Properties

Label 8001.2.a.l
Level $8001$
Weight $2$
Character orbit 8001.a
Self dual yes
Analytic conductor $63.888$
Analytic rank $1$
Dimension $7$
CM no
Inner twists $1$

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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: \(1\)
Dimension: \(7\)
Coefficient field: 7.7.118870813.1
Defining polynomial: \(x^{7} - 9 x^{5} - 3 x^{4} + 20 x^{3} + 7 x^{2} - 13 x - 4\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2667)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{4} q^{2} + ( \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{4} + ( 1 + \beta_{3} - \beta_{5} ) q^{5} + q^{7} + ( 1 + \beta_{5} + \beta_{6} ) q^{8} +O(q^{10})\) \( q -\beta_{4} q^{2} + ( \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{4} + ( 1 + \beta_{3} - \beta_{5} ) q^{5} + q^{7} + ( 1 + \beta_{5} + \beta_{6} ) q^{8} + ( -\beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{10} + ( 1 + \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{11} + ( -3 - \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{13} -\beta_{4} q^{14} + ( -\beta_{1} + 2 \beta_{5} ) q^{16} + ( \beta_{1} - \beta_{3} + \beta_{5} - 2 \beta_{6} ) q^{17} + ( -1 - 2 \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{5} - 2 \beta_{6} ) q^{19} + ( 1 + 2 \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} ) q^{20} + ( -2 - 3 \beta_{1} - \beta_{2} - 3 \beta_{3} - \beta_{4} - \beta_{6} ) q^{22} + ( -1 - 3 \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - \beta_{6} ) q^{23} + ( 1 + 2 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{25} + ( -1 - \beta_{1} - 2 \beta_{3} + 3 \beta_{4} - \beta_{6} ) q^{26} + ( \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{28} + ( 1 - \beta_{2} + \beta_{3} + 2 \beta_{4} + 3 \beta_{5} + \beta_{6} ) q^{29} + ( -5 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} ) q^{31} + ( -2 + 2 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} - 2 \beta_{5} ) q^{32} + ( -\beta_{2} + 2 \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} ) q^{34} + ( 1 + \beta_{3} - \beta_{5} ) q^{35} + ( -5 - 3 \beta_{1} - \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{37} + ( 1 + 2 \beta_{1} + \beta_{2} + 2 \beta_{3} + 3 \beta_{4} - 2 \beta_{5} ) q^{38} + ( -2 - 2 \beta_{1} + \beta_{3} + \beta_{5} + 2 \beta_{6} ) q^{40} + ( 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{6} ) q^{41} + ( -1 - \beta_{1} + 2 \beta_{2} + \beta_{5} - 3 \beta_{6} ) q^{43} + ( 1 - 3 \beta_{1} + \beta_{2} - 2 \beta_{3} - 3 \beta_{5} - 2 \beta_{6} ) q^{44} + ( -3 - \beta_{3} + 4 \beta_{4} - \beta_{6} ) q^{46} + ( -3 - \beta_{1} + 5 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} ) q^{47} + q^{49} + ( -3 - 2 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} + \beta_{4} + 2 \beta_{5} - 4 \beta_{6} ) q^{50} + ( -5 \beta_{1} - 2 \beta_{2} - 6 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} - 6 \beta_{6} ) q^{52} + ( \beta_{1} - 3 \beta_{2} + \beta_{3} - 3 \beta_{5} ) q^{53} + ( -3 - \beta_{1} + 3 \beta_{2} + 3 \beta_{3} - 3 \beta_{5} + 2 \beta_{6} ) q^{55} + ( 1 + \beta_{5} + \beta_{6} ) q^{56} + ( -3 + 3 \beta_{1} + 3 \beta_{2} + 6 \beta_{3} + 2 \beta_{5} + 2 \beta_{6} ) q^{58} + ( 6 - \beta_{1} - 2 \beta_{2} - 4 \beta_{3} - \beta_{5} - 3 \beta_{6} ) q^{59} + ( -3 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + 3 \beta_{5} - \beta_{6} ) q^{61} + ( 1 - 2 \beta_{2} + 5 \beta_{4} + \beta_{5} - \beta_{6} ) q^{62} + ( -2 + \beta_{1} - 4 \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{64} + ( -1 - \beta_{1} - 2 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} + 2 \beta_{6} ) q^{65} + ( -3 - \beta_{1} + \beta_{2} - 3 \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{67} + ( -3 - 2 \beta_{1} - 3 \beta_{2} + \beta_{3} + 3 \beta_{4} + \beta_{5} ) q^{68} + ( -\beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{70} + ( -2 - \beta_{1} + 5 \beta_{2} + \beta_{3} - \beta_{4} + 3 \beta_{5} - \beta_{6} ) q^{71} + ( -3 + 5 \beta_{1} - \beta_{5} + 2 \beta_{6} ) q^{73} + ( 2 + 2 \beta_{1} + 3 \beta_{2} + \beta_{3} + 4 \beta_{4} - 2 \beta_{5} + 3 \beta_{6} ) q^{74} + ( -5 - 4 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - \beta_{6} ) q^{76} + ( 1 + \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{77} + ( 1 + \beta_{2} - 4 \beta_{4} - 5 \beta_{5} - 2 \beta_{6} ) q^{79} + ( -2 - 2 \beta_{1} - \beta_{2} - \beta_{3} + 3 \beta_{5} + \beta_{6} ) q^{80} + ( -\beta_{1} + 2 \beta_{2} - 3 \beta_{4} + 4 \beta_{6} ) q^{82} + ( 5 - 3 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - \beta_{4} + 3 \beta_{5} - 2 \beta_{6} ) q^{83} + ( -5 + 2 \beta_{2} + \beta_{3} - 6 \beta_{4} - 2 \beta_{5} - 5 \beta_{6} ) q^{85} + ( -2 - \beta_{1} - 4 \beta_{2} - \beta_{3} + 4 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} ) q^{86} + ( 3 - 4 \beta_{2} - 6 \beta_{3} + 3 \beta_{4} - \beta_{5} - 3 \beta_{6} ) q^{88} + ( \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} + 7 \beta_{6} ) q^{89} + ( -3 - \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{91} + ( -6 + \beta_{1} - 3 \beta_{2} - 3 \beta_{3} + 4 \beta_{4} - \beta_{5} - 3 \beta_{6} ) q^{92} + ( -1 - 3 \beta_{2} - \beta_{3} + \beta_{4} + 6 \beta_{5} + 2 \beta_{6} ) q^{94} + ( -1 - 2 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} - \beta_{4} + 4 \beta_{5} - 3 \beta_{6} ) q^{95} + ( 4 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} + 5 \beta_{6} ) q^{97} -\beta_{4} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q + 2q^{2} + 4q^{4} + 8q^{5} + 7q^{7} + 9q^{8} + O(q^{10}) \) \( 7q + 2q^{2} + 4q^{4} + 8q^{5} + 7q^{7} + 9q^{8} + 3q^{11} - 23q^{13} + 2q^{14} + 2q^{16} - 3q^{17} - 9q^{19} + 9q^{20} - 19q^{22} - 12q^{23} + 3q^{25} - 18q^{26} + 4q^{28} + 9q^{29} - 33q^{31} - 10q^{32} - 2q^{34} + 8q^{35} - 33q^{37} + 3q^{38} - 9q^{40} + 3q^{41} - 9q^{43} - 2q^{44} - 32q^{46} - 11q^{47} + 7q^{49} - 29q^{50} - 21q^{52} - q^{53} - 16q^{55} + 9q^{56} - 5q^{58} + 30q^{59} - 19q^{61} - 3q^{62} - 21q^{64} - 14q^{65} - 30q^{67} - 24q^{68} - 8q^{71} - 20q^{73} + 9q^{74} - 42q^{76} + 3q^{77} + 8q^{79} - 12q^{80} + 10q^{82} + 34q^{83} - 28q^{85} - 24q^{86} - q^{88} + 12q^{89} - 23q^{91} - 60q^{92} - 3q^{94} - 12q^{95} + 7q^{97} + 2q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7} - 9 x^{5} - 3 x^{4} + 20 x^{3} + 7 x^{2} - 13 x - 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{5} - 7 \nu^{3} - 4 \nu^{2} + 7 \nu + 5 \)
\(\beta_{3}\)\(=\)\( \nu^{5} - 8 \nu^{3} - 3 \nu^{2} + 11 \nu + 4 \)
\(\beta_{4}\)\(=\)\( -\nu^{6} + 8 \nu^{4} + 3 \nu^{3} - 12 \nu^{2} - 4 \nu + 2 \)
\(\beta_{5}\)\(=\)\( \nu^{6} + 2 \nu^{5} - 8 \nu^{4} - 18 \nu^{3} + 6 \nu^{2} + 22 \nu + 4 \)
\(\beta_{6}\)\(=\)\( 2 \nu^{6} + \nu^{5} - 15 \nu^{4} - 14 \nu^{3} + 15 \nu^{2} + 16 \nu + 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{5} + \beta_{4} - 2 \beta_{3} + 4 \beta_{1} + 2\)
\(\nu^{4}\)\(=\)\(\beta_{6} + 6 \beta_{5} + 8 \beta_{4} - 7 \beta_{3} - 6 \beta_{2} + 3 \beta_{1} + 15\)
\(\nu^{5}\)\(=\)\(11 \beta_{5} + 11 \beta_{4} - 18 \beta_{3} - 3 \beta_{2} + 21 \beta_{1} + 21\)
\(\nu^{6}\)\(=\)\(8 \beta_{6} + 39 \beta_{5} + 54 \beta_{4} - 50 \beta_{3} - 36 \beta_{2} + 32 \beta_{1} + 92\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−0.301070
2.69855
−2.06168
−1.52532
1.20244
−1.14753
1.13462
−2.09968 0 2.40865 3.29054 0 1.00000 −0.858029 0 −6.90907
1.2 −0.840819 0 −1.29302 −2.74724 0 1.00000 2.76884 0 2.30993
1.3 −0.692358 0 −1.52064 2.78145 0 1.00000 2.43754 0 −1.92576
1.4 −0.246202 0 −1.93938 0.318209 0 1.00000 0.969884 0 −0.0783436
1.5 1.24280 0 −0.455452 3.33400 0 1.00000 −3.05163 0 4.14349
1.6 2.15625 0 2.64943 0.239094 0 1.00000 1.40032 0 0.515547
1.7 2.48001 0 4.15043 0.783950 0 1.00000 5.33307 0 1.94420
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8001.2.a.l 7
3.b odd 2 1 2667.2.a.j 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2667.2.a.j 7 3.b odd 2 1
8001.2.a.l 7 1.a even 1 1 trivial

Hecke kernels

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

\( T_{2}^{7} - 2 T_{2}^{6} - 7 T_{2}^{5} + 11 T_{2}^{4} + 14 T_{2}^{3} - 9 T_{2}^{2} - 11 T_{2} - 2 \)
\( T_{5}^{7} - 8 T_{5}^{6} + 13 T_{5}^{5} + 42 T_{5}^{4} - 149 T_{5}^{3} + 138 T_{5}^{2} - 46 T_{5} + 5 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -2 - 11 T - 9 T^{2} + 14 T^{3} + 11 T^{4} - 7 T^{5} - 2 T^{6} + T^{7} \)
$3$ \( T^{7} \)
$5$ \( 5 - 46 T + 138 T^{2} - 149 T^{3} + 42 T^{4} + 13 T^{5} - 8 T^{6} + T^{7} \)
$7$ \( ( -1 + T )^{7} \)
$11$ \( 1130 - 3011 T - 890 T^{2} + 645 T^{3} + 105 T^{4} - 46 T^{5} - 3 T^{6} + T^{7} \)
$13$ \( -5519 - 7825 T - 2853 T^{2} + 689 T^{3} + 738 T^{4} + 198 T^{5} + 23 T^{6} + T^{7} \)
$17$ \( -3898 - 1923 T + 2520 T^{2} + 1169 T^{3} - 160 T^{4} - 69 T^{5} + 3 T^{6} + T^{7} \)
$19$ \( 248 + 1117 T + 508 T^{2} - 310 T^{3} - 198 T^{4} - 6 T^{5} + 9 T^{6} + T^{7} \)
$23$ \( 31373 + 29113 T + 5963 T^{2} - 1516 T^{3} - 589 T^{4} - 11 T^{5} + 12 T^{6} + T^{7} \)
$29$ \( 53173 - 769 T - 15299 T^{2} + 1023 T^{3} + 914 T^{4} - 96 T^{5} - 9 T^{6} + T^{7} \)
$31$ \( -15973 - 5103 T + 10812 T^{2} + 8763 T^{3} + 2745 T^{4} + 428 T^{5} + 33 T^{6} + T^{7} \)
$37$ \( -1549 - 4929 T - 554 T^{2} + 3483 T^{3} + 1989 T^{4} + 394 T^{5} + 33 T^{6} + T^{7} \)
$41$ \( -5198 + 12853 T - 7909 T^{2} + 310 T^{3} + 688 T^{4} - 117 T^{5} - 3 T^{6} + T^{7} \)
$43$ \( -52792 - 5333 T + 12794 T^{2} + 1289 T^{3} - 763 T^{4} - 84 T^{5} + 9 T^{6} + T^{7} \)
$47$ \( -1279202 - 245633 T + 86717 T^{2} + 11556 T^{3} - 1763 T^{4} - 184 T^{5} + 11 T^{6} + T^{7} \)
$53$ \( -24569 + 19634 T + 24282 T^{2} + 4955 T^{3} - 457 T^{4} - 155 T^{5} + T^{6} + T^{7} \)
$59$ \( 108929 - 279635 T + 186633 T^{2} - 43956 T^{3} + 3231 T^{4} + 157 T^{5} - 30 T^{6} + T^{7} \)
$61$ \( -37165 + 41509 T + 23658 T^{2} - 2599 T^{3} - 1305 T^{4} + 19 T^{6} + T^{7} \)
$67$ \( -386 - 1049 T + 6545 T^{2} - 5794 T^{3} - 700 T^{4} + 206 T^{5} + 30 T^{6} + T^{7} \)
$71$ \( -981910 - 378537 T + 80787 T^{2} + 17712 T^{3} - 1621 T^{4} - 251 T^{5} + 8 T^{6} + T^{7} \)
$73$ \( 65465 + 44811 T - 4269 T^{2} - 8910 T^{3} - 1799 T^{4} - T^{5} + 20 T^{6} + T^{7} \)
$79$ \( -546464 - 374131 T + 29277 T^{2} + 25440 T^{3} + 934 T^{4} - 280 T^{5} - 8 T^{6} + T^{7} \)
$83$ \( -6353 - 10549 T + 6100 T^{2} + 1872 T^{3} - 1704 T^{4} + 381 T^{5} - 34 T^{6} + T^{7} \)
$89$ \( -9440161 - 4558287 T - 225933 T^{2} + 78939 T^{3} + 3749 T^{4} - 472 T^{5} - 12 T^{6} + T^{7} \)
$97$ \( 674 + 13315 T + 53310 T^{2} + 29021 T^{3} + 1232 T^{4} - 353 T^{5} - 7 T^{6} + T^{7} \)
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