# Properties

 Label 1936.4.a.m Level $1936$ Weight $4$ Character orbit 1936.a Self dual yes Analytic conductor $114.228$ 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] = [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: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 44) 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)

 $$f(q)$$ $$=$$ $$q + 5 q^{3} - 7 q^{5} - 26 q^{7} - 2 q^{9}+O(q^{10})$$ q + 5 * q^3 - 7 * q^5 - 26 * q^7 - 2 * q^9 $$q + 5 q^{3} - 7 q^{5} - 26 q^{7} - 2 q^{9} - 52 q^{13} - 35 q^{15} - 46 q^{17} - 96 q^{19} - 130 q^{21} - 27 q^{23} - 76 q^{25} - 145 q^{27} - 16 q^{29} + 293 q^{31} + 182 q^{35} - 29 q^{37} - 260 q^{39} + 472 q^{41} - 110 q^{43} + 14 q^{45} + 224 q^{47} + 333 q^{49} - 230 q^{51} + 754 q^{53} - 480 q^{57} - 825 q^{59} + 548 q^{61} + 52 q^{63} + 364 q^{65} + 123 q^{67} - 135 q^{69} - 1001 q^{71} + 1020 q^{73} - 380 q^{75} + 526 q^{79} - 671 q^{81} - 158 q^{83} + 322 q^{85} - 80 q^{87} - 1217 q^{89} + 1352 q^{91} + 1465 q^{93} + 672 q^{95} - 263 q^{97}+O(q^{100})$$ q + 5 * q^3 - 7 * q^5 - 26 * q^7 - 2 * q^9 - 52 * q^13 - 35 * q^15 - 46 * q^17 - 96 * q^19 - 130 * q^21 - 27 * q^23 - 76 * q^25 - 145 * q^27 - 16 * q^29 + 293 * q^31 + 182 * q^35 - 29 * q^37 - 260 * q^39 + 472 * q^41 - 110 * q^43 + 14 * q^45 + 224 * q^47 + 333 * q^49 - 230 * q^51 + 754 * q^53 - 480 * q^57 - 825 * q^59 + 548 * q^61 + 52 * q^63 + 364 * q^65 + 123 * q^67 - 135 * q^69 - 1001 * q^71 + 1020 * q^73 - 380 * q^75 + 526 * q^79 - 671 * q^81 - 158 * q^83 + 322 * q^85 - 80 * q^87 - 1217 * q^89 + 1352 * q^91 + 1465 * q^93 + 672 * q^95 - 263 * 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 5.00000 0 −7.00000 0 −26.0000 0 −2.00000 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
$$2$$ $$-1$$
$$11$$ $$-1$$

## 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 1936.4.a.m 1
4.b odd 2 1 484.4.a.a 1
11.b odd 2 1 176.4.a.e 1
33.d even 2 1 1584.4.a.p 1
44.c even 2 1 44.4.a.a 1
88.b odd 2 1 704.4.a.c 1
88.g even 2 1 704.4.a.j 1
132.d odd 2 1 396.4.a.e 1
220.g even 2 1 1100.4.a.d 1
220.i odd 4 2 1100.4.b.c 2
308.g odd 2 1 2156.4.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
44.4.a.a 1 44.c even 2 1
176.4.a.e 1 11.b odd 2 1
396.4.a.e 1 132.d odd 2 1
484.4.a.a 1 4.b odd 2 1
704.4.a.c 1 88.b odd 2 1
704.4.a.j 1 88.g even 2 1
1100.4.a.d 1 220.g even 2 1
1100.4.b.c 2 220.i odd 4 2
1584.4.a.p 1 33.d even 2 1
1936.4.a.m 1 1.a even 1 1 trivial
2156.4.a.b 1 308.g odd 2 1

## Hecke kernels

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

 $$T_{3} - 5$$ T3 - 5 $$T_{5} + 7$$ T5 + 7 $$T_{7} + 26$$ T7 + 26

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 5$$
$5$ $$T + 7$$
$7$ $$T + 26$$
$11$ $$T$$
$13$ $$T + 52$$
$17$ $$T + 46$$
$19$ $$T + 96$$
$23$ $$T + 27$$
$29$ $$T + 16$$
$31$ $$T - 293$$
$37$ $$T + 29$$
$41$ $$T - 472$$
$43$ $$T + 110$$
$47$ $$T - 224$$
$53$ $$T - 754$$
$59$ $$T + 825$$
$61$ $$T - 548$$
$67$ $$T - 123$$
$71$ $$T + 1001$$
$73$ $$T - 1020$$
$79$ $$T - 526$$
$83$ $$T + 158$$
$89$ $$T + 1217$$
$97$ $$T + 263$$