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Properties

Label 128.11.f.b
Level $128$
Weight $11$
Character orbit 128.f
Analytic conductor $81.326$
Analytic rank $0$
Dimension $38$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [128,11,Mod(31,128)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(128, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1]))
 
N = Newforms(chi, 11, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("128.31");
 
S:= CuspForms(chi, 11);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 128 = 2^{7} \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 128.f (of order \(4\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(81.3257283422\)
Analytic rank: \(0\)
Dimension: \(38\)
Relative dimension: \(19\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 16)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 38 q + 2 q^{3} + 2 q^{5} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 38 q + 2 q^{3} + 2 q^{5} - 4 q^{7} + 45906 q^{11} + 2 q^{13} - 4 q^{17} + 5107042 q^{19} + 118100 q^{21} - 8279748 q^{23} + 25871552 q^{27} + 29652306 q^{29} - 4 q^{33} - 14674460 q^{35} + 94618130 q^{37} - 279841732 q^{39} - 172486862 q^{43} - 19649346 q^{45} + 1049193778 q^{49} + 72924900 q^{51} - 607072398 q^{53} - 1427102468 q^{55} - 1543683854 q^{59} - 1673280158 q^{61} + 1098715212 q^{65} + 4830427746 q^{67} - 1698121420 q^{69} + 7572888316 q^{71} + 11656678318 q^{75} + 1336459028 q^{77} - 8523250762 q^{81} - 16141605438 q^{83} - 4779968748 q^{85} - 31832612676 q^{87} - 19874217404 q^{91} + 15001617056 q^{93} - 4 q^{97} + 25616384098 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1 0 −305.982 − 305.982i 0 −155.264 − 155.264i 0 8523.80 0 128200.i 0
31.2 0 −287.146 − 287.146i 0 3974.49 + 3974.49i 0 −11755.9 0 105857.i 0
31.3 0 −280.546 − 280.546i 0 −3102.01 − 3102.01i 0 −27408.1 0 98363.4i 0
31.4 0 −196.669 − 196.669i 0 1812.92 + 1812.92i 0 22339.6 0 18308.2i 0
31.5 0 −183.793 − 183.793i 0 −3496.01 − 3496.01i 0 26135.6 0 8510.59i 0
31.6 0 −164.886 − 164.886i 0 −897.573 − 897.573i 0 594.817 0 − 4674.41i 0
31.7 0 −127.462 − 127.462i 0 1086.76 + 1086.76i 0 −8881.75 0 − 26555.9i 0
31.8 0 −44.0645 − 44.0645i 0 126.270 + 126.270i 0 −19594.2 0 − 55165.6i 0
31.9 0 −32.7265 − 32.7265i 0 1645.29 + 1645.29i 0 16644.7 0 − 56907.0i 0
31.10 0 39.7294 + 39.7294i 0 4096.79 + 4096.79i 0 −7866.10 0 − 55892.1i 0
31.11 0 46.0373 + 46.0373i 0 −1933.37 − 1933.37i 0 −25927.9 0 − 54810.1i 0
31.12 0 61.4496 + 61.4496i 0 −2595.70 − 2595.70i 0 6393.57 0 − 51496.9i 0
31.13 0 107.144 + 107.144i 0 −3434.17 − 3434.17i 0 8771.09 0 − 36089.4i 0
31.14 0 126.065 + 126.065i 0 1655.31 + 1655.31i 0 29862.3 0 − 27264.3i 0
31.15 0 196.215 + 196.215i 0 1311.50 + 1311.50i 0 −1120.35 0 17952.0i 0
31.16 0 221.985 + 221.985i 0 −2579.78 − 2579.78i 0 4017.27 0 39505.3i 0
31.17 0 222.022 + 222.022i 0 2711.76 + 2711.76i 0 −24891.4 0 39538.9i 0
31.18 0 286.288 + 286.288i 0 1338.28 + 1338.28i 0 19657.4 0 104873.i 0
31.19 0 317.338 + 317.338i 0 −1564.48 − 1564.48i 0 −15496.4 0 142358.i 0
95.1 0 −305.982 + 305.982i 0 −155.264 + 155.264i 0 8523.80 0 − 128200.i 0
See all 38 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.19
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.f odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 128.11.f.b 38
4.b odd 2 1 128.11.f.a 38
8.b even 2 1 16.11.f.a ✓ 38
8.d odd 2 1 64.11.f.a 38
16.e even 4 1 64.11.f.a 38
16.e even 4 1 128.11.f.a 38
16.f odd 4 1 16.11.f.a ✓ 38
16.f odd 4 1 inner 128.11.f.b 38
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.11.f.a ✓ 38 8.b even 2 1
16.11.f.a ✓ 38 16.f odd 4 1
64.11.f.a 38 8.d odd 2 1
64.11.f.a 38 16.e even 4 1
128.11.f.a 38 4.b odd 2 1
128.11.f.a 38 16.e even 4 1
128.11.f.b 38 1.a even 1 1 trivial
128.11.f.b 38 16.f odd 4 1 inner