Properties

Label 1520.2.d.j
Level $1520$
Weight $2$
Character orbit 1520.d
Analytic conductor $12.137$
Analytic rank $0$
Dimension $6$
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: \(6\)
Coefficient field: 6.0.5161984.1
Defining polynomial: \(x^{6} - 4 x^{3} + 25 x^{2} - 20 x + 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 190)
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.

\(f(q)\) \(=\) \( q + \beta_{5} q^{3} + ( \beta_{2} + \beta_{3} ) q^{5} + ( -\beta_{4} + \beta_{5} ) q^{7} + ( -2 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{9} +O(q^{10})\) \( q + \beta_{5} q^{3} + ( \beta_{2} + \beta_{3} ) q^{5} + ( -\beta_{4} + \beta_{5} ) q^{7} + ( -2 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{9} + ( \beta_{1} - \beta_{2} ) q^{11} + ( \beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} ) q^{13} + ( -1 + 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{15} + ( \beta_{1} + \beta_{2} - \beta_{5} ) q^{17} + q^{19} + ( -3 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{21} + ( 2 \beta_{1} + 2 \beta_{2} + \beta_{4} - 3 \beta_{5} ) q^{23} + ( 2 + 2 \beta_{1} - \beta_{3} + \beta_{5} ) q^{25} + ( 2 \beta_{1} + 2 \beta_{2} - \beta_{5} ) q^{27} + ( 3 + 3 \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{29} + ( -2 - 3 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} ) q^{31} + ( -\beta_{1} - \beta_{2} - 2 \beta_{4} + 2 \beta_{5} ) q^{33} + ( -1 + \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{35} + ( 3 \beta_{4} - 2 \beta_{5} ) q^{37} + ( 5 + 2 \beta_{1} - 2 \beta_{2} - 5 \beta_{3} ) q^{39} + ( \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{41} + ( -\beta_{1} - \beta_{2} + 3 \beta_{4} ) q^{43} + ( 4 + \beta_{1} - \beta_{2} - 3 \beta_{3} + \beta_{4} ) q^{45} + ( 2 \beta_{1} + 2 \beta_{2} + 3 \beta_{4} - 2 \beta_{5} ) q^{47} + ( 2 - \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{49} + ( 3 + 2 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} ) q^{51} + ( -\beta_{1} - \beta_{2} - 3 \beta_{4} + \beta_{5} ) q^{53} + ( -1 - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{55} + \beta_{5} q^{57} + ( 1 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{59} + ( -10 - \beta_{1} + \beta_{2} ) q^{61} + ( \beta_{4} - 2 \beta_{5} ) q^{63} + ( -2 - 2 \beta_{1} - \beta_{2} + 4 \beta_{5} ) q^{65} + ( -2 \beta_{1} - 2 \beta_{2} - \beta_{5} ) q^{67} + ( 9 + 5 \beta_{1} - 5 \beta_{2} - 5 \beta_{3} ) q^{69} + ( 4 - \beta_{1} + \beta_{2} - 4 \beta_{3} ) q^{71} + ( 3 \beta_{1} + 3 \beta_{2} - 5 \beta_{5} ) q^{73} + ( -7 - 2 \beta_{1} - 2 \beta_{2} - \beta_{3} + 4 \beta_{5} ) q^{75} + ( \beta_{1} + \beta_{2} - 2 \beta_{4} + 2 \beta_{5} ) q^{77} + ( -2 + 6 \beta_{3} ) q^{79} + ( -5 - 2 \beta_{3} ) q^{81} + ( -2 \beta_{1} - 2 \beta_{2} - \beta_{4} ) q^{83} + ( -2 - \beta_{2} + \beta_{4} + 2 \beta_{5} ) q^{85} + ( -4 \beta_{1} - 4 \beta_{2} - 4 \beta_{4} + 9 \beta_{5} ) q^{87} + ( 4 - \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{89} + ( -1 - 3 \beta_{3} ) q^{91} + ( 5 \beta_{1} + 5 \beta_{2} + 2 \beta_{4} - 8 \beta_{5} ) q^{93} + ( \beta_{2} + \beta_{3} ) q^{95} + ( 4 \beta_{1} + 4 \beta_{2} + 3 \beta_{4} - 4 \beta_{5} ) q^{97} -4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 2q^{5} - 10q^{9} + O(q^{10}) \) \( 6q + 2q^{5} - 10q^{9} - 8q^{15} + 6q^{19} - 20q^{21} + 10q^{25} + 16q^{29} - 8q^{31} - 8q^{35} + 20q^{39} + 4q^{41} + 18q^{45} + 6q^{49} + 12q^{51} - 4q^{55} + 4q^{59} - 60q^{61} - 12q^{65} + 44q^{69} + 16q^{71} - 44q^{75} - 34q^{81} - 12q^{85} + 28q^{89} - 12q^{91} + 2q^{95} - 24q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 4 x^{3} + 25 x^{2} - 20 x + 8\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 5 \nu^{5} + 2 \nu^{4} + 25 \nu^{3} - 10 \nu^{2} + 121 \nu - 100 \)\()/121\)
\(\beta_{3}\)\(=\)\((\)\( -7 \nu^{5} - 27 \nu^{4} - 35 \nu^{3} + 14 \nu^{2} - 223 \)\()/121\)
\(\beta_{4}\)\(=\)\((\)\( -25 \nu^{5} - 10 \nu^{4} - 4 \nu^{3} + 50 \nu^{2} - 605 \nu + 258 \)\()/121\)
\(\beta_{5}\)\(=\)\((\)\( -45 \nu^{5} - 18 \nu^{4} + 17 \nu^{3} + 211 \nu^{2} - 968 \nu + 416 \)\()/121\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{5} - 2 \beta_{4} - \beta_{2} - \beta_{1}\)
\(\nu^{3}\)\(=\)\(\beta_{4} + 5 \beta_{2} + 2\)
\(\nu^{4}\)\(=\)\(-5 \beta_{3} - 7 \beta_{2} + 7 \beta_{1} - 15\)
\(\nu^{5}\)\(=\)\(2 \beta_{5} - 9 \beta_{4} + 2 \beta_{3} - 29 \beta_{1} + 16\)

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
0.432320 + 0.432320i
1.32001 1.32001i
−1.75233 1.75233i
−1.75233 + 1.75233i
1.32001 + 1.32001i
0.432320 0.432320i
0 2.76156i 0 −2.19388 + 0.432320i 0 0.761557i 0 −4.62620 0
609.2 0 2.12489i 0 1.80487 1.32001i 0 4.12489i 0 −1.51514 0
609.3 0 1.36333i 0 1.38900 1.75233i 0 0.636672i 0 1.14134 0
609.4 0 1.36333i 0 1.38900 + 1.75233i 0 0.636672i 0 1.14134 0
609.5 0 2.12489i 0 1.80487 + 1.32001i 0 4.12489i 0 −1.51514 0
609.6 0 2.76156i 0 −2.19388 0.432320i 0 0.761557i 0 −4.62620 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.j 6
4.b odd 2 1 190.2.b.b 6
5.b even 2 1 inner 1520.2.d.j 6
5.c odd 4 1 7600.2.a.bi 3
5.c odd 4 1 7600.2.a.cd 3
12.b even 2 1 1710.2.d.d 6
20.d odd 2 1 190.2.b.b 6
20.e even 4 1 950.2.a.i 3
20.e even 4 1 950.2.a.n 3
60.h even 2 1 1710.2.d.d 6
60.l odd 4 1 8550.2.a.ck 3
60.l odd 4 1 8550.2.a.cl 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.b.b 6 4.b odd 2 1
190.2.b.b 6 20.d odd 2 1
950.2.a.i 3 20.e even 4 1
950.2.a.n 3 20.e even 4 1
1520.2.d.j 6 1.a even 1 1 trivial
1520.2.d.j 6 5.b even 2 1 inner
1710.2.d.d 6 12.b even 2 1
1710.2.d.d 6 60.h even 2 1
7600.2.a.bi 3 5.c odd 4 1
7600.2.a.cd 3 5.c odd 4 1
8550.2.a.ck 3 60.l odd 4 1
8550.2.a.cl 3 60.l odd 4 1

Hecke kernels

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

\( T_{3}^{6} + 14 T_{3}^{4} + 57 T_{3}^{2} + 64 \)
\( T_{7}^{6} + 18 T_{7}^{4} + 17 T_{7}^{2} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \)
$3$ \( 64 + 57 T^{2} + 14 T^{4} + T^{6} \)
$5$ \( 125 - 50 T - 15 T^{2} + 24 T^{3} - 3 T^{4} - 2 T^{5} + T^{6} \)
$7$ \( 4 + 17 T^{2} + 18 T^{4} + T^{6} \)
$11$ \( ( 8 - 10 T + T^{3} )^{2} \)
$13$ \( 4 + 201 T^{2} + 38 T^{4} + T^{6} \)
$17$ \( 16 + 65 T^{2} + 18 T^{4} + T^{6} \)
$19$ \( ( -1 + T )^{6} \)
$23$ \( 14884 + 2401 T^{2} + 98 T^{4} + T^{6} \)
$29$ \( ( 410 - 51 T - 8 T^{2} + T^{3} )^{2} \)
$31$ \( ( -232 - 62 T + 4 T^{2} + T^{3} )^{2} \)
$37$ \( 256 + 1152 T^{2} + 116 T^{4} + T^{6} \)
$41$ \( ( -100 - 50 T - 2 T^{2} + T^{3} )^{2} \)
$43$ \( 21904 + 4276 T^{2} + 128 T^{4} + T^{6} \)
$47$ \( 4096 + 2816 T^{2} + 132 T^{4} + T^{6} \)
$53$ \( 11236 + 2537 T^{2} + 102 T^{4} + T^{6} \)
$59$ \( ( 80 - 29 T - 2 T^{2} + T^{3} )^{2} \)
$61$ \( ( 892 + 290 T + 30 T^{2} + T^{3} )^{2} \)
$67$ \( 4096 + 3977 T^{2} + 126 T^{4} + T^{6} \)
$71$ \( ( 1016 - 122 T - 8 T^{2} + T^{3} )^{2} \)
$73$ \( 26896 + 5745 T^{2} + 290 T^{4} + T^{6} \)
$79$ \( ( -880 - 228 T + T^{3} )^{2} \)
$83$ \( 64 + 880 T^{2} + 92 T^{4} + T^{6} \)
$89$ \( ( 20 + 46 T - 14 T^{2} + T^{3} )^{2} \)
$97$ \( 238144 + 19760 T^{2} + 300 T^{4} + T^{6} \)
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