Properties

Label 140.2.o.a
Level $140$
Weight $2$
Character orbit 140.o
Analytic conductor $1.118$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 140 = 2^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 140.o (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.11790562830\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32q + 2q^{2} - 2q^{4} - 4q^{8} - 16q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 32q + 2q^{2} - 2q^{4} - 4q^{8} - 16q^{9} - 30q^{12} + 2q^{14} - 14q^{16} - 12q^{21} - 8q^{22} + 36q^{24} + 16q^{25} + 30q^{26} + 2q^{28} - 40q^{29} + 2q^{32} + 60q^{36} + 8q^{37} - 60q^{38} - 62q^{42} - 18q^{44} + 12q^{45} + 2q^{46} - 16q^{49} + 4q^{50} - 36q^{52} - 8q^{53} + 12q^{54} - 4q^{56} + 48q^{57} + 2q^{58} + 14q^{60} + 24q^{61} + 4q^{64} + 4q^{65} + 24q^{66} + 60q^{68} - 4q^{70} + 4q^{72} - 72q^{73} + 38q^{74} - 40q^{77} + 120q^{78} - 36q^{81} + 42q^{82} - 20q^{84} + 28q^{86} + 4q^{88} - 60q^{89} - 4q^{92} - 8q^{93} + 18q^{94} - 60q^{96} + 78q^{98} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1 −1.39328 + 0.242400i −0.406021 + 0.703249i 1.88248 0.675465i 0.866025 0.500000i 0.395235 1.07825i 0.336411 2.62428i −2.45910 + 1.39743i 1.17029 + 2.02701i −1.08542 + 0.906567i
31.2 −1.27092 + 0.620297i −0.331177 + 0.573616i 1.23046 1.57669i −0.866025 + 0.500000i 0.0650866 0.934447i 1.68748 + 2.03775i −0.585797 + 2.76710i 1.28064 + 2.21814i 0.790498 1.17265i
31.3 −1.26796 0.626319i −1.49907 + 2.59647i 1.21545 + 1.58830i −0.866025 + 0.500000i 3.52698 2.35332i −2.06101 1.65899i −0.546365 2.77516i −2.99443 5.18651i 1.41125 0.0915727i
31.4 −0.950668 + 1.04701i 1.36859 2.37047i −0.192463 1.99072i −0.866025 + 0.500000i 1.18083 + 3.68646i −1.02440 2.43939i 2.26727 + 1.69100i −2.24609 3.89033i 0.299797 1.38207i
31.5 −0.894275 1.09557i 0.895374 1.55083i −0.400544 + 1.95948i 0.866025 0.500000i −2.49976 + 0.405928i 0.644798 2.56598i 2.50494 1.31349i −0.103389 0.179074i −1.32225 0.501653i
31.6 −0.501653 1.32225i −0.895374 + 1.55083i −1.49669 + 1.32662i 0.866025 0.500000i 2.49976 + 0.405928i −0.644798 + 2.56598i 2.50494 + 1.31349i −0.103389 0.179074i −1.09557 0.894275i
31.7 −0.397222 + 1.35728i 0.556469 0.963833i −1.68443 1.07828i 0.866025 0.500000i 1.08715 + 1.13814i 2.32410 + 1.26433i 2.13263 1.85793i 0.880685 + 1.52539i 0.334637 + 1.37405i
31.8 0.0915727 1.41125i 1.49907 2.59647i −1.98323 0.258463i −0.866025 + 0.500000i −3.52698 2.35332i 2.06101 + 1.65899i −0.546365 + 2.77516i −2.99443 5.18651i 0.626319 + 1.26796i
31.9 0.288532 + 1.38447i −0.450639 + 0.780530i −1.83350 + 0.798926i −0.866025 + 0.500000i −1.21064 0.398687i −2.29962 + 1.30833i −1.63511 2.30790i 1.09385 + 1.89460i −0.942109 1.05472i
31.10 0.569639 + 1.29442i −1.51353 + 2.62152i −1.35102 + 1.47470i 0.866025 0.500000i −4.25550 0.465823i 2.57616 0.602834i −2.67847 0.908739i −3.08156 5.33743i 1.14053 + 0.836177i
31.11 0.836177 + 1.14053i 1.51353 2.62152i −0.601615 + 1.90737i 0.866025 0.500000i 4.25550 0.465823i −2.57616 + 0.602834i −2.67847 + 0.908739i −3.08156 5.33743i 1.29442 + 0.569639i
31.12 0.906567 1.08542i 0.406021 0.703249i −0.356272 1.96801i 0.866025 0.500000i −0.395235 1.07825i −0.336411 + 2.62428i −2.45910 1.39743i 1.17029 + 2.02701i 0.242400 1.39328i
31.13 1.05472 + 0.942109i 0.450639 0.780530i 0.224860 + 1.98732i −0.866025 + 0.500000i 1.21064 0.398687i 2.29962 1.30833i −1.63511 + 2.30790i 1.09385 + 1.89460i −1.38447 0.288532i
31.14 1.17265 0.790498i 0.331177 0.573616i 0.750225 1.85396i −0.866025 + 0.500000i −0.0650866 0.934447i −1.68748 2.03775i −0.585797 2.76710i 1.28064 + 2.21814i −0.620297 + 1.27092i
31.15 1.37405 + 0.334637i −0.556469 + 0.963833i 1.77604 + 0.919616i 0.866025 0.500000i −1.08715 + 1.13814i −2.32410 1.26433i 2.13263 + 1.85793i 0.880685 + 1.52539i 1.35728 0.397222i
31.16 1.38207 0.299797i −1.36859 + 2.37047i 1.82024 0.828682i −0.866025 + 0.500000i −1.18083 + 3.68646i 1.02440 + 2.43939i 2.26727 1.69100i −2.24609 3.89033i −1.04701 + 0.950668i
131.1 −1.39328 0.242400i −0.406021 0.703249i 1.88248 + 0.675465i 0.866025 + 0.500000i 0.395235 + 1.07825i 0.336411 + 2.62428i −2.45910 1.39743i 1.17029 2.02701i −1.08542 0.906567i
131.2 −1.27092 0.620297i −0.331177 0.573616i 1.23046 + 1.57669i −0.866025 0.500000i 0.0650866 + 0.934447i 1.68748 2.03775i −0.585797 2.76710i 1.28064 2.21814i 0.790498 + 1.17265i
131.3 −1.26796 + 0.626319i −1.49907 2.59647i 1.21545 1.58830i −0.866025 0.500000i 3.52698 + 2.35332i −2.06101 + 1.65899i −0.546365 + 2.77516i −2.99443 + 5.18651i 1.41125 + 0.0915727i
131.4 −0.950668 1.04701i 1.36859 + 2.37047i −0.192463 + 1.99072i −0.866025 0.500000i 1.18083 3.68646i −1.02440 + 2.43939i 2.26727 1.69100i −2.24609 + 3.89033i 0.299797 + 1.38207i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 131.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.d odd 6 1 inner
28.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 140.2.o.a 32
4.b odd 2 1 inner 140.2.o.a 32
5.b even 2 1 700.2.p.c 32
5.c odd 4 1 700.2.t.c 32
5.c odd 4 1 700.2.t.d 32
7.b odd 2 1 980.2.o.f 32
7.c even 3 1 980.2.g.a 32
7.c even 3 1 980.2.o.f 32
7.d odd 6 1 inner 140.2.o.a 32
7.d odd 6 1 980.2.g.a 32
20.d odd 2 1 700.2.p.c 32
20.e even 4 1 700.2.t.c 32
20.e even 4 1 700.2.t.d 32
28.d even 2 1 980.2.o.f 32
28.f even 6 1 inner 140.2.o.a 32
28.f even 6 1 980.2.g.a 32
28.g odd 6 1 980.2.g.a 32
28.g odd 6 1 980.2.o.f 32
35.i odd 6 1 700.2.p.c 32
35.k even 12 1 700.2.t.c 32
35.k even 12 1 700.2.t.d 32
140.s even 6 1 700.2.p.c 32
140.x odd 12 1 700.2.t.c 32
140.x odd 12 1 700.2.t.d 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.o.a 32 1.a even 1 1 trivial
140.2.o.a 32 4.b odd 2 1 inner
140.2.o.a 32 7.d odd 6 1 inner
140.2.o.a 32 28.f even 6 1 inner
700.2.p.c 32 5.b even 2 1
700.2.p.c 32 20.d odd 2 1
700.2.p.c 32 35.i odd 6 1
700.2.p.c 32 140.s even 6 1
700.2.t.c 32 5.c odd 4 1
700.2.t.c 32 20.e even 4 1
700.2.t.c 32 35.k even 12 1
700.2.t.c 32 140.x odd 12 1
700.2.t.d 32 5.c odd 4 1
700.2.t.d 32 20.e even 4 1
700.2.t.d 32 35.k even 12 1
700.2.t.d 32 140.x odd 12 1
980.2.g.a 32 7.c even 3 1
980.2.g.a 32 7.d odd 6 1
980.2.g.a 32 28.f even 6 1
980.2.g.a 32 28.g odd 6 1
980.2.o.f 32 7.b odd 2 1
980.2.o.f 32 7.c even 3 1
980.2.o.f 32 28.d even 2 1
980.2.o.f 32 28.g odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(140, [\chi])\).