# Properties

 Label 3267.1.h.a Level $3267$ Weight $1$ Character orbit 3267.h Analytic conductor $1.630$ Analytic rank $0$ Dimension $4$ Projective image $S_{4}$ CM/RM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3267 = 3^{3} \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3267.h (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.63044539627$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} - 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1089) Projective image: $$S_{4}$$ Projective field: Galois closure of 4.2.107811.1

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + \beta_{2} q^{4} -\beta_{2} q^{5} + \beta_{1} q^{7} +O(q^{10})$$ $$q + \beta_{1} q^{2} + \beta_{2} q^{4} -\beta_{2} q^{5} + \beta_{1} q^{7} -\beta_{3} q^{10} + 2 \beta_{2} q^{14} + ( 1 - \beta_{2} ) q^{16} + ( 1 - \beta_{2} ) q^{20} + \beta_{3} q^{28} + \beta_{1} q^{29} -\beta_{2} q^{31} + ( \beta_{1} - \beta_{3} ) q^{32} -\beta_{3} q^{35} - q^{37} + ( -1 + \beta_{2} ) q^{47} + \beta_{2} q^{49} - q^{53} + 2 \beta_{2} q^{58} + \beta_{2} q^{59} + \beta_{1} q^{61} -\beta_{3} q^{62} + q^{64} + \beta_{2} q^{67} + ( 2 - 2 \beta_{2} ) q^{70} + q^{71} -\beta_{3} q^{73} -\beta_{1} q^{74} - q^{80} -\beta_{1} q^{83} + ( -\beta_{1} + \beta_{3} ) q^{94} + ( -1 + \beta_{2} ) q^{97} + \beta_{3} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{4} - 2 q^{5} + O(q^{10})$$ $$4 q + 2 q^{4} - 2 q^{5} + 4 q^{14} + 2 q^{16} + 2 q^{20} - 2 q^{31} - 4 q^{37} - 2 q^{47} + 2 q^{49} - 4 q^{53} + 4 q^{58} + 2 q^{59} + 4 q^{64} + 2 q^{67} + 4 q^{70} + 4 q^{71} - 4 q^{80} - 2 q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{3}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3267\mathbb{Z}\right)^\times$$.

 $$n$$ $$244$$ $$3026$$ $$\chi(n)$$ $$-1$$ $$-\beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
604.1
 −1.22474 + 0.707107i 1.22474 − 0.707107i −1.22474 − 0.707107i 1.22474 + 0.707107i
−1.22474 + 0.707107i 0 0.500000 0.866025i −0.500000 + 0.866025i 0 −1.22474 + 0.707107i 0 0 1.41421i
604.2 1.22474 0.707107i 0 0.500000 0.866025i −0.500000 + 0.866025i 0 1.22474 0.707107i 0 0 1.41421i
1693.1 −1.22474 0.707107i 0 0.500000 + 0.866025i −0.500000 0.866025i 0 −1.22474 0.707107i 0 0 1.41421i
1693.2 1.22474 + 0.707107i 0 0.500000 + 0.866025i −0.500000 0.866025i 0 1.22474 + 0.707107i 0 0 1.41421i
 $$n$$: 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
9.c even 3 1 inner
11.b odd 2 1 inner
99.h odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3267.1.h.a 4
3.b odd 2 1 1089.1.h.a 4
9.c even 3 1 inner 3267.1.h.a 4
9.d odd 6 1 1089.1.h.a 4
11.b odd 2 1 inner 3267.1.h.a 4
11.c even 5 4 3267.1.w.b 16
11.d odd 10 4 3267.1.w.b 16
33.d even 2 1 1089.1.h.a 4
33.f even 10 4 1089.1.s.b 16
33.h odd 10 4 1089.1.s.b 16
99.g even 6 1 1089.1.h.a 4
99.h odd 6 1 inner 3267.1.h.a 4
99.m even 15 4 3267.1.w.b 16
99.n odd 30 4 1089.1.s.b 16
99.o odd 30 4 3267.1.w.b 16
99.p even 30 4 1089.1.s.b 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1089.1.h.a 4 3.b odd 2 1
1089.1.h.a 4 9.d odd 6 1
1089.1.h.a 4 33.d even 2 1
1089.1.h.a 4 99.g even 6 1
1089.1.s.b 16 33.f even 10 4
1089.1.s.b 16 33.h odd 10 4
1089.1.s.b 16 99.n odd 30 4
1089.1.s.b 16 99.p even 30 4
3267.1.h.a 4 1.a even 1 1 trivial
3267.1.h.a 4 9.c even 3 1 inner
3267.1.h.a 4 11.b odd 2 1 inner
3267.1.h.a 4 99.h odd 6 1 inner
3267.1.w.b 16 11.c even 5 4
3267.1.w.b 16 11.d odd 10 4
3267.1.w.b 16 99.m even 15 4
3267.1.w.b 16 99.o odd 30 4

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(3267, [\chi])$$.

## Hecke characteristic polynomials

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