Properties

Label 460.2.e.b
Level $460$
Weight $2$
Character orbit 460.e
Analytic conductor $3.673$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 460 = 2^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 460.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.67311849298\)
Analytic rank: \(0\)
Dimension: \(32\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32q + 4q^{2} - 4q^{4} - 16q^{6} - 2q^{8} - 52q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 32q + 4q^{2} - 4q^{4} - 16q^{6} - 2q^{8} - 52q^{9} + 24q^{12} - 4q^{13} + 20q^{16} - 56q^{18} - 6q^{24} - 32q^{25} + 68q^{26} + 8q^{29} - 16q^{32} + 8q^{36} + 44q^{41} - 4q^{46} - 4q^{48} - 12q^{49} - 4q^{50} + 16q^{52} + 42q^{54} - 10q^{58} - 36q^{62} - 22q^{64} - 44q^{69} - 42q^{70} - 32q^{72} - 8q^{73} - 72q^{77} + 122q^{78} - 32q^{81} + 20q^{82} - 44q^{85} + 64q^{92} + 40q^{93} - 26q^{94} + 16q^{96} + 62q^{98} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
91.1 −1.39466 0.234385i 1.37989i 1.89013 + 0.653773i 1.00000i −0.323424 + 1.92446i −2.31854 −2.48284 1.35481i 1.09592 −0.234385 + 1.39466i
91.2 −1.39466 0.234385i 1.37989i 1.89013 + 0.653773i 1.00000i −0.323424 + 1.92446i 2.31854 −2.48284 1.35481i 1.09592 0.234385 1.39466i
91.3 −1.39466 + 0.234385i 1.37989i 1.89013 0.653773i 1.00000i −0.323424 1.92446i 2.31854 −2.48284 + 1.35481i 1.09592 0.234385 + 1.39466i
91.4 −1.39466 + 0.234385i 1.37989i 1.89013 0.653773i 1.00000i −0.323424 1.92446i −2.31854 −2.48284 + 1.35481i 1.09592 −0.234385 1.39466i
91.5 −1.05127 0.945957i 1.57817i 0.210332 + 1.98891i 1.00000i −1.49288 + 1.65908i 1.14830 1.66031 2.28984i 0.509388 −0.945957 + 1.05127i
91.6 −1.05127 0.945957i 1.57817i 0.210332 + 1.98891i 1.00000i −1.49288 + 1.65908i −1.14830 1.66031 2.28984i 0.509388 0.945957 1.05127i
91.7 −1.05127 + 0.945957i 1.57817i 0.210332 1.98891i 1.00000i −1.49288 1.65908i −1.14830 1.66031 + 2.28984i 0.509388 0.945957 + 1.05127i
91.8 −1.05127 + 0.945957i 1.57817i 0.210332 1.98891i 1.00000i −1.49288 1.65908i 1.14830 1.66031 + 2.28984i 0.509388 −0.945957 1.05127i
91.9 −0.265302 1.38911i 0.0612154i −1.85923 + 0.737066i 1.00000i −0.0850346 + 0.0162406i 0.810885 1.51712 + 2.38712i 2.99625 −1.38911 + 0.265302i
91.10 −0.265302 1.38911i 0.0612154i −1.85923 + 0.737066i 1.00000i −0.0850346 + 0.0162406i −0.810885 1.51712 + 2.38712i 2.99625 1.38911 0.265302i
91.11 −0.265302 + 1.38911i 0.0612154i −1.85923 0.737066i 1.00000i −0.0850346 0.0162406i −0.810885 1.51712 2.38712i 2.99625 1.38911 + 0.265302i
91.12 −0.265302 + 1.38911i 0.0612154i −1.85923 0.737066i 1.00000i −0.0850346 0.0162406i 0.810885 1.51712 2.38712i 2.99625 −1.38911 0.265302i
91.13 −0.151248 1.40610i 2.99817i −1.95425 + 0.425339i 1.00000i −4.21573 + 0.453465i 3.98264 0.893646 + 2.68354i −5.98900 −1.40610 + 0.151248i
91.14 −0.151248 1.40610i 2.99817i −1.95425 + 0.425339i 1.00000i −4.21573 + 0.453465i −3.98264 0.893646 + 2.68354i −5.98900 1.40610 0.151248i
91.15 −0.151248 + 1.40610i 2.99817i −1.95425 0.425339i 1.00000i −4.21573 0.453465i −3.98264 0.893646 2.68354i −5.98900 1.40610 + 0.151248i
91.16 −0.151248 + 1.40610i 2.99817i −1.95425 0.425339i 1.00000i −4.21573 0.453465i 3.98264 0.893646 2.68354i −5.98900 −1.40610 0.151248i
91.17 0.341935 1.37225i 2.41873i −1.76616 0.938444i 1.00000i 3.31911 + 0.827049i 2.44077 −1.89170 + 2.10273i −2.85025 −1.37225 0.341935i
91.18 0.341935 1.37225i 2.41873i −1.76616 0.938444i 1.00000i 3.31911 + 0.827049i −2.44077 −1.89170 + 2.10273i −2.85025 1.37225 + 0.341935i
91.19 0.341935 + 1.37225i 2.41873i −1.76616 + 0.938444i 1.00000i 3.31911 0.827049i −2.44077 −1.89170 2.10273i −2.85025 1.37225 0.341935i
91.20 0.341935 + 1.37225i 2.41873i −1.76616 + 0.938444i 1.00000i 3.31911 0.827049i 2.44077 −1.89170 2.10273i −2.85025 −1.37225 + 0.341935i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 91.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
23.b odd 2 1 inner
92.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 460.2.e.b 32
4.b odd 2 1 inner 460.2.e.b 32
23.b odd 2 1 inner 460.2.e.b 32
92.b even 2 1 inner 460.2.e.b 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
460.2.e.b 32 1.a even 1 1 trivial
460.2.e.b 32 4.b odd 2 1 inner
460.2.e.b 32 23.b odd 2 1 inner
460.2.e.b 32 92.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{3}^{16} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(460, [\chi])\).