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Properties

Label 945.2.m.b
Level $945$
Weight $2$
Character orbit 945.m
Analytic conductor $7.546$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [945,2,Mod(323,945)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(945, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("945.323");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 945 = 3^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 945.m (of order \(4\), degree \(2\), 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: \(48\)
Relative dimension: \(24\) over \(\Q(i)\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 16 q^{10} + 32 q^{13} - 32 q^{16} - 8 q^{22} - 8 q^{25} - 48 q^{31} + 40 q^{37} - 64 q^{40} + 16 q^{43} + 64 q^{46} - 96 q^{52} + 80 q^{55} + 32 q^{58} + 64 q^{61} + 48 q^{67} + 8 q^{70} - 32 q^{73} - 80 q^{76} + 32 q^{82} + 24 q^{85} - 120 q^{88}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
323.1 −1.90940 − 1.90940i 0 5.29158i −0.795853 − 2.08965i 0 0.707107 − 0.707107i 6.28493 − 6.28493i 0 −2.47036 + 5.50956i
323.2 −1.80190 − 1.80190i 0 4.49368i −0.148581 + 2.23113i 0 −0.707107 + 0.707107i 4.49335 − 4.49335i 0 4.28799 − 3.75254i
323.3 −1.49816 − 1.49816i 0 2.48894i −2.18340 + 0.482438i 0 0.707107 − 0.707107i 0.732509 − 0.732509i 0 3.99385 + 2.54831i
323.4 −1.32724 − 1.32724i 0 1.52313i −2.23324 − 0.112406i 0 −0.707107 + 0.707107i −0.632923 + 0.632923i 0 2.81486 + 3.11323i
323.5 −1.24880 − 1.24880i 0 1.11902i 0.657873 − 2.13710i 0 −0.707107 + 0.707107i −1.10017 + 1.10017i 0 −3.49037 + 1.84727i
323.6 −1.14990 − 1.14990i 0 0.644558i 2.23139 − 0.144494i 0 0.707107 − 0.707107i −1.55863 + 1.55863i 0 −2.73204 − 2.39974i
323.7 −1.09957 − 1.09957i 0 0.418095i 0.0740560 + 2.23484i 0 −0.707107 + 0.707107i −1.73941 + 1.73941i 0 2.37593 − 2.53879i
323.8 −0.721086 − 0.721086i 0 − 0.960071i 0.500303 − 2.17938i 0 0.707107 − 0.707107i −2.13446 + 2.13446i 0 −1.93228 + 1.21076i
323.9 −0.506632 − 0.506632i 0 − 1.48665i −1.31810 + 1.80627i 0 0.707107 − 0.707107i −1.76645 + 1.76645i 0 1.58291 − 0.247320i
323.10 −0.465126 − 0.465126i 0 − 1.56732i 1.83450 + 1.27851i 0 −0.707107 + 0.707107i −1.65925 + 1.65925i 0 −0.258605 − 1.44795i
323.11 −0.104020 − 0.104020i 0 − 1.97836i −1.68513 − 1.46981i 0 0.707107 − 0.707107i −0.413828 + 0.413828i 0 0.0223968 + 0.328176i
323.12 −0.0818359 − 0.0818359i 0 − 1.98661i 2.23154 − 0.142242i 0 −0.707107 + 0.707107i −0.326248 + 0.326248i 0 −0.194261 − 0.170980i
323.13 0.0818359 + 0.0818359i 0 − 1.98661i −2.23154 + 0.142242i 0 −0.707107 + 0.707107i 0.326248 − 0.326248i 0 −0.194261 − 0.170980i
323.14 0.104020 + 0.104020i 0 − 1.97836i 1.68513 + 1.46981i 0 0.707107 − 0.707107i 0.413828 − 0.413828i 0 0.0223968 + 0.328176i
323.15 0.465126 + 0.465126i 0 − 1.56732i −1.83450 − 1.27851i 0 −0.707107 + 0.707107i 1.65925 − 1.65925i 0 −0.258605 − 1.44795i
323.16 0.506632 + 0.506632i 0 − 1.48665i 1.31810 − 1.80627i 0 0.707107 − 0.707107i 1.76645 − 1.76645i 0 1.58291 − 0.247320i
323.17 0.721086 + 0.721086i 0 − 0.960071i −0.500303 + 2.17938i 0 0.707107 − 0.707107i 2.13446 − 2.13446i 0 −1.93228 + 1.21076i
323.18 1.09957 + 1.09957i 0 0.418095i −0.0740560 − 2.23484i 0 −0.707107 + 0.707107i 1.73941 − 1.73941i 0 2.37593 − 2.53879i
323.19 1.14990 + 1.14990i 0 0.644558i −2.23139 + 0.144494i 0 0.707107 − 0.707107i 1.55863 − 1.55863i 0 −2.73204 − 2.39974i
323.20 1.24880 + 1.24880i 0 1.11902i −0.657873 + 2.13710i 0 −0.707107 + 0.707107i 1.10017 − 1.10017i 0 −3.49037 + 1.84727i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 323.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 945.2.m.b ✓ 48
3.b odd 2 1 inner 945.2.m.b ✓ 48
5.c odd 4 1 inner 945.2.m.b ✓ 48
15.e even 4 1 inner 945.2.m.b ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
945.2.m.b ✓ 48 1.a even 1 1 trivial
945.2.m.b ✓ 48 3.b odd 2 1 inner
945.2.m.b ✓ 48 5.c odd 4 1 inner
945.2.m.b ✓ 48 15.e even 4 1 inner