Properties

Label 4056.2.c.h
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: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q + q^{3} + \beta q^{5} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + \beta q^{5} + q^{9} + \beta q^{15} - 2 q^{17} - 2 \beta q^{19} + q^{25} + q^{27} + 6 q^{29} + \beta q^{37} + 3 \beta q^{41} + 12 q^{43} + \beta q^{45} + 2 \beta q^{47} + 7 q^{49} - 2 q^{51} + 6 q^{53} - 2 \beta q^{57} + 4 \beta q^{59} - 2 q^{61} + 2 \beta q^{67} - 6 \beta q^{71} + 7 \beta q^{73} + q^{75} + q^{81} + 4 \beta q^{83} - 2 \beta q^{85} + 6 q^{87} + 9 \beta q^{89} + 8 q^{95} - 3 \beta q^{97} +O(q^{100}) \) 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 \( 2 q + 2 q^{3} + 2 q^{9} - 4 q^{17} + 2 q^{25} + 2 q^{27} + 12 q^{29} + 24 q^{43} + 14 q^{49} - 4 q^{51} + 12 q^{53} - 4 q^{61} + 2 q^{75} + 2 q^{81} + 12 q^{87} + 16 q^{95}+O(q^{100}) \) 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 2.00000i 0 0 0 1.00000 0
337.2 0 1.00000 0 2.00000i 0 0 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.h 2
13.b even 2 1 inner 4056.2.c.h 2
13.d odd 4 1 312.2.a.f 1
13.d odd 4 1 4056.2.a.m 1
39.f even 4 1 936.2.a.b 1
52.f even 4 1 624.2.a.d 1
52.f even 4 1 8112.2.a.f 1
65.g odd 4 1 7800.2.a.d 1
104.j odd 4 1 2496.2.a.c 1
104.m even 4 1 2496.2.a.s 1
156.l odd 4 1 1872.2.a.e 1
312.w odd 4 1 7488.2.a.br 1
312.y even 4 1 7488.2.a.bs 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
312.2.a.f 1 13.d odd 4 1
624.2.a.d 1 52.f even 4 1
936.2.a.b 1 39.f even 4 1
1872.2.a.e 1 156.l odd 4 1
2496.2.a.c 1 104.j odd 4 1
2496.2.a.s 1 104.m even 4 1
4056.2.a.m 1 13.d odd 4 1
4056.2.c.h 2 1.a even 1 1 trivial
4056.2.c.h 2 13.b even 2 1 inner
7488.2.a.br 1 312.w odd 4 1
7488.2.a.bs 1 312.y even 4 1
7800.2.a.d 1 65.g odd 4 1
8112.2.a.f 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 \(S_{2}^{\mathrm{new}}(4056, [\chi])\):

\( T_{5}^{2} + 4 \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display
\( T_{11} \) 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
$5$ \( T^{2} + 4 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( (T + 2)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 16 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( (T - 6)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 4 \) Copy content Toggle raw display
$41$ \( T^{2} + 36 \) Copy content Toggle raw display
$43$ \( (T - 12)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 16 \) Copy content Toggle raw display
$53$ \( (T - 6)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 64 \) Copy content Toggle raw display
$61$ \( (T + 2)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 16 \) Copy content Toggle raw display
$71$ \( T^{2} + 144 \) Copy content Toggle raw display
$73$ \( T^{2} + 196 \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 64 \) Copy content Toggle raw display
$89$ \( T^{2} + 324 \) Copy content Toggle raw display
$97$ \( T^{2} + 36 \) Copy content Toggle raw display
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