↑

Properties

Label 128.2.a.d.1.1
Level $128$
Weight $2$
Character 128.1
Self dual yes
Analytic conductor $1.022$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [128,2,Mod(1,128)] 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("128.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(128, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 128 = 2^{7} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 128.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,2,0,2] 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: \(1.02208514587\)
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 \(\chi\) \(=\) 128.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.00000 q^{3} +2.00000 q^{5} -4.00000 q^{7} +1.00000 q^{9} -2.00000 q^{11} +2.00000 q^{13} +4.00000 q^{15} -2.00000 q^{17} +2.00000 q^{19} -8.00000 q^{21} +4.00000 q^{23} -1.00000 q^{25} -4.00000 q^{27} -6.00000 q^{29} -4.00000 q^{33} -8.00000 q^{35} +10.0000 q^{37} +4.00000 q^{39} -6.00000 q^{41} +6.00000 q^{43} +2.00000 q^{45} -8.00000 q^{47} +9.00000 q^{49} -4.00000 q^{51} -6.00000 q^{53} -4.00000 q^{55} +4.00000 q^{57} +14.0000 q^{59} +2.00000 q^{61} -4.00000 q^{63} +4.00000 q^{65} +10.0000 q^{67} +8.00000 q^{69} +12.0000 q^{71} +14.0000 q^{73} -2.00000 q^{75} +8.00000 q^{77} -8.00000 q^{79} -11.0000 q^{81} -6.00000 q^{83} -4.00000 q^{85} -12.0000 q^{87} -2.00000 q^{89} -8.00000 q^{91} +4.00000 q^{95} -2.00000 q^{97} -2.00000 q^{99} +O(q^{100})\)

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
\(3\) 2.00000 1.15470 0.577350 − 0.816497i \(-0.304087\pi\)
0.577350 + 0.816497i \(0.304087\pi\)
\(4\) 0 0
\(5\) 2.00000 0.894427 0.447214 − 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 0 0
\(7\) −4.00000 −1.51186 −0.755929 − 0.654654i \(-0.772814\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 − 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 − 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) 4.00000 1.03280
\(16\) 0 0
\(17\) −2.00000 −0.485071 −0.242536 − 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) 2.00000 0.458831 0.229416 − 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 0 0
\(21\) −8.00000 −1.74574
\(22\) 0 0
\(23\) 4.00000 0.834058 0.417029 − 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) −4.00000 −0.769800
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 − 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) −4.00000 −0.696311
\(34\) 0 0
\(35\) −8.00000 −1.35225
\(36\) 0 0
\(37\) 10.0000 1.64399 0.821995 − 0.569495i \(-0.192861\pi\)
0.821995 + 0.569495i \(0.192861\pi\)
\(38\) 0 0
\(39\) 4.00000 0.640513
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 6.00000 0.914991 0.457496 − 0.889212i \(-0.348747\pi\)
0.457496 + 0.889212i \(0.348747\pi\)
\(44\) 0 0
\(45\) 2.00000 0.298142
\(46\) 0 0
\(47\) −8.00000 −1.16692 −0.583460 − 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 0 0
\(49\) 9.00000 1.28571
\(50\) 0 0
\(51\) −4.00000 −0.560112
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 − 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) −4.00000 −0.539360
\(56\) 0 0
\(57\) 4.00000 0.529813
\(58\) 0 0
\(59\) 14.0000 1.82264 0.911322 − 0.411693i \(-0.135063\pi\)
0.911322 + 0.411693i \(0.135063\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 − 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 0 0
\(63\) −4.00000 −0.503953
\(64\) 0 0
\(65\) 4.00000 0.496139
\(66\) 0 0
\(67\) 10.0000 1.22169 0.610847 − 0.791748i \(-0.290829\pi\)
0.610847 + 0.791748i \(0.290829\pi\)
\(68\) 0 0
\(69\) 8.00000 0.963087
\(70\) 0 0
\(71\) 12.0000 1.42414 0.712069 − 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 0 0
\(73\) 14.0000 1.63858 0.819288 − 0.573382i \(-0.194369\pi\)
0.819288 + 0.573382i \(0.194369\pi\)
\(74\) 0 0
\(75\) −2.00000 −0.230940
\(76\) 0 0
\(77\) 8.00000 0.911685
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 − 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) −6.00000 −0.658586 −0.329293 − 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) 0 0
\(85\) −4.00000 −0.433861
\(86\) 0 0
\(87\) −12.0000 −1.28654
\(88\) 0 0
\(89\) −2.00000 −0.212000 −0.106000 − 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 0 0
\(91\) −8.00000 −0.838628
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 4.00000 0.410391
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 − 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 0 0
\(99\) −2.00000 −0.201008
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 128.2.a.d.1.1 yes 1
3.2 odd 2 1152.2.a.c.1.1 1
4.3 odd 2 128.2.a.b.1.1 yes 1
5.2 odd 4 3200.2.c.e.2049.1 2
5.3 odd 4 3200.2.c.e.2049.2 2
5.4 even 2 3200.2.a.h.1.1 1
7.6 odd 2 6272.2.a.a.1.1 1
8.3 odd 2 128.2.a.c.1.1 yes 1
8.5 even 2 128.2.a.a.1.1 ✓ 1
12.11 even 2 1152.2.a.h.1.1 1
16.3 odd 4 256.2.b.a.129.1 2
16.5 even 4 256.2.b.c.129.1 2
16.11 odd 4 256.2.b.a.129.2 2
16.13 even 4 256.2.b.c.129.2 2
20.3 even 4 3200.2.c.k.2049.1 2
20.7 even 4 3200.2.c.k.2049.2 2
20.19 odd 2 3200.2.a.u.1.1 1
24.5 odd 2 1152.2.a.m.1.1 1
24.11 even 2 1152.2.a.r.1.1 1
28.27 even 2 6272.2.a.g.1.1 1
32.3 odd 8 1024.2.e.m.257.1 4
32.5 even 8 1024.2.e.i.769.1 4
32.11 odd 8 1024.2.e.m.769.1 4
32.13 even 8 1024.2.e.i.257.1 4
32.19 odd 8 1024.2.e.m.257.2 4
32.21 even 8 1024.2.e.i.769.2 4
32.27 odd 8 1024.2.e.m.769.2 4
32.29 even 8 1024.2.e.i.257.2 4
40.3 even 4 3200.2.c.f.2049.2 2
40.13 odd 4 3200.2.c.l.2049.1 2
40.19 odd 2 3200.2.a.e.1.1 1
40.27 even 4 3200.2.c.f.2049.1 2
40.29 even 2 3200.2.a.x.1.1 1
40.37 odd 4 3200.2.c.l.2049.2 2
48.5 odd 4 2304.2.d.r.1153.1 2
48.11 even 4 2304.2.d.b.1153.1 2
48.29 odd 4 2304.2.d.r.1153.2 2
48.35 even 4 2304.2.d.b.1153.2 2
56.13 odd 2 6272.2.a.h.1.1 1
56.27 even 2 6272.2.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
128.2.a.a.1.1 ✓ 1 8.5 even 2
128.2.a.b.1.1 yes 1 4.3 odd 2
128.2.a.c.1.1 yes 1 8.3 odd 2
128.2.a.d.1.1 yes 1 1.1 even 1 trivial
256.2.b.a.129.1 2 16.3 odd 4
256.2.b.a.129.2 2 16.11 odd 4
256.2.b.c.129.1 2 16.5 even 4
256.2.b.c.129.2 2 16.13 even 4
1024.2.e.i.257.1 4 32.13 even 8
1024.2.e.i.257.2 4 32.29 even 8
1024.2.e.i.769.1 4 32.5 even 8
1024.2.e.i.769.2 4 32.21 even 8
1024.2.e.m.257.1 4 32.3 odd 8
1024.2.e.m.257.2 4 32.19 odd 8
1024.2.e.m.769.1 4 32.11 odd 8
1024.2.e.m.769.2 4 32.27 odd 8
1152.2.a.c.1.1 1 3.2 odd 2
1152.2.a.h.1.1 1 12.11 even 2
1152.2.a.m.1.1 1 24.5 odd 2
1152.2.a.r.1.1 1 24.11 even 2
2304.2.d.b.1153.1 2 48.11 even 4
2304.2.d.b.1153.2 2 48.35 even 4
2304.2.d.r.1153.1 2 48.5 odd 4
2304.2.d.r.1153.2 2 48.29 odd 4
3200.2.a.e.1.1 1 40.19 odd 2
3200.2.a.h.1.1 1 5.4 even 2
3200.2.a.u.1.1 1 20.19 odd 2
3200.2.a.x.1.1 1 40.29 even 2
3200.2.c.e.2049.1 2 5.2 odd 4
3200.2.c.e.2049.2 2 5.3 odd 4
3200.2.c.f.2049.1 2 40.27 even 4
3200.2.c.f.2049.2 2 40.3 even 4
3200.2.c.k.2049.1 2 20.3 even 4
3200.2.c.k.2049.2 2 20.7 even 4
3200.2.c.l.2049.1 2 40.13 odd 4
3200.2.c.l.2049.2 2 40.37 odd 4
6272.2.a.a.1.1 1 7.6 odd 2
6272.2.a.b.1.1 1 56.27 even 2
6272.2.a.g.1.1 1 28.27 even 2
6272.2.a.h.1.1 1 56.13 odd 2