# Properties

 Label 1225.2.b.a Level $1225$ Weight $2$ Character orbit 1225.b Analytic conductor $9.782$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1225,2,Mod(99,1225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1225, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1225.99");

S:= CuspForms(chi, 2);

N := Newforms(S);

 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

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: $$9.78167424761$$ 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 245) 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.

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

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
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.a 2
5.b even 2 1 inner 1225.2.b.a 2
5.c odd 4 1 245.2.a.b yes 1
5.c odd 4 1 1225.2.a.h 1
7.b odd 2 1 1225.2.b.b 2
15.e even 4 1 2205.2.a.l 1
20.e even 4 1 3920.2.a.a 1
35.c odd 2 1 1225.2.b.b 2
35.f even 4 1 245.2.a.a 1
35.f even 4 1 1225.2.a.j 1
35.k even 12 2 245.2.e.d 2
35.l odd 12 2 245.2.e.c 2
105.k odd 4 1 2205.2.a.j 1
140.j odd 4 1 3920.2.a.bj 1

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

## Hecke kernels

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

 $$T_{2}^{2} + 4$$ T2^2 + 4 $$T_{19} + 6$$ T19 + 6 $$T_{31} - 6$$ T31 - 6

## Hecke characteristic polynomials

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