# Properties

 Label 288.2.w.a Level $288$ Weight $2$ Character orbit 288.w Analytic conductor $2.300$ Analytic rank $0$ Dimension $32$ CM no Inner twists $2$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$288 = 2^{5} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 288.w (of order $$8$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.29969157821$$ Analytic rank: $$0$$ Dimension: $$32$$ Relative dimension: $$8$$ over $$\Q(\zeta_{8})$$ Twist minimal: yes 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.

 $$\operatorname{Tr}(f)(q) =$$ $$32 q - 4 q^{2} - 4 q^{8}+O(q^{10})$$ 32 * q - 4 * q^2 - 4 * q^8 $$\operatorname{Tr}(f)(q) =$$ $$32 q - 4 q^{2} - 4 q^{8} + 8 q^{10} - 8 q^{11} + 12 q^{14} + 8 q^{16} + 32 q^{20} + 16 q^{22} - 36 q^{26} - 16 q^{29} - 24 q^{32} + 24 q^{35} + 32 q^{38} - 32 q^{40} + 8 q^{44} - 32 q^{46} + 8 q^{50} - 56 q^{52} - 16 q^{53} - 32 q^{55} + 40 q^{56} - 32 q^{58} + 32 q^{59} + 32 q^{61} - 68 q^{62} - 48 q^{64} - 16 q^{67} - 72 q^{68} - 48 q^{70} + 16 q^{71} + 60 q^{74} - 8 q^{76} - 16 q^{77} - 32 q^{79} + 96 q^{80} + 40 q^{82} - 40 q^{83} - 40 q^{86} + 40 q^{88} - 48 q^{91} - 16 q^{92} + 72 q^{94} - 80 q^{95} - 44 q^{98}+O(q^{100})$$ 32 * q - 4 * q^2 - 4 * q^8 + 8 * q^10 - 8 * q^11 + 12 * q^14 + 8 * q^16 + 32 * q^20 + 16 * q^22 - 36 * q^26 - 16 * q^29 - 24 * q^32 + 24 * q^35 + 32 * q^38 - 32 * q^40 + 8 * q^44 - 32 * q^46 + 8 * q^50 - 56 * q^52 - 16 * q^53 - 32 * q^55 + 40 * q^56 - 32 * q^58 + 32 * q^59 + 32 * q^61 - 68 * q^62 - 48 * q^64 - 16 * q^67 - 72 * q^68 - 48 * q^70 + 16 * q^71 + 60 * q^74 - 8 * q^76 - 16 * q^77 - 32 * q^79 + 96 * q^80 + 40 * q^82 - 40 * q^83 - 40 * q^86 + 40 * q^88 - 48 * q^91 - 16 * q^92 + 72 * q^94 - 80 * q^95 - 44 * q^98

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
35.1 −1.39224 0.248345i 0 1.87665 + 0.691511i 4.01952 1.66494i 0 2.72280 + 2.72280i −2.44101 1.42881i 0 −6.00960 + 1.31976i
35.2 −1.33776 0.458699i 0 1.57919 + 1.22726i −2.70076 + 1.11869i 0 −1.06647 1.06647i −1.54963 2.36614i 0 4.12611 0.257702i
35.3 −1.19734 + 0.752583i 0 0.867238 1.80219i −0.0913223 + 0.0378270i 0 −3.05457 3.05457i 0.317923 + 2.81050i 0 0.0808758 0.114019i
35.4 −0.345448 1.37137i 0 −1.76133 + 0.947475i −1.64859 + 0.682869i 0 2.51270 + 2.51270i 1.90779 + 2.08814i 0 1.50597 + 2.02494i
35.5 0.349793 + 1.37027i 0 −1.75529 + 0.958624i −2.97412 + 1.23192i 0 0.237717 + 0.237717i −1.92756 2.06990i 0 −2.72839 3.64443i
35.6 0.430512 1.34709i 0 −1.62932 1.15988i 1.78959 0.741273i 0 −2.33709 2.33709i −2.26391 + 1.69550i 0 −0.228123 2.72987i
35.7 1.16775 + 0.797726i 0 0.727265 + 1.86308i 1.65625 0.686041i 0 −0.456585 0.456585i −0.636970 + 2.75577i 0 2.48135 + 0.520112i
35.8 1.32473 0.495070i 0 1.50981 1.31167i −0.0505677 + 0.0209458i 0 1.44150 + 1.44150i 1.35072 2.48506i 0 −0.0566189 + 0.0527821i
107.1 −1.39224 + 0.248345i 0 1.87665 0.691511i 4.01952 + 1.66494i 0 2.72280 2.72280i −2.44101 + 1.42881i 0 −6.00960 1.31976i
107.2 −1.33776 + 0.458699i 0 1.57919 1.22726i −2.70076 1.11869i 0 −1.06647 + 1.06647i −1.54963 + 2.36614i 0 4.12611 + 0.257702i
107.3 −1.19734 0.752583i 0 0.867238 + 1.80219i −0.0913223 0.0378270i 0 −3.05457 + 3.05457i 0.317923 2.81050i 0 0.0808758 + 0.114019i
107.4 −0.345448 + 1.37137i 0 −1.76133 0.947475i −1.64859 0.682869i 0 2.51270 2.51270i 1.90779 2.08814i 0 1.50597 2.02494i
107.5 0.349793 1.37027i 0 −1.75529 0.958624i −2.97412 1.23192i 0 0.237717 0.237717i −1.92756 + 2.06990i 0 −2.72839 + 3.64443i
107.6 0.430512 + 1.34709i 0 −1.62932 + 1.15988i 1.78959 + 0.741273i 0 −2.33709 + 2.33709i −2.26391 1.69550i 0 −0.228123 + 2.72987i
107.7 1.16775 0.797726i 0 0.727265 1.86308i 1.65625 + 0.686041i 0 −0.456585 + 0.456585i −0.636970 2.75577i 0 2.48135 0.520112i
107.8 1.32473 + 0.495070i 0 1.50981 + 1.31167i −0.0505677 0.0209458i 0 1.44150 1.44150i 1.35072 + 2.48506i 0 −0.0566189 0.0527821i
179.1 −1.40684 + 0.144194i 0 1.95842 0.405715i 0.366958 + 0.885915i 0 1.21471 + 1.21471i −2.69668 + 0.853169i 0 −0.643996 1.19343i
179.2 −1.04926 0.948179i 0 0.201912 + 1.98978i −1.39555 3.36915i 0 −1.05755 1.05755i 1.67481 2.27926i 0 −1.73026 + 4.85836i
179.3 −0.850468 + 1.12991i 0 −0.553410 1.92191i 0.352978 + 0.852163i 0 −3.43393 3.43393i 2.64225 + 1.00922i 0 −1.26307 0.325903i
179.4 −0.337906 1.37325i 0 −1.77164 + 0.928061i 1.32268 + 3.19322i 0 −2.32913 2.32913i 1.87311 + 2.11931i 0 3.93816 2.89538i
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 251.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
96.o even 8 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 288.2.w.a 32
3.b odd 2 1 288.2.w.b yes 32
4.b odd 2 1 1152.2.w.b 32
12.b even 2 1 1152.2.w.a 32
32.g even 8 1 1152.2.w.a 32
32.h odd 8 1 288.2.w.b yes 32
96.o even 8 1 inner 288.2.w.a 32
96.p odd 8 1 1152.2.w.b 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.2.w.a 32 1.a even 1 1 trivial
288.2.w.a 32 96.o even 8 1 inner
288.2.w.b yes 32 3.b odd 2 1
288.2.w.b yes 32 32.h odd 8 1
1152.2.w.a 32 12.b even 2 1
1152.2.w.a 32 32.g even 8 1
1152.2.w.b 32 4.b odd 2 1
1152.2.w.b 32 96.p odd 8 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{32} - 16 T_{5}^{29} + 224 T_{5}^{27} + 896 T_{5}^{26} + 6080 T_{5}^{25} + 48512 T_{5}^{24} + 87168 T_{5}^{23} + 188416 T_{5}^{22} + 181376 T_{5}^{21} + 155648 T_{5}^{20} - 380672 T_{5}^{19} - 6281728 T_{5}^{18} + \cdots + 4096$$ acting on $$S_{2}^{\mathrm{new}}(288, [\chi])$$.