# Properties

 Label 3332.1.g.e Level $3332$ Weight $1$ Character orbit 3332.g Self dual yes Analytic conductor $1.663$ Analytic rank $0$ Dimension $1$ Projective image $D_{3}$ CM discriminant -68 Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3332 = 2^{2} \cdot 7^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3332.g (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$1.66288462209$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ 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 Artin image: $D_6$ Artin field: Galois closure of 6.0.44408896.1

## $q$-expansion

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

## Character values

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

 $$n$$ $$785$$ $$885$$ $$1667$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
883.1
 0
1.00000 1.00000 1.00000 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 $$\Q(\sqrt{-17})$$

## Twists

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

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

## Hecke kernels

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

 $$T_{3} - 1$$ T3 - 1 $$T_{5}$$ T5 $$T_{11} - 1$$ T11 - 1 $$T_{13} + 1$$ T13 + 1

## Hecke characteristic polynomials

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