Properties

Label 60.8.a.c.1.1
Level $60$
Weight $8$
Character 60.1
Self dual yes
Analytic conductor $18.743$
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] = [60,8,Mod(1,60)] 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("60.1"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(60, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 60 = 2^{2} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 60.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,27,0,-125] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(18.7431015290\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 60.1

$q$-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)
 
\(f(q)\) \(=\) \(q+27.0000 q^{3} -125.000 q^{5} +92.0000 q^{7} +729.000 q^{9} +3456.00 q^{11} +4610.00 q^{13} -3375.00 q^{15} +17502.0 q^{17} -1300.00 q^{19} +2484.00 q^{21} +14088.0 q^{23} +15625.0 q^{25} +19683.0 q^{27} +174306. q^{29} +189824. q^{31} +93312.0 q^{33} -11500.0 q^{35} +279506. q^{37} +124470. q^{39} +357690. q^{41} -283852. q^{43} -91125.0 q^{45} +101688. q^{47} -815079. q^{49} +472554. q^{51} -392574. q^{53} -432000. q^{55} -35100.0 q^{57} -539904. q^{59} -1.94634e6 q^{61} +67068.0 q^{63} -576250. q^{65} -1.85585e6 q^{67} +380376. q^{69} +1.68384e6 q^{71} -4.11005e6 q^{73} +421875. q^{75} +317952. q^{77} -4.56501e6 q^{79} +531441. q^{81} +5.44487e6 q^{83} -2.18775e6 q^{85} +4.70626e6 q^{87} -5.46123e6 q^{89} +424120. q^{91} +5.12525e6 q^{93} +162500. q^{95} -1.20747e7 q^{97} +2.51942e6 q^{99} +O(q^{100})\)

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
\(3\) 27.0000 0.577350
\(4\) 0 0
\(5\) −125.000 −0.447214
\(6\) 0 0
\(7\) 92.0000 0.101378 0.0506891 − 0.998714i \(-0.483858\pi\)
0.0506891 + 0.998714i \(0.483858\pi\)
\(8\) 0 0
\(9\) 729.000 0.333333
\(10\) 0 0
\(11\) 3456.00 0.782887 0.391444 − 0.920202i \(-0.371976\pi\)
0.391444 + 0.920202i \(0.371976\pi\)
\(12\) 0 0
\(13\) 4610.00 0.581968 0.290984 − 0.956728i \(-0.406017\pi\)
0.290984 + 0.956728i \(0.406017\pi\)
\(14\) 0 0
\(15\) −3375.00 −0.258199
\(16\) 0 0
\(17\) 17502.0 0.864005 0.432003 − 0.901872i \(-0.357807\pi\)
0.432003 + 0.901872i \(0.357807\pi\)
\(18\) 0 0
\(19\) −1300.00 −0.0434816 −0.0217408 − 0.999764i \(-0.506921\pi\)
−0.0217408 + 0.999764i \(0.506921\pi\)
\(20\) 0 0
\(21\) 2484.00 0.0585307
\(22\) 0 0
\(23\) 14088.0 0.241436 0.120718 − 0.992687i \(-0.461480\pi\)
0.120718 + 0.992687i \(0.461480\pi\)
\(24\) 0 0
\(25\) 15625.0 0.200000
\(26\) 0 0
\(27\) 19683.0 0.192450
\(28\) 0 0
\(29\) 174306. 1.32715 0.663574 − 0.748111i \(-0.269039\pi\)
0.663574 + 0.748111i \(0.269039\pi\)
\(30\) 0 0
\(31\) 189824. 1.14442 0.572210 − 0.820107i \(-0.306086\pi\)
0.572210 + 0.820107i \(0.306086\pi\)
\(32\) 0 0
\(33\) 93312.0 0.452000
\(34\) 0 0
\(35\) −11500.0 −0.0453377
\(36\) 0 0
\(37\) 279506. 0.907163 0.453581 − 0.891215i \(-0.350146\pi\)
0.453581 + 0.891215i \(0.350146\pi\)
\(38\) 0 0
\(39\) 124470. 0.335999
\(40\) 0 0
\(41\) 357690. 0.810519 0.405260 − 0.914202i \(-0.367181\pi\)
0.405260 + 0.914202i \(0.367181\pi\)
\(42\) 0 0
\(43\) −283852. −0.544443 −0.272221 − 0.962235i \(-0.587758\pi\)
−0.272221 + 0.962235i \(0.587758\pi\)
\(44\) 0 0
\(45\) −91125.0 −0.149071
\(46\) 0 0
\(47\) 101688. 0.142865 0.0714327 − 0.997445i \(-0.477243\pi\)
0.0714327 + 0.997445i \(0.477243\pi\)
\(48\) 0 0
\(49\) −815079. −0.989722
\(50\) 0 0
\(51\) 472554. 0.498834
\(52\) 0 0
\(53\) −392574. −0.362206 −0.181103 − 0.983464i \(-0.557967\pi\)
−0.181103 + 0.983464i \(0.557967\pi\)
\(54\) 0 0
\(55\) −432000. −0.350118
\(56\) 0 0
\(57\) −35100.0 −0.0251041
\(58\) 0 0
\(59\) −539904. −0.342243 −0.171121 − 0.985250i \(-0.554739\pi\)
−0.171121 + 0.985250i \(0.554739\pi\)
\(60\) 0 0
\(61\) −1.94634e6 −1.09790 −0.548951 − 0.835854i \(-0.684973\pi\)
−0.548951 + 0.835854i \(0.684973\pi\)
\(62\) 0 0
\(63\) 67068.0 0.0337927
\(64\) 0 0
\(65\) −576250. −0.260264
\(66\) 0 0
\(67\) −1.85585e6 −0.753844 −0.376922 − 0.926245i \(-0.623018\pi\)
−0.376922 + 0.926245i \(0.623018\pi\)
\(68\) 0 0
\(69\) 380376. 0.139393
\(70\) 0 0
\(71\) 1.68384e6 0.558337 0.279169 − 0.960242i \(-0.409941\pi\)
0.279169 + 0.960242i \(0.409941\pi\)
\(72\) 0 0
\(73\) −4.11005e6 −1.23656 −0.618282 − 0.785956i \(-0.712171\pi\)
−0.618282 + 0.785956i \(0.712171\pi\)
\(74\) 0 0
\(75\) 421875. 0.115470
\(76\) 0 0
\(77\) 317952. 0.0793677
\(78\) 0 0
\(79\) −4.56501e6 −1.04171 −0.520855 − 0.853645i \(-0.674387\pi\)
−0.520855 + 0.853645i \(0.674387\pi\)
\(80\) 0 0
\(81\) 531441. 0.111111
\(82\) 0 0
\(83\) 5.44487e6 1.04524 0.522618 − 0.852567i \(-0.324956\pi\)
0.522618 + 0.852567i \(0.324956\pi\)
\(84\) 0 0
\(85\) −2.18775e6 −0.386395
\(86\) 0 0
\(87\) 4.70626e6 0.766229
\(88\) 0 0
\(89\) −5.46123e6 −0.821156 −0.410578 − 0.911826i \(-0.634673\pi\)
−0.410578 + 0.911826i \(0.634673\pi\)
\(90\) 0 0
\(91\) 424120. 0.0589989
\(92\) 0 0
\(93\) 5.12525e6 0.660731
\(94\) 0 0
\(95\) 162500. 0.0194456
\(96\) 0 0
\(97\) −1.20747e7 −1.34331 −0.671653 − 0.740866i \(-0.734415\pi\)
−0.671653 + 0.740866i \(0.734415\pi\)
\(98\) 0 0
\(99\) 2.51942e6 0.260962
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 60.8.a.c.1.1 ✓ 1
3.2 odd 2 180.8.a.d.1.1 1
4.3 odd 2 240.8.a.b.1.1 1
5.2 odd 4 300.8.d.f.49.1 2
5.3 odd 4 300.8.d.f.49.2 2
5.4 even 2 300.8.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
60.8.a.c.1.1 ✓ 1 1.1 even 1 trivial
180.8.a.d.1.1 1 3.2 odd 2
240.8.a.b.1.1 1 4.3 odd 2
300.8.a.c.1.1 1 5.4 even 2
300.8.d.f.49.1 2 5.2 odd 4
300.8.d.f.49.2 2 5.3 odd 4