Properties

Label 1521.4.a.t
Level $1521$
Weight $4$
Character orbit 1521.a
Self dual yes
Analytic conductor $89.742$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 13)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta + 3) q^{2} + ( - 5 \beta + 5) q^{4} + (5 \beta + 5) q^{5} + ( - \beta + 8) q^{7} + ( - 7 \beta + 11) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta + 3) q^{2} + ( - 5 \beta + 5) q^{4} + (5 \beta + 5) q^{5} + ( - \beta + 8) q^{7} + ( - 7 \beta + 11) q^{8} + (5 \beta - 5) q^{10} + (15 \beta - 16) q^{11} + ( - 10 \beta + 28) q^{14} + (15 \beta + 21) q^{16} + ( - 16 \beta - 27) q^{17} + ( - 5 \beta - 68) q^{19} + ( - 25 \beta - 75) q^{20} + (46 \beta - 108) q^{22} + ( - 33 \beta - 56) q^{23} + 75 \beta q^{25} + ( - 40 \beta + 60) q^{28} + (60 \beta - 47) q^{29} + ( - 100 \beta - 20) q^{31} + (65 \beta - 85) q^{32} + ( - 5 \beta - 17) q^{34} + (30 \beta + 20) q^{35} + (44 \beta - 117) q^{37} + (58 \beta - 184) q^{38} + ( - 15 \beta - 85) q^{40} + (20 \beta - 279) q^{41} + ( - 97 \beta + 276) q^{43} + (80 \beta - 380) q^{44} + ( - 10 \beta - 36) q^{46} + ( - 140 \beta + 40) q^{47} + ( - 15 \beta - 275) q^{49} + (150 \beta - 300) q^{50} + (165 \beta - 355) q^{53} + (70 \beta + 220) q^{55} + ( - 60 \beta + 116) q^{56} + (167 \beta - 381) q^{58} + ( - 55 \beta + 432) q^{59} + (200 \beta + 151) q^{61} + ( - 180 \beta + 340) q^{62} + (95 \beta - 683) q^{64} + ( - 91 \beta - 192) q^{67} + (135 \beta + 185) q^{68} + (40 \beta - 60) q^{70} + (105 \beta - 116) q^{71} + ( - 85 \beta + 335) q^{73} + (205 \beta - 527) q^{74} + (340 \beta - 240) q^{76} + (121 \beta - 188) q^{77} + (40 \beta + 100) q^{79} + (255 \beta + 405) q^{80} + (319 \beta - 917) q^{82} + ( - 100 \beta - 80) q^{83} + ( - 295 \beta - 455) q^{85} + ( - 470 \beta + 1216) q^{86} + (172 \beta - 596) q^{88} + ( - 125 \beta - 398) q^{89} + (280 \beta + 380) q^{92} + ( - 320 \beta + 680) q^{94} + ( - 390 \beta - 440) q^{95} + (469 \beta - 442) q^{97} + (245 \beta - 765) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 5 q^{2} + 5 q^{4} + 15 q^{5} + 15 q^{7} + 15 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 5 q^{2} + 5 q^{4} + 15 q^{5} + 15 q^{7} + 15 q^{8} - 5 q^{10} - 17 q^{11} + 46 q^{14} + 57 q^{16} - 70 q^{17} - 141 q^{19} - 175 q^{20} - 170 q^{22} - 145 q^{23} + 75 q^{25} + 80 q^{28} - 34 q^{29} - 140 q^{31} - 105 q^{32} - 39 q^{34} + 70 q^{35} - 190 q^{37} - 310 q^{38} - 185 q^{40} - 538 q^{41} + 455 q^{43} - 680 q^{44} - 82 q^{46} - 60 q^{47} - 565 q^{49} - 450 q^{50} - 545 q^{53} + 510 q^{55} + 172 q^{56} - 595 q^{58} + 809 q^{59} + 502 q^{61} + 500 q^{62} - 1271 q^{64} - 475 q^{67} + 505 q^{68} - 80 q^{70} - 127 q^{71} + 585 q^{73} - 849 q^{74} - 140 q^{76} - 255 q^{77} + 240 q^{79} + 1065 q^{80} - 1515 q^{82} - 260 q^{83} - 1205 q^{85} + 1962 q^{86} - 1020 q^{88} - 921 q^{89} + 1040 q^{92} + 1040 q^{94} - 1270 q^{95} - 415 q^{97} - 1285 q^{98}+O(q^{100}) \) 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
2.56155
−1.56155
0.438447 0 −7.80776 17.8078 0 5.43845 −6.93087 0 7.80776
1.2 4.56155 0 12.8078 −2.80776 0 9.56155 21.9309 0 −12.8078
\(n\): 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.t 2
3.b odd 2 1 169.4.a.f 2
13.b even 2 1 1521.4.a.l 2
13.c even 3 2 117.4.g.d 4
39.d odd 2 1 169.4.a.j 2
39.f even 4 2 169.4.b.e 4
39.h odd 6 2 169.4.c.f 4
39.i odd 6 2 13.4.c.b 4
39.k even 12 4 169.4.e.g 8
156.p even 6 2 208.4.i.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.c.b 4 39.i odd 6 2
117.4.g.d 4 13.c even 3 2
169.4.a.f 2 3.b odd 2 1
169.4.a.j 2 39.d odd 2 1
169.4.b.e 4 39.f even 4 2
169.4.c.f 4 39.h odd 6 2
169.4.e.g 8 39.k even 12 4
208.4.i.e 4 156.p even 6 2
1521.4.a.l 2 13.b even 2 1
1521.4.a.t 2 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1521))\):

\( T_{2}^{2} - 5T_{2} + 2 \) Copy content Toggle raw display
\( T_{5}^{2} - 15T_{5} - 50 \) Copy content Toggle raw display
\( T_{7}^{2} - 15T_{7} + 52 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 5T + 2 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 15T - 50 \) Copy content Toggle raw display
$7$ \( T^{2} - 15T + 52 \) Copy content Toggle raw display
$11$ \( T^{2} + 17T - 884 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 70T + 137 \) Copy content Toggle raw display
$19$ \( T^{2} + 141T + 4864 \) Copy content Toggle raw display
$23$ \( T^{2} + 145T + 628 \) Copy content Toggle raw display
$29$ \( T^{2} + 34T - 15011 \) Copy content Toggle raw display
$31$ \( T^{2} + 140T - 37600 \) Copy content Toggle raw display
$37$ \( T^{2} + 190T + 797 \) Copy content Toggle raw display
$41$ \( T^{2} + 538T + 70661 \) Copy content Toggle raw display
$43$ \( T^{2} - 455T + 11768 \) Copy content Toggle raw display
$47$ \( T^{2} + 60T - 82400 \) Copy content Toggle raw display
$53$ \( T^{2} + 545T - 41450 \) Copy content Toggle raw display
$59$ \( T^{2} - 809T + 150764 \) Copy content Toggle raw display
$61$ \( T^{2} - 502T - 106999 \) Copy content Toggle raw display
$67$ \( T^{2} + 475T + 21212 \) Copy content Toggle raw display
$71$ \( T^{2} + 127T - 42824 \) Copy content Toggle raw display
$73$ \( T^{2} - 585T + 54850 \) Copy content Toggle raw display
$79$ \( T^{2} - 240T + 7600 \) Copy content Toggle raw display
$83$ \( T^{2} + 260T - 25600 \) Copy content Toggle raw display
$89$ \( T^{2} + 921T + 145654 \) Copy content Toggle raw display
$97$ \( T^{2} + 415T - 891778 \) Copy content Toggle raw display
show more
show less