Properties

Label 126.3.s.a
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + 2 \beta_{2} q^{4} + 3 \beta_{1} q^{5} + ( -7 + 7 \beta_{2} ) q^{7} + 2 \beta_{3} q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + 2 \beta_{2} q^{4} + 3 \beta_{1} q^{5} + ( -7 + 7 \beta_{2} ) q^{7} + 2 \beta_{3} q^{8} + 6 \beta_{2} q^{10} + ( 9 \beta_{1} - 9 \beta_{3} ) q^{11} - q^{13} + ( -7 \beta_{1} + 7 \beta_{3} ) q^{14} + ( -4 + 4 \beta_{2} ) q^{16} + ( 12 \beta_{1} - 12 \beta_{3} ) q^{17} + ( -23 + 23 \beta_{2} ) q^{19} + 6 \beta_{3} q^{20} + 18 q^{22} -12 \beta_{1} q^{23} -7 \beta_{2} q^{25} -\beta_{1} q^{26} -14 q^{28} -24 \beta_{3} q^{29} -47 \beta_{2} q^{31} + ( -4 \beta_{1} + 4 \beta_{3} ) q^{32} + 24 q^{34} + ( -21 \beta_{1} + 21 \beta_{3} ) q^{35} + ( 55 - 55 \beta_{2} ) q^{37} + ( -23 \beta_{1} + 23 \beta_{3} ) q^{38} + ( -12 + 12 \beta_{2} ) q^{40} + 33 \beta_{3} q^{41} + 23 q^{43} + 18 \beta_{1} q^{44} -24 \beta_{2} q^{46} -3 \beta_{1} q^{47} -49 \beta_{2} q^{49} -7 \beta_{3} q^{50} -2 \beta_{2} q^{52} + ( 36 \beta_{1} - 36 \beta_{3} ) q^{53} + 54 q^{55} -14 \beta_{1} q^{56} + ( 48 - 48 \beta_{2} ) q^{58} + ( -60 \beta_{1} + 60 \beta_{3} ) q^{59} + ( -104 + 104 \beta_{2} ) q^{61} -47 \beta_{3} q^{62} -8 q^{64} -3 \beta_{1} q^{65} + 97 \beta_{2} q^{67} + 24 \beta_{1} q^{68} -42 q^{70} -69 \beta_{3} q^{71} -65 \beta_{2} q^{73} + ( 55 \beta_{1} - 55 \beta_{3} ) q^{74} -46 q^{76} + 63 \beta_{3} q^{77} + ( -113 + 113 \beta_{2} ) q^{79} + ( -12 \beta_{1} + 12 \beta_{3} ) q^{80} + ( -66 + 66 \beta_{2} ) q^{82} + 21 \beta_{3} q^{83} + 72 q^{85} + 23 \beta_{1} q^{86} + 36 \beta_{2} q^{88} + 96 \beta_{1} q^{89} + ( 7 - 7 \beta_{2} ) q^{91} -24 \beta_{3} q^{92} -6 \beta_{2} q^{94} + ( -69 \beta_{1} + 69 \beta_{3} ) q^{95} + 104 q^{97} -49 \beta_{3} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{4} - 14q^{7} + O(q^{10}) \) \( 4q + 4q^{4} - 14q^{7} + 12q^{10} - 4q^{13} - 8q^{16} - 46q^{19} + 72q^{22} - 14q^{25} - 56q^{28} - 94q^{31} + 96q^{34} + 110q^{37} - 24q^{40} + 92q^{43} - 48q^{46} - 98q^{49} - 4q^{52} + 216q^{55} + 96q^{58} - 208q^{61} - 32q^{64} + 194q^{67} - 168q^{70} - 130q^{73} - 184q^{76} - 226q^{79} - 132q^{82} + 288q^{85} + 72q^{88} + 14q^{91} - 12q^{94} + 416q^{97} + O(q^{100}) \)

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 −3.67423 + 2.12132i 0 −3.50000 6.06218i 2.82843i 0 3.00000 5.19615i
53.2 1.22474 0.707107i 0 1.00000 1.73205i 3.67423 2.12132i 0 −3.50000 6.06218i 2.82843i 0 3.00000 5.19615i
107.1 −1.22474 0.707107i 0 1.00000 + 1.73205i −3.67423 2.12132i 0 −3.50000 + 6.06218i 2.82843i 0 3.00000 + 5.19615i
107.2 1.22474 + 0.707107i 0 1.00000 + 1.73205i 3.67423 + 2.12132i 0 −3.50000 + 6.06218i 2.82843i 0 3.00000 + 5.19615i
\(n\): 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.a 4
3.b odd 2 1 inner 126.3.s.a 4
4.b odd 2 1 1008.3.dc.b 4
7.b odd 2 1 882.3.s.a 4
7.c even 3 1 inner 126.3.s.a 4
7.c even 3 1 882.3.b.b 2
7.d odd 6 1 882.3.b.e 2
7.d odd 6 1 882.3.s.a 4
12.b even 2 1 1008.3.dc.b 4
21.c even 2 1 882.3.s.a 4
21.g even 6 1 882.3.b.e 2
21.g even 6 1 882.3.s.a 4
21.h odd 6 1 inner 126.3.s.a 4
21.h odd 6 1 882.3.b.b 2
28.g odd 6 1 1008.3.dc.b 4
84.n even 6 1 1008.3.dc.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.3.s.a 4 1.a even 1 1 trivial
126.3.s.a 4 3.b odd 2 1 inner
126.3.s.a 4 7.c even 3 1 inner
126.3.s.a 4 21.h odd 6 1 inner
882.3.b.b 2 7.c even 3 1
882.3.b.b 2 21.h odd 6 1
882.3.b.e 2 7.d odd 6 1
882.3.b.e 2 21.g even 6 1
882.3.s.a 4 7.b odd 2 1
882.3.s.a 4 7.d odd 6 1
882.3.s.a 4 21.c even 2 1
882.3.s.a 4 21.g even 6 1
1008.3.dc.b 4 4.b odd 2 1
1008.3.dc.b 4 12.b even 2 1
1008.3.dc.b 4 28.g odd 6 1
1008.3.dc.b 4 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 18 T_{5}^{2} + 324 \) acting on \(S_{3}^{\mathrm{new}}(126, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 - 2 T^{2} + T^{4} \)
$3$ \( T^{4} \)
$5$ \( 324 - 18 T^{2} + T^{4} \)
$7$ \( ( 49 + 7 T + T^{2} )^{2} \)
$11$ \( 26244 - 162 T^{2} + T^{4} \)
$13$ \( ( 1 + T )^{4} \)
$17$ \( 82944 - 288 T^{2} + T^{4} \)
$19$ \( ( 529 + 23 T + T^{2} )^{2} \)
$23$ \( 82944 - 288 T^{2} + T^{4} \)
$29$ \( ( 1152 + T^{2} )^{2} \)
$31$ \( ( 2209 + 47 T + T^{2} )^{2} \)
$37$ \( ( 3025 - 55 T + T^{2} )^{2} \)
$41$ \( ( 2178 + T^{2} )^{2} \)
$43$ \( ( -23 + T )^{4} \)
$47$ \( 324 - 18 T^{2} + T^{4} \)
$53$ \( 6718464 - 2592 T^{2} + T^{4} \)
$59$ \( 51840000 - 7200 T^{2} + T^{4} \)
$61$ \( ( 10816 + 104 T + T^{2} )^{2} \)
$67$ \( ( 9409 - 97 T + T^{2} )^{2} \)
$71$ \( ( 9522 + T^{2} )^{2} \)
$73$ \( ( 4225 + 65 T + T^{2} )^{2} \)
$79$ \( ( 12769 + 113 T + T^{2} )^{2} \)
$83$ \( ( 882 + T^{2} )^{2} \)
$89$ \( 339738624 - 18432 T^{2} + T^{4} \)
$97$ \( ( -104 + T )^{4} \)
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