↑

Properties

Label 392.3.k.j
Level $392$
Weight $3$
Character orbit 392.k
Analytic conductor $10.681$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [392,3,Mod(67,392)]
 
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([3, 3, 4]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("392.67");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 392 = 2^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 392.k (of order \(6\), 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: \(10.6812263629\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.796594176.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 18x^{6} - 40x^{5} + 83x^{4} - 104x^{3} + 22x^{2} + 24x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 56)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{4} - \beta_{3} - \beta_{2}) q^{2} + ( - \beta_{6} + \beta_{3} + 2 \beta_1 + 2) q^{3} + ( - \beta_{7} + \beta_{5} + 3 \beta_1 + 3) q^{4} + ( - \beta_{6} - 2 \beta_{5} + \cdots + 2 \beta_{2}) q^{5}+ \cdots + ( - 4 \beta_{6} - 3 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{4} - \beta_{3} - \beta_{2}) q^{2} + ( - \beta_{6} + \beta_{3} + 2 \beta_1 + 2) q^{3} + ( - \beta_{7} + \beta_{5} + 3 \beta_1 + 3) q^{4} + ( - \beta_{6} - 2 \beta_{5} + \cdots + 2 \beta_{2}) q^{5}+ \cdots + (34 \beta_{3} - 60) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{3} + 12 q^{4} - 8 q^{6} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{3} + 12 q^{4} - 8 q^{6} + 12 q^{9} - 28 q^{10} - 16 q^{11} - 24 q^{12} - 8 q^{16} + 72 q^{17} - 16 q^{18} + 8 q^{19} - 112 q^{20} - 48 q^{22} - 40 q^{24} + 68 q^{25} + 28 q^{26} + 128 q^{27} - 16 q^{33} - 32 q^{34} + 72 q^{36} - 76 q^{38} + 56 q^{40} + 80 q^{41} - 160 q^{43} + 48 q^{44} - 224 q^{46} - 32 q^{48} + 224 q^{50} - 176 q^{51} - 56 q^{52} - 16 q^{54} - 272 q^{57} + 168 q^{58} + 184 q^{59} - 56 q^{60} + 224 q^{62} + 288 q^{64} - 168 q^{65} - 32 q^{66} + 224 q^{67} - 216 q^{68} + 160 q^{72} + 232 q^{73} + 280 q^{74} + 88 q^{75} + 48 q^{76} - 336 q^{80} + 52 q^{81} - 48 q^{82} - 176 q^{83} - 8 q^{86} - 240 q^{88} + 312 q^{89} - 616 q^{90} + 112 q^{92} - 112 q^{94} + 176 q^{96} + 272 q^{97} - 480 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 18x^{6} - 40x^{5} + 83x^{4} - 104x^{3} + 22x^{2} + 24x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{6} - 6\nu^{5} + 31\nu^{4} - 52\nu^{3} + 125\nu^{2} - 100\nu - 34 ) / 18 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -23\nu^{7} - 176\nu^{6} + 410\nu^{5} - 3150\nu^{4} + 5009\nu^{3} - 13942\nu^{2} + 13100\nu + 2464 ) / 1026 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 46\nu^{7} - 161\nu^{6} + 719\nu^{5} - 1395\nu^{4} + 2807\nu^{3} - 2896\nu^{2} - 1576\nu + 1228 ) / 1026 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 101\nu^{7} - 211\nu^{6} + 1285\nu^{5} - 1317\nu^{4} + 4276\nu^{3} - 614\nu^{2} - 4184\nu - 2860 ) / 1026 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 176\nu^{7} - 616\nu^{6} + 2974\nu^{5} - 5895\nu^{4} + 13684\nu^{3} - 14939\nu^{2} + 5836\nu - 610 ) / 1026 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 124\nu^{7} - 434\nu^{6} + 2072\nu^{5} - 4095\nu^{4} + 9128\nu^{3} - 9814\nu^{2} + 1127\nu + 946 ) / 513 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 248\nu^{7} - 868\nu^{6} + 4144\nu^{5} - 8190\nu^{4} + 18256\nu^{3} - 19628\nu^{2} + 3280\nu + 1379 ) / 513 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{7} - 2\beta_{6} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} - 4\beta_{4} - 4\beta_{3} - 4\beta_{2} - 4\beta _1 - 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -7\beta_{7} + 12\beta_{6} + 6\beta_{5} - 6\beta_{4} - 2\beta_{3} - 6\beta_{2} - 6\beta _1 - 11 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -15\beta_{7} + 16\beta_{6} + 12\beta_{5} + 8\beta_{4} + 24\beta_{3} + 24\beta_{2} + 44\beta _1 + 37 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 19\beta_{7} - 12\beta_{6} - 40\beta_{5} + 30\beta_{4} - 18\beta_{3} + 70\beta_{2} + 120\beta _1 + 111 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 95\beta_{7} - 72\beta_{6} - 150\beta_{5} + 60\beta_{4} - 228\beta_{3} - 68\beta_{2} - 210\beta _1 - 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 105\beta_{7} - 380\beta_{6} + 98\beta_{5} + 98\beta_{4} + 126\beta_{3} - 490\beta_{2} - 1162\beta _1 - 419 ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
−0.207107 − 2.54762i
1.20711 + 2.54762i
−0.207107 + 0.0981308i
1.20711 − 0.0981308i
−0.207107 + 2.54762i
1.20711 − 2.54762i
−0.207107 − 0.0981308i
1.20711 + 0.0981308i
−1.97374 + 0.323042i 1.70711 − 2.95680i 3.79129 − 1.27520i −1.34221 + 0.774923i −2.41421 + 6.38741i 0 −7.07107 + 3.74166i −1.32843 − 2.30090i 2.39883 − 1.96308i
67.2 −1.26663 + 1.54779i 0.292893 − 0.507306i −0.791288 − 3.92095i 7.82295 − 4.51658i 0.414214 + 1.09591i 0 7.07107 + 3.74166i 4.32843 + 7.49706i −2.91809 + 17.8291i
67.3 1.26663 − 1.54779i 1.70711 − 2.95680i −0.791288 − 3.92095i 1.34221 − 0.774923i −2.41421 − 6.38741i 0 −7.07107 − 3.74166i −1.32843 − 2.30090i 0.500665 − 3.05899i
67.4 1.97374 − 0.323042i 0.292893 − 0.507306i 3.79129 − 1.27520i −7.82295 + 4.51658i 0.414214 − 1.09591i 0 7.07107 − 3.74166i 4.32843 + 7.49706i −13.9814 + 11.4417i
275.1 −1.97374 − 0.323042i 1.70711 + 2.95680i 3.79129 + 1.27520i −1.34221 − 0.774923i −2.41421 − 6.38741i 0 −7.07107 − 3.74166i −1.32843 + 2.30090i 2.39883 + 1.96308i
275.2 −1.26663 − 1.54779i 0.292893 + 0.507306i −0.791288 + 3.92095i 7.82295 + 4.51658i 0.414214 − 1.09591i 0 7.07107 − 3.74166i 4.32843 − 7.49706i −2.91809 − 17.8291i
275.3 1.26663 + 1.54779i 1.70711 + 2.95680i −0.791288 + 3.92095i 1.34221 + 0.774923i −2.41421 + 6.38741i 0 −7.07107 + 3.74166i −1.32843 + 2.30090i 0.500665 + 3.05899i
275.4 1.97374 + 0.323042i 0.292893 + 0.507306i 3.79129 + 1.27520i −7.82295 − 4.51658i 0.414214 + 1.09591i 0 7.07107 + 3.74166i 4.32843 − 7.49706i −13.9814 − 11.4417i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 67.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
8.d odd 2 1 inner
56.k odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 392.3.k.j 8
7.b odd 2 1 392.3.k.i 8
7.c even 3 1 392.3.g.h 4
7.c even 3 1 inner 392.3.k.j 8
7.d odd 6 1 56.3.g.a ✓ 4
7.d odd 6 1 392.3.k.i 8
8.d odd 2 1 inner 392.3.k.j 8
21.g even 6 1 504.3.g.a 4
28.f even 6 1 224.3.g.a 4
28.g odd 6 1 1568.3.g.h 4
56.e even 2 1 392.3.k.i 8
56.j odd 6 1 224.3.g.a 4
56.k odd 6 1 392.3.g.h 4
56.k odd 6 1 inner 392.3.k.j 8
56.m even 6 1 56.3.g.a ✓ 4
56.m even 6 1 392.3.k.i 8
56.p even 6 1 1568.3.g.h 4
84.j odd 6 1 2016.3.g.a 4
112.v even 12 2 1792.3.d.g 8
112.x odd 12 2 1792.3.d.g 8
168.ba even 6 1 2016.3.g.a 4
168.be odd 6 1 504.3.g.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.3.g.a ✓ 4 7.d odd 6 1
56.3.g.a ✓ 4 56.m even 6 1
224.3.g.a 4 28.f even 6 1
224.3.g.a 4 56.j odd 6 1
392.3.g.h 4 7.c even 3 1
392.3.g.h 4 56.k odd 6 1
392.3.k.i 8 7.b odd 2 1
392.3.k.i 8 7.d odd 6 1
392.3.k.i 8 56.e even 2 1
392.3.k.i 8 56.m even 6 1
392.3.k.j 8 1.a even 1 1 trivial
392.3.k.j 8 7.c even 3 1 inner
392.3.k.j 8 8.d odd 2 1 inner
392.3.k.j 8 56.k odd 6 1 inner
504.3.g.a 4 21.g even 6 1
504.3.g.a 4 168.be odd 6 1
1568.3.g.h 4 28.g odd 6 1
1568.3.g.h 4 56.p even 6 1
1792.3.d.g 8 112.v even 12 2
1792.3.d.g 8 112.x odd 12 2
2016.3.g.a 4 84.j odd 6 1
2016.3.g.a 4 168.ba even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(392, [\chi])\):

\( T_{3}^{4} - 4T_{3}^{3} + 14T_{3}^{2} - 8T_{3} + 4 \) Copy content Toggle raw display
\( T_{5}^{8} - 84T_{5}^{6} + 6860T_{5}^{4} - 16464T_{5}^{2} + 38416 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 6 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$3$ \( (T^{4} - 4 T^{3} + 14 T^{2} + \cdots + 4)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} - 84 T^{6} + \cdots + 38416 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} + 8 T^{3} + \cdots + 3136)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 84 T^{2} + 196)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 36 T^{3} + \cdots + 85264)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 4 T^{3} + \cdots + 515524)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 567647723776 \) Copy content Toggle raw display
$29$ \( (T^{2} + 504)^{4} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 377801998336 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 821386940416 \) Copy content Toggle raw display
$41$ \( (T^{2} - 20 T - 188)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 40 T + 392)^{4} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 2517630976 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 377801998336 \) Copy content Toggle raw display
$59$ \( (T^{4} - 92 T^{3} + \cdots + 3511876)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 108554434576 \) Copy content Toggle raw display
$67$ \( (T^{4} - 112 T^{3} + \cdots + 6885376)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 10752 T^{2} + 3211264)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 116 T^{3} + \cdots + 10471696)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 96717311574016 \) Copy content Toggle raw display
$83$ \( (T^{2} + 44 T + 146)^{4} \) Copy content Toggle raw display
$89$ \( (T^{4} - 156 T^{3} + \cdots + 24324624)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 68 T - 15772)^{4} \) Copy content Toggle raw display
show more
show less