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Properties

Label 100.3.h.b
Level $100$
Weight $3$
Character orbit 100.h
Analytic conductor $2.725$
Analytic rank $0$
Dimension $104$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [100,3,Mod(19,100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(100, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 9]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("100.19");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 100 = 2^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 100.h (of order \(10\), degree \(4\), 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: \(2.72480264360\)
Analytic rank: \(0\)
Dimension: \(104\)
Relative dimension: \(26\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 104 q - 5 q^{2} - 11 q^{4} + 5 q^{6} + 10 q^{8} - 96 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 104 q - 5 q^{2} - 11 q^{4} + 5 q^{6} + 10 q^{8} - 96 q^{9} - 5 q^{10} - 5 q^{12} - 10 q^{13} - 45 q^{14} + 89 q^{16} - 10 q^{17} + 155 q^{20} - 60 q^{21} + 70 q^{22} + 80 q^{24} - 60 q^{25} - 62 q^{26} - 85 q^{28} + 58 q^{29} + 35 q^{30} - 10 q^{33} - 187 q^{34} + 269 q^{36} - 410 q^{37} - 485 q^{38} - 230 q^{40} - 142 q^{41} - 535 q^{42} - 140 q^{44} + 60 q^{45} + 105 q^{46} - 70 q^{48} + 904 q^{49} + 285 q^{50} - 220 q^{52} + 510 q^{53} + 245 q^{54} - 240 q^{56} + 250 q^{58} + 1000 q^{60} + 138 q^{61} + 320 q^{62} + 124 q^{64} - 510 q^{65} - 410 q^{66} - 210 q^{69} - 170 q^{70} - 185 q^{72} - 10 q^{73} - 12 q^{74} - 40 q^{76} - 500 q^{77} + 755 q^{78} - 130 q^{80} + 324 q^{81} + 450 q^{84} - 490 q^{85} - 465 q^{86} + 1150 q^{88} - 162 q^{89} + 795 q^{90} + 930 q^{92} + 465 q^{94} - 700 q^{96} - 210 q^{97} - 200 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −1.99643 + 0.119417i 3.51088 − 2.55080i 3.97148 − 0.476814i −0.215508 − 4.99535i −6.70463 + 5.51177i 2.45065 −7.87185 + 1.42619i 3.03853 − 9.35164i 1.02678 + 9.94715i
19.2 −1.98436 − 0.249604i 0.964687 − 0.700886i 3.87540 + 0.990609i 4.11154 + 2.84522i −2.08923 + 1.15002i −4.12530 −7.44293 − 2.93304i −2.34177 + 7.20724i −7.44861 − 6.67220i
19.3 −1.93705 − 0.497815i −4.58342 + 3.33005i 3.50436 + 1.92859i −0.534476 + 4.97135i 10.5361 − 4.16879i −0.848461 −5.82805 − 5.48031i 7.13737 − 21.9666i 3.51013 − 9.36371i
19.4 −1.68534 + 1.07686i −3.51088 + 2.55080i 1.68073 − 3.62976i −0.215508 − 4.99535i 3.17016 − 8.07971i −2.45065 1.07615 + 7.92729i 3.03853 − 9.35164i 5.74252 + 8.18679i
19.5 −1.56208 − 1.24896i −0.874232 + 0.635166i 0.880212 + 3.90195i −4.85105 − 1.21134i 2.15892 + 0.0996945i 12.5983 3.49840 − 7.19452i −2.42031 + 7.44894i 6.06484 + 7.95096i
19.6 −1.54210 − 1.27355i −1.40730 + 1.02246i 0.756140 + 3.92788i 1.89815 − 4.62569i 3.47236 + 0.215528i −9.70036 3.83631 − 7.02017i −1.84609 + 5.68168i −8.81818 + 4.71589i
19.7 −1.45867 + 1.36831i −0.964687 + 0.700886i 0.255438 − 3.99184i 4.11154 + 2.84522i 0.448128 − 2.34236i 4.12530 5.08948 + 6.17229i −2.34177 + 7.20724i −9.89053 + 1.47564i
19.8 −1.27450 + 1.54131i 4.58342 − 3.33005i −0.751293 − 3.92881i −0.534476 + 4.97135i −0.708926 + 11.3086i 0.848461 7.01305 + 3.84930i 7.13737 − 21.9666i −6.98122 − 7.15979i
19.9 −1.16882 − 1.62292i 3.20468 − 2.32834i −1.26773 + 3.79379i 4.46886 + 2.24260i −7.52439 − 2.47953i 9.75139 7.63876 − 2.37684i 2.06767 − 6.36363i −1.58373 − 9.87379i
19.10 −0.529634 + 1.92860i 0.874232 − 0.635166i −3.43898 − 2.04290i −4.85105 − 1.21134i 0.761958 + 2.02245i −12.5983 5.76133 − 5.55041i −2.42031 + 7.44894i 4.90546 − 8.71415i
19.11 −0.512971 − 1.93310i 3.71517 − 2.69923i −3.47372 + 1.98324i −3.49258 − 3.57797i −7.12364 − 5.79715i −8.45630 5.61572 + 5.69769i 3.73549 − 11.4967i −5.12497 + 8.58689i
19.12 −0.499011 + 1.93675i 1.40730 − 1.02246i −3.50198 − 1.93291i 1.89815 − 4.62569i 1.27800 + 3.23581i 9.70036 5.49109 − 5.81790i −1.84609 + 5.68168i 8.01160 + 5.98450i
19.13 −0.375929 − 1.96435i −0.801540 + 0.582353i −3.71736 + 1.47691i −1.47715 + 4.77682i 1.44527 + 1.35558i −4.49935 4.29863 + 6.74698i −2.47782 + 7.62595i 9.93866 + 1.10590i
19.14 −0.206447 − 1.98932i −3.97732 + 2.88969i −3.91476 + 0.821376i 4.55404 − 2.06416i 6.56962 + 7.31558i 7.33448 2.44217 + 7.61812i 4.68760 − 14.4270i −5.04643 − 8.63328i
19.15 0.00833349 + 1.99998i −3.20468 + 2.32834i −3.99986 + 0.0333337i 4.46886 + 2.24260i −4.68334 − 6.38990i −9.75139 −0.0999996 − 7.99937i 2.06767 − 6.36363i −4.44793 + 8.95634i
19.16 0.721244 + 1.86542i −3.71517 + 2.69923i −2.95962 + 2.69085i −3.49258 − 3.57797i −7.71474 − 4.98356i 8.45630 −7.15418 − 3.58018i 3.73549 − 11.4967i 4.15543 − 9.09574i
19.17 0.828572 − 1.82029i 1.65383 − 1.20158i −2.62694 − 3.01649i 4.75867 − 1.53462i −0.816904 − 4.00604i −1.33970 −7.66750 + 2.28241i −1.48979 + 4.58512i 1.14944 − 9.93372i
19.18 0.850484 + 1.81016i 0.801540 − 0.582353i −2.55335 + 3.07902i −1.47715 + 4.77682i 1.73585 + 0.955633i 4.49935 −7.74511 − 2.00331i −2.47782 + 7.62595i −9.90310 + 1.38873i
19.19 1.00227 + 1.73074i 3.97732 − 2.88969i −1.99090 + 3.46934i 4.55404 − 2.06416i 8.98765 + 3.98744i −7.33448 −7.99994 + 0.0314907i 4.68760 − 14.4270i 8.13690 + 5.81300i
19.20 1.03678 − 1.71029i −2.86831 + 2.08395i −1.85019 − 3.54638i −4.61115 − 1.93321i 0.590359 + 7.06623i −3.59926 −7.98357 − 0.512450i 1.10320 − 3.39529i −8.08709 + 5.88209i
See next 80 embeddings (of 104 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.26
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
25.e even 10 1 inner
100.h odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 100.3.h.b ✓ 104
4.b odd 2 1 inner 100.3.h.b ✓ 104
25.e even 10 1 inner 100.3.h.b ✓ 104
100.h odd 10 1 inner 100.3.h.b ✓ 104
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
100.3.h.b ✓ 104 1.a even 1 1 trivial
100.3.h.b ✓ 104 4.b odd 2 1 inner
100.3.h.b ✓ 104 25.e even 10 1 inner
100.3.h.b ✓ 104 100.h odd 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{104} + 165 T_{3}^{102} + 15165 T_{3}^{100} + 1027915 T_{3}^{98} + 57623685 T_{3}^{96} + \cdots + 11\!\cdots\!00 \) acting on \(S_{3}^{\mathrm{new}}(100, [\chi])\). Copy content Toggle raw display