Properties

Label 4018.2.a.bj
Level 4018
Weight 2
Character orbit 4018.a
Self dual yes
Analytic conductor 32.084
Analytic rank 0
Dimension 4
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: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.11348.1
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + \beta_{2} q^{3} + q^{4} + ( -1 + \beta_{2} - \beta_{3} ) q^{5} -\beta_{2} q^{6} - q^{8} + ( 2 - \beta_{1} + \beta_{3} ) q^{9} +O(q^{10})\) \( q - q^{2} + \beta_{2} q^{3} + q^{4} + ( -1 + \beta_{2} - \beta_{3} ) q^{5} -\beta_{2} q^{6} - q^{8} + ( 2 - \beta_{1} + \beta_{3} ) q^{9} + ( 1 - \beta_{2} + \beta_{3} ) q^{10} + ( 1 - \beta_{3} ) q^{11} + \beta_{2} q^{12} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{13} + ( 3 - \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{15} + q^{16} + ( -2 \beta_{1} - \beta_{2} ) q^{17} + ( -2 + \beta_{1} - \beta_{3} ) q^{18} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{19} + ( -1 + \beta_{2} - \beta_{3} ) q^{20} + ( -1 + \beta_{3} ) q^{22} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{23} -\beta_{2} q^{24} + ( 3 - 2 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{25} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{26} + ( 3 + \beta_{1} + 2 \beta_{2} + 3 \beta_{3} ) q^{27} + ( 4 + \beta_{1} + 2 \beta_{2} ) q^{29} + ( -3 + \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{30} + ( -1 - \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{31} - q^{32} + ( -2 - 2 \beta_{3} ) q^{33} + ( 2 \beta_{1} + \beta_{2} ) q^{34} + ( 2 - \beta_{1} + \beta_{3} ) q^{36} + ( -2 + 2 \beta_{2} ) q^{37} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{38} + ( 4 + 2 \beta_{2} ) q^{39} + ( 1 - \beta_{2} + \beta_{3} ) q^{40} - q^{41} + ( -3 + \beta_{1} + \beta_{3} ) q^{43} + ( 1 - \beta_{3} ) q^{44} + ( -8 + 3 \beta_{1} + \beta_{2} ) q^{45} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{46} + ( -1 - \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{47} + \beta_{2} q^{48} + ( -3 + 2 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{50} + ( -3 + 3 \beta_{1} + 4 \beta_{2} + \beta_{3} ) q^{51} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{52} + ( 8 + \beta_{1} - 2 \beta_{2} + 4 \beta_{3} ) q^{53} + ( -3 - \beta_{1} - 2 \beta_{2} - 3 \beta_{3} ) q^{54} + ( 3 - \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{55} + 4 q^{57} + ( -4 - \beta_{1} - 2 \beta_{2} ) q^{58} + ( -4 + \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{59} + ( 3 - \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{60} + ( -5 + 2 \beta_{1} + \beta_{2} + 3 \beta_{3} ) q^{61} + ( 1 + \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{62} + q^{64} + ( 4 + 2 \beta_{2} - 4 \beta_{3} ) q^{65} + ( 2 + 2 \beta_{3} ) q^{66} + ( 1 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{67} + ( -2 \beta_{1} - \beta_{2} ) q^{68} + ( 4 + 2 \beta_{2} ) q^{69} + ( 3 - 5 \beta_{1} - \beta_{3} ) q^{71} + ( -2 + \beta_{1} - \beta_{3} ) q^{72} + ( -2 + 4 \beta_{1} ) q^{73} + ( 2 - 2 \beta_{2} ) q^{74} + ( -7 + 3 \beta_{1} + 5 \beta_{2} - 3 \beta_{3} ) q^{75} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{76} + ( -4 - 2 \beta_{2} ) q^{78} + ( -7 - \beta_{1} + \beta_{3} ) q^{79} + ( -1 + \beta_{2} - \beta_{3} ) q^{80} + ( 9 + 4 \beta_{2} + 4 \beta_{3} ) q^{81} + q^{82} + ( 4 - \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{83} + ( -7 + 5 \beta_{1} + 4 \beta_{2} + \beta_{3} ) q^{85} + ( 3 - \beta_{1} - \beta_{3} ) q^{86} + ( 9 - 3 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{87} + ( -1 + \beta_{3} ) q^{88} + ( -9 - \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{89} + ( 8 - 3 \beta_{1} - \beta_{2} ) q^{90} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{92} + ( -11 + 3 \beta_{1} - 3 \beta_{3} ) q^{93} + ( 1 + \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{94} + ( 6 - 2 \beta_{3} ) q^{95} -\beta_{2} q^{96} + ( 1 + \beta_{1} + 3 \beta_{3} ) q^{97} + ( -7 - 4 \beta_{2} - \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{2} + q^{3} + 4q^{4} - 3q^{5} - q^{6} - 4q^{8} + 9q^{9} + O(q^{10}) \) \( 4q - 4q^{2} + q^{3} + 4q^{4} - 3q^{5} - q^{6} - 4q^{8} + 9q^{9} + 3q^{10} + 4q^{11} + q^{12} + 6q^{13} + 11q^{15} + 4q^{16} + q^{17} - 9q^{18} - 2q^{19} - 3q^{20} - 4q^{22} + 6q^{23} - q^{24} + 13q^{25} - 6q^{26} + 13q^{27} + 17q^{29} - 11q^{30} - 5q^{31} - 4q^{32} - 8q^{33} - q^{34} + 9q^{36} - 6q^{37} + 2q^{38} + 18q^{39} + 3q^{40} - 4q^{41} - 13q^{43} + 4q^{44} - 34q^{45} - 6q^{46} - 6q^{47} + q^{48} - 13q^{50} - 11q^{51} + 6q^{52} + 29q^{53} - 13q^{54} + 16q^{55} + 16q^{57} - 17q^{58} - 16q^{59} + 11q^{60} - 21q^{61} + 5q^{62} + 4q^{64} + 18q^{65} + 8q^{66} + 4q^{67} + q^{68} + 18q^{69} + 17q^{71} - 9q^{72} - 12q^{73} + 6q^{74} - 26q^{75} - 2q^{76} - 18q^{78} - 27q^{79} - 3q^{80} + 40q^{81} + 4q^{82} + 18q^{83} - 29q^{85} + 13q^{86} + 41q^{87} - 4q^{88} - 37q^{89} + 34q^{90} + 6q^{92} - 47q^{93} + 6q^{94} + 24q^{95} - q^{96} + 3q^{97} - 32q^{99} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{2}\)\(=\)\( \nu^{3} - \nu^{2} - 5 \nu + 1 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 2 \nu^{2} - 3 \nu + 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} - \beta_{2} + \beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{1} + 3\)
\(\nu^{3}\)\(=\)\((\)\(5 \beta_{3} - 3 \beta_{2} + 7 \beta_{1} + 9\)\()/2\)

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
0.723742
2.64119
−1.77571
−0.589216
−1.00000 −2.76342 1.00000 −3.92368 2.76342 0 −1.00000 4.63646 3.92368
1.2 −1.00000 −0.757235 1.00000 −1.30651 0.757235 0 −1.00000 −2.42659 1.30651
1.3 −1.00000 1.12631 1.00000 3.70458 −1.12631 0 −1.00000 −1.73143 −3.70458
1.4 −1.00000 3.39434 1.00000 −1.47439 −3.39434 0 −1.00000 8.52156 1.47439
\(n\): 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.bj 4
7.b odd 2 1 574.2.a.m 4
21.c even 2 1 5166.2.a.bx 4
28.d even 2 1 4592.2.a.ba 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
574.2.a.m 4 7.b odd 2 1
4018.2.a.bj 4 1.a even 1 1 trivial
4592.2.a.ba 4 28.d even 2 1
5166.2.a.bx 4 21.c even 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4018))\):

\( T_{3}^{4} - T_{3}^{3} - 10 T_{3}^{2} + 4 T_{3} + 8 \)
\( T_{5}^{4} + 3 T_{5}^{3} - 12 T_{5}^{2} - 40 T_{5} - 28 \)
\( T_{11}^{4} - 4 T_{11}^{3} - 8 T_{11}^{2} + 28 T_{11} - 16 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{4} \)
$3$ \( 1 - T + 2 T^{2} - 5 T^{3} + 2 T^{4} - 15 T^{5} + 18 T^{6} - 27 T^{7} + 81 T^{8} \)
$5$ \( 1 + 3 T + 8 T^{2} + 5 T^{3} + 2 T^{4} + 25 T^{5} + 200 T^{6} + 375 T^{7} + 625 T^{8} \)
$7$ \( \)
$11$ \( 1 - 4 T + 36 T^{2} - 104 T^{3} + 534 T^{4} - 1144 T^{5} + 4356 T^{6} - 5324 T^{7} + 14641 T^{8} \)
$13$ \( 1 - 6 T + 44 T^{2} - 170 T^{3} + 774 T^{4} - 2210 T^{5} + 7436 T^{6} - 13182 T^{7} + 28561 T^{8} \)
$17$ \( 1 - T + 10 T^{2} - 23 T^{3} - 86 T^{4} - 391 T^{5} + 2890 T^{6} - 4913 T^{7} + 83521 T^{8} \)
$19$ \( 1 + 2 T + 56 T^{2} + 106 T^{3} + 1438 T^{4} + 2014 T^{5} + 20216 T^{6} + 13718 T^{7} + 130321 T^{8} \)
$23$ \( 1 - 6 T + 84 T^{2} - 350 T^{3} + 2774 T^{4} - 8050 T^{5} + 44436 T^{6} - 73002 T^{7} + 279841 T^{8} \)
$29$ \( 1 - 17 T + 178 T^{2} - 1231 T^{3} + 7294 T^{4} - 35699 T^{5} + 149698 T^{6} - 414613 T^{7} + 707281 T^{8} \)
$31$ \( 1 + 5 T + 68 T^{2} + 289 T^{3} + 2326 T^{4} + 8959 T^{5} + 65348 T^{6} + 148955 T^{7} + 923521 T^{8} \)
$37$ \( 1 + 6 T + 120 T^{2} + 546 T^{3} + 6174 T^{4} + 20202 T^{5} + 164280 T^{6} + 303918 T^{7} + 1874161 T^{8} \)
$41$ \( ( 1 + T )^{4} \)
$43$ \( 1 + 13 T + 220 T^{2} + 1709 T^{3} + 15190 T^{4} + 73487 T^{5} + 406780 T^{6} + 1033591 T^{7} + 3418801 T^{8} \)
$47$ \( 1 + 6 T + 108 T^{2} + 190 T^{3} + 4550 T^{4} + 8930 T^{5} + 238572 T^{6} + 622938 T^{7} + 4879681 T^{8} \)
$53$ \( 1 - 29 T + 362 T^{2} - 2619 T^{3} + 16622 T^{4} - 138807 T^{5} + 1016858 T^{6} - 4317433 T^{7} + 7890481 T^{8} \)
$59$ \( 1 + 16 T + 264 T^{2} + 2444 T^{3} + 23526 T^{4} + 144196 T^{5} + 918984 T^{6} + 3286064 T^{7} + 12117361 T^{8} \)
$61$ \( 1 + 21 T + 276 T^{2} + 2979 T^{3} + 26922 T^{4} + 181719 T^{5} + 1026996 T^{6} + 4766601 T^{7} + 13845841 T^{8} \)
$67$ \( 1 - 4 T + 172 T^{2} - 384 T^{3} + 14822 T^{4} - 25728 T^{5} + 772108 T^{6} - 1203052 T^{7} + 20151121 T^{8} \)
$71$ \( 1 - 17 T + 84 T^{2} + 1019 T^{3} - 16170 T^{4} + 72349 T^{5} + 423444 T^{6} - 6084487 T^{7} + 25411681 T^{8} \)
$73$ \( 1 + 12 T + 116 T^{2} + 276 T^{3} + 3382 T^{4} + 20148 T^{5} + 618164 T^{6} + 4668204 T^{7} + 28398241 T^{8} \)
$79$ \( 1 + 27 T + 548 T^{2} + 7039 T^{3} + 74358 T^{4} + 556081 T^{5} + 3420068 T^{6} + 13312053 T^{7} + 38950081 T^{8} \)
$83$ \( 1 - 18 T + 412 T^{2} - 4590 T^{3} + 54630 T^{4} - 380970 T^{5} + 2838268 T^{6} - 10292166 T^{7} + 47458321 T^{8} \)
$89$ \( 1 + 37 T + 814 T^{2} + 11891 T^{3} + 130434 T^{4} + 1058299 T^{5} + 6447694 T^{6} + 26083853 T^{7} + 62742241 T^{8} \)
$97$ \( 1 - 3 T + 290 T^{2} - 621 T^{3} + 37786 T^{4} - 60237 T^{5} + 2728610 T^{6} - 2738019 T^{7} + 88529281 T^{8} \)
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