# Properties

 Label 912.2.a.e Level $912$ Weight $2$ Character orbit 912.a Self dual yes Analytic conductor $7.282$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [912,2,Mod(1,912)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(912, 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("912.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$912 = 2^{4} \cdot 3 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 912.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: $$7.28235666434$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 456) 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)

 $$f(q)$$ $$=$$ $$q - q^{3} + 2 q^{5} + q^{9}+O(q^{10})$$ q - q^3 + 2 * q^5 + q^9 $$q - q^{3} + 2 q^{5} + q^{9} + 2 q^{13} - 2 q^{15} + 2 q^{17} + q^{19} - q^{25} - q^{27} + 2 q^{29} + 4 q^{31} + 2 q^{37} - 2 q^{39} + 6 q^{41} + 4 q^{43} + 2 q^{45} - 7 q^{49} - 2 q^{51} + 10 q^{53} - q^{57} + 4 q^{59} - 2 q^{61} + 4 q^{65} + 12 q^{67} - 6 q^{73} + q^{75} + 4 q^{79} + q^{81} + 8 q^{83} + 4 q^{85} - 2 q^{87} + 6 q^{89} - 4 q^{93} + 2 q^{95} - 14 q^{97}+O(q^{100})$$ q - q^3 + 2 * q^5 + q^9 + 2 * q^13 - 2 * q^15 + 2 * q^17 + q^19 - q^25 - q^27 + 2 * q^29 + 4 * q^31 + 2 * q^37 - 2 * q^39 + 6 * q^41 + 4 * q^43 + 2 * q^45 - 7 * q^49 - 2 * q^51 + 10 * q^53 - q^57 + 4 * q^59 - 2 * q^61 + 4 * q^65 + 12 * q^67 - 6 * q^73 + q^75 + 4 * q^79 + q^81 + 8 * q^83 + 4 * q^85 - 2 * q^87 + 6 * q^89 - 4 * q^93 + 2 * q^95 - 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 2.00000 0 0 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
$$2$$ $$1$$
$$3$$ $$1$$
$$19$$ $$-1$$

## 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 912.2.a.e 1
3.b odd 2 1 2736.2.a.e 1
4.b odd 2 1 456.2.a.d 1
8.b even 2 1 3648.2.a.w 1
8.d odd 2 1 3648.2.a.d 1
12.b even 2 1 1368.2.a.c 1
76.d even 2 1 8664.2.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
456.2.a.d 1 4.b odd 2 1
912.2.a.e 1 1.a even 1 1 trivial
1368.2.a.c 1 12.b even 2 1
2736.2.a.e 1 3.b odd 2 1
3648.2.a.d 1 8.d odd 2 1
3648.2.a.w 1 8.b even 2 1
8664.2.a.f 1 76.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(912))$$:

 $$T_{5} - 2$$ T5 - 2 $$T_{7}$$ T7 $$T_{11}$$ T11

## Hecke characteristic polynomials

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