Properties

Label 4018.2.a.z
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.

\(f(q)\) \(=\) \( q + q^{2} + ( -1 + \beta ) q^{3} + q^{4} + q^{5} + ( -1 + \beta ) q^{6} + q^{8} -2 \beta q^{9} +O(q^{10})\) \( q + q^{2} + ( -1 + \beta ) q^{3} + q^{4} + q^{5} + ( -1 + \beta ) q^{6} + q^{8} -2 \beta q^{9} + q^{10} + ( 2 - \beta ) q^{11} + ( -1 + \beta ) q^{12} + ( -2 - 3 \beta ) q^{13} + ( -1 + \beta ) q^{15} + q^{16} + ( -5 + 2 \beta ) q^{17} -2 \beta q^{18} + 2 \beta q^{19} + q^{20} + ( 2 - \beta ) q^{22} -4 q^{23} + ( -1 + \beta ) q^{24} -4 q^{25} + ( -2 - 3 \beta ) q^{26} + ( -1 - \beta ) q^{27} + ( 1 + 2 \beta ) q^{29} + ( -1 + \beta ) q^{30} + ( -7 - \beta ) q^{31} + q^{32} + ( -4 + 3 \beta ) q^{33} + ( -5 + 2 \beta ) q^{34} -2 \beta q^{36} -\beta q^{37} + 2 \beta q^{38} + ( -4 + \beta ) q^{39} + q^{40} + q^{41} + ( 1 - 5 \beta ) q^{43} + ( 2 - \beta ) q^{44} -2 \beta q^{45} -4 q^{46} + ( -10 + \beta ) q^{47} + ( -1 + \beta ) q^{48} -4 q^{50} + ( 9 - 7 \beta ) q^{51} + ( -2 - 3 \beta ) q^{52} + ( 3 + 4 \beta ) q^{53} + ( -1 - \beta ) q^{54} + ( 2 - \beta ) q^{55} + ( 4 - 2 \beta ) q^{57} + ( 1 + 2 \beta ) q^{58} + ( 2 + 2 \beta ) q^{59} + ( -1 + \beta ) q^{60} + ( 3 - 4 \beta ) q^{61} + ( -7 - \beta ) q^{62} + q^{64} + ( -2 - 3 \beta ) q^{65} + ( -4 + 3 \beta ) q^{66} + ( 6 - 2 \beta ) q^{67} + ( -5 + 2 \beta ) q^{68} + ( 4 - 4 \beta ) q^{69} + ( -3 + 3 \beta ) q^{71} -2 \beta q^{72} -4 q^{73} -\beta q^{74} + ( 4 - 4 \beta ) q^{75} + 2 \beta q^{76} + ( -4 + \beta ) q^{78} + ( 1 - 3 \beta ) q^{79} + q^{80} + ( -1 + 6 \beta ) q^{81} + q^{82} + ( 6 - 5 \beta ) q^{83} + ( -5 + 2 \beta ) q^{85} + ( 1 - 5 \beta ) q^{86} + ( 3 - \beta ) q^{87} + ( 2 - \beta ) q^{88} + ( -5 - 2 \beta ) q^{89} -2 \beta q^{90} -4 q^{92} + ( 5 - 6 \beta ) q^{93} + ( -10 + \beta ) q^{94} + 2 \beta q^{95} + ( -1 + \beta ) q^{96} + ( -15 + 2 \beta ) q^{97} + ( 4 - 4 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} - 2q^{3} + 2q^{4} + 2q^{5} - 2q^{6} + 2q^{8} + O(q^{10}) \) \( 2q + 2q^{2} - 2q^{3} + 2q^{4} + 2q^{5} - 2q^{6} + 2q^{8} + 2q^{10} + 4q^{11} - 2q^{12} - 4q^{13} - 2q^{15} + 2q^{16} - 10q^{17} + 2q^{20} + 4q^{22} - 8q^{23} - 2q^{24} - 8q^{25} - 4q^{26} - 2q^{27} + 2q^{29} - 2q^{30} - 14q^{31} + 2q^{32} - 8q^{33} - 10q^{34} - 8q^{39} + 2q^{40} + 2q^{41} + 2q^{43} + 4q^{44} - 8q^{46} - 20q^{47} - 2q^{48} - 8q^{50} + 18q^{51} - 4q^{52} + 6q^{53} - 2q^{54} + 4q^{55} + 8q^{57} + 2q^{58} + 4q^{59} - 2q^{60} + 6q^{61} - 14q^{62} + 2q^{64} - 4q^{65} - 8q^{66} + 12q^{67} - 10q^{68} + 8q^{69} - 6q^{71} - 8q^{73} + 8q^{75} - 8q^{78} + 2q^{79} + 2q^{80} - 2q^{81} + 2q^{82} + 12q^{83} - 10q^{85} + 2q^{86} + 6q^{87} + 4q^{88} - 10q^{89} - 8q^{92} + 10q^{93} - 20q^{94} - 2q^{96} - 30q^{97} + 8q^{99} + O(q^{100}) \)

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T )^{2} \)
$3$ \( 1 + 2 T + 5 T^{2} + 6 T^{3} + 9 T^{4} \)
$5$ \( ( 1 - T + 5 T^{2} )^{2} \)
$7$ 1
$11$ \( 1 - 4 T + 24 T^{2} - 44 T^{3} + 121 T^{4} \)
$13$ \( 1 + 4 T + 12 T^{2} + 52 T^{3} + 169 T^{4} \)
$17$ \( 1 + 10 T + 51 T^{2} + 170 T^{3} + 289 T^{4} \)
$19$ \( 1 + 30 T^{2} + 361 T^{4} \)
$23$ \( ( 1 + 4 T + 23 T^{2} )^{2} \)
$29$ \( 1 - 2 T + 51 T^{2} - 58 T^{3} + 841 T^{4} \)
$31$ \( 1 + 14 T + 109 T^{2} + 434 T^{3} + 961 T^{4} \)
$37$ \( 1 + 72 T^{2} + 1369 T^{4} \)
$41$ \( ( 1 - T )^{2} \)
$43$ \( 1 - 2 T + 37 T^{2} - 86 T^{3} + 1849 T^{4} \)
$47$ \( 1 + 20 T + 192 T^{2} + 940 T^{3} + 2209 T^{4} \)
$53$ \( 1 - 6 T + 83 T^{2} - 318 T^{3} + 2809 T^{4} \)
$59$ \( 1 - 4 T + 114 T^{2} - 236 T^{3} + 3481 T^{4} \)
$61$ \( 1 - 6 T + 99 T^{2} - 366 T^{3} + 3721 T^{4} \)
$67$ \( 1 - 12 T + 162 T^{2} - 804 T^{3} + 4489 T^{4} \)
$71$ \( 1 + 6 T + 133 T^{2} + 426 T^{3} + 5041 T^{4} \)
$73$ \( ( 1 + 4 T + 73 T^{2} )^{2} \)
$79$ \( 1 - 2 T + 141 T^{2} - 158 T^{3} + 6241 T^{4} \)
$83$ \( 1 - 12 T + 152 T^{2} - 996 T^{3} + 6889 T^{4} \)
$89$ \( 1 + 10 T + 195 T^{2} + 890 T^{3} + 7921 T^{4} \)
$97$ \( 1 + 30 T + 411 T^{2} + 2910 T^{3} + 9409 T^{4} \)
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