# Properties

 Label 74.4.f.b Level $74$ Weight $4$ Character orbit 74.f Analytic conductor $4.366$ Analytic rank $0$ Dimension $30$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$74 = 2 \cdot 37$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 74.f (of order $$9$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.36614134042$$ Analytic rank: $$0$$ Dimension: $$30$$ Relative dimension: $$5$$ over $$\Q(\zeta_{9})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$30q + 6q^{3} - 6q^{5} - 36q^{6} - 75q^{7} - 120q^{8} - 48q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$30q + 6q^{3} - 6q^{5} - 36q^{6} - 75q^{7} - 120q^{8} - 48q^{9} + 54q^{10} - 21q^{11} + 24q^{12} - 159q^{13} + 6q^{14} - 69q^{15} - 171q^{17} + 192q^{18} - 45q^{19} - 24q^{20} + 579q^{21} + 42q^{22} + 459q^{23} + 48q^{24} - 204q^{25} + 264q^{26} + 435q^{27} + 96q^{28} + 273q^{29} - 138q^{30} - 258q^{31} - 318q^{33} - 558q^{34} - 753q^{35} + 1296q^{36} - 891q^{37} - 2556q^{38} - 504q^{39} + 96q^{40} + 648q^{41} + 1158q^{42} - 216q^{43} + 84q^{44} + 1731q^{45} - 6q^{46} + 48q^{47} + 144q^{48} + 1731q^{49} + 240q^{50} + 972q^{51} + 48q^{52} - 135q^{53} + 360q^{54} + 765q^{55} + 408q^{56} - 405q^{57} - 762q^{58} + 1836q^{59} + 108q^{60} - 684q^{61} - 204q^{62} - 3264q^{63} - 960q^{64} - 1350q^{65} + 252q^{66} + 1095q^{67} - 432q^{68} - 5823q^{69} - 1146q^{70} + 4179q^{71} + 768q^{72} - 2658q^{73} - 1692q^{74} - 10416q^{75} - 180q^{76} + 267q^{77} + 1530q^{78} + 4866q^{79} - 864q^{80} - 2076q^{81} + 1800q^{82} - 186q^{83} + 504q^{84} + 753q^{85} + 648q^{86} - 525q^{87} - 168q^{88} + 687q^{89} + 768q^{90} + 3990q^{91} + 132q^{92} + 3753q^{93} + 6210q^{94} + 9000q^{95} - 384q^{96} + 7428q^{97} - 1542q^{98} + 6324q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1 −1.87939 + 0.684040i −6.73393 2.45095i 3.06418 2.57115i −2.75980 15.6516i 14.3322 5.01435 + 28.4378i −4.00000 + 6.92820i 18.6555 + 15.6538i 15.8931 + 27.5276i
7.2 −1.87939 + 0.684040i −1.99226 0.725122i 3.06418 2.57115i 1.44178 + 8.17673i 4.24023 −1.26639 7.18203i −4.00000 + 6.92820i −17.2399 14.4660i −8.30287 14.3810i
7.3 −1.87939 + 0.684040i −1.18036 0.429617i 3.06418 2.57115i 0.00632031 + 0.0358443i 2.51223 −0.240077 1.36154i −4.00000 + 6.92820i −19.4745 16.3411i −0.0363972 0.0630418i
7.4 −1.87939 + 0.684040i 6.42413 + 2.33819i 3.06418 2.57115i −3.55870 20.1824i −13.6728 −4.31979 24.4987i −4.00000 + 6.92820i 15.1191 + 12.6865i 20.4938 + 35.4962i
7.5 −1.87939 + 0.684040i 6.95420 + 2.53112i 3.06418 2.57115i 2.65487 + 15.0565i −14.8010 −1.00418 5.69497i −4.00000 + 6.92820i 21.2711 + 17.8486i −15.2888 26.4809i
9.1 0.347296 + 1.96962i −1.62017 + 9.18845i −3.75877 + 1.36808i −9.08898 7.62656i −18.6604 21.4230 + 17.9761i −4.00000 6.92820i −56.4309 20.5392i 11.8648 20.5505i
9.2 0.347296 + 1.96962i −1.04722 + 5.93905i −3.75877 + 1.36808i 13.7217 + 11.5139i −12.0613 −10.6649 8.94892i −4.00000 6.92820i −8.80400 3.20439i −17.9124 + 31.0252i
9.3 0.347296 + 1.96962i −0.267400 + 1.51650i −3.75877 + 1.36808i −13.2707 11.1355i −3.07979 −24.3073 20.3962i −4.00000 6.92820i 23.1434 + 8.42352i 17.3237 30.0056i
9.4 0.347296 + 1.96962i 0.551654 3.12858i −3.75877 + 1.36808i 7.17496 + 6.02051i 6.35370 7.01886 + 5.88953i −4.00000 6.92820i 15.8880 + 5.78275i −9.36625 + 16.2228i
9.5 0.347296 + 1.96962i 1.33010 7.54337i −3.75877 + 1.36808i −4.89924 4.11095i 15.3195 −6.34775 5.32640i −4.00000 6.92820i −29.7615 10.8323i 6.39551 11.0773i
33.1 0.347296 1.96962i −1.62017 9.18845i −3.75877 1.36808i −9.08898 + 7.62656i −18.6604 21.4230 17.9761i −4.00000 + 6.92820i −56.4309 + 20.5392i 11.8648 + 20.5505i
33.2 0.347296 1.96962i −1.04722 5.93905i −3.75877 1.36808i 13.7217 11.5139i −12.0613 −10.6649 + 8.94892i −4.00000 + 6.92820i −8.80400 + 3.20439i −17.9124 31.0252i
33.3 0.347296 1.96962i −0.267400 1.51650i −3.75877 1.36808i −13.2707 + 11.1355i −3.07979 −24.3073 + 20.3962i −4.00000 + 6.92820i 23.1434 8.42352i 17.3237 + 30.0056i
33.4 0.347296 1.96962i 0.551654 + 3.12858i −3.75877 1.36808i 7.17496 6.02051i 6.35370 7.01886 5.88953i −4.00000 + 6.92820i 15.8880 5.78275i −9.36625 16.2228i
33.5 0.347296 1.96962i 1.33010 + 7.54337i −3.75877 1.36808i −4.89924 + 4.11095i 15.3195 −6.34775 + 5.32640i −4.00000 + 6.92820i −29.7615 + 10.8323i 6.39551 + 11.0773i
49.1 1.53209 1.28558i −7.21877 6.05727i 0.694593 3.93923i 11.5648 4.20923i −18.8469 −26.9521 + 9.80975i −4.00000 6.92820i 10.7317 + 60.8623i 12.3070 21.3163i
49.2 1.53209 1.28558i −2.66631 2.23730i 0.694593 3.93923i −17.8476 + 6.49601i −6.96124 −11.9434 + 4.34706i −4.00000 6.92820i −2.58480 14.6591i −18.9931 + 32.8969i
49.3 1.53209 1.28558i −0.840966 0.705654i 0.694593 3.93923i 5.39277 1.96281i −2.19561 14.5717 5.30365i −4.00000 6.92820i −4.47923 25.4029i 5.73887 9.94001i
49.4 1.53209 1.28558i 4.80507 + 4.03193i 0.694593 3.93923i 14.5989 5.31355i 12.5451 −23.2945 + 8.47851i −4.00000 6.92820i 2.14372 + 12.1576i 15.5358 26.9088i
49.5 1.53209 1.28558i 6.50223 + 5.45602i 0.694593 3.93923i −8.13091 + 2.95941i 16.9761 24.8125 9.03100i −4.00000 6.92820i 7.82235 + 44.3628i −8.65273 + 14.9870i
See all 30 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 71.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.f even 9 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 74.4.f.b 30
37.f even 9 1 inner 74.4.f.b 30

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.4.f.b 30 1.a even 1 1 trivial
74.4.f.b 30 37.f even 9 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$24\!\cdots\!95$$$$T_{3}^{16} -$$$$15\!\cdots\!85$$$$T_{3}^{15} +$$$$73\!\cdots\!80$$$$T_{3}^{14} +$$$$44\!\cdots\!38$$$$T_{3}^{13} -$$$$32\!\cdots\!63$$$$T_{3}^{12} -$$$$99\!\cdots\!30$$$$T_{3}^{11} +$$$$80\!\cdots\!97$$$$T_{3}^{10} +$$$$24\!\cdots\!63$$$$T_{3}^{9} +$$$$15\!\cdots\!41$$$$T_{3}^{8} +$$$$73\!\cdots\!35$$$$T_{3}^{7} +$$$$24\!\cdots\!13$$$$T_{3}^{6} +$$$$58\!\cdots\!28$$$$T_{3}^{5} +$$$$99\!\cdots\!76$$$$T_{3}^{4} +$$$$12\!\cdots\!44$$$$T_{3}^{3} +$$$$10\!\cdots\!72$$$$T_{3}^{2} +$$$$58\!\cdots\!00$$$$T_{3} +$$$$15\!\cdots\!44$$">$$T_{3}^{30} - \cdots$$ acting on $$S_{4}^{\mathrm{new}}(74, [\chi])$$.