Properties

 Label 825.2.a.b Level $825$ Weight $2$ Character orbit 825.a Self dual yes Analytic conductor $6.588$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [825,2,Mod(1,825)]

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([0, 0, 0]))

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

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

chi := DirichletCharacter("825.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$825 = 3 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 825.a (trivial)

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: yes Analytic conductor: $$6.58765816676$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - q^{3} - 2 q^{4} + q^{7} + q^{9}+O(q^{10})$$ q - q^3 - 2 * q^4 + q^7 + q^9 $$q - q^{3} - 2 q^{4} + q^{7} + q^{9} - q^{11} + 2 q^{12} + q^{13} + 4 q^{16} + 6 q^{17} - 7 q^{19} - q^{21} - 6 q^{23} - q^{27} - 2 q^{28} - 6 q^{29} - 7 q^{31} + q^{33} - 2 q^{36} - 2 q^{37} - q^{39} - 6 q^{41} + q^{43} + 2 q^{44} - 4 q^{48} - 6 q^{49} - 6 q^{51} - 2 q^{52} + 6 q^{53} + 7 q^{57} + 5 q^{61} + q^{63} - 8 q^{64} - 5 q^{67} - 12 q^{68} + 6 q^{69} - 12 q^{71} - 14 q^{73} + 14 q^{76} - q^{77} - 4 q^{79} + q^{81} + 6 q^{83} + 2 q^{84} + 6 q^{87} + 6 q^{89} + q^{91} + 12 q^{92} + 7 q^{93} - 17 q^{97} - q^{99}+O(q^{100})$$ q - q^3 - 2 * q^4 + q^7 + q^9 - q^11 + 2 * q^12 + q^13 + 4 * q^16 + 6 * q^17 - 7 * q^19 - q^21 - 6 * q^23 - q^27 - 2 * q^28 - 6 * q^29 - 7 * q^31 + q^33 - 2 * q^36 - 2 * q^37 - q^39 - 6 * q^41 + q^43 + 2 * q^44 - 4 * q^48 - 6 * q^49 - 6 * q^51 - 2 * q^52 + 6 * q^53 + 7 * q^57 + 5 * q^61 + q^63 - 8 * q^64 - 5 * q^67 - 12 * q^68 + 6 * q^69 - 12 * q^71 - 14 * q^73 + 14 * q^76 - q^77 - 4 * q^79 + q^81 + 6 * q^83 + 2 * q^84 + 6 * q^87 + 6 * q^89 + q^91 + 12 * q^92 + 7 * q^93 - 17 * q^97 - q^99

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 −1.00000 −2.00000 0 0 1.00000 0 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$+1$$
$$5$$ $$+1$$
$$11$$ $$+1$$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 825.2.a.b 1
3.b odd 2 1 2475.2.a.f 1
5.b even 2 1 825.2.a.c yes 1
5.c odd 4 2 825.2.c.b 2
11.b odd 2 1 9075.2.a.i 1
15.d odd 2 1 2475.2.a.e 1
15.e even 4 2 2475.2.c.h 2
55.d odd 2 1 9075.2.a.l 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.2.a.b 1 1.a even 1 1 trivial
825.2.a.c yes 1 5.b even 2 1
825.2.c.b 2 5.c odd 4 2
2475.2.a.e 1 15.d odd 2 1
2475.2.a.f 1 3.b odd 2 1
2475.2.c.h 2 15.e even 4 2
9075.2.a.i 1 11.b odd 2 1
9075.2.a.l 1 55.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(825))$$:

 $$T_{2}$$ T2 $$T_{7} - 1$$ T7 - 1

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 1$$
$5$ $$T$$
$7$ $$T - 1$$
$11$ $$T + 1$$
$13$ $$T - 1$$
$17$ $$T - 6$$
$19$ $$T + 7$$
$23$ $$T + 6$$
$29$ $$T + 6$$
$31$ $$T + 7$$
$37$ $$T + 2$$
$41$ $$T + 6$$
$43$ $$T - 1$$
$47$ $$T$$
$53$ $$T - 6$$
$59$ $$T$$
$61$ $$T - 5$$
$67$ $$T + 5$$
$71$ $$T + 12$$
$73$ $$T + 14$$
$79$ $$T + 4$$
$83$ $$T - 6$$
$89$ $$T - 6$$
$97$ $$T + 17$$