# Properties

 Label 225.4.b.g Level $225$ Weight $4$ Character orbit 225.b Analytic conductor $13.275$ Analytic rank $0$ Dimension $2$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$225 = 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 225.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$13.2754297513$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2\cdot 5$$ Twist minimal: no (minimal twist has level 9) 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.

 $$f(q)$$ $$=$$ $$q + 8 q^{4} + 2 \beta q^{7}+O(q^{10})$$ q + 8 * q^4 + 2*b * q^7 $$q + 8 q^{4} + 2 \beta q^{7} + 7 \beta q^{13} + 64 q^{16} - 56 q^{19} + 16 \beta q^{28} + 308 q^{31} + 11 \beta q^{37} + 52 \beta q^{43} - 57 q^{49} + 56 \beta q^{52} + 182 q^{61} + 512 q^{64} - 88 \beta q^{67} - 119 \beta q^{73} - 448 q^{76} - 884 q^{79} - 1400 q^{91} - 133 \beta q^{97} +O(q^{100})$$ q + 8 * q^4 + 2*b * q^7 + 7*b * q^13 + 64 * q^16 - 56 * q^19 + 16*b * q^28 + 308 * q^31 + 11*b * q^37 + 52*b * q^43 - 57 * q^49 + 56*b * q^52 + 182 * q^61 + 512 * q^64 - 88*b * q^67 - 119*b * q^73 - 448 * q^76 - 884 * q^79 - 1400 * q^91 - 133*b * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 16 q^{4}+O(q^{10})$$ 2 * q + 16 * q^4 $$2 q + 16 q^{4} + 128 q^{16} - 112 q^{19} + 616 q^{31} - 114 q^{49} + 364 q^{61} + 1024 q^{64} - 896 q^{76} - 1768 q^{79} - 2800 q^{91}+O(q^{100})$$ 2 * q + 16 * q^4 + 128 * q^16 - 112 * q^19 + 616 * q^31 - 114 * q^49 + 364 * q^61 + 1024 * q^64 - 896 * q^76 - 1768 * q^79 - 2800 * 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 8.00000 0 0 20.0000i 0 0 0
199.2 0 0 8.00000 0 0 20.0000i 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 225.4.b.g 2
3.b odd 2 1 CM 225.4.b.g 2
5.b even 2 1 inner 225.4.b.g 2
5.c odd 4 1 9.4.a.a 1
5.c odd 4 1 225.4.a.d 1
15.d odd 2 1 inner 225.4.b.g 2
15.e even 4 1 9.4.a.a 1
15.e even 4 1 225.4.a.d 1
20.e even 4 1 144.4.a.d 1
35.f even 4 1 441.4.a.f 1
35.k even 12 2 441.4.e.j 2
35.l odd 12 2 441.4.e.i 2
40.i odd 4 1 576.4.a.m 1
40.k even 4 1 576.4.a.l 1
45.k odd 12 2 81.4.c.b 2
45.l even 12 2 81.4.c.b 2
55.e even 4 1 1089.4.a.g 1
60.l odd 4 1 144.4.a.d 1
65.h odd 4 1 1521.4.a.g 1
105.k odd 4 1 441.4.a.f 1
105.w odd 12 2 441.4.e.j 2
105.x even 12 2 441.4.e.i 2
120.q odd 4 1 576.4.a.l 1
120.w even 4 1 576.4.a.m 1
165.l odd 4 1 1089.4.a.g 1
195.s even 4 1 1521.4.a.g 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.4.a.a 1 5.c odd 4 1
9.4.a.a 1 15.e even 4 1
81.4.c.b 2 45.k odd 12 2
81.4.c.b 2 45.l even 12 2
144.4.a.d 1 20.e even 4 1
144.4.a.d 1 60.l odd 4 1
225.4.a.d 1 5.c odd 4 1
225.4.a.d 1 15.e even 4 1
225.4.b.g 2 1.a even 1 1 trivial
225.4.b.g 2 3.b odd 2 1 CM
225.4.b.g 2 5.b even 2 1 inner
225.4.b.g 2 15.d odd 2 1 inner
441.4.a.f 1 35.f even 4 1
441.4.a.f 1 105.k odd 4 1
441.4.e.i 2 35.l odd 12 2
441.4.e.i 2 105.x even 12 2
441.4.e.j 2 35.k even 12 2
441.4.e.j 2 105.w odd 12 2
576.4.a.l 1 40.k even 4 1
576.4.a.l 1 120.q odd 4 1
576.4.a.m 1 40.i odd 4 1
576.4.a.m 1 120.w even 4 1
1089.4.a.g 1 55.e even 4 1
1089.4.a.g 1 165.l odd 4 1
1521.4.a.g 1 65.h odd 4 1
1521.4.a.g 1 195.s even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(225, [\chi])$$:

 $$T_{2}$$ T2 $$T_{7}^{2} + 400$$ T7^2 + 400 $$T_{11}$$ T11

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 400$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 4900$$
$17$ $$T^{2}$$
$19$ $$(T + 56)^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$(T - 308)^{2}$$
$37$ $$T^{2} + 12100$$
$41$ $$T^{2}$$
$43$ $$T^{2} + 270400$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$(T - 182)^{2}$$
$67$ $$T^{2} + 774400$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 1416100$$
$79$ $$(T + 884)^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$T^{2} + 1768900$$