Properties

Label 522.4.a.j
Level $522$
Weight $4$
Character orbit 522.a
Self dual yes
Analytic conductor $30.799$
Analytic rank $0$
Dimension $2$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
Defining polynomial: \( x^{2} - 6 \) 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 \(\beta = \sqrt{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{2} + 4 q^{4} + (6 \beta + 5) q^{5} + (8 \beta - 8) q^{7} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} + 4 q^{4} + (6 \beta + 5) q^{5} + (8 \beta - 8) q^{7} + 8 q^{8} + (12 \beta + 10) q^{10} + ( - 3 \beta + 45) q^{11} + ( - 8 \beta - 25) q^{13} + (16 \beta - 16) q^{14} + 16 q^{16} + (16 \beta + 22) q^{17} + (4 \beta + 54) q^{19} + (24 \beta + 20) q^{20} + ( - 6 \beta + 90) q^{22} + ( - 78 \beta + 14) q^{23} + (60 \beta + 116) q^{25} + ( - 16 \beta - 50) q^{26} + (32 \beta - 32) q^{28} - 29 q^{29} + ( - 109 \beta + 33) q^{31} + 32 q^{32} + (32 \beta + 44) q^{34} + ( - 8 \beta + 248) q^{35} + ( - 4 \beta + 20) q^{37} + (8 \beta + 108) q^{38} + (48 \beta + 40) q^{40} + ( - 80 \beta - 152) q^{41} + ( - 53 \beta - 65) q^{43} + ( - 12 \beta + 180) q^{44} + ( - 156 \beta + 28) q^{46} + ( - 99 \beta + 257) q^{47} + ( - 128 \beta + 105) q^{49} + (120 \beta + 232) q^{50} + ( - 32 \beta - 100) q^{52} + (52 \beta + 479) q^{53} + (255 \beta + 117) q^{55} + (64 \beta - 64) q^{56} - 58 q^{58} + (250 \beta + 90) q^{59} + (12 \beta + 514) q^{61} + ( - 218 \beta + 66) q^{62} + 64 q^{64} + ( - 190 \beta - 413) q^{65} + (20 \beta - 456) q^{67} + (64 \beta + 88) q^{68} + ( - 16 \beta + 496) q^{70} + (34 \beta - 398) q^{71} + ( - 176 \beta - 428) q^{73} + ( - 8 \beta + 40) q^{74} + (16 \beta + 216) q^{76} + (384 \beta - 504) q^{77} + (361 \beta - 159) q^{79} + (96 \beta + 80) q^{80} + ( - 160 \beta - 304) q^{82} + ( - 38 \beta + 914) q^{83} + (212 \beta + 686) q^{85} + ( - 106 \beta - 130) q^{86} + ( - 24 \beta + 360) q^{88} + (72 \beta + 472) q^{89} + ( - 136 \beta - 184) q^{91} + ( - 312 \beta + 56) q^{92} + ( - 198 \beta + 514) q^{94} + (344 \beta + 414) q^{95} + ( - 100 \beta + 184) q^{97} + ( - 256 \beta + 210) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{2} + 8 q^{4} + 10 q^{5} - 16 q^{7} + 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{2} + 8 q^{4} + 10 q^{5} - 16 q^{7} + 16 q^{8} + 20 q^{10} + 90 q^{11} - 50 q^{13} - 32 q^{14} + 32 q^{16} + 44 q^{17} + 108 q^{19} + 40 q^{20} + 180 q^{22} + 28 q^{23} + 232 q^{25} - 100 q^{26} - 64 q^{28} - 58 q^{29} + 66 q^{31} + 64 q^{32} + 88 q^{34} + 496 q^{35} + 40 q^{37} + 216 q^{38} + 80 q^{40} - 304 q^{41} - 130 q^{43} + 360 q^{44} + 56 q^{46} + 514 q^{47} + 210 q^{49} + 464 q^{50} - 200 q^{52} + 958 q^{53} + 234 q^{55} - 128 q^{56} - 116 q^{58} + 180 q^{59} + 1028 q^{61} + 132 q^{62} + 128 q^{64} - 826 q^{65} - 912 q^{67} + 176 q^{68} + 992 q^{70} - 796 q^{71} - 856 q^{73} + 80 q^{74} + 432 q^{76} - 1008 q^{77} - 318 q^{79} + 160 q^{80} - 608 q^{82} + 1828 q^{83} + 1372 q^{85} - 260 q^{86} + 720 q^{88} + 944 q^{89} - 368 q^{91} + 112 q^{92} + 1028 q^{94} + 828 q^{95} + 368 q^{97} + 420 q^{98}+O(q^{100}) \) 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
−2.44949
2.44949
2.00000 0 4.00000 −9.69694 0 −27.5959 8.00000 0 −19.3939
1.2 2.00000 0 4.00000 19.6969 0 11.5959 8.00000 0 39.3939
\(n\): 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.j 2
3.b odd 2 1 58.4.a.c 2
12.b even 2 1 464.4.a.e 2
15.d odd 2 1 1450.4.a.g 2
24.f even 2 1 1856.4.a.i 2
24.h odd 2 1 1856.4.a.l 2
87.d odd 2 1 1682.4.a.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
58.4.a.c 2 3.b odd 2 1
464.4.a.e 2 12.b even 2 1
522.4.a.j 2 1.a even 1 1 trivial
1450.4.a.g 2 15.d odd 2 1
1682.4.a.c 2 87.d odd 2 1
1856.4.a.i 2 24.f even 2 1
1856.4.a.l 2 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 10T_{5} - 191 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(522))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 10T - 191 \) Copy content Toggle raw display
$7$ \( T^{2} + 16T - 320 \) Copy content Toggle raw display
$11$ \( T^{2} - 90T + 1971 \) Copy content Toggle raw display
$13$ \( T^{2} + 50T + 241 \) Copy content Toggle raw display
$17$ \( T^{2} - 44T - 1052 \) Copy content Toggle raw display
$19$ \( T^{2} - 108T + 2820 \) Copy content Toggle raw display
$23$ \( T^{2} - 28T - 36308 \) Copy content Toggle raw display
$29$ \( (T + 29)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 66T - 70197 \) Copy content Toggle raw display
$37$ \( T^{2} - 40T + 304 \) Copy content Toggle raw display
$41$ \( T^{2} + 304T - 15296 \) Copy content Toggle raw display
$43$ \( T^{2} + 130T - 12629 \) Copy content Toggle raw display
$47$ \( T^{2} - 514T + 7243 \) Copy content Toggle raw display
$53$ \( T^{2} - 958T + 213217 \) Copy content Toggle raw display
$59$ \( T^{2} - 180T - 366900 \) Copy content Toggle raw display
$61$ \( T^{2} - 1028 T + 263332 \) Copy content Toggle raw display
$67$ \( T^{2} + 912T + 205536 \) Copy content Toggle raw display
$71$ \( T^{2} + 796T + 151468 \) Copy content Toggle raw display
$73$ \( T^{2} + 856T - 2672 \) Copy content Toggle raw display
$79$ \( T^{2} + 318T - 756645 \) Copy content Toggle raw display
$83$ \( T^{2} - 1828 T + 826732 \) Copy content Toggle raw display
$89$ \( T^{2} - 944T + 191680 \) Copy content Toggle raw display
$97$ \( T^{2} - 368T - 26144 \) Copy content Toggle raw display
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