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 \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.599456581593\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Defining polynomial: \(x^{2} + 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

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

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 Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 22.3.b.a 2
3.b odd 2 1 198.3.d.b 2
4.b odd 2 1 176.3.h.c 2
5.b even 2 1 550.3.d.a 2
5.c odd 4 2 550.3.c.a 4
7.b odd 2 1 1078.3.d.a 2
8.b even 2 1 704.3.h.d 2
8.d odd 2 1 704.3.h.e 2
11.b odd 2 1 inner 22.3.b.a 2
11.c even 5 4 242.3.d.b 8
11.d odd 10 4 242.3.d.b 8
12.b even 2 1 1584.3.j.d 2
33.d even 2 1 198.3.d.b 2
44.c even 2 1 176.3.h.c 2
55.d odd 2 1 550.3.d.a 2
55.e even 4 2 550.3.c.a 4
77.b even 2 1 1078.3.d.a 2
88.b odd 2 1 704.3.h.d 2
88.g even 2 1 704.3.h.e 2
132.d odd 2 1 1584.3.j.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.3.b.a 2 1.a even 1 1 trivial
22.3.b.a 2 11.b odd 2 1 inner
176.3.h.c 2 4.b odd 2 1
176.3.h.c 2 44.c even 2 1
198.3.d.b 2 3.b odd 2 1
198.3.d.b 2 33.d even 2 1
242.3.d.b 8 11.c even 5 4
242.3.d.b 8 11.d odd 10 4
550.3.c.a 4 5.c odd 4 2
550.3.c.a 4 55.e even 4 2
550.3.d.a 2 5.b even 2 1
550.3.d.a 2 55.d odd 2 1
704.3.h.d 2 8.b even 2 1
704.3.h.d 2 88.b odd 2 1
704.3.h.e 2 8.d odd 2 1
704.3.h.e 2 88.g even 2 1
1078.3.d.a 2 7.b odd 2 1
1078.3.d.a 2 77.b even 2 1
1584.3.j.d 2 12.b even 2 1
1584.3.j.d 2 132.d odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(22, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T^{2} \)
$3$ \( ( 1 - T + 9 T^{2} )^{2} \)
$5$ \( ( 1 + T + 25 T^{2} )^{2} \)
$7$ \( 1 - 26 T^{2} + 2401 T^{4} \)
$11$ \( 1 - 14 T + 121 T^{2} \)
$13$ \( 1 - 266 T^{2} + 28561 T^{4} \)
$17$ \( 1 + 70 T^{2} + 83521 T^{4} \)
$19$ \( 1 - 74 T^{2} + 130321 T^{4} \)
$23$ \( ( 1 - 17 T + 529 T^{2} )^{2} \)
$29$ \( 1 - 530 T^{2} + 707281 T^{4} \)
$31$ \( ( 1 - 17 T + 961 T^{2} )^{2} \)
$37$ \( ( 1 - 47 T + 1369 T^{2} )^{2} \)
$41$ \( 1 - 3290 T^{2} + 2825761 T^{4} \)
$43$ \( 1 - 3410 T^{2} + 3418801 T^{4} \)
$47$ \( ( 1 + 58 T + 2209 T^{2} )^{2} \)
$53$ \( ( 1 - 2 T + 2809 T^{2} )^{2} \)
$59$ \( ( 1 + 55 T + 3481 T^{2} )^{2} \)
$61$ \( 1 - 242 T^{2} + 13845841 T^{4} \)
$67$ \( ( 1 - 89 T + 4489 T^{2} )^{2} \)
$71$ \( ( 1 + 7 T + 5041 T^{2} )^{2} \)
$73$ \( 1 + 5542 T^{2} + 28398241 T^{4} \)
$79$ \( 1 - 11330 T^{2} + 38950081 T^{4} \)
$83$ \( 1 - 12626 T^{2} + 47458321 T^{4} \)
$89$ \( ( 1 + 97 T + 7921 T^{2} )^{2} \)
$97$ \( ( 1 + 121 T + 9409 T^{2} )^{2} \)
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