# Properties

 Label 1911.4.a.f Level $1911$ Weight $4$ Character orbit 1911.a Self dual yes Analytic conductor $112.753$ 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] = [1911,4,Mod(1,1911)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1911, 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("1911.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1911 = 3 \cdot 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1911.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: $$112.752650021$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 39) 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)

 $$f(q)$$ $$=$$ $$q + 3 q^{3} - 8 q^{4} + 12 q^{5} + 9 q^{9}+O(q^{10})$$ q + 3 * q^3 - 8 * q^4 + 12 * q^5 + 9 * q^9 $$q + 3 q^{3} - 8 q^{4} + 12 q^{5} + 9 q^{9} - 36 q^{11} - 24 q^{12} - 13 q^{13} + 36 q^{15} + 64 q^{16} + 78 q^{17} - 74 q^{19} - 96 q^{20} - 96 q^{23} + 19 q^{25} + 27 q^{27} + 18 q^{29} + 214 q^{31} - 108 q^{33} - 72 q^{36} - 286 q^{37} - 39 q^{39} + 384 q^{41} + 524 q^{43} + 288 q^{44} + 108 q^{45} - 300 q^{47} + 192 q^{48} + 234 q^{51} + 104 q^{52} + 558 q^{53} - 432 q^{55} - 222 q^{57} - 576 q^{59} - 288 q^{60} - 74 q^{61} - 512 q^{64} - 156 q^{65} + 38 q^{67} - 624 q^{68} - 288 q^{69} - 456 q^{71} + 682 q^{73} + 57 q^{75} + 592 q^{76} + 704 q^{79} + 768 q^{80} + 81 q^{81} + 888 q^{83} + 936 q^{85} + 54 q^{87} + 1020 q^{89} + 768 q^{92} + 642 q^{93} - 888 q^{95} - 110 q^{97} - 324 q^{99}+O(q^{100})$$ q + 3 * q^3 - 8 * q^4 + 12 * q^5 + 9 * q^9 - 36 * q^11 - 24 * q^12 - 13 * q^13 + 36 * q^15 + 64 * q^16 + 78 * q^17 - 74 * q^19 - 96 * q^20 - 96 * q^23 + 19 * q^25 + 27 * q^27 + 18 * q^29 + 214 * q^31 - 108 * q^33 - 72 * q^36 - 286 * q^37 - 39 * q^39 + 384 * q^41 + 524 * q^43 + 288 * q^44 + 108 * q^45 - 300 * q^47 + 192 * q^48 + 234 * q^51 + 104 * q^52 + 558 * q^53 - 432 * q^55 - 222 * q^57 - 576 * q^59 - 288 * q^60 - 74 * q^61 - 512 * q^64 - 156 * q^65 + 38 * q^67 - 624 * q^68 - 288 * q^69 - 456 * q^71 + 682 * q^73 + 57 * q^75 + 592 * q^76 + 704 * q^79 + 768 * q^80 + 81 * q^81 + 888 * q^83 + 936 * q^85 + 54 * q^87 + 1020 * q^89 + 768 * q^92 + 642 * q^93 - 888 * q^95 - 110 * q^97 - 324 * 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 −8.00000 12.0000 0 0 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
$$3$$ $$-1$$
$$7$$ $$-1$$
$$13$$ $$1$$

## 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 1911.4.a.f 1
7.b odd 2 1 39.4.a.a 1
21.c even 2 1 117.4.a.a 1
28.d even 2 1 624.4.a.g 1
35.c odd 2 1 975.4.a.e 1
56.e even 2 1 2496.4.a.f 1
56.h odd 2 1 2496.4.a.o 1
84.h odd 2 1 1872.4.a.m 1
91.b odd 2 1 507.4.a.c 1
91.i even 4 2 507.4.b.b 2
273.g even 2 1 1521.4.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.a.a 1 7.b odd 2 1
117.4.a.a 1 21.c even 2 1
507.4.a.c 1 91.b odd 2 1
507.4.b.b 2 91.i even 4 2
624.4.a.g 1 28.d even 2 1
975.4.a.e 1 35.c odd 2 1
1521.4.a.f 1 273.g even 2 1
1872.4.a.m 1 84.h odd 2 1
1911.4.a.f 1 1.a even 1 1 trivial
2496.4.a.f 1 56.e even 2 1
2496.4.a.o 1 56.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1911))$$:

 $$T_{2}$$ T2 $$T_{5} - 12$$ T5 - 12

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 3$$
$5$ $$T - 12$$
$7$ $$T$$
$11$ $$T + 36$$
$13$ $$T + 13$$
$17$ $$T - 78$$
$19$ $$T + 74$$
$23$ $$T + 96$$
$29$ $$T - 18$$
$31$ $$T - 214$$
$37$ $$T + 286$$
$41$ $$T - 384$$
$43$ $$T - 524$$
$47$ $$T + 300$$
$53$ $$T - 558$$
$59$ $$T + 576$$
$61$ $$T + 74$$
$67$ $$T - 38$$
$71$ $$T + 456$$
$73$ $$T - 682$$
$79$ $$T - 704$$
$83$ $$T - 888$$
$89$ $$T - 1020$$
$97$ $$T + 110$$