# Properties

 Label 525.2.j.c Level 525 Weight 2 Character orbit 525.j Analytic conductor 4.192 Analytic rank 0 Dimension 32 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.j (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.19214610612$$ Analytic rank: $$0$$ Dimension: $$32$$ Relative dimension: $$16$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$32q - 16q^{6} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32q - 16q^{6} - 16q^{16} - 8q^{21} + 16q^{31} + 48q^{36} + 144q^{46} - 64q^{51} - 112q^{61} - 192q^{76} - 64q^{81} + 64q^{91} + 360q^{96} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
218.1 −1.90099 1.90099i 0.394708 1.68648i 5.22756i 0 −3.95632 + 2.45565i −0.707107 + 0.707107i 6.13557 6.13557i −2.68841 1.33133i 0
218.2 −1.90099 1.90099i 1.68648 0.394708i 5.22756i 0 −3.95632 2.45565i 0.707107 0.707107i 6.13557 6.13557i 2.68841 1.33133i 0
218.3 −1.27211 1.27211i −1.54904 + 0.774907i 1.23654i 0 2.95632 + 0.984783i 0.707107 0.707107i −0.971203 + 0.971203i 1.79904 2.40072i 0
218.4 −1.27211 1.27211i −0.774907 + 1.54904i 1.23654i 0 2.95632 0.984783i −0.707107 + 0.707107i −0.971203 + 0.971203i −1.79904 2.40072i 0
218.5 −1.11527 1.11527i −1.51282 0.843438i 0.487636i 0 0.746534 + 2.62785i 0.707107 0.707107i −1.68669 + 1.68669i 1.57722 + 2.55193i 0
218.6 −1.11527 1.11527i 0.843438 + 1.51282i 0.487636i 0 0.746534 2.62785i −0.707107 + 0.707107i −1.68669 + 1.68669i −1.57722 + 2.55193i 0
218.7 −0.723969 0.723969i 0.994020 1.41842i 0.951738i 0 −1.74653 + 0.307254i −0.707107 + 0.707107i −2.13697 + 2.13697i −1.02385 2.81988i 0
218.8 −0.723969 0.723969i 1.41842 0.994020i 0.951738i 0 −1.74653 0.307254i 0.707107 0.707107i −2.13697 + 2.13697i 1.02385 2.81988i 0
218.9 0.723969 + 0.723969i −1.41842 + 0.994020i 0.951738i 0 −1.74653 0.307254i −0.707107 + 0.707107i 2.13697 2.13697i 1.02385 2.81988i 0
218.10 0.723969 + 0.723969i −0.994020 + 1.41842i 0.951738i 0 −1.74653 + 0.307254i 0.707107 0.707107i 2.13697 2.13697i −1.02385 2.81988i 0
218.11 1.11527 + 1.11527i −0.843438 1.51282i 0.487636i 0 0.746534 2.62785i 0.707107 0.707107i 1.68669 1.68669i −1.57722 + 2.55193i 0
218.12 1.11527 + 1.11527i 1.51282 + 0.843438i 0.487636i 0 0.746534 + 2.62785i −0.707107 + 0.707107i 1.68669 1.68669i 1.57722 + 2.55193i 0
218.13 1.27211 + 1.27211i 0.774907 1.54904i 1.23654i 0 2.95632 0.984783i 0.707107 0.707107i 0.971203 0.971203i −1.79904 2.40072i 0
218.14 1.27211 + 1.27211i 1.54904 0.774907i 1.23654i 0 2.95632 + 0.984783i −0.707107 + 0.707107i 0.971203 0.971203i 1.79904 2.40072i 0
218.15 1.90099 + 1.90099i −1.68648 + 0.394708i 5.22756i 0 −3.95632 2.45565i −0.707107 + 0.707107i −6.13557 + 6.13557i 2.68841 1.33133i 0
218.16 1.90099 + 1.90099i −0.394708 + 1.68648i 5.22756i 0 −3.95632 + 2.45565i 0.707107 0.707107i −6.13557 + 6.13557i −2.68841 1.33133i 0
407.1 −1.90099 + 1.90099i 0.394708 + 1.68648i 5.22756i 0 −3.95632 2.45565i −0.707107 0.707107i 6.13557 + 6.13557i −2.68841 + 1.33133i 0
407.2 −1.90099 + 1.90099i 1.68648 + 0.394708i 5.22756i 0 −3.95632 + 2.45565i 0.707107 + 0.707107i 6.13557 + 6.13557i 2.68841 + 1.33133i 0
407.3 −1.27211 + 1.27211i −1.54904 0.774907i 1.23654i 0 2.95632 0.984783i 0.707107 + 0.707107i −0.971203 0.971203i 1.79904 + 2.40072i 0
407.4 −1.27211 + 1.27211i −0.774907 1.54904i 1.23654i 0 2.95632 + 0.984783i −0.707107 0.707107i −0.971203 0.971203i −1.79904 + 2.40072i 0
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 407.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.2.j.c 32
3.b odd 2 1 inner 525.2.j.c 32
5.b even 2 1 inner 525.2.j.c 32
5.c odd 4 2 inner 525.2.j.c 32
15.d odd 2 1 inner 525.2.j.c 32
15.e even 4 2 inner 525.2.j.c 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.2.j.c 32 1.a even 1 1 trivial
525.2.j.c 32 3.b odd 2 1 inner
525.2.j.c 32 5.b even 2 1 inner
525.2.j.c 32 5.c odd 4 2 inner
525.2.j.c 32 15.d odd 2 1 inner
525.2.j.c 32 15.e even 4 2 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{16} + 70 T_{2}^{12} + 1011 T_{2}^{8} + 4414 T_{2}^{4} + 3721$$ acting on $$S_{2}^{\mathrm{new}}(525, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database