# Properties

 Label 1280.3.e.k Level $1280$ Weight $3$ Character orbit 1280.e Analytic conductor $34.877$ Analytic rank $0$ Dimension $24$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1280 = 2^{8} \cdot 5$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1280.e (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$34.8774738381$$ Analytic rank: $$0$$ Dimension: $$24$$ Twist minimal: no (minimal twist has level 640) 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.

 $$\operatorname{Tr}(f)(q) =$$ $$24 q - 72 q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24 q - 72 q^{9} - 16 q^{11} + 48 q^{19} - 24 q^{25} - 112 q^{35} - 80 q^{41} + 168 q^{49} - 576 q^{51} + 496 q^{59} + 32 q^{65} + 224 q^{75} + 184 q^{81} - 144 q^{89} - 864 q^{91} + 368 q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
639.1 0 4.06912i 0 4.37877 + 2.41379i 0 −13.1670 0 −7.55776 0
639.2 0 4.06912i 0 4.37877 2.41379i 0 −13.1670 0 −7.55776 0
639.3 0 1.73799i 0 −3.35996 3.70279i 0 −7.32638 0 5.97938 0
639.4 0 1.73799i 0 −3.35996 + 3.70279i 0 −7.32638 0 5.97938 0
639.5 0 0.933412i 0 0.723593 + 4.94736i 0 6.91082 0 8.12874 0
639.6 0 0.933412i 0 0.723593 4.94736i 0 6.91082 0 8.12874 0
639.7 0 5.46407i 0 3.70957 3.35247i 0 −5.17181 0 −20.8560 0
639.8 0 5.46407i 0 3.70957 + 3.35247i 0 −5.17181 0 −20.8560 0
639.9 0 4.20423i 0 −1.52118 4.76298i 0 3.29905 0 −8.67551 0
639.10 0 4.20423i 0 −1.52118 + 4.76298i 0 3.29905 0 −8.67551 0
639.11 0 2.00470i 0 4.99385 0.247851i 0 4.85433 0 4.98119 0
639.12 0 2.00470i 0 4.99385 + 0.247851i 0 4.85433 0 4.98119 0
639.13 0 2.00470i 0 −4.99385 + 0.247851i 0 −4.85433 0 4.98119 0
639.14 0 2.00470i 0 −4.99385 0.247851i 0 −4.85433 0 4.98119 0
639.15 0 4.20423i 0 1.52118 + 4.76298i 0 −3.29905 0 −8.67551 0
639.16 0 4.20423i 0 1.52118 4.76298i 0 −3.29905 0 −8.67551 0
639.17 0 5.46407i 0 −3.70957 + 3.35247i 0 5.17181 0 −20.8560 0
639.18 0 5.46407i 0 −3.70957 3.35247i 0 5.17181 0 −20.8560 0
639.19 0 0.933412i 0 −0.723593 4.94736i 0 −6.91082 0 8.12874 0
639.20 0 0.933412i 0 −0.723593 + 4.94736i 0 −6.91082 0 8.12874 0
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 639.24 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
8.d odd 2 1 inner
40.e odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1280.3.e.k 24
4.b odd 2 1 1280.3.e.l 24
5.b even 2 1 inner 1280.3.e.k 24
8.b even 2 1 1280.3.e.l 24
8.d odd 2 1 inner 1280.3.e.k 24
16.e even 4 1 640.3.h.a 24
16.e even 4 1 640.3.h.b yes 24
16.f odd 4 1 640.3.h.a 24
16.f odd 4 1 640.3.h.b yes 24
20.d odd 2 1 1280.3.e.l 24
40.e odd 2 1 inner 1280.3.e.k 24
40.f even 2 1 1280.3.e.l 24
80.k odd 4 1 640.3.h.a 24
80.k odd 4 1 640.3.h.b yes 24
80.q even 4 1 640.3.h.a 24
80.q even 4 1 640.3.h.b yes 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
640.3.h.a 24 16.e even 4 1
640.3.h.a 24 16.f odd 4 1
640.3.h.a 24 80.k odd 4 1
640.3.h.a 24 80.q even 4 1
640.3.h.b yes 24 16.e even 4 1
640.3.h.b yes 24 16.f odd 4 1
640.3.h.b yes 24 80.k odd 4 1
640.3.h.b yes 24 80.q even 4 1
1280.3.e.k 24 1.a even 1 1 trivial
1280.3.e.k 24 5.b even 2 1 inner
1280.3.e.k 24 8.d odd 2 1 inner
1280.3.e.k 24 40.e odd 2 1 inner
1280.3.e.l 24 4.b odd 2 1
1280.3.e.l 24 8.b even 2 1
1280.3.e.l 24 20.d odd 2 1
1280.3.e.l 24 40.f even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(1280, [\chi])$$:

 $$T_{3}^{12} + 72 T_{3}^{10} + 1840 T_{3}^{8} + 20320 T_{3}^{6} + 93824 T_{3}^{4} + 173568 T_{3}^{2} + 92416$$ $$T_{7}^{12} - 336 T_{7}^{10} + 38144 T_{7}^{8} - 2008032 T_{7}^{6} + 52816128 T_{7}^{4} - 661713920 T_{7}^{2} + 3048806656$$ $$T_{11}^{6} + 4 T_{11}^{5} - 348 T_{11}^{4} - 1184 T_{11}^{3} + 29168 T_{11}^{2} + 87872 T_{11} - 4672$$ $$T_{13}^{12} - 1216 T_{13}^{10} + 564864 T_{13}^{8} - 126646784 T_{13}^{6} + 14349455360 T_{13}^{4} - 771603464192 T_{13}^{2} +$$$$15\!\cdots\!36$$