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Properties

Label 229.2.b.a
Level 229
Weight 2
Character orbit 229.b
Analytic conductor 1.829
Analytic rank 0
Dimension 2
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 229 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 229.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(1.82857420629\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-5}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + q^{3} -3 q^{4} + 3 q^{5} + \beta q^{6} -\beta q^{8} -2 q^{9} +O(q^{10})\) \( q + \beta q^{2} + q^{3} -3 q^{4} + 3 q^{5} + \beta q^{6} -\beta q^{8} -2 q^{9} + 3 \beta q^{10} + 3 q^{11} -3 q^{12} + 3 q^{15} - q^{16} -3 q^{17} -2 \beta q^{18} - q^{19} -9 q^{20} + 3 \beta q^{22} -2 \beta q^{23} -\beta q^{24} + 4 q^{25} -5 q^{27} -2 \beta q^{29} + 3 \beta q^{30} -3 \beta q^{32} + 3 q^{33} -3 \beta q^{34} + 6 q^{36} + 2 q^{37} -\beta q^{38} -3 \beta q^{40} -2 \beta q^{41} - q^{43} -9 q^{44} -6 q^{45} + 10 q^{46} + 4 \beta q^{47} - q^{48} + 7 q^{49} + 4 \beta q^{50} -3 q^{51} + 6 q^{53} -5 \beta q^{54} + 9 q^{55} - q^{57} + 10 q^{58} + 4 \beta q^{59} -9 q^{60} + 5 q^{61} + 13 q^{64} + 3 \beta q^{66} + 6 \beta q^{67} + 9 q^{68} -2 \beta q^{69} -15 q^{71} + 2 \beta q^{72} -6 \beta q^{73} + 2 \beta q^{74} + 4 q^{75} + 3 q^{76} -6 \beta q^{79} -3 q^{80} + q^{81} + 10 q^{82} -9 q^{83} -9 q^{85} -\beta q^{86} -2 \beta q^{87} -3 \beta q^{88} -8 \beta q^{89} -6 \beta q^{90} + 6 \beta q^{92} -20 q^{94} -3 q^{95} -3 \beta q^{96} -7 q^{97} + 7 \beta q^{98} -6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} - 6q^{4} + 6q^{5} - 4q^{9} + O(q^{10}) \) \( 2q + 2q^{3} - 6q^{4} + 6q^{5} - 4q^{9} + 6q^{11} - 6q^{12} + 6q^{15} - 2q^{16} - 6q^{17} - 2q^{19} - 18q^{20} + 8q^{25} - 10q^{27} + 6q^{33} + 12q^{36} + 4q^{37} - 2q^{43} - 18q^{44} - 12q^{45} + 20q^{46} - 2q^{48} + 14q^{49} - 6q^{51} + 12q^{53} + 18q^{55} - 2q^{57} + 20q^{58} - 18q^{60} + 10q^{61} + 26q^{64} + 18q^{68} - 30q^{71} + 8q^{75} + 6q^{76} - 6q^{80} + 2q^{81} + 20q^{82} - 18q^{83} - 18q^{85} - 40q^{94} - 6q^{95} - 14q^{97} - 12q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/229\mathbb{Z}\right)^\times\).

\(n\) \(6\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
228.1
− 2.23607i
2.23607i
− 2.23607i 1.00000 −3.00000 3.00000 − 2.23607i 0 2.23607i −2.00000 − 6.70820i
228.2 2.23607i 1.00000 −3.00000 3.00000 2.23607i 0 − 2.23607i −2.00000 6.70820i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
229.b Even 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \( T_{2}^{2} + 5 \) acting on \(S_{2}^{\mathrm{new}}(229, [\chi])\).