# Properties

 Label 5415.2.a.j 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 15) 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)

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} - q^{4} + q^{5} + q^{6} - 3 q^{8} + q^{9}+O(q^{10})$$ q + q^2 + q^3 - q^4 + q^5 + q^6 - 3 * q^8 + q^9 $$q + q^{2} + q^{3} - q^{4} + q^{5} + q^{6} - 3 q^{8} + q^{9} + q^{10} - 4 q^{11} - q^{12} + 2 q^{13} + q^{15} - q^{16} + 2 q^{17} + q^{18} - q^{20} - 4 q^{22} - 3 q^{24} + q^{25} + 2 q^{26} + q^{27} + 2 q^{29} + q^{30} + 5 q^{32} - 4 q^{33} + 2 q^{34} - q^{36} + 10 q^{37} + 2 q^{39} - 3 q^{40} - 10 q^{41} + 4 q^{43} + 4 q^{44} + q^{45} + 8 q^{47} - q^{48} - 7 q^{49} + q^{50} + 2 q^{51} - 2 q^{52} + 10 q^{53} + q^{54} - 4 q^{55} + 2 q^{58} + 4 q^{59} - q^{60} - 2 q^{61} + 7 q^{64} + 2 q^{65} - 4 q^{66} - 12 q^{67} - 2 q^{68} + 8 q^{71} - 3 q^{72} + 10 q^{73} + 10 q^{74} + q^{75} + 2 q^{78} - q^{80} + q^{81} - 10 q^{82} + 12 q^{83} + 2 q^{85} + 4 q^{86} + 2 q^{87} + 12 q^{88} + 6 q^{89} + q^{90} + 8 q^{94} + 5 q^{96} - 2 q^{97} - 7 q^{98} - 4 q^{99}+O(q^{100})$$ q + q^2 + q^3 - q^4 + q^5 + q^6 - 3 * q^8 + q^9 + q^10 - 4 * q^11 - q^12 + 2 * q^13 + q^15 - q^16 + 2 * q^17 + q^18 - q^20 - 4 * q^22 - 3 * q^24 + q^25 + 2 * q^26 + q^27 + 2 * q^29 + q^30 + 5 * q^32 - 4 * q^33 + 2 * q^34 - q^36 + 10 * q^37 + 2 * q^39 - 3 * q^40 - 10 * q^41 + 4 * q^43 + 4 * q^44 + q^45 + 8 * q^47 - q^48 - 7 * q^49 + q^50 + 2 * q^51 - 2 * q^52 + 10 * q^53 + q^54 - 4 * q^55 + 2 * q^58 + 4 * q^59 - q^60 - 2 * q^61 + 7 * q^64 + 2 * q^65 - 4 * q^66 - 12 * q^67 - 2 * q^68 + 8 * q^71 - 3 * q^72 + 10 * q^73 + 10 * q^74 + q^75 + 2 * q^78 - q^80 + q^81 - 10 * q^82 + 12 * q^83 + 2 * q^85 + 4 * q^86 + 2 * q^87 + 12 * q^88 + 6 * q^89 + q^90 + 8 * q^94 + 5 * q^96 - 2 * q^97 - 7 * q^98 - 4 * 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 0 −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.j 1
19.b odd 2 1 15.2.a.a 1
57.d even 2 1 45.2.a.a 1
76.d even 2 1 240.2.a.d 1
95.d odd 2 1 75.2.a.b 1
95.g even 4 2 75.2.b.b 2
133.c even 2 1 735.2.a.c 1
133.o even 6 2 735.2.i.d 2
133.r odd 6 2 735.2.i.e 2
152.b even 2 1 960.2.a.a 1
152.g odd 2 1 960.2.a.l 1
171.l even 6 2 405.2.e.c 2
171.o odd 6 2 405.2.e.f 2
209.d even 2 1 1815.2.a.d 1
228.b odd 2 1 720.2.a.c 1
247.d odd 2 1 2535.2.a.j 1
285.b even 2 1 225.2.a.b 1
285.j odd 4 2 225.2.b.b 2
304.j odd 4 2 3840.2.k.m 2
304.m even 4 2 3840.2.k.r 2
323.c odd 2 1 4335.2.a.c 1
380.d even 2 1 1200.2.a.e 1
380.j odd 4 2 1200.2.f.h 2
399.h odd 2 1 2205.2.a.i 1
437.b even 2 1 7935.2.a.d 1
456.l odd 2 1 2880.2.a.bc 1
456.p even 2 1 2880.2.a.y 1
627.b odd 2 1 5445.2.a.c 1
665.g even 2 1 3675.2.a.j 1
741.d even 2 1 7605.2.a.g 1
760.b odd 2 1 4800.2.a.t 1
760.p even 2 1 4800.2.a.bz 1
760.t even 4 2 4800.2.f.bf 2
760.y odd 4 2 4800.2.f.c 2
1045.e even 2 1 9075.2.a.g 1
1140.p odd 2 1 3600.2.a.u 1
1140.w even 4 2 3600.2.f.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.2.a.a 1 19.b odd 2 1
45.2.a.a 1 57.d even 2 1
75.2.a.b 1 95.d odd 2 1
75.2.b.b 2 95.g even 4 2
225.2.a.b 1 285.b even 2 1
225.2.b.b 2 285.j odd 4 2
240.2.a.d 1 76.d even 2 1
405.2.e.c 2 171.l even 6 2
405.2.e.f 2 171.o odd 6 2
720.2.a.c 1 228.b odd 2 1
735.2.a.c 1 133.c even 2 1
735.2.i.d 2 133.o even 6 2
735.2.i.e 2 133.r odd 6 2
960.2.a.a 1 152.b even 2 1
960.2.a.l 1 152.g odd 2 1
1200.2.a.e 1 380.d even 2 1
1200.2.f.h 2 380.j odd 4 2
1815.2.a.d 1 209.d even 2 1
2205.2.a.i 1 399.h odd 2 1
2535.2.a.j 1 247.d odd 2 1
2880.2.a.y 1 456.p even 2 1
2880.2.a.bc 1 456.l odd 2 1
3600.2.a.u 1 1140.p odd 2 1
3600.2.f.e 2 1140.w even 4 2
3675.2.a.j 1 665.g even 2 1
3840.2.k.m 2 304.j odd 4 2
3840.2.k.r 2 304.m even 4 2
4335.2.a.c 1 323.c odd 2 1
4800.2.a.t 1 760.b odd 2 1
4800.2.a.bz 1 760.p even 2 1
4800.2.f.c 2 760.y odd 4 2
4800.2.f.bf 2 760.t even 4 2
5415.2.a.j 1 1.a even 1 1 trivial
5445.2.a.c 1 627.b odd 2 1
7605.2.a.g 1 741.d even 2 1
7935.2.a.d 1 437.b even 2 1
9075.2.a.g 1 1045.e even 2 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

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