# Properties

 Label 825.2.d.a Level $825$ Weight $2$ Character orbit 825.d Analytic conductor $6.588$ Analytic rank $0$ Dimension $4$ CM discriminant -11 Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [825,2,Mod(824,825)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(825, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("825.824");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$825 = 3 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 825.d (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: $$6.58765816676$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{11})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 5x^{2} + 9$$ x^4 - 5*x^2 + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{23}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 33) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} - \beta_1) q^{3} + 2 q^{4} + ( - \beta_{3} + 2) q^{9}+O(q^{10})$$ q + (b2 - b1) * q^3 + 2 * q^4 + (-b3 + 2) * q^9 $$q + (\beta_{2} - \beta_1) q^{3} + 2 q^{4} + ( - \beta_{3} + 2) q^{9} + (2 \beta_{3} + 1) q^{11} + (2 \beta_{2} - 2 \beta_1) q^{12} + 4 q^{16} + ( - \beta_{2} + 2 \beta_1) q^{23} + (5 \beta_{2} - 2 \beta_1) q^{27} + 5 q^{31} + ( - 5 \beta_{2} - \beta_1) q^{33} + ( - 2 \beta_{3} + 4) q^{36} + 7 \beta_{2} q^{37} + (4 \beta_{3} + 2) q^{44} + ( - 2 \beta_{2} + 4 \beta_1) q^{47} + (4 \beta_{2} - 4 \beta_1) q^{48} - 7 q^{49} + ( - 4 \beta_{2} + 8 \beta_1) q^{53} + (2 \beta_{3} + 1) q^{59} + 8 q^{64} + 13 \beta_{2} q^{67} + (\beta_{3} - 5) q^{69} + (10 \beta_{3} + 5) q^{71} + ( - 5 \beta_{3} + 1) q^{81} + ( - 10 \beta_{3} - 5) q^{89} + ( - 2 \beta_{2} + 4 \beta_1) q^{92} + (5 \beta_{2} - 5 \beta_1) q^{93} - 17 \beta_{2} q^{97} + (5 \beta_{3} + 8) q^{99}+O(q^{100})$$ q + (b2 - b1) * q^3 + 2 * q^4 + (-b3 + 2) * q^9 + (2*b3 + 1) * q^11 + (2*b2 - 2*b1) * q^12 + 4 * q^16 + (-b2 + 2*b1) * q^23 + (5*b2 - 2*b1) * q^27 + 5 * q^31 + (-5*b2 - b1) * q^33 + (-2*b3 + 4) * q^36 + 7*b2 * q^37 + (4*b3 + 2) * q^44 + (-2*b2 + 4*b1) * q^47 + (4*b2 - 4*b1) * q^48 - 7 * q^49 + (-4*b2 + 8*b1) * q^53 + (2*b3 + 1) * q^59 + 8 * q^64 + 13*b2 * q^67 + (b3 - 5) * q^69 + (10*b3 + 5) * q^71 + (-5*b3 + 1) * q^81 + (-10*b3 - 5) * q^89 + (-2*b2 + 4*b1) * q^92 + (5*b2 - 5*b1) * q^93 - 17*b2 * q^97 + (5*b3 + 8) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 8 q^{4} + 10 q^{9}+O(q^{10})$$ 4 * q + 8 * q^4 + 10 * q^9 $$4 q + 8 q^{4} + 10 q^{9} + 16 q^{16} + 20 q^{31} + 20 q^{36} - 28 q^{49} + 32 q^{64} - 22 q^{69} + 14 q^{81} + 22 q^{99}+O(q^{100})$$ 4 * q + 8 * q^4 + 10 * q^9 + 16 * q^16 + 20 * q^31 + 20 * q^36 - 28 * q^49 + 32 * q^64 - 22 * q^69 + 14 * q^81 + 22 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 5x^{2} + 9$$ :

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/825\mathbb{Z}\right)^\times$$.

 $$n$$ $$376$$ $$551$$ $$727$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
824.1
 1.65831 − 0.500000i 1.65831 + 0.500000i −1.65831 − 0.500000i −1.65831 + 0.500000i
0 −1.65831 0.500000i 2.00000 0 0 0 0 2.50000 + 1.65831i 0
824.2 0 −1.65831 + 0.500000i 2.00000 0 0 0 0 2.50000 1.65831i 0
824.3 0 1.65831 0.500000i 2.00000 0 0 0 0 2.50000 1.65831i 0
824.4 0 1.65831 + 0.500000i 2.00000 0 0 0 0 2.50000 + 1.65831i 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
11.b odd 2 1 CM by $$\Q(\sqrt{-11})$$
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner
33.d even 2 1 inner
55.d odd 2 1 inner
165.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 825.2.d.a 4
3.b odd 2 1 inner 825.2.d.a 4
5.b even 2 1 inner 825.2.d.a 4
5.c odd 4 1 33.2.d.a 2
5.c odd 4 1 825.2.f.a 2
11.b odd 2 1 CM 825.2.d.a 4
15.d odd 2 1 inner 825.2.d.a 4
15.e even 4 1 33.2.d.a 2
15.e even 4 1 825.2.f.a 2
20.e even 4 1 528.2.b.a 2
33.d even 2 1 inner 825.2.d.a 4
40.i odd 4 1 2112.2.b.e 2
40.k even 4 1 2112.2.b.f 2
45.k odd 12 2 891.2.g.a 4
45.l even 12 2 891.2.g.a 4
55.d odd 2 1 inner 825.2.d.a 4
55.e even 4 1 33.2.d.a 2
55.e even 4 1 825.2.f.a 2
55.k odd 20 4 363.2.f.c 8
55.l even 20 4 363.2.f.c 8
60.l odd 4 1 528.2.b.a 2
120.q odd 4 1 2112.2.b.f 2
120.w even 4 1 2112.2.b.e 2
165.d even 2 1 inner 825.2.d.a 4
165.l odd 4 1 33.2.d.a 2
165.l odd 4 1 825.2.f.a 2
165.u odd 20 4 363.2.f.c 8
165.v even 20 4 363.2.f.c 8
220.i odd 4 1 528.2.b.a 2
440.t even 4 1 2112.2.b.e 2
440.w odd 4 1 2112.2.b.f 2
495.bd odd 12 2 891.2.g.a 4
495.bf even 12 2 891.2.g.a 4
660.q even 4 1 528.2.b.a 2
1320.bn odd 4 1 2112.2.b.e 2
1320.bt even 4 1 2112.2.b.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.2.d.a 2 5.c odd 4 1
33.2.d.a 2 15.e even 4 1
33.2.d.a 2 55.e even 4 1
33.2.d.a 2 165.l odd 4 1
363.2.f.c 8 55.k odd 20 4
363.2.f.c 8 55.l even 20 4
363.2.f.c 8 165.u odd 20 4
363.2.f.c 8 165.v even 20 4
528.2.b.a 2 20.e even 4 1
528.2.b.a 2 60.l odd 4 1
528.2.b.a 2 220.i odd 4 1
528.2.b.a 2 660.q even 4 1
825.2.d.a 4 1.a even 1 1 trivial
825.2.d.a 4 3.b odd 2 1 inner
825.2.d.a 4 5.b even 2 1 inner
825.2.d.a 4 11.b odd 2 1 CM
825.2.d.a 4 15.d odd 2 1 inner
825.2.d.a 4 33.d even 2 1 inner
825.2.d.a 4 55.d odd 2 1 inner
825.2.d.a 4 165.d even 2 1 inner
825.2.f.a 2 5.c odd 4 1
825.2.f.a 2 15.e even 4 1
825.2.f.a 2 55.e even 4 1
825.2.f.a 2 165.l odd 4 1
891.2.g.a 4 45.k odd 12 2
891.2.g.a 4 45.l even 12 2
891.2.g.a 4 495.bd odd 12 2
891.2.g.a 4 495.bf even 12 2
2112.2.b.e 2 40.i odd 4 1
2112.2.b.e 2 120.w even 4 1
2112.2.b.e 2 440.t even 4 1
2112.2.b.e 2 1320.bn odd 4 1
2112.2.b.f 2 40.k even 4 1
2112.2.b.f 2 120.q odd 4 1
2112.2.b.f 2 440.w odd 4 1
2112.2.b.f 2 1320.bt even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(825, [\chi])$$:

 $$T_{2}$$ T2 $$T_{7}$$ T7 $$T_{23}^{2} - 11$$ T23^2 - 11

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} - 5T^{2} + 9$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$(T^{2} + 11)^{2}$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$T^{4}$$
$23$ $$(T^{2} - 11)^{2}$$
$29$ $$T^{4}$$
$31$ $$(T - 5)^{4}$$
$37$ $$(T^{2} + 49)^{2}$$
$41$ $$T^{4}$$
$43$ $$T^{4}$$
$47$ $$(T^{2} - 44)^{2}$$
$53$ $$(T^{2} - 176)^{2}$$
$59$ $$(T^{2} + 11)^{2}$$
$61$ $$T^{4}$$
$67$ $$(T^{2} + 169)^{2}$$
$71$ $$(T^{2} + 275)^{2}$$
$73$ $$T^{4}$$
$79$ $$T^{4}$$
$83$ $$T^{4}$$
$89$ $$(T^{2} + 275)^{2}$$
$97$ $$(T^{2} + 289)^{2}$$