Properties

Label 3744.2.m.f
Level $3744$
Weight $2$
Character orbit 3744.m
Analytic conductor $29.896$
Analytic rank $0$
Dimension $8$
CM discriminant -39
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 3744 = 2^{5} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3744.m (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(29.8959905168\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.151613669376.21
Defining polynomial: \( x^{8} + 5x^{4} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{41}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 936)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{5}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{5} + (\beta_{6} - \beta_{2}) q^{11} - \beta_1 q^{13} + ( - 2 \beta_{3} + 5) q^{25} + ( - 2 \beta_{5} - \beta_{4}) q^{41} + \beta_{7} q^{43} + (\beta_{5} - 3 \beta_{4}) q^{47} + 7 q^{49} + ( - \beta_{3} - 8) q^{55} + (\beta_{6} + 3 \beta_{2}) q^{59} + 2 \beta_1 q^{61} + ( - 2 \beta_{5} + 3 \beta_{4}) q^{65} + ( - 3 \beta_{5} + \beta_{4}) q^{71} - 3 \beta_{3} q^{79} + (3 \beta_{6} + \beta_{2}) q^{83} + ( - 2 \beta_{5} - 3 \beta_{4}) q^{89}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 40 q^{25} + 56 q^{49} - 64 q^{55}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 5x^{4} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{6} + 9\nu^{2} ) / 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} + 4\nu^{5} - 3\nu^{3} + 12\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{6} - \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} + 4\nu^{5} + 3\nu^{3} + 12\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} + 5\nu^{3} + 8\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{7} - 5\nu^{3} + 8\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( 4\nu^{4} + 10 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{6} + \beta_{5} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 2\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{6} + \beta_{5} + 2\beta_{4} - 2\beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{7} - 10 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -3\beta_{6} - 3\beta_{5} + 4\beta_{4} + 4\beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -9\beta_{3} - 2\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -3\beta_{6} + 3\beta_{5} - 10\beta_{4} + 10\beta_{2} ) / 4 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3744\mathbb{Z}\right)^\times\).

\(n\) \(703\) \(2017\) \(2081\) \(2341\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1585.1
−0.752986 1.19709i
−0.752986 + 1.19709i
1.19709 + 0.752986i
1.19709 0.752986i
−1.19709 0.752986i
−1.19709 + 0.752986i
0.752986 + 1.19709i
0.752986 1.19709i
0 0 0 −4.11439 0 0 0 0 0
1585.2 0 0 0 −4.11439 0 0 0 0 0
1585.3 0 0 0 −1.75265 0 0 0 0 0
1585.4 0 0 0 −1.75265 0 0 0 0 0
1585.5 0 0 0 1.75265 0 0 0 0 0
1585.6 0 0 0 1.75265 0 0 0 0 0
1585.7 0 0 0 4.11439 0 0 0 0 0
1585.8 0 0 0 4.11439 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1585.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
39.d odd 2 1 CM by \(\Q(\sqrt{-39}) \)
3.b odd 2 1 inner
8.b even 2 1 inner
13.b even 2 1 inner
24.h odd 2 1 inner
104.e even 2 1 inner
312.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3744.2.m.f 8
3.b odd 2 1 inner 3744.2.m.f 8
4.b odd 2 1 936.2.m.g 8
8.b even 2 1 inner 3744.2.m.f 8
8.d odd 2 1 936.2.m.g 8
12.b even 2 1 936.2.m.g 8
13.b even 2 1 inner 3744.2.m.f 8
24.f even 2 1 936.2.m.g 8
24.h odd 2 1 inner 3744.2.m.f 8
39.d odd 2 1 CM 3744.2.m.f 8
52.b odd 2 1 936.2.m.g 8
104.e even 2 1 inner 3744.2.m.f 8
104.h odd 2 1 936.2.m.g 8
156.h even 2 1 936.2.m.g 8
312.b odd 2 1 inner 3744.2.m.f 8
312.h even 2 1 936.2.m.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
936.2.m.g 8 4.b odd 2 1
936.2.m.g 8 8.d odd 2 1
936.2.m.g 8 12.b even 2 1
936.2.m.g 8 24.f even 2 1
936.2.m.g 8 52.b odd 2 1
936.2.m.g 8 104.h odd 2 1
936.2.m.g 8 156.h even 2 1
936.2.m.g 8 312.h even 2 1
3744.2.m.f 8 1.a even 1 1 trivial
3744.2.m.f 8 3.b odd 2 1 inner
3744.2.m.f 8 8.b even 2 1 inner
3744.2.m.f 8 13.b even 2 1 inner
3744.2.m.f 8 24.h odd 2 1 inner
3744.2.m.f 8 39.d odd 2 1 CM
3744.2.m.f 8 104.e even 2 1 inner
3744.2.m.f 8 312.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 20T_{5}^{2} + 52 \) acting on \(S_{2}^{\mathrm{new}}(3744, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} - 20 T^{2} + 52)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} - 44 T^{2} + 52)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 13)^{4} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( (T^{4} + 164 T^{2} + 6292)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 156)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} + 188 T^{2} + 8788)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( (T^{4} - 236 T^{2} + 52)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 52)^{4} \) Copy content Toggle raw display
$67$ \( T^{8} \) Copy content Toggle raw display
$71$ \( (T^{4} + 284 T^{2} + 6292)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} \) Copy content Toggle raw display
$79$ \( (T^{2} - 108)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 332 T^{2} + 27508)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 356 T^{2} + 6292)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} \) Copy content Toggle raw display
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