# Properties

 Label 234.6.b.c Level $234$ Weight $6$ Character orbit 234.b Analytic conductor $37.530$ Analytic rank $0$ Dimension $6$ 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: $$6$$ Coefficient field: $$\mathbb{Q}[x]/(x^{6} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 2x^{5} + 2x^{4} - 2946x^{3} + 131769x^{2} - 1332936x + 6741792$$ x^6 - 2*x^5 + 2*x^4 - 2946*x^3 + 131769*x^2 - 1332936*x + 6741792 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{11}\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 78) 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 a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} - 16 q^{4} + ( - \beta_{3} + 3 \beta_1) q^{5} + (\beta_{5} + 14 \beta_1) q^{7} + 16 \beta_1 q^{8}+O(q^{10})$$ q - b1 * q^2 - 16 * q^4 + (-b3 + 3*b1) * q^5 + (b5 + 14*b1) * q^7 + 16*b1 * q^8 $$q - \beta_1 q^{2} - 16 q^{4} + ( - \beta_{3} + 3 \beta_1) q^{5} + (\beta_{5} + 14 \beta_1) q^{7} + 16 \beta_1 q^{8} + (\beta_{4} + 53) q^{10} + ( - \beta_{5} + 7 \beta_{3} - 11 \beta_1) q^{11} + ( - 3 \beta_{5} + 2 \beta_{4} + \cdots + 88) q^{13}+ \cdots + (232 \beta_{5} + 544 \beta_{3} + 649 \beta_1) q^{98}+O(q^{100})$$ q - b1 * q^2 - 16 * q^4 + (-b3 + 3*b1) * q^5 + (b5 + 14*b1) * q^7 + 16*b1 * q^8 + (b4 + 53) * q^10 + (-b5 + 7*b3 - 11*b1) * q^11 + (-3*b5 + 2*b4 + b3 + b2 + 15*b1 + 88) * q^13 + (-2*b2 + 226) * q^14 + 256 * q^16 + (2*b4 - 5*b2 + 137) * q^17 + (-5*b5 - 34*b3 - 16*b1) * q^19 + (16*b3 - 48*b1) * q^20 + (-7*b4 + 2*b2 - 213) * q^22 + (-4*b4 - 8*b2 + 68) * q^23 + (-14*b4 + 5*b2 + 126) * q^25 + (8*b5 - b4 + 32*b3 + 6*b2 - 79*b1 + 229) * q^26 + (-16*b5 - 224*b1) * q^28 + (-14*b4 + 5*b2 - 3125) * q^29 + (b5 - 66*b3 + 596*b1) * q^31 - 256*b1 * q^32 + (-40*b5 + 32*b3 - 122*b1) * q^34 + (-28*b4 - 8*b2 - 1012) * q^35 + (-6*b5 - 112*b3 + 612*b1) * q^37 + (34*b4 + 10*b2 - 96) * q^38 + (-16*b4 - 848) * q^40 + (-54*b5 + b3 + 1521*b1) * q^41 + (10*b4 + 59*b2 - 4017) * q^43 + (16*b5 - 112*b3 + 176*b1) * q^44 + (-64*b5 - 64*b3 - 80*b1) * q^46 + (-163*b5 + 7*b3 - 311*b1) * q^47 + (34*b4 + 29*b2 - 508) * q^49 + (40*b5 - 224*b3 - 201*b1) * q^50 + (48*b5 - 32*b4 - 16*b3 - 16*b2 - 240*b1 - 1408) * q^52 + (-132*b4 + 36*b2 + 7122) * q^53 + (102*b4 - 27*b2 + 20733) * q^55 + (32*b2 - 3616) * q^56 + (40*b5 - 224*b3 + 3050*b1) * q^58 + (-189*b5 - 263*b3 + 87*b1) * q^59 + (-154*b4 - 5*b2 - 483) * q^61 + (66*b4 - 2*b2 + 9868) * q^62 - 4096 * q^64 + (112*b5 + 38*b4 - 319*b3 + 19*b2 + 5693*b1 + 2855) * q^65 + (-121*b5 + 68*b3 - 8858*b1) * q^67 + (-32*b4 + 80*b2 - 2192) * q^68 + (-64*b5 - 448*b3 + 880*b1) * q^70 + (-27*b5 - 651*b3 - 5937*b1) * q^71 + (-56*b5 - 242*b3 + 5510*b1) * q^73 + (112*b4 + 12*b2 + 10340) * q^74 + (80*b5 + 544*b3 + 256*b1) * q^76 + (162*b4 + 75*b2 + 18975) * q^77 + (16*b4 + 140*b2 - 28180) * q^79 + (-256*b3 + 768*b1) * q^80 + (-b4 + 108*b2 + 24223) * q^82 + (-73*b5 + 201*b3 - 3329*b1) * q^83 + (-80*b5 - 1034*b3 + 6446*b1) * q^85 + (472*b5 + 160*b3 + 4008*b1) * q^86 + (112*b4 - 32*b2 + 3408) * q^88 + (194*b5 + 853*b3 - 875*b1) * q^89 + (-149*b5 - 74*b4 - 1168*b3 + 41*b2 - 5638*b1 + 39397) * q^91 + (64*b4 + 128*b2 - 1088) * q^92 + (-7*b4 + 326*b2 - 5337) * q^94 + (-288*b4 + 210*b2 - 94362) * q^95 + (-556*b5 + 82*b3 + 8722*b1) * q^97 + (232*b5 + 544*b3 + 649*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 96 q^{4}+O(q^{10})$$ 6 * q - 96 * q^4 $$6 q - 96 q^{4} + 320 q^{10} + 530 q^{13} + 1360 q^{14} + 1536 q^{16} + 836 q^{17} - 1296 q^{22} + 416 q^{23} + 718 q^{25} + 1360 q^{26} - 18788 q^{29} - 6112 q^{35} - 528 q^{38} - 5120 q^{40} - 24200 q^{43} - 3038 q^{49} - 8480 q^{52} + 42396 q^{53} + 124656 q^{55} - 21760 q^{56} - 3196 q^{61} + 59344 q^{62} - 24576 q^{64} + 17168 q^{65} - 13376 q^{68} + 62240 q^{74} + 114024 q^{77} - 169328 q^{79} + 145120 q^{82} + 20736 q^{88} + 236152 q^{91} - 6656 q^{92} - 32688 q^{94} - 567168 q^{95}+O(q^{100})$$ 6 * q - 96 * q^4 + 320 * q^10 + 530 * q^13 + 1360 * q^14 + 1536 * q^16 + 836 * q^17 - 1296 * q^22 + 416 * q^23 + 718 * q^25 + 1360 * q^26 - 18788 * q^29 - 6112 * q^35 - 528 * q^38 - 5120 * q^40 - 24200 * q^43 - 3038 * q^49 - 8480 * q^52 + 42396 * q^53 + 124656 * q^55 - 21760 * q^56 - 3196 * q^61 + 59344 * q^62 - 24576 * q^64 + 17168 * q^65 - 13376 * q^68 + 62240 * q^74 + 114024 * q^77 - 169328 * q^79 + 145120 * q^82 + 20736 * q^88 + 236152 * q^91 - 6656 * q^92 - 32688 * q^94 - 567168 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2x^{5} + 2x^{4} - 2946x^{3} + 131769x^{2} - 1332936x + 6741792$$ :

 $$\beta_{1}$$ $$=$$ $$( 11164\nu^{5} + 33517\nu^{4} + 136007\nu^{3} - 12505719\nu^{2} + 1388809431\nu - 7854226236 ) / 1807933527$$ (11164*v^5 + 33517*v^4 + 136007*v^3 - 12505719*v^2 + 1388809431*v - 7854226236) / 1807933527 $$\beta_{2}$$ $$=$$ $$( -1108\nu^{5} - 131752\nu^{4} - 404420\nu^{3} + 1632084\nu^{2} + 4068576\nu - 9481422087 ) / 35449677$$ (-1108*v^5 - 131752*v^4 - 404420*v^3 + 1632084*v^2 + 4068576*v - 9481422087) / 35449677 $$\beta_{3}$$ $$=$$ $$( 383264 \nu^{5} + 1222625 \nu^{4} - 21601613 \nu^{3} - 1608343917 \nu^{2} + 37929979503 \nu - 221669454492 ) / 1807933527$$ (383264*v^5 + 1222625*v^4 - 21601613*v^3 - 1608343917*v^2 + 37929979503*v - 221669454492) / 1807933527 $$\beta_{4}$$ $$=$$ $$( -8786\nu^{5} - 276878\nu^{4} - 3206890\nu^{3} + 12941778\nu^{2} + 32262192\nu - 10639840401 ) / 35449677$$ (-8786*v^5 - 276878*v^4 - 3206890*v^3 + 12941778*v^2 + 32262192*v - 10639840401) / 35449677 $$\beta_{5}$$ $$=$$ $$( 307633 \nu^{5} + 917590 \nu^{4} + 5937014 \nu^{3} - 1150320390 \nu^{2} + 39082121901 \nu - 220426926804 ) / 1205289018$$ (307633*v^5 + 917590*v^4 + 5937014*v^3 - 1150320390*v^2 + 39082121901*v - 220426926804) / 1205289018
 $$\nu$$ $$=$$ $$( 4\beta_{5} + \beta_{4} - 4\beta_{3} - 2\beta_{2} + 2\beta _1 + 15 ) / 48$$ (4*b5 + b4 - 4*b3 - 2*b2 + 2*b1 + 15) / 48 $$\nu^{2}$$ $$=$$ $$( -8\beta_{5} - \beta_{3} + 365\beta_1 ) / 6$$ (-8*b5 - b3 + 365*b1) / 6 $$\nu^{3}$$ $$=$$ $$( 1388\beta_{5} - 365\beta_{4} - 1460\beta_{3} + 694\beta_{2} - 18386\beta _1 + 71025 ) / 48$$ (1388*b5 - 365*b4 - 1460*b3 + 694*b2 - 18386*b1 + 71025) / 48 $$\nu^{4}$$ $$=$$ $$( 554\beta_{4} - 4393\beta_{2} - 1008681 ) / 12$$ (554*b4 - 4393*b2 - 1008681) / 12 $$\nu^{5}$$ $$=$$ $$( -586204\beta_{5} - 126607\beta_{4} + 506428\beta_{3} + 293102\beta_{2} + 11019394\beta _1 + 43151439 ) / 48$$ (-586204*b5 - 126607*b4 + 506428*b3 + 293102*b2 + 11019394*b1 + 43151439) / 48

## 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
 6.10758 + 6.10758i −15.0768 − 15.0768i 9.96927 + 9.96927i 9.96927 − 9.96927i −15.0768 + 15.0768i 6.10758 − 6.10758i
4.00000i 0 −16.0000 37.1158i 0 176.407i 64.0000i 0 −148.463
181.2 4.00000i 0 −16.0000 9.73803i 0 105.184i 64.0000i 0 −38.9521
181.3 4.00000i 0 −16.0000 86.8538i 0 98.7774i 64.0000i 0 347.415
181.4 4.00000i 0 −16.0000 86.8538i 0 98.7774i 64.0000i 0 347.415
181.5 4.00000i 0 −16.0000 9.73803i 0 105.184i 64.0000i 0 −38.9521
181.6 4.00000i 0 −16.0000 37.1158i 0 176.407i 64.0000i 0 −148.463
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 181.6 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.c 6
3.b odd 2 1 78.6.b.a 6
12.b even 2 1 624.6.c.d 6
13.b even 2 1 inner 234.6.b.c 6
39.d odd 2 1 78.6.b.a 6
39.f even 4 1 1014.6.a.o 3
39.f even 4 1 1014.6.a.q 3
156.h even 2 1 624.6.c.d 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.6.b.a 6 3.b odd 2 1
78.6.b.a 6 39.d odd 2 1
234.6.b.c 6 1.a even 1 1 trivial
234.6.b.c 6 13.b even 2 1 inner
624.6.c.d 6 12.b even 2 1
624.6.c.d 6 156.h even 2 1
1014.6.a.o 3 39.f even 4 1
1014.6.a.q 3 39.f even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{6} + 9016T_{5}^{4} + 11237904T_{5}^{2} + 985457664$$ acting on $$S_{6}^{\mathrm{new}}(234, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 16)^{3}$$
$3$ $$T^{6}$$
$5$ $$T^{6} + 9016 T^{4} + \cdots + 985457664$$
$7$ $$T^{6} + \cdots + 3359273140224$$
$11$ $$T^{6} + \cdots + 871026613185600$$
$13$ $$T^{6} + \cdots + 51\!\cdots\!57$$
$17$ $$(T^{3} - 418 T^{2} + \cdots + 1783218312)^{2}$$
$19$ $$T^{6} + \cdots + 224220196832256$$
$23$ $$(T^{3} - 208 T^{2} + \cdots - 2602266624)^{2}$$
$29$ $$(T^{3} + 9394 T^{2} + \cdots + 2546571960)^{2}$$
$31$ $$T^{6} + \cdots + 18\!\cdots\!00$$
$37$ $$T^{6} + \cdots + 15\!\cdots\!44$$
$41$ $$T^{6} + \cdots + 40\!\cdots\!00$$
$43$ $$(T^{3} + \cdots - 1729964364544)^{2}$$
$47$ $$T^{6} + \cdots + 20\!\cdots\!24$$
$53$ $$(T^{3} + \cdots + 26960052479832)^{2}$$
$59$ $$T^{6} + \cdots + 11\!\cdots\!04$$
$61$ $$(T^{3} + \cdots + 17380383329800)^{2}$$
$67$ $$T^{6} + \cdots + 94\!\cdots\!84$$
$71$ $$T^{6} + \cdots + 22\!\cdots\!00$$
$73$ $$T^{6} + \cdots + 11\!\cdots\!16$$
$79$ $$(T^{3} + \cdots - 35835411156480)^{2}$$
$83$ $$T^{6} + \cdots + 11\!\cdots\!56$$
$89$ $$T^{6} + \cdots + 35\!\cdots\!16$$
$97$ $$T^{6} + \cdots + 30\!\cdots\!64$$