Properties

Label 192.9.e.k
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: \( x^{16} + 76 x^{14} - 248 x^{13} + 4938 x^{12} - 55200 x^{11} + 274396 x^{10} + 6509208 x^{9} + \cdots + 889067248974921 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{110}\cdot 3^{30} \)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} - \beta_{3} q^{5} + (\beta_{2} - 3 \beta_1) q^{7} + (\beta_{6} - \beta_{4} - \beta_{3} - 435) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} - \beta_{3} q^{5} + (\beta_{2} - 3 \beta_1) q^{7} + (\beta_{6} - \beta_{4} - \beta_{3} - 435) q^{9} + (\beta_{11} + 23 \beta_1) q^{11} + (\beta_{5} + 6366) q^{13} + ( - \beta_{11} + \beta_{10} + \cdots - \beta_1) q^{15}+ \cdots + (768 \beta_{14} + 69 \beta_{13} + \cdots - 149349 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 6960 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 6960 q^{9} + 101856 q^{13} + 279648 q^{21} - 1773424 q^{25} + 2428992 q^{33} - 4039456 q^{37} - 10154112 q^{45} + 2562992 q^{49} - 15023520 q^{57} + 28444384 q^{61} + 47615616 q^{69} - 12069600 q^{73} + 13926672 q^{81} - 136312832 q^{85} + 24251616 q^{93} + 141063840 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 76 x^{14} - 248 x^{13} + 4938 x^{12} - 55200 x^{11} + 274396 x^{10} + 6509208 x^{9} + \cdots + 889067248974921 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 69\!\cdots\!29 \nu^{15} + \cdots - 14\!\cdots\!98 ) / 93\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 37\!\cdots\!27 \nu^{15} + \cdots - 17\!\cdots\!14 ) / 93\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 27\!\cdots\!49 \nu^{15} + \cdots + 33\!\cdots\!83 ) / 17\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 78\!\cdots\!84 \nu^{15} + \cdots + 10\!\cdots\!48 ) / 42\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 10\!\cdots\!80 \nu^{15} + \cdots - 38\!\cdots\!60 ) / 33\!\cdots\!95 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 13\!\cdots\!42 \nu^{15} + \cdots - 20\!\cdots\!90 ) / 35\!\cdots\!15 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 24\!\cdots\!92 \nu^{15} + \cdots + 56\!\cdots\!34 ) / 46\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 10\!\cdots\!76 \nu^{15} + \cdots + 28\!\cdots\!68 ) / 17\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 53\!\cdots\!51 \nu^{15} + \cdots + 81\!\cdots\!08 ) / 46\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 85\!\cdots\!49 \nu^{15} + \cdots + 17\!\cdots\!08 ) / 46\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 24\!\cdots\!11 \nu^{15} + \cdots - 48\!\cdots\!22 ) / 93\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 15\!\cdots\!72 \nu^{15} + \cdots + 27\!\cdots\!04 ) / 58\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 17\!\cdots\!91 \nu^{15} + \cdots + 55\!\cdots\!74 ) / 62\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 35\!\cdots\!73 \nu^{15} + \cdots + 65\!\cdots\!66 ) / 93\!\cdots\!85 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 79\!\cdots\!67 \nu^{15} + \cdots - 88\!\cdots\!29 ) / 17\!\cdots\!75 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 26 \beta_{14} + 11 \beta_{13} - 2 \beta_{10} + 11 \beta_{9} - 295 \beta_{7} + 162 \beta_{4} + \cdots - 1424 \beta_1 ) / 331776 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 144 \beta_{15} - 53 \beta_{14} - 11 \beta_{13} - 48 \beta_{12} + 243 \beta_{11} + 29 \beta_{10} + \cdots - 6303744 ) / 663552 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 54 \beta_{15} - 8 \beta_{14} + 736 \beta_{13} + 198 \beta_{12} - 378 \beta_{11} - 1258 \beta_{10} + \cdots + 7713792 ) / 165888 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 50616 \beta_{15} - 7886 \beta_{14} + 10822 \beta_{13} + 16872 \beta_{12} + 25650 \beta_{11} + \cdots - 340070400 ) / 663552 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 291600 \beta_{15} + 284170 \beta_{14} - 541646 \beta_{13} - 499878 \beta_{11} + \cdots + 7537950720 ) / 663552 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 2213064 \beta_{15} + 2377397 \beta_{14} + 106673 \beta_{13} - 3079608 \beta_{12} + \cdots - 1826592768 ) / 331776 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 24709104 \beta_{15} - 43501237 \beta_{14} + 1450013 \beta_{13} + 1901592 \beta_{12} + \cdots - 1523600160768 ) / 331776 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 385661880 \beta_{15} - 566122184 \beta_{14} - 158624264 \beta_{13} + 591594072 \beta_{12} + \cdots + 29862138212352 ) / 663552 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 2565577584 \beta_{15} + 39147358 \beta_{14} + 6237097804 \beta_{13} - 708689376 \beta_{12} + \cdots + 351689641426944 ) / 663552 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 145771209936 \beta_{15} + 12649855435 \beta_{14} - 26493212831 \beta_{13} + \cdots + 11\!\cdots\!76 ) / 663552 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 322961764356 \beta_{15} + 1019953767314 \beta_{14} - 108286243708 \beta_{13} + \cdots + 11\!\cdots\!32 ) / 331776 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 12926986326000 \beta_{15} - 2944971881701 \beta_{14} + 4960966817021 \beta_{13} + \cdots - 62\!\cdots\!32 ) / 331776 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 33911563300056 \beta_{15} - 141010275783499 \beta_{14} - 31828334548342 \beta_{13} + \cdots + 40\!\cdots\!68 ) / 331776 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 654737630093136 \beta_{15} + \cdots + 16\!\cdots\!52 ) / 663552 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 29\!\cdots\!02 \beta_{15} + \cdots - 19\!\cdots\!88 ) / 165888 \) 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
−2.48948 + 10.2518i
−2.48948 10.2518i
−7.66393 + 8.82528i
−7.66393 8.82528i
−5.52055 0.0941314i
−5.52055 + 0.0941314i
7.58882 + 0.912295i
7.58882 0.912295i
−7.58882 + 2.32651i
−7.58882 2.32651i
5.52055 + 1.32008i
5.52055 1.32008i
7.66393 + 7.41107i
7.66393 7.41107i
2.48948 + 11.6660i
2.48948 11.6660i
0 −75.2171 30.0563i 0 517.842i 0 −3763.79 0 4754.24 + 4521.50i 0
65.2 0 −75.2171 + 30.0563i 0 517.842i 0 −3763.79 0 4754.24 4521.50i 0
65.3 0 −70.4557 39.9624i 0 249.117i 0 2035.26 0 3367.02 + 5631.16i 0
65.4 0 −70.4557 + 39.9624i 0 249.117i 0 2035.26 0 3367.02 5631.16i 0
65.5 0 −40.3363 70.2423i 0 1140.75i 0 1656.42 0 −3306.96 + 5666.63i 0
65.6 0 −40.3363 + 70.2423i 0 1140.75i 0 1656.42 0 −3306.96 5666.63i 0
65.7 0 −1.83135 80.9793i 0 611.814i 0 1627.21 0 −6554.29 + 296.603i 0
65.8 0 −1.83135 + 80.9793i 0 611.814i 0 1627.21 0 −6554.29 296.603i 0
65.9 0 1.83135 80.9793i 0 611.814i 0 −1627.21 0 −6554.29 296.603i 0
65.10 0 1.83135 + 80.9793i 0 611.814i 0 −1627.21 0 −6554.29 + 296.603i 0
65.11 0 40.3363 70.2423i 0 1140.75i 0 −1656.42 0 −3306.96 5666.63i 0
65.12 0 40.3363 + 70.2423i 0 1140.75i 0 −1656.42 0 −3306.96 + 5666.63i 0
65.13 0 70.4557 39.9624i 0 249.117i 0 −2035.26 0 3367.02 5631.16i 0
65.14 0 70.4557 + 39.9624i 0 249.117i 0 −2035.26 0 3367.02 + 5631.16i 0
65.15 0 75.2171 30.0563i 0 517.842i 0 3763.79 0 4754.24 4521.50i 0
65.16 0 75.2171 + 30.0563i 0 517.842i 0 3763.79 0 4754.24 + 4521.50i 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.k 16
3.b odd 2 1 inner 192.9.e.k 16
4.b odd 2 1 inner 192.9.e.k 16
8.b even 2 1 96.9.e.a 16
8.d odd 2 1 96.9.e.a 16
12.b even 2 1 inner 192.9.e.k 16
24.f even 2 1 96.9.e.a 16
24.h odd 2 1 96.9.e.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
96.9.e.a 16 8.b even 2 1
96.9.e.a 16 8.d odd 2 1
96.9.e.a 16 24.f even 2 1
96.9.e.a 16 24.h odd 2 1
192.9.e.k 16 1.a even 1 1 trivial
192.9.e.k 16 3.b odd 2 1 inner
192.9.e.k 16 4.b odd 2 1 inner
192.9.e.k 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 \(S_{9}^{\mathrm{new}}(192, [\chi])\):

\( T_{5}^{8} + 2005856T_{5}^{6} + 1057075569024T_{5}^{4} + 188737602250496000T_{5}^{2} + 8106353661468797440000 \) Copy content Toggle raw display
\( T_{7}^{8} - 23699952 T_{7}^{6} + 164655419114592 T_{7}^{4} + \cdots + 42\!\cdots\!04 \) 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
$5$ \( (T^{8} + \cdots + 81\!\cdots\!00)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} + \cdots + 42\!\cdots\!04)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + \cdots + 49\!\cdots\!84)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + \cdots + 36\!\cdots\!24)^{4} \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots + 14\!\cdots\!76)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots + 39\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + \cdots + 30\!\cdots\!00)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots + 46\!\cdots\!76)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 38\!\cdots\!84)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots + 42\!\cdots\!76)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 34\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 73\!\cdots\!84)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 95\!\cdots\!64)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 14\!\cdots\!24)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots + 24\!\cdots\!04)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 94\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 37\!\cdots\!44)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots + 15\!\cdots\!00)^{4} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 32\!\cdots\!84)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 46\!\cdots\!16)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 26\!\cdots\!00)^{4} \) Copy content Toggle raw display
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