# Properties

 Label 234.3.bb.b Level $234$ Weight $3$ Character orbit 234.bb Analytic conductor $6.376$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [234,3,Mod(19,234)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(234, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([0, 5]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("234.19");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$234 = 2 \cdot 3^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 234.bb (of order $$12$$, degree $$4$$, 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: $$6.37603818603$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 26) Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-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 $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\zeta_{12}^{3} - \zeta_{12}^{2} - \zeta_{12}) q^{2} + 2 \zeta_{12} q^{4} + ( - \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 2 \zeta_{12} + 1) q^{5} + (7 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 4 \zeta_{12} - 4) q^{7} + ( - 2 \zeta_{12}^{3} - 2) q^{8}+O(q^{10})$$ q + (z^3 - z^2 - z) * q^2 + 2*z * q^4 + (-z^3 - 2*z^2 + 2*z + 1) * q^5 + (7*z^3 - 3*z^2 - 4*z - 4) * q^7 + (-2*z^3 - 2) * q^8 $$q + (\zeta_{12}^{3} - \zeta_{12}^{2} - \zeta_{12}) q^{2} + 2 \zeta_{12} q^{4} + ( - \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 2 \zeta_{12} + 1) q^{5} + (7 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 4 \zeta_{12} - 4) q^{7} + ( - 2 \zeta_{12}^{3} - 2) q^{8} + (2 \zeta_{12}^{2} - 4) q^{10} + (5 \zeta_{12}^{3} + 5 \zeta_{12}^{2} - 4 \zeta_{12} - 1) q^{11} + ( - 2 \zeta_{12}^{3} + 12 \zeta_{12}^{2} + 8 \zeta_{12} - 9) q^{13} + ( - 7 \zeta_{12}^{3} + 14 \zeta_{12} + 1) q^{14} + 4 \zeta_{12}^{2} q^{16} + ( - 13 \zeta_{12}^{2} + 6 \zeta_{12} - 13) q^{17} + ( - 17 \zeta_{12}^{3} - 5 \zeta_{12}^{2} + 12 \zeta_{12} - 12) q^{19} + ( - 4 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 2 \zeta_{12} + 2) q^{20} + ( - 2 \zeta_{12}^{3} - 9 \zeta_{12}^{2} + \zeta_{12} + 9) q^{22} + (21 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 21 \zeta_{12} - 4) q^{23} + 19 \zeta_{12}^{3} q^{25} + ( - 15 \zeta_{12}^{3} - \zeta_{12}^{2} - 5 \zeta_{12} + 4) q^{26} + ( - 6 \zeta_{12}^{3} + 6 \zeta_{12}^{2} - 8 \zeta_{12} - 14) q^{28} + (16 \zeta_{12}^{3} + 27 \zeta_{12}^{2} + 16 \zeta_{12}) q^{29} + (37 \zeta_{12}^{3} + 10 \zeta_{12}^{2} - 10 \zeta_{12} - 37) q^{31} + ( - 4 \zeta_{12}^{2} - 4 \zeta_{12} + 4) q^{32} + ( - 19 \zeta_{12}^{3} + 26 \zeta_{12}^{2} + 26 \zeta_{12} - 19) q^{34} + (2 \zeta_{12}^{3} + 21 \zeta_{12}^{2} - \zeta_{12} - 21) q^{35} + (6 \zeta_{12}^{3} + 6 \zeta_{12}^{2} - 13 \zeta_{12} + 7) q^{37} + ( - 7 \zeta_{12}^{3} + 34 \zeta_{12}^{2} - 17) q^{38} + (4 \zeta_{12}^{3} - 8 \zeta_{12}) q^{40} + (26 \zeta_{12}^{3} - 26 \zeta_{12}^{2} - 13 \zeta_{12} + 13) q^{41} + (20 \zeta_{12}^{2} - 39 \zeta_{12} + 20) q^{43} + (10 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 2 \zeta_{12} - 10) q^{44} + ( - 4 \zeta_{12}^{3} - 19 \zeta_{12}^{2} + 23 \zeta_{12} + 23) q^{46} + ( - 33 \zeta_{12}^{3} - 33) q^{47} + ( - 25 \zeta_{12}^{3} - 7 \zeta_{12}^{2} + 25 \zeta_{12} + 14) q^{49} + ( - 19 \zeta_{12}^{3} - 19 \zeta_{12}^{2} + 19 \zeta_{12}) q^{50} + (24 \zeta_{12}^{3} + 12 \zeta_{12}^{2} - 18 \zeta_{12} + 4) q^{52} + (22 \zeta_{12}^{3} - 44 \zeta_{12} + 42) q^{53} + (9 \zeta_{12}^{3} + 3 \zeta_{12}^{2} + 9 \zeta_{12}) q^{55} + (14 \zeta_{12}^{2} + 2 \zeta_{12} + 14) q^{56} + ( - 32 \zeta_{12}^{3} - 43 \zeta_{12}^{2} - 11 \zeta_{12} + 11) q^{58} + (13 \zeta_{12}^{3} - 23 \zeta_{12}^{2} + 10 \zeta_{12} + 10) q^{59} + ( - 12 \zeta_{12}^{3} - 39 \zeta_{12}^{2} + 6 \zeta_{12} + 39) q^{61} + ( - 64 \zeta_{12}^{3} - 10 \zeta_{12}^{2} + 64 \zeta_{12} + 20) q^{62} + 8 \zeta_{12}^{3} q^{64} + (7 \zeta_{12}^{3} + 10 \zeta_{12}^{2} - 2 \zeta_{12} + 25) q^{65} + (23 \zeta_{12}^{3} - 23 \zeta_{12}^{2} + 10 \zeta_{12} + 33) q^{67} + ( - 26 \zeta_{12}^{3} + 12 \zeta_{12}^{2} - 26 \zeta_{12}) q^{68} + ( - 22 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 2 \zeta_{12} + 22) q^{70} + ( - 25 \zeta_{12}^{3} - 29 \zeta_{12}^{2} - 4 \zeta_{12} + 4) q^{71} + ( - 7 \zeta_{12}^{3} - 54 \zeta_{12}^{2} - 54 \zeta_{12} - 7) q^{73} + (14 \zeta_{12}^{3} - 19 \zeta_{12}^{2} - 7 \zeta_{12} + 19) q^{74} + ( - 10 \zeta_{12}^{3} - 10 \zeta_{12}^{2} - 24 \zeta_{12} + 34) q^{76} + ( - 15 \zeta_{12}^{3} - 64 \zeta_{12}^{2} + 32) q^{77} + ( - 36 \zeta_{12}^{3} + 72 \zeta_{12} + 48) q^{79} + (4 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 4 \zeta_{12} + 8) q^{80} + ( - 13 \zeta_{12}^{2} + 39 \zeta_{12} - 13) q^{82} + (25 \zeta_{12}^{3} - 46 \zeta_{12}^{2} + 46 \zeta_{12} - 25) q^{83} + ( - 12 \zeta_{12}^{3} + 45 \zeta_{12}^{2} - 33 \zeta_{12} - 33) q^{85} + (59 \zeta_{12}^{3} - 40 \zeta_{12}^{2} - 40 \zeta_{12} + 59) q^{86} + ( - 18 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 18 \zeta_{12} + 4) q^{88} + (56 \zeta_{12}^{3} + 56 \zeta_{12}^{2} - 43 \zeta_{12} - 13) q^{89} + ( - 37 \zeta_{12}^{3} - 25 \zeta_{12}^{2} - 86 \zeta_{12} + 22) q^{91} + (4 \zeta_{12}^{3} - 8 \zeta_{12} - 42) q^{92} + 66 \zeta_{12}^{2} q^{94} + (7 \zeta_{12}^{2} - 51 \zeta_{12} + 7) q^{95} + (69 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 71 \zeta_{12} + 71) q^{97} + (14 \zeta_{12}^{3} + 18 \zeta_{12}^{2} - 32 \zeta_{12} - 32) q^{98}+O(q^{100})$$ q + (z^3 - z^2 - z) * q^2 + 2*z * q^4 + (-z^3 - 2*z^2 + 2*z + 1) * q^5 + (7*z^3 - 3*z^2 - 4*z - 4) * q^7 + (-2*z^3 - 2) * q^8 + (2*z^2 - 4) * q^10 + (5*z^3 + 5*z^2 - 4*z - 1) * q^11 + (-2*z^3 + 12*z^2 + 8*z - 9) * q^13 + (-7*z^3 + 14*z + 1) * q^14 + 4*z^2 * q^16 + (-13*z^2 + 6*z - 13) * q^17 + (-17*z^3 - 5*z^2 + 12*z - 12) * q^19 + (-4*z^3 + 2*z^2 + 2*z + 2) * q^20 + (-2*z^3 - 9*z^2 + z + 9) * q^22 + (21*z^3 + 2*z^2 - 21*z - 4) * q^23 + 19*z^3 * q^25 + (-15*z^3 - z^2 - 5*z + 4) * q^26 + (-6*z^3 + 6*z^2 - 8*z - 14) * q^28 + (16*z^3 + 27*z^2 + 16*z) * q^29 + (37*z^3 + 10*z^2 - 10*z - 37) * q^31 + (-4*z^2 - 4*z + 4) * q^32 + (-19*z^3 + 26*z^2 + 26*z - 19) * q^34 + (2*z^3 + 21*z^2 - z - 21) * q^35 + (6*z^3 + 6*z^2 - 13*z + 7) * q^37 + (-7*z^3 + 34*z^2 - 17) * q^38 + (4*z^3 - 8*z) * q^40 + (26*z^3 - 26*z^2 - 13*z + 13) * q^41 + (20*z^2 - 39*z + 20) * q^43 + (10*z^3 + 2*z^2 - 2*z - 10) * q^44 + (-4*z^3 - 19*z^2 + 23*z + 23) * q^46 + (-33*z^3 - 33) * q^47 + (-25*z^3 - 7*z^2 + 25*z + 14) * q^49 + (-19*z^3 - 19*z^2 + 19*z) * q^50 + (24*z^3 + 12*z^2 - 18*z + 4) * q^52 + (22*z^3 - 44*z + 42) * q^53 + (9*z^3 + 3*z^2 + 9*z) * q^55 + (14*z^2 + 2*z + 14) * q^56 + (-32*z^3 - 43*z^2 - 11*z + 11) * q^58 + (13*z^3 - 23*z^2 + 10*z + 10) * q^59 + (-12*z^3 - 39*z^2 + 6*z + 39) * q^61 + (-64*z^3 - 10*z^2 + 64*z + 20) * q^62 + 8*z^3 * q^64 + (7*z^3 + 10*z^2 - 2*z + 25) * q^65 + (23*z^3 - 23*z^2 + 10*z + 33) * q^67 + (-26*z^3 + 12*z^2 - 26*z) * q^68 + (-22*z^3 - 2*z^2 + 2*z + 22) * q^70 + (-25*z^3 - 29*z^2 - 4*z + 4) * q^71 + (-7*z^3 - 54*z^2 - 54*z - 7) * q^73 + (14*z^3 - 19*z^2 - 7*z + 19) * q^74 + (-10*z^3 - 10*z^2 - 24*z + 34) * q^76 + (-15*z^3 - 64*z^2 + 32) * q^77 + (-36*z^3 + 72*z + 48) * q^79 + (4*z^3 - 4*z^2 + 4*z + 8) * q^80 + (-13*z^2 + 39*z - 13) * q^82 + (25*z^3 - 46*z^2 + 46*z - 25) * q^83 + (-12*z^3 + 45*z^2 - 33*z - 33) * q^85 + (59*z^3 - 40*z^2 - 40*z + 59) * q^86 + (-18*z^3 - 2*z^2 + 18*z + 4) * q^88 + (56*z^3 + 56*z^2 - 43*z - 13) * q^89 + (-37*z^3 - 25*z^2 - 86*z + 22) * q^91 + (4*z^3 - 8*z - 42) * q^92 + 66*z^2 * q^94 + (7*z^2 - 51*z + 7) * q^95 + (69*z^3 - 2*z^2 - 71*z + 71) * q^97 + (14*z^3 + 18*z^2 - 32*z - 32) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{2} - 22 q^{7} - 8 q^{8}+O(q^{10})$$ 4 * q - 2 * q^2 - 22 * q^7 - 8 * q^8 $$4 q - 2 q^{2} - 22 q^{7} - 8 q^{8} - 12 q^{10} + 6 q^{11} - 12 q^{13} + 4 q^{14} + 8 q^{16} - 78 q^{17} - 58 q^{19} + 12 q^{20} + 18 q^{22} - 12 q^{23} + 14 q^{26} - 44 q^{28} + 54 q^{29} - 128 q^{31} + 8 q^{32} - 24 q^{34} - 42 q^{35} + 40 q^{37} + 120 q^{43} - 36 q^{44} + 54 q^{46} - 132 q^{47} + 42 q^{49} - 38 q^{50} + 40 q^{52} + 168 q^{53} + 6 q^{55} + 84 q^{56} - 42 q^{58} - 6 q^{59} + 78 q^{61} + 60 q^{62} + 120 q^{65} + 86 q^{67} + 24 q^{68} + 84 q^{70} - 42 q^{71} - 136 q^{73} + 38 q^{74} + 116 q^{76} + 192 q^{79} + 24 q^{80} - 78 q^{82} - 192 q^{83} - 42 q^{85} + 156 q^{86} + 12 q^{88} + 60 q^{89} + 38 q^{91} - 168 q^{92} + 132 q^{94} + 42 q^{95} + 280 q^{97} - 92 q^{98}+O(q^{100})$$ 4 * q - 2 * q^2 - 22 * q^7 - 8 * q^8 - 12 * q^10 + 6 * q^11 - 12 * q^13 + 4 * q^14 + 8 * q^16 - 78 * q^17 - 58 * q^19 + 12 * q^20 + 18 * q^22 - 12 * q^23 + 14 * q^26 - 44 * q^28 + 54 * q^29 - 128 * q^31 + 8 * q^32 - 24 * q^34 - 42 * q^35 + 40 * q^37 + 120 * q^43 - 36 * q^44 + 54 * q^46 - 132 * q^47 + 42 * q^49 - 38 * q^50 + 40 * q^52 + 168 * q^53 + 6 * q^55 + 84 * q^56 - 42 * q^58 - 6 * q^59 + 78 * q^61 + 60 * q^62 + 120 * q^65 + 86 * q^67 + 24 * q^68 + 84 * q^70 - 42 * q^71 - 136 * q^73 + 38 * q^74 + 116 * q^76 + 192 * q^79 + 24 * q^80 - 78 * q^82 - 192 * q^83 - 42 * q^85 + 156 * q^86 + 12 * q^88 + 60 * q^89 + 38 * q^91 - 168 * q^92 + 132 * q^94 + 42 * q^95 + 280 * q^97 - 92 * q^98

## Character values

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

 $$n$$ $$145$$ $$209$$ $$\chi(n)$$ $$\zeta_{12}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1
 −0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 + 0.500000i 0.866025 − 0.500000i
0.366025 + 1.36603i 0 −1.73205 + 1.00000i −1.73205 + 1.73205i 0 −2.03590 + 7.59808i −2.00000 2.00000i 0 −3.00000 1.73205i
37.1 0.366025 1.36603i 0 −1.73205 1.00000i −1.73205 1.73205i 0 −2.03590 7.59808i −2.00000 + 2.00000i 0 −3.00000 + 1.73205i
145.1 −1.36603 0.366025i 0 1.73205 + 1.00000i 1.73205 1.73205i 0 −8.96410 + 2.40192i −2.00000 2.00000i 0 −3.00000 + 1.73205i
163.1 −1.36603 + 0.366025i 0 1.73205 1.00000i 1.73205 + 1.73205i 0 −8.96410 2.40192i −2.00000 + 2.00000i 0 −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
13.f odd 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 234.3.bb.b 4
3.b odd 2 1 26.3.f.a 4
12.b even 2 1 208.3.bd.c 4
13.f odd 12 1 inner 234.3.bb.b 4
39.d odd 2 1 338.3.f.d 4
39.f even 4 1 338.3.f.c 4
39.f even 4 1 338.3.f.f 4
39.h odd 6 1 338.3.d.e 4
39.h odd 6 1 338.3.f.c 4
39.i odd 6 1 338.3.d.d 4
39.i odd 6 1 338.3.f.f 4
39.k even 12 1 26.3.f.a 4
39.k even 12 1 338.3.d.d 4
39.k even 12 1 338.3.d.e 4
39.k even 12 1 338.3.f.d 4
156.v odd 12 1 208.3.bd.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.3.f.a 4 3.b odd 2 1
26.3.f.a 4 39.k even 12 1
208.3.bd.c 4 12.b even 2 1
208.3.bd.c 4 156.v odd 12 1
234.3.bb.b 4 1.a even 1 1 trivial
234.3.bb.b 4 13.f odd 12 1 inner
338.3.d.d 4 39.i odd 6 1
338.3.d.d 4 39.k even 12 1
338.3.d.e 4 39.h odd 6 1
338.3.d.e 4 39.k even 12 1
338.3.f.c 4 39.f even 4 1
338.3.f.c 4 39.h odd 6 1
338.3.f.d 4 39.d odd 2 1
338.3.f.d 4 39.k even 12 1
338.3.f.f 4 39.f even 4 1
338.3.f.f 4 39.i odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + 36$$ acting on $$S_{3}^{\mathrm{new}}(234, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 2 T^{3} + 2 T^{2} + 4 T + 4$$
$3$ $$T^{4}$$
$5$ $$T^{4} + 36$$
$7$ $$T^{4} + 22 T^{3} + 221 T^{2} + \cdots + 5329$$
$11$ $$T^{4} - 6 T^{3} + 45 T^{2} + \cdots + 1521$$
$13$ $$T^{4} + 12 T^{3} + 182 T^{2} + \cdots + 28561$$
$17$ $$T^{4} + 78 T^{3} + 2499 T^{2} + \cdots + 221841$$
$19$ $$T^{4} + 58 T^{3} + 1325 T^{2} + \cdots + 167281$$
$23$ $$T^{4} + 12 T^{3} - 381 T^{2} + \cdots + 184041$$
$29$ $$T^{4} - 54 T^{3} + 2955 T^{2} + \cdots + 1521$$
$31$ $$T^{4} + 128 T^{3} + 8192 T^{2} + \cdots + 3602404$$
$37$ $$T^{4} - 40 T^{3} + 401 T^{2} + \cdots + 11449$$
$41$ $$T^{4} + 1521 T^{2} + 39546 T + 257049$$
$43$ $$T^{4} - 120 T^{3} + 4479 T^{2} + \cdots + 103041$$
$47$ $$(T^{2} + 66 T + 2178)^{2}$$
$53$ $$(T^{2} - 84 T + 312)^{2}$$
$59$ $$T^{4} + 6 T^{3} + 1305 T^{2} + \cdots + 84681$$
$61$ $$T^{4} - 78 T^{3} + 4671 T^{2} + \cdots + 1996569$$
$67$ $$T^{4} - 86 T^{3} + 4985 T^{2} + \cdots + 2399401$$
$71$ $$T^{4} + 42 T^{3} + 3357 T^{2} + \cdots + 154449$$
$73$ $$T^{4} + 136 T^{3} + 9248 T^{2} + \cdots + 4251844$$
$79$ $$(T^{2} - 96 T - 1584)^{2}$$
$83$ $$T^{4} + 192 T^{3} + 18432 T^{2} + \cdots + 2056356$$
$89$ $$T^{4} - 60 T^{3} + 5661 T^{2} + \cdots + 21594609$$
$97$ $$T^{4} - 280 T^{3} + \cdots + 20043529$$