Properties

Label 1620.3.t.e
Level $1620$
Weight $3$
Character orbit 1620.t
Analytic conductor $44.142$
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: \( k \) \(=\) \( 3 \)
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: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \( x^{16} - 81 x^{14} + 4781 x^{12} - 127980 x^{10} + 2502300 x^{8} - 12798000 x^{6} + 47810000 x^{4} - 81000000 x^{2} + 100000000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{8}\cdot 5^{2} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{5} + (\beta_{7} - \beta_{3}) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{5} + (\beta_{7} - \beta_{3}) q^{7} + \beta_{14} q^{11} + \beta_{10} q^{13} + ( - \beta_{8} + \beta_{5} + \beta_{2}) q^{17} + (\beta_{11} - 4) q^{19} + ( - \beta_{15} - \beta_{12} - 2 \beta_{5} + 3 \beta_{4}) q^{23} + (\beta_{11} + \beta_{9} - \beta_{7} - \beta_{3} + \beta_1 - 1) q^{25} + 3 \beta_{14} q^{29} + ( - \beta_{9} - 5 \beta_1) q^{31} + ( - 3 \beta_{14} - 3 \beta_{13} - \beta_{8} + 2 \beta_{5} + 2 \beta_{2}) q^{35} + (4 \beta_{10} + 5 \beta_{7} - 5 \beta_{6} - \beta_{3}) q^{37} + ( - \beta_{15} + 2 \beta_{13} - \beta_{12} + \beta_{5} - 4 \beta_{4}) q^{41} + ( - \beta_{7} + 10 \beta_{3}) q^{43} + (5 \beta_{8} + 5 \beta_{4} - 5 \beta_{2}) q^{47} + (\beta_{9} + 13 \beta_1) q^{49} + ( - \beta_{15} - 8 \beta_{8} + 3 \beta_{5} + 3 \beta_{2}) q^{53} + (10 \beta_{10} - 5 \beta_{7} + 5 \beta_{6} + 15 \beta_{3} + 5) q^{55} + ( - 2 \beta_{15} + 6 \beta_{13} - 2 \beta_{12} + 2 \beta_{5} - 8 \beta_{4}) q^{59} + ( - \beta_{11} - \beta_{9} - 14 \beta_1 + 14) q^{61} + ( - \beta_{14} - \beta_{2}) q^{65} + ( - 4 \beta_{10} - 5 \beta_{6}) q^{67} + ( - \beta_{15} - 8 \beta_{14} - 8 \beta_{13} + 4 \beta_{8} + \beta_{5} + \beta_{2}) q^{71} + ( - 26 \beta_{10} - 5 \beta_{7} + 5 \beta_{6} - 21 \beta_{3}) q^{73} + ( - 3 \beta_{15} - 3 \beta_{12} + \beta_{5} + 16 \beta_{4}) q^{77} + (39 \beta_1 - 39) q^{79} + ( - \beta_{12} + 10 \beta_{8} + 10 \beta_{4} - 5 \beta_{2}) q^{83} + ( - 15 \beta_{10} + 5 \beta_{6} - 20 \beta_1) q^{85} + ( - 2 \beta_{15} + 6 \beta_{14} + 6 \beta_{13} + 8 \beta_{8} + 2 \beta_{5} + 2 \beta_{2}) q^{89} + ( - \beta_{11} + 4) q^{91} + ( - 5 \beta_{15} + \beta_{13} - 5 \beta_{12} + \beta_{5} + 7 \beta_{4}) q^{95} + (11 \beta_{7} - 31 \beta_{3}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 72 q^{19} - 12 q^{25} - 44 q^{31} + 108 q^{49} + 80 q^{55} + 116 q^{61} - 312 q^{79} - 160 q^{85} + 72 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 81 x^{14} + 4781 x^{12} - 127980 x^{10} + 2502300 x^{8} - 12798000 x^{6} + 47810000 x^{4} - 81000000 x^{2} + 100000000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 632961 \nu^{14} + 50273041 \nu^{12} - 2950251741 \nu^{10} + 76602463980 \nu^{8} - 1477520862300 \nu^{6} + \cdots + 47651681000000 ) / 33801501000000 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 15295558 \nu^{15} - 530247135 \nu^{14} + 1450181698 \nu^{13} + 42510698685 \nu^{12} - 89296431798 \nu^{11} + \cdots + 11\!\cdots\!00 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 5723692 \nu^{15} - 484193627 \nu^{13} + 28937202402 \nu^{11} - 823417036860 \nu^{9} + 16517265879225 \nu^{7} + \cdots - 538300135750000 \nu ) / 84503752500000 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 11936858 \nu^{15} - 25077327 \nu^{14} - 912947588 \nu^{13} + 2063365787 \nu^{12} + 52737306888 \nu^{11} - 122351168187 \nu^{10} + \cdots + 21\!\cdots\!00 ) / 202809006000000 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 11936858 \nu^{15} + 28867737 \nu^{14} + 912947588 \nu^{13} - 2598371397 \nu^{12} - 52737306888 \nu^{11} + 157953964797 \nu^{10} + \cdots - 32\!\cdots\!00 ) / 202809006000000 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 49573 \nu^{15} - 3789458 \nu^{13} + 219014628 \nu^{11} - 5288389155 \nu^{9} + 96551212470 \nu^{7} - 105596338500 \nu^{5} + \cdots + 4876383680000 \nu ) / 699341400000 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 74603797 \nu^{15} + 5848406957 \nu^{13} - 341824861857 \nu^{11} + 8688485918460 \nu^{9} - 165931986692100 \nu^{7} + \cdots + 53\!\cdots\!00 \nu ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 37489924 \nu^{15} - 46184325 \nu^{14} + 3007459819 \nu^{13} + 3530774450 \nu^{12} - 176491483119 \nu^{11} + \cdots - 29\!\cdots\!00 ) / 507022515000000 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 13903311 \nu^{14} + 1179265591 \nu^{12} - 70523061291 \nu^{10} + 2013931248180 \nu^{8} - 40454623104300 \nu^{6} + \cdots + 13\!\cdots\!00 ) / 16900750500000 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 3033031 \nu^{15} - 231900866 \nu^{13} + 13399999116 \nu^{11} - 323560168785 \nu^{9} + 5900808103350 \nu^{7} + \cdots + 98073590300000 \nu ) / 33801501000000 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 267 \nu^{14} + 20412 \nu^{12} - 1179612 \nu^{10} + 28483245 \nu^{8} - 519639900 \nu^{6} + 568741500 \nu^{4} - 969150000 \nu^{2} + \cdots - 8142100000 ) / 234050000 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 15295558 \nu^{15} + 280253655 \nu^{14} + 1450181698 \nu^{13} - 22404842005 \nu^{12} - 89296431798 \nu^{11} + \cdots - 61\!\cdots\!00 ) / 202809006000000 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 1220101 \nu^{15} + 93282961 \nu^{13} - 5390433636 \nu^{11} + 130158935235 \nu^{9} - 2374557287775 \nu^{7} + \cdots - 64414896925000 \nu ) / 5070225150000 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 55567627 \nu^{15} + 4603416787 \nu^{13} - 273502998687 \nu^{11} + 7564122707460 \nu^{9} - 149974944707100 \nu^{7} + \cdots + 48\!\cdots\!00 \nu ) / 202809006000000 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 37489924 \nu^{15} - 75538305 \nu^{14} - 3007459819 \nu^{13} + 5776503730 \nu^{12} + 176491483119 \nu^{11} - 333729928980 \nu^{10} + \cdots - 28\!\cdots\!00 ) / 101404503000000 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{15} + 5\beta_{13} + \beta_{12} - 5\beta_{10} + 15\beta_{6} - \beta_{5} + 4\beta_{4} ) / 30 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -12\beta_{12} - 5\beta_{11} - 5\beta_{9} + 87\beta_{8} + 87\beta_{4} - 27\beta_{2} - 605\beta _1 + 605 ) / 30 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 41 \beta_{15} + 295 \beta_{14} + 295 \beta_{13} + 145 \beta_{10} - 164 \beta_{8} - 465 \beta_{7} + 465 \beta_{6} - 41 \beta_{5} + 610 \beta_{3} - 41 \beta_{2} ) / 30 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -204\beta_{15} - 204\beta_{12} - 135\beta_{9} + 309\beta_{5} + 1329\beta_{4} - 7435\beta_1 ) / 10 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 13895 \beta_{14} - 1141 \beta_{12} - 4564 \beta_{8} - 18465 \beta_{7} - 4564 \beta_{4} + 33610 \beta_{3} - 1141 \beta_{2} ) / 30 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -29412\beta_{15} + 23405\beta_{11} - 176787\beta_{8} + 29727\beta_{5} + 29727\beta_{2} - 912305 ) / 30 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 28541 \beta_{15} - 645895 \beta_{13} - 28541 \beta_{12} - 933145 \beta_{10} - 774465 \beta_{6} + 28541 \beta_{5} - 114164 \beta_{4} ) / 30 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 463404 \beta_{12} + 405135 \beta_{11} + 405135 \beta_{9} - 2642529 \beta_{8} - 2642529 \beta_{4} + 325509 \beta_{2} + 12931435 \beta _1 - 12931435 ) / 10 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 602941 \beta_{15} - 29923895 \beta_{14} - 29923895 \beta_{13} - 49851145 \beta_{10} + 2411764 \beta_{8} + 33480465 \beta_{7} - 33480465 \beta_{6} + 602941 \beta_{5} + \cdots + 602941 \beta_{2} ) / 30 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 65091012 \beta_{15} + 65091012 \beta_{12} + 60017405 \beta_{9} - 33423327 \beta_{5} - 358878387 \beta_{4} + 1693056305 \beta_1 ) / 30 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 1383741895 \beta_{14} + 6867341 \beta_{12} + 27469364 \beta_{8} + 1478286465 \beta_{7} + 27469364 \beta_{4} - 3976455610 \beta_{3} + 6867341 \beta_{2} ) / 30 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 1009970604 \beta_{15} - 961173135 \beta_{11} + 5450049729 \beta_{8} - 400196709 \beta_{5} - 400196709 \beta_{2} + 25083439435 ) / 10 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 297208259 \beta_{15} + 63911359895 \beta_{13} - 297208259 \beta_{12} + 121023687145 \beta_{10} + 66234492465 \beta_{6} + 297208259 \beta_{5} - 1188833036 \beta_{4} ) / 30 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 140527452612 \beta_{12} - 136344821405 \beta_{11} - 136344821405 \beta_{9} + 748004139987 \beta_{8} + 748004139987 \beta_{4} - 45366876927 \beta_{2} + \cdots + 3386746380305 ) / 30 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 31699323859 \beta_{15} + 2949684177895 \beta_{14} + 2949684177895 \beta_{13} + 5750640405145 \beta_{10} + 126797295436 \beta_{8} + \cdots + 31699323859 \beta_{2} ) / 30 \) 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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
269.1
−1.59914 + 0.923267i
4.69001 2.70778i
−1.27560 + 0.736469i
1.27560 0.736469i
5.87958 3.39457i
−5.87958 + 3.39457i
1.59914 0.923267i
−4.69001 + 2.70778i
−1.59914 0.923267i
4.69001 + 2.70778i
−1.27560 0.736469i
1.27560 + 0.736469i
5.87958 + 3.39457i
−5.87958 3.39457i
1.59914 + 0.923267i
−4.69001 2.70778i
0 0 0 −4.82247 + 1.32053i 0 −5.42313 3.13104i 0 0 0
269.2 0 0 0 −3.55485 + 3.51612i 0 5.42313 + 3.13104i 0 0 0
269.3 0 0 0 −2.77776 4.15741i 0 −8.02120 4.63104i 0 0 0
269.4 0 0 0 −2.21154 4.48432i 0 8.02120 + 4.63104i 0 0 0
269.5 0 0 0 2.21154 + 4.48432i 0 8.02120 + 4.63104i 0 0 0
269.6 0 0 0 2.77776 + 4.15741i 0 −8.02120 4.63104i 0 0 0
269.7 0 0 0 3.55485 3.51612i 0 5.42313 + 3.13104i 0 0 0
269.8 0 0 0 4.82247 1.32053i 0 −5.42313 3.13104i 0 0 0
1349.1 0 0 0 −4.82247 1.32053i 0 −5.42313 + 3.13104i 0 0 0
1349.2 0 0 0 −3.55485 3.51612i 0 5.42313 3.13104i 0 0 0
1349.3 0 0 0 −2.77776 + 4.15741i 0 −8.02120 + 4.63104i 0 0 0
1349.4 0 0 0 −2.21154 + 4.48432i 0 8.02120 4.63104i 0 0 0
1349.5 0 0 0 2.21154 4.48432i 0 8.02120 4.63104i 0 0 0
1349.6 0 0 0 2.77776 4.15741i 0 −8.02120 + 4.63104i 0 0 0
1349.7 0 0 0 3.55485 + 3.51612i 0 5.42313 3.13104i 0 0 0
1349.8 0 0 0 4.82247 + 1.32053i 0 −5.42313 + 3.13104i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1349.8
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 1620.3.t.e 16
3.b odd 2 1 inner 1620.3.t.e 16
5.b even 2 1 inner 1620.3.t.e 16
9.c even 3 1 540.3.b.c 8
9.c even 3 1 inner 1620.3.t.e 16
9.d odd 6 1 540.3.b.c 8
9.d odd 6 1 inner 1620.3.t.e 16
15.d odd 2 1 inner 1620.3.t.e 16
36.f odd 6 1 2160.3.c.n 8
36.h even 6 1 2160.3.c.n 8
45.h odd 6 1 540.3.b.c 8
45.h odd 6 1 inner 1620.3.t.e 16
45.j even 6 1 540.3.b.c 8
45.j even 6 1 inner 1620.3.t.e 16
45.k odd 12 1 2700.3.g.o 4
45.k odd 12 1 2700.3.g.p 4
45.l even 12 1 2700.3.g.o 4
45.l even 12 1 2700.3.g.p 4
180.n even 6 1 2160.3.c.n 8
180.p odd 6 1 2160.3.c.n 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
540.3.b.c 8 9.c even 3 1
540.3.b.c 8 9.d odd 6 1
540.3.b.c 8 45.h odd 6 1
540.3.b.c 8 45.j even 6 1
1620.3.t.e 16 1.a even 1 1 trivial
1620.3.t.e 16 3.b odd 2 1 inner
1620.3.t.e 16 5.b even 2 1 inner
1620.3.t.e 16 9.c even 3 1 inner
1620.3.t.e 16 9.d odd 6 1 inner
1620.3.t.e 16 15.d odd 2 1 inner
1620.3.t.e 16 45.h odd 6 1 inner
1620.3.t.e 16 45.j even 6 1 inner
2160.3.c.n 8 36.f odd 6 1
2160.3.c.n 8 36.h even 6 1
2160.3.c.n 8 180.n even 6 1
2160.3.c.n 8 180.p odd 6 1
2700.3.g.o 4 45.k odd 12 1
2700.3.g.o 4 45.l even 12 1
2700.3.g.p 4 45.k odd 12 1
2700.3.g.p 4 45.l even 12 1

Hecke kernels

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

\( T_{7}^{8} - 125T_{7}^{6} + 12261T_{7}^{4} - 420500T_{7}^{2} + 11316496 \) Copy content Toggle raw display
\( T_{17}^{4} - 265T_{17}^{2} + 4000 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + 6 T^{14} + \cdots + 152587890625 \) Copy content Toggle raw display
$7$ \( (T^{8} - 125 T^{6} + 12261 T^{4} + \cdots + 11316496)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} - 235 T^{6} + 54975 T^{4} + \cdots + 62500)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 9 T^{2} + 81)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} - 265 T^{2} + 4000)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 9 T - 522)^{8} \) Copy content Toggle raw display
$23$ \( (T^{8} + 2635 T^{6} + \cdots + 2966145062500)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} - 2115 T^{6} + 4452975 T^{4} + \cdots + 410062500)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 11 T^{3} + 633 T^{2} + \cdots + 262144)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + 3053 T^{2} + 2208196)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} - 2890 T^{6} + \cdots + 2560000000000)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} - 1745 T^{6} + \cdots + 319794774016)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + 6625 T^{6} + \cdots + 6250000000000)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 5310 T^{2} + 6561000)^{4} \) Copy content Toggle raw display
$59$ \( (T^{8} - 15460 T^{6} + \cdots + 35\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 29 T^{3} + 1173 T^{2} + \cdots + 110224)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} - 3773 T^{6} + \cdots + 1607509551376)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 18990 T^{2} + 50625000)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} + 12953 T^{2} + 11999296)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 39 T + 1521)^{8} \) Copy content Toggle raw display
$83$ \( (T^{8} + 10090 T^{6} + \cdots + 623201296000000)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 15460 T^{2} + 59536000)^{4} \) Copy content Toggle raw display
$97$ \( (T^{8} - 26285 T^{6} + \cdots + 4275978808336)^{2} \) Copy content Toggle raw display
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