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Properties

Label 462.2.y.b
Level $462$
Weight $2$
Character orbit 462.y
Analytic conductor $3.689$
Analytic rank $0$
Dimension $24$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [462,2,Mod(25,462)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("462.25"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(462, base_ring=CyclotomicField(30)) chi = DirichletCharacter(H, H._module([0, 20, 24])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 462 = 2 \cdot 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 462.y (of order \(15\), degree \(8\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.68908857338\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(3\) over \(\Q(\zeta_{15})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 3 q^{2} + 3 q^{3} + 3 q^{4} + 5 q^{5} - 6 q^{6} - 5 q^{7} - 6 q^{8} + 3 q^{9} + 10 q^{10} - 2 q^{11} - 12 q^{12} + 10 q^{13} + 7 q^{14} + 20 q^{15} + 3 q^{16} + 8 q^{17} + 3 q^{18} + 7 q^{19} - 10 q^{20}+ \cdots - 6 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1 0.913545 + 0.406737i −0.978148 − 0.207912i 0.669131 + 0.743145i −0.225801 + 2.14835i −0.809017 − 0.587785i −2.17096 + 1.51227i 0.309017 + 0.951057i 0.913545 + 0.406737i −1.08009 + 1.87078i
25.2 0.913545 + 0.406737i −0.978148 − 0.207912i 0.669131 + 0.743145i −0.122006 + 1.16081i −0.809017 − 0.587785i 2.38913 − 1.13668i 0.309017 + 0.951057i 0.913545 + 0.406737i −0.583600 + 1.01083i
25.3 0.913545 + 0.406737i −0.978148 − 0.207912i 0.669131 + 0.743145i 0.258529 − 2.45974i −0.809017 − 0.587785i −0.613647 + 2.57360i 0.309017 + 0.951057i 0.913545 + 0.406737i 1.23664 − 2.14193i
37.1 0.913545 − 0.406737i −0.978148 + 0.207912i 0.669131 − 0.743145i −0.225801 − 2.14835i −0.809017 + 0.587785i −2.17096 − 1.51227i 0.309017 − 0.951057i 0.913545 − 0.406737i −1.08009 − 1.87078i
37.2 0.913545 − 0.406737i −0.978148 + 0.207912i 0.669131 − 0.743145i −0.122006 − 1.16081i −0.809017 + 0.587785i 2.38913 + 1.13668i 0.309017 − 0.951057i 0.913545 − 0.406737i −0.583600 − 1.01083i
37.3 0.913545 − 0.406737i −0.978148 + 0.207912i 0.669131 − 0.743145i 0.258529 + 2.45974i −0.809017 + 0.587785i −0.613647 − 2.57360i 0.309017 − 0.951057i 0.913545 − 0.406737i 1.23664 + 2.14193i
163.1 0.669131 + 0.743145i 0.913545 + 0.406737i −0.104528 + 0.994522i 0.282912 + 0.0601348i 0.309017 + 0.951057i 1.34567 − 2.27797i −0.809017 + 0.587785i 0.669131 + 0.743145i 0.144616 + 0.250483i
163.2 0.669131 + 0.743145i 0.913545 + 0.406737i −0.104528 + 0.994522i 1.46774 + 0.311978i 0.309017 + 0.951057i 1.41096 + 2.23812i −0.809017 + 0.587785i 0.669131 + 0.743145i 0.750265 + 1.29950i
163.3 0.669131 + 0.743145i 0.913545 + 0.406737i −0.104528 + 0.994522i 3.97552 + 0.845024i 0.309017 + 0.951057i −2.27849 − 1.34480i −0.809017 + 0.587785i 0.669131 + 0.743145i 2.03217 + 3.51982i
235.1 −0.104528 + 0.994522i 0.669131 + 0.743145i −0.978148 − 0.207912i −2.25946 − 1.00598i −0.809017 + 0.587785i −1.01628 − 2.44278i 0.309017 − 0.951057i −0.104528 + 0.994522i 1.23664 − 2.14193i
235.2 −0.104528 + 0.994522i 0.669131 + 0.743145i −0.978148 − 0.207912i 1.06629 + 0.474743i −0.809017 + 0.587785i −1.26472 + 2.32389i 0.309017 − 0.951057i −0.104528 + 0.994522i −0.583600 + 1.01083i
235.3 −0.104528 + 0.994522i 0.669131 + 0.743145i −0.978148 − 0.207912i 1.97343 + 0.878627i −0.809017 + 0.587785i 0.867452 − 2.49951i 0.309017 − 0.951057i −0.104528 + 0.994522i −1.08009 + 1.87078i
247.1 −0.978148 − 0.207912i −0.104528 + 0.994522i 0.913545 + 0.406737i −2.71957 − 3.02039i 0.309017 − 0.951057i −1.98308 + 1.75140i −0.809017 − 0.587785i −0.978148 − 0.207912i 2.03217 + 3.51982i
247.2 −0.978148 − 0.207912i −0.104528 + 0.994522i 0.913545 + 0.406737i −1.00405 − 1.11511i 0.309017 − 0.951057i 2.56459 − 0.650286i −0.809017 − 0.587785i −0.978148 − 0.207912i 0.750265 + 1.29950i
247.3 −0.978148 − 0.207912i −0.104528 + 0.994522i 0.913545 + 0.406737i −0.193534 − 0.214942i 0.309017 − 0.951057i −1.75065 − 1.98374i −0.809017 − 0.587785i −0.978148 − 0.207912i 0.144616 + 0.250483i
289.1 −0.104528 − 0.994522i 0.669131 − 0.743145i −0.978148 + 0.207912i −2.25946 + 1.00598i −0.809017 − 0.587785i −1.01628 + 2.44278i 0.309017 + 0.951057i −0.104528 − 0.994522i 1.23664 + 2.14193i
289.2 −0.104528 − 0.994522i 0.669131 − 0.743145i −0.978148 + 0.207912i 1.06629 − 0.474743i −0.809017 − 0.587785i −1.26472 − 2.32389i 0.309017 + 0.951057i −0.104528 − 0.994522i −0.583600 − 1.01083i
289.3 −0.104528 − 0.994522i 0.669131 − 0.743145i −0.978148 + 0.207912i 1.97343 − 0.878627i −0.809017 − 0.587785i 0.867452 + 2.49951i 0.309017 + 0.951057i −0.104528 − 0.994522i −1.08009 − 1.87078i
361.1 −0.978148 + 0.207912i −0.104528 − 0.994522i 0.913545 − 0.406737i −2.71957 + 3.02039i 0.309017 + 0.951057i −1.98308 − 1.75140i −0.809017 + 0.587785i −0.978148 + 0.207912i 2.03217 − 3.51982i
361.2 −0.978148 + 0.207912i −0.104528 − 0.994522i 0.913545 − 0.406737i −1.00405 + 1.11511i 0.309017 + 0.951057i 2.56459 + 0.650286i −0.809017 + 0.587785i −0.978148 + 0.207912i 0.750265 − 1.29950i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.3
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
11.c even 5 1 inner
77.m even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 462.2.y.b ✓ 24
7.c even 3 1 inner 462.2.y.b ✓ 24
11.c even 5 1 inner 462.2.y.b ✓ 24
77.m even 15 1 inner 462.2.y.b ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.y.b ✓ 24 1.a even 1 1 trivial
462.2.y.b ✓ 24 7.c even 3 1 inner
462.2.y.b ✓ 24 11.c even 5 1 inner
462.2.y.b ✓ 24 77.m even 15 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{24} - 5 T_{5}^{23} + T_{5}^{22} - 38 T_{5}^{21} + 340 T_{5}^{20} - 747 T_{5}^{19} + 1932 T_{5}^{18} + \cdots + 14641 \) acting on \(S_{2}^{\mathrm{new}}(462, [\chi])\). Copy content Toggle raw display