# Properties

 Label 2925.2.a.d Level $2925$ Weight $2$ Character orbit 2925.a Self dual yes Analytic conductor $23.356$ Analytic rank $0$ 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] = [2925,2,Mod(1,2925)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2925, 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("2925.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2925 = 3^{2} \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2925.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: $$23.3562425912$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 195) 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)

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{4} + 3 q^{8}+O(q^{10})$$ q - q^2 - q^4 + 3 * q^8 $$q - q^{2} - q^{4} + 3 q^{8} - 4 q^{11} - q^{13} - q^{16} + 2 q^{17} - 4 q^{19} + 4 q^{22} + 8 q^{23} + q^{26} + 2 q^{29} - 8 q^{31} - 5 q^{32} - 2 q^{34} - 6 q^{37} + 4 q^{38} + 6 q^{41} + 4 q^{43} + 4 q^{44} - 8 q^{46} - 8 q^{47} - 7 q^{49} + q^{52} + 6 q^{53} - 2 q^{58} + 12 q^{59} - 2 q^{61} + 8 q^{62} + 7 q^{64} + 4 q^{67} - 2 q^{68} + 6 q^{73} + 6 q^{74} + 4 q^{76} + 16 q^{79} - 6 q^{82} - 4 q^{83} - 4 q^{86} - 12 q^{88} - 10 q^{89} - 8 q^{92} + 8 q^{94} - 18 q^{97} + 7 q^{98}+O(q^{100})$$ q - q^2 - q^4 + 3 * q^8 - 4 * q^11 - q^13 - q^16 + 2 * q^17 - 4 * q^19 + 4 * q^22 + 8 * q^23 + q^26 + 2 * q^29 - 8 * q^31 - 5 * q^32 - 2 * q^34 - 6 * q^37 + 4 * q^38 + 6 * q^41 + 4 * q^43 + 4 * q^44 - 8 * q^46 - 8 * q^47 - 7 * q^49 + q^52 + 6 * q^53 - 2 * q^58 + 12 * q^59 - 2 * q^61 + 8 * q^62 + 7 * q^64 + 4 * q^67 - 2 * q^68 + 6 * q^73 + 6 * q^74 + 4 * q^76 + 16 * q^79 - 6 * q^82 - 4 * q^83 - 4 * q^86 - 12 * q^88 - 10 * q^89 - 8 * q^92 + 8 * q^94 - 18 * q^97 + 7 * q^98

## 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
−1.00000 0 −1.00000 0 0 0 3.00000 0 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
$$3$$ $$-1$$
$$5$$ $$1$$
$$13$$ $$1$$

## 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 2925.2.a.d 1
3.b odd 2 1 975.2.a.i 1
5.b even 2 1 585.2.a.g 1
5.c odd 4 2 2925.2.c.f 2
15.d odd 2 1 195.2.a.a 1
15.e even 4 2 975.2.c.e 2
20.d odd 2 1 9360.2.a.o 1
60.h even 2 1 3120.2.a.k 1
65.d even 2 1 7605.2.a.h 1
105.g even 2 1 9555.2.a.b 1
195.e odd 2 1 2535.2.a.k 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.a.a 1 15.d odd 2 1
585.2.a.g 1 5.b even 2 1
975.2.a.i 1 3.b odd 2 1
975.2.c.e 2 15.e even 4 2
2535.2.a.k 1 195.e odd 2 1
2925.2.a.d 1 1.a even 1 1 trivial
2925.2.c.f 2 5.c odd 4 2
3120.2.a.k 1 60.h even 2 1
7605.2.a.h 1 65.d even 2 1
9360.2.a.o 1 20.d odd 2 1
9555.2.a.b 1 105.g even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2925))$$:

 $$T_{2} + 1$$ T2 + 1 $$T_{7}$$ T7 $$T_{11} + 4$$ T11 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T$$
$11$ $$T + 4$$
$13$ $$T + 1$$
$17$ $$T - 2$$
$19$ $$T + 4$$
$23$ $$T - 8$$
$29$ $$T - 2$$
$31$ $$T + 8$$
$37$ $$T + 6$$
$41$ $$T - 6$$
$43$ $$T - 4$$
$47$ $$T + 8$$
$53$ $$T - 6$$
$59$ $$T - 12$$
$61$ $$T + 2$$
$67$ $$T - 4$$
$71$ $$T$$
$73$ $$T - 6$$
$79$ $$T - 16$$
$83$ $$T + 4$$
$89$ $$T + 10$$
$97$ $$T + 18$$