Properties

Label 33.4.a.b
Level $33$
Weight $4$
Character orbit 33.a
Self dual yes
Analytic conductor $1.947$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 33 = 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 33.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.94706303019\)
Analytic rank: \(1\)
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

\(f(q)\) \(=\) \( q - q^{2} - 3 q^{3} - 7 q^{4} - 4 q^{5} + 3 q^{6} - 26 q^{7} + 15 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - 3 q^{3} - 7 q^{4} - 4 q^{5} + 3 q^{6} - 26 q^{7} + 15 q^{8} + 9 q^{9} + 4 q^{10} + 11 q^{11} + 21 q^{12} - 32 q^{13} + 26 q^{14} + 12 q^{15} + 41 q^{16} + 74 q^{17} - 9 q^{18} - 60 q^{19} + 28 q^{20} + 78 q^{21} - 11 q^{22} - 182 q^{23} - 45 q^{24} - 109 q^{25} + 32 q^{26} - 27 q^{27} + 182 q^{28} - 90 q^{29} - 12 q^{30} - 8 q^{31} - 161 q^{32} - 33 q^{33} - 74 q^{34} + 104 q^{35} - 63 q^{36} - 66 q^{37} + 60 q^{38} + 96 q^{39} - 60 q^{40} + 422 q^{41} - 78 q^{42} + 408 q^{43} - 77 q^{44} - 36 q^{45} + 182 q^{46} - 506 q^{47} - 123 q^{48} + 333 q^{49} + 109 q^{50} - 222 q^{51} + 224 q^{52} + 348 q^{53} + 27 q^{54} - 44 q^{55} - 390 q^{56} + 180 q^{57} + 90 q^{58} - 200 q^{59} - 84 q^{60} + 132 q^{61} + 8 q^{62} - 234 q^{63} - 167 q^{64} + 128 q^{65} + 33 q^{66} - 1036 q^{67} - 518 q^{68} + 546 q^{69} - 104 q^{70} + 762 q^{71} + 135 q^{72} - 542 q^{73} + 66 q^{74} + 327 q^{75} + 420 q^{76} - 286 q^{77} - 96 q^{78} - 550 q^{79} - 164 q^{80} + 81 q^{81} - 422 q^{82} - 132 q^{83} - 546 q^{84} - 296 q^{85} - 408 q^{86} + 270 q^{87} + 165 q^{88} + 570 q^{89} + 36 q^{90} + 832 q^{91} + 1274 q^{92} + 24 q^{93} + 506 q^{94} + 240 q^{95} + 483 q^{96} + 14 q^{97} - 333 q^{98} + 99 q^{99}+O(q^{100}) \) 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 −3.00000 −7.00000 −4.00000 3.00000 −26.0000 15.0000 9.00000 4.00000
\(n\): 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 33.4.a.b 1
3.b odd 2 1 99.4.a.a 1
4.b odd 2 1 528.4.a.h 1
5.b even 2 1 825.4.a.f 1
5.c odd 4 2 825.4.c.f 2
7.b odd 2 1 1617.4.a.d 1
8.b even 2 1 2112.4.a.u 1
8.d odd 2 1 2112.4.a.h 1
11.b odd 2 1 363.4.a.d 1
12.b even 2 1 1584.4.a.l 1
15.d odd 2 1 2475.4.a.e 1
33.d even 2 1 1089.4.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.4.a.b 1 1.a even 1 1 trivial
99.4.a.a 1 3.b odd 2 1
363.4.a.d 1 11.b odd 2 1
528.4.a.h 1 4.b odd 2 1
825.4.a.f 1 5.b even 2 1
825.4.c.f 2 5.c odd 4 2
1089.4.a.e 1 33.d even 2 1
1584.4.a.l 1 12.b even 2 1
1617.4.a.d 1 7.b odd 2 1
2112.4.a.h 1 8.d odd 2 1
2112.4.a.u 1 8.b even 2 1
2475.4.a.e 1 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 1 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(33))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 1 \) Copy content Toggle raw display
$3$ \( T + 3 \) Copy content Toggle raw display
$5$ \( T + 4 \) Copy content Toggle raw display
$7$ \( T + 26 \) Copy content Toggle raw display
$11$ \( T - 11 \) Copy content Toggle raw display
$13$ \( T + 32 \) Copy content Toggle raw display
$17$ \( T - 74 \) Copy content Toggle raw display
$19$ \( T + 60 \) Copy content Toggle raw display
$23$ \( T + 182 \) Copy content Toggle raw display
$29$ \( T + 90 \) Copy content Toggle raw display
$31$ \( T + 8 \) Copy content Toggle raw display
$37$ \( T + 66 \) Copy content Toggle raw display
$41$ \( T - 422 \) Copy content Toggle raw display
$43$ \( T - 408 \) Copy content Toggle raw display
$47$ \( T + 506 \) Copy content Toggle raw display
$53$ \( T - 348 \) Copy content Toggle raw display
$59$ \( T + 200 \) Copy content Toggle raw display
$61$ \( T - 132 \) Copy content Toggle raw display
$67$ \( T + 1036 \) Copy content Toggle raw display
$71$ \( T - 762 \) Copy content Toggle raw display
$73$ \( T + 542 \) Copy content Toggle raw display
$79$ \( T + 550 \) Copy content Toggle raw display
$83$ \( T + 132 \) Copy content Toggle raw display
$89$ \( T - 570 \) Copy content Toggle raw display
$97$ \( T - 14 \) Copy content Toggle raw display
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