Properties

 Label 176.2.m.c 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 22) 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.

 $$f(q)$$ $$=$$ $$q + (\zeta_{10}^{3} - 2 \zeta_{10}^{2} + \zeta_{10}) q^{3} + ( - 2 \zeta_{10}^{2} - 2) q^{5} - 2 \zeta_{10}^{3} q^{7} + ( - \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10} + 1) q^{9} +O(q^{10})$$ q + (z^3 - 2*z^2 + z) * q^3 + (-2*z^2 - 2) * q^5 - 2*z^3 * q^7 + (-z^3 - 2*z^2 + 2*z + 1) * q^9 $$q + (\zeta_{10}^{3} - 2 \zeta_{10}^{2} + \zeta_{10}) q^{3} + ( - 2 \zeta_{10}^{2} - 2) q^{5} - 2 \zeta_{10}^{3} q^{7} + ( - \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10} + 1) q^{9} + ( - 2 \zeta_{10}^{3} + \zeta_{10}^{2} - 4 \zeta_{10} + 2) q^{11} + (2 \zeta_{10}^{2} - 2 \zeta_{10}) q^{13} + (2 \zeta_{10} - 2) q^{15} + (\zeta_{10}^{2} - \zeta_{10} + 1) q^{17} + (3 \zeta_{10}^{3} + \zeta_{10}^{2} + 3 \zeta_{10}) q^{19} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2) q^{21} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2}) q^{23} + (4 \zeta_{10}^{3} - \zeta_{10}^{2} + 4 \zeta_{10}) q^{25} + ( - 2 \zeta_{10}^{2} + \zeta_{10} - 2) q^{27} + ( - 2 \zeta_{10}^{3} - 4 \zeta_{10} + 4) q^{29} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} + 2) q^{31} + (3 \zeta_{10}^{3} - 4 \zeta_{10} + 3) q^{33} + (4 \zeta_{10}^{3} - 4) q^{35} + (6 \zeta_{10} - 6) q^{37} + (4 \zeta_{10}^{2} - 6 \zeta_{10} + 4) q^{39} + (\zeta_{10}^{3} + 4 \zeta_{10}^{2} + \zeta_{10}) q^{41} + ( - 9 \zeta_{10}^{3} + 9 \zeta_{10}^{2} + 3) q^{43} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} - 8) q^{45} + (4 \zeta_{10}^{3} + 4 \zeta_{10}) q^{47} + 3 \zeta_{10} q^{49} + (\zeta_{10}^{3} - 2 \zeta_{10} + 2) q^{51} + ( - 4 \zeta_{10}^{3} + 8 \zeta_{10}^{2} - 8 \zeta_{10} + 4) q^{53} + (10 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + 6 \zeta_{10} - 6) q^{55} + ( - \zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} + 1) q^{57} + ( - 8 \zeta_{10}^{3} - \zeta_{10} + 1) q^{59} + (4 \zeta_{10}^{2} - 4 \zeta_{10} + 4) q^{61} + ( - 6 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 6 \zeta_{10}) q^{63} + 4 q^{65} + (5 \zeta_{10}^{3} - 5 \zeta_{10}^{2} - 8) q^{67} + (4 \zeta_{10}^{3} - 6 \zeta_{10}^{2} + 4 \zeta_{10}) q^{69} + ( - 2 \zeta_{10}^{2} - 2 \zeta_{10} - 2) q^{71} + ( - 11 \zeta_{10}^{3} + \zeta_{10} - 1) q^{73} + (\zeta_{10}^{3} - 6 \zeta_{10}^{2} + 6 \zeta_{10} - 1) q^{75} + (4 \zeta_{10}^{3} - 8 \zeta_{10}^{2} + 4 \zeta_{10} - 6) q^{77} + ( - 6 \zeta_{10}^{3} - 6 \zeta_{10}^{2} + 6 \zeta_{10} + 6) q^{79} + ( - 4 \zeta_{10}^{3} - 6 \zeta_{10} + 6) q^{81} + (7 \zeta_{10}^{2} - 2 \zeta_{10} + 7) q^{83} - 2 \zeta_{10}^{2} q^{85} + (6 \zeta_{10}^{3} - 6 \zeta_{10}^{2} + 2) q^{87} + (5 \zeta_{10}^{3} - 5 \zeta_{10}^{2} - 5) q^{89} + (4 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + 4 \zeta_{10}) q^{91} + (2 \zeta_{10}^{2} - 4 \zeta_{10} + 2) q^{93} + ( - 14 \zeta_{10}^{3} - 8 \zeta_{10} + 8) q^{95} + (9 \zeta_{10}^{3} - 12 \zeta_{10}^{2} + 12 \zeta_{10} - 9) q^{97} + (4 \zeta_{10}^{3} - 9 \zeta_{10}^{2} - 4 \zeta_{10} + 1) q^{99} +O(q^{100})$$ q + (z^3 - 2*z^2 + z) * q^3 + (-2*z^2 - 2) * q^5 - 2*z^3 * q^7 + (-z^3 - 2*z^2 + 2*z + 1) * q^9 + (-2*z^3 + z^2 - 4*z + 2) * q^11 + (2*z^2 - 2*z) * q^13 + (2*z - 2) * q^15 + (z^2 - z + 1) * q^17 + (3*z^3 + z^2 + 3*z) * q^19 + (-2*z^3 + 2*z^2 - 2) * q^21 + (2*z^3 - 2*z^2) * q^23 + (4*z^3 - z^2 + 4*z) * q^25 + (-2*z^2 + z - 2) * q^27 + (-2*z^3 - 4*z + 4) * q^29 + (-2*z^3 + 2*z^2 - 2*z + 2) * q^31 + (3*z^3 - 4*z + 3) * q^33 + (4*z^3 - 4) * q^35 + (6*z - 6) * q^37 + (4*z^2 - 6*z + 4) * q^39 + (z^3 + 4*z^2 + z) * q^41 + (-9*z^3 + 9*z^2 + 3) * q^43 + (2*z^3 - 2*z^2 - 8) * q^45 + (4*z^3 + 4*z) * q^47 + 3*z * q^49 + (z^3 - 2*z + 2) * q^51 + (-4*z^3 + 8*z^2 - 8*z + 4) * q^53 + (10*z^3 - 4*z^2 + 6*z - 6) * q^55 + (-z^3 - z^2 + z + 1) * q^57 + (-8*z^3 - z + 1) * q^59 + (4*z^2 - 4*z + 4) * q^61 + (-6*z^3 + 4*z^2 - 6*z) * q^63 + 4 * q^65 + (5*z^3 - 5*z^2 - 8) * q^67 + (4*z^3 - 6*z^2 + 4*z) * q^69 + (-2*z^2 - 2*z - 2) * q^71 + (-11*z^3 + z - 1) * q^73 + (z^3 - 6*z^2 + 6*z - 1) * q^75 + (4*z^3 - 8*z^2 + 4*z - 6) * q^77 + (-6*z^3 - 6*z^2 + 6*z + 6) * q^79 + (-4*z^3 - 6*z + 6) * q^81 + (7*z^2 - 2*z + 7) * q^83 - 2*z^2 * q^85 + (6*z^3 - 6*z^2 + 2) * q^87 + (5*z^3 - 5*z^2 - 5) * q^89 + (4*z^3 - 4*z^2 + 4*z) * q^91 + (2*z^2 - 4*z + 2) * q^93 + (-14*z^3 - 8*z + 8) * q^95 + (9*z^3 - 12*z^2 + 12*z - 9) * q^97 + (4*z^3 - 9*z^2 - 4*z + 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{3} - 6 q^{5} - 2 q^{7} + 7 q^{9}+O(q^{10})$$ 4 * q + 4 * q^3 - 6 * q^5 - 2 * q^7 + 7 * q^9 $$4 q + 4 q^{3} - 6 q^{5} - 2 q^{7} + 7 q^{9} + q^{11} - 4 q^{13} - 6 q^{15} + 2 q^{17} + 5 q^{19} - 12 q^{21} + 4 q^{23} + 9 q^{25} - 5 q^{27} + 10 q^{29} + 2 q^{31} + 11 q^{33} - 12 q^{35} - 18 q^{37} + 6 q^{39} - 2 q^{41} - 6 q^{43} - 28 q^{45} + 8 q^{47} + 3 q^{49} + 7 q^{51} - 4 q^{53} - 4 q^{55} + 5 q^{57} - 5 q^{59} + 8 q^{61} - 16 q^{63} + 16 q^{65} - 22 q^{67} + 14 q^{69} - 8 q^{71} - 14 q^{73} + 9 q^{75} - 8 q^{77} + 30 q^{79} + 14 q^{81} + 19 q^{83} + 2 q^{85} + 20 q^{87} - 10 q^{89} + 12 q^{91} + 2 q^{93} + 10 q^{95} - 3 q^{97} + 13 q^{99}+O(q^{100})$$ 4 * q + 4 * q^3 - 6 * q^5 - 2 * q^7 + 7 * q^9 + q^11 - 4 * q^13 - 6 * q^15 + 2 * q^17 + 5 * q^19 - 12 * q^21 + 4 * q^23 + 9 * q^25 - 5 * q^27 + 10 * q^29 + 2 * q^31 + 11 * q^33 - 12 * q^35 - 18 * q^37 + 6 * q^39 - 2 * q^41 - 6 * q^43 - 28 * q^45 + 8 * q^47 + 3 * q^49 + 7 * q^51 - 4 * q^53 - 4 * q^55 + 5 * q^57 - 5 * q^59 + 8 * q^61 - 16 * q^63 + 16 * q^65 - 22 * q^67 + 14 * q^69 - 8 * q^71 - 14 * q^73 + 9 * q^75 - 8 * q^77 + 30 * q^79 + 14 * q^81 + 19 * q^83 + 2 * q^85 + 20 * q^87 - 10 * q^89 + 12 * q^91 + 2 * q^93 + 10 * q^95 - 3 * q^97 + 13 * 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 2.11803 + 1.53884i 0 −0.381966 + 1.17557i 0 −1.61803 + 1.17557i 0 1.19098 + 3.66547i 0
81.1 0 −0.118034 + 0.363271i 0 −2.61803 + 1.90211i 0 0.618034 + 1.90211i 0 2.30902 + 1.67760i 0
97.1 0 2.11803 1.53884i 0 −0.381966 1.17557i 0 −1.61803 1.17557i 0 1.19098 3.66547i 0
113.1 0 −0.118034 0.363271i 0 −2.61803 1.90211i 0 0.618034 1.90211i 0 2.30902 1.67760i 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.c 4
4.b odd 2 1 22.2.c.a 4
8.b even 2 1 704.2.m.a 4
8.d odd 2 1 704.2.m.h 4
11.c even 5 1 inner 176.2.m.c 4
11.c even 5 1 1936.2.a.o 2
11.d odd 10 1 1936.2.a.n 2
12.b even 2 1 198.2.f.e 4
20.d odd 2 1 550.2.h.h 4
20.e even 4 2 550.2.ba.c 8
44.c even 2 1 242.2.c.c 4
44.g even 10 1 242.2.a.d 2
44.g even 10 1 242.2.c.c 4
44.g even 10 2 242.2.c.d 4
44.h odd 10 1 22.2.c.a 4
44.h odd 10 1 242.2.a.f 2
44.h odd 10 2 242.2.c.a 4
88.k even 10 1 7744.2.a.bn 2
88.l odd 10 1 704.2.m.h 4
88.l odd 10 1 7744.2.a.bm 2
88.o even 10 1 704.2.m.a 4
88.o even 10 1 7744.2.a.cz 2
88.p odd 10 1 7744.2.a.cy 2
132.n odd 10 1 2178.2.a.x 2
132.o even 10 1 198.2.f.e 4
132.o even 10 1 2178.2.a.p 2
220.n odd 10 1 550.2.h.h 4
220.n odd 10 1 6050.2.a.bs 2
220.o even 10 1 6050.2.a.ci 2
220.v even 20 2 550.2.ba.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.2.c.a 4 4.b odd 2 1
22.2.c.a 4 44.h odd 10 1
176.2.m.c 4 1.a even 1 1 trivial
176.2.m.c 4 11.c even 5 1 inner
198.2.f.e 4 12.b even 2 1
198.2.f.e 4 132.o even 10 1
242.2.a.d 2 44.g even 10 1
242.2.a.f 2 44.h odd 10 1
242.2.c.a 4 44.h odd 10 2
242.2.c.c 4 44.c even 2 1
242.2.c.c 4 44.g even 10 1
242.2.c.d 4 44.g even 10 2
550.2.h.h 4 20.d odd 2 1
550.2.h.h 4 220.n odd 10 1
550.2.ba.c 8 20.e even 4 2
550.2.ba.c 8 220.v even 20 2
704.2.m.a 4 8.b even 2 1
704.2.m.a 4 88.o even 10 1
704.2.m.h 4 8.d odd 2 1
704.2.m.h 4 88.l odd 10 1
1936.2.a.n 2 11.d odd 10 1
1936.2.a.o 2 11.c even 5 1
2178.2.a.p 2 132.o even 10 1
2178.2.a.x 2 132.n odd 10 1
6050.2.a.bs 2 220.n odd 10 1
6050.2.a.ci 2 220.o even 10 1
7744.2.a.bm 2 88.l odd 10 1
7744.2.a.bn 2 88.k even 10 1
7744.2.a.cy 2 88.p odd 10 1
7744.2.a.cz 2 88.o even 10 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} - 4 T^{3} + 6 T^{2} + T + 1$$
$5$ $$T^{4} + 6 T^{3} + 16 T^{2} + 16 T + 16$$
$7$ $$T^{4} + 2 T^{3} + 4 T^{2} + 8 T + 16$$
$11$ $$T^{4} - T^{3} + 21 T^{2} - 11 T + 121$$
$13$ $$T^{4} + 4 T^{3} + 16 T^{2} + 24 T + 16$$
$17$ $$T^{4} - 2 T^{3} + 4 T^{2} - 3 T + 1$$
$19$ $$T^{4} - 5 T^{3} + 40 T^{2} - 50 T + 25$$
$23$ $$(T^{2} - 2 T - 4)^{2}$$
$29$ $$T^{4} - 10 T^{3} + 60 T^{2} + \cdots + 400$$
$31$ $$T^{4} - 2 T^{3} + 4 T^{2} - 8 T + 16$$
$37$ $$T^{4} + 18 T^{3} + 144 T^{2} + \cdots + 1296$$
$41$ $$T^{4} + 2 T^{3} + 24 T^{2} + 133 T + 361$$
$43$ $$(T^{2} + 3 T - 99)^{2}$$
$47$ $$T^{4} - 8 T^{3} + 64 T^{2} - 192 T + 256$$
$53$ $$T^{4} + 4 T^{3} + 96 T^{2} - 256 T + 256$$
$59$ $$T^{4} + 5 T^{3} + 60 T^{2} + \cdots + 3025$$
$61$ $$T^{4} - 8 T^{3} + 64 T^{2} - 192 T + 256$$
$67$ $$(T^{2} + 11 T - 1)^{2}$$
$71$ $$T^{4} + 8 T^{3} + 24 T^{2} - 8 T + 16$$
$73$ $$T^{4} + 14 T^{3} + 136 T^{2} + \cdots + 17161$$
$79$ $$T^{4} - 30 T^{3} + 540 T^{2} + \cdots + 32400$$
$83$ $$T^{4} - 19 T^{3} + 186 T^{2} + \cdots + 3481$$
$89$ $$(T^{2} + 5 T - 25)^{2}$$
$97$ $$T^{4} + 3 T^{3} + 144 T^{2} + \cdots + 9801$$