# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$392 = 2^{3} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 392.h (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

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

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$28q + 4q^{2} + 8q^{4} - 20q^{8} + 64q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$28q + 4q^{2} + 8q^{4} - 20q^{8} + 64q^{9} + 28q^{15} - 32q^{16} + 84q^{18} - 92q^{22} - 60q^{23} + 64q^{25} - 44q^{30} - 176q^{32} + 256q^{36} + 40q^{39} + 84q^{44} - 136q^{46} + 400q^{50} + 124q^{57} + 44q^{58} + 124q^{60} - 520q^{64} + 104q^{65} - 136q^{71} - 192q^{72} + 276q^{74} - 956q^{78} + 324q^{79} + 36q^{81} - 336q^{86} - 100q^{88} + 1020q^{92} - 580q^{95} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
293.1 −1.95479 0.422821i −3.86988 3.64244 + 1.65306i −4.67764 7.56482 + 1.63627i 0 −6.42128 4.77149i 5.97596 9.14383 + 1.97781i
293.2 −1.95479 0.422821i 3.86988 3.64244 + 1.65306i 4.67764 −7.56482 1.63627i 0 −6.42128 4.77149i 5.97596 −9.14383 1.97781i
293.3 −1.95479 + 0.422821i −3.86988 3.64244 1.65306i −4.67764 7.56482 1.63627i 0 −6.42128 + 4.77149i 5.97596 9.14383 1.97781i
293.4 −1.95479 + 0.422821i 3.86988 3.64244 1.65306i 4.67764 −7.56482 + 1.63627i 0 −6.42128 + 4.77149i 5.97596 −9.14383 + 1.97781i
293.5 −1.61618 1.17812i −2.33563 1.22408 + 3.80810i 3.10110 3.77480 + 2.75165i 0 2.50807 7.59668i −3.54484 −5.01193 3.65346i
293.6 −1.61618 1.17812i 2.33563 1.22408 + 3.80810i −3.10110 −3.77480 2.75165i 0 2.50807 7.59668i −3.54484 5.01193 + 3.65346i
293.7 −1.61618 + 1.17812i −2.33563 1.22408 3.80810i 3.10110 3.77480 2.75165i 0 2.50807 + 7.59668i −3.54484 −5.01193 + 3.65346i
293.8 −1.61618 + 1.17812i 2.33563 1.22408 3.80810i −3.10110 −3.77480 + 2.75165i 0 2.50807 + 7.59668i −3.54484 5.01193 3.65346i
293.9 −0.621472 1.90099i −3.40276 −3.22755 + 2.36283i 4.31716 2.11472 + 6.46862i 0 6.49755 + 4.66711i 2.57876 −2.68300 8.20689i
293.10 −0.621472 1.90099i 3.40276 −3.22755 + 2.36283i −4.31716 −2.11472 6.46862i 0 6.49755 + 4.66711i 2.57876 2.68300 + 8.20689i
293.11 −0.621472 + 1.90099i −3.40276 −3.22755 2.36283i 4.31716 2.11472 6.46862i 0 6.49755 4.66711i 2.57876 −2.68300 + 8.20689i
293.12 −0.621472 + 1.90099i 3.40276 −3.22755 2.36283i −4.31716 −2.11472 + 6.46862i 0 6.49755 4.66711i 2.57876 2.68300 8.20689i
293.13 0.324499 1.97350i −0.253256 −3.78940 1.28080i 3.57178 −0.0821813 + 0.499801i 0 −3.75731 + 7.06276i −8.93586 1.15904 7.04890i
293.14 0.324499 1.97350i 0.253256 −3.78940 1.28080i −3.57178 0.0821813 0.499801i 0 −3.75731 + 7.06276i −8.93586 −1.15904 + 7.04890i
293.15 0.324499 + 1.97350i −0.253256 −3.78940 + 1.28080i 3.57178 −0.0821813 0.499801i 0 −3.75731 7.06276i −8.93586 1.15904 + 7.04890i
293.16 0.324499 + 1.97350i 0.253256 −3.78940 + 1.28080i −3.57178 0.0821813 + 0.499801i 0 −3.75731 7.06276i −8.93586 −1.15904 7.04890i
293.17 1.28255 1.53463i −3.89635 −0.710153 3.93646i −8.85969 −4.99725 + 5.97944i 0 −6.95179 3.95886i 6.18155 −11.3630 + 13.5963i
293.18 1.28255 1.53463i 3.89635 −0.710153 3.93646i 8.85969 4.99725 5.97944i 0 −6.95179 3.95886i 6.18155 11.3630 13.5963i
293.19 1.28255 + 1.53463i −3.89635 −0.710153 + 3.93646i −8.85969 −4.99725 5.97944i 0 −6.95179 + 3.95886i 6.18155 −11.3630 13.5963i
293.20 1.28255 + 1.53463i 3.89635 −0.710153 + 3.93646i 8.85969 4.99725 + 5.97944i 0 −6.95179 + 3.95886i 6.18155 11.3630 + 13.5963i
See all 28 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 293.28 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.b odd 2 1 inner
8.b even 2 1 inner
56.h odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 392.3.h.a 28
4.b odd 2 1 1568.3.h.a 28
7.b odd 2 1 inner 392.3.h.a 28
7.c even 3 1 56.3.j.a 28
7.c even 3 1 392.3.j.e 28
7.d odd 6 1 56.3.j.a 28
7.d odd 6 1 392.3.j.e 28
8.b even 2 1 inner 392.3.h.a 28
8.d odd 2 1 1568.3.h.a 28
28.d even 2 1 1568.3.h.a 28
28.f even 6 1 224.3.n.a 28
28.g odd 6 1 224.3.n.a 28
56.e even 2 1 1568.3.h.a 28
56.h odd 2 1 inner 392.3.h.a 28
56.j odd 6 1 56.3.j.a 28
56.j odd 6 1 392.3.j.e 28
56.k odd 6 1 224.3.n.a 28
56.m even 6 1 224.3.n.a 28
56.p even 6 1 56.3.j.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 7.c even 3 1
56.3.j.a 28 7.d odd 6 1
56.3.j.a 28 56.j odd 6 1
56.3.j.a 28 56.p even 6 1
224.3.n.a 28 28.f even 6 1
224.3.n.a 28 28.g odd 6 1
224.3.n.a 28 56.k odd 6 1
224.3.n.a 28 56.m even 6 1
392.3.h.a 28 1.a even 1 1 trivial
392.3.h.a 28 7.b odd 2 1 inner
392.3.h.a 28 8.b even 2 1 inner
392.3.h.a 28 56.h odd 2 1 inner
392.3.j.e 28 7.c even 3 1
392.3.j.e 28 7.d odd 6 1
392.3.j.e 28 56.j odd 6 1
392.3.j.e 28 56.p even 6 1
1568.3.h.a 28 4.b odd 2 1
1568.3.h.a 28 8.d odd 2 1
1568.3.h.a 28 28.d even 2 1
1568.3.h.a 28 56.e even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{14} - 79 T_{3}^{12} + 2333 T_{3}^{10} - 32667 T_{3}^{8} + 220483 T_{3}^{6} - 618077 T_{3}^{4} + 407087 T_{3}^{2} - 23625$$ acting on $$S_{3}^{\mathrm{new}}(392, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database