# Properties

 Label 43.7.h.a Level $43$ Weight $7$ Character orbit 43.h Analytic conductor $9.892$ Analytic rank $0$ Dimension $252$ CM no Inner twists $2$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$43$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 43.h (of order $$42$$, degree $$12$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$9.89232559565$$ Analytic rank: $$0$$ Dimension: $$252$$ Relative dimension: $$21$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$252 q - 14 q^{2} - 11 q^{3} + 1180 q^{4} - 11 q^{5} - 118 q^{6} + 126 q^{7} - 2254 q^{8} - 8178 q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$252 q - 14 q^{2} - 11 q^{3} + 1180 q^{4} - 11 q^{5} - 118 q^{6} + 126 q^{7} - 2254 q^{8} - 8178 q^{9} + 967 q^{10} + 4090 q^{11} - 484 q^{12} + 1527 q^{13} + 7812 q^{14} - 8923 q^{15} - 13760 q^{16} - 3655 q^{17} + 5596 q^{18} + 38605 q^{19} + 26085 q^{20} + 24873 q^{21} - 25214 q^{22} + 27991 q^{23} - 20480 q^{24} - 24146 q^{25} - 34923 q^{26} - 14 q^{27} - 77634 q^{28} - 112735 q^{29} + 375072 q^{30} + 290647 q^{31} - 344064 q^{32} - 152174 q^{33} - 198311 q^{34} - 182471 q^{35} - 930891 q^{36} + 282762 q^{37} + 487342 q^{38} + 857290 q^{39} + 745459 q^{40} + 27674 q^{41} + 148802 q^{43} - 1255832 q^{44} - 191310 q^{45} - 1344785 q^{46} + 69166 q^{47} - 871241 q^{48} + 1786604 q^{49} + 837705 q^{50} - 182756 q^{51} + 2906416 q^{52} + 7779 q^{53} + 487154 q^{54} - 529592 q^{55} - 2464007 q^{56} - 767855 q^{57} - 1523780 q^{58} + 1531338 q^{59} - 1858647 q^{60} + 646957 q^{61} - 448559 q^{62} - 2007847 q^{63} + 3937366 q^{64} - 1473542 q^{65} + 1513319 q^{66} - 575523 q^{67} + 2717157 q^{68} + 967041 q^{69} + 4624473 q^{70} - 2149319 q^{71} - 9986371 q^{72} - 514565 q^{73} - 882972 q^{74} - 6655138 q^{75} + 1152658 q^{76} - 341803 q^{77} + 6322388 q^{78} - 2476742 q^{79} + 10750038 q^{80} + 4308941 q^{81} + 5079172 q^{82} + 2318609 q^{83} + 5578407 q^{84} + 585292 q^{86} - 1990710 q^{87} - 2864890 q^{88} - 4561993 q^{89} - 17681545 q^{90} - 4648506 q^{91} - 1991714 q^{92} - 8868294 q^{93} - 10200218 q^{94} + 10091737 q^{95} + 6250318 q^{96} - 1676750 q^{97} + 7512700 q^{98} + 18452884 q^{99} + O(q^{100})$$

## 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 −6.81728 + 14.1562i −27.0740 2.02892i −114.020 142.977i 9.10871 29.5297i 213.293 369.435i −416.520 + 240.478i 1820.95 415.620i 8.02997 + 1.21032i 355.933 + 330.257i
3.2 −6.27248 + 13.0249i 36.1802 + 2.71133i −90.4018 113.360i 22.1660 71.8603i −262.254 + 454.238i 414.759 239.462i 1141.53 260.547i 580.795 + 87.5408i 796.941 + 739.453i
3.3 −5.20719 + 10.8128i 3.59499 + 0.269407i −49.8995 62.5720i −53.9072 + 174.763i −21.6329 + 37.4692i 85.2807 49.2368i 187.588 42.8158i −708.006 106.715i −1608.98 1492.91i
3.4 −4.58142 + 9.51343i −35.2682 2.64299i −29.6125 37.1329i 22.3499 72.4567i 186.723 323.413i 336.875 194.495i −169.911 + 38.7811i 516.004 + 77.7751i 586.917 + 544.580i
3.5 −4.53656 + 9.42027i 21.7428 + 1.62939i −28.2577 35.4340i 33.0785 107.238i −113.987 + 197.431i −303.689 + 175.335i −190.398 + 43.4570i −250.765 37.7967i 860.145 + 798.098i
3.6 −2.91348 + 6.04990i 50.5591 + 3.78888i 11.7904 + 14.7847i −30.3679 + 98.4503i −170.225 + 294.839i −190.382 + 109.917i −542.775 + 123.885i 1821.01 + 274.473i −507.139 470.556i
3.7 −2.62546 + 5.45182i −13.6379 1.02202i 17.0740 + 21.4102i 29.1602 94.5350i 41.3776 71.6680i −270.866 + 156.384i −539.110 + 123.048i −535.910 80.7755i 438.829 + 407.174i
3.8 −2.45963 + 5.10748i −50.4681 3.78206i 19.8668 + 24.9121i −59.6400 + 193.348i 143.450 248.462i −466.054 + 269.077i −529.815 + 120.927i 1811.86 + 273.095i −840.829 780.176i
3.9 −0.910519 + 1.89071i 23.8619 + 1.78820i 37.1576 + 46.5942i 47.7974 154.955i −25.1077 + 43.4878i 360.408 208.082i −252.868 + 57.7154i −154.664 23.3118i 249.455 + 231.461i
3.10 −0.806603 + 1.67493i −16.6260 1.24594i 37.7486 + 47.3352i −23.7056 + 76.8517i 15.4974 26.8423i 289.941 167.398i −225.726 + 51.5205i −445.988 67.2218i −109.600 101.694i
3.11 −0.711983 + 1.47845i 20.0817 + 1.50492i 38.2245 + 47.9320i −40.7360 + 132.063i −16.5228 + 28.6183i 72.9518 42.1187i −200.468 + 45.7555i −319.847 48.2092i −166.245 154.253i
3.12 1.05720 2.19531i −37.2864 2.79423i 36.2017 + 45.3954i 67.3700 218.408i −45.5535 + 78.9010i −230.704 + 133.197i 289.962 66.1820i 661.611 + 99.7219i −408.249 378.799i
3.13 1.68242 3.49359i 5.89314 + 0.441629i 30.5287 + 38.2818i −19.8674 + 64.4087i 11.4576 19.8452i −541.820 + 312.820i 427.047 97.4707i −686.324 103.447i 191.592 + 177.771i
3.14 2.66970 5.54370i −33.1314 2.48285i 16.2981 + 20.4372i −16.0047 + 51.8861i −102.215 + 177.042i 211.134 121.898i 540.730 123.418i 370.667 + 55.8691i 244.913 + 227.246i
3.15 2.75934 5.72982i 42.1286 + 3.15710i 14.6864 + 18.4162i 13.8848 45.0135i 134.337 232.678i −37.7866 + 21.8161i 542.857 123.904i 1043.99 + 157.357i −219.606 203.765i
3.16 3.92333 8.14687i 7.09953 + 0.532036i −11.0757 13.8885i 43.9242 142.399i 32.1882 55.7516i −72.9514 + 42.1185i 407.599 93.0319i −670.737 101.097i −987.776 916.522i
3.17 4.47573 9.29395i 30.6638 + 2.29794i −26.4420 33.1572i −67.0797 + 217.467i 158.600 274.703i 521.839 301.284i 217.132 49.5589i 214.131 + 32.2751i 1720.90 + 1596.76i
3.18 5.49027 11.4007i −8.01752 0.600830i −59.9287 75.1482i −46.2364 + 149.895i −50.8682 + 88.1064i −276.764 + 159.790i −396.227 + 90.4362i −656.938 99.0175i 1455.05 + 1350.09i
3.19 5.75579 11.9520i −1.56239 0.117085i −69.8182 87.5493i 36.5881 118.616i −10.3922 + 17.9998i 348.105 200.979i −620.528 + 141.631i −718.430 108.286i −1207.10 1120.03i
3.20 5.93639 12.3270i −47.2449 3.54052i −76.8118 96.3189i 8.18721 26.5423i −324.108 + 561.372i −116.163 + 67.0667i −789.619 + 180.225i 1498.69 + 225.891i −278.585 258.489i
See next 80 embeddings (of 252 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 34.21 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 43.7.h.a 252
43.h odd 42 1 inner 43.7.h.a 252

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.7.h.a 252 1.a even 1 1 trivial
43.7.h.a 252 43.h odd 42 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{7}^{\mathrm{new}}(43, [\chi])$$.