Properties

Label 912.6.k.d
Level $912$
Weight $6$
Character orbit 912.k
Analytic conductor $146.270$
Analytic rank $0$
Dimension $34$
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] = [912,6,Mod(607,912)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(912, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("912.607");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 912.k (of order \(2\), degree \(1\), 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: \(146.270043669\)
Analytic rank: \(0\)
Dimension: \(34\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 34 q + 306 q^{3} - 22 q^{5} + 2754 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 34 q + 306 q^{3} - 22 q^{5} + 2754 q^{9} - 198 q^{15} - 574 q^{17} + 80 q^{19} + 24372 q^{25} + 24786 q^{27} + 10880 q^{31} - 1782 q^{45} - 120940 q^{49} - 5166 q^{51} + 720 q^{57} + 94808 q^{59} - 27910 q^{61} + 44608 q^{67} + 108152 q^{71} - 112446 q^{73} + 219348 q^{75} + 88958 q^{77} - 12408 q^{79} + 223074 q^{81} + 70462 q^{85} - 228248 q^{91} + 97920 q^{93} + 43576 q^{95}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
607.1 0 9.00000 0 −97.4348 0 158.656i 0 81.0000 0
607.2 0 9.00000 0 −97.4348 0 158.656i 0 81.0000 0
607.3 0 9.00000 0 −86.4267 0 12.7350i 0 81.0000 0
607.4 0 9.00000 0 −86.4267 0 12.7350i 0 81.0000 0
607.5 0 9.00000 0 −75.7862 0 122.693i 0 81.0000 0
607.6 0 9.00000 0 −75.7862 0 122.693i 0 81.0000 0
607.7 0 9.00000 0 −63.9110 0 214.330i 0 81.0000 0
607.8 0 9.00000 0 −63.9110 0 214.330i 0 81.0000 0
607.9 0 9.00000 0 −47.0512 0 145.543i 0 81.0000 0
607.10 0 9.00000 0 −47.0512 0 145.543i 0 81.0000 0
607.11 0 9.00000 0 −38.6945 0 219.678i 0 81.0000 0
607.12 0 9.00000 0 −38.6945 0 219.678i 0 81.0000 0
607.13 0 9.00000 0 −35.6115 0 92.6135i 0 81.0000 0
607.14 0 9.00000 0 −35.6115 0 92.6135i 0 81.0000 0
607.15 0 9.00000 0 −14.7152 0 127.969i 0 81.0000 0
607.16 0 9.00000 0 −14.7152 0 127.969i 0 81.0000 0
607.17 0 9.00000 0 −10.8850 0 36.4238i 0 81.0000 0
607.18 0 9.00000 0 −10.8850 0 36.4238i 0 81.0000 0
607.19 0 9.00000 0 18.1600 0 5.76476i 0 81.0000 0
607.20 0 9.00000 0 18.1600 0 5.76476i 0 81.0000 0
See all 34 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 607.34
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
76.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 912.6.k.d yes 34
4.b odd 2 1 912.6.k.c 34
19.b odd 2 1 912.6.k.c 34
76.d even 2 1 inner 912.6.k.d yes 34
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
912.6.k.c 34 4.b odd 2 1
912.6.k.c 34 19.b odd 2 1
912.6.k.d yes 34 1.a even 1 1 trivial
912.6.k.d yes 34 76.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(912, [\chi])\):

\( T_{5}^{17} + 11 T_{5}^{16} - 32595 T_{5}^{15} - 436217 T_{5}^{14} + 420187667 T_{5}^{13} + 6388075161 T_{5}^{12} - 2747590673713 T_{5}^{11} - 44895804542675 T_{5}^{10} + \cdots + 13\!\cdots\!40 \) Copy content Toggle raw display
\( T_{31}^{17} - 5440 T_{31}^{16} - 226121244 T_{31}^{15} + 942057892560 T_{31}^{14} + \cdots + 19\!\cdots\!20 \) Copy content Toggle raw display