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Properties

Label 192.9.g.e
Level $192$
Weight $9$
Character orbit 192.g
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(127,192)]
 
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([1, 0, 0]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("192.127");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 192 = 2^{6} \cdot 3 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 192.g (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: \( x^{8} - 3x^{7} - 40x^{6} - 395x^{5} + 403x^{4} + 8998x^{3} + 74584x^{2} + 217224x + 269328 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{52}\cdot 3^{10} \)
Twist minimal: no (minimal twist has level 12)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + (\beta_{2} + 42) q^{5} + ( - \beta_{6} - 7 \beta_1) q^{7} - 2187 q^{9} + (\beta_{6} + \beta_{3} - 53 \beta_1) q^{11} + (\beta_{5} - 2 \beta_{4} + 4 \beta_{2} + 358) q^{13} + ( - \beta_{7} + 2 \beta_{6} + \cdots - 41 \beta_1) q^{15}+ \cdots + ( - 2187 \beta_{6} + \cdots + 115911 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 336 q^{5} - 17496 q^{9} + 2864 q^{13} - 193200 q^{17} - 121824 q^{21} - 579048 q^{25} - 2063472 q^{29} - 920160 q^{33} - 7470352 q^{37} - 8865456 q^{41} - 734832 q^{45} - 18923896 q^{49} - 8706672 q^{53}+ \cdots - 123291632 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} - 40x^{6} - 395x^{5} + 403x^{4} + 8998x^{3} + 74584x^{2} + 217224x + 269328 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 132651 \nu^{7} - 1870587 \nu^{6} - 11854782 \nu^{5} - 35450757 \nu^{4} + 699206391 \nu^{3} + \cdots + 27533830392 ) / 288974000 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 45295091 \nu^{7} - 142748667 \nu^{6} - 2981454662 \nu^{5} - 5512648637 \nu^{4} + \cdots + 7731529798072 ) / 13999758250 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2560781331 \nu^{7} - 371896053747 \nu^{6} - 505291504542 \nu^{5} + 723699624483 \nu^{4} + \cdots + 41\!\cdots\!52 ) / 195996615500 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 400898393 \nu^{7} - 8193891441 \nu^{6} + 15030256174 \nu^{5} + 271224370249 \nu^{4} + \cdots - 100807526678744 ) / 13999758250 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 450113942 \nu^{7} - 2257851546 \nu^{6} + 42760953644 \nu^{5} + 243321179594 \nu^{4} + \cdots - 224223260670064 ) / 6999879125 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 16710368107 \nu^{7} - 23662158459 \nu^{6} - 300166613374 \nu^{5} - 5698255258949 \nu^{4} + \cdots + 816278663265144 ) / 156797292400 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 50408211761 \nu^{7} - 329076666657 \nu^{6} - 1378090939802 \nu^{5} - 13686907413727 \nu^{4} + \cdots + 31\!\cdots\!12 ) / 156797292400 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - 38 \beta_{7} + 103 \beta_{6} + 42 \beta_{5} - 27 \beta_{4} + 25 \beta_{3} + 1209 \beta_{2} + \cdots + 82944 ) / 221184 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 182 \beta_{7} - 391 \beta_{6} + 210 \beta_{5} - 351 \beta_{4} - 25 \beta_{3} + 6693 \beta_{2} + \cdots + 2460672 ) / 221184 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 10 \beta_{7} + 101 \beta_{6} + 594 \beta_{5} - 423 \beta_{4} - 181 \beta_{3} + 6525 \beta_{2} + \cdots + 10865664 ) / 55296 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 3242 \beta_{7} + 7969 \beta_{6} + 18846 \beta_{5} - 12321 \beta_{4} + 1375 \beta_{3} + \cdots + 217009152 ) / 221184 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 21066 \beta_{7} + 8439 \beta_{6} + 144942 \beta_{5} - 146097 \beta_{4} + 10473 \beta_{3} + \cdots + 2084189184 ) / 221184 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 17266 \beta_{7} + 88721 \beta_{6} + 308556 \beta_{5} - 273546 \beta_{4} - 48001 \beta_{3} + \cdots + 4497527808 ) / 55296 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 231486 \beta_{7} + 2673645 \beta_{6} + 10260582 \beta_{5} - 8117037 \beta_{4} - 484653 \beta_{3} + \cdots + 135172979712 ) / 221184 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
−3.64622 + 4.31154i
−1.11858 − 4.64627i
−1.97054 + 1.25304i
8.23534 − 0.0522875i
−3.64622 − 4.31154i
−1.11858 + 4.64627i
−1.97054 − 1.25304i
8.23534 + 0.0522875i
0 − 46.7654i 0 −904.196 0 − 888.085i 0 −2187.00 0
127.2 0 − 46.7654i 0 159.249 0 707.133i 0 −2187.00 0
127.3 0 − 46.7654i 0 374.901 0 − 4472.52i 0 −2187.00 0
127.4 0 − 46.7654i 0 538.046 0 3350.97i 0 −2187.00 0
127.5 0 46.7654i 0 −904.196 0 888.085i 0 −2187.00 0
127.6 0 46.7654i 0 159.249 0 − 707.133i 0 −2187.00 0
127.7 0 46.7654i 0 374.901 0 4472.52i 0 −2187.00 0
127.8 0 46.7654i 0 538.046 0 − 3350.97i 0 −2187.00 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 127.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.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.g.e 8
4.b odd 2 1 inner 192.9.g.e 8
8.b even 2 1 12.9.d.a ✓ 8
8.d odd 2 1 12.9.d.a ✓ 8
24.f even 2 1 36.9.d.c 8
24.h odd 2 1 36.9.d.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.9.d.a ✓ 8 8.b even 2 1
12.9.d.a ✓ 8 8.d odd 2 1
36.9.d.c 8 24.f even 2 1
36.9.d.c 8 24.h odd 2 1
192.9.g.e 8 1.a even 1 1 trivial
192.9.g.e 8 4.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 168T_{5}^{3} - 622376T_{5}^{2} + 281724000T_{5} - 29045327600 \) acting on \(S_{9}^{\mathrm{new}}(192, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} + 2187)^{4} \) Copy content Toggle raw display
$5$ \( (T^{4} - 168 T^{3} + \cdots - 29045327600)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 88\!\cdots\!36 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 71\!\cdots\!04 \) Copy content Toggle raw display
$13$ \( (T^{4} + \cdots + 46\!\cdots\!76)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + \cdots + 31\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 20\!\cdots\!56 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 83\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots - 15\!\cdots\!24)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 17\!\cdots\!64 \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots - 48\!\cdots\!84)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots - 44\!\cdots\!92)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 21\!\cdots\!16 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots - 39\!\cdots\!44)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 15\!\cdots\!96 \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots + 34\!\cdots\!04)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 10\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 30\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots + 42\!\cdots\!28)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 29\!\cdots\!24 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 25\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots - 16\!\cdots\!88)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 30\!\cdots\!48)^{2} \) Copy content Toggle raw display
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