# Properties

 Label 1225.1.g.b Level $1225$ Weight $1$ Character orbit 1225.g Analytic conductor $0.611$ Analytic rank $0$ Dimension $4$ Projective image $D_{6}$ CM discriminant -7 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: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(i, \sqrt{6})$$ Defining polynomial: $$x^{4} + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$D_{6}$$ Projective field Galois closure of 6.2.153125.1

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + 2 \beta_{2} q^{4} -\beta_{3} q^{8} + \beta_{2} q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{2} + 2 \beta_{2} q^{4} -\beta_{3} q^{8} + \beta_{2} q^{9} - q^{11} - q^{16} -\beta_{3} q^{18} + \beta_{1} q^{22} + \beta_{3} q^{23} + \beta_{2} q^{29} -2 q^{36} + \beta_{1} q^{37} + \beta_{3} q^{43} -2 \beta_{2} q^{44} + 3 q^{46} -\beta_{3} q^{58} + \beta_{2} q^{64} + \beta_{1} q^{67} + q^{71} + \beta_{1} q^{72} -3 \beta_{2} q^{74} -\beta_{2} q^{79} - q^{81} + 3 q^{86} + \beta_{3} q^{88} -2 \beta_{1} q^{92} -\beta_{2} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q - 4q^{11} - 4q^{16} - 8q^{36} + 12q^{46} + 4q^{71} - 4q^{81} + 12q^{86} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/3$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$3 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$3 \beta_{3}$$

## Character values

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

 $$n$$ $$101$$ $$1177$$ $$\chi(n)$$ $$1$$ $$\beta_{2}$$

## 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
 1.22474 − 1.22474i −1.22474 + 1.22474i 1.22474 + 1.22474i −1.22474 − 1.22474i
−1.22474 + 1.22474i 0 2.00000i 0 0 0 1.22474 + 1.22474i 1.00000i 0
393.2 1.22474 1.22474i 0 2.00000i 0 0 0 −1.22474 1.22474i 1.00000i 0
932.1 −1.22474 1.22474i 0 2.00000i 0 0 0 1.22474 1.22474i 1.00000i 0
932.2 1.22474 + 1.22474i 0 2.00000i 0 0 0 −1.22474 + 1.22474i 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
7.b odd 2 1 CM by $$\Q(\sqrt{-7})$$
5.b even 2 1 inner
5.c odd 4 2 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 1225.1.g.b 4
5.b even 2 1 inner 1225.1.g.b 4
5.c odd 4 2 inner 1225.1.g.b 4
7.b odd 2 1 CM 1225.1.g.b 4
7.c even 3 2 1225.1.q.b 8
7.d odd 6 2 1225.1.q.b 8
35.c odd 2 1 inner 1225.1.g.b 4
35.f even 4 2 inner 1225.1.g.b 4
35.i odd 6 2 1225.1.q.b 8
35.j even 6 2 1225.1.q.b 8
35.k even 12 4 1225.1.q.b 8
35.l odd 12 4 1225.1.q.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1225.1.g.b 4 1.a even 1 1 trivial
1225.1.g.b 4 5.b even 2 1 inner
1225.1.g.b 4 5.c odd 4 2 inner
1225.1.g.b 4 7.b odd 2 1 CM
1225.1.g.b 4 35.c odd 2 1 inner
1225.1.g.b 4 35.f even 4 2 inner
1225.1.q.b 8 7.c even 3 2
1225.1.q.b 8 7.d odd 6 2
1225.1.q.b 8 35.i odd 6 2
1225.1.q.b 8 35.j even 6 2
1225.1.q.b 8 35.k even 12 4
1225.1.q.b 8 35.l odd 12 4

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 9$$ acting on $$S_{1}^{\mathrm{new}}(1225, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$9 + T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$( 1 + T )^{4}$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$T^{4}$$
$23$ $$9 + T^{4}$$
$29$ $$( 1 + T^{2} )^{2}$$
$31$ $$T^{4}$$
$37$ $$9 + T^{4}$$
$41$ $$T^{4}$$
$43$ $$9 + T^{4}$$
$47$ $$T^{4}$$
$53$ $$T^{4}$$
$59$ $$T^{4}$$
$61$ $$T^{4}$$
$67$ $$9 + T^{4}$$
$71$ $$( -1 + T )^{4}$$
$73$ $$T^{4}$$
$79$ $$( 1 + T^{2} )^{2}$$
$83$ $$T^{4}$$
$89$ $$T^{4}$$
$97$ $$T^{4}$$