Properties

Label 192.9.e.g
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: \(\Q(\sqrt{-110}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 110 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2\cdot 3 \)
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 \(\beta = 6\sqrt{-110}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta + 51) q^{3} + 18 \beta q^{5} - 3094 q^{7} + ( - 102 \beta - 1359) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta + 51) q^{3} + 18 \beta q^{5} - 3094 q^{7} + ( - 102 \beta - 1359) q^{9} + 18 \beta q^{11} + 7294 q^{13} + (918 \beta + 71280) q^{15} + 936 \beta q^{17} + 80326 q^{19} + (3094 \beta - 157794) q^{21} - 1548 \beta q^{23} - 892415 q^{25} + ( - 3843 \beta - 473229) q^{27} + 13734 \beta q^{29} + 435914 q^{31} + (918 \beta + 71280) q^{33} - 55692 \beta q^{35} - 1159298 q^{37} + ( - 7294 \beta + 371994) q^{39} - 43164 \beta q^{41} - 990266 q^{43} + ( - 24462 \beta + 7270560) q^{45} - 106488 \beta q^{47} + 3808035 q^{49} + (47736 \beta + 3706560) q^{51} - 160038 \beta q^{53} - 1283040 q^{55} + ( - 80326 \beta + 4096626) q^{57} - 25398 \beta q^{59} - 19369154 q^{61} + (315588 \beta + 4204746) q^{63} + 131292 \beta q^{65} + 28024294 q^{67} + ( - 78948 \beta - 6130080) q^{69} - 534852 \beta q^{71} - 25230142 q^{73} + (892415 \beta - 45513165) q^{75} - 55692 \beta q^{77} - 63401398 q^{79} + (277236 \beta - 39352959) q^{81} - 755514 \beta q^{83} - 66718080 q^{85} + (700434 \beta + 54386640) q^{87} - 1244196 \beta q^{89} - 22567636 q^{91} + ( - 435914 \beta + 22231614) q^{93} + 1445868 \beta q^{95} + 19550306 q^{97} + ( - 24462 \beta + 7270560) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 102 q^{3} - 6188 q^{7} - 2718 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 102 q^{3} - 6188 q^{7} - 2718 q^{9} + 14588 q^{13} + 142560 q^{15} + 160652 q^{19} - 315588 q^{21} - 1784830 q^{25} - 946458 q^{27} + 871828 q^{31} + 142560 q^{33} - 2318596 q^{37} + 743988 q^{39} - 1980532 q^{43} + 14541120 q^{45} + 7616070 q^{49} + 7413120 q^{51} - 2566080 q^{55} + 8193252 q^{57} - 38738308 q^{61} + 8409492 q^{63} + 56048588 q^{67} - 12260160 q^{69} - 50460284 q^{73} - 91026330 q^{75} - 126802796 q^{79} - 78705918 q^{81} - 133436160 q^{85} + 108773280 q^{87} - 45135272 q^{91} + 44463228 q^{93} + 39100612 q^{97} + 14541120 q^{99}+O(q^{100}) \) 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
10.4881i
10.4881i
0 51.0000 62.9285i 0 1132.71i 0 −3094.00 0 −1359.00 6418.71i 0
65.2 0 51.0000 + 62.9285i 0 1132.71i 0 −3094.00 0 −1359.00 + 6418.71i 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.g 2
3.b odd 2 1 inner 192.9.e.g 2
4.b odd 2 1 192.9.e.d 2
8.b even 2 1 12.9.c.b 2
8.d odd 2 1 48.9.e.c 2
12.b even 2 1 192.9.e.d 2
24.f even 2 1 48.9.e.c 2
24.h odd 2 1 12.9.c.b 2
40.f even 2 1 300.9.g.d 2
40.i odd 4 2 300.9.b.c 4
72.j odd 6 2 324.9.g.f 4
72.n even 6 2 324.9.g.f 4
120.i odd 2 1 300.9.g.d 2
120.w even 4 2 300.9.b.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.9.c.b 2 8.b even 2 1
12.9.c.b 2 24.h odd 2 1
48.9.e.c 2 8.d odd 2 1
48.9.e.c 2 24.f even 2 1
192.9.e.d 2 4.b odd 2 1
192.9.e.d 2 12.b even 2 1
192.9.e.g 2 1.a even 1 1 trivial
192.9.e.g 2 3.b odd 2 1 inner
300.9.b.c 4 40.i odd 4 2
300.9.b.c 4 120.w even 4 2
300.9.g.d 2 40.f even 2 1
300.9.g.d 2 120.i odd 2 1
324.9.g.f 4 72.j odd 6 2
324.9.g.f 4 72.n even 6 2

Hecke kernels

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

\( T_{5}^{2} + 1283040 \) Copy content Toggle raw display
\( T_{7} + 3094 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 102T + 6561 \) Copy content Toggle raw display
$5$ \( T^{2} + 1283040 \) Copy content Toggle raw display
$7$ \( (T + 3094)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 1283040 \) Copy content Toggle raw display
$13$ \( (T - 7294)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 3469340160 \) Copy content Toggle raw display
$19$ \( (T - 80326)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 9489363840 \) Copy content Toggle raw display
$29$ \( T^{2} + 746946113760 \) Copy content Toggle raw display
$31$ \( (T - 435914)^{2} \) Copy content Toggle raw display
$37$ \( (T + 1159298)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 7377998348160 \) Copy content Toggle raw display
$43$ \( (T + 990266)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 44905188810240 \) Copy content Toggle raw display
$53$ \( T^{2} + 101424159318240 \) Copy content Toggle raw display
$59$ \( T^{2} + 2554431279840 \) Copy content Toggle raw display
$61$ \( (T + 19369154)^{2} \) Copy content Toggle raw display
$67$ \( (T - 28024294)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 11\!\cdots\!40 \) Copy content Toggle raw display
$73$ \( (T + 25230142)^{2} \) Copy content Toggle raw display
$79$ \( (T + 63401398)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 22\!\cdots\!60 \) Copy content Toggle raw display
$89$ \( T^{2} + 61\!\cdots\!60 \) Copy content Toggle raw display
$97$ \( (T - 19550306)^{2} \) Copy content Toggle raw display
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