Properties

Label 133.1.m.a
Level 133
Weight 1
Character orbit 133.m
Analytic conductor 0.066
Analytic rank 0
Dimension 4
Projective image \(A_{4}\)
CM/RM no
Inner twists 4

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(0.0663756466802\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(A_{4}\)
Projective field Galois closure of 4.0.17689.1
Artin image $\SL(2,3):C_2$
Artin field Galois closure of \(\mathbb{Q}[x]/(x^{16} - \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{12}^{2} q^{2} + \zeta_{12}^{5} q^{3} -\zeta_{12}^{5} q^{5} + \zeta_{12} q^{6} -\zeta_{12}^{3} q^{7} - q^{8} +O(q^{10})\) \( q -\zeta_{12}^{2} q^{2} + \zeta_{12}^{5} q^{3} -\zeta_{12}^{5} q^{5} + \zeta_{12} q^{6} -\zeta_{12}^{3} q^{7} - q^{8} -\zeta_{12} q^{10} + \zeta_{12} q^{13} + \zeta_{12}^{5} q^{14} + \zeta_{12}^{4} q^{15} + \zeta_{12}^{2} q^{16} + \zeta_{12}^{5} q^{17} + \zeta_{12}^{3} q^{19} + \zeta_{12}^{2} q^{21} + \zeta_{12}^{4} q^{23} -\zeta_{12}^{5} q^{24} -\zeta_{12}^{3} q^{26} -\zeta_{12}^{3} q^{27} -\zeta_{12}^{4} q^{29} + q^{30} + \zeta_{12} q^{34} -\zeta_{12}^{2} q^{35} -\zeta_{12}^{5} q^{38} - q^{39} + \zeta_{12}^{5} q^{40} + \zeta_{12}^{5} q^{41} -\zeta_{12}^{4} q^{42} -\zeta_{12}^{2} q^{43} + q^{46} + \zeta_{12} q^{47} -\zeta_{12} q^{48} - q^{49} -\zeta_{12}^{4} q^{51} -\zeta_{12}^{4} q^{53} + \zeta_{12}^{5} q^{54} + \zeta_{12}^{3} q^{56} -\zeta_{12}^{2} q^{57} - q^{58} -\zeta_{12}^{5} q^{59} -\zeta_{12} q^{61} + q^{64} + q^{65} + \zeta_{12}^{4} q^{67} -\zeta_{12}^{3} q^{69} + \zeta_{12}^{4} q^{70} + \zeta_{12}^{2} q^{71} -\zeta_{12}^{5} q^{73} + \zeta_{12}^{2} q^{78} -\zeta_{12}^{2} q^{79} + \zeta_{12} q^{80} + \zeta_{12}^{2} q^{81} + \zeta_{12} q^{82} + \zeta_{12}^{4} q^{85} + \zeta_{12}^{4} q^{86} + \zeta_{12}^{3} q^{87} -\zeta_{12} q^{89} -\zeta_{12}^{4} q^{91} -\zeta_{12}^{3} q^{94} + \zeta_{12}^{2} q^{95} + \zeta_{12}^{5} q^{97} + \zeta_{12}^{2} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{2} - 4q^{8} + O(q^{10}) \) \( 4q - 2q^{2} - 4q^{8} - 2q^{15} + 2q^{16} + 2q^{21} - 2q^{23} + 2q^{29} + 4q^{30} - 2q^{35} - 4q^{39} + 2q^{42} - 2q^{43} + 4q^{46} - 4q^{49} + 2q^{51} + 2q^{53} - 2q^{57} - 4q^{58} + 4q^{64} + 4q^{65} - 2q^{67} - 2q^{70} + 2q^{71} + 2q^{78} - 2q^{79} + 2q^{81} - 2q^{85} - 2q^{86} + 2q^{91} + 2q^{95} + 2q^{98} + O(q^{100}) \)

Character values

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

\(n\) \(78\) \(115\)
\(\chi(n)\) \(\zeta_{12}^{4}\) \(-1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
83.1
0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
−0.500000 0.866025i −0.866025 + 0.500000i 0 0.866025 0.500000i 0.866025 + 0.500000i 1.00000i −1.00000 0 −0.866025 0.500000i
83.2 −0.500000 0.866025i 0.866025 0.500000i 0 −0.866025 + 0.500000i −0.866025 0.500000i 1.00000i −1.00000 0 0.866025 + 0.500000i
125.1 −0.500000 + 0.866025i −0.866025 0.500000i 0 0.866025 + 0.500000i 0.866025 0.500000i 1.00000i −1.00000 0 −0.866025 + 0.500000i
125.2 −0.500000 + 0.866025i 0.866025 + 0.500000i 0 −0.866025 0.500000i −0.866025 + 0.500000i 1.00000i −1.00000 0 0.866025 0.500000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
19.c even 3 1 inner
133.m 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.m.a 4
3.b odd 2 1 1197.1.ci.c 4
4.b odd 2 1 2128.1.cy.a 4
5.b even 2 1 3325.1.bc.a 4
5.c odd 4 1 3325.1.bi.a 4
5.c odd 4 1 3325.1.bi.b 4
7.b odd 2 1 inner 133.1.m.a 4
7.c even 3 1 931.1.k.a 4
7.c even 3 1 931.1.t.a 4
7.d odd 6 1 931.1.k.a 4
7.d odd 6 1 931.1.t.a 4
19.b odd 2 1 2527.1.m.d 4
19.c even 3 1 inner 133.1.m.a 4
19.c even 3 1 2527.1.d.e 2
19.d odd 6 1 2527.1.d.b 2
19.d odd 6 1 2527.1.m.d 4
19.e even 9 6 2527.1.y.e 12
19.f odd 18 6 2527.1.y.d 12
21.c even 2 1 1197.1.ci.c 4
28.d even 2 1 2128.1.cy.a 4
35.c odd 2 1 3325.1.bc.a 4
35.f even 4 1 3325.1.bi.a 4
35.f even 4 1 3325.1.bi.b 4
57.h odd 6 1 1197.1.ci.c 4
76.g odd 6 1 2128.1.cy.a 4
95.i even 6 1 3325.1.bc.a 4
95.m odd 12 1 3325.1.bi.a 4
95.m odd 12 1 3325.1.bi.b 4
133.c even 2 1 2527.1.m.d 4
133.g even 3 1 931.1.t.a 4
133.h even 3 1 931.1.k.a 4
133.k odd 6 1 931.1.t.a 4
133.m odd 6 1 inner 133.1.m.a 4
133.m odd 6 1 2527.1.d.e 2
133.p even 6 1 2527.1.d.b 2
133.p even 6 1 2527.1.m.d 4
133.t odd 6 1 931.1.k.a 4
133.y odd 18 6 2527.1.y.e 12
133.ba even 18 6 2527.1.y.d 12
399.z even 6 1 1197.1.ci.c 4
532.u even 6 1 2128.1.cy.a 4
665.bh odd 6 1 3325.1.bc.a 4
665.ci even 12 1 3325.1.bi.a 4
665.ci even 12 1 3325.1.bi.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
133.1.m.a 4 1.a even 1 1 trivial
133.1.m.a 4 7.b odd 2 1 inner
133.1.m.a 4 19.c even 3 1 inner
133.1.m.a 4 133.m odd 6 1 inner
931.1.k.a 4 7.c even 3 1
931.1.k.a 4 7.d odd 6 1
931.1.k.a 4 133.h even 3 1
931.1.k.a 4 133.t odd 6 1
931.1.t.a 4 7.c even 3 1
931.1.t.a 4 7.d odd 6 1
931.1.t.a 4 133.g even 3 1
931.1.t.a 4 133.k odd 6 1
1197.1.ci.c 4 3.b odd 2 1
1197.1.ci.c 4 21.c even 2 1
1197.1.ci.c 4 57.h odd 6 1
1197.1.ci.c 4 399.z even 6 1
2128.1.cy.a 4 4.b odd 2 1
2128.1.cy.a 4 28.d even 2 1
2128.1.cy.a 4 76.g odd 6 1
2128.1.cy.a 4 532.u even 6 1
2527.1.d.b 2 19.d odd 6 1
2527.1.d.b 2 133.p even 6 1
2527.1.d.e 2 19.c even 3 1
2527.1.d.e 2 133.m odd 6 1
2527.1.m.d 4 19.b odd 2 1
2527.1.m.d 4 19.d odd 6 1
2527.1.m.d 4 133.c even 2 1
2527.1.m.d 4 133.p even 6 1
2527.1.y.d 12 19.f odd 18 6
2527.1.y.d 12 133.ba even 18 6
2527.1.y.e 12 19.e even 9 6
2527.1.y.e 12 133.y odd 18 6
3325.1.bc.a 4 5.b even 2 1
3325.1.bc.a 4 35.c odd 2 1
3325.1.bc.a 4 95.i even 6 1
3325.1.bc.a 4 665.bh odd 6 1
3325.1.bi.a 4 5.c odd 4 1
3325.1.bi.a 4 35.f even 4 1
3325.1.bi.a 4 95.m odd 12 1
3325.1.bi.a 4 665.ci even 12 1
3325.1.bi.b 4 5.c odd 4 1
3325.1.bi.b 4 35.f even 4 1
3325.1.bi.b 4 95.m odd 12 1
3325.1.bi.b 4 665.ci even 12 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \)
$3$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
$5$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
$7$ \( ( 1 + T^{2} )^{2} \)
$11$ \( ( 1 + T^{2} )^{4} \)
$13$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
$17$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
$19$ \( ( 1 + T^{2} )^{2} \)
$23$ \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \)
$29$ \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \)
$31$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
$37$ \( ( 1 + T^{2} )^{4} \)
$41$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
$43$ \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \)
$47$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
$53$ \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \)
$59$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
$61$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
$67$ \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \)
$71$ \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \)
$73$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
$79$ \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \)
$83$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
$89$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
$97$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
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