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Properties

Label 147.6.c.d
Level $147$
Weight $6$
Character orbit 147.c
Analytic conductor $23.576$
Analytic rank $0$
Dimension $40$
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] = [147,6,Mod(146,147)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(147, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("147.146");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 147.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: \(23.5764215125\)
Analytic rank: \(0\)
Dimension: \(40\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 640 q^{4} + 440 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 640 q^{4} + 440 q^{9} - 3176 q^{15} + 3552 q^{16} - 4544 q^{18} + 21088 q^{22} + 30104 q^{25} + 44664 q^{30} - 59320 q^{36} - 26224 q^{37} - 16216 q^{39} + 21584 q^{43} - 27680 q^{46} + 95784 q^{51} - 130304 q^{57} - 131376 q^{58} + 329208 q^{60} + 53968 q^{64} - 187632 q^{67} + 606736 q^{72} + 70312 q^{78} - 597424 q^{79} + 755256 q^{81} + 108112 q^{85} - 1673072 q^{88} + 590480 q^{93} - 258480 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
146.1 − 8.23628i 13.6946 + 7.44699i −35.8363 −95.6857 61.3355 − 112.793i 0 31.5969i 132.085 + 203.967i 788.094i
146.2 8.23628i 13.6946 − 7.44699i −35.8363 −95.6857 61.3355 + 112.793i 0 − 31.5969i 132.085 − 203.967i − 788.094i
146.3 − 9.64154i 2.14483 − 15.4402i −60.9593 −95.0200 −148.867 − 20.6794i 0 279.213i −233.799 − 66.2331i 916.140i
146.4 9.64154i 2.14483 + 15.4402i −60.9593 −95.0200 −148.867 + 20.6794i 0 − 279.213i −233.799 + 66.2331i − 916.140i
146.5 − 7.35335i 15.5875 − 0.174581i −22.0717 89.2354 −1.28375 − 114.620i 0 − 73.0063i 242.939 − 5.44254i − 656.179i
146.6 7.35335i 15.5875 + 0.174581i −22.0717 89.2354 −1.28375 + 114.620i 0 73.0063i 242.939 + 5.44254i 656.179i
146.7 − 4.29037i −1.28755 − 15.5352i 13.5927 −75.7260 −66.6517 + 5.52407i 0 − 195.610i −239.684 + 40.0047i 324.892i
146.8 4.29037i −1.28755 + 15.5352i 13.5927 −75.7260 −66.6517 − 5.52407i 0 195.610i −239.684 − 40.0047i − 324.892i
146.9 − 3.81342i −6.23210 − 14.2885i 17.4578 63.5678 −54.4880 + 23.7656i 0 − 188.603i −165.322 + 178.095i − 242.411i
146.10 3.81342i −6.23210 + 14.2885i 17.4578 63.5678 −54.4880 − 23.7656i 0 188.603i −165.322 − 178.095i 242.411i
146.11 − 8.31267i −14.0092 + 6.83685i −37.1006 28.6894 56.8325 + 116.454i 0 42.3992i 149.515 − 191.558i − 238.485i
146.12 8.31267i −14.0092 − 6.83685i −37.1006 28.6894 56.8325 − 116.454i 0 − 42.3992i 149.515 + 191.558i 238.485i
146.13 − 10.7583i 14.9806 − 4.31048i −83.7406 −26.3120 −46.3734 − 161.166i 0 556.640i 205.839 − 129.148i 283.072i
146.14 10.7583i 14.9806 + 4.31048i −83.7406 −26.3120 −46.3734 + 161.166i 0 − 556.640i 205.839 + 129.148i − 283.072i
146.15 − 1.52115i −15.4465 − 2.09880i 29.6861 −32.5134 −3.19260 + 23.4966i 0 − 93.8341i 234.190 + 64.8384i 49.4578i
146.16 1.52115i −15.4465 + 2.09880i 29.6861 −32.5134 −3.19260 − 23.4966i 0 93.8341i 234.190 − 64.8384i − 49.4578i
146.17 − 6.54010i 2.32648 − 15.4139i −10.7729 13.6838 −100.808 − 15.2154i 0 − 138.827i −232.175 − 71.7202i − 89.4937i
146.18 6.54010i 2.32648 + 15.4139i −10.7729 13.6838 −100.808 + 15.2154i 0 138.827i −232.175 + 71.7202i 89.4937i
146.19 − 1.50172i −11.3889 − 10.6440i 29.7448 9.66598 −15.9842 + 17.1029i 0 − 92.7234i 16.4124 + 242.445i − 14.5156i
146.20 1.50172i −11.3889 + 10.6440i 29.7448 9.66598 −15.9842 − 17.1029i 0 92.7234i 16.4124 − 242.445i 14.5156i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 146.40
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 147.6.c.d ✓ 40
3.b odd 2 1 inner 147.6.c.d ✓ 40
7.b odd 2 1 inner 147.6.c.d ✓ 40
21.c even 2 1 inner 147.6.c.d ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.6.c.d ✓ 40 1.a even 1 1 trivial
147.6.c.d ✓ 40 3.b odd 2 1 inner
147.6.c.d ✓ 40 7.b odd 2 1 inner
147.6.c.d ✓ 40 21.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{20} + 480 T_{2}^{18} + 96836 T_{2}^{16} + 10697796 T_{2}^{14} + 706034311 T_{2}^{12} + \cdots + 162931412267008 \) acting on \(S_{6}^{\mathrm{new}}(147, [\chi])\). Copy content Toggle raw display