Properties

Label 1520.2.d.h
Level $1520$
Weight $2$
Character orbit 1520.d
Analytic conductor $12.137$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

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: \(6\)
Coefficient field: 6.0.16516096.1
Defining polynomial: \(x^{6} + 9 x^{4} + 13 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 95)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{4} q^{3} + \beta_{3} q^{5} + ( \beta_{1} + \beta_{4} ) q^{7} + ( -3 - \beta_{2} - \beta_{5} ) q^{9} +O(q^{10})\) \( q -\beta_{4} q^{3} + \beta_{3} q^{5} + ( \beta_{1} + \beta_{4} ) q^{7} + ( -3 - \beta_{2} - \beta_{5} ) q^{9} + ( \beta_{2} + \beta_{3} ) q^{11} + ( -\beta_{1} - \beta_{2} + \beta_{3} ) q^{13} + ( -2 + \beta_{2} - \beta_{4} - \beta_{5} ) q^{15} + ( 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{17} - q^{19} + ( 4 + \beta_{2} + \beta_{5} ) q^{21} + ( -\beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} ) q^{23} + ( -\beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{25} + ( 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} ) q^{27} -6 q^{29} + ( \beta_{2} + 2 \beta_{3} - \beta_{5} ) q^{31} + ( 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} ) q^{33} + ( 1 - 2 \beta_{1} + \beta_{5} ) q^{35} + ( \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{37} + ( -2 + 2 \beta_{3} - 2 \beta_{5} ) q^{39} + ( 2 + \beta_{2} + 2 \beta_{3} - \beta_{5} ) q^{41} + ( \beta_{1} + \beta_{4} ) q^{43} + ( -2 + 2 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{45} + ( -3 \beta_{1} + \beta_{4} ) q^{47} + ( 1 + \beta_{2} + \beta_{3} ) q^{49} + ( -2 \beta_{3} + 2 \beta_{5} ) q^{51} + ( -3 \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{53} + ( 5 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{55} + \beta_{4} q^{57} + ( 4 + \beta_{2} + \beta_{5} ) q^{59} + ( -2 + 2 \beta_{2} + 3 \beta_{3} - \beta_{5} ) q^{61} + ( \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} ) q^{63} + ( -4 + \beta_{1} - 2 \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{65} + ( -2 \beta_{1} - 5 \beta_{4} ) q^{67} + ( 4 + \beta_{2} + 2 \beta_{3} - \beta_{5} ) q^{69} + ( -8 + \beta_{2} + \beta_{5} ) q^{71} + ( -\beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{73} + ( 6 + 2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{75} + ( -4 \beta_{1} - \beta_{2} + \beta_{3} ) q^{77} + ( -4 - \beta_{2} - 2 \beta_{3} + \beta_{5} ) q^{79} + ( 7 - \beta_{2} - 4 \beta_{3} + 3 \beta_{5} ) q^{81} + ( -\beta_{1} + 3 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} ) q^{83} + ( 3 - 3 \beta_{1} + 3 \beta_{2} - \beta_{3} - 3 \beta_{4} + \beta_{5} ) q^{85} + 6 \beta_{4} q^{87} + ( -2 + 2 \beta_{2} - 2 \beta_{3} + 4 \beta_{5} ) q^{89} + ( 4 - \beta_{2} - 2 \beta_{3} + \beta_{5} ) q^{91} + ( 2 \beta_{1} + 6 \beta_{2} - 6 \beta_{3} - 2 \beta_{4} ) q^{93} -\beta_{3} q^{95} + ( \beta_{1} + \beta_{2} - \beta_{3} ) q^{97} + ( -4 + \beta_{2} - \beta_{3} + 2 \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - q^{5} - 14q^{9} + O(q^{10}) \) \( 6q - q^{5} - 14q^{9} - 2q^{11} - 10q^{15} - 6q^{19} + 20q^{21} + 3q^{25} - 36q^{29} + 3q^{35} - 8q^{39} + 12q^{41} - 15q^{45} + 4q^{49} - 4q^{51} + 33q^{55} + 20q^{59} - 14q^{61} - 20q^{65} + 24q^{69} - 52q^{71} + 34q^{75} - 24q^{79} + 38q^{81} + 13q^{85} - 24q^{89} + 24q^{91} + q^{95} - 30q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} + 9 x^{4} + 13 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{5} - 8 \nu^{3} - 5 \nu \)\()/2\)
\(\beta_{2}\)\(=\)\((\)\( 3 \nu^{5} + \nu^{4} + 26 \nu^{3} + 6 \nu^{2} + 33 \nu - 1 \)\()/4\)
\(\beta_{3}\)\(=\)\((\)\( -3 \nu^{5} + \nu^{4} - 26 \nu^{3} + 6 \nu^{2} - 33 \nu - 1 \)\()/4\)
\(\beta_{4}\)\(=\)\( \nu^{5} + 9 \nu^{3} + 12 \nu \)
\(\beta_{5}\)\(=\)\((\)\( -3 \nu^{5} + 3 \nu^{4} - 26 \nu^{3} + 26 \nu^{2} - 33 \nu + 21 \)\()/4\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{4} - \beta_{3} + \beta_{2} + \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{5} - 2 \beta_{3} - \beta_{2} - 6\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(9 \beta_{4} + 7 \beta_{3} - 7 \beta_{2} - 3 \beta_{1}\)\()/2\)
\(\nu^{4}\)\(=\)\(-3 \beta_{5} + 8 \beta_{3} + 5 \beta_{2} + 19\)
\(\nu^{5}\)\(=\)\((\)\(-67 \beta_{4} - 51 \beta_{3} + 51 \beta_{2} + 15 \beta_{1}\)\()/2\)

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
0.285442i
2.68667i
1.30397i
1.30397i
2.68667i
0.285442i
0 3.21789i 0 −0.370556 2.20515i 0 2.59637i 0 −7.35482 0
609.2 0 2.31446i 0 1.94827 + 1.09737i 0 1.45033i 0 −2.35673 0
609.3 0 0.537080i 0 −2.07772 0.826491i 0 3.18676i 0 2.71155 0
609.4 0 0.537080i 0 −2.07772 + 0.826491i 0 3.18676i 0 2.71155 0
609.5 0 2.31446i 0 1.94827 1.09737i 0 1.45033i 0 −2.35673 0
609.6 0 3.21789i 0 −0.370556 + 2.20515i 0 2.59637i 0 −7.35482 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 609.6
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.h 6
4.b odd 2 1 95.2.b.b 6
5.b even 2 1 inner 1520.2.d.h 6
5.c odd 4 2 7600.2.a.ck 6
12.b even 2 1 855.2.c.d 6
20.d odd 2 1 95.2.b.b 6
20.e even 4 2 475.2.a.j 6
60.h even 2 1 855.2.c.d 6
60.l odd 4 2 4275.2.a.br 6
76.d even 2 1 1805.2.b.e 6
380.d even 2 1 1805.2.b.e 6
380.j odd 4 2 9025.2.a.bx 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.b.b 6 4.b odd 2 1
95.2.b.b 6 20.d odd 2 1
475.2.a.j 6 20.e even 4 2
855.2.c.d 6 12.b even 2 1
855.2.c.d 6 60.h even 2 1
1520.2.d.h 6 1.a even 1 1 trivial
1520.2.d.h 6 5.b even 2 1 inner
1805.2.b.e 6 76.d even 2 1
1805.2.b.e 6 380.d even 2 1
4275.2.a.br 6 60.l odd 4 2
7600.2.a.ck 6 5.c odd 4 2
9025.2.a.bx 6 380.j odd 4 2

Hecke kernels

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

\( T_{3}^{6} + 16 T_{3}^{4} + 60 T_{3}^{2} + 16 \)
\( T_{7}^{6} + 19 T_{7}^{4} + 104 T_{7}^{2} + 144 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \)
$3$ \( 16 + 60 T^{2} + 16 T^{4} + T^{6} \)
$5$ \( 125 + 25 T - 5 T^{2} - 2 T^{3} - T^{4} + T^{5} + T^{6} \)
$7$ \( 144 + 104 T^{2} + 19 T^{4} + T^{6} \)
$11$ \( ( -12 - 16 T + T^{2} + T^{3} )^{2} \)
$13$ \( 576 + 236 T^{2} + 28 T^{4} + T^{6} \)
$17$ \( 5184 + 1008 T^{2} + 59 T^{4} + T^{6} \)
$19$ \( ( 1 + T )^{6} \)
$23$ \( 64 + 208 T^{2} + 36 T^{4} + T^{6} \)
$29$ \( ( 6 + T )^{6} \)
$31$ \( ( -128 - 56 T + T^{3} )^{2} \)
$37$ \( 1296 + 764 T^{2} + 56 T^{4} + T^{6} \)
$41$ \( ( -24 - 44 T - 6 T^{2} + T^{3} )^{2} \)
$43$ \( 144 + 104 T^{2} + 19 T^{4} + T^{6} \)
$47$ \( 85264 + 7464 T^{2} + 187 T^{4} + T^{6} \)
$53$ \( 64 + 2476 T^{2} + 156 T^{4} + T^{6} \)
$59$ \( ( 48 + 8 T - 10 T^{2} + T^{3} )^{2} \)
$61$ \( ( -776 - 104 T + 7 T^{2} + T^{3} )^{2} \)
$67$ \( 484416 + 28556 T^{2} + 340 T^{4} + T^{6} \)
$71$ \( ( 432 + 200 T + 26 T^{2} + T^{3} )^{2} \)
$73$ \( 5184 + 1616 T^{2} + 131 T^{4} + T^{6} \)
$79$ \( ( -32 - 8 T + 12 T^{2} + T^{3} )^{2} \)
$83$ \( 141376 + 11728 T^{2} + 228 T^{4} + T^{6} \)
$89$ \( ( -3456 - 284 T + 12 T^{2} + T^{3} )^{2} \)
$97$ \( 576 + 236 T^{2} + 28 T^{4} + T^{6} \)
show more
show less