Properties

Label 192.9.e.c
Level $192$
Weight $9$
Character orbit 192.e
Analytic conductor $78.217$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: no (minimal twist has level 6)
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 \(\beta = 12\sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 3 \beta - 63) q^{3} + 34 \beta q^{5} - 2786 q^{7} + (378 \beta + 1377) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 3 \beta - 63) q^{3} + 34 \beta q^{5} - 2786 q^{7} + (378 \beta + 1377) q^{9} - 1322 \beta q^{11} + 13150 q^{13} + ( - 2142 \beta + 29376) q^{15} + 3912 \beta q^{17} + 144002 q^{19} + (8358 \beta + 175518) q^{21} + 2908 \beta q^{23} + 57697 q^{25} + ( - 27945 \beta + 239841) q^{27} - 36970 \beta q^{29} - 728738 q^{31} + (83286 \beta - 1142208) q^{33} - 94724 \beta q^{35} + 1964446 q^{37} + ( - 39450 \beta - 828450) q^{39} - 58108 \beta q^{41} - 78142 q^{43} + (46818 \beta - 3701376) q^{45} - 207400 \beta q^{47} + 1996995 q^{49} + ( - 246456 \beta + 3379968) q^{51} + 30762 \beta q^{53} + 12945024 q^{55} + ( - 432006 \beta - 9072126) q^{57} + 294878 \beta q^{59} - 17578274 q^{61} + ( - 1053108 \beta - 3836322) q^{63} + 447100 \beta q^{65} - 17136766 q^{67} + ( - 183204 \beta + 2512512) q^{69} + 1525620 \beta q^{71} + 28139330 q^{73} + ( - 173091 \beta - 3634911) q^{75} + 3683092 \beta q^{77} - 9182498 q^{79} + (1041012 \beta - 39254463) q^{81} + 5132914 \beta q^{83} - 38306304 q^{85} + (2329110 \beta - 31942080) q^{87} + 4787868 \beta q^{89} - 36635900 q^{91} + (2186214 \beta + 45910494) q^{93} + 4896068 \beta q^{95} - 128722558 q^{97} + ( - 1820394 \beta + 143918208) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 126 q^{3} - 5572 q^{7} + 2754 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 126 q^{3} - 5572 q^{7} + 2754 q^{9} + 26300 q^{13} + 58752 q^{15} + 288004 q^{19} + 351036 q^{21} + 115394 q^{25} + 479682 q^{27} - 1457476 q^{31} - 2284416 q^{33} + 3928892 q^{37} - 1656900 q^{39} - 156284 q^{43} - 7402752 q^{45} + 3993990 q^{49} + 6759936 q^{51} + 25890048 q^{55} - 18144252 q^{57} - 35156548 q^{61} - 7672644 q^{63} - 34273532 q^{67} + 5025024 q^{69} + 56278660 q^{73} - 7269822 q^{75} - 18364996 q^{79} - 78508926 q^{81} - 76612608 q^{85} - 63884160 q^{87} - 73271800 q^{91} + 91820988 q^{93} - 257445116 q^{97} + 287836416 q^{99}+O(q^{100}) \) 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
1.41421i
1.41421i
0 −63.0000 50.9117i 0 576.999i 0 −2786.00 0 1377.00 + 6414.87i 0
65.2 0 −63.0000 + 50.9117i 0 576.999i 0 −2786.00 0 1377.00 6414.87i 0
\(n\): e.g. 2-40 or 990-1000
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.c 2
3.b odd 2 1 inner 192.9.e.c 2
4.b odd 2 1 192.9.e.h 2
8.b even 2 1 48.9.e.d 2
8.d odd 2 1 6.9.b.a 2
12.b even 2 1 192.9.e.h 2
24.f even 2 1 6.9.b.a 2
24.h odd 2 1 48.9.e.d 2
40.e odd 2 1 150.9.d.a 2
40.k even 4 2 150.9.b.a 4
72.l even 6 2 162.9.d.a 4
72.p odd 6 2 162.9.d.a 4
120.m even 2 1 150.9.d.a 2
120.q odd 4 2 150.9.b.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.9.b.a 2 8.d odd 2 1
6.9.b.a 2 24.f even 2 1
48.9.e.d 2 8.b even 2 1
48.9.e.d 2 24.h odd 2 1
150.9.b.a 4 40.k even 4 2
150.9.b.a 4 120.q odd 4 2
150.9.d.a 2 40.e odd 2 1
150.9.d.a 2 120.m even 2 1
162.9.d.a 4 72.l even 6 2
162.9.d.a 4 72.p odd 6 2
192.9.e.c 2 1.a even 1 1 trivial
192.9.e.c 2 3.b odd 2 1 inner
192.9.e.h 2 4.b odd 2 1
192.9.e.h 2 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 \(S_{9}^{\mathrm{new}}(192, [\chi])\):

\( T_{5}^{2} + 332928 \) Copy content Toggle raw display
\( T_{7} + 2786 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 126T + 6561 \) Copy content Toggle raw display
$5$ \( T^{2} + 332928 \) Copy content Toggle raw display
$7$ \( (T + 2786)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 503332992 \) Copy content Toggle raw display
$13$ \( (T - 13150)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 4407478272 \) Copy content Toggle raw display
$19$ \( (T - 144002)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 2435461632 \) Copy content Toggle raw display
$29$ \( T^{2} + 393632899200 \) Copy content Toggle raw display
$31$ \( (T + 728738)^{2} \) Copy content Toggle raw display
$37$ \( (T - 1964446)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 972443423232 \) Copy content Toggle raw display
$43$ \( (T + 78142)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 12388250880000 \) Copy content Toggle raw display
$53$ \( T^{2} + 272534585472 \) Copy content Toggle raw display
$59$ \( T^{2} + 25042474046592 \) Copy content Toggle raw display
$61$ \( (T + 17578274)^{2} \) Copy content Toggle raw display
$67$ \( (T + 17136766)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 670324718707200 \) Copy content Toggle raw display
$73$ \( (T - 28139330)^{2} \) Copy content Toggle raw display
$79$ \( (T + 9182498)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 75\!\cdots\!48 \) Copy content Toggle raw display
$89$ \( T^{2} + 66\!\cdots\!12 \) Copy content Toggle raw display
$97$ \( (T + 128722558)^{2} \) Copy content Toggle raw display
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