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Properties

Label 19.8.e.a
Level $19$
Weight $8$
Character orbit 19.e
Analytic conductor $5.935$
Analytic rank $0$
Dimension $66$
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] = [19,8,Mod(4,19)]
 
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([2]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("19.4");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 19 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 19.e (of order \(9\), 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: \(5.93531548420\)
Analytic rank: \(0\)
Dimension: \(66\)
Relative dimension: \(11\) over \(\Q(\zeta_{9})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 66 q - 6 q^{2} + 33 q^{3} + 72 q^{4} - 6 q^{5} - 876 q^{6} + 588 q^{7} - 4611 q^{8} + 5025 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 66 q - 6 q^{2} + 33 q^{3} + 72 q^{4} - 6 q^{5} - 876 q^{6} + 588 q^{7} - 4611 q^{8} + 5025 q^{9} - 45 q^{10} - 5796 q^{11} - 10371 q^{12} + 32673 q^{13} - 39819 q^{14} - 49377 q^{15} - 39168 q^{16} + 91407 q^{17} + 256836 q^{18} + 41484 q^{19} - 172518 q^{20} - 75012 q^{21} - 231942 q^{22} - 123690 q^{23} + 44544 q^{24} + 422232 q^{25} + 389013 q^{26} - 439350 q^{27} - 240864 q^{28} + 649293 q^{29} + 819918 q^{30} - 677607 q^{31} - 1210293 q^{32} - 1370571 q^{33} - 1253856 q^{34} - 213654 q^{35} + 3328071 q^{36} + 1507992 q^{37} + 2882532 q^{38} - 277944 q^{39} - 1069794 q^{40} + 477039 q^{41} - 4778763 q^{42} - 3007047 q^{43} - 1415295 q^{44} + 817341 q^{45} + 1063032 q^{46} + 1903425 q^{47} + 9423153 q^{48} - 70491 q^{49} - 6797778 q^{50} - 9524817 q^{51} + 766773 q^{52} - 2301807 q^{53} + 5147835 q^{54} + 4962510 q^{55} + 19668414 q^{56} + 6133224 q^{57} - 9760788 q^{58} + 5008692 q^{59} - 20058444 q^{60} - 3904872 q^{61} + 4822044 q^{62} + 10908864 q^{63} - 4993305 q^{64} - 9445368 q^{65} - 13670268 q^{66} - 8852934 q^{67} - 839274 q^{68} - 10321572 q^{69} + 4173267 q^{70} + 5475522 q^{71} - 21010854 q^{72} + 19169496 q^{73} + 8211435 q^{74} + 19921302 q^{75} + 45571194 q^{76} + 5739030 q^{77} + 20452029 q^{78} + 9628437 q^{79} + 15051621 q^{80} - 4640520 q^{81} - 66242697 q^{82} - 16475364 q^{83} - 54312333 q^{84} - 32222184 q^{85} - 43366296 q^{86} - 25937349 q^{87} - 54939 q^{88} + 25239300 q^{89} + 99525738 q^{90} + 26838687 q^{91} + 104550252 q^{92} - 26920467 q^{93} - 119336550 q^{94} + 38007867 q^{95} + 49202922 q^{96} + 81045072 q^{97} + 88124517 q^{98} + 34256625 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −20.4728 − 7.45148i −4.63945 − 26.3116i 265.556 + 222.828i 384.427 − 322.572i −101.078 + 573.243i −238.816 + 413.641i −2381.93 − 4125.62i 1384.33 − 503.855i −10273.9 + 3739.40i
4.2 −16.1237 − 5.86856i −1.10168 − 6.24791i 127.481 + 106.969i −376.180 + 315.652i −18.9031 + 107.205i 461.686 − 799.664i −329.571 − 570.833i 2017.29 − 734.232i 7917.85 − 2881.86i
4.3 −14.5692 − 5.30275i 15.4254 + 87.4821i 86.0882 + 72.2366i 40.6287 − 34.0916i 239.159 − 1356.34i −203.416 + 352.328i 121.087 + 209.728i −5360.06 + 1950.90i −772.706 + 281.242i
4.4 −8.72597 − 3.17599i 1.07731 + 6.10973i −31.9981 − 26.8496i 41.8327 − 35.1018i 10.0039 − 56.7348i −434.677 + 752.883i 788.243 + 1365.28i 2018.94 − 734.834i −476.514 + 173.437i
4.5 −8.08875 − 2.94406i −14.2479 − 80.8039i −41.2933 − 34.6492i 63.0220 − 52.8818i −122.644 + 695.549i 204.570 − 354.325i 782.905 + 1356.03i −4271.15 + 1554.57i −665.457 + 242.206i
4.6 −0.187302 − 0.0681723i 4.20494 + 23.8474i −98.0233 − 82.2513i 159.085 − 133.488i 0.838140 − 4.75333i 433.332 − 750.553i 25.5093 + 44.1834i 1504.09 − 547.444i −38.8972 + 14.1574i
4.7 5.64598 + 2.05497i −6.62088 − 37.5489i −70.3995 − 59.0722i −287.758 + 241.457i 39.7804 − 225.606i −681.236 + 1179.94i −660.616 − 1144.22i 689.025 − 250.785i −2120.86 + 771.930i
4.8 6.77842 + 2.46714i 12.1486 + 68.8984i −58.1936 − 48.8302i −297.538 + 249.664i −87.6335 + 496.994i 330.309 − 572.112i −735.649 − 1274.18i −2544.29 + 926.046i −2632.79 + 958.257i
4.9 12.6181 + 4.59260i −9.09979 − 51.6075i 40.0701 + 33.6228i 154.381 − 129.541i 122.191 − 692.979i 257.216 − 445.511i −508.193 − 880.216i −525.417 + 191.236i 2542.92 − 925.547i
4.10 14.3741 + 5.23173i 10.4622 + 59.3343i 81.1888 + 68.1255i 309.155 − 259.412i −160.036 + 907.611i −763.591 + 1322.58i −168.381 − 291.644i −1356.00 + 493.542i 5800.99 − 2111.39i
4.11 19.4676 + 7.08563i 1.26187 + 7.15643i 230.728 + 193.603i −192.996 + 161.943i −26.1422 + 148.260i 294.273 − 509.696i 1794.02 + 3107.34i 2005.49 − 729.937i −4904.64 + 1785.14i
5.1 −20.4728 + 7.45148i −4.63945 + 26.3116i 265.556 − 222.828i 384.427 + 322.572i −101.078 − 573.243i −238.816 − 413.641i −2381.93 + 4125.62i 1384.33 + 503.855i −10273.9 − 3739.40i
5.2 −16.1237 + 5.86856i −1.10168 + 6.24791i 127.481 − 106.969i −376.180 − 315.652i −18.9031 − 107.205i 461.686 + 799.664i −329.571 + 570.833i 2017.29 + 734.232i 7917.85 + 2881.86i
5.3 −14.5692 + 5.30275i 15.4254 − 87.4821i 86.0882 − 72.2366i 40.6287 + 34.0916i 239.159 + 1356.34i −203.416 − 352.328i 121.087 − 209.728i −5360.06 − 1950.90i −772.706 − 281.242i
5.4 −8.72597 + 3.17599i 1.07731 − 6.10973i −31.9981 + 26.8496i 41.8327 + 35.1018i 10.0039 + 56.7348i −434.677 − 752.883i 788.243 − 1365.28i 2018.94 + 734.834i −476.514 − 173.437i
5.5 −8.08875 + 2.94406i −14.2479 + 80.8039i −41.2933 + 34.6492i 63.0220 + 52.8818i −122.644 − 695.549i 204.570 + 354.325i 782.905 − 1356.03i −4271.15 − 1554.57i −665.457 − 242.206i
5.6 −0.187302 + 0.0681723i 4.20494 − 23.8474i −98.0233 + 82.2513i 159.085 + 133.488i 0.838140 + 4.75333i 433.332 + 750.553i 25.5093 − 44.1834i 1504.09 + 547.444i −38.8972 − 14.1574i
5.7 5.64598 − 2.05497i −6.62088 + 37.5489i −70.3995 + 59.0722i −287.758 − 241.457i 39.7804 + 225.606i −681.236 − 1179.94i −660.616 + 1144.22i 689.025 + 250.785i −2120.86 − 771.930i
5.8 6.77842 − 2.46714i 12.1486 − 68.8984i −58.1936 + 48.8302i −297.538 − 249.664i −87.6335 − 496.994i 330.309 + 572.112i −735.649 + 1274.18i −2544.29 − 926.046i −2632.79 − 958.257i
5.9 12.6181 − 4.59260i −9.09979 + 51.6075i 40.0701 − 33.6228i 154.381 + 129.541i 122.191 + 692.979i 257.216 + 445.511i −508.193 + 880.216i −525.417 − 191.236i 2542.92 + 925.547i
See all 66 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.11
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 19.8.e.a ✓ 66
19.e even 9 1 inner 19.8.e.a ✓ 66
19.e even 9 1 361.8.a.j 33
19.f odd 18 1 361.8.a.i 33
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.8.e.a ✓ 66 1.a even 1 1 trivial
19.8.e.a ✓ 66 19.e even 9 1 inner
361.8.a.i 33 19.f odd 18 1
361.8.a.j 33 19.e even 9 1

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(19, [\chi])\).