# Properties

 Label 3328.1.bv.b Level $3328$ Weight $1$ Character orbit 3328.bv Analytic conductor $1.661$ Analytic rank $0$ Dimension $4$ Projective image $D_{12}$ CM discriminant -4 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3328 = 2^{8} \cdot 13$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3328.bv (of order $$12$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.66088836204$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 416) Projective image: $$D_{12}$$ Projective field: Galois closure of 12.0.469804094334435328.7

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + (\zeta_{12}^{2} - \zeta_{12}) q^{5} - \zeta_{12}^{2} q^{9} +O(q^{10})$$ q + (z^2 - z) * q^5 - z^2 * q^9 $$q + (\zeta_{12}^{2} - \zeta_{12}) q^{5} - \zeta_{12}^{2} q^{9} + \zeta_{12}^{4} q^{13} - \zeta_{12}^{5} q^{17} + (\zeta_{12}^{4} - \zeta_{12}^{3} + \zeta_{12}^{2}) q^{25} + (\zeta_{12}^{2} + 1) q^{29} + ( - \zeta_{12}^{5} + 1) q^{37} + ( - \zeta_{12}^{5} + 1) q^{41} + ( - \zeta_{12}^{4} + \zeta_{12}^{3}) q^{45} - \zeta_{12} q^{49} + \zeta_{12}^{3} q^{53} + \zeta_{12}^{5} q^{61} + ( - \zeta_{12}^{5} - 1) q^{65} + ( - \zeta_{12}^{5} - \zeta_{12}^{4}) q^{73} + \zeta_{12}^{4} q^{81} + (\zeta_{12} - 1) q^{85} + (\zeta_{12}^{4} + \zeta_{12}) q^{89} + ( - \zeta_{12}^{5} - \zeta_{12}^{2}) q^{97} +O(q^{100})$$ q + (z^2 - z) * q^5 - z^2 * q^9 + z^4 * q^13 - z^5 * q^17 + (z^4 - z^3 + z^2) * q^25 + (z^2 + 1) * q^29 + (-z^5 + 1) * q^37 + (-z^5 + 1) * q^41 + (-z^4 + z^3) * q^45 - z * q^49 + z^3 * q^53 + z^5 * q^61 + (-z^5 - 1) * q^65 + (-z^5 - z^4) * q^73 + z^4 * q^81 + (z - 1) * q^85 + (z^4 + z) * q^89 + (-z^5 - z^2) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{5} - 2 q^{9}+O(q^{10})$$ 4 * q + 2 * q^5 - 2 * q^9 $$4 q + 2 q^{5} - 2 q^{9} - 2 q^{13} + 6 q^{29} + 4 q^{37} + 4 q^{41} + 2 q^{45} - 4 q^{65} + 2 q^{73} - 2 q^{81} - 4 q^{85} - 2 q^{89} - 2 q^{97}+O(q^{100})$$ 4 * q + 2 * q^5 - 2 * q^9 - 2 * q^13 + 6 * q^29 + 4 * q^37 + 4 * q^41 + 2 * q^45 - 4 * q^65 + 2 * q^73 - 2 * q^81 - 4 * q^85 - 2 * q^89 - 2 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3328\mathbb{Z}\right)^\times$$.

 $$n$$ $$261$$ $$769$$ $$1535$$ $$\chi(n)$$ $$-1$$ $$-\zeta_{12}^{5}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2177.1
 −0.866025 − 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 + 0.500000i
0 0 0 1.36603 + 1.36603i 0 0 0 −0.500000 0.866025i 0
2433.1 0 0 0 −0.366025 0.366025i 0 0 0 −0.500000 + 0.866025i 0
2689.1 0 0 0 1.36603 1.36603i 0 0 0 −0.500000 + 0.866025i 0
2945.1 0 0 0 −0.366025 + 0.366025i 0 0 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
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
104.u even 12 1 inner
104.x odd 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3328.1.bv.b 4
4.b odd 2 1 CM 3328.1.bv.b 4
8.b even 2 1 3328.1.bv.a 4
8.d odd 2 1 3328.1.bv.a 4
13.f odd 12 1 3328.1.bv.a 4
16.e even 4 1 416.1.bl.a 4
16.e even 4 1 832.1.bl.a 4
16.f odd 4 1 416.1.bl.a 4
16.f odd 4 1 832.1.bl.a 4
48.i odd 4 1 3744.1.gs.c 4
48.k even 4 1 3744.1.gs.c 4
52.l even 12 1 3328.1.bv.a 4
104.u even 12 1 inner 3328.1.bv.b 4
104.x odd 12 1 inner 3328.1.bv.b 4
208.be odd 12 1 416.1.bl.a 4
208.bf even 12 1 416.1.bl.a 4
208.bk even 12 1 832.1.bl.a 4
208.bl odd 12 1 832.1.bl.a 4
624.cy odd 12 1 3744.1.gs.c 4
624.da even 12 1 3744.1.gs.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
416.1.bl.a 4 16.e even 4 1
416.1.bl.a 4 16.f odd 4 1
416.1.bl.a 4 208.be odd 12 1
416.1.bl.a 4 208.bf even 12 1
832.1.bl.a 4 16.e even 4 1
832.1.bl.a 4 16.f odd 4 1
832.1.bl.a 4 208.bk even 12 1
832.1.bl.a 4 208.bl odd 12 1
3328.1.bv.a 4 8.b even 2 1
3328.1.bv.a 4 8.d odd 2 1
3328.1.bv.a 4 13.f odd 12 1
3328.1.bv.a 4 52.l even 12 1
3328.1.bv.b 4 1.a even 1 1 trivial
3328.1.bv.b 4 4.b odd 2 1 CM
3328.1.bv.b 4 104.u even 12 1 inner
3328.1.bv.b 4 104.x odd 12 1 inner
3744.1.gs.c 4 48.i odd 4 1
3744.1.gs.c 4 48.k even 4 1
3744.1.gs.c 4 624.cy odd 12 1
3744.1.gs.c 4 624.da even 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} - 2T_{5}^{3} + 2T_{5}^{2} + 2T_{5} + 1$$ acting on $$S_{1}^{\mathrm{new}}(3328, [\chi])$$.

## Hecke characteristic polynomials

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