# Properties

 Label 896.2.u.c Level $896$ Weight $2$ Character orbit 896.u Analytic conductor $7.155$ Analytic rank $0$ Dimension $52$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$896 = 2^{7} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 896.u (of order $$8$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.15459602111$$ Analytic rank: $$0$$ Dimension: $$52$$ Relative dimension: $$13$$ over $$\Q(\zeta_{8})$$ Twist minimal: no (minimal twist has level 224) Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$52 q + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$52 q + 20 q^{23} + 24 q^{27} - 48 q^{33} + 24 q^{39} + 44 q^{43} + 40 q^{45} - 16 q^{51} - 36 q^{53} - 32 q^{55} - 32 q^{61} - 68 q^{63} + 80 q^{65} - 28 q^{67} - 32 q^{69} - 32 q^{75} - 12 q^{77} + 64 q^{85} + 56 q^{87} + 64 q^{95} - 72 q^{97} + 64 q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
113.1 0 −1.23966 2.99281i 0 2.25399 + 0.933632i 0 0.707107 + 0.707107i 0 −5.29885 + 5.29885i 0
113.2 0 −1.19114 2.87567i 0 −3.07070 1.27193i 0 0.707107 + 0.707107i 0 −4.72935 + 4.72935i 0
113.3 0 −0.738703 1.78339i 0 −0.509656 0.211106i 0 0.707107 + 0.707107i 0 −0.513466 + 0.513466i 0
113.4 0 −0.694865 1.67755i 0 −0.286541 0.118689i 0 0.707107 + 0.707107i 0 −0.210023 + 0.210023i 0
113.5 0 −0.424336 1.02444i 0 3.70330 + 1.53396i 0 0.707107 + 0.707107i 0 1.25191 1.25191i 0
113.6 0 −0.264188 0.637807i 0 −1.89692 0.785729i 0 0.707107 + 0.707107i 0 1.78432 1.78432i 0
113.7 0 −0.0427122 0.103116i 0 2.38883 + 0.989485i 0 0.707107 + 0.707107i 0 2.11251 2.11251i 0
113.8 0 0.439285 + 1.06053i 0 −3.40891 1.41202i 0 0.707107 + 0.707107i 0 1.18957 1.18957i 0
113.9 0 0.587552 + 1.41848i 0 2.93480 + 1.21564i 0 0.707107 + 0.707107i 0 0.454461 0.454461i 0
113.10 0 0.622222 + 1.50218i 0 −0.781648 0.323769i 0 0.707107 + 0.707107i 0 0.251948 0.251948i 0
113.11 0 0.728858 + 1.75962i 0 −1.90564 0.789342i 0 0.707107 + 0.707107i 0 −0.443707 + 0.443707i 0
113.12 0 0.951433 + 2.29696i 0 2.43243 + 1.00754i 0 0.707107 + 0.707107i 0 −2.24949 + 2.24949i 0
113.13 0 1.26626 + 3.05702i 0 −1.85332 0.767672i 0 0.707107 + 0.707107i 0 −5.62065 + 5.62065i 0
337.1 0 −2.91760 1.20851i 0 −0.629460 1.51965i 0 −0.707107 + 0.707107i 0 4.93055 + 4.93055i 0
337.2 0 −2.47594 1.02557i 0 −1.23251 2.97553i 0 −0.707107 + 0.707107i 0 2.95715 + 2.95715i 0
337.3 0 −2.18713 0.905938i 0 0.797931 + 1.92638i 0 −0.707107 + 0.707107i 0 1.84149 + 1.84149i 0
337.4 0 −1.74965 0.724727i 0 1.44353 + 3.48498i 0 −0.707107 + 0.707107i 0 0.414709 + 0.414709i 0
337.5 0 −0.999166 0.413868i 0 0.523077 + 1.26282i 0 −0.707107 + 0.707107i 0 −1.29427 1.29427i 0
337.6 0 −0.496926 0.205834i 0 −0.334218 0.806875i 0 −0.707107 + 0.707107i 0 −1.91675 1.91675i 0
337.7 0 0.301186 + 0.124755i 0 −0.107860 0.260396i 0 −0.707107 + 0.707107i 0 −2.04617 2.04617i 0
See all 52 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 785.13 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
32.g even 8 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 896.2.u.c 52
4.b odd 2 1 224.2.u.c 52
32.g even 8 1 inner 896.2.u.c 52
32.h odd 8 1 224.2.u.c 52

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.2.u.c 52 4.b odd 2 1
224.2.u.c 52 32.h odd 8 1
896.2.u.c 52 1.a even 1 1 trivial
896.2.u.c 52 32.g even 8 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$20\!\cdots\!72$$$$T_{3}^{20} -$$$$15\!\cdots\!40$$$$T_{3}^{19} +$$$$28\!\cdots\!12$$$$T_{3}^{18} +$$$$10\!\cdots\!92$$$$T_{3}^{17} +$$$$13\!\cdots\!00$$$$T_{3}^{16} + 418291314944 T_{3}^{15} + 193510615808 T_{3}^{14} -$$$$12\!\cdots\!32$$$$T_{3}^{13} +$$$$28\!\cdots\!08$$$$T_{3}^{12} -$$$$47\!\cdots\!96$$$$T_{3}^{11} +$$$$37\!\cdots\!48$$$$T_{3}^{10} + 773593018368 T_{3}^{9} + 768512428032 T_{3}^{8} - 566562717696 T_{3}^{7} - 485462818816 T_{3}^{6} + 161995554816 T_{3}^{5} + 145292263424 T_{3}^{4} - 68025581568 T_{3}^{3} + 6817841152 T_{3}^{2} + 17825792 T_{3} + 151519232$$">$$T_{3}^{52} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(896, [\chi])$$.