Properties

Label 1620.4.i.u
Level $1620$
Weight $4$
Character orbit 1620.i
Analytic conductor $95.583$
Analytic rank $0$
Dimension $6$
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: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \( x^{6} - x^{5} + 91x^{4} - 294x^{3} + 8292x^{2} - 17280x + 36864 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (5 \beta_{2} + 5) q^{5} + ( - \beta_{4} + 5 \beta_{2}) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (5 \beta_{2} + 5) q^{5} + ( - \beta_{4} + 5 \beta_{2}) q^{7} + (\beta_{5} - 8 \beta_{2}) q^{11} + (\beta_{4} - \beta_{3} - 11 \beta_{2} - 11) q^{13} + ( - 2 \beta_{3} + 2 \beta_1 - 14) q^{17} + ( - 3 \beta_{3} + \beta_1 + 7) q^{19} + ( - 2 \beta_{5} - 3 \beta_{4} + 3 \beta_{3} - 11 \beta_{2} + 2 \beta_1 - 11) q^{23} + 25 \beta_{2} q^{25} + ( - 2 \beta_{5} + 5 \beta_{4} - 74 \beta_{2}) q^{29} + (3 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} - 44 \beta_{2} - 3 \beta_1 - 44) q^{31} + ( - 5 \beta_{3} - 25) q^{35} + ( - 8 \beta_{3} - 2 \beta_1 + 58) q^{37} + ( - 6 \beta_{4} + 6 \beta_{3} - 33 \beta_{2} - 33) q^{41} + ( - 6 \beta_{5} + 2 \beta_{4} - 40 \beta_{2}) q^{43} + ( - 2 \beta_{5} + 3 \beta_{4} - 179 \beta_{2}) q^{47} + ( - 8 \beta_{5} + 17 \beta_{4} - 17 \beta_{3} - 164 \beta_{2} + 8 \beta_1 - 164) q^{49} + (9 \beta_{3} - 10 \beta_1 - 89) q^{53} + (5 \beta_1 + 40) q^{55} + (9 \beta_{5} + 5 \beta_{4} - 5 \beta_{3} - 75 \beta_{2} - 9 \beta_1 - 75) q^{59} + (12 \beta_{5} + 6 \beta_{4} - 160 \beta_{2}) q^{61} + (5 \beta_{4} - 55 \beta_{2}) q^{65} + (10 \beta_{5} - 20 \beta_{4} + 20 \beta_{3} - 4 \beta_{2} - 10 \beta_1 - 4) q^{67} + ( - 8 \beta_{3} + \beta_1 - 190) q^{71} + (2 \beta_{3} - 10 \beta_1 + 354) q^{73} + (2 \beta_{5} + 16 \beta_{4} - 16 \beta_{3} + 104 \beta_{2} - 2 \beta_1 + 104) q^{77} + (4 \beta_{5} - 26 \beta_{4} - 342 \beta_{2}) q^{79} + (18 \beta_{5} - 32 \beta_{4} - 234 \beta_{2}) q^{83} + ( - 10 \beta_{5} + 10 \beta_{4} - 10 \beta_{3} - 70 \beta_{2} + 10 \beta_1 - 70) q^{85} + (11 \beta_{3} + 8 \beta_1 - 380) q^{89} + (23 \beta_{3} - 8 \beta_1 + 537) q^{91} + ( - 5 \beta_{5} + 15 \beta_{4} - 15 \beta_{3} + 35 \beta_{2} + 5 \beta_1 + 35) q^{95} + ( - 2 \beta_{5} + 26 \beta_{4} - 852 \beta_{2}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 15 q^{5} - 15 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 15 q^{5} - 15 q^{7} + 24 q^{11} - 33 q^{13} - 84 q^{17} + 42 q^{19} - 33 q^{23} - 75 q^{25} + 222 q^{29} - 132 q^{31} - 150 q^{35} + 348 q^{37} - 99 q^{41} + 120 q^{43} + 537 q^{47} - 492 q^{49} - 534 q^{53} + 240 q^{55} - 225 q^{59} + 480 q^{61} + 165 q^{65} - 12 q^{67} - 1140 q^{71} + 2124 q^{73} + 312 q^{77} + 1026 q^{79} + 702 q^{83} - 210 q^{85} - 2280 q^{89} + 3222 q^{91} + 105 q^{95} + 2556 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} + 91x^{4} - 294x^{3} + 8292x^{2} - 17280x + 36864 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 79\nu^{5} - 7189\nu^{4} - 83093\nu^{3} - 655068\nu^{2} + 1365120\nu - 29852544 ) / 1474584 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -1365\nu^{5} + 1333\nu^{4} - 121303\nu^{3} + 136318\nu^{2} - 11053236\nu - 559104 ) / 23593344 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -91\nu^{5} + 8281\nu^{4} - 16279\nu^{3} + 754572\nu^{2} - 1572480\nu + 45773136 ) / 1474584 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 6585\nu^{5} + 15175\nu^{4} + 585187\nu^{3} - 657622\nu^{2} + 53085092\nu + 2697216 ) / 3932224 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -660\nu^{5} - 1381\nu^{4} - 58652\nu^{3} + 65912\nu^{2} - 3110265\nu - 270336 ) / 368646 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} + \beta_{4} - 2\beta_{2} ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} - 11\beta_{4} + 11\beta_{3} - 362\beta_{2} - \beta _1 - 362 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -79\beta_{3} - 91\beta _1 + 610 ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 11\beta_{5} + 1103\beta_{4} + 31586\beta_{2} ) / 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -7987\beta_{5} - 8119\beta_{4} + 8119\beta_{3} - 92818\beta_{2} + 7987\beta _1 - 92818 ) / 6 \) Copy content Toggle raw display

Character values

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

\(n\) \(811\) \(1297\) \(1541\)
\(\chi(n)\) \(1\) \(1\) \(-1 - \beta_{2}\)

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
1.09880 1.90317i
4.38373 7.59284i
−4.98253 + 8.62999i
1.09880 + 1.90317i
4.38373 + 7.59284i
−4.98253 8.62999i
0 0 0 2.50000 + 4.33013i 0 −16.8420 + 29.1713i 0 0 0
541.2 0 0 0 2.50000 + 4.33013i 0 −0.474792 + 0.822364i 0 0 0
541.3 0 0 0 2.50000 + 4.33013i 0 9.81683 17.0033i 0 0 0
1081.1 0 0 0 2.50000 4.33013i 0 −16.8420 29.1713i 0 0 0
1081.2 0 0 0 2.50000 4.33013i 0 −0.474792 0.822364i 0 0 0
1081.3 0 0 0 2.50000 4.33013i 0 9.81683 + 17.0033i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1081.3
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.u 6
3.b odd 2 1 1620.4.i.s 6
9.c even 3 1 1620.4.a.d 3
9.c even 3 1 inner 1620.4.i.u 6
9.d odd 6 1 1620.4.a.f yes 3
9.d odd 6 1 1620.4.i.s 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1620.4.a.d 3 9.c even 3 1
1620.4.a.f yes 3 9.d odd 6 1
1620.4.i.s 6 3.b odd 2 1
1620.4.i.s 6 9.d odd 6 1
1620.4.i.u 6 1.a even 1 1 trivial
1620.4.i.u 6 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 \(S_{4}^{\mathrm{new}}(1620, [\chi])\):

\( T_{7}^{6} + 15T_{7}^{5} + 873T_{7}^{4} - 8464T_{7}^{3} + 429324T_{7}^{2} + 406944T_{7} + 394384 \) Copy content Toggle raw display
\( T_{11}^{6} - 24T_{11}^{5} + 2721T_{11}^{4} + 86184T_{11}^{3} + 4184577T_{11}^{2} + 37220040T_{11} + 301091904 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( (T^{2} - 5 T + 25)^{3} \) Copy content Toggle raw display
$7$ \( T^{6} + 15 T^{5} + 873 T^{4} + \cdots + 394384 \) Copy content Toggle raw display
$11$ \( T^{6} - 24 T^{5} + \cdots + 301091904 \) Copy content Toggle raw display
$13$ \( T^{6} + 33 T^{5} + 1449 T^{4} + \cdots + 14137600 \) Copy content Toggle raw display
$17$ \( (T^{3} + 42 T^{2} - 10884 T - 429480)^{2} \) Copy content Toggle raw display
$19$ \( (T^{3} - 21 T^{2} - 8121 T + 311725)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 33 T^{5} + \cdots + 301915083024 \) Copy content Toggle raw display
$29$ \( T^{6} - 222 T^{5} + \cdots + 124366254336 \) Copy content Toggle raw display
$31$ \( T^{6} + 132 T^{5} + \cdots + 1300238478400 \) Copy content Toggle raw display
$37$ \( (T^{3} - 174 T^{2} - 48600 T - 849920)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} + 99 T^{5} + \cdots + 2076996910041 \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots + 163840000000000 \) Copy content Toggle raw display
$47$ \( T^{6} - 537 T^{5} + \cdots + 9594134553600 \) Copy content Toggle raw display
$53$ \( (T^{3} + 267 T^{2} - 251220 T + 14075604)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots + 162393249576321 \) Copy content Toggle raw display
$61$ \( T^{6} - 480 T^{5} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{6} + 12 T^{5} + \cdots + 34\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( (T^{3} + 570 T^{2} + 61227 T + 1353402)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} - 1062 T^{2} + 143196 T + 79568728)^{2} \) Copy content Toggle raw display
$79$ \( T^{6} - 1026 T^{5} + \cdots + 69\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{6} - 702 T^{5} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{3} + 1140 T^{2} + 179253 T - 2459646)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} - 2556 T^{5} + \cdots + 88\!\cdots\!00 \) Copy content Toggle raw display
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