# Properties

 Label 234.2.h.a Level $234$ Weight $2$ Character orbit 234.h Analytic conductor $1.868$ 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] = [234,2,Mod(55,234)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(234, 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("234.55");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$234 = 2 \cdot 3^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 234.h (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: $$1.86849940730$$ 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: no (minimal twist has level 78) 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.

 $$f(q)$$ $$=$$ $$q + (\zeta_{6} - 1) q^{2} - \zeta_{6} q^{4} - 3 q^{5} - 2 \zeta_{6} q^{7} + q^{8} +O(q^{10})$$ q + (z - 1) * q^2 - z * q^4 - 3 * q^5 - 2*z * q^7 + q^8 $$q + (\zeta_{6} - 1) q^{2} - \zeta_{6} q^{4} - 3 q^{5} - 2 \zeta_{6} q^{7} + q^{8} + ( - 3 \zeta_{6} + 3) q^{10} + ( - 6 \zeta_{6} + 6) q^{11} + ( - \zeta_{6} - 3) q^{13} + 2 q^{14} + (\zeta_{6} - 1) q^{16} - 3 \zeta_{6} q^{17} - 2 \zeta_{6} q^{19} + 3 \zeta_{6} q^{20} + 6 \zeta_{6} q^{22} + (6 \zeta_{6} - 6) q^{23} + 4 q^{25} + ( - 3 \zeta_{6} + 4) q^{26} + (2 \zeta_{6} - 2) q^{28} + ( - 3 \zeta_{6} + 3) q^{29} - 4 q^{31} - \zeta_{6} q^{32} + 3 q^{34} + 6 \zeta_{6} q^{35} + ( - 7 \zeta_{6} + 7) q^{37} + 2 q^{38} - 3 q^{40} + (3 \zeta_{6} - 3) q^{41} + 10 \zeta_{6} q^{43} - 6 q^{44} - 6 \zeta_{6} q^{46} - 6 q^{47} + ( - 3 \zeta_{6} + 3) q^{49} + (4 \zeta_{6} - 4) q^{50} + (4 \zeta_{6} - 1) q^{52} - 3 q^{53} + (18 \zeta_{6} - 18) q^{55} - 2 \zeta_{6} q^{56} + 3 \zeta_{6} q^{58} + 7 \zeta_{6} q^{61} + ( - 4 \zeta_{6} + 4) q^{62} + q^{64} + (3 \zeta_{6} + 9) q^{65} + ( - 10 \zeta_{6} + 10) q^{67} + (3 \zeta_{6} - 3) q^{68} - 6 q^{70} + 6 \zeta_{6} q^{71} - 13 q^{73} + 7 \zeta_{6} q^{74} + (2 \zeta_{6} - 2) q^{76} - 12 q^{77} - 4 q^{79} + ( - 3 \zeta_{6} + 3) q^{80} - 3 \zeta_{6} q^{82} + 6 q^{83} + 9 \zeta_{6} q^{85} - 10 q^{86} + ( - 6 \zeta_{6} + 6) q^{88} + ( - 18 \zeta_{6} + 18) q^{89} + (8 \zeta_{6} - 2) q^{91} + 6 q^{92} + ( - 6 \zeta_{6} + 6) q^{94} + 6 \zeta_{6} q^{95} - 14 \zeta_{6} q^{97} + 3 \zeta_{6} q^{98} +O(q^{100})$$ q + (z - 1) * q^2 - z * q^4 - 3 * q^5 - 2*z * q^7 + q^8 + (-3*z + 3) * q^10 + (-6*z + 6) * q^11 + (-z - 3) * q^13 + 2 * q^14 + (z - 1) * q^16 - 3*z * q^17 - 2*z * q^19 + 3*z * q^20 + 6*z * q^22 + (6*z - 6) * q^23 + 4 * q^25 + (-3*z + 4) * q^26 + (2*z - 2) * q^28 + (-3*z + 3) * q^29 - 4 * q^31 - z * q^32 + 3 * q^34 + 6*z * q^35 + (-7*z + 7) * q^37 + 2 * q^38 - 3 * q^40 + (3*z - 3) * q^41 + 10*z * q^43 - 6 * q^44 - 6*z * q^46 - 6 * q^47 + (-3*z + 3) * q^49 + (4*z - 4) * q^50 + (4*z - 1) * q^52 - 3 * q^53 + (18*z - 18) * q^55 - 2*z * q^56 + 3*z * q^58 + 7*z * q^61 + (-4*z + 4) * q^62 + q^64 + (3*z + 9) * q^65 + (-10*z + 10) * q^67 + (3*z - 3) * q^68 - 6 * q^70 + 6*z * q^71 - 13 * q^73 + 7*z * q^74 + (2*z - 2) * q^76 - 12 * q^77 - 4 * q^79 + (-3*z + 3) * q^80 - 3*z * q^82 + 6 * q^83 + 9*z * q^85 - 10 * q^86 + (-6*z + 6) * q^88 + (-18*z + 18) * q^89 + (8*z - 2) * q^91 + 6 * q^92 + (-6*z + 6) * q^94 + 6*z * q^95 - 14*z * q^97 + 3*z * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} - q^{4} - 6 q^{5} - 2 q^{7} + 2 q^{8}+O(q^{10})$$ 2 * q - q^2 - q^4 - 6 * q^5 - 2 * q^7 + 2 * q^8 $$2 q - q^{2} - q^{4} - 6 q^{5} - 2 q^{7} + 2 q^{8} + 3 q^{10} + 6 q^{11} - 7 q^{13} + 4 q^{14} - q^{16} - 3 q^{17} - 2 q^{19} + 3 q^{20} + 6 q^{22} - 6 q^{23} + 8 q^{25} + 5 q^{26} - 2 q^{28} + 3 q^{29} - 8 q^{31} - q^{32} + 6 q^{34} + 6 q^{35} + 7 q^{37} + 4 q^{38} - 6 q^{40} - 3 q^{41} + 10 q^{43} - 12 q^{44} - 6 q^{46} - 12 q^{47} + 3 q^{49} - 4 q^{50} + 2 q^{52} - 6 q^{53} - 18 q^{55} - 2 q^{56} + 3 q^{58} + 7 q^{61} + 4 q^{62} + 2 q^{64} + 21 q^{65} + 10 q^{67} - 3 q^{68} - 12 q^{70} + 6 q^{71} - 26 q^{73} + 7 q^{74} - 2 q^{76} - 24 q^{77} - 8 q^{79} + 3 q^{80} - 3 q^{82} + 12 q^{83} + 9 q^{85} - 20 q^{86} + 6 q^{88} + 18 q^{89} + 4 q^{91} + 12 q^{92} + 6 q^{94} + 6 q^{95} - 14 q^{97} + 3 q^{98}+O(q^{100})$$ 2 * q - q^2 - q^4 - 6 * q^5 - 2 * q^7 + 2 * q^8 + 3 * q^10 + 6 * q^11 - 7 * q^13 + 4 * q^14 - q^16 - 3 * q^17 - 2 * q^19 + 3 * q^20 + 6 * q^22 - 6 * q^23 + 8 * q^25 + 5 * q^26 - 2 * q^28 + 3 * q^29 - 8 * q^31 - q^32 + 6 * q^34 + 6 * q^35 + 7 * q^37 + 4 * q^38 - 6 * q^40 - 3 * q^41 + 10 * q^43 - 12 * q^44 - 6 * q^46 - 12 * q^47 + 3 * q^49 - 4 * q^50 + 2 * q^52 - 6 * q^53 - 18 * q^55 - 2 * q^56 + 3 * q^58 + 7 * q^61 + 4 * q^62 + 2 * q^64 + 21 * q^65 + 10 * q^67 - 3 * q^68 - 12 * q^70 + 6 * q^71 - 26 * q^73 + 7 * q^74 - 2 * q^76 - 24 * q^77 - 8 * q^79 + 3 * q^80 - 3 * q^82 + 12 * q^83 + 9 * q^85 - 20 * q^86 + 6 * q^88 + 18 * q^89 + 4 * q^91 + 12 * q^92 + 6 * q^94 + 6 * q^95 - 14 * q^97 + 3 * q^98

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/234\mathbb{Z}\right)^\times$$.

 $$n$$ $$145$$ $$209$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
55.1
 0.5 − 0.866025i 0.5 + 0.866025i
−0.500000 0.866025i 0 −0.500000 + 0.866025i −3.00000 0 −1.00000 + 1.73205i 1.00000 0 1.50000 + 2.59808i
217.1 −0.500000 + 0.866025i 0 −0.500000 0.866025i −3.00000 0 −1.00000 1.73205i 1.00000 0 1.50000 2.59808i
 $$n$$: 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
13.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 234.2.h.a 2
3.b odd 2 1 78.2.e.a 2
4.b odd 2 1 1872.2.t.c 2
12.b even 2 1 624.2.q.g 2
13.c even 3 1 inner 234.2.h.a 2
13.c even 3 1 3042.2.a.i 1
13.e even 6 1 3042.2.a.h 1
13.f odd 12 2 3042.2.b.h 2
15.d odd 2 1 1950.2.i.m 2
15.e even 4 2 1950.2.z.g 4
39.d odd 2 1 1014.2.e.a 2
39.f even 4 2 1014.2.i.b 4
39.h odd 6 1 1014.2.a.f 1
39.h odd 6 1 1014.2.e.a 2
39.i odd 6 1 78.2.e.a 2
39.i odd 6 1 1014.2.a.c 1
39.k even 12 2 1014.2.b.c 2
39.k even 12 2 1014.2.i.b 4
52.j odd 6 1 1872.2.t.c 2
156.p even 6 1 624.2.q.g 2
156.p even 6 1 8112.2.a.m 1
156.r even 6 1 8112.2.a.c 1
195.x odd 6 1 1950.2.i.m 2
195.bl even 12 2 1950.2.z.g 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.e.a 2 3.b odd 2 1
78.2.e.a 2 39.i odd 6 1
234.2.h.a 2 1.a even 1 1 trivial
234.2.h.a 2 13.c even 3 1 inner
624.2.q.g 2 12.b even 2 1
624.2.q.g 2 156.p even 6 1
1014.2.a.c 1 39.i odd 6 1
1014.2.a.f 1 39.h odd 6 1
1014.2.b.c 2 39.k even 12 2
1014.2.e.a 2 39.d odd 2 1
1014.2.e.a 2 39.h odd 6 1
1014.2.i.b 4 39.f even 4 2
1014.2.i.b 4 39.k even 12 2
1872.2.t.c 2 4.b odd 2 1
1872.2.t.c 2 52.j odd 6 1
1950.2.i.m 2 15.d odd 2 1
1950.2.i.m 2 195.x odd 6 1
1950.2.z.g 4 15.e even 4 2
1950.2.z.g 4 195.bl even 12 2
3042.2.a.h 1 13.e even 6 1
3042.2.a.i 1 13.c even 3 1
3042.2.b.h 2 13.f odd 12 2
8112.2.a.c 1 156.r even 6 1
8112.2.a.m 1 156.p even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(234, [\chi])$$:

 $$T_{5} + 3$$ T5 + 3 $$T_{7}^{2} + 2T_{7} + 4$$ T7^2 + 2*T7 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T + 1$$
$3$ $$T^{2}$$
$5$ $$(T + 3)^{2}$$
$7$ $$T^{2} + 2T + 4$$
$11$ $$T^{2} - 6T + 36$$
$13$ $$T^{2} + 7T + 13$$
$17$ $$T^{2} + 3T + 9$$
$19$ $$T^{2} + 2T + 4$$
$23$ $$T^{2} + 6T + 36$$
$29$ $$T^{2} - 3T + 9$$
$31$ $$(T + 4)^{2}$$
$37$ $$T^{2} - 7T + 49$$
$41$ $$T^{2} + 3T + 9$$
$43$ $$T^{2} - 10T + 100$$
$47$ $$(T + 6)^{2}$$
$53$ $$(T + 3)^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2} - 7T + 49$$
$67$ $$T^{2} - 10T + 100$$
$71$ $$T^{2} - 6T + 36$$
$73$ $$(T + 13)^{2}$$
$79$ $$(T + 4)^{2}$$
$83$ $$(T - 6)^{2}$$
$89$ $$T^{2} - 18T + 324$$
$97$ $$T^{2} + 14T + 196$$