Properties

Label 240.3.c.c
Level $240$
Weight $3$
Character orbit 240.c
Analytic conductor $6.540$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [240,3,Mod(209,240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(240, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 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("240.209");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 240.c (of order \(2\), degree \(1\), not 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: \(6.53952634465\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-17})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 16x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 30)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (\beta_{3} + 2 \beta_{2}) q^{5} + ( - \beta_{2} - 2 \beta_1) q^{7} + (\beta_{3} - 8) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + (\beta_{3} + 2 \beta_{2}) q^{5} + ( - \beta_{2} - 2 \beta_1) q^{7} + (\beta_{3} - 8) q^{9} + 4 \beta_{3} q^{11} + ( - 2 \beta_{3} + 9 \beta_{2} + \cdots - 2) q^{15}+ \cdots + ( - 32 \beta_{3} - 68) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 32 q^{9} - 8 q^{15} - 48 q^{19} + 68 q^{21} - 36 q^{25} + 128 q^{31} - 68 q^{45} + 60 q^{49} - 32 q^{51} - 272 q^{55} - 64 q^{61} - 68 q^{69} + 272 q^{75} + 288 q^{79} + 188 q^{81} + 128 q^{85} - 272 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 16x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 7\nu ) / 9 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} + 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 9\beta_{2} - 7\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/240\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(97\) \(161\) \(181\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
209.1
−0.707107 2.91548i
−0.707107 + 2.91548i
0.707107 2.91548i
0.707107 + 2.91548i
0 −0.707107 2.91548i 0 2.82843 + 4.12311i 0 5.83095i 0 −8.00000 + 4.12311i 0
209.2 0 −0.707107 + 2.91548i 0 2.82843 4.12311i 0 5.83095i 0 −8.00000 4.12311i 0
209.3 0 0.707107 2.91548i 0 −2.82843 4.12311i 0 5.83095i 0 −8.00000 4.12311i 0
209.4 0 0.707107 + 2.91548i 0 −2.82843 + 4.12311i 0 5.83095i 0 −8.00000 + 4.12311i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 240.3.c.c 4
3.b odd 2 1 inner 240.3.c.c 4
4.b odd 2 1 30.3.b.a 4
5.b even 2 1 inner 240.3.c.c 4
5.c odd 4 2 1200.3.l.t 4
8.b even 2 1 960.3.c.e 4
8.d odd 2 1 960.3.c.f 4
12.b even 2 1 30.3.b.a 4
15.d odd 2 1 inner 240.3.c.c 4
15.e even 4 2 1200.3.l.t 4
20.d odd 2 1 30.3.b.a 4
20.e even 4 2 150.3.d.d 4
24.f even 2 1 960.3.c.f 4
24.h odd 2 1 960.3.c.e 4
36.f odd 6 2 810.3.j.c 8
36.h even 6 2 810.3.j.c 8
40.e odd 2 1 960.3.c.f 4
40.f even 2 1 960.3.c.e 4
60.h even 2 1 30.3.b.a 4
60.l odd 4 2 150.3.d.d 4
120.i odd 2 1 960.3.c.e 4
120.m even 2 1 960.3.c.f 4
180.n even 6 2 810.3.j.c 8
180.p odd 6 2 810.3.j.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.3.b.a 4 4.b odd 2 1
30.3.b.a 4 12.b even 2 1
30.3.b.a 4 20.d odd 2 1
30.3.b.a 4 60.h even 2 1
150.3.d.d 4 20.e even 4 2
150.3.d.d 4 60.l odd 4 2
240.3.c.c 4 1.a even 1 1 trivial
240.3.c.c 4 3.b odd 2 1 inner
240.3.c.c 4 5.b even 2 1 inner
240.3.c.c 4 15.d odd 2 1 inner
810.3.j.c 8 36.f odd 6 2
810.3.j.c 8 36.h even 6 2
810.3.j.c 8 180.n even 6 2
810.3.j.c 8 180.p odd 6 2
960.3.c.e 4 8.b even 2 1
960.3.c.e 4 24.h odd 2 1
960.3.c.e 4 40.f even 2 1
960.3.c.e 4 120.i odd 2 1
960.3.c.f 4 8.d odd 2 1
960.3.c.f 4 24.f even 2 1
960.3.c.f 4 40.e odd 2 1
960.3.c.f 4 120.m even 2 1
1200.3.l.t 4 5.c odd 4 2
1200.3.l.t 4 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(240, [\chi])\):

\( T_{7}^{2} + 34 \) Copy content Toggle raw display
\( T_{17}^{2} - 128 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 16T^{2} + 81 \) Copy content Toggle raw display
$5$ \( T^{4} + 18T^{2} + 625 \) Copy content Toggle raw display
$7$ \( (T^{2} + 34)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 272)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} - 128)^{2} \) Copy content Toggle raw display
$19$ \( (T + 12)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} - 578)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( (T - 32)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 544)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 3332)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 1666)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 1250)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 4608)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 272)^{2} \) Copy content Toggle raw display
$61$ \( (T + 16)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 34)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 13600)^{2} \) Copy content Toggle raw display
$79$ \( (T - 72)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 1922)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 4352)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 26656)^{2} \) Copy content Toggle raw display
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