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Properties

Label 98.8.g.a
Level $98$
Weight $8$
Character orbit 98.g
Analytic conductor $30.614$
Analytic rank $0$
Dimension $192$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [98,8,Mod(9,98)]
 
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([2]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("98.9");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 98 = 2 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 98.g (of order \(21\), 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: \(30.6137324974\)
Analytic rank: \(0\)
Dimension: \(192\)
Relative dimension: \(16\) over \(\Q(\zeta_{21})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{21}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 192 q - 128 q^{2} + 259 q^{3} + 1024 q^{4} - 14 q^{5} - 728 q^{6} + 1848 q^{7} + 16384 q^{8} + 12105 q^{9} + 112 q^{10} + 12789 q^{11} + 3584 q^{12} - 9870 q^{13} + 22176 q^{14} - 8071 q^{15} + 65536 q^{16}+ \cdots + 175838922 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1 5.86441 + 5.44138i −84.7851 − 12.7793i 4.78273 + 63.8210i −115.488 − 294.260i −427.678 − 536.291i −102.397 − 901.697i −319.227 + 400.298i 4935.37 + 1522.36i 923.907 − 2354.08i
9.2 5.86441 + 5.44138i −75.6682 − 11.4051i 4.78273 + 63.8210i 90.3509 + 230.210i −381.690 − 478.624i 123.140 + 899.099i −319.227 + 400.298i 3505.76 + 1081.38i −722.807 + 1841.68i
9.3 5.86441 + 5.44138i −66.5794 − 10.0352i 4.78273 + 63.8210i −18.5758 − 47.3305i −335.844 − 421.135i 858.678 + 293.624i −319.227 + 400.298i 2242.28 + 691.651i 148.607 − 378.644i
9.4 5.86441 + 5.44138i −56.3081 − 8.48708i 4.78273 + 63.8210i 81.0622 + 206.543i −284.033 − 356.166i −899.426 − 120.729i −319.227 + 400.298i 1008.73 + 311.153i −648.498 + 1652.35i
9.5 5.86441 + 5.44138i −44.0648 − 6.64170i 4.78273 + 63.8210i −119.922 − 305.555i −222.274 − 278.723i −714.362 + 559.670i −319.227 + 400.298i −192.244 − 59.2995i 959.373 − 2444.44i
9.6 5.86441 + 5.44138i −24.5910 − 3.70650i 4.78273 + 63.8210i −94.1241 − 239.824i −124.044 − 155.546i 95.2065 − 902.485i −319.227 + 400.298i −1498.86 − 462.336i 752.993 − 1918.59i
9.7 5.86441 + 5.44138i −11.4009 − 1.71841i 4.78273 + 63.8210i 116.184 + 296.031i −57.5090 − 72.1140i −834.528 − 356.519i −319.227 + 400.298i −1962.81 − 605.447i −929.468 + 2368.25i
9.8 5.86441 + 5.44138i −5.67930 − 0.856016i 4.78273 + 63.8210i −10.0713 − 25.6613i −28.6478 − 35.9233i 878.717 − 226.714i −319.227 + 400.298i −2058.32 − 634.906i 80.5704 − 205.290i
9.9 5.86441 + 5.44138i 6.05998 + 0.913395i 4.78273 + 63.8210i 53.6783 + 136.770i 30.5681 + 38.3312i 400.369 + 814.400i −319.227 + 400.298i −2053.95 − 633.559i −429.426 + 1094.16i
9.10 5.86441 + 5.44138i 19.6534 + 2.96227i 4.78273 + 63.8210i 186.155 + 474.315i 99.1368 + 124.314i 149.871 + 895.032i −319.227 + 400.298i −1712.36 − 528.192i −1489.24 + 3794.52i
9.11 5.86441 + 5.44138i 22.4554 + 3.38461i 4.78273 + 63.8210i −188.461 − 480.190i 113.271 + 142.037i 385.316 + 821.629i −319.227 + 400.298i −1597.05 − 492.624i 1507.69 − 3841.52i
9.12 5.86441 + 5.44138i 40.1709 + 6.05479i 4.78273 + 63.8210i 87.4053 + 222.705i 202.633 + 254.093i −183.592 − 888.728i −319.227 + 400.298i −512.795 − 158.176i −699.242 + 1781.64i
9.13 5.86441 + 5.44138i 50.0661 + 7.54625i 4.78273 + 63.8210i −87.2444 − 222.295i 252.546 + 316.683i −812.549 − 404.113i −319.227 + 400.298i 359.830 + 110.993i 697.955 − 1778.36i
9.14 5.86441 + 5.44138i 69.9795 + 10.5477i 4.78273 + 63.8210i 142.350 + 362.702i 352.995 + 442.641i 512.540 − 748.896i −319.227 + 400.298i 2696.04 + 831.618i −1138.80 + 2901.62i
9.15 5.86441 + 5.44138i 75.6338 + 11.4000i 4.78273 + 63.8210i −138.922 − 353.967i 381.517 + 478.407i 835.662 − 353.854i −319.227 + 400.298i 3500.68 + 1079.82i 1111.38 − 2831.74i
9.16 5.86441 + 5.44138i 79.3124 + 11.9544i 4.78273 + 63.8210i 24.7543 + 63.0729i 400.072 + 501.675i −250.767 + 872.157i −319.227 + 400.298i 4057.71 + 1251.64i −198.034 + 504.583i
11.1 5.86441 − 5.44138i −84.7851 + 12.7793i 4.78273 − 63.8210i −115.488 + 294.260i −427.678 + 536.291i −102.397 + 901.697i −319.227 − 400.298i 4935.37 − 1522.36i 923.907 + 2354.08i
11.2 5.86441 − 5.44138i −75.6682 + 11.4051i 4.78273 − 63.8210i 90.3509 − 230.210i −381.690 + 478.624i 123.140 − 899.099i −319.227 − 400.298i 3505.76 − 1081.38i −722.807 − 1841.68i
11.3 5.86441 − 5.44138i −66.5794 + 10.0352i 4.78273 − 63.8210i −18.5758 + 47.3305i −335.844 + 421.135i 858.678 − 293.624i −319.227 − 400.298i 2242.28 − 691.651i 148.607 + 378.644i
11.4 5.86441 − 5.44138i −56.3081 + 8.48708i 4.78273 − 63.8210i 81.0622 − 206.543i −284.033 + 356.166i −899.426 + 120.729i −319.227 − 400.298i 1008.73 − 311.153i −648.498 − 1652.35i
See next 80 embeddings (of 192 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.16
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
49.g even 21 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 98.8.g.a ✓ 192
49.g even 21 1 inner 98.8.g.a ✓ 192
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
98.8.g.a ✓ 192 1.a even 1 1 trivial
98.8.g.a ✓ 192 49.g even 21 1 inner