Properties

 Label 114.2.h.b Level $114$ Weight $2$ Character orbit 114.h 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(65,114)]

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([3, 1]))

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

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

chi := DirichletCharacter("114.65");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$114 = 2 \cdot 3 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 114.h (of order $$6$$, 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: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

 $$f(q)$$ $$=$$ $$q + (\zeta_{6} - 1) q^{2} + ( - \zeta_{6} - 1) q^{3} - \zeta_{6} q^{4} + (2 \zeta_{6} + 2) q^{5} + ( - \zeta_{6} + 2) q^{6} + q^{7} + q^{8} + 3 \zeta_{6} q^{9} +O(q^{10})$$ q + (z - 1) * q^2 + (-z - 1) * q^3 - z * q^4 + (2*z + 2) * q^5 + (-z + 2) * q^6 + q^7 + q^8 + 3*z * q^9 $$q + (\zeta_{6} - 1) q^{2} + ( - \zeta_{6} - 1) q^{3} - \zeta_{6} q^{4} + (2 \zeta_{6} + 2) q^{5} + ( - \zeta_{6} + 2) q^{6} + q^{7} + q^{8} + 3 \zeta_{6} q^{9} + (2 \zeta_{6} - 4) q^{10} + (4 \zeta_{6} - 2) q^{11} + (2 \zeta_{6} - 1) q^{12} + ( - 3 \zeta_{6} + 6) q^{13} + (\zeta_{6} - 1) q^{14} - 6 \zeta_{6} q^{15} + (\zeta_{6} - 1) q^{16} + ( - 2 \zeta_{6} - 2) q^{17} - 3 q^{18} + ( - 2 \zeta_{6} - 3) q^{19} + ( - 4 \zeta_{6} + 2) q^{20} + ( - \zeta_{6} - 1) q^{21} + ( - 2 \zeta_{6} - 2) q^{22} + ( - \zeta_{6} - 1) q^{24} + 7 \zeta_{6} q^{25} + (6 \zeta_{6} - 3) q^{26} + ( - 6 \zeta_{6} + 3) q^{27} - \zeta_{6} q^{28} - 6 \zeta_{6} q^{29} + 6 q^{30} + ( - 2 \zeta_{6} + 1) q^{31} - \zeta_{6} q^{32} + ( - 6 \zeta_{6} + 6) q^{33} + ( - 2 \zeta_{6} + 4) q^{34} + (2 \zeta_{6} + 2) q^{35} + ( - 3 \zeta_{6} + 3) q^{36} + (6 \zeta_{6} - 3) q^{37} + ( - 3 \zeta_{6} + 5) q^{38} - 9 q^{39} + (2 \zeta_{6} + 2) q^{40} + (12 \zeta_{6} - 12) q^{41} + ( - \zeta_{6} + 2) q^{42} + ( - \zeta_{6} + 1) q^{43} + ( - 2 \zeta_{6} + 4) q^{44} + (12 \zeta_{6} - 6) q^{45} + ( - 4 \zeta_{6} + 8) q^{47} + ( - \zeta_{6} + 2) q^{48} - 6 q^{49} - 7 q^{50} + 6 \zeta_{6} q^{51} + ( - 3 \zeta_{6} - 3) q^{52} - 12 \zeta_{6} q^{53} + (3 \zeta_{6} + 3) q^{54} + (12 \zeta_{6} - 12) q^{55} + q^{56} + (7 \zeta_{6} + 1) q^{57} + 6 q^{58} + (6 \zeta_{6} - 6) q^{60} - 7 \zeta_{6} q^{61} + (\zeta_{6} + 1) q^{62} + 3 \zeta_{6} q^{63} + q^{64} + 18 q^{65} + 6 \zeta_{6} q^{66} + ( - 5 \zeta_{6} + 10) q^{67} + (4 \zeta_{6} - 2) q^{68} + (2 \zeta_{6} - 4) q^{70} + (6 \zeta_{6} - 6) q^{71} + 3 \zeta_{6} q^{72} + ( - 7 \zeta_{6} + 7) q^{73} + ( - 3 \zeta_{6} - 3) q^{74} + ( - 14 \zeta_{6} + 7) q^{75} + (5 \zeta_{6} - 2) q^{76} + (4 \zeta_{6} - 2) q^{77} + ( - 9 \zeta_{6} + 9) q^{78} + (3 \zeta_{6} + 3) q^{79} + (2 \zeta_{6} - 4) q^{80} + (9 \zeta_{6} - 9) q^{81} - 12 \zeta_{6} q^{82} + ( - 12 \zeta_{6} + 6) q^{83} + (2 \zeta_{6} - 1) q^{84} - 12 \zeta_{6} q^{85} + \zeta_{6} q^{86} + (12 \zeta_{6} - 6) q^{87} + (4 \zeta_{6} - 2) q^{88} + ( - 6 \zeta_{6} - 6) q^{90} + ( - 3 \zeta_{6} + 6) q^{91} + (3 \zeta_{6} - 3) q^{93} + (8 \zeta_{6} - 4) q^{94} + ( - 14 \zeta_{6} - 2) q^{95} + (2 \zeta_{6} - 1) q^{96} + (4 \zeta_{6} + 4) q^{97} + ( - 6 \zeta_{6} + 6) q^{98} + (6 \zeta_{6} - 12) q^{99} +O(q^{100})$$ q + (z - 1) * q^2 + (-z - 1) * q^3 - z * q^4 + (2*z + 2) * q^5 + (-z + 2) * q^6 + q^7 + q^8 + 3*z * q^9 + (2*z - 4) * q^10 + (4*z - 2) * q^11 + (2*z - 1) * q^12 + (-3*z + 6) * q^13 + (z - 1) * q^14 - 6*z * q^15 + (z - 1) * q^16 + (-2*z - 2) * q^17 - 3 * q^18 + (-2*z - 3) * q^19 + (-4*z + 2) * q^20 + (-z - 1) * q^21 + (-2*z - 2) * q^22 + (-z - 1) * q^24 + 7*z * q^25 + (6*z - 3) * q^26 + (-6*z + 3) * q^27 - z * q^28 - 6*z * q^29 + 6 * q^30 + (-2*z + 1) * q^31 - z * q^32 + (-6*z + 6) * q^33 + (-2*z + 4) * q^34 + (2*z + 2) * q^35 + (-3*z + 3) * q^36 + (6*z - 3) * q^37 + (-3*z + 5) * q^38 - 9 * q^39 + (2*z + 2) * q^40 + (12*z - 12) * q^41 + (-z + 2) * q^42 + (-z + 1) * q^43 + (-2*z + 4) * q^44 + (12*z - 6) * q^45 + (-4*z + 8) * q^47 + (-z + 2) * q^48 - 6 * q^49 - 7 * q^50 + 6*z * q^51 + (-3*z - 3) * q^52 - 12*z * q^53 + (3*z + 3) * q^54 + (12*z - 12) * q^55 + q^56 + (7*z + 1) * q^57 + 6 * q^58 + (6*z - 6) * q^60 - 7*z * q^61 + (z + 1) * q^62 + 3*z * q^63 + q^64 + 18 * q^65 + 6*z * q^66 + (-5*z + 10) * q^67 + (4*z - 2) * q^68 + (2*z - 4) * q^70 + (6*z - 6) * q^71 + 3*z * q^72 + (-7*z + 7) * q^73 + (-3*z - 3) * q^74 + (-14*z + 7) * q^75 + (5*z - 2) * q^76 + (4*z - 2) * q^77 + (-9*z + 9) * q^78 + (3*z + 3) * q^79 + (2*z - 4) * q^80 + (9*z - 9) * q^81 - 12*z * q^82 + (-12*z + 6) * q^83 + (2*z - 1) * q^84 - 12*z * q^85 + z * q^86 + (12*z - 6) * q^87 + (4*z - 2) * q^88 + (-6*z - 6) * q^90 + (-3*z + 6) * q^91 + (3*z - 3) * q^93 + (8*z - 4) * q^94 + (-14*z - 2) * q^95 + (2*z - 1) * q^96 + (4*z + 4) * q^97 + (-6*z + 6) * q^98 + (6*z - 12) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} - 3 q^{3} - q^{4} + 6 q^{5} + 3 q^{6} + 2 q^{7} + 2 q^{8} + 3 q^{9}+O(q^{10})$$ 2 * q - q^2 - 3 * q^3 - q^4 + 6 * q^5 + 3 * q^6 + 2 * q^7 + 2 * q^8 + 3 * q^9 $$2 q - q^{2} - 3 q^{3} - q^{4} + 6 q^{5} + 3 q^{6} + 2 q^{7} + 2 q^{8} + 3 q^{9} - 6 q^{10} + 9 q^{13} - q^{14} - 6 q^{15} - q^{16} - 6 q^{17} - 6 q^{18} - 8 q^{19} - 3 q^{21} - 6 q^{22} - 3 q^{24} + 7 q^{25} - q^{28} - 6 q^{29} + 12 q^{30} - q^{32} + 6 q^{33} + 6 q^{34} + 6 q^{35} + 3 q^{36} + 7 q^{38} - 18 q^{39} + 6 q^{40} - 12 q^{41} + 3 q^{42} + q^{43} + 6 q^{44} + 12 q^{47} + 3 q^{48} - 12 q^{49} - 14 q^{50} + 6 q^{51} - 9 q^{52} - 12 q^{53} + 9 q^{54} - 12 q^{55} + 2 q^{56} + 9 q^{57} + 12 q^{58} - 6 q^{60} - 7 q^{61} + 3 q^{62} + 3 q^{63} + 2 q^{64} + 36 q^{65} + 6 q^{66} + 15 q^{67} - 6 q^{70} - 6 q^{71} + 3 q^{72} + 7 q^{73} - 9 q^{74} + q^{76} + 9 q^{78} + 9 q^{79} - 6 q^{80} - 9 q^{81} - 12 q^{82} - 12 q^{85} + q^{86} - 18 q^{90} + 9 q^{91} - 3 q^{93} - 18 q^{95} + 12 q^{97} + 6 q^{98} - 18 q^{99}+O(q^{100})$$ 2 * q - q^2 - 3 * q^3 - q^4 + 6 * q^5 + 3 * q^6 + 2 * q^7 + 2 * q^8 + 3 * q^9 - 6 * q^10 + 9 * q^13 - q^14 - 6 * q^15 - q^16 - 6 * q^17 - 6 * q^18 - 8 * q^19 - 3 * q^21 - 6 * q^22 - 3 * q^24 + 7 * q^25 - q^28 - 6 * q^29 + 12 * q^30 - q^32 + 6 * q^33 + 6 * q^34 + 6 * q^35 + 3 * q^36 + 7 * q^38 - 18 * q^39 + 6 * q^40 - 12 * q^41 + 3 * q^42 + q^43 + 6 * q^44 + 12 * q^47 + 3 * q^48 - 12 * q^49 - 14 * q^50 + 6 * q^51 - 9 * q^52 - 12 * q^53 + 9 * q^54 - 12 * q^55 + 2 * q^56 + 9 * q^57 + 12 * q^58 - 6 * q^60 - 7 * q^61 + 3 * q^62 + 3 * q^63 + 2 * q^64 + 36 * q^65 + 6 * q^66 + 15 * q^67 - 6 * q^70 - 6 * q^71 + 3 * q^72 + 7 * q^73 - 9 * q^74 + q^76 + 9 * q^78 + 9 * q^79 - 6 * q^80 - 9 * q^81 - 12 * q^82 - 12 * q^85 + q^86 - 18 * q^90 + 9 * q^91 - 3 * q^93 - 18 * q^95 + 12 * q^97 + 6 * q^98 - 18 * q^99

Character values

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

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

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
65.1
 0.5 + 0.866025i 0.5 − 0.866025i
−0.500000 + 0.866025i −1.50000 0.866025i −0.500000 0.866025i 3.00000 + 1.73205i 1.50000 0.866025i 1.00000 1.00000 1.50000 + 2.59808i −3.00000 + 1.73205i
107.1 −0.500000 0.866025i −1.50000 + 0.866025i −0.500000 + 0.866025i 3.00000 1.73205i 1.50000 + 0.866025i 1.00000 1.00000 1.50000 2.59808i −3.00000 1.73205i
 $$n$$: 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.f even 6 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 114.2.h.b 2
3.b odd 2 1 114.2.h.c yes 2
4.b odd 2 1 912.2.bn.f 2
12.b even 2 1 912.2.bn.c 2
19.d odd 6 1 114.2.h.c yes 2
57.f even 6 1 inner 114.2.h.b 2
76.f even 6 1 912.2.bn.c 2
228.n odd 6 1 912.2.bn.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.2.h.b 2 1.a even 1 1 trivial
114.2.h.b 2 57.f even 6 1 inner
114.2.h.c yes 2 3.b odd 2 1
114.2.h.c yes 2 19.d odd 6 1
912.2.bn.c 2 12.b even 2 1
912.2.bn.c 2 76.f even 6 1
912.2.bn.f 2 4.b odd 2 1
912.2.bn.f 2 228.n odd 6 1

Hecke kernels

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

 $$T_{5}^{2} - 6T_{5} + 12$$ T5^2 - 6*T5 + 12 $$T_{7} - 1$$ T7 - 1 $$T_{17}^{2} + 6T_{17} + 12$$ T17^2 + 6*T17 + 12

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T + 1$$
$3$ $$T^{2} + 3T + 3$$
$5$ $$T^{2} - 6T + 12$$
$7$ $$(T - 1)^{2}$$
$11$ $$T^{2} + 12$$
$13$ $$T^{2} - 9T + 27$$
$17$ $$T^{2} + 6T + 12$$
$19$ $$T^{2} + 8T + 19$$
$23$ $$T^{2}$$
$29$ $$T^{2} + 6T + 36$$
$31$ $$T^{2} + 3$$
$37$ $$T^{2} + 27$$
$41$ $$T^{2} + 12T + 144$$
$43$ $$T^{2} - T + 1$$
$47$ $$T^{2} - 12T + 48$$
$53$ $$T^{2} + 12T + 144$$
$59$ $$T^{2}$$
$61$ $$T^{2} + 7T + 49$$
$67$ $$T^{2} - 15T + 75$$
$71$ $$T^{2} + 6T + 36$$
$73$ $$T^{2} - 7T + 49$$
$79$ $$T^{2} - 9T + 27$$
$83$ $$T^{2} + 108$$
$89$ $$T^{2}$$
$97$ $$T^{2} - 12T + 48$$