Properties

Label 37.7.g.a
Level $37$
Weight $7$
Character orbit 37.g
Analytic conductor $8.512$
Analytic rank $0$
Dimension $72$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 37 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 37.g (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.51200109393\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(18\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 72q - 2q^{2} - 6q^{3} + 264q^{4} + 52q^{5} - 132q^{6} - 2q^{7} - 1968q^{8} + 8094q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 72q - 2q^{2} - 6q^{3} + 264q^{4} + 52q^{5} - 132q^{6} - 2q^{7} - 1968q^{8} + 8094q^{9} + 1864q^{10} + 1178q^{12} + 2898q^{13} + 10168q^{14} + 1418q^{15} + 23668q^{16} - 16734q^{17} + 30222q^{18} - 20756q^{19} + 27910q^{20} - 22326q^{21} + 40188q^{22} + 23388q^{23} + 51426q^{24} - 54060q^{25} + 31064q^{26} - 84486q^{28} - 101426q^{29} - 280062q^{30} - 112740q^{31} + 66416q^{32} + 27376q^{33} + 45094q^{34} + 152710q^{35} - 23168q^{37} - 67600q^{38} + 386674q^{39} - 473862q^{40} - 566622q^{41} - 569640q^{42} - 33332q^{43} + 667736q^{44} + 399362q^{45} + 303310q^{46} + 352424q^{47} - 402226q^{49} - 190836q^{50} - 525822q^{51} - 290936q^{52} + 365094q^{53} + 1191884q^{54} - 343392q^{55} + 2217442q^{56} + 73378q^{57} - 581028q^{58} - 576604q^{59} + 2105904q^{60} - 213462q^{61} - 423156q^{62} - 1483688q^{63} + 2273880q^{65} - 846680q^{66} + 1591698q^{67} - 3165948q^{68} + 2464258q^{69} + 2158358q^{70} - 1388114q^{71} + 1672988q^{72} - 3228324q^{74} + 1857268q^{75} - 3488628q^{76} + 846288q^{77} - 6537300q^{78} - 781076q^{79} - 3635536q^{80} - 2236868q^{81} + 3720360q^{82} - 944002q^{83} + 6632744q^{84} - 1174862q^{86} - 1073982q^{87} - 427452q^{88} + 3381078q^{89} + 412204q^{90} - 2879114q^{91} + 696698q^{92} - 543710q^{93} - 3545780q^{94} - 268230q^{95} - 9026178q^{96} - 1842138q^{97} - 7266130q^{98} + 8295480q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
8.1 −13.9637 3.74156i 7.40363 + 4.27449i 125.560 + 72.4919i −72.2350 + 19.3553i −87.3887 87.3887i 180.588 312.787i −827.827 827.827i −327.958 568.039i 1081.09
8.2 −13.9349 3.73385i −44.2547 25.5505i 124.814 + 72.0615i −58.7930 + 15.7535i 521.284 + 521.284i −230.098 + 398.542i −817.340 817.340i 941.155 + 1630.13i 878.096
8.3 −13.0056 3.48484i 38.1387 + 22.0194i 101.576 + 58.6446i 226.086 60.5794i −419.282 419.282i −208.015 + 360.292i −507.354 507.354i 605.206 + 1048.25i −3151.48
8.4 −9.93212 2.66130i −3.04848 1.76004i 36.1388 + 20.8647i −34.6811 + 9.29276i 25.5938 + 25.5938i −217.964 + 377.524i 161.925 + 161.925i −358.305 620.602i 369.187
8.5 −9.59622 2.57130i −21.8555 12.6183i 30.0502 + 17.3495i 173.019 46.3602i 177.285 + 177.285i 188.845 327.090i 205.837 + 205.837i −46.0571 79.7733i −1779.53
8.6 −8.85799 2.37349i 32.4379 + 18.7280i 17.4050 + 10.0488i −153.103 + 41.0238i −242.884 242.884i 76.7196 132.882i 284.686 + 284.686i 336.977 + 583.662i 1453.55
8.7 −4.02062 1.07732i −24.6756 14.2465i −40.4209 23.3370i −157.005 + 42.0695i 83.8633 + 83.8633i 63.0530 109.211i 325.747 + 325.747i 41.4241 + 71.7487i 676.582
8.8 −3.54127 0.948881i 11.5842 + 6.68812i −43.7854 25.2795i 59.9765 16.0707i −34.6765 34.6765i −141.297 + 244.734i 296.982 + 296.982i −275.038 476.380i −227.643
8.9 −2.34959 0.629570i 32.2680 + 18.6299i −50.3014 29.0415i 85.5401 22.9204i −64.0877 64.0877i 207.449 359.312i 209.985 + 209.985i 329.649 + 570.970i −215.414
8.10 1.46393 + 0.392259i −38.8847 22.4501i −53.4364 30.8515i 78.2708 20.9726i −48.1182 48.1182i −50.3376 + 87.1872i −134.712 134.712i 643.512 + 1114.60i 122.810
8.11 3.45422 + 0.925557i 32.3417 + 18.6725i −44.3506 25.6058i −181.632 + 48.6682i 94.4331 + 94.4331i −292.762 + 507.079i −291.332 291.332i 332.825 + 576.470i −672.444
8.12 3.89838 + 1.04457i −2.90484 1.67711i −41.3194 23.8558i 124.161 33.2689i −9.57232 9.57232i −66.0477 + 114.398i −318.804 318.804i −358.875 621.589i 518.779
8.13 5.90851 + 1.58318i 1.50469 + 0.868733i −23.0216 13.2915i −121.426 + 32.5360i 7.51511 + 7.51511i 268.634 465.288i −391.802 391.802i −362.991 628.718i −768.957
8.14 10.0194 + 2.68468i 36.8050 + 21.2494i 37.7546 + 21.7976i 48.6688 13.0408i 311.715 + 311.715i 32.6865 56.6146i −149.662 149.662i 538.570 + 932.831i 522.641
8.15 10.2697 + 2.75175i −23.6115 13.6321i 42.4685 + 24.5192i −109.253 + 29.2742i −204.970 204.970i −217.179 + 376.165i −112.480 112.480i 7.16866 + 12.4165i −1202.55
8.16 11.7663 + 3.15278i 2.99680 + 1.73020i 73.0812 + 42.1934i 187.344 50.1987i 29.8064 + 29.8064i −71.1800 + 123.287i 175.603 + 175.603i −358.513 620.962i 2362.62
8.17 12.6486 + 3.38919i −37.5401 21.6738i 93.0755 + 53.7372i 72.3779 19.3936i −401.374 401.374i 296.773 514.026i 402.548 + 402.548i 575.006 + 995.940i 981.210
8.18 14.9428 + 4.00391i 13.6514 + 7.88162i 151.830 + 87.6593i −121.406 + 32.5306i 172.432 + 172.432i 6.42619 11.1305i 1217.70 + 1217.70i −240.260 416.143i −1944.40
14.1 −13.9637 + 3.74156i 7.40363 4.27449i 125.560 72.4919i −72.2350 19.3553i −87.3887 + 87.3887i 180.588 + 312.787i −827.827 + 827.827i −327.958 + 568.039i 1081.09
14.2 −13.9349 + 3.73385i −44.2547 + 25.5505i 124.814 72.0615i −58.7930 15.7535i 521.284 521.284i −230.098 398.542i −817.340 + 817.340i 941.155 1630.13i 878.096
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.g odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 37.7.g.a 72
37.g odd 12 1 inner 37.7.g.a 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.7.g.a 72 1.a even 1 1 trivial
37.7.g.a 72 37.g odd 12 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(37, [\chi])\).