Properties

Label 1620.4.i.i
Level $1620$
Weight $4$
Character orbit 1620.i
Analytic conductor $95.583$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(95.5830942093\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 180)
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.

\(f(q)\) \(=\) \( q + 5 \zeta_{6} q^{5} + (2 \zeta_{6} - 2) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 5 \zeta_{6} q^{5} + (2 \zeta_{6} - 2) q^{7} + ( - 30 \zeta_{6} + 30) q^{11} + 4 \zeta_{6} q^{13} - 90 q^{17} - 28 q^{19} + 120 \zeta_{6} q^{23} + (25 \zeta_{6} - 25) q^{25} + ( - 210 \zeta_{6} + 210) q^{29} + 4 \zeta_{6} q^{31} - 10 q^{35} + 200 q^{37} + 240 \zeta_{6} q^{41} + ( - 136 \zeta_{6} + 136) q^{43} + (120 \zeta_{6} - 120) q^{47} + 339 \zeta_{6} q^{49} + 30 q^{53} + 150 q^{55} - 450 \zeta_{6} q^{59} + ( - 166 \zeta_{6} + 166) q^{61} + (20 \zeta_{6} - 20) q^{65} - 908 \zeta_{6} q^{67} + 1020 q^{71} - 250 q^{73} + 60 \zeta_{6} q^{77} + ( - 916 \zeta_{6} + 916) q^{79} + (1140 \zeta_{6} - 1140) q^{83} - 450 \zeta_{6} q^{85} + 420 q^{89} - 8 q^{91} - 140 \zeta_{6} q^{95} + (1538 \zeta_{6} - 1538) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 5 q^{5} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 5 q^{5} - 2 q^{7} + 30 q^{11} + 4 q^{13} - 180 q^{17} - 56 q^{19} + 120 q^{23} - 25 q^{25} + 210 q^{29} + 4 q^{31} - 20 q^{35} + 400 q^{37} + 240 q^{41} + 136 q^{43} - 120 q^{47} + 339 q^{49} + 60 q^{53} + 300 q^{55} - 450 q^{59} + 166 q^{61} - 20 q^{65} - 908 q^{67} + 2040 q^{71} - 500 q^{73} + 60 q^{77} + 916 q^{79} - 1140 q^{83} - 450 q^{85} + 840 q^{89} - 16 q^{91} - 140 q^{95} - 1538 q^{97}+O(q^{100}) \) 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 2.50000 + 4.33013i 0 −1.00000 + 1.73205i 0 0 0
1081.1 0 0 0 2.50000 4.33013i 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 1620.4.i.i 2
3.b odd 2 1 1620.4.i.c 2
9.c even 3 1 180.4.a.b 1
9.c even 3 1 inner 1620.4.i.i 2
9.d odd 6 1 180.4.a.e yes 1
9.d odd 6 1 1620.4.i.c 2
36.f odd 6 1 720.4.a.h 1
36.h even 6 1 720.4.a.w 1
45.h odd 6 1 900.4.a.j 1
45.j even 6 1 900.4.a.i 1
45.k odd 12 2 900.4.d.d 2
45.l even 12 2 900.4.d.i 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.4.a.b 1 9.c even 3 1
180.4.a.e yes 1 9.d odd 6 1
720.4.a.h 1 36.f odd 6 1
720.4.a.w 1 36.h even 6 1
900.4.a.i 1 45.j even 6 1
900.4.a.j 1 45.h odd 6 1
900.4.d.d 2 45.k odd 12 2
900.4.d.i 2 45.l even 12 2
1620.4.i.c 2 3.b odd 2 1
1620.4.i.c 2 9.d odd 6 1
1620.4.i.i 2 1.a even 1 1 trivial
1620.4.i.i 2 9.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(1620, [\chi])\):

\( T_{7}^{2} + 2T_{7} + 4 \) Copy content Toggle raw display
\( T_{11}^{2} - 30T_{11} + 900 \) 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
$5$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$7$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$11$ \( T^{2} - 30T + 900 \) Copy content Toggle raw display
$13$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$17$ \( (T + 90)^{2} \) Copy content Toggle raw display
$19$ \( (T + 28)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 120T + 14400 \) Copy content Toggle raw display
$29$ \( T^{2} - 210T + 44100 \) Copy content Toggle raw display
$31$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$37$ \( (T - 200)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 240T + 57600 \) Copy content Toggle raw display
$43$ \( T^{2} - 136T + 18496 \) Copy content Toggle raw display
$47$ \( T^{2} + 120T + 14400 \) Copy content Toggle raw display
$53$ \( (T - 30)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 450T + 202500 \) Copy content Toggle raw display
$61$ \( T^{2} - 166T + 27556 \) Copy content Toggle raw display
$67$ \( T^{2} + 908T + 824464 \) Copy content Toggle raw display
$71$ \( (T - 1020)^{2} \) Copy content Toggle raw display
$73$ \( (T + 250)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 916T + 839056 \) Copy content Toggle raw display
$83$ \( T^{2} + 1140 T + 1299600 \) Copy content Toggle raw display
$89$ \( (T - 420)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 1538 T + 2365444 \) Copy content Toggle raw display
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