Properties

Label 69.8.g.a
Level $69$
Weight $8$
Character orbit 69.g
Analytic conductor $21.555$
Analytic rank $0$
Dimension $540$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 69 = 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 69.g (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(21.5545667584\)
Analytic rank: \(0\)
Dimension: \(540\)
Relative dimension: \(54\) over \(\Q(\zeta_{22})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 540q - 35q^{3} + 3362q^{4} - 1280q^{6} - 22q^{7} - 659q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 540q - 35q^{3} + 3362q^{4} - 1280q^{6} - 22q^{7} - 659q^{9} - 22q^{10} + 4756q^{12} + 4970q^{13} + 3828q^{15} - 174622q^{16} + 250050q^{18} - 22q^{19} - 221738q^{21} + 94934q^{24} - 788392q^{25} + 1020934q^{27} - 22q^{28} - 1480875q^{30} + 738458q^{31} + 1215192q^{33} + 1815924q^{34} - 394502q^{36} - 905234q^{37} - 1635803q^{39} + 10260228q^{40} - 11q^{42} - 1694330q^{43} - 6978726q^{46} + 134426q^{48} + 7785740q^{49} - 11q^{51} + 16219654q^{52} - 10697559q^{54} - 1371960q^{55} + 7508908q^{57} - 21860696q^{58} - 21971851q^{60} - 11102850q^{61} - 9022926q^{63} + 2572060q^{64} + 16582830q^{66} + 6031564q^{67} + 12019470q^{69} - 2949828q^{70} + 12828949q^{72} - 1809596q^{73} - 16620656q^{75} - 42651158q^{76} - 47106384q^{78} - 1156914q^{79} + 48837101q^{81} + 47208588q^{82} + 124183367q^{84} + 5833252q^{85} - 17916287q^{87} + 360426q^{88} - 142298860q^{90} - 51670516q^{93} - 18401348q^{94} + 36462337q^{96} - 63860500q^{97} + 135683977q^{99} + O(q^{100}) \)

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.

Label \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −6.22368 + 21.1959i 46.1239 + 7.71935i −302.852 194.631i −24.2322 + 168.538i −450.679 + 929.594i 1127.70 515.004i 3873.27 3356.20i 2067.82 + 712.093i −3421.51 1562.55i
5.2 −6.12233 + 20.8507i −3.35885 46.6446i −289.589 186.108i −62.0721 + 431.721i 993.138 + 215.539i −1340.25 + 612.072i 3551.28 3077.20i −2164.44 + 313.345i −8621.67 3937.38i
5.3 −5.98416 + 20.3802i −23.3594 + 40.5134i −271.861 174.714i 41.5571 289.036i −685.885 718.507i 104.054 47.5200i 3132.84 2714.62i −1095.68 1892.74i 5641.92 + 2576.58i
5.4 −5.50277 + 18.7407i −46.6006 3.92169i −213.253 137.049i −12.7497 + 88.6762i 329.928 851.749i 98.7958 45.1185i 1852.45 1605.15i 2156.24 + 365.506i −1591.69 726.903i
5.5 −5.36309 + 18.2650i −20.6078 41.9800i −197.168 126.712i 51.4914 358.131i 877.287 151.259i 605.621 276.578i 1530.35 1326.05i −1337.64 + 1730.23i 6265.11 + 2861.18i
5.6 −5.35232 + 18.2283i 37.5882 27.8231i −195.945 125.926i 46.1891 321.252i 305.985 + 834.089i −583.063 + 266.276i 1506.40 1305.30i 638.749 2091.64i 5608.68 + 2561.40i
5.7 −5.10680 + 17.3922i 36.1324 + 29.6892i −168.727 108.434i 33.4826 232.877i −700.880 + 476.805i −1036.98 + 473.573i 994.090 861.384i 424.107 + 2145.48i 3879.24 + 1771.59i
5.8 −4.92412 + 16.7700i 15.0326 + 44.2834i −149.305 95.9528i −35.8560 + 249.384i −816.655 + 34.0408i −759.793 + 346.986i 653.576 566.327i −1735.04 + 1331.39i −4005.61 1829.30i
5.9 −4.91020 + 16.7226i −24.1906 + 40.0227i −147.855 95.0204i −72.4009 + 503.559i −550.503 601.048i 752.211 343.523i 629.012 545.042i −1016.63 1936.34i −8065.31 3683.30i
5.10 −4.40655 + 15.0073i 25.2068 39.3905i −98.1221 63.0593i −11.4970 + 79.9631i 480.072 + 551.864i 722.763 330.075i −134.306 + 116.377i −916.230 1985.82i −1149.37 524.900i
5.11 −3.89917 + 13.2794i 28.6223 + 36.9833i −53.4575 34.3551i 9.28121 64.5522i −602.718 + 235.882i 1579.08 721.143i −674.170 + 584.172i −548.528 + 2117.09i 821.024 + 374.949i
5.12 −3.89080 + 13.2509i −38.0703 27.1597i −52.7665 33.9110i −31.2611 + 217.426i 508.014 398.791i 454.818 207.708i −681.296 + 590.346i 711.698 + 2067.96i −2759.45 1260.20i
5.13 −3.86607 + 13.1666i −43.3698 + 17.4946i −50.7333 32.6043i 2.55481 17.7691i −62.6737 638.670i −1461.32 + 667.362i −702.031 + 608.313i 1574.88 1517.47i 224.082 + 102.335i
5.14 −3.60512 + 12.2779i 46.6247 3.62518i −30.0694 19.3245i −56.5273 + 393.156i −123.578 + 585.522i −611.212 + 279.131i −892.188 + 773.086i 2160.72 338.046i −4623.34 2111.41i
5.15 −3.58658 + 12.2148i −5.91340 + 46.3900i −28.6569 18.4167i 47.6282 331.261i −545.435 238.613i 343.311 156.785i −903.757 + 783.110i −2117.06 548.645i 3875.46 + 1769.87i
5.16 −3.09645 + 10.5455i −20.3936 42.0844i 6.06031 + 3.89472i 37.8324 263.130i 506.950 84.7496i −1413.39 + 645.472i −1123.04 + 973.116i −1355.20 + 1716.51i 2657.70 + 1213.73i
5.17 −2.84941 + 9.70420i −46.0191 + 8.32112i 21.6281 + 13.8995i 66.9720 465.800i 50.3775 470.289i 593.781 271.171i −1174.89 + 1018.05i 2048.52 765.861i 4329.39 + 1977.17i
5.18 −2.54280 + 8.65999i 46.3387 6.30245i 39.1508 + 25.1607i 56.4455 392.587i −63.2512 + 417.319i 136.586 62.3769i −1190.54 + 1031.61i 2107.56 584.095i 3256.27 + 1487.09i
5.19 −2.50658 + 8.53663i −3.74386 46.6153i 41.0893 + 26.4065i −49.8381 + 346.631i 407.322 + 84.8851i 206.284 94.2068i −1189.08 + 1030.34i −2158.97 + 349.042i −2834.14 1294.31i
5.20 −1.67700 + 5.71134i 44.7400 + 13.6137i 77.8734 + 50.0462i 16.5134 114.853i −152.781 + 232.695i 97.1619 44.3723i −992.241 + 859.782i 1816.34 + 1218.15i 628.272 + 286.922i
See next 80 embeddings (of 540 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 65.54
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
23.d odd 22 1 inner
69.g even 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 69.8.g.a 540
3.b odd 2 1 inner 69.8.g.a 540
23.d odd 22 1 inner 69.8.g.a 540
69.g even 22 1 inner 69.8.g.a 540
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
69.8.g.a 540 1.a even 1 1 trivial
69.8.g.a 540 3.b odd 2 1 inner
69.8.g.a 540 23.d odd 22 1 inner
69.8.g.a 540 69.g even 22 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(69, [\chi])\).