Properties

Label 1521.4.a.bb
Level $1521$
Weight $4$
Character orbit 1521.a
Self dual yes
Analytic conductor $89.742$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1521 = 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1521.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(89.7419051187\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \( x^{4} - 2x^{3} - 25x^{2} + 24x + 78 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 39)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + \beta_1 + 5) q^{4} + ( - \beta_{2} - \beta_1 - 1) q^{5} + ( - \beta_{3} - \beta_1 + 4) q^{7} + (\beta_{3} + 2 \beta_{2} + 5 \beta_1 + 11) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} + \beta_1 + 5) q^{4} + ( - \beta_{2} - \beta_1 - 1) q^{5} + ( - \beta_{3} - \beta_1 + 4) q^{7} + (\beta_{3} + 2 \beta_{2} + 5 \beta_1 + 11) q^{8} + ( - \beta_{3} - 2 \beta_{2} - 9 \beta_1 - 11) q^{10} + (\beta_{3} - \beta_{2} + 4 \beta_1 + 8) q^{11} + ( - 7 \beta_{2} + 6 \beta_1 - 13) q^{14} + (2 \beta_{3} + 5 \beta_{2} + 19 \beta_1 + 21) q^{16} + (\beta_{3} - 19 \beta_1 - 15) q^{17} + ( - \beta_{3} + 7 \beta_{2} - 6 \beta_1 - 28) q^{19} + ( - 2 \beta_{3} - 9 \beta_{2} - 23 \beta_1 - 105) q^{20} + ( - \beta_{3} + 9 \beta_{2} + 2 \beta_1 + 54) q^{22} + ( - 3 \beta_{3} - \beta_{2} + 4 \beta_1 - 28) q^{23} + (2 \beta_{3} + 5 \beta_{2} + 19 \beta_1 - 24) q^{25} + (\beta_{3} - \beta_{2} - 48 \beta_1 + 60) q^{28} + (6 \beta_{3} - \beta_{2} - 11 \beta_1 - 43) q^{29} + (7 \beta_{2} - 53 \beta_1 + 20) q^{31} + ( - 3 \beta_{3} + 20 \beta_{2} + 29 \beta_1 + 149) q^{32} + ( - 13 \beta_{2} - 37 \beta_1 - 247) q^{34} + ( - 5 \beta_{3} + \beta_{2} + 44 \beta_1 - 44) q^{35} + (\beta_{3} + 2 \beta_{2} - 21 \beta_1 - 15) q^{37} + (7 \beta_{3} - 5 \beta_{2} + 18 \beta_1 - 92) q^{38} + ( - \beta_{3} - 28 \beta_{2} - 113 \beta_1 - 193) q^{40} + (3 \beta_{3} + 43 \beta_1 - 285) q^{41} + ( - \beta_{3} - 14 \beta_{2} + 57 \beta_1 + 84) q^{43} + (\beta_{3} + 13 \beta_{2} + 90 \beta_1 - 56) q^{44} + ( - \beta_{3} - 15 \beta_{2} - 22 \beta_1 + 54) q^{46} + (3 \beta_{3} - 47 \beta_{2} - 8 \beta_1 - 20) q^{47} + ( - 2 \beta_{3} - 29 \beta_{2} - 43 \beta_1 + 289) q^{49} + (5 \beta_{3} + 36 \beta_{2} + 24 \beta_1 + 237) q^{50} + ( - 2 \beta_{3} - 23 \beta_{2} - 37 \beta_1 - 47) q^{53} + (3 \beta_{3} - 17 \beta_{2} - 74 \beta_1 + 88) q^{55} + ( - \beta_{3} + 13 \beta_{2} - 46 \beta_1 - 518) q^{56} + ( - \beta_{3} + 24 \beta_{2} - 79 \beta_1 - 141) q^{58} + (14 \beta_{3} + 24 \beta_{2} + 10 \beta_1 + 72) q^{59} + ( - 3 \beta_{3} - 31 \beta_{2} + 26 \beta_1 - 245) q^{61} + (7 \beta_{3} - 46 \beta_{2} + 16 \beta_1 - 703) q^{62} + (4 \beta_{3} - 9 \beta_{2} + 175 \beta_1 + 169) q^{64} + (\beta_{3} + 14 \beta_{2} - 129 \beta_1 + 348) q^{67} + ( - 21 \beta_{3} - 50 \beta_{2} - 223 \beta_1 - 335) q^{68} + (\beta_{3} + 15 \beta_{2} + 22 \beta_1 + 570) q^{70} + (11 \beta_{3} + 49 \beta_{2} - 76 \beta_1 + 304) q^{71} + (2 \beta_{3} + 24 \beta_{2} + 194 \beta_1 - 335) q^{73} + (2 \beta_{3} - 13 \beta_{2} - 25 \beta_1 - 277) q^{74} + (3 \beta_{3} - \beta_{2} - 82 \beta_1 + 468) q^{76} + ( - 16 \beta_{3} + 2 \beta_{2} + 98 \beta_1 - 676) q^{77} + ( - 2 \beta_{3} - 7 \beta_{2} - 21 \beta_1 - 176) q^{79} + ( - 12 \beta_{3} - 75 \beta_{2} - 315 \beta_1 - 573) q^{80} + (61 \beta_{2} - 251 \beta_1 + 559) q^{82} + (5 \beta_{3} + \beta_{2} - 138 \beta_1 - 32) q^{83} + (25 \beta_{3} + 50 \beta_{2} + 147 \beta_1 + 275) q^{85} + ( - 14 \beta_{3} + 37 \beta_{2} + 46 \beta_1 + 769) q^{86} + (21 \beta_{3} + 37 \beta_{2} + 106 \beta_1 + 712) q^{88} + ( - 28 \beta_{3} - 50 \beta_{2} - 146 \beta_1 + 478) q^{89} + (9 \beta_{3} - 35 \beta_{2} - 102 \beta_1 - 32) q^{92} + ( - 47 \beta_{3} - 37 \beta_{2} - 366 \beta_1 - 10) q^{94} + ( - 7 \beta_{3} + 29 \beta_{2} + 58 \beta_1 - 580) q^{95} + (22 \beta_{3} + 63 \beta_{2} - 71 \beta_1 - 506) q^{97} + ( - 29 \beta_{3} - 84 \beta_{2} + 49 \beta_1 - 501) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} + 22 q^{4} - 6 q^{5} + 14 q^{7} + 54 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} + 22 q^{4} - 6 q^{5} + 14 q^{7} + 54 q^{8} - 62 q^{10} + 40 q^{11} - 40 q^{14} + 122 q^{16} - 98 q^{17} - 124 q^{19} - 466 q^{20} + 220 q^{22} - 104 q^{23} - 58 q^{25} + 144 q^{28} - 194 q^{29} - 26 q^{31} + 654 q^{32} - 1062 q^{34} - 88 q^{35} - 102 q^{37} - 332 q^{38} - 998 q^{40} - 1054 q^{41} + 450 q^{43} - 44 q^{44} + 172 q^{46} - 96 q^{47} + 1070 q^{49} + 996 q^{50} - 262 q^{53} + 204 q^{55} - 2164 q^{56} - 722 q^{58} + 308 q^{59} - 928 q^{61} - 2780 q^{62} + 1026 q^{64} + 1134 q^{67} - 1786 q^{68} + 2324 q^{70} + 1064 q^{71} - 952 q^{73} - 1158 q^{74} + 1708 q^{76} - 2508 q^{77} - 746 q^{79} - 2922 q^{80} + 1734 q^{82} - 404 q^{83} + 1394 q^{85} + 3168 q^{86} + 3060 q^{88} + 1620 q^{89} - 332 q^{92} - 772 q^{94} - 2204 q^{95} - 2166 q^{97} - 1906 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 25x^{2} + 24x + 78 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 13 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 2\nu^{2} - 19\nu + 15 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 13 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 2\beta_{2} + 21\beta _1 + 11 \) 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
−4.22605
−1.46610
2.36176
5.33039
−4.22605 0 9.85953 −5.85953 0 24.1254 −7.85849 0 24.7627
1.2 −1.46610 0 −5.85055 9.85055 0 −29.9396 20.3063 0 −14.4419
1.3 2.36176 0 −2.42208 6.42208 0 29.4938 −24.6145 0 15.1674
1.4 5.33039 0 20.4131 −16.4131 0 −9.67968 66.1667 0 −87.4882
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(13\) \(1\)

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 1521.4.a.bb 4
3.b odd 2 1 507.4.a.i 4
13.b even 2 1 1521.4.a.v 4
13.e even 6 2 117.4.g.e 8
39.d odd 2 1 507.4.a.m 4
39.f even 4 2 507.4.b.h 8
39.h odd 6 2 39.4.e.c 8
156.r even 6 2 624.4.q.i 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.e.c 8 39.h odd 6 2
117.4.g.e 8 13.e even 6 2
507.4.a.i 4 3.b odd 2 1
507.4.a.m 4 39.d odd 2 1
507.4.b.h 8 39.f even 4 2
624.4.q.i 8 156.r even 6 2
1521.4.a.v 4 13.b even 2 1
1521.4.a.bb 4 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1521))\):

\( T_{2}^{4} - 2T_{2}^{3} - 25T_{2}^{2} + 24T_{2} + 78 \) Copy content Toggle raw display
\( T_{5}^{4} + 6T_{5}^{3} - 203T_{5}^{2} - 156T_{5} + 6084 \) Copy content Toggle raw display
\( T_{7}^{4} - 14T_{7}^{3} - 1123T_{7}^{2} + 12652T_{7} + 206212 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 2 T^{3} - 25 T^{2} + 24 T + 78 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 6 T^{3} - 203 T^{2} + \cdots + 6084 \) Copy content Toggle raw display
$7$ \( T^{4} - 14 T^{3} - 1123 T^{2} + \cdots + 206212 \) Copy content Toggle raw display
$11$ \( T^{4} - 40 T^{3} - 1256 T^{2} + \cdots + 27408 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} + 98 T^{3} - 7255 T^{2} + \cdots + 22571952 \) Copy content Toggle raw display
$19$ \( T^{4} + 124 T^{3} - 5932 T^{2} + \cdots + 1720384 \) Copy content Toggle raw display
$23$ \( T^{4} + 104 T^{3} - 6824 T^{2} + \cdots - 2571504 \) Copy content Toggle raw display
$29$ \( T^{4} + 194 T^{3} + \cdots - 274591068 \) Copy content Toggle raw display
$31$ \( T^{4} + 26 T^{3} + \cdots + 328187792 \) Copy content Toggle raw display
$37$ \( T^{4} + 102 T^{3} + \cdots + 27176708 \) Copy content Toggle raw display
$41$ \( T^{4} + 1054 T^{3} + \cdots + 1021233552 \) Copy content Toggle raw display
$43$ \( T^{4} - 450 T^{3} + \cdots - 2362804828 \) Copy content Toggle raw display
$47$ \( T^{4} + 96 T^{3} + \cdots + 42871452048 \) Copy content Toggle raw display
$53$ \( T^{4} + 262 T^{3} + \cdots + 744728256 \) Copy content Toggle raw display
$59$ \( T^{4} - 308 T^{3} + \cdots - 2116598016 \) Copy content Toggle raw display
$61$ \( T^{4} + 928 T^{3} + \cdots - 5230543711 \) Copy content Toggle raw display
$67$ \( T^{4} - 1134 T^{3} + \cdots + 10235224388 \) Copy content Toggle raw display
$71$ \( T^{4} - 1064 T^{3} + \cdots - 99058755696 \) Copy content Toggle raw display
$73$ \( T^{4} + 952 T^{3} + \cdots - 120133390247 \) Copy content Toggle raw display
$79$ \( T^{4} + 746 T^{3} + \cdots + 680937616 \) Copy content Toggle raw display
$83$ \( T^{4} + 404 T^{3} + \cdots + 58964273856 \) Copy content Toggle raw display
$89$ \( T^{4} - 1620 T^{3} + \cdots - 268616132736 \) Copy content Toggle raw display
$97$ \( T^{4} + 2166 T^{3} + \cdots - 445091164 \) Copy content Toggle raw display
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