# Properties

 Label 1008.4.bt.d Level $1008$ Weight $4$ Character orbit 1008.bt Analytic conductor $59.474$ Analytic rank $0$ Dimension $48$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1008 = 2^{4} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1008.bt (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$59.4739252858$$ Analytic rank: $$0$$ Dimension: $$48$$ Relative dimension: $$24$$ over $$\Q(\zeta_{6})$$ Twist minimal: no (minimal twist has level 504) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$48q + 24q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$48q + 24q^{7} - 540q^{19} - 924q^{25} - 648q^{31} - 132q^{37} + 792q^{43} + 672q^{49} + 12q^{67} + 2412q^{73} - 1680q^{79} + 480q^{85} - 1404q^{91} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1 0 0 0 −10.9253 18.9231i 0 12.1367 + 13.9893i 0 0 0
17.2 0 0 0 −9.74197 16.8736i 0 18.4446 1.67197i 0 0 0
17.3 0 0 0 −8.48442 14.6954i 0 −3.52711 + 18.1813i 0 0 0
17.4 0 0 0 −7.89184 13.6691i 0 −10.6185 15.1739i 0 0 0
17.5 0 0 0 −7.29543 12.6361i 0 −18.4933 0.998739i 0 0 0
17.6 0 0 0 −5.74088 9.94350i 0 −17.2542 6.73007i 0 0 0
17.7 0 0 0 −5.00025 8.66069i 0 −1.56940 + 18.4536i 0 0 0
17.8 0 0 0 −3.34783 5.79860i 0 12.7404 13.4418i 0 0 0
17.9 0 0 0 −3.30930 5.73188i 0 4.31556 18.0104i 0 0 0
17.10 0 0 0 −2.08296 3.60779i 0 18.2790 + 2.97950i 0 0 0
17.11 0 0 0 −1.56246 2.70625i 0 −17.1949 + 6.88015i 0 0 0
17.12 0 0 0 −1.35769 2.35159i 0 8.74105 + 16.3277i 0 0 0
17.13 0 0 0 1.35769 + 2.35159i 0 8.74105 + 16.3277i 0 0 0
17.14 0 0 0 1.56246 + 2.70625i 0 −17.1949 + 6.88015i 0 0 0
17.15 0 0 0 2.08296 + 3.60779i 0 18.2790 + 2.97950i 0 0 0
17.16 0 0 0 3.30930 + 5.73188i 0 4.31556 18.0104i 0 0 0
17.17 0 0 0 3.34783 + 5.79860i 0 12.7404 13.4418i 0 0 0
17.18 0 0 0 5.00025 + 8.66069i 0 −1.56940 + 18.4536i 0 0 0
17.19 0 0 0 5.74088 + 9.94350i 0 −17.2542 6.73007i 0 0 0
17.20 0 0 0 7.29543 + 12.6361i 0 −18.4933 0.998739i 0 0 0
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 593.24 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.d odd 6 1 inner
21.g even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.4.bt.d 48
3.b odd 2 1 inner 1008.4.bt.d 48
4.b odd 2 1 504.4.bl.a 48
7.d odd 6 1 inner 1008.4.bt.d 48
12.b even 2 1 504.4.bl.a 48
21.g even 6 1 inner 1008.4.bt.d 48
28.f even 6 1 504.4.bl.a 48
84.j odd 6 1 504.4.bl.a 48

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.4.bl.a 48 4.b odd 2 1
504.4.bl.a 48 12.b even 2 1
504.4.bl.a 48 28.f even 6 1
504.4.bl.a 48 84.j odd 6 1
1008.4.bt.d 48 1.a even 1 1 trivial
1008.4.bt.d 48 3.b odd 2 1 inner
1008.4.bt.d 48 7.d odd 6 1 inner
1008.4.bt.d 48 21.g even 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$43\!\cdots\!06$$$$T_{5}^{38} +$$$$15\!\cdots\!97$$$$T_{5}^{36} +$$$$44\!\cdots\!78$$$$T_{5}^{34} +$$$$10\!\cdots\!70$$$$T_{5}^{32} +$$$$20\!\cdots\!54$$$$T_{5}^{30} +$$$$31\!\cdots\!41$$$$T_{5}^{28} +$$$$40\!\cdots\!62$$$$T_{5}^{26} +$$$$42\!\cdots\!01$$$$T_{5}^{24} +$$$$36\!\cdots\!68$$$$T_{5}^{22} +$$$$24\!\cdots\!04$$$$T_{5}^{20} +$$$$12\!\cdots\!56$$$$T_{5}^{18} +$$$$54\!\cdots\!48$$$$T_{5}^{16} +$$$$17\!\cdots\!00$$$$T_{5}^{14} +$$$$42\!\cdots\!60$$$$T_{5}^{12} +$$$$77\!\cdots\!84$$$$T_{5}^{10} +$$$$10\!\cdots\!88$$$$T_{5}^{8} +$$$$10\!\cdots\!60$$$$T_{5}^{6} +$$$$69\!\cdots\!48$$$$T_{5}^{4} +$$$$30\!\cdots\!84$$$$T_{5}^{2} +$$$$80\!\cdots\!56$$">$$T_{5}^{48} + \cdots$$ acting on $$S_{4}^{\mathrm{new}}(1008, [\chi])$$.