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Properties

Label 98.7.h.a
Level $98$
Weight $7$
Character orbit 98.h
Analytic conductor $22.545$
Analytic rank $0$
Dimension $336$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [98,7,Mod(3,98)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("98.3"); S:= CuspForms(chi, 7); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(98, base_ring=CyclotomicField(42)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 7, names="a")
 
Level: \( N \) \(=\) \( 98 = 2 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 98.h (of order \(42\), degree \(12\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(22.5453001947\)
Analytic rank: \(0\)
Dimension: \(336\)
Relative dimension: \(28\) over \(\Q(\zeta_{42})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{42}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 336 q + 896 q^{4} + 336 q^{5} - 784 q^{6} - 652 q^{7} - 11732 q^{9} + 2016 q^{10} + 9772 q^{11} - 2064 q^{14} + 3948 q^{15} + 28672 q^{16} + 11032 q^{17} + 5600 q^{18} + 32004 q^{19} + 25088 q^{20} - 9756 q^{21}+ \cdots + 7151816 q^{99}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 −2.06668 + 5.26582i −52.8527 − 3.96076i −23.4577 − 21.7655i −44.0561 + 64.6185i 130.086 − 270.127i −71.2258 + 335.523i 163.093 − 78.5413i 2056.86 + 310.022i −249.219 − 365.537i
3.2 −2.06668 + 5.26582i −41.1635 − 3.08478i −23.4577 − 21.7655i 104.098 − 152.683i 101.316 − 210.384i −19.6271 − 342.438i 163.093 − 78.5413i 964.063 + 145.309i 588.865 + 863.706i
3.3 −2.06668 + 5.26582i −33.6973 − 2.52526i −23.4577 − 21.7655i −66.4096 + 97.4050i 82.9391 − 172.225i 212.606 − 269.161i 163.093 − 78.5413i 408.272 + 61.5371i −375.670 − 551.006i
3.4 −2.06668 + 5.26582i −29.7262 − 2.22767i −23.4577 − 21.7655i 124.432 − 182.509i 73.1651 − 151.929i 136.163 + 314.815i 163.093 − 78.5413i 157.828 + 23.7888i 703.895 + 1032.42i
3.5 −2.06668 + 5.26582i −24.6197 − 1.84499i −23.4577 − 21.7655i −51.6733 + 75.7908i 60.5965 − 125.830i 286.909 + 187.969i 163.093 − 78.5413i −118.131 − 17.8054i −292.308 − 428.737i
3.6 −2.06668 + 5.26582i −11.9092 − 0.892473i −23.4577 − 21.7655i 35.0497 − 51.4084i 29.3122 − 60.8673i −328.937 + 97.2087i 163.093 − 78.5413i −579.825 − 87.3945i 198.271 + 290.810i
3.7 −2.06668 + 5.26582i −10.5573 − 0.791157i −23.4577 − 21.7655i −130.889 + 191.979i 25.9846 − 53.9575i −312.934 − 140.432i 163.093 − 78.5413i −610.028 − 91.9469i −740.420 − 1086.00i
3.8 −2.06668 + 5.26582i 8.19795 + 0.614352i −23.4577 − 21.7655i 65.1207 − 95.5145i −20.1776 + 41.8993i 37.8699 − 340.903i 163.093 − 78.5413i −654.029 − 98.5790i 368.378 + 540.311i
3.9 −2.06668 + 5.26582i 11.4270 + 0.856333i −23.4577 − 21.7655i −14.7182 + 21.5876i −28.1252 + 58.4025i 11.4188 + 342.810i 163.093 − 78.5413i −591.015 − 89.0812i −83.2587 − 122.118i
3.10 −2.06668 + 5.26582i 14.7555 + 1.10577i −23.4577 − 21.7655i −11.5446 + 16.9328i −36.3178 + 75.4146i 342.323 − 21.5414i 163.093 − 78.5413i −504.355 − 76.0193i −65.3060 − 95.7863i
3.11 −2.06668 + 5.26582i 30.7582 + 2.30501i −23.4577 − 21.7655i −62.0936 + 91.0746i −75.7052 + 157.203i −234.238 − 250.562i 163.093 − 78.5413i 219.897 + 33.1442i −351.255 − 515.196i
3.12 −2.06668 + 5.26582i 36.5368 + 2.73805i −23.4577 − 21.7655i 129.707 − 190.245i −89.9280 + 186.737i 323.897 + 112.869i 163.093 − 78.5413i 606.583 + 91.4277i 733.732 + 1076.19i
3.13 −2.06668 + 5.26582i 43.4502 + 3.25615i −23.4577 − 21.7655i −10.5632 + 15.4934i −106.944 + 222.072i −156.550 + 305.190i 163.093 − 78.5413i 1156.46 + 174.309i −59.7545 − 87.6438i
3.14 −2.06668 + 5.26582i 48.2782 + 3.61795i −23.4577 − 21.7655i −99.3986 + 145.791i −118.827 + 246.747i 280.579 − 197.292i 163.093 − 78.5413i 1596.84 + 240.685i −562.283 − 824.718i
3.15 2.06668 − 5.26582i −47.7441 − 3.57792i −23.4577 − 21.7655i −9.18434 + 13.4710i −117.512 + 244.017i 342.199 − 23.4251i −163.093 + 78.5413i 1545.84 + 232.997i 51.9545 + 76.2032i
3.16 2.06668 − 5.26582i −38.3941 − 2.87724i −23.4577 − 21.7655i −120.584 + 176.865i −94.4994 + 196.230i −287.248 + 187.450i −163.093 + 78.5413i 744.970 + 112.286i 682.128 + 1000.50i
3.17 2.06668 − 5.26582i −31.5552 − 2.36474i −23.4577 − 21.7655i 82.7859 − 121.425i −77.6669 + 161.277i −64.1407 + 336.950i −163.093 + 78.5413i 269.283 + 40.5879i −468.308 − 686.881i
3.18 2.06668 − 5.26582i −26.2643 − 1.96824i −23.4577 − 21.7655i −16.4360 + 24.1072i −64.6443 + 134.235i −1.57940 − 342.996i −163.093 + 78.5413i −34.9190 − 5.26320i 92.9761 + 136.371i
3.19 2.06668 − 5.26582i −14.5392 − 1.08956i −23.4577 − 21.7655i 6.62526 − 9.71747i −35.7853 + 74.3089i −134.567 − 315.501i −163.093 + 78.5413i −510.657 − 76.9691i −37.4781 − 54.9703i
3.20 2.06668 − 5.26582i −4.42405 − 0.331536i −23.4577 − 21.7655i −40.7642 + 59.7902i −10.8889 + 22.6110i −223.680 + 260.032i −163.093 + 78.5413i −701.395 − 105.718i 230.597 + 338.224i
See next 80 embeddings (of 336 total)
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 3.28
Significant digits:
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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 98.7.h.a ✓ 336
49.h odd 42 1 inner 98.7.h.a ✓ 336
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
98.7.h.a ✓ 336 1.a even 1 1 trivial
98.7.h.a ✓ 336 49.h odd 42 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(98, [\chi])\).