LMFDB - Embedded newform 4598.2.a.bn.1.1

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:	$3$
Coefficient field:	3.3.568.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 6x - 2 $ x^3 - x^2 - 6*x - 2
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			1.1
Root			$-1.76156$ 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)

$f(q)$	$=$	\(q+1.00000 q^{2} -2.76156 q^{3} +1.00000 q^{4} +1.76156 q^{5} -2.76156 q^{6} -3.62620 q^{7} +1.00000 q^{8} +4.62620 q^{9} +1.76156 q^{10} -2.76156 q^{12} +2.62620 q^{13} -3.62620 q^{14} -4.86464 q^{15} +1.00000 q^{16} -4.62620 q^{17} +4.62620 q^{18} -1.00000 q^{19} +1.76156 q^{20} +10.0140 q^{21} +3.62620 q^{23} -2.76156 q^{24} -1.89692 q^{25} +2.62620 q^{26} -4.49084 q^{27} -3.62620 q^{28} +5.38776 q^{29} -4.86464 q^{30} -1.72928 q^{31} +1.00000 q^{32} -4.62620 q^{34} -6.38776 q^{35} +4.62620 q^{36} -2.61224 q^{37} -1.00000 q^{38} -7.25240 q^{39} +1.76156 q^{40} +5.01395 q^{41} +10.0140 q^{42} +7.25240 q^{43} +8.14931 q^{45} +3.62620 q^{46} -7.42003 q^{47} -2.76156 q^{48} +6.14931 q^{49} -1.89692 q^{50} +12.7755 q^{51} +2.62620 q^{52} -5.11704 q^{53} -4.49084 q^{54} -3.62620 q^{56} +2.76156 q^{57} +5.38776 q^{58} -14.0140 q^{59} -4.86464 q^{60} +4.59392 q^{61} -1.72928 q^{62} -16.7755 q^{63} +1.00000 q^{64} +4.62620 q^{65} -14.5048 q^{67} -4.62620 q^{68} -10.0140 q^{69} -6.38776 q^{70} -0.864641 q^{71} +4.62620 q^{72} +5.79383 q^{73} -2.61224 q^{74} +5.23844 q^{75} -1.00000 q^{76} -7.25240 q^{78} -6.38776 q^{79} +1.76156 q^{80} -1.47689 q^{81} +5.01395 q^{82} +6.38776 q^{83} +10.0140 q^{84} -8.14931 q^{85} +7.25240 q^{86} -14.8786 q^{87} +16.2663 q^{89} +8.14931 q^{90} -9.52311 q^{91} +3.62620 q^{92} +4.77551 q^{93} -7.42003 q^{94} -1.76156 q^{95} -2.76156 q^{96} +1.10308 q^{97} +6.14931 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} - 2 q^{3} + 3 q^{4} - q^{5} - 2 q^{6} - 2 q^{7} + 3 q^{8} + 5 q^{9} - q^{10} - 2 q^{12} - q^{13} - 2 q^{14} - 12 q^{15} + 3 q^{16} - 5 q^{17} + 5 q^{18} - 3 q^{19} - q^{20} + 6 q^{21}+ \cdots - 3 q^{98}+O(q^{100}) $ 3 * q + 3 * q^2 - 2 * q^3 + 3 * q^4 - q^5 - 2 * q^6 - 2 * q^7 + 3 * q^8 + 5 * q^9 - q^10 - 2 * q^12 - q^13 - 2 * q^14 - 12 * q^15 + 3 * q^16 - 5 * q^17 + 5 * q^18 - 3 * q^19 - q^20 + 6 * q^21 + 2 * q^23 - 2 * q^24 - 2 * q^25 - q^26 - 2 * q^27 - 2 * q^28 + q^29 - 12 * q^30 + 3 * q^32 - 5 * q^34 - 4 * q^35 + 5 * q^36 - 23 * q^37 - 3 * q^38 - 4 * q^39 - q^40 - 9 * q^41 + 6 * q^42 + 4 * q^43 + 3 * q^45 + 2 * q^46 - 6 * q^47 - 2 * q^48 - 3 * q^49 - 2 * q^50 + 8 * q^51 - q^52 + 5 * q^53 - 2 * q^54 - 2 * q^56 + 2 * q^57 + q^58 - 18 * q^59 - 12 * q^60 + 6 * q^61 - 20 * q^63 + 3 * q^64 + 5 * q^65 - 8 * q^67 - 5 * q^68 - 6 * q^69 - 4 * q^70 + 5 * q^72 + 10 * q^73 - 23 * q^74 + 22 * q^75 - 3 * q^76 - 4 * q^78 - 4 * q^79 - q^80 - 17 * q^81 - 9 * q^82 + 4 * q^83 + 6 * q^84 - 3 * q^85 + 4 * q^86 - 18 * q^87 + 7 * q^89 + 3 * q^90 - 16 * q^91 + 2 * q^92 - 16 * q^93 - 6 * q^94 + q^95 - 2 * q^96 + 7 * q^97 - 3 * 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
$3$	−2.76156	−1.59439	−0.797193	−	0.603725i	$-0.793683\pi$
$3$	−2.76156	−1.59439	−0.797193	+	0.603725i	$0.793683\pi$
$4$	1.00000	0.500000
$5$	1.76156	0.787792	0.393896	−	0.919155i	$-0.371127\pi$
$5$	1.76156	0.787792	0.393896	+	0.919155i	$0.371127\pi$
$6$	−2.76156	−1.12740
$7$	−3.62620	−1.37057	−0.685287	−	0.728273i	$-0.740323\pi$
$7$	−3.62620	−1.37057	−0.685287	+	0.728273i	$0.740323\pi$
$8$	1.00000	0.353553
$9$	4.62620	1.54207
$10$	1.76156	0.557053
$11$	0	0
$11$	0	0
$12$	−2.76156	−0.797193
$13$	2.62620	0.728376	0.364188	−	0.931325i	$-0.381347\pi$
$13$	2.62620	0.728376	0.364188	+	0.931325i	$0.381347\pi$
$14$	−3.62620	−0.969142
$15$	−4.86464	−1.25604
$16$	1.00000	0.250000
$17$	−4.62620	−1.12202	−0.561009	−	0.827810i	$-0.689587\pi$
$17$	−4.62620	−1.12202	−0.561009	+	0.827810i	$0.689587\pi$
$18$	4.62620	1.09041
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	1.76156	0.393896
$21$	10.0140	2.18522
$22$	0	0
$23$	3.62620	0.756115	0.378057	−	0.925782i	$-0.376592\pi$
$23$	3.62620	0.756115	0.378057	+	0.925782i	$0.376592\pi$
$24$	−2.76156	−0.563700
$25$	−1.89692	−0.379383
$26$	2.62620	0.515040
$27$	−4.49084	−0.864262
$28$	−3.62620	−0.685287
$29$	5.38776	1.00048	0.500241	−	0.865886i	$-0.333245\pi$
$29$	5.38776	1.00048	0.500241	+	0.865886i	$0.333245\pi$
$30$	−4.86464	−0.888158
$31$	−1.72928	−0.310588	−0.155294	−	0.987868i	$-0.549633\pi$
$31$	−1.72928	−0.310588	−0.155294	+	0.987868i	$0.549633\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−4.62620	−0.793386
$35$	−6.38776	−1.07973
$36$	4.62620	0.771033
$37$	−2.61224	−0.429450	−0.214725	−	0.976675i	$-0.568886\pi$
$37$	−2.61224	−0.429450	−0.214725	+	0.976675i	$0.568886\pi$
$38$	−1.00000	−0.162221
$39$	−7.25240	−1.16131
$40$	1.76156	0.278527
$41$	5.01395	0.783048	0.391524	−	0.920168i	$-0.371948\pi$
$41$	5.01395	0.783048	0.391524	+	0.920168i	$0.371948\pi$
$42$	10.0140	1.54519
$43$	7.25240	1.10598	0.552990	−	0.833188i	$-0.313487\pi$
$43$	7.25240	1.10598	0.552990	+	0.833188i	$0.313487\pi$
$44$	0	0
$45$	8.14931	1.21483
$46$	3.62620	0.534654
$47$	−7.42003	−1.08232	−0.541161	−	0.840919i	$-0.682015\pi$
$47$	−7.42003	−1.08232	−0.541161	+	0.840919i	$0.682015\pi$
$48$	−2.76156	−0.398596
$49$	6.14931	0.878473
$50$	−1.89692	−0.268264
$51$	12.7755	1.78893
$52$	2.62620	0.364188
$53$	−5.11704	−0.702879	−0.351440	−	0.936211i	$-0.614308\pi$
$53$	−5.11704	−0.702879	−0.351440	+	0.936211i	$0.614308\pi$
$54$	−4.49084	−0.611126
$55$	0	0
$56$	−3.62620	−0.484571
$57$	2.76156	0.365777
$58$	5.38776	0.707447
$59$	−14.0140	−1.82446	−0.912231	−	0.409677i	$-0.865641\pi$
$59$	−14.0140	−1.82446	−0.912231	+	0.409677i	$0.865641\pi$
$60$	−4.86464	−0.628022
$61$	4.59392	0.588192	0.294096	−	0.955776i	$-0.404982\pi$
$61$	4.59392	0.588192	0.294096	+	0.955776i	$0.404982\pi$
$62$	−1.72928	−0.219619
$63$	−16.7755	−2.11352
$64$	1.00000	0.125000
$65$	4.62620	0.573809
$66$	0	0
$67$	−14.5048	−1.77204	−0.886021	−	0.463645i	$-0.846541\pi$
$67$	−14.5048	−1.77204	−0.886021	+	0.463645i	$0.846541\pi$
$68$	−4.62620	−0.561009
$69$	−10.0140	−1.20554
$70$	−6.38776	−0.763483
$71$	−0.864641	−0.102614	−0.0513070	−	0.998683i	$-0.516339\pi$
$71$	−0.864641	−0.102614	−0.0513070	+	0.998683i	$0.516339\pi$
$72$	4.62620	0.545203
$73$	5.79383	0.678117	0.339058	−	0.940765i	$-0.389892\pi$
$73$	5.79383	0.678117	0.339058	+	0.940765i	$0.389892\pi$
$74$	−2.61224	−0.303667
$75$	5.23844	0.604883
$76$	−1.00000	−0.114708
$77$	0	0
$78$	−7.25240	−0.821172
$79$	−6.38776	−0.718679	−0.359339	−	0.933207i	$-0.616998\pi$
$79$	−6.38776	−0.718679	−0.359339	+	0.933207i	$0.616998\pi$
$80$	1.76156	0.196948
$81$	−1.47689	−0.164098
$82$	5.01395	0.553699
$83$	6.38776	0.701147	0.350574	−	0.936535i	$-0.385987\pi$
$83$	6.38776	0.701147	0.350574	+	0.936535i	$0.385987\pi$
$84$	10.0140	1.09261
$85$	−8.14931	−0.883917
$86$	7.25240	0.782046
$87$	−14.8786	−1.59515
$88$	0	0
$89$	16.2663	1.72423	0.862115	−	0.506713i	$-0.169140\pi$
$89$	16.2663	1.72423	0.862115	+	0.506713i	$0.169140\pi$
$90$	8.14931	0.859013
$91$	−9.52311	−0.998294
$92$	3.62620	0.378057
$93$	4.77551	0.495197
$94$	−7.42003	−0.765318
$95$	−1.76156	−0.180732
$96$	−2.76156	−0.281850
$97$	1.10308	0.112001	0.0560006	−	0.998431i	$-0.482165\pi$
$97$	1.10308	0.112001	0.0560006	+	0.998431i	$0.482165\pi$
$98$	6.14931	0.621174
$99$	0	0
$100$	−1.89692	−0.189692
$101$	17.3694	1.72832	0.864162	−	0.503214i	$-0.167849\pi$
$101$	17.3694	1.72832	0.864162	+	0.503214i	$0.167849\pi$
$102$	12.7755	1.26496
$103$	12.8925	1.27034	0.635170	−	0.772372i	$-0.280930\pi$
$103$	12.8925	1.27034	0.635170	+	0.772372i	$0.280930\pi$
$104$	2.62620	0.257520
$105$	17.6402	1.72150
$106$	−5.11704	−0.497011
$107$	−3.50916	−0.339243	−0.169622	−	0.985509i	$-0.554255\pi$
$107$	−3.50916	−0.339243	−0.169622	+	0.985509i	$0.554255\pi$
$108$	−4.49084	−0.432131
$109$	−16.1633	−1.54816	−0.774080	−	0.633088i	$-0.781787\pi$
$109$	−16.1633	−1.54816	−0.774080	+	0.633088i	$0.781787\pi$
$110$	0	0
$111$	7.21386	0.684710
$112$	−3.62620	−0.342644
$113$	−15.7616	−1.48272	−0.741361	−	0.671106i	$-0.765819\pi$
$113$	−15.7616	−1.48272	−0.741361	+	0.671106i	$0.765819\pi$
$114$	2.76156	0.258644
$115$	6.38776	0.595661
$116$	5.38776	0.500241
$117$	12.1493	1.12320
$118$	−14.0140	−1.29009
$119$	16.7755	1.53781
$120$	−4.86464	−0.444079
$121$	0	0
$122$	4.59392	0.415914
$123$	−13.8463	−1.24848
$124$	−1.72928	−0.155294
$125$	−12.1493	−1.08667
$126$	−16.7755	−1.49448
$127$	−16.7755	−1.48859	−0.744293	−	0.667853i	$-0.767213\pi$
$127$	−16.7755	−1.48859	−0.744293	+	0.667853i	$0.767213\pi$
$128$	1.00000	0.0883883
$129$	−20.0279	−1.76336
$130$	4.62620	0.405744
$131$	−20.9817	−1.83318	−0.916589	−	0.399831i	$-0.869069\pi$
$131$	−20.9817	−1.83318	−0.916589	+	0.399831i	$0.869069\pi$
$132$	0	0
$133$	3.62620	0.314431
$134$	−14.5048	−1.25302
$135$	−7.91087	−0.680859
$136$	−4.62620	−0.396693
$137$	−5.25240	−0.448742	−0.224371	−	0.974504i	$-0.572033\pi$
$137$	−5.25240	−0.448742	−0.224371	+	0.974504i	$0.572033\pi$
$138$	−10.0140	−0.852444
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	−6.38776	−0.539864
$141$	20.4908	1.72564
$142$	−0.864641	−0.0725591
$143$	0	0
$144$	4.62620	0.385517
$145$	9.49084	0.788171
$146$	5.79383	0.479501
$147$	−16.9817	−1.40063
$148$	−2.61224	−0.214725
$149$	−12.0848	−0.990022	−0.495011	−	0.868887i	$-0.664836\pi$
$149$	−12.0848	−0.990022	−0.495011	+	0.868887i	$0.664836\pi$
$150$	5.23844	0.427717
$151$	19.5756	1.59304	0.796520	−	0.604612i	$-0.206672\pi$
$151$	19.5756	1.59304	0.796520	+	0.604612i	$0.206672\pi$
$152$	−1.00000	−0.0811107
$153$	−21.4017	−1.73023
$154$	0	0
$155$	−3.04623	−0.244679
$156$	−7.25240	−0.580656
$157$	−17.5756	−1.40269	−0.701343	−	0.712824i	$-0.747416\pi$
$157$	−17.5756	−1.40269	−0.701343	+	0.712824i	$0.747416\pi$
$158$	−6.38776	−0.508183
$159$	14.1310	1.12066
$160$	1.76156	0.139263
$161$	−13.1493	−1.03631
$162$	−1.47689	−0.116035
$163$	−15.2524	−1.19466	−0.597330	−	0.801996i	$-0.703772\pi$
$163$	−15.2524	−1.19466	−0.597330	+	0.801996i	$0.703772\pi$
$164$	5.01395	0.391524
$165$	0	0
$166$	6.38776	0.495786
$167$	2.47689	0.191667	0.0958336	−	0.995397i	$-0.469448\pi$
$167$	2.47689	0.191667	0.0958336	+	0.995397i	$0.469448\pi$
$168$	10.0140	0.772593
$169$	−6.10308	−0.469468
$170$	−8.14931	−0.625024
$171$	−4.62620	−0.353774
$172$	7.25240	0.552990
$173$	−7.06018	−0.536776	−0.268388	−	0.963311i	$-0.586491\pi$
$173$	−7.06018	−0.536776	−0.268388	+	0.963311i	$0.586491\pi$
$174$	−14.8786	−1.12794
$175$	6.87859	0.519973
$176$	0	0
$177$	38.7003	2.90890
$178$	16.2663	1.21921
$179$	−5.44461	−0.406949	−0.203475	−	0.979080i	$-0.565223\pi$
$179$	−5.44461	−0.406949	−0.203475	+	0.979080i	$0.565223\pi$
$180$	8.14931	0.607414
$181$	3.40171	0.252847	0.126424	−	0.991976i	$-0.459650\pi$
$181$	3.40171	0.252847	0.126424	+	0.991976i	$0.459650\pi$
$182$	−9.52311	−0.705900
$183$	−12.6864	−0.937804
$184$	3.62620	0.267327
$185$	−4.60162	−0.338318
$186$	4.77551	0.350157
$187$	0	0
$188$	−7.42003	−0.541161
$189$	16.2847	1.18454
$190$	−1.76156	−0.127797
$191$	7.83237	0.566730	0.283365	−	0.959012i	$-0.408549\pi$
$191$	7.83237	0.566730	0.283365	+	0.959012i	$0.408549\pi$
$192$	−2.76156	−0.199298
$193$	7.94315	0.571760	0.285880	−	0.958265i	$-0.407714\pi$
$193$	7.94315	0.571760	0.285880	+	0.958265i	$0.407714\pi$
$194$	1.10308	0.0791968
$195$	−12.7755	−0.914873
$196$	6.14931	0.439237
$197$	−8.21386	−0.585214	−0.292607	−	0.956233i	$-0.594523\pi$
$197$	−8.21386	−0.585214	−0.292607	+	0.956233i	$0.594523\pi$
$198$	0	0
$199$	−20.0279	−1.41974	−0.709870	−	0.704332i	$-0.751246\pi$
$199$	−20.0279	−1.41974	−0.709870	+	0.704332i	$0.751246\pi$
$200$	−1.89692	−0.134132
$201$	40.0558	2.82532
$202$	17.3694	1.22211
$203$	−19.5371	−1.37123
$204$	12.7755	0.894465
$205$	8.83237	0.616879
$206$	12.8925	0.898266
$207$	16.7755	1.16598
$208$	2.62620	0.182094
$209$	0	0
$210$	17.6402	1.21729
$211$	−20.3126	−1.39838	−0.699188	−	0.714938i	$-0.746455\pi$
$211$	−20.3126	−1.39838	−0.699188	+	0.714938i	$0.746455\pi$
$212$	−5.11704	−0.351440
$213$	2.38776	0.163606
$214$	−3.50916	−0.239881
$215$	12.7755	0.871283
$216$	−4.49084	−0.305563
$217$	6.27072	0.425684
$218$	−16.1633	−1.09471
$219$	−16.0000	−1.08118
$220$	0	0
$221$	−12.1493	−0.817251
$222$	7.21386	0.484163
$223$	14.4157	0.965344	0.482672	−	0.875801i	$-0.339666\pi$
$223$	14.4157	0.965344	0.482672	+	0.875801i	$0.339666\pi$
$224$	−3.62620	−0.242286
$225$	−8.77551	−0.585034
$226$	−15.7616	−1.04844
$227$	−20.3632	−1.35155	−0.675776	−	0.737107i	$-0.736191\pi$
$227$	−20.3632	−1.35155	−0.675776	+	0.737107i	$0.736191\pi$
$228$	2.76156	0.182889
$229$	−19.6724	−1.29999	−0.649995	−	0.759938i	$-0.725229\pi$
$229$	−19.6724	−1.29999	−0.649995	+	0.759938i	$0.725229\pi$
$230$	6.38776	0.421196
$231$	0	0
$232$	5.38776	0.353723
$233$	−4.25240	−0.278584	−0.139292	−	0.990251i	$-0.544483\pi$
$233$	−4.25240	−0.278584	−0.139292	+	0.990251i	$0.544483\pi$
$234$	12.1493	0.794225
$235$	−13.0708	−0.852646
$236$	−14.0140	−0.912231
$237$	17.6402	1.14585
$238$	16.7755	1.08739
$239$	−20.8140	−1.34635	−0.673174	−	0.739484i	$-0.735070\pi$
$239$	−20.8140	−1.34635	−0.673174	+	0.739484i	$0.735070\pi$
$240$	−4.86464	−0.314011
$241$	30.6864	1.97668	0.988342	−	0.152252i	$-0.0486524\pi$
$241$	30.6864	1.97668	0.988342	+	0.152252i	$0.0486524\pi$
$242$	0	0
$243$	17.5510	1.12590
$244$	4.59392	0.294096
$245$	10.8324	0.692054
$246$	−13.8463	−0.882809
$247$	−2.62620	−0.167101
$248$	−1.72928	−0.109810
$249$	−17.6402	−1.11790
$250$	−12.1493	−0.768390
$251$	16.5693	1.04585	0.522924	−	0.852379i	$-0.324841\pi$
$251$	16.5693	1.04585	0.522924	+	0.852379i	$0.324841\pi$
$252$	−16.7755	−1.05676
$253$	0	0
$254$	−16.7755	−1.05259
$255$	22.5048	1.40930
$256$	1.00000	0.0625000
$257$	−17.1676	−1.07089	−0.535444	−	0.844571i	$-0.679856\pi$
$257$	−17.1676	−1.07089	−0.535444	+	0.844571i	$0.679856\pi$
$258$	−20.0279	−1.24688
$259$	9.47252	0.588594
$260$	4.62620	0.286905
$261$	24.9248	1.54281
$262$	−20.9817	−1.29625
$263$	−31.0741	−1.91611	−0.958057	−	0.286579i	$-0.907482\pi$
$263$	−31.0741	−1.91611	−0.958057	+	0.286579i	$0.907482\pi$
$264$	0	0
$265$	−9.01395	−0.553723
$266$	3.62620	0.222336
$267$	−44.9205	−2.74909
$268$	−14.5048	−0.886021
$269$	0.406077	0.0247590	0.0123795	−	0.999923i	$-0.496059\pi$
$269$	0.406077	0.0247590	0.0123795	+	0.999923i	$0.496059\pi$
$270$	−7.91087	−0.481440
$271$	−3.79383	−0.230459	−0.115229	−	0.993339i	$-0.536760\pi$
$271$	−3.79383	−0.230459	−0.115229	+	0.993339i	$0.536760\pi$
$272$	−4.62620	−0.280504
$273$	26.2986	1.59167
$274$	−5.25240	−0.317309
$275$	0	0
$276$	−10.0140	−0.602769
$277$	23.3126	1.40072	0.700359	−	0.713791i	$-0.253023\pi$
$277$	23.3126	1.40072	0.700359	+	0.713791i	$0.253023\pi$
$278$	4.00000	0.239904
$279$	−8.00000	−0.478947
$280$	−6.38776	−0.381741
$281$	11.6402	0.694393	0.347197	−	0.937792i	$-0.387134\pi$
$281$	11.6402	0.694393	0.347197	+	0.937792i	$0.387134\pi$
$282$	20.4908	1.22021
$283$	−5.31695	−0.316060	−0.158030	−	0.987434i	$-0.550514\pi$
$283$	−5.31695	−0.316060	−0.158030	+	0.987434i	$0.550514\pi$
$284$	−0.864641	−0.0513070
$285$	4.86464	0.288156
$286$	0	0
$287$	−18.1816	−1.07323
$288$	4.62620	0.272601
$289$	4.40171	0.258924
$290$	9.49084	0.557321
$291$	−3.04623	−0.178573
$292$	5.79383	0.339058
$293$	−2.84632	−0.166284	−0.0831419	−	0.996538i	$-0.526495\pi$
$293$	−2.84632	−0.166284	−0.0831419	+	0.996538i	$0.526495\pi$
$294$	−16.9817	−0.990392
$295$	−24.6864	−1.43730
$296$	−2.61224	−0.151834
$297$	0	0
$298$	−12.0848	−0.700051
$299$	9.52311	0.550736
$300$	5.23844	0.302442
$301$	−26.2986	−1.51583
$302$	19.5756	1.12645
$303$	−47.9667	−2.75561
$304$	−1.00000	−0.0573539
$305$	8.09246	0.463373
$306$	−21.4017	−1.22345
$307$	−17.6016	−1.00458	−0.502289	−	0.864700i	$-0.667509\pi$
$307$	−17.6016	−1.00458	−0.502289	+	0.864700i	$0.667509\pi$
$308$	0	0
$309$	−35.6035	−2.02541
$310$	−3.04623	−0.173014
$311$	13.7572	0.780099	0.390049	−	0.920794i	$-0.372458\pi$
$311$	13.7572	0.780099	0.390049	+	0.920794i	$0.372458\pi$
$312$	−7.25240	−0.410586
$313$	−20.4586	−1.15639	−0.578193	−	0.815900i	$-0.696242\pi$
$313$	−20.4586	−1.15639	−0.578193	+	0.815900i	$0.696242\pi$
$314$	−17.5756	−0.991849
$315$	−29.5510	−1.66501
$316$	−6.38776	−0.359339
$317$	−14.3632	−0.806716	−0.403358	−	0.915042i	$-0.632157\pi$
$317$	−14.3632	−0.806716	−0.403358	+	0.915042i	$0.632157\pi$
$318$	14.1310	0.792427
$319$	0	0
$320$	1.76156	0.0984740
$321$	9.69075	0.540885
$322$	−13.1493	−0.732783
$323$	4.62620	0.257409
$324$	−1.47689	−0.0820492
$325$	−4.98168	−0.276334
$326$	−15.2524	−0.844752
$327$	44.6358	2.46836
$328$	5.01395	0.276849
$329$	26.9065	1.48340
$330$	0	0
$331$	−24.1064	−1.32501	−0.662504	−	0.749058i	$-0.730506\pi$
$331$	−24.1064	−1.32501	−0.662504	+	0.749058i	$0.730506\pi$
$332$	6.38776	0.350574
$333$	−12.0848	−0.662241
$334$	2.47689	0.135529
$335$	−25.5510	−1.39600
$336$	10.0140	0.546306
$337$	6.09683	0.332115	0.166058	−	0.986116i	$-0.446896\pi$
$337$	6.09683	0.332115	0.166058	+	0.986116i	$0.446896\pi$
$338$	−6.10308	−0.331964
$339$	43.5264	2.36403
$340$	−8.14931	−0.441959
$341$	0	0
$342$	−4.62620	−0.250156
$343$	3.08476	0.166561
$344$	7.25240	0.391023
$345$	−17.6402	−0.949714
$346$	−7.06018	−0.379558
$347$	27.1633	1.45820	0.729100	−	0.684407i	$-0.239939\pi$
$347$	27.1633	1.45820	0.729100	+	0.684407i	$0.239939\pi$
$348$	−14.8786	−0.797576
$349$	18.2139	0.974966	0.487483	−	0.873133i	$-0.337915\pi$
$349$	18.2139	0.974966	0.487483	+	0.873133i	$0.337915\pi$
$350$	6.87859	0.367676
$351$	−11.7938	−0.629508
$352$	0	0
$353$	−26.5510	−1.41317	−0.706584	−	0.707629i	$-0.749765\pi$
$353$	−26.5510	−1.41317	−0.706584	+	0.707629i	$0.749765\pi$
$354$	38.7003	2.05690
$355$	−1.52311	−0.0808385
$356$	16.2663	0.862115
$357$	−46.3265	−2.45186
$358$	−5.44461	−0.287757
$359$	−22.4277	−1.18369	−0.591845	−	0.806052i	$-0.701600\pi$
$359$	−22.4277	−1.18369	−0.591845	+	0.806052i	$0.701600\pi$
$360$	8.14931	0.429506
$361$	1.00000	0.0526316
$362$	3.40171	0.178790
$363$	0	0
$364$	−9.52311	−0.499147
$365$	10.2062	0.534215
$366$	−12.6864	−0.663128
$367$	−21.4846	−1.12149	−0.560743	−	0.827990i	$-0.689484\pi$
$367$	−21.4846	−1.12149	−0.560743	+	0.827990i	$0.689484\pi$
$368$	3.62620	0.189029
$369$	23.1955	1.20751
$370$	−4.60162	−0.239227
$371$	18.5554	0.963348
$372$	4.77551	0.247599
$373$	−0.632456	−0.0327473	−0.0163737	−	0.999866i	$-0.505212\pi$
$373$	−0.632456	−0.0327473	−0.0163737	+	0.999866i	$0.505212\pi$
$374$	0	0
$375$	33.5510	1.73257
$376$	−7.42003	−0.382659
$377$	14.1493	0.728727
$378$	16.2847	0.837593
$379$	33.4725	1.71937	0.859684	−	0.510826i	$-0.170661\pi$
$379$	33.4725	1.71937	0.859684	+	0.510826i	$0.170661\pi$
$380$	−1.76156	−0.0903660
$381$	46.3265	2.37338
$382$	7.83237	0.400739
$383$	−13.0096	−0.664759	−0.332379	−	0.943146i	$-0.607851\pi$
$383$	−13.0096	−0.664759	−0.332379	+	0.943146i	$0.607851\pi$
$384$	−2.76156	−0.140925
$385$	0	0
$386$	7.94315	0.404295
$387$	33.5510	1.70549
$388$	1.10308	0.0560006
$389$	−10.8078	−0.547976	−0.273988	−	0.961733i	$-0.588343\pi$
$389$	−10.8078	−0.547976	−0.273988	+	0.961733i	$0.588343\pi$
$390$	−12.7755	−0.646913
$391$	−16.7755	−0.848374
$392$	6.14931	0.310587
$393$	57.9421	2.92279
$394$	−8.21386	−0.413808
$395$	−11.2524	−0.566169
$396$	0	0
$397$	−21.7370	−1.09095	−0.545474	−	0.838128i	$-0.683650\pi$
$397$	−21.7370	−1.09095	−0.545474	+	0.838128i	$0.683650\pi$
$398$	−20.0279	−1.00391
$399$	−10.0140	−0.501325
$400$	−1.89692	−0.0948458
$401$	15.9311	0.795560	0.397780	−	0.917481i	$-0.369781\pi$
$401$	15.9311	0.795560	0.397780	+	0.917481i	$0.369781\pi$
$402$	40.0558	1.99780
$403$	−4.54144	−0.226225
$404$	17.3694	0.864162
$405$	−2.60162	−0.129275
$406$	−19.5371	−0.969608
$407$	0	0
$408$	12.7755	0.632482
$409$	−38.4113	−1.89932	−0.949658	−	0.313288i	$-0.898569\pi$
$409$	−38.4113	−1.89932	−0.949658	+	0.313288i	$0.898569\pi$
$410$	8.83237	0.436199
$411$	14.5048	0.715469
$412$	12.8925	0.635170
$413$	50.8174	2.50056
$414$	16.7755	0.824471
$415$	11.2524	0.552358
$416$	2.62620	0.128760
$417$	−11.0462	−0.540936
$418$	0	0
$419$	5.07081	0.247725	0.123863	−	0.992299i	$-0.460472\pi$
$419$	5.07081	0.247725	0.123863	+	0.992299i	$0.460472\pi$
$420$	17.6402	0.860751
$421$	6.16327	0.300379	0.150190	−	0.988657i	$-0.452012\pi$
$421$	6.16327	0.300379	0.150190	+	0.988657i	$0.452012\pi$
$422$	−20.3126	−0.988801
$423$	−34.3265	−1.66901
$424$	−5.11704	−0.248505
$425$	8.77551	0.425675
$426$	2.38776	0.115687
$427$	−16.6585	−0.806160
$428$	−3.50916	−0.169622
$429$	0	0
$430$	12.7755	0.616090
$431$	−25.6681	−1.23639	−0.618193	−	0.786026i	$-0.712135\pi$
$431$	−25.6681	−1.23639	−0.618193	+	0.786026i	$0.712135\pi$
$432$	−4.49084	−0.216066
$433$	29.5433	1.41976	0.709881	−	0.704322i	$-0.248749\pi$
$433$	29.5433	1.41976	0.709881	+	0.704322i	$0.248749\pi$
$434$	6.27072	0.301004
$435$	−26.2095	−1.25665
$436$	−16.1633	−0.774080
$437$	−3.62620	−0.173465
$438$	−16.0000	−0.764510
$439$	11.5877	0.553049	0.276525	−	0.961007i	$-0.410817\pi$
$439$	11.5877	0.553049	0.276525	+	0.961007i	$0.410817\pi$
$440$	0	0
$441$	28.4479	1.35466
$442$	−12.1493	−0.577884
$443$	−3.97209	−0.188720	−0.0943599	−	0.995538i	$-0.530080\pi$
$443$	−3.97209	−0.188720	−0.0943599	+	0.995538i	$0.530080\pi$
$444$	7.21386	0.342355
$445$	28.6541	1.35833
$446$	14.4157	0.682601
$447$	33.3728	1.57848
$448$	−3.62620	−0.171322
$449$	7.03853	0.332169	0.166084	−	0.986112i	$-0.446888\pi$
$449$	7.03853	0.332169	0.166084	+	0.986112i	$0.446888\pi$
$450$	−8.77551	−0.413682
$451$	0	0
$452$	−15.7616	−0.741361
$453$	−54.0591	−2.53992
$454$	−20.3632	−0.955691
$455$	−16.7755	−0.786448
$456$	2.76156	0.129322
$457$	9.29862	0.434971	0.217486	−	0.976064i	$-0.430214\pi$
$457$	9.29862	0.434971	0.217486	+	0.976064i	$0.430214\pi$
$458$	−19.6724	−0.919232
$459$	20.7755	0.969718
$460$	6.38776	0.297831
$461$	−12.3309	−0.574307	−0.287154	−	0.957885i	$-0.592709\pi$
$461$	−12.3309	−0.574307	−0.287154	+	0.957885i	$0.592709\pi$
$462$	0	0
$463$	−28.1570	−1.30857	−0.654284	−	0.756249i	$-0.727030\pi$
$463$	−28.1570	−1.30857	−0.654284	+	0.756249i	$0.727030\pi$
$464$	5.38776	0.250120
$465$	8.41233	0.390113
$466$	−4.25240	−0.196988
$467$	35.4865	1.64212	0.821059	−	0.570843i	$-0.193384\pi$
$467$	35.4865	1.64212	0.821059	+	0.570843i	$0.193384\pi$
$468$	12.1493	0.561602
$469$	52.5972	2.42872
$470$	−13.0708	−0.602911
$471$	48.5360	2.23642
$472$	−14.0140	−0.645044
$473$	0	0
$474$	17.6402	0.810239
$475$	1.89692	0.0870365
$476$	16.7755	0.768904
$477$	−23.6724	−1.08389
$478$	−20.8140	−0.952012
$479$	−0.195541	−0.00893450	−0.00446725	−	0.999990i	$-0.501422\pi$
$479$	−0.195541	−0.00893450	−0.00446725	+	0.999990i	$0.501422\pi$
$480$	−4.86464	−0.222039
$481$	−6.86027	−0.312801
$482$	30.6864	1.39773
$483$	36.3126	1.65228
$484$	0	0
$485$	1.94315	0.0882337
$486$	17.5510	0.796130
$487$	−24.0279	−1.08881	−0.544404	−	0.838823i	$-0.683244\pi$
$487$	−24.0279	−1.08881	−0.544404	+	0.838823i	$0.683244\pi$
$488$	4.59392	0.207957
$489$	42.1204	1.90475
$490$	10.8324	0.489356
$491$	−4.23407	−0.191081	−0.0955405	−	0.995426i	$-0.530458\pi$
$491$	−4.23407	−0.191081	−0.0955405	+	0.995426i	$0.530458\pi$
$492$	−13.8463	−0.624240
$493$	−24.9248	−1.12256
$494$	−2.62620	−0.118158
$495$	0	0
$496$	−1.72928	−0.0776470
$497$	3.13536	0.140640
$498$	−17.6402	−0.790474
$499$	33.2278	1.48748	0.743741	−	0.668468i	$-0.233050\pi$
$499$	33.2278	1.48748	0.743741	+	0.668468i	$0.233050\pi$
$500$	−12.1493	−0.543334
$501$	−6.84006	−0.305591
$502$	16.5693	0.739526
$503$	21.1493	0.943001	0.471501	−	0.881866i	$-0.343713\pi$
$503$	21.1493	0.943001	0.471501	+	0.881866i	$0.343713\pi$
$504$	−16.7755	−0.747241
$505$	30.5972	1.36156
$506$	0	0
$507$	16.8540	0.748513
$508$	−16.7755	−0.744293
$509$	35.8078	1.58715	0.793576	−	0.608471i	$-0.208217\pi$
$509$	35.8078	1.58715	0.793576	+	0.608471i	$0.208217\pi$
$510$	22.5048	0.996529
$511$	−21.0096	−0.929409
$512$	1.00000	0.0441942
$513$	4.49084	0.198275
$514$	−17.1676	−0.757232
$515$	22.7110	1.00076
$516$	−20.0279	−0.881679
$517$	0	0
$518$	9.47252	0.416198
$519$	19.4971	0.855828
$520$	4.62620	0.202872
$521$	−7.96336	−0.348881	−0.174440	−	0.984668i	$-0.555812\pi$
$521$	−7.96336	−0.348881	−0.174440	+	0.984668i	$0.555812\pi$
$522$	24.9248	1.09093
$523$	27.3030	1.19388	0.596938	−	0.802287i	$-0.296384\pi$
$523$	27.3030	1.19388	0.596938	+	0.802287i	$0.296384\pi$
$524$	−20.9817	−0.916589
$525$	−18.9956	−0.829037
$526$	−31.0741	−1.35490
$527$	8.00000	0.348485
$528$	0	0
$529$	−9.85069	−0.428291
$530$	−9.01395	−0.391541
$531$	−64.8313	−2.81344
$532$	3.62620	0.157216
$533$	13.1676	0.570354
$534$	−44.9205	−1.94390
$535$	−6.18159	−0.267253
$536$	−14.5048	−0.626512
$537$	15.0356	0.648834
$538$	0.406077	0.0175072
$539$	0	0
$540$	−7.91087	−0.340430
$541$	31.7851	1.36655	0.683274	−	0.730162i	$-0.260555\pi$
$541$	31.7851	1.36655	0.683274	+	0.730162i	$0.260555\pi$
$542$	−3.79383	−0.162959
$543$	−9.39401	−0.403136
$544$	−4.62620	−0.198347
$545$	−28.4725	−1.21963
$546$	26.2986	1.12548
$547$	1.44461	0.0617671	0.0308835	−	0.999523i	$-0.490168\pi$
$547$	1.44461	0.0617671	0.0308835	+	0.999523i	$0.490168\pi$
$548$	−5.25240	−0.224371
$549$	21.2524	0.907030
$550$	0	0
$551$	−5.38776	−0.229526
$552$	−10.0140	−0.426222
$553$	23.1633	0.985002
$554$	23.3126	0.990457
$555$	12.7076	0.539409
$556$	4.00000	0.169638
$557$	26.3632	1.11704	0.558522	−	0.829490i	$-0.311369\pi$
$557$	26.3632	1.11704	0.558522	+	0.829490i	$0.311369\pi$
$558$	−8.00000	−0.338667
$559$	19.0462	0.805570
$560$	−6.38776	−0.269932
$561$	0	0
$562$	11.6402	0.491010
$563$	−11.7711	−0.496094	−0.248047	−	0.968748i	$-0.579789\pi$
$563$	−11.7711	−0.496094	−0.248047	+	0.968748i	$0.579789\pi$
$564$	20.4908	0.862820
$565$	−27.7649	−1.16808
$566$	−5.31695	−0.223488
$567$	5.35548	0.224909
$568$	−0.864641	−0.0362795
$569$	−21.2524	−0.890947	−0.445473	−	0.895295i	$-0.646965\pi$
$569$	−21.2524	−0.890947	−0.445473	+	0.895295i	$0.646965\pi$
$570$	4.86464	0.203757
$571$	4.14494	0.173460	0.0867302	−	0.996232i	$-0.472358\pi$
$571$	4.14494	0.173460	0.0867302	+	0.996232i	$0.472358\pi$
$572$	0	0
$573$	−21.6295	−0.903586
$574$	−18.1816	−0.758885
$575$	−6.87859	−0.286857
$576$	4.62620	0.192758
$577$	0.690749	0.0287563	0.0143781	−	0.999897i	$-0.495423\pi$
$577$	0.690749	0.0287563	0.0143781	+	0.999897i	$0.495423\pi$
$578$	4.40171	0.183087
$579$	−21.9354	−0.911606
$580$	9.49084	0.394086
$581$	−23.1633	−0.960974
$582$	−3.04623	−0.126270
$583$	0	0
$584$	5.79383	0.239750
$585$	21.4017	0.884852
$586$	−2.84632	−0.117580
$587$	−1.30488	−0.0538583	−0.0269291	−	0.999637i	$-0.508573\pi$
$587$	−1.30488	−0.0538583	−0.0269291	+	0.999637i	$0.508573\pi$
$588$	−16.9817	−0.700313
$589$	1.72928	0.0712538
$590$	−24.6864	−1.01632
$591$	22.6831	0.933056
$592$	−2.61224	−0.107363
$593$	−21.6743	−0.890057	−0.445029	−	0.895516i	$-0.646807\pi$
$593$	−21.6743	−0.890057	−0.445029	+	0.895516i	$0.646807\pi$
$594$	0	0
$595$	29.5510	1.21147
$596$	−12.0848	−0.495011
$597$	55.3082	2.26361
$598$	9.52311	0.389429
$599$	18.6339	0.761360	0.380680	−	0.924707i	$-0.375690\pi$
$599$	18.6339	0.761360	0.380680	+	0.924707i	$0.375690\pi$
$600$	5.23844	0.213859
$601$	37.1310	1.51460	0.757302	−	0.653064i	$-0.226517\pi$
$601$	37.1310	1.51460	0.757302	+	0.653064i	$0.226517\pi$
$602$	−26.2986	−1.07185
$603$	−67.1020	−2.73261
$604$	19.5756	0.796520
$605$	0	0
$606$	−47.9667	−1.94851
$607$	14.7509	0.598722	0.299361	−	0.954140i	$-0.403227\pi$
$607$	14.7509	0.598722	0.299361	+	0.954140i	$0.403227\pi$
$608$	−1.00000	−0.0405554
$609$	53.9527	2.18627
$610$	8.09246	0.327654
$611$	−19.4865	−0.788338
$612$	−21.4017	−0.865113
$613$	0.238443	0.00963061	0.00481531	−	0.999988i	$-0.498467\pi$
$613$	0.238443	0.00963061	0.00481531	+	0.999988i	$0.498467\pi$
$614$	−17.6016	−0.710344
$615$	−24.3911	−0.983543
$616$	0	0
$617$	−36.8217	−1.48239	−0.741194	−	0.671291i	$-0.765740\pi$
$617$	−36.8217	−1.48239	−0.741194	+	0.671291i	$0.765740\pi$
$618$	−35.6035	−1.43218
$619$	5.75719	0.231401	0.115700	−	0.993284i	$-0.463089\pi$
$619$	5.75719	0.231401	0.115700	+	0.993284i	$0.463089\pi$
$620$	−3.04623	−0.122339
$621$	−16.2847	−0.653481
$622$	13.7572	0.551613
$623$	−58.9850	−2.36318
$624$	−7.25240	−0.290328
$625$	−11.9171	−0.476685
$626$	−20.4586	−0.817689
$627$	0	0
$628$	−17.5756	−0.701343
$629$	12.0848	0.481851
$630$	−29.5510	−1.17734
$631$	23.4479	0.933448	0.466724	−	0.884403i	$-0.345434\pi$
$631$	23.4479	0.933448	0.466724	+	0.884403i	$0.345434\pi$
$632$	−6.38776	−0.254091
$633$	56.0943	2.22955
$634$	−14.3632	−0.570435
$635$	−29.5510	−1.17270
$636$	14.1310	0.560330
$637$	16.1493	0.639859
$638$	0	0
$639$	−4.00000	−0.158238
$640$	1.76156	0.0696317
$641$	−7.45419	−0.294423	−0.147211	−	0.989105i	$-0.547030\pi$
$641$	−7.45419	−0.294423	−0.147211	+	0.989105i	$0.547030\pi$
$642$	9.69075	0.382463
$643$	38.7110	1.52661	0.763306	−	0.646038i	$-0.223575\pi$
$643$	38.7110	1.52661	0.763306	+	0.646038i	$0.223575\pi$
$644$	−13.1493	−0.518155
$645$	−35.2803	−1.38916
$646$	4.62620	0.182015
$647$	3.42003	0.134455	0.0672276	−	0.997738i	$-0.478585\pi$
$647$	3.42003	0.134455	0.0672276	+	0.997738i	$0.478585\pi$
$648$	−1.47689	−0.0580175
$649$	0	0
$650$	−4.98168	−0.195397
$651$	−17.3169	−0.678705
$652$	−15.2524	−0.597330
$653$	32.4556	1.27009	0.635044	−	0.772476i	$-0.280982\pi$
$653$	32.4556	1.27009	0.635044	+	0.772476i	$0.280982\pi$
$654$	44.6358	1.74540
$655$	−36.9604	−1.44416
$656$	5.01395	0.195762
$657$	26.8034	1.04570
$658$	26.9065	1.04892
$659$	34.4277	1.34111	0.670557	−	0.741858i	$-0.266055\pi$
$659$	34.4277	1.34111	0.670557	+	0.741858i	$0.266055\pi$
$660$	0	0
$661$	33.4436	1.30080	0.650402	−	0.759590i	$-0.274600\pi$
$661$	33.4436	1.30080	0.650402	+	0.759590i	$0.274600\pi$
$662$	−24.1064	−0.936922
$663$	33.5510	1.30301
$664$	6.38776	0.247893
$665$	6.38776	0.247707
$666$	−12.0848	−0.468275
$667$	19.5371	0.756478
$668$	2.47689	0.0958336
$669$	−39.8097	−1.53913
$670$	−25.5510	−0.987122
$671$	0	0
$672$	10.0140	0.386297
$673$	−9.54769	−0.368037	−0.184018	−	0.982923i	$-0.558911\pi$
$673$	−9.54769	−0.368037	−0.184018	+	0.982923i	$0.558911\pi$
$674$	6.09683	0.234841
$675$	8.51875	0.327887
$676$	−6.10308	−0.234734
$677$	−32.1912	−1.23721	−0.618604	−	0.785703i	$-0.712301\pi$
$677$	−32.1912	−1.23721	−0.618604	+	0.785703i	$0.712301\pi$
$678$	43.5264	1.67162
$679$	−4.00000	−0.153506
$680$	−8.14931	−0.312512
$681$	56.2341	2.15489
$682$	0	0
$683$	−43.1020	−1.64925	−0.824627	−	0.565677i	$-0.808615\pi$
$683$	−43.1020	−1.64925	−0.824627	+	0.565677i	$0.808615\pi$
$684$	−4.62620	−0.176887
$685$	−9.25240	−0.353516
$686$	3.08476	0.117777
$687$	54.3265	2.07269
$688$	7.25240	0.276495
$689$	−13.4384	−0.511960
$690$	−17.6402	−0.671549
$691$	22.7509	0.865486	0.432743	−	0.901517i	$-0.357546\pi$
$691$	22.7509	0.865486	0.432743	+	0.901517i	$0.357546\pi$
$692$	−7.06018	−0.268388
$693$	0	0
$694$	27.1633	1.03110
$695$	7.04623	0.267279
$696$	−14.8786	−0.563972
$697$	−23.1955	−0.878594
$698$	18.2139	0.689405
$699$	11.7432	0.444170
$700$	6.87859	0.259986
$701$	−30.3063	−1.14465	−0.572327	−	0.820026i	$-0.693959\pi$
$701$	−30.3063	−1.14465	−0.572327	+	0.820026i	$0.693959\pi$
$702$	−11.7938	−0.445130
$703$	2.61224	0.0985227
$704$	0	0
$705$	36.0958	1.35945
$706$	−26.5510	−0.999261
$707$	−62.9850	−2.36879
$708$	38.7003	1.45445
$709$	0.655146	0.0246045	0.0123023	−	0.999924i	$-0.496084\pi$
$709$	0.655146	0.0246045	0.0123023	+	0.999924i	$0.496084\pi$
$710$	−1.52311	−0.0571615
$711$	−29.5510	−1.10825
$712$	16.2663	0.609607
$713$	−6.27072	−0.234840
$714$	−46.3265	−1.73373
$715$	0	0
$716$	−5.44461	−0.203475
$717$	57.4792	2.14660
$718$	−22.4277	−0.836995
$719$	−20.0558	−0.747956	−0.373978	−	0.927438i	$-0.622006\pi$
$719$	−20.0558	−0.747956	−0.373978	+	0.927438i	$0.622006\pi$
$720$	8.14931	0.303707
$721$	−46.7509	−1.74110
$722$	1.00000	0.0372161
$723$	−84.7422	−3.15160
$724$	3.40171	0.126424
$725$	−10.2201	−0.379566
$726$	0	0
$727$	−30.0539	−1.11464	−0.557319	−	0.830298i	$-0.688170\pi$
$727$	−30.0539	−1.11464	−0.557319	+	0.830298i	$0.688170\pi$
$728$	−9.52311	−0.352950
$729$	−44.0375	−1.63102
$730$	10.2062	0.377747
$731$	−33.5510	−1.24093
$732$	−12.6864	−0.468902
$733$	−13.2726	−0.490235	−0.245117	−	0.969493i	$-0.578827\pi$
$733$	−13.2726	−0.490235	−0.245117	+	0.969493i	$0.578827\pi$
$734$	−21.4846	−0.793010
$735$	−29.9142	−1.10340
$736$	3.62620	0.133663
$737$	0	0
$738$	23.1955	0.853840
$739$	6.00333	0.220836	0.110418	−	0.993885i	$-0.464781\pi$
$739$	6.00333	0.220836	0.110418	+	0.993885i	$0.464781\pi$
$740$	−4.60162	−0.169159
$741$	7.25240	0.266423
$742$	18.5554	0.681190
$743$	39.4865	1.44862	0.724309	−	0.689475i	$-0.242159\pi$
$743$	39.4865	1.44862	0.724309	+	0.689475i	$0.242159\pi$
$744$	4.77551	0.175079
$745$	−21.2880	−0.779932
$746$	−0.632456	−0.0231558
$747$	29.5510	1.08122
$748$	0	0
$749$	12.7249	0.464958
$750$	33.5510	1.22511
$751$	0.904612	0.0330098	0.0165049	−	0.999864i	$-0.494746\pi$
$751$	0.904612	0.0330098	0.0165049	+	0.999864i	$0.494746\pi$
$752$	−7.42003	−0.270581
$753$	−45.7572	−1.66748
$754$	14.1493	0.515288
$755$	34.4835	1.25498
$756$	16.2847	0.592268
$757$	−30.5004	−1.10856	−0.554278	−	0.832331i	$-0.687006\pi$
$757$	−30.5004	−1.10856	−0.554278	+	0.832331i	$0.687006\pi$
$758$	33.4725	1.21578
$759$	0	0
$760$	−1.76156	−0.0638984
$761$	7.23407	0.262235	0.131117	−	0.991367i	$-0.458143\pi$
$761$	7.23407	0.262235	0.131117	+	0.991367i	$0.458143\pi$
$762$	46.3265	1.67823
$763$	58.6112	2.12187
$764$	7.83237	0.283365
$765$	−37.7003	−1.36306
$766$	−13.0096	−0.470055
$767$	−36.8034	−1.32889
$768$	−2.76156	−0.0996491
$769$	4.83048	0.174191	0.0870957	−	0.996200i	$-0.472241\pi$
$769$	4.83048	0.174191	0.0870957	+	0.996200i	$0.472241\pi$
$770$	0	0
$771$	47.4094	1.70741
$772$	7.94315	0.285880
$773$	24.9186	0.896259	0.448129	−	0.893969i	$-0.352090\pi$
$773$	24.9186	0.896259	0.448129	+	0.893969i	$0.352090\pi$
$774$	33.5510	1.20597
$775$	3.28030	0.117832
$776$	1.10308	0.0395984
$777$	−26.1589	−0.938445
$778$	−10.8078	−0.387478
$779$	−5.01395	−0.179644
$780$	−12.7755	−0.457437
$781$	0	0
$782$	−16.7755	−0.599891
$783$	−24.1955	−0.864678
$784$	6.14931	0.219618
$785$	−30.9604	−1.10502
$786$	57.9421	2.06673
$787$	53.3222	1.90073	0.950365	−	0.311137i	$-0.100710\pi$
$787$	53.3222	1.90073	0.950365	+	0.311137i	$0.100710\pi$
$788$	−8.21386	−0.292607
$789$	85.8130	3.05502
$790$	−11.2524	−0.400342
$791$	57.1545	2.03218
$792$	0	0
$793$	12.0646	0.428425
$794$	−21.7370	−0.771416
$795$	24.8925	0.882848
$796$	−20.0279	−0.709870
$797$	−10.0558	−0.356195	−0.178098	−	0.984013i	$-0.556994\pi$
$797$	−10.0558	−0.356195	−0.178098	+	0.984013i	$0.556994\pi$
$798$	−10.0140	−0.354490
$799$	34.3265	1.21439
$800$	−1.89692	−0.0670661
$801$	75.2514	2.65888
$802$	15.9311	0.562546
$803$	0	0
$804$	40.0558	1.41266
$805$	−23.1633	−0.816398
$806$	−4.54144	−0.159965
$807$	−1.12141	−0.0394754
$808$	17.3694	0.611055
$809$	−25.3188	−0.890163	−0.445081	−	0.895490i	$-0.646825\pi$
$809$	−25.3188	−0.890163	−0.445081	+	0.895490i	$0.646825\pi$
$810$	−2.60162	−0.0914116
$811$	7.50916	0.263682	0.131841	−	0.991271i	$-0.457911\pi$
$811$	7.50916	0.263682	0.131841	+	0.991271i	$0.457911\pi$
$812$	−19.5371	−0.685617
$813$	10.4769	0.367440
$814$	0	0
$815$	−26.8680	−0.941144
$816$	12.7755	0.447232
$817$	−7.25240	−0.253729
$818$	−38.4113	−1.34302
$819$	−44.0558	−1.53943
$820$	8.83237	0.308440
$821$	−20.7230	−0.723239	−0.361619	−	0.932326i	$-0.617776\pi$
$821$	−20.7230	−0.723239	−0.361619	+	0.932326i	$0.617776\pi$
$822$	14.5048	0.505913
$823$	9.95273	0.346930	0.173465	−	0.984840i	$-0.444504\pi$
$823$	9.95273	0.346930	0.173465	+	0.984840i	$0.444504\pi$
$824$	12.8925	0.449133
$825$	0	0
$826$	50.8174	1.76816
$827$	45.4007	1.57874	0.789368	−	0.613920i	$-0.210408\pi$
$827$	45.4007	1.57874	0.789368	+	0.613920i	$0.210408\pi$
$828$	16.7755	0.582989
$829$	−19.2081	−0.667123	−0.333562	−	0.942728i	$-0.608251\pi$
$829$	−19.2081	−0.667123	−0.333562	+	0.942728i	$0.608251\pi$
$830$	11.2524	0.390576
$831$	−64.3790	−2.23328
$832$	2.62620	0.0910470
$833$	−28.4479	−0.985663
$834$	−11.0462	−0.382500
$835$	4.36318	0.150994
$836$	0	0
$837$	7.76593	0.268430
$838$	5.07081	0.175168
$839$	−2.59392	−0.0895522	−0.0447761	−	0.998997i	$-0.514257\pi$
$839$	−2.59392	−0.0895522	−0.0447761	+	0.998997i	$0.514257\pi$
$840$	17.6402	0.608643
$841$	0.0279066	0.000962298 0
$842$	6.16327	0.212400
$843$	−32.1449	−1.10713
$844$	−20.3126	−0.699188
$845$	−10.7509	−0.369843
$846$	−34.3265	−1.18017
$847$	0	0
$848$	−5.11704	−0.175720
$849$	14.6831	0.503921
$850$	8.77551	0.300998
$851$	−9.47252	−0.324714
$852$	2.38776	0.0818031
$853$	−43.3372	−1.48384	−0.741918	−	0.670491i	$-0.766084\pi$
$853$	−43.3372	−1.48384	−0.741918	+	0.670491i	$0.766084\pi$
$854$	−16.6585	−0.570041
$855$	−8.14931	−0.278701
$856$	−3.50916	−0.119941
$857$	5.22449	0.178465	0.0892326	−	0.996011i	$-0.471559\pi$
$857$	5.22449	0.178465	0.0892326	+	0.996011i	$0.471559\pi$
$858$	0	0
$859$	15.8830	0.541920	0.270960	−	0.962591i	$-0.412659\pi$
$859$	15.8830	0.541920	0.270960	+	0.962591i	$0.412659\pi$
$860$	12.7755	0.435641
$861$	50.2095	1.71113
$862$	−25.6681	−0.874258
$863$	42.4556	1.44521	0.722603	−	0.691263i	$-0.242945\pi$
$863$	42.4556	1.44521	0.722603	+	0.691263i	$0.242945\pi$
$864$	−4.49084	−0.152781
$865$	−12.4369	−0.422868
$866$	29.5433	1.00392
$867$	−12.1556	−0.412825
$868$	6.27072	0.212842
$869$	0	0
$870$	−26.2095	−0.888585
$871$	−38.0925	−1.29071
$872$	−16.1633	−0.547357
$873$	5.10308	0.172713
$874$	−3.62620	−0.122658
$875$	44.0558	1.48936
$876$	−16.0000	−0.540590
$877$	40.7957	1.37757	0.688787	−	0.724964i	$-0.258144\pi$
$877$	40.7957	1.37757	0.688787	+	0.724964i	$0.258144\pi$
$878$	11.5877	0.391065
$879$	7.86027	0.265120
$880$	0	0
$881$	3.85258	0.129797	0.0648983	−	0.997892i	$-0.479328\pi$
$881$	3.85258	0.129797	0.0648983	+	0.997892i	$0.479328\pi$
$882$	28.4479	0.957892
$883$	−32.4523	−1.09211	−0.546054	−	0.837750i	$-0.683870\pi$
$883$	−32.4523	−1.09211	−0.546054	+	0.837750i	$0.683870\pi$
$884$	−12.1493	−0.408626
$885$	68.1729	2.29161
$886$	−3.97209	−0.133445
$887$	32.8155	1.10184	0.550918	−	0.834559i	$-0.314278\pi$
$887$	32.8155	1.10184	0.550918	+	0.834559i	$0.314278\pi$
$888$	7.21386	0.242081
$889$	60.8313	2.04022
$890$	28.6541	0.960488
$891$	0	0
$892$	14.4157	0.482672
$893$	7.42003	0.248302
$894$	33.3728	1.11615
$895$	−9.59099	−0.320592
$896$	−3.62620	−0.121143
$897$	−26.2986	−0.878086
$898$	7.03853	0.234879
$899$	−9.31695	−0.310738
$900$	−8.77551	−0.292517
$901$	23.6724	0.788643
$902$	0	0
$903$	72.6252	2.41681
$904$	−15.7616	−0.524222
$905$	5.99230	0.199191
$906$	−54.0591	−1.79599
$907$	−12.9032	−0.428443	−0.214221	−	0.976785i	$-0.568721\pi$
$907$	−12.9032	−0.428443	−0.214221	+	0.976785i	$0.568721\pi$
$908$	−20.3632	−0.675776
$909$	80.3544	2.66519
$910$	−16.7755	−0.556103
$911$	39.3973	1.30529	0.652646	−	0.757663i	$-0.273659\pi$
$911$	39.3973	1.30529	0.652646	+	0.757663i	$0.273659\pi$
$912$	2.76156	0.0914443
$913$	0	0
$914$	9.29862	0.307571
$915$	−22.3478	−0.738795
$916$	−19.6724	−0.649995
$917$	76.0837	2.51251
$918$	20.7755	0.685694
$919$	−6.44898	−0.212732	−0.106366	−	0.994327i	$-0.533922\pi$
$919$	−6.44898	−0.212732	−0.106366	+	0.994327i	$0.533922\pi$
$920$	6.38776	0.210598
$921$	48.6079	1.60168
$922$	−12.3309	−0.406097
$923$	−2.27072	−0.0747416
$924$	0	0
$925$	4.95521	0.162926
$926$	−28.1570	−0.925297
$927$	59.6435	1.95895
$928$	5.38776	0.176862
$929$	40.1666	1.31782	0.658912	−	0.752220i	$-0.271017\pi$
$929$	40.1666	1.31782	0.658912	+	0.752220i	$0.271017\pi$
$930$	8.41233	0.275851
$931$	−6.14931	−0.201536
$932$	−4.25240	−0.139292
$933$	−37.9913	−1.24378
$934$	35.4865	1.16115
$935$	0	0
$936$	12.1493	0.397113
$937$	−46.9634	−1.53423	−0.767113	−	0.641512i	$-0.778307\pi$
$937$	−46.9634	−1.53423	−0.767113	+	0.641512i	$0.778307\pi$
$938$	52.5972	1.71736
$939$	56.4975	1.84373
$940$	−13.0708	−0.426323
$941$	24.3063	0.792363	0.396182	−	0.918172i	$-0.370335\pi$
$941$	24.3063	0.792363	0.396182	+	0.918172i	$0.370335\pi$
$942$	48.5360	1.58139
$943$	18.1816	0.592074
$944$	−14.0140	−0.456115
$945$	28.6864	0.933168
$946$	0	0
$947$	−46.0192	−1.49542	−0.747711	−	0.664024i	$-0.768847\pi$
$947$	−46.0192	−1.49542	−0.747711	+	0.664024i	$0.768847\pi$
$948$	17.6402	0.572925
$949$	15.2158	0.493924
$950$	1.89692	0.0615441
$951$	39.6647	1.28622
$952$	16.7755	0.543697
$953$	52.5562	1.70246	0.851232	−	0.524790i	$-0.175856\pi$
$953$	52.5562	1.70246	0.851232	+	0.524790i	$0.175856\pi$
$954$	−23.6724	−0.766423
$955$	13.7972	0.446466
$956$	−20.8140	−0.673174
$957$	0	0
$958$	−0.195541	−0.00631765
$959$	19.0462	0.615035
$960$	−4.86464	−0.157006
$961$	−28.0096	−0.903535
$962$	−6.86027	−0.221184
$963$	−16.2341	−0.523136
$964$	30.6864	0.988342
$965$	13.9923	0.450428
$966$	36.3126	1.16834
$967$	2.87859	0.0925693	0.0462847	−	0.998928i	$-0.485262\pi$
$967$	2.87859	0.0925693	0.0462847	+	0.998928i	$0.485262\pi$
$968$	0	0
$969$	−12.7755	−0.410409
$970$	1.94315	0.0623906
$971$	−6.81737	−0.218780	−0.109390	−	0.993999i	$-0.534890\pi$
$971$	−6.81737	−0.218780	−0.109390	+	0.993999i	$0.534890\pi$
$972$	17.5510	0.562949
$973$	−14.5048	−0.465002
$974$	−24.0279	−0.769904
$975$	13.7572	0.440583
$976$	4.59392	0.147048
$977$	44.8969	1.43638	0.718190	−	0.695847i	$-0.244971\pi$
$977$	44.8969	1.43638	0.718190	+	0.695847i	$0.244971\pi$
$978$	42.1204	1.34686
$979$	0	0
$980$	10.8324	0.346027
$981$	−74.7745	−2.38737
$982$	−4.23407	−0.135115
$983$	35.7818	1.14126	0.570631	−	0.821207i	$-0.306699\pi$
$983$	35.7818	1.14126	0.570631	+	0.821207i	$0.306699\pi$
$984$	−13.8463	−0.441405
$985$	−14.4692	−0.461027
$986$	−24.9248	−0.793768
$987$	−74.3038	−2.36512
$988$	−2.62620	−0.0835505
$989$	26.2986	0.836248
$990$	0	0
$991$	−17.2278	−0.547260	−0.273630	−	0.961835i	$-0.588224\pi$
$991$	−17.2278	−0.547260	−0.273630	+	0.961835i	$0.588224\pi$
$992$	−1.72928	−0.0549048
$993$	66.5712	2.11257
$994$	3.13536	0.0994476
$995$	−35.2803	−1.11846
$996$	−17.6402	−0.558950
$997$	−3.19221	−0.101098	−0.0505492	−	0.998722i	$-0.516097\pi$
$997$	−3.19221	−0.101098	−0.0505492	+	0.998722i	$0.516097\pi$
$998$	33.2278	1.05181
$999$	11.7312	0.371158

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4598.2.a.bn.1.1	yes	3
11.10	odd	2		4598.2.a.bk.1.1	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4598.2.a.bk.1.1	✓	3	11.10	odd	2
4598.2.a.bn.1.1	yes	3	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.76156 q^{3} +1.00000 q^{4} +1.76156 q^{5} -2.76156 q^{6} -3.62620 q^{7} +1.00000 q^{8} +4.62620 q^{9} +1.76156 q^{10} -2.76156 q^{12} +2.62620 q^{13} -3.62620 q^{14} -4.86464 q^{15} +1.00000 q^{16} -4.62620 q^{17} +4.62620 q^{18} -1.00000 q^{19} +1.76156 q^{20} +10.0140 q^{21} +3.62620 q^{23} -2.76156 q^{24} -1.89692 q^{25} +2.62620 q^{26} -4.49084 q^{27} -3.62620 q^{28} +5.38776 q^{29} -4.86464 q^{30} -1.72928 q^{31} +1.00000 q^{32} -4.62620 q^{34} -6.38776 q^{35} +4.62620 q^{36} -2.61224 q^{37} -1.00000 q^{38} -7.25240 q^{39} +1.76156 q^{40} +5.01395 q^{41} +10.0140 q^{42} +7.25240 q^{43} +8.14931 q^{45} +3.62620 q^{46} -7.42003 q^{47} -2.76156 q^{48} +6.14931 q^{49} -1.89692 q^{50} +12.7755 q^{51} +2.62620 q^{52} -5.11704 q^{53} -4.49084 q^{54} -3.62620 q^{56} +2.76156 q^{57} +5.38776 q^{58} -14.0140 q^{59} -4.86464 q^{60} +4.59392 q^{61} -1.72928 q^{62} -16.7755 q^{63} +1.00000 q^{64} +4.62620 q^{65} -14.5048 q^{67} -4.62620 q^{68} -10.0140 q^{69} -6.38776 q^{70} -0.864641 q^{71} +4.62620 q^{72} +5.79383 q^{73} -2.61224 q^{74} +5.23844 q^{75} -1.00000 q^{76} -7.25240 q^{78} -6.38776 q^{79} +1.76156 q^{80} -1.47689 q^{81} +5.01395 q^{82} +6.38776 q^{83} +10.0140 q^{84} -8.14931 q^{85} +7.25240 q^{86} -14.8786 q^{87} +16.2663 q^{89} +8.14931 q^{90} -9.52311 q^{91} +3.62620 q^{92} +4.77551 q^{93} -7.42003 q^{94} -1.76156 q^{95} -2.76156 q^{96} +1.10308 q^{97} +6.14931 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} - 2 q^{3} + 3 q^{4} - q^{5} - 2 q^{6} - 2 q^{7} + 3 q^{8} + 5 q^{9} - q^{10} - 2 q^{12} - q^{13} - 2 q^{14} - 12 q^{15} + 3 q^{16} - 5 q^{17} + 5 q^{18} - 3 q^{19} - q^{20} + 6 q^{21}+ \cdots - 3 q^{98}+O(q^{100}) \) 3 * q + 3 * q^2 - 2 * q^3 + 3 * q^4 - q^5 - 2 * q^6 - 2 * q^7 + 3 * q^8 + 5 * q^9 - q^10 - 2 * q^12 - q^13 - 2 * q^14 - 12 * q^15 + 3 * q^16 - 5 * q^17 + 5 * q^18 - 3 * q^19 - q^20 + 6 * q^21 + 2 * q^23 - 2 * q^24 - 2 * q^25 - q^26 - 2 * q^27 - 2 * q^28 + q^29 - 12 * q^30 + 3 * q^32 - 5 * q^34 - 4 * q^35 + 5 * q^36 - 23 * q^37 - 3 * q^38 - 4 * q^39 - q^40 - 9 * q^41 + 6 * q^42 + 4 * q^43 + 3 * q^45 + 2 * q^46 - 6 * q^47 - 2 * q^48 - 3 * q^49 - 2 * q^50 + 8 * q^51 - q^52 + 5 * q^53 - 2 * q^54 - 2 * q^56 + 2 * q^57 + q^58 - 18 * q^59 - 12 * q^60 + 6 * q^61 - 20 * q^63 + 3 * q^64 + 5 * q^65 - 8 * q^67 - 5 * q^68 - 6 * q^69 - 4 * q^70 + 5 * q^72 + 10 * q^73 - 23 * q^74 + 22 * q^75 - 3 * q^76 - 4 * q^78 - 4 * q^79 - q^80 - 17 * q^81 - 9 * q^82 + 4 * q^83 + 6 * q^84 - 3 * q^85 + 4 * q^86 - 18 * q^87 + 7 * q^89 + 3 * q^90 - 16 * q^91 + 2 * q^92 - 16 * q^93 - 6 * q^94 + q^95 - 2 * q^96 + 7 * q^97 - 3 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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