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Properties

Label 120.3.i.e
Level $120$
Weight $3$
Character orbit 120.i
Analytic conductor $3.270$
Analytic rank $0$
Dimension $4$
CM discriminant -24
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [120,3,Mod(29,120)]
 
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([0, 1, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("120.29");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 120 = 2^{3} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 120.i (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: \(3.26976317232\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
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.

\(f(q)\) \(=\) \( q - 2 \beta_1 q^{2} + 3 \beta_1 q^{3} - 4 q^{4} + ( - \beta_{3} - \beta_1) q^{5} + 6 q^{6} - 2 \beta_{2} q^{7} + 8 \beta_1 q^{8} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 \beta_1 q^{2} + 3 \beta_1 q^{3} - 4 q^{4} + ( - \beta_{3} - \beta_1) q^{5} + 6 q^{6} - 2 \beta_{2} q^{7} + 8 \beta_1 q^{8} - 9 q^{9} + (2 \beta_{2} - 2) q^{10} - 4 \beta_{3} q^{11} - 12 \beta_1 q^{12} - 4 \beta_{3} q^{14} + ( - 3 \beta_{2} + 3) q^{15} + 16 q^{16} + 18 \beta_1 q^{18} + (4 \beta_{3} + 4 \beta_1) q^{20} + 6 \beta_{3} q^{21} + 8 \beta_{2} q^{22} - 24 q^{24} + (2 \beta_{2} + 23) q^{25} - 27 \beta_1 q^{27} + 8 \beta_{2} q^{28} + 6 \beta_{3} q^{29} + ( - 6 \beta_{3} - 6 \beta_1) q^{30} - 38 q^{31} - 32 \beta_1 q^{32} - 12 \beta_{2} q^{33} + ( - 2 \beta_{3} + 48 \beta_1) q^{35} + 36 q^{36} + ( - 8 \beta_{2} + 8) q^{40} - 12 \beta_{2} q^{42} + 16 \beta_{3} q^{44} + (9 \beta_{3} + 9 \beta_1) q^{45} + 48 \beta_1 q^{48} - 47 q^{49} + (4 \beta_{3} - 46 \beta_1) q^{50} - 94 \beta_1 q^{53} - 54 q^{54} + (4 \beta_{2} + 96) q^{55} + 16 \beta_{3} q^{56} - 12 \beta_{2} q^{58} - 24 \beta_{3} q^{59} + (12 \beta_{2} - 12) q^{60} + 76 \beta_1 q^{62} + 18 \beta_{2} q^{63} - 64 q^{64} - 24 \beta_{3} q^{66} + (4 \beta_{2} + 96) q^{70} - 72 \beta_1 q^{72} + 28 \beta_{2} q^{73} + ( - 6 \beta_{3} + 69 \beta_1) q^{75} + 192 \beta_1 q^{77} - 58 q^{79} + ( - 16 \beta_{3} - 16 \beta_1) q^{80} + 81 q^{81} - 134 \beta_1 q^{83} - 24 \beta_{3} q^{84} + 18 \beta_{2} q^{87} - 32 \beta_{2} q^{88} + ( - 18 \beta_{2} + 18) q^{90} - 114 \beta_1 q^{93} + 96 q^{96} + 8 \beta_{2} q^{97} + 94 \beta_1 q^{98} + 36 \beta_{3} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 16 q^{4} + 24 q^{6} - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 16 q^{4} + 24 q^{6} - 36 q^{9} - 8 q^{10} + 12 q^{15} + 64 q^{16} - 96 q^{24} + 92 q^{25} - 152 q^{31} + 144 q^{36} + 32 q^{40} - 188 q^{49} - 216 q^{54} + 384 q^{55} - 48 q^{60} - 256 q^{64} + 384 q^{70} - 232 q^{79} + 324 q^{81} + 72 q^{90} + 384 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

\(\beta_{1}\)\(=\) \( ( \nu^{2} ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2\nu^{3} + 6\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{3} + 6\nu ) / 3 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 3\beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{3} + 3\beta_{2} ) / 4 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1
1.22474 + 1.22474i
−1.22474 − 1.22474i
1.22474 − 1.22474i
−1.22474 + 1.22474i
− 2.00000i 3.00000i −4.00000 −4.89898 − 1.00000i 6.00000 − 9.79796i 8.00000i −9.00000 −2.00000 + 9.79796i
29.2 − 2.00000i 3.00000i −4.00000 4.89898 − 1.00000i 6.00000 9.79796i 8.00000i −9.00000 −2.00000 − 9.79796i
29.3 2.00000i − 3.00000i −4.00000 −4.89898 + 1.00000i 6.00000 9.79796i − 8.00000i −9.00000 −2.00000 − 9.79796i
29.4 2.00000i − 3.00000i −4.00000 4.89898 + 1.00000i 6.00000 − 9.79796i − 8.00000i −9.00000 −2.00000 + 9.79796i
\(n\): 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 \(\Q(\sqrt{-6}) \)
3.b odd 2 1 inner
5.b even 2 1 inner
8.b even 2 1 inner
15.d odd 2 1 inner
40.f even 2 1 inner
120.i odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 120.3.i.e ✓ 4
3.b odd 2 1 inner 120.3.i.e ✓ 4
4.b odd 2 1 480.3.i.e 4
5.b even 2 1 inner 120.3.i.e ✓ 4
8.b even 2 1 inner 120.3.i.e ✓ 4
8.d odd 2 1 480.3.i.e 4
12.b even 2 1 480.3.i.e 4
15.d odd 2 1 inner 120.3.i.e ✓ 4
20.d odd 2 1 480.3.i.e 4
24.f even 2 1 480.3.i.e 4
24.h odd 2 1 CM 120.3.i.e ✓ 4
40.e odd 2 1 480.3.i.e 4
40.f even 2 1 inner 120.3.i.e ✓ 4
60.h even 2 1 480.3.i.e 4
120.i odd 2 1 inner 120.3.i.e ✓ 4
120.m even 2 1 480.3.i.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.3.i.e ✓ 4 1.a even 1 1 trivial
120.3.i.e ✓ 4 3.b odd 2 1 inner
120.3.i.e ✓ 4 5.b even 2 1 inner
120.3.i.e ✓ 4 8.b even 2 1 inner
120.3.i.e ✓ 4 15.d odd 2 1 inner
120.3.i.e ✓ 4 24.h odd 2 1 CM
120.3.i.e ✓ 4 40.f even 2 1 inner
120.3.i.e ✓ 4 120.i odd 2 1 inner
480.3.i.e 4 4.b odd 2 1
480.3.i.e 4 8.d odd 2 1
480.3.i.e 4 12.b even 2 1
480.3.i.e 4 20.d odd 2 1
480.3.i.e 4 24.f even 2 1
480.3.i.e 4 40.e odd 2 1
480.3.i.e 4 60.h even 2 1
480.3.i.e 4 120.m even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(120, [\chi])\):

\( T_{7}^{2} + 96 \) Copy content Toggle raw display
\( T_{11}^{2} - 384 \) Copy content Toggle raw display
\( T_{13} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} - 46T^{2} + 625 \) Copy content Toggle raw display
$7$ \( (T^{2} + 96)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 384)^{2} \) 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
$29$ \( (T^{2} - 864)^{2} \) Copy content Toggle raw display
$31$ \( (T + 38)^{4} \) 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
$53$ \( (T^{2} + 8836)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 13824)^{2} \) 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
$73$ \( (T^{2} + 18816)^{2} \) Copy content Toggle raw display
$79$ \( (T + 58)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 17956)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + 1536)^{2} \) Copy content Toggle raw display
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