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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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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [37,9,Mod(8,37)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(37, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("37.8");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 37 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 37.g (of order \(12\), degree \(4\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
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.

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

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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 8.25
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 \(S_{9}^{\mathrm{new}}(37, [\chi])\).