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.

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

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 \(T_{2}^{2} \) \(\mathstrut +\mathstrut 5 \) acting on \(S_{2}^{\mathrm{new}}(229, [\chi])\).