Properties

Label 133.1.r.a
Level 133
Weight 1
Character orbit 133.r
Analytic conductor 0.066
Analytic rank 0
Dimension 2
Projective image \(D_{3}\)
CM discriminant -19
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 133 = 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 133.r (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.0663756466802\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{3}\)
Projective field Galois closure of 3.1.931.1
Artin image $C_3\times S_3$
Artin field Galois closure of 6.0.336091.1

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{6} q^{4} -\zeta_{6}^{2} q^{5} + \zeta_{6}^{2} q^{7} + \zeta_{6}^{2} q^{9} +O(q^{10})\) \( q -\zeta_{6} q^{4} -\zeta_{6}^{2} q^{5} + \zeta_{6}^{2} q^{7} + \zeta_{6}^{2} q^{9} + \zeta_{6} q^{11} + \zeta_{6}^{2} q^{16} -2 \zeta_{6} q^{17} + \zeta_{6}^{2} q^{19} - q^{20} -\zeta_{6}^{2} q^{23} + q^{28} + \zeta_{6} q^{35} + q^{36} - q^{43} -\zeta_{6}^{2} q^{44} + \zeta_{6} q^{45} -\zeta_{6}^{2} q^{47} -\zeta_{6} q^{49} + q^{55} -\zeta_{6}^{2} q^{61} -\zeta_{6} q^{63} + q^{64} + 2 \zeta_{6}^{2} q^{68} + \zeta_{6} q^{73} + q^{76} - q^{77} + \zeta_{6} q^{80} -\zeta_{6} q^{81} - q^{83} -2 q^{85} - q^{92} + \zeta_{6} q^{95} - q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{4} + q^{5} - q^{7} - q^{9} + O(q^{10}) \) \( 2q - q^{4} + q^{5} - q^{7} - q^{9} + q^{11} - q^{16} - 2q^{17} - q^{19} - 2q^{20} + q^{23} + 2q^{28} + q^{35} + 2q^{36} - 2q^{43} + q^{44} + q^{45} + q^{47} - q^{49} + 2q^{55} + q^{61} - q^{63} + 2q^{64} - 2q^{68} + q^{73} + 2q^{76} - 2q^{77} + q^{80} - q^{81} - 2q^{83} - 4q^{85} - 2q^{92} + q^{95} - 2q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/133\mathbb{Z}\right)^\times\).

\(n\) \(78\) \(115\)
\(\chi(n)\) \(-1\) \(\zeta_{6}^{2}\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
18.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0 −0.500000 + 0.866025i 0.500000 + 0.866025i 0 −0.500000 0.866025i 0 −0.500000 0.866025i 0
37.1 0 0 −0.500000 0.866025i 0.500000 0.866025i 0 −0.500000 + 0.866025i 0 −0.500000 + 0.866025i 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
19.b odd 2 1 CM by \(\Q(\sqrt{-19}) \)
7.c even 3 1 inner
133.r odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 133.1.r.a 2
3.b odd 2 1 1197.1.cz.a 2
4.b odd 2 1 2128.1.cl.c 2
5.b even 2 1 3325.1.bm.a 2
5.c odd 4 2 3325.1.y.a 4
7.b odd 2 1 931.1.r.a 2
7.c even 3 1 inner 133.1.r.a 2
7.c even 3 1 931.1.b.a 1
7.d odd 6 1 931.1.b.b 1
7.d odd 6 1 931.1.r.a 2
19.b odd 2 1 CM 133.1.r.a 2
19.c even 3 1 2527.1.j.a 2
19.c even 3 1 2527.1.n.a 2
19.d odd 6 1 2527.1.j.a 2
19.d odd 6 1 2527.1.n.a 2
19.e even 9 3 2527.1.bd.a 6
19.e even 9 3 2527.1.be.a 6
19.f odd 18 3 2527.1.bd.a 6
19.f odd 18 3 2527.1.be.a 6
21.h odd 6 1 1197.1.cz.a 2
28.g odd 6 1 2128.1.cl.c 2
35.j even 6 1 3325.1.bm.a 2
35.l odd 12 2 3325.1.y.a 4
57.d even 2 1 1197.1.cz.a 2
76.d even 2 1 2128.1.cl.c 2
95.d odd 2 1 3325.1.bm.a 2
95.g even 4 2 3325.1.y.a 4
133.c even 2 1 931.1.r.a 2
133.g even 3 1 2527.1.j.a 2
133.h even 3 1 2527.1.n.a 2
133.j odd 6 1 2527.1.n.a 2
133.n odd 6 1 2527.1.j.a 2
133.o even 6 1 931.1.b.b 1
133.o even 6 1 931.1.r.a 2
133.r odd 6 1 inner 133.1.r.a 2
133.r odd 6 1 931.1.b.a 1
133.u even 9 3 2527.1.be.a 6
133.w even 9 3 2527.1.bd.a 6
133.bd odd 18 3 2527.1.be.a 6
133.be odd 18 3 2527.1.bd.a 6
399.w even 6 1 1197.1.cz.a 2
532.t even 6 1 2128.1.cl.c 2
665.x odd 6 1 3325.1.bm.a 2
665.ck even 12 2 3325.1.y.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
133.1.r.a 2 1.a even 1 1 trivial
133.1.r.a 2 7.c even 3 1 inner
133.1.r.a 2 19.b odd 2 1 CM
133.1.r.a 2 133.r odd 6 1 inner
931.1.b.a 1 7.c even 3 1
931.1.b.a 1 133.r odd 6 1
931.1.b.b 1 7.d odd 6 1
931.1.b.b 1 133.o even 6 1
931.1.r.a 2 7.b odd 2 1
931.1.r.a 2 7.d odd 6 1
931.1.r.a 2 133.c even 2 1
931.1.r.a 2 133.o even 6 1
1197.1.cz.a 2 3.b odd 2 1
1197.1.cz.a 2 21.h odd 6 1
1197.1.cz.a 2 57.d even 2 1
1197.1.cz.a 2 399.w even 6 1
2128.1.cl.c 2 4.b odd 2 1
2128.1.cl.c 2 28.g odd 6 1
2128.1.cl.c 2 76.d even 2 1
2128.1.cl.c 2 532.t even 6 1
2527.1.j.a 2 19.c even 3 1
2527.1.j.a 2 19.d odd 6 1
2527.1.j.a 2 133.g even 3 1
2527.1.j.a 2 133.n odd 6 1
2527.1.n.a 2 19.c even 3 1
2527.1.n.a 2 19.d odd 6 1
2527.1.n.a 2 133.h even 3 1
2527.1.n.a 2 133.j odd 6 1
2527.1.bd.a 6 19.e even 9 3
2527.1.bd.a 6 19.f odd 18 3
2527.1.bd.a 6 133.w even 9 3
2527.1.bd.a 6 133.be odd 18 3
2527.1.be.a 6 19.e even 9 3
2527.1.be.a 6 19.f odd 18 3
2527.1.be.a 6 133.u even 9 3
2527.1.be.a 6 133.bd odd 18 3
3325.1.y.a 4 5.c odd 4 2
3325.1.y.a 4 35.l odd 12 2
3325.1.y.a 4 95.g even 4 2
3325.1.y.a 4 665.ck even 12 2
3325.1.bm.a 2 5.b even 2 1
3325.1.bm.a 2 35.j even 6 1
3325.1.bm.a 2 95.d odd 2 1
3325.1.bm.a 2 665.x odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(133, [\chi])\).

Hecke characteristic polynomials

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