Properties

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

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(2.87461447277\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Defining polynomial: \(x^{2} + 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
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 \(\beta = \sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} -2 q^{4} + q^{5} + 3 \beta q^{7} -2 \beta q^{8} +O(q^{10})\) \( q + \beta q^{2} -2 q^{4} + q^{5} + 3 \beta q^{7} -2 \beta q^{8} + \beta q^{10} -\beta q^{11} + 3 \beta q^{13} -6 q^{14} + 4 q^{16} + 2 \beta q^{17} -4 q^{19} -2 q^{20} + 2 q^{22} -6 q^{23} + q^{25} -6 q^{26} -6 \beta q^{28} + 6 q^{29} + 6 \beta q^{31} + 4 \beta q^{32} -4 q^{34} + 3 \beta q^{35} -3 \beta q^{37} -4 \beta q^{38} -2 \beta q^{40} -7 \beta q^{41} + 8 q^{43} + 2 \beta q^{44} -6 \beta q^{46} -11 q^{49} + \beta q^{50} -6 \beta q^{52} + 12 q^{53} -\beta q^{55} + 12 q^{56} + 6 \beta q^{58} -\beta q^{59} -6 \beta q^{61} -12 q^{62} -8 q^{64} + 3 \beta q^{65} + 8 q^{67} -4 \beta q^{68} -6 q^{70} + 14 q^{73} + 6 q^{74} + 8 q^{76} + 6 q^{77} + 6 \beta q^{79} + 4 q^{80} + 14 q^{82} + 2 \beta q^{83} + 2 \beta q^{85} + 8 \beta q^{86} -4 q^{88} + 5 \beta q^{89} -18 q^{91} + 12 q^{92} -4 q^{95} -10 q^{97} -11 \beta q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{4} + 2q^{5} + O(q^{10}) \) \( 2q - 4q^{4} + 2q^{5} - 12q^{14} + 8q^{16} - 8q^{19} - 4q^{20} + 4q^{22} - 12q^{23} + 2q^{25} - 12q^{26} + 12q^{29} - 8q^{34} + 16q^{43} - 22q^{49} + 24q^{53} + 24q^{56} - 24q^{62} - 16q^{64} + 16q^{67} - 12q^{70} + 28q^{73} + 12q^{74} + 16q^{76} + 12q^{77} + 8q^{80} + 28q^{82} - 8q^{88} - 36q^{91} + 24q^{92} - 8q^{95} - 20q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
251.1
1.41421i
1.41421i
1.41421i 0 −2.00000 1.00000 0 4.24264i 2.82843i 0 1.41421i
251.2 1.41421i 0 −2.00000 1.00000 0 4.24264i 2.82843i 0 1.41421i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
24.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 360.2.b.b yes 2
3.b odd 2 1 360.2.b.a 2
4.b odd 2 1 1440.2.b.b 2
5.b even 2 1 1800.2.b.a 2
5.c odd 4 2 1800.2.m.b 4
8.b even 2 1 1440.2.b.a 2
8.d odd 2 1 360.2.b.a 2
12.b even 2 1 1440.2.b.a 2
15.d odd 2 1 1800.2.b.b 2
15.e even 4 2 1800.2.m.a 4
20.d odd 2 1 7200.2.b.a 2
20.e even 4 2 7200.2.m.a 4
24.f even 2 1 inner 360.2.b.b yes 2
24.h odd 2 1 1440.2.b.b 2
40.e odd 2 1 1800.2.b.b 2
40.f even 2 1 7200.2.b.b 2
40.i odd 4 2 7200.2.m.b 4
40.k even 4 2 1800.2.m.a 4
60.h even 2 1 7200.2.b.b 2
60.l odd 4 2 7200.2.m.b 4
120.i odd 2 1 7200.2.b.a 2
120.m even 2 1 1800.2.b.a 2
120.q odd 4 2 1800.2.m.b 4
120.w even 4 2 7200.2.m.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.b.a 2 3.b odd 2 1
360.2.b.a 2 8.d odd 2 1
360.2.b.b yes 2 1.a even 1 1 trivial
360.2.b.b yes 2 24.f even 2 1 inner
1440.2.b.a 2 8.b even 2 1
1440.2.b.a 2 12.b even 2 1
1440.2.b.b 2 4.b odd 2 1
1440.2.b.b 2 24.h odd 2 1
1800.2.b.a 2 5.b even 2 1
1800.2.b.a 2 120.m even 2 1
1800.2.b.b 2 15.d odd 2 1
1800.2.b.b 2 40.e odd 2 1
1800.2.m.a 4 15.e even 4 2
1800.2.m.a 4 40.k even 4 2
1800.2.m.b 4 5.c odd 4 2
1800.2.m.b 4 120.q odd 4 2
7200.2.b.a 2 20.d odd 2 1
7200.2.b.a 2 120.i odd 2 1
7200.2.b.b 2 40.f even 2 1
7200.2.b.b 2 60.h even 2 1
7200.2.m.a 4 20.e even 4 2
7200.2.m.a 4 120.w even 4 2
7200.2.m.b 4 40.i odd 4 2
7200.2.m.b 4 60.l odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(360, [\chi])\):

\( T_{7}^{2} + 18 \)
\( T_{23} + 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 2 + T^{2} \)
$3$ \( T^{2} \)
$5$ \( ( -1 + T )^{2} \)
$7$ \( 18 + T^{2} \)
$11$ \( 2 + T^{2} \)
$13$ \( 18 + T^{2} \)
$17$ \( 8 + T^{2} \)
$19$ \( ( 4 + T )^{2} \)
$23$ \( ( 6 + T )^{2} \)
$29$ \( ( -6 + T )^{2} \)
$31$ \( 72 + T^{2} \)
$37$ \( 18 + T^{2} \)
$41$ \( 98 + T^{2} \)
$43$ \( ( -8 + T )^{2} \)
$47$ \( T^{2} \)
$53$ \( ( -12 + T )^{2} \)
$59$ \( 2 + T^{2} \)
$61$ \( 72 + T^{2} \)
$67$ \( ( -8 + T )^{2} \)
$71$ \( T^{2} \)
$73$ \( ( -14 + T )^{2} \)
$79$ \( 72 + T^{2} \)
$83$ \( 8 + T^{2} \)
$89$ \( 50 + T^{2} \)
$97$ \( ( 10 + T )^{2} \)
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