# Properties

 Label 2112.4.a.u Level $2112$ Weight $4$ Character orbit 2112.a Self dual yes Analytic conductor $124.612$ 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] = [2112,4,Mod(1,2112)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2112, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2112.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2112 = 2^{6} \cdot 3 \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2112.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: $$124.612033932$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 33) 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)

 $$f(q)$$ $$=$$ $$q + 3 q^{3} + 4 q^{5} - 26 q^{7} + 9 q^{9}+O(q^{10})$$ q + 3 * q^3 + 4 * q^5 - 26 * q^7 + 9 * q^9 $$q + 3 q^{3} + 4 q^{5} - 26 q^{7} + 9 q^{9} - 11 q^{11} + 32 q^{13} + 12 q^{15} + 74 q^{17} + 60 q^{19} - 78 q^{21} - 182 q^{23} - 109 q^{25} + 27 q^{27} + 90 q^{29} - 8 q^{31} - 33 q^{33} - 104 q^{35} + 66 q^{37} + 96 q^{39} + 422 q^{41} - 408 q^{43} + 36 q^{45} - 506 q^{47} + 333 q^{49} + 222 q^{51} - 348 q^{53} - 44 q^{55} + 180 q^{57} + 200 q^{59} - 132 q^{61} - 234 q^{63} + 128 q^{65} + 1036 q^{67} - 546 q^{69} + 762 q^{71} - 542 q^{73} - 327 q^{75} + 286 q^{77} - 550 q^{79} + 81 q^{81} + 132 q^{83} + 296 q^{85} + 270 q^{87} + 570 q^{89} - 832 q^{91} - 24 q^{93} + 240 q^{95} + 14 q^{97} - 99 q^{99}+O(q^{100})$$ q + 3 * q^3 + 4 * q^5 - 26 * q^7 + 9 * q^9 - 11 * q^11 + 32 * q^13 + 12 * q^15 + 74 * q^17 + 60 * q^19 - 78 * q^21 - 182 * q^23 - 109 * q^25 + 27 * q^27 + 90 * q^29 - 8 * q^31 - 33 * q^33 - 104 * q^35 + 66 * q^37 + 96 * q^39 + 422 * q^41 - 408 * q^43 + 36 * q^45 - 506 * q^47 + 333 * q^49 + 222 * q^51 - 348 * q^53 - 44 * q^55 + 180 * q^57 + 200 * q^59 - 132 * q^61 - 234 * q^63 + 128 * q^65 + 1036 * q^67 - 546 * q^69 + 762 * q^71 - 542 * q^73 - 327 * q^75 + 286 * q^77 - 550 * q^79 + 81 * q^81 + 132 * q^83 + 296 * q^85 + 270 * q^87 + 570 * q^89 - 832 * q^91 - 24 * q^93 + 240 * q^95 + 14 * q^97 - 99 * 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 3.00000 0 4.00000 0 −26.0000 0 9.00000 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
$$2$$ $$1$$
$$3$$ $$-1$$
$$11$$ $$1$$

## 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 2112.4.a.u 1
4.b odd 2 1 2112.4.a.h 1
8.b even 2 1 33.4.a.b 1
8.d odd 2 1 528.4.a.h 1
24.f even 2 1 1584.4.a.l 1
24.h odd 2 1 99.4.a.a 1
40.f even 2 1 825.4.a.f 1
40.i odd 4 2 825.4.c.f 2
56.h odd 2 1 1617.4.a.d 1
88.b odd 2 1 363.4.a.d 1
120.i odd 2 1 2475.4.a.e 1
264.m even 2 1 1089.4.a.e 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.4.a.b 1 8.b even 2 1
99.4.a.a 1 24.h odd 2 1
363.4.a.d 1 88.b odd 2 1
528.4.a.h 1 8.d odd 2 1
825.4.a.f 1 40.f even 2 1
825.4.c.f 2 40.i odd 4 2
1089.4.a.e 1 264.m even 2 1
1584.4.a.l 1 24.f even 2 1
1617.4.a.d 1 56.h odd 2 1
2112.4.a.h 1 4.b odd 2 1
2112.4.a.u 1 1.a even 1 1 trivial
2475.4.a.e 1 120.i odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(2112))$$:

 $$T_{5} - 4$$ T5 - 4 $$T_{7} + 26$$ T7 + 26

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 3$$
$5$ $$T - 4$$
$7$ $$T + 26$$
$11$ $$T + 11$$
$13$ $$T - 32$$
$17$ $$T - 74$$
$19$ $$T - 60$$
$23$ $$T + 182$$
$29$ $$T - 90$$
$31$ $$T + 8$$
$37$ $$T - 66$$
$41$ $$T - 422$$
$43$ $$T + 408$$
$47$ $$T + 506$$
$53$ $$T + 348$$
$59$ $$T - 200$$
$61$ $$T + 132$$
$67$ $$T - 1036$$
$71$ $$T - 762$$
$73$ $$T + 542$$
$79$ $$T + 550$$
$83$ $$T - 132$$
$89$ $$T - 570$$
$97$ $$T - 14$$