Properties

Label 1620.2.r.c
Level $1620$
Weight $2$
Character orbit 1620.r
Analytic conductor $12.936$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1620.r (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.9357651274\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 60)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 2 \zeta_{12} - \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{5} + 4 \zeta_{12} q^{7} +O(q^{10})\) \( q + ( 2 \zeta_{12} - \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{5} + 4 \zeta_{12} q^{7} + ( 4 - 4 \zeta_{12}^{2} ) q^{11} -4 \zeta_{12}^{3} q^{17} + ( 4 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{23} + ( 3 - 4 \zeta_{12} - 3 \zeta_{12}^{2} ) q^{25} + ( -6 + 6 \zeta_{12}^{2} ) q^{29} -4 \zeta_{12}^{2} q^{31} + ( 8 - 4 \zeta_{12}^{3} ) q^{35} + 8 \zeta_{12}^{3} q^{37} + 10 \zeta_{12}^{2} q^{41} -4 \zeta_{12} q^{43} -4 \zeta_{12} q^{47} + 9 \zeta_{12}^{2} q^{49} -12 \zeta_{12}^{3} q^{53} + ( -4 - 8 \zeta_{12}^{3} ) q^{55} + 4 \zeta_{12}^{2} q^{59} + ( -2 + 2 \zeta_{12}^{2} ) q^{61} + ( 4 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{67} -8 \zeta_{12}^{3} q^{73} + ( 16 \zeta_{12} - 16 \zeta_{12}^{3} ) q^{77} + ( -12 + 12 \zeta_{12}^{2} ) q^{79} -4 \zeta_{12} q^{83} + ( -4 \zeta_{12} - 8 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{85} + 10 q^{89} + 8 \zeta_{12} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{5} + O(q^{10}) \) \( 4q - 2q^{5} + 8q^{11} + 6q^{25} - 12q^{29} - 8q^{31} + 32q^{35} + 20q^{41} + 18q^{49} - 16q^{55} + 8q^{59} - 4q^{61} - 24q^{79} - 16q^{85} + 40q^{89} + O(q^{100}) \)

Character values

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

\(n\) \(811\) \(1297\) \(1541\)
\(\chi(n)\) \(1\) \(-1\) \(-\zeta_{12}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
0 0 0 −2.23205 0.133975i 0 −3.46410 + 2.00000i 0 0 0
109.2 0 0 0 1.23205 + 1.86603i 0 3.46410 2.00000i 0 0 0
1189.1 0 0 0 −2.23205 + 0.133975i 0 −3.46410 2.00000i 0 0 0
1189.2 0 0 0 1.23205 1.86603i 0 3.46410 + 2.00000i 0 0 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
9.c even 3 1 inner
45.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1620.2.r.c 4
3.b odd 2 1 1620.2.r.d 4
5.b even 2 1 inner 1620.2.r.c 4
9.c even 3 1 60.2.d.a 2
9.c even 3 1 inner 1620.2.r.c 4
9.d odd 6 1 180.2.d.a 2
9.d odd 6 1 1620.2.r.d 4
15.d odd 2 1 1620.2.r.d 4
36.f odd 6 1 240.2.f.b 2
36.h even 6 1 720.2.f.c 2
45.h odd 6 1 180.2.d.a 2
45.h odd 6 1 1620.2.r.d 4
45.j even 6 1 60.2.d.a 2
45.j even 6 1 inner 1620.2.r.c 4
45.k odd 12 1 300.2.a.a 1
45.k odd 12 1 300.2.a.d 1
45.l even 12 1 900.2.a.a 1
45.l even 12 1 900.2.a.h 1
63.g even 3 1 2940.2.bb.d 4
63.h even 3 1 2940.2.bb.d 4
63.k odd 6 1 2940.2.bb.e 4
63.l odd 6 1 2940.2.k.c 2
63.t odd 6 1 2940.2.bb.e 4
72.j odd 6 1 2880.2.f.l 2
72.l even 6 1 2880.2.f.p 2
72.n even 6 1 960.2.f.f 2
72.p odd 6 1 960.2.f.c 2
144.v odd 12 1 3840.2.d.b 2
144.v odd 12 1 3840.2.d.be 2
144.x even 12 1 3840.2.d.o 2
144.x even 12 1 3840.2.d.r 2
180.n even 6 1 720.2.f.c 2
180.p odd 6 1 240.2.f.b 2
180.v odd 12 1 3600.2.a.d 1
180.v odd 12 1 3600.2.a.bm 1
180.x even 12 1 1200.2.a.a 1
180.x even 12 1 1200.2.a.s 1
315.q odd 6 1 2940.2.bb.e 4
315.r even 6 1 2940.2.bb.d 4
315.bg odd 6 1 2940.2.k.c 2
315.bn odd 6 1 2940.2.bb.e 4
315.bo even 6 1 2940.2.bb.d 4
360.z odd 6 1 960.2.f.c 2
360.bd even 6 1 2880.2.f.p 2
360.bh odd 6 1 2880.2.f.l 2
360.bk even 6 1 960.2.f.f 2
360.bo even 12 1 4800.2.a.bf 1
360.bo even 12 1 4800.2.a.bk 1
360.bu odd 12 1 4800.2.a.bj 1
360.bu odd 12 1 4800.2.a.bn 1
720.ce even 12 1 3840.2.d.o 2
720.ce even 12 1 3840.2.d.r 2
720.cz odd 12 1 3840.2.d.b 2
720.cz odd 12 1 3840.2.d.be 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.2.d.a 2 9.c even 3 1
60.2.d.a 2 45.j even 6 1
180.2.d.a 2 9.d odd 6 1
180.2.d.a 2 45.h odd 6 1
240.2.f.b 2 36.f odd 6 1
240.2.f.b 2 180.p odd 6 1
300.2.a.a 1 45.k odd 12 1
300.2.a.d 1 45.k odd 12 1
720.2.f.c 2 36.h even 6 1
720.2.f.c 2 180.n even 6 1
900.2.a.a 1 45.l even 12 1
900.2.a.h 1 45.l even 12 1
960.2.f.c 2 72.p odd 6 1
960.2.f.c 2 360.z odd 6 1
960.2.f.f 2 72.n even 6 1
960.2.f.f 2 360.bk even 6 1
1200.2.a.a 1 180.x even 12 1
1200.2.a.s 1 180.x even 12 1
1620.2.r.c 4 1.a even 1 1 trivial
1620.2.r.c 4 5.b even 2 1 inner
1620.2.r.c 4 9.c even 3 1 inner
1620.2.r.c 4 45.j even 6 1 inner
1620.2.r.d 4 3.b odd 2 1
1620.2.r.d 4 9.d odd 6 1
1620.2.r.d 4 15.d odd 2 1
1620.2.r.d 4 45.h odd 6 1
2880.2.f.l 2 72.j odd 6 1
2880.2.f.l 2 360.bh odd 6 1
2880.2.f.p 2 72.l even 6 1
2880.2.f.p 2 360.bd even 6 1
2940.2.k.c 2 63.l odd 6 1
2940.2.k.c 2 315.bg odd 6 1
2940.2.bb.d 4 63.g even 3 1
2940.2.bb.d 4 63.h even 3 1
2940.2.bb.d 4 315.r even 6 1
2940.2.bb.d 4 315.bo even 6 1
2940.2.bb.e 4 63.k odd 6 1
2940.2.bb.e 4 63.t odd 6 1
2940.2.bb.e 4 315.q odd 6 1
2940.2.bb.e 4 315.bn odd 6 1
3600.2.a.d 1 180.v odd 12 1
3600.2.a.bm 1 180.v odd 12 1
3840.2.d.b 2 144.v odd 12 1
3840.2.d.b 2 720.cz odd 12 1
3840.2.d.o 2 144.x even 12 1
3840.2.d.o 2 720.ce even 12 1
3840.2.d.r 2 144.x even 12 1
3840.2.d.r 2 720.ce even 12 1
3840.2.d.be 2 144.v odd 12 1
3840.2.d.be 2 720.cz odd 12 1
4800.2.a.bf 1 360.bo even 12 1
4800.2.a.bj 1 360.bu odd 12 1
4800.2.a.bk 1 360.bo even 12 1
4800.2.a.bn 1 360.bu odd 12 1

Hecke kernels

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

\( T_{7}^{4} - 16 T_{7}^{2} + 256 \)
\( T_{11}^{2} - 4 T_{11} + 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( 25 + 10 T - T^{2} + 2 T^{3} + T^{4} \)
$7$ \( 256 - 16 T^{2} + T^{4} \)
$11$ \( ( 16 - 4 T + T^{2} )^{2} \)
$13$ \( T^{4} \)
$17$ \( ( 16 + T^{2} )^{2} \)
$19$ \( T^{4} \)
$23$ \( 256 - 16 T^{2} + T^{4} \)
$29$ \( ( 36 + 6 T + T^{2} )^{2} \)
$31$ \( ( 16 + 4 T + T^{2} )^{2} \)
$37$ \( ( 64 + T^{2} )^{2} \)
$41$ \( ( 100 - 10 T + T^{2} )^{2} \)
$43$ \( 256 - 16 T^{2} + T^{4} \)
$47$ \( 256 - 16 T^{2} + T^{4} \)
$53$ \( ( 144 + T^{2} )^{2} \)
$59$ \( ( 16 - 4 T + T^{2} )^{2} \)
$61$ \( ( 4 + 2 T + T^{2} )^{2} \)
$67$ \( 256 - 16 T^{2} + T^{4} \)
$71$ \( T^{4} \)
$73$ \( ( 64 + T^{2} )^{2} \)
$79$ \( ( 144 + 12 T + T^{2} )^{2} \)
$83$ \( 256 - 16 T^{2} + T^{4} \)
$89$ \( ( -10 + T )^{4} \)
$97$ \( 4096 - 64 T^{2} + T^{4} \)
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