Properties

Label 392.6.a.d
Level $392$
Weight $6$
Character orbit 392.a
Self dual yes
Analytic conductor $62.870$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [392,6,Mod(1,392)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(392, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("392.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 392 = 2^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 392.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: \(62.8704573667\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{345}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 86 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 56)
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 \(\beta = \sqrt{345}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 3) q^{3} + (3 \beta - 41) q^{5} + (6 \beta + 111) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 3) q^{3} + (3 \beta - 41) q^{5} + (6 \beta + 111) q^{9} + ( - 6 \beta + 170) q^{11} + ( - 39 \beta - 455) q^{13} + ( - 32 \beta + 912) q^{15} + (6 \beta - 1608) q^{17} + ( - 45 \beta + 337) q^{19} + ( - 72 \beta - 552) q^{23} + ( - 246 \beta + 1661) q^{25} + ( - 114 \beta + 1674) q^{27} + ( - 114 \beta + 4032) q^{29} + ( - 210 \beta + 3106) q^{31} + (152 \beta - 1560) q^{33} + (390 \beta - 4256) q^{37} + ( - 572 \beta - 14820) q^{39} + ( - 678 \beta + 652) q^{41} + (798 \beta - 5002) q^{43} + (87 \beta + 1659) q^{45} + (714 \beta + 6374) q^{47} + ( - 1590 \beta - 2754) q^{51} + ( - 768 \beta - 5610) q^{53} + (756 \beta - 13180) q^{55} + (202 \beta - 14514) q^{57} + ( - 2253 \beta + 6009) q^{59} + (75 \beta - 51369) q^{61} + (234 \beta - 21710) q^{65} + (1944 \beta + 12068) q^{67} + ( - 768 \beta - 26496) q^{69} + (1740 \beta + 44860) q^{71} + (1716 \beta + 27794) q^{73} + (923 \beta - 79887) q^{75} + (1956 \beta + 24412) q^{79} + ( - 126 \beta - 61281) q^{81} + (3447 \beta - 17891) q^{83} + ( - 5070 \beta + 72138) q^{85} + (3690 \beta - 27234) q^{87} + ( - 4728 \beta + 9150) q^{89} + (2476 \beta - 63132) q^{93} + (2856 \beta - 60392) q^{95} + (462 \beta + 34992) q^{97} + (354 \beta + 6450) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{3} - 82 q^{5} + 222 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 6 q^{3} - 82 q^{5} + 222 q^{9} + 340 q^{11} - 910 q^{13} + 1824 q^{15} - 3216 q^{17} + 674 q^{19} - 1104 q^{23} + 3322 q^{25} + 3348 q^{27} + 8064 q^{29} + 6212 q^{31} - 3120 q^{33} - 8512 q^{37} - 29640 q^{39} + 1304 q^{41} - 10004 q^{43} + 3318 q^{45} + 12748 q^{47} - 5508 q^{51} - 11220 q^{53} - 26360 q^{55} - 29028 q^{57} + 12018 q^{59} - 102738 q^{61} - 43420 q^{65} + 24136 q^{67} - 52992 q^{69} + 89720 q^{71} + 55588 q^{73} - 159774 q^{75} + 48824 q^{79} - 122562 q^{81} - 35782 q^{83} + 144276 q^{85} - 54468 q^{87} + 18300 q^{89} - 126264 q^{93} - 120784 q^{95} + 69984 q^{97} + 12900 q^{99}+O(q^{100}) \) 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
−8.78709
9.78709
0 −15.5742 0 −96.7225 0 0 0 −0.445054 0
1.2 0 21.5742 0 14.7225 0 0 0 222.445 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(7\) \( -1 \)

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 392.6.a.d 2
4.b odd 2 1 784.6.a.u 2
7.b odd 2 1 56.6.a.e 2
7.c even 3 2 392.6.i.i 4
7.d odd 6 2 392.6.i.j 4
21.c even 2 1 504.6.a.i 2
28.d even 2 1 112.6.a.i 2
56.e even 2 1 448.6.a.t 2
56.h odd 2 1 448.6.a.v 2
84.h odd 2 1 1008.6.a.bd 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.6.a.e 2 7.b odd 2 1
112.6.a.i 2 28.d even 2 1
392.6.a.d 2 1.a even 1 1 trivial
392.6.i.i 4 7.c even 3 2
392.6.i.j 4 7.d odd 6 2
448.6.a.t 2 56.e even 2 1
448.6.a.v 2 56.h odd 2 1
504.6.a.i 2 21.c even 2 1
784.6.a.u 2 4.b odd 2 1
1008.6.a.bd 2 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 6T_{3} - 336 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(392))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 6T - 336 \) Copy content Toggle raw display
$5$ \( T^{2} + 82T - 1424 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 340T + 16480 \) Copy content Toggle raw display
$13$ \( T^{2} + 910T - 317720 \) Copy content Toggle raw display
$17$ \( T^{2} + 3216 T + 2573244 \) Copy content Toggle raw display
$19$ \( T^{2} - 674T - 585056 \) Copy content Toggle raw display
$23$ \( T^{2} + 1104 T - 1483776 \) Copy content Toggle raw display
$29$ \( T^{2} - 8064 T + 11773404 \) Copy content Toggle raw display
$31$ \( T^{2} - 6212 T - 5567264 \) Copy content Toggle raw display
$37$ \( T^{2} + 8512 T - 34360964 \) Copy content Toggle raw display
$41$ \( T^{2} - 1304 T - 158165876 \) Copy content Toggle raw display
$43$ \( T^{2} + 10004 T - 194677376 \) Copy content Toggle raw display
$47$ \( T^{2} - 12748 T - 135251744 \) Copy content Toggle raw display
$53$ \( T^{2} + 11220 T - 172017180 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots - 1715115024 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots + 2636833536 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots - 1158165296 \) Copy content Toggle raw display
$71$ \( T^{2} - 89720 T + 967897600 \) Copy content Toggle raw display
$73$ \( T^{2} - 55588 T - 243399884 \) Copy content Toggle raw display
$79$ \( T^{2} - 48824 T - 724002176 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots - 3779136224 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots - 7628401980 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots + 1150801884 \) Copy content Toggle raw display
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