Properties

Label 1520.2.a.n
Level $1520$
Weight $2$
Character orbit 1520.a
Self dual yes
Analytic conductor $12.137$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1520.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(12.1372611072\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 190)
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.

\(f(q)\) \(=\) \( q + \beta q^{3} + q^{5} + \beta q^{7} + (\beta + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{3} + q^{5} + \beta q^{7} + (\beta + 1) q^{9} - 4 q^{11} + (3 \beta - 2) q^{13} + \beta q^{15} + ( - \beta + 6) q^{17} + q^{19} + (\beta + 4) q^{21} - 3 \beta q^{23} + q^{25} + ( - \beta + 4) q^{27} + ( - 3 \beta + 2) q^{29} + 2 \beta q^{31} - 4 \beta q^{33} + \beta q^{35} - 6 q^{37} + (\beta + 12) q^{39} + (4 \beta + 2) q^{41} + ( - 2 \beta + 8) q^{43} + (\beta + 1) q^{45} + ( - 4 \beta + 4) q^{47} + (\beta - 3) q^{49} + (5 \beta - 4) q^{51} + ( - \beta - 2) q^{53} - 4 q^{55} + \beta q^{57} - \beta q^{59} + (2 \beta + 6) q^{61} + (2 \beta + 4) q^{63} + (3 \beta - 2) q^{65} + \beta q^{67} + ( - 3 \beta - 12) q^{69} - 4 \beta q^{71} + ( - 3 \beta + 6) q^{73} + \beta q^{75} - 4 \beta q^{77} + 2 \beta q^{79} - 7 q^{81} + (2 \beta - 8) q^{83} + ( - \beta + 6) q^{85} + ( - \beta - 12) q^{87} + 2 q^{89} + (\beta + 12) q^{91} + (2 \beta + 8) q^{93} + q^{95} + 6 q^{97} + ( - 4 \beta - 4) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} + 2 q^{5} + q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{3} + 2 q^{5} + q^{7} + 3 q^{9} - 8 q^{11} - q^{13} + q^{15} + 11 q^{17} + 2 q^{19} + 9 q^{21} - 3 q^{23} + 2 q^{25} + 7 q^{27} + q^{29} + 2 q^{31} - 4 q^{33} + q^{35} - 12 q^{37} + 25 q^{39} + 8 q^{41} + 14 q^{43} + 3 q^{45} + 4 q^{47} - 5 q^{49} - 3 q^{51} - 5 q^{53} - 8 q^{55} + q^{57} - q^{59} + 14 q^{61} + 10 q^{63} - q^{65} + q^{67} - 27 q^{69} - 4 q^{71} + 9 q^{73} + q^{75} - 4 q^{77} + 2 q^{79} - 14 q^{81} - 14 q^{83} + 11 q^{85} - 25 q^{87} + 4 q^{89} + 25 q^{91} + 18 q^{93} + 2 q^{95} + 12 q^{97} - 12 q^{99}+O(q^{100}) \) 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
0 −1.56155 0 1.00000 0 −1.56155 0 −0.561553 0
1.2 0 2.56155 0 1.00000 0 2.56155 0 3.56155 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)
\(19\) \(-1\)

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 1520.2.a.n 2
4.b odd 2 1 190.2.a.d 2
5.b even 2 1 7600.2.a.y 2
8.b even 2 1 6080.2.a.bb 2
8.d odd 2 1 6080.2.a.bh 2
12.b even 2 1 1710.2.a.w 2
20.d odd 2 1 950.2.a.h 2
20.e even 4 2 950.2.b.f 4
28.d even 2 1 9310.2.a.bc 2
60.h even 2 1 8550.2.a.br 2
76.d even 2 1 3610.2.a.t 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.a.d 2 4.b odd 2 1
950.2.a.h 2 20.d odd 2 1
950.2.b.f 4 20.e even 4 2
1520.2.a.n 2 1.a even 1 1 trivial
1710.2.a.w 2 12.b even 2 1
3610.2.a.t 2 76.d even 2 1
6080.2.a.bb 2 8.b even 2 1
6080.2.a.bh 2 8.d odd 2 1
7600.2.a.y 2 5.b even 2 1
8550.2.a.br 2 60.h even 2 1
9310.2.a.bc 2 28.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1520))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - T - 4 \) Copy content Toggle raw display
$5$ \( (T - 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - T - 4 \) Copy content Toggle raw display
$11$ \( (T + 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + T - 38 \) Copy content Toggle raw display
$17$ \( T^{2} - 11T + 26 \) Copy content Toggle raw display
$19$ \( (T - 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 3T - 36 \) Copy content Toggle raw display
$29$ \( T^{2} - T - 38 \) Copy content Toggle raw display
$31$ \( T^{2} - 2T - 16 \) Copy content Toggle raw display
$37$ \( (T + 6)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 8T - 52 \) Copy content Toggle raw display
$43$ \( T^{2} - 14T + 32 \) Copy content Toggle raw display
$47$ \( T^{2} - 4T - 64 \) Copy content Toggle raw display
$53$ \( T^{2} + 5T + 2 \) Copy content Toggle raw display
$59$ \( T^{2} + T - 4 \) Copy content Toggle raw display
$61$ \( T^{2} - 14T + 32 \) Copy content Toggle raw display
$67$ \( T^{2} - T - 4 \) Copy content Toggle raw display
$71$ \( T^{2} + 4T - 64 \) Copy content Toggle raw display
$73$ \( T^{2} - 9T - 18 \) Copy content Toggle raw display
$79$ \( T^{2} - 2T - 16 \) Copy content Toggle raw display
$83$ \( T^{2} + 14T + 32 \) Copy content Toggle raw display
$89$ \( (T - 2)^{2} \) Copy content Toggle raw display
$97$ \( (T - 6)^{2} \) Copy content Toggle raw display
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