# Properties

 Label 322.2.i.e Level $322$ Weight $2$ Character orbit 322.i Analytic conductor $2.571$ Analytic rank $0$ Dimension $40$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$322 = 2 \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 322.i (of order $$11$$, degree $$10$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.57118294509$$ Analytic rank: $$0$$ Dimension: $$40$$ Relative dimension: $$4$$ over $$\Q(\zeta_{11})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$40q + 4q^{2} + 2q^{3} - 4q^{4} - 9q^{5} - 2q^{6} - 4q^{7} + 4q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$40q + 4q^{2} + 2q^{3} - 4q^{4} - 9q^{5} - 2q^{6} - 4q^{7} + 4q^{8} - 2q^{10} + 2q^{12} + 4q^{13} + 4q^{14} - 11q^{15} - 4q^{16} - 21q^{17} + 4q^{19} + 2q^{20} + 2q^{21} - 11q^{23} - 2q^{24} + 5q^{25} - 15q^{26} + 95q^{27} - 4q^{28} + 22q^{30} - 12q^{31} + 4q^{32} - 8q^{33} + 10q^{34} - 9q^{35} + 41q^{37} + 7q^{38} - 38q^{39} - 13q^{40} + 25q^{41} - 2q^{42} - 59q^{43} - 6q^{45} + 11q^{46} - 60q^{47} + 2q^{48} - 4q^{49} + 61q^{50} - 87q^{51} + 4q^{52} + 47q^{53} - 29q^{54} - 30q^{55} + 4q^{56} + 20q^{57} - 96q^{59} - 11q^{60} + 84q^{61} - 65q^{62} - 4q^{64} + 52q^{65} - 3q^{66} + 11q^{67} - 10q^{68} - 42q^{69} - 2q^{70} - 13q^{71} + 11q^{72} - 48q^{73} - 41q^{74} + 23q^{75} + 26q^{76} - 83q^{78} + 80q^{79} - 9q^{80} + 2q^{81} + 30q^{82} - 20q^{83} + 2q^{84} - 71q^{85} - 40q^{86} + 99q^{87} - 30q^{89} + 116q^{90} + 26q^{91} - 22q^{92} + 20q^{93} - 17q^{94} - 4q^{95} - 2q^{96} - 56q^{97} + 4q^{98} + 138q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
29.1 −0.415415 + 0.909632i −2.45665 0.721337i −0.654861 0.755750i −3.26944 + 2.10114i 1.67668 1.93499i −0.142315 + 0.989821i 0.959493 0.281733i 2.99103 + 1.92222i −0.553091 3.84683i
29.2 −0.415415 + 0.909632i −0.589310 0.173037i −0.654861 0.755750i 3.36368 2.16170i 0.402208 0.464173i −0.142315 + 0.989821i 0.959493 0.281733i −2.20642 1.41798i 0.569033 + 3.95771i
29.3 −0.415415 + 0.909632i −0.427601 0.125555i −0.654861 0.755750i −0.0300272 + 0.0192973i 0.291841 0.336802i −0.142315 + 0.989821i 0.959493 0.281733i −2.35668 1.51455i −0.00507970 0.0353301i
29.4 −0.415415 + 0.909632i 3.20046 + 0.939739i −0.654861 0.755750i −3.28037 + 2.10817i −2.18434 + 2.52086i −0.142315 + 0.989821i 0.959493 0.281733i 6.83607 + 4.39327i −0.554941 3.85970i
71.1 −0.841254 0.540641i −0.459025 3.19259i 0.415415 + 0.909632i −1.51548 0.444984i −1.33989 + 2.93395i −0.654861 + 0.755750i 0.142315 0.989821i −7.10346 + 2.08576i 1.03432 + 1.19367i
71.2 −0.841254 0.540641i −0.0909948 0.632882i 0.415415 + 0.909632i −0.867979 0.254862i −0.265612 + 0.581610i −0.654861 + 0.755750i 0.142315 0.989821i 2.48622 0.730020i 0.592402 + 0.683668i
71.3 −0.841254 0.540641i 0.0894077 + 0.621844i 0.415415 + 0.909632i −2.40216 0.705337i 0.260980 0.571466i −0.654861 + 0.755750i 0.142315 0.989821i 2.49978 0.734002i 1.63949 + 1.89207i
71.4 −0.841254 0.540641i 0.274220 + 1.90724i 0.415415 + 0.909632i 3.76154 + 1.10449i 0.800444 1.75273i −0.654861 + 0.755750i 0.142315 0.989821i −0.683890 + 0.200808i −2.56727 2.96279i
85.1 0.654861 0.755750i −1.71143 + 1.09987i −0.142315 0.989821i 0.475695 + 1.04163i −0.289523 + 2.01368i −0.959493 + 0.281733i −0.841254 0.540641i 0.473040 1.03581i 1.09872 + 0.322614i
85.2 0.654861 0.755750i −1.01640 + 0.653203i −0.142315 0.989821i −1.23775 2.71030i −0.171945 + 1.19590i −0.959493 + 0.281733i −0.841254 0.540641i −0.639842 + 1.40106i −2.85887 0.839440i
85.3 0.654861 0.755750i 2.09228 1.34463i −0.142315 0.989821i −0.762523 1.66969i 0.353951 2.46178i −0.959493 + 0.281733i −0.841254 0.540641i 1.32337 2.89778i −1.76122 0.517140i
85.4 0.654861 0.755750i 2.24991 1.44593i −0.142315 0.989821i 1.60528 + 3.51508i 0.380617 2.64725i −0.959493 + 0.281733i −0.841254 0.540641i 1.72514 3.77752i 3.70776 + 1.08870i
127.1 −0.841254 + 0.540641i −0.459025 + 3.19259i 0.415415 0.909632i −1.51548 + 0.444984i −1.33989 2.93395i −0.654861 0.755750i 0.142315 + 0.989821i −7.10346 2.08576i 1.03432 1.19367i
127.2 −0.841254 + 0.540641i −0.0909948 + 0.632882i 0.415415 0.909632i −0.867979 + 0.254862i −0.265612 0.581610i −0.654861 0.755750i 0.142315 + 0.989821i 2.48622 + 0.730020i 0.592402 0.683668i
127.3 −0.841254 + 0.540641i 0.0894077 0.621844i 0.415415 0.909632i −2.40216 + 0.705337i 0.260980 + 0.571466i −0.654861 0.755750i 0.142315 + 0.989821i 2.49978 + 0.734002i 1.63949 1.89207i
127.4 −0.841254 + 0.540641i 0.274220 1.90724i 0.415415 0.909632i 3.76154 1.10449i 0.800444 + 1.75273i −0.654861 0.755750i 0.142315 + 0.989821i −0.683890 0.200808i −2.56727 + 2.96279i
141.1 0.142315 + 0.989821i −1.36463 + 2.98813i −0.959493 + 0.281733i −1.28840 + 1.48690i −3.15192 0.925489i 0.841254 0.540641i −0.415415 0.909632i −5.10212 5.88817i −1.65512 1.06368i
141.2 0.142315 + 0.989821i −0.605793 + 1.32650i −0.959493 + 0.281733i 2.76217 3.18771i −1.39921 0.410846i 0.841254 0.540641i −0.415415 0.909632i 0.571960 + 0.660077i 3.54836 + 2.28039i
141.3 0.142315 + 0.989821i 0.527208 1.15442i −0.959493 + 0.281733i −2.43563 + 2.81087i 1.21770 + 0.357550i 0.841254 0.540641i −0.415415 0.909632i 0.909835 + 1.05001i −3.12889 2.01081i
141.4 0.142315 + 0.989821i 0.744281 1.62975i −0.959493 + 0.281733i 1.17446 1.35540i 1.71908 + 0.504768i 0.841254 0.540641i −0.415415 0.909632i −0.137541 0.158731i 1.50875 + 0.969614i
See all 40 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 239.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.c even 11 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 322.2.i.e 40
23.c even 11 1 inner 322.2.i.e 40
23.c even 11 1 7406.2.a.bt 20
23.d odd 22 1 7406.2.a.bs 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
322.2.i.e 40 1.a even 1 1 trivial
322.2.i.e 40 23.c even 11 1 inner
7406.2.a.bs 20 23.d odd 22 1
7406.2.a.bt 20 23.c even 11 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{40} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(322, [\chi])$$.