Properties

Label 600.4.a.a.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,-20,0,9,0,16] 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)
 
\(f(q)\) \(=\) \(q-3.00000 q^{3} -20.0000 q^{7} +9.00000 q^{9} +16.0000 q^{11} -58.0000 q^{13} -38.0000 q^{17} +4.00000 q^{19} +60.0000 q^{21} +80.0000 q^{23} -27.0000 q^{27} +82.0000 q^{29} -8.00000 q^{31} -48.0000 q^{33} -426.000 q^{37} +174.000 q^{39} -246.000 q^{41} +524.000 q^{43} +464.000 q^{47} +57.0000 q^{49} +114.000 q^{51} +702.000 q^{53} -12.0000 q^{57} -592.000 q^{59} +574.000 q^{61} -180.000 q^{63} +172.000 q^{67} -240.000 q^{69} +768.000 q^{71} +558.000 q^{73} -320.000 q^{77} +408.000 q^{79} +81.0000 q^{81} -164.000 q^{83} -246.000 q^{87} -510.000 q^{89} +1160.00 q^{91} +24.0000 q^{93} -514.000 q^{97} +144.000 q^{99} +O(q^{100})\)

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
\(7\) −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
\(11\) 16.0000 0.438562 0.219281 − 0.975662i \(-0.429629\pi\)
0.219281 + 0.975662i \(0.429629\pi\)
\(12\) 0 0
\(13\) −58.0000 −1.23741 −0.618704 − 0.785624i \(-0.712342\pi\)
−0.618704 + 0.785624i \(0.712342\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −38.0000 −0.542138 −0.271069 − 0.962560i \(-0.587377\pi\)
−0.271069 + 0.962560i \(0.587377\pi\)
\(18\) 0 0
\(19\) 4.00000 0.0482980 0.0241490 − 0.999708i \(-0.492312\pi\)
0.0241490 + 0.999708i \(0.492312\pi\)
\(20\) 0 0
\(21\) 60.0000 0.623480
\(22\) 0 0
\(23\) 80.0000 0.725268 0.362634 − 0.931932i \(-0.381878\pi\)
0.362634 + 0.931932i \(0.381878\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) 82.0000 0.525070 0.262535 − 0.964923i \(-0.415442\pi\)
0.262535 + 0.964923i \(0.415442\pi\)
\(30\) 0 0
\(31\) −8.00000 −0.0463498 −0.0231749 − 0.999731i \(-0.507377\pi\)
−0.0231749 + 0.999731i \(0.507377\pi\)
\(32\) 0 0
\(33\) −48.0000 −0.253204
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −426.000 −1.89281 −0.946405 − 0.322982i \(-0.895315\pi\)
−0.946405 + 0.322982i \(0.895315\pi\)
\(38\) 0 0
\(39\) 174.000 0.714418
\(40\) 0 0
\(41\) −246.000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 524.000 1.85835 0.929177 − 0.369634i \(-0.120517\pi\)
0.929177 + 0.369634i \(0.120517\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 464.000 1.44003 0.720014 − 0.693959i \(-0.244135\pi\)
0.720014 + 0.693959i \(0.244135\pi\)
\(48\) 0 0
\(49\) 57.0000 0.166181
\(50\) 0 0
\(51\) 114.000 0.313004
\(52\) 0 0
\(53\) 702.000 1.81938 0.909690 − 0.415288i \(-0.136319\pi\)
0.909690 + 0.415288i \(0.136319\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −12.0000 −0.0278849
\(58\) 0 0
\(59\) −592.000 −1.30630 −0.653151 − 0.757228i \(-0.726553\pi\)
−0.653151 + 0.757228i \(0.726553\pi\)
\(60\) 0 0
\(61\) 574.000 1.20481 0.602403 − 0.798192i \(-0.294210\pi\)
0.602403 + 0.798192i \(0.294210\pi\)
\(62\) 0 0
\(63\) −180.000 −0.359966
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 172.000 0.313629 0.156815 − 0.987628i \(-0.449878\pi\)
0.156815 + 0.987628i \(0.449878\pi\)
\(68\) 0 0
\(69\) −240.000 −0.418733
\(70\) 0 0
\(71\) 768.000 1.28373 0.641865 − 0.766818i \(-0.278161\pi\)
0.641865 + 0.766818i \(0.278161\pi\)
\(72\) 0 0
\(73\) 558.000 0.894643 0.447322 − 0.894373i \(-0.352378\pi\)
0.447322 + 0.894373i \(0.352378\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −320.000 −0.473602
\(78\) 0 0
\(79\) 408.000 0.581058 0.290529 − 0.956866i \(-0.406169\pi\)
0.290529 + 0.956866i \(0.406169\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −164.000 −0.216884 −0.108442 − 0.994103i \(-0.534586\pi\)
−0.108442 + 0.994103i \(0.534586\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −246.000 −0.303149
\(88\) 0 0
\(89\) −510.000 −0.607415 −0.303707 − 0.952765i \(-0.598224\pi\)
−0.303707 + 0.952765i \(0.598224\pi\)
\(90\) 0 0
\(91\) 1160.00 1.33628
\(92\) 0 0
\(93\) 24.0000 0.0267600
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −514.000 −0.538029 −0.269014 − 0.963136i \(-0.586698\pi\)
−0.269014 + 0.963136i \(0.586698\pi\)
\(98\) 0 0
\(99\) 144.000 0.146187
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.a.1.1 1
3.2 odd 2 1800.4.a.e.1.1 1
4.3 odd 2 1200.4.a.bj.1.1 1
5.2 odd 4 600.4.f.f.49.2 2
5.3 odd 4 600.4.f.f.49.1 2
5.4 even 2 120.4.a.e.1.1 ✓ 1
15.2 even 4 1800.4.f.k.649.1 2
15.8 even 4 1800.4.f.k.649.2 2
15.14 odd 2 360.4.a.m.1.1 1
20.3 even 4 1200.4.f.h.49.2 2
20.7 even 4 1200.4.f.h.49.1 2
20.19 odd 2 240.4.a.a.1.1 1
40.19 odd 2 960.4.a.bd.1.1 1
40.29 even 2 960.4.a.q.1.1 1
60.59 even 2 720.4.a.s.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
120.4.a.e.1.1 ✓ 1 5.4 even 2
240.4.a.a.1.1 1 20.19 odd 2
360.4.a.m.1.1 1 15.14 odd 2
600.4.a.a.1.1 1 1.1 even 1 trivial
600.4.f.f.49.1 2 5.3 odd 4
600.4.f.f.49.2 2 5.2 odd 4
720.4.a.s.1.1 1 60.59 even 2
960.4.a.q.1.1 1 40.29 even 2
960.4.a.bd.1.1 1 40.19 odd 2
1200.4.a.bj.1.1 1 4.3 odd 2
1200.4.f.h.49.1 2 20.7 even 4
1200.4.f.h.49.2 2 20.3 even 4
1800.4.a.e.1.1 1 3.2 odd 2
1800.4.f.k.649.1 2 15.2 even 4
1800.4.f.k.649.2 2 15.8 even 4