# Properties

 Label 1225.2.b.j Level $1225$ Weight $2$ Character orbit 1225.b Analytic conductor $9.782$ Analytic rank $0$ Dimension $4$ CM no 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: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ 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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} + ( - \beta_{2} - \beta_1) q^{3} + (\beta_{3} + 2) q^{6} + 2 \beta_{2} q^{8} - 2 \beta_{3} q^{9}+O(q^{10})$$ q + b2 * q^2 + (-b2 - b1) * q^3 + (b3 + 2) * q^6 + 2*b2 * q^8 - 2*b3 * q^9 $$q + \beta_{2} q^{2} + ( - \beta_{2} - \beta_1) q^{3} + (\beta_{3} + 2) q^{6} + 2 \beta_{2} q^{8} - 2 \beta_{3} q^{9} + ( - 2 \beta_{3} - 3) q^{11} + (\beta_{2} - 3 \beta_1) q^{13} - 4 q^{16} + ( - 3 \beta_{2} - \beta_1) q^{17} - 4 \beta_1 q^{18} - 6 q^{19} + ( - 3 \beta_{2} - 4 \beta_1) q^{22} + (\beta_{2} - 6 \beta_1) q^{23} + (2 \beta_{3} + 4) q^{24} + (3 \beta_{3} - 2) q^{26} + ( - \beta_{2} + \beta_1) q^{27} + ( - 4 \beta_{3} + 3) q^{29} + ( - 3 \beta_{3} + 6) q^{31} + (5 \beta_{2} + 7 \beta_1) q^{33} + (\beta_{3} + 6) q^{34} + ( - 3 \beta_{2} - 2 \beta_1) q^{37} - 6 \beta_{2} q^{38} + ( - 2 \beta_{3} - 1) q^{39} + (3 \beta_{3} - 2) q^{41} - 2 \beta_1 q^{43} + (6 \beta_{3} - 2) q^{46} + (3 \beta_{2} - 3 \beta_1) q^{47} + (4 \beta_{2} + 4 \beta_1) q^{48} + ( - 4 \beta_{3} - 7) q^{51} + 3 \beta_{2} q^{53} + ( - \beta_{3} + 2) q^{54} + (6 \beta_{2} + 6 \beta_1) q^{57} + (3 \beta_{2} - 8 \beta_1) q^{58} + ( - 3 \beta_{3} - 2) q^{59} + 2 \beta_{3} q^{61} + (6 \beta_{2} - 6 \beta_1) q^{62} - 8 q^{64} + ( - 7 \beta_{3} - 10) q^{66} + (3 \beta_{2} - 4 \beta_1) q^{67} + ( - 5 \beta_{3} - 4) q^{69} + ( - 2 \beta_{3} - 6) q^{71} - 8 \beta_1 q^{72} - 6 \beta_{2} q^{73} + (2 \beta_{3} + 6) q^{74} + ( - \beta_{2} - 4 \beta_1) q^{78} + (6 \beta_{3} + 7) q^{79} + ( - 6 \beta_{3} - 1) q^{81} + ( - 2 \beta_{2} + 6 \beta_1) q^{82} + 2 \beta_{3} q^{86} + (\beta_{2} + 5 \beta_1) q^{87} + ( - 6 \beta_{2} - 8 \beta_1) q^{88} + 8 q^{89} - 3 \beta_{2} q^{93} + (3 \beta_{3} - 6) q^{94} + ( - 3 \beta_{2} + 9 \beta_1) q^{97} + (6 \beta_{3} + 8) q^{99}+O(q^{100})$$ q + b2 * q^2 + (-b2 - b1) * q^3 + (b3 + 2) * q^6 + 2*b2 * q^8 - 2*b3 * q^9 + (-2*b3 - 3) * q^11 + (b2 - 3*b1) * q^13 - 4 * q^16 + (-3*b2 - b1) * q^17 - 4*b1 * q^18 - 6 * q^19 + (-3*b2 - 4*b1) * q^22 + (b2 - 6*b1) * q^23 + (2*b3 + 4) * q^24 + (3*b3 - 2) * q^26 + (-b2 + b1) * q^27 + (-4*b3 + 3) * q^29 + (-3*b3 + 6) * q^31 + (5*b2 + 7*b1) * q^33 + (b3 + 6) * q^34 + (-3*b2 - 2*b1) * q^37 - 6*b2 * q^38 + (-2*b3 - 1) * q^39 + (3*b3 - 2) * q^41 - 2*b1 * q^43 + (6*b3 - 2) * q^46 + (3*b2 - 3*b1) * q^47 + (4*b2 + 4*b1) * q^48 + (-4*b3 - 7) * q^51 + 3*b2 * q^53 + (-b3 + 2) * q^54 + (6*b2 + 6*b1) * q^57 + (3*b2 - 8*b1) * q^58 + (-3*b3 - 2) * q^59 + 2*b3 * q^61 + (6*b2 - 6*b1) * q^62 - 8 * q^64 + (-7*b3 - 10) * q^66 + (3*b2 - 4*b1) * q^67 + (-5*b3 - 4) * q^69 + (-2*b3 - 6) * q^71 - 8*b1 * q^72 - 6*b2 * q^73 + (2*b3 + 6) * q^74 + (-b2 - 4*b1) * q^78 + (6*b3 + 7) * q^79 + (-6*b3 - 1) * q^81 + (-2*b2 + 6*b1) * q^82 + 2*b3 * q^86 + (b2 + 5*b1) * q^87 + (-6*b2 - 8*b1) * q^88 + 8 * q^89 - 3*b2 * q^93 + (3*b3 - 6) * q^94 + (-3*b2 + 9*b1) * q^97 + (6*b3 + 8) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 8 q^{6}+O(q^{10})$$ 4 * q + 8 * q^6 $$4 q + 8 q^{6} - 12 q^{11} - 16 q^{16} - 24 q^{19} + 16 q^{24} - 8 q^{26} + 12 q^{29} + 24 q^{31} + 24 q^{34} - 4 q^{39} - 8 q^{41} - 8 q^{46} - 28 q^{51} + 8 q^{54} - 8 q^{59} - 32 q^{64} - 40 q^{66} - 16 q^{69} - 24 q^{71} + 24 q^{74} + 28 q^{79} - 4 q^{81} + 32 q^{89} - 24 q^{94} + 32 q^{99}+O(q^{100})$$ 4 * q + 8 * q^6 - 12 * q^11 - 16 * q^16 - 24 * q^19 + 16 * q^24 - 8 * q^26 + 12 * q^29 + 24 * q^31 + 24 * q^34 - 4 * q^39 - 8 * q^41 - 8 * q^46 - 28 * q^51 + 8 * q^54 - 8 * q^59 - 32 * q^64 - 40 * q^66 - 16 * q^69 - 24 * q^71 + 24 * q^74 + 28 * q^79 - 4 * q^81 + 32 * q^89 - 24 * q^94 + 32 * q^99

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{8}^{2}$$ v^2 $$\beta_{2}$$ $$=$$ $$\zeta_{8}^{3} + \zeta_{8}$$ v^3 + v $$\beta_{3}$$ $$=$$ $$-\zeta_{8}^{3} + \zeta_{8}$$ -v^3 + v
 $$\zeta_{8}$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 2$$ (b3 + b2) / 2 $$\zeta_{8}^{2}$$ $$=$$ $$\beta_1$$ b1 $$\zeta_{8}^{3}$$ $$=$$ $$( -\beta_{3} + \beta_{2} ) / 2$$ (-b3 + b2) / 2

## 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
 −0.707107 − 0.707107i 0.707107 − 0.707107i 0.707107 + 0.707107i −0.707107 + 0.707107i
1.41421i 0.414214i 0 0 0.585786 0 2.82843i 2.82843 0
99.2 1.41421i 2.41421i 0 0 3.41421 0 2.82843i −2.82843 0
99.3 1.41421i 2.41421i 0 0 3.41421 0 2.82843i −2.82843 0
99.4 1.41421i 0.414214i 0 0 0.585786 0 2.82843i 2.82843 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.j 4
5.b even 2 1 inner 1225.2.b.j 4
5.c odd 4 1 245.2.a.f yes 2
5.c odd 4 1 1225.2.a.p 2
7.b odd 2 1 1225.2.b.i 4
15.e even 4 1 2205.2.a.t 2
20.e even 4 1 3920.2.a.br 2
35.c odd 2 1 1225.2.b.i 4
35.f even 4 1 245.2.a.e 2
35.f even 4 1 1225.2.a.r 2
35.k even 12 2 245.2.e.g 4
35.l odd 12 2 245.2.e.f 4
105.k odd 4 1 2205.2.a.v 2
140.j odd 4 1 3920.2.a.bw 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
245.2.a.e 2 35.f even 4 1
245.2.a.f yes 2 5.c odd 4 1
245.2.e.f 4 35.l odd 12 2
245.2.e.g 4 35.k even 12 2
1225.2.a.p 2 5.c odd 4 1
1225.2.a.r 2 35.f even 4 1
1225.2.b.i 4 7.b odd 2 1
1225.2.b.i 4 35.c odd 2 1
1225.2.b.j 4 1.a even 1 1 trivial
1225.2.b.j 4 5.b even 2 1 inner
2205.2.a.t 2 15.e even 4 1
2205.2.a.v 2 105.k odd 4 1
3920.2.a.br 2 20.e even 4 1
3920.2.a.bw 2 140.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 $$S_{2}^{\mathrm{new}}(1225, [\chi])$$:

 $$T_{2}^{2} + 2$$ T2^2 + 2 $$T_{19} + 6$$ T19 + 6 $$T_{31}^{2} - 12T_{31} + 18$$ T31^2 - 12*T31 + 18

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 2)^{2}$$
$3$ $$T^{4} + 6T^{2} + 1$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$(T^{2} + 6 T + 1)^{2}$$
$13$ $$T^{4} + 22T^{2} + 49$$
$17$ $$T^{4} + 38T^{2} + 289$$
$19$ $$(T + 6)^{4}$$
$23$ $$T^{4} + 76T^{2} + 1156$$
$29$ $$(T^{2} - 6 T - 23)^{2}$$
$31$ $$(T^{2} - 12 T + 18)^{2}$$
$37$ $$T^{4} + 44T^{2} + 196$$
$41$ $$(T^{2} + 4 T - 14)^{2}$$
$43$ $$(T^{2} + 4)^{2}$$
$47$ $$T^{4} + 54T^{2} + 81$$
$53$ $$(T^{2} + 18)^{2}$$
$59$ $$(T^{2} + 4 T - 14)^{2}$$
$61$ $$(T^{2} - 8)^{2}$$
$67$ $$T^{4} + 68T^{2} + 4$$
$71$ $$(T^{2} + 12 T + 28)^{2}$$
$73$ $$(T^{2} + 72)^{2}$$
$79$ $$(T^{2} - 14 T - 23)^{2}$$
$83$ $$T^{4}$$
$89$ $$(T - 8)^{4}$$
$97$ $$T^{4} + 198T^{2} + 3969$$