# Properties

 Label 56.3.j.a Level 56 Weight 3 Character orbit 56.j Analytic conductor 1.526 Analytic rank 0 Dimension 28 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$56 = 2^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 56.j (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.52588948042$$ Analytic rank: $$0$$ Dimension: $$28$$ Relative dimension: $$14$$ over $$\Q(\zeta_{6})$$ Twist minimal: yes 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.

 $$\operatorname{Tr}(f)(q) =$$ $$28q - 2q^{2} - 4q^{4} - 4q^{7} - 20q^{8} - 32q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$28q - 2q^{2} - 4q^{4} - 4q^{7} - 20q^{8} - 32q^{9} + 24q^{10} - 18q^{12} + 24q^{14} + 28q^{15} + 16q^{16} - 6q^{17} - 42q^{18} - 92q^{22} + 30q^{23} - 30q^{24} - 32q^{25} - 30q^{26} - 42q^{28} + 22q^{30} - 6q^{31} + 88q^{32} - 6q^{33} + 256q^{36} + 6q^{38} - 20q^{39} + 102q^{40} + 18q^{42} - 42q^{44} + 68q^{46} - 294q^{47} - 20q^{49} + 400q^{50} - 168q^{52} + 330q^{54} - 96q^{56} + 124q^{57} - 22q^{58} - 62q^{60} + 432q^{63} - 520q^{64} - 52q^{65} - 306q^{66} - 456q^{68} - 324q^{70} - 136q^{71} + 96q^{72} + 234q^{73} - 138q^{74} - 956q^{78} - 162q^{79} + 276q^{80} - 18q^{81} - 642q^{82} + 504q^{84} + 168q^{86} + 48q^{87} + 50q^{88} - 150q^{89} + 1020q^{92} + 618q^{94} + 290q^{95} + 1044q^{96} + 424q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1 −1.97030 + 0.343404i −1.94818 3.37434i 3.76415 1.35322i −4.42985 + 7.67272i 4.99725 + 5.97944i −6.92329 + 1.03347i −6.95179 + 3.95886i −3.09078 + 5.35338i 6.09327 16.6388i
5.2 −1.87135 0.705725i −0.126628 0.219326i 3.00390 + 2.64132i 1.78589 3.09325i 0.0821813 + 0.499801i 2.89466 6.37346i −3.75731 7.06276i 4.46793 7.73868i −5.52501 + 4.52821i
5.3 −1.70738 + 1.04157i 2.78005 + 4.81519i 1.83027 3.55670i −1.52921 + 2.64866i −9.76195 5.32573i 0.608243 6.97352i 0.579586 + 7.97898i −10.9574 + 18.9787i −0.147833 6.11504i
5.4 −1.61426 + 1.18075i −0.455431 0.788830i 1.21166 3.81207i 3.17251 5.49495i 1.66660 + 0.735624i 3.79106 + 5.88455i 2.54518 + 7.58433i 4.08516 7.07571i 1.36692 + 12.6162i
5.5 −1.33557 1.48871i 1.70138 + 2.94687i −0.432496 + 3.97655i −2.15858 + 3.73877i 2.11472 6.46862i −1.43197 + 6.85197i 6.49755 4.66711i −1.28938 + 2.23327i 8.44888 1.77990i
5.6 −0.215431 + 1.98836i 0.455431 + 0.788830i −3.90718 0.856711i −3.17251 + 5.49495i −1.66660 + 0.735624i 3.79106 + 5.88455i 2.54518 7.58433i 4.08516 7.07571i −10.2425 7.49189i
5.7 −0.212190 1.98871i −1.16781 2.02271i −3.90995 + 0.843971i 1.55055 2.68563i −3.77480 + 2.75165i −6.89374 1.21502i 2.50807 + 7.59668i 1.77242 3.06992i −5.66995 2.51373i
5.8 −0.0483365 + 1.99942i −2.78005 4.81519i −3.99533 0.193289i 1.52921 2.64866i 9.76195 5.32573i 0.608243 6.97352i 0.579586 7.97898i −10.9574 + 18.9787i 5.22186 + 3.18554i
5.9 0.611223 1.90431i 1.93494 + 3.35141i −3.25281 2.32792i 2.33882 4.05096i 7.56482 1.63627i 6.95505 + 0.792023i −6.42128 + 4.77149i −2.98798 + 5.17534i −6.28474 6.92988i
5.10 0.687752 + 1.87803i 1.94818 + 3.37434i −3.05399 + 2.58324i 4.42985 7.67272i −4.99725 + 5.97944i −6.92329 + 1.03347i −6.95179 3.95886i −3.09078 + 5.35338i 17.4562 + 3.04246i
5.11 1.34357 1.48149i −1.93494 3.35141i −0.389632 3.98098i −2.33882 + 4.05096i −7.56482 1.63627i 6.95505 + 0.792023i −6.42128 4.77149i −2.98798 + 5.17534i 2.85908 + 8.90769i
5.12 1.54685 + 1.26777i 0.126628 + 0.219326i 0.785498 + 3.92212i −1.78589 + 3.09325i −0.0821813 + 0.499801i 2.89466 6.37346i −3.75731 + 7.06276i 4.46793 7.73868i −6.68405 + 2.52070i
5.13 1.82837 0.810594i 1.16781 + 2.02271i 2.68588 2.96413i −1.55055 + 2.68563i 3.77480 + 2.75165i −6.89374 1.21502i 2.50807 7.59668i 1.77242 3.06992i −0.658023 + 6.16719i
5.14 1.95704 + 0.412286i −1.70138 2.94687i 3.66004 + 1.61372i 2.15858 3.73877i −2.11472 6.46862i −1.43197 + 6.85197i 6.49755 + 4.66711i −1.28938 + 2.23327i 5.76588 6.42699i
45.1 −1.97030 0.343404i −1.94818 + 3.37434i 3.76415 + 1.35322i −4.42985 7.67272i 4.99725 5.97944i −6.92329 1.03347i −6.95179 3.95886i −3.09078 5.35338i 6.09327 + 16.6388i
45.2 −1.87135 + 0.705725i −0.126628 + 0.219326i 3.00390 2.64132i 1.78589 + 3.09325i 0.0821813 0.499801i 2.89466 + 6.37346i −3.75731 + 7.06276i 4.46793 + 7.73868i −5.52501 4.52821i
45.3 −1.70738 1.04157i 2.78005 4.81519i 1.83027 + 3.55670i −1.52921 2.64866i −9.76195 + 5.32573i 0.608243 + 6.97352i 0.579586 7.97898i −10.9574 18.9787i −0.147833 + 6.11504i
45.4 −1.61426 1.18075i −0.455431 + 0.788830i 1.21166 + 3.81207i 3.17251 + 5.49495i 1.66660 0.735624i 3.79106 5.88455i 2.54518 7.58433i 4.08516 + 7.07571i 1.36692 12.6162i
45.5 −1.33557 + 1.48871i 1.70138 2.94687i −0.432496 3.97655i −2.15858 3.73877i 2.11472 + 6.46862i −1.43197 6.85197i 6.49755 + 4.66711i −1.28938 2.23327i 8.44888 + 1.77990i
45.6 −0.215431 1.98836i 0.455431 0.788830i −3.90718 + 0.856711i −3.17251 5.49495i −1.66660 0.735624i 3.79106 5.88455i 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 45.14 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 56.3.j.a 28
4.b odd 2 1 224.3.n.a 28
7.b odd 2 1 392.3.j.e 28
7.c even 3 1 392.3.h.a 28
7.c even 3 1 392.3.j.e 28
7.d odd 6 1 inner 56.3.j.a 28
7.d odd 6 1 392.3.h.a 28
8.b even 2 1 inner 56.3.j.a 28
8.d odd 2 1 224.3.n.a 28
28.f even 6 1 224.3.n.a 28
28.f even 6 1 1568.3.h.a 28
28.g odd 6 1 1568.3.h.a 28
56.h odd 2 1 392.3.j.e 28
56.j odd 6 1 inner 56.3.j.a 28
56.j odd 6 1 392.3.h.a 28
56.k odd 6 1 1568.3.h.a 28
56.m even 6 1 224.3.n.a 28
56.m even 6 1 1568.3.h.a 28
56.p even 6 1 392.3.h.a 28
56.p even 6 1 392.3.j.e 28

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.3.j.a 28 1.a even 1 1 trivial
56.3.j.a 28 7.d odd 6 1 inner
56.3.j.a 28 8.b even 2 1 inner
56.3.j.a 28 56.j odd 6 1 inner
224.3.n.a 28 4.b odd 2 1
224.3.n.a 28 8.d odd 2 1
224.3.n.a 28 28.f even 6 1
224.3.n.a 28 56.m even 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 7.b odd 2 1
392.3.j.e 28 7.c even 3 1
392.3.j.e 28 56.h odd 2 1
392.3.j.e 28 56.p even 6 1
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 is the entire newspace $$S_{3}^{\mathrm{new}}(56, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database