# Properties

 Label 3332.1.o.a Level $3332$ Weight $1$ Character orbit 3332.o Analytic conductor $1.663$ Analytic rank $0$ Dimension $2$ Projective image $D_{3}$ CM discriminant -68 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3332 = 2^{2} \cdot 7^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3332.o (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.66288462209$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 476) Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.3332.1

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q + \zeta_{6}^{2} q^{2} -\zeta_{6} q^{3} -\zeta_{6} q^{4} + q^{6} + q^{8} +O(q^{10})$$ $$q + \zeta_{6}^{2} q^{2} -\zeta_{6} q^{3} -\zeta_{6} q^{4} + q^{6} + q^{8} + \zeta_{6} q^{11} + \zeta_{6}^{2} q^{12} + q^{13} + \zeta_{6}^{2} q^{16} + \zeta_{6} q^{17} - q^{22} + 2 \zeta_{6}^{2} q^{23} -\zeta_{6} q^{24} -\zeta_{6} q^{25} + \zeta_{6}^{2} q^{26} - q^{27} + 2 \zeta_{6} q^{31} -\zeta_{6} q^{32} -\zeta_{6}^{2} q^{33} - q^{34} -\zeta_{6} q^{39} -\zeta_{6}^{2} q^{44} -2 \zeta_{6} q^{46} + q^{48} + q^{50} -\zeta_{6}^{2} q^{51} -\zeta_{6} q^{52} + \zeta_{6} q^{53} -\zeta_{6}^{2} q^{54} -2 q^{62} + q^{64} + \zeta_{6} q^{66} -\zeta_{6}^{2} q^{68} + 2 q^{69} - q^{71} + \zeta_{6}^{2} q^{75} + q^{78} -\zeta_{6}^{2} q^{79} + \zeta_{6} q^{81} + \zeta_{6} q^{88} + \zeta_{6}^{2} q^{89} + 2 q^{92} -2 \zeta_{6}^{2} q^{93} + \zeta_{6}^{2} q^{96} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} - q^{3} - q^{4} + 2 q^{6} + 2 q^{8} + O(q^{10})$$ $$2 q - q^{2} - q^{3} - q^{4} + 2 q^{6} + 2 q^{8} + q^{11} - q^{12} + 2 q^{13} - q^{16} + q^{17} - 2 q^{22} - 2 q^{23} - q^{24} - q^{25} - q^{26} - 2 q^{27} + 2 q^{31} - q^{32} + q^{33} - 2 q^{34} - q^{39} + q^{44} - 2 q^{46} + 2 q^{48} + 2 q^{50} + q^{51} - q^{52} + q^{53} + q^{54} - 4 q^{62} + 2 q^{64} + q^{66} + q^{68} + 4 q^{69} - 2 q^{71} - q^{75} + 2 q^{78} + q^{79} + q^{81} + q^{88} - q^{89} + 4 q^{92} + 2 q^{93} - q^{96} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3332\mathbb{Z}\right)^\times$$.

 $$n$$ $$785$$ $$885$$ $$1667$$ $$\chi(n)$$ $$-1$$ $$\zeta_{6}^{2}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
67.1
 0.5 − 0.866025i 0.5 + 0.866025i
−0.500000 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i 0 1.00000 0 1.00000 0 0
2039.1 −0.500000 + 0.866025i −0.500000 0.866025i −0.500000 0.866025i 0 1.00000 0 1.00000 0 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
68.d odd 2 1 CM by $$\Q(\sqrt{-17})$$
7.c even 3 1 inner
476.o odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3332.1.o.a 2
4.b odd 2 1 3332.1.o.b 2
7.b odd 2 1 476.1.o.b yes 2
7.c even 3 1 3332.1.g.d 1
7.c even 3 1 inner 3332.1.o.a 2
7.d odd 6 1 476.1.o.b yes 2
7.d odd 6 1 3332.1.g.b 1
17.b even 2 1 3332.1.o.b 2
28.d even 2 1 476.1.o.a 2
28.f even 6 1 476.1.o.a 2
28.f even 6 1 3332.1.g.e 1
28.g odd 6 1 3332.1.g.c 1
28.g odd 6 1 3332.1.o.b 2
68.d odd 2 1 CM 3332.1.o.a 2
119.d odd 2 1 476.1.o.a 2
119.h odd 6 1 476.1.o.a 2
119.h odd 6 1 3332.1.g.e 1
119.j even 6 1 3332.1.g.c 1
119.j even 6 1 3332.1.o.b 2
476.e even 2 1 476.1.o.b yes 2
476.o odd 6 1 3332.1.g.d 1
476.o odd 6 1 inner 3332.1.o.a 2
476.q even 6 1 476.1.o.b yes 2
476.q even 6 1 3332.1.g.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
476.1.o.a 2 28.d even 2 1
476.1.o.a 2 28.f even 6 1
476.1.o.a 2 119.d odd 2 1
476.1.o.a 2 119.h odd 6 1
476.1.o.b yes 2 7.b odd 2 1
476.1.o.b yes 2 7.d odd 6 1
476.1.o.b yes 2 476.e even 2 1
476.1.o.b yes 2 476.q even 6 1
3332.1.g.b 1 7.d odd 6 1
3332.1.g.b 1 476.q even 6 1
3332.1.g.c 1 28.g odd 6 1
3332.1.g.c 1 119.j even 6 1
3332.1.g.d 1 7.c even 3 1
3332.1.g.d 1 476.o odd 6 1
3332.1.g.e 1 28.f even 6 1
3332.1.g.e 1 119.h odd 6 1
3332.1.o.a 2 1.a even 1 1 trivial
3332.1.o.a 2 7.c even 3 1 inner
3332.1.o.a 2 68.d odd 2 1 CM
3332.1.o.a 2 476.o odd 6 1 inner
3332.1.o.b 2 4.b odd 2 1
3332.1.o.b 2 17.b even 2 1
3332.1.o.b 2 28.g odd 6 1
3332.1.o.b 2 119.j even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(3332, [\chi])$$:

 $$T_{3}^{2} + T_{3} + 1$$ $$T_{5}$$ $$T_{13} - 1$$

## Hecke characteristic polynomials

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