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

\(f(q)\) \(=\) \( q - q^{2} - 3q^{3} - 7q^{4} - 4q^{5} + 3q^{6} - 26q^{7} + 15q^{8} + 9q^{9} + O(q^{10}) \) \( q - q^{2} - 3q^{3} - 7q^{4} - 4q^{5} + 3q^{6} - 26q^{7} + 15q^{8} + 9q^{9} + 4q^{10} + 11q^{11} + 21q^{12} - 32q^{13} + 26q^{14} + 12q^{15} + 41q^{16} + 74q^{17} - 9q^{18} - 60q^{19} + 28q^{20} + 78q^{21} - 11q^{22} - 182q^{23} - 45q^{24} - 109q^{25} + 32q^{26} - 27q^{27} + 182q^{28} - 90q^{29} - 12q^{30} - 8q^{31} - 161q^{32} - 33q^{33} - 74q^{34} + 104q^{35} - 63q^{36} - 66q^{37} + 60q^{38} + 96q^{39} - 60q^{40} + 422q^{41} - 78q^{42} + 408q^{43} - 77q^{44} - 36q^{45} + 182q^{46} - 506q^{47} - 123q^{48} + 333q^{49} + 109q^{50} - 222q^{51} + 224q^{52} + 348q^{53} + 27q^{54} - 44q^{55} - 390q^{56} + 180q^{57} + 90q^{58} - 200q^{59} - 84q^{60} + 132q^{61} + 8q^{62} - 234q^{63} - 167q^{64} + 128q^{65} + 33q^{66} - 1036q^{67} - 518q^{68} + 546q^{69} - 104q^{70} + 762q^{71} + 135q^{72} - 542q^{73} + 66q^{74} + 327q^{75} + 420q^{76} - 286q^{77} - 96q^{78} - 550q^{79} - 164q^{80} + 81q^{81} - 422q^{82} - 132q^{83} - 546q^{84} - 296q^{85} - 408q^{86} + 270q^{87} + 165q^{88} + 570q^{89} + 36q^{90} + 832q^{91} + 1274q^{92} + 24q^{93} + 506q^{94} + 240q^{95} + 483q^{96} + 14q^{97} - 333q^{98} + 99q^{99} + O(q^{100}) \)

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 \(S_{4}^{\mathrm{new}}(\Gamma_0(33))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + 8 T^{2} \)
$3$ \( 1 + 3 T \)
$5$ \( 1 + 4 T + 125 T^{2} \)
$7$ \( 1 + 26 T + 343 T^{2} \)
$11$ \( 1 - 11 T \)
$13$ \( 1 + 32 T + 2197 T^{2} \)
$17$ \( 1 - 74 T + 4913 T^{2} \)
$19$ \( 1 + 60 T + 6859 T^{2} \)
$23$ \( 1 + 182 T + 12167 T^{2} \)
$29$ \( 1 + 90 T + 24389 T^{2} \)
$31$ \( 1 + 8 T + 29791 T^{2} \)
$37$ \( 1 + 66 T + 50653 T^{2} \)
$41$ \( 1 - 422 T + 68921 T^{2} \)
$43$ \( 1 - 408 T + 79507 T^{2} \)
$47$ \( 1 + 506 T + 103823 T^{2} \)
$53$ \( 1 - 348 T + 148877 T^{2} \)
$59$ \( 1 + 200 T + 205379 T^{2} \)
$61$ \( 1 - 132 T + 226981 T^{2} \)
$67$ \( 1 + 1036 T + 300763 T^{2} \)
$71$ \( 1 - 762 T + 357911 T^{2} \)
$73$ \( 1 + 542 T + 389017 T^{2} \)
$79$ \( 1 + 550 T + 493039 T^{2} \)
$83$ \( 1 + 132 T + 571787 T^{2} \)
$89$ \( 1 - 570 T + 704969 T^{2} \)
$97$ \( 1 - 14 T + 912673 T^{2} \)
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