LMFDB - Embedded newform 210.4.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [210,4,Mod(1,210)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(210, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("210.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 210 = 2 \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	210.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:	$12.3904011012$
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	$\chi$	$=$	210.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -5.00000 q^{5} +6.00000 q^{6} +7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$

\(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -5.00000 q^{5} +6.00000 q^{6} +7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +10.0000 q^{10} +12.0000 q^{11} -12.0000 q^{12} +2.00000 q^{13} -14.0000 q^{14} +15.0000 q^{15} +16.0000 q^{16} -18.0000 q^{17} -18.0000 q^{18} +56.0000 q^{19} -20.0000 q^{20} -21.0000 q^{21} -24.0000 q^{22} -156.000 q^{23} +24.0000 q^{24} +25.0000 q^{25} -4.00000 q^{26} -27.0000 q^{27} +28.0000 q^{28} -186.000 q^{29} -30.0000 q^{30} -52.0000 q^{31} -32.0000 q^{32} -36.0000 q^{33} +36.0000 q^{34} -35.0000 q^{35} +36.0000 q^{36} -178.000 q^{37} -112.000 q^{38} -6.00000 q^{39} +40.0000 q^{40} -138.000 q^{41} +42.0000 q^{42} -412.000 q^{43} +48.0000 q^{44} -45.0000 q^{45} +312.000 q^{46} -456.000 q^{47} -48.0000 q^{48} +49.0000 q^{49} -50.0000 q^{50} +54.0000 q^{51} +8.00000 q^{52} -198.000 q^{53} +54.0000 q^{54} -60.0000 q^{55} -56.0000 q^{56} -168.000 q^{57} +372.000 q^{58} +348.000 q^{59} +60.0000 q^{60} +110.000 q^{61} +104.000 q^{62} +63.0000 q^{63} +64.0000 q^{64} -10.0000 q^{65} +72.0000 q^{66} -196.000 q^{67} -72.0000 q^{68} +468.000 q^{69} +70.0000 q^{70} -936.000 q^{71} -72.0000 q^{72} +542.000 q^{73} +356.000 q^{74} -75.0000 q^{75} +224.000 q^{76} +84.0000 q^{77} +12.0000 q^{78} +992.000 q^{79} -80.0000 q^{80} +81.0000 q^{81} +276.000 q^{82} -276.000 q^{83} -84.0000 q^{84} +90.0000 q^{85} +824.000 q^{86} +558.000 q^{87} -96.0000 q^{88} +630.000 q^{89} +90.0000 q^{90} +14.0000 q^{91} -624.000 q^{92} +156.000 q^{93} +912.000 q^{94} -280.000 q^{95} +96.0000 q^{96} +110.000 q^{97} -98.0000 q^{98} +108.000 q^{99} +O(q^{100})\)

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
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	6.00000	0.408248
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	−8.00000	−0.353553
$9$	9.00000	0.333333
$10$	10.0000	0.316228
$11$	12.0000	0.328921	0.164461	−	0.986384i	$-0.447412\pi$
$11$	12.0000	0.328921	0.164461	+	0.986384i	$0.447412\pi$
$12$	−12.0000	−0.288675
$13$	2.00000	0.0426692	0.0213346	−	0.999772i	$-0.493208\pi$
$13$	2.00000	0.0426692	0.0213346	+	0.999772i	$0.493208\pi$
$14$	−14.0000	−0.267261
$15$	15.0000	0.258199
$16$	16.0000	0.250000
$17$	−18.0000	−0.256802	−0.128401	−	0.991722i	$-0.540985\pi$
$17$	−18.0000	−0.256802	−0.128401	+	0.991722i	$0.540985\pi$
$18$	−18.0000	−0.235702
$19$	56.0000	0.676173	0.338086	−	0.941115i	$-0.390220\pi$
$19$	56.0000	0.676173	0.338086	+	0.941115i	$0.390220\pi$
$20$	−20.0000	−0.223607
$21$	−21.0000	−0.218218
$22$	−24.0000	−0.232583
$23$	−156.000	−1.41427	−0.707136	−	0.707078i	$-0.750013\pi$
$23$	−156.000	−1.41427	−0.707136	+	0.707078i	$0.750013\pi$
$24$	24.0000	0.204124
$25$	25.0000	0.200000
$26$	−4.00000	−0.0301717
$27$	−27.0000	−0.192450
$28$	28.0000	0.188982
$29$	−186.000	−1.19101	−0.595506	−	0.803351i	$-0.703048\pi$
$29$	−186.000	−1.19101	−0.595506	+	0.803351i	$0.703048\pi$
$30$	−30.0000	−0.182574
$31$	−52.0000	−0.301273	−0.150637	−	0.988589i	$-0.548132\pi$
$31$	−52.0000	−0.301273	−0.150637	+	0.988589i	$0.548132\pi$
$32$	−32.0000	−0.176777
$33$	−36.0000	−0.189903
$34$	36.0000	0.181587
$35$	−35.0000	−0.169031
$36$	36.0000	0.166667
$37$	−178.000	−0.790892	−0.395446	−	0.918489i	$-0.629410\pi$
$37$	−178.000	−0.790892	−0.395446	+	0.918489i	$0.629410\pi$
$38$	−112.000	−0.478126
$39$	−6.00000	−0.0246351
$40$	40.0000	0.158114
$41$	−138.000	−0.525658	−0.262829	−	0.964842i	$-0.584656\pi$
$41$	−138.000	−0.525658	−0.262829	+	0.964842i	$0.584656\pi$
$42$	42.0000	0.154303
$43$	−412.000	−1.46115	−0.730575	−	0.682833i	$-0.760748\pi$
$43$	−412.000	−1.46115	−0.730575	+	0.682833i	$0.760748\pi$
$44$	48.0000	0.164461
$45$	−45.0000	−0.149071
$46$	312.000	1.00004
$47$	−456.000	−1.41520	−0.707600	−	0.706613i	$-0.750222\pi$
$47$	−456.000	−1.41520	−0.707600	+	0.706613i	$0.750222\pi$
$48$	−48.0000	−0.144338
$49$	49.0000	0.142857
$50$	−50.0000	−0.141421
$51$	54.0000	0.148265
$52$	8.00000	0.0213346
$53$	−198.000	−0.513158	−0.256579	−	0.966523i	$-0.582595\pi$
$53$	−198.000	−0.513158	−0.256579	+	0.966523i	$0.582595\pi$
$54$	54.0000	0.136083
$55$	−60.0000	−0.147098
$56$	−56.0000	−0.133631
$57$	−168.000	−0.390388
$58$	372.000	0.842172
$59$	348.000	0.767894	0.383947	−	0.923355i	$-0.374565\pi$
$59$	348.000	0.767894	0.383947	+	0.923355i	$0.374565\pi$
$60$	60.0000	0.129099
$61$	110.000	0.230886	0.115443	−	0.993314i	$-0.463171\pi$
$61$	110.000	0.230886	0.115443	+	0.993314i	$0.463171\pi$
$62$	104.000	0.213032
$63$	63.0000	0.125988
$64$	64.0000	0.125000
$65$	−10.0000	−0.0190823
$66$	72.0000	0.134282
$67$	−196.000	−0.357391	−0.178696	−	0.983904i	$-0.557188\pi$
$67$	−196.000	−0.357391	−0.178696	+	0.983904i	$0.557188\pi$
$68$	−72.0000	−0.128401
$69$	468.000	0.816530
$70$	70.0000	0.119523
$71$	−936.000	−1.56455	−0.782273	−	0.622936i	$-0.785940\pi$
$71$	−936.000	−1.56455	−0.782273	+	0.622936i	$0.785940\pi$
$72$	−72.0000	−0.117851
$73$	542.000	0.868990	0.434495	−	0.900674i	$-0.356927\pi$
$73$	542.000	0.868990	0.434495	+	0.900674i	$0.356927\pi$
$74$	356.000	0.559245
$75$	−75.0000	−0.115470
$76$	224.000	0.338086
$77$	84.0000	0.124321
$78$	12.0000	0.0174196
$79$	992.000	1.41277	0.706384	−	0.707829i	$-0.250325\pi$
$79$	992.000	1.41277	0.706384	+	0.707829i	$0.250325\pi$
$80$	−80.0000	−0.111803
$81$	81.0000	0.111111
$82$	276.000	0.371696
$83$	−276.000	−0.364999	−0.182500	−	0.983206i	$-0.558419\pi$
$83$	−276.000	−0.364999	−0.182500	+	0.983206i	$0.558419\pi$
$84$	−84.0000	−0.109109
$85$	90.0000	0.114846
$86$	824.000	1.03319
$87$	558.000	0.687631
$88$	−96.0000	−0.116291
$89$	630.000	0.750336	0.375168	−	0.926957i	$-0.377585\pi$
$89$	630.000	0.750336	0.375168	+	0.926957i	$0.377585\pi$
$90$	90.0000	0.105409
$91$	14.0000	0.0161275
$92$	−624.000	−0.707136
$93$	156.000	0.173940
$94$	912.000	1.00070
$95$	−280.000	−0.302394
$96$	96.0000	0.102062
$97$	110.000	0.115142	0.0575712	−	0.998341i	$-0.481664\pi$
$97$	110.000	0.115142	0.0575712	+	0.998341i	$0.481664\pi$
$98$	−98.0000	−0.101015
$99$	108.000	0.109640
$100$	100.000	0.100000
$101$	570.000	0.561556	0.280778	−	0.959773i	$-0.409408\pi$
$101$	570.000	0.561556	0.280778	+	0.959773i	$0.409408\pi$
$102$	−108.000	−0.104839
$103$	−304.000	−0.290816	−0.145408	−	0.989372i	$-0.546449\pi$
$103$	−304.000	−0.290816	−0.145408	+	0.989372i	$0.546449\pi$
$104$	−16.0000	−0.0150859
$105$	105.000	0.0975900
$106$	396.000	0.362858
$107$	216.000	0.195154	0.0975771	−	0.995228i	$-0.468891\pi$
$107$	216.000	0.195154	0.0975771	+	0.995228i	$0.468891\pi$
$108$	−108.000	−0.0962250
$109$	614.000	0.539546	0.269773	−	0.962924i	$-0.413051\pi$
$109$	614.000	0.539546	0.269773	+	0.962924i	$0.413051\pi$
$110$	120.000	0.104014
$111$	534.000	0.456622
$112$	112.000	0.0944911
$113$	−498.000	−0.414583	−0.207292	−	0.978279i	$-0.566465\pi$
$113$	−498.000	−0.414583	−0.207292	+	0.978279i	$0.566465\pi$
$114$	336.000	0.276046
$115$	780.000	0.632482
$116$	−744.000	−0.595506
$117$	18.0000	0.0142231
$118$	−696.000	−0.542983
$119$	−126.000	−0.0970622
$120$	−120.000	−0.0912871
$121$	−1187.00	−0.891811
$122$	−220.000	−0.163261
$123$	414.000	0.303489
$124$	−208.000	−0.150637
$125$	−125.000	−0.0894427
$126$	−126.000	−0.0890871
$127$	−1888.00	−1.31916	−0.659578	−	0.751636i	$-0.729265\pi$
$127$	−1888.00	−1.31916	−0.659578	+	0.751636i	$0.729265\pi$
$128$	−128.000	−0.0883883
$129$	1236.00	0.843595
$130$	20.0000	0.0134932
$131$	−2892.00	−1.92882	−0.964409	−	0.264414i	$-0.914821\pi$
$131$	−2892.00	−1.92882	−0.964409	+	0.264414i	$0.914821\pi$
$132$	−144.000	−0.0949514
$133$	392.000	0.255569
$134$	392.000	0.252714
$135$	135.000	0.0860663
$136$	144.000	0.0907934
$137$	822.000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	822.000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	−936.000	−0.577374
$139$	−376.000	−0.229438	−0.114719	−	0.993398i	$-0.536597\pi$
$139$	−376.000	−0.229438	−0.114719	+	0.993398i	$0.536597\pi$
$140$	−140.000	−0.0845154
$141$	1368.00	0.817067
$142$	1872.00	1.10630
$143$	24.0000	0.0140348
$144$	144.000	0.0833333
$145$	930.000	0.532637
$146$	−1084.00	−0.614469
$147$	−147.000	−0.0824786
$148$	−712.000	−0.395446
$149$	3390.00	1.86389	0.931945	−	0.362600i	$-0.118111\pi$
$149$	3390.00	1.86389	0.931945	+	0.362600i	$0.118111\pi$
$150$	150.000	0.0816497
$151$	−2968.00	−1.59955	−0.799776	−	0.600298i	$-0.795049\pi$
$151$	−2968.00	−1.59955	−0.799776	+	0.600298i	$0.795049\pi$
$152$	−448.000	−0.239063
$153$	−162.000	−0.0856008
$154$	−168.000	−0.0879080
$155$	260.000	0.134734
$156$	−24.0000	−0.0123176
$157$	1874.00	0.952621	0.476310	−	0.879277i	$-0.341974\pi$
$157$	1874.00	0.952621	0.476310	+	0.879277i	$0.341974\pi$
$158$	−1984.00	−0.998978
$159$	594.000	0.296272
$160$	160.000	0.0790569
$161$	−1092.00	−0.534544
$162$	−162.000	−0.0785674
$163$	452.000	0.217199	0.108599	−	0.994086i	$-0.465363\pi$
$163$	452.000	0.217199	0.108599	+	0.994086i	$0.465363\pi$
$164$	−552.000	−0.262829
$165$	180.000	0.0849272
$166$	552.000	0.258093
$167$	−1416.00	−0.656128	−0.328064	−	0.944656i	$-0.606396\pi$
$167$	−1416.00	−0.656128	−0.328064	+	0.944656i	$0.606396\pi$
$168$	168.000	0.0771517
$169$	−2193.00	−0.998179
$170$	−180.000	−0.0812081
$171$	504.000	0.225391
$172$	−1648.00	−0.730575
$173$	426.000	0.187215	0.0936075	−	0.995609i	$-0.470160\pi$
$173$	426.000	0.187215	0.0936075	+	0.995609i	$0.470160\pi$
$174$	−1116.00	−0.486228
$175$	175.000	0.0755929
$176$	192.000	0.0822304
$177$	−1044.00	−0.443344
$178$	−1260.00	−0.530567
$179$	2700.00	1.12742	0.563708	−	0.825974i	$-0.309374\pi$
$179$	2700.00	1.12742	0.563708	+	0.825974i	$0.309374\pi$
$180$	−180.000	−0.0745356
$181$	−1978.00	−0.812285	−0.406142	−	0.913810i	$-0.633126\pi$
$181$	−1978.00	−0.812285	−0.406142	+	0.913810i	$0.633126\pi$
$182$	−28.0000	−0.0114038
$183$	−330.000	−0.133302
$184$	1248.00	0.500021
$185$	890.000	0.353698
$186$	−312.000	−0.122994
$187$	−216.000	−0.0844678
$188$	−1824.00	−0.707600
$189$	−189.000	−0.0727393
$190$	560.000	0.213825
$191$	−2328.00	−0.881928	−0.440964	−	0.897525i	$-0.645363\pi$
$191$	−2328.00	−0.881928	−0.440964	+	0.897525i	$0.645363\pi$
$192$	−192.000	−0.0721688
$193$	−3166.00	−1.18080	−0.590398	−	0.807112i	$-0.701029\pi$
$193$	−3166.00	−1.18080	−0.590398	+	0.807112i	$0.701029\pi$
$194$	−220.000	−0.0814179
$195$	30.0000	0.0110172
$196$	196.000	0.0714286
$197$	−414.000	−0.149727	−0.0748637	−	0.997194i	$-0.523852\pi$
$197$	−414.000	−0.149727	−0.0748637	+	0.997194i	$0.523852\pi$
$198$	−216.000	−0.0775275
$199$	−1636.00	−0.582779	−0.291389	−	0.956605i	$-0.594118\pi$
$199$	−1636.00	−0.582779	−0.291389	+	0.956605i	$0.594118\pi$
$200$	−200.000	−0.0707107
$201$	588.000	0.206340
$202$	−1140.00	−0.397080
$203$	−1302.00	−0.450160
$204$	216.000	0.0741325
$205$	690.000	0.235081
$206$	608.000	0.205638
$207$	−1404.00	−0.471424
$208$	32.0000	0.0106673
$209$	672.000	0.222408
$210$	−210.000	−0.0690066
$211$	−2860.00	−0.933130	−0.466565	−	0.884487i	$-0.654509\pi$
$211$	−2860.00	−0.933130	−0.466565	+	0.884487i	$0.654509\pi$
$212$	−792.000	−0.256579
$213$	2808.00	0.903291
$214$	−432.000	−0.137995
$215$	2060.00	0.653446
$216$	216.000	0.0680414
$217$	−364.000	−0.113871
$218$	−1228.00	−0.381517
$219$	−1626.00	−0.501712
$220$	−240.000	−0.0735491
$221$	−36.0000	−0.0109576
$222$	−1068.00	−0.322880
$223$	−1096.00	−0.329119	−0.164560	−	0.986367i	$-0.552620\pi$
$223$	−1096.00	−0.329119	−0.164560	+	0.986367i	$0.552620\pi$
$224$	−224.000	−0.0668153
$225$	225.000	0.0666667
$226$	996.000	0.293155
$227$	6276.00	1.83503	0.917517	−	0.397696i	$-0.130190\pi$
$227$	6276.00	1.83503	0.917517	+	0.397696i	$0.130190\pi$
$228$	−672.000	−0.195194
$229$	−754.000	−0.217580	−0.108790	−	0.994065i	$-0.534698\pi$
$229$	−754.000	−0.217580	−0.108790	+	0.994065i	$0.534698\pi$
$230$	−1560.00	−0.447232
$231$	−252.000	−0.0717765
$232$	1488.00	0.421086
$233$	3870.00	1.08812	0.544060	−	0.839046i	$-0.316886\pi$
$233$	3870.00	1.08812	0.544060	+	0.839046i	$0.316886\pi$
$234$	−36.0000	−0.0100572
$235$	2280.00	0.632897
$236$	1392.00	0.383947
$237$	−2976.00	−0.815662
$238$	252.000	0.0686333
$239$	744.000	0.201361	0.100681	−	0.994919i	$-0.467898\pi$
$239$	744.000	0.201361	0.100681	+	0.994919i	$0.467898\pi$
$240$	240.000	0.0645497
$241$	5474.00	1.46312	0.731559	−	0.681778i	$-0.238793\pi$
$241$	5474.00	1.46312	0.731559	+	0.681778i	$0.238793\pi$
$242$	2374.00	0.630605
$243$	−243.000	−0.0641500
$244$	440.000	0.115443
$245$	−245.000	−0.0638877
$246$	−828.000	−0.214599
$247$	112.000	0.0288518
$248$	416.000	0.106516
$249$	828.000	0.210732
$250$	250.000	0.0632456
$251$	1980.00	0.497914	0.248957	−	0.968514i	$-0.419912\pi$
$251$	1980.00	0.497914	0.248957	+	0.968514i	$0.419912\pi$
$252$	252.000	0.0629941
$253$	−1872.00	−0.465184
$254$	3776.00	0.932785
$255$	−270.000	−0.0663061
$256$	256.000	0.0625000
$257$	−210.000	−0.0509706	−0.0254853	−	0.999675i	$-0.508113\pi$
$257$	−210.000	−0.0509706	−0.0254853	+	0.999675i	$0.508113\pi$
$258$	−2472.00	−0.596512
$259$	−1246.00	−0.298929
$260$	−40.0000	−0.00954113
$261$	−1674.00	−0.397004
$262$	5784.00	1.36388
$263$	−1428.00	−0.334807	−0.167404	−	0.985888i	$-0.553538\pi$
$263$	−1428.00	−0.334807	−0.167404	+	0.985888i	$0.553538\pi$
$264$	288.000	0.0671408
$265$	990.000	0.229491
$266$	−784.000	−0.180715
$267$	−1890.00	−0.433206
$268$	−784.000	−0.178696
$269$	4122.00	0.934285	0.467143	−	0.884182i	$-0.345283\pi$
$269$	4122.00	0.934285	0.467143	+	0.884182i	$0.345283\pi$
$270$	−270.000	−0.0608581
$271$	5780.00	1.29561	0.647804	−	0.761807i	$-0.275687\pi$
$271$	5780.00	1.29561	0.647804	+	0.761807i	$0.275687\pi$
$272$	−288.000	−0.0642006
$273$	−42.0000	−0.00931119
$274$	−1644.00	−0.362473
$275$	300.000	0.0657843
$276$	1872.00	0.408265
$277$	4574.00	0.992148	0.496074	−	0.868280i	$-0.334774\pi$
$277$	4574.00	0.992148	0.496074	+	0.868280i	$0.334774\pi$
$278$	752.000	0.162237
$279$	−468.000	−0.100424
$280$	280.000	0.0597614
$281$	3450.00	0.732419	0.366210	−	0.930532i	$-0.380655\pi$
$281$	3450.00	0.732419	0.366210	+	0.930532i	$0.380655\pi$
$282$	−2736.00	−0.577753
$283$	−700.000	−0.147034	−0.0735171	−	0.997294i	$-0.523422\pi$
$283$	−700.000	−0.147034	−0.0735171	+	0.997294i	$0.523422\pi$
$284$	−3744.00	−0.782273
$285$	840.000	0.174587
$286$	−48.0000	−0.00992412
$287$	−966.000	−0.198680
$288$	−288.000	−0.0589256
$289$	−4589.00	−0.934053
$290$	−1860.00	−0.376631
$291$	−330.000	−0.0664775
$292$	2168.00	0.434495
$293$	7170.00	1.42961	0.714805	−	0.699324i	$-0.246515\pi$
$293$	7170.00	1.42961	0.714805	+	0.699324i	$0.246515\pi$
$294$	294.000	0.0583212
$295$	−1740.00	−0.343413
$296$	1424.00	0.279623
$297$	−324.000	−0.0633010
$298$	−6780.00	−1.31797
$299$	−312.000	−0.0603459
$300$	−300.000	−0.0577350
$301$	−2884.00	−0.552262
$302$	5936.00	1.13105
$303$	−1710.00	−0.324214
$304$	896.000	0.169043
$305$	−550.000	−0.103255
$306$	324.000	0.0605289
$307$	6644.00	1.23516	0.617578	−	0.786509i	$-0.288114\pi$
$307$	6644.00	1.23516	0.617578	+	0.786509i	$0.288114\pi$
$308$	336.000	0.0621603
$309$	912.000	0.167902
$310$	−520.000	−0.0952710
$311$	−5376.00	−0.980209	−0.490104	−	0.871664i	$-0.663041\pi$
$311$	−5376.00	−0.980209	−0.490104	+	0.871664i	$0.663041\pi$
$312$	48.0000	0.00870982
$313$	2126.00	0.383925	0.191963	−	0.981402i	$-0.438515\pi$
$313$	2126.00	0.383925	0.191963	+	0.981402i	$0.438515\pi$
$314$	−3748.00	−0.673605
$315$	−315.000	−0.0563436
$316$	3968.00	0.706384
$317$	1074.00	0.190290	0.0951449	−	0.995463i	$-0.469669\pi$
$317$	1074.00	0.190290	0.0951449	+	0.995463i	$0.469669\pi$
$318$	−1188.00	−0.209496
$319$	−2232.00	−0.391749
$320$	−320.000	−0.0559017
$321$	−648.000	−0.112672
$322$	2184.00	0.377980
$323$	−1008.00	−0.173643
$324$	324.000	0.0555556
$325$	50.0000	0.00853385
$326$	−904.000	−0.153583
$327$	−1842.00	−0.311507
$328$	1104.00	0.185848
$329$	−3192.00	−0.534896
$330$	−360.000	−0.0600526
$331$	−2788.00	−0.462968	−0.231484	−	0.972839i	$-0.574358\pi$
$331$	−2788.00	−0.462968	−0.231484	+	0.972839i	$0.574358\pi$
$332$	−1104.00	−0.182500
$333$	−1602.00	−0.263631
$334$	2832.00	0.463953
$335$	980.000	0.159830
$336$	−336.000	−0.0545545
$337$	−6334.00	−1.02384	−0.511921	−	0.859032i	$-0.671066\pi$
$337$	−6334.00	−1.02384	−0.511921	+	0.859032i	$0.671066\pi$
$338$	4386.00	0.705819
$339$	1494.00	0.239360
$340$	360.000	0.0574228
$341$	−624.000	−0.0990953
$342$	−1008.00	−0.159375
$343$	343.000	0.0539949
$344$	3296.00	0.516594
$345$	−2340.00	−0.365163
$346$	−852.000	−0.132381
$347$	−7032.00	−1.08789	−0.543945	−	0.839121i	$-0.683070\pi$
$347$	−7032.00	−1.08789	−0.543945	+	0.839121i	$0.683070\pi$
$348$	2232.00	0.343815
$349$	−1474.00	−0.226079	−0.113039	−	0.993591i	$-0.536059\pi$
$349$	−1474.00	−0.226079	−0.113039	+	0.993591i	$0.536059\pi$
$350$	−350.000	−0.0534522
$351$	−54.0000	−0.00821170
$352$	−384.000	−0.0581456
$353$	7950.00	1.19868	0.599342	−	0.800493i	$-0.295429\pi$
$353$	7950.00	1.19868	0.599342	+	0.800493i	$0.295429\pi$
$354$	2088.00	0.313491
$355$	4680.00	0.699686
$356$	2520.00	0.375168
$357$	378.000	0.0560389
$358$	−5400.00	−0.797204
$359$	6624.00	0.973820	0.486910	−	0.873452i	$-0.338124\pi$
$359$	6624.00	0.973820	0.486910	+	0.873452i	$0.338124\pi$
$360$	360.000	0.0527046
$361$	−3723.00	−0.542790
$362$	3956.00	0.574372
$363$	3561.00	0.514887
$364$	56.0000	0.00806373
$365$	−2710.00	−0.388624
$366$	660.000	0.0942589
$367$	1784.00	0.253744	0.126872	−	0.991919i	$-0.459506\pi$
$367$	1784.00	0.253744	0.126872	+	0.991919i	$0.459506\pi$
$368$	−2496.00	−0.353568
$369$	−1242.00	−0.175219
$370$	−1780.00	−0.250102
$371$	−1386.00	−0.193956
$372$	624.000	0.0869701
$373$	−1978.00	−0.274576	−0.137288	−	0.990531i	$-0.543839\pi$
$373$	−1978.00	−0.274576	−0.137288	+	0.990531i	$0.543839\pi$
$374$	432.000	0.0597278
$375$	375.000	0.0516398
$376$	3648.00	0.500349
$377$	−372.000	−0.0508196
$378$	378.000	0.0514344
$379$	−10780.0	−1.46103	−0.730516	−	0.682895i	$-0.760721\pi$
$379$	−10780.0	−1.46103	−0.730516	+	0.682895i	$0.760721\pi$
$380$	−1120.00	−0.151197
$381$	5664.00	0.761616
$382$	4656.00	0.623617
$383$	−5880.00	−0.784475	−0.392238	−	0.919864i	$-0.628299\pi$
$383$	−5880.00	−0.784475	−0.392238	+	0.919864i	$0.628299\pi$
$384$	384.000	0.0510310
$385$	−420.000	−0.0555979
$386$	6332.00	0.834949
$387$	−3708.00	−0.487050
$388$	440.000	0.0575712
$389$	6438.00	0.839125	0.419562	−	0.907726i	$-0.362184\pi$
$389$	6438.00	0.839125	0.419562	+	0.907726i	$0.362184\pi$
$390$	−60.0000	−0.00779030
$391$	2808.00	0.363188
$392$	−392.000	−0.0505076
$393$	8676.00	1.11360
$394$	828.000	0.105873
$395$	−4960.00	−0.631809
$396$	432.000	0.0548202
$397$	2954.00	0.373443	0.186722	−	0.982413i	$-0.440214\pi$
$397$	2954.00	0.373443	0.186722	+	0.982413i	$0.440214\pi$
$398$	3272.00	0.412087
$399$	−1176.00	−0.147553
$400$	400.000	0.0500000
$401$	−5574.00	−0.694145	−0.347073	−	0.937838i	$-0.612824\pi$
$401$	−5574.00	−0.694145	−0.347073	+	0.937838i	$0.612824\pi$
$402$	−1176.00	−0.145904
$403$	−104.000	−0.0128551
$404$	2280.00	0.280778
$405$	−405.000	−0.0496904
$406$	2604.00	0.318311
$407$	−2136.00	−0.260141
$408$	−432.000	−0.0524196
$409$	−15406.0	−1.86254	−0.931269	−	0.364333i	$-0.881297\pi$
$409$	−15406.0	−1.86254	−0.931269	+	0.364333i	$0.881297\pi$
$410$	−1380.00	−0.166228
$411$	−2466.00	−0.295958
$412$	−1216.00	−0.145408
$413$	2436.00	0.290237
$414$	2808.00	0.333347
$415$	1380.00	0.163233
$416$	−64.0000	−0.00754293
$417$	1128.00	0.132466
$418$	−1344.00	−0.157266
$419$	2940.00	0.342789	0.171394	−	0.985203i	$-0.445173\pi$
$419$	2940.00	0.342789	0.171394	+	0.985203i	$0.445173\pi$
$420$	420.000	0.0487950
$421$	254.000	0.0294043	0.0147021	−	0.999892i	$-0.495320\pi$
$421$	254.000	0.0294043	0.0147021	+	0.999892i	$0.495320\pi$
$422$	5720.00	0.659823
$423$	−4104.00	−0.471734
$424$	1584.00	0.181429
$425$	−450.000	−0.0513605
$426$	−5616.00	−0.638723
$427$	770.000	0.0872668
$428$	864.000	0.0975771
$429$	−72.0000	−0.00810301
$430$	−4120.00	−0.462056
$431$	−13248.0	−1.48059	−0.740294	−	0.672283i	$-0.765314\pi$
$431$	−13248.0	−1.48059	−0.740294	+	0.672283i	$0.765314\pi$
$432$	−432.000	−0.0481125
$433$	16598.0	1.84215	0.921073	−	0.389391i	$-0.127314\pi$
$433$	16598.0	1.84215	0.921073	+	0.389391i	$0.127314\pi$
$434$	728.000	0.0805187
$435$	−2790.00	−0.307518
$436$	2456.00	0.269773
$437$	−8736.00	−0.956292
$438$	3252.00	0.354764
$439$	−6532.00	−0.710149	−0.355074	−	0.934838i	$-0.615544\pi$
$439$	−6532.00	−0.710149	−0.355074	+	0.934838i	$0.615544\pi$
$440$	480.000	0.0520071
$441$	441.000	0.0476190
$442$	72.0000	0.00774817
$443$	12216.0	1.31016	0.655079	−	0.755561i	$-0.272635\pi$
$443$	12216.0	1.31016	0.655079	+	0.755561i	$0.272635\pi$
$444$	2136.00	0.228311
$445$	−3150.00	−0.335560
$446$	2192.00	0.232722
$447$	−10170.0	−1.07612
$448$	448.000	0.0472456
$449$	306.000	0.0321627	0.0160813	−	0.999871i	$-0.494881\pi$
$449$	306.000	0.0321627	0.0160813	+	0.999871i	$0.494881\pi$
$450$	−450.000	−0.0471405
$451$	−1656.00	−0.172900
$452$	−1992.00	−0.207292
$453$	8904.00	0.923502
$454$	−12552.0	−1.29757
$455$	−70.0000	−0.00721242
$456$	1344.00	0.138023
$457$	−6046.00	−0.618862	−0.309431	−	0.950922i	$-0.600139\pi$
$457$	−6046.00	−0.618862	−0.309431	+	0.950922i	$0.600139\pi$
$458$	1508.00	0.153852
$459$	486.000	0.0494217
$460$	3120.00	0.316241
$461$	7122.00	0.719533	0.359766	−	0.933042i	$-0.382856\pi$
$461$	7122.00	0.719533	0.359766	+	0.933042i	$0.382856\pi$
$462$	504.000	0.0507537
$463$	−11248.0	−1.12903	−0.564513	−	0.825424i	$-0.690936\pi$
$463$	−11248.0	−1.12903	−0.564513	+	0.825424i	$0.690936\pi$
$464$	−2976.00	−0.297753
$465$	−780.000	−0.0777885
$466$	−7740.00	−0.769418
$467$	−18252.0	−1.80857	−0.904285	−	0.426930i	$-0.859595\pi$
$467$	−18252.0	−1.80857	−0.904285	+	0.426930i	$0.859595\pi$
$468$	72.0000	0.00711154
$469$	−1372.00	−0.135081
$470$	−4560.00	−0.447526
$471$	−5622.00	−0.549996
$472$	−2784.00	−0.271491
$473$	−4944.00	−0.480603
$474$	5952.00	0.576760
$475$	1400.00	0.135235
$476$	−504.000	−0.0485311
$477$	−1782.00	−0.171053
$478$	−1488.00	−0.142384
$479$	−18168.0	−1.73302	−0.866511	−	0.499159i	$-0.833642\pi$
$479$	−18168.0	−1.73302	−0.866511	+	0.499159i	$0.833642\pi$
$480$	−480.000	−0.0456435
$481$	−356.000	−0.0337468
$482$	−10948.0	−1.03458
$483$	3276.00	0.308619
$484$	−4748.00	−0.445905
$485$	−550.000	−0.0514932
$486$	486.000	0.0453609
$487$	19856.0	1.84756	0.923780	−	0.382925i	$-0.125083\pi$
$487$	19856.0	1.84756	0.923780	+	0.382925i	$0.125083\pi$
$488$	−880.000	−0.0816306
$489$	−1356.00	−0.125400
$490$	490.000	0.0451754
$491$	11220.0	1.03127	0.515633	−	0.856810i	$-0.327557\pi$
$491$	11220.0	1.03127	0.515633	+	0.856810i	$0.327557\pi$
$492$	1656.00	0.151744
$493$	3348.00	0.305855
$494$	−224.000	−0.0204013
$495$	−540.000	−0.0490327
$496$	−832.000	−0.0753184
$497$	−6552.00	−0.591343
$498$	−1656.00	−0.149010
$499$	−9268.00	−0.831448	−0.415724	−	0.909491i	$-0.636472\pi$
$499$	−9268.00	−0.831448	−0.415724	+	0.909491i	$0.636472\pi$
$500$	−500.000	−0.0447214
$501$	4248.00	0.378816
$502$	−3960.00	−0.352079
$503$	18576.0	1.64665	0.823323	−	0.567573i	$-0.192118\pi$
$503$	18576.0	1.64665	0.823323	+	0.567573i	$0.192118\pi$
$504$	−504.000	−0.0445435
$505$	−2850.00	−0.251135
$506$	3744.00	0.328935
$507$	6579.00	0.576299
$508$	−7552.00	−0.659578
$509$	−11190.0	−0.974436	−0.487218	−	0.873280i	$-0.661988\pi$
$509$	−11190.0	−0.974436	−0.487218	+	0.873280i	$0.661988\pi$
$510$	540.000	0.0468855
$511$	3794.00	0.328448
$512$	−512.000	−0.0441942
$513$	−1512.00	−0.130129
$514$	420.000	0.0360416
$515$	1520.00	0.130057
$516$	4944.00	0.421797
$517$	−5472.00	−0.465490
$518$	2492.00	0.211375
$519$	−1278.00	−0.108089
$520$	80.0000	0.00674660
$521$	−306.000	−0.0257315	−0.0128657	−	0.999917i	$-0.504095\pi$
$521$	−306.000	−0.0257315	−0.0128657	+	0.999917i	$0.504095\pi$
$522$	3348.00	0.280724
$523$	17444.0	1.45846	0.729228	−	0.684270i	$-0.239879\pi$
$523$	17444.0	1.45846	0.729228	+	0.684270i	$0.239879\pi$
$524$	−11568.0	−0.964409
$525$	−525.000	−0.0436436
$526$	2856.00	0.236744
$527$	936.000	0.0773677
$528$	−576.000	−0.0474757
$529$	12169.0	1.00016
$530$	−1980.00	−0.162275
$531$	3132.00	0.255965
$532$	1568.00	0.127785
$533$	−276.000	−0.0224294
$534$	3780.00	0.306323
$535$	−1080.00	−0.0872756
$536$	1568.00	0.126357
$537$	−8100.00	−0.650914
$538$	−8244.00	−0.660640
$539$	588.000	0.0469888
$540$	540.000	0.0430331
$541$	−538.000	−0.0427549	−0.0213775	−	0.999771i	$-0.506805\pi$
$541$	−538.000	−0.0427549	−0.0213775	+	0.999771i	$0.506805\pi$
$542$	−11560.0	−0.916134
$543$	5934.00	0.468973
$544$	576.000	0.0453967
$545$	−3070.00	−0.241292
$546$	84.0000	0.00658401
$547$	19820.0	1.54925	0.774627	−	0.632418i	$-0.217938\pi$
$547$	19820.0	1.54925	0.774627	+	0.632418i	$0.217938\pi$
$548$	3288.00	0.256307
$549$	990.000	0.0769621
$550$	−600.000	−0.0465165
$551$	−10416.0	−0.805329
$552$	−3744.00	−0.288687
$553$	6944.00	0.533976
$554$	−9148.00	−0.701555
$555$	−2670.00	−0.204208
$556$	−1504.00	−0.114719
$557$	−16686.0	−1.26932	−0.634658	−	0.772794i	$-0.718859\pi$
$557$	−16686.0	−1.26932	−0.634658	+	0.772794i	$0.718859\pi$
$558$	936.000	0.0710108
$559$	−824.000	−0.0623461
$560$	−560.000	−0.0422577
$561$	648.000	0.0487675
$562$	−6900.00	−0.517898
$563$	−16788.0	−1.25671	−0.628357	−	0.777925i	$-0.716272\pi$
$563$	−16788.0	−1.25671	−0.628357	+	0.777925i	$0.716272\pi$
$564$	5472.00	0.408533
$565$	2490.00	0.185407
$566$	1400.00	0.103969
$567$	567.000	0.0419961
$568$	7488.00	0.553151
$569$	15906.0	1.17191	0.585953	−	0.810345i	$-0.300720\pi$
$569$	15906.0	1.17191	0.585953	+	0.810345i	$0.300720\pi$
$570$	−1680.00	−0.123452
$571$	17084.0	1.25209	0.626045	−	0.779787i	$-0.284673\pi$
$571$	17084.0	1.25209	0.626045	+	0.779787i	$0.284673\pi$
$572$	96.0000	0.00701742
$573$	6984.00	0.509181
$574$	1932.00	0.140488
$575$	−3900.00	−0.282854
$576$	576.000	0.0416667
$577$	25382.0	1.83131	0.915656	−	0.401964i	$-0.131672\pi$
$577$	25382.0	1.83131	0.915656	+	0.401964i	$0.131672\pi$
$578$	9178.00	0.660475
$579$	9498.00	0.681733
$580$	3720.00	0.266318
$581$	−1932.00	−0.137957
$582$	660.000	0.0470067
$583$	−2376.00	−0.168789
$584$	−4336.00	−0.307235
$585$	−90.0000	−0.00636076
$586$	−14340.0	−1.01089
$587$	13764.0	0.967804	0.483902	−	0.875122i	$-0.339219\pi$
$587$	13764.0	0.967804	0.483902	+	0.875122i	$0.339219\pi$
$588$	−588.000	−0.0412393
$589$	−2912.00	−0.203713
$590$	3480.00	0.242829
$591$	1242.00	0.0864451
$592$	−2848.00	−0.197723
$593$	−13266.0	−0.918667	−0.459333	−	0.888264i	$-0.651912\pi$
$593$	−13266.0	−0.918667	−0.459333	+	0.888264i	$0.651912\pi$
$594$	648.000	0.0447605
$595$	630.000	0.0434075
$596$	13560.0	0.931945
$597$	4908.00	0.336467
$598$	624.000	0.0426710
$599$	9600.00	0.654834	0.327417	−	0.944880i	$-0.393822\pi$
$599$	9600.00	0.654834	0.327417	+	0.944880i	$0.393822\pi$
$600$	600.000	0.0408248
$601$	−19582.0	−1.32906	−0.664531	−	0.747261i	$-0.731369\pi$
$601$	−19582.0	−1.32906	−0.664531	+	0.747261i	$0.731369\pi$
$602$	5768.00	0.390509
$603$	−1764.00	−0.119130
$604$	−11872.0	−0.799776
$605$	5935.00	0.398830
$606$	3420.00	0.229254
$607$	3944.00	0.263727	0.131863	−	0.991268i	$-0.457904\pi$
$607$	3944.00	0.263727	0.131863	+	0.991268i	$0.457904\pi$
$608$	−1792.00	−0.119532
$609$	3906.00	0.259900
$610$	1100.00	0.0730126
$611$	−912.000	−0.0603855
$612$	−648.000	−0.0428004
$613$	20846.0	1.37351	0.686755	−	0.726889i	$-0.259034\pi$
$613$	20846.0	1.37351	0.686755	+	0.726889i	$0.259034\pi$
$614$	−13288.0	−0.873388
$615$	−2070.00	−0.135724
$616$	−672.000	−0.0439540
$617$	15342.0	1.00105	0.500523	−	0.865723i	$-0.333141\pi$
$617$	15342.0	1.00105	0.500523	+	0.865723i	$0.333141\pi$
$618$	−1824.00	−0.118725
$619$	−5128.00	−0.332975	−0.166488	−	0.986044i	$-0.553243\pi$
$619$	−5128.00	−0.332975	−0.166488	+	0.986044i	$0.553243\pi$
$620$	1040.00	0.0673668
$621$	4212.00	0.272177
$622$	10752.0	0.693112
$623$	4410.00	0.283600
$624$	−96.0000	−0.00615878
$625$	625.000	0.0400000
$626$	−4252.00	−0.271476
$627$	−2016.00	−0.128407
$628$	7496.00	0.476310
$629$	3204.00	0.203103
$630$	630.000	0.0398410
$631$	31016.0	1.95678	0.978389	−	0.206771i	$-0.0662954\pi$
$631$	31016.0	1.95678	0.978389	+	0.206771i	$0.0662954\pi$
$632$	−7936.00	−0.499489
$633$	8580.00	0.538743
$634$	−2148.00	−0.134555
$635$	9440.00	0.589945
$636$	2376.00	0.148136
$637$	98.0000	0.00609561
$638$	4464.00	0.277009
$639$	−8424.00	−0.521515
$640$	640.000	0.0395285
$641$	15474.0	0.953489	0.476744	−	0.879042i	$-0.341817\pi$
$641$	15474.0	0.953489	0.476744	+	0.879042i	$0.341817\pi$
$642$	1296.00	0.0796714
$643$	24644.0	1.51145	0.755727	−	0.654887i	$-0.227284\pi$
$643$	24644.0	1.51145	0.755727	+	0.654887i	$0.227284\pi$
$644$	−4368.00	−0.267272
$645$	−6180.00	−0.377267
$646$	2016.00	0.122784
$647$	−16632.0	−1.01062	−0.505310	−	0.862938i	$-0.668622\pi$
$647$	−16632.0	−1.01062	−0.505310	+	0.862938i	$0.668622\pi$
$648$	−648.000	−0.0392837
$649$	4176.00	0.252577
$650$	−100.000	−0.00603434
$651$	1092.00	0.0657432
$652$	1808.00	0.108599
$653$	−10542.0	−0.631762	−0.315881	−	0.948799i	$-0.602300\pi$
$653$	−10542.0	−0.631762	−0.315881	+	0.948799i	$0.602300\pi$
$654$	3684.00	0.220269
$655$	14460.0	0.862594
$656$	−2208.00	−0.131415
$657$	4878.00	0.289663
$658$	6384.00	0.378228
$659$	15276.0	0.902987	0.451494	−	0.892274i	$-0.350891\pi$
$659$	15276.0	0.902987	0.451494	+	0.892274i	$0.350891\pi$
$660$	720.000	0.0424636
$661$	1478.00	0.0869706	0.0434853	−	0.999054i	$-0.486154\pi$
$661$	1478.00	0.0869706	0.0434853	+	0.999054i	$0.486154\pi$
$662$	5576.00	0.327368
$663$	108.000	0.00632635
$664$	2208.00	0.129047
$665$	−1960.00	−0.114294
$666$	3204.00	0.186415
$667$	29016.0	1.68441
$668$	−5664.00	−0.328064
$669$	3288.00	0.190017
$670$	−1960.00	−0.113017
$671$	1320.00	0.0759434
$672$	672.000	0.0385758
$673$	−19366.0	−1.10922	−0.554610	−	0.832111i	$-0.687132\pi$
$673$	−19366.0	−1.10922	−0.554610	+	0.832111i	$0.687132\pi$
$674$	12668.0	0.723966
$675$	−675.000	−0.0384900
$676$	−8772.00	−0.499090
$677$	−21390.0	−1.21430	−0.607152	−	0.794585i	$-0.707688\pi$
$677$	−21390.0	−1.21430	−0.607152	+	0.794585i	$0.707688\pi$
$678$	−2988.00	−0.169253
$679$	770.000	0.0435197
$680$	−720.000	−0.0406040
$681$	−18828.0	−1.05946
$682$	1248.00	0.0700710
$683$	−21672.0	−1.21414	−0.607069	−	0.794649i	$-0.707655\pi$
$683$	−21672.0	−1.21414	−0.607069	+	0.794649i	$0.707655\pi$
$684$	2016.00	0.112695
$685$	−4110.00	−0.229248
$686$	−686.000	−0.0381802
$687$	2262.00	0.125620
$688$	−6592.00	−0.365287
$689$	−396.000	−0.0218961
$690$	4680.00	0.258210
$691$	−5992.00	−0.329879	−0.164940	−	0.986304i	$-0.552743\pi$
$691$	−5992.00	−0.329879	−0.164940	+	0.986304i	$0.552743\pi$
$692$	1704.00	0.0936075
$693$	756.000	0.0414402
$694$	14064.0	0.769254
$695$	1880.00	0.102608
$696$	−4464.00	−0.243114
$697$	2484.00	0.134990
$698$	2948.00	0.159862
$699$	−11610.0	−0.628227
$700$	700.000	0.0377964
$701$	17766.0	0.957222	0.478611	−	0.878027i	$-0.341140\pi$
$701$	17766.0	0.957222	0.478611	+	0.878027i	$0.341140\pi$
$702$	108.000	0.00580655
$703$	−9968.00	−0.534780
$704$	768.000	0.0411152
$705$	−6840.00	−0.365403
$706$	−15900.0	−0.847598
$707$	3990.00	0.212248
$708$	−4176.00	−0.221672
$709$	−24514.0	−1.29851	−0.649254	−	0.760571i	$-0.724919\pi$
$709$	−24514.0	−1.29851	−0.649254	+	0.760571i	$0.724919\pi$
$710$	−9360.00	−0.494753
$711$	8928.00	0.470923
$712$	−5040.00	−0.265284
$713$	8112.00	0.426082
$714$	−756.000	−0.0396255
$715$	−120.000	−0.00627657
$716$	10800.0	0.563708
$717$	−2232.00	−0.116256
$718$	−13248.0	−0.688595
$719$	−13176.0	−0.683424	−0.341712	−	0.939805i	$-0.611007\pi$
$719$	−13176.0	−0.683424	−0.341712	+	0.939805i	$0.611007\pi$
$720$	−720.000	−0.0372678
$721$	−2128.00	−0.109918
$722$	7446.00	0.383811
$723$	−16422.0	−0.844731
$724$	−7912.00	−0.406142
$725$	−4650.00	−0.238202
$726$	−7122.00	−0.364080
$727$	20792.0	1.06071	0.530353	−	0.847777i	$-0.322060\pi$
$727$	20792.0	1.06071	0.530353	+	0.847777i	$0.322060\pi$
$728$	−112.000	−0.00570192
$729$	729.000	0.0370370
$730$	5420.00	0.274799
$731$	7416.00	0.375227
$732$	−1320.00	−0.0666511
$733$	−21742.0	−1.09558	−0.547789	−	0.836616i	$-0.684530\pi$
$733$	−21742.0	−1.09558	−0.547789	+	0.836616i	$0.684530\pi$
$734$	−3568.00	−0.179424
$735$	735.000	0.0368856
$736$	4992.00	0.250010
$737$	−2352.00	−0.117554
$738$	2484.00	0.123899
$739$	39044.0	1.94351	0.971757	−	0.235984i	$-0.0758313\pi$
$739$	39044.0	1.94351	0.971757	+	0.235984i	$0.0758313\pi$
$740$	3560.00	0.176849
$741$	−336.000	−0.0166576
$742$	2772.00	0.137147
$743$	−31116.0	−1.53639	−0.768193	−	0.640218i	$-0.778844\pi$
$743$	−31116.0	−1.53639	−0.768193	+	0.640218i	$0.778844\pi$
$744$	−1248.00	−0.0614972
$745$	−16950.0	−0.833557
$746$	3956.00	0.194155
$747$	−2484.00	−0.121666
$748$	−864.000	−0.0422339
$749$	1512.00	0.0737614
$750$	−750.000	−0.0365148
$751$	−29320.0	−1.42464	−0.712318	−	0.701857i	$-0.752355\pi$
$751$	−29320.0	−1.42464	−0.712318	+	0.701857i	$0.752355\pi$
$752$	−7296.00	−0.353800
$753$	−5940.00	−0.287471
$754$	744.000	0.0359349
$755$	14840.0	0.715342
$756$	−756.000	−0.0363696
$757$	−2266.00	−0.108797	−0.0543984	−	0.998519i	$-0.517324\pi$
$757$	−2266.00	−0.108797	−0.0543984	+	0.998519i	$0.517324\pi$
$758$	21560.0	1.03311
$759$	5616.00	0.268574
$760$	2240.00	0.106912
$761$	−29946.0	−1.42647	−0.713234	−	0.700926i	$-0.752770\pi$
$761$	−29946.0	−1.42647	−0.713234	+	0.700926i	$0.752770\pi$
$762$	−11328.0	−0.538543
$763$	4298.00	0.203929
$764$	−9312.00	−0.440964
$765$	810.000	0.0382818
$766$	11760.0	0.554708
$767$	696.000	0.0327655
$768$	−768.000	−0.0360844
$769$	−23110.0	−1.08370	−0.541852	−	0.840474i	$-0.682277\pi$
$769$	−23110.0	−1.08370	−0.541852	+	0.840474i	$0.682277\pi$
$770$	840.000	0.0393136
$771$	630.000	0.0294279
$772$	−12664.0	−0.590398
$773$	−31950.0	−1.48663	−0.743313	−	0.668944i	$-0.766747\pi$
$773$	−31950.0	−1.48663	−0.743313	+	0.668944i	$0.766747\pi$
$774$	7416.00	0.344396
$775$	−1300.00	−0.0602547
$776$	−880.000	−0.0407090
$777$	3738.00	0.172587
$778$	−12876.0	−0.593351
$779$	−7728.00	−0.355436
$780$	120.000	0.00550858
$781$	−11232.0	−0.514613
$782$	−5616.00	−0.256813
$783$	5022.00	0.229210
$784$	784.000	0.0357143
$785$	−9370.00	−0.426025
$786$	−17352.0	−0.787437
$787$	6284.00	0.284626	0.142313	−	0.989822i	$-0.454546\pi$
$787$	6284.00	0.284626	0.142313	+	0.989822i	$0.454546\pi$
$788$	−1656.00	−0.0748637
$789$	4284.00	0.193301
$790$	9920.00	0.446757
$791$	−3486.00	−0.156698
$792$	−864.000	−0.0387638
$793$	220.000	0.00985174
$794$	−5908.00	−0.264064
$795$	−2970.00	−0.132497
$796$	−6544.00	−0.291389
$797$	5946.00	0.264264	0.132132	−	0.991232i	$-0.457818\pi$
$797$	5946.00	0.264264	0.132132	+	0.991232i	$0.457818\pi$
$798$	2352.00	0.104336
$799$	8208.00	0.363427
$800$	−800.000	−0.0353553
$801$	5670.00	0.250112
$802$	11148.0	0.490835
$803$	6504.00	0.285830
$804$	2352.00	0.103170
$805$	5460.00	0.239056
$806$	208.000	0.00908993
$807$	−12366.0	−0.539410
$808$	−4560.00	−0.198540
$809$	27090.0	1.17730	0.588649	−	0.808389i	$-0.299660\pi$
$809$	27090.0	1.17730	0.588649	+	0.808389i	$0.299660\pi$
$810$	810.000	0.0351364
$811$	−20104.0	−0.870465	−0.435232	−	0.900318i	$-0.643334\pi$
$811$	−20104.0	−0.870465	−0.435232	+	0.900318i	$0.643334\pi$
$812$	−5208.00	−0.225080
$813$	−17340.0	−0.748020
$814$	4272.00	0.183948
$815$	−2260.00	−0.0971342
$816$	864.000	0.0370662
$817$	−23072.0	−0.987989
$818$	30812.0	1.31701
$819$	126.000	0.00537582
$820$	2760.00	0.117541
$821$	7302.00	0.310404	0.155202	−	0.987883i	$-0.450397\pi$
$821$	7302.00	0.310404	0.155202	+	0.987883i	$0.450397\pi$
$822$	4932.00	0.209274
$823$	−24136.0	−1.02227	−0.511135	−	0.859500i	$-0.670775\pi$
$823$	−24136.0	−1.02227	−0.511135	+	0.859500i	$0.670775\pi$
$824$	2432.00	0.102819
$825$	−900.000	−0.0379806
$826$	−4872.00	−0.205228
$827$	22680.0	0.953641	0.476820	−	0.879001i	$-0.341789\pi$
$827$	22680.0	0.953641	0.476820	+	0.879001i	$0.341789\pi$
$828$	−5616.00	−0.235712
$829$	−20338.0	−0.852072	−0.426036	−	0.904706i	$-0.640090\pi$
$829$	−20338.0	−0.852072	−0.426036	+	0.904706i	$0.640090\pi$
$830$	−2760.00	−0.115423
$831$	−13722.0	−0.572817
$832$	128.000	0.00533366
$833$	−882.000	−0.0366861
$834$	−2256.00	−0.0936677
$835$	7080.00	0.293429
$836$	2688.00	0.111204
$837$	1404.00	0.0579801
$838$	−5880.00	−0.242388
$839$	6600.00	0.271582	0.135791	−	0.990738i	$-0.456642\pi$
$839$	6600.00	0.271582	0.135791	+	0.990738i	$0.456642\pi$
$840$	−840.000	−0.0345033
$841$	10207.0	0.418508
$842$	−508.000	−0.0207920
$843$	−10350.0	−0.422862
$844$	−11440.0	−0.466565
$845$	10965.0	0.446399
$846$	8208.00	0.333566
$847$	−8309.00	−0.337073
$848$	−3168.00	−0.128290
$849$	2100.00	0.0848902
$850$	900.000	0.0363173
$851$	27768.0	1.11854
$852$	11232.0	0.451646
$853$	−40174.0	−1.61258	−0.806290	−	0.591520i	$-0.798528\pi$
$853$	−40174.0	−1.61258	−0.806290	+	0.591520i	$0.798528\pi$
$854$	−1540.00	−0.0617069
$855$	−2520.00	−0.100798
$856$	−1728.00	−0.0689975
$857$	−20778.0	−0.828195	−0.414097	−	0.910233i	$-0.635903\pi$
$857$	−20778.0	−0.828195	−0.414097	+	0.910233i	$0.635903\pi$
$858$	144.000	0.00572970
$859$	7400.00	0.293929	0.146964	−	0.989142i	$-0.453050\pi$
$859$	7400.00	0.293929	0.146964	+	0.989142i	$0.453050\pi$
$860$	8240.00	0.326723
$861$	2898.00	0.114708
$862$	26496.0	1.04693
$863$	684.000	0.0269799	0.0134899	−	0.999909i	$-0.495706\pi$
$863$	684.000	0.0269799	0.0134899	+	0.999909i	$0.495706\pi$
$864$	864.000	0.0340207
$865$	−2130.00	−0.0837251
$866$	−33196.0	−1.30259
$867$	13767.0	0.539275
$868$	−1456.00	−0.0569353
$869$	11904.0	0.464690
$870$	5580.00	0.217448
$871$	−392.000	−0.0152496
$872$	−4912.00	−0.190758
$873$	990.000	0.0383808
$874$	17472.0	0.676200
$875$	−875.000	−0.0338062
$876$	−6504.00	−0.250856
$877$	−9754.00	−0.375563	−0.187782	−	0.982211i	$-0.560130\pi$
$877$	−9754.00	−0.375563	−0.187782	+	0.982211i	$0.560130\pi$
$878$	13064.0	0.502151
$879$	−21510.0	−0.825386
$880$	−960.000	−0.0367745
$881$	14310.0	0.547237	0.273619	−	0.961838i	$-0.411779\pi$
$881$	14310.0	0.547237	0.273619	+	0.961838i	$0.411779\pi$
$882$	−882.000	−0.0336718
$883$	−14092.0	−0.537071	−0.268535	−	0.963270i	$-0.586540\pi$
$883$	−14092.0	−0.537071	−0.268535	+	0.963270i	$0.586540\pi$
$884$	−144.000	−0.00547878
$885$	5220.00	0.198269
$886$	−24432.0	−0.926421
$887$	45600.0	1.72615	0.863077	−	0.505073i	$-0.168534\pi$
$887$	45600.0	1.72615	0.863077	+	0.505073i	$0.168534\pi$
$888$	−4272.00	−0.161440
$889$	−13216.0	−0.498594
$890$	6300.00	0.237277
$891$	972.000	0.0365468
$892$	−4384.00	−0.164560
$893$	−25536.0	−0.956920
$894$	20340.0	0.760930
$895$	−13500.0	−0.504196
$896$	−896.000	−0.0334077
$897$	936.000	0.0348407
$898$	−612.000	−0.0227424
$899$	9672.00	0.358820
$900$	900.000	0.0333333
$901$	3564.00	0.131780
$902$	3312.00	0.122259
$903$	8652.00	0.318849
$904$	3984.00	0.146577
$905$	9890.00	0.363265
$906$	−17808.0	−0.653015
$907$	8012.00	0.293312	0.146656	−	0.989188i	$-0.453149\pi$
$907$	8012.00	0.293312	0.146656	+	0.989188i	$0.453149\pi$
$908$	25104.0	0.917517
$909$	5130.00	0.187185
$910$	140.000	0.00509995
$911$	2136.00	0.0776826	0.0388413	−	0.999245i	$-0.487633\pi$
$911$	2136.00	0.0776826	0.0388413	+	0.999245i	$0.487633\pi$
$912$	−2688.00	−0.0975971
$913$	−3312.00	−0.120056
$914$	12092.0	0.437602
$915$	1650.00	0.0596146
$916$	−3016.00	−0.108790
$917$	−20244.0	−0.729025
$918$	−972.000	−0.0349464
$919$	−15280.0	−0.548466	−0.274233	−	0.961663i	$-0.588424\pi$
$919$	−15280.0	−0.548466	−0.274233	+	0.961663i	$0.588424\pi$
$920$	−6240.00	−0.223616
$921$	−19932.0	−0.713118
$922$	−14244.0	−0.508787
$923$	−1872.00	−0.0667580
$924$	−1008.00	−0.0358883
$925$	−4450.00	−0.158178
$926$	22496.0	0.798342
$927$	−2736.00	−0.0969385
$928$	5952.00	0.210543
$929$	20910.0	0.738466	0.369233	−	0.929337i	$-0.379620\pi$
$929$	20910.0	0.738466	0.369233	+	0.929337i	$0.379620\pi$
$930$	1560.00	0.0550047
$931$	2744.00	0.0965961
$932$	15480.0	0.544060
$933$	16128.0	0.565924
$934$	36504.0	1.27885
$935$	1080.00	0.0377752
$936$	−144.000	−0.00502862
$937$	−38122.0	−1.32913	−0.664563	−	0.747232i	$-0.731382\pi$
$937$	−38122.0	−1.32913	−0.664563	+	0.747232i	$0.731382\pi$
$938$	2744.00	0.0955168
$939$	−6378.00	−0.221659
$940$	9120.00	0.316449
$941$	42810.0	1.48307	0.741534	−	0.670916i	$-0.234099\pi$
$941$	42810.0	1.48307	0.741534	+	0.670916i	$0.234099\pi$
$942$	11244.0	0.388906
$943$	21528.0	0.743423
$944$	5568.00	0.191973
$945$	945.000	0.0325300
$946$	9888.00	0.339838
$947$	39864.0	1.36790	0.683952	−	0.729527i	$-0.260260\pi$
$947$	39864.0	1.36790	0.683952	+	0.729527i	$0.260260\pi$
$948$	−11904.0	−0.407831
$949$	1084.00	0.0370792
$950$	−2800.00	−0.0956253
$951$	−3222.00	−0.109864
$952$	1008.00	0.0343167
$953$	−23850.0	−0.810679	−0.405340	−	0.914166i	$-0.632847\pi$
$953$	−23850.0	−0.810679	−0.405340	+	0.914166i	$0.632847\pi$
$954$	3564.00	0.120953
$955$	11640.0	0.394410
$956$	2976.00	0.100681
$957$	6696.00	0.226177
$958$	36336.0	1.22543
$959$	5754.00	0.193750
$960$	960.000	0.0322749
$961$	−27087.0	−0.909234
$962$	712.000	0.0238626
$963$	1944.00	0.0650514
$964$	21896.0	0.731559
$965$	15830.0	0.528068
$966$	−6552.00	−0.218227
$967$	−12832.0	−0.426731	−0.213366	−	0.976972i	$-0.568443\pi$
$967$	−12832.0	−0.426731	−0.213366	+	0.976972i	$0.568443\pi$
$968$	9496.00	0.315303
$969$	3024.00	0.100253
$970$	1100.00	0.0364112
$971$	−15804.0	−0.522322	−0.261161	−	0.965295i	$-0.584105\pi$
$971$	−15804.0	−0.522322	−0.261161	+	0.965295i	$0.584105\pi$
$972$	−972.000	−0.0320750
$973$	−2632.00	−0.0867195
$974$	−39712.0	−1.30642
$975$	−150.000	−0.00492702
$976$	1760.00	0.0577215
$977$	−33114.0	−1.08435	−0.542175	−	0.840265i	$-0.682399\pi$
$977$	−33114.0	−1.08435	−0.542175	+	0.840265i	$0.682399\pi$
$978$	2712.00	0.0886710
$979$	7560.00	0.246801
$980$	−980.000	−0.0319438
$981$	5526.00	0.179849
$982$	−22440.0	−0.729215
$983$	−58632.0	−1.90241	−0.951206	−	0.308558i	$-0.900154\pi$
$983$	−58632.0	−1.90241	−0.951206	+	0.308558i	$0.900154\pi$
$984$	−3312.00	−0.107299
$985$	2070.00	0.0669601
$986$	−6696.00	−0.216272
$987$	9576.00	0.308822
$988$	448.000	0.0144259
$989$	64272.0	2.06646
$990$	1080.00	0.0346714
$991$	55784.0	1.78813	0.894065	−	0.447937i	$-0.147841\pi$
$991$	55784.0	1.78813	0.894065	+	0.447937i	$0.147841\pi$
$992$	1664.00	0.0532581
$993$	8364.00	0.267295
$994$	13104.0	0.418142
$995$	8180.00	0.260627
$996$	3312.00	0.105366
$997$	−23326.0	−0.740965	−0.370482	−	0.928840i	$-0.620808\pi$
$997$	−23326.0	−0.740965	−0.370482	+	0.928840i	$0.620808\pi$
$998$	18536.0	0.587923
$999$	4806.00	0.152207

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	210.4.a.a.1.1	✓	1
3.2	odd	2		630.4.a.v.1.1		1
4.3	odd	2		1680.4.a.n.1.1		1
5.2	odd	4		1050.4.g.o.799.1		2
5.3	odd	4		1050.4.g.o.799.2		2
5.4	even	2		1050.4.a.t.1.1		1
7.6	odd	2		1470.4.a.n.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.4.a.a.1.1	✓	1	1.1	even	1	trivial
630.4.a.v.1.1		1	3.2	odd	2
1050.4.a.t.1.1		1	5.4	even	2
1050.4.g.o.799.1		2	5.2	odd	4
1050.4.g.o.799.2		2	5.3	odd	4
1470.4.a.n.1.1		1	7.6	odd	2
1680.4.a.n.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:	\( N \)	\(=\)	\( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	210.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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