Properties

Label 740.2.a.e
Level $740$
Weight $2$
Character orbit 740.a
Self dual yes
Analytic conductor $5.909$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(3\)
Coefficient field: 3.3.148.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) 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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} - q^{5} + ( - \beta_{2} + 2 \beta_1) q^{7} + ( - \beta_{2} - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} - q^{5} + ( - \beta_{2} + 2 \beta_1) q^{7} + ( - \beta_{2} - \beta_1) q^{9} + (2 \beta_1 - 2) q^{11} + ( - \beta_{2} - 3 \beta_1 + 3) q^{13} + \beta_{2} q^{15} + (3 \beta_{2} + \beta_1 + 3) q^{17} + ( - \beta_{2} + 6) q^{19} + ( - \beta_{2} - 3 \beta_1 + 5) q^{21} - 2 \beta_1 q^{23} + q^{25} + (2 \beta_{2} + 2) q^{27} + (2 \beta_{2} + 4 \beta_1 - 2) q^{29} - 3 \beta_{2} q^{31} + (2 \beta_{2} - 2 \beta_1 + 2) q^{33} + (\beta_{2} - 2 \beta_1) q^{35} - q^{37} + ( - 4 \beta_{2} + 2 \beta_1) q^{39} + ( - 2 \beta_{2} - 2) q^{41} + (2 \beta_{2} + 4 \beta_1 + 2) q^{43} + (\beta_{2} + \beta_1) q^{45} + (\beta_{2} - 4 \beta_1 + 6) q^{47} + (3 \beta_{2} - \beta_1 + 8) q^{49} + (2 \beta_1 - 8) q^{51} + (2 \beta_{2} + 6 \beta_1) q^{53} + ( - 2 \beta_1 + 2) q^{55} + ( - 7 \beta_{2} - \beta_1 + 3) q^{57} + ( - 5 \beta_{2} - 4 \beta_1 + 4) q^{59} + (8 \beta_{2} + 4 \beta_1 + 2) q^{61} + ( - 3 \beta_{2} - 4 \beta_1) q^{63} + (\beta_{2} + 3 \beta_1 - 3) q^{65} + (7 \beta_{2} + 2 \beta_1 + 6) q^{67} + (2 \beta_1 - 2) q^{69} + (2 \beta_{2} - 2 \beta_1 - 10) q^{71} + 2 \beta_1 q^{73} - \beta_{2} q^{75} + (6 \beta_{2} - 2 \beta_1 + 10) q^{77} + (3 \beta_{2} - 2 \beta_1 - 2) q^{79} + (3 \beta_{2} + 5 \beta_1 - 6) q^{81} + ( - \beta_{2} + 2 \beta_1 - 6) q^{83} + ( - 3 \beta_{2} - \beta_1 - 3) q^{85} + (4 \beta_{2} - 2 \beta_1 - 2) q^{87} + ( - 6 \beta_{2} - 6 \beta_1) q^{89} + ( - 10 \beta_{2} - 10) q^{91} + ( - 3 \beta_{2} - 3 \beta_1 + 9) q^{93} + (\beta_{2} - 6) q^{95} + ( - 2 \beta_{2} + 6) q^{97} + ( - 2 \beta_1 - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{5} + 2 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{5} + 2 q^{7} - q^{9} - 4 q^{11} + 6 q^{13} + 10 q^{17} + 18 q^{19} + 12 q^{21} - 2 q^{23} + 3 q^{25} + 6 q^{27} - 2 q^{29} + 4 q^{33} - 2 q^{35} - 3 q^{37} + 2 q^{39} - 6 q^{41} + 10 q^{43} + q^{45} + 14 q^{47} + 23 q^{49} - 22 q^{51} + 6 q^{53} + 4 q^{55} + 8 q^{57} + 8 q^{59} + 10 q^{61} - 4 q^{63} - 6 q^{65} + 20 q^{67} - 4 q^{69} - 32 q^{71} + 2 q^{73} + 28 q^{77} - 8 q^{79} - 13 q^{81} - 16 q^{83} - 10 q^{85} - 8 q^{87} - 6 q^{89} - 30 q^{91} + 24 q^{93} - 18 q^{95} + 18 q^{97} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 2 \) 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
−1.48119
2.17009
0.311108
0 −1.67513 0 −1.00000 0 −4.63752 0 −0.193937 0
1.2 0 −0.539189 0 −1.00000 0 3.80098 0 −2.70928 0
1.3 0 2.21432 0 −1.00000 0 2.83654 0 1.90321 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.e 3
3.b odd 2 1 6660.2.a.q 3
4.b odd 2 1 2960.2.a.t 3
5.b even 2 1 3700.2.a.i 3
5.c odd 4 2 3700.2.d.h 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
740.2.a.e 3 1.a even 1 1 trivial
2960.2.a.t 3 4.b odd 2 1
3700.2.a.i 3 5.b even 2 1
3700.2.d.h 6 5.c odd 4 2
6660.2.a.q 3 3.b odd 2 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} - 4T - 2 \) Copy content Toggle raw display
$5$ \( (T + 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} - 2 T^{2} + \cdots + 50 \) Copy content Toggle raw display
$11$ \( T^{3} + 4 T^{2} + \cdots - 16 \) Copy content Toggle raw display
$13$ \( T^{3} - 6 T^{2} + \cdots + 100 \) Copy content Toggle raw display
$17$ \( T^{3} - 10T^{2} + 148 \) Copy content Toggle raw display
$19$ \( T^{3} - 18 T^{2} + \cdots - 194 \) Copy content Toggle raw display
$23$ \( T^{3} + 2 T^{2} + \cdots - 8 \) Copy content Toggle raw display
$29$ \( T^{3} + 2 T^{2} + \cdots - 184 \) Copy content Toggle raw display
$31$ \( T^{3} - 36T - 54 \) Copy content Toggle raw display
$37$ \( (T + 1)^{3} \) Copy content Toggle raw display
$41$ \( T^{3} + 6 T^{2} + \cdots - 40 \) Copy content Toggle raw display
$43$ \( T^{3} - 10 T^{2} + \cdots - 8 \) Copy content Toggle raw display
$47$ \( T^{3} - 14T^{2} + 74 \) Copy content Toggle raw display
$53$ \( T^{3} - 6 T^{2} + \cdots - 200 \) Copy content Toggle raw display
$59$ \( T^{3} - 8 T^{2} + \cdots + 158 \) Copy content Toggle raw display
$61$ \( T^{3} - 10 T^{2} + \cdots + 2056 \) Copy content Toggle raw display
$67$ \( T^{3} - 20 T^{2} + \cdots + 1850 \) Copy content Toggle raw display
$71$ \( T^{3} + 32 T^{2} + \cdots + 736 \) Copy content Toggle raw display
$73$ \( T^{3} - 2 T^{2} + \cdots + 8 \) Copy content Toggle raw display
$79$ \( T^{3} + 8 T^{2} + \cdots - 262 \) Copy content Toggle raw display
$83$ \( T^{3} + 16 T^{2} + \cdots + 74 \) Copy content Toggle raw display
$89$ \( T^{3} + 6 T^{2} + \cdots - 216 \) Copy content Toggle raw display
$97$ \( T^{3} - 18 T^{2} + \cdots - 136 \) Copy content Toggle raw display
show more
show less