Properties

Label 124.2.d.b
Level $124$
Weight $2$
Character orbit 124.d
Analytic conductor $0.990$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 124 = 2^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 124.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.990144985064\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{6}, \sqrt{-7})\)
Defining polynomial: \( x^{4} - 2x^{3} - 7x^{2} + 8x + 58 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + \beta_{3} q^{3} + ( - \beta_1 - 2) q^{4} - 2 q^{5} + (\beta_{3} + \beta_{2}) q^{6} + (\beta_1 - 2) q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + \beta_{3} q^{3} + ( - \beta_1 - 2) q^{4} - 2 q^{5} + (\beta_{3} + \beta_{2}) q^{6} + (\beta_1 - 2) q^{8} + 3 q^{9} + 2 \beta_1 q^{10} + \beta_{3} q^{11} + ( - \beta_{3} + \beta_{2}) q^{12} + ( - \beta_{3} - 2 \beta_{2}) q^{13} - 2 \beta_{3} q^{15} + (3 \beta_1 + 2) q^{16} - 3 \beta_1 q^{18} + ( - 4 \beta_1 - 2) q^{19} + (2 \beta_1 + 4) q^{20} + (\beta_{3} + \beta_{2}) q^{22} - 2 \beta_{3} q^{23} + ( - 3 \beta_{3} - \beta_{2}) q^{24} - q^{25} + (3 \beta_{3} - \beta_{2}) q^{26} + (\beta_{3} + 2 \beta_{2}) q^{29} + ( - 2 \beta_{3} - 2 \beta_{2}) q^{30} + (2 \beta_{3} + 2 \beta_1 + 1) q^{31} + (\beta_1 + 6) q^{32} + 6 q^{33} + ( - 3 \beta_1 - 6) q^{36} + ( - \beta_{3} - 2 \beta_{2}) q^{37} + ( - 2 \beta_1 - 8) q^{38} + (12 \beta_1 + 6) q^{39} + ( - 2 \beta_1 + 4) q^{40} - 8 q^{41} + \beta_{3} q^{43} + ( - \beta_{3} + \beta_{2}) q^{44} - 6 q^{45} + ( - 2 \beta_{3} - 2 \beta_{2}) q^{46} + ( - 8 \beta_1 - 4) q^{47} + ( - \beta_{3} - 3 \beta_{2}) q^{48} + 7 q^{49} + \beta_1 q^{50} + (5 \beta_{3} + 3 \beta_{2}) q^{52} + (\beta_{3} + 2 \beta_{2}) q^{53} - 2 \beta_{3} q^{55} + (2 \beta_{3} + 4 \beta_{2}) q^{57} + ( - 3 \beta_{3} + \beta_{2}) q^{58} + (4 \beta_1 + 2) q^{59} + (2 \beta_{3} - 2 \beta_{2}) q^{60} + (\beta_{3} + 2 \beta_{2}) q^{61} + (2 \beta_{3} + 2 \beta_{2} + \beta_1 + 4) q^{62} + ( - 5 \beta_1 + 2) q^{64} + (2 \beta_{3} + 4 \beta_{2}) q^{65} - 6 \beta_1 q^{66} + ( - 4 \beta_1 - 2) q^{67} - 12 q^{69} + ( - 4 \beta_1 - 2) q^{71} + (3 \beta_1 - 6) q^{72} + ( - 2 \beta_{3} - 4 \beta_{2}) q^{73} + (3 \beta_{3} - \beta_{2}) q^{74} - \beta_{3} q^{75} + (6 \beta_1 - 4) q^{76} + (6 \beta_1 + 24) q^{78} - 4 \beta_{3} q^{79} + ( - 6 \beta_1 - 4) q^{80} - 9 q^{81} + 8 \beta_1 q^{82} + 5 \beta_{3} q^{83} + (\beta_{3} + \beta_{2}) q^{86} + ( - 12 \beta_1 - 6) q^{87} + ( - 3 \beta_{3} - \beta_{2}) q^{88} + ( - 2 \beta_{3} - 4 \beta_{2}) q^{89} + 6 \beta_1 q^{90} + (2 \beta_{3} - 2 \beta_{2}) q^{92} + ( - \beta_{3} - 2 \beta_{2} + 12) q^{93} + ( - 4 \beta_1 - 16) q^{94} + (8 \beta_1 + 4) q^{95} + (5 \beta_{3} - \beta_{2}) q^{96} + 4 q^{97} - 7 \beta_1 q^{98} + 3 \beta_{3} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} - 6 q^{4} - 8 q^{5} - 10 q^{8} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} - 6 q^{4} - 8 q^{5} - 10 q^{8} + 12 q^{9} - 4 q^{10} + 2 q^{16} + 6 q^{18} + 12 q^{20} - 4 q^{25} + 22 q^{32} + 24 q^{33} - 18 q^{36} - 28 q^{38} + 20 q^{40} - 32 q^{41} - 24 q^{45} + 28 q^{49} - 2 q^{50} + 14 q^{62} + 18 q^{64} + 12 q^{66} - 48 q^{69} - 30 q^{72} - 28 q^{76} + 84 q^{78} - 4 q^{80} - 36 q^{81} - 16 q^{82} - 12 q^{90} + 48 q^{93} - 56 q^{94} + 16 q^{97} + 14 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 7x^{2} + 8x + 58 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -2\nu^{3} + 3\nu^{2} - 16 ) / 31 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 14\nu^{2} - 31\nu - 54 ) / 31 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{3} + 3\nu^{2} + 31\nu - 16 ) / 31 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_{3} - \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{3} + 2\beta_{2} - \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{3} + 3\beta_{2} - 17\beta _1 - 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/124\mathbb{Z}\right)^\times\).

\(n\) \(63\) \(65\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
123.1
−1.94949 1.32288i
2.94949 1.32288i
−1.94949 + 1.32288i
2.94949 + 1.32288i
0.500000 1.32288i −2.44949 −1.50000 1.32288i −2.00000 −1.22474 + 3.24037i 0 −2.50000 + 1.32288i 3.00000 −1.00000 + 2.64575i
123.2 0.500000 1.32288i 2.44949 −1.50000 1.32288i −2.00000 1.22474 3.24037i 0 −2.50000 + 1.32288i 3.00000 −1.00000 + 2.64575i
123.3 0.500000 + 1.32288i −2.44949 −1.50000 + 1.32288i −2.00000 −1.22474 3.24037i 0 −2.50000 1.32288i 3.00000 −1.00000 2.64575i
123.4 0.500000 + 1.32288i 2.44949 −1.50000 + 1.32288i −2.00000 1.22474 + 3.24037i 0 −2.50000 1.32288i 3.00000 −1.00000 2.64575i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
31.b odd 2 1 inner
124.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 124.2.d.b 4
3.b odd 2 1 1116.2.g.b 4
4.b odd 2 1 inner 124.2.d.b 4
8.b even 2 1 1984.2.h.e 4
8.d odd 2 1 1984.2.h.e 4
12.b even 2 1 1116.2.g.b 4
31.b odd 2 1 inner 124.2.d.b 4
93.c even 2 1 1116.2.g.b 4
124.d even 2 1 inner 124.2.d.b 4
248.b even 2 1 1984.2.h.e 4
248.g odd 2 1 1984.2.h.e 4
372.b odd 2 1 1116.2.g.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
124.2.d.b 4 1.a even 1 1 trivial
124.2.d.b 4 4.b odd 2 1 inner
124.2.d.b 4 31.b odd 2 1 inner
124.2.d.b 4 124.d even 2 1 inner
1116.2.g.b 4 3.b odd 2 1
1116.2.g.b 4 12.b even 2 1
1116.2.g.b 4 93.c even 2 1
1116.2.g.b 4 372.b odd 2 1
1984.2.h.e 4 8.b even 2 1
1984.2.h.e 4 8.d odd 2 1
1984.2.h.e 4 248.b even 2 1
1984.2.h.e 4 248.g odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(124, [\chi])\):

\( T_{3}^{2} - 6 \) Copy content Toggle raw display
\( T_{5} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 2)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} - 6)^{2} \) Copy content Toggle raw display
$5$ \( (T + 2)^{4} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} - 6)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 42)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 28)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 24)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 42)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} - 34T^{2} + 961 \) Copy content Toggle raw display
$37$ \( (T^{2} + 42)^{2} \) Copy content Toggle raw display
$41$ \( (T + 8)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} - 6)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 112)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 42)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 28)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 42)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 28)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 28)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 168)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 96)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 150)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 168)^{2} \) Copy content Toggle raw display
$97$ \( (T - 4)^{4} \) Copy content Toggle raw display
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