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: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
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.

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

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 \(S_{2}^{\mathrm{new}}(1620, [\chi])\):

\( T_{7}^{2} + 2T_{7} + 4 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{17} + 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$7$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$17$ \( (T + 6)^{2} \) Copy content Toggle raw display
$19$ \( (T + 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$29$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$31$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$37$ \( (T - 2)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$43$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$47$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$53$ \( (T + 6)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 12T + 144 \) Copy content Toggle raw display
$61$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$67$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$71$ \( (T + 12)^{2} \) Copy content Toggle raw display
$73$ \( (T - 2)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$83$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$89$ \( (T + 6)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
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