Properties

Label 1620.4.i.b
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 540)
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.

\(f(q)\) \(=\) \( q - 5 \zeta_{6} q^{5} + (17 \zeta_{6} - 17) q^{7} +O(q^{10}) \) Copy content Toggle raw display \( q - 5 \zeta_{6} q^{5} + (17 \zeta_{6} - 17) q^{7} + ( - 30 \zeta_{6} + 30) q^{11} + 61 \zeta_{6} q^{13} - 120 q^{17} - 43 q^{19} - 90 \zeta_{6} q^{23} + (25 \zeta_{6} - 25) q^{25} + ( - 90 \zeta_{6} + 90) q^{29} - 8 \zeta_{6} q^{31} + 85 q^{35} + 317 q^{37} + 30 \zeta_{6} q^{41} + ( - 220 \zeta_{6} + 220) q^{43} + ( - 180 \zeta_{6} + 180) q^{47} + 54 \zeta_{6} q^{49} - 630 q^{53} - 150 q^{55} + 840 \zeta_{6} q^{59} + (599 \zeta_{6} - 599) q^{61} + ( - 305 \zeta_{6} + 305) q^{65} - 107 \zeta_{6} q^{67} - 210 q^{71} - 421 q^{73} + 510 \zeta_{6} q^{77} + (353 \zeta_{6} - 353) q^{79} + ( - 1350 \zeta_{6} + 1350) q^{83} + 600 \zeta_{6} q^{85} + 1020 q^{89} - 1037 q^{91} + 215 \zeta_{6} q^{95} + ( - 997 \zeta_{6} + 997) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 5 q^{5} - 17 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 5 q^{5} - 17 q^{7} + 30 q^{11} + 61 q^{13} - 240 q^{17} - 86 q^{19} - 90 q^{23} - 25 q^{25} + 90 q^{29} - 8 q^{31} + 170 q^{35} + 634 q^{37} + 30 q^{41} + 220 q^{43} + 180 q^{47} + 54 q^{49} - 1260 q^{53} - 300 q^{55} + 840 q^{59} - 599 q^{61} + 305 q^{65} - 107 q^{67} - 420 q^{71} - 842 q^{73} + 510 q^{77} - 353 q^{79} + 1350 q^{83} + 600 q^{85} + 2040 q^{89} - 2074 q^{91} + 215 q^{95} + 997 q^{97}+O(q^{100}) \) 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 −8.50000 + 14.7224i 0 0 0
1081.1 0 0 0 −2.50000 + 4.33013i 0 −8.50000 14.7224i 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.b 2
3.b odd 2 1 1620.4.i.h 2
9.c even 3 1 540.4.a.d yes 1
9.c even 3 1 inner 1620.4.i.b 2
9.d odd 6 1 540.4.a.b 1
9.d odd 6 1 1620.4.i.h 2
36.f odd 6 1 2160.4.a.k 1
36.h even 6 1 2160.4.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
540.4.a.b 1 9.d odd 6 1
540.4.a.d yes 1 9.c even 3 1
1620.4.i.b 2 1.a even 1 1 trivial
1620.4.i.b 2 9.c even 3 1 inner
1620.4.i.h 2 3.b odd 2 1
1620.4.i.h 2 9.d odd 6 1
2160.4.a.a 1 36.h even 6 1
2160.4.a.k 1 36.f odd 6 1

Hecke kernels

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

\( T_{7}^{2} + 17T_{7} + 289 \) 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
$7$ \( T^{2} + 17T + 289 \) Copy content Toggle raw display
$11$ \( T^{2} - 30T + 900 \) Copy content Toggle raw display
$13$ \( T^{2} - 61T + 3721 \) Copy content Toggle raw display
$17$ \( (T + 120)^{2} \) Copy content Toggle raw display
$19$ \( (T + 43)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 90T + 8100 \) Copy content Toggle raw display
$29$ \( T^{2} - 90T + 8100 \) Copy content Toggle raw display
$31$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$37$ \( (T - 317)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 30T + 900 \) Copy content Toggle raw display
$43$ \( T^{2} - 220T + 48400 \) Copy content Toggle raw display
$47$ \( T^{2} - 180T + 32400 \) Copy content Toggle raw display
$53$ \( (T + 630)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 840T + 705600 \) Copy content Toggle raw display
$61$ \( T^{2} + 599T + 358801 \) Copy content Toggle raw display
$67$ \( T^{2} + 107T + 11449 \) Copy content Toggle raw display
$71$ \( (T + 210)^{2} \) Copy content Toggle raw display
$73$ \( (T + 421)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 353T + 124609 \) Copy content Toggle raw display
$83$ \( T^{2} - 1350 T + 1822500 \) Copy content Toggle raw display
$89$ \( (T - 1020)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 997T + 994009 \) Copy content Toggle raw display
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