# Properties

 Label 525.2.m.c Level 525 Weight 2 Character orbit 525.m 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.m (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.19214610612$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$12$$ over $$\Q(i)$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$24q + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q + 48q^{11} + 24q^{16} + 12q^{21} - 24q^{36} - 120q^{46} + 48q^{51} - 96q^{56} - 96q^{71} - 24q^{81} - 120q^{86} + 108q^{91} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
118.1 −1.68361 + 1.68361i −0.707107 + 0.707107i 3.66908i 0 2.38098i 0.901937 + 2.48727i 2.81008 + 2.81008i 1.00000i 0
118.2 −1.68361 + 1.68361i 0.707107 0.707107i 3.66908i 0 2.38098i −2.48727 0.901937i 2.81008 + 2.81008i 1.00000i 0
118.3 −1.11266 + 1.11266i −0.707107 + 0.707107i 0.476024i 0 1.57354i −1.62049 2.09141i −1.69567 1.69567i 1.00000i 0
118.4 −1.11266 + 1.11266i 0.707107 0.707107i 0.476024i 0 1.57354i 2.09141 + 1.62049i −1.69567 1.69567i 1.00000i 0
118.5 −0.653796 + 0.653796i −0.707107 + 0.707107i 1.14510i 0 0.924607i 0.741188 2.53981i −2.05625 2.05625i 1.00000i 0
118.6 −0.653796 + 0.653796i 0.707107 0.707107i 1.14510i 0 0.924607i 2.53981 0.741188i −2.05625 2.05625i 1.00000i 0
118.7 0.653796 0.653796i −0.707107 + 0.707107i 1.14510i 0 0.924607i −2.53981 + 0.741188i 2.05625 + 2.05625i 1.00000i 0
118.8 0.653796 0.653796i 0.707107 0.707107i 1.14510i 0 0.924607i −0.741188 + 2.53981i 2.05625 + 2.05625i 1.00000i 0
118.9 1.11266 1.11266i −0.707107 + 0.707107i 0.476024i 0 1.57354i −2.09141 1.62049i 1.69567 + 1.69567i 1.00000i 0
118.10 1.11266 1.11266i 0.707107 0.707107i 0.476024i 0 1.57354i 1.62049 + 2.09141i 1.69567 + 1.69567i 1.00000i 0
118.11 1.68361 1.68361i −0.707107 + 0.707107i 3.66908i 0 2.38098i 2.48727 + 0.901937i −2.81008 2.81008i 1.00000i 0
118.12 1.68361 1.68361i 0.707107 0.707107i 3.66908i 0 2.38098i −0.901937 2.48727i −2.81008 2.81008i 1.00000i 0
307.1 −1.68361 1.68361i −0.707107 0.707107i 3.66908i 0 2.38098i 0.901937 2.48727i 2.81008 2.81008i 1.00000i 0
307.2 −1.68361 1.68361i 0.707107 + 0.707107i 3.66908i 0 2.38098i −2.48727 + 0.901937i 2.81008 2.81008i 1.00000i 0
307.3 −1.11266 1.11266i −0.707107 0.707107i 0.476024i 0 1.57354i −1.62049 + 2.09141i −1.69567 + 1.69567i 1.00000i 0
307.4 −1.11266 1.11266i 0.707107 + 0.707107i 0.476024i 0 1.57354i 2.09141 1.62049i −1.69567 + 1.69567i 1.00000i 0
307.5 −0.653796 0.653796i −0.707107 0.707107i 1.14510i 0 0.924607i 0.741188 + 2.53981i −2.05625 + 2.05625i 1.00000i 0
307.6 −0.653796 0.653796i 0.707107 + 0.707107i 1.14510i 0 0.924607i 2.53981 + 0.741188i −2.05625 + 2.05625i 1.00000i 0
307.7 0.653796 + 0.653796i −0.707107 0.707107i 1.14510i 0 0.924607i −2.53981 0.741188i 2.05625 2.05625i 1.00000i 0
307.8 0.653796 + 0.653796i 0.707107 + 0.707107i 1.14510i 0 0.924607i −0.741188 2.53981i 2.05625 2.05625i 1.00000i 0
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 307.12 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.b odd 2 1 inner
35.c odd 2 1 inner
35.f even 4 2 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.2.m.c 24
5.b even 2 1 inner 525.2.m.c 24
5.c odd 4 2 inner 525.2.m.c 24
7.b odd 2 1 inner 525.2.m.c 24
35.c odd 2 1 inner 525.2.m.c 24
35.f even 4 2 inner 525.2.m.c 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.2.m.c 24 1.a even 1 1 trivial
525.2.m.c 24 5.b even 2 1 inner
525.2.m.c 24 5.c odd 4 2 inner
525.2.m.c 24 7.b odd 2 1 inner
525.2.m.c 24 35.c odd 2 1 inner
525.2.m.c 24 35.f even 4 2 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{12} + 39 T_{2}^{8} + 225 T_{2}^{4} + 144$$ acting on $$S_{2}^{\mathrm{new}}(525, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database