Properties

Label 32.7.d.a
Level $32$
Weight $7$
Character orbit 32.d
Self dual yes
Analytic conductor $7.362$
Analytic rank $0$
Dimension $1$
CM discriminant -8
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 32 = 2^{5} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 32.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(7.36173067584\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 8)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

\(f(q)\) \(=\) \( q - 46 q^{3} + 1387 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 46 q^{3} + 1387 q^{9} + 2338 q^{11} - 1726 q^{17} + 2482 q^{19} + 15625 q^{25} - 30268 q^{27} - 107548 q^{33} + 134642 q^{41} + 74914 q^{43} + 117649 q^{49} + 79396 q^{51} - 114172 q^{57} - 304958 q^{59} + 596626 q^{67} - 593134 q^{73} - 718750 q^{75} + 381205 q^{81} - 678926 q^{83} - 357262 q^{89} + 1822754 q^{97} + 3242806 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/32\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(31\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
15.1
0
0 −46.0000 0 0 0 0 0 1387.00 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
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 32.7.d.a 1
3.b odd 2 1 288.7.b.a 1
4.b odd 2 1 8.7.d.a 1
8.b even 2 1 8.7.d.a 1
8.d odd 2 1 CM 32.7.d.a 1
12.b even 2 1 72.7.b.a 1
16.e even 4 2 256.7.c.d 2
16.f odd 4 2 256.7.c.d 2
24.f even 2 1 288.7.b.a 1
24.h odd 2 1 72.7.b.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.7.d.a 1 4.b odd 2 1
8.7.d.a 1 8.b even 2 1
32.7.d.a 1 1.a even 1 1 trivial
32.7.d.a 1 8.d odd 2 1 CM
72.7.b.a 1 12.b even 2 1
72.7.b.a 1 24.h odd 2 1
256.7.c.d 2 16.e even 4 2
256.7.c.d 2 16.f odd 4 2
288.7.b.a 1 3.b odd 2 1
288.7.b.a 1 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 46 \) acting on \(S_{7}^{\mathrm{new}}(32, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 46 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T - 2338 \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T + 1726 \) Copy content Toggle raw display
$19$ \( T - 2482 \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T \) Copy content Toggle raw display
$31$ \( T \) Copy content Toggle raw display
$37$ \( T \) Copy content Toggle raw display
$41$ \( T - 134642 \) Copy content Toggle raw display
$43$ \( T - 74914 \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T \) Copy content Toggle raw display
$59$ \( T + 304958 \) Copy content Toggle raw display
$61$ \( T \) Copy content Toggle raw display
$67$ \( T - 596626 \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T + 593134 \) Copy content Toggle raw display
$79$ \( T \) Copy content Toggle raw display
$83$ \( T + 678926 \) Copy content Toggle raw display
$89$ \( T + 357262 \) Copy content Toggle raw display
$97$ \( T - 1822754 \) Copy content Toggle raw display
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