Properties

Label 4018.2.a.u
Level 4018
Weight 2
Character orbit 4018.a
Self dual yes
Analytic conductor 32.084
Analytic rank 1
Dimension 2
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: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{33})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} -\beta q^{3} + q^{4} + \beta q^{5} + \beta q^{6} - q^{8} + ( 5 + \beta ) q^{9} +O(q^{10})\) \( q - q^{2} -\beta q^{3} + q^{4} + \beta q^{5} + \beta q^{6} - q^{8} + ( 5 + \beta ) q^{9} -\beta q^{10} + ( -2 + \beta ) q^{11} -\beta q^{12} + ( 3 - \beta ) q^{13} + ( -8 - \beta ) q^{15} + q^{16} + ( -2 + 2 \beta ) q^{17} + ( -5 - \beta ) q^{18} + ( 2 + \beta ) q^{19} + \beta q^{20} + ( 2 - \beta ) q^{22} + ( -2 - 2 \beta ) q^{23} + \beta q^{24} + ( 3 + \beta ) q^{25} + ( -3 + \beta ) q^{26} + ( -8 - 3 \beta ) q^{27} + ( 3 - 3 \beta ) q^{29} + ( 8 + \beta ) q^{30} + ( 4 - 2 \beta ) q^{31} - q^{32} + ( -8 + \beta ) q^{33} + ( 2 - 2 \beta ) q^{34} + ( 5 + \beta ) q^{36} -2 q^{37} + ( -2 - \beta ) q^{38} + ( 8 - 2 \beta ) q^{39} -\beta q^{40} + q^{41} + ( -3 - \beta ) q^{43} + ( -2 + \beta ) q^{44} + ( 8 + 6 \beta ) q^{45} + ( 2 + 2 \beta ) q^{46} -8 q^{47} -\beta q^{48} + ( -3 - \beta ) q^{50} -16 q^{51} + ( 3 - \beta ) q^{52} -10 q^{53} + ( 8 + 3 \beta ) q^{54} + ( 8 - \beta ) q^{55} + ( -8 - 3 \beta ) q^{57} + ( -3 + 3 \beta ) q^{58} + ( -3 + \beta ) q^{59} + ( -8 - \beta ) q^{60} + ( -6 + \beta ) q^{61} + ( -4 + 2 \beta ) q^{62} + q^{64} + ( -8 + 2 \beta ) q^{65} + ( 8 - \beta ) q^{66} -8 q^{67} + ( -2 + 2 \beta ) q^{68} + ( 16 + 4 \beta ) q^{69} + ( -5 - 2 \beta ) q^{71} + ( -5 - \beta ) q^{72} + ( -3 - 3 \beta ) q^{73} + 2 q^{74} + ( -8 - 4 \beta ) q^{75} + ( 2 + \beta ) q^{76} + ( -8 + 2 \beta ) q^{78} + ( -8 + \beta ) q^{79} + \beta q^{80} + ( 9 + 8 \beta ) q^{81} - q^{82} + ( -3 + \beta ) q^{83} + 16 q^{85} + ( 3 + \beta ) q^{86} + 24 q^{87} + ( 2 - \beta ) q^{88} + 2 q^{89} + ( -8 - 6 \beta ) q^{90} + ( -2 - 2 \beta ) q^{92} + ( 16 - 2 \beta ) q^{93} + 8 q^{94} + ( 8 + 3 \beta ) q^{95} + \beta q^{96} -8 q^{97} + ( -2 + 4 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} - q^{3} + 2q^{4} + q^{5} + q^{6} - 2q^{8} + 11q^{9} + O(q^{10}) \) \( 2q - 2q^{2} - q^{3} + 2q^{4} + q^{5} + q^{6} - 2q^{8} + 11q^{9} - q^{10} - 3q^{11} - q^{12} + 5q^{13} - 17q^{15} + 2q^{16} - 2q^{17} - 11q^{18} + 5q^{19} + q^{20} + 3q^{22} - 6q^{23} + q^{24} + 7q^{25} - 5q^{26} - 19q^{27} + 3q^{29} + 17q^{30} + 6q^{31} - 2q^{32} - 15q^{33} + 2q^{34} + 11q^{36} - 4q^{37} - 5q^{38} + 14q^{39} - q^{40} + 2q^{41} - 7q^{43} - 3q^{44} + 22q^{45} + 6q^{46} - 16q^{47} - q^{48} - 7q^{50} - 32q^{51} + 5q^{52} - 20q^{53} + 19q^{54} + 15q^{55} - 19q^{57} - 3q^{58} - 5q^{59} - 17q^{60} - 11q^{61} - 6q^{62} + 2q^{64} - 14q^{65} + 15q^{66} - 16q^{67} - 2q^{68} + 36q^{69} - 12q^{71} - 11q^{72} - 9q^{73} + 4q^{74} - 20q^{75} + 5q^{76} - 14q^{78} - 15q^{79} + q^{80} + 26q^{81} - 2q^{82} - 5q^{83} + 32q^{85} + 7q^{86} + 48q^{87} + 3q^{88} + 4q^{89} - 22q^{90} - 6q^{92} + 30q^{93} + 16q^{94} + 19q^{95} + q^{96} - 16q^{97} + O(q^{100}) \)

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
3.37228
−2.37228
−1.00000 −3.37228 1.00000 3.37228 3.37228 0 −1.00000 8.37228 −3.37228
1.2 −1.00000 2.37228 1.00000 −2.37228 −2.37228 0 −1.00000 2.62772 2.37228
\(n\): 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.u 2
7.b odd 2 1 4018.2.a.v 2
7.d odd 6 2 574.2.e.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
574.2.e.d 4 7.d odd 6 2
4018.2.a.u 2 1.a even 1 1 trivial
4018.2.a.v 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4018))\):

\( T_{3}^{2} + T_{3} - 8 \)
\( T_{5}^{2} - T_{5} - 8 \)
\( T_{11}^{2} + 3 T_{11} - 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( 1 + T - 2 T^{2} + 3 T^{3} + 9 T^{4} \)
$5$ \( 1 - T + 2 T^{2} - 5 T^{3} + 25 T^{4} \)
$7$ 1
$11$ \( 1 + 3 T + 16 T^{2} + 33 T^{3} + 121 T^{4} \)
$13$ \( 1 - 5 T + 24 T^{2} - 65 T^{3} + 169 T^{4} \)
$17$ \( 1 + 2 T + 2 T^{2} + 34 T^{3} + 289 T^{4} \)
$19$ \( 1 - 5 T + 36 T^{2} - 95 T^{3} + 361 T^{4} \)
$23$ \( 1 + 6 T + 22 T^{2} + 138 T^{3} + 529 T^{4} \)
$29$ \( 1 - 3 T - 14 T^{2} - 87 T^{3} + 841 T^{4} \)
$31$ \( 1 - 6 T + 38 T^{2} - 186 T^{3} + 961 T^{4} \)
$37$ \( ( 1 + 2 T + 37 T^{2} )^{2} \)
$41$ \( ( 1 - T )^{2} \)
$43$ \( 1 + 7 T + 90 T^{2} + 301 T^{3} + 1849 T^{4} \)
$47$ \( ( 1 + 8 T + 47 T^{2} )^{2} \)
$53$ \( ( 1 + 10 T + 53 T^{2} )^{2} \)
$59$ \( 1 + 5 T + 116 T^{2} + 295 T^{3} + 3481 T^{4} \)
$61$ \( 1 + 11 T + 144 T^{2} + 671 T^{3} + 3721 T^{4} \)
$67$ \( ( 1 + 8 T + 67 T^{2} )^{2} \)
$71$ \( 1 + 12 T + 145 T^{2} + 852 T^{3} + 5041 T^{4} \)
$73$ \( 1 + 9 T + 92 T^{2} + 657 T^{3} + 5329 T^{4} \)
$79$ \( 1 + 15 T + 206 T^{2} + 1185 T^{3} + 6241 T^{4} \)
$83$ \( 1 + 5 T + 164 T^{2} + 415 T^{3} + 6889 T^{4} \)
$89$ \( ( 1 - 2 T + 89 T^{2} )^{2} \)
$97$ \( ( 1 + 8 T + 97 T^{2} )^{2} \)
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