LMFDB - Embedded newform 7.4.a.a.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7,4,Mod(1,7)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("7.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$N$	$=$	$7$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	7.a (trivial)

Newform invariants

comment:select newform

sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)

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

Self dual:	yes
Analytic conductor:	$0.413013370040$
Analytic rank:	$0$
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	$\chi$	$=$	7.1

$q$ -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} -2.00000 q^{3} -7.00000 q^{4} +16.0000 q^{5} +2.00000 q^{6} -7.00000 q^{7} +15.0000 q^{8} -23.0000 q^{9} -16.0000 q^{10} -8.00000 q^{11} +14.0000 q^{12} +28.0000 q^{13} +7.00000 q^{14} -32.0000 q^{15} +41.0000 q^{16} +54.0000 q^{17} +23.0000 q^{18} -110.000 q^{19} -112.000 q^{20} +14.0000 q^{21} +8.00000 q^{22} +48.0000 q^{23} -30.0000 q^{24} +131.000 q^{25} -28.0000 q^{26} +100.000 q^{27} +49.0000 q^{28} -110.000 q^{29} +32.0000 q^{30} +12.0000 q^{31} -161.000 q^{32} +16.0000 q^{33} -54.0000 q^{34} -112.000 q^{35} +161.000 q^{36} -246.000 q^{37} +110.000 q^{38} -56.0000 q^{39} +240.000 q^{40} +182.000 q^{41} -14.0000 q^{42} +128.000 q^{43} +56.0000 q^{44} -368.000 q^{45} -48.0000 q^{46} +324.000 q^{47} -82.0000 q^{48} +49.0000 q^{49} -131.000 q^{50} -108.000 q^{51} -196.000 q^{52} -162.000 q^{53} -100.000 q^{54} -128.000 q^{55} -105.000 q^{56} +220.000 q^{57} +110.000 q^{58} +810.000 q^{59} +224.000 q^{60} -488.000 q^{61} -12.0000 q^{62} +161.000 q^{63} -167.000 q^{64} +448.000 q^{65} -16.0000 q^{66} +244.000 q^{67} -378.000 q^{68} -96.0000 q^{69} +112.000 q^{70} -768.000 q^{71} -345.000 q^{72} -702.000 q^{73} +246.000 q^{74} -262.000 q^{75} +770.000 q^{76} +56.0000 q^{77} +56.0000 q^{78} +440.000 q^{79} +656.000 q^{80} +421.000 q^{81} -182.000 q^{82} -1302.00 q^{83} -98.0000 q^{84} +864.000 q^{85} -128.000 q^{86} +220.000 q^{87} -120.000 q^{88} +730.000 q^{89} +368.000 q^{90} -196.000 q^{91} -336.000 q^{92} -24.0000 q^{93} -324.000 q^{94} -1760.00 q^{95} +322.000 q^{96} +294.000 q^{97} -49.0000 q^{98} +184.000 q^{99} +O(q^{100})

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	$0.556567\pi$
$3$	−2.00000	−0.384900	−0.192450	−	0.981307i	$-0.561643\pi$
$3$	−2.00000	−0.384900	−0.192450	+	0.981307i	$0.561643\pi$
$4$	−7.00000	−0.875000
$5$	16.0000	1.43108	0.715542	−	0.698570i	$-0.246180\pi$
$5$	16.0000	1.43108	0.715542	+	0.698570i	$0.246180\pi$
$6$	2.00000	0.136083
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	15.0000	0.662913
$9$	−23.0000	−0.851852
$10$	−16.0000	−0.505964
$11$	−8.00000	−0.219281	−0.109640	−	0.993971i	$-0.534970\pi$
$11$	−8.00000	−0.219281	−0.109640	+	0.993971i	$0.534970\pi$
$12$	14.0000	0.336788
$13$	28.0000	0.597369	0.298685	−	0.954352i	$-0.403452\pi$
$13$	28.0000	0.597369	0.298685	+	0.954352i	$0.403452\pi$
$14$	7.00000	0.133631
$15$	−32.0000	−0.550824
$16$	41.0000	0.640625
$17$	54.0000	0.770407	0.385204	−	0.922832i	$-0.374131\pi$
$17$	54.0000	0.770407	0.385204	+	0.922832i	$0.374131\pi$
$18$	23.0000	0.301175
$19$	−110.000	−1.32820	−0.664098	−	0.747645i	$-0.731184\pi$
$19$	−110.000	−1.32820	−0.664098	+	0.747645i	$0.731184\pi$
$20$	−112.000	−1.25220
$21$	14.0000	0.145479
$22$	8.00000	0.0775275
$23$	48.0000	0.435161	0.217580	−	0.976042i	$-0.430184\pi$
$23$	48.0000	0.435161	0.217580	+	0.976042i	$0.430184\pi$
$24$	−30.0000	−0.255155
$25$	131.000	1.04800
$26$	−28.0000	−0.211202
$27$	100.000	0.712778
$28$	49.0000	0.330719
$29$	−110.000	−0.704362	−0.352181	−	0.935932i	$-0.614560\pi$
$29$	−110.000	−0.704362	−0.352181	+	0.935932i	$0.614560\pi$
$30$	32.0000	0.194746
$31$	12.0000	0.0695246	0.0347623	−	0.999396i	$-0.488933\pi$
$31$	12.0000	0.0695246	0.0347623	+	0.999396i	$0.488933\pi$
$32$	−161.000	−0.889408
$33$	16.0000	0.0844013
$34$	−54.0000	−0.272380
$35$	−112.000	−0.540899
$36$	161.000	0.745370
$37$	−246.000	−1.09303	−0.546516	−	0.837449i	$-0.684046\pi$
$37$	−246.000	−1.09303	−0.546516	+	0.837449i	$0.684046\pi$
$38$	110.000	0.469588
$39$	−56.0000	−0.229928
$40$	240.000	0.948683
$41$	182.000	0.693259	0.346630	−	0.938002i	$-0.387326\pi$
$41$	182.000	0.693259	0.346630	+	0.938002i	$0.387326\pi$
$42$	−14.0000	−0.0514344
$43$	128.000	0.453949	0.226975	−	0.973901i	$-0.427117\pi$
$43$	128.000	0.453949	0.226975	+	0.973901i	$0.427117\pi$
$44$	56.0000	0.191871
$45$	−368.000	−1.21907
$46$	−48.0000	−0.153852
$47$	324.000	1.00554	0.502769	−	0.864421i	$-0.332315\pi$
$47$	324.000	1.00554	0.502769	+	0.864421i	$0.332315\pi$
$48$	−82.0000	−0.246577
$49$	49.0000	0.142857
$50$	−131.000	−0.370524
$51$	−108.000	−0.296530
$52$	−196.000	−0.522698
$53$	−162.000	−0.419857	−0.209928	−	0.977717i	$-0.567323\pi$
$53$	−162.000	−0.419857	−0.209928	+	0.977717i	$0.567323\pi$
$54$	−100.000	−0.252005
$55$	−128.000	−0.313809
$56$	−105.000	−0.250557
$57$	220.000	0.511223
$58$	110.000	0.249029
$59$	810.000	1.78734	0.893670	−	0.448725i	$-0.148122\pi$
$59$	810.000	1.78734	0.893670	+	0.448725i	$0.148122\pi$
$60$	224.000	0.481971
$61$	−488.000	−1.02430	−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−488.000	−1.02430	−0.512148	+	0.858898i	$0.671150\pi$
$62$	−12.0000	−0.0245807
$63$	161.000	0.321970
$64$	−167.000	−0.326172
$65$	448.000	0.854886
$66$	−16.0000	−0.0298404
$67$	244.000	0.444916	0.222458	−	0.974942i	$-0.428592\pi$
$67$	244.000	0.444916	0.222458	+	0.974942i	$0.428592\pi$
$68$	−378.000	−0.674106
$69$	−96.0000	−0.167493
$70$	112.000	0.191237
$71$	−768.000	−1.28373	−0.641865	−	0.766818i	$-0.721839\pi$
$71$	−768.000	−1.28373	−0.641865	+	0.766818i	$0.721839\pi$
$72$	−345.000	−0.564703
$73$	−702.000	−1.12552	−0.562759	−	0.826621i	$-0.690260\pi$
$73$	−702.000	−1.12552	−0.562759	+	0.826621i	$0.690260\pi$
$74$	246.000	0.386445
$75$	−262.000	−0.403375
$76$	770.000	1.16217
$77$	56.0000	0.0828804
$78$	56.0000	0.0812917
$79$	440.000	0.626631	0.313316	−	0.949649i	$-0.398560\pi$
$79$	440.000	0.626631	0.313316	+	0.949649i	$0.398560\pi$
$80$	656.000	0.916788
$81$	421.000	0.577503
$82$	−182.000	−0.245104
$83$	−1302.00	−1.72184	−0.860922	−	0.508737i	$-0.830113\pi$
$83$	−1302.00	−1.72184	−0.860922	+	0.508737i	$0.830113\pi$
$84$	−98.0000	−0.127294
$85$	864.000	1.10252
$86$	−128.000	−0.160495
$87$	220.000	0.271109
$88$	−120.000	−0.145364
$89$	730.000	0.869436	0.434718	−	0.900567i	$-0.356848\pi$
$89$	730.000	0.869436	0.434718	+	0.900567i	$0.356848\pi$
$90$	368.000	0.431007
$91$	−196.000	−0.225784
$92$	−336.000	−0.380765
$93$	−24.0000	−0.0267600
$94$	−324.000	−0.355511
$95$	−1760.00	−1.90076
$96$	322.000	0.342333
$97$	294.000	0.307744	0.153872	−	0.988091i	$-0.450826\pi$
$97$	294.000	0.307744	0.153872	+	0.988091i	$0.450826\pi$
$98$	−49.0000	−0.0505076
$99$	184.000	0.186795
$100$	−917.000	−0.917000
$101$	−688.000	−0.677808	−0.338904	−	0.940821i	$-0.610056\pi$
$101$	−688.000	−0.677808	−0.338904	+	0.940821i	$0.610056\pi$
$102$	108.000	0.104839
$103$	1388.00	1.32780	0.663901	−	0.747820i	$-0.268899\pi$
$103$	1388.00	1.32780	0.663901	+	0.747820i	$0.268899\pi$
$104$	420.000	0.396004
$105$	224.000	0.208192
$106$	162.000	0.148442
$107$	244.000	0.220452	0.110226	−	0.993907i	$-0.464843\pi$
$107$	244.000	0.220452	0.110226	+	0.993907i	$0.464843\pi$
$108$	−700.000	−0.623681
$109$	90.0000	0.0790866	0.0395433	−	0.999218i	$-0.487410\pi$
$109$	90.0000	0.0790866	0.0395433	+	0.999218i	$0.487410\pi$
$110$	128.000	0.110948
$111$	492.000	0.420708
$112$	−287.000	−0.242133
$113$	1318.00	1.09723	0.548615	−	0.836075i	$-0.315155\pi$
$113$	1318.00	1.09723	0.548615	+	0.836075i	$0.315155\pi$
$114$	−220.000	−0.180745
$115$	768.000	0.622751
$116$	770.000	0.616316
$117$	−644.000	−0.508870
$118$	−810.000	−0.631920
$119$	−378.000	−0.291187
$120$	−480.000	−0.365148
$121$	−1267.00	−0.951916
$122$	488.000	0.362143
$123$	−364.000	−0.266836
$124$	−84.0000	−0.0608341
$125$	96.0000	0.0686920
$126$	−161.000	−0.113833
$127$	−1776.00	−1.24090	−0.620451	−	0.784245i	$-0.713050\pi$
$127$	−1776.00	−1.24090	−0.620451	+	0.784245i	$0.713050\pi$
$128$	1455.00	1.00473
$129$	−256.000	−0.174725
$130$	−448.000	−0.302248
$131$	−1118.00	−0.745650	−0.372825	−	0.927902i	$-0.621611\pi$
$131$	−1118.00	−0.745650	−0.372825	+	0.927902i	$0.621611\pi$
$132$	−112.000	−0.0738511
$133$	770.000	0.502011
$134$	−244.000	−0.157301
$135$	1600.00	1.02004
$136$	810.000	0.510713
$137$	2274.00	1.41811	0.709054	−	0.705154i	$-0.249122\pi$
$137$	2274.00	1.41811	0.709054	+	0.705154i	$0.249122\pi$
$138$	96.0000	0.0592178
$139$	−210.000	−0.128144	−0.0640718	−	0.997945i	$-0.520409\pi$
$139$	−210.000	−0.128144	−0.0640718	+	0.997945i	$0.520409\pi$
$140$	784.000	0.473286
$141$	−648.000	−0.387032
$142$	768.000	0.453867
$143$	−224.000	−0.130992
$144$	−943.000	−0.545718
$145$	−1760.00	−1.00800
$146$	702.000	0.397931
$147$	−98.0000	−0.0549857
$148$	1722.00	0.956402
$149$	−2010.00	−1.10514	−0.552569	−	0.833467i	$-0.686352\pi$
$149$	−2010.00	−1.10514	−0.552569	+	0.833467i	$0.686352\pi$
$150$	262.000	0.142615
$151$	1112.00	0.599293	0.299647	−	0.954050i	$-0.403131\pi$
$151$	1112.00	0.599293	0.299647	+	0.954050i	$0.403131\pi$
$152$	−1650.00	−0.880478
$153$	−1242.00	−0.656273
$154$	−56.0000	−0.0293027
$155$	192.000	0.0994956
$156$	392.000	0.201187
$157$	124.000	0.0630336	0.0315168	−	0.999503i	$-0.489966\pi$
$157$	124.000	0.0630336	0.0315168	+	0.999503i	$0.489966\pi$
$158$	−440.000	−0.221548
$159$	324.000	0.161603
$160$	−2576.00	−1.27282
$161$	−336.000	−0.164475
$162$	−421.000	−0.204178
$163$	2008.00	0.964900	0.482450	−	0.875924i	$-0.339747\pi$
$163$	2008.00	0.964900	0.482450	+	0.875924i	$0.339747\pi$
$164$	−1274.00	−0.606602
$165$	256.000	0.120785
$166$	1302.00	0.608764
$167$	2884.00	1.33635	0.668176	−	0.744004i	$-0.267076\pi$
$167$	2884.00	1.33635	0.668176	+	0.744004i	$0.267076\pi$
$168$	210.000	0.0964396
$169$	−1413.00	−0.643150
$170$	−864.000	−0.389799
$171$	2530.00	1.13143
$172$	−896.000	−0.397206
$173$	2228.00	0.979143	0.489571	−	0.871963i	$-0.337153\pi$
$173$	2228.00	0.979143	0.489571	+	0.871963i	$0.337153\pi$
$174$	−220.000	−0.0958515
$175$	−917.000	−0.396107
$176$	−328.000	−0.140477
$177$	−1620.00	−0.687947
$178$	−730.000	−0.307392
$179$	−820.000	−0.342400	−0.171200	−	0.985236i	$-0.554764\pi$
$179$	−820.000	−0.342400	−0.171200	+	0.985236i	$0.554764\pi$
$180$	2576.00	1.06669
$181$	3892.00	1.59829	0.799144	−	0.601140i	$-0.205287\pi$
$181$	3892.00	1.59829	0.799144	+	0.601140i	$0.205287\pi$
$182$	196.000	0.0798268
$183$	976.000	0.394251
$184$	720.000	0.288473
$185$	−3936.00	−1.56422
$186$	24.0000	0.00946110
$187$	−432.000	−0.168936
$188$	−2268.00	−0.879845
$189$	−700.000	−0.269405
$190$	1760.00	0.672020
$191$	−5048.00	−1.91236	−0.956179	−	0.292782i	$-0.905419\pi$
$191$	−5048.00	−1.91236	−0.956179	+	0.292782i	$0.905419\pi$
$192$	334.000	0.125544
$193$	−2962.00	−1.10471	−0.552356	−	0.833608i	$-0.686271\pi$
$193$	−2962.00	−1.10471	−0.552356	+	0.833608i	$0.686271\pi$
$194$	−294.000	−0.108804
$195$	−896.000	−0.329046
$196$	−343.000	−0.125000
$197$	3334.00	1.20577	0.602887	−	0.797826i	$-0.294017\pi$
$197$	3334.00	1.20577	0.602887	+	0.797826i	$0.294017\pi$
$198$	−184.000	−0.0660420
$199$	1860.00	0.662572	0.331286	−	0.943530i	$-0.392517\pi$
$199$	1860.00	0.662572	0.331286	+	0.943530i	$0.392517\pi$
$200$	1965.00	0.694732
$201$	−488.000	−0.171248
$202$	688.000	0.239641
$203$	770.000	0.266224
$204$	756.000	0.259464
$205$	2912.00	0.992112
$206$	−1388.00	−0.469449
$207$	−1104.00	−0.370692
$208$	1148.00	0.382690
$209$	880.000	0.291248
$210$	−224.000	−0.0736070
$211$	−4268.00	−1.39252	−0.696259	−	0.717791i	$-0.745153\pi$
$211$	−4268.00	−1.39252	−0.696259	+	0.717791i	$0.745153\pi$
$212$	1134.00	0.367375
$213$	1536.00	0.494108
$214$	−244.000	−0.0779416
$215$	2048.00	0.649639
$216$	1500.00	0.472510
$217$	−84.0000	−0.0262778
$218$	−90.0000	−0.0279613
$219$	1404.00	0.433212
$220$	896.000	0.274583
$221$	1512.00	0.460218
$222$	−492.000	−0.148743
$223$	−5432.00	−1.63118	−0.815591	−	0.578629i	$-0.803588\pi$
$223$	−5432.00	−1.63118	−0.815591	+	0.578629i	$0.803588\pi$
$224$	1127.00	0.336165
$225$	−3013.00	−0.892741
$226$	−1318.00	−0.387929
$227$	−2046.00	−0.598228	−0.299114	−	0.954217i	$-0.596691\pi$
$227$	−2046.00	−0.598228	−0.299114	+	0.954217i	$0.596691\pi$
$228$	−1540.00	−0.447320
$229$	−2980.00	−0.859930	−0.429965	−	0.902846i	$-0.641474\pi$
$229$	−2980.00	−0.859930	−0.429965	+	0.902846i	$0.641474\pi$
$230$	−768.000	−0.220176
$231$	−112.000	−0.0319007
$232$	−1650.00	−0.466930
$233$	4458.00	1.25345	0.626724	−	0.779241i	$-0.284395\pi$
$233$	4458.00	1.25345	0.626724	+	0.779241i	$0.284395\pi$
$234$	644.000	0.179913
$235$	5184.00	1.43901
$236$	−5670.00	−1.56392
$237$	−880.000	−0.241190
$238$	378.000	0.102950
$239$	4440.00	1.20167	0.600836	−	0.799372i	$-0.294834\pi$
$239$	4440.00	1.20167	0.600836	+	0.799372i	$0.294834\pi$
$240$	−1312.00	−0.352872
$241$	3302.00	0.882575	0.441287	−	0.897366i	$-0.354522\pi$
$241$	3302.00	0.882575	0.441287	+	0.897366i	$0.354522\pi$
$242$	1267.00	0.336553
$243$	−3542.00	−0.935059
$244$	3416.00	0.896258
$245$	784.000	0.204441
$246$	364.000	0.0943406
$247$	−3080.00	−0.793424
$248$	180.000	0.0460888
$249$	2604.00	0.662738
$250$	−96.0000	−0.0242863
$251$	1582.00	0.397829	0.198914	−	0.980017i	$-0.436258\pi$
$251$	1582.00	0.397829	0.198914	+	0.980017i	$0.436258\pi$
$252$	−1127.00	−0.281724
$253$	−384.000	−0.0954224
$254$	1776.00	0.438725
$255$	−1728.00	−0.424359
$256$	−119.000	−0.0290527
$257$	2354.00	0.571356	0.285678	−	0.958326i	$-0.407781\pi$
$257$	2354.00	0.571356	0.285678	+	0.958326i	$0.407781\pi$
$258$	256.000	0.0617747
$259$	1722.00	0.413127
$260$	−3136.00	−0.748025
$261$	2530.00	0.600012
$262$	1118.00	0.263627
$263$	−3872.00	−0.907824	−0.453912	−	0.891046i	$-0.649972\pi$
$263$	−3872.00	−0.907824	−0.453912	+	0.891046i	$0.649972\pi$
$264$	240.000	0.0559507
$265$	−2592.00	−0.600850
$266$	−770.000	−0.177488
$267$	−1460.00	−0.334646
$268$	−1708.00	−0.389301
$269$	180.000	0.0407985	0.0203992	−	0.999792i	$-0.493506\pi$
$269$	180.000	0.0407985	0.0203992	+	0.999792i	$0.493506\pi$
$270$	−1600.00	−0.360640
$271$	2032.00	0.455480	0.227740	−	0.973722i	$-0.426866\pi$
$271$	2032.00	0.455480	0.227740	+	0.973722i	$0.426866\pi$
$272$	2214.00	0.493542
$273$	392.000	0.0869045
$274$	−2274.00	−0.501377
$275$	−1048.00	−0.229806
$276$	672.000	0.146557
$277$	−5426.00	−1.17696	−0.588478	−	0.808513i	$-0.700273\pi$
$277$	−5426.00	−1.17696	−0.588478	+	0.808513i	$0.700273\pi$
$278$	210.000	0.0453056
$279$	−276.000	−0.0592247
$280$	−1680.00	−0.358569
$281$	842.000	0.178753	0.0893764	−	0.995998i	$-0.471513\pi$
$281$	842.000	0.178753	0.0893764	+	0.995998i	$0.471513\pi$
$282$	648.000	0.136836
$283$	−3782.00	−0.794405	−0.397202	−	0.917731i	$-0.630019\pi$
$283$	−3782.00	−0.794405	−0.397202	+	0.917731i	$0.630019\pi$
$284$	5376.00	1.12326
$285$	3520.00	0.731603
$286$	224.000	0.0463126
$287$	−1274.00	−0.262027
$288$	3703.00	0.757644
$289$	−1997.00	−0.406473
$290$	1760.00	0.356382
$291$	−588.000	−0.118451
$292$	4914.00	0.984829
$293$	−4312.00	−0.859760	−0.429880	−	0.902886i	$-0.641444\pi$
$293$	−4312.00	−0.859760	−0.429880	+	0.902886i	$0.641444\pi$
$294$	98.0000	0.0194404
$295$	12960.0	2.55783
$296$	−3690.00	−0.724584
$297$	−800.000	−0.156299
$298$	2010.00	0.390725
$299$	1344.00	0.259952
$300$	1834.00	0.352953
$301$	−896.000	−0.171577
$302$	−1112.00	−0.211882
$303$	1376.00	0.260888
$304$	−4510.00	−0.850876
$305$	−7808.00	−1.46585
$306$	1242.00	0.232027
$307$	2674.00	0.497112	0.248556	−	0.968618i	$-0.420044\pi$
$307$	2674.00	0.497112	0.248556	+	0.968618i	$0.420044\pi$
$308$	−392.000	−0.0725204
$309$	−2776.00	−0.511072
$310$	−192.000	−0.0351770
$311$	−3768.00	−0.687021	−0.343511	−	0.939149i	$-0.611616\pi$
$311$	−3768.00	−0.687021	−0.343511	+	0.939149i	$0.611616\pi$
$312$	−840.000	−0.152422
$313$	2438.00	0.440268	0.220134	−	0.975470i	$-0.429351\pi$
$313$	2438.00	0.440268	0.220134	+	0.975470i	$0.429351\pi$
$314$	−124.000	−0.0222857
$315$	2576.00	0.460766
$316$	−3080.00	−0.548302
$317$	−3186.00	−0.564491	−0.282245	−	0.959342i	$-0.591079\pi$
$317$	−3186.00	−0.564491	−0.282245	+	0.959342i	$0.591079\pi$
$318$	−324.000	−0.0571353
$319$	880.000	0.154453
$320$	−2672.00	−0.466779
$321$	−488.000	−0.0848520
$322$	336.000	0.0581508
$323$	−5940.00	−1.02325
$324$	−2947.00	−0.505316
$325$	3668.00	0.626043
$326$	−2008.00	−0.341144
$327$	−180.000	−0.0304404
$328$	2730.00	0.459570
$329$	−2268.00	−0.380057
$330$	−256.000	−0.0427040
$331$	8672.00	1.44005	0.720025	−	0.693949i	$-0.244131\pi$
$331$	8672.00	1.44005	0.720025	+	0.693949i	$0.244131\pi$
$332$	9114.00	1.50661
$333$	5658.00	0.931101
$334$	−2884.00	−0.472471
$335$	3904.00	0.636711
$336$	574.000	0.0931972
$337$	814.000	0.131577	0.0657884	−	0.997834i	$-0.479044\pi$
$337$	814.000	0.131577	0.0657884	+	0.997834i	$0.479044\pi$
$338$	1413.00	0.227388
$339$	−2636.00	−0.422324
$340$	−6048.00	−0.964703
$341$	−96.0000	−0.0152454
$342$	−2530.00	−0.400020
$343$	−343.000	−0.0539949
$344$	1920.00	0.300929
$345$	−1536.00	−0.239697
$346$	−2228.00	−0.346179
$347$	9344.00	1.44557	0.722784	−	0.691074i	$-0.242862\pi$
$347$	9344.00	1.44557	0.722784	+	0.691074i	$0.242862\pi$
$348$	−1540.00	−0.237220
$349$	−5180.00	−0.794496	−0.397248	−	0.917711i	$-0.630035\pi$
$349$	−5180.00	−0.794496	−0.397248	+	0.917711i	$0.630035\pi$
$350$	917.000	0.140045
$351$	2800.00	0.425792
$352$	1288.00	0.195030
$353$	12178.0	1.83617	0.918087	−	0.396379i	$-0.129733\pi$
$353$	12178.0	1.83617	0.918087	+	0.396379i	$0.129733\pi$
$354$	1620.00	0.243226
$355$	−12288.0	−1.83712
$356$	−5110.00	−0.760757
$357$	756.000	0.112078
$358$	820.000	0.121057
$359$	440.000	0.0646861	0.0323431	−	0.999477i	$-0.489703\pi$
$359$	440.000	0.0646861	0.0323431	+	0.999477i	$0.489703\pi$
$360$	−5520.00	−0.808138
$361$	5241.00	0.764106
$362$	−3892.00	−0.565080
$363$	2534.00	0.366393
$364$	1372.00	0.197561
$365$	−11232.0	−1.61071
$366$	−976.000	−0.139389
$367$	−9816.00	−1.39616	−0.698080	−	0.716019i	$-0.745962\pi$
$367$	−9816.00	−1.39616	−0.698080	+	0.716019i	$0.745962\pi$
$368$	1968.00	0.278775
$369$	−4186.00	−0.590554
$370$	3936.00	0.553035
$371$	1134.00	0.158691
$372$	168.000	0.0234150
$373$	−442.000	−0.0613563	−0.0306781	−	0.999529i	$-0.509767\pi$
$373$	−442.000	−0.0613563	−0.0306781	+	0.999529i	$0.509767\pi$
$374$	432.000	0.0597278
$375$	−192.000	−0.0264396
$376$	4860.00	0.666583
$377$	−3080.00	−0.420764
$378$	700.000	0.0952490
$379$	−3960.00	−0.536706	−0.268353	−	0.963321i	$-0.586479\pi$
$379$	−3960.00	−0.536706	−0.268353	+	0.963321i	$0.586479\pi$
$380$	12320.0	1.66316
$381$	3552.00	0.477623
$382$	5048.00	0.676121
$383$	6708.00	0.894942	0.447471	−	0.894298i	$-0.352325\pi$
$383$	6708.00	0.894942	0.447471	+	0.894298i	$0.352325\pi$
$384$	−2910.00	−0.386720
$385$	896.000	0.118609
$386$	2962.00	0.390575
$387$	−2944.00	−0.386697
$388$	−2058.00	−0.269276
$389$	−13350.0	−1.74003	−0.870015	−	0.493025i	$-0.835891\pi$
$389$	−13350.0	−1.74003	−0.870015	+	0.493025i	$0.835891\pi$
$390$	896.000	0.116335
$391$	2592.00	0.335251
$392$	735.000	0.0947018
$393$	2236.00	0.287001
$394$	−3334.00	−0.426306
$395$	7040.00	0.896762
$396$	−1288.00	−0.163446
$397$	−1356.00	−0.171425	−0.0857125	−	0.996320i	$-0.527317\pi$
$397$	−1356.00	−0.171425	−0.0857125	+	0.996320i	$0.527317\pi$
$398$	−1860.00	−0.234255
$399$	−1540.00	−0.193224
$400$	5371.00	0.671375
$401$	6222.00	0.774843	0.387421	−	0.921903i	$-0.373366\pi$
$401$	6222.00	0.774843	0.387421	+	0.921903i	$0.373366\pi$
$402$	488.000	0.0605453
$403$	336.000	0.0415319
$404$	4816.00	0.593082
$405$	6736.00	0.826456
$406$	−770.000	−0.0941243
$407$	1968.00	0.239681
$408$	−1620.00	−0.196573
$409$	5150.00	0.622619	0.311309	−	0.950309i	$-0.399232\pi$
$409$	5150.00	0.622619	0.311309	+	0.950309i	$0.399232\pi$
$410$	−2912.00	−0.350764
$411$	−4548.00	−0.545830
$412$	−9716.00	−1.16183
$413$	−5670.00	−0.675551
$414$	1104.00	0.131060
$415$	−20832.0	−2.46410
$416$	−4508.00	−0.531305
$417$	420.000	0.0493225
$418$	−880.000	−0.102972
$419$	2310.00	0.269334	0.134667	−	0.990891i	$-0.457004\pi$
$419$	2310.00	0.269334	0.134667	+	0.990891i	$0.457004\pi$
$420$	−1568.00	−0.182168
$421$	1262.00	0.146095	0.0730476	−	0.997328i	$-0.476727\pi$
$421$	1262.00	0.146095	0.0730476	+	0.997328i	$0.476727\pi$
$422$	4268.00	0.492329
$423$	−7452.00	−0.856569
$424$	−2430.00	−0.278328
$425$	7074.00	0.807387
$426$	−1536.00	−0.174694
$427$	3416.00	0.387147
$428$	−1708.00	−0.192896
$429$	448.000	0.0504188
$430$	−2048.00	−0.229682
$431$	−4488.00	−0.501576	−0.250788	−	0.968042i	$-0.580690\pi$
$431$	−4488.00	−0.501576	−0.250788	+	0.968042i	$0.580690\pi$
$432$	4100.00	0.456623
$433$	17038.0	1.89098	0.945490	−	0.325652i	$-0.105584\pi$
$433$	17038.0	1.89098	0.945490	+	0.325652i	$0.105584\pi$
$434$	84.0000	0.00929062
$435$	3520.00	0.387979
$436$	−630.000	−0.0692008
$437$	−5280.00	−0.577979
$438$	−1404.00	−0.153164
$439$	16200.0	1.76124	0.880619	−	0.473824i	$-0.157127\pi$
$439$	16200.0	1.76124	0.880619	+	0.473824i	$0.157127\pi$
$440$	−1920.00	−0.208028
$441$	−1127.00	−0.121693
$442$	−1512.00	−0.162712
$443$	−8772.00	−0.940791	−0.470395	−	0.882456i	$-0.655889\pi$
$443$	−8772.00	−0.940791	−0.470395	+	0.882456i	$0.655889\pi$
$444$	−3444.00	−0.368119
$445$	11680.0	1.24424
$446$	5432.00	0.576710
$447$	4020.00	0.425368
$448$	1169.00	0.123281
$449$	2130.00	0.223877	0.111939	−	0.993715i	$-0.464294\pi$
$449$	2130.00	0.223877	0.111939	+	0.993715i	$0.464294\pi$
$450$	3013.00	0.315632
$451$	−1456.00	−0.152019
$452$	−9226.00	−0.960076
$453$	−2224.00	−0.230668
$454$	2046.00	0.211506
$455$	−3136.00	−0.323116
$456$	3300.00	0.338896
$457$	10534.0	1.07825	0.539124	−	0.842226i	$-0.318755\pi$
$457$	10534.0	1.07825	0.539124	+	0.842226i	$0.318755\pi$
$458$	2980.00	0.304031
$459$	5400.00	0.549129
$460$	−5376.00	−0.544907
$461$	−9268.00	−0.936342	−0.468171	−	0.883638i	$-0.655087\pi$
$461$	−9268.00	−0.936342	−0.468171	+	0.883638i	$0.655087\pi$
$462$	112.000	0.0112786
$463$	−9392.00	−0.942728	−0.471364	−	0.881939i	$-0.656238\pi$
$463$	−9392.00	−0.942728	−0.471364	+	0.881939i	$0.656238\pi$
$464$	−4510.00	−0.451232
$465$	−384.000	−0.0382959
$466$	−4458.00	−0.443161
$467$	−10806.0	−1.07075	−0.535377	−	0.844613i	$-0.679830\pi$
$467$	−10806.0	−1.07075	−0.535377	+	0.844613i	$0.679830\pi$
$468$	4508.00	0.445261
$469$	−1708.00	−0.168162
$470$	−5184.00	−0.508766
$471$	−248.000	−0.0242616
$472$	12150.0	1.18485
$473$	−1024.00	−0.0995424
$474$	880.000	0.0852737
$475$	−14410.0	−1.39195
$476$	2646.00	0.254788
$477$	3726.00	0.357656
$478$	−4440.00	−0.424855
$479$	4940.00	0.471220	0.235610	−	0.971848i	$-0.424291\pi$
$479$	4940.00	0.471220	0.235610	+	0.971848i	$0.424291\pi$
$480$	5152.00	0.489907
$481$	−6888.00	−0.652943
$482$	−3302.00	−0.312037
$483$	672.000	0.0633065
$484$	8869.00	0.832926
$485$	4704.00	0.440407
$486$	3542.00	0.330593
$487$	−5216.00	−0.485338	−0.242669	−	0.970109i	$-0.578023\pi$
$487$	−5216.00	−0.485338	−0.242669	+	0.970109i	$0.578023\pi$
$488$	−7320.00	−0.679018
$489$	−4016.00	−0.371390
$490$	−784.000	−0.0722806
$491$	4412.00	0.405521	0.202760	−	0.979228i	$-0.435009\pi$
$491$	4412.00	0.405521	0.202760	+	0.979228i	$0.435009\pi$
$492$	2548.00	0.233481
$493$	−5940.00	−0.542645
$494$	3080.00	0.280518
$495$	2944.00	0.267319
$496$	492.000	0.0445392
$497$	5376.00	0.485204
$498$	−2604.00	−0.234313
$499$	19060.0	1.70991	0.854953	−	0.518706i	$-0.173586\pi$
$499$	19060.0	1.70991	0.854953	+	0.518706i	$0.173586\pi$
$500$	−672.000	−0.0601055
$501$	−5768.00	−0.514362
$502$	−1582.00	−0.140654
$503$	12768.0	1.13180	0.565902	−	0.824473i	$-0.308528\pi$
$503$	12768.0	1.13180	0.565902	+	0.824473i	$0.308528\pi$
$504$	2415.00	0.213438
$505$	−11008.0	−0.969999
$506$	384.000	0.0337369
$507$	2826.00	0.247548
$508$	12432.0	1.08579
$509$	−5500.00	−0.478945	−0.239473	−	0.970903i	$-0.576975\pi$
$509$	−5500.00	−0.478945	−0.239473	+	0.970903i	$0.576975\pi$
$510$	1728.00	0.150034
$511$	4914.00	0.425406
$512$	−11521.0	−0.994455
$513$	−11000.0	−0.946709
$514$	−2354.00	−0.202005
$515$	22208.0	1.90020
$516$	1792.00	0.152884
$517$	−2592.00	−0.220495
$518$	−1722.00	−0.146062
$519$	−4456.00	−0.376872
$520$	6720.00	0.566714
$521$	−7338.00	−0.617051	−0.308526	−	0.951216i	$-0.599836\pi$
$521$	−7338.00	−0.617051	−0.308526	+	0.951216i	$0.599836\pi$
$522$	−2530.00	−0.212136
$523$	−17582.0	−1.46999	−0.734997	−	0.678070i	$-0.762817\pi$
$523$	−17582.0	−1.46999	−0.734997	+	0.678070i	$0.762817\pi$
$524$	7826.00	0.652444
$525$	1834.00	0.152462
$526$	3872.00	0.320964
$527$	648.000	0.0535623
$528$	656.000	0.0540696
$529$	−9863.00	−0.810635
$530$	2592.00	0.212433
$531$	−18630.0	−1.52255
$532$	−5390.00	−0.439260
$533$	5096.00	0.414132
$534$	1460.00	0.118315
$535$	3904.00	0.315485
$536$	3660.00	0.294940
$537$	1640.00	0.131790
$538$	−180.000	−0.0144244
$539$	−392.000	−0.0313259
$540$	−11200.0	−0.892539
$541$	−1618.00	−0.128583	−0.0642914	−	0.997931i	$-0.520479\pi$
$541$	−1618.00	−0.128583	−0.0642914	+	0.997931i	$0.520479\pi$
$542$	−2032.00	−0.161037
$543$	−7784.00	−0.615181
$544$	−8694.00	−0.685206
$545$	1440.00	0.113179
$546$	−392.000	−0.0307254
$547$	16144.0	1.26192	0.630958	−	0.775817i	$-0.282662\pi$
$547$	16144.0	1.26192	0.630958	+	0.775817i	$0.282662\pi$
$548$	−15918.0	−1.24085
$549$	11224.0	0.872548
$550$	1048.00	0.0812489
$551$	12100.0	0.935531
$552$	−1440.00	−0.111033
$553$	−3080.00	−0.236844
$554$	5426.00	0.416117
$555$	7872.00	0.602068
$556$	1470.00	0.112126
$557$	4654.00	0.354033	0.177016	−	0.984208i	$-0.443355\pi$
$557$	4654.00	0.354033	0.177016	+	0.984208i	$0.443355\pi$
$558$	276.000	0.0209391
$559$	3584.00	0.271175
$560$	−4592.00	−0.346513
$561$	864.000	0.0650234
$562$	−842.000	−0.0631986
$563$	10078.0	0.754418	0.377209	−	0.926128i	$-0.376884\pi$
$563$	10078.0	0.754418	0.377209	+	0.926128i	$0.376884\pi$
$564$	4536.00	0.338653
$565$	21088.0	1.57023
$566$	3782.00	0.280865
$567$	−2947.00	−0.218276
$568$	−11520.0	−0.851001
$569$	−5930.00	−0.436904	−0.218452	−	0.975848i	$-0.570101\pi$
$569$	−5930.00	−0.436904	−0.218452	+	0.975848i	$0.570101\pi$
$570$	−3520.00	−0.258661
$571$	−19048.0	−1.39603	−0.698016	−	0.716082i	$-0.745933\pi$
$571$	−19048.0	−1.39603	−0.698016	+	0.716082i	$0.745933\pi$
$572$	1568.00	0.114618
$573$	10096.0	0.736067
$574$	1274.00	0.0926406
$575$	6288.00	0.456048
$576$	3841.00	0.277850
$577$	−14366.0	−1.03651	−0.518253	−	0.855227i	$-0.673418\pi$
$577$	−14366.0	−1.03651	−0.518253	+	0.855227i	$0.673418\pi$
$578$	1997.00	0.143710
$579$	5924.00	0.425204
$580$	12320.0	0.882000
$581$	9114.00	0.650796
$582$	588.000	0.0418787
$583$	1296.00	0.0920666
$584$	−10530.0	−0.746121
$585$	−10304.0	−0.728236
$586$	4312.00	0.303971
$587$	−3626.00	−0.254959	−0.127480	−	0.991841i	$-0.540689\pi$
$587$	−3626.00	−0.254959	−0.127480	+	0.991841i	$0.540689\pi$
$588$	686.000	0.0481125
$589$	−1320.00	−0.0923424
$590$	−12960.0	−0.904330
$591$	−6668.00	−0.464103
$592$	−10086.0	−0.700223
$593$	−1062.00	−0.0735432	−0.0367716	−	0.999324i	$-0.511707\pi$
$593$	−1062.00	−0.0735432	−0.0367716	+	0.999324i	$0.511707\pi$
$594$	800.000	0.0552599
$595$	−6048.00	−0.416712
$596$	14070.0	0.966996
$597$	−3720.00	−0.255024
$598$	−1344.00	−0.0919068
$599$	−10200.0	−0.695761	−0.347880	−	0.937539i	$-0.613098\pi$
$599$	−10200.0	−0.695761	−0.347880	+	0.937539i	$0.613098\pi$
$600$	−3930.00	−0.267403
$601$	−25158.0	−1.70751	−0.853757	−	0.520671i	$-0.825682\pi$
$601$	−25158.0	−1.70751	−0.853757	+	0.520671i	$0.825682\pi$
$602$	896.000	0.0606615
$603$	−5612.00	−0.379002
$604$	−7784.00	−0.524382
$605$	−20272.0	−1.36227
$606$	−1376.00	−0.0922379
$607$	25664.0	1.71609	0.858047	−	0.513570i	$-0.171677\pi$
$607$	25664.0	1.71609	0.858047	+	0.513570i	$0.171677\pi$
$608$	17710.0	1.18131
$609$	−1540.00	−0.102470
$610$	7808.00	0.518257
$611$	9072.00	0.600677
$612$	8694.00	0.574239
$613$	19018.0	1.25307	0.626533	−	0.779395i	$-0.284473\pi$
$613$	19018.0	1.25307	0.626533	+	0.779395i	$0.284473\pi$
$614$	−2674.00	−0.175755
$615$	−5824.00	−0.381864
$616$	840.000	0.0549425
$617$	17334.0	1.13102	0.565511	−	0.824741i	$-0.308679\pi$
$617$	17334.0	1.13102	0.565511	+	0.824741i	$0.308679\pi$
$618$	2776.00	0.180691
$619$	18730.0	1.21619	0.608096	−	0.793864i	$-0.291934\pi$
$619$	18730.0	1.21619	0.608096	+	0.793864i	$0.291934\pi$
$620$	−1344.00	−0.0870586
$621$	4800.00	0.310173
$622$	3768.00	0.242899
$623$	−5110.00	−0.328616
$624$	−2296.00	−0.147297
$625$	−14839.0	−0.949696
$626$	−2438.00	−0.155658
$627$	−1760.00	−0.112101
$628$	−868.000	−0.0551544
$629$	−13284.0	−0.842079
$630$	−2576.00	−0.162905
$631$	−6928.00	−0.437083	−0.218541	−	0.975828i	$-0.570130\pi$
$631$	−6928.00	−0.437083	−0.218541	+	0.975828i	$0.570130\pi$
$632$	6600.00	0.415402
$633$	8536.00	0.535980
$634$	3186.00	0.199578
$635$	−28416.0	−1.77583
$636$	−2268.00	−0.141403
$637$	1372.00	0.0853385
$638$	−880.000	−0.0546074
$639$	17664.0	1.09355
$640$	23280.0	1.43785
$641$	16302.0	1.00451	0.502255	−	0.864720i	$-0.332504\pi$
$641$	16302.0	1.00451	0.502255	+	0.864720i	$0.332504\pi$
$642$	488.000	0.0299997
$643$	4718.00	0.289362	0.144681	−	0.989478i	$-0.453784\pi$
$643$	4718.00	0.289362	0.144681	+	0.989478i	$0.453784\pi$
$644$	2352.00	0.143916
$645$	−4096.00	−0.250046
$646$	5940.00	0.361774
$647$	−21436.0	−1.30253	−0.651264	−	0.758851i	$-0.725761\pi$
$647$	−21436.0	−1.30253	−0.651264	+	0.758851i	$0.725761\pi$
$648$	6315.00	0.382834
$649$	−6480.00	−0.391930
$650$	−3668.00	−0.221340
$651$	168.000	0.0101143
$652$	−14056.0	−0.844287
$653$	4458.00	0.267159	0.133580	−	0.991038i	$-0.457353\pi$
$653$	4458.00	0.267159	0.133580	+	0.991038i	$0.457353\pi$
$654$	180.000	0.0107623
$655$	−17888.0	−1.06709
$656$	7462.00	0.444119
$657$	16146.0	0.958775
$658$	2268.00	0.134371
$659$	−26640.0	−1.57473	−0.787365	−	0.616487i	$-0.788555\pi$
$659$	−26640.0	−1.57473	−0.787365	+	0.616487i	$0.788555\pi$
$660$	−1792.00	−0.105687
$661$	7432.00	0.437324	0.218662	−	0.975801i	$-0.429831\pi$
$661$	7432.00	0.437324	0.218662	+	0.975801i	$0.429831\pi$
$662$	−8672.00	−0.509134
$663$	−3024.00	−0.177138
$664$	−19530.0	−1.14143
$665$	12320.0	0.718420
$666$	−5658.00	−0.329194
$667$	−5280.00	−0.306510
$668$	−20188.0	−1.16931
$669$	10864.0	0.627842
$670$	−3904.00	−0.225111
$671$	3904.00	0.224608
$672$	−2254.00	−0.129390
$673$	58.0000	0.00332204	0.00166102	−	0.999999i	$-0.499471\pi$
$673$	58.0000	0.00332204	0.00166102	+	0.999999i	$0.499471\pi$
$674$	−814.000	−0.0465194
$675$	13100.0	0.746991
$676$	9891.00	0.562756
$677$	−21516.0	−1.22146	−0.610729	−	0.791840i	$-0.709124\pi$
$677$	−21516.0	−1.22146	−0.610729	+	0.791840i	$0.709124\pi$
$678$	2636.00	0.149314
$679$	−2058.00	−0.116316
$680$	12960.0	0.730873
$681$	4092.00	0.230258
$682$	96.0000	0.00539007
$683$	18108.0	1.01447	0.507235	−	0.861808i	$-0.330668\pi$
$683$	18108.0	1.01447	0.507235	+	0.861808i	$0.330668\pi$
$684$	−17710.0	−0.989998
$685$	36384.0	2.02943
$686$	343.000	0.0190901
$687$	5960.00	0.330987
$688$	5248.00	0.290811
$689$	−4536.00	−0.250810
$690$	1536.00	0.0847457
$691$	−10078.0	−0.554827	−0.277413	−	0.960751i	$-0.589477\pi$
$691$	−10078.0	−0.554827	−0.277413	+	0.960751i	$0.589477\pi$
$692$	−15596.0	−0.856750
$693$	−1288.00	−0.0706018
$694$	−9344.00	−0.511086
$695$	−3360.00	−0.183384
$696$	3300.00	0.179722
$697$	9828.00	0.534092
$698$	5180.00	0.280897
$699$	−8916.00	−0.482452
$700$	6419.00	0.346593
$701$	18762.0	1.01089	0.505443	−	0.862860i	$-0.331329\pi$
$701$	18762.0	1.01089	0.505443	+	0.862860i	$0.331329\pi$
$702$	−2800.00	−0.150540
$703$	27060.0	1.45176
$704$	1336.00	0.0715233
$705$	−10368.0	−0.553874
$706$	−12178.0	−0.649186
$707$	4816.00	0.256187
$708$	11340.0	0.601954
$709$	6810.00	0.360726	0.180363	−	0.983600i	$-0.442273\pi$
$709$	6810.00	0.360726	0.180363	+	0.983600i	$0.442273\pi$
$710$	12288.0	0.649522
$711$	−10120.0	−0.533797
$712$	10950.0	0.576360
$713$	576.000	0.0302544
$714$	−756.000	−0.0396255
$715$	−3584.00	−0.187460
$716$	5740.00	0.299600
$717$	−8880.00	−0.462524
$718$	−440.000	−0.0228700
$719$	4860.00	0.252083	0.126041	−	0.992025i	$-0.459773\pi$
$719$	4860.00	0.252083	0.126041	+	0.992025i	$0.459773\pi$
$720$	−15088.0	−0.780967
$721$	−9716.00	−0.501862
$722$	−5241.00	−0.270152
$723$	−6604.00	−0.339703
$724$	−27244.0	−1.39850
$725$	−14410.0	−0.738171
$726$	−2534.00	−0.129539
$727$	−13636.0	−0.695641	−0.347821	−	0.937561i	$-0.613078\pi$
$727$	−13636.0	−0.695641	−0.347821	+	0.937561i	$0.613078\pi$
$728$	−2940.00	−0.149675
$729$	−4283.00	−0.217599
$730$	11232.0	0.569473
$731$	6912.00	0.349726
$732$	−6832.00	−0.344970
$733$	2088.00	0.105214	0.0526071	−	0.998615i	$-0.483247\pi$
$733$	2088.00	0.105214	0.0526071	+	0.998615i	$0.483247\pi$
$734$	9816.00	0.493617
$735$	−1568.00	−0.0786892
$736$	−7728.00	−0.387035
$737$	−1952.00	−0.0975615
$738$	4186.00	0.208792
$739$	−5160.00	−0.256852	−0.128426	−	0.991719i	$-0.540992\pi$
$739$	−5160.00	−0.256852	−0.128426	+	0.991719i	$0.540992\pi$
$740$	27552.0	1.36869
$741$	6160.00	0.305389
$742$	−1134.00	−0.0561057
$743$	−28152.0	−1.39004	−0.695018	−	0.718992i	$-0.744604\pi$
$743$	−28152.0	−1.39004	−0.695018	+	0.718992i	$0.744604\pi$
$744$	−360.000	−0.0177396
$745$	−32160.0	−1.58155
$746$	442.000	0.0216927
$747$	29946.0	1.46676
$748$	3024.00	0.147819
$749$	−1708.00	−0.0833230
$750$	192.000	0.00934780
$751$	−16808.0	−0.816688	−0.408344	−	0.912828i	$-0.633894\pi$
$751$	−16808.0	−0.816688	−0.408344	+	0.912828i	$0.633894\pi$
$752$	13284.0	0.644172
$753$	−3164.00	−0.153124
$754$	3080.00	0.148763
$755$	17792.0	0.857639
$756$	4900.00	0.235729
$757$	21674.0	1.04063	0.520314	−	0.853975i	$-0.325815\pi$
$757$	21674.0	1.04063	0.520314	+	0.853975i	$0.325815\pi$
$758$	3960.00	0.189754
$759$	768.000	0.0367281
$760$	−26400.0	−1.26004
$761$	7422.00	0.353544	0.176772	−	0.984252i	$-0.443434\pi$
$761$	7422.00	0.353544	0.176772	+	0.984252i	$0.443434\pi$
$762$	−3552.00	−0.168865
$763$	−630.000	−0.0298919
$764$	35336.0	1.67331
$765$	−19872.0	−0.939181
$766$	−6708.00	−0.316410
$767$	22680.0	1.06770
$768$	238.000	0.0111824
$769$	13790.0	0.646658	0.323329	−	0.946287i	$-0.395198\pi$
$769$	13790.0	0.646658	0.323329	+	0.946287i	$0.395198\pi$
$770$	−896.000	−0.0419345
$771$	−4708.00	−0.219915
$772$	20734.0	0.966623
$773$	−6232.00	−0.289973	−0.144987	−	0.989434i	$-0.546314\pi$
$773$	−6232.00	−0.289973	−0.144987	+	0.989434i	$0.546314\pi$
$774$	2944.00	0.136718
$775$	1572.00	0.0728618
$776$	4410.00	0.204007
$777$	−3444.00	−0.159013
$778$	13350.0	0.615194
$779$	−20020.0	−0.920784
$780$	6272.00	0.287915
$781$	6144.00	0.281498
$782$	−2592.00	−0.118529
$783$	−11000.0	−0.502054
$784$	2009.00	0.0915179
$785$	1984.00	0.0902064
$786$	−2236.00	−0.101470
$787$	−1766.00	−0.0799887	−0.0399943	−	0.999200i	$-0.512734\pi$
$787$	−1766.00	−0.0799887	−0.0399943	+	0.999200i	$0.512734\pi$
$788$	−23338.0	−1.05505
$789$	7744.00	0.349422
$790$	−7040.00	−0.317053
$791$	−9226.00	−0.414714
$792$	2760.00	0.123829
$793$	−13664.0	−0.611883
$794$	1356.00	0.0606079
$795$	5184.00	0.231267
$796$	−13020.0	−0.579751
$797$	1204.00	0.0535105	0.0267552	−	0.999642i	$-0.491483\pi$
$797$	1204.00	0.0535105	0.0267552	+	0.999642i	$0.491483\pi$
$798$	1540.00	0.0683150
$799$	17496.0	0.774673
$800$	−21091.0	−0.932099
$801$	−16790.0	−0.740631
$802$	−6222.00	−0.273948
$803$	5616.00	0.246805
$804$	3416.00	0.149842
$805$	−5376.00	−0.235378
$806$	−336.000	−0.0146837
$807$	−360.000	−0.0157033
$808$	−10320.0	−0.449327
$809$	−7050.00	−0.306384	−0.153192	−	0.988196i	$-0.548955\pi$
$809$	−7050.00	−0.306384	−0.153192	+	0.988196i	$0.548955\pi$
$810$	−6736.00	−0.292196
$811$	23282.0	1.00807	0.504033	−	0.863684i	$-0.331849\pi$
$811$	23282.0	1.00807	0.504033	+	0.863684i	$0.331849\pi$
$812$	−5390.00	−0.232946
$813$	−4064.00	−0.175315
$814$	−1968.00	−0.0847400
$815$	32128.0	1.38085
$816$	−4428.00	−0.189964
$817$	−14080.0	−0.602934
$818$	−5150.00	−0.220129
$819$	4508.00	0.192335
$820$	−20384.0	−0.868098
$821$	10142.0	0.431131	0.215565	−	0.976489i	$-0.430841\pi$
$821$	10142.0	0.431131	0.215565	+	0.976489i	$0.430841\pi$
$822$	4548.00	0.192980
$823$	−9192.00	−0.389323	−0.194662	−	0.980870i	$-0.562361\pi$
$823$	−9192.00	−0.389323	−0.194662	+	0.980870i	$0.562361\pi$
$824$	20820.0	0.880217
$825$	2096.00	0.0884525
$826$	5670.00	0.238843
$827$	−46716.0	−1.96430	−0.982149	−	0.188104i	$-0.939766\pi$
$827$	−46716.0	−1.96430	−0.982149	+	0.188104i	$0.939766\pi$
$828$	7728.00	0.324356
$829$	11240.0	0.470906	0.235453	−	0.971886i	$-0.424343\pi$
$829$	11240.0	0.470906	0.235453	+	0.971886i	$0.424343\pi$
$830$	20832.0	0.871192
$831$	10852.0	0.453010
$832$	−4676.00	−0.194845
$833$	2646.00	0.110058
$834$	−420.000	−0.0174381
$835$	46144.0	1.91243
$836$	−6160.00	−0.254842
$837$	1200.00	0.0495556
$838$	−2310.00	−0.0952239
$839$	700.000	0.0288042	0.0144021	−	0.999896i	$-0.495416\pi$
$839$	700.000	0.0288042	0.0144021	+	0.999896i	$0.495416\pi$
$840$	3360.00	0.138013
$841$	−12289.0	−0.503875
$842$	−1262.00	−0.0516525
$843$	−1684.00	−0.0688019
$844$	29876.0	1.21845
$845$	−22608.0	−0.920401
$846$	7452.00	0.302843
$847$	8869.00	0.359790
$848$	−6642.00	−0.268971
$849$	7564.00	0.305767
$850$	−7074.00	−0.285454
$851$	−11808.0	−0.475644
$852$	−10752.0	−0.432344
$853$	−37492.0	−1.50493	−0.752463	−	0.658635i	$-0.771134\pi$
$853$	−37492.0	−1.50493	−0.752463	+	0.658635i	$0.771134\pi$
$854$	−3416.00	−0.136877
$855$	40480.0	1.61917
$856$	3660.00	0.146140
$857$	28894.0	1.15169	0.575846	−	0.817558i	$-0.304673\pi$
$857$	28894.0	1.15169	0.575846	+	0.817558i	$0.304673\pi$
$858$	−448.000	−0.0178257
$859$	−2770.00	−0.110025	−0.0550123	−	0.998486i	$-0.517520\pi$
$859$	−2770.00	−0.110025	−0.0550123	+	0.998486i	$0.517520\pi$
$860$	−14336.0	−0.568434
$861$	2548.00	0.100854
$862$	4488.00	0.177334
$863$	17688.0	0.697690	0.348845	−	0.937180i	$-0.386574\pi$
$863$	17688.0	0.697690	0.348845	+	0.937180i	$0.386574\pi$
$864$	−16100.0	−0.633950
$865$	35648.0	1.40124
$866$	−17038.0	−0.668562
$867$	3994.00	0.156451
$868$	588.000	0.0229931
$869$	−3520.00	−0.137408
$870$	−3520.00	−0.137171
$871$	6832.00	0.265779
$872$	1350.00	0.0524275
$873$	−6762.00	−0.262152
$874$	5280.00	0.204346
$875$	−672.000	−0.0259631
$876$	−9828.00	−0.379061
$877$	−33566.0	−1.29241	−0.646205	−	0.763164i	$-0.723645\pi$
$877$	−33566.0	−1.29241	−0.646205	+	0.763164i	$0.723645\pi$
$878$	−16200.0	−0.622692
$879$	8624.00	0.330922
$880$	−5248.00	−0.201034
$881$	−16758.0	−0.640853	−0.320426	−	0.947273i	$-0.603826\pi$
$881$	−16758.0	−0.640853	−0.320426	+	0.947273i	$0.603826\pi$
$882$	1127.00	0.0430250
$883$	11468.0	0.437066	0.218533	−	0.975830i	$-0.429873\pi$
$883$	11468.0	0.437066	0.218533	+	0.975830i	$0.429873\pi$
$884$	−10584.0	−0.402691
$885$	−25920.0	−0.984510
$886$	8772.00	0.332620
$887$	−50356.0	−1.90619	−0.953094	−	0.302674i	$-0.902121\pi$
$887$	−50356.0	−1.90619	−0.953094	+	0.302674i	$0.902121\pi$
$888$	7380.00	0.278893
$889$	12432.0	0.469017
$890$	−11680.0	−0.439904
$891$	−3368.00	−0.126636
$892$	38024.0	1.42728
$893$	−35640.0	−1.33555
$894$	−4020.00	−0.150390
$895$	−13120.0	−0.490004
$896$	−10185.0	−0.379751
$897$	−2688.00	−0.100055
$898$	−2130.00	−0.0791526
$899$	−1320.00	−0.0489705
$900$	21091.0	0.781148
$901$	−8748.00	−0.323461
$902$	1456.00	0.0537467
$903$	1792.00	0.0660399
$904$	19770.0	0.727368
$905$	62272.0	2.28728
$906$	2224.00	0.0815535
$907$	−8716.00	−0.319085	−0.159542	−	0.987191i	$-0.551002\pi$
$907$	−8716.00	−0.319085	−0.159542	+	0.987191i	$0.551002\pi$
$908$	14322.0	0.523450
$909$	15824.0	0.577392
$910$	3136.00	0.114239
$911$	7632.00	0.277563	0.138781	−	0.990323i	$-0.455682\pi$
$911$	7632.00	0.277563	0.138781	+	0.990323i	$0.455682\pi$
$912$	9020.00	0.327502
$913$	10416.0	0.377568
$914$	−10534.0	−0.381219
$915$	15616.0	0.564207
$916$	20860.0	0.752439
$917$	7826.00	0.281829
$918$	−5400.00	−0.194147
$919$	−23080.0	−0.828443	−0.414221	−	0.910176i	$-0.635946\pi$
$919$	−23080.0	−0.828443	−0.414221	+	0.910176i	$0.635946\pi$
$920$	11520.0	0.412830
$921$	−5348.00	−0.191338
$922$	9268.00	0.331047
$923$	−21504.0	−0.766861
$924$	784.000	0.0279131
$925$	−32226.0	−1.14550
$926$	9392.00	0.333305
$927$	−31924.0	−1.13109
$928$	17710.0	0.626465
$929$	45110.0	1.59312	0.796561	−	0.604558i	$-0.206650\pi$
$929$	45110.0	1.59312	0.796561	+	0.604558i	$0.206650\pi$
$930$	384.000	0.0135396
$931$	−5390.00	−0.189742
$932$	−31206.0	−1.09677
$933$	7536.00	0.264435
$934$	10806.0	0.378569
$935$	−6912.00	−0.241761
$936$	−9660.00	−0.337337
$937$	16674.0	0.581340	0.290670	−	0.956823i	$-0.406122\pi$
$937$	16674.0	0.581340	0.290670	+	0.956823i	$0.406122\pi$
$938$	1708.00	0.0594543
$939$	−4876.00	−0.169459
$940$	−36288.0	−1.25913
$941$	43832.0	1.51847	0.759236	−	0.650815i	$-0.225573\pi$
$941$	43832.0	1.51847	0.759236	+	0.650815i	$0.225573\pi$
$942$	248.000	0.00857779
$943$	8736.00	0.301679
$944$	33210.0	1.14501
$945$	−11200.0	−0.385541
$946$	1024.00	0.0351936
$947$	−736.000	−0.0252553	−0.0126277	−	0.999920i	$-0.504020\pi$
$947$	−736.000	−0.0252553	−0.0126277	+	0.999920i	$0.504020\pi$
$948$	6160.00	0.211042
$949$	−19656.0	−0.672351
$950$	14410.0	0.492129
$951$	6372.00	0.217273
$952$	−5670.00	−0.193031
$953$	38138.0	1.29634	0.648169	−	0.761496i	$-0.275535\pi$
$953$	38138.0	1.29634	0.648169	+	0.761496i	$0.275535\pi$
$954$	−3726.00	−0.126450
$955$	−80768.0	−2.73674
$956$	−31080.0	−1.05146
$957$	−1760.00	−0.0594490
$958$	−4940.00	−0.166601
$959$	−15918.0	−0.535995
$960$	5344.00	0.179663
$961$	−29647.0	−0.995166
$962$	6888.00	0.230850
$963$	−5612.00	−0.187792
$964$	−23114.0	−0.772253
$965$	−47392.0	−1.58094
$966$	−672.000	−0.0223822
$967$	26224.0	0.872086	0.436043	−	0.899926i	$-0.356380\pi$
$967$	26224.0	0.872086	0.436043	+	0.899926i	$0.356380\pi$
$968$	−19005.0	−0.631037
$969$	11880.0	0.393850
$970$	−4704.00	−0.155708
$971$	18762.0	0.620084	0.310042	−	0.950723i	$-0.399657\pi$
$971$	18762.0	0.620084	0.310042	+	0.950723i	$0.399657\pi$
$972$	24794.0	0.818177
$973$	1470.00	0.0484337
$974$	5216.00	0.171593
$975$	−7336.00	−0.240964
$976$	−20008.0	−0.656189
$977$	38394.0	1.25725	0.628625	−	0.777709i	$-0.283618\pi$
$977$	38394.0	1.25725	0.628625	+	0.777709i	$0.283618\pi$
$978$	4016.00	0.131306
$979$	−5840.00	−0.190651
$980$	−5488.00	−0.178885
$981$	−2070.00	−0.0673700
$982$	−4412.00	−0.143373
$983$	5388.00	0.174822	0.0874112	−	0.996172i	$-0.472141\pi$
$983$	5388.00	0.174822	0.0874112	+	0.996172i	$0.472141\pi$
$984$	−5460.00	−0.176889
$985$	53344.0	1.72556
$986$	5940.00	0.191854
$987$	4536.00	0.146284
$988$	21560.0	0.694246
$989$	6144.00	0.197541
$990$	−2944.00	−0.0945116
$991$	25472.0	0.816493	0.408247	−	0.912872i	$-0.366140\pi$
$991$	25472.0	0.816493	0.408247	+	0.912872i	$0.366140\pi$
$992$	−1932.00	−0.0618357
$993$	−17344.0	−0.554275
$994$	−5376.00	−0.171546
$995$	29760.0	0.948196
$996$	−18228.0	−0.579896
$997$	−17096.0	−0.543065	−0.271532	−	0.962429i	$-0.587530\pi$
$997$	−17096.0	−0.543065	−0.271532	+	0.962429i	$0.587530\pi$
$998$	−19060.0	−0.604543
$999$	−24600.0	−0.779089

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7.4.a.a.1.1	✓	1
3.2	odd	2		63.4.a.b.1.1		1
4.3	odd	2		112.4.a.f.1.1		1
5.2	odd	4		175.4.b.b.99.1		2
5.3	odd	4		175.4.b.b.99.2		2
5.4	even	2		175.4.a.b.1.1		1
7.2	even	3		49.4.c.c.18.1		2
7.3	odd	6		49.4.c.b.30.1		2
7.4	even	3		49.4.c.c.30.1		2
7.5	odd	6		49.4.c.b.18.1		2
7.6	odd	2		49.4.a.b.1.1		1
8.3	odd	2		448.4.a.e.1.1		1
8.5	even	2		448.4.a.i.1.1		1
11.10	odd	2		847.4.a.b.1.1		1
12.11	even	2		1008.4.a.c.1.1		1
13.12	even	2		1183.4.a.b.1.1		1
15.14	odd	2		1575.4.a.e.1.1		1
17.16	even	2		2023.4.a.a.1.1		1
21.2	odd	6		441.4.e.h.361.1		2
21.5	even	6		441.4.e.e.361.1		2
21.11	odd	6		441.4.e.h.226.1		2
21.17	even	6		441.4.e.e.226.1		2
21.20	even	2		441.4.a.i.1.1		1
28.27	even	2		784.4.a.g.1.1		1
35.34	odd	2		1225.4.a.j.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7.4.a.a.1.1	✓	1	1.1	even	1	trivial
49.4.a.b.1.1		1	7.6	odd	2
49.4.c.b.18.1		2	7.5	odd	6
49.4.c.b.30.1		2	7.3	odd	6
49.4.c.c.18.1		2	7.2	even	3
49.4.c.c.30.1		2	7.4	even	3
63.4.a.b.1.1		1	3.2	odd	2
112.4.a.f.1.1		1	4.3	odd	2
175.4.a.b.1.1		1	5.4	even	2
175.4.b.b.99.1		2	5.2	odd	4
175.4.b.b.99.2		2	5.3	odd	4
441.4.a.i.1.1		1	21.20	even	2
441.4.e.e.226.1		2	21.17	even	6
441.4.e.e.361.1		2	21.5	even	6
441.4.e.h.226.1		2	21.11	odd	6
441.4.e.h.361.1		2	21.2	odd	6
448.4.a.e.1.1		1	8.3	odd	2
448.4.a.i.1.1		1	8.5	even	2
784.4.a.g.1.1		1	28.27	even	2
847.4.a.b.1.1		1	11.10	odd	2
1008.4.a.c.1.1		1	12.11	even	2
1183.4.a.b.1.1		1	13.12	even	2
1225.4.a.j.1.1		1	35.34	odd	2
1575.4.a.e.1.1		1	15.14	odd	2
2023.4.a.a.1.1		1	17.16	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

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

Label	7.4.a.a.1.1

Level	$7$
Weight	$4$
Character	7.1
Self dual	yes
Analytic conductor	$0.413$
Analytic rank	$0$
Dimension	$1$
CM	no
Inner twists	$1$