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Properties

Label 945.2.bl.j
Level $945$
Weight $2$
Character orbit 945.bl
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(251,945)]
 
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([1, 0, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("945.251");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 945 = 3^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 945.bl (of order \(6\), 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 \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 315)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 6 q^{2} + 18 q^{4} + 12 q^{5} + 9 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 6 q^{2} + 18 q^{4} + 12 q^{5} + 9 q^{7} - 9 q^{11} + 3 q^{13} - 18 q^{14} - 18 q^{16} - 18 q^{17} - 18 q^{20} - 9 q^{22} - 9 q^{23} - 12 q^{25} + 18 q^{26} - 9 q^{28} - 9 q^{29} - 42 q^{31} - 18 q^{32} - 39 q^{34} + 9 q^{35} + 12 q^{38} - 6 q^{40} + 33 q^{41} + 18 q^{43} - 30 q^{46} + 9 q^{49} + 6 q^{50} + 129 q^{52} + 9 q^{56} - 15 q^{58} - 12 q^{59} - 15 q^{61} - 12 q^{62} - 60 q^{64} + 3 q^{65} - 15 q^{67} - 9 q^{68} - 9 q^{70} + 18 q^{74} + 54 q^{76} + 45 q^{77} + 21 q^{79} - 36 q^{80} + 30 q^{83} - 9 q^{85} + 102 q^{86} - 9 q^{88} - 102 q^{89} + 42 q^{91} + 3 q^{92} - 156 q^{94} + 18 q^{95} - 45 q^{97} - 3 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
251.1 −2.32000 − 1.33945i 0 2.58825 + 4.48298i 0.500000 + 0.866025i 0 2.62997 − 0.288507i − 8.50953i 0 − 2.67890i
251.2 −1.99878 − 1.15400i 0 1.66342 + 2.88113i 0.500000 + 0.866025i 0 −2.42508 − 1.05782i − 3.06234i 0 − 2.30799i
251.3 −1.94805 − 1.12471i 0 1.52994 + 2.64993i 0.500000 + 0.866025i 0 0.315547 + 2.62687i − 2.38412i 0 − 2.24942i
251.4 −1.58089 − 0.912729i 0 0.666147 + 1.15380i 0.500000 + 0.866025i 0 0.317742 − 2.62660i 1.21887i 0 − 1.82546i
251.5 −1.02342 − 0.590871i 0 −0.301744 − 0.522636i 0.500000 + 0.866025i 0 2.45705 + 0.981280i 3.07665i 0 − 1.18174i
251.6 −0.963349 − 0.556190i 0 −0.381305 − 0.660440i 0.500000 + 0.866025i 0 2.64533 + 0.0472671i 3.07307i 0 − 1.11238i
251.7 −0.552767 − 0.319140i 0 −0.796299 − 1.37923i 0.500000 + 0.866025i 0 −1.77287 − 1.96391i 2.29308i 0 − 0.638280i
251.8 0.334847 + 0.193324i 0 −0.925251 − 1.60258i 0.500000 + 0.866025i 0 1.06999 − 2.41974i − 1.48879i 0 0.386648i
251.9 1.13192 + 0.653515i 0 −0.145837 − 0.252598i 0.500000 + 0.866025i 0 −2.56224 + 0.659483i − 2.99529i 0 1.30703i
251.10 1.41561 + 0.817305i 0 0.335974 + 0.581925i 0.500000 + 0.866025i 0 2.11432 + 1.59049i − 2.17085i 0 1.63461i
251.11 2.21303 + 1.27769i 0 2.26501 + 3.92311i 0.500000 + 0.866025i 0 1.04917 − 2.42884i 6.46517i 0 2.55539i
251.12 2.29184 + 1.32320i 0 2.50170 + 4.33307i 0.500000 + 0.866025i 0 −1.33893 + 2.28195i 7.94816i 0 2.64639i
881.1 −2.32000 + 1.33945i 0 2.58825 − 4.48298i 0.500000 − 0.866025i 0 2.62997 + 0.288507i 8.50953i 0 2.67890i
881.2 −1.99878 + 1.15400i 0 1.66342 − 2.88113i 0.500000 − 0.866025i 0 −2.42508 + 1.05782i 3.06234i 0 2.30799i
881.3 −1.94805 + 1.12471i 0 1.52994 − 2.64993i 0.500000 − 0.866025i 0 0.315547 − 2.62687i 2.38412i 0 2.24942i
881.4 −1.58089 + 0.912729i 0 0.666147 − 1.15380i 0.500000 − 0.866025i 0 0.317742 + 2.62660i − 1.21887i 0 1.82546i
881.5 −1.02342 + 0.590871i 0 −0.301744 + 0.522636i 0.500000 − 0.866025i 0 2.45705 − 0.981280i − 3.07665i 0 1.18174i
881.6 −0.963349 + 0.556190i 0 −0.381305 + 0.660440i 0.500000 − 0.866025i 0 2.64533 − 0.0472671i − 3.07307i 0 1.11238i
881.7 −0.552767 + 0.319140i 0 −0.796299 + 1.37923i 0.500000 − 0.866025i 0 −1.77287 + 1.96391i − 2.29308i 0 0.638280i
881.8 0.334847 − 0.193324i 0 −0.925251 + 1.60258i 0.500000 − 0.866025i 0 1.06999 + 2.41974i 1.48879i 0 − 0.386648i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 251.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.o even 6 1 inner

Twists

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

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(945, [\chi])\):

\( T_{2}^{24} + 6 T_{2}^{23} - 3 T_{2}^{22} - 90 T_{2}^{21} - 48 T_{2}^{20} + 954 T_{2}^{19} + 1571 T_{2}^{18} + \cdots + 14161 \) Copy content Toggle raw display
\( T_{11}^{24} + 9 T_{11}^{23} - 36 T_{11}^{22} - 567 T_{11}^{21} + 1074 T_{11}^{20} + 21777 T_{11}^{19} + \cdots + 207475216 \) Copy content Toggle raw display
\( T_{13}^{24} - 3 T_{13}^{23} - 69 T_{13}^{22} + 216 T_{13}^{21} + 3150 T_{13}^{20} - 11916 T_{13}^{19} + \cdots + 10732176 \) Copy content Toggle raw display
\( T_{17}^{12} + 9 T_{17}^{11} - 57 T_{17}^{10} - 681 T_{17}^{9} + 174 T_{17}^{8} + 15678 T_{17}^{7} + \cdots + 410004 \) Copy content Toggle raw display