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Properties

Label 98.8.e.b
Level $98$
Weight $8$
Character orbit 98.e
Analytic conductor $30.614$
Analytic rank $0$
Dimension $102$
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(15,98)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(98, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([10]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("98.15");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 98 = 2 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 98.e (of order \(7\), degree \(6\), 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: \(102\)
Relative dimension: \(17\) over \(\Q(\zeta_{7})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 102 q + 136 q^{2} - 145 q^{3} - 1088 q^{4} - 28 q^{5} + 1160 q^{6} - 673 q^{7} + 8704 q^{8} - 16446 q^{9} + 224 q^{10} - 14973 q^{11} + 3712 q^{12} - 21458 q^{13} + 5384 q^{14} + 8071 q^{15} - 69632 q^{16}+ \cdots - 175838922 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
15.1 1.78017 − 7.79942i −55.7774 + 69.9426i −57.6620 − 27.7686i 91.4953 − 114.731i 446.219 + 559.541i −813.026 + 403.153i −319.227 + 400.298i −1294.20 − 5670.27i −731.962 − 917.852i
15.2 1.78017 − 7.79942i −52.5259 + 65.8654i −57.6620 − 27.7686i −152.040 + 190.653i 420.208 + 526.924i 679.141 − 601.922i −319.227 + 400.298i −1092.63 − 4787.12i 1216.32 + 1525.22i
15.3 1.78017 − 7.79942i −38.4819 + 48.2548i −57.6620 − 27.7686i −55.7583 + 69.9187i 307.855 + 386.038i 328.861 + 845.809i −319.227 + 400.298i −361.014 − 1581.71i 446.066 + 559.349i
15.4 1.78017 − 7.79942i −33.9176 + 42.5313i −57.6620 − 27.7686i 325.129 − 407.699i 271.341 + 340.250i 879.931 + 221.957i −319.227 + 400.298i −171.856 − 752.950i −2601.03 − 3261.59i
15.5 1.78017 − 7.79942i −32.5928 + 40.8701i −57.6620 − 27.7686i −312.974 + 392.457i 260.742 + 326.960i −741.847 − 522.691i −319.227 + 400.298i −121.419 − 531.971i 2503.79 + 3139.66i
15.6 1.78017 − 7.79942i −27.4194 + 34.3829i −57.6620 − 27.7686i 195.595 − 245.269i 219.356 + 275.063i −226.683 − 878.725i −319.227 + 400.298i 56.2955 + 246.647i −1564.76 − 1962.15i
15.7 1.78017 − 7.79942i −11.6444 + 14.6016i −57.6620 − 27.7686i −57.8585 + 72.5522i 93.1551 + 116.813i 209.537 − 882.971i −319.227 + 400.298i 409.038 + 1792.11i 462.868 + 580.418i
15.8 1.78017 − 7.79942i −10.2478 + 12.8504i −57.6620 − 27.7686i −258.215 + 323.791i 81.9827 + 102.803i 397.750 + 815.683i −319.227 + 400.298i 426.539 + 1868.79i 2065.72 + 2590.33i
15.9 1.78017 − 7.79942i −9.65569 + 12.1078i −57.6620 − 27.7686i 9.08707 − 11.3948i 77.2455 + 96.8628i −905.628 − 58.1530i −319.227 + 400.298i 433.286 + 1898.35i −72.6966 − 91.1586i
15.10 1.78017 − 7.79942i 7.56516 − 9.48641i −57.6620 − 27.7686i 43.7139 − 54.8155i −60.5213 − 75.8913i 838.976 − 345.923i −319.227 + 400.298i 453.893 + 1988.64i −349.711 − 438.524i
15.11 1.78017 − 7.79942i 18.3592 − 23.0217i −57.6620 − 27.7686i 219.842 − 275.673i −146.874 − 184.174i −662.230 + 620.480i −319.227 + 400.298i 293.714 + 1286.85i −1758.73 − 2205.38i
15.12 1.78017 − 7.79942i 29.1030 − 36.4940i −57.6620 − 27.7686i −180.044 + 225.768i −232.824 − 291.952i 904.138 + 77.9552i −319.227 + 400.298i 1.82441 + 7.99325i 1440.35 + 1806.14i
15.13 1.78017 − 7.79942i 29.1375 − 36.5372i −57.6620 − 27.7686i 197.304 − 247.412i −233.100 − 292.298i 463.570 + 780.157i −319.227 + 400.298i 0.676716 + 2.96488i −1578.44 − 1979.30i
15.14 1.78017 − 7.79942i 29.3763 − 36.8367i −57.6620 − 27.7686i −202.735 + 254.222i −235.010 − 294.693i −898.131 + 130.015i −319.227 + 400.298i −7.32243 − 32.0817i 1621.88 + 2033.77i
15.15 1.78017 − 7.79942i 35.7645 − 44.8473i −57.6620 − 27.7686i 282.977 − 354.842i −286.116 − 358.778i −218.313 − 880.842i −319.227 + 400.298i −245.526 − 1075.72i −2263.82 − 2838.74i
15.16 1.78017 − 7.79942i 48.7234 − 61.0972i −57.6620 − 27.7686i −110.281 + 138.288i −389.787 − 488.777i −502.110 + 755.929i −319.227 + 400.298i −872.244 − 3821.55i 882.248 + 1106.30i
15.17 1.78017 − 7.79942i 51.6871 − 64.8135i −57.6620 − 27.7686i −39.9684 + 50.1188i −413.497 − 518.508i 27.7587 − 907.068i −319.227 + 400.298i −1042.59 − 4567.88i 319.747 + 400.951i
29.1 7.20775 + 3.47107i −19.2316 − 84.2593i 39.9033 + 50.0372i −89.6368 − 392.724i 153.853 − 674.074i −412.536 + 808.305i 113.931 + 499.163i −4759.35 + 2291.98i 717.094 − 3141.79i
29.2 7.20775 + 3.47107i −18.3841 − 80.5460i 39.9033 + 50.0372i 36.1367 + 158.325i 147.073 − 644.368i 905.041 − 66.6577i 113.931 + 499.163i −4179.26 + 2012.63i −289.093 + 1266.60i
29.3 7.20775 + 3.47107i −13.2057 − 57.8581i 39.9033 + 50.0372i 107.513 + 471.047i 105.646 − 462.865i 100.323 + 901.930i 113.931 + 499.163i −1202.75 + 579.212i −860.106 + 3768.37i
See next 80 embeddings (of 102 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 15.17
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
49.e even 7 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 98.8.e.b ✓ 102
49.e even 7 1 inner 98.8.e.b ✓ 102
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
98.8.e.b ✓ 102 1.a even 1 1 trivial
98.8.e.b ✓ 102 49.e even 7 1 inner