# Properties

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

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,2,Mod(199,225)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(225, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("225.199");

S:= CuspForms(chi, 2);

N := Newforms(S);

 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

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$1.79663404548$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 15) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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

 $$f(q)$$ $$=$$ $$q + i q^{2} + q^{4} + 3 i q^{8}+O(q^{10})$$ q + i * q^2 + q^4 + 3*i * q^8 $$q + i q^{2} + q^{4} + 3 i q^{8} + 4 q^{11} + 2 i q^{13} - q^{16} - 2 i q^{17} - 4 q^{19} + 4 i q^{22} - 2 q^{26} - 2 q^{29} + 5 i q^{32} + 2 q^{34} - 10 i q^{37} - 4 i q^{38} - 10 q^{41} - 4 i q^{43} + 4 q^{44} - 8 i q^{47} + 7 q^{49} + 2 i q^{52} - 10 i q^{53} - 2 i q^{58} - 4 q^{59} - 2 q^{61} - 7 q^{64} + 12 i q^{67} - 2 i q^{68} + 8 q^{71} - 10 i q^{73} + 10 q^{74} - 4 q^{76} - 10 i q^{82} + 12 i q^{83} + 4 q^{86} + 12 i q^{88} - 6 q^{89} + 8 q^{94} + 2 i q^{97} + 7 i q^{98} +O(q^{100})$$ q + i * q^2 + q^4 + 3*i * q^8 + 4 * q^11 + 2*i * q^13 - q^16 - 2*i * q^17 - 4 * q^19 + 4*i * q^22 - 2 * q^26 - 2 * q^29 + 5*i * q^32 + 2 * q^34 - 10*i * q^37 - 4*i * q^38 - 10 * q^41 - 4*i * q^43 + 4 * q^44 - 8*i * q^47 + 7 * q^49 + 2*i * q^52 - 10*i * q^53 - 2*i * q^58 - 4 * q^59 - 2 * q^61 - 7 * q^64 + 12*i * q^67 - 2*i * q^68 + 8 * q^71 - 10*i * q^73 + 10 * q^74 - 4 * q^76 - 10*i * q^82 + 12*i * q^83 + 4 * q^86 + 12*i * q^88 - 6 * q^89 + 8 * q^94 + 2*i * q^97 + 7*i * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{4}+O(q^{10})$$ 2 * q + 2 * q^4 $$2 q + 2 q^{4} + 8 q^{11} - 2 q^{16} - 8 q^{19} - 4 q^{26} - 4 q^{29} + 4 q^{34} - 20 q^{41} + 8 q^{44} + 14 q^{49} - 8 q^{59} - 4 q^{61} - 14 q^{64} + 16 q^{71} + 20 q^{74} - 8 q^{76} + 8 q^{86} - 12 q^{89} + 16 q^{94}+O(q^{100})$$ 2 * q + 2 * q^4 + 8 * q^11 - 2 * q^16 - 8 * q^19 - 4 * q^26 - 4 * q^29 + 4 * q^34 - 20 * q^41 + 8 * q^44 + 14 * q^49 - 8 * q^59 - 4 * q^61 - 14 * q^64 + 16 * q^71 + 20 * q^74 - 8 * q^76 + 8 * q^86 - 12 * q^89 + 16 * 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

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
1.00000i 0 1.00000 0 0 0 3.00000i 0 0
199.2 1.00000i 0 1.00000 0 0 0 3.00000i 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.b 2
3.b odd 2 1 75.2.b.b 2
4.b odd 2 1 3600.2.f.e 2
5.b even 2 1 inner 225.2.b.b 2
5.c odd 4 1 45.2.a.a 1
5.c odd 4 1 225.2.a.b 1
12.b even 2 1 1200.2.f.h 2
15.d odd 2 1 75.2.b.b 2
15.e even 4 1 15.2.a.a 1
15.e even 4 1 75.2.a.b 1
20.d odd 2 1 3600.2.f.e 2
20.e even 4 1 720.2.a.c 1
20.e even 4 1 3600.2.a.u 1
24.f even 2 1 4800.2.f.c 2
24.h odd 2 1 4800.2.f.bf 2
35.f even 4 1 2205.2.a.i 1
40.i odd 4 1 2880.2.a.y 1
40.k even 4 1 2880.2.a.bc 1
45.k odd 12 2 405.2.e.c 2
45.l even 12 2 405.2.e.f 2
55.e even 4 1 5445.2.a.c 1
60.h even 2 1 1200.2.f.h 2
60.l odd 4 1 240.2.a.d 1
60.l odd 4 1 1200.2.a.e 1
65.h odd 4 1 7605.2.a.g 1
105.k odd 4 1 735.2.a.c 1
105.k odd 4 1 3675.2.a.j 1
105.w odd 12 2 735.2.i.d 2
105.x even 12 2 735.2.i.e 2
120.i odd 2 1 4800.2.f.bf 2
120.m even 2 1 4800.2.f.c 2
120.q odd 4 1 960.2.a.a 1
120.q odd 4 1 4800.2.a.bz 1
120.w even 4 1 960.2.a.l 1
120.w even 4 1 4800.2.a.t 1
165.l odd 4 1 1815.2.a.d 1
165.l odd 4 1 9075.2.a.g 1
195.s even 4 1 2535.2.a.j 1
240.z odd 4 1 3840.2.k.r 2
240.bb even 4 1 3840.2.k.m 2
240.bd odd 4 1 3840.2.k.r 2
240.bf even 4 1 3840.2.k.m 2
255.o even 4 1 4335.2.a.c 1
285.j odd 4 1 5415.2.a.j 1
345.l odd 4 1 7935.2.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.2.a.a 1 15.e even 4 1
45.2.a.a 1 5.c odd 4 1
75.2.a.b 1 15.e even 4 1
75.2.b.b 2 3.b odd 2 1
75.2.b.b 2 15.d odd 2 1
225.2.a.b 1 5.c odd 4 1
225.2.b.b 2 1.a even 1 1 trivial
225.2.b.b 2 5.b even 2 1 inner
240.2.a.d 1 60.l odd 4 1
405.2.e.c 2 45.k odd 12 2
405.2.e.f 2 45.l even 12 2
720.2.a.c 1 20.e even 4 1
735.2.a.c 1 105.k odd 4 1
735.2.i.d 2 105.w odd 12 2
735.2.i.e 2 105.x even 12 2
960.2.a.a 1 120.q odd 4 1
960.2.a.l 1 120.w even 4 1
1200.2.a.e 1 60.l odd 4 1
1200.2.f.h 2 12.b even 2 1
1200.2.f.h 2 60.h even 2 1
1815.2.a.d 1 165.l odd 4 1
2205.2.a.i 1 35.f even 4 1
2535.2.a.j 1 195.s even 4 1
2880.2.a.y 1 40.i odd 4 1
2880.2.a.bc 1 40.k even 4 1
3600.2.a.u 1 20.e even 4 1
3600.2.f.e 2 4.b odd 2 1
3600.2.f.e 2 20.d odd 2 1
3675.2.a.j 1 105.k odd 4 1
3840.2.k.m 2 240.bb even 4 1
3840.2.k.m 2 240.bf even 4 1
3840.2.k.r 2 240.z odd 4 1
3840.2.k.r 2 240.bd odd 4 1
4335.2.a.c 1 255.o even 4 1
4800.2.a.t 1 120.w even 4 1
4800.2.a.bz 1 120.q odd 4 1
4800.2.f.c 2 24.f even 2 1
4800.2.f.c 2 120.m even 2 1
4800.2.f.bf 2 24.h odd 2 1
4800.2.f.bf 2 120.i odd 2 1
5415.2.a.j 1 285.j odd 4 1
5445.2.a.c 1 55.e even 4 1
7605.2.a.g 1 65.h odd 4 1
7935.2.a.d 1 345.l odd 4 1
9075.2.a.g 1 165.l odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(225, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$(T - 4)^{2}$$
$13$ $$T^{2} + 4$$
$17$ $$T^{2} + 4$$
$19$ $$(T + 4)^{2}$$
$23$ $$T^{2}$$
$29$ $$(T + 2)^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2} + 100$$
$41$ $$(T + 10)^{2}$$
$43$ $$T^{2} + 16$$
$47$ $$T^{2} + 64$$
$53$ $$T^{2} + 100$$
$59$ $$(T + 4)^{2}$$
$61$ $$(T + 2)^{2}$$
$67$ $$T^{2} + 144$$
$71$ $$(T - 8)^{2}$$
$73$ $$T^{2} + 100$$
$79$ $$T^{2}$$
$83$ $$T^{2} + 144$$
$89$ $$(T + 6)^{2}$$
$97$ $$T^{2} + 4$$