Properties

Label 490.4.a.d.1.1
Level $490$
Weight $4$
Character 490.1
Self dual yes
Analytic conductor $28.911$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [490,4,Mod(1,490)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("490.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(490, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 490 = 2 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 490.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-2,1,4,5,-2,0,-8,-26,-10,-65] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(28.9109359028\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 70)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 490.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{2} +1.00000 q^{3} +4.00000 q^{4} +5.00000 q^{5} -2.00000 q^{6} -8.00000 q^{8} -26.0000 q^{9} -10.0000 q^{10} -65.0000 q^{11} +4.00000 q^{12} -13.0000 q^{13} +5.00000 q^{15} +16.0000 q^{16} +73.0000 q^{17} +52.0000 q^{18} +142.000 q^{19} +20.0000 q^{20} +130.000 q^{22} +130.000 q^{23} -8.00000 q^{24} +25.0000 q^{25} +26.0000 q^{26} -53.0000 q^{27} +111.000 q^{29} -10.0000 q^{30} -256.000 q^{31} -32.0000 q^{32} -65.0000 q^{33} -146.000 q^{34} -104.000 q^{36} -266.000 q^{37} -284.000 q^{38} -13.0000 q^{39} -40.0000 q^{40} +424.000 q^{41} +534.000 q^{43} -260.000 q^{44} -130.000 q^{45} -260.000 q^{46} +269.000 q^{47} +16.0000 q^{48} -50.0000 q^{50} +73.0000 q^{51} -52.0000 q^{52} -132.000 q^{53} +106.000 q^{54} -325.000 q^{55} +142.000 q^{57} -222.000 q^{58} +224.000 q^{59} +20.0000 q^{60} +572.000 q^{61} +512.000 q^{62} +64.0000 q^{64} -65.0000 q^{65} +130.000 q^{66} -108.000 q^{67} +292.000 q^{68} +130.000 q^{69} +560.000 q^{71} +208.000 q^{72} -586.000 q^{73} +532.000 q^{74} +25.0000 q^{75} +568.000 q^{76} +26.0000 q^{78} +57.0000 q^{79} +80.0000 q^{80} +649.000 q^{81} -848.000 q^{82} -252.000 q^{83} +365.000 q^{85} -1068.00 q^{86} +111.000 q^{87} +520.000 q^{88} +184.000 q^{89} +260.000 q^{90} +520.000 q^{92} -256.000 q^{93} -538.000 q^{94} +710.000 q^{95} -32.0000 q^{96} +605.000 q^{97} +1690.00 q^{99} +O(q^{100})\)

Coefficient data

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



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −0.707107
\(3\) 1.00000 0.192450 0.0962250 − 0.995360i \(-0.469323\pi\)
0.0962250 + 0.995360i \(0.469323\pi\)
\(4\) 4.00000 0.500000
\(5\) 5.00000 0.447214
\(6\) −2.00000 −0.136083
\(7\) 0 0
\(8\) −8.00000 −0.353553
\(9\) −26.0000 −0.962963
\(10\) −10.0000 −0.316228
\(11\) −65.0000 −1.78166 −0.890829 − 0.454339i \(-0.849876\pi\)
−0.890829 + 0.454339i \(0.849876\pi\)
\(12\) 4.00000 0.0962250
\(13\) −13.0000 −0.277350 −0.138675 − 0.990338i \(-0.544284\pi\)
−0.138675 + 0.990338i \(0.544284\pi\)
\(14\) 0 0
\(15\) 5.00000 0.0860663
\(16\) 16.0000 0.250000
\(17\) 73.0000 1.04148 0.520738 − 0.853716i \(-0.325657\pi\)
0.520738 + 0.853716i \(0.325657\pi\)
\(18\) 52.0000 0.680918
\(19\) 142.000 1.71458 0.857290 − 0.514833i \(-0.172146\pi\)
0.857290 + 0.514833i \(0.172146\pi\)
\(20\) 20.0000 0.223607
\(21\) 0 0
\(22\) 130.000 1.25982
\(23\) 130.000 1.17856 0.589280 − 0.807929i \(-0.299412\pi\)
0.589280 + 0.807929i \(0.299412\pi\)
\(24\) −8.00000 −0.0680414
\(25\) 25.0000 0.200000
\(26\) 26.0000 0.196116
\(27\) −53.0000 −0.377772
\(28\) 0 0
\(29\) 111.000 0.710765 0.355382 − 0.934721i \(-0.384351\pi\)
0.355382 + 0.934721i \(0.384351\pi\)
\(30\) −10.0000 −0.0608581
\(31\) −256.000 −1.48319 −0.741596 − 0.670847i \(-0.765931\pi\)
−0.741596 + 0.670847i \(0.765931\pi\)
\(32\) −32.0000 −0.176777
\(33\) −65.0000 −0.342880
\(34\) −146.000 −0.736435
\(35\) 0 0
\(36\) −104.000 −0.481481
\(37\) −266.000 −1.18190 −0.590948 − 0.806710i \(-0.701246\pi\)
−0.590948 + 0.806710i \(0.701246\pi\)
\(38\) −284.000 −1.21239
\(39\) −13.0000 −0.0533761
\(40\) −40.0000 −0.158114
\(41\) 424.000 1.61507 0.807533 − 0.589823i \(-0.200802\pi\)
0.807533 + 0.589823i \(0.200802\pi\)
\(42\) 0 0
\(43\) 534.000 1.89382 0.946910 − 0.321500i \(-0.104187\pi\)
0.946910 + 0.321500i \(0.104187\pi\)
\(44\) −260.000 −0.890829
\(45\) −130.000 −0.430650
\(46\) −260.000 −0.833368
\(47\) 269.000 0.834844 0.417422 − 0.908713i \(-0.362934\pi\)
0.417422 + 0.908713i \(0.362934\pi\)
\(48\) 16.0000 0.0481125
\(49\) 0 0
\(50\) −50.0000 −0.141421
\(51\) 73.0000 0.200432
\(52\) −52.0000 −0.138675
\(53\) −132.000 −0.342106 −0.171053 − 0.985262i \(-0.554717\pi\)
−0.171053 + 0.985262i \(0.554717\pi\)
\(54\) 106.000 0.267125
\(55\) −325.000 −0.796782
\(56\) 0 0
\(57\) 142.000 0.329971
\(58\) −222.000 −0.502587
\(59\) 224.000 0.494277 0.247138 − 0.968980i \(-0.420510\pi\)
0.247138 + 0.968980i \(0.420510\pi\)
\(60\) 20.0000 0.0430331
\(61\) 572.000 1.20061 0.600304 − 0.799772i \(-0.295046\pi\)
0.600304 + 0.799772i \(0.295046\pi\)
\(62\) 512.000 1.04878
\(63\) 0 0
\(64\) 64.0000 0.125000
\(65\) −65.0000 −0.124035
\(66\) 130.000 0.242453
\(67\) −108.000 −0.196930 −0.0984649 − 0.995141i \(-0.531393\pi\)
−0.0984649 + 0.995141i \(0.531393\pi\)
\(68\) 292.000 0.520738
\(69\) 130.000 0.226814
\(70\) 0 0
\(71\) 560.000 0.936053 0.468027 − 0.883714i \(-0.344965\pi\)
0.468027 + 0.883714i \(0.344965\pi\)
\(72\) 208.000 0.340459
\(73\) −586.000 −0.939536 −0.469768 − 0.882790i \(-0.655662\pi\)
−0.469768 + 0.882790i \(0.655662\pi\)
\(74\) 532.000 0.835726
\(75\) 25.0000 0.0384900
\(76\) 568.000 0.857290
\(77\) 0 0
\(78\) 26.0000 0.0377426
\(79\) 57.0000 0.0811772 0.0405886 − 0.999176i \(-0.487077\pi\)
0.0405886 + 0.999176i \(0.487077\pi\)
\(80\) 80.0000 0.111803
\(81\) 649.000 0.890261
\(82\) −848.000 −1.14202
\(83\) −252.000 −0.333260 −0.166630 − 0.986019i \(-0.553289\pi\)
−0.166630 + 0.986019i \(0.553289\pi\)
\(84\) 0 0
\(85\) 365.000 0.465762
\(86\) −1068.00 −1.33913
\(87\) 111.000 0.136787
\(88\) 520.000 0.629911
\(89\) 184.000 0.219146 0.109573 − 0.993979i \(-0.465052\pi\)
0.109573 + 0.993979i \(0.465052\pi\)
\(90\) 260.000 0.304516
\(91\) 0 0
\(92\) 520.000 0.589280
\(93\) −256.000 −0.285440
\(94\) −538.000 −0.590324
\(95\) 710.000 0.766784
\(96\) −32.0000 −0.0340207
\(97\) 605.000 0.633283 0.316641 − 0.948545i \(-0.397445\pi\)
0.316641 + 0.948545i \(0.397445\pi\)
\(98\) 0 0
\(99\) 1690.00 1.71567
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 490.4.a.d.1.1 1
5.4 even 2 2450.4.a.bc.1.1 1
7.2 even 3 490.4.e.n.361.1 2
7.3 odd 6 490.4.e.o.471.1 2
7.4 even 3 490.4.e.n.471.1 2
7.5 odd 6 490.4.e.o.361.1 2
7.6 odd 2 70.4.a.c.1.1 ✓ 1
21.20 even 2 630.4.a.x.1.1 1
28.27 even 2 560.4.a.i.1.1 1
35.13 even 4 350.4.c.h.99.2 2
35.27 even 4 350.4.c.h.99.1 2
35.34 odd 2 350.4.a.r.1.1 1
56.13 odd 2 2240.4.a.v.1.1 1
56.27 even 2 2240.4.a.r.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
70.4.a.c.1.1 ✓ 1 7.6 odd 2
350.4.a.r.1.1 1 35.34 odd 2
350.4.c.h.99.1 2 35.27 even 4
350.4.c.h.99.2 2 35.13 even 4
490.4.a.d.1.1 1 1.1 even 1 trivial
490.4.e.n.361.1 2 7.2 even 3
490.4.e.n.471.1 2 7.4 even 3
490.4.e.o.361.1 2 7.5 odd 6
490.4.e.o.471.1 2 7.3 odd 6
560.4.a.i.1.1 1 28.27 even 2
630.4.a.x.1.1 1 21.20 even 2
2240.4.a.r.1.1 1 56.27 even 2
2240.4.a.v.1.1 1 56.13 odd 2
2450.4.a.bc.1.1 1 5.4 even 2