Properties

Label 522.4.a.k
Level $522$
Weight $4$
Character orbit 522.a
Self dual yes
Analytic conductor $30.799$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 522 = 2 \cdot 3^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 522.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(30.7989970230\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.19816.1
Defining polynomial: \( x^{3} - x^{2} - 42x - 54 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 58)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 q^{2} + 4 q^{4} + ( - \beta_{2} - 2 \beta_1 - 6) q^{5} + ( - 4 \beta_{2} + 8) q^{7} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{2} + 4 q^{4} + ( - \beta_{2} - 2 \beta_1 - 6) q^{5} + ( - 4 \beta_{2} + 8) q^{7} - 8 q^{8} + (2 \beta_{2} + 4 \beta_1 + 12) q^{10} + (2 \beta_{2} - \beta_1 - 3) q^{11} + (11 \beta_{2} + 8 \beta_1 - 4) q^{13} + (8 \beta_{2} - 16) q^{14} + 16 q^{16} + (16 \beta_{2} + 12 \beta_1 + 18) q^{17} + (10 \beta_{2} - 8 \beta_1 - 52) q^{19} + ( - 4 \beta_{2} - 8 \beta_1 - 24) q^{20} + ( - 4 \beta_{2} + 2 \beta_1 + 6) q^{22} + ( - 12 \beta_{2} + 6 \beta_1 + 66) q^{23} + (11 \beta_{2} + 40 \beta_1 + 13) q^{25} + ( - 22 \beta_{2} - 16 \beta_1 + 8) q^{26} + ( - 16 \beta_{2} + 32) q^{28} - 29 q^{29} + ( - 36 \beta_{2} - 11 \beta_1 - 25) q^{31} - 32 q^{32} + ( - 32 \beta_{2} - 24 \beta_1 - 36) q^{34} + ( - 12 \beta_{2} - 24 \beta_1) q^{35} + (38 \beta_{2} - 12 \beta_1 - 10) q^{37} + ( - 20 \beta_{2} + 16 \beta_1 + 104) q^{38} + (8 \beta_{2} + 16 \beta_1 + 48) q^{40} + (6 \beta_{2} - 4 \beta_1 - 186) q^{41} + (2 \beta_{2} - 15 \beta_1 + 11) q^{43} + (8 \beta_{2} - 4 \beta_1 - 12) q^{44} + (24 \beta_{2} - 12 \beta_1 - 132) q^{46} + ( - 2 \beta_{2} - 33 \beta_1 - 207) q^{47} + ( - 80 \beta_{2} - 64 \beta_1 + 201) q^{49} + ( - 22 \beta_{2} - 80 \beta_1 - 26) q^{50} + (44 \beta_{2} + 32 \beta_1 - 16) q^{52} + (27 \beta_{2} + 116 \beta_1 - 276) q^{53} + (8 \beta_{2} + 25 \beta_1 + 39) q^{55} + (32 \beta_{2} - 64) q^{56} + 58 q^{58} + ( - 4 \beta_{2} + 86 \beta_1 - 90) q^{59} + (40 \beta_{2} - 80 \beta_1 + 134) q^{61} + (72 \beta_{2} + 22 \beta_1 + 50) q^{62} + 64 q^{64} + ( - 9 \beta_{2} - 90 \beta_1 - 468) q^{65} + ( - 72 \beta_{2} + 36 \beta_1 - 88) q^{67} + (64 \beta_{2} + 48 \beta_1 + 72) q^{68} + (24 \beta_{2} + 48 \beta_1) q^{70} + (4 \beta_{2} - 2 \beta_1 + 18) q^{71} + (30 \beta_{2} + 40 \beta_1 - 178) q^{73} + ( - 76 \beta_{2} + 24 \beta_1 + 20) q^{74} + (40 \beta_{2} - 32 \beta_1 - 208) q^{76} + (24 \beta_{2} + 28 \beta_1 - 300) q^{77} + ( - 38 \beta_{2} - 37 \beta_1 - 691) q^{79} + ( - 16 \beta_{2} - 32 \beta_1 - 96) q^{80} + ( - 12 \beta_{2} + 8 \beta_1 + 372) q^{82} + ( - 12 \beta_{2} - 162 \beta_1 + 150) q^{83} + ( - 38 \beta_{2} - 184 \beta_1 - 840) q^{85} + ( - 4 \beta_{2} + 30 \beta_1 - 22) q^{86} + ( - 16 \beta_{2} + 8 \beta_1 + 24) q^{88} + ( - 154 \beta_{2} - 68 \beta_1 - 282) q^{89} + (244 \beta_{2} + 208 \beta_1 - 1064) q^{91} + ( - 48 \beta_{2} + 24 \beta_1 + 264) q^{92} + (4 \beta_{2} + 66 \beta_1 + 414) q^{94} + (86 \beta_{2} + 244 \beta_1 + 552) q^{95} + ( - 34 \beta_{2} + 176 \beta_1 + 14) q^{97} + (160 \beta_{2} + 128 \beta_1 - 402) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 6 q^{2} + 12 q^{4} - 20 q^{5} + 24 q^{7} - 24 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 6 q^{2} + 12 q^{4} - 20 q^{5} + 24 q^{7} - 24 q^{8} + 40 q^{10} - 10 q^{11} - 4 q^{13} - 48 q^{14} + 48 q^{16} + 66 q^{17} - 164 q^{19} - 80 q^{20} + 20 q^{22} + 204 q^{23} + 79 q^{25} + 8 q^{26} + 96 q^{28} - 87 q^{29} - 86 q^{31} - 96 q^{32} - 132 q^{34} - 24 q^{35} - 42 q^{37} + 328 q^{38} + 160 q^{40} - 562 q^{41} + 18 q^{43} - 40 q^{44} - 408 q^{46} - 654 q^{47} + 539 q^{49} - 158 q^{50} - 16 q^{52} - 712 q^{53} + 142 q^{55} - 192 q^{56} + 174 q^{58} - 184 q^{59} + 322 q^{61} + 172 q^{62} + 192 q^{64} - 1494 q^{65} - 228 q^{67} + 264 q^{68} + 48 q^{70} + 52 q^{71} - 494 q^{73} + 84 q^{74} - 656 q^{76} - 872 q^{77} - 2110 q^{79} - 320 q^{80} + 1124 q^{82} + 288 q^{83} - 2704 q^{85} - 36 q^{86} + 80 q^{88} - 914 q^{89} - 2984 q^{91} + 816 q^{92} + 1308 q^{94} + 1900 q^{95} + 218 q^{97} - 1078 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 42x - 54 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} - 4\nu - 27 ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 3\beta_{2} + 4\beta _1 + 27 \) 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
7.53003
−5.13291
−1.39712
−2.00000 0 4.00000 −20.9205 0 8.55839 −8.00000 0 41.8409
1.2 −2.00000 0 4.00000 −2.36031 0 −18.5045 −8.00000 0 4.72062
1.3 −2.00000 0 4.00000 3.28077 0 33.9461 −8.00000 0 −6.56153
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(29\) \(1\)

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 522.4.a.k 3
3.b odd 2 1 58.4.a.d 3
12.b even 2 1 464.4.a.i 3
15.d odd 2 1 1450.4.a.h 3
24.f even 2 1 1856.4.a.s 3
24.h odd 2 1 1856.4.a.r 3
87.d odd 2 1 1682.4.a.d 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
58.4.a.d 3 3.b odd 2 1
464.4.a.i 3 12.b even 2 1
522.4.a.k 3 1.a even 1 1 trivial
1450.4.a.h 3 15.d odd 2 1
1682.4.a.d 3 87.d odd 2 1
1856.4.a.r 3 24.h odd 2 1
1856.4.a.s 3 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{3} + 20T_{5}^{2} - 27T_{5} - 162 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(522))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 2)^{3} \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( T^{3} + 20 T^{2} - 27 T - 162 \) Copy content Toggle raw display
$7$ \( T^{3} - 24 T^{2} - 496 T + 5376 \) Copy content Toggle raw display
$11$ \( T^{3} + 10 T^{2} - 233 T - 2424 \) Copy content Toggle raw display
$13$ \( T^{3} + 4 T^{2} - 5619 T + 131706 \) Copy content Toggle raw display
$17$ \( T^{3} - 66 T^{2} - 10660 T + 679368 \) Copy content Toggle raw display
$19$ \( T^{3} + 164 T^{2} - 124 T - 664448 \) Copy content Toggle raw display
$23$ \( T^{3} - 204 T^{2} + 4284 T + 677376 \) Copy content Toggle raw display
$29$ \( (T + 29)^{3} \) Copy content Toggle raw display
$31$ \( T^{3} + 86 T^{2} - 48089 T - 4766172 \) Copy content Toggle raw display
$37$ \( T^{3} + 42 T^{2} - 79456 T - 7684896 \) Copy content Toggle raw display
$41$ \( T^{3} + 562 T^{2} + 102432 T + 5982048 \) Copy content Toggle raw display
$43$ \( T^{3} - 18 T^{2} - 10369 T + 196488 \) Copy content Toggle raw display
$47$ \( T^{3} + 654 T^{2} + 98015 T + 3425124 \) Copy content Toggle raw display
$53$ \( T^{3} + 712 T^{2} + \cdots - 252120546 \) Copy content Toggle raw display
$59$ \( T^{3} + 184 T^{2} + \cdots - 57362928 \) Copy content Toggle raw display
$61$ \( T^{3} - 322 T^{2} - 388372 T - 5254424 \) Copy content Toggle raw display
$67$ \( T^{3} + 228 T^{2} + \cdots + 47608192 \) Copy content Toggle raw display
$71$ \( T^{3} - 52 T^{2} - 164 T + 672 \) Copy content Toggle raw display
$73$ \( T^{3} + 494 T^{2} + 6112 T - 9410208 \) Copy content Toggle raw display
$79$ \( T^{3} + 2110 T^{2} + \cdots + 285187172 \) Copy content Toggle raw display
$83$ \( T^{3} - 288 T^{2} + \cdots + 437606064 \) Copy content Toggle raw display
$89$ \( T^{3} + 914 T^{2} + \cdots - 598011552 \) Copy content Toggle raw display
$97$ \( T^{3} - 218 T^{2} + \cdots - 17006112 \) Copy content Toggle raw display
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