Properties

Label 1521.4.a.o
Level $1521$
Weight $4$
Character orbit 1521.a
Self dual yes
Analytic conductor $89.742$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \( x^{2} - 3 \) 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 \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} - 5 q^{4} + \beta q^{5} + 8 \beta q^{7} - 13 \beta q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} - 5 q^{4} + \beta q^{5} + 8 \beta q^{7} - 13 \beta q^{8} + 3 q^{10} - 8 \beta q^{11} + 24 q^{14} + q^{16} + 117 q^{17} - 66 \beta q^{19} - 5 \beta q^{20} - 24 q^{22} - 78 q^{23} - 122 q^{25} - 40 \beta q^{28} + 141 q^{29} + 90 \beta q^{31} + 105 \beta q^{32} + 117 \beta q^{34} + 24 q^{35} - 83 \beta q^{37} - 198 q^{38} - 39 q^{40} + 157 \beta q^{41} - 104 q^{43} + 40 \beta q^{44} - 78 \beta q^{46} + 174 \beta q^{47} - 151 q^{49} - 122 \beta q^{50} - 93 q^{53} - 24 q^{55} - 312 q^{56} + 141 \beta q^{58} - 164 \beta q^{59} + 145 q^{61} + 270 q^{62} + 307 q^{64} + 454 \beta q^{67} - 585 q^{68} + 24 \beta q^{70} - 610 \beta q^{71} - 265 \beta q^{73} - 249 q^{74} + 330 \beta q^{76} - 192 q^{77} + 1276 q^{79} + \beta q^{80} + 471 q^{82} + 456 \beta q^{83} + 117 \beta q^{85} - 104 \beta q^{86} + 312 q^{88} + 564 \beta q^{89} + 390 q^{92} + 522 q^{94} - 198 q^{95} + 116 \beta q^{97} - 151 \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 10 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 10 q^{4} + 6 q^{10} + 48 q^{14} + 2 q^{16} + 234 q^{17} - 48 q^{22} - 156 q^{23} - 244 q^{25} + 282 q^{29} + 48 q^{35} - 396 q^{38} - 78 q^{40} - 208 q^{43} - 302 q^{49} - 186 q^{53} - 48 q^{55} - 624 q^{56} + 290 q^{61} + 540 q^{62} + 614 q^{64} - 1170 q^{68} - 498 q^{74} - 384 q^{77} + 2552 q^{79} + 942 q^{82} + 624 q^{88} + 780 q^{92} + 1044 q^{94} - 396 q^{95}+O(q^{100}) \) 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
−1.73205
1.73205
−1.73205 0 −5.00000 −1.73205 0 −13.8564 22.5167 0 3.00000
1.2 1.73205 0 −5.00000 1.73205 0 13.8564 −22.5167 0 3.00000
\(n\): 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.o 2
3.b odd 2 1 169.4.a.i 2
13.b even 2 1 inner 1521.4.a.o 2
13.f odd 12 2 117.4.q.a 2
39.d odd 2 1 169.4.a.i 2
39.f even 4 2 169.4.b.d 2
39.h odd 6 2 169.4.c.h 4
39.i odd 6 2 169.4.c.h 4
39.k even 12 2 13.4.e.b 2
39.k even 12 2 169.4.e.a 2
156.v odd 12 2 208.4.w.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.e.b 2 39.k even 12 2
117.4.q.a 2 13.f odd 12 2
169.4.a.i 2 3.b odd 2 1
169.4.a.i 2 39.d odd 2 1
169.4.b.d 2 39.f even 4 2
169.4.c.h 4 39.h odd 6 2
169.4.c.h 4 39.i odd 6 2
169.4.e.a 2 39.k even 12 2
208.4.w.b 2 156.v odd 12 2
1521.4.a.o 2 1.a even 1 1 trivial
1521.4.a.o 2 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1521))\):

\( T_{2}^{2} - 3 \) Copy content Toggle raw display
\( T_{5}^{2} - 3 \) Copy content Toggle raw display
\( T_{7}^{2} - 192 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 3 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 3 \) Copy content Toggle raw display
$7$ \( T^{2} - 192 \) Copy content Toggle raw display
$11$ \( T^{2} - 192 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( (T - 117)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 13068 \) Copy content Toggle raw display
$23$ \( (T + 78)^{2} \) Copy content Toggle raw display
$29$ \( (T - 141)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 24300 \) Copy content Toggle raw display
$37$ \( T^{2} - 20667 \) Copy content Toggle raw display
$41$ \( T^{2} - 73947 \) Copy content Toggle raw display
$43$ \( (T + 104)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 90828 \) Copy content Toggle raw display
$53$ \( (T + 93)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 80688 \) Copy content Toggle raw display
$61$ \( (T - 145)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 618348 \) Copy content Toggle raw display
$71$ \( T^{2} - 1116300 \) Copy content Toggle raw display
$73$ \( T^{2} - 210675 \) Copy content Toggle raw display
$79$ \( (T - 1276)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 623808 \) Copy content Toggle raw display
$89$ \( T^{2} - 954288 \) Copy content Toggle raw display
$97$ \( T^{2} - 40368 \) Copy content Toggle raw display
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