Properties

Label 124.2.d.a
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: \(\Q(i, \sqrt{6})\)
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
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.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{2} - \beta_{3} q^{3} - 2 \beta_1 q^{4} + q^{5} + (\beta_{3} - \beta_{2}) q^{6} - 3 \beta_1 q^{7} + (2 \beta_1 + 2) q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{2} - \beta_{3} q^{3} - 2 \beta_1 q^{4} + q^{5} + (\beta_{3} - \beta_{2}) q^{6} - 3 \beta_1 q^{7} + (2 \beta_1 + 2) q^{8} + 3 q^{9} + (\beta_1 - 1) q^{10} + 2 \beta_{3} q^{11} + 2 \beta_{2} q^{12} - \beta_{2} q^{13} + (3 \beta_1 + 3) q^{14} - \beta_{3} q^{15} - 4 q^{16} - 3 \beta_{2} q^{17} + (3 \beta_1 - 3) q^{18} + \beta_1 q^{19} - 2 \beta_1 q^{20} + 3 \beta_{2} q^{21} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{22} - \beta_{3} q^{23} + ( - 2 \beta_{3} - 2 \beta_{2}) q^{24} - 4 q^{25} + (\beta_{3} + \beta_{2}) q^{26} - 6 q^{28} + \beta_{2} q^{29} + (\beta_{3} - \beta_{2}) q^{30} + (\beta_{3} - 5 \beta_1) q^{31} + ( - 4 \beta_1 + 4) q^{32} - 12 q^{33} + (3 \beta_{3} + 3 \beta_{2}) q^{34} - 3 \beta_1 q^{35} - 6 \beta_1 q^{36} + 2 \beta_{2} q^{37} + ( - \beta_1 - 1) q^{38} + 6 \beta_1 q^{39} + (2 \beta_1 + 2) q^{40} + 7 q^{41} + ( - 3 \beta_{3} - 3 \beta_{2}) q^{42} - \beta_{3} q^{43} - 4 \beta_{2} q^{44} + 3 q^{45} + (\beta_{3} - \beta_{2}) q^{46} + 2 \beta_1 q^{47} + 4 \beta_{3} q^{48} - 2 q^{49} + ( - 4 \beta_1 + 4) q^{50} + 18 \beta_1 q^{51} - 2 \beta_{3} q^{52} + 4 \beta_{2} q^{53} + 2 \beta_{3} q^{55} + ( - 6 \beta_1 + 6) q^{56} - \beta_{2} q^{57} + ( - \beta_{3} - \beta_{2}) q^{58} + 11 \beta_1 q^{59} + 2 \beta_{2} q^{60} - 5 \beta_{2} q^{61} + ( - \beta_{3} + \beta_{2} + 5 \beta_1 + 5) q^{62} - 9 \beta_1 q^{63} + 8 \beta_1 q^{64} - \beta_{2} q^{65} + ( - 12 \beta_1 + 12) q^{66} - 8 \beta_1 q^{67} - 6 \beta_{3} q^{68} + 6 q^{69} + (3 \beta_1 + 3) q^{70} - 5 \beta_1 q^{71} + (6 \beta_1 + 6) q^{72} + 4 \beta_{2} q^{73} + ( - 2 \beta_{3} - 2 \beta_{2}) q^{74} + 4 \beta_{3} q^{75} + 2 q^{76} - 6 \beta_{2} q^{77} + ( - 6 \beta_1 - 6) q^{78} - 5 \beta_{3} q^{79} - 4 q^{80} - 9 q^{81} + (7 \beta_1 - 7) q^{82} + 4 \beta_{3} q^{83} + 6 \beta_{3} q^{84} - 3 \beta_{2} q^{85} + (\beta_{3} - \beta_{2}) q^{86} - 6 \beta_1 q^{87} + (4 \beta_{3} + 4 \beta_{2}) q^{88} + \beta_{2} q^{89} + (3 \beta_1 - 3) q^{90} - 3 \beta_{3} q^{91} + 2 \beta_{2} q^{92} + (5 \beta_{2} - 6) q^{93} + ( - 2 \beta_1 - 2) q^{94} + \beta_1 q^{95} + ( - 4 \beta_{3} + 4 \beta_{2}) q^{96} + 13 q^{97} + ( - 2 \beta_1 + 2) q^{98} + 6 \beta_{3} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} + 4 q^{5} + 8 q^{8} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} + 4 q^{5} + 8 q^{8} + 12 q^{9} - 4 q^{10} + 12 q^{14} - 16 q^{16} - 12 q^{18} - 16 q^{25} - 24 q^{28} + 16 q^{32} - 48 q^{33} - 4 q^{38} + 8 q^{40} + 28 q^{41} + 12 q^{45} - 8 q^{49} + 16 q^{50} + 24 q^{56} + 20 q^{62} + 48 q^{66} + 24 q^{69} + 12 q^{70} + 24 q^{72} + 8 q^{76} - 24 q^{78} - 16 q^{80} - 36 q^{81} - 28 q^{82} - 12 q^{90} - 24 q^{93} - 8 q^{94} + 52 q^{97} + 8 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{2} ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 3\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 3\nu ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 3\beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{3} + 3\beta_{2} ) / 2 \) 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
1.22474 1.22474i
−1.22474 + 1.22474i
1.22474 + 1.22474i
−1.22474 1.22474i
−1.00000 1.00000i −2.44949 2.00000i 1.00000 2.44949 + 2.44949i 3.00000i 2.00000 2.00000i 3.00000 −1.00000 1.00000i
123.2 −1.00000 1.00000i 2.44949 2.00000i 1.00000 −2.44949 2.44949i 3.00000i 2.00000 2.00000i 3.00000 −1.00000 1.00000i
123.3 −1.00000 + 1.00000i −2.44949 2.00000i 1.00000 2.44949 2.44949i 3.00000i 2.00000 + 2.00000i 3.00000 −1.00000 + 1.00000i
123.4 −1.00000 + 1.00000i 2.44949 2.00000i 1.00000 −2.44949 + 2.44949i 3.00000i 2.00000 + 2.00000i 3.00000 −1.00000 + 1.00000i
\(n\): 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.a 4
3.b odd 2 1 1116.2.g.d 4
4.b odd 2 1 inner 124.2.d.a 4
8.b even 2 1 1984.2.h.d 4
8.d odd 2 1 1984.2.h.d 4
12.b even 2 1 1116.2.g.d 4
31.b odd 2 1 inner 124.2.d.a 4
93.c even 2 1 1116.2.g.d 4
124.d even 2 1 inner 124.2.d.a 4
248.b even 2 1 1984.2.h.d 4
248.g odd 2 1 1984.2.h.d 4
372.b odd 2 1 1116.2.g.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
124.2.d.a 4 1.a even 1 1 trivial
124.2.d.a 4 4.b odd 2 1 inner
124.2.d.a 4 31.b odd 2 1 inner
124.2.d.a 4 124.d even 2 1 inner
1116.2.g.d 4 3.b odd 2 1
1116.2.g.d 4 12.b even 2 1
1116.2.g.d 4 93.c even 2 1
1116.2.g.d 4 372.b odd 2 1
1984.2.h.d 4 8.b even 2 1
1984.2.h.d 4 8.d odd 2 1
1984.2.h.d 4 248.b even 2 1
1984.2.h.d 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
\( T_{5} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2 T + 2)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} - 6)^{2} \) Copy content Toggle raw display
$5$ \( (T - 1)^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 24)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 6)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 54)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 6)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 6)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 38T^{2} + 961 \) Copy content Toggle raw display
$37$ \( (T^{2} + 24)^{2} \) Copy content Toggle raw display
$41$ \( (T - 7)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} - 6)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 96)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 121)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 150)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 25)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 96)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 150)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 96)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 6)^{2} \) Copy content Toggle raw display
$97$ \( (T - 13)^{4} \) Copy content Toggle raw display
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