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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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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [69,8,Mod(68,69)]
 
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, 1]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("69.68");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 69 = 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 69.c (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: \(21.5545667584\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 24 q^{3} - 2616 q^{4} + 4416 q^{6} + 648 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 24 q^{3} - 2616 q^{4} + 4416 q^{6} + 648 q^{9} + 16668 q^{12} - 4992 q^{13} + 76296 q^{16} + 100548 q^{18} - 620268 q^{24} + 1257120 q^{25} - 105408 q^{27} - 738480 q^{31} - 1227096 q^{36} + 3832248 q^{39} - 489768 q^{46} + 733404 q^{48} - 5569584 q^{49} - 5646984 q^{52} + 1260252 q^{54} - 305760 q^{55} + 16920336 q^{58} + 6032664 q^{64} + 4496304 q^{69} + 23272152 q^{70} - 26866080 q^{72} - 2542752 q^{73} + 487176 q^{75} - 47724612 q^{78} + 25279128 q^{81} - 68451480 q^{82} - 18046992 q^{85} + 42338184 q^{87} - 59017272 q^{93} + 106197744 q^{94} + 67778460 q^{96}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
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 \( T_{2}^{24} + 2190 T_{2}^{22} + 2066337 T_{2}^{20} + 1101779876 T_{2}^{18} + 366191315088 T_{2}^{16} + \cdots + 75\!\cdots\!12 \) acting on \(S_{8}^{\mathrm{new}}(69, [\chi])\). Copy content Toggle raw display