# Properties

 Label 114.2.b.b Level $114$ Weight $2$ Character orbit 114.b Analytic conductor $0.910$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [114,2,Mod(113,114)]

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([1, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("114.113");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$114 = 2 \cdot 3 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 114.b (of order $$2$$, degree $$1$$, minimal)

## 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: no Analytic conductor: $$0.910294583043$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + (\zeta_{6} + 1) q^{3} + q^{4} + ( - 4 \zeta_{6} + 2) q^{5} + ( - \zeta_{6} - 1) q^{6} + q^{7} - q^{8} + 3 \zeta_{6} q^{9}+O(q^{10})$$ q - q^2 + (z + 1) * q^3 + q^4 + (-4*z + 2) * q^5 + (-z - 1) * q^6 + q^7 - q^8 + 3*z * q^9 $$q - q^{2} + (\zeta_{6} + 1) q^{3} + q^{4} + ( - 4 \zeta_{6} + 2) q^{5} + ( - \zeta_{6} - 1) q^{6} + q^{7} - q^{8} + 3 \zeta_{6} q^{9} + (4 \zeta_{6} - 2) q^{10} + (4 \zeta_{6} - 2) q^{11} + (\zeta_{6} + 1) q^{12} + ( - 2 \zeta_{6} + 1) q^{13} - q^{14} + ( - 6 \zeta_{6} + 6) q^{15} + q^{16} + ( - 2 \zeta_{6} + 1) q^{17} - 3 \zeta_{6} q^{18} + (2 \zeta_{6} - 5) q^{19} + ( - 4 \zeta_{6} + 2) q^{20} + (\zeta_{6} + 1) q^{21} + ( - 4 \zeta_{6} + 2) q^{22} + (6 \zeta_{6} - 3) q^{23} + ( - \zeta_{6} - 1) q^{24} - 7 q^{25} + (2 \zeta_{6} - 1) q^{26} + (6 \zeta_{6} - 3) q^{27} + q^{28} - 9 q^{29} + (6 \zeta_{6} - 6) q^{30} + ( - 12 \zeta_{6} + 6) q^{31} - q^{32} + (6 \zeta_{6} - 6) q^{33} + (2 \zeta_{6} - 1) q^{34} + ( - 4 \zeta_{6} + 2) q^{35} + 3 \zeta_{6} q^{36} + (8 \zeta_{6} - 4) q^{37} + ( - 2 \zeta_{6} + 5) q^{38} + ( - 3 \zeta_{6} + 3) q^{39} + (4 \zeta_{6} - 2) q^{40} + ( - \zeta_{6} - 1) q^{42} + 2 q^{43} + (4 \zeta_{6} - 2) q^{44} + ( - 6 \zeta_{6} + 12) q^{45} + ( - 6 \zeta_{6} + 3) q^{46} + ( - 4 \zeta_{6} + 2) q^{47} + (\zeta_{6} + 1) q^{48} - 6 q^{49} + 7 q^{50} + ( - 3 \zeta_{6} + 3) q^{51} + ( - 2 \zeta_{6} + 1) q^{52} + 9 q^{53} + ( - 6 \zeta_{6} + 3) q^{54} + 12 q^{55} - q^{56} + ( - \zeta_{6} - 7) q^{57} + 9 q^{58} + 3 q^{59} + ( - 6 \zeta_{6} + 6) q^{60} - 8 q^{61} + (12 \zeta_{6} - 6) q^{62} + 3 \zeta_{6} q^{63} + q^{64} - 6 q^{65} + ( - 6 \zeta_{6} + 6) q^{66} + (10 \zeta_{6} - 5) q^{67} + ( - 2 \zeta_{6} + 1) q^{68} + (9 \zeta_{6} - 9) q^{69} + (4 \zeta_{6} - 2) q^{70} + 12 q^{71} - 3 \zeta_{6} q^{72} + 11 q^{73} + ( - 8 \zeta_{6} + 4) q^{74} + ( - 7 \zeta_{6} - 7) q^{75} + (2 \zeta_{6} - 5) q^{76} + (4 \zeta_{6} - 2) q^{77} + (3 \zeta_{6} - 3) q^{78} + (8 \zeta_{6} - 4) q^{79} + ( - 4 \zeta_{6} + 2) q^{80} + (9 \zeta_{6} - 9) q^{81} + ( - 12 \zeta_{6} + 6) q^{83} + (\zeta_{6} + 1) q^{84} - 6 q^{85} - 2 q^{86} + ( - 9 \zeta_{6} - 9) q^{87} + ( - 4 \zeta_{6} + 2) q^{88} - 6 q^{89} + (6 \zeta_{6} - 12) q^{90} + ( - 2 \zeta_{6} + 1) q^{91} + (6 \zeta_{6} - 3) q^{92} + ( - 18 \zeta_{6} + 18) q^{93} + (4 \zeta_{6} - 2) q^{94} + (16 \zeta_{6} - 2) q^{95} + ( - \zeta_{6} - 1) q^{96} + ( - 16 \zeta_{6} + 8) q^{97} + 6 q^{98} + (6 \zeta_{6} - 12) q^{99} +O(q^{100})$$ q - q^2 + (z + 1) * q^3 + q^4 + (-4*z + 2) * q^5 + (-z - 1) * q^6 + q^7 - q^8 + 3*z * q^9 + (4*z - 2) * q^10 + (4*z - 2) * q^11 + (z + 1) * q^12 + (-2*z + 1) * q^13 - q^14 + (-6*z + 6) * q^15 + q^16 + (-2*z + 1) * q^17 - 3*z * q^18 + (2*z - 5) * q^19 + (-4*z + 2) * q^20 + (z + 1) * q^21 + (-4*z + 2) * q^22 + (6*z - 3) * q^23 + (-z - 1) * q^24 - 7 * q^25 + (2*z - 1) * q^26 + (6*z - 3) * q^27 + q^28 - 9 * q^29 + (6*z - 6) * q^30 + (-12*z + 6) * q^31 - q^32 + (6*z - 6) * q^33 + (2*z - 1) * q^34 + (-4*z + 2) * q^35 + 3*z * q^36 + (8*z - 4) * q^37 + (-2*z + 5) * q^38 + (-3*z + 3) * q^39 + (4*z - 2) * q^40 + (-z - 1) * q^42 + 2 * q^43 + (4*z - 2) * q^44 + (-6*z + 12) * q^45 + (-6*z + 3) * q^46 + (-4*z + 2) * q^47 + (z + 1) * q^48 - 6 * q^49 + 7 * q^50 + (-3*z + 3) * q^51 + (-2*z + 1) * q^52 + 9 * q^53 + (-6*z + 3) * q^54 + 12 * q^55 - q^56 + (-z - 7) * q^57 + 9 * q^58 + 3 * q^59 + (-6*z + 6) * q^60 - 8 * q^61 + (12*z - 6) * q^62 + 3*z * q^63 + q^64 - 6 * q^65 + (-6*z + 6) * q^66 + (10*z - 5) * q^67 + (-2*z + 1) * q^68 + (9*z - 9) * q^69 + (4*z - 2) * q^70 + 12 * q^71 - 3*z * q^72 + 11 * q^73 + (-8*z + 4) * q^74 + (-7*z - 7) * q^75 + (2*z - 5) * q^76 + (4*z - 2) * q^77 + (3*z - 3) * q^78 + (8*z - 4) * q^79 + (-4*z + 2) * q^80 + (9*z - 9) * q^81 + (-12*z + 6) * q^83 + (z + 1) * q^84 - 6 * q^85 - 2 * q^86 + (-9*z - 9) * q^87 + (-4*z + 2) * q^88 - 6 * q^89 + (6*z - 12) * q^90 + (-2*z + 1) * q^91 + (6*z - 3) * q^92 + (-18*z + 18) * q^93 + (4*z - 2) * q^94 + (16*z - 2) * q^95 + (-z - 1) * q^96 + (-16*z + 8) * q^97 + 6 * q^98 + (6*z - 12) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 3 q^{3} + 2 q^{4} - 3 q^{6} + 2 q^{7} - 2 q^{8} + 3 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 + 3 * q^3 + 2 * q^4 - 3 * q^6 + 2 * q^7 - 2 * q^8 + 3 * q^9 $$2 q - 2 q^{2} + 3 q^{3} + 2 q^{4} - 3 q^{6} + 2 q^{7} - 2 q^{8} + 3 q^{9} + 3 q^{12} - 2 q^{14} + 6 q^{15} + 2 q^{16} - 3 q^{18} - 8 q^{19} + 3 q^{21} - 3 q^{24} - 14 q^{25} + 2 q^{28} - 18 q^{29} - 6 q^{30} - 2 q^{32} - 6 q^{33} + 3 q^{36} + 8 q^{38} + 3 q^{39} - 3 q^{42} + 4 q^{43} + 18 q^{45} + 3 q^{48} - 12 q^{49} + 14 q^{50} + 3 q^{51} + 18 q^{53} + 24 q^{55} - 2 q^{56} - 15 q^{57} + 18 q^{58} + 6 q^{59} + 6 q^{60} - 16 q^{61} + 3 q^{63} + 2 q^{64} - 12 q^{65} + 6 q^{66} - 9 q^{69} + 24 q^{71} - 3 q^{72} + 22 q^{73} - 21 q^{75} - 8 q^{76} - 3 q^{78} - 9 q^{81} + 3 q^{84} - 12 q^{85} - 4 q^{86} - 27 q^{87} - 12 q^{89} - 18 q^{90} + 18 q^{93} + 12 q^{95} - 3 q^{96} + 12 q^{98} - 18 q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 + 3 * q^3 + 2 * q^4 - 3 * q^6 + 2 * q^7 - 2 * q^8 + 3 * q^9 + 3 * q^12 - 2 * q^14 + 6 * q^15 + 2 * q^16 - 3 * q^18 - 8 * q^19 + 3 * q^21 - 3 * q^24 - 14 * q^25 + 2 * q^28 - 18 * q^29 - 6 * q^30 - 2 * q^32 - 6 * q^33 + 3 * q^36 + 8 * q^38 + 3 * q^39 - 3 * q^42 + 4 * q^43 + 18 * q^45 + 3 * q^48 - 12 * q^49 + 14 * q^50 + 3 * q^51 + 18 * q^53 + 24 * q^55 - 2 * q^56 - 15 * q^57 + 18 * q^58 + 6 * q^59 + 6 * q^60 - 16 * q^61 + 3 * q^63 + 2 * q^64 - 12 * q^65 + 6 * q^66 - 9 * q^69 + 24 * q^71 - 3 * q^72 + 22 * q^73 - 21 * q^75 - 8 * q^76 - 3 * q^78 - 9 * q^81 + 3 * q^84 - 12 * q^85 - 4 * q^86 - 27 * q^87 - 12 * q^89 - 18 * q^90 + 18 * q^93 + 12 * q^95 - 3 * q^96 + 12 * q^98 - 18 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/114\mathbb{Z}\right)^\times$$.

 $$n$$ $$77$$ $$97$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
113.1
 0.5 − 0.866025i 0.5 + 0.866025i
−1.00000 1.50000 0.866025i 1.00000 3.46410i −1.50000 + 0.866025i 1.00000 −1.00000 1.50000 2.59808i 3.46410i
113.2 −1.00000 1.50000 + 0.866025i 1.00000 3.46410i −1.50000 0.866025i 1.00000 −1.00000 1.50000 + 2.59808i 3.46410i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
57.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 114.2.b.b 2
3.b odd 2 1 114.2.b.c yes 2
4.b odd 2 1 912.2.f.a 2
12.b even 2 1 912.2.f.e 2
19.b odd 2 1 114.2.b.c yes 2
57.d even 2 1 inner 114.2.b.b 2
76.d even 2 1 912.2.f.e 2
228.b odd 2 1 912.2.f.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.2.b.b 2 1.a even 1 1 trivial
114.2.b.b 2 57.d even 2 1 inner
114.2.b.c yes 2 3.b odd 2 1
114.2.b.c yes 2 19.b odd 2 1
912.2.f.a 2 4.b odd 2 1
912.2.f.a 2 228.b odd 2 1
912.2.f.e 2 12.b even 2 1
912.2.f.e 2 76.d even 2 1

## Hecke kernels

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

 $$T_{5}^{2} + 12$$ T5^2 + 12 $$T_{29} + 9$$ T29 + 9

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{2}$$
$3$ $$T^{2} - 3T + 3$$
$5$ $$T^{2} + 12$$
$7$ $$(T - 1)^{2}$$
$11$ $$T^{2} + 12$$
$13$ $$T^{2} + 3$$
$17$ $$T^{2} + 3$$
$19$ $$T^{2} + 8T + 19$$
$23$ $$T^{2} + 27$$
$29$ $$(T + 9)^{2}$$
$31$ $$T^{2} + 108$$
$37$ $$T^{2} + 48$$
$41$ $$T^{2}$$
$43$ $$(T - 2)^{2}$$
$47$ $$T^{2} + 12$$
$53$ $$(T - 9)^{2}$$
$59$ $$(T - 3)^{2}$$
$61$ $$(T + 8)^{2}$$
$67$ $$T^{2} + 75$$
$71$ $$(T - 12)^{2}$$
$73$ $$(T - 11)^{2}$$
$79$ $$T^{2} + 48$$
$83$ $$T^{2} + 108$$
$89$ $$(T + 6)^{2}$$
$97$ $$T^{2} + 192$$