Properties

Label 8001.2.a.ba
Level $8001$
Weight $2$
Character orbit 8001.a
Self dual yes
Analytic conductor $63.888$
Analytic rank $0$
Dimension $40$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8001.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.8883066572\)
Analytic rank: \(0\)
Dimension: \(40\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40q + 54q^{4} + 40q^{7} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 40q + 54q^{4} + 40q^{7} + 20q^{10} + 10q^{13} + 90q^{16} + 38q^{19} + 14q^{22} + 84q^{25} + 54q^{28} + 66q^{31} + 22q^{34} + 40q^{37} + 26q^{40} + 38q^{43} + 28q^{46} + 40q^{49} + 28q^{52} + 60q^{55} + 42q^{58} + 54q^{61} + 124q^{64} + 48q^{67} + 20q^{70} + 16q^{76} + 102q^{79} + 48q^{82} + 104q^{85} + 48q^{88} + 10q^{91} - 10q^{94} + 8q^{97} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −2.76162 0 5.62653 2.49672 0 1.00000 −10.0151 0 −6.89498
1.2 −2.68554 0 5.21214 −3.72554 0 1.00000 −8.62633 0 10.0051
1.3 −2.66267 0 5.08984 −1.08542 0 1.00000 −8.22723 0 2.89013
1.4 −2.63730 0 4.95534 −2.98743 0 1.00000 −7.79413 0 7.87873
1.5 −2.52993 0 4.40057 4.10792 0 1.00000 −6.07329 0 −10.3928
1.6 −2.31568 0 3.36240 1.76797 0 1.00000 −3.15488 0 −4.09407
1.7 −2.26712 0 3.13982 −1.21847 0 1.00000 −2.58412 0 2.76242
1.8 −2.10140 0 2.41588 −0.560790 0 1.00000 −0.873928 0 1.17844
1.9 −2.09235 0 2.37795 −3.96692 0 1.00000 −0.790800 0 8.30020
1.10 −1.67982 0 0.821796 −1.48302 0 1.00000 1.97917 0 2.49121
1.11 −1.50508 0 0.265269 2.12799 0 1.00000 2.61091 0 −3.20280
1.12 −1.42650 0 0.0348943 4.07722 0 1.00000 2.80322 0 −5.81614
1.13 −1.20189 0 −0.555461 0.607848 0 1.00000 3.07138 0 −0.730567
1.14 −1.13347 0 −0.715244 −1.11166 0 1.00000 3.07765 0 1.26004
1.15 −1.07579 0 −0.842680 1.06884 0 1.00000 3.05812 0 −1.14985
1.16 −0.997785 0 −1.00442 −3.25439 0 1.00000 2.99777 0 3.24718
1.17 −0.476800 0 −1.77266 −3.87912 0 1.00000 1.79880 0 1.84956
1.18 −0.308449 0 −1.90486 1.85496 0 1.00000 1.20445 0 −0.572160
1.19 −0.301452 0 −1.90913 −3.64431 0 1.00000 1.17841 0 1.09858
1.20 −0.0449988 0 −1.99798 2.40582 0 1.00000 0.179904 0 −0.108259
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.40
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(7\) \(-1\)
\(127\) \(1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8001.2.a.ba 40
3.b odd 2 1 inner 8001.2.a.ba 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8001.2.a.ba 40 1.a even 1 1 trivial
8001.2.a.ba 40 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8001))\):

\(T_{2}^{40} - \cdots\)
\(15\!\cdots\!60\)\( T_{5}^{22} + \)\(73\!\cdots\!60\)\( T_{5}^{20} - \)\(28\!\cdots\!04\)\( T_{5}^{18} + \)\(81\!\cdots\!96\)\( T_{5}^{16} - \)\(17\!\cdots\!24\)\( T_{5}^{14} + \)\(29\!\cdots\!12\)\( T_{5}^{12} - \)\(36\!\cdots\!20\)\( T_{5}^{10} + \)\(31\!\cdots\!40\)\( T_{5}^{8} - \)\(18\!\cdots\!92\)\( T_{5}^{6} + \)\(68\!\cdots\!48\)\( T_{5}^{4} - \)\(14\!\cdots\!12\)\( T_{5}^{2} + \)\(12\!\cdots\!84\)\( \)">\(T_{5}^{40} - \cdots\)

Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database