Properties

Label 124.2.d.c
Level $124$
Weight $2$
Character orbit 124.d
Analytic conductor $0.990$
Analytic rank $0$
Dimension $6$
CM discriminant -31
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: \(6\)
Coefficient field: 6.0.21717639.1
Defining polynomial: \( x^{6} - x^{3} + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{5} q^{2} - \beta_{3} q^{4} + ( - \beta_{5} + \beta_{4} + \beta_{3}) q^{5} + ( - \beta_{5} - \beta_{4} + \beta_{3}) q^{7} + ( - \beta_{2} + 1) q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{2} - \beta_{3} q^{4} + ( - \beta_{5} + \beta_{4} + \beta_{3}) q^{5} + ( - \beta_{5} - \beta_{4} + \beta_{3}) q^{7} + ( - \beta_{2} + 1) q^{8} - 3 q^{9} + ( - \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{10} + (\beta_{4} - \beta_{3} + \beta_{2} + \beta_1 - 3) q^{14} + ( - \beta_{5} + \beta_{4} - \beta_1) q^{16} + 3 \beta_{5} q^{18} + (3 \beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_1) q^{19} + ( - \beta_{5} - \beta_{4} - \beta_{2} + \beta_1 - 3) q^{20} + (3 \beta_{5} - \beta_{4} + \beta_{3} - 2 \beta_1 + 5) q^{25} + (3 \beta_{5} - \beta_{4} - \beta_{2} + \beta_1 + 5) q^{28} + ( - 2 \beta_{2} + 1) q^{31} + ( - 2 \beta_{4} - \beta_{3} - 2 \beta_1) q^{32} + ( - \beta_{5} + \beta_{4} - 3 \beta_{3} + 4 \beta_{2} - 2 \beta_1 - 2) q^{35} + 3 \beta_{3} q^{36} + (\beta_{4} + 3 \beta_{3} + \beta_{2} + \beta_1 + 5) q^{38} + (3 \beta_{5} + 3 \beta_{4} - \beta_{3} + \beta_1) q^{40} + ( - \beta_{5} - \beta_{4} - 3 \beta_{3} + 2 \beta_1) q^{41} + (3 \beta_{5} - 3 \beta_{4} - 3 \beta_{3}) q^{45} + ( - 4 \beta_{2} + 2) q^{47} + ( - 5 \beta_{5} + 3 \beta_{4} + \beta_{3} + 2 \beta_1 - 7) q^{49} + ( - 5 \beta_{5} - \beta_{4} + 3 \beta_{3} + \beta_{2} - \beta_1 - 7) q^{50} + ( - 5 \beta_{5} + 3 \beta_{4} + 3 \beta_{3} + \beta_1) q^{56} + ( - 5 \beta_{5} - 3 \beta_{4} + \beta_{3} - 2 \beta_1) q^{59} + ( - \beta_{5} + 2 \beta_{4} - 2 \beta_1) q^{62} + (3 \beta_{5} + 3 \beta_{4} - 3 \beta_{3}) q^{63} + ( - \beta_{2} - 7) q^{64} + (4 \beta_{2} - 2) q^{67} + (2 \beta_{5} - 7 \beta_{4} - \beta_{3} - 3 \beta_{2} + \beta_1 + 1) q^{70} + ( - 5 \beta_{5} - \beta_{4} - 3 \beta_{3} - 4 \beta_1) q^{71} + (3 \beta_{2} - 3) q^{72} + ( - 5 \beta_{5} - \beta_{4} + 3 \beta_{2} + \beta_1 + 1) q^{76} + ( - 2 \beta_{4} + 3 \beta_{3} - \beta_{2} - 2 \beta_1 + 9) q^{80} + 9 q^{81} + (3 \beta_{4} - \beta_{3} - 3 \beta_{2} + 3 \beta_1 + 5) q^{82} + (3 \beta_{4} + 3 \beta_{3} - 3 \beta_{2} + 3 \beta_1 - 3) q^{90} + ( - 2 \beta_{5} + 4 \beta_{4} - 4 \beta_1) q^{94} + (7 \beta_{5} + 3 \beta_{4} + \beta_{3} - 4 \beta_{2} + 4 \beta_1 + 2) q^{95} + ( - \beta_{5} + 3 \beta_{4} + 5 \beta_{3} - 2 \beta_1) q^{97} + (7 \beta_{5} - \beta_{4} - 5 \beta_{3} + \beta_{2} - \beta_1 + 9) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{8} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 3 q^{8} - 18 q^{9} + 9 q^{10} - 15 q^{14} - 21 q^{20} + 30 q^{25} + 27 q^{28} + 33 q^{38} - 42 q^{49} - 39 q^{50} - 45 q^{64} - 3 q^{70} - 9 q^{72} + 15 q^{76} + 51 q^{80} + 54 q^{81} + 21 q^{82} - 27 q^{90} + 57 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{3} + 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} + \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{4} - \nu ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\nu^{2} + \nu \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} - \nu^{2} ) / 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{4} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{4} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{4} + 4\beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 8\beta_{5} - \beta_{4} + \beta_1 ) / 2 \) 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
−1.18073 + 0.778374i
−1.18073 0.778374i
−0.0837246 + 1.41173i
−0.0837246 1.41173i
1.26446 + 0.633359i
1.26446 0.633359i
−1.18073 0.778374i 0 0.788267 + 1.83811i −3.93800 0 5.23296i 0.500000 2.78388i −3.00000 4.64974 + 3.06524i
123.2 −1.18073 + 0.778374i 0 0.788267 1.83811i −3.93800 0 5.23296i 0.500000 + 2.78388i −3.00000 4.64974 3.06524i
123.3 −0.0837246 1.41173i 0 −1.98598 + 0.236394i 3.80451 0 3.29625i 0.500000 + 2.78388i −3.00000 −0.318531 5.37095i
123.4 −0.0837246 + 1.41173i 0 −1.98598 0.236394i 3.80451 0 3.29625i 0.500000 2.78388i −3.00000 −0.318531 + 5.37095i
123.5 1.26446 0.633359i 0 1.19771 1.60171i 0.133492 0 1.93671i 0.500000 2.78388i −3.00000 0.168795 0.0845483i
123.6 1.26446 + 0.633359i 0 1.19771 + 1.60171i 0.133492 0 1.93671i 0.500000 + 2.78388i −3.00000 0.168795 + 0.0845483i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 123.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.b odd 2 1 CM by \(\Q(\sqrt{-31}) \)
4.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.c 6
3.b odd 2 1 1116.2.g.f 6
4.b odd 2 1 inner 124.2.d.c 6
8.b even 2 1 1984.2.h.f 6
8.d odd 2 1 1984.2.h.f 6
12.b even 2 1 1116.2.g.f 6
31.b odd 2 1 CM 124.2.d.c 6
93.c even 2 1 1116.2.g.f 6
124.d even 2 1 inner 124.2.d.c 6
248.b even 2 1 1984.2.h.f 6
248.g odd 2 1 1984.2.h.f 6
372.b odd 2 1 1116.2.g.f 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
124.2.d.c 6 1.a even 1 1 trivial
124.2.d.c 6 4.b odd 2 1 inner
124.2.d.c 6 31.b odd 2 1 CM
124.2.d.c 6 124.d even 2 1 inner
1116.2.g.f 6 3.b odd 2 1
1116.2.g.f 6 12.b even 2 1
1116.2.g.f 6 93.c even 2 1
1116.2.g.f 6 372.b odd 2 1
1984.2.h.f 6 8.b even 2 1
1984.2.h.f 6 8.d odd 2 1
1984.2.h.f 6 248.b even 2 1
1984.2.h.f 6 248.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 \(S_{2}^{\mathrm{new}}(124, [\chi])\):

\( T_{3} \) Copy content Toggle raw display
\( T_{5}^{3} - 15T_{5} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - T^{3} + 8 \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( (T^{3} - 15 T + 2)^{2} \) Copy content Toggle raw display
$7$ \( T^{6} + 42 T^{4} + 441 T^{2} + \cdots + 1116 \) Copy content Toggle raw display
$11$ \( T^{6} \) Copy content Toggle raw display
$13$ \( T^{6} \) Copy content Toggle raw display
$17$ \( T^{6} \) Copy content Toggle raw display
$19$ \( T^{6} + 114 T^{4} + 3249 T^{2} + \cdots + 3100 \) Copy content Toggle raw display
$23$ \( T^{6} \) Copy content Toggle raw display
$29$ \( T^{6} \) Copy content Toggle raw display
$31$ \( (T^{2} + 31)^{3} \) Copy content Toggle raw display
$37$ \( T^{6} \) Copy content Toggle raw display
$41$ \( (T^{3} - 123 T + 278)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} \) Copy content Toggle raw display
$47$ \( (T^{2} + 124)^{3} \) Copy content Toggle raw display
$53$ \( T^{6} \) Copy content Toggle raw display
$59$ \( T^{6} + 354 T^{4} + 31329 T^{2} + \cdots + 273916 \) Copy content Toggle raw display
$61$ \( T^{6} \) Copy content Toggle raw display
$67$ \( (T^{2} + 124)^{3} \) Copy content Toggle raw display
$71$ \( T^{6} + 426 T^{4} + 45369 T^{2} + \cdots + 431644 \) Copy content Toggle raw display
$73$ \( T^{6} \) Copy content Toggle raw display
$79$ \( T^{6} \) Copy content Toggle raw display
$83$ \( T^{6} \) Copy content Toggle raw display
$89$ \( T^{6} \) Copy content Toggle raw display
$97$ \( (T^{3} - 291 T - 1906)^{2} \) Copy content Toggle raw display
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