Properties

Label 120.2.w.b
Level $120$
Weight $2$
Character orbit 120.w
Analytic conductor $0.958$
Analytic rank $0$
Dimension $4$
CM discriminant -24
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 120 = 2^{3} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 120.w (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.958204824255\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
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: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + 1) q^{2} + \beta_1 q^{3} - 2 \beta_{2} q^{4} + ( - \beta_{3} - \beta_{2} - 1) q^{5} + ( - \beta_{3} + \beta_1) q^{6} + (2 \beta_{3} + \beta_{2} - 1) q^{7} + ( - 2 \beta_{2} - 2) q^{8} + 3 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + 1) q^{2} + \beta_1 q^{3} - 2 \beta_{2} q^{4} + ( - \beta_{3} - \beta_{2} - 1) q^{5} + ( - \beta_{3} + \beta_1) q^{6} + (2 \beta_{3} + \beta_{2} - 1) q^{7} + ( - 2 \beta_{2} - 2) q^{8} + 3 \beta_{2} q^{9} + ( - \beta_{3} - \beta_1 - 2) q^{10} + (\beta_{3} - \beta_1 + 4) q^{11} - 2 \beta_{3} q^{12} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{14} + ( - \beta_{3} - \beta_1 + 3) q^{15} - 4 q^{16} + (3 \beta_{2} + 3) q^{18} + (2 \beta_{2} - 2 \beta_1 - 2) q^{20} + (\beta_{3} - \beta_1 - 6) q^{21} + (2 \beta_{3} - 4 \beta_{2} + 4) q^{22} + ( - 2 \beta_{3} - 2 \beta_1) q^{24} + (2 \beta_{3} - \beta_{2} - 2 \beta_1) q^{25} + 3 \beta_{3} q^{27} + (2 \beta_{2} + 4 \beta_1 + 2) q^{28} + ( - 3 \beta_{3} + 2 \beta_{2} - 3 \beta_1) q^{29} + ( - 3 \beta_{2} - 2 \beta_1 + 3) q^{30} + ( - 2 \beta_{3} + 2 \beta_1) q^{31} + (4 \beta_{2} - 4) q^{32} + ( - 3 \beta_{2} + 4 \beta_1 - 3) q^{33} + ( - \beta_{3} + 6 \beta_{2} + 3 \beta_1 + 2) q^{35} + 6 q^{36} + (2 \beta_{3} + 4 \beta_{2} - 2 \beta_1) q^{40} + (2 \beta_{3} + 6 \beta_{2} - 6) q^{42} + (2 \beta_{3} - 8 \beta_{2} + 2 \beta_1) q^{44} + ( - 3 \beta_{2} + 3 \beta_1 + 3) q^{45} - 4 \beta_1 q^{48} + ( - 4 \beta_{3} - 7 \beta_{2} - 4 \beta_1) q^{49} + (4 \beta_{3} - \beta_{2} - 1) q^{50} + 2 \beta_1 q^{53} + (3 \beta_{3} + 3 \beta_1) q^{54} + ( - 4 \beta_{3} - \beta_{2} + 2 \beta_1 - 7) q^{55} + ( - 4 \beta_{3} + 4 \beta_1 + 4) q^{56} + (2 \beta_{2} - 6 \beta_1 + 2) q^{58} + ( - 3 \beta_{3} - 8 \beta_{2} - 3 \beta_1) q^{59} + (2 \beta_{3} - 6 \beta_{2} - 2 \beta_1) q^{60} - 4 \beta_{3} q^{62} + ( - 3 \beta_{2} - 6 \beta_1 - 3) q^{63} + 8 \beta_{2} q^{64} + ( - 4 \beta_{3} + 4 \beta_1 - 6) q^{66} + ( - 4 \beta_{3} + 4 \beta_{2} + 2 \beta_1 + 8) q^{70} + ( - 6 \beta_{2} + 6) q^{72} + (7 \beta_{2} + 4 \beta_1 + 7) q^{73} + ( - \beta_{3} - 6 \beta_{2} - 6) q^{75} + (6 \beta_{3} - 2 \beta_{2} + 2) q^{77} + (6 \beta_{3} + 6 \beta_1) q^{79} + (4 \beta_{3} + 4 \beta_{2} + 4) q^{80} - 9 q^{81} + ( - 4 \beta_{2} - 4) q^{83} + (2 \beta_{3} + 12 \beta_{2} + 2 \beta_1) q^{84} + (2 \beta_{3} - 9 \beta_{2} + 9) q^{87} + ( - 8 \beta_{2} + 4 \beta_1 - 8) q^{88} + ( - 3 \beta_{3} - 6 \beta_{2} + 3 \beta_1) q^{90} + (6 \beta_{2} + 6) q^{93} + (4 \beta_{3} - 4 \beta_1) q^{96} + ( - 8 \beta_{3} + \beta_{2} - 1) q^{97} + ( - 7 \beta_{2} - 8 \beta_1 - 7) q^{98} + ( - 3 \beta_{3} + 12 \beta_{2} - 3 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} - 4 q^{5} - 4 q^{7} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{2} - 4 q^{5} - 4 q^{7} - 8 q^{8} - 8 q^{10} + 16 q^{11} + 12 q^{15} - 16 q^{16} + 12 q^{18} - 8 q^{20} - 24 q^{21} + 16 q^{22} + 8 q^{28} + 12 q^{30} - 16 q^{32} - 12 q^{33} + 8 q^{35} + 24 q^{36} - 24 q^{42} + 12 q^{45} - 4 q^{50} - 28 q^{55} + 16 q^{56} + 8 q^{58} - 12 q^{63} - 24 q^{66} + 32 q^{70} + 24 q^{72} + 28 q^{73} - 24 q^{75} + 8 q^{77} + 16 q^{80} - 36 q^{81} - 16 q^{83} + 36 q^{87} - 32 q^{88} + 24 q^{93} - 4 q^{97} - 28 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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

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

Character values

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

\(n\) \(31\) \(41\) \(61\) \(97\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(-\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1
−1.22474 1.22474i
1.22474 + 1.22474i
−1.22474 + 1.22474i
1.22474 1.22474i
1.00000 1.00000i −1.22474 1.22474i 2.00000i −2.22474 + 0.224745i −2.44949 1.44949 1.44949i −2.00000 2.00000i 3.00000i −2.00000 + 2.44949i
53.2 1.00000 1.00000i 1.22474 + 1.22474i 2.00000i 0.224745 2.22474i 2.44949 −3.44949 + 3.44949i −2.00000 2.00000i 3.00000i −2.00000 2.44949i
77.1 1.00000 + 1.00000i −1.22474 + 1.22474i 2.00000i −2.22474 0.224745i −2.44949 1.44949 + 1.44949i −2.00000 + 2.00000i 3.00000i −2.00000 2.44949i
77.2 1.00000 + 1.00000i 1.22474 1.22474i 2.00000i 0.224745 + 2.22474i 2.44949 −3.44949 3.44949i −2.00000 + 2.00000i 3.00000i −2.00000 + 2.44949i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
24.h odd 2 1 CM by \(\Q(\sqrt{-6}) \)
5.c odd 4 1 inner
120.w even 4 1 inner

Twists

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

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(120, [\chi])\):

\( T_{7}^{4} + 4T_{7}^{3} + 8T_{7}^{2} - 40T_{7} + 100 \) Copy content Toggle raw display
\( T_{11}^{2} - 8T_{11} + 10 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2 T + 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} + 9 \) Copy content Toggle raw display
$5$ \( T^{4} + 4 T^{3} + 8 T^{2} + 20 T + 25 \) Copy content Toggle raw display
$7$ \( T^{4} + 4 T^{3} + 8 T^{2} - 40 T + 100 \) Copy content Toggle raw display
$11$ \( (T^{2} - 8 T + 10)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} + 116T^{2} + 2500 \) Copy content Toggle raw display
$31$ \( (T^{2} - 24)^{2} \) 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
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} + 144 \) Copy content Toggle raw display
$59$ \( T^{4} + 236T^{2} + 100 \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} - 28 T^{3} + 392 T^{2} + \cdots + 2500 \) Copy content Toggle raw display
$79$ \( (T^{2} + 216)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 8 T + 32)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( T^{4} + 4 T^{3} + 8 T^{2} + \cdots + 36100 \) Copy content Toggle raw display
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