Properties

Label 360.2.bi.b
Level $360$
Weight $2$
Character orbit 360.bi
Analytic conductor $2.875$
Analytic rank $0$
Dimension $32$
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.bi (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.87461447277\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 32q - 2q^{5} + 4q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 32q - 2q^{5} + 4q^{9} + 16q^{11} - 10q^{15} + 8q^{19} - 4q^{21} - 6q^{25} + 20q^{29} - 12q^{31} + 4q^{35} - 28q^{39} - 8q^{41} + 38q^{45} + 36q^{49} - 84q^{51} + 20q^{55} - 20q^{61} + 10q^{65} - 4q^{69} + 16q^{71} - 10q^{75} + 4q^{79} - 52q^{81} + 36q^{85} - 96q^{89} - 8q^{91} - 32q^{95} - 28q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1 0 −1.72704 + 0.131617i 0 2.23514 0.0643013i 0 −1.19102 + 0.687633i 0 2.96535 0.454616i 0
49.2 0 −1.56081 + 0.750911i 0 −2.18001 0.497547i 0 2.51643 1.45286i 0 1.87227 2.34406i 0
49.3 0 −1.54437 0.784174i 0 0.554267 2.16628i 0 −0.608912 + 0.351555i 0 1.77014 + 2.42211i 0
49.4 0 −1.27766 + 1.16944i 0 0.307984 + 2.21476i 0 −3.28422 + 1.89614i 0 0.264813 2.98829i 0
49.5 0 −1.19594 1.25289i 0 −2.14796 0.621508i 0 −1.98012 + 1.14322i 0 −0.139458 + 2.99676i 0
49.6 0 −1.11432 1.32601i 0 −0.355011 + 2.20771i 0 2.19623 1.26799i 0 −0.516592 + 2.95519i 0
49.7 0 −0.557359 + 1.63992i 0 0.668629 + 2.13376i 0 3.90215 2.25291i 0 −2.37870 1.82805i 0
49.8 0 −0.284761 1.70848i 0 1.16287 1.90990i 0 3.55262 2.05111i 0 −2.83782 + 0.973019i 0
49.9 0 0.284761 + 1.70848i 0 1.07259 1.96203i 0 −3.55262 + 2.05111i 0 −2.83782 + 0.973019i 0
49.10 0 0.557359 1.63992i 0 −2.18221 + 0.487830i 0 −3.90215 + 2.25291i 0 −2.37870 1.82805i 0
49.11 0 1.11432 + 1.32601i 0 −1.73442 + 1.41130i 0 −2.19623 + 1.26799i 0 −0.516592 + 2.95519i 0
49.12 0 1.19594 + 1.25289i 0 1.61222 + 1.54943i 0 1.98012 1.14322i 0 −0.139458 + 2.99676i 0
49.13 0 1.27766 1.16944i 0 −2.07203 + 0.840657i 0 3.28422 1.89614i 0 0.264813 2.98829i 0
49.14 0 1.54437 + 0.784174i 0 1.59892 1.56315i 0 0.608912 0.351555i 0 1.77014 + 2.42211i 0
49.15 0 1.56081 0.750911i 0 1.52089 + 1.63917i 0 −2.51643 + 1.45286i 0 1.87227 2.34406i 0
49.16 0 1.72704 0.131617i 0 −1.06189 1.96784i 0 1.19102 0.687633i 0 2.96535 0.454616i 0
169.1 0 −1.72704 0.131617i 0 2.23514 + 0.0643013i 0 −1.19102 0.687633i 0 2.96535 + 0.454616i 0
169.2 0 −1.56081 0.750911i 0 −2.18001 + 0.497547i 0 2.51643 + 1.45286i 0 1.87227 + 2.34406i 0
169.3 0 −1.54437 + 0.784174i 0 0.554267 + 2.16628i 0 −0.608912 0.351555i 0 1.77014 2.42211i 0
169.4 0 −1.27766 1.16944i 0 0.307984 2.21476i 0 −3.28422 1.89614i 0 0.264813 + 2.98829i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 169.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
9.c even 3 1 inner
45.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 360.2.bi.b 32
3.b odd 2 1 1080.2.bi.b 32
4.b odd 2 1 720.2.by.f 32
5.b even 2 1 inner 360.2.bi.b 32
9.c even 3 1 inner 360.2.bi.b 32
9.c even 3 1 3240.2.f.k 16
9.d odd 6 1 1080.2.bi.b 32
9.d odd 6 1 3240.2.f.i 16
12.b even 2 1 2160.2.by.f 32
15.d odd 2 1 1080.2.bi.b 32
20.d odd 2 1 720.2.by.f 32
36.f odd 6 1 720.2.by.f 32
36.h even 6 1 2160.2.by.f 32
45.h odd 6 1 1080.2.bi.b 32
45.h odd 6 1 3240.2.f.i 16
45.j even 6 1 inner 360.2.bi.b 32
45.j even 6 1 3240.2.f.k 16
60.h even 2 1 2160.2.by.f 32
180.n even 6 1 2160.2.by.f 32
180.p odd 6 1 720.2.by.f 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.bi.b 32 1.a even 1 1 trivial
360.2.bi.b 32 5.b even 2 1 inner
360.2.bi.b 32 9.c even 3 1 inner
360.2.bi.b 32 45.j even 6 1 inner
720.2.by.f 32 4.b odd 2 1
720.2.by.f 32 20.d odd 2 1
720.2.by.f 32 36.f odd 6 1
720.2.by.f 32 180.p odd 6 1
1080.2.bi.b 32 3.b odd 2 1
1080.2.bi.b 32 9.d odd 6 1
1080.2.bi.b 32 15.d odd 2 1
1080.2.bi.b 32 45.h odd 6 1
2160.2.by.f 32 12.b even 2 1
2160.2.by.f 32 36.h even 6 1
2160.2.by.f 32 60.h even 2 1
2160.2.by.f 32 180.n even 6 1
3240.2.f.i 16 9.d odd 6 1
3240.2.f.i 16 45.h odd 6 1
3240.2.f.k 16 9.c even 3 1
3240.2.f.k 16 45.j even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(30\!\cdots\!22\)\( T_{7}^{10} + \)\(75\!\cdots\!65\)\( T_{7}^{8} - \)\(11\!\cdots\!52\)\( T_{7}^{6} + \)\(12\!\cdots\!92\)\( T_{7}^{4} - \)\(54\!\cdots\!48\)\( T_{7}^{2} + \)\(17\!\cdots\!96\)\( \)">\(T_{7}^{32} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(360, [\chi])\).