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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [128,2,Mod(65,128)] 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("128.65"); 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, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 128 = 2^{7} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 128.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,0,0,-10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.02208514587\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 65.1
Root \(-1.41421i\) of defining polynomial
Character \(\chi\) \(=\) 128.65
Dual form 128.2.b.a.65.2

$q$-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)
 
\(f(q)\) \(=\) \(q-2.82843i q^{3} -5.00000 q^{9} -2.82843i q^{11} +6.00000 q^{17} +8.48528i q^{19} +5.00000 q^{25} +5.65685i q^{27} -8.00000 q^{33} -6.00000 q^{41} +8.48528i q^{43} -7.00000 q^{49} -16.9706i q^{51} +24.0000 q^{57} -14.1421i q^{59} +8.48528i q^{67} -2.00000 q^{73} -14.1421i q^{75} +1.00000 q^{81} -2.82843i q^{83} -18.0000 q^{89} -10.0000 q^{97} +14.1421i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 10 q^{9} + 12 q^{17} + 10 q^{25} - 16 q^{33} - 12 q^{41} - 14 q^{49} + 48 q^{57} - 4 q^{73} + 2 q^{81} - 36 q^{89} - 20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/128\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(127\)
\(\chi(n)\) \(-1\) \(1\)

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
\(3\) − 2.82843i − 1.63299i −0.577350 − 0.816497i \(-0.695913\pi\)
0.577350 − 0.816497i \(-0.304087\pi\)
\(4\) 0 0
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) −5.00000 −1.66667
\(10\) 0 0
\(11\) − 2.82843i − 0.852803i −0.904534 − 0.426401i \(-0.859781\pi\)
0.904534 − 0.426401i \(-0.140219\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6.00000 1.45521 0.727607 − 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 0 0
\(19\) 8.48528i 1.94666i 0.229416 + 0.973329i \(0.426318\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 5.00000 1.00000
\(26\) 0 0
\(27\) 5.65685i 1.08866i
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) −8.00000 −1.39262
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 8.48528i 1.29399i 0.762493 + 0.646997i \(0.223975\pi\)
−0.762493 + 0.646997i \(0.776025\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) − 16.9706i − 2.37635i
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 24.0000 3.17888
\(58\) 0 0
\(59\) − 14.1421i − 1.84115i −0.390567 − 0.920575i \(-0.627721\pi\)
0.390567 − 0.920575i \(-0.372279\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 8.48528i 1.03664i 0.855186 + 0.518321i \(0.173443\pi\)
−0.855186 + 0.518321i \(0.826557\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −2.00000 −0.234082 −0.117041 − 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) 0 0
\(75\) − 14.1421i − 1.63299i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) − 2.82843i − 0.310460i −0.987878 − 0.155230i \(-0.950388\pi\)
0.987878 − 0.155230i \(-0.0496119\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −18.0000 −1.90800 −0.953998 − 0.299813i \(-0.903076\pi\)
−0.953998 + 0.299813i \(0.903076\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −10.0000 −1.01535 −0.507673 − 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 0 0
\(99\) 14.1421i 1.42134i
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.b.a.65.1 ✓ 2
3.2 odd 2 1152.2.d.c.577.2 2
4.3 odd 2 inner 128.2.b.a.65.2 yes 2
5.2 odd 4 3200.2.f.o.449.1 4
5.3 odd 4 3200.2.f.o.449.3 4
5.4 even 2 3200.2.d.c.1601.2 2
8.3 odd 2 CM 128.2.b.a.65.1 ✓ 2
8.5 even 2 inner 128.2.b.a.65.2 yes 2
12.11 even 2 1152.2.d.c.577.1 2
16.3 odd 4 256.2.a.e.1.1 2
16.5 even 4 256.2.a.e.1.1 2
16.11 odd 4 256.2.a.e.1.2 2
16.13 even 4 256.2.a.e.1.2 2
20.3 even 4 3200.2.f.o.449.2 4
20.7 even 4 3200.2.f.o.449.4 4
20.19 odd 2 3200.2.d.c.1601.1 2
24.5 odd 2 1152.2.d.c.577.1 2
24.11 even 2 1152.2.d.c.577.2 2
32.3 odd 8 1024.2.e.a.769.1 2
32.5 even 8 1024.2.e.f.257.1 2
32.11 odd 8 1024.2.e.f.257.1 2
32.13 even 8 1024.2.e.a.769.1 2
32.19 odd 8 1024.2.e.f.769.1 2
32.21 even 8 1024.2.e.a.257.1 2
32.27 odd 8 1024.2.e.a.257.1 2
32.29 even 8 1024.2.e.f.769.1 2
40.3 even 4 3200.2.f.o.449.3 4
40.13 odd 4 3200.2.f.o.449.2 4
40.19 odd 2 3200.2.d.c.1601.2 2
40.27 even 4 3200.2.f.o.449.1 4
40.29 even 2 3200.2.d.c.1601.1 2
40.37 odd 4 3200.2.f.o.449.4 4
48.5 odd 4 2304.2.a.t.1.1 2
48.11 even 4 2304.2.a.t.1.2 2
48.29 odd 4 2304.2.a.t.1.2 2
48.35 even 4 2304.2.a.t.1.1 2
80.19 odd 4 6400.2.a.by.1.2 2
80.29 even 4 6400.2.a.by.1.1 2
80.59 odd 4 6400.2.a.by.1.1 2
80.69 even 4 6400.2.a.by.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
128.2.b.a.65.1 ✓ 2 1.1 even 1 trivial
128.2.b.a.65.1 ✓ 2 8.3 odd 2 CM
128.2.b.a.65.2 yes 2 4.3 odd 2 inner
128.2.b.a.65.2 yes 2 8.5 even 2 inner
256.2.a.e.1.1 2 16.3 odd 4
256.2.a.e.1.1 2 16.5 even 4
256.2.a.e.1.2 2 16.11 odd 4
256.2.a.e.1.2 2 16.13 even 4
1024.2.e.a.257.1 2 32.21 even 8
1024.2.e.a.257.1 2 32.27 odd 8
1024.2.e.a.769.1 2 32.3 odd 8
1024.2.e.a.769.1 2 32.13 even 8
1024.2.e.f.257.1 2 32.5 even 8
1024.2.e.f.257.1 2 32.11 odd 8
1024.2.e.f.769.1 2 32.19 odd 8
1024.2.e.f.769.1 2 32.29 even 8
1152.2.d.c.577.1 2 12.11 even 2
1152.2.d.c.577.1 2 24.5 odd 2
1152.2.d.c.577.2 2 3.2 odd 2
1152.2.d.c.577.2 2 24.11 even 2
2304.2.a.t.1.1 2 48.5 odd 4
2304.2.a.t.1.1 2 48.35 even 4
2304.2.a.t.1.2 2 48.11 even 4
2304.2.a.t.1.2 2 48.29 odd 4
3200.2.d.c.1601.1 2 20.19 odd 2
3200.2.d.c.1601.1 2 40.29 even 2
3200.2.d.c.1601.2 2 5.4 even 2
3200.2.d.c.1601.2 2 40.19 odd 2
3200.2.f.o.449.1 4 5.2 odd 4
3200.2.f.o.449.1 4 40.27 even 4
3200.2.f.o.449.2 4 20.3 even 4
3200.2.f.o.449.2 4 40.13 odd 4
3200.2.f.o.449.3 4 5.3 odd 4
3200.2.f.o.449.3 4 40.3 even 4
3200.2.f.o.449.4 4 20.7 even 4
3200.2.f.o.449.4 4 40.37 odd 4
6400.2.a.by.1.1 2 80.29 even 4
6400.2.a.by.1.1 2 80.59 odd 4
6400.2.a.by.1.2 2 80.19 odd 4
6400.2.a.by.1.2 2 80.69 even 4