# Properties

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

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [176,2,Mod(49,176)]

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([0, 0, 4]))

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

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

chi := DirichletCharacter("176.49");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$176 = 2^{4} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 176.m (of order $$5$$, degree $$4$$, 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: $$1.40536707557$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} + x^{2} - x + 1$$ x^4 - x^3 + x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 88) Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-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 primitive root of unity $$\zeta_{10}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10}) q^{3} + (\zeta_{10}^{2} + 1) q^{5} + ( - \zeta_{10} + 1) q^{7} + ( - 2 \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 2) q^{9}+O(q^{10})$$ q + (-z^3 + z^2 - z) * q^3 + (z^2 + 1) * q^5 + (-z + 1) * q^7 + (-2*z^3 + z^2 - z + 2) * q^9 $$q + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10}) q^{3} + (\zeta_{10}^{2} + 1) q^{5} + ( - \zeta_{10} + 1) q^{7} + ( - 2 \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 2) q^{9} + ( - 2 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 2 \zeta_{10} + 3) q^{11} + ( - \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 4 \zeta_{10} + 1) q^{13} - \zeta_{10}^{3} q^{15} + ( - 3 \zeta_{10}^{2} - 3) q^{17} + (3 \zeta_{10}^{3} - 3 \zeta_{10}^{2} + 3 \zeta_{10}) q^{19} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - 1) q^{21} + 4 q^{23} + (\zeta_{10}^{3} - 4 \zeta_{10}^{2} + \zeta_{10}) q^{25} + ( - 4 \zeta_{10}^{2} + 3 \zeta_{10} - 4) q^{27} + (4 \zeta_{10}^{3} + 7 \zeta_{10} - 7) q^{29} + (7 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10} - 7) q^{31} + ( - \zeta_{10}^{3} - 3 \zeta_{10}^{2} + 3 \zeta_{10} - 2) q^{33} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{35} + (8 \zeta_{10}^{3} - \zeta_{10} + 1) q^{37} + ( - 4 \zeta_{10}^{2} + 7 \zeta_{10} - 4) q^{39} + (\zeta_{10}^{3} - 9 \zeta_{10}^{2} + \zeta_{10}) q^{41} + (4 \zeta_{10}^{3} - 4 \zeta_{10}^{2} - 8) q^{43} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} + 3) q^{45} + ( - 9 \zeta_{10}^{3} + 5 \zeta_{10}^{2} - 9 \zeta_{10}) q^{47} + (\zeta_{10}^{2} + 5 \zeta_{10} + 1) q^{49} + 3 \zeta_{10}^{3} q^{51} + ( - \zeta_{10}^{3} + 1) q^{53} + (3 \zeta_{10}^{2} + 2 \zeta_{10} + 1) q^{55} + ( - 3 \zeta_{10}^{3} + 6 \zeta_{10}^{2} - 6 \zeta_{10} + 3) q^{57} + ( - 4 \zeta_{10}^{3} - \zeta_{10} + 1) q^{59} + ( - 3 \zeta_{10}^{2} - 3) q^{61} + ( - \zeta_{10}^{3} - \zeta_{10}) q^{63} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - 2) q^{65} + (4 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + 8) q^{67} + ( - 4 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 4 \zeta_{10}) q^{69} + (5 \zeta_{10}^{2} - 2 \zeta_{10} + 5) q^{71} + (8 \zeta_{10}^{3} + 3 \zeta_{10} - 3) q^{73} + ( - \zeta_{10}^{3} + 5 \zeta_{10}^{2} - 5 \zeta_{10} + 1) q^{75} + ( - 4 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 3 \zeta_{10} + 1) q^{77} + ( - \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10} + 1) q^{79} + ( - 2 \zeta_{10}^{3} - 6 \zeta_{10} + 6) q^{81} + (\zeta_{10}^{2} + 10 \zeta_{10} + 1) q^{83} + ( - 3 \zeta_{10}^{3} - 3 \zeta_{10}^{2} - 3 \zeta_{10}) q^{85} + (3 \zeta_{10}^{3} - 3 \zeta_{10}^{2} + 7) q^{87} + ( - 4 \zeta_{10}^{3} + 4 \zeta_{10}^{2} + 2) q^{89} + ( - 4 \zeta_{10}^{3} + 7 \zeta_{10}^{2} - 4 \zeta_{10}) q^{91} + (2 \zeta_{10}^{2} + 3 \zeta_{10} + 2) q^{93} + 3 \zeta_{10}^{3} q^{95} + ( - \zeta_{10}^{3} + 8 \zeta_{10}^{2} - 8 \zeta_{10} + 1) q^{97} + ( - 6 \zeta_{10}^{3} + 3 \zeta_{10}^{2} - \zeta_{10} + 6) q^{99} +O(q^{100})$$ q + (-z^3 + z^2 - z) * q^3 + (z^2 + 1) * q^5 + (-z + 1) * q^7 + (-2*z^3 + z^2 - z + 2) * q^9 + (-2*z^3 + 4*z^2 - 2*z + 3) * q^11 + (-z^3 + 4*z^2 - 4*z + 1) * q^13 - z^3 * q^15 + (-3*z^2 - 3) * q^17 + (3*z^3 - 3*z^2 + 3*z) * q^19 + (-z^3 + z^2 - 1) * q^21 + 4 * q^23 + (z^3 - 4*z^2 + z) * q^25 + (-4*z^2 + 3*z - 4) * q^27 + (4*z^3 + 7*z - 7) * q^29 + (7*z^3 - 2*z^2 + 2*z - 7) * q^31 + (-z^3 - 3*z^2 + 3*z - 2) * q^33 + (-z^3 + z^2 - z + 1) * q^35 + (8*z^3 - z + 1) * q^37 + (-4*z^2 + 7*z - 4) * q^39 + (z^3 - 9*z^2 + z) * q^41 + (4*z^3 - 4*z^2 - 8) * q^43 + (-2*z^3 + 2*z^2 + 3) * q^45 + (-9*z^3 + 5*z^2 - 9*z) * q^47 + (z^2 + 5*z + 1) * q^49 + 3*z^3 * q^51 + (-z^3 + 1) * q^53 + (3*z^2 + 2*z + 1) * q^55 + (-3*z^3 + 6*z^2 - 6*z + 3) * q^57 + (-4*z^3 - z + 1) * q^59 + (-3*z^2 - 3) * q^61 + (-z^3 - z) * q^63 + (-z^3 + z^2 - 2) * q^65 + (4*z^3 - 4*z^2 + 8) * q^67 + (-4*z^3 + 4*z^2 - 4*z) * q^69 + (5*z^2 - 2*z + 5) * q^71 + (8*z^3 + 3*z - 3) * q^73 + (-z^3 + 5*z^2 - 5*z + 1) * q^75 + (-4*z^3 + 4*z^2 - 3*z + 1) * q^77 + (-z^3 - 2*z^2 + 2*z + 1) * q^79 + (-2*z^3 - 6*z + 6) * q^81 + (z^2 + 10*z + 1) * q^83 + (-3*z^3 - 3*z^2 - 3*z) * q^85 + (3*z^3 - 3*z^2 + 7) * q^87 + (-4*z^3 + 4*z^2 + 2) * q^89 + (-4*z^3 + 7*z^2 - 4*z) * q^91 + (2*z^2 + 3*z + 2) * q^93 + 3*z^3 * q^95 + (-z^3 + 8*z^2 - 8*z + 1) * q^97 + (-6*z^3 + 3*z^2 - z + 6) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 3 q^{3} + 3 q^{5} + 3 q^{7} + 4 q^{9}+O(q^{10})$$ 4 * q - 3 * q^3 + 3 * q^5 + 3 * q^7 + 4 * q^9 $$4 q - 3 q^{3} + 3 q^{5} + 3 q^{7} + 4 q^{9} + 4 q^{11} - 5 q^{13} - q^{15} - 9 q^{17} + 9 q^{19} - 6 q^{21} + 16 q^{23} + 6 q^{25} - 9 q^{27} - 17 q^{29} - 17 q^{31} - 3 q^{33} + q^{35} + 11 q^{37} - 5 q^{39} + 11 q^{41} - 24 q^{43} + 8 q^{45} - 23 q^{47} + 8 q^{49} + 3 q^{51} + 3 q^{53} + 3 q^{55} - 3 q^{57} - q^{59} - 9 q^{61} - 2 q^{63} - 10 q^{65} + 40 q^{67} - 12 q^{69} + 13 q^{71} - q^{73} - 7 q^{75} - 7 q^{77} + 7 q^{79} + 16 q^{81} + 13 q^{83} - 3 q^{85} + 34 q^{87} - 15 q^{91} + 9 q^{93} + 3 q^{95} - 13 q^{97} + 14 q^{99}+O(q^{100})$$ 4 * q - 3 * q^3 + 3 * q^5 + 3 * q^7 + 4 * q^9 + 4 * q^11 - 5 * q^13 - q^15 - 9 * q^17 + 9 * q^19 - 6 * q^21 + 16 * q^23 + 6 * q^25 - 9 * q^27 - 17 * q^29 - 17 * q^31 - 3 * q^33 + q^35 + 11 * q^37 - 5 * q^39 + 11 * q^41 - 24 * q^43 + 8 * q^45 - 23 * q^47 + 8 * q^49 + 3 * q^51 + 3 * q^53 + 3 * q^55 - 3 * q^57 - q^59 - 9 * q^61 - 2 * q^63 - 10 * q^65 + 40 * q^67 - 12 * q^69 + 13 * q^71 - q^73 - 7 * q^75 - 7 * q^77 + 7 * q^79 + 16 * q^81 + 13 * q^83 - 3 * q^85 + 34 * q^87 - 15 * q^91 + 9 * q^93 + 3 * q^95 - 13 * q^97 + 14 * q^99

## Character values

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

 $$n$$ $$111$$ $$133$$ $$145$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\zeta_{10}^{3}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 −0.309017 + 0.951057i 0.809017 − 0.587785i −0.309017 − 0.951057i 0.809017 + 0.587785i
0 −1.30902 0.951057i 0 0.190983 0.587785i 0 1.30902 0.951057i 0 −0.118034 0.363271i 0
81.1 0 −0.190983 + 0.587785i 0 1.30902 0.951057i 0 0.190983 + 0.587785i 0 2.11803 + 1.53884i 0
97.1 0 −1.30902 + 0.951057i 0 0.190983 + 0.587785i 0 1.30902 + 0.951057i 0 −0.118034 + 0.363271i 0
113.1 0 −0.190983 0.587785i 0 1.30902 + 0.951057i 0 0.190983 0.587785i 0 2.11803 1.53884i 0
 $$n$$: 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
11.c even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 176.2.m.a 4
4.b odd 2 1 88.2.i.a 4
8.b even 2 1 704.2.m.g 4
8.d odd 2 1 704.2.m.b 4
11.c even 5 1 inner 176.2.m.a 4
11.c even 5 1 1936.2.a.t 2
11.d odd 10 1 1936.2.a.u 2
12.b even 2 1 792.2.r.b 4
44.c even 2 1 968.2.i.k 4
44.g even 10 1 968.2.a.h 2
44.g even 10 2 968.2.i.c 4
44.g even 10 1 968.2.i.k 4
44.h odd 10 1 88.2.i.a 4
44.h odd 10 1 968.2.a.i 2
44.h odd 10 2 968.2.i.d 4
88.k even 10 1 7744.2.a.cq 2
88.l odd 10 1 704.2.m.b 4
88.l odd 10 1 7744.2.a.cr 2
88.o even 10 1 704.2.m.g 4
88.o even 10 1 7744.2.a.cb 2
88.p odd 10 1 7744.2.a.cc 2
132.n odd 10 1 8712.2.a.bm 2
132.o even 10 1 792.2.r.b 4
132.o even 10 1 8712.2.a.bp 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
88.2.i.a 4 4.b odd 2 1
88.2.i.a 4 44.h odd 10 1
176.2.m.a 4 1.a even 1 1 trivial
176.2.m.a 4 11.c even 5 1 inner
704.2.m.b 4 8.d odd 2 1
704.2.m.b 4 88.l odd 10 1
704.2.m.g 4 8.b even 2 1
704.2.m.g 4 88.o even 10 1
792.2.r.b 4 12.b even 2 1
792.2.r.b 4 132.o even 10 1
968.2.a.h 2 44.g even 10 1
968.2.a.i 2 44.h odd 10 1
968.2.i.c 4 44.g even 10 2
968.2.i.d 4 44.h odd 10 2
968.2.i.k 4 44.c even 2 1
968.2.i.k 4 44.g even 10 1
1936.2.a.t 2 11.c even 5 1
1936.2.a.u 2 11.d odd 10 1
7744.2.a.cb 2 88.o even 10 1
7744.2.a.cc 2 88.p odd 10 1
7744.2.a.cq 2 88.k even 10 1
7744.2.a.cr 2 88.l odd 10 1
8712.2.a.bm 2 132.n odd 10 1
8712.2.a.bp 2 132.o even 10 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + 3T_{3}^{3} + 4T_{3}^{2} + 2T_{3} + 1$$ acting on $$S_{2}^{\mathrm{new}}(176, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 3 T^{3} + 4 T^{2} + 2 T + 1$$
$5$ $$T^{4} - 3 T^{3} + 4 T^{2} - 2 T + 1$$
$7$ $$T^{4} - 3 T^{3} + 4 T^{2} - 2 T + 1$$
$11$ $$T^{4} - 4 T^{3} + 6 T^{2} - 44 T + 121$$
$13$ $$T^{4} + 5 T^{3} + 40 T^{2} + 50 T + 25$$
$17$ $$T^{4} + 9 T^{3} + 36 T^{2} + 54 T + 81$$
$19$ $$T^{4} - 9 T^{3} + 36 T^{2} - 54 T + 81$$
$23$ $$(T - 4)^{4}$$
$29$ $$T^{4} + 17 T^{3} + 184 T^{2} + \cdots + 3721$$
$31$ $$T^{4} + 17 T^{3} + 114 T^{2} + \cdots + 121$$
$37$ $$T^{4} - 11 T^{3} + 76 T^{2} + \cdots + 5041$$
$41$ $$T^{4} - 11 T^{3} + 76 T^{2} + \cdots + 5041$$
$43$ $$(T^{2} + 12 T + 16)^{2}$$
$47$ $$T^{4} + 23 T^{3} + 304 T^{2} + \cdots + 10201$$
$53$ $$T^{4} - 3 T^{3} + 4 T^{2} - 2 T + 1$$
$59$ $$T^{4} + T^{3} + 16 T^{2} + 66 T + 121$$
$61$ $$T^{4} + 9 T^{3} + 36 T^{2} + 54 T + 81$$
$67$ $$(T^{2} - 20 T + 80)^{2}$$
$71$ $$T^{4} - 13 T^{3} + 94 T^{2} + \cdots + 961$$
$73$ $$T^{4} + T^{3} + 76 T^{2} - 434 T + 961$$
$79$ $$T^{4} - 7 T^{3} + 34 T^{2} - 88 T + 121$$
$83$ $$T^{4} - 13 T^{3} + 114 T^{2} + \cdots + 11881$$
$89$ $$(T^{2} - 20)^{2}$$
$97$ $$T^{4} + 13 T^{3} + 204 T^{2} + \cdots + 1681$$