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Properties

Label 1176.2.c.e
Level $1176$
Weight $2$
Character orbit 1176.c
Analytic conductor $9.390$
Analytic rank $0$
Dimension $16$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1176,2,Mod(589,1176)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1176, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1176.589");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1176 = 2^{3} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1176.c (of order \(2\), degree \(1\), 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: \(9.39040727770\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + x^{14} - 2 x^{13} - 2 x^{12} - 4 x^{11} - 2 x^{10} + 16 x^{9} + 8 x^{8} + 32 x^{7} - 8 x^{6} + \cdots + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 168)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{6} q^{2} + \beta_{5} q^{3} - \beta_{2} q^{4} - \beta_{4} q^{5} - \beta_1 q^{6} + ( - \beta_{13} - \beta_{12}) q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{2} + \beta_{5} q^{3} - \beta_{2} q^{4} - \beta_{4} q^{5} - \beta_1 q^{6} + ( - \beta_{13} - \beta_{12}) q^{8} - q^{9} + ( - \beta_{15} + \beta_{14} + \cdots - \beta_{9}) q^{10}+ \cdots - \beta_{15} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 2 q^{2} + 2 q^{4} - 8 q^{8} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 2 q^{2} + 2 q^{4} - 8 q^{8} - 16 q^{9} - 6 q^{10} + 10 q^{16} + 2 q^{18} - 20 q^{20} - 6 q^{22} + 8 q^{23} + 6 q^{24} - 16 q^{25} - 6 q^{26} + 8 q^{30} + 24 q^{31} + 8 q^{32} - 12 q^{34} - 2 q^{36} + 26 q^{38} + 6 q^{40} - 20 q^{44} - 16 q^{46} + 24 q^{47} - 8 q^{48} + 26 q^{50} - 44 q^{52} - 32 q^{55} - 8 q^{57} - 34 q^{58} + 22 q^{60} + 50 q^{62} - 10 q^{64} - 12 q^{66} - 16 q^{68} - 40 q^{71} + 8 q^{72} - 8 q^{73} - 10 q^{74} - 16 q^{76} + 6 q^{78} - 8 q^{79} + 56 q^{80} + 16 q^{81} + 22 q^{86} - 24 q^{87} - 50 q^{88} + 6 q^{90} + 32 q^{92} + 48 q^{94} - 24 q^{95} - 10 q^{96} - 24 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + x^{14} - 2 x^{13} - 2 x^{12} - 4 x^{11} - 2 x^{10} + 16 x^{9} + 8 x^{8} + 32 x^{7} - 8 x^{6} + \cdots + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - \nu^{15} + 2 \nu^{14} - \nu^{13} + 4 \nu^{12} + 6 \nu^{11} - 16 \nu^{10} - 14 \nu^{9} - 20 \nu^{8} + \cdots - 384 ) / 256 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{15} - 2 \nu^{14} + \nu^{13} + 4 \nu^{12} - 6 \nu^{11} + 8 \nu^{10} - 2 \nu^{9} + 20 \nu^{8} + \cdots + 384 ) / 256 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - \nu^{15} + 3 \nu^{13} - 2 \nu^{12} + 6 \nu^{11} + 2 \nu^{9} - 16 \nu^{7} + 8 \nu^{6} - 24 \nu^{5} + \cdots - 128 ) / 128 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - \nu^{14} + \nu^{13} - \nu^{12} + \nu^{11} - 4 \nu^{9} + 2 \nu^{8} - 10 \nu^{7} + 8 \nu^{6} + \cdots - 64 ) / 32 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - \nu^{14} + 2 \nu^{13} - 5 \nu^{12} + 2 \nu^{10} - 4 \nu^{9} + 18 \nu^{8} - 4 \nu^{7} + 16 \nu^{6} + \cdots - 64 ) / 64 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 3 \nu^{15} + 2 \nu^{14} - 3 \nu^{13} + 16 \nu^{12} - 6 \nu^{11} - 16 \nu^{10} - 10 \nu^{9} + \cdots + 896 ) / 256 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( \nu^{15} + \nu^{13} - 2 \nu^{12} - 2 \nu^{11} - 4 \nu^{10} - 2 \nu^{9} + 16 \nu^{8} + 8 \nu^{7} + \cdots + 64 \nu ) / 64 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( \nu^{15} + \nu^{13} - 2 \nu^{11} + 2 \nu^{10} - 6 \nu^{9} + 8 \nu^{8} - 8 \nu^{7} + 28 \nu^{6} + \cdots - 64 ) / 64 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( \nu^{15} - 4 \nu^{14} + 5 \nu^{13} - 6 \nu^{12} + 10 \nu^{11} + 12 \nu^{10} - 26 \nu^{9} + 24 \nu^{8} + \cdots - 512 ) / 128 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 3 \nu^{15} - \nu^{13} + 2 \nu^{12} - 10 \nu^{11} + 4 \nu^{10} + 18 \nu^{9} + 16 \nu^{8} + 32 \nu^{7} + \cdots + 512 ) / 128 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 3 \nu^{15} + 2 \nu^{14} - 7 \nu^{13} + 8 \nu^{12} + 14 \nu^{11} + 22 \nu^{9} - 52 \nu^{8} + \cdots + 128 ) / 128 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 7 \nu^{15} + 2 \nu^{14} + \nu^{13} + 8 \nu^{12} + 26 \nu^{11} - 2 \nu^{9} - 84 \nu^{8} - 72 \nu^{7} + \cdots - 640 ) / 256 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{15} + \beta_{13} + \beta_{8} - \beta_{6} + \beta_{4} + \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{15} + \beta_{14} - \beta_{12} + 2 \beta_{11} - \beta_{10} + \beta_{8} + \beta_{7} + \cdots + \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{15} + \beta_{14} - \beta_{12} + 2 \beta_{11} + \beta_{10} + \beta_{8} + \beta_{7} - \beta_{6} + \cdots + \beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{15} + \beta_{14} + 2 \beta_{13} + \beta_{12} + \beta_{10} + 2 \beta_{9} + 3 \beta_{8} - \beta_{7} + \cdots - 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - \beta_{15} + \beta_{14} - 2 \beta_{13} - \beta_{12} + \beta_{10} - 2 \beta_{9} + \beta_{8} + \cdots - 5 \beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - \beta_{15} + 3 \beta_{14} + 2 \beta_{13} - 5 \beta_{12} + 4 \beta_{11} - \beta_{10} - 2 \beta_{9} + \cdots - 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 3 \beta_{15} + 3 \beta_{14} + 4 \beta_{13} + \beta_{12} + 8 \beta_{11} - 5 \beta_{10} + 2 \beta_{9} + \cdots - 8 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 5 \beta_{15} + 7 \beta_{14} - 2 \beta_{13} + 7 \beta_{12} + 7 \beta_{10} + 2 \beta_{9} - 3 \beta_{8} + \cdots + 8 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 7 \beta_{15} - 5 \beta_{14} - 4 \beta_{13} + 5 \beta_{12} - 12 \beta_{11} + 11 \beta_{10} - 2 \beta_{9} + \cdots - 24 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - \beta_{15} - 13 \beta_{14} + 10 \beta_{13} + 7 \beta_{12} - 8 \beta_{11} - 5 \beta_{10} + 10 \beta_{9} + \cdots - 24 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - \beta_{15} - 17 \beta_{14} - 12 \beta_{13} + 13 \beta_{12} - 4 \beta_{11} - \beta_{10} - 2 \beta_{9} + \cdots + 56 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 15 \beta_{15} - 13 \beta_{14} + 46 \beta_{13} + 43 \beta_{12} - 24 \beta_{11} + 19 \beta_{10} + 10 \beta_{9} + \cdots + 8 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1176\mathbb{Z}\right)^\times\).

\(n\) \(295\) \(589\) \(785\) \(1081\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
589.1
−0.236856 + 1.39424i
−0.236856 − 1.39424i
0.462163 − 1.33656i
0.462163 + 1.33656i
−0.865014 + 1.11882i
−0.865014 − 1.11882i
1.36937 − 0.353295i
1.36937 + 0.353295i
−1.39783 − 0.214614i
−1.39783 + 0.214614i
1.30955 + 0.533922i
1.30955 − 0.533922i
−0.925802 − 1.06906i
−0.925802 + 1.06906i
0.284419 + 1.38532i
0.284419 − 1.38532i
−1.39424 − 0.236856i 1.00000i 1.88780 + 0.660467i − 3.56550i 0.236856 − 1.39424i 0 −2.47560 − 1.36799i −1.00000 −0.844510 + 4.97116i
589.2 −1.39424 + 0.236856i − 1.00000i 1.88780 − 0.660467i 3.56550i 0.236856 + 1.39424i 0 −2.47560 + 1.36799i −1.00000 −0.844510 − 4.97116i
589.3 −1.33656 − 0.462163i − 1.00000i 1.57281 + 1.23542i − 1.42114i −0.462163 + 1.33656i 0 −1.53120 − 2.37811i −1.00000 −0.656797 + 1.89944i
589.4 −1.33656 + 0.462163i 1.00000i 1.57281 − 1.23542i 1.42114i −0.462163 − 1.33656i 0 −1.53120 + 2.37811i −1.00000 −0.656797 − 1.89944i
589.5 −1.11882 − 0.865014i 1.00000i 0.503501 + 1.93558i 3.57776i 0.865014 − 1.11882i 0 1.11098 − 2.60110i −1.00000 3.09481 − 4.00286i
589.6 −1.11882 + 0.865014i − 1.00000i 0.503501 − 1.93558i − 3.57776i 0.865014 + 1.11882i 0 1.11098 + 2.60110i −1.00000 3.09481 + 4.00286i
589.7 −0.353295 − 1.36937i − 1.00000i −1.75037 + 0.967585i 0.677172i −1.36937 + 0.353295i 0 1.94338 + 2.05506i −1.00000 0.927301 − 0.239241i
589.8 −0.353295 + 1.36937i 1.00000i −1.75037 − 0.967585i − 0.677172i −1.36937 − 0.353295i 0 1.94338 − 2.05506i −1.00000 0.927301 + 0.239241i
589.9 0.214614 − 1.39783i 1.00000i −1.90788 − 0.599989i − 0.0464447i 1.39783 + 0.214614i 0 −1.24814 + 2.53814i −1.00000 −0.0649220 − 0.00996767i
589.10 0.214614 + 1.39783i − 1.00000i −1.90788 + 0.599989i 0.0464447i 1.39783 − 0.214614i 0 −1.24814 − 2.53814i −1.00000 −0.0649220 + 0.00996767i
589.11 0.533922 − 1.30955i − 1.00000i −1.42985 − 1.39840i − 3.38908i −1.30955 − 0.533922i 0 −2.59471 + 1.12583i −1.00000 −4.43818 − 1.80951i
589.12 0.533922 + 1.30955i 1.00000i −1.42985 + 1.39840i 3.38908i −1.30955 + 0.533922i 0 −2.59471 − 1.12583i −1.00000 −4.43818 + 1.80951i
589.13 1.06906 − 0.925802i 1.00000i 0.285780 − 1.97948i − 1.80422i 0.925802 + 1.06906i 0 −1.52709 − 2.38076i −1.00000 −1.67035 − 1.92881i
589.14 1.06906 + 0.925802i − 1.00000i 0.285780 + 1.97948i 1.80422i 0.925802 − 1.06906i 0 −1.52709 + 2.38076i −1.00000 −1.67035 + 1.92881i
589.15 1.38532 − 0.284419i − 1.00000i 1.83821 − 0.788022i 2.29465i −0.284419 − 1.38532i 0 2.32238 − 1.61448i −1.00000 0.652642 + 3.17882i
589.16 1.38532 + 0.284419i 1.00000i 1.83821 + 0.788022i − 2.29465i −0.284419 + 1.38532i 0 2.32238 + 1.61448i −1.00000 0.652642 − 3.17882i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 589.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1176.2.c.e 16
4.b odd 2 1 4704.2.c.e 16
7.b odd 2 1 1176.2.c.f 16
7.c even 3 2 168.2.bc.a ✓ 32
8.b even 2 1 inner 1176.2.c.e 16
8.d odd 2 1 4704.2.c.e 16
21.h odd 6 2 504.2.cj.e 32
28.d even 2 1 4704.2.c.f 16
28.g odd 6 2 672.2.bk.a 32
56.e even 2 1 4704.2.c.f 16
56.h odd 2 1 1176.2.c.f 16
56.k odd 6 2 672.2.bk.a 32
56.p even 6 2 168.2.bc.a ✓ 32
84.n even 6 2 2016.2.cr.e 32
168.s odd 6 2 504.2.cj.e 32
168.v even 6 2 2016.2.cr.e 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.bc.a ✓ 32 7.c even 3 2
168.2.bc.a ✓ 32 56.p even 6 2
504.2.cj.e 32 21.h odd 6 2
504.2.cj.e 32 168.s odd 6 2
672.2.bk.a 32 28.g odd 6 2
672.2.bk.a 32 56.k odd 6 2
1176.2.c.e 16 1.a even 1 1 trivial
1176.2.c.e 16 8.b even 2 1 inner
1176.2.c.f 16 7.b odd 2 1
1176.2.c.f 16 56.h odd 2 1
2016.2.cr.e 32 84.n even 6 2
2016.2.cr.e 32 168.v even 6 2
4704.2.c.e 16 4.b odd 2 1
4704.2.c.e 16 8.d odd 2 1
4704.2.c.f 16 28.d even 2 1
4704.2.c.f 16 56.e even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1176, [\chi])\):

\( T_{5}^{16} + 48T_{5}^{14} + 902T_{5}^{12} + 8384T_{5}^{10} + 40313T_{5}^{8} + 96864T_{5}^{6} + 101584T_{5}^{4} + 29888T_{5}^{2} + 64 \) Copy content Toggle raw display
\( T_{17}^{8} - 76T_{17}^{6} - 104T_{17}^{5} + 1544T_{17}^{4} + 3744T_{17}^{3} - 1184T_{17}^{2} - 896T_{17} + 256 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 2 T^{15} + \cdots + 256 \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$5$ \( T^{16} + 48 T^{14} + \cdots + 64 \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( T^{16} + 76 T^{14} + \cdots + 2304 \) Copy content Toggle raw display
$13$ \( T^{16} + 108 T^{14} + \cdots + 1149184 \) Copy content Toggle raw display
$17$ \( (T^{8} - 76 T^{6} + \cdots + 256)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} + 140 T^{14} + \cdots + 11723776 \) Copy content Toggle raw display
$23$ \( (T^{8} - 4 T^{7} + \cdots + 25472)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} + 192 T^{14} + \cdots + 589824 \) Copy content Toggle raw display
$31$ \( (T^{8} - 12 T^{7} + \cdots - 145241)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 13545235456 \) Copy content Toggle raw display
$41$ \( (T^{8} - 148 T^{6} + \cdots - 29696)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 326726560000 \) Copy content Toggle raw display
$47$ \( (T^{8} - 12 T^{7} + \cdots - 2304)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 1973491776 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 3515066944 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 10909384704 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 499498389504 \) Copy content Toggle raw display
$71$ \( (T^{8} + 20 T^{7} + \cdots + 8590080)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 4 T^{7} + \cdots - 1022064)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 4 T^{7} + \cdots - 37585)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 29873665600 \) Copy content Toggle raw display
$89$ \( (T^{8} - 336 T^{6} + \cdots + 1114112)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + 12 T^{7} + \cdots - 2509616)^{2} \) Copy content Toggle raw display
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