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Properties

Label 192.11.e.k
Level $192$
Weight $11$
Character orbit 192.e
Analytic conductor $121.989$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [192,11,Mod(65,192)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(192, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 11, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("192.65");
 
S:= CuspForms(chi, 11);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 192 = 2^{6} \cdot 3 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 192.e (of order \(2\), degree \(1\), 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: \(121.988592513\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 10188 x^{18} + 45921062 x^{16} + 120525675724 x^{14} + 203898078666673 x^{12} + \cdots + 53\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{176}\cdot 3^{47} \)
Twist minimal: no (minimal twist has level 96)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{3} - \beta_{6} q^{5} + ( - \beta_{4} - 10 \beta_{3}) q^{7} + (\beta_{6} - \beta_{5} + \beta_{2} - 5014) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{3} - \beta_{6} q^{5} + ( - \beta_{4} - 10 \beta_{3}) q^{7} + (\beta_{6} - \beta_{5} + \beta_{2} - 5014) q^{9} + ( - \beta_{11} - 47 \beta_{3}) q^{11} + (\beta_1 - 26394) q^{13} + ( - \beta_{12} + \beta_{11} + \cdots - \beta_{3}) q^{15}+ \cdots + ( - 459 \beta_{19} + \cdots + 2802546 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 100284 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 100284 q^{9} - 527880 q^{13} + 11589912 q^{21} - 56425772 q^{25} - 55706832 q^{33} + 148642168 q^{37} + 377224224 q^{45} + 1363854748 q^{49} + 1399004568 q^{57} - 1635155656 q^{61} + 435726432 q^{69} + 654515880 q^{73} - 9447541740 q^{81} + 19066455296 q^{85} - 2169686088 q^{93} - 15378184440 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 10188 x^{18} + 45921062 x^{16} + 120525675724 x^{14} + 203898078666673 x^{12} + \cdots + 53\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 37\!\cdots\!45 \nu^{18} + \cdots - 30\!\cdots\!20 ) / 21\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 15\!\cdots\!12 \nu^{19} + \cdots - 12\!\cdots\!00 \nu ) / 45\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 14\!\cdots\!61 \nu^{19} + \cdots + 19\!\cdots\!00 ) / 40\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 14\!\cdots\!61 \nu^{19} + \cdots + 17\!\cdots\!60 ) / 40\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 80\!\cdots\!21 \nu^{19} + \cdots + 96\!\cdots\!00 ) / 18\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 10\!\cdots\!59 \nu^{19} + \cdots - 75\!\cdots\!60 \nu ) / 14\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 14\!\cdots\!43 \nu^{19} + \cdots + 12\!\cdots\!00 ) / 36\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 55\!\cdots\!57 \nu^{19} + \cdots - 86\!\cdots\!00 ) / 67\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 14\!\cdots\!19 \nu^{19} + \cdots - 14\!\cdots\!00 ) / 90\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 32\!\cdots\!33 \nu^{19} + \cdots - 79\!\cdots\!00 ) / 18\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 27\!\cdots\!91 \nu^{19} + \cdots - 32\!\cdots\!00 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 60\!\cdots\!09 \nu^{19} + \cdots - 68\!\cdots\!60 ) / 13\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 45\!\cdots\!53 \nu^{19} + \cdots - 33\!\cdots\!00 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 33\!\cdots\!91 \nu^{19} + \cdots + 10\!\cdots\!00 ) / 72\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 14\!\cdots\!53 \nu^{19} + \cdots + 96\!\cdots\!00 ) / 18\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 38\!\cdots\!07 \nu^{19} + \cdots - 32\!\cdots\!00 ) / 46\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 78\!\cdots\!39 \nu^{19} + \cdots - 73\!\cdots\!00 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 13\!\cdots\!17 \nu^{19} + \cdots + 13\!\cdots\!00 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 12\!\cdots\!39 \nu^{19} + \cdots + 11\!\cdots\!00 ) / 13\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 100 \beta_{19} + 28 \beta_{18} + 65 \beta_{17} + 36 \beta_{16} + 148 \beta_{12} + \cdots - 3888 \beta_{2} ) / 31850496 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 14 \beta_{19} + 14 \beta_{18} + 41 \beta_{17} + 194 \beta_{16} - 204 \beta_{14} + \cdots - 8112321624 ) / 7962624 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 130348 \beta_{19} - 35236 \beta_{18} - 104099 \beta_{17} + 39396 \beta_{16} - 30240 \beta_{15} + \cdots - 85248 ) / 31850496 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 2658 \beta_{19} - 2658 \beta_{18} - 93615 \beta_{17} - 208494 \beta_{16} + 203790 \beta_{14} + \cdots + 4758950668434 ) / 3981312 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 173947268 \beta_{19} + 53825420 \beta_{18} + 168102697 \beta_{17} - 121685580 \beta_{16} + \cdots + 272789760 ) / 31850496 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 30425222 \beta_{19} - 30425222 \beta_{18} + 501820483 \beta_{17} + 760239190 \beta_{16} + \cdots - 12\!\cdots\!88 ) / 7962624 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 235679122700 \beta_{19} - 87654589316 \beta_{18} - 269952899803 \beta_{17} + \cdots - 591714864384 ) / 31850496 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 52711013820 \beta_{19} + 52711013820 \beta_{18} - 545250275742 \beta_{17} - 668812440924 \beta_{16} + \cdots + 85\!\cdots\!10 ) / 3981312 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 322514248050532 \beta_{19} + 144281194783084 \beta_{18} + 430918930266593 \beta_{17} + \cdots + 11\!\cdots\!92 ) / 31850496 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 241784160829930 \beta_{19} - 241784160829930 \beta_{18} + \cdots - 24\!\cdots\!56 ) / 7962624 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 44\!\cdots\!88 \beta_{19} + \cdots - 20\!\cdots\!92 ) / 31850496 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 79\!\cdots\!54 \beta_{19} + \cdots + 60\!\cdots\!18 ) / 1327104 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 61\!\cdots\!00 \beta_{19} + \cdots + 35\!\cdots\!96 ) / 31850496 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 86\!\cdots\!26 \beta_{19} + \cdots - 54\!\cdots\!04 ) / 7962624 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 85\!\cdots\!24 \beta_{19} + \cdots - 61\!\cdots\!04 ) / 31850496 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 75\!\cdots\!12 \beta_{19} + \cdots + 41\!\cdots\!22 ) / 3981312 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 11\!\cdots\!40 \beta_{19} + \cdots + 10\!\cdots\!68 ) / 31850496 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( - 25\!\cdots\!14 \beta_{19} + \cdots - 12\!\cdots\!20 ) / 7962624 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( - 16\!\cdots\!96 \beta_{19} + \cdots - 17\!\cdots\!48 ) / 31850496 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/192\mathbb{Z}\right)^\times\).

\(n\) \(65\) \(127\) \(133\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
− 37.5624i
37.5624i
23.2047i
− 23.2047i
− 26.0163i
26.0163i
32.2047i
− 32.2047i
− 38.9030i
38.9030i
− 36.0746i
36.0746i
29.3763i
− 29.3763i
− 23.1879i
23.1879i
26.0332i
− 26.0332i
− 40.3908i
40.3908i
0 −234.518 − 63.6437i 0 1237.30i 0 6835.61 0 50948.0 + 29851.1i 0
65.2 0 −234.518 + 63.6437i 0 − 1237.30i 0 6835.61 0 50948.0 − 29851.1i 0
65.3 0 −197.704 − 141.287i 0 5217.91i 0 −7041.29 0 19125.0 + 55866.1i 0
65.4 0 −197.704 + 141.287i 0 − 5217.91i 0 −7041.29 0 19125.0 − 55866.1i 0
65.5 0 −168.749 − 174.851i 0 − 5349.27i 0 −28157.9 0 −2096.84 + 59011.8i 0
65.6 0 −168.749 + 174.851i 0 5349.27i 0 −28157.9 0 −2096.84 − 59011.8i 0
65.7 0 −108.418 − 217.473i 0 − 1857.49i 0 7904.44 0 −35540.0 + 47156.0i 0
65.8 0 −108.418 + 217.473i 0 1857.49i 0 7904.44 0 −35540.0 − 47156.0i 0
65.9 0 −27.7660 − 241.408i 0 − 1453.33i 0 28314.1 0 −57507.1 + 13405.9i 0
65.10 0 −27.7660 + 241.408i 0 1453.33i 0 28314.1 0 −57507.1 − 13405.9i 0
65.11 0 27.7660 − 241.408i 0 1453.33i 0 −28314.1 0 −57507.1 − 13405.9i 0
65.12 0 27.7660 + 241.408i 0 − 1453.33i 0 −28314.1 0 −57507.1 + 13405.9i 0
65.13 0 108.418 − 217.473i 0 1857.49i 0 −7904.44 0 −35540.0 − 47156.0i 0
65.14 0 108.418 + 217.473i 0 − 1857.49i 0 −7904.44 0 −35540.0 + 47156.0i 0
65.15 0 168.749 − 174.851i 0 5349.27i 0 28157.9 0 −2096.84 − 59011.8i 0
65.16 0 168.749 + 174.851i 0 − 5349.27i 0 28157.9 0 −2096.84 + 59011.8i 0
65.17 0 197.704 − 141.287i 0 − 5217.91i 0 7041.29 0 19125.0 − 55866.1i 0
65.18 0 197.704 + 141.287i 0 5217.91i 0 7041.29 0 19125.0 + 55866.1i 0
65.19 0 234.518 − 63.6437i 0 − 1237.30i 0 −6835.61 0 50948.0 − 29851.1i 0
65.20 0 234.518 + 63.6437i 0 1237.30i 0 −6835.61 0 50948.0 + 29851.1i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 65.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 192.11.e.k 20
3.b odd 2 1 inner 192.11.e.k 20
4.b odd 2 1 inner 192.11.e.k 20
8.b even 2 1 96.11.e.a ✓ 20
8.d odd 2 1 96.11.e.a ✓ 20
12.b even 2 1 inner 192.11.e.k 20
24.f even 2 1 96.11.e.a ✓ 20
24.h odd 2 1 96.11.e.a ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
96.11.e.a ✓ 20 8.b even 2 1
96.11.e.a ✓ 20 8.d odd 2 1
96.11.e.a ✓ 20 24.f even 2 1
96.11.e.a ✓ 20 24.h odd 2 1
192.11.e.k 20 1.a even 1 1 trivial
192.11.e.k 20 3.b odd 2 1 inner
192.11.e.k 20 4.b odd 2 1 inner
192.11.e.k 20 12.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{11}^{\mathrm{new}}(192, [\chi])\):

\( T_{5}^{10} + 62934568 T_{5}^{8} + \cdots + 86\!\cdots\!00 \) Copy content Toggle raw display
\( T_{7}^{10} - 1753339932 T_{7}^{8} + \cdots - 92\!\cdots\!68 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} + \cdots + 51\!\cdots\!01 \) Copy content Toggle raw display
$5$ \( (T^{10} + \cdots + 86\!\cdots\!00)^{2} \) Copy content Toggle raw display
$7$ \( (T^{10} + \cdots - 92\!\cdots\!68)^{2} \) Copy content Toggle raw display
$11$ \( (T^{10} + \cdots + 10\!\cdots\!56)^{2} \) Copy content Toggle raw display
$13$ \( (T^{5} + \cdots - 17\!\cdots\!56)^{4} \) Copy content Toggle raw display
$17$ \( (T^{10} + \cdots + 46\!\cdots\!52)^{2} \) Copy content Toggle raw display
$19$ \( (T^{10} + \cdots - 57\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( (T^{10} + \cdots + 26\!\cdots\!00)^{2} \) Copy content Toggle raw display
$29$ \( (T^{10} + \cdots + 52\!\cdots\!28)^{2} \) Copy content Toggle raw display
$31$ \( (T^{10} + \cdots - 23\!\cdots\!92)^{2} \) Copy content Toggle raw display
$37$ \( (T^{5} + \cdots + 76\!\cdots\!92)^{4} \) Copy content Toggle raw display
$41$ \( (T^{10} + \cdots + 15\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( (T^{10} + \cdots - 22\!\cdots\!48)^{2} \) Copy content Toggle raw display
$47$ \( (T^{10} + \cdots + 22\!\cdots\!16)^{2} \) Copy content Toggle raw display
$53$ \( (T^{10} + \cdots + 22\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{10} + \cdots + 34\!\cdots\!16)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} + \cdots + 11\!\cdots\!48)^{4} \) Copy content Toggle raw display
$67$ \( (T^{10} + \cdots - 25\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots + 36\!\cdots\!24)^{2} \) Copy content Toggle raw display
$73$ \( (T^{5} + \cdots - 65\!\cdots\!00)^{4} \) Copy content Toggle raw display
$79$ \( (T^{10} + \cdots - 20\!\cdots\!52)^{2} \) Copy content Toggle raw display
$83$ \( (T^{10} + \cdots + 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots + 48\!\cdots\!92)^{2} \) Copy content Toggle raw display
$97$ \( (T^{5} + \cdots - 29\!\cdots\!00)^{4} \) Copy content Toggle raw display
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