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 $4$

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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:

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 \(S_{4}^{\mathrm{new}}(1008, [\chi])\).