Properties

Label 60.4.a.b.1.1
Level $60$
Weight $4$
Character 60.1
Self dual yes
Analytic conductor $3.540$
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] = [60,4,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, 4); 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, 4, names="a")
 
Level: \( N \) \(=\) \( 60 = 2^{2} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 60.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,-3,0,5] 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: \(3.54011460034\)
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)
 
\(f(q)\) \(=\) \(q-3.00000 q^{3} +5.00000 q^{5} +32.0000 q^{7} +9.00000 q^{9} +36.0000 q^{11} -10.0000 q^{13} -15.0000 q^{15} -78.0000 q^{17} +140.000 q^{19} -96.0000 q^{21} -192.000 q^{23} +25.0000 q^{25} -27.0000 q^{27} +6.00000 q^{29} -16.0000 q^{31} -108.000 q^{33} +160.000 q^{35} -34.0000 q^{37} +30.0000 q^{39} -390.000 q^{41} -52.0000 q^{43} +45.0000 q^{45} +408.000 q^{47} +681.000 q^{49} +234.000 q^{51} -114.000 q^{53} +180.000 q^{55} -420.000 q^{57} +516.000 q^{59} -58.0000 q^{61} +288.000 q^{63} -50.0000 q^{65} -892.000 q^{67} +576.000 q^{69} -120.000 q^{71} -646.000 q^{73} -75.0000 q^{75} +1152.00 q^{77} -1168.00 q^{79} +81.0000 q^{81} -732.000 q^{83} -390.000 q^{85} -18.0000 q^{87} -1590.00 q^{89} -320.000 q^{91} +48.0000 q^{93} +700.000 q^{95} +194.000 q^{97} +324.000 q^{99} +O(q^{100})\)

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
\(5\) 5.00000 0.447214
\(6\) 0 0
\(7\) 32.0000 1.72784 0.863919 − 0.503631i \(-0.168003\pi\)
0.863919 + 0.503631i \(0.168003\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 36.0000 0.986764 0.493382 − 0.869813i \(-0.335760\pi\)
0.493382 + 0.869813i \(0.335760\pi\)
\(12\) 0 0
\(13\) −10.0000 −0.213346 −0.106673 − 0.994294i \(-0.534020\pi\)
−0.106673 + 0.994294i \(0.534020\pi\)
\(14\) 0 0
\(15\) −15.0000 −0.258199
\(16\) 0 0
\(17\) −78.0000 −1.11281 −0.556405 − 0.830911i \(-0.687820\pi\)
−0.556405 + 0.830911i \(0.687820\pi\)
\(18\) 0 0
\(19\) 140.000 1.69043 0.845216 − 0.534425i \(-0.179472\pi\)
0.845216 + 0.534425i \(0.179472\pi\)
\(20\) 0 0
\(21\) −96.0000 −0.997567
\(22\) 0 0
\(23\) −192.000 −1.74064 −0.870321 − 0.492485i \(-0.836089\pi\)
−0.870321 + 0.492485i \(0.836089\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) 6.00000 0.0384197 0.0192099 − 0.999815i \(-0.493885\pi\)
0.0192099 + 0.999815i \(0.493885\pi\)
\(30\) 0 0
\(31\) −16.0000 −0.0926995 −0.0463498 − 0.998925i \(-0.514759\pi\)
−0.0463498 + 0.998925i \(0.514759\pi\)
\(32\) 0 0
\(33\) −108.000 −0.569709
\(34\) 0 0
\(35\) 160.000 0.772712
\(36\) 0 0
\(37\) −34.0000 −0.151069 −0.0755347 − 0.997143i \(-0.524066\pi\)
−0.0755347 + 0.997143i \(0.524066\pi\)
\(38\) 0 0
\(39\) 30.0000 0.123176
\(40\) 0 0
\(41\) −390.000 −1.48556 −0.742778 − 0.669538i \(-0.766492\pi\)
−0.742778 + 0.669538i \(0.766492\pi\)
\(42\) 0 0
\(43\) −52.0000 −0.184417 −0.0922084 − 0.995740i \(-0.529393\pi\)
−0.0922084 + 0.995740i \(0.529393\pi\)
\(44\) 0 0
\(45\) 45.0000 0.149071
\(46\) 0 0
\(47\) 408.000 1.26623 0.633116 − 0.774057i \(-0.281776\pi\)
0.633116 + 0.774057i \(0.281776\pi\)
\(48\) 0 0
\(49\) 681.000 1.98542
\(50\) 0 0
\(51\) 234.000 0.642481
\(52\) 0 0
\(53\) −114.000 −0.295455 −0.147727 − 0.989028i \(-0.547196\pi\)
−0.147727 + 0.989028i \(0.547196\pi\)
\(54\) 0 0
\(55\) 180.000 0.441294
\(56\) 0 0
\(57\) −420.000 −0.975971
\(58\) 0 0
\(59\) 516.000 1.13860 0.569301 − 0.822129i \(-0.307214\pi\)
0.569301 + 0.822129i \(0.307214\pi\)
\(60\) 0 0
\(61\) −58.0000 −0.121740 −0.0608700 − 0.998146i \(-0.519388\pi\)
−0.0608700 + 0.998146i \(0.519388\pi\)
\(62\) 0 0
\(63\) 288.000 0.575946
\(64\) 0 0
\(65\) −50.0000 −0.0954113
\(66\) 0 0
\(67\) −892.000 −1.62649 −0.813247 − 0.581918i \(-0.802302\pi\)
−0.813247 + 0.581918i \(0.802302\pi\)
\(68\) 0 0
\(69\) 576.000 1.00496
\(70\) 0 0
\(71\) −120.000 −0.200583 −0.100291 − 0.994958i \(-0.531978\pi\)
−0.100291 + 0.994958i \(0.531978\pi\)
\(72\) 0 0
\(73\) −646.000 −1.03573 −0.517867 − 0.855461i \(-0.673274\pi\)
−0.517867 + 0.855461i \(0.673274\pi\)
\(74\) 0 0
\(75\) −75.0000 −0.115470
\(76\) 0 0
\(77\) 1152.00 1.70497
\(78\) 0 0
\(79\) −1168.00 −1.66342 −0.831711 − 0.555209i \(-0.812638\pi\)
−0.831711 + 0.555209i \(0.812638\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −732.000 −0.968041 −0.484021 − 0.875057i \(-0.660824\pi\)
−0.484021 + 0.875057i \(0.660824\pi\)
\(84\) 0 0
\(85\) −390.000 −0.497664
\(86\) 0 0
\(87\) −18.0000 −0.0221816
\(88\) 0 0
\(89\) −1590.00 −1.89370 −0.946852 − 0.321669i \(-0.895756\pi\)
−0.946852 + 0.321669i \(0.895756\pi\)
\(90\) 0 0
\(91\) −320.000 −0.368628
\(92\) 0 0
\(93\) 48.0000 0.0535201
\(94\) 0 0
\(95\) 700.000 0.755984
\(96\) 0 0
\(97\) 194.000 0.203069 0.101535 − 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 0 0
\(99\) 324.000 0.328921
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 60.4.a.b.1.1 ✓ 1
3.2 odd 2 180.4.a.c.1.1 1
4.3 odd 2 240.4.a.j.1.1 1
5.2 odd 4 300.4.d.d.49.2 2
5.3 odd 4 300.4.d.d.49.1 2
5.4 even 2 300.4.a.e.1.1 1
8.3 odd 2 960.4.a.a.1.1 1
8.5 even 2 960.4.a.bb.1.1 1
9.2 odd 6 1620.4.i.g.1081.1 2
9.4 even 3 1620.4.i.a.541.1 2
9.5 odd 6 1620.4.i.g.541.1 2
9.7 even 3 1620.4.i.a.1081.1 2
12.11 even 2 720.4.a.c.1.1 1
15.2 even 4 900.4.d.b.649.2 2
15.8 even 4 900.4.d.b.649.1 2
15.14 odd 2 900.4.a.b.1.1 1
20.3 even 4 1200.4.f.e.49.2 2
20.7 even 4 1200.4.f.e.49.1 2
20.19 odd 2 1200.4.a.s.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
60.4.a.b.1.1 ✓ 1 1.1 even 1 trivial
180.4.a.c.1.1 1 3.2 odd 2
240.4.a.j.1.1 1 4.3 odd 2
300.4.a.e.1.1 1 5.4 even 2
300.4.d.d.49.1 2 5.3 odd 4
300.4.d.d.49.2 2 5.2 odd 4
720.4.a.c.1.1 1 12.11 even 2
900.4.a.b.1.1 1 15.14 odd 2
900.4.d.b.649.1 2 15.8 even 4
900.4.d.b.649.2 2 15.2 even 4
960.4.a.a.1.1 1 8.3 odd 2
960.4.a.bb.1.1 1 8.5 even 2
1200.4.a.s.1.1 1 20.19 odd 2
1200.4.f.e.49.1 2 20.7 even 4
1200.4.f.e.49.2 2 20.3 even 4
1620.4.i.a.541.1 2 9.4 even 3
1620.4.i.a.1081.1 2 9.7 even 3
1620.4.i.g.541.1 2 9.5 odd 6
1620.4.i.g.1081.1 2 9.2 odd 6