Properties

Label 1521.2.a.m
Level $1521$
Weight $2$
Character orbit 1521.a
Self dual yes
Analytic conductor $12.145$
Analytic rank $0$
Dimension $2$
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: \(0\)
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 39)
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.

\(f(q)\) \(=\) \( q + \beta q^{2} + (\beta + 2) q^{4} + (\beta - 2) q^{5} + (\beta + 1) q^{7} + (\beta + 4) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} + (\beta + 2) q^{4} + (\beta - 2) q^{5} + (\beta + 1) q^{7} + (\beta + 4) q^{8} + ( - \beta + 4) q^{10} - 2 q^{11} + (2 \beta + 4) q^{14} + 3 \beta q^{16} - \beta q^{17} + (2 \beta - 4) q^{19} + \beta q^{20} - 2 \beta q^{22} - 2 q^{23} + ( - 3 \beta + 3) q^{25} + (4 \beta + 6) q^{28} + (3 \beta - 2) q^{29} + (\beta - 1) q^{31} + (\beta + 4) q^{32} + ( - \beta - 4) q^{34} + 2 q^{35} + (\beta - 6) q^{37} + ( - 2 \beta + 8) q^{38} + (3 \beta - 4) q^{40} + \beta q^{41} + ( - \beta + 3) q^{43} + ( - 2 \beta - 4) q^{44} - 2 \beta q^{46} + ( - 4 \beta + 2) q^{47} + (3 \beta - 2) q^{49} - 12 q^{50} + ( - 3 \beta - 4) q^{53} + ( - 2 \beta + 4) q^{55} + (6 \beta + 8) q^{56} + (\beta + 12) q^{58} + ( - 2 \beta - 6) q^{59} + (2 \beta + 7) q^{61} + 4 q^{62} + ( - \beta + 4) q^{64} + (\beta - 3) q^{67} + ( - 3 \beta - 4) q^{68} + 2 \beta q^{70} + 14 q^{71} + ( - 2 \beta + 7) q^{73} + ( - 5 \beta + 4) q^{74} + 2 \beta q^{76} + ( - 2 \beta - 2) q^{77} + (\beta + 7) q^{79} + ( - 3 \beta + 12) q^{80} + (\beta + 4) q^{82} + ( - 2 \beta - 4) q^{83} + (\beta - 4) q^{85} + (2 \beta - 4) q^{86} + ( - 2 \beta - 8) q^{88} + (2 \beta + 8) q^{89} + ( - 2 \beta - 4) q^{92} + ( - 2 \beta - 16) q^{94} + ( - 6 \beta + 16) q^{95} + ( - \beta + 7) q^{97} + (\beta + 12) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + 5 q^{4} - 3 q^{5} + 3 q^{7} + 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} + 5 q^{4} - 3 q^{5} + 3 q^{7} + 9 q^{8} + 7 q^{10} - 4 q^{11} + 10 q^{14} + 3 q^{16} - q^{17} - 6 q^{19} + q^{20} - 2 q^{22} - 4 q^{23} + 3 q^{25} + 16 q^{28} - q^{29} - q^{31} + 9 q^{32} - 9 q^{34} + 4 q^{35} - 11 q^{37} + 14 q^{38} - 5 q^{40} + q^{41} + 5 q^{43} - 10 q^{44} - 2 q^{46} - q^{49} - 24 q^{50} - 11 q^{53} + 6 q^{55} + 22 q^{56} + 25 q^{58} - 14 q^{59} + 16 q^{61} + 8 q^{62} + 7 q^{64} - 5 q^{67} - 11 q^{68} + 2 q^{70} + 28 q^{71} + 12 q^{73} + 3 q^{74} + 2 q^{76} - 6 q^{77} + 15 q^{79} + 21 q^{80} + 9 q^{82} - 10 q^{83} - 7 q^{85} - 6 q^{86} - 18 q^{88} + 18 q^{89} - 10 q^{92} - 34 q^{94} + 26 q^{95} + 13 q^{97} + 25 q^{98}+O(q^{100}) \) 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
−1.56155
2.56155
−1.56155 0 0.438447 −3.56155 0 −0.561553 2.43845 0 5.56155
1.2 2.56155 0 4.56155 0.561553 0 3.56155 6.56155 0 1.43845
\(n\): 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.m 2
3.b odd 2 1 507.2.a.d 2
12.b even 2 1 8112.2.a.bo 2
13.b even 2 1 1521.2.a.g 2
13.d odd 4 2 1521.2.b.h 4
13.e even 6 2 117.2.g.c 4
39.d odd 2 1 507.2.a.g 2
39.f even 4 2 507.2.b.d 4
39.h odd 6 2 39.2.e.b 4
39.i odd 6 2 507.2.e.g 4
39.k even 12 4 507.2.j.g 8
52.i odd 6 2 1872.2.t.r 4
156.h even 2 1 8112.2.a.bk 2
156.r even 6 2 624.2.q.h 4
195.y odd 6 2 975.2.i.k 4
195.bf even 12 4 975.2.bb.i 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.e.b 4 39.h odd 6 2
117.2.g.c 4 13.e even 6 2
507.2.a.d 2 3.b odd 2 1
507.2.a.g 2 39.d odd 2 1
507.2.b.d 4 39.f even 4 2
507.2.e.g 4 39.i odd 6 2
507.2.j.g 8 39.k even 12 4
624.2.q.h 4 156.r even 6 2
975.2.i.k 4 195.y odd 6 2
975.2.bb.i 8 195.bf even 12 4
1521.2.a.g 2 13.b even 2 1
1521.2.a.m 2 1.a even 1 1 trivial
1521.2.b.h 4 13.d odd 4 2
1872.2.t.r 4 52.i odd 6 2
8112.2.a.bk 2 156.h even 2 1
8112.2.a.bo 2 12.b even 2 1

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T - 4 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 3T - 2 \) Copy content Toggle raw display
$7$ \( T^{2} - 3T - 2 \) Copy content Toggle raw display
$11$ \( (T + 2)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + T - 4 \) Copy content Toggle raw display
$19$ \( T^{2} + 6T - 8 \) Copy content Toggle raw display
$23$ \( (T + 2)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + T - 38 \) Copy content Toggle raw display
$31$ \( T^{2} + T - 4 \) Copy content Toggle raw display
$37$ \( T^{2} + 11T + 26 \) Copy content Toggle raw display
$41$ \( T^{2} - T - 4 \) Copy content Toggle raw display
$43$ \( T^{2} - 5T + 2 \) Copy content Toggle raw display
$47$ \( T^{2} - 68 \) Copy content Toggle raw display
$53$ \( T^{2} + 11T - 8 \) Copy content Toggle raw display
$59$ \( T^{2} + 14T + 32 \) Copy content Toggle raw display
$61$ \( T^{2} - 16T + 47 \) Copy content Toggle raw display
$67$ \( T^{2} + 5T + 2 \) Copy content Toggle raw display
$71$ \( (T - 14)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 12T + 19 \) Copy content Toggle raw display
$79$ \( T^{2} - 15T + 52 \) Copy content Toggle raw display
$83$ \( T^{2} + 10T + 8 \) Copy content Toggle raw display
$89$ \( T^{2} - 18T + 64 \) Copy content Toggle raw display
$97$ \( T^{2} - 13T + 38 \) Copy content Toggle raw display
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