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Properties

Label 176.2.q.b
Level $176$
Weight $2$
Character orbit 176.q
Analytic conductor $1.405$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

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: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{10})\)
Coefficient field: 16.0.4526322734619140625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 3 x^{15} + 7 x^{14} - 6 x^{13} + 3 x^{12} + 6 x^{11} + 14 x^{10} - 48 x^{9} + 113 x^{8} + \cdots + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{8} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{11} - \beta_{9} + \cdots + \beta_{2}) q^{3}+ \cdots + (\beta_{15} - \beta_{13} + 2 \beta_{3} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{11} - \beta_{9} + \cdots + \beta_{2}) q^{3}+ \cdots + (\beta_{14} - 5 \beta_{12} + \cdots - 3 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 6 q^{5} + 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 6 q^{5} + 22 q^{9} + 10 q^{13} + 10 q^{17} - 42 q^{25} - 30 q^{29} - 48 q^{33} - 6 q^{37} - 70 q^{41} - 56 q^{45} + 38 q^{49} + 6 q^{53} + 20 q^{57} + 30 q^{61} + 8 q^{69} + 10 q^{73} + 10 q^{77} + 8 q^{81} + 90 q^{85} + 52 q^{89} + 82 q^{93} + 76 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 3 x^{15} + 7 x^{14} - 6 x^{13} + 3 x^{12} + 6 x^{11} + 14 x^{10} - 48 x^{9} + 113 x^{8} + \cdots + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 5 \nu^{15} - 271 \nu^{14} + 335 \nu^{13} + 26 \nu^{12} - 2081 \nu^{11} + 5050 \nu^{10} + \cdots + 17472 ) / 15296 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 45 \nu^{15} + 907 \nu^{14} - 331 \nu^{13} - 1200 \nu^{12} + 7083 \nu^{11} - 6652 \nu^{10} + \cdots + 111360 ) / 30592 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 25 \nu^{15} - 79 \nu^{14} + 237 \nu^{13} - 130 \nu^{12} - 589 \nu^{11} + 2952 \nu^{10} + \cdots + 19712 ) / 15296 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 79 \nu^{15} + 649 \nu^{14} - 2425 \nu^{13} + 6090 \nu^{12} - 8993 \nu^{11} + 6250 \nu^{10} + \cdots - 113920 ) / 30592 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 103 \nu^{15} + 468 \nu^{14} - 1404 \nu^{13} + 2189 \nu^{12} - 1681 \nu^{11} - 1977 \nu^{10} + \cdots - 5056 ) / 15296 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 257 \nu^{15} + 2075 \nu^{14} - 2879 \nu^{13} + 4782 \nu^{12} + 5245 \nu^{11} - 4318 \nu^{10} + \cdots + 261376 ) / 30592 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 34 \nu^{15} - 117 \nu^{14} + 351 \nu^{13} - 1085 \nu^{12} + 958 \nu^{11} - 1119 \nu^{10} + \cdots - 17856 ) / 3824 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 285 \nu^{15} - 1585 \nu^{14} + 5233 \nu^{13} - 10468 \nu^{12} + 12355 \nu^{11} - 2296 \nu^{10} + \cdots + 93440 ) / 30592 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 153 \nu^{15} - 310 \nu^{14} + 452 \nu^{13} + 461 \nu^{12} - 1443 \nu^{11} - 103 \nu^{10} + \cdots - 49664 ) / 15296 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 286 \nu^{15} - 731 \nu^{14} + 1715 \nu^{13} - 1333 \nu^{12} + 658 \nu^{11} + 1821 \nu^{10} + \cdots - 31552 ) / 15296 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 201 \nu^{15} + 426 \nu^{14} - 561 \nu^{13} + 580 \nu^{12} - 902 \nu^{11} + 4203 \nu^{10} + \cdots + 53248 ) / 7648 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 429 \nu^{15} - 1216 \nu^{14} + 2214 \nu^{13} - 2597 \nu^{12} + 509 \nu^{11} - 973 \nu^{10} + \cdots - 39680 ) / 15296 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - \nu^{15} + 4 \nu^{14} - 5 \nu^{13} + 2 \nu^{12} + 14 \nu^{11} - 21 \nu^{10} + 15 \nu^{9} + \cdots + 128 ) / 32 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 909 \nu^{15} - 2615 \nu^{14} + 4499 \nu^{13} - 1774 \nu^{12} - 4721 \nu^{11} + 7022 \nu^{10} + \cdots - 197888 ) / 15296 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 1991 \nu^{15} - 5429 \nu^{14} + 11985 \nu^{13} - 9914 \nu^{12} + 8037 \nu^{11} + 442 \nu^{10} + \cdots - 161408 ) / 30592 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{8} + \beta_{5} + \beta_{4} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{14} - \beta_{12} - 2\beta_{10} + \beta_{8} + \beta_{5} - \beta_{3} + \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{14} - 2\beta_{9} + \beta_{7} + 4\beta_{5} + 4\beta_{3} + 2\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - \beta_{15} + 2 \beta_{13} + \beta_{12} + 3 \beta_{10} + \beta_{9} - 2 \beta_{7} - \beta_{6} + \cdots - 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 3 \beta_{15} - 5 \beta_{14} + \beta_{12} + 2 \beta_{10} + \beta_{9} - 3 \beta_{8} - \beta_{7} - 3 \beta_{6} + \cdots + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 4 \beta_{14} + 4 \beta_{12} - \beta_{11} + 6 \beta_{9} - \beta_{7} + 2 \beta_{6} - 9 \beta_{5} + \cdots - 9 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( \beta_{15} + 2 \beta_{14} - \beta_{13} - 2 \beta_{12} - \beta_{11} - 12 \beta_{10} - 3 \beta_{8} + \cdots - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( \beta_{14} - \beta_{13} + 4 \beta_{12} + 4 \beta_{11} + 4 \beta_{10} - 8 \beta_{9} - \beta_{8} + \cdots - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - \beta_{12} + 9 \beta_{11} + 8 \beta_{10} - \beta_{9} - 4 \beta_{7} - 2 \beta_{6} - 8 \beta_{4} + \cdots - 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 6 \beta_{15} - 2 \beta_{14} - 15 \beta_{13} - 10 \beta_{12} + 8 \beta_{10} - 25 \beta_{9} - 6 \beta_{8} + \cdots + 10 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 27 \beta_{15} + 31 \beta_{14} - 31 \beta_{12} - 30 \beta_{11} - 4 \beta_{9} + 19 \beta_{8} + 3 \beta_{7} + \cdots - 9 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 28 \beta_{14} - 62 \beta_{12} - 28 \beta_{11} - 31 \beta_{10} - 6 \beta_{6} + 17 \beta_{5} - 28 \beta_{4} + \cdots + 80 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 6 \beta_{14} + 26 \beta_{13} + 41 \beta_{12} + 41 \beta_{11} + 41 \beta_{10} + 88 \beta_{9} + \cdots + 26 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 20 \beta_{15} + 41 \beta_{13} + 83 \beta_{12} + 94 \beta_{11} - 52 \beta_{10} + 83 \beta_{9} + 9 \beta_{7} + \cdots + 9 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 9 \beta_{14} + 135 \beta_{12} - 9 \beta_{10} - 135 \beta_{9} - 20 \beta_{7} - 18 \beta_{6} + \cdots - 91 ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
0.589191 − 1.28563i
0.818796 + 1.15307i
0.0153379 + 1.41413i
−1.23234 − 0.693782i
−1.23816 − 0.683337i
1.06903 − 0.925833i
1.21087 − 0.730613i
0.267278 + 1.38873i
0.589191 + 1.28563i
0.818796 − 1.15307i
0.0153379 − 1.41413i
−1.23234 + 0.693782i
−1.23816 + 0.683337i
1.06903 + 0.925833i
1.21087 + 0.730613i
0.267278 − 1.38873i
0 −2.94730 + 0.957636i 0 −0.540641 − 0.392798i 0 0.818875 − 2.52024i 0 5.34247 − 3.88153i 0
63.2 0 −1.30002 + 0.422403i 0 1.84966 + 1.34385i 0 −0.199199 + 0.613071i 0 −0.915415 + 0.665088i 0
63.3 0 1.30002 − 0.422403i 0 1.84966 + 1.34385i 0 0.199199 − 0.613071i 0 −0.915415 + 0.665088i 0
63.4 0 2.94730 − 0.957636i 0 −0.540641 − 0.392798i 0 −0.818875 + 2.52024i 0 5.34247 − 3.88153i 0
79.1 0 −0.930415 − 1.28061i 0 1.10004 + 3.38558i 0 2.06453 + 1.49997i 0 0.152771 − 0.470181i 0
79.2 0 −0.0876593 − 0.120653i 0 −0.909057 − 2.79779i 0 −3.71180 − 2.69678i 0 0.920178 − 2.83202i 0
79.3 0 0.0876593 + 0.120653i 0 −0.909057 − 2.79779i 0 3.71180 + 2.69678i 0 0.920178 − 2.83202i 0
79.4 0 0.930415 + 1.28061i 0 1.10004 + 3.38558i 0 −2.06453 − 1.49997i 0 0.152771 − 0.470181i 0
95.1 0 −2.94730 − 0.957636i 0 −0.540641 + 0.392798i 0 0.818875 + 2.52024i 0 5.34247 + 3.88153i 0
95.2 0 −1.30002 − 0.422403i 0 1.84966 − 1.34385i 0 −0.199199 − 0.613071i 0 −0.915415 − 0.665088i 0
95.3 0 1.30002 + 0.422403i 0 1.84966 − 1.34385i 0 0.199199 + 0.613071i 0 −0.915415 − 0.665088i 0
95.4 0 2.94730 + 0.957636i 0 −0.540641 + 0.392798i 0 −0.818875 − 2.52024i 0 5.34247 + 3.88153i 0
127.1 0 −0.930415 + 1.28061i 0 1.10004 − 3.38558i 0 2.06453 − 1.49997i 0 0.152771 + 0.470181i 0
127.2 0 −0.0876593 + 0.120653i 0 −0.909057 + 2.79779i 0 −3.71180 + 2.69678i 0 0.920178 + 2.83202i 0
127.3 0 0.0876593 − 0.120653i 0 −0.909057 + 2.79779i 0 3.71180 − 2.69678i 0 0.920178 + 2.83202i 0
127.4 0 0.930415 − 1.28061i 0 1.10004 − 3.38558i 0 −2.06453 + 1.49997i 0 0.152771 + 0.470181i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 63.4
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.b ✓ 16
4.b odd 2 1 inner 176.2.q.b ✓ 16
8.b even 2 1 704.2.u.b 16
8.d odd 2 1 704.2.u.b 16
11.c even 5 1 1936.2.e.f 16
11.d odd 10 1 inner 176.2.q.b ✓ 16
11.d odd 10 1 1936.2.e.f 16
44.g even 10 1 inner 176.2.q.b ✓ 16
44.g even 10 1 1936.2.e.f 16
44.h odd 10 1 1936.2.e.f 16
88.k even 10 1 704.2.u.b 16
88.p odd 10 1 704.2.u.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
176.2.q.b ✓ 16 1.a even 1 1 trivial
176.2.q.b ✓ 16 4.b odd 2 1 inner
176.2.q.b ✓ 16 11.d odd 10 1 inner
176.2.q.b ✓ 16 44.g even 10 1 inner
704.2.u.b 16 8.b even 2 1
704.2.u.b 16 8.d odd 2 1
704.2.u.b 16 88.k even 10 1
704.2.u.b 16 88.p odd 10 1
1936.2.e.f 16 11.c even 5 1
1936.2.e.f 16 11.d odd 10 1
1936.2.e.f 16 44.g even 10 1
1936.2.e.f 16 44.h odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} - 17T_{3}^{14} + 120T_{3}^{12} - 227T_{3}^{10} + 699T_{3}^{8} - 1583T_{3}^{6} + 2000T_{3}^{4} + 27T_{3}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(176, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} - 17 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( (T^{8} - 3 T^{7} + \cdots + 256)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} - 5 T^{14} + \cdots + 160000 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 214358881 \) Copy content Toggle raw display
$13$ \( (T^{8} - 5 T^{7} + \cdots + 400)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} - 5 T^{7} + \cdots + 9025)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} + 10 T^{14} + \cdots + 9150625 \) Copy content Toggle raw display
$23$ \( (T^{8} + 44 T^{6} + \cdots + 256)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 15 T^{7} + \cdots + 336400)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 274760478976 \) Copy content Toggle raw display
$37$ \( (T^{8} + 3 T^{7} + 26 T^{6} + \cdots + 16)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + 35 T^{7} + \cdots + 42025)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} - 235 T^{6} + \cdots + 5382400)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 3072924856576 \) Copy content Toggle raw display
$53$ \( (T^{8} - 3 T^{7} + \cdots + 156816)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 1998607065841 \) Copy content Toggle raw display
$61$ \( (T^{8} - 15 T^{7} + \cdots + 70224400)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + 301 T^{6} + \cdots + 1547536)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 9508194263296 \) Copy content Toggle raw display
$73$ \( (T^{8} - 5 T^{7} + \cdots + 156025)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 6887475360000 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 88223850625 \) Copy content Toggle raw display
$89$ \( (T^{4} - 13 T^{3} + \cdots - 164)^{4} \) Copy content Toggle raw display
$97$ \( (T^{8} - 38 T^{7} + \cdots + 11881)^{2} \) Copy content Toggle raw display
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