# Properties

 Label 1225.1.i.b Level $1225$ Weight $1$ Character orbit 1225.i Analytic conductor $0.611$ Analytic rank $0$ Dimension $2$ Projective image $D_{3}$ CM discriminant -7 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1225 = 5^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1225.i (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.611354640475$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 175) Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.175.1 Artin image: $C_3\times S_3$ Artin field: Galois closure of 6.0.10504375.1

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\zeta_{6}^{2} q^{2} + q^{8} + \zeta_{6}^{2} q^{9} +O(q^{10})$$ $$q -\zeta_{6}^{2} q^{2} + q^{8} + \zeta_{6}^{2} q^{9} + \zeta_{6} q^{11} -\zeta_{6}^{2} q^{16} + \zeta_{6} q^{18} + q^{22} -\zeta_{6}^{2} q^{23} - q^{29} -\zeta_{6}^{2} q^{37} - q^{43} -\zeta_{6} q^{46} -2 \zeta_{6} q^{53} + \zeta_{6}^{2} q^{58} + q^{64} + \zeta_{6} q^{67} - q^{71} + \zeta_{6}^{2} q^{72} -\zeta_{6} q^{74} -\zeta_{6}^{2} q^{79} -\zeta_{6} q^{81} + \zeta_{6}^{2} q^{86} + \zeta_{6} q^{88} - q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} + 2 q^{8} - q^{9} + O(q^{10})$$ $$2 q + q^{2} + 2 q^{8} - q^{9} + q^{11} + q^{16} + q^{18} + 2 q^{22} + q^{23} - 2 q^{29} + q^{37} - 2 q^{43} - q^{46} - 2 q^{53} - q^{58} + 2 q^{64} + q^{67} - 2 q^{71} - q^{72} - q^{74} + q^{79} - q^{81} - q^{86} + q^{88} - 2 q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$101$$ $$1177$$ $$\chi(n)$$ $$-\zeta_{6}^{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
276.1
 0.5 − 0.866025i 0.5 + 0.866025i
0.500000 + 0.866025i 0 0 0 0 0 1.00000 −0.500000 0.866025i 0
901.1 0.500000 0.866025i 0 0 0 0 0 1.00000 −0.500000 + 0.866025i 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 $$\Q(\sqrt{-7})$$
7.c even 3 1 inner
7.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1225.1.i.b 2
5.b even 2 1 1225.1.i.a 2
5.c odd 4 2 1225.1.j.a 4
7.b odd 2 1 CM 1225.1.i.b 2
7.c even 3 1 175.1.d.a 1
7.c even 3 1 inner 1225.1.i.b 2
7.d odd 6 1 175.1.d.a 1
7.d odd 6 1 inner 1225.1.i.b 2
21.g even 6 1 1575.1.h.c 1
21.h odd 6 1 1575.1.h.c 1
28.f even 6 1 2800.1.f.a 1
28.g odd 6 1 2800.1.f.a 1
35.c odd 2 1 1225.1.i.a 2
35.f even 4 2 1225.1.j.a 4
35.i odd 6 1 175.1.d.b yes 1
35.i odd 6 1 1225.1.i.a 2
35.j even 6 1 175.1.d.b yes 1
35.j even 6 1 1225.1.i.a 2
35.k even 12 2 175.1.c.a 2
35.k even 12 2 1225.1.j.a 4
35.l odd 12 2 175.1.c.a 2
35.l odd 12 2 1225.1.j.a 4
105.o odd 6 1 1575.1.h.a 1
105.p even 6 1 1575.1.h.a 1
105.w odd 12 2 1575.1.e.a 2
105.x even 12 2 1575.1.e.a 2
140.p odd 6 1 2800.1.f.b 1
140.s even 6 1 2800.1.f.b 1
140.w even 12 2 2800.1.p.a 2
140.x odd 12 2 2800.1.p.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
175.1.c.a 2 35.k even 12 2
175.1.c.a 2 35.l odd 12 2
175.1.d.a 1 7.c even 3 1
175.1.d.a 1 7.d odd 6 1
175.1.d.b yes 1 35.i odd 6 1
175.1.d.b yes 1 35.j even 6 1
1225.1.i.a 2 5.b even 2 1
1225.1.i.a 2 35.c odd 2 1
1225.1.i.a 2 35.i odd 6 1
1225.1.i.a 2 35.j even 6 1
1225.1.i.b 2 1.a even 1 1 trivial
1225.1.i.b 2 7.b odd 2 1 CM
1225.1.i.b 2 7.c even 3 1 inner
1225.1.i.b 2 7.d odd 6 1 inner
1225.1.j.a 4 5.c odd 4 2
1225.1.j.a 4 35.f even 4 2
1225.1.j.a 4 35.k even 12 2
1225.1.j.a 4 35.l odd 12 2
1575.1.e.a 2 105.w odd 12 2
1575.1.e.a 2 105.x even 12 2
1575.1.h.a 1 105.o odd 6 1
1575.1.h.a 1 105.p even 6 1
1575.1.h.c 1 21.g even 6 1
1575.1.h.c 1 21.h odd 6 1
2800.1.f.a 1 28.f even 6 1
2800.1.f.a 1 28.g odd 6 1
2800.1.f.b 1 140.p odd 6 1
2800.1.f.b 1 140.s even 6 1
2800.1.p.a 2 140.w even 12 2
2800.1.p.a 2 140.x odd 12 2

## Hecke kernels

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

## Hecke characteristic polynomials

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