# Properties

 Label 114.2.i.b Level $114$ Weight $2$ Character orbit 114.i Analytic conductor $0.910$ Analytic rank $0$ Dimension $6$ 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(25,114)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(114, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([0, 14]))

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

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

chi := DirichletCharacter("114.25");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$114 = 2 \cdot 3 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 114.i (of order $$9$$, degree $$6$$, 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: $$6$$ Coefficient field: $$\Q(\zeta_{18})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - x^{3} + 1$$ x^6 - x^3 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

## $q$-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 $$\zeta_{18}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

 $$n$$ $$77$$ $$97$$ $$\chi(n)$$ $$1$$ $$\zeta_{18}^{2}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
25.1
 −0.766044 + 0.642788i 0.939693 − 0.342020i −0.173648 + 0.984808i 0.939693 + 0.342020i −0.766044 − 0.642788i −0.173648 − 0.984808i
0.766044 0.642788i −0.939693 0.342020i 0.173648 0.984808i −0.0812519 0.460802i −0.939693 + 0.342020i 2.20574 3.82045i −0.500000 0.866025i 0.766044 + 0.642788i −0.358441 0.300767i
43.1 −0.939693 + 0.342020i 0.173648 0.984808i 0.766044 0.642788i −2.97178 2.49362i 0.173648 + 0.984808i −0.613341 1.06234i −0.500000 + 0.866025i −0.939693 0.342020i 3.64543 + 1.32683i
55.1 0.173648 0.984808i 0.766044 + 0.642788i −0.939693 0.342020i 1.55303 0.565258i 0.766044 0.642788i −0.0923963 0.160035i −0.500000 + 0.866025i 0.173648 + 0.984808i −0.286989 1.62760i
61.1 −0.939693 0.342020i 0.173648 + 0.984808i 0.766044 + 0.642788i −2.97178 + 2.49362i 0.173648 0.984808i −0.613341 + 1.06234i −0.500000 0.866025i −0.939693 + 0.342020i 3.64543 1.32683i
73.1 0.766044 + 0.642788i −0.939693 + 0.342020i 0.173648 + 0.984808i −0.0812519 + 0.460802i −0.939693 0.342020i 2.20574 + 3.82045i −0.500000 + 0.866025i 0.766044 0.642788i −0.358441 + 0.300767i
85.1 0.173648 + 0.984808i 0.766044 0.642788i −0.939693 + 0.342020i 1.55303 + 0.565258i 0.766044 + 0.642788i −0.0923963 + 0.160035i −0.500000 0.866025i 0.173648 0.984808i −0.286989 + 1.62760i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 25.1 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.e even 9 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 114.2.i.b 6
3.b odd 2 1 342.2.u.d 6
4.b odd 2 1 912.2.bo.c 6
19.e even 9 1 inner 114.2.i.b 6
19.e even 9 1 2166.2.a.t 3
19.f odd 18 1 2166.2.a.n 3
57.j even 18 1 6498.2.a.bt 3
57.l odd 18 1 342.2.u.d 6
57.l odd 18 1 6498.2.a.bo 3
76.l odd 18 1 912.2.bo.c 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.2.i.b 6 1.a even 1 1 trivial
114.2.i.b 6 19.e even 9 1 inner
342.2.u.d 6 3.b odd 2 1
342.2.u.d 6 57.l odd 18 1
912.2.bo.c 6 4.b odd 2 1
912.2.bo.c 6 76.l odd 18 1
2166.2.a.n 3 19.f odd 18 1
2166.2.a.t 3 19.e even 9 1
6498.2.a.bo 3 57.l odd 18 1
6498.2.a.bt 3 57.j even 18 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{6} + 3T_{5}^{5} - 30T_{5}^{3} + 36T_{5}^{2} + 9$$ acting on $$S_{2}^{\mathrm{new}}(114, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} + T^{3} + 1$$
$3$ $$T^{6} + T^{3} + 1$$
$5$ $$T^{6} + 3 T^{5} + \cdots + 9$$
$7$ $$T^{6} - 3 T^{5} + \cdots + 1$$
$11$ $$T^{6} - 12 T^{5} + \cdots + 2601$$
$13$ $$T^{6} + 21 T^{5} + \cdots + 361$$
$17$ $$T^{6} - 3 T^{5} + \cdots + 2601$$
$19$ $$T^{6} - 6 T^{5} + \cdots + 6859$$
$23$ $$T^{6} + 15 T^{5} + \cdots + 9$$
$29$ $$T^{6} - 15 T^{5} + \cdots + 2601$$
$31$ $$T^{6} - 3 T^{5} + \cdots + 289$$
$37$ $$(T^{3} - 3 T^{2} - 9 T + 19)^{2}$$
$41$ $$T^{6} + 9 T^{5} + \cdots + 729$$
$43$ $$T^{6} + 9 T^{5} + \cdots + 1$$
$47$ $$T^{6} + 21 T^{5} + \cdots + 25281$$
$53$ $$T^{6} - 30 T^{5} + \cdots + 47961$$
$59$ $$T^{6} - 27 T^{5} + \cdots + 23409$$
$61$ $$T^{6} + 9 T^{5} + \cdots + 1$$
$67$ $$T^{6} + 15 T^{5} + \cdots + 7921$$
$71$ $$T^{6} - 9 T^{5} + \cdots + 6561$$
$73$ $$T^{6} - 12 T^{5} + \cdots + 546121$$
$79$ $$T^{6} - 15 T^{5} + \cdots + 5329$$
$83$ $$T^{6} + 3 T^{5} + \cdots + 3583449$$
$89$ $$T^{6} + 48 T^{5} + \cdots + 3583449$$
$97$ $$T^{6} - 18 T^{5} + \cdots + 1630729$$