Properties

Label 1305.1.l.a
Level $1305$
Weight $1$
Character orbit 1305.l
Analytic conductor $0.651$
Analytic rank $0$
Dimension $2$
Projective image $D_{4}$
RM discriminant 5
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1305,1,Mod(244,1305)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1305, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 2, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1305.244");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1305 = 3^{2} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1305.l (of order \(4\), degree \(2\), 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: \(0.651279841486\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 145)
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.0.121945.1
Artin image: $C_4^2:C_2^2$
Artin field: Galois closure of 16.0.1524467112922265625.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q - i q^{4} - i q^{5} +O(q^{10}) \) Copy content Toggle raw display \( q - i q^{4} - i q^{5} + ( - i + 1) q^{11} - q^{16} + (i - 1) q^{19} - q^{20} - q^{25} + i q^{29} + ( - i + 1) q^{31} + ( - i - 1) q^{41} + ( - i - 1) q^{44} + q^{49} + ( - i - 1) q^{55} + ( - i + 1) q^{61} + i q^{64} + 2 i q^{71} + (i + 1) q^{76} + (i - 1) q^{79} + i q^{80} + ( - i + 1) q^{89} + (i + 1) q^{95} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{11} - 2 q^{16} - 2 q^{19} - 2 q^{20} - 2 q^{25} + 2 q^{31} - 2 q^{41} - 2 q^{44} + 2 q^{49} - 2 q^{55} + 2 q^{61} + 2 q^{76} - 2 q^{79} + 2 q^{89} + 2 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
244.1
1.00000i
1.00000i
0 0 1.00000i 1.00000i 0 0 0 0 0
829.1 0 0 1.00000i 1.00000i 0 0 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 RM by \(\Q(\sqrt{5}) \)
29.c odd 4 1 inner
145.f odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1305.1.l.a 2
3.b odd 2 1 145.1.f.a 2
5.b even 2 1 RM 1305.1.l.a 2
12.b even 2 1 2320.1.bj.a 2
15.d odd 2 1 145.1.f.a 2
15.e even 4 2 725.1.g.a 2
29.c odd 4 1 inner 1305.1.l.a 2
60.h even 2 1 2320.1.bj.a 2
87.f even 4 1 145.1.f.a 2
145.f odd 4 1 inner 1305.1.l.a 2
348.k odd 4 1 2320.1.bj.a 2
435.i odd 4 1 725.1.g.a 2
435.l even 4 1 145.1.f.a 2
435.t odd 4 1 725.1.g.a 2
1740.bm odd 4 1 2320.1.bj.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
145.1.f.a 2 3.b odd 2 1
145.1.f.a 2 15.d odd 2 1
145.1.f.a 2 87.f even 4 1
145.1.f.a 2 435.l even 4 1
725.1.g.a 2 15.e even 4 2
725.1.g.a 2 435.i odd 4 1
725.1.g.a 2 435.t odd 4 1
1305.1.l.a 2 1.a even 1 1 trivial
1305.1.l.a 2 5.b even 2 1 RM
1305.1.l.a 2 29.c odd 4 1 inner
1305.1.l.a 2 145.f odd 4 1 inner
2320.1.bj.a 2 12.b even 2 1
2320.1.bj.a 2 60.h even 2 1
2320.1.bj.a 2 348.k odd 4 1
2320.1.bj.a 2 1740.bm odd 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1305, [\chi])\).

Hecke characteristic polynomials

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