Properties

Label 392.6.i.a
Level $392$
Weight $6$
Character orbit 392.i
Analytic conductor $62.870$
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] = [392,6,Mod(177,392)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(392, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("392.177");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 392 = 2^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 392.i (of order \(3\), degree \(2\), 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: \(62.8704573667\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 56)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (30 \zeta_{6} - 30) q^{3} - 32 \zeta_{6} q^{5} - 657 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (30 \zeta_{6} - 30) q^{3} - 32 \zeta_{6} q^{5} - 657 \zeta_{6} q^{9} + ( - 624 \zeta_{6} + 624) q^{11} - 708 q^{13} + 960 q^{15} + (934 \zeta_{6} - 934) q^{17} - 1858 \zeta_{6} q^{19} + 1120 \zeta_{6} q^{23} + ( - 2101 \zeta_{6} + 2101) q^{25} + 12420 q^{27} - 1174 q^{29} + (2908 \zeta_{6} - 2908) q^{31} + 18720 \zeta_{6} q^{33} + 12462 \zeta_{6} q^{37} + ( - 21240 \zeta_{6} + 21240) q^{39} + 2662 q^{41} - 7144 q^{43} + (21024 \zeta_{6} - 21024) q^{45} + 7468 \zeta_{6} q^{47} - 28020 \zeta_{6} q^{51} + ( - 27274 \zeta_{6} + 27274) q^{53} - 19968 q^{55} + 55740 q^{57} + (2490 \zeta_{6} - 2490) q^{59} + 11096 \zeta_{6} q^{61} + 22656 \zeta_{6} q^{65} + (39756 \zeta_{6} - 39756) q^{67} - 33600 q^{69} - 69888 q^{71} + (16450 \zeta_{6} - 16450) q^{73} + 63030 \zeta_{6} q^{75} - 78376 \zeta_{6} q^{79} + (212949 \zeta_{6} - 212949) q^{81} + 109818 q^{83} + 29888 q^{85} + ( - 35220 \zeta_{6} + 35220) q^{87} + 56966 \zeta_{6} q^{89} - 87240 \zeta_{6} q^{93} + (59456 \zeta_{6} - 59456) q^{95} - 115946 q^{97} - 409968 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 30 q^{3} - 32 q^{5} - 657 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 30 q^{3} - 32 q^{5} - 657 q^{9} + 624 q^{11} - 1416 q^{13} + 1920 q^{15} - 934 q^{17} - 1858 q^{19} + 1120 q^{23} + 2101 q^{25} + 24840 q^{27} - 2348 q^{29} - 2908 q^{31} + 18720 q^{33} + 12462 q^{37} + 21240 q^{39} + 5324 q^{41} - 14288 q^{43} - 21024 q^{45} + 7468 q^{47} - 28020 q^{51} + 27274 q^{53} - 39936 q^{55} + 111480 q^{57} - 2490 q^{59} + 11096 q^{61} + 22656 q^{65} - 39756 q^{67} - 67200 q^{69} - 139776 q^{71} - 16450 q^{73} + 63030 q^{75} - 78376 q^{79} - 212949 q^{81} + 219636 q^{83} + 59776 q^{85} + 35220 q^{87} + 56966 q^{89} - 87240 q^{93} - 59456 q^{95} - 231892 q^{97} - 819936 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/392\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(295\) \(297\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
177.1
0.500000 0.866025i
0.500000 + 0.866025i
0 −15.0000 25.9808i 0 −16.0000 + 27.7128i 0 0 0 −328.500 + 568.979i 0
361.1 0 −15.0000 + 25.9808i 0 −16.0000 27.7128i 0 0 0 −328.500 568.979i 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
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 392.6.i.a 2
7.b odd 2 1 392.6.i.f 2
7.c even 3 1 56.6.a.b 1
7.c even 3 1 inner 392.6.i.a 2
7.d odd 6 1 392.6.a.a 1
7.d odd 6 1 392.6.i.f 2
21.h odd 6 1 504.6.a.b 1
28.f even 6 1 784.6.a.n 1
28.g odd 6 1 112.6.a.a 1
56.k odd 6 1 448.6.a.p 1
56.p even 6 1 448.6.a.a 1
84.n even 6 1 1008.6.a.h 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.6.a.b 1 7.c even 3 1
112.6.a.a 1 28.g odd 6 1
392.6.a.a 1 7.d odd 6 1
392.6.i.a 2 1.a even 1 1 trivial
392.6.i.a 2 7.c even 3 1 inner
392.6.i.f 2 7.b odd 2 1
392.6.i.f 2 7.d odd 6 1
448.6.a.a 1 56.p even 6 1
448.6.a.p 1 56.k odd 6 1
504.6.a.b 1 21.h odd 6 1
784.6.a.n 1 28.f even 6 1
1008.6.a.h 1 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 30T_{3} + 900 \) acting on \(S_{6}^{\mathrm{new}}(392, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 30T + 900 \) Copy content Toggle raw display
$5$ \( T^{2} + 32T + 1024 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 624T + 389376 \) Copy content Toggle raw display
$13$ \( (T + 708)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 934T + 872356 \) Copy content Toggle raw display
$19$ \( T^{2} + 1858 T + 3452164 \) Copy content Toggle raw display
$23$ \( T^{2} - 1120 T + 1254400 \) Copy content Toggle raw display
$29$ \( (T + 1174)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 2908 T + 8456464 \) Copy content Toggle raw display
$37$ \( T^{2} - 12462 T + 155301444 \) Copy content Toggle raw display
$41$ \( (T - 2662)^{2} \) Copy content Toggle raw display
$43$ \( (T + 7144)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 7468 T + 55771024 \) Copy content Toggle raw display
$53$ \( T^{2} - 27274 T + 743871076 \) Copy content Toggle raw display
$59$ \( T^{2} + 2490 T + 6200100 \) Copy content Toggle raw display
$61$ \( T^{2} - 11096 T + 123121216 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 1580539536 \) Copy content Toggle raw display
$71$ \( (T + 69888)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 16450 T + 270602500 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots + 6142797376 \) Copy content Toggle raw display
$83$ \( (T - 109818)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 3245125156 \) Copy content Toggle raw display
$97$ \( (T + 115946)^{2} \) Copy content Toggle raw display
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