# Properties

 Label 234.6.b.a 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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ 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 $$\beta = 2i$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 \beta q^{2} - 16 q^{4} + 34 \beta q^{5} - 41 \beta q^{7} - 32 \beta q^{8} +O(q^{10})$$ q + 2*b * q^2 - 16 * q^4 + 34*b * q^5 - 41*b * q^7 - 32*b * q^8 $$q + 2 \beta q^{2} - 16 q^{4} + 34 \beta q^{5} - 41 \beta q^{7} - 32 \beta q^{8} - 272 q^{10} + 195 \beta q^{11} + (169 \beta + 507) q^{13} + 328 q^{14} + 256 q^{16} - 1738 q^{17} + 537 \beta q^{19} - 544 \beta q^{20} - 1560 q^{22} - 2104 q^{23} - 1499 q^{25} + (1014 \beta - 1352) q^{26} + 656 \beta q^{28} + 1690 q^{29} + 715 \beta q^{31} + 512 \beta q^{32} - 3476 \beta q^{34} + 5576 q^{35} - 4426 \beta q^{37} - 4296 q^{38} + 4352 q^{40} + 3380 \beta q^{41} - 16916 q^{43} - 3120 \beta q^{44} - 4208 \beta q^{46} - 12579 \beta q^{47} + 10083 q^{49} - 2998 \beta q^{50} + ( - 2704 \beta - 8112) q^{52} - 38214 q^{53} - 26520 q^{55} - 5248 q^{56} + 3380 \beta q^{58} + 10643 \beta q^{59} - 5458 q^{61} - 5720 q^{62} - 4096 q^{64} + (17238 \beta - 22984) q^{65} - 22271 \beta q^{67} + 27808 q^{68} + 11152 \beta q^{70} - 8895 \beta q^{71} - 15532 \beta q^{73} + 35408 q^{74} - 8592 \beta q^{76} + 31980 q^{77} - 45360 q^{79} + 8704 \beta q^{80} - 27040 q^{82} - 62273 \beta q^{83} - 59092 \beta q^{85} - 33832 \beta q^{86} + 24960 q^{88} - 9372 \beta q^{89} + ( - 20787 \beta + 27716) q^{91} + 33664 q^{92} + 100632 q^{94} - 73032 q^{95} + 60744 \beta q^{97} + 20166 \beta q^{98} +O(q^{100})$$ q + 2*b * q^2 - 16 * q^4 + 34*b * q^5 - 41*b * q^7 - 32*b * q^8 - 272 * q^10 + 195*b * q^11 + (169*b + 507) * q^13 + 328 * q^14 + 256 * q^16 - 1738 * q^17 + 537*b * q^19 - 544*b * q^20 - 1560 * q^22 - 2104 * q^23 - 1499 * q^25 + (1014*b - 1352) * q^26 + 656*b * q^28 + 1690 * q^29 + 715*b * q^31 + 512*b * q^32 - 3476*b * q^34 + 5576 * q^35 - 4426*b * q^37 - 4296 * q^38 + 4352 * q^40 + 3380*b * q^41 - 16916 * q^43 - 3120*b * q^44 - 4208*b * q^46 - 12579*b * q^47 + 10083 * q^49 - 2998*b * q^50 + (-2704*b - 8112) * q^52 - 38214 * q^53 - 26520 * q^55 - 5248 * q^56 + 3380*b * q^58 + 10643*b * q^59 - 5458 * q^61 - 5720 * q^62 - 4096 * q^64 + (17238*b - 22984) * q^65 - 22271*b * q^67 + 27808 * q^68 + 11152*b * q^70 - 8895*b * q^71 - 15532*b * q^73 + 35408 * q^74 - 8592*b * q^76 + 31980 * q^77 - 45360 * q^79 + 8704*b * q^80 - 27040 * q^82 - 62273*b * q^83 - 59092*b * q^85 - 33832*b * q^86 + 24960 * q^88 - 9372*b * q^89 + (-20787*b + 27716) * q^91 + 33664 * q^92 + 100632 * q^94 - 73032 * q^95 + 60744*b * q^97 + 20166*b * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 32 q^{4}+O(q^{10})$$ 2 * q - 32 * q^4 $$2 q - 32 q^{4} - 544 q^{10} + 1014 q^{13} + 656 q^{14} + 512 q^{16} - 3476 q^{17} - 3120 q^{22} - 4208 q^{23} - 2998 q^{25} - 2704 q^{26} + 3380 q^{29} + 11152 q^{35} - 8592 q^{38} + 8704 q^{40} - 33832 q^{43} + 20166 q^{49} - 16224 q^{52} - 76428 q^{53} - 53040 q^{55} - 10496 q^{56} - 10916 q^{61} - 11440 q^{62} - 8192 q^{64} - 45968 q^{65} + 55616 q^{68} + 70816 q^{74} + 63960 q^{77} - 90720 q^{79} - 54080 q^{82} + 49920 q^{88} + 55432 q^{91} + 67328 q^{92} + 201264 q^{94} - 146064 q^{95}+O(q^{100})$$ 2 * q - 32 * q^4 - 544 * q^10 + 1014 * q^13 + 656 * q^14 + 512 * q^16 - 3476 * q^17 - 3120 * q^22 - 4208 * q^23 - 2998 * q^25 - 2704 * q^26 + 3380 * q^29 + 11152 * q^35 - 8592 * q^38 + 8704 * q^40 - 33832 * q^43 + 20166 * q^49 - 16224 * q^52 - 76428 * q^53 - 53040 * q^55 - 10496 * q^56 - 10916 * q^61 - 11440 * q^62 - 8192 * q^64 - 45968 * q^65 + 55616 * q^68 + 70816 * q^74 + 63960 * q^77 - 90720 * q^79 - 54080 * q^82 + 49920 * q^88 + 55432 * q^91 + 67328 * q^92 + 201264 * q^94 - 146064 * 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 68.0000i 0 82.0000i 64.0000i 0 −272.000
181.2 4.00000i 0 −16.0000 68.0000i 0 82.0000i 64.0000i 0 −272.000
 $$n$$: 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.a 2
3.b odd 2 1 26.6.b.b 2
12.b even 2 1 208.6.f.a 2
13.b even 2 1 inner 234.6.b.a 2
39.d odd 2 1 26.6.b.b 2
39.f even 4 1 338.6.a.b 1
39.f even 4 1 338.6.a.e 1
156.h even 2 1 208.6.f.a 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 4624$$ acting on $$S_{6}^{\mathrm{new}}(234, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 16$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 4624$$
$7$ $$T^{2} + 6724$$
$11$ $$T^{2} + 152100$$
$13$ $$T^{2} - 1014 T + 371293$$
$17$ $$(T + 1738)^{2}$$
$19$ $$T^{2} + 1153476$$
$23$ $$(T + 2104)^{2}$$
$29$ $$(T - 1690)^{2}$$
$31$ $$T^{2} + 2044900$$
$37$ $$T^{2} + 78357904$$
$41$ $$T^{2} + 45697600$$
$43$ $$(T + 16916)^{2}$$
$47$ $$T^{2} + 632924964$$
$53$ $$(T + 38214)^{2}$$
$59$ $$T^{2} + 453093796$$
$61$ $$(T + 5458)^{2}$$
$67$ $$T^{2} + 1983989764$$
$71$ $$T^{2} + 316484100$$
$73$ $$T^{2} + 964972096$$
$79$ $$(T + 45360)^{2}$$
$83$ $$T^{2} + 15511706116$$
$89$ $$T^{2} + 351337536$$
$97$ $$T^{2} + 14759334144$$