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Properties

Label 456.2.p.b
Level $456$
Weight $2$
Character orbit 456.p
Analytic conductor $3.641$
Analytic rank $0$
Dimension $64$
Inner twists $8$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [456,2,Mod(341,456)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(456, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1, 1])) 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("456.341"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 456 = 2^{3} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 456.p (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [64] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.64117833217\)
Analytic rank: \(0\)
Dimension: \(64\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 64 q + 16 q^{4} + 2 q^{6} - 8 q^{7} - 4 q^{9} - 56 q^{16} - 2 q^{24} + 112 q^{25} - 20 q^{28} - 38 q^{36} - 4 q^{39} + 14 q^{42} - 24 q^{49} + 80 q^{54} - 48 q^{55} - 12 q^{57} + 12 q^{58} - 92 q^{63} + 64 q^{64}+ \cdots - 6 q^{96}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
341.1 −1.39926 − 0.205108i −1.01233 + 1.40541i 1.91586 + 0.573998i −1.38187 1.70478 − 1.75890i 1.61031 −2.56306 − 1.19613i −0.950371 − 2.84549i 1.93360 + 0.283433i
341.2 −1.39926 − 0.205108i 1.01233 + 1.40541i 1.91586 + 0.573998i 1.38187 −1.12825 − 2.17418i 1.61031 −2.56306 − 1.19613i −0.950371 + 2.84549i −1.93360 − 0.283433i
341.3 −1.39926 + 0.205108i −1.01233 − 1.40541i 1.91586 − 0.573998i −1.38187 1.70478 + 1.75890i 1.61031 −2.56306 + 1.19613i −0.950371 + 2.84549i 1.93360 − 0.283433i
341.4 −1.39926 + 0.205108i 1.01233 − 1.40541i 1.91586 − 0.573998i 1.38187 −1.12825 + 2.17418i 1.61031 −2.56306 + 1.19613i −0.950371 − 2.84549i −1.93360 + 0.283433i
341.5 −1.34810 − 0.427337i −0.346436 − 1.69705i 1.63477 + 1.15219i 3.35151 −0.258182 + 2.43585i −3.22924 −1.71146 − 2.25187i −2.75996 + 1.17584i −4.51819 − 1.43223i
341.6 −1.34810 − 0.427337i 0.346436 − 1.69705i 1.63477 + 1.15219i −3.35151 −1.19224 + 2.13976i −3.22924 −1.71146 − 2.25187i −2.75996 − 1.17584i 4.51819 + 1.43223i
341.7 −1.34810 + 0.427337i −0.346436 + 1.69705i 1.63477 − 1.15219i 3.35151 −0.258182 − 2.43585i −3.22924 −1.71146 + 2.25187i −2.75996 − 1.17584i −4.51819 + 1.43223i
341.8 −1.34810 + 0.427337i 0.346436 + 1.69705i 1.63477 − 1.15219i −3.35151 −1.19224 − 2.13976i −3.22924 −1.71146 + 2.25187i −2.75996 + 1.17584i 4.51819 − 1.43223i
341.9 −1.24348 − 0.673618i −1.47930 + 0.900922i 1.09248 + 1.67526i 0.886774 2.44636 − 0.123791i −3.55534 −0.229986 − 2.81906i 1.37668 − 2.66547i −1.10268 − 0.597347i
341.10 −1.24348 − 0.673618i 1.47930 + 0.900922i 1.09248 + 1.67526i −0.886774 −1.23260 − 2.11676i −3.55534 −0.229986 − 2.81906i 1.37668 + 2.66547i 1.10268 + 0.597347i
341.11 −1.24348 + 0.673618i −1.47930 − 0.900922i 1.09248 − 1.67526i 0.886774 2.44636 + 0.123791i −3.55534 −0.229986 + 2.81906i 1.37668 + 2.66547i −1.10268 + 0.597347i
341.12 −1.24348 + 0.673618i 1.47930 − 0.900922i 1.09248 − 1.67526i −0.886774 −1.23260 + 2.11676i −3.55534 −0.229986 + 2.81906i 1.37668 − 2.66547i 1.10268 − 0.597347i
341.13 −1.20225 − 0.744704i −1.60049 − 0.662144i 0.890831 + 1.79065i −2.20279 1.43109 + 1.98796i 1.65376 0.262497 − 2.81622i 2.12313 + 2.11951i 2.64831 + 1.64043i
341.14 −1.20225 − 0.744704i 1.60049 − 0.662144i 0.890831 + 1.79065i 2.20279 −2.41730 − 0.395826i 1.65376 0.262497 − 2.81622i 2.12313 − 2.11951i −2.64831 − 1.64043i
341.15 −1.20225 + 0.744704i −1.60049 + 0.662144i 0.890831 − 1.79065i −2.20279 1.43109 − 1.98796i 1.65376 0.262497 + 2.81622i 2.12313 − 2.11951i 2.64831 − 1.64043i
341.16 −1.20225 + 0.744704i 1.60049 + 0.662144i 0.890831 − 1.79065i 2.20279 −2.41730 + 0.395826i 1.65376 0.262497 + 2.81622i 2.12313 + 2.11951i −2.64831 + 1.64043i
341.17 −0.968690 − 1.03036i −0.749561 + 1.56146i −0.123280 + 1.99620i 3.14901 2.33496 − 0.740254i 4.20433 2.17622 − 1.80667i −1.87632 − 2.34082i −3.05041 − 3.24461i
341.18 −0.968690 − 1.03036i 0.749561 + 1.56146i −0.123280 + 1.99620i −3.14901 0.882773 − 2.28489i 4.20433 2.17622 − 1.80667i −1.87632 + 2.34082i 3.05041 + 3.24461i
341.19 −0.968690 + 1.03036i −0.749561 − 1.56146i −0.123280 − 1.99620i 3.14901 2.33496 + 0.740254i 4.20433 2.17622 + 1.80667i −1.87632 + 2.34082i −3.05041 + 3.24461i
341.20 −0.968690 + 1.03036i 0.749561 − 1.56146i −0.123280 − 1.99620i −3.14901 0.882773 + 2.28489i 4.20433 2.17622 + 1.80667i −1.87632 − 2.34082i 3.05041 − 3.24461i
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 341.64
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.b even 2 1 inner
19.b odd 2 1 inner
24.h odd 2 1 inner
57.d even 2 1 inner
152.g odd 2 1 inner
456.p even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 456.2.p.b ✓ 64
3.b odd 2 1 inner 456.2.p.b ✓ 64
4.b odd 2 1 1824.2.p.b 64
8.b even 2 1 inner 456.2.p.b ✓ 64
8.d odd 2 1 1824.2.p.b 64
12.b even 2 1 1824.2.p.b 64
19.b odd 2 1 inner 456.2.p.b ✓ 64
24.f even 2 1 1824.2.p.b 64
24.h odd 2 1 inner 456.2.p.b ✓ 64
57.d even 2 1 inner 456.2.p.b ✓ 64
76.d even 2 1 1824.2.p.b 64
152.b even 2 1 1824.2.p.b 64
152.g odd 2 1 inner 456.2.p.b ✓ 64
228.b odd 2 1 1824.2.p.b 64
456.l odd 2 1 1824.2.p.b 64
456.p even 2 1 inner 456.2.p.b ✓ 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
456.2.p.b ✓ 64 1.a even 1 1 trivial
456.2.p.b ✓ 64 3.b odd 2 1 inner
456.2.p.b ✓ 64 8.b even 2 1 inner
456.2.p.b ✓ 64 19.b odd 2 1 inner
456.2.p.b ✓ 64 24.h odd 2 1 inner
456.2.p.b ✓ 64 57.d even 2 1 inner
456.2.p.b ✓ 64 152.g odd 2 1 inner
456.2.p.b ✓ 64 456.p even 2 1 inner
1824.2.p.b 64 4.b odd 2 1
1824.2.p.b 64 8.d odd 2 1
1824.2.p.b 64 12.b even 2 1
1824.2.p.b 64 24.f even 2 1
1824.2.p.b 64 76.d even 2 1
1824.2.p.b 64 152.b even 2 1
1824.2.p.b 64 228.b odd 2 1
1824.2.p.b 64 456.l odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{16} - 54 T_{5}^{14} + 1174 T_{5}^{12} - 13344 T_{5}^{10} + 86113 T_{5}^{8} - 320322 T_{5}^{6} + \cdots + 246784 \) acting on \(S_{2}^{\mathrm{new}}(456, [\chi])\). Copy content Toggle raw display