LMFDB - Embedded newform 1734.2.a.s.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1734,2,Mod(1,1734)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1734, 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("1734.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$1734 = 2 \cdot 3 \cdot 17^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1734.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:	$13.8460597105$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{3} - 3x - 1$ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$1$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.87939$ of defining polynomial
Character	$\chi$	$=$	1734.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +0.120615 q^{5} +1.00000 q^{6} -0.305407 q^{7} +1.00000 q^{8} +1.00000 q^{9} +0.120615 q^{10} -1.41147 q^{11} +1.00000 q^{12} +5.75877 q^{13} -0.305407 q^{14} +0.120615 q^{15} +1.00000 q^{16} +1.00000 q^{18} +1.30541 q^{19} +0.120615 q^{20} -0.305407 q^{21} -1.41147 q^{22} +4.12836 q^{23} +1.00000 q^{24} -4.98545 q^{25} +5.75877 q^{26} +1.00000 q^{27} -0.305407 q^{28} +8.35504 q^{29} +0.120615 q^{30} -5.61587 q^{31} +1.00000 q^{32} -1.41147 q^{33} -0.0368366 q^{35} +1.00000 q^{36} -2.93582 q^{37} +1.30541 q^{38} +5.75877 q^{39} +0.120615 q^{40} -4.36959 q^{41} -0.305407 q^{42} +6.00000 q^{43} -1.41147 q^{44} +0.120615 q^{45} +4.12836 q^{46} +13.4338 q^{47} +1.00000 q^{48} -6.90673 q^{49} -4.98545 q^{50} +5.75877 q^{52} +7.53983 q^{53} +1.00000 q^{54} -0.170245 q^{55} -0.305407 q^{56} +1.30541 q^{57} +8.35504 q^{58} +3.26857 q^{59} +0.120615 q^{60} -14.9513 q^{61} -5.61587 q^{62} -0.305407 q^{63} +1.00000 q^{64} +0.694593 q^{65} -1.41147 q^{66} +13.1480 q^{67} +4.12836 q^{69} -0.0368366 q^{70} -8.36959 q^{71} +1.00000 q^{72} +7.37464 q^{73} -2.93582 q^{74} -4.98545 q^{75} +1.30541 q^{76} +0.431074 q^{77} +5.75877 q^{78} -2.21894 q^{79} +0.120615 q^{80} +1.00000 q^{81} -4.36959 q^{82} -6.75877 q^{83} -0.305407 q^{84} +6.00000 q^{86} +8.35504 q^{87} -1.41147 q^{88} -5.88713 q^{89} +0.120615 q^{90} -1.75877 q^{91} +4.12836 q^{92} -5.61587 q^{93} +13.4338 q^{94} +0.157451 q^{95} +1.00000 q^{96} +1.08647 q^{97} -6.90673 q^{98} -1.41147 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$3 q + 3 q^{2} + 3 q^{3} + 3 q^{4} + 6 q^{5} + 3 q^{6} - 3 q^{7} + 3 q^{8} + 3 q^{9} + 6 q^{10} + 6 q^{11} + 3 q^{12} + 6 q^{13} - 3 q^{14} + 6 q^{15} + 3 q^{16} + 3 q^{18} + 6 q^{19} + 6 q^{20} - 3 q^{21}+ \cdots + 6 q^{99}+O(q^{100})$ 3 * q + 3 * q^2 + 3 * q^3 + 3 * q^4 + 6 * q^5 + 3 * q^6 - 3 * q^7 + 3 * q^8 + 3 * q^9 + 6 * q^10 + 6 * q^11 + 3 * q^12 + 6 * q^13 - 3 * q^14 + 6 * q^15 + 3 * q^16 + 3 * q^18 + 6 * q^19 + 6 * q^20 - 3 * q^21 + 6 * q^22 - 6 * q^23 + 3 * q^24 + 3 * q^25 + 6 * q^26 + 3 * q^27 - 3 * q^28 + 6 * q^30 - 6 * q^31 + 3 * q^32 + 6 * q^33 - 12 * q^35 + 3 * q^36 - 18 * q^37 + 6 * q^38 + 6 * q^39 + 6 * q^40 - 6 * q^41 - 3 * q^42 + 18 * q^43 + 6 * q^44 + 6 * q^45 - 6 * q^46 + 24 * q^47 + 3 * q^48 + 6 * q^49 + 3 * q^50 + 6 * q^52 - 6 * q^53 + 3 * q^54 + 21 * q^55 - 3 * q^56 + 6 * q^57 + 6 * q^60 - 6 * q^61 - 6 * q^62 - 3 * q^63 + 3 * q^64 + 6 * q^66 + 24 * q^67 - 6 * q^69 - 12 * q^70 - 18 * q^71 + 3 * q^72 - 18 * q^74 + 3 * q^75 + 6 * q^76 - 6 * q^77 + 6 * q^78 - 24 * q^79 + 6 * q^80 + 3 * q^81 - 6 * q^82 - 9 * q^83 - 3 * q^84 + 18 * q^86 + 6 * q^88 + 12 * q^89 + 6 * q^90 + 6 * q^91 - 6 * q^92 - 6 * q^93 + 24 * q^94 + 18 * q^95 + 3 * q^96 - 12 * q^97 + 6 * q^98 + 6 * q^99

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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	0.120615	0.0539406	0.0269703	−	0.999636i	$-0.491414\pi$
$5$	0.120615	0.0539406	0.0269703	+	0.999636i	$0.491414\pi$
$6$	1.00000	0.408248
$7$	−0.305407	−0.115433	−0.0577166	−	0.998333i	$-0.518382\pi$
$7$	−0.305407	−0.115433	−0.0577166	+	0.998333i	$0.518382\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	0.120615	0.0381417
$11$	−1.41147	−0.425575	−0.212788	−	0.977098i	$-0.568254\pi$
$11$	−1.41147	−0.425575	−0.212788	+	0.977098i	$0.568254\pi$
$12$	1.00000	0.288675
$13$	5.75877	1.59720	0.798598	−	0.601865i	$-0.205576\pi$
$13$	5.75877	1.59720	0.798598	+	0.601865i	$0.205576\pi$
$14$	−0.305407	−0.0816235
$15$	0.120615	0.0311426
$16$	1.00000	0.250000
$17$	0	0
$17$	0	0
$18$	1.00000	0.235702
$19$	1.30541	0.299481	0.149740	−	0.988725i	$-0.452156\pi$
$19$	1.30541	0.299481	0.149740	+	0.988725i	$0.452156\pi$
$20$	0.120615	0.0269703
$21$	−0.305407	−0.0666453
$22$	−1.41147	−0.300927
$23$	4.12836	0.860822	0.430411	−	0.902633i	$-0.358369\pi$
$23$	4.12836	0.860822	0.430411	+	0.902633i	$0.358369\pi$
$24$	1.00000	0.204124
$25$	−4.98545	−0.997090
$26$	5.75877	1.12939
$27$	1.00000	0.192450
$28$	−0.305407	−0.0577166
$29$	8.35504	1.55149	0.775746	−	0.631046i	$-0.217374\pi$
$29$	8.35504	1.55149	0.775746	+	0.631046i	$0.217374\pi$
$30$	0.120615	0.0220211
$31$	−5.61587	−1.00864	−0.504320	−	0.863517i	$-0.668257\pi$
$31$	−5.61587	−1.00864	−0.504320	+	0.863517i	$0.668257\pi$
$32$	1.00000	0.176777
$33$	−1.41147	−0.245706
$34$	0	0
$35$	−0.0368366	−0.00622653
$36$	1.00000	0.166667
$37$	−2.93582	−0.482646	−0.241323	−	0.970445i	$-0.577581\pi$
$37$	−2.93582	−0.482646	−0.241323	+	0.970445i	$0.577581\pi$
$38$	1.30541	0.211765
$39$	5.75877	0.922141
$40$	0.120615	0.0190709
$41$	−4.36959	−0.682415	−0.341207	−	0.939988i	$-0.610836\pi$
$41$	−4.36959	−0.682415	−0.341207	+	0.939988i	$0.610836\pi$
$42$	−0.305407	−0.0471254
$43$	6.00000	0.914991	0.457496	−	0.889212i	$-0.348747\pi$
$43$	6.00000	0.914991	0.457496	+	0.889212i	$0.348747\pi$
$44$	−1.41147	−0.212788
$45$	0.120615	0.0179802
$46$	4.12836	0.608693
$47$	13.4338	1.95952	0.979758	−	0.200186i	$-0.0641547\pi$
$47$	13.4338	1.95952	0.979758	+	0.200186i	$0.0641547\pi$
$48$	1.00000	0.144338
$49$	−6.90673	−0.986675
$50$	−4.98545	−0.705049
$51$	0	0
$52$	5.75877	0.798598
$53$	7.53983	1.03568	0.517838	−	0.855479i	$-0.326737\pi$
$53$	7.53983	1.03568	0.517838	+	0.855479i	$0.326737\pi$
$54$	1.00000	0.136083
$55$	−0.170245	−0.0229558
$56$	−0.305407	−0.0408118
$57$	1.30541	0.172905
$58$	8.35504	1.09707
$59$	3.26857	0.425532	0.212766	−	0.977103i	$-0.431753\pi$
$59$	3.26857	0.425532	0.212766	+	0.977103i	$0.431753\pi$
$60$	0.120615	0.0155713
$61$	−14.9513	−1.91432	−0.957159	−	0.289562i	$-0.906490\pi$
$61$	−14.9513	−1.91432	−0.957159	+	0.289562i	$0.906490\pi$
$62$	−5.61587	−0.713216
$63$	−0.305407	−0.0384777
$64$	1.00000	0.125000
$65$	0.694593	0.0861536
$66$	−1.41147	−0.173740
$67$	13.1480	1.60628	0.803139	−	0.595791i	$-0.203161\pi$
$67$	13.1480	1.60628	0.803139	+	0.595791i	$0.203161\pi$
$68$	0	0
$69$	4.12836	0.496996
$70$	−0.0368366	−0.00440282
$71$	−8.36959	−0.993287	−0.496644	−	0.867955i	$-0.665434\pi$
$71$	−8.36959	−0.993287	−0.496644	+	0.867955i	$0.665434\pi$
$72$	1.00000	0.117851
$73$	7.37464	0.863136	0.431568	−	0.902080i	$-0.357960\pi$
$73$	7.37464	0.863136	0.431568	+	0.902080i	$0.357960\pi$
$74$	−2.93582	−0.341282
$75$	−4.98545	−0.575670
$76$	1.30541	0.149740
$77$	0.431074	0.0491255
$78$	5.75877	0.652052
$79$	−2.21894	−0.249650	−0.124825	−	0.992179i	$-0.539837\pi$
$79$	−2.21894	−0.249650	−0.124825	+	0.992179i	$0.539837\pi$
$80$	0.120615	0.0134851
$81$	1.00000	0.111111
$82$	−4.36959	−0.482540
$83$	−6.75877	−0.741871	−0.370936	−	0.928659i	$-0.620963\pi$
$83$	−6.75877	−0.741871	−0.370936	+	0.928659i	$0.620963\pi$
$84$	−0.305407	−0.0333227
$85$	0	0
$86$	6.00000	0.646997
$87$	8.35504	0.895754
$88$	−1.41147	−0.150464
$89$	−5.88713	−0.624034	−0.312017	−	0.950077i	$-0.601005\pi$
$89$	−5.88713	−0.624034	−0.312017	+	0.950077i	$0.601005\pi$
$90$	0.120615	0.0127139
$91$	−1.75877	−0.184369
$92$	4.12836	0.430411
$93$	−5.61587	−0.582338
$94$	13.4338	1.38559
$95$	0.157451	0.0161542
$96$	1.00000	0.102062
$97$	1.08647	0.110314	0.0551570	−	0.998478i	$-0.482434\pi$
$97$	1.08647	0.110314	0.0551570	+	0.998478i	$0.482434\pi$
$98$	−6.90673	−0.697685
$99$	−1.41147	−0.141858
$100$	−4.98545	−0.498545
$101$	−10.2490	−1.01981	−0.509905	−	0.860231i	$-0.670320\pi$
$101$	−10.2490	−1.01981	−0.509905	+	0.860231i	$0.670320\pi$
$102$	0	0
$103$	1.16250	0.114545	0.0572725	−	0.998359i	$-0.481760\pi$
$103$	1.16250	0.114545	0.0572725	+	0.998359i	$0.481760\pi$
$104$	5.75877	0.564694
$105$	−0.0368366	−0.00359489
$106$	7.53983	0.732333
$107$	11.9855	1.15868	0.579339	−	0.815087i	$-0.303311\pi$
$107$	11.9855	1.15868	0.579339	+	0.815087i	$0.303311\pi$
$108$	1.00000	0.0962250
$109$	7.60132	0.728074	0.364037	−	0.931384i	$-0.381398\pi$
$109$	7.60132	0.728074	0.364037	+	0.931384i	$0.381398\pi$
$110$	−0.170245	−0.0162322
$111$	−2.93582	−0.278656
$112$	−0.305407	−0.0288583
$113$	−19.1925	−1.80548	−0.902741	−	0.430185i	$-0.858448\pi$
$113$	−19.1925	−1.80548	−0.902741	+	0.430185i	$0.858448\pi$
$114$	1.30541	0.122263
$115$	0.497941	0.0464332
$116$	8.35504	0.775746
$117$	5.75877	0.532399
$118$	3.26857	0.300896
$119$	0	0
$120$	0.120615	0.0110106
$121$	−9.00774	−0.818886
$122$	−14.9513	−1.35363
$123$	−4.36959	−0.393992
$124$	−5.61587	−0.504320
$125$	−1.20439	−0.107724
$126$	−0.305407	−0.0272078
$127$	−5.69459	−0.505313	−0.252657	−	0.967556i	$-0.581304\pi$
$127$	−5.69459	−0.505313	−0.252657	+	0.967556i	$0.581304\pi$
$128$	1.00000	0.0883883
$129$	6.00000	0.528271
$130$	0.694593	0.0609198
$131$	6.63041	0.579302	0.289651	−	0.957132i	$-0.406461\pi$
$131$	6.63041	0.579302	0.289651	+	0.957132i	$0.406461\pi$
$132$	−1.41147	−0.122853
$133$	−0.398681	−0.0345700
$134$	13.1480	1.13581
$135$	0.120615	0.0103809
$136$	0	0
$137$	−18.3405	−1.56693	−0.783467	−	0.621434i	$-0.786551\pi$
$137$	−18.3405	−1.56693	−0.783467	+	0.621434i	$0.786551\pi$
$138$	4.12836	0.351429
$139$	−12.9067	−1.09473	−0.547367	−	0.836893i	$-0.684370\pi$
$139$	−12.9067	−1.09473	−0.547367	+	0.836893i	$0.684370\pi$
$140$	−0.0368366	−0.00311326
$141$	13.4338	1.13133
$142$	−8.36959	−0.702360
$143$	−8.12836	−0.679727
$144$	1.00000	0.0833333
$145$	1.00774	0.0836883
$146$	7.37464	0.610329
$147$	−6.90673	−0.569657
$148$	−2.93582	−0.241323
$149$	7.49525	0.614035	0.307017	−	0.951704i	$-0.400669\pi$
$149$	7.49525	0.614035	0.307017	+	0.951704i	$0.400669\pi$
$150$	−4.98545	−0.407060
$151$	−12.6040	−1.02570	−0.512850	−	0.858478i	$-0.671410\pi$
$151$	−12.6040	−1.02570	−0.512850	+	0.858478i	$0.671410\pi$
$152$	1.30541	0.105883
$153$	0	0
$154$	0.431074	0.0347370
$155$	−0.677356	−0.0544066
$156$	5.75877	0.461071
$157$	−16.2567	−1.29743	−0.648713	−	0.761033i	$-0.724693\pi$
$157$	−16.2567	−1.29743	−0.648713	+	0.761033i	$0.724693\pi$
$158$	−2.21894	−0.176529
$159$	7.53983	0.597947
$160$	0.120615	0.00953543
$161$	−1.26083	−0.0993673
$162$	1.00000	0.0785674
$163$	−15.7588	−1.23432	−0.617161	−	0.786837i	$-0.711717\pi$
$163$	−15.7588	−1.23432	−0.617161	+	0.786837i	$0.711717\pi$
$164$	−4.36959	−0.341207
$165$	−0.170245	−0.0132535
$166$	−6.75877	−0.524582
$167$	13.1925	1.02087	0.510434	−	0.859917i	$-0.329485\pi$
$167$	13.1925	1.02087	0.510434	+	0.859917i	$0.329485\pi$
$168$	−0.305407	−0.0235627
$169$	20.1634	1.55103
$170$	0	0
$171$	1.30541	0.0998270
$172$	6.00000	0.457496
$173$	20.4020	1.55113	0.775567	−	0.631265i	$-0.217464\pi$
$173$	20.4020	1.55113	0.775567	+	0.631265i	$0.217464\pi$
$174$	8.35504	0.633394
$175$	1.52259	0.115097
$176$	−1.41147	−0.106394
$177$	3.26857	0.245681
$178$	−5.88713	−0.441259
$179$	−7.66550	−0.572946	−0.286473	−	0.958088i	$-0.592483\pi$
$179$	−7.66550	−0.572946	−0.286473	+	0.958088i	$0.592483\pi$
$180$	0.120615	0.00899009
$181$	7.67499	0.570478	0.285239	−	0.958456i	$-0.407927\pi$
$181$	7.67499	0.570478	0.285239	+	0.958456i	$0.407927\pi$
$182$	−1.75877	−0.130369
$183$	−14.9513	−1.10523
$184$	4.12836	0.304346
$185$	−0.354103	−0.0260342
$186$	−5.61587	−0.411775
$187$	0	0
$188$	13.4338	0.979758
$189$	−0.305407	−0.0222151
$190$	0.157451	0.0114227
$191$	21.0351	1.52205	0.761023	−	0.648725i	$-0.224698\pi$
$191$	21.0351	1.52205	0.761023	+	0.648725i	$0.224698\pi$
$192$	1.00000	0.0721688
$193$	−22.8016	−1.64129	−0.820647	−	0.571435i	$-0.806387\pi$
$193$	−22.8016	−1.64129	−0.820647	+	0.571435i	$0.806387\pi$
$194$	1.08647	0.0780037
$195$	0.694593	0.0497408
$196$	−6.90673	−0.493338
$197$	1.15745	0.0824650	0.0412325	−	0.999150i	$-0.486872\pi$
$197$	1.15745	0.0824650	0.0412325	+	0.999150i	$0.486872\pi$
$198$	−1.41147	−0.100309
$199$	−7.39693	−0.524354	−0.262177	−	0.965020i	$-0.584440\pi$
$199$	−7.39693	−0.524354	−0.262177	+	0.965020i	$0.584440\pi$
$200$	−4.98545	−0.352525
$201$	13.1480	0.927385
$202$	−10.2490	−0.721115
$203$	−2.55169	−0.179093
$204$	0	0
$205$	−0.527036	−0.0368098
$206$	1.16250	0.0809955
$207$	4.12836	0.286941
$208$	5.75877	0.399299
$209$	−1.84255	−0.127452
$210$	−0.0368366	−0.00254197
$211$	−20.0155	−1.37792	−0.688961	−	0.724798i	$-0.741933\pi$
$211$	−20.0155	−1.37792	−0.688961	+	0.724798i	$0.741933\pi$
$212$	7.53983	0.517838
$213$	−8.36959	−0.573475
$214$	11.9855	0.819309
$215$	0.723689	0.0493551
$216$	1.00000	0.0680414
$217$	1.71513	0.116430
$218$	7.60132	0.514826
$219$	7.37464	0.498332
$220$	−0.170245	−0.0114779
$221$	0	0
$222$	−2.93582	−0.197039
$223$	−18.5202	−1.24021	−0.620103	−	0.784520i	$-0.712909\pi$
$223$	−18.5202	−1.24021	−0.620103	+	0.784520i	$0.712909\pi$
$224$	−0.305407	−0.0204059
$225$	−4.98545	−0.332363
$226$	−19.1925	−1.27667
$227$	26.1480	1.73550	0.867750	−	0.497000i	$-0.165565\pi$
$227$	26.1480	1.73550	0.867750	+	0.497000i	$0.165565\pi$
$228$	1.30541	0.0864527
$229$	−2.93582	−0.194005	−0.0970023	−	0.995284i	$-0.530925\pi$
$229$	−2.93582	−0.194005	−0.0970023	+	0.995284i	$0.530925\pi$
$230$	0.497941	0.0328332
$231$	0.431074	0.0283626
$232$	8.35504	0.548535
$233$	−1.41828	−0.0929147	−0.0464573	−	0.998920i	$-0.514793\pi$
$233$	−1.41828	−0.0929147	−0.0464573	+	0.998920i	$0.514793\pi$
$234$	5.75877	0.376463
$235$	1.62031	0.105697
$236$	3.26857	0.212766
$237$	−2.21894	−0.144136
$238$	0	0
$239$	−23.1634	−1.49832	−0.749159	−	0.662390i	$-0.769542\pi$
$239$	−23.1634	−1.49832	−0.749159	+	0.662390i	$0.769542\pi$
$240$	0.120615	0.00778565
$241$	4.08141	0.262907	0.131453	−	0.991322i	$-0.458036\pi$
$241$	4.08141	0.262907	0.131453	+	0.991322i	$0.458036\pi$
$242$	−9.00774	−0.579040
$243$	1.00000	0.0641500
$244$	−14.9513	−0.957159
$245$	−0.833053	−0.0532218
$246$	−4.36959	−0.278595
$247$	7.51754	0.478330
$248$	−5.61587	−0.356608
$249$	−6.75877	−0.428320
$250$	−1.20439	−0.0761725
$251$	−14.8530	−0.937512	−0.468756	−	0.883328i	$-0.655298\pi$
$251$	−14.8530	−0.937512	−0.468756	+	0.883328i	$0.655298\pi$
$252$	−0.305407	−0.0192389
$253$	−5.82707	−0.366345
$254$	−5.69459	−0.357311
$255$	0	0
$256$	1.00000	0.0625000
$257$	−4.00000	−0.249513	−0.124757	−	0.992187i	$-0.539815\pi$
$257$	−4.00000	−0.249513	−0.124757	+	0.992187i	$0.539815\pi$
$258$	6.00000	0.373544
$259$	0.896622	0.0557133
$260$	0.694593	0.0430768
$261$	8.35504	0.517164
$262$	6.63041	0.409628
$263$	9.88713	0.609666	0.304833	−	0.952406i	$-0.401399\pi$
$263$	9.88713	0.609666	0.304833	+	0.952406i	$0.401399\pi$
$264$	−1.41147	−0.0868702
$265$	0.909415	0.0558649
$266$	−0.398681	−0.0244447
$267$	−5.88713	−0.360286
$268$	13.1480	0.803139
$269$	−17.8803	−1.09018	−0.545091	−	0.838377i	$-0.683505\pi$
$269$	−17.8803	−1.09018	−0.545091	+	0.838377i	$0.683505\pi$
$270$	0.120615	0.00734038
$271$	18.3259	1.11322	0.556611	−	0.830773i	$-0.312101\pi$
$271$	18.3259	1.11322	0.556611	+	0.830773i	$0.312101\pi$
$272$	0	0
$273$	−1.75877	−0.106446
$274$	−18.3405	−1.10799
$275$	7.03684	0.424337
$276$	4.12836	0.248498
$277$	−14.5817	−0.876131	−0.438065	−	0.898943i	$-0.644336\pi$
$277$	−14.5817	−0.876131	−0.438065	+	0.898943i	$0.644336\pi$
$278$	−12.9067	−0.774094
$279$	−5.61587	−0.336213
$280$	−0.0368366	−0.00220141
$281$	−16.8384	−1.00450	−0.502248	−	0.864723i	$-0.667494\pi$
$281$	−16.8384	−1.00450	−0.502248	+	0.864723i	$0.667494\pi$
$282$	13.4338	0.799969
$283$	30.3114	1.80183	0.900913	−	0.434000i	$-0.142898\pi$
$283$	30.3114	1.80183	0.900913	+	0.434000i	$0.142898\pi$
$284$	−8.36959	−0.496644
$285$	0.157451	0.00932662
$286$	−8.12836	−0.480640
$287$	1.33450	0.0787732
$288$	1.00000	0.0589256
$289$	0	0
$290$	1.00774	0.0591766
$291$	1.08647	0.0636898
$292$	7.37464	0.431568
$293$	−2.47472	−0.144575	−0.0722873	−	0.997384i	$-0.523030\pi$
$293$	−2.47472	−0.144575	−0.0722873	+	0.997384i	$0.523030\pi$
$294$	−6.90673	−0.402808
$295$	0.394238	0.0229534
$296$	−2.93582	−0.170641
$297$	−1.41147	−0.0819020
$298$	7.49525	0.434188
$299$	23.7743	1.37490
$300$	−4.98545	−0.287835
$301$	−1.83244	−0.105620
$302$	−12.6040	−0.725279
$303$	−10.2490	−0.588788
$304$	1.30541	0.0748702
$305$	−1.80335	−0.103259
$306$	0	0
$307$	−18.4243	−1.05153	−0.525764	−	0.850630i	$-0.676221\pi$
$307$	−18.4243	−1.05153	−0.525764	+	0.850630i	$0.676221\pi$
$308$	0.431074	0.0245627
$309$	1.16250	0.0661325
$310$	−0.677356	−0.0384713
$311$	−21.5175	−1.22015	−0.610074	−	0.792345i	$-0.708860\pi$
$311$	−21.5175	−1.22015	−0.610074	+	0.792345i	$0.708860\pi$
$312$	5.75877	0.326026
$313$	26.5134	1.49863	0.749314	−	0.662215i	$-0.230384\pi$
$313$	26.5134	1.49863	0.749314	+	0.662215i	$0.230384\pi$
$314$	−16.2567	−0.917419
$315$	−0.0368366	−0.00207551
$316$	−2.21894	−0.124825
$317$	12.8425	0.721309	0.360655	−	0.932699i	$-0.382553\pi$
$317$	12.8425	0.721309	0.360655	+	0.932699i	$0.382553\pi$
$318$	7.53983	0.422813
$319$	−11.7929	−0.660277
$320$	0.120615	0.00674257
$321$	11.9855	0.668963
$322$	−1.26083	−0.0702633
$323$	0	0
$324$	1.00000	0.0555556
$325$	−28.7101	−1.59255
$326$	−15.7588	−0.872798
$327$	7.60132	0.420354
$328$	−4.36959	−0.241270
$329$	−4.10277	−0.226193
$330$	−0.170245	−0.00937166
$331$	26.4534	1.45401	0.727004	−	0.686633i	$-0.240912\pi$
$331$	26.4534	1.45401	0.727004	+	0.686633i	$0.240912\pi$
$332$	−6.75877	−0.370936
$333$	−2.93582	−0.160882
$334$	13.1925	0.721863
$335$	1.58584	0.0866436
$336$	−0.305407	−0.0166613
$337$	5.73917	0.312633	0.156316	−	0.987707i	$-0.450038\pi$
$337$	5.73917	0.312633	0.156316	+	0.987707i	$0.450038\pi$
$338$	20.1634	1.09675
$339$	−19.1925	−1.04240
$340$	0	0
$341$	7.92665	0.429252
$342$	1.30541	0.0705883
$343$	4.24722	0.229328
$344$	6.00000	0.323498
$345$	0.497941	0.0268082
$346$	20.4020	1.09682
$347$	−4.08141	−0.219102	−0.109551	−	0.993981i	$-0.534941\pi$
$347$	−4.08141	−0.219102	−0.109551	+	0.993981i	$0.534941\pi$
$348$	8.35504	0.447877
$349$	−7.34461	−0.393148	−0.196574	−	0.980489i	$-0.562982\pi$
$349$	−7.34461	−0.393148	−0.196574	+	0.980489i	$0.562982\pi$
$350$	1.52259	0.0813860
$351$	5.75877	0.307380
$352$	−1.41147	−0.0752318
$353$	12.4243	0.661277	0.330639	−	0.943757i	$-0.392736\pi$
$353$	12.4243	0.661277	0.330639	+	0.943757i	$0.392736\pi$
$354$	3.26857	0.173723
$355$	−1.00950	−0.0535785
$356$	−5.88713	−0.312017
$357$	0	0
$358$	−7.66550	−0.405134
$359$	6.93582	0.366059	0.183029	−	0.983107i	$-0.441410\pi$
$359$	6.93582	0.366059	0.183029	+	0.983107i	$0.441410\pi$
$360$	0.120615	0.00635696
$361$	−17.2959	−0.910311
$362$	7.67499	0.403389
$363$	−9.00774	−0.472784
$364$	−1.75877	−0.0921846
$365$	0.889490	0.0465580
$366$	−14.9513	−0.781517
$367$	−27.2294	−1.42136	−0.710681	−	0.703515i	$-0.751613\pi$
$367$	−27.2294	−1.42136	−0.710681	+	0.703515i	$0.751613\pi$
$368$	4.12836	0.215205
$369$	−4.36959	−0.227472
$370$	−0.354103	−0.0184090
$371$	−2.30272	−0.119551
$372$	−5.61587	−0.291169
$373$	−26.9668	−1.39629	−0.698144	−	0.715958i	$-0.745990\pi$
$373$	−26.9668	−1.39629	−0.698144	+	0.715958i	$0.745990\pi$
$374$	0	0
$375$	−1.20439	−0.0621946
$376$	13.4338	0.692793
$377$	48.1147	2.47804
$378$	−0.305407	−0.0157085
$379$	−13.7297	−0.705246	−0.352623	−	0.935765i	$-0.614710\pi$
$379$	−13.7297	−0.705246	−0.352623	+	0.935765i	$0.614710\pi$
$380$	0.157451	0.00807709
$381$	−5.69459	−0.291743
$382$	21.0351	1.07625
$383$	−28.6709	−1.46501	−0.732507	−	0.680760i	$-0.761650\pi$
$383$	−28.6709	−1.46501	−0.732507	+	0.680760i	$0.761650\pi$
$384$	1.00000	0.0510310
$385$	0.0519939	0.00264986
$386$	−22.8016	−1.16057
$387$	6.00000	0.304997
$388$	1.08647	0.0551570
$389$	1.77332	0.0899108	0.0449554	−	0.998989i	$-0.485685\pi$
$389$	1.77332	0.0899108	0.0449554	+	0.998989i	$0.485685\pi$
$390$	0.694593	0.0351721
$391$	0	0
$392$	−6.90673	−0.348842
$393$	6.63041	0.334460
$394$	1.15745	0.0583116
$395$	−0.267637	−0.0134663
$396$	−1.41147	−0.0709292
$397$	−3.47296	−0.174303	−0.0871515	−	0.996195i	$-0.527776\pi$
$397$	−3.47296	−0.174303	−0.0871515	+	0.996195i	$0.527776\pi$
$398$	−7.39693	−0.370774
$399$	−0.398681	−0.0199590
$400$	−4.98545	−0.249273
$401$	10.7101	0.534836	0.267418	−	0.963581i	$-0.413830\pi$
$401$	10.7101	0.534836	0.267418	+	0.963581i	$0.413830\pi$
$402$	13.1480	0.655760
$403$	−32.3405	−1.61099
$404$	−10.2490	−0.509905
$405$	0.120615	0.00599340
$406$	−2.55169	−0.126638
$407$	4.14384	0.205402
$408$	0	0
$409$	10.8999	0.538966	0.269483	−	0.963005i	$-0.413147\pi$
$409$	10.8999	0.538966	0.269483	+	0.963005i	$0.413147\pi$
$410$	−0.527036	−0.0260285
$411$	−18.3405	−0.904670
$412$	1.16250	0.0572725
$413$	−0.998245	−0.0491204
$414$	4.12836	0.202898
$415$	−0.815207	−0.0400170
$416$	5.75877	0.282347
$417$	−12.9067	−0.632045
$418$	−1.84255	−0.0901220
$419$	28.7469	1.40438	0.702189	−	0.711990i	$-0.252206\pi$
$419$	28.7469	1.40438	0.702189	+	0.711990i	$0.252206\pi$
$420$	−0.0368366	−0.00179744
$421$	−14.5972	−0.711424	−0.355712	−	0.934596i	$-0.615762\pi$
$421$	−14.5972	−0.711424	−0.355712	+	0.934596i	$0.615762\pi$
$422$	−20.0155	−0.974338
$423$	13.4338	0.653172
$424$	7.53983	0.366166
$425$	0	0
$426$	−8.36959	−0.405508
$427$	4.56624	0.220976
$428$	11.9855	0.579339
$429$	−8.12836	−0.392441
$430$	0.723689	0.0348994
$431$	32.0547	1.54402	0.772010	−	0.635611i	$-0.219252\pi$
$431$	32.0547	1.54402	0.772010	+	0.635611i	$0.219252\pi$
$432$	1.00000	0.0481125
$433$	7.84018	0.376775	0.188388	−	0.982095i	$-0.439674\pi$
$433$	7.84018	0.376775	0.188388	+	0.982095i	$0.439674\pi$
$434$	1.71513	0.0823287
$435$	1.00774	0.0483175
$436$	7.60132	0.364037
$437$	5.38919	0.257800
$438$	7.37464	0.352374
$439$	−19.3500	−0.923524	−0.461762	−	0.887004i	$-0.652783\pi$
$439$	−19.3500	−0.923524	−0.461762	+	0.887004i	$0.652783\pi$
$440$	−0.170245	−0.00811609
$441$	−6.90673	−0.328892
$442$	0	0
$443$	25.1438	1.19462	0.597310	−	0.802011i	$-0.296236\pi$
$443$	25.1438	1.19462	0.597310	+	0.802011i	$0.296236\pi$
$444$	−2.93582	−0.139328
$445$	−0.710074	−0.0336607
$446$	−18.5202	−0.876958
$447$	7.49525	0.354513
$448$	−0.305407	−0.0144291
$449$	14.3250	0.676039	0.338019	−	0.941139i	$-0.390243\pi$
$449$	14.3250	0.676039	0.338019	+	0.941139i	$0.390243\pi$
$450$	−4.98545	−0.235016
$451$	6.16756	0.290419
$452$	−19.1925	−0.902741
$453$	−12.6040	−0.592188
$454$	26.1480	1.22718
$455$	−0.212134	−0.00994498
$456$	1.30541	0.0611313
$457$	24.8239	1.16121	0.580606	−	0.814185i	$-0.302816\pi$
$457$	24.8239	1.16121	0.580606	+	0.814185i	$0.302816\pi$
$458$	−2.93582	−0.137182
$459$	0	0
$460$	0.497941	0.0232166
$461$	−21.2618	−0.990259	−0.495130	−	0.868819i	$-0.664879\pi$
$461$	−21.2618	−0.990259	−0.495130	+	0.868819i	$0.664879\pi$
$462$	0.431074	0.0200554
$463$	−0.120615	−0.00560544	−0.00280272	−	0.999996i	$-0.500892\pi$
$463$	−0.120615	−0.00560544	−0.00280272	+	0.999996i	$0.500892\pi$
$464$	8.35504	0.387873
$465$	−0.677356	−0.0314117
$466$	−1.41828	−0.0657006
$467$	−10.5963	−0.490337	−0.245168	−	0.969481i	$-0.578843\pi$
$467$	−10.5963	−0.490337	−0.245168	+	0.969481i	$0.578843\pi$
$468$	5.75877	0.266199
$469$	−4.01548	−0.185418
$470$	1.62031	0.0747393
$471$	−16.2567	−0.749070
$472$	3.26857	0.150448
$473$	−8.46884	−0.389398
$474$	−2.21894	−0.101919
$475$	−6.50805	−0.298610
$476$	0	0
$477$	7.53983	0.345225
$478$	−23.1634	−1.05947
$479$	−21.6560	−0.989488	−0.494744	−	0.869039i	$-0.664738\pi$
$479$	−21.6560	−0.989488	−0.494744	+	0.869039i	$0.664738\pi$
$480$	0.120615	0.00550529
$481$	−16.9067	−0.770880
$482$	4.08141	0.185903
$483$	−1.26083	−0.0573697
$484$	−9.00774	−0.409443
$485$	0.131044	0.00595040
$486$	1.00000	0.0453609
$487$	10.4320	0.472719	0.236360	−	0.971666i	$-0.424046\pi$
$487$	10.4320	0.472719	0.236360	+	0.971666i	$0.424046\pi$
$488$	−14.9513	−0.676814
$489$	−15.7588	−0.712636
$490$	−0.833053	−0.0376335
$491$	7.43613	0.335588	0.167794	−	0.985822i	$-0.446336\pi$
$491$	7.43613	0.335588	0.167794	+	0.985822i	$0.446336\pi$
$492$	−4.36959	−0.196996
$493$	0	0
$494$	7.51754	0.338230
$495$	−0.170245	−0.00765193
$496$	−5.61587	−0.252160
$497$	2.55613	0.114658
$498$	−6.75877	−0.302868
$499$	−3.47296	−0.155471	−0.0777356	−	0.996974i	$-0.524769\pi$
$499$	−3.47296	−0.155471	−0.0777356	+	0.996974i	$0.524769\pi$
$500$	−1.20439	−0.0538621
$501$	13.1925	0.589399
$502$	−14.8530	−0.662921
$503$	21.1581	0.943391	0.471696	−	0.881761i	$-0.343642\pi$
$503$	21.1581	0.943391	0.471696	+	0.881761i	$0.343642\pi$
$504$	−0.305407	−0.0136039
$505$	−1.23618	−0.0550092
$506$	−5.82707	−0.259045
$507$	20.1634	0.895490
$508$	−5.69459	−0.252657
$509$	−40.0110	−1.77346	−0.886729	−	0.462290i	$-0.847028\pi$
$509$	−40.0110	−1.77346	−0.886729	+	0.462290i	$0.847028\pi$
$510$	0	0
$511$	−2.25227	−0.0996345
$512$	1.00000	0.0441942
$513$	1.30541	0.0576351
$514$	−4.00000	−0.176432
$515$	0.140215	0.00617862
$516$	6.00000	0.264135
$517$	−18.9614	−0.833922
$518$	0.896622	0.0393953
$519$	20.4020	0.895547
$520$	0.694593	0.0304599
$521$	23.8033	1.04284	0.521422	−	0.853299i	$-0.325402\pi$
$521$	23.8033	1.04284	0.521422	+	0.853299i	$0.325402\pi$
$522$	8.35504	0.365690
$523$	3.26083	0.142586	0.0712931	−	0.997455i	$-0.477287\pi$
$523$	3.26083	0.142586	0.0712931	+	0.997455i	$0.477287\pi$
$524$	6.63041	0.289651
$525$	1.52259	0.0664514
$526$	9.88713	0.431099
$527$	0	0
$528$	−1.41147	−0.0614265
$529$	−5.95668	−0.258986
$530$	0.909415	0.0395025
$531$	3.26857	0.141844
$532$	−0.398681	−0.0172850
$533$	−25.1634	−1.08995
$534$	−5.88713	−0.254761
$535$	1.44562	0.0624997
$536$	13.1480	0.567905
$537$	−7.66550	−0.330791
$538$	−17.8803	−0.770875
$539$	9.74867	0.419905
$540$	0.120615	0.00519043
$541$	0.340489	0.0146388	0.00731939	−	0.999973i	$-0.497670\pi$
$541$	0.340489	0.0146388	0.00731939	+	0.999973i	$0.497670\pi$
$542$	18.3259	0.787167
$543$	7.67499	0.329365
$544$	0	0
$545$	0.916831	0.0392727
$546$	−1.75877	−0.0752684
$547$	−39.2181	−1.67685	−0.838423	−	0.545020i	$-0.816522\pi$
$547$	−39.2181	−1.67685	−0.838423	+	0.545020i	$0.816522\pi$
$548$	−18.3405	−0.783467
$549$	−14.9513	−0.638106
$550$	7.03684	0.300052
$551$	10.9067	0.464642
$552$	4.12836	0.175714
$553$	0.677681	0.0288179
$554$	−14.5817	−0.619518
$555$	−0.354103	−0.0150309
$556$	−12.9067	−0.547367
$557$	−12.6750	−0.537057	−0.268528	−	0.963272i	$-0.586537\pi$
$557$	−12.6750	−0.537057	−0.268528	+	0.963272i	$0.586537\pi$
$558$	−5.61587	−0.237739
$559$	34.5526	1.46142
$560$	−0.0368366	−0.00155663
$561$	0	0
$562$	−16.8384	−0.710286
$563$	15.9855	0.673706	0.336853	−	0.941557i	$-0.390637\pi$
$563$	15.9855	0.673706	0.336853	+	0.941557i	$0.390637\pi$
$564$	13.4338	0.565663
$565$	−2.31490	−0.0973887
$566$	30.3114	1.27408
$567$	−0.305407	−0.0128259
$568$	−8.36959	−0.351180
$569$	−11.9864	−0.502495	−0.251248	−	0.967923i	$-0.580841\pi$
$569$	−11.9864	−0.502495	−0.251248	+	0.967923i	$0.580841\pi$
$570$	0.157451	0.00659491
$571$	24.3560	1.01927	0.509633	−	0.860392i	$-0.329781\pi$
$571$	24.3560	1.01927	0.509633	+	0.860392i	$0.329781\pi$
$572$	−8.12836	−0.339864
$573$	21.0351	0.878753
$574$	1.33450	0.0557011
$575$	−20.5817	−0.858317
$576$	1.00000	0.0416667
$577$	−9.90673	−0.412422	−0.206211	−	0.978508i	$-0.566113\pi$
$577$	−9.90673	−0.412422	−0.206211	+	0.978508i	$0.566113\pi$
$578$	0	0
$579$	−22.8016	−0.947602
$580$	1.00774	0.0418442
$581$	2.06418	0.0856365
$582$	1.08647	0.0450355
$583$	−10.6423	−0.440758
$584$	7.37464	0.305165
$585$	0.694593	0.0287179
$586$	−2.47472	−0.102230
$587$	18.6186	0.768470	0.384235	−	0.923235i	$-0.374465\pi$
$587$	18.6186	0.768470	0.384235	+	0.923235i	$0.374465\pi$
$588$	−6.90673	−0.284829
$589$	−7.33099	−0.302068
$590$	0.394238	0.0162305
$591$	1.15745	0.0476112
$592$	−2.93582	−0.120662
$593$	−18.8384	−0.773602	−0.386801	−	0.922163i	$-0.626420\pi$
$593$	−18.8384	−0.773602	−0.386801	+	0.922163i	$0.626420\pi$
$594$	−1.41147	−0.0579135
$595$	0	0
$596$	7.49525	0.307017
$597$	−7.39693	−0.302736
$598$	23.7743	0.972201
$599$	14.5270	0.593559	0.296779	−	0.954946i	$-0.404087\pi$
$599$	14.5270	0.593559	0.296779	+	0.954946i	$0.404087\pi$
$600$	−4.98545	−0.203530
$601$	−9.98545	−0.407315	−0.203658	−	0.979042i	$-0.565283\pi$
$601$	−9.98545	−0.407315	−0.203658	+	0.979042i	$0.565283\pi$
$602$	−1.83244	−0.0746848
$603$	13.1480	0.535426
$604$	−12.6040	−0.512850
$605$	−1.08647	−0.0441711
$606$	−10.2490	−0.416336
$607$	−15.2618	−0.619456	−0.309728	−	0.950825i	$-0.600238\pi$
$607$	−15.2618	−0.619456	−0.309728	+	0.950825i	$0.600238\pi$
$608$	1.30541	0.0529413
$609$	−2.55169	−0.103400
$610$	−1.80335	−0.0730154
$611$	77.3620	3.12973
$612$	0	0
$613$	32.1985	1.30049	0.650243	−	0.759726i	$-0.274667\pi$
$613$	32.1985	1.30049	0.650243	+	0.759726i	$0.274667\pi$
$614$	−18.4243	−0.743543
$615$	−0.527036	−0.0212522
$616$	0.431074	0.0173685
$617$	−6.82295	−0.274682	−0.137341	−	0.990524i	$-0.543856\pi$
$617$	−6.82295	−0.274682	−0.137341	+	0.990524i	$0.543856\pi$
$618$	1.16250	0.0467628
$619$	9.98639	0.401387	0.200693	−	0.979654i	$-0.435681\pi$
$619$	9.98639	0.401387	0.200693	+	0.979654i	$0.435681\pi$
$620$	−0.677356	−0.0272033
$621$	4.12836	0.165665
$622$	−21.5175	−0.862775
$623$	1.79797	0.0720342
$624$	5.75877	0.230535
$625$	24.7820	0.991280
$626$	26.5134	1.05969
$627$	−1.84255	−0.0735843
$628$	−16.2567	−0.648713
$629$	0	0
$630$	−0.0368366	−0.00146761
$631$	48.0901	1.91444	0.957218	−	0.289367i	$-0.0934449\pi$
$631$	48.0901	1.91444	0.957218	+	0.289367i	$0.0934449\pi$
$632$	−2.21894	−0.0882647
$633$	−20.0155	−0.795544
$634$	12.8425	0.510043
$635$	−0.686852	−0.0272569
$636$	7.53983	0.298974
$637$	−39.7743	−1.57591
$638$	−11.7929	−0.466886
$639$	−8.36959	−0.331096
$640$	0.120615	0.00476772
$641$	−38.5918	−1.52429	−0.762143	−	0.647409i	$-0.775853\pi$
$641$	−38.5918	−1.52429	−0.762143	+	0.647409i	$0.775853\pi$
$642$	11.9855	0.473028
$643$	32.3506	1.27578	0.637891	−	0.770126i	$-0.279807\pi$
$643$	32.3506	1.27578	0.637891	+	0.770126i	$0.279807\pi$
$644$	−1.26083	−0.0496837
$645$	0.723689	0.0284952
$646$	0	0
$647$	10.9804	0.431684	0.215842	−	0.976428i	$-0.430750\pi$
$647$	10.9804	0.431684	0.215842	+	0.976428i	$0.430750\pi$
$648$	1.00000	0.0392837
$649$	−4.61350	−0.181096
$650$	−28.7101	−1.12610
$651$	1.71513	0.0672211
$652$	−15.7588	−0.617161
$653$	5.34998	0.209361	0.104681	−	0.994506i	$-0.466618\pi$
$653$	5.34998	0.209361	0.104681	+	0.994506i	$0.466618\pi$
$654$	7.60132	0.297235
$655$	0.799726	0.0312479
$656$	−4.36959	−0.170604
$657$	7.37464	0.287712
$658$	−4.10277	−0.159943
$659$	−12.9290	−0.503643	−0.251821	−	0.967774i	$-0.581030\pi$
$659$	−12.9290	−0.503643	−0.251821	+	0.967774i	$0.581030\pi$
$660$	−0.170245	−0.00662676
$661$	−12.3114	−0.478858	−0.239429	−	0.970914i	$-0.576960\pi$
$661$	−12.3114	−0.478858	−0.239429	+	0.970914i	$0.576960\pi$
$662$	26.4534	1.02814
$663$	0	0
$664$	−6.75877	−0.262291
$665$	−0.0480868	−0.00186473
$666$	−2.93582	−0.113761
$667$	34.4926	1.33556
$668$	13.1925	0.510434
$669$	−18.5202	−0.716033
$670$	1.58584	0.0612662
$671$	21.1034	0.814687
$672$	−0.305407	−0.0117813
$673$	−11.6459	−0.448916	−0.224458	−	0.974484i	$-0.572061\pi$
$673$	−11.6459	−0.448916	−0.224458	+	0.974484i	$0.572061\pi$
$674$	5.73917	0.221065
$675$	−4.98545	−0.191890
$676$	20.1634	0.775517
$677$	19.3250	0.742720	0.371360	−	0.928489i	$-0.378892\pi$
$677$	19.3250	0.742720	0.371360	+	0.928489i	$0.378892\pi$
$678$	−19.1925	−0.737085
$679$	−0.331815	−0.0127339
$680$	0	0
$681$	26.1480	1.00199
$682$	7.92665	0.303527
$683$	34.0624	1.30336	0.651681	−	0.758493i	$-0.274064\pi$
$683$	34.0624	1.30336	0.651681	+	0.758493i	$0.274064\pi$
$684$	1.30541	0.0499135
$685$	−2.21213	−0.0845213
$686$	4.24722	0.162159
$687$	−2.93582	−0.112009
$688$	6.00000	0.228748
$689$	43.4201	1.65418
$690$	0.497941	0.0189563
$691$	−26.9905	−1.02677	−0.513384	−	0.858159i	$-0.671608\pi$
$691$	−26.9905	−1.02677	−0.513384	+	0.858159i	$0.671608\pi$
$692$	20.4020	0.775567
$693$	0.431074	0.0163752
$694$	−4.08141	−0.154928
$695$	−1.55674	−0.0590506
$696$	8.35504	0.316697
$697$	0	0
$698$	−7.34461	−0.277998
$699$	−1.41828	−0.0536443
$700$	1.52259	0.0575486
$701$	47.7383	1.80305	0.901526	−	0.432724i	$-0.142448\pi$
$701$	47.7383	1.80305	0.901526	+	0.432724i	$0.142448\pi$
$702$	5.75877	0.217351
$703$	−3.83244	−0.144543
$704$	−1.41147	−0.0531969
$705$	1.62031	0.0610244
$706$	12.4243	0.467593
$707$	3.13011	0.117720
$708$	3.26857	0.122840
$709$	40.5580	1.52319	0.761594	−	0.648055i	$-0.224417\pi$
$709$	40.5580	1.52319	0.761594	+	0.648055i	$0.224417\pi$
$710$	−1.00950	−0.0378857
$711$	−2.21894	−0.0832168
$712$	−5.88713	−0.220629
$713$	−23.1843	−0.868259
$714$	0	0
$715$	−0.980400	−0.0366649
$716$	−7.66550	−0.286473
$717$	−23.1634	−0.865054
$718$	6.93582	0.258843
$719$	−2.63640	−0.0983212	−0.0491606	−	0.998791i	$-0.515655\pi$
$719$	−2.63640	−0.0983212	−0.0491606	+	0.998791i	$0.515655\pi$
$720$	0.120615	0.00449505
$721$	−0.355037	−0.0132223
$722$	−17.2959	−0.643687
$723$	4.08141	0.151789
$724$	7.67499	0.285239
$725$	−41.6536	−1.54698
$726$	−9.00774	−0.334309
$727$	−18.2645	−0.677391	−0.338696	−	0.940896i	$-0.609986\pi$
$727$	−18.2645	−0.677391	−0.338696	+	0.940896i	$0.609986\pi$
$728$	−1.75877	−0.0651844
$729$	1.00000	0.0370370
$730$	0.889490	0.0329215
$731$	0	0
$732$	−14.9513	−0.552616
$733$	11.4201	0.421813	0.210906	−	0.977506i	$-0.432358\pi$
$733$	11.4201	0.421813	0.210906	+	0.977506i	$0.432358\pi$
$734$	−27.2294	−1.00505
$735$	−0.833053	−0.0307276
$736$	4.12836	0.152173
$737$	−18.5580	−0.683593
$738$	−4.36959	−0.160847
$739$	−6.95130	−0.255708	−0.127854	−	0.991793i	$-0.540809\pi$
$739$	−6.95130	−0.255708	−0.127854	+	0.991793i	$0.540809\pi$
$740$	−0.354103	−0.0130171
$741$	7.51754	0.276164
$742$	−2.30272	−0.0845355
$743$	36.4742	1.33811	0.669055	−	0.743213i	$-0.266699\pi$
$743$	36.4742	1.33811	0.669055	+	0.743213i	$0.266699\pi$
$744$	−5.61587	−0.205888
$745$	0.904038	0.0331214
$746$	−26.9668	−0.987324
$747$	−6.75877	−0.247290
$748$	0	0
$749$	−3.66044	−0.133750
$750$	−1.20439	−0.0439782
$751$	−7.22937	−0.263803	−0.131902	−	0.991263i	$-0.542108\pi$
$751$	−7.22937	−0.263803	−0.131902	+	0.991263i	$0.542108\pi$
$752$	13.4338	0.489879
$753$	−14.8530	−0.541273
$754$	48.1147	1.75224
$755$	−1.52023	−0.0553268
$756$	−0.305407	−0.0111076
$757$	−21.9608	−0.798179	−0.399089	−	0.916912i	$-0.630674\pi$
$757$	−21.9608	−0.798179	−0.399089	+	0.916912i	$0.630674\pi$
$758$	−13.7297	−0.498684
$759$	−5.82707	−0.211509
$760$	0.157451	0.00571136
$761$	40.3013	1.46092	0.730460	−	0.682955i	$-0.239306\pi$
$761$	40.3013	1.46092	0.730460	+	0.682955i	$0.239306\pi$
$762$	−5.69459	−0.206293
$763$	−2.32150	−0.0840439
$764$	21.0351	0.761023
$765$	0	0
$766$	−28.6709	−1.03592
$767$	18.8229	0.679657
$768$	1.00000	0.0360844
$769$	−29.6382	−1.06878	−0.534390	−	0.845238i	$-0.679458\pi$
$769$	−29.6382	−1.06878	−0.534390	+	0.845238i	$0.679458\pi$
$770$	0.0519939	0.00187373
$771$	−4.00000	−0.144056
$772$	−22.8016	−0.820647
$773$	−50.0479	−1.80010	−0.900048	−	0.435790i	$-0.856469\pi$
$773$	−50.0479	−1.80010	−0.900048	+	0.435790i	$0.856469\pi$
$774$	6.00000	0.215666
$775$	27.9976	1.00570
$776$	1.08647	0.0390019
$777$	0.896622	0.0321661
$778$	1.77332	0.0635765
$779$	−5.70409	−0.204370
$780$	0.694593	0.0248704
$781$	11.8135	0.422719
$782$	0	0
$783$	8.35504	0.298585
$784$	−6.90673	−0.246669
$785$	−1.96080	−0.0699839
$786$	6.63041	0.236499
$787$	44.8040	1.59709	0.798544	−	0.601936i	$-0.205604\pi$
$787$	44.8040	1.59709	0.798544	+	0.601936i	$0.205604\pi$
$788$	1.15745	0.0412325
$789$	9.88713	0.351991
$790$	−0.267637	−0.00952210
$791$	5.86154	0.208412
$792$	−1.41147	−0.0501545
$793$	−86.1011	−3.05754
$794$	−3.47296	−0.123251
$795$	0.909415	0.0322536
$796$	−7.39693	−0.262177
$797$	−50.1735	−1.77724	−0.888619	−	0.458646i	$-0.848335\pi$
$797$	−50.1735	−1.77724	−0.888619	+	0.458646i	$0.848335\pi$
$798$	−0.398681	−0.0141132
$799$	0	0
$800$	−4.98545	−0.176262
$801$	−5.88713	−0.208011
$802$	10.7101	0.378186
$803$	−10.4091	−0.367330
$804$	13.1480	0.463693
$805$	−0.152075	−0.00535993
$806$	−32.3405	−1.13915
$807$	−17.8803	−0.629417
$808$	−10.2490	−0.360558
$809$	51.3073	1.80387	0.901934	−	0.431874i	$-0.142148\pi$
$809$	51.3073	1.80387	0.901934	+	0.431874i	$0.142148\pi$
$810$	0.120615	0.00423797
$811$	28.3250	0.994626	0.497313	−	0.867571i	$-0.334320\pi$
$811$	28.3250	0.994626	0.497313	+	0.867571i	$0.334320\pi$
$812$	−2.55169	−0.0895467
$813$	18.3259	0.642719
$814$	4.14384	0.145241
$815$	−1.90074	−0.0665800
$816$	0	0
$817$	7.83244	0.274023
$818$	10.8999	0.381107
$819$	−1.75877	−0.0614564
$820$	−0.527036	−0.0184049
$821$	26.1985	0.914335	0.457167	−	0.889381i	$-0.348864\pi$
$821$	26.1985	0.914335	0.457167	+	0.889381i	$0.348864\pi$
$822$	−18.3405	−0.639698
$823$	17.0865	0.595597	0.297798	−	0.954629i	$-0.403748\pi$
$823$	17.0865	0.595597	0.297798	+	0.954629i	$0.403748\pi$
$824$	1.16250	0.0404977
$825$	7.03684	0.244991
$826$	−0.998245	−0.0347334
$827$	42.8907	1.49146	0.745729	−	0.666250i	$-0.232102\pi$
$827$	42.8907	1.49146	0.745729	+	0.666250i	$0.232102\pi$
$828$	4.12836	0.143470
$829$	−36.4944	−1.26750	−0.633752	−	0.773536i	$-0.718486\pi$
$829$	−36.4944	−1.26750	−0.633752	+	0.773536i	$0.718486\pi$
$830$	−0.815207	−0.0282963
$831$	−14.5817	−0.505834
$832$	5.75877	0.199649
$833$	0	0
$834$	−12.9067	−0.446923
$835$	1.59121	0.0550662
$836$	−1.84255	−0.0637259
$837$	−5.61587	−0.194113
$838$	28.7469	0.993046
$839$	−15.4047	−0.531828	−0.265914	−	0.963997i	$-0.585674\pi$
$839$	−15.4047	−0.531828	−0.265914	+	0.963997i	$0.585674\pi$
$840$	−0.0368366	−0.00127098
$841$	40.8066	1.40713
$842$	−14.5972	−0.503053
$843$	−16.8384	−0.579946
$844$	−20.0155	−0.688961
$845$	2.43201	0.0836636
$846$	13.4338	0.461862
$847$	2.75103	0.0945265
$848$	7.53983	0.258919
$849$	30.3114	1.04028
$850$	0	0
$851$	−12.1201	−0.415472
$852$	−8.36959	−0.286737
$853$	26.1676	0.895960	0.447980	−	0.894044i	$-0.352144\pi$
$853$	26.1676	0.895960	0.447980	+	0.894044i	$0.352144\pi$
$854$	4.56624	0.156253
$855$	0.157451	0.00538472
$856$	11.9855	0.409654
$857$	29.8735	1.02046	0.510230	−	0.860038i	$-0.329560\pi$
$857$	29.8735	1.02046	0.510230	+	0.860038i	$0.329560\pi$
$858$	−8.12836	−0.277497
$859$	40.7648	1.39088	0.695438	−	0.718586i	$-0.255210\pi$
$859$	40.7648	1.39088	0.695438	+	0.718586i	$0.255210\pi$
$860$	0.723689	0.0246776
$861$	1.33450	0.0454797
$862$	32.0547	1.09179
$863$	5.18716	0.176573	0.0882864	−	0.996095i	$-0.471861\pi$
$863$	5.18716	0.176573	0.0882864	+	0.996095i	$0.471861\pi$
$864$	1.00000	0.0340207
$865$	2.46078	0.0836690
$866$	7.84018	0.266420
$867$	0	0
$868$	1.71513	0.0582152
$869$	3.13198	0.106245
$870$	1.00774	0.0341656
$871$	75.7161	2.56554
$872$	7.60132	0.257413
$873$	1.08647	0.0367713
$874$	5.38919	0.182292
$875$	0.367830	0.0124349
$876$	7.37464	0.249166
$877$	25.8817	0.873965	0.436982	−	0.899470i	$-0.356047\pi$
$877$	25.8817	0.873965	0.436982	+	0.899470i	$0.356047\pi$
$878$	−19.3500	−0.653030
$879$	−2.47472	−0.0834702
$880$	−0.170245	−0.00573894
$881$	−41.6715	−1.40395	−0.701974	−	0.712203i	$-0.747698\pi$
$881$	−41.6715	−1.40395	−0.701974	+	0.712203i	$0.747698\pi$
$882$	−6.90673	−0.232562
$883$	−49.0215	−1.64970	−0.824852	−	0.565349i	$-0.808742\pi$
$883$	−49.0215	−1.64970	−0.824852	+	0.565349i	$0.808742\pi$
$884$	0	0
$885$	0.394238	0.0132522
$886$	25.1438	0.844724
$887$	−49.5194	−1.66270	−0.831350	−	0.555750i	$-0.812431\pi$
$887$	−49.5194	−1.66270	−0.831350	+	0.555750i	$0.812431\pi$
$888$	−2.93582	−0.0985197
$889$	1.73917	0.0583299
$890$	−0.710074	−0.0238017
$891$	−1.41147	−0.0472862
$892$	−18.5202	−0.620103
$893$	17.5365	0.586838
$894$	7.49525	0.250679
$895$	−0.924572	−0.0309050
$896$	−0.305407	−0.0102029
$897$	23.7743	0.793799
$898$	14.3250	0.478032
$899$	−46.9208	−1.56490
$900$	−4.98545	−0.166182
$901$	0	0
$902$	6.16756	0.205357
$903$	−1.83244	−0.0609799
$904$	−19.1925	−0.638334
$905$	0.925717	0.0307719
$906$	−12.6040	−0.418740
$907$	19.1189	0.634831	0.317416	−	0.948287i	$-0.397185\pi$
$907$	19.1189	0.634831	0.317416	+	0.948287i	$0.397185\pi$
$908$	26.1480	0.867750
$909$	−10.2490	−0.339937
$910$	−0.212134	−0.00703216
$911$	−33.7606	−1.11854	−0.559270	−	0.828986i	$-0.688918\pi$
$911$	−33.7606	−1.11854	−0.559270	+	0.828986i	$0.688918\pi$
$912$	1.30541	0.0432264
$913$	9.53983	0.315722
$914$	24.8239	0.821101
$915$	−1.80335	−0.0596168
$916$	−2.93582	−0.0970023
$917$	−2.02498	−0.0668706
$918$	0	0
$919$	−13.9513	−0.460211	−0.230106	−	0.973166i	$-0.573907\pi$
$919$	−13.9513	−0.460211	−0.230106	+	0.973166i	$0.573907\pi$
$920$	0.497941	0.0164166
$921$	−18.4243	−0.607100
$922$	−21.2618	−0.700219
$923$	−48.1985	−1.58647
$924$	0.431074	0.0141813
$925$	14.6364	0.481242
$926$	−0.120615	−0.00396365
$927$	1.16250	0.0381816
$928$	8.35504	0.274268
$929$	−39.8479	−1.30737	−0.653684	−	0.756768i	$-0.726777\pi$
$929$	−39.8479	−1.30737	−0.653684	+	0.756768i	$0.726777\pi$
$930$	−0.677356	−0.0222114
$931$	−9.01609	−0.295490
$932$	−1.41828	−0.0464573
$933$	−21.5175	−0.704453
$934$	−10.5963	−0.346720
$935$	0	0
$936$	5.75877	0.188231
$937$	34.8907	1.13983	0.569916	−	0.821703i	$-0.306976\pi$
$937$	34.8907	1.13983	0.569916	+	0.821703i	$0.306976\pi$
$938$	−4.01548	−0.131110
$939$	26.5134	0.865233
$940$	1.62031	0.0528487
$941$	−21.2026	−0.691186	−0.345593	−	0.938384i	$-0.612322\pi$
$941$	−21.2026	−0.691186	−0.345593	+	0.938384i	$0.612322\pi$
$942$	−16.2567	−0.529672
$943$	−18.0392	−0.587437
$944$	3.26857	0.106383
$945$	−0.0368366	−0.00119830
$946$	−8.46884	−0.275346
$947$	43.2695	1.40607	0.703035	−	0.711155i	$-0.251828\pi$
$947$	43.2695	1.40607	0.703035	+	0.711155i	$0.251828\pi$
$948$	−2.21894	−0.0720678
$949$	42.4688	1.37860
$950$	−6.50805	−0.211149
$951$	12.8425	0.416448
$952$	0	0
$953$	37.6323	1.21903	0.609515	−	0.792775i	$-0.291364\pi$
$953$	37.6323	1.21903	0.609515	+	0.792775i	$0.291364\pi$
$954$	7.53983	0.244111
$955$	2.53714	0.0821000
$956$	−23.1634	−0.749159
$957$	−11.7929	−0.381211
$958$	−21.6560	−0.699674
$959$	5.60132	0.180876
$960$	0.120615	0.00389282
$961$	0.537962	0.0173536
$962$	−16.9067	−0.545095
$963$	11.9855	0.386226
$964$	4.08141	0.131453
$965$	−2.75021	−0.0885323
$966$	−1.26083	−0.0405665
$967$	2.56860	0.0826006	0.0413003	−	0.999147i	$-0.486850\pi$
$967$	2.56860	0.0826006	0.0413003	+	0.999147i	$0.486850\pi$
$968$	−9.00774	−0.289520
$969$	0	0
$970$	0.131044	0.00420757
$971$	39.9777	1.28295	0.641473	−	0.767146i	$-0.278324\pi$
$971$	39.9777	1.28295	0.641473	+	0.767146i	$0.278324\pi$
$972$	1.00000	0.0320750
$973$	3.94181	0.126369
$974$	10.4320	0.334263
$975$	−28.7101	−0.919458
$976$	−14.9513	−0.478580
$977$	57.4938	1.83939	0.919695	−	0.392633i	$-0.128436\pi$
$977$	57.4938	1.83939	0.919695	+	0.392633i	$0.128436\pi$
$978$	−15.7588	−0.503910
$979$	8.30953	0.265574
$980$	−0.833053	−0.0266109
$981$	7.60132	0.242691
$982$	7.43613	0.237296
$983$	51.6377	1.64699	0.823493	−	0.567327i	$-0.192022\pi$
$983$	51.6377	1.64699	0.823493	+	0.567327i	$0.192022\pi$
$984$	−4.36959	−0.139297
$985$	0.139606	0.00444821
$986$	0	0
$987$	−4.10277	−0.130593
$988$	7.51754	0.239165
$989$	24.7701	0.787644
$990$	−0.170245	−0.00541073
$991$	47.5972	1.51197	0.755987	−	0.654586i	$-0.227157\pi$
$991$	47.5972	1.51197	0.755987	+	0.654586i	$0.227157\pi$
$992$	−5.61587	−0.178304
$993$	26.4534	0.839472
$994$	2.55613	0.0810756
$995$	−0.892178	−0.0282840
$996$	−6.75877	−0.214160
$997$	−1.38919	−0.0439959	−0.0219980	−	0.999758i	$-0.507003\pi$
$997$	−1.38919	−0.0439959	−0.0219980	+	0.999758i	$0.507003\pi$
$998$	−3.47296	−0.109935
$999$	−2.93582	−0.0928853

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1734.2.a.s.1.1	yes	3
3.2	odd	2		5202.2.a.bf.1.3		3
17.2	even	8		1734.2.f.o.1483.6		12
17.4	even	4		1734.2.b.i.577.3		6
17.8	even	8		1734.2.f.o.829.1		12
17.9	even	8		1734.2.f.o.829.6		12
17.13	even	4		1734.2.b.i.577.4		6
17.15	even	8		1734.2.f.o.1483.1		12
17.16	even	2		1734.2.a.r.1.3	✓	3
51.50	odd	2		5202.2.a.bk.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1734.2.a.r.1.3	✓	3	17.16	even	2
1734.2.a.s.1.1	yes	3	1.1	even	1	trivial
1734.2.b.i.577.3		6	17.4	even	4
1734.2.b.i.577.4		6	17.13	even	4
1734.2.f.o.829.1		12	17.8	even	8
1734.2.f.o.829.6		12	17.9	even	8
1734.2.f.o.1483.1		12	17.15	even	8
1734.2.f.o.1483.6		12	17.2	even	8
5202.2.a.bf.1.3		3	3.2	odd	2
5202.2.a.bk.1.1		3	51.50	odd	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 $\F_{q}$
Belyi maps

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	1734.2.a.s.1.1

Level	$1734$
Weight	$2$
Character	1734.1
Self dual	yes
Analytic conductor	$13.846$
Analytic rank	$0$
Dimension	$3$
CM	no
Inner twists	$1$