Properties

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

Newform invariants

Self dual: no
Analytic conductor: \(3.43325133094\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} + 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 42)
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + 2 q^{4} + ( 2 \beta_{2} - \beta_{3} ) q^{5} + ( 2 + 3 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{7} + 2 \beta_{1} q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + 2 q^{4} + ( 2 \beta_{2} - \beta_{3} ) q^{5} + ( 2 + 3 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{7} + 2 \beta_{1} q^{8} + ( 2 \beta_{2} - 2 \beta_{3} ) q^{10} + ( 6 - 3 \beta_{1} ) q^{11} + ( 8 \beta_{2} - 2 \beta_{3} ) q^{13} + ( 6 + 2 \beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{14} + 4 q^{16} + ( 2 \beta_{2} + 11 \beta_{3} ) q^{17} + ( -2 \beta_{2} + 8 \beta_{3} ) q^{19} + ( 4 \beta_{2} - 2 \beta_{3} ) q^{20} + ( -6 + 6 \beta_{1} ) q^{22} + ( -6 - 9 \beta_{1} ) q^{23} + ( 7 - 12 \beta_{1} ) q^{25} + ( 4 \beta_{2} - 8 \beta_{3} ) q^{26} + ( 4 + 6 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} ) q^{28} -30 q^{29} + ( -12 \beta_{2} - 12 \beta_{3} ) q^{31} + 4 \beta_{1} q^{32} + ( -22 \beta_{2} - 2 \beta_{3} ) q^{34} + ( 6 + 9 \beta_{1} + 10 \beta_{2} - 8 \beta_{3} ) q^{35} + ( -20 - 36 \beta_{1} ) q^{37} + ( -16 \beta_{2} + 2 \beta_{3} ) q^{38} + ( 4 \beta_{2} - 4 \beta_{3} ) q^{40} + ( -14 \beta_{2} + 7 \beta_{3} ) q^{41} + ( -32 + 30 \beta_{1} ) q^{43} + ( 12 - 6 \beta_{1} ) q^{44} + ( -18 - 6 \beta_{1} ) q^{46} + ( -28 \beta_{2} - 4 \beta_{3} ) q^{47} + ( -5 + 24 \beta_{1} - 20 \beta_{2} + 2 \beta_{3} ) q^{49} + ( -24 + 7 \beta_{1} ) q^{50} + ( 16 \beta_{2} - 4 \beta_{3} ) q^{52} + ( 54 - 12 \beta_{1} ) q^{53} + 6 \beta_{2} q^{55} + ( 12 + 4 \beta_{1} - 8 \beta_{2} - 2 \beta_{3} ) q^{56} -30 \beta_{1} q^{58} + ( 28 \beta_{2} - 20 \beta_{3} ) q^{59} + ( 4 \beta_{2} - 4 \beta_{3} ) q^{61} + ( 24 \beta_{2} + 12 \beta_{3} ) q^{62} + 8 q^{64} + ( -60 - 36 \beta_{1} ) q^{65} + ( 44 + 12 \beta_{1} ) q^{67} + ( 4 \beta_{2} + 22 \beta_{3} ) q^{68} + ( 18 + 6 \beta_{1} + 16 \beta_{2} - 10 \beta_{3} ) q^{70} + ( 30 + 57 \beta_{1} ) q^{71} + ( 4 \beta_{2} + 26 \beta_{3} ) q^{73} + ( -72 - 20 \beta_{1} ) q^{74} + ( -4 \beta_{2} + 16 \beta_{3} ) q^{76} + ( -6 + 12 \beta_{1} + 18 \beta_{2} + 15 \beta_{3} ) q^{77} + ( 32 - 72 \beta_{1} ) q^{79} + ( 8 \beta_{2} - 4 \beta_{3} ) q^{80} + ( -14 \beta_{2} + 14 \beta_{3} ) q^{82} + ( -32 \beta_{2} - 20 \beta_{3} ) q^{83} + ( 54 + 60 \beta_{1} ) q^{85} + ( 60 - 32 \beta_{1} ) q^{86} + ( -12 + 12 \beta_{1} ) q^{88} + ( 54 \beta_{2} + 21 \beta_{3} ) q^{89} + ( 42 \beta_{1} + 28 \beta_{2} - 28 \beta_{3} ) q^{91} + ( -12 - 18 \beta_{1} ) q^{92} + ( 8 \beta_{2} + 28 \beta_{3} ) q^{94} + ( 60 + 54 \beta_{1} ) q^{95} + ( 44 \beta_{2} + 10 \beta_{3} ) q^{97} + ( 48 - 5 \beta_{1} - 4 \beta_{2} + 20 \beta_{3} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 8q^{4} + 8q^{7} + O(q^{10}) \) \( 4q + 8q^{4} + 8q^{7} + 24q^{11} + 24q^{14} + 16q^{16} - 24q^{22} - 24q^{23} + 28q^{25} + 16q^{28} - 120q^{29} + 24q^{35} - 80q^{37} - 128q^{43} + 48q^{44} - 72q^{46} - 20q^{49} - 96q^{50} + 216q^{53} + 48q^{56} + 32q^{64} - 240q^{65} + 176q^{67} + 72q^{70} + 120q^{71} - 288q^{74} - 24q^{77} + 128q^{79} + 216q^{85} + 240q^{86} - 48q^{88} - 48q^{92} + 240q^{95} + 192q^{98} + O(q^{100}) \)

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

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

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 1.22474i
0.707107 + 1.22474i
−0.707107 + 1.22474i
−0.707107 1.22474i
−1.41421 0 2.00000 1.01461i 0 −2.24264 6.63103i −2.82843 0 1.43488i
55.2 −1.41421 0 2.00000 1.01461i 0 −2.24264 + 6.63103i −2.82843 0 1.43488i
55.3 1.41421 0 2.00000 5.91359i 0 6.24264 + 3.16693i 2.82843 0 8.36308i
55.4 1.41421 0 2.00000 5.91359i 0 6.24264 3.16693i 2.82843 0 8.36308i
\(n\): 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 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 126.3.c.b 4
3.b odd 2 1 42.3.c.a 4
4.b odd 2 1 1008.3.f.g 4
7.b odd 2 1 inner 126.3.c.b 4
7.c even 3 1 882.3.n.a 4
7.c even 3 1 882.3.n.d 4
7.d odd 6 1 882.3.n.a 4
7.d odd 6 1 882.3.n.d 4
12.b even 2 1 336.3.f.c 4
15.d odd 2 1 1050.3.f.a 4
15.e even 4 2 1050.3.h.a 8
21.c even 2 1 42.3.c.a 4
21.g even 6 1 294.3.g.b 4
21.g even 6 1 294.3.g.c 4
21.h odd 6 1 294.3.g.b 4
21.h odd 6 1 294.3.g.c 4
24.f even 2 1 1344.3.f.e 4
24.h odd 2 1 1344.3.f.f 4
28.d even 2 1 1008.3.f.g 4
84.h odd 2 1 336.3.f.c 4
105.g even 2 1 1050.3.f.a 4
105.k odd 4 2 1050.3.h.a 8
168.e odd 2 1 1344.3.f.e 4
168.i even 2 1 1344.3.f.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.3.c.a 4 3.b odd 2 1
42.3.c.a 4 21.c even 2 1
126.3.c.b 4 1.a even 1 1 trivial
126.3.c.b 4 7.b odd 2 1 inner
294.3.g.b 4 21.g even 6 1
294.3.g.b 4 21.h odd 6 1
294.3.g.c 4 21.g even 6 1
294.3.g.c 4 21.h odd 6 1
336.3.f.c 4 12.b even 2 1
336.3.f.c 4 84.h odd 2 1
882.3.n.a 4 7.c even 3 1
882.3.n.a 4 7.d odd 6 1
882.3.n.d 4 7.c even 3 1
882.3.n.d 4 7.d odd 6 1
1008.3.f.g 4 4.b odd 2 1
1008.3.f.g 4 28.d even 2 1
1050.3.f.a 4 15.d odd 2 1
1050.3.f.a 4 105.g even 2 1
1050.3.h.a 8 15.e even 4 2
1050.3.h.a 8 105.k odd 4 2
1344.3.f.e 4 24.f even 2 1
1344.3.f.e 4 168.e odd 2 1
1344.3.f.f 4 24.h odd 2 1
1344.3.f.f 4 168.i even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 36 T_{5}^{2} + 36 \) acting on \(S_{3}^{\mathrm{new}}(126, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -2 + T^{2} )^{2} \)
$3$ \( T^{4} \)
$5$ \( 36 + 36 T^{2} + T^{4} \)
$7$ \( 2401 - 392 T + 42 T^{2} - 8 T^{3} + T^{4} \)
$11$ \( ( 18 - 12 T + T^{2} )^{2} \)
$13$ \( 28224 + 432 T^{2} + T^{4} \)
$17$ \( 509796 + 1476 T^{2} + T^{4} \)
$19$ \( 138384 + 792 T^{2} + T^{4} \)
$23$ \( ( -126 + 12 T + T^{2} )^{2} \)
$29$ \( ( 30 + T )^{4} \)
$31$ \( 186624 + 2592 T^{2} + T^{4} \)
$37$ \( ( -2192 + 40 T + T^{2} )^{2} \)
$41$ \( 86436 + 1764 T^{2} + T^{4} \)
$43$ \( ( -776 + 64 T + T^{2} )^{2} \)
$47$ \( 5089536 + 4896 T^{2} + T^{4} \)
$53$ \( ( 2628 - 108 T + T^{2} )^{2} \)
$59$ \( 2304 + 9504 T^{2} + T^{4} \)
$61$ \( 2304 + 288 T^{2} + T^{4} \)
$67$ \( ( 1648 - 88 T + T^{2} )^{2} \)
$71$ \( ( -5598 - 60 T + T^{2} )^{2} \)
$73$ \( 16064064 + 8208 T^{2} + T^{4} \)
$79$ \( ( -9344 - 64 T + T^{2} )^{2} \)
$83$ \( 451584 + 10944 T^{2} + T^{4} \)
$89$ \( 37234404 + 22788 T^{2} + T^{4} \)
$97$ \( 27123264 + 12816 T^{2} + T^{4} \)
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