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Properties

Label 462.2.y.d
Level $462$
Weight $2$
Character orbit 462.y
Analytic conductor $3.689$
Analytic rank $0$
Dimension $40$
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 = [40,5] 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: \(40\)
Relative dimension: \(5\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 5 q^{2} - 5 q^{3} + 5 q^{4} - 3 q^{5} + 10 q^{6} - 7 q^{7} - 10 q^{8} + 5 q^{9} + 12 q^{10} + 2 q^{11} + 20 q^{12} + 10 q^{13} + 5 q^{14} - 6 q^{15} + 5 q^{16} + 5 q^{18} - 13 q^{19} + 6 q^{20} - 2 q^{21}+ \cdots - 4 q^{99}+O(q^{100}) \) 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 −0.305851 + 2.90998i 0.809017 + 0.587785i −2.53596 − 0.754243i 0.309017 + 0.951057i 0.913545 + 0.406737i −1.46300 + 2.53399i
25.2 0.913545 + 0.406737i 0.978148 + 0.207912i 0.669131 + 0.743145i −0.0415637 + 0.395452i 0.809017 + 0.587785i −0.168521 + 2.64038i 0.309017 + 0.951057i 0.913545 + 0.406737i −0.198815 + 0.344358i
25.3 0.913545 + 0.406737i 0.978148 + 0.207912i 0.669131 + 0.743145i 0.0926977 − 0.881959i 0.809017 + 0.587785i 1.77880 − 1.95854i 0.309017 + 0.951057i 0.913545 + 0.406737i 0.443409 − 0.768006i
25.4 0.913545 + 0.406737i 0.978148 + 0.207912i 0.669131 + 0.743145i 0.111151 − 1.05753i 0.809017 + 0.587785i 2.10806 + 1.59878i 0.309017 + 0.951057i 0.913545 + 0.406737i 0.531680 − 0.920896i
25.5 0.913545 + 0.406737i 0.978148 + 0.207912i 0.669131 + 0.743145i 0.457151 − 4.34950i 0.809017 + 0.587785i −2.04249 − 1.68174i 0.309017 + 0.951057i 0.913545 + 0.406737i 2.18673 − 3.78753i
37.1 0.913545 − 0.406737i 0.978148 − 0.207912i 0.669131 − 0.743145i −0.305851 − 2.90998i 0.809017 − 0.587785i −2.53596 + 0.754243i 0.309017 − 0.951057i 0.913545 − 0.406737i −1.46300 − 2.53399i
37.2 0.913545 − 0.406737i 0.978148 − 0.207912i 0.669131 − 0.743145i −0.0415637 − 0.395452i 0.809017 − 0.587785i −0.168521 − 2.64038i 0.309017 − 0.951057i 0.913545 − 0.406737i −0.198815 − 0.344358i
37.3 0.913545 − 0.406737i 0.978148 − 0.207912i 0.669131 − 0.743145i 0.0926977 + 0.881959i 0.809017 − 0.587785i 1.77880 + 1.95854i 0.309017 − 0.951057i 0.913545 − 0.406737i 0.443409 + 0.768006i
37.4 0.913545 − 0.406737i 0.978148 − 0.207912i 0.669131 − 0.743145i 0.111151 + 1.05753i 0.809017 − 0.587785i 2.10806 − 1.59878i 0.309017 − 0.951057i 0.913545 − 0.406737i 0.531680 + 0.920896i
37.5 0.913545 − 0.406737i 0.978148 − 0.207912i 0.669131 − 0.743145i 0.457151 + 4.34950i 0.809017 − 0.587785i −2.04249 + 1.68174i 0.309017 − 0.951057i 0.913545 − 0.406737i 2.18673 + 3.78753i
163.1 0.669131 + 0.743145i −0.913545 − 0.406737i −0.104528 + 0.994522i −2.21083 − 0.469926i −0.309017 − 0.951057i 2.39247 + 1.12965i −0.809017 + 0.587785i 0.669131 + 0.743145i −1.13011 − 1.95741i
163.2 0.669131 + 0.743145i −0.913545 − 0.406737i −0.104528 + 0.994522i −1.55422 − 0.330360i −0.309017 − 0.951057i −1.63560 − 2.07962i −0.809017 + 0.587785i 0.669131 + 0.743145i −0.794473 − 1.37607i
163.3 0.669131 + 0.743145i −0.913545 − 0.406737i −0.104528 + 0.994522i −0.408781 − 0.0868890i −0.309017 − 0.951057i −2.24058 − 1.40705i −0.809017 + 0.587785i 0.669131 + 0.743145i −0.208957 − 0.361923i
163.4 0.669131 + 0.743145i −0.913545 − 0.406737i −0.104528 + 0.994522i 3.44957 + 0.733230i −0.309017 − 0.951057i −2.03992 + 1.68486i −0.809017 + 0.587785i 0.669131 + 0.743145i 1.76332 + 3.05416i
163.5 0.669131 + 0.743145i −0.913545 − 0.406737i −0.104528 + 0.994522i 3.65870 + 0.777681i −0.309017 − 0.951057i 2.31914 − 1.27341i −0.809017 + 0.587785i 0.669131 + 0.743145i 1.87022 + 3.23932i
235.1 −0.104528 + 0.994522i −0.669131 − 0.743145i −0.978148 − 0.207912i −3.99535 − 1.77885i 0.809017 − 0.587785i 2.64091 + 0.160013i 0.309017 − 0.951057i −0.104528 + 0.994522i 2.18673 − 3.78753i
235.2 −0.104528 + 0.994522i −0.669131 − 0.743145i −0.978148 − 0.207912i −0.971427 − 0.432507i 0.809017 − 0.587785i −2.64519 − 0.0543560i 0.309017 − 0.951057i −0.104528 + 0.994522i 0.531680 − 0.920896i
235.3 −0.104528 + 0.994522i −0.669131 − 0.743145i −0.978148 − 0.207912i −0.810148 − 0.360701i 0.809017 − 0.587785i −0.287883 + 2.63004i 0.309017 − 0.951057i −0.104528 + 0.994522i 0.443409 − 0.768006i
235.4 −0.104528 + 0.994522i −0.669131 − 0.743145i −0.978148 − 0.207912i 0.363254 + 0.161731i 0.809017 − 0.587785i −1.41564 − 2.23517i 0.309017 − 0.951057i −0.104528 + 0.994522i −0.198815 + 0.344358i
235.5 −0.104528 + 0.994522i −0.669131 − 0.743145i −0.978148 − 0.207912i 2.67304 + 1.19011i 0.809017 − 0.587785i 2.49497 − 0.880407i 0.309017 − 0.951057i −0.104528 + 0.994522i −1.46300 + 2.53399i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.5
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.d ✓ 40
7.c even 3 1 inner 462.2.y.d ✓ 40
11.c even 5 1 inner 462.2.y.d ✓ 40
77.m even 15 1 inner 462.2.y.d ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.y.d ✓ 40 1.a even 1 1 trivial
462.2.y.d ✓ 40 7.c even 3 1 inner
462.2.y.d ✓ 40 11.c even 5 1 inner
462.2.y.d ✓ 40 77.m even 15 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{40} + 3 T_{5}^{39} - 14 T_{5}^{38} - 113 T_{5}^{37} - 214 T_{5}^{36} - 96 T_{5}^{35} + \cdots + 81450625 \) acting on \(S_{2}^{\mathrm{new}}(462, [\chi])\). Copy content Toggle raw display