LMFDB - Embedded newform 4598.2.a.bt.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4598,2,Mod(1,4598)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4598, 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("4598.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4598 = 2 \cdot 11^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4598.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:	$36.7152148494$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.7488.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 4x^{2} + 2x + 1 $ x^4 - 2x^3 - 4x^2 + 2*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$3.05896$ of defining polynomial
Character	$\chi$	$=$	4598.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} +0.326909 q^{3} +1.00000 q^{4} +2.05896 q^{5} +0.326909 q^{6} -1.00000 q^{7} +1.00000 q^{8} -2.89313 q^{9} +2.05896 q^{10} +0.326909 q^{12} -6.11792 q^{13} -1.00000 q^{14} +0.673091 q^{15} +1.00000 q^{16} +0.239314 q^{17} -2.89313 q^{18} +1.00000 q^{19} +2.05896 q^{20} -0.326909 q^{21} -7.61066 q^{23} +0.326909 q^{24} -0.760686 q^{25} -6.11792 q^{26} -1.92651 q^{27} -1.00000 q^{28} +9.90893 q^{29} +0.673091 q^{30} -3.80037 q^{31} +1.00000 q^{32} +0.239314 q^{34} -2.05896 q^{35} -2.89313 q^{36} -11.4162 q^{37} +1.00000 q^{38} -2.00000 q^{39} +2.05896 q^{40} -6.77690 q^{41} -0.326909 q^{42} +6.97136 q^{43} -5.95684 q^{45} -7.61066 q^{46} +4.21068 q^{47} +0.326909 q^{48} -6.00000 q^{49} -0.760686 q^{50} +0.0782337 q^{51} -6.11792 q^{52} -8.68414 q^{53} -1.92651 q^{54} -1.00000 q^{56} +0.326909 q^{57} +9.90893 q^{58} +13.2551 q^{59} +0.673091 q^{60} +6.50854 q^{61} -3.80037 q^{62} +2.89313 q^{63} +1.00000 q^{64} -12.5965 q^{65} -9.28891 q^{67} +0.239314 q^{68} -2.48799 q^{69} -2.05896 q^{70} -8.28416 q^{71} -2.89313 q^{72} -5.37176 q^{73} -11.4162 q^{74} -0.248675 q^{75} +1.00000 q^{76} -2.00000 q^{78} +13.2311 q^{79} +2.05896 q^{80} +8.04960 q^{81} -6.77690 q^{82} -13.6983 q^{83} -0.326909 q^{84} +0.492737 q^{85} +6.97136 q^{86} +3.23931 q^{87} -13.5486 q^{89} -5.95684 q^{90} +6.11792 q^{91} -7.61066 q^{92} -1.24237 q^{93} +4.21068 q^{94} +2.05896 q^{95} +0.326909 q^{96} +17.9789 q^{97} -6.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} - 2 q^{3} + 4 q^{4} - 2 q^{5} - 2 q^{6} - 4 q^{7} + 4 q^{8} - 2 q^{10} - 2 q^{12} - 4 q^{13} - 4 q^{14} + 6 q^{15} + 4 q^{16} - 4 q^{17} + 4 q^{19} - 2 q^{20} + 2 q^{21} - 8 q^{23} - 2 q^{24}+ \cdots - 24 q^{98}+O(q^{100}) $ 4 * q + 4 * q^2 - 2 * q^3 + 4 * q^4 - 2 * q^5 - 2 * q^6 - 4 * q^7 + 4 * q^8 - 2 * q^10 - 2 * q^12 - 4 * q^13 - 4 * q^14 + 6 * q^15 + 4 * q^16 - 4 * q^17 + 4 * q^19 - 2 * q^20 + 2 * q^21 - 8 * q^23 - 2 * q^24 - 8 * q^25 - 4 * q^26 - 14 * q^27 - 4 * q^28 + 2 * q^29 + 6 * q^30 + 4 * q^32 - 4 * q^34 + 2 * q^35 - 10 * q^37 + 4 * q^38 - 8 * q^39 - 2 * q^40 - 2 * q^41 + 2 * q^42 + 16 * q^43 - 8 * q^45 - 8 * q^46 - 2 * q^48 - 24 * q^49 - 8 * q^50 - 4 * q^52 - 6 * q^53 - 14 * q^54 - 4 * q^56 - 2 * q^57 + 2 * q^58 + 22 * q^59 + 6 * q^60 - 2 * q^61 + 4 * q^64 - 20 * q^65 - 20 * q^67 - 4 * q^68 - 2 * q^69 + 2 * q^70 - 10 * q^71 - 10 * q^74 + 2 * q^75 + 4 * q^76 - 8 * q^78 + 6 * q^79 - 2 * q^80 + 20 * q^81 - 2 * q^82 - 34 * q^83 + 2 * q^84 + 16 * q^86 + 8 * q^87 - 2 * q^89 - 8 * q^90 + 4 * q^91 - 8 * q^92 - 40 * q^93 - 2 * q^95 - 2 * q^96 - 8 * q^97 - 24 * q^98

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0.326909	0.188741	0.0943704	−	0.995537i	$-0.469916\pi$
$3$	0.326909	0.188741	0.0943704	+	0.995537i	$0.469916\pi$
$4$	1.00000	0.500000
$5$	2.05896	0.920795	0.460397	−	0.887713i	$-0.347707\pi$
$5$	2.05896	0.920795	0.460397	+	0.887713i	$0.347707\pi$
$6$	0.326909	0.133460
$7$	−1.00000	−0.377964	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−1.00000	−0.377964	−0.188982	+	0.981981i	$0.560519\pi$
$8$	1.00000	0.353553
$9$	−2.89313	−0.964377
$10$	2.05896	0.651100
$11$	0	0
$11$	0	0
$12$	0.326909	0.0943704
$13$	−6.11792	−1.69681	−0.848403	−	0.529351i	$-0.822435\pi$
$13$	−6.11792	−1.69681	−0.848403	+	0.529351i	$0.822435\pi$
$14$	−1.00000	−0.267261
$15$	0.673091	0.173791
$16$	1.00000	0.250000
$17$	0.239314	0.0580421	0.0290210	−	0.999579i	$-0.490761\pi$
$17$	0.239314	0.0580421	0.0290210	+	0.999579i	$0.490761\pi$
$18$	−2.89313	−0.681917
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	2.05896	0.460397
$21$	−0.326909	−0.0713373
$22$	0	0
$23$	−7.61066	−1.58693	−0.793466	−	0.608615i	$-0.791725\pi$
$23$	−7.61066	−1.58693	−0.793466	+	0.608615i	$0.791725\pi$
$24$	0.326909	0.0667299
$25$	−0.760686	−0.152137
$26$	−6.11792	−1.19982
$27$	−1.92651	−0.370758
$28$	−1.00000	−0.188982
$29$	9.90893	1.84004	0.920021	−	0.391869i	$-0.128171\pi$
$29$	9.90893	1.84004	0.920021	+	0.391869i	$0.128171\pi$
$30$	0.673091	0.122889
$31$	−3.80037	−0.682567	−0.341283	−	0.939960i	$-0.610862\pi$
$31$	−3.80037	−0.682567	−0.341283	+	0.939960i	$0.610862\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	0.239314	0.0410420
$35$	−2.05896	−0.348028
$36$	−2.89313	−0.482188
$37$	−11.4162	−1.87681	−0.938405	−	0.345537i	$-0.887697\pi$
$37$	−11.4162	−1.87681	−0.938405	+	0.345537i	$0.887697\pi$
$38$	1.00000	0.162221
$39$	−2.00000	−0.320256
$40$	2.05896	0.325550
$41$	−6.77690	−1.05837	−0.529187	−	0.848505i	$-0.677503\pi$
$41$	−6.77690	−1.05837	−0.529187	+	0.848505i	$0.677503\pi$
$42$	−0.326909	−0.0504431
$43$	6.97136	1.06312	0.531562	−	0.847020i	$-0.321605\pi$
$43$	6.97136	1.06312	0.531562	+	0.847020i	$0.321605\pi$
$44$	0	0
$45$	−5.95684	−0.887993
$46$	−7.61066	−1.12213
$47$	4.21068	0.614191	0.307095	−	0.951679i	$-0.400643\pi$
$47$	4.21068	0.614191	0.307095	+	0.951679i	$0.400643\pi$
$48$	0.326909	0.0471852
$49$	−6.00000	−0.857143
$50$	−0.760686	−0.107577
$51$	0.0782337	0.0109549
$52$	−6.11792	−0.848403
$53$	−8.68414	−1.19286	−0.596429	−	0.802666i	$-0.703414\pi$
$53$	−8.68414	−1.19286	−0.596429	+	0.802666i	$0.703414\pi$
$54$	−1.92651	−0.262165
$55$	0	0
$56$	−1.00000	−0.133631
$57$	0.326909	0.0433001
$58$	9.90893	1.30111
$59$	13.2551	1.72567	0.862834	−	0.505487i	$-0.168687\pi$
$59$	13.2551	1.72567	0.862834	+	0.505487i	$0.168687\pi$
$60$	0.673091	0.0868957
$61$	6.50854	0.833333	0.416666	−	0.909060i	$-0.363198\pi$
$61$	6.50854	0.833333	0.416666	+	0.909060i	$0.363198\pi$
$62$	−3.80037	−0.482648
$63$	2.89313	0.364500
$64$	1.00000	0.125000
$65$	−12.5965	−1.56241
$66$	0	0
$67$	−9.28891	−1.13482	−0.567411	−	0.823435i	$-0.692055\pi$
$67$	−9.28891	−1.13482	−0.567411	+	0.823435i	$0.692055\pi$
$68$	0.239314	0.0290210
$69$	−2.48799	−0.299519
$70$	−2.05896	−0.246093
$71$	−8.28416	−0.983149	−0.491575	−	0.870835i	$-0.663579\pi$
$71$	−8.28416	−0.983149	−0.491575	+	0.870835i	$0.663579\pi$
$72$	−2.89313	−0.340959
$73$	−5.37176	−0.628717	−0.314358	−	0.949304i	$-0.601789\pi$
$73$	−5.37176	−0.628717	−0.314358	+	0.949304i	$0.601789\pi$
$74$	−11.4162	−1.32711
$75$	−0.248675	−0.0287145
$76$	1.00000	0.114708
$77$	0	0
$78$	−2.00000	−0.226455
$79$	13.2311	1.48861	0.744307	−	0.667838i	$-0.232780\pi$
$79$	13.2311	1.48861	0.744307	+	0.667838i	$0.232780\pi$
$80$	2.05896	0.230199
$81$	8.04960	0.894400
$82$	−6.77690	−0.748383
$83$	−13.6983	−1.50358	−0.751789	−	0.659404i	$-0.770809\pi$
$83$	−13.6983	−1.50358	−0.751789	+	0.659404i	$0.770809\pi$
$84$	−0.326909	−0.0356686
$85$	0.492737	0.0534448
$86$	6.97136	0.751742
$87$	3.23931	0.347291
$88$	0	0
$89$	−13.5486	−1.43615	−0.718076	−	0.695964i	$-0.754977\pi$
$89$	−13.5486	−1.43615	−0.718076	+	0.695964i	$0.754977\pi$
$90$	−5.95684	−0.627906
$91$	6.11792	0.641332
$92$	−7.61066	−0.793466
$93$	−1.24237	−0.128828
$94$	4.21068	0.434298
$95$	2.05896	0.211245
$96$	0.326909	0.0333650
$97$	17.9789	1.82548	0.912742	−	0.408536i	$-0.133961\pi$
$97$	17.9789	1.82548	0.912742	+	0.408536i	$0.133961\pi$
$98$	−6.00000	−0.606092
$99$	0	0
$100$	−0.760686	−0.0760686
$101$	0.755938	0.0752186	0.0376093	−	0.999293i	$-0.488026\pi$
$101$	0.755938	0.0752186	0.0376093	+	0.999293i	$0.488026\pi$
$102$	0.0782337	0.00774629
$103$	−10.2842	−1.01333	−0.506664	−	0.862143i	$-0.669122\pi$
$103$	−10.2842	−1.01333	−0.506664	+	0.862143i	$0.669122\pi$
$104$	−6.11792	−0.599911
$105$	−0.673091	−0.0656870
$106$	−8.68414	−0.843478
$107$	−19.7478	−1.90910	−0.954548	−	0.298056i	$-0.903662\pi$
$107$	−19.7478	−1.90910	−0.954548	+	0.298056i	$0.903662\pi$
$108$	−1.92651	−0.185379
$109$	0.462413	0.0442912	0.0221456	−	0.999755i	$-0.492950\pi$
$109$	0.462413	0.0442912	0.0221456	+	0.999755i	$0.492950\pi$
$110$	0	0
$111$	−3.73205	−0.354231
$112$	−1.00000	−0.0944911
$113$	−1.12728	−0.106046	−0.0530228	−	0.998593i	$-0.516886\pi$
$113$	−1.12728	−0.106046	−0.0530228	+	0.998593i	$0.516886\pi$
$114$	0.326909	0.0306178
$115$	−15.6700	−1.46124
$116$	9.90893	0.920021
$117$	17.6999	1.63636
$118$	13.2551	1.22023
$119$	−0.239314	−0.0219378
$120$	0.673091	0.0614446
$121$	0	0
$122$	6.50854	0.589255
$123$	−2.21543	−0.199758
$124$	−3.80037	−0.341283
$125$	−11.8610	−1.06088
$126$	2.89313	0.257741
$127$	−1.70689	−0.151462	−0.0757311	−	0.997128i	$-0.524129\pi$
$127$	−1.70689	−0.151462	−0.0757311	+	0.997128i	$0.524129\pi$
$128$	1.00000	0.0883883
$129$	2.27900	0.200655
$130$	−12.5965	−1.10479
$131$	−4.75763	−0.415676	−0.207838	−	0.978163i	$-0.566643\pi$
$131$	−4.75763	−0.415676	−0.207838	+	0.978163i	$0.566643\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	−9.28891	−0.802440
$135$	−3.96662	−0.341392
$136$	0.239314	0.0205210
$137$	6.66067	0.569059	0.284530	−	0.958667i	$-0.408163\pi$
$137$	6.66067	0.569059	0.284530	+	0.958667i	$0.408163\pi$
$138$	−2.48799	−0.211792
$139$	9.90418	0.840062	0.420031	−	0.907510i	$-0.362019\pi$
$139$	9.90418	0.840062	0.420031	+	0.907510i	$0.362019\pi$
$140$	−2.05896	−0.174014
$141$	1.37651	0.115923
$142$	−8.28416	−0.695192
$143$	0	0
$144$	−2.89313	−0.241094
$145$	20.4021	1.69430
$146$	−5.37176	−0.444570
$147$	−1.96145	−0.161778
$148$	−11.4162	−0.938405
$149$	15.6427	1.28150	0.640749	−	0.767751i	$-0.278624\pi$
$149$	15.6427	1.28150	0.640749	+	0.767751i	$0.278624\pi$
$150$	−0.248675	−0.0203042
$151$	7.72383	0.628556	0.314278	−	0.949331i	$-0.398238\pi$
$151$	7.72383	0.628556	0.314278	+	0.949331i	$0.398238\pi$
$152$	1.00000	0.0811107
$153$	−0.692366	−0.0559745
$154$	0	0
$155$	−7.82481	−0.628504
$156$	−2.00000	−0.160128
$157$	−8.20132	−0.654536	−0.327268	−	0.944932i	$-0.606128\pi$
$157$	−8.20132	−0.654536	−0.327268	+	0.944932i	$0.606128\pi$
$158$	13.2311	1.05261
$159$	−2.83892	−0.225141
$160$	2.05896	0.162775
$161$	7.61066	0.599804
$162$	8.04960	0.632436
$163$	−3.24237	−0.253962	−0.126981	−	0.991905i	$-0.540529\pi$
$163$	−3.24237	−0.253962	−0.126981	+	0.991905i	$0.540529\pi$
$164$	−6.77690	−0.529187
$165$	0	0
$166$	−13.6983	−1.06319
$167$	−18.8278	−1.45694	−0.728468	−	0.685079i	$-0.759767\pi$
$167$	−18.8278	−1.45694	−0.728468	+	0.685079i	$0.759767\pi$
$168$	−0.326909	−0.0252215
$169$	24.4289	1.87915
$170$	0.492737	0.0377912
$171$	−2.89313	−0.221243
$172$	6.97136	0.531562
$173$	−5.34143	−0.406102	−0.203051	−	0.979168i	$-0.565086\pi$
$173$	−5.34143	−0.406102	−0.203051	+	0.979168i	$0.565086\pi$
$174$	3.23931	0.245572
$175$	0.760686	0.0575025
$176$	0	0
$177$	4.33321	0.325704
$178$	−13.5486	−1.01551
$179$	17.7349	1.32557	0.662783	−	0.748811i	$-0.269375\pi$
$179$	17.7349	1.32557	0.662783	+	0.748811i	$0.269375\pi$
$180$	−5.95684	−0.443997
$181$	3.94273	0.293061	0.146530	−	0.989206i	$-0.453189\pi$
$181$	3.94273	0.293061	0.146530	+	0.989206i	$0.453189\pi$
$182$	6.11792	0.453490
$183$	2.12770	0.157284
$184$	−7.61066	−0.561065
$185$	−23.5055	−1.72816
$186$	−1.24237	−0.0910953
$187$	0	0
$188$	4.21068	0.307095
$189$	1.92651	0.140133
$190$	2.05896	0.149373
$191$	23.2183	1.68001	0.840007	−	0.542576i	$-0.182551\pi$
$191$	23.2183	1.68001	0.840007	+	0.542576i	$0.182551\pi$
$192$	0.326909	0.0235926
$193$	22.6431	1.62988	0.814942	−	0.579542i	$-0.196769\pi$
$193$	22.6431	1.62988	0.814942	+	0.579542i	$0.196769\pi$
$194$	17.9789	1.29081
$195$	−4.11792	−0.294890
$196$	−6.00000	−0.428571
$197$	11.7534	0.837397	0.418699	−	0.908125i	$-0.362486\pi$
$197$	11.7534	0.837397	0.418699	+	0.908125i	$0.362486\pi$
$198$	0	0
$199$	−2.14012	−0.151709	−0.0758544	−	0.997119i	$-0.524168\pi$
$199$	−2.14012	−0.151709	−0.0758544	+	0.997119i	$0.524168\pi$
$200$	−0.760686	−0.0537886
$201$	−3.03662	−0.214187
$202$	0.755938	0.0531876
$203$	−9.90893	−0.695470
$204$	0.0782337	0.00547745
$205$	−13.9534	−0.974545
$206$	−10.2842	−0.716532
$207$	22.0186	1.53040
$208$	−6.11792	−0.424201
$209$	0	0
$210$	−0.673091	−0.0464477
$211$	−26.3268	−1.81241	−0.906206	−	0.422836i	$-0.861035\pi$
$211$	−26.3268	−1.81241	−0.906206	+	0.422836i	$0.861035\pi$
$212$	−8.68414	−0.596429
$213$	−2.70816	−0.185560
$214$	−19.7478	−1.34994
$215$	14.3538	0.978918
$216$	−1.92651	−0.131083
$217$	3.80037	0.257986
$218$	0.462413	0.0313186
$219$	−1.75607	−0.118664
$220$	0	0
$221$	−1.46410	−0.0984861
$222$	−3.73205	−0.250479
$223$	6.11359	0.409396	0.204698	−	0.978825i	$-0.434379\pi$
$223$	6.11359	0.409396	0.204698	+	0.978825i	$0.434379\pi$
$224$	−1.00000	−0.0668153
$225$	2.20077	0.146718
$226$	−1.12728	−0.0749855
$227$	5.67784	0.376851	0.188426	−	0.982087i	$-0.439662\pi$
$227$	5.67784	0.376851	0.188426	+	0.982087i	$0.439662\pi$
$228$	0.326909	0.0216500
$229$	1.27397	0.0841863	0.0420932	−	0.999114i	$-0.486597\pi$
$229$	1.27397	0.0841863	0.0420932	+	0.999114i	$0.486597\pi$
$230$	−15.6700	−1.03325
$231$	0	0
$232$	9.90893	0.650553
$233$	−6.32512	−0.414372	−0.207186	−	0.978302i	$-0.566431\pi$
$233$	−6.32512	−0.414372	−0.207186	+	0.978302i	$0.566431\pi$
$234$	17.6999	1.15708
$235$	8.66962	0.565543
$236$	13.2551	0.862834
$237$	4.32536	0.280962
$238$	−0.239314	−0.0155124
$239$	−12.3145	−0.796558	−0.398279	−	0.917264i	$-0.630392\pi$
$239$	−12.3145	−0.796558	−0.398279	+	0.917264i	$0.630392\pi$
$240$	0.673091	0.0434479
$241$	10.6345	0.685031	0.342516	−	0.939512i	$-0.388721\pi$
$241$	10.6345	0.685031	0.342516	+	0.939512i	$0.388721\pi$
$242$	0	0
$243$	8.41103	0.539568
$244$	6.50854	0.416666
$245$	−12.3538	−0.789253
$246$	−2.21543	−0.141250
$247$	−6.11792	−0.389274
$248$	−3.80037	−0.241324
$249$	−4.47808	−0.283786
$250$	−11.8610	−0.750157
$251$	−17.4068	−1.09871	−0.549355	−	0.835589i	$-0.685126\pi$
$251$	−17.4068	−1.09871	−0.549355	+	0.835589i	$0.685126\pi$
$252$	2.89313	0.182250
$253$	0	0
$254$	−1.70689	−0.107100
$255$	0.161080	0.0100872
$256$	1.00000	0.0625000
$257$	10.1752	0.634711	0.317355	−	0.948307i	$-0.397205\pi$
$257$	10.1752	0.634711	0.317355	+	0.948307i	$0.397205\pi$
$258$	2.27900	0.141884
$259$	11.4162	0.709368
$260$	−12.5965	−0.781205
$261$	−28.6678	−1.77449
$262$	−4.75763	−0.293927
$263$	1.02977	0.0634985	0.0317492	−	0.999496i	$-0.489892\pi$
$263$	1.02977	0.0634985	0.0317492	+	0.999496i	$0.489892\pi$
$264$	0	0
$265$	−17.8803	−1.09838
$266$	−1.00000	−0.0613139
$267$	−4.42917	−0.271061
$268$	−9.28891	−0.567411
$269$	13.9055	0.847830	0.423915	−	0.905702i	$-0.360655\pi$
$269$	13.9055	0.847830	0.423915	+	0.905702i	$0.360655\pi$
$270$	−3.96662	−0.241401
$271$	17.5690	1.06724	0.533622	−	0.845723i	$-0.320831\pi$
$271$	17.5690	1.06724	0.533622	+	0.845723i	$0.320831\pi$
$272$	0.239314	0.0145105
$273$	2.00000	0.121046
$274$	6.66067	0.402386
$275$	0	0
$276$	−2.48799	−0.149759
$277$	−22.7162	−1.36488	−0.682441	−	0.730941i	$-0.739081\pi$
$277$	−22.7162	−1.36488	−0.682441	+	0.730941i	$0.739081\pi$
$278$	9.90418	0.594013
$279$	10.9950	0.658252
$280$	−2.05896	−0.123046
$281$	−1.25215	−0.0746971	−0.0373485	−	0.999302i	$-0.511891\pi$
$281$	−1.25215	−0.0746971	−0.0373485	+	0.999302i	$0.511891\pi$
$282$	1.37651	0.0819698
$283$	0.0103291	0.000614000 0	0.000307000	−	1.00000i	$-0.499902\pi$
$283$	0.0103291	0.000614000 0	0.000307000		1.00000i	$0.499902\pi$
$284$	−8.28416	−0.491575
$285$	0.673091	0.0398705
$286$	0	0
$287$	6.77690	0.400028
$288$	−2.89313	−0.170479
$289$	−16.9427	−0.996631
$290$	20.4021	1.19805
$291$	5.87747	0.344543
$292$	−5.37176	−0.314358
$293$	11.8307	0.691157	0.345578	−	0.938390i	$-0.387683\pi$
$293$	11.8307	0.691157	0.345578	+	0.938390i	$0.387683\pi$
$294$	−1.96145	−0.114394
$295$	27.2917	1.58899
$296$	−11.4162	−0.663553
$297$	0	0
$298$	15.6427	0.906155
$299$	46.5614	2.69271
$300$	−0.248675	−0.0143573
$301$	−6.97136	−0.401823
$302$	7.72383	0.444456
$303$	0.247123	0.0141968
$304$	1.00000	0.0573539
$305$	13.4008	0.767328
$306$	−0.692366	−0.0395799
$307$	−25.9451	−1.48077	−0.740384	−	0.672185i	$-0.765356\pi$
$307$	−25.9451	−1.48077	−0.740384	+	0.672185i	$0.765356\pi$
$308$	0	0
$309$	−3.36198	−0.191256
$310$	−7.82481	−0.444419
$311$	17.7512	1.00658	0.503289	−	0.864118i	$-0.332123\pi$
$311$	17.7512	1.00658	0.503289	+	0.864118i	$0.332123\pi$
$312$	−2.00000	−0.113228
$313$	7.63162	0.431365	0.215682	−	0.976464i	$-0.430802\pi$
$313$	7.63162	0.431365	0.215682	+	0.976464i	$0.430802\pi$
$314$	−8.20132	−0.462827
$315$	5.95684	0.335630
$316$	13.2311	0.744307
$317$	12.5298	0.703742	0.351871	−	0.936048i	$-0.385546\pi$
$317$	12.5298	0.703742	0.351871	+	0.936048i	$0.385546\pi$
$318$	−2.83892	−0.159199
$319$	0	0
$320$	2.05896	0.115099
$321$	−6.45574	−0.360324
$322$	7.61066	0.424125
$323$	0.239314	0.0133158
$324$	8.04960	0.447200
$325$	4.65382	0.258147
$326$	−3.24237	−0.179578
$327$	0.151167	0.00835955
$328$	−6.77690	−0.374192
$329$	−4.21068	−0.232142
$330$	0	0
$331$	−34.3093	−1.88581	−0.942905	−	0.333060i	$-0.891919\pi$
$331$	−34.3093	−1.88581	−0.942905	+	0.333060i	$0.891919\pi$
$332$	−13.6983	−0.751789
$333$	33.0285	1.80995
$334$	−18.8278	−1.03021
$335$	−19.1255	−1.04494
$336$	−0.326909	−0.0178343
$337$	1.03380	0.0563147	0.0281573	−	0.999604i	$-0.491036\pi$
$337$	1.03380	0.0563147	0.0281573	+	0.999604i	$0.491036\pi$
$338$	24.4289	1.32876
$339$	−0.368517	−0.0200151
$340$	0.492737	0.0267224
$341$	0	0
$342$	−2.89313	−0.156443
$343$	13.0000	0.701934
$344$	6.97136	0.375871
$345$	−5.12267	−0.275795
$346$	−5.34143	−0.287157
$347$	2.59025	0.139052	0.0695258	−	0.997580i	$-0.477851\pi$
$347$	2.59025	0.139052	0.0695258	+	0.997580i	$0.477851\pi$
$348$	3.23931	0.173645
$349$	11.1278	0.595659	0.297830	−	0.954619i	$-0.403737\pi$
$349$	11.1278	0.595659	0.297830	+	0.954619i	$0.403737\pi$
$350$	0.760686	0.0406604
$351$	11.7863	0.629104
$352$	0	0
$353$	−28.1896	−1.50038	−0.750191	−	0.661221i	$-0.770038\pi$
$353$	−28.1896	−1.50038	−0.750191	+	0.661221i	$0.770038\pi$
$354$	4.33321	0.230307
$355$	−17.0568	−0.905279
$356$	−13.5486	−0.718076
$357$	−0.0782337	−0.00414057
$358$	17.7349	0.937317
$359$	−22.4469	−1.18470	−0.592352	−	0.805679i	$-0.701800\pi$
$359$	−22.4469	−1.18470	−0.592352	+	0.805679i	$0.701800\pi$
$360$	−5.95684	−0.313953
$361$	1.00000	0.0526316
$362$	3.94273	0.207225
$363$	0	0
$364$	6.11792	0.320666
$365$	−11.0602	−0.578919
$366$	2.12770	0.111216
$367$	−7.10729	−0.370997	−0.185499	−	0.982645i	$-0.559390\pi$
$367$	−7.10729	−0.370997	−0.185499	+	0.982645i	$0.559390\pi$
$368$	−7.61066	−0.396733
$369$	19.6065	1.02067
$370$	−23.5055	−1.22199
$371$	8.68414	0.450858
$372$	−1.24237	−0.0644141
$373$	6.36546	0.329591	0.164795	−	0.986328i	$-0.447304\pi$
$373$	6.36546	0.329591	0.164795	+	0.986328i	$0.447304\pi$
$374$	0	0
$375$	−3.87747	−0.200232
$376$	4.21068	0.217149
$377$	−60.6220	−3.12219
$378$	1.92651	0.0990892
$379$	21.0623	1.08190	0.540950	−	0.841055i	$-0.318065\pi$
$379$	21.0623	1.08190	0.540950	+	0.841055i	$0.318065\pi$
$380$	2.05896	0.105622
$381$	−0.557997	−0.0285871
$382$	23.2183	1.18795
$383$	−24.8640	−1.27049	−0.635245	−	0.772311i	$-0.719101\pi$
$383$	−24.8640	−1.27049	−0.635245	+	0.772311i	$0.719101\pi$
$384$	0.326909	0.0166825
$385$	0	0
$386$	22.6431	1.15250
$387$	−20.1691	−1.02525
$388$	17.9789	0.912742
$389$	−11.6841	−0.592409	−0.296205	−	0.955124i	$-0.595721\pi$
$389$	−11.6841	−0.592409	−0.296205	+	0.955124i	$0.595721\pi$
$390$	−4.11792	−0.208519
$391$	−1.82133	−0.0921088
$392$	−6.00000	−0.303046
$393$	−1.55531	−0.0784549
$394$	11.7534	0.592129
$395$	27.2423	1.37071
$396$	0	0
$397$	−24.7751	−1.24343	−0.621714	−	0.783245i	$-0.713563\pi$
$397$	−24.7751	−1.24343	−0.621714	+	0.783245i	$0.713563\pi$
$398$	−2.14012	−0.107274
$399$	−0.326909	−0.0163659
$400$	−0.760686	−0.0380343
$401$	−15.5124	−0.774654	−0.387327	−	0.921942i	$-0.626601\pi$
$401$	−15.5124	−0.774654	−0.387327	+	0.921942i	$0.626601\pi$
$402$	−3.03662	−0.151453
$403$	23.2504	1.15818
$404$	0.755938	0.0376093
$405$	16.5738	0.823559
$406$	−9.90893	−0.491772
$407$	0	0
$408$	0.0782337	0.00387314
$409$	−36.1316	−1.78659	−0.893297	−	0.449467i	$-0.851614\pi$
$409$	−36.1316	−1.78659	−0.893297	+	0.449467i	$0.851614\pi$
$410$	−13.9534	−0.689107
$411$	2.17743	0.107405
$412$	−10.2842	−0.506664
$413$	−13.2551	−0.652241
$414$	22.0186	1.08216
$415$	−28.2041	−1.38449
$416$	−6.11792	−0.299956
$417$	3.23776	0.158554
$418$	0	0
$419$	−29.9947	−1.46534	−0.732669	−	0.680585i	$-0.761726\pi$
$419$	−29.9947	−1.46534	−0.732669	+	0.680585i	$0.761726\pi$
$420$	−0.673091	−0.0328435
$421$	30.1891	1.47132	0.735662	−	0.677348i	$-0.236871\pi$
$421$	30.1891	1.47132	0.735662	+	0.677348i	$0.236871\pi$
$422$	−26.3268	−1.28157
$423$	−12.1820	−0.592311
$424$	−8.68414	−0.421739
$425$	−0.182043	−0.00883037
$426$	−2.70816	−0.131211
$427$	−6.50854	−0.314970
$428$	−19.7478	−0.954548
$429$	0	0
$430$	14.3538	0.692200
$431$	6.56694	0.316318	0.158159	−	0.987414i	$-0.449444\pi$
$431$	6.56694	0.316318	0.158159	+	0.987414i	$0.449444\pi$
$432$	−1.92651	−0.0926895
$433$	25.2584	1.21384	0.606919	−	0.794763i	$-0.292405\pi$
$433$	25.2584	1.21384	0.606919	+	0.794763i	$0.292405\pi$
$434$	3.80037	0.182424
$435$	6.66962	0.319784
$436$	0.462413	0.0221456
$437$	−7.61066	−0.364067
$438$	−1.75607	−0.0839084
$439$	27.5247	1.31368	0.656842	−	0.754028i	$-0.271892\pi$
$439$	27.5247	1.31368	0.656842	+	0.754028i	$0.271892\pi$
$440$	0	0
$441$	17.3588	0.826609
$442$	−1.46410	−0.0696402
$443$	−17.1569	−0.815148	−0.407574	−	0.913172i	$-0.633625\pi$
$443$	−17.1569	−0.815148	−0.407574	+	0.913172i	$0.633625\pi$
$444$	−3.73205	−0.177115
$445$	−27.8961	−1.32240
$446$	6.11359	0.289487
$447$	5.11372	0.241871
$448$	−1.00000	−0.0472456
$449$	−37.2186	−1.75645	−0.878226	−	0.478245i	$-0.841273\pi$
$449$	−37.2186	−1.75645	−0.878226	+	0.478245i	$0.841273\pi$
$450$	2.20077	0.103745
$451$	0	0
$452$	−1.12728	−0.0530228
$453$	2.52498	0.118634
$454$	5.67784	0.266474
$455$	12.5965	0.590535
$456$	0.326909	0.0153089
$457$	1.31069	0.0613117	0.0306559	−	0.999530i	$-0.490240\pi$
$457$	1.31069	0.0613117	0.0306559	+	0.999530i	$0.490240\pi$
$458$	1.27397	0.0595287
$459$	−0.461041	−0.0215196
$460$	−15.6700	−0.730619
$461$	19.3338	0.900463	0.450232	−	0.892912i	$-0.351341\pi$
$461$	19.3338	0.900463	0.450232	+	0.892912i	$0.351341\pi$
$462$	0	0
$463$	2.81631	0.130885	0.0654424	−	0.997856i	$-0.479154\pi$
$463$	2.81631	0.130885	0.0654424	+	0.997856i	$0.479154\pi$
$464$	9.90893	0.460010
$465$	−2.55800	−0.118624
$466$	−6.32512	−0.293006
$467$	19.0030	0.879352	0.439676	−	0.898156i	$-0.355093\pi$
$467$	19.0030	0.879352	0.439676	+	0.898156i	$0.355093\pi$
$468$	17.6999	0.818180
$469$	9.28891	0.428922
$470$	8.66962	0.399900
$471$	−2.68108	−0.123538
$472$	13.2551	0.610116
$473$	0	0
$474$	4.32536	0.198670
$475$	−0.760686	−0.0349027
$476$	−0.239314	−0.0109689
$477$	25.1244	1.15037
$478$	−12.3145	−0.563252
$479$	34.7289	1.58680	0.793402	−	0.608698i	$-0.208308\pi$
$479$	34.7289	1.58680	0.793402	+	0.608698i	$0.208308\pi$
$480$	0.673091	0.0307223
$481$	69.8433	3.18458
$482$	10.6345	0.484390
$483$	2.48799	0.113207
$484$	0	0
$485$	37.0179	1.68090
$486$	8.41103	0.381532
$487$	−16.5313	−0.749104	−0.374552	−	0.927206i	$-0.622203\pi$
$487$	−16.5313	−0.749104	−0.374552	+	0.927206i	$0.622203\pi$
$488$	6.50854	0.294628
$489$	−1.05996	−0.0479330
$490$	−12.3538	−0.558086
$491$	−27.1874	−1.22695	−0.613475	−	0.789714i	$-0.710229\pi$
$491$	−27.1874	−1.22695	−0.613475	+	0.789714i	$0.710229\pi$
$492$	−2.21543	−0.0998792
$493$	2.37134	0.106800
$494$	−6.11792	−0.275258
$495$	0	0
$496$	−3.80037	−0.170642
$497$	8.28416	0.371596
$498$	−4.47808	−0.200667
$499$	−2.60980	−0.116831	−0.0584153	−	0.998292i	$-0.518605\pi$
$499$	−2.60980	−0.116831	−0.0584153	+	0.998292i	$0.518605\pi$
$500$	−11.8610	−0.530441
$501$	−6.15496	−0.274983
$502$	−17.4068	−0.776905
$503$	11.5111	0.513253	0.256626	−	0.966511i	$-0.417389\pi$
$503$	11.5111	0.513253	0.256626	+	0.966511i	$0.417389\pi$
$504$	2.89313	0.128870
$505$	1.55645	0.0692609
$506$	0	0
$507$	7.98603	0.354672
$508$	−1.70689	−0.0757311
$509$	−20.5685	−0.911681	−0.455841	−	0.890061i	$-0.650661\pi$
$509$	−20.5685	−0.911681	−0.455841	+	0.890061i	$0.650661\pi$
$510$	0.161080	0.00713274
$511$	5.37176	0.237633
$512$	1.00000	0.0441942
$513$	−1.92651	−0.0850577
$514$	10.1752	0.448808
$515$	−21.1747	−0.933068
$516$	2.27900	0.100327
$517$	0	0
$518$	11.4162	0.501599
$519$	−1.74616	−0.0766479
$520$	−12.5965	−0.552395
$521$	14.6328	0.641073	0.320536	−	0.947236i	$-0.396137\pi$
$521$	14.6328	0.641073	0.320536	+	0.947236i	$0.396137\pi$
$522$	−28.6678	−1.25476
$523$	−21.7222	−0.949844	−0.474922	−	0.880028i	$-0.657524\pi$
$523$	−21.7222	−0.949844	−0.474922	+	0.880028i	$0.657524\pi$
$524$	−4.75763	−0.207838
$525$	0.248675	0.0108531
$526$	1.02977	0.0449002
$527$	−0.909481	−0.0396176
$528$	0	0
$529$	34.9221	1.51835
$530$	−17.8803	−0.776670
$531$	−38.3488	−1.66419
$532$	−1.00000	−0.0433555
$533$	41.4605	1.79585
$534$	−4.42917	−0.191669
$535$	−40.6600	−1.75789
$536$	−9.28891	−0.401220
$537$	5.79768	0.250188
$538$	13.9055	0.599507
$539$	0	0
$540$	−3.96662	−0.170696
$541$	7.31796	0.314624	0.157312	−	0.987549i	$-0.449717\pi$
$541$	7.31796	0.314624	0.157312	+	0.987549i	$0.449717\pi$
$542$	17.5690	0.754655
$543$	1.28891	0.0553125
$544$	0.239314	0.0102605
$545$	0.952090	0.0407831
$546$	2.00000	0.0855921
$547$	−10.1644	−0.434599	−0.217300	−	0.976105i	$-0.569725\pi$
$547$	−10.1644	−0.434599	−0.217300	+	0.976105i	$0.569725\pi$
$548$	6.66067	0.284530
$549$	−18.8300	−0.803647
$550$	0	0
$551$	9.90893	0.422135
$552$	−2.48799	−0.105896
$553$	−13.2311	−0.562643
$554$	−22.7162	−0.965117
$555$	−7.68414	−0.326174
$556$	9.90418	0.420031
$557$	−6.23584	−0.264221	−0.132110	−	0.991235i	$-0.542175\pi$
$557$	−6.23584	−0.264221	−0.132110	+	0.991235i	$0.542175\pi$
$558$	10.9950	0.465454
$559$	−42.6502	−1.80391
$560$	−2.05896	−0.0870069
$561$	0	0
$562$	−1.25215	−0.0528188
$563$	12.4704	0.525565	0.262782	−	0.964855i	$-0.415360\pi$
$563$	12.4704	0.525565	0.262782	+	0.964855i	$0.415360\pi$
$564$	1.37651	0.0579614
$565$	−2.32102	−0.0976462
$566$	0.0103291	0.000434164 0
$567$	−8.04960	−0.338051
$568$	−8.28416	−0.347596
$569$	6.33861	0.265728	0.132864	−	0.991134i	$-0.457583\pi$
$569$	6.33861	0.265728	0.132864	+	0.991134i	$0.457583\pi$
$570$	0.673091	0.0281927
$571$	21.6888	0.907646	0.453823	−	0.891092i	$-0.350060\pi$
$571$	21.6888	0.907646	0.453823	+	0.891092i	$0.350060\pi$
$572$	0	0
$573$	7.59025	0.317087
$574$	6.77690	0.282862
$575$	5.78932	0.241431
$576$	−2.89313	−0.120547
$577$	10.1622	0.423059	0.211529	−	0.977372i	$-0.432156\pi$
$577$	10.1622	0.423059	0.211529	+	0.977372i	$0.432156\pi$
$578$	−16.9427	−0.704725
$579$	7.40222	0.307626
$580$	20.4021	0.847150
$581$	13.6983	0.568299
$582$	5.87747	0.243629
$583$	0	0
$584$	−5.37176	−0.222285
$585$	36.4435	1.50675
$586$	11.8307	0.488722
$587$	−24.2773	−1.00203	−0.501016	−	0.865438i	$-0.667040\pi$
$587$	−24.2773	−1.00203	−0.501016	+	0.865438i	$0.667040\pi$
$588$	−1.96145	−0.0808889
$589$	−3.80037	−0.156592
$590$	27.2917	1.12358
$591$	3.84230	0.158051
$592$	−11.4162	−0.469203
$593$	−15.5115	−0.636979	−0.318490	−	0.947926i	$-0.603176\pi$
$593$	−15.5115	−0.636979	−0.318490	+	0.947926i	$0.603176\pi$
$594$	0	0
$595$	−0.492737	−0.0202003
$596$	15.6427	0.640749
$597$	−0.699623	−0.0286336
$598$	46.5614	1.90404
$599$	−6.71530	−0.274380	−0.137190	−	0.990545i	$-0.543807\pi$
$599$	−6.71530	−0.274380	−0.137190	+	0.990545i	$0.543807\pi$
$600$	−0.248675	−0.0101521
$601$	18.3912	0.750193	0.375096	−	0.926986i	$-0.377610\pi$
$601$	18.3912	0.750193	0.375096	+	0.926986i	$0.377610\pi$
$602$	−6.97136	−0.284132
$603$	26.8740	1.09440
$604$	7.72383	0.314278
$605$	0	0
$606$	0.247123	0.0100387
$607$	−3.35974	−0.136368	−0.0681838	−	0.997673i	$-0.521720\pi$
$607$	−3.35974	−0.136368	−0.0681838	+	0.997673i	$0.521720\pi$
$608$	1.00000	0.0405554
$609$	−3.23931	−0.131264
$610$	13.4008	0.542583
$611$	−25.7606	−1.04216
$612$	−0.692366	−0.0279872
$613$	19.0516	0.769487	0.384743	−	0.923024i	$-0.374290\pi$
$613$	19.0516	0.769487	0.384743	+	0.923024i	$0.374290\pi$
$614$	−25.9451	−1.04706
$615$	−4.56147	−0.183936
$616$	0	0
$617$	−20.5000	−0.825299	−0.412650	−	0.910890i	$-0.635397\pi$
$617$	−20.5000	−0.825299	−0.412650	+	0.910890i	$0.635397\pi$
$618$	−3.36198	−0.135239
$619$	−6.78351	−0.272652	−0.136326	−	0.990664i	$-0.543529\pi$
$619$	−6.78351	−0.272652	−0.136326	+	0.990664i	$0.543529\pi$
$620$	−7.82481	−0.314252
$621$	14.6620	0.588367
$622$	17.7512	0.711758
$623$	13.5486	0.542815
$624$	−2.00000	−0.0800641
$625$	−20.6179	−0.824717
$626$	7.63162	0.305021
$627$	0	0
$628$	−8.20132	−0.327268
$629$	−2.73205	−0.108934
$630$	5.95684	0.237326
$631$	−0.184271	−0.00733571	−0.00366786	−	0.999993i	$-0.501168\pi$
$631$	−0.184271	−0.00733571	−0.00366786	+	0.999993i	$0.501168\pi$
$632$	13.2311	0.526304
$633$	−8.60646	−0.342076
$634$	12.5298	0.497621
$635$	−3.51442	−0.139466
$636$	−2.83892	−0.112571
$637$	36.7075	1.45440
$638$	0	0
$639$	23.9672	0.948127
$640$	2.05896	0.0813875
$641$	16.9380	0.669010	0.334505	−	0.942394i	$-0.391431\pi$
$641$	16.9380	0.669010	0.334505	+	0.942394i	$0.391431\pi$
$642$	−6.45574	−0.254788
$643$	0.934052	0.0368354	0.0184177	−	0.999830i	$-0.494137\pi$
$643$	0.934052	0.0368354	0.0184177	+	0.999830i	$0.494137\pi$
$644$	7.61066	0.299902
$645$	4.69237	0.184762
$646$	0.239314	0.00941567
$647$	29.1175	1.14473	0.572364	−	0.820000i	$-0.306027\pi$
$647$	29.1175	1.14473	0.572364	+	0.820000i	$0.306027\pi$
$648$	8.04960	0.316218
$649$	0	0
$650$	4.65382	0.182538
$651$	1.24237	0.0486925
$652$	−3.24237	−0.126981
$653$	2.11414	0.0827326	0.0413663	−	0.999144i	$-0.486829\pi$
$653$	2.11414	0.0827326	0.0413663	+	0.999144i	$0.486829\pi$
$654$	0.151167	0.00591109
$655$	−9.79576	−0.382752
$656$	−6.77690	−0.264594
$657$	15.5412	0.606320
$658$	−4.21068	−0.164149
$659$	−30.7961	−1.19964	−0.599822	−	0.800133i	$-0.704762\pi$
$659$	−30.7961	−1.19964	−0.599822	+	0.800133i	$0.704762\pi$
$660$	0	0
$661$	−2.31197	−0.0899251	−0.0449625	−	0.998989i	$-0.514317\pi$
$661$	−2.31197	−0.0899251	−0.0449625	+	0.998989i	$0.514317\pi$
$662$	−34.3093	−1.33347
$663$	−0.478627	−0.0185883
$664$	−13.6983	−0.531595
$665$	−2.05896	−0.0798430
$666$	33.0285	1.27983
$667$	−75.4134	−2.92002
$668$	−18.8278	−0.728468
$669$	1.99858	0.0772697
$670$	−19.1255	−0.738882
$671$	0	0
$672$	−0.326909	−0.0126108
$673$	30.8157	1.18786	0.593928	−	0.804518i	$-0.297576\pi$
$673$	30.8157	1.18786	0.593928	+	0.804518i	$0.297576\pi$
$674$	1.03380	0.0398205
$675$	1.46547	0.0564061
$676$	24.4289	0.939574
$677$	−29.4745	−1.13280	−0.566399	−	0.824131i	$-0.691664\pi$
$677$	−29.4745	−1.13280	−0.566399	+	0.824131i	$0.691664\pi$
$678$	−0.368517	−0.0141528
$679$	−17.9789	−0.689968
$680$	0.492737	0.0188956
$681$	1.85613	0.0711272
$682$	0	0
$683$	35.8855	1.37312	0.686559	−	0.727074i	$-0.259120\pi$
$683$	35.8855	1.37312	0.686559	+	0.727074i	$0.259120\pi$
$684$	−2.89313	−0.110622
$685$	13.7140	0.523987
$686$	13.0000	0.496342
$687$	0.416472	0.0158894
$688$	6.97136	0.265781
$689$	53.1289	2.02405
$690$	−5.12267	−0.195017
$691$	24.3791	0.927423	0.463711	−	0.885986i	$-0.346517\pi$
$691$	24.3791	0.927423	0.463711	+	0.885986i	$0.346517\pi$
$692$	−5.34143	−0.203051
$693$	0	0
$694$	2.59025	0.0983244
$695$	20.3923	0.773524
$696$	3.23931	0.122786
$697$	−1.62180	−0.0614302
$698$	11.1278	0.421195
$699$	−2.06774	−0.0782090
$700$	0.760686	0.0287512
$701$	−17.4077	−0.657478	−0.328739	−	0.944421i	$-0.606624\pi$
$701$	−17.4077	−0.657478	−0.328739	+	0.944421i	$0.606624\pi$
$702$	11.7863	0.444844
$703$	−11.4162	−0.430570
$704$	0	0
$705$	2.83417	0.106741
$706$	−28.1896	−1.06093
$707$	−0.755938	−0.0284300
$708$	4.33321	0.162852
$709$	16.4309	0.617074	0.308537	−	0.951212i	$-0.400161\pi$
$709$	16.4309	0.617074	0.308537	+	0.951212i	$0.400161\pi$
$710$	−17.0568	−0.640129
$711$	−38.2793	−1.43558
$712$	−13.5486	−0.507757
$713$	28.9233	1.08319
$714$	−0.0782337	−0.00292782
$715$	0	0
$716$	17.7349	0.662783
$717$	−4.02571	−0.150343
$718$	−22.4469	−0.837712
$719$	−17.2596	−0.643674	−0.321837	−	0.946795i	$-0.604300\pi$
$719$	−17.2596	−0.643674	−0.321837	+	0.946795i	$0.604300\pi$
$720$	−5.95684	−0.221998
$721$	10.2842	0.383002
$722$	1.00000	0.0372161
$723$	3.47652	0.129293
$724$	3.94273	0.146530
$725$	−7.53759	−0.279939
$726$	0	0
$727$	−16.6519	−0.617583	−0.308792	−	0.951130i	$-0.599925\pi$
$727$	−16.6519	−0.617583	−0.308792	+	0.951130i	$0.599925\pi$
$728$	6.11792	0.226745
$729$	−21.3992	−0.792561
$730$	−11.0602	−0.409358
$731$	1.66834	0.0617059
$732$	2.12770	0.0786419
$733$	37.2550	1.37604	0.688022	−	0.725690i	$-0.258479\pi$
$733$	37.2550	1.37604	0.688022	+	0.725690i	$0.258479\pi$
$734$	−7.10729	−0.262335
$735$	−4.03855	−0.148964
$736$	−7.61066	−0.280532
$737$	0	0
$738$	19.6065	0.721724
$739$	17.6970	0.650995	0.325498	−	0.945543i	$-0.394468\pi$
$739$	17.6970	0.650995	0.325498	+	0.945543i	$0.394468\pi$
$740$	−23.5055	−0.864078
$741$	−2.00000	−0.0734718
$742$	8.68414	0.318805
$743$	−13.0381	−0.478323	−0.239161	−	0.970980i	$-0.576872\pi$
$743$	−13.0381	−0.478323	−0.239161	+	0.970980i	$0.576872\pi$
$744$	−1.24237	−0.0455476
$745$	32.2076	1.18000
$746$	6.36546	0.233056
$747$	39.6308	1.45002
$748$	0	0
$749$	19.7478	0.721571
$750$	−3.87747	−0.141585
$751$	−28.7414	−1.04879	−0.524395	−	0.851475i	$-0.675709\pi$
$751$	−28.7414	−1.04879	−0.524395	+	0.851475i	$0.675709\pi$
$752$	4.21068	0.153548
$753$	−5.69044	−0.207371
$754$	−60.6220	−2.20772
$755$	15.9030	0.578771
$756$	1.92651	0.0700667
$757$	−47.5277	−1.72742	−0.863711	−	0.503987i	$-0.831866\pi$
$757$	−47.5277	−1.72742	−0.863711	+	0.503987i	$0.831866\pi$
$758$	21.0623	0.765018
$759$	0	0
$760$	2.05896	0.0746863
$761$	15.3362	0.555936	0.277968	−	0.960590i	$-0.410339\pi$
$761$	15.3362	0.555936	0.277968	+	0.960590i	$0.410339\pi$
$762$	−0.557997	−0.0202141
$763$	−0.462413	−0.0167405
$764$	23.2183	0.840007
$765$	−1.42555	−0.0515410
$766$	−24.8640	−0.898372
$767$	−81.0937	−2.92812
$768$	0.326909	0.0117963
$769$	4.81672	0.173695	0.0868477	−	0.996222i	$-0.472321\pi$
$769$	4.81672	0.173695	0.0868477	+	0.996222i	$0.472321\pi$
$770$	0	0
$771$	3.32636	0.119796
$772$	22.6431	0.814942
$773$	38.2891	1.37716	0.688581	−	0.725159i	$-0.258234\pi$
$773$	38.2891	1.37716	0.688581	+	0.725159i	$0.258234\pi$
$774$	−20.1691	−0.724962
$775$	2.89089	0.103844
$776$	17.9789	0.645406
$777$	3.73205	0.133887
$778$	−11.6841	−0.418897
$779$	−6.77690	−0.242808
$780$	−4.11792	−0.147445
$781$	0	0
$782$	−1.82133	−0.0651308
$783$	−19.0897	−0.682210
$784$	−6.00000	−0.214286
$785$	−16.8862	−0.602694
$786$	−1.55531	−0.0554760
$787$	−48.0759	−1.71372	−0.856861	−	0.515548i	$-0.827589\pi$
$787$	−48.0759	−1.71372	−0.856861	+	0.515548i	$0.827589\pi$
$788$	11.7534	0.418699
$789$	0.336641	0.0119847
$790$	27.2423	0.969237
$791$	1.12728	0.0400815
$792$	0	0
$793$	−39.8187	−1.41400
$794$	−24.7751	−0.879236
$795$	−5.84522	−0.207309
$796$	−2.14012	−0.0758544
$797$	12.8137	0.453883	0.226942	−	0.973908i	$-0.427127\pi$
$797$	12.8137	0.453883	0.226942	+	0.973908i	$0.427127\pi$
$798$	−0.326909	−0.0115724
$799$	1.00767	0.0356489
$800$	−0.760686	−0.0268943
$801$	39.1980	1.38499
$802$	−15.5124	−0.547763
$803$	0	0
$804$	−3.03662	−0.107093
$805$	15.6700	0.552296
$806$	23.2504	0.818959
$807$	4.54581	0.160020
$808$	0.755938	0.0265938
$809$	10.4695	0.368090	0.184045	−	0.982918i	$-0.441081\pi$
$809$	10.4695	0.368090	0.184045	+	0.982918i	$0.441081\pi$
$810$	16.5738	0.582344
$811$	−37.4640	−1.31554	−0.657769	−	0.753220i	$-0.728500\pi$
$811$	−37.4640	−1.31554	−0.657769	+	0.753220i	$0.728500\pi$
$812$	−9.90893	−0.347735
$813$	5.74347	0.201432
$814$	0	0
$815$	−6.67592	−0.233847
$816$	0.0782337	0.00273873
$817$	6.97136	0.243897
$818$	−36.1316	−1.26331
$819$	−17.6999	−0.618486
$820$	−13.9534	−0.487273
$821$	−1.99474	−0.0696168	−0.0348084	−	0.999394i	$-0.511082\pi$
$821$	−1.99474	−0.0696168	−0.0348084	+	0.999394i	$0.511082\pi$
$822$	2.17743	0.0759466
$823$	−39.5721	−1.37940	−0.689698	−	0.724097i	$-0.742257\pi$
$823$	−39.5721	−1.37940	−0.689698	+	0.724097i	$0.742257\pi$
$824$	−10.2842	−0.358266
$825$	0	0
$826$	−13.2551	−0.461204
$827$	15.1252	0.525954	0.262977	−	0.964802i	$-0.415296\pi$
$827$	15.1252	0.525954	0.262977	+	0.964802i	$0.415296\pi$
$828$	22.0186	0.765200
$829$	4.54877	0.157985	0.0789927	−	0.996875i	$-0.474830\pi$
$829$	4.54877	0.157985	0.0789927	+	0.996875i	$0.474830\pi$
$830$	−28.2041	−0.978980
$831$	−7.42610	−0.257609
$832$	−6.11792	−0.212101
$833$	−1.43588	−0.0497504
$834$	3.23776	0.112115
$835$	−38.7656	−1.34154
$836$	0	0
$837$	7.32147	0.253067
$838$	−29.9947	−1.03615
$839$	22.7135	0.784157	0.392079	−	0.919932i	$-0.371756\pi$
$839$	22.7135	0.784157	0.392079	+	0.919932i	$0.371756\pi$
$840$	−0.673091	−0.0232239
$841$	69.1869	2.38575
$842$	30.1891	1.04038
$843$	−0.409339	−0.0140984
$844$	−26.3268	−0.906206
$845$	50.2982	1.73031
$846$	−12.1820	−0.418827
$847$	0	0
$848$	−8.68414	−0.298215
$849$	0.00337666	0.000115887 0
$850$	−0.182043	−0.00624401
$851$	86.8847	2.97837
$852$	−2.70816	−0.0927802
$853$	−8.10525	−0.277518	−0.138759	−	0.990326i	$-0.544311\pi$
$853$	−8.10525	−0.277518	−0.138759	+	0.990326i	$0.544311\pi$
$854$	−6.50854	−0.222717
$855$	−5.95684	−0.203720
$856$	−19.7478	−0.674968
$857$	−12.0630	−0.412064	−0.206032	−	0.978545i	$-0.566055\pi$
$857$	−12.0630	−0.412064	−0.206032	+	0.978545i	$0.566055\pi$
$858$	0	0
$859$	28.7979	0.982570	0.491285	−	0.870999i	$-0.336527\pi$
$859$	28.7979	0.982570	0.491285	+	0.870999i	$0.336527\pi$
$860$	14.3538	0.489459
$861$	2.21543	0.0755015
$862$	6.56694	0.223671
$863$	−18.7423	−0.637995	−0.318997	−	0.947756i	$-0.603346\pi$
$863$	−18.7423	−0.637995	−0.318997	+	0.947756i	$0.603346\pi$
$864$	−1.92651	−0.0655414
$865$	−10.9978	−0.373936
$866$	25.2584	0.858314
$867$	−5.53872	−0.188105
$868$	3.80037	0.128993
$869$	0	0
$870$	6.66962	0.226121
$871$	56.8288	1.92557
$872$	0.462413	0.0156593
$873$	−52.0154	−1.76046
$874$	−7.61066	−0.257434
$875$	11.8610	0.400976
$876$	−1.75607	−0.0593322
$877$	44.6022	1.50611	0.753054	−	0.657959i	$-0.228580\pi$
$877$	44.6022	1.50611	0.753054	+	0.657959i	$0.228580\pi$
$878$	27.5247	0.928915
$879$	3.86756	0.130449
$880$	0	0
$881$	9.17519	0.309120	0.154560	−	0.987983i	$-0.450604\pi$
$881$	9.17519	0.309120	0.154560	+	0.987983i	$0.450604\pi$
$882$	17.3588	0.584501
$883$	15.1520	0.509906	0.254953	−	0.966953i	$-0.417940\pi$
$883$	15.1520	0.509906	0.254953	+	0.966953i	$0.417940\pi$
$884$	−1.46410	−0.0492431
$885$	8.92190	0.299906
$886$	−17.1569	−0.576397
$887$	−4.03883	−0.135611	−0.0678053	−	0.997699i	$-0.521600\pi$
$887$	−4.03883	−0.135611	−0.0678053	+	0.997699i	$0.521600\pi$
$888$	−3.73205	−0.125239
$889$	1.70689	0.0572473
$890$	−27.8961	−0.935079
$891$	0	0
$892$	6.11359	0.204698
$893$	4.21068	0.140905
$894$	5.11372	0.171028
$895$	36.5154	1.22057
$896$	−1.00000	−0.0334077
$897$	15.2213	0.508225
$898$	−37.2186	−1.24200
$899$	−37.6576	−1.25595
$900$	2.20077	0.0733588
$901$	−2.07823	−0.0692360
$902$	0	0
$903$	−2.27900	−0.0758403
$904$	−1.12728	−0.0374928
$905$	8.11792	0.269849
$906$	2.52498	0.0838870
$907$	12.0131	0.398890	0.199445	−	0.979909i	$-0.436086\pi$
$907$	12.0131	0.398890	0.199445	+	0.979909i	$0.436086\pi$
$908$	5.67784	0.188426
$909$	−2.18703	−0.0725391
$910$	12.5965	0.417571
$911$	21.8314	0.723307	0.361654	−	0.932313i	$-0.382212\pi$
$911$	21.8314	0.723307	0.361654	+	0.932313i	$0.382212\pi$
$912$	0.326909	0.0108250
$913$	0	0
$914$	1.31069	0.0433539
$915$	4.38084	0.144826
$916$	1.27397	0.0420932
$917$	4.75763	0.157111
$918$	−0.461041	−0.0152166
$919$	−26.5793	−0.876769	−0.438385	−	0.898788i	$-0.644449\pi$
$919$	−26.5793	−0.876769	−0.438385	+	0.898788i	$0.644449\pi$
$920$	−15.6700	−0.516626
$921$	−8.48169	−0.279481
$922$	19.3338	0.636724
$923$	50.6818	1.66821
$924$	0	0
$925$	8.68414	0.285533
$926$	2.81631	0.0925495
$927$	29.7534	0.977231
$928$	9.90893	0.325277
$929$	46.5896	1.52856	0.764278	−	0.644887i	$-0.223096\pi$
$929$	46.5896	1.52856	0.764278	+	0.644887i	$0.223096\pi$
$930$	−2.55800	−0.0838800
$931$	−6.00000	−0.196642
$932$	−6.32512	−0.207186
$933$	5.80302	0.189982
$934$	19.0030	0.621796
$935$	0	0
$936$	17.6999	0.578541
$937$	40.0189	1.30736	0.653681	−	0.756770i	$-0.273224\pi$
$937$	40.0189	1.30736	0.653681	+	0.756770i	$0.273224\pi$
$938$	9.28891	0.303294
$939$	2.49484	0.0814161
$940$	8.66962	0.282772
$941$	10.7298	0.349782	0.174891	−	0.984588i	$-0.444043\pi$
$941$	10.7298	0.349782	0.174891	+	0.984588i	$0.444043\pi$
$942$	−2.68108	−0.0873543
$943$	51.5767	1.67957
$944$	13.2551	0.431417
$945$	3.96662	0.129034
$946$	0	0
$947$	58.1190	1.88861	0.944306	−	0.329068i	$-0.106735\pi$
$947$	58.1190	1.88861	0.944306	+	0.329068i	$0.106735\pi$
$948$	4.32536	0.140481
$949$	32.8640	1.06681
$950$	−0.760686	−0.0246799
$951$	4.09609	0.132825
$952$	−0.239314	−0.00775620
$953$	31.4693	1.01939	0.509695	−	0.860355i	$-0.329758\pi$
$953$	31.4693	1.01939	0.509695	+	0.860355i	$0.329758\pi$
$954$	25.1244	0.813431
$955$	47.8054	1.54695
$956$	−12.3145	−0.398279
$957$	0	0
$958$	34.7289	1.12204
$959$	−6.66067	−0.215084
$960$	0.673091	0.0217239
$961$	−16.5572	−0.534102
$962$	69.8433	2.25184
$963$	57.1331	1.84109
$964$	10.6345	0.342516
$965$	46.6212	1.50079
$966$	2.48799	0.0800497
$967$	28.9626	0.931373	0.465686	−	0.884950i	$-0.345807\pi$
$967$	28.9626	0.931373	0.465686	+	0.884950i	$0.345807\pi$
$968$	0	0
$969$	0.0782337	0.00251323
$970$	37.0179	1.18857
$971$	−4.45629	−0.143009	−0.0715046	−	0.997440i	$-0.522780\pi$
$971$	−4.45629	−0.143009	−0.0715046	+	0.997440i	$0.522780\pi$
$972$	8.41103	0.269784
$973$	−9.90418	−0.317513
$974$	−16.5313	−0.529697
$975$	1.52137	0.0487229
$976$	6.50854	0.208333
$977$	4.61527	0.147656	0.0738278	−	0.997271i	$-0.476478\pi$
$977$	4.61527	0.147656	0.0738278	+	0.997271i	$0.476478\pi$
$978$	−1.05996	−0.0338938
$979$	0	0
$980$	−12.3538	−0.394626
$981$	−1.33782	−0.0427134
$982$	−27.1874	−0.867584
$983$	3.49665	0.111526	0.0557630	−	0.998444i	$-0.482241\pi$
$983$	3.49665	0.111526	0.0557630	+	0.998444i	$0.482241\pi$
$984$	−2.21543	−0.0706252
$985$	24.1998	0.771071
$986$	2.37134	0.0755189
$987$	−1.37651	−0.0438147
$988$	−6.11792	−0.194637
$989$	−53.0567	−1.68710
$990$	0	0
$991$	−41.3012	−1.31198	−0.655988	−	0.754771i	$-0.727748\pi$
$991$	−41.3012	−1.31198	−0.655988	+	0.754771i	$0.727748\pi$
$992$	−3.80037	−0.120662
$993$	−11.2160	−0.355929
$994$	8.28416	0.262758
$995$	−4.40641	−0.139693
$996$	−4.47808	−0.141893
$997$	−34.1333	−1.08101	−0.540506	−	0.841340i	$-0.681767\pi$
$997$	−34.1333	−1.08101	−0.540506	+	0.841340i	$0.681767\pi$
$998$	−2.60980	−0.0826118
$999$	21.9935	0.695842

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4598.2.a.bt.1.3	yes	4
11.10	odd	2		4598.2.a.bq.1.3	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4598.2.a.bq.1.3	✓	4	11.10	odd	2
4598.2.a.bt.1.3	yes	4	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:	\( N \)	\(=\)	\( 4598 = 2 \cdot 11^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4598.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +0.326909 q^{3} +1.00000 q^{4} +2.05896 q^{5} +0.326909 q^{6} -1.00000 q^{7} +1.00000 q^{8} -2.89313 q^{9} +2.05896 q^{10} +0.326909 q^{12} -6.11792 q^{13} -1.00000 q^{14} +0.673091 q^{15} +1.00000 q^{16} +0.239314 q^{17} -2.89313 q^{18} +1.00000 q^{19} +2.05896 q^{20} -0.326909 q^{21} -7.61066 q^{23} +0.326909 q^{24} -0.760686 q^{25} -6.11792 q^{26} -1.92651 q^{27} -1.00000 q^{28} +9.90893 q^{29} +0.673091 q^{30} -3.80037 q^{31} +1.00000 q^{32} +0.239314 q^{34} -2.05896 q^{35} -2.89313 q^{36} -11.4162 q^{37} +1.00000 q^{38} -2.00000 q^{39} +2.05896 q^{40} -6.77690 q^{41} -0.326909 q^{42} +6.97136 q^{43} -5.95684 q^{45} -7.61066 q^{46} +4.21068 q^{47} +0.326909 q^{48} -6.00000 q^{49} -0.760686 q^{50} +0.0782337 q^{51} -6.11792 q^{52} -8.68414 q^{53} -1.92651 q^{54} -1.00000 q^{56} +0.326909 q^{57} +9.90893 q^{58} +13.2551 q^{59} +0.673091 q^{60} +6.50854 q^{61} -3.80037 q^{62} +2.89313 q^{63} +1.00000 q^{64} -12.5965 q^{65} -9.28891 q^{67} +0.239314 q^{68} -2.48799 q^{69} -2.05896 q^{70} -8.28416 q^{71} -2.89313 q^{72} -5.37176 q^{73} -11.4162 q^{74} -0.248675 q^{75} +1.00000 q^{76} -2.00000 q^{78} +13.2311 q^{79} +2.05896 q^{80} +8.04960 q^{81} -6.77690 q^{82} -13.6983 q^{83} -0.326909 q^{84} +0.492737 q^{85} +6.97136 q^{86} +3.23931 q^{87} -13.5486 q^{89} -5.95684 q^{90} +6.11792 q^{91} -7.61066 q^{92} -1.24237 q^{93} +4.21068 q^{94} +2.05896 q^{95} +0.326909 q^{96} +17.9789 q^{97} -6.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} - 2 q^{3} + 4 q^{4} - 2 q^{5} - 2 q^{6} - 4 q^{7} + 4 q^{8} - 2 q^{10} - 2 q^{12} - 4 q^{13} - 4 q^{14} + 6 q^{15} + 4 q^{16} - 4 q^{17} + 4 q^{19} - 2 q^{20} + 2 q^{21} - 8 q^{23} - 2 q^{24}+ \cdots - 24 q^{98}+O(q^{100}) \) 4 * q + 4 * q^2 - 2 * q^3 + 4 * q^4 - 2 * q^5 - 2 * q^6 - 4 * q^7 + 4 * q^8 - 2 * q^10 - 2 * q^12 - 4 * q^13 - 4 * q^14 + 6 * q^15 + 4 * q^16 - 4 * q^17 + 4 * q^19 - 2 * q^20 + 2 * q^21 - 8 * q^23 - 2 * q^24 - 8 * q^25 - 4 * q^26 - 14 * q^27 - 4 * q^28 + 2 * q^29 + 6 * q^30 + 4 * q^32 - 4 * q^34 + 2 * q^35 - 10 * q^37 + 4 * q^38 - 8 * q^39 - 2 * q^40 - 2 * q^41 + 2 * q^42 + 16 * q^43 - 8 * q^45 - 8 * q^46 - 2 * q^48 - 24 * q^49 - 8 * q^50 - 4 * q^52 - 6 * q^53 - 14 * q^54 - 4 * q^56 - 2 * q^57 + 2 * q^58 + 22 * q^59 + 6 * q^60 - 2 * q^61 + 4 * q^64 - 20 * q^65 - 20 * q^67 - 4 * q^68 - 2 * q^69 + 2 * q^70 - 10 * q^71 - 10 * q^74 + 2 * q^75 + 4 * q^76 - 8 * q^78 + 6 * q^79 - 2 * q^80 + 20 * q^81 - 2 * q^82 - 34 * q^83 + 2 * q^84 + 16 * q^86 + 8 * q^87 - 2 * q^89 - 8 * q^90 + 4 * q^91 - 8 * q^92 - 40 * q^93 - 2 * q^95 - 2 * q^96 - 8 * q^97 - 24 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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