Properties

Label 4056.2.c.i
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: \(\Z[a_1, \ldots, a_{31}]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q + q^{3} + 3 i q^{5} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + 3 i q^{5} + q^{9} + 3 i q^{15} + q^{17} - 4 q^{23} - 4 q^{25} + q^{27} + 3 q^{29} + 8 i q^{31} + 5 i q^{37} + 3 i q^{41} - 4 q^{43} + 3 i q^{45} + 8 i q^{47} + 7 q^{49} + q^{51} - 13 q^{53} - 12 i q^{59} + 15 q^{61} + 12 i q^{67} - 4 q^{69} + 8 i q^{71} - 3 i q^{73} - 4 q^{75} - 4 q^{79} + q^{81} + 12 i q^{83} + 3 i q^{85} + 3 q^{87} - 10 i q^{89} + 8 i q^{93} + 2 i q^{97} +O(q^{100}) \) 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 \( 2 q + 2 q^{3} + 2 q^{9} + 2 q^{17} - 8 q^{23} - 8 q^{25} + 2 q^{27} + 6 q^{29} - 8 q^{43} + 14 q^{49} + 2 q^{51} - 26 q^{53} + 30 q^{61} - 8 q^{69} - 8 q^{75} - 8 q^{79} + 2 q^{81} + 6 q^{87}+O(q^{100}) \) 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 3.00000i 0 0 0 1.00000 0
337.2 0 1.00000 0 3.00000i 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.i 2
13.b even 2 1 inner 4056.2.c.i 2
13.d odd 4 1 4056.2.a.l 1
13.d odd 4 1 4056.2.a.q 1
13.f odd 12 2 312.2.q.a 2
39.k even 12 2 936.2.t.a 2
52.f even 4 1 8112.2.a.b 1
52.f even 4 1 8112.2.a.n 1
52.l even 12 2 624.2.q.f 2
156.v odd 12 2 1872.2.t.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
312.2.q.a 2 13.f odd 12 2
624.2.q.f 2 52.l even 12 2
936.2.t.a 2 39.k even 12 2
1872.2.t.b 2 156.v odd 12 2
4056.2.a.l 1 13.d odd 4 1
4056.2.a.q 1 13.d odd 4 1
4056.2.c.i 2 1.a even 1 1 trivial
4056.2.c.i 2 13.b even 2 1 inner
8112.2.a.b 1 52.f even 4 1
8112.2.a.n 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 \(S_{2}^{\mathrm{new}}(4056, [\chi])\):

\( T_{5}^{2} + 9 \) 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
$5$ \( T^{2} + 9 \) 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
$17$ \( (T - 1)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( (T + 4)^{2} \) Copy content Toggle raw display
$29$ \( (T - 3)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 64 \) Copy content Toggle raw display
$37$ \( T^{2} + 25 \) Copy content Toggle raw display
$41$ \( T^{2} + 9 \) Copy content Toggle raw display
$43$ \( (T + 4)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 64 \) Copy content Toggle raw display
$53$ \( (T + 13)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 144 \) Copy content Toggle raw display
$61$ \( (T - 15)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 144 \) Copy content Toggle raw display
$71$ \( T^{2} + 64 \) Copy content Toggle raw display
$73$ \( T^{2} + 9 \) Copy content Toggle raw display
$79$ \( (T + 4)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 144 \) Copy content Toggle raw display
$89$ \( T^{2} + 100 \) Copy content Toggle raw display
$97$ \( T^{2} + 4 \) Copy content Toggle raw display
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