Properties

Label 1260.2.k.b
Level $1260$
Weight $2$
Character orbit 1260.k
Analytic conductor $10.061$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1260,2,Mod(1009,1260)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1260, 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("1260.1009");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1260 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1260.k (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: \(10.0611506547\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 140)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 2 i - 1) q^{5} + i q^{7} - 4 i q^{13} - 4 i q^{17} - 4 q^{19} + 8 i q^{23} + (4 i - 3) q^{25} + 2 q^{29} - 8 q^{31} + ( - i + 2) q^{35} - 8 i q^{37} - 6 q^{41} - 8 i q^{43} - 8 i q^{47} - q^{49} + \cdots - 12 i q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} - 8 q^{19} - 6 q^{25} + 4 q^{29} - 16 q^{31} + 4 q^{35} - 12 q^{41} - 2 q^{49} - 8 q^{59} - 12 q^{61} - 16 q^{65} - 24 q^{71} + 8 q^{79} - 16 q^{85} - 20 q^{89} + 8 q^{91} + 8 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1260\mathbb{Z}\right)^\times\).

\(n\) \(281\) \(631\) \(757\) \(1081\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1009.1
1.00000i
1.00000i
0 0 0 −1.00000 2.00000i 0 1.00000i 0 0 0
1009.2 0 0 0 −1.00000 + 2.00000i 0 1.00000i 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 1260.2.k.b 2
3.b odd 2 1 140.2.e.b 2
4.b odd 2 1 5040.2.t.g 2
5.b even 2 1 inner 1260.2.k.b 2
5.c odd 4 1 6300.2.a.g 1
5.c odd 4 1 6300.2.a.y 1
12.b even 2 1 560.2.g.c 2
15.d odd 2 1 140.2.e.b 2
15.e even 4 1 700.2.a.f 1
15.e even 4 1 700.2.a.h 1
20.d odd 2 1 5040.2.t.g 2
21.c even 2 1 980.2.e.a 2
21.g even 6 2 980.2.q.e 4
21.h odd 6 2 980.2.q.d 4
24.f even 2 1 2240.2.g.c 2
24.h odd 2 1 2240.2.g.d 2
60.h even 2 1 560.2.g.c 2
60.l odd 4 1 2800.2.a.o 1
60.l odd 4 1 2800.2.a.s 1
105.g even 2 1 980.2.e.a 2
105.k odd 4 1 4900.2.a.l 1
105.k odd 4 1 4900.2.a.m 1
105.o odd 6 2 980.2.q.d 4
105.p even 6 2 980.2.q.e 4
120.i odd 2 1 2240.2.g.d 2
120.m even 2 1 2240.2.g.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.e.b 2 3.b odd 2 1
140.2.e.b 2 15.d odd 2 1
560.2.g.c 2 12.b even 2 1
560.2.g.c 2 60.h even 2 1
700.2.a.f 1 15.e even 4 1
700.2.a.h 1 15.e even 4 1
980.2.e.a 2 21.c even 2 1
980.2.e.a 2 105.g even 2 1
980.2.q.d 4 21.h odd 6 2
980.2.q.d 4 105.o odd 6 2
980.2.q.e 4 21.g even 6 2
980.2.q.e 4 105.p even 6 2
1260.2.k.b 2 1.a even 1 1 trivial
1260.2.k.b 2 5.b even 2 1 inner
2240.2.g.c 2 24.f even 2 1
2240.2.g.c 2 120.m even 2 1
2240.2.g.d 2 24.h odd 2 1
2240.2.g.d 2 120.i odd 2 1
2800.2.a.o 1 60.l odd 4 1
2800.2.a.s 1 60.l odd 4 1
4900.2.a.l 1 105.k odd 4 1
4900.2.a.m 1 105.k odd 4 1
5040.2.t.g 2 4.b odd 2 1
5040.2.t.g 2 20.d odd 2 1
6300.2.a.g 1 5.c odd 4 1
6300.2.a.y 1 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11} \) acting on \(S_{2}^{\mathrm{new}}(1260, [\chi])\). 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
$5$ \( T^{2} + 2T + 5 \) Copy content Toggle raw display
$7$ \( T^{2} + 1 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 16 \) Copy content Toggle raw display
$17$ \( T^{2} + 16 \) Copy content Toggle raw display
$19$ \( (T + 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 64 \) Copy content Toggle raw display
$29$ \( (T - 2)^{2} \) Copy content Toggle raw display
$31$ \( (T + 8)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 64 \) Copy content Toggle raw display
$41$ \( (T + 6)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 64 \) Copy content Toggle raw display
$47$ \( T^{2} + 64 \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( (T + 4)^{2} \) Copy content Toggle raw display
$61$ \( (T + 6)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 64 \) Copy content Toggle raw display
$71$ \( (T + 12)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 16 \) Copy content Toggle raw display
$79$ \( (T - 4)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( (T + 10)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 144 \) Copy content Toggle raw display
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