Properties

Label 740.2.a.f
Level $740$
Weight $2$
Character orbit 740.a
Self dual yes
Analytic conductor $5.909$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [740,2,Mod(1,740)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(740, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("740.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 740 = 2^{2} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 740.a (trivial)

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: yes
Analytic conductor: \(5.90892974957\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.286164.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 11x^{2} - 3x + 12 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{3} + q^{5} + ( - \beta_1 - 1) q^{7} + (\beta_{2} + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{3} + q^{5} + ( - \beta_1 - 1) q^{7} + (\beta_{2} + 3) q^{9} + (\beta_{3} - \beta_{2} + 2) q^{11} + ( - \beta_{3} - 2) q^{13} + ( - \beta_1 + 1) q^{15} + ( - \beta_{3} + 2) q^{17} + (\beta_{3} - \beta_{2} - \beta_1 + 1) q^{19} + (\beta_{2} + 2 \beta_1 + 4) q^{21} + 2 \beta_1 q^{23} + q^{25} + ( - \beta_{3} + \beta_{2} - 2 \beta_1 + 2) q^{27} + 2 \beta_1 q^{29} + ( - \beta_{3} - \beta_{2} - \beta_1 - 1) q^{31} + (3 \beta_{3} - 3 \beta_{2} - 2 \beta_1) q^{33} + ( - \beta_1 - 1) q^{35} + q^{37} + ( - 2 \beta_{3} + 2 \beta_{2} + \cdots - 2) q^{39}+ \cdots + (6 \beta_{3} - 4 \beta_{2} + 2 \beta_1 - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 3 q^{3} + 4 q^{5} - 5 q^{7} + 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 3 q^{3} + 4 q^{5} - 5 q^{7} + 13 q^{9} + 5 q^{11} - 6 q^{13} + 3 q^{15} + 10 q^{17} + 19 q^{21} + 2 q^{23} + 4 q^{25} + 9 q^{27} + 2 q^{29} - 4 q^{31} - 11 q^{33} - 5 q^{35} + 4 q^{37} + 2 q^{39} + 29 q^{41} + 4 q^{43} + 13 q^{45} - 9 q^{47} + q^{49} + 14 q^{51} - 3 q^{53} + 5 q^{55} + 8 q^{57} - 10 q^{59} + 8 q^{61} - 8 q^{63} - 6 q^{65} + 14 q^{67} - 44 q^{69} - 17 q^{71} + 7 q^{73} + 3 q^{75} - 21 q^{77} - 24 q^{79} + 28 q^{81} + 31 q^{83} + 10 q^{85} - 44 q^{87} - 4 q^{89} + 14 q^{91} + 18 q^{93} + 12 q^{97} - 22 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 11x^{2} - 3x + 12 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2\nu - 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 2\nu^{2} - 7\nu + 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2\beta _1 + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 2\beta_{2} + 11\beta _1 + 8 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
3.84956
0.914132
−1.51951
−2.24418
0 −2.84956 0 1.00000 0 −4.84956 0 5.11999 0
1.2 0 0.0858680 0 1.00000 0 −1.91413 0 −2.99263 0
1.3 0 2.51951 0 1.00000 0 0.519509 0 3.34793 0
1.4 0 3.24418 0 1.00000 0 1.24418 0 7.52471 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(5\) \( -1 \)
\(37\) \( -1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 740.2.a.f 4
3.b odd 2 1 6660.2.a.r 4
4.b odd 2 1 2960.2.a.v 4
5.b even 2 1 3700.2.a.j 4
5.c odd 4 2 3700.2.d.i 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
740.2.a.f 4 1.a even 1 1 trivial
2960.2.a.v 4 4.b odd 2 1
3700.2.a.j 4 5.b even 2 1
3700.2.d.i 8 5.c odd 4 2
6660.2.a.r 4 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 3T_{3}^{3} - 8T_{3}^{2} + 24T_{3} - 2 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(740))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 3 T^{3} + \cdots - 2 \) Copy content Toggle raw display
$5$ \( (T - 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 5 T^{3} + \cdots + 6 \) Copy content Toggle raw display
$11$ \( T^{4} - 5 T^{3} + \cdots - 240 \) Copy content Toggle raw display
$13$ \( T^{4} + 6 T^{3} + \cdots + 160 \) Copy content Toggle raw display
$17$ \( T^{4} - 10 T^{3} + \cdots + 48 \) Copy content Toggle raw display
$19$ \( T^{4} - 38 T^{2} + \cdots + 64 \) Copy content Toggle raw display
$23$ \( T^{4} - 2 T^{3} + \cdots + 192 \) Copy content Toggle raw display
$29$ \( T^{4} - 2 T^{3} + \cdots + 192 \) Copy content Toggle raw display
$31$ \( T^{4} + 4 T^{3} + \cdots + 148 \) Copy content Toggle raw display
$37$ \( (T - 1)^{4} \) Copy content Toggle raw display
$41$ \( T^{4} - 29 T^{3} + \cdots - 168 \) Copy content Toggle raw display
$43$ \( T^{4} - 4 T^{3} + \cdots + 1296 \) Copy content Toggle raw display
$47$ \( T^{4} + 9 T^{3} + \cdots + 54 \) Copy content Toggle raw display
$53$ \( T^{4} + 3 T^{3} + \cdots + 72 \) Copy content Toggle raw display
$59$ \( T^{4} + 10 T^{3} + \cdots - 1968 \) Copy content Toggle raw display
$61$ \( (T - 2)^{4} \) Copy content Toggle raw display
$67$ \( T^{4} - 14 T^{3} + \cdots - 8 \) Copy content Toggle raw display
$71$ \( T^{4} + 17 T^{3} + \cdots - 288 \) Copy content Toggle raw display
$73$ \( T^{4} - 7 T^{3} + \cdots + 360 \) Copy content Toggle raw display
$79$ \( T^{4} + 24 T^{3} + \cdots - 332 \) Copy content Toggle raw display
$83$ \( T^{4} - 31 T^{3} + \cdots - 5238 \) Copy content Toggle raw display
$89$ \( T^{4} + 4 T^{3} + \cdots + 3504 \) Copy content Toggle raw display
$97$ \( T^{4} - 12 T^{3} + \cdots + 400 \) Copy content Toggle raw display
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