LMFDB - Embedded newform 4598.2.a.bu.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4598,2,Mod(1,4598)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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

Newform invariants

comment:select newform

sage:traces = [4,4,1,4,3,1,-1,4,5,3,0,1,-9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$36.7152148494$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.33452.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 8x^{2} + 7x - 1 $ x^4 - x^3 - 8x^2 + 7x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.94749$ 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)

$f(q)$	$=$	\(q+1.00000 q^{2} +2.94749 q^{3} +1.00000 q^{4} +2.36624 q^{5} +2.94749 q^{6} -1.66073 q^{7} +1.00000 q^{8} +5.68769 q^{9} +2.36624 q^{10} +2.94749 q^{12} -0.633765 q^{13} -1.66073 q^{14} +6.97445 q^{15} +1.00000 q^{16} +0.581254 q^{17} +5.68769 q^{18} -1.00000 q^{19} +2.36624 q^{20} -4.89498 q^{21} +3.60822 q^{23} +2.94749 q^{24} +0.599069 q^{25} -0.633765 q^{26} +7.92194 q^{27} -1.66073 q^{28} +9.13696 q^{29} +6.97445 q^{30} +1.02555 q^{31} +1.00000 q^{32} +0.581254 q^{34} -3.92967 q^{35} +5.68769 q^{36} +2.15895 q^{37} -1.00000 q^{38} -1.86801 q^{39} +2.36624 q^{40} -4.40093 q^{41} -4.89498 q^{42} -9.58267 q^{43} +13.4584 q^{45} +3.60822 q^{46} +8.73247 q^{47} +2.94749 q^{48} -4.24198 q^{49} +0.599069 q^{50} +1.71324 q^{51} -0.633765 q^{52} -12.5302 q^{53} +7.92194 q^{54} -1.66073 q^{56} -2.94749 q^{57} +9.13696 q^{58} +6.47623 q^{59} +6.97445 q^{60} -1.32146 q^{61} +1.02555 q^{62} -9.44571 q^{63} +1.00000 q^{64} -1.49964 q^{65} -0.393199 q^{67} +0.581254 q^{68} +10.6352 q^{69} -3.92967 q^{70} -0.312308 q^{71} +5.68769 q^{72} +9.31372 q^{73} +2.15895 q^{74} +1.76575 q^{75} -1.00000 q^{76} -1.86801 q^{78} -11.2704 q^{79} +2.36624 q^{80} +6.28676 q^{81} -4.40093 q^{82} +6.90413 q^{83} -4.89498 q^{84} +1.37538 q^{85} -9.58267 q^{86} +26.9311 q^{87} +0.519595 q^{89} +13.4584 q^{90} +1.05251 q^{91} +3.60822 q^{92} +3.02279 q^{93} +8.73247 q^{94} -2.36624 q^{95} +2.94749 q^{96} -8.37894 q^{97} -4.24198 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} + q^{3} + 4 q^{4} + 3 q^{5} + q^{6} - q^{7} + 4 q^{8} + 5 q^{9} + 3 q^{10} + q^{12} - 9 q^{13} - q^{14} + 5 q^{15} + 4 q^{16} - 2 q^{17} + 5 q^{18} - 4 q^{19} + 3 q^{20} + 2 q^{21} - 2 q^{23}+ \cdots - 7 q^{98}+O(q^{100}) $ 4 * q + 4 * q^2 + q^3 + 4 * q^4 + 3 * q^5 + q^6 - q^7 + 4 * q^8 + 5 * q^9 + 3 * q^10 + q^12 - 9 * q^13 - q^14 + 5 * q^15 + 4 * q^16 - 2 * q^17 + 5 * q^18 - 4 * q^19 + 3 * q^20 + 2 * q^21 - 2 * q^23 + q^24 + 15 * q^25 - 9 * q^26 - 2 * q^27 - q^28 + 5 * q^29 + 5 * q^30 + 27 * q^31 + 4 * q^32 - 2 * q^34 + 12 * q^35 + 5 * q^36 + 6 * q^37 - 4 * q^38 + 2 * q^39 + 3 * q^40 - 5 * q^41 + 2 * q^42 + q^43 + 11 * q^45 - 2 * q^46 + 22 * q^47 + q^48 - 7 * q^49 + 15 * q^50 + 12 * q^51 - 9 * q^52 - 2 * q^54 - q^56 - q^57 + 5 * q^58 + 5 * q^60 + 6 * q^61 + 27 * q^62 - 30 * q^63 + 4 * q^64 + 26 * q^65 + 17 * q^67 - 2 * q^68 + 14 * q^69 + 12 * q^70 - 19 * q^71 + 5 * q^72 + 20 * q^73 + 6 * q^74 + 23 * q^75 - 4 * q^76 + 2 * q^78 + 12 * q^79 + 3 * q^80 + 20 * q^81 - 5 * q^82 - 23 * q^83 + 2 * q^84 - 30 * q^85 + q^86 + 45 * q^87 + 16 * q^89 + 11 * q^90 + 15 * q^91 - 2 * q^92 - 12 * q^93 + 22 * q^94 - 3 * q^95 + q^96 + 8 * q^97 - 7 * 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
$3$	2.94749	1.70173	0.850867	−	0.525381i	$-0.176077\pi$
$3$	2.94749	1.70173	0.850867	+	0.525381i	$0.176077\pi$
$4$	1.00000	0.500000
$5$	2.36624	1.05821	0.529106	−	0.848556i	$-0.322527\pi$
$5$	2.36624	1.05821	0.529106	+	0.848556i	$0.322527\pi$
$6$	2.94749	1.20331
$7$	−1.66073	−0.627696	−0.313848	−	0.949473i	$-0.601618\pi$
$7$	−1.66073	−0.627696	−0.313848	+	0.949473i	$0.601618\pi$
$8$	1.00000	0.353553
$9$	5.68769	1.89590
$10$	2.36624	0.748269
$11$	0	0
$11$	0	0
$12$	2.94749	0.850867
$13$	−0.633765	−0.175775	−0.0878874	−	0.996130i	$-0.528012\pi$
$13$	−0.633765	−0.175775	−0.0878874	+	0.996130i	$0.528012\pi$
$14$	−1.66073	−0.443848
$15$	6.97445	1.80080
$16$	1.00000	0.250000
$17$	0.581254	0.140975	0.0704874	−	0.997513i	$-0.477545\pi$
$17$	0.581254	0.140975	0.0704874	+	0.997513i	$0.477545\pi$
$18$	5.68769	1.34060
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	2.36624	0.529106
$21$	−4.89498	−1.06817
$22$	0	0
$23$	3.60822	0.752365	0.376183	−	0.926546i	$-0.377237\pi$
$23$	3.60822	0.752365	0.376183	+	0.926546i	$0.377237\pi$
$24$	2.94749	0.601654
$25$	0.599069	0.119814
$26$	−0.633765	−0.124291
$27$	7.92194	1.52458
$28$	−1.66073	−0.313848
$29$	9.13696	1.69669	0.848345	−	0.529443i	$-0.177599\pi$
$29$	9.13696	1.69669	0.848345	+	0.529443i	$0.177599\pi$
$30$	6.97445	1.27335
$31$	1.02555	0.184194	0.0920969	−	0.995750i	$-0.470643\pi$
$31$	1.02555	0.184194	0.0920969	+	0.995750i	$0.470643\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	0.581254	0.0996842
$35$	−3.92967	−0.664236
$36$	5.68769	0.947949
$37$	2.15895	0.354929	0.177464	−	0.984127i	$-0.443211\pi$
$37$	2.15895	0.354929	0.177464	+	0.984127i	$0.443211\pi$
$38$	−1.00000	−0.162221
$39$	−1.86801	−0.299122
$40$	2.36624	0.374135
$41$	−4.40093	−0.687310	−0.343655	−	0.939096i	$-0.611665\pi$
$41$	−4.40093	−0.687310	−0.343655	+	0.939096i	$0.611665\pi$
$42$	−4.89498	−0.755312
$43$	−9.58267	−1.46134	−0.730672	−	0.682729i	$-0.760793\pi$
$43$	−9.58267	−1.46134	−0.730672	+	0.682729i	$0.760793\pi$
$44$	0	0
$45$	13.4584	2.00626
$46$	3.60822	0.532003
$47$	8.73247	1.27376	0.636881	−	0.770962i	$-0.280224\pi$
$47$	8.73247	1.27376	0.636881	+	0.770962i	$0.280224\pi$
$48$	2.94749	0.425433
$49$	−4.24198	−0.605997
$50$	0.599069	0.0847212
$51$	1.71324	0.239901
$52$	−0.633765	−0.0878874
$53$	−12.5302	−1.72115	−0.860575	−	0.509324i	$-0.829895\pi$
$53$	−12.5302	−1.72115	−0.860575	+	0.509324i	$0.829895\pi$
$54$	7.92194	1.07804
$55$	0	0
$56$	−1.66073	−0.221924
$57$	−2.94749	−0.390404
$58$	9.13696	1.19974
$59$	6.47623	0.843134	0.421567	−	0.906797i	$-0.361480\pi$
$59$	6.47623	0.843134	0.421567	+	0.906797i	$0.361480\pi$
$60$	6.97445	0.900398
$61$	−1.32146	−0.169195	−0.0845976	−	0.996415i	$-0.526960\pi$
$61$	−1.32146	−0.169195	−0.0845976	+	0.996415i	$0.526960\pi$
$62$	1.02555	0.130245
$63$	−9.44571	−1.19005
$64$	1.00000	0.125000
$65$	−1.49964	−0.186007
$66$	0	0
$67$	−0.393199	−0.0480369	−0.0240184	−	0.999712i	$-0.507646\pi$
$67$	−0.393199	−0.0480369	−0.0240184	+	0.999712i	$0.507646\pi$
$68$	0.581254	0.0704874
$69$	10.6352	1.28033
$70$	−3.92967	−0.469686
$71$	−0.312308	−0.0370642	−0.0185321	−	0.999828i	$-0.505899\pi$
$71$	−0.312308	−0.0370642	−0.0185321	+	0.999828i	$0.505899\pi$
$72$	5.68769	0.670301
$73$	9.31372	1.09009	0.545044	−	0.838407i	$-0.316513\pi$
$73$	9.31372	1.09009	0.545044	+	0.838407i	$0.316513\pi$
$74$	2.15895	0.250973
$75$	1.76575	0.203891
$76$	−1.00000	−0.114708
$77$	0	0
$78$	−1.86801	−0.211511
$79$	−11.2704	−1.26801	−0.634007	−	0.773327i	$-0.718591\pi$
$79$	−11.2704	−1.26801	−0.634007	+	0.773327i	$0.718591\pi$
$80$	2.36624	0.264553
$81$	6.28676	0.698529
$82$	−4.40093	−0.486001
$83$	6.90413	0.757826	0.378913	−	0.925432i	$-0.376298\pi$
$83$	6.90413	0.757826	0.378913	+	0.925432i	$0.376298\pi$
$84$	−4.89498	−0.534086
$85$	1.37538	0.149181
$86$	−9.58267	−1.03333
$87$	26.9311	2.88732
$88$	0	0
$89$	0.519595	0.0550769	0.0275385	−	0.999621i	$-0.491233\pi$
$89$	0.519595	0.0550769	0.0275385	+	0.999621i	$0.491233\pi$
$90$	13.4584	1.41864
$91$	1.05251	0.110333
$92$	3.60822	0.376183
$93$	3.02279	0.313449
$94$	8.73247	0.900686
$95$	−2.36624	−0.242771
$96$	2.94749	0.300827
$97$	−8.37894	−0.850753	−0.425376	−	0.905017i	$-0.639858\pi$
$97$	−8.37894	−0.850753	−0.425376	+	0.905017i	$0.639858\pi$
$98$	−4.24198	−0.428505
$99$	0	0
$100$	0.599069	0.0599069
$101$	16.5379	1.64558	0.822791	−	0.568344i	$-0.192416\pi$
$101$	16.5379	1.64558	0.822791	+	0.568344i	$0.192416\pi$
$102$	1.71324	0.169636
$103$	2.79627	0.275525	0.137762	−	0.990465i	$-0.456009\pi$
$103$	2.79627	0.275525	0.137762	+	0.990465i	$0.456009\pi$
$104$	−0.633765	−0.0621457
$105$	−11.5827	−1.13035
$106$	−12.5302	−1.21704
$107$	−18.9681	−1.83372	−0.916859	−	0.399210i	$-0.869284\pi$
$107$	−18.9681	−1.83372	−0.916859	+	0.399210i	$0.869284\pi$
$108$	7.92194	0.762289
$109$	−17.0732	−1.63531	−0.817656	−	0.575707i	$-0.804727\pi$
$109$	−17.0732	−1.63531	−0.817656	+	0.575707i	$0.804727\pi$
$110$	0	0
$111$	6.36348	0.603995
$112$	−1.66073	−0.156924
$113$	21.1114	1.98599	0.992997	−	0.118137i	$-0.0376922\pi$
$113$	21.1114	1.98599	0.992997	+	0.118137i	$0.0376922\pi$
$114$	−2.94749	−0.276058
$115$	8.53789	0.796162
$116$	9.13696	0.848345
$117$	−3.60466	−0.333251
$118$	6.47623	0.596185
$119$	−0.965305	−0.0884893
$120$	6.97445	0.636677
$121$	0	0
$122$	−1.32146	−0.119639
$123$	−12.9717	−1.16962
$124$	1.02555	0.0920969
$125$	−10.4136	−0.931424
$126$	−9.44571	−0.841491
$127$	−4.62745	−0.410620	−0.205310	−	0.978697i	$-0.565820\pi$
$127$	−4.62745	−0.410620	−0.205310	+	0.978697i	$0.565820\pi$
$128$	1.00000	0.0883883
$129$	−28.2448	−2.48682
$130$	−1.49964	−0.131527
$131$	15.7763	1.37838	0.689191	−	0.724579i	$-0.257966\pi$
$131$	15.7763	1.37838	0.689191	+	0.724579i	$0.257966\pi$
$132$	0	0
$133$	1.66073	0.144003
$134$	−0.393199	−0.0339672
$135$	18.7452	1.61333
$136$	0.581254	0.0498421
$137$	−20.6366	−1.76310	−0.881552	−	0.472087i	$-0.843501\pi$
$137$	−20.6366	−1.76310	−0.881552	+	0.472087i	$0.843501\pi$
$138$	10.6352	0.905327
$139$	9.56721	0.811480	0.405740	−	0.913989i	$-0.367014\pi$
$139$	9.56721	0.811480	0.405740	+	0.913989i	$0.367014\pi$
$140$	−3.92967	−0.332118
$141$	25.7389	2.16760
$142$	−0.312308	−0.0262083
$143$	0	0
$144$	5.68769	0.473974
$145$	21.6202	1.79546
$146$	9.31372	0.770809
$147$	−12.5032	−1.03125
$148$	2.15895	0.177464
$149$	−18.3096	−1.49998	−0.749988	−	0.661451i	$-0.769941\pi$
$149$	−18.3096	−1.49998	−0.749988	+	0.661451i	$0.769941\pi$
$150$	1.76575	0.144173
$151$	−6.25562	−0.509075	−0.254538	−	0.967063i	$-0.581923\pi$
$151$	−6.25562	−0.509075	−0.254538	+	0.967063i	$0.581923\pi$
$152$	−1.00000	−0.0811107
$153$	3.30599	0.267274
$154$	0	0
$155$	2.42669	0.194916
$156$	−1.86801	−0.149561
$157$	−6.15619	−0.491318	−0.245659	−	0.969356i	$-0.579004\pi$
$157$	−6.15619	−0.491318	−0.245659	+	0.969356i	$0.579004\pi$
$158$	−11.2704	−0.896622
$159$	−36.9325	−2.92894
$160$	2.36624	0.187067
$161$	−5.99227	−0.472257
$162$	6.28676	0.493935
$163$	−11.8439	−0.927685	−0.463842	−	0.885918i	$-0.653530\pi$
$163$	−11.8439	−0.927685	−0.463842	+	0.885918i	$0.653530\pi$
$164$	−4.40093	−0.343655
$165$	0	0
$166$	6.90413	0.535864
$167$	4.89142	0.378509	0.189255	−	0.981928i	$-0.439393\pi$
$167$	4.89142	0.378509	0.189255	+	0.981928i	$0.439393\pi$
$168$	−4.89498	−0.377656
$169$	−12.5983	−0.969103
$170$	1.37538	0.105487
$171$	−5.68769	−0.434949
$172$	−9.58267	−0.730672
$173$	18.1206	1.37768	0.688840	−	0.724913i	$-0.258120\pi$
$173$	18.1206	1.37768	0.688840	+	0.724913i	$0.258120\pi$
$174$	26.9311	2.04164
$175$	−0.994891	−0.0752067
$176$	0	0
$177$	19.0886	1.43479
$178$	0.519595	0.0389453
$179$	3.07947	0.230171	0.115085	−	0.993356i	$-0.463286\pi$
$179$	3.07947	0.230171	0.115085	+	0.993356i	$0.463286\pi$
$180$	13.4584	1.00313
$181$	0.0693910	0.00515779	0.00257889	−	0.999997i	$-0.499179\pi$
$181$	0.0693910	0.00515779	0.00257889	+	0.999997i	$0.499179\pi$
$182$	1.05251	0.0780173
$183$	−3.89498	−0.287925
$184$	3.60822	0.266001
$185$	5.10858	0.375590
$186$	3.02279	0.221642
$187$	0	0
$188$	8.73247	0.636881
$189$	−13.1562	−0.956972
$190$	−2.36624	−0.171665
$191$	13.6776	0.989677	0.494838	−	0.868985i	$-0.335227\pi$
$191$	13.6776	0.989677	0.494838	+	0.868985i	$0.335227\pi$
$192$	2.94749	0.212717
$193$	−5.61830	−0.404414	−0.202207	−	0.979343i	$-0.564811\pi$
$193$	−5.61830	−0.404414	−0.202207	+	0.979343i	$0.564811\pi$
$194$	−8.37894	−0.601573
$195$	−4.42016	−0.316534
$196$	−4.24198	−0.302999
$197$	−5.46494	−0.389361	−0.194680	−	0.980867i	$-0.562367\pi$
$197$	−5.46494	−0.389361	−0.194680	+	0.980867i	$0.562367\pi$
$198$	0	0
$199$	−13.5571	−0.961039	−0.480519	−	0.876984i	$-0.659552\pi$
$199$	−13.5571	−0.961039	−0.480519	+	0.876984i	$0.659552\pi$
$200$	0.599069	0.0423606
$201$	−1.15895	−0.0817459
$202$	16.5379	1.16360
$203$	−15.1740	−1.06501
$204$	1.71324	0.119951
$205$	−10.4136	−0.727320
$206$	2.79627	0.194826
$207$	20.5224	1.42641
$208$	−0.633765	−0.0439437
$209$	0	0
$210$	−11.5827	−0.799280
$211$	−14.3562	−0.988318	−0.494159	−	0.869371i	$-0.664524\pi$
$211$	−14.3562	−0.988318	−0.494159	+	0.869371i	$0.664524\pi$
$212$	−12.5302	−0.860575
$213$	−0.920526	−0.0630734
$214$	−18.9681	−1.29664
$215$	−22.6749	−1.54641
$216$	7.92194	0.539020
$217$	−1.70316	−0.115618
$218$	−17.0732	−1.15634
$219$	27.4521	1.85504
$220$	0	0
$221$	−0.368378	−0.0247798
$222$	6.36348	0.427089
$223$	−5.16251	−0.345707	−0.172854	−	0.984948i	$-0.555299\pi$
$223$	−5.16251	−0.345707	−0.172854	+	0.984948i	$0.555299\pi$
$224$	−1.66073	−0.110962
$225$	3.40732	0.227155
$226$	21.1114	1.40431
$227$	18.3435	1.21750	0.608751	−	0.793361i	$-0.291671\pi$
$227$	18.3435	1.21750	0.608751	+	0.793361i	$0.291671\pi$
$228$	−2.94749	−0.195202
$229$	18.3344	1.21157	0.605785	−	0.795629i	$-0.292859\pi$
$229$	18.3344	1.21157	0.605785	+	0.795629i	$0.292859\pi$
$230$	8.53789	0.562972
$231$	0	0
$232$	9.13696	0.599871
$233$	14.6429	0.959289	0.479645	−	0.877463i	$-0.340766\pi$
$233$	14.6429	0.959289	0.479645	+	0.877463i	$0.340766\pi$
$234$	−3.60466	−0.235644
$235$	20.6631	1.34791
$236$	6.47623	0.421567
$237$	−33.2193	−2.15782
$238$	−0.965305	−0.0625714
$239$	12.5868	0.814175	0.407088	−	0.913389i	$-0.366544\pi$
$239$	12.5868	0.814175	0.407088	+	0.913389i	$0.366544\pi$
$240$	6.97445	0.450199
$241$	2.79627	0.180124	0.0900619	−	0.995936i	$-0.471294\pi$
$241$	2.79627	0.180124	0.0900619	+	0.995936i	$0.471294\pi$
$242$	0	0
$243$	−5.23567	−0.335868
$244$	−1.32146	−0.0845976
$245$	−10.0375	−0.641274
$246$	−12.9717	−0.827045
$247$	0.633765	0.0403255
$248$	1.02555	0.0651223
$249$	20.3498	1.28962
$250$	−10.4136	−0.658616
$251$	−1.72891	−0.109128	−0.0545640	−	0.998510i	$-0.517377\pi$
$251$	−1.72891	−0.109128	−0.0545640	+	0.998510i	$0.517377\pi$
$252$	−9.44571	−0.595024
$253$	0	0
$254$	−4.62745	−0.290352
$255$	4.05393	0.253867
$256$	1.00000	0.0625000
$257$	21.5414	1.34372	0.671859	−	0.740679i	$-0.265496\pi$
$257$	21.5414	1.34372	0.671859	+	0.740679i	$0.265496\pi$
$258$	−28.2448	−1.75845
$259$	−3.58543	−0.222788
$260$	−1.49964	−0.0930035
$261$	51.9682	3.21675
$262$	15.7763	0.974664
$263$	22.7920	1.40541	0.702707	−	0.711479i	$-0.251974\pi$
$263$	22.7920	1.40541	0.702707	+	0.711479i	$0.251974\pi$
$264$	0	0
$265$	−29.6493	−1.82134
$266$	1.66073	0.101826
$267$	1.53150	0.0937263
$268$	−0.393199	−0.0240184
$269$	−18.6429	−1.13668	−0.568339	−	0.822794i	$-0.692414\pi$
$269$	−18.6429	−1.13668	−0.568339	+	0.822794i	$0.692414\pi$
$270$	18.7452	1.14080
$271$	3.33571	0.202630	0.101315	−	0.994854i	$-0.467695\pi$
$271$	3.33571	0.202630	0.101315	+	0.994854i	$0.467695\pi$
$272$	0.581254	0.0352437
$273$	3.10226	0.187758
$274$	−20.6366	−1.24670
$275$	0	0
$276$	10.6352	0.640163
$277$	−23.5883	−1.41728	−0.708641	−	0.705570i	$-0.750691\pi$
$277$	−23.5883	−1.41728	−0.708641	+	0.705570i	$0.750691\pi$
$278$	9.56721	0.573803
$279$	5.83300	0.349212
$280$	−3.92967	−0.234843
$281$	−31.6366	−1.88728	−0.943641	−	0.330972i	$-0.892623\pi$
$281$	−31.6366	−1.88728	−0.943641	+	0.330972i	$0.892623\pi$
$282$	25.7389	1.53273
$283$	−6.06757	−0.360680	−0.180340	−	0.983604i	$-0.557720\pi$
$283$	−6.06757	−0.360680	−0.180340	+	0.983604i	$0.557720\pi$
$284$	−0.312308	−0.0185321
$285$	−6.97445	−0.413131
$286$	0	0
$287$	7.30875	0.431422
$288$	5.68769	0.335150
$289$	−16.6621	−0.980126
$290$	21.6202	1.26958
$291$	−24.6968	−1.44775
$292$	9.31372	0.545044
$293$	−16.8312	−0.983288	−0.491644	−	0.870796i	$-0.663604\pi$
$293$	−16.8312	−0.983288	−0.491644	+	0.870796i	$0.663604\pi$
$294$	−12.5032	−0.729201
$295$	15.3243	0.892215
$296$	2.15895	0.125486
$297$	0	0
$298$	−18.3096	−1.06064
$299$	−2.28676	−0.132247
$300$	1.76575	0.101946
$301$	15.9142	0.917280
$302$	−6.25562	−0.359971
$303$	48.7453	2.80034
$304$	−1.00000	−0.0573539
$305$	−3.12688	−0.179044
$306$	3.30599	0.188991
$307$	5.10858	0.291562	0.145781	−	0.989317i	$-0.453430\pi$
$307$	5.10858	0.291562	0.145781	+	0.989317i	$0.453430\pi$
$308$	0	0
$309$	8.24198	0.468870
$310$	2.42669	0.137827
$311$	−24.5635	−1.39287	−0.696435	−	0.717620i	$-0.745231\pi$
$311$	−24.5635	−1.39287	−0.696435	+	0.717620i	$0.745231\pi$
$312$	−1.86801	−0.105756
$313$	−24.2868	−1.37277	−0.686387	−	0.727237i	$-0.740804\pi$
$313$	−24.2868	−1.37277	−0.686387	+	0.727237i	$0.740804\pi$
$314$	−6.15619	−0.347414
$315$	−22.3508	−1.25932
$316$	−11.2704	−0.634007
$317$	−28.6225	−1.60760	−0.803801	−	0.594898i	$-0.797192\pi$
$317$	−28.6225	−1.60760	−0.803801	+	0.594898i	$0.797192\pi$
$318$	−36.9325	−2.07107
$319$	0	0
$320$	2.36624	0.132277
$321$	−55.9084	−3.12050
$322$	−5.99227	−0.333936
$323$	−0.581254	−0.0323418
$324$	6.28676	0.349264
$325$	−0.379669	−0.0210602
$326$	−11.8439	−0.655972
$327$	−50.3229	−2.78287
$328$	−4.40093	−0.243001
$329$	−14.5023	−0.799535
$330$	0	0
$331$	−2.40960	−0.132444	−0.0662218	−	0.997805i	$-0.521094\pi$
$331$	−2.40960	−0.132444	−0.0662218	+	0.997805i	$0.521094\pi$
$332$	6.90413	0.378913
$333$	12.2794	0.672909
$334$	4.89142	0.267647
$335$	−0.930401	−0.0508332
$336$	−4.89498	−0.267043
$337$	17.5195	0.954346	0.477173	−	0.878809i	$-0.341662\pi$
$337$	17.5195	0.954346	0.477173	+	0.878809i	$0.341662\pi$
$338$	−12.5983	−0.685259
$339$	62.2257	3.37963
$340$	1.37538	0.0745906
$341$	0	0
$342$	−5.68769	−0.307555
$343$	18.6699	1.00808
$344$	−9.58267	−0.516663
$345$	25.1653	1.35486
$346$	18.1206	0.974167
$347$	32.1534	1.72609	0.863043	−	0.505130i	$-0.168555\pi$
$347$	32.1534	1.72609	0.863043	+	0.505130i	$0.168555\pi$
$348$	26.9311	1.44366
$349$	0.407455	0.0218106	0.0109053	−	0.999941i	$-0.496529\pi$
$349$	0.407455	0.0218106	0.0109053	+	0.999941i	$0.496529\pi$
$350$	−0.994891	−0.0531792
$351$	−5.02065	−0.267982
$352$	0	0
$353$	3.94158	0.209789	0.104895	−	0.994483i	$-0.466549\pi$
$353$	3.94158	0.209789	0.104895	+	0.994483i	$0.466549\pi$
$354$	19.0886	1.01455
$355$	−0.738995	−0.0392218
$356$	0.519595	0.0275385
$357$	−2.84522	−0.150585
$358$	3.07947	0.162755
$359$	−20.0781	−1.05968	−0.529842	−	0.848097i	$-0.677749\pi$
$359$	−20.0781	−1.05968	−0.529842	+	0.848097i	$0.677749\pi$
$360$	13.4584	0.709321
$361$	1.00000	0.0526316
$362$	0.0693910	0.00364711
$363$	0	0
$364$	1.05251	0.0551666
$365$	22.0385	1.15355
$366$	−3.89498	−0.203594
$367$	−20.5918	−1.07488	−0.537442	−	0.843301i	$-0.680609\pi$
$367$	−20.5918	−1.07488	−0.537442	+	0.843301i	$0.680609\pi$
$368$	3.60822	0.188091
$369$	−25.0311	−1.30307
$370$	5.10858	0.265582
$371$	20.8092	1.08036
$372$	3.02279	0.156724
$373$	−20.8813	−1.08119	−0.540597	−	0.841282i	$-0.681802\pi$
$373$	−20.8813	−1.08119	−0.540597	+	0.841282i	$0.681802\pi$
$374$	0	0
$375$	−30.6941	−1.58504
$376$	8.73247	0.450343
$377$	−5.79068	−0.298235
$378$	−13.1562	−0.676681
$379$	−0.753845	−0.0387224	−0.0193612	−	0.999813i	$-0.506163\pi$
$379$	−0.753845	−0.0387224	−0.0193612	+	0.999813i	$0.506163\pi$
$380$	−2.36624	−0.121385
$381$	−13.6394	−0.698765
$382$	13.6776	0.699807
$383$	−18.3142	−0.935812	−0.467906	−	0.883778i	$-0.654991\pi$
$383$	−18.3142	−0.935812	−0.467906	+	0.883778i	$0.654991\pi$
$384$	2.94749	0.150413
$385$	0	0
$386$	−5.61830	−0.285964
$387$	−54.5033	−2.77056
$388$	−8.37894	−0.425376
$389$	−26.1206	−1.32436	−0.662182	−	0.749343i	$-0.730370\pi$
$389$	−26.1206	−1.32436	−0.662182	+	0.749343i	$0.730370\pi$
$390$	−4.42016	−0.223824
$391$	2.09729	0.106065
$392$	−4.24198	−0.214252
$393$	46.5005	2.34564
$394$	−5.46494	−0.275320
$395$	−26.6683	−1.34183
$396$	0	0
$397$	−34.9079	−1.75198	−0.875988	−	0.482332i	$-0.839790\pi$
$397$	−34.9079	−1.75198	−0.875988	+	0.482332i	$0.839790\pi$
$398$	−13.5571	−0.679557
$399$	4.89498	0.245055
$400$	0.599069	0.0299535
$401$	6.35353	0.317280	0.158640	−	0.987336i	$-0.449289\pi$
$401$	6.35353	0.317280	0.158640	+	0.987336i	$0.449289\pi$
$402$	−1.15895	−0.0578031
$403$	−0.649956	−0.0323766
$404$	16.5379	0.822791
$405$	14.8760	0.739192
$406$	−15.1740	−0.753073
$407$	0	0
$408$	1.71324	0.0848180
$409$	26.1816	1.29460	0.647299	−	0.762237i	$-0.275899\pi$
$409$	26.1816	1.29460	0.647299	+	0.762237i	$0.275899\pi$
$410$	−10.4136	−0.514293
$411$	−60.8261	−3.00033
$412$	2.79627	0.137762
$413$	−10.7553	−0.529232
$414$	20.5224	1.00862
$415$	16.3368	0.801941
$416$	−0.633765	−0.0310729
$417$	28.1992	1.38092
$418$	0	0
$419$	8.15060	0.398183	0.199091	−	0.979981i	$-0.436201\pi$
$419$	8.15060	0.398183	0.199091	+	0.979981i	$0.436201\pi$
$420$	−11.5827	−0.565176
$421$	10.9531	0.533820	0.266910	−	0.963721i	$-0.413997\pi$
$421$	10.9531	0.533820	0.266910	+	0.963721i	$0.413997\pi$
$422$	−14.3562	−0.698847
$423$	49.6676	2.41492
$424$	−12.5302	−0.608518
$425$	0.348211	0.0168907
$426$	−0.920526	−0.0445996
$427$	2.19458	0.106203
$428$	−18.9681	−0.916859
$429$	0	0
$430$	−22.6749	−1.09348
$431$	17.8510	0.859852	0.429926	−	0.902864i	$-0.358540\pi$
$431$	17.8510	0.859852	0.429926	+	0.902864i	$0.358540\pi$
$432$	7.92194	0.381145
$433$	39.4138	1.89411	0.947054	−	0.321074i	$-0.104044\pi$
$433$	39.4138	1.89411	0.947054	+	0.321074i	$0.104044\pi$
$434$	−1.70316	−0.0817541
$435$	63.7253	3.05539
$436$	−17.0732	−0.817656
$437$	−3.60822	−0.172604
$438$	27.4521	1.31171
$439$	16.9993	0.811331	0.405666	−	0.914022i	$-0.367040\pi$
$439$	16.9993	0.811331	0.405666	+	0.914022i	$0.367040\pi$
$440$	0	0
$441$	−24.1271	−1.14891
$442$	−0.368378	−0.0175220
$443$	16.2668	0.772859	0.386430	−	0.922319i	$-0.373708\pi$
$443$	16.2668	0.772859	0.386430	+	0.922319i	$0.373708\pi$
$444$	6.36348	0.301997
$445$	1.22948	0.0582831
$446$	−5.16251	−0.244452
$447$	−53.9672	−2.55256
$448$	−1.66073	−0.0784620
$449$	16.5534	0.781201	0.390601	−	0.920560i	$-0.372267\pi$
$449$	16.5534	0.781201	0.390601	+	0.920560i	$0.372267\pi$
$450$	3.40732	0.160623
$451$	0	0
$452$	21.1114	0.992997
$453$	−18.4384	−0.866311
$454$	18.3435	0.860904
$455$	2.49049	0.116756
$456$	−2.94749	−0.138029
$457$	−11.5805	−0.541714	−0.270857	−	0.962620i	$-0.587307\pi$
$457$	−11.5805	−0.541714	−0.270857	+	0.962620i	$0.587307\pi$
$458$	18.3344	0.856709
$459$	4.60466	0.214927
$460$	8.53789	0.398081
$461$	−5.42648	−0.252736	−0.126368	−	0.991983i	$-0.540332\pi$
$461$	−5.42648	−0.252736	−0.126368	+	0.991983i	$0.540332\pi$
$462$	0	0
$463$	26.9168	1.25093	0.625466	−	0.780252i	$-0.284909\pi$
$463$	26.9168	1.25093	0.625466	+	0.780252i	$0.284909\pi$
$464$	9.13696	0.424173
$465$	7.15263	0.331695
$466$	14.6429	0.678320
$467$	12.5581	0.581118	0.290559	−	0.956857i	$-0.406159\pi$
$467$	12.5581	0.581118	0.290559	+	0.956857i	$0.406159\pi$
$468$	−3.60466	−0.166625
$469$	0.652996	0.0301526
$470$	20.6631	0.953117
$471$	−18.1453	−0.836092
$472$	6.47623	0.298093
$473$	0	0
$474$	−33.2193	−1.52581
$475$	−0.599069	−0.0274872
$476$	−0.965305	−0.0442447
$477$	−71.2677	−3.26312
$478$	12.5868	0.575709
$479$	40.2322	1.83826	0.919128	−	0.393960i	$-0.128895\pi$
$479$	40.2322	1.83826	0.919128	+	0.393960i	$0.128895\pi$
$480$	6.97445	0.318339
$481$	−1.36827	−0.0623875
$482$	2.79627	0.127367
$483$	−17.6621	−0.803655
$484$	0	0
$485$	−19.8265	−0.900277
$486$	−5.23567	−0.237495
$487$	−14.2795	−0.647066	−0.323533	−	0.946217i	$-0.604871\pi$
$487$	−14.2795	−0.647066	−0.323533	+	0.946217i	$0.604871\pi$
$488$	−1.32146	−0.0598195
$489$	−34.9097	−1.57867
$490$	−10.0375	−0.453449
$491$	14.0094	0.632233	0.316117	−	0.948720i	$-0.397621\pi$
$491$	14.0094	0.632233	0.316117	+	0.948720i	$0.397621\pi$
$492$	−12.9717	−0.584809
$493$	5.31089	0.239191
$494$	0.633765	0.0285144
$495$	0	0
$496$	1.02555	0.0460484
$497$	0.518659	0.0232651
$498$	20.3498	0.911898
$499$	18.1506	0.812533	0.406266	−	0.913755i	$-0.366831\pi$
$499$	18.1506	0.812533	0.406266	+	0.913755i	$0.366831\pi$
$500$	−10.4136	−0.465712
$501$	14.4174	0.644122
$502$	−1.72891	−0.0771651
$503$	−29.8672	−1.33171	−0.665855	−	0.746081i	$-0.731933\pi$
$503$	−29.8672	−1.33171	−0.665855	+	0.746081i	$0.731933\pi$
$504$	−9.44571	−0.420745
$505$	39.1325	1.74138
$506$	0	0
$507$	−37.1335	−1.64916
$508$	−4.62745	−0.205310
$509$	24.2045	1.07285	0.536423	−	0.843949i	$-0.319775\pi$
$509$	24.2045	1.07285	0.536423	+	0.843949i	$0.319775\pi$
$510$	4.05393	0.179511
$511$	−15.4676	−0.684245
$512$	1.00000	0.0441942
$513$	−7.92194	−0.349762
$514$	21.5414	0.950153
$515$	6.61664	0.291564
$516$	−28.2448	−1.24341
$517$	0	0
$518$	−3.58543	−0.157535
$519$	53.4102	2.34445
$520$	−1.49964	−0.0657634
$521$	−6.36899	−0.279031	−0.139515	−	0.990220i	$-0.544554\pi$
$521$	−6.36899	−0.279031	−0.139515	+	0.990220i	$0.544554\pi$
$522$	51.9682	2.27459
$523$	−21.2749	−0.930284	−0.465142	−	0.885236i	$-0.653997\pi$
$523$	−21.2749	−0.930284	−0.465142	+	0.885236i	$0.653997\pi$
$524$	15.7763	0.689191
$525$	−2.93243	−0.127982
$526$	22.7920	0.993778
$527$	0.596103	0.0259667
$528$	0	0
$529$	−9.98077	−0.433946
$530$	−29.6493	−1.28788
$531$	36.8348	1.59849
$532$	1.66073	0.0720017
$533$	2.78915	0.120812
$534$	1.53150	0.0662745
$535$	−44.8831	−1.94046
$536$	−0.393199	−0.0169836
$537$	9.07672	0.391689
$538$	−18.6429	−0.803753
$539$	0	0
$540$	18.7452	0.806664
$541$	−34.2668	−1.47324	−0.736622	−	0.676304i	$-0.763580\pi$
$541$	−34.2668	−1.47324	−0.736622	+	0.676304i	$0.763580\pi$
$542$	3.33571	0.143281
$543$	0.204529	0.00877718
$544$	0.581254	0.0249211
$545$	−40.3991	−1.73051
$546$	3.10226	0.132765
$547$	−31.2732	−1.33715	−0.668573	−	0.743647i	$-0.733094\pi$
$547$	−31.2732	−1.33715	−0.668573	+	0.743647i	$0.733094\pi$
$548$	−20.6366	−0.881552
$549$	−7.51604	−0.320777
$550$	0	0
$551$	−9.13696	−0.389248
$552$	10.6352	0.452663
$553$	18.7170	0.795928
$554$	−23.5883	−1.00217
$555$	15.0575	0.639155
$556$	9.56721	0.405740
$557$	24.7198	1.04741	0.523707	−	0.851899i	$-0.324549\pi$
$557$	24.7198	1.04741	0.523707	+	0.851899i	$0.324549\pi$
$558$	5.83300	0.246931
$559$	6.07316	0.256867
$560$	−3.92967	−0.166059
$561$	0	0
$562$	−31.6366	−1.33451
$563$	8.94251	0.376882	0.188441	−	0.982085i	$-0.439657\pi$
$563$	8.94251	0.376882	0.188441	+	0.982085i	$0.439657\pi$
$564$	25.7389	1.08380
$565$	49.9546	2.10160
$566$	−6.06757	−0.255039
$567$	−10.4406	−0.438464
$568$	−0.312308	−0.0131042
$569$	−0.764409	−0.0320457	−0.0160228	−	0.999872i	$-0.505100\pi$
$569$	−0.764409	−0.0320457	−0.0160228	+	0.999872i	$0.505100\pi$
$570$	−6.97445	−0.292128
$571$	4.11866	0.172361	0.0861804	−	0.996280i	$-0.472534\pi$
$571$	4.11866	0.172361	0.0861804	+	0.996280i	$0.472534\pi$
$572$	0	0
$573$	40.3146	1.68417
$574$	7.30875	0.305061
$575$	2.16157	0.0901438
$576$	5.68769	0.236987
$577$	26.8538	1.11794	0.558968	−	0.829189i	$-0.311197\pi$
$577$	26.8538	1.11794	0.558968	+	0.829189i	$0.311197\pi$
$578$	−16.6621	−0.693054
$579$	−16.5599	−0.688205
$580$	21.6202	0.897730
$581$	−11.4659	−0.475685
$582$	−24.6968	−1.02372
$583$	0	0
$584$	9.31372	0.385405
$585$	−8.52947	−0.352650
$586$	−16.8312	−0.695289
$587$	11.4832	0.473964	0.236982	−	0.971514i	$-0.423842\pi$
$587$	11.4832	0.473964	0.236982	+	0.971514i	$0.423842\pi$
$588$	−12.5032	−0.515623
$589$	−1.02555	−0.0422570
$590$	15.3243	0.630891
$591$	−16.1079	−0.662589
$592$	2.15895	0.0887322
$593$	32.3508	1.32849	0.664245	−	0.747515i	$-0.268753\pi$
$593$	32.3508	1.32849	0.664245	+	0.747515i	$0.268753\pi$
$594$	0	0
$595$	−2.28414	−0.0936405
$596$	−18.3096	−0.749988
$597$	−39.9595	−1.63543
$598$	−2.28676	−0.0935126
$599$	6.06380	0.247760	0.123880	−	0.992297i	$-0.460466\pi$
$599$	6.06380	0.247760	0.123880	+	0.992297i	$0.460466\pi$
$600$	1.76575	0.0720864
$601$	−23.0923	−0.941953	−0.470976	−	0.882146i	$-0.656098\pi$
$601$	−23.0923	−0.941953	−0.470976	+	0.882146i	$0.656098\pi$
$602$	15.9142	0.648615
$603$	−2.23639	−0.0910730
$604$	−6.25562	−0.254538
$605$	0	0
$606$	48.7453	1.98014
$607$	−11.1808	−0.453815	−0.226907	−	0.973916i	$-0.572861\pi$
$607$	−11.1808	−0.453815	−0.226907	+	0.973916i	$0.572861\pi$
$608$	−1.00000	−0.0405554
$609$	−44.7252	−1.81236
$610$	−3.12688	−0.126604
$611$	−5.53433	−0.223895
$612$	3.30599	0.133637
$613$	−33.8355	−1.36660	−0.683302	−	0.730136i	$-0.739457\pi$
$613$	−33.8355	−1.36660	−0.683302	+	0.730136i	$0.739457\pi$
$614$	5.10858	0.206166
$615$	−30.6941	−1.23770
$616$	0	0
$617$	24.9927	1.00617	0.503085	−	0.864237i	$-0.332198\pi$
$617$	24.9927	1.00617	0.503085	+	0.864237i	$0.332198\pi$
$618$	8.24198	0.331541
$619$	7.68493	0.308884	0.154442	−	0.988002i	$-0.450642\pi$
$619$	7.68493	0.308884	0.154442	+	0.988002i	$0.450642\pi$
$620$	2.42669	0.0974581
$621$	28.5841	1.14704
$622$	−24.5635	−0.984907
$623$	−0.862906	−0.0345716
$624$	−1.86801	−0.0747804
$625$	−27.6365	−1.10546
$626$	−24.2868	−0.970697
$627$	0	0
$628$	−6.15619	−0.245659
$629$	1.25490	0.0500360
$630$	−22.3508	−0.890476
$631$	25.7004	1.02312	0.511558	−	0.859249i	$-0.329068\pi$
$631$	25.7004	1.02312	0.511558	+	0.859249i	$0.329068\pi$
$632$	−11.2704	−0.448311
$633$	−42.3146	−1.68185
$634$	−28.6225	−1.13675
$635$	−10.9496	−0.434523
$636$	−36.9325	−1.46447
$637$	2.68842	0.106519
$638$	0	0
$639$	−1.77631	−0.0702699
$640$	2.36624	0.0935337
$641$	4.03846	0.159510	0.0797548	−	0.996815i	$-0.474586\pi$
$641$	4.03846	0.159510	0.0797548	+	0.996815i	$0.474586\pi$
$642$	−55.9084	−2.20653
$643$	5.44194	0.214609	0.107305	−	0.994226i	$-0.465778\pi$
$643$	5.44194	0.214609	0.107305	+	0.994226i	$0.465778\pi$
$644$	−5.99227	−0.236128
$645$	−66.8339	−2.63158
$646$	−0.581254	−0.0228691
$647$	−32.4046	−1.27395	−0.636977	−	0.770882i	$-0.719816\pi$
$647$	−32.4046	−1.27395	−0.636977	+	0.770882i	$0.719816\pi$
$648$	6.28676	0.246967
$649$	0	0
$650$	−0.379669	−0.0148918
$651$	−5.02003	−0.196751
$652$	−11.8439	−0.463842
$653$	24.7672	0.969217	0.484609	−	0.874731i	$-0.338962\pi$
$653$	24.7672	0.969217	0.484609	+	0.874731i	$0.338962\pi$
$654$	−50.3229	−1.96778
$655$	37.3305	1.45862
$656$	−4.40093	−0.171827
$657$	52.9736	2.06670
$658$	−14.5023	−0.565357
$659$	27.5600	1.07358	0.536792	−	0.843715i	$-0.319636\pi$
$659$	27.5600	1.07358	0.536792	+	0.843715i	$0.319636\pi$
$660$	0	0
$661$	−7.53372	−0.293028	−0.146514	−	0.989209i	$-0.546805\pi$
$661$	−7.53372	−0.293028	−0.146514	+	0.989209i	$0.546805\pi$
$662$	−2.40960	−0.0936517
$663$	−1.08579	−0.0421686
$664$	6.90413	0.267932
$665$	3.92967	0.152386
$666$	12.2794	0.475818
$667$	32.9681	1.27653
$668$	4.89142	0.189255
$669$	−15.2164	−0.588301
$670$	−0.930401	−0.0359445
$671$	0	0
$672$	−4.89498	−0.188828
$673$	8.79534	0.339035	0.169518	−	0.985527i	$-0.445779\pi$
$673$	8.79534	0.339035	0.169518	+	0.985527i	$0.445779\pi$
$674$	17.5195	0.674824
$675$	4.74579	0.182666
$676$	−12.5983	−0.484552
$677$	−15.8310	−0.608434	−0.304217	−	0.952603i	$-0.598395\pi$
$677$	−15.8310	−0.608434	−0.304217	+	0.952603i	$0.598395\pi$
$678$	62.2257	2.38976
$679$	13.9151	0.534014
$680$	1.37538	0.0527435
$681$	54.0673	2.07186
$682$	0	0
$683$	38.6640	1.47944	0.739719	−	0.672916i	$-0.234958\pi$
$683$	38.6640	1.47944	0.739719	+	0.672916i	$0.234958\pi$
$684$	−5.68769	−0.217474
$685$	−48.8310	−1.86574
$686$	18.6699	0.712819
$687$	54.0404	2.06177
$688$	−9.58267	−0.365336
$689$	7.94117	0.302535
$690$	25.1653	0.958028
$691$	−27.4678	−1.04492	−0.522462	−	0.852663i	$-0.674986\pi$
$691$	−27.4678	−1.04492	−0.522462	+	0.852663i	$0.674986\pi$
$692$	18.1206	0.688840
$693$	0	0
$694$	32.1534	1.22053
$695$	22.6383	0.858718
$696$	26.9311	1.02082
$697$	−2.55806	−0.0968933
$698$	0.407455	0.0154224
$699$	43.1598	1.63245
$700$	−0.994891	−0.0376034
$701$	1.50892	0.0569911	0.0284955	−	0.999594i	$-0.490928\pi$
$701$	1.50892	0.0569911	0.0284955	+	0.999594i	$0.490928\pi$
$702$	−5.02065	−0.189492
$703$	−2.15895	−0.0814263
$704$	0	0
$705$	60.9042	2.29378
$706$	3.94158	0.148343
$707$	−27.4649	−1.03293
$708$	19.0886	0.717394
$709$	16.2959	0.612006	0.306003	−	0.952031i	$-0.401008\pi$
$709$	16.2959	0.612006	0.306003	+	0.952031i	$0.401008\pi$
$710$	−0.738995	−0.0277340
$711$	−64.1023	−2.40403
$712$	0.519595	0.0194726
$713$	3.70040	0.138581
$714$	−2.84522	−0.106480
$715$	0	0
$716$	3.07947	0.115085
$717$	37.0996	1.38551
$718$	−20.0781	−0.749309
$719$	40.6785	1.51705	0.758526	−	0.651643i	$-0.225920\pi$
$719$	40.6785	1.51705	0.758526	+	0.651643i	$0.225920\pi$
$720$	13.4584	0.501566
$721$	−4.64385	−0.172946
$722$	1.00000	0.0372161
$723$	8.24198	0.306523
$724$	0.0693910	0.00257889
$725$	5.47367	0.203287
$726$	0	0
$727$	27.7545	1.02936	0.514679	−	0.857383i	$-0.327911\pi$
$727$	27.7545	1.02936	0.514679	+	0.857383i	$0.327911\pi$
$728$	1.05251	0.0390087
$729$	−34.2924	−1.27009
$730$	22.0385	0.815680
$731$	−5.56996	−0.206013
$732$	−3.89498	−0.143963
$733$	6.89854	0.254803	0.127402	−	0.991851i	$-0.459336\pi$
$733$	6.89854	0.254803	0.127402	+	0.991851i	$0.459336\pi$
$734$	−20.5918	−0.760058
$735$	−29.5855	−1.09128
$736$	3.60822	0.133001
$737$	0	0
$738$	−25.0311	−0.921409
$739$	−49.2039	−1.80999	−0.904997	−	0.425418i	$-0.860127\pi$
$739$	−49.2039	−1.80999	−0.904997	+	0.425418i	$0.860127\pi$
$740$	5.10858	0.187795
$741$	1.86801	0.0686232
$742$	20.8092	0.763929
$743$	−10.4046	−0.381709	−0.190854	−	0.981618i	$-0.561126\pi$
$743$	−10.4046	−0.381709	−0.190854	+	0.981618i	$0.561126\pi$
$744$	3.02279	0.110821
$745$	−43.3247	−1.58729
$746$	−20.8813	−0.764520
$747$	39.2685	1.43676
$748$	0	0
$749$	31.5009	1.15102
$750$	−30.6941	−1.12079
$751$	37.7103	1.37607	0.688035	−	0.725677i	$-0.258474\pi$
$751$	37.7103	1.37607	0.688035	+	0.725677i	$0.258474\pi$
$752$	8.73247	0.318440
$753$	−5.09595	−0.185707
$754$	−5.79068	−0.210884
$755$	−14.8023	−0.538710
$756$	−13.1562	−0.478486
$757$	14.3972	0.523274	0.261637	−	0.965166i	$-0.415738\pi$
$757$	14.3972	0.523274	0.261637	+	0.965166i	$0.415738\pi$
$758$	−0.753845	−0.0273809
$759$	0	0
$760$	−2.36624	−0.0858324
$761$	39.7311	1.44025	0.720126	−	0.693843i	$-0.244084\pi$
$761$	39.7311	1.44025	0.720126	+	0.693843i	$0.244084\pi$
$762$	−13.6394	−0.494102
$763$	28.3539	1.02648
$764$	13.6776	0.494838
$765$	7.82276	0.282832
$766$	−18.3142	−0.661719
$767$	−4.10441	−0.148202
$768$	2.94749	0.106358
$769$	2.27170	0.0819197	0.0409598	−	0.999161i	$-0.486958\pi$
$769$	2.27170	0.0819197	0.0409598	+	0.999161i	$0.486958\pi$
$770$	0	0
$771$	63.4932	2.28665
$772$	−5.61830	−0.202207
$773$	3.64157	0.130978	0.0654891	−	0.997853i	$-0.479139\pi$
$773$	3.64157	0.130978	0.0654891	+	0.997853i	$0.479139\pi$
$774$	−54.5033	−1.95908
$775$	0.614374	0.0220690
$776$	−8.37894	−0.300786
$777$	−10.5680	−0.379125
$778$	−26.1206	−0.936467
$779$	4.40093	0.157680
$780$	−4.42016	−0.158267
$781$	0	0
$782$	2.09729	0.0749989
$783$	72.3825	2.58674
$784$	−4.24198	−0.151499
$785$	−14.5670	−0.519918
$786$	46.5005	1.65862
$787$	−0.279643	−0.00996821	−0.00498410	−	0.999988i	$-0.501586\pi$
$787$	−0.279643	−0.00996821	−0.00498410	+	0.999988i	$0.501586\pi$
$788$	−5.46494	−0.194680
$789$	67.1791	2.39164
$790$	−26.6683	−0.948816
$791$	−35.0603	−1.24660
$792$	0	0
$793$	0.837492	0.0297402
$794$	−34.9079	−1.23883
$795$	−87.3910	−3.09944
$796$	−13.5571	−0.480519
$797$	−3.66014	−0.129649	−0.0648243	−	0.997897i	$-0.520649\pi$
$797$	−3.66014	−0.129649	−0.0648243	+	0.997897i	$0.520649\pi$
$798$	4.89498	0.173280
$799$	5.07578	0.179568
$800$	0.599069	0.0211803
$801$	2.95530	0.104420
$802$	6.35353	0.224351
$803$	0	0
$804$	−1.15895	−0.0408730
$805$	−14.1791	−0.499748
$806$	−0.649956	−0.0228937
$807$	−54.9498	−1.93432
$808$	16.5379	0.581801
$809$	27.0990	0.952749	0.476375	−	0.879242i	$-0.341951\pi$
$809$	27.0990	0.952749	0.476375	+	0.879242i	$0.341951\pi$
$810$	14.8760	0.522688
$811$	31.4212	1.10335	0.551673	−	0.834060i	$-0.313990\pi$
$811$	31.4212	1.10335	0.551673	+	0.834060i	$0.313990\pi$
$812$	−15.1740	−0.532503
$813$	9.83198	0.344823
$814$	0	0
$815$	−28.0254	−0.981687
$816$	1.71324	0.0599754
$817$	9.58267	0.335255
$818$	26.1816	0.915418
$819$	5.98636	0.209180
$820$	−10.4136	−0.363660
$821$	−20.1304	−0.702557	−0.351279	−	0.936271i	$-0.614253\pi$
$821$	−20.1304	−0.702557	−0.351279	+	0.936271i	$0.614253\pi$
$822$	−60.8261	−2.12156
$823$	−7.23922	−0.252344	−0.126172	−	0.992008i	$-0.540269\pi$
$823$	−7.23922	−0.252344	−0.126172	+	0.992008i	$0.540269\pi$
$824$	2.79627	0.0974128
$825$	0	0
$826$	−10.7553	−0.374223
$827$	23.1088	0.803571	0.401786	−	0.915734i	$-0.368390\pi$
$827$	23.1088	0.803571	0.401786	+	0.915734i	$0.368390\pi$
$828$	20.5224	0.713204
$829$	21.7081	0.753955	0.376977	−	0.926223i	$-0.376963\pi$
$829$	21.7081	0.753955	0.376977	+	0.926223i	$0.376963\pi$
$830$	16.3368	0.567058
$831$	−69.5261	−2.41183
$832$	−0.633765	−0.0219718
$833$	−2.46567	−0.0854303
$834$	28.1992	0.976460
$835$	11.5742	0.400543
$836$	0	0
$837$	8.12433	0.280818
$838$	8.15060	0.281558
$839$	−21.1935	−0.731681	−0.365841	−	0.930678i	$-0.619218\pi$
$839$	−21.1935	−0.731681	−0.365841	+	0.930678i	$0.619218\pi$
$840$	−11.5827	−0.399640
$841$	54.4840	1.87876
$842$	10.9531	0.377468
$843$	−93.2485	−3.21165
$844$	−14.3562	−0.494159
$845$	−29.8106	−1.02552
$846$	49.6676	1.70761
$847$	0	0
$848$	−12.5302	−0.430287
$849$	−17.8841	−0.613780
$850$	0.348211	0.0119435
$851$	7.78996	0.267036
$852$	−0.920526	−0.0315367
$853$	30.5835	1.04716	0.523579	−	0.851977i	$-0.324596\pi$
$853$	30.5835	1.04716	0.523579	+	0.851977i	$0.324596\pi$
$854$	2.19458	0.0750970
$855$	−13.4584	−0.460268
$856$	−18.9681	−0.648318
$857$	48.3757	1.65248	0.826241	−	0.563317i	$-0.190475\pi$
$857$	48.3757	1.65248	0.826241	+	0.563317i	$0.190475\pi$
$858$	0	0
$859$	−57.0941	−1.94802	−0.974012	−	0.226495i	$-0.927273\pi$
$859$	−57.0941	−1.94802	−0.974012	+	0.226495i	$0.927273\pi$
$860$	−22.6749	−0.773206
$861$	21.5425	0.734165
$862$	17.8510	0.608007
$863$	3.31963	0.113002	0.0565008	−	0.998403i	$-0.482006\pi$
$863$	3.31963	0.113002	0.0565008	+	0.998403i	$0.482006\pi$
$864$	7.92194	0.269510
$865$	42.8775	1.45788
$866$	39.4138	1.33934
$867$	−49.1115	−1.66791
$868$	−1.70316	−0.0578089
$869$	0	0
$870$	63.7253	2.16049
$871$	0.249195	0.00844367
$872$	−17.0732	−0.578170
$873$	−47.6568	−1.61294
$874$	−3.60822	−0.122050
$875$	17.2942	0.584651
$876$	27.4521	0.927521
$877$	4.46676	0.150832	0.0754159	−	0.997152i	$-0.475972\pi$
$877$	4.46676	0.150832	0.0754159	+	0.997152i	$0.475972\pi$
$878$	16.9993	0.573698
$879$	−49.6097	−1.67329
$880$	0	0
$881$	6.10320	0.205622	0.102811	−	0.994701i	$-0.467216\pi$
$881$	6.10320	0.205622	0.102811	+	0.994701i	$0.467216\pi$
$882$	−24.1271	−0.812401
$883$	6.98526	0.235073	0.117536	−	0.993069i	$-0.462500\pi$
$883$	6.98526	0.235073	0.117536	+	0.993069i	$0.462500\pi$
$884$	−0.368378	−0.0123899
$885$	45.1682	1.51831
$886$	16.2668	0.546494
$887$	39.3670	1.32182	0.660908	−	0.750467i	$-0.270171\pi$
$887$	39.3670	1.32182	0.660908	+	0.750467i	$0.270171\pi$
$888$	6.36348	0.213544
$889$	7.68493	0.257744
$890$	1.22948	0.0412124
$891$	0	0
$892$	−5.16251	−0.172854
$893$	−8.73247	−0.292221
$894$	−53.9672	−1.80493
$895$	7.28676	0.243570
$896$	−1.66073	−0.0554810
$897$	−6.74020	−0.225049
$898$	16.5534	0.552393
$899$	9.37039	0.312520
$900$	3.40732	0.113577
$901$	−7.28320	−0.242639
$902$	0	0
$903$	46.9070	1.56097
$904$	21.1114	0.702155
$905$	0.164195	0.00545804
$906$	−18.4384	−0.612574
$907$	56.7300	1.88369	0.941844	−	0.336050i	$-0.109091\pi$
$907$	56.7300	1.88369	0.941844	+	0.336050i	$0.109091\pi$
$908$	18.3435	0.608751
$909$	94.0624	3.11985
$910$	2.49049	0.0825589
$911$	−47.8355	−1.58486	−0.792431	−	0.609962i	$-0.791185\pi$
$911$	−47.8355	−1.58486	−0.792431	+	0.609962i	$0.791185\pi$
$912$	−2.94749	−0.0976011
$913$	0	0
$914$	−11.5805	−0.383050
$915$	−9.21643	−0.304686
$916$	18.3344	0.605785
$917$	−26.2002	−0.865206
$918$	4.60466	0.151976
$919$	−0.295588	−0.00975054	−0.00487527	−	0.999988i	$-0.501552\pi$
$919$	−0.295588	−0.00975054	−0.00487527	+	0.999988i	$0.501552\pi$
$920$	8.53789	0.281486
$921$	15.0575	0.496161
$922$	−5.42648	−0.178712
$923$	0.197930	0.00651495
$924$	0	0
$925$	1.29336	0.0425254
$926$	26.9168	0.884542
$927$	15.9043	0.522367
$928$	9.13696	0.299935
$929$	−18.6254	−0.611080	−0.305540	−	0.952179i	$-0.598837\pi$
$929$	−18.6254	−0.611080	−0.305540	+	0.952179i	$0.598837\pi$
$930$	7.15263	0.234544
$931$	4.24198	0.139025
$932$	14.6429	0.479645
$933$	−72.4007	−2.37029
$934$	12.5581	0.410912
$935$	0	0
$936$	−3.60466	−0.117822
$937$	50.5786	1.65233	0.826165	−	0.563428i	$-0.190518\pi$
$937$	50.5786	1.65233	0.826165	+	0.563428i	$0.190518\pi$
$938$	0.652996	0.0213211
$939$	−71.5852	−2.33609
$940$	20.6631	0.673955
$941$	−47.1169	−1.53597	−0.767983	−	0.640470i	$-0.778739\pi$
$941$	−47.1169	−1.53597	−0.767983	+	0.640470i	$0.778739\pi$
$942$	−18.1453	−0.591206
$943$	−15.8795	−0.517108
$944$	6.47623	0.210783
$945$	−31.1306	−1.01268
$946$	0	0
$947$	8.28227	0.269138	0.134569	−	0.990904i	$-0.457035\pi$
$947$	8.28227	0.269138	0.134569	+	0.990904i	$0.457035\pi$
$948$	−33.2193	−1.07891
$949$	−5.90271	−0.191610
$950$	−0.599069	−0.0194364
$951$	−84.3646	−2.73571
$952$	−0.965305	−0.0312857
$953$	10.0765	0.326410	0.163205	−	0.986592i	$-0.447817\pi$
$953$	10.0765	0.326410	0.163205	+	0.986592i	$0.447817\pi$
$954$	−71.2677	−2.30738
$955$	32.3644	1.04729
$956$	12.5868	0.407088
$957$	0	0
$958$	40.2322	1.29984
$959$	34.2718	1.10669
$960$	6.97445	0.225099
$961$	−29.9483	−0.966073
$962$	−1.36827	−0.0441147
$963$	−107.885	−3.47654
$964$	2.79627	0.0900619
$965$	−13.2942	−0.427956
$966$	−17.6621	−0.568270
$967$	−48.6402	−1.56416	−0.782082	−	0.623175i	$-0.785842\pi$
$967$	−48.6402	−1.56416	−0.782082	+	0.623175i	$0.785842\pi$
$968$	0	0
$969$	−1.71324	−0.0550372
$970$	−19.8265	−0.636592
$971$	−4.89578	−0.157113	−0.0785565	−	0.996910i	$-0.525031\pi$
$971$	−4.89578	−0.157113	−0.0785565	+	0.996910i	$0.525031\pi$
$972$	−5.23567	−0.167934
$973$	−15.8885	−0.509363
$974$	−14.2795	−0.457545
$975$	−1.11907	−0.0358389
$976$	−1.32146	−0.0422988
$977$	−36.1052	−1.15511	−0.577553	−	0.816353i	$-0.695992\pi$
$977$	−36.1052	−1.15511	−0.577553	+	0.816353i	$0.695992\pi$
$978$	−34.9097	−1.11629
$979$	0	0
$980$	−10.0375	−0.320637
$981$	−97.1069	−3.10038
$982$	14.0094	0.447057
$983$	−18.7371	−0.597621	−0.298811	−	0.954312i	$-0.596590\pi$
$983$	−18.7371	−0.597621	−0.298811	+	0.954312i	$0.596590\pi$
$984$	−12.9717	−0.413523
$985$	−12.9313	−0.412027
$986$	5.31089	0.169133
$987$	−42.7453	−1.36060
$988$	0.633765	0.0201627
$989$	−34.5764	−1.09946
$990$	0	0
$991$	−8.60669	−0.273400	−0.136700	−	0.990612i	$-0.543650\pi$
$991$	−8.60669	−0.273400	−0.136700	+	0.990612i	$0.543650\pi$
$992$	1.02555	0.0325612
$993$	−7.10226	−0.225384
$994$	0.518659	0.0164509
$995$	−32.0793	−1.01698
$996$	20.3498	0.644809
$997$	13.3500	0.422798	0.211399	−	0.977400i	$-0.432198\pi$
$997$	13.3500	0.422798	0.211399	+	0.977400i	$0.432198\pi$
$998$	18.1506	0.574547
$999$	17.1031	0.541117

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4598.2.a.bu.1.4	yes	4
11.10	odd	2		4598.2.a.br.1.4	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4598.2.a.br.1.4	✓	4	11.10	odd	2
4598.2.a.bu.1.4	yes	4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +2.94749 q^{3} +1.00000 q^{4} +2.36624 q^{5} +2.94749 q^{6} -1.66073 q^{7} +1.00000 q^{8} +5.68769 q^{9} +2.36624 q^{10} +2.94749 q^{12} -0.633765 q^{13} -1.66073 q^{14} +6.97445 q^{15} +1.00000 q^{16} +0.581254 q^{17} +5.68769 q^{18} -1.00000 q^{19} +2.36624 q^{20} -4.89498 q^{21} +3.60822 q^{23} +2.94749 q^{24} +0.599069 q^{25} -0.633765 q^{26} +7.92194 q^{27} -1.66073 q^{28} +9.13696 q^{29} +6.97445 q^{30} +1.02555 q^{31} +1.00000 q^{32} +0.581254 q^{34} -3.92967 q^{35} +5.68769 q^{36} +2.15895 q^{37} -1.00000 q^{38} -1.86801 q^{39} +2.36624 q^{40} -4.40093 q^{41} -4.89498 q^{42} -9.58267 q^{43} +13.4584 q^{45} +3.60822 q^{46} +8.73247 q^{47} +2.94749 q^{48} -4.24198 q^{49} +0.599069 q^{50} +1.71324 q^{51} -0.633765 q^{52} -12.5302 q^{53} +7.92194 q^{54} -1.66073 q^{56} -2.94749 q^{57} +9.13696 q^{58} +6.47623 q^{59} +6.97445 q^{60} -1.32146 q^{61} +1.02555 q^{62} -9.44571 q^{63} +1.00000 q^{64} -1.49964 q^{65} -0.393199 q^{67} +0.581254 q^{68} +10.6352 q^{69} -3.92967 q^{70} -0.312308 q^{71} +5.68769 q^{72} +9.31372 q^{73} +2.15895 q^{74} +1.76575 q^{75} -1.00000 q^{76} -1.86801 q^{78} -11.2704 q^{79} +2.36624 q^{80} +6.28676 q^{81} -4.40093 q^{82} +6.90413 q^{83} -4.89498 q^{84} +1.37538 q^{85} -9.58267 q^{86} +26.9311 q^{87} +0.519595 q^{89} +13.4584 q^{90} +1.05251 q^{91} +3.60822 q^{92} +3.02279 q^{93} +8.73247 q^{94} -2.36624 q^{95} +2.94749 q^{96} -8.37894 q^{97} -4.24198 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} + q^{3} + 4 q^{4} + 3 q^{5} + q^{6} - q^{7} + 4 q^{8} + 5 q^{9} + 3 q^{10} + q^{12} - 9 q^{13} - q^{14} + 5 q^{15} + 4 q^{16} - 2 q^{17} + 5 q^{18} - 4 q^{19} + 3 q^{20} + 2 q^{21} - 2 q^{23}+ \cdots - 7 q^{98}+O(q^{100}) \) 4 * q + 4 * q^2 + q^3 + 4 * q^4 + 3 * q^5 + q^6 - q^7 + 4 * q^8 + 5 * q^9 + 3 * q^10 + q^12 - 9 * q^13 - q^14 + 5 * q^15 + 4 * q^16 - 2 * q^17 + 5 * q^18 - 4 * q^19 + 3 * q^20 + 2 * q^21 - 2 * q^23 + q^24 + 15 * q^25 - 9 * q^26 - 2 * q^27 - q^28 + 5 * q^29 + 5 * q^30 + 27 * q^31 + 4 * q^32 - 2 * q^34 + 12 * q^35 + 5 * q^36 + 6 * q^37 - 4 * q^38 + 2 * q^39 + 3 * q^40 - 5 * q^41 + 2 * q^42 + q^43 + 11 * q^45 - 2 * q^46 + 22 * q^47 + q^48 - 7 * q^49 + 15 * q^50 + 12 * q^51 - 9 * q^52 - 2 * q^54 - q^56 - q^57 + 5 * q^58 + 5 * q^60 + 6 * q^61 + 27 * q^62 - 30 * q^63 + 4 * q^64 + 26 * q^65 + 17 * q^67 - 2 * q^68 + 14 * q^69 + 12 * q^70 - 19 * q^71 + 5 * q^72 + 20 * q^73 + 6 * q^74 + 23 * q^75 - 4 * q^76 + 2 * q^78 + 12 * q^79 + 3 * q^80 + 20 * q^81 - 5 * q^82 - 23 * q^83 + 2 * q^84 - 30 * q^85 + q^86 + 45 * q^87 + 16 * q^89 + 11 * q^90 + 15 * q^91 - 2 * q^92 - 12 * q^93 + 22 * q^94 - 3 * q^95 + q^96 + 8 * q^97 - 7 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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