Properties

Label 740.2.bb.a
Level $740$
Weight $2$
Character orbit 740.bb
Analytic conductor $5.909$
Analytic rank $0$
Dimension $36$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [740,2,Mod(269,740)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(740, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("740.269");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 740 = 2^{2} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 740.bb (of order \(6\), 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: \(5.90892974957\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(18\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 2 q^{5} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q - 2 q^{5} + 12 q^{9} - 12 q^{11} + 3 q^{15} - 10 q^{19} + 4 q^{21} + 8 q^{25} + 4 q^{29} + 40 q^{31} - q^{35} + 4 q^{39} + 22 q^{41} + 22 q^{45} + 24 q^{49} + 40 q^{51} - 6 q^{55} - 14 q^{59} + 12 q^{61} + 9 q^{65} + 2 q^{69} - 24 q^{71} - 62 q^{75} - 12 q^{79} + 26 q^{81} - 24 q^{85} + 30 q^{91} + 9 q^{95} - 16 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
269.1 0 −2.52616 + 1.45848i 0 −1.70568 + 1.44591i 0 1.93278 1.11589i 0 2.75432 4.77062i 0
269.2 0 −2.34777 + 1.35549i 0 2.18160 + 0.490514i 0 −3.51625 + 2.03011i 0 2.17468 3.76666i 0
269.3 0 −2.31618 + 1.33725i 0 −1.42198 1.72568i 0 −0.105556 + 0.0609429i 0 2.07646 3.59654i 0
269.4 0 −1.69975 + 0.981350i 0 2.05545 + 0.880424i 0 4.10335 2.36907i 0 0.426096 0.738020i 0
269.5 0 −1.39171 + 0.803502i 0 −2.13362 + 0.669088i 0 −4.16585 + 2.40515i 0 −0.208768 + 0.361596i 0
269.6 0 −1.00394 + 0.579623i 0 1.49599 1.66193i 0 −0.211890 + 0.122335i 0 −0.828074 + 1.43427i 0
269.7 0 −0.954843 + 0.551279i 0 0.369612 2.20531i 0 −0.195612 + 0.112936i 0 −0.892183 + 1.54531i 0
269.8 0 −0.755229 + 0.436031i 0 0.0540287 + 2.23542i 0 0.453259 0.261689i 0 −1.11975 + 1.93947i 0
269.9 0 −0.419315 + 0.242092i 0 −2.16774 0.548549i 0 2.37885 1.37343i 0 −1.38278 + 2.39505i 0
269.10 0 0.419315 0.242092i 0 0.608812 2.15159i 0 −2.37885 + 1.37343i 0 −1.38278 + 2.39505i 0
269.11 0 0.755229 0.436031i 0 1.90891 + 1.16450i 0 −0.453259 + 0.261689i 0 −1.11975 + 1.93947i 0
269.12 0 0.954843 0.551279i 0 −2.09466 0.782561i 0 0.195612 0.112936i 0 −0.892183 + 1.54531i 0
269.13 0 1.00394 0.579623i 0 −2.18727 + 0.464599i 0 0.211890 0.122335i 0 −0.828074 + 1.43427i 0
269.14 0 1.39171 0.803502i 0 1.64626 1.51322i 0 4.16585 2.40515i 0 −0.208768 + 0.361596i 0
269.15 0 1.69975 0.981350i 0 −0.265253 + 2.22028i 0 −4.10335 + 2.36907i 0 0.426096 0.738020i 0
269.16 0 2.31618 1.33725i 0 −0.783499 2.09431i 0 0.105556 0.0609429i 0 2.07646 3.59654i 0
269.17 0 2.34777 1.35549i 0 −0.666005 + 2.13458i 0 3.51625 2.03011i 0 2.17468 3.76666i 0
269.18 0 2.52616 1.45848i 0 2.10504 0.754203i 0 −1.93278 + 1.11589i 0 2.75432 4.77062i 0
729.1 0 −2.52616 1.45848i 0 −1.70568 1.44591i 0 1.93278 + 1.11589i 0 2.75432 + 4.77062i 0
729.2 0 −2.34777 1.35549i 0 2.18160 0.490514i 0 −3.51625 2.03011i 0 2.17468 + 3.76666i 0
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 269.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
37.c even 3 1 inner
185.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 740.2.bb.a 36
5.b even 2 1 inner 740.2.bb.a 36
37.c even 3 1 inner 740.2.bb.a 36
185.n even 6 1 inner 740.2.bb.a 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
740.2.bb.a 36 1.a even 1 1 trivial
740.2.bb.a 36 5.b even 2 1 inner
740.2.bb.a 36 37.c even 3 1 inner
740.2.bb.a 36 185.n even 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(740, [\chi])\).