# Properties

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

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [825,2,Mod(626,825)]

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, 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("825.626");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$825 = 3 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 825.f (of order $$2$$, degree $$1$$, 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: $$2$$ Coefficient field: $$\Q(\sqrt{-11})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 3$$ x^2 - x + 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{-11})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta q^{3} - 2 q^{4} + (\beta - 3) q^{9} +O(q^{10})$$ q - b * q^3 - 2 * q^4 + (b - 3) * q^9 $$q - \beta q^{3} - 2 q^{4} + (\beta - 3) q^{9} + (2 \beta - 1) q^{11} + 2 \beta q^{12} + 4 q^{16} + (2 \beta - 1) q^{23} + (2 \beta + 3) q^{27} + 5 q^{31} + ( - \beta + 6) q^{33} + ( - 2 \beta + 6) q^{36} + 7 q^{37} + ( - 4 \beta + 2) q^{44} + ( - 4 \beta + 2) q^{47} - 4 \beta q^{48} + 7 q^{49} + (8 \beta - 4) q^{53} + ( - 2 \beta + 1) q^{59} - 8 q^{64} + 13 q^{67} + ( - \beta + 6) q^{69} + (10 \beta - 5) q^{71} + ( - 5 \beta + 6) q^{81} + (10 \beta - 5) q^{89} + ( - 4 \beta + 2) q^{92} - 5 \beta q^{93} - 17 q^{97} + ( - 5 \beta - 3) q^{99} +O(q^{100})$$ q - b * q^3 - 2 * q^4 + (b - 3) * q^9 + (2*b - 1) * q^11 + 2*b * q^12 + 4 * q^16 + (2*b - 1) * q^23 + (2*b + 3) * q^27 + 5 * q^31 + (-b + 6) * q^33 + (-2*b + 6) * q^36 + 7 * q^37 + (-4*b + 2) * q^44 + (-4*b + 2) * q^47 - 4*b * q^48 + 7 * q^49 + (8*b - 4) * q^53 + (-2*b + 1) * q^59 - 8 * q^64 + 13 * q^67 + (-b + 6) * q^69 + (10*b - 5) * q^71 + (-5*b + 6) * q^81 + (10*b - 5) * q^89 + (-4*b + 2) * q^92 - 5*b * q^93 - 17 * q^97 + (-5*b - 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{3} - 4 q^{4} - 5 q^{9}+O(q^{10})$$ 2 * q - q^3 - 4 * q^4 - 5 * q^9 $$2 q - q^{3} - 4 q^{4} - 5 q^{9} + 2 q^{12} + 8 q^{16} + 8 q^{27} + 10 q^{31} + 11 q^{33} + 10 q^{36} + 14 q^{37} - 4 q^{48} + 14 q^{49} - 16 q^{64} + 26 q^{67} + 11 q^{69} + 7 q^{81} - 5 q^{93} - 34 q^{97} - 11 q^{99}+O(q^{100})$$ 2 * q - q^3 - 4 * q^4 - 5 * q^9 + 2 * q^12 + 8 * q^16 + 8 * q^27 + 10 * q^31 + 11 * q^33 + 10 * q^36 + 14 * q^37 - 4 * q^48 + 14 * q^49 - 16 * q^64 + 26 * q^67 + 11 * q^69 + 7 * q^81 - 5 * q^93 - 34 * q^97 - 11 * q^99

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
626.1
 0.5 + 1.65831i 0.5 − 1.65831i
0 −0.500000 1.65831i −2.00000 0 0 0 0 −2.50000 + 1.65831i 0
626.2 0 −0.500000 + 1.65831i −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
33.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.f.a 2
3.b odd 2 1 inner 825.2.f.a 2
5.b even 2 1 33.2.d.a 2
5.c odd 4 2 825.2.d.a 4
11.b odd 2 1 CM 825.2.f.a 2
15.d odd 2 1 33.2.d.a 2
15.e even 4 2 825.2.d.a 4
20.d odd 2 1 528.2.b.a 2
33.d even 2 1 inner 825.2.f.a 2
40.e odd 2 1 2112.2.b.f 2
40.f even 2 1 2112.2.b.e 2
45.h odd 6 2 891.2.g.a 4
45.j even 6 2 891.2.g.a 4
55.d odd 2 1 33.2.d.a 2
55.e even 4 2 825.2.d.a 4
55.h odd 10 4 363.2.f.c 8
55.j even 10 4 363.2.f.c 8
60.h even 2 1 528.2.b.a 2
120.i odd 2 1 2112.2.b.e 2
120.m even 2 1 2112.2.b.f 2
165.d even 2 1 33.2.d.a 2
165.l odd 4 2 825.2.d.a 4
165.o odd 10 4 363.2.f.c 8
165.r even 10 4 363.2.f.c 8
220.g even 2 1 528.2.b.a 2
440.c even 2 1 2112.2.b.f 2
440.o odd 2 1 2112.2.b.e 2
495.o odd 6 2 891.2.g.a 4
495.r even 6 2 891.2.g.a 4
660.g odd 2 1 528.2.b.a 2
1320.b odd 2 1 2112.2.b.f 2
1320.u even 2 1 2112.2.b.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.2.d.a 2 5.b even 2 1
33.2.d.a 2 15.d odd 2 1
33.2.d.a 2 55.d odd 2 1
33.2.d.a 2 165.d even 2 1
363.2.f.c 8 55.h odd 10 4
363.2.f.c 8 55.j even 10 4
363.2.f.c 8 165.o odd 10 4
363.2.f.c 8 165.r even 10 4
528.2.b.a 2 20.d odd 2 1
528.2.b.a 2 60.h even 2 1
528.2.b.a 2 220.g even 2 1
528.2.b.a 2 660.g odd 2 1
825.2.d.a 4 5.c odd 4 2
825.2.d.a 4 15.e even 4 2
825.2.d.a 4 55.e even 4 2
825.2.d.a 4 165.l odd 4 2
825.2.f.a 2 1.a even 1 1 trivial
825.2.f.a 2 3.b odd 2 1 inner
825.2.f.a 2 11.b odd 2 1 CM
825.2.f.a 2 33.d even 2 1 inner
891.2.g.a 4 45.h odd 6 2
891.2.g.a 4 45.j even 6 2
891.2.g.a 4 495.o odd 6 2
891.2.g.a 4 495.r even 6 2
2112.2.b.e 2 40.f even 2 1
2112.2.b.e 2 120.i odd 2 1
2112.2.b.e 2 440.o odd 2 1
2112.2.b.e 2 1320.u even 2 1
2112.2.b.f 2 40.e odd 2 1
2112.2.b.f 2 120.m even 2 1
2112.2.b.f 2 440.c even 2 1
2112.2.b.f 2 1320.b odd 2 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 $$T_{23}^{2} + 11$$ T23^2 + 11 $$T_{37} - 7$$ T37 - 7

## Hecke characteristic polynomials

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