Properties

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

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Newspace parameters

Level: \( N \) \(=\) \( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 126.s (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.43325133094\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} - 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + 2 \beta_{2} q^{4} + \beta_{1} q^{5} + ( 5 + 3 \beta_{2} ) q^{7} + 2 \beta_{3} q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + 2 \beta_{2} q^{4} + \beta_{1} q^{5} + ( 5 + 3 \beta_{2} ) q^{7} + 2 \beta_{3} q^{8} + 2 \beta_{2} q^{10} + ( -5 \beta_{1} + 5 \beta_{3} ) q^{11} + 15 q^{13} + ( 5 \beta_{1} + 3 \beta_{3} ) q^{14} + ( -4 + 4 \beta_{2} ) q^{16} + ( -8 \beta_{1} + 8 \beta_{3} ) q^{17} + ( 13 - 13 \beta_{2} ) q^{19} + 2 \beta_{3} q^{20} -10 q^{22} -16 \beta_{1} q^{23} -23 \beta_{2} q^{25} + 15 \beta_{1} q^{26} + ( -6 + 16 \beta_{2} ) q^{28} -16 \beta_{3} q^{29} -3 \beta_{2} q^{31} + ( -4 \beta_{1} + 4 \beta_{3} ) q^{32} -16 q^{34} + ( 5 \beta_{1} + 3 \beta_{3} ) q^{35} + ( -17 + 17 \beta_{2} ) q^{37} + ( 13 \beta_{1} - 13 \beta_{3} ) q^{38} + ( -4 + 4 \beta_{2} ) q^{40} -57 \beta_{3} q^{41} -85 q^{43} -10 \beta_{1} q^{44} -32 \beta_{2} q^{46} + 51 \beta_{1} q^{47} + ( 16 + 39 \beta_{2} ) q^{49} -23 \beta_{3} q^{50} + 30 \beta_{2} q^{52} + ( 24 \beta_{1} - 24 \beta_{3} ) q^{53} -10 q^{55} + ( -6 \beta_{1} + 16 \beta_{3} ) q^{56} + ( 32 - 32 \beta_{2} ) q^{58} + ( -64 \beta_{1} + 64 \beta_{3} ) q^{59} + ( 72 - 72 \beta_{2} ) q^{61} -3 \beta_{3} q^{62} -8 q^{64} + 15 \beta_{1} q^{65} -43 \beta_{2} q^{67} -16 \beta_{1} q^{68} + ( -6 + 16 \beta_{2} ) q^{70} + 37 \beta_{3} q^{71} + 95 \beta_{2} q^{73} + ( -17 \beta_{1} + 17 \beta_{3} ) q^{74} + 26 q^{76} + ( -40 \beta_{1} + 25 \beta_{3} ) q^{77} + ( -69 + 69 \beta_{2} ) q^{79} + ( -4 \beta_{1} + 4 \beta_{3} ) q^{80} + ( 114 - 114 \beta_{2} ) q^{82} + 43 \beta_{3} q^{83} -16 q^{85} -85 \beta_{1} q^{86} -20 \beta_{2} q^{88} + 96 \beta_{1} q^{89} + ( 75 + 45 \beta_{2} ) q^{91} -32 \beta_{3} q^{92} + 102 \beta_{2} q^{94} + ( 13 \beta_{1} - 13 \beta_{3} ) q^{95} + 16 q^{97} + ( 16 \beta_{1} + 39 \beta_{3} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{4} + 26 q^{7} + O(q^{10}) \) \( 4 q + 4 q^{4} + 26 q^{7} + 4 q^{10} + 60 q^{13} - 8 q^{16} + 26 q^{19} - 40 q^{22} - 46 q^{25} + 8 q^{28} - 6 q^{31} - 64 q^{34} - 34 q^{37} - 8 q^{40} - 340 q^{43} - 64 q^{46} + 142 q^{49} + 60 q^{52} - 40 q^{55} + 64 q^{58} + 144 q^{61} - 32 q^{64} - 86 q^{67} + 8 q^{70} + 190 q^{73} + 104 q^{76} - 138 q^{79} + 228 q^{82} - 64 q^{85} - 40 q^{88} + 390 q^{91} + 204 q^{94} + 64 q^{97} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/2\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(2 \beta_{2}\)
\(\nu^{3}\)\(=\)\(2 \beta_{3}\)

Character values

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

\(n\) \(29\) \(73\)
\(\chi(n)\) \(-1\) \(-1 + \beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1
−1.22474 + 0.707107i
1.22474 0.707107i
−1.22474 0.707107i
1.22474 + 0.707107i
−1.22474 + 0.707107i 0 1.00000 1.73205i −1.22474 + 0.707107i 0 6.50000 2.59808i 2.82843i 0 1.00000 1.73205i
53.2 1.22474 0.707107i 0 1.00000 1.73205i 1.22474 0.707107i 0 6.50000 2.59808i 2.82843i 0 1.00000 1.73205i
107.1 −1.22474 0.707107i 0 1.00000 + 1.73205i −1.22474 0.707107i 0 6.50000 + 2.59808i 2.82843i 0 1.00000 + 1.73205i
107.2 1.22474 + 0.707107i 0 1.00000 + 1.73205i 1.22474 + 0.707107i 0 6.50000 + 2.59808i 2.82843i 0 1.00000 + 1.73205i
\(n\): 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
7.c even 3 1 inner
21.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 126.3.s.b 4
3.b odd 2 1 inner 126.3.s.b 4
4.b odd 2 1 1008.3.dc.a 4
7.b odd 2 1 882.3.s.c 4
7.c even 3 1 inner 126.3.s.b 4
7.c even 3 1 882.3.b.c 2
7.d odd 6 1 882.3.b.d 2
7.d odd 6 1 882.3.s.c 4
12.b even 2 1 1008.3.dc.a 4
21.c even 2 1 882.3.s.c 4
21.g even 6 1 882.3.b.d 2
21.g even 6 1 882.3.s.c 4
21.h odd 6 1 inner 126.3.s.b 4
21.h odd 6 1 882.3.b.c 2
28.g odd 6 1 1008.3.dc.a 4
84.n even 6 1 1008.3.dc.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.3.s.b 4 1.a even 1 1 trivial
126.3.s.b 4 3.b odd 2 1 inner
126.3.s.b 4 7.c even 3 1 inner
126.3.s.b 4 21.h odd 6 1 inner
882.3.b.c 2 7.c even 3 1
882.3.b.c 2 21.h odd 6 1
882.3.b.d 2 7.d odd 6 1
882.3.b.d 2 21.g even 6 1
882.3.s.c 4 7.b odd 2 1
882.3.s.c 4 7.d odd 6 1
882.3.s.c 4 21.c even 2 1
882.3.s.c 4 21.g even 6 1
1008.3.dc.a 4 4.b odd 2 1
1008.3.dc.a 4 12.b even 2 1
1008.3.dc.a 4 28.g odd 6 1
1008.3.dc.a 4 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 2 T_{5}^{2} + 4 \) acting on \(S_{3}^{\mathrm{new}}(126, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 - 2 T^{2} + T^{4} \)
$3$ \( T^{4} \)
$5$ \( 4 - 2 T^{2} + T^{4} \)
$7$ \( ( 49 - 13 T + T^{2} )^{2} \)
$11$ \( 2500 - 50 T^{2} + T^{4} \)
$13$ \( ( -15 + T )^{4} \)
$17$ \( 16384 - 128 T^{2} + T^{4} \)
$19$ \( ( 169 - 13 T + T^{2} )^{2} \)
$23$ \( 262144 - 512 T^{2} + T^{4} \)
$29$ \( ( 512 + T^{2} )^{2} \)
$31$ \( ( 9 + 3 T + T^{2} )^{2} \)
$37$ \( ( 289 + 17 T + T^{2} )^{2} \)
$41$ \( ( 6498 + T^{2} )^{2} \)
$43$ \( ( 85 + T )^{4} \)
$47$ \( 27060804 - 5202 T^{2} + T^{4} \)
$53$ \( 1327104 - 1152 T^{2} + T^{4} \)
$59$ \( 67108864 - 8192 T^{2} + T^{4} \)
$61$ \( ( 5184 - 72 T + T^{2} )^{2} \)
$67$ \( ( 1849 + 43 T + T^{2} )^{2} \)
$71$ \( ( 2738 + T^{2} )^{2} \)
$73$ \( ( 9025 - 95 T + T^{2} )^{2} \)
$79$ \( ( 4761 + 69 T + T^{2} )^{2} \)
$83$ \( ( 3698 + T^{2} )^{2} \)
$89$ \( 339738624 - 18432 T^{2} + T^{4} \)
$97$ \( ( -16 + T )^{4} \)
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