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Properties

Label 69.9.b.a
Level $69$
Weight $9$
Character orbit 69.b
Analytic conductor $28.109$
Analytic rank $0$
Dimension $58$
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] = [69,9,Mod(47,69)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(69, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("69.47");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 69 = 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 69.b (of order \(2\), degree \(1\), 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: \(28.1091240942\)
Analytic rank: \(0\)
Dimension: \(58\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 58 q - 112 q^{3} - 7168 q^{4} - 3409 q^{6} + 6996 q^{7} - 2924 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 58 q - 112 q^{3} - 7168 q^{4} - 3409 q^{6} + 6996 q^{7} - 2924 q^{9} - 1640 q^{10} - 8117 q^{12} - 131624 q^{13} - 110534 q^{15} + 981608 q^{16} - 764275 q^{18} - 51984 q^{19} - 465698 q^{21} + 107820 q^{22} - 776384 q^{24} - 6119198 q^{25} + 2169458 q^{27} - 2088388 q^{28} - 275266 q^{30} + 1761928 q^{31} - 1296054 q^{33} + 1065148 q^{34} + 6538683 q^{36} - 3753560 q^{37} - 3455142 q^{39} - 1718376 q^{40} + 15281204 q^{42} + 17086408 q^{43} - 5501106 q^{45} + 21011911 q^{48} + 28425318 q^{49} - 29296854 q^{51} + 53928442 q^{52} - 7273416 q^{54} + 3946264 q^{55} - 19348988 q^{57} + 38920170 q^{58} + 14801436 q^{60} - 28182292 q^{61} - 61798166 q^{63} - 108920570 q^{64} + 16525334 q^{66} - 113053100 q^{67} + 15476576 q^{70} + 112152660 q^{72} - 203835496 q^{73} - 105597476 q^{75} - 145466092 q^{76} - 26695313 q^{78} + 8414240 q^{79} - 8331316 q^{81} + 49159978 q^{82} + 140561238 q^{84} + 21675792 q^{85} + 18667742 q^{87} + 391284868 q^{88} - 45728650 q^{90} + 582399940 q^{91} - 127172458 q^{93} - 503093126 q^{94} + 582228969 q^{96} - 645252224 q^{97} - 296258454 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1 − 31.1952i 27.4981 − 76.1896i −717.138 109.487i −2376.75 − 857.806i −2734.93 14385.3i −5048.71 − 4190.13i 3415.46
47.2 − 30.8109i −19.7061 + 78.5663i −693.314 1192.28i 2420.70 + 607.164i 493.255 13474.1i −5784.34 − 3096.48i 36735.2
47.3 − 29.6557i 63.6881 + 50.0483i −623.462 − 755.944i 1484.22 − 1888.72i 17.8658 10897.4i 1551.34 + 6374.96i −22418.1
47.4 − 28.5854i −76.7528 − 25.8845i −561.123 88.9806i −739.918 + 2194.01i −507.999 8722.05i 5220.99 + 3973.41i 2543.54
47.5 − 27.7665i 80.7112 − 6.83349i −514.977 399.298i −189.742 − 2241.07i 1638.64 7190.87i 6467.61 − 1103.08i 11087.1
47.6 − 27.6581i −50.9463 + 62.9720i −508.972 − 873.391i 1741.69 + 1409.08i 3356.23 6996.72i −1369.95 − 6416.38i −24156.4
47.7 − 26.1150i −23.1297 − 77.6274i −425.993 − 963.091i −2027.24 + 604.032i 2760.07 4439.36i −5491.04 + 3591.00i −25151.1
47.8 − 25.2942i −52.7207 + 61.4942i −383.795 − 364.741i 1555.44 + 1333.52i −3954.16 3232.46i −1002.06 − 6484.03i −9225.81
47.9 − 23.8755i 52.3086 − 61.8451i −314.041 122.410i −1476.58 − 1248.90i 3171.11 1385.77i −1088.63 − 6470.05i 2922.59
47.10 − 22.9210i −44.9091 − 67.4105i −269.374 1084.44i −1545.12 + 1029.36i 2612.54 306.546i −2527.35 + 6054.69i 24856.5
47.11 − 22.0670i 71.7067 − 37.6716i −230.954 − 1159.22i −831.300 − 1582.35i −2758.31 − 552.692i 3722.70 − 5402.61i −25580.5
47.12 − 20.1201i 80.5859 − 8.18005i −148.819 1143.54i −164.584 − 1621.40i −3304.21 − 2156.49i 6427.17 − 1318.39i 23008.2
47.13 − 20.0949i −78.5777 + 19.6607i −147.806 218.291i 395.079 + 1579.01i 1465.83 − 2174.16i 5787.92 − 3089.78i 4386.53
47.14 − 19.8530i 8.17434 + 80.5865i −138.141 87.0307i 1599.88 − 162.285i 4113.85 − 2339.85i −6427.36 + 1317.48i 1727.82
47.15 − 19.1918i −11.6618 − 80.1561i −112.327 352.131i −1538.34 + 223.811i −2986.30 − 2757.35i −6289.01 + 1869.53i 6758.05
47.16 − 16.7518i −70.2210 − 40.3734i −24.6240 − 838.736i −676.329 + 1176.33i −2444.82 − 3875.97i 3300.98 + 5670.12i −14050.4
47.17 − 15.7222i −72.2122 + 36.6933i 8.81373 887.410i 576.899 + 1135.33i 312.611 − 4163.44i 3868.20 − 5299.41i 13952.0
47.18 − 13.6580i 47.5962 − 65.5408i 69.4583 − 266.455i −895.158 − 650.070i −888.827 − 4445.12i −2030.20 − 6238.99i −3639.26
47.19 − 12.5445i 79.9987 + 12.6967i 98.6367 − 406.624i 159.273 − 1003.54i 1636.76 − 4448.72i 6238.59 + 2031.43i −5100.87
47.20 − 11.3478i 1.84705 + 80.9789i 127.228 − 980.954i 918.929 − 20.9599i −829.187 − 4348.78i −6554.18 + 299.144i −11131.6
See all 58 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.58
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 69.9.b.a ✓ 58
3.b odd 2 1 inner 69.9.b.a ✓ 58
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
69.9.b.a ✓ 58 1.a even 1 1 trivial
69.9.b.a ✓ 58 3.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(69, [\chi])\).