# Properties

 Label 1225.2.b.b Level $1225$ Weight $2$ Character orbit 1225.b Analytic conductor $9.782$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$9.78167424761$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 245) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 i q^{2} -3 i q^{3} -2 q^{4} + 6 q^{6} -6 q^{9} +O(q^{10})$$ $$q + 2 i q^{2} -3 i q^{3} -2 q^{4} + 6 q^{6} -6 q^{9} + q^{11} + 6 i q^{12} -3 i q^{13} -4 q^{16} -3 i q^{17} -12 i q^{18} + 6 q^{19} + 2 i q^{22} -4 i q^{23} + 6 q^{26} + 9 i q^{27} + q^{29} -6 q^{31} -8 i q^{32} -3 i q^{33} + 6 q^{34} + 12 q^{36} + 12 i q^{38} -9 q^{39} -6 q^{41} -6 i q^{43} -2 q^{44} + 8 q^{46} -9 i q^{47} + 12 i q^{48} -9 q^{51} + 6 i q^{52} -10 i q^{53} -18 q^{54} -18 i q^{57} + 2 i q^{58} -6 q^{59} -12 i q^{62} + 8 q^{64} + 6 q^{66} + 14 i q^{67} + 6 i q^{68} -12 q^{69} -8 q^{71} -6 i q^{73} -12 q^{76} -18 i q^{78} + q^{79} + 9 q^{81} -12 i q^{82} -12 i q^{83} + 12 q^{86} -3 i q^{87} + 12 q^{89} + 8 i q^{92} + 18 i q^{93} + 18 q^{94} -24 q^{96} -15 i q^{97} -6 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{4} + 12q^{6} - 12q^{9} + O(q^{10})$$ $$2q - 4q^{4} + 12q^{6} - 12q^{9} + 2q^{11} - 8q^{16} + 12q^{19} + 12q^{26} + 2q^{29} - 12q^{31} + 12q^{34} + 24q^{36} - 18q^{39} - 12q^{41} - 4q^{44} + 16q^{46} - 18q^{51} - 36q^{54} - 12q^{59} + 16q^{64} + 12q^{66} - 24q^{69} - 16q^{71} - 24q^{76} + 2q^{79} + 18q^{81} + 24q^{86} + 24q^{89} + 36q^{94} - 48q^{96} - 12q^{99} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
99.1
 − 1.00000i 1.00000i
2.00000i 3.00000i −2.00000 0 6.00000 0 0 −6.00000 0
99.2 2.00000i 3.00000i −2.00000 0 6.00000 0 0 −6.00000 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 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1225.2.b.b 2
5.b even 2 1 inner 1225.2.b.b 2
5.c odd 4 1 245.2.a.a 1
5.c odd 4 1 1225.2.a.j 1
7.b odd 2 1 1225.2.b.a 2
15.e even 4 1 2205.2.a.j 1
20.e even 4 1 3920.2.a.bj 1
35.c odd 2 1 1225.2.b.a 2
35.f even 4 1 245.2.a.b yes 1
35.f even 4 1 1225.2.a.h 1
35.k even 12 2 245.2.e.c 2
35.l odd 12 2 245.2.e.d 2
105.k odd 4 1 2205.2.a.l 1
140.j odd 4 1 3920.2.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
245.2.a.a 1 5.c odd 4 1
245.2.a.b yes 1 35.f even 4 1
245.2.e.c 2 35.k even 12 2
245.2.e.d 2 35.l odd 12 2
1225.2.a.h 1 35.f even 4 1
1225.2.a.j 1 5.c odd 4 1
1225.2.b.a 2 7.b odd 2 1
1225.2.b.a 2 35.c odd 2 1
1225.2.b.b 2 1.a even 1 1 trivial
1225.2.b.b 2 5.b even 2 1 inner
2205.2.a.j 1 15.e even 4 1
2205.2.a.l 1 105.k odd 4 1
3920.2.a.a 1 140.j odd 4 1
3920.2.a.bj 1 20.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1225, [\chi])$$:

 $$T_{2}^{2} + 4$$ $$T_{19} - 6$$ $$T_{31} + 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 + T^{2}$$
$3$ $$9 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$( -1 + T )^{2}$$
$13$ $$9 + T^{2}$$
$17$ $$9 + T^{2}$$
$19$ $$( -6 + T )^{2}$$
$23$ $$16 + T^{2}$$
$29$ $$( -1 + T )^{2}$$
$31$ $$( 6 + T )^{2}$$
$37$ $$T^{2}$$
$41$ $$( 6 + T )^{2}$$
$43$ $$36 + T^{2}$$
$47$ $$81 + T^{2}$$
$53$ $$100 + T^{2}$$
$59$ $$( 6 + T )^{2}$$
$61$ $$T^{2}$$
$67$ $$196 + T^{2}$$
$71$ $$( 8 + T )^{2}$$
$73$ $$36 + T^{2}$$
$79$ $$( -1 + T )^{2}$$
$83$ $$144 + T^{2}$$
$89$ $$( -12 + T )^{2}$$
$97$ $$225 + T^{2}$$