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Properties

Label 2535.1.f.d
Level $2535$
Weight $1$
Character orbit 2535.f
Self dual yes
Analytic conductor $1.265$
Analytic rank $0$
Dimension $3$
Projective image $D_{7}$
CM discriminant -15
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2535,1,Mod(1184,2535)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2535, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2535.1184");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2535 = 3 \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2535.f (of order \(2\), degree \(1\), 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: yes
Analytic conductor: \(1.26512980702\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{7}\)
Projective field: Galois closure of 7.1.16290480375.1

$q$-expansion

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

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + q^{3} + (\beta_{2} + 1) q^{4} - q^{5} + \beta_1 q^{6} + (\beta_{2} + 1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + q^{3} + (\beta_{2} + 1) q^{4} - q^{5} + \beta_1 q^{6} + (\beta_{2} + 1) q^{8} + q^{9} - \beta_1 q^{10} + (\beta_{2} + 1) q^{12} - q^{15} + \beta_1 q^{16} + ( - \beta_{2} + \beta_1 - 1) q^{17} + \beta_1 q^{18} - \beta_{2} q^{19} + ( - \beta_{2} - 1) q^{20} - \beta_1 q^{23} + (\beta_{2} + 1) q^{24} + q^{25} + q^{27} - \beta_1 q^{30} + (\beta_{2} - \beta_1 + 1) q^{31} + q^{32} + ( - \beta_1 + 1) q^{34} + (\beta_{2} + 1) q^{36} + ( - \beta_{2} - 1) q^{38} + ( - \beta_{2} - 1) q^{40} - q^{45} + ( - \beta_{2} - 2) q^{46} - \beta_{2} q^{47} + \beta_1 q^{48} + q^{49} + \beta_1 q^{50} + ( - \beta_{2} + \beta_1 - 1) q^{51} + \beta_{2} q^{53} + \beta_1 q^{54} - \beta_{2} q^{57} + ( - \beta_{2} - 1) q^{60} + ( - \beta_{2} + \beta_1 - 1) q^{61} + (\beta_1 - 1) q^{62} - q^{68} - \beta_1 q^{69} + (\beta_{2} + 1) q^{72} + q^{75} + ( - \beta_1 - 1) q^{76} - \beta_1 q^{79} - \beta_1 q^{80} + q^{81} + (\beta_{2} - \beta_1 + 1) q^{83} + (\beta_{2} - \beta_1 + 1) q^{85} - \beta_1 q^{90} + ( - \beta_{2} - \beta_1 - 1) q^{92} + (\beta_{2} - \beta_1 + 1) q^{93} + ( - \beta_{2} - 1) q^{94} + \beta_{2} q^{95} + q^{96} + \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{2} + 3 q^{3} + 2 q^{4} - 3 q^{5} + q^{6} + 2 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + q^{2} + 3 q^{3} + 2 q^{4} - 3 q^{5} + q^{6} + 2 q^{8} + 3 q^{9} - q^{10} + 2 q^{12} - 3 q^{15} + q^{16} - q^{17} + q^{18} + q^{19} - 2 q^{20} - q^{23} + 2 q^{24} + 3 q^{25} + 3 q^{27} - q^{30} + q^{31} + 3 q^{32} + 2 q^{34} + 2 q^{36} - 2 q^{38} - 2 q^{40} - 3 q^{45} - 5 q^{46} + q^{47} + q^{48} + 3 q^{49} + q^{50} - q^{51} - q^{53} + q^{54} + q^{57} - 2 q^{60} - q^{61} - 2 q^{62} - 3 q^{68} - q^{69} + 2 q^{72} + 3 q^{75} - 4 q^{76} - q^{79} - q^{80} + 3 q^{81} + q^{83} + q^{85} - q^{90} - 3 q^{92} + q^{93} - 2 q^{94} - q^{95} + 3 q^{96} + q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of \(\nu = \zeta_{14} + \zeta_{14}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2535\mathbb{Z}\right)^\times\).

\(n\) \(1522\) \(1691\) \(1861\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1184.1
−1.24698
0.445042
1.80194
−1.24698 1.00000 0.554958 −1.00000 −1.24698 0 0.554958 1.00000 1.24698
1184.2 0.445042 1.00000 −0.801938 −1.00000 0.445042 0 −0.801938 1.00000 −0.445042
1184.3 1.80194 1.00000 2.24698 −1.00000 1.80194 0 2.24698 1.00000 −1.80194
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 CM by \(\Q(\sqrt{-15}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2535.1.f.d yes 3
3.b odd 2 1 2535.1.f.a ✓ 3
5.b even 2 1 2535.1.f.a ✓ 3
13.b even 2 1 2535.1.f.b yes 3
13.c even 3 2 2535.1.x.a 6
13.d odd 4 2 2535.1.e.b 6
13.e even 6 2 2535.1.x.c 6
13.f odd 12 4 2535.1.y.b 12
15.d odd 2 1 CM 2535.1.f.d yes 3
39.d odd 2 1 2535.1.f.c yes 3
39.f even 4 2 2535.1.e.a 6
39.h odd 6 2 2535.1.x.b 6
39.i odd 6 2 2535.1.x.d 6
39.k even 12 4 2535.1.y.c 12
65.d even 2 1 2535.1.f.c yes 3
65.g odd 4 2 2535.1.e.a 6
65.l even 6 2 2535.1.x.b 6
65.n even 6 2 2535.1.x.d 6
65.s odd 12 4 2535.1.y.c 12
195.e odd 2 1 2535.1.f.b yes 3
195.n even 4 2 2535.1.e.b 6
195.x odd 6 2 2535.1.x.a 6
195.y odd 6 2 2535.1.x.c 6
195.bh even 12 4 2535.1.y.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2535.1.e.a 6 39.f even 4 2
2535.1.e.a 6 65.g odd 4 2
2535.1.e.b 6 13.d odd 4 2
2535.1.e.b 6 195.n even 4 2
2535.1.f.a ✓ 3 3.b odd 2 1
2535.1.f.a ✓ 3 5.b even 2 1
2535.1.f.b yes 3 13.b even 2 1
2535.1.f.b yes 3 195.e odd 2 1
2535.1.f.c yes 3 39.d odd 2 1
2535.1.f.c yes 3 65.d even 2 1
2535.1.f.d yes 3 1.a even 1 1 trivial
2535.1.f.d yes 3 15.d odd 2 1 CM
2535.1.x.a 6 13.c even 3 2
2535.1.x.a 6 195.x odd 6 2
2535.1.x.b 6 39.h odd 6 2
2535.1.x.b 6 65.l even 6 2
2535.1.x.c 6 13.e even 6 2
2535.1.x.c 6 195.y odd 6 2
2535.1.x.d 6 39.i odd 6 2
2535.1.x.d 6 65.n even 6 2
2535.1.y.b 12 13.f odd 12 4
2535.1.y.b 12 195.bh even 12 4
2535.1.y.c 12 39.k even 12 4
2535.1.y.c 12 65.s odd 12 4

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(2535, [\chi])\):

\( T_{2}^{3} - T_{2}^{2} - 2T_{2} + 1 \) Copy content Toggle raw display
\( T_{17}^{3} + T_{17}^{2} - 2T_{17} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - T^{2} - 2T + 1 \) Copy content Toggle raw display
$3$ \( (T - 1)^{3} \) Copy content Toggle raw display
$5$ \( (T + 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} \) Copy content Toggle raw display
$11$ \( T^{3} \) Copy content Toggle raw display
$13$ \( T^{3} \) Copy content Toggle raw display
$17$ \( T^{3} + T^{2} - 2T - 1 \) Copy content Toggle raw display
$19$ \( T^{3} - T^{2} - 2T + 1 \) Copy content Toggle raw display
$23$ \( T^{3} + T^{2} - 2T - 1 \) Copy content Toggle raw display
$29$ \( T^{3} \) Copy content Toggle raw display
$31$ \( T^{3} - T^{2} - 2T + 1 \) Copy content Toggle raw display
$37$ \( T^{3} \) Copy content Toggle raw display
$41$ \( T^{3} \) Copy content Toggle raw display
$43$ \( T^{3} \) Copy content Toggle raw display
$47$ \( T^{3} - T^{2} - 2T + 1 \) Copy content Toggle raw display
$53$ \( T^{3} + T^{2} - 2T - 1 \) Copy content Toggle raw display
$59$ \( T^{3} \) Copy content Toggle raw display
$61$ \( T^{3} + T^{2} - 2T - 1 \) Copy content Toggle raw display
$67$ \( T^{3} \) Copy content Toggle raw display
$71$ \( T^{3} \) Copy content Toggle raw display
$73$ \( T^{3} \) Copy content Toggle raw display
$79$ \( T^{3} + T^{2} - 2T - 1 \) Copy content Toggle raw display
$83$ \( T^{3} - T^{2} - 2T + 1 \) Copy content Toggle raw display
$89$ \( T^{3} \) Copy content Toggle raw display
$97$ \( T^{3} \) Copy content Toggle raw display
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