Properties

Label 232.3.b.c
Level $232$
Weight $3$
Character orbit 232.b
Analytic conductor $6.322$
Analytic rank $0$
Dimension $56$
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] = [232,3,Mod(115,232)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(232, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("232.115");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 232 = 2^{3} \cdot 29 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 232.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: \(6.32154213316\)
Analytic rank: \(0\)
Dimension: \(56\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 56 q - 4 q^{4} - 184 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 56 q - 4 q^{4} - 184 q^{9} - 44 q^{16} - 12 q^{20} - 68 q^{22} - 72 q^{24} - 280 q^{25} + 116 q^{28} + 8 q^{30} + 32 q^{33} - 116 q^{34} + 96 q^{35} - 112 q^{36} - 176 q^{38} + 156 q^{42} - 424 q^{49} + 32 q^{51} - 56 q^{52} + 160 q^{54} + 32 q^{57} + 296 q^{58} + 512 q^{59} + 640 q^{62} - 304 q^{64} + 192 q^{65} + 352 q^{67} + 472 q^{74} + 44 q^{78} + 424 q^{80} + 280 q^{81} - 76 q^{82} + 128 q^{83} + 412 q^{86} - 84 q^{88} + 96 q^{91} - 92 q^{92} + 276 q^{94} + 300 q^{96}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
115.1 −1.99351 0.160993i 4.31476i 3.94816 + 0.641884i 9.09838i −0.694647 + 8.60151i 8.77758i −7.76736 1.91523i −9.61712 1.46478 18.1377i
115.2 −1.99351 + 0.160993i 4.31476i 3.94816 0.641884i 9.09838i −0.694647 8.60151i 8.77758i −7.76736 + 1.91523i −9.61712 1.46478 + 18.1377i
115.3 −1.92780 0.532521i 4.75153i 3.43284 + 2.05319i 5.77072i −2.53029 + 9.16001i 4.89588i −5.52447 5.78621i −13.5770 −3.07303 + 11.1248i
115.4 −1.92780 + 0.532521i 4.75153i 3.43284 2.05319i 5.77072i −2.53029 9.16001i 4.89588i −5.52447 + 5.78621i −13.5770 −3.07303 11.1248i
115.5 −1.85899 0.737665i 4.63447i 2.91170 + 2.74263i 1.58058i 3.41869 8.61545i 8.48084i −3.38969 7.24638i −12.4783 −1.16594 + 2.93828i
115.6 −1.85899 + 0.737665i 4.63447i 2.91170 2.74263i 1.58058i 3.41869 + 8.61545i 8.48084i −3.38969 + 7.24638i −12.4783 −1.16594 2.93828i
115.7 −1.78690 0.898321i 3.19007i 2.38604 + 3.21042i 3.89874i 2.86570 5.70034i 12.5367i −1.37963 7.88014i −1.17652 3.50232 6.96667i
115.8 −1.78690 + 0.898321i 3.19007i 2.38604 3.21042i 3.89874i 2.86570 + 5.70034i 12.5367i −1.37963 + 7.88014i −1.17652 3.50232 + 6.96667i
115.9 −1.75422 0.960583i 0.800506i 2.15456 + 3.37014i 9.11788i −0.768952 + 1.40426i 7.90968i −0.542273 7.98160i 8.35919 −8.75848 + 15.9948i
115.10 −1.75422 + 0.960583i 0.800506i 2.15456 3.37014i 9.11788i −0.768952 1.40426i 7.90968i −0.542273 + 7.98160i 8.35919 −8.75848 15.9948i
115.11 −1.64262 1.14096i 1.23647i 1.39641 + 3.74834i 6.52788i 1.41077 2.03106i 8.84035i 1.98293 7.75035i 7.47113 7.44806 10.7228i
115.12 −1.64262 + 1.14096i 1.23647i 1.39641 3.74834i 6.52788i 1.41077 + 2.03106i 8.84035i 1.98293 + 7.75035i 7.47113 7.44806 + 10.7228i
115.13 −1.63691 1.14914i 3.33686i 1.35897 + 3.76207i 0.938063i −3.83451 + 5.46215i 5.06710i 2.09861 7.71983i −2.13463 1.07796 1.53553i
115.14 −1.63691 + 1.14914i 3.33686i 1.35897 3.76207i 0.938063i −3.83451 5.46215i 5.06710i 2.09861 + 7.71983i −2.13463 1.07796 + 1.53553i
115.15 −1.23306 1.57467i 2.41553i −0.959149 + 3.88330i 3.79912i −3.80365 + 2.97848i 1.38473i 7.29759 3.27799i 3.16523 5.98235 4.68452i
115.16 −1.23306 + 1.57467i 2.41553i −0.959149 3.88330i 3.79912i −3.80365 2.97848i 1.38473i 7.29759 + 3.27799i 3.16523 5.98235 + 4.68452i
115.17 −1.02899 1.71499i 0.366308i −1.88236 + 3.52941i 4.93433i 0.628213 0.376927i 6.63118i 7.98982 0.403509i 8.86582 −8.46232 + 5.07738i
115.18 −1.02899 + 1.71499i 0.366308i −1.88236 3.52941i 4.93433i 0.628213 + 0.376927i 6.63118i 7.98982 + 0.403509i 8.86582 −8.46232 5.07738i
115.19 −0.951200 1.75932i 3.37464i −2.19044 + 3.34694i 4.09203i 5.93709 3.20996i 0.0802561i 7.97189 + 0.670088i −2.38822 −7.19920 + 3.89234i
115.20 −0.951200 + 1.75932i 3.37464i −2.19044 3.34694i 4.09203i 5.93709 + 3.20996i 0.0802561i 7.97189 0.670088i −2.38822 −7.19920 3.89234i
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 115.56
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
29.b even 2 1 inner
232.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 232.3.b.c 56
4.b odd 2 1 928.3.b.c 56
8.b even 2 1 928.3.b.c 56
8.d odd 2 1 inner 232.3.b.c 56
29.b even 2 1 inner 232.3.b.c 56
116.d odd 2 1 928.3.b.c 56
232.b odd 2 1 inner 232.3.b.c 56
232.g even 2 1 928.3.b.c 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
232.3.b.c 56 1.a even 1 1 trivial
232.3.b.c 56 8.d odd 2 1 inner
232.3.b.c 56 29.b even 2 1 inner
232.3.b.c 56 232.b odd 2 1 inner
928.3.b.c 56 4.b odd 2 1
928.3.b.c 56 8.b even 2 1
928.3.b.c 56 116.d odd 2 1
928.3.b.c 56 232.g even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(232, [\chi])\):

\( T_{3}^{28} + 172 T_{3}^{26} + 13002 T_{3}^{24} + 569432 T_{3}^{22} + 16036759 T_{3}^{20} + 304789316 T_{3}^{18} + 3991348352 T_{3}^{16} + 36115653072 T_{3}^{14} + 222992195775 T_{3}^{12} + \cdots + 89548674228 \) Copy content Toggle raw display
\( T_{31}^{28} - 12300 T_{31}^{26} + 67923754 T_{31}^{24} - 222275550672 T_{31}^{22} + 479106557841303 T_{31}^{20} + \cdots + 59\!\cdots\!00 \) Copy content Toggle raw display