Properties

Label 1440.4.a.bb
Level $1440$
Weight $4$
Character orbit 1440.a
Self dual yes
Analytic conductor $84.963$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1440.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(84.9627504083\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 160)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 5 q^{5} -7 \beta q^{7} +O(q^{10})\) \( q + 5 q^{5} -7 \beta q^{7} + 2 \beta q^{11} -62 q^{13} + 46 q^{17} + 24 \beta q^{19} -43 \beta q^{23} + 25 q^{25} + 90 q^{29} -34 \beta q^{31} -35 \beta q^{35} -214 q^{37} + 10 q^{41} -15 \beta q^{43} -89 \beta q^{47} + 637 q^{49} + 678 q^{53} + 10 \beta q^{55} + 92 \beta q^{59} + 250 q^{61} -310 q^{65} + 11 \beta q^{67} + 82 \beta q^{71} + 522 q^{73} -280 q^{77} + 196 \beta q^{79} -85 \beta q^{83} + 230 q^{85} -970 q^{89} + 434 \beta q^{91} + 120 \beta q^{95} -934 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 10 q^{5} + O(q^{10}) \) \( 2 q + 10 q^{5} - 124 q^{13} + 92 q^{17} + 50 q^{25} + 180 q^{29} - 428 q^{37} + 20 q^{41} + 1274 q^{49} + 1356 q^{53} + 500 q^{61} - 620 q^{65} + 1044 q^{73} - 560 q^{77} + 460 q^{85} - 1940 q^{89} - 1868 q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
1.61803
−0.618034
0 0 0 5.00000 0 −31.3050 0 0 0
1.2 0 0 0 5.00000 0 31.3050 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(5\) \(-1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1440.4.a.bb 2
3.b odd 2 1 160.4.a.d 2
4.b odd 2 1 inner 1440.4.a.bb 2
12.b even 2 1 160.4.a.d 2
15.d odd 2 1 800.4.a.o 2
15.e even 4 2 800.4.c.l 4
24.f even 2 1 320.4.a.q 2
24.h odd 2 1 320.4.a.q 2
48.i odd 4 2 1280.4.d.v 4
48.k even 4 2 1280.4.d.v 4
60.h even 2 1 800.4.a.o 2
60.l odd 4 2 800.4.c.l 4
120.i odd 2 1 1600.4.a.cg 2
120.m even 2 1 1600.4.a.cg 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.4.a.d 2 3.b odd 2 1
160.4.a.d 2 12.b even 2 1
320.4.a.q 2 24.f even 2 1
320.4.a.q 2 24.h odd 2 1
800.4.a.o 2 15.d odd 2 1
800.4.a.o 2 60.h even 2 1
800.4.c.l 4 15.e even 4 2
800.4.c.l 4 60.l odd 4 2
1280.4.d.v 4 48.i odd 4 2
1280.4.d.v 4 48.k even 4 2
1440.4.a.bb 2 1.a even 1 1 trivial
1440.4.a.bb 2 4.b odd 2 1 inner
1600.4.a.cg 2 120.i odd 2 1
1600.4.a.cg 2 120.m even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1440))\):

\( T_{7}^{2} - 980 \)
\( T_{11}^{2} - 80 \)
\( T_{17} - 46 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( ( -5 + T )^{2} \)
$7$ \( -980 + T^{2} \)
$11$ \( -80 + T^{2} \)
$13$ \( ( 62 + T )^{2} \)
$17$ \( ( -46 + T )^{2} \)
$19$ \( -11520 + T^{2} \)
$23$ \( -36980 + T^{2} \)
$29$ \( ( -90 + T )^{2} \)
$31$ \( -23120 + T^{2} \)
$37$ \( ( 214 + T )^{2} \)
$41$ \( ( -10 + T )^{2} \)
$43$ \( -4500 + T^{2} \)
$47$ \( -158420 + T^{2} \)
$53$ \( ( -678 + T )^{2} \)
$59$ \( -169280 + T^{2} \)
$61$ \( ( -250 + T )^{2} \)
$67$ \( -2420 + T^{2} \)
$71$ \( -134480 + T^{2} \)
$73$ \( ( -522 + T )^{2} \)
$79$ \( -768320 + T^{2} \)
$83$ \( -144500 + T^{2} \)
$89$ \( ( 970 + T )^{2} \)
$97$ \( ( 934 + T )^{2} \)
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