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 $2$

Related objects

Downloads

Learn more

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: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + 2 q^{4} + ( - \beta_{3} + 2 \beta_{2}) q^{5} + (2 \beta_{3} + \beta_{2} + 3 \beta_1 + 2) q^{7} + 2 \beta_1 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + 2 q^{4} + ( - \beta_{3} + 2 \beta_{2}) q^{5} + (2 \beta_{3} + \beta_{2} + 3 \beta_1 + 2) q^{7} + 2 \beta_1 q^{8} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{10} + ( - 3 \beta_1 + 6) q^{11} + ( - 2 \beta_{3} + 8 \beta_{2}) q^{13} + ( - \beta_{3} - 4 \beta_{2} + 2 \beta_1 + 6) q^{14} + 4 q^{16} + (11 \beta_{3} + 2 \beta_{2}) q^{17} + (8 \beta_{3} - 2 \beta_{2}) q^{19} + ( - 2 \beta_{3} + 4 \beta_{2}) q^{20} + (6 \beta_1 - 6) q^{22} + ( - 9 \beta_1 - 6) q^{23} + ( - 12 \beta_1 + 7) q^{25} + ( - 8 \beta_{3} + 4 \beta_{2}) q^{26} + (4 \beta_{3} + 2 \beta_{2} + 6 \beta_1 + 4) q^{28} - 30 q^{29} + ( - 12 \beta_{3} - 12 \beta_{2}) q^{31} + 4 \beta_1 q^{32} + ( - 2 \beta_{3} - 22 \beta_{2}) q^{34} + ( - 8 \beta_{3} + 10 \beta_{2} + 9 \beta_1 + 6) q^{35} + ( - 36 \beta_1 - 20) q^{37} + (2 \beta_{3} - 16 \beta_{2}) q^{38} + ( - 4 \beta_{3} + 4 \beta_{2}) q^{40} + (7 \beta_{3} - 14 \beta_{2}) q^{41} + (30 \beta_1 - 32) q^{43} + ( - 6 \beta_1 + 12) q^{44} + ( - 6 \beta_1 - 18) q^{46} + ( - 4 \beta_{3} - 28 \beta_{2}) q^{47} + (2 \beta_{3} - 20 \beta_{2} + 24 \beta_1 - 5) q^{49} + (7 \beta_1 - 24) q^{50} + ( - 4 \beta_{3} + 16 \beta_{2}) q^{52} + ( - 12 \beta_1 + 54) q^{53} + 6 \beta_{2} q^{55} + ( - 2 \beta_{3} - 8 \beta_{2} + 4 \beta_1 + 12) q^{56} - 30 \beta_1 q^{58} + ( - 20 \beta_{3} + 28 \beta_{2}) q^{59} + ( - 4 \beta_{3} + 4 \beta_{2}) q^{61} + (12 \beta_{3} + 24 \beta_{2}) q^{62} + 8 q^{64} + ( - 36 \beta_1 - 60) q^{65} + (12 \beta_1 + 44) q^{67} + (22 \beta_{3} + 4 \beta_{2}) q^{68} + ( - 10 \beta_{3} + 16 \beta_{2} + 6 \beta_1 + 18) q^{70} + (57 \beta_1 + 30) q^{71} + (26 \beta_{3} + 4 \beta_{2}) q^{73} + ( - 20 \beta_1 - 72) q^{74} + (16 \beta_{3} - 4 \beta_{2}) q^{76} + (15 \beta_{3} + 18 \beta_{2} + 12 \beta_1 - 6) q^{77} + ( - 72 \beta_1 + 32) q^{79} + ( - 4 \beta_{3} + 8 \beta_{2}) q^{80} + (14 \beta_{3} - 14 \beta_{2}) q^{82} + ( - 20 \beta_{3} - 32 \beta_{2}) q^{83} + (60 \beta_1 + 54) q^{85} + ( - 32 \beta_1 + 60) q^{86} + (12 \beta_1 - 12) q^{88} + (21 \beta_{3} + 54 \beta_{2}) q^{89} + ( - 28 \beta_{3} + 28 \beta_{2} + 42 \beta_1) q^{91} + ( - 18 \beta_1 - 12) q^{92} + (28 \beta_{3} + 8 \beta_{2}) q^{94} + (54 \beta_1 + 60) q^{95} + (10 \beta_{3} + 44 \beta_{2}) q^{97} + (20 \beta_{3} - 4 \beta_{2} - 5 \beta_1 + 48) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{4} + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{4} + 8 q^{7} + 24 q^{11} + 24 q^{14} + 16 q^{16} - 24 q^{22} - 24 q^{23} + 28 q^{25} + 16 q^{28} - 120 q^{29} + 24 q^{35} - 80 q^{37} - 128 q^{43} + 48 q^{44} - 72 q^{46} - 20 q^{49} - 96 q^{50} + 216 q^{53} + 48 q^{56} + 32 q^{64} - 240 q^{65} + 176 q^{67} + 72 q^{70} + 120 q^{71} - 288 q^{74} - 24 q^{77} + 128 q^{79} + 216 q^{85} + 240 q^{86} - 48 q^{88} - 48 q^{92} + 240 q^{95} + 192 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 4\nu ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_1 \) Copy content Toggle raw display

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 \( T_{5}^{4} + 36T_{5}^{2} + 36 \) acting on \(S_{3}^{\mathrm{new}}(126, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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