# Properties

 Label 1936.4.a.a Level $1936$ Weight $4$ Character orbit 1936.a Self dual yes Analytic conductor $114.228$ Analytic rank $1$ Dimension $1$ CM discriminant -11 Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1936,4,Mod(1,1936)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1936, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1936.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1936 = 2^{4} \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1936.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: $$114.227697771$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 121) Fricke sign: $$-1$$ Sato-Tate group: $N(\mathrm{U}(1))$

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - 8 q^{3} + 18 q^{5} + 37 q^{9}+O(q^{10})$$ q - 8 * q^3 + 18 * q^5 + 37 * q^9 $$q - 8 q^{3} + 18 q^{5} + 37 q^{9} - 144 q^{15} + 108 q^{23} + 199 q^{25} - 80 q^{27} - 340 q^{31} - 434 q^{37} + 666 q^{45} + 36 q^{47} - 343 q^{49} - 738 q^{53} + 720 q^{59} + 416 q^{67} - 864 q^{69} - 612 q^{71} - 1592 q^{75} - 359 q^{81} + 1674 q^{89} + 2720 q^{93} - 34 q^{97}+O(q^{100})$$ q - 8 * q^3 + 18 * q^5 + 37 * q^9 - 144 * q^15 + 108 * q^23 + 199 * q^25 - 80 * q^27 - 340 * q^31 - 434 * q^37 + 666 * q^45 + 36 * q^47 - 343 * q^49 - 738 * q^53 + 720 * q^59 + 416 * q^67 - 864 * q^69 - 612 * q^71 - 1592 * q^75 - 359 * q^81 + 1674 * q^89 + 2720 * q^93 - 34 * 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 −8.00000 0 18.0000 0 0 0 37.0000 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
$$2$$ $$-1$$
$$11$$ $$1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 CM by $$\Q(\sqrt{-11})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1936.4.a.a 1
4.b odd 2 1 121.4.a.a 1
11.b odd 2 1 CM 1936.4.a.a 1
12.b even 2 1 1089.4.a.f 1
44.c even 2 1 121.4.a.a 1
44.g even 10 4 121.4.c.a 4
44.h odd 10 4 121.4.c.a 4
132.d odd 2 1 1089.4.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
121.4.a.a 1 4.b odd 2 1
121.4.a.a 1 44.c even 2 1
121.4.c.a 4 44.g even 10 4
121.4.c.a 4 44.h odd 10 4
1089.4.a.f 1 12.b even 2 1
1089.4.a.f 1 132.d odd 2 1
1936.4.a.a 1 1.a even 1 1 trivial
1936.4.a.a 1 11.b odd 2 1 CM

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1936))$$:

 $$T_{3} + 8$$ T3 + 8 $$T_{5} - 18$$ T5 - 18 $$T_{7}$$ T7

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 8$$
$5$ $$T - 18$$
$7$ $$T$$
$11$ $$T$$
$13$ $$T$$
$17$ $$T$$
$19$ $$T$$
$23$ $$T - 108$$
$29$ $$T$$
$31$ $$T + 340$$
$37$ $$T + 434$$
$41$ $$T$$
$43$ $$T$$
$47$ $$T - 36$$
$53$ $$T + 738$$
$59$ $$T - 720$$
$61$ $$T$$
$67$ $$T - 416$$
$71$ $$T + 612$$
$73$ $$T$$
$79$ $$T$$
$83$ $$T$$
$89$ $$T - 1674$$
$97$ $$T + 34$$