Properties

Label 33.2.a.a
Level $33$
Weight $2$
Character orbit 33.a
Self dual yes
Analytic conductor $0.264$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 33 = 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 33.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.263506326670\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
1.00000 −1.00000 −1.00000 −2.00000 −1.00000 4.00000 −3.00000 1.00000 −2.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(11\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 33.2.a.a 1
3.b odd 2 1 99.2.a.b 1
4.b odd 2 1 528.2.a.g 1
5.b even 2 1 825.2.a.a 1
5.c odd 4 2 825.2.c.a 2
7.b odd 2 1 1617.2.a.j 1
8.b even 2 1 2112.2.a.bb 1
8.d odd 2 1 2112.2.a.j 1
9.c even 3 2 891.2.e.e 2
9.d odd 6 2 891.2.e.g 2
11.b odd 2 1 363.2.a.b 1
11.c even 5 4 363.2.e.e 4
11.d odd 10 4 363.2.e.g 4
12.b even 2 1 1584.2.a.o 1
13.b even 2 1 5577.2.a.a 1
15.d odd 2 1 2475.2.a.g 1
15.e even 4 2 2475.2.c.d 2
17.b even 2 1 9537.2.a.m 1
21.c even 2 1 4851.2.a.b 1
24.f even 2 1 6336.2.a.n 1
24.h odd 2 1 6336.2.a.x 1
33.d even 2 1 1089.2.a.j 1
44.c even 2 1 5808.2.a.t 1
55.d odd 2 1 9075.2.a.q 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.2.a.a 1 1.a even 1 1 trivial
99.2.a.b 1 3.b odd 2 1
363.2.a.b 1 11.b odd 2 1
363.2.e.e 4 11.c even 5 4
363.2.e.g 4 11.d odd 10 4
528.2.a.g 1 4.b odd 2 1
825.2.a.a 1 5.b even 2 1
825.2.c.a 2 5.c odd 4 2
891.2.e.e 2 9.c even 3 2
891.2.e.g 2 9.d odd 6 2
1089.2.a.j 1 33.d even 2 1
1584.2.a.o 1 12.b even 2 1
1617.2.a.j 1 7.b odd 2 1
2112.2.a.j 1 8.d odd 2 1
2112.2.a.bb 1 8.b even 2 1
2475.2.a.g 1 15.d odd 2 1
2475.2.c.d 2 15.e even 4 2
4851.2.a.b 1 21.c even 2 1
5577.2.a.a 1 13.b even 2 1
5808.2.a.t 1 44.c even 2 1
6336.2.a.n 1 24.f even 2 1
6336.2.a.x 1 24.h odd 2 1
9075.2.a.q 1 55.d odd 2 1
9537.2.a.m 1 17.b even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(\Gamma_0(33))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 1 \) Copy content Toggle raw display
$3$ \( T + 1 \) Copy content Toggle raw display
$5$ \( T + 2 \) Copy content Toggle raw display
$7$ \( T - 4 \) Copy content Toggle raw display
$11$ \( T - 1 \) Copy content Toggle raw display
$13$ \( T + 2 \) Copy content Toggle raw display
$17$ \( T + 2 \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T - 8 \) Copy content Toggle raw display
$29$ \( T + 6 \) Copy content Toggle raw display
$31$ \( T + 8 \) Copy content Toggle raw display
$37$ \( T - 6 \) Copy content Toggle raw display
$41$ \( T + 2 \) Copy content Toggle raw display
$43$ \( T \) Copy content Toggle raw display
$47$ \( T - 8 \) Copy content Toggle raw display
$53$ \( T - 6 \) Copy content Toggle raw display
$59$ \( T + 4 \) Copy content Toggle raw display
$61$ \( T - 6 \) Copy content Toggle raw display
$67$ \( T + 4 \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T + 14 \) Copy content Toggle raw display
$79$ \( T + 4 \) Copy content Toggle raw display
$83$ \( T - 12 \) Copy content Toggle raw display
$89$ \( T + 6 \) Copy content Toggle raw display
$97$ \( T - 2 \) Copy content Toggle raw display
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