LMFDB - Embedded newform 4598.2.a.bo.1.2

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:	$0$
Dimension:	$3$
Coefficient field:	3.3.621.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 3 $ x^3 - 6*x - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 418)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.523976$ 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)

$f(q)$	$=$	\(q+1.00000 q^{2} -0.523976 q^{3} +1.00000 q^{4} +2.72545 q^{5} -0.523976 q^{6} +4.67750 q^{7} +1.00000 q^{8} -2.72545 q^{9} +2.72545 q^{10} -0.523976 q^{12} +2.67750 q^{13} +4.67750 q^{14} -1.42807 q^{15} +1.00000 q^{16} -0.201472 q^{17} -2.72545 q^{18} +1.00000 q^{19} +2.72545 q^{20} -2.45090 q^{21} -1.79853 q^{23} -0.523976 q^{24} +2.42807 q^{25} +2.67750 q^{26} +3.00000 q^{27} +4.67750 q^{28} +4.92692 q^{29} -1.42807 q^{30} -2.57193 q^{31} +1.00000 q^{32} -0.201472 q^{34} +12.7483 q^{35} -2.72545 q^{36} +4.40294 q^{37} +1.00000 q^{38} -1.40294 q^{39} +2.72545 q^{40} -4.97487 q^{41} -2.45090 q^{42} +11.7734 q^{43} -7.42807 q^{45} -1.79853 q^{46} -12.4989 q^{47} -0.523976 q^{48} +14.8790 q^{49} +2.42807 q^{50} +0.105567 q^{51} +2.67750 q^{52} -4.10557 q^{53} +3.00000 q^{54} +4.67750 q^{56} -0.523976 q^{57} +4.92692 q^{58} -5.24943 q^{59} -1.42807 q^{60} -1.59706 q^{61} -2.57193 q^{62} -12.7483 q^{63} +1.00000 q^{64} +7.29738 q^{65} +9.08044 q^{67} -0.201472 q^{68} +0.942386 q^{69} +12.7483 q^{70} +12.8214 q^{71} -2.72545 q^{72} -2.20147 q^{73} +4.40294 q^{74} -1.27225 q^{75} +1.00000 q^{76} -1.40294 q^{78} -11.8538 q^{79} +2.72545 q^{80} +6.60442 q^{81} -4.97487 q^{82} +13.6775 q^{83} -2.45090 q^{84} -0.549103 q^{85} +11.7734 q^{86} -2.58159 q^{87} -5.45090 q^{89} -7.42807 q^{90} +12.5240 q^{91} -1.79853 q^{92} +1.34763 q^{93} -12.4989 q^{94} +2.72545 q^{95} -0.523976 q^{96} +16.8059 q^{97} +14.8790 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 3 q^{4} - 3 q^{5} + 6 q^{7} + 3 q^{8} + 3 q^{9} - 3 q^{10} + 6 q^{14} - 9 q^{15} + 3 q^{16} + 9 q^{17} + 3 q^{18} + 3 q^{19} - 3 q^{20} + 15 q^{21} - 15 q^{23} + 12 q^{25} + 9 q^{27} + 6 q^{28}+ \cdots + 27 q^{98}+O(q^{100}) $ 3 * q + 3 * q^2 + 3 * q^4 - 3 * q^5 + 6 * q^7 + 3 * q^8 + 3 * q^9 - 3 * q^10 + 6 * q^14 - 9 * q^15 + 3 * q^16 + 9 * q^17 + 3 * q^18 + 3 * q^19 - 3 * q^20 + 15 * q^21 - 15 * q^23 + 12 * q^25 + 9 * q^27 + 6 * q^28 - 6 * q^29 - 9 * q^30 - 3 * q^31 + 3 * q^32 + 9 * q^34 + 3 * q^36 - 6 * q^37 + 3 * q^38 + 15 * q^39 - 3 * q^40 + 9 * q^41 + 15 * q^42 + 21 * q^43 - 27 * q^45 - 15 * q^46 - 12 * q^47 + 27 * q^49 + 12 * q^50 - 3 * q^51 - 9 * q^53 + 9 * q^54 + 6 * q^56 - 6 * q^58 - 3 * q^59 - 9 * q^60 - 24 * q^61 - 3 * q^62 + 3 * q^64 + 6 * q^65 + 9 * q^68 + 3 * q^69 + 21 * q^71 + 3 * q^72 + 3 * q^73 - 6 * q^74 + 36 * q^75 + 3 * q^76 + 15 * q^78 + 6 * q^79 - 3 * q^80 - 9 * q^81 + 9 * q^82 + 33 * q^83 + 15 * q^84 - 24 * q^85 + 21 * q^86 - 6 * q^87 + 6 * q^89 - 27 * q^90 + 36 * q^91 - 15 * q^92 + 36 * q^93 - 12 * q^94 - 3 * q^95 + 12 * q^97 + 27 * 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
$3$	−0.523976	−0.302518	−0.151259	−	0.988494i	$-0.548333\pi$
$3$	−0.523976	−0.302518	−0.151259	+	0.988494i	$0.548333\pi$
$4$	1.00000	0.500000
$5$	2.72545	1.21886	0.609429	−	0.792841i	$-0.291399\pi$
$5$	2.72545	1.21886	0.609429	+	0.792841i	$0.291399\pi$
$6$	−0.523976	−0.213912
$7$	4.67750	1.76793	0.883964	−	0.467556i	$-0.154865\pi$
$7$	4.67750	1.76793	0.883964	+	0.467556i	$0.154865\pi$
$8$	1.00000	0.353553
$9$	−2.72545	−0.908483
$10$	2.72545	0.861863
$11$	0	0
$11$	0	0
$12$	−0.523976	−0.151259
$13$	2.67750	0.742604	0.371302	−	0.928512i	$-0.378911\pi$
$13$	2.67750	0.742604	0.371302	+	0.928512i	$0.378911\pi$
$14$	4.67750	1.25011
$15$	−1.42807	−0.368726
$16$	1.00000	0.250000
$17$	−0.201472	−0.0488642	−0.0244321	−	0.999701i	$-0.507778\pi$
$17$	−0.201472	−0.0488642	−0.0244321	+	0.999701i	$0.507778\pi$
$18$	−2.72545	−0.642394
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	2.72545	0.609429
$21$	−2.45090	−0.534830
$22$	0	0
$23$	−1.79853	−0.375019	−0.187509	−	0.982263i	$-0.560042\pi$
$23$	−1.79853	−0.375019	−0.187509	+	0.982263i	$0.560042\pi$
$24$	−0.523976	−0.106956
$25$	2.42807	0.485614
$26$	2.67750	0.525100
$27$	3.00000	0.577350
$28$	4.67750	0.883964
$29$	4.92692	0.914906	0.457453	−	0.889234i	$-0.348762\pi$
$29$	4.92692	0.914906	0.457453	+	0.889234i	$0.348762\pi$
$30$	−1.42807	−0.260729
$31$	−2.57193	−0.461932	−0.230966	−	0.972962i	$-0.574189\pi$
$31$	−2.57193	−0.461932	−0.230966	+	0.972962i	$0.574189\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−0.201472	−0.0345522
$35$	12.7483	2.15485
$36$	−2.72545	−0.454241
$37$	4.40294	0.723840	0.361920	−	0.932209i	$-0.382121\pi$
$37$	4.40294	0.723840	0.361920	+	0.932209i	$0.382121\pi$
$38$	1.00000	0.162221
$39$	−1.40294	−0.224651
$40$	2.72545	0.430931
$41$	−4.97487	−0.776945	−0.388472	−	0.921460i	$-0.626997\pi$
$41$	−4.97487	−0.776945	−0.388472	+	0.921460i	$0.626997\pi$
$42$	−2.45090	−0.378182
$43$	11.7734	1.79543	0.897713	−	0.440580i	$-0.145227\pi$
$43$	11.7734	1.79543	0.897713	+	0.440580i	$0.145227\pi$
$44$	0	0
$45$	−7.42807	−1.10731
$46$	−1.79853	−0.265178
$47$	−12.4989	−1.82314	−0.911572	−	0.411140i	$-0.865131\pi$
$47$	−12.4989	−1.82314	−0.911572	+	0.411140i	$0.865131\pi$
$48$	−0.523976	−0.0756295
$49$	14.8790	2.12557
$50$	2.42807	0.343381
$51$	0.105567	0.0147823
$52$	2.67750	0.371302
$53$	−4.10557	−0.563943	−0.281971	−	0.959423i	$-0.590988\pi$
$53$	−4.10557	−0.563943	−0.281971	+	0.959423i	$0.590988\pi$
$54$	3.00000	0.408248
$55$	0	0
$56$	4.67750	0.625057
$57$	−0.523976	−0.0694024
$58$	4.92692	0.646936
$59$	−5.24943	−0.683417	−0.341708	−	0.939806i	$-0.611006\pi$
$59$	−5.24943	−0.683417	−0.341708	+	0.939806i	$0.611006\pi$
$60$	−1.42807	−0.184363
$61$	−1.59706	−0.204482	−0.102241	−	0.994760i	$-0.532601\pi$
$61$	−1.59706	−0.204482	−0.102241	+	0.994760i	$0.532601\pi$
$62$	−2.57193	−0.326635
$63$	−12.7483	−1.60613
$64$	1.00000	0.125000
$65$	7.29738	0.905128
$66$	0	0
$67$	9.08044	1.10935	0.554676	−	0.832066i	$-0.312842\pi$
$67$	9.08044	1.10935	0.554676	+	0.832066i	$0.312842\pi$
$68$	−0.201472	−0.0244321
$69$	0.942386	0.113450
$70$	12.7483	1.52371
$71$	12.8214	1.52161	0.760807	−	0.648978i	$-0.224803\pi$
$71$	12.8214	1.52161	0.760807	+	0.648978i	$0.224803\pi$
$72$	−2.72545	−0.321197
$73$	−2.20147	−0.257663	−0.128831	−	0.991667i	$-0.541123\pi$
$73$	−2.20147	−0.257663	−0.128831	+	0.991667i	$0.541123\pi$
$74$	4.40294	0.511832
$75$	−1.27225	−0.146907
$76$	1.00000	0.114708
$77$	0	0
$78$	−1.40294	−0.158852
$79$	−11.8538	−1.33366	−0.666831	−	0.745209i	$-0.732350\pi$
$79$	−11.8538	−1.33366	−0.666831	+	0.745209i	$0.732350\pi$
$80$	2.72545	0.304714
$81$	6.60442	0.733824
$82$	−4.97487	−0.549383
$83$	13.6775	1.50130	0.750650	−	0.660700i	$-0.229740\pi$
$83$	13.6775	1.50130	0.750650	+	0.660700i	$0.229740\pi$
$84$	−2.45090	−0.267415
$85$	−0.549103	−0.0595585
$86$	11.7734	1.26956
$87$	−2.58159	−0.276776
$88$	0	0
$89$	−5.45090	−0.577794	−0.288897	−	0.957360i	$-0.593289\pi$
$89$	−5.45090	−0.577794	−0.288897	+	0.957360i	$0.593289\pi$
$90$	−7.42807	−0.782987
$91$	12.5240	1.31287
$92$	−1.79853	−0.187509
$93$	1.34763	0.139743
$94$	−12.4989	−1.28916
$95$	2.72545	0.279625
$96$	−0.523976	−0.0534781
$97$	16.8059	1.70638	0.853190	−	0.521601i	$-0.174665\pi$
$97$	16.8059	1.70638	0.853190	+	0.521601i	$0.174665\pi$
$98$	14.8790	1.50300
$99$	0	0
$100$	2.42807	0.242807
$101$	−7.45090	−0.741392	−0.370696	−	0.928754i	$-0.620881\pi$
$101$	−7.45090	−0.741392	−0.370696	+	0.928754i	$0.620881\pi$
$102$	0.105567	0.0104527
$103$	−14.4834	−1.42709	−0.713545	−	0.700609i	$-0.752912\pi$
$103$	−14.4834	−1.42709	−0.713545	+	0.700609i	$0.752912\pi$
$104$	2.67750	0.262550
$105$	−6.67980	−0.651881
$106$	−4.10557	−0.398768
$107$	−1.15352	−0.111515	−0.0557575	−	0.998444i	$-0.517757\pi$
$107$	−1.15352	−0.111515	−0.0557575	+	0.998444i	$0.517757\pi$
$108$	3.00000	0.288675
$109$	9.55646	0.915343	0.457672	−	0.889121i	$-0.348684\pi$
$109$	9.55646	0.915343	0.457672	+	0.889121i	$0.348684\pi$
$110$	0	0
$111$	−2.30704	−0.218974
$112$	4.67750	0.441982
$113$	−8.00000	−0.752577	−0.376288	−	0.926503i	$-0.622800\pi$
$113$	−8.00000	−0.752577	−0.376288	+	0.926503i	$0.622800\pi$
$114$	−0.523976	−0.0490749
$115$	−4.90179	−0.457095
$116$	4.92692	0.457453
$117$	−7.29738	−0.674643
$118$	−5.24943	−0.483249
$119$	−0.942386	−0.0863884
$120$	−1.42807	−0.130364
$121$	0	0
$122$	−1.59706	−0.144591
$123$	2.60672	0.235040
$124$	−2.57193	−0.230966
$125$	−7.00966	−0.626963
$126$	−12.7483	−1.13571
$127$	−5.75794	−0.510934	−0.255467	−	0.966818i	$-0.582229\pi$
$127$	−5.75794	−0.510934	−0.255467	+	0.966818i	$0.582229\pi$
$128$	1.00000	0.0883883
$129$	−6.16898	−0.543149
$130$	7.29738	0.640022
$131$	−1.42807	−0.124771	−0.0623856	−	0.998052i	$-0.519871\pi$
$131$	−1.42807	−0.124771	−0.0623856	+	0.998052i	$0.519871\pi$
$132$	0	0
$133$	4.67750	0.405590
$134$	9.08044	0.784431
$135$	8.17635	0.703708
$136$	−0.201472	−0.0172761
$137$	−14.5313	−1.24150	−0.620748	−	0.784010i	$-0.713171\pi$
$137$	−14.5313	−1.24150	−0.620748	+	0.784010i	$0.713171\pi$
$138$	0.942386	0.0802212
$139$	−16.1763	−1.37206	−0.686030	−	0.727573i	$-0.740648\pi$
$139$	−16.1763	−1.37206	−0.686030	+	0.727573i	$0.740648\pi$
$140$	12.7483	1.07743
$141$	6.54910	0.551534
$142$	12.8214	1.07594
$143$	0	0
$144$	−2.72545	−0.227121
$145$	13.4281	1.11514
$146$	−2.20147	−0.182195
$147$	−7.79623	−0.643022
$148$	4.40294	0.361920
$149$	−6.30704	−0.516693	−0.258346	−	0.966052i	$-0.583178\pi$
$149$	−6.30704	−0.516693	−0.258346	+	0.966052i	$0.583178\pi$
$150$	−1.27225	−0.103879
$151$	−0.498850	−0.0405959	−0.0202979	−	0.999794i	$-0.506461\pi$
$151$	−0.498850	−0.0405959	−0.0202979	+	0.999794i	$0.506461\pi$
$152$	1.00000	0.0811107
$153$	0.549103	0.0443923
$154$	0	0
$155$	−7.00966	−0.563030
$156$	−1.40294	−0.112325
$157$	−13.6272	−1.08757	−0.543786	−	0.839224i	$-0.683010\pi$
$157$	−13.6272	−1.08757	−0.543786	+	0.839224i	$0.683010\pi$
$158$	−11.8538	−0.943041
$159$	2.15122	0.170603
$160$	2.72545	0.215466
$161$	−8.41261	−0.663006
$162$	6.60442	0.518892
$163$	15.9497	1.24928	0.624640	−	0.780913i	$-0.285246\pi$
$163$	15.9497	1.24928	0.624640	+	0.780913i	$0.285246\pi$
$164$	−4.97487	−0.388472
$165$	0	0
$166$	13.6775	1.06158
$167$	−18.8059	−1.45524	−0.727622	−	0.685979i	$-0.759374\pi$
$167$	−18.8059	−1.45524	−0.727622	+	0.685979i	$0.759374\pi$
$168$	−2.45090	−0.189091
$169$	−5.83102	−0.448540
$170$	−0.549103	−0.0421142
$171$	−2.72545	−0.208420
$172$	11.7734	0.897713
$173$	3.22430	0.245139	0.122569	−	0.992460i	$-0.460887\pi$
$173$	3.22430	0.245139	0.122569	+	0.992460i	$0.460887\pi$
$174$	−2.58159	−0.195710
$175$	11.3573	0.858531
$176$	0	0
$177$	2.75057	0.206746
$178$	−5.45090	−0.408562
$179$	13.3778	0.999905	0.499953	−	0.866053i	$-0.333351\pi$
$179$	13.3778	0.999905	0.499953	+	0.866053i	$0.333351\pi$
$180$	−7.42807	−0.553656
$181$	15.9497	1.18554	0.592768	−	0.805373i	$-0.298035\pi$
$181$	15.9497	1.18554	0.592768	+	0.805373i	$0.298035\pi$
$182$	12.5240	0.928339
$183$	0.836819	0.0618595
$184$	−1.79853	−0.132589
$185$	12.0000	0.882258
$186$	1.34763	0.0988130
$187$	0	0
$188$	−12.4989	−0.911572
$189$	14.0325	1.02071
$190$	2.72545	0.197725
$191$	−24.0553	−1.74058	−0.870291	−	0.492538i	$-0.836069\pi$
$191$	−24.0553	−1.74058	−0.870291	+	0.492538i	$0.836069\pi$
$192$	−0.523976	−0.0378147
$193$	−13.0325	−0.938099	−0.469050	−	0.883172i	$-0.655403\pi$
$193$	−13.0325	−0.938099	−0.469050	+	0.883172i	$0.655403\pi$
$194$	16.8059	1.20659
$195$	−3.82365	−0.273818
$196$	14.8790	1.06278
$197$	5.59706	0.398774	0.199387	−	0.979921i	$-0.436105\pi$
$197$	5.59706	0.398774	0.199387	+	0.979921i	$0.436105\pi$
$198$	0	0
$199$	−7.39558	−0.524259	−0.262129	−	0.965033i	$-0.584425\pi$
$199$	−7.39558	−0.524259	−0.262129	+	0.965033i	$0.584425\pi$
$200$	2.42807	0.171691
$201$	−4.75794	−0.335599
$202$	−7.45090	−0.524243
$203$	23.0457	1.61749
$204$	0.105567	0.00739115
$205$	−13.5588	−0.946985
$206$	−14.4834	−1.00911
$207$	4.90179	0.340698
$208$	2.67750	0.185651
$209$	0	0
$210$	−6.67980	−0.460950
$211$	14.2974	0.984272	0.492136	−	0.870518i	$-0.336216\pi$
$211$	14.2974	0.984272	0.492136	+	0.870518i	$0.336216\pi$
$212$	−4.10557	−0.281971
$213$	−6.71809	−0.460316
$214$	−1.15352	−0.0788530
$215$	32.0878	2.18837
$216$	3.00000	0.204124
$217$	−12.0302	−0.816662
$218$	9.55646	0.647245
$219$	1.15352	0.0779476
$220$	0	0
$221$	−0.539441	−0.0362868
$222$	−2.30704	−0.154838
$223$	−0.402945	−0.0269832	−0.0134916	−	0.999909i	$-0.504295\pi$
$223$	−0.402945	−0.0269832	−0.0134916	+	0.999909i	$0.504295\pi$
$224$	4.67750	0.312528
$225$	−6.61758	−0.441172
$226$	−8.00000	−0.532152
$227$	21.0074	1.39431	0.697154	−	0.716922i	$-0.254449\pi$
$227$	21.0074	1.39431	0.697154	+	0.716922i	$0.254449\pi$
$228$	−0.523976	−0.0347012
$229$	−14.3299	−0.946944	−0.473472	−	0.880809i	$-0.657000\pi$
$229$	−14.3299	−0.946944	−0.473472	+	0.880809i	$0.657000\pi$
$230$	−4.90179	−0.323215
$231$	0	0
$232$	4.92692	0.323468
$233$	24.4029	1.59869	0.799345	−	0.600872i	$-0.205180\pi$
$233$	24.4029	1.59869	0.799345	+	0.600872i	$0.205180\pi$
$234$	−7.29738	−0.477045
$235$	−34.0650	−2.22215
$236$	−5.24943	−0.341708
$237$	6.21113	0.403456
$238$	−0.942386	−0.0610858
$239$	12.6849	0.820515	0.410258	−	0.911970i	$-0.365439\pi$
$239$	12.6849	0.820515	0.410258	+	0.911970i	$0.365439\pi$
$240$	−1.42807	−0.0921816
$241$	23.9343	1.54174	0.770871	−	0.636991i	$-0.219821\pi$
$241$	23.9343	1.54174	0.770871	+	0.636991i	$0.219821\pi$
$242$	0	0
$243$	−12.4606	−0.799345
$244$	−1.59706	−0.102241
$245$	40.5519	2.59076
$246$	2.60672	0.166198
$247$	2.67750	0.170365
$248$	−2.57193	−0.163318
$249$	−7.16669	−0.454170
$250$	−7.00966	−0.443330
$251$	18.4989	1.16764	0.583819	−	0.811884i	$-0.301558\pi$
$251$	18.4989	1.16764	0.583819	+	0.811884i	$0.301558\pi$
$252$	−12.7483	−0.803066
$253$	0	0
$254$	−5.75794	−0.361285
$255$	0.287717	0.0180175
$256$	1.00000	0.0625000
$257$	27.7579	1.73149	0.865746	−	0.500483i	$-0.166844\pi$
$257$	27.7579	1.73149	0.865746	+	0.500483i	$0.166844\pi$
$258$	−6.16898	−0.384064
$259$	20.5948	1.27970
$260$	7.29738	0.452564
$261$	−13.4281	−0.831177
$262$	−1.42807	−0.0882265
$263$	−20.2723	−1.25004	−0.625020	−	0.780608i	$-0.714909\pi$
$263$	−20.2723	−1.25004	−0.625020	+	0.780608i	$0.714909\pi$
$264$	0	0
$265$	−11.1895	−0.687366
$266$	4.67750	0.286796
$267$	2.85614	0.174793
$268$	9.08044	0.554676
$269$	−15.6427	−0.953753	−0.476876	−	0.878970i	$-0.658231\pi$
$269$	−15.6427	−0.953753	−0.476876	+	0.878970i	$0.658231\pi$
$270$	8.17635	0.497597
$271$	12.3395	0.749573	0.374786	−	0.927111i	$-0.377716\pi$
$271$	12.3395	0.749573	0.374786	+	0.927111i	$0.377716\pi$
$272$	−0.201472	−0.0122161
$273$	−6.56227	−0.397167
$274$	−14.5313	−0.877870
$275$	0	0
$276$	0.942386	0.0567250
$277$	−8.49885	−0.510646	−0.255323	−	0.966856i	$-0.582182\pi$
$277$	−8.49885	−0.510646	−0.255323	+	0.966856i	$0.582182\pi$
$278$	−16.1763	−0.970193
$279$	7.00966	0.419657
$280$	12.7483	0.761855
$281$	7.12839	0.425244	0.212622	−	0.977134i	$-0.431800\pi$
$281$	7.12839	0.425244	0.212622	+	0.977134i	$0.431800\pi$
$282$	6.54910	0.389993
$283$	−11.5240	−0.685029	−0.342515	−	0.939512i	$-0.611279\pi$
$283$	−11.5240	−0.685029	−0.342515	+	0.939512i	$0.611279\pi$
$284$	12.8214	0.760807
$285$	−1.42807	−0.0845916
$286$	0	0
$287$	−23.2700	−1.37358
$288$	−2.72545	−0.160599
$289$	−16.9594	−0.997612
$290$	13.4281	0.788523
$291$	−8.80589	−0.516210
$292$	−2.20147	−0.128831
$293$	−10.6775	−0.623786	−0.311893	−	0.950117i	$-0.600963\pi$
$293$	−10.6775	−0.623786	−0.311893	+	0.950117i	$0.600963\pi$
$294$	−7.79623	−0.454685
$295$	−14.3070	−0.832988
$296$	4.40294	0.255916
$297$	0	0
$298$	−6.30704	−0.365357
$299$	−4.81555	−0.278490
$300$	−1.27225	−0.0734535
$301$	55.0700	3.17418
$302$	−0.498850	−0.0287056
$303$	3.90409	0.224284
$304$	1.00000	0.0573539
$305$	−4.35269	−0.249234
$306$	0.549103	0.0313901
$307$	30.2568	1.72685	0.863423	−	0.504481i	$-0.168316\pi$
$307$	30.2568	1.72685	0.863423	+	0.504481i	$0.168316\pi$
$308$	0	0
$309$	7.58895	0.431720
$310$	−7.00966	−0.398122
$311$	8.79623	0.498788	0.249394	−	0.968402i	$-0.419769\pi$
$311$	8.79623	0.498788	0.249394	+	0.968402i	$0.419769\pi$
$312$	−1.40294	−0.0794261
$313$	10.1860	0.575747	0.287874	−	0.957668i	$-0.407052\pi$
$313$	10.1860	0.575747	0.287874	+	0.957668i	$0.407052\pi$
$314$	−13.6272	−0.769030
$315$	−34.7448	−1.95765
$316$	−11.8538	−0.666831
$317$	−12.0097	−0.674530	−0.337265	−	0.941410i	$-0.609502\pi$
$317$	−12.0097	−0.674530	−0.337265	+	0.941410i	$0.609502\pi$
$318$	2.15122	0.120634
$319$	0	0
$320$	2.72545	0.152357
$321$	0.604417	0.0337353
$322$	−8.41261	−0.468816
$323$	−0.201472	−0.0112102
$324$	6.60442	0.366912
$325$	6.50115	0.360619
$326$	15.9497	0.883375
$327$	−5.00736	−0.276908
$328$	−4.97487	−0.274691
$329$	−58.4633	−3.22319
$330$	0	0
$331$	26.1667	1.43825	0.719126	−	0.694880i	$-0.244543\pi$
$331$	26.1667	1.43825	0.719126	+	0.694880i	$0.244543\pi$
$332$	13.6775	0.750650
$333$	−12.0000	−0.657596
$334$	−18.8059	−1.02901
$335$	24.7483	1.35214
$336$	−2.45090	−0.133707
$337$	−17.3778	−0.946630	−0.473315	−	0.880893i	$-0.656943\pi$
$337$	−17.3778	−0.946630	−0.473315	+	0.880893i	$0.656943\pi$
$338$	−5.83102	−0.317165
$339$	4.19181	0.227668
$340$	−0.549103	−0.0297793
$341$	0	0
$342$	−2.72545	−0.147375
$343$	36.8538	1.98992
$344$	11.7734	0.634779
$345$	2.56842	0.138279
$346$	3.22430	0.173339
$347$	12.4029	0.665825	0.332912	−	0.942958i	$-0.391969\pi$
$347$	12.4029	0.665825	0.332912	+	0.942958i	$0.391969\pi$
$348$	−2.58159	−0.138388
$349$	23.8036	1.27418	0.637088	−	0.770791i	$-0.280139\pi$
$349$	23.8036	1.27418	0.637088	+	0.770791i	$0.280139\pi$
$350$	11.3573	0.607073
$351$	8.03249	0.428742
$352$	0	0
$353$	−24.2664	−1.29157	−0.645786	−	0.763518i	$-0.723470\pi$
$353$	−24.2664	−1.29157	−0.645786	+	0.763518i	$0.723470\pi$
$354$	2.75057	0.146191
$355$	34.9439	1.85463
$356$	−5.45090	−0.288897
$357$	0.493788	0.0261340
$358$	13.3778	0.707040
$359$	−20.3395	−1.07348	−0.536740	−	0.843748i	$-0.680344\pi$
$359$	−20.3395	−1.07348	−0.536740	+	0.843748i	$0.680344\pi$
$360$	−7.42807	−0.391494
$361$	1.00000	0.0526316
$362$	15.9497	0.838300
$363$	0	0
$364$	12.5240	0.656435
$365$	−6.00000	−0.314054
$366$	0.836819	0.0437412
$367$	−23.0170	−1.20148	−0.600739	−	0.799445i	$-0.705127\pi$
$367$	−23.0170	−1.20148	−0.600739	+	0.799445i	$0.705127\pi$
$368$	−1.79853	−0.0937547
$369$	13.5588	0.705841
$370$	12.0000	0.623850
$371$	−19.2038	−0.997010
$372$	1.34763	0.0698714
$373$	18.5890	0.962499	0.481250	−	0.876584i	$-0.340183\pi$
$373$	18.5890	0.962499	0.481250	+	0.876584i	$0.340183\pi$
$374$	0	0
$375$	3.67290	0.189668
$376$	−12.4989	−0.644579
$377$	13.1918	0.679413
$378$	14.0325	0.721753
$379$	−10.5816	−0.543540	−0.271770	−	0.962362i	$-0.587609\pi$
$379$	−10.5816	−0.543540	−0.271770	+	0.962362i	$0.587609\pi$
$380$	2.72545	0.139813
$381$	3.01702	0.154567
$382$	−24.0553	−1.23078
$383$	−26.0878	−1.33302	−0.666512	−	0.745494i	$-0.732214\pi$
$383$	−26.0878	−1.33302	−0.666512	+	0.745494i	$0.732214\pi$
$384$	−0.523976	−0.0267391
$385$	0	0
$386$	−13.0325	−0.663336
$387$	−32.0878	−1.63111
$388$	16.8059	0.853190
$389$	−9.46636	−0.479964	−0.239982	−	0.970777i	$-0.577141\pi$
$389$	−9.46636	−0.479964	−0.239982	+	0.970777i	$0.577141\pi$
$390$	−3.82365	−0.193618
$391$	0.362354	0.0183250
$392$	14.8790	0.751501
$393$	0.748275	0.0377455
$394$	5.59706	0.281976
$395$	−32.3070	−1.62554
$396$	0	0
$397$	−25.1667	−1.26308	−0.631540	−	0.775343i	$-0.717577\pi$
$397$	−25.1667	−1.26308	−0.631540	+	0.775343i	$0.717577\pi$
$398$	−7.39558	−0.370707
$399$	−2.45090	−0.122698
$400$	2.42807	0.121404
$401$	−16.0650	−0.802247	−0.401123	−	0.916024i	$-0.631380\pi$
$401$	−16.0650	−0.802247	−0.401123	+	0.916024i	$0.631380\pi$
$402$	−4.75794	−0.237304
$403$	−6.88633	−0.343033
$404$	−7.45090	−0.370696
$405$	18.0000	0.894427
$406$	23.0457	1.14374
$407$	0	0
$408$	0.105567	0.00522633
$409$	19.5313	0.965763	0.482881	−	0.875686i	$-0.339590\pi$
$409$	19.5313	0.965763	0.482881	+	0.875686i	$0.339590\pi$
$410$	−13.5588	−0.669620
$411$	7.61408	0.375575
$412$	−14.4834	−0.713545
$413$	−24.5542	−1.20823
$414$	4.90179	0.240910
$415$	37.2773	1.82987
$416$	2.67750	0.131275
$417$	8.47602	0.415073
$418$	0	0
$419$	24.9018	1.21653	0.608266	−	0.793733i	$-0.291865\pi$
$419$	24.9018	1.21653	0.608266	+	0.793733i	$0.291865\pi$
$420$	−6.67980	−0.325941
$421$	26.6044	1.29662	0.648310	−	0.761377i	$-0.275476\pi$
$421$	26.6044	1.29662	0.648310	+	0.761377i	$0.275476\pi$
$422$	14.2974	0.695985
$423$	34.0650	1.65630
$424$	−4.10557	−0.199384
$425$	−0.489189	−0.0237292
$426$	−6.71809	−0.325492
$427$	−7.47022	−0.361509
$428$	−1.15352	−0.0557575
$429$	0	0
$430$	32.0878	1.54741
$431$	32.5948	1.57003	0.785017	−	0.619474i	$-0.212654\pi$
$431$	32.5948	1.57003	0.785017	+	0.619474i	$0.212654\pi$
$432$	3.00000	0.144338
$433$	19.7386	0.948577	0.474289	−	0.880369i	$-0.342705\pi$
$433$	19.7386	0.948577	0.474289	+	0.880369i	$0.342705\pi$
$434$	−12.0302	−0.577468
$435$	−7.03599	−0.337350
$436$	9.55646	0.457672
$437$	−1.79853	−0.0860352
$438$	1.15352	0.0551173
$439$	24.4029	1.16469	0.582345	−	0.812942i	$-0.302135\pi$
$439$	24.4029	1.16469	0.582345	+	0.812942i	$0.302135\pi$
$440$	0	0
$441$	−40.5519	−1.93104
$442$	−0.539441	−0.0256586
$443$	−16.5948	−0.788441	−0.394220	−	0.919016i	$-0.628985\pi$
$443$	−16.5948	−0.788441	−0.394220	+	0.919016i	$0.628985\pi$
$444$	−2.30704	−0.109487
$445$	−14.8561	−0.704249
$446$	−0.402945	−0.0190800
$447$	3.30474	0.156309
$448$	4.67750	0.220991
$449$	−20.9520	−0.988788	−0.494394	−	0.869238i	$-0.664610\pi$
$449$	−20.9520	−0.988788	−0.494394	+	0.869238i	$0.664610\pi$
$450$	−6.61758	−0.311956
$451$	0	0
$452$	−8.00000	−0.376288
$453$	0.261386	0.0122810
$454$	21.0074	0.985924
$455$	34.1335	1.60020
$456$	−0.523976	−0.0245374
$457$	−23.3144	−1.09060	−0.545301	−	0.838240i	$-0.683585\pi$
$457$	−23.3144	−1.09060	−0.545301	+	0.838240i	$0.683585\pi$
$458$	−14.3299	−0.669591
$459$	−0.604417	−0.0282118
$460$	−4.90179	−0.228547
$461$	−20.8059	−0.969027	−0.484513	−	0.874784i	$-0.661003\pi$
$461$	−20.8059	−0.969027	−0.484513	+	0.874784i	$0.661003\pi$
$462$	0	0
$463$	−9.29002	−0.431744	−0.215872	−	0.976422i	$-0.569259\pi$
$463$	−9.29002	−0.431744	−0.215872	+	0.976422i	$0.569259\pi$
$464$	4.92692	0.228727
$465$	3.67290	0.170327
$466$	24.4029	1.13044
$467$	12.0457	0.557406	0.278703	−	0.960377i	$-0.410095\pi$
$467$	12.0457	0.557406	0.278703	+	0.960377i	$0.410095\pi$
$468$	−7.29738	−0.337321
$469$	42.4737	1.96125
$470$	−34.0650	−1.57130
$471$	7.14035	0.329010
$472$	−5.24943	−0.241624
$473$	0	0
$474$	6.21113	0.285287
$475$	2.42807	0.111408
$476$	−0.942386	−0.0431942
$477$	11.1895	0.512333
$478$	12.6849	0.580192
$479$	−0.188307	−0.00860396	−0.00430198	−	0.999991i	$-0.501369\pi$
$479$	−0.188307	−0.00860396	−0.00430198	+	0.999991i	$0.501369\pi$
$480$	−1.42807	−0.0651822
$481$	11.7889	0.537526
$482$	23.9343	1.09018
$483$	4.40801	0.200571
$484$	0	0
$485$	45.8036	2.07983
$486$	−12.4606	−0.565222
$487$	−33.3852	−1.51283	−0.756413	−	0.654094i	$-0.773050\pi$
$487$	−33.3852	−1.51283	−0.756413	+	0.654094i	$0.773050\pi$
$488$	−1.59706	−0.0722953
$489$	−8.35729	−0.377930
$490$	40.5519	1.83195
$491$	−12.5217	−0.565095	−0.282548	−	0.959253i	$-0.591180\pi$
$491$	−12.5217	−0.565095	−0.282548	+	0.959253i	$0.591180\pi$
$492$	2.60672	0.117520
$493$	−0.992638	−0.0447062
$494$	2.67750	0.120466
$495$	0	0
$496$	−2.57193	−0.115483
$497$	59.9718	2.69010
$498$	−7.16669	−0.321147
$499$	16.3070	0.730003	0.365002	−	0.931007i	$-0.381068\pi$
$499$	16.3070	0.730003	0.365002	+	0.931007i	$0.381068\pi$
$500$	−7.00966	−0.313482
$501$	9.85384	0.440237
$502$	18.4989	0.825644
$503$	18.9726	0.845945	0.422973	−	0.906142i	$-0.360987\pi$
$503$	18.9726	0.845945	0.422973	+	0.906142i	$0.360987\pi$
$504$	−12.7483	−0.567853
$505$	−20.3070	−0.903651
$506$	0	0
$507$	3.05531	0.135691
$508$	−5.75794	−0.255467
$509$	−30.6907	−1.36034	−0.680170	−	0.733055i	$-0.738094\pi$
$509$	−30.6907	−1.36034	−0.680170	+	0.733055i	$0.738094\pi$
$510$	0.287717	0.0127403
$511$	−10.2974	−0.455529
$512$	1.00000	0.0441942
$513$	3.00000	0.132453
$514$	27.7579	1.22435
$515$	−39.4737	−1.73942
$516$	−6.16898	−0.271574
$517$	0	0
$518$	20.5948	0.904882
$519$	−1.68946	−0.0741589
$520$	7.29738	0.320011
$521$	−42.1106	−1.84490	−0.922450	−	0.386116i	$-0.873816\pi$
$521$	−42.1106	−1.84490	−0.922450	+	0.386116i	$0.873816\pi$
$522$	−13.4281	−0.587731
$523$	−37.8133	−1.65346	−0.826729	−	0.562600i	$-0.809801\pi$
$523$	−37.8133	−1.65346	−0.826729	+	0.562600i	$0.809801\pi$
$524$	−1.42807	−0.0623856
$525$	−5.95095	−0.259721
$526$	−20.2723	−0.883912
$527$	0.518173	0.0225720
$528$	0	0
$529$	−19.7653	−0.859361
$530$	−11.1895	−0.486041
$531$	14.3070	0.620873
$532$	4.67750	0.202795
$533$	−13.3202	−0.576962
$534$	2.85614	0.123597
$535$	−3.14386	−0.135921
$536$	9.08044	0.392215
$537$	−7.00966	−0.302489
$538$	−15.6427	−0.674405
$539$	0	0
$540$	8.17635	0.351854
$541$	−13.5661	−0.583253	−0.291627	−	0.956532i	$-0.594196\pi$
$541$	−13.5661	−0.583253	−0.291627	+	0.956532i	$0.594196\pi$
$542$	12.3395	0.530028
$543$	−8.35729	−0.358646
$544$	−0.201472	−0.00863806
$545$	26.0457	1.11567
$546$	−6.56227	−0.280839
$547$	−46.4177	−1.98468	−0.992338	−	0.123552i	$-0.960571\pi$
$547$	−46.4177	−1.98468	−0.992338	+	0.123552i	$0.960571\pi$
$548$	−14.5313	−0.620748
$549$	4.35269	0.185768
$550$	0	0
$551$	4.92692	0.209894
$552$	0.942386	0.0401106
$553$	−55.4463	−2.35782
$554$	−8.49885	−0.361082
$555$	−6.28772	−0.266899
$556$	−16.1763	−0.686030
$557$	38.2065	1.61886	0.809431	−	0.587214i	$-0.199775\pi$
$557$	38.2065	1.61886	0.809431	+	0.587214i	$0.199775\pi$
$558$	7.00966	0.296743
$559$	31.5232	1.33329
$560$	12.7483	0.538713
$561$	0	0
$562$	7.12839	0.300693
$563$	7.01702	0.295732	0.147866	−	0.989007i	$-0.452760\pi$
$563$	7.01702	0.295732	0.147866	+	0.989007i	$0.452760\pi$
$564$	6.54910	0.275767
$565$	−21.8036	−0.917284
$566$	−11.5240	−0.484389
$567$	30.8921	1.29735
$568$	12.8214	0.537972
$569$	41.4235	1.73656	0.868281	−	0.496072i	$-0.165225\pi$
$569$	41.4235	1.73656	0.868281	+	0.496072i	$0.165225\pi$
$570$	−1.42807	−0.0598153
$571$	−12.1381	−0.507962	−0.253981	−	0.967209i	$-0.581740\pi$
$571$	−12.1381	−0.507962	−0.253981	+	0.967209i	$0.581740\pi$
$572$	0	0
$573$	12.6044	0.526557
$574$	−23.2700	−0.971269
$575$	−4.36695	−0.182115
$576$	−2.72545	−0.113560
$577$	12.8384	0.534469	0.267234	−	0.963632i	$-0.413890\pi$
$577$	12.8384	0.534469	0.267234	+	0.963632i	$0.413890\pi$
$578$	−16.9594	−0.705418
$579$	6.82872	0.283792
$580$	13.4281	0.557570
$581$	63.9764	2.65419
$582$	−8.80589	−0.365016
$583$	0	0
$584$	−2.20147	−0.0910976
$585$	−19.8886	−0.822294
$586$	−10.6775	−0.441083
$587$	4.00000	0.165098	0.0825488	−	0.996587i	$-0.473694\pi$
$587$	4.00000	0.165098	0.0825488	+	0.996587i	$0.473694\pi$
$588$	−7.79623	−0.321511
$589$	−2.57193	−0.105974
$590$	−14.3070	−0.589011
$591$	−2.93272	−0.120636
$592$	4.40294	0.180960
$593$	−40.4177	−1.65975	−0.829877	−	0.557946i	$-0.811590\pi$
$593$	−40.4177	−1.65975	−0.829877	+	0.557946i	$0.811590\pi$
$594$	0	0
$595$	−2.56842	−0.105295
$596$	−6.30704	−0.258346
$597$	3.87511	0.158598
$598$	−4.81555	−0.196923
$599$	0.679795	0.0277757	0.0138878	−	0.999904i	$-0.495579\pi$
$599$	0.679795	0.0277757	0.0138878	+	0.999904i	$0.495579\pi$
$600$	−1.27225	−0.0519395
$601$	−21.5240	−0.877981	−0.438991	−	0.898492i	$-0.644664\pi$
$601$	−21.5240	−0.877981	−0.438991	+	0.898492i	$0.644664\pi$
$602$	55.0700	2.24449
$603$	−24.7483	−1.00783
$604$	−0.498850	−0.0202979
$605$	0	0
$606$	3.90409	0.158593
$607$	−13.3550	−0.542062	−0.271031	−	0.962571i	$-0.587365\pi$
$607$	−13.3550	−0.542062	−0.271031	+	0.962571i	$0.587365\pi$
$608$	1.00000	0.0405554
$609$	−12.0754	−0.489319
$610$	−4.35269	−0.176235
$611$	−33.4656	−1.35387
$612$	0.549103	0.0221962
$613$	−9.11293	−0.368068	−0.184034	−	0.982920i	$-0.558916\pi$
$613$	−9.11293	−0.368068	−0.184034	+	0.982920i	$0.558916\pi$
$614$	30.2568	1.22106
$615$	7.10447	0.286480
$616$	0	0
$617$	0.329866	0.0132799	0.00663995	−	0.999978i	$-0.497886\pi$
$617$	0.329866	0.0132799	0.00663995	+	0.999978i	$0.497886\pi$
$618$	7.58895	0.305272
$619$	8.92112	0.358570	0.179285	−	0.983797i	$-0.442622\pi$
$619$	8.92112	0.358570	0.179285	+	0.983797i	$0.442622\pi$
$620$	−7.00966	−0.281515
$621$	−5.39558	−0.216517
$622$	8.79623	0.352697
$623$	−25.4966	−1.02150
$624$	−1.40294	−0.0561627
$625$	−31.2448	−1.24979
$626$	10.1860	0.407115
$627$	0	0
$628$	−13.6272	−0.543786
$629$	−0.887072	−0.0353699
$630$	−34.7448	−1.38426
$631$	44.6597	1.77788	0.888938	−	0.458028i	$-0.151444\pi$
$631$	44.6597	1.77788	0.888938	+	0.458028i	$0.151444\pi$
$632$	−11.8538	−0.471521
$633$	−7.49149	−0.297760
$634$	−12.0097	−0.476965
$635$	−15.6930	−0.622756
$636$	2.15122	0.0853014
$637$	39.8384	1.57845
$638$	0	0
$639$	−34.9439	−1.38236
$640$	2.72545	0.107733
$641$	−6.93272	−0.273826	−0.136913	−	0.990583i	$-0.543718\pi$
$641$	−6.93272	−0.273826	−0.136913	+	0.990583i	$0.543718\pi$
$642$	0.604417	0.0238544
$643$	23.4966	0.926614	0.463307	−	0.886198i	$-0.346663\pi$
$643$	23.4966	0.926614	0.463307	+	0.886198i	$0.346663\pi$
$644$	−8.41261	−0.331503
$645$	−16.8133	−0.662021
$646$	−0.201472	−0.00792682
$647$	20.9880	0.825125	0.412562	−	0.910929i	$-0.364634\pi$
$647$	20.9880	0.825125	0.412562	+	0.910929i	$0.364634\pi$
$648$	6.60442	0.259446
$649$	0	0
$650$	6.50115	0.254996
$651$	6.30353	0.247055
$652$	15.9497	0.624640
$653$	−44.1335	−1.72708	−0.863538	−	0.504284i	$-0.831756\pi$
$653$	−44.1335	−1.72708	−0.863538	+	0.504284i	$0.831756\pi$
$654$	−5.00736	−0.195803
$655$	−3.89213	−0.152078
$656$	−4.97487	−0.194236
$657$	6.00000	0.234082
$658$	−58.4633	−2.27914
$659$	−24.0862	−0.938267	−0.469133	−	0.883127i	$-0.655434\pi$
$659$	−24.0862	−0.938267	−0.469133	+	0.883127i	$0.655434\pi$
$660$	0	0
$661$	−9.65237	−0.375434	−0.187717	−	0.982223i	$-0.560109\pi$
$661$	−9.65237	−0.375434	−0.187717	+	0.982223i	$0.560109\pi$
$662$	26.1667	1.01700
$663$	0.282655	0.0109774
$664$	13.6775	0.530790
$665$	12.7483	0.494357
$666$	−12.0000	−0.464991
$667$	−8.86120	−0.343107
$668$	−18.8059	−0.727622
$669$	0.211133	0.00816289
$670$	24.7483	0.956109
$671$	0	0
$672$	−2.45090	−0.0945454
$673$	−24.6176	−0.948938	−0.474469	−	0.880272i	$-0.657360\pi$
$673$	−24.6176	−0.948938	−0.474469	+	0.880272i	$0.657360\pi$
$674$	−17.3778	−0.669369
$675$	7.28421	0.280369
$676$	−5.83102	−0.224270
$677$	−21.5719	−0.829077	−0.414538	−	0.910032i	$-0.636057\pi$
$677$	−21.5719	−0.829077	−0.414538	+	0.910032i	$0.636057\pi$
$678$	4.19181	0.160986
$679$	78.6095	3.01675
$680$	−0.549103	−0.0210571
$681$	−11.0074	−0.421803
$682$	0	0
$683$	−0.997701	−0.0381759	−0.0190880	−	0.999818i	$-0.506076\pi$
$683$	−0.997701	−0.0381759	−0.0190880	+	0.999818i	$0.506076\pi$
$684$	−2.72545	−0.104210
$685$	−39.6044	−1.51321
$686$	36.8538	1.40709
$687$	7.50851	0.286468
$688$	11.7734	0.448857
$689$	−10.9926	−0.418786
$690$	2.56842	0.0977783
$691$	−9.00230	−0.342464	−0.171232	−	0.985231i	$-0.554775\pi$
$691$	−9.00230	−0.342464	−0.171232	+	0.985231i	$0.554775\pi$
$692$	3.22430	0.122569
$693$	0	0
$694$	12.4029	0.470809
$695$	−44.0878	−1.67235
$696$	−2.58159	−0.0978549
$697$	1.00230	0.0379648
$698$	23.8036	0.900979
$699$	−12.7866	−0.483632
$700$	11.3573	0.429265
$701$	15.0936	0.570078	0.285039	−	0.958516i	$-0.407994\pi$
$701$	15.0936	0.570078	0.285039	+	0.958516i	$0.407994\pi$
$702$	8.03249	0.303167
$703$	4.40294	0.166060
$704$	0	0
$705$	17.8492	0.672241
$706$	−24.2664	−0.913280
$707$	−34.8515	−1.31073
$708$	2.75057	0.103373
$709$	−22.9246	−0.860952	−0.430476	−	0.902602i	$-0.641654\pi$
$709$	−22.9246	−0.860952	−0.430476	+	0.902602i	$0.641654\pi$
$710$	34.9439	1.31142
$711$	32.3070	1.21161
$712$	−5.45090	−0.204281
$713$	4.62569	0.173233
$714$	0.493788	0.0184796
$715$	0	0
$716$	13.3778	0.499953
$717$	−6.64657	−0.248221
$718$	−20.3395	−0.759064
$719$	6.70032	0.249880	0.124940	−	0.992164i	$-0.460126\pi$
$719$	6.70032	0.249880	0.124940	+	0.992164i	$0.460126\pi$
$720$	−7.42807	−0.276828
$721$	−67.7460	−2.52299
$722$	1.00000	0.0372161
$723$	−12.5410	−0.466405
$724$	15.9497	0.592768
$725$	11.9629	0.444291
$726$	0	0
$727$	−18.4583	−0.684579	−0.342289	−	0.939595i	$-0.611202\pi$
$727$	−18.4583	−0.684579	−0.342289	+	0.939595i	$0.611202\pi$
$728$	12.5240	0.464169
$729$	−13.2842	−0.492008
$730$	−6.00000	−0.222070
$731$	−2.37201	−0.0877321
$732$	0.836819	0.0309297
$733$	−41.8995	−1.54759	−0.773797	−	0.633434i	$-0.781645\pi$
$733$	−41.8995	−1.54759	−0.773797	+	0.633434i	$0.781645\pi$
$734$	−23.0170	−0.849574
$735$	−21.2482	−0.783752
$736$	−1.79853	−0.0662946
$737$	0	0
$738$	13.5588	0.499105
$739$	34.8361	1.28147	0.640733	−	0.767764i	$-0.278631\pi$
$739$	34.8361	1.28147	0.640733	+	0.767764i	$0.278631\pi$
$740$	12.0000	0.441129
$741$	−1.40294	−0.0515385
$742$	−19.2038	−0.704993
$743$	25.7889	0.946102	0.473051	−	0.881035i	$-0.343153\pi$
$743$	25.7889	0.946102	0.473051	+	0.881035i	$0.343153\pi$
$744$	1.34763	0.0494065
$745$	−17.1895	−0.629775
$746$	18.5890	0.680590
$747$	−37.2773	−1.36391
$748$	0	0
$749$	−5.39558	−0.197150
$750$	3.67290	0.134115
$751$	−33.5011	−1.22247	−0.611237	−	0.791447i	$-0.709328\pi$
$751$	−33.5011	−1.22247	−0.611237	+	0.791447i	$0.709328\pi$
$752$	−12.4989	−0.455786
$753$	−9.69296	−0.353231
$754$	13.1918	0.480417
$755$	−1.35959	−0.0494806
$756$	14.0325	0.510357
$757$	7.53134	0.273731	0.136866	−	0.990590i	$-0.456297\pi$
$757$	7.53134	0.273731	0.136866	+	0.990590i	$0.456297\pi$
$758$	−10.5816	−0.384341
$759$	0	0
$760$	2.72545	0.0988624
$761$	−38.1969	−1.38464	−0.692318	−	0.721593i	$-0.743410\pi$
$761$	−38.1969	−1.38464	−0.692318	+	0.721593i	$0.743410\pi$
$762$	3.01702	0.109295
$763$	44.7003	1.61826
$764$	−24.0553	−0.870291
$765$	1.49655	0.0541079
$766$	−26.0878	−0.942591
$767$	−14.0553	−0.507508
$768$	−0.523976	−0.0189074
$769$	−16.6501	−0.600417	−0.300208	−	0.953874i	$-0.597056\pi$
$769$	−16.6501	−0.600417	−0.300208	+	0.953874i	$0.597056\pi$
$770$	0	0
$771$	−14.5445	−0.523808
$772$	−13.0325	−0.469050
$773$	−23.6330	−0.850022	−0.425011	−	0.905188i	$-0.639730\pi$
$773$	−23.6330	−0.850022	−0.425011	+	0.905188i	$0.639730\pi$
$774$	−32.0878	−1.15337
$775$	−6.24483	−0.224321
$776$	16.8059	0.603296
$777$	−10.7912	−0.387131
$778$	−9.46636	−0.339386
$779$	−4.97487	−0.178243
$780$	−3.82365	−0.136909
$781$	0	0
$782$	0.362354	0.0129577
$783$	14.7808	0.528221
$784$	14.8790	0.531392
$785$	−37.1404	−1.32560
$786$	0.748275	0.0266901
$787$	20.3933	0.726942	0.363471	−	0.931606i	$-0.381592\pi$
$787$	20.3933	0.726942	0.363471	+	0.931606i	$0.381592\pi$
$788$	5.59706	0.199387
$789$	10.6222	0.378160
$790$	−32.3070	−1.14943
$791$	−37.4200	−1.33050
$792$	0	0
$793$	−4.27611	−0.151849
$794$	−25.1667	−0.893132
$795$	5.86304	0.207941
$796$	−7.39558	−0.262129
$797$	−9.70262	−0.343685	−0.171842	−	0.985124i	$-0.554972\pi$
$797$	−9.70262	−0.343685	−0.171842	+	0.985124i	$0.554972\pi$
$798$	−2.45090	−0.0867608
$799$	2.51817	0.0890865
$800$	2.42807	0.0858453
$801$	14.8561	0.524916
$802$	−16.0650	−0.567274
$803$	0	0
$804$	−4.75794	−0.167799
$805$	−22.9281	−0.808110
$806$	−6.88633	−0.242561
$807$	8.19641	0.288527
$808$	−7.45090	−0.262122
$809$	27.0723	0.951813	0.475906	−	0.879496i	$-0.342120\pi$
$809$	27.0723	0.951813	0.475906	+	0.879496i	$0.342120\pi$
$810$	18.0000	0.632456
$811$	−51.9548	−1.82438	−0.912190	−	0.409767i	$-0.865610\pi$
$811$	−51.9548	−1.82438	−0.912190	+	0.409767i	$0.865610\pi$
$812$	23.0457	0.808744
$813$	−6.46562	−0.226759
$814$	0	0
$815$	43.4702	1.52270
$816$	0.105567	0.00369558
$817$	11.7734	0.411899
$818$	19.5313	0.682897
$819$	−34.1335	−1.19272
$820$	−13.5588	−0.473493
$821$	25.6574	0.895451	0.447725	−	0.894171i	$-0.352234\pi$
$821$	25.6574	0.895451	0.447725	+	0.894171i	$0.352234\pi$
$822$	7.61408	0.265571
$823$	−41.8635	−1.45927	−0.729635	−	0.683837i	$-0.760310\pi$
$823$	−41.8635	−1.45927	−0.729635	+	0.683837i	$0.760310\pi$
$824$	−14.4834	−0.504553
$825$	0	0
$826$	−24.5542	−0.854349
$827$	−6.15582	−0.214059	−0.107029	−	0.994256i	$-0.534134\pi$
$827$	−6.15582	−0.214059	−0.107029	+	0.994256i	$0.534134\pi$
$828$	4.90179	0.170349
$829$	−0.247126	−0.00858303	−0.00429151	−	0.999991i	$-0.501366\pi$
$829$	−0.247126	−0.00858303	−0.00429151	+	0.999991i	$0.501366\pi$
$830$	37.2773	1.29391
$831$	4.45320	0.154480
$832$	2.67750	0.0928255
$833$	−2.99770	−0.103864
$834$	8.47602	0.293501
$835$	−51.2545	−1.77373
$836$	0	0
$837$	−7.71579	−0.266697
$838$	24.9018	0.860218
$839$	16.9320	0.584557	0.292278	−	0.956333i	$-0.405587\pi$
$839$	16.9320	0.584557	0.292278	+	0.956333i	$0.405587\pi$
$840$	−6.67980	−0.230475
$841$	−4.72545	−0.162947
$842$	26.6044	0.916849
$843$	−3.73511	−0.128644
$844$	14.2974	0.492136
$845$	−15.8921	−0.546706
$846$	34.0650	1.17118
$847$	0	0
$848$	−4.10557	−0.140986
$849$	6.03829	0.207234
$850$	−0.489189	−0.0167790
$851$	−7.91882	−0.271454
$852$	−6.71809	−0.230158
$853$	−21.6930	−0.742753	−0.371376	−	0.928482i	$-0.621114\pi$
$853$	−21.6930	−0.742753	−0.371376	+	0.928482i	$0.621114\pi$
$854$	−7.47022	−0.255626
$855$	−7.42807	−0.254035
$856$	−1.15352	−0.0394265
$857$	−34.7328	−1.18645	−0.593225	−	0.805037i	$-0.702146\pi$
$857$	−34.7328	−1.18645	−0.593225	+	0.805037i	$0.702146\pi$
$858$	0	0
$859$	−15.3047	−0.522191	−0.261095	−	0.965313i	$-0.584084\pi$
$859$	−15.3047	−0.522191	−0.261095	+	0.965313i	$0.584084\pi$
$860$	32.0878	1.09418
$861$	12.1929	0.415533
$862$	32.5948	1.11018
$863$	35.8457	1.22020	0.610102	−	0.792323i	$-0.291129\pi$
$863$	35.8457	1.22020	0.610102	+	0.792323i	$0.291129\pi$
$864$	3.00000	0.102062
$865$	8.78766	0.298789
$866$	19.7386	0.670745
$867$	8.88633	0.301796
$868$	−12.0302	−0.408331
$869$	0	0
$870$	−7.03599	−0.238542
$871$	24.3128	0.823809
$872$	9.55646	0.323623
$873$	−45.8036	−1.55022
$874$	−1.79853	−0.0608361
$875$	−32.7877	−1.10843
$876$	1.15352	0.0389738
$877$	45.4907	1.53611	0.768057	−	0.640382i	$-0.221224\pi$
$877$	45.4907	1.53611	0.768057	+	0.640382i	$0.221224\pi$
$878$	24.4029	0.823559
$879$	5.59476	0.188706
$880$	0	0
$881$	31.3322	1.05561	0.527804	−	0.849366i	$-0.323016\pi$
$881$	31.3322	1.05561	0.527804	+	0.849366i	$0.323016\pi$
$882$	−40.5519	−1.36545
$883$	35.3550	1.18979	0.594895	−	0.803803i	$-0.297194\pi$
$883$	35.3550	1.18979	0.594895	+	0.803803i	$0.297194\pi$
$884$	−0.539441	−0.0181434
$885$	7.49655	0.251994
$886$	−16.5948	−0.557512
$887$	16.6141	0.557846	0.278923	−	0.960313i	$-0.410023\pi$
$887$	16.6141	0.557846	0.278923	+	0.960313i	$0.410023\pi$
$888$	−2.30704	−0.0774192
$889$	−26.9327	−0.903295
$890$	−14.8561	−0.497979
$891$	0	0
$892$	−0.402945	−0.0134916
$893$	−12.4989	−0.418258
$894$	3.30474	0.110527
$895$	36.4606	1.21874
$896$	4.67750	0.156264
$897$	2.52323	0.0842484
$898$	−20.9520	−0.699179
$899$	−12.6717	−0.422625
$900$	−6.61758	−0.220586
$901$	0.827158	0.0275566
$902$	0	0
$903$	−28.8554	−0.960248
$904$	−8.00000	−0.266076
$905$	43.4702	1.44500
$906$	0.261386	0.00868396
$907$	54.8612	1.82164	0.910818	−	0.412808i	$-0.135452\pi$
$907$	54.8612	1.82164	0.910818	+	0.412808i	$0.135452\pi$
$908$	21.0074	0.697154
$909$	20.3070	0.673542
$910$	34.1335	1.13151
$911$	2.98298	0.0988304	0.0494152	−	0.998778i	$-0.484264\pi$
$911$	2.98298	0.0988304	0.0494152	+	0.998778i	$0.484264\pi$
$912$	−0.523976	−0.0173506
$913$	0	0
$914$	−23.3144	−0.771172
$915$	2.28071	0.0753979
$916$	−14.3299	−0.473472
$917$	−6.67980	−0.220586
$918$	−0.604417	−0.0199487
$919$	26.6701	0.879767	0.439883	−	0.898055i	$-0.355020\pi$
$919$	26.6701	0.879767	0.439883	+	0.898055i	$0.355020\pi$
$920$	−4.90179	−0.161607
$921$	−15.8538	−0.522402
$922$	−20.8059	−0.685205
$923$	34.3291	1.12996
$924$	0	0
$925$	10.6907	0.351507
$926$	−9.29002	−0.305289
$927$	39.4737	1.29649
$928$	4.92692	0.161734
$929$	43.7595	1.43570	0.717851	−	0.696197i	$-0.245126\pi$
$929$	43.7595	1.43570	0.717851	+	0.696197i	$0.245126\pi$
$930$	3.67290	0.120439
$931$	14.8790	0.487638
$932$	24.4029	0.799345
$933$	−4.60902	−0.150892
$934$	12.0457	0.394146
$935$	0	0
$936$	−7.29738	−0.238522
$937$	−47.7823	−1.56098	−0.780490	−	0.625168i	$-0.785030\pi$
$937$	−47.7823	−1.56098	−0.780490	+	0.625168i	$0.785030\pi$
$938$	42.4737	1.38682
$939$	−5.33723	−0.174174
$940$	−34.0650	−1.11108
$941$	−36.2664	−1.18225	−0.591126	−	0.806579i	$-0.701316\pi$
$941$	−36.2664	−1.18225	−0.591126	+	0.806579i	$0.701316\pi$
$942$	7.14035	0.232645
$943$	8.94745	0.291369
$944$	−5.24943	−0.170854
$945$	38.2448	1.24410
$946$	0	0
$947$	35.5468	1.15512	0.577558	−	0.816350i	$-0.304006\pi$
$947$	35.5468	1.15512	0.577558	+	0.816350i	$0.304006\pi$
$948$	6.21113	0.201728
$949$	−5.89443	−0.191341
$950$	2.42807	0.0787770
$951$	6.29278	0.204057
$952$	−0.942386	−0.0305429
$953$	−23.2088	−0.751808	−0.375904	−	0.926659i	$-0.622668\pi$
$953$	−23.2088	−0.751808	−0.375904	+	0.926659i	$0.622668\pi$
$954$	11.1895	0.362274
$955$	−65.5615	−2.12152
$956$	12.6849	0.410258
$957$	0	0
$958$	−0.188307	−0.00608392
$959$	−67.9703	−2.19487
$960$	−1.42807	−0.0460908
$961$	−24.3852	−0.786619
$962$	11.7889	0.380088
$963$	3.14386	0.101309
$964$	23.9343	0.770871
$965$	−35.5194	−1.14341
$966$	4.40801	0.141825
$967$	6.09591	0.196031	0.0980156	−	0.995185i	$-0.468751\pi$
$967$	6.09591	0.196031	0.0980156	+	0.995185i	$0.468751\pi$
$968$	0	0
$969$	0.105567	0.00339129
$970$	45.8036	1.47066
$971$	34.9822	1.12263	0.561317	−	0.827601i	$-0.310295\pi$
$971$	34.9822	1.12263	0.561317	+	0.827601i	$0.310295\pi$
$972$	−12.4606	−0.399673
$973$	−75.6648	−2.42570
$974$	−33.3852	−1.06973
$975$	−3.40645	−0.109094
$976$	−1.59706	−0.0511205
$977$	−10.9018	−0.348779	−0.174390	−	0.984677i	$-0.555795\pi$
$977$	−10.9018	−0.348779	−0.174390	+	0.984677i	$0.555795\pi$
$978$	−8.35729	−0.267237
$979$	0	0
$980$	40.5519	1.29538
$981$	−26.0457	−0.831574
$982$	−12.5217	−0.399583
$983$	−25.4281	−0.811030	−0.405515	−	0.914089i	$-0.632908\pi$
$983$	−25.4281	−0.811030	−0.405515	+	0.914089i	$0.632908\pi$
$984$	2.60672	0.0830991
$985$	15.2545	0.486048
$986$	−0.992638	−0.0316120
$987$	30.6334	0.975072
$988$	2.67750	0.0851825
$989$	−21.1748	−0.673319
$990$	0	0
$991$	49.3925	1.56901	0.784503	−	0.620125i	$-0.212918\pi$
$991$	49.3925	1.56901	0.784503	+	0.620125i	$0.212918\pi$
$992$	−2.57193	−0.0816588
$993$	−13.7107	−0.435097
$994$	59.9718	1.90219
$995$	−20.1563	−0.638997
$996$	−7.16669	−0.227085
$997$	−20.7409	−0.656871	−0.328436	−	0.944526i	$-0.606521\pi$
$997$	−20.7409	−0.656871	−0.328436	+	0.944526i	$0.606521\pi$
$998$	16.3070	0.516190
$999$	13.2088	0.417909

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4598.2.a.bo.1.2		3
11.10	odd	2		418.2.a.g.1.2	✓	3
33.32	even	2		3762.2.a.bg.1.1		3
44.43	even	2		3344.2.a.q.1.2		3
209.208	even	2		7942.2.a.bi.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
418.2.a.g.1.2	✓	3	11.10	odd	2
3344.2.a.q.1.2		3	44.43	even	2
3762.2.a.bg.1.1		3	33.32	even	2
4598.2.a.bo.1.2		3	1.1	even	1	trivial
7942.2.a.bi.1.2		3	209.208	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4598 = 2 \cdot 11^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4598.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -0.523976 q^{3} +1.00000 q^{4} +2.72545 q^{5} -0.523976 q^{6} +4.67750 q^{7} +1.00000 q^{8} -2.72545 q^{9} +2.72545 q^{10} -0.523976 q^{12} +2.67750 q^{13} +4.67750 q^{14} -1.42807 q^{15} +1.00000 q^{16} -0.201472 q^{17} -2.72545 q^{18} +1.00000 q^{19} +2.72545 q^{20} -2.45090 q^{21} -1.79853 q^{23} -0.523976 q^{24} +2.42807 q^{25} +2.67750 q^{26} +3.00000 q^{27} +4.67750 q^{28} +4.92692 q^{29} -1.42807 q^{30} -2.57193 q^{31} +1.00000 q^{32} -0.201472 q^{34} +12.7483 q^{35} -2.72545 q^{36} +4.40294 q^{37} +1.00000 q^{38} -1.40294 q^{39} +2.72545 q^{40} -4.97487 q^{41} -2.45090 q^{42} +11.7734 q^{43} -7.42807 q^{45} -1.79853 q^{46} -12.4989 q^{47} -0.523976 q^{48} +14.8790 q^{49} +2.42807 q^{50} +0.105567 q^{51} +2.67750 q^{52} -4.10557 q^{53} +3.00000 q^{54} +4.67750 q^{56} -0.523976 q^{57} +4.92692 q^{58} -5.24943 q^{59} -1.42807 q^{60} -1.59706 q^{61} -2.57193 q^{62} -12.7483 q^{63} +1.00000 q^{64} +7.29738 q^{65} +9.08044 q^{67} -0.201472 q^{68} +0.942386 q^{69} +12.7483 q^{70} +12.8214 q^{71} -2.72545 q^{72} -2.20147 q^{73} +4.40294 q^{74} -1.27225 q^{75} +1.00000 q^{76} -1.40294 q^{78} -11.8538 q^{79} +2.72545 q^{80} +6.60442 q^{81} -4.97487 q^{82} +13.6775 q^{83} -2.45090 q^{84} -0.549103 q^{85} +11.7734 q^{86} -2.58159 q^{87} -5.45090 q^{89} -7.42807 q^{90} +12.5240 q^{91} -1.79853 q^{92} +1.34763 q^{93} -12.4989 q^{94} +2.72545 q^{95} -0.523976 q^{96} +16.8059 q^{97} +14.8790 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{4} - 3 q^{5} + 6 q^{7} + 3 q^{8} + 3 q^{9} - 3 q^{10} + 6 q^{14} - 9 q^{15} + 3 q^{16} + 9 q^{17} + 3 q^{18} + 3 q^{19} - 3 q^{20} + 15 q^{21} - 15 q^{23} + 12 q^{25} + 9 q^{27} + 6 q^{28}+ \cdots + 27 q^{98}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^4 - 3 * q^5 + 6 * q^7 + 3 * q^8 + 3 * q^9 - 3 * q^10 + 6 * q^14 - 9 * q^15 + 3 * q^16 + 9 * q^17 + 3 * q^18 + 3 * q^19 - 3 * q^20 + 15 * q^21 - 15 * q^23 + 12 * q^25 + 9 * q^27 + 6 * q^28 - 6 * q^29 - 9 * q^30 - 3 * q^31 + 3 * q^32 + 9 * q^34 + 3 * q^36 - 6 * q^37 + 3 * q^38 + 15 * q^39 - 3 * q^40 + 9 * q^41 + 15 * q^42 + 21 * q^43 - 27 * q^45 - 15 * q^46 - 12 * q^47 + 27 * q^49 + 12 * q^50 - 3 * q^51 - 9 * q^53 + 9 * q^54 + 6 * q^56 - 6 * q^58 - 3 * q^59 - 9 * q^60 - 24 * q^61 - 3 * q^62 + 3 * q^64 + 6 * q^65 + 9 * q^68 + 3 * q^69 + 21 * q^71 + 3 * q^72 + 3 * q^73 - 6 * q^74 + 36 * q^75 + 3 * q^76 + 15 * q^78 + 6 * q^79 - 3 * q^80 - 9 * q^81 + 9 * q^82 + 33 * q^83 + 15 * q^84 - 24 * q^85 + 21 * q^86 - 6 * q^87 + 6 * q^89 - 27 * q^90 + 36 * q^91 - 15 * q^92 + 36 * q^93 - 12 * q^94 - 3 * q^95 + 12 * q^97 + 27 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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