# Properties

 Label 15.9.d.b Level $15$ Weight $9$ Character orbit 15.d Self dual yes Analytic conductor $6.111$ 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] = [15,9,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, 9, names="a")

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

chi := DirichletCharacter("15.14");

S:= CuspForms(chi, 9);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$15 = 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$9$$ 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: $$6.11067915092$$ 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)

 $$f(q)$$ $$=$$ $$q + 17 q^{2} + 81 q^{3} + 33 q^{4} + 625 q^{5} + 1377 q^{6} - 3791 q^{8} + 6561 q^{9}+O(q^{10})$$ q + 17 * q^2 + 81 * q^3 + 33 * q^4 + 625 * q^5 + 1377 * q^6 - 3791 * q^8 + 6561 * q^9 $$q + 17 q^{2} + 81 q^{3} + 33 q^{4} + 625 q^{5} + 1377 q^{6} - 3791 q^{8} + 6561 q^{9} + 10625 q^{10} + 2673 q^{12} + 50625 q^{15} - 72895 q^{16} - 21118 q^{17} + 111537 q^{18} - 203998 q^{19} + 20625 q^{20} - 550078 q^{23} - 307071 q^{24} + 390625 q^{25} + 531441 q^{27} + 860625 q^{30} + 1831682 q^{31} - 268719 q^{32} - 359006 q^{34} + 216513 q^{36} - 3467966 q^{38} - 2369375 q^{40} + 4100625 q^{45} - 9351326 q^{46} + 8065922 q^{47} - 5904495 q^{48} + 5764801 q^{49} + 6640625 q^{50} - 1710558 q^{51} - 12619678 q^{53} + 9034497 q^{54} - 16523838 q^{57} + 1670625 q^{60} + 14324642 q^{61} + 31138594 q^{62} + 14092897 q^{64} - 696894 q^{68} - 44556318 q^{69} - 24872751 q^{72} + 31640625 q^{75} - 6731934 q^{76} - 69617278 q^{79} - 45559375 q^{80} + 43046721 q^{81} + 3847202 q^{83} - 13198750 q^{85} + 69710625 q^{90} - 18152574 q^{92} + 148366242 q^{93} + 137120674 q^{94} - 127498750 q^{95} - 21766239 q^{96} + 98001617 q^{98}+O(q^{100})$$ q + 17 * q^2 + 81 * q^3 + 33 * q^4 + 625 * q^5 + 1377 * q^6 - 3791 * q^8 + 6561 * q^9 + 10625 * q^10 + 2673 * q^12 + 50625 * q^15 - 72895 * q^16 - 21118 * q^17 + 111537 * q^18 - 203998 * q^19 + 20625 * q^20 - 550078 * q^23 - 307071 * q^24 + 390625 * q^25 + 531441 * q^27 + 860625 * q^30 + 1831682 * q^31 - 268719 * q^32 - 359006 * q^34 + 216513 * q^36 - 3467966 * q^38 - 2369375 * q^40 + 4100625 * q^45 - 9351326 * q^46 + 8065922 * q^47 - 5904495 * q^48 + 5764801 * q^49 + 6640625 * q^50 - 1710558 * q^51 - 12619678 * q^53 + 9034497 * q^54 - 16523838 * q^57 + 1670625 * q^60 + 14324642 * q^61 + 31138594 * q^62 + 14092897 * q^64 - 696894 * q^68 - 44556318 * q^69 - 24872751 * q^72 + 31640625 * q^75 - 6731934 * q^76 - 69617278 * q^79 - 45559375 * q^80 + 43046721 * q^81 + 3847202 * q^83 - 13198750 * q^85 + 69710625 * q^90 - 18152574 * q^92 + 148366242 * q^93 + 137120674 * q^94 - 127498750 * q^95 - 21766239 * q^96 + 98001617 * 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
17.0000 81.0000 33.0000 625.000 1377.00 0 −3791.00 6561.00 10625.0
 $$n$$: 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 15.9.d.b yes 1
3.b odd 2 1 15.9.d.a 1
4.b odd 2 1 240.9.c.a 1
5.b even 2 1 15.9.d.a 1
5.c odd 4 2 75.9.c.d 2
12.b even 2 1 240.9.c.b 1
15.d odd 2 1 CM 15.9.d.b yes 1
15.e even 4 2 75.9.c.d 2
20.d odd 2 1 240.9.c.b 1
60.h even 2 1 240.9.c.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.9.d.a 1 3.b odd 2 1
15.9.d.a 1 5.b even 2 1
15.9.d.b yes 1 1.a even 1 1 trivial
15.9.d.b yes 1 15.d odd 2 1 CM
75.9.c.d 2 5.c odd 4 2
75.9.c.d 2 15.e even 4 2
240.9.c.a 1 4.b odd 2 1
240.9.c.a 1 60.h even 2 1
240.9.c.b 1 12.b even 2 1
240.9.c.b 1 20.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} - 17$$ acting on $$S_{9}^{\mathrm{new}}(15, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 17$$
$3$ $$T - 81$$
$5$ $$T - 625$$
$7$ $$T$$
$11$ $$T$$
$13$ $$T$$
$17$ $$T + 21118$$
$19$ $$T + 203998$$
$23$ $$T + 550078$$
$29$ $$T$$
$31$ $$T - 1831682$$
$37$ $$T$$
$41$ $$T$$
$43$ $$T$$
$47$ $$T - 8065922$$
$53$ $$T + 12619678$$
$59$ $$T$$
$61$ $$T - 14324642$$
$67$ $$T$$
$71$ $$T$$
$73$ $$T$$
$79$ $$T + 69617278$$
$83$ $$T - 3847202$$
$89$ $$T$$
$97$ $$T$$