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Properties

Label 264.2.z.b
Level $264$
Weight $2$
Character orbit 264.z
Analytic conductor $2.108$
Analytic rank $0$
Dimension $48$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [264,2,Mod(19,264)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(264, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([5, 5, 0, 3])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("264.19"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 264 = 2^{3} \cdot 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 264.z (of order \(10\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [48,0,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.10805061336\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(12\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 12 q^{3} + 4 q^{4} - 12 q^{9} + 4 q^{11} + 6 q^{12} - 16 q^{14} + 20 q^{16} - 5 q^{18} + 25 q^{20} + 3 q^{22} + 20 q^{24} - 4 q^{25} + 4 q^{26} + 12 q^{27} - 25 q^{28} - 10 q^{30} - 4 q^{33} + 4 q^{36}+ \cdots - 16 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −1.40421 − 0.167908i −0.309017 − 0.951057i 1.94361 + 0.471557i −1.03968 + 1.43099i 0.274235 + 1.38737i −0.430909 + 1.32620i −2.65006 − 0.988513i −0.809017 + 0.587785i 1.70020 − 1.83484i
19.2 −1.32746 − 0.487693i −0.309017 − 0.951057i 1.52431 + 1.29479i 2.61989 − 3.60596i −0.0536155 + 1.41320i −0.177135 + 0.545166i −1.39201 − 2.46218i −0.809017 + 0.587785i −5.23640 + 3.50908i
19.3 −1.23741 + 0.684696i −0.309017 − 0.951057i 1.06238 − 1.69450i −0.409298 + 0.563350i 1.03357 + 0.965267i −0.510340 + 1.57067i −0.154385 + 2.82421i −0.809017 + 0.587785i 0.120747 − 0.977341i
19.4 −0.565882 + 1.29606i −0.309017 − 0.951057i −1.35955 − 1.46684i −2.02566 + 2.78809i 1.40750 + 0.137681i 1.45266 − 4.47083i 2.67046 − 0.932009i −0.809017 + 0.587785i −2.46725 − 4.20312i
19.5 −0.0536155 − 1.41320i −0.309017 − 0.951057i −1.99425 + 0.151538i −2.61989 + 3.60596i −1.32746 + 0.487693i 0.177135 − 0.545166i 0.321076 + 2.81014i −0.809017 + 0.587785i 5.23640 + 3.50908i
19.6 0.0942405 + 1.41107i −0.309017 − 0.951057i −1.98224 + 0.265960i −0.509926 + 0.701853i 1.31289 − 0.525673i −0.919860 + 2.83104i −0.562095 − 2.77201i −0.809017 + 0.587785i −1.03842 − 0.653398i
19.7 0.274235 − 1.38737i −0.309017 − 0.951057i −1.84959 − 0.760930i 1.03968 − 1.43099i −1.40421 + 0.167908i 0.430909 − 1.32620i −1.56291 + 2.35739i −0.809017 + 0.587785i −1.70020 − 1.83484i
19.8 0.445083 + 1.34235i −0.309017 − 0.951057i −1.60380 + 1.19491i 1.48262 − 2.04066i 1.13911 − 0.838108i 1.07225 − 3.30005i −2.31782 − 1.62103i −0.809017 + 0.587785i 3.39916 + 1.08194i
19.9 1.03357 − 0.965267i −0.309017 − 0.951057i 0.136519 − 1.99534i 0.409298 − 0.563350i −1.23741 − 0.684696i 0.510340 − 1.57067i −1.78493 − 2.19409i −0.809017 + 0.587785i −0.120747 − 0.977341i
19.10 1.13911 + 0.838108i −0.309017 − 0.951057i 0.595151 + 1.90940i −1.48262 + 2.04066i 0.445083 − 1.34235i −1.07225 + 3.30005i −0.922336 + 2.67382i −0.809017 + 0.587785i −3.39916 + 1.08194i
19.11 1.31289 + 0.525673i −0.309017 − 0.951057i 1.44734 + 1.38030i 0.509926 − 0.701853i 0.0942405 − 1.41107i 0.919860 − 2.83104i 1.17460 + 2.57300i −0.809017 + 0.587785i 1.03842 − 0.653398i
19.12 1.40750 − 0.137681i −0.309017 − 0.951057i 1.96209 − 0.387571i 2.02566 − 2.78809i −0.565882 − 1.29606i −1.45266 + 4.47083i 2.70827 − 0.815646i −0.809017 + 0.587785i 2.46725 − 4.20312i
139.1 −1.40421 + 0.167908i −0.309017 + 0.951057i 1.94361 − 0.471557i −1.03968 − 1.43099i 0.274235 − 1.38737i −0.430909 − 1.32620i −2.65006 + 0.988513i −0.809017 − 0.587785i 1.70020 + 1.83484i
139.2 −1.32746 + 0.487693i −0.309017 + 0.951057i 1.52431 − 1.29479i 2.61989 + 3.60596i −0.0536155 − 1.41320i −0.177135 − 0.545166i −1.39201 + 2.46218i −0.809017 − 0.587785i −5.23640 − 3.50908i
139.3 −1.23741 − 0.684696i −0.309017 + 0.951057i 1.06238 + 1.69450i −0.409298 − 0.563350i 1.03357 − 0.965267i −0.510340 − 1.57067i −0.154385 − 2.82421i −0.809017 − 0.587785i 0.120747 + 0.977341i
139.4 −0.565882 − 1.29606i −0.309017 + 0.951057i −1.35955 + 1.46684i −2.02566 − 2.78809i 1.40750 − 0.137681i 1.45266 + 4.47083i 2.67046 + 0.932009i −0.809017 − 0.587785i −2.46725 + 4.20312i
139.5 −0.0536155 + 1.41320i −0.309017 + 0.951057i −1.99425 − 0.151538i −2.61989 − 3.60596i −1.32746 − 0.487693i 0.177135 + 0.545166i 0.321076 − 2.81014i −0.809017 − 0.587785i 5.23640 − 3.50908i
139.6 0.0942405 − 1.41107i −0.309017 + 0.951057i −1.98224 − 0.265960i −0.509926 − 0.701853i 1.31289 + 0.525673i −0.919860 − 2.83104i −0.562095 + 2.77201i −0.809017 − 0.587785i −1.03842 + 0.653398i
139.7 0.274235 + 1.38737i −0.309017 + 0.951057i −1.84959 + 0.760930i 1.03968 + 1.43099i −1.40421 − 0.167908i 0.430909 + 1.32620i −1.56291 − 2.35739i −0.809017 − 0.587785i −1.70020 + 1.83484i
139.8 0.445083 − 1.34235i −0.309017 + 0.951057i −1.60380 − 1.19491i 1.48262 + 2.04066i 1.13911 + 0.838108i 1.07225 + 3.30005i −2.31782 + 1.62103i −0.809017 − 0.587785i 3.39916 − 1.08194i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.12
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
11.d odd 10 1 inner
88.k even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 264.2.z.b ✓ 48
3.b odd 2 1 792.2.bp.d 48
4.b odd 2 1 1056.2.bp.a 48
8.b even 2 1 1056.2.bp.a 48
8.d odd 2 1 inner 264.2.z.b ✓ 48
11.d odd 10 1 inner 264.2.z.b ✓ 48
24.f even 2 1 792.2.bp.d 48
33.f even 10 1 792.2.bp.d 48
44.g even 10 1 1056.2.bp.a 48
88.k even 10 1 inner 264.2.z.b ✓ 48
88.p odd 10 1 1056.2.bp.a 48
264.r odd 10 1 792.2.bp.d 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
264.2.z.b ✓ 48 1.a even 1 1 trivial
264.2.z.b ✓ 48 8.d odd 2 1 inner
264.2.z.b ✓ 48 11.d odd 10 1 inner
264.2.z.b ✓ 48 88.k even 10 1 inner
792.2.bp.d 48 3.b odd 2 1
792.2.bp.d 48 24.f even 2 1
792.2.bp.d 48 33.f even 10 1
792.2.bp.d 48 264.r odd 10 1
1056.2.bp.a 48 4.b odd 2 1
1056.2.bp.a 48 8.b even 2 1
1056.2.bp.a 48 44.g even 10 1
1056.2.bp.a 48 88.p odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{48} - 28 T_{5}^{46} + 778 T_{5}^{44} - 19176 T_{5}^{42} + 368015 T_{5}^{40} + \cdots + 4308219140625 \) acting on \(S_{2}^{\mathrm{new}}(264, [\chi])\). Copy content Toggle raw display