Properties

Label 760.2.d.b
Level $760$
Weight $2$
Character orbit 760.d
Analytic conductor $6.069$
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] = [760,2,Mod(609,760)]
 
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, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("760.609");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 760 = 2^{3} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 760.d (of order \(2\), degree \(1\), 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: \(6.06863055362\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (\beta_{2} - 1) q^{5} + (\beta_{2} + 2 \beta_1) q^{7} + (\beta_{3} - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + (\beta_{2} - 1) q^{5} + (\beta_{2} + 2 \beta_1) q^{7} + (\beta_{3} - 3) q^{9} + (\beta_{2} + \beta_1) q^{13} + ( - \beta_{3} - \beta_1 + 2) q^{15} + ( - 2 \beta_{2} - 2 \beta_1) q^{17} - q^{19} + (\beta_{3} - 10) q^{21} - \beta_{2} q^{23} + ( - 2 \beta_{2} - 3) q^{25} + (4 \beta_{2} - 2 \beta_1) q^{27} - 2 q^{29} + (\beta_{3} - 6) q^{31} + ( - 2 \beta_{3} - \beta_{2} - 2 \beta_1) q^{35} + (3 \beta_{2} - \beta_1) q^{37} - 4 q^{39} + ( - \beta_{3} + 8) q^{41} + ( - \beta_{2} + 2 \beta_1) q^{43} + ( - \beta_{3} - \beta_{2} + 4 \beta_1 + 3) q^{45} + (5 \beta_{2} + 2 \beta_1) q^{47} - 13 q^{49} + 8 q^{51} + ( - 5 \beta_{2} - \beta_1) q^{53} - \beta_1 q^{57} + (\beta_{3} + 6) q^{59} - \beta_{3} q^{61} + (7 \beta_{2} - 6 \beta_1) q^{63} + ( - \beta_{3} - \beta_{2} - \beta_1 - 2) q^{65} + \beta_1 q^{67} + (\beta_{3} - 2) q^{69} + (\beta_{3} - 6) q^{71} + ( - 2 \beta_{2} - 2 \beta_1) q^{73} + (2 \beta_{3} - 3 \beta_1 - 4) q^{75} + (\beta_{3} + 2) q^{79} + ( - 3 \beta_{3} + 11) q^{81} + (5 \beta_{2} + 4 \beta_1) q^{83} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 4) q^{85} - 2 \beta_1 q^{87} + (2 \beta_{3} - 2) q^{89} + ( - \beta_{3} - 10) q^{91} + (4 \beta_{2} - 8 \beta_1) q^{93} + ( - \beta_{2} + 1) q^{95} + ( - \beta_{2} + 5 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{5} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{5} - 12 q^{9} + 8 q^{15} - 4 q^{19} - 40 q^{21} - 12 q^{25} - 8 q^{29} - 24 q^{31} - 16 q^{39} + 32 q^{41} + 12 q^{45} - 52 q^{49} + 32 q^{51} + 24 q^{59} - 8 q^{65} - 8 q^{69} - 24 q^{71} - 16 q^{75} + 8 q^{79} + 44 q^{81} + 16 q^{85} - 8 q^{89} - 40 q^{91} + 4 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 3x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\nu^{3} + 4\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 4\nu^{2} + 6 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 6 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{2} - 2\beta_1 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/760\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(381\) \(401\) \(457\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
609.1
1.61803i
0.618034i
0.618034i
1.61803i
0 3.23607i 0 −1.00000 + 2.00000i 0 4.47214i 0 −7.47214 0
609.2 0 1.23607i 0 −1.00000 2.00000i 0 4.47214i 0 1.47214 0
609.3 0 1.23607i 0 −1.00000 + 2.00000i 0 4.47214i 0 1.47214 0
609.4 0 3.23607i 0 −1.00000 2.00000i 0 4.47214i 0 −7.47214 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.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 760.2.d.b 4
4.b odd 2 1 1520.2.d.c 4
5.b even 2 1 inner 760.2.d.b 4
5.c odd 4 1 3800.2.a.k 2
5.c odd 4 1 3800.2.a.q 2
20.d odd 2 1 1520.2.d.c 4
20.e even 4 1 7600.2.a.w 2
20.e even 4 1 7600.2.a.be 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
760.2.d.b 4 1.a even 1 1 trivial
760.2.d.b 4 5.b even 2 1 inner
1520.2.d.c 4 4.b odd 2 1
1520.2.d.c 4 20.d odd 2 1
3800.2.a.k 2 5.c odd 4 1
3800.2.a.q 2 5.c odd 4 1
7600.2.a.w 2 20.e even 4 1
7600.2.a.be 2 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 12T_{3}^{2} + 16 \) acting on \(S_{2}^{\mathrm{new}}(760, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 12T^{2} + 16 \) Copy content Toggle raw display
$5$ \( (T^{2} + 2 T + 5)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 20)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 12T^{2} + 16 \) Copy content Toggle raw display
$17$ \( T^{4} + 48T^{2} + 256 \) Copy content Toggle raw display
$19$ \( (T + 1)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$29$ \( (T + 2)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 12 T + 16)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 108T^{2} + 1936 \) Copy content Toggle raw display
$41$ \( (T^{2} - 16 T + 44)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 72T^{2} + 16 \) Copy content Toggle raw display
$47$ \( T^{4} + 168T^{2} + 1936 \) Copy content Toggle raw display
$53$ \( T^{4} + 172T^{2} + 5776 \) Copy content Toggle raw display
$59$ \( (T^{2} - 12 T + 16)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 20)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 12T^{2} + 16 \) Copy content Toggle raw display
$71$ \( (T^{2} + 12 T + 16)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 48T^{2} + 256 \) Copy content Toggle raw display
$79$ \( (T^{2} - 4 T - 16)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 232T^{2} + 1936 \) Copy content Toggle raw display
$89$ \( (T^{2} + 4 T - 76)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 348T^{2} + 5776 \) Copy content Toggle raw display
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