Properties

Label 37.6.e.a
Level $37$
Weight $6$
Character orbit 37.e
Analytic conductor $5.934$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(5.93420133308\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32q - 6q^{2} + 18q^{3} + 298q^{4} + 144q^{5} - 52q^{7} - 1490q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 32q - 6q^{2} + 18q^{3} + 298q^{4} + 144q^{5} - 52q^{7} - 1490q^{9} + 668q^{10} - 348q^{11} + 134q^{12} + 222q^{13} - 4134q^{15} - 6998q^{16} - 624q^{17} + 7632q^{18} + 2154q^{19} + 4806q^{20} - 130q^{21} - 8214q^{22} + 24642q^{24} + 15808q^{25} + 4332q^{26} - 30384q^{27} - 9048q^{28} + 7780q^{30} + 35088q^{32} - 924q^{33} - 5982q^{34} + 27072q^{35} - 57468q^{36} - 46062q^{37} - 48048q^{38} - 31896q^{39} + 57956q^{40} - 11136q^{41} + 50886q^{42} - 43686q^{44} + 42866q^{46} + 63708q^{47} - 39260q^{48} - 52426q^{49} - 29292q^{50} + 132684q^{52} - 85398q^{53} + 235314q^{54} - 65346q^{55} + 121836q^{56} - 96270q^{57} - 121896q^{58} + 40980q^{59} - 74616q^{61} + 89346q^{62} + 232304q^{63} - 321132q^{64} + 24066q^{65} + 68018q^{67} - 11052q^{69} - 230194q^{70} - 32544q^{71} + 117876q^{72} - 179176q^{73} - 89166q^{74} - 379288q^{75} + 196110q^{76} + 22428q^{77} - 288138q^{78} + 217218q^{79} - 6200q^{81} - 127434q^{83} + 1109800q^{84} + 218576q^{85} - 80364q^{86} + 457230q^{87} - 164844q^{89} - 360436q^{90} + 167160q^{91} - 984606q^{92} + 532392q^{93} - 369822q^{94} + 187398q^{95} + 1476018q^{96} - 174684q^{98} + 194298q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −9.33962 + 5.39223i 14.0993 24.4208i 42.1523 73.0099i −41.0060 23.6748i 304.108i −18.9093 + 32.7518i 564.077i −276.083 478.189i 510.641
11.2 −9.18598 + 5.30353i −6.26292 + 10.8477i 40.2548 69.7233i 63.9646 + 36.9300i 132.862i 113.255 196.163i 514.543i 43.0517 + 74.5678i −783.436
11.3 −7.60377 + 4.39004i −8.72617 + 15.1142i 22.5449 39.0489i −72.9853 42.1381i 153.233i −77.9335 + 134.985i 114.930i −30.7921 53.3336i 739.952
11.4 −6.08219 + 3.51156i 3.15654 5.46728i 8.66206 15.0031i 30.9238 + 17.8538i 44.3374i −47.9780 + 83.1003i 103.070i 101.573 + 175.929i −250.779
11.5 −5.33497 + 3.08014i 3.71739 6.43870i 2.97457 5.15211i −14.5389 8.39402i 45.8003i 41.9646 72.6849i 160.481i 93.8621 + 162.574i 103.419
11.6 −3.47160 + 2.00433i −13.3195 + 23.0700i −7.96531 + 13.7963i 85.3532 + 49.2787i 106.786i −91.8980 + 159.172i 192.138i −233.316 404.115i −395.083
11.7 −2.20874 + 1.27522i 14.9045 25.8153i −12.7476 + 22.0796i 67.9636 + 39.2388i 76.0257i 34.5008 59.7572i 146.638i −322.785 559.081i −200.152
11.8 −1.48416 + 0.856882i −9.17590 + 15.8931i −14.5315 + 25.1693i −29.0287 16.7598i 31.4507i 80.8273 139.997i 104.648i −46.8942 81.2231i 57.4445
11.9 0.312322 0.180319i 9.26592 16.0490i −15.9350 + 27.6002i −90.8164 52.4329i 6.68329i −48.1376 + 83.3769i 23.0340i −50.2145 86.9741i −37.8186
11.10 0.987887 0.570357i 2.15443 3.73158i −15.3494 + 26.5859i 19.1329 + 11.0464i 4.91517i −51.7707 + 89.6695i 71.5213i 112.217 + 194.365i 25.2015
11.11 4.33309 2.50171i −3.26958 + 5.66309i −3.48290 + 6.03256i 54.4047 + 31.4106i 32.7182i 22.9091 39.6798i 194.962i 100.120 + 173.412i 314.320
11.12 5.18743 2.99497i 9.51908 16.4875i 1.93964 3.35955i −4.99548 2.88414i 114.037i 118.181 204.696i 168.441i −59.7259 103.448i −34.5516
11.13 5.44256 3.14226i −10.4803 + 18.1524i 3.74762 6.49106i −46.0737 26.6006i 131.727i −56.9410 + 98.6246i 154.001i −98.1728 170.040i −334.345
11.14 7.81751 4.51344i 10.9075 18.8923i 24.7424 42.8550i 44.4487 + 25.6624i 196.921i −121.740 + 210.859i 157.833i −116.446 201.690i 463.304
11.15 8.27772 4.77914i 2.34354 4.05913i 29.6804 51.4080i −63.0654 36.4108i 44.8004i 27.0685 46.8840i 261.522i 110.516 + 191.419i −696.050
11.16 9.35252 5.39968i −9.83385 + 17.0327i 42.3131 73.2884i 68.3185 + 39.4437i 212.399i 50.6016 87.6446i 568.329i −71.9093 124.551i 851.933
27.1 −9.33962 5.39223i 14.0993 + 24.4208i 42.1523 + 73.0099i −41.0060 + 23.6748i 304.108i −18.9093 32.7518i 564.077i −276.083 + 478.189i 510.641
27.2 −9.18598 5.30353i −6.26292 10.8477i 40.2548 + 69.7233i 63.9646 36.9300i 132.862i 113.255 + 196.163i 514.543i 43.0517 74.5678i −783.436
27.3 −7.60377 4.39004i −8.72617 15.1142i 22.5449 + 39.0489i −72.9853 + 42.1381i 153.233i −77.9335 134.985i 114.930i −30.7921 + 53.3336i 739.952
27.4 −6.08219 3.51156i 3.15654 + 5.46728i 8.66206 + 15.0031i 30.9238 17.8538i 44.3374i −47.9780 83.1003i 103.070i 101.573 175.929i −250.779
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 27.16
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 37.6.e.a 32
37.e even 6 1 inner 37.6.e.a 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.6.e.a 32 1.a even 1 1 trivial
37.6.e.a 32 37.e even 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(37, [\chi])\).