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Properties

Label 256.6.e.c
Level $256$
Weight $6$
Character orbit 256.e
Analytic conductor $41.058$
Analytic rank $0$
Dimension $24$
Inner twists $4$

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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] = [256,6,Mod(65,256)] 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("256.65"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(256, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 3])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 256 = 2^{8} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 256.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24,0,0,0,-160] 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: no
Analytic conductor: \(41.0582578721\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 160 q^{5} - 1040 q^{13} + 384 q^{17} + 7376 q^{21} - 14624 q^{29} + 78208 q^{33} - 35248 q^{37} - 118288 q^{45} + 100072 q^{49} - 104144 q^{53} - 174640 q^{61} + 464496 q^{65} - 29968 q^{69} - 667920 q^{77}+ \cdots + 833152 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1 0 −17.5695 − 17.5695i 0 −43.4890 + 43.4890i 0 227.856i 0 374.378i 0
65.2 0 −14.1218 − 14.1218i 0 −23.0034 + 23.0034i 0 − 38.9237i 0 155.852i 0
65.3 0 −13.0057 − 13.0057i 0 −4.08320 + 4.08320i 0 − 44.6895i 0 95.2962i 0
65.4 0 −9.76896 − 9.76896i 0 −57.3733 + 57.3733i 0 − 89.3741i 0 − 52.1348i 0
65.5 0 −8.37595 − 8.37595i 0 57.0449 − 57.0449i 0 33.9058i 0 − 102.687i 0
65.6 0 −4.14100 − 4.14100i 0 30.9040 − 30.9040i 0 − 106.094i 0 − 208.704i 0
65.7 0 4.14100 + 4.14100i 0 30.9040 − 30.9040i 0 106.094i 0 − 208.704i 0
65.8 0 8.37595 + 8.37595i 0 57.0449 − 57.0449i 0 − 33.9058i 0 − 102.687i 0
65.9 0 9.76896 + 9.76896i 0 −57.3733 + 57.3733i 0 89.3741i 0 − 52.1348i 0
65.10 0 13.0057 + 13.0057i 0 −4.08320 + 4.08320i 0 44.6895i 0 95.2962i 0
65.11 0 14.1218 + 14.1218i 0 −23.0034 + 23.0034i 0 38.9237i 0 155.852i 0
65.12 0 17.5695 + 17.5695i 0 −43.4890 + 43.4890i 0 − 227.856i 0 374.378i 0
193.1 0 −17.5695 + 17.5695i 0 −43.4890 − 43.4890i 0 − 227.856i 0 − 374.378i 0
193.2 0 −14.1218 + 14.1218i 0 −23.0034 − 23.0034i 0 38.9237i 0 − 155.852i 0
193.3 0 −13.0057 + 13.0057i 0 −4.08320 − 4.08320i 0 44.6895i 0 − 95.2962i 0
193.4 0 −9.76896 + 9.76896i 0 −57.3733 − 57.3733i 0 89.3741i 0 52.1348i 0
193.5 0 −8.37595 + 8.37595i 0 57.0449 + 57.0449i 0 − 33.9058i 0 102.687i 0
193.6 0 −4.14100 + 4.14100i 0 30.9040 + 30.9040i 0 106.094i 0 208.704i 0
193.7 0 4.14100 − 4.14100i 0 30.9040 + 30.9040i 0 − 106.094i 0 208.704i 0
193.8 0 8.37595 − 8.37595i 0 57.0449 + 57.0449i 0 33.9058i 0 102.687i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 65.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
16.e even 4 1 inner
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 256.6.e.c ✓ 24
4.b odd 2 1 inner 256.6.e.c ✓ 24
8.b even 2 1 256.6.e.d yes 24
8.d odd 2 1 256.6.e.d yes 24
16.e even 4 1 inner 256.6.e.c ✓ 24
16.e even 4 1 256.6.e.d yes 24
16.f odd 4 1 inner 256.6.e.c ✓ 24
16.f odd 4 1 256.6.e.d yes 24
32.g even 8 1 1024.6.a.h 12
32.g even 8 1 1024.6.a.j 12
32.h odd 8 1 1024.6.a.h 12
32.h odd 8 1 1024.6.a.j 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
256.6.e.c ✓ 24 1.a even 1 1 trivial
256.6.e.c ✓ 24 4.b odd 2 1 inner
256.6.e.c ✓ 24 16.e even 4 1 inner
256.6.e.c ✓ 24 16.f odd 4 1 inner
256.6.e.d yes 24 8.b even 2 1
256.6.e.d yes 24 8.d odd 2 1
256.6.e.d yes 24 16.e even 4 1
256.6.e.d yes 24 16.f odd 4 1
1024.6.a.h 12 32.g even 8 1
1024.6.a.h 12 32.h odd 8 1
1024.6.a.j 12 32.g even 8 1
1024.6.a.j 12 32.h odd 8 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(256, [\chi])\):

\( T_{3}^{24} + 711976 T_{3}^{20} + 160754791152 T_{3}^{16} + \cdots + 58\!\cdots\!16 \) Copy content Toggle raw display
\( T_{5}^{12} + 80 T_{5}^{11} + 3200 T_{5}^{10} - 5600 T_{5}^{9} + 46819948 T_{5}^{8} + \cdots + 10\!\cdots\!96 \) Copy content Toggle raw display