Properties

Label 1620.4.i.l
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 60)
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.

\(f(q)\) \(=\) \( q + 5 \zeta_{6} q^{5} + ( - 28 \zeta_{6} + 28) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 5 \zeta_{6} q^{5} + ( - 28 \zeta_{6} + 28) q^{7} + ( - 24 \zeta_{6} + 24) q^{11} + 70 \zeta_{6} q^{13} + 102 q^{17} + 20 q^{19} + 72 \zeta_{6} q^{23} + (25 \zeta_{6} - 25) q^{25} + (306 \zeta_{6} - 306) q^{29} + 136 \zeta_{6} q^{31} + 140 q^{35} - 214 q^{37} + 150 \zeta_{6} q^{41} + ( - 292 \zeta_{6} + 292) q^{43} + ( - 72 \zeta_{6} + 72) q^{47} - 441 \zeta_{6} q^{49} - 414 q^{53} + 120 q^{55} + 744 \zeta_{6} q^{59} + ( - 418 \zeta_{6} + 418) q^{61} + (350 \zeta_{6} - 350) q^{65} - 188 \zeta_{6} q^{67} + 480 q^{71} + 434 q^{73} - 672 \zeta_{6} q^{77} + (1352 \zeta_{6} - 1352) q^{79} + ( - 612 \zeta_{6} + 612) q^{83} + 510 \zeta_{6} q^{85} - 30 q^{89} + 1960 q^{91} + 100 \zeta_{6} q^{95} + ( - 286 \zeta_{6} + 286) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 5 q^{5} + 28 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 5 q^{5} + 28 q^{7} + 24 q^{11} + 70 q^{13} + 204 q^{17} + 40 q^{19} + 72 q^{23} - 25 q^{25} - 306 q^{29} + 136 q^{31} + 280 q^{35} - 428 q^{37} + 150 q^{41} + 292 q^{43} + 72 q^{47} - 441 q^{49} - 828 q^{53} + 240 q^{55} + 744 q^{59} + 418 q^{61} - 350 q^{65} - 188 q^{67} + 960 q^{71} + 868 q^{73} - 672 q^{77} - 1352 q^{79} + 612 q^{83} + 510 q^{85} - 60 q^{89} + 3920 q^{91} + 100 q^{95} + 286 q^{97}+O(q^{100}) \) 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 14.0000 24.2487i 0 0 0
1081.1 0 0 0 2.50000 4.33013i 0 14.0000 + 24.2487i 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.l 2
3.b odd 2 1 1620.4.i.f 2
9.c even 3 1 60.4.a.a 1
9.c even 3 1 inner 1620.4.i.l 2
9.d odd 6 1 180.4.a.d 1
9.d odd 6 1 1620.4.i.f 2
36.f odd 6 1 240.4.a.i 1
36.h even 6 1 720.4.a.bb 1
45.h odd 6 1 900.4.a.q 1
45.j even 6 1 300.4.a.i 1
45.k odd 12 2 300.4.d.b 2
45.l even 12 2 900.4.d.h 2
72.n even 6 1 960.4.a.bc 1
72.p odd 6 1 960.4.a.r 1
180.p odd 6 1 1200.4.a.a 1
180.x even 12 2 1200.4.f.n 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.4.a.a 1 9.c even 3 1
180.4.a.d 1 9.d odd 6 1
240.4.a.i 1 36.f odd 6 1
300.4.a.i 1 45.j even 6 1
300.4.d.b 2 45.k odd 12 2
720.4.a.bb 1 36.h even 6 1
900.4.a.q 1 45.h odd 6 1
900.4.d.h 2 45.l even 12 2
960.4.a.r 1 72.p odd 6 1
960.4.a.bc 1 72.n even 6 1
1200.4.a.a 1 180.p odd 6 1
1200.4.f.n 2 180.x even 12 2
1620.4.i.f 2 3.b odd 2 1
1620.4.i.f 2 9.d odd 6 1
1620.4.i.l 2 1.a even 1 1 trivial
1620.4.i.l 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 \(S_{4}^{\mathrm{new}}(1620, [\chi])\):

\( T_{7}^{2} - 28T_{7} + 784 \) Copy content Toggle raw display
\( T_{11}^{2} - 24T_{11} + 576 \) 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
$7$ \( T^{2} - 28T + 784 \) Copy content Toggle raw display
$11$ \( T^{2} - 24T + 576 \) Copy content Toggle raw display
$13$ \( T^{2} - 70T + 4900 \) Copy content Toggle raw display
$17$ \( (T - 102)^{2} \) Copy content Toggle raw display
$19$ \( (T - 20)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 72T + 5184 \) Copy content Toggle raw display
$29$ \( T^{2} + 306T + 93636 \) Copy content Toggle raw display
$31$ \( T^{2} - 136T + 18496 \) Copy content Toggle raw display
$37$ \( (T + 214)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 150T + 22500 \) Copy content Toggle raw display
$43$ \( T^{2} - 292T + 85264 \) Copy content Toggle raw display
$47$ \( T^{2} - 72T + 5184 \) Copy content Toggle raw display
$53$ \( (T + 414)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 744T + 553536 \) Copy content Toggle raw display
$61$ \( T^{2} - 418T + 174724 \) Copy content Toggle raw display
$67$ \( T^{2} + 188T + 35344 \) Copy content Toggle raw display
$71$ \( (T - 480)^{2} \) Copy content Toggle raw display
$73$ \( (T - 434)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 1352 T + 1827904 \) Copy content Toggle raw display
$83$ \( T^{2} - 612T + 374544 \) Copy content Toggle raw display
$89$ \( (T + 30)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 286T + 81796 \) Copy content Toggle raw display
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