Properties

 Label 825.4.a.e Level $825$ Weight $4$ Character orbit 825.a Self dual yes Analytic conductor $48.677$ 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] = [825,4,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, 4, names="a")

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

chi := DirichletCharacter("825.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$825 = 3 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ 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: $$48.6765757547$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 165) 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)

 $$f(q)$$ $$=$$ $$q + 3 q^{3} - 8 q^{4} - 2 q^{7} + 9 q^{9}+O(q^{10})$$ q + 3 * q^3 - 8 * q^4 - 2 * q^7 + 9 * q^9 $$q + 3 q^{3} - 8 q^{4} - 2 q^{7} + 9 q^{9} - 11 q^{11} - 24 q^{12} + 22 q^{13} + 64 q^{16} - 72 q^{17} + 122 q^{19} - 6 q^{21} - 72 q^{23} + 27 q^{27} + 16 q^{28} + 96 q^{29} - 112 q^{31} - 33 q^{33} - 72 q^{36} - 266 q^{37} + 66 q^{39} - 96 q^{41} + 382 q^{43} + 88 q^{44} - 360 q^{47} + 192 q^{48} - 339 q^{49} - 216 q^{51} - 176 q^{52} - 318 q^{53} + 366 q^{57} + 660 q^{59} - 430 q^{61} - 18 q^{63} - 512 q^{64} - 380 q^{67} + 576 q^{68} - 216 q^{69} + 168 q^{71} - 218 q^{73} - 976 q^{76} + 22 q^{77} - 706 q^{79} + 81 q^{81} - 1068 q^{83} + 48 q^{84} + 288 q^{87} - 6 q^{89} - 44 q^{91} + 576 q^{92} - 336 q^{93} - 686 q^{97} - 99 q^{99}+O(q^{100})$$ q + 3 * q^3 - 8 * q^4 - 2 * q^7 + 9 * q^9 - 11 * q^11 - 24 * q^12 + 22 * q^13 + 64 * q^16 - 72 * q^17 + 122 * q^19 - 6 * q^21 - 72 * q^23 + 27 * q^27 + 16 * q^28 + 96 * q^29 - 112 * q^31 - 33 * q^33 - 72 * q^36 - 266 * q^37 + 66 * q^39 - 96 * q^41 + 382 * q^43 + 88 * q^44 - 360 * q^47 + 192 * q^48 - 339 * q^49 - 216 * q^51 - 176 * q^52 - 318 * q^53 + 366 * q^57 + 660 * q^59 - 430 * q^61 - 18 * q^63 - 512 * q^64 - 380 * q^67 + 576 * q^68 - 216 * q^69 + 168 * q^71 - 218 * q^73 - 976 * q^76 + 22 * q^77 - 706 * q^79 + 81 * q^81 - 1068 * q^83 + 48 * q^84 + 288 * q^87 - 6 * q^89 - 44 * q^91 + 576 * q^92 - 336 * q^93 - 686 * q^97 - 99 * 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 3.00000 −8.00000 0 0 −2.00000 0 9.00000 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 825.4.a.e 1
3.b odd 2 1 2475.4.a.f 1
5.b even 2 1 165.4.a.a 1
5.c odd 4 2 825.4.c.g 2
15.d odd 2 1 495.4.a.c 1
55.d odd 2 1 1815.4.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.4.a.a 1 5.b even 2 1
495.4.a.c 1 15.d odd 2 1
825.4.a.e 1 1.a even 1 1 trivial
825.4.c.g 2 5.c odd 4 2
1815.4.a.f 1 55.d odd 2 1
2475.4.a.f 1 3.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 $$S_{4}^{\mathrm{new}}(\Gamma_0(825))$$:

 $$T_{2}$$ T2 $$T_{7} + 2$$ T7 + 2

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 3$$
$5$ $$T$$
$7$ $$T + 2$$
$11$ $$T + 11$$
$13$ $$T - 22$$
$17$ $$T + 72$$
$19$ $$T - 122$$
$23$ $$T + 72$$
$29$ $$T - 96$$
$31$ $$T + 112$$
$37$ $$T + 266$$
$41$ $$T + 96$$
$43$ $$T - 382$$
$47$ $$T + 360$$
$53$ $$T + 318$$
$59$ $$T - 660$$
$61$ $$T + 430$$
$67$ $$T + 380$$
$71$ $$T - 168$$
$73$ $$T + 218$$
$79$ $$T + 706$$
$83$ $$T + 1068$$
$89$ $$T + 6$$
$97$ $$T + 686$$