Properties

Label 1620.4.i.m
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: \(\Q(\sqrt{-3}, \sqrt{41})\)
Defining polynomial: \( x^{4} - x^{3} + 11x^{2} + 10x + 100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3^{2} \)
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.

\(f(q)\) \(=\) \( q + (5 \beta_1 - 5) q^{5} + (\beta_{2} - 7 \beta_1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (5 \beta_1 - 5) q^{5} + (\beta_{2} - 7 \beta_1) q^{7} + (5 \beta_{2} - 10 \beta_1) q^{11} + ( - 4 \beta_{3} - 4 \beta_{2} + 7 \beta_1 - 3) q^{13} + ( - 5 \beta_{3} + 25) q^{17} + ( - 13 \beta_{3} + 28) q^{19} + (5 \beta_{3} + 5 \beta_{2} + 50 \beta_1 - 55) q^{23} - 25 \beta_1 q^{25} + ( - 5 \beta_{2} - 50 \beta_1) q^{29} + (\beta_{3} + \beta_{2} - 36 \beta_1 + 35) q^{31} + (5 \beta_{3} + 30) q^{35} + (15 \beta_{3} - 82) q^{37} + ( - 30 \beta_{3} - 30 \beta_{2} + 210 \beta_1 - 180) q^{41} + (7 \beta_{2} - 88 \beta_1) q^{43} + ( - 25 \beta_{2} - 160 \beta_1) q^{47} + ( - 13 \beta_{3} - 13 \beta_{2} - 202 \beta_1 + 215) q^{49} + ( - 10 \beta_{3} + 380) q^{53} + (25 \beta_{3} + 25) q^{55} + (20 \beta_{3} + 20 \beta_{2} + 440 \beta_1 - 460) q^{59} + ( - 51 \beta_{2} + 25 \beta_1) q^{61} + (20 \beta_{2} - 35 \beta_1) q^{65} + (63 \beta_{3} + 63 \beta_{2} - 199 \beta_1 + 136) q^{67} + ( - 30 \beta_{3} + 840) q^{71} + ( - 15 \beta_{3} + 86) q^{73} + ( - 40 \beta_{3} - 40 \beta_{2} + 530 \beta_1 - 490) q^{77} + (60 \beta_{2} - 203 \beta_1) q^{79} + (100 \beta_{2} - 350 \beta_1) q^{83} + (25 \beta_{3} + 25 \beta_{2} + 100 \beta_1 - 125) q^{85} + (60 \beta_{3} + 720) q^{89} + (31 \beta_{3} + 386) q^{91} + (65 \beta_{3} + 65 \beta_{2} + 75 \beta_1 - 140) q^{95} + (29 \beta_{2} + 297 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 10 q^{5} - 13 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 10 q^{5} - 13 q^{7} - 15 q^{11} - 10 q^{13} + 90 q^{17} + 86 q^{19} - 105 q^{23} - 50 q^{25} - 105 q^{29} + 71 q^{31} + 130 q^{35} - 298 q^{37} - 390 q^{41} - 169 q^{43} - 345 q^{47} + 417 q^{49} + 1500 q^{53} + 150 q^{55} - 900 q^{59} - q^{61} - 50 q^{65} + 335 q^{67} + 3300 q^{71} + 314 q^{73} - 1020 q^{77} - 346 q^{79} - 600 q^{83} - 225 q^{85} + 3000 q^{89} + 1606 q^{91} - 215 q^{95} + 623 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 11x^{2} + 10x + 100 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{3} + 11\nu^{2} - 11\nu + 100 ) / 110 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 11\nu^{2} + 341\nu - 100 ) / 110 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3\nu^{3} + 52 ) / 11 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + \beta_{2} + 31\beta _1 - 32 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 11\beta_{3} - 52 ) / 3 \) Copy content Toggle raw display

Character values

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

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

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
−1.35078 + 2.33962i
1.85078 3.20565i
−1.35078 2.33962i
1.85078 + 3.20565i
0 0 0 −2.50000 4.33013i 0 −8.05234 + 13.9471i 0 0 0
541.2 0 0 0 −2.50000 4.33013i 0 1.55234 2.68874i 0 0 0
1081.1 0 0 0 −2.50000 + 4.33013i 0 −8.05234 13.9471i 0 0 0
1081.2 0 0 0 −2.50000 + 4.33013i 0 1.55234 + 2.68874i 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.m 4
3.b odd 2 1 1620.4.i.p 4
9.c even 3 1 540.4.a.j yes 2
9.c even 3 1 inner 1620.4.i.m 4
9.d odd 6 1 540.4.a.g 2
9.d odd 6 1 1620.4.i.p 4
36.f odd 6 1 2160.4.a.z 2
36.h even 6 1 2160.4.a.u 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
540.4.a.g 2 9.d odd 6 1
540.4.a.j yes 2 9.c even 3 1
1620.4.i.m 4 1.a even 1 1 trivial
1620.4.i.m 4 9.c even 3 1 inner
1620.4.i.p 4 3.b odd 2 1
1620.4.i.p 4 9.d odd 6 1
2160.4.a.u 2 36.h even 6 1
2160.4.a.z 2 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 \(S_{4}^{\mathrm{new}}(1620, [\chi])\):

\( T_{7}^{4} + 13T_{7}^{3} + 219T_{7}^{2} - 650T_{7} + 2500 \) Copy content Toggle raw display
\( T_{11}^{4} + 15T_{11}^{3} + 2475T_{11}^{2} - 33750T_{11} + 5062500 \) 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
$7$ \( T^{4} + 13 T^{3} + 219 T^{2} + \cdots + 2500 \) Copy content Toggle raw display
$11$ \( T^{4} + 15 T^{3} + 2475 T^{2} + \cdots + 5062500 \) Copy content Toggle raw display
$13$ \( T^{4} + 10 T^{3} + 1551 T^{2} + \cdots + 2105401 \) Copy content Toggle raw display
$17$ \( (T^{2} - 45 T - 1800)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 43 T - 15128)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 105 T^{3} + 10575 T^{2} + \cdots + 202500 \) Copy content Toggle raw display
$29$ \( T^{4} + 105 T^{3} + 10575 T^{2} + \cdots + 202500 \) Copy content Toggle raw display
$31$ \( T^{4} - 71 T^{3} + 3873 T^{2} + \cdots + 1364224 \) Copy content Toggle raw display
$37$ \( (T^{2} + 149 T - 15206)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 390 T^{3} + \cdots + 2025000000 \) Copy content Toggle raw display
$43$ \( T^{4} + 169 T^{3} + 25941 T^{2} + \cdots + 6864400 \) Copy content Toggle raw display
$47$ \( T^{4} + 345 T^{3} + \cdots + 778410000 \) Copy content Toggle raw display
$53$ \( (T^{2} - 750 T + 131400)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 900 T^{3} + \cdots + 27423360000 \) Copy content Toggle raw display
$61$ \( T^{4} + T^{3} + 239943 T^{2} + \cdots + 57572163364 \) Copy content Toggle raw display
$67$ \( T^{4} - 335 T^{3} + \cdots + 114300791056 \) Copy content Toggle raw display
$71$ \( (T^{2} - 1650 T + 597600)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 157 T - 14594)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + 346 T^{3} + \cdots + 91307313241 \) Copy content Toggle raw display
$83$ \( T^{4} + 600 T^{3} + \cdots + 693056250000 \) Copy content Toggle raw display
$89$ \( (T^{2} - 1500 T + 230400)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} - 623 T^{3} + \cdots + 378302500 \) Copy content Toggle raw display
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