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Properties

Label 192.8.d.d
Level $192$
Weight $8$
Character orbit 192.d
Analytic conductor $59.978$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [192,8,Mod(97,192)]
 
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, 1, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("192.97");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 192 = 2^{6} \cdot 3 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 192.d (of order \(2\), degree \(1\), 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: \(59.9779248930\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 5249 x^{10} + 20722017 x^{8} - 34316449184 x^{6} + 42622339324672 x^{4} + \cdots + 58\!\cdots\!56 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{60}\cdot 3^{4} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 27 \beta_{3} q^{3} + \beta_{2} q^{5} + ( - \beta_{9} + 2 \beta_{7}) q^{7} - 729 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 27 \beta_{3} q^{3} + \beta_{2} q^{5} + ( - \beta_{9} + 2 \beta_{7}) q^{7} - 729 q^{9} + ( - \beta_{10} + 1264 \beta_{3}) q^{11} + ( - 7 \beta_{5} - 7 \beta_{2} - 7 \beta_1) q^{13} + 27 \beta_{7} q^{15} + ( - \beta_{6} + 2 \beta_{4} + 406) q^{17} + (\beta_{11} - 6 \beta_{10} - 708 \beta_{3}) q^{19} + (27 \beta_{5} - 54 \beta_{2}) q^{21} + ( - 6 \beta_{9} - 84 \beta_{8} + 20 \beta_{7}) q^{23} + (\beta_{6} - 15 \beta_{4} - 36831) q^{25} - 19683 \beta_{3} q^{27} + ( - 42 \beta_{5} - 161 \beta_{2} - 129 \beta_1) q^{29} + (77 \beta_{9} - 221 \beta_{8} + 2 \beta_{7}) q^{31} + ( - 27 \beta_{4} - 34128) q^{33} + ( - 2 \beta_{11} + \cdots - 175188 \beta_{3}) q^{35}+ \cdots + (729 \beta_{10} - 921456 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 8748 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 8748 q^{9} + 4872 q^{17} - 441972 q^{25} - 409536 q^{33} + 356664 q^{41} + 6446076 q^{49} + 229392 q^{57} + 11543616 q^{65} + 26806872 q^{73} + 6377292 q^{81} + 45367560 q^{89} + 82412136 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 5249 x^{10} + 20722017 x^{8} - 34316449184 x^{6} + 42622339324672 x^{4} + \cdots + 58\!\cdots\!56 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 1474281714143 \nu^{10} + \cdots - 40\!\cdots\!52 ) / 13\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 19\!\cdots\!41 \nu^{10} + \cdots + 54\!\cdots\!80 ) / 17\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 19414442957 \nu^{11} + 99062097423877 \nu^{9} + \cdots + 87\!\cdots\!12 \nu ) / 95\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 203548750673 \nu^{10} - 803570331829809 \nu^{8} + \cdots + 81\!\cdots\!52 ) / 12\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 96\!\cdots\!51 \nu^{10} + \cdots - 26\!\cdots\!20 ) / 51\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3352042416341 \nu^{10} + \cdots + 21\!\cdots\!72 ) / 21\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 20\!\cdots\!47 \nu^{11} + \cdots - 24\!\cdots\!88 \nu ) / 49\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 272027128 \nu^{11} + 1430492210747 \nu^{9} + \cdots + 28\!\cdots\!32 \nu ) / 59\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 67\!\cdots\!71 \nu^{11} + \cdots + 72\!\cdots\!56 \nu ) / 89\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 29\!\cdots\!11 \nu^{11} + \cdots - 13\!\cdots\!76 \nu ) / 22\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 50\!\cdots\!77 \nu^{11} + \cdots + 22\!\cdots\!32 \nu ) / 22\!\cdots\!60 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2\beta_{11} + 6\beta_{10} + 96\beta_{9} + 3\beta_{8} + 96\beta_{7} + 512\beta_{3} ) / 3072 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 10\beta_{6} - 2784\beta_{5} - 414\beta_{4} + 6432\beta_{2} + 11865\beta _1 + 2687488 ) / 3072 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2629\beta_{11} + 8079\beta_{10} + 23286016\beta_{3} ) / 768 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 86890 \beta_{6} - 10218720 \beta_{5} + 1268670 \beta_{4} + 13973280 \beta_{2} + 31220841 \beta _1 - 7112720896 ) / 3072 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 14160842 \beta_{11} + 55133406 \beta_{10} - 755625696 \beta_{9} + 806357337 \beta_{8} + \cdots + 203750400512 \beta_{3} ) / 3072 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -193892885\beta_{6} + 1915817727\beta_{4} - 10078694514944 ) / 768 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 39952340618 \beta_{11} - 183638777118 \beta_{10} - 2300402881248 \beta_{9} + \cdots - 751792357908992 \beta_{3} ) / 3072 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 1449700881130 \beta_{6} + 114079976559840 \beta_{5} + 11764825020990 \beta_{4} + \cdots - 59\!\cdots\!56 ) / 3072 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -58512378674341\beta_{11} - 299876975856303\beta_{10} - 1295135022994944256\beta_{3} ) / 768 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 50\!\cdots\!30 \beta_{6} + \cdots + 17\!\cdots\!88 ) / 3072 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 35\!\cdots\!78 \beta_{11} + \cdots - 86\!\cdots\!52 \beta_{3} ) / 3072 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1
9.63634 + 5.56354i
−38.2732 + 22.0970i
−48.7756 − 28.1606i
48.7756 − 28.1606i
38.2732 + 22.0970i
−9.63634 + 5.56354i
−9.63634 − 5.56354i
38.2732 − 22.0970i
48.7756 + 28.1606i
−48.7756 + 28.1606i
−38.2732 − 22.0970i
9.63634 − 5.56354i
0 − 27.0000i 0 − 446.148i 0 −1632.95 0 −729.000 0
97.2 0 − 27.0000i 0 − 358.749i 0 134.638 0 −729.000 0
97.3 0 − 27.0000i 0 − 130.839i 0 1182.16 0 −729.000 0
97.4 0 − 27.0000i 0 130.839i 0 −1182.16 0 −729.000 0
97.5 0 − 27.0000i 0 358.749i 0 −134.638 0 −729.000 0
97.6 0 − 27.0000i 0 446.148i 0 1632.95 0 −729.000 0
97.7 0 27.0000i 0 − 446.148i 0 1632.95 0 −729.000 0
97.8 0 27.0000i 0 − 358.749i 0 −134.638 0 −729.000 0
97.9 0 27.0000i 0 − 130.839i 0 −1182.16 0 −729.000 0
97.10 0 27.0000i 0 130.839i 0 1182.16 0 −729.000 0
97.11 0 27.0000i 0 358.749i 0 134.638 0 −729.000 0
97.12 0 27.0000i 0 446.148i 0 −1632.95 0 −729.000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 97.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 192.8.d.d ✓ 12
3.b odd 2 1 576.8.d.i 12
4.b odd 2 1 inner 192.8.d.d ✓ 12
8.b even 2 1 inner 192.8.d.d ✓ 12
8.d odd 2 1 inner 192.8.d.d ✓ 12
12.b even 2 1 576.8.d.i 12
24.f even 2 1 576.8.d.i 12
24.h odd 2 1 576.8.d.i 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
192.8.d.d ✓ 12 1.a even 1 1 trivial
192.8.d.d ✓ 12 4.b odd 2 1 inner
192.8.d.d ✓ 12 8.b even 2 1 inner
192.8.d.d ✓ 12 8.d odd 2 1 inner
576.8.d.i 12 3.b odd 2 1
576.8.d.i 12 12.b even 2 1
576.8.d.i 12 24.f even 2 1
576.8.d.i 12 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} + 344868T_{5}^{4} + 31228398000T_{5}^{2} + 438547825080000 \) acting on \(S_{8}^{\mathrm{new}}(192, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{2} + 729)^{6} \) Copy content Toggle raw display
$5$ \( (T^{6} + \cdots + 438547825080000)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} + \cdots - 67\!\cdots\!88)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} + \cdots + 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + \cdots + 22\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( (T^{3} + \cdots + 4049573334120)^{4} \) Copy content Toggle raw display
$19$ \( (T^{6} + \cdots + 75\!\cdots\!36)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} + \cdots - 30\!\cdots\!00)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots + 37\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots - 77\!\cdots\!68)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots + 83\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{3} + \cdots - 42\!\cdots\!44)^{4} \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots + 72\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots - 22\!\cdots\!88)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots + 52\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots + 25\!\cdots\!16)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots + 55\!\cdots\!92)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots + 11\!\cdots\!56)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots - 99\!\cdots\!12)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} + \cdots - 75\!\cdots\!16)^{4} \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots - 46\!\cdots\!52)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots + 59\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{3} + \cdots - 34\!\cdots\!76)^{4} \) Copy content Toggle raw display
$97$ \( (T^{3} + \cdots + 12\!\cdots\!00)^{4} \) Copy content Toggle raw display
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