# Properties

 Label 5415.2.a.g Level $5415$ Weight $2$ Character orbit 5415.a Self dual yes Analytic conductor $43.239$ 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] = [5415,2,Mod(1,5415)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5415, 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("5415.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$5415 = 3 \cdot 5 \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5415.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: $$43.2389926945$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 285) 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)

 $$f(q)$$ $$=$$ $$q + q^{3} - 2 q^{4} + q^{5} + 2 q^{7} + q^{9}+O(q^{10})$$ q + q^3 - 2 * q^4 + q^5 + 2 * q^7 + q^9 $$q + q^{3} - 2 q^{4} + q^{5} + 2 q^{7} + q^{9} - 3 q^{11} - 2 q^{12} - 4 q^{13} + q^{15} + 4 q^{16} - 2 q^{20} + 2 q^{21} - 6 q^{23} + q^{25} + q^{27} - 4 q^{28} + 3 q^{29} + 5 q^{31} - 3 q^{33} + 2 q^{35} - 2 q^{36} + 8 q^{37} - 4 q^{39} - 6 q^{41} - 4 q^{43} + 6 q^{44} + q^{45} + 6 q^{47} + 4 q^{48} - 3 q^{49} + 8 q^{52} - 6 q^{53} - 3 q^{55} - 9 q^{59} - 2 q^{60} - 7 q^{61} + 2 q^{63} - 8 q^{64} - 4 q^{65} + 2 q^{67} - 6 q^{69} - 9 q^{71} - 4 q^{73} + q^{75} - 6 q^{77} - 7 q^{79} + 4 q^{80} + q^{81} - 4 q^{84} + 3 q^{87} + 3 q^{89} - 8 q^{91} + 12 q^{92} + 5 q^{93} - 10 q^{97} - 3 q^{99}+O(q^{100})$$ q + q^3 - 2 * q^4 + q^5 + 2 * q^7 + q^9 - 3 * q^11 - 2 * q^12 - 4 * q^13 + q^15 + 4 * q^16 - 2 * q^20 + 2 * q^21 - 6 * q^23 + q^25 + q^27 - 4 * q^28 + 3 * q^29 + 5 * q^31 - 3 * q^33 + 2 * q^35 - 2 * q^36 + 8 * q^37 - 4 * q^39 - 6 * q^41 - 4 * q^43 + 6 * q^44 + q^45 + 6 * q^47 + 4 * q^48 - 3 * q^49 + 8 * q^52 - 6 * q^53 - 3 * q^55 - 9 * q^59 - 2 * q^60 - 7 * q^61 + 2 * q^63 - 8 * q^64 - 4 * q^65 + 2 * q^67 - 6 * q^69 - 9 * q^71 - 4 * q^73 + q^75 - 6 * q^77 - 7 * q^79 + 4 * q^80 + q^81 - 4 * q^84 + 3 * q^87 + 3 * q^89 - 8 * q^91 + 12 * q^92 + 5 * q^93 - 10 * q^97 - 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
0 1.00000 −2.00000 1.00000 0 2.00000 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 5415.2.a.g 1
19.b odd 2 1 5415.2.a.f 1
19.c even 3 2 285.2.i.b 2
57.h odd 6 2 855.2.k.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
285.2.i.b 2 19.c even 3 2
855.2.k.c 2 57.h odd 6 2
5415.2.a.f 1 19.b odd 2 1
5415.2.a.g 1 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(5415))$$:

 $$T_{2}$$ T2 $$T_{7} - 2$$ T7 - 2 $$T_{11} + 3$$ T11 + 3 $$T_{13} + 4$$ T13 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 1$$
$5$ $$T - 1$$
$7$ $$T - 2$$
$11$ $$T + 3$$
$13$ $$T + 4$$
$17$ $$T$$
$19$ $$T$$
$23$ $$T + 6$$
$29$ $$T - 3$$
$31$ $$T - 5$$
$37$ $$T - 8$$
$41$ $$T + 6$$
$43$ $$T + 4$$
$47$ $$T - 6$$
$53$ $$T + 6$$
$59$ $$T + 9$$
$61$ $$T + 7$$
$67$ $$T - 2$$
$71$ $$T + 9$$
$73$ $$T + 4$$
$79$ $$T + 7$$
$83$ $$T$$
$89$ $$T - 3$$
$97$ $$T + 10$$