# Properties

 Label 43.2.g.a Level $43$ Weight $2$ Character orbit 43.g Analytic conductor $0.343$ Analytic rank $0$ Dimension $36$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$43$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 43.g (of order $$21$$, degree $$12$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.343356728692$$ Analytic rank: $$0$$ Dimension: $$36$$ Relative dimension: $$3$$ over $$\Q(\zeta_{21})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{21}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$36q - 10q^{2} - 16q^{3} - 18q^{4} - 17q^{5} - 4q^{6} + 6q^{7} + 18q^{8} - q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$36q - 10q^{2} - 16q^{3} - 18q^{4} - 17q^{5} - 4q^{6} + 6q^{7} + 18q^{8} - q^{9} - 7q^{10} - 4q^{11} + 2q^{12} + 18q^{14} - 3q^{15} - 10q^{16} - 10q^{17} + 11q^{18} + 10q^{19} - 3q^{20} - 21q^{21} - 3q^{22} + 4q^{23} + 31q^{24} - 2q^{25} - 15q^{26} - 4q^{27} + 20q^{28} + 9q^{29} + 88q^{30} + 40q^{31} + 48q^{32} - 11q^{33} - 42q^{34} + 11q^{35} - 47q^{36} - 19q^{37} - 21q^{38} - q^{39} - 97q^{40} - 28q^{41} + 2q^{42} - 8q^{43} + 14q^{44} - 46q^{45} - 61q^{46} - 30q^{47} - 97q^{48} + 6q^{49} - 3q^{50} + 57q^{51} - 8q^{52} - 24q^{53} + 6q^{54} + 14q^{55} + 39q^{56} + 52q^{57} + 64q^{58} - q^{59} + 111q^{60} - 14q^{61} + 33q^{62} + 47q^{63} + 48q^{64} + 38q^{65} + 79q^{66} + 66q^{67} + 66q^{68} - 7q^{69} + 47q^{70} - 33q^{71} + 26q^{72} + 29q^{73} - 40q^{74} - 55q^{75} - 39q^{76} - 27q^{77} - 126q^{78} - 17q^{79} + 8q^{80} + 38q^{81} - 54q^{82} - 23q^{83} - 155q^{84} - 56q^{85} - 45q^{86} - 86q^{87} - 17q^{88} - 19q^{89} - 127q^{90} - 13q^{91} - 18q^{92} - 30q^{93} + 44q^{94} + q^{95} - 36q^{96} - 31q^{97} - 5q^{98} - 31q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
9.1 −1.44188 1.80806i 2.66973 + 0.402397i −0.745019 + 3.26414i −2.09843 1.43068i −3.12187 5.40724i −1.09365 + 1.89425i 2.80884 1.35267i 4.09883 + 1.26432i 0.438918 + 5.85695i
9.2 −1.11243 1.39495i −2.93359 0.442167i −0.263330 + 1.15372i −0.492700 0.335917i 2.64662 + 4.58409i 2.18154 3.77854i −1.31271 + 0.632167i 5.54371 + 1.71001i 0.0795092 + 1.06098i
9.3 1.06016 + 1.32939i −1.14296 0.172273i −0.198316 + 0.868879i −1.45700 0.993363i −0.982695 1.70208i −0.297283 + 0.514909i 1.69861 0.818009i −1.59004 0.490464i −0.224073 2.99004i
10.1 −0.483758 2.11948i −0.240184 0.222858i −2.45624 + 1.18286i 0.0260188 0.00392170i −0.356153 + 0.616875i 1.56464 + 2.71003i 0.984367 + 1.23436i −0.216168 2.88456i −0.0208988 0.0532492i
10.2 0.188565 + 0.826155i −0.0129300 0.0119973i 1.15496 0.556200i −3.39819 + 0.512194i 0.00747346 0.0129444i −0.134521 0.232998i 1.73398 + 2.17435i −0.224167 2.99130i −1.06393 2.71085i
10.3 0.581275 + 2.54673i −1.40948 1.30780i −4.34603 + 2.09294i 2.95419 0.445272i 2.51133 4.34976i −0.339884 0.588696i −4.59900 5.76696i 0.0520854 + 0.695032i 2.85118 + 7.26470i
13.1 −0.483758 + 2.11948i −0.240184 + 0.222858i −2.45624 1.18286i 0.0260188 + 0.00392170i −0.356153 0.616875i 1.56464 2.71003i 0.984367 1.23436i −0.216168 + 2.88456i −0.0208988 + 0.0532492i
13.2 0.188565 0.826155i −0.0129300 + 0.0119973i 1.15496 + 0.556200i −3.39819 0.512194i 0.00747346 + 0.0129444i −0.134521 + 0.232998i 1.73398 2.17435i −0.224167 + 2.99130i −1.06393 + 2.71085i
13.3 0.581275 2.54673i −1.40948 + 1.30780i −4.34603 2.09294i 2.95419 + 0.445272i 2.51133 + 4.34976i −0.339884 + 0.588696i −4.59900 + 5.76696i 0.0520854 0.695032i 2.85118 7.26470i
14.1 −2.05399 + 0.989151i −0.239882 3.20100i 1.99349 2.49975i −0.131558 + 0.0405802i 3.65898 + 6.33755i 0.934721 1.61898i −0.607386 + 2.66113i −7.22235 + 1.08859i 0.230079 0.213482i
14.2 −0.982954 + 0.473366i 0.109070 + 1.45544i −0.504856 + 0.633069i 1.29085 0.398175i −0.796166 1.37900i −0.108163 + 0.187343i 0.682116 2.98855i 0.860089 0.129637i −1.08036 + 1.00243i
14.3 0.993512 0.478450i −0.0482746 0.644179i −0.488828 + 0.612971i −3.17479 + 0.979294i −0.356169 0.616903i 1.23273 2.13515i −0.683135 + 2.99301i 2.55386 0.384932i −2.68565 + 2.49192i
15.1 −1.72039 + 2.15730i −0.528255 + 1.34597i −1.24917 5.47296i 0.0684907 + 0.913945i −1.99486 3.45520i −0.971539 + 1.68276i 8.98382 + 4.32638i 0.666571 + 0.618487i −2.08949 1.42459i
15.2 −0.651405 + 0.816837i 0.922165 2.34964i 0.202149 + 0.885672i −0.0373507 0.498411i 1.31857 + 2.28383i −1.65334 + 2.86367i −2.73775 1.31843i −2.47126 2.29299i 0.431451 + 0.294158i
15.3 −0.0594739 + 0.0745779i −0.863608 + 2.20044i 0.443017 + 1.94098i −0.284956 3.80248i −0.112742 0.195275i 1.30981 2.26866i −0.342987 0.165174i −1.89695 1.76011i 0.300528 + 0.204897i
17.1 −0.515822 2.25996i 1.28860 0.397480i −3.03942 + 1.46371i −1.48781 + 3.79089i −1.56298 2.70715i 1.38920 2.40617i 1.98513 + 2.48927i −0.976224 + 0.665578i 9.33472 + 1.40698i
17.2 −0.178150 0.780524i −0.711521 + 0.219475i 1.22446 0.589667i 0.511972 1.30448i 0.298063 + 0.516260i −2.37928 + 4.12103i −1.67671 2.10253i −2.02062 + 1.37764i −1.10939 0.167213i
17.3 0.377390 + 1.65345i −1.63701 + 0.504949i −0.789549 + 0.380227i 0.140805 0.358765i −1.45270 2.51615i 1.74586 3.02391i 1.18819 + 1.48995i −0.0539035 + 0.0367508i 0.646339 + 0.0974200i
23.1 −1.72039 2.15730i −0.528255 1.34597i −1.24917 + 5.47296i 0.0684907 0.913945i −1.99486 + 3.45520i −0.971539 1.68276i 8.98382 4.32638i 0.666571 0.618487i −2.08949 + 1.42459i
23.2 −0.651405 0.816837i 0.922165 + 2.34964i 0.202149 0.885672i −0.0373507 + 0.498411i 1.31857 2.28383i −1.65334 2.86367i −2.73775 + 1.31843i −2.47126 + 2.29299i 0.431451 0.294158i
See all 36 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 40.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
43.g even 21 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 43.2.g.a 36
3.b odd 2 1 387.2.y.c 36
4.b odd 2 1 688.2.bg.c 36
43.g even 21 1 inner 43.2.g.a 36
43.g even 21 1 1849.2.a.n 18
43.h odd 42 1 1849.2.a.o 18
129.o odd 42 1 387.2.y.c 36
172.o odd 42 1 688.2.bg.c 36

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.2.g.a 36 1.a even 1 1 trivial
43.2.g.a 36 43.g even 21 1 inner
387.2.y.c 36 3.b odd 2 1
387.2.y.c 36 129.o odd 42 1
688.2.bg.c 36 4.b odd 2 1
688.2.bg.c 36 172.o odd 42 1
1849.2.a.n 18 43.g even 21 1
1849.2.a.o 18 43.h odd 42 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(43, [\chi])$$.