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Properties

Label 600.4.a.q.1.1
Level $600$
Weight $4$
Character 600.1
Self dual yes
Analytic conductor $35.401$
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] = [600,4,Mod(1,600)] 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("600.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(600, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 600.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,3,0,0,0,16,0,9,0,-28] 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: \(35.4011460034\)
Analytic rank: \(0\)
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 \(\chi\) \(=\) 600.1

$q$-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)
 
\(f(q)\) \(=\) \(q+3.00000 q^{3} +16.0000 q^{7} +9.00000 q^{9} -28.0000 q^{11} +26.0000 q^{13} +62.0000 q^{17} -68.0000 q^{19} +48.0000 q^{21} +208.000 q^{23} +27.0000 q^{27} -58.0000 q^{29} +160.000 q^{31} -84.0000 q^{33} -270.000 q^{37} +78.0000 q^{39} +282.000 q^{41} -76.0000 q^{43} +280.000 q^{47} -87.0000 q^{49} +186.000 q^{51} +210.000 q^{53} -204.000 q^{57} +196.000 q^{59} +742.000 q^{61} +144.000 q^{63} -836.000 q^{67} +624.000 q^{69} -504.000 q^{71} +1062.00 q^{73} -448.000 q^{77} +768.000 q^{79} +81.0000 q^{81} +1052.00 q^{83} -174.000 q^{87} -726.000 q^{89} +416.000 q^{91} +480.000 q^{93} +1406.00 q^{97} -252.000 q^{99} +O(q^{100})\)

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
\(7\) 16.0000 0.863919 0.431959 − 0.901893i \(-0.357822\pi\)
0.431959 + 0.901893i \(0.357822\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −28.0000 −0.767483 −0.383742 − 0.923440i \(-0.625365\pi\)
−0.383742 + 0.923440i \(0.625365\pi\)
\(12\) 0 0
\(13\) 26.0000 0.554700 0.277350 − 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 62.0000 0.884542 0.442271 − 0.896882i \(-0.354173\pi\)
0.442271 + 0.896882i \(0.354173\pi\)
\(18\) 0 0
\(19\) −68.0000 −0.821067 −0.410533 − 0.911846i \(-0.634657\pi\)
−0.410533 + 0.911846i \(0.634657\pi\)
\(20\) 0 0
\(21\) 48.0000 0.498784
\(22\) 0 0
\(23\) 208.000 1.88570 0.942848 − 0.333224i \(-0.108136\pi\)
0.942848 + 0.333224i \(0.108136\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) −58.0000 −0.371391 −0.185695 − 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) 160.000 0.926995 0.463498 − 0.886098i \(-0.346594\pi\)
0.463498 + 0.886098i \(0.346594\pi\)
\(32\) 0 0
\(33\) −84.0000 −0.443107
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −270.000 −1.19967 −0.599834 − 0.800124i \(-0.704767\pi\)
−0.599834 + 0.800124i \(0.704767\pi\)
\(38\) 0 0
\(39\) 78.0000 0.320256
\(40\) 0 0
\(41\) 282.000 1.07417 0.537085 − 0.843528i \(-0.319525\pi\)
0.537085 + 0.843528i \(0.319525\pi\)
\(42\) 0 0
\(43\) −76.0000 −0.269532 −0.134766 − 0.990877i \(-0.543028\pi\)
−0.134766 + 0.990877i \(0.543028\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 280.000 0.868983 0.434491 − 0.900676i \(-0.356928\pi\)
0.434491 + 0.900676i \(0.356928\pi\)
\(48\) 0 0
\(49\) −87.0000 −0.253644
\(50\) 0 0
\(51\) 186.000 0.510690
\(52\) 0 0
\(53\) 210.000 0.544259 0.272129 − 0.962261i \(-0.412272\pi\)
0.272129 + 0.962261i \(0.412272\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −204.000 −0.474043
\(58\) 0 0
\(59\) 196.000 0.432492 0.216246 − 0.976339i \(-0.430619\pi\)
0.216246 + 0.976339i \(0.430619\pi\)
\(60\) 0 0
\(61\) 742.000 1.55743 0.778716 − 0.627376i \(-0.215871\pi\)
0.778716 + 0.627376i \(0.215871\pi\)
\(62\) 0 0
\(63\) 144.000 0.287973
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −836.000 −1.52438 −0.762191 − 0.647352i \(-0.775877\pi\)
−0.762191 + 0.647352i \(0.775877\pi\)
\(68\) 0 0
\(69\) 624.000 1.08871
\(70\) 0 0
\(71\) −504.000 −0.842448 −0.421224 − 0.906957i \(-0.638399\pi\)
−0.421224 + 0.906957i \(0.638399\pi\)
\(72\) 0 0
\(73\) 1062.00 1.70271 0.851354 − 0.524591i \(-0.175782\pi\)
0.851354 + 0.524591i \(0.175782\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −448.000 −0.663043
\(78\) 0 0
\(79\) 768.000 1.09376 0.546878 − 0.837212i \(-0.315816\pi\)
0.546878 + 0.837212i \(0.315816\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 1052.00 1.39123 0.695614 − 0.718415i \(-0.255132\pi\)
0.695614 + 0.718415i \(0.255132\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −174.000 −0.214423
\(88\) 0 0
\(89\) −726.000 −0.864672 −0.432336 − 0.901712i \(-0.642311\pi\)
−0.432336 + 0.901712i \(0.642311\pi\)
\(90\) 0 0
\(91\) 416.000 0.479216
\(92\) 0 0
\(93\) 480.000 0.535201
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1406.00 1.47173 0.735864 − 0.677129i \(-0.236776\pi\)
0.735864 + 0.677129i \(0.236776\pi\)
\(98\) 0 0
\(99\) −252.000 −0.255828
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 600.4.a.q.1.1 1
3.2 odd 2 1800.4.a.bb.1.1 1
4.3 odd 2 1200.4.a.c.1.1 1
5.2 odd 4 600.4.f.c.49.1 2
5.3 odd 4 600.4.f.c.49.2 2
5.4 even 2 120.4.a.c.1.1 ✓ 1
15.2 even 4 1800.4.f.r.649.2 2
15.8 even 4 1800.4.f.r.649.1 2
15.14 odd 2 360.4.a.b.1.1 1
20.3 even 4 1200.4.f.o.49.1 2
20.7 even 4 1200.4.f.o.49.2 2
20.19 odd 2 240.4.a.l.1.1 1
40.19 odd 2 960.4.a.h.1.1 1
40.29 even 2 960.4.a.u.1.1 1
60.59 even 2 720.4.a.l.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
120.4.a.c.1.1 ✓ 1 5.4 even 2
240.4.a.l.1.1 1 20.19 odd 2
360.4.a.b.1.1 1 15.14 odd 2
600.4.a.q.1.1 1 1.1 even 1 trivial
600.4.f.c.49.1 2 5.2 odd 4
600.4.f.c.49.2 2 5.3 odd 4
720.4.a.l.1.1 1 60.59 even 2
960.4.a.h.1.1 1 40.19 odd 2
960.4.a.u.1.1 1 40.29 even 2
1200.4.a.c.1.1 1 4.3 odd 2
1200.4.f.o.49.1 2 20.3 even 4
1200.4.f.o.49.2 2 20.7 even 4
1800.4.a.bb.1.1 1 3.2 odd 2
1800.4.f.r.649.1 2 15.8 even 4
1800.4.f.r.649.2 2 15.2 even 4