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Properties

Label 1183.2.c.f
Level $1183$
Weight $2$
Character orbit 1183.c
Analytic conductor $9.446$
Analytic rank $0$
Dimension $6$
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] = [1183,2,Mod(337,1183)] 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("1183.337"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1183, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1183 = 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1183.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(9.44630255912\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.399424.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 91)
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 a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{5} q^{2} + ( - \beta_{3} - \beta_1 - 1) q^{3} + ( - \beta_1 - 1) q^{4} + (\beta_{5} - \beta_{2}) q^{5} + (2 \beta_{5} - 2 \beta_{2}) q^{6} - \beta_{2} q^{7} + ( - \beta_{4} + \beta_{2}) q^{8}+ \cdots + ( - 7 \beta_{5} - 3 \beta_{4} + 7 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 4 q^{3} - 6 q^{4} + 14 q^{9} + 16 q^{10} + 24 q^{12} - 2 q^{14} - 2 q^{16} - 8 q^{17} - 8 q^{22} - 20 q^{23} + 10 q^{25} - 16 q^{27} + 48 q^{29} - 40 q^{30} - 4 q^{35} + 2 q^{36} + 20 q^{38} - 4 q^{40}+ \cdots - 12 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{4} + 2\nu^{3} - \nu^{2} + 2\nu - 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} - 3\nu^{3} + 4\nu^{2} - 2\nu + 8 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{5} + 2\nu^{4} - 3\nu^{3} + 6\nu^{2} - 2\nu + 4 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} - 2\nu^{4} + 3\nu^{3} - 6\nu^{2} + 10\nu - 8 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3\nu^{5} - 2\nu^{4} + 5\nu^{3} - 6\nu^{2} + 2\nu - 12 ) / 4 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{4} + \beta_{3} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + 2\beta_{3} + \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{4} + \beta_{3} - 2\beta_{2} + 2\beta _1 + 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -\beta_{5} + 2\beta_{3} - 5\beta_{2} - \beta _1 + 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 4\beta_{5} + \beta_{4} + 3\beta_{3} + 2\beta_{2} - 2\beta _1 + 5 ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1183\mathbb{Z}\right)^\times\).

\(n\) \(339\) \(1016\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
337.1
−0.671462 − 1.24464i
1.40680 + 0.144584i
0.264658 + 1.38923i
0.264658 − 1.38923i
1.40680 − 0.144584i
−0.671462 + 1.24464i
− 2.34292i −1.14637 −3.48929 1.34292i 2.68585i − 1.00000i 3.48929i −1.68585 3.14637
337.2 − 1.81361i −3.10278 −1.28917 2.81361i 5.62721i 1.00000i − 1.28917i 6.62721 5.10278
337.3 − 0.470683i 2.24914 1.77846 − 0.529317i − 1.05863i − 1.00000i − 1.77846i 2.05863 −0.249141
337.4 0.470683i 2.24914 1.77846 0.529317i 1.05863i 1.00000i 1.77846i 2.05863 −0.249141
337.5 1.81361i −3.10278 −1.28917 − 2.81361i − 5.62721i − 1.00000i 1.28917i 6.62721 5.10278
337.6 2.34292i −1.14637 −3.48929 − 1.34292i − 2.68585i 1.00000i − 3.48929i −1.68585 3.14637
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 337.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1183.2.c.f 6
13.b even 2 1 inner 1183.2.c.f 6
13.d odd 4 1 91.2.a.d ✓ 3
13.d odd 4 1 1183.2.a.i 3
39.f even 4 1 819.2.a.i 3
52.f even 4 1 1456.2.a.t 3
65.g odd 4 1 2275.2.a.m 3
91.i even 4 1 637.2.a.j 3
91.i even 4 1 8281.2.a.bg 3
91.z odd 12 2 637.2.e.j 6
91.bb even 12 2 637.2.e.i 6
104.j odd 4 1 5824.2.a.by 3
104.m even 4 1 5824.2.a.bs 3
273.o odd 4 1 5733.2.a.x 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.a.d ✓ 3 13.d odd 4 1
637.2.a.j 3 91.i even 4 1
637.2.e.i 6 91.bb even 12 2
637.2.e.j 6 91.z odd 12 2
819.2.a.i 3 39.f even 4 1
1183.2.a.i 3 13.d odd 4 1
1183.2.c.f 6 1.a even 1 1 trivial
1183.2.c.f 6 13.b even 2 1 inner
1456.2.a.t 3 52.f even 4 1
2275.2.a.m 3 65.g odd 4 1
5733.2.a.x 3 273.o odd 4 1
5824.2.a.bs 3 104.m even 4 1
5824.2.a.by 3 104.j odd 4 1
8281.2.a.bg 3 91.i even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + 9T_{2}^{4} + 20T_{2}^{2} + 4 \) acting on \(S_{2}^{\mathrm{new}}(1183, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 9 T^{4} + \cdots + 4 \) Copy content Toggle raw display
$3$ \( (T^{3} + 2 T^{2} - 6 T - 8)^{2} \) Copy content Toggle raw display
$5$ \( T^{6} + 10 T^{4} + \cdots + 4 \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$11$ \( T^{6} + 16 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$13$ \( T^{6} \) Copy content Toggle raw display
$17$ \( (T^{3} + 4 T^{2} - 10 T + 4)^{2} \) Copy content Toggle raw display
$19$ \( T^{6} + 14 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$23$ \( (T^{3} + 10 T^{2} + \cdots - 136)^{2} \) Copy content Toggle raw display
$29$ \( (T^{3} - 24 T^{2} + \cdots - 454)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} + 54 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$37$ \( T^{6} + 116 T^{4} + \cdots + 15376 \) Copy content Toggle raw display
$41$ \( T^{6} + 60 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$43$ \( (T^{3} + 10 T^{2} + \cdots - 628)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + 222 T^{4} + \cdots + 295936 \) Copy content Toggle raw display
$53$ \( (T^{3} - 8 T^{2} - 35 T - 22)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + 328 T^{4} + \cdots + 473344 \) Copy content Toggle raw display
$61$ \( (T + 2)^{6} \) Copy content Toggle raw display
$67$ \( T^{6} + 392 T^{4} + \cdots + 952576 \) Copy content Toggle raw display
$71$ \( T^{6} + 80 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$73$ \( T^{6} + 298 T^{4} + \cdots + 75076 \) Copy content Toggle raw display
$79$ \( (T^{3} + 14 T^{2} + \cdots - 16)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 686 T^{4} + \cdots + 10679824 \) Copy content Toggle raw display
$89$ \( T^{6} + 194 T^{4} + \cdots + 178084 \) Copy content Toggle raw display
$97$ \( T^{6} + 42 T^{4} + \cdots + 484 \) Copy content Toggle raw display
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