Properties

Label 4032.2.v.c
Level 4032
Weight 2
Character orbit 4032.v
Analytic conductor 32.196
Analytic rank 0
Dimension 12
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4032.v (of order \(4\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(32.1956820950\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: 12.0.653473922154496.1
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: no (minimal twist has level 1008)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \beta_{3} - \beta_{5} ) q^{5} + q^{7} +O(q^{10})\) \( q + ( \beta_{3} - \beta_{5} ) q^{5} + q^{7} + ( \beta_{1} + \beta_{3} + \beta_{5} ) q^{11} + ( 1 - \beta_{2} - \beta_{8} ) q^{13} + ( -\beta_{1} - 3 \beta_{3} - \beta_{7} - \beta_{10} ) q^{17} -\beta_{10} q^{23} + \beta_{2} q^{25} + ( \beta_{1} - \beta_{6} - \beta_{10} ) q^{29} + ( 2 \beta_{2} + \beta_{4} + \beta_{8} ) q^{31} + ( \beta_{3} - \beta_{5} ) q^{35} + ( -1 - \beta_{2} ) q^{37} + ( \beta_{1} - 3 \beta_{5} + \beta_{6} - \beta_{7} ) q^{41} + ( 2 - 2 \beta_{2} + 2 \beta_{8} + \beta_{11} ) q^{43} + ( 2 \beta_{5} - 2 \beta_{6} ) q^{47} + q^{49} + ( 3 \beta_{3} - 3 \beta_{5} - \beta_{6} + \beta_{7} + \beta_{10} ) q^{53} + ( -4 + \beta_{4} - \beta_{8} + \beta_{9} + \beta_{11} ) q^{55} + ( 2 \beta_{1} + \beta_{3} + \beta_{5} - \beta_{6} - \beta_{10} ) q^{59} + ( -3 + 3 \beta_{2} - \beta_{8} ) q^{61} + ( 2 \beta_{3} + 2 \beta_{10} ) q^{65} + ( -4 - 4 \beta_{2} + \beta_{4} ) q^{67} + ( -\beta_{1} + 2 \beta_{3} - \beta_{7} - 3 \beta_{10} ) q^{71} + ( -4 \beta_{2} + \beta_{9} - \beta_{11} ) q^{73} + ( \beta_{1} + \beta_{3} + \beta_{5} ) q^{77} + ( -4 \beta_{2} - 2 \beta_{4} - 2 \beta_{8} ) q^{79} + ( -5 \beta_{3} + 5 \beta_{5} - \beta_{6} + \beta_{10} ) q^{83} + ( 6 + 6 \beta_{2} - 2 \beta_{9} ) q^{85} + ( \beta_{1} - 7 \beta_{5} + 3 \beta_{6} - \beta_{7} ) q^{89} + ( 1 - \beta_{2} - \beta_{8} ) q^{91} + ( -6 + 3 \beta_{4} - 3 \beta_{8} - \beta_{9} - \beta_{11} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 12q^{7} + O(q^{10}) \) \( 12q + 12q^{7} + 16q^{13} - 12q^{37} + 20q^{43} + 12q^{49} - 32q^{55} - 32q^{61} - 44q^{67} + 64q^{85} + 16q^{91} - 56q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 4 x^{10} + 13 x^{8} - 28 x^{6} + 52 x^{4} - 64 x^{2} + 64\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 3 \nu^{11} + 6 \nu^{9} - 5 \nu^{7} + 46 \nu^{5} - 32 \nu^{3} + 32 \nu \)\()/64\)
\(\beta_{2}\)\(=\)\((\)\( -3 \nu^{10} + 10 \nu^{8} - 27 \nu^{6} + 34 \nu^{4} - 64 \nu^{2} + 32 \)\()/64\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{11} - 4 \nu^{9} + 5 \nu^{7} - 12 \nu^{5} + 12 \nu^{3} - 16 \nu \)\()/32\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{10} + 18 \nu^{8} - 55 \nu^{6} + 138 \nu^{4} - 256 \nu^{2} + 352 \)\()/64\)
\(\beta_{5}\)\(=\)\((\)\( -3 \nu^{11} + 6 \nu^{9} - 19 \nu^{7} + 30 \nu^{5} - 24 \nu^{3} \)\()/64\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{11} - 18 \nu^{9} + 49 \nu^{7} - 90 \nu^{5} + 136 \nu^{3} - 192 \nu \)\()/64\)
\(\beta_{7}\)\(=\)\((\)\( -3 \nu^{11} + 10 \nu^{9} - 27 \nu^{7} + 98 \nu^{5} - 128 \nu^{3} + 224 \nu \)\()/64\)
\(\beta_{8}\)\(=\)\((\)\( -7 \nu^{10} + 18 \nu^{8} - 31 \nu^{6} + 74 \nu^{4} - 160 \nu^{2} + 96 \)\()/64\)
\(\beta_{9}\)\(=\)\((\)\( -11 \nu^{10} + 10 \nu^{8} - 67 \nu^{6} + 98 \nu^{4} - 160 \nu^{2} + 96 \)\()/64\)
\(\beta_{10}\)\(=\)\((\)\( 3 \nu^{11} - 12 \nu^{9} + 31 \nu^{7} - 68 \nu^{5} + 116 \nu^{3} - 80 \nu \)\()/32\)
\(\beta_{11}\)\(=\)\((\)\( -13 \nu^{10} + 54 \nu^{8} - 117 \nu^{6} + 286 \nu^{4} - 320 \nu^{2} + 416 \)\()/64\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{10} + \beta_{7} - \beta_{6} - \beta_{5} - \beta_{3} - \beta_{1}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{11} - 2 \beta_{8} - \beta_{4} + 2\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(3 \beta_{10} + \beta_{7} - \beta_{6} + 5 \beta_{5} - \beta_{3} + \beta_{1}\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(2 \beta_{11} + \beta_{9} - \beta_{8} - 10 \beta_{2} - 8\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(\beta_{10} + 3 \beta_{7} + 3 \beta_{6} + 7 \beta_{5} + 3 \beta_{3} + 5 \beta_{1}\)\()/4\)
\(\nu^{6}\)\(=\)\((\)\(\beta_{11} - 2 \beta_{9} + 8 \beta_{8} - \beta_{4} - 16 \beta_{2} - 2\)\()/4\)
\(\nu^{7}\)\(=\)\((\)\(-3 \beta_{10} + 3 \beta_{7} + 9 \beta_{6} - 13 \beta_{5} - 15 \beta_{3} + 3 \beta_{1}\)\()/4\)
\(\nu^{8}\)\(=\)\((\)\(2 \beta_{11} - 13 \beta_{9} + 5 \beta_{8} - 4 \beta_{4} + 26 \beta_{2} + 8\)\()/4\)
\(\nu^{9}\)\(=\)\((\)\(-\beta_{10} - 11 \beta_{7} - 3 \beta_{6} - 23 \beta_{5} - 51 \beta_{3} + 3 \beta_{1}\)\()/4\)
\(\nu^{10}\)\(=\)\((\)\(-\beta_{11} - 14 \beta_{9} - 24 \beta_{8} + 17 \beta_{4} + 32 \beta_{2} - 46\)\()/4\)
\(\nu^{11}\)\(=\)\((\)\(3 \beta_{10} - 19 \beta_{7} - 25 \beta_{6} - 19 \beta_{5} + 31 \beta_{3} + 29 \beta_{1}\)\()/4\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4032\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(1793\) \(3781\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1583.1
1.35489 + 0.405301i
−1.16947 0.795191i
−0.892524 + 1.09700i
0.892524 1.09700i
1.16947 + 0.795191i
−1.35489 0.405301i
1.35489 0.405301i
−1.16947 + 0.795191i
−0.892524 1.09700i
0.892524 + 1.09700i
1.16947 0.795191i
−1.35489 + 0.405301i
0 0 0 −1.41421 1.41421i 0 1.00000 0 0 0
1583.2 0 0 0 −1.41421 1.41421i 0 1.00000 0 0 0
1583.3 0 0 0 −1.41421 1.41421i 0 1.00000 0 0 0
1583.4 0 0 0 1.41421 + 1.41421i 0 1.00000 0 0 0
1583.5 0 0 0 1.41421 + 1.41421i 0 1.00000 0 0 0
1583.6 0 0 0 1.41421 + 1.41421i 0 1.00000 0 0 0
3599.1 0 0 0 −1.41421 + 1.41421i 0 1.00000 0 0 0
3599.2 0 0 0 −1.41421 + 1.41421i 0 1.00000 0 0 0
3599.3 0 0 0 −1.41421 + 1.41421i 0 1.00000 0 0 0
3599.4 0 0 0 1.41421 1.41421i 0 1.00000 0 0 0
3599.5 0 0 0 1.41421 1.41421i 0 1.00000 0 0 0
3599.6 0 0 0 1.41421 1.41421i 0 1.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3599.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
16.f odd 4 1 inner
48.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4032.2.v.c 12
3.b odd 2 1 inner 4032.2.v.c 12
4.b odd 2 1 1008.2.v.c 12
12.b even 2 1 1008.2.v.c 12
16.e even 4 1 1008.2.v.c 12
16.f odd 4 1 inner 4032.2.v.c 12
48.i odd 4 1 1008.2.v.c 12
48.k even 4 1 inner 4032.2.v.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1008.2.v.c 12 4.b odd 2 1
1008.2.v.c 12 12.b even 2 1
1008.2.v.c 12 16.e even 4 1
1008.2.v.c 12 48.i odd 4 1
4032.2.v.c 12 1.a even 1 1 trivial
4032.2.v.c 12 3.b odd 2 1 inner
4032.2.v.c 12 16.f odd 4 1 inner
4032.2.v.c 12 48.k even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4032, [\chi])\):

\( T_{5}^{4} + 16 \)
\( T_{11}^{12} + 1056 T_{11}^{8} + 53504 T_{11}^{4} + 65536 \)