Properties

Label 4018.2.a.bf
Level 4018
Weight 2
Character orbit 4018.a
Self dual yes
Analytic conductor 32.084
Analytic rank 1
Dimension 3
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 4018 = 2 \cdot 7^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4018.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.0838915322\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.321.1
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 574)
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.

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} + ( 1 - \beta_{2} ) q^{5} - q^{6} + q^{8} -2 q^{9} +O(q^{10})\) \( q + q^{2} - q^{3} + q^{4} + ( 1 - \beta_{2} ) q^{5} - q^{6} + q^{8} -2 q^{9} + ( 1 - \beta_{2} ) q^{10} + ( -2 + \beta_{1} ) q^{11} - q^{12} + ( 1 + 2 \beta_{2} ) q^{13} + ( -1 + \beta_{2} ) q^{15} + q^{16} + ( -2 + 2 \beta_{1} + \beta_{2} ) q^{17} -2 q^{18} + ( 2 - 2 \beta_{1} + \beta_{2} ) q^{19} + ( 1 - \beta_{2} ) q^{20} + ( -2 + \beta_{1} ) q^{22} + ( -2 - 3 \beta_{1} ) q^{23} - q^{24} + ( -\beta_{1} - 4 \beta_{2} ) q^{25} + ( 1 + 2 \beta_{2} ) q^{26} + 5 q^{27} + ( -3 - 2 \beta_{1} + \beta_{2} ) q^{29} + ( -1 + \beta_{2} ) q^{30} + ( -3 - 2 \beta_{1} ) q^{31} + q^{32} + ( 2 - \beta_{1} ) q^{33} + ( -2 + 2 \beta_{1} + \beta_{2} ) q^{34} -2 q^{36} + ( -6 + \beta_{1} + \beta_{2} ) q^{37} + ( 2 - 2 \beta_{1} + \beta_{2} ) q^{38} + ( -1 - 2 \beta_{2} ) q^{39} + ( 1 - \beta_{2} ) q^{40} + q^{41} + ( -5 + 4 \beta_{1} + 2 \beta_{2} ) q^{43} + ( -2 + \beta_{1} ) q^{44} + ( -2 + 2 \beta_{2} ) q^{45} + ( -2 - 3 \beta_{1} ) q^{46} + ( -1 + 5 \beta_{1} ) q^{47} - q^{48} + ( -\beta_{1} - 4 \beta_{2} ) q^{50} + ( 2 - 2 \beta_{1} - \beta_{2} ) q^{51} + ( 1 + 2 \beta_{2} ) q^{52} + ( -2 + \beta_{1} + 3 \beta_{2} ) q^{53} + 5 q^{54} + ( -1 + 2 \beta_{2} ) q^{55} + ( -2 + 2 \beta_{1} - \beta_{2} ) q^{57} + ( -3 - 2 \beta_{1} + \beta_{2} ) q^{58} + ( 1 + 3 \beta_{1} - \beta_{2} ) q^{59} + ( -1 + \beta_{2} ) q^{60} + ( 5 - 6 \beta_{1} - \beta_{2} ) q^{61} + ( -3 - 2 \beta_{1} ) q^{62} + q^{64} + ( -7 + 2 \beta_{1} + 5 \beta_{2} ) q^{65} + ( 2 - \beta_{1} ) q^{66} + ( -7 - \beta_{1} + \beta_{2} ) q^{67} + ( -2 + 2 \beta_{1} + \beta_{2} ) q^{68} + ( 2 + 3 \beta_{1} ) q^{69} + ( -1 + \beta_{1} - 2 \beta_{2} ) q^{71} -2 q^{72} + ( -3 - 2 \beta_{1} - 3 \beta_{2} ) q^{73} + ( -6 + \beta_{1} + \beta_{2} ) q^{74} + ( \beta_{1} + 4 \beta_{2} ) q^{75} + ( 2 - 2 \beta_{1} + \beta_{2} ) q^{76} + ( -1 - 2 \beta_{2} ) q^{78} + ( -2 + 2 \beta_{1} - 5 \beta_{2} ) q^{79} + ( 1 - \beta_{2} ) q^{80} + q^{81} + q^{82} + ( 3 \beta_{1} - \beta_{2} ) q^{83} + ( -4 + \beta_{1} + 5 \beta_{2} ) q^{85} + ( -5 + 4 \beta_{1} + 2 \beta_{2} ) q^{86} + ( 3 + 2 \beta_{1} - \beta_{2} ) q^{87} + ( -2 + \beta_{1} ) q^{88} + ( -5 + 2 \beta_{1} - 3 \beta_{2} ) q^{89} + ( -2 + 2 \beta_{2} ) q^{90} + ( -2 - 3 \beta_{1} ) q^{92} + ( 3 + 2 \beta_{1} ) q^{93} + ( -1 + 5 \beta_{1} ) q^{94} + ( -4 + \beta_{1} + \beta_{2} ) q^{95} - q^{96} + ( -1 - 3 \beta_{1} - 3 \beta_{2} ) q^{97} + ( 4 - 2 \beta_{1} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{2} - 3q^{3} + 3q^{4} + 4q^{5} - 3q^{6} + 3q^{8} - 6q^{9} + O(q^{10}) \) \( 3q + 3q^{2} - 3q^{3} + 3q^{4} + 4q^{5} - 3q^{6} + 3q^{8} - 6q^{9} + 4q^{10} - 5q^{11} - 3q^{12} + q^{13} - 4q^{15} + 3q^{16} - 5q^{17} - 6q^{18} + 3q^{19} + 4q^{20} - 5q^{22} - 9q^{23} - 3q^{24} + 3q^{25} + q^{26} + 15q^{27} - 12q^{29} - 4q^{30} - 11q^{31} + 3q^{32} + 5q^{33} - 5q^{34} - 6q^{36} - 18q^{37} + 3q^{38} - q^{39} + 4q^{40} + 3q^{41} - 13q^{43} - 5q^{44} - 8q^{45} - 9q^{46} + 2q^{47} - 3q^{48} + 3q^{50} + 5q^{51} + q^{52} - 8q^{53} + 15q^{54} - 5q^{55} - 3q^{57} - 12q^{58} + 7q^{59} - 4q^{60} + 10q^{61} - 11q^{62} + 3q^{64} - 24q^{65} + 5q^{66} - 23q^{67} - 5q^{68} + 9q^{69} - 6q^{72} - 8q^{73} - 18q^{74} - 3q^{75} + 3q^{76} - q^{78} + q^{79} + 4q^{80} + 3q^{81} + 3q^{82} + 4q^{83} - 16q^{85} - 13q^{86} + 12q^{87} - 5q^{88} - 10q^{89} - 8q^{90} - 9q^{92} + 11q^{93} + 2q^{94} - 12q^{95} - 3q^{96} - 3q^{97} + 10q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 4 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 3\)

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
−1.69963
2.46050
0.239123
1.00000 −1.00000 1.00000 −0.588364 −1.00000 0 1.00000 −2.00000 −0.588364
1.2 1.00000 −1.00000 1.00000 0.406421 −1.00000 0 1.00000 −2.00000 0.406421
1.3 1.00000 −1.00000 1.00000 4.18194 −1.00000 0 1.00000 −2.00000 4.18194
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 4018.2.a.bf 3
7.b odd 2 1 4018.2.a.bh 3
7.c even 3 2 574.2.e.e 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
574.2.e.e 6 7.c even 3 2
4018.2.a.bf 3 1.a even 1 1 trivial
4018.2.a.bh 3 7.b odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(1\)
\(41\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4018))\):

\( T_{3} + 1 \)
\( T_{5}^{3} - 4 T_{5}^{2} - T_{5} + 1 \)
\( T_{11}^{3} + 5 T_{11}^{2} + 4 T_{11} - 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T )^{3} \)
$3$ \( ( 1 + T + 3 T^{2} )^{3} \)
$5$ \( 1 - 4 T + 14 T^{2} - 39 T^{3} + 70 T^{4} - 100 T^{5} + 125 T^{6} \)
$7$ 1
$11$ \( 1 + 5 T + 37 T^{2} + 107 T^{3} + 407 T^{4} + 605 T^{5} + 1331 T^{6} \)
$13$ \( 1 - T + 14 T^{2} + 23 T^{3} + 182 T^{4} - 169 T^{5} + 2197 T^{6} \)
$17$ \( 1 + 5 T + 39 T^{2} + 107 T^{3} + 663 T^{4} + 1445 T^{5} + 4913 T^{6} \)
$19$ \( 1 - 3 T + 33 T^{2} - 141 T^{3} + 627 T^{4} - 1083 T^{5} + 6859 T^{6} \)
$23$ \( 1 + 9 T + 57 T^{2} + 335 T^{3} + 1311 T^{4} + 4761 T^{5} + 12167 T^{6} \)
$29$ \( 1 + 12 T + 108 T^{2} + 599 T^{3} + 3132 T^{4} + 10092 T^{5} + 24389 T^{6} \)
$31$ \( 1 + 11 T + 116 T^{2} + 671 T^{3} + 3596 T^{4} + 10571 T^{5} + 29791 T^{6} \)
$37$ \( 1 + 18 T + 210 T^{2} + 1493 T^{3} + 7770 T^{4} + 24642 T^{5} + 50653 T^{6} \)
$41$ \( ( 1 - T )^{3} \)
$43$ \( 1 + 13 T + 104 T^{2} + 577 T^{3} + 4472 T^{4} + 24037 T^{5} + 79507 T^{6} \)
$47$ \( 1 - 2 T + 34 T^{2} - 167 T^{3} + 1598 T^{4} - 4418 T^{5} + 103823 T^{6} \)
$53$ \( 1 + 8 T + 124 T^{2} + 875 T^{3} + 6572 T^{4} + 22472 T^{5} + 148877 T^{6} \)
$59$ \( 1 - 7 T + 143 T^{2} - 609 T^{3} + 8437 T^{4} - 24367 T^{5} + 205379 T^{6} \)
$61$ \( 1 - 10 T + 64 T^{2} - 269 T^{3} + 3904 T^{4} - 37210 T^{5} + 226981 T^{6} \)
$67$ \( 1 + 23 T + 365 T^{2} + 3425 T^{3} + 24455 T^{4} + 103247 T^{5} + 300763 T^{6} \)
$71$ \( 1 + 180 T^{2} + 9 T^{3} + 12780 T^{4} + 357911 T^{6} \)
$73$ \( 1 + 8 T + 176 T^{2} + 911 T^{3} + 12848 T^{4} + 42632 T^{5} + 389017 T^{6} \)
$79$ \( 1 - T + 45 T^{2} - 167 T^{3} + 3555 T^{4} - 6241 T^{5} + 493039 T^{6} \)
$83$ \( 1 - 4 T + 204 T^{2} - 487 T^{3} + 16932 T^{4} - 27556 T^{5} + 571787 T^{6} \)
$89$ \( 1 + 10 T + 216 T^{2} + 1657 T^{3} + 19224 T^{4} + 79210 T^{5} + 704969 T^{6} \)
$97$ \( 1 + 3 T + 213 T^{2} + 529 T^{3} + 20661 T^{4} + 28227 T^{5} + 912673 T^{6} \)
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