# Properties

 Label 43.7.f.a Level $43$ Weight $7$ Character orbit 43.f Analytic conductor $9.892$ Analytic rank $0$ Dimension $126$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$43$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 43.f (of order $$14$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$9.89232559565$$ Analytic rank: $$0$$ Dimension: $$126$$ Relative dimension: $$21$$ over $$\Q(\zeta_{14})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$126q - 7q^{2} - 7q^{3} + 725q^{4} - 7q^{5} + 244q^{6} + 2233q^{8} + 6312q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$126q - 7q^{2} - 7q^{3} + 725q^{4} - 7q^{5} + 244q^{6} + 2233q^{8} + 6312q^{9} + 1955q^{10} - 985q^{11} - 455q^{12} - 4485q^{13} - 6771q^{14} + 5017q^{15} - 41935q^{16} - 10409q^{17} - 5110q^{18} - 33397q^{19} - 24507q^{20} - 5448q^{21} + 25193q^{22} - 29017q^{23} - 1885q^{24} - 2080q^{25} + 34713q^{26} - 7q^{27} + 28665q^{28} + 17353q^{29} - 290052q^{30} + 13547q^{31} + 458283q^{32} - 94801q^{33} - 191527q^{34} + 183008q^{35} + 956352q^{36} + 304607q^{38} - 381031q^{39} + 66419q^{40} + 313447q^{41} - 323228q^{43} + 281588q^{44} - 1237551q^{45} + 80948q^{46} - 180133q^{47} - 762055q^{48} - 1861754q^{49} + 999215q^{51} + 471797q^{52} + 1030011q^{53} - 92099q^{54} + 324233q^{55} - 842713q^{56} + 331037q^{57} + 849461q^{58} - 369609q^{59} + 553323q^{60} - 594727q^{61} - 886711q^{62} + 1914136q^{63} - 3250219q^{64} + 1473521q^{65} - 427559q^{66} + 122535q^{67} - 602346q^{68} + 4320057q^{69} - 1694994q^{70} - 770707q^{71} - 391202q^{72} + 476945q^{73} - 3943812q^{74} + 2061367q^{75} - 2044462q^{76} + 3016216q^{77} + 239350q^{78} + 5077268q^{79} + 2388946q^{81} + 5820437q^{82} - 1324235q^{83} + 3759618q^{84} - 4967947q^{86} - 3537894q^{87} - 7289555q^{88} - 4331495q^{89} - 12825206q^{90} + 1802976q^{91} - 12038260q^{92} + 6249341q^{94} - 5313259q^{95} + 7069598q^{96} + 788975q^{97} + 4724783q^{98} - 2216971q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2.1 −6.68851 13.8888i 4.36630 9.06671i −108.260 + 135.754i 1.12402 + 0.256549i −155.130 114.871i 1647.71 + 376.078i 391.383 + 490.779i −3.95482 17.3272i
2.2 −5.62328 11.6769i −22.1545 + 46.0043i −64.8243 + 81.2871i 43.4728 + 9.92237i 661.767 306.680i 505.038 + 115.272i −1171.05 1468.45i −128.597 563.421i
2.3 −5.10721 10.6052i −7.80492 + 16.2071i −46.4840 + 58.2891i −223.373 50.9834i 211.741 193.213i 121.123 + 27.6454i 252.771 + 316.965i 600.121 + 2629.30i
2.4 −5.02991 10.4447i 20.6257 42.8298i −43.8887 + 55.0347i 127.013 + 28.9898i −551.091 619.099i 72.2435 + 16.4891i −954.446 1196.84i −336.072 1472.43i
2.5 −4.79242 9.95156i −6.97408 + 14.4818i −36.1629 + 45.3469i 117.045 + 26.7147i 177.540 466.480i −64.6015 14.7449i 293.438 + 367.960i −295.075 1292.81i
2.6 −4.35509 9.04344i 15.2129 31.5900i −22.9137 + 28.7328i −93.5256 21.3466i −351.936 345.453i −266.657 60.8626i −311.968 391.196i 214.266 + 938.759i
2.7 −3.64714 7.57336i 2.80632 5.82738i −4.15080 + 5.20494i 190.781 + 43.5445i −54.3678 593.422i −469.925 107.257i 428.441 + 537.248i −366.026 1603.66i
2.8 −2.72088 5.64997i −8.85233 + 18.3821i 15.3844 19.2915i −52.8314 12.0584i 127.944 8.48034i −542.136 123.739i 194.988 + 244.507i 75.6182 + 331.305i
2.9 −1.12993 2.34633i 12.1637 25.2581i 35.6748 44.7348i −135.516 30.9307i −73.0081 261.823i −307.765 70.2454i −35.4943 44.5085i 80.5506 + 352.915i
2.10 −0.716329 1.48747i 3.40007 7.06031i 38.2039 47.9062i 69.5466 + 15.8736i −12.9376 168.130i −201.638 46.0227i 416.237 + 521.944i −26.2068 114.819i
2.11 −0.546272 1.13435i −16.4415 + 34.1411i 38.9150 48.7979i −89.4697 20.4209i 47.7093 356.575i −155.169 35.4164i −440.768 552.706i 25.7105 + 112.645i
2.12 −0.0322332 0.0669329i −17.7426 + 36.8430i 39.8999 50.0329i 157.975 + 36.0568i 3.03791 203.317i −9.27029 2.11588i −588.078 737.427i −2.67865 11.7360i
2.13 1.11483 + 2.31496i 18.6979 38.8265i 35.7871 44.8756i 137.550 + 31.3949i 110.727 167.161i 304.102 + 69.4092i −703.365 881.992i 80.6666 + 353.424i
2.14 2.61426 + 5.42856i 1.01637 2.11052i 17.2684 21.6539i −102.842 23.4731i 14.1142 486.006i 538.641 + 122.941i 451.103 + 565.665i −141.431 619.650i
2.15 2.79658 + 5.80715i −16.1345 + 33.5037i 14.0012 17.5569i −177.830 40.5886i −239.682 522.371i 543.278 + 124.000i −407.649 511.176i −261.612 1146.20i
2.16 3.53371 + 7.33783i −9.42884 + 19.5792i −1.45324 + 1.82231i 197.653 + 45.1129i −176.987 222.322i 489.664 + 111.763i 160.083 + 200.737i 367.416 + 1609.76i
2.17 3.58164 + 7.43735i 3.11543 6.46926i −2.58264 + 3.23853i 50.0201 + 11.4168i 59.2725 578.950i 481.727 + 109.951i 422.379 + 529.646i 94.2435 + 412.908i
2.18 4.32423 + 8.97936i 20.4277 42.4185i −22.0266 + 27.6205i −183.723 41.9336i 469.225 28.9921i 278.591 + 63.5867i −927.514 1163.07i −417.925 1831.05i
2.19 5.62961 + 11.6900i −16.3881 + 34.0303i −65.0604 + 81.5832i 15.2899 + 3.48982i −490.074 124.671i −510.398 116.495i −434.969 545.434i 45.2802 + 198.386i
2.20 5.84606 + 12.1395i 12.7263 26.4263i −73.2871 + 91.8991i 119.457 + 27.2654i 395.200 91.2949i −703.346 160.534i −81.8698 102.661i 367.368 + 1609.55i
See next 80 embeddings (of 126 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 39.21 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.f odd 14 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 43.7.f.a 126
43.f odd 14 1 inner 43.7.f.a 126

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.7.f.a 126 1.a even 1 1 trivial
43.7.f.a 126 43.f odd 14 1 inner

## Hecke kernels

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