LMFDB - Embedded newform 51.4.a.b.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:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
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)

$f(q)$

$=$

$q-1.00000 q^{2} +3.00000 q^{3} -7.00000 q^{4} -20.0000 q^{5} -3.00000 q^{6} -2.00000 q^{7} +15.0000 q^{8} +9.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} +3.00000 q^{3} -7.00000 q^{4} -20.0000 q^{5} -3.00000 q^{6} -2.00000 q^{7} +15.0000 q^{8} +9.00000 q^{9} +20.0000 q^{10} -48.0000 q^{11} -21.0000 q^{12} -14.0000 q^{13} +2.00000 q^{14} -60.0000 q^{15} +41.0000 q^{16} -17.0000 q^{17} -9.00000 q^{18} +92.0000 q^{19} +140.000 q^{20} -6.00000 q^{21} +48.0000 q^{22} -122.000 q^{23} +45.0000 q^{24} +275.000 q^{25} +14.0000 q^{26} +27.0000 q^{27} +14.0000 q^{28} -36.0000 q^{29} +60.0000 q^{30} -182.000 q^{31} -161.000 q^{32} -144.000 q^{33} +17.0000 q^{34} +40.0000 q^{35} -63.0000 q^{36} +76.0000 q^{37} -92.0000 q^{38} -42.0000 q^{39} -300.000 q^{40} +294.000 q^{41} +6.00000 q^{42} -428.000 q^{43} +336.000 q^{44} -180.000 q^{45} +122.000 q^{46} -12.0000 q^{47} +123.000 q^{48} -339.000 q^{49} -275.000 q^{50} -51.0000 q^{51} +98.0000 q^{52} -234.000 q^{53} -27.0000 q^{54} +960.000 q^{55} -30.0000 q^{56} +276.000 q^{57} +36.0000 q^{58} -540.000 q^{59} +420.000 q^{60} -820.000 q^{61} +182.000 q^{62} -18.0000 q^{63} -167.000 q^{64} +280.000 q^{65} +144.000 q^{66} +700.000 q^{67} +119.000 q^{68} -366.000 q^{69} -40.0000 q^{70} +794.000 q^{71} +135.000 q^{72} -1038.00 q^{73} -76.0000 q^{74} +825.000 q^{75} -644.000 q^{76} +96.0000 q^{77} +42.0000 q^{78} +858.000 q^{79} -820.000 q^{80} +81.0000 q^{81} -294.000 q^{82} +1052.00 q^{83} +42.0000 q^{84} +340.000 q^{85} +428.000 q^{86} -108.000 q^{87} -720.000 q^{88} +1102.00 q^{89} +180.000 q^{90} +28.0000 q^{91} +854.000 q^{92} -546.000 q^{93} +12.0000 q^{94} -1840.00 q^{95} -483.000 q^{96} +710.000 q^{97} +339.000 q^{98} -432.000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

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

$2$	−1.00000	−0.353553	−0.176777	−	0.984251i	$-0.556567\pi$
$2$	−1.00000	−0.353553	−0.176777	+	0.984251i	$0.556567\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	−7.00000	−0.875000
$5$	−20.0000	−1.78885	−0.894427	−	0.447214i	$-0.852416\pi$
$5$	−20.0000	−1.78885	−0.894427	+	0.447214i	$0.852416\pi$
$6$	−3.00000	−0.204124
$7$	−2.00000	−0.107990	−0.0539949	−	0.998541i	$-0.517195\pi$
$7$	−2.00000	−0.107990	−0.0539949	+	0.998541i	$0.517195\pi$
$8$	15.0000	0.662913
$9$	9.00000	0.333333
$10$	20.0000	0.632456
$11$	−48.0000	−1.31569	−0.657843	−	0.753155i	$-0.728531\pi$
$11$	−48.0000	−1.31569	−0.657843	+	0.753155i	$0.728531\pi$
$12$	−21.0000	−0.505181
$13$	−14.0000	−0.298685	−0.149342	−	0.988786i	$-0.547716\pi$
$13$	−14.0000	−0.298685	−0.149342	+	0.988786i	$0.547716\pi$
$14$	2.00000	0.0381802
$15$	−60.0000	−1.03280
$16$	41.0000	0.640625
$17$	−17.0000	−0.242536
$17$	−17.0000	−0.242536
$18$	−9.00000	−0.117851
$19$	92.0000	1.11086	0.555428	−	0.831565i	$-0.312555\pi$
$19$	92.0000	1.11086	0.555428	+	0.831565i	$0.312555\pi$
$20$	140.000	1.56525
$21$	−6.00000	−0.0623480
$22$	48.0000	0.465165
$23$	−122.000	−1.10603	−0.553016	−	0.833170i	$-0.686523\pi$
$23$	−122.000	−1.10603	−0.553016	+	0.833170i	$0.686523\pi$
$24$	45.0000	0.382733
$25$	275.000	2.20000
$26$	14.0000	0.105601
$27$	27.0000	0.192450
$28$	14.0000	0.0944911
$29$	−36.0000	−0.230518	−0.115259	−	0.993335i	$-0.536770\pi$
$29$	−36.0000	−0.230518	−0.115259	+	0.993335i	$0.536770\pi$
$30$	60.0000	0.365148
$31$	−182.000	−1.05446	−0.527228	−	0.849724i	$-0.676769\pi$
$31$	−182.000	−1.05446	−0.527228	+	0.849724i	$0.676769\pi$
$32$	−161.000	−0.889408
$33$	−144.000	−0.759612
$34$	17.0000	0.0857493
$35$	40.0000	0.193178
$36$	−63.0000	−0.291667
$37$	76.0000	0.337684	0.168842	−	0.985643i	$-0.445997\pi$
$37$	76.0000	0.337684	0.168842	+	0.985643i	$0.445997\pi$
$38$	−92.0000	−0.392747
$39$	−42.0000	−0.172446
$40$	−300.000	−1.18585
$41$	294.000	1.11988	0.559940	−	0.828533i	$-0.310824\pi$
$41$	294.000	1.11988	0.559940	+	0.828533i	$0.310824\pi$
$42$	6.00000	0.0220433
$43$	−428.000	−1.51789	−0.758946	−	0.651153i	$-0.774286\pi$
$43$	−428.000	−1.51789	−0.758946	+	0.651153i	$0.774286\pi$
$44$	336.000	1.15123
$45$	−180.000	−0.596285
$46$	122.000	0.391042
$47$	−12.0000	−0.0372421	−0.0186211	−	0.999827i	$-0.505928\pi$
$47$	−12.0000	−0.0372421	−0.0186211	+	0.999827i	$0.505928\pi$
$48$	123.000	0.369865
$49$	−339.000	−0.988338
$50$	−275.000	−0.777817
$51$	−51.0000	−0.140028
$52$	98.0000	0.261349
$53$	−234.000	−0.606460	−0.303230	−	0.952917i	$-0.598065\pi$
$53$	−234.000	−0.606460	−0.303230	+	0.952917i	$0.598065\pi$
$54$	−27.0000	−0.0680414
$55$	960.000	2.35357
$56$	−30.0000	−0.0715878
$57$	276.000	0.641353
$58$	36.0000	0.0815005
$59$	−540.000	−1.19156	−0.595780	−	0.803148i	$-0.703157\pi$
$59$	−540.000	−1.19156	−0.595780	+	0.803148i	$0.703157\pi$
$60$	420.000	0.903696
$61$	−820.000	−1.72115	−0.860576	−	0.509322i	$-0.829896\pi$
$61$	−820.000	−1.72115	−0.860576	+	0.509322i	$0.829896\pi$
$62$	182.000	0.372807
$63$	−18.0000	−0.0359966
$64$	−167.000	−0.326172
$65$	280.000	0.534303
$66$	144.000	0.268563
$67$	700.000	1.27640	0.638199	−	0.769872i	$-0.279680\pi$
$67$	700.000	1.27640	0.638199	+	0.769872i	$0.279680\pi$
$68$	119.000	0.212219
$69$	−366.000	−0.638568
$70$	−40.0000	−0.0682988
$71$	794.000	1.32719	0.663595	−	0.748092i	$-0.269030\pi$
$71$	794.000	1.32719	0.663595	+	0.748092i	$0.269030\pi$
$72$	135.000	0.220971
$73$	−1038.00	−1.66423	−0.832114	−	0.554604i	$-0.812870\pi$
$73$	−1038.00	−1.66423	−0.832114	+	0.554604i	$0.812870\pi$
$74$	−76.0000	−0.119389
$75$	825.000	1.27017
$76$	−644.000	−0.971998
$77$	96.0000	0.142081
$78$	42.0000	0.0609688
$79$	858.000	1.22193	0.610965	−	0.791657i	$-0.290781\pi$
$79$	858.000	1.22193	0.610965	+	0.791657i	$0.290781\pi$
$80$	−820.000	−1.14598
$81$	81.0000	0.111111
$82$	−294.000	−0.395937
$83$	1052.00	1.39123	0.695614	−	0.718415i	$-0.255132\pi$
$83$	1052.00	1.39123	0.695614	+	0.718415i	$0.255132\pi$
$84$	42.0000	0.0545545
$85$	340.000	0.433861
$86$	428.000	0.536656
$87$	−108.000	−0.133090
$88$	−720.000	−0.872185
$89$	1102.00	1.31249	0.656246	−	0.754547i	$-0.272143\pi$
$89$	1102.00	1.31249	0.656246	+	0.754547i	$0.272143\pi$
$90$	180.000	0.210819
$91$	28.0000	0.0322549
$92$	854.000	0.967779
$93$	−546.000	−0.608791
$94$	12.0000	0.0131671
$95$	−1840.00	−1.98716
$96$	−483.000	−0.513500
$97$	710.000	0.743192	0.371596	−	0.928395i	$-0.378811\pi$
$97$	710.000	0.743192	0.371596	+	0.928395i	$0.378811\pi$
$98$	339.000	0.349430
$99$	−432.000	−0.438562
$100$	−1925.00	−1.92500
$101$	−1210.00	−1.19207	−0.596037	−	0.802957i	$-0.703259\pi$
$101$	−1210.00	−1.19207	−0.596037	+	0.802957i	$0.703259\pi$
$102$	51.0000	0.0495074
$103$	644.000	0.616070	0.308035	−	0.951375i	$-0.400329\pi$
$103$	644.000	0.616070	0.308035	+	0.951375i	$0.400329\pi$
$104$	−210.000	−0.198002
$105$	120.000	0.111531
$106$	234.000	0.214416
$107$	−256.000	−0.231294	−0.115647	−	0.993290i	$-0.536894\pi$
$107$	−256.000	−0.231294	−0.115647	+	0.993290i	$0.536894\pi$
$108$	−189.000	−0.168394
$109$	−1248.00	−1.09667	−0.548334	−	0.836260i	$-0.684738\pi$
$109$	−1248.00	−1.09667	−0.548334	+	0.836260i	$0.684738\pi$
$110$	−960.000	−0.832113
$111$	228.000	0.194962
$112$	−82.0000	−0.0691810
$113$	−350.000	−0.291374	−0.145687	−	0.989331i	$-0.546539\pi$
$113$	−350.000	−0.291374	−0.145687	+	0.989331i	$0.546539\pi$
$114$	−276.000	−0.226752
$115$	2440.00	1.97853
$116$	252.000	0.201704
$117$	−126.000	−0.0995616
$118$	540.000	0.421280
$119$	34.0000	0.0261914
$120$	−900.000	−0.684653
$121$	973.000	0.731029
$122$	820.000	0.608519
$123$	882.000	0.646563
$124$	1274.00	0.922650
$125$	−3000.00	−2.14663
$126$	18.0000	0.0127267
$127$	−328.000	−0.229176	−0.114588	−	0.993413i	$-0.536555\pi$
$127$	−328.000	−0.229176	−0.114588	+	0.993413i	$0.536555\pi$
$128$	1455.00	1.00473
$129$	−1284.00	−0.876356
$130$	−280.000	−0.188905
$131$	−1952.00	−1.30189	−0.650943	−	0.759127i	$-0.725626\pi$
$131$	−1952.00	−1.30189	−0.650943	+	0.759127i	$0.725626\pi$
$132$	1008.00	0.664660
$133$	−184.000	−0.119961
$134$	−700.000	−0.451275
$135$	−540.000	−0.344265
$136$	−255.000	−0.160780
$137$	2634.00	1.64261	0.821306	−	0.570488i	$-0.193246\pi$
$137$	2634.00	1.64261	0.821306	+	0.570488i	$0.193246\pi$
$138$	366.000	0.225768
$139$	1864.00	1.13743	0.568714	−	0.822536i	$-0.307441\pi$
$139$	1864.00	1.13743	0.568714	+	0.822536i	$0.307441\pi$
$140$	−280.000	−0.169031
$141$	−36.0000	−0.0215018
$142$	−794.000	−0.469232
$143$	672.000	0.392975
$144$	369.000	0.213542
$145$	720.000	0.412364
$146$	1038.00	0.588394
$147$	−1017.00	−0.570617
$148$	−532.000	−0.295474
$149$	−286.000	−0.157249	−0.0786243	−	0.996904i	$-0.525053\pi$
$149$	−286.000	−0.157249	−0.0786243	+	0.996904i	$0.525053\pi$
$150$	−825.000	−0.449073
$151$	−1624.00	−0.875227	−0.437613	−	0.899163i	$-0.644176\pi$
$151$	−1624.00	−0.875227	−0.437613	+	0.899163i	$0.644176\pi$
$152$	1380.00	0.736400
$153$	−153.000	−0.0808452
$154$	−96.0000	−0.0502331
$155$	3640.00	1.88627
$156$	294.000	0.150890
$157$	2542.00	1.29219	0.646095	−	0.763257i	$-0.276401\pi$
$157$	2542.00	1.29219	0.646095	+	0.763257i	$0.276401\pi$
$158$	−858.000	−0.432018
$159$	−702.000	−0.350140
$160$	3220.00	1.59102
$161$	244.000	0.119440
$162$	−81.0000	−0.0392837
$163$	684.000	0.328681	0.164341	−	0.986404i	$-0.447450\pi$
$163$	684.000	0.328681	0.164341	+	0.986404i	$0.447450\pi$
$164$	−2058.00	−0.979895
$165$	2880.00	1.35883
$166$	−1052.00	−0.491874
$167$	542.000	0.251145	0.125573	−	0.992084i	$-0.459923\pi$
$167$	542.000	0.251145	0.125573	+	0.992084i	$0.459923\pi$
$168$	−90.0000	−0.0413313
$169$	−2001.00	−0.910787
$170$	−340.000	−0.153393
$171$	828.000	0.370285
$172$	2996.00	1.32816
$173$	−836.000	−0.367398	−0.183699	−	0.982983i	$-0.558807\pi$
$173$	−836.000	−0.367398	−0.183699	+	0.982983i	$0.558807\pi$
$174$	108.000	0.0470544
$175$	−550.000	−0.237578
$176$	−1968.00	−0.842861
$177$	−1620.00	−0.687947
$178$	−1102.00	−0.464036
$179$	−3444.00	−1.43808	−0.719041	−	0.694968i	$-0.755419\pi$
$179$	−3444.00	−1.43808	−0.719041	+	0.694968i	$0.755419\pi$
$180$	1260.00	0.521749
$181$	−3284.00	−1.34861	−0.674303	−	0.738454i	$-0.735556\pi$
$181$	−3284.00	−1.34861	−0.674303	+	0.738454i	$0.735556\pi$
$182$	−28.0000	−0.0114038
$183$	−2460.00	−0.993707
$184$	−1830.00	−0.733203
$185$	−1520.00	−0.604068
$186$	546.000	0.215240
$187$	816.000	0.319101
$188$	84.0000	0.0325869
$189$	−54.0000	−0.0207827
$190$	1840.00	0.702566
$191$	−340.000	−0.128804	−0.0644019	−	0.997924i	$-0.520514\pi$
$191$	−340.000	−0.128804	−0.0644019	+	0.997924i	$0.520514\pi$
$192$	−501.000	−0.188315
$193$	−1498.00	−0.558696	−0.279348	−	0.960190i	$-0.590118\pi$
$193$	−1498.00	−0.558696	−0.279348	+	0.960190i	$0.590118\pi$
$194$	−710.000	−0.262758
$195$	840.000	0.308480
$196$	2373.00	0.864796
$197$	−1176.00	−0.425312	−0.212656	−	0.977127i	$-0.568211\pi$
$197$	−1176.00	−0.425312	−0.212656	+	0.977127i	$0.568211\pi$
$198$	432.000	0.155055
$199$	−2450.00	−0.872743	−0.436372	−	0.899767i	$-0.643737\pi$
$199$	−2450.00	−0.872743	−0.436372	+	0.899767i	$0.643737\pi$
$200$	4125.00	1.45841
$201$	2100.00	0.736928
$202$	1210.00	0.421462
$203$	72.0000	0.0248936
$204$	357.000	0.122525
$205$	−5880.00	−2.00330
$206$	−644.000	−0.217814
$207$	−1098.00	−0.368678
$208$	−574.000	−0.191345
$209$	−4416.00	−1.46154
$210$	−120.000	−0.0394323
$211$	1064.00	0.347151	0.173575	−	0.984821i	$-0.444468\pi$
$211$	1064.00	0.347151	0.173575	+	0.984821i	$0.444468\pi$
$212$	1638.00	0.530652
$213$	2382.00	0.766253
$214$	256.000	0.0817748
$215$	8560.00	2.71529
$216$	405.000	0.127578
$217$	364.000	0.113871
$218$	1248.00	0.387730
$219$	−3114.00	−0.960843
$220$	−6720.00	−2.05937
$221$	238.000	0.0724417
$222$	−228.000	−0.0689295
$223$	−2244.00	−0.673854	−0.336927	−	0.941531i	$-0.609387\pi$
$223$	−2244.00	−0.673854	−0.336927	+	0.941531i	$0.609387\pi$
$224$	322.000	0.0960470
$225$	2475.00	0.733333
$226$	350.000	0.103016
$227$	516.000	0.150873	0.0754364	−	0.997151i	$-0.475965\pi$
$227$	516.000	0.150873	0.0754364	+	0.997151i	$0.475965\pi$
$228$	−1932.00	−0.561183
$229$	−2922.00	−0.843193	−0.421596	−	0.906784i	$-0.638530\pi$
$229$	−2922.00	−0.843193	−0.421596	+	0.906784i	$0.638530\pi$
$230$	−2440.00	−0.699517
$231$	288.000	0.0820303
$232$	−540.000	−0.152814
$233$	3114.00	0.875558	0.437779	−	0.899083i	$-0.355765\pi$
$233$	3114.00	0.875558	0.437779	+	0.899083i	$0.355765\pi$
$234$	126.000	0.0352003
$235$	240.000	0.0666207
$236$	3780.00	1.04261
$237$	2574.00	0.705482
$238$	−34.0000	−0.00926005
$239$	4124.00	1.11615	0.558074	−	0.829791i	$-0.311541\pi$
$239$	4124.00	1.11615	0.558074	+	0.829791i	$0.311541\pi$
$240$	−2460.00	−0.661635
$241$	−2034.00	−0.543658	−0.271829	−	0.962346i	$-0.587628\pi$
$241$	−2034.00	−0.543658	−0.271829	+	0.962346i	$0.587628\pi$
$242$	−973.000	−0.258458
$243$	243.000	0.0641500
$244$	5740.00	1.50601
$245$	6780.00	1.76799
$246$	−882.000	−0.228595
$247$	−1288.00	−0.331795
$248$	−2730.00	−0.699013
$249$	3156.00	0.803226
$250$	3000.00	0.758947
$251$	996.000	0.250466	0.125233	−	0.992127i	$-0.460032\pi$
$251$	996.000	0.250466	0.125233	+	0.992127i	$0.460032\pi$
$252$	126.000	0.0314970
$253$	5856.00	1.45519
$254$	328.000	0.0810258
$255$	1020.00	0.250490
$256$	−119.000	−0.0290527
$257$	−3246.00	−0.787860	−0.393930	−	0.919141i	$-0.628885\pi$
$257$	−3246.00	−0.787860	−0.393930	+	0.919141i	$0.628885\pi$
$258$	1284.00	0.309839
$259$	−152.000	−0.0364665
$260$	−1960.00	−0.467516
$261$	−324.000	−0.0768395
$262$	1952.00	0.460286
$263$	932.000	0.218516	0.109258	−	0.994013i	$-0.465153\pi$
$263$	932.000	0.218516	0.109258	+	0.994013i	$0.465153\pi$
$264$	−2160.00	−0.503556
$265$	4680.00	1.08487
$266$	184.000	0.0424126
$267$	3306.00	0.757767
$268$	−4900.00	−1.11685
$269$	−3884.00	−0.880341	−0.440170	−	0.897914i	$-0.645082\pi$
$269$	−3884.00	−0.880341	−0.440170	+	0.897914i	$0.645082\pi$
$270$	540.000	0.121716
$271$	−1936.00	−0.433962	−0.216981	−	0.976176i	$-0.569621\pi$
$271$	−1936.00	−0.433962	−0.216981	+	0.976176i	$0.569621\pi$
$272$	−697.000	−0.155374
$273$	84.0000	0.0186224
$274$	−2634.00	−0.580751
$275$	−13200.0	−2.89451
$276$	2562.00	0.558747
$277$	872.000	0.189146	0.0945729	−	0.995518i	$-0.469851\pi$
$277$	872.000	0.189146	0.0945729	+	0.995518i	$0.469851\pi$
$278$	−1864.00	−0.402141
$279$	−1638.00	−0.351486
$280$	600.000	0.128060
$281$	3198.00	0.678921	0.339460	−	0.940620i	$-0.389756\pi$
$281$	3198.00	0.678921	0.339460	+	0.940620i	$0.389756\pi$
$282$	36.0000	0.00760202
$283$	−1936.00	−0.406655	−0.203327	−	0.979111i	$-0.565176\pi$
$283$	−1936.00	−0.406655	−0.203327	+	0.979111i	$0.565176\pi$
$284$	−5558.00	−1.16129
$285$	−5520.00	−1.14729
$286$	−672.000	−0.138938
$287$	−588.000	−0.120936
$288$	−1449.00	−0.296469
$289$	289.000	0.0588235
$290$	−720.000	−0.145793
$291$	2130.00	0.429082
$292$	7266.00	1.45620
$293$	−5718.00	−1.14010	−0.570050	−	0.821610i	$-0.693076\pi$
$293$	−5718.00	−1.14010	−0.570050	+	0.821610i	$0.693076\pi$
$294$	1017.00	0.201744
$295$	10800.0	2.13153
$296$	1140.00	0.223855
$297$	−1296.00	−0.253204
$298$	286.000	0.0555958
$299$	1708.00	0.330355
$300$	−5775.00	−1.11140
$301$	856.000	0.163917
$302$	1624.00	0.309439
$303$	−3630.00	−0.688244
$304$	3772.00	0.711642
$305$	16400.0	3.07889
$306$	153.000	0.0285831
$307$	684.000	0.127159	0.0635797	−	0.997977i	$-0.479748\pi$
$307$	684.000	0.127159	0.0635797	+	0.997977i	$0.479748\pi$
$308$	−672.000	−0.124321
$309$	1932.00	0.355688
$310$	−3640.00	−0.666897
$311$	−290.000	−0.0528759	−0.0264379	−	0.999650i	$-0.508416\pi$
$311$	−290.000	−0.0528759	−0.0264379	+	0.999650i	$0.508416\pi$
$312$	−630.000	−0.114316
$313$	−11050.0	−1.99547	−0.997736	−	0.0672478i	$-0.978578\pi$
$313$	−11050.0	−1.99547	−0.997736	+	0.0672478i	$0.978578\pi$
$314$	−2542.00	−0.456858
$315$	360.000	0.0643927
$316$	−6006.00	−1.06919
$317$	992.000	0.175761	0.0878806	−	0.996131i	$-0.471991\pi$
$317$	992.000	0.175761	0.0878806	+	0.996131i	$0.471991\pi$
$318$	702.000	0.123793
$319$	1728.00	0.303290
$320$	3340.00	0.583474
$321$	−768.000	−0.133538
$322$	−244.000	−0.0422285
$323$	−1564.00	−0.269422
$324$	−567.000	−0.0972222
$325$	−3850.00	−0.657106
$326$	−684.000	−0.116206
$327$	−3744.00	−0.633161
$328$	4410.00	0.742383
$329$	24.0000	0.00402177
$330$	−2880.00	−0.480421
$331$	2860.00	0.474924	0.237462	−	0.971397i	$-0.423684\pi$
$331$	2860.00	0.474924	0.237462	+	0.971397i	$0.423684\pi$
$332$	−7364.00	−1.21733
$333$	684.000	0.112561
$334$	−542.000	−0.0887932
$335$	−14000.0	−2.28329
$336$	−246.000	−0.0399417
$337$	6298.00	1.01802	0.509012	−	0.860760i	$-0.330011\pi$
$337$	6298.00	1.01802	0.509012	+	0.860760i	$0.330011\pi$
$338$	2001.00	0.322012
$339$	−1050.00	−0.168225
$340$	−2380.00	−0.379628
$341$	8736.00	1.38733
$342$	−828.000	−0.130916
$343$	1364.00	0.214720
$344$	−6420.00	−1.00623
$345$	7320.00	1.14231
$346$	836.000	0.129895
$347$	−3508.00	−0.542707	−0.271353	−	0.962480i	$-0.587471\pi$
$347$	−3508.00	−0.542707	−0.271353	+	0.962480i	$0.587471\pi$
$348$	756.000	0.116454
$349$	−2406.00	−0.369026	−0.184513	−	0.982830i	$-0.559071\pi$
$349$	−2406.00	−0.369026	−0.184513	+	0.982830i	$0.559071\pi$
$350$	550.000	0.0839964
$351$	−378.000	−0.0574819
$352$	7728.00	1.17018
$353$	−1842.00	−0.277733	−0.138867	−	0.990311i	$-0.544346\pi$
$353$	−1842.00	−0.277733	−0.138867	+	0.990311i	$0.544346\pi$
$354$	1620.00	0.243226
$355$	−15880.0	−2.37415
$356$	−7714.00	−1.14843
$357$	102.000	0.0151216
$358$	3444.00	0.508439
$359$	3264.00	0.479853	0.239927	−	0.970791i	$-0.422877\pi$
$359$	3264.00	0.479853	0.239927	+	0.970791i	$0.422877\pi$
$360$	−2700.00	−0.395285
$361$	1605.00	0.233999
$362$	3284.00	0.476804
$363$	2919.00	0.422060
$364$	−196.000	−0.0282231
$365$	20760.0	2.97706
$366$	2460.00	0.351329
$367$	7354.00	1.04598	0.522991	−	0.852338i	$-0.324816\pi$
$367$	7354.00	1.04598	0.522991	+	0.852338i	$0.324816\pi$
$368$	−5002.00	−0.708552
$369$	2646.00	0.373293
$370$	1520.00	0.213570
$371$	468.000	0.0654915
$372$	3822.00	0.532692
$373$	11322.0	1.57166	0.785832	−	0.618440i	$-0.212235\pi$
$373$	11322.0	1.57166	0.785832	+	0.618440i	$0.212235\pi$
$374$	−816.000	−0.112819
$375$	−9000.00	−1.23935
$376$	−180.000	−0.0246883
$377$	504.000	0.0688523
$378$	54.0000	0.00734778
$379$	12796.0	1.73426	0.867132	−	0.498078i	$-0.165961\pi$
$379$	12796.0	1.73426	0.867132	+	0.498078i	$0.165961\pi$
$380$	12880.0	1.73876
$381$	−984.000	−0.132315
$382$	340.000	0.0455390
$383$	−420.000	−0.0560339	−0.0280170	−	0.999607i	$-0.508919\pi$
$383$	−420.000	−0.0560339	−0.0280170	+	0.999607i	$0.508919\pi$
$384$	4365.00	0.580079
$385$	−1920.00	−0.254162
$386$	1498.00	0.197529
$387$	−3852.00	−0.505964
$388$	−4970.00	−0.650293
$389$	6426.00	0.837561	0.418780	−	0.908088i	$-0.362458\pi$
$389$	6426.00	0.837561	0.418780	+	0.908088i	$0.362458\pi$
$390$	−840.000	−0.109064
$391$	2074.00	0.268252
$392$	−5085.00	−0.655182
$393$	−5856.00	−0.751644
$394$	1176.00	0.150371
$395$	−17160.0	−2.18586
$396$	3024.00	0.383742
$397$	−13212.0	−1.67026	−0.835128	−	0.550056i	$-0.814606\pi$
$397$	−13212.0	−1.67026	−0.835128	+	0.550056i	$0.814606\pi$
$398$	2450.00	0.308561
$399$	−552.000	−0.0692596
$400$	11275.0	1.40938
$401$	−6242.00	−0.777333	−0.388667	−	0.921378i	$-0.627064\pi$
$401$	−6242.00	−0.777333	−0.388667	+	0.921378i	$0.627064\pi$
$402$	−2100.00	−0.260543
$403$	2548.00	0.314950
$404$	8470.00	1.04306
$405$	−1620.00	−0.198762
$406$	−72.0000	−0.00880123
$407$	−3648.00	−0.444287
$408$	−765.000	−0.0928263
$409$	−5194.00	−0.627938	−0.313969	−	0.949433i	$-0.601659\pi$
$409$	−5194.00	−0.627938	−0.313969	+	0.949433i	$0.601659\pi$
$410$	5880.00	0.708274
$411$	7902.00	0.948362
$412$	−4508.00	−0.539061
$413$	1080.00	0.128676
$414$	1098.00	0.130347
$415$	−21040.0	−2.48871
$416$	2254.00	0.265653
$417$	5592.00	0.656694
$418$	4416.00	0.516731
$419$	15956.0	1.86039	0.930193	−	0.367071i	$-0.119639\pi$
$419$	15956.0	1.86039	0.930193	+	0.367071i	$0.119639\pi$
$420$	−840.000	−0.0975900
$421$	−15254.0	−1.76588	−0.882939	−	0.469488i	$-0.844438\pi$
$421$	−15254.0	−1.76588	−0.882939	+	0.469488i	$0.844438\pi$
$422$	−1064.00	−0.122736
$423$	−108.000	−0.0124140
$424$	−3510.00	−0.402030
$425$	−4675.00	−0.533578
$426$	−2382.00	−0.270911
$427$	1640.00	0.185867
$428$	1792.00	0.202382
$429$	2016.00	0.226884
$430$	−8560.00	−0.960000
$431$	−5538.00	−0.618924	−0.309462	−	0.950912i	$-0.600149\pi$
$431$	−5538.00	−0.618924	−0.309462	+	0.950912i	$0.600149\pi$
$432$	1107.00	0.123288
$433$	11342.0	1.25880	0.629402	−	0.777080i	$-0.283300\pi$
$433$	11342.0	1.25880	0.629402	+	0.777080i	$0.283300\pi$
$434$	−364.000	−0.0402594
$435$	2160.00	0.238078
$436$	8736.00	0.959584
$437$	−11224.0	−1.22864
$438$	3114.00	0.339709
$439$	982.000	0.106762	0.0533808	−	0.998574i	$-0.483000\pi$
$439$	982.000	0.106762	0.0533808	+	0.998574i	$0.483000\pi$
$440$	14400.0	1.56021
$441$	−3051.00	−0.329446
$442$	−238.000	−0.0256120
$443$	−2492.00	−0.267265	−0.133633	−	0.991031i	$-0.542664\pi$
$443$	−2492.00	−0.267265	−0.133633	+	0.991031i	$0.542664\pi$
$444$	−1596.00	−0.170592
$445$	−22040.0	−2.34786
$446$	2244.00	0.238243
$447$	−858.000	−0.0907875
$448$	334.000	0.0352233
$449$	−5498.00	−0.577877	−0.288938	−	0.957348i	$-0.593302\pi$
$449$	−5498.00	−0.577877	−0.288938	+	0.957348i	$0.593302\pi$
$450$	−2475.00	−0.259272
$451$	−14112.0	−1.47341
$452$	2450.00	0.254952
$453$	−4872.00	−0.505312
$454$	−516.000	−0.0533416
$455$	−560.000	−0.0576994
$456$	4140.00	0.425161
$457$	4998.00	0.511590	0.255795	−	0.966731i	$-0.417663\pi$
$457$	4998.00	0.511590	0.255795	+	0.966731i	$0.417663\pi$
$458$	2922.00	0.298114
$459$	−459.000	−0.0466760
$460$	−17080.0	−1.73122
$461$	−7586.00	−0.766411	−0.383205	−	0.923663i	$-0.625180\pi$
$461$	−7586.00	−0.766411	−0.383205	+	0.923663i	$0.625180\pi$
$462$	−288.000	−0.0290021
$463$	5900.00	0.592217	0.296108	−	0.955154i	$-0.404311\pi$
$463$	5900.00	0.592217	0.296108	+	0.955154i	$0.404311\pi$
$464$	−1476.00	−0.147676
$465$	10920.0	1.08904
$466$	−3114.00	−0.309556
$467$	−19404.0	−1.92272	−0.961360	−	0.275295i	$-0.911224\pi$
$467$	−19404.0	−1.92272	−0.961360	+	0.275295i	$0.911224\pi$
$468$	882.000	0.0871164
$469$	−1400.00	−0.137838
$470$	−240.000	−0.0235540
$471$	7626.00	0.746046
$472$	−8100.00	−0.789900
$473$	20544.0	1.99707
$474$	−2574.00	−0.249426
$475$	25300.0	2.44388
$476$	−238.000	−0.0229175
$477$	−2106.00	−0.202153
$478$	−4124.00	−0.394618
$479$	2586.00	0.246675	0.123338	−	0.992365i	$-0.460640\pi$
$479$	2586.00	0.246675	0.123338	+	0.992365i	$0.460640\pi$
$480$	9660.00	0.918576
$481$	−1064.00	−0.100861
$482$	2034.00	0.192212
$483$	732.000	0.0689589
$484$	−6811.00	−0.639651
$485$	−14200.0	−1.32946
$486$	−243.000	−0.0226805
$487$	10106.0	0.940342	0.470171	−	0.882575i	$-0.344192\pi$
$487$	10106.0	0.940342	0.470171	+	0.882575i	$0.344192\pi$
$488$	−12300.0	−1.14097
$489$	2052.00	0.189764
$490$	−6780.00	−0.625080
$491$	−76.0000	−0.00698540	−0.00349270	−	0.999994i	$-0.501112\pi$
$491$	−76.0000	−0.00698540	−0.00349270	+	0.999994i	$0.501112\pi$
$492$	−6174.00	−0.565743
$493$	612.000	0.0559089
$494$	1288.00	0.117307
$495$	8640.00	0.784523
$496$	−7462.00	−0.675511
$497$	−1588.00	−0.143323
$498$	−3156.00	−0.283983
$499$	8096.00	0.726306	0.363153	−	0.931730i	$-0.381700\pi$
$499$	8096.00	0.726306	0.363153	+	0.931730i	$0.381700\pi$
$500$	21000.0	1.87830
$501$	1626.00	0.144999
$502$	−996.000	−0.0885531
$503$	15942.0	1.41316	0.706579	−	0.707634i	$-0.250237\pi$
$503$	15942.0	1.41316	0.706579	+	0.707634i	$0.250237\pi$
$504$	−270.000	−0.0238626
$505$	24200.0	2.13245
$506$	−5856.00	−0.514488
$507$	−6003.00	−0.525843
$508$	2296.00	0.200529
$509$	−13742.0	−1.19667	−0.598333	−	0.801247i	$-0.704170\pi$
$509$	−13742.0	−1.19667	−0.598333	+	0.801247i	$0.704170\pi$
$510$	−1020.00	−0.0885615
$511$	2076.00	0.179720
$512$	−11521.0	−0.994455
$513$	2484.00	0.213784
$514$	3246.00	0.278550
$515$	−12880.0	−1.10206
$516$	8988.00	0.766811
$517$	576.000	0.0489989
$518$	152.000	0.0128929
$519$	−2508.00	−0.212117
$520$	4200.00	0.354197
$521$	11942.0	1.00420	0.502100	−	0.864809i	$-0.332561\pi$
$521$	11942.0	1.00420	0.502100	+	0.864809i	$0.332561\pi$
$522$	324.000	0.0271668
$523$	−6012.00	−0.502651	−0.251325	−	0.967903i	$-0.580866\pi$
$523$	−6012.00	−0.502651	−0.251325	+	0.967903i	$0.580866\pi$
$524$	13664.0	1.13915
$525$	−1650.00	−0.137166
$526$	−932.000	−0.0772569
$527$	3094.00	0.255743
$528$	−5904.00	−0.486626
$529$	2717.00	0.223309
$530$	−4680.00	−0.383559
$531$	−4860.00	−0.397187
$532$	1288.00	0.104966
$533$	−4116.00	−0.334491
$534$	−3306.00	−0.267911
$535$	5120.00	0.413751
$536$	10500.0	0.846140
$537$	−10332.0	−0.830277
$538$	3884.00	0.311247
$539$	16272.0	1.30034
$540$	3780.00	0.301232
$541$	6420.00	0.510198	0.255099	−	0.966915i	$-0.417892\pi$
$541$	6420.00	0.510198	0.255099	+	0.966915i	$0.417892\pi$
$542$	1936.00	0.153429
$543$	−9852.00	−0.778618
$544$	2737.00	0.215713
$545$	24960.0	1.96178
$546$	−84.0000	−0.00658401
$547$	1576.00	0.123190	0.0615950	−	0.998101i	$-0.480381\pi$
$547$	1576.00	0.123190	0.0615950	+	0.998101i	$0.480381\pi$
$548$	−18438.0	−1.43729
$549$	−7380.00	−0.573717
$550$	13200.0	1.02336
$551$	−3312.00	−0.256072
$552$	−5490.00	−0.423315
$553$	−1716.00	−0.131956
$554$	−872.000	−0.0668732
$555$	−4560.00	−0.348759
$556$	−13048.0	−0.995249
$557$	15318.0	1.16525	0.582625	−	0.812741i	$-0.302026\pi$
$557$	15318.0	1.16525	0.582625	+	0.812741i	$0.302026\pi$
$558$	1638.00	0.124269
$559$	5992.00	0.453371
$560$	1640.00	0.123755
$561$	2448.00	0.184233
$562$	−3198.00	−0.240035
$563$	13220.0	0.989621	0.494810	−	0.869001i	$-0.335237\pi$
$563$	13220.0	0.989621	0.494810	+	0.869001i	$0.335237\pi$
$564$	252.000	0.0188140
$565$	7000.00	0.521225
$566$	1936.00	0.143774
$567$	−162.000	−0.0119989
$568$	11910.0	0.879811
$569$	2794.00	0.205853	0.102927	−	0.994689i	$-0.467179\pi$
$569$	2794.00	0.205853	0.102927	+	0.994689i	$0.467179\pi$
$570$	5520.00	0.405627
$571$	−14756.0	−1.08147	−0.540735	−	0.841193i	$-0.681854\pi$
$571$	−14756.0	−1.08147	−0.540735	+	0.841193i	$0.681854\pi$
$572$	−4704.00	−0.343853
$573$	−1020.00	−0.0743649
$574$	588.000	0.0427572
$575$	−33550.0	−2.43327
$576$	−1503.00	−0.108724
$577$	−2846.00	−0.205339	−0.102669	−	0.994716i	$-0.532738\pi$
$577$	−2846.00	−0.205339	−0.102669	+	0.994716i	$0.532738\pi$
$578$	−289.000	−0.0207973
$579$	−4494.00	−0.322563
$580$	−5040.00	−0.360818
$581$	−2104.00	−0.150239
$582$	−2130.00	−0.151703
$583$	11232.0	0.797911
$584$	−15570.0	−1.10324
$585$	2520.00	0.178101
$586$	5718.00	0.403086
$587$	−16820.0	−1.18268	−0.591342	−	0.806421i	$-0.701402\pi$
$587$	−16820.0	−1.18268	−0.591342	+	0.806421i	$0.701402\pi$
$588$	7119.00	0.499290
$589$	−16744.0	−1.17135
$590$	−10800.0	−0.753608
$591$	−3528.00	−0.245554
$592$	3116.00	0.216329
$593$	13314.0	0.921991	0.460995	−	0.887403i	$-0.347492\pi$
$593$	13314.0	0.921991	0.460995	+	0.887403i	$0.347492\pi$
$594$	1296.00	0.0895211
$595$	−680.000	−0.0468526
$596$	2002.00	0.137592
$597$	−7350.00	−0.503879
$598$	−1708.00	−0.116798
$599$	2880.00	0.196450	0.0982250	−	0.995164i	$-0.468683\pi$
$599$	2880.00	0.196450	0.0982250	+	0.995164i	$0.468683\pi$
$600$	12375.0	0.842012
$601$	−24854.0	−1.68688	−0.843441	−	0.537222i	$-0.819474\pi$
$601$	−24854.0	−1.68688	−0.843441	+	0.537222i	$0.819474\pi$
$602$	−856.000	−0.0579534
$603$	6300.00	0.425466
$604$	11368.0	0.765823
$605$	−19460.0	−1.30770
$606$	3630.00	0.243331
$607$	6122.00	0.409365	0.204682	−	0.978828i	$-0.434384\pi$
$607$	6122.00	0.409365	0.204682	+	0.978828i	$0.434384\pi$
$608$	−14812.0	−0.988003
$609$	216.000	0.0143724
$610$	−16400.0	−1.08855
$611$	168.000	0.0111237
$612$	1071.00	0.0707396
$613$	17398.0	1.14633	0.573164	−	0.819441i	$-0.305716\pi$
$613$	17398.0	1.14633	0.573164	+	0.819441i	$0.305716\pi$
$614$	−684.000	−0.0449576
$615$	−17640.0	−1.15661
$616$	1440.00	0.0941871
$617$	2922.00	0.190657	0.0953284	−	0.995446i	$-0.469610\pi$
$617$	2922.00	0.190657	0.0953284	+	0.995446i	$0.469610\pi$
$618$	−1932.00	−0.125755
$619$	9660.00	0.627251	0.313625	−	0.949547i	$-0.398456\pi$
$619$	9660.00	0.627251	0.313625	+	0.949547i	$0.398456\pi$
$620$	−25480.0	−1.65049
$621$	−3294.00	−0.212856
$622$	290.000	0.0186944
$623$	−2204.00	−0.141736
$624$	−1722.00	−0.110473
$625$	25625.0	1.64000
$626$	11050.0	0.705506
$627$	−13248.0	−0.843818
$628$	−17794.0	−1.13067
$629$	−1292.00	−0.0819005
$630$	−360.000	−0.0227663
$631$	2788.00	0.175893	0.0879465	−	0.996125i	$-0.471970\pi$
$631$	2788.00	0.175893	0.0879465	+	0.996125i	$0.471970\pi$
$632$	12870.0	0.810033
$633$	3192.00	0.200428
$634$	−992.000	−0.0621409
$635$	6560.00	0.409962
$636$	4914.00	0.306372
$637$	4746.00	0.295202
$638$	−1728.00	−0.107229
$639$	7146.00	0.442397
$640$	−29100.0	−1.79731
$641$	−16290.0	−1.00377	−0.501885	−	0.864934i	$-0.667360\pi$
$641$	−16290.0	−1.00377	−0.501885	+	0.864934i	$0.667360\pi$
$642$	768.000	0.0472127
$643$	−16588.0	−1.01737	−0.508683	−	0.860954i	$-0.669868\pi$
$643$	−16588.0	−1.01737	−0.508683	+	0.860954i	$0.669868\pi$
$644$	−1708.00	−0.104510
$645$	25680.0	1.56767
$646$	1564.00	0.0952550
$647$	22364.0	1.35892	0.679459	−	0.733714i	$-0.262215\pi$
$647$	22364.0	1.35892	0.679459	+	0.733714i	$0.262215\pi$
$648$	1215.00	0.0736570
$649$	25920.0	1.56772
$650$	3850.00	0.232322
$651$	1092.00	0.0657432
$652$	−4788.00	−0.287596
$653$	−19356.0	−1.15997	−0.579984	−	0.814628i	$-0.696941\pi$
$653$	−19356.0	−1.15997	−0.579984	+	0.814628i	$0.696941\pi$
$654$	3744.00	0.223856
$655$	39040.0	2.32888
$656$	12054.0	0.717423
$657$	−9342.00	−0.554743
$658$	−24.0000	−0.00142191
$659$	−4220.00	−0.249450	−0.124725	−	0.992191i	$-0.539805\pi$
$659$	−4220.00	−0.249450	−0.124725	+	0.992191i	$0.539805\pi$
$660$	−20160.0	−1.18898
$661$	−12070.0	−0.710240	−0.355120	−	0.934821i	$-0.615560\pi$
$661$	−12070.0	−0.710240	−0.355120	+	0.934821i	$0.615560\pi$
$662$	−2860.00	−0.167911
$663$	714.000	0.0418242
$664$	15780.0	0.922263
$665$	3680.00	0.214593
$666$	−684.000	−0.0397965
$667$	4392.00	0.254961
$668$	−3794.00	−0.219752
$669$	−6732.00	−0.389050
$670$	14000.0	0.807264
$671$	39360.0	2.26449
$672$	966.000	0.0554528
$673$	−5914.00	−0.338734	−0.169367	−	0.985553i	$-0.554172\pi$
$673$	−5914.00	−0.338734	−0.169367	+	0.985553i	$0.554172\pi$
$674$	−6298.00	−0.359926
$675$	7425.00	0.423390
$676$	14007.0	0.796939
$677$	−27624.0	−1.56821	−0.784104	−	0.620630i	$-0.786877\pi$
$677$	−27624.0	−1.56821	−0.784104	+	0.620630i	$0.786877\pi$
$678$	1050.00	0.0594764
$679$	−1420.00	−0.0802571
$680$	5100.00	0.287612
$681$	1548.00	0.0871064
$682$	−8736.00	−0.490497
$683$	29132.0	1.63207	0.816036	−	0.578001i	$-0.196167\pi$
$683$	29132.0	1.63207	0.816036	+	0.578001i	$0.196167\pi$
$684$	−5796.00	−0.323999
$685$	−52680.0	−2.93839
$686$	−1364.00	−0.0759151
$687$	−8766.00	−0.486818
$688$	−17548.0	−0.972400
$689$	3276.00	0.181140
$690$	−7320.00	−0.403866
$691$	−20336.0	−1.11956	−0.559781	−	0.828640i	$-0.689115\pi$
$691$	−20336.0	−1.11956	−0.559781	+	0.828640i	$0.689115\pi$
$692$	5852.00	0.321473
$693$	864.000	0.0473602
$694$	3508.00	0.191876
$695$	−37280.0	−2.03469
$696$	−1620.00	−0.0882269
$697$	−4998.00	−0.271611
$698$	2406.00	0.130471
$699$	9342.00	0.505503
$700$	3850.00	0.207880
$701$	17142.0	0.923601	0.461801	−	0.886984i	$-0.347204\pi$
$701$	17142.0	0.923601	0.461801	+	0.886984i	$0.347204\pi$
$702$	378.000	0.0203229
$703$	6992.00	0.375118
$704$	8016.00	0.429140
$705$	720.000	0.0384635
$706$	1842.00	0.0981935
$707$	2420.00	0.128732
$708$	11340.0	0.601954
$709$	22784.0	1.20687	0.603435	−	0.797412i	$-0.293798\pi$
$709$	22784.0	1.20687	0.603435	+	0.797412i	$0.293798\pi$
$710$	15880.0	0.839388
$711$	7722.00	0.407310
$712$	16530.0	0.870067
$713$	22204.0	1.16626
$714$	−102.000	−0.00534629
$715$	−13440.0	−0.702976
$716$	24108.0	1.25832
$717$	12372.0	0.644408
$718$	−3264.00	−0.169654
$719$	58.0000	0.00300839	0.00150420	−	0.999999i	$-0.499521\pi$
$719$	58.0000	0.00300839	0.00150420	+	0.999999i	$0.499521\pi$
$720$	−7380.00	−0.381995
$721$	−1288.00	−0.0665293
$722$	−1605.00	−0.0827312
$723$	−6102.00	−0.313881
$724$	22988.0	1.18003
$725$	−9900.00	−0.507140
$726$	−2919.00	−0.149221
$727$	712.000	0.0363227	0.0181614	−	0.999835i	$-0.494219\pi$
$727$	712.000	0.0363227	0.0181614	+	0.999835i	$0.494219\pi$
$728$	420.000	0.0213822
$729$	729.000	0.0370370
$730$	−20760.0	−1.05255
$731$	7276.00	0.368143
$732$	17220.0	0.869494
$733$	23050.0	1.16149	0.580744	−	0.814086i	$-0.302762\pi$
$733$	23050.0	1.16149	0.580744	+	0.814086i	$0.302762\pi$
$734$	−7354.00	−0.369811
$735$	20340.0	1.02075
$736$	19642.0	0.983714
$737$	−33600.0	−1.67934
$738$	−2646.00	−0.131979
$739$	−38708.0	−1.92679	−0.963394	−	0.268088i	$-0.913608\pi$
$739$	−38708.0	−1.92679	−0.963394	+	0.268088i	$0.913608\pi$
$740$	10640.0	0.528560
$741$	−3864.00	−0.191562
$742$	−468.000	−0.0231547
$743$	−11034.0	−0.544816	−0.272408	−	0.962182i	$-0.587820\pi$
$743$	−11034.0	−0.544816	−0.272408	+	0.962182i	$0.587820\pi$
$744$	−8190.00	−0.403575
$745$	5720.00	0.281295
$746$	−11322.0	−0.555667
$747$	9468.00	0.463743
$748$	−5712.00	−0.279213
$749$	512.000	0.0249774
$750$	9000.00	0.438178
$751$	7502.00	0.364516	0.182258	−	0.983251i	$-0.441659\pi$
$751$	7502.00	0.364516	0.182258	+	0.983251i	$0.441659\pi$
$752$	−492.000	−0.0238582
$753$	2988.00	0.144607
$754$	−504.000	−0.0243430
$755$	32480.0	1.56565
$756$	378.000	0.0181848
$757$	−20954.0	−1.00606	−0.503029	−	0.864269i	$-0.667781\pi$
$757$	−20954.0	−1.00606	−0.503029	+	0.864269i	$0.667781\pi$
$758$	−12796.0	−0.613155
$759$	17568.0	0.840155
$760$	−27600.0	−1.31731
$761$	8186.00	0.389937	0.194969	−	0.980809i	$-0.437539\pi$
$761$	8186.00	0.389937	0.194969	+	0.980809i	$0.437539\pi$
$762$	984.000	0.0467803
$763$	2496.00	0.118429
$764$	2380.00	0.112703
$765$	3060.00	0.144620
$766$	420.000	0.0198110
$767$	7560.00	0.355901
$768$	−357.000	−0.0167736
$769$	−5798.00	−0.271887	−0.135944	−	0.990717i	$-0.543407\pi$
$769$	−5798.00	−0.271887	−0.135944	+	0.990717i	$0.543407\pi$
$770$	1920.00	0.0898597
$771$	−9738.00	−0.454871
$772$	10486.0	0.488859
$773$	−39950.0	−1.85886	−0.929432	−	0.368994i	$-0.879702\pi$
$773$	−39950.0	−1.85886	−0.929432	+	0.368994i	$0.879702\pi$
$774$	3852.00	0.178885
$775$	−50050.0	−2.31981
$776$	10650.0	0.492671
$777$	−456.000	−0.0210539
$778$	−6426.00	−0.296122
$779$	27048.0	1.24402
$780$	−5880.00	−0.269920
$781$	−38112.0	−1.74616
$782$	−2074.00	−0.0948415
$783$	−972.000	−0.0443633
$784$	−13899.0	−0.633154
$785$	−50840.0	−2.31154
$786$	5856.00	0.265746
$787$	20656.0	0.935587	0.467793	−	0.883838i	$-0.345049\pi$
$787$	20656.0	0.935587	0.467793	+	0.883838i	$0.345049\pi$
$788$	8232.00	0.372148
$789$	2796.00	0.126160
$790$	17160.0	0.772817
$791$	700.000	0.0314654
$792$	−6480.00	−0.290728
$793$	11480.0	0.514082
$794$	13212.0	0.590524
$795$	14040.0	0.626349
$796$	17150.0	0.763650
$797$	−3722.00	−0.165420	−0.0827102	−	0.996574i	$-0.526358\pi$
$797$	−3722.00	−0.165420	−0.0827102	+	0.996574i	$0.526358\pi$
$798$	552.000	0.0244870
$799$	204.000	0.00903254
$800$	−44275.0	−1.95670
$801$	9918.00	0.437497
$802$	6242.00	0.274829
$803$	49824.0	2.18960
$804$	−14700.0	−0.644812
$805$	−4880.00	−0.213661
$806$	−2548.00	−0.111352
$807$	−11652.0	−0.508265
$808$	−18150.0	−0.790241
$809$	6518.00	0.283264	0.141632	−	0.989919i	$-0.454765\pi$
$809$	6518.00	0.283264	0.141632	+	0.989919i	$0.454765\pi$
$810$	1620.00	0.0702728
$811$	−33068.0	−1.43178	−0.715891	−	0.698212i	$-0.753979\pi$
$811$	−33068.0	−1.43178	−0.715891	+	0.698212i	$0.753979\pi$
$812$	−504.000	−0.0217819
$813$	−5808.00	−0.250548
$814$	3648.00	0.157079
$815$	−13680.0	−0.587963
$816$	−2091.00	−0.0897054
$817$	−39376.0	−1.68616
$818$	5194.00	0.222010
$819$	252.000	0.0107516
$820$	41160.0	1.75289
$821$	−5328.00	−0.226490	−0.113245	−	0.993567i	$-0.536125\pi$
$821$	−5328.00	−0.226490	−0.113245	+	0.993567i	$0.536125\pi$
$822$	−7902.00	−0.335297
$823$	25874.0	1.09588	0.547941	−	0.836517i	$-0.315412\pi$
$823$	25874.0	1.09588	0.547941	+	0.836517i	$0.315412\pi$
$824$	9660.00	0.408401
$825$	−39600.0	−1.67115
$826$	−1080.00	−0.0454940
$827$	29184.0	1.22712	0.613559	−	0.789649i	$-0.289737\pi$
$827$	29184.0	1.22712	0.613559	+	0.789649i	$0.289737\pi$
$828$	7686.00	0.322593
$829$	−9394.00	−0.393567	−0.196784	−	0.980447i	$-0.563050\pi$
$829$	−9394.00	−0.393567	−0.196784	+	0.980447i	$0.563050\pi$
$830$	21040.0	0.879890
$831$	2616.00	0.109203
$832$	2338.00	0.0974226
$833$	5763.00	0.239707
$834$	−5592.00	−0.232176
$835$	−10840.0	−0.449262
$836$	30912.0	1.27884
$837$	−4914.00	−0.202930
$838$	−15956.0	−0.657746
$839$	−32062.0	−1.31931	−0.659656	−	0.751567i	$-0.729298\pi$
$839$	−32062.0	−1.31931	−0.659656	+	0.751567i	$0.729298\pi$
$840$	1800.00	0.0739356
$841$	−23093.0	−0.946861
$842$	15254.0	0.624332
$843$	9594.00	0.391975
$844$	−7448.00	−0.303757
$845$	40020.0	1.62927
$846$	108.000	0.00438903
$847$	−1946.00	−0.0789437
$848$	−9594.00	−0.388513
$849$	−5808.00	−0.234782
$850$	4675.00	0.188648
$851$	−9272.00	−0.373490
$852$	−16674.0	−0.670472
$853$	9152.00	0.367361	0.183680	−	0.982986i	$-0.441199\pi$
$853$	9152.00	0.367361	0.183680	+	0.982986i	$0.441199\pi$
$854$	−1640.00	−0.0657139
$855$	−16560.0	−0.662386
$856$	−3840.00	−0.153328
$857$	−19706.0	−0.785466	−0.392733	−	0.919653i	$-0.628470\pi$
$857$	−19706.0	−0.785466	−0.392733	+	0.919653i	$0.628470\pi$
$858$	−2016.00	−0.0802157
$859$	28684.0	1.13933	0.569666	−	0.821877i	$-0.307073\pi$
$859$	28684.0	1.13933	0.569666	+	0.821877i	$0.307073\pi$
$860$	−59920.0	−2.37588
$861$	−1764.00	−0.0698223
$862$	5538.00	0.218823
$863$	−6320.00	−0.249288	−0.124644	−	0.992202i	$-0.539779\pi$
$863$	−6320.00	−0.249288	−0.124644	+	0.992202i	$0.539779\pi$
$864$	−4347.00	−0.171167
$865$	16720.0	0.657222
$866$	−11342.0	−0.445054
$867$	867.000	0.0339618
$868$	−2548.00	−0.0996368
$869$	−41184.0	−1.60768
$870$	−2160.00	−0.0841734
$871$	−9800.00	−0.381240
$872$	−18720.0	−0.726994
$873$	6390.00	0.247731
$874$	11224.0	0.434391
$875$	6000.00	0.231814
$876$	21798.0	0.840738
$877$	49272.0	1.89715	0.948573	−	0.316558i	$-0.102527\pi$
$877$	49272.0	1.89715	0.948573	+	0.316558i	$0.102527\pi$
$878$	−982.000	−0.0377459
$879$	−17154.0	−0.658237
$880$	39360.0	1.50776
$881$	−16462.0	−0.629533	−0.314767	−	0.949169i	$-0.601926\pi$
$881$	−16462.0	−0.629533	−0.314767	+	0.949169i	$0.601926\pi$
$882$	3051.00	0.116477
$883$	21300.0	0.811780	0.405890	−	0.913922i	$-0.366962\pi$
$883$	21300.0	0.811780	0.405890	+	0.913922i	$0.366962\pi$
$884$	−1666.00	−0.0633865
$885$	32400.0	1.23064
$886$	2492.00	0.0944925
$887$	−16590.0	−0.628002	−0.314001	−	0.949423i	$-0.601670\pi$
$887$	−16590.0	−0.628002	−0.314001	+	0.949423i	$0.601670\pi$
$888$	3420.00	0.129243
$889$	656.000	0.0247486
$890$	22040.0	0.830093
$891$	−3888.00	−0.146187
$892$	15708.0	0.589622
$893$	−1104.00	−0.0413706
$894$	858.000	0.0320982
$895$	68880.0	2.57252
$896$	−2910.00	−0.108500
$897$	5124.00	0.190731
$898$	5498.00	0.204310
$899$	6552.00	0.243072
$900$	−17325.0	−0.641667
$901$	3978.00	0.147088
$902$	14112.0	0.520929
$903$	2568.00	0.0946375
$904$	−5250.00	−0.193155
$905$	65680.0	2.41246
$906$	4872.00	0.178655
$907$	−36184.0	−1.32466	−0.662332	−	0.749211i	$-0.730433\pi$
$907$	−36184.0	−1.32466	−0.662332	+	0.749211i	$0.730433\pi$
$908$	−3612.00	−0.132014
$909$	−10890.0	−0.397358
$910$	560.000	0.0203998
$911$	15626.0	0.568290	0.284145	−	0.958781i	$-0.408290\pi$
$911$	15626.0	0.568290	0.284145	+	0.958781i	$0.408290\pi$
$912$	11316.0	0.410866
$913$	−50496.0	−1.83042
$914$	−4998.00	−0.180874
$915$	49200.0	1.77760
$916$	20454.0	0.737794
$917$	3904.00	0.140590
$918$	459.000	0.0165025
$919$	36672.0	1.31632	0.658160	−	0.752878i	$-0.271335\pi$
$919$	36672.0	1.31632	0.658160	+	0.752878i	$0.271335\pi$
$920$	36600.0	1.31159
$921$	2052.00	0.0734155
$922$	7586.00	0.270967
$923$	−11116.0	−0.396411
$924$	−2016.00	−0.0717765
$925$	20900.0	0.742906
$926$	−5900.00	−0.209380
$927$	5796.00	0.205357
$928$	5796.00	0.205025
$929$	−15810.0	−0.558352	−0.279176	−	0.960240i	$-0.590061\pi$
$929$	−15810.0	−0.558352	−0.279176	+	0.960240i	$0.590061\pi$
$930$	−10920.0	−0.385033
$931$	−31188.0	−1.09790
$932$	−21798.0	−0.766113
$933$	−870.000	−0.0305279
$934$	19404.0	0.679784
$935$	−16320.0	−0.570825
$936$	−1890.00	−0.0660006
$937$	48646.0	1.69605	0.848023	−	0.529959i	$-0.177793\pi$
$937$	48646.0	1.69605	0.848023	+	0.529959i	$0.177793\pi$
$938$	1400.00	0.0487331
$939$	−33150.0	−1.15209
$940$	−1680.00	−0.0582931
$941$	43872.0	1.51986	0.759929	−	0.650006i	$-0.225234\pi$
$941$	43872.0	1.51986	0.759929	+	0.650006i	$0.225234\pi$
$942$	−7626.00	−0.263767
$943$	−35868.0	−1.23862
$944$	−22140.0	−0.763343
$945$	1080.00	0.0371771
$946$	−20544.0	−0.706071
$947$	38552.0	1.32288	0.661442	−	0.749996i	$-0.269945\pi$
$947$	38552.0	1.32288	0.661442	+	0.749996i	$0.269945\pi$
$948$	−18018.0	−0.617297
$949$	14532.0	0.497080
$950$	−25300.0	−0.864043
$951$	2976.00	0.101476
$952$	510.000	0.0173626
$953$	−52954.0	−1.79995	−0.899973	−	0.435946i	$-0.856414\pi$
$953$	−52954.0	−1.79995	−0.899973	+	0.435946i	$0.856414\pi$
$954$	2106.00	0.0714720
$955$	6800.00	0.230411
$956$	−28868.0	−0.976630
$957$	5184.00	0.175104
$958$	−2586.00	−0.0872128
$959$	−5268.00	−0.177385
$960$	10020.0	0.336869
$961$	3333.00	0.111879
$962$	1064.00	0.0356598
$963$	−2304.00	−0.0770980
$964$	14238.0	0.475700
$965$	29960.0	0.999426
$966$	−732.000	−0.0243807
$967$	−46428.0	−1.54398	−0.771988	−	0.635638i	$-0.780737\pi$
$967$	−46428.0	−1.54398	−0.771988	+	0.635638i	$0.780737\pi$
$968$	14595.0	0.484609
$969$	−4692.00	−0.155551
$970$	14200.0	0.470036
$971$	−40980.0	−1.35439	−0.677194	−	0.735804i	$-0.736804\pi$
$971$	−40980.0	−1.35439	−0.677194	+	0.735804i	$0.736804\pi$
$972$	−1701.00	−0.0561313
$973$	−3728.00	−0.122831
$974$	−10106.0	−0.332461
$975$	−11550.0	−0.379381
$976$	−33620.0	−1.10261
$977$	−10206.0	−0.334206	−0.167103	−	0.985939i	$-0.553441\pi$
$977$	−10206.0	−0.334206	−0.167103	+	0.985939i	$0.553441\pi$
$978$	−2052.00	−0.0670917
$979$	−52896.0	−1.72683
$980$	−47460.0	−1.54699
$981$	−11232.0	−0.365556
$982$	76.0000	0.00246971
$983$	−44934.0	−1.45796	−0.728979	−	0.684536i	$-0.760005\pi$
$983$	−44934.0	−1.45796	−0.728979	+	0.684536i	$0.760005\pi$
$984$	13230.0	0.428615
$985$	23520.0	0.760822
$986$	−612.000	−0.0197668
$987$	72.0000	0.00232197
$988$	9016.00	0.290321
$989$	52216.0	1.67884
$990$	−8640.00	−0.277371
$991$	20526.0	0.657951	0.328976	−	0.944338i	$-0.393297\pi$
$991$	20526.0	0.657951	0.328976	+	0.944338i	$0.393297\pi$
$992$	29302.0	0.937842
$993$	8580.00	0.274197
$994$	1588.00	0.0506723
$995$	49000.0	1.56121
$996$	−22092.0	−0.702823
$997$	29260.0	0.929462	0.464731	−	0.885452i	$-0.346151\pi$
$997$	29260.0	0.929462	0.464731	+	0.885452i	$0.346151\pi$
$998$	−8096.00	−0.256788
$999$	2052.00	0.0649874

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	51.4.a.b.1.1	✓	1
3.2	odd	2		153.4.a.c.1.1		1
4.3	odd	2		816.4.a.a.1.1		1
5.4	even	2		1275.4.a.e.1.1		1
7.6	odd	2		2499.4.a.d.1.1		1
12.11	even	2		2448.4.a.r.1.1		1
17.16	even	2		867.4.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
51.4.a.b.1.1	✓	1	1.1	even	1	trivial
153.4.a.c.1.1		1	3.2	odd	2
816.4.a.a.1.1		1	4.3	odd	2
867.4.a.c.1.1		1	17.16	even	2
1275.4.a.e.1.1		1	5.4	even	2
2448.4.a.r.1.1		1	12.11	even	2
2499.4.a.d.1.1		1	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 51 = 3 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	51.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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