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Properties

Label 63.9.t.a
Level $63$
Weight $9$
Character orbit 63.t
Analytic conductor $25.665$
Analytic rank $0$
Dimension $124$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [63,9,Mod(40,63)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(63, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 5]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("63.40");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 63.t (of order \(6\), degree \(2\), 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: \(25.6648524339\)
Analytic rank: \(0\)
Dimension: \(124\)
Relative dimension: \(62\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 124 q - 2 q^{2} - 3 q^{3} + 15358 q^{4} - 3 q^{5} - 768 q^{6} - 924 q^{7} - 520 q^{8} - 9375 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 124 q - 2 q^{2} - 3 q^{3} + 15358 q^{4} - 3 q^{5} - 768 q^{6} - 924 q^{7} - 520 q^{8} - 9375 q^{9} - 6 q^{10} + 5695 q^{11} + 82374 q^{12} - 5055 q^{13} - 23678 q^{14} + 45306 q^{15} + 1834494 q^{16} + 31719 q^{17} + 125922 q^{18} - 6 q^{19} - 393984 q^{20} - 566727 q^{21} - 514 q^{22} + 126358 q^{23} - 216294 q^{24} + 4218751 q^{25} + 1261818 q^{26} - 1070415 q^{27} - 538372 q^{28} - 5870 q^{29} - 1998387 q^{30} + 8473518 q^{32} - 3 q^{33} - 768 q^{34} - 5323482 q^{35} - 1640586 q^{36} - 273495 q^{37} - 9654033 q^{38} - 9685383 q^{39} - 1173414 q^{40} + 2696130 q^{41} - 15501978 q^{42} + 2262654 q^{43} - 3391473 q^{44} - 17240307 q^{45} + 3708158 q^{46} + 21068061 q^{48} + 3561898 q^{49} + 11780161 q^{50} - 22345944 q^{51} - 2981379 q^{52} - 791852 q^{53} - 558513 q^{54} + 51302270 q^{56} - 25516971 q^{57} - 7853311 q^{58} - 16440267 q^{60} + 39150753 q^{63} + 201195000 q^{64} + 38402556 q^{65} + 84034194 q^{66} - 22149892 q^{67} - 10331514 q^{68} + 38545338 q^{69} - 36991557 q^{70} - 64113686 q^{71} - 5871294 q^{72} - 6 q^{73} + 63412127 q^{74} - 15951354 q^{75} - 198144 q^{76} - 124510400 q^{77} + 130524897 q^{78} + 61378916 q^{79} + 157375911 q^{80} + 25718685 q^{81} - 774 q^{82} - 315836370 q^{83} - 352567425 q^{84} - 28991877 q^{85} + 58390001 q^{86} - 245754993 q^{87} + 16645633 q^{88} - 10489479 q^{89} - 744050877 q^{90} - 3718725 q^{91} + 292919220 q^{92} - 313797534 q^{93} - 274876566 q^{95} + 169454607 q^{96} + 86346849 q^{97} + 86833963 q^{98} - 134954982 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
40.1 −29.6830 13.8097 + 79.8141i 625.078 −329.320 + 190.133i −409.913 − 2369.12i 2399.89 + 73.1404i −10955.3 −6179.58 + 2204.42i 9775.18 − 5643.70i
40.2 −29.5954 −56.0939 + 58.4335i 619.887 308.278 − 177.984i 1660.12 − 1729.36i −1990.54 − 1342.59i −10769.4 −267.939 − 6555.53i −9123.61 + 5267.52i
40.3 −29.5450 65.7281 − 47.3372i 616.908 −865.453 + 499.669i −1941.94 + 1398.58i −1181.14 + 2090.39i −10663.0 2079.37 − 6222.77i 25569.8 − 14762.7i
40.4 −29.3800 38.5954 − 71.2137i 607.184 672.175 − 388.080i −1133.93 + 2092.26i 187.280 − 2393.68i −10317.8 −3581.79 − 5497.05i −19748.5 + 11401.8i
40.5 −29.3240 −52.4778 − 61.7015i 603.895 238.840 − 137.894i 1538.86 + 1809.33i −490.954 + 2350.27i −10201.7 −1053.15 + 6475.92i −7003.74 + 4043.61i
40.6 −27.9654 −79.2246 − 16.8662i 526.066 −618.032 + 356.821i 2215.55 + 471.671i 1250.14 − 2049.87i −7552.51 5992.06 + 2672.44i 17283.5 − 9978.65i
40.7 −27.5655 63.6137 + 50.1428i 503.859 987.028 − 569.861i −1753.55 − 1382.21i −1588.44 + 1800.46i −6832.35 1532.40 + 6379.53i −27207.9 + 15708.5i
40.8 −26.1147 80.9984 − 0.509089i 425.977 47.9396 − 27.6780i −2115.25 + 13.2947i 2400.99 + 6.76237i −4438.90 6560.48 − 82.4708i −1251.93 + 722.801i
40.9 −23.5375 28.4456 + 75.8410i 298.015 −731.704 + 422.449i −669.538 − 1785.11i −2399.54 + 83.7525i −988.929 −4942.70 + 4314.68i 17222.5 − 9943.41i
40.10 −23.0569 −7.14242 − 80.6845i 275.622 −452.402 + 261.194i 164.682 + 1860.34i −1280.74 − 2030.89i −452.430 −6458.97 + 1152.57i 10431.0 − 6022.34i
40.11 −22.4632 79.6943 + 14.4853i 248.595 −198.716 + 114.728i −1790.19 − 325.387i −1319.56 − 2005.88i 166.343 6141.35 + 2308.80i 4463.78 − 2577.17i
40.12 −22.2896 −57.1483 + 57.4028i 240.828 47.2999 − 27.3086i 1273.81 − 1279.49i 565.338 + 2333.49i 338.174 −29.1537 − 6560.94i −1054.30 + 608.699i
40.13 −21.2841 −80.5657 − 8.37706i 197.014 852.621 − 492.261i 1714.77 + 178.298i 2296.10 − 701.934i 1255.47 6420.65 + 1349.81i −18147.3 + 10477.3i
40.14 −19.1785 −17.6149 − 79.0615i 111.816 −704.773 + 406.901i 337.828 + 1516.28i 2178.76 + 1008.86i 2765.23 −5940.43 + 2785.32i 13516.5 − 7803.77i
40.15 −18.9708 33.2052 − 73.8811i 103.890 368.502 − 212.755i −629.929 + 1401.58i 1880.60 + 1492.70i 2885.64 −4355.83 − 4906.48i −6990.77 + 4036.13i
40.16 −18.2668 −80.9917 − 1.16117i 77.6768 −573.909 + 331.346i 1479.46 + 21.2108i −2212.70 + 932.075i 3257.40 6558.30 + 188.090i 10483.5 − 6052.64i
40.17 −17.0768 40.0841 + 70.3865i 35.6165 326.449 − 188.475i −684.508 − 1201.98i 1475.79 − 1893.90i 3763.44 −3347.53 + 5642.76i −5574.69 + 3218.55i
40.18 −16.1934 −25.1449 + 76.9983i 6.22567 617.276 − 356.385i 407.181 − 1246.86i −730.324 − 2287.23i 4044.69 −5296.47 − 3872.23i −9995.79 + 5771.07i
40.19 −16.0881 −46.4102 − 66.3859i 2.82745 632.603 − 365.233i 746.652 + 1068.02i −2372.22 − 370.625i 4073.07 −2253.19 + 6161.97i −10177.4 + 5875.91i
40.20 −14.6593 69.4882 − 41.6219i −41.1036 389.791 − 225.046i −1018.65 + 610.150i −2040.04 + 1266.11i 4355.34 3096.23 − 5784.47i −5714.07 + 3299.02i
See next 80 embeddings (of 124 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 40.62
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.t odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 63.9.t.a yes 124
7.d odd 6 1 63.9.k.a ✓ 124
9.c even 3 1 63.9.k.a ✓ 124
63.t odd 6 1 inner 63.9.t.a yes 124
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.9.k.a ✓ 124 7.d odd 6 1
63.9.k.a ✓ 124 9.c even 3 1
63.9.t.a yes 124 1.a even 1 1 trivial
63.9.t.a yes 124 63.t odd 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(63, [\chi])\).