Properties

Label 392.3.j.e
Level 392
Weight 3
Character orbit 392.j
Analytic conductor 10.681
Analytic rank 0
Dimension 28
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 392 = 2^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 392.j (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(10.6812263629\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(14\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 56)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 28q - 2q^{2} - 4q^{4} - 20q^{8} - 32q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 28q - 2q^{2} - 4q^{4} - 20q^{8} - 32q^{9} - 24q^{10} + 18q^{12} + 28q^{15} + 16q^{16} + 6q^{17} - 42q^{18} - 92q^{22} + 30q^{23} + 30q^{24} - 32q^{25} + 30q^{26} + 22q^{30} + 6q^{31} + 88q^{32} + 6q^{33} + 256q^{36} - 6q^{38} - 20q^{39} - 102q^{40} - 42q^{44} + 68q^{46} + 294q^{47} + 400q^{50} + 168q^{52} - 330q^{54} + 124q^{57} - 22q^{58} - 62q^{60} - 520q^{64} - 52q^{65} + 306q^{66} + 456q^{68} - 136q^{71} + 96q^{72} - 234q^{73} - 138q^{74} - 956q^{78} - 162q^{79} - 276q^{80} - 18q^{81} + 642q^{82} + 168q^{86} - 48q^{87} + 50q^{88} + 150q^{89} + 1020q^{92} - 618q^{94} + 290q^{95} - 1044q^{96} + O(q^{100}) \)

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.

Label \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
117.1 −1.97030 + 0.343404i 1.94818 + 3.37434i 3.76415 1.35322i 4.42985 7.67272i −4.99725 5.97944i 0 −6.95179 + 3.95886i −3.09078 + 5.35338i −6.09327 + 16.6388i
117.2 −1.87135 0.705725i 0.126628 + 0.219326i 3.00390 + 2.64132i −1.78589 + 3.09325i −0.0821813 0.499801i 0 −3.75731 7.06276i 4.46793 7.73868i 5.52501 4.52821i
117.3 −1.70738 + 1.04157i −2.78005 4.81519i 1.83027 3.55670i 1.52921 2.64866i 9.76195 + 5.32573i 0 0.579586 + 7.97898i −10.9574 + 18.9787i 0.147833 + 6.11504i
117.4 −1.61426 + 1.18075i 0.455431 + 0.788830i 1.21166 3.81207i −3.17251 + 5.49495i −1.66660 0.735624i 0 2.54518 + 7.58433i 4.08516 7.07571i −1.36692 12.6162i
117.5 −1.33557 1.48871i −1.70138 2.94687i −0.432496 + 3.97655i 2.15858 3.73877i −2.11472 + 6.46862i 0 6.49755 4.66711i −1.28938 + 2.23327i −8.44888 + 1.77990i
117.6 −0.215431 + 1.98836i −0.455431 0.788830i −3.90718 0.856711i 3.17251 5.49495i 1.66660 0.735624i 0 2.54518 7.58433i 4.08516 7.07571i 10.2425 + 7.49189i
117.7 −0.212190 1.98871i 1.16781 + 2.02271i −3.90995 + 0.843971i −1.55055 + 2.68563i 3.77480 2.75165i 0 2.50807 + 7.59668i 1.77242 3.06992i 5.66995 + 2.51373i
117.8 −0.0483365 + 1.99942i 2.78005 + 4.81519i −3.99533 0.193289i −1.52921 + 2.64866i −9.76195 + 5.32573i 0 0.579586 7.97898i −10.9574 + 18.9787i −5.22186 3.18554i
117.9 0.611223 1.90431i −1.93494 3.35141i −3.25281 2.32792i −2.33882 + 4.05096i −7.56482 + 1.63627i 0 −6.42128 + 4.77149i −2.98798 + 5.17534i 6.28474 + 6.92988i
117.10 0.687752 + 1.87803i −1.94818 3.37434i −3.05399 + 2.58324i −4.42985 + 7.67272i 4.99725 5.97944i 0 −6.95179 3.95886i −3.09078 + 5.35338i −17.4562 3.04246i
117.11 1.34357 1.48149i 1.93494 + 3.35141i −0.389632 3.98098i 2.33882 4.05096i 7.56482 + 1.63627i 0 −6.42128 4.77149i −2.98798 + 5.17534i −2.85908 8.90769i
117.12 1.54685 + 1.26777i −0.126628 0.219326i 0.785498 + 3.92212i 1.78589 3.09325i 0.0821813 0.499801i 0 −3.75731 + 7.06276i 4.46793 7.73868i 6.68405 2.52070i
117.13 1.82837 0.810594i −1.16781 2.02271i 2.68588 2.96413i 1.55055 2.68563i −3.77480 2.75165i 0 2.50807 7.59668i 1.77242 3.06992i 0.658023 6.16719i
117.14 1.95704 + 0.412286i 1.70138 + 2.94687i 3.66004 + 1.61372i −2.15858 + 3.73877i 2.11472 + 6.46862i 0 6.49755 + 4.66711i −1.28938 + 2.23327i −5.76588 + 6.42699i
325.1 −1.97030 0.343404i 1.94818 3.37434i 3.76415 + 1.35322i 4.42985 + 7.67272i −4.99725 + 5.97944i 0 −6.95179 3.95886i −3.09078 5.35338i −6.09327 16.6388i
325.2 −1.87135 + 0.705725i 0.126628 0.219326i 3.00390 2.64132i −1.78589 3.09325i −0.0821813 + 0.499801i 0 −3.75731 + 7.06276i 4.46793 + 7.73868i 5.52501 + 4.52821i
325.3 −1.70738 1.04157i −2.78005 + 4.81519i 1.83027 + 3.55670i 1.52921 + 2.64866i 9.76195 5.32573i 0 0.579586 7.97898i −10.9574 18.9787i 0.147833 6.11504i
325.4 −1.61426 1.18075i 0.455431 0.788830i 1.21166 + 3.81207i −3.17251 5.49495i −1.66660 + 0.735624i 0 2.54518 7.58433i 4.08516 + 7.07571i −1.36692 + 12.6162i
325.5 −1.33557 + 1.48871i −1.70138 + 2.94687i −0.432496 3.97655i 2.15858 + 3.73877i −2.11472 6.46862i 0 6.49755 + 4.66711i −1.28938 2.23327i −8.44888 1.77990i
325.6 −0.215431 1.98836i −0.455431 + 0.788830i −3.90718 + 0.856711i 3.17251 + 5.49495i 1.66660 + 0.735624i 0 2.54518 + 7.58433i 4.08516 + 7.07571i 10.2425 7.49189i
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 325.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
8.b even 2 1 inner
56.j 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.j.e 28
7.b odd 2 1 56.3.j.a 28
7.c even 3 1 56.3.j.a 28
7.c even 3 1 392.3.h.a 28
7.d odd 6 1 392.3.h.a 28
7.d odd 6 1 inner 392.3.j.e 28
8.b even 2 1 inner 392.3.j.e 28
28.d even 2 1 224.3.n.a 28
28.f even 6 1 1568.3.h.a 28
28.g odd 6 1 224.3.n.a 28
28.g odd 6 1 1568.3.h.a 28
56.e even 2 1 224.3.n.a 28
56.h odd 2 1 56.3.j.a 28
56.j odd 6 1 392.3.h.a 28
56.j odd 6 1 inner 392.3.j.e 28
56.k odd 6 1 224.3.n.a 28
56.k odd 6 1 1568.3.h.a 28
56.m even 6 1 1568.3.h.a 28
56.p even 6 1 56.3.j.a 28
56.p even 6 1 392.3.h.a 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.3.j.a 28 7.b odd 2 1
56.3.j.a 28 7.c even 3 1
56.3.j.a 28 56.h odd 2 1
56.3.j.a 28 56.p even 6 1
224.3.n.a 28 28.d even 2 1
224.3.n.a 28 28.g odd 6 1
224.3.n.a 28 56.e even 2 1
224.3.n.a 28 56.k odd 6 1
392.3.h.a 28 7.c even 3 1
392.3.h.a 28 7.d odd 6 1
392.3.h.a 28 56.j odd 6 1
392.3.h.a 28 56.p even 6 1
392.3.j.e 28 1.a even 1 1 trivial
392.3.j.e 28 7.d odd 6 1 inner
392.3.j.e 28 8.b even 2 1 inner
392.3.j.e 28 56.j odd 6 1 inner
1568.3.h.a 28 28.f even 6 1
1568.3.h.a 28 28.g odd 6 1
1568.3.h.a 28 56.k odd 6 1
1568.3.h.a 28 56.m even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{3}^{28} + \cdots\) acting on \(S_{3}^{\mathrm{new}}(392, [\chi])\).

Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database