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Properties

Label 240.3.c.d
Level $240$
Weight $3$
Character orbit 240.c
Analytic conductor $6.540$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

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: \(\Q(i, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 60)
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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} + ( - \beta_{3} + \beta_1) q^{5} + (2 \beta_{3} - 2 \beta_{2}) q^{7} + (\beta_{3} - \beta_{2} - 2 \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{3} + ( - \beta_{3} + \beta_1) q^{5} + (2 \beta_{3} - 2 \beta_{2}) q^{7} + (\beta_{3} - \beta_{2} - 2 \beta_1 + 1) q^{9} + (\beta_{3} - \beta_{2} - 2 \beta_1) q^{11} + (3 \beta_{3} - 3 \beta_{2}) q^{13} + ( - \beta_{3} + 5 \beta_{2} + \beta_1 - 5) q^{15} + (7 \beta_{3} + 7 \beta_{2}) q^{17} + 6 q^{19} + (2 \beta_{3} - 2 \beta_{2} - 4 \beta_1 - 16) q^{21} + (\beta_{3} + \beta_{2}) q^{23} + (5 \beta_{3} - 5 \beta_{2} - 15) q^{25} + (2 \beta_{3} - 9 \beta_{2}) q^{27} + ( - 3 \beta_{3} + 3 \beta_{2} + 6 \beta_1) q^{29} - 34 q^{31} + (\beta_{3} - 9 \beta_{2}) q^{33} + (6 \beta_{3} + 10 \beta_{2} + 4 \beta_1) q^{35} + ( - 11 \beta_{3} + 11 \beta_{2}) q^{37} + (3 \beta_{3} - 3 \beta_{2} - 6 \beta_1 - 24) q^{39} + (2 \beta_{3} - 2 \beta_{2} - 4 \beta_1) q^{41} + (7 \beta_{3} - 7 \beta_{2}) q^{43} + ( - 6 \beta_{3} + 5 \beta_{2} + \cdots + 40) q^{45}+ \cdots + (\beta_{3} - \beta_{2} - 2 \beta_1 - 80) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{9} - 20 q^{15} + 24 q^{19} - 64 q^{21} - 60 q^{25} - 136 q^{31} - 96 q^{39} + 160 q^{45} - 60 q^{49} + 280 q^{51} + 160 q^{55} + 296 q^{61} + 40 q^{69} - 160 q^{75} + 312 q^{79} - 316 q^{81} - 280 q^{85} - 384 q^{91} - 320 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 3x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 4\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\nu^{3} + 2\nu^{2} + 4\nu + 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -2\nu^{3} + 2\nu^{2} - 4\nu + 3 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + \beta_{2} - 6 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{3} + \beta_{2} - 2\beta_1 ) / 4 \) Copy content Toggle raw display

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

Character values

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

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

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
− 1.61803i
1.61803i
0.618034i
− 0.618034i
0 −2.23607 − 2.00000i 0 2.23607 − 4.47214i 0 − 8.00000i 0 1.00000 + 8.94427i 0
209.2 0 −2.23607 + 2.00000i 0 2.23607 + 4.47214i 0 8.00000i 0 1.00000 − 8.94427i 0
209.3 0 2.23607 − 2.00000i 0 −2.23607 + 4.47214i 0 − 8.00000i 0 1.00000 − 8.94427i 0
209.4 0 2.23607 + 2.00000i 0 −2.23607 − 4.47214i 0 8.00000i 0 1.00000 + 8.94427i 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.d 4
3.b odd 2 1 inner 240.3.c.d 4
4.b odd 2 1 60.3.b.a ✓ 4
5.b even 2 1 inner 240.3.c.d 4
5.c odd 4 1 1200.3.l.h 2
5.c odd 4 1 1200.3.l.q 2
8.b even 2 1 960.3.c.g 4
8.d odd 2 1 960.3.c.h 4
12.b even 2 1 60.3.b.a ✓ 4
15.d odd 2 1 inner 240.3.c.d 4
15.e even 4 1 1200.3.l.h 2
15.e even 4 1 1200.3.l.q 2
20.d odd 2 1 60.3.b.a ✓ 4
20.e even 4 1 300.3.g.e 2
20.e even 4 1 300.3.g.h 2
24.f even 2 1 960.3.c.h 4
24.h odd 2 1 960.3.c.g 4
36.f odd 6 2 1620.3.t.b 8
36.h even 6 2 1620.3.t.b 8
40.e odd 2 1 960.3.c.h 4
40.f even 2 1 960.3.c.g 4
60.h even 2 1 60.3.b.a ✓ 4
60.l odd 4 1 300.3.g.e 2
60.l odd 4 1 300.3.g.h 2
120.i odd 2 1 960.3.c.g 4
120.m even 2 1 960.3.c.h 4
180.n even 6 2 1620.3.t.b 8
180.p odd 6 2 1620.3.t.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.3.b.a ✓ 4 4.b odd 2 1
60.3.b.a ✓ 4 12.b even 2 1
60.3.b.a ✓ 4 20.d odd 2 1
60.3.b.a ✓ 4 60.h even 2 1
240.3.c.d 4 1.a even 1 1 trivial
240.3.c.d 4 3.b odd 2 1 inner
240.3.c.d 4 5.b even 2 1 inner
240.3.c.d 4 15.d odd 2 1 inner
300.3.g.e 2 20.e even 4 1
300.3.g.e 2 60.l odd 4 1
300.3.g.h 2 20.e even 4 1
300.3.g.h 2 60.l odd 4 1
960.3.c.g 4 8.b even 2 1
960.3.c.g 4 24.h odd 2 1
960.3.c.g 4 40.f even 2 1
960.3.c.g 4 120.i odd 2 1
960.3.c.h 4 8.d odd 2 1
960.3.c.h 4 24.f even 2 1
960.3.c.h 4 40.e odd 2 1
960.3.c.h 4 120.m even 2 1
1200.3.l.h 2 5.c odd 4 1
1200.3.l.h 2 15.e even 4 1
1200.3.l.q 2 5.c odd 4 1
1200.3.l.q 2 15.e even 4 1
1620.3.t.b 8 36.f odd 6 2
1620.3.t.b 8 36.h even 6 2
1620.3.t.b 8 180.n even 6 2
1620.3.t.b 8 180.p odd 6 2

Hecke kernels

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

\( T_{7}^{2} + 64 \) Copy content Toggle raw display
\( T_{17}^{2} - 980 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 2T^{2} + 81 \) Copy content Toggle raw display
$5$ \( T^{4} + 30T^{2} + 625 \) Copy content Toggle raw display
$7$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 80)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 144)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 980)^{2} \) Copy content Toggle raw display
$19$ \( (T - 6)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} - 20)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 720)^{2} \) Copy content Toggle raw display
$31$ \( (T + 34)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 1936)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 320)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 784)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 20)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 1620)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 9680)^{2} \) Copy content Toggle raw display
$61$ \( (T - 74)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 8464)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 2880)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 3136)^{2} \) Copy content Toggle raw display
$79$ \( (T - 78)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 10580)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 320)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 1024)^{2} \) Copy content Toggle raw display
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