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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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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [69,8,Mod(5,69)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(69, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([11, 1]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("69.5");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 69 = 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 69.g (of order \(22\), degree \(10\), 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: \(21.5545667584\)
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.

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

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
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 5.54
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 \(S_{8}^{\mathrm{new}}(69, [\chi])\).