Properties

Label 1620.2.d.d
Level $1620$
Weight $2$
Character orbit 1620.d
Analytic conductor $12.936$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1620.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.9357651274\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.301925376.2
Defining polynomial: \(x^{6} + 14 x^{4} + 43 x^{2} + 36\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 180)
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.

\(f(q)\) \(=\) \( q -\beta_{3} q^{5} + ( \beta_{1} + \beta_{3} - \beta_{4} ) q^{7} +O(q^{10})\) \( q -\beta_{3} q^{5} + ( \beta_{1} + \beta_{3} - \beta_{4} ) q^{7} + \beta_{2} q^{11} + ( -\beta_{1} - \beta_{5} ) q^{13} + ( \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{17} + ( -\beta_{2} - \beta_{3} - \beta_{4} ) q^{19} + ( -3 \beta_{1} - \beta_{3} + \beta_{4} ) q^{23} + ( \beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} ) q^{25} -3 q^{29} + ( -2 + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} ) q^{31} + ( 3 - \beta_{1} + \beta_{4} + 2 \beta_{5} ) q^{35} + ( -\beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{37} + ( 3 - \beta_{2} + \beta_{3} + \beta_{4} ) q^{41} + ( -\beta_{1} + 2 \beta_{3} - 2 \beta_{4} + 3 \beta_{5} ) q^{43} + ( 2 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{47} + ( -\beta_{2} - \beta_{3} - \beta_{4} ) q^{49} + ( 2 \beta_{1} - 4 \beta_{5} ) q^{53} + ( -1 - 3 \beta_{1} - \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{55} + ( -6 - \beta_{3} - \beta_{4} ) q^{59} + ( -1 - \beta_{2} - \beta_{3} - \beta_{4} ) q^{61} + ( 3 - 2 \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{65} + ( 2 \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{67} + ( -\beta_{2} - \beta_{3} - \beta_{4} ) q^{71} + ( 6 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{73} + ( 5 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{77} + ( -3 \beta_{3} - 3 \beta_{4} ) q^{79} + ( \beta_{1} - \beta_{3} + \beta_{4} + 4 \beta_{5} ) q^{83} + ( 2 + \beta_{1} + \beta_{3} - \beta_{4} + 3 \beta_{5} ) q^{85} + ( -9 + \beta_{3} + \beta_{4} ) q^{89} + ( \beta_{2} - 2 \beta_{3} - 2 \beta_{4} ) q^{91} + ( 6 + 4 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{95} + ( -2 \beta_{1} - \beta_{3} + \beta_{4} - 4 \beta_{5} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + q^{5} + O(q^{10}) \) \( 6q + q^{5} + 2q^{11} + 3q^{25} - 18q^{29} - 6q^{31} + 17q^{35} + 14q^{41} - 3q^{55} - 34q^{59} - 6q^{61} + 15q^{65} + 6q^{79} + 12q^{85} - 56q^{89} + 6q^{91} + 36q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} + 14 x^{4} + 43 x^{2} + 36\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + 5 \)
\(\beta_{3}\)\(=\)\((\)\( \nu^{5} + 6 \nu^{4} + 14 \nu^{3} + 72 \nu^{2} + 31 \nu + 114 \)\()/12\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{5} + 6 \nu^{4} - 14 \nu^{3} + 72 \nu^{2} - 31 \nu + 114 \)\()/12\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{5} - 12 \nu^{3} - 19 \nu \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 5\)
\(\nu^{3}\)\(=\)\(\beta_{5} - 3 \beta_{4} + 3 \beta_{3} - 6 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{4} + \beta_{3} - 12 \beta_{2} + 41\)
\(\nu^{5}\)\(=\)\(-14 \beta_{5} + 36 \beta_{4} - 36 \beta_{3} + 53 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times\).

\(n\) \(811\) \(1297\) \(1541\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
649.1
1.20590i
1.20590i
3.17695i
3.17695i
1.56613i
1.56613i
0 0 0 −1.83216 1.28187i 0 3.76963i 0 0 0
649.2 0 0 0 −1.83216 + 1.28187i 0 3.76963i 0 0 0
649.3 0 0 0 0.123563 2.23265i 0 1.28835i 0 0 0
649.4 0 0 0 0.123563 + 2.23265i 0 1.28835i 0 0 0
649.5 0 0 0 2.20860 0.349411i 0 2.26496i 0 0 0
649.6 0 0 0 2.20860 + 0.349411i 0 2.26496i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 649.6
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 1620.2.d.d 6
3.b odd 2 1 1620.2.d.c 6
5.b even 2 1 inner 1620.2.d.d 6
5.c odd 4 2 8100.2.a.bd 6
9.c even 3 2 540.2.r.a 12
9.d odd 6 2 180.2.r.a 12
15.d odd 2 1 1620.2.d.c 6
15.e even 4 2 8100.2.a.bc 6
36.f odd 6 2 2160.2.by.e 12
36.h even 6 2 720.2.by.e 12
45.h odd 6 2 180.2.r.a 12
45.j even 6 2 540.2.r.a 12
45.k odd 12 4 2700.2.i.f 12
45.l even 12 4 900.2.i.f 12
180.n even 6 2 720.2.by.e 12
180.p odd 6 2 2160.2.by.e 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.2.r.a 12 9.d odd 6 2
180.2.r.a 12 45.h odd 6 2
540.2.r.a 12 9.c even 3 2
540.2.r.a 12 45.j even 6 2
720.2.by.e 12 36.h even 6 2
720.2.by.e 12 180.n even 6 2
900.2.i.f 12 45.l even 12 4
1620.2.d.c 6 3.b odd 2 1
1620.2.d.c 6 15.d odd 2 1
1620.2.d.d 6 1.a even 1 1 trivial
1620.2.d.d 6 5.b even 2 1 inner
2160.2.by.e 12 36.f odd 6 2
2160.2.by.e 12 180.p odd 6 2
2700.2.i.f 12 45.k odd 12 4
8100.2.a.bc 6 15.e even 4 2
8100.2.a.bd 6 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1620, [\chi])\):

\( T_{7}^{6} + 21 T_{7}^{4} + 105 T_{7}^{2} + 121 \)
\( T_{11}^{3} - T_{11}^{2} - 22 T_{11} + 46 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \)
$3$ \( T^{6} \)
$5$ \( 125 - 25 T - 5 T^{2} - 6 T^{3} - T^{4} - T^{5} + T^{6} \)
$7$ \( 121 + 105 T^{2} + 21 T^{4} + T^{6} \)
$11$ \( ( 46 - 22 T - T^{2} + T^{3} )^{2} \)
$13$ \( 324 + 360 T^{2} + 39 T^{4} + T^{6} \)
$17$ \( 64 + 108 T^{2} + 36 T^{4} + T^{6} \)
$19$ \( ( 72 - 42 T + T^{3} )^{2} \)
$23$ \( 28561 + 2841 T^{2} + 93 T^{4} + T^{6} \)
$29$ \( ( 3 + T )^{6} \)
$31$ \( ( -358 - 78 T + 3 T^{2} + T^{3} )^{2} \)
$37$ \( 256 + 396 T^{2} + 60 T^{4} + T^{6} \)
$41$ \( ( 97 - 19 T - 7 T^{2} + T^{3} )^{2} \)
$43$ \( 91204 + 14088 T^{2} + 231 T^{4} + T^{6} \)
$47$ \( 961 + 957 T^{2} + 105 T^{4} + T^{6} \)
$53$ \( 331776 + 20016 T^{2} + 264 T^{4} + T^{6} \)
$59$ \( ( 88 + 80 T + 17 T^{2} + T^{3} )^{2} \)
$61$ \( ( 31 - 39 T + 3 T^{2} + T^{3} )^{2} \)
$67$ \( 14641 + 2541 T^{2} + 105 T^{4} + T^{6} \)
$71$ \( ( 72 - 42 T + T^{3} )^{2} \)
$73$ \( 369664 + 30144 T^{2} + 384 T^{4} + T^{6} \)
$79$ \( ( 108 - 144 T - 3 T^{2} + T^{3} )^{2} \)
$83$ \( 1907161 + 49641 T^{2} + 405 T^{4} + T^{6} \)
$89$ \( ( 662 + 245 T + 28 T^{2} + T^{3} )^{2} \)
$97$ \( 55696 + 8568 T^{2} + 339 T^{4} + T^{6} \)
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