Properties

Label 120.2.m.a
Level $120$
Weight $2$
Character orbit 120.m
Analytic conductor $0.958$
Analytic rank $0$
Dimension $4$
CM discriminant -15
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 120 = 2^{3} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 120.m (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.958204824255\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{3}, \sqrt{-5})\)
Defining polynomial: \( x^{4} + x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{3} - \beta_1) q^{3} + \beta_{2} q^{4} + (\beta_{3} + \beta_1) q^{5} + ( - \beta_{2} - 2) q^{6} + (2 \beta_{3} - \beta_1) q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{3} - \beta_1) q^{3} + \beta_{2} q^{4} + (\beta_{3} + \beta_1) q^{5} + ( - \beta_{2} - 2) q^{6} + (2 \beta_{3} - \beta_1) q^{8} + 3 q^{9} + (\beta_{2} - 2) q^{10} + ( - 2 \beta_{3} - \beta_1) q^{12} + ( - 2 \beta_{2} - 1) q^{15} + ( - \beta_{2} - 4) q^{16} + ( - 4 \beta_{3} + 4 \beta_1) q^{17} + 3 \beta_1 q^{18} + 4 q^{19} + (2 \beta_{3} - 3 \beta_1) q^{20} + ( - 4 \beta_{3} - 4 \beta_1) q^{23} + ( - \beta_{2} + 4) q^{24} - 5 q^{25} + (3 \beta_{3} - 3 \beta_1) q^{27} + ( - 4 \beta_{3} + \beta_1) q^{30} + (4 \beta_{2} + 2) q^{31} + ( - 2 \beta_{3} - 3 \beta_1) q^{32} + (4 \beta_{2} + 8) q^{34} + 3 \beta_{2} q^{36} + 4 \beta_1 q^{38} + ( - 3 \beta_{2} - 4) q^{40} + (3 \beta_{3} + 3 \beta_1) q^{45} + ( - 4 \beta_{2} + 8) q^{46} + (4 \beta_{3} + 4 \beta_1) q^{47} + ( - 2 \beta_{3} + 5 \beta_1) q^{48} - 7 q^{49} - 5 \beta_1 q^{50} - 12 q^{51} + ( - 2 \beta_{3} - 2 \beta_1) q^{53} + ( - 3 \beta_{2} - 6) q^{54} + (4 \beta_{3} - 4 \beta_1) q^{57} + (\beta_{2} + 8) q^{60} + ( - 8 \beta_{2} - 4) q^{61} + (8 \beta_{3} - 2 \beta_1) q^{62} + ( - 3 \beta_{2} + 4) q^{64} + (8 \beta_{3} + 4 \beta_1) q^{68} + (8 \beta_{2} + 4) q^{69} + (6 \beta_{3} - 3 \beta_1) q^{72} + ( - 5 \beta_{3} + 5 \beta_1) q^{75} + 4 \beta_{2} q^{76} + ( - 4 \beta_{2} - 2) q^{79} + ( - 6 \beta_{3} - \beta_1) q^{80} + 9 q^{81} + ( - 2 \beta_{3} + 2 \beta_1) q^{83} + (8 \beta_{2} + 4) q^{85} + (3 \beta_{2} - 6) q^{90} + ( - 8 \beta_{3} + 12 \beta_1) q^{92} + ( - 6 \beta_{3} - 6 \beta_1) q^{93} + (4 \beta_{2} - 8) q^{94} + (4 \beta_{3} + 4 \beta_1) q^{95} + (5 \beta_{2} + 4) q^{96} - 7 \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{4} - 6 q^{6} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{4} - 6 q^{6} + 12 q^{9} - 10 q^{10} - 14 q^{16} + 16 q^{19} + 18 q^{24} - 20 q^{25} + 24 q^{34} - 6 q^{36} - 10 q^{40} + 40 q^{46} - 28 q^{49} - 48 q^{51} - 18 q^{54} + 30 q^{60} + 22 q^{64} - 8 q^{76} + 36 q^{81} - 30 q^{90} - 40 q^{94} + 6 q^{96}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + \nu ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} - \beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/120\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(41\) \(61\) \(97\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
59.1
−0.866025 1.11803i
−0.866025 + 1.11803i
0.866025 1.11803i
0.866025 + 1.11803i
−0.866025 1.11803i 1.73205 −0.500000 + 1.93649i 2.23607i −1.50000 1.93649i 0 2.59808 1.11803i 3.00000 −2.50000 + 1.93649i
59.2 −0.866025 + 1.11803i 1.73205 −0.500000 1.93649i 2.23607i −1.50000 + 1.93649i 0 2.59808 + 1.11803i 3.00000 −2.50000 1.93649i
59.3 0.866025 1.11803i −1.73205 −0.500000 1.93649i 2.23607i −1.50000 + 1.93649i 0 −2.59808 1.11803i 3.00000 −2.50000 1.93649i
59.4 0.866025 + 1.11803i −1.73205 −0.500000 + 1.93649i 2.23607i −1.50000 1.93649i 0 −2.59808 + 1.11803i 3.00000 −2.50000 + 1.93649i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 CM by \(\Q(\sqrt{-15}) \)
3.b odd 2 1 inner
5.b even 2 1 inner
8.d odd 2 1 inner
24.f even 2 1 inner
40.e odd 2 1 inner
120.m even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 120.2.m.a 4
3.b odd 2 1 inner 120.2.m.a 4
4.b odd 2 1 480.2.m.a 4
5.b even 2 1 inner 120.2.m.a 4
5.c odd 4 2 600.2.b.c 4
8.b even 2 1 480.2.m.a 4
8.d odd 2 1 inner 120.2.m.a 4
12.b even 2 1 480.2.m.a 4
15.d odd 2 1 CM 120.2.m.a 4
15.e even 4 2 600.2.b.c 4
20.d odd 2 1 480.2.m.a 4
20.e even 4 2 2400.2.b.c 4
24.f even 2 1 inner 120.2.m.a 4
24.h odd 2 1 480.2.m.a 4
40.e odd 2 1 inner 120.2.m.a 4
40.f even 2 1 480.2.m.a 4
40.i odd 4 2 2400.2.b.c 4
40.k even 4 2 600.2.b.c 4
60.h even 2 1 480.2.m.a 4
60.l odd 4 2 2400.2.b.c 4
120.i odd 2 1 480.2.m.a 4
120.m even 2 1 inner 120.2.m.a 4
120.q odd 4 2 600.2.b.c 4
120.w even 4 2 2400.2.b.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.2.m.a 4 1.a even 1 1 trivial
120.2.m.a 4 3.b odd 2 1 inner
120.2.m.a 4 5.b even 2 1 inner
120.2.m.a 4 8.d odd 2 1 inner
120.2.m.a 4 15.d odd 2 1 CM
120.2.m.a 4 24.f even 2 1 inner
120.2.m.a 4 40.e odd 2 1 inner
120.2.m.a 4 120.m even 2 1 inner
480.2.m.a 4 4.b odd 2 1
480.2.m.a 4 8.b even 2 1
480.2.m.a 4 12.b even 2 1
480.2.m.a 4 20.d odd 2 1
480.2.m.a 4 24.h odd 2 1
480.2.m.a 4 40.f even 2 1
480.2.m.a 4 60.h even 2 1
480.2.m.a 4 120.i odd 2 1
600.2.b.c 4 5.c odd 4 2
600.2.b.c 4 15.e even 4 2
600.2.b.c 4 40.k even 4 2
600.2.b.c 4 120.q odd 4 2
2400.2.b.c 4 20.e even 4 2
2400.2.b.c 4 40.i odd 4 2
2400.2.b.c 4 60.l odd 4 2
2400.2.b.c 4 120.w even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} \) acting on \(S_{2}^{\mathrm{new}}(120, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + T^{2} + 4 \) Copy content Toggle raw display
$3$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 5)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} - 48)^{2} \) Copy content Toggle raw display
$19$ \( (T - 4)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} + 80)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 60)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} + 80)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 20)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 240)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 60)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( T^{4} \) Copy content Toggle raw display
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