# Properties

 Label 225.2.b.a Level $225$ Weight $2$ Character orbit 225.b Analytic conductor $1.797$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# 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: $$1$$ Twist minimal: no (minimal twist has level 75) 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.

 $$f(q)$$ $$=$$ $$q + 2 i q^{2} - 2 q^{4} + 3 i q^{7}+O(q^{10})$$ q + 2*i * q^2 - 2 * q^4 + 3*i * q^7 $$q + 2 i q^{2} - 2 q^{4} + 3 i q^{7} - 2 q^{11} + i q^{13} - 6 q^{14} - 4 q^{16} + 2 i q^{17} + 5 q^{19} - 4 i q^{22} - 6 i q^{23} - 2 q^{26} - 6 i q^{28} + 10 q^{29} - 3 q^{31} - 8 i q^{32} - 4 q^{34} - 2 i q^{37} + 10 i q^{38} + 8 q^{41} + i q^{43} + 4 q^{44} + 12 q^{46} + 2 i q^{47} - 2 q^{49} - 2 i q^{52} + 4 i q^{53} + 20 i q^{58} - 10 q^{59} + 7 q^{61} - 6 i q^{62} + 8 q^{64} + 3 i q^{67} - 4 i q^{68} + 8 q^{71} - 14 i q^{73} + 4 q^{74} - 10 q^{76} - 6 i q^{77} + 16 i q^{82} - 6 i q^{83} - 2 q^{86} - 3 q^{91} + 12 i q^{92} - 4 q^{94} - 17 i q^{97} - 4 i q^{98} +O(q^{100})$$ q + 2*i * q^2 - 2 * q^4 + 3*i * q^7 - 2 * q^11 + i * q^13 - 6 * q^14 - 4 * q^16 + 2*i * q^17 + 5 * q^19 - 4*i * q^22 - 6*i * q^23 - 2 * q^26 - 6*i * q^28 + 10 * q^29 - 3 * q^31 - 8*i * q^32 - 4 * q^34 - 2*i * q^37 + 10*i * q^38 + 8 * q^41 + i * q^43 + 4 * q^44 + 12 * q^46 + 2*i * q^47 - 2 * q^49 - 2*i * q^52 + 4*i * q^53 + 20*i * q^58 - 10 * q^59 + 7 * q^61 - 6*i * q^62 + 8 * q^64 + 3*i * q^67 - 4*i * q^68 + 8 * q^71 - 14*i * q^73 + 4 * q^74 - 10 * q^76 - 6*i * q^77 + 16*i * q^82 - 6*i * q^83 - 2 * q^86 - 3 * q^91 + 12*i * q^92 - 4 * q^94 - 17*i * q^97 - 4*i * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{4}+O(q^{10})$$ 2 * q - 4 * q^4 $$2 q - 4 q^{4} - 4 q^{11} - 12 q^{14} - 8 q^{16} + 10 q^{19} - 4 q^{26} + 20 q^{29} - 6 q^{31} - 8 q^{34} + 16 q^{41} + 8 q^{44} + 24 q^{46} - 4 q^{49} - 20 q^{59} + 14 q^{61} + 16 q^{64} + 16 q^{71} + 8 q^{74} - 20 q^{76} - 4 q^{86} - 6 q^{91} - 8 q^{94}+O(q^{100})$$ 2 * q - 4 * q^4 - 4 * q^11 - 12 * q^14 - 8 * q^16 + 10 * q^19 - 4 * q^26 + 20 * q^29 - 6 * q^31 - 8 * q^34 + 16 * q^41 + 8 * q^44 + 24 * q^46 - 4 * q^49 - 20 * q^59 + 14 * q^61 + 16 * q^64 + 16 * q^71 + 8 * q^74 - 20 * q^76 - 4 * q^86 - 6 * q^91 - 8 * q^94

## 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
2.00000i 0 −2.00000 0 0 3.00000i 0 0 0
199.2 2.00000i 0 −2.00000 0 0 3.00000i 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
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 225.2.b.a 2
3.b odd 2 1 75.2.b.a 2
4.b odd 2 1 3600.2.f.p 2
5.b even 2 1 inner 225.2.b.a 2
5.c odd 4 1 225.2.a.a 1
5.c odd 4 1 225.2.a.e 1
12.b even 2 1 1200.2.f.d 2
15.d odd 2 1 75.2.b.a 2
15.e even 4 1 75.2.a.a 1
15.e even 4 1 75.2.a.c yes 1
20.d odd 2 1 3600.2.f.p 2
20.e even 4 1 3600.2.a.j 1
20.e even 4 1 3600.2.a.bk 1
24.f even 2 1 4800.2.f.y 2
24.h odd 2 1 4800.2.f.l 2
60.h even 2 1 1200.2.f.d 2
60.l odd 4 1 1200.2.a.c 1
60.l odd 4 1 1200.2.a.p 1
105.k odd 4 1 3675.2.a.b 1
105.k odd 4 1 3675.2.a.q 1
120.i odd 2 1 4800.2.f.l 2
120.m even 2 1 4800.2.f.y 2
120.q odd 4 1 4800.2.a.be 1
120.q odd 4 1 4800.2.a.br 1
120.w even 4 1 4800.2.a.bb 1
120.w even 4 1 4800.2.a.bq 1
165.l odd 4 1 9075.2.a.a 1
165.l odd 4 1 9075.2.a.s 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.2.a.a 1 15.e even 4 1
75.2.a.c yes 1 15.e even 4 1
75.2.b.a 2 3.b odd 2 1
75.2.b.a 2 15.d odd 2 1
225.2.a.a 1 5.c odd 4 1
225.2.a.e 1 5.c odd 4 1
225.2.b.a 2 1.a even 1 1 trivial
225.2.b.a 2 5.b even 2 1 inner
1200.2.a.c 1 60.l odd 4 1
1200.2.a.p 1 60.l odd 4 1
1200.2.f.d 2 12.b even 2 1
1200.2.f.d 2 60.h even 2 1
3600.2.a.j 1 20.e even 4 1
3600.2.a.bk 1 20.e even 4 1
3600.2.f.p 2 4.b odd 2 1
3600.2.f.p 2 20.d odd 2 1
3675.2.a.b 1 105.k odd 4 1
3675.2.a.q 1 105.k odd 4 1
4800.2.a.bb 1 120.w even 4 1
4800.2.a.be 1 120.q odd 4 1
4800.2.a.bq 1 120.w even 4 1
4800.2.a.br 1 120.q odd 4 1
4800.2.f.l 2 24.h odd 2 1
4800.2.f.l 2 120.i odd 2 1
4800.2.f.y 2 24.f even 2 1
4800.2.f.y 2 120.m even 2 1
9075.2.a.a 1 165.l odd 4 1
9075.2.a.s 1 165.l odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 4$$ acting on $$S_{2}^{\mathrm{new}}(225, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 4$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 9$$
$11$ $$(T + 2)^{2}$$
$13$ $$T^{2} + 1$$
$17$ $$T^{2} + 4$$
$19$ $$(T - 5)^{2}$$
$23$ $$T^{2} + 36$$
$29$ $$(T - 10)^{2}$$
$31$ $$(T + 3)^{2}$$
$37$ $$T^{2} + 4$$
$41$ $$(T - 8)^{2}$$
$43$ $$T^{2} + 1$$
$47$ $$T^{2} + 4$$
$53$ $$T^{2} + 16$$
$59$ $$(T + 10)^{2}$$
$61$ $$(T - 7)^{2}$$
$67$ $$T^{2} + 9$$
$71$ $$(T - 8)^{2}$$
$73$ $$T^{2} + 196$$
$79$ $$T^{2}$$
$83$ $$T^{2} + 36$$
$89$ $$T^{2}$$
$97$ $$T^{2} + 289$$