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Properties

Label 2535.1.e.b
Level $2535$
Weight $1$
Character orbit 2535.e
Analytic conductor $1.265$
Analytic rank $0$
Dimension $6$
Projective image $D_{7}$
CM discriminant -15
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2535,1,Mod(2534,2535)]
 
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, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2535.2534");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2535 = 3 \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2535.e (of order \(2\), degree \(1\), not 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: no
Analytic conductor: \(1.26512980702\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.153664.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 5x^{4} + 6x^{2} + 1 \) 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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 5x^{4} + 6x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 3\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} + 3\nu^{2} + 1 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{5} + 4\nu^{3} + 3\nu \) 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
\(\nu^{3}\)\(=\) \( \beta_{3} - 3\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} - 3\beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{5} - 4\beta_{3} + 9\beta_1 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2534.1
− 0.445042i
1.80194i
1.24698i
− 1.24698i
− 1.80194i
0.445042i
− 1.80194i 1.00000 −2.24698 1.00000i − 1.80194i 0 2.24698i 1.00000 1.80194
2534.2 − 1.24698i 1.00000 −0.554958 − 1.00000i − 1.24698i 0 − 0.554958i 1.00000 −1.24698
2534.3 − 0.445042i 1.00000 0.801938 1.00000i − 0.445042i 0 − 0.801938i 1.00000 0.445042
2534.4 0.445042i 1.00000 0.801938 − 1.00000i 0.445042i 0 0.801938i 1.00000 0.445042
2534.5 1.24698i 1.00000 −0.554958 1.00000i 1.24698i 0 0.554958i 1.00000 −1.24698
2534.6 1.80194i 1.00000 −2.24698 − 1.00000i 1.80194i 0 − 2.24698i 1.00000 1.80194
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2534.6
Significant digits:
Format:

Inner twists

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

Twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{17}^{3} - T_{17}^{2} - 2T_{17} + 1 \) acting on \(S_{1}^{\mathrm{new}}(2535, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 5 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( (T - 1)^{6} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( T^{6} \) Copy content Toggle raw display
$13$ \( T^{6} \) Copy content Toggle raw display
$17$ \( (T^{3} - T^{2} - 2 T + 1)^{2} \) Copy content Toggle raw display
$19$ \( T^{6} + 5 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$23$ \( (T^{3} - T^{2} - 2 T + 1)^{2} \) Copy content Toggle raw display
$29$ \( T^{6} \) Copy content Toggle raw display
$31$ \( T^{6} + 5 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$37$ \( T^{6} \) Copy content Toggle raw display
$41$ \( T^{6} \) Copy content Toggle raw display
$43$ \( T^{6} \) Copy content Toggle raw display
$47$ \( T^{6} + 5 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$53$ \( (T^{3} + T^{2} - 2 T - 1)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} \) Copy content Toggle raw display
$61$ \( (T^{3} + T^{2} - 2 T - 1)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} \) Copy content Toggle raw display
$71$ \( T^{6} \) Copy content Toggle raw display
$73$ \( T^{6} \) Copy content Toggle raw display
$79$ \( (T^{3} + T^{2} - 2 T - 1)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 5 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$89$ \( T^{6} \) Copy content Toggle raw display
$97$ \( T^{6} \) Copy content Toggle raw display
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