# Properties

 Label 3525.1.l.a Level $3525$ Weight $1$ Character orbit 3525.l Analytic conductor $1.759$ Analytic rank $0$ Dimension $4$ Projective image $D_{2}$ CM/RM discs -15, -47, 705 Inner twists $16$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3525 = 3 \cdot 5^{2} \cdot 47$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3525.l (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.75920416953$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{2}$$ Projective field: Galois closure of $$\Q(\sqrt{-15}, \sqrt{-47})$$ Artin image: $OD_{16}:C_2$ Artin field: Galois closure of $$\mathbb{Q}[x]/(x^{16} - \cdots)$$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -2 \zeta_{8} q^{2} + \zeta_{8}^{3} q^{3} + 3 \zeta_{8}^{2} q^{4} + 2 q^{6} -4 \zeta_{8}^{3} q^{8} -\zeta_{8}^{2} q^{9} +O(q^{10})$$ $$q -2 \zeta_{8} q^{2} + \zeta_{8}^{3} q^{3} + 3 \zeta_{8}^{2} q^{4} + 2 q^{6} -4 \zeta_{8}^{3} q^{8} -\zeta_{8}^{2} q^{9} -3 \zeta_{8} q^{12} -5 q^{16} -2 \zeta_{8} q^{17} + 2 \zeta_{8}^{3} q^{18} + 4 \zeta_{8}^{2} q^{24} + \zeta_{8} q^{27} + 6 \zeta_{8} q^{32} + 4 \zeta_{8}^{2} q^{34} + 3 q^{36} + \zeta_{8} q^{47} -5 \zeta_{8}^{3} q^{48} + \zeta_{8}^{2} q^{49} + 2 q^{51} + 2 \zeta_{8}^{3} q^{53} -2 \zeta_{8}^{2} q^{54} -2 q^{61} -7 \zeta_{8}^{2} q^{64} -6 \zeta_{8}^{3} q^{68} -4 \zeta_{8} q^{72} -2 \zeta_{8}^{2} q^{79} - q^{81} + 2 \zeta_{8}^{3} q^{83} -2 \zeta_{8}^{2} q^{94} -6 q^{96} -2 \zeta_{8}^{3} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 8q^{6} + O(q^{10})$$ $$4q + 8q^{6} - 20q^{16} + 12q^{36} + 8q^{51} - 8q^{61} - 4q^{81} - 24q^{96} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3525\mathbb{Z}\right)^\times$$.

 $$n$$ $$1552$$ $$2026$$ $$2351$$ $$\chi(n)$$ $$-\zeta_{8}^{2}$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1268.1
 0.707107 + 0.707107i −0.707107 − 0.707107i 0.707107 − 0.707107i −0.707107 + 0.707107i
−1.41421 1.41421i −0.707107 + 0.707107i 3.00000i 0 2.00000 0 2.82843 2.82843i 1.00000i 0
1268.2 1.41421 + 1.41421i 0.707107 0.707107i 3.00000i 0 2.00000 0 −2.82843 + 2.82843i 1.00000i 0
1832.1 −1.41421 + 1.41421i −0.707107 0.707107i 3.00000i 0 2.00000 0 2.82843 + 2.82843i 1.00000i 0
1832.2 1.41421 1.41421i 0.707107 + 0.707107i 3.00000i 0 2.00000 0 −2.82843 2.82843i 1.00000i 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
15.d odd 2 1 CM by $$\Q(\sqrt{-15})$$
47.b odd 2 1 CM by $$\Q(\sqrt{-47})$$
705.g even 2 1 RM by $$\Q(\sqrt{705})$$
3.b odd 2 1 inner
5.b even 2 1 inner
5.c odd 4 2 inner
15.e even 4 2 inner
141.c even 2 1 inner
235.b odd 2 1 inner
235.e even 4 2 inner
705.l odd 4 2 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3525.1.l.a 4
3.b odd 2 1 inner 3525.1.l.a 4
5.b even 2 1 inner 3525.1.l.a 4
5.c odd 4 2 inner 3525.1.l.a 4
15.d odd 2 1 CM 3525.1.l.a 4
15.e even 4 2 inner 3525.1.l.a 4
47.b odd 2 1 CM 3525.1.l.a 4
141.c even 2 1 inner 3525.1.l.a 4
235.b odd 2 1 inner 3525.1.l.a 4
235.e even 4 2 inner 3525.1.l.a 4
705.g even 2 1 RM 3525.1.l.a 4
705.l odd 4 2 inner 3525.1.l.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3525.1.l.a 4 1.a even 1 1 trivial
3525.1.l.a 4 3.b odd 2 1 inner
3525.1.l.a 4 5.b even 2 1 inner
3525.1.l.a 4 5.c odd 4 2 inner
3525.1.l.a 4 15.d odd 2 1 CM
3525.1.l.a 4 15.e even 4 2 inner
3525.1.l.a 4 47.b odd 2 1 CM
3525.1.l.a 4 141.c even 2 1 inner
3525.1.l.a 4 235.b odd 2 1 inner
3525.1.l.a 4 235.e even 4 2 inner
3525.1.l.a 4 705.g even 2 1 RM
3525.1.l.a 4 705.l odd 4 2 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 16$$ acting on $$S_{1}^{\mathrm{new}}(3525, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$16 + T^{4}$$
$3$ $$1 + T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$T^{4}$$
$13$ $$T^{4}$$
$17$ $$16 + T^{4}$$
$19$ $$T^{4}$$
$23$ $$T^{4}$$
$29$ $$T^{4}$$
$31$ $$T^{4}$$
$37$ $$T^{4}$$
$41$ $$T^{4}$$
$43$ $$T^{4}$$
$47$ $$1 + T^{4}$$
$53$ $$16 + T^{4}$$
$59$ $$T^{4}$$
$61$ $$( 2 + T )^{4}$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$T^{4}$$
$79$ $$( 4 + T^{2} )^{2}$$
$83$ $$16 + T^{4}$$
$89$ $$T^{4}$$
$97$ $$T^{4}$$