Properties

Label 1520.2.d.c
Level $1520$
Weight $2$
Character orbit 1520.d
Analytic conductor $12.137$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1520.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.1372611072\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Defining polynomial: \(x^{4} + 3 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 760)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + ( -1 - \beta_{2} ) q^{5} + ( 2 \beta_{1} + \beta_{2} ) q^{7} + ( -3 + \beta_{3} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + ( -1 - \beta_{2} ) q^{5} + ( 2 \beta_{1} + \beta_{2} ) q^{7} + ( -3 + \beta_{3} ) q^{9} + ( -\beta_{1} - \beta_{2} ) q^{13} + ( -2 - \beta_{1} + \beta_{3} ) q^{15} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{17} + q^{19} + ( -10 + \beta_{3} ) q^{21} -\beta_{2} q^{23} + ( -3 + 2 \beta_{2} ) q^{25} + ( -2 \beta_{1} + 4 \beta_{2} ) q^{27} -2 q^{29} + ( 6 - \beta_{3} ) q^{31} + ( -2 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{35} + ( \beta_{1} - 3 \beta_{2} ) q^{37} + 4 q^{39} + ( 8 - \beta_{3} ) q^{41} + ( 2 \beta_{1} - \beta_{2} ) q^{43} + ( 3 - 4 \beta_{1} + \beta_{2} - \beta_{3} ) q^{45} + ( 2 \beta_{1} + 5 \beta_{2} ) q^{47} -13 q^{49} -8 q^{51} + ( \beta_{1} + 5 \beta_{2} ) q^{53} + \beta_{1} q^{57} + ( -6 - \beta_{3} ) q^{59} -\beta_{3} q^{61} + ( -6 \beta_{1} + 7 \beta_{2} ) q^{63} + ( -2 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{65} + \beta_{1} q^{67} + ( -2 + \beta_{3} ) q^{69} + ( 6 - \beta_{3} ) q^{71} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{73} + ( 4 - 3 \beta_{1} - 2 \beta_{3} ) q^{75} + ( -2 - \beta_{3} ) q^{79} + ( 11 - 3 \beta_{3} ) q^{81} + ( 4 \beta_{1} + 5 \beta_{2} ) q^{83} + ( 4 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{85} -2 \beta_{1} q^{87} + ( -2 + 2 \beta_{3} ) q^{89} + ( 10 + \beta_{3} ) q^{91} + ( 8 \beta_{1} - 4 \beta_{2} ) q^{93} + ( -1 - \beta_{2} ) q^{95} + ( -5 \beta_{1} + \beta_{2} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{5} - 12q^{9} + O(q^{10}) \) \( 4q - 4q^{5} - 12q^{9} - 8q^{15} + 4q^{19} - 40q^{21} - 12q^{25} - 8q^{29} + 24q^{31} + 16q^{39} + 32q^{41} + 12q^{45} - 52q^{49} - 32q^{51} - 24q^{59} - 8q^{65} - 8q^{69} + 24q^{71} + 16q^{75} - 8q^{79} + 44q^{81} + 16q^{85} - 8q^{89} + 40q^{91} - 4q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 3 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu \)
\(\beta_{2}\)\(=\)\( 2 \nu^{3} + 4 \nu \)
\(\beta_{3}\)\(=\)\( 4 \nu^{2} + 6 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} - 6\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{2} - 2 \beta_{1}\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1520\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(401\) \(1141\) \(1217\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
609.1
1.61803i
0.618034i
0.618034i
1.61803i
0 3.23607i 0 −1.00000 2.00000i 0 4.47214i 0 −7.47214 0
609.2 0 1.23607i 0 −1.00000 + 2.00000i 0 4.47214i 0 1.47214 0
609.3 0 1.23607i 0 −1.00000 2.00000i 0 4.47214i 0 1.47214 0
609.4 0 3.23607i 0 −1.00000 + 2.00000i 0 4.47214i 0 −7.47214 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1520.2.d.c 4
4.b odd 2 1 760.2.d.b 4
5.b even 2 1 inner 1520.2.d.c 4
5.c odd 4 1 7600.2.a.w 2
5.c odd 4 1 7600.2.a.be 2
20.d odd 2 1 760.2.d.b 4
20.e even 4 1 3800.2.a.k 2
20.e even 4 1 3800.2.a.q 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
760.2.d.b 4 4.b odd 2 1
760.2.d.b 4 20.d odd 2 1
1520.2.d.c 4 1.a even 1 1 trivial
1520.2.d.c 4 5.b even 2 1 inner
3800.2.a.k 2 20.e even 4 1
3800.2.a.q 2 20.e even 4 1
7600.2.a.w 2 5.c odd 4 1
7600.2.a.be 2 5.c odd 4 1

Hecke kernels

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

\( T_{3}^{4} + 12 T_{3}^{2} + 16 \)
\( T_{7}^{2} + 20 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 16 + 12 T^{2} + T^{4} \)
$5$ \( ( 5 + 2 T + T^{2} )^{2} \)
$7$ \( ( 20 + T^{2} )^{2} \)
$11$ \( T^{4} \)
$13$ \( 16 + 12 T^{2} + T^{4} \)
$17$ \( 256 + 48 T^{2} + T^{4} \)
$19$ \( ( -1 + T )^{4} \)
$23$ \( ( 4 + T^{2} )^{2} \)
$29$ \( ( 2 + T )^{4} \)
$31$ \( ( 16 - 12 T + T^{2} )^{2} \)
$37$ \( 1936 + 108 T^{2} + T^{4} \)
$41$ \( ( 44 - 16 T + T^{2} )^{2} \)
$43$ \( 16 + 72 T^{2} + T^{4} \)
$47$ \( 1936 + 168 T^{2} + T^{4} \)
$53$ \( 5776 + 172 T^{2} + T^{4} \)
$59$ \( ( 16 + 12 T + T^{2} )^{2} \)
$61$ \( ( -20 + T^{2} )^{2} \)
$67$ \( 16 + 12 T^{2} + T^{4} \)
$71$ \( ( 16 - 12 T + T^{2} )^{2} \)
$73$ \( 256 + 48 T^{2} + T^{4} \)
$79$ \( ( -16 + 4 T + T^{2} )^{2} \)
$83$ \( 1936 + 232 T^{2} + T^{4} \)
$89$ \( ( -76 + 4 T + T^{2} )^{2} \)
$97$ \( 5776 + 348 T^{2} + T^{4} \)
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