# Properties

 Label 525.2.bc.c Level 525 Weight 2 Character orbit 525.bc Analytic conductor 4.192 Analytic rank 0 Dimension 24 CM no Inner twists 8

# Related objects

## Newspace parameters

 Level: $$N$$ = $$525 = 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 525.bc (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.19214610612$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$6$$ over $$\Q(\zeta_{12})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$24q + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q + 36q^{16} + 12q^{21} + 12q^{26} - 24q^{31} - 24q^{36} - 24q^{46} + 24q^{51} - 36q^{56} - 180q^{61} - 72q^{66} - 96q^{71} + 12q^{81} + 120q^{86} - 12q^{91} - 108q^{96} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
82.1 −2.53443 + 0.679097i −0.258819 + 0.965926i 4.23009 2.44224i 0 2.62383i 1.00739 2.44646i −5.35166 + 5.35166i −0.866025 0.500000i 0
82.2 −1.32591 + 0.355276i 0.258819 0.965926i −0.100242 + 0.0578747i 0 1.37268i −1.82946 1.91130i 2.05361 2.05361i −0.866025 0.500000i 0
82.3 −0.464515 + 0.124466i −0.258819 + 0.965926i −1.53177 + 0.884367i 0 0.480901i −2.38856 1.13788i 1.28155 1.28155i −0.866025 0.500000i 0
82.4 0.464515 0.124466i 0.258819 0.965926i −1.53177 + 0.884367i 0 0.480901i 2.38856 + 1.13788i −1.28155 + 1.28155i −0.866025 0.500000i 0
82.5 1.32591 0.355276i −0.258819 + 0.965926i −0.100242 + 0.0578747i 0 1.37268i 1.82946 + 1.91130i −2.05361 + 2.05361i −0.866025 0.500000i 0
82.6 2.53443 0.679097i 0.258819 0.965926i 4.23009 2.44224i 0 2.62383i −1.00739 + 2.44646i 5.35166 5.35166i −0.866025 0.500000i 0
157.1 −0.679097 + 2.53443i 0.965926 0.258819i −4.23009 2.44224i 0 2.62383i 2.44646 1.00739i 5.35166 5.35166i 0.866025 0.500000i 0
157.2 −0.355276 + 1.32591i −0.965926 + 0.258819i 0.100242 + 0.0578747i 0 1.37268i 1.91130 + 1.82946i −2.05361 + 2.05361i 0.866025 0.500000i 0
157.3 −0.124466 + 0.464515i 0.965926 0.258819i 1.53177 + 0.884367i 0 0.480901i 1.13788 + 2.38856i −1.28155 + 1.28155i 0.866025 0.500000i 0
157.4 0.124466 0.464515i −0.965926 + 0.258819i 1.53177 + 0.884367i 0 0.480901i −1.13788 2.38856i 1.28155 1.28155i 0.866025 0.500000i 0
157.5 0.355276 1.32591i 0.965926 0.258819i 0.100242 + 0.0578747i 0 1.37268i −1.91130 1.82946i 2.05361 2.05361i 0.866025 0.500000i 0
157.6 0.679097 2.53443i −0.965926 + 0.258819i −4.23009 2.44224i 0 2.62383i −2.44646 + 1.00739i −5.35166 + 5.35166i 0.866025 0.500000i 0
418.1 −0.679097 2.53443i 0.965926 + 0.258819i −4.23009 + 2.44224i 0 2.62383i 2.44646 + 1.00739i 5.35166 + 5.35166i 0.866025 + 0.500000i 0
418.2 −0.355276 1.32591i −0.965926 0.258819i 0.100242 0.0578747i 0 1.37268i 1.91130 1.82946i −2.05361 2.05361i 0.866025 + 0.500000i 0
418.3 −0.124466 0.464515i 0.965926 + 0.258819i 1.53177 0.884367i 0 0.480901i 1.13788 2.38856i −1.28155 1.28155i 0.866025 + 0.500000i 0
418.4 0.124466 + 0.464515i −0.965926 0.258819i 1.53177 0.884367i 0 0.480901i −1.13788 + 2.38856i 1.28155 + 1.28155i 0.866025 + 0.500000i 0
418.5 0.355276 + 1.32591i 0.965926 + 0.258819i 0.100242 0.0578747i 0 1.37268i −1.91130 + 1.82946i 2.05361 + 2.05361i 0.866025 + 0.500000i 0
418.6 0.679097 + 2.53443i −0.965926 0.258819i −4.23009 + 2.44224i 0 2.62383i −2.44646 1.00739i −5.35166 5.35166i 0.866025 + 0.500000i 0
493.1 −2.53443 0.679097i −0.258819 0.965926i 4.23009 + 2.44224i 0 2.62383i 1.00739 + 2.44646i −5.35166 5.35166i −0.866025 + 0.500000i 0
493.2 −1.32591 0.355276i 0.258819 + 0.965926i −0.100242 0.0578747i 0 1.37268i −1.82946 + 1.91130i 2.05361 + 2.05361i −0.866025 + 0.500000i 0
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 493.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
5.c odd 4 2 inner
7.d odd 6 1 inner
35.i odd 6 1 inner
35.k even 12 2 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.2.bc.c 24
5.b even 2 1 inner 525.2.bc.c 24
5.c odd 4 2 inner 525.2.bc.c 24
7.d odd 6 1 inner 525.2.bc.c 24
35.i odd 6 1 inner 525.2.bc.c 24
35.k even 12 2 inner 525.2.bc.c 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.2.bc.c 24 1.a even 1 1 trivial
525.2.bc.c 24 5.b even 2 1 inner
525.2.bc.c 24 5.c odd 4 2 inner
525.2.bc.c 24 7.d odd 6 1 inner
525.2.bc.c 24 35.i odd 6 1 inner
525.2.bc.c 24 35.k even 12 2 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{24} - 51 T_{2}^{20} + 2430 T_{2}^{16} - 8703 T_{2}^{12} + 28782 T_{2}^{8} - 1539 T_{2}^{4} + 81$$ acting on $$S_{2}^{\mathrm{new}}(525, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database