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Properties

Label 231.3.f.a
Level $231$
Weight $3$
Character orbit 231.f
Analytic conductor $6.294$
Analytic rank $0$
Dimension $28$
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] = [231,3,Mod(34,231)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(231, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("231.34");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 231 = 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 231.f (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: \(6.29429410672\)
Analytic rank: \(0\)
Dimension: \(28\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 28 q + 64 q^{4} - 8 q^{7} + 24 q^{8} - 84 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q + 64 q^{4} - 8 q^{7} + 24 q^{8} - 84 q^{9} + 52 q^{14} + 112 q^{16} + 48 q^{21} - 8 q^{23} - 156 q^{25} - 96 q^{28} + 56 q^{29} - 96 q^{30} - 104 q^{32} - 88 q^{35} - 192 q^{36} - 96 q^{37} + 48 q^{39} + 84 q^{42} - 208 q^{43} + 176 q^{44} + 128 q^{46} - 164 q^{49} - 48 q^{50} + 48 q^{51} - 256 q^{53} + 244 q^{56} - 240 q^{57} + 220 q^{58} + 396 q^{60} + 24 q^{63} + 676 q^{64} - 152 q^{65} - 416 q^{67} - 340 q^{70} + 16 q^{71} - 72 q^{72} + 256 q^{74} - 44 q^{77} - 204 q^{78} + 224 q^{79} + 252 q^{81} + 904 q^{85} + 344 q^{86} - 132 q^{88} + 368 q^{91} - 928 q^{92} + 96 q^{93} + 776 q^{95} + 696 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
34.1 −3.92240 − 1.73205i 11.3853 5.97295i 6.79380i −4.18539 − 5.61093i −28.9680 −3.00000 − 23.4283i
34.2 −3.92240 1.73205i 11.3853 − 5.97295i − 6.79380i −4.18539 + 5.61093i −28.9680 −3.00000 23.4283i
34.3 −3.03669 − 1.73205i 5.22148 − 3.38408i 5.25970i 0.382320 + 6.98955i −3.70926 −3.00000 10.2764i
34.4 −3.03669 1.73205i 5.22148 3.38408i − 5.25970i 0.382320 − 6.98955i −3.70926 −3.00000 − 10.2764i
34.5 −2.83123 − 1.73205i 4.01587 7.94590i 4.90384i −0.929011 + 6.93808i −0.0449219 −3.00000 − 22.4967i
34.6 −2.83123 1.73205i 4.01587 − 7.94590i − 4.90384i −0.929011 − 6.93808i −0.0449219 −3.00000 22.4967i
34.7 −2.07029 − 1.73205i 0.286086 − 5.22448i 3.58584i −0.100388 − 6.99928i 7.68887 −3.00000 10.8162i
34.8 −2.07029 1.73205i 0.286086 5.22448i − 3.58584i −0.100388 + 6.99928i 7.68887 −3.00000 − 10.8162i
34.9 −1.97752 − 1.73205i −0.0894120 3.03683i 3.42517i −4.92879 − 4.97061i 8.08690 −3.00000 − 6.00540i
34.10 −1.97752 1.73205i −0.0894120 − 3.03683i − 3.42517i −4.92879 + 4.97061i 8.08690 −3.00000 6.00540i
34.11 −1.23200 − 1.73205i −2.48217 1.05077i 2.13389i 6.99962 − 0.0726870i 7.98605 −3.00000 − 1.29456i
34.12 −1.23200 1.73205i −2.48217 − 1.05077i − 2.13389i 6.99962 + 0.0726870i 7.98605 −3.00000 1.29456i
34.13 −0.597216 − 1.73205i −3.64333 − 9.46349i 1.03441i −5.23933 + 4.64213i 4.56472 −3.00000 5.65174i
34.14 −0.597216 1.73205i −3.64333 9.46349i − 1.03441i −5.23933 − 4.64213i 4.56472 −3.00000 − 5.65174i
34.15 0.0392230 − 1.73205i −3.99846 1.64004i − 0.0679363i 4.76441 + 5.12839i −0.313724 −3.00000 0.0643273i
34.16 0.0392230 1.73205i −3.99846 − 1.64004i 0.0679363i 4.76441 − 5.12839i −0.313724 −3.00000 − 0.0643273i
34.17 1.37590 − 1.73205i −2.10691 3.74223i − 2.38312i −5.63148 + 4.15769i −8.40247 −3.00000 5.14892i
34.18 1.37590 1.73205i −2.10691 − 3.74223i 2.38312i −5.63148 − 4.15769i −8.40247 −3.00000 − 5.14892i
34.19 1.85814 − 1.73205i −0.547320 − 6.07606i − 3.21839i 6.77078 + 1.77668i −8.44955 −3.00000 − 11.2902i
34.20 1.85814 1.73205i −0.547320 6.07606i 3.21839i 6.77078 − 1.77668i −8.44955 −3.00000 11.2902i
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 34.28
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 231.3.f.a ✓ 28
3.b odd 2 1 693.3.f.d 28
7.b odd 2 1 inner 231.3.f.a ✓ 28
21.c even 2 1 693.3.f.d 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.3.f.a ✓ 28 1.a even 1 1 trivial
231.3.f.a ✓ 28 7.b odd 2 1 inner
693.3.f.d 28 3.b odd 2 1
693.3.f.d 28 21.c even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(231, [\chi])\).