Properties

Label 43.5.f.a
Level 43
Weight 5
Character orbit 43.f
Analytic conductor 4.445
Analytic rank 0
Dimension 78
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 43 \)
Weight: \( k \) = \( 5 \)
Character orbit: \([\chi]\) = 43.f (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.44490841261\)
Analytic rank: \(0\)
Dimension: \(78\)
Relative dimension: \(13\) over \(\Q(\zeta_{14})\)
Coefficient ring index: multiple of None
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.

\(\operatorname{Tr}(f)(q) = \) \( 78q - 7q^{2} - 7q^{3} + 69q^{4} - 7q^{5} - 140q^{6} - 343q^{8} + 486q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 78q - 7q^{2} - 7q^{3} + 69q^{4} - 7q^{5} - 140q^{6} - 343q^{8} + 486q^{9} - 189q^{10} - 26q^{11} - 119q^{12} + 608q^{13} + 45q^{14} - 797q^{15} + 593q^{16} - 188q^{17} - 574q^{18} + 1421q^{19} - 2107q^{20} + 2952q^{21} - 2359q^{22} + 652q^{23} - 253q^{24} + 1264q^{25} - 2023q^{26} - 7q^{27} + 1785q^{28} - 2317q^{29} - 252q^{30} + 5714q^{31} - 6517q^{32} + 9023q^{33} - 679q^{34} + 1374q^{35} - 8544q^{36} + 95q^{38} - 10339q^{39} - 5901q^{40} - 3611q^{41} + 7001q^{43} + 8148q^{44} + 8862q^{45} - 2660q^{46} + 6268q^{47} + 11081q^{48} - 1074q^{49} - 17353q^{51} + 8277q^{52} - 200q^{53} - 8147q^{54} + 9863q^{55} + 15927q^{56} - 25075q^{57} + 13045q^{58} + 1468q^{59} - 357q^{60} - 8239q^{61} + 10073q^{62} - 875q^{63} + 1173q^{64} - 19159q^{65} - 24551q^{66} + 42658q^{67} + 3910q^{68} - 903q^{69} + 37870q^{70} + 8960q^{71} - 40586q^{72} + 13342q^{73} + 37116q^{74} - 44996q^{75} - 60438q^{76} + 2625q^{77} - 13562q^{78} - 48318q^{79} + 1048q^{81} - 57211q^{82} + 17014q^{83} - 50382q^{84} + 21461q^{86} - 6174q^{87} + 34909q^{88} + 51590q^{89} - 6494q^{90} + 35840q^{91} + 105420q^{92} - 38115q^{94} + 65915q^{95} + 101606q^{96} - 18427q^{97} + 63455q^{98} + 36377q^{99} + O(q^{100}) \)

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 −3.00827 6.24674i −2.99008 + 6.20897i −19.9962 + 25.0745i 23.3607 + 5.33193i 47.7808 14.3857i 108.636 + 24.7953i 20.8920 + 26.1977i −36.9682 161.968i
2.2 −2.76035 5.73192i 3.68382 7.64953i −15.2596 + 19.1349i −21.3617 4.87567i −54.0152 26.8619i 52.5622 + 11.9970i 5.55784 + 6.96931i 31.0188 + 135.902i
2.3 −1.63624 3.39769i −3.03818 + 6.30885i 1.10883 1.39043i −20.2554 4.62315i 26.4067 82.0298i −65.3642 14.9189i 19.9317 + 24.9935i 17.4346 + 76.3860i
2.4 −1.48659 3.08695i −6.44806 + 13.3895i 2.65657 3.33123i −2.46437 0.562476i 50.9184 87.1493i −67.6781 15.4471i −87.1994 109.345i 1.92718 + 8.44355i
2.5 −1.40281 2.91297i 3.45350 7.17127i 3.45835 4.33663i 33.8936 + 7.73599i −25.7343 18.1404i −67.9172 15.5017i 11.0023 + 13.7964i −25.0116 109.583i
2.6 −0.495110 1.02811i 6.66668 13.8435i 9.16397 11.4913i −12.7517 2.91049i −17.5333 56.8875i −34.1514 7.79484i −96.6955 121.252i 3.32119 + 14.5511i
2.7 0.149318 + 0.310062i 0.205202 0.426106i 9.90199 12.4167i −43.8253 10.0028i 0.162760 47.2346i 10.6967 + 2.44146i 50.3632 + 63.1535i −3.44241 15.0822i
2.8 0.607375 + 1.26123i −1.76429 + 3.66358i 8.75405 10.9772i 19.0995 + 4.35934i −5.69219 6.49631i 40.9979 + 9.35750i 40.1936 + 50.4011i 6.10245 + 26.7366i
2.9 1.76935 + 3.67410i −7.08052 + 14.7028i −0.392560 + 0.492254i −5.20176 1.18727i −66.5476 28.1577i 61.1080 + 13.9475i −115.537 144.879i −4.84161 21.2125i
2.10 1.83585 + 3.81219i 5.21899 10.8373i −1.18660 + 1.48795i −1.25492 0.286428i 50.8953 55.1541i 58.1513 + 13.2726i −39.7075 49.7916i −1.21194 5.30985i
2.11 2.55909 + 5.31401i 3.37270 7.00348i −11.7139 + 14.6887i 17.8072 + 4.06438i 45.8475 82.2284i −16.0293 3.65859i 12.8291 + 16.0872i 23.9721 + 105.029i
2.12 2.81569 + 5.84684i −1.11756 + 2.32063i −16.2816 + 20.4164i −37.2979 8.51300i −16.7150 16.8082i −63.9865 14.6045i 46.3663 + 58.1415i −55.2451 242.045i
2.13 3.21422 + 6.67439i −3.25251 + 6.75392i −24.2405 + 30.3966i 40.7370 + 9.29795i −55.5326 88.1833i −165.237 37.7142i 15.4662 + 19.3939i 68.8794 + 301.780i
8.1 −7.22912 1.65000i −7.05737 + 1.61080i 35.1222 + 16.9140i 14.3521 + 11.4455i 53.6764 6.07845i −133.238 106.254i −25.7666 + 12.4086i −84.8684 106.422i
8.2 −6.68339 1.52544i 15.1977 3.46877i 27.9252 + 13.4481i 1.95321 + 1.55764i −106.863 79.0264i −80.3660 64.0897i 145.958 70.2899i −10.6780 13.3898i
8.3 −4.81647 1.09933i 3.75314 0.856629i 7.57434 + 3.64761i −6.30959 5.03173i −19.0186 27.6828i 29.3285 + 23.3887i −59.6263 + 28.7145i 24.8584 + 31.1715i
8.4 −4.61843 1.05413i −9.66855 + 2.20678i 5.80324 + 2.79469i −23.4239 18.6799i 46.9798 27.2594i 35.4032 + 28.2331i 15.6326 7.52825i 88.4905 + 110.964i
8.5 −2.98221 0.680671i 8.81268 2.01144i −5.98522 2.88233i 26.0327 + 20.7604i −27.6504 60.4753i 54.1521 + 43.1848i 0.638992 0.307722i −63.5041 79.6316i
8.6 −2.02384 0.461929i −16.6744 + 3.80583i −10.5329 5.07240i 21.9239 + 17.4837i 35.5044 46.4861i 44.9418 + 35.8399i 190.574 91.7754i −36.2943 45.5116i
8.7 −0.600398 0.137037i 11.2678 2.57179i −14.0738 6.77759i −38.2585 30.5101i −7.11757 7.54973i 15.2248 + 12.1414i 47.3697 22.8121i 18.7893 + 23.5611i
See all 78 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 39.13
Significant digits:
Format:

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 43.5.f.a 78
43.f odd 14 1 inner 43.5.f.a 78
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.5.f.a 78 1.a even 1 1 trivial
43.5.f.a 78 43.f odd 14 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(43, [\chi])\).

Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database