LMFDB - Embedded newform 51.4.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [51,4,Mod(1,51)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(51, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("51.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$51 = 3 \cdot 17$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	51.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:	$3.00909741029$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - 2$ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$3$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	51.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.24264 q^{2} -3.00000 q^{3} +10.0000 q^{4} -13.9706 q^{5} +12.7279 q^{6} +12.9706 q^{7} -8.48528 q^{8} +9.00000 q^{9} +59.2721 q^{10} +49.9706 q^{11} -30.0000 q^{12} +32.9411 q^{13} -55.0294 q^{14} +41.9117 q^{15} -44.0000 q^{16} -17.0000 q^{17} -38.1838 q^{18} +54.8823 q^{19} -139.706 q^{20} -38.9117 q^{21} -212.007 q^{22} -82.0294 q^{23} +25.4558 q^{24} +70.1766 q^{25} -139.757 q^{26} -27.0000 q^{27} +129.706 q^{28} +289.882 q^{29} -177.816 q^{30} +232.059 q^{31} +254.558 q^{32} -149.912 q^{33} +72.1249 q^{34} -181.206 q^{35} +90.0000 q^{36} -227.529 q^{37} -232.846 q^{38} -98.8234 q^{39} +118.544 q^{40} +437.735 q^{41} +165.088 q^{42} -158.882 q^{43} +499.706 q^{44} -125.735 q^{45} +348.021 q^{46} +159.088 q^{47} +132.000 q^{48} -174.765 q^{49} -297.734 q^{50} +51.0000 q^{51} +329.411 q^{52} +376.087 q^{53} +114.551 q^{54} -698.117 q^{55} -110.059 q^{56} -164.647 q^{57} -1229.87 q^{58} -185.294 q^{59} +419.117 q^{60} +861.852 q^{61} -984.542 q^{62} +116.735 q^{63} -728.000 q^{64} -460.206 q^{65} +636.021 q^{66} -178.530 q^{67} -170.000 q^{68} +246.088 q^{69} +768.792 q^{70} -1161.59 q^{71} -76.3675 q^{72} +383.088 q^{73} +965.324 q^{74} -210.530 q^{75} +548.823 q^{76} +648.146 q^{77} +419.272 q^{78} +254.000 q^{79} +614.705 q^{80} +81.0000 q^{81} -1857.15 q^{82} +447.088 q^{83} -389.117 q^{84} +237.500 q^{85} +674.080 q^{86} -869.647 q^{87} -424.014 q^{88} -1213.15 q^{89} +533.449 q^{90} +427.265 q^{91} -820.294 q^{92} -696.177 q^{93} -674.955 q^{94} -766.736 q^{95} -763.675 q^{96} +291.383 q^{97} +741.463 q^{98} +449.735 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 6 q^{3} + 20 q^{4} + 6 q^{5} - 8 q^{7} + 18 q^{9} + 144 q^{10} + 66 q^{11} - 60 q^{12} - 2 q^{13} - 144 q^{14} - 18 q^{15} - 88 q^{16} - 34 q^{17} - 26 q^{19} + 60 q^{20} + 24 q^{21} - 144 q^{22}+ \cdots + 594 q^{99}+O(q^{100})$ 2 * q - 6 * q^3 + 20 * q^4 + 6 * q^5 - 8 * q^7 + 18 * q^9 + 144 * q^10 + 66 * q^11 - 60 * q^12 - 2 * q^13 - 144 * q^14 - 18 * q^15 - 88 * q^16 - 34 * q^17 - 26 * q^19 + 60 * q^20 + 24 * q^21 - 144 * q^22 - 198 * q^23 + 344 * q^25 - 288 * q^26 - 54 * q^27 - 80 * q^28 + 444 * q^29 - 432 * q^30 + 532 * q^31 - 198 * q^33 - 600 * q^35 + 180 * q^36 + 88 * q^37 - 576 * q^38 + 6 * q^39 + 288 * q^40 + 570 * q^41 + 432 * q^42 - 182 * q^43 + 660 * q^44 + 54 * q^45 - 144 * q^46 + 420 * q^47 + 264 * q^48 - 78 * q^49 + 864 * q^50 + 102 * q^51 - 20 * q^52 - 300 * q^53 - 378 * q^55 - 288 * q^56 + 78 * q^57 - 576 * q^58 + 444 * q^59 - 180 * q^60 + 400 * q^61 + 288 * q^62 - 72 * q^63 - 1456 * q^64 - 1158 * q^65 + 432 * q^66 - 968 * q^67 - 340 * q^68 + 594 * q^69 - 1008 * q^70 - 1848 * q^71 + 868 * q^73 + 2304 * q^74 - 1032 * q^75 - 260 * q^76 + 312 * q^77 + 864 * q^78 + 508 * q^79 - 264 * q^80 + 162 * q^81 - 1296 * q^82 + 996 * q^83 + 240 * q^84 - 102 * q^85 + 576 * q^86 - 1332 * q^87 - 288 * q^88 - 288 * q^89 + 1296 * q^90 + 1160 * q^91 - 1980 * q^92 - 1596 * q^93 + 432 * q^94 - 2382 * q^95 + 1024 * q^97 + 1152 * q^98 + 594 * 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

)

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

$2$	−4.24264	−1.50000	−0.750000	−	0.661438i	$-0.769947\pi$
$2$	−4.24264	−1.50000	−0.750000	+	0.661438i	$0.769947\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	10.0000	1.25000
$5$	−13.9706	−1.24957	−0.624783	−	0.780799i	$-0.714812\pi$
$5$	−13.9706	−1.24957	−0.624783	+	0.780799i	$0.714812\pi$
$6$	12.7279	0.866025
$7$	12.9706	0.700345	0.350172	−	0.936685i	$-0.386123\pi$
$7$	12.9706	0.700345	0.350172	+	0.936685i	$0.386123\pi$
$8$	−8.48528	−0.375000
$9$	9.00000	0.333333
$10$	59.2721	1.87435
$11$	49.9706	1.36970	0.684850	−	0.728684i	$-0.259868\pi$
$11$	49.9706	1.36970	0.684850	+	0.728684i	$0.259868\pi$
$12$	−30.0000	−0.721688
$13$	32.9411	0.702786	0.351393	−	0.936228i	$-0.385708\pi$
$13$	32.9411	0.702786	0.351393	+	0.936228i	$0.385708\pi$
$14$	−55.0294	−1.05052
$15$	41.9117	0.721437
$16$	−44.0000	−0.687500
$17$	−17.0000	−0.242536
$17$	−17.0000	−0.242536
$18$	−38.1838	−0.500000
$19$	54.8823	0.662676	0.331338	−	0.943512i	$-0.392500\pi$
$19$	54.8823	0.662676	0.331338	+	0.943512i	$0.392500\pi$
$20$	−139.706	−1.56196
$21$	−38.9117	−0.404344
$22$	−212.007	−2.05455
$23$	−82.0294	−0.743666	−0.371833	−	0.928300i	$-0.621271\pi$
$23$	−82.0294	−0.743666	−0.371833	+	0.928300i	$0.621271\pi$
$24$	25.4558	0.216506
$25$	70.1766	0.561413
$26$	−139.757	−1.05418
$27$	−27.0000	−0.192450
$28$	129.706	0.875431
$29$	289.882	1.85620	0.928100	−	0.372332i	$-0.121442\pi$
$29$	289.882	1.85620	0.928100	+	0.372332i	$0.121442\pi$
$30$	−177.816	−1.08216
$31$	232.059	1.34448	0.672242	−	0.740331i	$-0.265331\pi$
$31$	232.059	1.34448	0.672242	+	0.740331i	$0.265331\pi$
$32$	254.558	1.40625
$33$	−149.912	−0.790796
$34$	72.1249	0.363803
$35$	−181.206	−0.875126
$36$	90.0000	0.416667
$37$	−227.529	−1.01096	−0.505480	−	0.862838i	$-0.668685\pi$
$37$	−227.529	−1.01096	−0.505480	+	0.862838i	$0.668685\pi$
$38$	−232.846	−0.994015
$39$	−98.8234	−0.405754
$40$	118.544	0.468587
$41$	437.735	1.66738	0.833692	−	0.552230i	$-0.186223\pi$
$41$	437.735	1.66738	0.833692	+	0.552230i	$0.186223\pi$
$42$	165.088	0.606516
$43$	−158.882	−0.563472	−0.281736	−	0.959492i	$-0.590910\pi$
$43$	−158.882	−0.563472	−0.281736	+	0.959492i	$0.590910\pi$
$44$	499.706	1.71212
$45$	−125.735	−0.416522
$46$	348.021	1.11550
$47$	159.088	0.493732	0.246866	−	0.969050i	$-0.420599\pi$
$47$	159.088	0.493732	0.246866	+	0.969050i	$0.420599\pi$
$48$	132.000	0.396928
$49$	−174.765	−0.509517
$50$	−297.734	−0.842119
$51$	51.0000	0.140028
$52$	329.411	0.878483
$53$	376.087	0.974709	0.487355	−	0.873204i	$-0.337962\pi$
$53$	376.087	0.974709	0.487355	+	0.873204i	$0.337962\pi$
$54$	114.551	0.288675
$55$	−698.117	−1.71153
$56$	−110.059	−0.262629
$57$	−164.647	−0.382596
$58$	−1229.87	−2.78430
$59$	−185.294	−0.408867	−0.204434	−	0.978880i	$-0.565535\pi$
$59$	−185.294	−0.408867	−0.204434	+	0.978880i	$0.565535\pi$
$60$	419.117	0.901796
$61$	861.852	1.80900	0.904499	−	0.426477i	$-0.140245\pi$
$61$	861.852	1.80900	0.904499	+	0.426477i	$0.140245\pi$
$62$	−984.542	−2.01673
$63$	116.735	0.233448
$64$	−728.000	−1.42188
$65$	−460.206	−0.878177
$66$	636.021	1.18619
$67$	−178.530	−0.325536	−0.162768	−	0.986664i	$-0.552042\pi$
$67$	−178.530	−0.325536	−0.162768	+	0.986664i	$0.552042\pi$
$68$	−170.000	−0.303170
$69$	246.088	0.429356
$70$	768.792	1.31269
$71$	−1161.59	−1.94162	−0.970811	−	0.239847i	$-0.922903\pi$
$71$	−1161.59	−1.94162	−0.970811	+	0.239847i	$0.922903\pi$
$72$	−76.3675	−0.125000
$73$	383.088	0.614207	0.307103	−	0.951676i	$-0.400640\pi$
$73$	383.088	0.614207	0.307103	+	0.951676i	$0.400640\pi$
$74$	965.324	1.51644
$75$	−210.530	−0.324132
$76$	548.823	0.828346
$77$	648.146	0.959261
$78$	419.272	0.608631
$79$	254.000	0.361737	0.180869	−	0.983507i	$-0.442109\pi$
$79$	254.000	0.361737	0.180869	+	0.983507i	$0.442109\pi$
$80$	614.705	0.859076
$81$	81.0000	0.111111
$82$	−1857.15	−2.50108
$83$	447.088	0.591257	0.295628	−	0.955303i	$-0.404471\pi$
$83$	447.088	0.591257	0.295628	+	0.955303i	$0.404471\pi$
$84$	−389.117	−0.505430
$85$	237.500	0.303064
$86$	674.080	0.845209
$87$	−869.647	−1.07168
$88$	−424.014	−0.513637
$89$	−1213.15	−1.44487	−0.722433	−	0.691440i	$-0.756976\pi$
$89$	−1213.15	−1.44487	−0.722433	+	0.691440i	$0.756976\pi$
$90$	533.449	0.624783
$91$	427.265	0.492193
$92$	−820.294	−0.929583
$93$	−696.177	−0.776238
$94$	−674.955	−0.740598
$95$	−766.736	−0.828057
$96$	−763.675	−0.811899
$97$	291.383	0.305004	0.152502	−	0.988303i	$-0.451267\pi$
$97$	291.383	0.305004	0.152502	+	0.988303i	$0.451267\pi$
$98$	741.463	0.764276
$99$	449.735	0.456566
$100$	701.766	0.701766
$101$	1568.53	1.54529	0.772645	−	0.634838i	$-0.218933\pi$
$101$	1568.53	1.54529	0.772645	+	0.634838i	$0.218933\pi$
$102$	−216.375	−0.210042
$103$	412.647	0.394750	0.197375	−	0.980328i	$-0.436758\pi$
$103$	412.647	0.394750	0.197375	+	0.980328i	$0.436758\pi$
$104$	−279.515	−0.263545
$105$	543.618	0.505254
$106$	−1595.60	−1.46206
$107$	−239.382	−0.216280	−0.108140	−	0.994136i	$-0.534489\pi$
$107$	−239.382	−0.216280	−0.108140	+	0.994136i	$0.534489\pi$
$108$	−270.000	−0.240563
$109$	−759.647	−0.667532	−0.333766	−	0.942656i	$-0.608319\pi$
$109$	−759.647	−0.667532	−0.333766	+	0.942656i	$0.608319\pi$
$110$	2961.86	2.56729
$111$	682.587	0.583678
$112$	−570.705	−0.481487
$113$	−292.383	−0.243408	−0.121704	−	0.992566i	$-0.538836\pi$
$113$	−292.383	−0.243408	−0.121704	+	0.992566i	$0.538836\pi$
$114$	698.537	0.573895
$115$	1146.00	0.929259
$116$	2898.82	2.32025
$117$	296.470	0.234262
$118$	786.134	0.613301
$119$	−220.500	−0.169859
$120$	−355.632	−0.270539
$121$	1166.06	0.876076
$122$	−3656.53	−2.71350
$123$	−1313.21	−0.962664
$124$	2320.59	1.68061
$125$	765.913	0.548043
$126$	−495.265	−0.350172
$127$	−2646.41	−1.84906	−0.924531	−	0.381107i	$-0.875543\pi$
$127$	−2646.41	−1.84906	−0.924531	+	0.381107i	$0.875543\pi$
$128$	1052.17	0.726562
$129$	476.647	0.325321
$130$	1952.49	1.31727
$131$	−1964.15	−1.30999	−0.654994	−	0.755634i	$-0.727329\pi$
$131$	−1964.15	−1.30999	−0.654994	+	0.755634i	$0.727329\pi$
$132$	−1499.12	−0.988495
$133$	711.854	0.464102
$134$	757.438	0.488304
$135$	377.205	0.240479
$136$	144.250	0.0909509
$137$	−2083.15	−1.29909	−0.649544	−	0.760324i	$-0.725040\pi$
$137$	−2083.15	−1.29909	−0.649544	+	0.760324i	$0.725040\pi$
$138$	−1044.06	−0.644034
$139$	1715.73	1.04695	0.523477	−	0.852040i	$-0.324635\pi$
$139$	1715.73	1.04695	0.523477	+	0.852040i	$0.324635\pi$
$140$	−1812.06	−1.09391
$141$	−477.265	−0.285056
$142$	4928.20	2.91243
$143$	1646.09	0.962606
$144$	−396.000	−0.229167
$145$	−4049.82	−2.31944
$146$	−1625.31	−0.921310
$147$	524.294	0.294170
$148$	−2275.29	−1.26370
$149$	−1679.62	−0.923487	−0.461743	−	0.887014i	$-0.652776\pi$
$149$	−1679.62	−0.923487	−0.461743	+	0.887014i	$0.652776\pi$
$150$	893.203	0.486198
$151$	644.353	0.347263	0.173632	−	0.984811i	$-0.444450\pi$
$151$	644.353	0.347263	0.173632	+	0.984811i	$0.444450\pi$
$152$	−465.691	−0.248504
$153$	−153.000	−0.0808452
$154$	−2749.85	−1.43889
$155$	−3241.99	−1.68002
$156$	−988.234	−0.507192
$157$	2130.53	1.08302	0.541512	−	0.840693i	$-0.317852\pi$
$157$	2130.53	1.08302	0.541512	+	0.840693i	$0.317852\pi$
$158$	−1077.63	−0.542606
$159$	−1128.26	−0.562749
$160$	−3556.32	−1.75720
$161$	−1063.97	−0.520822
$162$	−343.654	−0.166667
$163$	3582.41	1.72145	0.860724	−	0.509072i	$-0.170011\pi$
$163$	3582.41	1.72145	0.860724	+	0.509072i	$0.170011\pi$
$164$	4377.35	2.08423
$165$	2094.35	0.988151
$166$	−1896.84	−0.886885
$167$	−1861.79	−0.862694	−0.431347	−	0.902186i	$-0.641962\pi$
$167$	−1861.79	−0.862694	−0.431347	+	0.902186i	$0.641962\pi$
$168$	330.177	0.151629
$169$	−1111.88	−0.506091
$170$	−1007.63	−0.454596
$171$	493.940	0.220892
$172$	−1588.82	−0.704341
$173$	2612.50	1.14812	0.574059	−	0.818814i	$-0.305368\pi$
$173$	2612.50	1.14812	0.574059	+	0.818814i	$0.305368\pi$
$174$	3689.60	1.60752
$175$	910.230	0.393183
$176$	−2198.70	−0.941668
$177$	555.881	0.236060
$178$	5146.94	2.16730
$179$	126.646	0.0528824	0.0264412	−	0.999650i	$-0.491583\pi$
$179$	126.646	0.0528824	0.0264412	+	0.999650i	$0.491583\pi$
$180$	−1257.35	−0.520652
$181$	1783.26	0.732314	0.366157	−	0.930553i	$-0.380673\pi$
$181$	1783.26	0.732314	0.366157	+	0.930553i	$0.380673\pi$
$182$	−1812.73	−0.738289
$183$	−2585.56	−1.04443
$184$	696.043	0.278875
$185$	3178.71	1.26326
$186$	2953.63	1.16436
$187$	−849.500	−0.332201
$188$	1590.88	0.617165
$189$	−350.205	−0.134781
$190$	3252.99	1.24209
$191$	2144.44	0.812388	0.406194	−	0.913787i	$-0.366856\pi$
$191$	2144.44	0.812388	0.406194	+	0.913787i	$0.366856\pi$
$192$	2184.00	0.820920
$193$	3205.82	1.19565	0.597823	−	0.801628i	$-0.296032\pi$
$193$	3205.82	1.19565	0.597823	+	0.801628i	$0.296032\pi$
$194$	−1236.23	−0.457507
$195$	1380.62	0.507016
$196$	−1747.65	−0.636897
$197$	2768.32	1.00119	0.500596	−	0.865681i	$-0.333114\pi$
$197$	2768.32	1.00119	0.500596	+	0.865681i	$0.333114\pi$
$198$	−1908.06	−0.684850
$199$	−712.792	−0.253912	−0.126956	−	0.991908i	$-0.540521\pi$
$199$	−712.792	−0.253912	−0.126956	+	0.991908i	$0.540521\pi$
$200$	−595.468	−0.210530
$201$	535.590	0.187948
$202$	−6654.70	−2.31794
$203$	3759.94	1.29998
$204$	510.000	0.175035
$205$	−6115.41	−2.08350
$206$	−1750.71	−0.592126
$207$	−738.265	−0.247889
$208$	−1449.41	−0.483166
$209$	2742.50	0.907667
$210$	−2306.38	−0.757881
$211$	787.591	0.256967	0.128483	−	0.991712i	$-0.458989\pi$
$211$	787.591	0.256967	0.128483	+	0.991712i	$0.458989\pi$
$212$	3760.87	1.21839
$213$	3484.76	1.12100
$214$	1015.61	0.324419
$215$	2219.67	0.704096
$216$	229.103	0.0721688
$217$	3009.93	0.941602
$218$	3222.91	1.00130
$219$	−1149.26	−0.354612
$220$	−6981.17	−2.13941
$221$	−559.999	−0.170451
$222$	−2895.97	−0.875517
$223$	−2926.53	−0.878811	−0.439405	−	0.898289i	$-0.644811\pi$
$223$	−2926.53	−0.878811	−0.439405	+	0.898289i	$0.644811\pi$
$224$	3301.77	0.984860
$225$	631.590	0.187138
$226$	1240.47	0.365111
$227$	−6212.03	−1.81633	−0.908164	−	0.418614i	$-0.862516\pi$
$227$	−6212.03	−1.81633	−0.908164	+	0.418614i	$0.862516\pi$
$228$	−1646.47	−0.478246
$229$	−4516.35	−1.30327	−0.651635	−	0.758533i	$-0.725916\pi$
$229$	−4516.35	−1.30327	−0.651635	+	0.758533i	$0.725916\pi$
$230$	−4862.06	−1.39389
$231$	−1944.44	−0.553830
$232$	−2459.73	−0.696075
$233$	3547.26	0.997376	0.498688	−	0.866782i	$-0.333815\pi$
$233$	3547.26	0.997376	0.498688	+	0.866782i	$0.333815\pi$
$234$	−1257.82	−0.351393
$235$	−2222.55	−0.616951
$236$	−1852.94	−0.511084
$237$	−762.000	−0.208849
$238$	935.500	0.254788
$239$	726.969	0.196752	0.0983760	−	0.995149i	$-0.468635\pi$
$239$	726.969	0.196752	0.0983760	+	0.995149i	$0.468635\pi$
$240$	−1844.11	−0.495988
$241$	1689.67	0.451623	0.225812	−	0.974171i	$-0.427497\pi$
$241$	1689.67	0.451623	0.225812	+	0.974171i	$0.427497\pi$
$242$	−4947.16	−1.31411
$243$	−243.000	−0.0641500
$244$	8618.52	2.26125
$245$	2441.56	0.636675
$246$	5571.46	1.44400
$247$	1807.88	0.465720
$248$	−1969.08	−0.504182
$249$	−1341.26	−0.341362
$250$	−3249.50	−0.822065
$251$	911.707	0.229269	0.114634	−	0.993408i	$-0.463430\pi$
$251$	911.707	0.229269	0.114634	+	0.993408i	$0.463430\pi$
$252$	1167.35	0.291810
$253$	−4099.06	−1.01860
$254$	11227.8	2.77359
$255$	−712.499	−0.174974
$256$	1360.00	0.332031
$257$	−3123.56	−0.758141	−0.379070	−	0.925368i	$-0.623756\pi$
$257$	−3123.56	−0.758141	−0.379070	+	0.925368i	$0.623756\pi$
$258$	−2022.24	−0.487981
$259$	−2951.18	−0.708021
$260$	−4602.06	−1.09772
$261$	2608.94	0.618733
$262$	8333.17	1.96498
$263$	137.288	0.0321884	0.0160942	−	0.999870i	$-0.494877\pi$
$263$	137.288	0.0321884	0.0160942	+	0.999870i	$0.494877\pi$
$264$	1272.04	0.296549
$265$	−5254.15	−1.21796
$266$	−3020.14	−0.696153
$267$	3639.44	0.834194
$268$	−1785.30	−0.406920
$269$	−2030.26	−0.460175	−0.230087	−	0.973170i	$-0.573901\pi$
$269$	−2030.26	−0.460175	−0.230087	+	0.973170i	$0.573901\pi$
$270$	−1600.35	−0.360718
$271$	−1187.23	−0.266123	−0.133061	−	0.991108i	$-0.542481\pi$
$271$	−1187.23	−0.266123	−0.133061	+	0.991108i	$0.542481\pi$
$272$	748.000	0.166743
$273$	−1281.79	−0.284168
$274$	8838.04	1.94863
$275$	3506.77	0.768967
$276$	2460.88	0.536695
$277$	3027.91	0.656786	0.328393	−	0.944541i	$-0.393493\pi$
$277$	3027.91	0.656786	0.328393	+	0.944541i	$0.393493\pi$
$278$	−7279.24	−1.57043
$279$	2088.53	0.448161
$280$	1537.58	0.328172
$281$	−5519.90	−1.17185	−0.585925	−	0.810365i	$-0.699269\pi$
$281$	−5519.90	−1.17185	−0.585925	+	0.810365i	$0.699269\pi$
$282$	2024.86	0.427585
$283$	−5888.17	−1.23680	−0.618402	−	0.785862i	$-0.712220\pi$
$283$	−5888.17	−1.23680	−0.618402	+	0.785862i	$0.712220\pi$
$284$	−11615.9	−2.42703
$285$	2300.21	0.478079
$286$	−6983.75	−1.44391
$287$	5677.67	1.16774
$288$	2291.03	0.468750
$289$	289.000	0.0588235
$290$	17181.9	3.47916
$291$	−874.148	−0.176094
$292$	3830.88	0.767758
$293$	−2873.06	−0.572852	−0.286426	−	0.958102i	$-0.592467\pi$
$293$	−2873.06	−0.572852	−0.286426	+	0.958102i	$0.592467\pi$
$294$	−2224.39	−0.441255
$295$	2588.65	0.510906
$296$	1930.65	0.379110
$297$	−1349.21	−0.263599
$298$	7126.01	1.38523
$299$	−2702.14	−0.522638
$300$	−2105.30	−0.405165
$301$	−2060.79	−0.394625
$302$	−2733.76	−0.520895
$303$	−4705.58	−0.892174
$304$	−2414.82	−0.455590
$305$	−12040.6	−2.26046
$306$	649.124	0.121268
$307$	318.234	0.0591614	0.0295807	−	0.999562i	$-0.490583\pi$
$307$	318.234	0.0591614	0.0295807	+	0.999562i	$0.490583\pi$
$308$	6481.46	1.19908
$309$	−1237.94	−0.227909
$310$	13754.6	2.52003
$311$	−1940.05	−0.353731	−0.176866	−	0.984235i	$-0.556596\pi$
$311$	−1940.05	−0.353731	−0.176866	+	0.984235i	$0.556596\pi$
$312$	838.544	0.152158
$313$	−5487.84	−0.991026	−0.495513	−	0.868600i	$-0.665020\pi$
$313$	−5487.84	−0.991026	−0.495513	+	0.868600i	$0.665020\pi$
$314$	−9039.07	−1.62454
$315$	−1630.85	−0.291709
$316$	2540.00	0.452171
$317$	1337.29	0.236940	0.118470	−	0.992958i	$-0.462201\pi$
$317$	1337.29	0.236940	0.118470	+	0.992958i	$0.462201\pi$
$318$	4786.81	0.844123
$319$	14485.6	2.54243
$320$	10170.6	1.77673
$321$	718.145	0.124869
$322$	4514.03	0.781234
$323$	−932.998	−0.160723
$324$	810.000	0.138889
$325$	2311.70	0.394553
$326$	−15198.9	−2.58217
$327$	2278.94	0.385400
$328$	−3714.31	−0.625269
$329$	2063.46	0.345783
$330$	−8885.58	−1.48223
$331$	−4826.46	−0.801470	−0.400735	−	0.916194i	$-0.631245\pi$
$331$	−4826.46	−0.801470	−0.400735	+	0.916194i	$0.631245\pi$
$332$	4470.88	0.739071
$333$	−2047.76	−0.336987
$334$	7898.92	1.29404
$335$	2494.16	0.406778
$336$	1712.11	0.277987
$337$	−8265.21	−1.33601	−0.668004	−	0.744158i	$-0.732851\pi$
$337$	−8265.21	−1.33601	−0.668004	+	0.744158i	$0.732851\pi$
$338$	4717.32	0.759137
$339$	877.148	0.140531
$340$	2375.00	0.378830
$341$	11596.1	1.84154
$342$	−2095.61	−0.331338
$343$	−6715.70	−1.05718
$344$	1348.16	0.211302
$345$	−3437.99	−0.536508
$346$	−11083.9	−1.72218
$347$	5841.35	0.903688	0.451844	−	0.892097i	$-0.350766\pi$
$347$	5841.35	0.903688	0.451844	+	0.892097i	$0.350766\pi$
$348$	−8696.47	−1.33960
$349$	−3873.11	−0.594048	−0.297024	−	0.954870i	$-0.595994\pi$
$349$	−3873.11	−0.594048	−0.297024	+	0.954870i	$0.595994\pi$
$350$	−3861.78	−0.589774
$351$	−889.410	−0.135251
$352$	12720.4	1.92614
$353$	4020.29	0.606171	0.303085	−	0.952963i	$-0.401983\pi$
$353$	4020.29	0.606171	0.303085	+	0.952963i	$0.401983\pi$
$354$	−2358.40	−0.354089
$355$	16228.0	2.42618
$356$	−12131.5	−1.80608
$357$	661.499	0.0980679
$358$	−537.313	−0.0793237
$359$	−2272.06	−0.334024	−0.167012	−	0.985955i	$-0.553412\pi$
$359$	−2272.06	−0.334024	−0.167012	+	0.985955i	$0.553412\pi$
$360$	1066.90	0.156196
$361$	−3846.94	−0.560860
$362$	−7565.75	−1.09847
$363$	−3498.17	−0.505803
$364$	4272.65	0.615241
$365$	−5351.96	−0.767491
$366$	10969.6	1.56664
$367$	7353.58	1.04592	0.522962	−	0.852356i	$-0.324827\pi$
$367$	7353.58	1.04592	0.522962	+	0.852356i	$0.324827\pi$
$368$	3609.30	0.511270
$369$	3939.62	0.555795
$370$	−13486.1	−1.89489
$371$	4878.07	0.682632
$372$	−6961.77	−0.970298
$373$	320.701	0.0445182	0.0222591	−	0.999752i	$-0.492914\pi$
$373$	320.701	0.0445182	0.0222591	+	0.999752i	$0.492914\pi$
$374$	3604.12	0.498301
$375$	−2297.74	−0.316413
$376$	−1349.91	−0.185150
$377$	9549.05	1.30451
$378$	1485.79	0.202172
$379$	−700.903	−0.0949946	−0.0474973	−	0.998871i	$-0.515125\pi$
$379$	−700.903	−0.0949946	−0.0474973	+	0.998871i	$0.515125\pi$
$380$	−7667.36	−1.03507
$381$	7939.23	1.06756
$382$	−9098.08	−1.21858
$383$	8217.97	1.09639	0.548196	−	0.836350i	$-0.315315\pi$
$383$	8217.97	1.09639	0.548196	+	0.836350i	$0.315315\pi$
$384$	−3156.52	−0.419481
$385$	−9054.97	−1.19866
$386$	−13601.1	−1.79347
$387$	−1429.94	−0.187824
$388$	2913.83	0.381256
$389$	−3800.59	−0.495366	−0.247683	−	0.968841i	$-0.579669\pi$
$389$	−3800.59	−0.495366	−0.247683	+	0.968841i	$0.579669\pi$
$390$	−5857.47	−0.760524
$391$	1394.50	0.180366
$392$	1482.93	0.191069
$393$	5892.44	0.756322
$394$	−11745.0	−1.50179
$395$	−3548.52	−0.452014
$396$	4497.35	0.570708
$397$	−5955.38	−0.752876	−0.376438	−	0.926442i	$-0.622851\pi$
$397$	−5955.38	−0.752876	−0.376438	+	0.926442i	$0.622851\pi$
$398$	3024.12	0.380868
$399$	−2135.56	−0.267949
$400$	−3087.77	−0.385971
$401$	581.967	0.0724739	0.0362370	−	0.999343i	$-0.488463\pi$
$401$	581.967	0.0724739	0.0362370	+	0.999343i	$0.488463\pi$
$402$	−2272.31	−0.281922
$403$	7644.28	0.944885
$404$	15685.3	1.93161
$405$	−1131.62	−0.138841
$406$	−15952.1	−1.94997
$407$	−11369.8	−1.38471
$408$	−432.749	−0.0525105
$409$	−11357.7	−1.37311	−0.686555	−	0.727078i	$-0.740878\pi$
$409$	−11357.7	−1.37311	−0.686555	+	0.727078i	$0.740878\pi$
$410$	25945.5	3.12526
$411$	6249.44	0.750029
$412$	4126.47	0.493438
$413$	−2403.36	−0.286348
$414$	3132.19	0.371833
$415$	−6246.08	−0.738814
$416$	8385.44	0.988294
$417$	−5147.20	−0.604459
$418$	−11635.4	−1.36150
$419$	−20.9420	−0.00244173	−0.00122086	−	0.999999i	$-0.500389\pi$
$419$	−20.9420	−0.00244173	−0.00122086	+	0.999999i	$0.500389\pi$
$420$	5436.18	0.631568
$421$	4455.28	0.515765	0.257882	−	0.966176i	$-0.416975\pi$
$421$	4455.28	0.515765	0.257882	+	0.966176i	$0.416975\pi$
$422$	−3341.47	−0.385450
$423$	1431.79	0.164577
$424$	−3191.21	−0.365516
$425$	−1193.00	−0.136163
$426$	−14784.6	−1.68149
$427$	11178.7	1.26692
$428$	−2393.82	−0.270349
$429$	−4938.26	−0.555761
$430$	−9417.28	−1.05614
$431$	−2410.99	−0.269451	−0.134725	−	0.990883i	$-0.543015\pi$
$431$	−2410.99	−0.269451	−0.134725	+	0.990883i	$0.543015\pi$
$432$	1188.00	0.132309
$433$	−1653.71	−0.183539	−0.0917693	−	0.995780i	$-0.529252\pi$
$433$	−1653.71	−0.183539	−0.0917693	+	0.995780i	$0.529252\pi$
$434$	−12770.1	−1.41240
$435$	12149.5	1.33913
$436$	−7596.47	−0.834415
$437$	−4501.96	−0.492810
$438$	4875.92	0.531919
$439$	14852.5	1.61474	0.807371	−	0.590044i	$-0.200890\pi$
$439$	14852.5	1.61474	0.807371	+	0.590044i	$0.200890\pi$
$440$	5923.72	0.641823
$441$	−1572.88	−0.169839
$442$	2375.88	0.255676
$443$	−5393.26	−0.578423	−0.289212	−	0.957265i	$-0.593393\pi$
$443$	−5393.26	−0.578423	−0.289212	+	0.957265i	$0.593393\pi$
$444$	6825.87	0.729598
$445$	16948.3	1.80546
$446$	12416.2	1.31822
$447$	5038.85	0.533175
$448$	−9442.57	−0.995802
$449$	6144.82	0.645862	0.322931	−	0.946422i	$-0.395332\pi$
$449$	6144.82	0.645862	0.322931	+	0.946422i	$0.395332\pi$
$450$	−2679.61	−0.280706
$451$	21873.9	2.28381
$452$	−2923.83	−0.304259
$453$	−1933.06	−0.200492
$454$	26355.4	2.72449
$455$	−5969.13	−0.615027
$456$	1397.07	0.143474
$457$	6041.18	0.618369	0.309184	−	0.951002i	$-0.399944\pi$
$457$	6041.18	0.618369	0.309184	+	0.951002i	$0.399944\pi$
$458$	19161.2	1.95490
$459$	459.000	0.0466760
$460$	11460.0	1.16157
$461$	13829.8	1.39722	0.698610	−	0.715503i	$-0.253802\pi$
$461$	13829.8	1.39722	0.698610	+	0.715503i	$0.253802\pi$
$462$	8249.56	0.830745
$463$	12585.3	1.26326	0.631629	−	0.775271i	$-0.282387\pi$
$463$	12585.3	1.26326	0.631629	+	0.775271i	$0.282387\pi$
$464$	−12754.8	−1.27614
$465$	9725.98	0.969960
$466$	−15049.7	−1.49606
$467$	−6561.91	−0.650212	−0.325106	−	0.945678i	$-0.605400\pi$
$467$	−6561.91	−0.650212	−0.325106	+	0.945678i	$0.605400\pi$
$468$	2964.70	0.292828
$469$	−2315.63	−0.227987
$470$	9429.49	0.925426
$471$	−6391.59	−0.625284
$472$	1572.27	0.153325
$473$	−7939.44	−0.771788
$474$	3232.89	0.313274
$475$	3851.45	0.372035
$476$	−2205.00	−0.212323
$477$	3384.79	0.324903
$478$	−3084.27	−0.295128
$479$	−16609.4	−1.58435	−0.792175	−	0.610294i	$-0.791051\pi$
$479$	−16609.4	−1.58435	−0.792175	+	0.610294i	$0.791051\pi$
$480$	10669.0	1.01452
$481$	−7495.06	−0.710489
$482$	−7168.66	−0.677435
$483$	3191.90	0.300697
$484$	11660.6	1.09509
$485$	−4070.78	−0.381123
$486$	1030.96	0.0962250
$487$	−7999.46	−0.744333	−0.372166	−	0.928166i	$-0.621385\pi$
$487$	−7999.46	−0.744333	−0.372166	+	0.928166i	$0.621385\pi$
$488$	−7313.06	−0.678374
$489$	−10747.2	−0.993879
$490$	−10358.7	−0.955013
$491$	−11669.1	−1.07255	−0.536274	−	0.844044i	$-0.680169\pi$
$491$	−11669.1	−1.07255	−0.536274	+	0.844044i	$0.680169\pi$
$492$	−13132.1	−1.20333
$493$	−4928.00	−0.450194
$494$	−7670.20	−0.698580
$495$	−6283.05	−0.570509
$496$	−10210.6	−0.924333
$497$	−15066.4	−1.35980
$498$	5690.51	0.512043
$499$	−20896.2	−1.87464	−0.937319	−	0.348473i	$-0.886700\pi$
$499$	−20896.2	−1.87464	−0.937319	+	0.348473i	$0.886700\pi$
$500$	7659.13	0.685054
$501$	5585.38	0.498077
$502$	−3868.05	−0.343903
$503$	17429.1	1.54498	0.772490	−	0.635028i	$-0.219011\pi$
$503$	17429.1	1.54498	0.772490	+	0.635028i	$0.219011\pi$
$504$	−990.530	−0.0875431
$505$	−21913.2	−1.93094
$506$	17390.8	1.52790
$507$	3335.65	0.292192
$508$	−26464.1	−2.31133
$509$	1020.29	0.0888481	0.0444240	−	0.999013i	$-0.485855\pi$
$509$	1020.29	0.0888481	0.0444240	+	0.999013i	$0.485855\pi$
$510$	3022.88	0.262461
$511$	4968.87	0.430156
$512$	−14187.4	−1.22461
$513$	−1481.82	−0.127532
$514$	13252.1	1.13721
$515$	−5764.91	−0.493266
$516$	4766.47	0.406651
$517$	7949.73	0.676265
$518$	12520.8	1.06203
$519$	−7837.49	−0.662866
$520$	3904.98	0.329317
$521$	−5281.92	−0.444155	−0.222078	−	0.975029i	$-0.571284\pi$
$521$	−5281.92	−0.444155	−0.222078	+	0.975029i	$0.571284\pi$
$522$	−11068.8	−0.928100
$523$	−15906.1	−1.32988	−0.664938	−	0.746898i	$-0.731542\pi$
$523$	−15906.1	−1.32988	−0.664938	+	0.746898i	$0.731542\pi$
$524$	−19641.5	−1.63748
$525$	−2730.69	−0.227004
$526$	−582.465	−0.0482827
$527$	−3945.00	−0.326085
$528$	6596.11	0.543672
$529$	−5438.17	−0.446961
$530$	22291.5	1.82694
$531$	−1667.64	−0.136289
$532$	7118.54	0.580127
$533$	14419.5	1.17181
$534$	−15440.8	−1.25129
$535$	3344.30	0.270255
$536$	1514.88	0.122076
$537$	−379.938	−0.0305317
$538$	8613.66	0.690262
$539$	−8733.08	−0.697886
$540$	3772.05	0.300599
$541$	21923.7	1.74228	0.871139	−	0.491036i	$-0.163382\pi$
$541$	21923.7	1.74228	0.871139	+	0.491036i	$0.163382\pi$
$542$	5037.01	0.399184
$543$	−5349.79	−0.422802
$544$	−4327.49	−0.341066
$545$	10612.7	0.834124
$546$	5438.19	0.426251
$547$	4960.74	0.387762	0.193881	−	0.981025i	$-0.437892\pi$
$547$	4960.74	0.387762	0.193881	+	0.981025i	$0.437892\pi$
$548$	−20831.5	−1.62386
$549$	7756.67	0.602999
$550$	−14877.9	−1.15345
$551$	15909.4	1.23006
$552$	−2088.13	−0.161008
$553$	3294.52	0.253341
$554$	−12846.3	−0.985179
$555$	−9536.12	−0.729344
$556$	17157.3	1.30869
$557$	22404.1	1.70429	0.852146	−	0.523305i	$-0.175301\pi$
$557$	22404.1	1.70429	0.852146	+	0.523305i	$0.175301\pi$
$558$	−8860.88	−0.672242
$559$	−5233.76	−0.396001
$560$	7973.07	0.601649
$561$	2548.50	0.191796
$562$	23419.0	1.75778
$563$	−17361.5	−1.29965	−0.649824	−	0.760085i	$-0.725157\pi$
$563$	−17361.5	−1.29965	−0.649824	+	0.760085i	$0.725157\pi$
$564$	−4772.65	−0.356321
$565$	4084.75	0.304154
$566$	24981.4	1.85521
$567$	1050.62	0.0778161
$568$	9856.40	0.728108
$569$	1980.78	0.145938	0.0729690	−	0.997334i	$-0.476753\pi$
$569$	1980.78	0.145938	0.0729690	+	0.997334i	$0.476753\pi$
$570$	−9758.96	−0.717119
$571$	−19164.5	−1.40457	−0.702284	−	0.711897i	$-0.747836\pi$
$571$	−19164.5	−1.40457	−0.702284	+	0.711897i	$0.747836\pi$
$572$	16460.9	1.20326
$573$	−6433.31	−0.469032
$574$	−24088.3	−1.75161
$575$	−5756.55	−0.417504
$576$	−6552.00	−0.473958
$577$	8729.71	0.629848	0.314924	−	0.949117i	$-0.398021\pi$
$577$	8729.71	0.629848	0.314924	+	0.949117i	$0.398021\pi$
$578$	−1226.12	−0.0882353
$579$	−9617.45	−0.690307
$580$	−40498.2	−2.89930
$581$	5798.99	0.414084
$582$	3708.70	0.264142
$583$	18793.3	1.33506
$584$	−3250.61	−0.230328
$585$	−4141.85	−0.292726
$586$	12189.3	0.859279
$587$	2927.87	0.205871	0.102935	−	0.994688i	$-0.467177\pi$
$587$	2927.87	0.205871	0.102935	+	0.994688i	$0.467177\pi$
$588$	5242.94	0.367713
$589$	12735.9	0.890958
$590$	−10982.7	−0.766359
$591$	−8304.97	−0.578039
$592$	10011.3	0.695035
$593$	2154.97	0.149231	0.0746157	−	0.997212i	$-0.476227\pi$
$593$	2154.97	0.149231	0.0746157	+	0.997212i	$0.476227\pi$
$594$	5724.19	0.395398
$595$	3080.50	0.212249
$596$	−16796.2	−1.15436
$597$	2138.38	0.146596
$598$	11464.2	0.783958
$599$	7065.12	0.481925	0.240963	−	0.970534i	$-0.422537\pi$
$599$	7065.12	0.481925	0.240963	+	0.970534i	$0.422537\pi$
$600$	1786.41	0.121549
$601$	10656.5	0.723272	0.361636	−	0.932319i	$-0.382218\pi$
$601$	10656.5	0.723272	0.361636	+	0.932319i	$0.382218\pi$
$602$	8743.20	0.591937
$603$	−1606.77	−0.108512
$604$	6443.53	0.434079
$605$	−16290.5	−1.09471
$606$	19964.1	1.33826
$607$	82.8667	0.00554111	0.00277056	−	0.999996i	$-0.499118\pi$
$607$	82.8667	0.00554111	0.00277056	+	0.999996i	$0.499118\pi$
$608$	13970.7	0.931889
$609$	−11279.8	−0.750543
$610$	51083.8	3.39069
$611$	5240.55	0.346988
$612$	−1530.00	−0.101057
$613$	−15588.4	−1.02710	−0.513549	−	0.858060i	$-0.671670\pi$
$613$	−15588.4	−1.02710	−0.513549	+	0.858060i	$0.671670\pi$
$614$	−1350.15	−0.0887422
$615$	18346.2	1.20291
$616$	−5499.70	−0.359723
$617$	−4430.58	−0.289090	−0.144545	−	0.989498i	$-0.546172\pi$
$617$	−4430.58	−0.289090	−0.144545	+	0.989498i	$0.546172\pi$
$618$	5252.14	0.341864
$619$	3111.78	0.202056	0.101028	−	0.994884i	$-0.467787\pi$
$619$	3111.78	0.202056	0.101028	+	0.994884i	$0.467787\pi$
$620$	−32419.9	−2.10003
$621$	2214.79	0.143119
$622$	8230.95	0.530597
$623$	−15735.2	−1.01190
$624$	4348.23	0.278956
$625$	−19472.3	−1.24623
$626$	23282.9	1.48654
$627$	−8227.49	−0.524042
$628$	21305.3	1.35378
$629$	3867.99	0.245194
$630$	6919.13	0.437563
$631$	−14808.2	−0.934236	−0.467118	−	0.884195i	$-0.654708\pi$
$631$	−14808.2	−0.934236	−0.467118	+	0.884195i	$0.654708\pi$
$632$	−2155.26	−0.135651
$633$	−2362.77	−0.148360
$634$	−5673.66	−0.355410
$635$	36971.8	2.31052
$636$	−11282.6	−0.703436
$637$	−5756.94	−0.358082
$638$	−61457.1	−3.81365
$639$	−10454.3	−0.647207
$640$	−14699.5	−0.907887
$641$	−9233.91	−0.568982	−0.284491	−	0.958679i	$-0.591825\pi$
$641$	−9233.91	−0.568982	−0.284491	+	0.958679i	$0.591825\pi$
$642$	−3046.83	−0.187304
$643$	11712.7	0.718355	0.359177	−	0.933269i	$-0.383057\pi$
$643$	11712.7	0.718355	0.359177	+	0.933269i	$0.383057\pi$
$644$	−10639.7	−0.651028
$645$	−6659.02	−0.406510
$646$	3958.38	0.241084
$647$	−12099.6	−0.735216	−0.367608	−	0.929981i	$-0.619823\pi$
$647$	−12099.6	−0.735216	−0.367608	+	0.929981i	$0.619823\pi$
$648$	−687.308	−0.0416667
$649$	−9259.22	−0.560025
$650$	−9807.70	−0.591830
$651$	−9029.80	−0.543634
$652$	35824.1	2.15181
$653$	−8335.37	−0.499523	−0.249761	−	0.968307i	$-0.580352\pi$
$653$	−8335.37	−0.499523	−0.249761	+	0.968307i	$0.580352\pi$
$654$	−9668.72	−0.578099
$655$	27440.2	1.63691
$656$	−19260.3	−1.14633
$657$	3447.79	0.204736
$658$	−8754.54	−0.518674
$659$	7751.68	0.458213	0.229107	−	0.973401i	$-0.426420\pi$
$659$	7751.68	0.458213	0.229107	+	0.973401i	$0.426420\pi$
$660$	20943.5	1.23519
$661$	13808.0	0.812510	0.406255	−	0.913760i	$-0.366834\pi$
$661$	13808.0	0.812510	0.406255	+	0.913760i	$0.366834\pi$
$662$	20476.9	1.20220
$663$	1680.00	0.0984098
$664$	−3793.67	−0.221721
$665$	−9945.00	−0.579925
$666$	8687.91	0.505480
$667$	−23778.9	−1.38039
$668$	−18617.9	−1.07837
$669$	8779.58	0.507382
$670$	−10581.8	−0.610167
$671$	43067.2	2.47778
$672$	−9905.30	−0.568609
$673$	−8406.44	−0.481493	−0.240746	−	0.970588i	$-0.577392\pi$
$673$	−8406.44	−0.481493	−0.240746	+	0.970588i	$0.577392\pi$
$674$	35066.3	2.00401
$675$	−1894.77	−0.108044
$676$	−11118.8	−0.632614
$677$	−19257.9	−1.09327	−0.546633	−	0.837372i	$-0.684091\pi$
$677$	−19257.9	−1.09327	−0.546633	+	0.837372i	$0.684091\pi$
$678$	−3721.42	−0.210797
$679$	3779.40	0.213608
$680$	−2015.25	−0.113649
$681$	18636.1	1.04866
$682$	−49198.1	−2.76231
$683$	−3451.93	−0.193388	−0.0966942	−	0.995314i	$-0.530827\pi$
$683$	−3451.93	−0.193388	−0.0966942	+	0.995314i	$0.530827\pi$
$684$	4939.40	0.276115
$685$	29102.7	1.62330
$686$	28492.3	1.58577
$687$	13549.0	0.752443
$688$	6990.82	0.387387
$689$	12388.7	0.685012
$690$	14586.2	0.804762
$691$	−26090.5	−1.43637	−0.718184	−	0.695853i	$-0.755026\pi$
$691$	−26090.5	−1.43637	−0.718184	+	0.695853i	$0.755026\pi$
$692$	26125.0	1.43515
$693$	5833.32	0.319754
$694$	−24782.7	−1.35553
$695$	−23969.8	−1.30824
$696$	7379.20	0.401879
$697$	−7441.50	−0.404400
$698$	16432.2	0.891072
$699$	−10641.8	−0.575835
$700$	9102.30	0.491478
$701$	22283.9	1.20064	0.600321	−	0.799759i	$-0.295039\pi$
$701$	22283.9	1.20064	0.600321	+	0.799759i	$0.295039\pi$
$702$	3773.45	0.202877
$703$	−12487.3	−0.669940
$704$	−36378.6	−1.94754
$705$	6667.66	0.356197
$706$	−17056.6	−0.909256
$707$	20344.7	1.08224
$708$	5558.81	0.295074
$709$	−4561.83	−0.241640	−0.120820	−	0.992674i	$-0.538552\pi$
$709$	−4561.83	−0.241640	−0.120820	+	0.992674i	$0.538552\pi$
$710$	−68849.7	−3.63927
$711$	2286.00	0.120579
$712$	10293.9	0.541825
$713$	−19035.7	−0.999847
$714$	−2806.50	−0.147102
$715$	−22996.8	−1.20284
$716$	1266.46	0.0661031
$717$	−2180.91	−0.113595
$718$	9639.52	0.501036
$719$	12458.7	0.646218	0.323109	−	0.946362i	$-0.395272\pi$
$719$	12458.7	0.646218	0.323109	+	0.946362i	$0.395272\pi$
$720$	5532.34	0.286359
$721$	5352.26	0.276461
$722$	16321.2	0.841290
$723$	−5069.01	−0.260745
$724$	17832.6	0.915393
$725$	20343.0	1.04209
$726$	14841.5	0.758704
$727$	19361.4	0.987721	0.493860	−	0.869541i	$-0.335585\pi$
$727$	19361.4	0.987721	0.493860	+	0.869541i	$0.335585\pi$
$728$	−3625.46	−0.184572
$729$	729.000	0.0370370
$730$	22706.4	1.15124
$731$	2701.00	0.136662
$732$	−25855.6	−1.30553
$733$	21638.4	1.09036	0.545180	−	0.838319i	$-0.316461\pi$
$733$	21638.4	1.09036	0.545180	+	0.838319i	$0.316461\pi$
$734$	−31198.6	−1.56889
$735$	−7324.68	−0.367585
$736$	−20881.3	−1.04578
$737$	−8921.24	−0.445886
$738$	−16714.4	−0.833692
$739$	−33520.7	−1.66858	−0.834290	−	0.551326i	$-0.814122\pi$
$739$	−33520.7	−1.66858	−0.834290	+	0.551326i	$0.814122\pi$
$740$	31787.1	1.57908
$741$	−5423.65	−0.268884
$742$	−20695.9	−1.02395
$743$	−28486.4	−1.40655	−0.703274	−	0.710919i	$-0.748279\pi$
$743$	−28486.4	−1.40655	−0.703274	+	0.710919i	$0.748279\pi$
$744$	5907.25	0.291089
$745$	23465.2	1.15396
$746$	−1360.62	−0.0667773
$747$	4023.79	0.197086
$748$	−8495.00	−0.415251
$749$	−3104.92	−0.151470
$750$	9748.49	0.474619
$751$	−29427.4	−1.42985	−0.714927	−	0.699199i	$-0.753540\pi$
$751$	−29427.4	−1.42985	−0.714927	+	0.699199i	$0.753540\pi$
$752$	−6999.89	−0.339441
$753$	−2735.12	−0.132368
$754$	−40513.2	−1.95677
$755$	−9001.98	−0.433928
$756$	−3502.05	−0.168477
$757$	30790.8	1.47835	0.739174	−	0.673514i	$-0.235216\pi$
$757$	30790.8	1.47835	0.739174	+	0.673514i	$0.235216\pi$
$758$	2973.68	0.142492
$759$	12297.2	0.588088
$760$	6505.97	0.310522
$761$	−17677.5	−0.842062	−0.421031	−	0.907046i	$-0.638332\pi$
$761$	−17677.5	−0.842062	−0.421031	+	0.907046i	$0.638332\pi$
$762$	−33683.3	−1.60133
$763$	−9853.05	−0.467502
$764$	21444.4	1.01548
$765$	2137.50	0.101021
$766$	−34865.9	−1.64459
$767$	−6103.78	−0.287346
$768$	−4080.00	−0.191698
$769$	6309.56	0.295876	0.147938	−	0.988997i	$-0.452736\pi$
$769$	6309.56	0.295876	0.147938	+	0.988997i	$0.452736\pi$
$770$	38417.0	1.79799
$771$	9370.67	0.437713
$772$	32058.2	1.49456
$773$	10323.3	0.480343	0.240171	−	0.970731i	$-0.422796\pi$
$773$	10323.3	0.480343	0.240171	+	0.970731i	$0.422796\pi$
$774$	6066.72	0.281736
$775$	16285.1	0.754811
$776$	−2472.46	−0.114377
$777$	8853.54	0.408776
$778$	16124.5	0.743049
$779$	24023.9	1.10494
$780$	13806.2	0.633770
$781$	−58045.2	−2.65944
$782$	−5916.36	−0.270548
$783$	−7826.82	−0.357226
$784$	7689.64	0.350293
$785$	−29764.7	−1.35331
$786$	−24999.5	−1.13448
$787$	11118.3	0.503589	0.251795	−	0.967781i	$-0.418979\pi$
$787$	11118.3	0.503589	0.251795	+	0.967781i	$0.418979\pi$
$788$	27683.2	1.25149
$789$	−411.865	−0.0185840
$790$	15055.1	0.678021
$791$	−3792.37	−0.170469
$792$	−3816.13	−0.171212
$793$	28390.4	1.27134
$794$	25266.5	1.12931
$795$	15762.5	0.703191
$796$	−7127.92	−0.317390
$797$	32556.2	1.44692	0.723462	−	0.690364i	$-0.242550\pi$
$797$	32556.2	1.44692	0.723462	+	0.690364i	$0.242550\pi$
$798$	9060.42	0.401924
$799$	−2704.50	−0.119748
$800$	17864.1	0.789487
$801$	−10918.3	−0.481622
$802$	−2469.08	−0.108711
$803$	19143.1	0.841279
$804$	5355.90	0.234935
$805$	14864.2	0.650802
$806$	−32431.9	−1.41733
$807$	6090.77	0.265682
$808$	−13309.4	−0.579484
$809$	39644.3	1.72289	0.861445	−	0.507851i	$-0.169560\pi$
$809$	39644.3	1.72289	0.861445	+	0.507851i	$0.169560\pi$
$810$	4801.04	0.208261
$811$	−7839.62	−0.339440	−0.169720	−	0.985492i	$-0.554286\pi$
$811$	−7839.62	−0.339440	−0.169720	+	0.985492i	$0.554286\pi$
$812$	37599.4	1.62497
$813$	3561.70	0.153646
$814$	48237.8	2.07707
$815$	−50048.3	−2.15106
$816$	−2244.00	−0.0962693
$817$	−8719.82	−0.373400
$818$	48186.6	2.05967
$819$	3845.38	0.164064
$820$	−61154.1	−2.60438
$821$	344.345	0.0146379	0.00731895	−	0.999973i	$-0.497670\pi$
$821$	344.345	0.0146379	0.00731895	+	0.999973i	$0.497670\pi$
$822$	−26514.1	−1.12504
$823$	−43172.9	−1.82857	−0.914284	−	0.405074i	$-0.867246\pi$
$823$	−43172.9	−1.82857	−0.914284	+	0.405074i	$0.867246\pi$
$824$	−3501.42	−0.148031
$825$	−10520.3	−0.443963
$826$	10196.6	0.429522
$827$	−21158.3	−0.889656	−0.444828	−	0.895616i	$-0.646735\pi$
$827$	−21158.3	−0.889656	−0.444828	+	0.895616i	$0.646735\pi$
$828$	−7382.65	−0.309861
$829$	−10514.0	−0.440490	−0.220245	−	0.975445i	$-0.570686\pi$
$829$	−10514.0	−0.440490	−0.220245	+	0.975445i	$0.570686\pi$
$830$	26499.9	1.10822
$831$	−9083.74	−0.379195
$832$	−23981.1	−0.999275
$833$	2971.00	0.123576
$834$	21837.7	0.906689
$835$	26010.3	1.07799
$836$	27425.0	1.13458
$837$	−6265.59	−0.258746
$838$	88.8493	0.00366259
$839$	−10036.2	−0.412978	−0.206489	−	0.978449i	$-0.566204\pi$
$839$	−10036.2	−0.412978	−0.206489	+	0.978449i	$0.566204\pi$
$840$	−4612.75	−0.189470
$841$	59642.7	2.44548
$842$	−18902.1	−0.773647
$843$	16559.7	0.676568
$844$	7875.91	0.321209
$845$	15533.6	0.632394
$846$	−6074.59	−0.246866
$847$	15124.4	0.613555
$848$	−16547.8	−0.670113
$849$	17664.5	0.714069
$850$	5061.48	0.204244
$851$	18664.1	0.751817
$852$	34847.6	1.40124
$853$	9343.14	0.375033	0.187516	−	0.982261i	$-0.439956\pi$
$853$	9343.14	0.375033	0.187516	+	0.982261i	$0.439956\pi$
$854$	−47427.2	−1.90038
$855$	−6900.62	−0.276019
$856$	2031.22	0.0811048
$857$	25235.0	1.00585	0.502923	−	0.864331i	$-0.332258\pi$
$857$	25235.0	1.00585	0.502923	+	0.864331i	$0.332258\pi$
$858$	20951.3	0.833641
$859$	−39717.7	−1.57759	−0.788795	−	0.614656i	$-0.789295\pi$
$859$	−39717.7	−1.57759	−0.788795	+	0.614656i	$0.789295\pi$
$860$	22196.7	0.880119
$861$	−17033.0	−0.674197
$862$	10229.0	0.404176
$863$	−26812.4	−1.05760	−0.528798	−	0.848748i	$-0.677357\pi$
$863$	−26812.4	−1.05760	−0.528798	+	0.848748i	$0.677357\pi$
$864$	−6873.08	−0.270633
$865$	−36498.0	−1.43465
$866$	7016.10	0.275308
$867$	−867.000	−0.0339618
$868$	30099.3	1.17700
$869$	12692.5	0.495471
$870$	−51545.8	−2.00870
$871$	−5880.97	−0.228782
$872$	6445.82	0.250324
$873$	2622.44	0.101668
$874$	19100.2	0.739215
$875$	9934.33	0.383819
$876$	−11492.6	−0.443266
$877$	−33850.8	−1.30338	−0.651688	−	0.758487i	$-0.725939\pi$
$877$	−33850.8	−1.30338	−0.651688	+	0.758487i	$0.725939\pi$
$878$	−63013.9	−2.42211
$879$	8619.17	0.330736
$880$	30717.1	1.17668
$881$	−19939.4	−0.762514	−0.381257	−	0.924469i	$-0.624509\pi$
$881$	−19939.4	−0.762514	−0.381257	+	0.924469i	$0.624509\pi$
$882$	6673.17	0.254759
$883$	31421.5	1.19753	0.598765	−	0.800925i	$-0.295658\pi$
$883$	31421.5	1.19753	0.598765	+	0.800925i	$0.295658\pi$
$884$	−5599.99	−0.213063
$885$	−7765.96	−0.294972
$886$	22881.7	0.867635
$887$	14713.6	0.556971	0.278486	−	0.960440i	$-0.410168\pi$
$887$	14713.6	0.556971	0.278486	+	0.960440i	$0.410168\pi$
$888$	−5791.94	−0.218879
$889$	−34325.4	−1.29498
$890$	−71905.7	−2.70818
$891$	4047.62	0.152189
$892$	−29265.3	−1.09851
$893$	8731.12	0.327185
$894$	−21378.0	−0.799763
$895$	−1769.31	−0.0660801
$896$	13647.3	0.508844
$897$	8106.43	0.301745
$898$	−26070.3	−0.968793
$899$	67269.7	2.49563
$900$	6315.90	0.233922
$901$	−6393.49	−0.236402
$902$	−92803.0	−3.42572
$903$	6182.38	0.227837
$904$	2480.95	0.0912778
$905$	−24913.2	−0.915075
$906$	8201.28	0.300739
$907$	36405.1	1.33276	0.666380	−	0.745612i	$-0.267843\pi$
$907$	36405.1	1.33276	0.666380	+	0.745612i	$0.267843\pi$
$908$	−62120.3	−2.27041
$909$	14116.8	0.515097
$910$	25324.9	0.922540
$911$	1574.12	0.0572481	0.0286241	−	0.999590i	$-0.490887\pi$
$911$	1574.12	0.0572481	0.0286241	+	0.999590i	$0.490887\pi$
$912$	7244.46	0.263035
$913$	22341.3	0.809844
$914$	−25630.6	−0.927553
$915$	36121.7	1.30508
$916$	−45163.5	−1.62909
$917$	−25476.1	−0.917442
$918$	−1947.37	−0.0700140
$919$	44823.7	1.60892	0.804460	−	0.594007i	$-0.202455\pi$
$919$	44823.7	1.60892	0.804460	+	0.594007i	$0.202455\pi$
$920$	−9724.11	−0.348472
$921$	−954.701	−0.0341569
$922$	−58674.9	−2.09583
$923$	−38264.0	−1.36455
$924$	−19444.4	−0.692287
$925$	−15967.2	−0.567566
$926$	−53394.9	−1.89489
$927$	3713.82	0.131583
$928$	73792.0	2.61028
$929$	−10654.7	−0.376287	−0.188143	−	0.982142i	$-0.560247\pi$
$929$	−10654.7	−0.376287	−0.188143	+	0.982142i	$0.560247\pi$
$930$	−41263.8	−1.45494
$931$	−9591.47	−0.337645
$932$	35472.6	1.24672
$933$	5820.16	0.204227
$934$	27839.8	0.975318
$935$	11868.0	0.415107
$936$	−2515.63	−0.0878483
$937$	46738.5	1.62954	0.814770	−	0.579784i	$-0.196863\pi$
$937$	46738.5	1.62954	0.814770	+	0.579784i	$0.196863\pi$
$938$	9824.40	0.341981
$939$	16463.5	0.572169
$940$	−22225.5	−0.771188
$941$	−31346.7	−1.08594	−0.542972	−	0.839751i	$-0.682701\pi$
$941$	−31346.7	−1.08594	−0.542972	+	0.839751i	$0.682701\pi$
$942$	27117.2	0.937926
$943$	−35907.2	−1.23998
$944$	8152.91	0.281096
$945$	4892.56	0.168418
$946$	33684.2	1.15768
$947$	−18191.9	−0.624242	−0.312121	−	0.950042i	$-0.601039\pi$
$947$	−18191.9	−0.624242	−0.312121	+	0.950042i	$0.601039\pi$
$948$	−7620.00	−0.261061
$949$	12619.4	0.431656
$950$	−16340.3	−0.558053
$951$	−4011.88	−0.136797
$952$	1871.00	0.0636969
$953$	47969.6	1.63052	0.815261	−	0.579094i	$-0.196593\pi$
$953$	47969.6	1.63052	0.815261	+	0.579094i	$0.196593\pi$
$954$	−14360.4	−0.487355
$955$	−29959.0	−1.01513
$956$	7269.69	0.245940
$957$	−43456.7	−1.46788
$958$	70467.8	2.37653
$959$	−27019.6	−0.909810
$960$	−30511.7	−1.02579
$961$	24060.3	0.807637
$962$	31798.9	1.06573
$963$	−2154.44	−0.0720932
$964$	16896.7	0.564529
$965$	−44787.1	−1.49404
$966$	−13542.1	−0.451046
$967$	51466.6	1.71154	0.855768	−	0.517360i	$-0.173085\pi$
$967$	51466.6	1.71154	0.855768	+	0.517360i	$0.173085\pi$
$968$	−9894.32	−0.328528
$969$	2798.99	0.0927933
$970$	17270.9	0.571684
$971$	−4007.62	−0.132452	−0.0662259	−	0.997805i	$-0.521096\pi$
$971$	−4007.62	−0.132452	−0.0662259	+	0.997805i	$0.521096\pi$
$972$	−2430.00	−0.0801875
$973$	22254.0	0.733229
$974$	33938.8	1.11650
$975$	−6935.09	−0.227796
$976$	−37921.5	−1.24369
$977$	−37362.2	−1.22346	−0.611731	−	0.791066i	$-0.709526\pi$
$977$	−37362.2	−1.22346	−0.611731	+	0.791066i	$0.709526\pi$
$978$	45596.6	1.49082
$979$	−60621.6	−1.97903
$980$	24415.6	0.795844
$981$	−6836.82	−0.222511
$982$	49508.0	1.60882
$983$	−38659.1	−1.25436	−0.627179	−	0.778875i	$-0.715790\pi$
$983$	−38659.1	−1.25436	−0.627179	+	0.778875i	$0.715790\pi$
$984$	11142.9	0.360999
$985$	−38675.0	−1.25106
$986$	20907.7	0.675292
$987$	−6190.39	−0.199638
$988$	18078.8	0.582150
$989$	13033.0	0.419035
$990$	26656.7	0.855764
$991$	46.8701	0.00150240	0.000751200	−	1.00000i	$-0.499761\pi$
$991$	46.8701	0.00150240	0.000751200		1.00000i	$0.499761\pi$
$992$	59072.5	1.89068
$993$	14479.4	0.462729
$994$	63921.5	2.03971
$995$	9958.11	0.317280
$996$	−13412.6	−0.426703
$997$	−16913.4	−0.537265	−0.268633	−	0.963243i	$-0.586572\pi$
$997$	−16913.4	−0.537265	−0.268633	+	0.963243i	$0.586572\pi$
$998$	88655.2	2.81196
$999$	6143.28	0.194559

Display

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p

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a_n

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a_n

instead) (See

a_n

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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	51.4.a.d.1.1	✓	2
3.2	odd	2		153.4.a.e.1.2		2
4.3	odd	2		816.4.a.o.1.1		2
5.4	even	2		1275.4.a.m.1.2		2
7.6	odd	2		2499.4.a.l.1.1		2
12.11	even	2		2448.4.a.v.1.2		2
17.16	even	2		867.4.a.j.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
51.4.a.d.1.1	✓	2	1.1	even	1	trivial
153.4.a.e.1.2		2	3.2	odd	2
816.4.a.o.1.1		2	4.3	odd	2
867.4.a.j.1.1		2	17.16	even	2
1275.4.a.m.1.2		2	5.4	even	2
2448.4.a.v.1.2		2	12.11	even	2
2499.4.a.l.1.1		2	7.6	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

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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	51.4.a.d.1.1

Level	$51$
Weight	$4$
Character	51.1
Self dual	yes
Analytic conductor	$3.009$
Analytic rank	$0$
Dimension	$2$
CM	no
Inner twists	$1$