Properties

Label 75.6.i.a
Level $75$
Weight $6$
Character orbit 75.i
Analytic conductor $12.029$
Analytic rank $0$
Dimension $96$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 75.i (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.0287864860\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(24\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 96q + 352q^{4} - 120q^{5} + 72q^{6} - 1230q^{8} + 1944q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 96q + 352q^{4} - 120q^{5} + 72q^{6} - 1230q^{8} + 1944q^{9} - 890q^{10} + 474q^{11} + 2748q^{14} - 780q^{16} + 1910q^{17} - 5522q^{19} - 19220q^{20} + 1764q^{21} + 6170q^{22} + 21980q^{23} + 13824q^{24} + 11790q^{25} - 11028q^{26} - 9120q^{28} - 24304q^{29} - 22860q^{30} + 1566q^{31} - 3690q^{33} + 24224q^{34} + 10250q^{35} - 28512q^{36} + 16490q^{37} + 68730q^{38} - 12168q^{39} - 106480q^{40} + 26126q^{41} - 43560q^{42} - 10814q^{44} + 9720q^{45} + 22956q^{46} + 66440q^{47} - 123480q^{49} + 329620q^{50} - 83232q^{51} + 251240q^{52} - 91690q^{53} - 5832q^{54} - 123470q^{55} + 116100q^{56} - 483290q^{58} - 97428q^{59} + 119610q^{60} + 91560q^{61} - 138410q^{62} + 47790q^{63} + 127738q^{64} - 235130q^{65} + 118296q^{66} + 163960q^{67} - 73332q^{69} + 209520q^{70} + 155672q^{71} - 99630q^{72} + 18980q^{73} + 100588q^{74} - 464032q^{76} + 119560q^{77} - 112020q^{79} - 66300q^{80} - 157464q^{81} + 148010q^{83} - 61128q^{84} + 346980q^{85} - 295956q^{86} - 371160q^{87} - 189090q^{88} - 494082q^{89} - 144180q^{90} - 79494q^{91} + 599810q^{92} + 234622q^{94} + 605800q^{95} + 210744q^{96} + 372140q^{97} + 891060q^{98} + 25596q^{99} + O(q^{100}) \)

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 −10.3671 3.36849i −5.29007 + 7.28115i 70.2423 + 51.0340i −12.9290 54.3860i 79.3693 57.6652i 32.5585i −351.272 483.484i −25.0304 77.0356i −49.1617 + 607.379i
4.2 −9.95394 3.23423i 5.29007 7.28115i 62.7322 + 45.5776i 5.06496 + 55.6718i −76.2060 + 55.3669i 213.332i −280.164 385.613i −25.0304 77.0356i 129.639 570.535i
4.3 −8.71565 2.83189i 5.29007 7.28115i 42.0544 + 30.5543i −15.1679 53.8046i −66.7257 + 48.4791i 6.33250i −107.635 148.146i −25.0304 77.0356i −20.1705 + 511.896i
4.4 −8.39483 2.72764i −5.29007 + 7.28115i 37.1445 + 26.9871i 47.9744 + 28.6959i 64.2696 46.6946i 34.5534i −72.1854 99.3547i −25.0304 77.0356i −324.465 371.754i
4.5 −7.50815 2.43955i −5.29007 + 7.28115i 24.5324 + 17.8239i −44.0632 + 34.4011i 57.4814 41.7627i 98.6919i 7.77820 + 10.7058i −25.0304 77.0356i 414.756 150.794i
4.6 −6.37631 2.07179i 5.29007 7.28115i 10.4765 + 7.61160i −54.0427 + 14.2963i −48.8161 + 35.4670i 23.6018i 75.0733 + 103.329i −25.0304 77.0356i 374.212 + 20.8074i
4.7 −4.73235 1.53763i −5.29007 + 7.28115i −5.85773 4.25589i −49.1135 26.6995i 36.2302 26.3228i 228.046i 114.769 + 157.966i −25.0304 77.0356i 191.368 + 201.870i
4.8 −4.43125 1.43980i 5.29007 7.28115i −8.32560 6.04890i 23.0763 + 50.9165i −33.9250 + 24.6480i 165.435i 115.821 + 159.414i −25.0304 77.0356i −28.9472 258.849i
4.9 −3.40554 1.10653i 5.29007 7.28115i −15.5152 11.2725i 54.3939 12.8960i −26.0723 + 18.9427i 236.745i 107.716 + 148.259i −25.0304 77.0356i −199.510 16.2706i
4.10 −2.43009 0.789586i −5.29007 + 7.28115i −20.6066 14.9716i −22.3194 + 51.2528i 18.6045 13.5169i 116.776i 86.3149 + 118.802i −25.0304 77.0356i 94.7066 106.926i
4.11 −1.31908 0.428594i 5.29007 7.28115i −24.3323 17.6784i −5.76663 55.6035i −10.0987 + 7.33711i 70.8359i 50.6068 + 69.6542i −25.0304 77.0356i −16.2247 + 75.8168i
4.12 −0.454745 0.147756i −5.29007 + 7.28115i −25.7036 18.6747i 33.1741 + 44.9942i 3.48146 2.52943i 200.826i 17.9228 + 24.6686i −25.0304 77.0356i −8.43759 25.3625i
4.13 −0.222946 0.0724396i −5.29007 + 7.28115i −25.8441 18.7768i −31.5757 46.1300i 1.70684 1.24009i 161.430i 8.81087 + 12.1271i −25.0304 77.0356i 3.69805 + 12.5718i
4.14 1.67454 + 0.544090i −5.29007 + 7.28115i −23.3805 16.9869i 55.3000 8.17979i −12.8200 + 9.31428i 116.620i −63.0265 86.7486i −25.0304 77.0356i 97.0524 + 16.3908i
4.15 2.74584 + 0.892176i 5.29007 7.28115i −19.1449 13.9096i −10.5312 + 54.9008i 21.0217 15.2732i 106.453i −94.4636 130.018i −25.0304 77.0356i −77.8982 + 141.353i
4.16 3.94639 + 1.28226i 5.29007 7.28115i −11.9588 8.68855i 54.1701 + 13.8059i 30.2130 21.9510i 87.4923i −114.101 157.047i −25.0304 77.0356i 196.073 + 123.943i
4.17 4.90117 + 1.59249i −5.29007 + 7.28115i −4.40306 3.19901i −30.2724 + 46.9955i −37.5227 + 27.2618i 117.152i −113.417 156.105i −25.0304 77.0356i −223.210 + 182.125i
4.18 5.18267 + 1.68395i 5.29007 7.28115i −1.86417 1.35440i −41.4891 37.4654i 39.6778 28.8276i 121.054i −109.879 151.235i −25.0304 77.0356i −151.934 264.036i
4.19 6.05984 + 1.96896i −5.29007 + 7.28115i 6.95629 + 5.05404i 12.4663 54.4940i −46.3933 + 33.7067i 98.3688i −87.6432 120.630i −25.0304 77.0356i 182.840 305.679i
4.20 7.19120 + 2.33656i 5.29007 7.28115i 20.3652 + 14.7962i −52.6473 18.7952i 55.0548 39.9996i 237.709i −30.3428 41.7633i −25.0304 77.0356i −334.681 258.174i
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 64.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.e even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 75.6.i.a 96
25.e even 10 1 inner 75.6.i.a 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.6.i.a 96 1.a even 1 1 trivial
75.6.i.a 96 25.e even 10 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(75, [\chi])\).