# Properties

 Label 3528.1.n.a Level $3528$ Weight $1$ Character orbit 3528.n Analytic conductor $1.761$ Analytic rank $0$ Dimension $4$ Projective image $D_{4}$ CM discriminant -7 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3528 = 2^{3} \cdot 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3528.n (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.76070136457$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$D_{4}$$ Projective field Galois closure of 4.2.84672.1

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\zeta_{8} q^{2} + \zeta_{8}^{2} q^{4} -\zeta_{8}^{3} q^{8} +O(q^{10})$$ $$q -\zeta_{8} q^{2} + \zeta_{8}^{2} q^{4} -\zeta_{8}^{3} q^{8} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{11} - q^{16} + ( 1 + \zeta_{8}^{2} ) q^{22} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{23} - q^{25} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{29} + \zeta_{8} q^{32} + 2 \zeta_{8}^{2} q^{37} -2 \zeta_{8}^{2} q^{43} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{44} + ( -1 + \zeta_{8}^{2} ) q^{46} + \zeta_{8} q^{50} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{53} + ( 1 + \zeta_{8}^{2} ) q^{58} -\zeta_{8}^{2} q^{64} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{71} -2 \zeta_{8}^{3} q^{74} -2 q^{79} + 2 \zeta_{8}^{3} q^{86} + ( -1 + \zeta_{8}^{2} ) q^{88} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{92} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q - 4q^{16} + 4q^{22} - 4q^{25} - 4q^{46} + 4q^{58} - 8q^{79} - 4q^{88} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3528\mathbb{Z}\right)^\times$$.

 $$n$$ $$785$$ $$1081$$ $$1765$$ $$2647$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
197.1
 0.707107 + 0.707107i 0.707107 − 0.707107i −0.707107 + 0.707107i −0.707107 − 0.707107i
−0.707107 0.707107i 0 1.00000i 0 0 0 0.707107 0.707107i 0 0
197.2 −0.707107 + 0.707107i 0 1.00000i 0 0 0 0.707107 + 0.707107i 0 0
197.3 0.707107 0.707107i 0 1.00000i 0 0 0 −0.707107 0.707107i 0 0
197.4 0.707107 + 0.707107i 0 1.00000i 0 0 0 −0.707107 + 0.707107i 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
7.b odd 2 1 CM by $$\Q(\sqrt{-7})$$
3.b odd 2 1 inner
8.b even 2 1 inner
21.c even 2 1 inner
24.h odd 2 1 inner
56.h odd 2 1 inner
168.i even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3528.1.n.a 4
3.b odd 2 1 inner 3528.1.n.a 4
7.b odd 2 1 CM 3528.1.n.a 4
7.c even 3 2 3528.1.bd.a 8
7.d odd 6 2 3528.1.bd.a 8
8.b even 2 1 inner 3528.1.n.a 4
21.c even 2 1 inner 3528.1.n.a 4
21.g even 6 2 3528.1.bd.a 8
21.h odd 6 2 3528.1.bd.a 8
24.h odd 2 1 inner 3528.1.n.a 4
56.h odd 2 1 inner 3528.1.n.a 4
56.j odd 6 2 3528.1.bd.a 8
56.p even 6 2 3528.1.bd.a 8
168.i even 2 1 inner 3528.1.n.a 4
168.s odd 6 2 3528.1.bd.a 8
168.ba even 6 2 3528.1.bd.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3528.1.n.a 4 1.a even 1 1 trivial
3528.1.n.a 4 3.b odd 2 1 inner
3528.1.n.a 4 7.b odd 2 1 CM
3528.1.n.a 4 8.b even 2 1 inner
3528.1.n.a 4 21.c even 2 1 inner
3528.1.n.a 4 24.h odd 2 1 inner
3528.1.n.a 4 56.h odd 2 1 inner
3528.1.n.a 4 168.i even 2 1 inner
3528.1.bd.a 8 7.c even 3 2
3528.1.bd.a 8 7.d odd 6 2
3528.1.bd.a 8 21.g even 6 2
3528.1.bd.a 8 21.h odd 6 2
3528.1.bd.a 8 56.j odd 6 2
3528.1.bd.a 8 56.p even 6 2
3528.1.bd.a 8 168.s odd 6 2
3528.1.bd.a 8 168.ba even 6 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(3528, [\chi])$$.

## Hecke characteristic polynomials

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