Properties

Label 180.3.c.a
Level $180$
Weight $3$
Character orbit 180.c
Analytic conductor $4.905$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(4.90464475849\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
Defining polynomial: \(x^{4} - x^{3} + x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 20)
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 \(\zeta_{10}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \zeta_{10} q^{2} + 4 \zeta_{10}^{2} q^{4} + ( -1 - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{5} + ( -4 + 8 \zeta_{10} - 2 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) q^{7} + 8 \zeta_{10}^{3} q^{8} +O(q^{10})\) \( q + 2 \zeta_{10} q^{2} + 4 \zeta_{10}^{2} q^{4} + ( -1 - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{5} + ( -4 + 8 \zeta_{10} - 2 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) q^{7} + 8 \zeta_{10}^{3} q^{8} + ( -4 + 2 \zeta_{10} - 4 \zeta_{10}^{2} ) q^{10} + ( -8 + 16 \zeta_{10} - 12 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{11} + ( -2 + 4 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{13} + ( -12 + 4 \zeta_{10} + 4 \zeta_{10}^{2} + 8 \zeta_{10}^{3} ) q^{14} + ( -16 + 16 \zeta_{10} - 16 \zeta_{10}^{2} + 16 \zeta_{10}^{3} ) q^{16} + ( 14 + 16 \zeta_{10}^{2} - 16 \zeta_{10}^{3} ) q^{17} + ( -8 + 16 \zeta_{10} - 8 \zeta_{10}^{2} + 8 \zeta_{10}^{3} ) q^{19} + ( -8 \zeta_{10} + 4 \zeta_{10}^{2} - 8 \zeta_{10}^{3} ) q^{20} + ( -8 - 8 \zeta_{10} + 24 \zeta_{10}^{2} - 16 \zeta_{10}^{3} ) q^{22} + ( 4 - 8 \zeta_{10} - 2 \zeta_{10}^{2} - 10 \zeta_{10}^{3} ) q^{23} + 5 q^{25} + ( 8 - 12 \zeta_{10} + 8 \zeta_{10}^{2} ) q^{26} + ( -16 - 8 \zeta_{10} - 8 \zeta_{10}^{2} + 24 \zeta_{10}^{3} ) q^{28} + ( -2 - 8 \zeta_{10}^{2} + 8 \zeta_{10}^{3} ) q^{29} + ( 8 - 16 \zeta_{10} - 12 \zeta_{10}^{2} - 28 \zeta_{10}^{3} ) q^{31} -32 q^{32} + ( 32 - 4 \zeta_{10} + 32 \zeta_{10}^{2} ) q^{34} + ( -10 \zeta_{10}^{2} - 10 \zeta_{10}^{3} ) q^{35} + ( 14 + 20 \zeta_{10}^{2} - 20 \zeta_{10}^{3} ) q^{37} + ( -16 + 16 \zeta_{10}^{2} ) q^{38} + ( 16 - 16 \zeta_{10} - 8 \zeta_{10}^{3} ) q^{40} + ( 34 + 12 \zeta_{10}^{2} - 12 \zeta_{10}^{3} ) q^{41} + ( 8 - 16 \zeta_{10} + 14 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{43} + ( 32 - 48 \zeta_{10} + 16 \zeta_{10}^{2} + 16 \zeta_{10}^{3} ) q^{44} + ( 20 - 12 \zeta_{10} + 4 \zeta_{10}^{2} - 24 \zeta_{10}^{3} ) q^{46} + ( -12 + 24 \zeta_{10} - 38 \zeta_{10}^{2} - 14 \zeta_{10}^{3} ) q^{47} + ( -11 - 20 \zeta_{10}^{2} + 20 \zeta_{10}^{3} ) q^{49} + 10 \zeta_{10} q^{50} + ( 16 \zeta_{10} - 24 \zeta_{10}^{2} + 16 \zeta_{10}^{3} ) q^{52} + ( 34 - 20 \zeta_{10}^{2} + 20 \zeta_{10}^{3} ) q^{53} + ( -16 + 32 \zeta_{10} - 28 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{55} + ( -48 + 16 \zeta_{10} - 64 \zeta_{10}^{2} + 32 \zeta_{10}^{3} ) q^{56} + ( -16 + 12 \zeta_{10} - 16 \zeta_{10}^{2} ) q^{58} + ( -24 + 48 \zeta_{10} + 48 \zeta_{10}^{3} ) q^{59} + ( 6 - 52 \zeta_{10}^{2} + 52 \zeta_{10}^{3} ) q^{61} + ( 56 - 40 \zeta_{10} + 24 \zeta_{10}^{2} - 80 \zeta_{10}^{3} ) q^{62} -64 \zeta_{10} q^{64} + ( -6 + 8 \zeta_{10}^{2} - 8 \zeta_{10}^{3} ) q^{65} + ( 40 - 80 \zeta_{10} + 18 \zeta_{10}^{2} - 62 \zeta_{10}^{3} ) q^{67} + ( 64 \zeta_{10} - 8 \zeta_{10}^{2} + 64 \zeta_{10}^{3} ) q^{68} + ( 20 - 20 \zeta_{10} + 20 \zeta_{10}^{2} - 40 \zeta_{10}^{3} ) q^{70} + ( 40 - 80 \zeta_{10} + 44 \zeta_{10}^{2} - 36 \zeta_{10}^{3} ) q^{71} + ( 98 + 64 \zeta_{10}^{2} - 64 \zeta_{10}^{3} ) q^{73} + ( 40 - 12 \zeta_{10} + 40 \zeta_{10}^{2} ) q^{74} + ( -32 \zeta_{10} + 32 \zeta_{10}^{3} ) q^{76} + ( -40 + 40 \zeta_{10}^{2} - 40 \zeta_{10}^{3} ) q^{77} + ( -32 + 64 \zeta_{10} + 8 \zeta_{10}^{2} + 72 \zeta_{10}^{3} ) q^{79} + ( 16 + 16 \zeta_{10} - 16 \zeta_{10}^{2} - 16 \zeta_{10}^{3} ) q^{80} + ( 24 + 44 \zeta_{10} + 24 \zeta_{10}^{2} ) q^{82} + ( 24 - 48 \zeta_{10} + 50 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{83} + ( -46 - 12 \zeta_{10}^{2} + 12 \zeta_{10}^{3} ) q^{85} + ( 4 + 12 \zeta_{10} - 28 \zeta_{10}^{2} + 24 \zeta_{10}^{3} ) q^{86} + ( -32 + 96 \zeta_{10} - 128 \zeta_{10}^{2} + 64 \zeta_{10}^{3} ) q^{88} + ( -18 - 80 \zeta_{10}^{2} + 80 \zeta_{10}^{3} ) q^{89} + ( 16 - 32 \zeta_{10} + 28 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{91} + ( 48 - 8 \zeta_{10} + 24 \zeta_{10}^{2} - 40 \zeta_{10}^{3} ) q^{92} + ( 28 - 52 \zeta_{10} + 76 \zeta_{10}^{2} - 104 \zeta_{10}^{3} ) q^{94} + ( -8 + 16 \zeta_{10} - 24 \zeta_{10}^{2} - 8 \zeta_{10}^{3} ) q^{95} + ( -54 + 24 \zeta_{10}^{2} - 24 \zeta_{10}^{3} ) q^{97} + ( -40 + 18 \zeta_{10} - 40 \zeta_{10}^{2} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{2} - 4q^{4} + 8q^{8} + O(q^{10}) \) \( 4q + 2q^{2} - 4q^{4} + 8q^{8} - 10q^{10} - 16q^{13} - 40q^{14} - 16q^{16} + 24q^{17} - 20q^{20} - 80q^{22} + 20q^{25} + 12q^{26} - 40q^{28} + 8q^{29} - 128q^{32} + 92q^{34} + 16q^{37} - 80q^{38} + 40q^{40} + 112q^{41} + 80q^{44} + 40q^{46} - 4q^{49} + 10q^{50} + 56q^{52} + 176q^{53} - 80q^{56} - 36q^{58} + 128q^{61} + 80q^{62} - 64q^{64} - 40q^{65} + 136q^{68} + 264q^{73} + 108q^{74} - 240q^{77} + 80q^{80} + 116q^{82} - 160q^{85} + 80q^{86} + 160q^{88} + 88q^{89} + 120q^{92} - 120q^{94} - 264q^{97} - 102q^{98} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
91.1
−0.309017 0.951057i
−0.309017 + 0.951057i
0.809017 0.587785i
0.809017 + 0.587785i
−0.618034 1.90211i 0 −3.23607 + 2.35114i 2.23607 0 5.25731i 6.47214 + 4.70228i 0 −1.38197 4.25325i
91.2 −0.618034 + 1.90211i 0 −3.23607 2.35114i 2.23607 0 5.25731i 6.47214 4.70228i 0 −1.38197 + 4.25325i
91.3 1.61803 1.17557i 0 1.23607 3.80423i −2.23607 0 8.50651i −2.47214 7.60845i 0 −3.61803 + 2.62866i
91.4 1.61803 + 1.17557i 0 1.23607 + 3.80423i −2.23607 0 8.50651i −2.47214 + 7.60845i 0 −3.61803 2.62866i
\(n\): 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 180.3.c.a 4
3.b odd 2 1 20.3.b.a 4
4.b odd 2 1 inner 180.3.c.a 4
5.b even 2 1 900.3.c.k 4
5.c odd 4 2 900.3.f.e 8
8.b even 2 1 2880.3.e.e 4
8.d odd 2 1 2880.3.e.e 4
12.b even 2 1 20.3.b.a 4
15.d odd 2 1 100.3.b.f 4
15.e even 4 2 100.3.d.b 8
20.d odd 2 1 900.3.c.k 4
20.e even 4 2 900.3.f.e 8
24.f even 2 1 320.3.b.c 4
24.h odd 2 1 320.3.b.c 4
48.i odd 4 2 1280.3.g.e 8
48.k even 4 2 1280.3.g.e 8
60.h even 2 1 100.3.b.f 4
60.l odd 4 2 100.3.d.b 8
120.i odd 2 1 1600.3.b.s 4
120.m even 2 1 1600.3.b.s 4
120.q odd 4 2 1600.3.h.n 8
120.w even 4 2 1600.3.h.n 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.3.b.a 4 3.b odd 2 1
20.3.b.a 4 12.b even 2 1
100.3.b.f 4 15.d odd 2 1
100.3.b.f 4 60.h even 2 1
100.3.d.b 8 15.e even 4 2
100.3.d.b 8 60.l odd 4 2
180.3.c.a 4 1.a even 1 1 trivial
180.3.c.a 4 4.b odd 2 1 inner
320.3.b.c 4 24.f even 2 1
320.3.b.c 4 24.h odd 2 1
900.3.c.k 4 5.b even 2 1
900.3.c.k 4 20.d odd 2 1
900.3.f.e 8 5.c odd 4 2
900.3.f.e 8 20.e even 4 2
1280.3.g.e 8 48.i odd 4 2
1280.3.g.e 8 48.k even 4 2
1600.3.b.s 4 120.i odd 2 1
1600.3.b.s 4 120.m even 2 1
1600.3.h.n 8 120.q odd 4 2
1600.3.h.n 8 120.w even 4 2
2880.3.e.e 4 8.b even 2 1
2880.3.e.e 4 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} + 100 T_{7}^{2} + 2000 \) acting on \(S_{3}^{\mathrm{new}}(180, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 16 - 8 T + 4 T^{2} - 2 T^{3} + T^{4} \)
$3$ \( T^{4} \)
$5$ \( ( -5 + T^{2} )^{2} \)
$7$ \( 2000 + 100 T^{2} + T^{4} \)
$11$ \( 1280 + 400 T^{2} + T^{4} \)
$13$ \( ( -4 + 8 T + T^{2} )^{2} \)
$17$ \( ( -284 - 12 T + T^{2} )^{2} \)
$19$ \( 20480 + 320 T^{2} + T^{4} \)
$23$ \( 80 + 260 T^{2} + T^{4} \)
$29$ \( ( -76 - 4 T + T^{2} )^{2} \)
$31$ \( 154880 + 2320 T^{2} + T^{4} \)
$37$ \( ( -484 - 8 T + T^{2} )^{2} \)
$41$ \( ( 604 - 56 T + T^{2} )^{2} \)
$43$ \( 2000 + 500 T^{2} + T^{4} \)
$47$ \( 3561680 + 4100 T^{2} + T^{4} \)
$53$ \( ( 1436 - 88 T + T^{2} )^{2} \)
$59$ \( 1658880 + 5760 T^{2} + T^{4} \)
$61$ \( ( -2356 - 64 T + T^{2} )^{2} \)
$67$ \( 19920080 + 10420 T^{2} + T^{4} \)
$71$ \( 10138880 + 8080 T^{2} + T^{4} \)
$73$ \( ( -764 - 132 T + T^{2} )^{2} \)
$79$ \( 2478080 + 13120 T^{2} + T^{4} \)
$83$ \( 2620880 + 6260 T^{2} + T^{4} \)
$89$ \( ( -7516 - 44 T + T^{2} )^{2} \)
$97$ \( ( 3636 + 132 T + T^{2} )^{2} \)
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