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Properties

Label 19.9.f.a
Level $19$
Weight $9$
Character orbit 19.f
Analytic conductor $7.740$
Analytic rank $0$
Dimension $72$
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] = [19,9,Mod(2,19)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(19, base_ring=CyclotomicField(18))
 
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("19.2");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 19 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 19.f (of order \(18\), degree \(6\), 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: \(7.74019359116\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(12\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 72 q - 6 q^{2} + 162 q^{3} - 720 q^{4} - 6 q^{5} - 720 q^{6} + 3282 q^{7} - 9 q^{8} + 8562 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q - 6 q^{2} + 162 q^{3} - 720 q^{4} - 6 q^{5} - 720 q^{6} + 3282 q^{7} - 9 q^{8} + 8562 q^{9} - 2793 q^{10} + 10482 q^{11} - 124425 q^{12} + 108465 q^{13} + 91905 q^{14} - 236277 q^{15} - 207120 q^{16} + 40269 q^{17} + 188304 q^{19} + 1700646 q^{20} + 386850 q^{21} - 1237254 q^{22} - 1339566 q^{23} - 1289928 q^{24} + 59748 q^{25} - 842907 q^{26} + 4849875 q^{27} - 2065044 q^{28} - 1312269 q^{29} + 299940 q^{30} + 3498759 q^{31} + 9109881 q^{32} - 1670298 q^{33} - 12448236 q^{34} - 5923416 q^{35} - 6763797 q^{36} + 5067372 q^{38} + 18303516 q^{39} + 15014562 q^{40} - 11716206 q^{41} - 9228981 q^{42} + 12974721 q^{43} - 2631387 q^{44} - 31264995 q^{45} - 30318354 q^{46} + 1571889 q^{47} + 55683057 q^{48} + 1898838 q^{49} + 30782124 q^{50} + 12474372 q^{51} - 839439 q^{52} - 49160157 q^{53} - 65504571 q^{54} - 27903114 q^{55} + 18105312 q^{57} + 85061964 q^{58} + 51376881 q^{59} + 61200294 q^{60} - 51972462 q^{61} - 85396644 q^{62} + 41912118 q^{63} - 35891445 q^{64} + 2267586 q^{65} + 53209500 q^{66} + 35494521 q^{67} + 13755102 q^{68} + 54287784 q^{69} + 98126001 q^{70} - 5729388 q^{71} - 580668798 q^{72} - 255453906 q^{73} + 159588693 q^{74} + 128877954 q^{76} + 345629838 q^{77} + 120889431 q^{78} + 73477881 q^{79} - 93973551 q^{80} + 114909627 q^{81} + 67089591 q^{82} + 55994352 q^{83} + 143496459 q^{84} - 29709876 q^{85} - 592637892 q^{86} - 151341879 q^{87} - 412023537 q^{88} - 7283562 q^{89} + 350808846 q^{90} + 81065763 q^{91} + 496900692 q^{92} - 609570561 q^{93} - 222935505 q^{95} + 89754174 q^{96} - 180436449 q^{97} + 472058181 q^{98} + 865983036 q^{99}+O(q^{100}) \) 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 −27.0408 − 4.76802i 43.2321 − 51.5220i 467.907 + 170.304i −872.668 + 317.625i −1414.68 + 1187.06i 536.781 + 929.732i −5753.07 − 3321.53i 353.804 + 2006.52i 25112.0 − 4427.93i
2.2 −25.8887 − 4.56487i −45.3255 + 54.0168i 408.823 + 148.799i 502.442 − 182.874i 1420.00 − 1191.52i 126.399 + 218.929i −4076.51 − 2353.57i 275.889 + 1564.64i −13842.3 + 2440.78i
2.3 −17.6477 − 3.11176i 88.1568 − 105.061i 61.1966 + 22.2738i 955.890 − 347.916i −1882.69 + 1579.76i 678.261 + 1174.78i 2962.23 + 1710.24i −2126.93 − 12062.4i −17951.9 + 3165.40i
2.4 −13.6035 − 2.39867i 15.5877 − 18.5766i −61.2592 − 22.2965i −36.1678 + 13.1640i −256.606 + 215.318i −1384.17 − 2397.46i 3842.32 + 2218.36i 1037.19 + 5882.19i 523.585 − 92.3221i
2.5 −10.0012 − 1.76348i −70.0946 + 83.5355i −143.647 − 52.2832i −433.854 + 157.910i 848.344 − 711.845i 171.924 + 297.780i 3595.94 + 2076.12i −925.619 − 5249.44i 4617.53 − 814.196i
2.6 −1.18052 − 0.208158i 63.0914 − 75.1894i −239.211 − 87.0657i −753.457 + 274.236i −90.1321 + 75.6298i 994.028 + 1721.71i 530.033 + 306.015i −533.617 − 3026.29i 946.557 − 166.904i
2.7 3.54293 + 0.624714i −19.5543 + 23.3039i −228.399 − 83.1305i 773.341 − 281.473i −83.8379 + 70.3484i 2158.31 + 3738.30i −1554.86 − 897.701i 978.604 + 5549.94i 2915.73 − 514.122i
2.8 7.58647 + 1.33770i 27.9671 − 33.3299i −184.796 − 67.2603i 295.009 − 107.375i 256.757 − 215.445i −1865.56 − 3231.24i −3019.86 − 1743.52i 810.583 + 4597.05i 2381.71 − 419.960i
2.9 14.3082 + 2.52292i −93.7652 + 111.745i −42.2029 − 15.3606i 404.530 − 147.237i −1623.53 + 1362.30i −1320.16 − 2286.59i −3786.18 − 2185.95i −2555.73 − 14494.2i 6159.55 − 1086.10i
2.10 18.3368 + 3.23327i −30.1192 + 35.8947i 85.2225 + 31.0185i −993.378 + 361.560i −668.347 + 560.810i 465.610 + 806.461i −2665.61 − 1538.99i 758.045 + 4299.09i −19384.4 + 3417.99i
2.11 21.4759 + 3.78678i 79.9115 − 95.2348i 206.314 + 75.0920i 64.9468 − 23.6387i 2076.80 − 1742.65i 256.522 + 444.310i −688.296 − 397.388i −1544.51 − 8759.37i 1484.31 − 261.723i
2.12 28.1724 + 4.96755i −19.0758 + 22.7336i 528.446 + 192.339i 353.774 − 128.763i −650.341 + 545.701i 44.1972 + 76.5518i 7589.89 + 4382.03i 986.373 + 5594.00i 10606.3 − 1870.18i
3.1 −19.5948 + 23.3522i −44.0282 − 120.966i −116.914 − 663.050i −107.009 + 606.875i 3687.55 + 1342.16i −1489.58 − 2580.03i 11016.2 + 6360.18i −7668.38 + 6434.54i −12075.0 − 14390.5i
3.2 −16.5036 + 19.6683i 23.6465 + 64.9681i −70.0169 − 397.085i 121.428 − 688.654i −1668.06 − 607.125i −892.850 − 1546.46i 3273.28 + 1889.83i 1364.32 − 1144.80i 11540.6 + 13753.6i
3.3 −14.7997 + 17.6376i 25.5124 + 70.0948i −47.6002 − 269.954i −204.279 + 1158.52i −1613.88 − 587.406i 1812.74 + 3139.75i 361.273 + 208.581i 763.617 − 640.751i −17410.3 − 20748.8i
3.4 −10.9746 + 13.0790i −26.6938 − 73.3406i −6.16460 − 34.9612i 90.6436 − 514.065i 1252.17 + 455.754i 1689.23 + 2925.83i −3260.30 − 1882.34i 359.732 − 301.851i 5728.67 + 6827.17i
3.5 −6.34622 + 7.56314i −20.1988 − 55.4957i 27.5275 + 156.116i −27.4521 + 155.689i 547.908 + 199.422i −956.918 − 1657.43i −3544.29 − 2046.29i 2354.23 − 1975.44i −1003.28 − 1195.66i
3.6 −2.04200 + 2.43356i 36.3630 + 99.9066i 42.7015 + 242.172i −2.12030 + 12.0248i −317.382 − 115.518i −453.880 − 786.144i −1380.84 − 797.228i −3633.05 + 3048.49i −24.9335 − 29.7146i
3.7 6.68795 − 7.97039i −14.7756 − 40.5957i 25.6555 + 145.500i −158.021 + 896.184i −422.383 − 153.735i 332.389 + 575.714i 3638.00 + 2100.40i 3596.32 − 3017.67i 6086.09 + 7253.12i
3.8 6.72219 − 8.01120i 9.24008 + 25.3869i 25.4625 + 144.405i 179.746 − 1019.39i 265.493 + 96.6316i 1532.60 + 2654.54i 3646.56 + 2105.34i 4466.90 − 3748.18i −6958.26 − 8292.54i
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.f odd 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 19.9.f.a ✓ 72
19.f odd 18 1 inner 19.9.f.a ✓ 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.9.f.a ✓ 72 1.a even 1 1 trivial
19.9.f.a ✓ 72 19.f odd 18 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(19, [\chi])\).