Properties

Label 69.8.c.b
Level $69$
Weight $8$
Character orbit 69.c
Analytic conductor $21.555$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 69 = 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 69.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(21.5545667584\)
Analytic rank: \(0\)
Dimension: \(48\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 48q + 24q^{3} - 2616q^{4} + 4416q^{6} + 648q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 48q + 24q^{3} - 2616q^{4} + 4416q^{6} + 648q^{9} + 16668q^{12} - 4992q^{13} + 76296q^{16} + 100548q^{18} - 620268q^{24} + 1257120q^{25} - 105408q^{27} - 738480q^{31} - 1227096q^{36} + 3832248q^{39} - 489768q^{46} + 733404q^{48} - 5569584q^{49} - 5646984q^{52} + 1260252q^{54} - 305760q^{55} + 16920336q^{58} + 6032664q^{64} + 4496304q^{69} + 23272152q^{70} - 26866080q^{72} - 2542752q^{73} + 487176q^{75} - 47724612q^{78} + 25279128q^{81} - 68451480q^{82} - 18046992q^{85} + 42338184q^{87} - 59017272q^{93} + 106197744q^{94} + 67778460q^{96} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
68.1 20.7157i 11.8197 + 45.2471i −301.139 −470.966 937.323 244.852i 481.517i 3586.70i −1907.59 + 1069.61i 9756.38i
68.2 20.7157i 11.8197 + 45.2471i −301.139 470.966 937.323 244.852i 481.517i 3586.70i −1907.59 + 1069.61i 9756.38i
68.3 20.0107i −46.5037 4.94059i −272.428 −368.180 −98.8646 + 930.570i 90.4608i 2890.10i 2138.18 + 459.511i 7367.54i
68.4 20.0107i −46.5037 4.94059i −272.428 368.180 −98.8646 + 930.570i 90.4608i 2890.10i 2138.18 + 459.511i 7367.54i
68.5 19.2454i 46.1752 + 7.40622i −242.386 −153.787 142.536 888.661i 1442.95i 2201.42i 2077.30 + 683.967i 2959.69i
68.6 19.2454i 46.1752 + 7.40622i −242.386 153.787 142.536 888.661i 1442.95i 2201.42i 2077.30 + 683.967i 2959.69i
68.7 16.5464i −28.9535 + 36.7246i −145.782 −13.0387 607.659 + 479.074i 1232.42i 294.228i −510.394 2126.61i 215.743i
68.8 16.5464i −28.9535 + 36.7246i −145.782 13.0387 607.659 + 479.074i 1232.42i 294.228i −510.394 2126.61i 215.743i
68.9 14.5803i −25.2080 39.3898i −84.5859 −276.798 −574.316 + 367.541i 978.900i 632.992i −916.113 + 1985.88i 4035.81i
68.10 14.5803i −25.2080 39.3898i −84.5859 276.798 −574.316 + 367.541i 978.900i 632.992i −916.113 + 1985.88i 4035.81i
68.11 13.3830i 20.4540 42.0551i −51.1052 −506.030 −562.825 273.736i 1171.85i 1029.08i −1350.27 1720.39i 6772.20i
68.12 13.3830i 20.4540 42.0551i −51.1052 506.030 −562.825 273.736i 1171.85i 1029.08i −1350.27 1720.39i 6772.20i
68.13 11.0616i 16.6400 + 43.7048i 5.64089 −185.789 483.445 184.066i 711.199i 1478.28i −1633.22 + 1454.50i 2055.12i
68.14 11.0616i 16.6400 + 43.7048i 5.64089 185.789 483.445 184.066i 711.199i 1478.28i −1633.22 + 1454.50i 2055.12i
68.15 9.65974i 43.8138 + 16.3509i 34.6893 −384.010 157.946 423.230i 402.824i 1571.54i 1652.29 + 1432.79i 3709.43i
68.16 9.65974i 43.8138 + 16.3509i 34.6893 384.010 157.946 423.230i 402.824i 1571.54i 1652.29 + 1432.79i 3709.43i
68.17 8.54844i −46.3879 + 5.92986i 54.9242 −270.970 50.6910 + 396.544i 1358.42i 1563.72i 2116.67 550.147i 2316.37i
68.18 8.54844i −46.3879 + 5.92986i 54.9242 270.970 50.6910 + 396.544i 1358.42i 1563.72i 2116.67 550.147i 2316.37i
68.19 4.24759i −28.7023 + 36.9213i 109.958 −462.789 156.826 + 121.915i 525.773i 1010.75i −539.361 2119.45i 1965.74i
68.20 4.24759i −28.7023 + 36.9213i 109.958 462.789 156.826 + 121.915i 525.773i 1010.75i −539.361 2119.45i 1965.74i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 68.48
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 69.8.c.b 48
3.b odd 2 1 inner 69.8.c.b 48
23.b odd 2 1 inner 69.8.c.b 48
69.c even 2 1 inner 69.8.c.b 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
69.8.c.b 48 1.a even 1 1 trivial
69.8.c.b 48 3.b odd 2 1 inner
69.8.c.b 48 23.b odd 2 1 inner
69.8.c.b 48 69.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(78\!\cdots\!08\)\( T_{2}^{14} + \)\(11\!\cdots\!16\)\( T_{2}^{12} + \)\(10\!\cdots\!76\)\( T_{2}^{10} + \)\(57\!\cdots\!40\)\( T_{2}^{8} + \)\(19\!\cdots\!28\)\( T_{2}^{6} + \)\(32\!\cdots\!28\)\( T_{2}^{4} + \)\(26\!\cdots\!44\)\( T_{2}^{2} + \)\(75\!\cdots\!12\)\( \)">\(T_{2}^{24} + \cdots\) acting on \(S_{8}^{\mathrm{new}}(69, [\chi])\).