Properties

Label 240.4.a.g.1.1
Level $240$
Weight $4$
Character 240.1
Self dual yes
Analytic conductor $14.160$
Analytic rank $1$
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] = [240,4,Mod(1,240)] 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("240.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(240, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 240.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,3,0,-5,0,-20,0,9,0,56] 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: \(14.1604584014\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 120)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 240.1

$q$-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)
 
\(f(q)\) \(=\) \(q+3.00000 q^{3} -5.00000 q^{5} -20.0000 q^{7} +9.00000 q^{9} +56.0000 q^{11} -86.0000 q^{13} -15.0000 q^{15} -106.000 q^{17} -4.00000 q^{19} -60.0000 q^{21} -136.000 q^{23} +25.0000 q^{25} +27.0000 q^{27} -206.000 q^{29} +152.000 q^{31} +168.000 q^{33} +100.000 q^{35} +282.000 q^{37} -258.000 q^{39} -246.000 q^{41} -412.000 q^{43} -45.0000 q^{45} -40.0000 q^{47} +57.0000 q^{49} -318.000 q^{51} -126.000 q^{53} -280.000 q^{55} -12.0000 q^{57} -56.0000 q^{59} -2.00000 q^{61} -180.000 q^{63} +430.000 q^{65} +388.000 q^{67} -408.000 q^{69} +672.000 q^{71} +1170.00 q^{73} +75.0000 q^{75} -1120.00 q^{77} -408.000 q^{79} +81.0000 q^{81} -668.000 q^{83} +530.000 q^{85} -618.000 q^{87} +66.0000 q^{89} +1720.00 q^{91} +456.000 q^{93} +20.0000 q^{95} -926.000 q^{97} +504.000 q^{99} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 3.00000 0.577350
\(4\) 0 0
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) −20.0000 −1.07990 −0.539949 − 0.841698i \(-0.681557\pi\)
−0.539949 + 0.841698i \(0.681557\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 56.0000 1.53497 0.767483 − 0.641069i \(-0.221509\pi\)
0.767483 + 0.641069i \(0.221509\pi\)
\(12\) 0 0
\(13\) −86.0000 −1.83478 −0.917389 − 0.397992i \(-0.869707\pi\)
−0.917389 + 0.397992i \(0.869707\pi\)
\(14\) 0 0
\(15\) −15.0000 −0.258199
\(16\) 0 0
\(17\) −106.000 −1.51228 −0.756140 − 0.654409i \(-0.772917\pi\)
−0.756140 + 0.654409i \(0.772917\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.0482980 −0.0241490 − 0.999708i \(-0.507688\pi\)
−0.0241490 + 0.999708i \(0.507688\pi\)
\(20\) 0 0
\(21\) −60.0000 −0.623480
\(22\) 0 0
\(23\) −136.000 −1.23295 −0.616477 − 0.787373i \(-0.711441\pi\)
−0.616477 + 0.787373i \(0.711441\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) −206.000 −1.31908 −0.659539 − 0.751671i \(-0.729248\pi\)
−0.659539 + 0.751671i \(0.729248\pi\)
\(30\) 0 0
\(31\) 152.000 0.880645 0.440323 − 0.897840i \(-0.354864\pi\)
0.440323 + 0.897840i \(0.354864\pi\)
\(32\) 0 0
\(33\) 168.000 0.886214
\(34\) 0 0
\(35\) 100.000 0.482945
\(36\) 0 0
\(37\) 282.000 1.25299 0.626493 − 0.779427i \(-0.284490\pi\)
0.626493 + 0.779427i \(0.284490\pi\)
\(38\) 0 0
\(39\) −258.000 −1.05931
\(40\) 0 0
\(41\) −246.000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) −412.000 −1.46115 −0.730575 − 0.682833i \(-0.760748\pi\)
−0.730575 + 0.682833i \(0.760748\pi\)
\(44\) 0 0
\(45\) −45.0000 −0.149071
\(46\) 0 0
\(47\) −40.0000 −0.124140 −0.0620702 − 0.998072i \(-0.519770\pi\)
−0.0620702 + 0.998072i \(0.519770\pi\)
\(48\) 0 0
\(49\) 57.0000 0.166181
\(50\) 0 0
\(51\) −318.000 −0.873116
\(52\) 0 0
\(53\) −126.000 −0.326555 −0.163278 − 0.986580i \(-0.552207\pi\)
−0.163278 + 0.986580i \(0.552207\pi\)
\(54\) 0 0
\(55\) −280.000 −0.686458
\(56\) 0 0
\(57\) −12.0000 −0.0278849
\(58\) 0 0
\(59\) −56.0000 −0.123569 −0.0617846 − 0.998090i \(-0.519679\pi\)
−0.0617846 + 0.998090i \(0.519679\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.00419793 −0.00209897 − 0.999998i \(-0.500668\pi\)
−0.00209897 + 0.999998i \(0.500668\pi\)
\(62\) 0 0
\(63\) −180.000 −0.359966
\(64\) 0 0
\(65\) 430.000 0.820537
\(66\) 0 0
\(67\) 388.000 0.707489 0.353744 − 0.935342i \(-0.384908\pi\)
0.353744 + 0.935342i \(0.384908\pi\)
\(68\) 0 0
\(69\) −408.000 −0.711847
\(70\) 0 0
\(71\) 672.000 1.12326 0.561632 − 0.827387i \(-0.310174\pi\)
0.561632 + 0.827387i \(0.310174\pi\)
\(72\) 0 0
\(73\) 1170.00 1.87586 0.937932 − 0.346818i \(-0.112738\pi\)
0.937932 + 0.346818i \(0.112738\pi\)
\(74\) 0 0
\(75\) 75.0000 0.115470
\(76\) 0 0
\(77\) −1120.00 −1.65761
\(78\) 0 0
\(79\) −408.000 −0.581058 −0.290529 − 0.956866i \(-0.593831\pi\)
−0.290529 + 0.956866i \(0.593831\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −668.000 −0.883404 −0.441702 − 0.897162i \(-0.645625\pi\)
−0.441702 + 0.897162i \(0.645625\pi\)
\(84\) 0 0
\(85\) 530.000 0.676313
\(86\) 0 0
\(87\) −618.000 −0.761570
\(88\) 0 0
\(89\) 66.0000 0.0786066 0.0393033 − 0.999227i \(-0.487486\pi\)
0.0393033 + 0.999227i \(0.487486\pi\)
\(90\) 0 0
\(91\) 1720.00 1.98137
\(92\) 0 0
\(93\) 456.000 0.508441
\(94\) 0 0
\(95\) 20.0000 0.0215995
\(96\) 0 0
\(97\) −926.000 −0.969289 −0.484645 − 0.874711i \(-0.661051\pi\)
−0.484645 + 0.874711i \(0.661051\pi\)
\(98\) 0 0
\(99\) 504.000 0.511656
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 240.4.a.g.1.1 1
3.2 odd 2 720.4.a.q.1.1 1
4.3 odd 2 120.4.a.b.1.1 ✓ 1
5.2 odd 4 1200.4.f.t.49.1 2
5.3 odd 4 1200.4.f.t.49.2 2
5.4 even 2 1200.4.a.p.1.1 1
8.3 odd 2 960.4.a.bj.1.1 1
8.5 even 2 960.4.a.k.1.1 1
12.11 even 2 360.4.a.n.1.1 1
20.3 even 4 600.4.f.a.49.1 2
20.7 even 4 600.4.f.a.49.2 2
20.19 odd 2 600.4.a.i.1.1 1
60.23 odd 4 1800.4.f.v.649.1 2
60.47 odd 4 1800.4.f.v.649.2 2
60.59 even 2 1800.4.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
120.4.a.b.1.1 ✓ 1 4.3 odd 2
240.4.a.g.1.1 1 1.1 even 1 trivial
360.4.a.n.1.1 1 12.11 even 2
600.4.a.i.1.1 1 20.19 odd 2
600.4.f.a.49.1 2 20.3 even 4
600.4.f.a.49.2 2 20.7 even 4
720.4.a.q.1.1 1 3.2 odd 2
960.4.a.k.1.1 1 8.5 even 2
960.4.a.bj.1.1 1 8.3 odd 2
1200.4.a.p.1.1 1 5.4 even 2
1200.4.f.t.49.1 2 5.2 odd 4
1200.4.f.t.49.2 2 5.3 odd 4
1800.4.a.f.1.1 1 60.59 even 2
1800.4.f.v.649.1 2 60.23 odd 4
1800.4.f.v.649.2 2 60.47 odd 4