Properties

Label 1620.4.i.r
Level $1620$
Weight $4$
Character orbit 1620.i
Analytic conductor $95.583$
Analytic rank $0$
Dimension $4$
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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-7})\)
Defining polynomial: \( x^{4} - x^{3} - x^{2} - 2x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{3} \)
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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 5 \beta_1 q^{5} + (\beta_{3} + \beta_1 + 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - 5 \beta_1 q^{5} + (\beta_{3} + \beta_1 + 1) q^{7} + (\beta_{3} - 21 \beta_1 - 21) q^{11} + (5 \beta_{3} - 5 \beta_{2} - 7 \beta_1) q^{13} + (7 \beta_{2} + 12) q^{17} + (\beta_{2} - 70) q^{19} + ( - 8 \beta_{3} + 8 \beta_{2} + 63 \beta_1) q^{23} + ( - 25 \beta_1 - 25) q^{25} + (11 \beta_{3} + 63 \beta_1 + 63) q^{29} + ( - 17 \beta_{3} + 17 \beta_{2} - 28 \beta_1) q^{31} + (5 \beta_{2} + 5) q^{35} - 286 q^{37} + (21 \beta_{3} - 21 \beta_{2} - 33 \beta_1) q^{41} + (7 \beta_{3} + 265 \beta_1 + 265) q^{43} + ( - 14 \beta_{3} + 198 \beta_1 + 198) q^{47} + (2 \beta_{3} - 2 \beta_{2} - 153 \beta_1) q^{49} + ( - 25 \beta_{2} - 252) q^{53} + (5 \beta_{2} - 105) q^{55} + (7 \beta_{3} - 7 \beta_{2} + 219 \beta_1) q^{59} + ( - 42 \beta_{3} + 301 \beta_1 + 301) q^{61} + (25 \beta_{3} - 35 \beta_1 - 35) q^{65} + (42 \beta_{3} - 42 \beta_{2} - 460 \beta_1) q^{67} + ( - 15 \beta_{2} + 399) q^{71} + ( - 27 \beta_{2} - 385) q^{73} + ( - 20 \beta_{3} + 20 \beta_{2} + 168 \beta_1) q^{77} + (21 \beta_{3} + 862 \beta_1 + 862) q^{79} + ( - 28 \beta_{3} + 243 \beta_1 + 243) q^{83} + ( - 35 \beta_{3} + 35 \beta_{2} - 60 \beta_1) q^{85} + (63 \beta_{2} - 147) q^{89} + (2 \beta_{2} - 938) q^{91} + ( - 5 \beta_{3} + 5 \beta_{2} + 350 \beta_1) q^{95} + (80 \beta_{3} - 56 \beta_1 - 56) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 10 q^{5} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 10 q^{5} + 2 q^{7} - 42 q^{11} + 14 q^{13} + 48 q^{17} - 280 q^{19} - 126 q^{23} - 50 q^{25} + 126 q^{29} + 56 q^{31} + 20 q^{35} - 1144 q^{37} + 66 q^{41} + 530 q^{43} + 396 q^{47} + 306 q^{49} - 1008 q^{53} - 420 q^{55} - 438 q^{59} + 602 q^{61} - 70 q^{65} + 920 q^{67} + 1596 q^{71} - 1540 q^{73} - 336 q^{77} + 1724 q^{79} + 486 q^{83} + 120 q^{85} - 588 q^{89} - 3752 q^{91} - 700 q^{95} - 112 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - x^{2} - 2x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + \nu^{2} - \nu - 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -3\nu^{3} + 3\nu^{2} + 9\nu + 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 15\nu^{3} + 3\nu^{2} + 9\nu - 42 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} - 9\beta_1 ) / 18 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{3} + 2\beta_{2} + 27\beta _1 + 27 ) / 18 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{3} - \beta_{2} + 45 ) / 18 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times\).

\(n\) \(811\) \(1297\) \(1541\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{1}\)

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
−0.895644 + 1.09445i
1.39564 0.228425i
−0.895644 1.09445i
1.39564 + 0.228425i
0 0 0 2.50000 + 4.33013i 0 −6.37386 + 11.0399i 0 0 0
541.2 0 0 0 2.50000 + 4.33013i 0 7.37386 12.7719i 0 0 0
1081.1 0 0 0 2.50000 4.33013i 0 −6.37386 11.0399i 0 0 0
1081.2 0 0 0 2.50000 4.33013i 0 7.37386 + 12.7719i 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.r 4
3.b odd 2 1 1620.4.i.o 4
9.c even 3 1 540.4.a.e 2
9.c even 3 1 inner 1620.4.i.r 4
9.d odd 6 1 540.4.a.h yes 2
9.d odd 6 1 1620.4.i.o 4
36.f odd 6 1 2160.4.a.x 2
36.h even 6 1 2160.4.a.bc 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
540.4.a.e 2 9.c even 3 1
540.4.a.h yes 2 9.d odd 6 1
1620.4.i.o 4 3.b odd 2 1
1620.4.i.o 4 9.d odd 6 1
1620.4.i.r 4 1.a even 1 1 trivial
1620.4.i.r 4 9.c even 3 1 inner
2160.4.a.x 2 36.f odd 6 1
2160.4.a.bc 2 36.h even 6 1

Hecke kernels

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

\( T_{7}^{4} - 2T_{7}^{3} + 192T_{7}^{2} + 376T_{7} + 35344 \) Copy content Toggle raw display
\( T_{11}^{4} + 42T_{11}^{3} + 1512T_{11}^{2} + 10584T_{11} + 63504 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} - 5 T + 25)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} - 2 T^{3} + 192 T^{2} + \cdots + 35344 \) Copy content Toggle raw display
$11$ \( T^{4} + 42 T^{3} + 1512 T^{2} + \cdots + 63504 \) Copy content Toggle raw display
$13$ \( T^{4} - 14 T^{3} + 4872 T^{2} + \cdots + 21864976 \) Copy content Toggle raw display
$17$ \( (T^{2} - 24 T - 9117)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 140 T + 4711)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 126 T^{3} + \cdots + 66048129 \) Copy content Toggle raw display
$29$ \( T^{4} - 126 T^{3} + \cdots + 357210000 \) Copy content Toggle raw display
$31$ \( T^{4} - 56 T^{3} + \cdots + 2898422569 \) Copy content Toggle raw display
$37$ \( (T + 286)^{4} \) Copy content Toggle raw display
$41$ \( T^{4} - 66 T^{3} + \cdots + 6766707600 \) Copy content Toggle raw display
$43$ \( T^{4} - 530 T^{3} + \cdots + 3716609296 \) Copy content Toggle raw display
$47$ \( T^{4} - 396 T^{3} + 154656 T^{2} + \cdots + 4665600 \) Copy content Toggle raw display
$53$ \( (T^{2} + 504 T - 54621)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 438 T^{3} + \cdots + 1497690000 \) Copy content Toggle raw display
$61$ \( T^{4} - 602 T^{3} + \cdots + 58949412025 \) Copy content Toggle raw display
$67$ \( T^{4} - 920 T^{3} + \cdots + 14834265616 \) Copy content Toggle raw display
$71$ \( (T^{2} - 798 T + 116676)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 770 T + 10444)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} - 1724 T^{3} + \cdots + 435197493025 \) Copy content Toggle raw display
$83$ \( T^{4} - 486 T^{3} + \cdots + 7943622129 \) Copy content Toggle raw display
$89$ \( (T^{2} + 294 T - 728532)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 112 T^{3} + \cdots + 1455555383296 \) Copy content Toggle raw display
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