# Properties

 Label 1225.1.q.a Level $1225$ Weight $1$ Character orbit 1225.q Analytic conductor $0.611$ Analytic rank $0$ Dimension $4$ Projective image $D_{2}$ CM/RM discs -7, -35, 5 Inner twists $16$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1225 = 5^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1225.q (of order $$12$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.611354640475$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$D_{2}$$ Projective field Galois closure of $$\Q(\sqrt{5}, \sqrt{-7})$$ Artin image $C_3\times OD_{16}$ Artin field Galois closure of $$\mathbb{Q}[x]/(x^{24} - \cdots)$$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \zeta_{12} q^{4} -\zeta_{12}^{5} q^{9} +O(q^{10})$$ $$q + \zeta_{12} q^{4} -\zeta_{12}^{5} q^{9} + 2 \zeta_{12}^{4} q^{11} + \zeta_{12}^{2} q^{16} -2 \zeta_{12}^{3} q^{29} + q^{36} + 2 \zeta_{12}^{5} q^{44} + \zeta_{12}^{3} q^{64} -2 q^{71} -2 \zeta_{12}^{5} q^{79} -\zeta_{12}^{4} q^{81} + 2 \zeta_{12}^{3} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q - 4q^{11} + 2q^{16} + 4q^{36} - 8q^{71} + 2q^{81} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1225\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$1177$$ $$\chi(n)$$ $$-\zeta_{12}^{2}$$ $$\zeta_{12}^{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
18.1
 −0.866025 − 0.500000i 0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i
0 0 −0.866025 0.500000i 0 0 0 0 −0.866025 + 0.500000i 0
557.1 0 0 0.866025 + 0.500000i 0 0 0 0 0.866025 0.500000i 0
618.1 0 0 0.866025 0.500000i 0 0 0 0 0.866025 + 0.500000i 0
1157.1 0 0 −0.866025 + 0.500000i 0 0 0 0 −0.866025 0.500000i 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
5.b even 2 1 RM by $$\Q(\sqrt{5})$$
7.b odd 2 1 CM by $$\Q(\sqrt{-7})$$
35.c odd 2 1 CM by $$\Q(\sqrt{-35})$$
5.c odd 4 2 inner
7.c even 3 1 inner
7.d odd 6 1 inner
35.f even 4 2 inner
35.i odd 6 1 inner
35.j even 6 1 inner
35.k even 12 2 inner
35.l odd 12 2 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1225.1.q.a 4
5.b even 2 1 RM 1225.1.q.a 4
5.c odd 4 2 inner 1225.1.q.a 4
7.b odd 2 1 CM 1225.1.q.a 4
7.c even 3 1 1225.1.g.a 2
7.c even 3 1 inner 1225.1.q.a 4
7.d odd 6 1 1225.1.g.a 2
7.d odd 6 1 inner 1225.1.q.a 4
35.c odd 2 1 CM 1225.1.q.a 4
35.f even 4 2 inner 1225.1.q.a 4
35.i odd 6 1 1225.1.g.a 2
35.i odd 6 1 inner 1225.1.q.a 4
35.j even 6 1 1225.1.g.a 2
35.j even 6 1 inner 1225.1.q.a 4
35.k even 12 2 1225.1.g.a 2
35.k even 12 2 inner 1225.1.q.a 4
35.l odd 12 2 1225.1.g.a 2
35.l odd 12 2 inner 1225.1.q.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1225.1.g.a 2 7.c even 3 1
1225.1.g.a 2 7.d odd 6 1
1225.1.g.a 2 35.i odd 6 1
1225.1.g.a 2 35.j even 6 1
1225.1.g.a 2 35.k even 12 2
1225.1.g.a 2 35.l odd 12 2
1225.1.q.a 4 1.a even 1 1 trivial
1225.1.q.a 4 5.b even 2 1 RM
1225.1.q.a 4 5.c odd 4 2 inner
1225.1.q.a 4 7.b odd 2 1 CM
1225.1.q.a 4 7.c even 3 1 inner
1225.1.q.a 4 7.d odd 6 1 inner
1225.1.q.a 4 35.c odd 2 1 CM
1225.1.q.a 4 35.f even 4 2 inner
1225.1.q.a 4 35.i odd 6 1 inner
1225.1.q.a 4 35.j even 6 1 inner
1225.1.q.a 4 35.k even 12 2 inner
1225.1.q.a 4 35.l odd 12 2 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}$$ acting on $$S_{1}^{\mathrm{new}}(1225, [\chi])$$.