Properties

Label 1620.3.t.d
Level $1620$
Weight $3$
Character orbit 1620.t
Analytic conductor $44.142$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1620.t (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(44.1418028264\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.154550410641.1
Defining polynomial: \(x^{8} - 15 x^{6} + 221 x^{4} - 60 x^{2} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 540)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\beta_{2} + \beta_{7} ) q^{5} + \beta_{1} q^{7} +O(q^{10})\) \( q + ( -\beta_{2} + \beta_{7} ) q^{5} + \beta_{1} q^{7} + ( 3 \beta_{1} + 2 \beta_{4} + 2 \beta_{5} ) q^{11} + ( -3 \beta_{3} + 3 \beta_{4} + 3 \beta_{6} + 3 \beta_{7} ) q^{13} + ( -1 + \beta_{3} - \beta_{5} + \beta_{7} ) q^{17} + ( 3 + 3 \beta_{3} - 3 \beta_{5} + 3 \beta_{7} ) q^{19} + ( 7 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{7} ) q^{23} + ( -3 + 5 \beta_{1} - 3 \beta_{2} + 3 \beta_{5} ) q^{25} + ( -9 \beta_{1} - 3 \beta_{4} - 3 \beta_{5} ) q^{29} + ( 7 \beta_{2} - 6 \beta_{3} + 6 \beta_{4} - 6 \beta_{7} ) q^{31} + ( -10 + 5 \beta_{3} ) q^{35} + ( 2 \beta_{1} + 6 \beta_{3} + 6 \beta_{5} - 2 \beta_{6} - 6 \beta_{7} ) q^{37} + ( -\beta_{3} + \beta_{4} + 3 \beta_{6} + \beta_{7} ) q^{41} + ( -\beta_{1} - 9 \beta_{4} - 9 \beta_{5} ) q^{43} + ( 14 + 14 \beta_{2} + 4 \beta_{4} - 4 \beta_{5} ) q^{47} + ( 21 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} - 3 \beta_{7} ) q^{49} + ( 51 + 6 \beta_{3} - 6 \beta_{5} + 6 \beta_{7} ) q^{53} + ( 14 + 10 \beta_{1} + 15 \beta_{3} + 6 \beta_{5} - 10 \beta_{6} - 6 \beta_{7} ) q^{55} + ( 11 \beta_{3} - 11 \beta_{4} - 9 \beta_{6} - 11 \beta_{7} ) q^{59} + ( -11 - 11 \beta_{2} - 3 \beta_{4} + 3 \beta_{5} ) q^{61} + ( 36 + 15 \beta_{1} + 36 \beta_{2} + 15 \beta_{4} + 9 \beta_{5} ) q^{65} + ( -6 \beta_{3} + 6 \beta_{4} - 8 \beta_{6} + 6 \beta_{7} ) q^{67} + ( 3 \beta_{1} + 15 \beta_{3} + 15 \beta_{5} - 3 \beta_{6} - 15 \beta_{7} ) q^{71} + ( -17 \beta_{1} - 12 \beta_{3} - 12 \beta_{5} + 17 \beta_{6} + 12 \beta_{7} ) q^{73} + ( -44 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{7} ) q^{77} + ( -3 - 3 \beta_{2} - 9 \beta_{4} + 9 \beta_{5} ) q^{79} + ( -67 - 67 \beta_{2} + 10 \beta_{4} - 10 \beta_{5} ) q^{83} + ( -25 \beta_{2} + 5 \beta_{6} ) q^{85} + ( -9 \beta_{1} + 11 \beta_{3} + 11 \beta_{5} + 9 \beta_{6} - 11 \beta_{7} ) q^{89} + ( 24 + 6 \beta_{3} - 6 \beta_{5} + 6 \beta_{7} ) q^{91} + ( -81 \beta_{2} + 15 \beta_{6} + 6 \beta_{7} ) q^{95} + ( -28 \beta_{1} - 3 \beta_{4} - 3 \beta_{5} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 3 q^{5} + O(q^{10}) \) \( 8 q + 3 q^{5} - 12 q^{17} + 12 q^{19} - 30 q^{23} - 9 q^{25} - 16 q^{31} - 90 q^{35} + 48 q^{47} - 78 q^{49} + 384 q^{53} + 94 q^{55} - 38 q^{61} + 138 q^{65} + 174 q^{77} + 6 q^{79} - 288 q^{83} + 100 q^{85} + 168 q^{91} + 318 q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 15 x^{6} + 221 x^{4} - 60 x^{2} + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{7} - 3034 \nu \)\()/221\)
\(\beta_{2}\)\(=\)\((\)\( -15 \nu^{6} + 221 \nu^{4} - 3315 \nu^{2} + 16 \)\()/884\)
\(\beta_{3}\)\(=\)\((\)\( 60 \nu^{7} - \nu^{6} - 884 \nu^{5} + 13039 \nu^{3} - 64 \nu - 1708 \)\()/442\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{7} + 29 \nu^{6} - 442 \nu^{4} + 6409 \nu^{2} + 3476 \nu - 1740 \)\()/442\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{7} - 29 \nu^{6} + 442 \nu^{4} - 6409 \nu^{2} + 3476 \nu + 1740 \)\()/442\)
\(\beta_{6}\)\(=\)\((\)\( 103 \nu^{7} - 1547 \nu^{5} + 22763 \nu^{3} - 6180 \nu \)\()/442\)
\(\beta_{7}\)\(=\)\((\)\( -59 \nu^{7} - 30 \nu^{6} + 884 \nu^{5} + 442 \nu^{4} - 13039 \nu^{3} - 6409 \nu^{2} + 3540 \nu + 32 \)\()/442\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{5} + \beta_{4} + \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{7} - \beta_{4} + \beta_{3} - 8 \beta_{2}\)
\(\nu^{3}\)\(=\)\(-7 \beta_{7} - 8 \beta_{6} + 7 \beta_{5} + 7 \beta_{3} + 8 \beta_{1}\)
\(\nu^{4}\)\(=\)\(15 \beta_{5} - 15 \beta_{4} - 116 \beta_{2} - 116\)
\(\nu^{5}\)\(=\)\(-103 \beta_{7} - 118 \beta_{6} - 103 \beta_{4} + 103 \beta_{3}\)
\(\nu^{6}\)\(=\)\(-221 \beta_{7} + 221 \beta_{5} - 221 \beta_{3} - 1708\)
\(\nu^{7}\)\(=\)\(-1517 \beta_{5} - 1517 \beta_{4} - 1738 \beta_{1}\)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
269.1
−0.451318 0.260569i
−3.32360 1.91888i
0.451318 + 0.260569i
3.32360 + 1.91888i
−0.451318 + 0.260569i
−3.32360 + 1.91888i
0.451318 0.260569i
3.32360 1.91888i
0 0 0 −4.98136 0.431312i 0 6.19588 + 3.57719i 0 0 0
269.2 0 0 0 0.0689865 + 4.99952i 0 −2.42096 1.39774i 0 0 0
269.3 0 0 0 2.11715 4.52964i 0 −6.19588 3.57719i 0 0 0
269.4 0 0 0 4.29522 + 2.55951i 0 2.42096 + 1.39774i 0 0 0
1349.1 0 0 0 −4.98136 + 0.431312i 0 6.19588 3.57719i 0 0 0
1349.2 0 0 0 0.0689865 4.99952i 0 −2.42096 + 1.39774i 0 0 0
1349.3 0 0 0 2.11715 + 4.52964i 0 −6.19588 + 3.57719i 0 0 0
1349.4 0 0 0 4.29522 2.55951i 0 2.42096 1.39774i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1349.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner
15.d odd 2 1 inner
45.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1620.3.t.d 8
3.b odd 2 1 1620.3.t.a 8
5.b even 2 1 1620.3.t.a 8
9.c even 3 1 540.3.b.a 4
9.c even 3 1 inner 1620.3.t.d 8
9.d odd 6 1 540.3.b.b yes 4
9.d odd 6 1 1620.3.t.a 8
15.d odd 2 1 inner 1620.3.t.d 8
36.f odd 6 1 2160.3.c.h 4
36.h even 6 1 2160.3.c.l 4
45.h odd 6 1 540.3.b.a 4
45.h odd 6 1 inner 1620.3.t.d 8
45.j even 6 1 540.3.b.b yes 4
45.j even 6 1 1620.3.t.a 8
45.k odd 12 2 2700.3.g.s 8
45.l even 12 2 2700.3.g.s 8
180.n even 6 1 2160.3.c.h 4
180.p odd 6 1 2160.3.c.l 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
540.3.b.a 4 9.c even 3 1
540.3.b.a 4 45.h odd 6 1
540.3.b.b yes 4 9.d odd 6 1
540.3.b.b yes 4 45.j even 6 1
1620.3.t.a 8 3.b odd 2 1
1620.3.t.a 8 5.b even 2 1
1620.3.t.a 8 9.d odd 6 1
1620.3.t.a 8 45.j even 6 1
1620.3.t.d 8 1.a even 1 1 trivial
1620.3.t.d 8 9.c even 3 1 inner
1620.3.t.d 8 15.d odd 2 1 inner
1620.3.t.d 8 45.h odd 6 1 inner
2160.3.c.h 4 36.f odd 6 1
2160.3.c.h 4 180.n even 6 1
2160.3.c.l 4 36.h even 6 1
2160.3.c.l 4 180.p odd 6 1
2700.3.g.s 8 45.k odd 12 2
2700.3.g.s 8 45.l even 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1620, [\chi])\):

\( T_{7}^{8} - 59 T_{7}^{6} + 3081 T_{7}^{4} - 23600 T_{7}^{2} + 160000 \)
\( T_{17}^{2} + 3 T_{17} - 50 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( T^{8} \)
$5$ \( 390625 - 46875 T + 5625 T^{2} + 3750 T^{3} - 850 T^{4} + 150 T^{5} + 9 T^{6} - 3 T^{7} + T^{8} \)
$7$ \( 160000 - 23600 T^{2} + 3081 T^{4} - 59 T^{6} + T^{8} \)
$11$ \( 71639296 - 3004720 T^{2} + 117561 T^{4} - 355 T^{6} + T^{8} \)
$13$ \( 26873856 - 2799360 T^{2} + 286416 T^{4} - 540 T^{6} + T^{8} \)
$17$ \( ( -50 + 3 T + T^{2} )^{4} \)
$19$ \( ( -468 - 3 T + T^{2} )^{4} \)
$23$ \( ( 16 + 60 T + 221 T^{2} + 15 T^{3} + T^{4} )^{2} \)
$29$ \( ( 2509056 - 1584 T^{2} + T^{4} )^{2} \)
$31$ \( ( 3478225 - 14920 T + 1929 T^{2} + 8 T^{3} + T^{4} )^{2} \)
$37$ \( ( 1216 + T^{2} )^{4} \)
$41$ \( ( 30976 - 176 T^{2} + T^{4} )^{2} \)
$43$ \( 93790392487936 - 64111681280 T^{2} + 34139856 T^{4} - 6620 T^{6} + T^{8} \)
$47$ \( ( 478864 + 16608 T + 1268 T^{2} - 24 T^{3} + T^{4} )^{2} \)
$53$ \( ( 423 - 96 T + T^{2} )^{4} \)
$59$ \( 17048839192576 - 28407685120 T^{2} + 43205376 T^{4} - 6880 T^{6} + T^{8} \)
$61$ \( ( 144400 - 7220 T + 741 T^{2} + 19 T^{3} + T^{4} )^{2} \)
$67$ \( 293434556416 - 6160166912 T^{2} + 128780688 T^{4} - 11372 T^{6} + T^{8} \)
$71$ \( ( 68558400 + 16956 T^{2} + T^{4} )^{2} \)
$73$ \( ( 6091024 + 11795 T^{2} + T^{4} )^{2} \)
$79$ \( ( 17892900 + 12690 T + 4239 T^{2} - 3 T^{3} + T^{4} )^{2} \)
$83$ \( ( 1681 - 5904 T + 20777 T^{2} + 144 T^{3} + T^{4} )^{2} \)
$89$ \( ( 19430464 + 24700 T^{2} + T^{4} )^{2} \)
$97$ \( 71077471001645056 - 10534840621760 T^{2} + 1294831641 T^{4} - 39515 T^{6} + T^{8} \)
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