Properties

 Label 336.4.h.b Level $336$ Weight $4$ Character orbit 336.h Analytic conductor $19.825$ Analytic rank $0$ Dimension $24$ CM no Inner twists $4$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$336 = 2^{4} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 336.h (of order $$2$$, degree $$1$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$19.8246417619$$ Analytic rank: $$0$$ Dimension: $$24$$ Twist minimal: yes 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.

 $$\operatorname{Tr}(f)(q) =$$ $$24 q - 136 q^{9}+O(q^{10})$$ 24 * q - 136 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$24 q - 136 q^{9} + 96 q^{13} - 112 q^{21} + 168 q^{25} - 320 q^{33} - 864 q^{37} + 592 q^{45} - 1176 q^{49} - 1120 q^{57} - 2304 q^{61} + 1168 q^{69} + 3312 q^{73} - 968 q^{81} - 5808 q^{85} + 1776 q^{93} + 6480 q^{97}+O(q^{100})$$ 24 * q - 136 * q^9 + 96 * q^13 - 112 * q^21 + 168 * q^25 - 320 * q^33 - 864 * q^37 + 592 * q^45 - 1176 * q^49 - 1120 * q^57 - 2304 * q^61 + 1168 * q^69 + 3312 * q^73 - 968 * q^81 - 5808 * q^85 + 1776 * q^93 + 6480 * q^97

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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
239.1 0 −5.19367 0.160438i 0 7.59851i 0 7.00000i 0 26.9485 + 1.66652i 0
239.2 0 −5.19367 + 0.160438i 0 7.59851i 0 7.00000i 0 26.9485 1.66652i 0
239.3 0 −4.13038 3.15276i 0 1.18345i 0 7.00000i 0 7.12016 + 26.0443i 0
239.4 0 −4.13038 + 3.15276i 0 1.18345i 0 7.00000i 0 7.12016 26.0443i 0
239.5 0 −3.03558 4.21726i 0 3.22455i 0 7.00000i 0 −8.57055 + 25.6036i 0
239.6 0 −3.03558 + 4.21726i 0 3.22455i 0 7.00000i 0 −8.57055 25.6036i 0
239.7 0 −2.33643 4.64124i 0 12.0055i 0 7.00000i 0 −16.0822 + 21.6878i 0
239.8 0 −2.33643 + 4.64124i 0 12.0055i 0 7.00000i 0 −16.0822 21.6878i 0
239.9 0 −1.99829 4.79655i 0 16.7287i 0 7.00000i 0 −19.0137 + 19.1697i 0
239.10 0 −1.99829 + 4.79655i 0 16.7287i 0 7.00000i 0 −19.0137 19.1697i 0
239.11 0 −1.13969 5.06963i 0 14.6453i 0 7.00000i 0 −24.4022 + 11.5556i 0
239.12 0 −1.13969 + 5.06963i 0 14.6453i 0 7.00000i 0 −24.4022 11.5556i 0
239.13 0 1.13969 5.06963i 0 14.6453i 0 7.00000i 0 −24.4022 11.5556i 0
239.14 0 1.13969 + 5.06963i 0 14.6453i 0 7.00000i 0 −24.4022 + 11.5556i 0
239.15 0 1.99829 4.79655i 0 16.7287i 0 7.00000i 0 −19.0137 19.1697i 0
239.16 0 1.99829 + 4.79655i 0 16.7287i 0 7.00000i 0 −19.0137 + 19.1697i 0
239.17 0 2.33643 4.64124i 0 12.0055i 0 7.00000i 0 −16.0822 21.6878i 0
239.18 0 2.33643 + 4.64124i 0 12.0055i 0 7.00000i 0 −16.0822 + 21.6878i 0
239.19 0 3.03558 4.21726i 0 3.22455i 0 7.00000i 0 −8.57055 25.6036i 0
239.20 0 3.03558 + 4.21726i 0 3.22455i 0 7.00000i 0 −8.57055 + 25.6036i 0
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 239.24 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.4.h.b 24
3.b odd 2 1 inner 336.4.h.b 24
4.b odd 2 1 inner 336.4.h.b 24
12.b even 2 1 inner 336.4.h.b 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.4.h.b 24 1.a even 1 1 trivial
336.4.h.b 24 3.b odd 2 1 inner
336.4.h.b 24 4.b odd 2 1 inner
336.4.h.b 24 12.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{12} + 708T_{5}^{10} + 176364T_{5}^{8} + 18224416T_{5}^{6} + 693441840T_{5}^{4} + 6129603648T_{5}^{2} + 7274042944$$ acting on $$S_{4}^{\mathrm{new}}(336, [\chi])$$.