Properties

Label 240.2.o.a
Level $240$
Weight $2$
Character orbit 240.o
Analytic conductor $1.916$
Analytic rank $0$
Dimension $4$
CM discriminant -20
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [240,2,Mod(239,240)]
 
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([1, 0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("240.239");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 240.o (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: \(1.91640964851\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 4x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3 \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - \beta_{2} q^{5} + ( - \beta_{3} + \beta_1) q^{7} + (\beta_{2} - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - \beta_{2} q^{5} + ( - \beta_{3} + \beta_1) q^{7} + (\beta_{2} - 2) q^{9} - \beta_{3} q^{15} + (3 \beta_{2} + 3) q^{21} + (\beta_{3} + 5 \beta_1) q^{23} - 5 q^{25} + (\beta_{3} - 2 \beta_1) q^{27} - 4 \beta_{2} q^{29} + ( - \beta_{3} - 5 \beta_1) q^{35} - 2 \beta_{2} q^{41} + (3 \beta_{3} - 3 \beta_1) q^{43} + (2 \beta_{2} + 5) q^{45} + ( - \beta_{3} - 5 \beta_1) q^{47} + 11 q^{49} - 8 q^{61} + (3 \beta_{3} + 3 \beta_1) q^{63} + (\beta_{3} - \beta_1) q^{67} + (3 \beta_{2} - 15) q^{69} - 5 \beta_1 q^{75} + ( - 4 \beta_{2} - 1) q^{81} + (\beta_{3} + 5 \beta_1) q^{83} - 4 \beta_{3} q^{87} + 8 \beta_{2} q^{89}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{9} + 12 q^{21} - 20 q^{25} + 20 q^{45} + 44 q^{49} - 32 q^{61} - 60 q^{69} - 4 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 4x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 2\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 2\beta_1 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
239.1
−0.707107 1.58114i
−0.707107 + 1.58114i
0.707107 1.58114i
0.707107 + 1.58114i
0 −0.707107 1.58114i 0 2.23607i 0 −4.24264 0 −2.00000 + 2.23607i 0
239.2 0 −0.707107 + 1.58114i 0 2.23607i 0 −4.24264 0 −2.00000 2.23607i 0
239.3 0 0.707107 1.58114i 0 2.23607i 0 4.24264 0 −2.00000 2.23607i 0
239.4 0 0.707107 + 1.58114i 0 2.23607i 0 4.24264 0 −2.00000 + 2.23607i 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
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)
3.b odd 2 1 inner
4.b odd 2 1 inner
5.b even 2 1 inner
12.b even 2 1 inner
15.d odd 2 1 inner
60.h even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 240.2.o.a 4
3.b odd 2 1 inner 240.2.o.a 4
4.b odd 2 1 inner 240.2.o.a 4
5.b even 2 1 inner 240.2.o.a 4
5.c odd 4 2 1200.2.h.l 4
8.b even 2 1 960.2.o.b 4
8.d odd 2 1 960.2.o.b 4
12.b even 2 1 inner 240.2.o.a 4
15.d odd 2 1 inner 240.2.o.a 4
15.e even 4 2 1200.2.h.l 4
20.d odd 2 1 CM 240.2.o.a 4
20.e even 4 2 1200.2.h.l 4
24.f even 2 1 960.2.o.b 4
24.h odd 2 1 960.2.o.b 4
40.e odd 2 1 960.2.o.b 4
40.f even 2 1 960.2.o.b 4
60.h even 2 1 inner 240.2.o.a 4
60.l odd 4 2 1200.2.h.l 4
120.i odd 2 1 960.2.o.b 4
120.m even 2 1 960.2.o.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
240.2.o.a 4 1.a even 1 1 trivial
240.2.o.a 4 3.b odd 2 1 inner
240.2.o.a 4 4.b odd 2 1 inner
240.2.o.a 4 5.b even 2 1 inner
240.2.o.a 4 12.b even 2 1 inner
240.2.o.a 4 15.d odd 2 1 inner
240.2.o.a 4 20.d odd 2 1 CM
240.2.o.a 4 60.h even 2 1 inner
960.2.o.b 4 8.b even 2 1
960.2.o.b 4 8.d odd 2 1
960.2.o.b 4 24.f even 2 1
960.2.o.b 4 24.h odd 2 1
960.2.o.b 4 40.e odd 2 1
960.2.o.b 4 40.f even 2 1
960.2.o.b 4 120.i odd 2 1
960.2.o.b 4 120.m even 2 1
1200.2.h.l 4 5.c odd 4 2
1200.2.h.l 4 15.e even 4 2
1200.2.h.l 4 20.e even 4 2
1200.2.h.l 4 60.l odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} - 18 \) acting on \(S_{2}^{\mathrm{new}}(240, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 4T^{2} + 9 \) Copy content Toggle raw display
$5$ \( (T^{2} + 5)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - 18)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} \) 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
$23$ \( (T^{2} + 90)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 80)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 20)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 162)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 90)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( (T + 8)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} - 18)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 90)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 320)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} \) Copy content Toggle raw display
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