Properties

Label 1521.2.a.o
Level $1521$
Weight $2$
Character orbit 1521.a
Self dual yes
Analytic conductor $12.145$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1521 = 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1521.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(12.1452461474\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
Defining polynomial: \( x^{3} - x^{2} - 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 169)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} - 1) q^{2} + (\beta_{2} + \beta_1) q^{4} + ( - \beta_{2} + \beta_1 - 2) q^{5} + ( - \beta_{2} - \beta_1 + 1) q^{7} + (\beta_{2} - 2 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} - 1) q^{2} + (\beta_{2} + \beta_1) q^{4} + ( - \beta_{2} + \beta_1 - 2) q^{5} + ( - \beta_{2} - \beta_1 + 1) q^{7} + (\beta_{2} - 2 \beta_1) q^{8} + (\beta_{2} + 2) q^{10} + (\beta_{2} - \beta_1 - 2) q^{11} + (2 \beta_1 + 1) q^{14} + ( - \beta_1 + 1) q^{16} + (2 \beta_{2} + \beta_1 + 1) q^{17} + (\beta_{2} + 2 \beta_1 + 1) q^{19} + ( - 3 \beta_1 + 1) q^{20} + (3 \beta_{2} + 2) q^{22} + (2 \beta_{2} - 2 \beta_1 + 3) q^{23} + (2 \beta_{2} - 3 \beta_1) q^{25} + ( - \beta_{2} - 5) q^{28} + (3 \beta_{2} - 5 \beta_1 + 3) q^{29} + ( - 3 \beta_{2} + 5 \beta_1 - 1) q^{31} + ( - 2 \beta_{2} + 5 \beta_1) q^{32} + ( - 2 \beta_{2} - 3 \beta_1 - 4) q^{34} + ( - \beta_{2} + 4 \beta_1 - 3) q^{35} + (\beta_{2} - 2 \beta_1 - 3) q^{37} + ( - 3 \beta_{2} - 3 \beta_1 - 4) q^{38} + (3 \beta_1 - 2) q^{40} + (6 \beta_{2} - 4 \beta_1 + 1) q^{41} + (3 \beta_{2} - 2 \beta_1 + 6) q^{43} + ( - 4 \beta_{2} - \beta_1 - 1) q^{44} + ( - \beta_{2} - 3) q^{46} + (\beta_{2} + \beta_1 - 6) q^{47} + ( - \beta_1 - 1) q^{49} + (3 \beta_{2} + \beta_1 + 1) q^{50} + ( - 7 \beta_{2} + 4 \beta_1 - 4) q^{53} + (2 \beta_{2} - \beta_1 + 3) q^{55} + (5 \beta_{2} - 3 \beta_1 + 4) q^{56} + (2 \beta_{2} + 2 \beta_1 - 1) q^{58} + ( - 4 \beta_1 - 5) q^{59} + ( - \beta_{2} + 6 \beta_1 - 1) q^{61} + ( - 4 \beta_{2} - 2 \beta_1 - 1) q^{62} + ( - 5 \beta_{2} - \beta_1 - 5) q^{64} + (5 \beta_{2} - 6 \beta_1 + 4) q^{67} + (3 \beta_{2} + 3 \beta_1 + 7) q^{68} + ( - \beta_{2} - 3 \beta_1) q^{70} + (3 \beta_1 - 10) q^{71} + (3 \beta_{2} + 6 \beta_1 - 4) q^{73} + (5 \beta_{2} + \beta_1 + 4) q^{74} + (5 \beta_{2} + 2 \beta_1 + 8) q^{76} + (5 \beta_{2} - 1) q^{77} + ( - 8 \beta_{2} - \beta_1 - 4) q^{79} + ( - \beta_{2} + 3 \beta_1 - 3) q^{80} + (3 \beta_{2} - 2 \beta_1 - 3) q^{82} + ( - 2 \beta_{2} + 9 \beta_1 - 6) q^{83} + ( - \beta_{2} - 3 \beta_1 - 1) q^{85} + ( - 4 \beta_{2} - \beta_1 - 7) q^{86} + ( - 4 \beta_{2} + 5 \beta_1 + 2) q^{88} + (7 \beta_1 - 6) q^{89} + ( - \beta_{2} + 5 \beta_1 - 2) q^{92} + (5 \beta_{2} - 2 \beta_1 + 4) q^{94} + ( - \beta_{2} - 4 \beta_1) q^{95} + ( - 7 \beta_{2} + \beta_1 - 5) q^{97} + (2 \beta_{2} + \beta_1 + 2) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{2} - 4 q^{5} + 3 q^{7} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 2 q^{2} - 4 q^{5} + 3 q^{7} - 3 q^{8} + 5 q^{10} - 8 q^{11} + 5 q^{14} + 2 q^{16} + 2 q^{17} + 4 q^{19} + 3 q^{22} + 5 q^{23} - 5 q^{25} - 14 q^{28} + q^{29} + 5 q^{31} + 7 q^{32} - 13 q^{34} - 4 q^{35} - 12 q^{37} - 12 q^{38} - 3 q^{40} - 7 q^{41} + 13 q^{43} - 8 q^{46} - 18 q^{47} - 4 q^{49} + q^{50} - q^{53} + 6 q^{55} + 4 q^{56} - 3 q^{58} - 19 q^{59} + 4 q^{61} - q^{62} - 11 q^{64} + q^{67} + 21 q^{68} - 2 q^{70} - 27 q^{71} - 9 q^{73} + 8 q^{74} + 21 q^{76} - 8 q^{77} - 5 q^{79} - 5 q^{80} - 14 q^{82} - 7 q^{83} - 5 q^{85} - 18 q^{86} + 15 q^{88} - 11 q^{89} + 5 q^{94} - 3 q^{95} - 7 q^{97} + 5 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of \(\nu = \zeta_{14} + \zeta_{14}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) 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
1.80194
−1.24698
0.445042
−2.24698 0 3.04892 −1.44504 0 −2.04892 −2.35690 0 3.24698
1.2 −0.554958 0 −1.69202 −2.80194 0 2.69202 2.04892 0 1.55496
1.3 0.801938 0 −1.35690 0.246980 0 2.35690 −2.69202 0 0.198062
\(n\): 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 1521.2.a.o 3
3.b odd 2 1 169.2.a.c yes 3
12.b even 2 1 2704.2.a.ba 3
13.b even 2 1 1521.2.a.r 3
13.d odd 4 2 1521.2.b.l 6
15.d odd 2 1 4225.2.a.bb 3
21.c even 2 1 8281.2.a.bj 3
39.d odd 2 1 169.2.a.b 3
39.f even 4 2 169.2.b.b 6
39.h odd 6 2 169.2.c.c 6
39.i odd 6 2 169.2.c.b 6
39.k even 12 4 169.2.e.b 12
156.h even 2 1 2704.2.a.z 3
156.l odd 4 2 2704.2.f.o 6
195.e odd 2 1 4225.2.a.bg 3
273.g even 2 1 8281.2.a.bf 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
169.2.a.b 3 39.d odd 2 1
169.2.a.c yes 3 3.b odd 2 1
169.2.b.b 6 39.f even 4 2
169.2.c.b 6 39.i odd 6 2
169.2.c.c 6 39.h odd 6 2
169.2.e.b 12 39.k even 12 4
1521.2.a.o 3 1.a even 1 1 trivial
1521.2.a.r 3 13.b even 2 1
1521.2.b.l 6 13.d odd 4 2
2704.2.a.z 3 156.h even 2 1
2704.2.a.ba 3 12.b even 2 1
2704.2.f.o 6 156.l odd 4 2
4225.2.a.bb 3 15.d odd 2 1
4225.2.a.bg 3 195.e odd 2 1
8281.2.a.bf 3 273.g even 2 1
8281.2.a.bj 3 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1521))\):

\( T_{2}^{3} + 2T_{2}^{2} - T_{2} - 1 \) Copy content Toggle raw display
\( T_{5}^{3} + 4T_{5}^{2} + 3T_{5} - 1 \) Copy content Toggle raw display
\( T_{7}^{3} - 3T_{7}^{2} - 4T_{7} + 13 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} + 2T^{2} - T - 1 \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( T^{3} + 4 T^{2} + 3 T - 1 \) Copy content Toggle raw display
$7$ \( T^{3} - 3 T^{2} - 4 T + 13 \) Copy content Toggle raw display
$11$ \( T^{3} + 8 T^{2} + 19 T + 13 \) Copy content Toggle raw display
$13$ \( T^{3} \) Copy content Toggle raw display
$17$ \( T^{3} - 2 T^{2} - 15 T - 13 \) Copy content Toggle raw display
$19$ \( T^{3} - 4 T^{2} - 11 T + 1 \) Copy content Toggle raw display
$23$ \( T^{3} - 5T^{2} - T + 13 \) Copy content Toggle raw display
$29$ \( T^{3} - T^{2} - 44 T - 83 \) Copy content Toggle raw display
$31$ \( T^{3} - 5 T^{2} - 36 T + 167 \) Copy content Toggle raw display
$37$ \( T^{3} + 12 T^{2} + 41 T + 29 \) Copy content Toggle raw display
$41$ \( T^{3} + 7 T^{2} - 49 T + 49 \) Copy content Toggle raw display
$43$ \( T^{3} - 13 T^{2} + 40 T + 13 \) Copy content Toggle raw display
$47$ \( T^{3} + 18 T^{2} + 101 T + 167 \) Copy content Toggle raw display
$53$ \( T^{3} + T^{2} - 86 T - 337 \) Copy content Toggle raw display
$59$ \( T^{3} + 19 T^{2} + 83 T + 1 \) Copy content Toggle raw display
$61$ \( T^{3} - 4 T^{2} - 67 T + 239 \) Copy content Toggle raw display
$67$ \( T^{3} - T^{2} - 72 T - 41 \) Copy content Toggle raw display
$71$ \( T^{3} + 27 T^{2} + 222 T + 547 \) Copy content Toggle raw display
$73$ \( T^{3} + 9 T^{2} - 120 T - 911 \) Copy content Toggle raw display
$79$ \( T^{3} + 5 T^{2} - 162 T + 127 \) Copy content Toggle raw display
$83$ \( T^{3} + 7 T^{2} - 140 T + 203 \) Copy content Toggle raw display
$89$ \( T^{3} + 11 T^{2} - 74 T - 281 \) Copy content Toggle raw display
$97$ \( T^{3} + 7 T^{2} - 84 T - 301 \) Copy content Toggle raw display
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