Properties

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

Related objects

Downloads

Learn more

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 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.

\(f(q)\) \(=\) \( q + 5 \zeta_{6} q^{5} + ( - 16 \zeta_{6} + 16) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 5 \zeta_{6} q^{5} + ( - 16 \zeta_{6} + 16) q^{7} + (60 \zeta_{6} - 60) q^{11} - 86 \zeta_{6} q^{13} - 18 q^{17} + 44 q^{19} + 48 \zeta_{6} q^{23} + (25 \zeta_{6} - 25) q^{25} + (186 \zeta_{6} - 186) q^{29} - 176 \zeta_{6} q^{31} + 80 q^{35} + 254 q^{37} + 186 \zeta_{6} q^{41} + ( - 100 \zeta_{6} + 100) q^{43} + ( - 168 \zeta_{6} + 168) q^{47} + 87 \zeta_{6} q^{49} + 498 q^{53} - 300 q^{55} - 252 \zeta_{6} q^{59} + ( - 58 \zeta_{6} + 58) q^{61} + ( - 430 \zeta_{6} + 430) q^{65} + 1036 \zeta_{6} q^{67} - 168 q^{71} + 506 q^{73} + 960 \zeta_{6} q^{77} + (272 \zeta_{6} - 272) q^{79} + ( - 948 \zeta_{6} + 948) q^{83} - 90 \zeta_{6} q^{85} + 1014 q^{89} - 1376 q^{91} + 220 \zeta_{6} q^{95} + ( - 766 \zeta_{6} + 766) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 5 q^{5} + 16 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 5 q^{5} + 16 q^{7} - 60 q^{11} - 86 q^{13} - 36 q^{17} + 88 q^{19} + 48 q^{23} - 25 q^{25} - 186 q^{29} - 176 q^{31} + 160 q^{35} + 508 q^{37} + 186 q^{41} + 100 q^{43} + 168 q^{47} + 87 q^{49} + 996 q^{53} - 600 q^{55} - 252 q^{59} + 58 q^{61} + 430 q^{65} + 1036 q^{67} - 336 q^{71} + 1012 q^{73} + 960 q^{77} - 272 q^{79} + 948 q^{83} - 90 q^{85} + 2028 q^{89} - 2752 q^{91} + 220 q^{95} + 766 q^{97}+O(q^{100}) \) 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 8.00000 13.8564i 0 0 0
1081.1 0 0 0 2.50000 4.33013i 0 8.00000 + 13.8564i 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.j 2
3.b odd 2 1 1620.4.i.d 2
9.c even 3 1 180.4.a.a 1
9.c even 3 1 inner 1620.4.i.j 2
9.d odd 6 1 20.4.a.a 1
9.d odd 6 1 1620.4.i.d 2
36.f odd 6 1 720.4.a.k 1
36.h even 6 1 80.4.a.c 1
45.h odd 6 1 100.4.a.a 1
45.j even 6 1 900.4.a.m 1
45.k odd 12 2 900.4.d.k 2
45.l even 12 2 100.4.c.a 2
63.i even 6 1 980.4.i.n 2
63.j odd 6 1 980.4.i.e 2
63.n odd 6 1 980.4.i.e 2
63.o even 6 1 980.4.a.c 1
63.s even 6 1 980.4.i.n 2
72.j odd 6 1 320.4.a.d 1
72.l even 6 1 320.4.a.k 1
99.g even 6 1 2420.4.a.d 1
144.u even 12 2 1280.4.d.c 2
144.w odd 12 2 1280.4.d.n 2
180.n even 6 1 400.4.a.o 1
180.v odd 12 2 400.4.c.j 2
360.bd even 6 1 1600.4.a.p 1
360.bh odd 6 1 1600.4.a.bl 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.4.a.a 1 9.d odd 6 1
80.4.a.c 1 36.h even 6 1
100.4.a.a 1 45.h odd 6 1
100.4.c.a 2 45.l even 12 2
180.4.a.a 1 9.c even 3 1
320.4.a.d 1 72.j odd 6 1
320.4.a.k 1 72.l even 6 1
400.4.a.o 1 180.n even 6 1
400.4.c.j 2 180.v odd 12 2
720.4.a.k 1 36.f odd 6 1
900.4.a.m 1 45.j even 6 1
900.4.d.k 2 45.k odd 12 2
980.4.a.c 1 63.o even 6 1
980.4.i.e 2 63.j odd 6 1
980.4.i.e 2 63.n odd 6 1
980.4.i.n 2 63.i even 6 1
980.4.i.n 2 63.s even 6 1
1280.4.d.c 2 144.u even 12 2
1280.4.d.n 2 144.w odd 12 2
1600.4.a.p 1 360.bd even 6 1
1600.4.a.bl 1 360.bh odd 6 1
1620.4.i.d 2 3.b odd 2 1
1620.4.i.d 2 9.d odd 6 1
1620.4.i.j 2 1.a even 1 1 trivial
1620.4.i.j 2 9.c even 3 1 inner
2420.4.a.d 1 99.g even 6 1

Hecke kernels

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

\( T_{7}^{2} - 16T_{7} + 256 \) Copy content Toggle raw display
\( T_{11}^{2} + 60T_{11} + 3600 \) 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
$7$ \( T^{2} - 16T + 256 \) Copy content Toggle raw display
$11$ \( T^{2} + 60T + 3600 \) Copy content Toggle raw display
$13$ \( T^{2} + 86T + 7396 \) Copy content Toggle raw display
$17$ \( (T + 18)^{2} \) Copy content Toggle raw display
$19$ \( (T - 44)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 48T + 2304 \) Copy content Toggle raw display
$29$ \( T^{2} + 186T + 34596 \) Copy content Toggle raw display
$31$ \( T^{2} + 176T + 30976 \) Copy content Toggle raw display
$37$ \( (T - 254)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 186T + 34596 \) Copy content Toggle raw display
$43$ \( T^{2} - 100T + 10000 \) Copy content Toggle raw display
$47$ \( T^{2} - 168T + 28224 \) Copy content Toggle raw display
$53$ \( (T - 498)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 252T + 63504 \) Copy content Toggle raw display
$61$ \( T^{2} - 58T + 3364 \) Copy content Toggle raw display
$67$ \( T^{2} - 1036 T + 1073296 \) Copy content Toggle raw display
$71$ \( (T + 168)^{2} \) Copy content Toggle raw display
$73$ \( (T - 506)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 272T + 73984 \) Copy content Toggle raw display
$83$ \( T^{2} - 948T + 898704 \) Copy content Toggle raw display
$89$ \( (T - 1014)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 766T + 586756 \) Copy content Toggle raw display
show more
show less