Properties

Label 8019.2.a.b
Level 8019
Weight 2
Character orbit 8019.a
Self dual yes
Analytic conductor 64.032
Analytic rank 1
Dimension 3
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 8019 = 3^{6} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8019.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.0320373809\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{18})^+\)
Defining polynomial: \(x^{3} - 3 x - 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 297)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + ( -2 + \beta^{2} ) q^{4} + ( -3 - \beta + 2 \beta^{2} ) q^{5} + ( -3 + \beta^{2} ) q^{7} + ( 1 - \beta ) q^{8} +O(q^{10})\) \( q + \beta q^{2} + ( -2 + \beta^{2} ) q^{4} + ( -3 - \beta + 2 \beta^{2} ) q^{5} + ( -3 + \beta^{2} ) q^{7} + ( 1 - \beta ) q^{8} + ( 2 + 3 \beta - \beta^{2} ) q^{10} + q^{11} + ( -7 - 3 \beta + 2 \beta^{2} ) q^{13} + q^{14} + ( 4 + \beta - 3 \beta^{2} ) q^{16} + ( -4 - \beta ) q^{17} + ( 6 - 2 \beta - \beta^{2} ) q^{19} + ( 5 + \beta - \beta^{2} ) q^{20} + \beta q^{22} + ( 4 - 2 \beta^{2} ) q^{23} + ( -2 \beta + \beta^{2} ) q^{25} + ( 2 - \beta - 3 \beta^{2} ) q^{26} + ( 6 + \beta - 2 \beta^{2} ) q^{28} + ( -6 + 4 \beta + 2 \beta^{2} ) q^{29} + ( -3 + 2 \beta + 2 \beta^{2} ) q^{31} + ( -5 - 3 \beta + \beta^{2} ) q^{32} + ( -4 \beta - \beta^{2} ) q^{34} + ( 8 + 2 \beta - 3 \beta^{2} ) q^{35} + ( 8 + 2 \beta - 4 \beta^{2} ) q^{37} + ( -1 + 3 \beta - 2 \beta^{2} ) q^{38} + ( -5 - 4 \beta + 3 \beta^{2} ) q^{40} + ( 5 - \beta - 3 \beta^{2} ) q^{41} + ( 7 + 6 \beta - 4 \beta^{2} ) q^{43} + ( -2 + \beta^{2} ) q^{44} + ( -2 - 2 \beta ) q^{46} + ( -7 + 2 \beta + \beta^{2} ) q^{47} + ( 2 + \beta - 3 \beta^{2} ) q^{49} + ( 1 + 3 \beta - 2 \beta^{2} ) q^{50} + ( 11 - \beta - 5 \beta^{2} ) q^{52} + ( 11 - 2 \beta^{2} ) q^{53} + ( -3 - \beta + 2 \beta^{2} ) q^{55} + ( -4 + \beta^{2} ) q^{56} + ( 2 + 4 \beta^{2} ) q^{58} + ( -7 - 2 \beta + 2 \beta^{2} ) q^{59} + ( -6 - 2 \beta + 2 \beta^{2} ) q^{61} + ( 2 + 3 \beta + 2 \beta^{2} ) q^{62} + ( -7 - 4 \beta + 3 \beta^{2} ) q^{64} + ( 13 - 4 \beta - 5 \beta^{2} ) q^{65} + ( 15 + \beta - 8 \beta^{2} ) q^{67} + ( 7 - \beta - 4 \beta^{2} ) q^{68} + ( -3 - \beta + 2 \beta^{2} ) q^{70} + ( 5 + \beta - 6 \beta^{2} ) q^{71} + ( 5 + 7 \beta - 2 \beta^{2} ) q^{73} + ( -4 - 4 \beta + 2 \beta^{2} ) q^{74} + ( -14 - 3 \beta + 5 \beta^{2} ) q^{76} + ( -3 + \beta^{2} ) q^{77} + ( 3 + \beta - 5 \beta^{2} ) q^{79} + ( -7 + 2 \beta - 2 \beta^{2} ) q^{80} + ( -3 - 4 \beta - \beta^{2} ) q^{82} + ( 17 + 4 \beta - 9 \beta^{2} ) q^{83} + ( 10 + \beta - 7 \beta^{2} ) q^{85} + ( -4 - 5 \beta + 6 \beta^{2} ) q^{86} + ( 1 - \beta ) q^{88} + ( 12 + 6 \beta - 9 \beta^{2} ) q^{89} + ( 18 + 2 \beta - 7 \beta^{2} ) q^{91} + ( -8 - 2 \beta + 2 \beta^{2} ) q^{92} + ( 1 - 4 \beta + 2 \beta^{2} ) q^{94} + ( -21 - 11 \beta + 11 \beta^{2} ) q^{95} + ( 17 - 2 \beta - 4 \beta^{2} ) q^{97} + ( -3 - 7 \beta + \beta^{2} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{5} - 3q^{7} + 3q^{8} + O(q^{10}) \) \( 3q + 3q^{5} - 3q^{7} + 3q^{8} + 3q^{11} - 9q^{13} + 3q^{14} - 6q^{16} - 12q^{17} + 12q^{19} + 9q^{20} + 6q^{25} - 12q^{26} + 6q^{28} - 6q^{29} + 3q^{31} - 9q^{32} - 6q^{34} + 6q^{35} - 15q^{38} + 3q^{40} - 3q^{41} - 3q^{43} - 6q^{46} - 15q^{47} - 12q^{49} - 9q^{50} + 3q^{52} + 21q^{53} + 3q^{55} - 6q^{56} + 30q^{58} - 9q^{59} - 6q^{61} + 18q^{62} - 3q^{64} + 9q^{65} - 3q^{67} - 3q^{68} + 3q^{70} - 21q^{71} + 3q^{73} - 12q^{76} - 3q^{77} - 21q^{79} - 33q^{80} - 15q^{82} - 3q^{83} - 12q^{85} + 24q^{86} + 3q^{88} - 18q^{89} + 12q^{91} - 12q^{92} + 15q^{94} + 3q^{95} + 27q^{97} - 3q^{98} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.53209
−0.347296
1.87939
−1.53209 0 0.347296 3.22668 0 −0.652704 2.53209 0 −4.94356
1.2 −0.347296 0 −1.87939 −2.41147 0 −2.87939 1.34730 0 0.837496
1.3 1.87939 0 1.53209 2.18479 0 0.532089 −0.879385 0 4.10607
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(11\) \(-1\)

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 8019.2.a.b 3
3.b odd 2 1 8019.2.a.a 3
27.e even 9 2 891.2.j.a 6
27.f odd 18 2 297.2.j.a 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
297.2.j.a 6 27.f odd 18 2
891.2.j.a 6 27.e even 9 2
8019.2.a.a 3 3.b odd 2 1
8019.2.a.b 3 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} - 3 T_{2} - 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8019))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 3 T^{2} - T^{3} + 6 T^{4} + 8 T^{6} \)
$3$ 1
$5$ \( 1 - 3 T + 9 T^{2} - 13 T^{3} + 45 T^{4} - 75 T^{5} + 125 T^{6} \)
$7$ \( 1 + 3 T + 21 T^{2} + 41 T^{3} + 147 T^{4} + 147 T^{5} + 343 T^{6} \)
$11$ \( ( 1 - T )^{3} \)
$13$ \( 1 + 9 T + 45 T^{2} + 161 T^{3} + 585 T^{4} + 1521 T^{5} + 2197 T^{6} \)
$17$ \( 1 + 12 T + 96 T^{2} + 461 T^{3} + 1632 T^{4} + 3468 T^{5} + 4913 T^{6} \)
$19$ \( 1 - 12 T + 84 T^{2} - 399 T^{3} + 1596 T^{4} - 4332 T^{5} + 6859 T^{6} \)
$23$ \( 1 + 57 T^{2} - 8 T^{3} + 1311 T^{4} + 12167 T^{6} \)
$29$ \( 1 + 6 T + 15 T^{2} - 108 T^{3} + 435 T^{4} + 5046 T^{5} + 24389 T^{6} \)
$31$ \( 1 - 3 T + 60 T^{2} - 223 T^{3} + 1860 T^{4} - 2883 T^{5} + 29791 T^{6} \)
$37$ \( 1 + 75 T^{2} - 72 T^{3} + 2775 T^{4} + 50653 T^{6} \)
$41$ \( 1 + 3 T + 87 T^{2} + 227 T^{3} + 3567 T^{4} + 5043 T^{5} + 68921 T^{6} \)
$43$ \( 1 + 3 T + 48 T^{2} + 471 T^{3} + 2064 T^{4} + 5547 T^{5} + 79507 T^{6} \)
$47$ \( 1 + 15 T + 195 T^{2} + 1393 T^{3} + 9165 T^{4} + 33135 T^{5} + 103823 T^{6} \)
$53$ \( 1 - 21 T + 294 T^{2} - 2493 T^{3} + 15582 T^{4} - 58989 T^{5} + 148877 T^{6} \)
$59$ \( 1 + 9 T + 192 T^{2} + 1045 T^{3} + 11328 T^{4} + 31329 T^{5} + 205379 T^{6} \)
$61$ \( 1 + 6 T + 183 T^{2} + 708 T^{3} + 11163 T^{4} + 22326 T^{5} + 226981 T^{6} \)
$67$ \( 1 + 3 T + 33 T^{2} - 425 T^{3} + 2211 T^{4} + 13467 T^{5} + 300763 T^{6} \)
$71$ \( 1 + 21 T + 267 T^{2} + 2385 T^{3} + 18957 T^{4} + 105861 T^{5} + 357911 T^{6} \)
$73$ \( 1 - 3 T + 105 T^{2} - 169 T^{3} + 7665 T^{4} - 15987 T^{5} + 389017 T^{6} \)
$79$ \( 1 + 21 T + 321 T^{2} + 3049 T^{3} + 25359 T^{4} + 131061 T^{5} + 493039 T^{6} \)
$83$ \( 1 + 3 T + 69 T^{2} - 585 T^{3} + 5727 T^{4} + 20667 T^{5} + 571787 T^{6} \)
$89$ \( 1 + 18 T + 186 T^{2} + 1827 T^{3} + 16554 T^{4} + 142578 T^{5} + 704969 T^{6} \)
$97$ \( 1 - 27 T + 450 T^{2} - 5075 T^{3} + 43650 T^{4} - 254043 T^{5} + 912673 T^{6} \)
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