# Properties

 Label 462.2.n.e Level $462$ Weight $2$ Character orbit 462.n Analytic conductor $3.689$ Analytic rank $0$ Dimension $24$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$462 = 2 \cdot 3 \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 462.n (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.68908857338$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$12$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$24 q - 12 q^{2} - 12 q^{4} + 24 q^{8} - 8 q^{9}+O(q^{10})$$ 24 * q - 12 * q^2 - 12 * q^4 + 24 * q^8 - 8 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$24 q - 12 q^{2} - 12 q^{4} + 24 q^{8} - 8 q^{9} - 4 q^{11} + 4 q^{15} - 12 q^{16} + 4 q^{17} - 8 q^{18} + 8 q^{22} + 20 q^{25} - 12 q^{27} + 40 q^{29} - 2 q^{30} - 28 q^{31} - 12 q^{32} + 32 q^{33} - 8 q^{34} - 32 q^{35} + 16 q^{36} - 16 q^{37} - 8 q^{41} + 6 q^{42} - 4 q^{44} - 28 q^{45} - 8 q^{49} - 40 q^{50} - 34 q^{51} + 6 q^{54} + 8 q^{55} + 40 q^{57} - 20 q^{58} - 2 q^{60} + 56 q^{62} - 14 q^{63} + 24 q^{64} - 44 q^{65} + 32 q^{66} + 4 q^{68} - 20 q^{69} + 64 q^{70} - 8 q^{72} - 16 q^{74} - 26 q^{75} + 24 q^{81} + 4 q^{82} + 144 q^{83} - 6 q^{84} + 8 q^{87} - 4 q^{88} + 56 q^{90} - 32 q^{91} - 22 q^{93} - 44 q^{95} + 64 q^{97} - 8 q^{98} - 52 q^{99}+O(q^{100})$$ 24 * q - 12 * q^2 - 12 * q^4 + 24 * q^8 - 8 * q^9 - 4 * q^11 + 4 * q^15 - 12 * q^16 + 4 * q^17 - 8 * q^18 + 8 * q^22 + 20 * q^25 - 12 * q^27 + 40 * q^29 - 2 * q^30 - 28 * q^31 - 12 * q^32 + 32 * q^33 - 8 * q^34 - 32 * q^35 + 16 * q^36 - 16 * q^37 - 8 * q^41 + 6 * q^42 - 4 * q^44 - 28 * q^45 - 8 * q^49 - 40 * q^50 - 34 * q^51 + 6 * q^54 + 8 * q^55 + 40 * q^57 - 20 * q^58 - 2 * q^60 + 56 * q^62 - 14 * q^63 + 24 * q^64 - 44 * q^65 + 32 * q^66 + 4 * q^68 - 20 * q^69 + 64 * q^70 - 8 * q^72 - 16 * q^74 - 26 * q^75 + 24 * q^81 + 4 * q^82 + 144 * q^83 - 6 * q^84 + 8 * q^87 - 4 * q^88 + 56 * q^90 - 32 * q^91 - 22 * q^93 - 44 * q^95 + 64 * q^97 - 8 * q^98 - 52 * q^99

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
65.1 −0.500000 + 0.866025i −1.72951 0.0938734i −0.500000 0.866025i −3.23623 1.86844i 0.946049 1.45086i −0.830341 + 2.51208i 1.00000 2.98238 + 0.324709i 3.23623 1.86844i
65.2 −0.500000 + 0.866025i −1.64448 0.543759i −0.500000 0.866025i 1.63087 + 0.941585i 1.29315 1.15228i −1.76745 1.96879i 1.00000 2.40865 + 1.78841i −1.63087 + 0.941585i
65.3 −0.500000 + 0.866025i −1.20836 1.24091i −0.500000 0.866025i −1.00577 0.580683i 1.67884 0.426020i 2.19957 1.47034i 1.00000 −0.0797091 + 2.99894i 1.00577 0.580683i
65.4 −0.500000 + 0.866025i −0.993776 + 1.41859i −0.500000 0.866025i 0.246102 + 0.142087i −0.731651 1.56993i 2.09441 + 1.61662i 1.00000 −1.02482 2.81953i −0.246102 + 0.142087i
65.5 −0.500000 + 0.866025i −0.470476 1.66693i −0.500000 0.866025i 1.00577 + 0.580683i 1.67884 + 0.426020i −2.19957 + 1.47034i 1.00000 −2.55730 + 1.56850i −1.00577 + 0.580683i
65.6 −0.500000 + 0.866025i 0.125098 + 1.72753i −0.500000 0.866025i 3.72427 + 2.15021i −1.55863 0.755426i −2.62857 0.301041i 1.00000 −2.96870 + 0.432220i −3.72427 + 2.15021i
65.7 −0.500000 + 0.866025i 0.301239 + 1.70565i −0.500000 0.866025i −1.38736 0.800992i −1.62776 0.591946i 0.229478 2.63578i 1.00000 −2.81851 + 1.02762i 1.38736 0.800992i
65.8 −0.500000 + 0.866025i 0.351333 1.69604i −0.500000 0.866025i −1.63087 0.941585i 1.29315 + 1.15228i 1.76745 + 1.96879i 1.00000 −2.75313 1.19175i 1.63087 0.941585i
65.9 −0.500000 + 0.866025i 0.783456 1.54473i −0.500000 0.866025i 3.23623 + 1.86844i 0.946049 + 1.45086i 0.830341 2.51208i 1.00000 −1.77239 2.42046i −3.23623 + 1.86844i
65.10 −0.500000 + 0.866025i 1.32652 + 1.11371i −0.500000 0.866025i 1.38736 + 0.800992i −1.62776 + 0.591946i −0.229478 + 2.63578i 1.00000 0.519310 + 2.95471i −1.38736 + 0.800992i
65.11 −0.500000 + 0.866025i 1.43353 + 0.972102i −0.500000 0.866025i −3.72427 2.15021i −1.55863 + 0.755426i 2.62857 + 0.301041i 1.00000 1.11004 + 2.78708i 3.72427 2.15021i
65.12 −0.500000 + 0.866025i 1.72543 0.151338i −0.500000 0.866025i −0.246102 0.142087i −0.731651 + 1.56993i −2.09441 1.61662i 1.00000 2.95419 0.522246i 0.246102 0.142087i
263.1 −0.500000 0.866025i −1.72951 + 0.0938734i −0.500000 + 0.866025i −3.23623 + 1.86844i 0.946049 + 1.45086i −0.830341 2.51208i 1.00000 2.98238 0.324709i 3.23623 + 1.86844i
263.2 −0.500000 0.866025i −1.64448 + 0.543759i −0.500000 + 0.866025i 1.63087 0.941585i 1.29315 + 1.15228i −1.76745 + 1.96879i 1.00000 2.40865 1.78841i −1.63087 0.941585i
263.3 −0.500000 0.866025i −1.20836 + 1.24091i −0.500000 + 0.866025i −1.00577 + 0.580683i 1.67884 + 0.426020i 2.19957 + 1.47034i 1.00000 −0.0797091 2.99894i 1.00577 + 0.580683i
263.4 −0.500000 0.866025i −0.993776 1.41859i −0.500000 + 0.866025i 0.246102 0.142087i −0.731651 + 1.56993i 2.09441 1.61662i 1.00000 −1.02482 + 2.81953i −0.246102 0.142087i
263.5 −0.500000 0.866025i −0.470476 + 1.66693i −0.500000 + 0.866025i 1.00577 0.580683i 1.67884 0.426020i −2.19957 1.47034i 1.00000 −2.55730 1.56850i −1.00577 0.580683i
263.6 −0.500000 0.866025i 0.125098 1.72753i −0.500000 + 0.866025i 3.72427 2.15021i −1.55863 + 0.755426i −2.62857 + 0.301041i 1.00000 −2.96870 0.432220i −3.72427 2.15021i
263.7 −0.500000 0.866025i 0.301239 1.70565i −0.500000 + 0.866025i −1.38736 + 0.800992i −1.62776 + 0.591946i 0.229478 + 2.63578i 1.00000 −2.81851 1.02762i 1.38736 + 0.800992i
263.8 −0.500000 0.866025i 0.351333 + 1.69604i −0.500000 + 0.866025i −1.63087 + 0.941585i 1.29315 1.15228i 1.76745 1.96879i 1.00000 −2.75313 + 1.19175i 1.63087 + 0.941585i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 263.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
33.d even 2 1 inner
231.l even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 462.2.n.e 24
3.b odd 2 1 462.2.n.f yes 24
7.c even 3 1 inner 462.2.n.e 24
11.b odd 2 1 462.2.n.f yes 24
21.h odd 6 1 462.2.n.f yes 24
33.d even 2 1 inner 462.2.n.e 24
77.h odd 6 1 462.2.n.f yes 24
231.l even 6 1 inner 462.2.n.e 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.n.e 24 1.a even 1 1 trivial
462.2.n.e 24 7.c even 3 1 inner
462.2.n.e 24 33.d even 2 1 inner
462.2.n.e 24 231.l even 6 1 inner
462.2.n.f yes 24 3.b odd 2 1
462.2.n.f yes 24 11.b odd 2 1
462.2.n.f yes 24 21.h odd 6 1
462.2.n.f yes 24 77.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(462, [\chi])$$:

 $$T_{5}^{24} - 40 T_{5}^{22} + 1079 T_{5}^{20} - 15752 T_{5}^{18} + 164601 T_{5}^{16} - 922588 T_{5}^{14} + 3683208 T_{5}^{12} - 9220072 T_{5}^{10} + 16606208 T_{5}^{8} - 16802592 T_{5}^{6} + 11401616 T_{5}^{4} + \cdots + 65536$$ T5^24 - 40*T5^22 + 1079*T5^20 - 15752*T5^18 + 164601*T5^16 - 922588*T5^14 + 3683208*T5^12 - 9220072*T5^10 + 16606208*T5^8 - 16802592*T5^6 + 11401616*T5^4 - 912384*T5^2 + 65536 $$T_{17}^{12} - 2 T_{17}^{11} + 47 T_{17}^{10} + 58 T_{17}^{9} + 1365 T_{17}^{8} + 1830 T_{17}^{7} + 21444 T_{17}^{6} + 41728 T_{17}^{5} + 223744 T_{17}^{4} + 218112 T_{17}^{3} + 540672 T_{17}^{2} - 294912 T_{17} + 589824$$ T17^12 - 2*T17^11 + 47*T17^10 + 58*T17^9 + 1365*T17^8 + 1830*T17^7 + 21444*T17^6 + 41728*T17^5 + 223744*T17^4 + 218112*T17^3 + 540672*T17^2 - 294912*T17 + 589824