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Properties

Label 126.3.n.c
Level $126$
Weight $3$
Character orbit 126.n
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.n (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: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 14)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + 2 \beta_{2} q^{4} + ( - 4 \beta_{3} - \beta_{2} - 2 \beta_1 + 1) q^{5} + ( - 5 \beta_{3} + 2 \beta_{2} - 4 \beta_1 + 3) q^{7} + 2 \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + 2 \beta_{2} q^{4} + ( - 4 \beta_{3} - \beta_{2} - 2 \beta_1 + 1) q^{5} + ( - 5 \beta_{3} + 2 \beta_{2} - 4 \beta_1 + 3) q^{7} + 2 \beta_{3} q^{8} + ( - \beta_{3} + 4 \beta_{2} + \beta_1 + 8) q^{10} + ( - 3 \beta_{3} + 9 \beta_{2} - 3 \beta_1) q^{11} + (2 \beta_{3} + 12 \beta_{2} + 4 \beta_1 + 6) q^{13} + (2 \beta_{3} + 2 \beta_{2} + 3 \beta_1 + 10) q^{14} + ( - 4 \beta_{2} - 4) q^{16} + (2 \beta_{3} + 5 \beta_{2} - 2 \beta_1 + 10) q^{17} + ( - 2 \beta_{3} - \beta_{2} - \beta_1 + 1) q^{19} + (4 \beta_{3} + 4 \beta_{2} + 8 \beta_1 + 2) q^{20} + (9 \beta_{3} + 6) q^{22} + ( - 15 \beta_{2} - 9 \beta_1 - 15) q^{23} + ( - 12 \beta_{3} - 2 \beta_{2} - 12 \beta_1) q^{25} + (12 \beta_{3} + 4 \beta_{2} + 6 \beta_1 - 4) q^{26} + (2 \beta_{3} + 2 \beta_{2} + 10 \beta_1 - 4) q^{28} + ( - 6 \beta_{3} - 12) q^{29} + (15 \beta_{3} - 7 \beta_{2} - 15 \beta_1 - 14) q^{31} + ( - 4 \beta_{3} - 4 \beta_1) q^{32} + (5 \beta_{3} - 8 \beta_{2} + 10 \beta_1 - 4) q^{34} + ( - 14 \beta_{3} - 35 \beta_{2} - 7 \beta_1 - 7) q^{35} + ( - 31 \beta_{2} + 24 \beta_1 - 31) q^{37} + ( - \beta_{3} + 2 \beta_{2} + \beta_1 + 4) q^{38} + (4 \beta_{3} + 8 \beta_{2} + 2 \beta_1 - 8) q^{40} + (10 \beta_{3} + 4 \beta_{2} + 20 \beta_1 + 2) q^{41} + (6 \beta_{3} - 2) q^{43} + ( - 18 \beta_{2} + 6 \beta_1 - 18) q^{44} + ( - 15 \beta_{3} - 18 \beta_{2} - 15 \beta_1) q^{46} + (2 \beta_{3} + 29 \beta_{2} + \beta_1 - 29) q^{47} + ( - 26 \beta_{3} - 40 \beta_{2} - 4 \beta_1 - 25) q^{49} + ( - 2 \beta_{3} + 24) q^{50} + (4 \beta_{3} - 12 \beta_{2} - 4 \beta_1 - 24) q^{52} + ( - 12 \beta_{3} - 39 \beta_{2} - 12 \beta_1) q^{53} + (15 \beta_{3} - 6 \beta_{2} + 30 \beta_1 - 3) q^{55} + (2 \beta_{3} + 16 \beta_{2} - 4 \beta_1 - 4) q^{56} + (12 \beta_{2} - 12 \beta_1 + 12) q^{58} + (25 \beta_{3} + 13 \beta_{2} - 25 \beta_1 + 26) q^{59} + (64 \beta_{3} + 7 \beta_{2} + 32 \beta_1 - 7) q^{61} + ( - 7 \beta_{3} - 60 \beta_{2} - 14 \beta_1 - 30) q^{62} + 8 q^{64} + (42 \beta_{2} + 42 \beta_1 + 42) q^{65} + ( - 45 \beta_{3} + 29 \beta_{2} - 45 \beta_1) q^{67} + ( - 8 \beta_{3} + 10 \beta_{2} - 4 \beta_1 - 10) q^{68} + ( - 35 \beta_{3} + 14 \beta_{2} - 7 \beta_1 + 28) q^{70} + (30 \beta_{3} + 6) q^{71} + (16 \beta_{3} + 53 \beta_{2} - 16 \beta_1 + 106) q^{73} + ( - 31 \beta_{3} + 48 \beta_{2} - 31 \beta_1) q^{74} + (2 \beta_{3} + 4 \beta_{2} + 4 \beta_1 + 2) q^{76} + ( - 21 \beta_{2} + 42 \beta_1 - 42) q^{77} + (55 \beta_{2} - 15 \beta_1 + 55) q^{79} + (8 \beta_{3} - 4 \beta_{2} - 8 \beta_1 - 8) q^{80} + (4 \beta_{3} + 20 \beta_{2} + 2 \beta_1 - 20) q^{82} + (4 \beta_{3} + 136 \beta_{2} + 8 \beta_1 + 68) q^{83} + ( - 24 \beta_{3} - 9) q^{85} + ( - 12 \beta_{2} - 2 \beta_1 - 12) q^{86} + ( - 18 \beta_{3} + 12 \beta_{2} - 18 \beta_1) q^{88} + (48 \beta_{3} - 63 \beta_{2} + 24 \beta_1 + 63) q^{89} + ( - 8 \beta_{3} + 48 \beta_{2} + 44 \beta_1 + 30) q^{91} + ( - 18 \beta_{3} + 30) q^{92} + (29 \beta_{3} - 2 \beta_{2} - 29 \beta_1 - 4) q^{94} + ( - 9 \beta_{3} - 15 \beta_{2} - 9 \beta_1) q^{95} + (26 \beta_{3} + 44 \beta_{2} + 52 \beta_1 + 22) q^{97} + ( - 40 \beta_{3} + 44 \beta_{2} - 25 \beta_1 + 52) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} + 6 q^{5} + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} + 6 q^{5} + 8 q^{7} + 24 q^{10} - 18 q^{11} + 36 q^{14} - 8 q^{16} + 30 q^{17} + 6 q^{19} + 24 q^{22} - 30 q^{23} + 4 q^{25} - 24 q^{26} - 20 q^{28} - 48 q^{29} - 42 q^{31} + 42 q^{35} - 62 q^{37} + 12 q^{38} - 48 q^{40} - 8 q^{43} - 36 q^{44} + 36 q^{46} - 174 q^{47} - 20 q^{49} + 96 q^{50} - 72 q^{52} + 78 q^{53} - 48 q^{56} + 24 q^{58} + 78 q^{59} - 42 q^{61} + 32 q^{64} + 84 q^{65} - 58 q^{67} - 60 q^{68} + 84 q^{70} + 24 q^{71} + 318 q^{73} - 96 q^{74} - 126 q^{77} + 110 q^{79} - 24 q^{80} - 120 q^{82} - 36 q^{85} - 24 q^{86} - 24 q^{88} + 378 q^{89} + 24 q^{91} + 120 q^{92} - 12 q^{94} + 30 q^{95} + 120 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

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

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
−0.707107 + 1.22474i
0.707107 − 1.22474i
−0.707107 − 1.22474i
0.707107 + 1.22474i
−0.707107 + 1.22474i 0 −1.00000 − 1.73205i −2.74264 − 1.58346i 0 −2.24264 − 6.63103i 2.82843 0 3.87868 − 2.23936i
19.2 0.707107 − 1.22474i 0 −1.00000 − 1.73205i 5.74264 + 3.31552i 0 6.24264 + 3.16693i −2.82843 0 8.12132 − 4.68885i
73.1 −0.707107 − 1.22474i 0 −1.00000 + 1.73205i −2.74264 + 1.58346i 0 −2.24264 + 6.63103i 2.82843 0 3.87868 + 2.23936i
73.2 0.707107 + 1.22474i 0 −1.00000 + 1.73205i 5.74264 − 3.31552i 0 6.24264 − 3.16693i −2.82843 0 8.12132 + 4.68885i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d 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.n.c 4
3.b odd 2 1 14.3.d.a ✓ 4
4.b odd 2 1 1008.3.cg.l 4
7.b odd 2 1 882.3.n.b 4
7.c even 3 1 882.3.c.f 4
7.c even 3 1 882.3.n.b 4
7.d odd 6 1 inner 126.3.n.c 4
7.d odd 6 1 882.3.c.f 4
12.b even 2 1 112.3.s.b 4
15.d odd 2 1 350.3.k.a 4
15.e even 4 2 350.3.i.a 8
21.c even 2 1 98.3.d.a 4
21.g even 6 1 14.3.d.a ✓ 4
21.g even 6 1 98.3.b.b 4
21.h odd 6 1 98.3.b.b 4
21.h odd 6 1 98.3.d.a 4
24.f even 2 1 448.3.s.c 4
24.h odd 2 1 448.3.s.d 4
28.f even 6 1 1008.3.cg.l 4
84.h odd 2 1 784.3.s.c 4
84.j odd 6 1 112.3.s.b 4
84.j odd 6 1 784.3.c.e 4
84.n even 6 1 784.3.c.e 4
84.n even 6 1 784.3.s.c 4
105.p even 6 1 350.3.k.a 4
105.w odd 12 2 350.3.i.a 8
168.ba even 6 1 448.3.s.d 4
168.be odd 6 1 448.3.s.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.3.d.a ✓ 4 3.b odd 2 1
14.3.d.a ✓ 4 21.g even 6 1
98.3.b.b 4 21.g even 6 1
98.3.b.b 4 21.h odd 6 1
98.3.d.a 4 21.c even 2 1
98.3.d.a 4 21.h odd 6 1
112.3.s.b 4 12.b even 2 1
112.3.s.b 4 84.j odd 6 1
126.3.n.c 4 1.a even 1 1 trivial
126.3.n.c 4 7.d odd 6 1 inner
350.3.i.a 8 15.e even 4 2
350.3.i.a 8 105.w odd 12 2
350.3.k.a 4 15.d odd 2 1
350.3.k.a 4 105.p even 6 1
448.3.s.c 4 24.f even 2 1
448.3.s.c 4 168.be odd 6 1
448.3.s.d 4 24.h odd 2 1
448.3.s.d 4 168.ba even 6 1
784.3.c.e 4 84.j odd 6 1
784.3.c.e 4 84.n even 6 1
784.3.s.c 4 84.h odd 2 1
784.3.s.c 4 84.n even 6 1
882.3.c.f 4 7.c even 3 1
882.3.c.f 4 7.d odd 6 1
882.3.n.b 4 7.b odd 2 1
882.3.n.b 4 7.c even 3 1
1008.3.cg.l 4 4.b odd 2 1
1008.3.cg.l 4 28.f even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 6T_{5}^{3} - 9T_{5}^{2} + 126T_{5} + 441 \) acting on \(S_{3}^{\mathrm{new}}(126, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 2T^{2} + 4 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 6 T^{3} - 9 T^{2} + 126 T + 441 \) Copy content Toggle raw display
$7$ \( T^{4} - 8 T^{3} + 42 T^{2} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( T^{4} + 18 T^{3} + 261 T^{2} + \cdots + 3969 \) Copy content Toggle raw display
$13$ \( T^{4} + 264T^{2} + 7056 \) Copy content Toggle raw display
$17$ \( T^{4} - 30 T^{3} + 351 T^{2} + \cdots + 2601 \) Copy content Toggle raw display
$19$ \( T^{4} - 6 T^{3} + 9 T^{2} + 18 T + 9 \) Copy content Toggle raw display
$23$ \( T^{4} + 30 T^{3} + 837 T^{2} + \cdots + 3969 \) Copy content Toggle raw display
$29$ \( (T^{2} + 24 T + 72)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 42 T^{3} - 615 T^{2} + \cdots + 1447209 \) Copy content Toggle raw display
$37$ \( T^{4} + 62 T^{3} + 4035 T^{2} + \cdots + 36481 \) Copy content Toggle raw display
$41$ \( T^{4} + 1224 T^{2} + 345744 \) Copy content Toggle raw display
$43$ \( (T^{2} + 4 T - 68)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 174 T^{3} + 12609 T^{2} + \cdots + 6335289 \) Copy content Toggle raw display
$53$ \( T^{4} - 78 T^{3} + 4851 T^{2} + \cdots + 1520289 \) Copy content Toggle raw display
$59$ \( T^{4} - 78 T^{3} - 1215 T^{2} + \cdots + 10517049 \) Copy content Toggle raw display
$61$ \( T^{4} + 42 T^{3} - 5409 T^{2} + \cdots + 35964009 \) Copy content Toggle raw display
$67$ \( T^{4} + 58 T^{3} + 6573 T^{2} + \cdots + 10297681 \) Copy content Toggle raw display
$71$ \( (T^{2} - 12 T - 1764)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} - 318 T^{3} + \cdots + 47485881 \) Copy content Toggle raw display
$79$ \( T^{4} - 110 T^{3} + 9525 T^{2} + \cdots + 6630625 \) Copy content Toggle raw display
$83$ \( T^{4} + 27936 T^{2} + \cdots + 189778176 \) Copy content Toggle raw display
$89$ \( T^{4} - 378 T^{3} + \cdots + 71419401 \) Copy content Toggle raw display
$97$ \( T^{4} + 11016 T^{2} + \cdots + 6780816 \) Copy content Toggle raw display
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