# Properties

 Label 230.2.g.d Level $230$ Weight $2$ Character orbit 230.g Analytic conductor $1.837$ Analytic rank $0$ Dimension $30$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$230 = 2 \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 230.g (of order $$11$$, degree $$10$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.83655924649$$ Analytic rank: $$0$$ Dimension: $$30$$ Relative dimension: $$3$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$30q - 3q^{2} - q^{3} - 3q^{4} + 3q^{5} - q^{6} + 8q^{7} - 3q^{8} + 12q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$30q - 3q^{2} - q^{3} - 3q^{4} + 3q^{5} - q^{6} + 8q^{7} - 3q^{8} + 12q^{9} + 3q^{10} - 3q^{11} - q^{12} + 12q^{13} - 3q^{14} + q^{15} - 3q^{16} - 26q^{17} + 12q^{18} - 25q^{19} + 3q^{20} - 11q^{21} - 14q^{22} + 14q^{23} + 10q^{24} - 3q^{25} - 10q^{26} - 52q^{27} - 3q^{28} + 4q^{29} + q^{30} + 5q^{31} - 3q^{32} + 9q^{33} + 7q^{34} + 3q^{35} + 12q^{36} - 20q^{37} + 19q^{38} + 41q^{39} + 3q^{40} + 32q^{41} + 22q^{42} - 46q^{43} - 3q^{44} - 56q^{45} + 3q^{46} + 80q^{47} - q^{48} - 41q^{49} - 3q^{50} - 34q^{51} - 21q^{52} + 29q^{53} + 47q^{54} + 3q^{55} + 8q^{56} - 33q^{57} - 7q^{58} - 3q^{59} + q^{60} - q^{61} - 28q^{62} - 8q^{63} - 3q^{64} + 21q^{65} - 46q^{66} - 63q^{67} - 4q^{68} + 78q^{69} - 8q^{70} + 55q^{71} - 32q^{72} - 36q^{73} - 20q^{74} - q^{75} - 3q^{76} - 9q^{77} - 80q^{78} - 29q^{79} + 3q^{80} + 15q^{81} - 23q^{82} - 63q^{83} + 4q^{85} + 31q^{86} + 63q^{87} - 3q^{88} - 95q^{89} + 10q^{90} - 12q^{91} + 14q^{92} + 126q^{93} + 3q^{94} - 19q^{95} - q^{96} + 11q^{97} + 58q^{98} + 97q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
31.1 −0.142315 + 0.989821i −0.844960 1.85020i −0.959493 0.281733i 0.654861 + 0.755750i 1.95162 0.573048i 1.31877 + 0.847525i 0.415415 0.909632i −0.744718 + 0.859450i −0.841254 + 0.540641i
31.2 −0.142315 + 0.989821i −0.0488649 0.106999i −0.959493 0.281733i 0.654861 + 0.755750i 0.112864 0.0331400i 1.43095 + 0.919619i 0.415415 0.909632i 1.95552 2.25679i −0.841254 + 0.540641i
31.3 −0.142315 + 0.989821i 1.30924 + 2.86684i −0.959493 0.281733i 0.654861 + 0.755750i −3.02398 + 0.887921i −0.493060 0.316870i 0.415415 0.909632i −4.54006 + 5.23951i −0.841254 + 0.540641i
41.1 −0.959493 0.281733i −1.80420 + 2.08215i 0.841254 + 0.540641i 0.142315 0.989821i 2.31772 1.48951i 1.84799 + 4.04654i −0.654861 0.755750i −0.653295 4.54377i −0.415415 + 0.909632i
41.2 −0.959493 0.281733i −0.858654 + 0.990939i 0.841254 + 0.540641i 0.142315 0.989821i 1.10305 0.708888i −1.50220 3.28936i −0.654861 0.755750i 0.182270 + 1.26772i −0.415415 + 0.909632i
41.3 −0.959493 0.281733i 2.00799 2.31734i 0.841254 + 0.540641i 0.142315 0.989821i −2.57952 + 1.65776i 0.414762 + 0.908202i −0.654861 0.755750i −0.911115 6.33694i −0.415415 + 0.909632i
71.1 0.841254 + 0.540641i −0.369448 2.56957i 0.415415 + 0.909632i 0.959493 + 0.281733i 1.07841 2.36139i 1.04688 1.20816i −0.142315 + 0.989821i −3.58770 + 1.05344i 0.654861 + 0.755750i
71.2 0.841254 + 0.540641i 0.00806642 + 0.0561032i 0.415415 + 0.909632i 0.959493 + 0.281733i −0.0235458 + 0.0515580i −3.27862 + 3.78373i −0.142315 + 0.989821i 2.87540 0.844293i 0.654861 + 0.755750i
71.3 0.841254 + 0.540641i 0.219067 + 1.52364i 0.415415 + 0.909632i 0.959493 + 0.281733i −0.639452 + 1.40020i 2.43457 2.80964i −0.142315 + 0.989821i 0.604987 0.177640i 0.654861 + 0.755750i
81.1 0.841254 0.540641i −0.369448 + 2.56957i 0.415415 0.909632i 0.959493 0.281733i 1.07841 + 2.36139i 1.04688 + 1.20816i −0.142315 0.989821i −3.58770 1.05344i 0.654861 0.755750i
81.2 0.841254 0.540641i 0.00806642 0.0561032i 0.415415 0.909632i 0.959493 0.281733i −0.0235458 0.0515580i −3.27862 3.78373i −0.142315 0.989821i 2.87540 + 0.844293i 0.654861 0.755750i
81.3 0.841254 0.540641i 0.219067 1.52364i 0.415415 0.909632i 0.959493 0.281733i −0.639452 1.40020i 2.43457 + 2.80964i −0.142315 0.989821i 0.604987 + 0.177640i 0.654861 0.755750i
101.1 −0.959493 + 0.281733i −1.80420 2.08215i 0.841254 0.540641i 0.142315 + 0.989821i 2.31772 + 1.48951i 1.84799 4.04654i −0.654861 + 0.755750i −0.653295 + 4.54377i −0.415415 0.909632i
101.2 −0.959493 + 0.281733i −0.858654 0.990939i 0.841254 0.540641i 0.142315 + 0.989821i 1.10305 + 0.708888i −1.50220 + 3.28936i −0.654861 + 0.755750i 0.182270 1.26772i −0.415415 0.909632i
101.3 −0.959493 + 0.281733i 2.00799 + 2.31734i 0.841254 0.540641i 0.142315 + 0.989821i −2.57952 1.65776i 0.414762 0.908202i −0.654861 + 0.755750i −0.911115 + 6.33694i −0.415415 0.909632i
121.1 0.415415 0.909632i −3.22429 0.946737i −0.654861 0.755750i −0.841254 + 0.540641i −2.20060 + 2.53963i −0.324589 + 2.25757i −0.959493 + 0.281733i 6.97598 + 4.48319i 0.142315 + 0.989821i
121.2 0.415415 0.909632i −0.732525 0.215089i −0.654861 0.755750i −0.841254 + 0.540641i −0.499953 + 0.576977i 0.575516 4.00280i −0.959493 + 0.281733i −2.03343 1.30681i 0.142315 + 0.989821i
121.3 0.415415 0.909632i 2.99732 + 0.880093i −0.654861 0.755750i −0.841254 + 0.540641i 2.04569 2.36086i −0.352734 + 2.45332i −0.959493 + 0.281733i 5.68562 + 3.65393i 0.142315 + 0.989821i
131.1 −0.654861 + 0.755750i −0.925739 + 0.594936i −0.142315 0.989821i −0.415415 0.909632i 0.156607 1.08923i 3.86758 1.13562i 0.841254 + 0.540641i −0.743202 + 1.62739i 0.959493 + 0.281733i
131.2 −0.654861 + 0.755750i −0.578669 + 0.371888i −0.142315 0.989821i −0.415415 0.909632i 0.0978934 0.680863i −2.88525 + 0.847187i 0.841254 + 0.540641i −1.04969 + 2.29850i 0.959493 + 0.281733i
See all 30 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 211.3 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 230.2.g.d 30
23.c even 11 1 inner 230.2.g.d 30
23.c even 11 1 5290.2.a.bk 15
23.d odd 22 1 5290.2.a.bl 15

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.2.g.d 30 1.a even 1 1 trivial
230.2.g.d 30 23.c even 11 1 inner
5290.2.a.bk 15 23.c even 11 1
5290.2.a.bl 15 23.d odd 22 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{30} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(230, [\chi])$$.