Properties

Label 74.6.f.a
Level $74$
Weight $6$
Character orbit 74.f
Analytic conductor $11.868$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 74 = 2 \cdot 37 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 74.f (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(11.8684026662\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(8\) over \(\Q(\zeta_{9})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48q - 3q^{3} - 72q^{5} + 114q^{7} - 1536q^{8} - 93q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 48q - 3q^{3} - 72q^{5} + 114q^{7} - 1536q^{8} - 93q^{9} - 648q^{10} - 9q^{11} - 48q^{12} + 288q^{13} - 660q^{14} - 3678q^{15} - 2475q^{17} + 744q^{18} + 945q^{19} - 1152q^{20} - 2073q^{21} - 60q^{22} - 5739q^{23} - 192q^{24} - 3834q^{25} - 5304q^{26} - 1077q^{27} - 5232q^{28} - 11640q^{29} - 14712q^{30} - 20394q^{31} - 62013q^{33} + 7812q^{34} + 27999q^{35} + 62208q^{36} + 13605q^{37} + 60696q^{38} + 25590q^{39} + 9216q^{40} - 18267q^{41} - 8292q^{42} + 2208q^{43} - 240q^{44} - 48132q^{45} - 40308q^{46} - 663q^{47} + 58326q^{49} + 61776q^{50} + 8748q^{51} - 31536q^{52} + 20940q^{53} + 37836q^{54} - 158868q^{55} + 13632q^{56} + 164910q^{57} - 28068q^{58} + 111258q^{59} - 31824q^{60} + 88758q^{61} + 116100q^{62} - 202488q^{63} - 98304q^{64} + 155997q^{65} - 53532q^{66} + 52503q^{67} + 32544q^{68} + 298758q^{69} - 5760q^{70} + 160434q^{71} + 11904q^{72} - 477210q^{73} + 8844q^{74} + 808230q^{75} + 15120q^{76} - 510765q^{77} + 26004q^{78} - 378768q^{79} + 82944q^{80} + 55476q^{81} - 149064q^{82} - 230082q^{83} - 42336q^{84} + 55653q^{85} - 149904q^{86} - 453051q^{87} - 576q^{88} + 492789q^{89} + 12276q^{90} + 220839q^{91} + 176160q^{92} - 109767q^{93} - 6768q^{94} - 630843q^{95} + 6144q^{96} - 632190q^{97} - 194628q^{98} - 414765q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −3.75877 + 1.36808i −23.1101 8.41139i 12.2567 10.2846i 6.03710 + 34.2381i 98.3730 19.4454 + 110.280i −32.0000 + 55.4256i 277.176 + 232.578i −69.5325 120.434i
7.2 −3.75877 + 1.36808i −20.0307 7.29057i 12.2567 10.2846i −15.8864 90.0960i 85.2648 −21.3019 120.809i −32.0000 + 55.4256i 161.927 + 135.873i 182.972 + 316.916i
7.3 −3.75877 + 1.36808i −13.8557 5.04306i 12.2567 10.2846i 11.2687 + 63.9081i 58.9797 −40.1269 227.571i −32.0000 + 55.4256i −19.6009 16.4471i −129.788 224.799i
7.4 −3.75877 + 1.36808i −2.35676 0.857791i 12.2567 10.2846i −1.87431 10.6298i 10.0321 20.8226 + 118.091i −32.0000 + 55.4256i −181.330 152.154i 21.5875 + 37.3906i
7.5 −3.75877 + 1.36808i 7.77948 + 2.83150i 12.2567 10.2846i −14.8075 83.9778i −33.1150 8.29469 + 47.0415i −32.0000 + 55.4256i −133.646 112.142i 170.547 + 295.395i
7.6 −3.75877 + 1.36808i 9.99337 + 3.63729i 12.2567 10.2846i 0.180282 + 1.02243i −42.5389 −23.4499 132.991i −32.0000 + 55.4256i −99.5112 83.4998i −2.07641 3.59644i
7.7 −3.75877 + 1.36808i 13.8016 + 5.02336i 12.2567 10.2846i 19.0297 + 107.923i −58.7493 6.51125 + 36.9271i −32.0000 + 55.4256i −20.8996 17.5369i −219.175 379.623i
7.8 −3.75877 + 1.36808i 27.4525 + 9.99188i 12.2567 10.2846i −2.40304 13.6283i −116.857 22.7293 + 128.904i −32.0000 + 55.4256i 467.651 + 392.406i 27.6772 + 47.9382i
9.1 0.694593 + 3.93923i −4.94687 + 28.0551i −15.0351 + 5.47232i 9.88188 + 8.29188i −113.952 −125.671 105.450i −32.0000 55.4256i −534.272 194.459i −25.7998 + 44.6865i
9.2 0.694593 + 3.93923i −2.94557 + 16.7052i −15.0351 + 5.47232i 46.4233 + 38.9537i −67.8514 193.520 + 162.382i −32.0000 55.4256i −42.0404 15.3014i −121.203 + 209.929i
9.3 0.694593 + 3.93923i −2.84323 + 16.1248i −15.0351 + 5.47232i −37.4537 31.4274i −65.4941 −21.7905 18.2844i −32.0000 55.4256i −23.5794 8.58219i 97.7847 169.368i
9.4 0.694593 + 3.93923i 0.515998 2.92637i −15.0351 + 5.47232i 33.8915 + 28.4384i 11.8861 −11.2213 9.41575i −32.0000 55.4256i 220.048 + 80.0909i −88.4845 + 153.260i
9.5 0.694593 + 3.93923i 0.677795 3.84397i −15.0351 + 5.47232i −48.1268 40.3832i 15.6131 36.5328 + 30.6547i −32.0000 55.4256i 214.029 + 77.9001i 125.650 217.633i
9.6 0.694593 + 3.93923i 1.57867 8.95310i −15.0351 + 5.47232i 22.3419 + 18.7471i 36.3648 −145.430 122.031i −32.0000 55.4256i 150.680 + 54.8429i −58.3305 + 101.031i
9.7 0.694593 + 3.93923i 3.82058 21.6676i −15.0351 + 5.47232i −48.1841 40.4312i 88.0075 134.280 + 112.674i −32.0000 55.4256i −226.543 82.4550i 125.800 217.891i
9.8 0.694593 + 3.93923i 4.40867 25.0028i −15.0351 + 5.47232i 68.9775 + 57.8790i 101.554 55.3841 + 46.4727i −32.0000 55.4256i −377.359 137.347i −180.088 + 311.921i
33.1 0.694593 3.93923i −4.94687 28.0551i −15.0351 5.47232i 9.88188 8.29188i −113.952 −125.671 + 105.450i −32.0000 + 55.4256i −534.272 + 194.459i −25.7998 44.6865i
33.2 0.694593 3.93923i −2.94557 16.7052i −15.0351 5.47232i 46.4233 38.9537i −67.8514 193.520 162.382i −32.0000 + 55.4256i −42.0404 + 15.3014i −121.203 209.929i
33.3 0.694593 3.93923i −2.84323 16.1248i −15.0351 5.47232i −37.4537 + 31.4274i −65.4941 −21.7905 + 18.2844i −32.0000 + 55.4256i −23.5794 + 8.58219i 97.7847 + 169.368i
33.4 0.694593 3.93923i 0.515998 + 2.92637i −15.0351 5.47232i 33.8915 28.4384i 11.8861 −11.2213 + 9.41575i −32.0000 + 55.4256i 220.048 80.0909i −88.4845 153.260i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 71.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.f even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 74.6.f.a 48
37.f even 9 1 inner 74.6.f.a 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.6.f.a 48 1.a even 1 1 trivial
74.6.f.a 48 37.f even 9 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(18\!\cdots\!21\)\( T_{3}^{39} + \)\(53\!\cdots\!63\)\( T_{3}^{38} + \)\(35\!\cdots\!26\)\( T_{3}^{37} + \)\(21\!\cdots\!39\)\( T_{3}^{36} + \)\(30\!\cdots\!51\)\( T_{3}^{35} + \)\(33\!\cdots\!60\)\( T_{3}^{34} + \)\(69\!\cdots\!19\)\( T_{3}^{33} + \)\(18\!\cdots\!49\)\( T_{3}^{32} + \)\(33\!\cdots\!59\)\( T_{3}^{31} + \)\(15\!\cdots\!46\)\( T_{3}^{30} + \)\(20\!\cdots\!51\)\( T_{3}^{29} - \)\(45\!\cdots\!87\)\( T_{3}^{28} - \)\(77\!\cdots\!18\)\( T_{3}^{27} - \)\(10\!\cdots\!67\)\( T_{3}^{26} - \)\(85\!\cdots\!80\)\( T_{3}^{25} + \)\(51\!\cdots\!76\)\( T_{3}^{24} + \)\(62\!\cdots\!53\)\( T_{3}^{23} + \)\(78\!\cdots\!41\)\( T_{3}^{22} - \)\(12\!\cdots\!73\)\( T_{3}^{21} - \)\(26\!\cdots\!69\)\( T_{3}^{20} - \)\(15\!\cdots\!83\)\( T_{3}^{19} + \)\(37\!\cdots\!62\)\( T_{3}^{18} + \)\(80\!\cdots\!61\)\( T_{3}^{17} - \)\(48\!\cdots\!22\)\( T_{3}^{16} + \)\(23\!\cdots\!33\)\( T_{3}^{15} + \)\(70\!\cdots\!74\)\( T_{3}^{14} - \)\(18\!\cdots\!05\)\( T_{3}^{13} + \)\(21\!\cdots\!94\)\( T_{3}^{12} - \)\(14\!\cdots\!57\)\( T_{3}^{11} + \)\(64\!\cdots\!11\)\( T_{3}^{10} - \)\(17\!\cdots\!10\)\( T_{3}^{9} + \)\(24\!\cdots\!79\)\( T_{3}^{8} + \)\(64\!\cdots\!04\)\( T_{3}^{7} - \)\(27\!\cdots\!79\)\( T_{3}^{6} + \)\(46\!\cdots\!18\)\( T_{3}^{5} + \)\(11\!\cdots\!04\)\( T_{3}^{4} - \)\(56\!\cdots\!72\)\( T_{3}^{3} + \)\(12\!\cdots\!52\)\( T_{3}^{2} - \)\(54\!\cdots\!44\)\( T_{3} + \)\(10\!\cdots\!24\)\( \)">\(T_{3}^{48} + \cdots\) acting on \(S_{6}^{\mathrm{new}}(74, [\chi])\).