Properties

Label 1620.2.r.h
Level $1620$
Weight $2$
Character orbit 1620.r
Analytic conductor $12.936$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(12.9357651274\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Defining polynomial: \(x^{16} + 3 x^{14} - 11 x^{12} - 90 x^{10} - 450 x^{8} - 2250 x^{6} - 6875 x^{4} + 46875 x^{2} + 390625\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{14} q^{5} + ( \beta_{2} - \beta_{12} ) q^{7} +O(q^{10})\) \( q + \beta_{14} q^{5} + ( \beta_{2} - \beta_{12} ) q^{7} + ( -\beta_{1} - \beta_{4} - \beta_{8} - \beta_{14} ) q^{11} + ( -\beta_{3} + \beta_{9} + \beta_{13} ) q^{13} + ( -\beta_{4} + \beta_{7} + \beta_{10} + \beta_{15} ) q^{17} + ( -1 - \beta_{9} + \beta_{11} + \beta_{13} ) q^{19} + ( 2 \beta_{1} - 2 \beta_{10} - 2 \beta_{14} ) q^{23} + ( -\beta_{2} - \beta_{6} - \beta_{11} + \beta_{12} ) q^{25} + 3 \beta_{8} q^{29} + ( 1 + \beta_{3} - \beta_{6} - \beta_{9} + \beta_{13} ) q^{31} + ( -\beta_{7} + 2 \beta_{10} - \beta_{15} ) q^{35} + ( -\beta_{2} + \beta_{9} + \beta_{11} + \beta_{13} ) q^{37} + ( \beta_{1} + \beta_{5} - \beta_{10} + \beta_{14} ) q^{41} + ( -\beta_{2} + \beta_{12} ) q^{43} + ( -2 \beta_{1} + 2 \beta_{4} + 2 \beta_{14} + \beta_{15} ) q^{47} + ( 7 + 2 \beta_{3} - 7 \beta_{6} - 2 \beta_{9} + 2 \beta_{13} ) q^{49} + ( -\beta_{7} - \beta_{15} ) q^{53} + ( 5 + \beta_{2} + \beta_{9} + \beta_{11} + \beta_{13} ) q^{55} + ( \beta_{1} - 3 \beta_{5} - \beta_{10} + \beta_{14} ) q^{59} + ( \beta_{3} - 4 \beta_{6} - \beta_{11} ) q^{61} + ( -3 \beta_{1} - 5 \beta_{8} - \beta_{15} ) q^{65} + ( -2 \beta_{3} + 2 \beta_{9} + 2 \beta_{12} + 2 \beta_{13} ) q^{67} + ( 3 \beta_{4} - \beta_{5} - \beta_{8} + 3 \beta_{10} ) q^{71} + ( \beta_{2} + \beta_{9} + \beta_{11} + \beta_{13} ) q^{73} + ( 2 \beta_{1} + \beta_{7} - 2 \beta_{10} - 2 \beta_{14} ) q^{77} + 6 \beta_{6} q^{79} + ( -2 \beta_{1} + 2 \beta_{4} + 2 \beta_{14} - \beta_{15} ) q^{83} + ( -2 + 2 \beta_{6} - 2 \beta_{12} + 3 \beta_{13} ) q^{85} + ( 2 \beta_{4} + 3 \beta_{5} + 3 \beta_{8} + 2 \beta_{10} ) q^{89} + ( 4 - 2 \beta_{9} + 2 \beta_{11} + 2 \beta_{13} ) q^{91} + ( 3 \beta_{1} + 5 \beta_{5} - \beta_{7} - 3 \beta_{10} - 3 \beta_{14} ) q^{95} + ( -\beta_{2} - 2 \beta_{3} - 2 \beta_{11} + \beta_{12} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + O(q^{10}) \) \( 16q - 24q^{19} - 6q^{25} + 4q^{31} + 48q^{49} + 80q^{55} - 28q^{61} + 48q^{79} - 22q^{85} + 48q^{91} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} + 3 x^{14} - 11 x^{12} - 90 x^{10} - 450 x^{8} - 2250 x^{6} - 6875 x^{4} + 46875 x^{2} + 390625\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{12} - 72 \nu^{6} - 4625 \)\()/4500\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{14} + 178 \nu^{12} + 639 \nu^{10} + 1485 \nu^{8} + 7425 \nu^{6} - 64125 \nu^{4} - 597500 \nu^{2} - 2140625 \)\()/562500\)
\(\beta_{4}\)\(=\)\((\)\( -7 \nu^{15} + 204 \nu^{13} + 1377 \nu^{11} + 9405 \nu^{9} - 11475 \nu^{7} - 57375 \nu^{5} - 317500 \nu^{3} - 3984375 \nu \)\()/5625000\)
\(\beta_{5}\)\(=\)\((\)\( 7 \nu^{15} - 304 \nu^{13} - 927 \nu^{11} - 3555 \nu^{9} - 17775 \nu^{7} + 113625 \nu^{5} + 1048750 \nu^{3} + 4390625 \nu \)\()/5625000\)
\(\beta_{6}\)\(=\)\((\)\( 9 \nu^{14} + 52 \nu^{12} + 351 \nu^{10} - 585 \nu^{8} - 2925 \nu^{6} - 14625 \nu^{4} - 146250 \nu^{2} + 109375 \)\()/1125000\)
\(\beta_{7}\)\(=\)\((\)\( -8 \nu^{15} - 149 \nu^{13} + 963 \nu^{11} - 405 \nu^{9} - 2025 \nu^{7} - 235125 \nu^{5} + 476875 \nu^{3} + 1187500 \nu \)\()/2812500\)
\(\beta_{8}\)\(=\)\((\)\( \nu^{15} + 8 \nu^{13} + 9 \nu^{11} + 45 \nu^{9} + 945 \nu^{7} - 3375 \nu^{5} - 12500 \nu^{3} - 94375 \nu \)\()/225000\)
\(\beta_{9}\)\(=\)\((\)\( \nu^{14} + 8 \nu^{12} + 9 \nu^{10} + 45 \nu^{8} + 945 \nu^{6} - 3375 \nu^{4} - 12500 \nu^{2} - 49375 \)\()/45000\)
\(\beta_{10}\)\(=\)\((\)\( -9 \nu^{15} - 52 \nu^{13} - 351 \nu^{11} + 585 \nu^{9} + 2925 \nu^{7} + 14625 \nu^{5} + 146250 \nu^{3} + 1015625 \nu \)\()/1125000\)
\(\beta_{11}\)\(=\)\((\)\( 31 \nu^{14} + 18 \nu^{12} - 441 \nu^{10} - 3465 \nu^{8} - 17325 \nu^{6} - 176625 \nu^{4} + 40000 \nu^{2} + 2390625 \)\()/1125000\)
\(\beta_{12}\)\(=\)\((\)\( -16 \nu^{14} + 52 \nu^{12} + 351 \nu^{10} + 1215 \nu^{8} - 2925 \nu^{6} - 14625 \nu^{4} + 194375 \nu^{2} - 1015625 \)\()/562500\)
\(\beta_{13}\)\(=\)\((\)\( 17 \nu^{14} + 76 \nu^{12} + 513 \nu^{10} + 3195 \nu^{8} - 4275 \nu^{6} - 21375 \nu^{4} - 426250 \nu^{2} - 1484375 \)\()/562500\)
\(\beta_{14}\)\(=\)\((\)\( \nu^{15} + 3 \nu^{13} - 11 \nu^{11} - 90 \nu^{9} - 450 \nu^{7} - 2250 \nu^{5} - 6875 \nu^{3} + 46875 \nu \)\()/78125\)
\(\beta_{15}\)\(=\)\((\)\( -13 \nu^{15} + 41 \nu^{13} - 117 \nu^{11} - 585 \nu^{9} - 225 \nu^{7} + 43875 \nu^{5} + 162500 \nu^{3} - 456250 \nu \)\()/562500\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{12} + \beta_{9} + \beta_{6} - \beta_{3} - 1\)
\(\nu^{3}\)\(=\)\(\beta_{15} + 5 \beta_{8} + \beta_{7} + 5 \beta_{5} + 2 \beta_{4}\)
\(\nu^{4}\)\(=\)\(-3 \beta_{12} - 5 \beta_{11} + 7 \beta_{6} - 2 \beta_{3} + 3 \beta_{2}\)
\(\nu^{5}\)\(=\)\(-9 \beta_{14} - 10 \beta_{10} - 8 \beta_{7} + 10 \beta_{5} + 10 \beta_{1}\)
\(\nu^{6}\)\(=\)\(-10 \beta_{13} - 10 \beta_{11} + 26 \beta_{9} - 14 \beta_{2} + 9\)
\(\nu^{7}\)\(=\)\(-4 \beta_{15} - 58 \beta_{14} + 130 \beta_{8} - 58 \beta_{4} + 45 \beta_{1}\)
\(\nu^{8}\)\(=\)\(130 \beta_{13} + 57 \beta_{12} + 37 \beta_{9} - 383 \beta_{6} - 37 \beta_{3} + 383\)
\(\nu^{9}\)\(=\)\(-73 \beta_{15} + 290 \beta_{10} + 185 \beta_{8} - 73 \beta_{7} + 185 \beta_{5} + 504 \beta_{4}\)
\(\nu^{10}\)\(=\)\(509 \beta_{12} - 185 \beta_{11} + 2479 \beta_{6} - 144 \beta_{3} - 509 \beta_{2}\)
\(\nu^{11}\)\(=\)\(-1573 \beta_{14} - 2520 \beta_{10} + 324 \beta_{7} + 720 \beta_{5} + 2520 \beta_{1}\)
\(\nu^{12}\)\(=\)\(-720 \beta_{13} - 720 \beta_{11} + 1872 \beta_{9} + 3492 \beta_{2} + 5273\)
\(\nu^{13}\)\(=\)\(4212 \beta_{15} + 4824 \beta_{14} + 9360 \beta_{8} + 4824 \beta_{4} + 7865 \beta_{1}\)
\(\nu^{14}\)\(=\)\(9360 \beta_{13} - 4771 \beta_{12} + 16289 \beta_{9} + 31049 \beta_{6} - 16289 \beta_{3} - 31049\)
\(\nu^{15}\)\(=\)\(-14131 \beta_{15} - 24120 \beta_{10} + 81445 \beta_{8} - 14131 \beta_{7} + 81445 \beta_{5} + 18538 \beta_{4}\)

Character values

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

\(n\) \(811\) \(1297\) \(1541\)
\(\chi(n)\) \(1\) \(-1\) \(-1 + \beta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1
2.23368 + 0.103355i
1.76348 + 1.37482i
1.02733 1.98610i
0.308893 + 2.21463i
−0.308893 2.21463i
−1.02733 + 1.98610i
−1.76348 1.37482i
−2.23368 0.103355i
2.23368 0.103355i
1.76348 1.37482i
1.02733 + 1.98610i
0.308893 2.21463i
−0.308893 + 2.21463i
−1.02733 1.98610i
−1.76348 + 1.37482i
−2.23368 + 0.103355i
0 0 0 −2.23368 + 0.103355i 0 −1.10979 + 0.640739i 0 0 0
109.2 0 0 0 −1.76348 + 1.37482i 0 −4.27415 + 2.46768i 0 0 0
109.3 0 0 0 −1.02733 1.98610i 0 1.10979 0.640739i 0 0 0
109.4 0 0 0 −0.308893 + 2.21463i 0 4.27415 2.46768i 0 0 0
109.5 0 0 0 0.308893 2.21463i 0 4.27415 2.46768i 0 0 0
109.6 0 0 0 1.02733 + 1.98610i 0 1.10979 0.640739i 0 0 0
109.7 0 0 0 1.76348 1.37482i 0 −4.27415 + 2.46768i 0 0 0
109.8 0 0 0 2.23368 0.103355i 0 −1.10979 + 0.640739i 0 0 0
1189.1 0 0 0 −2.23368 0.103355i 0 −1.10979 0.640739i 0 0 0
1189.2 0 0 0 −1.76348 1.37482i 0 −4.27415 2.46768i 0 0 0
1189.3 0 0 0 −1.02733 + 1.98610i 0 1.10979 + 0.640739i 0 0 0
1189.4 0 0 0 −0.308893 2.21463i 0 4.27415 + 2.46768i 0 0 0
1189.5 0 0 0 0.308893 + 2.21463i 0 4.27415 + 2.46768i 0 0 0
1189.6 0 0 0 1.02733 1.98610i 0 1.10979 + 0.640739i 0 0 0
1189.7 0 0 0 1.76348 + 1.37482i 0 −4.27415 2.46768i 0 0 0
1189.8 0 0 0 2.23368 + 0.103355i 0 −1.10979 0.640739i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1189.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1620.2.r.h 16
3.b odd 2 1 inner 1620.2.r.h 16
5.b even 2 1 inner 1620.2.r.h 16
9.c even 3 1 1620.2.d.e 8
9.c even 3 1 inner 1620.2.r.h 16
9.d odd 6 1 1620.2.d.e 8
9.d odd 6 1 inner 1620.2.r.h 16
15.d odd 2 1 inner 1620.2.r.h 16
45.h odd 6 1 1620.2.d.e 8
45.h odd 6 1 inner 1620.2.r.h 16
45.j even 6 1 1620.2.d.e 8
45.j even 6 1 inner 1620.2.r.h 16
45.k odd 12 2 8100.2.a.be 8
45.l even 12 2 8100.2.a.be 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1620.2.d.e 8 9.c even 3 1
1620.2.d.e 8 9.d odd 6 1
1620.2.d.e 8 45.h odd 6 1
1620.2.d.e 8 45.j even 6 1
1620.2.r.h 16 1.a even 1 1 trivial
1620.2.r.h 16 3.b odd 2 1 inner
1620.2.r.h 16 5.b even 2 1 inner
1620.2.r.h 16 9.c even 3 1 inner
1620.2.r.h 16 9.d odd 6 1 inner
1620.2.r.h 16 15.d odd 2 1 inner
1620.2.r.h 16 45.h odd 6 1 inner
1620.2.r.h 16 45.j even 6 1 inner
8100.2.a.be 8 45.k odd 12 2
8100.2.a.be 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 \(S_{2}^{\mathrm{new}}(1620, [\chi])\):

\( T_{7}^{8} - 26 T_{7}^{6} + 636 T_{7}^{4} - 1040 T_{7}^{2} + 1600 \)
\( T_{11}^{8} + 23 T_{11}^{6} + 429 T_{11}^{4} + 2300 T_{11}^{2} + 10000 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \)
$3$ \( T^{16} \)
$5$ \( 390625 + 46875 T^{2} - 6875 T^{4} - 2250 T^{6} - 450 T^{8} - 90 T^{10} - 11 T^{12} + 3 T^{14} + T^{16} \)
$7$ \( ( 1600 - 1040 T^{2} + 636 T^{4} - 26 T^{6} + T^{8} )^{2} \)
$11$ \( ( 10000 + 2300 T^{2} + 429 T^{4} + 23 T^{6} + T^{8} )^{2} \)
$13$ \( ( 129600 - 18360 T^{2} + 2241 T^{4} - 51 T^{6} + T^{8} )^{2} \)
$17$ \( ( 1690 + 83 T^{2} + T^{4} )^{4} \)
$19$ \( ( -30 + 3 T + T^{2} )^{8} \)
$23$ \( ( 409600 - 43520 T^{2} + 3984 T^{4} - 68 T^{6} + T^{8} )^{2} \)
$29$ \( ( 729 + 27 T^{2} + T^{4} )^{4} \)
$31$ \( ( 1024 + 32 T + 33 T^{2} - T^{3} + T^{4} )^{4} \)
$37$ \( ( 1690 + 89 T^{2} + T^{4} )^{4} \)
$41$ \( ( 256 + 560 T^{2} + 1209 T^{4} + 35 T^{6} + T^{8} )^{2} \)
$43$ \( ( 1600 - 1040 T^{2} + 636 T^{4} - 26 T^{6} + T^{8} )^{2} \)
$47$ \( ( 16000000 - 680000 T^{2} + 24900 T^{4} - 170 T^{6} + T^{8} )^{2} \)
$53$ \( ( 360 + 78 T^{2} + T^{4} )^{4} \)
$59$ \( ( 4096 + 3776 T^{2} + 3417 T^{4} + 59 T^{6} + T^{8} )^{2} \)
$61$ \( ( 400 - 140 T + 69 T^{2} + 7 T^{3} + T^{4} )^{4} \)
$67$ \( ( 16000000 - 1040000 T^{2} + 63600 T^{4} - 260 T^{6} + T^{8} )^{2} \)
$71$ \( ( 6084 - 231 T^{2} + T^{4} )^{4} \)
$73$ \( ( 250 + 65 T^{2} + T^{4} )^{4} \)
$79$ \( ( 36 - 6 T + T^{2} )^{8} \)
$83$ \( ( 6553600 - 312320 T^{2} + 12324 T^{4} - 122 T^{6} + T^{8} )^{2} \)
$89$ \( ( 961 - 110 T^{2} + T^{4} )^{4} \)
$97$ \( ( 25600 - 32960 T^{2} + 42276 T^{4} - 206 T^{6} + T^{8} )^{2} \)
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