# 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 $4$

# Related objects

## 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.

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

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$T_{3}^{16} + 37T_{3}^{14} + 558T_{3}^{12} + 4419T_{3}^{10} + 19735T_{3}^{8} + 49530T_{3}^{6} + 64840T_{3}^{4} + 34400T_{3}^{2} + 128$$ acting on $$S_{2}^{\mathrm{new}}(460, [\chi])$$.