Properties

Label 6014.2.a.c
Level 6014
Weight 2
Character orbit 6014.a
Self dual Yes
Analytic conductor 48.022
Analytic rank 0
Dimension 2
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6014 = 2 \cdot 31 \cdot 97 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6014.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(48.0220317756\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q\) \(- q^{2}\) \( + 2 \beta q^{3} \) \(+ q^{4}\) \( + ( 2 + \beta ) q^{5} \) \( -2 \beta q^{6} \) \( + ( -2 + \beta ) q^{7} \) \(- q^{8}\) \( + 5 q^{9} \) \(+O(q^{10})\) \( q\) \(- q^{2}\) \( + 2 \beta q^{3} \) \(+ q^{4}\) \( + ( 2 + \beta ) q^{5} \) \( -2 \beta q^{6} \) \( + ( -2 + \beta ) q^{7} \) \(- q^{8}\) \( + 5 q^{9} \) \( + ( -2 - \beta ) q^{10} \) \( + 4 q^{11} \) \( + 2 \beta q^{12} \) \( + ( 2 - \beta ) q^{14} \) \( + ( 4 + 4 \beta ) q^{15} \) \(+ q^{16}\) \( -\beta q^{17} \) \( -5 q^{18} \) \( -6 q^{19} \) \( + ( 2 + \beta ) q^{20} \) \( + ( 4 - 4 \beta ) q^{21} \) \( -4 q^{22} \) \( + 4 \beta q^{23} \) \( -2 \beta q^{24} \) \( + ( 1 + 4 \beta ) q^{25} \) \( + 4 \beta q^{27} \) \( + ( -2 + \beta ) q^{28} \) \( + ( 4 - 4 \beta ) q^{29} \) \( + ( -4 - 4 \beta ) q^{30} \) \(- q^{31}\) \(- q^{32}\) \( + 8 \beta q^{33} \) \( + \beta q^{34} \) \( -2 q^{35} \) \( + 5 q^{36} \) \( + ( -4 - 4 \beta ) q^{37} \) \( + 6 q^{38} \) \( + ( -2 - \beta ) q^{40} \) \( + ( 6 - 2 \beta ) q^{41} \) \( + ( -4 + 4 \beta ) q^{42} \) \( + 6 \beta q^{43} \) \( + 4 q^{44} \) \( + ( 10 + 5 \beta ) q^{45} \) \( -4 \beta q^{46} \) \( + 4 q^{47} \) \( + 2 \beta q^{48} \) \( + ( -1 - 4 \beta ) q^{49} \) \( + ( -1 - 4 \beta ) q^{50} \) \( -4 q^{51} \) \( + ( 8 + 2 \beta ) q^{53} \) \( -4 \beta q^{54} \) \( + ( 8 + 4 \beta ) q^{55} \) \( + ( 2 - \beta ) q^{56} \) \( -12 \beta q^{57} \) \( + ( -4 + 4 \beta ) q^{58} \) \( + ( 10 + 2 \beta ) q^{59} \) \( + ( 4 + 4 \beta ) q^{60} \) \( + ( -4 - 6 \beta ) q^{61} \) \(+ q^{62}\) \( + ( -10 + 5 \beta ) q^{63} \) \(+ q^{64}\) \( -8 \beta q^{66} \) \( + ( 2 + 6 \beta ) q^{67} \) \( -\beta q^{68} \) \( + 16 q^{69} \) \( + 2 q^{70} \) \( + ( 2 - 3 \beta ) q^{71} \) \( -5 q^{72} \) \( + ( 2 + 4 \beta ) q^{73} \) \( + ( 4 + 4 \beta ) q^{74} \) \( + ( 16 + 2 \beta ) q^{75} \) \( -6 q^{76} \) \( + ( -8 + 4 \beta ) q^{77} \) \( + ( -2 + 2 \beta ) q^{79} \) \( + ( 2 + \beta ) q^{80} \) \(+ q^{81}\) \( + ( -6 + 2 \beta ) q^{82} \) \( + ( -4 + 9 \beta ) q^{83} \) \( + ( 4 - 4 \beta ) q^{84} \) \( + ( -2 - 2 \beta ) q^{85} \) \( -6 \beta q^{86} \) \( + ( -16 + 8 \beta ) q^{87} \) \( -4 q^{88} \) \( + ( -2 + 6 \beta ) q^{89} \) \( + ( -10 - 5 \beta ) q^{90} \) \( + 4 \beta q^{92} \) \( -2 \beta q^{93} \) \( -4 q^{94} \) \( + ( -12 - 6 \beta ) q^{95} \) \( -2 \beta q^{96} \) \(+ q^{97}\) \( + ( 1 + 4 \beta ) q^{98} \) \( + 20 q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 2q^{2} \) \(\mathstrut +\mathstrut 2q^{4} \) \(\mathstrut +\mathstrut 4q^{5} \) \(\mathstrut -\mathstrut 4q^{7} \) \(\mathstrut -\mathstrut 2q^{8} \) \(\mathstrut +\mathstrut 10q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 2q^{2} \) \(\mathstrut +\mathstrut 2q^{4} \) \(\mathstrut +\mathstrut 4q^{5} \) \(\mathstrut -\mathstrut 4q^{7} \) \(\mathstrut -\mathstrut 2q^{8} \) \(\mathstrut +\mathstrut 10q^{9} \) \(\mathstrut -\mathstrut 4q^{10} \) \(\mathstrut +\mathstrut 8q^{11} \) \(\mathstrut +\mathstrut 4q^{14} \) \(\mathstrut +\mathstrut 8q^{15} \) \(\mathstrut +\mathstrut 2q^{16} \) \(\mathstrut -\mathstrut 10q^{18} \) \(\mathstrut -\mathstrut 12q^{19} \) \(\mathstrut +\mathstrut 4q^{20} \) \(\mathstrut +\mathstrut 8q^{21} \) \(\mathstrut -\mathstrut 8q^{22} \) \(\mathstrut +\mathstrut 2q^{25} \) \(\mathstrut -\mathstrut 4q^{28} \) \(\mathstrut +\mathstrut 8q^{29} \) \(\mathstrut -\mathstrut 8q^{30} \) \(\mathstrut -\mathstrut 2q^{31} \) \(\mathstrut -\mathstrut 2q^{32} \) \(\mathstrut -\mathstrut 4q^{35} \) \(\mathstrut +\mathstrut 10q^{36} \) \(\mathstrut -\mathstrut 8q^{37} \) \(\mathstrut +\mathstrut 12q^{38} \) \(\mathstrut -\mathstrut 4q^{40} \) \(\mathstrut +\mathstrut 12q^{41} \) \(\mathstrut -\mathstrut 8q^{42} \) \(\mathstrut +\mathstrut 8q^{44} \) \(\mathstrut +\mathstrut 20q^{45} \) \(\mathstrut +\mathstrut 8q^{47} \) \(\mathstrut -\mathstrut 2q^{49} \) \(\mathstrut -\mathstrut 2q^{50} \) \(\mathstrut -\mathstrut 8q^{51} \) \(\mathstrut +\mathstrut 16q^{53} \) \(\mathstrut +\mathstrut 16q^{55} \) \(\mathstrut +\mathstrut 4q^{56} \) \(\mathstrut -\mathstrut 8q^{58} \) \(\mathstrut +\mathstrut 20q^{59} \) \(\mathstrut +\mathstrut 8q^{60} \) \(\mathstrut -\mathstrut 8q^{61} \) \(\mathstrut +\mathstrut 2q^{62} \) \(\mathstrut -\mathstrut 20q^{63} \) \(\mathstrut +\mathstrut 2q^{64} \) \(\mathstrut +\mathstrut 4q^{67} \) \(\mathstrut +\mathstrut 32q^{69} \) \(\mathstrut +\mathstrut 4q^{70} \) \(\mathstrut +\mathstrut 4q^{71} \) \(\mathstrut -\mathstrut 10q^{72} \) \(\mathstrut +\mathstrut 4q^{73} \) \(\mathstrut +\mathstrut 8q^{74} \) \(\mathstrut +\mathstrut 32q^{75} \) \(\mathstrut -\mathstrut 12q^{76} \) \(\mathstrut -\mathstrut 16q^{77} \) \(\mathstrut -\mathstrut 4q^{79} \) \(\mathstrut +\mathstrut 4q^{80} \) \(\mathstrut +\mathstrut 2q^{81} \) \(\mathstrut -\mathstrut 12q^{82} \) \(\mathstrut -\mathstrut 8q^{83} \) \(\mathstrut +\mathstrut 8q^{84} \) \(\mathstrut -\mathstrut 4q^{85} \) \(\mathstrut -\mathstrut 32q^{87} \) \(\mathstrut -\mathstrut 8q^{88} \) \(\mathstrut -\mathstrut 4q^{89} \) \(\mathstrut -\mathstrut 20q^{90} \) \(\mathstrut -\mathstrut 8q^{94} \) \(\mathstrut -\mathstrut 24q^{95} \) \(\mathstrut +\mathstrut 2q^{97} \) \(\mathstrut +\mathstrut 2q^{98} \) \(\mathstrut +\mathstrut 40q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −2.82843 1.00000 0.585786 2.82843 −3.41421 −1.00000 5.00000 −0.585786
1.2 −1.00000 2.82843 1.00000 3.41421 −2.82843 −0.585786 −1.00000 5.00000 −3.41421
\(n\): 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
\(2\) \(1\)
\(31\) \(1\)
\(97\) \(-1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{3}^{2} \) \(\mathstrut -\mathstrut 8 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6014))\).