Properties

Label 1521.4.a.ba
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: 4.4.1520092.1
Defining polynomial: \( x^{4} - x^{3} - 40x^{2} - 9x + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 117)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + 14) q^{4} - \beta_{3} q^{5} + ( - 2 \beta_{2} - 8) q^{7} + (2 \beta_{3} + 9 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} + 14) q^{4} - \beta_{3} q^{5} + ( - 2 \beta_{2} - 8) q^{7} + (2 \beta_{3} + 9 \beta_1) q^{8} + ( - 10 \beta_{2} + 4) q^{10} + ( - \beta_{3} + 2 \beta_1) q^{11} + ( - 4 \beta_{3} - 14 \beta_1) q^{14} + (21 \beta_{2} + 78) q^{16} + 8 \beta_1 q^{17} + (14 \beta_{2} - 28) q^{19} + ( - 12 \beta_{3} - 26 \beta_1) q^{20} + ( - 8 \beta_{2} + 48) q^{22} + (2 \beta_{3} + 32 \beta_1) q^{23} + ( - 12 \beta_{2} + 171) q^{25} + ( - 38 \beta_{2} - 228) q^{28} + ( - 10 \beta_{3} + 20 \beta_1) q^{29} + ( - 2 \beta_{2} + 152) q^{31} + (26 \beta_{3} + 69 \beta_1) q^{32} + (8 \beta_{2} + 176) q^{34} + (4 \beta_{3} + 52 \beta_1) q^{35} + ( - 32 \beta_{2} - 30) q^{37} + (28 \beta_{3} + 14 \beta_1) q^{38} + ( - 66 \beta_{2} - 556) q^{40} + (9 \beta_{3} + 4 \beta_1) q^{41} + (8 \beta_{2} + 216) q^{43} + ( - 8 \beta_{3} + 8 \beta_1) q^{44} + (52 \beta_{2} + 696) q^{46} + ( - 5 \beta_{3} + 54 \beta_1) q^{47} + (36 \beta_{2} - 47) q^{49} + ( - 24 \beta_{3} + 135 \beta_1) q^{50} + ( - 8 \beta_{3} + 52 \beta_1) q^{53} + ( - 32 \beta_{2} + 304) q^{55} + ( - 44 \beta_{3} - 230 \beta_1) q^{56} + ( - 80 \beta_{2} + 480) q^{58} + ( - 15 \beta_{3} + 46 \beta_1) q^{59} + (44 \beta_{2} + 142) q^{61} + ( - 4 \beta_{3} + 146 \beta_1) q^{62} + (161 \beta_{2} + 790) q^{64} + (34 \beta_{2} - 780) q^{67} + (16 \beta_{3} + 136 \beta_1) q^{68} + (92 \beta_{2} + 1128) q^{70} + ( - 29 \beta_{3} - 98 \beta_1) q^{71} + (136 \beta_{2} + 10) q^{73} + ( - 64 \beta_{3} - 126 \beta_1) q^{74} + (182 \beta_{2} + 420) q^{76} + ( - 4 \beta_{3} + 24 \beta_1) q^{77} + (112 \beta_{2} - 236) q^{79} + ( - 36 \beta_{3} - 546 \beta_1) q^{80} + (94 \beta_{2} + 52) q^{82} + ( - 5 \beta_{3} + 2 \beta_1) q^{83} + ( - 80 \beta_{2} + 32) q^{85} + (16 \beta_{3} + 240 \beta_1) q^{86} + ( - 8 \beta_{2} - 176) q^{88} + (7 \beta_{3} + 204 \beta_1) q^{89} + (88 \beta_{3} + 596 \beta_1) q^{92} + (4 \beta_{2} + 1208) q^{94} + (56 \beta_{3} - 364 \beta_1) q^{95} + (104 \beta_{2} + 34) q^{97} + (72 \beta_{3} + 61 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 58 q^{4} - 36 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 58 q^{4} - 36 q^{7} - 4 q^{10} + 354 q^{16} - 84 q^{19} + 176 q^{22} + 660 q^{25} - 988 q^{28} + 604 q^{31} + 720 q^{34} - 184 q^{37} - 2356 q^{40} + 880 q^{43} + 2888 q^{46} - 116 q^{49} + 1152 q^{55} + 1760 q^{58} + 656 q^{61} + 3482 q^{64} - 3052 q^{67} + 4696 q^{70} + 312 q^{73} + 2044 q^{76} - 720 q^{79} + 396 q^{82} - 32 q^{85} - 720 q^{88} + 4840 q^{94} + 344 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 40x^{2} - 9x + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} - \nu^{2} - 31\nu - 9 ) / 9 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + \nu^{2} + 49\nu + 9 ) / 9 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 10\nu^{2} + 22\nu - 171 ) / 9 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{3} + \beta_{2} + 3\beta _1 + 40 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 16\beta_{2} + 26\beta _1 + 29 \) 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
1.32145
−5.49403
−1.63814
6.81072
−5.48928 0 22.1322 14.0859 0 −24.2643 −77.5754 0 −77.3217
1.2 −3.85588 0 6.86783 −19.5342 0 6.26434 4.36551 0 75.3217
1.3 3.85588 0 6.86783 19.5342 0 6.26434 −4.36551 0 75.3217
1.4 5.48928 0 22.1322 −14.0859 0 −24.2643 77.5754 0 −77.3217
\(n\): 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
3.b odd 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.ba 4
3.b odd 2 1 inner 1521.4.a.ba 4
13.b even 2 1 117.4.a.g 4
39.d odd 2 1 117.4.a.g 4
52.b odd 2 1 1872.4.a.bo 4
156.h even 2 1 1872.4.a.bo 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
117.4.a.g 4 13.b even 2 1
117.4.a.g 4 39.d odd 2 1
1521.4.a.ba 4 1.a even 1 1 trivial
1521.4.a.ba 4 3.b odd 2 1 inner
1872.4.a.bo 4 52.b odd 2 1
1872.4.a.bo 4 156.h even 2 1

Hecke kernels

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

\( T_{2}^{4} - 45T_{2}^{2} + 448 \) Copy content Toggle raw display
\( T_{5}^{4} - 580T_{5}^{2} + 75712 \) Copy content Toggle raw display
\( T_{7}^{2} + 18T_{7} - 152 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 45T^{2} + 448 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 580 T^{2} + 75712 \) Copy content Toggle raw display
$7$ \( (T^{2} + 18 T - 152)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} - 752T^{2} + 7168 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} - 2880 T^{2} + \cdots + 1835008 \) Copy content Toggle raw display
$19$ \( (T^{2} + 42 T - 10976)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 48656 T^{2} + \cdots + 295386112 \) Copy content Toggle raw display
$29$ \( T^{4} - 75200 T^{2} + \cdots + 71680000 \) Copy content Toggle raw display
$31$ \( (T^{2} - 302 T + 22568)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 92 T - 57532)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} - 47844 T^{2} + \cdots + 569017792 \) Copy content Toggle raw display
$43$ \( (T^{2} - 440 T + 44672)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 144640 T^{2} + \cdots + 4778706688 \) Copy content Toggle raw display
$53$ \( T^{4} - 157136 T^{2} + \cdots + 3798925312 \) Copy content Toggle raw display
$59$ \( T^{4} - 222960 T^{2} + \cdots + 375897088 \) Copy content Toggle raw display
$61$ \( (T^{2} - 328 T - 85876)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 1526 T + 514832)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} - 931328 T^{2} + \cdots + 31867295488 \) Copy content Toggle raw display
$73$ \( (T^{2} - 156 T - 1071308)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 360 T - 698288)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} - 14640 T^{2} + \cdots + 39251968 \) Copy content Toggle raw display
$89$ \( T^{4} - 1906852 T^{2} + \cdots + 626946109888 \) Copy content Toggle raw display
$97$ \( (T^{2} - 172 T - 622636)^{2} \) Copy content Toggle raw display
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