Properties

Label 99.8.m.a
Level $99$
Weight $8$
Character orbit 99.m
Analytic conductor $30.926$
Analytic rank $0$
Dimension $656$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [99,8,Mod(4,99)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(99, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([10, 6]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("99.4");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 99 = 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 99.m (of order \(15\), degree \(8\), 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: \(30.9261175229\)
Analytic rank: \(0\)
Dimension: \(656\)
Relative dimension: \(82\) over \(\Q(\zeta_{15})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 656 q + 13 q^{2} - 45 q^{3} + 5117 q^{4} - 74 q^{5} - 1294 q^{6} - 3 q^{7} - 6668 q^{8} + 1163 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 656 q + 13 q^{2} - 45 q^{3} + 5117 q^{4} - 74 q^{5} - 1294 q^{6} - 3 q^{7} - 6668 q^{8} + 1163 q^{9} - 544 q^{10} + 3131 q^{11} + 18650 q^{12} - 3 q^{13} - 515 q^{14} - 21416 q^{15} + 311549 q^{16} - 6704 q^{17} - 77334 q^{18} + 6270 q^{19} - 84099 q^{20} + 256832 q^{21} + 34996 q^{22} + 221694 q^{23} - 53994 q^{24} + 1120250 q^{25} - 133804 q^{26} + 29295 q^{27} - 66060 q^{28} + 487120 q^{29} + 647118 q^{30} + 99324 q^{31} - 1802760 q^{32} - 1299668 q^{33} - 177208 q^{34} - 1389642 q^{35} + 154614 q^{36} - 151794 q^{37} - 471605 q^{38} - 224855 q^{39} + 106583 q^{40} + 975505 q^{41} - 5925059 q^{42} - 720650 q^{43} - 3117934 q^{44} - 1511840 q^{45} + 756 q^{46} + 158821 q^{47} - 6239960 q^{48} + 8235427 q^{49} + 3717606 q^{50} - 748887 q^{51} - 65283 q^{52} - 8401720 q^{53} - 29174618 q^{54} - 1333598 q^{55} + 4218834 q^{56} + 4108974 q^{57} + 1037869 q^{58} - 3883405 q^{59} + 9556791 q^{60} - 3 q^{61} - 1378934 q^{62} - 8591443 q^{63} - 35716364 q^{64} - 20951300 q^{65} - 23570487 q^{66} + 3711370 q^{67} + 5780506 q^{68} - 16405140 q^{69} - 7188604 q^{70} + 28467282 q^{71} + 20534207 q^{72} + 9608280 q^{73} + 43858953 q^{74} - 13191764 q^{75} + 434296 q^{76} + 6119027 q^{77} + 22508624 q^{78} - 9090579 q^{79} + 5818856 q^{80} + 16699595 q^{81} - 5685788 q^{82} + 49843106 q^{83} + 48240411 q^{84} + 2791747 q^{85} - 27455591 q^{86} - 121536 q^{87} - 21667796 q^{88} + 16511344 q^{89} - 141533716 q^{90} + 22130348 q^{91} - 55193155 q^{92} - 69307496 q^{93} - 5201099 q^{94} + 7179039 q^{95} + 28278755 q^{96} - 26736381 q^{97} + 197330000 q^{98} + 29658991 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −2.34818 + 22.3414i −46.7312 1.78684i −368.423 78.3108i 17.3772 + 165.333i 149.654 1039.85i 518.907 576.305i 1726.14 5312.50i 2180.61 + 167.003i −3734.58
4.2 −2.26625 + 21.5619i 44.1521 + 15.4141i −334.578 71.1168i 54.0291 + 514.053i −432.418 + 917.072i −408.404 + 453.579i 1434.09 4413.68i 1711.81 + 1361.13i −11206.4
4.3 −2.24266 + 21.3375i 45.1220 + 12.2885i −325.056 69.0927i −43.1214 410.273i −363.398 + 935.231i −319.671 + 355.030i 1354.62 4169.09i 1884.99 + 1108.96i 8850.90
4.4 −2.19218 + 20.8572i −9.85754 45.7146i −305.015 64.8330i −53.7246 511.155i 975.090 105.386i 655.464 727.966i 1191.35 3666.60i −1992.66 + 901.268i 10779.1
4.5 −2.14837 + 20.4404i −30.8731 + 35.1263i −287.992 61.2145i −28.2903 269.164i −651.668 706.522i −915.851 + 1017.16i 1057.01 3253.13i −280.708 2168.91i 5562.60
4.6 −2.13229 + 20.2874i −0.235807 + 46.7648i −281.828 59.9043i −3.72766 35.4663i −948.231 104.500i 702.013 779.664i 1009.37 3106.52i −2186.89 22.0549i 727.466
4.7 −2.04230 + 19.4312i 33.0489 33.0873i −248.199 52.7563i 13.5289 + 128.719i 575.430 + 709.756i 724.816 804.990i 759.197 2336.57i −2.53557 2187.00i −2528.80
4.8 −2.02291 + 19.2467i −8.36332 46.0115i −241.139 51.2558i 34.1422 + 324.841i 902.486 67.8893i 187.510 208.251i 708.825 2181.54i −2047.11 + 769.618i −6321.18
4.9 −2.01446 + 19.1663i −29.6154 36.1930i −238.086 50.6068i 12.9425 + 123.139i 753.344 494.708i −1042.21 + 1157.49i 687.279 2115.23i −432.859 + 2143.74i −2386.20
4.10 −1.98777 + 18.9123i 25.6286 39.1174i −228.522 48.5739i −11.4151 108.607i 688.858 + 562.453i −854.394 + 948.900i 620.711 1910.35i −873.346 2005.05i 2076.70
4.11 −1.86345 + 17.7296i 26.5285 + 38.5128i −185.662 39.4637i −8.46615 80.5501i −732.250 + 398.573i 171.770 190.770i 340.506 1047.97i −779.474 + 2043.38i 1443.89
4.12 −1.86085 + 17.7048i −0.345438 + 46.7641i −184.795 39.2793i 29.5692 + 281.332i −827.307 93.1369i −334.557 + 371.563i 335.152 1031.49i −2186.76 32.3082i −5035.95
4.13 −1.74350 + 16.5883i −44.5516 + 14.2182i −146.930 31.2310i 31.5299 + 299.987i −158.180 763.826i −37.8314 + 42.0161i 114.491 352.367i 1782.69 1266.89i −5031.26
4.14 −1.70543 + 16.2261i −45.5868 10.4327i −135.175 28.7324i −34.0746 324.198i 247.027 721.905i −155.649 + 172.866i 51.4014 158.197i 1969.32 + 951.188i 5318.59
4.15 −1.68021 + 15.9861i 45.0374 12.5948i −127.530 27.1073i −5.98543 56.9476i 125.670 + 741.135i 281.013 312.096i 11.8169 36.3685i 1869.74 1134.48i 920.427
4.16 −1.59052 + 15.1328i −36.1926 + 29.6158i −101.268 21.5252i −34.4217 327.501i −390.603 594.800i 1126.92 1251.57i −115.056 + 354.105i 432.815 2143.74i 5010.75
4.17 −1.53055 + 14.5622i −34.7710 + 31.2727i −84.5129 17.9638i 38.7925 + 369.086i −402.181 554.207i −323.317 + 359.080i −188.226 + 579.299i 231.041 2174.76i −5434.09
4.18 −1.46369 + 13.9261i 35.1415 + 30.8557i −66.5905 14.1542i −36.3863 346.193i −481.136 + 444.220i 85.8218 95.3147i −259.288 + 798.006i 282.846 + 2168.63i 4874.36
4.19 −1.46138 + 13.9041i −40.5122 23.3615i −65.9852 14.0256i 27.5332 + 261.961i 384.024 529.146i 857.350 952.184i −261.552 + 804.974i 1095.48 + 1892.85i −3682.56
4.20 −1.38295 + 13.1579i −25.5408 39.1749i −46.0139 9.78055i −17.4056 165.604i 550.779 281.885i 30.8976 34.3153i −330.989 + 1018.68i −882.339 + 2001.11i 2203.06
See next 80 embeddings (of 656 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.82
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner
11.c even 5 1 inner
99.m even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 99.8.m.a 656
9.c even 3 1 inner 99.8.m.a 656
11.c even 5 1 inner 99.8.m.a 656
99.m even 15 1 inner 99.8.m.a 656
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.8.m.a 656 1.a even 1 1 trivial
99.8.m.a 656 9.c even 3 1 inner
99.8.m.a 656 11.c even 5 1 inner
99.8.m.a 656 99.m even 15 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(99, [\chi])\).