# Properties

 Label 288.2.v.c Level $288$ Weight $2$ Character orbit 288.v Analytic conductor $2.300$ Analytic rank $0$ Dimension $32$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$288 = 2^{5} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 288.v (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.

 $$\operatorname{Tr}(f)(q) =$$ $$32q + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32q - 8q^{10} - 8q^{16} - 24q^{22} - 40q^{28} + 48q^{31} - 40q^{34} - 72q^{40} + 16q^{43} - 32q^{46} - 8q^{52} + 32q^{55} - 32q^{58} - 32q^{61} + 72q^{64} + 16q^{67} + 120q^{70} + 72q^{76} + 120q^{82} + 128q^{88} - 48q^{91} + 80q^{94} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
37.1 −1.41364 0.0402136i 0 1.99677 + 0.113695i −1.51282 0.626632i 0 −1.32530 1.32530i −2.81814 0.241022i 0 2.11339 + 0.946670i
37.2 −1.06920 + 0.925636i 0 0.286395 1.97939i 3.41317 + 1.41378i 0 −1.42999 1.42999i 1.52598 + 2.38147i 0 −4.95802 + 1.64773i
37.3 −0.835415 1.14109i 0 −0.604162 + 1.90656i −0.823699 0.341187i 0 0.760681 + 0.760681i 2.68028 0.903371i 0 0.298806 + 1.22495i
37.4 −0.716976 + 1.21899i 0 −0.971891 1.74798i −2.42249 1.00343i 0 3.40882 + 3.40882i 2.82760 + 0.0685293i 0 2.96004 2.23356i
37.5 0.716976 1.21899i 0 −0.971891 1.74798i 2.42249 + 1.00343i 0 3.40882 + 3.40882i −2.82760 0.0685293i 0 2.96004 2.23356i
37.6 0.835415 + 1.14109i 0 −0.604162 + 1.90656i 0.823699 + 0.341187i 0 0.760681 + 0.760681i −2.68028 + 0.903371i 0 0.298806 + 1.22495i
37.7 1.06920 0.925636i 0 0.286395 1.97939i −3.41317 1.41378i 0 −1.42999 1.42999i −1.52598 2.38147i 0 −4.95802 + 1.64773i
37.8 1.41364 + 0.0402136i 0 1.99677 + 0.113695i 1.51282 + 0.626632i 0 −1.32530 1.32530i 2.81814 + 0.241022i 0 2.11339 + 0.946670i
109.1 −1.41364 + 0.0402136i 0 1.99677 0.113695i −1.51282 + 0.626632i 0 −1.32530 + 1.32530i −2.81814 + 0.241022i 0 2.11339 0.946670i
109.2 −1.06920 0.925636i 0 0.286395 + 1.97939i 3.41317 1.41378i 0 −1.42999 + 1.42999i 1.52598 2.38147i 0 −4.95802 1.64773i
109.3 −0.835415 + 1.14109i 0 −0.604162 1.90656i −0.823699 + 0.341187i 0 0.760681 0.760681i 2.68028 + 0.903371i 0 0.298806 1.22495i
109.4 −0.716976 1.21899i 0 −0.971891 + 1.74798i −2.42249 + 1.00343i 0 3.40882 3.40882i 2.82760 0.0685293i 0 2.96004 + 2.23356i
109.5 0.716976 + 1.21899i 0 −0.971891 + 1.74798i 2.42249 1.00343i 0 3.40882 3.40882i −2.82760 + 0.0685293i 0 2.96004 + 2.23356i
109.6 0.835415 1.14109i 0 −0.604162 1.90656i 0.823699 0.341187i 0 0.760681 0.760681i −2.68028 0.903371i 0 0.298806 1.22495i
109.7 1.06920 + 0.925636i 0 0.286395 + 1.97939i −3.41317 + 1.41378i 0 −1.42999 + 1.42999i −1.52598 + 2.38147i 0 −4.95802 1.64773i
109.8 1.41364 0.0402136i 0 1.99677 0.113695i 1.51282 0.626632i 0 −1.32530 + 1.32530i 2.81814 0.241022i 0 2.11339 0.946670i
181.1 −1.39967 0.202304i 0 1.91815 + 0.566318i 1.53803 3.71314i 0 −1.56292 1.56292i −2.57020 1.18071i 0 −2.90392 + 4.88602i
181.2 −1.14926 + 0.824131i 0 0.641617 1.89429i −0.549515 + 1.32665i 0 0.197478 + 0.197478i 0.823754 + 2.70581i 0 −0.461792 1.97754i
181.3 −0.582293 1.28877i 0 −1.32187 + 1.50089i 0.445570 1.07570i 0 2.57533 + 2.57533i 2.70402 + 0.829636i 0 −1.64578 + 0.0521344i
181.4 −0.165832 + 1.40446i 0 −1.94500 0.465809i 0.805403 1.94441i 0 −2.62411 2.62411i 0.976752 2.65442i 0 2.59728 + 1.45360i
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 253.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
32.g even 8 1 inner
96.p odd 8 1 inner

## Twists

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

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{32} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(288, [\chi])$$.