Properties

Label 1620.4.i.q
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: \(\Q(\sqrt{-3}, \sqrt{-23})\)
Defining polynomial: \( x^{4} - x^{3} - 5x^{2} - 6x + 36 \) 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.

\(f(q)\) \(=\) \( q - 5 \beta_1 q^{5} + ( - \beta_{3} - 5 \beta_1 - 5) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - 5 \beta_1 q^{5} + ( - \beta_{3} - 5 \beta_1 - 5) q^{7} + ( - \beta_{3} + 9 \beta_1 + 9) q^{11} + (\beta_{3} - \beta_{2} + 17 \beta_1) q^{13} + ( - \beta_{2} - 12) q^{17} + ( - \beta_{2} + 14) q^{19} + ( - 4 \beta_{3} + 4 \beta_{2} - 15 \beta_1) q^{23} + ( - 25 \beta_1 - 25) q^{25} + ( - 5 \beta_{3} - 21 \beta_1 - 21) q^{29} + ( - 7 \beta_{3} + 7 \beta_{2} + 128 \beta_1) q^{31} + ( - 5 \beta_{2} - 25) q^{35} + 74 q^{37} + (9 \beta_{3} - 9 \beta_{2} + 111 \beta_1) q^{41} + (5 \beta_{3} - 209 \beta_1 - 209) q^{43} + ( - 10 \beta_{3} - 6 \beta_1 - 6) q^{47} + (10 \beta_{3} - 10 \beta_{2} + 303 \beta_1) q^{49} + ( - 17 \beta_{2} + 24) q^{53} + ( - 5 \beta_{2} + 45) q^{55} + ( - 7 \beta_{3} + 7 \beta_{2} + 177 \beta_1) q^{59} + ( - 6 \beta_{3} - 419 \beta_1 - 419) q^{61} + (5 \beta_{3} + 85 \beta_1 + 85) q^{65} + (6 \beta_{3} - 6 \beta_{2} + 620 \beta_1) q^{67} + (27 \beta_{2} - 231) q^{71} + (21 \beta_{2} + 443) q^{73} + ( - 4 \beta_{3} + 4 \beta_{2} + 576 \beta_1) q^{77} + (3 \beta_{3} - 578 \beta_1 - 578) q^{79} + (28 \beta_{3} + 573 \beta_1 + 573) q^{83} + (5 \beta_{3} - 5 \beta_{2} + 60 \beta_1) q^{85} + (3 \beta_{2} - 75) q^{89} + (22 \beta_{2} + 706) q^{91} + (5 \beta_{3} - 5 \beta_{2} - 70 \beta_1) q^{95} + (16 \beta_{3} - 392 \beta_1 - 392) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 10 q^{5} - 10 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 10 q^{5} - 10 q^{7} + 18 q^{11} - 34 q^{13} - 48 q^{17} + 56 q^{19} + 30 q^{23} - 50 q^{25} - 42 q^{29} - 256 q^{31} - 100 q^{35} + 296 q^{37} - 222 q^{41} - 418 q^{43} - 12 q^{47} - 606 q^{49} + 96 q^{53} + 180 q^{55} - 354 q^{59} - 838 q^{61} + 170 q^{65} - 1240 q^{67} - 924 q^{71} + 1772 q^{73} - 1152 q^{77} - 1156 q^{79} + 1146 q^{83} - 120 q^{85} - 300 q^{89} + 2824 q^{91} + 140 q^{95} - 784 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 5x^{2} - 6x + 36 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 5\nu^{2} - 5\nu - 36 ) / 30 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\nu^{3} + \nu^{2} + 11\nu + 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 13\nu^{3} + 5\nu^{2} + 55\nu - 138 ) / 10 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} - 9\beta_1 ) / 18 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{3} + 2\beta_{2} + 99\beta _1 + 99 ) / 18 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 10\beta_{3} - 5\beta_{2} + 153 ) / 18 \) 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
2.32666 0.765945i
−1.82666 + 1.63197i
2.32666 + 0.765945i
−1.82666 1.63197i
0 0 0 2.50000 + 4.33013i 0 −14.9599 + 25.9114i 0 0 0
541.2 0 0 0 2.50000 + 4.33013i 0 9.95994 17.2511i 0 0 0
1081.1 0 0 0 2.50000 4.33013i 0 −14.9599 25.9114i 0 0 0
1081.2 0 0 0 2.50000 4.33013i 0 9.95994 + 17.2511i 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.q 4
3.b odd 2 1 1620.4.i.n 4
9.c even 3 1 540.4.a.f 2
9.c even 3 1 inner 1620.4.i.q 4
9.d odd 6 1 540.4.a.i yes 2
9.d odd 6 1 1620.4.i.n 4
36.f odd 6 1 2160.4.a.v 2
36.h even 6 1 2160.4.a.ba 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
540.4.a.f 2 9.c even 3 1
540.4.a.i yes 2 9.d odd 6 1
1620.4.i.n 4 3.b odd 2 1
1620.4.i.n 4 9.d odd 6 1
1620.4.i.q 4 1.a even 1 1 trivial
1620.4.i.q 4 9.c even 3 1 inner
2160.4.a.v 2 36.f odd 6 1
2160.4.a.ba 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 \(S_{4}^{\mathrm{new}}(1620, [\chi])\):

\( T_{7}^{4} + 10T_{7}^{3} + 696T_{7}^{2} - 5960T_{7} + 355216 \) Copy content Toggle raw display
\( T_{11}^{4} - 18T_{11}^{3} + 864T_{11}^{2} + 9720T_{11} + 291600 \) 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
$7$ \( T^{4} + 10 T^{3} + 696 T^{2} + \cdots + 355216 \) Copy content Toggle raw display
$11$ \( T^{4} - 18 T^{3} + 864 T^{2} + \cdots + 291600 \) Copy content Toggle raw display
$13$ \( T^{4} + 34 T^{3} + 1488 T^{2} + \cdots + 110224 \) Copy content Toggle raw display
$17$ \( (T^{2} + 24 T - 477)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 28 T - 425)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 30 T^{3} + 10611 T^{2} + \cdots + 94303521 \) Copy content Toggle raw display
$29$ \( T^{4} + 42 T^{3} + \cdots + 227527056 \) Copy content Toggle raw display
$31$ \( T^{4} + 256 T^{3} + \cdots + 197262025 \) Copy content Toggle raw display
$37$ \( (T - 74)^{4} \) Copy content Toggle raw display
$41$ \( T^{4} + 222 T^{3} + \cdots + 1442480400 \) Copy content Toggle raw display
$43$ \( T^{4} + 418 T^{3} + \cdots + 792760336 \) Copy content Toggle raw display
$47$ \( T^{4} + 12 T^{3} + \cdots + 3851940096 \) Copy content Toggle raw display
$53$ \( (T^{2} - 48 T - 178893)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 354 T^{3} + 124416 T^{2} + \cdots + 810000 \) Copy content Toggle raw display
$61$ \( T^{4} + 838 T^{3} + \cdots + 23471772025 \) Copy content Toggle raw display
$67$ \( T^{4} + 1240 T^{3} + \cdots + 131075857936 \) Copy content Toggle raw display
$71$ \( (T^{2} + 462 T - 399348)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 886 T - 77612)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + 1156 T^{3} + \cdots + 107908965025 \) Copy content Toggle raw display
$83$ \( T^{4} - 1146 T^{3} + \cdots + 25133346225 \) Copy content Toggle raw display
$89$ \( (T^{2} + 150 T + 36)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 784 T^{3} + \cdots + 28217344 \) Copy content Toggle raw display
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