# Properties

 Label 896.2.bh.a Level $896$ Weight $2$ Character orbit 896.bh Analytic conductor $7.155$ Analytic rank $0$ Dimension $240$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$896 = 2^{7} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 896.bh (of order $$24$$, degree $$8$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.15459602111$$ Analytic rank: $$0$$ Dimension: $$240$$ Relative dimension: $$30$$ over $$\Q(\zeta_{24})$$ Twist minimal: no (minimal twist has level 224) Sato-Tate group: $\mathrm{SU}(2)[C_{24}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$240 q + 4 q^{3} - 4 q^{5} + 8 q^{7} - 4 q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$240 q + 4 q^{3} - 4 q^{5} + 8 q^{7} - 4 q^{9} + 4 q^{11} - 16 q^{13} + 4 q^{19} - 8 q^{21} + 12 q^{23} - 4 q^{25} + 16 q^{27} - 16 q^{29} + 56 q^{31} - 8 q^{33} + 32 q^{35} - 4 q^{37} + 4 q^{39} - 16 q^{41} + 8 q^{45} + 28 q^{51} - 20 q^{53} + 16 q^{55} - 16 q^{57} + 36 q^{59} - 4 q^{61} + 16 q^{63} - 8 q^{65} - 36 q^{67} - 16 q^{69} - 48 q^{71} - 4 q^{73} - 16 q^{75} - 8 q^{77} + 96 q^{83} - 56 q^{85} + 4 q^{87} - 4 q^{89} + 56 q^{91} + 20 q^{93} + 8 q^{95} - 32 q^{97} - 8 q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
81.1 0 −2.53727 + 1.94692i 0 0.157528 0.205294i 0 −0.893273 2.49039i 0 1.87081 6.98194i 0
81.2 0 −2.44576 + 1.87670i 0 0.0746500 0.0972858i 0 −1.07291 + 2.41844i 0 1.68329 6.28212i 0
81.3 0 −2.34780 + 1.80153i 0 −2.10755 + 2.74661i 0 2.53918 0.743361i 0 1.49020 5.56151i 0
81.4 0 −1.87086 + 1.43556i 0 −0.351264 + 0.457777i 0 −2.57326 + 0.615072i 0 0.662818 2.47367i 0
81.5 0 −1.80239 + 1.38302i 0 −0.194347 + 0.253278i 0 1.82678 + 1.91386i 0 0.559406 2.08773i 0
81.6 0 −1.59186 + 1.22147i 0 1.50255 1.95816i 0 2.25138 1.38970i 0 0.265550 0.991047i 0
81.7 0 −1.56993 + 1.20465i 0 −1.98898 + 2.59208i 0 −1.59198 2.11320i 0 0.237043 0.884657i 0
81.8 0 −1.42891 + 1.09644i 0 1.65207 2.15302i 0 −0.399279 2.61545i 0 0.0631376 0.235633i 0
81.9 0 −1.32101 + 1.01365i 0 2.54115 3.31169i 0 1.39322 + 2.24921i 0 −0.0588677 + 0.219697i 0
81.10 0 −0.867605 + 0.665736i 0 −0.00636490 + 0.00829490i 0 0.529064 + 2.59231i 0 −0.466924 + 1.74259i 0
81.11 0 −0.678599 + 0.520708i 0 −2.43841 + 3.17779i 0 −1.24705 + 2.33343i 0 −0.587096 + 2.19107i 0
81.12 0 −0.653842 + 0.501711i 0 −1.21118 + 1.57844i 0 1.14530 2.38501i 0 −0.600661 + 2.24170i 0
81.13 0 −0.194135 + 0.148965i 0 2.33985 3.04935i 0 −2.62216 0.352546i 0 −0.760959 + 2.83994i 0
81.14 0 −0.163619 + 0.125549i 0 −0.947705 + 1.23507i 0 2.63443 0.244488i 0 −0.765449 + 2.85669i 0
81.15 0 −0.112695 + 0.0864738i 0 1.22200 1.59255i 0 1.93515 1.80422i 0 −0.771235 + 2.87829i 0
81.16 0 −0.102520 + 0.0786667i 0 1.38225 1.80138i 0 −2.39854 + 1.11670i 0 −0.772135 + 2.88165i 0
81.17 0 0.360007 0.276243i 0 0.0970846 0.126523i 0 −2.18584 1.49067i 0 −0.723162 + 2.69888i 0
81.18 0 0.440708 0.338167i 0 −1.41512 + 1.84422i 0 −0.974187 2.45987i 0 −0.696591 + 2.59971i 0
81.19 0 0.656890 0.504050i 0 1.17870 1.53611i 0 0.376163 + 2.61887i 0 −0.599018 + 2.23557i 0
81.20 0 1.11575 0.856146i 0 0.209020 0.272400i 0 −1.04743 + 2.42958i 0 −0.264542 + 0.987286i 0
See next 80 embeddings (of 240 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 849.30 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.c even 3 1 inner
32.g even 8 1 inner
224.bd even 24 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 896.2.bh.a 240
4.b odd 2 1 224.2.bd.a 240
7.c even 3 1 inner 896.2.bh.a 240
28.g odd 6 1 224.2.bd.a 240
32.g even 8 1 inner 896.2.bh.a 240
32.h odd 8 1 224.2.bd.a 240
224.bd even 24 1 inner 896.2.bh.a 240
224.bf odd 24 1 224.2.bd.a 240

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.2.bd.a 240 4.b odd 2 1
224.2.bd.a 240 28.g odd 6 1
224.2.bd.a 240 32.h odd 8 1
224.2.bd.a 240 224.bf odd 24 1
896.2.bh.a 240 1.a even 1 1 trivial
896.2.bh.a 240 7.c even 3 1 inner
896.2.bh.a 240 32.g even 8 1 inner
896.2.bh.a 240 224.bd even 24 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(896, [\chi])$$.