# Properties

 Label 3525.1.l.b Level $3525$ Weight $1$ Character orbit 3525.l Analytic conductor $1.759$ Analytic rank $0$ Dimension $8$ Projective image $D_{6}$ CM discriminant -47 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: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{6}$$ Projective field: Galois closure of 6.0.186384375.1

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \zeta_{24}^{9} q^{2} -\zeta_{24}^{7} q^{3} + \zeta_{24}^{4} q^{6} + ( -\zeta_{24} - \zeta_{24}^{5} ) q^{7} -\zeta_{24}^{3} q^{8} -\zeta_{24}^{2} q^{9} +O(q^{10})$$ $$q + \zeta_{24}^{9} q^{2} -\zeta_{24}^{7} q^{3} + \zeta_{24}^{4} q^{6} + ( -\zeta_{24} - \zeta_{24}^{5} ) q^{7} -\zeta_{24}^{3} q^{8} -\zeta_{24}^{2} q^{9} + ( \zeta_{24}^{2} - \zeta_{24}^{10} ) q^{14} + q^{16} + \zeta_{24}^{9} q^{17} -\zeta_{24}^{11} q^{18} + ( -1 + \zeta_{24}^{8} ) q^{21} + \zeta_{24}^{10} q^{24} + \zeta_{24}^{9} q^{27} -\zeta_{24}^{6} q^{34} + ( -\zeta_{24}^{5} - \zeta_{24}^{9} ) q^{42} + \zeta_{24}^{9} q^{47} -\zeta_{24}^{7} q^{48} + ( \zeta_{24}^{2} + \zeta_{24}^{6} + \zeta_{24}^{10} ) q^{49} + \zeta_{24}^{4} q^{51} + 2 \zeta_{24}^{3} q^{53} -\zeta_{24}^{6} q^{54} + ( \zeta_{24}^{4} + \zeta_{24}^{8} ) q^{56} + ( -\zeta_{24}^{2} + \zeta_{24}^{10} ) q^{59} -2 q^{61} + ( \zeta_{24}^{3} + \zeta_{24}^{7} ) q^{63} + \zeta_{24}^{6} q^{64} + ( -\zeta_{24}^{4} - \zeta_{24}^{8} ) q^{71} + \zeta_{24}^{5} q^{72} + 2 \zeta_{24}^{6} q^{79} + \zeta_{24}^{4} q^{81} -\zeta_{24}^{3} q^{83} -\zeta_{24}^{6} q^{94} + ( -\zeta_{24}^{3} - \zeta_{24}^{7} + \zeta_{24}^{11} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 4 q^{6} + O(q^{10})$$ $$8 q + 4 q^{6} + 8 q^{16} - 12 q^{21} + 4 q^{51} - 16 q^{61} + 4 q^{81} + O(q^{100})$$

## Character values

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

 $$n$$ $$1552$$ $$2026$$ $$2351$$ $$\chi(n)$$ $$\zeta_{24}^{6}$$ $$-1$$ $$-1$$

## 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
 −0.258819 + 0.965926i 0.965926 − 0.258819i −0.965926 + 0.258819i 0.258819 − 0.965926i −0.258819 − 0.965926i 0.965926 + 0.258819i −0.965926 − 0.258819i 0.258819 + 0.965926i
−0.707107 0.707107i −0.965926 0.258819i 0 0 0.500000 + 0.866025i 1.22474 1.22474i −0.707107 + 0.707107i 0.866025 + 0.500000i 0
1268.2 −0.707107 0.707107i 0.258819 + 0.965926i 0 0 0.500000 0.866025i −1.22474 + 1.22474i −0.707107 + 0.707107i −0.866025 + 0.500000i 0
1268.3 0.707107 + 0.707107i −0.258819 0.965926i 0 0 0.500000 0.866025i 1.22474 1.22474i 0.707107 0.707107i −0.866025 + 0.500000i 0
1268.4 0.707107 + 0.707107i 0.965926 + 0.258819i 0 0 0.500000 + 0.866025i −1.22474 + 1.22474i 0.707107 0.707107i 0.866025 + 0.500000i 0
1832.1 −0.707107 + 0.707107i −0.965926 + 0.258819i 0 0 0.500000 0.866025i 1.22474 + 1.22474i −0.707107 0.707107i 0.866025 0.500000i 0
1832.2 −0.707107 + 0.707107i 0.258819 0.965926i 0 0 0.500000 + 0.866025i −1.22474 1.22474i −0.707107 0.707107i −0.866025 0.500000i 0
1832.3 0.707107 0.707107i −0.258819 + 0.965926i 0 0 0.500000 + 0.866025i 1.22474 + 1.22474i 0.707107 + 0.707107i −0.866025 0.500000i 0
1832.4 0.707107 0.707107i 0.965926 0.258819i 0 0 0.500000 0.866025i −1.22474 1.22474i 0.707107 + 0.707107i 0.866025 0.500000i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1832.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
47.b odd 2 1 CM by $$\Q(\sqrt{-47})$$
3.b odd 2 1 inner
5.b even 2 1 inner
5.c odd 4 2 inner
15.d odd 2 1 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.g even 2 1 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.b 8
3.b odd 2 1 inner 3525.1.l.b 8
5.b even 2 1 inner 3525.1.l.b 8
5.c odd 4 2 inner 3525.1.l.b 8
15.d odd 2 1 inner 3525.1.l.b 8
15.e even 4 2 inner 3525.1.l.b 8
47.b odd 2 1 CM 3525.1.l.b 8
141.c even 2 1 inner 3525.1.l.b 8
235.b odd 2 1 inner 3525.1.l.b 8
235.e even 4 2 inner 3525.1.l.b 8
705.g even 2 1 inner 3525.1.l.b 8
705.l odd 4 2 inner 3525.1.l.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3525.1.l.b 8 1.a even 1 1 trivial
3525.1.l.b 8 3.b odd 2 1 inner
3525.1.l.b 8 5.b even 2 1 inner
3525.1.l.b 8 5.c odd 4 2 inner
3525.1.l.b 8 15.d odd 2 1 inner
3525.1.l.b 8 15.e even 4 2 inner
3525.1.l.b 8 47.b odd 2 1 CM
3525.1.l.b 8 141.c even 2 1 inner
3525.1.l.b 8 235.b odd 2 1 inner
3525.1.l.b 8 235.e even 4 2 inner
3525.1.l.b 8 705.g even 2 1 inner
3525.1.l.b 8 705.l odd 4 2 inner

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{4} )^{2}$$
$3$ $$1 - T^{4} + T^{8}$$
$5$ $$T^{8}$$
$7$ $$( 9 + T^{4} )^{2}$$
$11$ $$T^{8}$$
$13$ $$T^{8}$$
$17$ $$( 1 + T^{4} )^{2}$$
$19$ $$T^{8}$$
$23$ $$T^{8}$$
$29$ $$T^{8}$$
$31$ $$T^{8}$$
$37$ $$T^{8}$$
$41$ $$T^{8}$$
$43$ $$T^{8}$$
$47$ $$( 1 + T^{4} )^{2}$$
$53$ $$( 16 + T^{4} )^{2}$$
$59$ $$( -3 + T^{2} )^{4}$$
$61$ $$( 2 + T )^{8}$$
$67$ $$T^{8}$$
$71$ $$( 3 + T^{2} )^{4}$$
$73$ $$T^{8}$$
$79$ $$( 4 + T^{2} )^{4}$$
$83$ $$( 1 + T^{4} )^{2}$$
$89$ $$T^{8}$$
$97$ $$T^{8}$$