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Properties

Label 37.8.c.a
Level $37$
Weight $8$
Character orbit 37.c
Analytic conductor $11.558$
Analytic rank $0$
Dimension $42$
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] = [37,8,Mod(10,37)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(37, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("37.10");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 37 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 37.c (of order \(3\), degree \(2\), 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: \(11.5582459429\)
Analytic rank: \(0\)
Dimension: \(42\)
Relative dimension: \(21\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 42 q - 16 q^{2} - 68 q^{3} - 1126 q^{4} - 337 q^{5} + 252 q^{6} - 158 q^{7} + 6264 q^{8} - 12773 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 42 q - 16 q^{2} - 68 q^{3} - 1126 q^{4} - 337 q^{5} + 252 q^{6} - 158 q^{7} + 6264 q^{8} - 12773 q^{9} + 10028 q^{10} + 14876 q^{11} - 4836 q^{12} - 5922 q^{13} + 20176 q^{14} - 8460 q^{15} - 36726 q^{16} - 19095 q^{17} - 7338 q^{18} - 34334 q^{19} - 75668 q^{20} - 32764 q^{21} + 18526 q^{22} - 99896 q^{23} - 203760 q^{24} - 322660 q^{25} + 101516 q^{26} + 846616 q^{27} - 73490 q^{28} + 168186 q^{29} - 88790 q^{30} - 38052 q^{31} - 172292 q^{32} - 54536 q^{33} + 90724 q^{34} - 247046 q^{35} + 352004 q^{36} + 740185 q^{37} + 1994976 q^{38} - 591380 q^{39} - 38962 q^{40} - 292827 q^{41} - 1499386 q^{42} + 630980 q^{43} - 1727390 q^{44} - 994286 q^{45} + 576988 q^{46} + 3394668 q^{47} - 79840 q^{48} + 728851 q^{49} - 69656 q^{50} - 3970200 q^{51} - 2043140 q^{52} + 584438 q^{53} - 2160774 q^{54} + 2019874 q^{55} - 658986 q^{56} + 4295748 q^{57} - 5684872 q^{58} - 1773152 q^{59} + 20208544 q^{60} - 3596569 q^{61} - 1538738 q^{62} + 11795092 q^{63} - 9622852 q^{64} - 5027290 q^{65} - 17464952 q^{66} - 3249754 q^{67} + 5608668 q^{68} + 9599384 q^{69} - 5102080 q^{70} - 1162530 q^{71} - 4730172 q^{72} - 5266532 q^{73} + 4052290 q^{74} + 35543072 q^{75} - 13781226 q^{76} + 121180 q^{77} - 7012890 q^{78} - 13545718 q^{79} - 17840888 q^{80} - 16689077 q^{81} - 43173544 q^{82} + 7359048 q^{83} + 41186680 q^{84} + 21047382 q^{85} + 6732838 q^{86} + 7672800 q^{87} + 42023140 q^{88} + 2955725 q^{89} - 14596808 q^{90} - 19887752 q^{91} + 13352900 q^{92} + 9156784 q^{93} + 690474 q^{94} - 5667270 q^{95} + 29986904 q^{96} + 52065982 q^{97} - 53664462 q^{98} - 35534038 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1 −11.0007 + 19.0538i 11.3160 + 19.5999i −178.031 − 308.359i 50.3125 + 87.1438i −497.937 −276.316 − 478.593i 5017.70 837.396 − 1450.41i −2213.89
10.2 −9.34255 + 16.1818i −43.2405 − 74.8948i −110.566 − 191.507i −171.836 − 297.629i 1615.91 −666.202 − 1153.90i 1740.20 −2645.99 + 4582.98i 6421.56
10.3 −8.90423 + 15.4226i 0.464514 + 0.804563i −94.5706 − 163.801i −251.123 − 434.958i −16.5446 772.514 + 1338.03i 1088.83 1093.07 − 1893.25i 8944.22
10.4 −8.43866 + 14.6162i −26.6641 − 46.1836i −78.4219 − 135.831i 196.505 + 340.357i 900.038 448.397 + 776.647i 486.805 −328.451 + 568.894i −6632.96
10.5 −7.57692 + 13.1236i 12.8050 + 22.1789i −50.8193 − 88.0217i 31.2763 + 54.1721i −388.090 −403.425 − 698.752i −399.475 765.563 − 1325.99i −947.911
10.6 −7.43537 + 12.8784i 41.2618 + 71.4676i −46.5696 − 80.6609i 62.4377 + 108.145i −1227.19 308.286 + 533.967i −518.407 −2311.58 + 4003.77i −1856.99
10.7 −5.02707 + 8.70714i −17.3307 − 30.0177i 13.4571 + 23.3084i 14.1829 + 24.5655i 348.491 −329.971 − 571.526i −1557.53 492.792 − 853.541i −285.194
10.8 −3.42160 + 5.92638i 28.0949 + 48.6617i 40.5853 + 70.2959i −242.165 − 419.442i −384.517 −642.579 − 1112.98i −1431.40 −485.141 + 840.289i 3314.36
10.9 −2.85295 + 4.94146i −22.7033 − 39.3233i 47.7213 + 82.6558i −48.1581 − 83.4123i 259.086 330.602 + 572.619i −1274.94 62.6168 − 108.455i 549.571
10.10 −1.26811 + 2.19642i 13.5559 + 23.4795i 60.7838 + 105.281i 253.342 + 438.801i −68.7612 −185.616 − 321.497i −632.957 725.976 − 1257.43i −1285.06
10.11 −1.18492 + 2.05234i 20.9206 + 36.2356i 61.1919 + 105.988i −19.7131 − 34.1440i −99.1570 666.527 + 1154.46i −593.369 218.155 − 377.855i 93.4336
10.12 0.746546 − 1.29305i −28.6079 − 49.5504i 62.8853 + 108.921i −130.514 − 226.056i −85.4285 −159.585 − 276.409i 378.903 −543.330 + 941.074i −389.737
10.13 2.32071 − 4.01959i −46.0982 − 79.8444i 53.2286 + 92.1946i 217.309 + 376.390i −427.923 19.1096 + 33.0988i 1088.22 −3156.59 + 5467.37i 2017.24
10.14 3.77156 − 6.53253i 9.73006 + 16.8530i 35.5507 + 61.5757i −147.558 − 255.578i 146.790 91.4021 + 158.313i 1501.84 904.152 − 1566.04i −2226.09
10.15 4.09845 − 7.09873i 41.0784 + 71.1498i 30.4054 + 52.6637i 21.4152 + 37.0921i 673.431 −128.692 − 222.901i 1547.66 −2281.37 + 3951.44i 351.076
10.16 4.93583 − 8.54911i −6.61210 − 11.4525i 15.2752 + 26.4574i 56.3382 + 97.5806i −130.545 −783.646 − 1357.32i 1565.16 1006.06 − 1742.55i 1112.30
10.17 6.29543 − 10.9040i −9.44168 − 16.3535i −15.2650 − 26.4397i 125.405 + 217.208i −237.758 817.371 + 1415.73i 1227.23 915.209 − 1585.19i 3157.91
10.18 7.43768 − 12.8824i −34.7930 − 60.2632i −46.6382 − 80.7798i −238.285 − 412.723i −1035.12 249.349 + 431.886i 516.525 −1327.60 + 2299.47i −7089.17
10.19 9.33278 − 16.1648i 28.0828 + 48.6408i −110.202 − 190.875i 166.579 + 288.523i 1048.36 1.55453 + 2.69252i −1724.76 −483.783 + 837.937i 6218.56
10.20 9.53293 − 16.5115i 19.4337 + 33.6601i −117.754 − 203.955i −179.660 − 311.180i 741.040 35.6214 + 61.6981i −2049.72 338.163 − 585.716i −6850.74
See all 42 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 10.21
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 37.8.c.a ✓ 42
37.c even 3 1 inner 37.8.c.a ✓ 42
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.8.c.a ✓ 42 1.a even 1 1 trivial
37.8.c.a ✓ 42 37.c even 3 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(37, [\chi])\).