Properties

Label 1620.2.i.h
Level 1620
Weight 2
Character orbit 1620.i
Analytic conductor 12.936
Analytic rank 1
Dimension 2
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.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.9357651274\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 20)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{6} q^{5} + ( -2 + 2 \zeta_{6} ) q^{7} +O(q^{10})\) \( q + \zeta_{6} q^{5} + ( -2 + 2 \zeta_{6} ) q^{7} -2 \zeta_{6} q^{13} -6 q^{17} -4 q^{19} -6 \zeta_{6} q^{23} + ( -1 + \zeta_{6} ) q^{25} + ( -6 + 6 \zeta_{6} ) q^{29} + 4 \zeta_{6} q^{31} -2 q^{35} + 2 q^{37} -6 \zeta_{6} q^{41} + ( 10 - 10 \zeta_{6} ) q^{43} + ( 6 - 6 \zeta_{6} ) q^{47} + 3 \zeta_{6} q^{49} -6 q^{53} -12 \zeta_{6} q^{59} + ( -2 + 2 \zeta_{6} ) q^{61} + ( 2 - 2 \zeta_{6} ) q^{65} -2 \zeta_{6} q^{67} -12 q^{71} + 2 q^{73} + ( -8 + 8 \zeta_{6} ) q^{79} + ( -6 + 6 \zeta_{6} ) q^{83} -6 \zeta_{6} q^{85} -6 q^{89} + 4 q^{91} -4 \zeta_{6} q^{95} + ( -2 + 2 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{5} - 2q^{7} + O(q^{10}) \) \( 2q + q^{5} - 2q^{7} - 2q^{13} - 12q^{17} - 8q^{19} - 6q^{23} - q^{25} - 6q^{29} + 4q^{31} - 4q^{35} + 4q^{37} - 6q^{41} + 10q^{43} + 6q^{47} + 3q^{49} - 12q^{53} - 12q^{59} - 2q^{61} + 2q^{65} - 2q^{67} - 24q^{71} + 4q^{73} - 8q^{79} - 6q^{83} - 6q^{85} - 12q^{89} + 8q^{91} - 4q^{95} - 2q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(811\) \(1297\) \(1541\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
541.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0 0 0.500000 + 0.866025i 0 −1.00000 + 1.73205i 0 0 0
1081.1 0 0 0 0.500000 0.866025i 0 −1.00000 1.73205i 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
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1620.2.i.h 2
3.b odd 2 1 1620.2.i.b 2
9.c even 3 1 20.2.a.a 1
9.c even 3 1 inner 1620.2.i.h 2
9.d odd 6 1 180.2.a.a 1
9.d odd 6 1 1620.2.i.b 2
36.f odd 6 1 80.2.a.b 1
36.h even 6 1 720.2.a.h 1
45.h odd 6 1 900.2.a.b 1
45.j even 6 1 100.2.a.a 1
45.k odd 12 2 100.2.c.a 2
45.l even 12 2 900.2.d.c 2
63.g even 3 1 980.2.i.i 2
63.h even 3 1 980.2.i.i 2
63.k odd 6 1 980.2.i.c 2
63.l odd 6 1 980.2.a.h 1
63.o even 6 1 8820.2.a.g 1
63.t odd 6 1 980.2.i.c 2
72.j odd 6 1 2880.2.a.m 1
72.l even 6 1 2880.2.a.f 1
72.n even 6 1 320.2.a.f 1
72.p odd 6 1 320.2.a.a 1
99.h odd 6 1 2420.2.a.a 1
117.t even 6 1 3380.2.a.c 1
117.y odd 12 2 3380.2.f.b 2
144.v odd 12 2 1280.2.d.g 2
144.x even 12 2 1280.2.d.c 2
153.h even 6 1 5780.2.a.f 1
153.n even 12 2 5780.2.c.a 2
171.o odd 6 1 7220.2.a.f 1
180.n even 6 1 3600.2.a.be 1
180.p odd 6 1 400.2.a.c 1
180.v odd 12 2 3600.2.f.j 2
180.x even 12 2 400.2.c.b 2
252.bi even 6 1 3920.2.a.h 1
315.bg odd 6 1 4900.2.a.e 1
315.cb even 12 2 4900.2.e.f 2
360.z odd 6 1 1600.2.a.w 1
360.bk even 6 1 1600.2.a.c 1
360.bo even 12 2 1600.2.c.e 2
360.bu odd 12 2 1600.2.c.d 2
396.k even 6 1 9680.2.a.ba 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.2.a.a 1 9.c even 3 1
80.2.a.b 1 36.f odd 6 1
100.2.a.a 1 45.j even 6 1
100.2.c.a 2 45.k odd 12 2
180.2.a.a 1 9.d odd 6 1
320.2.a.a 1 72.p odd 6 1
320.2.a.f 1 72.n even 6 1
400.2.a.c 1 180.p odd 6 1
400.2.c.b 2 180.x even 12 2
720.2.a.h 1 36.h even 6 1
900.2.a.b 1 45.h odd 6 1
900.2.d.c 2 45.l even 12 2
980.2.a.h 1 63.l odd 6 1
980.2.i.c 2 63.k odd 6 1
980.2.i.c 2 63.t odd 6 1
980.2.i.i 2 63.g even 3 1
980.2.i.i 2 63.h even 3 1
1280.2.d.c 2 144.x even 12 2
1280.2.d.g 2 144.v odd 12 2
1600.2.a.c 1 360.bk even 6 1
1600.2.a.w 1 360.z odd 6 1
1600.2.c.d 2 360.bu odd 12 2
1600.2.c.e 2 360.bo even 12 2
1620.2.i.b 2 3.b odd 2 1
1620.2.i.b 2 9.d odd 6 1
1620.2.i.h 2 1.a even 1 1 trivial
1620.2.i.h 2 9.c even 3 1 inner
2420.2.a.a 1 99.h odd 6 1
2880.2.a.f 1 72.l even 6 1
2880.2.a.m 1 72.j odd 6 1
3380.2.a.c 1 117.t even 6 1
3380.2.f.b 2 117.y odd 12 2
3600.2.a.be 1 180.n even 6 1
3600.2.f.j 2 180.v odd 12 2
3920.2.a.h 1 252.bi even 6 1
4900.2.a.e 1 315.bg odd 6 1
4900.2.e.f 2 315.cb even 12 2
5780.2.a.f 1 153.h even 6 1
5780.2.c.a 2 153.n even 12 2
7220.2.a.f 1 171.o odd 6 1
8820.2.a.g 1 63.o even 6 1
9680.2.a.ba 1 396.k even 6 1

Hecke kernels

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

\( T_{7}^{2} + 2 T_{7} + 4 \)
\( T_{11} \)
\( T_{17} + 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 - T + T^{2} \)
$7$ \( 1 + 2 T - 3 T^{2} + 14 T^{3} + 49 T^{4} \)
$11$ \( 1 - 11 T^{2} + 121 T^{4} \)
$13$ \( ( 1 - 5 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} ) \)
$17$ \( ( 1 + 6 T + 17 T^{2} )^{2} \)
$19$ \( ( 1 + 4 T + 19 T^{2} )^{2} \)
$23$ \( 1 + 6 T + 13 T^{2} + 138 T^{3} + 529 T^{4} \)
$29$ \( 1 + 6 T + 7 T^{2} + 174 T^{3} + 841 T^{4} \)
$31$ \( ( 1 - 11 T + 31 T^{2} )( 1 + 7 T + 31 T^{2} ) \)
$37$ \( ( 1 - 2 T + 37 T^{2} )^{2} \)
$41$ \( 1 + 6 T - 5 T^{2} + 246 T^{3} + 1681 T^{4} \)
$43$ \( 1 - 10 T + 57 T^{2} - 430 T^{3} + 1849 T^{4} \)
$47$ \( 1 - 6 T - 11 T^{2} - 282 T^{3} + 2209 T^{4} \)
$53$ \( ( 1 + 6 T + 53 T^{2} )^{2} \)
$59$ \( 1 + 12 T + 85 T^{2} + 708 T^{3} + 3481 T^{4} \)
$61$ \( 1 + 2 T - 57 T^{2} + 122 T^{3} + 3721 T^{4} \)
$67$ \( 1 + 2 T - 63 T^{2} + 134 T^{3} + 4489 T^{4} \)
$71$ \( ( 1 + 12 T + 71 T^{2} )^{2} \)
$73$ \( ( 1 - 2 T + 73 T^{2} )^{2} \)
$79$ \( 1 + 8 T - 15 T^{2} + 632 T^{3} + 6241 T^{4} \)
$83$ \( 1 + 6 T - 47 T^{2} + 498 T^{3} + 6889 T^{4} \)
$89$ \( ( 1 + 6 T + 89 T^{2} )^{2} \)
$97$ \( 1 + 2 T - 93 T^{2} + 194 T^{3} + 9409 T^{4} \)
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