Properties

Label 4018.2.a.w
Level 4018
Weight 2
Character orbit 4018.a
Self dual yes
Analytic conductor 32.084
Analytic rank 0
Dimension 2
CM no
Inner twists 1

Related objects

Downloads

Learn more about

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: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

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

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
−0.618034
1.61803
−1.00000 1.00000 1.00000 −2.23607 −1.00000 0 −1.00000 −2.00000 2.23607
1.2 −1.00000 1.00000 1.00000 2.23607 −1.00000 0 −1.00000 −2.00000 −2.23607
\(n\): 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.w yes 2
7.b odd 2 1 4018.2.a.t 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4018.2.a.t 2 7.b odd 2 1
4018.2.a.w yes 2 1.a even 1 1 trivial

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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4018))\):

\( T_{3} - 1 \)
\( T_{5}^{2} - 5 \)
\( T_{11}^{2} - 20 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( ( 1 - T + 3 T^{2} )^{2} \)
$5$ \( 1 + 5 T^{2} + 25 T^{4} \)
$7$ 1
$11$ \( 1 + 2 T^{2} + 121 T^{4} \)
$13$ \( 1 - 4 T + 10 T^{2} - 52 T^{3} + 169 T^{4} \)
$17$ \( ( 1 - 3 T + 17 T^{2} )^{2} \)
$19$ \( ( 1 + 4 T + 19 T^{2} )^{2} \)
$23$ \( 1 - 8 T + 42 T^{2} - 184 T^{3} + 529 T^{4} \)
$29$ \( 1 - 8 T + 69 T^{2} - 232 T^{3} + 841 T^{4} \)
$31$ \( 1 - 8 T + 73 T^{2} - 248 T^{3} + 961 T^{4} \)
$37$ \( 1 + 8 T + 70 T^{2} + 296 T^{3} + 1369 T^{4} \)
$41$ \( ( 1 + T )^{2} \)
$43$ \( ( 1 - 9 T + 43 T^{2} )^{2} \)
$47$ \( 1 + 8 T + 30 T^{2} + 376 T^{3} + 2209 T^{4} \)
$53$ \( 1 + 61 T^{2} + 2809 T^{4} \)
$59$ \( 1 - 16 T + 162 T^{2} - 944 T^{3} + 3481 T^{4} \)
$61$ \( 1 - 16 T + 141 T^{2} - 976 T^{3} + 3721 T^{4} \)
$67$ \( 1 - 4 T - 42 T^{2} - 268 T^{3} + 4489 T^{4} \)
$71$ \( 1 + 137 T^{2} + 5041 T^{4} \)
$73$ \( ( 1 - 6 T + 73 T^{2} )^{2} \)
$79$ \( 1 - 8 T + 129 T^{2} - 632 T^{3} + 6241 T^{4} \)
$83$ \( 1 - 12 T + 122 T^{2} - 996 T^{3} + 6889 T^{4} \)
$89$ \( ( 1 + 3 T + 89 T^{2} )^{2} \)
$97$ \( ( 1 - T + 97 T^{2} )^{2} \)
show more
show less