Properties

Label 4018.2.a.bc
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

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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: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
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.

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

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 −0.414214 1.00000 −1.82843 −0.414214 0 1.00000 −2.82843 −1.82843
1.2 1.00000 2.41421 1.00000 3.82843 2.41421 0 1.00000 2.82843 3.82843
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(1\)
\(41\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4018))\):

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