# Properties

 Label 471.2.e.b Level $471$ Weight $2$ Character orbit 471.e Analytic conductor $3.761$ Analytic rank $0$ Dimension $22$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$471 = 3 \cdot 157$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 471.e (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.76095393520$$ Analytic rank: $$0$$ Dimension: $$22$$ Relative dimension: $$11$$ over $$\Q(\zeta_{3})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$22 q + 2 q^{2} + 11 q^{3} + 30 q^{4} - 4 q^{5} + q^{6} + 4 q^{7} - 11 q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$22 q + 2 q^{2} + 11 q^{3} + 30 q^{4} - 4 q^{5} + q^{6} + 4 q^{7} - 11 q^{9} - 5 q^{10} + 15 q^{12} + 3 q^{13} - 14 q^{14} + 4 q^{15} + 54 q^{16} - q^{17} - q^{18} - 22 q^{19} - 7 q^{20} + 2 q^{21} - 22 q^{22} - 10 q^{23} - 15 q^{25} - 10 q^{26} - 22 q^{27} - 38 q^{28} + 22 q^{29} + 5 q^{30} - 6 q^{31} + 32 q^{32} + 17 q^{34} - 11 q^{35} - 15 q^{36} + 8 q^{37} + 14 q^{38} + 6 q^{39} + 32 q^{40} - 7 q^{42} + q^{43} - 12 q^{44} + 8 q^{45} + 24 q^{46} + 7 q^{47} + 27 q^{48} + 22 q^{49} + 13 q^{50} + q^{51} + 17 q^{52} + 30 q^{53} - 2 q^{54} + 31 q^{55} - 82 q^{56} + 22 q^{57} - 90 q^{58} - 16 q^{59} + 7 q^{60} + 8 q^{61} - 28 q^{62} - 2 q^{63} - 32 q^{64} - 68 q^{65} + 22 q^{66} - 38 q^{67} - 8 q^{68} - 5 q^{69} + 43 q^{70} + 45 q^{71} - 4 q^{73} + 3 q^{74} - 30 q^{75} - 33 q^{76} + 21 q^{77} - 20 q^{78} + 26 q^{79} - 12 q^{80} - 11 q^{81} + 16 q^{82} + 8 q^{83} - 19 q^{84} - 28 q^{85} - 16 q^{86} + 11 q^{87} - 65 q^{88} + 15 q^{89} + 10 q^{90} - 3 q^{91} - 18 q^{92} - 12 q^{93} - 28 q^{94} - 5 q^{95} + 16 q^{96} - 35 q^{97} + 90 q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
169.1 −2.48879 0.500000 + 0.866025i 4.19406 −1.91720 3.32070i −1.24439 2.15535i 3.28748 −5.46054 −0.500000 + 0.866025i 4.77151 + 8.26450i
169.2 −2.44281 0.500000 + 0.866025i 3.96732 0.189821 + 0.328779i −1.22140 2.11553i −0.758173 −4.80578 −0.500000 + 0.866025i −0.463696 0.803145i
169.3 −2.18058 0.500000 + 0.866025i 2.75495 1.44529 + 2.50331i −1.09029 1.88844i −2.39860 −1.64623 −0.500000 + 0.866025i −3.15157 5.45869i
169.4 −0.685532 0.500000 + 0.866025i −1.53005 −0.295622 0.512033i −0.342766 0.593688i −1.98864 2.41996 −0.500000 + 0.866025i 0.202658 + 0.351015i
169.5 −0.479742 0.500000 + 0.866025i −1.76985 1.73849 + 3.01115i −0.239871 0.415469i 2.28663 1.80855 −0.500000 + 0.866025i −0.834027 1.44458i
169.6 0.267371 0.500000 + 0.866025i −1.92851 −0.990152 1.71499i 0.133686 + 0.231550i 4.13452 −1.05037 −0.500000 + 0.866025i −0.264738 0.458540i
169.7 0.537815 0.500000 + 0.866025i −1.71075 −0.249194 0.431617i 0.268908 + 0.465762i 1.57818 −1.99570 −0.500000 + 0.866025i −0.134021 0.232130i
169.8 1.32447 0.500000 + 0.866025i −0.245772 −2.05322 3.55628i 0.662236 + 1.14703i −3.69381 −2.97446 −0.500000 + 0.866025i −2.71944 4.71020i
169.9 2.06264 0.500000 + 0.866025i 2.25447 1.20043 + 2.07920i 1.03132 + 1.78630i 1.31301 0.524880 −0.500000 + 0.866025i 2.47604 + 4.28863i
169.10 2.33676 0.500000 + 0.866025i 3.46043 −1.34656 2.33230i 1.16838 + 2.02369i 2.66781 3.41268 −0.500000 + 0.866025i −3.14657 5.45003i
169.11 2.74840 0.500000 + 0.866025i 5.55371 0.277926 + 0.481381i 1.37420 + 2.38018i −4.42840 9.76701 −0.500000 + 0.866025i 0.763851 + 1.32303i
301.1 −2.48879 0.500000 0.866025i 4.19406 −1.91720 + 3.32070i −1.24439 + 2.15535i 3.28748 −5.46054 −0.500000 0.866025i 4.77151 8.26450i
301.2 −2.44281 0.500000 0.866025i 3.96732 0.189821 0.328779i −1.22140 + 2.11553i −0.758173 −4.80578 −0.500000 0.866025i −0.463696 + 0.803145i
301.3 −2.18058 0.500000 0.866025i 2.75495 1.44529 2.50331i −1.09029 + 1.88844i −2.39860 −1.64623 −0.500000 0.866025i −3.15157 + 5.45869i
301.4 −0.685532 0.500000 0.866025i −1.53005 −0.295622 + 0.512033i −0.342766 + 0.593688i −1.98864 2.41996 −0.500000 0.866025i 0.202658 0.351015i
301.5 −0.479742 0.500000 0.866025i −1.76985 1.73849 3.01115i −0.239871 + 0.415469i 2.28663 1.80855 −0.500000 0.866025i −0.834027 + 1.44458i
301.6 0.267371 0.500000 0.866025i −1.92851 −0.990152 + 1.71499i 0.133686 0.231550i 4.13452 −1.05037 −0.500000 0.866025i −0.264738 + 0.458540i
301.7 0.537815 0.500000 0.866025i −1.71075 −0.249194 + 0.431617i 0.268908 0.465762i 1.57818 −1.99570 −0.500000 0.866025i −0.134021 + 0.232130i
301.8 1.32447 0.500000 0.866025i −0.245772 −2.05322 + 3.55628i 0.662236 1.14703i −3.69381 −2.97446 −0.500000 0.866025i −2.71944 + 4.71020i
301.9 2.06264 0.500000 0.866025i 2.25447 1.20043 2.07920i 1.03132 1.78630i 1.31301 0.524880 −0.500000 0.866025i 2.47604 4.28863i
See all 22 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 301.11 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
157.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 471.2.e.b 22
157.c even 3 1 inner 471.2.e.b 22

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
471.2.e.b 22 1.a even 1 1 trivial
471.2.e.b 22 157.c even 3 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{11} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(471, [\chi])$$.