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Properties

Label 98.3.h.a
Level $98$
Weight $3$
Character orbit 98.h
Analytic conductor $2.670$
Analytic rank $0$
Dimension $120$
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] = [98,3,Mod(3,98)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(98, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("98.3");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 98 = 2 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 98.h (of order \(42\), degree \(12\), 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: \(2.67030659073\)
Analytic rank: \(0\)
Dimension: \(120\)
Relative dimension: \(10\) over \(\Q(\zeta_{42})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{42}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 120 q + 6 q^{3} + 20 q^{4} + 6 q^{5} - 28 q^{6} - 8 q^{7} + 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 120 q + 6 q^{3} + 20 q^{4} + 6 q^{5} - 28 q^{6} - 8 q^{7} + 28 q^{9} - 24 q^{10} - 76 q^{11} - 12 q^{12} + 36 q^{14} - 72 q^{15} + 40 q^{16} + 128 q^{17} + 8 q^{18} - 6 q^{19} + 28 q^{20} - 54 q^{21} + 104 q^{22} - 136 q^{23} - 24 q^{24} - 48 q^{25} - 136 q^{26} + 210 q^{27} + 20 q^{28} - 100 q^{29} + 4 q^{30} + 42 q^{31} + 90 q^{33} + 182 q^{35} + 224 q^{36} - 460 q^{37} - 268 q^{38} - 578 q^{39} - 176 q^{40} - 392 q^{41} - 688 q^{42} + 96 q^{43} - 124 q^{44} - 830 q^{45} - 180 q^{46} - 272 q^{47} + 132 q^{49} + 32 q^{50} + 318 q^{51} - 12 q^{52} + 936 q^{53} + 540 q^{54} + 1246 q^{55} + 176 q^{56} - 718 q^{57} + 776 q^{58} + 498 q^{59} + 324 q^{60} + 896 q^{61} + 392 q^{62} + 1182 q^{63} - 160 q^{64} + 28 q^{65} + 144 q^{66} + 122 q^{67} - 60 q^{68} + 56 q^{69} - 308 q^{70} + 638 q^{71} + 16 q^{72} - 444 q^{73} + 192 q^{74} - 1056 q^{75} + 84 q^{77} - 368 q^{78} - 158 q^{79} - 24 q^{80} - 790 q^{81} + 120 q^{82} - 1876 q^{83} + 408 q^{84} + 460 q^{85} - 428 q^{86} - 724 q^{87} - 160 q^{88} - 826 q^{89} + 672 q^{90} - 500 q^{91} + 40 q^{92} + 592 q^{93} + 292 q^{94} - 214 q^{95} + 48 q^{96} - 384 q^{98} + 176 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 −0.516670 + 1.31645i −5.70981 − 0.427891i −1.46610 − 1.36035i 1.14093 − 1.67344i 3.51339 − 7.29562i 6.77649 + 1.75474i 2.54832 − 1.22721i 23.5193 + 3.54497i 1.61352 + 2.36660i
3.2 −0.516670 + 1.31645i −1.33273 − 0.0998745i −1.46610 − 1.36035i 4.10019 − 6.01387i 0.820064 − 1.70288i −5.55976 − 4.25313i 2.54832 − 1.22721i −7.13327 − 1.07517i 5.79854 + 8.50490i
3.3 −0.516670 + 1.31645i −1.13707 − 0.0852119i −1.46610 − 1.36035i −3.86782 + 5.67305i 0.699670 − 1.45288i 0.289084 − 6.99403i 2.54832 − 1.22721i −7.61380 − 1.14760i −5.46992 − 8.02290i
3.4 −0.516670 + 1.31645i 3.50642 + 0.262770i −1.46610 − 1.36035i 2.17724 − 3.19343i −2.15759 + 4.48027i 6.99777 − 0.176820i 2.54832 − 1.22721i 3.32644 + 0.501380i 3.07908 + 4.51619i
3.5 −0.516670 + 1.31645i 4.81155 + 0.360576i −1.46610 − 1.36035i −4.79179 + 7.02827i −2.96067 + 6.14789i −5.88839 + 3.78508i 2.54832 − 1.22721i 14.1215 + 2.12848i −6.77662 − 9.93947i
3.6 0.516670 − 1.31645i −3.09176 − 0.231696i −1.46610 − 1.36035i 1.68520 − 2.47173i −1.90244 + 3.95046i −6.82949 − 1.53561i −2.54832 + 1.22721i 0.605847 + 0.0913168i −2.38323 − 3.49555i
3.7 0.516670 − 1.31645i −2.96982 − 0.222557i −1.46610 − 1.36035i −4.31436 + 6.32800i −1.82740 + 3.79464i 3.93232 + 5.79110i −2.54832 + 1.22721i −0.129176 − 0.0194702i 6.10143 + 8.94915i
3.8 0.516670 − 1.31645i 0.524575 + 0.0393115i −1.46610 − 1.36035i 1.91894 − 2.81457i 0.322784 − 0.670268i 5.78048 − 3.94792i −2.54832 + 1.22721i −8.62584 − 1.30014i −2.71379 − 3.98040i
3.9 0.516670 − 1.31645i 3.38979 + 0.254030i −1.46610 − 1.36035i 3.27752 − 4.80724i 2.08582 − 4.33125i −4.89024 + 5.00855i −2.54832 + 1.22721i 2.52667 + 0.380834i −4.63512 − 6.79847i
3.10 0.516670 − 1.31645i 5.46328 + 0.409416i −1.46610 − 1.36035i −2.85867 + 4.19290i 3.36169 − 6.98063i 3.67224 − 5.95942i −2.54832 + 1.22721i 20.7803 + 3.13213i 4.04277 + 5.92965i
5.1 −1.39842 + 0.210778i −2.20141 + 3.22887i 1.91115 − 0.589510i 8.14983 + 0.610745i 2.39791 − 4.97932i −6.62905 + 2.24850i −2.54832 + 1.22721i −2.29135 − 5.83826i −11.5256 + 0.863724i
5.2 −1.39842 + 0.210778i −0.503030 + 0.737810i 1.91115 − 0.589510i −1.58444 − 0.118737i 0.547933 − 1.13779i 6.87662 + 1.30847i −2.54832 + 1.22721i 2.99675 + 7.63558i 2.24074 − 0.167920i
5.3 −1.39842 + 0.210778i 0.908589 − 1.33266i 1.91115 − 0.589510i 3.18673 + 0.238812i −0.989693 + 2.05512i −2.88180 − 6.37928i −2.54832 + 1.22721i 2.33763 + 5.95619i −4.50671 + 0.337731i
5.4 −1.39842 + 0.210778i 1.01702 − 1.49170i 1.91115 − 0.589510i −9.03850 − 0.677342i −1.10780 + 2.30038i −5.54951 + 4.26649i −2.54832 + 1.22721i 2.09724 + 5.34369i 12.7824 − 0.957906i
5.5 −1.39842 + 0.210778i 3.34338 − 4.90384i 1.91115 − 0.589510i 5.12533 + 0.384091i −3.64182 + 7.56233i 1.76189 + 6.77464i −2.54832 + 1.22721i −9.58137 − 24.4129i −7.24832 + 0.543186i
5.6 1.39842 − 0.210778i −2.55119 + 3.74190i 1.91115 − 0.589510i −9.71864 − 0.728311i −2.77892 + 5.77048i 2.85328 + 6.39209i 2.54832 − 1.22721i −4.20522 − 10.7147i −13.7442 + 1.02999i
5.7 1.39842 − 0.210778i −1.23808 + 1.81593i 1.91115 − 0.589510i 5.24975 + 0.393415i −1.34860 + 2.80039i −3.33019 + 6.15710i 2.54832 − 1.22721i 1.52330 + 3.88131i 7.42427 − 0.556372i
5.8 1.39842 − 0.210778i −0.824985 + 1.21003i 1.91115 − 0.589510i 3.02159 + 0.226437i −0.898626 + 1.86602i 4.50565 − 5.35715i 2.54832 − 1.22721i 2.50450 + 6.38135i 4.27317 − 0.320230i
5.9 1.39842 − 0.210778i 1.77776 − 2.60750i 1.91115 − 0.589510i −4.65738 − 0.349023i 1.93645 − 4.02108i 6.62748 − 2.25313i 2.54832 − 1.22721i −0.350537 − 0.893154i −6.58654 + 0.493593i
5.10 1.39842 − 0.210778i 2.22334 − 3.26103i 1.91115 − 0.589510i 1.44145 + 0.108022i 2.42180 − 5.02892i −6.57588 + 2.39955i 2.54832 − 1.22721i −2.40306 − 6.12289i 2.03852 − 0.152766i
See next 80 embeddings (of 120 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
49.h odd 42 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 98.3.h.a ✓ 120
49.h odd 42 1 inner 98.3.h.a ✓ 120
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
98.3.h.a ✓ 120 1.a even 1 1 trivial
98.3.h.a ✓ 120 49.h odd 42 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(98, [\chi])\).