Properties

Label 462.4.e.b
Level $462$
Weight $4$
Character orbit 462.e
Analytic conductor $27.259$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [462,4,Mod(307,462)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(462, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("462.307");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 462 = 2 \cdot 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 462.e (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: \(27.2588824227\)
Analytic rank: \(0\)
Dimension: \(24\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 96 q^{4} + 144 q^{6} - 36 q^{7} - 216 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 96 q^{4} + 144 q^{6} - 36 q^{7} - 216 q^{9} - 56 q^{10} - 28 q^{11} + 32 q^{14} + 84 q^{15} + 384 q^{16} + 204 q^{17} + 244 q^{19} - 48 q^{21} - 64 q^{22} + 56 q^{23} - 576 q^{24} - 1096 q^{25} + 144 q^{28} + 96 q^{33} + 16 q^{35} + 864 q^{36} - 696 q^{37} + 224 q^{40} - 1276 q^{41} - 216 q^{42} + 112 q^{44} + 836 q^{49} + 168 q^{53} - 1296 q^{54} - 712 q^{55} - 128 q^{56} - 336 q^{58} - 336 q^{60} - 616 q^{61} + 1048 q^{62} + 324 q^{63} - 1536 q^{64} - 168 q^{66} - 1544 q^{67} - 816 q^{68} - 168 q^{70} - 944 q^{71} - 2212 q^{73} - 976 q^{76} - 1888 q^{77} + 1944 q^{81} + 236 q^{83} + 192 q^{84} - 336 q^{86} + 504 q^{87} + 256 q^{88} + 504 q^{90} - 508 q^{91} - 224 q^{92} - 1572 q^{93} - 584 q^{94} + 2304 q^{96} - 2008 q^{98} + 252 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
307.1 2.00000i 3.00000i −4.00000 21.6833i 6.00000 14.4011 + 11.6451i 8.00000i −9.00000 −43.3665
307.2 2.00000i 3.00000i −4.00000 18.5526i 6.00000 −11.3241 + 14.6549i 8.00000i −9.00000 −37.1052
307.3 2.00000i 3.00000i −4.00000 12.2530i 6.00000 8.77074 16.3118i 8.00000i −9.00000 −24.5060
307.4 2.00000i 3.00000i −4.00000 8.14463i 6.00000 −16.3752 8.65177i 8.00000i −9.00000 −16.2893
307.5 2.00000i 3.00000i −4.00000 6.71871i 6.00000 −17.1062 + 7.09771i 8.00000i −9.00000 −13.4374
307.6 2.00000i 3.00000i −4.00000 6.45926i 6.00000 17.9060 4.73009i 8.00000i −9.00000 −12.9185
307.7 2.00000i 3.00000i −4.00000 3.90709i 6.00000 −9.50728 15.8938i 8.00000i −9.00000 −7.81419
307.8 2.00000i 3.00000i −4.00000 7.44234i 6.00000 5.85787 + 17.5694i 8.00000i −9.00000 14.8847
307.9 2.00000i 3.00000i −4.00000 7.94213i 6.00000 −16.9601 + 7.44012i 8.00000i −9.00000 15.8843
307.10 2.00000i 3.00000i −4.00000 12.6816i 6.00000 2.63864 18.3313i 8.00000i −9.00000 25.3632
307.11 2.00000i 3.00000i −4.00000 17.5107i 6.00000 18.3856 + 2.22927i 8.00000i −9.00000 35.0213
307.12 2.00000i 3.00000i −4.00000 18.1418i 6.00000 −14.6872 + 11.2822i 8.00000i −9.00000 36.2836
307.13 2.00000i 3.00000i −4.00000 18.1418i 6.00000 −14.6872 11.2822i 8.00000i −9.00000 36.2836
307.14 2.00000i 3.00000i −4.00000 17.5107i 6.00000 18.3856 2.22927i 8.00000i −9.00000 35.0213
307.15 2.00000i 3.00000i −4.00000 12.6816i 6.00000 2.63864 + 18.3313i 8.00000i −9.00000 25.3632
307.16 2.00000i 3.00000i −4.00000 7.94213i 6.00000 −16.9601 7.44012i 8.00000i −9.00000 15.8843
307.17 2.00000i 3.00000i −4.00000 7.44234i 6.00000 5.85787 17.5694i 8.00000i −9.00000 14.8847
307.18 2.00000i 3.00000i −4.00000 3.90709i 6.00000 −9.50728 + 15.8938i 8.00000i −9.00000 −7.81419
307.19 2.00000i 3.00000i −4.00000 6.45926i 6.00000 17.9060 + 4.73009i 8.00000i −9.00000 −12.9185
307.20 2.00000i 3.00000i −4.00000 6.71871i 6.00000 −17.1062 7.09771i 8.00000i −9.00000 −13.4374
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 307.24
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 462.4.e.b yes 24
7.b odd 2 1 462.4.e.a 24
11.b odd 2 1 462.4.e.a 24
77.b even 2 1 inner 462.4.e.b yes 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.4.e.a 24 7.b odd 2 1
462.4.e.a 24 11.b odd 2 1
462.4.e.b yes 24 1.a even 1 1 trivial
462.4.e.b yes 24 77.b even 2 1 inner