Properties

Label 4018.2.a.y
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: \(\Q(\sqrt{2}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + ( -1 + \beta ) q^{3} + q^{4} + ( -1 + 2 \beta ) q^{5} + ( -1 + \beta ) q^{6} + q^{8} -2 \beta q^{9} +O(q^{10})\) \( q + q^{2} + ( -1 + \beta ) q^{3} + q^{4} + ( -1 + 2 \beta ) q^{5} + ( -1 + \beta ) q^{6} + q^{8} -2 \beta q^{9} + ( -1 + 2 \beta ) q^{10} + ( -2 - 3 \beta ) q^{11} + ( -1 + \beta ) q^{12} + ( -2 + \beta ) q^{13} + ( 5 - 3 \beta ) q^{15} + q^{16} + ( -1 - 2 \beta ) q^{17} -2 \beta q^{18} -2 \beta q^{19} + ( -1 + 2 \beta ) q^{20} + ( -2 - 3 \beta ) q^{22} + 4 \beta q^{23} + ( -1 + \beta ) q^{24} + ( 4 - 4 \beta ) q^{25} + ( -2 + \beta ) q^{26} + ( -1 - \beta ) q^{27} -5 q^{29} + ( 5 - 3 \beta ) q^{30} + ( 7 - \beta ) q^{31} + q^{32} + ( -4 + \beta ) q^{33} + ( -1 - 2 \beta ) q^{34} -2 \beta q^{36} + ( 8 - \beta ) q^{37} -2 \beta q^{38} + ( 4 - 3 \beta ) q^{39} + ( -1 + 2 \beta ) q^{40} + q^{41} + ( -3 + 3 \beta ) q^{43} + ( -2 - 3 \beta ) q^{44} + ( -8 + 2 \beta ) q^{45} + 4 \beta q^{46} + ( -6 - 3 \beta ) q^{47} + ( -1 + \beta ) q^{48} + ( 4 - 4 \beta ) q^{50} + ( -3 + \beta ) q^{51} + ( -2 + \beta ) q^{52} + ( 1 - 6 \beta ) q^{53} + ( -1 - \beta ) q^{54} + ( -10 - \beta ) q^{55} + ( -4 + 2 \beta ) q^{57} -5 q^{58} + ( -2 - 6 \beta ) q^{59} + ( 5 - 3 \beta ) q^{60} + ( -3 - 6 \beta ) q^{61} + ( 7 - \beta ) q^{62} + q^{64} + ( 6 - 5 \beta ) q^{65} + ( -4 + \beta ) q^{66} + ( -10 + 2 \beta ) q^{67} + ( -1 - 2 \beta ) q^{68} + ( 8 - 4 \beta ) q^{69} + ( -1 + 7 \beta ) q^{71} -2 \beta q^{72} + ( 8 - \beta ) q^{74} + ( -12 + 8 \beta ) q^{75} -2 \beta q^{76} + ( 4 - 3 \beta ) q^{78} + ( -5 - 3 \beta ) q^{79} + ( -1 + 2 \beta ) q^{80} + ( -1 + 6 \beta ) q^{81} + q^{82} + ( -10 - 3 \beta ) q^{83} -7 q^{85} + ( -3 + 3 \beta ) q^{86} + ( 5 - 5 \beta ) q^{87} + ( -2 - 3 \beta ) q^{88} + ( -1 - 2 \beta ) q^{89} + ( -8 + 2 \beta ) q^{90} + 4 \beta q^{92} + ( -9 + 8 \beta ) q^{93} + ( -6 - 3 \beta ) q^{94} + ( -8 + 2 \beta ) q^{95} + ( -1 + \beta ) q^{96} + ( -3 + 2 \beta ) q^{97} + ( 12 + 4 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} - 2q^{3} + 2q^{4} - 2q^{5} - 2q^{6} + 2q^{8} + O(q^{10}) \) \( 2q + 2q^{2} - 2q^{3} + 2q^{4} - 2q^{5} - 2q^{6} + 2q^{8} - 2q^{10} - 4q^{11} - 2q^{12} - 4q^{13} + 10q^{15} + 2q^{16} - 2q^{17} - 2q^{20} - 4q^{22} - 2q^{24} + 8q^{25} - 4q^{26} - 2q^{27} - 10q^{29} + 10q^{30} + 14q^{31} + 2q^{32} - 8q^{33} - 2q^{34} + 16q^{37} + 8q^{39} - 2q^{40} + 2q^{41} - 6q^{43} - 4q^{44} - 16q^{45} - 12q^{47} - 2q^{48} + 8q^{50} - 6q^{51} - 4q^{52} + 2q^{53} - 2q^{54} - 20q^{55} - 8q^{57} - 10q^{58} - 4q^{59} + 10q^{60} - 6q^{61} + 14q^{62} + 2q^{64} + 12q^{65} - 8q^{66} - 20q^{67} - 2q^{68} + 16q^{69} - 2q^{71} + 16q^{74} - 24q^{75} + 8q^{78} - 10q^{79} - 2q^{80} - 2q^{81} + 2q^{82} - 20q^{83} - 14q^{85} - 6q^{86} + 10q^{87} - 4q^{88} - 2q^{89} - 16q^{90} - 18q^{93} - 12q^{94} - 16q^{95} - 2q^{96} - 6q^{97} + 24q^{99} + O(q^{100}) \)

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
−1.41421
1.41421
1.00000 −2.41421 1.00000 −3.82843 −2.41421 0 1.00000 2.82843 −3.82843
1.2 1.00000 0.414214 1.00000 1.82843 0.414214 0 1.00000 −2.82843 1.82843
\(n\): 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.y 2
7.b odd 2 1 4018.2.a.bc yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4018.2.a.y 2 1.a even 1 1 trivial
4018.2.a.bc yes 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4018))\):

\( T_{3}^{2} + 2 T_{3} - 1 \)
\( T_{5}^{2} + 2 T_{5} - 7 \)
\( T_{11}^{2} + 4 T_{11} - 14 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T )^{2} \)
$3$ \( 1 + 2 T + 5 T^{2} + 6 T^{3} + 9 T^{4} \)
$5$ \( 1 + 2 T + 3 T^{2} + 10 T^{3} + 25 T^{4} \)
$7$ 1
$11$ \( 1 + 4 T + 8 T^{2} + 44 T^{3} + 121 T^{4} \)
$13$ \( 1 + 4 T + 28 T^{2} + 52 T^{3} + 169 T^{4} \)
$17$ \( 1 + 2 T + 27 T^{2} + 34 T^{3} + 289 T^{4} \)
$19$ \( 1 + 30 T^{2} + 361 T^{4} \)
$23$ \( 1 + 14 T^{2} + 529 T^{4} \)
$29$ \( ( 1 + 5 T + 29 T^{2} )^{2} \)
$31$ \( 1 - 14 T + 109 T^{2} - 434 T^{3} + 961 T^{4} \)
$37$ \( 1 - 16 T + 136 T^{2} - 592 T^{3} + 1369 T^{4} \)
$41$ \( ( 1 - T )^{2} \)
$43$ \( 1 + 6 T + 77 T^{2} + 258 T^{3} + 1849 T^{4} \)
$47$ \( 1 + 12 T + 112 T^{2} + 564 T^{3} + 2209 T^{4} \)
$53$ \( 1 - 2 T + 35 T^{2} - 106 T^{3} + 2809 T^{4} \)
$59$ \( 1 + 4 T + 50 T^{2} + 236 T^{3} + 3481 T^{4} \)
$61$ \( 1 + 6 T + 59 T^{2} + 366 T^{3} + 3721 T^{4} \)
$67$ \( 1 + 20 T + 226 T^{2} + 1340 T^{3} + 4489 T^{4} \)
$71$ \( 1 + 2 T + 45 T^{2} + 142 T^{3} + 5041 T^{4} \)
$73$ \( ( 1 + 73 T^{2} )^{2} \)
$79$ \( 1 + 10 T + 165 T^{2} + 790 T^{3} + 6241 T^{4} \)
$83$ \( 1 + 20 T + 248 T^{2} + 1660 T^{3} + 6889 T^{4} \)
$89$ \( 1 + 2 T + 171 T^{2} + 178 T^{3} + 7921 T^{4} \)
$97$ \( 1 + 6 T + 195 T^{2} + 582 T^{3} + 9409 T^{4} \)
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