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Properties

Label 2400.4.a.g.1.1
Level $2400$
Weight $4$
Character 2400.1
Self dual yes
Analytic conductor $141.605$
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] = [2400,4,Mod(1,2400)] 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("2400.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2400, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 2400 = 2^{5} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2400.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,-3,0,0,0,12,0,9,0,-24,0,-38] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(141.604584014\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 480)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2400.1

$q$-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)
 
\(f(q)\) \(=\) \(q-3.00000 q^{3} +12.0000 q^{7} +9.00000 q^{9} -24.0000 q^{11} -38.0000 q^{13} +6.00000 q^{17} +104.000 q^{19} -36.0000 q^{21} -100.000 q^{23} -27.0000 q^{27} +230.000 q^{29} -56.0000 q^{31} +72.0000 q^{33} -190.000 q^{37} +114.000 q^{39} +202.000 q^{41} +148.000 q^{43} -124.000 q^{47} -199.000 q^{49} -18.0000 q^{51} -206.000 q^{53} -312.000 q^{57} -128.000 q^{59} +190.000 q^{61} +108.000 q^{63} +204.000 q^{67} +300.000 q^{69} -440.000 q^{71} -1210.00 q^{73} -288.000 q^{77} +816.000 q^{79} +81.0000 q^{81} +1412.00 q^{83} -690.000 q^{87} -214.000 q^{89} -456.000 q^{91} +168.000 q^{93} -1202.00 q^{97} -216.000 q^{99} +O(q^{100})\)

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
\(7\) 12.0000 0.647939 0.323970 − 0.946068i \(-0.394982\pi\)
0.323970 + 0.946068i \(0.394982\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −24.0000 −0.657843 −0.328921 − 0.944357i \(-0.606685\pi\)
−0.328921 + 0.944357i \(0.606685\pi\)
\(12\) 0 0
\(13\) −38.0000 −0.810716 −0.405358 − 0.914158i \(-0.632853\pi\)
−0.405358 + 0.914158i \(0.632853\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6.00000 0.0856008 0.0428004 − 0.999084i \(-0.486372\pi\)
0.0428004 + 0.999084i \(0.486372\pi\)
\(18\) 0 0
\(19\) 104.000 1.25575 0.627875 − 0.778314i \(-0.283925\pi\)
0.627875 + 0.778314i \(0.283925\pi\)
\(20\) 0 0
\(21\) −36.0000 −0.374088
\(22\) 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
\(29\) 230.000 1.47276 0.736378 − 0.676570i \(-0.236535\pi\)
0.736378 + 0.676570i \(0.236535\pi\)
\(30\) 0 0
\(31\) −56.0000 −0.324448 −0.162224 − 0.986754i \(-0.551867\pi\)
−0.162224 + 0.986754i \(0.551867\pi\)
\(32\) 0 0
\(33\) 72.0000 0.379806
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −190.000 −0.844211 −0.422106 − 0.906547i \(-0.638709\pi\)
−0.422106 + 0.906547i \(0.638709\pi\)
\(38\) 0 0
\(39\) 114.000 0.468067
\(40\) 0 0
\(41\) 202.000 0.769441 0.384721 − 0.923033i \(-0.374298\pi\)
0.384721 + 0.923033i \(0.374298\pi\)
\(42\) 0 0
\(43\) 148.000 0.524879 0.262439 − 0.964948i \(-0.415473\pi\)
0.262439 + 0.964948i \(0.415473\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −124.000 −0.384835 −0.192418 − 0.981313i \(-0.561633\pi\)
−0.192418 + 0.981313i \(0.561633\pi\)
\(48\) 0 0
\(49\) −199.000 −0.580175
\(50\) 0 0
\(51\) −18.0000 −0.0494217
\(52\) 0 0
\(53\) −206.000 −0.533892 −0.266946 − 0.963711i \(-0.586015\pi\)
−0.266946 + 0.963711i \(0.586015\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −312.000 −0.725007
\(58\) 0 0
\(59\) −128.000 −0.282444 −0.141222 − 0.989978i \(-0.545103\pi\)
−0.141222 + 0.989978i \(0.545103\pi\)
\(60\) 0 0
\(61\) 190.000 0.398803 0.199402 − 0.979918i \(-0.436100\pi\)
0.199402 + 0.979918i \(0.436100\pi\)
\(62\) 0 0
\(63\) 108.000 0.215980
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 204.000 0.371979 0.185989 − 0.982552i \(-0.440451\pi\)
0.185989 + 0.982552i \(0.440451\pi\)
\(68\) 0 0
\(69\) 300.000 0.523417
\(70\) 0 0
\(71\) −440.000 −0.735470 −0.367735 − 0.929931i \(-0.619867\pi\)
−0.367735 + 0.929931i \(0.619867\pi\)
\(72\) 0 0
\(73\) −1210.00 −1.94000 −0.969999 − 0.243111i \(-0.921832\pi\)
−0.969999 + 0.243111i \(0.921832\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −288.000 −0.426242
\(78\) 0 0
\(79\) 816.000 1.16212 0.581058 − 0.813862i \(-0.302639\pi\)
0.581058 + 0.813862i \(0.302639\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 1412.00 1.86731 0.933657 − 0.358167i \(-0.116598\pi\)
0.933657 + 0.358167i \(0.116598\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −690.000 −0.850296
\(88\) 0 0
\(89\) −214.000 −0.254876 −0.127438 − 0.991847i \(-0.540675\pi\)
−0.127438 + 0.991847i \(0.540675\pi\)
\(90\) 0 0
\(91\) −456.000 −0.525294
\(92\) 0 0
\(93\) 168.000 0.187320
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1202.00 −1.25819 −0.629096 − 0.777328i \(-0.716575\pi\)
−0.629096 + 0.777328i \(0.716575\pi\)
\(98\) 0 0
\(99\) −216.000 −0.219281
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 2400.4.a.g.1.1 1
4.3 odd 2 2400.4.a.p.1.1 1
5.4 even 2 480.4.a.j.1.1 yes 1
15.14 odd 2 1440.4.a.d.1.1 1
20.19 odd 2 480.4.a.e.1.1 ✓ 1
40.19 odd 2 960.4.a.y.1.1 1
40.29 even 2 960.4.a.d.1.1 1
60.59 even 2 1440.4.a.g.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
480.4.a.e.1.1 ✓ 1 20.19 odd 2
480.4.a.j.1.1 yes 1 5.4 even 2
960.4.a.d.1.1 1 40.29 even 2
960.4.a.y.1.1 1 40.19 odd 2
1440.4.a.d.1.1 1 15.14 odd 2
1440.4.a.g.1.1 1 60.59 even 2
2400.4.a.g.1.1 1 1.1 even 1 trivial
2400.4.a.p.1.1 1 4.3 odd 2