Properties

Label 1521.4.a.bc
Level $1521$
Weight $4$
Character orbit 1521.a
Self dual yes
Analytic conductor $89.742$
Analytic rank $0$
Dimension $8$
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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \( x^{8} - 52x^{6} + 805x^{4} - 4210x^{2} + 4992 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + 5) q^{4} + \beta_{3} q^{5} + (\beta_{5} - 3) q^{7} + (\beta_{4} + \beta_{3} + 5 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} + 5) q^{4} + \beta_{3} q^{5} + (\beta_{5} - 3) q^{7} + (\beta_{4} + \beta_{3} + 5 \beta_1) q^{8} + (\beta_{6} + \beta_{5} + 3 \beta_{2} + 4) q^{10} + (\beta_{7} + \beta_{3} + \beta_1) q^{11} + (\beta_{7} - \beta_{4} + 2 \beta_{3} - 7 \beta_1) q^{14} + (2 \beta_{6} + 9 \beta_{2} + 25) q^{16} + (\beta_{7} - 2 \beta_{4} - 13 \beta_1) q^{17} + ( - \beta_{6} - \beta_{5} + 6 \beta_{2} + 31) q^{19} + (\beta_{7} + 3 \beta_{4} + 6 \beta_{3} + 18 \beta_1) q^{20} + (\beta_{6} + 7 \beta_{5} + 8 \beta_{2} + 15) q^{22} + ( - \beta_{7} - 2 \beta_{4} + 5 \beta_{3} - 11 \beta_1) q^{23} + ( - \beta_{6} - 6 \beta_{5} + 46) q^{25} + (\beta_{6} + \beta_{5} - 6 \beta_{2} - 57) q^{28} + (6 \beta_{4} - \beta_{3} + 10 \beta_1) q^{29} + (3 \beta_{6} + 6 \beta_{5} + 8 \beta_{2} + 28) q^{31} + (3 \beta_{4} + 19 \beta_{3} + 45 \beta_1) q^{32} + ( - 2 \beta_{6} + 8 \beta_{5} - 27 \beta_{2} - 163) q^{34} + ( - \beta_{7} - 17 \beta_{3} + 27 \beta_1) q^{35} + ( - 2 \beta_{6} + 7 \beta_{5} + 12 \beta_{2} + 126) q^{37} + ( - \beta_{7} + 6 \beta_{4} - 5 \beta_{3} + 89 \beta_1) q^{38} + (\beta_{6} + \beta_{5} + 43 \beta_{2} + 212) q^{40} + (\beta_{7} - 4 \beta_{4} + 6 \beta_{3} + 49 \beta_1) q^{41} + ( - 7 \beta_{5} - 26 \beta_{2} + 11) q^{43} + ( - \beta_{7} + 2 \beta_{4} + 23 \beta_{3} + 37 \beta_1) q^{44} + (3 \beta_{6} + \beta_{5} - 18 \beta_{2} - 113) q^{46} + (\beta_{7} + 2 \beta_{4} + 11 \beta_{3} - 33 \beta_1) q^{47} + ( - 7 \beta_{6} + 8 \beta_{5} - 10 \beta_{2} + 37) q^{49} + ( - 6 \beta_{7} + 5 \beta_{4} - 21 \beta_{3} + 76 \beta_1) q^{50} + ( - 6 \beta_{7} + 2 \beta_{4} + 19 \beta_{3} - 60 \beta_1) q^{53} + ( - 11 \beta_{6} - 7 \beta_{5} + 24 \beta_{2} + 167) q^{55} + ( - 7 \beta_{7} + 2 \beta_{4} - 11 \beta_{3} - 59 \beta_1) q^{56} + (5 \beta_{6} - 7 \beta_{5} + 61 \beta_{2} + 102) q^{58} + ( - 6 \beta_{7} - 4 \beta_{4} + 6 \beta_{3} + 2 \beta_1) q^{59} + ( - 3 \beta_{6} - 15 \beta_{5} - 10 \beta_{2} + 148) q^{61} + (6 \beta_{7} + 5 \beta_{4} + 47 \beta_{3} + 50 \beta_1) q^{62} + (6 \beta_{6} + 16 \beta_{5} + 57 \beta_{2} + 449) q^{64} + ( - 8 \beta_{6} + \beta_{5} - 30 \beta_{2} - 273) q^{67} + ( - 21 \beta_{4} - 29 \beta_{3} - 295 \beta_1) q^{68} + ( - 17 \beta_{6} - 23 \beta_{5} - 28 \beta_{2} + 285) q^{70} + (\beta_{7} - 14 \beta_{4} + 11 \beta_{3} + 131 \beta_1) q^{71} + (12 \beta_{6} - 8 \beta_{5} + 6 \beta_{2} - 273) q^{73} + (7 \beta_{7} + 3 \beta_{4} + 8 \beta_{3} + 206 \beta_1) q^{74} + (9 \beta_{6} - 9 \beta_{5} + 76 \beta_{2} + 867) q^{76} + (6 \beta_{7} - 26 \beta_{4} - 28 \beta_{3} + 176 \beta_1) q^{77} + ( - 5 \beta_{6} + 16 \beta_{2} + 234) q^{79} + ( - 7 \beta_{7} + 19 \beta_{4} + 6 \beta_{3} + 402 \beta_1) q^{80} + (2 \beta_{6} + 16 \beta_{5} + 35 \beta_{2} + 675) q^{82} + (7 \beta_{7} - 8 \beta_{4} - 13 \beta_{3} + 51 \beta_1) q^{83} + ( - 18 \beta_{6} - 19 \beta_{5} - 74 \beta_{2} - 102) q^{85} + ( - 7 \beta_{7} - 19 \beta_{4} - 40 \beta_{3} - 169 \beta_1) q^{86} + (17 \beta_{6} - 41 \beta_{5} + 56 \beta_{2} + 447) q^{88} + (8 \beta_{7} - 24 \beta_{4} + 32 \beta_{3} - 44 \beta_1) q^{89} + (9 \beta_{7} - 29 \beta_{3} - 191 \beta_1) q^{92} + (13 \beta_{6} + 15 \beta_{5} + 22 \beta_{2} - 395) q^{94} + (15 \beta_{7} + 22 \beta_{4} + 65 \beta_{3} - 39 \beta_1) q^{95} + ( - \beta_{6} + 48 \beta_{5} + 58 \beta_{2} - 558) q^{97} + (8 \beta_{7} - 25 \beta_{4} - 57 \beta_{3} - 33 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 40 q^{4} - 22 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 40 q^{4} - 22 q^{7} + 36 q^{10} + 204 q^{16} + 244 q^{19} + 136 q^{22} + 354 q^{25} - 452 q^{28} + 242 q^{31} - 1292 q^{34} + 1018 q^{37} + 1700 q^{40} + 74 q^{43} - 896 q^{46} + 298 q^{49} + 1300 q^{55} + 812 q^{58} + 1148 q^{61} + 3636 q^{64} - 2198 q^{67} + 2200 q^{70} - 2176 q^{73} + 6936 q^{76} + 1862 q^{79} + 5436 q^{82} - 890 q^{85} + 3528 q^{88} - 3104 q^{94} - 4370 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 52x^{6} + 805x^{4} - 4210x^{2} + 4992 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 13 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} - 35\nu^{3} + 210\nu ) / 16 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{5} + 51\nu^{3} - 546\nu ) / 16 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{6} - 43\nu^{4} + 426\nu^{2} - 688 ) / 16 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{4} - 33\nu^{2} + 156 ) / 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} - 46\nu^{5} + 547\nu^{3} - 1590\nu ) / 16 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 13 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} + \beta_{3} + 21\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{6} + 33\beta_{2} + 273 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 35\beta_{4} + 51\beta_{3} + 525\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 86\beta_{6} + 16\beta_{5} + 993\beta_{2} + 6889 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 16\beta_{7} + 1063\beta_{4} + 1799\beta_{3} + 14253\beta_1 \) 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
−5.39589
−3.69212
−2.75628
−1.28670
1.28670
2.75628
3.69212
5.39589
−5.39589 0 21.1156 −13.0421 0 −6.42494 −70.7705 0 70.3740
1.2 −3.69212 0 5.63172 18.7574 0 −24.1383 8.74396 0 −69.2546
1.3 −2.75628 0 −0.402937 −0.313209 0 28.5660 23.1608 0 0.863291
1.4 −1.28670 0 −6.34441 −12.4484 0 −9.00273 18.4569 0 16.0173
1.5 1.28670 0 −6.34441 12.4484 0 −9.00273 −18.4569 0 16.0173
1.6 2.75628 0 −0.402937 0.313209 0 28.5660 −23.1608 0 0.863291
1.7 3.69212 0 5.63172 −18.7574 0 −24.1383 −8.74396 0 −69.2546
1.8 5.39589 0 21.1156 13.0421 0 −6.42494 70.7705 0 70.3740
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
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.bc 8
3.b odd 2 1 inner 1521.4.a.bc 8
13.b even 2 1 1521.4.a.bd 8
13.c even 3 2 117.4.g.f 16
39.d odd 2 1 1521.4.a.bd 8
39.i odd 6 2 117.4.g.f 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
117.4.g.f 16 13.c even 3 2
117.4.g.f 16 39.i odd 6 2
1521.4.a.bc 8 1.a even 1 1 trivial
1521.4.a.bc 8 3.b odd 2 1 inner
1521.4.a.bd 8 13.b even 2 1
1521.4.a.bd 8 39.d odd 2 1

Hecke kernels

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

\( T_{2}^{8} - 52T_{2}^{6} + 805T_{2}^{4} - 4210T_{2}^{2} + 4992 \) Copy content Toggle raw display
\( T_{5}^{8} - 677T_{5}^{6} + 140795T_{5}^{4} - 9287935T_{5}^{2} + 909792 \) Copy content Toggle raw display
\( T_{7}^{4} + 11T_{7}^{3} - 700T_{7}^{2} - 10894T_{7} - 39884 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 52 T^{6} + 805 T^{4} + \cdots + 4992 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - 677 T^{6} + 140795 T^{4} + \cdots + 909792 \) Copy content Toggle raw display
$7$ \( (T^{4} + 11 T^{3} - 700 T^{2} + \cdots - 39884)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} - 9720 T^{6} + \cdots + 17384221390848 \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( T^{8} - 29425 T^{6} + \cdots + 30103640387808 \) Copy content Toggle raw display
$19$ \( (T^{4} - 122 T^{3} - 8174 T^{2} + \cdots + 14520480)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 203262208261632 \) Copy content Toggle raw display
$29$ \( T^{8} - 139365 T^{6} + \cdots + 24\!\cdots\!48 \) Copy content Toggle raw display
$31$ \( (T^{4} - 121 T^{3} - 54236 T^{2} + \cdots - 349762400)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 509 T^{3} + 30151 T^{2} + \cdots - 1178984222)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} - 225537 T^{6} + \cdots + 14\!\cdots\!92 \) Copy content Toggle raw display
$43$ \( (T^{4} - 37 T^{3} - 146172 T^{2} + \cdots + 1323590892)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 157456 T^{6} + \cdots + 36\!\cdots\!52 \) Copy content Toggle raw display
$53$ \( T^{8} - 733517 T^{6} + \cdots + 44\!\cdots\!28 \) Copy content Toggle raw display
$59$ \( T^{8} - 445056 T^{6} + \cdots + 18\!\cdots\!88 \) Copy content Toggle raw display
$61$ \( (T^{4} - 574 T^{3} - 70694 T^{2} + \cdots + 684529261)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 1099 T^{3} + \cdots + 2700497644)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} - 1791632 T^{6} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{4} + 1088 T^{3} + \cdots + 20997802657)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 931 T^{3} + 176852 T^{2} + \cdots - 9861650240)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} - 870232 T^{6} + \cdots + 23\!\cdots\!32 \) Copy content Toggle raw display
$89$ \( T^{8} - 3056192 T^{6} + \cdots + 14\!\cdots\!68 \) Copy content Toggle raw display
$97$ \( (T^{4} + 2185 T^{3} + \cdots + 291511171744)^{2} \) Copy content Toggle raw display
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