Properties

Label 3744.2.m.d
Level $3744$
Weight $2$
Character orbit 3744.m
Analytic conductor $29.896$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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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: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 312)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{5} - 4 q^{11} + ( - \beta + 3) q^{13} + 6 q^{17} + 6 q^{19} - q^{25} + 3 \beta q^{29} - 6 q^{37} - \beta q^{41} - 6 \beta q^{43} + 4 \beta q^{47} + 7 q^{49} + 3 \beta q^{53} - 8 q^{55} + 4 q^{59} - 4 \beta q^{61} + ( - 2 \beta + 6) q^{65} + 6 q^{67} + 2 \beta q^{71} - 6 \beta q^{73} - 6 q^{79} + 8 q^{83} + 12 q^{85} + \beta q^{89} + 12 q^{95} + 6 \beta q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{5} - 8 q^{11} + 6 q^{13} + 12 q^{17} + 12 q^{19} - 2 q^{25} - 12 q^{37} + 14 q^{49} - 16 q^{55} + 8 q^{59} + 12 q^{65} + 12 q^{67} - 12 q^{79} + 16 q^{83} + 24 q^{85} + 24 q^{95}+O(q^{100}) \) 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
1.00000i
1.00000i
0 0 0 2.00000 0 0 0 0 0
1585.2 0 0 0 2.00000 0 0 0 0 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
104.e even 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.d 2
3.b odd 2 1 1248.2.m.a 2
4.b odd 2 1 936.2.m.d 2
8.b even 2 1 3744.2.m.a 2
8.d odd 2 1 936.2.m.a 2
12.b even 2 1 312.2.m.a 2
13.b even 2 1 3744.2.m.a 2
24.f even 2 1 312.2.m.b yes 2
24.h odd 2 1 1248.2.m.b 2
39.d odd 2 1 1248.2.m.b 2
52.b odd 2 1 936.2.m.a 2
104.e even 2 1 inner 3744.2.m.d 2
104.h odd 2 1 936.2.m.d 2
156.h even 2 1 312.2.m.b yes 2
312.b odd 2 1 1248.2.m.a 2
312.h even 2 1 312.2.m.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
312.2.m.a 2 12.b even 2 1
312.2.m.a 2 312.h even 2 1
312.2.m.b yes 2 24.f even 2 1
312.2.m.b yes 2 156.h even 2 1
936.2.m.a 2 8.d odd 2 1
936.2.m.a 2 52.b odd 2 1
936.2.m.d 2 4.b odd 2 1
936.2.m.d 2 104.h odd 2 1
1248.2.m.a 2 3.b odd 2 1
1248.2.m.a 2 312.b odd 2 1
1248.2.m.b 2 24.h odd 2 1
1248.2.m.b 2 39.d odd 2 1
3744.2.m.a 2 8.b even 2 1
3744.2.m.a 2 13.b even 2 1
3744.2.m.d 2 1.a even 1 1 trivial
3744.2.m.d 2 104.e even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 2 \) acting on \(S_{2}^{\mathrm{new}}(3744, [\chi])\). 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
$5$ \( (T - 2)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( (T + 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 6T + 13 \) Copy content Toggle raw display
$17$ \( (T - 6)^{2} \) Copy content Toggle raw display
$19$ \( (T - 6)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 36 \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( (T + 6)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 4 \) Copy content Toggle raw display
$43$ \( T^{2} + 144 \) Copy content Toggle raw display
$47$ \( T^{2} + 64 \) Copy content Toggle raw display
$53$ \( T^{2} + 36 \) Copy content Toggle raw display
$59$ \( (T - 4)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 64 \) Copy content Toggle raw display
$67$ \( (T - 6)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 16 \) Copy content Toggle raw display
$73$ \( T^{2} + 144 \) Copy content Toggle raw display
$79$ \( (T + 6)^{2} \) Copy content Toggle raw display
$83$ \( (T - 8)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 4 \) Copy content Toggle raw display
$97$ \( T^{2} + 144 \) Copy content Toggle raw display
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