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

\(f(q)\) \(=\) \( q - 5 q^{2} + 3 q^{3} + 17 q^{4} - 14 q^{5} - 15 q^{6} - 32 q^{7} - 45 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 5 q^{2} + 3 q^{3} + 17 q^{4} - 14 q^{5} - 15 q^{6} - 32 q^{7} - 45 q^{8} + 9 q^{9} + 70 q^{10} - 11 q^{11} + 51 q^{12} - 38 q^{13} + 160 q^{14} - 42 q^{15} + 89 q^{16} - 2 q^{17} - 45 q^{18} + 72 q^{19} - 238 q^{20} - 96 q^{21} + 55 q^{22} + 68 q^{23} - 135 q^{24} + 71 q^{25} + 190 q^{26} + 27 q^{27} - 544 q^{28} - 54 q^{29} + 210 q^{30} - 152 q^{31} - 85 q^{32} - 33 q^{33} + 10 q^{34} + 448 q^{35} + 153 q^{36} + 174 q^{37} - 360 q^{38} - 114 q^{39} + 630 q^{40} + 94 q^{41} + 480 q^{42} - 528 q^{43} - 187 q^{44} - 126 q^{45} - 340 q^{46} - 340 q^{47} + 267 q^{48} + 681 q^{49} - 355 q^{50} - 6 q^{51} - 646 q^{52} - 438 q^{53} - 135 q^{54} + 154 q^{55} + 1440 q^{56} + 216 q^{57} + 270 q^{58} + 20 q^{59} - 714 q^{60} + 570 q^{61} + 760 q^{62} - 288 q^{63} - 287 q^{64} + 532 q^{65} + 165 q^{66} - 460 q^{67} - 34 q^{68} + 204 q^{69} - 2240 q^{70} - 1092 q^{71} - 405 q^{72} + 562 q^{73} - 870 q^{74} + 213 q^{75} + 1224 q^{76} + 352 q^{77} + 570 q^{78} - 16 q^{79} - 1246 q^{80} + 81 q^{81} - 470 q^{82} + 372 q^{83} - 1632 q^{84} + 28 q^{85} + 2640 q^{86} - 162 q^{87} + 495 q^{88} - 966 q^{89} + 630 q^{90} + 1216 q^{91} + 1156 q^{92} - 456 q^{93} + 1700 q^{94} - 1008 q^{95} - 255 q^{96} - 526 q^{97} - 3405 q^{98} - 99 q^{99}+O(q^{100}) \) 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
−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 \(S_{4}^{\mathrm{new}}(\Gamma_0(33))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 5 \) Copy content Toggle raw display
$3$ \( T - 3 \) Copy content Toggle raw display
$5$ \( T + 14 \) Copy content Toggle raw display
$7$ \( T + 32 \) Copy content Toggle raw display
$11$ \( T + 11 \) Copy content Toggle raw display
$13$ \( T + 38 \) Copy content Toggle raw display
$17$ \( T + 2 \) Copy content Toggle raw display
$19$ \( T - 72 \) Copy content Toggle raw display
$23$ \( T - 68 \) Copy content Toggle raw display
$29$ \( T + 54 \) Copy content Toggle raw display
$31$ \( T + 152 \) Copy content Toggle raw display
$37$ \( T - 174 \) Copy content Toggle raw display
$41$ \( T - 94 \) Copy content Toggle raw display
$43$ \( T + 528 \) Copy content Toggle raw display
$47$ \( T + 340 \) Copy content Toggle raw display
$53$ \( T + 438 \) Copy content Toggle raw display
$59$ \( T - 20 \) Copy content Toggle raw display
$61$ \( T - 570 \) Copy content Toggle raw display
$67$ \( T + 460 \) Copy content Toggle raw display
$71$ \( T + 1092 \) Copy content Toggle raw display
$73$ \( T - 562 \) Copy content Toggle raw display
$79$ \( T + 16 \) Copy content Toggle raw display
$83$ \( T - 372 \) Copy content Toggle raw display
$89$ \( T + 966 \) Copy content Toggle raw display
$97$ \( T + 526 \) Copy content Toggle raw display
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