Properties

Label 4018.2.a.v
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: \( 1 \)
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.

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

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
−2.37228
3.37228
−1.00000 −2.37228 1.00000 2.37228 2.37228 0 −1.00000 2.62772 −2.37228
1.2 −1.00000 3.37228 1.00000 −3.37228 −3.37228 0 −1.00000 8.37228 3.37228
\(n\): 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4018))\):

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