LMFDB - Embedded newform 138.4.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [138,4,Mod(1,138)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(138, 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("138.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 138 = 2 \cdot 3 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	138.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:	$8.14226358079$
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	$\chi$	$=$	138.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} +2.00000 q^{5} -6.00000 q^{6} -32.0000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$

\(q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} +2.00000 q^{5} -6.00000 q^{6} -32.0000 q^{7} -8.00000 q^{8} +9.00000 q^{9} -4.00000 q^{10} -48.0000 q^{11} +12.0000 q^{12} +22.0000 q^{13} +64.0000 q^{14} +6.00000 q^{15} +16.0000 q^{16} +42.0000 q^{17} -18.0000 q^{18} -144.000 q^{19} +8.00000 q^{20} -96.0000 q^{21} +96.0000 q^{22} -23.0000 q^{23} -24.0000 q^{24} -121.000 q^{25} -44.0000 q^{26} +27.0000 q^{27} -128.000 q^{28} +174.000 q^{29} -12.0000 q^{30} -304.000 q^{31} -32.0000 q^{32} -144.000 q^{33} -84.0000 q^{34} -64.0000 q^{35} +36.0000 q^{36} -318.000 q^{37} +288.000 q^{38} +66.0000 q^{39} -16.0000 q^{40} +74.0000 q^{41} +192.000 q^{42} +192.000 q^{43} -192.000 q^{44} +18.0000 q^{45} +46.0000 q^{46} +392.000 q^{47} +48.0000 q^{48} +681.000 q^{49} +242.000 q^{50} +126.000 q^{51} +88.0000 q^{52} -734.000 q^{53} -54.0000 q^{54} -96.0000 q^{55} +256.000 q^{56} -432.000 q^{57} -348.000 q^{58} +156.000 q^{59} +24.0000 q^{60} +706.000 q^{61} +608.000 q^{62} -288.000 q^{63} +64.0000 q^{64} +44.0000 q^{65} +288.000 q^{66} +192.000 q^{67} +168.000 q^{68} -69.0000 q^{69} +128.000 q^{70} +624.000 q^{71} -72.0000 q^{72} -406.000 q^{73} +636.000 q^{74} -363.000 q^{75} -576.000 q^{76} +1536.00 q^{77} -132.000 q^{78} +696.000 q^{79} +32.0000 q^{80} +81.0000 q^{81} -148.000 q^{82} -800.000 q^{83} -384.000 q^{84} +84.0000 q^{85} -384.000 q^{86} +522.000 q^{87} +384.000 q^{88} -102.000 q^{89} -36.0000 q^{90} -704.000 q^{91} -92.0000 q^{92} -912.000 q^{93} -784.000 q^{94} -288.000 q^{95} -96.0000 q^{96} -918.000 q^{97} -1362.00 q^{98} -432.000 q^{99} +O(q^{100})\)

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 $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	4.00000	0.500000
$5$	2.00000	0.178885	0.0894427	−	0.995992i	$-0.471491\pi$
$5$	2.00000	0.178885	0.0894427	+	0.995992i	$0.471491\pi$
$6$	−6.00000	−0.408248
$7$	−32.0000	−1.72784	−0.863919	−	0.503631i	$-0.831997\pi$
$7$	−32.0000	−1.72784	−0.863919	+	0.503631i	$0.831997\pi$
$8$	−8.00000	−0.353553
$9$	9.00000	0.333333
$10$	−4.00000	−0.126491
$11$	−48.0000	−1.31569	−0.657843	−	0.753155i	$-0.728531\pi$
$11$	−48.0000	−1.31569	−0.657843	+	0.753155i	$0.728531\pi$
$12$	12.0000	0.288675
$13$	22.0000	0.469362	0.234681	−	0.972072i	$-0.424595\pi$
$13$	22.0000	0.469362	0.234681	+	0.972072i	$0.424595\pi$
$14$	64.0000	1.22177
$15$	6.00000	0.103280
$16$	16.0000	0.250000
$17$	42.0000	0.599206	0.299603	−	0.954064i	$-0.403146\pi$
$17$	42.0000	0.599206	0.299603	+	0.954064i	$0.403146\pi$
$18$	−18.0000	−0.235702
$19$	−144.000	−1.73873	−0.869365	−	0.494171i	$-0.835472\pi$
$19$	−144.000	−1.73873	−0.869365	+	0.494171i	$0.835472\pi$
$20$	8.00000	0.0894427
$21$	−96.0000	−0.997567
$22$	96.0000	0.930330
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	−24.0000	−0.204124
$25$	−121.000	−0.968000
$26$	−44.0000	−0.331889
$27$	27.0000	0.192450
$28$	−128.000	−0.863919
$29$	174.000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	174.000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	−12.0000	−0.0730297
$31$	−304.000	−1.76129	−0.880645	−	0.473776i	$-0.842891\pi$
$31$	−304.000	−1.76129	−0.880645	+	0.473776i	$0.842891\pi$
$32$	−32.0000	−0.176777
$33$	−144.000	−0.759612
$34$	−84.0000	−0.423702
$35$	−64.0000	−0.309085
$36$	36.0000	0.166667
$37$	−318.000	−1.41294	−0.706471	−	0.707742i	$-0.749714\pi$
$37$	−318.000	−1.41294	−0.706471	+	0.707742i	$0.749714\pi$
$38$	288.000	1.22947
$39$	66.0000	0.270986
$40$	−16.0000	−0.0632456
$41$	74.0000	0.281875	0.140937	−	0.990019i	$-0.454988\pi$
$41$	74.0000	0.281875	0.140937	+	0.990019i	$0.454988\pi$
$42$	192.000	0.705387
$43$	192.000	0.680924	0.340462	−	0.940258i	$-0.389417\pi$
$43$	192.000	0.680924	0.340462	+	0.940258i	$0.389417\pi$
$44$	−192.000	−0.657843
$45$	18.0000	0.0596285
$46$	46.0000	0.147442
$47$	392.000	1.21658	0.608288	−	0.793716i	$-0.291857\pi$
$47$	392.000	1.21658	0.608288	+	0.793716i	$0.291857\pi$
$48$	48.0000	0.144338
$49$	681.000	1.98542
$50$	242.000	0.684479
$51$	126.000	0.345952
$52$	88.0000	0.234681
$53$	−734.000	−1.90231	−0.951157	−	0.308707i	$-0.900104\pi$
$53$	−734.000	−1.90231	−0.951157	+	0.308707i	$0.900104\pi$
$54$	−54.0000	−0.136083
$55$	−96.0000	−0.235357
$56$	256.000	0.610883
$57$	−432.000	−1.00386
$58$	−348.000	−0.787839
$59$	156.000	0.344228	0.172114	−	0.985077i	$-0.444940\pi$
$59$	156.000	0.344228	0.172114	+	0.985077i	$0.444940\pi$
$60$	24.0000	0.0516398
$61$	706.000	1.48187	0.740935	−	0.671577i	$-0.234383\pi$
$61$	706.000	1.48187	0.740935	+	0.671577i	$0.234383\pi$
$62$	608.000	1.24542
$63$	−288.000	−0.575946
$64$	64.0000	0.125000
$65$	44.0000	0.0839620
$66$	288.000	0.537127
$67$	192.000	0.350098	0.175049	−	0.984560i	$-0.443992\pi$
$67$	192.000	0.350098	0.175049	+	0.984560i	$0.443992\pi$
$68$	168.000	0.299603
$69$	−69.0000	−0.120386
$70$	128.000	0.218556
$71$	624.000	1.04303	0.521515	−	0.853242i	$-0.325367\pi$
$71$	624.000	1.04303	0.521515	+	0.853242i	$0.325367\pi$
$72$	−72.0000	−0.117851
$73$	−406.000	−0.650941	−0.325471	−	0.945552i	$-0.605523\pi$
$73$	−406.000	−0.650941	−0.325471	+	0.945552i	$0.605523\pi$
$74$	636.000	0.999101
$75$	−363.000	−0.558875
$76$	−576.000	−0.869365
$77$	1536.00	2.27329
$78$	−132.000	−0.191616
$79$	696.000	0.991217	0.495608	−	0.868546i	$-0.334945\pi$
$79$	696.000	0.991217	0.495608	+	0.868546i	$0.334945\pi$
$80$	32.0000	0.0447214
$81$	81.0000	0.111111
$82$	−148.000	−0.199315
$83$	−800.000	−1.05797	−0.528984	−	0.848632i	$-0.677427\pi$
$83$	−800.000	−1.05797	−0.528984	+	0.848632i	$0.677427\pi$
$84$	−384.000	−0.498784
$85$	84.0000	0.107189
$86$	−384.000	−0.481486
$87$	522.000	0.643268
$88$	384.000	0.465165
$89$	−102.000	−0.121483	−0.0607415	−	0.998154i	$-0.519347\pi$
$89$	−102.000	−0.121483	−0.0607415	+	0.998154i	$0.519347\pi$
$90$	−36.0000	−0.0421637
$91$	−704.000	−0.810981
$92$	−92.0000	−0.104257
$93$	−912.000	−1.01688
$94$	−784.000	−0.860249
$95$	−288.000	−0.311033
$96$	−96.0000	−0.102062
$97$	−918.000	−0.960915	−0.480458	−	0.877018i	$-0.659529\pi$
$97$	−918.000	−0.960915	−0.480458	+	0.877018i	$0.659529\pi$
$98$	−1362.00	−1.40391
$99$	−432.000	−0.438562
$100$	−484.000	−0.484000
$101$	366.000	0.360578	0.180289	−	0.983614i	$-0.442297\pi$
$101$	366.000	0.360578	0.180289	+	0.983614i	$0.442297\pi$
$102$	−252.000	−0.244625
$103$	−856.000	−0.818876	−0.409438	−	0.912338i	$-0.634275\pi$
$103$	−856.000	−0.818876	−0.409438	+	0.912338i	$0.634275\pi$
$104$	−176.000	−0.165944
$105$	−192.000	−0.178450
$106$	1468.00	1.34514
$107$	1096.00	0.990227	0.495114	−	0.868828i	$-0.335126\pi$
$107$	1096.00	0.990227	0.495114	+	0.868828i	$0.335126\pi$
$108$	108.000	0.0962250
$109$	1634.00	1.43586	0.717930	−	0.696115i	$-0.245090\pi$
$109$	1634.00	1.43586	0.717930	+	0.696115i	$0.245090\pi$
$110$	192.000	0.166423
$111$	−954.000	−0.815763
$112$	−512.000	−0.431959
$113$	−1006.00	−0.837491	−0.418746	−	0.908104i	$-0.637530\pi$
$113$	−1006.00	−0.837491	−0.418746	+	0.908104i	$0.637530\pi$
$114$	864.000	0.709833
$115$	−46.0000	−0.0373002
$116$	696.000	0.557086
$117$	198.000	0.156454
$118$	−312.000	−0.243406
$119$	−1344.00	−1.03533
$120$	−48.0000	−0.0365148
$121$	973.000	0.731029
$122$	−1412.00	−1.04784
$123$	222.000	0.162740
$124$	−1216.00	−0.880645
$125$	−492.000	−0.352047
$126$	576.000	0.407255
$127$	−1512.00	−1.05644	−0.528222	−	0.849107i	$-0.677141\pi$
$127$	−1512.00	−1.05644	−0.528222	+	0.849107i	$0.677141\pi$
$128$	−128.000	−0.0883883
$129$	576.000	0.393132
$130$	−88.0000	−0.0593701
$131$	−1180.00	−0.787001	−0.393500	−	0.919324i	$-0.628736\pi$
$131$	−1180.00	−0.787001	−0.393500	+	0.919324i	$0.628736\pi$
$132$	−576.000	−0.379806
$133$	4608.00	3.00424
$134$	−384.000	−0.247556
$135$	54.0000	0.0344265
$136$	−336.000	−0.211851
$137$	−1614.00	−1.00652	−0.503260	−	0.864135i	$-0.667866\pi$
$137$	−1614.00	−1.00652	−0.503260	+	0.864135i	$0.667866\pi$
$138$	138.000	0.0851257
$139$	−420.000	−0.256287	−0.128144	−	0.991756i	$-0.540902\pi$
$139$	−420.000	−0.256287	−0.128144	+	0.991756i	$0.540902\pi$
$140$	−256.000	−0.154542
$141$	1176.00	0.702391
$142$	−1248.00	−0.737534
$143$	−1056.00	−0.617533
$144$	144.000	0.0833333
$145$	348.000	0.199309
$146$	812.000	0.460285
$147$	2043.00	1.14628
$148$	−1272.00	−0.706471
$149$	2850.00	1.56699	0.783494	−	0.621400i	$-0.213436\pi$
$149$	2850.00	1.56699	0.783494	+	0.621400i	$0.213436\pi$
$150$	726.000	0.395184
$151$	−360.000	−0.194016	−0.0970079	−	0.995284i	$-0.530927\pi$
$151$	−360.000	−0.194016	−0.0970079	+	0.995284i	$0.530927\pi$
$152$	1152.00	0.614734
$153$	378.000	0.199735
$154$	−3072.00	−1.60746
$155$	−608.000	−0.315069
$156$	264.000	0.135493
$157$	2266.00	1.15189	0.575944	−	0.817489i	$-0.304635\pi$
$157$	2266.00	1.15189	0.575944	+	0.817489i	$0.304635\pi$
$158$	−1392.00	−0.700896
$159$	−2202.00	−1.09830
$160$	−64.0000	−0.0316228
$161$	736.000	0.360279
$162$	−162.000	−0.0785674
$163$	1452.00	0.697726	0.348863	−	0.937174i	$-0.386568\pi$
$163$	1452.00	0.697726	0.348863	+	0.937174i	$0.386568\pi$
$164$	296.000	0.140937
$165$	−288.000	−0.135883
$166$	1600.00	0.748097
$167$	1888.00	0.874837	0.437419	−	0.899258i	$-0.355893\pi$
$167$	1888.00	0.874837	0.437419	+	0.899258i	$0.355893\pi$
$168$	768.000	0.352693
$169$	−1713.00	−0.779700
$170$	−168.000	−0.0757942
$171$	−1296.00	−0.579577
$172$	768.000	0.340462
$173$	−210.000	−0.0922890	−0.0461445	−	0.998935i	$-0.514693\pi$
$173$	−210.000	−0.0922890	−0.0461445	+	0.998935i	$0.514693\pi$
$174$	−1044.00	−0.454859
$175$	3872.00	1.67255
$176$	−768.000	−0.328921
$177$	468.000	0.198740
$178$	204.000	0.0859014
$179$	−3092.00	−1.29110	−0.645550	−	0.763718i	$-0.723372\pi$
$179$	−3092.00	−1.29110	−0.645550	+	0.763718i	$0.723372\pi$
$180$	72.0000	0.0298142
$181$	−166.000	−0.0681695	−0.0340848	−	0.999419i	$-0.510852\pi$
$181$	−166.000	−0.0681695	−0.0340848	+	0.999419i	$0.510852\pi$
$182$	1408.00	0.573450
$183$	2118.00	0.855558
$184$	184.000	0.0737210
$185$	−636.000	−0.252755
$186$	1824.00	0.719044
$187$	−2016.00	−0.788366
$188$	1568.00	0.608288
$189$	−864.000	−0.332522
$190$	576.000	0.219934
$191$	360.000	0.136381	0.0681903	−	0.997672i	$-0.478278\pi$
$191$	360.000	0.136381	0.0681903	+	0.997672i	$0.478278\pi$
$192$	192.000	0.0721688
$193$	−1774.00	−0.661634	−0.330817	−	0.943695i	$-0.607324\pi$
$193$	−1774.00	−0.661634	−0.330817	+	0.943695i	$0.607324\pi$
$194$	1836.00	0.679470
$195$	132.000	0.0484755
$196$	2724.00	0.992711
$197$	−4938.00	−1.78588	−0.892939	−	0.450178i	$-0.851361\pi$
$197$	−4938.00	−1.78588	−0.892939	+	0.450178i	$0.851361\pi$
$198$	864.000	0.310110
$199$	−2512.00	−0.894829	−0.447414	−	0.894327i	$-0.647655\pi$
$199$	−2512.00	−0.894829	−0.447414	+	0.894327i	$0.647655\pi$
$200$	968.000	0.342240
$201$	576.000	0.202129
$202$	−732.000	−0.254967
$203$	−5568.00	−1.92511
$204$	504.000	0.172976
$205$	148.000	0.0504233
$206$	1712.00	0.579032
$207$	−207.000	−0.0695048
$208$	352.000	0.117340
$209$	6912.00	2.28762
$210$	384.000	0.126183
$211$	−652.000	−0.212728	−0.106364	−	0.994327i	$-0.533921\pi$
$211$	−652.000	−0.212728	−0.106364	+	0.994327i	$0.533921\pi$
$212$	−2936.00	−0.951157
$213$	1872.00	0.602194
$214$	−2192.00	−0.700196
$215$	384.000	0.121807
$216$	−216.000	−0.0680414
$217$	9728.00	3.04322
$218$	−3268.00	−1.01531
$219$	−1218.00	−0.375821
$220$	−384.000	−0.117679
$221$	924.000	0.281244
$222$	1908.00	0.576831
$223$	−3120.00	−0.936909	−0.468454	−	0.883488i	$-0.655189\pi$
$223$	−3120.00	−0.936909	−0.468454	+	0.883488i	$0.655189\pi$
$224$	1024.00	0.305441
$225$	−1089.00	−0.322667
$226$	2012.00	0.592196
$227$	−488.000	−0.142686	−0.0713429	−	0.997452i	$-0.522728\pi$
$227$	−488.000	−0.142686	−0.0713429	+	0.997452i	$0.522728\pi$
$228$	−1728.00	−0.501928
$229$	−4566.00	−1.31760	−0.658799	−	0.752319i	$-0.728935\pi$
$229$	−4566.00	−1.31760	−0.658799	+	0.752319i	$0.728935\pi$
$230$	92.0000	0.0263752
$231$	4608.00	1.31249
$232$	−1392.00	−0.393919
$233$	−838.000	−0.235619	−0.117809	−	0.993036i	$-0.537587\pi$
$233$	−838.000	−0.235619	−0.117809	+	0.993036i	$0.537587\pi$
$234$	−396.000	−0.110630
$235$	784.000	0.217628
$236$	624.000	0.172114
$237$	2088.00	0.572279
$238$	2688.00	0.732089
$239$	5248.00	1.42036	0.710178	−	0.704023i	$-0.248615\pi$
$239$	5248.00	1.42036	0.710178	+	0.704023i	$0.248615\pi$
$240$	96.0000	0.0258199
$241$	−2550.00	−0.681577	−0.340788	−	0.940140i	$-0.610694\pi$
$241$	−2550.00	−0.681577	−0.340788	+	0.940140i	$0.610694\pi$
$242$	−1946.00	−0.516916
$243$	243.000	0.0641500
$244$	2824.00	0.740935
$245$	1362.00	0.355163
$246$	−444.000	−0.115075
$247$	−3168.00	−0.816093
$248$	2432.00	0.622710
$249$	−2400.00	−0.610819
$250$	984.000	0.248934
$251$	−6648.00	−1.67179	−0.835893	−	0.548893i	$-0.815049\pi$
$251$	−6648.00	−1.67179	−0.835893	+	0.548893i	$0.815049\pi$
$252$	−1152.00	−0.287973
$253$	1104.00	0.274339
$254$	3024.00	0.747018
$255$	252.000	0.0618857
$256$	256.000	0.0625000
$257$	−4014.00	−0.974266	−0.487133	−	0.873328i	$-0.661957\pi$
$257$	−4014.00	−0.974266	−0.487133	+	0.873328i	$0.661957\pi$
$258$	−1152.00	−0.277986
$259$	10176.0	2.44134
$260$	176.000	0.0419810
$261$	1566.00	0.371391
$262$	2360.00	0.556493
$263$	−1864.00	−0.437031	−0.218516	−	0.975833i	$-0.570121\pi$
$263$	−1864.00	−0.437031	−0.218516	+	0.975833i	$0.570121\pi$
$264$	1152.00	0.268563
$265$	−1468.00	−0.340296
$266$	−9216.00	−2.12432
$267$	−306.000	−0.0701382
$268$	768.000	0.175049
$269$	−4626.00	−1.04852	−0.524261	−	0.851558i	$-0.675658\pi$
$269$	−4626.00	−1.04852	−0.524261	+	0.851558i	$0.675658\pi$
$270$	−108.000	−0.0243432
$271$	−1352.00	−0.303056	−0.151528	−	0.988453i	$-0.548419\pi$
$271$	−1352.00	−0.303056	−0.151528	+	0.988453i	$0.548419\pi$
$272$	672.000	0.149801
$273$	−2112.00	−0.468220
$274$	3228.00	0.711718
$275$	5808.00	1.27358
$276$	−276.000	−0.0601929
$277$	−4138.00	−0.897575	−0.448788	−	0.893638i	$-0.648144\pi$
$277$	−4138.00	−0.897575	−0.448788	+	0.893638i	$0.648144\pi$
$278$	840.000	0.181222
$279$	−2736.00	−0.587097
$280$	512.000	0.109278
$281$	1514.00	0.321415	0.160708	−	0.987002i	$-0.448622\pi$
$281$	1514.00	0.321415	0.160708	+	0.987002i	$0.448622\pi$
$282$	−2352.00	−0.496665
$283$	−7632.00	−1.60309	−0.801546	−	0.597932i	$-0.795989\pi$
$283$	−7632.00	−1.60309	−0.801546	+	0.597932i	$0.795989\pi$
$284$	2496.00	0.521515
$285$	−864.000	−0.179575
$286$	2112.00	0.436661
$287$	−2368.00	−0.487034
$288$	−288.000	−0.0589256
$289$	−3149.00	−0.640953
$290$	−696.000	−0.140933
$291$	−2754.00	−0.554785
$292$	−1624.00	−0.325471
$293$	354.000	0.0705833	0.0352916	−	0.999377i	$-0.488764\pi$
$293$	354.000	0.0705833	0.0352916	+	0.999377i	$0.488764\pi$
$294$	−4086.00	−0.810545
$295$	312.000	0.0615774
$296$	2544.00	0.499551
$297$	−1296.00	−0.253204
$298$	−5700.00	−1.10803
$299$	−506.000	−0.0978687
$300$	−1452.00	−0.279438
$301$	−6144.00	−1.17653
$302$	720.000	0.137190
$303$	1098.00	0.208180
$304$	−2304.00	−0.434682
$305$	1412.00	0.265085
$306$	−756.000	−0.141234
$307$	4732.00	0.879705	0.439853	−	0.898070i	$-0.355031\pi$
$307$	4732.00	0.879705	0.439853	+	0.898070i	$0.355031\pi$
$308$	6144.00	1.13665
$309$	−2568.00	−0.472778
$310$	1216.00	0.222788
$311$	9992.00	1.82185	0.910923	−	0.412576i	$-0.135371\pi$
$311$	9992.00	1.82185	0.910923	+	0.412576i	$0.135371\pi$
$312$	−528.000	−0.0958081
$313$	5794.00	1.04631	0.523157	−	0.852236i	$-0.324754\pi$
$313$	5794.00	1.04631	0.523157	+	0.852236i	$0.324754\pi$
$314$	−4532.00	−0.814508
$315$	−576.000	−0.103028
$316$	2784.00	0.495608
$317$	2846.00	0.504250	0.252125	−	0.967695i	$-0.418871\pi$
$317$	2846.00	0.504250	0.252125	+	0.967695i	$0.418871\pi$
$318$	4404.00	0.776617
$319$	−8352.00	−1.46590
$320$	128.000	0.0223607
$321$	3288.00	0.571708
$322$	−1472.00	−0.254756
$323$	−6048.00	−1.04186
$324$	324.000	0.0555556
$325$	−2662.00	−0.454342
$326$	−2904.00	−0.493367
$327$	4902.00	0.828995
$328$	−592.000	−0.0996577
$329$	−12544.0	−2.10205
$330$	576.000	0.0960841
$331$	3388.00	0.562602	0.281301	−	0.959620i	$-0.409234\pi$
$331$	3388.00	0.562602	0.281301	+	0.959620i	$0.409234\pi$
$332$	−3200.00	−0.528984
$333$	−2862.00	−0.470981
$334$	−3776.00	−0.618603
$335$	384.000	0.0626273
$336$	−1536.00	−0.249392
$337$	4850.00	0.783965	0.391983	−	0.919973i	$-0.371789\pi$
$337$	4850.00	0.783965	0.391983	+	0.919973i	$0.371789\pi$
$338$	3426.00	0.551331
$339$	−3018.00	−0.483526
$340$	336.000	0.0535946
$341$	14592.0	2.31731
$342$	2592.00	0.409823
$343$	−10816.0	−1.70265
$344$	−1536.00	−0.240743
$345$	−138.000	−0.0215353
$346$	420.000	0.0652582
$347$	−11836.0	−1.83109	−0.915547	−	0.402211i	$-0.868242\pi$
$347$	−11836.0	−1.83109	−0.915547	+	0.402211i	$0.868242\pi$
$348$	2088.00	0.321634
$349$	2750.00	0.421788	0.210894	−	0.977509i	$-0.432362\pi$
$349$	2750.00	0.421788	0.210894	+	0.977509i	$0.432362\pi$
$350$	−7744.00	−1.18267
$351$	594.000	0.0903287
$352$	1536.00	0.232583
$353$	−2782.00	−0.419464	−0.209732	−	0.977759i	$-0.567259\pi$
$353$	−2782.00	−0.419464	−0.209732	+	0.977759i	$0.567259\pi$
$354$	−936.000	−0.140531
$355$	1248.00	0.186583
$356$	−408.000	−0.0607415
$357$	−4032.00	−0.597748
$358$	6184.00	0.912946
$359$	9888.00	1.45367	0.726837	−	0.686810i	$-0.240990\pi$
$359$	9888.00	1.45367	0.726837	+	0.686810i	$0.240990\pi$
$360$	−144.000	−0.0210819
$361$	13877.0	2.02318
$362$	332.000	0.0482031
$363$	2919.00	0.422060
$364$	−2816.00	−0.405490
$365$	−812.000	−0.116444
$366$	−4236.00	−0.604971
$367$	−3464.00	−0.492696	−0.246348	−	0.969181i	$-0.579231\pi$
$367$	−3464.00	−0.492696	−0.246348	+	0.969181i	$0.579231\pi$
$368$	−368.000	−0.0521286
$369$	666.000	0.0939582
$370$	1272.00	0.178725
$371$	23488.0	3.28689
$372$	−3648.00	−0.508441
$373$	3250.00	0.451149	0.225575	−	0.974226i	$-0.427574\pi$
$373$	3250.00	0.451149	0.225575	+	0.974226i	$0.427574\pi$
$374$	4032.00	0.557459
$375$	−1476.00	−0.203254
$376$	−3136.00	−0.430125
$377$	3828.00	0.522950
$378$	1728.00	0.235129
$379$	−4440.00	−0.601761	−0.300881	−	0.953662i	$-0.597281\pi$
$379$	−4440.00	−0.601761	−0.300881	+	0.953662i	$0.597281\pi$
$380$	−1152.00	−0.155517
$381$	−4536.00	−0.609938
$382$	−720.000	−0.0964356
$383$	7904.00	1.05451	0.527253	−	0.849709i	$-0.323222\pi$
$383$	7904.00	1.05451	0.527253	+	0.849709i	$0.323222\pi$
$384$	−384.000	−0.0510310
$385$	3072.00	0.406659
$386$	3548.00	0.467846
$387$	1728.00	0.226975
$388$	−3672.00	−0.480458
$389$	−5430.00	−0.707743	−0.353871	−	0.935294i	$-0.615135\pi$
$389$	−5430.00	−0.707743	−0.353871	+	0.935294i	$0.615135\pi$
$390$	−264.000	−0.0342773
$391$	−966.000	−0.124943
$392$	−5448.00	−0.701953
$393$	−3540.00	−0.454375
$394$	9876.00	1.26281
$395$	1392.00	0.177314
$396$	−1728.00	−0.219281
$397$	−3066.00	−0.387602	−0.193801	−	0.981041i	$-0.562082\pi$
$397$	−3066.00	−0.387602	−0.193801	+	0.981041i	$0.562082\pi$
$398$	5024.00	0.632740
$399$	13824.0	1.73450
$400$	−1936.00	−0.242000
$401$	5882.00	0.732501	0.366251	−	0.930516i	$-0.380641\pi$
$401$	5882.00	0.732501	0.366251	+	0.930516i	$0.380641\pi$
$402$	−1152.00	−0.142927
$403$	−6688.00	−0.826682
$404$	1464.00	0.180289
$405$	162.000	0.0198762
$406$	11136.0	1.36126
$407$	15264.0	1.85899
$408$	−1008.00	−0.122312
$409$	−3718.00	−0.449495	−0.224747	−	0.974417i	$-0.572156\pi$
$409$	−3718.00	−0.449495	−0.224747	+	0.974417i	$0.572156\pi$
$410$	−296.000	−0.0356546
$411$	−4842.00	−0.581115
$412$	−3424.00	−0.409438
$413$	−4992.00	−0.594771
$414$	414.000	0.0491473
$415$	−1600.00	−0.189255
$416$	−704.000	−0.0829722
$417$	−1260.00	−0.147968
$418$	−13824.0	−1.61759
$419$	9104.00	1.06148	0.530739	−	0.847535i	$-0.321914\pi$
$419$	9104.00	1.06148	0.530739	+	0.847535i	$0.321914\pi$
$420$	−768.000	−0.0892251
$421$	11554.0	1.33755	0.668774	−	0.743466i	$-0.266819\pi$
$421$	11554.0	1.33755	0.668774	+	0.743466i	$0.266819\pi$
$422$	1304.00	0.150421
$423$	3528.00	0.405525
$424$	5872.00	0.672570
$425$	−5082.00	−0.580031
$426$	−3744.00	−0.425815
$427$	−22592.0	−2.56043
$428$	4384.00	0.495114
$429$	−3168.00	−0.356533
$430$	−768.000	−0.0861308
$431$	12984.0	1.45108	0.725542	−	0.688178i	$-0.241589\pi$
$431$	12984.0	1.45108	0.725542	+	0.688178i	$0.241589\pi$
$432$	432.000	0.0481125
$433$	−3014.00	−0.334512	−0.167256	−	0.985914i	$-0.553491\pi$
$433$	−3014.00	−0.334512	−0.167256	+	0.985914i	$0.553491\pi$
$434$	−19456.0	−2.15188
$435$	1044.00	0.115071
$436$	6536.00	0.717930
$437$	3312.00	0.362550
$438$	2436.00	0.265746
$439$	560.000	0.0608823	0.0304412	−	0.999537i	$-0.490309\pi$
$439$	560.000	0.0608823	0.0304412	+	0.999537i	$0.490309\pi$
$440$	768.000	0.0832113
$441$	6129.00	0.661808
$442$	−1848.00	−0.198870
$443$	8388.00	0.899607	0.449804	−	0.893128i	$-0.351494\pi$
$443$	8388.00	0.899607	0.449804	+	0.893128i	$0.351494\pi$
$444$	−3816.00	−0.407881
$445$	−204.000	−0.0217315
$446$	6240.00	0.662495
$447$	8550.00	0.904700
$448$	−2048.00	−0.215980
$449$	8338.00	0.876380	0.438190	−	0.898882i	$-0.355620\pi$
$449$	8338.00	0.876380	0.438190	+	0.898882i	$0.355620\pi$
$450$	2178.00	0.228160
$451$	−3552.00	−0.370858
$452$	−4024.00	−0.418746
$453$	−1080.00	−0.112015
$454$	976.000	0.100894
$455$	−1408.00	−0.145073
$456$	3456.00	0.354917
$457$	−4606.00	−0.471465	−0.235733	−	0.971818i	$-0.575749\pi$
$457$	−4606.00	−0.471465	−0.235733	+	0.971818i	$0.575749\pi$
$458$	9132.00	0.931682
$459$	1134.00	0.115317
$460$	−184.000	−0.0186501
$461$	−2466.00	−0.249139	−0.124569	−	0.992211i	$-0.539755\pi$
$461$	−2466.00	−0.249139	−0.124569	+	0.992211i	$0.539755\pi$
$462$	−9216.00	−0.928067
$463$	4568.00	0.458516	0.229258	−	0.973366i	$-0.426370\pi$
$463$	4568.00	0.458516	0.229258	+	0.973366i	$0.426370\pi$
$464$	2784.00	0.278543
$465$	−1824.00	−0.181905
$466$	1676.00	0.166608
$467$	−8352.00	−0.827590	−0.413795	−	0.910370i	$-0.635797\pi$
$467$	−8352.00	−0.827590	−0.413795	+	0.910370i	$0.635797\pi$
$468$	792.000	0.0782270
$469$	−6144.00	−0.604912
$470$	−1568.00	−0.153886
$471$	6798.00	0.665043
$472$	−1248.00	−0.121703
$473$	−9216.00	−0.895882
$474$	−4176.00	−0.404663
$475$	17424.0	1.68309
$476$	−5376.00	−0.517665
$477$	−6606.00	−0.634105
$478$	−10496.0	−1.00434
$479$	13872.0	1.32323	0.661616	−	0.749843i	$-0.269871\pi$
$479$	13872.0	1.32323	0.661616	+	0.749843i	$0.269871\pi$
$480$	−192.000	−0.0182574
$481$	−6996.00	−0.663181
$482$	5100.00	0.481947
$483$	2208.00	0.208007
$484$	3892.00	0.365515
$485$	−1836.00	−0.171894
$486$	−486.000	−0.0453609
$487$	14048.0	1.30714	0.653568	−	0.756867i	$-0.273271\pi$
$487$	14048.0	1.30714	0.653568	+	0.756867i	$0.273271\pi$
$488$	−5648.00	−0.523920
$489$	4356.00	0.402833
$490$	−2724.00	−0.251138
$491$	260.000	0.0238974	0.0119487	−	0.999929i	$-0.496197\pi$
$491$	260.000	0.0238974	0.0119487	+	0.999929i	$0.496197\pi$
$492$	888.000	0.0813702
$493$	7308.00	0.667618
$494$	6336.00	0.577065
$495$	−864.000	−0.0784523
$496$	−4864.00	−0.440323
$497$	−19968.0	−1.80219
$498$	4800.00	0.431914
$499$	−14996.0	−1.34532	−0.672658	−	0.739953i	$-0.734848\pi$
$499$	−14996.0	−1.34532	−0.672658	+	0.739953i	$0.734848\pi$
$500$	−1968.00	−0.176023
$501$	5664.00	0.505088
$502$	13296.0	1.18213
$503$	−16272.0	−1.44241	−0.721205	−	0.692721i	$-0.756412\pi$
$503$	−16272.0	−1.44241	−0.721205	+	0.692721i	$0.756412\pi$
$504$	2304.00	0.203628
$505$	732.000	0.0645021
$506$	−2208.00	−0.193987
$507$	−5139.00	−0.450160
$508$	−6048.00	−0.528222
$509$	−18770.0	−1.63451	−0.817255	−	0.576276i	$-0.804505\pi$
$509$	−18770.0	−1.63451	−0.817255	+	0.576276i	$0.804505\pi$
$510$	−504.000	−0.0437598
$511$	12992.0	1.12472
$512$	−512.000	−0.0441942
$513$	−3888.00	−0.334619
$514$	8028.00	0.688910
$515$	−1712.00	−0.146485
$516$	2304.00	0.196566
$517$	−18816.0	−1.60063
$518$	−20352.0	−1.72628
$519$	−630.000	−0.0532831
$520$	−352.000	−0.0296850
$521$	4706.00	0.395727	0.197863	−	0.980230i	$-0.436600\pi$
$521$	4706.00	0.395727	0.197863	+	0.980230i	$0.436600\pi$
$522$	−3132.00	−0.262613
$523$	−7768.00	−0.649466	−0.324733	−	0.945806i	$-0.605275\pi$
$523$	−7768.00	−0.649466	−0.324733	+	0.945806i	$0.605275\pi$
$524$	−4720.00	−0.393500
$525$	11616.0	0.965645
$526$	3728.00	0.309028
$527$	−12768.0	−1.05538
$528$	−2304.00	−0.189903
$529$	529.000	0.0434783
$530$	2936.00	0.240626
$531$	1404.00	0.114743
$532$	18432.0	1.50212
$533$	1628.00	0.132301
$534$	612.000	0.0495952
$535$	2192.00	0.177137
$536$	−1536.00	−0.123778
$537$	−9276.00	−0.745417
$538$	9252.00	0.741416
$539$	−32688.0	−2.61219
$540$	216.000	0.0172133
$541$	5534.00	0.439788	0.219894	−	0.975524i	$-0.429429\pi$
$541$	5534.00	0.439788	0.219894	+	0.975524i	$0.429429\pi$
$542$	2704.00	0.214293
$543$	−498.000	−0.0393577
$544$	−1344.00	−0.105926
$545$	3268.00	0.256855
$546$	4224.00	0.331082
$547$	−19868.0	−1.55301	−0.776503	−	0.630113i	$-0.783008\pi$
$547$	−19868.0	−1.55301	−0.776503	+	0.630113i	$0.783008\pi$
$548$	−6456.00	−0.503260
$549$	6354.00	0.493956
$550$	−11616.0	−0.900560
$551$	−25056.0	−1.93724
$552$	552.000	0.0425628
$553$	−22272.0	−1.71266
$554$	8276.00	0.634681
$555$	−1908.00	−0.145928
$556$	−1680.00	−0.128144
$557$	−21390.0	−1.62715	−0.813576	−	0.581459i	$-0.802482\pi$
$557$	−21390.0	−1.62715	−0.813576	+	0.581459i	$0.802482\pi$
$558$	5472.00	0.415140
$559$	4224.00	0.319600
$560$	−1024.00	−0.0772712
$561$	−6048.00	−0.455164
$562$	−3028.00	−0.227275
$563$	808.000	0.0604852	0.0302426	−	0.999543i	$-0.490372\pi$
$563$	808.000	0.0604852	0.0302426	+	0.999543i	$0.490372\pi$
$564$	4704.00	0.351195
$565$	−2012.00	−0.149815
$566$	15264.0	1.13356
$567$	−2592.00	−0.191982
$568$	−4992.00	−0.368767
$569$	−22942.0	−1.69030	−0.845148	−	0.534532i	$-0.820488\pi$
$569$	−22942.0	−1.69030	−0.845148	+	0.534532i	$0.820488\pi$
$570$	1728.00	0.126979
$571$	−20504.0	−1.50274	−0.751371	−	0.659880i	$-0.770607\pi$
$571$	−20504.0	−1.50274	−0.751371	+	0.659880i	$0.770607\pi$
$572$	−4224.00	−0.308766
$573$	1080.00	0.0787393
$574$	4736.00	0.344385
$575$	2783.00	0.201842
$576$	576.000	0.0416667
$577$	18578.0	1.34040	0.670201	−	0.742179i	$-0.266208\pi$
$577$	18578.0	1.34040	0.670201	+	0.742179i	$0.266208\pi$
$578$	6298.00	0.453222
$579$	−5322.00	−0.381994
$580$	1392.00	0.0996546
$581$	25600.0	1.82800
$582$	5508.00	0.392292
$583$	35232.0	2.50285
$584$	3248.00	0.230142
$585$	396.000	0.0279873
$586$	−708.000	−0.0499099
$587$	22364.0	1.57251	0.786253	−	0.617905i	$-0.212018\pi$
$587$	22364.0	1.57251	0.786253	+	0.617905i	$0.212018\pi$
$588$	8172.00	0.573142
$589$	43776.0	3.06241
$590$	−624.000	−0.0435418
$591$	−14814.0	−1.03108
$592$	−5088.00	−0.353236
$593$	−12222.0	−0.846370	−0.423185	−	0.906043i	$-0.639088\pi$
$593$	−12222.0	−0.846370	−0.423185	+	0.906043i	$0.639088\pi$
$594$	2592.00	0.179042
$595$	−2688.00	−0.185205
$596$	11400.0	0.783494
$597$	−7536.00	−0.516630
$598$	1012.00	0.0692036
$599$	15416.0	1.05155	0.525777	−	0.850623i	$-0.323775\pi$
$599$	15416.0	1.05155	0.525777	+	0.850623i	$0.323775\pi$
$600$	2904.00	0.197592
$601$	18394.0	1.24843	0.624215	−	0.781252i	$-0.285419\pi$
$601$	18394.0	1.24843	0.624215	+	0.781252i	$0.285419\pi$
$602$	12288.0	0.831929
$603$	1728.00	0.116699
$604$	−1440.00	−0.0970079
$605$	1946.00	0.130770
$606$	−2196.00	−0.147205
$607$	−448.000	−0.0299568	−0.0149784	−	0.999888i	$-0.504768\pi$
$607$	−448.000	−0.0299568	−0.0149784	+	0.999888i	$0.504768\pi$
$608$	4608.00	0.307367
$609$	−16704.0	−1.11146
$610$	−2824.00	−0.187443
$611$	8624.00	0.571014
$612$	1512.00	0.0998676
$613$	−2182.00	−0.143769	−0.0718843	−	0.997413i	$-0.522901\pi$
$613$	−2182.00	−0.143769	−0.0718843	+	0.997413i	$0.522901\pi$
$614$	−9464.00	−0.622046
$615$	444.000	0.0291119
$616$	−12288.0	−0.803730
$617$	15002.0	0.978862	0.489431	−	0.872042i	$-0.337205\pi$
$617$	15002.0	0.978862	0.489431	+	0.872042i	$0.337205\pi$
$618$	5136.00	0.334305
$619$	12048.0	0.782310	0.391155	−	0.920325i	$-0.372076\pi$
$619$	12048.0	0.782310	0.391155	+	0.920325i	$0.372076\pi$
$620$	−2432.00	−0.157535
$621$	−621.000	−0.0401286
$622$	−19984.0	−1.28824
$623$	3264.00	0.209903
$624$	1056.00	0.0677465
$625$	14141.0	0.905024
$626$	−11588.0	−0.739856
$627$	20736.0	1.32076
$628$	9064.00	0.575944
$629$	−13356.0	−0.846643
$630$	1152.00	0.0728520
$631$	−23768.0	−1.49951	−0.749754	−	0.661717i	$-0.769828\pi$
$631$	−23768.0	−1.49951	−0.749754	+	0.661717i	$0.769828\pi$
$632$	−5568.00	−0.350448
$633$	−1956.00	−0.122818
$634$	−5692.00	−0.356559
$635$	−3024.00	−0.188982
$636$	−8808.00	−0.549151
$637$	14982.0	0.931881
$638$	16704.0	1.03655
$639$	5616.00	0.347677
$640$	−256.000	−0.0158114
$641$	−21094.0	−1.29979	−0.649893	−	0.760026i	$-0.725186\pi$
$641$	−21094.0	−1.29979	−0.649893	+	0.760026i	$0.725186\pi$
$642$	−6576.00	−0.404259
$643$	11528.0	0.707029	0.353515	−	0.935429i	$-0.384986\pi$
$643$	11528.0	0.707029	0.353515	+	0.935429i	$0.384986\pi$
$644$	2944.00	0.180140
$645$	1152.00	0.0703255
$646$	12096.0	0.736704
$647$	−7976.00	−0.484651	−0.242325	−	0.970195i	$-0.577910\pi$
$647$	−7976.00	−0.484651	−0.242325	+	0.970195i	$0.577910\pi$
$648$	−648.000	−0.0392837
$649$	−7488.00	−0.452896
$650$	5324.00	0.321268
$651$	29184.0	1.75701
$652$	5808.00	0.348863
$653$	21238.0	1.27275	0.636376	−	0.771379i	$-0.280433\pi$
$653$	21238.0	1.27275	0.636376	+	0.771379i	$0.280433\pi$
$654$	−9804.00	−0.586188
$655$	−2360.00	−0.140783
$656$	1184.00	0.0704686
$657$	−3654.00	−0.216980
$658$	25088.0	1.48637
$659$	−4080.00	−0.241175	−0.120587	−	0.992703i	$-0.538478\pi$
$659$	−4080.00	−0.241175	−0.120587	+	0.992703i	$0.538478\pi$
$660$	−1152.00	−0.0679417
$661$	−31654.0	−1.86263	−0.931315	−	0.364216i	$-0.881337\pi$
$661$	−31654.0	−1.86263	−0.931315	+	0.364216i	$0.881337\pi$
$662$	−6776.00	−0.397820
$663$	2772.00	0.162376
$664$	6400.00	0.374048
$665$	9216.00	0.537415
$666$	5724.00	0.333034
$667$	−4002.00	−0.232321
$668$	7552.00	0.437419
$669$	−9360.00	−0.540925
$670$	−768.000	−0.0442842
$671$	−33888.0	−1.94967
$672$	3072.00	0.176347
$673$	4370.00	0.250299	0.125149	−	0.992138i	$-0.460059\pi$
$673$	4370.00	0.250299	0.125149	+	0.992138i	$0.460059\pi$
$674$	−9700.00	−0.554347
$675$	−3267.00	−0.186292
$676$	−6852.00	−0.389850
$677$	25410.0	1.44252	0.721260	−	0.692665i	$-0.243563\pi$
$677$	25410.0	1.44252	0.721260	+	0.692665i	$0.243563\pi$
$678$	6036.00	0.341904
$679$	29376.0	1.66031
$680$	−672.000	−0.0378971
$681$	−1464.00	−0.0823797
$682$	−29184.0	−1.63858
$683$	1404.00	0.0786568	0.0393284	−	0.999226i	$-0.487478\pi$
$683$	1404.00	0.0786568	0.0393284	+	0.999226i	$0.487478\pi$
$684$	−5184.00	−0.289788
$685$	−3228.00	−0.180052
$686$	21632.0	1.20396
$687$	−13698.0	−0.760715
$688$	3072.00	0.170231
$689$	−16148.0	−0.892873
$690$	276.000	0.0152277
$691$	−7900.00	−0.434921	−0.217460	−	0.976069i	$-0.569777\pi$
$691$	−7900.00	−0.434921	−0.217460	+	0.976069i	$0.569777\pi$
$692$	−840.000	−0.0461445
$693$	13824.0	0.757764
$694$	23672.0	1.29478
$695$	−840.000	−0.0458461
$696$	−4176.00	−0.227429
$697$	3108.00	0.168901
$698$	−5500.00	−0.298249
$699$	−2514.00	−0.136035
$700$	15488.0	0.836273
$701$	−3518.00	−0.189548	−0.0947739	−	0.995499i	$-0.530213\pi$
$701$	−3518.00	−0.189548	−0.0947739	+	0.995499i	$0.530213\pi$
$702$	−1188.00	−0.0638720
$703$	45792.0	2.45673
$704$	−3072.00	−0.164461
$705$	2352.00	0.125647
$706$	5564.00	0.296606
$707$	−11712.0	−0.623020
$708$	1872.00	0.0993702
$709$	−13358.0	−0.707574	−0.353787	−	0.935326i	$-0.615106\pi$
$709$	−13358.0	−0.707574	−0.353787	+	0.935326i	$0.615106\pi$
$710$	−2496.00	−0.131934
$711$	6264.00	0.330406
$712$	816.000	0.0429507
$713$	6992.00	0.367254
$714$	8064.00	0.422672
$715$	−2112.00	−0.110468
$716$	−12368.0	−0.645550
$717$	15744.0	0.820042
$718$	−19776.0	−1.02790
$719$	9168.00	0.475534	0.237767	−	0.971322i	$-0.423585\pi$
$719$	9168.00	0.475534	0.237767	+	0.971322i	$0.423585\pi$
$720$	288.000	0.0149071
$721$	27392.0	1.41488
$722$	−27754.0	−1.43061
$723$	−7650.00	−0.393508
$724$	−664.000	−0.0340848
$725$	−21054.0	−1.07852
$726$	−5838.00	−0.298441
$727$	4704.00	0.239975	0.119987	−	0.992775i	$-0.461715\pi$
$727$	4704.00	0.239975	0.119987	+	0.992775i	$0.461715\pi$
$728$	5632.00	0.286725
$729$	729.000	0.0370370
$730$	1624.00	0.0823383
$731$	8064.00	0.408013
$732$	8472.00	0.427779
$733$	5322.00	0.268175	0.134088	−	0.990969i	$-0.457190\pi$
$733$	5322.00	0.268175	0.134088	+	0.990969i	$0.457190\pi$
$734$	6928.00	0.348388
$735$	4086.00	0.205054
$736$	736.000	0.0368605
$737$	−9216.00	−0.460618
$738$	−1332.00	−0.0664385
$739$	−5636.00	−0.280546	−0.140273	−	0.990113i	$-0.544798\pi$
$739$	−5636.00	−0.280546	−0.140273	+	0.990113i	$0.544798\pi$
$740$	−2544.00	−0.126377
$741$	−9504.00	−0.471172
$742$	−46976.0	−2.32418
$743$	15992.0	0.789623	0.394811	−	0.918762i	$-0.370810\pi$
$743$	15992.0	0.789623	0.394811	+	0.918762i	$0.370810\pi$
$744$	7296.00	0.359522
$745$	5700.00	0.280311
$746$	−6500.00	−0.319011
$747$	−7200.00	−0.352656
$748$	−8064.00	−0.394183
$749$	−35072.0	−1.71095
$750$	2952.00	0.143722
$751$	−12176.0	−0.591623	−0.295811	−	0.955246i	$-0.595590\pi$
$751$	−12176.0	−0.591623	−0.295811	+	0.955246i	$0.595590\pi$
$752$	6272.00	0.304144
$753$	−19944.0	−0.965206
$754$	−7656.00	−0.369781
$755$	−720.000	−0.0347066
$756$	−3456.00	−0.166261
$757$	−23798.0	−1.14261	−0.571303	−	0.820739i	$-0.693562\pi$
$757$	−23798.0	−1.14261	−0.571303	+	0.820739i	$0.693562\pi$
$758$	8880.00	0.425509
$759$	3312.00	0.158390
$760$	2304.00	0.109967
$761$	36282.0	1.72828	0.864140	−	0.503251i	$-0.167863\pi$
$761$	36282.0	1.72828	0.864140	+	0.503251i	$0.167863\pi$
$762$	9072.00	0.431291
$763$	−52288.0	−2.48093
$764$	1440.00	0.0681903
$765$	756.000	0.0357297
$766$	−15808.0	−0.745648
$767$	3432.00	0.161568
$768$	768.000	0.0360844
$769$	−16886.0	−0.791840	−0.395920	−	0.918285i	$-0.629574\pi$
$769$	−16886.0	−0.791840	−0.395920	+	0.918285i	$0.629574\pi$
$770$	−6144.00	−0.287551
$771$	−12042.0	−0.562493
$772$	−7096.00	−0.330817
$773$	13722.0	0.638481	0.319241	−	0.947674i	$-0.396572\pi$
$773$	13722.0	0.638481	0.319241	+	0.947674i	$0.396572\pi$
$774$	−3456.00	−0.160495
$775$	36784.0	1.70493
$776$	7344.00	0.339735
$777$	30528.0	1.40951
$778$	10860.0	0.500450
$779$	−10656.0	−0.490104
$780$	528.000	0.0242377
$781$	−29952.0	−1.37230
$782$	1932.00	0.0883481
$783$	4698.00	0.214423
$784$	10896.0	0.496356
$785$	4532.00	0.206056
$786$	7080.00	0.321292
$787$	15224.0	0.689551	0.344776	−	0.938685i	$-0.387955\pi$
$787$	15224.0	0.689551	0.344776	+	0.938685i	$0.387955\pi$
$788$	−19752.0	−0.892939
$789$	−5592.00	−0.252320
$790$	−2784.00	−0.125380
$791$	32192.0	1.44705
$792$	3456.00	0.155055
$793$	15532.0	0.695533
$794$	6132.00	0.274076
$795$	−4404.00	−0.196470
$796$	−10048.0	−0.447414
$797$	20946.0	0.930923	0.465461	−	0.885068i	$-0.345888\pi$
$797$	20946.0	0.930923	0.465461	+	0.885068i	$0.345888\pi$
$798$	−27648.0	−1.22648
$799$	16464.0	0.728979
$800$	3872.00	0.171120
$801$	−918.000	−0.0404943
$802$	−11764.0	−0.517957
$803$	19488.0	0.856434
$804$	2304.00	0.101064
$805$	1472.00	0.0644487
$806$	13376.0	0.584553
$807$	−13878.0	−0.605364
$808$	−2928.00	−0.127484
$809$	23594.0	1.02537	0.512683	−	0.858578i	$-0.328652\pi$
$809$	23594.0	1.02537	0.512683	+	0.858578i	$0.328652\pi$
$810$	−324.000	−0.0140546
$811$	30028.0	1.30015	0.650077	−	0.759868i	$-0.274736\pi$
$811$	30028.0	1.30015	0.650077	+	0.759868i	$0.274736\pi$
$812$	−22272.0	−0.962554
$813$	−4056.00	−0.174969
$814$	−30528.0	−1.31450
$815$	2904.00	0.124813
$816$	2016.00	0.0864879
$817$	−27648.0	−1.18394
$818$	7436.00	0.317841
$819$	−6336.00	−0.270327
$820$	592.000	0.0252116
$821$	−5490.00	−0.233377	−0.116688	−	0.993169i	$-0.537228\pi$
$821$	−5490.00	−0.233377	−0.116688	+	0.993169i	$0.537228\pi$
$822$	9684.00	0.410910
$823$	−18504.0	−0.783729	−0.391864	−	0.920023i	$-0.628170\pi$
$823$	−18504.0	−0.783729	−0.391864	+	0.920023i	$0.628170\pi$
$824$	6848.00	0.289516
$825$	17424.0	0.735304
$826$	9984.00	0.420566
$827$	−6936.00	−0.291643	−0.145821	−	0.989311i	$-0.546582\pi$
$827$	−6936.00	−0.291643	−0.145821	+	0.989311i	$0.546582\pi$
$828$	−828.000	−0.0347524
$829$	2046.00	0.0857184	0.0428592	−	0.999081i	$-0.486353\pi$
$829$	2046.00	0.0857184	0.0428592	+	0.999081i	$0.486353\pi$
$830$	3200.00	0.133824
$831$	−12414.0	−0.518215
$832$	1408.00	0.0586702
$833$	28602.0	1.18968
$834$	2520.00	0.104629
$835$	3776.00	0.156496
$836$	27648.0	1.14381
$837$	−8208.00	−0.338961
$838$	−18208.0	−0.750579
$839$	23112.0	0.951031	0.475515	−	0.879707i	$-0.342262\pi$
$839$	23112.0	0.951031	0.475515	+	0.879707i	$0.342262\pi$
$840$	1536.00	0.0630917
$841$	5887.00	0.241379
$842$	−23108.0	−0.945789
$843$	4542.00	0.185569
$844$	−2608.00	−0.106364
$845$	−3426.00	−0.139477
$846$	−7056.00	−0.286750
$847$	−31136.0	−1.26310
$848$	−11744.0	−0.475579
$849$	−22896.0	−0.925546
$850$	10164.0	0.410144
$851$	7314.00	0.294619
$852$	7488.00	0.301097
$853$	−17346.0	−0.696267	−0.348133	−	0.937445i	$-0.613184\pi$
$853$	−17346.0	−0.696267	−0.348133	+	0.937445i	$0.613184\pi$
$854$	45184.0	1.81050
$855$	−2592.00	−0.103678
$856$	−8768.00	−0.350098
$857$	−31862.0	−1.26999	−0.634997	−	0.772514i	$-0.718999\pi$
$857$	−31862.0	−1.26999	−0.634997	+	0.772514i	$0.718999\pi$
$858$	6336.00	0.252107
$859$	−18748.0	−0.744672	−0.372336	−	0.928098i	$-0.621443\pi$
$859$	−18748.0	−0.744672	−0.372336	+	0.928098i	$0.621443\pi$
$860$	1536.00	0.0609037
$861$	−7104.00	−0.281189
$862$	−25968.0	−1.02607
$863$	−19472.0	−0.768059	−0.384029	−	0.923321i	$-0.625464\pi$
$863$	−19472.0	−0.768059	−0.384029	+	0.923321i	$0.625464\pi$
$864$	−864.000	−0.0340207
$865$	−420.000	−0.0165092
$866$	6028.00	0.236536
$867$	−9447.00	−0.370054
$868$	38912.0	1.52161
$869$	−33408.0	−1.30413
$870$	−2088.00	−0.0813676
$871$	4224.00	0.164322
$872$	−13072.0	−0.507653
$873$	−8262.00	−0.320305
$874$	−6624.00	−0.256362
$875$	15744.0	0.608279
$876$	−4872.00	−0.187911
$877$	32398.0	1.24744	0.623719	−	0.781649i	$-0.285621\pi$
$877$	32398.0	1.24744	0.623719	+	0.781649i	$0.285621\pi$
$878$	−1120.00	−0.0430503
$879$	1062.00	0.0407513
$880$	−1536.00	−0.0588393
$881$	36290.0	1.38779	0.693894	−	0.720077i	$-0.255894\pi$
$881$	36290.0	1.38779	0.693894	+	0.720077i	$0.255894\pi$
$882$	−12258.0	−0.467969
$883$	−27276.0	−1.03954	−0.519768	−	0.854307i	$-0.673982\pi$
$883$	−27276.0	−1.03954	−0.519768	+	0.854307i	$0.673982\pi$
$884$	3696.00	0.140622
$885$	936.000	0.0355517
$886$	−16776.0	−0.636118
$887$	8032.00	0.304045	0.152023	−	0.988377i	$-0.451421\pi$
$887$	8032.00	0.304045	0.152023	+	0.988377i	$0.451421\pi$
$888$	7632.00	0.288416
$889$	48384.0	1.82536
$890$	408.000	0.0153665
$891$	−3888.00	−0.146187
$892$	−12480.0	−0.468454
$893$	−56448.0	−2.11530
$894$	−17100.0	−0.639720
$895$	−6184.00	−0.230959
$896$	4096.00	0.152721
$897$	−1518.00	−0.0565045
$898$	−16676.0	−0.619694
$899$	−52896.0	−1.96238
$900$	−4356.00	−0.161333
$901$	−30828.0	−1.13988
$902$	7104.00	0.262237
$903$	−18432.0	−0.679268
$904$	8048.00	0.296098
$905$	−332.000	−0.0121945
$906$	2160.00	0.0792066
$907$	23872.0	0.873932	0.436966	−	0.899478i	$-0.356053\pi$
$907$	23872.0	0.873932	0.436966	+	0.899478i	$0.356053\pi$
$908$	−1952.00	−0.0713429
$909$	3294.00	0.120193
$910$	2816.00	0.102582
$911$	−38872.0	−1.41371	−0.706853	−	0.707360i	$-0.749886\pi$
$911$	−38872.0	−1.41371	−0.706853	+	0.707360i	$0.749886\pi$
$912$	−6912.00	−0.250964
$913$	38400.0	1.39195
$914$	9212.00	0.333376
$915$	4236.00	0.153047
$916$	−18264.0	−0.658799
$917$	37760.0	1.35981
$918$	−2268.00	−0.0815416
$919$	−43616.0	−1.56557	−0.782785	−	0.622292i	$-0.786202\pi$
$919$	−43616.0	−1.56557	−0.782785	+	0.622292i	$0.786202\pi$
$920$	368.000	0.0131876
$921$	14196.0	0.507898
$922$	4932.00	0.176168
$923$	13728.0	0.489559
$924$	18432.0	0.656243
$925$	38478.0	1.36773
$926$	−9136.00	−0.324220
$927$	−7704.00	−0.272959
$928$	−5568.00	−0.196960
$929$	12882.0	0.454946	0.227473	−	0.973784i	$-0.426954\pi$
$929$	12882.0	0.454946	0.227473	+	0.973784i	$0.426954\pi$
$930$	3648.00	0.128626
$931$	−98064.0	−3.45211
$932$	−3352.00	−0.117809
$933$	29976.0	1.05184
$934$	16704.0	0.585194
$935$	−4032.00	−0.141027
$936$	−1584.00	−0.0553148
$937$	27578.0	0.961509	0.480755	−	0.876855i	$-0.340363\pi$
$937$	27578.0	0.961509	0.480755	+	0.876855i	$0.340363\pi$
$938$	12288.0	0.427737
$939$	17382.0	0.604090
$940$	3136.00	0.108814
$941$	46410.0	1.60778	0.803891	−	0.594777i	$-0.202760\pi$
$941$	46410.0	1.60778	0.803891	+	0.594777i	$0.202760\pi$
$942$	−13596.0	−0.470256
$943$	−1702.00	−0.0587749
$944$	2496.00	0.0860571
$945$	−1728.00	−0.0594834
$946$	18432.0	0.633484
$947$	34828.0	1.19510	0.597549	−	0.801832i	$-0.296141\pi$
$947$	34828.0	1.19510	0.597549	+	0.801832i	$0.296141\pi$
$948$	8352.00	0.286140
$949$	−8932.00	−0.305527
$950$	−34848.0	−1.19012
$951$	8538.00	0.291129
$952$	10752.0	0.366044
$953$	−26334.0	−0.895112	−0.447556	−	0.894256i	$-0.647706\pi$
$953$	−26334.0	−0.895112	−0.447556	+	0.894256i	$0.647706\pi$
$954$	13212.0	0.448380
$955$	720.000	0.0243965
$956$	20992.0	0.710178
$957$	−25056.0	−0.846338
$958$	−27744.0	−0.935666
$959$	51648.0	1.73910
$960$	384.000	0.0129099
$961$	62625.0	2.10214
$962$	13992.0	0.468940
$963$	9864.00	0.330076
$964$	−10200.0	−0.340788
$965$	−3548.00	−0.118357
$966$	−4416.00	−0.147083
$967$	808.000	0.0268702	0.0134351	−	0.999910i	$-0.495723\pi$
$967$	808.000	0.0268702	0.0134351	+	0.999910i	$0.495723\pi$
$968$	−7784.00	−0.258458
$969$	−18144.0	−0.601516
$970$	3672.00	0.121547
$971$	−30224.0	−0.998902	−0.499451	−	0.866342i	$-0.666465\pi$
$971$	−30224.0	−0.998902	−0.499451	+	0.866342i	$0.666465\pi$
$972$	972.000	0.0320750
$973$	13440.0	0.442823
$974$	−28096.0	−0.924285
$975$	−7986.00	−0.262315
$976$	11296.0	0.370467
$977$	16554.0	0.542077	0.271039	−	0.962568i	$-0.412633\pi$
$977$	16554.0	0.542077	0.271039	+	0.962568i	$0.412633\pi$
$978$	−8712.00	−0.284846
$979$	4896.00	0.159833
$980$	5448.00	0.177582
$981$	14706.0	0.478620
$982$	−520.000	−0.0168980
$983$	−47440.0	−1.53927	−0.769634	−	0.638485i	$-0.779561\pi$
$983$	−47440.0	−1.53927	−0.769634	+	0.638485i	$0.779561\pi$
$984$	−1776.00	−0.0575374
$985$	−9876.00	−0.319468
$986$	−14616.0	−0.472077
$987$	−37632.0	−1.21362
$988$	−12672.0	−0.408047
$989$	−4416.00	−0.141982
$990$	1728.00	0.0554742
$991$	−17920.0	−0.574417	−0.287209	−	0.957868i	$-0.592727\pi$
$991$	−17920.0	−0.574417	−0.287209	+	0.957868i	$0.592727\pi$
$992$	9728.00	0.311355
$993$	10164.0	0.324819
$994$	39936.0	1.27434
$995$	−5024.00	−0.160072
$996$	−9600.00	−0.305409
$997$	5206.00	0.165372	0.0826859	−	0.996576i	$-0.473650\pi$
$997$	5206.00	0.165372	0.0826859	+	0.996576i	$0.473650\pi$
$998$	29992.0	0.951283
$999$	−8586.00	−0.271921

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.4.a.b.1.1	✓	1
3.2	odd	2		414.4.a.c.1.1		1
4.3	odd	2		1104.4.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.4.a.b.1.1	✓	1	1.1	even	1	trivial
414.4.a.c.1.1		1	3.2	odd	2
1104.4.a.c.1.1		1	4.3	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:	\( N \)	\(=\)	\( 138 = 2 \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	138.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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