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Properties

Label 2900.2.s.a
Level $2900$
Weight $2$
Character orbit 2900.s
Analytic conductor $23.157$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2900,2,Mod(157,2900)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2900, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 1, 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("2900.157"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 2900 = 2^{2} \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2900.s (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(23.1566165862\)
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: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q - 2 q^{3} + (3 i + 3) q^{7} + q^{9} + (2 i + 2) q^{11} + ( - 2 i - 2) q^{13} - 6 i q^{17} + ( - 4 i + 4) q^{19} + ( - 6 i - 6) q^{21} + (5 i - 5) q^{23} + 4 q^{27} + ( - 5 i + 2) q^{29} + ( - 6 i - 6) q^{31} + \cdots + (2 i + 2) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{3} + 6 q^{7} + 2 q^{9} + 4 q^{11} - 4 q^{13} + 8 q^{19} - 12 q^{21} - 10 q^{23} + 8 q^{27} + 4 q^{29} - 12 q^{31} - 8 q^{33} - 12 q^{37} + 8 q^{39} + 14 q^{41} - 8 q^{43} + 16 q^{47} - 8 q^{53}+ \cdots + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2900\mathbb{Z}\right)^\times\).

\(n\) \(901\) \(1277\) \(1451\)
\(\chi(n)\) \(i\) \(i\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
157.1
1.00000i
− 1.00000i
0 −2.00000 0 0 0 3.00000 + 3.00000i 0 1.00000 0
1293.1 0 −2.00000 0 0 0 3.00000 − 3.00000i 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2900.2.s.a yes 2
5.b even 2 1 2900.2.s.b yes 2
5.c odd 4 1 2900.2.j.a ✓ 2
5.c odd 4 1 2900.2.j.b yes 2
29.c odd 4 1 2900.2.j.b yes 2
145.e even 4 1 inner 2900.2.s.a yes 2
145.f odd 4 1 2900.2.j.a ✓ 2
145.j even 4 1 2900.2.s.b yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2900.2.j.a ✓ 2 5.c odd 4 1
2900.2.j.a ✓ 2 145.f odd 4 1
2900.2.j.b yes 2 5.c odd 4 1
2900.2.j.b yes 2 29.c odd 4 1
2900.2.s.a yes 2 1.a even 1 1 trivial
2900.2.s.a yes 2 145.e even 4 1 inner
2900.2.s.b yes 2 5.b even 2 1
2900.2.s.b yes 2 145.j even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 2 \) acting on \(S_{2}^{\mathrm{new}}(2900, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T + 2)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 6T + 18 \) Copy content Toggle raw display
$11$ \( T^{2} - 4T + 8 \) Copy content Toggle raw display
$13$ \( T^{2} + 4T + 8 \) Copy content Toggle raw display
$17$ \( T^{2} + 36 \) Copy content Toggle raw display
$19$ \( T^{2} - 8T + 32 \) Copy content Toggle raw display
$23$ \( T^{2} + 10T + 50 \) Copy content Toggle raw display
$29$ \( T^{2} - 4T + 29 \) Copy content Toggle raw display
$31$ \( T^{2} + 12T + 72 \) Copy content Toggle raw display
$37$ \( (T + 6)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 14T + 98 \) Copy content Toggle raw display
$43$ \( (T + 4)^{2} \) Copy content Toggle raw display
$47$ \( (T - 8)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 8T + 32 \) Copy content Toggle raw display
$59$ \( T^{2} + 16 \) Copy content Toggle raw display
$61$ \( T^{2} - 18T + 162 \) Copy content Toggle raw display
$67$ \( T^{2} + 6T + 18 \) Copy content Toggle raw display
$71$ \( T^{2} + 144 \) Copy content Toggle raw display
$73$ \( T^{2} + 196 \) Copy content Toggle raw display
$79$ \( T^{2} - 16T + 128 \) Copy content Toggle raw display
$83$ \( T^{2} + 6T + 18 \) Copy content Toggle raw display
$89$ \( T^{2} + 14T + 98 \) Copy content Toggle raw display
$97$ \( (T + 2)^{2} \) Copy content Toggle raw display
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