Properties

Label 3040.2.a.h
Level $3040$
Weight $2$
Character orbit 3040.a
Self dual yes
Analytic conductor $24.275$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3040,2,Mod(1,3040)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3040, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("3040.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3040 = 2^{5} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3040.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: \(24.2745222145\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) 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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} - q^{5} + ( - \beta - 2) q^{7} + (\beta + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{3} - q^{5} + ( - \beta - 2) q^{7} + (\beta + 1) q^{9} + 4 q^{11} + \beta q^{13} - \beta q^{15} + ( - \beta + 2) q^{17} + q^{19} + ( - 3 \beta - 4) q^{21} + (3 \beta - 2) q^{23} + q^{25} + ( - \beta + 4) q^{27} + (\beta - 6) q^{29} + 2 \beta q^{31} + 4 \beta q^{33} + (\beta + 2) q^{35} + ( - 2 \beta + 4) q^{37} + (\beta + 4) q^{39} + (4 \beta - 2) q^{41} + 2 q^{43} + ( - \beta - 1) q^{45} + ( - 2 \beta + 2) q^{47} + (5 \beta + 1) q^{49} + (\beta - 4) q^{51} + (\beta + 4) q^{53} - 4 q^{55} + \beta q^{57} - 5 \beta q^{59} + (2 \beta - 10) q^{61} + ( - 4 \beta - 6) q^{63} - \beta q^{65} + (\beta + 8) q^{67} + (\beta + 12) q^{69} + 8 q^{71} + (5 \beta + 2) q^{73} + \beta q^{75} + ( - 4 \beta - 8) q^{77} - 2 \beta q^{79} - 7 q^{81} + 14 q^{83} + (\beta - 2) q^{85} + ( - 5 \beta + 4) q^{87} - 10 q^{89} + ( - 3 \beta - 4) q^{91} + (2 \beta + 8) q^{93} - q^{95} + ( - 2 \beta + 4) q^{97} + (4 \beta + 4) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} - 2 q^{5} - 5 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{3} - 2 q^{5} - 5 q^{7} + 3 q^{9} + 8 q^{11} + q^{13} - q^{15} + 3 q^{17} + 2 q^{19} - 11 q^{21} - q^{23} + 2 q^{25} + 7 q^{27} - 11 q^{29} + 2 q^{31} + 4 q^{33} + 5 q^{35} + 6 q^{37} + 9 q^{39} + 4 q^{43} - 3 q^{45} + 2 q^{47} + 7 q^{49} - 7 q^{51} + 9 q^{53} - 8 q^{55} + q^{57} - 5 q^{59} - 18 q^{61} - 16 q^{63} - q^{65} + 17 q^{67} + 25 q^{69} + 16 q^{71} + 9 q^{73} + q^{75} - 20 q^{77} - 2 q^{79} - 14 q^{81} + 28 q^{83} - 3 q^{85} + 3 q^{87} - 20 q^{89} - 11 q^{91} + 18 q^{93} - 2 q^{95} + 6 q^{97} + 12 q^{99}+O(q^{100}) \) 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
−1.56155
2.56155
0 −1.56155 0 −1.00000 0 −0.438447 0 −0.561553 0
1.2 0 2.56155 0 −1.00000 0 −4.56155 0 3.56155 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(1\)
\(19\) \(-1\)

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 3040.2.a.h yes 2
4.b odd 2 1 3040.2.a.e 2
8.b even 2 1 6080.2.a.bc 2
8.d odd 2 1 6080.2.a.bi 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3040.2.a.e 2 4.b odd 2 1
3040.2.a.h yes 2 1.a even 1 1 trivial
6080.2.a.bc 2 8.b even 2 1
6080.2.a.bi 2 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3040))\):

\( T_{3}^{2} - T_{3} - 4 \) Copy content Toggle raw display
\( T_{7}^{2} + 5T_{7} + 2 \) Copy content Toggle raw display
\( T_{11} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - T - 4 \) Copy content Toggle raw display
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 5T + 2 \) Copy content Toggle raw display
$11$ \( (T - 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - T - 4 \) Copy content Toggle raw display
$17$ \( T^{2} - 3T - 2 \) Copy content Toggle raw display
$19$ \( (T - 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + T - 38 \) Copy content Toggle raw display
$29$ \( T^{2} + 11T + 26 \) Copy content Toggle raw display
$31$ \( T^{2} - 2T - 16 \) Copy content Toggle raw display
$37$ \( T^{2} - 6T - 8 \) Copy content Toggle raw display
$41$ \( T^{2} - 68 \) Copy content Toggle raw display
$43$ \( (T - 2)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 2T - 16 \) Copy content Toggle raw display
$53$ \( T^{2} - 9T + 16 \) Copy content Toggle raw display
$59$ \( T^{2} + 5T - 100 \) Copy content Toggle raw display
$61$ \( T^{2} + 18T + 64 \) Copy content Toggle raw display
$67$ \( T^{2} - 17T + 68 \) Copy content Toggle raw display
$71$ \( (T - 8)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 9T - 86 \) Copy content Toggle raw display
$79$ \( T^{2} + 2T - 16 \) Copy content Toggle raw display
$83$ \( (T - 14)^{2} \) Copy content Toggle raw display
$89$ \( (T + 10)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 6T - 8 \) Copy content Toggle raw display
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