# Properties

 Label 1225.1.q.b Level $1225$ Weight $1$ Character orbit 1225.q Analytic conductor $0.611$ Analytic rank $0$ Dimension $8$ Projective image $D_{6}$ CM discriminant -7 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: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{12})$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$3^{2}$$ Twist minimal: yes Projective image $$D_{6}$$ Projective field Galois closure of 6.2.153125.1

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + ( \zeta_{24}^{5} + \zeta_{24}^{9} ) q^{2} + ( -\zeta_{24}^{2} - \zeta_{24}^{6} + \zeta_{24}^{10} ) q^{4} + ( -\zeta_{24}^{7} - \zeta_{24}^{11} ) q^{8} -\zeta_{24}^{10} q^{9} +O(q^{10})$$ $$q + ( \zeta_{24}^{5} + \zeta_{24}^{9} ) q^{2} + ( -\zeta_{24}^{2} - \zeta_{24}^{6} + \zeta_{24}^{10} ) q^{4} + ( -\zeta_{24}^{7} - \zeta_{24}^{11} ) q^{8} -\zeta_{24}^{10} q^{9} -\zeta_{24}^{8} q^{11} + \zeta_{24}^{4} q^{16} + ( \zeta_{24}^{3} + \zeta_{24}^{7} ) q^{18} + ( \zeta_{24} + \zeta_{24}^{5} ) q^{22} + ( \zeta_{24}^{3} - \zeta_{24}^{11} ) q^{23} + \zeta_{24}^{6} q^{29} + ( -1 - \zeta_{24}^{4} + \zeta_{24}^{8} ) q^{36} + ( -\zeta_{24}^{5} - \zeta_{24}^{9} ) q^{37} + ( \zeta_{24}^{7} + \zeta_{24}^{11} ) q^{43} + ( -\zeta_{24}^{2} + \zeta_{24}^{6} + \zeta_{24}^{10} ) q^{44} + ( -1 + \zeta_{24}^{4} + 2 \zeta_{24}^{8} ) q^{46} + ( -\zeta_{24}^{3} + \zeta_{24}^{11} ) q^{58} + \zeta_{24}^{6} q^{64} + ( -\zeta_{24} + \zeta_{24}^{9} ) q^{67} + q^{71} + ( -\zeta_{24}^{5} - \zeta_{24}^{9} ) q^{72} + ( 2 \zeta_{24}^{2} + \zeta_{24}^{6} - \zeta_{24}^{10} ) q^{74} + \zeta_{24}^{10} q^{79} -\zeta_{24}^{8} q^{81} + ( -1 - 2 \zeta_{24}^{4} - \zeta_{24}^{8} ) q^{86} + ( -\zeta_{24}^{3} - \zeta_{24}^{7} ) q^{88} + ( -2 \zeta_{24} - 2 \zeta_{24}^{5} ) q^{92} -\zeta_{24}^{6} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q + 4q^{11} + 4q^{16} - 16q^{36} - 12q^{46} + 8q^{71} + 4q^{81} - 12q^{86} + O(q^{100})$$

## Character values

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

 $$n$$ $$101$$ $$1177$$ $$\chi(n)$$ $$-\zeta_{24}^{4}$$ $$\zeta_{24}^{6}$$

## 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
 −0.258819 + 0.965926i 0.258819 − 0.965926i 0.965926 + 0.258819i −0.965926 − 0.258819i 0.965926 − 0.258819i −0.965926 + 0.258819i −0.258819 − 0.965926i 0.258819 + 0.965926i
−1.67303 0.448288i 0 1.73205 + 1.00000i 0 0 0 −1.22474 1.22474i −0.866025 + 0.500000i 0
18.2 1.67303 + 0.448288i 0 1.73205 + 1.00000i 0 0 0 1.22474 + 1.22474i −0.866025 + 0.500000i 0
557.1 −0.448288 + 1.67303i 0 −1.73205 1.00000i 0 0 0 1.22474 1.22474i 0.866025 0.500000i 0
557.2 0.448288 1.67303i 0 −1.73205 1.00000i 0 0 0 −1.22474 + 1.22474i 0.866025 0.500000i 0
618.1 −0.448288 1.67303i 0 −1.73205 + 1.00000i 0 0 0 1.22474 + 1.22474i 0.866025 + 0.500000i 0
618.2 0.448288 + 1.67303i 0 −1.73205 + 1.00000i 0 0 0 −1.22474 1.22474i 0.866025 + 0.500000i 0
1157.1 −1.67303 + 0.448288i 0 1.73205 1.00000i 0 0 0 −1.22474 + 1.22474i −0.866025 0.500000i 0
1157.2 1.67303 0.448288i 0 1.73205 1.00000i 0 0 0 1.22474 1.22474i −0.866025 0.500000i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1157.2 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
7.c even 3 1 inner
7.d odd 6 1 inner
35.c odd 2 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.b 8
5.b even 2 1 inner 1225.1.q.b 8
5.c odd 4 2 inner 1225.1.q.b 8
7.b odd 2 1 CM 1225.1.q.b 8
7.c even 3 1 1225.1.g.b 4
7.c even 3 1 inner 1225.1.q.b 8
7.d odd 6 1 1225.1.g.b 4
7.d odd 6 1 inner 1225.1.q.b 8
35.c odd 2 1 inner 1225.1.q.b 8
35.f even 4 2 inner 1225.1.q.b 8
35.i odd 6 1 1225.1.g.b 4
35.i odd 6 1 inner 1225.1.q.b 8
35.j even 6 1 1225.1.g.b 4
35.j even 6 1 inner 1225.1.q.b 8
35.k even 12 2 1225.1.g.b 4
35.k even 12 2 inner 1225.1.q.b 8
35.l odd 12 2 1225.1.g.b 4
35.l odd 12 2 inner 1225.1.q.b 8

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} - 9 T_{2}^{4} + 81$$ acting on $$S_{1}^{\mathrm{new}}(1225, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$81 - 9 T^{4} + T^{8}$$
$3$ $$T^{8}$$
$5$ $$T^{8}$$
$7$ $$T^{8}$$
$11$ $$( 1 - T + T^{2} )^{4}$$
$13$ $$T^{8}$$
$17$ $$T^{8}$$
$19$ $$T^{8}$$
$23$ $$81 - 9 T^{4} + T^{8}$$
$29$ $$( 1 + T^{2} )^{4}$$
$31$ $$T^{8}$$
$37$ $$81 - 9 T^{4} + T^{8}$$
$41$ $$T^{8}$$
$43$ $$( 9 + T^{4} )^{2}$$
$47$ $$T^{8}$$
$53$ $$T^{8}$$
$59$ $$T^{8}$$
$61$ $$T^{8}$$
$67$ $$81 - 9 T^{4} + T^{8}$$
$71$ $$( -1 + T )^{8}$$
$73$ $$T^{8}$$
$79$ $$( 1 - T^{2} + T^{4} )^{2}$$
$83$ $$T^{8}$$
$89$ $$T^{8}$$
$97$ $$T^{8}$$