Properties

Label 63.4.c.a
Level $63$
Weight $4$
Character orbit 63.c
Analytic conductor $3.717$
Analytic rank $0$
Dimension $4$
CM discriminant -7
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 63.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.71712033036\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{7})\)
Defining polynomial: \(x^{4} + 8 x^{2} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + ( -8 - 5 \beta_{3} ) q^{4} -7 \beta_{3} q^{7} + ( -5 \beta_{1} - 10 \beta_{2} ) q^{8} +O(q^{10})\) \( q + \beta_{2} q^{2} + ( -8 - 5 \beta_{3} ) q^{4} -7 \beta_{3} q^{7} + ( -5 \beta_{1} - 10 \beta_{2} ) q^{8} + ( -\beta_{1} + 13 \beta_{2} ) q^{11} + ( -7 \beta_{1} - 14 \beta_{2} ) q^{14} + ( 111 + 40 \beta_{3} ) q^{16} + ( -205 - 59 \beta_{3} ) q^{22} + ( 25 \beta_{1} + 7 \beta_{2} ) q^{23} -125 q^{25} + ( 245 + 56 \beta_{3} ) q^{28} + ( 37 \beta_{1} - 11 \beta_{2} ) q^{29} + 111 \beta_{2} q^{32} + 4 \beta_{3} q^{37} + 202 \beta_{3} q^{43} + ( -67 \beta_{1} - 219 \beta_{2} ) q^{44} + ( -187 - 185 \beta_{3} ) q^{46} + 343 q^{49} -125 \beta_{2} q^{50} + ( -85 \beta_{1} + 67 \beta_{2} ) q^{53} + 245 \beta_{2} q^{56} + ( 65 - 167 \beta_{3} ) q^{58} + ( -888 - 235 \beta_{3} ) q^{64} + 740 q^{67} + ( 141 \beta_{1} - 53 \beta_{2} ) q^{71} + ( 4 \beta_{1} + 8 \beta_{2} ) q^{74} + ( -105 \beta_{1} - 161 \beta_{2} ) q^{77} -1384 q^{79} + ( 202 \beta_{1} + 404 \beta_{2} ) q^{86} + ( 2065 + 1025 \beta_{3} ) q^{88} + ( 15 \beta_{1} - 501 \beta_{2} ) q^{92} + 343 \beta_{2} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 32q^{4} + O(q^{10}) \) \( 4q - 32q^{4} + 444q^{16} - 820q^{22} - 500q^{25} + 980q^{28} - 748q^{46} + 1372q^{49} + 260q^{58} - 3552q^{64} + 2960q^{67} - 5536q^{79} + 8260q^{88} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 8 x^{2} + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 3 \nu \)
\(\beta_{2}\)\(=\)\( \nu^{3} + 6 \nu \)
\(\beta_{3}\)\(=\)\( \nu^{2} + 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/3\)
\(\nu^{2}\)\(=\)\(\beta_{3} - 4\)
\(\nu^{3}\)\(=\)\(\beta_{2} - 2 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/63\mathbb{Z}\right)^\times\).

\(n\) \(10\) \(29\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
62.1
1.16372i
2.57794i
2.57794i
1.16372i
5.40636i 0 −21.2288 0 0 −18.5203 71.5195i 0 0
62.2 1.66471i 0 5.22876 0 0 18.5203 22.0220i 0 0
62.3 1.66471i 0 5.22876 0 0 18.5203 22.0220i 0 0
62.4 5.40636i 0 −21.2288 0 0 −18.5203 71.5195i 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
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
3.b odd 2 1 inner
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 63.4.c.a 4
3.b odd 2 1 inner 63.4.c.a 4
4.b odd 2 1 1008.4.k.a 4
7.b odd 2 1 CM 63.4.c.a 4
7.c even 3 2 441.4.p.b 8
7.d odd 6 2 441.4.p.b 8
12.b even 2 1 1008.4.k.a 4
21.c even 2 1 inner 63.4.c.a 4
21.g even 6 2 441.4.p.b 8
21.h odd 6 2 441.4.p.b 8
28.d even 2 1 1008.4.k.a 4
84.h odd 2 1 1008.4.k.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.4.c.a 4 1.a even 1 1 trivial
63.4.c.a 4 3.b odd 2 1 inner
63.4.c.a 4 7.b odd 2 1 CM
63.4.c.a 4 21.c even 2 1 inner
441.4.p.b 8 7.c even 3 2
441.4.p.b 8 7.d odd 6 2
441.4.p.b 8 21.g even 6 2
441.4.p.b 8 21.h odd 6 2
1008.4.k.a 4 4.b odd 2 1
1008.4.k.a 4 12.b even 2 1
1008.4.k.a 4 28.d even 2 1
1008.4.k.a 4 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 32 T_{2}^{2} + 81 \) acting on \(S_{4}^{\mathrm{new}}(63, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 81 + 32 T^{2} + T^{4} \)
$3$ \( T^{4} \)
$5$ \( T^{4} \)
$7$ \( ( -343 + T^{2} )^{2} \)
$11$ \( 3849444 + 5324 T^{2} + T^{4} \)
$13$ \( T^{4} \)
$17$ \( T^{4} \)
$19$ \( T^{4} \)
$23$ \( 516834756 + 48668 T^{2} + T^{4} \)
$29$ \( 450373284 + 97556 T^{2} + T^{4} \)
$31$ \( T^{4} \)
$37$ \( ( -112 + T^{2} )^{2} \)
$41$ \( T^{4} \)
$43$ \( ( -285628 + T^{2} )^{2} \)
$47$ \( T^{4} \)
$53$ \( 2534719716 + 595508 T^{2} + T^{4} \)
$59$ \( T^{4} \)
$61$ \( T^{4} \)
$67$ \( ( -740 + T )^{4} \)
$71$ \( 58795580484 + 1431644 T^{2} + T^{4} \)
$73$ \( T^{4} \)
$79$ \( ( 1384 + T )^{4} \)
$83$ \( T^{4} \)
$89$ \( T^{4} \)
$97$ \( T^{4} \)
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