↑

Properties

Label 176.3.n.a
Level $176$
Weight $3$
Character orbit 176.n
Analytic conductor $4.796$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [176,3,Mod(17,176)]
 
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, 9]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("176.17");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 176 = 2^{4} \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 176.n (of order \(10\), 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: \(4.79565265274\)
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 11)
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 primitive root of unity \(\zeta_{10}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 2 \zeta_{10}^{3} + 3 \zeta_{10}^{2} + \cdots + 2) q^{3}+ \cdots + (4 \zeta_{10}^{3} + 5 \zeta_{10} - 5) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \zeta_{10}^{3} + 3 \zeta_{10}^{2} + \cdots + 2) q^{3}+ \cdots + (69 \zeta_{10}^{3} - 7 \zeta_{10}^{2} + \cdots - 28) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{5} - 10 q^{7} - 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{5} - 10 q^{7} - 11 q^{9} - q^{11} - 20 q^{13} - 25 q^{19} + 20 q^{23} + 9 q^{25} - 15 q^{27} - 40 q^{29} + 58 q^{31} + 65 q^{33} - 80 q^{35} + 90 q^{37} - 50 q^{39} - 80 q^{41} - 24 q^{45} + 30 q^{47} - 109 q^{49} + 195 q^{51} + 120 q^{53} + 76 q^{55} + 45 q^{57} - 23 q^{59} + 10 q^{61} - 90 q^{63} + 230 q^{67} - 10 q^{69} - 148 q^{71} + 300 q^{73} - 45 q^{75} - 200 q^{77} - 70 q^{79} - 116 q^{81} - 225 q^{83} + 260 q^{85} + 122 q^{89} + 80 q^{91} - 200 q^{93} + 100 q^{95} - 165 q^{97} - 31 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(111\) \(133\) \(145\)
\(\chi(n)\) \(1\) \(1\) \(\zeta_{10}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
0.809017 − 0.587785i
−0.309017 + 0.951057i
0.809017 + 0.587785i
−0.309017 − 0.951057i
0 1.11803 + 0.812299i 0 1.23607 − 3.80423i 0 −5.85410 − 8.05748i 0 −2.19098 − 6.74315i 0
129.1 0 −1.11803 − 3.44095i 0 −3.23607 − 2.35114i 0 0.854102 + 0.277515i 0 −3.30902 + 2.40414i 0
145.1 0 1.11803 − 0.812299i 0 1.23607 + 3.80423i 0 −5.85410 + 8.05748i 0 −2.19098 + 6.74315i 0
161.1 0 −1.11803 + 3.44095i 0 −3.23607 + 2.35114i 0 0.854102 − 0.277515i 0 −3.30902 − 2.40414i 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.d odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 176.3.n.a 4
4.b odd 2 1 11.3.d.a ✓ 4
11.d odd 10 1 inner 176.3.n.a 4
12.b even 2 1 99.3.k.a 4
20.d odd 2 1 275.3.x.e 4
20.e even 4 2 275.3.q.d 8
44.c even 2 1 121.3.d.d 4
44.g even 10 1 11.3.d.a ✓ 4
44.g even 10 1 121.3.b.b 4
44.g even 10 1 121.3.d.a 4
44.g even 10 1 121.3.d.c 4
44.h odd 10 1 121.3.b.b 4
44.h odd 10 1 121.3.d.a 4
44.h odd 10 1 121.3.d.c 4
44.h odd 10 1 121.3.d.d 4
132.n odd 10 1 99.3.k.a 4
132.n odd 10 1 1089.3.c.e 4
132.o even 10 1 1089.3.c.e 4
220.o even 10 1 275.3.x.e 4
220.w odd 20 2 275.3.q.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.3.d.a ✓ 4 4.b odd 2 1
11.3.d.a ✓ 4 44.g even 10 1
99.3.k.a 4 12.b even 2 1
99.3.k.a 4 132.n odd 10 1
121.3.b.b 4 44.g even 10 1
121.3.b.b 4 44.h odd 10 1
121.3.d.a 4 44.g even 10 1
121.3.d.a 4 44.h odd 10 1
121.3.d.c 4 44.g even 10 1
121.3.d.c 4 44.h odd 10 1
121.3.d.d 4 44.c even 2 1
121.3.d.d 4 44.h odd 10 1
176.3.n.a 4 1.a even 1 1 trivial
176.3.n.a 4 11.d odd 10 1 inner
275.3.q.d 8 20.e even 4 2
275.3.q.d 8 220.w odd 20 2
275.3.x.e 4 20.d odd 2 1
275.3.x.e 4 220.o even 10 1
1089.3.c.e 4 132.n odd 10 1
1089.3.c.e 4 132.o even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 10T_{3}^{2} - 25T_{3} + 25 \) acting on \(S_{3}^{\mathrm{new}}(176, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 10 T^{2} + \cdots + 25 \) Copy content Toggle raw display
$5$ \( T^{4} + 4 T^{3} + \cdots + 256 \) Copy content Toggle raw display
$7$ \( T^{4} + 10 T^{3} + \cdots + 80 \) Copy content Toggle raw display
$11$ \( T^{4} + T^{3} + \cdots + 14641 \) Copy content Toggle raw display
$13$ \( T^{4} + 20 T^{3} + \cdots + 2000 \) Copy content Toggle raw display
$17$ \( T^{4} - 10985 T + 142805 \) Copy content Toggle raw display
$19$ \( T^{4} + 25 T^{3} + \cdots + 605 \) Copy content Toggle raw display
$23$ \( (T^{2} - 10 T + 20)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} + 40 T^{3} + \cdots + 9680 \) Copy content Toggle raw display
$31$ \( T^{4} - 58 T^{3} + \cdots + 55696 \) Copy content Toggle raw display
$37$ \( T^{4} - 90 T^{3} + \cdots + 2624400 \) Copy content Toggle raw display
$41$ \( T^{4} + 80 T^{3} + \cdots + 8405 \) Copy content Toggle raw display
$43$ \( T^{4} + 1625 T^{2} + 581405 \) Copy content Toggle raw display
$47$ \( T^{4} - 30 T^{3} + \cdots + 384400 \) Copy content Toggle raw display
$53$ \( T^{4} - 120 T^{3} + \cdots + 810000 \) Copy content Toggle raw display
$59$ \( T^{4} + 23 T^{3} + \cdots + 5041 \) Copy content Toggle raw display
$61$ \( T^{4} - 10 T^{3} + \cdots + 403280 \) Copy content Toggle raw display
$67$ \( (T^{2} - 115 T + 2945)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 148 T^{3} + \cdots + 22619536 \) Copy content Toggle raw display
$73$ \( T^{4} - 300 T^{3} + \cdots + 93787805 \) Copy content Toggle raw display
$79$ \( T^{4} + 70 T^{3} + \cdots + 67280 \) Copy content Toggle raw display
$83$ \( T^{4} + 225 T^{3} + \cdots + 22281605 \) Copy content Toggle raw display
$89$ \( (T^{2} - 61 T - 7681)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 165 T^{3} + \cdots + 31416025 \) Copy content Toggle raw display
show more
show less