Properties

Label 124.3.c.a
Level $124$
Weight $3$
Character orbit 124.c
Analytic conductor $3.379$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 124 = 2^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 124.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.37875527807\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{-3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{3} - 6 q^{5} - 10 q^{7} - 3 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{3} - 6 q^{5} - 10 q^{7} - 3 q^{9} + 5 \beta q^{11} - 5 \beta q^{13} + 6 \beta q^{15} - 4 \beta q^{17} - 14 q^{19} + 10 \beta q^{21} - 10 \beta q^{23} + 11 q^{25} - 6 \beta q^{27} - 5 \beta q^{29} - 31 q^{31} + 60 q^{33} + 60 q^{35} + \beta q^{37} - 60 q^{39} + 54 q^{41} + 21 \beta q^{43} + 18 q^{45} + 30 q^{47} + 51 q^{49} - 48 q^{51} + 9 \beta q^{53} - 30 \beta q^{55} + 14 \beta q^{57} - 6 q^{59} - 5 \beta q^{61} + 30 q^{63} + 30 \beta q^{65} - 110 q^{67} - 120 q^{69} + 66 q^{71} - 26 \beta q^{73} - 11 \beta q^{75} - 50 \beta q^{77} - 20 \beta q^{79} - 99 q^{81} + 11 \beta q^{83} + 24 \beta q^{85} - 60 q^{87} + 10 \beta q^{89} + 50 \beta q^{91} + 31 \beta q^{93} + 84 q^{95} + 110 q^{97} - 15 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 12 q^{5} - 20 q^{7} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 12 q^{5} - 20 q^{7} - 6 q^{9} - 28 q^{19} + 22 q^{25} - 62 q^{31} + 120 q^{33} + 120 q^{35} - 120 q^{39} + 108 q^{41} + 36 q^{45} + 60 q^{47} + 102 q^{49} - 96 q^{51} - 12 q^{59} + 60 q^{63} - 220 q^{67} - 240 q^{69} + 132 q^{71} - 198 q^{81} - 120 q^{87} + 168 q^{95} + 220 q^{97}+O(q^{100}) \) 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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
61.1
0.500000 + 0.866025i
0.500000 0.866025i
0 3.46410i 0 −6.00000 0 −10.0000 0 −3.00000 0
61.2 0 3.46410i 0 −6.00000 0 −10.0000 0 −3.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 124.3.c.a 2
3.b odd 2 1 1116.3.h.c 2
4.b odd 2 1 496.3.e.a 2
5.b even 2 1 3100.3.d.a 2
5.c odd 4 2 3100.3.f.a 4
31.b odd 2 1 inner 124.3.c.a 2
93.c even 2 1 1116.3.h.c 2
124.d even 2 1 496.3.e.a 2
155.c odd 2 1 3100.3.d.a 2
155.f even 4 2 3100.3.f.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
124.3.c.a 2 1.a even 1 1 trivial
124.3.c.a 2 31.b odd 2 1 inner
496.3.e.a 2 4.b odd 2 1
496.3.e.a 2 124.d even 2 1
1116.3.h.c 2 3.b odd 2 1
1116.3.h.c 2 93.c even 2 1
3100.3.d.a 2 5.b even 2 1
3100.3.d.a 2 155.c odd 2 1
3100.3.f.a 4 5.c odd 4 2
3100.3.f.a 4 155.f even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 12 \) acting on \(S_{3}^{\mathrm{new}}(124, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 12 \) Copy content Toggle raw display
$5$ \( (T + 6)^{2} \) Copy content Toggle raw display
$7$ \( (T + 10)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 300 \) Copy content Toggle raw display
$13$ \( T^{2} + 300 \) Copy content Toggle raw display
$17$ \( T^{2} + 192 \) Copy content Toggle raw display
$19$ \( (T + 14)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 1200 \) Copy content Toggle raw display
$29$ \( T^{2} + 300 \) Copy content Toggle raw display
$31$ \( (T + 31)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 12 \) Copy content Toggle raw display
$41$ \( (T - 54)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 5292 \) Copy content Toggle raw display
$47$ \( (T - 30)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 972 \) Copy content Toggle raw display
$59$ \( (T + 6)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 300 \) Copy content Toggle raw display
$67$ \( (T + 110)^{2} \) Copy content Toggle raw display
$71$ \( (T - 66)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 8112 \) Copy content Toggle raw display
$79$ \( T^{2} + 4800 \) Copy content Toggle raw display
$83$ \( T^{2} + 1452 \) Copy content Toggle raw display
$89$ \( T^{2} + 1200 \) Copy content Toggle raw display
$97$ \( (T - 110)^{2} \) Copy content Toggle raw display
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