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Properties

Label 2499.1.q.f
Level $2499$
Weight $1$
Character orbit 2499.q
Analytic conductor $1.247$
Analytic rank $0$
Dimension $8$
Projective image $D_{10}$
CM discriminant -119
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2499,1,Mod(1733,2499)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2499, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 4, 3]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2499.1733");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2499 = 3 \cdot 7^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2499.q (of order \(6\), degree \(2\), 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: \(1.24716346657\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{15})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{10}\)
Projective field: Galois closure of 10.2.341108199621.1

$q$-expansion

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

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + (\zeta_{30}^{8} - \zeta_{30}^{2}) q^{2} - \zeta_{30}^{7} q^{3} + ( - \zeta_{30}^{10} + \cdots - \zeta_{30}) q^{4} + \cdots + \zeta_{30}^{14} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{30}^{8} - \zeta_{30}^{2}) q^{2} - \zeta_{30}^{7} q^{3} + ( - \zeta_{30}^{10} + \cdots - \zeta_{30}) q^{4} + \cdots + ( - \zeta_{30}^{9} - \zeta_{30}^{6}) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + q^{3} + 6 q^{4} - 2 q^{5} + 10 q^{6} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + q^{3} + 6 q^{4} - 2 q^{5} + 10 q^{6} + q^{9} + q^{12} + 4 q^{15} - 4 q^{16} - 4 q^{17} - 16 q^{20} + 5 q^{24} - 2 q^{25} - 2 q^{27} - 5 q^{30} + 8 q^{36} - 4 q^{41} + 4 q^{43} + 3 q^{45} + 8 q^{48} + q^{51} + 3 q^{60} - 12 q^{64} - 2 q^{67} + 6 q^{68} + 5 q^{72} - 2 q^{75} - 12 q^{80} + q^{81} + 4 q^{85} + 10 q^{90} - 5 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2499\mathbb{Z}\right)^\times\).

\(n\) \(52\) \(1618\) \(1667\)
\(\chi(n)\) \(-\zeta_{30}^{5}\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1733.1
0.913545 − 0.406737i
−0.978148 − 0.207912i
0.669131 + 0.743145i
−0.104528 − 0.994522i
0.913545 + 0.406737i
−0.978148 + 0.207912i
0.669131 − 0.743145i
−0.104528 + 0.994522i
−1.64728 + 0.951057i −0.978148 − 0.207912i 1.30902 − 2.26728i −0.809017 − 1.40126i 1.80902 − 0.587785i 0 3.07768i 0.913545 + 0.406737i 2.66535 + 1.53884i
1733.2 −1.01807 + 0.587785i −0.104528 − 0.994522i 0.190983 − 0.330792i 0.309017 + 0.535233i 0.690983 + 0.951057i 0 − 0.726543i −0.978148 + 0.207912i −0.629204 − 0.363271i
1733.3 1.01807 − 0.587785i 0.913545 − 0.406737i 0.190983 − 0.330792i 0.309017 + 0.535233i 0.690983 − 0.951057i 0 0.726543i 0.669131 − 0.743145i 0.629204 + 0.363271i
1733.4 1.64728 − 0.951057i 0.669131 + 0.743145i 1.30902 − 2.26728i −0.809017 − 1.40126i 1.80902 + 0.587785i 0 − 3.07768i −0.104528 + 0.994522i −2.66535 − 1.53884i
2039.1 −1.64728 − 0.951057i −0.978148 + 0.207912i 1.30902 + 2.26728i −0.809017 + 1.40126i 1.80902 + 0.587785i 0 − 3.07768i 0.913545 − 0.406737i 2.66535 − 1.53884i
2039.2 −1.01807 − 0.587785i −0.104528 + 0.994522i 0.190983 + 0.330792i 0.309017 − 0.535233i 0.690983 − 0.951057i 0 0.726543i −0.978148 − 0.207912i −0.629204 + 0.363271i
2039.3 1.01807 + 0.587785i 0.913545 + 0.406737i 0.190983 + 0.330792i 0.309017 − 0.535233i 0.690983 + 0.951057i 0 − 0.726543i 0.669131 + 0.743145i 0.629204 − 0.363271i
2039.4 1.64728 + 0.951057i 0.669131 − 0.743145i 1.30902 + 2.26728i −0.809017 + 1.40126i 1.80902 − 0.587785i 0 3.07768i −0.104528 − 0.994522i −2.66535 + 1.53884i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1733.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
119.d odd 2 1 CM by \(\Q(\sqrt{-119}) \)
7.c even 3 1 inner
21.c even 2 1 inner
21.g even 6 1 inner
51.c odd 2 1 inner
119.h odd 6 1 inner
357.q odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2499.1.q.f 8
3.b odd 2 1 2499.1.q.e 8
7.b odd 2 1 2499.1.q.e 8
7.c even 3 1 2499.1.e.e ✓ 4
7.c even 3 1 inner 2499.1.q.f 8
7.d odd 6 1 2499.1.e.f yes 4
7.d odd 6 1 2499.1.q.e 8
17.b even 2 1 2499.1.q.e 8
21.c even 2 1 inner 2499.1.q.f 8
21.g even 6 1 2499.1.e.e ✓ 4
21.g even 6 1 inner 2499.1.q.f 8
21.h odd 6 1 2499.1.e.f yes 4
21.h odd 6 1 2499.1.q.e 8
51.c odd 2 1 inner 2499.1.q.f 8
119.d odd 2 1 CM 2499.1.q.f 8
119.h odd 6 1 2499.1.e.e ✓ 4
119.h odd 6 1 inner 2499.1.q.f 8
119.j even 6 1 2499.1.e.f yes 4
119.j even 6 1 2499.1.q.e 8
357.c even 2 1 2499.1.q.e 8
357.q odd 6 1 2499.1.e.e ✓ 4
357.q odd 6 1 inner 2499.1.q.f 8
357.s even 6 1 2499.1.e.f yes 4
357.s even 6 1 2499.1.q.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2499.1.e.e ✓ 4 7.c even 3 1
2499.1.e.e ✓ 4 21.g even 6 1
2499.1.e.e ✓ 4 119.h odd 6 1
2499.1.e.e ✓ 4 357.q odd 6 1
2499.1.e.f yes 4 7.d odd 6 1
2499.1.e.f yes 4 21.h odd 6 1
2499.1.e.f yes 4 119.j even 6 1
2499.1.e.f yes 4 357.s even 6 1
2499.1.q.e 8 3.b odd 2 1
2499.1.q.e 8 7.b odd 2 1
2499.1.q.e 8 7.d odd 6 1
2499.1.q.e 8 17.b even 2 1
2499.1.q.e 8 21.h odd 6 1
2499.1.q.e 8 119.j even 6 1
2499.1.q.e 8 357.c even 2 1
2499.1.q.e 8 357.s even 6 1
2499.1.q.f 8 1.a even 1 1 trivial
2499.1.q.f 8 7.c even 3 1 inner
2499.1.q.f 8 21.c even 2 1 inner
2499.1.q.f 8 21.g even 6 1 inner
2499.1.q.f 8 51.c odd 2 1 inner
2499.1.q.f 8 119.d odd 2 1 CM
2499.1.q.f 8 119.h odd 6 1 inner
2499.1.q.f 8 357.q odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(2499, [\chi])\):

\( T_{2}^{8} - 5T_{2}^{6} + 20T_{2}^{4} - 25T_{2}^{2} + 25 \) Copy content Toggle raw display
\( T_{5}^{4} + T_{5}^{3} + 2T_{5}^{2} - T_{5} + 1 \) Copy content Toggle raw display
\( T_{41}^{2} + T_{41} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 5 T^{6} + \cdots + 25 \) Copy content Toggle raw display
$3$ \( T^{8} - T^{7} + T^{5} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( (T^{4} + T^{3} + 2 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( T^{8} - 5 T^{6} + \cdots + 25 \) Copy content Toggle raw display
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( (T^{2} + T - 1)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} - T - 1)^{4} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( T^{8} - 5 T^{6} + \cdots + 25 \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( T^{8} - 5 T^{6} + \cdots + 25 \) Copy content Toggle raw display
$67$ \( (T^{4} + T^{3} + 2 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( T^{8} - 5 T^{6} + \cdots + 25 \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( T^{8} \) Copy content Toggle raw display
$89$ \( T^{8} \) Copy content Toggle raw display
$97$ \( (T^{4} + 5 T^{2} + 5)^{2} \) Copy content Toggle raw display
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