Properties

Label 23.2.a.a
Level 23
Weight 2
Character orbit 23.a
Self dual Yes
Analytic conductor 0.184
Analytic rank 0
Dimension 2
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 23 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 23.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(0.183655924649\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta q^{2} + ( -1 + 2 \beta ) q^{3} + ( -1 + \beta ) q^{4} -2 \beta q^{5} + ( -2 - \beta ) q^{6} + ( 2 - 2 \beta ) q^{7} + ( -1 + 2 \beta ) q^{8} + 2 q^{9} +O(q^{10})\) \( q -\beta q^{2} + ( -1 + 2 \beta ) q^{3} + ( -1 + \beta ) q^{4} -2 \beta q^{5} + ( -2 - \beta ) q^{6} + ( 2 - 2 \beta ) q^{7} + ( -1 + 2 \beta ) q^{8} + 2 q^{9} + ( 2 + 2 \beta ) q^{10} + ( -4 + 2 \beta ) q^{11} + ( 3 - \beta ) q^{12} + 3 q^{13} + 2 q^{14} + ( -4 - 2 \beta ) q^{15} -3 \beta q^{16} + ( 2 + 2 \beta ) q^{17} -2 \beta q^{18} -2 q^{19} -2 q^{20} + ( -6 + 2 \beta ) q^{21} + ( -2 + 2 \beta ) q^{22} + q^{23} + 5 q^{24} + ( -1 + 4 \beta ) q^{25} -3 \beta q^{26} + ( 1 - 2 \beta ) q^{27} + ( -4 + 2 \beta ) q^{28} -3 q^{29} + ( 2 + 6 \beta ) q^{30} + ( 3 - 6 \beta ) q^{31} + ( 5 - \beta ) q^{32} + ( 8 - 6 \beta ) q^{33} + ( -2 - 4 \beta ) q^{34} + 4 q^{35} + ( -2 + 2 \beta ) q^{36} + 2 \beta q^{37} + 2 \beta q^{38} + ( -3 + 6 \beta ) q^{39} + ( -4 - 2 \beta ) q^{40} + ( -1 + 4 \beta ) q^{41} + ( -2 + 4 \beta ) q^{42} + ( 6 - 4 \beta ) q^{44} -4 \beta q^{45} -\beta q^{46} + ( -1 + 2 \beta ) q^{47} + ( -6 - 3 \beta ) q^{48} + ( 1 - 4 \beta ) q^{49} + ( -4 - 3 \beta ) q^{50} + ( 2 + 6 \beta ) q^{51} + ( -3 + 3 \beta ) q^{52} + ( -2 - 4 \beta ) q^{53} + ( 2 + \beta ) q^{54} + ( -4 + 4 \beta ) q^{55} + ( -6 + 2 \beta ) q^{56} + ( 2 - 4 \beta ) q^{57} + 3 \beta q^{58} + ( 4 - 4 \beta ) q^{59} + ( 2 - 4 \beta ) q^{60} + ( -2 + 8 \beta ) q^{61} + ( 6 + 3 \beta ) q^{62} + ( 4 - 4 \beta ) q^{63} + ( 1 + 2 \beta ) q^{64} -6 \beta q^{65} + ( 6 - 2 \beta ) q^{66} + ( -4 - 2 \beta ) q^{67} + 2 \beta q^{68} + ( -1 + 2 \beta ) q^{69} -4 \beta q^{70} + ( 11 - 2 \beta ) q^{71} + ( -2 + 4 \beta ) q^{72} + ( 9 + 4 \beta ) q^{73} + ( -2 - 2 \beta ) q^{74} + ( 9 + 2 \beta ) q^{75} + ( 2 - 2 \beta ) q^{76} + ( -12 + 8 \beta ) q^{77} + ( -6 - 3 \beta ) q^{78} + ( -6 + 8 \beta ) q^{79} + ( 6 + 6 \beta ) q^{80} -11 q^{81} + ( -4 - 3 \beta ) q^{82} + ( -10 - 2 \beta ) q^{83} + ( 8 - 6 \beta ) q^{84} + ( -4 - 8 \beta ) q^{85} + ( 3 - 6 \beta ) q^{87} + ( 8 - 6 \beta ) q^{88} + ( -8 + 4 \beta ) q^{89} + ( 4 + 4 \beta ) q^{90} + ( 6 - 6 \beta ) q^{91} + ( -1 + \beta ) q^{92} -15 q^{93} + ( -2 - \beta ) q^{94} + 4 \beta q^{95} + ( -7 + 9 \beta ) q^{96} + ( 14 - 6 \beta ) q^{97} + ( 4 + 3 \beta ) q^{98} + ( -8 + 4 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} - q^{4} - 2q^{5} - 5q^{6} + 2q^{7} + 4q^{9} + O(q^{10}) \) \( 2q - q^{2} - q^{4} - 2q^{5} - 5q^{6} + 2q^{7} + 4q^{9} + 6q^{10} - 6q^{11} + 5q^{12} + 6q^{13} + 4q^{14} - 10q^{15} - 3q^{16} + 6q^{17} - 2q^{18} - 4q^{19} - 4q^{20} - 10q^{21} - 2q^{22} + 2q^{23} + 10q^{24} + 2q^{25} - 3q^{26} - 6q^{28} - 6q^{29} + 10q^{30} + 9q^{32} + 10q^{33} - 8q^{34} + 8q^{35} - 2q^{36} + 2q^{37} + 2q^{38} - 10q^{40} + 2q^{41} + 8q^{44} - 4q^{45} - q^{46} - 15q^{48} - 2q^{49} - 11q^{50} + 10q^{51} - 3q^{52} - 8q^{53} + 5q^{54} - 4q^{55} - 10q^{56} + 3q^{58} + 4q^{59} + 4q^{61} + 15q^{62} + 4q^{63} + 4q^{64} - 6q^{65} + 10q^{66} - 10q^{67} + 2q^{68} - 4q^{70} + 20q^{71} + 22q^{73} - 6q^{74} + 20q^{75} + 2q^{76} - 16q^{77} - 15q^{78} - 4q^{79} + 18q^{80} - 22q^{81} - 11q^{82} - 22q^{83} + 10q^{84} - 16q^{85} + 10q^{88} - 12q^{89} + 12q^{90} + 6q^{91} - q^{92} - 30q^{93} - 5q^{94} + 4q^{95} - 5q^{96} + 22q^{97} + 11q^{98} - 12q^{99} + O(q^{100}) \)

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
1.61803
−0.618034
−1.61803 2.23607 0.618034 −3.23607 −3.61803 −1.23607 2.23607 2.00000 5.23607
1.2 0.618034 −2.23607 −1.61803 1.23607 −1.38197 3.23607 −2.23607 2.00000 0.763932
\(n\): 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
\(23\) \(-1\)

Hecke kernels

There are no other newforms in \(S_{2}^{\mathrm{new}}(\Gamma_0(23))\).