Properties

Label 37.9.g.a
Level $37$
Weight $9$
Character orbit 37.g
Analytic conductor $15.073$
Analytic rank $0$
Dimension $100$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 37 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 37.g (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(15.0730085723\)
Analytic rank: \(0\)
Dimension: \(100\)
Relative dimension: \(25\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 100q - 10q^{2} - 6q^{3} + 552q^{4} - 1642q^{5} - 516q^{6} - 2q^{7} - 7440q^{8} + 117258q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 100q - 10q^{2} - 6q^{3} + 552q^{4} - 1642q^{5} - 516q^{6} - 2q^{7} - 7440q^{8} + 117258q^{9} + 10744q^{10} - 62086q^{12} - 63598q^{13} - 62648q^{14} + 53318q^{15} + 1056692q^{16} - 53050q^{17} - 150954q^{18} - 161388q^{19} - 922330q^{20} + 589656q^{21} - 713044q^{22} + 32888q^{23} - 1193742q^{24} + 1859250q^{25} + 900088q^{26} - 7001094q^{28} + 2288858q^{29} + 9620658q^{30} + 5782072q^{31} - 4920832q^{32} - 719444q^{33} - 5308306q^{34} - 10575422q^{35} + 14052530q^{37} - 10994096q^{38} + 3417088q^{39} + 9928698q^{40} + 16320936q^{41} + 39067848q^{42} - 9546800q^{43} - 19056984q^{44} - 12333970q^{45} - 15608210q^{46} - 844736q^{47} - 54639310q^{49} - 52548004q^{50} + 25866216q^{51} + 31025896q^{52} - 17856728q^{53} + 8699372q^{54} - 14191504q^{55} - 23168766q^{56} + 74736160q^{57} + 109856484q^{58} + 499856q^{59} - 286204560q^{60} - 34778024q^{61} + 6537756q^{62} + 154042660q^{63} - 107386782q^{65} - 14426072q^{66} + 84112848q^{67} + 247473908q^{68} - 74383808q^{69} - 55414618q^{70} - 57046730q^{71} - 172153588q^{72} + 315393932q^{74} + 164078200q^{75} - 141128756q^{76} + 272235456q^{77} + 655339260q^{78} + 18584140q^{79} + 43263824q^{80} - 337954862q^{81} - 363936872q^{82} - 86956118q^{83} - 505940536q^{84} + 41750098q^{86} + 195776154q^{87} + 425134052q^{88} + 253234298q^{89} - 139130828q^{90} - 327702612q^{91} - 572151958q^{92} - 608615132q^{93} - 502568260q^{94} - 115562040q^{95} + 1165361934q^{96} - 123302218q^{97} + 537492294q^{98} + 533594376q^{99} + O(q^{100}) \)

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 −29.9817 8.03357i 96.7293 + 55.8467i 612.661 + 353.720i −928.181 + 248.705i −2451.46 2451.46i −1962.55 + 3399.23i −9908.26 9908.26i 2957.20 + 5122.03i 29826.4
8.2 −29.0187 7.77554i −54.3802 31.3964i 559.924 + 323.272i 923.655 247.493i 1333.92 + 1333.92i −635.915 + 1101.44i −8296.40 8296.40i −1309.03 2267.30i −28727.7
8.3 −26.9838 7.23030i −71.9894 41.5631i 454.148 + 262.202i −763.969 + 204.705i 1642.04 + 1642.04i 1794.23 3107.71i −5301.94 5301.94i 174.479 + 302.206i 22094.9
8.4 −23.7708 6.36938i 94.2638 + 54.4232i 302.782 + 174.811i 460.453 123.378i −1894.09 1894.09i 1918.35 3322.68i −1629.17 1629.17i 2643.27 + 4578.28i −11731.2
8.5 −22.4552 6.01684i −2.68847 1.55219i 246.329 + 142.218i 152.334 40.8178i 51.0307 + 51.0307i −434.094 + 751.872i −467.444 467.444i −3275.68 5673.65i −3666.28
8.6 −19.4436 5.20990i −78.9822 45.6004i 129.209 + 74.5989i −557.634 + 149.418i 1298.13 + 1298.13i −564.669 + 978.035i 1520.19 + 1520.19i 878.292 + 1521.25i 11620.9
8.7 −17.9701 4.81507i 55.1301 + 31.8294i 78.0367 + 45.0545i −19.5258 + 5.23193i −837.432 837.432i −774.650 + 1341.73i 2182.30 + 2182.30i −1254.28 2172.48i 376.073
8.8 −14.5407 3.89616i −127.969 73.8828i −25.4520 14.6947i 528.293 141.556i 1572.89 + 1572.89i −30.8038 + 53.3538i 3037.83 + 3037.83i 7636.83 + 13227.4i −8233.25
8.9 −9.75824 2.61471i 46.3249 + 26.7457i −133.316 76.9700i −1105.12 + 296.117i −382.117 382.117i 938.117 1624.87i 2928.42 + 2928.42i −1849.84 3204.01i 11558.3
8.10 −6.85365 1.83643i 29.3470 + 16.9435i −178.102 102.827i 1138.20 304.979i −170.019 170.019i −1165.96 + 2019.51i 2316.23 + 2316.23i −2706.34 4687.51i −8360.89
8.11 −6.32937 1.69595i 134.573 + 77.6955i −184.518 106.531i 44.9983 12.0573i −719.992 719.992i −750.951 + 1300.69i 2173.37 + 2173.37i 8792.70 + 15229.4i −305.259
8.12 −5.47162 1.46612i −38.0479 21.9670i −193.913 111.956i 326.464 87.4758i 175.978 + 175.978i 1762.78 3053.23i 1922.29 + 1922.29i −2315.40 4010.40i −1914.54
8.13 −3.23067 0.865656i −66.6884 38.5025i −212.015 122.407i −678.195 + 181.722i 182.118 + 182.118i −2300.40 + 3984.40i 1184.43 + 1184.43i −315.607 546.648i 2348.34
8.14 5.19180 + 1.39114i −60.8932 35.1567i −196.683 113.555i −61.3978 + 16.4515i −267.238 267.238i −417.786 + 723.627i −1836.14 1836.14i −808.514 1400.39i −341.651
8.15 7.07517 + 1.89579i 65.9616 + 38.0829i −175.238 101.174i −317.418 + 85.0520i 394.492 + 394.492i 582.766 1009.38i −2373.96 2373.96i −379.880 657.971i −2407.03
8.16 10.3785 + 2.78090i −134.250 77.5093i −121.724 70.2771i −903.574 + 242.112i −1177.76 1177.76i 1653.64 2864.19i −3012.84 3012.84i 8734.88 + 15129.2i −10051.0
8.17 11.5756 + 3.10167i 46.5342 + 26.8665i −97.3286 56.1927i −129.992 + 34.8313i 455.330 + 455.330i −1422.98 + 2464.68i −3121.67 3121.67i −1836.88 3181.57i −1612.77
8.18 13.2543 + 3.55149i 86.8010 + 50.1146i −58.6381 33.8547i 964.131 258.338i 972.509 + 972.509i 1465.21 2537.82i −3140.90 3140.90i 1742.45 + 3018.00i 13696.4
8.19 14.0121 + 3.75453i −97.4491 56.2622i −39.4602 22.7824i 1028.99 275.718i −1154.23 1154.23i −1032.28 + 1787.96i −3093.32 3093.32i 3050.38 + 5283.41i 15453.5
8.20 21.1757 + 5.67401i −53.5012 30.8889i 194.514 + 112.303i 280.857 75.2555i −957.661 957.661i 727.522 1260.11i −486.675 486.675i −1372.25 2376.80i 6374.36
See all 100 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.25
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 37.9.g.a 100
37.g odd 12 1 inner 37.9.g.a 100
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.9.g.a 100 1.a even 1 1 trivial
37.9.g.a 100 37.g odd 12 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(37, [\chi])\).