Properties

Label 180.3.f.e
Level $180$
Weight $3$
Character orbit 180.f
Analytic conductor $4.905$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 180 = 2^{2} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 180.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.90464475849\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 60)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \zeta_{12} q^{2} + 4 \zeta_{12}^{2} q^{4} -5 \zeta_{12}^{3} q^{5} + ( 12 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{7} + 8 \zeta_{12}^{3} q^{8} +O(q^{10})\) \( q + 2 \zeta_{12} q^{2} + 4 \zeta_{12}^{2} q^{4} -5 \zeta_{12}^{3} q^{5} + ( 12 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{7} + 8 \zeta_{12}^{3} q^{8} + ( 10 - 10 \zeta_{12}^{2} ) q^{10} + ( 6 - 12 \zeta_{12}^{2} ) q^{11} + 18 \zeta_{12}^{3} q^{13} + ( 12 + 12 \zeta_{12}^{2} ) q^{14} + ( -16 + 16 \zeta_{12}^{2} ) q^{16} + 10 \zeta_{12}^{3} q^{17} + ( 8 - 16 \zeta_{12}^{2} ) q^{19} + ( 20 \zeta_{12} - 20 \zeta_{12}^{3} ) q^{20} + ( 12 \zeta_{12} - 24 \zeta_{12}^{3} ) q^{22} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{23} -25 q^{25} + ( -36 + 36 \zeta_{12}^{2} ) q^{26} + ( 24 \zeta_{12} + 24 \zeta_{12}^{3} ) q^{28} -36 q^{29} + ( 4 - 8 \zeta_{12}^{2} ) q^{31} + ( -32 \zeta_{12} + 32 \zeta_{12}^{3} ) q^{32} + ( -20 + 20 \zeta_{12}^{2} ) q^{34} + ( 30 - 60 \zeta_{12}^{2} ) q^{35} -54 \zeta_{12}^{3} q^{37} + ( 16 \zeta_{12} - 32 \zeta_{12}^{3} ) q^{38} + 40 q^{40} -18 q^{41} + ( -24 \zeta_{12} + 12 \zeta_{12}^{3} ) q^{43} + ( 48 - 24 \zeta_{12}^{2} ) q^{44} + ( 8 + 8 \zeta_{12}^{2} ) q^{46} + 59 q^{49} -50 \zeta_{12} q^{50} + ( -72 \zeta_{12} + 72 \zeta_{12}^{3} ) q^{52} + 26 \zeta_{12}^{3} q^{53} + ( -60 \zeta_{12} + 30 \zeta_{12}^{3} ) q^{55} + ( -48 + 96 \zeta_{12}^{2} ) q^{56} -72 \zeta_{12} q^{58} + ( 18 - 36 \zeta_{12}^{2} ) q^{59} -74 q^{61} + ( 8 \zeta_{12} - 16 \zeta_{12}^{3} ) q^{62} -64 q^{64} + 90 q^{65} + ( 48 \zeta_{12} - 24 \zeta_{12}^{3} ) q^{67} + ( -40 \zeta_{12} + 40 \zeta_{12}^{3} ) q^{68} + ( 60 \zeta_{12} - 120 \zeta_{12}^{3} ) q^{70} + ( -60 + 120 \zeta_{12}^{2} ) q^{71} + 36 \zeta_{12}^{3} q^{73} + ( 108 - 108 \zeta_{12}^{2} ) q^{74} + ( 64 - 32 \zeta_{12}^{2} ) q^{76} -108 \zeta_{12}^{3} q^{77} + ( 52 - 104 \zeta_{12}^{2} ) q^{79} + 80 \zeta_{12} q^{80} -36 \zeta_{12} q^{82} + ( -104 \zeta_{12} + 52 \zeta_{12}^{3} ) q^{83} + 50 q^{85} + ( -24 - 24 \zeta_{12}^{2} ) q^{86} + ( 96 \zeta_{12} - 48 \zeta_{12}^{3} ) q^{88} -18 q^{89} + ( -108 + 216 \zeta_{12}^{2} ) q^{91} + ( 16 \zeta_{12} + 16 \zeta_{12}^{3} ) q^{92} + ( -80 \zeta_{12} + 40 \zeta_{12}^{3} ) q^{95} + 72 \zeta_{12}^{3} q^{97} + 118 \zeta_{12} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 8q^{4} + O(q^{10}) \) \( 4q + 8q^{4} + 20q^{10} + 72q^{14} - 32q^{16} - 100q^{25} - 72q^{26} - 144q^{29} - 40q^{34} + 160q^{40} - 72q^{41} + 144q^{44} + 48q^{46} + 236q^{49} - 296q^{61} - 256q^{64} + 360q^{65} + 216q^{74} + 192q^{76} + 200q^{85} - 144q^{86} - 72q^{89} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/180\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(91\) \(101\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
−0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
0.866025 + 0.500000i
−1.73205 1.00000i 0 2.00000 + 3.46410i 5.00000i 0 −10.3923 8.00000i 0 5.00000 8.66025i
19.2 −1.73205 + 1.00000i 0 2.00000 3.46410i 5.00000i 0 −10.3923 8.00000i 0 5.00000 + 8.66025i
19.3 1.73205 1.00000i 0 2.00000 3.46410i 5.00000i 0 10.3923 8.00000i 0 5.00000 + 8.66025i
19.4 1.73205 + 1.00000i 0 2.00000 + 3.46410i 5.00000i 0 10.3923 8.00000i 0 5.00000 8.66025i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
20.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 180.3.f.e 4
3.b odd 2 1 60.3.f.a 4
4.b odd 2 1 inner 180.3.f.e 4
5.b even 2 1 inner 180.3.f.e 4
5.c odd 4 1 900.3.c.f 2
5.c odd 4 1 900.3.c.j 2
12.b even 2 1 60.3.f.a 4
15.d odd 2 1 60.3.f.a 4
15.e even 4 1 300.3.c.a 2
15.e even 4 1 300.3.c.c 2
20.d odd 2 1 inner 180.3.f.e 4
20.e even 4 1 900.3.c.f 2
20.e even 4 1 900.3.c.j 2
24.f even 2 1 960.3.j.b 4
24.h odd 2 1 960.3.j.b 4
60.h even 2 1 60.3.f.a 4
60.l odd 4 1 300.3.c.a 2
60.l odd 4 1 300.3.c.c 2
120.i odd 2 1 960.3.j.b 4
120.m even 2 1 960.3.j.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.3.f.a 4 3.b odd 2 1
60.3.f.a 4 12.b even 2 1
60.3.f.a 4 15.d odd 2 1
60.3.f.a 4 60.h even 2 1
180.3.f.e 4 1.a even 1 1 trivial
180.3.f.e 4 4.b odd 2 1 inner
180.3.f.e 4 5.b even 2 1 inner
180.3.f.e 4 20.d odd 2 1 inner
300.3.c.a 2 15.e even 4 1
300.3.c.a 2 60.l odd 4 1
300.3.c.c 2 15.e even 4 1
300.3.c.c 2 60.l odd 4 1
900.3.c.f 2 5.c odd 4 1
900.3.c.f 2 20.e even 4 1
900.3.c.j 2 5.c odd 4 1
900.3.c.j 2 20.e even 4 1
960.3.j.b 4 24.f even 2 1
960.3.j.b 4 24.h odd 2 1
960.3.j.b 4 120.i odd 2 1
960.3.j.b 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 \(S_{3}^{\mathrm{new}}(180, [\chi])\):

\( T_{7}^{2} - 108 \)
\( T_{13}^{2} + 324 \)
\( T_{23}^{2} - 48 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 16 - 4 T^{2} + T^{4} \)
$3$ \( T^{4} \)
$5$ \( ( 25 + T^{2} )^{2} \)
$7$ \( ( -108 + T^{2} )^{2} \)
$11$ \( ( 108 + T^{2} )^{2} \)
$13$ \( ( 324 + T^{2} )^{2} \)
$17$ \( ( 100 + T^{2} )^{2} \)
$19$ \( ( 192 + T^{2} )^{2} \)
$23$ \( ( -48 + T^{2} )^{2} \)
$29$ \( ( 36 + T )^{4} \)
$31$ \( ( 48 + T^{2} )^{2} \)
$37$ \( ( 2916 + T^{2} )^{2} \)
$41$ \( ( 18 + T )^{4} \)
$43$ \( ( -432 + T^{2} )^{2} \)
$47$ \( T^{4} \)
$53$ \( ( 676 + T^{2} )^{2} \)
$59$ \( ( 972 + T^{2} )^{2} \)
$61$ \( ( 74 + T )^{4} \)
$67$ \( ( -1728 + T^{2} )^{2} \)
$71$ \( ( 10800 + T^{2} )^{2} \)
$73$ \( ( 1296 + T^{2} )^{2} \)
$79$ \( ( 8112 + T^{2} )^{2} \)
$83$ \( ( -8112 + T^{2} )^{2} \)
$89$ \( ( 18 + T )^{4} \)
$97$ \( ( 5184 + T^{2} )^{2} \)
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