Properties

Label 4003.2.a.a
Level 4003
Weight 2
Character orbit 4003.a
Self dual Yes
Analytic conductor 31.964
Analytic rank 1
Dimension 2
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4003 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4003.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(31.9641159291\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
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.

\(f(q)\)  \(=\)  \( q\) \( + \beta q^{2} \) \( + \beta q^{3} \) \( + \beta q^{5} \) \( + 2 q^{6} \) \(- q^{7}\) \( -2 \beta q^{8} \) \(- q^{9}\) \(+O(q^{10})\) \( q\) \( + \beta q^{2} \) \( + \beta q^{3} \) \( + \beta q^{5} \) \( + 2 q^{6} \) \(- q^{7}\) \( -2 \beta q^{8} \) \(- q^{9}\) \( + 2 q^{10} \) \( + ( -2 - 2 \beta ) q^{11} \) \( + ( 4 + 2 \beta ) q^{13} \) \( - \beta q^{14} \) \( + 2 q^{15} \) \( -4 q^{16} \) \( + ( -1 - 4 \beta ) q^{17} \) \( - \beta q^{18} \) \( + ( 1 + 4 \beta ) q^{19} \) \( - \beta q^{21} \) \( + ( -4 - 2 \beta ) q^{22} \) \( + ( -6 + 2 \beta ) q^{23} \) \( -4 q^{24} \) \( -3 q^{25} \) \( + ( 4 + 4 \beta ) q^{26} \) \( -4 \beta q^{27} \) \( + ( -4 - 2 \beta ) q^{29} \) \( + 2 \beta q^{30} \) \( + ( 2 - 6 \beta ) q^{31} \) \( + ( -4 - 2 \beta ) q^{33} \) \( + ( -8 - \beta ) q^{34} \) \( - \beta q^{35} \) \( + ( -7 - 2 \beta ) q^{37} \) \( + ( 8 + \beta ) q^{38} \) \( + ( 4 + 4 \beta ) q^{39} \) \( -4 q^{40} \) \( + ( 8 + 2 \beta ) q^{41} \) \( -2 q^{42} \) \( - \beta q^{45} \) \( + ( 4 - 6 \beta ) q^{46} \) \( + ( 4 + 5 \beta ) q^{47} \) \( -4 \beta q^{48} \) \( -6 q^{49} \) \( -3 \beta q^{50} \) \( + ( -8 - \beta ) q^{51} \) \( + ( -2 - 8 \beta ) q^{53} \) \( -8 q^{54} \) \( + ( -4 - 2 \beta ) q^{55} \) \( + 2 \beta q^{56} \) \( + ( 8 + \beta ) q^{57} \) \( + ( -4 - 4 \beta ) q^{58} \) \( + ( -6 + 2 \beta ) q^{59} \) \( + ( 10 + 2 \beta ) q^{61} \) \( + ( -12 + 2 \beta ) q^{62} \) \(+ q^{63}\) \( + 8 q^{64} \) \( + ( 4 + 4 \beta ) q^{65} \) \( + ( -4 - 4 \beta ) q^{66} \) \( + ( 7 - 4 \beta ) q^{67} \) \( + ( 4 - 6 \beta ) q^{69} \) \( -2 q^{70} \) \( + 6 q^{71} \) \( + 2 \beta q^{72} \) \( -3 q^{73} \) \( + ( -4 - 7 \beta ) q^{74} \) \( -3 \beta q^{75} \) \( + ( 2 + 2 \beta ) q^{77} \) \( + ( 8 + 4 \beta ) q^{78} \) \( + ( 3 - 6 \beta ) q^{79} \) \( -4 \beta q^{80} \) \( -5 q^{81} \) \( + ( 4 + 8 \beta ) q^{82} \) \( + ( 3 - 10 \beta ) q^{83} \) \( + ( -8 - \beta ) q^{85} \) \( + ( -4 - 4 \beta ) q^{87} \) \( + ( 8 + 4 \beta ) q^{88} \) \( + ( -9 + 2 \beta ) q^{89} \) \( -2 q^{90} \) \( + ( -4 - 2 \beta ) q^{91} \) \( + ( -12 + 2 \beta ) q^{93} \) \( + ( 10 + 4 \beta ) q^{94} \) \( + ( 8 + \beta ) q^{95} \) \( + ( 6 + 3 \beta ) q^{97} \) \( -6 \beta q^{98} \) \( + ( 2 + 2 \beta ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)  \(=\)  \(2q \) \(\mathstrut +\mathstrut 4q^{6} \) \(\mathstrut -\mathstrut 2q^{7} \) \(\mathstrut -\mathstrut 2q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut 4q^{6} \) \(\mathstrut -\mathstrut 2q^{7} \) \(\mathstrut -\mathstrut 2q^{9} \) \(\mathstrut +\mathstrut 4q^{10} \) \(\mathstrut -\mathstrut 4q^{11} \) \(\mathstrut +\mathstrut 8q^{13} \) \(\mathstrut +\mathstrut 4q^{15} \) \(\mathstrut -\mathstrut 8q^{16} \) \(\mathstrut -\mathstrut 2q^{17} \) \(\mathstrut +\mathstrut 2q^{19} \) \(\mathstrut -\mathstrut 8q^{22} \) \(\mathstrut -\mathstrut 12q^{23} \) \(\mathstrut -\mathstrut 8q^{24} \) \(\mathstrut -\mathstrut 6q^{25} \) \(\mathstrut +\mathstrut 8q^{26} \) \(\mathstrut -\mathstrut 8q^{29} \) \(\mathstrut +\mathstrut 4q^{31} \) \(\mathstrut -\mathstrut 8q^{33} \) \(\mathstrut -\mathstrut 16q^{34} \) \(\mathstrut -\mathstrut 14q^{37} \) \(\mathstrut +\mathstrut 16q^{38} \) \(\mathstrut +\mathstrut 8q^{39} \) \(\mathstrut -\mathstrut 8q^{40} \) \(\mathstrut +\mathstrut 16q^{41} \) \(\mathstrut -\mathstrut 4q^{42} \) \(\mathstrut +\mathstrut 8q^{46} \) \(\mathstrut +\mathstrut 8q^{47} \) \(\mathstrut -\mathstrut 12q^{49} \) \(\mathstrut -\mathstrut 16q^{51} \) \(\mathstrut -\mathstrut 4q^{53} \) \(\mathstrut -\mathstrut 16q^{54} \) \(\mathstrut -\mathstrut 8q^{55} \) \(\mathstrut +\mathstrut 16q^{57} \) \(\mathstrut -\mathstrut 8q^{58} \) \(\mathstrut -\mathstrut 12q^{59} \) \(\mathstrut +\mathstrut 20q^{61} \) \(\mathstrut -\mathstrut 24q^{62} \) \(\mathstrut +\mathstrut 2q^{63} \) \(\mathstrut +\mathstrut 16q^{64} \) \(\mathstrut +\mathstrut 8q^{65} \) \(\mathstrut -\mathstrut 8q^{66} \) \(\mathstrut +\mathstrut 14q^{67} \) \(\mathstrut +\mathstrut 8q^{69} \) \(\mathstrut -\mathstrut 4q^{70} \) \(\mathstrut +\mathstrut 12q^{71} \) \(\mathstrut -\mathstrut 6q^{73} \) \(\mathstrut -\mathstrut 8q^{74} \) \(\mathstrut +\mathstrut 4q^{77} \) \(\mathstrut +\mathstrut 16q^{78} \) \(\mathstrut +\mathstrut 6q^{79} \) \(\mathstrut -\mathstrut 10q^{81} \) \(\mathstrut +\mathstrut 8q^{82} \) \(\mathstrut +\mathstrut 6q^{83} \) \(\mathstrut -\mathstrut 16q^{85} \) \(\mathstrut -\mathstrut 8q^{87} \) \(\mathstrut +\mathstrut 16q^{88} \) \(\mathstrut -\mathstrut 18q^{89} \) \(\mathstrut -\mathstrut 4q^{90} \) \(\mathstrut -\mathstrut 8q^{91} \) \(\mathstrut -\mathstrut 24q^{93} \) \(\mathstrut +\mathstrut 20q^{94} \) \(\mathstrut +\mathstrut 16q^{95} \) \(\mathstrut +\mathstrut 12q^{97} \) \(\mathstrut +\mathstrut 4q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
−1.41421 −1.41421 0 −1.41421 2.00000 −1.00000 2.82843 −1.00000 2.00000
1.2 1.41421 1.41421 0 1.41421 2.00000 −1.00000 −2.82843 −1.00000 2.00000
\(n\): 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
\(4003\) \(1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{2} \) \(\mathstrut -\mathstrut 2 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4003))\).