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Properties

Label 1260.4.k.b
Level $1260$
Weight $4$
Character orbit 1260.k
Analytic conductor $74.342$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1260,4,Mod(1009,1260)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1260.1009"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage: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, 4, names="a")
 
Level: \( N \) \(=\) \( 1260 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1260.k (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(74.3424066072\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Copy content comment:defining polynomial
 
Copy content 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

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q + ( - 11 i + 2) q^{5} + 7 i q^{7} + 7 q^{11} + 3 i q^{13} + 61 i q^{17} - 48 q^{19} - 58 i q^{23} + ( - 44 i - 117) q^{25} + 219 q^{29} + 298 q^{31} + (14 i + 77) q^{35} + 170 i q^{37} - 50 q^{41} + 484 i q^{43} + \cdots + 1339 i q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{5} + 14 q^{11} - 96 q^{19} - 234 q^{25} + 438 q^{29} + 596 q^{31} + 154 q^{35} - 100 q^{41} - 98 q^{49} + 28 q^{55} - 1564 q^{59} + 976 q^{61} + 66 q^{65} + 480 q^{71} + 2130 q^{79} + 1342 q^{85}+ \cdots - 192 q^{95}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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 2.00000 − 11.0000i 0 7.00000i 0 0 0
1009.2 0 0 0 2.00000 + 11.0000i 0 − 7.00000i 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 1260.4.k.b 2
3.b odd 2 1 140.4.e.b ✓ 2
5.b even 2 1 inner 1260.4.k.b 2
12.b even 2 1 560.4.g.b 2
15.d odd 2 1 140.4.e.b ✓ 2
15.e even 4 1 700.4.a.c 1
15.e even 4 1 700.4.a.m 1
21.c even 2 1 980.4.e.b 2
60.h even 2 1 560.4.g.b 2
105.g even 2 1 980.4.e.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.4.e.b ✓ 2 3.b odd 2 1
140.4.e.b ✓ 2 15.d odd 2 1
560.4.g.b 2 12.b even 2 1
560.4.g.b 2 60.h even 2 1
700.4.a.c 1 15.e even 4 1
700.4.a.m 1 15.e even 4 1
980.4.e.b 2 21.c even 2 1
980.4.e.b 2 105.g even 2 1
1260.4.k.b 2 1.a even 1 1 trivial
1260.4.k.b 2 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11} - 7 \) acting on \(S_{4}^{\mathrm{new}}(1260, [\chi])\). 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
$5$ \( T^{2} - 4T + 125 \) Copy content Toggle raw display
$7$ \( T^{2} + 49 \) Copy content Toggle raw display
$11$ \( (T - 7)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 9 \) Copy content Toggle raw display
$17$ \( T^{2} + 3721 \) Copy content Toggle raw display
$19$ \( (T + 48)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 3364 \) Copy content Toggle raw display
$29$ \( (T - 219)^{2} \) Copy content Toggle raw display
$31$ \( (T - 298)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 28900 \) Copy content Toggle raw display
$41$ \( (T + 50)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 234256 \) Copy content Toggle raw display
$47$ \( T^{2} + 17161 \) Copy content Toggle raw display
$53$ \( T^{2} + 44100 \) Copy content Toggle raw display
$59$ \( (T + 782)^{2} \) Copy content Toggle raw display
$61$ \( (T - 488)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 244036 \) Copy content Toggle raw display
$71$ \( (T - 240)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 3364 \) Copy content Toggle raw display
$79$ \( (T - 1065)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 1073296 \) Copy content Toggle raw display
$89$ \( (T - 608)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 1792921 \) Copy content Toggle raw display
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