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Properties

Label 460.5.d.b
Level $460$
Weight $5$
Character orbit 460.d
Self dual yes
Analytic conductor $47.550$
Analytic rank $0$
Dimension $2$
CM discriminant -115
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [460,5,Mod(229,460)] 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("460.229"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(460, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 460 = 2^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 460.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,50] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(47.5501830186\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{345}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 86 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(-1 + 3\sqrt{345})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 25 q^{5} + (3 \beta + 10) q^{7} + 81 q^{9} + (11 \beta + 234) q^{17} + 529 q^{23} + 625 q^{25} + (19 \beta - 774) q^{29} + ( - 53 \beta - 470) q^{31} + (75 \beta + 250) q^{35} + ( - 85 \beta + 26) q^{37}+ \cdots + 11458 q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 50 q^{5} + 17 q^{7} + 162 q^{9} + 457 q^{17} + 1058 q^{23} + 1250 q^{25} - 1567 q^{29} - 887 q^{31} + 425 q^{35} + 137 q^{37} + 2273 q^{41} - 7324 q^{43} + 4050 q^{45} + 9315 q^{49} - 4583 q^{53}+ \cdots + 22916 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/460\mathbb{Z}\right)^\times\).

\(n\) \(231\) \(277\) \(281\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
229.1
−8.78709
9.78709
0 0 0 25.0000 0 −75.0838 0 81.0000 0
229.2 0 0 0 25.0000 0 92.0838 0 81.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
115.c odd 2 1 CM by \(\Q(\sqrt{-115}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 460.5.d.b yes 2
5.b even 2 1 460.5.d.a ✓ 2
23.b odd 2 1 460.5.d.a ✓ 2
115.c odd 2 1 CM 460.5.d.b yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
460.5.d.a ✓ 2 5.b even 2 1
460.5.d.a ✓ 2 23.b odd 2 1
460.5.d.b yes 2 1.a even 1 1 trivial
460.5.d.b yes 2 115.c odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{5}^{\mathrm{new}}(460, [\chi])\):

\( T_{3} \) Copy content Toggle raw display
\( T_{7}^{2} - 17T_{7} - 6914 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T - 25)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 17T - 6914 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 457T - 41714 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( (T - 529)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 1567 T + 333646 \) Copy content Toggle raw display
$31$ \( T^{2} + 887 T - 1983794 \) Copy content Toggle raw display
$37$ \( T^{2} - 137 T - 5603714 \) Copy content Toggle raw display
$41$ \( T^{2} - 2273 T - 3310754 \) Copy content Toggle raw display
$43$ \( (T + 3662)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 4583 T - 2667554 \) Copy content Toggle raw display
$59$ \( T^{2} - 6953 T + 11992126 \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 3343 T - 49277714 \) Copy content Toggle raw display
$71$ \( T^{2} - 9353 T + 11243566 \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 12097 T + 3962446 \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( (T - 11458)^{2} \) Copy content Toggle raw display
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