# Properties

 Label 234.6.b.b Level $234$ Weight $6$ Character orbit 234.b Analytic conductor $37.530$ 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] = [234,6,Mod(181,234)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(234, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("234.181");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$234 = 2 \cdot 3^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 234.b (of order $$2$$, degree $$1$$, 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: $$37.5298138362$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 26) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 4 i q^{2} - 16 q^{4} + 51 i q^{5} - 105 i q^{7} + 64 i q^{8} +O(q^{10})$$ q - 4*i * q^2 - 16 * q^4 + 51*i * q^5 - 105*i * q^7 + 64*i * q^8 $$q - 4 i q^{2} - 16 q^{4} + 51 i q^{5} - 105 i q^{7} + 64 i q^{8} + 204 q^{10} + 120 i q^{11} + ( - 117 i - 598) q^{13} - 420 q^{14} + 256 q^{16} + 1101 q^{17} + 1170 i q^{19} - 816 i q^{20} + 480 q^{22} - 1050 q^{23} + 524 q^{25} + (2392 i - 468) q^{26} + 1680 i q^{28} + 4104 q^{29} - 9624 i q^{31} - 1024 i q^{32} - 4404 i q^{34} + 5355 q^{35} - 8709 i q^{37} + 4680 q^{38} - 3264 q^{40} - 9480 i q^{41} + 9995 q^{43} - 1920 i q^{44} + 4200 i q^{46} - 2943 i q^{47} + 5782 q^{49} - 2096 i q^{50} + (1872 i + 9568) q^{52} + 750 q^{53} - 6120 q^{55} + 6720 q^{56} - 16416 i q^{58} - 40938 i q^{59} - 57920 q^{61} - 38496 q^{62} - 4096 q^{64} + ( - 30498 i + 5967) q^{65} - 22812 i q^{67} - 17616 q^{68} - 21420 i q^{70} + 63741 i q^{71} - 58866 i q^{73} - 34836 q^{74} - 18720 i q^{76} + 12600 q^{77} + 63202 q^{79} + 13056 i q^{80} - 37920 q^{82} + 55458 i q^{83} + 56151 i q^{85} - 39980 i q^{86} - 7680 q^{88} - 104778 i q^{89} + (62790 i - 12285) q^{91} + 16800 q^{92} - 11772 q^{94} - 59670 q^{95} - 160452 i q^{97} - 23128 i q^{98} +O(q^{100})$$ q - 4*i * q^2 - 16 * q^4 + 51*i * q^5 - 105*i * q^7 + 64*i * q^8 + 204 * q^10 + 120*i * q^11 + (-117*i - 598) * q^13 - 420 * q^14 + 256 * q^16 + 1101 * q^17 + 1170*i * q^19 - 816*i * q^20 + 480 * q^22 - 1050 * q^23 + 524 * q^25 + (2392*i - 468) * q^26 + 1680*i * q^28 + 4104 * q^29 - 9624*i * q^31 - 1024*i * q^32 - 4404*i * q^34 + 5355 * q^35 - 8709*i * q^37 + 4680 * q^38 - 3264 * q^40 - 9480*i * q^41 + 9995 * q^43 - 1920*i * q^44 + 4200*i * q^46 - 2943*i * q^47 + 5782 * q^49 - 2096*i * q^50 + (1872*i + 9568) * q^52 + 750 * q^53 - 6120 * q^55 + 6720 * q^56 - 16416*i * q^58 - 40938*i * q^59 - 57920 * q^61 - 38496 * q^62 - 4096 * q^64 + (-30498*i + 5967) * q^65 - 22812*i * q^67 - 17616 * q^68 - 21420*i * q^70 + 63741*i * q^71 - 58866*i * q^73 - 34836 * q^74 - 18720*i * q^76 + 12600 * q^77 + 63202 * q^79 + 13056*i * q^80 - 37920 * q^82 + 55458*i * q^83 + 56151*i * q^85 - 39980*i * q^86 - 7680 * q^88 - 104778*i * q^89 + (62790*i - 12285) * q^91 + 16800 * q^92 - 11772 * q^94 - 59670 * q^95 - 160452*i * q^97 - 23128*i * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 32 q^{4}+O(q^{10})$$ 2 * q - 32 * q^4 $$2 q - 32 q^{4} + 408 q^{10} - 1196 q^{13} - 840 q^{14} + 512 q^{16} + 2202 q^{17} + 960 q^{22} - 2100 q^{23} + 1048 q^{25} - 936 q^{26} + 8208 q^{29} + 10710 q^{35} + 9360 q^{38} - 6528 q^{40} + 19990 q^{43} + 11564 q^{49} + 19136 q^{52} + 1500 q^{53} - 12240 q^{55} + 13440 q^{56} - 115840 q^{61} - 76992 q^{62} - 8192 q^{64} + 11934 q^{65} - 35232 q^{68} - 69672 q^{74} + 25200 q^{77} + 126404 q^{79} - 75840 q^{82} - 15360 q^{88} - 24570 q^{91} + 33600 q^{92} - 23544 q^{94} - 119340 q^{95}+O(q^{100})$$ 2 * q - 32 * q^4 + 408 * q^10 - 1196 * q^13 - 840 * q^14 + 512 * q^16 + 2202 * q^17 + 960 * q^22 - 2100 * q^23 + 1048 * q^25 - 936 * q^26 + 8208 * q^29 + 10710 * q^35 + 9360 * q^38 - 6528 * q^40 + 19990 * q^43 + 11564 * q^49 + 19136 * q^52 + 1500 * q^53 - 12240 * q^55 + 13440 * q^56 - 115840 * q^61 - 76992 * q^62 - 8192 * q^64 + 11934 * q^65 - 35232 * q^68 - 69672 * q^74 + 25200 * q^77 + 126404 * q^79 - 75840 * q^82 - 15360 * q^88 - 24570 * q^91 + 33600 * q^92 - 23544 * q^94 - 119340 * q^95

## Character values

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

 $$n$$ $$145$$ $$209$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
181.1
 1.00000i − 1.00000i
4.00000i 0 −16.0000 51.0000i 0 105.000i 64.0000i 0 204.000
181.2 4.00000i 0 −16.0000 51.0000i 0 105.000i 64.0000i 0 204.000
 $$n$$: 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.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 234.6.b.b 2
3.b odd 2 1 26.6.b.a 2
12.b even 2 1 208.6.f.b 2
13.b even 2 1 inner 234.6.b.b 2
39.d odd 2 1 26.6.b.a 2
39.f even 4 1 338.6.a.a 1
39.f even 4 1 338.6.a.c 1
156.h even 2 1 208.6.f.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.6.b.a 2 3.b odd 2 1
26.6.b.a 2 39.d odd 2 1
208.6.f.b 2 12.b even 2 1
208.6.f.b 2 156.h even 2 1
234.6.b.b 2 1.a even 1 1 trivial
234.6.b.b 2 13.b even 2 1 inner
338.6.a.a 1 39.f even 4 1
338.6.a.c 1 39.f even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 2601$$ acting on $$S_{6}^{\mathrm{new}}(234, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 16$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 2601$$
$7$ $$T^{2} + 11025$$
$11$ $$T^{2} + 14400$$
$13$ $$T^{2} + 1196 T + 371293$$
$17$ $$(T - 1101)^{2}$$
$19$ $$T^{2} + 1368900$$
$23$ $$(T + 1050)^{2}$$
$29$ $$(T - 4104)^{2}$$
$31$ $$T^{2} + 92621376$$
$37$ $$T^{2} + 75846681$$
$41$ $$T^{2} + 89870400$$
$43$ $$(T - 9995)^{2}$$
$47$ $$T^{2} + 8661249$$
$53$ $$(T - 750)^{2}$$
$59$ $$T^{2} + 1675919844$$
$61$ $$(T + 57920)^{2}$$
$67$ $$T^{2} + 520387344$$
$71$ $$T^{2} + 4062915081$$
$73$ $$T^{2} + 3465205956$$
$79$ $$(T - 63202)^{2}$$
$83$ $$T^{2} + 3075589764$$
$89$ $$T^{2} + 10978429284$$
$97$ $$T^{2} + 25744844304$$