LMFDB - Embedded newform 1785.2.a.z.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1785,2,Mod(1,1785)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1785, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1785.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$1785 = 3 \cdot 5 \cdot 7 \cdot 17$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1785.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,-1,-4,3,4,1,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$14.2532967608$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.8468.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} - x^{3} - 5x^{2} + 3x + 4$ x^4 - x^3 - 5x^2 + 3x + 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$1$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.704624$ of defining polynomial
Character	$\chi$	$=$	1785.1

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.704624 q^{2} -1.00000 q^{3} -1.50350 q^{4} +1.00000 q^{5} -0.704624 q^{6} +1.00000 q^{7} -2.46865 q^{8} +1.00000 q^{9} +0.704624 q^{10} -5.17328 q^{11} +1.50350 q^{12} +4.03157 q^{13} +0.704624 q^{14} -1.00000 q^{15} +1.26753 q^{16} -1.00000 q^{17} +0.704624 q^{18} +5.94432 q^{19} -1.50350 q^{20} -1.00000 q^{21} -3.64522 q^{22} -4.66977 q^{23} +2.46865 q^{24} +1.00000 q^{25} +2.84074 q^{26} -1.00000 q^{27} -1.50350 q^{28} -10.3688 q^{29} -0.704624 q^{30} +5.51051 q^{31} +5.83045 q^{32} +5.17328 q^{33} -0.704624 q^{34} +1.00000 q^{35} -1.50350 q^{36} -8.21185 q^{37} +4.18851 q^{38} -4.03157 q^{39} -2.46865 q^{40} -7.67678 q^{41} -0.704624 q^{42} -0.937309 q^{43} +7.77805 q^{44} +1.00000 q^{45} -3.29044 q^{46} -11.4631 q^{47} -1.26753 q^{48} +1.00000 q^{49} +0.704624 q^{50} +1.00000 q^{51} -6.06148 q^{52} +6.18029 q^{53} -0.704624 q^{54} -5.17328 q^{55} -2.46865 q^{56} -5.94432 q^{57} -7.30611 q^{58} +7.61178 q^{59} +1.50350 q^{60} -4.69202 q^{61} +3.88284 q^{62} +1.00000 q^{63} +1.57320 q^{64} +4.03157 q^{65} +3.64522 q^{66} -1.40925 q^{67} +1.50350 q^{68} +4.66977 q^{69} +0.704624 q^{70} -9.17328 q^{71} -2.46865 q^{72} -3.29044 q^{73} -5.78627 q^{74} -1.00000 q^{75} -8.93731 q^{76} -5.17328 q^{77} -2.84074 q^{78} -12.0140 q^{79} +1.26753 q^{80} +1.00000 q^{81} -5.40925 q^{82} -14.7535 q^{83} +1.50350 q^{84} -1.00000 q^{85} -0.660451 q^{86} +10.3688 q^{87} +12.7710 q^{88} +9.00044 q^{89} +0.704624 q^{90} +4.03157 q^{91} +7.02103 q^{92} -5.51051 q^{93} -8.07715 q^{94} +5.94432 q^{95} -5.83045 q^{96} -8.15805 q^{97} +0.704624 q^{98} -5.17328 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q - q^{2} - 4 q^{3} + 3 q^{4} + 4 q^{5} + q^{6} + 4 q^{7} - 3 q^{8} + 4 q^{9} - q^{10} - 10 q^{11} - 3 q^{12} + q^{13} - q^{14} - 4 q^{15} - 7 q^{16} - 4 q^{17} - q^{18} - 8 q^{19} + 3 q^{20} - 4 q^{21}+ \cdots - 10 q^{99}+O(q^{100})$ 4 * q - q^2 - 4 * q^3 + 3 * q^4 + 4 * q^5 + q^6 + 4 * q^7 - 3 * q^8 + 4 * q^9 - q^10 - 10 * q^11 - 3 * q^12 + q^13 - q^14 - 4 * q^15 - 7 * q^16 - 4 * q^17 - q^18 - 8 * q^19 + 3 * q^20 - 4 * q^21 - 10 * q^22 - 17 * q^23 + 3 * q^24 + 4 * q^25 - 14 * q^26 - 4 * q^27 + 3 * q^28 - 10 * q^29 + q^30 - 5 * q^31 + 3 * q^32 + 10 * q^33 + q^34 + 4 * q^35 + 3 * q^36 + 11 * q^37 + 14 * q^38 - q^39 - 3 * q^40 - 11 * q^41 + q^42 + 10 * q^43 - 8 * q^44 + 4 * q^45 - 4 * q^46 - 13 * q^47 + 7 * q^48 + 4 * q^49 - q^50 + 4 * q^51 + 6 * q^52 - 4 * q^53 + q^54 - 10 * q^55 - 3 * q^56 + 8 * q^57 + 16 * q^58 - 16 * q^59 - 3 * q^60 - 7 * q^61 + 14 * q^62 + 4 * q^63 - 7 * q^64 + q^65 + 10 * q^66 + 2 * q^67 - 3 * q^68 + 17 * q^69 - q^70 - 26 * q^71 - 3 * q^72 - 4 * q^73 - 10 * q^74 - 4 * q^75 - 22 * q^76 - 10 * q^77 + 14 * q^78 - 12 * q^79 - 7 * q^80 + 4 * q^81 - 14 * q^82 - 17 * q^83 - 3 * q^84 - 4 * q^85 - 6 * q^86 + 10 * q^87 + 30 * q^88 - 8 * q^89 - q^90 + q^91 - 26 * q^92 + 5 * q^93 + 34 * q^94 - 8 * q^95 - 3 * q^96 - 14 * q^97 - q^98 - 10 * 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

)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$\alpha_n$			$\theta_n$
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$\alpha_p$			$\theta_p$

$2$	0.704624	0.498245	0.249122	−	0.968472i	$-0.419858\pi$
$2$	0.704624	0.498245	0.249122	+	0.968472i	$0.419858\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.50350	−0.751752
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−0.704624	−0.287662
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−2.46865	−0.872801
$9$	1.00000	0.333333
$10$	0.704624	0.222822
$11$	−5.17328	−1.55980	−0.779901	−	0.625903i	$-0.784731\pi$
$11$	−5.17328	−1.55980	−0.779901	+	0.625903i	$0.784731\pi$
$12$	1.50350	0.434024
$13$	4.03157	1.11815	0.559077	−	0.829115i	$-0.311155\pi$
$13$	4.03157	1.11815	0.559077	+	0.829115i	$0.311155\pi$
$14$	0.704624	0.188319
$15$	−1.00000	−0.258199
$16$	1.26753	0.316884
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0.704624	0.166082
$19$	5.94432	1.36372	0.681860	−	0.731483i	$-0.261171\pi$
$19$	5.94432	1.36372	0.681860	+	0.731483i	$0.261171\pi$
$20$	−1.50350	−0.336194
$21$	−1.00000	−0.218218
$22$	−3.64522	−0.777163
$23$	−4.66977	−0.973715	−0.486858	−	0.873481i	$-0.661857\pi$
$23$	−4.66977	−0.973715	−0.486858	+	0.873481i	$0.661857\pi$
$24$	2.46865	0.503912
$25$	1.00000	0.200000
$26$	2.84074	0.557115
$27$	−1.00000	−0.192450
$28$	−1.50350	−0.284136
$29$	−10.3688	−1.92544	−0.962719	−	0.270504i	$-0.912810\pi$
$29$	−10.3688	−1.92544	−0.962719	+	0.270504i	$0.912810\pi$
$30$	−0.704624	−0.128646
$31$	5.51051	0.989717	0.494859	−	0.868973i	$-0.335220\pi$
$31$	5.51051	0.989717	0.494859	+	0.868973i	$0.335220\pi$
$32$	5.83045	1.03069
$33$	5.17328	0.900552
$34$	−0.704624	−0.120842
$35$	1.00000	0.169031
$36$	−1.50350	−0.250584
$37$	−8.21185	−1.35002	−0.675010	−	0.737808i	$-0.735861\pi$
$37$	−8.21185	−1.35002	−0.675010	+	0.737808i	$0.735861\pi$
$38$	4.18851	0.679466
$39$	−4.03157	−0.645567
$40$	−2.46865	−0.390329
$41$	−7.67678	−1.19891	−0.599456	−	0.800408i	$-0.704616\pi$
$41$	−7.67678	−1.19891	−0.599456	+	0.800408i	$0.704616\pi$
$42$	−0.704624	−0.108726
$43$	−0.937309	−0.142938	−0.0714692	−	0.997443i	$-0.522769\pi$
$43$	−0.937309	−0.142938	−0.0714692	+	0.997443i	$0.522769\pi$
$44$	7.77805	1.17258
$45$	1.00000	0.149071
$46$	−3.29044	−0.485148
$47$	−11.4631	−1.67206	−0.836029	−	0.548685i	$-0.815129\pi$
$47$	−11.4631	−1.67206	−0.836029	+	0.548685i	$0.815129\pi$
$48$	−1.26753	−0.182953
$49$	1.00000	0.142857
$50$	0.704624	0.0996489
$51$	1.00000	0.140028
$52$	−6.06148	−0.840576
$53$	6.18029	0.848928	0.424464	−	0.905445i	$-0.360463\pi$
$53$	6.18029	0.848928	0.424464	+	0.905445i	$0.360463\pi$
$54$	−0.704624	−0.0958872
$55$	−5.17328	−0.697565
$56$	−2.46865	−0.329888
$57$	−5.94432	−0.787344
$58$	−7.30611	−0.959339
$59$	7.61178	0.990969	0.495485	−	0.868617i	$-0.334991\pi$
$59$	7.61178	0.990969	0.495485	+	0.868617i	$0.334991\pi$
$60$	1.50350	0.194102
$61$	−4.69202	−0.600751	−0.300376	−	0.953821i	$-0.597112\pi$
$61$	−4.69202	−0.600751	−0.300376	+	0.953821i	$0.597112\pi$
$62$	3.88284	0.493121
$63$	1.00000	0.125988
$64$	1.57320	0.196651
$65$	4.03157	0.500054
$66$	3.64522	0.448695
$67$	−1.40925	−0.172167	−0.0860836	−	0.996288i	$-0.527435\pi$
$67$	−1.40925	−0.172167	−0.0860836	+	0.996288i	$0.527435\pi$
$68$	1.50350	0.182327
$69$	4.66977	0.562175
$70$	0.704624	0.0842187
$71$	−9.17328	−1.08867	−0.544334	−	0.838869i	$-0.683218\pi$
$71$	−9.17328	−1.08867	−0.544334	+	0.838869i	$0.683218\pi$
$72$	−2.46865	−0.290934
$73$	−3.29044	−0.385117	−0.192558	−	0.981286i	$-0.561678\pi$
$73$	−3.29044	−0.385117	−0.192558	+	0.981286i	$0.561678\pi$
$74$	−5.78627	−0.672640
$75$	−1.00000	−0.115470
$76$	−8.93731	−1.02518
$77$	−5.17328	−0.589550
$78$	−2.84074	−0.321650
$79$	−12.0140	−1.35168	−0.675841	−	0.737047i	$-0.736220\pi$
$79$	−12.0140	−1.35168	−0.675841	+	0.737047i	$0.736220\pi$
$80$	1.26753	0.141715
$81$	1.00000	0.111111
$82$	−5.40925	−0.597352
$83$	−14.7535	−1.61941	−0.809703	−	0.586840i	$-0.800372\pi$
$83$	−14.7535	−1.61941	−0.809703	+	0.586840i	$0.800372\pi$
$84$	1.50350	0.164046
$85$	−1.00000	−0.108465
$86$	−0.660451	−0.0712183
$87$	10.3688	1.11165
$88$	12.7710	1.36140
$89$	9.00044	0.954045	0.477022	−	0.878891i	$-0.341716\pi$
$89$	9.00044	0.954045	0.477022	+	0.878891i	$0.341716\pi$
$90$	0.704624	0.0742739
$91$	4.03157	0.422623
$92$	7.02103	0.731993
$93$	−5.51051	−0.571414
$94$	−8.07715	−0.833094
$95$	5.94432	0.609874
$96$	−5.83045	−0.595067
$97$	−8.15805	−0.828324	−0.414162	−	0.910203i	$-0.635925\pi$
$97$	−8.15805	−0.828324	−0.414162	+	0.910203i	$0.635925\pi$
$98$	0.704624	0.0711778
$99$	−5.17328	−0.519934
$100$	−1.50350	−0.150350
$101$	−4.60477	−0.458192	−0.229096	−	0.973404i	$-0.573577\pi$
$101$	−4.60477	−0.458192	−0.229096	+	0.973404i	$0.573577\pi$
$102$	0.704624	0.0697682
$103$	4.39402	0.432955	0.216478	−	0.976288i	$-0.430543\pi$
$103$	4.39402	0.432955	0.216478	+	0.976288i	$0.430543\pi$
$104$	−9.95254	−0.975927
$105$	−1.00000	−0.0975900
$106$	4.35478	0.422974
$107$	−10.6838	−1.03284	−0.516421	−	0.856335i	$-0.672736\pi$
$107$	−10.6838	−1.03284	−0.516421	+	0.856335i	$0.672736\pi$
$108$	1.50350	0.144675
$109$	1.29044	0.123601	0.0618007	−	0.998089i	$-0.480316\pi$
$109$	1.29044	0.123601	0.0618007	+	0.998089i	$0.480316\pi$
$110$	−3.64522	−0.347558
$111$	8.21185	0.779435
$112$	1.26753	0.119771
$113$	15.4930	1.45746	0.728728	−	0.684803i	$-0.240112\pi$
$113$	15.4930	1.45746	0.728728	+	0.684803i	$0.240112\pi$
$114$	−4.18851	−0.392290
$115$	−4.66977	−0.435459
$116$	15.5895	1.44745
$117$	4.03157	0.372718
$118$	5.36344	0.493745
$119$	−1.00000	−0.0916698
$120$	2.46865	0.225356
$121$	15.7628	1.43298
$122$	−3.30611	−0.299321
$123$	7.67678	0.692192
$124$	−8.28508	−0.744022
$125$	1.00000	0.0894427
$126$	0.704624	0.0627729
$127$	−19.0907	−1.69403	−0.847014	−	0.531571i	$-0.821602\pi$
$127$	−19.0907	−1.69403	−0.847014	+	0.531571i	$0.821602\pi$
$128$	−10.5524	−0.932707
$129$	0.937309	0.0825255
$130$	2.84074	0.249149
$131$	2.87230	0.250954	0.125477	−	0.992097i	$-0.459954\pi$
$131$	2.87230	0.250954	0.125477	+	0.992097i	$0.459954\pi$
$132$	−7.77805	−0.676992
$133$	5.94432	0.515438
$134$	−0.992991	−0.0857814
$135$	−1.00000	−0.0860663
$136$	2.46865	0.211685
$137$	12.9988	1.11056	0.555281	−	0.831663i	$-0.312611\pi$
$137$	12.9988	1.11056	0.555281	+	0.831663i	$0.312611\pi$
$138$	3.29044	0.280101
$139$	−18.1803	−1.54203	−0.771016	−	0.636816i	$-0.780251\pi$
$139$	−18.1803	−1.54203	−0.771016	+	0.636816i	$0.780251\pi$
$140$	−1.50350	−0.127069
$141$	11.4631	0.965363
$142$	−6.46372	−0.542423
$143$	−20.8564	−1.74410
$144$	1.26753	0.105628
$145$	−10.3688	−0.861082
$146$	−2.31852	−0.191882
$147$	−1.00000	−0.0824786
$148$	12.3466	1.01488
$149$	−11.0468	−0.904989	−0.452494	−	0.891767i	$-0.649466\pi$
$149$	−11.0468	−0.904989	−0.452494	+	0.891767i	$0.649466\pi$
$150$	−0.704624	−0.0575323
$151$	−11.3934	−0.927178	−0.463589	−	0.886050i	$-0.653439\pi$
$151$	−11.3934	−0.927178	−0.463589	+	0.886050i	$0.653439\pi$
$152$	−14.6745	−1.19026
$153$	−1.00000	−0.0808452
$154$	−3.64522	−0.293740
$155$	5.51051	0.442615
$156$	6.06148	0.485306
$157$	−3.51984	−0.280914	−0.140457	−	0.990087i	$-0.544857\pi$
$157$	−3.51984	−0.280914	−0.140457	+	0.990087i	$0.544857\pi$
$158$	−8.46537	−0.673469
$159$	−6.18029	−0.490129
$160$	5.83045	0.460937
$161$	−4.66977	−0.368030
$162$	0.704624	0.0553605
$163$	18.3513	1.43738	0.718691	−	0.695330i	$-0.244742\pi$
$163$	18.3513	1.43738	0.718691	+	0.695330i	$0.244742\pi$
$164$	11.5421	0.901285
$165$	5.17328	0.402739
$166$	−10.3957	−0.806860
$167$	−16.4794	−1.27521	−0.637607	−	0.770362i	$-0.720075\pi$
$167$	−16.4794	−1.27521	−0.637607	+	0.770362i	$0.720075\pi$
$168$	2.46865	0.190461
$169$	3.25352	0.250271
$170$	−0.704624	−0.0540422
$171$	5.94432	0.454573
$172$	1.40925	0.107454
$173$	16.4303	1.24917	0.624585	−	0.780957i	$-0.285268\pi$
$173$	16.4303	1.24917	0.624585	+	0.780957i	$0.285268\pi$
$174$	7.30611	0.553875
$175$	1.00000	0.0755929
$176$	−6.55731	−0.494276
$177$	−7.61178	−0.572136
$178$	6.34193	0.475348
$179$	−24.5332	−1.83370	−0.916849	−	0.399235i	$-0.869276\pi$
$179$	−24.5332	−1.83370	−0.916849	+	0.399235i	$0.869276\pi$
$180$	−1.50350	−0.112065
$181$	−6.66155	−0.495149	−0.247575	−	0.968869i	$-0.579634\pi$
$181$	−6.66155	−0.495149	−0.247575	+	0.968869i	$0.579634\pi$
$182$	2.84074	0.210570
$183$	4.69202	0.346844
$184$	11.5281	0.849860
$185$	−8.21185	−0.603747
$186$	−3.88284	−0.284704
$187$	5.17328	0.378308
$188$	17.2348	1.25697
$189$	−1.00000	−0.0727393
$190$	4.18851	0.303867
$191$	22.6772	1.64087	0.820433	−	0.571742i	$-0.193732\pi$
$191$	22.6772	1.64087	0.820433	+	0.571742i	$0.193732\pi$
$192$	−1.57320	−0.113536
$193$	17.9677	1.29334	0.646670	−	0.762770i	$-0.276161\pi$
$193$	17.9677	1.29334	0.646670	+	0.762770i	$0.276161\pi$
$194$	−5.74836	−0.412708
$195$	−4.03157	−0.288706
$196$	−1.50350	−0.107393
$197$	7.02103	0.500227	0.250114	−	0.968216i	$-0.419532\pi$
$197$	7.02103	0.500227	0.250114	+	0.968216i	$0.419532\pi$
$198$	−3.64522	−0.259054
$199$	−11.8337	−0.838871	−0.419435	−	0.907785i	$-0.637772\pi$
$199$	−11.8337	−0.838871	−0.419435	+	0.907785i	$0.637772\pi$
$200$	−2.46865	−0.174560
$201$	1.40925	0.0994007
$202$	−3.24463	−0.228292
$203$	−10.3688	−0.727747
$204$	−1.50350	−0.105266
$205$	−7.67678	−0.536170
$206$	3.09613	0.215718
$207$	−4.66977	−0.324572
$208$	5.11015	0.354325
$209$	−30.7516	−2.12713
$210$	−0.704624	−0.0486237
$211$	1.62298	0.111730	0.0558652	−	0.998438i	$-0.482208\pi$
$211$	1.62298	0.111730	0.0558652	+	0.998438i	$0.482208\pi$
$212$	−9.29209	−0.638183
$213$	9.17328	0.628543
$214$	−7.52806	−0.514608
$215$	−0.937309	−0.0639240
$216$	2.46865	0.167971
$217$	5.51051	0.374078
$218$	0.909273	0.0615838
$219$	3.29044	0.222347
$220$	7.77805	0.524396
$221$	−4.03157	−0.271192
$222$	5.78627	0.388349
$223$	7.39170	0.494985	0.247492	−	0.968890i	$-0.420393\pi$
$223$	7.39170	0.494985	0.247492	+	0.968890i	$0.420393\pi$
$224$	5.83045	0.389563
$225$	1.00000	0.0666667
$226$	10.9167	0.726170
$227$	24.4237	1.62106	0.810529	−	0.585698i	$-0.199180\pi$
$227$	24.4237	1.62106	0.810529	+	0.585698i	$0.199180\pi$
$228$	8.93731	0.591888
$229$	15.9027	1.05088	0.525438	−	0.850832i	$-0.323901\pi$
$229$	15.9027	1.05088	0.525438	+	0.850832i	$0.323901\pi$
$230$	−3.29044	−0.216965
$231$	5.17328	0.340377
$232$	25.5970	1.68052
$233$	17.3629	1.13748	0.568740	−	0.822517i	$-0.307431\pi$
$233$	17.3629	1.13748	0.568740	+	0.822517i	$0.307431\pi$
$234$	2.84074	0.185705
$235$	−11.4631	−0.747767
$236$	−11.4443	−0.744963
$237$	12.0140	0.780394
$238$	−0.704624	−0.0456740
$239$	−12.5019	−0.808677	−0.404339	−	0.914609i	$-0.632498\pi$
$239$	−12.5019	−0.808677	−0.404339	+	0.914609i	$0.632498\pi$
$240$	−1.26753	−0.0818190
$241$	5.16396	0.332640	0.166320	−	0.986072i	$-0.446812\pi$
$241$	5.16396	0.332640	0.166320	+	0.986072i	$0.446812\pi$
$242$	11.1069	0.713976
$243$	−1.00000	−0.0641500
$244$	7.05447	0.451616
$245$	1.00000	0.0638877
$246$	5.40925	0.344881
$247$	23.9649	1.52485
$248$	−13.6036	−0.863827
$249$	14.7535	0.934965
$250$	0.704624	0.0445644
$251$	14.2516	0.899556	0.449778	−	0.893140i	$-0.351503\pi$
$251$	14.2516	0.899556	0.449778	+	0.893140i	$0.351503\pi$
$252$	−1.50350	−0.0947119
$253$	24.1580	1.51880
$254$	−13.4518	−0.844040
$255$	1.00000	0.0626224
$256$	−10.5819	−0.661367
$257$	18.2259	1.13690	0.568449	−	0.822718i	$-0.307544\pi$
$257$	18.2259	1.13690	0.568449	+	0.822718i	$0.307544\pi$
$258$	0.660451	0.0411179
$259$	−8.21185	−0.510260
$260$	−6.06148	−0.375917
$261$	−10.3688	−0.641813
$262$	2.02390	0.125037
$263$	−25.2430	−1.55655	−0.778274	−	0.627924i	$-0.783905\pi$
$263$	−25.2430	−1.55655	−0.778274	+	0.627924i	$0.783905\pi$
$264$	−12.7710	−0.786003
$265$	6.18029	0.379652
$266$	4.18851	0.256814
$267$	−9.00044	−0.550818
$268$	2.11881	0.129427
$269$	23.7679	1.44916	0.724579	−	0.689192i	$-0.242034\pi$
$269$	23.7679	1.44916	0.724579	+	0.689192i	$0.242034\pi$
$270$	−0.704624	−0.0428821
$271$	−4.30732	−0.261651	−0.130826	−	0.991405i	$-0.541763\pi$
$271$	−4.30732	−0.261651	−0.130826	+	0.991405i	$0.541763\pi$
$272$	−1.26753	−0.0768556
$273$	−4.03157	−0.244001
$274$	9.15926	0.553331
$275$	−5.17328	−0.311960
$276$	−7.02103	−0.422616
$277$	−3.03035	−0.182076	−0.0910381	−	0.995847i	$-0.529018\pi$
$277$	−3.03035	−0.182076	−0.0910381	+	0.995847i	$0.529018\pi$
$278$	−12.8103	−0.768309
$279$	5.51051	0.329906
$280$	−2.46865	−0.147530
$281$	−14.9672	−0.892870	−0.446435	−	0.894816i	$-0.647307\pi$
$281$	−14.9672	−0.892870	−0.446435	+	0.894816i	$0.647307\pi$
$282$	8.07715	0.480987
$283$	−3.38866	−0.201435	−0.100717	−	0.994915i	$-0.532114\pi$
$283$	−3.38866	−0.201435	−0.100717	+	0.994915i	$0.532114\pi$
$284$	13.7921	0.818409
$285$	−5.94432	−0.352111
$286$	−14.6959	−0.868989
$287$	−7.67678	−0.453146
$288$	5.83045	0.343562
$289$	1.00000	0.0588235
$290$	−7.30611	−0.429029
$291$	8.15805	0.478233
$292$	4.94719	0.289512
$293$	1.16748	0.0682052	0.0341026	−	0.999418i	$-0.489143\pi$
$293$	1.16748	0.0682052	0.0341026	+	0.999418i	$0.489143\pi$
$294$	−0.704624	−0.0410945
$295$	7.61178	0.443175
$296$	20.2722	1.17830
$297$	5.17328	0.300184
$298$	−7.78384	−0.450906
$299$	−18.8265	−1.08876
$300$	1.50350	0.0868049
$301$	−0.937309	−0.0540256
$302$	−8.02804	−0.461962
$303$	4.60477	0.264537
$304$	7.53463	0.432141
$305$	−4.69202	−0.268664
$306$	−0.704624	−0.0402807
$307$	−8.43850	−0.481611	−0.240805	−	0.970573i	$-0.577412\pi$
$307$	−8.43850	−0.481611	−0.240805	+	0.970573i	$0.577412\pi$
$308$	7.77805	0.443195
$309$	−4.39402	−0.249967
$310$	3.88284	0.220531
$311$	7.75636	0.439823	0.219911	−	0.975520i	$-0.429423\pi$
$311$	7.75636	0.439823	0.219911	+	0.975520i	$0.429423\pi$
$312$	9.95254	0.563452
$313$	25.2562	1.42757	0.713783	−	0.700367i	$-0.246980\pi$
$313$	25.2562	1.42757	0.713783	+	0.700367i	$0.246980\pi$
$314$	−2.48016	−0.139964
$315$	1.00000	0.0563436
$316$	18.0631	1.01613
$317$	20.5584	1.15468	0.577338	−	0.816505i	$-0.304092\pi$
$317$	20.5584	1.15468	0.577338	+	0.816505i	$0.304092\pi$
$318$	−4.35478	−0.244204
$319$	53.6407	3.00330
$320$	1.57320	0.0879448
$321$	10.6838	0.596311
$322$	−3.29044	−0.183369
$323$	−5.94432	−0.330751
$324$	−1.50350	−0.0835280
$325$	4.03157	0.223631
$326$	12.9307	0.716167
$327$	−1.29044	−0.0713613
$328$	18.9513	1.04641
$329$	−11.4631	−0.631979
$330$	3.64522	0.200663
$331$	8.59351	0.472342	0.236171	−	0.971712i	$-0.424107\pi$
$331$	8.59351	0.472342	0.236171	+	0.971712i	$0.424107\pi$
$332$	22.1819	1.21739
$333$	−8.21185	−0.450007
$334$	−11.6118	−0.635368
$335$	−1.40925	−0.0769955
$336$	−1.26753	−0.0691497
$337$	28.2797	1.54049	0.770246	−	0.637747i	$-0.220133\pi$
$337$	28.2797	1.54049	0.770246	+	0.637747i	$0.220133\pi$
$338$	2.29251	0.124696
$339$	−15.4930	−0.841462
$340$	1.50350	0.0815390
$341$	−28.5074	−1.54376
$342$	4.18851	0.226489
$343$	1.00000	0.0539949
$344$	2.31389	0.124757
$345$	4.66977	0.251412
$346$	11.5772	0.622393
$347$	4.00000	0.214731	0.107366	−	0.994220i	$-0.465758\pi$
$347$	4.00000	0.214731	0.107366	+	0.994220i	$0.465758\pi$
$348$	−15.5895	−0.835687
$349$	16.1721	0.865671	0.432835	−	0.901473i	$-0.357513\pi$
$349$	16.1721	0.865671	0.432835	+	0.901473i	$0.357513\pi$
$350$	0.704624	0.0376638
$351$	−4.03157	−0.215189
$352$	−30.1625	−1.60767
$353$	16.7787	0.893041	0.446520	−	0.894774i	$-0.352663\pi$
$353$	16.7787	0.893041	0.446520	+	0.894774i	$0.352663\pi$
$354$	−5.36344	−0.285064
$355$	−9.17328	−0.486867
$356$	−13.5322	−0.717205
$357$	1.00000	0.0529256
$358$	−17.2867	−0.913630
$359$	−20.8026	−1.09792	−0.548960	−	0.835849i	$-0.684976\pi$
$359$	−20.8026	−1.09792	−0.548960	+	0.835849i	$0.684976\pi$
$360$	−2.46865	−0.130110
$361$	16.3349	0.859733
$362$	−4.69389	−0.246705
$363$	−15.7628	−0.827333
$364$	−6.06148	−0.317708
$365$	−3.29044	−0.172229
$366$	3.30611	0.172813
$367$	−26.6885	−1.39313	−0.696564	−	0.717495i	$-0.745289\pi$
$367$	−26.6885	−1.39313	−0.696564	+	0.717495i	$0.745289\pi$
$368$	−5.91910	−0.308555
$369$	−7.67678	−0.399637
$370$	−5.78627	−0.300814
$371$	6.18029	0.320865
$372$	8.28508	0.429561
$373$	−13.1193	−0.679289	−0.339645	−	0.940554i	$-0.610307\pi$
$373$	−13.1193	−0.679289	−0.339645	+	0.940554i	$0.610307\pi$
$374$	3.64522	0.188490
$375$	−1.00000	−0.0516398
$376$	28.2983	1.45937
$377$	−41.8025	−2.15294
$378$	−0.704624	−0.0362420
$379$	37.0210	1.90164	0.950821	−	0.309740i	$-0.100242\pi$
$379$	37.0210	1.90164	0.950821	+	0.309740i	$0.100242\pi$
$380$	−8.93731	−0.458474
$381$	19.0907	0.978048
$382$	15.9789	0.817553
$383$	−6.00745	−0.306966	−0.153483	−	0.988151i	$-0.549049\pi$
$383$	−6.00745	−0.306966	−0.153483	+	0.988151i	$0.549049\pi$
$384$	10.5524	0.538498
$385$	−5.17328	−0.263655
$386$	12.6605	0.644400
$387$	−0.937309	−0.0476461
$388$	12.2657	0.622695
$389$	22.0865	1.11983	0.559914	−	0.828551i	$-0.310834\pi$
$389$	22.0865	1.11983	0.559914	+	0.828551i	$0.310834\pi$
$390$	−2.84074	−0.143846
$391$	4.66977	0.236161
$392$	−2.46865	−0.124686
$393$	−2.87230	−0.144889
$394$	4.94719	0.249236
$395$	−12.0140	−0.604491
$396$	7.77805	0.390862
$397$	−20.3288	−1.02027	−0.510136	−	0.860094i	$-0.670405\pi$
$397$	−20.3288	−1.02027	−0.510136	+	0.860094i	$0.670405\pi$
$398$	−8.33833	−0.417963
$399$	−5.94432	−0.297588
$400$	1.26753	0.0633767
$401$	−3.11015	−0.155313	−0.0776567	−	0.996980i	$-0.524744\pi$
$401$	−3.11015	−0.155313	−0.0776567	+	0.996980i	$0.524744\pi$
$402$	0.992991	0.0495259
$403$	22.2160	1.10666
$404$	6.92329	0.344447
$405$	1.00000	0.0496904
$406$	−7.30611	−0.362596
$407$	42.4822	2.10576
$408$	−2.46865	−0.122217
$409$	−25.8124	−1.27634	−0.638170	−	0.769896i	$-0.720308\pi$
$409$	−25.8124	−1.27634	−0.638170	+	0.769896i	$0.720308\pi$
$410$	−5.40925	−0.267144
$411$	−12.9988	−0.641183
$412$	−6.60642	−0.325475
$413$	7.61178	0.374551
$414$	−3.29044	−0.161716
$415$	−14.7535	−0.724220
$416$	23.5058	1.15247
$417$	18.1803	0.890293
$418$	−21.6683	−1.05983
$419$	−34.0678	−1.66432	−0.832161	−	0.554534i	$-0.812897\pi$
$419$	−34.0678	−1.66432	−0.832161	+	0.554534i	$0.812897\pi$
$420$	1.50350	0.0733635
$421$	−26.7053	−1.30153	−0.650767	−	0.759277i	$-0.725553\pi$
$421$	−26.7053	−1.30153	−0.650767	+	0.759277i	$0.725553\pi$
$422$	1.14359	0.0556690
$423$	−11.4631	−0.557353
$424$	−15.2570	−0.740945
$425$	−1.00000	−0.0485071
$426$	6.46372	0.313168
$427$	−4.69202	−0.227063
$428$	16.0631	0.776441
$429$	20.8564	1.00696
$430$	−0.660451	−0.0318498
$431$	−14.8865	−0.717060	−0.358530	−	0.933518i	$-0.616722\pi$
$431$	−14.8865	−0.717060	−0.358530	+	0.933518i	$0.616722\pi$
$432$	−1.26753	−0.0609843
$433$	−33.1966	−1.59533	−0.797664	−	0.603102i	$-0.793931\pi$
$433$	−33.1966	−1.59533	−0.797664	+	0.603102i	$0.793931\pi$
$434$	3.88284	0.186382
$435$	10.3688	0.497146
$436$	−1.94018	−0.0929177
$437$	−27.7586	−1.32788
$438$	2.31852	0.110783
$439$	−6.29165	−0.300284	−0.150142	−	0.988664i	$-0.547973\pi$
$439$	−6.29165	−0.300284	−0.150142	+	0.988664i	$0.547973\pi$
$440$	12.7710	0.608835
$441$	1.00000	0.0476190
$442$	−2.84074	−0.135120
$443$	29.7463	1.41329	0.706644	−	0.707569i	$-0.250208\pi$
$443$	29.7463	1.41329	0.706644	+	0.707569i	$0.250208\pi$
$444$	−12.3466	−0.585942
$445$	9.00044	0.426662
$446$	5.20837	0.246624
$447$	11.0468	0.522496
$448$	1.57320	0.0743269
$449$	3.34777	0.157991	0.0789956	−	0.996875i	$-0.474829\pi$
$449$	3.34777	0.157991	0.0789956	+	0.996875i	$0.474829\pi$
$450$	0.704624	0.0332163
$451$	39.7141	1.87007
$452$	−23.2937	−1.09565
$453$	11.3934	0.535307
$454$	17.2095	0.807684
$455$	4.03157	0.189003
$456$	14.6745	0.687195
$457$	−25.4930	−1.19251	−0.596255	−	0.802795i	$-0.703345\pi$
$457$	−25.4930	−1.19251	−0.596255	+	0.802795i	$0.703345\pi$
$458$	11.2054	0.523594
$459$	1.00000	0.0466760
$460$	7.02103	0.327357
$461$	−34.0986	−1.58813	−0.794065	−	0.607832i	$-0.792039\pi$
$461$	−34.0986	−1.58813	−0.794065	+	0.607832i	$0.792039\pi$
$462$	3.64522	0.169591
$463$	24.5252	1.13978	0.569891	−	0.821720i	$-0.306985\pi$
$463$	24.5252	1.13978	0.569891	+	0.821720i	$0.306985\pi$
$464$	−13.1428	−0.610140
$465$	−5.51051	−0.255544
$466$	12.2343	0.566744
$467$	0.0463582	0.00214520	0.00107260	−	0.999999i	$-0.499659\pi$
$467$	0.0463582	0.00214520	0.00107260	+	0.999999i	$0.499659\pi$
$468$	−6.06148	−0.280192
$469$	−1.40925	−0.0650731
$470$	−8.07715	−0.372571
$471$	3.51984	0.162186
$472$	−18.7909	−0.864919
$473$	4.84896	0.222956
$474$	8.46537	0.388827
$475$	5.94432	0.272744
$476$	1.50350	0.0689130
$477$	6.18029	0.282976
$478$	−8.80911	−0.402919
$479$	40.2866	1.84074	0.920370	−	0.391048i	$-0.127887\pi$
$479$	40.2866	1.84074	0.920370	+	0.391048i	$0.127887\pi$
$480$	−5.83045	−0.266122
$481$	−33.1066	−1.50953
$482$	3.63865	0.165736
$483$	4.66977	0.212482
$484$	−23.6995	−1.07725
$485$	−8.15805	−0.370438
$486$	−0.704624	−0.0319624
$487$	−19.6112	−0.888669	−0.444335	−	0.895861i	$-0.646560\pi$
$487$	−19.6112	−0.888669	−0.444335	+	0.895861i	$0.646560\pi$
$488$	11.5830	0.524336
$489$	−18.3513	−0.829872
$490$	0.704624	0.0318317
$491$	−7.52994	−0.339821	−0.169911	−	0.985459i	$-0.554348\pi$
$491$	−7.52994	−0.339821	−0.169911	+	0.985459i	$0.554348\pi$
$492$	−11.5421	−0.520357
$493$	10.3688	0.466987
$494$	16.8863	0.759749
$495$	−5.17328	−0.232522
$496$	6.98477	0.313625
$497$	−9.17328	−0.411478
$498$	10.3957	0.465841
$499$	−34.5186	−1.54527	−0.772633	−	0.634853i	$-0.781061\pi$
$499$	−34.5186	−1.54527	−0.772633	+	0.634853i	$0.781061\pi$
$500$	−1.50350	−0.0672388
$501$	16.4794	0.736245
$502$	10.0421	0.448199
$503$	1.44716	0.0645258	0.0322629	−	0.999479i	$-0.489729\pi$
$503$	1.44716	0.0645258	0.0322629	+	0.999479i	$0.489729\pi$
$504$	−2.46865	−0.109963
$505$	−4.60477	−0.204910
$506$	17.0223	0.756736
$507$	−3.25352	−0.144494
$508$	28.7030	1.27349
$509$	5.89764	0.261408	0.130704	−	0.991421i	$-0.458276\pi$
$509$	5.89764	0.261408	0.130704	+	0.991421i	$0.458276\pi$
$510$	0.704624	0.0312013
$511$	−3.29044	−0.145560
$512$	13.6485	0.603184
$513$	−5.94432	−0.262448
$514$	12.8424	0.566454
$515$	4.39402	0.193623
$516$	−1.40925	−0.0620387
$517$	59.3016	2.60808
$518$	−5.78627	−0.254234
$519$	−16.4303	−0.721209
$520$	−9.95254	−0.436448
$521$	−36.9283	−1.61786	−0.808929	−	0.587906i	$-0.799953\pi$
$521$	−36.9283	−1.61786	−0.808929	+	0.587906i	$0.799953\pi$
$522$	−7.30611	−0.319780
$523$	13.8332	0.604883	0.302441	−	0.953168i	$-0.402198\pi$
$523$	13.8332	0.604883	0.302441	+	0.953168i	$0.402198\pi$
$524$	−4.31852	−0.188656
$525$	−1.00000	−0.0436436
$526$	−17.7868	−0.775542
$527$	−5.51051	−0.240042
$528$	6.55731	0.285370
$529$	−1.19321	−0.0518785
$530$	4.35478	0.189160
$531$	7.61178	0.330323
$532$	−8.93731	−0.387481
$533$	−30.9495	−1.34057
$534$	−6.34193	−0.274442
$535$	−10.6838	−0.461901
$536$	3.47895	0.150268
$537$	24.5332	1.05869
$538$	16.7475	0.722035
$539$	−5.17328	−0.222829
$540$	1.50350	0.0647005
$541$	−6.89565	−0.296467	−0.148233	−	0.988952i	$-0.547359\pi$
$541$	−6.89565	−0.296467	−0.148233	+	0.988952i	$0.547359\pi$
$542$	−3.03504	−0.130366
$543$	6.66155	0.285875
$544$	−5.83045	−0.249978
$545$	1.29044	0.0552762
$546$	−2.84074	−0.121572
$547$	4.31859	0.184649	0.0923247	−	0.995729i	$-0.470570\pi$
$547$	4.31859	0.184649	0.0923247	+	0.995729i	$0.470570\pi$
$548$	−19.5437	−0.834867
$549$	−4.69202	−0.200250
$550$	−3.64522	−0.155433
$551$	−61.6354	−2.62576
$552$	−11.5281	−0.490667
$553$	−12.0140	−0.510888
$554$	−2.13526	−0.0907185
$555$	8.21185	0.348574
$556$	27.3341	1.15923
$557$	35.5198	1.50502	0.752512	−	0.658579i	$-0.228842\pi$
$557$	35.5198	1.50502	0.752512	+	0.658579i	$0.228842\pi$
$558$	3.88284	0.164374
$559$	−3.77882	−0.159827
$560$	1.26753	0.0535631
$561$	−5.17328	−0.218416
$562$	−10.5463	−0.444867
$563$	26.3282	1.10960	0.554801	−	0.831983i	$-0.312794\pi$
$563$	26.3282	1.10960	0.554801	+	0.831983i	$0.312794\pi$
$564$	−17.2348	−0.725714
$565$	15.4930	0.651794
$566$	−2.38773	−0.100364
$567$	1.00000	0.0419961
$568$	22.6457	0.950191
$569$	−4.63931	−0.194490	−0.0972450	−	0.995260i	$-0.531003\pi$
$569$	−4.63931	−0.194490	−0.0972450	+	0.995260i	$0.531003\pi$
$570$	−4.18851	−0.175437
$571$	−22.2492	−0.931101	−0.465550	−	0.885021i	$-0.654144\pi$
$571$	−22.2492	−0.931101	−0.465550	+	0.885021i	$0.654144\pi$
$572$	31.3577	1.31113
$573$	−22.6772	−0.947355
$574$	−5.40925	−0.225778
$575$	−4.66977	−0.194743
$576$	1.57320	0.0655502
$577$	−25.4389	−1.05904	−0.529518	−	0.848298i	$-0.677627\pi$
$577$	−25.4389	−1.05904	−0.529518	+	0.848298i	$0.677627\pi$
$578$	0.704624	0.0293085
$579$	−17.9677	−0.746710
$580$	15.5895	0.647320
$581$	−14.7535	−0.612078
$582$	5.74836	0.238277
$583$	−31.9724	−1.32416
$584$	8.12295	0.336130
$585$	4.03157	0.166685
$586$	0.822638	0.0339829
$587$	−2.93643	−0.121199	−0.0605997	−	0.998162i	$-0.519301\pi$
$587$	−2.93643	−0.121199	−0.0605997	+	0.998162i	$0.519301\pi$
$588$	1.50350	0.0620035
$589$	32.7562	1.34970
$590$	5.36344	0.220810
$591$	−7.02103	−0.288806
$592$	−10.4088	−0.427799
$593$	15.8662	0.651546	0.325773	−	0.945448i	$-0.394376\pi$
$593$	15.8662	0.651546	0.325773	+	0.945448i	$0.394376\pi$
$594$	3.64522	0.149565
$595$	−1.00000	−0.0409960
$596$	16.6089	0.680327
$597$	11.8337	0.484322
$598$	−13.2656	−0.542471
$599$	−31.2674	−1.27755	−0.638776	−	0.769393i	$-0.720559\pi$
$599$	−31.2674	−1.27755	−0.638776	+	0.769393i	$0.720559\pi$
$600$	2.46865	0.100782
$601$	−25.4436	−1.03786	−0.518932	−	0.854815i	$-0.673670\pi$
$601$	−25.4436	−1.03786	−0.518932	+	0.854815i	$0.673670\pi$
$602$	−0.660451	−0.0269180
$603$	−1.40925	−0.0573890
$604$	17.1300	0.697008
$605$	15.7628	0.640850
$606$	3.24463	0.131804
$607$	13.8301	0.561348	0.280674	−	0.959803i	$-0.409442\pi$
$607$	13.8301	0.561348	0.280674	+	0.959803i	$0.409442\pi$
$608$	34.6580	1.40557
$609$	10.3688	0.420165
$610$	−3.30611	−0.133860
$611$	−46.2141	−1.86962
$612$	1.50350	0.0607756
$613$	−17.4612	−0.705250	−0.352625	−	0.935765i	$-0.614711\pi$
$613$	−17.4612	−0.705250	−0.352625	+	0.935765i	$0.614711\pi$
$614$	−5.94597	−0.239960
$615$	7.67678	0.309558
$616$	12.7710	0.514560
$617$	−24.8592	−1.00079	−0.500396	−	0.865797i	$-0.666812\pi$
$617$	−24.8592	−1.00079	−0.500396	+	0.865797i	$0.666812\pi$
$618$	−3.09613	−0.124545
$619$	−10.6242	−0.427022	−0.213511	−	0.976941i	$-0.568490\pi$
$619$	−10.6242	−0.427022	−0.213511	+	0.976941i	$0.568490\pi$
$620$	−8.28508	−0.332737
$621$	4.66977	0.187392
$622$	5.46532	0.219139
$623$	9.00044	0.360595
$624$	−5.11015	−0.204570
$625$	1.00000	0.0400000
$626$	17.7961	0.711277
$627$	30.7516	1.22810
$628$	5.29209	0.211177
$629$	8.21185	0.327428
$630$	0.704624	0.0280729
$631$	33.2488	1.32361	0.661806	−	0.749675i	$-0.269790\pi$
$631$	33.2488	1.32361	0.661806	+	0.749675i	$0.269790\pi$
$632$	29.6585	1.17975
$633$	−1.62298	−0.0645075
$634$	14.4860	0.575311
$635$	−19.0907	−0.757592
$636$	9.29209	0.368455
$637$	4.03157	0.159736
$638$	37.7965	1.49638
$639$	−9.17328	−0.362889
$640$	−10.5524	−0.417119
$641$	−19.6134	−0.774684	−0.387342	−	0.921936i	$-0.626607\pi$
$641$	−19.6134	−0.774684	−0.387342	+	0.921936i	$0.626607\pi$
$642$	7.52806	0.297109
$643$	−20.9906	−0.827787	−0.413893	−	0.910325i	$-0.635831\pi$
$643$	−20.9906	−0.827787	−0.413893	+	0.910325i	$0.635831\pi$
$644$	7.02103	0.276667
$645$	0.937309	0.0369065
$646$	−4.18851	−0.164795
$647$	14.0206	0.551206	0.275603	−	0.961272i	$-0.411122\pi$
$647$	14.0206	0.551206	0.275603	+	0.961272i	$0.411122\pi$
$648$	−2.46865	−0.0969779
$649$	−39.3779	−1.54572
$650$	2.84074	0.111423
$651$	−5.51051	−0.215974
$652$	−27.5912	−1.08055
$653$	9.10993	0.356499	0.178250	−	0.983985i	$-0.442957\pi$
$653$	9.10993	0.356499	0.178250	+	0.983985i	$0.442957\pi$
$654$	−0.909273	−0.0355554
$655$	2.87230	0.112230
$656$	−9.73059	−0.379916
$657$	−3.29044	−0.128372
$658$	−8.07715	−0.314880
$659$	−4.69969	−0.183074	−0.0915369	−	0.995802i	$-0.529178\pi$
$659$	−4.69969	−0.183074	−0.0915369	+	0.995802i	$0.529178\pi$
$660$	−7.77805	−0.302760
$661$	−31.7797	−1.23609	−0.618044	−	0.786144i	$-0.712074\pi$
$661$	−31.7797	−1.23609	−0.618044	+	0.786144i	$0.712074\pi$
$662$	6.05519	0.235342
$663$	4.03157	0.156573
$664$	36.4213	1.41342
$665$	5.94432	0.230511
$666$	−5.78627	−0.224213
$667$	48.4200	1.87483
$668$	24.7768	0.958644
$669$	−7.39170	−0.285780
$670$	−0.992991	−0.0383626
$671$	24.2731	0.937053
$672$	−5.83045	−0.224914
$673$	11.2680	0.434348	0.217174	−	0.976133i	$-0.430316\pi$
$673$	11.2680	0.434348	0.217174	+	0.976133i	$0.430316\pi$
$674$	19.9265	0.767542
$675$	−1.00000	−0.0384900
$676$	−4.89168	−0.188141
$677$	6.08703	0.233943	0.116972	−	0.993135i	$-0.462681\pi$
$677$	6.08703	0.233943	0.116972	+	0.993135i	$0.462681\pi$
$678$	−10.9167	−0.419254
$679$	−8.15805	−0.313077
$680$	2.46865	0.0946686
$681$	−24.4237	−0.935918
$682$	−20.0870	−0.769172
$683$	−10.9579	−0.419292	−0.209646	−	0.977777i	$-0.567231\pi$
$683$	−10.9579	−0.419292	−0.209646	+	0.977777i	$0.567231\pi$
$684$	−8.93731	−0.341727
$685$	12.9988	0.496658
$686$	0.704624	0.0269027
$687$	−15.9027	−0.606724
$688$	−1.18807	−0.0452948
$689$	24.9162	0.949233
$690$	3.29044	0.125265
$691$	−11.9366	−0.454092	−0.227046	−	0.973884i	$-0.572907\pi$
$691$	−11.9366	−0.454092	−0.227046	+	0.973884i	$0.572907\pi$
$692$	−24.7030	−0.939067
$693$	−5.17328	−0.196517
$694$	2.81850	0.106989
$695$	−18.1803	−0.689618
$696$	−25.5970	−0.970251
$697$	7.67678	0.290779
$698$	11.3952	0.431316
$699$	−17.3629	−0.656725
$700$	−1.50350	−0.0568271
$701$	15.0913	0.569990	0.284995	−	0.958529i	$-0.408008\pi$
$701$	15.0913	0.569990	0.284995	+	0.958529i	$0.408008\pi$
$702$	−2.84074	−0.107217
$703$	−48.8139	−1.84105
$704$	−8.13862	−0.306736
$705$	11.4631	0.431724
$706$	11.8227	0.444953
$707$	−4.60477	−0.173180
$708$	11.4443	0.430105
$709$	28.2610	1.06137	0.530683	−	0.847571i	$-0.321936\pi$
$709$	28.2610	1.06137	0.530683	+	0.847571i	$0.321936\pi$
$710$	−6.46372	−0.242579
$711$	−12.0140	−0.450561
$712$	−22.2190	−0.832691
$713$	−25.7329	−0.963703
$714$	0.704624	0.0263699
$715$	−20.8564	−0.779986
$716$	36.8858	1.37849
$717$	12.5019	0.466890
$718$	−14.6580	−0.547033
$719$	23.2142	0.865745	0.432872	−	0.901455i	$-0.357500\pi$
$719$	23.2142	0.865745	0.432872	+	0.901455i	$0.357500\pi$
$720$	1.26753	0.0472382
$721$	4.39402	0.163642
$722$	11.5100	0.428357
$723$	−5.16396	−0.192050
$724$	10.0157	0.372230
$725$	−10.3688	−0.385088
$726$	−11.1069	−0.412214
$727$	40.9805	1.51988	0.759941	−	0.649992i	$-0.225228\pi$
$727$	40.9805	1.51988	0.759941	+	0.649992i	$0.225228\pi$
$728$	−9.95254	−0.368866
$729$	1.00000	0.0370370
$730$	−2.31852	−0.0858124
$731$	0.937309	0.0346676
$732$	−7.05447	−0.260741
$733$	26.6124	0.982953	0.491476	−	0.870891i	$-0.336457\pi$
$733$	26.6124	0.982953	0.491476	+	0.870891i	$0.336457\pi$
$734$	−18.8054	−0.694118
$735$	−1.00000	−0.0368856
$736$	−27.2269	−1.00360
$737$	7.29044	0.268547
$738$	−5.40925	−0.199117
$739$	−1.22074	−0.0449055	−0.0224528	−	0.999748i	$-0.507148\pi$
$739$	−1.22074	−0.0449055	−0.0224528	+	0.999748i	$0.507148\pi$
$740$	12.3466	0.453868
$741$	−23.9649	−0.880373
$742$	4.35478	0.159869
$743$	28.0069	1.02747	0.513737	−	0.857948i	$-0.328261\pi$
$743$	28.0069	1.02747	0.513737	+	0.857948i	$0.328261\pi$
$744$	13.6036	0.498731
$745$	−11.0468	−0.404723
$746$	−9.24414	−0.338452
$747$	−14.7535	−0.539802
$748$	−7.77805	−0.284394
$749$	−10.6838	−0.390377
$750$	−0.704624	−0.0257292
$751$	−16.7071	−0.609652	−0.304826	−	0.952408i	$-0.598598\pi$
$751$	−16.7071	−0.609652	−0.304826	+	0.952408i	$0.598598\pi$
$752$	−14.5298	−0.529848
$753$	−14.2516	−0.519359
$754$	−29.4551	−1.07269
$755$	−11.3934	−0.414647
$756$	1.50350	0.0546819
$757$	9.46713	0.344089	0.172044	−	0.985089i	$-0.444963\pi$
$757$	9.46713	0.344089	0.172044	+	0.985089i	$0.444963\pi$
$758$	26.0859	0.947483
$759$	−24.1580	−0.876882
$760$	−14.6745	−0.532299
$761$	27.0982	0.982308	0.491154	−	0.871073i	$-0.336575\pi$
$761$	27.0982	0.982308	0.491154	+	0.871073i	$0.336575\pi$
$762$	13.4518	0.487307
$763$	1.29044	0.0467170
$764$	−34.0953	−1.23352
$765$	−1.00000	−0.0361551
$766$	−4.23299	−0.152944
$767$	30.6874	1.10806
$768$	10.5819	0.381840
$769$	−45.8684	−1.65406	−0.827029	−	0.562159i	$-0.809971\pi$
$769$	−45.8684	−1.65406	−0.827029	+	0.562159i	$0.809971\pi$
$770$	−3.64522	−0.131365
$771$	−18.2259	−0.656389
$772$	−27.0145	−0.972272
$773$	−26.0700	−0.937674	−0.468837	−	0.883285i	$-0.655327\pi$
$773$	−26.0700	−0.937674	−0.468837	+	0.883285i	$0.655327\pi$
$774$	−0.660451	−0.0237394
$775$	5.51051	0.197943
$776$	20.1394	0.722962
$777$	8.21185	0.294599
$778$	15.5627	0.557949
$779$	−45.6332	−1.63498
$780$	6.06148	0.217036
$781$	47.4559	1.69811
$782$	3.29044	0.117666
$783$	10.3688	0.370551
$784$	1.26753	0.0452691
$785$	−3.51984	−0.125628
$786$	−2.02390	−0.0721900
$787$	−11.0677	−0.394521	−0.197261	−	0.980351i	$-0.563205\pi$
$787$	−11.0677	−0.394521	−0.197261	+	0.980351i	$0.563205\pi$
$788$	−10.5561	−0.376047
$789$	25.2430	0.898674
$790$	−8.46537	−0.301184
$791$	15.4930	0.550866
$792$	12.7710	0.453799
$793$	−18.9162	−0.671733
$794$	−14.3242	−0.508345
$795$	−6.18029	−0.219192
$796$	17.7921	0.630623
$797$	2.49528	0.0883874	0.0441937	−	0.999023i	$-0.485928\pi$
$797$	2.49528	0.0883874	0.0441937	+	0.999023i	$0.485928\pi$
$798$	−4.18851	−0.148272
$799$	11.4631	0.405534
$800$	5.83045	0.206137
$801$	9.00044	0.318015
$802$	−2.19149	−0.0773841
$803$	17.0223	0.600706
$804$	−2.11881	−0.0747247
$805$	−4.66977	−0.164588
$806$	15.6539	0.551386
$807$	−23.7679	−0.836671
$808$	11.3676	0.399910
$809$	−32.1671	−1.13094	−0.565468	−	0.824770i	$-0.691305\pi$
$809$	−32.1671	−1.13094	−0.565468	+	0.824770i	$0.691305\pi$
$810$	0.704624	0.0247580
$811$	21.7152	0.762525	0.381263	−	0.924467i	$-0.375489\pi$
$811$	21.7152	0.762525	0.381263	+	0.924467i	$0.375489\pi$
$812$	15.5895	0.547085
$813$	4.30732	0.151064
$814$	29.9340	1.04919
$815$	18.3513	0.642816
$816$	1.26753	0.0443726
$817$	−5.57166	−0.194928
$818$	−18.1880	−0.635929
$819$	4.03157	0.140874
$820$	11.5421	0.403067
$821$	26.5994	0.928324	0.464162	−	0.885750i	$-0.346356\pi$
$821$	26.5994	0.928324	0.464162	+	0.885750i	$0.346356\pi$
$822$	−9.15926	−0.319466
$823$	53.6454	1.86996	0.934980	−	0.354700i	$-0.115417\pi$
$823$	53.6454	1.86996	0.934980	+	0.354700i	$0.115417\pi$
$824$	−10.8473	−0.377884
$825$	5.17328	0.180110
$826$	5.36344	0.186618
$827$	12.5935	0.437919	0.218960	−	0.975734i	$-0.429734\pi$
$827$	12.5935	0.437919	0.218960	+	0.975734i	$0.429734\pi$
$828$	7.02103	0.243998
$829$	6.47194	0.224780	0.112390	−	0.993664i	$-0.464149\pi$
$829$	6.47194	0.224780	0.112390	+	0.993664i	$0.464149\pi$
$830$	−10.3957	−0.360839
$831$	3.03035	0.105122
$832$	6.34248	0.219886
$833$	−1.00000	−0.0346479
$834$	12.8103	0.443584
$835$	−16.4794	−0.570293
$836$	46.2352	1.59908
$837$	−5.51051	−0.190471
$838$	−24.0050	−0.829240
$839$	−2.69092	−0.0929007	−0.0464504	−	0.998921i	$-0.514791\pi$
$839$	−2.69092	−0.0929007	−0.0464504	+	0.998921i	$0.514791\pi$
$840$	2.46865	0.0851767
$841$	78.5120	2.70731
$842$	−18.8172	−0.648483
$843$	14.9672	0.515498
$844$	−2.44015	−0.0839935
$845$	3.25352	0.111924
$846$	−8.07715	−0.277698
$847$	15.7628	0.541617
$848$	7.83373	0.269011
$849$	3.38866	0.116299
$850$	−0.704624	−0.0241684
$851$	38.3475	1.31454
$852$	−13.7921	−0.472508
$853$	48.3288	1.65475	0.827373	−	0.561653i	$-0.189834\pi$
$853$	48.3288	1.65475	0.827373	+	0.561653i	$0.189834\pi$
$854$	−3.30611	−0.113133
$855$	5.94432	0.203291
$856$	26.3746	0.901465
$857$	36.6070	1.25047	0.625235	−	0.780436i	$-0.285003\pi$
$857$	36.6070	1.25047	0.625235	+	0.780436i	$0.285003\pi$
$858$	14.6959	0.501711
$859$	52.4803	1.79060	0.895302	−	0.445460i	$-0.146960\pi$
$859$	52.4803	1.79060	0.895302	+	0.445460i	$0.146960\pi$
$860$	1.40925	0.0480550
$861$	7.67678	0.261624
$862$	−10.4894	−0.357271
$863$	7.75089	0.263843	0.131922	−	0.991260i	$-0.457885\pi$
$863$	7.75089	0.263843	0.131922	+	0.991260i	$0.457885\pi$
$864$	−5.83045	−0.198356
$865$	16.4303	0.558646
$866$	−23.3911	−0.794864
$867$	−1.00000	−0.0339618
$868$	−8.28508	−0.281214
$869$	62.1519	2.10836
$870$	7.30611	0.247700
$871$	−5.68148	−0.192510
$872$	−3.18564	−0.107879
$873$	−8.15805	−0.276108
$874$	−19.5594	−0.661607
$875$	1.00000	0.0338062
$876$	−4.94719	−0.167150
$877$	−9.94957	−0.335973	−0.167986	−	0.985789i	$-0.553727\pi$
$877$	−9.94957	−0.335973	−0.167986	+	0.985789i	$0.553727\pi$
$878$	−4.43325	−0.149615
$879$	−1.16748	−0.0393783
$880$	−6.55731	−0.221047
$881$	−17.4836	−0.589037	−0.294518	−	0.955646i	$-0.595159\pi$
$881$	−17.4836	−0.589037	−0.294518	+	0.955646i	$0.595159\pi$
$882$	0.704624	0.0237259
$883$	−39.5331	−1.33039	−0.665197	−	0.746668i	$-0.731652\pi$
$883$	−39.5331	−1.33039	−0.665197	+	0.746668i	$0.731652\pi$
$884$	6.06148	0.203870
$885$	−7.61178	−0.255867
$886$	20.9599	0.704163
$887$	−22.5771	−0.758066	−0.379033	−	0.925383i	$-0.623743\pi$
$887$	−22.5771	−0.758066	−0.379033	+	0.925383i	$0.623743\pi$
$888$	−20.2722	−0.680291
$889$	−19.0907	−0.640282
$890$	6.34193	0.212582
$891$	−5.17328	−0.173311
$892$	−11.1135	−0.372106
$893$	−68.1400	−2.28022
$894$	7.78384	0.260331
$895$	−24.5332	−0.820054
$896$	−10.5524	−0.352530
$897$	18.8265	0.628599
$898$	2.35892	0.0787183
$899$	−57.1374	−1.90564
$900$	−1.50350	−0.0501168
$901$	−6.18029	−0.205895
$902$	27.9836	0.931750
$903$	0.937309	0.0311917
$904$	−38.2468	−1.27207
$905$	−6.66155	−0.221437
$906$	8.02804	0.266714
$907$	14.4187	0.478765	0.239382	−	0.970925i	$-0.423055\pi$
$907$	14.4187	0.478765	0.239382	+	0.970925i	$0.423055\pi$
$908$	−36.7212	−1.21863
$909$	−4.60477	−0.152731
$910$	2.84074	0.0941696
$911$	2.04089	0.0676177	0.0338088	−	0.999428i	$-0.489236\pi$
$911$	2.04089	0.0676177	0.0338088	+	0.999428i	$0.489236\pi$
$912$	−7.53463	−0.249497
$913$	76.3239	2.52595
$914$	−17.9630	−0.594162
$915$	4.69202	0.155113
$916$	−23.9097	−0.789999
$917$	2.87230	0.0948518
$918$	0.704624	0.0232561
$919$	−42.1403	−1.39008	−0.695040	−	0.718971i	$-0.744613\pi$
$919$	−42.1403	−1.39008	−0.695040	+	0.718971i	$0.744613\pi$
$920$	11.5281	0.380069
$921$	8.43850	0.278058
$922$	−24.0267	−0.791278
$923$	−36.9827	−1.21730
$924$	−7.77805	−0.255879
$925$	−8.21185	−0.270004
$926$	17.2810	0.567890
$927$	4.39402	0.144318
$928$	−60.4547	−1.98452
$929$	6.43215	0.211032	0.105516	−	0.994418i	$-0.466351\pi$
$929$	6.43215	0.211032	0.105516	+	0.994418i	$0.466351\pi$
$930$	−3.88284	−0.127323
$931$	5.94432	0.194817
$932$	−26.1052	−0.855104
$933$	−7.75636	−0.253932
$934$	0.0326651	0.00106883
$935$	5.17328	0.169184
$936$	−9.95254	−0.325309
$937$	−49.5407	−1.61843	−0.809213	−	0.587516i	$-0.800106\pi$
$937$	−49.5407	−1.61843	−0.809213	+	0.587516i	$0.800106\pi$
$938$	−0.992991	−0.0324223
$939$	−25.2562	−0.824206
$940$	17.2348	0.562136
$941$	−0.355003	−0.0115728	−0.00578638	−	0.999983i	$-0.501842\pi$
$941$	−0.355003	−0.0115728	−0.00578638	+	0.999983i	$0.501842\pi$
$942$	2.48016	0.0808081
$943$	35.8488	1.16740
$944$	9.64819	0.314022
$945$	−1.00000	−0.0325300
$946$	3.41670	0.111086
$947$	45.4096	1.47561	0.737806	−	0.675012i	$-0.235862\pi$
$947$	45.4096	1.47561	0.737806	+	0.675012i	$0.235862\pi$
$948$	−18.0631	−0.586663
$949$	−13.2656	−0.430620
$950$	4.18851	0.135893
$951$	−20.5584	−0.666652
$952$	2.46865	0.0800096
$953$	−3.38044	−0.109503	−0.0547516	−	0.998500i	$-0.517437\pi$
$953$	−3.38044	−0.109503	−0.0547516	+	0.998500i	$0.517437\pi$
$954$	4.35478	0.140991
$955$	22.6772	0.733818
$956$	18.7966	0.607925
$957$	−53.6407	−1.73396
$958$	28.3869	0.917139
$959$	12.9988	0.419753
$960$	−1.57320	−0.0507749
$961$	−0.634241	−0.0204594
$962$	−23.3277	−0.752116
$963$	−10.6838	−0.344280
$964$	−7.76403	−0.250063
$965$	17.9677	0.578399
$966$	3.29044	0.105868
$967$	−7.94675	−0.255550	−0.127775	−	0.991803i	$-0.540784\pi$
$967$	−7.94675	−0.255550	−0.127775	+	0.991803i	$0.540784\pi$
$968$	−38.9129	−1.25071
$969$	5.94432	0.190959
$970$	−5.74836	−0.184569
$971$	−13.9419	−0.447417	−0.223708	−	0.974656i	$-0.571816\pi$
$971$	−13.9419	−0.447417	−0.223708	+	0.974656i	$0.571816\pi$
$972$	1.50350	0.0482249
$973$	−18.1803	−0.582833
$974$	−13.8185	−0.442775
$975$	−4.03157	−0.129113
$976$	−5.94729	−0.190368
$977$	−43.0222	−1.37640	−0.688201	−	0.725520i	$-0.741599\pi$
$977$	−43.0222	−1.37640	−0.688201	+	0.725520i	$0.741599\pi$
$978$	−12.9307	−0.413479
$979$	−46.5618	−1.48812
$980$	−1.50350	−0.0480277
$981$	1.29044	0.0412005
$982$	−5.30578	−0.169314
$983$	−36.8433	−1.17512	−0.587559	−	0.809181i	$-0.699911\pi$
$983$	−36.8433	−1.17512	−0.587559	+	0.809181i	$0.699911\pi$
$984$	−18.9513	−0.604146
$985$	7.02103	0.223709
$986$	7.30611	0.232674
$987$	11.4631	0.364873
$988$	−36.0313	−1.14631
$989$	4.37702	0.139181
$990$	−3.64522	−0.115853
$991$	−35.5256	−1.12851	−0.564254	−	0.825601i	$-0.690836\pi$
$991$	−35.5256	−1.12851	−0.564254	+	0.825601i	$0.690836\pi$
$992$	32.1287	1.02009
$993$	−8.59351	−0.272707
$994$	−6.46372	−0.205017
$995$	−11.8337	−0.375154
$996$	−22.1819	−0.702862
$997$	−49.2025	−1.55826	−0.779129	−	0.626864i	$-0.784338\pi$
$997$	−49.2025	−1.55826	−0.779129	+	0.626864i	$0.784338\pi$
$998$	−24.3227	−0.769920
$999$	8.21185	0.259812

Display

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a_n

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a_n

instead) (See

a_n

instead) Display

a_n

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up to: 50 250 1000 (See only

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)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1785.2.a.z.1.3	✓	4
3.2	odd	2		5355.2.a.bl.1.2		4
5.4	even	2		8925.2.a.bu.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1785.2.a.z.1.3	✓	4	1.1	even	1	trivial
5355.2.a.bl.1.2		4	3.2	odd	2
8925.2.a.bu.1.2		4	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Introduction

L-functions

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Varieties

Fields

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Groups

Database

Properties

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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	1785.2.a.z.1.3

Level	$1785$
Weight	$2$
Character	1785.1
Self dual	yes
Analytic conductor	$14.253$
Analytic rank	$1$
Dimension	$4$
CM	no
Inner twists	$1$