Properties

Label 176.2.q.a
Level $176$
Weight $2$
Character orbit 176.q
Analytic conductor $1.405$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [176,2,Mod(63,176)]
 
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([5, 0, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("176.63");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 176 = 2^{4} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 176.q (of order \(10\), degree \(4\), 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: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: 8.0.484000000.6
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{6} + 16x^{4} - 66x^{2} + 121 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (\beta_{5} - \beta_{3} + \beta_{2}) q^{5} + (\beta_{7} - \beta_{4} + 2 \beta_1) q^{7} + (\beta_{3} - \beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + (\beta_{5} - \beta_{3} + \beta_{2}) q^{5} + (\beta_{7} - \beta_{4} + 2 \beta_1) q^{7} + (\beta_{3} - \beta_{2} - 1) q^{9} + ( - \beta_{7} + \beta_1) q^{11} + ( - 4 \beta_{5} + \beta_{3} - 2 \beta_{2} - 3) q^{13} + (\beta_{6} - \beta_1) q^{15} + ( - \beta_{5} + 3 \beta_{3} + \beta_{2} - 2) q^{17} + (2 \beta_{7} - 2 \beta_{6} + 2 \beta_{4} + \beta_1) q^{19} + (5 \beta_{5} - 4 \beta_{3} - \beta_{2} + 2) q^{21} + ( - 2 \beta_{7} - 2 \beta_{4} - 2 \beta_1) q^{23} + (3 \beta_{5} - 3 \beta_{3} + 4 \beta_{2} + 4) q^{25} + ( - 2 \beta_{7} + 2 \beta_{6} + \beta_{4} - 3 \beta_1) q^{27} + ( - \beta_{5} - 2 \beta_{3} + 4 \beta_{2} + 5) q^{29} + (3 \beta_{7} - \beta_{6}) q^{31} + ( - 2 \beta_{5} + 4 \beta_{3} - 7 \beta_{2} - 1) q^{33} + ( - \beta_{7} + 2 \beta_{6} + 2 \beta_{4} - 4 \beta_1) q^{35} + ( - \beta_{5} + 6 \beta_{3} - 1) q^{37} + ( - 2 \beta_{7} - 2 \beta_{6} - 3 \beta_{4} - \beta_1) q^{39} + ( - 4 \beta_{5} - 4 \beta_{3} - \beta_{2} + 1) q^{41} + (2 \beta_{7} - 4 \beta_{6} + 2 \beta_1) q^{43} + q^{45} + (4 \beta_{6} + 4 \beta_{4} - 5 \beta_1) q^{47} + (10 \beta_{5} - 9 \beta_{3} + 9 \beta_{2}) q^{49} + ( - 2 \beta_{7} + \beta_{6} + 2 \beta_{4} - 3 \beta_1) q^{51} + (9 \beta_{3} - 6 \beta_{2} - 9) q^{53} + (\beta_{7} + \beta_{6} - \beta_{4}) q^{55} + ( - 8 \beta_{5} - \beta_{3} - 4 \beta_{2} - 9) q^{57} + ( - 3 \beta_{7} - 2 \beta_{6} - 3 \beta_{4} + 2 \beta_1) q^{59} + ( - 3 \beta_{5} + 5 \beta_{3} - \beta_{2} - 6) q^{61} + ( - \beta_{6} + \beta_{4}) q^{63} + (3 \beta_{5} - 2 \beta_{3} - \beta_{2} + 1) q^{65} + (4 \beta_{7} + 4 \beta_1) q^{67} + (2 \beta_{5} - 2 \beta_{3} + 10 \beta_{2} + 10) q^{69} + (\beta_{7} - \beta_{6} - 3 \beta_{4} + 4 \beta_1) q^{71} + ( - 7 \beta_{5} + 2 \beta_{3} - 4 \beta_{2} + 3) q^{73} + ( - \beta_{7} + 4 \beta_{6}) q^{75} + (3 \beta_{5} - 6 \beta_{3} - 6 \beta_{2} + 7) q^{77} + (3 \beta_{7} - 3 \beta_{6} + 6 \beta_1) q^{79} + (2 \beta_{5} + 8 \beta_{3} + 2) q^{81} + (\beta_{7} + \beta_{6} + 3 \beta_{4} + 2 \beta_1) q^{83} + ( - 2 \beta_{5} - 2 \beta_{3} - 3 \beta_{2} + 3) q^{85} + ( - 5 \beta_{7} + 4 \beta_{6} - 3 \beta_{4} + \beta_1) q^{87} + (10 \beta_{5} + 10 \beta_{2} + 6) q^{89} + ( - 5 \beta_{6} - 5 \beta_{4} + 5 \beta_1) q^{91} + (3 \beta_{5} - 10 \beta_{3} + 10 \beta_{2}) q^{93} + (\beta_{6} + \beta_1) q^{95} + ( - \beta_{3} - 4 \beta_{2} + 1) q^{97} + (2 \beta_{7} - \beta_{6} + 2 \beta_{4}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 6 q^{5} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 6 q^{5} - 4 q^{9} - 10 q^{13} - 10 q^{17} + 12 q^{25} + 30 q^{29} + 18 q^{33} + 6 q^{37} + 10 q^{41} + 8 q^{45} - 56 q^{49} - 42 q^{53} - 50 q^{57} - 30 q^{61} + 52 q^{69} + 50 q^{73} + 50 q^{77} + 28 q^{81} + 30 q^{85} + 8 q^{89} - 46 q^{93} + 14 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{6} + 16x^{4} - 66x^{2} + 121 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -7\nu^{6} - 37\nu^{4} - 629\nu^{2} - 363 ) / 1991 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -28\nu^{6} - 148\nu^{4} - 525\nu^{2} + 539 ) / 1991 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -28\nu^{7} - 148\nu^{5} - 525\nu^{3} + 539\nu ) / 1991 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 40\nu^{6} - 73\nu^{4} + 750\nu^{2} - 2761 ) / 1991 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 61\nu^{7} + 38\nu^{5} + 646\nu^{3} - 1672\nu ) / 1991 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 68\nu^{7} + 75\nu^{5} + 1275\nu^{3} - 3300\nu ) / 1991 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 4\beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4\beta_{7} - 4\beta_{6} + \beta_{4} + 3\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -7\beta_{5} - 10\beta_{3} - 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -7\beta_{7} - 17\beta_{4} - 7\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 37\beta_{5} - 37\beta_{3} + 75\beta_{2} + 75 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -38\beta_{7} + 75\beta_{6} \) Copy content Toggle raw display

Character values

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

\(n\) \(111\) \(133\) \(145\)
\(\chi(n)\) \(-1\) \(1\) \(\beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
63.1
−1.46782 + 0.476925i
1.46782 0.476925i
−1.26313 1.73855i
1.26313 + 1.73855i
−1.46782 0.476925i
1.46782 + 0.476925i
−1.26313 + 1.73855i
1.26313 1.73855i
0 −1.46782 + 0.476925i 0 −1.30902 0.951057i 0 −1.46782 + 4.51750i 0 −0.500000 + 0.363271i 0
63.2 0 1.46782 0.476925i 0 −1.30902 0.951057i 0 1.46782 4.51750i 0 −0.500000 + 0.363271i 0
79.1 0 −1.26313 1.73855i 0 −0.190983 0.587785i 0 −1.26313 0.917716i 0 −0.500000 + 1.53884i 0
79.2 0 1.26313 + 1.73855i 0 −0.190983 0.587785i 0 1.26313 + 0.917716i 0 −0.500000 + 1.53884i 0
95.1 0 −1.46782 0.476925i 0 −1.30902 + 0.951057i 0 −1.46782 4.51750i 0 −0.500000 0.363271i 0
95.2 0 1.46782 + 0.476925i 0 −1.30902 + 0.951057i 0 1.46782 + 4.51750i 0 −0.500000 0.363271i 0
127.1 0 −1.26313 + 1.73855i 0 −0.190983 + 0.587785i 0 −1.26313 + 0.917716i 0 −0.500000 1.53884i 0
127.2 0 1.26313 1.73855i 0 −0.190983 + 0.587785i 0 1.26313 0.917716i 0 −0.500000 1.53884i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 63.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
11.d odd 10 1 inner
44.g even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 176.2.q.a 8
4.b odd 2 1 inner 176.2.q.a 8
8.b even 2 1 704.2.u.a 8
8.d odd 2 1 704.2.u.a 8
11.c even 5 1 1936.2.e.e 8
11.d odd 10 1 inner 176.2.q.a 8
11.d odd 10 1 1936.2.e.e 8
44.g even 10 1 inner 176.2.q.a 8
44.g even 10 1 1936.2.e.e 8
44.h odd 10 1 1936.2.e.e 8
88.k even 10 1 704.2.u.a 8
88.p odd 10 1 704.2.u.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
176.2.q.a 8 1.a even 1 1 trivial
176.2.q.a 8 4.b odd 2 1 inner
176.2.q.a 8 11.d odd 10 1 inner
176.2.q.a 8 44.g even 10 1 inner
704.2.u.a 8 8.b even 2 1
704.2.u.a 8 8.d odd 2 1
704.2.u.a 8 88.k even 10 1
704.2.u.a 8 88.p odd 10 1
1936.2.e.e 8 11.c even 5 1
1936.2.e.e 8 11.d odd 10 1
1936.2.e.e 8 44.g even 10 1
1936.2.e.e 8 44.h odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - T_{3}^{6} + 16T_{3}^{4} - 66T_{3}^{2} + 121 \) acting on \(S_{2}^{\mathrm{new}}(176, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - T^{6} + 16 T^{4} - 66 T^{2} + \cdots + 121 \) Copy content Toggle raw display
$5$ \( (T^{4} + 3 T^{3} + 4 T^{2} + 2 T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} + 35 T^{6} + 460 T^{4} + \cdots + 3025 \) Copy content Toggle raw display
$11$ \( T^{8} - 16 T^{6} + 286 T^{4} + \cdots + 14641 \) Copy content Toggle raw display
$13$ \( (T^{4} + 5 T^{3} + 125)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 5 T^{3} + 20 T^{2} - 20 T + 5)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} - 25 T^{6} + 1860 T^{4} + \cdots + 3025 \) Copy content Toggle raw display
$23$ \( (T^{4} + 52 T^{2} + 176)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 15 T^{3} + 120 T^{2} - 440 T + 605)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} - 39 T^{6} + 1336 T^{4} + \cdots + 1771561 \) Copy content Toggle raw display
$37$ \( (T^{4} - 3 T^{3} + 34 T^{2} - 232 T + 841)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - 5 T^{3} - 10 T^{2} - 190 T + 1805)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 80 T^{2} + 880)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 209 T^{6} + 16456 T^{4} + \cdots + 1771561 \) Copy content Toggle raw display
$53$ \( (T^{4} + 21 T^{3} + 306 T^{2} + 2376 T + 9801)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} - 141 T^{6} + 7456 T^{4} + \cdots + 1771561 \) Copy content Toggle raw display
$61$ \( (T^{4} + 15 T^{3} + 120 T^{2} + 440 T + 605)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 128 T^{2} + 2816)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + 45 T^{6} + 6400 T^{4} + \cdots + 75625 \) Copy content Toggle raw display
$73$ \( (T^{4} - 25 T^{3} + 200 T^{2} - 440 T + 605)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + 45 T^{6} + 4860 T^{4} + \cdots + 19847025 \) Copy content Toggle raw display
$83$ \( T^{8} + 185 T^{6} + 13060 T^{4} + \cdots + 3025 \) Copy content Toggle raw display
$89$ \( (T^{2} - 2 T - 124)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} - 7 T^{3} + 24 T^{2} - 38 T + 361)^{2} \) Copy content Toggle raw display
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