Properties

Label 1521.4.a.z
Level $1521$
Weight $4$
Character orbit 1521.a
Self dual yes
Analytic conductor $89.742$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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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: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{3}, \sqrt{17})\)
Defining polynomial: \( x^{4} - 2x^{3} - 13x^{2} + 14x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + 9 q^{4} + ( - 2 \beta_{3} + 3 \beta_1) q^{5} + ( - 3 \beta_{3} + 11 \beta_1) q^{7} + \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} + 9 q^{4} + ( - 2 \beta_{3} + 3 \beta_1) q^{5} + ( - 3 \beta_{3} + 11 \beta_1) q^{7} + \beta_{3} q^{8} + (3 \beta_{2} - 34) q^{10} + ( - \beta_{3} + 21 \beta_1) q^{11} + (11 \beta_{2} - 51) q^{14} - 55 q^{16} + ( - \beta_{2} + 36) q^{17} + (9 \beta_{3} + 37 \beta_1) q^{19} + ( - 18 \beta_{3} + 27 \beta_1) q^{20} + (21 \beta_{2} - 17) q^{22} + (7 \beta_{2} + 69) q^{23} + ( - 12 \beta_{2} - 30) q^{25} + ( - 27 \beta_{3} + 99 \beta_1) q^{28} + ( - 22 \beta_{2} - 3) q^{29} + (42 \beta_{3} - 78 \beta_1) q^{31} - 63 \beta_{3} q^{32} + (36 \beta_{3} - 17 \beta_1) q^{34} + ( - 31 \beta_{2} + 201) q^{35} + (45 \beta_{3} + 82 \beta_1) q^{37} + (37 \beta_{2} + 153) q^{38} + (3 \beta_{2} - 34) q^{40} + (\beta_{3} + 30 \beta_1) q^{41} + ( - 15 \beta_{2} + 235) q^{43} + ( - 9 \beta_{3} + 189 \beta_1) q^{44} + (69 \beta_{3} + 119 \beta_1) q^{46} + ( - 79 \beta_{3} - 111 \beta_1) q^{47} + ( - 66 \beta_{2} + 173) q^{49} + ( - 30 \beta_{3} - 204 \beta_1) q^{50} + (18 \beta_{2} + 567) q^{53} + ( - 45 \beta_{2} + 223) q^{55} + (11 \beta_{2} - 51) q^{56} + ( - 3 \beta_{3} - 374 \beta_1) q^{58} + (8 \beta_{3} + 360 \beta_1) q^{59} + (87 \beta_{2} + 80) q^{61} + ( - 78 \beta_{2} + 714) q^{62} - 631 q^{64} + ( - 21 \beta_{3} + 83 \beta_1) q^{67} + ( - 9 \beta_{2} + 324) q^{68} + (201 \beta_{3} - 527 \beta_1) q^{70} + ( - 17 \beta_{3} + 219 \beta_1) q^{71} + 225 \beta_1 q^{73} + (82 \beta_{2} + 765) q^{74} + (81 \beta_{3} + 333 \beta_1) q^{76} + ( - 74 \beta_{2} + 744) q^{77} + (126 \beta_{2} + 2) q^{79} + (110 \beta_{3} - 165 \beta_1) q^{80} + (30 \beta_{2} + 17) q^{82} + (241 \beta_{3} - 177 \beta_1) q^{83} + ( - 81 \beta_{3} + 142 \beta_1) q^{85} + (235 \beta_{3} - 255 \beta_1) q^{86} + (21 \beta_{2} - 17) q^{88} + (242 \beta_{3} + 42 \beta_1) q^{89} + (63 \beta_{2} + 621) q^{92} + ( - 111 \beta_{2} - 1343) q^{94} + ( - 47 \beta_{2} + 27) q^{95} + ( - 402 \beta_{3} - 56 \beta_1) q^{97} + (173 \beta_{3} - 1122 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 36 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 36 q^{4} - 136 q^{10} - 204 q^{14} - 220 q^{16} + 144 q^{17} - 68 q^{22} + 276 q^{23} - 120 q^{25} - 12 q^{29} + 804 q^{35} + 612 q^{38} - 136 q^{40} + 940 q^{43} + 692 q^{49} + 2268 q^{53} + 892 q^{55} - 204 q^{56} + 320 q^{61} + 2856 q^{62} - 2524 q^{64} + 1296 q^{68} + 3060 q^{74} + 2976 q^{77} + 8 q^{79} + 68 q^{82} - 68 q^{88} + 2484 q^{92} - 5372 q^{94} + 108 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 13x^{2} + 14x - 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{3} - 3\nu^{2} - 25\nu + 13 ) / 5 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 7 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -4\nu^{3} + 6\nu^{2} + 60\nu - 31 ) / 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + 2\beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 2\beta_{2} + 2\beta _1 + 15 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 14\beta_{3} + 3\beta_{2} + 33\beta _1 + 22 ) / 2 \) 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
−3.29360
0.170498
0.829502
4.29360
−4.12311 0 9.00000 3.05006 0 −6.68324 −4.12311 0 −12.5757
1.2 −4.12311 0 9.00000 13.4424 0 31.4219 −4.12311 0 −55.4243
1.3 4.12311 0 9.00000 −13.4424 0 −31.4219 4.12311 0 −55.4243
1.4 4.12311 0 9.00000 −3.05006 0 6.68324 4.12311 0 −12.5757
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1521.4.a.z 4
3.b odd 2 1 507.4.a.k 4
13.b even 2 1 inner 1521.4.a.z 4
13.f odd 12 2 117.4.q.d 4
39.d odd 2 1 507.4.a.k 4
39.f even 4 2 507.4.b.e 4
39.k even 12 2 39.4.j.b 4
156.v odd 12 2 624.4.bv.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.j.b 4 39.k even 12 2
117.4.q.d 4 13.f odd 12 2
507.4.a.k 4 3.b odd 2 1
507.4.a.k 4 39.d odd 2 1
507.4.b.e 4 39.f even 4 2
624.4.bv.c 4 156.v odd 12 2
1521.4.a.z 4 1.a even 1 1 trivial
1521.4.a.z 4 13.b even 2 1 inner

Hecke kernels

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

\( T_{2}^{2} - 17 \) Copy content Toggle raw display
\( T_{5}^{4} - 190T_{5}^{2} + 1681 \) Copy content Toggle raw display
\( T_{7}^{4} - 1032T_{7}^{2} + 44100 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 17)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 190T^{2} + 1681 \) Copy content Toggle raw display
$7$ \( T^{4} - 1032 T^{2} + 44100 \) Copy content Toggle raw display
$11$ \( T^{4} - 2680 T^{2} + \cdots + 1705636 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} - 72 T + 1245)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} - 10968 T^{2} + \cdots + 7452900 \) Copy content Toggle raw display
$23$ \( (T^{2} - 138 T + 2262)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 6 T - 24675)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} - 96480 T^{2} + \cdots + 137733696 \) Copy content Toggle raw display
$37$ \( T^{4} - 109194 T^{2} + \cdots + 203148009 \) Copy content Toggle raw display
$41$ \( T^{4} - 5434 T^{2} + \cdots + 7198489 \) Copy content Toggle raw display
$43$ \( (T^{2} - 470 T + 43750)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 286120 T^{2} + \cdots + 4779509956 \) Copy content Toggle raw display
$53$ \( (T^{2} - 1134 T + 304965)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 779776 T^{2} + \cdots + 150320594944 \) Copy content Toggle raw display
$61$ \( (T^{2} - 160 T - 379619)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 56328 T^{2} + \cdots + 173448900 \) Copy content Toggle raw display
$71$ \( T^{4} - 297592 T^{2} + \cdots + 19312660900 \) Copy content Toggle raw display
$73$ \( (T^{2} - 151875)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 4 T - 809672)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} - 2162728 T^{2} + \cdots + 798145692100 \) Copy content Toggle raw display
$89$ \( T^{4} - 2001760 T^{2} + \cdots + 980686167616 \) Copy content Toggle raw display
$97$ \( T^{4} - 5513352 T^{2} + \cdots + 7495877379600 \) Copy content Toggle raw display
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