# Properties

 Label 1425.2.a.i Level $1425$ Weight $2$ Character orbit 1425.a Self dual yes Analytic conductor $11.379$ 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] = [1425,2,Mod(1,1425)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1425, 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("1425.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1425 = 3 \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1425.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: $$11.3786822880$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 57) 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)

 $$f(q)$$ $$=$$ $$q + 2 q^{2} - q^{3} + 2 q^{4} - 2 q^{6} - 3 q^{7} + q^{9}+O(q^{10})$$ q + 2 * q^2 - q^3 + 2 * q^4 - 2 * q^6 - 3 * q^7 + q^9 $$q + 2 q^{2} - q^{3} + 2 q^{4} - 2 q^{6} - 3 q^{7} + q^{9} - 3 q^{11} - 2 q^{12} + 6 q^{13} - 6 q^{14} - 4 q^{16} - 3 q^{17} + 2 q^{18} - q^{19} + 3 q^{21} - 6 q^{22} - 4 q^{23} + 12 q^{26} - q^{27} - 6 q^{28} - 10 q^{29} + 2 q^{31} - 8 q^{32} + 3 q^{33} - 6 q^{34} + 2 q^{36} - 8 q^{37} - 2 q^{38} - 6 q^{39} - 8 q^{41} + 6 q^{42} + q^{43} - 6 q^{44} - 8 q^{46} - 3 q^{47} + 4 q^{48} + 2 q^{49} + 3 q^{51} + 12 q^{52} + 6 q^{53} - 2 q^{54} + q^{57} - 20 q^{58} + 7 q^{61} + 4 q^{62} - 3 q^{63} - 8 q^{64} + 6 q^{66} - 8 q^{67} - 6 q^{68} + 4 q^{69} + 12 q^{71} + 11 q^{73} - 16 q^{74} - 2 q^{76} + 9 q^{77} - 12 q^{78} + q^{81} - 16 q^{82} - 4 q^{83} + 6 q^{84} + 2 q^{86} + 10 q^{87} + 10 q^{89} - 18 q^{91} - 8 q^{92} - 2 q^{93} - 6 q^{94} + 8 q^{96} + 2 q^{97} + 4 q^{98} - 3 q^{99}+O(q^{100})$$ q + 2 * q^2 - q^3 + 2 * q^4 - 2 * q^6 - 3 * q^7 + q^9 - 3 * q^11 - 2 * q^12 + 6 * q^13 - 6 * q^14 - 4 * q^16 - 3 * q^17 + 2 * q^18 - q^19 + 3 * q^21 - 6 * q^22 - 4 * q^23 + 12 * q^26 - q^27 - 6 * q^28 - 10 * q^29 + 2 * q^31 - 8 * q^32 + 3 * q^33 - 6 * q^34 + 2 * q^36 - 8 * q^37 - 2 * q^38 - 6 * q^39 - 8 * q^41 + 6 * q^42 + q^43 - 6 * q^44 - 8 * q^46 - 3 * q^47 + 4 * q^48 + 2 * q^49 + 3 * q^51 + 12 * q^52 + 6 * q^53 - 2 * q^54 + q^57 - 20 * q^58 + 7 * q^61 + 4 * q^62 - 3 * q^63 - 8 * q^64 + 6 * q^66 - 8 * q^67 - 6 * q^68 + 4 * q^69 + 12 * q^71 + 11 * q^73 - 16 * q^74 - 2 * q^76 + 9 * q^77 - 12 * q^78 + q^81 - 16 * q^82 - 4 * q^83 + 6 * q^84 + 2 * q^86 + 10 * q^87 + 10 * q^89 - 18 * q^91 - 8 * q^92 - 2 * q^93 - 6 * q^94 + 8 * q^96 + 2 * q^97 + 4 * q^98 - 3 * 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 −1.00000 2.00000 0 −2.00000 −3.00000 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
$$3$$ $$+1$$
$$5$$ $$+1$$
$$19$$ $$+1$$

## 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 1425.2.a.i 1
3.b odd 2 1 4275.2.a.a 1
5.b even 2 1 57.2.a.b 1
5.c odd 4 2 1425.2.c.a 2
15.d odd 2 1 171.2.a.c 1
20.d odd 2 1 912.2.a.d 1
35.c odd 2 1 2793.2.a.a 1
40.e odd 2 1 3648.2.a.y 1
40.f even 2 1 3648.2.a.h 1
55.d odd 2 1 6897.2.a.g 1
60.h even 2 1 2736.2.a.h 1
65.d even 2 1 9633.2.a.p 1
95.d odd 2 1 1083.2.a.d 1
105.g even 2 1 8379.2.a.q 1
285.b even 2 1 3249.2.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.2.a.b 1 5.b even 2 1
171.2.a.c 1 15.d odd 2 1
912.2.a.d 1 20.d odd 2 1
1083.2.a.d 1 95.d odd 2 1
1425.2.a.i 1 1.a even 1 1 trivial
1425.2.c.a 2 5.c odd 4 2
2736.2.a.h 1 60.h even 2 1
2793.2.a.a 1 35.c odd 2 1
3249.2.a.a 1 285.b even 2 1
3648.2.a.h 1 40.f even 2 1
3648.2.a.y 1 40.e odd 2 1
4275.2.a.a 1 3.b odd 2 1
6897.2.a.g 1 55.d odd 2 1
8379.2.a.q 1 105.g even 2 1
9633.2.a.p 1 65.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1425))$$:

 $$T_{2} - 2$$ T2 - 2 $$T_{7} + 3$$ T7 + 3 $$T_{11} + 3$$ T11 + 3

## Hecke characteristic polynomials

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