Properties

Label 192.8.a.r
Level $192$
Weight $8$
Character orbit 192.a
Self dual yes
Analytic conductor $59.978$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [192,8,Mod(1,192)]
 
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, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("192.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 192 = 2^{6} \cdot 3 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 192.a (trivial)

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: yes
Analytic conductor: \(59.9779248930\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{235}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 235 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 96)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(\beta = 16\sqrt{235}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 27 q^{3} + (\beta - 90) q^{5} + (5 \beta + 516) q^{7} + 729 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 27 q^{3} + (\beta - 90) q^{5} + (5 \beta + 516) q^{7} + 729 q^{9} + (6 \beta - 1420) q^{11} + (42 \beta + 170) q^{13} + ( - 27 \beta + 2430) q^{15} + (78 \beta + 4890) q^{17} + (70 \beta - 16020) q^{19} + ( - 135 \beta - 13932) q^{21} + (330 \beta + 5568) q^{23} + ( - 180 \beta - 9865) q^{25} - 19683 q^{27} + ( - 5 \beta - 152106) q^{29} + ( - 383 \beta - 38820) q^{31} + ( - 162 \beta + 38340) q^{33} + (66 \beta + 254360) q^{35} + (12 \beta - 507910) q^{37} + ( - 1134 \beta - 4590) q^{39} + ( - 1030 \beta + 352050) q^{41} + ( - 1150 \beta + 197748) q^{43} + (729 \beta - 65610) q^{45} + (1710 \beta + 578744) q^{47} + (5160 \beta + 946713) q^{49} + ( - 2106 \beta - 132030) q^{51} + ( - 1897 \beta - 784290) q^{53} + ( - 1960 \beta + 488760) q^{55} + ( - 1890 \beta + 432540) q^{57} + (7128 \beta - 69620) q^{59} + (6960 \beta - 1301790) q^{61} + (3645 \beta + 376164) q^{63} + ( - 3610 \beta + 2511420) q^{65} + ( - 1960 \beta + 2644884) q^{67} + ( - 8910 \beta - 150336) q^{69} + ( - 5262 \beta + 2860880) q^{71} + ( - 9336 \beta - 595350) q^{73} + (4860 \beta + 266355) q^{75} + ( - 4004 \beta + 1072080) q^{77} + (16597 \beta - 199140) q^{79} + 531441 q^{81} + ( - 17490 \beta + 3493308) q^{83} + ( - 2130 \beta + 4252380) q^{85} + (135 \beta + 4106862) q^{87} + (22420 \beta - 4083366) q^{89} + (22522 \beta + 12721320) q^{91} + (10341 \beta + 1048140) q^{93} + ( - 22320 \beta + 5653000) q^{95} + (20148 \beta - 5180750) q^{97} + (4374 \beta - 1035180) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 54 q^{3} - 180 q^{5} + 1032 q^{7} + 1458 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 54 q^{3} - 180 q^{5} + 1032 q^{7} + 1458 q^{9} - 2840 q^{11} + 340 q^{13} + 4860 q^{15} + 9780 q^{17} - 32040 q^{19} - 27864 q^{21} + 11136 q^{23} - 19730 q^{25} - 39366 q^{27} - 304212 q^{29} - 77640 q^{31} + 76680 q^{33} + 508720 q^{35} - 1015820 q^{37} - 9180 q^{39} + 704100 q^{41} + 395496 q^{43} - 131220 q^{45} + 1157488 q^{47} + 1893426 q^{49} - 264060 q^{51} - 1568580 q^{53} + 977520 q^{55} + 865080 q^{57} - 139240 q^{59} - 2603580 q^{61} + 752328 q^{63} + 5022840 q^{65} + 5289768 q^{67} - 300672 q^{69} + 5721760 q^{71} - 1190700 q^{73} + 532710 q^{75} + 2144160 q^{77} - 398280 q^{79} + 1062882 q^{81} + 6986616 q^{83} + 8504760 q^{85} + 8213724 q^{87} - 8166732 q^{89} + 25442640 q^{91} + 2096280 q^{93} + 11306000 q^{95} - 10361500 q^{97} - 2070360 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−15.3297
15.3297
0 −27.0000 0 −335.275 0 −710.377 0 729.000 0
1.2 0 −27.0000 0 155.275 0 1742.38 0 729.000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(3\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 192.8.a.r 2
3.b odd 2 1 576.8.a.bm 2
4.b odd 2 1 192.8.a.u 2
8.b even 2 1 96.8.a.g yes 2
8.d odd 2 1 96.8.a.d 2
12.b even 2 1 576.8.a.bl 2
24.f even 2 1 288.8.a.i 2
24.h odd 2 1 288.8.a.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
96.8.a.d 2 8.d odd 2 1
96.8.a.g yes 2 8.b even 2 1
192.8.a.r 2 1.a even 1 1 trivial
192.8.a.u 2 4.b odd 2 1
288.8.a.i 2 24.f even 2 1
288.8.a.j 2 24.h odd 2 1
576.8.a.bl 2 12.b even 2 1
576.8.a.bm 2 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(192))\):

\( T_{5}^{2} + 180T_{5} - 52060 \) Copy content Toggle raw display
\( T_{7}^{2} - 1032T_{7} - 1237744 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T + 27)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 180T - 52060 \) Copy content Toggle raw display
$7$ \( T^{2} - 1032 T - 1237744 \) Copy content Toggle raw display
$11$ \( T^{2} + 2840 T - 149360 \) Copy content Toggle raw display
$13$ \( T^{2} - 340 T - 106093340 \) Copy content Toggle raw display
$17$ \( T^{2} - 9780 T - 342101340 \) Copy content Toggle raw display
$19$ \( T^{2} + 32040 T - 38143600 \) Copy content Toggle raw display
$23$ \( T^{2} + \cdots - 6520421376 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots + 23134731236 \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots - 7317817840 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots + 257963905060 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots + 60115458500 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots - 40457328496 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots + 159030761536 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots + 398618486660 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots - 3051785437040 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots - 1219589451900 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 6764300717456 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots + 6518885551360 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots - 4889157880860 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots - 16532041465840 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots - 6199749233136 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots - 13565931134044 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots + 2418705617860 \) Copy content Toggle raw display
show more
show less