# Properties

 Label 114.2.e.a Level $114$ Weight $2$ Character orbit 114.e 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(7,114)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(114, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2]))

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

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

chi := DirichletCharacter("114.7");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$114 = 2 \cdot 3 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 114.e (of order $$3$$, degree $$2$$, 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + (\zeta_{6} - 1) q^{2} + ( - \zeta_{6} + 1) q^{3} - \zeta_{6} q^{4} + ( - 4 \zeta_{6} + 4) q^{5} + \zeta_{6} q^{6} - 3 q^{7} + q^{8} - \zeta_{6} q^{9} +O(q^{10})$$ q + (z - 1) * q^2 + (-z + 1) * q^3 - z * q^4 + (-4*z + 4) * q^5 + z * q^6 - 3 * q^7 + q^8 - z * q^9 $$q + (\zeta_{6} - 1) q^{2} + ( - \zeta_{6} + 1) q^{3} - \zeta_{6} q^{4} + ( - 4 \zeta_{6} + 4) q^{5} + \zeta_{6} q^{6} - 3 q^{7} + q^{8} - \zeta_{6} q^{9} + 4 \zeta_{6} q^{10} + 2 q^{11} - q^{12} + 7 \zeta_{6} q^{13} + ( - 3 \zeta_{6} + 3) q^{14} - 4 \zeta_{6} q^{15} + (\zeta_{6} - 1) q^{16} + q^{18} + (2 \zeta_{6} - 5) q^{19} - 4 q^{20} + (3 \zeta_{6} - 3) q^{21} + (2 \zeta_{6} - 2) q^{22} + 4 \zeta_{6} q^{23} + ( - \zeta_{6} + 1) q^{24} - 11 \zeta_{6} q^{25} - 7 q^{26} - q^{27} + 3 \zeta_{6} q^{28} - 4 \zeta_{6} q^{29} + 4 q^{30} + q^{31} - \zeta_{6} q^{32} + ( - 2 \zeta_{6} + 2) q^{33} + (12 \zeta_{6} - 12) q^{35} + (\zeta_{6} - 1) q^{36} + 7 q^{37} + ( - 5 \zeta_{6} + 3) q^{38} + 7 q^{39} + ( - 4 \zeta_{6} + 4) q^{40} + (4 \zeta_{6} - 4) q^{41} - 3 \zeta_{6} q^{42} + (7 \zeta_{6} - 7) q^{43} - 2 \zeta_{6} q^{44} - 4 q^{45} - 4 q^{46} - 2 \zeta_{6} q^{47} + \zeta_{6} q^{48} + 2 q^{49} + 11 q^{50} + ( - 7 \zeta_{6} + 7) q^{52} + 4 \zeta_{6} q^{53} + ( - \zeta_{6} + 1) q^{54} + ( - 8 \zeta_{6} + 8) q^{55} - 3 q^{56} + (5 \zeta_{6} - 3) q^{57} + 4 q^{58} + ( - 6 \zeta_{6} + 6) q^{59} + (4 \zeta_{6} - 4) q^{60} + \zeta_{6} q^{61} + (\zeta_{6} - 1) q^{62} + 3 \zeta_{6} q^{63} + q^{64} + 28 q^{65} + 2 \zeta_{6} q^{66} - 3 \zeta_{6} q^{67} + 4 q^{69} - 12 \zeta_{6} q^{70} + (2 \zeta_{6} - 2) q^{71} - \zeta_{6} q^{72} + ( - 3 \zeta_{6} + 3) q^{73} + (7 \zeta_{6} - 7) q^{74} - 11 q^{75} + (3 \zeta_{6} + 2) q^{76} - 6 q^{77} + (7 \zeta_{6} - 7) q^{78} + (5 \zeta_{6} - 5) q^{79} + 4 \zeta_{6} q^{80} + (\zeta_{6} - 1) q^{81} - 4 \zeta_{6} q^{82} - 12 q^{83} + 3 q^{84} - 7 \zeta_{6} q^{86} - 4 q^{87} + 2 q^{88} - 18 \zeta_{6} q^{89} + ( - 4 \zeta_{6} + 4) q^{90} - 21 \zeta_{6} q^{91} + ( - 4 \zeta_{6} + 4) q^{92} + ( - \zeta_{6} + 1) q^{93} + 2 q^{94} + (20 \zeta_{6} - 12) q^{95} - q^{96} + (10 \zeta_{6} - 10) q^{97} + (2 \zeta_{6} - 2) q^{98} - 2 \zeta_{6} q^{99} +O(q^{100})$$ q + (z - 1) * q^2 + (-z + 1) * q^3 - z * q^4 + (-4*z + 4) * q^5 + z * q^6 - 3 * q^7 + q^8 - z * q^9 + 4*z * q^10 + 2 * q^11 - q^12 + 7*z * q^13 + (-3*z + 3) * q^14 - 4*z * q^15 + (z - 1) * q^16 + q^18 + (2*z - 5) * q^19 - 4 * q^20 + (3*z - 3) * q^21 + (2*z - 2) * q^22 + 4*z * q^23 + (-z + 1) * q^24 - 11*z * q^25 - 7 * q^26 - q^27 + 3*z * q^28 - 4*z * q^29 + 4 * q^30 + q^31 - z * q^32 + (-2*z + 2) * q^33 + (12*z - 12) * q^35 + (z - 1) * q^36 + 7 * q^37 + (-5*z + 3) * q^38 + 7 * q^39 + (-4*z + 4) * q^40 + (4*z - 4) * q^41 - 3*z * q^42 + (7*z - 7) * q^43 - 2*z * q^44 - 4 * q^45 - 4 * q^46 - 2*z * q^47 + z * q^48 + 2 * q^49 + 11 * q^50 + (-7*z + 7) * q^52 + 4*z * q^53 + (-z + 1) * q^54 + (-8*z + 8) * q^55 - 3 * q^56 + (5*z - 3) * q^57 + 4 * q^58 + (-6*z + 6) * q^59 + (4*z - 4) * q^60 + z * q^61 + (z - 1) * q^62 + 3*z * q^63 + q^64 + 28 * q^65 + 2*z * q^66 - 3*z * q^67 + 4 * q^69 - 12*z * q^70 + (2*z - 2) * q^71 - z * q^72 + (-3*z + 3) * q^73 + (7*z - 7) * q^74 - 11 * q^75 + (3*z + 2) * q^76 - 6 * q^77 + (7*z - 7) * q^78 + (5*z - 5) * q^79 + 4*z * q^80 + (z - 1) * q^81 - 4*z * q^82 - 12 * q^83 + 3 * q^84 - 7*z * q^86 - 4 * q^87 + 2 * q^88 - 18*z * q^89 + (-4*z + 4) * q^90 - 21*z * q^91 + (-4*z + 4) * q^92 + (-z + 1) * q^93 + 2 * q^94 + (20*z - 12) * q^95 - q^96 + (10*z - 10) * q^97 + (2*z - 2) * q^98 - 2*z * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} + q^{3} - q^{4} + 4 q^{5} + q^{6} - 6 q^{7} + 2 q^{8} - q^{9}+O(q^{10})$$ 2 * q - q^2 + q^3 - q^4 + 4 * q^5 + q^6 - 6 * q^7 + 2 * q^8 - q^9 $$2 q - q^{2} + q^{3} - q^{4} + 4 q^{5} + q^{6} - 6 q^{7} + 2 q^{8} - q^{9} + 4 q^{10} + 4 q^{11} - 2 q^{12} + 7 q^{13} + 3 q^{14} - 4 q^{15} - q^{16} + 2 q^{18} - 8 q^{19} - 8 q^{20} - 3 q^{21} - 2 q^{22} + 4 q^{23} + q^{24} - 11 q^{25} - 14 q^{26} - 2 q^{27} + 3 q^{28} - 4 q^{29} + 8 q^{30} + 2 q^{31} - q^{32} + 2 q^{33} - 12 q^{35} - q^{36} + 14 q^{37} + q^{38} + 14 q^{39} + 4 q^{40} - 4 q^{41} - 3 q^{42} - 7 q^{43} - 2 q^{44} - 8 q^{45} - 8 q^{46} - 2 q^{47} + q^{48} + 4 q^{49} + 22 q^{50} + 7 q^{52} + 4 q^{53} + q^{54} + 8 q^{55} - 6 q^{56} - q^{57} + 8 q^{58} + 6 q^{59} - 4 q^{60} + q^{61} - q^{62} + 3 q^{63} + 2 q^{64} + 56 q^{65} + 2 q^{66} - 3 q^{67} + 8 q^{69} - 12 q^{70} - 2 q^{71} - q^{72} + 3 q^{73} - 7 q^{74} - 22 q^{75} + 7 q^{76} - 12 q^{77} - 7 q^{78} - 5 q^{79} + 4 q^{80} - q^{81} - 4 q^{82} - 24 q^{83} + 6 q^{84} - 7 q^{86} - 8 q^{87} + 4 q^{88} - 18 q^{89} + 4 q^{90} - 21 q^{91} + 4 q^{92} + q^{93} + 4 q^{94} - 4 q^{95} - 2 q^{96} - 10 q^{97} - 2 q^{98} - 2 q^{99}+O(q^{100})$$ 2 * q - q^2 + q^3 - q^4 + 4 * q^5 + q^6 - 6 * q^7 + 2 * q^8 - q^9 + 4 * q^10 + 4 * q^11 - 2 * q^12 + 7 * q^13 + 3 * q^14 - 4 * q^15 - q^16 + 2 * q^18 - 8 * q^19 - 8 * q^20 - 3 * q^21 - 2 * q^22 + 4 * q^23 + q^24 - 11 * q^25 - 14 * q^26 - 2 * q^27 + 3 * q^28 - 4 * q^29 + 8 * q^30 + 2 * q^31 - q^32 + 2 * q^33 - 12 * q^35 - q^36 + 14 * q^37 + q^38 + 14 * q^39 + 4 * q^40 - 4 * q^41 - 3 * q^42 - 7 * q^43 - 2 * q^44 - 8 * q^45 - 8 * q^46 - 2 * q^47 + q^48 + 4 * q^49 + 22 * q^50 + 7 * q^52 + 4 * q^53 + q^54 + 8 * q^55 - 6 * q^56 - q^57 + 8 * q^58 + 6 * q^59 - 4 * q^60 + q^61 - q^62 + 3 * q^63 + 2 * q^64 + 56 * q^65 + 2 * q^66 - 3 * q^67 + 8 * q^69 - 12 * q^70 - 2 * q^71 - q^72 + 3 * q^73 - 7 * q^74 - 22 * q^75 + 7 * q^76 - 12 * q^77 - 7 * q^78 - 5 * q^79 + 4 * q^80 - q^81 - 4 * q^82 - 24 * q^83 + 6 * q^84 - 7 * q^86 - 8 * q^87 + 4 * q^88 - 18 * q^89 + 4 * q^90 - 21 * q^91 + 4 * q^92 + q^93 + 4 * q^94 - 4 * q^95 - 2 * q^96 - 10 * q^97 - 2 * q^98 - 2 * q^99

## Character values

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

 $$n$$ $$77$$ $$97$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1
 0.5 − 0.866025i 0.5 + 0.866025i
−0.500000 0.866025i 0.500000 + 0.866025i −0.500000 + 0.866025i 2.00000 + 3.46410i 0.500000 0.866025i −3.00000 1.00000 −0.500000 + 0.866025i 2.00000 3.46410i
49.1 −0.500000 + 0.866025i 0.500000 0.866025i −0.500000 0.866025i 2.00000 3.46410i 0.500000 + 0.866025i −3.00000 1.00000 −0.500000 0.866025i 2.00000 + 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
19.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 114.2.e.a 2
3.b odd 2 1 342.2.g.d 2
4.b odd 2 1 912.2.q.d 2
12.b even 2 1 2736.2.s.c 2
19.c even 3 1 inner 114.2.e.a 2
19.c even 3 1 2166.2.a.f 1
19.d odd 6 1 2166.2.a.c 1
57.f even 6 1 6498.2.a.x 1
57.h odd 6 1 342.2.g.d 2
57.h odd 6 1 6498.2.a.l 1
76.g odd 6 1 912.2.q.d 2
228.m even 6 1 2736.2.s.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.2.e.a 2 1.a even 1 1 trivial
114.2.e.a 2 19.c even 3 1 inner
342.2.g.d 2 3.b odd 2 1
342.2.g.d 2 57.h odd 6 1
912.2.q.d 2 4.b odd 2 1
912.2.q.d 2 76.g odd 6 1
2166.2.a.c 1 19.d odd 6 1
2166.2.a.f 1 19.c even 3 1
2736.2.s.c 2 12.b even 2 1
2736.2.s.c 2 228.m even 6 1
6498.2.a.l 1 57.h odd 6 1
6498.2.a.x 1 57.f even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 4T_{5} + 16$$ acting on $$S_{2}^{\mathrm{new}}(114, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T + 1$$
$3$ $$T^{2} - T + 1$$
$5$ $$T^{2} - 4T + 16$$
$7$ $$(T + 3)^{2}$$
$11$ $$(T - 2)^{2}$$
$13$ $$T^{2} - 7T + 49$$
$17$ $$T^{2}$$
$19$ $$T^{2} + 8T + 19$$
$23$ $$T^{2} - 4T + 16$$
$29$ $$T^{2} + 4T + 16$$
$31$ $$(T - 1)^{2}$$
$37$ $$(T - 7)^{2}$$
$41$ $$T^{2} + 4T + 16$$
$43$ $$T^{2} + 7T + 49$$
$47$ $$T^{2} + 2T + 4$$
$53$ $$T^{2} - 4T + 16$$
$59$ $$T^{2} - 6T + 36$$
$61$ $$T^{2} - T + 1$$
$67$ $$T^{2} + 3T + 9$$
$71$ $$T^{2} + 2T + 4$$
$73$ $$T^{2} - 3T + 9$$
$79$ $$T^{2} + 5T + 25$$
$83$ $$(T + 12)^{2}$$
$89$ $$T^{2} + 18T + 324$$
$97$ $$T^{2} + 10T + 100$$