Properties

Label 126.3.c.a
Level $126$
Weight $3$
Character orbit 126.c
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.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.43325133094\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-33})\)
Defining polynomial: \(x^{4} + 32 x^{2} + 289\)
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + 2 q^{4} -\beta_{2} q^{5} + ( -4 + \beta_{3} ) q^{7} + 2 \beta_{1} q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + 2 q^{4} -\beta_{2} q^{5} + ( -4 + \beta_{3} ) q^{7} + 2 \beta_{1} q^{8} + 2 \beta_{3} q^{10} + 9 \beta_{1} q^{11} -4 \beta_{3} q^{13} + ( -4 \beta_{1} - \beta_{2} ) q^{14} + 4 q^{16} -\beta_{2} q^{17} -2 \beta_{3} q^{19} -2 \beta_{2} q^{20} + 18 q^{22} + 15 \beta_{1} q^{23} -41 q^{25} + 4 \beta_{2} q^{26} + ( -8 + 2 \beta_{3} ) q^{28} -24 \beta_{1} q^{29} + 4 \beta_{1} q^{32} + 2 \beta_{3} q^{34} + ( -33 \beta_{1} + 4 \beta_{2} ) q^{35} + 16 q^{37} + 2 \beta_{2} q^{38} + 4 \beta_{3} q^{40} + 7 \beta_{2} q^{41} + 52 q^{43} + 18 \beta_{1} q^{44} + 30 q^{46} -4 \beta_{2} q^{47} + ( -17 - 8 \beta_{3} ) q^{49} -41 \beta_{1} q^{50} -8 \beta_{3} q^{52} -12 \beta_{1} q^{53} + 18 \beta_{3} q^{55} + ( -8 \beta_{1} - 2 \beta_{2} ) q^{56} -48 q^{58} + 4 \beta_{2} q^{59} + 4 \beta_{3} q^{61} + 8 q^{64} + 132 \beta_{1} q^{65} -52 q^{67} -2 \beta_{2} q^{68} + ( -66 - 8 \beta_{3} ) q^{70} -63 \beta_{1} q^{71} -8 \beta_{3} q^{73} + 16 \beta_{1} q^{74} -4 \beta_{3} q^{76} + ( -36 \beta_{1} - 9 \beta_{2} ) q^{77} + 104 q^{79} -4 \beta_{2} q^{80} -14 \beta_{3} q^{82} -20 \beta_{2} q^{83} -66 q^{85} + 52 \beta_{1} q^{86} + 36 q^{88} + 9 \beta_{2} q^{89} + ( 132 + 16 \beta_{3} ) q^{91} + 30 \beta_{1} q^{92} + 8 \beta_{3} q^{94} + 66 \beta_{1} q^{95} -16 \beta_{3} q^{97} + ( -17 \beta_{1} + 8 \beta_{2} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{4} - 16 q^{7} + O(q^{10}) \) \( 4 q + 8 q^{4} - 16 q^{7} + 16 q^{16} + 72 q^{22} - 164 q^{25} - 32 q^{28} + 64 q^{37} + 208 q^{43} + 120 q^{46} - 68 q^{49} - 192 q^{58} + 32 q^{64} - 208 q^{67} - 264 q^{70} + 416 q^{79} - 264 q^{85} + 144 q^{88} + 528 q^{91} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 32 x^{2} + 289\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + 15 \nu \)\()/17\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 49 \nu \)\()/17\)
\(\beta_{3}\)\(=\)\( \nu^{2} + 16 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} - \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{3} - 16\)
\(\nu^{3}\)\(=\)\((\)\(-15 \beta_{2} + 49 \beta_{1}\)\()/2\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
55.1
0.707107 + 4.06202i
0.707107 4.06202i
−0.707107 + 4.06202i
−0.707107 4.06202i
−1.41421 0 2.00000 8.12404i 0 −4.00000 + 5.74456i −2.82843 0 11.4891i
55.2 −1.41421 0 2.00000 8.12404i 0 −4.00000 5.74456i −2.82843 0 11.4891i
55.3 1.41421 0 2.00000 8.12404i 0 −4.00000 5.74456i 2.82843 0 11.4891i
55.4 1.41421 0 2.00000 8.12404i 0 −4.00000 + 5.74456i 2.82843 0 11.4891i
\(n\): 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.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 126.3.c.a 4
3.b odd 2 1 inner 126.3.c.a 4
4.b odd 2 1 1008.3.f.i 4
7.b odd 2 1 inner 126.3.c.a 4
7.c even 3 2 882.3.n.i 8
7.d odd 6 2 882.3.n.i 8
12.b even 2 1 1008.3.f.i 4
21.c even 2 1 inner 126.3.c.a 4
21.g even 6 2 882.3.n.i 8
21.h odd 6 2 882.3.n.i 8
28.d even 2 1 1008.3.f.i 4
84.h odd 2 1 1008.3.f.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.3.c.a 4 1.a even 1 1 trivial
126.3.c.a 4 3.b odd 2 1 inner
126.3.c.a 4 7.b odd 2 1 inner
126.3.c.a 4 21.c even 2 1 inner
882.3.n.i 8 7.c even 3 2
882.3.n.i 8 7.d odd 6 2
882.3.n.i 8 21.g even 6 2
882.3.n.i 8 21.h odd 6 2
1008.3.f.i 4 4.b odd 2 1
1008.3.f.i 4 12.b even 2 1
1008.3.f.i 4 28.d even 2 1
1008.3.f.i 4 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 66 \) acting on \(S_{3}^{\mathrm{new}}(126, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -2 + T^{2} )^{2} \)
$3$ \( T^{4} \)
$5$ \( ( 66 + T^{2} )^{2} \)
$7$ \( ( 49 + 8 T + T^{2} )^{2} \)
$11$ \( ( -162 + T^{2} )^{2} \)
$13$ \( ( 528 + T^{2} )^{2} \)
$17$ \( ( 66 + T^{2} )^{2} \)
$19$ \( ( 132 + T^{2} )^{2} \)
$23$ \( ( -450 + T^{2} )^{2} \)
$29$ \( ( -1152 + T^{2} )^{2} \)
$31$ \( T^{4} \)
$37$ \( ( -16 + T )^{4} \)
$41$ \( ( 3234 + T^{2} )^{2} \)
$43$ \( ( -52 + T )^{4} \)
$47$ \( ( 1056 + T^{2} )^{2} \)
$53$ \( ( -288 + T^{2} )^{2} \)
$59$ \( ( 1056 + T^{2} )^{2} \)
$61$ \( ( 528 + T^{2} )^{2} \)
$67$ \( ( 52 + T )^{4} \)
$71$ \( ( -7938 + T^{2} )^{2} \)
$73$ \( ( 2112 + T^{2} )^{2} \)
$79$ \( ( -104 + T )^{4} \)
$83$ \( ( 26400 + T^{2} )^{2} \)
$89$ \( ( 5346 + T^{2} )^{2} \)
$97$ \( ( 8448 + T^{2} )^{2} \)
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