Properties

Label 8019.2.a.a
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

Related objects

Downloads

Learn more about

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
1.87939
−0.347296
−1.53209
−1.87939 0 1.53209 −2.18479 0 0.532089 0.879385 0 4.10607
1.2 0.347296 0 −1.87939 2.41147 0 −2.87939 −1.34730 0 0.837496
1.3 1.53209 0 0.347296 −3.22668 0 −0.652704 −2.53209 0 −4.94356
\(n\): 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.a 3
3.b odd 2 1 8019.2.a.b 3
27.e even 9 2 297.2.j.a 6
27.f odd 18 2 891.2.j.a 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
297.2.j.a 6 27.e even 9 2
891.2.j.a 6 27.f odd 18 2
8019.2.a.a 3 1.a even 1 1 trivial
8019.2.a.b 3 3.b odd 2 1

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