Properties

Label 33.6.a.a
Level 33
Weight 6
Character orbit 33.a
Self dual yes
Analytic conductor 5.293
Analytic rank 0
Dimension 1
CM no
Inner twists 1

Related objects

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Newspace parameters

Level: \( N \) \(=\) \( 33 = 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 33.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.29266605383\)
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

\(f(q)\) \(=\) \( q - 2q^{2} - 9q^{3} - 28q^{4} + 46q^{5} + 18q^{6} + 148q^{7} + 120q^{8} + 81q^{9} + O(q^{10}) \) \( q - 2q^{2} - 9q^{3} - 28q^{4} + 46q^{5} + 18q^{6} + 148q^{7} + 120q^{8} + 81q^{9} - 92q^{10} + 121q^{11} + 252q^{12} + 574q^{13} - 296q^{14} - 414q^{15} + 656q^{16} - 722q^{17} - 162q^{18} + 2160q^{19} - 1288q^{20} - 1332q^{21} - 242q^{22} - 2536q^{23} - 1080q^{24} - 1009q^{25} - 1148q^{26} - 729q^{27} - 4144q^{28} + 4650q^{29} + 828q^{30} + 5032q^{31} - 5152q^{32} - 1089q^{33} + 1444q^{34} + 6808q^{35} - 2268q^{36} + 8118q^{37} - 4320q^{38} - 5166q^{39} + 5520q^{40} - 5138q^{41} + 2664q^{42} + 8304q^{43} - 3388q^{44} + 3726q^{45} + 5072q^{46} + 24728q^{47} - 5904q^{48} + 5097q^{49} + 2018q^{50} + 6498q^{51} - 16072q^{52} - 28746q^{53} + 1458q^{54} + 5566q^{55} + 17760q^{56} - 19440q^{57} - 9300q^{58} - 5860q^{59} + 11592q^{60} - 53658q^{61} - 10064q^{62} + 11988q^{63} - 10688q^{64} + 26404q^{65} + 2178q^{66} + 30908q^{67} + 20216q^{68} + 22824q^{69} - 13616q^{70} - 69648q^{71} + 9720q^{72} - 18446q^{73} - 16236q^{74} + 9081q^{75} - 60480q^{76} + 17908q^{77} + 10332q^{78} - 25300q^{79} + 30176q^{80} + 6561q^{81} + 10276q^{82} - 17556q^{83} + 37296q^{84} - 33212q^{85} - 16608q^{86} - 41850q^{87} + 14520q^{88} + 132570q^{89} - 7452q^{90} + 84952q^{91} + 71008q^{92} - 45288q^{93} - 49456q^{94} + 99360q^{95} + 46368q^{96} + 70658q^{97} - 10194q^{98} + 9801q^{99} + O(q^{100}) \)

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
−2.00000 −9.00000 −28.0000 46.0000 18.0000 148.000 120.000 81.0000 −92.0000
\(n\): 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 33.6.a.a 1
3.b odd 2 1 99.6.a.b 1
4.b odd 2 1 528.6.a.i 1
5.b even 2 1 825.6.a.b 1
11.b odd 2 1 363.6.a.c 1
33.d even 2 1 1089.6.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.6.a.a 1 1.a even 1 1 trivial
99.6.a.b 1 3.b odd 2 1
363.6.a.c 1 11.b odd 2 1
528.6.a.i 1 4.b odd 2 1
825.6.a.b 1 5.b even 2 1
1089.6.a.d 1 33.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 2 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(33))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T + 32 T^{2} \)
$3$ \( 1 + 9 T \)
$5$ \( 1 - 46 T + 3125 T^{2} \)
$7$ \( 1 - 148 T + 16807 T^{2} \)
$11$ \( 1 - 121 T \)
$13$ \( 1 - 574 T + 371293 T^{2} \)
$17$ \( 1 + 722 T + 1419857 T^{2} \)
$19$ \( 1 - 2160 T + 2476099 T^{2} \)
$23$ \( 1 + 2536 T + 6436343 T^{2} \)
$29$ \( 1 - 4650 T + 20511149 T^{2} \)
$31$ \( 1 - 5032 T + 28629151 T^{2} \)
$37$ \( 1 - 8118 T + 69343957 T^{2} \)
$41$ \( 1 + 5138 T + 115856201 T^{2} \)
$43$ \( 1 - 8304 T + 147008443 T^{2} \)
$47$ \( 1 - 24728 T + 229345007 T^{2} \)
$53$ \( 1 + 28746 T + 418195493 T^{2} \)
$59$ \( 1 + 5860 T + 714924299 T^{2} \)
$61$ \( 1 + 53658 T + 844596301 T^{2} \)
$67$ \( 1 - 30908 T + 1350125107 T^{2} \)
$71$ \( 1 + 69648 T + 1804229351 T^{2} \)
$73$ \( 1 + 18446 T + 2073071593 T^{2} \)
$79$ \( 1 + 25300 T + 3077056399 T^{2} \)
$83$ \( 1 + 17556 T + 3939040643 T^{2} \)
$89$ \( 1 - 132570 T + 5584059449 T^{2} \)
$97$ \( 1 - 70658 T + 8587340257 T^{2} \)
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