# Properties

 Label 832.2.w.i Level $832$ Weight $2$ Character orbit 832.w Analytic conductor $6.644$ Analytic rank $0$ Dimension $8$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [832,2,Mod(257,832)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(832, base_ring=CyclotomicField(6))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("832.257");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$832 = 2^{6} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 832.w (of order $$6$$, degree $$2$$, not 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: $$6.64355344817$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.195105024.2 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 4x^{7} + 5x^{6} + 4x^{5} - 20x^{4} + 12x^{3} + 45x^{2} - 108x + 81$$ x^8 - 4*x^7 + 5*x^6 + 4*x^5 - 20*x^4 + 12*x^3 + 45*x^2 - 108*x + 81 Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 104) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{5} + \beta_{4} - \beta_{2}) q^{3} + ( - \beta_{7} - \beta_{6} + \cdots - \beta_1) q^{5}+ \cdots + ( - 2 \beta_{7} - 2 \beta_{5} + \cdots - 2) q^{9}+O(q^{10})$$ q + (-b5 + b4 - b2) * q^3 + (-b7 - b6 - b5 - b3 - b1) * q^5 + (-b5 + b3) * q^7 + (-2*b7 - 2*b5 - b2 - b1 - 2) * q^9 $$q + ( - \beta_{5} + \beta_{4} - \beta_{2}) q^{3} + ( - \beta_{7} - \beta_{6} + \cdots - \beta_1) q^{5}+ \cdots + ( - 2 \beta_{7} - 2 \beta_{6} + \cdots - 2) q^{99}+O(q^{100})$$ q + (-b5 + b4 - b2) * q^3 + (-b7 - b6 - b5 - b3 - b1) * q^5 + (-b5 + b3) * q^7 + (-2*b7 - 2*b5 - b2 - b1 - 2) * q^9 + (-b6 + 1) * q^11 + (b6 + b5 + b4 - b1) * q^13 + (-2*b7 - b6 - b5 - b4 - b2 - 1) * q^15 + (b5 + 2*b2 + 1) * q^17 + (b5 - 2*b4 + b2 - 2*b1 - 1) * q^19 + (b7 - 2*b5 + b4 - 2*b2 + b1 - 1) * q^21 + (b5 - b4 + b2) * q^23 + (b7 - b4 - b1 - 3) * q^25 + (-2*b7 - b6 - b5 - 2*b4 + b3 + 2*b1 - 5) * q^27 + (b6 - 2*b5 - b4 + 2*b3 + b2 - 1) * q^29 + (b6 + 5*b5 + b4 + b3 - 2*b2 + 2) * q^31 + (-b5 + 2*b4 - b2 + b1 + 1) * q^33 + (-4*b7 + 4*b5 - 2*b1 + 4) * q^35 + (-2*b7 - b6 + 2*b5 + b4 + b2 + 5) * q^37 + (-2*b7 - b6 - 4*b5 - b3 - b2 - 4*b1) * q^39 + (2*b7 + b5 + 2) * q^41 + (2*b6 + b5 + b3 - 2*b2) * q^43 + (4*b5 - 4*b4 - b3 + 2*b2 + 3*b1 - 3) * q^45 + (2*b7 - b6 + 3*b5 - b4 - b3 + 2*b2 + 2*b1 + 2) * q^47 + (b7 - b4 + b2 + 2*b1) * q^49 + (2*b7 + b4 - 2*b1 + 9) * q^51 + (b7 - b6 - b5 - 2*b4 + b3 - b1 - 4) * q^53 + (-2*b7 + 4*b5 - 4*b1) * q^55 + (b7 - 2*b6 + 8*b5 + b4 - 2*b3 - 2*b2 + b1 + 5) * q^57 + (-3*b5 + 3*b3 + 2*b1) * q^59 + (4*b7 + 2*b6 + b5 + b3 - b2 + 2*b1) * q^61 + (-2*b7 - 2*b6 - 4*b5 - 6) * q^63 + (b7 + 2*b6 - 6*b5 + b4 - 3*b2 + 2*b1 - 2) * q^65 + (-2*b7 - 2*b6 - 5*b5 - b4 - b2 - 8) * q^67 + (2*b7 + 5*b5 + b2 + b1 + 5) * q^69 + (-5*b5 + 2*b4 - b2 + 2*b1 + 5) * q^71 + (-3*b7 + b4 - 2*b2 - 3*b1) * q^73 + (-b6 + 6*b5 - 2*b3 + 1) * q^75 + (-b7 - b4 + b1 + 7) * q^77 + (-2*b7 - 2*b6 - 2*b5 - 2*b4 + 2*b3 + 2*b1 - 4) * q^79 + (2*b7 + 2*b6 + 7*b5 - 6*b4 + 4*b3 + 6*b2 + 4*b1 - 2) * q^81 + (4*b7 + 2*b6 - 2*b5 + 2*b3 + 4*b1 - 2) * q^83 + (4*b4 - b3 - 2*b2 - 3*b1 + 1) * q^85 + (4*b7 + b5 - 3*b2 + 2*b1 + 1) * q^87 + (b7 - 2*b6 - b5 + 3*b4 + 3*b2) * q^89 + (-2*b6 - b5 + 2*b4 - 3*b3 - 2*b1 - 4) * q^91 + (-2*b7 - 2*b5 + 2*b4 + 2*b2 - 4) * q^93 + (4*b7 + 6*b6 - 2*b5 + 3*b3 + b2 + 2*b1 - 5) * q^95 + (-b5 - 2*b4 - 2*b3 + b2 - 5*b1 + 3) * q^97 + (-2*b7 - 2*b6 - 6*b5 - 2*b3 - 2*b1 - 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 2 q^{3} + 6 q^{7} - 6 q^{9}+O(q^{10})$$ 8 * q + 2 * q^3 + 6 * q^7 - 6 * q^9 $$8 q + 2 q^{3} + 6 q^{7} - 6 q^{9} + 6 q^{11} - 6 q^{13} - 6 q^{19} - 2 q^{23} - 20 q^{25} - 28 q^{27} + 8 q^{29} + 6 q^{33} + 16 q^{35} + 24 q^{37} + 14 q^{39} + 12 q^{41} + 6 q^{43} - 30 q^{45} + 2 q^{49} + 68 q^{51} - 20 q^{53} - 16 q^{55} + 18 q^{59} + 4 q^{61} - 36 q^{63} + 14 q^{65} - 42 q^{67} + 18 q^{69} + 54 q^{71} - 22 q^{75} + 60 q^{77} - 16 q^{79} - 20 q^{81} - 6 q^{85} + 10 q^{87} - 18 q^{89} - 46 q^{91} - 36 q^{93} - 16 q^{95} + 30 q^{97}+O(q^{100})$$ 8 * q + 2 * q^3 + 6 * q^7 - 6 * q^9 + 6 * q^11 - 6 * q^13 - 6 * q^19 - 2 * q^23 - 20 * q^25 - 28 * q^27 + 8 * q^29 + 6 * q^33 + 16 * q^35 + 24 * q^37 + 14 * q^39 + 12 * q^41 + 6 * q^43 - 30 * q^45 + 2 * q^49 + 68 * q^51 - 20 * q^53 - 16 * q^55 + 18 * q^59 + 4 * q^61 - 36 * q^63 + 14 * q^65 - 42 * q^67 + 18 * q^69 + 54 * q^71 - 22 * q^75 + 60 * q^77 - 16 * q^79 - 20 * q^81 - 6 * q^85 + 10 * q^87 - 18 * q^89 - 46 * q^91 - 36 * q^93 - 16 * q^95 + 30 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4x^{7} + 5x^{6} + 4x^{5} - 20x^{4} + 12x^{3} + 45x^{2} - 108x + 81$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{7} - \nu^{6} + 2\nu^{5} + 10\nu^{4} + 10\nu^{3} + 42\nu^{2} - 45\nu - 27 ) / 108$$ (v^7 - v^6 + 2*v^5 + 10*v^4 + 10*v^3 + 42*v^2 - 45*v - 27) / 108 $$\beta_{2}$$ $$=$$ $$( -\nu^{7} + \nu^{6} - 2\nu^{5} - 10\nu^{4} - 10\nu^{3} + 66\nu^{2} + 45\nu - 81 ) / 108$$ (-v^7 + v^6 - 2*v^5 - 10*v^4 - 10*v^3 + 66*v^2 + 45*v - 81) / 108 $$\beta_{3}$$ $$=$$ $$( \nu^{7} - \nu^{6} + 2\nu^{5} + 10\nu^{4} + 10\nu^{3} - 66\nu^{2} + 171\nu - 27 ) / 108$$ (v^7 - v^6 + 2*v^5 + 10*v^4 + 10*v^3 - 66*v^2 + 171*v - 27) / 108 $$\beta_{4}$$ $$=$$ $$( \nu^{7} + 2\nu^{6} - 10\nu^{5} - 2\nu^{4} + 22\nu^{3} - 18\nu^{2} - 9\nu + 54 ) / 54$$ (v^7 + 2*v^6 - 10*v^5 - 2*v^4 + 22*v^3 - 18*v^2 - 9*v + 54) / 54 $$\beta_{5}$$ $$=$$ $$( -5\nu^{7} + 8\nu^{6} - 4\nu^{5} - 26\nu^{4} + 52\nu^{3} + 18\nu^{2} - 153\nu + 162 ) / 108$$ (-5*v^7 + 8*v^6 - 4*v^5 - 26*v^4 + 52*v^3 + 18*v^2 - 153*v + 162) / 108 $$\beta_{6}$$ $$=$$ $$( 16\nu^{7} - 37\nu^{6} + 26\nu^{5} + 64\nu^{4} - 158\nu^{3} - 24\nu^{2} + 450\nu - 729 ) / 108$$ (16*v^7 - 37*v^6 + 26*v^5 + 64*v^4 - 158*v^3 - 24*v^2 + 450*v - 729) / 108 $$\beta_{7}$$ $$=$$ $$( 19\nu^{7} - 49\nu^{6} + 14\nu^{5} + 130\nu^{4} - 218\nu^{3} - 150\nu^{2} + 801\nu - 891 ) / 108$$ (19*v^7 - 49*v^6 + 14*v^5 + 130*v^4 - 218*v^3 - 150*v^2 + 801*v - 891) / 108
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} + 1 ) / 2$$ (b3 + b2 + 1) / 2 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 1$$ b2 + b1 + 1 $$\nu^{3}$$ $$=$$ $$( \beta_{6} + 5\beta_{5} + \beta_{4} + 3\beta_{3} + 4\beta_1 ) / 2$$ (b6 + 5*b5 + b4 + 3*b3 + 4*b1) / 2 $$\nu^{4}$$ $$=$$ $$\beta_{7} - 2\beta_{6} - 2\beta_{5} - \beta_{4} - \beta_{2} + 4\beta _1 - 1$$ b7 - 2*b6 - 2*b5 - b4 - b2 + 4*b1 - 1 $$\nu^{5}$$ $$=$$ $$( -4\beta_{7} + 3\beta_{6} - 5\beta_{5} - 7\beta_{4} + 8\beta_{3} - 5\beta_{2} + 4\beta _1 + 1 ) / 2$$ (-4*b7 + 3*b6 - 5*b5 - 7*b4 + 8*b3 - 5*b2 + 4*b1 + 1) / 2 $$\nu^{6}$$ $$=$$ $$-4\beta_{7} - 4\beta_{6} - 28\beta_{5} - 2\beta_{4} + 4\beta_{3} - 15$$ -4*b7 - 4*b6 - 28*b5 - 2*b4 + 4*b3 - 15 $$\nu^{7}$$ $$=$$ $$( -20\beta_{7} + 16\beta_{6} - 56\beta_{5} + 20\beta_{4} + 7\beta_{3} - 9\beta_{2} + 4\beta _1 + 3 ) / 2$$ (-20*b7 + 16*b6 - 56*b5 + 20*b4 + 7*b3 - 9*b2 + 4*b1 + 3) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/832\mathbb{Z}\right)^\times$$.

 $$n$$ $$261$$ $$703$$ $$769$$ $$\chi(n)$$ $$1$$ $$1$$ $$1 + \beta_{5}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
257.1
 0.560908 + 1.63871i −1.58726 − 0.693255i 1.72124 + 0.193255i 1.30512 − 1.13871i 0.560908 − 1.63871i −1.58726 + 0.693255i 1.72124 − 0.193255i 1.30512 + 1.13871i
0 −1.13871 1.97231i 0 1.12182i 0 2.26053 + 1.30512i 0 −1.09334 + 1.89372i 0
257.2 0 −0.193255 0.334727i 0 3.17452i 0 −2.98127 1.72124i 0 1.42531 2.46870i 0
257.3 0 0.693255 + 1.20075i 0 3.44247i 0 2.74922 + 1.58726i 0 0.538796 0.933222i 0
257.4 0 1.63871 + 2.83834i 0 2.61023i 0 0.971521 + 0.560908i 0 −3.87076 + 6.70436i 0
641.1 0 −1.13871 + 1.97231i 0 1.12182i 0 2.26053 1.30512i 0 −1.09334 1.89372i 0
641.2 0 −0.193255 + 0.334727i 0 3.17452i 0 −2.98127 + 1.72124i 0 1.42531 + 2.46870i 0
641.3 0 0.693255 1.20075i 0 3.44247i 0 2.74922 1.58726i 0 0.538796 + 0.933222i 0
641.4 0 1.63871 2.83834i 0 2.61023i 0 0.971521 0.560908i 0 −3.87076 6.70436i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 257.4 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.e even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 832.2.w.i 8
4.b odd 2 1 832.2.w.g 8
8.b even 2 1 208.2.w.c 8
8.d odd 2 1 104.2.o.a 8
13.e even 6 1 inner 832.2.w.i 8
24.f even 2 1 936.2.bi.b 8
24.h odd 2 1 1872.2.by.n 8
52.i odd 6 1 832.2.w.g 8
104.h odd 2 1 1352.2.o.f 8
104.m even 4 1 1352.2.i.k 8
104.m even 4 1 1352.2.i.l 8
104.n odd 6 1 1352.2.f.f 8
104.n odd 6 1 1352.2.o.f 8
104.p odd 6 1 104.2.o.a 8
104.p odd 6 1 1352.2.f.f 8
104.r even 6 1 2704.2.f.q 8
104.s even 6 1 208.2.w.c 8
104.s even 6 1 2704.2.f.q 8
104.u even 12 1 1352.2.a.k 4
104.u even 12 1 1352.2.a.l 4
104.u even 12 1 1352.2.i.k 8
104.u even 12 1 1352.2.i.l 8
104.x odd 12 1 2704.2.a.bd 4
104.x odd 12 1 2704.2.a.be 4
312.ba even 6 1 936.2.bi.b 8
312.bg odd 6 1 1872.2.by.n 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
104.2.o.a 8 8.d odd 2 1
104.2.o.a 8 104.p odd 6 1
208.2.w.c 8 8.b even 2 1
208.2.w.c 8 104.s even 6 1
832.2.w.g 8 4.b odd 2 1
832.2.w.g 8 52.i odd 6 1
832.2.w.i 8 1.a even 1 1 trivial
832.2.w.i 8 13.e even 6 1 inner
936.2.bi.b 8 24.f even 2 1
936.2.bi.b 8 312.ba even 6 1
1352.2.a.k 4 104.u even 12 1
1352.2.a.l 4 104.u even 12 1
1352.2.f.f 8 104.n odd 6 1
1352.2.f.f 8 104.p odd 6 1
1352.2.i.k 8 104.m even 4 1
1352.2.i.k 8 104.u even 12 1
1352.2.i.l 8 104.m even 4 1
1352.2.i.l 8 104.u even 12 1
1352.2.o.f 8 104.h odd 2 1
1352.2.o.f 8 104.n odd 6 1
1872.2.by.n 8 24.h odd 2 1
1872.2.by.n 8 312.bg odd 6 1
2704.2.a.bd 4 104.x odd 12 1
2704.2.a.be 4 104.x odd 12 1
2704.2.f.q 8 104.r even 6 1
2704.2.f.q 8 104.s even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{8} - 2T_{3}^{7} + 11T_{3}^{6} - 2T_{3}^{5} + 61T_{3}^{4} - 40T_{3}^{3} + 92T_{3}^{2} + 32T_{3} + 16$$ acting on $$S_{2}^{\mathrm{new}}(832, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8} - 2 T^{7} + \cdots + 16$$
$5$ $$T^{8} + 30 T^{6} + \cdots + 1024$$
$7$ $$T^{8} - 6 T^{7} + \cdots + 1024$$
$11$ $$T^{8} - 6 T^{7} + \cdots + 1024$$
$13$ $$T^{8} + 6 T^{7} + \cdots + 28561$$
$17$ $$T^{8} + 34 T^{6} + \cdots + 9409$$
$19$ $$T^{8} + 6 T^{7} + \cdots + 80656$$
$23$ $$T^{8} + 2 T^{7} + \cdots + 16$$
$29$ $$T^{8} - 8 T^{7} + \cdots + 3721$$
$31$ $$T^{8} + 192 T^{6} + \cdots + 256$$
$37$ $$T^{8} - 24 T^{7} + \cdots + 89401$$
$41$ $$(T^{4} - 6 T^{3} + \cdots + 169)^{2}$$
$43$ $$T^{8} - 6 T^{7} + \cdots + 692224$$
$47$ $$T^{8} + 192 T^{6} + \cdots + 135424$$
$53$ $$(T^{4} + 10 T^{3} + \cdots - 1724)^{2}$$
$59$ $$T^{8} - 18 T^{7} + \cdots + 43264$$
$61$ $$T^{8} - 4 T^{7} + \cdots + 896809$$
$67$ $$T^{8} + 42 T^{7} + \cdots + 55696$$
$71$ $$T^{8} - 54 T^{7} + \cdots + 1430416$$
$73$ $$T^{8} + 198 T^{6} + \cdots + 20736$$
$79$ $$(T^{4} + 8 T^{3} + \cdots + 3328)^{2}$$
$83$ $$T^{8} + 248 T^{6} + \cdots + 65536$$
$89$ $$T^{8} + 18 T^{7} + \cdots + 186267904$$
$97$ $$T^{8} - 30 T^{7} + \cdots + 55830784$$