$	0	0
$76$	26.0206	2.98477
$77$	24.6855	2.81318
$78$	0	0
$79$	−9.58543	−1.07844	−0.539222	−	0.842163i	$-0.681282\pi$
$79$	−9.58543	−1.07844	−0.539222	+	0.842163i	$0.681282\pi$
$80$	−6.03086	−0.674271
$81$	0	0
$82$	20.2360	2.23469
$83$	−4.32336	−0.474551	−0.237275	−	0.971442i	$-0.576254\pi$
$83$	−4.32336	−0.474551	−0.237275	+	0.971442i	$0.576254\pi$
$84$	0	0
$85$	8.55551	0.927975
$86$	−1.36420	−0.147105
$87$	0	0
$88$	−23.8416	−2.54152
$89$	2.94180	0.311830	0.155915	−	0.987770i	$-0.450167\pi$
$89$	2.94180	0.311830	0.155915	+	0.987770i	$0.450167\pi$
$90$	0	0
$91$	13.7472	1.44110
$92$	9.61127	1.00204
$93$	0	0
$94$	−19.9124	−2.05380
$95$	−9.63803	−0.988841
$96$	0	0
$97$	−18.4986	−1.87825	−0.939123	−	0.343582i	$-0.888360\pi$
$97$	−18.4986	−1.87825	−0.939123	+	0.343582i	$0.888360\pi$
$98$	46.6562	4.71299
$99$	0	0
$100$	−11.2034	−1.12034
$101$	8.57364	0.853109	0.426555	−	0.904462i	$-0.359727\pi$
$101$	8.57364	0.853109	0.426555	+	0.904462i	$0.359727\pi$
$102$	0	0
$103$	7.39821	0.728967	0.364484	−	0.931210i	$-0.381246\pi$
$103$	7.39821	0.728967	0.364484	+	0.931210i	$0.381246\pi$
$104$	−13.2772	−1.30194
$105$	0	0
$106$	−3.82471	−0.371489
$107$	−13.5559	−1.31050	−0.655250	−	0.755412i	$-0.727437\pi$
$107$	−13.5559	−1.31050	−0.655250	+	0.755412i	$0.727437\pi$
$108$	0	0
$109$	−12.1413	−1.16293	−0.581463	−	0.813573i	$-0.697519\pi$
$109$	−12.1413	−1.16293	−0.581463	+	0.813573i	$0.697519\pi$
$110$	17.6178	1.67979
$111$	0	0
$112$	−20.7161	−1.95749
$113$	−17.3031	−1.62774	−0.813869	−	0.581048i	$-0.802643\pi$
$113$	−17.3031	−1.62774	−0.813869	+	0.581048i	$0.802643\pi$
$114$	0	0
$115$	−3.56002	−0.331973
$116$	−5.85796	−0.543898
$117$	0	0
$118$	−25.7585	−2.37127
$119$	29.3884	2.69403
$120$	0	0
$121$	12.4092	1.12811
$122$	−1.38639	−0.125518
$123$	0	0
$124$	28.8497	2.59078
$125$	11.5763	1.03542
$126$	0	0
$127$	2.69840	0.239444	0.119722	−	0.992807i	$-0.461800\pi$
$127$	2.69840	0.239444	0.119722	+	0.992807i	$0.461800\pi$
$128$	−19.5124	−1.72467
$129$	0	0
$130$	9.81121	0.860500
$131$	3.98729	0.348371	0.174186	−	0.984713i	$-0.444271\pi$
$131$	3.98729	0.348371	0.174186	+	0.984713i	$0.444271\pi$
$132$	0	0
$133$	−33.1068	−2.87073
$134$	21.2461	1.83539
$135$	0	0
$136$	−28.3837	−2.43388
$137$	−9.83648	−0.840387	−0.420194	−	0.907434i	$-0.638038\pi$
$137$	−9.83648	−0.840387	−0.420194	+	0.907434i	$0.638038\pi$
$138$	0	0
$139$	−18.5514	−1.57351	−0.786756	−	0.617264i	$-0.788241\pi$
$139$	−18.5514	−1.57351	−0.786756	+	0.617264i	$0.788241\pi$
$140$	30.3890	2.56834
$141$	0	0
$142$	−8.94033	−0.750256
$143$	13.0364	1.09016
$144$	0	0
$145$	2.16979	0.180191
$146$	−4.56008	−0.377395
$147$	0	0
$148$	13.9075	1.14319
$149$	−3.71647	−0.304465	−0.152232	−	0.988345i	$-0.548646\pi$
$149$	−3.71647	−0.304465	−0.152232	+	0.988345i	$0.548646\pi$
$150$	0	0
$151$	17.4813	1.42261	0.711304	−	0.702884i	$-0.248105\pi$
$151$	17.4813	1.42261	0.711304	+	0.702884i	$0.248105\pi$
$152$	31.9751	2.59352
$153$	0	0
$154$	60.5175	4.87664
$155$	−10.6859	−0.858316
$156$	0	0
$157$	−1.91857	−0.153119	−0.0765595	−	0.997065i	$-0.524394\pi$
$157$	−1.91857	−0.153119	−0.0765595	+	0.997065i	$0.524394\pi$
$158$	−23.4990	−1.86948
$159$	0	0
$160$	−0.146542	−0.0115852
$161$	−12.2287	−0.963759
$162$	0	0
$163$	−18.7990	−1.47245	−0.736226	−	0.676736i	$-0.763394\pi$
$163$	−18.7990	−1.47245	−0.736226	+	0.676736i	$0.763394\pi$
$164$	33.1005	2.58472
$165$	0	0
$166$	−10.5989	−0.822633
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	−5.74012	−0.441548
$170$	20.9741	1.60864
$171$	0	0
$172$	−2.23145	−0.170146
$173$	23.0233	1.75043	0.875214	−	0.483736i	$-0.160721\pi$
$173$	23.0233	1.75043	0.875214	+	0.483736i	$0.160721\pi$
$174$	0	0
$175$	14.2544	1.07753
$176$	−19.6450	−1.48080
$177$	0	0
$178$	7.21193	0.540557
$179$	14.1422	1.05704	0.528520	−	0.848921i	$-0.322747\pi$
$179$	14.1422	1.05704	0.528520	+	0.848921i	$0.322747\pi$
$180$	0	0
$181$	−5.73493	−0.426274	−0.213137	−	0.977022i	$-0.568368\pi$
$181$	−5.73493	−0.426274	−0.213137	+	0.977022i	$0.568368\pi$
$182$	33.7017	2.49814
$183$	0	0
$184$	11.8107	0.870695
$185$	−5.15133	−0.378734
$186$	0	0
$187$	27.8689	2.03797
$188$	−32.5711	−2.37549
$189$	0	0
$190$	−23.6280	−1.71415
$191$	19.1032	1.38226	0.691128	−	0.722732i	$-0.257114\pi$
$191$	19.1032	1.38226	0.691128	+	0.722732i	$0.257114\pi$
$192$	0	0
$193$	−2.71444	−0.195390	−0.0976949	−	0.995216i	$-0.531147\pi$
$193$	−2.71444	−0.195390	−0.0976949	+	0.995216i	$0.531147\pi$
$194$	−45.3499	−3.25594
$195$	0	0
$196$	76.3167	5.45119
$197$	−13.6028	−0.969157	−0.484579	−	0.874748i	$-0.661027\pi$
$197$	−13.6028	−0.969157	−0.484579	+	0.874748i	$0.661027\pi$
$198$	0	0
$199$	−12.9433	−0.917524	−0.458762	−	0.888559i	$-0.651707\pi$
$199$	−12.9433	−0.917524	−0.458762	+	0.888559i	$0.651707\pi$
$200$	−13.7671	−0.973481
$201$	0	0
$202$	21.0186	1.47886
$203$	7.45327	0.523117
$204$	0	0
$205$	−12.2604	−0.856306
$206$	18.1370	1.26366
$207$	0	0
$208$	−10.9402	−0.758565
$209$	−31.3951	−2.17165
$210$	0	0
$211$	−8.02853	−0.552707	−0.276354	−	0.961056i	$-0.589126\pi$
$211$	−8.02853	−0.552707	−0.276354	+	0.961056i	$0.589126\pi$
$212$	−6.25617	−0.429676
$213$	0	0
$214$	−33.2328	−2.27175
$215$	0.826529	0.0563688
$216$	0	0
$217$	−36.7065	−2.49180
$218$	−29.7648	−2.01593
$219$	0	0
$220$	28.8178	1.94290
$221$	15.5200	1.04399
$222$	0	0
$223$	10.8752	0.728259	0.364130	−	0.931348i	$-0.381366\pi$
$223$	10.8752	0.728259	0.364130	+	0.931348i	$0.381366\pi$
$224$	−0.503375	−0.0336332
$225$	0	0
$226$	−42.4192	−2.82168
$227$	22.9345	1.52221	0.761107	−	0.648626i	$-0.224656\pi$
$227$	22.9345	1.52221	0.761107	+	0.648626i	$0.224656\pi$
$228$	0	0
$229$	−15.3613	−1.01511	−0.507553	−	0.861621i	$-0.669450\pi$
$229$	−15.3613	−1.01511	−0.507553	+	0.861621i	$0.669450\pi$
$230$	−8.72751	−0.575475
$231$	0	0
$232$	−7.19847	−0.472603
$233$	−7.12266	−0.466621	−0.233310	−	0.972402i	$-0.574956\pi$
$233$	−7.12266	−0.466621	−0.233310	+	0.972402i	$0.574956\pi$
$234$	0	0
$235$	12.0643	0.786991
$236$	−42.1338	−2.74268
$237$	0	0
$238$	72.0466	4.67009
$239$	2.26815	0.146714	0.0733572	−	0.997306i	$-0.476629\pi$
$239$	2.26815	0.146714	0.0733572	+	0.997306i	$0.476629\pi$
$240$	0	0
$241$	16.9464	1.09161	0.545806	−	0.837911i	$-0.316223\pi$
$241$	16.9464	1.09161	0.545806	+	0.837911i	$0.316223\pi$
$242$	30.4216	1.95557
$243$	0	0
$244$	−2.26775	−0.145178
$245$	−28.2677	−1.80596
$246$	0	0
$247$	−17.4837	−1.11246
$248$	35.4516	2.25118
$249$	0	0
$250$	28.3798	1.79489
$251$	−6.81717	−0.430296	−0.215148	−	0.976581i	$-0.569023\pi$
$251$	−6.81717	−0.430296	−0.215148	+	0.976581i	$0.569023\pi$
$252$	0	0
$253$	−11.5965	−0.729064
$254$	6.61523	0.415076
$255$	0	0
$256$	−32.0779	−2.00487
$257$	−14.7749	−0.921634	−0.460817	−	0.887495i	$-0.652444\pi$
$257$	−14.7749	−0.921634	−0.460817	+	0.887495i	$0.652444\pi$
$258$	0	0
$259$	−17.6949	−1.09951
$260$	16.0484	0.995281
$261$	0	0
$262$	9.77500	0.603902
$263$	13.8198	0.852166	0.426083	−	0.904684i	$-0.359893\pi$
$263$	13.8198	0.852166	0.426083	+	0.904684i	$0.359893\pi$
$264$	0	0
$265$	2.31729	0.142350
$266$	−81.1627	−4.97640
$267$	0	0
$268$	34.7528	2.12287
$269$	18.3067	1.11618	0.558091	−	0.829780i	$-0.311534\pi$
$269$	18.3067	1.11618	0.558091	+	0.829780i	$0.311534\pi$
$270$	0	0
$271$	17.0332	1.03469	0.517346	−	0.855776i	$-0.326920\pi$
$271$	17.0332	1.03469	0.517346	+	0.855776i	$0.326920\pi$
$272$	−23.3876	−1.41808
$273$	0	0
$274$	−24.1145	−1.45681
$275$	13.5174	0.815130
$276$	0	0
$277$	−18.7596	−1.12716	−0.563578	−	0.826063i	$-0.690575\pi$
$277$	−18.7596	−1.12716	−0.563578	+	0.826063i	$0.690575\pi$
$278$	−45.4795	−2.72768
$279$	0	0
$280$	37.3431	2.23168
$281$	−3.93782	−0.234911	−0.117455	−	0.993078i	$-0.537474\pi$
$281$	−3.93782	−0.234911	−0.117455	+	0.993078i	$0.537474\pi$
$282$	0	0
$283$	2.03436	0.120930	0.0604650	−	0.998170i	$-0.480742\pi$
$283$	2.03436	0.120930	0.0604650	+	0.998170i	$0.480742\pi$
$284$	−14.6239	−0.867769
$285$	0	0
$286$	31.9592	1.88979
$287$	−42.1149	−2.48596
$288$	0	0
$289$	16.1782	0.951657
$290$	5.31932	0.312361
$291$	0	0
$292$	−7.45902	−0.436506
$293$	12.3149	0.719443	0.359722	−	0.933060i	$-0.382872\pi$
$293$	12.3149	0.719443	0.359722	+	0.933060i	$0.382872\pi$
$294$	0	0
$295$	15.6064	0.908638
$296$	17.0900	0.993338
$297$	0	0
$298$	−9.11105	−0.527789
$299$	−6.45799	−0.373475
$300$	0	0
$301$	2.83914	0.163646
$302$	42.8561	2.46609
$303$	0	0
$304$	26.3469	1.51110
$305$	0.839976	0.0480969
$306$	0	0
$307$	28.3795	1.61970	0.809851	−	0.586635i	$-0.199548\pi$
$307$	28.3795	1.61970	0.809851	+	0.586635i	$0.199548\pi$
$308$	98.9898	5.64047
$309$	0	0
$310$	−26.1970	−1.48789
$311$	−2.32773	−0.131994	−0.0659968	−	0.997820i	$-0.521023\pi$
$311$	−2.32773	−0.131994	−0.0659968	+	0.997820i	$0.521023\pi$
$312$	0	0
$313$	−21.3332	−1.20583	−0.602913	−	0.797807i	$-0.705993\pi$
$313$	−21.3332	−1.20583	−0.602913	+	0.797807i	$0.705993\pi$
$314$	−4.70346	−0.265432
$315$	0	0
$316$	−38.4379	−2.16230
$317$	−14.4947	−0.814106	−0.407053	−	0.913405i	$-0.633444\pi$
$317$	−14.4947	−0.814106	−0.407053	+	0.913405i	$0.633444\pi$
$318$	0	0
$319$	7.06791	0.395727
$320$	11.7025	0.654188
$321$	0	0
$322$	−29.9792	−1.67067
$323$	−37.3762	−2.07967
$324$	0	0
$325$	7.52774	0.417564
$326$	−46.0865	−2.55249
$327$	0	0
$328$	40.6751	2.24591
$329$	41.4413	2.28473
$330$	0	0
$331$	4.02128	0.221029	0.110515	−	0.993874i	$-0.464750\pi$
$331$	4.02128	0.221029	0.110515	+	0.993874i	$0.464750\pi$
$332$	−17.3368	−0.951483
$333$	0	0
$334$	2.45154	0.134142
$335$	−12.8724	−0.703297
$336$	0	0
$337$	−27.9488	−1.52247	−0.761234	−	0.648477i	$-0.775406\pi$
$337$	−27.9488	−1.52247	−0.761234	+	0.648477i	$0.775406\pi$
$338$	−14.0721	−0.765423
$339$	0	0
$340$	34.3079	1.86061
$341$	−34.8086	−1.88499
$342$	0	0
$343$	−61.3855	−3.31451
$344$	−2.74208	−0.147843
$345$	0	0
$346$	56.4424	3.03436
$347$	29.1522	1.56497	0.782487	−	0.622667i	$-0.213951\pi$
$347$	29.1522	1.56497	0.782487	+	0.622667i	$0.213951\pi$
$348$	0	0
$349$	19.3594	1.03628	0.518142	−	0.855295i	$-0.326624\pi$
$349$	19.3594	1.03628	0.518142	+	0.855295i	$0.326624\pi$
$350$	34.9452	1.86790
$351$	0	0
$352$	−0.477349	−0.0254428
$353$	20.5367	1.09306	0.546530	−	0.837439i	$-0.315948\pi$
$353$	20.5367	1.09306	0.546530	+	0.837439i	$0.315948\pi$
$354$	0	0
$355$	5.41670	0.287488
$356$	11.7967	0.625225
$357$	0	0
$358$	34.6702	1.83238
$359$	−1.03832	−0.0548006	−0.0274003	−	0.999625i	$-0.508723\pi$
$359$	−1.03832	−0.0548006	−0.0274003	+	0.999625i	$0.508723\pi$
$360$	0	0
$361$	23.1054	1.21607
$362$	−14.0594	−0.738946
$363$	0	0
$364$	55.1267	2.88942
$365$	2.76282	0.144613
$366$	0	0
$367$	17.5630	0.916781	0.458390	−	0.888751i	$-0.348426\pi$
$367$	17.5630	0.916781	0.458390	+	0.888751i	$0.348426\pi$
$368$	9.73178	0.507304
$369$	0	0
$370$	−12.6287	−0.656534
$371$	7.95993	0.413259
$372$	0	0
$373$	10.9248	0.565665	0.282832	−	0.959169i	$-0.408726\pi$
$373$	10.9248	0.565665	0.282832	+	0.959169i	$0.408726\pi$
$374$	68.3216	3.53283
$375$	0	0
$376$	−40.0246	−2.06411
$377$	3.93607	0.202718
$378$	0	0
$379$	3.31793	0.170431	0.0852154	−	0.996363i	$-0.472842\pi$
$379$	3.31793	0.170431	0.0852154	+	0.996363i	$0.472842\pi$
$380$	−38.6489	−1.98264
$381$	0	0
$382$	46.8321	2.39614
$383$	20.5296	1.04902	0.524508	−	0.851406i	$-0.324249\pi$
$383$	20.5296	1.04902	0.524508	+	0.851406i	$0.324249\pi$
$384$	0	0
$385$	−36.6658	−1.86866
$386$	−6.65456	−0.338708
$387$	0	0
$388$	−74.1799	−3.76592
$389$	−16.9080	−0.857272	−0.428636	−	0.903477i	$-0.641006\pi$
$389$	−16.9080	−0.857272	−0.428636	+	0.903477i	$0.641006\pi$
$390$	0	0
$391$	−13.8057	−0.698185
$392$	93.7807	4.73664
$393$	0	0
$394$	−33.3477	−1.68003
$395$	14.2374	0.716362
$396$	0	0
$397$	−2.48004	−0.124470	−0.0622348	−	0.998062i	$-0.519823\pi$
$397$	−2.48004	−0.124470	−0.0622348	+	0.998062i	$0.519823\pi$
$398$	−31.7309	−1.59053
$399$	0	0
$400$	−11.3438	−0.567192
$401$	−29.7283	−1.48456	−0.742280	−	0.670089i	$-0.766256\pi$
$401$	−29.7283	−1.48456	−0.742280	+	0.670089i	$0.766256\pi$
$402$	0	0
$403$	−19.3847	−0.965619
$404$	34.3806	1.71050
$405$	0	0
$406$	18.2720	0.906823
$407$	−16.7801	−0.831757
$408$	0	0
$409$	18.2001	0.899936	0.449968	−	0.893045i	$-0.351435\pi$
$409$	18.2001	0.899936	0.449968	+	0.893045i	$0.351435\pi$
$410$	−30.0569	−1.48440
$411$	0	0
$412$	29.6671	1.46159
$413$	53.6082	2.63789
$414$	0	0
$415$	6.42157	0.315222
$416$	−0.265832	−0.0130335
$417$	0	0
$418$	−76.9663	−3.76455
$419$	13.0669	0.638361	0.319181	−	0.947694i	$-0.396592\pi$
$419$	13.0669	0.638361	0.319181	+	0.947694i	$0.396592\pi$
$420$	0	0
$421$	−8.35108	−0.407007	−0.203503	−	0.979074i	$-0.565233\pi$
$421$	−8.35108	−0.407007	−0.203503	+	0.979074i	$0.565233\pi$
$422$	−19.6823	−0.958117
$423$	0	0
$424$	−7.68781	−0.373353
$425$	16.0926	0.780606
$426$	0	0
$427$	2.88534	0.139631
$428$	−54.3597	−2.62757
$429$	0	0
$430$	2.02627	0.0977152
$431$	−1.44427	−0.0695682	−0.0347841	−	0.999395i	$-0.511074\pi$
$431$	−1.44427	−0.0695682	−0.0347841	+	0.999395i	$0.511074\pi$
$432$	0	0
$433$	4.21901	0.202753	0.101376	−	0.994848i	$-0.467675\pi$
$433$	4.21901	0.202753	0.101376	+	0.994848i	$0.467675\pi$
$434$	−89.9873	−4.31953
$435$	0	0
$436$	−48.6870	−2.33169
$437$	15.5525	0.743979
$438$	0	0
$439$	31.9361	1.52422	0.762112	−	0.647445i	$-0.224162\pi$
$439$	31.9361	1.52422	0.762112	+	0.647445i	$0.224162\pi$
$440$	35.4124	1.68822
$441$	0	0
$442$	38.0478	1.80975
$443$	9.09117	0.431934	0.215967	−	0.976401i	$-0.430710\pi$
$443$	9.09117	0.431934	0.215967	+	0.976401i	$0.430710\pi$
$444$	0	0
$445$	−4.36950	−0.207134
$446$	26.6610	1.26244
$447$	0	0
$448$	40.1982	1.89919
$449$	20.5322	0.968974	0.484487	−	0.874799i	$-0.339006\pi$
$449$	20.5322	0.968974	0.484487	+	0.874799i	$0.339006\pi$
$450$	0	0
$451$	−39.9374	−1.88058
$452$	−69.3860	−3.26364
$453$	0	0
$454$	56.2247	2.63876
$455$	−20.4189	−0.957254
$456$	0	0
$457$	−20.1726	−0.943634	−0.471817	−	0.881696i	$-0.656402\pi$
$457$	−20.1726	−0.943634	−0.471817	+	0.881696i	$0.656402\pi$
$458$	−37.6589	−1.75968
$459$	0	0
$460$	−14.2758	−0.665612
$461$	−37.2942	−1.73696	−0.868482	−	0.495720i	$-0.834904\pi$
$461$	−37.2942	−1.73696	−0.868482	+	0.495720i	$0.834904\pi$
$462$	0	0
$463$	21.8842	1.01705	0.508523	−	0.861048i	$-0.330192\pi$
$463$	21.8842	1.01705	0.508523	+	0.861048i	$0.330192\pi$
$464$	−5.93141	−0.275359
$465$	0	0
$466$	−17.4615	−0.808887
$467$	−1.01174	−0.0468178	−0.0234089	−	0.999726i	$-0.507452\pi$
$467$	−1.01174	−0.0468178	−0.0234089	+	0.999726i	$0.507452\pi$
$468$	0	0
$469$	−44.2171	−2.04176
$470$	29.5762	1.36425
$471$	0	0
$472$	−51.7756	−2.38316
$473$	2.69235	0.123794
$474$	0	0
$475$	−18.1288	−0.831806
$476$	117.848	5.40157
$477$	0	0
$478$	5.56045	0.254329
$479$	−6.93401	−0.316823	−0.158411	−	0.987373i	$-0.550637\pi$
$479$	−6.93401	−0.316823	−0.158411	+	0.987373i	$0.550637\pi$
$480$	0	0
$481$	−9.34469	−0.426081
$482$	41.5447	1.89231
$483$	0	0
$484$	49.7613	2.26188
$485$	27.4763	1.24763
$486$	0	0
$487$	28.3450	1.28444	0.642218	−	0.766522i	$-0.278014\pi$
$487$	28.3450	1.28444	0.642218	+	0.766522i	$0.278014\pi$
$488$	−2.78670	−0.126148
$489$	0	0
$490$	−69.2994	−3.13063
$491$	7.56911	0.341589	0.170795	−	0.985307i	$-0.445367\pi$
$491$	7.56911	0.341589	0.170795	+	0.985307i	$0.445367\pi$
$492$	0	0
$493$	8.41442	0.378967
$494$	−42.8620	−1.92845
$495$	0	0
$496$	29.2115	1.31163
$497$	18.6065	0.834614
$498$	0	0
$499$	−34.1096	−1.52695	−0.763477	−	0.645835i	$-0.776510\pi$
$499$	−34.1096	−1.52695	−0.763477	+	0.645835i	$0.776510\pi$
$500$	46.4214	2.07603
$501$	0	0
$502$	−16.7126	−0.745918
$503$	−26.3017	−1.17273	−0.586367	−	0.810045i	$-0.699443\pi$
$503$	−26.3017	−1.17273	−0.586367	+	0.810045i	$0.699443\pi$
$504$	0	0
$505$	−12.7346	−0.566682
$506$	−28.4292	−1.26383
$507$	0	0
$508$	10.8207	0.480090
$509$	−6.05796	−0.268514	−0.134257	−	0.990947i	$-0.542865\pi$
$509$	−6.05796	−0.268514	−0.134257	+	0.990947i	$0.542865\pi$
$510$	0	0
$511$	9.49036	0.419829
$512$	−39.6153	−1.75076
$513$	0	0
$514$	−36.2213	−1.59765
$515$	−10.9887	−0.484220
$516$	0	0
$517$	39.2986	1.72835
$518$	−43.3798	−1.90600
$519$	0	0
$520$	19.7209	0.864818
$521$	2.11603	0.0927049	0.0463525	−	0.998925i	$-0.485240\pi$
$521$	2.11603	0.0927049	0.0463525	+	0.998925i	$0.485240\pi$
$522$	0	0
$523$	−10.1691	−0.444664	−0.222332	−	0.974971i	$-0.571367\pi$
$523$	−10.1691	−0.444664	−0.222332	+	0.974971i	$0.571367\pi$
$524$	15.9892	0.698491
$525$	0	0
$526$	33.8798	1.47723
$527$	−41.4400	−1.80515
$528$	0	0
$529$	−17.2553	−0.750232
$530$	5.68092	0.246763
$531$	0	0
$532$	−132.760	−5.75586
$533$	−22.2408	−0.963357
$534$	0	0
$535$	20.1348	0.870505
$536$	42.7055	1.84460
$537$	0	0
$538$	44.8796	1.93490
$539$	−92.0798	−3.96616
$540$	0	0
$541$	−6.25241	−0.268812	−0.134406	−	0.990926i	$-0.542913\pi$
$541$	−6.25241	−0.268812	−0.134406	+	0.990926i	$0.542913\pi$
$542$	41.7575	1.79364
$543$	0	0
$544$	−0.568289	−0.0243652
$545$	18.0337	0.772478
$546$	0	0
$547$	7.31119	0.312604	0.156302	−	0.987709i	$-0.450043\pi$
$547$	7.31119	0.312604	0.156302	+	0.987709i	$0.450043\pi$
$548$	−39.4446	−1.68499
$549$	0	0
$550$	33.1384	1.41303
$551$	−9.47910	−0.403823
$552$	0	0
$553$	48.9058	2.07969
$554$	−45.9899	−1.95392
$555$	0	0
$556$	−74.3919	−3.15492
$557$	−29.8225	−1.26362	−0.631811	−	0.775123i	$-0.717688\pi$
$557$	−29.8225	−1.26362	−0.631811	+	0.775123i	$0.717688\pi$
$558$	0	0
$559$	1.49935	0.0634158
$560$	30.7701	1.30027
$561$	0	0
$562$	−9.65371	−0.407217
$563$	−4.98167	−0.209953	−0.104976	−	0.994475i	$-0.533477\pi$
$563$	−4.98167	−0.209953	−0.104976	+	0.994475i	$0.533477\pi$
$564$	0	0
$565$	25.7006	1.08123
$566$	4.98730	0.209632
$567$	0	0
$568$	−17.9704	−0.754021
$569$	−4.90783	−0.205747	−0.102874	−	0.994694i	$-0.532804\pi$
$569$	−4.90783	−0.205747	−0.102874	+	0.994694i	$0.532804\pi$
$570$	0	0
$571$	37.7052	1.57792	0.788958	−	0.614448i	$-0.210621\pi$
$571$	37.7052	1.57792	0.788958	+	0.614448i	$0.210621\pi$
$572$	52.2765	2.18579
$573$	0	0
$574$	−103.246	−4.30941
$575$	−6.69626	−0.279253
$576$	0	0
$577$	−47.5025	−1.97756	−0.988778	−	0.149394i	$-0.952268\pi$
$577$	−47.5025	−1.97756	−0.988778	+	0.149394i	$0.952268\pi$
$578$	39.6614	1.64970
$579$	0	0
$580$	8.70093	0.361287
$581$	22.0582	0.915129
$582$	0	0
$583$	7.54838	0.312622
$584$	−9.16592	−0.379288
$585$	0	0
$586$	30.1904	1.24715
$587$	−24.5930	−1.01506	−0.507532	−	0.861633i	$-0.669442\pi$
$587$	−24.5930	−1.01506	−0.507532	+	0.861633i	$0.669442\pi$
$588$	0	0
$589$	46.6834	1.92356
$590$	38.2596	1.57512
$591$	0	0
$592$	14.0819	0.578761
$593$	−4.05926	−0.166694	−0.0833469	−	0.996521i	$-0.526561\pi$
$593$	−4.05926	−0.166694	−0.0833469	+	0.996521i	$0.526561\pi$
$594$	0	0
$595$	−43.6510	−1.78952
$596$	−14.9032	−0.610457
$597$	0	0
$598$	−15.8320	−0.647418
$599$	−21.1425	−0.863859	−0.431929	−	0.901907i	$-0.642167\pi$
$599$	−21.1425	−0.863859	−0.431929	+	0.901907i	$0.642167\pi$
$600$	0	0
$601$	33.5075	1.36680	0.683400	−	0.730044i	$-0.260500\pi$
$601$	33.5075	1.36680	0.683400	+	0.730044i	$0.260500\pi$
$602$	6.96027	0.283679
$603$	0	0
$604$	70.1007	2.85236
$605$	−18.4316	−0.749351
$606$	0	0
$607$	−3.59160	−0.145779	−0.0728894	−	0.997340i	$-0.523222\pi$
$607$	−3.59160	−0.145779	−0.0728894	+	0.997340i	$0.523222\pi$
$608$	0.640194	0.0259633
$609$	0	0
$610$	2.05923	0.0833759
$611$	21.8851	0.885377
$612$	0	0
$613$	11.2200	0.453171	0.226585	−	0.973991i	$-0.427244\pi$
$613$	11.2200	0.453171	0.226585	+	0.973991i	$0.427244\pi$
$614$	69.5733	2.80775
$615$	0	0
$616$	121.642	4.90111
$617$	−13.0328	−0.524682	−0.262341	−	0.964975i	$-0.584495\pi$
$617$	−13.0328	−0.524682	−0.262341	+	0.964975i	$0.584495\pi$
$618$	0	0
$619$	36.1895	1.45458	0.727290	−	0.686330i	$-0.240779\pi$
$619$	36.1895	1.45458	0.727290	+	0.686330i	$0.240779\pi$
$620$	−42.8510	−1.72094
$621$	0	0
$622$	−5.70653	−0.228811
$623$	−15.0093	−0.601337
$624$	0	0
$625$	−3.22536	−0.129014
$626$	−52.2992	−2.09030
$627$	0	0
$628$	−7.69355	−0.307006
$629$	−19.9768	−0.796528
$630$	0	0
$631$	−43.4131	−1.72825	−0.864124	−	0.503279i	$-0.832127\pi$
$631$	−43.4131	−1.72825	−0.864124	+	0.503279i	$0.832127\pi$
$632$	−47.2339	−1.87886
$633$	0	0
$634$	−35.5344	−1.41125
$635$	−4.00798	−0.159052
$636$	0	0
$637$	−51.2786	−2.03173
$638$	17.3273	0.685993
$639$	0	0
$640$	28.9821	1.14562
$641$	25.6251	1.01213	0.506066	−	0.862495i	$-0.331099\pi$
$641$	25.6251	1.01213	0.506066	+	0.862495i	$0.331099\pi$
$642$	0	0
$643$	8.66473	0.341704	0.170852	−	0.985297i	$-0.445348\pi$
$643$	8.66473	0.341704	0.170852	+	0.985297i	$0.445348\pi$
$644$	−49.0376	−1.93235
$645$	0	0
$646$	−91.6292	−3.60510
$647$	−7.90300	−0.310699	−0.155349	−	0.987860i	$-0.549650\pi$
$647$	−7.90300	−0.310699	−0.155349	+	0.987860i	$0.549650\pi$
$648$	0	0
$649$	50.8365	1.99551
$650$	18.4545	0.723847
$651$	0	0
$652$	−75.3847	−2.95229
$653$	9.39559	0.367678	0.183839	−	0.982956i	$-0.441148\pi$
$653$	9.39559	0.367678	0.183839	+	0.982956i	$0.441148\pi$
$654$	0	0
$655$	−5.92240	−0.231407
$656$	33.5155	1.30856
$657$	0	0
$658$	101.595	3.96058
$659$	−13.2222	−0.515063	−0.257531	−	0.966270i	$-0.582909\pi$
$659$	−13.2222	−0.515063	−0.257531	+	0.966270i	$0.582909\pi$
$660$	0	0
$661$	−48.1553	−1.87302	−0.936512	−	0.350635i	$-0.885966\pi$
$661$	−48.1553	−1.87302	−0.936512	+	0.350635i	$0.885966\pi$
$662$	9.85831	0.383154
$663$	0	0
$664$	−21.3041	−0.826761
$665$	49.1742	1.90689
$666$	0	0
$667$	−3.50131	−0.135571
$668$	4.01004	0.155153
$669$	0	0
$670$	−31.5573	−1.21916
$671$	2.73615	0.105628
$672$	0	0
$673$	−26.6592	−1.02764	−0.513818	−	0.857899i	$-0.671770\pi$
$673$	−26.6592	−1.02764	−0.513818	+	0.857899i	$0.671770\pi$
$674$	−68.5175	−2.63920
$675$	0	0
$676$	−23.0181	−0.885311
$677$	42.9295	1.64991	0.824957	−	0.565195i	$-0.191199\pi$
$677$	42.9295	1.64991	0.824957	+	0.565195i	$0.191199\pi$
$678$	0	0
$679$	94.3815	3.62203
$680$	42.1588	1.61672
$681$	0	0
$682$	−85.3346	−3.26763
$683$	−45.0876	−1.72523	−0.862614	−	0.505863i	$-0.831174\pi$
$683$	−45.0876	−1.72523	−0.862614	+	0.505863i	$0.831174\pi$
$684$	0	0
$685$	14.6103	0.558231
$686$	−150.489	−5.74569
$687$	0	0
$688$	−2.25943	−0.0861399
$689$	4.20364	0.160146
$690$	0	0
$691$	−30.3920	−1.15617	−0.578083	−	0.815978i	$-0.696199\pi$
$691$	−30.3920	−1.15617	−0.578083	+	0.815978i	$0.696199\pi$
$692$	92.3242	3.50964
$693$	0	0
$694$	71.4678	2.71288
$695$	27.5548	1.04521
$696$	0	0
$697$	−47.5458	−1.80093
$698$	47.4602	1.79640
$699$	0	0
$700$	57.1607	2.16047
$701$	−6.19544	−0.233998	−0.116999	−	0.993132i	$-0.537328\pi$
$701$	−6.19544	−0.233998	−0.116999	+	0.993132i	$0.537328\pi$
$702$	0	0
$703$	22.5045	0.848773
$704$	38.1199	1.43670
$705$	0	0
$706$	50.3466	1.89482
$707$	−43.7436	−1.64515
$708$	0	0
$709$	−41.4782	−1.55775	−0.778873	−	0.627182i	$-0.784208\pi$
$709$	−41.4782	−1.55775	−0.778873	+	0.627182i	$0.784208\pi$
$710$	13.2792	0.498361
$711$	0	0
$712$	14.4962	0.543269
$713$	17.2435	0.645775
$714$	0	0
$715$	−19.3632	−0.724143
$716$	56.7109	2.11939
$717$	0	0
$718$	−2.54549	−0.0949968
$719$	15.9450	0.594647	0.297323	−	0.954777i	$-0.403906\pi$
$719$	15.9450	0.594647	0.297323	+	0.954777i	$0.403906\pi$
$720$	0	0
$721$	−37.7464	−1.40575
$722$	56.6438	2.10806
$723$	0	0
$724$	−22.9973	−0.854687
$725$	4.08130	0.151576
$726$	0	0
$727$	−1.09256	−0.0405206	−0.0202603	−	0.999795i	$-0.506449\pi$
$727$	−1.09256	−0.0405206	−0.0202603	+	0.999795i	$0.506449\pi$
$728$	67.7417	2.51067
$729$	0	0
$730$	6.77316	0.250686
$731$	3.20527	0.118551
$732$	0	0
$733$	−38.0026	−1.40366	−0.701828	−	0.712346i	$-0.747633\pi$
$733$	−38.0026	−1.40366	−0.701828	+	0.712346i	$0.747633\pi$
$734$	43.0563	1.58924
$735$	0	0
$736$	0.236469	0.00871638
$737$	−41.9310	−1.54455
$738$	0	0
$739$	26.2659	0.966205	0.483103	−	0.875564i	$-0.339510\pi$
$739$	26.2659	0.966205	0.483103	+	0.875564i	$0.339510\pi$
$740$	−20.6570	−0.759368
$741$	0	0
$742$	19.5141	0.716384
$743$	43.0889	1.58078	0.790389	−	0.612605i	$-0.209879\pi$
$743$	43.0889	1.58078	0.790389	+	0.612605i	$0.209879\pi$
$744$	0	0
$745$	5.52013	0.202242
$746$	26.7826	0.980579
$747$	0	0
$748$	111.755	4.08617
$749$	69.1636	2.52718
$750$	0	0
$751$	−22.7833	−0.831375	−0.415688	−	0.909507i	$-0.636459\pi$
$751$	−22.7833	−0.831375	−0.415688	+	0.909507i	$0.636459\pi$
$752$	−32.9795	−1.20264
$753$	0	0
$754$	9.64942	0.351411
$755$	−25.9653	−0.944974
$756$	0	0
$757$	41.1190	1.49450	0.747248	−	0.664545i	$-0.231375\pi$
$757$	41.1190	1.49450	0.747248	+	0.664545i	$0.231375\pi$
$758$	8.13404	0.295442
$759$	0	0
$760$	−47.4931	−1.72276
$761$	15.9685	0.578856	0.289428	−	0.957200i	$-0.406535\pi$
$761$	15.9685	0.578856	0.289428	+	0.957200i	$0.406535\pi$
$762$	0	0
$763$	61.9461	2.24260
$764$	76.6044	2.77145
$765$	0	0
$766$	50.3292	1.81847
$767$	28.3105	1.02223
$768$	0	0
$769$	−1.27225	−0.0458787	−0.0229393	−	0.999737i	$-0.507302\pi$
$769$	−1.27225	−0.0458787	−0.0229393	+	0.999737i	$0.507302\pi$
$770$	−89.8877	−3.23933
$771$	0	0
$772$	−10.8850	−0.391760
$773$	23.5735	0.847879	0.423940	−	0.905690i	$-0.360647\pi$
$773$	23.5735	0.847879	0.423940	+	0.905690i	$0.360647\pi$
$774$	0	0
$775$	−20.0999	−0.722009
$776$	−91.1550	−3.27227
$777$	0	0
$778$	−41.4507	−1.48608
$779$	53.5618	1.91905
$780$	0	0
$781$	17.6445	0.631368
$782$	−33.8452	−1.21030
$783$	0	0
$784$	77.2736	2.75977
$785$	2.84969	0.101710
$786$	0	0
$787$	−2.29170	−0.0816903	−0.0408451	−	0.999165i	$-0.513005\pi$
$787$	−2.29170	−0.0816903	−0.0408451	+	0.999165i	$0.513005\pi$
$788$	−54.5476	−1.94318
$789$	0	0
$790$	34.9035	1.24181
$791$	88.2821	3.13895
$792$	0	0
$793$	1.52374	0.0541097
$794$	−6.07991	−0.215768
$795$	0	0
$796$	−51.9030	−1.83965
$797$	−12.1742	−0.431232	−0.215616	−	0.976478i	$-0.569176\pi$
$797$	−12.1742	−0.431232	−0.215616	+	0.976478i	$0.569176\pi$
$798$	0	0
$799$	46.7854	1.65515
$800$	−0.275640	−0.00974536
$801$	0	0
$802$	−72.8801	−2.57348
$803$	8.99967	0.317592
$804$	0	0
$805$	18.1636	0.640181
$806$	−47.5222	−1.67390
$807$	0	0
$808$	42.2481	1.48628
$809$	18.9132	0.664953	0.332477	−	0.943111i	$-0.392116\pi$
$809$	18.9132	0.664953	0.332477	+	0.943111i	$0.392116\pi$
$810$	0	0
$811$	−19.3598	−0.679814	−0.339907	−	0.940459i	$-0.610396\pi$
$811$	−19.3598	−0.679814	−0.339907	+	0.940459i	$0.610396\pi$
$812$	29.8879	1.04886
$813$	0	0
$814$	−41.1370	−1.44185
$815$	27.9225	0.978083
$816$	0	0
$817$	−3.61083	−0.126327
$818$	44.6182	1.56004
$819$	0	0
$820$	−49.1648	−1.71691
$821$	−9.85285	−0.343867	−0.171933	−	0.985109i	$-0.555001\pi$
$821$	−9.85285	−0.343867	−0.171933	+	0.985109i	$0.555001\pi$
$822$	0	0
$823$	−33.3072	−1.16102	−0.580508	−	0.814255i	$-0.697146\pi$
$823$	−33.3072	−1.16102	−0.580508	+	0.814255i	$0.697146\pi$
$824$	36.4560	1.27000
$825$	0	0
$826$	131.423	4.57278
$827$	−14.9827	−0.520998	−0.260499	−	0.965474i	$-0.583887\pi$
$827$	−14.9827	−0.520998	−0.260499	+	0.965474i	$0.583887\pi$
$828$	0	0
$829$	6.98763	0.242690	0.121345	−	0.992610i	$-0.461279\pi$
$829$	6.98763	0.242690	0.121345	+	0.992610i	$0.461279\pi$
$830$	15.7427	0.546438
$831$	0	0
$832$	21.2287	0.735972
$833$	−109.622	−3.79818
$834$	0	0
$835$	−1.48532	−0.0514015
$836$	−125.896	−4.35419
$837$	0	0
$838$	32.0341	1.10660
$839$	1.67464	0.0578150	0.0289075	−	0.999582i	$-0.490797\pi$
$839$	1.67464	0.0578150	0.0289075	+	0.999582i	$0.490797\pi$
$840$	0	0
$841$	−26.8660	−0.926414
$842$	−20.4730	−0.705546
$843$	0	0
$844$	−32.1947	−1.10819
$845$	8.52591	0.293300
$846$	0	0
$847$	−63.3130	−2.17546
$848$	−6.33462	−0.217532
$849$	0	0
$850$	39.4516	1.35318
$851$	8.31252	0.284950
$852$	0	0
$853$	−37.6200	−1.28808	−0.644042	−	0.764990i	$-0.722744\pi$
$853$	−37.6200	−1.28808	−0.644042	+	0.764990i	$0.722744\pi$
$854$	7.07351	0.242050
$855$	0	0
$856$	−66.7992	−2.28315
$857$	−40.7489	−1.39196	−0.695978	−	0.718063i	$-0.745029\pi$
$857$	−40.7489	−1.39196	−0.695978	+	0.718063i	$0.745029\pi$
$858$	0	0
$859$	13.6973	0.467347	0.233674	−	0.972315i	$-0.424925\pi$
$859$	13.6973	0.467347	0.233674	+	0.972315i	$0.424925\pi$
$860$	3.31441	0.113020
$861$	0	0
$862$	−3.54069	−0.120596
$863$	28.4588	0.968750	0.484375	−	0.874861i	$-0.339047\pi$
$863$	28.4588	0.968750	0.484375	+	0.874861i	$0.339047\pi$
$864$	0	0
$865$	−34.1969	−1.16273
$866$	10.3431	0.351471
$867$	0	0
$868$	−147.194	−4.99610
$869$	46.3772	1.57324
$870$	0	0
$871$	−23.3510	−0.791220
$872$	−59.8284	−2.02604
$873$	0	0
$874$	38.1276	1.28969
$875$	−59.0635	−1.99671
$876$	0	0
$877$	9.98414	0.337140	0.168570	−	0.985690i	$-0.446085\pi$
$877$	9.98414	0.337140	0.168570	+	0.985690i	$0.446085\pi$
$878$	78.2924	2.64224
$879$	0	0
$880$	29.1791	0.983629
$881$	3.20716	0.108052	0.0540260	−	0.998540i	$-0.482795\pi$
$881$	3.20716	0.108052	0.0540260	+	0.998540i	$0.482795\pi$
$882$	0	0
$883$	−25.5472	−0.859731	−0.429865	−	0.902893i	$-0.641439\pi$
$883$	−25.5472	−0.859731	−0.429865	+	0.902893i	$0.641439\pi$
$884$	62.2356	2.09321
$885$	0	0
$886$	22.2873	0.748758
$887$	−34.6900	−1.16478	−0.582389	−	0.812911i	$-0.697882\pi$
$887$	−34.6900	−1.16478	−0.582389	+	0.812911i	$0.697882\pi$
$888$	0	0
$889$	−13.7675	−0.461747
$890$	−10.7120	−0.359067
$891$	0	0
$892$	43.6101	1.46017
$893$	−52.7051	−1.76371
$894$	0	0
$895$	−21.0057	−0.702144
$896$	99.5542	3.32587
$897$	0	0
$898$	50.3354	1.67971
$899$	−10.5097	−0.350519
$900$	0	0
$901$	8.98642	0.299381
$902$	−97.9080	−3.25998
$903$	0	0
$904$	−85.2641	−2.83584
$905$	8.51820	0.283154
$906$	0	0
$907$	−12.3199	−0.409076	−0.204538	−	0.978859i	$-0.565569\pi$
$907$	−12.3199	−0.409076	−0.204538	+	0.978859i	$0.565569\pi$
$908$	91.9681	3.05207
$909$	0	0
$910$	−50.0578	−1.65940
$911$	−36.0044	−1.19288	−0.596440	−	0.802658i	$-0.703418\pi$
$911$	−36.0044	−1.19288	−0.596440	+	0.802658i	$0.703418\pi$
$912$	0	0
$913$	20.9177	0.692276
$914$	−49.4539	−1.63579
$915$	0	0
$916$	−61.5995	−2.03530
$917$	−20.3436	−0.671804
$918$	0	0
$919$	11.1744	0.368608	0.184304	−	0.982869i	$-0.440997\pi$
$919$	11.1744	0.368608	0.184304	+	0.982869i	$0.440997\pi$
$920$	−17.5426	−0.578363
$921$	0	0
$922$	−91.4282	−3.01103
$923$	9.82607	0.323429
$924$	0	0
$925$	−9.68948	−0.318588
$926$	53.6500	1.76305
$927$	0	0
$928$	−0.144125	−0.00473115
$929$	−53.0839	−1.74163	−0.870813	−	0.491615i	$-0.836407\pi$
$929$	−53.0839	−1.74163	−0.870813	+	0.491615i	$0.836407\pi$
$930$	0	0
$931$	123.492	4.04730
$932$	−28.5621	−0.935583
$933$	0	0
$934$	−2.48032	−0.0811587
$935$	−41.3941	−1.35373
$936$	0	0
$937$	−36.0060	−1.17626	−0.588132	−	0.808765i	$-0.700136\pi$
$937$	−36.0060	−1.17626	−0.588132	+	0.808765i	$0.700136\pi$
$938$	−108.400	−3.53938
$939$	0	0
$940$	48.3784	1.57793
$941$	−36.4244	−1.18740	−0.593700	−	0.804686i	$-0.702333\pi$
$941$	−36.4244	−1.18740	−0.593700	+	0.804686i	$0.702333\pi$
$942$	0	0
$943$	19.7842	0.644263
$944$	−42.6621	−1.38853
$945$	0	0
$946$	6.60040	0.214598
$947$	−16.0753	−0.522378	−0.261189	−	0.965288i	$-0.584115\pi$
$947$	−16.0753	−0.522378	−0.261189	+	0.965288i	$0.584115\pi$
$948$	0	0
$949$	5.01185	0.162692
$950$	−44.4434	−1.44193
$951$	0	0
$952$	144.816	4.69352
$953$	37.9202	1.22836	0.614178	−	0.789167i	$-0.289488\pi$
$953$	37.9202	1.22836	0.614178	+	0.789167i	$0.289488\pi$
$954$	0	0
$955$	−28.3743	−0.918170
$956$	9.09536	0.294165
$957$	0	0
$958$	−16.9990	−0.549212
$959$	50.1867	1.62061
$960$	0	0
$961$	20.7591	0.669650
$962$	−22.9089	−0.738611
$963$	0	0
$964$	67.9556	2.18870
$965$	4.03181	0.129789
$966$	0	0
$967$	−3.18241	−0.102340	−0.0511698	−	0.998690i	$-0.516295\pi$
$967$	−3.18241	−0.102340	−0.0511698	+	0.998690i	$0.516295\pi$
$968$	61.1485	1.96539
$969$	0	0
$970$	67.3591	2.16277
$971$	39.6821	1.27346	0.636729	−	0.771088i	$-0.280287\pi$
$971$	39.6821	1.27346	0.636729	+	0.771088i	$0.280287\pi$
$972$	0	0
$973$	94.6513	3.03438
$974$	69.4889	2.22657
$975$	0	0
$976$	−2.29619	−0.0734991
$977$	−37.8889	−1.21217	−0.606087	−	0.795399i	$-0.707262\pi$
$977$	−37.8889	−1.21217	−0.606087	+	0.795399i	$0.707262\pi$
$978$	0	0
$979$	−14.2333	−0.454899
$980$	−113.355	−3.62098
$981$	0	0
$982$	18.5560	0.592144
$983$	27.1622	0.866341	0.433170	−	0.901312i	$-0.357395\pi$
$983$	27.1622	0.866341	0.433170	+	0.901312i	$0.357395\pi$
$984$	0	0
$985$	20.2044	0.643767
$986$	20.6283	0.656938
$987$	0	0
$988$	−70.1103	−2.23051
$989$	−1.33374	−0.0424104
$990$	0	0
$991$	−16.1557	−0.513204	−0.256602	−	0.966517i	$-0.582603\pi$
$991$	−16.1557	−0.513204	−0.256602	+	0.966517i	$0.582603\pi$
$992$	0.709800	0.0225362
$993$	0	0
$994$	45.6145	1.44680
$995$	19.2249	0.609469
$996$	0	0
$997$	16.9034	0.535335	0.267667	−	0.963511i	$-0.413747\pi$
$997$	16.9034	0.535335	0.267667	+	0.963511i	$0.413747\pi$
$998$	−83.6209	−2.64697
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1503.2.a.e.1.8		8
3.2	odd	2		501.2.a.e.1.1	✓	8
12.11	even	2		8016.2.a.x.1.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
501.2.a.e.1.1	✓	8	3.2	odd	2
1503.2.a.e.1.8		8	1.1	even	1	trivial
8016.2.a.x.1.4		8	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1503 = 3^{2} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1503.a (trivial)

\(f(q)\)	\(=\)	\(q+2.45154 q^{2} +4.01004 q^{4} -1.48532 q^{5} -5.10210 q^{7} +4.92768 q^{8} +O(q^{10})\)	\(q+2.45154 q^{2} +4.01004 q^{4} -1.48532 q^{5} -5.10210 q^{7} +4.92768 q^{8} -3.64131 q^{10} -4.83831 q^{11} -2.69442 q^{13} -12.5080 q^{14} +4.06032 q^{16} -5.76005 q^{17} +6.48887 q^{19} -5.95618 q^{20} -11.8613 q^{22} +2.39680 q^{23} -2.79383 q^{25} -6.60546 q^{26} -20.4596 q^{28} -1.46082 q^{29} +7.19438 q^{31} +0.0986604 q^{32} -14.1210 q^{34} +7.57824 q^{35} +3.46817 q^{37} +15.9077 q^{38} -7.31917 q^{40} +8.25442 q^{41} -0.556466 q^{43} -19.4018 q^{44} +5.87585 q^{46} -8.12240 q^{47} +19.0314 q^{49} -6.84918 q^{50} -10.8047 q^{52} -1.56013 q^{53} +7.18642 q^{55} -25.1415 q^{56} -3.58127 q^{58} -10.5071 q^{59} -0.565519 q^{61} +17.6373 q^{62} -7.87876 q^{64} +4.00206 q^{65} +8.66646 q^{67} -23.0980 q^{68} +18.5783 q^{70} -3.64683 q^{71} -1.86009 q^{73} +8.50235 q^{74} +26.0206 q^{76} +24.6855 q^{77} -9.58543 q^{79} -6.03086 q^{80} +20.2360 q^{82} -4.32336 q^{83} +8.55551 q^{85} -1.36420 q^{86} -23.8416 q^{88} +2.94180 q^{89} +13.7472 q^{91} +9.61127 q^{92} -19.9124 q^{94} -9.63803 q^{95} -18.4986 q^{97} +46.6562 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 3 q^{2} + 9 q^{4} - 7 q^{5} - 4 q^{7} - 3 q^{8}+O(q^{10}) \) 8 * q - 3 * q^2 + 9 * q^4 - 7 * q^5 - 4 * q^7 - 3 * q^8	\( 8 q - 3 q^{2} + 9 q^{4} - 7 q^{5} - 4 q^{7} - 3 q^{8} - q^{10} - 13 q^{11} - 5 q^{14} + 7 q^{16} - 11 q^{17} + 12 q^{19} - 9 q^{20} - 17 q^{22} - 7 q^{23} - 5 q^{25} - 3 q^{26} - 27 q^{28} - q^{29} - 2 q^{31} - 4 q^{32} - 14 q^{34} + 4 q^{35} - 9 q^{37} - 22 q^{40} - 4 q^{41} + 2 q^{43} - 3 q^{44} - 5 q^{46} - 17 q^{47} - 2 q^{49} + 4 q^{50} - 36 q^{52} - 9 q^{53} + 7 q^{55} - 9 q^{56} - 29 q^{58} - 29 q^{59} - 12 q^{61} + 34 q^{62} - 5 q^{64} - 8 q^{65} - 26 q^{68} + 5 q^{70} - 13 q^{71} - 20 q^{73} + 17 q^{74} + 30 q^{76} + 22 q^{77} + 8 q^{79} + 34 q^{80} + 15 q^{82} - 33 q^{83} - 31 q^{85} - 11 q^{86} - 44 q^{88} - 4 q^{89} + q^{91} + 33 q^{92} - 2 q^{94} - 3 q^{95} - 31 q^{97} + 57 q^{98}+O(q^{100}) \) 8 * q - 3 * q^2 + 9 * q^4 - 7 * q^5 - 4 * q^7 - 3 * q^8 - q^10 - 13 * q^11 - 5 * q^14 + 7 * q^16 - 11 * q^17 + 12 * q^19 - 9 * q^20 - 17 * q^22 - 7 * q^23 - 5 * q^25 - 3 * q^26 - 27 * q^28 - q^29 - 2 * q^31 - 4 * q^32 - 14 * q^34 + 4 * q^35 - 9 * q^37 - 22 * q^40 - 4 * q^41 + 2 * q^43 - 3 * q^44 - 5 * q^46 - 17 * q^47 - 2 * q^49 + 4 * q^50 - 36 * q^52 - 9 * q^53 + 7 * q^55 - 9 * q^56 - 29 * q^58 - 29 * q^59 - 12 * q^61 + 34 * q^62 - 5 * q^64 - 8 * q^65 - 26 * q^68 + 5 * q^70 - 13 * q^71 - 20 * q^73 + 17 * q^74 + 30 * q^76 + 22 * q^77 + 8 * q^79 + 34 * q^80 + 15 * q^82 - 33 * q^83 - 31 * q^85 - 11 * q^86 - 44 * q^88 - 4 * q^89 + q^91 + 33 * q^92 - 2 * q^94 - 3 * q^95 - 31 * q^97 + 57 * q^98

Introduction

L-functions

Modular forms

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Database

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