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Properties

Label 1170.4.a.z
Level $1170$
Weight $4$
Character orbit 1170.a
Self dual yes
Analytic conductor $69.032$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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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] = [1170,4,Mod(1,1170)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1170, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1170.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: \( N \) \(=\) \( 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1170.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,4,0,8,10,0,0,16,0,20,-54] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(69.0322347067\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 130)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(\beta = \sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{2} + 4 q^{4} + 5 q^{5} - 8 \beta q^{7} + 8 q^{8} + 10 q^{10} + (5 \beta - 27) q^{11} - 13 q^{13} - 16 \beta q^{14} + 16 q^{16} + (16 \beta - 34) q^{17} + (59 \beta - 21) q^{19} + 20 q^{20} + (10 \beta - 54) q^{22} + \cdots - 46 q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{2} + 8 q^{4} + 10 q^{5} + 16 q^{8} + 20 q^{10} - 54 q^{11} - 26 q^{13} + 32 q^{16} - 68 q^{17} - 42 q^{19} + 40 q^{20} - 108 q^{22} - 294 q^{23} + 50 q^{25} - 52 q^{26} - 312 q^{29} + 170 q^{31}+ \cdots - 92 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
1.61803
−0.618034
2.00000 0 4.00000 5.00000 0 −17.8885 8.00000 0 10.0000
1.2 2.00000 0 4.00000 5.00000 0 17.8885 8.00000 0 10.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( -1 \)
\(5\) \( -1 \)
\(13\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1170.4.a.z 2
3.b odd 2 1 130.4.a.e ✓ 2
12.b even 2 1 1040.4.a.j 2
15.d odd 2 1 650.4.a.s 2
15.e even 4 2 650.4.b.k 4
39.d odd 2 1 1690.4.a.t 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.4.a.e ✓ 2 3.b odd 2 1
650.4.a.s 2 15.d odd 2 1
650.4.b.k 4 15.e even 4 2
1040.4.a.j 2 12.b even 2 1
1170.4.a.z 2 1.a even 1 1 trivial
1690.4.a.t 2 39.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1170))\):

\( T_{7}^{2} - 320 \) Copy content Toggle raw display
\( T_{11}^{2} + 54T_{11} + 604 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T - 5)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 320 \) Copy content Toggle raw display
$11$ \( T^{2} + 54T + 604 \) Copy content Toggle raw display
$13$ \( (T + 13)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 68T - 124 \) Copy content Toggle raw display
$19$ \( T^{2} + 42T - 16964 \) Copy content Toggle raw display
$23$ \( T^{2} + 294T + 21204 \) Copy content Toggle raw display
$29$ \( T^{2} + 312T + 23356 \) Copy content Toggle raw display
$31$ \( T^{2} - 170T + 7180 \) Copy content Toggle raw display
$37$ \( T^{2} + 408T + 35836 \) Copy content Toggle raw display
$41$ \( T^{2} + 140T - 23980 \) Copy content Toggle raw display
$43$ \( T^{2} + 322T + 24116 \) Copy content Toggle raw display
$47$ \( T^{2} - 184T - 49856 \) Copy content Toggle raw display
$53$ \( T^{2} - 116T - 275116 \) Copy content Toggle raw display
$59$ \( T^{2} - 382T - 106324 \) Copy content Toggle raw display
$61$ \( T^{2} - 48T - 64404 \) Copy content Toggle raw display
$67$ \( T^{2} + 1620 T + 619120 \) Copy content Toggle raw display
$71$ \( T^{2} + 930T + 195100 \) Copy content Toggle raw display
$73$ \( T^{2} - 456T - 168516 \) Copy content Toggle raw display
$79$ \( T^{2} - 532T - 35824 \) Copy content Toggle raw display
$83$ \( T^{2} - 120 T - 1307120 \) Copy content Toggle raw display
$89$ \( T^{2} + 292T - 147964 \) Copy content Toggle raw display
$97$ \( T^{2} + 1092 T + 205636 \) Copy content Toggle raw display
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