LMFDB - Embedded newform 38.4.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [38,4,Mod(1,38)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(38, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("38.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 38 = 2 \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	38.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:	$2.24207258022$
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	$\chi$	$=$	38.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-2.00000 q^{2} -2.00000 q^{3} +4.00000 q^{4} -9.00000 q^{5} +4.00000 q^{6} -31.0000 q^{7} -8.00000 q^{8} -23.0000 q^{9} +O(q^{10})$

\(q-2.00000 q^{2} -2.00000 q^{3} +4.00000 q^{4} -9.00000 q^{5} +4.00000 q^{6} -31.0000 q^{7} -8.00000 q^{8} -23.0000 q^{9} +18.0000 q^{10} +57.0000 q^{11} -8.00000 q^{12} -52.0000 q^{13} +62.0000 q^{14} +18.0000 q^{15} +16.0000 q^{16} +69.0000 q^{17} +46.0000 q^{18} +19.0000 q^{19} -36.0000 q^{20} +62.0000 q^{21} -114.000 q^{22} -72.0000 q^{23} +16.0000 q^{24} -44.0000 q^{25} +104.000 q^{26} +100.000 q^{27} -124.000 q^{28} -150.000 q^{29} -36.0000 q^{30} +32.0000 q^{31} -32.0000 q^{32} -114.000 q^{33} -138.000 q^{34} +279.000 q^{35} -92.0000 q^{36} -226.000 q^{37} -38.0000 q^{38} +104.000 q^{39} +72.0000 q^{40} -258.000 q^{41} -124.000 q^{42} -67.0000 q^{43} +228.000 q^{44} +207.000 q^{45} +144.000 q^{46} +579.000 q^{47} -32.0000 q^{48} +618.000 q^{49} +88.0000 q^{50} -138.000 q^{51} -208.000 q^{52} -432.000 q^{53} -200.000 q^{54} -513.000 q^{55} +248.000 q^{56} -38.0000 q^{57} +300.000 q^{58} -330.000 q^{59} +72.0000 q^{60} -13.0000 q^{61} -64.0000 q^{62} +713.000 q^{63} +64.0000 q^{64} +468.000 q^{65} +228.000 q^{66} -856.000 q^{67} +276.000 q^{68} +144.000 q^{69} -558.000 q^{70} +642.000 q^{71} +184.000 q^{72} -487.000 q^{73} +452.000 q^{74} +88.0000 q^{75} +76.0000 q^{76} -1767.00 q^{77} -208.000 q^{78} -700.000 q^{79} -144.000 q^{80} +421.000 q^{81} +516.000 q^{82} -12.0000 q^{83} +248.000 q^{84} -621.000 q^{85} +134.000 q^{86} +300.000 q^{87} -456.000 q^{88} -600.000 q^{89} -414.000 q^{90} +1612.00 q^{91} -288.000 q^{92} -64.0000 q^{93} -1158.00 q^{94} -171.000 q^{95} +64.0000 q^{96} +1424.00 q^{97} -1236.00 q^{98} -1311.00 q^{99} +O(q^{100})\)

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 $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	−2.00000	−0.384900	−0.192450	−	0.981307i	$-0.561643\pi$
$3$	−2.00000	−0.384900	−0.192450	+	0.981307i	$0.561643\pi$
$4$	4.00000	0.500000
$5$	−9.00000	−0.804984	−0.402492	−	0.915423i	$-0.631856\pi$
$5$	−9.00000	−0.804984	−0.402492	+	0.915423i	$0.631856\pi$
$6$	4.00000	0.272166
$7$	−31.0000	−1.67384	−0.836921	−	0.547323i	$-0.815647\pi$
$7$	−31.0000	−1.67384	−0.836921	+	0.547323i	$0.815647\pi$
$8$	−8.00000	−0.353553
$9$	−23.0000	−0.851852
$10$	18.0000	0.569210
$11$	57.0000	1.56238	0.781188	−	0.624295i	$-0.214614\pi$
$11$	57.0000	1.56238	0.781188	+	0.624295i	$0.214614\pi$
$12$	−8.00000	−0.192450
$13$	−52.0000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−52.0000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	62.0000	1.18359
$15$	18.0000	0.309839
$16$	16.0000	0.250000
$17$	69.0000	0.984409	0.492205	−	0.870480i	$-0.336191\pi$
$17$	69.0000	0.984409	0.492205	+	0.870480i	$0.336191\pi$
$18$	46.0000	0.602350
$19$	19.0000	0.229416
$19$	19.0000	0.229416
$20$	−36.0000	−0.402492
$21$	62.0000	0.644262
$22$	−114.000	−1.10477
$23$	−72.0000	−0.652741	−0.326370	−	0.945242i	$-0.605826\pi$
$23$	−72.0000	−0.652741	−0.326370	+	0.945242i	$0.605826\pi$
$24$	16.0000	0.136083
$25$	−44.0000	−0.352000
$26$	104.000	0.784465
$27$	100.000	0.712778
$28$	−124.000	−0.836921
$29$	−150.000	−0.960493	−0.480247	−	0.877134i	$-0.659453\pi$
$29$	−150.000	−0.960493	−0.480247	+	0.877134i	$0.659453\pi$
$30$	−36.0000	−0.219089
$31$	32.0000	0.185399	0.0926995	−	0.995694i	$-0.470450\pi$
$31$	32.0000	0.185399	0.0926995	+	0.995694i	$0.470450\pi$
$32$	−32.0000	−0.176777
$33$	−114.000	−0.601359
$34$	−138.000	−0.696082
$35$	279.000	1.34742
$36$	−92.0000	−0.425926
$37$	−226.000	−1.00417	−0.502083	−	0.864819i	$-0.667433\pi$
$37$	−226.000	−1.00417	−0.502083	+	0.864819i	$0.667433\pi$
$38$	−38.0000	−0.162221
$39$	104.000	0.427008
$40$	72.0000	0.284605
$41$	−258.000	−0.982752	−0.491376	−	0.870948i	$-0.663506\pi$
$41$	−258.000	−0.982752	−0.491376	+	0.870948i	$0.663506\pi$
$42$	−124.000	−0.455562
$43$	−67.0000	−0.237614	−0.118807	−	0.992917i	$-0.537907\pi$
$43$	−67.0000	−0.237614	−0.118807	+	0.992917i	$0.537907\pi$
$44$	228.000	0.781188
$45$	207.000	0.685728
$46$	144.000	0.461557
$47$	579.000	1.79693	0.898466	−	0.439043i	$-0.144682\pi$
$47$	579.000	1.79693	0.898466	+	0.439043i	$0.144682\pi$
$48$	−32.0000	−0.0962250
$49$	618.000	1.80175
$50$	88.0000	0.248902
$51$	−138.000	−0.378899
$52$	−208.000	−0.554700
$53$	−432.000	−1.11962	−0.559809	−	0.828622i	$-0.689126\pi$
$53$	−432.000	−1.11962	−0.559809	+	0.828622i	$0.689126\pi$
$54$	−200.000	−0.504010
$55$	−513.000	−1.25769
$56$	248.000	0.591793
$57$	−38.0000	−0.0883022
$58$	300.000	0.679171
$59$	−330.000	−0.728175	−0.364088	−	0.931365i	$-0.618619\pi$
$59$	−330.000	−0.728175	−0.364088	+	0.931365i	$0.618619\pi$
$60$	72.0000	0.154919
$61$	−13.0000	−0.0272865	−0.0136433	−	0.999907i	$-0.504343\pi$
$61$	−13.0000	−0.0272865	−0.0136433	+	0.999907i	$0.504343\pi$
$62$	−64.0000	−0.131097
$63$	713.000	1.42587
$64$	64.0000	0.125000
$65$	468.000	0.893050
$66$	228.000	0.425225
$67$	−856.000	−1.56085	−0.780426	−	0.625249i	$-0.784998\pi$
$67$	−856.000	−1.56085	−0.780426	+	0.625249i	$0.784998\pi$
$68$	276.000	0.492205
$69$	144.000	0.251240
$70$	−558.000	−0.952768
$71$	642.000	1.07312	0.536559	−	0.843863i	$-0.319724\pi$
$71$	642.000	1.07312	0.536559	+	0.843863i	$0.319724\pi$
$72$	184.000	0.301175
$73$	−487.000	−0.780809	−0.390404	−	0.920643i	$-0.627665\pi$
$73$	−487.000	−0.780809	−0.390404	+	0.920643i	$0.627665\pi$
$74$	452.000	0.710053
$75$	88.0000	0.135485
$76$	76.0000	0.114708
$77$	−1767.00	−2.61517
$78$	−208.000	−0.301941
$79$	−700.000	−0.996913	−0.498457	−	0.866915i	$-0.666100\pi$
$79$	−700.000	−0.996913	−0.498457	+	0.866915i	$0.666100\pi$
$80$	−144.000	−0.201246
$81$	421.000	0.577503
$82$	516.000	0.694911
$83$	−12.0000	−0.0158695	−0.00793477	−	0.999969i	$-0.502526\pi$
$83$	−12.0000	−0.0158695	−0.00793477	+	0.999969i	$0.502526\pi$
$84$	248.000	0.322131
$85$	−621.000	−0.792434
$86$	134.000	0.168019
$87$	300.000	0.369694
$88$	−456.000	−0.552384
$89$	−600.000	−0.714605	−0.357303	−	0.933989i	$-0.616304\pi$
$89$	−600.000	−0.714605	−0.357303	+	0.933989i	$0.616304\pi$
$90$	−414.000	−0.484883
$91$	1612.00	1.85696
$92$	−288.000	−0.326370
$93$	−64.0000	−0.0713601
$94$	−1158.00	−1.27062
$95$	−171.000	−0.184676
$96$	64.0000	0.0680414
$97$	1424.00	1.49057	0.745285	−	0.666746i	$-0.232313\pi$
$97$	1424.00	1.49057	0.745285	+	0.666746i	$0.232313\pi$
$98$	−1236.00	−1.27403
$99$	−1311.00	−1.33091
$100$	−176.000	−0.176000
$101$	1062.00	1.04627	0.523133	−	0.852251i	$-0.324763\pi$
$101$	1062.00	1.04627	0.523133	+	0.852251i	$0.324763\pi$
$102$	276.000	0.267922
$103$	1178.00	1.12691	0.563455	−	0.826147i	$-0.309472\pi$
$103$	1178.00	1.12691	0.563455	+	0.826147i	$0.309472\pi$
$104$	416.000	0.392232
$105$	−558.000	−0.518621
$106$	864.000	0.791690
$107$	114.000	0.102998	0.0514990	−	0.998673i	$-0.483600\pi$
$107$	114.000	0.102998	0.0514990	+	0.998673i	$0.483600\pi$
$108$	400.000	0.356389
$109$	1460.00	1.28296	0.641480	−	0.767140i	$-0.278321\pi$
$109$	1460.00	1.28296	0.641480	+	0.767140i	$0.278321\pi$
$110$	1026.00	0.889321
$111$	452.000	0.386504
$112$	−496.000	−0.418461
$113$	−822.000	−0.684312	−0.342156	−	0.939643i	$-0.611157\pi$
$113$	−822.000	−0.684312	−0.342156	+	0.939643i	$0.611157\pi$
$114$	76.0000	0.0624391
$115$	648.000	0.525446
$116$	−600.000	−0.480247
$117$	1196.00	0.945045
$118$	660.000	0.514898
$119$	−2139.00	−1.64775
$120$	−144.000	−0.109545
$121$	1918.00	1.44102
$122$	26.0000	0.0192945
$123$	516.000	0.378261
$124$	128.000	0.0926995
$125$	1521.00	1.08834
$126$	−1426.00	−1.00824
$127$	−2086.00	−1.45750	−0.728750	−	0.684780i	$-0.759898\pi$
$127$	−2086.00	−1.45750	−0.728750	+	0.684780i	$0.759898\pi$
$128$	−128.000	−0.0883883
$129$	134.000	0.0914577
$130$	−936.000	−0.631482
$131$	−93.0000	−0.0620263	−0.0310132	−	0.999519i	$-0.509873\pi$
$131$	−93.0000	−0.0620263	−0.0310132	+	0.999519i	$0.509873\pi$
$132$	−456.000	−0.300680
$133$	−589.000	−0.384006
$134$	1712.00	1.10369
$135$	−900.000	−0.573775
$136$	−552.000	−0.348041
$137$	1269.00	0.791372	0.395686	−	0.918386i	$-0.370507\pi$
$137$	1269.00	0.791372	0.395686	+	0.918386i	$0.370507\pi$
$138$	−288.000	−0.177654
$139$	−1975.00	−1.20516	−0.602580	−	0.798058i	$-0.705861\pi$
$139$	−1975.00	−1.20516	−0.602580	+	0.798058i	$0.705861\pi$
$140$	1116.00	0.673709
$141$	−1158.00	−0.691640
$142$	−1284.00	−0.758809
$143$	−2964.00	−1.73330
$144$	−368.000	−0.212963
$145$	1350.00	0.773182
$146$	974.000	0.552115
$147$	−1236.00	−0.693494
$148$	−904.000	−0.502083
$149$	−1695.00	−0.931945	−0.465973	−	0.884799i	$-0.654295\pi$
$149$	−1695.00	−0.931945	−0.465973	+	0.884799i	$0.654295\pi$
$150$	−176.000	−0.0958023
$151$	1802.00	0.971157	0.485578	−	0.874193i	$-0.338609\pi$
$151$	1802.00	0.971157	0.485578	+	0.874193i	$0.338609\pi$
$152$	−152.000	−0.0811107
$153$	−1587.00	−0.838571
$154$	3534.00	1.84921
$155$	−288.000	−0.149243
$156$	416.000	0.213504
$157$	−3226.00	−1.63989	−0.819945	−	0.572442i	$-0.805996\pi$
$157$	−3226.00	−1.63989	−0.819945	+	0.572442i	$0.805996\pi$
$158$	1400.00	0.704924
$159$	864.000	0.430941
$160$	288.000	0.142302
$161$	2232.00	1.09259
$162$	−842.000	−0.408357
$163$	1268.00	0.609309	0.304655	−	0.952463i	$-0.401459\pi$
$163$	1268.00	0.609309	0.304655	+	0.952463i	$0.401459\pi$
$164$	−1032.00	−0.491376
$165$	1026.00	0.484085
$166$	24.0000	0.0112215
$167$	654.000	0.303042	0.151521	−	0.988454i	$-0.451583\pi$
$167$	654.000	0.303042	0.151521	+	0.988454i	$0.451583\pi$
$168$	−496.000	−0.227781
$169$	507.000	0.230769
$170$	1242.00	0.560336
$171$	−437.000	−0.195428
$172$	−268.000	−0.118807
$173$	−1362.00	−0.598560	−0.299280	−	0.954165i	$-0.596747\pi$
$173$	−1362.00	−0.598560	−0.299280	+	0.954165i	$0.596747\pi$
$174$	−600.000	−0.261413
$175$	1364.00	0.589193
$176$	912.000	0.390594
$177$	660.000	0.280275
$178$	1200.00	0.505302
$179$	−210.000	−0.0876879	−0.0438440	−	0.999038i	$-0.513960\pi$
$179$	−210.000	−0.0876879	−0.0438440	+	0.999038i	$0.513960\pi$
$180$	828.000	0.342864
$181$	2.00000	0.000821319 0	0.000410660	−	1.00000i	$-0.499869\pi$
$181$	2.00000	0.000821319 0	0.000410660		1.00000i	$0.499869\pi$
$182$	−3224.00	−1.31307
$183$	26.0000	0.0105026
$184$	576.000	0.230779
$185$	2034.00	0.808339
$186$	128.000	0.0504592
$187$	3933.00	1.53802
$188$	2316.00	0.898466
$189$	−3100.00	−1.19308
$190$	342.000	0.130586
$191$	−2643.00	−1.00126	−0.500630	−	0.865661i	$-0.666898\pi$
$191$	−2643.00	−1.00126	−0.500630	+	0.865661i	$0.666898\pi$
$192$	−128.000	−0.0481125
$193$	3248.00	1.21138	0.605690	−	0.795701i	$-0.292897\pi$
$193$	3248.00	1.21138	0.605690	+	0.795701i	$0.292897\pi$
$194$	−2848.00	−1.05399
$195$	−936.000	−0.343735
$196$	2472.00	0.900875
$197$	−3126.00	−1.13055	−0.565275	−	0.824903i	$-0.691230\pi$
$197$	−3126.00	−1.13055	−0.565275	+	0.824903i	$0.691230\pi$
$198$	2622.00	0.941098
$199$	−2995.00	−1.06688	−0.533442	−	0.845837i	$-0.679102\pi$
$199$	−2995.00	−1.06688	−0.533442	+	0.845837i	$0.679102\pi$
$200$	352.000	0.124451
$201$	1712.00	0.600772
$202$	−2124.00	−0.739822
$203$	4650.00	1.60771
$204$	−552.000	−0.189450
$205$	2322.00	0.791100
$206$	−2356.00	−0.796846
$207$	1656.00	0.556038
$208$	−832.000	−0.277350
$209$	1083.00	0.358434
$210$	1116.00	0.366721
$211$	−4318.00	−1.40883	−0.704416	−	0.709788i	$-0.748791\pi$
$211$	−4318.00	−1.40883	−0.704416	+	0.709788i	$0.748791\pi$
$212$	−1728.00	−0.559809
$213$	−1284.00	−0.413043
$214$	−228.000	−0.0728307
$215$	603.000	0.191276
$216$	−800.000	−0.252005
$217$	−992.000	−0.310329
$218$	−2920.00	−0.907190
$219$	974.000	0.300533
$220$	−2052.00	−0.628845
$221$	−3588.00	−1.09210
$222$	−904.000	−0.273300
$223$	518.000	0.155551	0.0777754	−	0.996971i	$-0.475218\pi$
$223$	518.000	0.155551	0.0777754	+	0.996971i	$0.475218\pi$
$224$	992.000	0.295896
$225$	1012.00	0.299852
$226$	1644.00	0.483882
$227$	2844.00	0.831555	0.415777	−	0.909466i	$-0.363510\pi$
$227$	2844.00	0.831555	0.415777	+	0.909466i	$0.363510\pi$
$228$	−152.000	−0.0441511
$229$	1745.00	0.503550	0.251775	−	0.967786i	$-0.418986\pi$
$229$	1745.00	0.503550	0.251775	+	0.967786i	$0.418986\pi$
$230$	−1296.00	−0.371547
$231$	3534.00	1.00658
$232$	1200.00	0.339586
$233$	5283.00	1.48541	0.742706	−	0.669618i	$-0.233542\pi$
$233$	5283.00	1.48541	0.742706	+	0.669618i	$0.233542\pi$
$234$	−2392.00	−0.668248
$235$	−5211.00	−1.44650
$236$	−1320.00	−0.364088
$237$	1400.00	0.383712
$238$	4278.00	1.16513
$239$	465.000	0.125851	0.0629254	−	0.998018i	$-0.479957\pi$
$239$	465.000	0.125851	0.0629254	+	0.998018i	$0.479957\pi$
$240$	288.000	0.0774597
$241$	−7078.00	−1.89184	−0.945921	−	0.324396i	$-0.894839\pi$
$241$	−7078.00	−1.89184	−0.945921	+	0.324396i	$0.894839\pi$
$242$	−3836.00	−1.01896
$243$	−3542.00	−0.935059
$244$	−52.0000	−0.0136433
$245$	−5562.00	−1.45038
$246$	−1032.00	−0.267471
$247$	−988.000	−0.254514
$248$	−256.000	−0.0655485
$249$	24.0000	0.00610819
$250$	−3042.00	−0.769572
$251$	3567.00	0.897000	0.448500	−	0.893783i	$-0.351958\pi$
$251$	3567.00	0.897000	0.448500	+	0.893783i	$0.351958\pi$
$252$	2852.00	0.712933
$253$	−4104.00	−1.01983
$254$	4172.00	1.03061
$255$	1242.00	0.305008
$256$	256.000	0.0625000
$257$	−1896.00	−0.460192	−0.230096	−	0.973168i	$-0.573904\pi$
$257$	−1896.00	−0.460192	−0.230096	+	0.973168i	$0.573904\pi$
$258$	−268.000	−0.0646704
$259$	7006.00	1.68082
$260$	1872.00	0.446525
$261$	3450.00	0.818198
$262$	186.000	0.0438592
$263$	−57.0000	−0.0133641	−0.00668207	−	0.999978i	$-0.502127\pi$
$263$	−57.0000	−0.0133641	−0.00668207	+	0.999978i	$0.502127\pi$
$264$	912.000	0.212613
$265$	3888.00	0.901275
$266$	1178.00	0.271533
$267$	1200.00	0.275052
$268$	−3424.00	−0.780426
$269$	2700.00	0.611977	0.305989	−	0.952035i	$-0.401013\pi$
$269$	2700.00	0.611977	0.305989	+	0.952035i	$0.401013\pi$
$270$	1800.00	0.405720
$271$	3872.00	0.867923	0.433962	−	0.900931i	$-0.357115\pi$
$271$	3872.00	0.867923	0.433962	+	0.900931i	$0.357115\pi$
$272$	1104.00	0.246102
$273$	−3224.00	−0.714745
$274$	−2538.00	−0.559585
$275$	−2508.00	−0.549957
$276$	576.000	0.125620
$277$	−7711.00	−1.67260	−0.836298	−	0.548275i	$-0.815285\pi$
$277$	−7711.00	−1.67260	−0.836298	+	0.548275i	$0.815285\pi$
$278$	3950.00	0.852177
$279$	−736.000	−0.157932
$280$	−2232.00	−0.476384
$281$	−6858.00	−1.45592	−0.727961	−	0.685619i	$-0.759532\pi$
$281$	−6858.00	−1.45592	−0.727961	+	0.685619i	$0.759532\pi$
$282$	2316.00	0.489063
$283$	−1807.00	−0.379558	−0.189779	−	0.981827i	$-0.560777\pi$
$283$	−1807.00	−0.379558	−0.189779	+	0.981827i	$0.560777\pi$
$284$	2568.00	0.536559
$285$	342.000	0.0710819
$286$	5928.00	1.22563
$287$	7998.00	1.64497
$288$	736.000	0.150588
$289$	−152.000	−0.0309383
$290$	−2700.00	−0.546722
$291$	−2848.00	−0.573721
$292$	−1948.00	−0.390404
$293$	−3012.00	−0.600556	−0.300278	−	0.953852i	$-0.597079\pi$
$293$	−3012.00	−0.600556	−0.300278	+	0.953852i	$0.597079\pi$
$294$	2472.00	0.490374
$295$	2970.00	0.586170
$296$	1808.00	0.355027
$297$	5700.00	1.11363
$298$	3390.00	0.658985
$299$	3744.00	0.724151
$300$	352.000	0.0677424
$301$	2077.00	0.397729
$302$	−3604.00	−0.686712
$303$	−2124.00	−0.402708
$304$	304.000	0.0573539
$305$	117.000	0.0219652
$306$	3174.00	0.592959
$307$	−1096.00	−0.203753	−0.101876	−	0.994797i	$-0.532485\pi$
$307$	−1096.00	−0.203753	−0.101876	+	0.994797i	$0.532485\pi$
$308$	−7068.00	−1.30759
$309$	−2356.00	−0.433748
$310$	576.000	0.105531
$311$	1947.00	0.354998	0.177499	−	0.984121i	$-0.443199\pi$
$311$	1947.00	0.354998	0.177499	+	0.984121i	$0.443199\pi$
$312$	−832.000	−0.150970
$313$	7598.00	1.37209	0.686045	−	0.727559i	$-0.259345\pi$
$313$	7598.00	1.37209	0.686045	+	0.727559i	$0.259345\pi$
$314$	6452.00	1.15958
$315$	−6417.00	−1.14780
$316$	−2800.00	−0.498457
$317$	8334.00	1.47661	0.738303	−	0.674469i	$-0.235627\pi$
$317$	8334.00	1.47661	0.738303	+	0.674469i	$0.235627\pi$
$318$	−1728.00	−0.304721
$319$	−8550.00	−1.50065
$320$	−576.000	−0.100623
$321$	−228.000	−0.0396440
$322$	−4464.00	−0.772575
$323$	1311.00	0.225839
$324$	1684.00	0.288752
$325$	2288.00	0.390509
$326$	−2536.00	−0.430847
$327$	−2920.00	−0.493812
$328$	2064.00	0.347455
$329$	−17949.0	−3.00778
$330$	−2052.00	−0.342300
$331$	−8368.00	−1.38957	−0.694784	−	0.719219i	$-0.744500\pi$
$331$	−8368.00	−1.38957	−0.694784	+	0.719219i	$0.744500\pi$
$332$	−48.0000	−0.00793477
$333$	5198.00	0.855401
$334$	−1308.00	−0.214283
$335$	7704.00	1.25646
$336$	992.000	0.161066
$337$	−10336.0	−1.67074	−0.835368	−	0.549692i	$-0.814745\pi$
$337$	−10336.0	−1.67074	−0.835368	+	0.549692i	$0.814745\pi$
$338$	−1014.00	−0.163178
$339$	1644.00	0.263392
$340$	−2484.00	−0.396217
$341$	1824.00	0.289663
$342$	874.000	0.138189
$343$	−8525.00	−1.34200
$344$	536.000	0.0840093
$345$	−1296.00	−0.202244
$346$	2724.00	0.423246
$347$	6879.00	1.06422	0.532110	−	0.846675i	$-0.321399\pi$
$347$	6879.00	1.06422	0.532110	+	0.846675i	$0.321399\pi$
$348$	1200.00	0.184847
$349$	−6355.00	−0.974714	−0.487357	−	0.873203i	$-0.662039\pi$
$349$	−6355.00	−0.974714	−0.487357	+	0.873203i	$0.662039\pi$
$350$	−2728.00	−0.416622
$351$	−5200.00	−0.790756
$352$	−1824.00	−0.276192
$353$	7218.00	1.08832	0.544158	−	0.838983i	$-0.316849\pi$
$353$	7218.00	1.08832	0.544158	+	0.838983i	$0.316849\pi$
$354$	−1320.00	−0.198184
$355$	−5778.00	−0.863843
$356$	−2400.00	−0.357303
$357$	4278.00	0.634218
$358$	420.000	0.0620047
$359$	1665.00	0.244778	0.122389	−	0.992482i	$-0.460944\pi$
$359$	1665.00	0.244778	0.122389	+	0.992482i	$0.460944\pi$
$360$	−1656.00	−0.242441
$361$	361.000	0.0526316
$362$	−4.00000	−0.000580761 0
$363$	−3836.00	−0.554650
$364$	6448.00	0.928481
$365$	4383.00	0.628539
$366$	−52.0000	−0.00742646
$367$	13064.0	1.85813	0.929067	−	0.369911i	$-0.120612\pi$
$367$	13064.0	1.85813	0.929067	+	0.369911i	$0.120612\pi$
$368$	−1152.00	−0.163185
$369$	5934.00	0.837159
$370$	−4068.00	−0.571582
$371$	13392.0	1.87406
$372$	−256.000	−0.0356801
$373$	−10492.0	−1.45645	−0.728224	−	0.685339i	$-0.759654\pi$
$373$	−10492.0	−1.45645	−0.728224	+	0.685339i	$0.759654\pi$
$374$	−7866.00	−1.08754
$375$	−3042.00	−0.418902
$376$	−4632.00	−0.635312
$377$	7800.00	1.06557
$378$	6200.00	0.843634
$379$	7610.00	1.03140	0.515698	−	0.856770i	$-0.327532\pi$
$379$	7610.00	1.03140	0.515698	+	0.856770i	$0.327532\pi$
$380$	−684.000	−0.0923381
$381$	4172.00	0.560992
$382$	5286.00	0.707998
$383$	4008.00	0.534724	0.267362	−	0.963596i	$-0.413848\pi$
$383$	4008.00	0.534724	0.267362	+	0.963596i	$0.413848\pi$
$384$	256.000	0.0340207
$385$	15903.0	2.10517
$386$	−6496.00	−0.856574
$387$	1541.00	0.202412
$388$	5696.00	0.745285
$389$	−3525.00	−0.459446	−0.229723	−	0.973256i	$-0.573782\pi$
$389$	−3525.00	−0.459446	−0.229723	+	0.973256i	$0.573782\pi$
$390$	1872.00	0.243057
$391$	−4968.00	−0.642564
$392$	−4944.00	−0.637015
$393$	186.000	0.0238739
$394$	6252.00	0.799419
$395$	6300.00	0.802500
$396$	−5244.00	−0.665457
$397$	6629.00	0.838035	0.419018	−	0.907978i	$-0.362375\pi$
$397$	6629.00	0.838035	0.419018	+	0.907978i	$0.362375\pi$
$398$	5990.00	0.754401
$399$	1178.00	0.147804
$400$	−704.000	−0.0880000
$401$	−10848.0	−1.35093	−0.675465	−	0.737392i	$-0.736057\pi$
$401$	−10848.0	−1.35093	−0.675465	+	0.737392i	$0.736057\pi$
$402$	−3424.00	−0.424810
$403$	−1664.00	−0.205682
$404$	4248.00	0.523133
$405$	−3789.00	−0.464881
$406$	−9300.00	−1.13683
$407$	−12882.0	−1.56889
$408$	1104.00	0.133961
$409$	−3040.00	−0.367526	−0.183763	−	0.982971i	$-0.558828\pi$
$409$	−3040.00	−0.367526	−0.183763	+	0.982971i	$0.558828\pi$
$410$	−4644.00	−0.559392
$411$	−2538.00	−0.304599
$412$	4712.00	0.563455
$413$	10230.0	1.21885
$414$	−3312.00	−0.393179
$415$	108.000	0.0127747
$416$	1664.00	0.196116
$417$	3950.00	0.463867
$418$	−2166.00	−0.253451
$419$	−3900.00	−0.454719	−0.227360	−	0.973811i	$-0.573009\pi$
$419$	−3900.00	−0.454719	−0.227360	+	0.973811i	$0.573009\pi$
$420$	−2232.00	−0.259311
$421$	4412.00	0.510755	0.255377	−	0.966841i	$-0.417800\pi$
$421$	4412.00	0.510755	0.255377	+	0.966841i	$0.417800\pi$
$422$	8636.00	0.996194
$423$	−13317.0	−1.53072
$424$	3456.00	0.395845
$425$	−3036.00	−0.346512
$426$	2568.00	0.292066
$427$	403.000	0.0456734
$428$	456.000	0.0514990
$429$	5928.00	0.667148
$430$	−1206.00	−0.135252
$431$	432.000	0.0482801	0.0241400	−	0.999709i	$-0.492315\pi$
$431$	432.000	0.0482801	0.0241400	+	0.999709i	$0.492315\pi$
$432$	1600.00	0.178195
$433$	−2002.00	−0.222194	−0.111097	−	0.993810i	$-0.535436\pi$
$433$	−2002.00	−0.222194	−0.111097	+	0.993810i	$0.535436\pi$
$434$	1984.00	0.219436
$435$	−2700.00	−0.297598
$436$	5840.00	0.641480
$437$	−1368.00	−0.149749
$438$	−1948.00	−0.212509
$439$	−1690.00	−0.183734	−0.0918671	−	0.995771i	$-0.529283\pi$
$439$	−1690.00	−0.183734	−0.0918671	+	0.995771i	$0.529283\pi$
$440$	4104.00	0.444660
$441$	−14214.0	−1.53482
$442$	7176.00	0.772234
$443$	−1977.00	−0.212032	−0.106016	−	0.994364i	$-0.533809\pi$
$443$	−1977.00	−0.212032	−0.106016	+	0.994364i	$0.533809\pi$
$444$	1808.00	0.193252
$445$	5400.00	0.575246
$446$	−1036.00	−0.109991
$447$	3390.00	0.358706
$448$	−1984.00	−0.209230
$449$	−2760.00	−0.290095	−0.145047	−	0.989425i	$-0.546333\pi$
$449$	−2760.00	−0.290095	−0.145047	+	0.989425i	$0.546333\pi$
$450$	−2024.00	−0.212027
$451$	−14706.0	−1.53543
$452$	−3288.00	−0.342156
$453$	−3604.00	−0.373798
$454$	−5688.00	−0.587998
$455$	−14508.0	−1.49483
$456$	304.000	0.0312195
$457$	4499.00	0.460513	0.230256	−	0.973130i	$-0.426044\pi$
$457$	4499.00	0.460513	0.230256	+	0.973130i	$0.426044\pi$
$458$	−3490.00	−0.356063
$459$	6900.00	0.701665
$460$	2592.00	0.262723
$461$	−11643.0	−1.17629	−0.588144	−	0.808756i	$-0.700141\pi$
$461$	−11643.0	−1.17629	−0.588144	+	0.808756i	$0.700141\pi$
$462$	−7068.00	−0.711760
$463$	−1537.00	−0.154277	−0.0771387	−	0.997020i	$-0.524578\pi$
$463$	−1537.00	−0.154277	−0.0771387	+	0.997020i	$0.524578\pi$
$464$	−2400.00	−0.240123
$465$	576.000	0.0574438
$466$	−10566.0	−1.05034
$467$	−7641.00	−0.757138	−0.378569	−	0.925573i	$-0.623584\pi$
$467$	−7641.00	−0.757138	−0.378569	+	0.925573i	$0.623584\pi$
$468$	4784.00	0.472522
$469$	26536.0	2.61262
$470$	10422.0	1.02283
$471$	6452.00	0.631194
$472$	2640.00	0.257449
$473$	−3819.00	−0.371243
$474$	−2800.00	−0.271325
$475$	−836.000	−0.0807543
$476$	−8556.00	−0.823873
$477$	9936.00	0.953749
$478$	−930.000	−0.0889900
$479$	−8580.00	−0.818435	−0.409217	−	0.912437i	$-0.634198\pi$
$479$	−8580.00	−0.818435	−0.409217	+	0.912437i	$0.634198\pi$
$480$	−576.000	−0.0547723
$481$	11752.0	1.11402
$482$	14156.0	1.33773
$483$	−4464.00	−0.420536
$484$	7672.00	0.720511
$485$	−12816.0	−1.19989
$486$	7084.00	0.661187
$487$	12134.0	1.12904	0.564522	−	0.825418i	$-0.309061\pi$
$487$	12134.0	1.12904	0.564522	+	0.825418i	$0.309061\pi$
$488$	104.000	0.00964725
$489$	−2536.00	−0.234523
$490$	11124.0	1.02557
$491$	−5508.00	−0.506258	−0.253129	−	0.967433i	$-0.581460\pi$
$491$	−5508.00	−0.506258	−0.253129	+	0.967433i	$0.581460\pi$
$492$	2064.00	0.189131
$493$	−10350.0	−0.945518
$494$	1976.00	0.179969
$495$	11799.0	1.07136
$496$	512.000	0.0463498
$497$	−19902.0	−1.79623
$498$	−48.0000	−0.00431914
$499$	−11905.0	−1.06802	−0.534009	−	0.845479i	$-0.679315\pi$
$499$	−11905.0	−1.06802	−0.534009	+	0.845479i	$0.679315\pi$
$500$	6084.00	0.544170
$501$	−1308.00	−0.116641
$502$	−7134.00	−0.634275
$503$	9108.00	0.807367	0.403684	−	0.914899i	$-0.367730\pi$
$503$	9108.00	0.807367	0.403684	+	0.914899i	$0.367730\pi$
$504$	−5704.00	−0.504120
$505$	−9558.00	−0.842229
$506$	8208.00	0.721127
$507$	−1014.00	−0.0888231
$508$	−8344.00	−0.728750
$509$	−2520.00	−0.219444	−0.109722	−	0.993962i	$-0.534996\pi$
$509$	−2520.00	−0.219444	−0.109722	+	0.993962i	$0.534996\pi$
$510$	−2484.00	−0.215673
$511$	15097.0	1.30695
$512$	−512.000	−0.0441942
$513$	1900.00	0.163523
$514$	3792.00	0.325405
$515$	−10602.0	−0.907146
$516$	536.000	0.0457288
$517$	33003.0	2.80749
$518$	−14012.0	−1.18852
$519$	2724.00	0.230386
$520$	−3744.00	−0.315741
$521$	21612.0	1.81735	0.908675	−	0.417505i	$-0.137095\pi$
$521$	21612.0	1.81735	0.908675	+	0.417505i	$0.137095\pi$
$522$	−6900.00	−0.578553
$523$	−9022.00	−0.754311	−0.377155	−	0.926150i	$-0.623098\pi$
$523$	−9022.00	−0.754311	−0.377155	+	0.926150i	$0.623098\pi$
$524$	−372.000	−0.0310132
$525$	−2728.00	−0.226780
$526$	114.000	0.00944988
$527$	2208.00	0.182509
$528$	−1824.00	−0.150340
$529$	−6983.00	−0.573929
$530$	−7776.00	−0.637298
$531$	7590.00	0.620297
$532$	−2356.00	−0.192003
$533$	13416.0	1.09027
$534$	−2400.00	−0.194491
$535$	−1026.00	−0.0829119
$536$	6848.00	0.551844
$537$	420.000	0.0337511
$538$	−5400.00	−0.432733
$539$	35226.0	2.81501
$540$	−3600.00	−0.286888
$541$	−9253.00	−0.735337	−0.367669	−	0.929957i	$-0.619844\pi$
$541$	−9253.00	−0.735337	−0.367669	+	0.929957i	$0.619844\pi$
$542$	−7744.00	−0.613715
$543$	−4.00000	−0.000316126 0
$544$	−2208.00	−0.174021
$545$	−13140.0	−1.03276
$546$	6448.00	0.505401
$547$	13244.0	1.03523	0.517617	−	0.855613i	$-0.326819\pi$
$547$	13244.0	1.03523	0.517617	+	0.855613i	$0.326819\pi$
$548$	5076.00	0.395686
$549$	299.000	0.0232441
$550$	5016.00	0.388878
$551$	−2850.00	−0.220352
$552$	−1152.00	−0.0888268
$553$	21700.0	1.66868
$554$	15422.0	1.18270
$555$	−4068.00	−0.311130
$556$	−7900.00	−0.602580
$557$	1569.00	0.119355	0.0596774	−	0.998218i	$-0.480993\pi$
$557$	1569.00	0.119355	0.0596774	+	0.998218i	$0.480993\pi$
$558$	1472.00	0.111675
$559$	3484.00	0.263609
$560$	4464.00	0.336854
$561$	−7866.00	−0.591984
$562$	13716.0	1.02949
$563$	−15762.0	−1.17991	−0.589955	−	0.807436i	$-0.700854\pi$
$563$	−15762.0	−1.17991	−0.589955	+	0.807436i	$0.700854\pi$
$564$	−4632.00	−0.345820
$565$	7398.00	0.550861
$566$	3614.00	0.268388
$567$	−13051.0	−0.966650
$568$	−5136.00	−0.379405
$569$	−13800.0	−1.01674	−0.508371	−	0.861138i	$-0.669752\pi$
$569$	−13800.0	−1.01674	−0.508371	+	0.861138i	$0.669752\pi$
$570$	−684.000	−0.0502625
$571$	−4348.00	−0.318666	−0.159333	−	0.987225i	$-0.550934\pi$
$571$	−4348.00	−0.318666	−0.159333	+	0.987225i	$0.550934\pi$
$572$	−11856.0	−0.866651
$573$	5286.00	0.385385
$574$	−15996.0	−1.16317
$575$	3168.00	0.229765
$576$	−1472.00	−0.106481
$577$	3539.00	0.255339	0.127669	−	0.991817i	$-0.459250\pi$
$577$	3539.00	0.255339	0.127669	+	0.991817i	$0.459250\pi$
$578$	304.000	0.0218767
$579$	−6496.00	−0.466260
$580$	5400.00	0.386591
$581$	372.000	0.0265631
$582$	5696.00	0.405682
$583$	−24624.0	−1.74927
$584$	3896.00	0.276058
$585$	−10764.0	−0.760746
$586$	6024.00	0.424657
$587$	−6321.00	−0.444456	−0.222228	−	0.974995i	$-0.571333\pi$
$587$	−6321.00	−0.444456	−0.222228	+	0.974995i	$0.571333\pi$
$588$	−4944.00	−0.346747
$589$	608.000	0.0425335
$590$	−5940.00	−0.414485
$591$	6252.00	0.435149
$592$	−3616.00	−0.251042
$593$	13278.0	0.919498	0.459749	−	0.888049i	$-0.347939\pi$
$593$	13278.0	0.919498	0.459749	+	0.888049i	$0.347939\pi$
$594$	−11400.0	−0.787454
$595$	19251.0	1.32641
$596$	−6780.00	−0.465973
$597$	5990.00	0.410644
$598$	−7488.00	−0.512052
$599$	20400.0	1.39152	0.695761	−	0.718274i	$-0.255067\pi$
$599$	20400.0	1.39152	0.695761	+	0.718274i	$0.255067\pi$
$600$	−704.000	−0.0479011
$601$	−22198.0	−1.50661	−0.753307	−	0.657669i	$-0.771543\pi$
$601$	−22198.0	−1.50661	−0.753307	+	0.657669i	$0.771543\pi$
$602$	−4154.00	−0.281237
$603$	19688.0	1.32961
$604$	7208.00	0.485578
$605$	−17262.0	−1.16000
$606$	4248.00	0.284758
$607$	9824.00	0.656909	0.328455	−	0.944520i	$-0.393472\pi$
$607$	9824.00	0.656909	0.328455	+	0.944520i	$0.393472\pi$
$608$	−608.000	−0.0405554
$609$	−9300.00	−0.618810
$610$	−234.000	−0.0155318
$611$	−30108.0	−1.99352
$612$	−6348.00	−0.419285
$613$	−4327.00	−0.285099	−0.142550	−	0.989788i	$-0.545530\pi$
$613$	−4327.00	−0.285099	−0.142550	+	0.989788i	$0.545530\pi$
$614$	2192.00	0.144075
$615$	−4644.00	−0.304495
$616$	14136.0	0.924603
$617$	−14151.0	−0.923335	−0.461668	−	0.887053i	$-0.652749\pi$
$617$	−14151.0	−0.923335	−0.461668	+	0.887053i	$0.652749\pi$
$618$	4712.00	0.306706
$619$	22460.0	1.45839	0.729195	−	0.684306i	$-0.239895\pi$
$619$	22460.0	1.45839	0.729195	+	0.684306i	$0.239895\pi$
$620$	−1152.00	−0.0746217
$621$	−7200.00	−0.465259
$622$	−3894.00	−0.251021
$623$	18600.0	1.19614
$624$	1664.00	0.106752
$625$	−8189.00	−0.524096
$626$	−15196.0	−0.970215
$627$	−2166.00	−0.137961
$628$	−12904.0	−0.819945
$629$	−15594.0	−0.988511
$630$	12834.0	0.811617
$631$	−16363.0	−1.03233	−0.516165	−	0.856489i	$-0.672641\pi$
$631$	−16363.0	−1.03233	−0.516165	+	0.856489i	$0.672641\pi$
$632$	5600.00	0.352462
$633$	8636.00	0.542259
$634$	−16668.0	−1.04412
$635$	18774.0	1.17327
$636$	3456.00	0.215471
$637$	−32136.0	−1.99886
$638$	17100.0	1.06112
$639$	−14766.0	−0.914138
$640$	1152.00	0.0711512
$641$	5592.00	0.344572	0.172286	−	0.985047i	$-0.444885\pi$
$641$	5592.00	0.344572	0.172286	+	0.985047i	$0.444885\pi$
$642$	456.000	0.0280325
$643$	16553.0	1.01522	0.507610	−	0.861587i	$-0.330529\pi$
$643$	16553.0	1.01522	0.507610	+	0.861587i	$0.330529\pi$
$644$	8928.00	0.546293
$645$	−1206.00	−0.0736220
$646$	−2622.00	−0.159692
$647$	−4611.00	−0.280181	−0.140091	−	0.990139i	$-0.544739\pi$
$647$	−4611.00	−0.280181	−0.140091	+	0.990139i	$0.544739\pi$
$648$	−3368.00	−0.204178
$649$	−18810.0	−1.13768
$650$	−4576.00	−0.276132
$651$	1984.00	0.119446
$652$	5072.00	0.304655
$653$	16413.0	0.983599	0.491800	−	0.870708i	$-0.336339\pi$
$653$	16413.0	0.983599	0.491800	+	0.870708i	$0.336339\pi$
$654$	5840.00	0.349177
$655$	837.000	0.0499302
$656$	−4128.00	−0.245688
$657$	11201.0	0.665133
$658$	35898.0	2.12682
$659$	27390.0	1.61906	0.809532	−	0.587076i	$-0.199721\pi$
$659$	27390.0	1.61906	0.809532	+	0.587076i	$0.199721\pi$
$660$	4104.00	0.242042
$661$	26912.0	1.58359	0.791797	−	0.610784i	$-0.209146\pi$
$661$	26912.0	1.58359	0.791797	+	0.610784i	$0.209146\pi$
$662$	16736.0	0.982573
$663$	7176.00	0.420351
$664$	96.0000	0.00561073
$665$	5301.00	0.309119
$666$	−10396.0	−0.604860
$667$	10800.0	0.626953
$668$	2616.00	0.151521
$669$	−1036.00	−0.0598716
$670$	−15408.0	−0.888452
$671$	−741.000	−0.0426319
$672$	−1984.00	−0.113891
$673$	−21562.0	−1.23500	−0.617499	−	0.786571i	$-0.711854\pi$
$673$	−21562.0	−1.23500	−0.617499	+	0.786571i	$0.711854\pi$
$674$	20672.0	1.18139
$675$	−4400.00	−0.250898
$676$	2028.00	0.115385
$677$	−21966.0	−1.24700	−0.623502	−	0.781822i	$-0.714291\pi$
$677$	−21966.0	−1.24700	−0.623502	+	0.781822i	$0.714291\pi$
$678$	−3288.00	−0.186246
$679$	−44144.0	−2.49498
$680$	4968.00	0.280168
$681$	−5688.00	−0.320066
$682$	−3648.00	−0.204823
$683$	15348.0	0.859846	0.429923	−	0.902866i	$-0.358541\pi$
$683$	15348.0	0.859846	0.429923	+	0.902866i	$0.358541\pi$
$684$	−1748.00	−0.0977141
$685$	−11421.0	−0.637042
$686$	17050.0	0.948939
$687$	−3490.00	−0.193816
$688$	−1072.00	−0.0594035
$689$	22464.0	1.24210
$690$	2592.00	0.143008
$691$	8147.00	0.448519	0.224259	−	0.974529i	$-0.428004\pi$
$691$	8147.00	0.448519	0.224259	+	0.974529i	$0.428004\pi$
$692$	−5448.00	−0.299280
$693$	40641.0	2.22774
$694$	−13758.0	−0.752517
$695$	17775.0	0.970136
$696$	−2400.00	−0.130707
$697$	−17802.0	−0.967430
$698$	12710.0	0.689227
$699$	−10566.0	−0.571735
$700$	5456.00	0.294596
$701$	14982.0	0.807222	0.403611	−	0.914931i	$-0.367755\pi$
$701$	14982.0	0.807222	0.403611	+	0.914931i	$0.367755\pi$
$702$	10400.0	0.559149
$703$	−4294.00	−0.230372
$704$	3648.00	0.195297
$705$	10422.0	0.556759
$706$	−14436.0	−0.769555
$707$	−32922.0	−1.75129
$708$	2640.00	0.140137
$709$	21890.0	1.15952	0.579758	−	0.814789i	$-0.303147\pi$
$709$	21890.0	1.15952	0.579758	+	0.814789i	$0.303147\pi$
$710$	11556.0	0.610830
$711$	16100.0	0.849222
$712$	4800.00	0.252651
$713$	−2304.00	−0.121018
$714$	−8556.00	−0.448460
$715$	26676.0	1.39528
$716$	−840.000	−0.0438440
$717$	−930.000	−0.0484400
$718$	−3330.00	−0.173084
$719$	−27015.0	−1.40124	−0.700619	−	0.713536i	$-0.747093\pi$
$719$	−27015.0	−1.40124	−0.700619	+	0.713536i	$0.747093\pi$
$720$	3312.00	0.171432
$721$	−36518.0	−1.88627
$722$	−722.000	−0.0372161
$723$	14156.0	0.728171
$724$	8.00000	0.000410660 0
$725$	6600.00	0.338094
$726$	7672.00	0.392196
$727$	−13021.0	−0.664267	−0.332134	−	0.943232i	$-0.607768\pi$
$727$	−13021.0	−0.664267	−0.332134	+	0.943232i	$0.607768\pi$
$728$	−12896.0	−0.656535
$729$	−4283.00	−0.217599
$730$	−8766.00	−0.444444
$731$	−4623.00	−0.233909
$732$	104.000	0.00525130
$733$	−6262.00	−0.315542	−0.157771	−	0.987476i	$-0.550431\pi$
$733$	−6262.00	−0.315542	−0.157771	+	0.987476i	$0.550431\pi$
$734$	−26128.0	−1.31390
$735$	11124.0	0.558252
$736$	2304.00	0.115389
$737$	−48792.0	−2.43864
$738$	−11868.0	−0.591961
$739$	−10855.0	−0.540335	−0.270168	−	0.962813i	$-0.587079\pi$
$739$	−10855.0	−0.540335	−0.270168	+	0.962813i	$0.587079\pi$
$740$	8136.00	0.404169
$741$	1976.00	0.0979624
$742$	−26784.0	−1.32516
$743$	−14892.0	−0.735309	−0.367654	−	0.929962i	$-0.619839\pi$
$743$	−14892.0	−0.735309	−0.367654	+	0.929962i	$0.619839\pi$
$744$	512.000	0.0252296
$745$	15255.0	0.750201
$746$	20984.0	1.02986
$747$	276.000	0.0135185
$748$	15732.0	0.769009
$749$	−3534.00	−0.172403
$750$	6084.00	0.296208
$751$	28952.0	1.40676	0.703378	−	0.710816i	$-0.251674\pi$
$751$	28952.0	1.40676	0.703378	+	0.710816i	$0.251674\pi$
$752$	9264.00	0.449233
$753$	−7134.00	−0.345256
$754$	−15600.0	−0.753473
$755$	−16218.0	−0.781766
$756$	−12400.0	−0.596539
$757$	−3541.00	−0.170013	−0.0850065	−	0.996380i	$-0.527091\pi$
$757$	−3541.00	−0.170013	−0.0850065	+	0.996380i	$0.527091\pi$
$758$	−15220.0	−0.729308
$759$	8208.00	0.392532
$760$	1368.00	0.0652929
$761$	22617.0	1.07735	0.538676	−	0.842513i	$-0.318925\pi$
$761$	22617.0	1.07735	0.538676	+	0.842513i	$0.318925\pi$
$762$	−8344.00	−0.396681
$763$	−45260.0	−2.14747
$764$	−10572.0	−0.500630
$765$	14283.0	0.675037
$766$	−8016.00	−0.378107
$767$	17160.0	0.807838
$768$	−512.000	−0.0240563
$769$	11495.0	0.539038	0.269519	−	0.962995i	$-0.413135\pi$
$769$	11495.0	0.539038	0.269519	+	0.962995i	$0.413135\pi$
$770$	−31806.0	−1.48858
$771$	3792.00	0.177128
$772$	12992.0	0.605690
$773$	−14622.0	−0.680358	−0.340179	−	0.940361i	$-0.610488\pi$
$773$	−14622.0	−0.680358	−0.340179	+	0.940361i	$0.610488\pi$
$774$	−3082.00	−0.143127
$775$	−1408.00	−0.0652605
$776$	−11392.0	−0.526996
$777$	−14012.0	−0.646947
$778$	7050.00	0.324878
$779$	−4902.00	−0.225459
$780$	−3744.00	−0.171868
$781$	36594.0	1.67661
$782$	9936.00	0.454361
$783$	−15000.0	−0.684618
$784$	9888.00	0.450437
$785$	29034.0	1.32009
$786$	−372.000	−0.0168814
$787$	7124.00	0.322672	0.161336	−	0.986900i	$-0.448420\pi$
$787$	7124.00	0.322672	0.161336	+	0.986900i	$0.448420\pi$
$788$	−12504.0	−0.565275
$789$	114.000	0.00514386
$790$	−12600.0	−0.567453
$791$	25482.0	1.14543
$792$	10488.0	0.470549
$793$	676.000	0.0302717
$794$	−13258.0	−0.592580
$795$	−7776.00	−0.346901
$796$	−11980.0	−0.533442
$797$	−3576.00	−0.158932	−0.0794658	−	0.996838i	$-0.525321\pi$
$797$	−3576.00	−0.158932	−0.0794658	+	0.996838i	$0.525321\pi$
$798$	−2356.00	−0.104513
$799$	39951.0	1.76892
$800$	1408.00	0.0622254
$801$	13800.0	0.608738
$802$	21696.0	0.955252
$803$	−27759.0	−1.21992
$804$	6848.00	0.300386
$805$	−20088.0	−0.879514
$806$	3328.00	0.145439
$807$	−5400.00	−0.235550
$808$	−8496.00	−0.369911
$809$	42855.0	1.86242	0.931212	−	0.364477i	$-0.118752\pi$
$809$	42855.0	1.86242	0.931212	+	0.364477i	$0.118752\pi$
$810$	7578.00	0.328721
$811$	−15568.0	−0.674065	−0.337032	−	0.941493i	$-0.609423\pi$
$811$	−15568.0	−0.674065	−0.337032	+	0.941493i	$0.609423\pi$
$812$	18600.0	0.803857
$813$	−7744.00	−0.334064
$814$	25764.0	1.10937
$815$	−11412.0	−0.490485
$816$	−2208.00	−0.0947248
$817$	−1273.00	−0.0545124
$818$	6080.00	0.259880
$819$	−37076.0	−1.58186
$820$	9288.00	0.395550
$821$	2517.00	0.106996	0.0534981	−	0.998568i	$-0.482963\pi$
$821$	2517.00	0.106996	0.0534981	+	0.998568i	$0.482963\pi$
$822$	5076.00	0.215384
$823$	−9727.00	−0.411983	−0.205991	−	0.978554i	$-0.566042\pi$
$823$	−9727.00	−0.411983	−0.205991	+	0.978554i	$0.566042\pi$
$824$	−9424.00	−0.398423
$825$	5016.00	0.211678
$826$	−20460.0	−0.861858
$827$	28224.0	1.18675	0.593376	−	0.804925i	$-0.297795\pi$
$827$	28224.0	1.18675	0.593376	+	0.804925i	$0.297795\pi$
$828$	6624.00	0.278019
$829$	3080.00	0.129038	0.0645192	−	0.997916i	$-0.479449\pi$
$829$	3080.00	0.129038	0.0645192	+	0.997916i	$0.479449\pi$
$830$	−216.000	−0.00903310
$831$	15422.0	0.643782
$832$	−3328.00	−0.138675
$833$	42642.0	1.77366
$834$	−7900.00	−0.328003
$835$	−5886.00	−0.243944
$836$	4332.00	0.179217
$837$	3200.00	0.132148
$838$	7800.00	0.321535
$839$	26790.0	1.10238	0.551188	−	0.834381i	$-0.314175\pi$
$839$	26790.0	1.10238	0.551188	+	0.834381i	$0.314175\pi$
$840$	4464.00	0.183360
$841$	−1889.00	−0.0774530
$842$	−8824.00	−0.361158
$843$	13716.0	0.560385
$844$	−17272.0	−0.704416
$845$	−4563.00	−0.185766
$846$	26634.0	1.08238
$847$	−59458.0	−2.41204
$848$	−6912.00	−0.279905
$849$	3614.00	0.146092
$850$	6072.00	0.245021
$851$	16272.0	0.655461
$852$	−5136.00	−0.206522
$853$	19178.0	0.769803	0.384902	−	0.922958i	$-0.374235\pi$
$853$	19178.0	0.769803	0.384902	+	0.922958i	$0.374235\pi$
$854$	−806.000	−0.0322960
$855$	3933.00	0.157317
$856$	−912.000	−0.0364153
$857$	−2406.00	−0.0959013	−0.0479506	−	0.998850i	$-0.515269\pi$
$857$	−2406.00	−0.0959013	−0.0479506	+	0.998850i	$0.515269\pi$
$858$	−11856.0	−0.471745
$859$	9125.00	0.362446	0.181223	−	0.983442i	$-0.441994\pi$
$859$	9125.00	0.362446	0.181223	+	0.983442i	$0.441994\pi$
$860$	2412.00	0.0956378
$861$	−15996.0	−0.633150
$862$	−864.000	−0.0341392
$863$	8898.00	0.350975	0.175488	−	0.984482i	$-0.443850\pi$
$863$	8898.00	0.350975	0.175488	+	0.984482i	$0.443850\pi$
$864$	−3200.00	−0.126003
$865$	12258.0	0.481832
$866$	4004.00	0.157115
$867$	304.000	0.0119082
$868$	−3968.00	−0.155164
$869$	−39900.0	−1.55755
$870$	5400.00	0.210434
$871$	44512.0	1.73161
$872$	−11680.0	−0.453595
$873$	−32752.0	−1.26974
$874$	2736.00	0.105889
$875$	−47151.0	−1.82171
$876$	3896.00	0.150267
$877$	−15886.0	−0.611667	−0.305834	−	0.952085i	$-0.598935\pi$
$877$	−15886.0	−0.611667	−0.305834	+	0.952085i	$0.598935\pi$
$878$	3380.00	0.129920
$879$	6024.00	0.231154
$880$	−8208.00	−0.314422
$881$	−25683.0	−0.982159	−0.491080	−	0.871115i	$-0.663398\pi$
$881$	−25683.0	−0.982159	−0.491080	+	0.871115i	$0.663398\pi$
$882$	28428.0	1.08528
$883$	−28267.0	−1.07730	−0.538652	−	0.842528i	$-0.681066\pi$
$883$	−28267.0	−1.07730	−0.538652	+	0.842528i	$0.681066\pi$
$884$	−14352.0	−0.546052
$885$	−5940.00	−0.225617
$886$	3954.00	0.149929
$887$	−2466.00	−0.0933486	−0.0466743	−	0.998910i	$-0.514862\pi$
$887$	−2466.00	−0.0933486	−0.0466743	+	0.998910i	$0.514862\pi$
$888$	−3616.00	−0.136650
$889$	64666.0	2.43963
$890$	−10800.0	−0.406760
$891$	23997.0	0.902278
$892$	2072.00	0.0777754
$893$	11001.0	0.412245
$894$	−6780.00	−0.253643
$895$	1890.00	0.0705874
$896$	3968.00	0.147948
$897$	−7488.00	−0.278726
$898$	5520.00	0.205128
$899$	−4800.00	−0.178074
$900$	4048.00	0.149926
$901$	−29808.0	−1.10216
$902$	29412.0	1.08571
$903$	−4154.00	−0.153086
$904$	6576.00	0.241941
$905$	−18.0000	−0.000661149 0
$906$	7208.00	0.264315
$907$	29324.0	1.07353	0.536763	−	0.843733i	$-0.319647\pi$
$907$	29324.0	1.07353	0.536763	+	0.843733i	$0.319647\pi$
$908$	11376.0	0.415777
$909$	−24426.0	−0.891264
$910$	29016.0	1.05700
$911$	47142.0	1.71447	0.857236	−	0.514924i	$-0.172180\pi$
$911$	47142.0	1.71447	0.857236	+	0.514924i	$0.172180\pi$
$912$	−608.000	−0.0220755
$913$	−684.000	−0.0247942
$914$	−8998.00	−0.325632
$915$	−234.000	−0.00845443
$916$	6980.00	0.251775
$917$	2883.00	0.103822
$918$	−13800.0	−0.496152
$919$	−39940.0	−1.43362	−0.716811	−	0.697267i	$-0.754399\pi$
$919$	−39940.0	−1.43362	−0.716811	+	0.697267i	$0.754399\pi$
$920$	−5184.00	−0.185773
$921$	2192.00	0.0784244
$922$	23286.0	0.831761
$923$	−33384.0	−1.19052
$924$	14136.0	0.503290
$925$	9944.00	0.353467
$926$	3074.00	0.109091
$927$	−27094.0	−0.959961
$928$	4800.00	0.169793
$929$	−4410.00	−0.155745	−0.0778727	−	0.996963i	$-0.524813\pi$
$929$	−4410.00	−0.155745	−0.0778727	+	0.996963i	$0.524813\pi$
$930$	−1152.00	−0.0406189
$931$	11742.0	0.413350
$932$	21132.0	0.742706
$933$	−3894.00	−0.136639
$934$	15282.0	0.535377
$935$	−35397.0	−1.23808
$936$	−9568.00	−0.334124
$937$	−41671.0	−1.45286	−0.726431	−	0.687239i	$-0.758822\pi$
$937$	−41671.0	−1.45286	−0.726431	+	0.687239i	$0.758822\pi$
$938$	−53072.0	−1.84740
$939$	−15196.0	−0.528118
$940$	−20844.0	−0.723251
$941$	4062.00	0.140720	0.0703599	−	0.997522i	$-0.477585\pi$
$941$	4062.00	0.140720	0.0703599	+	0.997522i	$0.477585\pi$
$942$	−12904.0	−0.446322
$943$	18576.0	0.641482
$944$	−5280.00	−0.182044
$945$	27900.0	0.960410
$946$	7638.00	0.262508
$947$	−45036.0	−1.54538	−0.772689	−	0.634785i	$-0.781089\pi$
$947$	−45036.0	−1.54538	−0.772689	+	0.634785i	$0.781089\pi$
$948$	5600.00	0.191856
$949$	25324.0	0.866230
$950$	1672.00	0.0571019
$951$	−16668.0	−0.568346
$952$	17112.0	0.582566
$953$	26508.0	0.901027	0.450513	−	0.892770i	$-0.351241\pi$
$953$	26508.0	0.901027	0.450513	+	0.892770i	$0.351241\pi$
$954$	−19872.0	−0.674402
$955$	23787.0	0.805999
$956$	1860.00	0.0629254
$957$	17100.0	0.577601
$958$	17160.0	0.578721
$959$	−39339.0	−1.32463
$960$	1152.00	0.0387298
$961$	−28767.0	−0.965627
$962$	−23504.0	−0.787733
$963$	−2622.00	−0.0877391
$964$	−28312.0	−0.945921
$965$	−29232.0	−0.975141
$966$	8928.00	0.297364
$967$	−15976.0	−0.531286	−0.265643	−	0.964071i	$-0.585584\pi$
$967$	−15976.0	−0.531286	−0.265643	+	0.964071i	$0.585584\pi$
$968$	−15344.0	−0.509478
$969$	−2622.00	−0.0869255
$970$	25632.0	0.848447
$971$	−39468.0	−1.30442	−0.652208	−	0.758040i	$-0.726157\pi$
$971$	−39468.0	−1.30442	−0.652208	+	0.758040i	$0.726157\pi$
$972$	−14168.0	−0.467530
$973$	61225.0	2.01725
$974$	−24268.0	−0.798354
$975$	−4576.00	−0.150307
$976$	−208.000	−0.00682164
$977$	21804.0	0.713994	0.356997	−	0.934106i	$-0.383801\pi$
$977$	21804.0	0.713994	0.356997	+	0.934106i	$0.383801\pi$
$978$	5072.00	0.165833
$979$	−34200.0	−1.11648
$980$	−22248.0	−0.725190
$981$	−33580.0	−1.09289
$982$	11016.0	0.357978
$983$	11268.0	0.365609	0.182804	−	0.983149i	$-0.441483\pi$
$983$	11268.0	0.365609	0.182804	+	0.983149i	$0.441483\pi$
$984$	−4128.00	−0.133736
$985$	28134.0	0.910075
$986$	20700.0	0.668582
$987$	35898.0	1.15770
$988$	−3952.00	−0.127257
$989$	4824.00	0.155100
$990$	−23598.0	−0.757569
$991$	−778.000	−0.0249384	−0.0124692	−	0.999922i	$-0.503969\pi$
$991$	−778.000	−0.0249384	−0.0124692	+	0.999922i	$0.503969\pi$
$992$	−1024.00	−0.0327742
$993$	16736.0	0.534845
$994$	39804.0	1.27013
$995$	26955.0	0.858825
$996$	96.0000	0.00305409
$997$	389.000	0.0123568	0.00617841	−	0.999981i	$-0.498033\pi$
$997$	389.000	0.0123568	0.00617841	+	0.999981i	$0.498033\pi$
$998$	23810.0	0.755203
$999$	−22600.0	−0.715748

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.4.a.a.1.1	✓	1
3.2	odd	2		342.4.a.d.1.1		1
4.3	odd	2		304.4.a.a.1.1		1
5.2	odd	4		950.4.b.d.799.1		2
5.3	odd	4		950.4.b.d.799.2		2
5.4	even	2		950.4.a.d.1.1		1
7.6	odd	2		1862.4.a.a.1.1		1
8.3	odd	2		1216.4.a.b.1.1		1
8.5	even	2		1216.4.a.e.1.1		1
19.18	odd	2		722.4.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.4.a.a.1.1	✓	1	1.1	even	1	trivial
304.4.a.a.1.1		1	4.3	odd	2
342.4.a.d.1.1		1	3.2	odd	2
722.4.a.d.1.1		1	19.18	odd	2
950.4.a.d.1.1		1	5.4	even	2
950.4.b.d.799.1		2	5.2	odd	4
950.4.b.d.799.2		2	5.3	odd	4
1216.4.a.b.1.1		1	8.3	odd	2
1216.4.a.e.1.1		1	8.5	even	2
1862.4.a.a.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:	\( N \)	\(=\)	\( 38 = 2 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	38.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more