LMFDB - Embedded newform 985.2.a.g.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [985,2,Mod(1,985)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(985, 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("985.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$985 = 5 \cdot 197$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	985.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:	$7.86526459910$
Analytic rank:	$0$
Dimension:	$17$
Coefficient field:	$\mathbb{Q}[x]/(x^{17} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{17} - 6 x^{16} - 8 x^{15} + 106 x^{14} - 60 x^{13} - 698 x^{12} + 877 x^{11} + 2076 x^{10} - 3556 x^{9} + \cdots + 2$ x^17 - 6x^16 - 8x^15 + 106x^14 - 60x^13 - 698x^12 + 877x^11 + 2076x^10 - 3556x^9 - 2631x^8 + 5998x^7 + 1031x^6 - 4058x^5 - 119x^4 + 1073x^3 + 15x^2 - 75x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$2^{2}$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.51786$ of defining polynomial
Character	$\chi$	$=$	985.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.51786 q^{2} -0.903960 q^{3} +0.303910 q^{4} -1.00000 q^{5} +1.37209 q^{6} -1.33147 q^{7} +2.57443 q^{8} -2.18286 q^{9} +1.51786 q^{10} -0.806560 q^{11} -0.274722 q^{12} -1.88783 q^{13} +2.02099 q^{14} +0.903960 q^{15} -4.51546 q^{16} +1.37538 q^{17} +3.31328 q^{18} -2.12082 q^{19} -0.303910 q^{20} +1.20360 q^{21} +1.22425 q^{22} -5.95716 q^{23} -2.32719 q^{24} +1.00000 q^{25} +2.86547 q^{26} +4.68510 q^{27} -0.404647 q^{28} -7.30354 q^{29} -1.37209 q^{30} +2.40299 q^{31} +1.70498 q^{32} +0.729098 q^{33} -2.08763 q^{34} +1.33147 q^{35} -0.663391 q^{36} +10.2938 q^{37} +3.21912 q^{38} +1.70652 q^{39} -2.57443 q^{40} -4.29376 q^{41} -1.82690 q^{42} -7.67260 q^{43} -0.245121 q^{44} +2.18286 q^{45} +9.04216 q^{46} +12.0285 q^{47} +4.08179 q^{48} -5.22718 q^{49} -1.51786 q^{50} -1.24329 q^{51} -0.573729 q^{52} +1.46853 q^{53} -7.11134 q^{54} +0.806560 q^{55} -3.42779 q^{56} +1.91714 q^{57} +11.0858 q^{58} +6.68418 q^{59} +0.274722 q^{60} -4.43447 q^{61} -3.64741 q^{62} +2.90641 q^{63} +6.44299 q^{64} +1.88783 q^{65} -1.10667 q^{66} -0.323212 q^{67} +0.417990 q^{68} +5.38504 q^{69} -2.02099 q^{70} +14.6128 q^{71} -5.61962 q^{72} +12.3306 q^{73} -15.6245 q^{74} -0.903960 q^{75} -0.644538 q^{76} +1.07391 q^{77} -2.59027 q^{78} -8.17908 q^{79} +4.51546 q^{80} +2.31343 q^{81} +6.51734 q^{82} +16.9689 q^{83} +0.365785 q^{84} -1.37538 q^{85} +11.6460 q^{86} +6.60211 q^{87} -2.07644 q^{88} +4.47219 q^{89} -3.31328 q^{90} +2.51359 q^{91} -1.81044 q^{92} -2.17220 q^{93} -18.2576 q^{94} +2.12082 q^{95} -1.54124 q^{96} +0.0885854 q^{97} +7.93415 q^{98} +1.76060 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$17 q + 6 q^{2} + 5 q^{3} + 18 q^{4} - 17 q^{5} + 3 q^{6} + 7 q^{7} + 18 q^{8} + 22 q^{9} - 6 q^{10} + 7 q^{11} + 20 q^{12} + 3 q^{13} + 17 q^{14} - 5 q^{15} + 28 q^{16} + 6 q^{17} + 13 q^{18} - 23 q^{19}+ \cdots - 46 q^{99}+O(q^{100})$ 17 * q + 6 * q^2 + 5 * q^3 + 18 * q^4 - 17 * q^5 + 3 * q^6 + 7 * q^7 + 18 * q^8 + 22 * q^9 - 6 * q^10 + 7 * q^11 + 20 * q^12 + 3 * q^13 + 17 * q^14 - 5 * q^15 + 28 * q^16 + 6 * q^17 + 13 * q^18 - 23 * q^19 - 18 * q^20 - q^21 + 14 * q^22 + 49 * q^23 + q^24 + 17 * q^25 + 3 * q^26 + 29 * q^27 + 9 * q^28 + 19 * q^29 - 3 * q^30 - 18 * q^31 + 42 * q^32 + 14 * q^33 - 19 * q^34 - 7 * q^35 + 31 * q^36 + 10 * q^37 + 27 * q^38 + 21 * q^39 - 18 * q^40 + 14 * q^41 + 31 * q^42 + 22 * q^43 + 27 * q^44 - 22 * q^45 - 16 * q^46 + 51 * q^47 + 38 * q^48 + 8 * q^49 + 6 * q^50 + 7 * q^51 + 17 * q^52 + 37 * q^53 - 36 * q^54 - 7 * q^55 + 47 * q^56 - 11 * q^57 + 20 * q^58 + 23 * q^59 - 20 * q^60 - 15 * q^61 - 6 * q^62 + 37 * q^63 - 2 * q^64 - 3 * q^65 - 4 * q^66 + 24 * q^67 + 13 * q^68 + 6 * q^69 - 17 * q^70 + 68 * q^71 - 33 * q^72 - 10 * q^73 - 9 * q^74 + 5 * q^75 - 57 * q^76 - 16 * q^77 - 24 * q^78 + 3 * q^79 - 28 * q^80 + 33 * q^81 - 40 * q^82 + 29 * q^83 - 83 * q^84 - 6 * q^85 + 37 * q^86 + 23 * q^87 + 5 * q^88 - 11 * q^89 - 13 * q^90 - 82 * q^91 + 15 * q^93 - 39 * q^94 + 23 * q^95 + 2 * q^96 + 6 * q^97 + 15 * q^98 - 46 * 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

)

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

$2$	−1.51786	−1.07329	−0.536646	−	0.843808i	$-0.680309\pi$
$2$	−1.51786	−1.07329	−0.536646	+	0.843808i	$0.680309\pi$
$3$	−0.903960	−0.521902	−0.260951	−	0.965352i	$-0.584036\pi$
$3$	−0.903960	−0.521902	−0.260951	+	0.965352i	$0.584036\pi$
$4$	0.303910	0.151955
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	1.37209	0.560153
$7$	−1.33147	−0.503249	−0.251625	−	0.967825i	$-0.580965\pi$
$7$	−1.33147	−0.503249	−0.251625	+	0.967825i	$0.580965\pi$
$8$	2.57443	0.910200
$9$	−2.18286	−0.727619
$10$	1.51786	0.479991
$11$	−0.806560	−0.243187	−0.121594	−	0.992580i	$-0.538800\pi$
$11$	−0.806560	−0.243187	−0.121594	+	0.992580i	$0.538800\pi$
$12$	−0.274722	−0.0793055
$13$	−1.88783	−0.523589	−0.261795	−	0.965124i	$-0.584314\pi$
$13$	−1.88783	−0.523589	−0.261795	+	0.965124i	$0.584314\pi$
$14$	2.02099	0.540133
$15$	0.903960	0.233402
$16$	−4.51546	−1.12886
$17$	1.37538	0.333578	0.166789	−	0.985993i	$-0.446660\pi$
$17$	1.37538	0.333578	0.166789	+	0.985993i	$0.446660\pi$
$18$	3.31328	0.780947
$19$	−2.12082	−0.486550	−0.243275	−	0.969957i	$-0.578222\pi$
$19$	−2.12082	−0.486550	−0.243275	+	0.969957i	$0.578222\pi$
$20$	−0.303910	−0.0679563
$21$	1.20360	0.262647
$22$	1.22425	0.261011
$23$	−5.95716	−1.24215	−0.621077	−	0.783750i	$-0.713305\pi$
$23$	−5.95716	−1.24215	−0.621077	+	0.783750i	$0.713305\pi$
$24$	−2.32719	−0.475035
$25$	1.00000	0.200000
$26$	2.86547	0.561964
$27$	4.68510	0.901647
$28$	−0.404647	−0.0764712
$29$	−7.30354	−1.35623	−0.678117	−	0.734954i	$-0.737204\pi$
$29$	−7.30354	−1.35623	−0.678117	+	0.734954i	$0.737204\pi$
$30$	−1.37209	−0.250508
$31$	2.40299	0.431589	0.215795	−	0.976439i	$-0.430766\pi$
$31$	2.40299	0.431589	0.215795	+	0.976439i	$0.430766\pi$
$32$	1.70498	0.301401
$33$	0.729098	0.126920
$34$	−2.08763	−0.358026
$35$	1.33147	0.225060
$36$	−0.663391	−0.110565
$37$	10.2938	1.69228	0.846142	−	0.532957i	$-0.178919\pi$
$37$	10.2938	1.69228	0.846142	+	0.532957i	$0.178919\pi$
$38$	3.21912	0.522210
$39$	1.70652	0.273262
$40$	−2.57443	−0.407054
$41$	−4.29376	−0.670572	−0.335286	−	0.942116i	$-0.608833\pi$
$41$	−4.29376	−0.670572	−0.335286	+	0.942116i	$0.608833\pi$
$42$	−1.82690	−0.281896
$43$	−7.67260	−1.17006	−0.585030	−	0.811012i	$-0.698917\pi$
$43$	−7.67260	−1.17006	−0.585030	+	0.811012i	$0.698917\pi$
$44$	−0.245121	−0.0369535
$45$	2.18286	0.325401
$46$	9.04216	1.33319
$47$	12.0285	1.75453	0.877267	−	0.480003i	$-0.159364\pi$
$47$	12.0285	1.75453	0.877267	+	0.480003i	$0.159364\pi$
$48$	4.08179	0.589156
$49$	−5.22718	−0.746740
$50$	−1.51786	−0.214658
$51$	−1.24329	−0.174095
$52$	−0.573729	−0.0795620
$53$	1.46853	0.201718	0.100859	−	0.994901i	$-0.467841\pi$
$53$	1.46853	0.201718	0.100859	+	0.994901i	$0.467841\pi$
$54$	−7.11134	−0.967730
$55$	0.806560	0.108757
$56$	−3.42779	−0.458057
$57$	1.91714	0.253931
$58$	11.0858	1.45563
$59$	6.68418	0.870206	0.435103	−	0.900381i	$-0.356712\pi$
$59$	6.68418	0.870206	0.435103	+	0.900381i	$0.356712\pi$
$60$	0.274722	0.0354665
$61$	−4.43447	−0.567776	−0.283888	−	0.958857i	$-0.591624\pi$
$61$	−4.43447	−0.567776	−0.283888	+	0.958857i	$0.591624\pi$
$62$	−3.64741	−0.463221
$63$	2.90641	0.366174
$64$	6.44299	0.805373
$65$	1.88783	0.234156
$66$	−1.10667	−0.136222
$67$	−0.323212	−0.0394866	−0.0197433	−	0.999805i	$-0.506285\pi$
$67$	−0.323212	−0.0394866	−0.0197433	+	0.999805i	$0.506285\pi$
$68$	0.417990	0.0506888
$69$	5.38504	0.648282
$70$	−2.02099	−0.241555
$71$	14.6128	1.73422	0.867108	−	0.498120i	$-0.165976\pi$
$71$	14.6128	1.73422	0.867108	+	0.498120i	$0.165976\pi$
$72$	−5.61962	−0.662278
$73$	12.3306	1.44319	0.721593	−	0.692318i	$-0.243410\pi$
$73$	12.3306	1.44319	0.721593	+	0.692318i	$0.243410\pi$
$74$	−15.6245	−1.81631
$75$	−0.903960	−0.104380
$76$	−0.644538	−0.0739336
$77$	1.07391	0.122384
$78$	−2.59027	−0.293290
$79$	−8.17908	−0.920219	−0.460109	−	0.887862i	$-0.652190\pi$
$79$	−8.17908	−0.920219	−0.460109	+	0.887862i	$0.652190\pi$
$80$	4.51546	0.504844
$81$	2.31343	0.257048
$82$	6.51734	0.719720
$83$	16.9689	1.86257	0.931287	−	0.364287i	$-0.118687\pi$
$83$	16.9689	1.86257	0.931287	+	0.364287i	$0.118687\pi$
$84$	0.365785	0.0399104
$85$	−1.37538	−0.149181
$86$	11.6460	1.25582
$87$	6.60211	0.707821
$88$	−2.07644	−0.221349
$89$	4.47219	0.474051	0.237025	−	0.971503i	$-0.423828\pi$
$89$	4.47219	0.474051	0.237025	+	0.971503i	$0.423828\pi$
$90$	−3.31328	−0.349250
$91$	2.51359	0.263496
$92$	−1.81044	−0.188751
$93$	−2.17220	−0.225247
$94$	−18.2576	−1.88313
$95$	2.12082	0.217592
$96$	−1.54124	−0.157302
$97$	0.0885854	0.00899448	0.00449724	−	0.999990i	$-0.498568\pi$
$97$	0.0885854	0.00899448	0.00449724	+	0.999990i	$0.498568\pi$
$98$	7.93415	0.801470
$99$	1.76060	0.176947
$100$	0.303910	0.0303910
$101$	−0.794584	−0.0790641	−0.0395320	−	0.999218i	$-0.512587\pi$
$101$	−0.794584	−0.0790641	−0.0395320	+	0.999218i	$0.512587\pi$
$102$	1.88714	0.186855
$103$	0.305393	0.0300912	0.0150456	−	0.999887i	$-0.495211\pi$
$103$	0.305393	0.0300912	0.0150456	+	0.999887i	$0.495211\pi$
$104$	−4.86009	−0.476571
$105$	−1.20360	−0.117459
$106$	−2.22902	−0.216502
$107$	2.31112	0.223424	0.111712	−	0.993741i	$-0.464367\pi$
$107$	2.31112	0.223424	0.111712	+	0.993741i	$0.464367\pi$
$108$	1.42385	0.137010
$109$	−0.836157	−0.0800893	−0.0400446	−	0.999198i	$-0.512750\pi$
$109$	−0.836157	−0.0800893	−0.0400446	+	0.999198i	$0.512750\pi$
$110$	−1.22425	−0.116727
$111$	−9.30515	−0.883206
$112$	6.01221	0.568100
$113$	14.9830	1.40948	0.704742	−	0.709464i	$-0.251063\pi$
$113$	14.9830	1.40948	0.704742	+	0.709464i	$0.251063\pi$
$114$	−2.90995	−0.272542
$115$	5.95716	0.555508
$116$	−2.21962	−0.206086
$117$	4.12086	0.380973
$118$	−10.1457	−0.933985
$119$	−1.83128	−0.167873
$120$	2.32719	0.212442
$121$	−10.3495	−0.940860
$122$	6.73092	0.609389
$123$	3.88139	0.349973
$124$	0.730291	0.0655821
$125$	−1.00000	−0.0894427
$126$	−4.41154	−0.393011
$127$	−6.62133	−0.587548	−0.293774	−	0.955875i	$-0.594911\pi$
$127$	−6.62133	−0.587548	−0.293774	+	0.955875i	$0.594911\pi$
$128$	−13.1895	−1.16580
$129$	6.93572	0.610656
$130$	−2.86547	−0.251318
$131$	12.3464	1.07871	0.539357	−	0.842077i	$-0.318667\pi$
$131$	12.3464	1.07871	0.539357	+	0.842077i	$0.318667\pi$
$132$	0.221580	0.0192861
$133$	2.82381	0.244856
$134$	0.490592	0.0423807
$135$	−4.68510	−0.403229
$136$	3.54082	0.303622
$137$	13.6905	1.16966	0.584830	−	0.811156i	$-0.301161\pi$
$137$	13.6905	1.16966	0.584830	+	0.811156i	$0.301161\pi$
$138$	−8.17375	−0.695796
$139$	−14.6545	−1.24298	−0.621488	−	0.783424i	$-0.713471\pi$
$139$	−14.6545	−1.24298	−0.621488	+	0.783424i	$0.713471\pi$
$140$	0.404647	0.0341989
$141$	−10.8733	−0.915694
$142$	−22.1802	−1.86132
$143$	1.52265	0.127330
$144$	9.85660	0.821383
$145$	7.30354	0.606526
$146$	−18.7161	−1.54896
$147$	4.72516	0.389725
$148$	3.12837	0.257151
$149$	2.47047	0.202389	0.101195	−	0.994867i	$-0.467734\pi$
$149$	2.47047	0.202389	0.101195	+	0.994867i	$0.467734\pi$
$150$	1.37209	0.112031
$151$	15.0125	1.22170	0.610850	−	0.791746i	$-0.290828\pi$
$151$	15.0125	1.22170	0.610850	+	0.791746i	$0.290828\pi$
$152$	−5.45991	−0.442857
$153$	−3.00225	−0.242717
$154$	−1.63005	−0.131353
$155$	−2.40299	−0.193013
$156$	0.518629	0.0415235
$157$	10.7690	0.859457	0.429729	−	0.902958i	$-0.358609\pi$
$157$	10.7690	0.859457	0.429729	+	0.902958i	$0.358609\pi$
$158$	12.4147	0.987663
$159$	−1.32749	−0.105277
$160$	−1.70498	−0.134791
$161$	7.93179	0.625113
$162$	−3.51147	−0.275887
$163$	−5.02583	−0.393653	−0.196827	−	0.980438i	$-0.563064\pi$
$163$	−5.02583	−0.393653	−0.196827	+	0.980438i	$0.563064\pi$
$164$	−1.30491	−0.101897
$165$	−0.729098	−0.0567602
$166$	−25.7564	−1.99908
$167$	1.90454	0.147378	0.0736890	−	0.997281i	$-0.476523\pi$
$167$	1.90454	0.147378	0.0736890	+	0.997281i	$0.476523\pi$
$168$	3.09858	0.239061
$169$	−9.43610	−0.725854
$170$	2.08763	0.160114
$171$	4.62945	0.354023
$172$	−2.33178	−0.177796
$173$	4.80845	0.365580	0.182790	−	0.983152i	$-0.441487\pi$
$173$	4.80845	0.365580	0.182790	+	0.983152i	$0.441487\pi$
$174$	−10.0211	−0.759698
$175$	−1.33147	−0.100650
$176$	3.64199	0.274525
$177$	−6.04223	−0.454162
$178$	−6.78817	−0.508795
$179$	14.8504	1.10997	0.554985	−	0.831861i	$-0.312724\pi$
$179$	14.8504	1.10997	0.554985	+	0.831861i	$0.312724\pi$
$180$	0.663391	0.0494463
$181$	−16.8196	−1.25019	−0.625097	−	0.780547i	$-0.714941\pi$
$181$	−16.8196	−1.25019	−0.625097	+	0.780547i	$0.714941\pi$
$182$	−3.81529	−0.282808
$183$	4.00858	0.296323
$184$	−15.3363	−1.13061
$185$	−10.2938	−0.756813
$186$	3.29711	0.241756
$187$	−1.10932	−0.0811218
$188$	3.65557	0.266610
$189$	−6.23807	−0.453753
$190$	−3.21912	−0.233539
$191$	4.07306	0.294717	0.147358	−	0.989083i	$-0.452923\pi$
$191$	4.07306	0.294717	0.147358	+	0.989083i	$0.452923\pi$
$192$	−5.82420	−0.420326
$193$	−19.6778	−1.41644	−0.708219	−	0.705993i	$-0.750501\pi$
$193$	−19.6778	−1.41644	−0.708219	+	0.705993i	$0.750501\pi$
$194$	−0.134461	−0.00965370
$195$	−1.70652	−0.122207
$196$	−1.58859	−0.113471
$197$	1.00000	0.0712470
$197$	1.00000	0.0712470
$198$	−2.67236	−0.189916
$199$	1.05752	0.0749658	0.0374829	−	0.999297i	$-0.488066\pi$
$199$	1.05752	0.0749658	0.0374829	+	0.999297i	$0.488066\pi$
$200$	2.57443	0.182040
$201$	0.292171	0.0206081
$202$	1.20607	0.0848588
$203$	9.72447	0.682524
$204$	−0.377847	−0.0264546
$205$	4.29376	0.299889
$206$	−0.463544	−0.0322967
$207$	13.0036	0.903814
$208$	8.52441	0.591062
$209$	1.71057	0.118323
$210$	1.82690	0.126068
$211$	−6.51731	−0.448670	−0.224335	−	0.974512i	$-0.572021\pi$
$211$	−6.51731	−0.448670	−0.224335	+	0.974512i	$0.572021\pi$
$212$	0.446299	0.0306520
$213$	−13.2094	−0.905090
$214$	−3.50796	−0.239799
$215$	7.67260	0.523267
$216$	12.0615	0.820679
$217$	−3.19951	−0.217197
$218$	1.26917	0.0859592
$219$	−11.1464	−0.753201
$220$	0.245121	0.0165261
$221$	−2.59648	−0.174658
$222$	14.1240	0.947938
$223$	−16.8930	−1.13124	−0.565618	−	0.824667i	$-0.691362\pi$
$223$	−16.8930	−1.13124	−0.565618	+	0.824667i	$0.691362\pi$
$224$	−2.27014	−0.151680
$225$	−2.18286	−0.145524
$226$	−22.7422	−1.51279
$227$	−16.2143	−1.07618	−0.538091	−	0.842887i	$-0.680854\pi$
$227$	−16.2143	−1.07618	−0.538091	+	0.842887i	$0.680854\pi$
$228$	0.582637	0.0385861
$229$	10.8362	0.716076	0.358038	−	0.933707i	$-0.383446\pi$
$229$	10.8362	0.716076	0.358038	+	0.933707i	$0.383446\pi$
$230$	−9.04216	−0.596222
$231$	−0.970774	−0.0638722
$232$	−18.8025	−1.23444
$233$	−19.5258	−1.27918	−0.639588	−	0.768718i	$-0.720895\pi$
$233$	−19.5258	−1.27918	−0.639588	+	0.768718i	$0.720895\pi$
$234$	−6.25490	−0.408896
$235$	−12.0285	−0.784652
$236$	2.03139	0.132232
$237$	7.39356	0.480264
$238$	2.77963	0.180176
$239$	−12.4920	−0.808041	−0.404021	−	0.914750i	$-0.632388\pi$
$239$	−12.4920	−0.808041	−0.404021	+	0.914750i	$0.632388\pi$
$240$	−4.08179	−0.263479
$241$	15.4974	0.998274	0.499137	−	0.866523i	$-0.333650\pi$
$241$	15.4974	0.998274	0.499137	+	0.866523i	$0.333650\pi$
$242$	15.7091	1.00982
$243$	−16.1465	−1.03580
$244$	−1.34768	−0.0862763
$245$	5.22718	0.333952
$246$	−5.89141	−0.375623
$247$	4.00375	0.254752
$248$	6.18633	0.392832
$249$	−15.3392	−0.972080
$250$	1.51786	0.0959981
$251$	−5.94720	−0.375384	−0.187692	−	0.982228i	$-0.560101\pi$
$251$	−5.94720	−0.375384	−0.187692	+	0.982228i	$0.560101\pi$
$252$	0.883287	0.0556418
$253$	4.80481	0.302076
$254$	10.0503	0.630610
$255$	1.24329	0.0778576
$256$	7.13395	0.445872
$257$	−22.5011	−1.40358	−0.701789	−	0.712384i	$-0.747615\pi$
$257$	−22.5011	−1.40358	−0.701789	+	0.712384i	$0.747615\pi$
$258$	−10.5275	−0.655412
$259$	−13.7059	−0.851641
$260$	0.573729	0.0355812
$261$	15.9426	0.986821
$262$	−18.7402	−1.15777
$263$	12.3109	0.759120	0.379560	−	0.925167i	$-0.376075\pi$
$263$	12.3109	0.759120	0.379560	+	0.925167i	$0.376075\pi$
$264$	1.87702	0.115522
$265$	−1.46853	−0.0902108
$266$	−4.28616	−0.262802
$267$	−4.04268	−0.247408
$268$	−0.0982273	−0.00600018
$269$	−16.8702	−1.02860	−0.514298	−	0.857611i	$-0.671948\pi$
$269$	−16.8702	−1.02860	−0.514298	+	0.857611i	$0.671948\pi$
$270$	7.11134	0.432782
$271$	31.4441	1.91010	0.955048	−	0.296451i	$-0.0958032\pi$
$271$	31.4441	1.91010	0.955048	+	0.296451i	$0.0958032\pi$
$272$	−6.21046	−0.376564
$273$	−2.27219	−0.137519
$274$	−20.7803	−1.25539
$275$	−0.806560	−0.0486374
$276$	1.63656	0.0985096
$277$	9.07483	0.545254	0.272627	−	0.962120i	$-0.412108\pi$
$277$	9.07483	0.545254	0.272627	+	0.962120i	$0.412108\pi$
$278$	22.2435	1.33407
$279$	−5.24537	−0.314032
$280$	3.42779	0.204849
$281$	−15.4125	−0.919435	−0.459717	−	0.888065i	$-0.652049\pi$
$281$	−15.4125	−0.919435	−0.459717	+	0.888065i	$0.652049\pi$
$282$	16.5041	0.982807
$283$	19.1311	1.13722	0.568612	−	0.822606i	$-0.307481\pi$
$283$	19.1311	1.13722	0.568612	+	0.822606i	$0.307481\pi$
$284$	4.44096	0.263523
$285$	−1.91714	−0.113561
$286$	−2.31117	−0.136662
$287$	5.71702	0.337465
$288$	−3.72173	−0.219305
$289$	−15.1083	−0.888726
$290$	−11.0858	−0.650980
$291$	−0.0800777	−0.00469424
$292$	3.74738	0.219299
$293$	2.52011	0.147226	0.0736132	−	0.997287i	$-0.476547\pi$
$293$	2.52011	0.147226	0.0736132	+	0.997287i	$0.476547\pi$
$294$	−7.17215	−0.418289
$295$	−6.68418	−0.389168
$296$	26.5006	1.54032
$297$	−3.77881	−0.219269
$298$	−3.74984	−0.217223
$299$	11.2461	0.650379
$300$	−0.274722	−0.0158611
$301$	10.2158	0.588832
$302$	−22.7869	−1.31124
$303$	0.718273	0.0412637
$304$	9.57648	0.549249
$305$	4.43447	0.253917
$306$	4.55700	0.260507
$307$	−23.7851	−1.35749	−0.678743	−	0.734376i	$-0.737475\pi$
$307$	−23.7851	−1.35749	−0.678743	+	0.734376i	$0.737475\pi$
$308$	0.326372	0.0185968
$309$	−0.276063	−0.0157047
$310$	3.64741	0.207159
$311$	6.99938	0.396898	0.198449	−	0.980111i	$-0.436410\pi$
$311$	6.99938	0.396898	0.198449	+	0.980111i	$0.436410\pi$
$312$	4.39333	0.248723
$313$	8.97400	0.507240	0.253620	−	0.967304i	$-0.418379\pi$
$313$	8.97400	0.507240	0.253620	+	0.967304i	$0.418379\pi$
$314$	−16.3458	−0.922448
$315$	−2.90641	−0.163758
$316$	−2.48570	−0.139832
$317$	−21.7471	−1.22144	−0.610719	−	0.791847i	$-0.709120\pi$
$317$	−21.7471	−1.22144	−0.610719	+	0.791847i	$0.709120\pi$
$318$	2.01495	0.112993
$319$	5.89075	0.329819
$320$	−6.44299	−0.360174
$321$	−2.08916	−0.116605
$322$	−12.0394	−0.670929
$323$	−2.91693	−0.162302
$324$	0.703073	0.0390596
$325$	−1.88783	−0.104718
$326$	7.62852	0.422505
$327$	0.755852	0.0417987
$328$	−11.0540	−0.610355
$329$	−16.0156	−0.882968
$330$	1.10667	0.0609203
$331$	29.5759	1.62564	0.812820	−	0.582515i	$-0.197931\pi$
$331$	29.5759	1.62564	0.812820	+	0.582515i	$0.197931\pi$
$332$	5.15700	0.283027
$333$	−22.4698	−1.23134
$334$	−2.89084	−0.158180
$335$	0.323212	0.0176590
$336$	−5.43480	−0.296492
$337$	19.0248	1.03635	0.518173	−	0.855276i	$-0.326612\pi$
$337$	19.0248	1.03635	0.518173	+	0.855276i	$0.326612\pi$
$338$	14.3227	0.779053
$339$	−13.5441	−0.735612
$340$	−0.417990	−0.0226687
$341$	−1.93815	−0.104957
$342$	−7.02687	−0.379969
$343$	16.2802	0.879046
$344$	−19.7526	−1.06499
$345$	−5.38504	−0.289921
$346$	−7.29857	−0.392374
$347$	5.49276	0.294867	0.147434	−	0.989072i	$-0.452899\pi$
$347$	5.49276	0.294867	0.147434	+	0.989072i	$0.452899\pi$
$348$	2.00645	0.107557
$349$	−18.5259	−0.991667	−0.495834	−	0.868418i	$-0.665137\pi$
$349$	−18.5259	−0.991667	−0.495834	+	0.868418i	$0.665137\pi$
$350$	2.02099	0.108027
$351$	−8.84466	−0.472093
$352$	−1.37517	−0.0732969
$353$	−6.11767	−0.325610	−0.162805	−	0.986658i	$-0.552054\pi$
$353$	−6.11767	−0.325610	−0.162805	+	0.986658i	$0.552054\pi$
$354$	9.17129	0.487448
$355$	−14.6128	−0.775565
$356$	1.35914	0.0720343
$357$	1.65540	0.0876131
$358$	−22.5409	−1.19132
$359$	25.8526	1.36445	0.682224	−	0.731143i	$-0.261013\pi$
$359$	25.8526	1.36445	0.682224	+	0.731143i	$0.261013\pi$
$360$	5.61962	0.296180
$361$	−14.5021	−0.763270
$362$	25.5299	1.34182
$363$	9.35550	0.491036
$364$	0.763905	0.0400395
$365$	−12.3306	−0.645412
$366$	−6.08448	−0.318041
$367$	36.7172	1.91662	0.958312	−	0.285724i	$-0.0922341\pi$
$367$	36.7172	1.91662	0.958312	+	0.285724i	$0.0922341\pi$
$368$	26.8993	1.40222
$369$	9.37266	0.487921
$370$	15.6245	0.812281
$371$	−1.95530	−0.101514
$372$	−0.660154	−0.0342274
$373$	12.1980	0.631589	0.315795	−	0.948828i	$-0.397729\pi$
$373$	12.1980	0.631589	0.315795	+	0.948828i	$0.397729\pi$
$374$	1.68380	0.0870674
$375$	0.903960	0.0466803
$376$	30.9665	1.59698
$377$	13.7878	0.710110
$378$	9.46855	0.487009
$379$	27.8766	1.43192	0.715962	−	0.698139i	$-0.245988\pi$
$379$	27.8766	1.43192	0.715962	+	0.698139i	$0.245988\pi$
$380$	0.644538	0.0330641
$381$	5.98542	0.306642
$382$	−6.18236	−0.316317
$383$	4.14289	0.211692	0.105846	−	0.994383i	$-0.466245\pi$
$383$	4.14289	0.211692	0.105846	+	0.994383i	$0.466245\pi$
$384$	11.9228	0.608434
$385$	−1.07391	−0.0547316
$386$	29.8682	1.52025
$387$	16.7482	0.851357
$388$	0.0269220	0.00136676
$389$	37.4707	1.89984	0.949921	−	0.312492i	$-0.101164\pi$
$389$	37.4707	1.89984	0.949921	+	0.312492i	$0.101164\pi$
$390$	2.59027	0.131163
$391$	−8.19334	−0.414355
$392$	−13.4570	−0.679683
$393$	−11.1607	−0.562982
$394$	−1.51786	−0.0764689
$395$	8.17908	0.411534
$396$	0.535065	0.0268880
$397$	−5.12435	−0.257184	−0.128592	−	0.991698i	$-0.541046\pi$
$397$	−5.12435	−0.257184	−0.128592	+	0.991698i	$0.541046\pi$
$398$	−1.60517	−0.0804601
$399$	−2.55261	−0.127791
$400$	−4.51546	−0.225773
$401$	15.4588	0.771974	0.385987	−	0.922504i	$-0.373861\pi$
$401$	15.4588	0.771974	0.385987	+	0.922504i	$0.373861\pi$
$402$	−0.443475	−0.0221185
$403$	−4.53643	−0.225976
$404$	−0.241482	−0.0120142
$405$	−2.31343	−0.114955
$406$	−14.7604	−0.732547
$407$	−8.30254	−0.411542
$408$	−3.20076	−0.158461
$409$	3.23937	0.160177	0.0800884	−	0.996788i	$-0.474480\pi$
$409$	3.23937	0.160177	0.0800884	+	0.996788i	$0.474480\pi$
$410$	−6.51734	−0.321868
$411$	−12.3757	−0.610448
$412$	0.0928118	0.00457251
$413$	−8.89980	−0.437931
$414$	−19.7377	−0.970056
$415$	−16.9689	−0.832968
$416$	−3.21871	−0.157810
$417$	13.2470	0.648711
$418$	−2.59641	−0.126995
$419$	−5.63874	−0.275470	−0.137735	−	0.990469i	$-0.543982\pi$
$419$	−5.63874	−0.275470	−0.137735	+	0.990469i	$0.543982\pi$
$420$	−0.365785	−0.0178485
$421$	−26.4776	−1.29044	−0.645219	−	0.763997i	$-0.723234\pi$
$421$	−26.4776	−1.29044	−0.645219	+	0.763997i	$0.723234\pi$
$422$	9.89239	0.481554
$423$	−26.2564	−1.27663
$424$	3.78062	0.183603
$425$	1.37538	0.0667156
$426$	20.0500	0.971426
$427$	5.90437	0.285733
$428$	0.702371	0.0339504
$429$	−1.37641	−0.0664538
$430$	−11.6460	−0.561618
$431$	1.87412	0.0902732	0.0451366	−	0.998981i	$-0.485628\pi$
$431$	1.87412	0.0902732	0.0451366	+	0.998981i	$0.485628\pi$
$432$	−21.1554	−1.01784
$433$	−17.8045	−0.855631	−0.427815	−	0.903866i	$-0.640717\pi$
$433$	−17.8045	−0.855631	−0.427815	+	0.903866i	$0.640717\pi$
$434$	4.85642	0.233116
$435$	−6.60211	−0.316547
$436$	−0.254116	−0.0121700
$437$	12.6341	0.604370
$438$	16.9187	0.808404
$439$	18.6000	0.887730	0.443865	−	0.896094i	$-0.353607\pi$
$439$	18.6000	0.887730	0.443865	+	0.896094i	$0.353607\pi$
$440$	2.07644	0.0989902
$441$	11.4102	0.543342
$442$	3.94110	0.187459
$443$	19.5292	0.927863	0.463931	−	0.885871i	$-0.346438\pi$
$443$	19.5292	0.927863	0.463931	+	0.885871i	$0.346438\pi$
$444$	−2.82793	−0.134207
$445$	−4.47219	−0.212002
$446$	25.6412	1.21415
$447$	−2.23321	−0.105627
$448$	−8.57866	−0.405303
$449$	28.6675	1.35290	0.676452	−	0.736487i	$-0.263517\pi$
$449$	28.6675	1.35290	0.676452	+	0.736487i	$0.263517\pi$
$450$	3.31328	0.156189
$451$	3.46317	0.163075
$452$	4.55349	0.214178
$453$	−13.5707	−0.637607
$454$	24.6111	1.15506
$455$	−2.51359	−0.117839
$456$	4.93554	0.231128
$457$	−30.9029	−1.44558	−0.722789	−	0.691069i	$-0.757140\pi$
$457$	−30.9029	−1.44558	−0.722789	+	0.691069i	$0.757140\pi$
$458$	−16.4479	−0.768559
$459$	6.44377	0.300769
$460$	1.81044	0.0844122
$461$	22.1399	1.03116	0.515578	−	0.856843i	$-0.327577\pi$
$461$	22.1399	1.03116	0.515578	+	0.856843i	$0.327577\pi$
$462$	1.47350	0.0685535
$463$	32.5114	1.51093	0.755467	−	0.655186i	$-0.227410\pi$
$463$	32.5114	1.51093	0.755467	+	0.655186i	$0.227410\pi$
$464$	32.9788	1.53100
$465$	2.17220	0.100734
$466$	29.6375	1.37293
$467$	−23.5719	−1.09078	−0.545388	−	0.838184i	$-0.683618\pi$
$467$	−23.5719	−1.09078	−0.545388	+	0.838184i	$0.683618\pi$
$468$	1.25237	0.0578908
$469$	0.430348	0.0198716
$470$	18.2576	0.842160
$471$	−9.73472	−0.448552
$472$	17.2080	0.792061
$473$	6.18841	0.284543
$474$	−11.2224	−0.515463
$475$	−2.12082	−0.0973099
$476$	−0.556543	−0.0255091
$477$	−3.20558	−0.146773
$478$	18.9612	0.867264
$479$	−14.9314	−0.682233	−0.341116	−	0.940021i	$-0.610805\pi$
$479$	−14.9314	−0.682233	−0.341116	+	0.940021i	$0.610805\pi$
$480$	1.54124	0.0703475
$481$	−19.4329	−0.886062
$482$	−23.5229	−1.07144
$483$	−7.17003	−0.326247
$484$	−3.14530	−0.142968
$485$	−0.0885854	−0.00402246
$486$	24.5082	1.11172
$487$	41.7196	1.89049	0.945247	−	0.326354i	$-0.105820\pi$
$487$	41.7196	1.89049	0.945247	+	0.326354i	$0.105820\pi$
$488$	−11.4163	−0.516789
$489$	4.54315	0.205448
$490$	−7.93415	−0.358428
$491$	27.0022	1.21859	0.609296	−	0.792943i	$-0.291452\pi$
$491$	27.0022	1.21859	0.609296	+	0.792943i	$0.291452\pi$
$492$	1.17959	0.0531801
$493$	−10.0451	−0.452410
$494$	−6.07714	−0.273423
$495$	−1.76060	−0.0791333
$496$	−10.8506	−0.487206
$497$	−19.4565	−0.872743
$498$	23.2828	1.04333
$499$	−22.4640	−1.00563	−0.502814	−	0.864395i	$-0.667702\pi$
$499$	−22.4640	−1.00563	−0.502814	+	0.864395i	$0.667702\pi$
$500$	−0.303910	−0.0135913
$501$	−1.72163	−0.0769168
$502$	9.02704	0.402897
$503$	26.0943	1.16349	0.581743	−	0.813373i	$-0.302371\pi$
$503$	26.0943	1.16349	0.581743	+	0.813373i	$0.302371\pi$
$504$	7.48236	0.333291
$505$	0.794584	0.0353585
$506$	−7.29304	−0.324215
$507$	8.52986	0.378824
$508$	−2.01229	−0.0892808
$509$	−36.2219	−1.60551	−0.802753	−	0.596312i	$-0.796632\pi$
$509$	−36.2219	−1.60551	−0.802753	+	0.596312i	$0.796632\pi$
$510$	−1.88714	−0.0835639
$511$	−16.4178	−0.726282
$512$	15.5507	0.687251
$513$	−9.93625	−0.438696
$514$	34.1536	1.50645
$515$	−0.305393	−0.0134572
$516$	2.10783	0.0927922
$517$	−9.70169	−0.426680
$518$	20.8036	0.914059
$519$	−4.34665	−0.190797
$520$	4.86009	0.213129
$521$	23.5520	1.03183	0.515916	−	0.856639i	$-0.327452\pi$
$521$	23.5520	1.03183	0.515916	+	0.856639i	$0.327452\pi$
$522$	−24.1987	−1.05915
$523$	28.2491	1.23525	0.617623	−	0.786474i	$-0.288096\pi$
$523$	28.2491	1.23525	0.617623	+	0.786474i	$0.288096\pi$
$524$	3.75220	0.163916
$525$	1.20360	0.0525293
$526$	−18.6862	−0.814757
$527$	3.30501	0.143969
$528$	−3.29221	−0.143275
$529$	12.4878	0.542947
$530$	2.22902	0.0968225
$531$	−14.5906	−0.633178
$532$	0.858184	0.0372070
$533$	8.10588	0.351105
$534$	6.13623	0.265541
$535$	−2.31112	−0.0999183
$536$	−0.832088	−0.0359407
$537$	−13.4242	−0.579295
$538$	25.6067	1.10398
$539$	4.21604	0.181598
$540$	−1.42385	−0.0612726
$541$	−33.1032	−1.42322	−0.711609	−	0.702576i	$-0.752033\pi$
$541$	−33.1032	−1.42322	−0.711609	+	0.702576i	$0.752033\pi$
$542$	−47.7279	−2.05009
$543$	15.2043	0.652478
$544$	2.34499	0.100541
$545$	0.836157	0.0358170
$546$	3.44887	0.147598
$547$	25.8029	1.10325	0.551626	−	0.834092i	$-0.314008\pi$
$547$	25.8029	1.10325	0.551626	+	0.834092i	$0.314008\pi$
$548$	4.16068	0.177736
$549$	9.67981	0.413124
$550$	1.22425	0.0522021
$551$	15.4895	0.659875
$552$	13.8634	0.590066
$553$	10.8902	0.463099
$554$	−13.7744	−0.585216
$555$	9.30515	0.394982
$556$	−4.45363	−0.188876
$557$	12.1833	0.516223	0.258111	−	0.966115i	$-0.416900\pi$
$557$	12.1833	0.516223	0.258111	+	0.966115i	$0.416900\pi$
$558$	7.96176	0.337048
$559$	14.4845	0.612631
$560$	−6.01221	−0.254062
$561$	1.00278	0.0423376
$562$	23.3941	0.986822
$563$	−7.00996	−0.295434	−0.147717	−	0.989030i	$-0.547193\pi$
$563$	−7.00996	−0.295434	−0.147717	+	0.989030i	$0.547193\pi$
$564$	−3.30449	−0.139144
$565$	−14.9830	−0.630340
$566$	−29.0383	−1.22057
$567$	−3.08027	−0.129359
$568$	37.6196	1.57848
$569$	12.3333	0.517038	0.258519	−	0.966006i	$-0.416766\pi$
$569$	12.3333	0.517038	0.258519	+	0.966006i	$0.416766\pi$
$570$	2.90995	0.121885
$571$	−31.8611	−1.33334	−0.666672	−	0.745351i	$-0.732282\pi$
$571$	−31.8611	−1.33334	−0.666672	+	0.745351i	$0.732282\pi$
$572$	0.462747	0.0193484
$573$	−3.68189	−0.153813
$574$	−8.67766	−0.362198
$575$	−5.95716	−0.248431
$576$	−14.0641	−0.586005
$577$	19.3870	0.807092	0.403546	−	0.914959i	$-0.367777\pi$
$577$	19.3870	0.807092	0.403546	+	0.914959i	$0.367777\pi$
$578$	22.9324	0.953862
$579$	17.7879	0.739242
$580$	2.21962	0.0921646
$581$	−22.5936	−0.937338
$582$	0.121547	0.00503828
$583$	−1.18445	−0.0490551
$584$	31.7443	1.31359
$585$	−4.12086	−0.170377
$586$	−3.82518	−0.158017
$587$	−12.4958	−0.515758	−0.257879	−	0.966177i	$-0.583024\pi$
$587$	−12.4958	−0.515758	−0.257879	+	0.966177i	$0.583024\pi$
$588$	1.43602	0.0592206
$589$	−5.09630	−0.209990
$590$	10.1457	0.417691
$591$	−0.903960	−0.0371840
$592$	−46.4811	−1.91036
$593$	12.3522	0.507244	0.253622	−	0.967303i	$-0.418378\pi$
$593$	12.3522	0.507244	0.253622	+	0.967303i	$0.418378\pi$
$594$	5.73572	0.235339
$595$	1.83128	0.0750750
$596$	0.750801	0.0307540
$597$	−0.955958	−0.0391248
$598$	−17.0700	−0.698046
$599$	19.2089	0.784854	0.392427	−	0.919783i	$-0.371636\pi$
$599$	19.2089	0.784854	0.392427	+	0.919783i	$0.371636\pi$
$600$	−2.32719	−0.0950069
$601$	−1.12280	−0.0458001	−0.0229001	−	0.999738i	$-0.507290\pi$
$601$	−1.12280	−0.0458001	−0.0229001	+	0.999738i	$0.507290\pi$
$602$	−15.5063	−0.631988
$603$	0.705525	0.0287312
$604$	4.56244	0.185643
$605$	10.3495	0.420765
$606$	−1.09024	−0.0442880
$607$	2.93054	0.118947	0.0594736	−	0.998230i	$-0.481058\pi$
$607$	2.93054	0.118947	0.0594736	+	0.998230i	$0.481058\pi$
$608$	−3.61596	−0.146647
$609$	−8.79053	−0.356210
$610$	−6.73092	−0.272527
$611$	−22.7077	−0.918656
$612$	−0.912413	−0.0368821
$613$	−44.9751	−1.81653	−0.908264	−	0.418398i	$-0.862592\pi$
$613$	−44.9751	−1.81653	−0.908264	+	0.418398i	$0.862592\pi$
$614$	36.1025	1.45698
$615$	−3.88139	−0.156513
$616$	2.76472	0.111394
$617$	48.0319	1.93369	0.966846	−	0.255359i	$-0.0821938\pi$
$617$	48.0319	1.93369	0.966846	+	0.255359i	$0.0821938\pi$
$618$	0.419025	0.0168557
$619$	−40.2362	−1.61723	−0.808615	−	0.588338i	$-0.799783\pi$
$619$	−40.2362	−1.61723	−0.808615	+	0.588338i	$0.799783\pi$
$620$	−0.730291	−0.0293292
$621$	−27.9099	−1.11998
$622$	−10.6241	−0.425988
$623$	−5.95459	−0.238566
$624$	−7.70573	−0.308476
$625$	1.00000	0.0400000
$626$	−13.6213	−0.544417
$627$	−1.54629	−0.0617527
$628$	3.27280	0.130599
$629$	14.1578	0.564509
$630$	4.41154	0.175760
$631$	−14.0981	−0.561238	−0.280619	−	0.959819i	$-0.590540\pi$
$631$	−14.0981	−0.561238	−0.280619	+	0.959819i	$0.590540\pi$
$632$	−21.0565	−0.837583
$633$	5.89139	0.234162
$634$	33.0091	1.31096
$635$	6.62133	0.262759
$636$	−0.403437	−0.0159973
$637$	9.86802	0.390985
$638$	−8.94135	−0.353991
$639$	−31.8976	−1.26185
$640$	13.1895	0.521362
$641$	−25.3926	−1.00295	−0.501474	−	0.865173i	$-0.667209\pi$
$641$	−25.3926	−1.00295	−0.501474	+	0.865173i	$0.667209\pi$
$642$	3.17106	0.125152
$643$	−32.3660	−1.27639	−0.638195	−	0.769875i	$-0.720319\pi$
$643$	−32.3660	−1.27639	−0.638195	+	0.769875i	$0.720319\pi$
$644$	2.41055	0.0949890
$645$	−6.93572	−0.273094
$646$	4.42750	0.174198
$647$	−28.5573	−1.12270	−0.561352	−	0.827577i	$-0.689719\pi$
$647$	−28.5573	−1.12270	−0.561352	+	0.827577i	$0.689719\pi$
$648$	5.95577	0.233965
$649$	−5.39119	−0.211623
$650$	2.86547	0.112393
$651$	2.89223	0.113355
$652$	−1.52740	−0.0598175
$653$	0.634409	0.0248263	0.0124132	−	0.999923i	$-0.496049\pi$
$653$	0.634409	0.0248263	0.0124132	+	0.999923i	$0.496049\pi$
$654$	−1.14728	−0.0448622
$655$	−12.3464	−0.482415
$656$	19.3883	0.756985
$657$	−26.9159	−1.05009
$658$	24.3095	0.947682
$659$	29.7068	1.15721	0.578606	−	0.815607i	$-0.303597\pi$
$659$	29.7068	1.15721	0.578606	+	0.815607i	$0.303597\pi$
$660$	−0.221580	−0.00862499
$661$	18.8224	0.732108	0.366054	−	0.930594i	$-0.380709\pi$
$661$	18.8224	0.732108	0.366054	+	0.930594i	$0.380709\pi$
$662$	−44.8922	−1.74479
$663$	2.34711	0.0911542
$664$	43.6852	1.69531
$665$	−2.82381	−0.109503
$666$	34.1061	1.32158
$667$	43.5084	1.68465
$668$	0.578809	0.0223948
$669$	15.2706	0.590394
$670$	−0.490592	−0.0189532
$671$	3.57667	0.138076
$672$	2.05211	0.0791620
$673$	−33.2858	−1.28307	−0.641537	−	0.767092i	$-0.721703\pi$
$673$	−33.2858	−1.28307	−0.641537	+	0.767092i	$0.721703\pi$
$674$	−28.8770	−1.11230
$675$	4.68510	0.180329
$676$	−2.86772	−0.110297
$677$	−48.2704	−1.85518	−0.927590	−	0.373599i	$-0.878124\pi$
$677$	−48.2704	−1.85518	−0.927590	+	0.373599i	$0.878124\pi$
$678$	20.5580	0.789526
$679$	−0.117949	−0.00452647
$680$	−3.54082	−0.135784
$681$	14.6571	0.561661
$682$	2.94185	0.112649
$683$	12.6365	0.483521	0.241760	−	0.970336i	$-0.422275\pi$
$683$	12.6365	0.483521	0.241760	+	0.970336i	$0.422275\pi$
$684$	1.40693	0.0537954
$685$	−13.6905	−0.523088
$686$	−24.7110	−0.943472
$687$	−9.79549	−0.373721
$688$	34.6453	1.32084
$689$	−2.77233	−0.105617
$690$	8.17375	0.311169
$691$	35.8710	1.36460	0.682298	−	0.731074i	$-0.260981\pi$
$691$	35.8710	1.36460	0.682298	+	0.731074i	$0.260981\pi$
$692$	1.46133	0.0555516
$693$	−2.34420	−0.0890486
$694$	−8.33727	−0.316478
$695$	14.6545	0.555875
$696$	16.9967	0.644258
$697$	−5.90554	−0.223688
$698$	28.1197	1.06435
$699$	17.6505	0.667604
$700$	−0.404647	−0.0152942
$701$	−27.7577	−1.04839	−0.524197	−	0.851597i	$-0.675635\pi$
$701$	−27.7577	−1.04839	−0.524197	+	0.851597i	$0.675635\pi$
$702$	13.4250	0.506693
$703$	−21.8312	−0.823380
$704$	−5.19666	−0.195856
$705$	10.8733	0.409511
$706$	9.28578	0.349475
$707$	1.05797	0.0397889
$708$	−1.83629	−0.0690121
$709$	−21.2316	−0.797369	−0.398684	−	0.917088i	$-0.630533\pi$
$709$	−21.2316	−0.797369	−0.398684	+	0.917088i	$0.630533\pi$
$710$	22.1802	0.832407
$711$	17.8538	0.669568
$712$	11.5133	0.431481
$713$	−14.3150	−0.536100
$714$	−2.51267	−0.0940344
$715$	−1.52265	−0.0569438
$716$	4.51317	0.168665
$717$	11.2923	0.421718
$718$	−39.2407	−1.46445
$719$	17.0121	0.634443	0.317222	−	0.948351i	$-0.397250\pi$
$719$	17.0121	0.634443	0.317222	+	0.948351i	$0.397250\pi$
$720$	−9.85660	−0.367334
$721$	−0.406622	−0.0151434
$722$	22.0122	0.819211
$723$	−14.0090	−0.521001
$724$	−5.11165	−0.189973
$725$	−7.30354	−0.271247
$726$	−14.2004	−0.527025
$727$	−25.6783	−0.952356	−0.476178	−	0.879349i	$-0.657978\pi$
$727$	−25.6783	−0.952356	−0.476178	+	0.879349i	$0.657978\pi$
$728$	6.47107	0.239834
$729$	7.65554	0.283538
$730$	18.7161	0.692716
$731$	−10.5527	−0.390306
$732$	1.21825	0.0450277
$733$	17.6235	0.650938	0.325469	−	0.945553i	$-0.394478\pi$
$733$	17.6235	0.650938	0.325469	+	0.945553i	$0.394478\pi$
$734$	−55.7317	−2.05710
$735$	−4.72516	−0.174290
$736$	−10.1569	−0.374387
$737$	0.260690	0.00960264
$738$	−14.2264	−0.523681
$739$	21.6771	0.797405	0.398702	−	0.917080i	$-0.369461\pi$
$739$	21.6771	0.797405	0.398702	+	0.917080i	$0.369461\pi$
$740$	−3.12837	−0.115001
$741$	−3.61923	−0.132956
$742$	2.96788	0.108954
$743$	−0.582645	−0.0213752	−0.0106876	−	0.999943i	$-0.503402\pi$
$743$	−0.582645	−0.0213752	−0.0106876	+	0.999943i	$0.503402\pi$
$744$	−5.59220	−0.205020
$745$	−2.47047	−0.0905112
$746$	−18.5149	−0.677879
$747$	−37.0406	−1.35524
$748$	−0.337134	−0.0123269
$749$	−3.07719	−0.112438
$750$	−1.37209	−0.0501016
$751$	21.3291	0.778309	0.389154	−	0.921173i	$-0.372767\pi$
$751$	21.3291	0.778309	0.389154	+	0.921173i	$0.372767\pi$
$752$	−54.3141	−1.98063
$753$	5.37604	0.195914
$754$	−20.9281	−0.762155
$755$	−15.0125	−0.546361
$756$	−1.89581	−0.0689500
$757$	29.8220	1.08390	0.541950	−	0.840411i	$-0.317686\pi$
$757$	29.8220	1.08390	0.541950	+	0.840411i	$0.317686\pi$
$758$	−42.3129	−1.53687
$759$	−4.34336	−0.157654
$760$	5.45991	0.198052
$761$	3.08106	0.111688	0.0558442	−	0.998439i	$-0.482215\pi$
$761$	3.08106	0.111688	0.0558442	+	0.998439i	$0.482215\pi$
$762$	−9.08505	−0.329117
$763$	1.11332	0.0403049
$764$	1.23784	0.0447836
$765$	3.00225	0.108547
$766$	−6.28834	−0.227207
$767$	−12.6186	−0.455631
$768$	−6.44880	−0.232701
$769$	40.6837	1.46709	0.733546	−	0.679640i	$-0.237864\pi$
$769$	40.6837	1.46709	0.733546	+	0.679640i	$0.237864\pi$
$770$	1.63005	0.0587430
$771$	20.3401	0.732530
$772$	−5.98027	−0.215235
$773$	45.0497	1.62032	0.810162	−	0.586206i	$-0.199379\pi$
$773$	45.0497	1.62032	0.810162	+	0.586206i	$0.199379\pi$
$774$	−25.4214	−0.913755
$775$	2.40299	0.0863178
$776$	0.228057	0.00818678
$777$	12.3896	0.444473
$778$	−56.8754	−2.03908
$779$	9.10629	0.326267
$780$	−0.518629	−0.0185699
$781$	−11.7861	−0.421739
$782$	12.4364	0.444724
$783$	−34.2178	−1.22284
$784$	23.6031	0.842969
$785$	−10.7690	−0.384361
$786$	16.9404	0.604244
$787$	−38.0102	−1.35492	−0.677458	−	0.735561i	$-0.736918\pi$
$787$	−38.0102	−1.35492	−0.677458	+	0.735561i	$0.736918\pi$
$788$	0.303910	0.0108263
$789$	−11.1285	−0.396186
$790$	−12.4147	−0.441696
$791$	−19.9495	−0.709322
$792$	4.53256	0.161058
$793$	8.37152	0.297281
$794$	7.77806	0.276033
$795$	1.32749	0.0470812
$796$	0.321391	0.0113914
$797$	−28.5045	−1.00968	−0.504841	−	0.863212i	$-0.668449\pi$
$797$	−28.5045	−1.00968	−0.504841	+	0.863212i	$0.668449\pi$
$798$	3.87452	0.137157
$799$	16.5437	0.585274
$800$	1.70498	0.0602802
$801$	−9.76214	−0.344928
$802$	−23.4643	−0.828553
$803$	−9.94536	−0.350964
$804$	0.0887935	0.00313151
$805$	−7.93179	−0.279559
$806$	6.88568	0.242538
$807$	15.2500	0.536826
$808$	−2.04560	−0.0719641
$809$	12.4265	0.436891	0.218446	−	0.975849i	$-0.429901\pi$
$809$	12.4265	0.436891	0.218446	+	0.975849i	$0.429901\pi$
$810$	3.51147	0.123380
$811$	−9.69402	−0.340403	−0.170202	−	0.985409i	$-0.554442\pi$
$811$	−9.69402	−0.340403	−0.170202	+	0.985409i	$0.554442\pi$
$812$	2.95536	0.103713
$813$	−28.4243	−0.996882
$814$	12.6021	0.441704
$815$	5.02583	0.176047
$816$	5.61400	0.196529
$817$	16.2722	0.569292
$818$	−4.91693	−0.171916
$819$	−5.48681	−0.191725
$820$	1.30491	0.0455696
$821$	−6.47366	−0.225932	−0.112966	−	0.993599i	$-0.536035\pi$
$821$	−6.47366	−0.225932	−0.112966	+	0.993599i	$0.536035\pi$
$822$	18.7846	0.655188
$823$	−18.0348	−0.628653	−0.314326	−	0.949315i	$-0.601779\pi$
$823$	−18.0348	−0.628653	−0.314326	+	0.949315i	$0.601779\pi$
$824$	0.786213	0.0273890
$825$	0.729098	0.0253839
$826$	13.5087	0.470027
$827$	−1.73838	−0.0604493	−0.0302246	−	0.999543i	$-0.509622\pi$
$827$	−1.73838	−0.0604493	−0.0302246	+	0.999543i	$0.509622\pi$
$828$	3.95193	0.137339
$829$	34.0976	1.18426	0.592129	−	0.805843i	$-0.298288\pi$
$829$	34.0976	1.18426	0.592129	+	0.805843i	$0.298288\pi$
$830$	25.7564	0.894018
$831$	−8.20329	−0.284569
$832$	−12.1633	−0.421685
$833$	−7.18934	−0.249096
$834$	−20.1072	−0.696256
$835$	−1.90454	−0.0659095
$836$	0.519859	0.0179797
$837$	11.2582	0.389141
$838$	8.55883	0.295660
$839$	18.1097	0.625216	0.312608	−	0.949882i	$-0.398797\pi$
$839$	18.1097	0.625216	0.312608	+	0.949882i	$0.398797\pi$
$840$	−3.09858	−0.106911
$841$	24.3417	0.839371
$842$	40.1894	1.38502
$843$	13.9323	0.479855
$844$	−1.98067	−0.0681776
$845$	9.43610	0.324612
$846$	39.8537	1.37020
$847$	13.7800	0.473487
$848$	−6.63107	−0.227712
$849$	−17.2937	−0.593519
$850$	−2.08763	−0.0716053
$851$	−61.3216	−2.10208
$852$	−4.01445	−0.137533
$853$	22.5440	0.771893	0.385946	−	0.922521i	$-0.373875\pi$
$853$	22.5440	0.771893	0.385946	+	0.922521i	$0.373875\pi$
$854$	−8.96203	−0.306675
$855$	−4.62945	−0.158324
$856$	5.94982	0.203361
$857$	4.62439	0.157966	0.0789831	−	0.996876i	$-0.474833\pi$
$857$	4.62439	0.157966	0.0789831	+	0.996876i	$0.474833\pi$
$858$	2.08921	0.0713243
$859$	−11.1076	−0.378987	−0.189494	−	0.981882i	$-0.560685\pi$
$859$	−11.1076	−0.378987	−0.189494	+	0.981882i	$0.560685\pi$
$860$	2.33178	0.0795129
$861$	−5.16796	−0.176124
$862$	−2.84466	−0.0968895
$863$	−39.7499	−1.35310	−0.676551	−	0.736396i	$-0.736526\pi$
$863$	−39.7499	−1.35310	−0.676551	+	0.736396i	$0.736526\pi$
$864$	7.98801	0.271757
$865$	−4.80845	−0.163492
$866$	27.0248	0.918342
$867$	13.6573	0.463827
$868$	−0.972362	−0.0330041
$869$	6.59692	0.223785
$870$	10.0211	0.339747
$871$	0.610169	0.0206748
$872$	−2.15263	−0.0728972
$873$	−0.193369	−0.00654455
$874$	−19.1768	−0.648665
$875$	1.33147	0.0450120
$876$	−3.38749	−0.114453
$877$	−6.45187	−0.217864	−0.108932	−	0.994049i	$-0.534743\pi$
$877$	−6.45187	−0.217864	−0.108932	+	0.994049i	$0.534743\pi$
$878$	−28.2323	−0.952794
$879$	−2.27808	−0.0768377
$880$	−3.64199	−0.122771
$881$	−50.7803	−1.71083	−0.855416	−	0.517942i	$-0.826698\pi$
$881$	−50.7803	−1.71083	−0.855416	+	0.517942i	$0.826698\pi$
$882$	−17.3191	−0.583165
$883$	−13.8119	−0.464808	−0.232404	−	0.972619i	$-0.574659\pi$
$883$	−13.8119	−0.464808	−0.232404	+	0.972619i	$0.574659\pi$
$884$	−0.789094	−0.0265401
$885$	6.04223	0.203107
$886$	−29.6427	−0.995867
$887$	−28.1384	−0.944794	−0.472397	−	0.881386i	$-0.656611\pi$
$887$	−28.1384	−0.944794	−0.472397	+	0.881386i	$0.656611\pi$
$888$	−23.9555	−0.803894
$889$	8.81612	0.295683
$890$	6.78817	0.227540
$891$	−1.86592	−0.0625106
$892$	−5.13393	−0.171897
$893$	−25.5102	−0.853668
$894$	3.38971	0.113369
$895$	−14.8504	−0.496393
$896$	17.5615	0.586689
$897$	−10.1660	−0.339434
$898$	−43.5134	−1.45206
$899$	−17.5503	−0.585336
$900$	−0.663391	−0.0221130
$901$	2.01978	0.0672885
$902$	−5.25663	−0.175027
$903$	−9.23472	−0.307312
$904$	38.5728	1.28291
$905$	16.8196	0.559104
$906$	20.5985	0.684338
$907$	3.39438	0.112709	0.0563543	−	0.998411i	$-0.482052\pi$
$907$	3.39438	0.112709	0.0563543	+	0.998411i	$0.482052\pi$
$908$	−4.92769	−0.163531
$909$	1.73446	0.0575285
$910$	3.81529	0.126476
$911$	14.4778	0.479670	0.239835	−	0.970814i	$-0.422907\pi$
$911$	14.4778	0.479670	0.239835	+	0.970814i	$0.422907\pi$
$912$	−8.65675	−0.286654
$913$	−13.6864	−0.452954
$914$	46.9064	1.55153
$915$	−4.00858	−0.132520
$916$	3.29323	0.108811
$917$	−16.4389	−0.542862
$918$	−9.78077	−0.322813
$919$	9.28005	0.306121	0.153060	−	0.988217i	$-0.451087\pi$
$919$	9.28005	0.306121	0.153060	+	0.988217i	$0.451087\pi$
$920$	15.3363	0.505623
$921$	21.5008	0.708474
$922$	−33.6053	−1.10673
$923$	−27.5864	−0.908017
$924$	−0.295028	−0.00970570
$925$	10.2938	0.338457
$926$	−49.3479	−1.62167
$927$	−0.666628	−0.0218949
$928$	−12.4524	−0.408770
$929$	−15.6769	−0.514343	−0.257172	−	0.966366i	$-0.582791\pi$
$929$	−15.6769	−0.514343	−0.257172	+	0.966366i	$0.582791\pi$
$930$	−3.29711	−0.108116
$931$	11.0859	0.363326
$932$	−5.93407	−0.194377
$933$	−6.32716	−0.207142
$934$	35.7789	1.17072
$935$	1.10932	0.0362788
$936$	10.6089	0.346762
$937$	2.90179	0.0947975	0.0473987	−	0.998876i	$-0.484907\pi$
$937$	2.90179	0.0947975	0.0473987	+	0.998876i	$0.484907\pi$
$938$	−0.653209	−0.0213280
$939$	−8.11214	−0.264730
$940$	−3.65557	−0.119232
$941$	−58.3823	−1.90321	−0.951605	−	0.307324i	$-0.900566\pi$
$941$	−58.3823	−1.90321	−0.951605	+	0.307324i	$0.900566\pi$
$942$	14.7760	0.481427
$943$	25.5786	0.832954
$944$	−30.1821	−0.982345
$945$	6.23807	0.202925
$946$	−9.39316	−0.305398
$947$	55.5899	1.80643	0.903214	−	0.429191i	$-0.141201\pi$
$947$	55.5899	1.80643	0.903214	+	0.429191i	$0.141201\pi$
$948$	2.24698	0.0729784
$949$	−23.2780	−0.755637
$950$	3.21912	0.104442
$951$	19.6585	0.637471
$952$	−4.71450	−0.152798
$953$	−53.1604	−1.72204	−0.861018	−	0.508575i	$-0.830172\pi$
$953$	−53.1604	−1.72204	−0.861018	+	0.508575i	$0.830172\pi$
$954$	4.86563	0.157531
$955$	−4.07306	−0.131801
$956$	−3.79645	−0.122786
$957$	−5.32500	−0.172133
$958$	22.6638	0.732235
$959$	−18.2286	−0.588631
$960$	5.82420	0.187975
$961$	−25.2257	−0.813731
$962$	29.4964	0.951003
$963$	−5.04484	−0.162568
$964$	4.70981	0.151693
$965$	19.6778	0.633451
$966$	10.8831	0.350159
$967$	−14.6506	−0.471133	−0.235566	−	0.971858i	$-0.575694\pi$
$967$	−14.6506	−0.471133	−0.235566	+	0.971858i	$0.575694\pi$
$968$	−26.6440	−0.856371
$969$	2.63679	0.0847058
$970$	0.134461	0.00431727
$971$	−8.27515	−0.265562	−0.132781	−	0.991145i	$-0.542391\pi$
$971$	−8.27515	−0.265562	−0.132781	+	0.991145i	$0.542391\pi$
$972$	−4.90709	−0.157395
$973$	19.5120	0.625526
$974$	−63.3246	−2.02905
$975$	1.70652	0.0546524
$976$	20.0237	0.640942
$977$	26.5630	0.849824	0.424912	−	0.905235i	$-0.360305\pi$
$977$	26.5630	0.849824	0.424912	+	0.905235i	$0.360305\pi$
$978$	−6.89588	−0.220506
$979$	−3.60709	−0.115283
$980$	1.58859	0.0507457
$981$	1.82521	0.0582745
$982$	−40.9856	−1.30790
$983$	20.7737	0.662579	0.331290	−	0.943529i	$-0.392516\pi$
$983$	20.7737	0.662579	0.331290	+	0.943529i	$0.392516\pi$
$984$	9.99237	0.318545
$985$	−1.00000	−0.0318626
$986$	15.2471	0.485567
$987$	14.4775	0.460822
$988$	1.21678	0.0387108
$989$	45.7069	1.45339
$990$	2.67236	0.0849331
$991$	18.8231	0.597937	0.298968	−	0.954263i	$-0.403357\pi$
$991$	18.8231	0.597937	0.298968	+	0.954263i	$0.403357\pi$
$992$	4.09705	0.130081
$993$	−26.7355	−0.848424
$994$	29.5323	0.936708
$995$	−1.05752	−0.0335257
$996$	−4.66172	−0.147712
$997$	50.8349	1.60996	0.804979	−	0.593304i	$-0.202177\pi$
$997$	50.8349	1.60996	0.804979	+	0.593304i	$0.202177\pi$
$998$	34.0973	1.07933
$999$	48.2273	1.52584

Display

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a_n

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instead) (See

a_n

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a_n

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Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	985.2.a.g.1.4	✓	17
3.2	odd	2		8865.2.a.z.1.14		17
5.4	even	2		4925.2.a.l.1.14		17

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
985.2.a.g.1.4	✓	17	1.1	even	1	trivial
4925.2.a.l.1.14		17	5.4	even	2
8865.2.a.z.1.14		17	3.2	odd	2

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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

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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	985.2.a.g.1.4

Level	$985$
Weight	$2$
Character	985.1
Self dual	yes
Analytic conductor	$7.865$
Analytic rank	$0$
Dimension	$17$
CM	no
Inner twists	$1$