Properties

Label 1305.2.c.c
Level $1305$
Weight $2$
Character orbit 1305.c
Analytic conductor $10.420$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1305,2,Mod(784,1305)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1305, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1305.784");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1305 = 3^{2} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1305.c (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: \(10.4204774638\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-5}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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 \(\beta = \sqrt{-5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{4} + \beta q^{5} - 2 \beta q^{7} +O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{4} + \beta q^{5} - 2 \beta q^{7} - 5 q^{11} - 2 \beta q^{13} + 4 q^{16} - 2 \beta q^{17} - 4 q^{19} + 2 \beta q^{20} - 3 \beta q^{23} - 5 q^{25} - 4 \beta q^{28} - q^{29} - 2 q^{31} + 10 q^{35} - \beta q^{37} + 5 q^{41} + \beta q^{43} - 10 q^{44} + 4 \beta q^{47} - 13 q^{49} - 4 \beta q^{52} - \beta q^{53} - 5 \beta q^{55} + 10 q^{59} + 12 q^{61} + 8 q^{64} + 10 q^{65} + 4 \beta q^{67} - 4 \beta q^{68} - 10 q^{71} + 3 \beta q^{73} - 8 q^{76} + 10 \beta q^{77} + 14 q^{79} + 4 \beta q^{80} - \beta q^{83} + 10 q^{85} - 10 q^{89} - 20 q^{91} - 6 \beta q^{92} - 4 \beta q^{95} + 3 \beta q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{4} - 10 q^{11} + 8 q^{16} - 8 q^{19} - 10 q^{25} - 2 q^{29} - 4 q^{31} + 20 q^{35} + 10 q^{41} - 20 q^{44} - 26 q^{49} + 20 q^{59} + 24 q^{61} + 16 q^{64} + 20 q^{65} - 20 q^{71} - 16 q^{76} + 28 q^{79} + 20 q^{85} - 20 q^{89} - 40 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1305\mathbb{Z}\right)^\times\).

\(n\) \(146\) \(262\) \(901\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
784.1
2.23607i
2.23607i
0 0 2.00000 2.23607i 0 4.47214i 0 0 0
784.2 0 0 2.00000 2.23607i 0 4.47214i 0 0 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 1305.2.c.c 2
3.b odd 2 1 1305.2.c.d yes 2
5.b even 2 1 inner 1305.2.c.c 2
5.c odd 4 2 6525.2.a.u 2
15.d odd 2 1 1305.2.c.d yes 2
15.e even 4 2 6525.2.a.v 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1305.2.c.c 2 1.a even 1 1 trivial
1305.2.c.c 2 5.b even 2 1 inner
1305.2.c.d yes 2 3.b odd 2 1
1305.2.c.d yes 2 15.d odd 2 1
6525.2.a.u 2 5.c odd 4 2
6525.2.a.v 2 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1305, [\chi])\):

\( T_{2} \) Copy content Toggle raw display
\( T_{7}^{2} + 20 \) Copy content Toggle raw display
\( T_{11} + 5 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 5 \) Copy content Toggle raw display
$7$ \( T^{2} + 20 \) Copy content Toggle raw display
$11$ \( (T + 5)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 20 \) Copy content Toggle raw display
$17$ \( T^{2} + 20 \) Copy content Toggle raw display
$19$ \( (T + 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 45 \) Copy content Toggle raw display
$29$ \( (T + 1)^{2} \) Copy content Toggle raw display
$31$ \( (T + 2)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 5 \) Copy content Toggle raw display
$41$ \( (T - 5)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 5 \) Copy content Toggle raw display
$47$ \( T^{2} + 80 \) Copy content Toggle raw display
$53$ \( T^{2} + 5 \) Copy content Toggle raw display
$59$ \( (T - 10)^{2} \) Copy content Toggle raw display
$61$ \( (T - 12)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 80 \) Copy content Toggle raw display
$71$ \( (T + 10)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 45 \) Copy content Toggle raw display
$79$ \( (T - 14)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 5 \) Copy content Toggle raw display
$89$ \( (T + 10)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 45 \) Copy content Toggle raw display
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