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 $4$

Related objects

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: $$\Q(\zeta_{6})$$ Defining polynomial: $$x^{2} - x + 1$$ 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
 0.5 − 0.866025i 0.5 + 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

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

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