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Properties

Label 1740.3.bl.a
Level $1740$
Weight $3$
Character orbit 1740.bl
Analytic conductor $47.412$
Analytic rank $0$
Dimension $80$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1740,3,Mod(481,1740)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1740, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1740.481");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1740 = 2^{2} \cdot 3 \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1740.bl (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: \(47.4115659987\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(40\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 80 q - 56 q^{17} + 24 q^{19} + 144 q^{23} - 400 q^{25} - 64 q^{29} - 128 q^{31} - 48 q^{37} + 48 q^{39} + 56 q^{41} - 88 q^{43} - 56 q^{47} + 448 q^{49} - 64 q^{53} - 80 q^{55} - 16 q^{59} + 248 q^{61}+ \cdots - 312 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
481.1 0 −1.22474 + 1.22474i 0 − 2.23607i 0 −12.9313 0 − 3.00000i 0
481.2 0 −1.22474 + 1.22474i 0 2.23607i 0 −10.4655 0 − 3.00000i 0
481.3 0 −1.22474 + 1.22474i 0 − 2.23607i 0 −9.56934 0 − 3.00000i 0
481.4 0 −1.22474 + 1.22474i 0 2.23607i 0 −9.09417 0 − 3.00000i 0
481.5 0 −1.22474 + 1.22474i 0 − 2.23607i 0 8.51481 0 − 3.00000i 0
481.6 0 −1.22474 + 1.22474i 0 2.23607i 0 −7.04307 0 − 3.00000i 0
481.7 0 −1.22474 + 1.22474i 0 − 2.23607i 0 4.30700 0 − 3.00000i 0
481.8 0 −1.22474 + 1.22474i 0 − 2.23607i 0 −4.63403 0 − 3.00000i 0
481.9 0 −1.22474 + 1.22474i 0 2.23607i 0 1.59253 0 − 3.00000i 0
481.10 0 −1.22474 + 1.22474i 0 − 2.23607i 0 0.496482 0 − 3.00000i 0
481.11 0 −1.22474 + 1.22474i 0 2.23607i 0 −0.562325 0 − 3.00000i 0
481.12 0 −1.22474 + 1.22474i 0 − 2.23607i 0 −2.97647 0 − 3.00000i 0
481.13 0 −1.22474 + 1.22474i 0 2.23607i 0 −3.53400 0 − 3.00000i 0
481.14 0 −1.22474 + 1.22474i 0 2.23607i 0 4.92030 0 − 3.00000i 0
481.15 0 −1.22474 + 1.22474i 0 − 2.23607i 0 −5.25933 0 − 3.00000i 0
481.16 0 −1.22474 + 1.22474i 0 2.23607i 0 6.64250 0 − 3.00000i 0
481.17 0 −1.22474 + 1.22474i 0 − 2.23607i 0 8.97457 0 − 3.00000i 0
481.18 0 −1.22474 + 1.22474i 0 − 2.23607i 0 8.60551 0 − 3.00000i 0
481.19 0 −1.22474 + 1.22474i 0 2.23607i 0 10.4383 0 − 3.00000i 0
481.20 0 −1.22474 + 1.22474i 0 2.23607i 0 11.5776 0 − 3.00000i 0
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 481.40
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1740.3.bl.a ✓ 80
29.c odd 4 1 inner 1740.3.bl.a ✓ 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1740.3.bl.a ✓ 80 1.a even 1 1 trivial
1740.3.bl.a ✓ 80 29.c odd 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(1740, [\chi])\).