# Properties

 Label 5415.2.a.c 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)

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{3} - q^{4} - q^{5} - q^{6} - 2 q^{7} + 3 q^{8} + q^{9}+O(q^{10})$$ q - q^2 + q^3 - q^4 - q^5 - q^6 - 2 * q^7 + 3 * q^8 + q^9 $$q - q^{2} + q^{3} - q^{4} - q^{5} - q^{6} - 2 q^{7} + 3 q^{8} + q^{9} + q^{10} - 2 q^{11} - q^{12} + 4 q^{13} + 2 q^{14} - q^{15} - q^{16} + 2 q^{17} - q^{18} + q^{20} - 2 q^{21} + 2 q^{22} - 4 q^{23} + 3 q^{24} + q^{25} - 4 q^{26} + q^{27} + 2 q^{28} - 4 q^{29} + q^{30} - 5 q^{32} - 2 q^{33} - 2 q^{34} + 2 q^{35} - q^{36} + 4 q^{39} - 3 q^{40} + 2 q^{42} - 10 q^{43} + 2 q^{44} - q^{45} + 4 q^{46} + 12 q^{47} - q^{48} - 3 q^{49} - q^{50} + 2 q^{51} - 4 q^{52} + 2 q^{53} - q^{54} + 2 q^{55} - 6 q^{56} + 4 q^{58} - 4 q^{59} + q^{60} + 2 q^{61} - 2 q^{63} + 7 q^{64} - 4 q^{65} + 2 q^{66} + 16 q^{67} - 2 q^{68} - 4 q^{69} - 2 q^{70} + 3 q^{72} - 2 q^{73} + q^{75} + 4 q^{77} - 4 q^{78} + 8 q^{79} + q^{80} + q^{81} - 12 q^{83} + 2 q^{84} - 2 q^{85} + 10 q^{86} - 4 q^{87} - 6 q^{88} + q^{90} - 8 q^{91} + 4 q^{92} - 12 q^{94} - 5 q^{96} + 16 q^{97} + 3 q^{98} - 2 q^{99}+O(q^{100})$$ q - q^2 + q^3 - q^4 - q^5 - q^6 - 2 * q^7 + 3 * q^8 + q^9 + q^10 - 2 * q^11 - q^12 + 4 * q^13 + 2 * q^14 - q^15 - q^16 + 2 * q^17 - q^18 + q^20 - 2 * q^21 + 2 * q^22 - 4 * q^23 + 3 * q^24 + q^25 - 4 * q^26 + q^27 + 2 * q^28 - 4 * q^29 + q^30 - 5 * q^32 - 2 * q^33 - 2 * q^34 + 2 * q^35 - q^36 + 4 * q^39 - 3 * q^40 + 2 * q^42 - 10 * q^43 + 2 * q^44 - q^45 + 4 * q^46 + 12 * q^47 - q^48 - 3 * q^49 - q^50 + 2 * q^51 - 4 * q^52 + 2 * q^53 - q^54 + 2 * q^55 - 6 * q^56 + 4 * q^58 - 4 * q^59 + q^60 + 2 * q^61 - 2 * q^63 + 7 * q^64 - 4 * q^65 + 2 * q^66 + 16 * q^67 - 2 * q^68 - 4 * q^69 - 2 * q^70 + 3 * q^72 - 2 * q^73 + q^75 + 4 * q^77 - 4 * q^78 + 8 * q^79 + q^80 + q^81 - 12 * q^83 + 2 * q^84 - 2 * q^85 + 10 * q^86 - 4 * q^87 - 6 * q^88 + q^90 - 8 * q^91 + 4 * q^92 - 12 * q^94 - 5 * q^96 + 16 * q^97 + 3 * q^98 - 2 * 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
−1.00000 1.00000 −1.00000 −1.00000 −1.00000 −2.00000 3.00000 1.00000 1.00000
 $$n$$: 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.c 1
19.b odd 2 1 285.2.a.b 1
57.d even 2 1 855.2.a.b 1
76.d even 2 1 4560.2.a.v 1
95.d odd 2 1 1425.2.a.d 1
95.g even 4 2 1425.2.c.d 2
285.b even 2 1 4275.2.a.o 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
285.2.a.b 1 19.b odd 2 1
855.2.a.b 1 57.d even 2 1
1425.2.a.d 1 95.d odd 2 1
1425.2.c.d 2 95.g even 4 2
4275.2.a.o 1 285.b even 2 1
4560.2.a.v 1 76.d even 2 1
5415.2.a.c 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(5415))$$:

 $$T_{2} + 1$$ T2 + 1 $$T_{7} + 2$$ T7 + 2 $$T_{11} + 2$$ T11 + 2 $$T_{13} - 4$$ T13 - 4

## Hecke characteristic polynomials

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