Properties

Label 192.13.e.c
Level $192$
Weight $13$
Character orbit 192.e
Analytic conductor $175.487$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [192,13,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, 13, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("192.65");
 
S:= CuspForms(chi, 13);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 192 = 2^{6} \cdot 3 \)
Weight: \( k \) \(=\) \( 13 \)
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: \(175.486812917\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-26}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 26 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 3)
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 \(\beta = 18\sqrt{-26}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (3 \beta - 675) q^{3} + 230 \beta q^{5} - 40250 q^{7} + ( - 4050 \beta + 379809) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (3 \beta - 675) q^{3} + 230 \beta q^{5} - 40250 q^{7} + ( - 4050 \beta + 379809) q^{9} + 12650 \beta q^{11} - 1284050 q^{13} + ( - 155250 \beta - 5812560) q^{15} - 161736 \beta q^{17} + 53343578 q^{19} + ( - 120750 \beta + 27168750) q^{21} + 1170884 \beta q^{23} - 201488975 q^{25} + (3873177 \beta - 154019475) q^{27} + 1310050 \beta q^{29} - 66526202 q^{31} + ( - 8538750 \beta - 319690800) q^{33} - 9257500 \beta q^{35} - 2228726450 q^{37} + ( - 3852150 \beta + 866733750) q^{39} + 89469100 \beta q^{41} + 8977216250 q^{43} + (87356070 \beta + 7846956000) q^{45} - 11733464 \beta q^{47} - 12221224701 q^{49} + (109171800 \beta + 4087392192) q^{51} + 448279614 \beta q^{53} - 24509628000 q^{55} + (160030734 \beta - 36006915150) q^{57} + 502355650 \beta q^{59} + 40679935918 q^{61} + (163012500 \beta - 15287312250) q^{63} - 295331500 \beta q^{65} + 121176846650 q^{67} + ( - 790346700 \beta - 29590580448) q^{69} + 488726700 \beta q^{71} - 60956187550 q^{73} + ( - 604466925 \beta + 136005058125) q^{75} - 509162500 \beta q^{77} + 252324997702 q^{79} + ( - 3076452900 \beta + 6080216481) q^{81} + 4475910446 \beta q^{83} + 313366734720 q^{85} + ( - 884283750 \beta - 33107583600) q^{87} - 1225929900 \beta q^{89} + 51683012500 q^{91} + ( - 199578606 \beta + 44905186350) q^{93} + 12269022940 \beta q^{95} + 653817778850 q^{97} + (4804583850 \beta + 431582580000) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 1350 q^{3} - 80500 q^{7} + 759618 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 1350 q^{3} - 80500 q^{7} + 759618 q^{9} - 2568100 q^{13} - 11625120 q^{15} + 106687156 q^{19} + 54337500 q^{21} - 402977950 q^{25} - 308038950 q^{27} - 133052404 q^{31} - 639381600 q^{33} - 4457452900 q^{37} + 1733467500 q^{39} + 17954432500 q^{43} + 15693912000 q^{45} - 24442449402 q^{49} + 8174784384 q^{51} - 49019256000 q^{55} - 72013830300 q^{57} + 81359871836 q^{61} - 30574624500 q^{63} + 242353693300 q^{67} - 59181160896 q^{69} - 121912375100 q^{73} + 272010116250 q^{75} + 504649995404 q^{79} + 12160432962 q^{81} + 626733469440 q^{85} - 66215167200 q^{87} + 103366025000 q^{91} + 89810372700 q^{93} + 1307635557700 q^{97} + 863165160000 q^{99}+O(q^{100}) \) 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
5.09902i
5.09902i
0 −675.000 275.347i 0 21109.9i 0 −40250.0 0 379809. + 371719.i 0
65.2 0 −675.000 + 275.347i 0 21109.9i 0 −40250.0 0 379809. 371719.i 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 192.13.e.c 2
3.b odd 2 1 inner 192.13.e.c 2
4.b odd 2 1 192.13.e.d 2
8.b even 2 1 48.13.e.b 2
8.d odd 2 1 3.13.b.b 2
12.b even 2 1 192.13.e.d 2
24.f even 2 1 3.13.b.b 2
24.h odd 2 1 48.13.e.b 2
40.e odd 2 1 75.13.c.c 2
40.k even 4 2 75.13.d.b 4
72.l even 6 2 81.13.d.c 4
72.p odd 6 2 81.13.d.c 4
120.m even 2 1 75.13.c.c 2
120.q odd 4 2 75.13.d.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.13.b.b 2 8.d odd 2 1
3.13.b.b 2 24.f even 2 1
48.13.e.b 2 8.b even 2 1
48.13.e.b 2 24.h odd 2 1
75.13.c.c 2 40.e odd 2 1
75.13.c.c 2 120.m even 2 1
75.13.d.b 4 40.k even 4 2
75.13.d.b 4 120.q odd 4 2
81.13.d.c 4 72.l even 6 2
81.13.d.c 4 72.p odd 6 2
192.13.e.c 2 1.a even 1 1 trivial
192.13.e.c 2 3.b odd 2 1 inner
192.13.e.d 2 4.b odd 2 1
192.13.e.d 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 \(S_{13}^{\mathrm{new}}(192, [\chi])\):

\( T_{5}^{2} + 445629600 \) Copy content Toggle raw display
\( T_{7} + 40250 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 1350 T + 531441 \) Copy content Toggle raw display
$5$ \( T^{2} + 445629600 \) Copy content Toggle raw display
$7$ \( (T + 40250)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 1348029540000 \) Copy content Toggle raw display
$13$ \( (T + 1284050)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + \cdots + 220359487855104 \) Copy content Toggle raw display
$19$ \( (T - 53343578)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 11\!\cdots\!44 \) Copy content Toggle raw display
$29$ \( T^{2} + 14\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T + 66526202)^{2} \) Copy content Toggle raw display
$37$ \( (T + 2228726450)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 67\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T - 8977216250)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 11\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( T^{2} + 16\!\cdots\!04 \) Copy content Toggle raw display
$59$ \( T^{2} + 21\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T - 40679935918)^{2} \) Copy content Toggle raw display
$67$ \( (T - 121176846650)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 20\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T + 60956187550)^{2} \) Copy content Toggle raw display
$79$ \( (T - 252324997702)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 16\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( T^{2} + 12\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T - 653817778850)^{2} \) Copy content Toggle raw display
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