Properties

Label 1440.2.bi.d
Level $1440$
Weight $2$
Character orbit 1440.bi
Analytic conductor $11.498$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1440.bi (of order \(4\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(11.4984578911\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Defining polynomial: \(x^{16} + 5 x^{12} + 28 x^{8} + 80 x^{4} + 256\)
Coefficient ring: \(\Z[a_1, \ldots, a_{49}]\)
Coefficient ring index: \( 2^{18} \)
Twist minimal: no (minimal twist has level 360)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q -\beta_{9} q^{5} -\beta_{10} q^{7} +O(q^{10})\) \( q -\beta_{9} q^{5} -\beta_{10} q^{7} -\beta_{15} q^{11} + \beta_{7} q^{13} + \beta_{13} q^{17} -4 \beta_{4} q^{19} + ( -2 \beta_{5} + 2 \beta_{9} ) q^{23} + ( -1 - 2 \beta_{4} - \beta_{11} ) q^{25} + ( 2 \beta_{1} - \beta_{2} + \beta_{5} - 2 \beta_{9} ) q^{29} + ( \beta_{6} + \beta_{7} ) q^{31} + ( -\beta_{3} - \beta_{13} + \beta_{15} ) q^{35} + \beta_{6} q^{37} + ( \beta_{3} - \beta_{13} ) q^{41} + ( 2 - 2 \beta_{4} - 2 \beta_{11} ) q^{43} + ( -2 \beta_{1} + 2 \beta_{2} + 2 \beta_{5} + 2 \beta_{9} ) q^{47} + ( -5 \beta_{4} - \beta_{11} - \beta_{12} ) q^{49} + ( -\beta_{1} - \beta_{2} ) q^{53} + ( \beta_{6} + \beta_{7} - \beta_{8} ) q^{55} -\beta_{14} q^{59} + ( \beta_{6} + \beta_{7} - 2 \beta_{8} - 2 \beta_{10} ) q^{61} + ( \beta_{13} + 2 \beta_{14} + 2 \beta_{15} ) q^{65} + ( -2 - 2 \beta_{4} - 2 \beta_{12} ) q^{67} + ( 5 - 5 \beta_{4} ) q^{73} + ( -3 \beta_{1} + 3 \beta_{2} + \beta_{5} + \beta_{9} ) q^{77} + ( -\beta_{6} + \beta_{7} ) q^{79} + ( 2 \beta_{3} + \beta_{14} + \beta_{15} ) q^{83} + ( \beta_{7} + 2 \beta_{8} - 2 \beta_{10} ) q^{85} + ( -\beta_{3} - \beta_{13} - 4 \beta_{14} ) q^{89} + ( 4 - 2 \beta_{11} + 2 \beta_{12} ) q^{91} -4 \beta_{2} q^{95} + ( -3 - 3 \beta_{4} + 2 \beta_{12} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + O(q^{10}) \) \( 16 q - 24 q^{25} + 16 q^{43} - 16 q^{67} + 80 q^{73} + 32 q^{91} - 64 q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} + 5 x^{12} + 28 x^{8} + 80 x^{4} + 256\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{15} - 4 \nu^{13} + 11 \nu^{11} - 36 \nu^{9} - 44 \nu^{7} + 64 \nu^{5} + 144 \nu^{3} + 256 \nu \)\()/640\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{15} - 4 \nu^{13} - 11 \nu^{11} - 36 \nu^{9} + 44 \nu^{7} + 64 \nu^{5} - 144 \nu^{3} + 256 \nu \)\()/640\)
\(\beta_{3}\)\(=\)\((\)\( 3 \nu^{13} + 27 \nu^{9} + 112 \nu^{5} + 288 \nu \)\()/80\)
\(\beta_{4}\)\(=\)\((\)\( -3 \nu^{14} - 7 \nu^{10} - 12 \nu^{6} - 48 \nu^{2} \)\()/320\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{15} - 4 \nu^{13} + 11 \nu^{11} + 4 \nu^{9} + 36 \nu^{7} - 56 \nu^{5} + 224 \nu^{3} + 96 \nu \)\()/320\)
\(\beta_{6}\)\(=\)\((\)\( 3 \nu^{12} + 10 \nu^{10} + 7 \nu^{8} + 10 \nu^{6} + 92 \nu^{4} + 80 \nu^{2} + 128 \)\()/80\)
\(\beta_{7}\)\(=\)\((\)\( 3 \nu^{12} - 10 \nu^{10} + 7 \nu^{8} - 10 \nu^{6} + 92 \nu^{4} - 80 \nu^{2} + 128 \)\()/80\)
\(\beta_{8}\)\(=\)\((\)\( -5 \nu^{14} - 4 \nu^{12} - 25 \nu^{10} + 44 \nu^{8} - 140 \nu^{6} + 144 \nu^{4} - 80 \nu^{2} + 576 \)\()/320\)
\(\beta_{9}\)\(=\)\((\)\( -\nu^{15} + 4 \nu^{13} + 11 \nu^{11} - 4 \nu^{9} + 36 \nu^{7} + 56 \nu^{5} + 224 \nu^{3} - 96 \nu \)\()/320\)
\(\beta_{10}\)\(=\)\((\)\( 5 \nu^{14} - 4 \nu^{12} + 25 \nu^{10} + 44 \nu^{8} + 140 \nu^{6} + 144 \nu^{4} + 80 \nu^{2} + 576 \)\()/320\)
\(\beta_{11}\)\(=\)\((\)\( \nu^{14} + 4 \nu^{12} + 5 \nu^{10} + 20 \nu^{8} + 28 \nu^{6} + 48 \nu^{4} + 144 \nu^{2} + 192 \)\()/64\)
\(\beta_{12}\)\(=\)\((\)\( \nu^{14} - 4 \nu^{12} + 5 \nu^{10} - 20 \nu^{8} + 28 \nu^{6} - 48 \nu^{4} + 144 \nu^{2} - 192 \)\()/64\)
\(\beta_{13}\)\(=\)\((\)\( -\nu^{15} - 9 \nu^{11} - 24 \nu^{7} + 24 \nu^{3} \)\()/80\)
\(\beta_{14}\)\(=\)\((\)\( 11 \nu^{15} + 12 \nu^{13} + 39 \nu^{11} + 28 \nu^{9} + 164 \nu^{7} + 48 \nu^{5} + 496 \nu^{3} + 832 \nu \)\()/640\)
\(\beta_{15}\)\(=\)\((\)\( -11 \nu^{15} + 12 \nu^{13} - 39 \nu^{11} + 28 \nu^{9} - 164 \nu^{7} + 48 \nu^{5} - 496 \nu^{3} + 832 \nu \)\()/640\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{15} + \beta_{14} - \beta_{9} + \beta_{5} + \beta_{2} + \beta_{1}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{12} + \beta_{11} - \beta_{10} + \beta_{8}\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{15} + \beta_{14} + 2 \beta_{13} + \beta_{9} + \beta_{5} - \beta_{2} + \beta_{1}\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(\beta_{12} - \beta_{11} + \beta_{10} + \beta_{8} + 2 \beta_{7} + 2 \beta_{6} - 4\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(-3 \beta_{15} - 3 \beta_{14} + 3 \beta_{9} - 3 \beta_{5} + 2 \beta_{3} + 3 \beta_{2} + 3 \beta_{1}\)\()/4\)
\(\nu^{6}\)\(=\)\((\)\(\beta_{12} + \beta_{11} + 5 \beta_{10} - 5 \beta_{8} + 2 \beta_{7} - 2 \beta_{6} + 20 \beta_{4}\)\()/4\)
\(\nu^{7}\)\(=\)\((\)\(\beta_{15} - \beta_{14} - 2 \beta_{13} + 7 \beta_{9} + 7 \beta_{5} + 17 \beta_{2} - 17 \beta_{1}\)\()/4\)
\(\nu^{8}\)\(=\)\((\)\(-5 \beta_{12} + 5 \beta_{11} + 7 \beta_{10} + 7 \beta_{8} - 6 \beta_{7} - 6 \beta_{6} - 36\)\()/4\)
\(\nu^{9}\)\(=\)\((\)\(-5 \beta_{15} - 5 \beta_{14} - 11 \beta_{9} + 11 \beta_{5} + 6 \beta_{3} - 19 \beta_{2} - 19 \beta_{1}\)\()/4\)
\(\nu^{10}\)\(=\)\((\)\(-9 \beta_{12} - 9 \beta_{11} + 3 \beta_{10} - 3 \beta_{8} - 18 \beta_{7} + 18 \beta_{6} - 20 \beta_{4}\)\()/4\)
\(\nu^{11}\)\(=\)\((\)\(7 \beta_{15} - 7 \beta_{14} - 30 \beta_{13} + \beta_{9} + \beta_{5} - 41 \beta_{2} + 41 \beta_{1}\)\()/4\)
\(\nu^{12}\)\(=\)\((\)\(-19 \beta_{12} + 19 \beta_{11} - 47 \beta_{10} - 47 \beta_{8} + 6 \beta_{7} + 6 \beta_{6} + 36\)\()/4\)
\(\nu^{13}\)\(=\)\((\)\(61 \beta_{15} + 61 \beta_{14} + 83 \beta_{9} - 83 \beta_{5} - 22 \beta_{3} - 37 \beta_{2} - 37 \beta_{1}\)\()/4\)
\(\nu^{14}\)\(=\)\((\)\(\beta_{12} + \beta_{11} - 11 \beta_{10} + 11 \beta_{8} + 34 \beta_{7} - 34 \beta_{6} - 460 \beta_{4}\)\()/4\)
\(\nu^{15}\)\(=\)\((\)\(-111 \beta_{15} + 111 \beta_{14} + 46 \beta_{13} - 153 \beta_{9} - 153 \beta_{5} - 63 \beta_{2} + 63 \beta_{1}\)\()/4\)

Character values

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

\(n\) \(577\) \(641\) \(901\) \(991\)
\(\chi(n)\) \(\beta_{4}\) \(1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
847.1
−0.512386 1.31813i
−1.31813 0.512386i
0.788026 1.17431i
−1.17431 + 0.788026i
−0.788026 + 1.17431i
1.17431 0.788026i
0.512386 + 1.31813i
1.31813 + 0.512386i
−0.512386 + 1.31813i
−1.31813 + 0.512386i
0.788026 + 1.17431i
−1.17431 0.788026i
−0.788026 1.17431i
1.17431 + 0.788026i
0.512386 1.31813i
1.31813 0.512386i
0 0 0 −1.83051 + 1.28422i 0 −2.94984 + 2.94984i 0 0 0
847.2 0 0 0 −1.83051 + 1.28422i 0 2.94984 2.94984i 0 0 0
847.3 0 0 0 −0.386289 2.20245i 0 −1.51606 + 1.51606i 0 0 0
847.4 0 0 0 −0.386289 2.20245i 0 1.51606 1.51606i 0 0 0
847.5 0 0 0 0.386289 + 2.20245i 0 −1.51606 + 1.51606i 0 0 0
847.6 0 0 0 0.386289 + 2.20245i 0 1.51606 1.51606i 0 0 0
847.7 0 0 0 1.83051 1.28422i 0 −2.94984 + 2.94984i 0 0 0
847.8 0 0 0 1.83051 1.28422i 0 2.94984 2.94984i 0 0 0
1423.1 0 0 0 −1.83051 1.28422i 0 −2.94984 2.94984i 0 0 0
1423.2 0 0 0 −1.83051 1.28422i 0 2.94984 + 2.94984i 0 0 0
1423.3 0 0 0 −0.386289 + 2.20245i 0 −1.51606 1.51606i 0 0 0
1423.4 0 0 0 −0.386289 + 2.20245i 0 1.51606 + 1.51606i 0 0 0
1423.5 0 0 0 0.386289 2.20245i 0 −1.51606 1.51606i 0 0 0
1423.6 0 0 0 0.386289 2.20245i 0 1.51606 + 1.51606i 0 0 0
1423.7 0 0 0 1.83051 + 1.28422i 0 −2.94984 2.94984i 0 0 0
1423.8 0 0 0 1.83051 + 1.28422i 0 2.94984 + 2.94984i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1423.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
8.d odd 2 1 inner
15.e even 4 1 inner
24.f even 2 1 inner
40.k even 4 1 inner
120.q odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1440.2.bi.d 16
3.b odd 2 1 inner 1440.2.bi.d 16
4.b odd 2 1 360.2.w.d 16
5.c odd 4 1 inner 1440.2.bi.d 16
8.b even 2 1 360.2.w.d 16
8.d odd 2 1 inner 1440.2.bi.d 16
12.b even 2 1 360.2.w.d 16
15.e even 4 1 inner 1440.2.bi.d 16
20.e even 4 1 360.2.w.d 16
24.f even 2 1 inner 1440.2.bi.d 16
24.h odd 2 1 360.2.w.d 16
40.i odd 4 1 360.2.w.d 16
40.k even 4 1 inner 1440.2.bi.d 16
60.l odd 4 1 360.2.w.d 16
120.q odd 4 1 inner 1440.2.bi.d 16
120.w even 4 1 360.2.w.d 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.w.d 16 4.b odd 2 1
360.2.w.d 16 8.b even 2 1
360.2.w.d 16 12.b even 2 1
360.2.w.d 16 20.e even 4 1
360.2.w.d 16 24.h odd 2 1
360.2.w.d 16 40.i odd 4 1
360.2.w.d 16 60.l odd 4 1
360.2.w.d 16 120.w even 4 1
1440.2.bi.d 16 1.a even 1 1 trivial
1440.2.bi.d 16 3.b odd 2 1 inner
1440.2.bi.d 16 5.c odd 4 1 inner
1440.2.bi.d 16 8.d odd 2 1 inner
1440.2.bi.d 16 15.e even 4 1 inner
1440.2.bi.d 16 24.f even 2 1 inner
1440.2.bi.d 16 40.k even 4 1 inner
1440.2.bi.d 16 120.q odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} + 324 T_{7}^{4} + 6400 \) acting on \(S_{2}^{\mathrm{new}}(1440, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \)
$3$ \( T^{16} \)
$5$ \( ( 625 + 150 T^{2} + 18 T^{4} + 6 T^{6} + T^{8} )^{2} \)
$7$ \( ( 6400 + 324 T^{4} + T^{8} )^{2} \)
$11$ \( ( 40 - 18 T^{2} + T^{4} )^{4} \)
$13$ \( ( 102400 + 804 T^{4} + T^{8} )^{2} \)
$17$ \( ( 25600 + 1796 T^{4} + T^{8} )^{2} \)
$19$ \( ( 16 + T^{2} )^{8} \)
$23$ \( ( 16384 + 2880 T^{4} + T^{8} )^{2} \)
$29$ \( ( 800 - 58 T^{2} + T^{4} )^{4} \)
$31$ \( ( 1280 + 76 T^{2} + T^{4} )^{4} \)
$37$ \( ( 102400 + 804 T^{4} + T^{8} )^{2} \)
$41$ \( ( 640 - 92 T^{2} + T^{4} )^{4} \)
$43$ \( ( 6400 + 320 T + 8 T^{2} - 4 T^{3} + T^{4} )^{4} \)
$47$ \( ( 262144 + 11520 T^{4} + T^{8} )^{2} \)
$53$ \( ( 16384 + 420 T^{4} + T^{8} )^{2} \)
$59$ \( ( 40 + 18 T^{2} + T^{4} )^{4} \)
$61$ \( ( 8000 + 220 T^{2} + T^{4} )^{4} \)
$67$ \( ( 6400 - 320 T + 8 T^{2} + 4 T^{3} + T^{4} )^{4} \)
$71$ \( T^{16} \)
$73$ \( ( 50 - 10 T + T^{2} )^{8} \)
$79$ \( ( 1280 - 76 T^{2} + T^{4} )^{4} \)
$83$ \( ( 104857600 + 21136 T^{4} + T^{8} )^{2} \)
$89$ \( ( 2560 + 348 T^{2} + T^{4} )^{4} \)
$97$ \( ( 2500 - 800 T + 128 T^{2} + 16 T^{3} + T^{4} )^{4} \)
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