Properties

 Label 114.2.a.b Level $114$ Weight $2$ Character orbit 114.a Self dual yes Analytic conductor $0.910$ 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] = [114,2,Mod(1,114)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(114, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("114.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$114 = 2 \cdot 3 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 114.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: $$0.910294583043$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes 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)

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{3} + q^{4} + 2 q^{5} - q^{6} + q^{8} + q^{9}+O(q^{10})$$ q + q^2 - q^3 + q^4 + 2 * q^5 - q^6 + q^8 + q^9 $$q + q^{2} - q^{3} + q^{4} + 2 q^{5} - q^{6} + q^{8} + q^{9} + 2 q^{10} - 4 q^{11} - q^{12} + 2 q^{13} - 2 q^{15} + q^{16} - 6 q^{17} + q^{18} - q^{19} + 2 q^{20} - 4 q^{22} - 4 q^{23} - q^{24} - q^{25} + 2 q^{26} - q^{27} - 2 q^{29} - 2 q^{30} + 4 q^{31} + q^{32} + 4 q^{33} - 6 q^{34} + q^{36} + 10 q^{37} - q^{38} - 2 q^{39} + 2 q^{40} + 10 q^{41} + 4 q^{43} - 4 q^{44} + 2 q^{45} - 4 q^{46} - 4 q^{47} - q^{48} - 7 q^{49} - q^{50} + 6 q^{51} + 2 q^{52} - 10 q^{53} - q^{54} - 8 q^{55} + q^{57} - 2 q^{58} + 12 q^{59} - 2 q^{60} + 14 q^{61} + 4 q^{62} + q^{64} + 4 q^{65} + 4 q^{66} - 12 q^{67} - 6 q^{68} + 4 q^{69} + 8 q^{71} + q^{72} - 6 q^{73} + 10 q^{74} + q^{75} - q^{76} - 2 q^{78} - 4 q^{79} + 2 q^{80} + q^{81} + 10 q^{82} + 12 q^{83} - 12 q^{85} + 4 q^{86} + 2 q^{87} - 4 q^{88} - 6 q^{89} + 2 q^{90} - 4 q^{92} - 4 q^{93} - 4 q^{94} - 2 q^{95} - q^{96} + 10 q^{97} - 7 q^{98} - 4 q^{99}+O(q^{100})$$ q + q^2 - q^3 + q^4 + 2 * q^5 - q^6 + q^8 + q^9 + 2 * q^10 - 4 * q^11 - q^12 + 2 * q^13 - 2 * q^15 + q^16 - 6 * q^17 + q^18 - q^19 + 2 * q^20 - 4 * q^22 - 4 * q^23 - q^24 - q^25 + 2 * q^26 - q^27 - 2 * q^29 - 2 * q^30 + 4 * q^31 + q^32 + 4 * q^33 - 6 * q^34 + q^36 + 10 * q^37 - q^38 - 2 * q^39 + 2 * q^40 + 10 * q^41 + 4 * q^43 - 4 * q^44 + 2 * q^45 - 4 * q^46 - 4 * q^47 - q^48 - 7 * q^49 - q^50 + 6 * q^51 + 2 * q^52 - 10 * q^53 - q^54 - 8 * q^55 + q^57 - 2 * q^58 + 12 * q^59 - 2 * q^60 + 14 * q^61 + 4 * q^62 + q^64 + 4 * q^65 + 4 * q^66 - 12 * q^67 - 6 * q^68 + 4 * q^69 + 8 * q^71 + q^72 - 6 * q^73 + 10 * q^74 + q^75 - q^76 - 2 * q^78 - 4 * q^79 + 2 * q^80 + q^81 + 10 * q^82 + 12 * q^83 - 12 * q^85 + 4 * q^86 + 2 * q^87 - 4 * q^88 - 6 * q^89 + 2 * q^90 - 4 * q^92 - 4 * q^93 - 4 * q^94 - 2 * q^95 - q^96 + 10 * q^97 - 7 * q^98 - 4 * 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
1.00000 −1.00000 1.00000 2.00000 −1.00000 0 1.00000 1.00000 2.00000
 $$n$$: 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 114.2.a.b 1
3.b odd 2 1 342.2.a.b 1
4.b odd 2 1 912.2.a.k 1
5.b even 2 1 2850.2.a.j 1
5.c odd 4 2 2850.2.d.b 2
7.b odd 2 1 5586.2.a.y 1
8.b even 2 1 3648.2.a.x 1
8.d odd 2 1 3648.2.a.c 1
12.b even 2 1 2736.2.a.d 1
15.d odd 2 1 8550.2.a.ba 1
19.b odd 2 1 2166.2.a.d 1
57.d even 2 1 6498.2.a.p 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.2.a.b 1 1.a even 1 1 trivial
342.2.a.b 1 3.b odd 2 1
912.2.a.k 1 4.b odd 2 1
2166.2.a.d 1 19.b odd 2 1
2736.2.a.d 1 12.b even 2 1
2850.2.a.j 1 5.b even 2 1
2850.2.d.b 2 5.c odd 4 2
3648.2.a.c 1 8.d odd 2 1
3648.2.a.x 1 8.b even 2 1
5586.2.a.y 1 7.b odd 2 1
6498.2.a.p 1 57.d even 2 1
8550.2.a.ba 1 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(114))$$:

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

Hecke characteristic polynomials

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