# Properties

 Label 3332.1.g.h Level $3332$ Weight $1$ Character orbit 3332.g Analytic conductor $1.663$ Analytic rank $0$ Dimension $2$ Projective image $D_{2}$ CM/RM discs -4, -119, 476 Inner twists $8$

# 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: no Analytic conductor: $$1.66288462209$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{2}$$ Projective field: Galois closure of $$\Q(i, \sqrt{119})$$ Artin image: $D_4:C_2$ Artin field: Galois closure of 8.0.8704143616.1

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} - i q^{5} + q^{8} - q^{9} +O(q^{10})$$ q + q^2 + q^4 - z * q^5 + q^8 - q^9 $$q + q^{2} + q^{4} - i q^{5} + q^{8} - q^{9} - 2 i q^{10} + q^{16} - i q^{17} - q^{18} - 2 i q^{20} - 3 q^{25} + q^{32} - i q^{34} - q^{36} - 2 i q^{40} + i q^{41} + 2 i q^{45} - 3 q^{50} + q^{53} - i q^{61} + q^{64} - i q^{68} - q^{72} + i q^{73} - 2 i q^{80} + q^{81} + 2 i q^{82} - 2 q^{85} + 2 i q^{90} - i q^{97} +O(q^{100})$$ q + q^2 + q^4 - z * q^5 + q^8 - q^9 - 2*z * q^10 + q^16 - z * q^17 - q^18 - 2*z * q^20 - 3 * q^25 + q^32 - z * q^34 - q^36 - 2*z * q^40 + z * q^41 + 2*z * q^45 - 3 * q^50 + q^53 - z * q^61 + q^64 - z * q^68 - q^72 + z * q^73 - 2*z * q^80 + q^81 + 2*z * q^82 - 2 * q^85 + 2*z * q^90 - z * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 - 2 * q^9 $$2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 2 q^{9} + 2 q^{16} - 2 q^{18} - 6 q^{25} + 2 q^{32} - 2 q^{36} - 6 q^{50} + 4 q^{53} + 2 q^{64} - 2 q^{72} + 2 q^{81} - 4 q^{85}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 - 2 * q^9 + 2 * q^16 - 2 * q^18 - 6 * q^25 + 2 * q^32 - 2 * q^36 - 6 * q^50 + 4 * q^53 + 2 * q^64 - 2 * q^72 + 2 * q^81 - 4 * q^85

## 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
 1.00000i − 1.00000i
1.00000 0 1.00000 2.00000i 0 0 1.00000 −1.00000 2.00000i
883.2 1.00000 0 1.00000 2.00000i 0 0 1.00000 −1.00000 2.00000i
 $$n$$: 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
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
119.d odd 2 1 CM by $$\Q(\sqrt{-119})$$
476.e even 2 1 RM by $$\Q(\sqrt{119})$$
7.b odd 2 1 inner
17.b even 2 1 inner
28.d even 2 1 inner
68.d odd 2 1 inner

## Twists

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

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3332.1.g.h 2 1.a even 1 1 trivial
3332.1.g.h 2 4.b odd 2 1 CM
3332.1.g.h 2 7.b odd 2 1 inner
3332.1.g.h 2 17.b even 2 1 inner
3332.1.g.h 2 28.d even 2 1 inner
3332.1.g.h 2 68.d odd 2 1 inner
3332.1.g.h 2 119.d odd 2 1 CM
3332.1.g.h 2 476.e even 2 1 RM
3332.1.o.e 4 7.c even 3 2
3332.1.o.e 4 7.d odd 6 2
3332.1.o.e 4 28.f even 6 2
3332.1.o.e 4 28.g odd 6 2
3332.1.o.e 4 119.h odd 6 2
3332.1.o.e 4 119.j even 6 2
3332.1.o.e 4 476.o odd 6 2
3332.1.o.e 4 476.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 $$S_{1}^{\mathrm{new}}(3332, [\chi])$$:

 $$T_{3}$$ T3 $$T_{5}^{2} + 4$$ T5^2 + 4 $$T_{11}$$ T11 $$T_{13}$$ T13

## Hecke characteristic polynomials

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