Properties

Label 360.2.k.d
Level $360$
Weight $2$
Character orbit 360.k
Analytic conductor $2.875$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 360.k (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.87461447277\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{7})\)
Defining polynomial: \(x^{4} - 3 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} -\beta_{3} q^{5} + 2 q^{7} + ( \beta_{1} + 2 \beta_{3} ) q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} -\beta_{3} q^{5} + 2 q^{7} + ( \beta_{1} + 2 \beta_{3} ) q^{8} + ( 1 - \beta_{2} ) q^{10} -2 \beta_{3} q^{11} + 2 \beta_{1} q^{14} + ( -1 + 3 \beta_{2} ) q^{16} + ( -2 + 4 \beta_{2} ) q^{19} + ( \beta_{1} - 2 \beta_{3} ) q^{20} + ( 2 - 2 \beta_{2} ) q^{22} + ( -4 \beta_{1} + 2 \beta_{3} ) q^{23} - q^{25} + ( 2 + 2 \beta_{2} ) q^{28} -6 \beta_{3} q^{29} + 4 q^{31} + ( -\beta_{1} + 6 \beta_{3} ) q^{32} -2 \beta_{3} q^{35} + ( 4 - 8 \beta_{2} ) q^{37} + ( -2 \beta_{1} + 8 \beta_{3} ) q^{38} + ( 3 - \beta_{2} ) q^{40} + ( -8 \beta_{1} + 4 \beta_{3} ) q^{41} + ( -4 + 8 \beta_{2} ) q^{43} + ( 2 \beta_{1} - 4 \beta_{3} ) q^{44} + ( -6 - 2 \beta_{2} ) q^{46} + ( -4 \beta_{1} + 2 \beta_{3} ) q^{47} -3 q^{49} -\beta_{1} q^{50} + 2 \beta_{3} q^{53} -2 q^{55} + ( 2 \beta_{1} + 4 \beta_{3} ) q^{56} + ( 6 - 6 \beta_{2} ) q^{58} -10 \beta_{3} q^{59} + ( 4 - 8 \beta_{2} ) q^{61} + 4 \beta_{1} q^{62} + ( -7 + 5 \beta_{2} ) q^{64} + ( 2 - 2 \beta_{2} ) q^{70} + ( 8 \beta_{1} - 4 \beta_{3} ) q^{71} -14 q^{73} + ( 4 \beta_{1} - 16 \beta_{3} ) q^{74} + ( -10 + 6 \beta_{2} ) q^{76} -4 \beta_{3} q^{77} + 4 q^{79} + ( 3 \beta_{1} - 2 \beta_{3} ) q^{80} + ( -12 - 4 \beta_{2} ) q^{82} + 12 \beta_{3} q^{83} + ( -4 \beta_{1} + 16 \beta_{3} ) q^{86} + ( 6 - 2 \beta_{2} ) q^{88} + ( 8 \beta_{1} - 4 \beta_{3} ) q^{89} + ( -6 \beta_{1} - 4 \beta_{3} ) q^{92} + ( -6 - 2 \beta_{2} ) q^{94} + ( 4 \beta_{1} - 2 \beta_{3} ) q^{95} + 2 q^{97} -3 \beta_{1} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 6q^{4} + 8q^{7} + O(q^{10}) \) \( 4q + 6q^{4} + 8q^{7} + 2q^{10} + 2q^{16} + 4q^{22} - 4q^{25} + 12q^{28} + 16q^{31} + 10q^{40} - 28q^{46} - 12q^{49} - 8q^{55} + 12q^{58} - 18q^{64} + 4q^{70} - 56q^{73} - 28q^{76} + 16q^{79} - 56q^{82} + 20q^{88} - 28q^{94} + 8q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 3 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 1 \)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} - \nu \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 1\)
\(\nu^{3}\)\(=\)\(2 \beta_{3} + \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/360\mathbb{Z}\right)^\times\).

\(n\) \(181\) \(217\) \(271\) \(281\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
181.1
−1.32288 0.500000i
−1.32288 + 0.500000i
1.32288 0.500000i
1.32288 + 0.500000i
−1.32288 0.500000i 0 1.50000 + 1.32288i 1.00000i 0 2.00000 −1.32288 2.50000i 0 0.500000 1.32288i
181.2 −1.32288 + 0.500000i 0 1.50000 1.32288i 1.00000i 0 2.00000 −1.32288 + 2.50000i 0 0.500000 + 1.32288i
181.3 1.32288 0.500000i 0 1.50000 1.32288i 1.00000i 0 2.00000 1.32288 2.50000i 0 0.500000 + 1.32288i
181.4 1.32288 + 0.500000i 0 1.50000 + 1.32288i 1.00000i 0 2.00000 1.32288 + 2.50000i 0 0.500000 1.32288i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.b even 2 1 inner
24.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 360.2.k.d 4
3.b odd 2 1 inner 360.2.k.d 4
4.b odd 2 1 1440.2.k.d 4
5.b even 2 1 1800.2.k.n 4
5.c odd 4 1 1800.2.d.k 4
5.c odd 4 1 1800.2.d.o 4
8.b even 2 1 inner 360.2.k.d 4
8.d odd 2 1 1440.2.k.d 4
12.b even 2 1 1440.2.k.d 4
15.d odd 2 1 1800.2.k.n 4
15.e even 4 1 1800.2.d.k 4
15.e even 4 1 1800.2.d.o 4
20.d odd 2 1 7200.2.k.o 4
20.e even 4 1 7200.2.d.k 4
20.e even 4 1 7200.2.d.l 4
24.f even 2 1 1440.2.k.d 4
24.h odd 2 1 inner 360.2.k.d 4
40.e odd 2 1 7200.2.k.o 4
40.f even 2 1 1800.2.k.n 4
40.i odd 4 1 1800.2.d.k 4
40.i odd 4 1 1800.2.d.o 4
40.k even 4 1 7200.2.d.k 4
40.k even 4 1 7200.2.d.l 4
60.h even 2 1 7200.2.k.o 4
60.l odd 4 1 7200.2.d.k 4
60.l odd 4 1 7200.2.d.l 4
120.i odd 2 1 1800.2.k.n 4
120.m even 2 1 7200.2.k.o 4
120.q odd 4 1 7200.2.d.k 4
120.q odd 4 1 7200.2.d.l 4
120.w even 4 1 1800.2.d.k 4
120.w even 4 1 1800.2.d.o 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.k.d 4 1.a even 1 1 trivial
360.2.k.d 4 3.b odd 2 1 inner
360.2.k.d 4 8.b even 2 1 inner
360.2.k.d 4 24.h odd 2 1 inner
1440.2.k.d 4 4.b odd 2 1
1440.2.k.d 4 8.d odd 2 1
1440.2.k.d 4 12.b even 2 1
1440.2.k.d 4 24.f even 2 1
1800.2.d.k 4 5.c odd 4 1
1800.2.d.k 4 15.e even 4 1
1800.2.d.k 4 40.i odd 4 1
1800.2.d.k 4 120.w even 4 1
1800.2.d.o 4 5.c odd 4 1
1800.2.d.o 4 15.e even 4 1
1800.2.d.o 4 40.i odd 4 1
1800.2.d.o 4 120.w even 4 1
1800.2.k.n 4 5.b even 2 1
1800.2.k.n 4 15.d odd 2 1
1800.2.k.n 4 40.f even 2 1
1800.2.k.n 4 120.i odd 2 1
7200.2.d.k 4 20.e even 4 1
7200.2.d.k 4 40.k even 4 1
7200.2.d.k 4 60.l odd 4 1
7200.2.d.k 4 120.q odd 4 1
7200.2.d.l 4 20.e even 4 1
7200.2.d.l 4 40.k even 4 1
7200.2.d.l 4 60.l odd 4 1
7200.2.d.l 4 120.q odd 4 1
7200.2.k.o 4 20.d odd 2 1
7200.2.k.o 4 40.e odd 2 1
7200.2.k.o 4 60.h even 2 1
7200.2.k.o 4 120.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 \(S_{2}^{\mathrm{new}}(360, [\chi])\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 - 3 T^{2} + T^{4} \)
$3$ \( T^{4} \)
$5$ \( ( 1 + T^{2} )^{2} \)
$7$ \( ( -2 + T )^{4} \)
$11$ \( ( 4 + T^{2} )^{2} \)
$13$ \( T^{4} \)
$17$ \( T^{4} \)
$19$ \( ( 28 + T^{2} )^{2} \)
$23$ \( ( -28 + T^{2} )^{2} \)
$29$ \( ( 36 + T^{2} )^{2} \)
$31$ \( ( -4 + T )^{4} \)
$37$ \( ( 112 + T^{2} )^{2} \)
$41$ \( ( -112 + T^{2} )^{2} \)
$43$ \( ( 112 + T^{2} )^{2} \)
$47$ \( ( -28 + T^{2} )^{2} \)
$53$ \( ( 4 + T^{2} )^{2} \)
$59$ \( ( 100 + T^{2} )^{2} \)
$61$ \( ( 112 + T^{2} )^{2} \)
$67$ \( T^{4} \)
$71$ \( ( -112 + T^{2} )^{2} \)
$73$ \( ( 14 + T )^{4} \)
$79$ \( ( -4 + T )^{4} \)
$83$ \( ( 144 + T^{2} )^{2} \)
$89$ \( ( -112 + T^{2} )^{2} \)
$97$ \( ( -2 + T )^{4} \)
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