Properties

Label 22.3.b.a
Level 22
Weight 3
Character orbit 22.b
Analytic conductor 0.599
Analytic rank 0
Dimension 2
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 22 = 2 \cdot 11 \)
Weight: \( k \) = \( 3 \)
Character orbit: \([\chi]\) = 22.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(0.599456581593\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
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 \(\beta = \sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + \beta q^{2} \) \(+ q^{3}\) \( -2 q^{4} \) \(- q^{5}\) \( + \beta q^{6} \) \( -6 \beta q^{7} \) \( -2 \beta q^{8} \) \( -8 q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta q^{2} \) \(+ q^{3}\) \( -2 q^{4} \) \(- q^{5}\) \( + \beta q^{6} \) \( -6 \beta q^{7} \) \( -2 \beta q^{8} \) \( -8 q^{9} \) \( -\beta q^{10} \) \( + ( 7 + 6 \beta ) q^{11} \) \( -2 q^{12} \) \( + 6 \beta q^{13} \) \( + 12 q^{14} \) \(- q^{15}\) \( + 4 q^{16} \) \( + 18 \beta q^{17} \) \( -8 \beta q^{18} \) \( -18 \beta q^{19} \) \( + 2 q^{20} \) \( -6 \beta q^{21} \) \( + ( -12 + 7 \beta ) q^{22} \) \( + 17 q^{23} \) \( -2 \beta q^{24} \) \( -24 q^{25} \) \( -12 q^{26} \) \( -17 q^{27} \) \( + 12 \beta q^{28} \) \( -24 \beta q^{29} \) \( -\beta q^{30} \) \( + 17 q^{31} \) \( + 4 \beta q^{32} \) \( + ( 7 + 6 \beta ) q^{33} \) \( -36 q^{34} \) \( + 6 \beta q^{35} \) \( + 16 q^{36} \) \( + 47 q^{37} \) \( + 36 q^{38} \) \( + 6 \beta q^{39} \) \( + 2 \beta q^{40} \) \( + 6 \beta q^{41} \) \( + 12 q^{42} \) \( + 12 \beta q^{43} \) \( + ( -14 - 12 \beta ) q^{44} \) \( + 8 q^{45} \) \( + 17 \beta q^{46} \) \( -58 q^{47} \) \( + 4 q^{48} \) \( -23 q^{49} \) \( -24 \beta q^{50} \) \( + 18 \beta q^{51} \) \( -12 \beta q^{52} \) \( + 2 q^{53} \) \( -17 \beta q^{54} \) \( + ( -7 - 6 \beta ) q^{55} \) \( -24 q^{56} \) \( -18 \beta q^{57} \) \( + 48 q^{58} \) \( -55 q^{59} \) \( + 2 q^{60} \) \( -60 \beta q^{61} \) \( + 17 \beta q^{62} \) \( + 48 \beta q^{63} \) \( -8 q^{64} \) \( -6 \beta q^{65} \) \( + ( -12 + 7 \beta ) q^{66} \) \( + 89 q^{67} \) \( -36 \beta q^{68} \) \( + 17 q^{69} \) \( -12 q^{70} \) \( -7 q^{71} \) \( + 16 \beta q^{72} \) \( + 90 \beta q^{73} \) \( + 47 \beta q^{74} \) \( -24 q^{75} \) \( + 36 \beta q^{76} \) \( + ( 72 - 42 \beta ) q^{77} \) \( -12 q^{78} \) \( -24 \beta q^{79} \) \( -4 q^{80} \) \( + 55 q^{81} \) \( -12 q^{82} \) \( -24 \beta q^{83} \) \( + 12 \beta q^{84} \) \( -18 \beta q^{85} \) \( -24 q^{86} \) \( -24 \beta q^{87} \) \( + ( 24 - 14 \beta ) q^{88} \) \( -97 q^{89} \) \( + 8 \beta q^{90} \) \( + 72 q^{91} \) \( -34 q^{92} \) \( + 17 q^{93} \) \( -58 \beta q^{94} \) \( + 18 \beta q^{95} \) \( + 4 \beta q^{96} \) \( -121 q^{97} \) \( -23 \beta q^{98} \) \( + ( -56 - 48 \beta ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut +\mathstrut 2q^{3} \) \(\mathstrut -\mathstrut 4q^{4} \) \(\mathstrut -\mathstrut 2q^{5} \) \(\mathstrut -\mathstrut 16q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut 2q^{3} \) \(\mathstrut -\mathstrut 4q^{4} \) \(\mathstrut -\mathstrut 2q^{5} \) \(\mathstrut -\mathstrut 16q^{9} \) \(\mathstrut +\mathstrut 14q^{11} \) \(\mathstrut -\mathstrut 4q^{12} \) \(\mathstrut +\mathstrut 24q^{14} \) \(\mathstrut -\mathstrut 2q^{15} \) \(\mathstrut +\mathstrut 8q^{16} \) \(\mathstrut +\mathstrut 4q^{20} \) \(\mathstrut -\mathstrut 24q^{22} \) \(\mathstrut +\mathstrut 34q^{23} \) \(\mathstrut -\mathstrut 48q^{25} \) \(\mathstrut -\mathstrut 24q^{26} \) \(\mathstrut -\mathstrut 34q^{27} \) \(\mathstrut +\mathstrut 34q^{31} \) \(\mathstrut +\mathstrut 14q^{33} \) \(\mathstrut -\mathstrut 72q^{34} \) \(\mathstrut +\mathstrut 32q^{36} \) \(\mathstrut +\mathstrut 94q^{37} \) \(\mathstrut +\mathstrut 72q^{38} \) \(\mathstrut +\mathstrut 24q^{42} \) \(\mathstrut -\mathstrut 28q^{44} \) \(\mathstrut +\mathstrut 16q^{45} \) \(\mathstrut -\mathstrut 116q^{47} \) \(\mathstrut +\mathstrut 8q^{48} \) \(\mathstrut -\mathstrut 46q^{49} \) \(\mathstrut +\mathstrut 4q^{53} \) \(\mathstrut -\mathstrut 14q^{55} \) \(\mathstrut -\mathstrut 48q^{56} \) \(\mathstrut +\mathstrut 96q^{58} \) \(\mathstrut -\mathstrut 110q^{59} \) \(\mathstrut +\mathstrut 4q^{60} \) \(\mathstrut -\mathstrut 16q^{64} \) \(\mathstrut -\mathstrut 24q^{66} \) \(\mathstrut +\mathstrut 178q^{67} \) \(\mathstrut +\mathstrut 34q^{69} \) \(\mathstrut -\mathstrut 24q^{70} \) \(\mathstrut -\mathstrut 14q^{71} \) \(\mathstrut -\mathstrut 48q^{75} \) \(\mathstrut +\mathstrut 144q^{77} \) \(\mathstrut -\mathstrut 24q^{78} \) \(\mathstrut -\mathstrut 8q^{80} \) \(\mathstrut +\mathstrut 110q^{81} \) \(\mathstrut -\mathstrut 24q^{82} \) \(\mathstrut -\mathstrut 48q^{86} \) \(\mathstrut +\mathstrut 48q^{88} \) \(\mathstrut -\mathstrut 194q^{89} \) \(\mathstrut +\mathstrut 144q^{91} \) \(\mathstrut -\mathstrut 68q^{92} \) \(\mathstrut +\mathstrut 34q^{93} \) \(\mathstrut -\mathstrut 242q^{97} \) \(\mathstrut -\mathstrut 112q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/22\mathbb{Z}\right)^\times\).

\(n\) \(13\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
21.1
1.41421i
1.41421i
1.41421i 1.00000 −2.00000 −1.00000 1.41421i 8.48528i 2.82843i −8.00000 1.41421i
21.2 1.41421i 1.00000 −2.00000 −1.00000 1.41421i 8.48528i 2.82843i −8.00000 1.41421i
\(n\): 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
11.b Odd 1 yes

Hecke kernels

There are no other newforms in \(S_{3}^{\mathrm{new}}(22, [\chi])\).