# Properties

 Label 43.6.e.a Level $43$ Weight $6$ Character orbit 43.e Analytic conductor $6.897$ Analytic rank $0$ Dimension $108$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$43$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 43.e (of order $$7$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.89650425196$$ Analytic rank: $$0$$ Dimension: $$108$$ Relative dimension: $$18$$ over $$\Q(\zeta_{7})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$108q - 3q^{2} - 9q^{3} - 331q^{4} + 67q^{5} - 20q^{6} - 624q^{7} + 589q^{8} - 579q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$108q - 3q^{2} - 9q^{3} - 331q^{4} + 67q^{5} - 20q^{6} - 624q^{7} + 589q^{8} - 579q^{9} + 627q^{10} - 220q^{11} - 663q^{12} - 1266q^{13} + 2353q^{14} + 1471q^{15} - 4583q^{16} - 1490q^{17} + 4728q^{18} + 946q^{19} + 14589q^{20} + 776q^{21} + 963q^{22} - 3018q^{23} - 17761q^{24} - 11939q^{25} + 1315q^{26} + 11277q^{27} - 10787q^{28} + 7283q^{29} - 55324q^{30} + 18676q^{31} - 10965q^{32} + 8776q^{33} + 23815q^{34} - 13340q^{35} + 136240q^{36} + 4994q^{37} + 39351q^{38} - 21251q^{39} + 14719q^{40} - 10519q^{41} - 199326q^{42} + 7026q^{43} - 180280q^{44} + 65996q^{45} - 5266q^{46} + 34102q^{47} + 69385q^{48} + 232252q^{49} + 17850q^{50} + 4116q^{51} + 32385q^{52} + 46774q^{53} - 30267q^{54} + 83643q^{55} - 231565q^{56} + 79511q^{57} - 222889q^{58} - 142070q^{59} + 42019q^{60} + 4715q^{61} - 150121q^{62} - 53547q^{63} + 163605q^{64} - 106653q^{65} - 56111q^{66} + 370962q^{67} - 579312q^{68} - 245635q^{69} - 91242q^{70} + 16922q^{71} + 586300q^{72} + 97842q^{73} + 535946q^{74} + 245407q^{75} + 942280q^{76} + 43381q^{77} + 173684q^{78} + 187210q^{79} - 752974q^{80} - 791814q^{81} + 387227q^{82} - 444471q^{83} - 688442q^{84} + 255432q^{85} - 244075q^{86} - 1418850q^{87} + 196837q^{88} + 408972q^{89} - 495568q^{90} - 25764q^{91} + 200120q^{92} + 692536q^{93} + 161449q^{94} + 406105q^{95} + 1096556q^{96} + 266261q^{97} + 527485q^{98} + 212416q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4.1 −7.04063 + 8.82868i 3.19853 + 4.01083i −21.2543 93.1212i 78.2981 + 37.7064i −57.9300 141.803 646.213 + 311.200i 48.2164 211.250i −884.165 + 425.791i
4.2 −6.31600 + 7.92001i −18.2652 22.9039i −15.7141 68.8479i −46.4718 22.3797i 296.762 −129.944 352.466 + 169.739i −136.896 + 599.780i 470.763 226.708i
4.3 −5.37139 + 6.73550i 15.6252 + 19.5934i −9.39457 41.1603i −42.9010 20.6600i −215.900 −55.8448 79.3171 + 38.1971i −85.6808 + 375.392i 369.594 177.987i
4.4 −4.84915 + 6.08065i 1.26806 + 1.59010i −6.33931 27.7743i 3.40560 + 1.64005i −15.8178 −175.475 −24.6053 11.8493i 53.1522 232.875i −26.4869 + 12.7554i
4.5 −4.75577 + 5.96355i −3.77448 4.73304i −5.82590 25.5249i −49.8748 24.0184i 46.1763 187.876 −39.9875 19.2570i 45.9176 201.178i 380.428 183.205i
4.6 −3.83850 + 4.81333i −10.8788 13.6416i −1.31337 5.75427i 77.8725 + 37.5014i 107.420 −36.5256 −144.759 69.7122i −13.6719 + 59.9004i −479.421 + 230.877i
4.7 −2.14533 + 2.69016i 12.6069 + 15.8086i 4.48615 + 19.6551i 43.8967 + 21.1395i −69.5737 137.995 −161.703 77.8719i −36.9042 + 161.688i −151.042 + 72.7379i
4.8 −1.82630 + 2.29011i −7.90210 9.90892i 5.21144 + 22.8328i −7.91873 3.81346i 37.1242 25.4167 −146.258 70.4342i 18.3291 80.3048i 23.1953 11.1702i
4.9 −0.245859 + 0.308297i 4.65960 + 5.84295i 7.08607 + 31.0461i −95.9791 46.2211i −2.94697 −56.9405 −22.6824 10.9233i 41.6444 182.456i 37.8471 18.2262i
4.10 0.186655 0.234058i −17.0687 21.4035i 7.10073 + 31.1103i −36.3074 17.4847i −8.19562 50.2807 17.2382 + 8.30146i −112.696 + 493.753i −10.8694 + 5.23442i
4.11 0.240368 0.301412i 8.08269 + 10.1354i 7.08760 + 31.0528i 33.3160 + 16.0441i 4.99775 −195.731 22.1783 + 10.6805i 16.6767 73.0652i 12.8440 6.18535i
4.12 1.97079 2.47129i −9.09722 11.4076i 4.89739 + 21.4569i 83.4030 + 40.1648i −46.1201 −19.6284 153.810 + 74.0710i 6.69973 29.3534i 263.629 126.957i
4.13 3.11467 3.90567i 2.22486 + 2.78989i 1.56755 + 6.86790i −1.98003 0.953533i 17.8261 187.548 175.733 + 84.6285i 51.2391 224.493i −9.89133 + 4.76342i
4.14 3.56453 4.46978i 18.2022 + 22.8249i −0.152384 0.667640i −27.3877 13.1892i 166.905 −9.91522 161.301 + 77.6786i −135.581 + 594.021i −156.577 + 75.4035i
4.15 4.37339 5.48406i −9.64274 12.0916i −3.82767 16.7701i −28.9111 13.9228i −108.483 −153.633 93.5232 + 45.0384i 0.847858 3.71471i −202.793 + 97.6600i
4.16 5.75093 7.21143i 8.62525 + 10.8157i −11.8110 51.7472i 90.5645 + 43.6136i 127.600 −71.2640 −175.165 84.3549i 11.4876 50.3307i 835.346 402.282i
4.17 6.33304 7.94138i 4.01851 + 5.03906i −15.8374 69.3884i −53.4066 25.7193i 65.4665 −5.04361 −358.491 172.640i 44.8289 196.408i −542.472 + 261.241i
4.18 6.45965 8.10014i −16.9351 21.2360i −16.7646 73.4505i 36.1651 + 17.4162i −281.410 237.288 −404.550 194.821i −110.096 + 482.362i 374.687 180.440i
11.1 −7.04063 8.82868i 3.19853 4.01083i −21.2543 + 93.1212i 78.2981 37.7064i −57.9300 141.803 646.213 311.200i 48.2164 + 211.250i −884.165 425.791i
11.2 −6.31600 7.92001i −18.2652 + 22.9039i −15.7141 + 68.8479i −46.4718 + 22.3797i 296.762 −129.944 352.466 169.739i −136.896 599.780i 470.763 + 226.708i
See next 80 embeddings (of 108 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 41.18 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.e even 7 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 43.6.e.a 108
43.e even 7 1 inner 43.6.e.a 108

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.6.e.a 108 1.a even 1 1 trivial
43.6.e.a 108 43.e even 7 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{6}^{\mathrm{new}}(43, [\chi])$$.