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Properties

Label 19.11.d.a
Level $19$
Weight $11$
Character orbit 19.d
Analytic conductor $12.072$
Analytic rank $0$
Dimension $30$
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] = [19,11,Mod(8,19)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(19, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 11, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("19.8");
 
S:= CuspForms(chi, 11);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 19 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 19.d (of order \(6\), 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: \(12.0717878008\)
Analytic rank: \(0\)
Dimension: \(30\)
Relative dimension: \(15\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 30 q - 3 q^{2} + 63 q^{3} + 5533 q^{4} - 1112 q^{5} - 323 q^{6} + 42200 q^{7} + 200606 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 30 q - 3 q^{2} + 63 q^{3} + 5533 q^{4} - 1112 q^{5} - 323 q^{6} + 42200 q^{7} + 200606 q^{9} + 221040 q^{10} - 86506 q^{11} + 628284 q^{13} + 878670 q^{14} - 3889548 q^{15} - 1967731 q^{16} - 2000976 q^{17} - 8136009 q^{19} + 10495908 q^{20} - 3338094 q^{21} + 4347261 q^{22} + 12093422 q^{23} + 13290553 q^{24} - 21386387 q^{25} + 18172292 q^{26} + 31090344 q^{28} + 44094606 q^{29} + 20576704 q^{30} - 31803969 q^{32} - 259432347 q^{33} - 173705994 q^{34} + 70424362 q^{35} + 149553522 q^{36} + 72383122 q^{38} + 25380928 q^{39} + 362776464 q^{40} - 750067065 q^{41} + 179185900 q^{42} + 58831816 q^{43} + 270758031 q^{44} - 4911628 q^{45} + 610026508 q^{47} - 315030999 q^{48} + 1121238618 q^{49} + 67449894 q^{51} - 3886242 q^{52} - 736647402 q^{53} - 1125608647 q^{54} + 467517672 q^{55} - 2591625118 q^{57} + 597549092 q^{58} - 1106560407 q^{59} - 3949390842 q^{60} + 198936220 q^{61} + 1592748508 q^{62} + 1207300778 q^{63} + 7879320062 q^{64} - 479159809 q^{66} - 7076062875 q^{67} - 7205128044 q^{68} + 17312860482 q^{70} + 9282304896 q^{71} + 917385402 q^{72} - 5737614181 q^{73} + 9805073646 q^{74} - 4884108669 q^{76} - 6130649752 q^{77} - 11845710990 q^{78} - 17462074926 q^{79} - 4161488746 q^{80} + 3501633673 q^{81} + 17605681423 q^{82} + 4787381102 q^{83} + 16958165420 q^{85} - 42896417742 q^{86} - 7897074148 q^{87} + 13321907874 q^{89} + 24086239122 q^{90} - 17558898516 q^{91} + 7828478412 q^{92} + 16404448376 q^{93} - 40071944832 q^{95} + 17501291766 q^{96} + 14965952121 q^{97} - 34565406645 q^{98} + 1708269232 q^{99}+O(q^{100}) \) 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 −48.6637 − 28.0960i 21.8135 + 12.5941i 1066.77 + 1847.70i 1856.59 − 3215.71i −707.685 − 1225.75i −23537.5 − 62347.0i −29207.3 − 50588.5i −180697. + 104325.i
8.2 −48.4172 − 27.9537i 177.767 + 102.634i 1050.82 + 1820.07i −2527.14 + 4377.14i −5737.99 − 9938.48i 15874.8 − 60247.9i −8457.12 − 14648.2i 244714. − 141286.i
8.3 −41.7349 − 24.0957i −304.133 − 175.591i 649.202 + 1124.45i 508.841 − 881.339i 8461.97 + 14656.6i 22582.4 − 13223.9i 32140.0 + 55668.2i −42472.9 + 24521.7i
8.4 −28.2965 − 16.3370i 295.794 + 170.777i 21.7959 + 37.7516i 1012.38 − 1753.50i −5579.96 − 9664.78i 4902.54 32033.9i 28804.9 + 49891.6i −57293.8 + 33078.6i
8.5 −26.7545 − 15.4467i −184.691 − 106.631i −34.7979 − 60.2717i −1653.05 + 2863.17i 3294.21 + 5705.74i −18787.4 33784.9i −6783.97 − 11750.2i 88453.1 − 51068.4i
8.6 −13.1112 − 7.56973i 66.4701 + 38.3765i −397.398 − 688.314i 53.6429 − 92.9121i −581.000 − 1006.32i 4202.52 27535.6i −26579.0 − 46036.2i −1406.64 + 812.124i
8.7 −5.46934 − 3.15773i −204.258 − 117.928i −492.058 − 852.269i 2396.40 − 4150.69i 744.769 + 1289.98i 7557.62 12682.2i −1710.41 − 2962.52i −26213.5 + 15134.4i
8.8 4.52578 + 2.61296i 347.010 + 200.347i −498.345 − 863.159i −2551.01 + 4418.47i 1047.00 + 1813.45i −25879.4 − 10560.0i 50753.0 + 87906.8i −23090.6 + 13331.4i
8.9 11.6285 + 6.71370i −15.8484 − 9.15009i −421.852 − 730.670i −2034.49 + 3523.84i −122.862 − 212.803i 31810.1 − 25078.4i −29357.1 − 50847.9i −47316.0 + 27317.9i
8.10 12.5329 + 7.23588i −377.553 − 217.981i −407.284 − 705.437i −483.164 + 836.864i −3154.56 − 5463.86i −10811.2 − 26607.3i 65506.5 + 113461.i −12110.9 + 6992.23i
8.11 20.6177 + 11.9036i 90.1086 + 52.0242i −228.607 − 395.959i 1520.47 − 2633.53i 1238.56 + 2145.24i −17023.9 − 35263.7i −24111.5 − 41762.3i 62697.2 − 36198.2i
8.12 29.5297 + 17.0490i 355.086 + 205.009i 69.3339 + 120.090i 1277.91 − 2213.41i 6990.38 + 12107.7i 27141.4 − 30188.0i 54533.0 + 94453.9i 75472.5 − 43574.1i
8.13 37.3265 + 21.5505i −167.153 − 96.5056i 416.847 + 722.001i −1449.31 + 2510.28i −4159.48 − 7204.44i −4158.26 − 8202.36i −10897.9 − 18875.6i −108196. + 62466.8i
8.14 46.4194 + 26.8002i −252.062 − 145.528i 924.506 + 1601.29i 2223.85 − 3851.82i −7800.36 − 13510.6i 12602.1 44221.0i 12832.2 + 22226.0i 206460. − 119199.i
8.15 48.3669 + 27.9246i 183.147 + 105.740i 1047.57 + 1814.45i −707.920 + 1226.15i 5905.52 + 10228.7i −5375.89 59822.4i −7162.50 − 12405.8i −68479.8 + 39536.8i
12.1 −48.6637 + 28.0960i 21.8135 − 12.5941i 1066.77 − 1847.70i 1856.59 + 3215.71i −707.685 + 1225.75i −23537.5 62347.0i −29207.3 + 50588.5i −180697. − 104325.i
12.2 −48.4172 + 27.9537i 177.767 − 102.634i 1050.82 − 1820.07i −2527.14 − 4377.14i −5737.99 + 9938.48i 15874.8 60247.9i −8457.12 + 14648.2i 244714. + 141286.i
12.3 −41.7349 + 24.0957i −304.133 + 175.591i 649.202 − 1124.45i 508.841 + 881.339i 8461.97 − 14656.6i 22582.4 13223.9i 32140.0 − 55668.2i −42472.9 − 24521.7i
12.4 −28.2965 + 16.3370i 295.794 − 170.777i 21.7959 − 37.7516i 1012.38 + 1753.50i −5579.96 + 9664.78i 4902.54 − 32033.9i 28804.9 − 49891.6i −57293.8 − 33078.6i
12.5 −26.7545 + 15.4467i −184.691 + 106.631i −34.7979 + 60.2717i −1653.05 − 2863.17i 3294.21 − 5705.74i −18787.4 − 33784.9i −6783.97 + 11750.2i 88453.1 + 51068.4i
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 8.15
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 19.11.d.a ✓ 30
19.d odd 6 1 inner 19.11.d.a ✓ 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.11.d.a ✓ 30 1.a even 1 1 trivial
19.11.d.a ✓ 30 19.d odd 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{11}^{\mathrm{new}}(19, [\chi])\).