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: \( x^{8} - 15x^{6} + 221x^{4} - 60x^{2} + 16 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + (\beta_{7} - \beta_{2}) q^{5} + \beta_1 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{7} - \beta_{2}) q^{5} + \beta_1 q^{7} + (2 \beta_{5} + 2 \beta_{4} + 3 \beta_1) q^{11} + (3 \beta_{7} + 3 \beta_{6} + 3 \beta_{4} - 3 \beta_{3}) q^{13} + (\beta_{7} - \beta_{5} + \beta_{3} - 1) q^{17} + (3 \beta_{7} - 3 \beta_{5} + 3 \beta_{3} + 3) q^{19} + (\beta_{7} - \beta_{4} + \beta_{3} + 7 \beta_{2}) q^{23} + (3 \beta_{5} - 3 \beta_{2} + 5 \beta_1 - 3) q^{25} + ( - 3 \beta_{5} - 3 \beta_{4} - 9 \beta_1) q^{29} + ( - 6 \beta_{7} + 6 \beta_{4} - 6 \beta_{3} + 7 \beta_{2}) q^{31} + (5 \beta_{3} - 10) q^{35} + ( - 6 \beta_{7} - 2 \beta_{6} + 6 \beta_{5} + 6 \beta_{3} + 2 \beta_1) q^{37} + (\beta_{7} + 3 \beta_{6} + \beta_{4} - \beta_{3}) q^{41} + ( - 9 \beta_{5} - 9 \beta_{4} - \beta_1) q^{43} + ( - 4 \beta_{5} + 4 \beta_{4} + 14 \beta_{2} + 14) q^{47} + ( - 3 \beta_{7} + 3 \beta_{4} - 3 \beta_{3} + 21 \beta_{2}) q^{49} + (6 \beta_{7} - 6 \beta_{5} + 6 \beta_{3} + 51) q^{53} + ( - 6 \beta_{7} - 10 \beta_{6} + 6 \beta_{5} + 15 \beta_{3} + 10 \beta_1 + 14) q^{55} + ( - 11 \beta_{7} - 9 \beta_{6} - 11 \beta_{4} + 11 \beta_{3}) q^{59} + (3 \beta_{5} - 3 \beta_{4} - 11 \beta_{2} - 11) q^{61} + (9 \beta_{5} + 15 \beta_{4} + 36 \beta_{2} + 15 \beta_1 + 36) q^{65} + (6 \beta_{7} - 8 \beta_{6} + 6 \beta_{4} - 6 \beta_{3}) q^{67} + ( - 15 \beta_{7} - 3 \beta_{6} + 15 \beta_{5} + 15 \beta_{3} + 3 \beta_1) q^{71} + (12 \beta_{7} + 17 \beta_{6} - 12 \beta_{5} - 12 \beta_{3} - 17 \beta_1) q^{73} + (\beta_{7} - \beta_{4} + \beta_{3} - 44 \beta_{2}) q^{77} + (9 \beta_{5} - 9 \beta_{4} - 3 \beta_{2} - 3) q^{79} + ( - 10 \beta_{5} + 10 \beta_{4} - 67 \beta_{2} - 67) q^{83} + (5 \beta_{6} - 25 \beta_{2}) q^{85} + ( - 11 \beta_{7} + 9 \beta_{6} + 11 \beta_{5} + 11 \beta_{3} - 9 \beta_1) q^{89} + (6 \beta_{7} - 6 \beta_{5} + 6 \beta_{3} + 24) q^{91} + (6 \beta_{7} + 15 \beta_{6} - 81 \beta_{2}) q^{95} + ( - 3 \beta_{5} - 3 \beta_{4} - 28 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 3 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 3 q^{5} - 12 q^{17} + 12 q^{19} - 30 q^{23} - 9 q^{25} - 16 q^{31} - 90 q^{35} + 48 q^{47} - 78 q^{49} + 384 q^{53} + 94 q^{55} - 38 q^{61} + 138 q^{65} + 174 q^{77} + 6 q^{79} - 288 q^{83} + 100 q^{85} + 168 q^{91} + 318 q^{95}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( -\nu^{7} - 3034\nu ) / 221 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -15\nu^{6} + 221\nu^{4} - 3315\nu^{2} + 16 ) / 884 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 60\nu^{7} - \nu^{6} - 884\nu^{5} + 13039\nu^{3} - 64\nu - 1708 ) / 442 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} + 29\nu^{6} - 442\nu^{4} + 6409\nu^{2} + 3476\nu - 1740 ) / 442 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} - 29\nu^{6} + 442\nu^{4} - 6409\nu^{2} + 3476\nu + 1740 ) / 442 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 103\nu^{7} - 1547\nu^{5} + 22763\nu^{3} - 6180\nu ) / 442 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -59\nu^{7} - 30\nu^{6} + 884\nu^{5} + 442\nu^{4} - 13039\nu^{3} - 6409\nu^{2} + 3540\nu + 32 ) / 442 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} + \beta_{4} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} - \beta_{4} + \beta_{3} - 8\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -7\beta_{7} - 8\beta_{6} + 7\beta_{5} + 7\beta_{3} + 8\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 15\beta_{5} - 15\beta_{4} - 116\beta_{2} - 116 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -103\beta_{7} - 118\beta_{6} - 103\beta_{4} + 103\beta_{3} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -221\beta_{7} + 221\beta_{5} - 221\beta_{3} - 1708 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -1517\beta_{5} - 1517\beta_{4} - 1738\beta_1 \) Copy content Toggle raw display

Character values

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

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

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 \(S_{3}^{\mathrm{new}}(1620, [\chi])\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - 3 T^{7} + 9 T^{6} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( T^{8} - 59 T^{6} + 3081 T^{4} + \cdots + 160000 \) Copy content Toggle raw display
$11$ \( T^{8} - 355 T^{6} + \cdots + 71639296 \) Copy content Toggle raw display
$13$ \( T^{8} - 540 T^{6} + \cdots + 26873856 \) Copy content Toggle raw display
$17$ \( (T^{2} + 3 T - 50)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} - 3 T - 468)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} + 15 T^{3} + 221 T^{2} + 60 T + 16)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 1584 T^{2} + 2509056)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 8 T^{3} + 1929 T^{2} + \cdots + 3478225)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 1216)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} - 176 T^{2} + 30976)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} - 6620 T^{6} + \cdots + 93790392487936 \) Copy content Toggle raw display
$47$ \( (T^{4} - 24 T^{3} + 1268 T^{2} + \cdots + 478864)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 96 T + 423)^{4} \) Copy content Toggle raw display
$59$ \( T^{8} - 6880 T^{6} + \cdots + 17048839192576 \) Copy content Toggle raw display
$61$ \( (T^{4} + 19 T^{3} + 741 T^{2} + \cdots + 144400)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} - 11372 T^{6} + \cdots + 293434556416 \) Copy content Toggle raw display
$71$ \( (T^{4} + 16956 T^{2} + 68558400)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 11795 T^{2} + 6091024)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 3 T^{3} + 4239 T^{2} + \cdots + 17892900)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 144 T^{3} + 20777 T^{2} + \cdots + 1681)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 24700 T^{2} + 19430464)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} - 39515 T^{6} + \cdots + 71\!\cdots\!56 \) Copy content Toggle raw display
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