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Properties

Label 98.8.e.a
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.

\(\operatorname{Tr}(f)(q) = \) \( 102 q - 136 q^{2} + 199 q^{3} - 1088 q^{4} + 274 q^{5} + 1592 q^{6} - 343 q^{7} - 8704 q^{8} - 15026 q^{9} + 2192 q^{10} - 7365 q^{11} - 256 q^{12} + 22428 q^{13} - 2744 q^{14} - 36073 q^{15} - 69632 q^{16}+ \cdots + 12378850 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
15.1 −1.78017 + 7.79942i −56.7562 + 71.1701i −57.6620 − 27.7686i 225.835 − 283.188i −454.050 − 569.360i 867.502 − 266.427i 319.227 − 400.298i −1357.25 − 5946.52i 1806.68 + 2265.50i
15.2 −1.78017 + 7.79942i −54.1604 + 67.9150i −57.6620 − 27.7686i −294.379 + 369.139i −433.283 − 543.320i 501.476 + 756.349i 319.227 − 400.298i −1192.44 − 5224.43i −2355.03 − 2953.11i
15.3 −1.78017 + 7.79942i −38.9447 + 48.8351i −57.6620 − 27.7686i −143.458 + 179.891i −311.558 − 390.681i −649.820 − 633.464i 319.227 − 400.298i −381.526 − 1671.58i −1147.67 − 1439.13i
15.4 −1.78017 + 7.79942i −37.5566 + 47.0945i −57.6620 − 27.7686i 204.312 − 256.200i −300.453 − 376.756i −891.423 − 170.025i 319.227 − 400.298i −320.741 − 1405.26i 1634.50 + 2049.60i
15.5 −1.78017 + 7.79942i −25.0262 + 31.3819i −57.6620 − 27.7686i 102.644 − 128.712i −200.210 − 251.055i −345.305 + 839.230i 319.227 − 400.298i 128.142 + 561.427i 821.152 + 1029.69i
15.6 −1.78017 + 7.79942i −19.6631 + 24.6568i −57.6620 − 27.7686i 3.04150 − 3.81392i −157.305 − 197.254i 715.482 − 558.238i 319.227 − 400.298i 265.335 + 1162.51i 24.3320 + 30.5113i
15.7 −1.78017 + 7.79942i −15.8734 + 19.9046i −57.6620 − 27.7686i −58.6019 + 73.4844i −126.987 − 159.237i 842.259 + 337.851i 319.227 − 400.298i 342.424 + 1500.26i −468.815 − 587.875i
15.8 −1.78017 + 7.79942i −11.4446 + 14.3511i −57.6620 − 27.7686i −206.925 + 259.476i −91.5569 − 114.809i −432.215 + 797.956i 319.227 − 400.298i 411.679 + 1803.68i −1655.40 − 2075.81i
15.9 −1.78017 + 7.79942i −1.09526 + 1.37342i −57.6620 − 27.7686i 312.714 − 392.131i −8.76210 − 10.9873i 372.584 − 827.481i 319.227 − 400.298i 485.967 + 2129.16i 2501.71 + 3137.04i
15.10 −1.78017 + 7.79942i 15.2217 − 19.0875i −57.6620 − 27.7686i −295.340 + 370.344i 121.774 + 152.700i 660.360 − 622.469i 319.227 − 400.298i 354.024 + 1551.08i −2362.72 − 2962.75i
15.11 −1.78017 + 7.79942i 16.7315 − 20.9807i −57.6620 − 27.7686i −224.935 + 282.059i 133.852 + 167.845i −504.041 − 754.643i 319.227 − 400.298i 326.409 + 1430.09i −1799.48 − 2256.47i
15.12 −1.78017 + 7.79942i 22.2524 − 27.9036i −57.6620 − 27.7686i 308.589 − 386.959i 178.019 + 223.229i 396.272 + 816.401i 319.227 − 400.298i 203.211 + 890.324i 2468.72 + 3095.67i
15.13 −1.78017 + 7.79942i 22.5618 − 28.2916i −57.6620 − 27.7686i 84.6849 − 106.192i 180.494 + 226.333i −812.225 + 404.763i 319.227 − 400.298i 195.273 + 855.548i 677.479 + 849.532i
15.14 −1.78017 + 7.79942i 27.3099 − 34.2455i −57.6620 − 27.7686i 10.7890 − 13.5289i 218.479 + 273.964i 659.270 + 623.623i 319.227 − 400.298i 59.7271 + 261.682i 86.3117 + 108.231i
15.15 −1.78017 + 7.79942i 40.4768 − 50.7563i −57.6620 − 27.7686i 8.03553 − 10.0762i 323.815 + 406.051i −856.937 − 298.667i 319.227 − 400.298i −451.178 − 1976.74i 64.2843 + 80.6099i
15.16 −1.78017 + 7.79942i 51.2277 − 64.2375i −57.6620 − 27.7686i 97.8098 − 122.650i 409.822 + 513.900i 512.516 − 748.913i 319.227 − 400.298i −1015.53 − 4449.31i 782.478 + 981.197i
15.17 −1.78017 + 7.79942i 53.6172 − 67.2338i −57.6620 − 27.7686i −288.205 + 361.398i 428.937 + 537.870i −119.993 + 899.525i 319.227 − 400.298i −1158.93 − 5077.61i −2305.64 − 2891.18i
29.1 −7.20775 − 3.47107i −20.7448 − 90.8888i 39.9033 + 50.0372i −87.6751 − 384.130i −165.958 + 727.110i −339.848 − 841.455i −113.931 − 499.163i −5860.01 + 2822.03i −701.401 + 3073.04i
29.2 −7.20775 − 3.47107i −16.2091 − 71.0169i 39.9033 + 50.0372i 55.2644 + 242.129i −129.673 + 568.135i −534.226 + 733.584i −113.931 − 499.163i −2810.25 + 1353.34i 442.116 − 1937.03i
29.3 −7.20775 − 3.47107i −14.1864 − 62.1546i 39.9033 + 50.0372i 12.1770 + 53.3510i −113.491 + 497.237i −635.405 + 647.922i −113.931 − 499.163i −1691.52 + 814.594i 97.4161 − 426.808i
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:
Format:

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