Properties

Label 192.9.e.l
Level $192$
Weight $9$
Character orbit 192.e
Analytic conductor $78.217$
Analytic rank $0$
Dimension $16$
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [192,9,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, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("192.65");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 192 = 2^{6} \cdot 3 \)
Weight: \( k \) \(=\) \( 9 \)
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: \(78.2166931317\)
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: \( x^{16} - 8 x^{15} + 1632 x^{14} - 6200 x^{13} + 1101040 x^{12} - 214728 x^{11} + 414852536 x^{10} + \cdots + 46\!\cdots\!49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{110}\cdot 3^{24}\cdot 7^{2} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} - \beta_{3} q^{5} + (\beta_{2} - \beta_1) q^{7} + ( - \beta_{6} - \beta_{3} + 197) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} - \beta_{3} q^{5} + (\beta_{2} - \beta_1) q^{7} + ( - \beta_{6} - \beta_{3} + 197) q^{9} + (\beta_{12} + \beta_{5} + 14 \beta_1) q^{11} + ( - \beta_{10} - 3154) q^{13} + (\beta_{12} - \beta_{8} - 2 \beta_{5} + \cdots + \beta_1) q^{15}+ \cdots + ( - 201 \beta_{13} + \cdots - 89296 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 3152 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 3152 q^{9} - 50464 q^{13} + 62048 q^{21} - 89200 q^{25} + 1439936 q^{33} - 5092384 q^{37} - 6818176 q^{45} + 16955568 q^{49} + 6929632 q^{57} - 36642336 q^{61} - 45119616 q^{69} + 91112736 q^{73} + 52195088 q^{81} - 13526016 q^{85} + 54214624 q^{93} - 77696096 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{15} + 1632 x^{14} - 6200 x^{13} + 1101040 x^{12} - 214728 x^{11} + 414852536 x^{10} + \cdots + 46\!\cdots\!49 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 75\!\cdots\!28 \nu^{15} + \cdots - 57\!\cdots\!61 ) / 13\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 29\!\cdots\!96 \nu^{15} + \cdots - 45\!\cdots\!27 ) / 45\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 71\!\cdots\!26 \nu^{15} + \cdots + 34\!\cdots\!11 ) / 50\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 55\!\cdots\!60 \nu^{15} + \cdots + 29\!\cdots\!73 ) / 20\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 20\!\cdots\!64 \nu^{15} + \cdots + 26\!\cdots\!74 ) / 67\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 11\!\cdots\!97 \nu^{15} + \cdots + 69\!\cdots\!42 ) / 25\!\cdots\!94 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 12\!\cdots\!12 \nu^{15} + \cdots - 96\!\cdots\!39 ) / 68\!\cdots\!26 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 83\!\cdots\!92 \nu^{15} + \cdots + 41\!\cdots\!11 ) / 40\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 53\!\cdots\!96 \nu^{15} + \cdots - 92\!\cdots\!71 ) / 20\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 24\!\cdots\!35 \nu^{15} + \cdots - 96\!\cdots\!77 ) / 86\!\cdots\!89 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 51\!\cdots\!24 \nu^{15} + \cdots - 50\!\cdots\!63 ) / 16\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 32\!\cdots\!44 \nu^{15} + \cdots + 76\!\cdots\!52 ) / 10\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 99\!\cdots\!88 \nu^{15} + \cdots + 15\!\cdots\!75 ) / 22\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 12\!\cdots\!03 \nu^{15} + \cdots + 22\!\cdots\!12 ) / 25\!\cdots\!94 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 71\!\cdots\!86 \nu^{15} + \cdots - 44\!\cdots\!86 ) / 12\!\cdots\!97 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( - 125 \beta_{15} - 361 \beta_{14} + 121 \beta_{11} - 216 \beta_{10} + 125 \beta_{7} + \cdots + 1161216 ) / 2322432 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 70 \beta_{15} - 3101 \beta_{14} - 504 \beta_{13} - 856 \beta_{12} - 1939 \beta_{11} + \cdots - 464486400 ) / 2322432 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 6443 \beta_{15} + 36927 \beta_{14} + 1008 \beta_{13} + 392 \beta_{12} - 22583 \beta_{11} + \cdots - 970389504 ) / 774144 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 49000 \beta_{15} + 1709771 \beta_{14} + 207648 \beta_{13} + 299632 \beta_{12} + \cdots + 102679363584 ) / 2322432 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 512989 \beta_{15} - 20321284 \beta_{14} - 2478420 \beta_{13} - 465920 \beta_{12} + \cdots + 1569941968896 ) / 2322432 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 7823102 \beta_{15} - 183481214 \beta_{14} - 42685776 \beta_{13} + 52973208 \beta_{12} + \cdots - 7565010259968 ) / 774144 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 1962218005 \beta_{15} + 11168587 \beta_{14} + 1532670804 \beta_{13} + 219752456 \beta_{12} + \cdots - 540357140219904 ) / 2322432 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 144288658 \beta_{15} + 3149417332 \beta_{14} + 2889880992 \beta_{13} - 3068840872 \beta_{12} + \cdots + 115644902999040 ) / 82944 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 391907775589 \beta_{15} + 560358969649 \beta_{14} - 223964863920 \beta_{13} - 271440560160 \beta_{12} + \cdots + 42\!\cdots\!56 ) / 774144 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 11203839587290 \beta_{15} + 30088288859027 \beta_{14} - 43640997828120 \beta_{13} + \cdots + 27\!\cdots\!80 ) / 2322432 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 525940495735187 \beta_{15} - 639926092844923 \beta_{14} + 184683968541072 \beta_{13} + \cdots - 63\!\cdots\!04 ) / 2322432 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 29\!\cdots\!96 \beta_{15} + \cdots - 10\!\cdots\!44 ) / 774144 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 19\!\cdots\!65 \beta_{15} + \cdots - 15\!\cdots\!24 ) / 2322432 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 50\!\cdots\!34 \beta_{15} + \cdots + 12\!\cdots\!16 ) / 2322432 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 17\!\cdots\!11 \beta_{15} + \cdots + 41\!\cdots\!36 ) / 774144 \) Copy content Toggle raw display

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
−0.195942 20.0975i
−0.195942 + 20.0975i
3.86965 9.59065i
3.86965 + 9.59065i
5.31078 19.7044i
5.31078 + 19.7044i
−6.98449 7.49075i
−6.98449 + 7.49075i
−6.98449 + 10.3192i
−6.98449 10.3192i
5.31078 + 16.8759i
5.31078 16.8759i
3.86965 + 12.4191i
3.86965 12.4191i
−0.195942 + 17.2691i
−0.195942 17.2691i
0 −80.3709 10.0760i 0 969.842i 0 −878.697 0 6357.95 + 1619.63i 0
65.2 0 −80.3709 + 10.0760i 0 969.842i 0 −878.697 0 6357.95 1619.63i 0
65.3 0 −69.8984 40.9294i 0 427.907i 0 2483.92 0 3210.56 + 5721.80i 0
65.4 0 −69.8984 + 40.9294i 0 427.907i 0 2483.92 0 3210.56 5721.80i 0
65.5 0 −41.6091 69.4959i 0 590.447i 0 −595.059 0 −3098.37 + 5783.32i 0
65.6 0 −41.6091 + 69.4959i 0 590.447i 0 −595.059 0 −3098.37 5783.32i 0
65.7 0 −20.9626 78.2405i 0 335.371i 0 −4472.36 0 −5682.14 + 3280.25i 0
65.8 0 −20.9626 + 78.2405i 0 335.371i 0 −4472.36 0 −5682.14 3280.25i 0
65.9 0 20.9626 78.2405i 0 335.371i 0 4472.36 0 −5682.14 3280.25i 0
65.10 0 20.9626 + 78.2405i 0 335.371i 0 4472.36 0 −5682.14 + 3280.25i 0
65.11 0 41.6091 69.4959i 0 590.447i 0 595.059 0 −3098.37 5783.32i 0
65.12 0 41.6091 + 69.4959i 0 590.447i 0 595.059 0 −3098.37 + 5783.32i 0
65.13 0 69.8984 40.9294i 0 427.907i 0 −2483.92 0 3210.56 5721.80i 0
65.14 0 69.8984 + 40.9294i 0 427.907i 0 −2483.92 0 3210.56 + 5721.80i 0
65.15 0 80.3709 10.0760i 0 969.842i 0 878.697 0 6357.95 1619.63i 0
65.16 0 80.3709 + 10.0760i 0 969.842i 0 878.697 0 6357.95 + 1619.63i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 65.16
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 192.9.e.l 16
3.b odd 2 1 inner 192.9.e.l 16
4.b odd 2 1 inner 192.9.e.l 16
8.b even 2 1 96.9.e.b 16
8.d odd 2 1 96.9.e.b 16
12.b even 2 1 inner 192.9.e.l 16
24.f even 2 1 96.9.e.b 16
24.h odd 2 1 96.9.e.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
96.9.e.b 16 8.b even 2 1
96.9.e.b 16 8.d odd 2 1
96.9.e.b 16 24.f even 2 1
96.9.e.b 16 24.h odd 2 1
192.9.e.l 16 1.a even 1 1 trivial
192.9.e.l 16 3.b odd 2 1 inner
192.9.e.l 16 4.b odd 2 1 inner
192.9.e.l 16 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 \(S_{9}^{\mathrm{new}}(192, [\chi])\):

\( T_{5}^{8} + 1584800T_{5}^{6} + 729577765248T_{5}^{4} + 123476140339865600T_{5}^{2} + 6753289812415982080000 \) Copy content Toggle raw display
\( T_{7}^{8} - 27298096 T_{7}^{6} + 153158019312480 T_{7}^{4} + \cdots + 33\!\cdots\!92 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 34\!\cdots\!81 \) Copy content Toggle raw display
$5$ \( (T^{8} + \cdots + 67\!\cdots\!00)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} + \cdots + 33\!\cdots\!92)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + \cdots + 75\!\cdots\!00)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + \cdots + 41\!\cdots\!00)^{4} \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots + 39\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots + 43\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + \cdots + 10\!\cdots\!16)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots + 30\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 57\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots + 18\!\cdots\!00)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 42\!\cdots\!32)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 11\!\cdots\!52)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 51\!\cdots\!36)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 20\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots + 50\!\cdots\!20)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 33\!\cdots\!52)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 15\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots - 34\!\cdots\!00)^{4} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 10\!\cdots\!04)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 15\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 56\!\cdots\!00)^{4} \) Copy content Toggle raw display
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