Properties

Label 33.8.a.a
Level $33$
Weight $8$
Character orbit 33.a
Self dual yes
Analytic conductor $10.309$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Newspace parameters

Level: \( N \) \(=\) \( 33 = 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 33.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(10.3087058410\)
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

\(f(q)\) \(=\) \( q + 10q^{2} + 27q^{3} - 28q^{4} - 410q^{5} + 270q^{6} - 1028q^{7} - 1560q^{8} + 729q^{9} + O(q^{10}) \) \( q + 10q^{2} + 27q^{3} - 28q^{4} - 410q^{5} + 270q^{6} - 1028q^{7} - 1560q^{8} + 729q^{9} - 4100q^{10} - 1331q^{11} - 756q^{12} + 12958q^{13} - 10280q^{14} - 11070q^{15} - 12016q^{16} + 17062q^{17} + 7290q^{18} - 54168q^{19} + 11480q^{20} - 27756q^{21} - 13310q^{22} - 11488q^{23} - 42120q^{24} + 89975q^{25} + 129580q^{26} + 19683q^{27} + 28784q^{28} - 186654q^{29} - 110700q^{30} - 188672q^{31} + 79520q^{32} - 35937q^{33} + 170620q^{34} + 421480q^{35} - 20412q^{36} + 395886q^{37} - 541680q^{38} + 349866q^{39} + 639600q^{40} - 47546q^{41} - 277560q^{42} + 602088q^{43} + 37268q^{44} - 298890q^{45} - 114880q^{46} - 647200q^{47} - 324432q^{48} + 233241q^{49} + 899750q^{50} + 460674q^{51} - 362824q^{52} - 1312722q^{53} + 196830q^{54} + 545710q^{55} + 1603680q^{56} - 1462536q^{57} - 1866540q^{58} - 2681140q^{59} + 309960q^{60} + 551190q^{61} - 1886720q^{62} - 749412q^{63} + 2333248q^{64} - 5312780q^{65} - 359370q^{66} + 459260q^{67} - 477736q^{68} - 310176q^{69} + 4214800q^{70} - 18072q^{71} - 1137240q^{72} - 426062q^{73} + 3958860q^{74} + 2429325q^{75} + 1516704q^{76} + 1368268q^{77} + 3498660q^{78} + 297764q^{79} + 4926560q^{80} + 531441q^{81} - 475460q^{82} + 5684028q^{83} + 777168q^{84} - 6995420q^{85} + 6020880q^{86} - 5039658q^{87} + 2076360q^{88} - 6342966q^{89} - 2988900q^{90} - 13320824q^{91} + 321664q^{92} - 5094144q^{93} - 6472000q^{94} + 22208880q^{95} + 2147040q^{96} + 16651586q^{97} + 2332410q^{98} - 970299q^{99} + O(q^{100}) \)

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
10.0000 27.0000 −28.0000 −410.000 270.000 −1028.00 −1560.00 729.000 −4100.00
\(n\): 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 33.8.a.a 1
3.b odd 2 1 99.8.a.a 1
4.b odd 2 1 528.8.a.a 1
11.b odd 2 1 363.8.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.8.a.a 1 1.a even 1 1 trivial
99.8.a.a 1 3.b odd 2 1
363.8.a.a 1 11.b odd 2 1
528.8.a.a 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} - 10 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(33))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 10 T + 128 T^{2} \)
$3$ \( 1 - 27 T \)
$5$ \( 1 + 410 T + 78125 T^{2} \)
$7$ \( 1 + 1028 T + 823543 T^{2} \)
$11$ \( 1 + 1331 T \)
$13$ \( 1 - 12958 T + 62748517 T^{2} \)
$17$ \( 1 - 17062 T + 410338673 T^{2} \)
$19$ \( 1 + 54168 T + 893871739 T^{2} \)
$23$ \( 1 + 11488 T + 3404825447 T^{2} \)
$29$ \( 1 + 186654 T + 17249876309 T^{2} \)
$31$ \( 1 + 188672 T + 27512614111 T^{2} \)
$37$ \( 1 - 395886 T + 94931877133 T^{2} \)
$41$ \( 1 + 47546 T + 194754273881 T^{2} \)
$43$ \( 1 - 602088 T + 271818611107 T^{2} \)
$47$ \( 1 + 647200 T + 506623120463 T^{2} \)
$53$ \( 1 + 1312722 T + 1174711139837 T^{2} \)
$59$ \( 1 + 2681140 T + 2488651484819 T^{2} \)
$61$ \( 1 - 551190 T + 3142742836021 T^{2} \)
$67$ \( 1 - 459260 T + 6060711605323 T^{2} \)
$71$ \( 1 + 18072 T + 9095120158391 T^{2} \)
$73$ \( 1 + 426062 T + 11047398519097 T^{2} \)
$79$ \( 1 - 297764 T + 19203908986159 T^{2} \)
$83$ \( 1 - 5684028 T + 27136050989627 T^{2} \)
$89$ \( 1 + 6342966 T + 44231334895529 T^{2} \)
$97$ \( 1 - 16651586 T + 80798284478113 T^{2} \)
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