Properties

Label 1440.2.x.e
Level $1440$
Weight $2$
Character orbit 1440.x
Analytic conductor $11.498$
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: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1440.x (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(11.4984578911\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 160)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + 2 i ) q^{5} + ( 1 - i ) q^{7} +O(q^{10})\) \( q + ( -1 + 2 i ) q^{5} + ( 1 - i ) q^{7} -6 i q^{11} + ( -1 + i ) q^{13} + ( -1 - i ) q^{17} -4 q^{19} + ( 5 + 5 i ) q^{23} + ( -3 - 4 i ) q^{25} -8 i q^{29} -2 i q^{31} + ( 1 + 3 i ) q^{35} + ( -5 - 5 i ) q^{37} -6 q^{41} + ( -3 - 3 i ) q^{43} + ( 7 - 7 i ) q^{47} + 5 i q^{49} + ( 1 - i ) q^{53} + ( 12 + 6 i ) q^{55} + 4 q^{59} + 2 q^{61} + ( -1 - 3 i ) q^{65} + ( 7 - 7 i ) q^{67} -6 i q^{71} + ( 9 - 9 i ) q^{73} + ( -6 - 6 i ) q^{77} + 8 q^{79} + ( -5 - 5 i ) q^{83} + ( 3 - i ) q^{85} + 2 i q^{91} + ( 4 - 8 i ) q^{95} + ( -3 - 3 i ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} + 2 q^{7} + O(q^{10}) \) \( 2 q - 2 q^{5} + 2 q^{7} - 2 q^{13} - 2 q^{17} - 8 q^{19} + 10 q^{23} - 6 q^{25} + 2 q^{35} - 10 q^{37} - 12 q^{41} - 6 q^{43} + 14 q^{47} + 2 q^{53} + 24 q^{55} + 8 q^{59} + 4 q^{61} - 2 q^{65} + 14 q^{67} + 18 q^{73} - 12 q^{77} + 16 q^{79} - 10 q^{83} + 6 q^{85} + 8 q^{95} - 6 q^{97} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
1.00000i
1.00000i
0 0 0 −1.00000 + 2.00000i 0 1.00000 1.00000i 0 0 0
703.1 0 0 0 −1.00000 2.00000i 0 1.00000 + 1.00000i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1440.2.x.e 2
3.b odd 2 1 160.2.n.e yes 2
4.b odd 2 1 1440.2.x.b 2
5.c odd 4 1 1440.2.x.b 2
12.b even 2 1 160.2.n.b 2
15.d odd 2 1 800.2.n.c 2
15.e even 4 1 160.2.n.b 2
15.e even 4 1 800.2.n.h 2
20.e even 4 1 inner 1440.2.x.e 2
24.f even 2 1 320.2.n.f 2
24.h odd 2 1 320.2.n.c 2
48.i odd 4 1 1280.2.o.d 2
48.i odd 4 1 1280.2.o.n 2
48.k even 4 1 1280.2.o.e 2
48.k even 4 1 1280.2.o.k 2
60.h even 2 1 800.2.n.h 2
60.l odd 4 1 160.2.n.e yes 2
60.l odd 4 1 800.2.n.c 2
120.i odd 2 1 1600.2.n.j 2
120.m even 2 1 1600.2.n.e 2
120.q odd 4 1 320.2.n.c 2
120.q odd 4 1 1600.2.n.j 2
120.w even 4 1 320.2.n.f 2
120.w even 4 1 1600.2.n.e 2
240.z odd 4 1 1280.2.o.n 2
240.bb even 4 1 1280.2.o.e 2
240.bd odd 4 1 1280.2.o.d 2
240.bf even 4 1 1280.2.o.k 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.2.n.b 2 12.b even 2 1
160.2.n.b 2 15.e even 4 1
160.2.n.e yes 2 3.b odd 2 1
160.2.n.e yes 2 60.l odd 4 1
320.2.n.c 2 24.h odd 2 1
320.2.n.c 2 120.q odd 4 1
320.2.n.f 2 24.f even 2 1
320.2.n.f 2 120.w even 4 1
800.2.n.c 2 15.d odd 2 1
800.2.n.c 2 60.l odd 4 1
800.2.n.h 2 15.e even 4 1
800.2.n.h 2 60.h even 2 1
1280.2.o.d 2 48.i odd 4 1
1280.2.o.d 2 240.bd odd 4 1
1280.2.o.e 2 48.k even 4 1
1280.2.o.e 2 240.bb even 4 1
1280.2.o.k 2 48.k even 4 1
1280.2.o.k 2 240.bf even 4 1
1280.2.o.n 2 48.i odd 4 1
1280.2.o.n 2 240.z odd 4 1
1440.2.x.b 2 4.b odd 2 1
1440.2.x.b 2 5.c odd 4 1
1440.2.x.e 2 1.a even 1 1 trivial
1440.2.x.e 2 20.e even 4 1 inner
1600.2.n.e 2 120.m even 2 1
1600.2.n.e 2 120.w even 4 1
1600.2.n.j 2 120.i odd 2 1
1600.2.n.j 2 120.q odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1440, [\chi])\):

\( T_{7}^{2} - 2 T_{7} + 2 \)
\( T_{11}^{2} + 36 \)
\( T_{17}^{2} + 2 T_{17} + 2 \)
\( T_{19} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 5 + 2 T + T^{2} \)
$7$ \( 2 - 2 T + T^{2} \)
$11$ \( 36 + T^{2} \)
$13$ \( 2 + 2 T + T^{2} \)
$17$ \( 2 + 2 T + T^{2} \)
$19$ \( ( 4 + T )^{2} \)
$23$ \( 50 - 10 T + T^{2} \)
$29$ \( 64 + T^{2} \)
$31$ \( 4 + T^{2} \)
$37$ \( 50 + 10 T + T^{2} \)
$41$ \( ( 6 + T )^{2} \)
$43$ \( 18 + 6 T + T^{2} \)
$47$ \( 98 - 14 T + T^{2} \)
$53$ \( 2 - 2 T + T^{2} \)
$59$ \( ( -4 + T )^{2} \)
$61$ \( ( -2 + T )^{2} \)
$67$ \( 98 - 14 T + T^{2} \)
$71$ \( 36 + T^{2} \)
$73$ \( 162 - 18 T + T^{2} \)
$79$ \( ( -8 + T )^{2} \)
$83$ \( 50 + 10 T + T^{2} \)
$89$ \( T^{2} \)
$97$ \( 18 + 6 T + T^{2} \)
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