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Properties

Label 861.2.j.b
Level $861$
Weight $2$
Character orbit 861.j
Analytic conductor $6.875$
Analytic rank $0$
Dimension $48$
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] = [861,2,Mod(337,861)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(861, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("861.337");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 861 = 3 \cdot 7 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 861.j (of order \(4\), degree \(2\), 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: \(6.87511961403\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 60 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 60 q^{4} + 24 q^{10} - 8 q^{12} + 4 q^{13} + 116 q^{16} - 8 q^{17} + 4 q^{18} + 4 q^{19} + 8 q^{22} - 16 q^{23} - 48 q^{25} - 24 q^{26} - 8 q^{28} - 12 q^{29} - 8 q^{30} - 20 q^{31} - 68 q^{34} + 20 q^{37} + 52 q^{38} - 56 q^{40} - 16 q^{41} + 4 q^{42} - 20 q^{44} - 16 q^{45} + 36 q^{47} + 16 q^{48} + 12 q^{51} - 60 q^{52} + 24 q^{53} + 4 q^{55} + 8 q^{57} - 4 q^{58} + 32 q^{59} - 196 q^{64} + 44 q^{65} - 16 q^{66} - 52 q^{67} + 120 q^{68} - 8 q^{69} - 8 q^{70} + 16 q^{71} - 12 q^{72} - 32 q^{75} + 28 q^{76} - 32 q^{78} + 40 q^{79} - 48 q^{81} + 68 q^{82} + 48 q^{83} + 44 q^{85} - 72 q^{86} - 24 q^{88} + 12 q^{89} + 64 q^{92} - 24 q^{93} + 8 q^{94} + 12 q^{95} + 20 q^{96} - 12 q^{97} + 4 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
337.1 − 2.81088i −0.707107 − 0.707107i −5.90103 1.82118i −1.98759 + 1.98759i −0.707107 − 0.707107i 10.9653i 1.00000i 5.11911
337.2 − 2.79250i 0.707107 + 0.707107i −5.79805 − 1.51108i 1.97459 − 1.97459i 0.707107 + 0.707107i 10.6060i 1.00000i −4.21970
337.3 − 2.61024i 0.707107 + 0.707107i −4.81334 1.55909i 1.84572 − 1.84572i 0.707107 + 0.707107i 7.34347i 1.00000i 4.06960
337.4 − 2.25200i −0.707107 − 0.707107i −3.07150 − 1.07217i −1.59240 + 1.59240i −0.707107 − 0.707107i 2.41303i 1.00000i −2.41454
337.5 − 1.92217i −0.707107 − 0.707107i −1.69472 3.88161i −1.35918 + 1.35918i −0.707107 − 0.707107i − 0.586800i 1.00000i 7.46110
337.6 − 1.86430i 0.707107 + 0.707107i −1.47560 0.977615i 1.31826 − 1.31826i 0.707107 + 0.707107i − 0.977638i 1.00000i 1.82256
337.7 − 1.77892i 0.707107 + 0.707107i −1.16457 3.94806i 1.25789 − 1.25789i 0.707107 + 0.707107i − 1.48616i 1.00000i 7.02329
337.8 − 1.21536i −0.707107 − 0.707107i 0.522904 − 3.33192i −0.859388 + 0.859388i −0.707107 − 0.707107i − 3.06623i 1.00000i −4.04948
337.9 − 1.13733i −0.707107 − 0.707107i 0.706479 2.19198i −0.804214 + 0.804214i −0.707107 − 0.707107i − 3.07816i 1.00000i 2.49300
337.10 − 0.744410i 0.707107 + 0.707107i 1.44585 − 0.644003i 0.526377 − 0.526377i 0.707107 + 0.707107i − 2.56513i 1.00000i −0.479402
337.11 − 0.534010i 0.707107 + 0.707107i 1.71483 − 4.24277i 0.377602 − 0.377602i 0.707107 + 0.707107i − 1.98376i 1.00000i −2.26568
337.12 − 0.112799i −0.707107 − 0.707107i 1.98728 − 2.53611i −0.0797608 + 0.0797608i −0.707107 − 0.707107i − 0.449760i 1.00000i −0.286070
337.13 0.0529045i 0.707107 + 0.707107i 1.99720 3.30308i −0.0374091 + 0.0374091i 0.707107 + 0.707107i 0.211470i 1.00000i −0.174747
337.14 0.0845603i −0.707107 − 0.707107i 1.99285 1.27594i 0.0597931 − 0.0597931i −0.707107 − 0.707107i 0.337636i 1.00000i −0.107894
337.15 0.664218i −0.707107 − 0.707107i 1.55881 3.77039i 0.469673 − 0.469673i −0.707107 − 0.707107i 2.36383i 1.00000i −2.50436
337.16 0.726001i 0.707107 + 0.707107i 1.47292 − 1.66337i −0.513361 + 0.513361i 0.707107 + 0.707107i 2.52135i 1.00000i 1.20761
337.17 1.06965i −0.707107 − 0.707107i 0.855841 − 0.349076i 0.756359 − 0.756359i −0.707107 − 0.707107i 3.05476i 1.00000i 0.373390
337.18 1.39163i 0.707107 + 0.707107i 0.0633737 0.0920095i −0.984029 + 0.984029i 0.707107 + 0.707107i 2.87145i 1.00000i −0.128043
337.19 1.70936i −0.707107 − 0.707107i −0.921903 − 1.67523i 1.20870 − 1.20870i −0.707107 − 0.707107i 1.84285i 1.00000i 2.86357
337.20 2.00963i 0.707107 + 0.707107i −2.03859 3.18043i −1.42102 + 1.42102i 0.707107 + 0.707107i − 0.0775588i 1.00000i −6.39146
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 337.24
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
41.c even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 861.2.j.b ✓ 48
41.c even 4 1 inner 861.2.j.b ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
861.2.j.b ✓ 48 1.a even 1 1 trivial
861.2.j.b ✓ 48 41.c even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{48} + 78 T_{2}^{46} + 2827 T_{2}^{44} + 63228 T_{2}^{42} + 977467 T_{2}^{40} + 11087578 T_{2}^{38} + \cdots + 64 \) acting on \(S_{2}^{\mathrm{new}}(861, [\chi])\). Copy content Toggle raw display