# Properties

 Label 176.2.m.b 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 44) 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.

 $$f(q)$$ $$=$$ $$q + (\zeta_{10}^{3} + \zeta_{10}^{2} + \zeta_{10}) q^{3} + (\zeta_{10}^{2} + 1) q^{5} + ( - 2 \zeta_{10}^{3} - 3 \zeta_{10} + 3) q^{7} + (2 \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} - 2) q^{9}+O(q^{10})$$ q + (z^3 + z^2 + z) * q^3 + (z^2 + 1) * q^5 + (-2*z^3 - 3*z + 3) * 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} + ( - 2 \zeta_{10}^{3} - 3 \zeta_{10} + 3) q^{7} + (2 \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} - 2) q^{9} + (4 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10} - 1) q^{11} + (3 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + 4 \zeta_{10} - 3) q^{13} + (3 \zeta_{10}^{3} + 2 \zeta_{10} - 2) q^{15} + (\zeta_{10}^{2} - 4 \zeta_{10} + 1) q^{17} + ( - 3 \zeta_{10}^{3} + \zeta_{10}^{2} - 3 \zeta_{10}) q^{19} + ( - 5 \zeta_{10}^{3} + 5 \zeta_{10}^{2} + 7) q^{21} + ( - 4 \zeta_{10}^{3} + 4 \zeta_{10}^{2}) q^{23} + (\zeta_{10}^{3} - 4 \zeta_{10}^{2} + \zeta_{10}) q^{25} + ( - 2 \zeta_{10}^{2} + \zeta_{10} - 2) q^{27} + ( - 8 \zeta_{10}^{3} - \zeta_{10} + 1) q^{29} + (\zeta_{10}^{3} - 4 \zeta_{10}^{2} + 4 \zeta_{10} - 1) q^{31} + (3 \zeta_{10}^{3} - 3 \zeta_{10}^{2} - \zeta_{10} - 6) q^{33} + ( - 5 \zeta_{10}^{3} + 3 \zeta_{10}^{2} - 3 \zeta_{10} + 5) q^{35} + (3 \zeta_{10} - 3) q^{37} + ( - 2 \zeta_{10}^{2} - 3 \zeta_{10} - 2) q^{39} + (\zeta_{10}^{3} + 7 \zeta_{10}^{2} + \zeta_{10}) q^{41} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} - 5) q^{45} + (\zeta_{10}^{3} - 3 \zeta_{10}^{2} + \zeta_{10}) q^{47} + ( - 3 \zeta_{10}^{2} - 3 \zeta_{10} - 3) q^{49} + ( - 5 \zeta_{10}^{3} - 2 \zeta_{10} + 2) q^{51} + ( - \zeta_{10}^{3} - 4 \zeta_{10}^{2} + 4 \zeta_{10} + 1) q^{53} + (4 \zeta_{10}^{3} - \zeta_{10}^{2} - 3) q^{55} + ( - 7 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} + 7) q^{57} + ( - 2 \zeta_{10}^{3} + 5 \zeta_{10} - 5) q^{59} + (\zeta_{10}^{2} - 4 \zeta_{10} + 1) q^{61} + (3 \zeta_{10}^{3} + 10 \zeta_{10}^{2} + 3 \zeta_{10}) q^{63} + (3 \zeta_{10}^{3} - 3 \zeta_{10}^{2} - 2) q^{65} + (8 \zeta_{10}^{3} - 8 \zeta_{10}^{2} - 8) q^{67} + (4 \zeta_{10}^{3} + 4 \zeta_{10}) q^{69} + (\zeta_{10}^{2} - 8 \zeta_{10} + 1) q^{71} + (4 \zeta_{10}^{3} - 5 \zeta_{10} + 5) q^{73} + ( - 5 \zeta_{10}^{3} + 3 \zeta_{10}^{2} - 3 \zeta_{10} + 5) q^{75} + (4 \zeta_{10}^{3} + 4 \zeta_{10}^{2} + \zeta_{10} + 9) q^{77} + (9 \zeta_{10}^{3} - 12 \zeta_{10}^{2} + 12 \zeta_{10} - 9) q^{79} + (2 \zeta_{10}^{3} + 6 \zeta_{10} - 6) q^{81} + (13 \zeta_{10}^{2} - 8 \zeta_{10} + 13) q^{83} + ( - 3 \zeta_{10}^{3} + \zeta_{10}^{2} - 3 \zeta_{10}) q^{85} + ( - 9 \zeta_{10}^{3} + 9 \zeta_{10}^{2} + 17) q^{87} + ( - 4 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 2) q^{89} + (10 \zeta_{10}^{3} - 7 \zeta_{10}^{2} + 10 \zeta_{10}) q^{91} + (2 \zeta_{10}^{2} - \zeta_{10} + 2) q^{93} + ( - 5 \zeta_{10}^{3} - 2 \zeta_{10} + 2) q^{95} + ( - 9 \zeta_{10}^{3} + 9) q^{97} + ( - 8 \zeta_{10}^{3} + 3 \zeta_{10}^{2} - 13 \zeta_{10} + 4) q^{99}+O(q^{100})$$ q + (z^3 + z^2 + z) * q^3 + (z^2 + 1) * q^5 + (-2*z^3 - 3*z + 3) * q^7 + (2*z^3 + z^2 - z - 2) * q^9 + (4*z^3 - 2*z^2 + 2*z - 1) * q^11 + (3*z^3 - 4*z^2 + 4*z - 3) * q^13 + (3*z^3 + 2*z - 2) * q^15 + (z^2 - 4*z + 1) * q^17 + (-3*z^3 + z^2 - 3*z) * q^19 + (-5*z^3 + 5*z^2 + 7) * q^21 + (-4*z^3 + 4*z^2) * q^23 + (z^3 - 4*z^2 + z) * q^25 + (-2*z^2 + z - 2) * q^27 + (-8*z^3 - z + 1) * q^29 + (z^3 - 4*z^2 + 4*z - 1) * q^31 + (3*z^3 - 3*z^2 - z - 6) * q^33 + (-5*z^3 + 3*z^2 - 3*z + 5) * q^35 + (3*z - 3) * q^37 + (-2*z^2 - 3*z - 2) * q^39 + (z^3 + 7*z^2 + z) * q^41 + (2*z^3 - 2*z^2 - 5) * q^45 + (z^3 - 3*z^2 + z) * q^47 + (-3*z^2 - 3*z - 3) * q^49 + (-5*z^3 - 2*z + 2) * q^51 + (-z^3 - 4*z^2 + 4*z + 1) * q^53 + (4*z^3 - z^2 - 3) * q^55 + (-7*z^3 + 2*z^2 - 2*z + 7) * q^57 + (-2*z^3 + 5*z - 5) * q^59 + (z^2 - 4*z + 1) * q^61 + (3*z^3 + 10*z^2 + 3*z) * q^63 + (3*z^3 - 3*z^2 - 2) * q^65 + (8*z^3 - 8*z^2 - 8) * q^67 + (4*z^3 + 4*z) * q^69 + (z^2 - 8*z + 1) * q^71 + (4*z^3 - 5*z + 5) * q^73 + (-5*z^3 + 3*z^2 - 3*z + 5) * q^75 + (4*z^3 + 4*z^2 + z + 9) * q^77 + (9*z^3 - 12*z^2 + 12*z - 9) * q^79 + (2*z^3 + 6*z - 6) * q^81 + (13*z^2 - 8*z + 13) * q^83 + (-3*z^3 + z^2 - 3*z) * q^85 + (-9*z^3 + 9*z^2 + 17) * q^87 + (-4*z^3 + 4*z^2 - 2) * q^89 + (10*z^3 - 7*z^2 + 10*z) * q^91 + (2*z^2 - z + 2) * q^93 + (-5*z^3 - 2*z + 2) * q^95 + (-9*z^3 + 9) * q^97 + (-8*z^3 + 3*z^2 - 13*z + 4) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + q^{3} + 3 q^{5} + 7 q^{7} - 8 q^{9}+O(q^{10})$$ 4 * q + q^3 + 3 * q^5 + 7 * q^7 - 8 * q^9 $$4 q + q^{3} + 3 q^{5} + 7 q^{7} - 8 q^{9} + 4 q^{11} - q^{13} - 3 q^{15} - q^{17} - 7 q^{19} + 18 q^{21} - 8 q^{23} + 6 q^{25} - 5 q^{27} - 5 q^{29} + 5 q^{31} - 19 q^{33} + 9 q^{35} - 9 q^{37} - 9 q^{39} - 5 q^{41} - 16 q^{45} + 5 q^{47} - 12 q^{49} + q^{51} + 11 q^{53} - 7 q^{55} + 17 q^{57} - 17 q^{59} - q^{61} - 4 q^{63} - 2 q^{65} - 16 q^{67} + 8 q^{69} - 5 q^{71} + 19 q^{73} + 9 q^{75} + 37 q^{77} - 3 q^{79} - 16 q^{81} + 31 q^{83} - 7 q^{85} + 50 q^{87} - 16 q^{89} + 27 q^{91} + 5 q^{93} + q^{95} + 27 q^{97} - 8 q^{99}+O(q^{100})$$ 4 * q + q^3 + 3 * q^5 + 7 * q^7 - 8 * q^9 + 4 * q^11 - q^13 - 3 * q^15 - q^17 - 7 * q^19 + 18 * q^21 - 8 * q^23 + 6 * q^25 - 5 * q^27 - 5 * q^29 + 5 * q^31 - 19 * q^33 + 9 * q^35 - 9 * q^37 - 9 * q^39 - 5 * q^41 - 16 * q^45 + 5 * q^47 - 12 * q^49 + q^51 + 11 * q^53 - 7 * q^55 + 17 * q^57 - 17 * q^59 - q^61 - 4 * q^63 - 2 * q^65 - 16 * q^67 + 8 * q^69 - 5 * q^71 + 19 * q^73 + 9 * q^75 + 37 * q^77 - 3 * q^79 - 16 * q^81 + 31 * q^83 - 7 * q^85 + 50 * q^87 - 16 * q^89 + 27 * q^91 + 5 * q^93 + q^95 + 27 * q^97 - 8 * 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 −0.309017 0.224514i 0 0.190983 0.587785i 0 2.30902 1.67760i 0 −0.881966 2.71441i 0
81.1 0 0.809017 2.48990i 0 1.30902 0.951057i 0 1.19098 + 3.66547i 0 −3.11803 2.26538i 0
97.1 0 −0.309017 + 0.224514i 0 0.190983 + 0.587785i 0 2.30902 + 1.67760i 0 −0.881966 + 2.71441i 0
113.1 0 0.809017 + 2.48990i 0 1.30902 + 0.951057i 0 1.19098 3.66547i 0 −3.11803 + 2.26538i 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.b 4
4.b odd 2 1 44.2.e.a 4
8.b even 2 1 704.2.m.d 4
8.d odd 2 1 704.2.m.e 4
11.c even 5 1 inner 176.2.m.b 4
11.c even 5 1 1936.2.a.ba 2
11.d odd 10 1 1936.2.a.z 2
12.b even 2 1 396.2.j.a 4
20.d odd 2 1 1100.2.n.a 4
20.e even 4 2 1100.2.cb.a 8
44.c even 2 1 484.2.e.c 4
44.g even 10 1 484.2.a.c 2
44.g even 10 1 484.2.e.c 4
44.g even 10 2 484.2.e.d 4
44.h odd 10 1 44.2.e.a 4
44.h odd 10 1 484.2.a.b 2
44.h odd 10 2 484.2.e.e 4
88.k even 10 1 7744.2.a.db 2
88.l odd 10 1 704.2.m.e 4
88.l odd 10 1 7744.2.a.da 2
88.o even 10 1 704.2.m.d 4
88.o even 10 1 7744.2.a.bp 2
88.p odd 10 1 7744.2.a.bo 2
132.n odd 10 1 4356.2.a.u 2
132.o even 10 1 396.2.j.a 4
132.o even 10 1 4356.2.a.t 2
220.n odd 10 1 1100.2.n.a 4
220.v even 20 2 1100.2.cb.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
44.2.e.a 4 4.b odd 2 1
44.2.e.a 4 44.h odd 10 1
176.2.m.b 4 1.a even 1 1 trivial
176.2.m.b 4 11.c even 5 1 inner
396.2.j.a 4 12.b even 2 1
396.2.j.a 4 132.o even 10 1
484.2.a.b 2 44.h odd 10 1
484.2.a.c 2 44.g even 10 1
484.2.e.c 4 44.c even 2 1
484.2.e.c 4 44.g even 10 1
484.2.e.d 4 44.g even 10 2
484.2.e.e 4 44.h odd 10 2
704.2.m.d 4 8.b even 2 1
704.2.m.d 4 88.o even 10 1
704.2.m.e 4 8.d odd 2 1
704.2.m.e 4 88.l odd 10 1
1100.2.n.a 4 20.d odd 2 1
1100.2.n.a 4 220.n odd 10 1
1100.2.cb.a 8 20.e even 4 2
1100.2.cb.a 8 220.v even 20 2
1936.2.a.z 2 11.d odd 10 1
1936.2.a.ba 2 11.c even 5 1
4356.2.a.t 2 132.o even 10 1
4356.2.a.u 2 132.n odd 10 1
7744.2.a.bo 2 88.p odd 10 1
7744.2.a.bp 2 88.o even 10 1
7744.2.a.da 2 88.l odd 10 1
7744.2.a.db 2 88.k even 10 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} - T_{3}^{3} + 6T_{3}^{2} + 4T_{3} + 1$$ acting on $$S_{2}^{\mathrm{new}}(176, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} - T^{3} + 6 T^{2} + 4 T + 1$$
$5$ $$T^{4} - 3 T^{3} + 4 T^{2} - 2 T + 1$$
$7$ $$T^{4} - 7 T^{3} + 34 T^{2} - 88 T + 121$$
$11$ $$T^{4} - 4 T^{3} + 6 T^{2} - 44 T + 121$$
$13$ $$T^{4} + T^{3} + 16 T^{2} + 66 T + 121$$
$17$ $$T^{4} + T^{3} + 16 T^{2} + 66 T + 121$$
$19$ $$T^{4} + 7 T^{3} + 34 T^{2} + 88 T + 121$$
$23$ $$(T^{2} + 4 T - 16)^{2}$$
$29$ $$T^{4} + 5 T^{3} + 60 T^{2} + \cdots + 3025$$
$31$ $$T^{4} - 5 T^{3} + 40 T^{2} - 50 T + 25$$
$37$ $$T^{4} + 9 T^{3} + 36 T^{2} + 54 T + 81$$
$41$ $$T^{4} + 5 T^{3} + 60 T^{2} + \cdots + 3025$$
$43$ $$T^{4}$$
$47$ $$T^{4} - 5 T^{3} + 10 T^{2} + 25$$
$53$ $$T^{4} - 11 T^{3} + 96 T^{2} + \cdots + 841$$
$59$ $$T^{4} + 17 T^{3} + 114 T^{2} + \cdots + 121$$
$61$ $$T^{4} + T^{3} + 16 T^{2} + 66 T + 121$$
$67$ $$(T^{2} + 8 T - 64)^{2}$$
$71$ $$T^{4} + 5 T^{3} + 60 T^{2} + \cdots + 3025$$
$73$ $$T^{4} - 19 T^{3} + 136 T^{2} + \cdots + 121$$
$79$ $$T^{4} + 3 T^{3} + 144 T^{2} + \cdots + 9801$$
$83$ $$T^{4} - 31 T^{3} + 636 T^{2} + \cdots + 43681$$
$89$ $$(T^{2} + 8 T - 4)^{2}$$
$97$ $$T^{4} - 27 T^{3} + 324 T^{2} + \cdots + 6561$$