Properties

Label 192.11.e.d
Level $192$
Weight $11$
Character orbit 192.e
Analytic conductor $121.989$
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] = [192,11,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, 11, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("192.65");
 
S:= CuspForms(chi, 11);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 192 = 2^{6} \cdot 3 \)
Weight: \( k \) \(=\) \( 11 \)
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: \(121.988592513\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-5}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
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 \(\beta = 12\sqrt{-5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (9 \beta - 27) q^{3} + 106 \beta q^{5} - 17234 q^{7} + ( - 486 \beta - 57591) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (9 \beta - 27) q^{3} + 106 \beta q^{5} - 17234 q^{7} + ( - 486 \beta - 57591) q^{9} - 6962 \beta q^{11} + 169654 q^{13} + ( - 2862 \beta - 686880) q^{15} - 12792 \beta q^{17} - 949462 q^{19} + ( - 155106 \beta + 465318) q^{21} - 99044 \beta q^{23} + 1675705 q^{25} + ( - 505197 \beta + 4704237) q^{27} - 118594 \beta q^{29} + 29793118 q^{31} + (187974 \beta + 45113760) q^{33} - 1826804 \beta q^{35} + 60811846 q^{37} + (1526886 \beta - 4580658) q^{39} + 6770372 \beta q^{41} + 107419706 q^{43} + ( - 6104646 \beta + 37091520) q^{45} + 9987608 \beta q^{47} + 14535507 q^{49} + (345384 \beta + 82892160) q^{51} - 7158798 \beta q^{53} + 531339840 q^{55} + ( - 8545158 \beta + 25635474) q^{57} - 24192682 \beta q^{59} - 1030793642 q^{61} + (8375724 \beta + 992523294) q^{63} + 17983324 \beta q^{65} + 1876742474 q^{67} + (2674188 \beta + 641805120) q^{69} - 100003596 \beta q^{71} - 2846528494 q^{73} + (15081345 \beta - 45244035) q^{75} + 119983108 \beta q^{77} - 1488647618 q^{79} + (55978452 \beta + 3146662161) q^{81} + 47175562 \beta q^{83} + 976285440 q^{85} + (3202038 \beta + 768489120) q^{87} - 224371428 \beta q^{89} - 2923817036 q^{91} + (268138062 \beta - 804414186) q^{93} - 100642972 \beta q^{95} - 1592948926 q^{97} + (400948542 \beta - 2436143040) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 54 q^{3} - 34468 q^{7} - 115182 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 54 q^{3} - 34468 q^{7} - 115182 q^{9} + 339308 q^{13} - 1373760 q^{15} - 1898924 q^{19} + 930636 q^{21} + 3351410 q^{25} + 9408474 q^{27} + 59586236 q^{31} + 90227520 q^{33} + 121623692 q^{37} - 9161316 q^{39} + 214839412 q^{43} + 74183040 q^{45} + 29071014 q^{49} + 165784320 q^{51} + 1062679680 q^{55} + 51270948 q^{57} - 2061587284 q^{61} + 1985046588 q^{63} + 3753484948 q^{67} + 1283610240 q^{69} - 5693056988 q^{73} - 90488070 q^{75} - 2977295236 q^{79} + 6293324322 q^{81} + 1952570880 q^{85} + 1536978240 q^{87} - 5847634072 q^{91} - 1608828372 q^{93} - 3185897852 q^{97} - 4872286080 q^{99}+O(q^{100}) \) 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
2.23607i
2.23607i
0 −27.0000 241.495i 0 2844.28i 0 −17234.0 0 −57591.0 + 13040.7i 0
65.2 0 −27.0000 + 241.495i 0 2844.28i 0 −17234.0 0 −57591.0 13040.7i 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 192.11.e.d 2
3.b odd 2 1 inner 192.11.e.d 2
4.b odd 2 1 192.11.e.e 2
8.b even 2 1 48.11.e.c 2
8.d odd 2 1 3.11.b.a 2
12.b even 2 1 192.11.e.e 2
24.f even 2 1 3.11.b.a 2
24.h odd 2 1 48.11.e.c 2
40.e odd 2 1 75.11.c.d 2
40.k even 4 2 75.11.d.b 4
72.l even 6 2 81.11.d.d 4
72.p odd 6 2 81.11.d.d 4
120.m even 2 1 75.11.c.d 2
120.q odd 4 2 75.11.d.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.11.b.a 2 8.d odd 2 1
3.11.b.a 2 24.f even 2 1
48.11.e.c 2 8.b even 2 1
48.11.e.c 2 24.h odd 2 1
75.11.c.d 2 40.e odd 2 1
75.11.c.d 2 120.m even 2 1
75.11.d.b 4 40.k even 4 2
75.11.d.b 4 120.q odd 4 2
81.11.d.d 4 72.l even 6 2
81.11.d.d 4 72.p odd 6 2
192.11.e.d 2 1.a even 1 1 trivial
192.11.e.d 2 3.b odd 2 1 inner
192.11.e.e 2 4.b odd 2 1
192.11.e.e 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 \(S_{11}^{\mathrm{new}}(192, [\chi])\):

\( T_{5}^{2} + 8089920 \) Copy content Toggle raw display
\( T_{7} + 17234 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 54T + 59049 \) Copy content Toggle raw display
$5$ \( T^{2} + 8089920 \) Copy content Toggle raw display
$7$ \( (T + 17234)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 34897999680 \) Copy content Toggle raw display
$13$ \( (T - 169654)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 117817390080 \) Copy content Toggle raw display
$19$ \( (T + 949462)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 7062994033920 \) Copy content Toggle raw display
$29$ \( T^{2} + 10126466521920 \) Copy content Toggle raw display
$31$ \( (T - 29793118)^{2} \) Copy content Toggle raw display
$37$ \( (T - 60811846)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 33\!\cdots\!80 \) Copy content Toggle raw display
$43$ \( (T - 107419706)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 71\!\cdots\!80 \) Copy content Toggle raw display
$53$ \( T^{2} + 36\!\cdots\!80 \) Copy content Toggle raw display
$59$ \( T^{2} + 42\!\cdots\!80 \) Copy content Toggle raw display
$61$ \( (T + 1030793642)^{2} \) Copy content Toggle raw display
$67$ \( (T - 1876742474)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 72\!\cdots\!20 \) Copy content Toggle raw display
$73$ \( (T + 2846528494)^{2} \) Copy content Toggle raw display
$79$ \( (T + 1488647618)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 16\!\cdots\!80 \) Copy content Toggle raw display
$89$ \( T^{2} + 36\!\cdots\!80 \) Copy content Toggle raw display
$97$ \( (T + 1592948926)^{2} \) Copy content Toggle raw display
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