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Properties

Label 1740.2.bc.c
Level $1740$
Weight $2$
Character orbit 1740.bc
Analytic conductor $13.894$
Analytic rank $0$
Dimension $30$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1740,2,Mod(853,1740)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1740, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 3, 1])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1740.853"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 1740 = 2^{2} \cdot 3 \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1740.bc (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [30] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(13.8939699517\)
Analytic rank: \(0\)
Dimension: \(30\)
Relative dimension: \(15\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 30 q - 30 q^{9} + 4 q^{11} - 6 q^{13} - 6 q^{15} + 8 q^{19} - 6 q^{25} + 4 q^{29} - 4 q^{33} - 16 q^{35} - 6 q^{39} - 10 q^{41} - 6 q^{53} + 4 q^{55} + 8 q^{57} + 30 q^{61} - 2 q^{65} + 4 q^{67} + 12 q^{73}+ \cdots - 4 q^{99}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
853.1 0 1.00000i 0 −2.22455 + 0.226663i 0 2.93449 − 2.93449i 0 −1.00000 0
853.2 0 1.00000i 0 −2.01691 + 0.965450i 0 −2.62508 + 2.62508i 0 −1.00000 0
853.3 0 1.00000i 0 −1.98303 + 1.03325i 0 0.377598 − 0.377598i 0 −1.00000 0
853.4 0 1.00000i 0 −1.70944 − 1.44147i 0 −0.746381 + 0.746381i 0 −1.00000 0
853.5 0 1.00000i 0 −1.37520 − 1.76319i 0 0.501610 − 0.501610i 0 −1.00000 0
853.6 0 1.00000i 0 −0.833642 + 2.07486i 0 0.321435 − 0.321435i 0 −1.00000 0
853.7 0 1.00000i 0 0.0712321 − 2.23493i 0 −0.799559 + 0.799559i 0 −1.00000 0
853.8 0 1.00000i 0 0.118428 + 2.23293i 0 3.19123 − 3.19123i 0 −1.00000 0
853.9 0 1.00000i 0 0.304001 + 2.21531i 0 −0.0397037 + 0.0397037i 0 −1.00000 0
853.10 0 1.00000i 0 0.734667 + 2.11193i 0 −3.54291 + 3.54291i 0 −1.00000 0
853.11 0 1.00000i 0 0.964742 − 2.01724i 0 2.38076 − 2.38076i 0 −1.00000 0
853.12 0 1.00000i 0 1.73866 + 1.40608i 0 −0.403526 + 0.403526i 0 −1.00000 0
853.13 0 1.00000i 0 1.84878 − 1.25778i 0 −2.83016 + 2.83016i 0 −1.00000 0
853.14 0 1.00000i 0 2.12991 − 0.680807i 0 2.56441 − 2.56441i 0 −1.00000 0
853.15 0 1.00000i 0 2.23235 + 0.128942i 0 −1.28422 + 1.28422i 0 −1.00000 0
1177.1 0 − 1.00000i 0 −2.22455 − 0.226663i 0 2.93449 + 2.93449i 0 −1.00000 0
1177.2 0 − 1.00000i 0 −2.01691 − 0.965450i 0 −2.62508 − 2.62508i 0 −1.00000 0
1177.3 0 − 1.00000i 0 −1.98303 − 1.03325i 0 0.377598 + 0.377598i 0 −1.00000 0
1177.4 0 − 1.00000i 0 −1.70944 + 1.44147i 0 −0.746381 − 0.746381i 0 −1.00000 0
1177.5 0 − 1.00000i 0 −1.37520 + 1.76319i 0 0.501610 + 0.501610i 0 −1.00000 0
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 853.15
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
145.j even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1740.2.bc.c yes 30
5.c odd 4 1 1740.2.bb.c ✓ 30
29.c odd 4 1 1740.2.bb.c ✓ 30
145.j even 4 1 inner 1740.2.bc.c yes 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1740.2.bb.c ✓ 30 5.c odd 4 1
1740.2.bb.c ✓ 30 29.c odd 4 1
1740.2.bc.c yes 30 1.a even 1 1 trivial
1740.2.bc.c yes 30 145.j even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{30} + 1052 T_{7}^{26} - 212 T_{7}^{25} + 6672 T_{7}^{23} + 351910 T_{7}^{22} - 53020 T_{7}^{21} + \cdots + 41472 \) acting on \(S_{2}^{\mathrm{new}}(1740, [\chi])\). Copy content Toggle raw display