Properties

Label 464.1.h.b
Level $464$
Weight $1$
Character orbit 464.h
Self dual yes
Analytic conductor $0.232$
Analytic rank $0$
Dimension $2$
Projective image $D_{6}$
CM discriminant -116
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 464 = 2^{4} \cdot 29 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 464.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.231566165862\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.0.53824.1

$q$-expansion

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

\(f(q)\) \(=\) \( q - \beta q^{3} - q^{5} + 2 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{3} - q^{5} + 2 q^{9} + \beta q^{11} + q^{13} + \beta q^{15} - \beta q^{27} - q^{29} + \beta q^{31} - 3 q^{33} - \beta q^{39} + \beta q^{43} - 2 q^{45} - \beta q^{47} + q^{49} + q^{53} - \beta q^{55} - q^{65} - \beta q^{79} + q^{81} + \beta q^{87} - 3 q^{93} + 2 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{5} + 4 q^{9} + 2 q^{13} - 2 q^{29} - 6 q^{33} - 4 q^{45} + 2 q^{49} + 2 q^{53} - 2 q^{65} + 2 q^{81} - 6 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/464\mathbb{Z}\right)^\times\).

\(n\) \(117\) \(175\) \(321\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
463.1
1.73205
−1.73205
0 −1.73205 0 −1.00000 0 0 0 2.00000 0
463.2 0 1.73205 0 −1.00000 0 0 0 2.00000 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
116.d odd 2 1 CM by \(\Q(\sqrt{-29}) \)
4.b odd 2 1 inner
29.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 464.1.h.b 2
4.b odd 2 1 inner 464.1.h.b 2
8.b even 2 1 1856.1.h.d 2
8.d odd 2 1 1856.1.h.d 2
29.b even 2 1 inner 464.1.h.b 2
116.d odd 2 1 CM 464.1.h.b 2
232.b odd 2 1 1856.1.h.d 2
232.g even 2 1 1856.1.h.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
464.1.h.b 2 1.a even 1 1 trivial
464.1.h.b 2 4.b odd 2 1 inner
464.1.h.b 2 29.b even 2 1 inner
464.1.h.b 2 116.d odd 2 1 CM
1856.1.h.d 2 8.b even 2 1
1856.1.h.d 2 8.d odd 2 1
1856.1.h.d 2 232.b odd 2 1
1856.1.h.d 2 232.g even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 3 \) acting on \(S_{1}^{\mathrm{new}}(464, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 3 \) Copy content Toggle raw display
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 3 \) Copy content Toggle raw display
$13$ \( (T - 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( (T + 1)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 3 \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 3 \) Copy content Toggle raw display
$47$ \( T^{2} - 3 \) Copy content Toggle raw display
$53$ \( (T - 1)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 3 \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} \) Copy content Toggle raw display
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