# Properties

 Label 816.2.a.d Level $816$ Weight $2$ Character orbit 816.a Self dual yes Analytic conductor $6.516$ 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] = [816,2,Mod(1,816)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(816, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("816.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$816 = 2^{4} \cdot 3 \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 816.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: $$6.51579280494$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 102) 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)

 $$f(q)$$ $$=$$ $$q - q^{3} - 2 q^{7} + q^{9}+O(q^{10})$$ q - q^3 - 2 * q^7 + q^9 $$q - q^{3} - 2 q^{7} + q^{9} + 2 q^{13} - q^{17} + 4 q^{19} + 2 q^{21} + 6 q^{23} - 5 q^{25} - q^{27} + 10 q^{31} + 8 q^{37} - 2 q^{39} + 6 q^{41} + 4 q^{43} - 12 q^{47} - 3 q^{49} + q^{51} + 6 q^{53} - 4 q^{57} + 12 q^{59} + 8 q^{61} - 2 q^{63} + 4 q^{67} - 6 q^{69} - 6 q^{71} + 2 q^{73} + 5 q^{75} + 10 q^{79} + q^{81} - 12 q^{83} - 18 q^{89} - 4 q^{91} - 10 q^{93} + 14 q^{97}+O(q^{100})$$ q - q^3 - 2 * q^7 + q^9 + 2 * q^13 - q^17 + 4 * q^19 + 2 * q^21 + 6 * q^23 - 5 * q^25 - q^27 + 10 * q^31 + 8 * q^37 - 2 * q^39 + 6 * q^41 + 4 * q^43 - 12 * q^47 - 3 * q^49 + q^51 + 6 * q^53 - 4 * q^57 + 12 * q^59 + 8 * q^61 - 2 * q^63 + 4 * q^67 - 6 * q^69 - 6 * q^71 + 2 * q^73 + 5 * q^75 + 10 * q^79 + q^81 - 12 * q^83 - 18 * q^89 - 4 * q^91 - 10 * q^93 + 14 * q^97

## 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 −1.00000 0 0 0 −2.00000 0 1.00000 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
$$2$$ $$-1$$
$$3$$ $$1$$
$$17$$ $$1$$

## 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 816.2.a.d 1
3.b odd 2 1 2448.2.a.i 1
4.b odd 2 1 102.2.a.b 1
8.b even 2 1 3264.2.a.w 1
8.d odd 2 1 3264.2.a.i 1
12.b even 2 1 306.2.a.c 1
20.d odd 2 1 2550.2.a.u 1
20.e even 4 2 2550.2.d.g 2
24.f even 2 1 9792.2.a.bg 1
24.h odd 2 1 9792.2.a.ba 1
28.d even 2 1 4998.2.a.d 1
60.h even 2 1 7650.2.a.j 1
68.d odd 2 1 1734.2.a.b 1
68.f odd 4 2 1734.2.b.f 2
68.g odd 8 4 1734.2.f.b 4
204.h even 2 1 5202.2.a.j 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
102.2.a.b 1 4.b odd 2 1
306.2.a.c 1 12.b even 2 1
816.2.a.d 1 1.a even 1 1 trivial
1734.2.a.b 1 68.d odd 2 1
1734.2.b.f 2 68.f odd 4 2
1734.2.f.b 4 68.g odd 8 4
2448.2.a.i 1 3.b odd 2 1
2550.2.a.u 1 20.d odd 2 1
2550.2.d.g 2 20.e even 4 2
3264.2.a.i 1 8.d odd 2 1
3264.2.a.w 1 8.b even 2 1
4998.2.a.d 1 28.d even 2 1
5202.2.a.j 1 204.h even 2 1
7650.2.a.j 1 60.h even 2 1
9792.2.a.ba 1 24.h odd 2 1
9792.2.a.bg 1 24.f even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(816))$$:

 $$T_{5}$$ T5 $$T_{7} + 2$$ T7 + 2 $$T_{19} - 4$$ T19 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 1$$
$5$ $$T$$
$7$ $$T + 2$$
$11$ $$T$$
$13$ $$T - 2$$
$17$ $$T + 1$$
$19$ $$T - 4$$
$23$ $$T - 6$$
$29$ $$T$$
$31$ $$T - 10$$
$37$ $$T - 8$$
$41$ $$T - 6$$
$43$ $$T - 4$$
$47$ $$T + 12$$
$53$ $$T - 6$$
$59$ $$T - 12$$
$61$ $$T - 8$$
$67$ $$T - 4$$
$71$ $$T + 6$$
$73$ $$T - 2$$
$79$ $$T - 10$$
$83$ $$T + 12$$
$89$ $$T + 18$$
$97$ $$T - 14$$