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 $2$

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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: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + (\zeta_{10}^{3} + \zeta_{10}^{2} + \zeta_{10}) q^{3} + (\zeta_{10}^{2} + 1) q^{5} + ( - 2 \zeta_{10}^{3} - 3 \zeta_{10} + 3) q^{7} + (2 \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 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}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{3} + 3 q^{5} + 7 q^{7} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{3} + 3 q^{5} + 7 q^{7} - 8 q^{9} + 4 q^{11} - q^{13} - 3 q^{15} - q^{17} - 7 q^{19} + 18 q^{21} - 8 q^{23} + 6 q^{25} - 5 q^{27} - 5 q^{29} + 5 q^{31} - 19 q^{33} + 9 q^{35} - 9 q^{37} - 9 q^{39} - 5 q^{41} - 16 q^{45} + 5 q^{47} - 12 q^{49} + q^{51} + 11 q^{53} - 7 q^{55} + 17 q^{57} - 17 q^{59} - q^{61} - 4 q^{63} - 2 q^{65} - 16 q^{67} + 8 q^{69} - 5 q^{71} + 19 q^{73} + 9 q^{75} + 37 q^{77} - 3 q^{79} - 16 q^{81} + 31 q^{83} - 7 q^{85} + 50 q^{87} - 16 q^{89} + 27 q^{91} + 5 q^{93} + q^{95} + 27 q^{97} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:

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 \(S_{2}^{\mathrm{new}}(176, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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