# Properties

 Label 114.2.h.e Level $114$ Weight $2$ Character orbit 114.h Analytic conductor $0.910$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Learn more

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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 2x^{2} + 4$$ x^4 - 2*x^2 + 4 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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 basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} - 1) q^{2} + ( - \beta_{3} + 1) q^{3} - \beta_{2} q^{4} - \beta_1 q^{5} + (\beta_{2} + \beta_1 - 1) q^{6} + ( - \beta_{3} + 2 \beta_1 + 2) q^{7} + q^{8} + ( - 2 \beta_{3} - 1) q^{9}+O(q^{10})$$ q + (b2 - 1) * q^2 + (-b3 + 1) * q^3 - b2 * q^4 - b1 * q^5 + (b2 + b1 - 1) * q^6 + (-b3 + 2*b1 + 2) * q^7 + q^8 + (-2*b3 - 1) * q^9 $$q + (\beta_{2} - 1) q^{2} + ( - \beta_{3} + 1) q^{3} - \beta_{2} q^{4} - \beta_1 q^{5} + (\beta_{2} + \beta_1 - 1) q^{6} + ( - \beta_{3} + 2 \beta_1 + 2) q^{7} + q^{8} + ( - 2 \beta_{3} - 1) q^{9} + ( - \beta_{3} + \beta_1) q^{10} + (\beta_{3} - 2 \beta_{2} + 1) q^{11} + (\beta_{3} - \beta_{2} - \beta_1) q^{12} + (2 \beta_{2} - 4) q^{13} + (2 \beta_{3} + 2 \beta_{2} - \beta_1 - 2) q^{14} + (2 \beta_{2} - \beta_1 - 2) q^{15} + (\beta_{2} - 1) q^{16} + (2 \beta_{2} + 2 \beta_1 + 2) q^{17} + ( - \beta_{2} + 2 \beta_1 + 1) q^{18} + (3 \beta_{3} - \beta_{2} - 3 \beta_1) q^{19} + \beta_{3} q^{20} + ( - 3 \beta_{3} - 4 \beta_{2} + 2 \beta_1 + 4) q^{21} + (\beta_{2} - \beta_1 + 1) q^{22} + (5 \beta_{3} - 5 \beta_1) q^{23} + ( - \beta_{3} + 1) q^{24} - 3 \beta_{2} q^{25} + ( - 4 \beta_{2} + 2) q^{26} + ( - \beta_{3} - 5) q^{27} + ( - \beta_{3} - 2 \beta_{2} - \beta_1) q^{28} + (\beta_{3} + \beta_1) q^{29} + ( - \beta_{3} - 2 \beta_{2} + \beta_1) q^{30} + 3 \beta_{3} q^{31} - \beta_{2} q^{32} + (2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 3) q^{33} + (2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 4) q^{34} + ( - 2 \beta_{2} - 2 \beta_1 - 2) q^{35} + (2 \beta_{3} + \beta_{2} - 2 \beta_1) q^{36} + ( - 3 \beta_{3} + 4 \beta_{2} - 2) q^{37} + ( - 3 \beta_{3} + 1) q^{38} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 4) q^{39} - \beta_1 q^{40} + ( - 3 \beta_{2} + 3) q^{41} + (2 \beta_{3} + 4 \beta_{2} + \beta_1) q^{42} + (4 \beta_{3} - 4 \beta_{2} - 2 \beta_1 + 4) q^{43} + ( - \beta_{3} + \beta_{2} + \beta_1 - 2) q^{44} + (4 \beta_{2} + \beta_1 - 4) q^{45} - 5 \beta_{3} q^{46} + ( - 7 \beta_{3} + 2 \beta_{2} + 7 \beta_1 - 4) q^{47} + (\beta_{2} + \beta_1 - 1) q^{48} + ( - 4 \beta_{3} + 8 \beta_1 + 3) q^{49} + 3 q^{50} + ( - 4 \beta_{3} - 2 \beta_{2} + 4 \beta_1 + 6) q^{51} + (2 \beta_{2} + 2) q^{52} + ( - 2 \beta_{3} + 6 \beta_{2} - 2 \beta_1) q^{53} + ( - 5 \beta_{2} + \beta_1 + 5) q^{54} + (2 \beta_{3} - 2 \beta_{2} - \beta_1 + 2) q^{55} + ( - \beta_{3} + 2 \beta_1 + 2) q^{56} + (4 \beta_{3} + 5 \beta_{2} - 4 \beta_1) q^{57} + (\beta_{3} - 2 \beta_1) q^{58} + ( - 2 \beta_{3} + 9 \beta_{2} + \beta_1 - 9) q^{59} + (\beta_{3} + 2) q^{60} + (\beta_{3} + 4 \beta_{2} + \beta_1) q^{61} - 3 \beta_1 q^{62} + ( - 3 \beta_{3} - 8 \beta_{2} - 2 \beta_1 + 2) q^{63} + q^{64} + ( - 2 \beta_{3} + 4 \beta_1) q^{65} + ( - 2 \beta_{3} + 3 \beta_{2} - 1) q^{66} + (3 \beta_{3} + \beta_{2} - 3 \beta_1 - 2) q^{67} + ( - 2 \beta_{3} - 4 \beta_{2} + 2) q^{68} + (5 \beta_{3} + 10 \beta_{2} - 5 \beta_1) q^{69} + ( - 2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 4) q^{70} + (6 \beta_{2} - 6) q^{71} + ( - 2 \beta_{3} - 1) q^{72} + ( - 8 \beta_{3} - \beta_{2} + 4 \beta_1 + 1) q^{73} + ( - 2 \beta_{2} + 3 \beta_1 - 2) q^{74} + (3 \beta_{3} - 3 \beta_{2} - 3 \beta_1) q^{75} + (\beta_{2} + 3 \beta_1 - 1) q^{76} - \beta_{3} q^{77} + (2 \beta_{3} - 4 \beta_{2} - 4 \beta_1 + 2) q^{78} + 6 \beta_1 q^{79} + ( - \beta_{3} + \beta_1) q^{80} + (4 \beta_{3} - 7) q^{81} + 3 \beta_{2} q^{82} + (\beta_{3} - 18 \beta_{2} + 9) q^{83} + (\beta_{3} - 3 \beta_1 - 4) q^{84} + ( - 2 \beta_{3} - 4 \beta_{2} - 2 \beta_1) q^{85} + ( - 2 \beta_{3} + 4 \beta_{2} - 2 \beta_1) q^{86} + (\beta_{3} - 2 \beta_{2} + \beta_1 + 4) q^{87} + (\beta_{3} - 2 \beta_{2} + 1) q^{88} + ( - 2 \beta_{3} - 12 \beta_{2} - 2 \beta_1) q^{89} + (\beta_{3} - 4 \beta_{2} - \beta_1) q^{90} + (6 \beta_{3} + 4 \beta_{2} - 6 \beta_1 - 8) q^{91} + 5 \beta_1 q^{92} + (3 \beta_{3} + 6) q^{93} + (7 \beta_{3} - 4 \beta_{2} + 2) q^{94} + (\beta_{3} + 6) q^{95} + (\beta_{3} - \beta_{2} - \beta_1) q^{96} + ( - 3 \beta_{2} - 6 \beta_1 - 3) q^{97} + (8 \beta_{3} + 3 \beta_{2} - 4 \beta_1 - 3) q^{98} + (\beta_{3} + 2 \beta_{2} - 4 \beta_1 + 3) q^{99}+O(q^{100})$$ q + (b2 - 1) * q^2 + (-b3 + 1) * q^3 - b2 * q^4 - b1 * q^5 + (b2 + b1 - 1) * q^6 + (-b3 + 2*b1 + 2) * q^7 + q^8 + (-2*b3 - 1) * q^9 + (-b3 + b1) * q^10 + (b3 - 2*b2 + 1) * q^11 + (b3 - b2 - b1) * q^12 + (2*b2 - 4) * q^13 + (2*b3 + 2*b2 - b1 - 2) * q^14 + (2*b2 - b1 - 2) * q^15 + (b2 - 1) * q^16 + (2*b2 + 2*b1 + 2) * q^17 + (-b2 + 2*b1 + 1) * q^18 + (3*b3 - b2 - 3*b1) * q^19 + b3 * q^20 + (-3*b3 - 4*b2 + 2*b1 + 4) * q^21 + (b2 - b1 + 1) * q^22 + (5*b3 - 5*b1) * q^23 + (-b3 + 1) * q^24 - 3*b2 * q^25 + (-4*b2 + 2) * q^26 + (-b3 - 5) * q^27 + (-b3 - 2*b2 - b1) * q^28 + (b3 + b1) * q^29 + (-b3 - 2*b2 + b1) * q^30 + 3*b3 * q^31 - b2 * q^32 + (2*b3 - 2*b2 - 2*b1 + 3) * q^33 + (2*b3 + 2*b2 - 2*b1 - 4) * q^34 + (-2*b2 - 2*b1 - 2) * q^35 + (2*b3 + b2 - 2*b1) * q^36 + (-3*b3 + 4*b2 - 2) * q^37 + (-3*b3 + 1) * q^38 + (2*b3 + 2*b2 + 2*b1 - 4) * q^39 - b1 * q^40 + (-3*b2 + 3) * q^41 + (2*b3 + 4*b2 + b1) * q^42 + (4*b3 - 4*b2 - 2*b1 + 4) * q^43 + (-b3 + b2 + b1 - 2) * q^44 + (4*b2 + b1 - 4) * q^45 - 5*b3 * q^46 + (-7*b3 + 2*b2 + 7*b1 - 4) * q^47 + (b2 + b1 - 1) * q^48 + (-4*b3 + 8*b1 + 3) * q^49 + 3 * q^50 + (-4*b3 - 2*b2 + 4*b1 + 6) * q^51 + (2*b2 + 2) * q^52 + (-2*b3 + 6*b2 - 2*b1) * q^53 + (-5*b2 + b1 + 5) * q^54 + (2*b3 - 2*b2 - b1 + 2) * q^55 + (-b3 + 2*b1 + 2) * q^56 + (4*b3 + 5*b2 - 4*b1) * q^57 + (b3 - 2*b1) * q^58 + (-2*b3 + 9*b2 + b1 - 9) * q^59 + (b3 + 2) * q^60 + (b3 + 4*b2 + b1) * q^61 - 3*b1 * q^62 + (-3*b3 - 8*b2 - 2*b1 + 2) * q^63 + q^64 + (-2*b3 + 4*b1) * q^65 + (-2*b3 + 3*b2 - 1) * q^66 + (3*b3 + b2 - 3*b1 - 2) * q^67 + (-2*b3 - 4*b2 + 2) * q^68 + (5*b3 + 10*b2 - 5*b1) * q^69 + (-2*b3 - 2*b2 + 2*b1 + 4) * q^70 + (6*b2 - 6) * q^71 + (-2*b3 - 1) * q^72 + (-8*b3 - b2 + 4*b1 + 1) * q^73 + (-2*b2 + 3*b1 - 2) * q^74 + (3*b3 - 3*b2 - 3*b1) * q^75 + (b2 + 3*b1 - 1) * q^76 - b3 * q^77 + (2*b3 - 4*b2 - 4*b1 + 2) * q^78 + 6*b1 * q^79 + (-b3 + b1) * q^80 + (4*b3 - 7) * q^81 + 3*b2 * q^82 + (b3 - 18*b2 + 9) * q^83 + (b3 - 3*b1 - 4) * q^84 + (-2*b3 - 4*b2 - 2*b1) * q^85 + (-2*b3 + 4*b2 - 2*b1) * q^86 + (b3 - 2*b2 + b1 + 4) * q^87 + (b3 - 2*b2 + 1) * q^88 + (-2*b3 - 12*b2 - 2*b1) * q^89 + (b3 - 4*b2 - b1) * q^90 + (6*b3 + 4*b2 - 6*b1 - 8) * q^91 + 5*b1 * q^92 + (3*b3 + 6) * q^93 + (7*b3 - 4*b2 + 2) * q^94 + (b3 + 6) * q^95 + (b3 - b2 - b1) * q^96 + (-3*b2 - 6*b1 - 3) * q^97 + (8*b3 + 3*b2 - 4*b1 - 3) * q^98 + (b3 + 2*b2 - 4*b1 + 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{2} + 4 q^{3} - 2 q^{4} - 2 q^{6} + 8 q^{7} + 4 q^{8} - 4 q^{9}+O(q^{10})$$ 4 * q - 2 * q^2 + 4 * q^3 - 2 * q^4 - 2 * q^6 + 8 * q^7 + 4 * q^8 - 4 * q^9 $$4 q - 2 q^{2} + 4 q^{3} - 2 q^{4} - 2 q^{6} + 8 q^{7} + 4 q^{8} - 4 q^{9} - 2 q^{12} - 12 q^{13} - 4 q^{14} - 4 q^{15} - 2 q^{16} + 12 q^{17} + 2 q^{18} - 2 q^{19} + 8 q^{21} + 6 q^{22} + 4 q^{24} - 6 q^{25} - 20 q^{27} - 4 q^{28} - 4 q^{30} - 2 q^{32} + 8 q^{33} - 12 q^{34} - 12 q^{35} + 2 q^{36} + 4 q^{38} - 12 q^{39} + 6 q^{41} + 8 q^{42} + 8 q^{43} - 6 q^{44} - 8 q^{45} - 12 q^{47} - 2 q^{48} + 12 q^{49} + 12 q^{50} + 20 q^{51} + 12 q^{52} + 12 q^{53} + 10 q^{54} + 4 q^{55} + 8 q^{56} + 10 q^{57} - 18 q^{59} + 8 q^{60} + 8 q^{61} - 8 q^{63} + 4 q^{64} + 2 q^{66} - 6 q^{67} + 20 q^{69} + 12 q^{70} - 12 q^{71} - 4 q^{72} + 2 q^{73} - 12 q^{74} - 6 q^{75} - 2 q^{76} - 28 q^{81} + 6 q^{82} - 16 q^{84} - 8 q^{85} + 8 q^{86} + 12 q^{87} - 24 q^{89} - 8 q^{90} - 24 q^{91} + 24 q^{93} + 24 q^{95} - 2 q^{96} - 18 q^{97} - 6 q^{98} + 16 q^{99}+O(q^{100})$$ 4 * q - 2 * q^2 + 4 * q^3 - 2 * q^4 - 2 * q^6 + 8 * q^7 + 4 * q^8 - 4 * q^9 - 2 * q^12 - 12 * q^13 - 4 * q^14 - 4 * q^15 - 2 * q^16 + 12 * q^17 + 2 * q^18 - 2 * q^19 + 8 * q^21 + 6 * q^22 + 4 * q^24 - 6 * q^25 - 20 * q^27 - 4 * q^28 - 4 * q^30 - 2 * q^32 + 8 * q^33 - 12 * q^34 - 12 * q^35 + 2 * q^36 + 4 * q^38 - 12 * q^39 + 6 * q^41 + 8 * q^42 + 8 * q^43 - 6 * q^44 - 8 * q^45 - 12 * q^47 - 2 * q^48 + 12 * q^49 + 12 * q^50 + 20 * q^51 + 12 * q^52 + 12 * q^53 + 10 * q^54 + 4 * q^55 + 8 * q^56 + 10 * q^57 - 18 * q^59 + 8 * q^60 + 8 * q^61 - 8 * q^63 + 4 * q^64 + 2 * q^66 - 6 * q^67 + 20 * q^69 + 12 * q^70 - 12 * q^71 - 4 * q^72 + 2 * q^73 - 12 * q^74 - 6 * q^75 - 2 * q^76 - 28 * q^81 + 6 * q^82 - 16 * q^84 - 8 * q^85 + 8 * q^86 + 12 * q^87 - 24 * q^89 - 8 * q^90 - 24 * q^91 + 24 * q^93 + 24 * q^95 - 2 * q^96 - 18 * q^97 - 6 * q^98 + 16 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2x^{2} + 4$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 2$$ (v^2) / 2 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 2$$ (v^3) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$2\beta_{2}$$ 2*b2 $$\nu^{3}$$ $$=$$ $$2\beta_{3}$$ 2*b3

## Character values

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

 $$n$$ $$77$$ $$97$$ $$\chi(n)$$ $$-1$$ $$\beta_{2}$$

## 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
 1.22474 + 0.707107i −1.22474 − 0.707107i −1.22474 + 0.707107i 1.22474 − 0.707107i
−0.500000 + 0.866025i 1.00000 1.41421i −0.500000 0.866025i −1.22474 0.707107i 0.724745 + 1.57313i 4.44949 1.00000 −1.00000 2.82843i 1.22474 0.707107i
65.2 −0.500000 + 0.866025i 1.00000 + 1.41421i −0.500000 0.866025i 1.22474 + 0.707107i −1.72474 + 0.158919i −0.449490 1.00000 −1.00000 + 2.82843i −1.22474 + 0.707107i
107.1 −0.500000 0.866025i 1.00000 1.41421i −0.500000 + 0.866025i 1.22474 0.707107i −1.72474 0.158919i −0.449490 1.00000 −1.00000 2.82843i −1.22474 0.707107i
107.2 −0.500000 0.866025i 1.00000 + 1.41421i −0.500000 + 0.866025i −1.22474 + 0.707107i 0.724745 1.57313i 4.44949 1.00000 −1.00000 + 2.82843i 1.22474 + 0.707107i
 $$n$$: 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.e 4
3.b odd 2 1 114.2.h.f yes 4
4.b odd 2 1 912.2.bn.g 4
12.b even 2 1 912.2.bn.h 4
19.d odd 6 1 114.2.h.f yes 4
57.f even 6 1 inner 114.2.h.e 4
76.f even 6 1 912.2.bn.h 4
228.n odd 6 1 912.2.bn.g 4

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

## Hecke kernels

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

 $$T_{5}^{4} - 2T_{5}^{2} + 4$$ T5^4 - 2*T5^2 + 4 $$T_{7}^{2} - 4T_{7} - 2$$ T7^2 - 4*T7 - 2 $$T_{17}^{4} - 12T_{17}^{3} + 52T_{17}^{2} - 48T_{17} + 16$$ T17^4 - 12*T17^3 + 52*T17^2 - 48*T17 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + T + 1)^{2}$$
$3$ $$(T^{2} - 2 T + 3)^{2}$$
$5$ $$T^{4} - 2T^{2} + 4$$
$7$ $$(T^{2} - 4 T - 2)^{2}$$
$11$ $$T^{4} + 10T^{2} + 1$$
$13$ $$(T^{2} + 6 T + 12)^{2}$$
$17$ $$T^{4} - 12 T^{3} + 52 T^{2} - 48 T + 16$$
$19$ $$T^{4} + 2 T^{3} - 15 T^{2} + 38 T + 361$$
$23$ $$T^{4} - 50T^{2} + 2500$$
$29$ $$T^{4} + 6T^{2} + 36$$
$31$ $$(T^{2} + 18)^{2}$$
$37$ $$T^{4} + 60T^{2} + 36$$
$41$ $$(T^{2} - 3 T + 9)^{2}$$
$43$ $$T^{4} - 8 T^{3} + 72 T^{2} + 64 T + 64$$
$47$ $$T^{4} + 12 T^{3} - 38 T^{2} + \cdots + 7396$$
$53$ $$T^{4} - 12 T^{3} + 132 T^{2} + \cdots + 144$$
$59$ $$T^{4} + 18 T^{3} + 249 T^{2} + \cdots + 5625$$
$61$ $$T^{4} - 8 T^{3} + 54 T^{2} - 80 T + 100$$
$67$ $$T^{4} + 6 T^{3} - 3 T^{2} - 90 T + 225$$
$71$ $$(T^{2} + 6 T + 36)^{2}$$
$73$ $$T^{4} - 2 T^{3} + 99 T^{2} + \cdots + 9025$$
$79$ $$T^{4} - 72T^{2} + 5184$$
$83$ $$T^{4} + 490 T^{2} + 58081$$
$89$ $$T^{4} + 24 T^{3} + 456 T^{2} + \cdots + 14400$$
$97$ $$T^{4} + 18 T^{3} + 63 T^{2} + \cdots + 2025$$
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