# Properties

 Label 176.2.q.a Level $176$ Weight $2$ Character orbit 176.q Analytic conductor $1.405$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [176,2,Mod(63,176)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(176, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([5, 0, 3]))

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

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

chi := DirichletCharacter("176.63");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$176 = 2^{4} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 176.q (of order $$10$$, degree $$4$$, 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: $$1.40536707557$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{10})$$ Coefficient field: 8.0.484000000.6 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - x^{6} + 16x^{4} - 66x^{2} + 121$$ x^8 - x^6 + 16*x^4 - 66*x^2 + 121 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} + (\beta_{5} - \beta_{3} + \beta_{2}) q^{5} + (\beta_{7} - \beta_{4} + 2 \beta_1) q^{7} + (\beta_{3} - \beta_{2} - 1) q^{9}+O(q^{10})$$ q + b1 * q^3 + (b5 - b3 + b2) * q^5 + (b7 - b4 + 2*b1) * q^7 + (b3 - b2 - 1) * q^9 $$q + \beta_1 q^{3} + (\beta_{5} - \beta_{3} + \beta_{2}) q^{5} + (\beta_{7} - \beta_{4} + 2 \beta_1) q^{7} + (\beta_{3} - \beta_{2} - 1) q^{9} + ( - \beta_{7} + \beta_1) q^{11} + ( - 4 \beta_{5} + \beta_{3} - 2 \beta_{2} - 3) q^{13} + (\beta_{6} - \beta_1) q^{15} + ( - \beta_{5} + 3 \beta_{3} + \beta_{2} - 2) q^{17} + (2 \beta_{7} - 2 \beta_{6} + 2 \beta_{4} + \beta_1) q^{19} + (5 \beta_{5} - 4 \beta_{3} - \beta_{2} + 2) q^{21} + ( - 2 \beta_{7} - 2 \beta_{4} - 2 \beta_1) q^{23} + (3 \beta_{5} - 3 \beta_{3} + 4 \beta_{2} + 4) q^{25} + ( - 2 \beta_{7} + 2 \beta_{6} + \beta_{4} - 3 \beta_1) q^{27} + ( - \beta_{5} - 2 \beta_{3} + 4 \beta_{2} + 5) q^{29} + (3 \beta_{7} - \beta_{6}) q^{31} + ( - 2 \beta_{5} + 4 \beta_{3} - 7 \beta_{2} - 1) q^{33} + ( - \beta_{7} + 2 \beta_{6} + 2 \beta_{4} - 4 \beta_1) q^{35} + ( - \beta_{5} + 6 \beta_{3} - 1) q^{37} + ( - 2 \beta_{7} - 2 \beta_{6} - 3 \beta_{4} - \beta_1) q^{39} + ( - 4 \beta_{5} - 4 \beta_{3} - \beta_{2} + 1) q^{41} + (2 \beta_{7} - 4 \beta_{6} + 2 \beta_1) q^{43} + q^{45} + (4 \beta_{6} + 4 \beta_{4} - 5 \beta_1) q^{47} + (10 \beta_{5} - 9 \beta_{3} + 9 \beta_{2}) q^{49} + ( - 2 \beta_{7} + \beta_{6} + 2 \beta_{4} - 3 \beta_1) q^{51} + (9 \beta_{3} - 6 \beta_{2} - 9) q^{53} + (\beta_{7} + \beta_{6} - \beta_{4}) q^{55} + ( - 8 \beta_{5} - \beta_{3} - 4 \beta_{2} - 9) q^{57} + ( - 3 \beta_{7} - 2 \beta_{6} - 3 \beta_{4} + 2 \beta_1) q^{59} + ( - 3 \beta_{5} + 5 \beta_{3} - \beta_{2} - 6) q^{61} + ( - \beta_{6} + \beta_{4}) q^{63} + (3 \beta_{5} - 2 \beta_{3} - \beta_{2} + 1) q^{65} + (4 \beta_{7} + 4 \beta_1) q^{67} + (2 \beta_{5} - 2 \beta_{3} + 10 \beta_{2} + 10) q^{69} + (\beta_{7} - \beta_{6} - 3 \beta_{4} + 4 \beta_1) q^{71} + ( - 7 \beta_{5} + 2 \beta_{3} - 4 \beta_{2} + 3) q^{73} + ( - \beta_{7} + 4 \beta_{6}) q^{75} + (3 \beta_{5} - 6 \beta_{3} - 6 \beta_{2} + 7) q^{77} + (3 \beta_{7} - 3 \beta_{6} + 6 \beta_1) q^{79} + (2 \beta_{5} + 8 \beta_{3} + 2) q^{81} + (\beta_{7} + \beta_{6} + 3 \beta_{4} + 2 \beta_1) q^{83} + ( - 2 \beta_{5} - 2 \beta_{3} - 3 \beta_{2} + 3) q^{85} + ( - 5 \beta_{7} + 4 \beta_{6} - 3 \beta_{4} + \beta_1) q^{87} + (10 \beta_{5} + 10 \beta_{2} + 6) q^{89} + ( - 5 \beta_{6} - 5 \beta_{4} + 5 \beta_1) q^{91} + (3 \beta_{5} - 10 \beta_{3} + 10 \beta_{2}) q^{93} + (\beta_{6} + \beta_1) q^{95} + ( - \beta_{3} - 4 \beta_{2} + 1) q^{97} + (2 \beta_{7} - \beta_{6} + 2 \beta_{4}) q^{99}+O(q^{100})$$ q + b1 * q^3 + (b5 - b3 + b2) * q^5 + (b7 - b4 + 2*b1) * q^7 + (b3 - b2 - 1) * q^9 + (-b7 + b1) * q^11 + (-4*b5 + b3 - 2*b2 - 3) * q^13 + (b6 - b1) * q^15 + (-b5 + 3*b3 + b2 - 2) * q^17 + (2*b7 - 2*b6 + 2*b4 + b1) * q^19 + (5*b5 - 4*b3 - b2 + 2) * q^21 + (-2*b7 - 2*b4 - 2*b1) * q^23 + (3*b5 - 3*b3 + 4*b2 + 4) * q^25 + (-2*b7 + 2*b6 + b4 - 3*b1) * q^27 + (-b5 - 2*b3 + 4*b2 + 5) * q^29 + (3*b7 - b6) * q^31 + (-2*b5 + 4*b3 - 7*b2 - 1) * q^33 + (-b7 + 2*b6 + 2*b4 - 4*b1) * q^35 + (-b5 + 6*b3 - 1) * q^37 + (-2*b7 - 2*b6 - 3*b4 - b1) * q^39 + (-4*b5 - 4*b3 - b2 + 1) * q^41 + (2*b7 - 4*b6 + 2*b1) * q^43 + q^45 + (4*b6 + 4*b4 - 5*b1) * q^47 + (10*b5 - 9*b3 + 9*b2) * q^49 + (-2*b7 + b6 + 2*b4 - 3*b1) * q^51 + (9*b3 - 6*b2 - 9) * q^53 + (b7 + b6 - b4) * q^55 + (-8*b5 - b3 - 4*b2 - 9) * q^57 + (-3*b7 - 2*b6 - 3*b4 + 2*b1) * q^59 + (-3*b5 + 5*b3 - b2 - 6) * q^61 + (-b6 + b4) * q^63 + (3*b5 - 2*b3 - b2 + 1) * q^65 + (4*b7 + 4*b1) * q^67 + (2*b5 - 2*b3 + 10*b2 + 10) * q^69 + (b7 - b6 - 3*b4 + 4*b1) * q^71 + (-7*b5 + 2*b3 - 4*b2 + 3) * q^73 + (-b7 + 4*b6) * q^75 + (3*b5 - 6*b3 - 6*b2 + 7) * q^77 + (3*b7 - 3*b6 + 6*b1) * q^79 + (2*b5 + 8*b3 + 2) * q^81 + (b7 + b6 + 3*b4 + 2*b1) * q^83 + (-2*b5 - 2*b3 - 3*b2 + 3) * q^85 + (-5*b7 + 4*b6 - 3*b4 + b1) * q^87 + (10*b5 + 10*b2 + 6) * q^89 + (-5*b6 - 5*b4 + 5*b1) * q^91 + (3*b5 - 10*b3 + 10*b2) * q^93 + (b6 + b1) * q^95 + (-b3 - 4*b2 + 1) * q^97 + (2*b7 - b6 + 2*b4) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 6 q^{5} - 4 q^{9}+O(q^{10})$$ 8 * q - 6 * q^5 - 4 * q^9 $$8 q - 6 q^{5} - 4 q^{9} - 10 q^{13} - 10 q^{17} + 12 q^{25} + 30 q^{29} + 18 q^{33} + 6 q^{37} + 10 q^{41} + 8 q^{45} - 56 q^{49} - 42 q^{53} - 50 q^{57} - 30 q^{61} + 52 q^{69} + 50 q^{73} + 50 q^{77} + 28 q^{81} + 30 q^{85} + 8 q^{89} - 46 q^{93} + 14 q^{97}+O(q^{100})$$ 8 * q - 6 * q^5 - 4 * q^9 - 10 * q^13 - 10 * q^17 + 12 * q^25 + 30 * q^29 + 18 * q^33 + 6 * q^37 + 10 * q^41 + 8 * q^45 - 56 * q^49 - 42 * q^53 - 50 * q^57 - 30 * q^61 + 52 * q^69 + 50 * q^73 + 50 * q^77 + 28 * q^81 + 30 * q^85 + 8 * q^89 - 46 * q^93 + 14 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - x^{6} + 16x^{4} - 66x^{2} + 121$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -7\nu^{6} - 37\nu^{4} - 629\nu^{2} - 363 ) / 1991$$ (-7*v^6 - 37*v^4 - 629*v^2 - 363) / 1991 $$\beta_{3}$$ $$=$$ $$( -28\nu^{6} - 148\nu^{4} - 525\nu^{2} + 539 ) / 1991$$ (-28*v^6 - 148*v^4 - 525*v^2 + 539) / 1991 $$\beta_{4}$$ $$=$$ $$( -28\nu^{7} - 148\nu^{5} - 525\nu^{3} + 539\nu ) / 1991$$ (-28*v^7 - 148*v^5 - 525*v^3 + 539*v) / 1991 $$\beta_{5}$$ $$=$$ $$( 40\nu^{6} - 73\nu^{4} + 750\nu^{2} - 2761 ) / 1991$$ (40*v^6 - 73*v^4 + 750*v^2 - 2761) / 1991 $$\beta_{6}$$ $$=$$ $$( 61\nu^{7} + 38\nu^{5} + 646\nu^{3} - 1672\nu ) / 1991$$ (61*v^7 + 38*v^5 + 646*v^3 - 1672*v) / 1991 $$\beta_{7}$$ $$=$$ $$( 68\nu^{7} + 75\nu^{5} + 1275\nu^{3} - 3300\nu ) / 1991$$ (68*v^7 + 75*v^5 + 1275*v^3 - 3300*v) / 1991
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} - 4\beta_{2} - 1$$ b3 - 4*b2 - 1 $$\nu^{3}$$ $$=$$ $$4\beta_{7} - 4\beta_{6} + \beta_{4} + 3\beta_1$$ 4*b7 - 4*b6 + b4 + 3*b1 $$\nu^{4}$$ $$=$$ $$-7\beta_{5} - 10\beta_{3} - 7$$ -7*b5 - 10*b3 - 7 $$\nu^{5}$$ $$=$$ $$-7\beta_{7} - 17\beta_{4} - 7\beta_1$$ -7*b7 - 17*b4 - 7*b1 $$\nu^{6}$$ $$=$$ $$37\beta_{5} - 37\beta_{3} + 75\beta_{2} + 75$$ 37*b5 - 37*b3 + 75*b2 + 75 $$\nu^{7}$$ $$=$$ $$-38\beta_{7} + 75\beta_{6}$$ -38*b7 + 75*b6

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/176\mathbb{Z}\right)^\times$$.

 $$n$$ $$111$$ $$133$$ $$145$$ $$\chi(n)$$ $$-1$$ $$1$$ $$\beta_{3}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
63.1
 −1.46782 + 0.476925i 1.46782 − 0.476925i −1.26313 − 1.73855i 1.26313 + 1.73855i −1.46782 − 0.476925i 1.46782 + 0.476925i −1.26313 + 1.73855i 1.26313 − 1.73855i
0 −1.46782 + 0.476925i 0 −1.30902 0.951057i 0 −1.46782 + 4.51750i 0 −0.500000 + 0.363271i 0
63.2 0 1.46782 0.476925i 0 −1.30902 0.951057i 0 1.46782 4.51750i 0 −0.500000 + 0.363271i 0
79.1 0 −1.26313 1.73855i 0 −0.190983 0.587785i 0 −1.26313 0.917716i 0 −0.500000 + 1.53884i 0
79.2 0 1.26313 + 1.73855i 0 −0.190983 0.587785i 0 1.26313 + 0.917716i 0 −0.500000 + 1.53884i 0
95.1 0 −1.46782 0.476925i 0 −1.30902 + 0.951057i 0 −1.46782 4.51750i 0 −0.500000 0.363271i 0
95.2 0 1.46782 + 0.476925i 0 −1.30902 + 0.951057i 0 1.46782 + 4.51750i 0 −0.500000 0.363271i 0
127.1 0 −1.26313 + 1.73855i 0 −0.190983 + 0.587785i 0 −1.26313 + 0.917716i 0 −0.500000 1.53884i 0
127.2 0 1.26313 1.73855i 0 −0.190983 + 0.587785i 0 1.26313 0.917716i 0 −0.500000 1.53884i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 63.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
11.d odd 10 1 inner
44.g even 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 176.2.q.a 8
4.b odd 2 1 inner 176.2.q.a 8
8.b even 2 1 704.2.u.a 8
8.d odd 2 1 704.2.u.a 8
11.c even 5 1 1936.2.e.e 8
11.d odd 10 1 inner 176.2.q.a 8
11.d odd 10 1 1936.2.e.e 8
44.g even 10 1 inner 176.2.q.a 8
44.g even 10 1 1936.2.e.e 8
44.h odd 10 1 1936.2.e.e 8
88.k even 10 1 704.2.u.a 8
88.p odd 10 1 704.2.u.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
176.2.q.a 8 1.a even 1 1 trivial
176.2.q.a 8 4.b odd 2 1 inner
176.2.q.a 8 11.d odd 10 1 inner
176.2.q.a 8 44.g even 10 1 inner
704.2.u.a 8 8.b even 2 1
704.2.u.a 8 8.d odd 2 1
704.2.u.a 8 88.k even 10 1
704.2.u.a 8 88.p odd 10 1
1936.2.e.e 8 11.c even 5 1
1936.2.e.e 8 11.d odd 10 1
1936.2.e.e 8 44.g even 10 1
1936.2.e.e 8 44.h odd 10 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{8} - T_{3}^{6} + 16T_{3}^{4} - 66T_{3}^{2} + 121$$ acting on $$S_{2}^{\mathrm{new}}(176, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8} - T^{6} + 16 T^{4} - 66 T^{2} + \cdots + 121$$
$5$ $$(T^{4} + 3 T^{3} + 4 T^{2} + 2 T + 1)^{2}$$
$7$ $$T^{8} + 35 T^{6} + 460 T^{4} + \cdots + 3025$$
$11$ $$T^{8} - 16 T^{6} + 286 T^{4} + \cdots + 14641$$
$13$ $$(T^{4} + 5 T^{3} + 125)^{2}$$
$17$ $$(T^{4} + 5 T^{3} + 20 T^{2} - 20 T + 5)^{2}$$
$19$ $$T^{8} - 25 T^{6} + 1860 T^{4} + \cdots + 3025$$
$23$ $$(T^{4} + 52 T^{2} + 176)^{2}$$
$29$ $$(T^{4} - 15 T^{3} + 120 T^{2} - 440 T + 605)^{2}$$
$31$ $$T^{8} - 39 T^{6} + 1336 T^{4} + \cdots + 1771561$$
$37$ $$(T^{4} - 3 T^{3} + 34 T^{2} - 232 T + 841)^{2}$$
$41$ $$(T^{4} - 5 T^{3} - 10 T^{2} - 190 T + 1805)^{2}$$
$43$ $$(T^{4} - 80 T^{2} + 880)^{2}$$
$47$ $$T^{8} - 209 T^{6} + 16456 T^{4} + \cdots + 1771561$$
$53$ $$(T^{4} + 21 T^{3} + 306 T^{2} + 2376 T + 9801)^{2}$$
$59$ $$T^{8} - 141 T^{6} + 7456 T^{4} + \cdots + 1771561$$
$61$ $$(T^{4} + 15 T^{3} + 120 T^{2} + 440 T + 605)^{2}$$
$67$ $$(T^{4} + 128 T^{2} + 2816)^{2}$$
$71$ $$T^{8} + 45 T^{6} + 6400 T^{4} + \cdots + 75625$$
$73$ $$(T^{4} - 25 T^{3} + 200 T^{2} - 440 T + 605)^{2}$$
$79$ $$T^{8} + 45 T^{6} + 4860 T^{4} + \cdots + 19847025$$
$83$ $$T^{8} + 185 T^{6} + 13060 T^{4} + \cdots + 3025$$
$89$ $$(T^{2} - 2 T - 124)^{4}$$
$97$ $$(T^{4} - 7 T^{3} + 24 T^{2} - 38 T + 361)^{2}$$