Properties

Label 1440.3.j.b
Level $1440$
Weight $3$
Character orbit 1440.j
Analytic conductor $39.237$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1440.j (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(39.2371580679\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.1827904.1
Defining polynomial: \(x^{6} + 9 x^{4} + 14 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{9} \)
Twist minimal: no (minimal twist has level 160)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q -\beta_{4} q^{5} + ( 2 + \beta_{1} - \beta_{4} ) q^{7} +O(q^{10})\) \( q -\beta_{4} q^{5} + ( 2 + \beta_{1} - \beta_{4} ) q^{7} + ( -2 - 2 \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{4} ) q^{11} -\beta_{5} q^{13} + ( -1 - \beta_{1} - 4 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{17} + ( -\beta_{2} - 2 \beta_{5} ) q^{19} + ( 12 - 3 \beta_{1} - 2 \beta_{3} + 3 \beta_{4} ) q^{23} + ( -5 - 2 \beta_{1} - 4 \beta_{2} + 6 \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{25} + ( -10 + 8 \beta_{3} ) q^{29} + ( -2 - 2 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{31} + ( 15 - 5 \beta_{2} + 5 \beta_{3} - 4 \beta_{4} ) q^{35} + ( -1 - \beta_{1} + 8 \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{37} + ( 13 + \beta_{1} - 5 \beta_{3} - \beta_{4} ) q^{41} + ( -13 - 6 \beta_{1} + \beta_{3} + 6 \beta_{4} ) q^{43} + ( -42 - \beta_{1} - 8 \beta_{3} + \beta_{4} ) q^{47} + ( -12 + 7 \beta_{1} + 5 \beta_{3} - 7 \beta_{4} ) q^{49} + ( 8 + 8 \beta_{1} + 8 \beta_{2} - 8 \beta_{3} + 8 \beta_{4} + 3 \beta_{5} ) q^{53} + ( -55 - 7 \beta_{1} - 9 \beta_{2} + 11 \beta_{3} - 3 \beta_{4} - \beta_{5} ) q^{55} + ( 15 \beta_{2} - 2 \beta_{5} ) q^{59} + ( -17 + 7 \beta_{1} + \beta_{3} - 7 \beta_{4} ) q^{61} + ( -5 - \beta_{1} + 8 \beta_{2} + 13 \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{65} + ( 47 - 2 \beta_{1} + 13 \beta_{3} + 2 \beta_{4} ) q^{67} + ( -6 - 6 \beta_{1} + 2 \beta_{2} + 6 \beta_{3} - 6 \beta_{4} + 2 \beta_{5} ) q^{71} + ( -1 - \beta_{1} - 4 \beta_{2} + \beta_{3} - \beta_{4} + 3 \beta_{5} ) q^{73} + ( -11 - 11 \beta_{1} - 24 \beta_{2} + 11 \beta_{3} - 11 \beta_{4} + \beta_{5} ) q^{77} + ( 6 + 6 \beta_{1} + 24 \beta_{2} - 6 \beta_{3} + 6 \beta_{4} + 2 \beta_{5} ) q^{79} + ( 33 - 14 \beta_{1} + 3 \beta_{3} + 14 \beta_{4} ) q^{83} + ( -5 - 13 \beta_{1} - 16 \beta_{2} - 11 \beta_{3} + 3 \beta_{4} + \beta_{5} ) q^{85} + ( -10 + 16 \beta_{1} - 8 \beta_{3} - 16 \beta_{4} ) q^{89} + ( 2 + 2 \beta_{1} + 10 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{91} + ( -5 - 5 \beta_{1} + 15 \beta_{2} + 25 \beta_{3} - \beta_{4} + 5 \beta_{5} ) q^{95} + ( 5 + 5 \beta_{1} + 12 \beta_{2} - 5 \beta_{3} + 5 \beta_{4} - \beta_{5} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 2q^{5} + 12q^{7} + O(q^{10}) \) \( 6q + 2q^{5} + 12q^{7} + 68q^{23} - 10q^{25} - 44q^{29} + 108q^{35} + 68q^{41} - 76q^{43} - 268q^{47} - 62q^{49} - 288q^{55} - 100q^{61} + 308q^{67} + 204q^{83} - 32q^{85} - 76q^{89} + 32q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} + 9 x^{4} + 14 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 2 \nu^{5} + 2 \nu^{4} + 20 \nu^{3} + 10 \nu^{2} + 48 \nu - 7 \)\()/5\)
\(\beta_{2}\)\(=\)\((\)\( -4 \nu^{5} - 40 \nu^{3} - 76 \nu \)\()/5\)
\(\beta_{3}\)\(=\)\((\)\( 4 \nu^{4} + 30 \nu^{2} + 21 \)\()/5\)
\(\beta_{4}\)\(=\)\((\)\( 2 \nu^{5} + 2 \nu^{4} + 20 \nu^{3} + 20 \nu^{2} + 48 \nu + 23 \)\()/5\)
\(\beta_{5}\)\(=\)\((\)\( -28 \nu^{5} - 240 \nu^{3} - 312 \nu \)\()/5\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{4} - \beta_{3} + \beta_{2} + \beta_{1} + 1\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{4} - \beta_{1} - 6\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{5} - 11 \beta_{4} + 11 \beta_{3} - 18 \beta_{2} - 11 \beta_{1} - 11\)\()/8\)
\(\nu^{4}\)\(=\)\((\)\(-15 \beta_{4} + 5 \beta_{3} + 15 \beta_{1} + 69\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(-5 \beta_{5} + 36 \beta_{4} - 36 \beta_{3} + 66 \beta_{2} + 36 \beta_{1} + 36\)\()/4\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1440\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(641\) \(901\) \(991\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1279.1
0.273891i
0.273891i
1.37720i
1.37720i
2.65109i
2.65109i
0 0 0 −4.30219 2.54778i 0 −3.84997 0 0 0
1279.2 0 0 0 −4.30219 + 2.54778i 0 −3.84997 0 0 0
1279.3 0 0 0 1.54778 4.75441i 0 −0.206625 0 0 0
1279.4 0 0 0 1.54778 + 4.75441i 0 −0.206625 0 0 0
1279.5 0 0 0 3.75441 3.30219i 0 10.0566 0 0 0
1279.6 0 0 0 3.75441 + 3.30219i 0 10.0566 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1279.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1440.3.j.b 6
3.b odd 2 1 160.3.h.b yes 6
4.b odd 2 1 1440.3.j.a 6
5.b even 2 1 1440.3.j.a 6
12.b even 2 1 160.3.h.a 6
15.d odd 2 1 160.3.h.a 6
15.e even 4 1 800.3.b.h 6
15.e even 4 1 800.3.b.i 6
20.d odd 2 1 inner 1440.3.j.b 6
24.f even 2 1 320.3.h.g 6
24.h odd 2 1 320.3.h.f 6
48.i odd 4 1 1280.3.e.f 6
48.i odd 4 1 1280.3.e.h 6
48.k even 4 1 1280.3.e.g 6
48.k even 4 1 1280.3.e.i 6
60.h even 2 1 160.3.h.b yes 6
60.l odd 4 1 800.3.b.h 6
60.l odd 4 1 800.3.b.i 6
120.i odd 2 1 320.3.h.g 6
120.m even 2 1 320.3.h.f 6
120.q odd 4 1 1600.3.b.v 6
120.q odd 4 1 1600.3.b.w 6
120.w even 4 1 1600.3.b.v 6
120.w even 4 1 1600.3.b.w 6
240.t even 4 1 1280.3.e.f 6
240.t even 4 1 1280.3.e.h 6
240.bm odd 4 1 1280.3.e.g 6
240.bm odd 4 1 1280.3.e.i 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.3.h.a 6 12.b even 2 1
160.3.h.a 6 15.d odd 2 1
160.3.h.b yes 6 3.b odd 2 1
160.3.h.b yes 6 60.h even 2 1
320.3.h.f 6 24.h odd 2 1
320.3.h.f 6 120.m even 2 1
320.3.h.g 6 24.f even 2 1
320.3.h.g 6 120.i odd 2 1
800.3.b.h 6 15.e even 4 1
800.3.b.h 6 60.l odd 4 1
800.3.b.i 6 15.e even 4 1
800.3.b.i 6 60.l odd 4 1
1280.3.e.f 6 48.i odd 4 1
1280.3.e.f 6 240.t even 4 1
1280.3.e.g 6 48.k even 4 1
1280.3.e.g 6 240.bm odd 4 1
1280.3.e.h 6 48.i odd 4 1
1280.3.e.h 6 240.t even 4 1
1280.3.e.i 6 48.k even 4 1
1280.3.e.i 6 240.bm odd 4 1
1440.3.j.a 6 4.b odd 2 1
1440.3.j.a 6 5.b even 2 1
1440.3.j.b 6 1.a even 1 1 trivial
1440.3.j.b 6 20.d odd 2 1 inner
1600.3.b.v 6 120.q odd 4 1
1600.3.b.v 6 120.w even 4 1
1600.3.b.w 6 120.q odd 4 1
1600.3.b.w 6 120.w even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1440, [\chi])\):

\( T_{7}^{3} - 6 T_{7}^{2} - 40 T_{7} - 8 \)
\( T_{23}^{3} - 34 T_{23}^{2} - 152 T_{23} + 9160 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \)
$3$ \( T^{6} \)
$5$ \( 15625 - 1250 T + 175 T^{2} + 100 T^{3} + 7 T^{4} - 2 T^{5} + T^{6} \)
$7$ \( ( -8 - 40 T - 6 T^{2} + T^{3} )^{2} \)
$11$ \( 2560000 + 86784 T^{2} + 560 T^{4} + T^{6} \)
$13$ \( 692224 + 43264 T^{2} + 416 T^{4} + T^{6} \)
$17$ \( 6553600 + 270336 T^{2} + 1088 T^{4} + T^{6} \)
$19$ \( 11505664 + 533248 T^{2} + 1712 T^{4} + T^{6} \)
$23$ \( ( 9160 - 152 T - 34 T^{2} + T^{3} )^{2} \)
$29$ \( ( 2120 - 948 T + 22 T^{2} + T^{3} )^{2} \)
$31$ \( 419430400 + 1900544 T^{2} + 2560 T^{4} + T^{6} \)
$37$ \( 432640000 + 2985216 T^{2} + 5408 T^{4} + T^{6} \)
$41$ \( ( 5000 - 100 T - 34 T^{2} + T^{3} )^{2} \)
$43$ \( ( -6280 - 1408 T + 38 T^{2} + T^{3} )^{2} \)
$47$ \( ( 22216 + 4824 T + 134 T^{2} + T^{3} )^{2} \)
$53$ \( 50319462400 + 49973504 T^{2} + 12960 T^{4} + T^{6} \)
$59$ \( 25416011776 + 41968384 T^{2} + 12464 T^{4} + T^{6} \)
$61$ \( ( -81544 - 1732 T + 50 T^{2} + T^{3} )^{2} \)
$67$ \( ( 24760 + 4768 T - 154 T^{2} + T^{3} )^{2} \)
$71$ \( 14231535616 + 19738624 T^{2} + 8384 T^{4} + T^{6} \)
$73$ \( 3114532864 + 6447104 T^{2} + 4416 T^{4} + T^{6} \)
$79$ \( 37060870144 + 159514624 T^{2} + 25856 T^{4} + T^{6} \)
$83$ \( ( 483400 - 6880 T - 102 T^{2} + T^{3} )^{2} \)
$89$ \( ( -155000 - 13940 T + 38 T^{2} + T^{3} )^{2} \)
$97$ \( 44302336 + 1384448 T^{2} + 7488 T^{4} + T^{6} \)
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