Properties

Label 945.2.k.b
Level $945$
Weight $2$
Character orbit 945.k
Analytic conductor $7.546$
Analytic rank $0$
Dimension $24$
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] = [945,2,Mod(361,945)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(945, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("945.361");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 945 = 3^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 945.k (of order \(3\), 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: \(7.54586299101\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{3})\)
Twist minimal: no (minimal twist has level 315)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + q^{2} - 7 q^{4} + 24 q^{5} + 7 q^{7} - 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + q^{2} - 7 q^{4} + 24 q^{5} + 7 q^{7} - 12 q^{8} + q^{10} + 2 q^{11} - 4 q^{13} + 13 q^{14} - 5 q^{16} + 7 q^{17} - 2 q^{19} - 7 q^{20} + 19 q^{22} + 2 q^{23} + 24 q^{25} - 11 q^{26} - 28 q^{28} + 8 q^{31} - 17 q^{32} + q^{34} + 7 q^{35} + 17 q^{37} - 70 q^{38} - 12 q^{40} - 20 q^{41} + 31 q^{43} + 7 q^{44} - 10 q^{46} + 31 q^{47} - 11 q^{49} + q^{50} + 8 q^{52} - 8 q^{53} + 2 q^{55} - 45 q^{56} - 90 q^{58} + 21 q^{59} + 5 q^{61} - 14 q^{62} - 56 q^{64} - 4 q^{65} + 43 q^{67} - 96 q^{68} + 13 q^{70} - 24 q^{71} - 18 q^{73} + 18 q^{74} - 13 q^{76} - 5 q^{77} + 40 q^{79} - 5 q^{80} + 5 q^{82} + 60 q^{83} + 7 q^{85} + 24 q^{86} - 100 q^{88} + 4 q^{89} - 33 q^{91} + 18 q^{92} - 11 q^{94} - 2 q^{95} + 6 q^{97} + 15 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1 −1.28004 + 2.21710i 0 −2.27701 3.94390i 1.00000 0 −1.62783 2.08571i 6.53853 0 −1.28004 + 2.21710i
361.2 −0.859635 + 1.48893i 0 −0.477944 0.827824i 1.00000 0 0.594106 2.57818i −1.79511 0 −0.859635 + 1.48893i
361.3 −0.805191 + 1.39463i 0 −0.296664 0.513837i 1.00000 0 2.48389 0.911196i −2.26528 0 −0.805191 + 1.39463i
361.4 −0.517769 + 0.896802i 0 0.463831 + 0.803378i 1.00000 0 −1.07729 + 2.41649i −3.03170 0 −0.517769 + 0.896802i
361.5 −0.304907 + 0.528114i 0 0.814064 + 1.41000i 1.00000 0 1.83653 + 1.90451i −2.21248 0 −0.304907 + 0.528114i
361.6 −0.154039 + 0.266804i 0 0.952544 + 1.64985i 1.00000 0 2.20433 1.46319i −1.20307 0 −0.154039 + 0.266804i
361.7 0.148731 0.257610i 0 0.955758 + 1.65542i 1.00000 0 −2.64436 0.0857253i 1.16353 0 0.148731 0.257610i
361.8 0.259245 0.449026i 0 0.865584 + 1.49923i 1.00000 0 −1.91816 1.82226i 1.93458 0 0.259245 0.449026i
361.9 0.599034 1.03756i 0 0.282317 + 0.488988i 1.00000 0 2.04342 + 1.68061i 3.07261 0 0.599034 1.03756i
361.10 1.08176 1.87367i 0 −1.34041 2.32167i 1.00000 0 2.14928 1.54292i −1.47299 0 1.08176 1.87367i
361.11 1.16277 2.01398i 0 −1.70408 2.95156i 1.00000 0 0.459529 + 2.60554i −3.27475 0 1.16277 2.01398i
361.12 1.17004 2.02657i 0 −1.73798 3.01027i 1.00000 0 −1.00344 2.44808i −3.45385 0 1.17004 2.02657i
856.1 −1.28004 2.21710i 0 −2.27701 + 3.94390i 1.00000 0 −1.62783 + 2.08571i 6.53853 0 −1.28004 2.21710i
856.2 −0.859635 1.48893i 0 −0.477944 + 0.827824i 1.00000 0 0.594106 + 2.57818i −1.79511 0 −0.859635 1.48893i
856.3 −0.805191 1.39463i 0 −0.296664 + 0.513837i 1.00000 0 2.48389 + 0.911196i −2.26528 0 −0.805191 1.39463i
856.4 −0.517769 0.896802i 0 0.463831 0.803378i 1.00000 0 −1.07729 2.41649i −3.03170 0 −0.517769 0.896802i
856.5 −0.304907 0.528114i 0 0.814064 1.41000i 1.00000 0 1.83653 1.90451i −2.21248 0 −0.304907 0.528114i
856.6 −0.154039 0.266804i 0 0.952544 1.64985i 1.00000 0 2.20433 + 1.46319i −1.20307 0 −0.154039 0.266804i
856.7 0.148731 + 0.257610i 0 0.955758 1.65542i 1.00000 0 −2.64436 + 0.0857253i 1.16353 0 0.148731 + 0.257610i
856.8 0.259245 + 0.449026i 0 0.865584 1.49923i 1.00000 0 −1.91816 + 1.82226i 1.93458 0 0.259245 + 0.449026i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 361.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.g even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 945.2.k.b 24
3.b odd 2 1 315.2.k.b 24
7.c even 3 1 945.2.l.b 24
9.c even 3 1 945.2.l.b 24
9.d odd 6 1 315.2.l.b yes 24
21.h odd 6 1 315.2.l.b yes 24
63.g even 3 1 inner 945.2.k.b 24
63.n odd 6 1 315.2.k.b 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.k.b 24 3.b odd 2 1
315.2.k.b 24 63.n odd 6 1
315.2.l.b yes 24 9.d odd 6 1
315.2.l.b yes 24 21.h odd 6 1
945.2.k.b 24 1.a even 1 1 trivial
945.2.k.b 24 63.g even 3 1 inner
945.2.l.b 24 7.c even 3 1
945.2.l.b 24 9.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{24} - T_{2}^{23} + 16 T_{2}^{22} - 9 T_{2}^{21} + 157 T_{2}^{20} - 62 T_{2}^{19} + 930 T_{2}^{18} - 70 T_{2}^{17} + 3854 T_{2}^{16} + 284 T_{2}^{15} + 10997 T_{2}^{14} + 2832 T_{2}^{13} + 21841 T_{2}^{12} + 5217 T_{2}^{11} + 26201 T_{2}^{10} + \cdots + 9 \) acting on \(S_{2}^{\mathrm{new}}(945, [\chi])\). Copy content Toggle raw display