Properties

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

Related objects

Downloads

Learn more about

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: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 120)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -1 + i ) q^{2} -2 i q^{4} + i q^{5} + 2 q^{7} + ( 2 + 2 i ) q^{8} +O(q^{10})\) \( q + ( -1 + i ) q^{2} -2 i q^{4} + i q^{5} + 2 q^{7} + ( 2 + 2 i ) q^{8} + ( -1 - i ) q^{10} -4 i q^{11} + ( -2 + 2 i ) q^{14} -4 q^{16} + 6 q^{17} + 4 i q^{19} + 2 q^{20} + ( 4 + 4 i ) q^{22} + 4 q^{23} - q^{25} -4 i q^{28} + 6 i q^{29} + 10 q^{31} + ( 4 - 4 i ) q^{32} + ( -6 + 6 i ) q^{34} + 2 i q^{35} + 4 i q^{37} + ( -4 - 4 i ) q^{38} + ( -2 + 2 i ) q^{40} -10 q^{41} -4 i q^{43} -8 q^{44} + ( -4 + 4 i ) q^{46} + 4 q^{47} -3 q^{49} + ( 1 - i ) q^{50} + 10 i q^{53} + 4 q^{55} + ( 4 + 4 i ) q^{56} + ( -6 - 6 i ) q^{58} -8 i q^{59} -8 i q^{61} + ( -10 + 10 i ) q^{62} + 8 i q^{64} -12 i q^{67} -12 i q^{68} + ( -2 - 2 i ) q^{70} + 4 q^{71} + 10 q^{73} + ( -4 - 4 i ) q^{74} + 8 q^{76} -8 i q^{77} -14 q^{79} -4 i q^{80} + ( 10 - 10 i ) q^{82} + 6 i q^{85} + ( 4 + 4 i ) q^{86} + ( 8 - 8 i ) q^{88} -14 q^{89} -8 i q^{92} + ( -4 + 4 i ) q^{94} -4 q^{95} -10 q^{97} + ( 3 - 3 i ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 4q^{7} + 4q^{8} + O(q^{10}) \) \( 2q - 2q^{2} + 4q^{7} + 4q^{8} - 2q^{10} - 4q^{14} - 8q^{16} + 12q^{17} + 4q^{20} + 8q^{22} + 8q^{23} - 2q^{25} + 20q^{31} + 8q^{32} - 12q^{34} - 8q^{38} - 4q^{40} - 20q^{41} - 16q^{44} - 8q^{46} + 8q^{47} - 6q^{49} + 2q^{50} + 8q^{55} + 8q^{56} - 12q^{58} - 20q^{62} - 4q^{70} + 8q^{71} + 20q^{73} - 8q^{74} + 16q^{76} - 28q^{79} + 20q^{82} + 8q^{86} + 16q^{88} - 28q^{89} - 8q^{94} - 8q^{95} - 20q^{97} + 6q^{98} + O(q^{100}) \)

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b 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.k.b 2
3.b odd 2 1 120.2.k.a 2
4.b odd 2 1 1440.2.k.a 2
5.b even 2 1 1800.2.k.g 2
5.c odd 4 1 1800.2.d.c 2
5.c odd 4 1 1800.2.d.h 2
8.b even 2 1 inner 360.2.k.b 2
8.d odd 2 1 1440.2.k.a 2
12.b even 2 1 480.2.k.a 2
15.d odd 2 1 600.2.k.a 2
15.e even 4 1 600.2.d.a 2
15.e even 4 1 600.2.d.d 2
20.d odd 2 1 7200.2.k.f 2
20.e even 4 1 7200.2.d.e 2
20.e even 4 1 7200.2.d.f 2
24.f even 2 1 480.2.k.a 2
24.h odd 2 1 120.2.k.a 2
40.e odd 2 1 7200.2.k.f 2
40.f even 2 1 1800.2.k.g 2
40.i odd 4 1 1800.2.d.c 2
40.i odd 4 1 1800.2.d.h 2
40.k even 4 1 7200.2.d.e 2
40.k even 4 1 7200.2.d.f 2
48.i odd 4 1 3840.2.a.d 1
48.i odd 4 1 3840.2.a.w 1
48.k even 4 1 3840.2.a.m 1
48.k even 4 1 3840.2.a.r 1
60.h even 2 1 2400.2.k.b 2
60.l odd 4 1 2400.2.d.a 2
60.l odd 4 1 2400.2.d.d 2
120.i odd 2 1 600.2.k.a 2
120.m even 2 1 2400.2.k.b 2
120.q odd 4 1 2400.2.d.a 2
120.q odd 4 1 2400.2.d.d 2
120.w even 4 1 600.2.d.a 2
120.w even 4 1 600.2.d.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.2.k.a 2 3.b odd 2 1
120.2.k.a 2 24.h odd 2 1
360.2.k.b 2 1.a even 1 1 trivial
360.2.k.b 2 8.b even 2 1 inner
480.2.k.a 2 12.b even 2 1
480.2.k.a 2 24.f even 2 1
600.2.d.a 2 15.e even 4 1
600.2.d.a 2 120.w even 4 1
600.2.d.d 2 15.e even 4 1
600.2.d.d 2 120.w even 4 1
600.2.k.a 2 15.d odd 2 1
600.2.k.a 2 120.i odd 2 1
1440.2.k.a 2 4.b odd 2 1
1440.2.k.a 2 8.d odd 2 1
1800.2.d.c 2 5.c odd 4 1
1800.2.d.c 2 40.i odd 4 1
1800.2.d.h 2 5.c odd 4 1
1800.2.d.h 2 40.i odd 4 1
1800.2.k.g 2 5.b even 2 1
1800.2.k.g 2 40.f even 2 1
2400.2.d.a 2 60.l odd 4 1
2400.2.d.a 2 120.q odd 4 1
2400.2.d.d 2 60.l odd 4 1
2400.2.d.d 2 120.q odd 4 1
2400.2.k.b 2 60.h even 2 1
2400.2.k.b 2 120.m even 2 1
3840.2.a.d 1 48.i odd 4 1
3840.2.a.m 1 48.k even 4 1
3840.2.a.r 1 48.k even 4 1
3840.2.a.w 1 48.i odd 4 1
7200.2.d.e 2 20.e even 4 1
7200.2.d.e 2 40.k even 4 1
7200.2.d.f 2 20.e even 4 1
7200.2.d.f 2 40.k even 4 1
7200.2.k.f 2 20.d odd 2 1
7200.2.k.f 2 40.e odd 2 1

Hecke kernels

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

\( T_{7} - 2 \)
\( T_{11}^{2} + 16 \)
\( T_{17} - 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 2 + 2 T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( 1 + T^{2} \)
$7$ \( ( -2 + T )^{2} \)
$11$ \( 16 + T^{2} \)
$13$ \( T^{2} \)
$17$ \( ( -6 + T )^{2} \)
$19$ \( 16 + T^{2} \)
$23$ \( ( -4 + T )^{2} \)
$29$ \( 36 + T^{2} \)
$31$ \( ( -10 + T )^{2} \)
$37$ \( 16 + T^{2} \)
$41$ \( ( 10 + T )^{2} \)
$43$ \( 16 + T^{2} \)
$47$ \( ( -4 + T )^{2} \)
$53$ \( 100 + T^{2} \)
$59$ \( 64 + T^{2} \)
$61$ \( 64 + T^{2} \)
$67$ \( 144 + T^{2} \)
$71$ \( ( -4 + T )^{2} \)
$73$ \( ( -10 + T )^{2} \)
$79$ \( ( 14 + T )^{2} \)
$83$ \( T^{2} \)
$89$ \( ( 14 + T )^{2} \)
$97$ \( ( 10 + T )^{2} \)
show more
show less