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Properties

Label 37.9.i.a
Level $37$
Weight $9$
Character orbit 37.i
Analytic conductor $15.073$
Analytic rank $0$
Dimension $288$
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(2,37)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(37, base_ring=CyclotomicField(36))
 
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("37.2");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 37 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 37.i (of order \(36\), degree \(12\), 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: \(288\)
Relative dimension: \(24\) over \(\Q(\zeta_{36})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{36}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 288 q - 12 q^{2} - 12 q^{3} - 570 q^{4} + 1500 q^{5} - 12 q^{6} - 12 q^{7} + 12678 q^{8} - 23742 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 288 q - 12 q^{2} - 12 q^{3} - 570 q^{4} + 1500 q^{5} - 12 q^{6} - 12 q^{7} + 12678 q^{8} - 23742 q^{9} - 6 q^{10} - 18 q^{11} + 107508 q^{12} - 12 q^{13} + 142836 q^{14} + 71010 q^{15} - 417582 q^{16} + 231828 q^{17} + 196818 q^{18} - 103404 q^{19} + 453108 q^{20} - 589674 q^{21} - 1548 q^{22} + 621366 q^{23} + 2767368 q^{24} - 1859268 q^{25} + 1370874 q^{26} - 4251546 q^{27} + 1225716 q^{28} + 1210344 q^{29} + 3069756 q^{30} - 2023956 q^{31} + 5714502 q^{32} - 4434762 q^{33} - 5294604 q^{34} + 2615100 q^{35} - 3238464 q^{37} + 25178052 q^{38} + 491334 q^{39} + 21037044 q^{40} - 18060024 q^{41} - 56994828 q^{42} - 17266644 q^{43} - 11514474 q^{44} + 10051992 q^{45} + 10839408 q^{46} + 6573792 q^{47} + 79626222 q^{48} + 44974650 q^{49} - 6052038 q^{50} - 45284172 q^{51} - 85344366 q^{52} - 10219242 q^{53} + 10497732 q^{54} + 20295792 q^{55} - 24634380 q^{56} - 92198604 q^{57} + 41796288 q^{58} - 34338738 q^{59} - 123926112 q^{60} - 52089420 q^{61} + 16268856 q^{62} + 67972314 q^{63} + 318881916 q^{64} + 282478956 q^{65} + 337396860 q^{66} - 97364034 q^{67} - 379089786 q^{68} - 390533658 q^{69} - 221880618 q^{70} - 83873676 q^{71} - 116538234 q^{72} + 134947950 q^{74} + 509355720 q^{75} + 501450654 q^{76} + 13364910 q^{77} + 168805302 q^{78} - 163152588 q^{79} + 196042008 q^{80} - 100748910 q^{81} + 12277680 q^{82} - 62660352 q^{83} - 962741010 q^{84} - 348096546 q^{85} - 227357814 q^{86} + 434446392 q^{87} + 215619120 q^{88} + 209846184 q^{89} + 430930176 q^{90} + 353094534 q^{91} + 886501914 q^{92} + 59904984 q^{93} - 1073441292 q^{94} - 520422978 q^{95} - 1343735856 q^{96} + 81225726 q^{97} + 406841802 q^{98} + 508698396 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −29.4012 − 2.57227i −54.9560 + 65.4940i 605.705 + 106.802i 127.342 − 273.085i 1784.24 − 1784.24i −1584.36 + 576.661i −10235.7 − 2742.65i −129.998 − 737.254i −4446.46 + 7701.49i
2.2 −28.0441 − 2.45354i 12.8056 − 15.2611i 528.341 + 93.1608i −421.654 + 904.241i −396.565 + 396.565i 822.927 − 299.521i −7627.14 − 2043.69i 1070.39 + 6070.47i 14043.5 − 24324.1i
2.3 −26.3945 − 2.30922i 32.0645 − 38.2130i 439.226 + 77.4475i 274.224 − 588.076i −934.568 + 934.568i 4169.48 − 1517.57i −4862.63 − 1302.94i 707.206 + 4010.76i −8596.01 + 14888.7i
2.4 −25.7676 − 2.25438i 88.0224 − 104.901i 406.779 + 71.7261i 22.3393 − 47.9069i −2504.62 + 2504.62i −2164.27 + 787.732i −3923.95 − 1051.42i −2116.97 − 12005.9i −683.632 + 1184.09i
2.5 −20.0596 − 1.75499i −85.5692 + 101.977i 147.196 + 25.9546i −131.003 + 280.937i 1895.45 − 1895.45i 1438.42 − 523.542i 2072.08 + 555.212i −1938.00 − 10990.9i 3120.91 − 5405.58i
2.6 −18.8199 − 1.64653i −9.57263 + 11.4082i 99.3663 + 17.5210i 419.895 − 900.469i 198.940 − 198.940i −1984.82 + 722.417i 2830.29 + 758.374i 1100.79 + 6242.91i −9385.03 + 16255.3i
2.7 −17.0180 − 1.48888i −21.2988 + 25.3829i 35.2832 + 6.22138i −332.800 + 713.693i 400.254 − 400.254i −2033.37 + 740.086i 3633.04 + 973.470i 948.652 + 5380.07i 6726.18 − 11650.1i
2.8 −15.0926 − 1.32043i 44.8985 − 53.5080i −26.0686 − 4.59659i −8.25559 + 17.7042i −748.288 + 748.288i −1283.49 + 467.153i 4133.67 + 1107.61i 292.079 + 1656.46i 147.975 − 256.300i
2.9 −10.4180 − 0.911453i 68.7119 − 81.8876i −144.408 − 25.4630i −307.528 + 659.497i −790.474 + 790.474i 3572.86 − 1300.41i 4067.19 + 1089.80i −844.955 − 4791.98i 3804.92 − 6590.31i
2.10 −8.84669 − 0.773985i −41.8031 + 49.8190i −174.446 − 30.7595i 185.368 − 397.523i 408.378 − 408.378i 3020.67 − 1099.43i 3715.40 + 995.538i 404.872 + 2296.14i −1947.57 + 3373.29i
2.11 −2.03187 − 0.177766i −97.1505 + 115.779i −248.014 − 43.7315i 332.787 − 713.664i 217.979 − 217.979i −4214.95 + 1534.11i 1000.51 + 268.086i −2827.36 − 16034.7i −803.044 + 1390.91i
2.12 −1.60922 − 0.140789i 94.2654 − 112.341i −249.541 − 44.0008i 416.778 − 893.784i −167.510 + 167.510i 846.169 − 307.980i 794.816 + 212.970i −2595.26 − 14718.4i −796.524 + 1379.62i
2.13 −0.690091 − 0.0603752i 30.9268 − 36.8571i −251.638 − 44.3706i 9.06962 − 19.4499i −23.5675 + 23.5675i −1420.77 + 517.117i 342.270 + 91.7109i 737.326 + 4181.58i −7.43315 + 12.8746i
2.14 2.37198 + 0.207522i −51.2392 + 61.0644i −246.528 − 43.4695i −340.614 + 730.449i −134.211 + 134.211i −616.268 + 224.303i −1164.52 − 312.031i 35.8901 + 203.543i −959.514 + 1661.93i
2.15 10.9313 + 0.956362i 19.9368 − 23.7597i −133.533 − 23.5454i 382.082 − 819.378i 240.657 − 240.657i −548.565 + 199.661i −4150.54 − 1112.13i 972.256 + 5513.94i 4960.27 − 8591.43i
2.16 11.1001 + 0.971130i −51.7792 + 61.7081i −129.842 − 22.8947i 235.738 − 505.541i −634.679 + 634.679i 2193.08 − 798.217i −4174.30 − 1118.50i 12.5063 + 70.9265i 3107.65 − 5382.61i
2.17 11.2501 + 0.984257i 91.2150 − 108.706i −126.515 − 22.3079i −421.975 + 904.928i 1133.17 − 1133.17i −3663.29 + 1333.33i −4193.87 − 1123.74i −2357.47 − 13369.9i −5637.94 + 9765.20i
2.18 13.1042 + 1.14647i 22.6967 − 27.0489i −81.7063 − 14.4070i −319.591 + 685.366i 328.432 − 328.432i 3102.38 − 1129.18i −4306.91 − 1154.03i 922.804 + 5233.48i −4973.72 + 8614.74i
2.19 19.8531 + 1.73692i 57.4425 − 68.4573i 139.019 + 24.5127i 57.0668 − 122.380i 1259.32 − 1259.32i 2002.52 − 728.856i −2210.59 − 592.327i −247.454 − 1403.38i 1345.52 − 2330.51i
2.20 20.9122 + 1.82958i −21.2252 + 25.2952i 181.864 + 32.0674i 27.4475 − 58.8613i −490.145 + 490.145i −4048.54 + 1473.55i −1446.37 − 387.553i 949.968 + 5387.54i 681.679 − 1180.70i
See next 80 embeddings (of 288 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.i odd 36 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 37.9.i.a ✓ 288
37.i odd 36 1 inner 37.9.i.a ✓ 288
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.9.i.a ✓ 288 1.a even 1 1 trivial
37.9.i.a ✓ 288 37.i odd 36 1 inner

Hecke kernels

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