Properties

Label 33.4.a.a
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

\(f(q)\) \(=\) \( q - 5q^{2} + 3q^{3} + 17q^{4} - 14q^{5} - 15q^{6} - 32q^{7} - 45q^{8} + 9q^{9} + O(q^{10}) \) \( q - 5q^{2} + 3q^{3} + 17q^{4} - 14q^{5} - 15q^{6} - 32q^{7} - 45q^{8} + 9q^{9} + 70q^{10} - 11q^{11} + 51q^{12} - 38q^{13} + 160q^{14} - 42q^{15} + 89q^{16} - 2q^{17} - 45q^{18} + 72q^{19} - 238q^{20} - 96q^{21} + 55q^{22} + 68q^{23} - 135q^{24} + 71q^{25} + 190q^{26} + 27q^{27} - 544q^{28} - 54q^{29} + 210q^{30} - 152q^{31} - 85q^{32} - 33q^{33} + 10q^{34} + 448q^{35} + 153q^{36} + 174q^{37} - 360q^{38} - 114q^{39} + 630q^{40} + 94q^{41} + 480q^{42} - 528q^{43} - 187q^{44} - 126q^{45} - 340q^{46} - 340q^{47} + 267q^{48} + 681q^{49} - 355q^{50} - 6q^{51} - 646q^{52} - 438q^{53} - 135q^{54} + 154q^{55} + 1440q^{56} + 216q^{57} + 270q^{58} + 20q^{59} - 714q^{60} + 570q^{61} + 760q^{62} - 288q^{63} - 287q^{64} + 532q^{65} + 165q^{66} - 460q^{67} - 34q^{68} + 204q^{69} - 2240q^{70} - 1092q^{71} - 405q^{72} + 562q^{73} - 870q^{74} + 213q^{75} + 1224q^{76} + 352q^{77} + 570q^{78} - 16q^{79} - 1246q^{80} + 81q^{81} - 470q^{82} + 372q^{83} - 1632q^{84} + 28q^{85} + 2640q^{86} - 162q^{87} + 495q^{88} - 966q^{89} + 630q^{90} + 1216q^{91} + 1156q^{92} - 456q^{93} + 1700q^{94} - 1008q^{95} - 255q^{96} - 526q^{97} - 3405q^{98} - 99q^{99} + O(q^{100}) \)

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
−5.00000 3.00000 17.0000 −14.0000 −15.0000 −32.0000 −45.0000 9.00000 70.0000
\(n\): 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.a 1
3.b odd 2 1 99.4.a.b 1
4.b odd 2 1 528.4.a.a 1
5.b even 2 1 825.4.a.i 1
5.c odd 4 2 825.4.c.a 2
7.b odd 2 1 1617.4.a.a 1
8.b even 2 1 2112.4.a.l 1
8.d odd 2 1 2112.4.a.y 1
11.b odd 2 1 363.4.a.h 1
12.b even 2 1 1584.4.a.t 1
15.d odd 2 1 2475.4.a.b 1
33.d even 2 1 1089.4.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.4.a.a 1 1.a even 1 1 trivial
99.4.a.b 1 3.b odd 2 1
363.4.a.h 1 11.b odd 2 1
528.4.a.a 1 4.b odd 2 1
825.4.a.i 1 5.b even 2 1
825.4.c.a 2 5.c odd 4 2
1089.4.a.a 1 33.d even 2 1
1584.4.a.t 1 12.b even 2 1
1617.4.a.a 1 7.b odd 2 1
2112.4.a.l 1 8.b even 2 1
2112.4.a.y 1 8.d odd 2 1
2475.4.a.b 1 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 5 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(33))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 5 T + 8 T^{2} \)
$3$ \( 1 - 3 T \)
$5$ \( 1 + 14 T + 125 T^{2} \)
$7$ \( 1 + 32 T + 343 T^{2} \)
$11$ \( 1 + 11 T \)
$13$ \( 1 + 38 T + 2197 T^{2} \)
$17$ \( 1 + 2 T + 4913 T^{2} \)
$19$ \( 1 - 72 T + 6859 T^{2} \)
$23$ \( 1 - 68 T + 12167 T^{2} \)
$29$ \( 1 + 54 T + 24389 T^{2} \)
$31$ \( 1 + 152 T + 29791 T^{2} \)
$37$ \( 1 - 174 T + 50653 T^{2} \)
$41$ \( 1 - 94 T + 68921 T^{2} \)
$43$ \( 1 + 528 T + 79507 T^{2} \)
$47$ \( 1 + 340 T + 103823 T^{2} \)
$53$ \( 1 + 438 T + 148877 T^{2} \)
$59$ \( 1 - 20 T + 205379 T^{2} \)
$61$ \( 1 - 570 T + 226981 T^{2} \)
$67$ \( 1 + 460 T + 300763 T^{2} \)
$71$ \( 1 + 1092 T + 357911 T^{2} \)
$73$ \( 1 - 562 T + 389017 T^{2} \)
$79$ \( 1 + 16 T + 493039 T^{2} \)
$83$ \( 1 - 372 T + 571787 T^{2} \)
$89$ \( 1 + 966 T + 704969 T^{2} \)
$97$ \( 1 + 526 T + 912673 T^{2} \)
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