# Properties

 Label 234.4.h.b Level $234$ Weight $4$ Character orbit 234.h Analytic conductor $13.806$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [234,4,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, 4, names="a")

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

chi := DirichletCharacter("234.55");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$234 = 2 \cdot 3^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ 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: $$13.8064469413$$ 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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 26) 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.

 $$f(q)$$ $$=$$ $$q + (2 \zeta_{6} - 2) q^{2} - 4 \zeta_{6} q^{4} - 2 q^{5} + 5 \zeta_{6} q^{7} + 8 q^{8} +O(q^{10})$$ q + (2*z - 2) * q^2 - 4*z * q^4 - 2 * q^5 + 5*z * q^7 + 8 * q^8 $$q + (2 \zeta_{6} - 2) q^{2} - 4 \zeta_{6} q^{4} - 2 q^{5} + 5 \zeta_{6} q^{7} + 8 q^{8} + ( - 4 \zeta_{6} + 4) q^{10} + ( - 13 \zeta_{6} + 13) q^{11} + (52 \zeta_{6} - 39) q^{13} - 10 q^{14} + (16 \zeta_{6} - 16) q^{16} + 27 \zeta_{6} q^{17} - 75 \zeta_{6} q^{19} + 8 \zeta_{6} q^{20} + 26 \zeta_{6} q^{22} + (187 \zeta_{6} - 187) q^{23} - 121 q^{25} + ( - 78 \zeta_{6} - 26) q^{26} + ( - 20 \zeta_{6} + 20) q^{28} + (13 \zeta_{6} - 13) q^{29} - 104 q^{31} - 32 \zeta_{6} q^{32} - 54 q^{34} - 10 \zeta_{6} q^{35} + (423 \zeta_{6} - 423) q^{37} + 150 q^{38} - 16 q^{40} + ( - 195 \zeta_{6} + 195) q^{41} - 199 \zeta_{6} q^{43} - 52 q^{44} - 374 \zeta_{6} q^{46} - 388 q^{47} + ( - 318 \zeta_{6} + 318) q^{49} + ( - 242 \zeta_{6} + 242) q^{50} + ( - 52 \zeta_{6} + 208) q^{52} - 618 q^{53} + (26 \zeta_{6} - 26) q^{55} + 40 \zeta_{6} q^{56} - 26 \zeta_{6} q^{58} + 491 \zeta_{6} q^{59} - 175 \zeta_{6} q^{61} + ( - 208 \zeta_{6} + 208) q^{62} + 64 q^{64} + ( - 104 \zeta_{6} + 78) q^{65} + (817 \zeta_{6} - 817) q^{67} + ( - 108 \zeta_{6} + 108) q^{68} + 20 q^{70} + 79 \zeta_{6} q^{71} + 230 q^{73} - 846 \zeta_{6} q^{74} + (300 \zeta_{6} - 300) q^{76} + 65 q^{77} + 764 q^{79} + ( - 32 \zeta_{6} + 32) q^{80} + 390 \zeta_{6} q^{82} + 732 q^{83} - 54 \zeta_{6} q^{85} + 398 q^{86} + ( - 104 \zeta_{6} + 104) q^{88} + (1041 \zeta_{6} - 1041) q^{89} + (65 \zeta_{6} - 260) q^{91} + 748 q^{92} + ( - 776 \zeta_{6} + 776) q^{94} + 150 \zeta_{6} q^{95} + 97 \zeta_{6} q^{97} + 636 \zeta_{6} q^{98} +O(q^{100})$$ q + (2*z - 2) * q^2 - 4*z * q^4 - 2 * q^5 + 5*z * q^7 + 8 * q^8 + (-4*z + 4) * q^10 + (-13*z + 13) * q^11 + (52*z - 39) * q^13 - 10 * q^14 + (16*z - 16) * q^16 + 27*z * q^17 - 75*z * q^19 + 8*z * q^20 + 26*z * q^22 + (187*z - 187) * q^23 - 121 * q^25 + (-78*z - 26) * q^26 + (-20*z + 20) * q^28 + (13*z - 13) * q^29 - 104 * q^31 - 32*z * q^32 - 54 * q^34 - 10*z * q^35 + (423*z - 423) * q^37 + 150 * q^38 - 16 * q^40 + (-195*z + 195) * q^41 - 199*z * q^43 - 52 * q^44 - 374*z * q^46 - 388 * q^47 + (-318*z + 318) * q^49 + (-242*z + 242) * q^50 + (-52*z + 208) * q^52 - 618 * q^53 + (26*z - 26) * q^55 + 40*z * q^56 - 26*z * q^58 + 491*z * q^59 - 175*z * q^61 + (-208*z + 208) * q^62 + 64 * q^64 + (-104*z + 78) * q^65 + (817*z - 817) * q^67 + (-108*z + 108) * q^68 + 20 * q^70 + 79*z * q^71 + 230 * q^73 - 846*z * q^74 + (300*z - 300) * q^76 + 65 * q^77 + 764 * q^79 + (-32*z + 32) * q^80 + 390*z * q^82 + 732 * q^83 - 54*z * q^85 + 398 * q^86 + (-104*z + 104) * q^88 + (1041*z - 1041) * q^89 + (65*z - 260) * q^91 + 748 * q^92 + (-776*z + 776) * q^94 + 150*z * q^95 + 97*z * q^97 + 636*z * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 4 q^{4} - 4 q^{5} + 5 q^{7} + 16 q^{8}+O(q^{10})$$ 2 * q - 2 * q^2 - 4 * q^4 - 4 * q^5 + 5 * q^7 + 16 * q^8 $$2 q - 2 q^{2} - 4 q^{4} - 4 q^{5} + 5 q^{7} + 16 q^{8} + 4 q^{10} + 13 q^{11} - 26 q^{13} - 20 q^{14} - 16 q^{16} + 27 q^{17} - 75 q^{19} + 8 q^{20} + 26 q^{22} - 187 q^{23} - 242 q^{25} - 130 q^{26} + 20 q^{28} - 13 q^{29} - 208 q^{31} - 32 q^{32} - 108 q^{34} - 10 q^{35} - 423 q^{37} + 300 q^{38} - 32 q^{40} + 195 q^{41} - 199 q^{43} - 104 q^{44} - 374 q^{46} - 776 q^{47} + 318 q^{49} + 242 q^{50} + 364 q^{52} - 1236 q^{53} - 26 q^{55} + 40 q^{56} - 26 q^{58} + 491 q^{59} - 175 q^{61} + 208 q^{62} + 128 q^{64} + 52 q^{65} - 817 q^{67} + 108 q^{68} + 40 q^{70} + 79 q^{71} + 460 q^{73} - 846 q^{74} - 300 q^{76} + 130 q^{77} + 1528 q^{79} + 32 q^{80} + 390 q^{82} + 1464 q^{83} - 54 q^{85} + 796 q^{86} + 104 q^{88} - 1041 q^{89} - 455 q^{91} + 1496 q^{92} + 776 q^{94} + 150 q^{95} + 97 q^{97} + 636 q^{98}+O(q^{100})$$ 2 * q - 2 * q^2 - 4 * q^4 - 4 * q^5 + 5 * q^7 + 16 * q^8 + 4 * q^10 + 13 * q^11 - 26 * q^13 - 20 * q^14 - 16 * q^16 + 27 * q^17 - 75 * q^19 + 8 * q^20 + 26 * q^22 - 187 * q^23 - 242 * q^25 - 130 * q^26 + 20 * q^28 - 13 * q^29 - 208 * q^31 - 32 * q^32 - 108 * q^34 - 10 * q^35 - 423 * q^37 + 300 * q^38 - 32 * q^40 + 195 * q^41 - 199 * q^43 - 104 * q^44 - 374 * q^46 - 776 * q^47 + 318 * q^49 + 242 * q^50 + 364 * q^52 - 1236 * q^53 - 26 * q^55 + 40 * q^56 - 26 * q^58 + 491 * q^59 - 175 * q^61 + 208 * q^62 + 128 * q^64 + 52 * q^65 - 817 * q^67 + 108 * q^68 + 40 * q^70 + 79 * q^71 + 460 * q^73 - 846 * q^74 - 300 * q^76 + 130 * q^77 + 1528 * q^79 + 32 * q^80 + 390 * q^82 + 1464 * q^83 - 54 * q^85 + 796 * q^86 + 104 * q^88 - 1041 * q^89 - 455 * q^91 + 1496 * q^92 + 776 * q^94 + 150 * q^95 + 97 * q^97 + 636 * 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
−1.00000 1.73205i 0 −2.00000 + 3.46410i −2.00000 0 2.50000 4.33013i 8.00000 0 2.00000 + 3.46410i
217.1 −1.00000 + 1.73205i 0 −2.00000 3.46410i −2.00000 0 2.50000 + 4.33013i 8.00000 0 2.00000 3.46410i
 $$n$$: 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 234.4.h.b 2
3.b odd 2 1 26.4.c.a 2
12.b even 2 1 208.4.i.a 2
13.c even 3 1 inner 234.4.h.b 2
39.d odd 2 1 338.4.c.d 2
39.f even 4 2 338.4.e.d 4
39.h odd 6 1 338.4.a.d 1
39.h odd 6 1 338.4.c.d 2
39.i odd 6 1 26.4.c.a 2
39.i odd 6 1 338.4.a.a 1
39.k even 12 2 338.4.b.a 2
39.k even 12 2 338.4.e.d 4
156.p even 6 1 208.4.i.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.4.c.a 2 3.b odd 2 1
26.4.c.a 2 39.i odd 6 1
208.4.i.a 2 12.b even 2 1
208.4.i.a 2 156.p even 6 1
234.4.h.b 2 1.a even 1 1 trivial
234.4.h.b 2 13.c even 3 1 inner
338.4.a.a 1 39.i odd 6 1
338.4.a.d 1 39.h odd 6 1
338.4.b.a 2 39.k even 12 2
338.4.c.d 2 39.d odd 2 1
338.4.c.d 2 39.h odd 6 1
338.4.e.d 4 39.f even 4 2
338.4.e.d 4 39.k even 12 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} + 2$$ acting on $$S_{4}^{\mathrm{new}}(234, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2T + 4$$
$3$ $$T^{2}$$
$5$ $$(T + 2)^{2}$$
$7$ $$T^{2} - 5T + 25$$
$11$ $$T^{2} - 13T + 169$$
$13$ $$T^{2} + 26T + 2197$$
$17$ $$T^{2} - 27T + 729$$
$19$ $$T^{2} + 75T + 5625$$
$23$ $$T^{2} + 187T + 34969$$
$29$ $$T^{2} + 13T + 169$$
$31$ $$(T + 104)^{2}$$
$37$ $$T^{2} + 423T + 178929$$
$41$ $$T^{2} - 195T + 38025$$
$43$ $$T^{2} + 199T + 39601$$
$47$ $$(T + 388)^{2}$$
$53$ $$(T + 618)^{2}$$
$59$ $$T^{2} - 491T + 241081$$
$61$ $$T^{2} + 175T + 30625$$
$67$ $$T^{2} + 817T + 667489$$
$71$ $$T^{2} - 79T + 6241$$
$73$ $$(T - 230)^{2}$$
$79$ $$(T - 764)^{2}$$
$83$ $$(T - 732)^{2}$$
$89$ $$T^{2} + 1041 T + 1083681$$
$97$ $$T^{2} - 97T + 9409$$