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Properties

Label 740.2.k.b
Level $740$
Weight $2$
Character orbit 740.k
Analytic conductor $5.909$
Analytic rank $0$
Dimension $2$
CM discriminant -4
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [740,2,Mod(179,740)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(740, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 2, 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("740.179"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 740 = 2^{2} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 740.k (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.90892974957\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - i + 1) q^{2} - 2 i q^{4} + (i + 2) q^{5} + ( - 2 i - 2) q^{8} + 3 q^{9} + ( - i + 3) q^{10} + (5 i + 5) q^{13} - 4 q^{16} + ( - 3 i - 3) q^{17} + ( - 3 i + 3) q^{18} + ( - 4 i + 2) q^{20} + (4 i + 3) q^{25} + \cdots + (7 i - 7) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 4 q^{5} - 4 q^{8} + 6 q^{9} + 6 q^{10} + 10 q^{13} - 8 q^{16} - 6 q^{17} + 6 q^{18} + 4 q^{20} + 6 q^{25} + 20 q^{26} - 14 q^{29} - 8 q^{32} - 12 q^{34} + 12 q^{37} - 4 q^{40} + 12 q^{45}+ \cdots - 14 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/740\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(297\) \(371\)
\(\chi(n)\) \(i\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
179.1
1.00000i
− 1.00000i
1.00000 − 1.00000i 0 − 2.00000i 2.00000 + 1.00000i 0 0 −2.00000 − 2.00000i 3.00000 3.00000 − 1.00000i
339.1 1.00000 + 1.00000i 0 2.00000i 2.00000 − 1.00000i 0 0 −2.00000 + 2.00000i 3.00000 3.00000 + 1.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
185.j odd 4 1 inner
740.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 740.2.k.b yes 2
4.b odd 2 1 CM 740.2.k.b yes 2
5.b even 2 1 740.2.k.a ✓ 2
20.d odd 2 1 740.2.k.a ✓ 2
37.d odd 4 1 740.2.k.a ✓ 2
148.g even 4 1 740.2.k.a ✓ 2
185.j odd 4 1 inner 740.2.k.b yes 2
740.k even 4 1 inner 740.2.k.b yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
740.2.k.a ✓ 2 5.b even 2 1
740.2.k.a ✓ 2 20.d odd 2 1
740.2.k.a ✓ 2 37.d odd 4 1
740.2.k.a ✓ 2 148.g even 4 1
740.2.k.b yes 2 1.a even 1 1 trivial
740.2.k.b yes 2 4.b odd 2 1 CM
740.2.k.b yes 2 185.j odd 4 1 inner
740.2.k.b yes 2 740.k even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(740, [\chi])\):

\( T_{3} \) Copy content Toggle raw display
\( T_{13}^{2} - 10T_{13} + 50 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 4T + 5 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 10T + 50 \) Copy content Toggle raw display
$17$ \( T^{2} + 6T + 18 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 14T + 98 \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 12T + 37 \) Copy content Toggle raw display
$41$ \( T^{2} + 100 \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 196 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( (T + 16)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 6T + 18 \) Copy content Toggle raw display
$97$ \( T^{2} - 10T + 50 \) Copy content Toggle raw display
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