# Properties

 Label 1225.1.g.a Level 1225 Weight 1 Character orbit 1225.g Analytic conductor 0.611 Analytic rank 0 Dimension 2 Projective image $$D_{2}$$ CM/RM discs -7, -35, 5 Inner twists 8

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1225 = 5^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1225.g (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.611354640475$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ 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 $OD_{16}$ Artin field Galois closure of 8.4.9191328125.1

## $q$-expansion

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

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

## Character values

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

 $$n$$ $$101$$ $$1177$$ $$\chi(n)$$ $$1$$ $$i$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
393.1
 − 1.00000i 1.00000i
0 0 1.00000i 0 0 0 0 1.00000i 0
932.1 0 0 1.00000i 0 0 0 0 1.00000i 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
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 1225.1.g.a 2
5.b even 2 1 RM 1225.1.g.a 2
5.c odd 4 2 inner 1225.1.g.a 2
7.b odd 2 1 CM 1225.1.g.a 2
7.c even 3 2 1225.1.q.a 4
7.d odd 6 2 1225.1.q.a 4
35.c odd 2 1 CM 1225.1.g.a 2
35.f even 4 2 inner 1225.1.g.a 2
35.i odd 6 2 1225.1.q.a 4
35.j even 6 2 1225.1.q.a 4
35.k even 12 4 1225.1.q.a 4
35.l odd 12 4 1225.1.q.a 4

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

## Hecke kernels

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

## Hecke characteristic polynomials

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