# Properties

 Label 5415.2.a.k Level $5415$ Weight $2$ Character orbit 5415.a Self dual yes Analytic conductor $43.239$ 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] = [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: $$0$$ 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)

 $$f(q)$$ $$=$$ $$q + 2 q^{2} + q^{3} + 2 q^{4} - q^{5} + 2 q^{6} - 2 q^{7} + q^{9}+O(q^{10})$$ q + 2 * q^2 + q^3 + 2 * q^4 - q^5 + 2 * q^6 - 2 * q^7 + q^9 $$q + 2 q^{2} + q^{3} + 2 q^{4} - q^{5} + 2 q^{6} - 2 q^{7} + q^{9} - 2 q^{10} - 3 q^{11} + 2 q^{12} + 6 q^{13} - 4 q^{14} - q^{15} - 4 q^{16} + 6 q^{17} + 2 q^{18} - 2 q^{20} - 2 q^{21} - 6 q^{22} - 8 q^{23} + q^{25} + 12 q^{26} + q^{27} - 4 q^{28} + 7 q^{29} - 2 q^{30} + 9 q^{31} - 8 q^{32} - 3 q^{33} + 12 q^{34} + 2 q^{35} + 2 q^{36} - 2 q^{37} + 6 q^{39} + 6 q^{41} - 4 q^{42} + 10 q^{43} - 6 q^{44} - q^{45} - 16 q^{46} + 4 q^{47} - 4 q^{48} - 3 q^{49} + 2 q^{50} + 6 q^{51} + 12 q^{52} + 14 q^{53} + 2 q^{54} + 3 q^{55} + 14 q^{58} + 3 q^{59} - 2 q^{60} - 7 q^{61} + 18 q^{62} - 2 q^{63} - 8 q^{64} - 6 q^{65} - 6 q^{66} + 4 q^{67} + 12 q^{68} - 8 q^{69} + 4 q^{70} + 7 q^{71} + 2 q^{73} - 4 q^{74} + q^{75} + 6 q^{77} + 12 q^{78} + 5 q^{79} + 4 q^{80} + q^{81} + 12 q^{82} - 6 q^{83} - 4 q^{84} - 6 q^{85} + 20 q^{86} + 7 q^{87} + 3 q^{89} - 2 q^{90} - 12 q^{91} - 16 q^{92} + 9 q^{93} + 8 q^{94} - 8 q^{96} - 12 q^{97} - 6 q^{98} - 3 q^{99}+O(q^{100})$$ q + 2 * q^2 + q^3 + 2 * q^4 - q^5 + 2 * q^6 - 2 * q^7 + q^9 - 2 * q^10 - 3 * q^11 + 2 * q^12 + 6 * q^13 - 4 * q^14 - q^15 - 4 * q^16 + 6 * q^17 + 2 * q^18 - 2 * q^20 - 2 * q^21 - 6 * q^22 - 8 * q^23 + q^25 + 12 * q^26 + q^27 - 4 * q^28 + 7 * q^29 - 2 * q^30 + 9 * q^31 - 8 * q^32 - 3 * q^33 + 12 * q^34 + 2 * q^35 + 2 * q^36 - 2 * q^37 + 6 * q^39 + 6 * q^41 - 4 * q^42 + 10 * q^43 - 6 * q^44 - q^45 - 16 * q^46 + 4 * q^47 - 4 * q^48 - 3 * q^49 + 2 * q^50 + 6 * q^51 + 12 * q^52 + 14 * q^53 + 2 * q^54 + 3 * q^55 + 14 * q^58 + 3 * q^59 - 2 * q^60 - 7 * q^61 + 18 * q^62 - 2 * q^63 - 8 * q^64 - 6 * q^65 - 6 * q^66 + 4 * q^67 + 12 * q^68 - 8 * q^69 + 4 * q^70 + 7 * q^71 + 2 * q^73 - 4 * q^74 + q^75 + 6 * q^77 + 12 * q^78 + 5 * q^79 + 4 * q^80 + q^81 + 12 * q^82 - 6 * q^83 - 4 * q^84 - 6 * q^85 + 20 * q^86 + 7 * q^87 + 3 * q^89 - 2 * q^90 - 12 * q^91 - 16 * q^92 + 9 * q^93 + 8 * q^94 - 8 * q^96 - 12 * q^97 - 6 * 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 −1.00000 2.00000 −2.00000 0 1.00000 −2.00000
 $$n$$: 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.k 1
19.b odd 2 1 5415.2.a.a 1
19.c even 3 2 285.2.i.a 2
57.h odd 6 2 855.2.k.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
285.2.i.a 2 19.c even 3 2
855.2.k.d 2 57.h odd 6 2
5415.2.a.a 1 19.b odd 2 1
5415.2.a.k 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(5415))$$:

 $$T_{2} - 2$$ T2 - 2 $$T_{7} + 2$$ T7 + 2 $$T_{11} + 3$$ T11 + 3 $$T_{13} - 6$$ T13 - 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 2$$
$3$ $$T - 1$$
$5$ $$T + 1$$
$7$ $$T + 2$$
$11$ $$T + 3$$
$13$ $$T - 6$$
$17$ $$T - 6$$
$19$ $$T$$
$23$ $$T + 8$$
$29$ $$T - 7$$
$31$ $$T - 9$$
$37$ $$T + 2$$
$41$ $$T - 6$$
$43$ $$T - 10$$
$47$ $$T - 4$$
$53$ $$T - 14$$
$59$ $$T - 3$$
$61$ $$T + 7$$
$67$ $$T - 4$$
$71$ $$T - 7$$
$73$ $$T - 2$$
$79$ $$T - 5$$
$83$ $$T + 6$$
$89$ $$T - 3$$
$97$ $$T + 12$$