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Properties

Label 1200.4.a.h.1.1
Level $1200$
Weight $4$
Character 1200.1
Self dual yes
Analytic conductor $70.802$
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] = [1200,4,Mod(1,1200)] 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("1200.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1200, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1200.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,-3,0,0,0,-2,0,9,0,-70] 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: \(70.8022920069\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 30)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1200.1

$q$-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)
 
\(f(q)\) \(=\) \(q-3.00000 q^{3} -2.00000 q^{7} +9.00000 q^{9} -70.0000 q^{11} -54.0000 q^{13} +22.0000 q^{17} -24.0000 q^{19} +6.00000 q^{21} -100.000 q^{23} -27.0000 q^{27} +216.000 q^{29} -208.000 q^{31} +210.000 q^{33} +254.000 q^{37} +162.000 q^{39} -206.000 q^{41} +292.000 q^{43} -320.000 q^{47} -339.000 q^{49} -66.0000 q^{51} +402.000 q^{53} +72.0000 q^{57} +370.000 q^{59} -550.000 q^{61} -18.0000 q^{63} +728.000 q^{67} +300.000 q^{69} +540.000 q^{71} -604.000 q^{73} +140.000 q^{77} -792.000 q^{79} +81.0000 q^{81} +404.000 q^{83} -648.000 q^{87} -938.000 q^{89} +108.000 q^{91} +624.000 q^{93} -56.0000 q^{97} -630.000 q^{99} +O(q^{100})\)

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
\(5\) 0 0
\(6\) 0 0
\(7\) −2.00000 −0.107990 −0.0539949 − 0.998541i \(-0.517195\pi\)
−0.0539949 + 0.998541i \(0.517195\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −70.0000 −1.91871 −0.959354 − 0.282204i \(-0.908934\pi\)
−0.959354 + 0.282204i \(0.908934\pi\)
\(12\) 0 0
\(13\) −54.0000 −1.15207 −0.576035 − 0.817425i \(-0.695401\pi\)
−0.576035 + 0.817425i \(0.695401\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 22.0000 0.313870 0.156935 − 0.987609i \(-0.449839\pi\)
0.156935 + 0.987609i \(0.449839\pi\)
\(18\) 0 0
\(19\) −24.0000 −0.289788 −0.144894 − 0.989447i \(-0.546284\pi\)
−0.144894 + 0.989447i \(0.546284\pi\)
\(20\) 0 0
\(21\) 6.00000 0.0623480
\(22\) 0 0
\(23\) −100.000 −0.906584 −0.453292 − 0.891362i \(-0.649751\pi\)
−0.453292 + 0.891362i \(0.649751\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) 216.000 1.38311 0.691555 − 0.722324i \(-0.256926\pi\)
0.691555 + 0.722324i \(0.256926\pi\)
\(30\) 0 0
\(31\) −208.000 −1.20509 −0.602547 − 0.798084i \(-0.705847\pi\)
−0.602547 + 0.798084i \(0.705847\pi\)
\(32\) 0 0
\(33\) 210.000 1.10777
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 254.000 1.12858 0.564288 − 0.825578i \(-0.309151\pi\)
0.564288 + 0.825578i \(0.309151\pi\)
\(38\) 0 0
\(39\) 162.000 0.665148
\(40\) 0 0
\(41\) −206.000 −0.784678 −0.392339 − 0.919821i \(-0.628334\pi\)
−0.392339 + 0.919821i \(0.628334\pi\)
\(42\) 0 0
\(43\) 292.000 1.03557 0.517786 − 0.855510i \(-0.326756\pi\)
0.517786 + 0.855510i \(0.326756\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −320.000 −0.993123 −0.496562 − 0.868001i \(-0.665404\pi\)
−0.496562 + 0.868001i \(0.665404\pi\)
\(48\) 0 0
\(49\) −339.000 −0.988338
\(50\) 0 0
\(51\) −66.0000 −0.181213
\(52\) 0 0
\(53\) 402.000 1.04187 0.520933 − 0.853597i \(-0.325584\pi\)
0.520933 + 0.853597i \(0.325584\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 72.0000 0.167309
\(58\) 0 0
\(59\) 370.000 0.816439 0.408219 − 0.912884i \(-0.366150\pi\)
0.408219 + 0.912884i \(0.366150\pi\)
\(60\) 0 0
\(61\) −550.000 −1.15443 −0.577215 − 0.816592i \(-0.695861\pi\)
−0.577215 + 0.816592i \(0.695861\pi\)
\(62\) 0 0
\(63\) −18.0000 −0.0359966
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 728.000 1.32745 0.663727 − 0.747975i \(-0.268974\pi\)
0.663727 + 0.747975i \(0.268974\pi\)
\(68\) 0 0
\(69\) 300.000 0.523417
\(70\) 0 0
\(71\) 540.000 0.902623 0.451311 − 0.892367i \(-0.350956\pi\)
0.451311 + 0.892367i \(0.350956\pi\)
\(72\) 0 0
\(73\) −604.000 −0.968395 −0.484198 − 0.874959i \(-0.660888\pi\)
−0.484198 + 0.874959i \(0.660888\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 140.000 0.207201
\(78\) 0 0
\(79\) −792.000 −1.12794 −0.563968 − 0.825797i \(-0.690726\pi\)
−0.563968 + 0.825797i \(0.690726\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 404.000 0.534274 0.267137 − 0.963659i \(-0.413922\pi\)
0.267137 + 0.963659i \(0.413922\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −648.000 −0.798539
\(88\) 0 0
\(89\) −938.000 −1.11717 −0.558583 − 0.829449i \(-0.688655\pi\)
−0.558583 + 0.829449i \(0.688655\pi\)
\(90\) 0 0
\(91\) 108.000 0.124412
\(92\) 0 0
\(93\) 624.000 0.695761
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −56.0000 −0.0586179 −0.0293090 − 0.999570i \(-0.509331\pi\)
−0.0293090 + 0.999570i \(0.509331\pi\)
\(98\) 0 0
\(99\) −630.000 −0.639570
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 1200.4.a.h.1.1 1
4.3 odd 2 150.4.a.d.1.1 1
5.2 odd 4 240.4.f.d.49.2 2
5.3 odd 4 240.4.f.d.49.1 2
5.4 even 2 1200.4.a.bc.1.1 1
12.11 even 2 450.4.a.p.1.1 1
15.2 even 4 720.4.f.c.289.2 2
15.8 even 4 720.4.f.c.289.1 2
20.3 even 4 30.4.c.a.19.2 yes 2
20.7 even 4 30.4.c.a.19.1 ✓ 2
20.19 odd 2 150.4.a.f.1.1 1
40.3 even 4 960.4.f.c.769.1 2
40.13 odd 4 960.4.f.d.769.2 2
40.27 even 4 960.4.f.c.769.2 2
40.37 odd 4 960.4.f.d.769.1 2
60.23 odd 4 90.4.c.a.19.1 2
60.47 odd 4 90.4.c.a.19.2 2
60.59 even 2 450.4.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
30.4.c.a.19.1 ✓ 2 20.7 even 4
30.4.c.a.19.2 yes 2 20.3 even 4
90.4.c.a.19.1 2 60.23 odd 4
90.4.c.a.19.2 2 60.47 odd 4
150.4.a.d.1.1 1 4.3 odd 2
150.4.a.f.1.1 1 20.19 odd 2
240.4.f.d.49.1 2 5.3 odd 4
240.4.f.d.49.2 2 5.2 odd 4
450.4.a.e.1.1 1 60.59 even 2
450.4.a.p.1.1 1 12.11 even 2
720.4.f.c.289.1 2 15.8 even 4
720.4.f.c.289.2 2 15.2 even 4
960.4.f.c.769.1 2 40.3 even 4
960.4.f.c.769.2 2 40.27 even 4
960.4.f.d.769.1 2 40.37 odd 4
960.4.f.d.769.2 2 40.13 odd 4
1200.4.a.h.1.1 1 1.1 even 1 trivial
1200.4.a.bc.1.1 1 5.4 even 2