Properties

Label 760.2.a.i
Level $760$
Weight $2$
Character orbit 760.a
Self dual yes
Analytic conductor $6.069$
Analytic rank $1$
Dimension $3$
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] = [760,2,Mod(1,760)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(760, 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("760.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 760 = 2^{3} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 760.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: \(6.06863055362\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x + 2 \) 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.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} - q^{5} + ( - \beta_{2} + 2 \beta_1 - 1) q^{7} + \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} - q^{5} + ( - \beta_{2} + 2 \beta_1 - 1) q^{7} + \beta_{2} q^{9} + 2 \beta_{2} q^{11} + ( - 2 \beta_{2} + \beta_1 - 4) q^{13} + \beta_1 q^{15} + (\beta_{2} - 1) q^{17} + q^{19} + ( - \beta_{2} + 2 \beta_1 - 5) q^{21} + (\beta_{2} - 3) q^{23} + q^{25} + ( - \beta_{2} + 2 \beta_1 - 1) q^{27} + (\beta_{2} - 4 \beta_1 - 1) q^{29} + ( - 2 \beta_{2} + 2) q^{31} + ( - 2 \beta_{2} - 2 \beta_1 - 2) q^{33} + (\beta_{2} - 2 \beta_1 + 1) q^{35} + ( - \beta_{2} + \beta_1 - 7) q^{37} + (\beta_{2} + 6 \beta_1 - 1) q^{39} + (2 \beta_{2} + 2 \beta_1 - 8) q^{41} + (2 \beta_{2} - 4 \beta_1 - 2) q^{43} - \beta_{2} q^{45} - 4 \beta_{2} q^{47} + (\beta_{2} - 6 \beta_1 + 6) q^{49} + ( - \beta_{2} - 1) q^{51} + (2 \beta_{2} - 7 \beta_1) q^{53} - 2 \beta_{2} q^{55} - \beta_1 q^{57} + ( - 3 \beta_{2} - 4 \beta_1 + 5) q^{59} + (2 \beta_{2} + 2 \beta_1 - 6) q^{61} + (2 \beta_{2} - 2) q^{63} + (2 \beta_{2} - \beta_1 + 4) q^{65} + (4 \beta_{2} - \beta_1) q^{67} + ( - \beta_{2} + 2 \beta_1 - 1) q^{69} + (3 \beta_{2} + 4 \beta_1 - 3) q^{73} - \beta_1 q^{75} + (4 \beta_{2} - 4) q^{77} + (2 \beta_1 + 8) q^{79} + ( - 4 \beta_{2} + 2 \beta_1 - 5) q^{81} + ( - 2 \beta_1 - 4) q^{83} + ( - \beta_{2} + 1) q^{85} + (3 \beta_{2} + 11) q^{87} + ( - 6 \beta_{2} + 6 \beta_1 - 4) q^{89} + (\beta_{2} - 10 \beta_1 + 13) q^{91} + (2 \beta_{2} + 2) q^{93} - q^{95} + ( - 3 \beta_{2} - \beta_1 + 3) q^{97} + ( - 2 \beta_{2} + 4 \beta_1 + 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{3} - 3 q^{5} - q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{3} - 3 q^{5} - q^{7} - 11 q^{13} + q^{15} - 3 q^{17} + 3 q^{19} - 13 q^{21} - 9 q^{23} + 3 q^{25} - q^{27} - 7 q^{29} + 6 q^{31} - 8 q^{33} + q^{35} - 20 q^{37} + 3 q^{39} - 22 q^{41} - 10 q^{43} + 12 q^{49} - 3 q^{51} - 7 q^{53} - q^{57} + 11 q^{59} - 16 q^{61} - 6 q^{63} + 11 q^{65} - q^{67} - q^{69} - 5 q^{73} - q^{75} - 12 q^{77} + 26 q^{79} - 13 q^{81} - 14 q^{83} + 3 q^{85} + 33 q^{87} - 6 q^{89} + 29 q^{91} + 6 q^{93} - 3 q^{95} + 8 q^{97} + 28 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) 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
2.34292
0.470683
−1.81361
0 −2.34292 0 −1.00000 0 1.19656 0 2.48929 0
1.2 0 −0.470683 0 −1.00000 0 2.71982 0 −2.77846 0
1.3 0 1.81361 0 −1.00000 0 −4.91638 0 0.289169 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 760.2.a.i 3
3.b odd 2 1 6840.2.a.bm 3
4.b odd 2 1 1520.2.a.q 3
5.b even 2 1 3800.2.a.w 3
5.c odd 4 2 3800.2.d.n 6
8.b even 2 1 6080.2.a.bx 3
8.d odd 2 1 6080.2.a.br 3
20.d odd 2 1 7600.2.a.bp 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
760.2.a.i 3 1.a even 1 1 trivial
1520.2.a.q 3 4.b odd 2 1
3800.2.a.w 3 5.b even 2 1
3800.2.d.n 6 5.c odd 4 2
6080.2.a.br 3 8.d odd 2 1
6080.2.a.bx 3 8.b even 2 1
6840.2.a.bm 3 3.b odd 2 1
7600.2.a.bp 3 20.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(760))\):

\( T_{3}^{3} + T_{3}^{2} - 4T_{3} - 2 \) Copy content Toggle raw display
\( T_{7}^{3} + T_{7}^{2} - 16T_{7} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} + T^{2} - 4T - 2 \) Copy content Toggle raw display
$5$ \( (T + 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} + T^{2} - 16 T + 16 \) Copy content Toggle raw display
$11$ \( T^{3} - 28T + 16 \) Copy content Toggle raw display
$13$ \( T^{3} + 11 T^{2} + 16 T - 86 \) Copy content Toggle raw display
$17$ \( T^{3} + 3 T^{2} - 4 T - 4 \) Copy content Toggle raw display
$19$ \( (T - 1)^{3} \) Copy content Toggle raw display
$23$ \( T^{3} + 9 T^{2} + 20 T + 8 \) Copy content Toggle raw display
$29$ \( T^{3} + 7 T^{2} - 44 T - 292 \) Copy content Toggle raw display
$31$ \( T^{3} - 6 T^{2} - 16 T + 32 \) Copy content Toggle raw display
$37$ \( T^{3} + 20 T^{2} + 126 T + 244 \) Copy content Toggle raw display
$41$ \( T^{3} + 22 T^{2} + 100 T - 232 \) Copy content Toggle raw display
$43$ \( T^{3} + 10 T^{2} - 32 T - 352 \) Copy content Toggle raw display
$47$ \( T^{3} - 112T - 128 \) Copy content Toggle raw display
$53$ \( T^{3} + 7 T^{2} - 168 T - 1342 \) Copy content Toggle raw display
$59$ \( T^{3} - 11 T^{2} - 140 T + 1544 \) Copy content Toggle raw display
$61$ \( T^{3} + 16 T^{2} + 24 T - 352 \) Copy content Toggle raw display
$67$ \( T^{3} + T^{2} - 100 T + 262 \) Copy content Toggle raw display
$71$ \( T^{3} \) Copy content Toggle raw display
$73$ \( T^{3} + 5 T^{2} - 172 T - 1228 \) Copy content Toggle raw display
$79$ \( T^{3} - 26 T^{2} + 208 T - 496 \) Copy content Toggle raw display
$83$ \( T^{3} + 14 T^{2} + 48 T + 16 \) Copy content Toggle raw display
$89$ \( T^{3} + 6 T^{2} - 252 T - 1256 \) Copy content Toggle raw display
$97$ \( T^{3} - 8 T^{2} - 58 T + 292 \) Copy content Toggle raw display
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