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Properties

Label 120.4.m.a
Level $120$
Weight $4$
Character orbit 120.m
Analytic conductor $7.080$
Analytic rank $0$
Dimension $4$
CM discriminant -15
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [120,4,Mod(59,120)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(120, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("120.59");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 120 = 2^{3} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 120.m (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.08022920069\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{3}, \sqrt{-5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + (2 \beta_{3} - \beta_1) q^{3} + (\beta_{2} + 6) q^{4} - 5 \beta_1 q^{5} + (\beta_{2} + 14) q^{6} + (3 \beta_{3} + 8 \beta_1) q^{8} + 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} + (2 \beta_{3} - \beta_1) q^{3} + (\beta_{2} + 6) q^{4} - 5 \beta_1 q^{5} + (\beta_{2} + 14) q^{6} + (3 \beta_{3} + 8 \beta_1) q^{8} + 27 q^{9} + ( - 5 \beta_{2} + 10) q^{10} + (11 \beta_{3} + 8 \beta_1) q^{12} + ( - 10 \beta_{2} - 5) q^{15} + (11 \beta_{2} + 2) q^{16} + (8 \beta_{3} - 4 \beta_1) q^{17} + 27 \beta_{3} q^{18} - 164 q^{19} + (25 \beta_{3} - 40 \beta_1) q^{20} + 44 \beta_1 q^{23} + (19 \beta_{2} + 50) q^{24} - 125 q^{25} + (54 \beta_{3} - 27 \beta_1) q^{27} + (25 \beta_{3} - 80 \beta_1) q^{30} + ( - 44 \beta_{2} - 22) q^{31} + ( - 31 \beta_{3} + 88 \beta_1) q^{32} + (4 \beta_{2} + 56) q^{34} + (27 \beta_{2} + 162) q^{36} - 164 \beta_{3} q^{38} + ( - 15 \beta_{2} + 230) q^{40} - 135 \beta_1 q^{45} + (44 \beta_{2} - 88) q^{46} + 244 \beta_1 q^{47} + ( - 7 \beta_{3} + 152 \beta_1) q^{48} - 343 q^{49} - 125 \beta_{3} q^{50} + 108 q^{51} - 278 \beta_1 q^{53} + (27 \beta_{2} + 378) q^{54} + ( - 328 \beta_{3} + 164 \beta_1) q^{57} + ( - 55 \beta_{2} + 310) q^{60} + ( - 152 \beta_{2} - 76) q^{61} + (110 \beta_{3} - 352 \beta_1) q^{62} + (57 \beta_{2} - 362) q^{64} + (44 \beta_{3} + 32 \beta_1) q^{68} + (88 \beta_{2} + 44) q^{69} + (81 \beta_{3} + 216 \beta_1) q^{72} + ( - 250 \beta_{3} + 125 \beta_1) q^{75} + ( - 164 \beta_{2} - 984) q^{76} + (236 \beta_{2} + 118) q^{79} + (275 \beta_{3} - 120 \beta_1) q^{80} + 729 q^{81} + (316 \beta_{3} - 158 \beta_1) q^{83} + ( - 40 \beta_{2} - 20) q^{85} + ( - 135 \beta_{2} + 270) q^{90} + ( - 220 \beta_{3} + 352 \beta_1) q^{92} - 594 \beta_1 q^{93} + (244 \beta_{2} - 488) q^{94} + 820 \beta_1 q^{95} + (145 \beta_{2} - 346) q^{96} - 343 \beta_{3} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 22 q^{4} + 54 q^{6} + 108 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 22 q^{4} + 54 q^{6} + 108 q^{9} + 50 q^{10} - 14 q^{16} - 656 q^{19} + 162 q^{24} - 500 q^{25} + 216 q^{34} + 594 q^{36} + 950 q^{40} - 440 q^{46} - 1372 q^{49} + 432 q^{51} + 1458 q^{54} + 1350 q^{60} - 1562 q^{64} - 3608 q^{76} + 2916 q^{81} + 1350 q^{90} - 2440 q^{94} - 1674 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

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

Display \(\nu^j\) in terms of \(\beta_i\)

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

Display \(\beta_i\) in terms of \(\nu^j\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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
−2.59808 − 1.11803i −5.19615 5.50000 + 5.80948i 11.1803i 13.5000 + 5.80948i 0 −7.79423 − 21.2426i 27.0000 12.5000 − 29.0474i
59.2 −2.59808 + 1.11803i −5.19615 5.50000 − 5.80948i − 11.1803i 13.5000 − 5.80948i 0 −7.79423 + 21.2426i 27.0000 12.5000 + 29.0474i
59.3 2.59808 − 1.11803i 5.19615 5.50000 − 5.80948i 11.1803i 13.5000 − 5.80948i 0 7.79423 − 21.2426i 27.0000 12.5000 + 29.0474i
59.4 2.59808 + 1.11803i 5.19615 5.50000 + 5.80948i − 11.1803i 13.5000 + 5.80948i 0 7.79423 + 21.2426i 27.0000 12.5000 − 29.0474i
\(n\): 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 120.4.m.a ✓ 4
3.b odd 2 1 inner 120.4.m.a ✓ 4
4.b odd 2 1 480.4.m.a 4
5.b even 2 1 inner 120.4.m.a ✓ 4
8.b even 2 1 480.4.m.a 4
8.d odd 2 1 inner 120.4.m.a ✓ 4
12.b even 2 1 480.4.m.a 4
15.d odd 2 1 CM 120.4.m.a ✓ 4
20.d odd 2 1 480.4.m.a 4
24.f even 2 1 inner 120.4.m.a ✓ 4
24.h odd 2 1 480.4.m.a 4
40.e odd 2 1 inner 120.4.m.a ✓ 4
40.f even 2 1 480.4.m.a 4
60.h even 2 1 480.4.m.a 4
120.i odd 2 1 480.4.m.a 4
120.m even 2 1 inner 120.4.m.a ✓ 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.4.m.a ✓ 4 1.a even 1 1 trivial
120.4.m.a ✓ 4 3.b odd 2 1 inner
120.4.m.a ✓ 4 5.b even 2 1 inner
120.4.m.a ✓ 4 8.d odd 2 1 inner
120.4.m.a ✓ 4 15.d odd 2 1 CM
120.4.m.a ✓ 4 24.f even 2 1 inner
120.4.m.a ✓ 4 40.e odd 2 1 inner
120.4.m.a ✓ 4 120.m even 2 1 inner
480.4.m.a 4 4.b odd 2 1
480.4.m.a 4 8.b even 2 1
480.4.m.a 4 12.b even 2 1
480.4.m.a 4 20.d odd 2 1
480.4.m.a 4 24.h odd 2 1
480.4.m.a 4 40.f even 2 1
480.4.m.a 4 60.h even 2 1
480.4.m.a 4 120.i odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} \) acting on \(S_{4}^{\mathrm{new}}(120, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 11T^{2} + 64 \) Copy content Toggle raw display
$3$ \( (T^{2} - 27)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 125)^{2} \) 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
$17$ \( (T^{2} - 432)^{2} \) Copy content Toggle raw display
$19$ \( (T + 164)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} + 9680)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 65340)^{2} \) 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
$47$ \( (T^{2} + 297680)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 386420)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 779760)^{2} \) 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
$79$ \( (T^{2} + 1879740)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 674028)^{2} \) 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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