Properties

Label 192.9.e.i
Level $192$
Weight $9$
Character orbit 192.e
Analytic conductor $78.217$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} - 78x^{6} + 144x^{5} + 2079x^{4} + 936x^{3} - 658x^{2} + 2884x + 30633 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{42}\cdot 3^{10} \)
Twist minimal: no (minimal twist has level 24)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} - 7) q^{3} + (\beta_{3} - \beta_{2}) q^{5} + (\beta_1 + 198) q^{7} + ( - \beta_{5} + 2 \beta_{3} + \cdots + 41) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} - 7) q^{3} + (\beta_{3} - \beta_{2}) q^{5} + (\beta_1 + 198) q^{7} + ( - \beta_{5} + 2 \beta_{3} + \cdots + 41) q^{9}+ \cdots + (17025 \beta_{7} - 31716 \beta_{6} + \cdots + 46212688) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 56 q^{3} + 1584 q^{7} + 328 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 56 q^{3} + 1584 q^{7} + 328 q^{9} - 25232 q^{13} + 38336 q^{15} - 157936 q^{19} - 30480 q^{21} - 579704 q^{25} + 276040 q^{27} + 805552 q^{31} + 102848 q^{33} + 3985008 q^{37} - 2297104 q^{39} - 6962672 q^{43} - 8670592 q^{45} - 5884520 q^{49} + 15590144 q^{51} + 27101312 q^{55} + 36756688 q^{57} + 51583600 q^{61} - 69759312 q^{63} - 58200688 q^{67} - 94226048 q^{69} - 116854768 q^{73} + 143181896 q^{75} + 172454576 q^{79} + 194700040 q^{81} + 264333824 q^{85} - 242851008 q^{87} - 382128480 q^{91} - 313470352 q^{93} - 337326704 q^{97} + 369701504 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} - 78x^{6} + 144x^{5} + 2079x^{4} + 936x^{3} - 658x^{2} + 2884x + 30633 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 14134 \nu^{7} - 759906 \nu^{6} + 8175192 \nu^{5} + 32119854 \nu^{4} - 356737260 \nu^{3} + \cdots - 10631678772 ) / 6908733 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 90977 \nu^{7} + 306225 \nu^{6} + 8157405 \nu^{5} - 15590670 \nu^{4} - 246555357 \nu^{3} + \cdots + 77404449 ) / 41452398 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1348973 \nu^{7} - 7217253 \nu^{6} - 97646277 \nu^{5} + 350373210 \nu^{4} + 2225360013 \nu^{3} + \cdots + 7288059399 ) / 41452398 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 2071745 \nu^{7} + 76404537 \nu^{6} - 163202991 \nu^{5} - 5405297226 \nu^{4} + \cdots - 270823955163 ) / 41452398 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1067294 \nu^{7} - 14014314 \nu^{6} - 49856268 \nu^{5} + 883003110 \nu^{4} + 1518931968 \nu^{3} + \cdots - 41475425052 ) / 20726199 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 580555 \nu^{7} - 2572731 \nu^{6} - 41619615 \nu^{5} + 91789914 \nu^{4} + 1034741919 \nu^{3} + \cdots + 2039309301 ) / 6908733 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 17611 \nu^{7} + 68427 \nu^{6} + 1422267 \nu^{5} - 2154414 \nu^{4} - 41574963 \nu^{3} + \cdots - 10880913 ) / 170586 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 16\beta_{7} + 23\beta_{6} + 16\beta_{5} + 4\beta_{4} + 4\beta_{3} + 494\beta_{2} - 24\beta _1 + 13824 ) / 27648 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 14\beta_{7} + 5\beta_{6} + 10\beta_{5} + 4\beta_{4} - 2\beta_{3} - 370\beta_{2} + 18\beta _1 + 74304 ) / 3456 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 296 \beta_{7} + 583 \beta_{6} + 296 \beta_{5} + 56 \beta_{4} - 1024 \beta_{3} - 914 \beta_{2} + \cdots + 654336 ) / 9216 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3056 \beta_{7} + 2423 \beta_{6} + 1192 \beta_{5} + 358 \beta_{4} - 2186 \beta_{3} - 64282 \beta_{2} + \cdots + 5871744 ) / 6912 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 21170 \beta_{7} + 36652 \beta_{6} + 9650 \beta_{5} + 1715 \beta_{4} - 60295 \beta_{3} + \cdots + 29137536 ) / 6912 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 150456 \beta_{7} + 168071 \beta_{6} + 32536 \beta_{5} + 9064 \beta_{4} - 202784 \beta_{3} + \cdots + 130065408 ) / 4608 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 6424576 \beta_{7} + 10532849 \beta_{6} + 1181248 \beta_{5} + 172084 \beta_{4} - 15833948 \beta_{3} + \cdots + 4132200960 ) / 27648 \) 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
−1.50551 1.41421i
−1.50551 + 1.41421i
−5.49843 1.41421i
−5.49843 + 1.41421i
7.73966 + 1.41421i
7.73966 1.41421i
1.26427 1.41421i
1.26427 + 1.41421i
0 −80.7366 6.52720i 0 920.542i 0 −2012.50 0 6475.79 + 1053.97i 0
65.2 0 −80.7366 + 6.52720i 0 920.542i 0 −2012.50 0 6475.79 1053.97i 0
65.3 0 −28.6420 75.7670i 0 868.404i 0 3909.88 0 −4920.27 + 4340.23i 0
65.4 0 −28.6420 + 75.7670i 0 868.404i 0 3909.88 0 −4920.27 4340.23i 0
65.5 0 4.95410 80.8484i 0 404.296i 0 −262.245 0 −6511.91 801.062i 0
65.6 0 4.95410 + 80.8484i 0 404.296i 0 −262.245 0 −6511.91 + 801.062i 0
65.7 0 76.4245 26.8384i 0 295.589i 0 −843.136 0 5120.40 4102.23i 0
65.8 0 76.4245 + 26.8384i 0 295.589i 0 −843.136 0 5120.40 + 4102.23i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 65.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 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.i 8
3.b odd 2 1 inner 192.9.e.i 8
4.b odd 2 1 192.9.e.j 8
8.b even 2 1 24.9.e.a 8
8.d odd 2 1 48.9.e.e 8
12.b even 2 1 192.9.e.j 8
24.f even 2 1 48.9.e.e 8
24.h odd 2 1 24.9.e.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.9.e.a 8 8.b even 2 1
24.9.e.a 8 24.h odd 2 1
48.9.e.e 8 8.d odd 2 1
48.9.e.e 8 24.f even 2 1
192.9.e.i 8 1.a even 1 1 trivial
192.9.e.i 8 3.b odd 2 1 inner
192.9.e.j 8 4.b odd 2 1
192.9.e.j 8 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{9}^{\mathrm{new}}(192, [\chi])\):

\( T_{5}^{8} + 1852352T_{5}^{6} + 1055033650176T_{5}^{4} + 183162750181376000T_{5}^{2} + 9126569679992258560000 \) Copy content Toggle raw display
\( T_{7}^{4} - 792T_{7}^{3} - 9744840T_{7}^{2} - 9117353952T_{7} - 1739819024496 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + \cdots + 18\!\cdots\!41 \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 91\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{4} + \cdots - 1739819024496)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 54\!\cdots\!24 \) Copy content Toggle raw display
$13$ \( (T^{4} + \cdots + 50\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 68\!\cdots\!36 \) Copy content Toggle raw display
$19$ \( (T^{4} + \cdots + 14\!\cdots\!56)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 10\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 20\!\cdots\!76 \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots - 24\!\cdots\!88)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots + 13\!\cdots\!76)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 71\!\cdots\!76 \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots + 40\!\cdots\!44)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 14\!\cdots\!56 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 12\!\cdots\!44 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 36\!\cdots\!44 \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots - 77\!\cdots\!96)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots + 23\!\cdots\!36)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 11\!\cdots\!96 \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots + 58\!\cdots\!04)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots - 61\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 98\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 25\!\cdots\!36 \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots - 90\!\cdots\!04)^{2} \) Copy content Toggle raw display
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