# Properties

 Label 225.2.b.c Level $225$ Weight $2$ Character orbit 225.b Analytic conductor $1.797$ Analytic rank $0$ Dimension $2$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$225 = 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 225.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.79663404548$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$5$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + 2 q^{4} + \beta q^{7}+O(q^{10})$$ q + 2 * q^4 + b * q^7 $$q + 2 q^{4} + \beta q^{7} - \beta q^{13} + 4 q^{16} + q^{19} + 2 \beta q^{28} - 7 q^{31} - 2 \beta q^{37} - \beta q^{43} - 18 q^{49} - 2 \beta q^{52} - 13 q^{61} + 8 q^{64} + \beta q^{67} + 2 \beta q^{73} + 2 q^{76} + 4 q^{79} + 25 q^{91} + \beta q^{97} +O(q^{100})$$ q + 2 * q^4 + b * q^7 - b * q^13 + 4 * q^16 + q^19 + 2*b * q^28 - 7 * q^31 - 2*b * q^37 - b * q^43 - 18 * q^49 - 2*b * q^52 - 13 * q^61 + 8 * q^64 + b * q^67 + 2*b * q^73 + 2 * q^76 + 4 * q^79 + 25 * q^91 + b * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{4}+O(q^{10})$$ 2 * q + 4 * q^4 $$2 q + 4 q^{4} + 8 q^{16} + 2 q^{19} - 14 q^{31} - 36 q^{49} - 26 q^{61} + 16 q^{64} + 4 q^{76} + 8 q^{79} + 50 q^{91}+O(q^{100})$$ 2 * q + 4 * q^4 + 8 * q^16 + 2 * q^19 - 14 * q^31 - 36 * q^49 - 26 * q^61 + 16 * q^64 + 4 * q^76 + 8 * q^79 + 50 * q^91

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/225\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
199.1
 − 1.00000i 1.00000i
0 0 2.00000 0 0 5.00000i 0 0 0
199.2 0 0 2.00000 0 0 5.00000i 0 0 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
5.b even 2 1 inner
15.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.2.b.c 2
3.b odd 2 1 CM 225.2.b.c 2
4.b odd 2 1 3600.2.f.k 2
5.b even 2 1 inner 225.2.b.c 2
5.c odd 4 1 225.2.a.c 1
5.c odd 4 1 225.2.a.d yes 1
12.b even 2 1 3600.2.f.k 2
15.d odd 2 1 inner 225.2.b.c 2
15.e even 4 1 225.2.a.c 1
15.e even 4 1 225.2.a.d yes 1
20.d odd 2 1 3600.2.f.k 2
20.e even 4 1 3600.2.a.b 1
20.e even 4 1 3600.2.a.br 1
60.h even 2 1 3600.2.f.k 2
60.l odd 4 1 3600.2.a.b 1
60.l odd 4 1 3600.2.a.br 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
225.2.a.c 1 5.c odd 4 1
225.2.a.c 1 15.e even 4 1
225.2.a.d yes 1 5.c odd 4 1
225.2.a.d yes 1 15.e even 4 1
225.2.b.c 2 1.a even 1 1 trivial
225.2.b.c 2 3.b odd 2 1 CM
225.2.b.c 2 5.b even 2 1 inner
225.2.b.c 2 15.d odd 2 1 inner
3600.2.a.b 1 20.e even 4 1
3600.2.a.b 1 60.l odd 4 1
3600.2.a.br 1 20.e even 4 1
3600.2.a.br 1 60.l odd 4 1
3600.2.f.k 2 4.b odd 2 1
3600.2.f.k 2 12.b even 2 1
3600.2.f.k 2 20.d odd 2 1
3600.2.f.k 2 60.h even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}$$ acting on $$S_{2}^{\mathrm{new}}(225, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 25$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 25$$
$17$ $$T^{2}$$
$19$ $$(T - 1)^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$(T + 7)^{2}$$
$37$ $$T^{2} + 100$$
$41$ $$T^{2}$$
$43$ $$T^{2} + 25$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$(T + 13)^{2}$$
$67$ $$T^{2} + 25$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 100$$
$79$ $$(T - 4)^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$T^{2} + 25$$