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Properties

Label 98.8.g.b
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.

\(\operatorname{Tr}(f)(q) = \) \( 192 q + 128 q^{2} - 259 q^{3} + 1024 q^{4} - 238 q^{5} - 728 q^{6} - 168 q^{7} - 16384 q^{8} + 15797 q^{9} - 1904 q^{10} + 16593 q^{11} - 3584 q^{12} - 14210 q^{13} + 2016 q^{14} + 36073 q^{15} + 65536 q^{16}+ \cdots - 12378850 q^{99}+O(q^{100}) \) 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 −75.7140 − 11.4121i 4.78273 + 63.8210i 108.271 + 275.870i 381.921 + 478.914i 800.419 + 427.637i 319.227 − 400.298i 3512.54 + 1083.48i 866.168 − 2206.96i
9.2 −5.86441 − 5.44138i −73.5376 − 11.0840i 4.78273 + 63.8210i −166.596 − 424.481i 370.942 + 465.147i 411.362 − 808.903i 319.227 − 400.298i 3195.08 + 985.551i −1332.77 + 3395.85i
9.3 −5.86441 − 5.44138i −72.3491 − 10.9049i 4.78273 + 63.8210i −7.21605 − 18.3862i 364.948 + 457.630i −907.483 + 4.19463i 319.227 − 400.298i 3025.64 + 933.285i −57.7284 + 147.090i
9.4 −5.86441 − 5.44138i −55.4096 − 8.35166i 4.78273 + 63.8210i 26.0816 + 66.4548i 279.501 + 350.483i −189.144 − 887.563i 319.227 − 400.298i 910.641 + 280.896i 208.653 − 531.638i
9.5 −5.86441 − 5.44138i −37.9974 − 5.72719i 4.78273 + 63.8210i −101.292 − 258.087i 191.669 + 240.345i −331.590 + 844.743i 319.227 − 400.298i −678.835 − 209.393i −810.332 + 2064.69i
9.6 −5.86441 − 5.44138i −21.7480 − 3.27799i 4.78273 + 63.8210i 39.0505 + 99.4991i 109.703 + 137.563i 759.117 + 497.276i 319.227 − 400.298i −1627.61 − 502.050i 312.404 − 795.993i
9.7 −5.86441 − 5.44138i −17.9042 − 2.69862i 4.78273 + 63.8210i −184.621 − 470.406i 90.3132 + 113.249i 801.608 + 425.404i 319.227 − 400.298i −1776.56 − 547.997i −1476.97 + 3763.25i
9.8 −5.86441 − 5.44138i −10.9774 − 1.65457i 4.78273 + 63.8210i 178.263 + 454.206i 55.3727 + 69.4352i −405.835 − 811.690i 319.227 − 400.298i −1972.07 − 608.304i 1426.10 − 3633.65i
9.9 −5.86441 − 5.44138i 5.18439 + 0.781420i 4.78273 + 63.8210i 63.2085 + 161.053i −26.1514 − 32.7928i 584.371 − 694.300i 319.227 − 400.298i −2063.57 − 636.527i 505.668 − 1288.42i
9.10 −5.86441 − 5.44138i 9.93628 + 1.49765i 4.78273 + 63.8210i 58.1431 + 148.146i −50.1212 − 62.8499i −450.266 + 787.911i 319.227 − 400.298i −1993.35 − 614.867i 465.144 − 1185.17i
9.11 −5.86441 − 5.44138i 24.3984 + 3.67747i 4.78273 + 63.8210i −108.317 − 275.987i −123.072 − 154.327i −790.874 − 445.040i 319.227 − 400.298i −1508.08 − 465.180i −866.535 + 2207.89i
9.12 −5.86441 − 5.44138i 41.3624 + 6.23438i 4.78273 + 63.8210i −72.3162 − 184.259i −208.643 − 261.630i 663.372 − 619.258i 319.227 − 400.298i −417.856 − 128.892i −578.530 + 1474.07i
9.13 −5.86441 − 5.44138i 61.5884 + 9.28295i 4.78273 + 63.8210i 137.855 + 351.248i −310.668 − 389.565i 638.463 + 644.909i 319.227 − 400.298i 1617.12 + 498.814i 1102.84 − 2809.99i
9.14 −5.86441 − 5.44138i 65.6190 + 9.89047i 4.78273 + 63.8210i 106.681 + 271.820i −330.999 − 415.060i −906.956 − 31.2066i 319.227 − 400.298i 2118.19 + 653.375i 853.450 − 2174.56i
9.15 −5.86441 − 5.44138i 76.8199 + 11.5787i 4.78273 + 63.8210i −156.475 − 398.693i −387.499 − 485.909i −271.134 + 866.042i 319.227 − 400.298i 3677.39 + 1134.32i −1251.80 + 3189.54i
9.16 −5.86441 − 5.44138i 86.4739 + 13.0338i 4.78273 + 63.8210i −21.0780 − 53.7058i −436.197 − 546.973i 320.189 − 849.130i 319.227 − 400.298i 5218.02 + 1609.54i −168.624 + 429.646i
11.1 −5.86441 + 5.44138i −75.7140 + 11.4121i 4.78273 − 63.8210i 108.271 − 275.870i 381.921 − 478.914i 800.419 − 427.637i 319.227 + 400.298i 3512.54 − 1083.48i 866.168 + 2206.96i
11.2 −5.86441 + 5.44138i −73.5376 + 11.0840i 4.78273 − 63.8210i −166.596 + 424.481i 370.942 − 465.147i 411.362 + 808.903i 319.227 + 400.298i 3195.08 − 985.551i −1332.77 − 3395.85i
11.3 −5.86441 + 5.44138i −72.3491 + 10.9049i 4.78273 − 63.8210i −7.21605 + 18.3862i 364.948 − 457.630i −907.483 − 4.19463i 319.227 + 400.298i 3025.64 − 933.285i −57.7284 − 147.090i
11.4 −5.86441 + 5.44138i −55.4096 + 8.35166i 4.78273 − 63.8210i 26.0816 − 66.4548i 279.501 − 350.483i −189.144 + 887.563i 319.227 + 400.298i 910.641 − 280.896i 208.653 + 531.638i
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.b ✓ 192
49.g even 21 1 inner 98.8.g.b ✓ 192
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
98.8.g.b ✓ 192 1.a even 1 1 trivial
98.8.g.b ✓ 192 49.g even 21 1 inner