Properties

Label 1440.2.o.b
Level $1440$
Weight $2$
Character orbit 1440.o
Analytic conductor $11.498$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(11.4984578911\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: 12.0.426337261060096.1
Defining polynomial: \(x^{12} - 4 x^{9} - 3 x^{8} + 4 x^{7} + 8 x^{6} + 8 x^{5} - 12 x^{4} - 32 x^{3} + 64\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{2} q^{5} -\beta_{9} q^{7} +O(q^{10})\) \( q -\beta_{2} q^{5} -\beta_{9} q^{7} + ( \beta_{2} - \beta_{3} - \beta_{4} ) q^{11} + ( \beta_{7} - \beta_{8} ) q^{13} + \beta_{6} q^{17} + 2 \beta_{7} q^{19} + ( -\beta_{2} - \beta_{3} - 2 \beta_{5} + \beta_{11} ) q^{23} + ( -1 + \beta_{1} + \beta_{7} + \beta_{8} + \beta_{9} ) q^{25} + ( \beta_{2} + \beta_{3} + 2 \beta_{11} ) q^{29} + ( -\beta_{1} - 2 \beta_{7} + \beta_{8} - \beta_{10} ) q^{31} + ( -2 \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{11} ) q^{35} + ( \beta_{1} + \beta_{7} + \beta_{8} + \beta_{10} ) q^{37} + ( -\beta_{5} + 2 \beta_{11} ) q^{41} + ( 4 + 2 \beta_{1} - 2 \beta_{9} - 2 \beta_{10} ) q^{43} + ( -2 \beta_{2} - 2 \beta_{3} + 2 \beta_{5} ) q^{47} + ( -1 + \beta_{1} - 2 \beta_{9} - \beta_{10} ) q^{49} + ( 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 3 \beta_{6} ) q^{53} + ( -4 - \beta_{1} - 2 \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} ) q^{55} + ( \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{6} ) q^{59} + ( -2 + 2 \beta_{1} - 2 \beta_{10} ) q^{61} + ( \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} + 2 \beta_{6} ) q^{65} + ( -4 + 2 \beta_{1} - 2 \beta_{10} ) q^{67} + ( 2 \beta_{2} - 2 \beta_{3} + 4 \beta_{4} + 2 \beta_{6} ) q^{71} + ( \beta_{1} - 4 \beta_{8} + \beta_{10} ) q^{73} + ( -\beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{6} ) q^{77} + ( -3 \beta_{1} - 2 \beta_{7} - \beta_{8} - 3 \beta_{10} ) q^{79} + ( -2 \beta_{2} - 2 \beta_{3} - 4 \beta_{5} - 2 \beta_{11} ) q^{83} + ( 2 - \beta_{7} + 3 \beta_{8} + \beta_{9} + \beta_{10} ) q^{85} + ( \beta_{5} + 2 \beta_{11} ) q^{89} + ( \beta_{1} + 4 \beta_{7} - 3 \beta_{8} + \beta_{10} ) q^{91} + ( 2 \beta_{2} - 2 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{11} ) q^{95} + ( \beta_{1} + 4 \beta_{7} + 2 \beta_{8} + \beta_{10} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + O(q^{10}) \) \( 12q - 8q^{25} + 64q^{43} - 4q^{49} - 48q^{55} - 8q^{61} - 32q^{67} + 20q^{85} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 4 x^{9} - 3 x^{8} + 4 x^{7} + 8 x^{6} + 8 x^{5} - 12 x^{4} - 32 x^{3} + 64\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{11} + 12 \nu^{9} + 16 \nu^{8} + 11 \nu^{7} - 28 \nu^{6} - 44 \nu^{5} + 52 \nu^{4} + 100 \nu^{3} + 56 \nu^{2} - 96 \nu - 288 \)\()/80\)
\(\beta_{2}\)\(=\)\((\)\( -3 \nu^{11} + \nu^{10} + 22 \nu^{9} + 2 \nu^{8} - 19 \nu^{7} - 67 \nu^{6} - 22 \nu^{5} + 110 \nu^{4} + 172 \nu^{3} - 40 \nu^{2} - 272 \nu - 352 \)\()/160\)
\(\beta_{3}\)\(=\)\((\)\( -4 \nu^{11} + 5 \nu^{10} + 8 \nu^{9} - 6 \nu^{8} + 4 \nu^{7} - 27 \nu^{6} + 4 \nu^{5} + 18 \nu^{4} + 60 \nu^{3} + 104 \nu^{2} + 16 \nu - 32 \)\()/160\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{11} + 4 \nu^{10} + \nu^{9} + 2 \nu^{8} - 7 \nu^{7} - 20 \nu^{6} + \nu^{5} + 18 \nu^{4} + 18 \nu^{3} - 16 \nu^{2} - 72 \nu \)\()/80\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{10} - \nu^{9} + 2 \nu^{7} + 3 \nu^{6} - 5 \nu^{5} - 4 \nu^{4} - 2 \nu^{3} + 8 \nu^{2} + 24 \nu \)\()/16\)
\(\beta_{6}\)\(=\)\((\)\( -3 \nu^{11} - 8 \nu^{10} + 18 \nu^{9} + 16 \nu^{8} + 29 \nu^{7} - 60 \nu^{6} - 102 \nu^{5} + 4 \nu^{4} + 24 \nu^{3} + 352 \nu^{2} + 64 \nu - 320 \)\()/160\)
\(\beta_{7}\)\(=\)\((\)\( -\nu^{11} - 12 \nu^{10} - 20 \nu^{9} - 12 \nu^{8} + 35 \nu^{7} + 48 \nu^{6} + 36 \nu^{5} - 136 \nu^{4} - 164 \nu^{3} - 48 \nu^{2} + 432 \nu + 448 \)\()/160\)
\(\beta_{8}\)\(=\)\((\)\( -3 \nu^{11} - \nu^{10} + 4 \nu^{8} + 5 \nu^{7} - \nu^{6} - 12 \nu^{5} - 8 \nu^{4} - 12 \nu^{3} + 36 \nu^{2} + 16 \nu + 64 \)\()/40\)
\(\beta_{9}\)\(=\)\((\)\( -7 \nu^{11} - 12 \nu^{10} + 12 \nu^{9} + 44 \nu^{8} + 21 \nu^{7} - 40 \nu^{6} - 108 \nu^{5} - 24 \nu^{4} + 196 \nu^{3} + 208 \nu^{2} + 176 \nu - 320 \)\()/160\)
\(\beta_{10}\)\(=\)\((\)\( 2 \nu^{11} + \nu^{10} + 2 \nu^{9} - 3 \nu^{8} - 4 \nu^{7} + 3 \nu^{6} - 2 \nu^{5} + 5 \nu^{4} + 2 \nu^{3} - 30 \nu^{2} - 12 \nu - 72 \)\()/20\)
\(\beta_{11}\)\(=\)\((\)\( -11 \nu^{11} - 22 \nu^{10} - 30 \nu^{9} + 28 \nu^{8} + 85 \nu^{7} + 38 \nu^{6} - 94 \nu^{5} - 256 \nu^{4} - 64 \nu^{3} + 352 \nu^{2} + 352 \nu + 448 \)\()/160\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{11} + \beta_{9} + \beta_{7} + \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{10} - \beta_{8} - 2 \beta_{7} + \beta_{6} - 2 \beta_{4} + \beta_{3} - \beta_{2} - \beta_{1}\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{11} + \beta_{9} - 2 \beta_{8} - \beta_{7} - 2 \beta_{6} + \beta_{5} - \beta_{4} + 3 \beta_{3} + \beta_{2} + 4\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(-3 \beta_{11} - 3 \beta_{10} - 2 \beta_{9} + 6 \beta_{5} - \beta_{3} - \beta_{2} + 3 \beta_{1} + 2\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(-\beta_{11} - 4 \beta_{10} + \beta_{9} - 4 \beta_{8} + \beta_{7} - 7 \beta_{5} - 7 \beta_{4} + \beta_{3} - \beta_{2} - 8\)\()/4\)
\(\nu^{6}\)\(=\)\((\)\(\beta_{10} + 5 \beta_{8} - 6 \beta_{7} - 5 \beta_{6} - 14 \beta_{4} + 3 \beta_{3} - 3 \beta_{2} + \beta_{1}\)\()/4\)
\(\nu^{7}\)\(=\)\((\)\(3 \beta_{11} - 9 \beta_{9} - 2 \beta_{8} + 9 \beta_{7} - 2 \beta_{6} - \beta_{5} + \beta_{4} + 5 \beta_{3} - \beta_{2} + 16 \beta_{1} + 4\)\()/4\)
\(\nu^{8}\)\(=\)\((\)\(-5 \beta_{11} - 5 \beta_{10} + 10 \beta_{9} - 14 \beta_{5} - 15 \beta_{3} - 15 \beta_{2} + 5 \beta_{1} - 2\)\()/4\)
\(\nu^{9}\)\(=\)\((\)\(-3 \beta_{11} + 12 \beta_{10} + 7 \beta_{9} + 16 \beta_{8} + 7 \beta_{7} + 4 \beta_{6} - 33 \beta_{5} - 33 \beta_{4} + 7 \beta_{3} + \beta_{2} + 32\)\()/4\)
\(\nu^{10}\)\(=\)\((\)\(15 \beta_{10} + 11 \beta_{8} - 2 \beta_{7} - 11 \beta_{6} + 38 \beta_{4} + 37 \beta_{3} - 37 \beta_{2} + 15 \beta_{1}\)\()/4\)
\(\nu^{11}\)\(=\)\((\)\(\beta_{11} + \beta_{9} - 50 \beta_{8} - \beta_{7} + 22 \beta_{6} - 7 \beta_{5} + 7 \beta_{4} - 21 \beta_{3} - 23 \beta_{2} + 8 \beta_{1} + 100\)\()/4\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1439.1
−0.0912546 + 1.41127i
−0.0912546 1.41127i
−0.760198 + 1.19252i
−0.760198 1.19252i
−1.35818 + 0.394157i
−1.35818 0.394157i
−0.394157 1.35818i
−0.394157 + 1.35818i
1.19252 + 0.760198i
1.19252 0.760198i
1.41127 + 0.0912546i
1.41127 0.0912546i
0 0 0 −2.20963 0.342849i 0 −2.64002 0 0 0
1439.2 0 0 0 −2.20963 + 0.342849i 0 −2.64002 0 0 0
1439.3 0 0 0 −1.24561 1.85700i 0 −0.864641 0 0 0
1439.4 0 0 0 −1.24561 + 1.85700i 0 −0.864641 0 0 0
1439.5 0 0 0 −0.256912 2.22126i 0 3.50466 0 0 0
1439.6 0 0 0 −0.256912 + 2.22126i 0 3.50466 0 0 0
1439.7 0 0 0 0.256912 2.22126i 0 3.50466 0 0 0
1439.8 0 0 0 0.256912 + 2.22126i 0 3.50466 0 0 0
1439.9 0 0 0 1.24561 1.85700i 0 −0.864641 0 0 0
1439.10 0 0 0 1.24561 + 1.85700i 0 −0.864641 0 0 0
1439.11 0 0 0 2.20963 0.342849i 0 −2.64002 0 0 0
1439.12 0 0 0 2.20963 + 0.342849i 0 −2.64002 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1439.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
20.d odd 2 1 inner
60.h even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1440.2.o.b yes 12
3.b odd 2 1 inner 1440.2.o.b yes 12
4.b odd 2 1 1440.2.o.a 12
5.b even 2 1 1440.2.o.a 12
5.c odd 4 1 7200.2.h.l 12
5.c odd 4 1 7200.2.h.m 12
8.b even 2 1 2880.2.o.e 12
8.d odd 2 1 2880.2.o.f 12
12.b even 2 1 1440.2.o.a 12
15.d odd 2 1 1440.2.o.a 12
15.e even 4 1 7200.2.h.l 12
15.e even 4 1 7200.2.h.m 12
20.d odd 2 1 inner 1440.2.o.b yes 12
20.e even 4 1 7200.2.h.l 12
20.e even 4 1 7200.2.h.m 12
24.f even 2 1 2880.2.o.f 12
24.h odd 2 1 2880.2.o.e 12
40.e odd 2 1 2880.2.o.e 12
40.f even 2 1 2880.2.o.f 12
60.h even 2 1 inner 1440.2.o.b yes 12
60.l odd 4 1 7200.2.h.l 12
60.l odd 4 1 7200.2.h.m 12
120.i odd 2 1 2880.2.o.f 12
120.m even 2 1 2880.2.o.e 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1440.2.o.a 12 4.b odd 2 1
1440.2.o.a 12 5.b even 2 1
1440.2.o.a 12 12.b even 2 1
1440.2.o.a 12 15.d odd 2 1
1440.2.o.b yes 12 1.a even 1 1 trivial
1440.2.o.b yes 12 3.b odd 2 1 inner
1440.2.o.b yes 12 20.d odd 2 1 inner
1440.2.o.b yes 12 60.h even 2 1 inner
2880.2.o.e 12 8.b even 2 1
2880.2.o.e 12 24.h odd 2 1
2880.2.o.e 12 40.e odd 2 1
2880.2.o.e 12 120.m even 2 1
2880.2.o.f 12 8.d odd 2 1
2880.2.o.f 12 24.f even 2 1
2880.2.o.f 12 40.f even 2 1
2880.2.o.f 12 120.i odd 2 1
7200.2.h.l 12 5.c odd 4 1
7200.2.h.l 12 15.e even 4 1
7200.2.h.l 12 20.e even 4 1
7200.2.h.l 12 60.l odd 4 1
7200.2.h.m 12 5.c odd 4 1
7200.2.h.m 12 15.e even 4 1
7200.2.h.m 12 20.e even 4 1
7200.2.h.m 12 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{3} - 10 T_{7} - 8 \) acting on \(S_{2}^{\mathrm{new}}(1440, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \)
$3$ \( T^{12} \)
$5$ \( 15625 + 2500 T^{2} - 425 T^{4} - 152 T^{6} - 17 T^{8} + 4 T^{10} + T^{12} \)
$7$ \( ( -8 - 10 T + T^{3} )^{4} \)
$11$ \( ( -512 + 228 T^{2} - 28 T^{4} + T^{6} )^{2} \)
$13$ \( ( 16 + 52 T^{2} + 32 T^{4} + T^{6} )^{2} \)
$17$ \( ( -128 + 288 T^{2} - 34 T^{4} + T^{6} )^{2} \)
$19$ \( ( 4096 + 1600 T^{2} + 80 T^{4} + T^{6} )^{2} \)
$23$ \( ( 512 + 528 T^{2} + 88 T^{4} + T^{6} )^{2} \)
$29$ \( ( 3200 + 2144 T^{2} + 98 T^{4} + T^{6} )^{2} \)
$31$ \( ( 6400 + 1424 T^{2} + 88 T^{4} + T^{6} )^{2} \)
$37$ \( ( 64 + 132 T^{2} + 44 T^{4} + T^{6} )^{2} \)
$41$ \( ( 13448 + 2828 T^{2} + 102 T^{4} + T^{6} )^{2} \)
$43$ \( ( 512 + 8 T - 16 T^{2} + T^{3} )^{4} \)
$47$ \( ( 51200 + 5696 T^{2} + 176 T^{4} + T^{6} )^{2} \)
$53$ \( ( -991232 + 30464 T^{2} - 306 T^{4} + T^{6} )^{2} \)
$59$ \( ( -359552 + 15652 T^{2} - 220 T^{4} + T^{6} )^{2} \)
$61$ \( ( -328 - 100 T + 2 T^{2} + T^{3} )^{4} \)
$67$ \( ( -512 - 80 T + 8 T^{2} + T^{3} )^{4} \)
$71$ \( ( -204800 + 15616 T^{2} - 256 T^{4} + T^{6} )^{2} \)
$73$ \( ( 215296 + 15872 T^{2} + 276 T^{4} + T^{6} )^{2} \)
$79$ \( ( 215296 + 29584 T^{2} + 344 T^{4} + T^{6} )^{2} \)
$83$ \( ( 204800 + 15616 T^{2} + 256 T^{4} + T^{6} )^{2} \)
$89$ \( ( 200 + 1356 T^{2} + 118 T^{4} + T^{6} )^{2} \)
$97$ \( ( 719104 + 24704 T^{2} + 276 T^{4} + T^{6} )^{2} \)
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