Properties

Label 4056.2.c.f
Level $4056$
Weight $2$
Character orbit 4056.c
Analytic conductor $32.387$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4056 = 2^{3} \cdot 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4056.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(32.3873230598\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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.

\(f(q)\) \(=\) \( q - q^{3} + 2 \beta q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} + 2 \beta q^{7} + q^{9} + \beta q^{11} + 6 q^{17} - 2 \beta q^{19} - 2 \beta q^{21} - 4 q^{23} + 5 q^{25} - q^{27} + 10 q^{29} - 4 \beta q^{31} - \beta q^{33} + \beta q^{37} + 4 q^{43} - \beta q^{47} - 9 q^{49} - 6 q^{51} - 2 q^{53} + 2 \beta q^{57} - 5 \beta q^{59} + 10 q^{61} + 2 \beta q^{63} + 4 \beta q^{67} + 4 q^{69} + \beta q^{71} + 5 \beta q^{73} - 5 q^{75} - 8 q^{77} + 8 q^{79} + q^{81} + 3 \beta q^{83} - 10 q^{87} + 6 \beta q^{89} + 4 \beta q^{93} - \beta q^{97} + \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} + 2 q^{9} + 12 q^{17} - 8 q^{23} + 10 q^{25} - 2 q^{27} + 20 q^{29} + 8 q^{43} - 18 q^{49} - 12 q^{51} - 4 q^{53} + 20 q^{61} + 8 q^{69} - 10 q^{75} - 16 q^{77} + 16 q^{79} + 2 q^{81} - 20 q^{87}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4056\mathbb{Z}\right)^\times\).

\(n\) \(1015\) \(2029\) \(2705\) \(3889\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
337.1
1.00000i
1.00000i
0 −1.00000 0 0 0 4.00000i 0 1.00000 0
337.2 0 −1.00000 0 0 0 4.00000i 0 1.00000 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
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4056.2.c.f 2
13.b even 2 1 inner 4056.2.c.f 2
13.d odd 4 1 312.2.a.b 1
13.d odd 4 1 4056.2.a.f 1
39.f even 4 1 936.2.a.d 1
52.f even 4 1 624.2.a.g 1
52.f even 4 1 8112.2.a.y 1
65.g odd 4 1 7800.2.a.w 1
104.j odd 4 1 2496.2.a.u 1
104.m even 4 1 2496.2.a.j 1
156.l odd 4 1 1872.2.a.k 1
312.w odd 4 1 7488.2.a.bh 1
312.y even 4 1 7488.2.a.y 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
312.2.a.b 1 13.d odd 4 1
624.2.a.g 1 52.f even 4 1
936.2.a.d 1 39.f even 4 1
1872.2.a.k 1 156.l odd 4 1
2496.2.a.j 1 104.m even 4 1
2496.2.a.u 1 104.j odd 4 1
4056.2.a.f 1 13.d odd 4 1
4056.2.c.f 2 1.a even 1 1 trivial
4056.2.c.f 2 13.b even 2 1 inner
7488.2.a.y 1 312.y even 4 1
7488.2.a.bh 1 312.w odd 4 1
7800.2.a.w 1 65.g odd 4 1
8112.2.a.y 1 52.f even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4056, [\chi])\):

\( T_{5} \) Copy content Toggle raw display
\( T_{7}^{2} + 16 \) Copy content Toggle raw display
\( T_{11}^{2} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 16 \) Copy content Toggle raw display
$11$ \( T^{2} + 4 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( (T - 6)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 16 \) Copy content Toggle raw display
$23$ \( (T + 4)^{2} \) Copy content Toggle raw display
$29$ \( (T - 10)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 64 \) Copy content Toggle raw display
$37$ \( T^{2} + 4 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( (T - 4)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 4 \) Copy content Toggle raw display
$53$ \( (T + 2)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 100 \) Copy content Toggle raw display
$61$ \( (T - 10)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 64 \) Copy content Toggle raw display
$71$ \( T^{2} + 4 \) Copy content Toggle raw display
$73$ \( T^{2} + 100 \) Copy content Toggle raw display
$79$ \( (T - 8)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 36 \) Copy content Toggle raw display
$89$ \( T^{2} + 144 \) Copy content Toggle raw display
$97$ \( T^{2} + 4 \) Copy content Toggle raw display
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