Properties

Label 126.3.b.a
Level $126$
Weight $3$
Character orbit 126.b
Analytic conductor $3.433$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 126.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.43325133094\)
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: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} -2 q^{4} + ( -\beta_{1} + 2 \beta_{2} ) q^{5} -\beta_{3} q^{7} -2 \beta_{1} q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} -2 q^{4} + ( -\beta_{1} + 2 \beta_{2} ) q^{5} -\beta_{3} q^{7} -2 \beta_{1} q^{8} + ( 2 - 4 \beta_{3} ) q^{10} + ( -2 \beta_{1} + 4 \beta_{2} ) q^{11} + ( -8 - 4 \beta_{3} ) q^{13} -\beta_{2} q^{14} + 4 q^{16} + ( 13 \beta_{1} - 2 \beta_{2} ) q^{17} + 20 q^{19} + ( 2 \beta_{1} - 4 \beta_{2} ) q^{20} + ( 4 - 8 \beta_{3} ) q^{22} + ( -2 \beta_{1} + 4 \beta_{2} ) q^{23} + ( -33 + 8 \beta_{3} ) q^{25} + ( -8 \beta_{1} - 4 \beta_{2} ) q^{26} + 2 \beta_{3} q^{28} + ( -19 \beta_{1} - 4 \beta_{2} ) q^{29} + ( 4 + 8 \beta_{3} ) q^{31} + 4 \beta_{1} q^{32} + ( -26 + 4 \beta_{3} ) q^{34} + ( -14 \beta_{1} + \beta_{2} ) q^{35} + 38 q^{37} + 20 \beta_{1} q^{38} + ( -4 + 8 \beta_{3} ) q^{40} + ( 27 \beta_{1} + 6 \beta_{2} ) q^{41} + ( 20 + 24 \beta_{3} ) q^{43} + ( 4 \beta_{1} - 8 \beta_{2} ) q^{44} + ( 4 - 8 \beta_{3} ) q^{46} + 12 \beta_{1} q^{47} + 7 q^{49} + ( -33 \beta_{1} + 8 \beta_{2} ) q^{50} + ( 16 + 8 \beta_{3} ) q^{52} + ( -3 \beta_{1} - 24 \beta_{2} ) q^{53} + ( -116 + 16 \beta_{3} ) q^{55} + 2 \beta_{2} q^{56} + ( 38 + 8 \beta_{3} ) q^{58} + ( 20 \beta_{1} + 8 \beta_{2} ) q^{59} + ( 58 - 16 \beta_{3} ) q^{61} + ( 4 \beta_{1} + 8 \beta_{2} ) q^{62} -8 q^{64} + ( -48 \beta_{1} - 12 \beta_{2} ) q^{65} + ( -48 - 32 \beta_{3} ) q^{67} + ( -26 \beta_{1} + 4 \beta_{2} ) q^{68} + ( 28 - 2 \beta_{3} ) q^{70} + ( -2 \beta_{1} + 4 \beta_{2} ) q^{71} + ( -24 - 20 \beta_{3} ) q^{73} + 38 \beta_{1} q^{74} -40 q^{76} + ( -28 \beta_{1} + 2 \beta_{2} ) q^{77} + ( 76 - 16 \beta_{3} ) q^{79} + ( -4 \beta_{1} + 8 \beta_{2} ) q^{80} + ( -54 - 12 \beta_{3} ) q^{82} + ( 64 \beta_{1} - 8 \beta_{2} ) q^{83} + ( 82 - 56 \beta_{3} ) q^{85} + ( 20 \beta_{1} + 24 \beta_{2} ) q^{86} + ( -8 + 16 \beta_{3} ) q^{88} + ( 51 \beta_{1} - 18 \beta_{2} ) q^{89} + ( 28 + 8 \beta_{3} ) q^{91} + ( 4 \beta_{1} - 8 \beta_{2} ) q^{92} -24 q^{94} + ( -20 \beta_{1} + 40 \beta_{2} ) q^{95} + ( -72 - 44 \beta_{3} ) q^{97} + 7 \beta_{1} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 8q^{4} + O(q^{10}) \) \( 4q - 8q^{4} + 8q^{10} - 32q^{13} + 16q^{16} + 80q^{19} + 16q^{22} - 132q^{25} + 16q^{31} - 104q^{34} + 152q^{37} - 16q^{40} + 80q^{43} + 16q^{46} + 28q^{49} + 64q^{52} - 464q^{55} + 152q^{58} + 232q^{61} - 32q^{64} - 192q^{67} + 112q^{70} - 96q^{73} - 160q^{76} + 304q^{79} - 216q^{82} + 328q^{85} - 32q^{88} + 112q^{91} - 96q^{94} - 288q^{97} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + 5 \nu \)\()/3\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 11 \nu \)\()/3\)
\(\beta_{3}\)\(=\)\( \nu^{2} + 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} - \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{3} - 4\)
\(\nu^{3}\)\(=\)\((\)\(-5 \beta_{2} + 11 \beta_{1}\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/126\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(73\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
71.1
1.16372i
2.57794i
2.57794i
1.16372i
1.41421i 0 −2.00000 6.06910i 0 −2.64575 2.82843i 0 −8.58301
71.2 1.41421i 0 −2.00000 8.89753i 0 2.64575 2.82843i 0 12.5830
71.3 1.41421i 0 −2.00000 8.89753i 0 2.64575 2.82843i 0 12.5830
71.4 1.41421i 0 −2.00000 6.06910i 0 −2.64575 2.82843i 0 −8.58301
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 126.3.b.a 4
3.b odd 2 1 inner 126.3.b.a 4
4.b odd 2 1 1008.3.d.a 4
5.b even 2 1 3150.3.e.e 4
5.c odd 4 2 3150.3.c.b 8
7.b odd 2 1 882.3.b.f 4
7.c even 3 2 882.3.s.e 8
7.d odd 6 2 882.3.s.i 8
8.b even 2 1 4032.3.d.i 4
8.d odd 2 1 4032.3.d.j 4
9.c even 3 2 1134.3.q.c 8
9.d odd 6 2 1134.3.q.c 8
12.b even 2 1 1008.3.d.a 4
15.d odd 2 1 3150.3.e.e 4
15.e even 4 2 3150.3.c.b 8
21.c even 2 1 882.3.b.f 4
21.g even 6 2 882.3.s.i 8
21.h odd 6 2 882.3.s.e 8
24.f even 2 1 4032.3.d.j 4
24.h odd 2 1 4032.3.d.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.3.b.a 4 1.a even 1 1 trivial
126.3.b.a 4 3.b odd 2 1 inner
882.3.b.f 4 7.b odd 2 1
882.3.b.f 4 21.c even 2 1
882.3.s.e 8 7.c even 3 2
882.3.s.e 8 21.h odd 6 2
882.3.s.i 8 7.d odd 6 2
882.3.s.i 8 21.g even 6 2
1008.3.d.a 4 4.b odd 2 1
1008.3.d.a 4 12.b even 2 1
1134.3.q.c 8 9.c even 3 2
1134.3.q.c 8 9.d odd 6 2
3150.3.c.b 8 5.c odd 4 2
3150.3.c.b 8 15.e even 4 2
3150.3.e.e 4 5.b even 2 1
3150.3.e.e 4 15.d odd 2 1
4032.3.d.i 4 8.b even 2 1
4032.3.d.i 4 24.h odd 2 1
4032.3.d.j 4 8.d odd 2 1
4032.3.d.j 4 24.f even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(126, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 2 + T^{2} )^{2} \)
$3$ \( T^{4} \)
$5$ \( 2916 + 116 T^{2} + T^{4} \)
$7$ \( ( -7 + T^{2} )^{2} \)
$11$ \( 46656 + 464 T^{2} + T^{4} \)
$13$ \( ( -48 + 16 T + T^{2} )^{2} \)
$17$ \( 79524 + 788 T^{2} + T^{4} \)
$19$ \( ( -20 + T )^{4} \)
$23$ \( 46656 + 464 T^{2} + T^{4} \)
$29$ \( 248004 + 1892 T^{2} + T^{4} \)
$31$ \( ( -432 - 8 T + T^{2} )^{2} \)
$37$ \( ( -38 + T )^{4} \)
$41$ \( 910116 + 3924 T^{2} + T^{4} \)
$43$ \( ( -3632 - 40 T + T^{2} )^{2} \)
$47$ \( ( 288 + T^{2} )^{2} \)
$53$ \( 64738116 + 16164 T^{2} + T^{4} \)
$59$ \( 9216 + 3392 T^{2} + T^{4} \)
$61$ \( ( 1572 - 116 T + T^{2} )^{2} \)
$67$ \( ( -4864 + 96 T + T^{2} )^{2} \)
$71$ \( 46656 + 464 T^{2} + T^{4} \)
$73$ \( ( -2224 + 48 T + T^{2} )^{2} \)
$79$ \( ( 3984 - 152 T + T^{2} )^{2} \)
$83$ \( 53231616 + 18176 T^{2} + T^{4} \)
$89$ \( 443556 + 19476 T^{2} + T^{4} \)
$97$ \( ( -8368 + 144 T + T^{2} )^{2} \)
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