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Properties

Label 240.3.bg.a
Level $240$
Weight $3$
Character orbit 240.bg
Analytic conductor $6.540$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [240,3,Mod(97,240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(240, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("240.97");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 240.bg (of order \(4\), degree \(2\), 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: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (2 \beta_{3} - 3 \beta_{2} - \beta_1 - 1) q^{5} + (2 \beta_{3} + \beta_{2} - 1) q^{7} + 3 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + (2 \beta_{3} - 3 \beta_{2} - \beta_1 - 1) q^{5} + (2 \beta_{3} + \beta_{2} - 1) q^{7} + 3 \beta_{2} q^{9} + (3 \beta_{3} - 3 \beta_1 - 4) q^{11} + ( - 8 \beta_{2} + 2 \beta_1 - 8) q^{13} + ( - 3 \beta_{3} - 3 \beta_{2} + \cdots - 6) q^{15}+ \cdots + ( - 9 \beta_{3} - 12 \beta_{2} - 9 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{5} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{5} - 4 q^{7} - 16 q^{11} - 32 q^{13} - 24 q^{15} - 40 q^{17} - 24 q^{21} - 56 q^{23} + 16 q^{25} + 16 q^{31} - 36 q^{33} + 40 q^{35} + 64 q^{37} - 56 q^{41} + 8 q^{43} + 36 q^{45} - 128 q^{47} - 72 q^{51} + 56 q^{53} + 124 q^{55} - 72 q^{57} + 200 q^{61} - 12 q^{63} - 112 q^{65} + 200 q^{67} + 272 q^{71} + 76 q^{73} - 24 q^{75} + 88 q^{77} - 36 q^{81} + 16 q^{83} + 232 q^{85} + 84 q^{87} + 16 q^{91} - 72 q^{93} - 144 q^{95} - 20 q^{97}+O(q^{100}) \) 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

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

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

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

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

Character values

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

\(n\) \(31\) \(97\) \(161\) \(181\)
\(\chi(n)\) \(1\) \(-\beta_{2}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1
−1.22474 + 1.22474i
1.22474 − 1.22474i
−1.22474 − 1.22474i
1.22474 + 1.22474i
0 −1.22474 + 1.22474i 0 2.67423 + 4.22474i 0 1.44949 + 1.44949i 0 − 3.00000i 0
97.2 0 1.22474 − 1.22474i 0 −4.67423 + 1.77526i 0 −3.44949 − 3.44949i 0 − 3.00000i 0
193.1 0 −1.22474 − 1.22474i 0 2.67423 − 4.22474i 0 1.44949 − 1.44949i 0 3.00000i 0
193.2 0 1.22474 + 1.22474i 0 −4.67423 − 1.77526i 0 −3.44949 + 3.44949i 0 3.00000i 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
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 240.3.bg.a 4
3.b odd 2 1 720.3.bh.k 4
4.b odd 2 1 15.3.f.a ✓ 4
5.b even 2 1 1200.3.bg.k 4
5.c odd 4 1 inner 240.3.bg.a 4
5.c odd 4 1 1200.3.bg.k 4
8.b even 2 1 960.3.bg.h 4
8.d odd 2 1 960.3.bg.i 4
12.b even 2 1 45.3.g.b 4
15.e even 4 1 720.3.bh.k 4
20.d odd 2 1 75.3.f.c 4
20.e even 4 1 15.3.f.a ✓ 4
20.e even 4 1 75.3.f.c 4
36.f odd 6 2 405.3.l.h 8
36.h even 6 2 405.3.l.f 8
40.i odd 4 1 960.3.bg.h 4
40.k even 4 1 960.3.bg.i 4
60.h even 2 1 225.3.g.a 4
60.l odd 4 1 45.3.g.b 4
60.l odd 4 1 225.3.g.a 4
180.v odd 12 2 405.3.l.f 8
180.x even 12 2 405.3.l.h 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.3.f.a ✓ 4 4.b odd 2 1
15.3.f.a ✓ 4 20.e even 4 1
45.3.g.b 4 12.b even 2 1
45.3.g.b 4 60.l odd 4 1
75.3.f.c 4 20.d odd 2 1
75.3.f.c 4 20.e even 4 1
225.3.g.a 4 60.h even 2 1
225.3.g.a 4 60.l odd 4 1
240.3.bg.a 4 1.a even 1 1 trivial
240.3.bg.a 4 5.c odd 4 1 inner
405.3.l.f 8 36.h even 6 2
405.3.l.f 8 180.v odd 12 2
405.3.l.h 8 36.f odd 6 2
405.3.l.h 8 180.x even 12 2
720.3.bh.k 4 3.b odd 2 1
720.3.bh.k 4 15.e even 4 1
960.3.bg.h 4 8.b even 2 1
960.3.bg.h 4 40.i odd 4 1
960.3.bg.i 4 8.d odd 2 1
960.3.bg.i 4 40.k even 4 1
1200.3.bg.k 4 5.b even 2 1
1200.3.bg.k 4 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} + 4T_{7}^{3} + 8T_{7}^{2} - 40T_{7} + 100 \) acting on \(S_{3}^{\mathrm{new}}(240, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 9 \) Copy content Toggle raw display
$5$ \( T^{4} + 4 T^{3} + \cdots + 625 \) Copy content Toggle raw display
$7$ \( T^{4} + 4 T^{3} + \cdots + 100 \) Copy content Toggle raw display
$11$ \( (T^{2} + 8 T - 38)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 32 T^{3} + \cdots + 13456 \) Copy content Toggle raw display
$17$ \( T^{4} + 40 T^{3} + \cdots + 8464 \) Copy content Toggle raw display
$19$ \( T^{4} + 504 T^{2} + 32400 \) Copy content Toggle raw display
$23$ \( T^{4} + 56 T^{3} + \cdots + 144400 \) Copy content Toggle raw display
$29$ \( T^{4} + 1236T^{2} + 900 \) Copy content Toggle raw display
$31$ \( (T^{2} - 8 T - 200)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} - 64 T^{3} + \cdots + 211600 \) Copy content Toggle raw display
$41$ \( (T^{2} + 28 T - 20)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} - 8 T^{3} + \cdots + 1420864 \) Copy content Toggle raw display
$47$ \( T^{4} + 128 T^{3} + \cdots + 3055504 \) Copy content Toggle raw display
$53$ \( T^{4} - 56 T^{3} + \cdots + 1600 \) Copy content Toggle raw display
$59$ \( T^{4} + 14124 T^{2} + 19980900 \) Copy content Toggle raw display
$61$ \( (T^{2} - 100 T + 556)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 200 T^{3} + \cdots + 24522304 \) Copy content Toggle raw display
$71$ \( (T - 68)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} - 76 T^{3} + \cdots + 38316100 \) Copy content Toggle raw display
$79$ \( (T^{2} + 600)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} - 16 T^{3} + \cdots + 309136 \) Copy content Toggle raw display
$89$ \( T^{4} + 15624 T^{2} + 59907600 \) Copy content Toggle raw display
$97$ \( T^{4} + 20 T^{3} + \cdots + 515524 \) Copy content Toggle raw display
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