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Properties

Label 1340.4.d.a
Level $1340$
Weight $4$
Character orbit 1340.d
Analytic conductor $79.063$
Analytic rank $0$
Dimension $98$
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] = [1340,4,Mod(269,1340)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1340, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1340.269");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1340 = 2^{2} \cdot 5 \cdot 67 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1340.d (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: \(79.0625594077\)
Analytic rank: \(0\)
Dimension: \(98\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 98 q - 20 q^{5} - 814 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 98 q - 20 q^{5} - 814 q^{9} - 36 q^{11} - 256 q^{15} + 144 q^{19} - 104 q^{21} + 348 q^{25} - 164 q^{29} + 552 q^{31} + 756 q^{35} - 1464 q^{39} - 220 q^{41} + 1544 q^{45} - 5350 q^{49} + 1528 q^{51} - 688 q^{55} + 672 q^{59} - 1096 q^{61} + 468 q^{65} - 1264 q^{69} - 2432 q^{71} - 1996 q^{75} + 568 q^{79} + 7890 q^{81} - 1992 q^{85} + 6932 q^{89} + 1160 q^{91} + 4920 q^{95} + 884 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
269.1 0 − 10.2243i 0 −9.41008 − 6.03741i 0 31.2311i 0 −77.5353 0
269.2 0 − 9.96783i 0 −7.58521 − 8.21368i 0 − 33.1159i 0 −72.3576 0
269.3 0 − 9.61334i 0 −6.29866 + 9.23725i 0 10.9516i 0 −65.4163 0
269.4 0 − 9.51855i 0 3.44393 + 10.6367i 0 − 26.2611i 0 −63.6028 0
269.5 0 − 9.06847i 0 11.1696 + 0.489701i 0 2.05288i 0 −55.2371 0
269.6 0 − 8.98482i 0 9.51344 − 5.87320i 0 − 17.6753i 0 −53.7269 0
269.7 0 − 8.96113i 0 9.48309 − 5.92208i 0 29.3547i 0 −53.3019 0
269.8 0 − 8.94512i 0 −0.903920 − 11.1437i 0 1.22055i 0 −53.0151 0
269.9 0 − 8.64573i 0 −11.1744 − 0.365485i 0 12.0944i 0 −47.7486 0
269.10 0 − 8.40140i 0 9.92232 − 5.15244i 0 − 17.0728i 0 −43.5834 0
269.11 0 − 8.32588i 0 −6.03684 + 9.41045i 0 − 4.65559i 0 −42.3203 0
269.12 0 − 8.14194i 0 2.71302 − 10.8462i 0 5.99285i 0 −39.2912 0
269.13 0 − 7.44233i 0 −9.97521 + 5.04927i 0 − 12.1506i 0 −28.3883 0
269.14 0 − 7.31843i 0 8.39849 + 7.38007i 0 14.8547i 0 −26.5594 0
269.15 0 − 6.71646i 0 −3.64754 + 10.5686i 0 32.9140i 0 −18.1108 0
269.16 0 − 6.67341i 0 −10.6066 + 3.53561i 0 − 27.7934i 0 −17.5345 0
269.17 0 − 6.62288i 0 −11.0545 + 1.67297i 0 − 3.75782i 0 −16.8625 0
269.18 0 − 6.24118i 0 4.12289 + 10.3924i 0 21.7068i 0 −11.9524 0
269.19 0 − 6.15153i 0 −7.98011 − 7.83057i 0 − 18.4018i 0 −10.8413 0
269.20 0 − 6.00836i 0 1.65754 − 11.0568i 0 − 8.14036i 0 −9.10037 0
See all 98 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 269.98
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1340.4.d.a ✓ 98
5.b even 2 1 inner 1340.4.d.a ✓ 98
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1340.4.d.a ✓ 98 1.a even 1 1 trivial
1340.4.d.a ✓ 98 5.b even 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(1340, [\chi])\).