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Properties

Label 2700.3.g.h
Level $2700$
Weight $3$
Character orbit 2700.g
Analytic conductor $73.570$
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] = [2700,3,Mod(701,2700)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2700, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 3, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2700.701"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Level: \( N \) \(=\) \( 2700 = 2^{2} \cdot 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2700.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,-16,0,0,0,0,0,26] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(73.5696713773\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-5}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 540)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(\beta = 3\sqrt{-5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 8 q^{7} + 3 \beta q^{11} + 13 q^{13} + \beta q^{17} - 4 q^{19} - \beta q^{23} + \beta q^{29} + 17 q^{31} - 38 q^{37} + 10 \beta q^{41} + 49 q^{43} - 5 \beta q^{47} + 15 q^{49} - 14 \beta q^{53} + 14 \beta q^{59} + \cdots - 98 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 16 q^{7} + 26 q^{13} - 8 q^{19} + 34 q^{31} - 76 q^{37} + 98 q^{43} + 30 q^{49} - 92 q^{61} + 44 q^{67} - 184 q^{73} + 262 q^{79} - 208 q^{91} - 196 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2700\mathbb{Z}\right)^\times\).

\(n\) \(1001\) \(1351\) \(2377\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
701.1
− 2.23607i
2.23607i
0 0 0 0 0 −8.00000 0 0 0
701.2 0 0 0 0 0 −8.00000 0 0 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
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2700.3.g.h 2
3.b odd 2 1 inner 2700.3.g.h 2
5.b even 2 1 540.3.g.c ✓ 2
5.c odd 4 2 2700.3.b.j 4
15.d odd 2 1 540.3.g.c ✓ 2
15.e even 4 2 2700.3.b.j 4
20.d odd 2 1 2160.3.l.b 2
45.h odd 6 2 1620.3.o.a 4
45.j even 6 2 1620.3.o.a 4
60.h even 2 1 2160.3.l.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
540.3.g.c ✓ 2 5.b even 2 1
540.3.g.c ✓ 2 15.d odd 2 1
1620.3.o.a 4 45.h odd 6 2
1620.3.o.a 4 45.j even 6 2
2160.3.l.b 2 20.d odd 2 1
2160.3.l.b 2 60.h even 2 1
2700.3.b.j 4 5.c odd 4 2
2700.3.b.j 4 15.e even 4 2
2700.3.g.h 2 1.a even 1 1 trivial
2700.3.g.h 2 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(2700, [\chi])\):

\( T_{7} + 8 \) Copy content Toggle raw display
\( T_{11}^{2} + 405 \) Copy content Toggle raw display
\( T_{13} - 13 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T + 8)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 405 \) Copy content Toggle raw display
$13$ \( (T - 13)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 45 \) Copy content Toggle raw display
$19$ \( (T + 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 45 \) Copy content Toggle raw display
$29$ \( T^{2} + 45 \) Copy content Toggle raw display
$31$ \( (T - 17)^{2} \) Copy content Toggle raw display
$37$ \( (T + 38)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 4500 \) Copy content Toggle raw display
$43$ \( (T - 49)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 1125 \) Copy content Toggle raw display
$53$ \( T^{2} + 8820 \) Copy content Toggle raw display
$59$ \( T^{2} + 8820 \) Copy content Toggle raw display
$61$ \( (T + 46)^{2} \) Copy content Toggle raw display
$67$ \( (T - 22)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 18000 \) Copy content Toggle raw display
$73$ \( (T + 92)^{2} \) Copy content Toggle raw display
$79$ \( (T - 131)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 8820 \) Copy content Toggle raw display
$89$ \( T^{2} + 720 \) Copy content Toggle raw display
$97$ \( (T + 98)^{2} \) Copy content Toggle raw display
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