Properties

Label 74.5.i.b
Level $74$
Weight $5$
Character orbit 74.i
Analytic conductor $7.649$
Analytic rank $0$
Dimension $72$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 74 = 2 \cdot 37 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 74.i (of order \(36\), degree \(12\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.64937726820\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(6\) over \(\Q(\zeta_{36})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{36}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 72q + 114q^{5} - 576q^{8} - 210q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 72q + 114q^{5} - 576q^{8} - 210q^{9} - 36q^{10} - 720q^{11} - 240q^{12} + 144q^{13} + 288q^{14} + 1662q^{15} - 984q^{17} - 840q^{18} - 792q^{19} + 912q^{20} - 1974q^{21} - 432q^{22} + 522q^{23} + 480q^{24} - 414q^{25} - 1140q^{26} - 5832q^{27} - 1968q^{28} + 2196q^{29} + 4380q^{30} + 9240q^{31} + 7800q^{33} + 2916q^{34} + 1104q^{35} + 3156q^{37} + 3216q^{38} - 11526q^{39} - 5568q^{40} - 4932q^{41} - 4584q^{42} - 7224q^{43} - 9972q^{45} - 3120q^{46} - 954q^{47} + 3456q^{48} + 30318q^{49} + 1368q^{50} + 6480q^{51} + 1152q^{52} - 366q^{53} - 9036q^{54} + 7146q^{55} + 2112q^{56} - 336q^{57} - 18876q^{58} - 8310q^{59} + 19008q^{60} + 31680q^{61} - 10668q^{62} + 8064q^{63} + 29262q^{65} + 22140q^{66} - 37086q^{67} + 1584q^{68} - 5814q^{69} + 84q^{70} - 2448q^{71} + 6720q^{72} + 11856q^{74} - 26976q^{75} - 6336q^{76} - 55248q^{77} - 43320q^{78} - 42456q^{79} + 1152q^{80} + 7428q^{81} + 14568q^{82} - 84q^{83} - 32292q^{85} + 10392q^{86} + 41196q^{87} + 7680q^{88} - 29964q^{89} - 1872q^{90} + 26778q^{91} - 20832q^{92} - 120900q^{93} - 5148q^{94} - 49374q^{95} + 7680q^{96} + 65706q^{97} + 53184q^{98} + 18006q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 1.19534 2.56343i −4.50230 + 12.3700i −5.14230 6.12836i −4.41825 + 6.30992i 26.3277 + 26.3277i −6.53988 37.0895i −21.8564 + 5.85641i −70.6956 59.3207i 10.8937 + 18.8684i
5.2 1.19534 2.56343i −3.78011 + 10.3858i −5.14230 6.12836i 17.8200 25.4496i 22.1046 + 22.1046i 6.77159 + 38.4036i −21.8564 + 5.85641i −31.5252 26.4528i −43.9370 76.1012i
5.3 1.19534 2.56343i −0.0952242 + 0.261626i −5.14230 6.12836i −15.0442 + 21.4853i 0.556834 + 0.556834i 8.44350 + 47.8855i −21.8564 + 5.85641i 61.9902 + 52.0160i 37.0930 + 64.2469i
5.4 1.19534 2.56343i 1.39360 3.82888i −5.14230 6.12836i 17.8538 25.4979i −8.14921 8.14921i −0.975650 5.53319i −21.8564 + 5.85641i 49.3314 + 41.3940i −44.0205 76.2457i
5.5 1.19534 2.56343i 4.49387 12.3468i −5.14230 6.12836i 12.5035 17.8568i −26.2784 26.2784i −0.981299 5.56523i −21.8564 + 5.85641i −70.1991 58.9041i −30.8286 53.3967i
5.6 1.19534 2.56343i 4.86126 13.3562i −5.14230 6.12836i −24.2053 + 34.5687i −28.4268 28.4268i −9.53880 54.0972i −21.8564 + 5.85641i −92.7069 77.7903i 59.6806 + 103.370i
13.1 1.62232 + 2.31691i −16.5144 + 2.91194i −2.73616 + 7.51754i −18.8912 1.65276i −33.5383 33.5383i 24.5278 20.5813i −21.8564 + 5.85641i 188.131 68.4741i −26.8182 46.4505i
13.2 1.62232 + 2.31691i −10.4684 + 1.84587i −2.73616 + 7.51754i 45.0013 + 3.93711i −21.2599 21.2599i −54.8456 + 46.0209i −21.8564 + 5.85641i 30.0659 10.9431i 63.8846 + 110.651i
13.3 1.62232 + 2.31691i −3.79524 + 0.669204i −2.73616 + 7.51754i −15.5320 1.35888i −7.70758 7.70758i 14.2036 11.9183i −21.8564 + 5.85641i −62.1591 + 22.6240i −22.0495 38.1909i
13.4 1.62232 + 2.31691i 0.340097 0.0599682i −2.73616 + 7.51754i 27.1996 + 2.37966i 0.690686 + 0.690686i 70.2106 58.9137i −21.8564 + 5.85641i −76.0030 + 27.6628i 38.6130 + 66.8797i
13.5 1.62232 + 2.31691i 4.66660 0.822847i −2.73616 + 7.51754i −26.4978 2.31826i 9.47717 + 9.47717i −50.3999 + 42.2906i −21.8564 + 5.85641i −55.0150 + 20.0238i −37.6167 65.1540i
13.6 1.62232 + 2.31691i 13.1101 2.31167i −2.73616 + 7.51754i 20.9802 + 1.83553i 26.6247 + 26.6247i −8.95601 + 7.51498i −21.8564 + 5.85641i 90.4163 32.9088i 29.7838 + 51.5871i
15.1 1.19534 + 2.56343i −4.50230 12.3700i −5.14230 + 6.12836i −4.41825 6.30992i 26.3277 26.3277i −6.53988 + 37.0895i −21.8564 5.85641i −70.6956 + 59.3207i 10.8937 18.8684i
15.2 1.19534 + 2.56343i −3.78011 10.3858i −5.14230 + 6.12836i 17.8200 + 25.4496i 22.1046 22.1046i 6.77159 38.4036i −21.8564 5.85641i −31.5252 + 26.4528i −43.9370 + 76.1012i
15.3 1.19534 + 2.56343i −0.0952242 0.261626i −5.14230 + 6.12836i −15.0442 21.4853i 0.556834 0.556834i 8.44350 47.8855i −21.8564 5.85641i 61.9902 52.0160i 37.0930 64.2469i
15.4 1.19534 + 2.56343i 1.39360 + 3.82888i −5.14230 + 6.12836i 17.8538 + 25.4979i −8.14921 + 8.14921i −0.975650 + 5.53319i −21.8564 5.85641i 49.3314 41.3940i −44.0205 + 76.2457i
15.5 1.19534 + 2.56343i 4.49387 + 12.3468i −5.14230 + 6.12836i 12.5035 + 17.8568i −26.2784 + 26.2784i −0.981299 + 5.56523i −21.8564 5.85641i −70.1991 + 58.9041i −30.8286 + 53.3967i
15.6 1.19534 + 2.56343i 4.86126 + 13.3562i −5.14230 + 6.12836i −24.2053 34.5687i −28.4268 + 28.4268i −9.53880 + 54.0972i −21.8564 5.85641i −92.7069 + 77.7903i 59.6806 103.370i
17.1 −2.31691 1.62232i −11.8311 2.08614i 2.73616 + 7.51754i 2.11625 + 24.1888i 24.0272 + 24.0272i 7.38470 + 6.19650i 5.85641 21.8564i 59.5075 + 21.6590i 34.3388 59.4765i
17.2 −2.31691 1.62232i −11.3787 2.00637i 2.73616 + 7.51754i −2.56481 29.3159i 23.1085 + 23.1085i −47.6426 39.9769i 5.85641 21.8564i 49.3345 + 17.9563i −41.6173 + 72.0832i
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 69.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.i odd 36 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 74.5.i.b 72
37.i odd 36 1 inner 74.5.i.b 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.5.i.b 72 1.a even 1 1 trivial
74.5.i.b 72 37.i odd 36 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(62\!\cdots\!30\)\( T_{3}^{61} + \)\(14\!\cdots\!60\)\( T_{3}^{60} + \)\(65\!\cdots\!04\)\( T_{3}^{59} + \)\(16\!\cdots\!31\)\( T_{3}^{58} + \)\(78\!\cdots\!70\)\( T_{3}^{57} - \)\(10\!\cdots\!68\)\( T_{3}^{56} + \)\(25\!\cdots\!40\)\( T_{3}^{55} - \)\(86\!\cdots\!76\)\( T_{3}^{54} - \)\(44\!\cdots\!64\)\( T_{3}^{53} - \)\(58\!\cdots\!47\)\( T_{3}^{52} - \)\(37\!\cdots\!12\)\( T_{3}^{51} + \)\(42\!\cdots\!29\)\( T_{3}^{50} - \)\(53\!\cdots\!34\)\( T_{3}^{49} + \)\(30\!\cdots\!22\)\( T_{3}^{48} + \)\(13\!\cdots\!60\)\( T_{3}^{47} + \)\(21\!\cdots\!32\)\( T_{3}^{46} + \)\(12\!\cdots\!60\)\( T_{3}^{45} - \)\(29\!\cdots\!55\)\( T_{3}^{44} + \)\(99\!\cdots\!74\)\( T_{3}^{43} - \)\(59\!\cdots\!92\)\( T_{3}^{42} - \)\(26\!\cdots\!12\)\( T_{3}^{41} - \)\(69\!\cdots\!67\)\( T_{3}^{40} - \)\(22\!\cdots\!00\)\( T_{3}^{39} - \)\(24\!\cdots\!56\)\( T_{3}^{38} - \)\(18\!\cdots\!66\)\( T_{3}^{37} + \)\(56\!\cdots\!81\)\( T_{3}^{36} + \)\(23\!\cdots\!60\)\( T_{3}^{35} + \)\(10\!\cdots\!26\)\( T_{3}^{34} + \)\(26\!\cdots\!86\)\( T_{3}^{33} + \)\(77\!\cdots\!18\)\( T_{3}^{32} + \)\(64\!\cdots\!70\)\( T_{3}^{31} + \)\(18\!\cdots\!25\)\( T_{3}^{30} + \)\(53\!\cdots\!08\)\( T_{3}^{29} + \)\(95\!\cdots\!02\)\( T_{3}^{28} - \)\(88\!\cdots\!48\)\( T_{3}^{27} - \)\(80\!\cdots\!28\)\( T_{3}^{26} + \)\(69\!\cdots\!90\)\( T_{3}^{25} + \)\(69\!\cdots\!33\)\( T_{3}^{24} - \)\(48\!\cdots\!60\)\( T_{3}^{23} - \)\(14\!\cdots\!31\)\( T_{3}^{22} + \)\(12\!\cdots\!30\)\( T_{3}^{21} - \)\(25\!\cdots\!13\)\( T_{3}^{20} - \)\(96\!\cdots\!14\)\( T_{3}^{19} + \)\(40\!\cdots\!57\)\( T_{3}^{18} + \)\(12\!\cdots\!34\)\( T_{3}^{17} - \)\(66\!\cdots\!01\)\( T_{3}^{16} - \)\(13\!\cdots\!82\)\( T_{3}^{15} + \)\(11\!\cdots\!17\)\( T_{3}^{14} - \)\(17\!\cdots\!78\)\( T_{3}^{13} - \)\(46\!\cdots\!57\)\( T_{3}^{12} + \)\(16\!\cdots\!86\)\( T_{3}^{11} - \)\(10\!\cdots\!62\)\( T_{3}^{10} - \)\(18\!\cdots\!76\)\( T_{3}^{9} + \)\(82\!\cdots\!92\)\( T_{3}^{8} - \)\(65\!\cdots\!88\)\( T_{3}^{7} + \)\(26\!\cdots\!40\)\( T_{3}^{6} - \)\(73\!\cdots\!52\)\( T_{3}^{5} + \)\(19\!\cdots\!40\)\( T_{3}^{4} - \)\(45\!\cdots\!24\)\( T_{3}^{3} + \)\(57\!\cdots\!36\)\( T_{3}^{2} + \)\(11\!\cdots\!32\)\( T_{3} + \)\(19\!\cdots\!56\)\( \)">\(T_{3}^{72} + \cdots\) acting on \(S_{5}^{\mathrm{new}}(74, [\chi])\).