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: \( x^{2} + 2 \) Copy content Toggle raw display
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.

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

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

$p$ $F_p(T)$
$2$ \( T^{2} + 2 \) Copy content Toggle raw display
$3$ \( (T - 1)^{2} \) Copy content Toggle raw display
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 72 \) Copy content Toggle raw display
$11$ \( T^{2} - 14T + 121 \) Copy content Toggle raw display
$13$ \( T^{2} + 72 \) Copy content Toggle raw display
$17$ \( T^{2} + 648 \) Copy content Toggle raw display
$19$ \( T^{2} + 648 \) Copy content Toggle raw display
$23$ \( (T - 17)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 1152 \) Copy content Toggle raw display
$31$ \( (T - 17)^{2} \) Copy content Toggle raw display
$37$ \( (T - 47)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 72 \) Copy content Toggle raw display
$43$ \( T^{2} + 288 \) Copy content Toggle raw display
$47$ \( (T + 58)^{2} \) Copy content Toggle raw display
$53$ \( (T - 2)^{2} \) Copy content Toggle raw display
$59$ \( (T + 55)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 7200 \) Copy content Toggle raw display
$67$ \( (T - 89)^{2} \) Copy content Toggle raw display
$71$ \( (T + 7)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 16200 \) Copy content Toggle raw display
$79$ \( T^{2} + 1152 \) Copy content Toggle raw display
$83$ \( T^{2} + 1152 \) Copy content Toggle raw display
$89$ \( (T + 97)^{2} \) Copy content Toggle raw display
$97$ \( (T + 121)^{2} \) Copy content Toggle raw display
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