# Properties

 Label 16.16.a.d Level $16$ Weight $16$ Character orbit 16.a Self dual yes Analytic conductor $22.831$ 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] = [16,16,Mod(1,16)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(16, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 16, names="a")

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

chi := DirichletCharacter("16.1");

S:= CuspForms(chi, 16);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$16 = 2^{4}$$ Weight: $$k$$ $$=$$ $$16$$ Character orbit: $$[\chi]$$ $$=$$ 16.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: $$22.8309608160$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1) 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)

 $$f(q)$$ $$=$$ $$q + 3348 q^{3} + 52110 q^{5} - 2822456 q^{7} - 3139803 q^{9}+O(q^{10})$$ q + 3348 * q^3 + 52110 * q^5 - 2822456 * q^7 - 3139803 * q^9 $$q + 3348 q^{3} + 52110 q^{5} - 2822456 q^{7} - 3139803 q^{9} - 20586852 q^{11} - 190073338 q^{13} + 174464280 q^{15} + 1646527986 q^{17} - 1563257180 q^{19} - 9449582688 q^{21} - 9451116072 q^{23} - 27802126025 q^{25} - 58552201080 q^{27} - 36902568330 q^{29} - 71588483552 q^{31} - 68924780496 q^{33} - 147078182160 q^{35} - 1033652081554 q^{37} - 636365535624 q^{39} + 1641974018202 q^{41} + 492403109308 q^{43} - 163615134330 q^{45} + 3410684952624 q^{47} + 3218696361993 q^{49} + 5512575697128 q^{51} + 6797151655902 q^{53} - 1072780857720 q^{55} - 5233785038640 q^{57} - 9858856815540 q^{59} + 4931842626902 q^{61} + 8861955816168 q^{63} - 9904721643180 q^{65} + 28837826625364 q^{67} - 31642336609056 q^{69} - 125050114914552 q^{71} - 82171455513478 q^{73} - 93081517931700 q^{75} + 58105483948512 q^{77} + 25413078694480 q^{79} - 150980027970519 q^{81} + 281736730890468 q^{83} + 85800573350460 q^{85} - 123549798768840 q^{87} + 715618564776810 q^{89} + 536473633278128 q^{91} - 239678242932096 q^{93} - 81461331649800 q^{95} + 612786136081826 q^{97} + 64638659670156 q^{99}+O(q^{100})$$ q + 3348 * q^3 + 52110 * q^5 - 2822456 * q^7 - 3139803 * q^9 - 20586852 * q^11 - 190073338 * q^13 + 174464280 * q^15 + 1646527986 * q^17 - 1563257180 * q^19 - 9449582688 * q^21 - 9451116072 * q^23 - 27802126025 * q^25 - 58552201080 * q^27 - 36902568330 * q^29 - 71588483552 * q^31 - 68924780496 * q^33 - 147078182160 * q^35 - 1033652081554 * q^37 - 636365535624 * q^39 + 1641974018202 * q^41 + 492403109308 * q^43 - 163615134330 * q^45 + 3410684952624 * q^47 + 3218696361993 * q^49 + 5512575697128 * q^51 + 6797151655902 * q^53 - 1072780857720 * q^55 - 5233785038640 * q^57 - 9858856815540 * q^59 + 4931842626902 * q^61 + 8861955816168 * q^63 - 9904721643180 * q^65 + 28837826625364 * q^67 - 31642336609056 * q^69 - 125050114914552 * q^71 - 82171455513478 * q^73 - 93081517931700 * q^75 + 58105483948512 * q^77 + 25413078694480 * q^79 - 150980027970519 * q^81 + 281736730890468 * q^83 + 85800573350460 * q^85 - 123549798768840 * q^87 + 715618564776810 * q^89 + 536473633278128 * q^91 - 239678242932096 * q^93 - 81461331649800 * q^95 + 612786136081826 * q^97 + 64638659670156 * 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
0 3348.00 0 52110.0 0 −2.82246e6 0 −3.13980e6 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$

## 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 16.16.a.d 1
3.b odd 2 1 144.16.a.f 1
4.b odd 2 1 1.16.a.a 1
8.b even 2 1 64.16.a.c 1
8.d odd 2 1 64.16.a.i 1
12.b even 2 1 9.16.a.a 1
20.d odd 2 1 25.16.a.a 1
20.e even 4 2 25.16.b.a 2
28.d even 2 1 49.16.a.a 1
28.f even 6 2 49.16.c.b 2
28.g odd 6 2 49.16.c.c 2
44.c even 2 1 121.16.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.16.a.a 1 4.b odd 2 1
9.16.a.a 1 12.b even 2 1
16.16.a.d 1 1.a even 1 1 trivial
25.16.a.a 1 20.d odd 2 1
25.16.b.a 2 20.e even 4 2
49.16.a.a 1 28.d even 2 1
49.16.c.b 2 28.f even 6 2
49.16.c.c 2 28.g odd 6 2
64.16.a.c 1 8.b even 2 1
64.16.a.i 1 8.d odd 2 1
121.16.a.a 1 44.c even 2 1
144.16.a.f 1 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} - 3348$$ acting on $$S_{16}^{\mathrm{new}}(\Gamma_0(16))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 3348$$
$5$ $$T - 52110$$
$7$ $$T + 2822456$$
$11$ $$T + 20586852$$
$13$ $$T + 190073338$$
$17$ $$T - 1646527986$$
$19$ $$T + 1563257180$$
$23$ $$T + 9451116072$$
$29$ $$T + 36902568330$$
$31$ $$T + 71588483552$$
$37$ $$T + 1033652081554$$
$41$ $$T - 1641974018202$$
$43$ $$T - 492403109308$$
$47$ $$T - 3410684952624$$
$53$ $$T - 6797151655902$$
$59$ $$T + 9858856815540$$
$61$ $$T - 4931842626902$$
$67$ $$T - 28837826625364$$
$71$ $$T + 125050114914552$$
$73$ $$T + 82171455513478$$
$79$ $$T - 25413078694480$$
$83$ $$T - 281736730890468$$
$89$ $$T - 715618564776810$$
$97$ $$T - 612786136081826$$