# Properties

 Label 15.3.d.a Level $15$ Weight $3$ Character orbit 15.d Self dual yes Analytic conductor $0.409$ Analytic rank $0$ Dimension $1$ CM discriminant -15 Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [15,3,Mod(14,15)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(15, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("15.14");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$15 = 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 15.d (of order $$2$$, degree $$1$$, minimal)

## 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: $$0.408720396540$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - q^{2} + 3 q^{3} - 3 q^{4} - 5 q^{5} - 3 q^{6} + 7 q^{8} + 9 q^{9}+O(q^{10})$$ q - q^2 + 3 * q^3 - 3 * q^4 - 5 * q^5 - 3 * q^6 + 7 * q^8 + 9 * q^9 $$q - q^{2} + 3 q^{3} - 3 q^{4} - 5 q^{5} - 3 q^{6} + 7 q^{8} + 9 q^{9} + 5 q^{10} - 9 q^{12} - 15 q^{15} + 5 q^{16} + 14 q^{17} - 9 q^{18} - 22 q^{19} + 15 q^{20} - 34 q^{23} + 21 q^{24} + 25 q^{25} + 27 q^{27} + 15 q^{30} + 2 q^{31} - 33 q^{32} - 14 q^{34} - 27 q^{36} + 22 q^{38} - 35 q^{40} - 45 q^{45} + 34 q^{46} + 14 q^{47} + 15 q^{48} + 49 q^{49} - 25 q^{50} + 42 q^{51} + 86 q^{53} - 27 q^{54} - 66 q^{57} + 45 q^{60} - 118 q^{61} - 2 q^{62} + 13 q^{64} - 42 q^{68} - 102 q^{69} + 63 q^{72} + 75 q^{75} + 66 q^{76} + 98 q^{79} - 25 q^{80} + 81 q^{81} - 154 q^{83} - 70 q^{85} + 45 q^{90} + 102 q^{92} + 6 q^{93} - 14 q^{94} + 110 q^{95} - 99 q^{96} - 49 q^{98}+O(q^{100})$$ q - q^2 + 3 * q^3 - 3 * q^4 - 5 * q^5 - 3 * q^6 + 7 * q^8 + 9 * q^9 + 5 * q^10 - 9 * q^12 - 15 * q^15 + 5 * q^16 + 14 * q^17 - 9 * q^18 - 22 * q^19 + 15 * q^20 - 34 * q^23 + 21 * q^24 + 25 * q^25 + 27 * q^27 + 15 * q^30 + 2 * q^31 - 33 * q^32 - 14 * q^34 - 27 * q^36 + 22 * q^38 - 35 * q^40 - 45 * q^45 + 34 * q^46 + 14 * q^47 + 15 * q^48 + 49 * q^49 - 25 * q^50 + 42 * q^51 + 86 * q^53 - 27 * q^54 - 66 * q^57 + 45 * q^60 - 118 * q^61 - 2 * q^62 + 13 * q^64 - 42 * q^68 - 102 * q^69 + 63 * q^72 + 75 * q^75 + 66 * q^76 + 98 * q^79 - 25 * q^80 + 81 * q^81 - 154 * q^83 - 70 * q^85 + 45 * q^90 + 102 * q^92 + 6 * q^93 - 14 * q^94 + 110 * q^95 - 99 * q^96 - 49 * q^98

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/15\mathbb{Z}\right)^\times$$.

 $$n$$ $$7$$ $$11$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
14.1
 0
−1.00000 3.00000 −3.00000 −5.00000 −3.00000 0 7.00000 9.00000 5.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 CM by $$\Q(\sqrt{-15})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 15.3.d.a 1
3.b odd 2 1 15.3.d.b yes 1
4.b odd 2 1 240.3.c.a 1
5.b even 2 1 15.3.d.b yes 1
5.c odd 4 2 75.3.c.d 2
8.b even 2 1 960.3.c.b 1
8.d odd 2 1 960.3.c.d 1
9.c even 3 2 405.3.h.b 2
9.d odd 6 2 405.3.h.a 2
12.b even 2 1 240.3.c.b 1
15.d odd 2 1 CM 15.3.d.a 1
15.e even 4 2 75.3.c.d 2
20.d odd 2 1 240.3.c.b 1
20.e even 4 2 1200.3.l.l 2
24.f even 2 1 960.3.c.a 1
24.h odd 2 1 960.3.c.c 1
40.e odd 2 1 960.3.c.a 1
40.f even 2 1 960.3.c.c 1
45.h odd 6 2 405.3.h.b 2
45.j even 6 2 405.3.h.a 2
60.h even 2 1 240.3.c.a 1
60.l odd 4 2 1200.3.l.l 2
120.i odd 2 1 960.3.c.b 1
120.m even 2 1 960.3.c.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.3.d.a 1 1.a even 1 1 trivial
15.3.d.a 1 15.d odd 2 1 CM
15.3.d.b yes 1 3.b odd 2 1
15.3.d.b yes 1 5.b even 2 1
75.3.c.d 2 5.c odd 4 2
75.3.c.d 2 15.e even 4 2
240.3.c.a 1 4.b odd 2 1
240.3.c.a 1 60.h even 2 1
240.3.c.b 1 12.b even 2 1
240.3.c.b 1 20.d odd 2 1
405.3.h.a 2 9.d odd 6 2
405.3.h.a 2 45.j even 6 2
405.3.h.b 2 9.c even 3 2
405.3.h.b 2 45.h odd 6 2
960.3.c.a 1 24.f even 2 1
960.3.c.a 1 40.e odd 2 1
960.3.c.b 1 8.b even 2 1
960.3.c.b 1 120.i odd 2 1
960.3.c.c 1 24.h odd 2 1
960.3.c.c 1 40.f even 2 1
960.3.c.d 1 8.d odd 2 1
960.3.c.d 1 120.m even 2 1
1200.3.l.l 2 20.e even 4 2
1200.3.l.l 2 60.l odd 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} + 1$$ acting on $$S_{3}^{\mathrm{new}}(15, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T - 3$$
$5$ $$T + 5$$
$7$ $$T$$
$11$ $$T$$
$13$ $$T$$
$17$ $$T - 14$$
$19$ $$T + 22$$
$23$ $$T + 34$$
$29$ $$T$$
$31$ $$T - 2$$
$37$ $$T$$
$41$ $$T$$
$43$ $$T$$
$47$ $$T - 14$$
$53$ $$T - 86$$
$59$ $$T$$
$61$ $$T + 118$$
$67$ $$T$$
$71$ $$T$$
$73$ $$T$$
$79$ $$T - 98$$
$83$ $$T + 154$$
$89$ $$T$$
$97$ $$T$$