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Properties

Label 460.2.m.b
Level $460$
Weight $2$
Character orbit 460.m
Analytic conductor $3.673$
Analytic rank $0$
Dimension $50$
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] = [460,2,Mod(41,460)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(460, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([0, 0, 12]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("460.41");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 460 = 2^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 460.m (of order \(11\), degree \(10\), 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: \(3.67311849298\)
Analytic rank: \(0\)
Dimension: \(50\)
Relative dimension: \(5\) over \(\Q(\zeta_{11})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 50 q + 5 q^{5} - q^{7} - 25 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 50 q + 5 q^{5} - q^{7} - 25 q^{9} - 6 q^{13} + 12 q^{17} + 19 q^{19} + 39 q^{21} - 16 q^{23} - 5 q^{25} + 21 q^{27} - 6 q^{29} + 34 q^{31} + 50 q^{33} - 10 q^{35} + 7 q^{37} - 70 q^{39} - 51 q^{41} - 18 q^{43} - 74 q^{45} + 30 q^{47} - 16 q^{49} - 80 q^{51} - 23 q^{53} - 33 q^{55} + 27 q^{57} - 18 q^{59} + 76 q^{61} + 138 q^{63} + 6 q^{65} + 25 q^{67} - 30 q^{69} - 37 q^{71} + 20 q^{73} + 92 q^{77} + 18 q^{79} + 25 q^{81} - 22 q^{83} - 12 q^{85} - 109 q^{87} + 8 q^{89} + 110 q^{91} + 64 q^{93} + 3 q^{95} - 38 q^{97} - 126 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
41.1 0 −2.08327 + 2.40422i 0 0.142315 − 0.989821i 0 −0.796185 − 1.74340i 0 −1.01332 − 7.04777i 0
41.2 0 −0.671975 + 0.775501i 0 0.142315 − 0.989821i 0 −1.80075 − 3.94308i 0 0.277094 + 1.92723i 0
41.3 0 −0.249442 + 0.287871i 0 0.142315 − 0.989821i 0 0.948129 + 2.07611i 0 0.406296 + 2.82585i 0
41.4 0 1.10811 − 1.27883i 0 0.142315 − 0.989821i 0 −1.08165 − 2.36848i 0 0.0194536 + 0.135303i 0
41.5 0 1.89657 − 2.18876i 0 0.142315 − 0.989821i 0 1.51786 + 3.32365i 0 −0.766744 − 5.33282i 0
81.1 0 −0.431764 + 3.00299i 0 0.959493 − 0.281733i 0 −2.67351 − 3.08540i 0 −5.95302 − 1.74797i 0
81.2 0 −0.282231 + 1.96296i 0 0.959493 − 0.281733i 0 2.71449 + 3.13269i 0 −0.895066 − 0.262815i 0
81.3 0 0.0891733 − 0.620214i 0 0.959493 − 0.281733i 0 0.926390 + 1.06911i 0 2.50177 + 0.734585i 0
81.4 0 0.147968 − 1.02914i 0 0.959493 − 0.281733i 0 −0.636768 − 0.734870i 0 1.84124 + 0.540637i 0
81.5 0 0.476853 − 3.31659i 0 0.959493 − 0.281733i 0 −0.777545 − 0.897335i 0 −7.89387 − 2.31785i 0
101.1 0 −2.08327 − 2.40422i 0 0.142315 + 0.989821i 0 −0.796185 + 1.74340i 0 −1.01332 + 7.04777i 0
101.2 0 −0.671975 − 0.775501i 0 0.142315 + 0.989821i 0 −1.80075 + 3.94308i 0 0.277094 − 1.92723i 0
101.3 0 −0.249442 − 0.287871i 0 0.142315 + 0.989821i 0 0.948129 − 2.07611i 0 0.406296 − 2.82585i 0
101.4 0 1.10811 + 1.27883i 0 0.142315 + 0.989821i 0 −1.08165 + 2.36848i 0 0.0194536 − 0.135303i 0
101.5 0 1.89657 + 2.18876i 0 0.142315 + 0.989821i 0 1.51786 − 3.32365i 0 −0.766744 + 5.33282i 0
121.1 0 −2.70621 − 0.794614i 0 −0.841254 + 0.540641i 0 −0.171423 + 1.19227i 0 4.16838 + 2.67886i 0
121.2 0 −1.08763 − 0.319356i 0 −0.841254 + 0.540641i 0 0.0860775 − 0.598682i 0 −1.44281 − 0.927240i 0
121.3 0 −0.169429 − 0.0497487i 0 −0.841254 + 0.540641i 0 0.0167209 − 0.116296i 0 −2.49753 − 1.60506i 0
121.4 0 1.36238 + 0.400031i 0 −0.841254 + 0.540641i 0 −0.649858 + 4.51987i 0 −0.827709 − 0.531936i 0
121.5 0 2.60088 + 0.763688i 0 −0.841254 + 0.540641i 0 0.742558 − 5.16460i 0 3.65762 + 2.35061i 0
See all 50 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 41.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.c even 11 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 460.2.m.b ✓ 50
23.c even 11 1 inner 460.2.m.b ✓ 50
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
460.2.m.b ✓ 50 1.a even 1 1 trivial
460.2.m.b ✓ 50 23.c even 11 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{50} + 20 T_{3}^{48} - 7 T_{3}^{47} + 223 T_{3}^{46} - 115 T_{3}^{45} + 1959 T_{3}^{44} + \cdots + 118352641 \) acting on \(S_{2}^{\mathrm{new}}(460, [\chi])\). Copy content Toggle raw display