# Properties

 Label 925.2.a.e Level $925$ Weight $2$ Character orbit 925.a Self dual yes Analytic conductor $7.386$ 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] = [925,2,Mod(1,925)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(925, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("925.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$925 = 5^{2} \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 925.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: $$7.38616218697$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 37) 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)

 $$f(q)$$ $$=$$ $$q + 2 q^{2} + 3 q^{3} + 2 q^{4} + 6 q^{6} + q^{7} + 6 q^{9}+O(q^{10})$$ q + 2 * q^2 + 3 * q^3 + 2 * q^4 + 6 * q^6 + q^7 + 6 * q^9 $$q + 2 q^{2} + 3 q^{3} + 2 q^{4} + 6 q^{6} + q^{7} + 6 q^{9} - 5 q^{11} + 6 q^{12} + 2 q^{13} + 2 q^{14} - 4 q^{16} + 12 q^{18} + 3 q^{21} - 10 q^{22} - 2 q^{23} + 4 q^{26} + 9 q^{27} + 2 q^{28} + 6 q^{29} - 4 q^{31} - 8 q^{32} - 15 q^{33} + 12 q^{36} + q^{37} + 6 q^{39} - 9 q^{41} + 6 q^{42} - 2 q^{43} - 10 q^{44} - 4 q^{46} + 9 q^{47} - 12 q^{48} - 6 q^{49} + 4 q^{52} - q^{53} + 18 q^{54} + 12 q^{58} + 8 q^{59} - 8 q^{61} - 8 q^{62} + 6 q^{63} - 8 q^{64} - 30 q^{66} - 8 q^{67} - 6 q^{69} + 9 q^{71} + q^{73} + 2 q^{74} - 5 q^{77} + 12 q^{78} + 4 q^{79} + 9 q^{81} - 18 q^{82} + 15 q^{83} + 6 q^{84} - 4 q^{86} + 18 q^{87} + 4 q^{89} + 2 q^{91} - 4 q^{92} - 12 q^{93} + 18 q^{94} - 24 q^{96} - 4 q^{97} - 12 q^{98} - 30 q^{99}+O(q^{100})$$ q + 2 * q^2 + 3 * q^3 + 2 * q^4 + 6 * q^6 + q^7 + 6 * q^9 - 5 * q^11 + 6 * q^12 + 2 * q^13 + 2 * q^14 - 4 * q^16 + 12 * q^18 + 3 * q^21 - 10 * q^22 - 2 * q^23 + 4 * q^26 + 9 * q^27 + 2 * q^28 + 6 * q^29 - 4 * q^31 - 8 * q^32 - 15 * q^33 + 12 * q^36 + q^37 + 6 * q^39 - 9 * q^41 + 6 * q^42 - 2 * q^43 - 10 * q^44 - 4 * q^46 + 9 * q^47 - 12 * q^48 - 6 * q^49 + 4 * q^52 - q^53 + 18 * q^54 + 12 * q^58 + 8 * q^59 - 8 * q^61 - 8 * q^62 + 6 * q^63 - 8 * q^64 - 30 * q^66 - 8 * q^67 - 6 * q^69 + 9 * q^71 + q^73 + 2 * q^74 - 5 * q^77 + 12 * q^78 + 4 * q^79 + 9 * q^81 - 18 * q^82 + 15 * q^83 + 6 * q^84 - 4 * q^86 + 18 * q^87 + 4 * q^89 + 2 * q^91 - 4 * q^92 - 12 * q^93 + 18 * q^94 - 24 * q^96 - 4 * q^97 - 12 * q^98 - 30 * 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
2.00000 3.00000 2.00000 0 6.00000 1.00000 0 6.00000 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
$$5$$ $$+1$$
$$37$$ $$-1$$

## 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 925.2.a.e 1
3.b odd 2 1 8325.2.a.e 1
5.b even 2 1 37.2.a.a 1
5.c odd 4 2 925.2.b.b 2
15.d odd 2 1 333.2.a.d 1
20.d odd 2 1 592.2.a.e 1
35.c odd 2 1 1813.2.a.a 1
40.e odd 2 1 2368.2.a.b 1
40.f even 2 1 2368.2.a.q 1
55.d odd 2 1 4477.2.a.b 1
60.h even 2 1 5328.2.a.r 1
65.d even 2 1 6253.2.a.c 1
185.d even 2 1 1369.2.a.e 1
185.j odd 4 2 1369.2.b.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.2.a.a 1 5.b even 2 1
333.2.a.d 1 15.d odd 2 1
592.2.a.e 1 20.d odd 2 1
925.2.a.e 1 1.a even 1 1 trivial
925.2.b.b 2 5.c odd 4 2
1369.2.a.e 1 185.d even 2 1
1369.2.b.c 2 185.j odd 4 2
1813.2.a.a 1 35.c odd 2 1
2368.2.a.b 1 40.e odd 2 1
2368.2.a.q 1 40.f even 2 1
4477.2.a.b 1 55.d odd 2 1
5328.2.a.r 1 60.h even 2 1
6253.2.a.c 1 65.d even 2 1
8325.2.a.e 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(925))$$:

 $$T_{2} - 2$$ T2 - 2 $$T_{3} - 3$$ T3 - 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 2$$
$3$ $$T - 3$$
$5$ $$T$$
$7$ $$T - 1$$
$11$ $$T + 5$$
$13$ $$T - 2$$
$17$ $$T$$
$19$ $$T$$
$23$ $$T + 2$$
$29$ $$T - 6$$
$31$ $$T + 4$$
$37$ $$T - 1$$
$41$ $$T + 9$$
$43$ $$T + 2$$
$47$ $$T - 9$$
$53$ $$T + 1$$
$59$ $$T - 8$$
$61$ $$T + 8$$
$67$ $$T + 8$$
$71$ $$T - 9$$
$73$ $$T - 1$$
$79$ $$T - 4$$
$83$ $$T - 15$$
$89$ $$T - 4$$
$97$ $$T + 4$$