Properties

Label 37.10.e.a
Level $37$
Weight $10$
Character orbit 37.e
Analytic conductor $19.056$
Analytic rank $0$
Dimension $56$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 37 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 37.e (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(19.0563259381\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(28\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 56q - 54q^{2} - 75q^{3} + 7082q^{4} - 186q^{5} - 4233q^{7} - 173645q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 56q - 54q^{2} - 75q^{3} + 7082q^{4} - 186q^{5} - 4233q^{7} - 173645q^{9} + 97788q^{10} - 239916q^{11} + 184790q^{12} - 16221q^{13} + 385251q^{15} - 1932502q^{16} - 951654q^{17} + 2811312q^{18} - 1336047q^{19} - 3977274q^{20} + 1205555q^{21} + 1410426q^{22} - 4560318q^{24} + 7900208q^{25} + 6077964q^{26} + 586098q^{27} + 3406456q^{28} - 12007340q^{30} + 47328192q^{32} + 8003280q^{33} - 19428782q^{34} - 4325523q^{35} - 99800316q^{36} + 38883015q^{37} - 631920q^{38} + 6587955q^{39} + 25546036q^{40} + 40335342q^{41} - 91333530q^{42} - 51286086q^{44} - 92082750q^{46} - 56174724q^{47} + 388858084q^{48} - 46172317q^{49} - 250482252q^{50} - 3706980q^{52} - 197686869q^{53} - 331423182q^{54} + 339434874q^{55} + 308799468q^{56} + 360939639q^{57} + 152139640q^{58} + 26190393q^{59} + 761586312q^{61} + 443235042q^{62} - 7358848q^{63} - 2112154540q^{64} - 76474689q^{65} + 69911823q^{67} + 1126766970q^{69} + 788397646q^{70} - 47291505q^{71} + 1854440580q^{72} - 1332048936q^{73} - 1443954222q^{74} - 1421404708q^{75} - 1509740898q^{76} + 1440770232q^{77} - 58000938q^{78} + 803079201q^{79} - 1163206256q^{81} + 231204831q^{83} - 60315320q^{84} - 822210144q^{85} - 1325958588q^{86} + 762221460q^{87} + 2475878382q^{89} - 4026498436q^{90} + 2004728283q^{91} + 183172434q^{92} + 3380032662q^{93} + 3222389442q^{94} - 926002287q^{95} - 14442188094q^{96} + 2664119844q^{98} + 3971333010q^{99} + O(q^{100}) \)

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 −39.1578 + 22.6078i −118.702 + 205.598i 766.223 1327.14i −1675.70 967.463i 10734.4i 1849.99 3204.27i 46140.0i −18339.0 31764.0i 87488.7
11.2 −37.8505 + 21.8530i 76.7161 132.876i 699.105 1210.89i 838.490 + 484.103i 6705.90i 260.580 451.338i 38732.7i −1929.22 3341.51i −42316.3
11.3 −31.1498 + 17.9843i 57.1110 98.9191i 390.874 677.013i −1648.13 951.547i 4108.41i −5484.32 + 9499.11i 9702.46i 3318.18 + 5747.25i 68451.8
11.4 −30.7576 + 17.7579i −73.3239 + 127.001i 374.688 648.979i 2028.62 + 1171.23i 5208.32i −279.638 + 484.348i 8430.63i −911.280 1578.38i −83194.2
11.5 −29.7974 + 17.2035i 14.1784 24.5577i 335.923 581.835i −83.3280 48.1094i 975.673i 4286.41 7424.29i 5499.82i 9439.45 + 16349.6i 3310.61
11.6 −27.6882 + 15.9858i −44.3839 + 76.8751i 255.090 441.829i −158.315 91.4030i 2838.04i −1394.49 + 2415.33i 58.1829i 5901.64 + 10221.9i 5844.59
11.7 −23.8055 + 13.7441i 131.157 227.170i 121.802 210.967i −1058.66 611.220i 7210.54i 4250.53 7362.13i 7377.74i −24562.7 42543.9i 33602.7
11.8 −21.0748 + 12.1675i 98.7278 171.002i 40.0973 69.4506i 2085.31 + 1203.95i 4805.09i −3856.72 + 6680.04i 10508.0i −9652.86 16719.2i −58596.6
11.9 −17.2642 + 9.96751i −121.025 + 209.622i −57.2974 + 99.2421i −231.945 133.913i 4825.28i −4145.43 + 7180.09i 12491.2i −19452.7 33693.1i 5339.13
11.10 −16.0906 + 9.28991i −49.1825 + 85.1865i −83.3953 + 144.445i −2260.08 1304.86i 1827.60i 2409.21 4172.87i 12611.8i 5003.67 + 8666.61i 48487.9
11.11 −11.7800 + 6.80116i 32.8904 56.9678i −163.488 + 283.170i 1285.87 + 742.395i 894.771i 2908.10 5036.98i 11412.0i 7677.95 + 13298.6i −20196.6
11.12 −11.2408 + 6.48986i 62.0101 107.405i −171.763 + 297.503i −199.442 115.148i 1609.75i −1112.71 + 1927.27i 11104.5i 2151.00 + 3725.64i 2989.18
11.13 −9.56308 + 5.52125i −112.816 + 195.403i −195.032 + 337.805i 615.768 + 355.514i 2491.54i 5957.98 10319.5i 9961.03i −15613.4 27043.2i −7851.52
11.14 −3.82302 + 2.20722i −29.2527 + 50.6671i −246.256 + 426.528i 111.286 + 64.2508i 258.269i −3702.66 + 6413.20i 4434.37i 8130.06 + 14081.7i −567.263
11.15 2.40508 1.38857i 61.5559 106.618i −252.144 + 436.726i −1906.94 1100.97i 341.899i 241.891 418.967i 2822.38i 2263.23 + 3920.03i −6115.11
11.16 4.44570 2.56673i −64.5524 + 111.808i −242.824 + 420.583i 2105.55 + 1215.64i 662.753i −1802.14 + 3121.40i 5121.38i 1507.48 + 2611.03i 12480.9
11.17 7.51322 4.33776i 133.176 230.668i −218.368 + 378.224i −134.122 77.4352i 2310.74i −3416.07 + 5916.81i 8230.77i −25630.2 44392.8i −1343.58
11.18 10.8373 6.25693i −17.7229 + 30.6970i −177.702 + 307.788i −53.3178 30.7830i 443.564i 3924.98 6798.27i 10854.6i 9213.30 + 15957.9i −770.429
11.19 11.4704 6.62244i −93.3595 + 161.703i −168.287 + 291.481i −1327.40 766.378i 2473.07i −1020.89 + 1768.24i 11239.3i −7590.47 13147.1i −20301.2
11.20 12.9927 7.50134i 92.3544 159.963i −143.460 + 248.480i 1347.77 + 778.135i 2771.13i 3647.45 6317.57i 11985.9i −7217.17 12500.5i 23348.2
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 27.28
Significant digits:
Format:

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 37.10.e.a 56
37.e even 6 1 inner 37.10.e.a 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.10.e.a 56 1.a even 1 1 trivial
37.10.e.a 56 37.e even 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{10}^{\mathrm{new}}(37, [\chi])\).