# Properties

 Label 43.3.h.a Level $43$ Weight $3$ Character orbit 43.h Analytic conductor $1.172$ Analytic rank $0$ Dimension $72$ CM no Inner twists $2$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$43$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 43.h (of order $$42$$, degree $$12$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.17166513675$$ Analytic rank: $$0$$ Dimension: $$72$$ Relative dimension: $$6$$ over $$\Q(\zeta_{42})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{42}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$72q - 14q^{2} - 14q^{3} + 12q^{4} - 11q^{5} + 2q^{6} - 30q^{7} - 42q^{8} + 54q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$72q - 14q^{2} - 14q^{3} + 12q^{4} - 11q^{5} + 2q^{6} - 30q^{7} - 42q^{8} + 54q^{9} - 13q^{10} - 42q^{11} + 20q^{12} - 24q^{13} - 108q^{14} - 43q^{15} - 40q^{16} - 7q^{17} + 16q^{18} - 38q^{19} - 55q^{20} + 3q^{21} - 98q^{22} + 30q^{23} + 268q^{24} + 49q^{25} - 79q^{26} - 14q^{27} + 66q^{28} + 27q^{29} + 132q^{30} + 330q^{31} + 56q^{32} + 142q^{33} + 109q^{34} - 31q^{35} + 9q^{36} + 69q^{37} + 262q^{38} + 49q^{39} + 239q^{40} - 94q^{41} - 19q^{43} - 64q^{44} - 420q^{45} - 9q^{46} - 66q^{47} - 221q^{48} - 6q^{49} - 495q^{50} - 560q^{51} - 452q^{52} + 16q^{53} - 394q^{54} + 328q^{55} - 1015q^{56} - 590q^{57} - 420q^{58} - 245q^{59} + 873q^{60} - 50q^{61} - 191q^{62} - 379q^{63} - 306q^{64} - 182q^{65} + 551q^{66} + 599q^{67} + 757q^{68} - 213q^{69} - 287q^{70} + 367q^{71} + 1337q^{72} + 486q^{73} + 1656q^{74} + 1337q^{75} + 746q^{76} + 79q^{77} + 1040q^{78} + 261q^{79} + 138q^{80} + 506q^{81} + 364q^{82} - 220q^{83} - 45q^{84} - 284q^{86} + 30q^{87} - 490q^{88} - 564q^{89} - 145q^{90} - 145q^{91} - 406q^{92} - 798q^{93} - 1666q^{94} - 353q^{95} - 506q^{96} - 99q^{97} - 500q^{98} - 2012q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1 −1.23133 + 2.55689i −4.45977 0.334214i −2.52755 3.16944i 1.18875 3.85384i 6.34601 10.9916i −8.83061 + 5.09836i 0.149040 0.0340174i 10.8784 + 1.63965i 8.39010 + 7.78488i
3.2 −1.03664 + 2.15260i 3.30208 + 0.247456i −1.06511 1.33560i 0.257183 0.833768i −3.95573 + 6.85153i −1.23199 + 0.711287i −5.33806 + 1.21838i 1.94300 + 0.292860i 1.52816 + 1.41793i
3.3 −0.402412 + 0.835617i −2.72881 0.204496i 1.95764 + 2.45480i −2.67009 + 8.65623i 1.26898 2.19794i 8.78777 5.07362i −6.45589 + 1.47352i −1.49492 0.225323i −6.15881 5.71454i
3.4 0.304460 0.632217i 0.576841 + 0.0432282i 2.18696 + 2.74236i 1.20897 3.91938i 0.202954 0.351527i −0.834967 + 0.482068i 5.13606 1.17227i −8.56860 1.29151i −2.10982 1.95763i
3.5 1.25989 2.61618i −5.25280 0.393643i −2.76312 3.46484i 1.10235 3.57375i −7.64776 + 13.2463i 6.79133 3.92097i −1.22212 + 0.278940i 18.5375 + 2.79407i −7.96072 7.38647i
3.6 1.31976 2.74051i 1.72784 + 0.129483i −3.27466 4.10629i −1.51671 + 4.91705i 2.63518 4.56426i −4.74658 + 2.74044i −3.71320 + 0.847514i −5.93082 0.893928i 11.4735 + 10.6459i
5.1 −3.19097 0.728319i −0.352152 + 1.14165i 6.04798 + 2.91256i 2.28850 0.898170i 1.95519 3.38649i 10.0706 5.81424i −6.94183 5.53593i 6.25680 + 4.26581i −7.95670 + 1.19928i
5.2 −1.60332 0.365947i −0.914301 + 2.96409i −1.16717 0.562078i −3.68671 + 1.44693i 2.55061 4.41779i −8.88846 + 5.13175i 6.80869 + 5.42975i −0.513742 0.350263i 6.44047 0.970744i
5.3 −0.948568 0.216505i 1.34540 4.36170i −2.75097 1.32480i −6.23473 + 2.44695i −2.22054 + 3.84608i 9.41041 5.43310i 5.36543 + 4.27879i −9.77813 6.66661i 6.44385 0.971254i
5.4 −0.187226 0.0427331i 0.614362 1.99171i −3.57065 1.71953i 8.48040 3.32831i −0.200136 + 0.346646i −4.12743 + 2.38297i 1.19561 + 0.953468i 3.84667 + 2.62262i −1.72998 + 0.260753i
5.5 1.81675 + 0.414660i −1.31342 + 4.25801i −0.475255 0.228871i 2.73180 1.07215i −4.15178 + 7.19109i 6.14989 3.55064i −6.59618 5.26028i −8.96941 6.11524i 5.40757 0.815060i
5.6 2.50298 + 0.571289i 0.445320 1.44369i 2.33467 + 1.12432i −3.49724 + 1.37257i 1.93939 3.35913i −3.35697 + 1.93815i −2.82762 2.25495i 5.55021 + 3.78407i −9.53767 + 1.43757i
12.1 −1.28258 2.66331i −1.59066 2.33308i −2.95426 + 3.70453i −1.17011 + 1.26107i −4.17355 + 7.22881i 1.73703 1.00288i 2.12764 + 0.485621i 0.375040 0.955585i 4.85940 + 1.49893i
12.2 −0.963431 2.00058i 1.57068 + 2.30376i −0.580179 + 0.727522i 4.59096 4.94788i 3.09563 5.36179i −4.41636 + 2.54979i −6.64480 1.51663i 0.447774 1.14091i −14.3217 4.41766i
12.3 −0.236569 0.491241i 2.13665 + 3.13389i 2.30861 2.89490i −5.17829 + 5.58087i 1.03403 1.79099i 8.57608 4.95140i −4.09450 0.934543i −1.96793 + 5.01420i 3.96658 + 1.22353i
12.4 0.185466 + 0.385124i −2.30187 3.37622i 2.38004 2.98447i −0.146154 + 0.157516i 0.873345 1.51268i 0.871816 0.503343i 3.25776 + 0.743563i −2.81219 + 7.16534i −0.0877700 0.0270734i
12.5 0.704959 + 1.46386i 0.920816 + 1.35059i 0.848033 1.06340i 0.650793 0.701388i −1.32794 + 2.30006i −11.1429 + 6.43335i 8.49061 + 1.93793i 2.31188 5.89057i 1.48552 + 0.458221i
12.6 1.47143 + 3.05545i −0.584290 0.856996i −4.67672 + 5.86443i −0.262700 + 0.283123i 1.75877 3.04628i 4.87516 2.81467i −11.5749 2.64189i 2.89502 7.37640i −1.25161 0.386071i
18.1 −1.28258 + 2.66331i −1.59066 + 2.33308i −2.95426 3.70453i −1.17011 1.26107i −4.17355 7.22881i 1.73703 + 1.00288i 2.12764 0.485621i 0.375040 + 0.955585i 4.85940 1.49893i
18.2 −0.963431 + 2.00058i 1.57068 2.30376i −0.580179 0.727522i 4.59096 + 4.94788i 3.09563 + 5.36179i −4.41636 2.54979i −6.64480 + 1.51663i 0.447774 + 1.14091i −14.3217 + 4.41766i
See all 72 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 34.6 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.h odd 42 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 43.3.h.a 72
3.b odd 2 1 387.3.bn.b 72
43.h odd 42 1 inner 43.3.h.a 72
129.n even 42 1 387.3.bn.b 72

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.3.h.a 72 1.a even 1 1 trivial
43.3.h.a 72 43.h odd 42 1 inner
387.3.bn.b 72 3.b odd 2 1
387.3.bn.b 72 129.n even 42 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(43, [\chi])$$.