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Properties

Label 19.10.e.a
Level $19$
Weight $10$
Character orbit 19.e
Analytic conductor $9.786$
Analytic rank $0$
Dimension $84$
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] = [19,10,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, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("19.4");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 19 \)
Weight: \( k \) \(=\) \( 10 \)
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: \(9.78568088711\)
Analytic rank: \(0\)
Dimension: \(84\)
Relative dimension: \(14\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 84 q - 6 q^{2} + 213 q^{3} - 1032 q^{4} - 6 q^{5} + 4668 q^{6} + 5715 q^{7} + 36861 q^{8} - 62421 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 84 q - 6 q^{2} + 213 q^{3} - 1032 q^{4} - 6 q^{5} + 4668 q^{6} + 5715 q^{7} + 36861 q^{8} - 62421 q^{9} - 60477 q^{10} + 11295 q^{11} + 373245 q^{12} - 405456 q^{13} + 251421 q^{14} + 336048 q^{15} + 250848 q^{16} - 1109112 q^{17} - 2718636 q^{18} - 2055102 q^{19} + 4320714 q^{20} + 6478617 q^{21} + 3977466 q^{22} + 4778646 q^{23} - 12027696 q^{24} - 11630970 q^{25} - 9164091 q^{26} + 9159258 q^{27} + 18844368 q^{28} - 11163270 q^{29} - 31527234 q^{30} + 14008185 q^{31} + 39067659 q^{32} + 13852056 q^{33} + 52272000 q^{34} - 556095 q^{35} - 138087633 q^{36} - 69738432 q^{37} - 36719772 q^{38} + 31440864 q^{39} + 72353238 q^{40} + 21081261 q^{41} + 278228397 q^{42} - 2650620 q^{43} - 45518607 q^{44} - 61014765 q^{45} - 63830328 q^{46} - 68926458 q^{47} - 261869319 q^{48} - 163636377 q^{49} + 314983974 q^{50} - 85145391 q^{51} + 456660285 q^{52} + 161166480 q^{53} + 159418851 q^{54} - 373457277 q^{55} - 745391202 q^{56} - 252057498 q^{57} + 574863468 q^{58} + 95772153 q^{59} - 357214644 q^{60} + 623252928 q^{61} + 907803708 q^{62} - 137368179 q^{63} - 866999625 q^{64} - 153878493 q^{65} - 782341476 q^{66} - 494022567 q^{67} - 821601474 q^{68} + 1193576865 q^{69} + 2528982243 q^{70} + 1179609600 q^{71} + 1908922506 q^{72} - 652208256 q^{73} - 2845641837 q^{74} - 4030728588 q^{75} - 1483780614 q^{76} - 958458222 q^{77} + 1805806725 q^{78} + 2000761536 q^{79} + 2544691341 q^{80} + 494126322 q^{81} + 675000663 q^{82} - 1226988585 q^{83} + 1236853875 q^{84} - 591903282 q^{85} - 3032971176 q^{86} - 601158705 q^{87} + 2100827301 q^{88} + 974713722 q^{89} + 915811338 q^{90} + 4649924571 q^{91} + 2420259036 q^{92} - 5301596115 q^{93} - 14821292934 q^{94} + 2090479674 q^{95} + 9265625706 q^{96} + 1629254199 q^{97} - 2513934243 q^{98} - 230051604 q^{99}+O(q^{100}) \) 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 −37.1255 − 13.5126i 23.8217 + 135.100i 803.500 + 674.217i −60.2331 + 50.5416i 941.151 − 5337.53i 1475.00 − 2554.78i −10605.9 − 18369.9i 811.556 − 295.382i 2919.13 − 1062.48i
4.2 −36.8078 − 13.3970i −43.1817 − 244.896i 783.124 + 657.119i −962.522 + 807.652i −1691.43 + 9592.59i 83.0798 − 143.899i −9994.16 − 17310.4i −39613.3 + 14418.1i 46248.4 − 16833.0i
4.3 −27.3759 − 9.96400i −3.09587 − 17.5575i 257.942 + 216.439i −58.0769 + 48.7323i −90.1913 + 511.500i −5103.16 + 8838.93i 2553.21 + 4422.29i 18197.3 − 6623.27i 2075.47 − 755.411i
4.4 −24.2695 − 8.83336i −19.4747 − 110.447i 118.763 + 99.6543i 1548.70 − 1299.52i −502.974 + 2852.50i 4850.18 − 8400.77i 4609.68 + 7984.19i 6676.79 − 2430.15i −49065.3 + 17858.3i
4.5 −13.2468 − 4.82143i 31.6453 + 179.469i −239.984 − 201.370i −1449.37 + 1216.17i 446.101 − 2529.97i 2251.38 − 3899.50i 5816.93 + 10075.2i −12711.8 + 4626.73i 25063.1 − 9122.24i
4.6 −10.0828 − 3.66983i 33.0850 + 187.634i −304.020 − 255.103i 1739.50 − 1459.62i 354.998 − 2013.29i −1079.55 + 1869.83i 4876.03 + 8445.53i −15616.0 + 5683.75i −22895.6 + 8333.31i
4.7 −6.37917 − 2.32183i −19.6296 − 111.325i −356.912 − 299.485i −1049.28 + 880.454i −133.257 + 755.739i 756.062 − 1309.54i 3319.32 + 5749.24i 6487.99 − 2361.44i 8737.83 − 3180.31i
4.8 4.82564 + 1.75639i −37.8964 − 214.921i −372.013 − 312.156i 1325.34 − 1112.09i 194.611 − 1103.69i −4799.34 + 8312.70i −2561.58 − 4436.79i −26259.0 + 9557.48i 8348.86 − 3038.74i
4.9 14.5258 + 5.28695i 28.8772 + 163.771i −209.168 − 175.513i −320.003 + 268.514i −446.385 + 2531.57i −4250.37 + 7361.86i −6067.66 − 10509.5i −7491.03 + 2726.51i −6067.91 + 2208.54i
4.10 15.8650 + 5.77439i 2.44302 + 13.8551i −173.860 − 145.886i 510.781 − 428.596i −41.2460 + 233.918i 2707.15 − 4688.92i −6237.98 − 10804.5i 18310.0 − 6664.29i 10578.4 − 3850.24i
4.11 27.8690 + 10.1435i −41.7136 − 236.569i 281.577 + 236.271i −1018.90 + 854.955i 1237.12 − 7016.08i 2734.11 − 4735.62i −2141.68 − 3709.50i −35729.0 + 13004.3i −37067.8 + 13491.6i
4.12 32.6060 + 11.8676i 6.30624 + 35.7645i 530.099 + 444.806i −1727.69 + 1449.70i −218.818 + 1240.98i −2859.56 + 4952.91i 3122.78 + 5408.82i 17256.6 − 6280.90i −73537.6 + 26765.5i
4.13 35.4682 + 12.9094i 46.3184 + 262.685i 699.128 + 586.638i 389.878 − 327.147i −1748.26 + 9914.90i 5610.72 − 9718.06i 7561.11 + 13096.2i −48361.8 + 17602.3i 18051.5 − 6570.23i
4.14 37.3967 + 13.6113i −11.8725 − 67.3325i 821.033 + 688.928i 1348.25 − 1131.32i 472.489 − 2679.62i −1975.12 + 3421.01i 11138.8 + 19292.9i 14103.3 − 5133.17i 65819.0 − 23956.2i
5.1 −37.1255 + 13.5126i 23.8217 − 135.100i 803.500 − 674.217i −60.2331 − 50.5416i 941.151 + 5337.53i 1475.00 + 2554.78i −10605.9 + 18369.9i 811.556 + 295.382i 2919.13 + 1062.48i
5.2 −36.8078 + 13.3970i −43.1817 + 244.896i 783.124 − 657.119i −962.522 − 807.652i −1691.43 − 9592.59i 83.0798 + 143.899i −9994.16 + 17310.4i −39613.3 − 14418.1i 46248.4 + 16833.0i
5.3 −27.3759 + 9.96400i −3.09587 + 17.5575i 257.942 − 216.439i −58.0769 − 48.7323i −90.1913 − 511.500i −5103.16 − 8838.93i 2553.21 − 4422.29i 18197.3 + 6623.27i 2075.47 + 755.411i
5.4 −24.2695 + 8.83336i −19.4747 + 110.447i 118.763 − 99.6543i 1548.70 + 1299.52i −502.974 − 2852.50i 4850.18 + 8400.77i 4609.68 − 7984.19i 6676.79 + 2430.15i −49065.3 − 17858.3i
5.5 −13.2468 + 4.82143i 31.6453 − 179.469i −239.984 + 201.370i −1449.37 − 1216.17i 446.101 + 2529.97i 2251.38 + 3899.50i 5816.93 − 10075.2i −12711.8 − 4626.73i 25063.1 + 9122.24i
5.6 −10.0828 + 3.66983i 33.0850 − 187.634i −304.020 + 255.103i 1739.50 + 1459.62i 354.998 + 2013.29i −1079.55 − 1869.83i 4876.03 − 8445.53i −15616.0 − 5683.75i −22895.6 − 8333.31i
See all 84 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.14
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 19.10.e.a ✓ 84
19.e even 9 1 inner 19.10.e.a ✓ 84
19.e even 9 1 361.10.a.i 42
19.f odd 18 1 361.10.a.j 42
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.10.e.a ✓ 84 1.a even 1 1 trivial
19.10.e.a ✓ 84 19.e even 9 1 inner
361.10.a.i 42 19.e even 9 1
361.10.a.j 42 19.f odd 18 1

Hecke kernels

This newform subspace is the entire newspace \(S_{10}^{\mathrm{new}}(19, [\chi])\).