Properties

Label 2715.1.dx.b
Level $2715$
Weight $1$
Character orbit 2715.dx
Analytic conductor $1.355$
Analytic rank $0$
Dimension $24$
Projective image $D_{90}$
CM discriminant -15
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2715,1,Mod(359,2715)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2715.359"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2715, base_ring=CyclotomicField(90)) chi = DirichletCharacter(H, H._module([45, 45, 73])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 2715 = 3 \cdot 5 \cdot 181 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2715.dx (of order \(90\), degree \(24\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.35496150930\)
Analytic rank: \(0\)
Dimension: \(24\)
Coefficient field: \(\Q(\zeta_{45})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{24} - x^{21} + x^{15} - x^{12} + x^{9} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{90}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{90} - \cdots)\)

$q$-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)
 

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

\(f(q)\) \(=\) \( q + (\zeta_{90}^{9} - \zeta_{90}^{5}) q^{2} - \zeta_{90}^{32} q^{3} + (\zeta_{90}^{18} + \cdots + \zeta_{90}^{10}) q^{4} + \zeta_{90}^{12} q^{5} + ( - \zeta_{90}^{41} + \zeta_{90}^{37}) q^{6} + (\zeta_{90}^{27} + \cdots - \zeta_{90}^{15}) q^{8} + \cdots + ( - \zeta_{90}^{39} + \zeta_{90}^{35}) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 6 q^{2} - 6 q^{4} + 3 q^{5} - 6 q^{8} - 3 q^{10} - 3 q^{12} - 9 q^{16} - 3 q^{17} + 3 q^{18} - 12 q^{20} - 3 q^{23} + 3 q^{24} + 3 q^{25} - 3 q^{27} - 21 q^{32} - 12 q^{34} - 3 q^{36} + 3 q^{38}+ \cdots + 3 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2715\mathbb{Z}\right)^\times\).

\(n\) \(1087\) \(1811\) \(2536\)
\(\chi(n)\) \(-1\) \(-1\) \(-\zeta_{90}^{14}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
359.1
0.990268 + 0.139173i
0.559193 0.829038i
−0.615661 + 0.788011i
−0.997564 + 0.0697565i
−0.719340 0.694658i
0.848048 + 0.529919i
0.848048 0.529919i
−0.374607 0.927184i
0.0348995 + 0.999391i
0.961262 + 0.275637i
−0.241922 0.970296i
0.961262 0.275637i
0.438371 0.898794i
−0.719340 + 0.694658i
0.0348995 0.999391i
−0.997564 0.0697565i
−0.882948 + 0.469472i
−0.374607 + 0.927184i
−0.882948 0.469472i
−0.241922 + 0.970296i
0.457027 0.308269i 0.241922 + 0.970296i −0.260762 + 0.645409i −0.104528 + 0.994522i 0.409677 + 0.368875i 0 0.194401 + 0.914583i −0.882948 + 0.469472i 0.258808 + 0.486747i
374.1 0.982665 + 1.57259i −0.990268 0.139173i −1.06905 + 2.19187i 0.669131 + 0.743145i −0.754239 1.69405i 0 −2.65323 + 0.278865i 0.961262 + 0.275637i −0.511133 + 1.78253i
509.1 −0.135369 1.93586i 0.719340 0.694658i −2.73898 + 0.384938i −0.104528 + 0.994522i −1.44214 1.29851i 0 0.712489 + 3.35200i 0.0348995 0.999391i 1.93941 + 0.0677257i
554.1 −0.130676 0.245765i 0.615661 + 0.788011i 0.515869 0.764807i 0.669131 0.743145i 0.113214 0.254282i 0 −0.532196 0.0559360i −0.241922 + 0.970296i −0.270078 0.0673380i
779.1 1.57506 + 1.23057i −0.848048 + 0.529919i 0.724587 + 2.90616i −0.978148 + 0.207912i −1.98783 0.208930i 0 −1.62199 + 3.64306i 0.438371 0.898794i −1.79649 0.876208i
824.1 −1.24871 + 1.29308i −0.559193 + 0.829038i −0.0778720 2.22996i 0.913545 + 0.406737i −0.373740 1.75831i 0 1.64489 + 1.48106i −0.374607 0.927184i −1.66669 + 0.673388i
1229.1 −1.24871 1.29308i −0.559193 0.829038i −0.0778720 + 2.22996i 0.913545 0.406737i −0.373740 + 1.75831i 0 1.64489 1.48106i −0.374607 + 0.927184i −1.66669 0.673388i
1304.1 −1.24871 0.609036i −0.961262 0.275637i 0.572689 + 0.733008i −0.104528 + 0.994522i 1.03246 + 0.929634i 0 0.0201618 + 0.0948538i 0.848048 + 0.529919i 0.736226 1.17821i
1319.1 −0.135369 + 0.0337512i −0.438371 + 0.898794i −0.865762 + 0.460334i 0.913545 0.406737i 0.0290064 0.136464i 0 0.205339 0.184888i −0.615661 0.788011i −0.109938 + 0.0858927i
1439.1 0.982665 + 0.397023i 0.882948 0.469472i 0.0886642 + 0.0856220i −0.978148 0.207912i 1.05403 0.110783i 0 −0.377942 0.848871i 0.559193 0.829038i −0.878646 0.592654i
1559.1 −0.130676 0.929805i −0.0348995 + 0.999391i 0.113800 0.0326316i −0.978148 0.207912i 0.933799 0.0981463i 0 −0.427114 0.959315i −0.997564 0.0697565i −0.0654974 + 0.936656i
1649.1 0.982665 0.397023i 0.882948 + 0.469472i 0.0886642 0.0856220i −0.978148 + 0.207912i 1.05403 + 0.110783i 0 −0.377942 + 0.848871i 0.559193 + 0.829038i −0.878646 + 0.592654i
1814.1 1.57506 + 0.0550024i 0.374607 0.927184i 1.48023 + 0.103508i 0.669131 0.743145i 0.641026 1.43977i 0 0.758371 + 0.0797080i −0.719340 0.694658i 1.09480 1.13369i
1889.1 1.57506 1.23057i −0.848048 0.529919i 0.724587 2.90616i −0.978148 0.207912i −1.98783 + 0.208930i 0 −1.62199 3.64306i 0.438371 + 0.898794i −1.79649 + 0.876208i
1904.1 −0.135369 0.0337512i −0.438371 0.898794i −0.865762 0.460334i 0.913545 + 0.406737i 0.0290064 + 0.136464i 0 0.205339 + 0.184888i −0.615661 + 0.788011i −0.109938 0.0858927i
2024.1 −0.130676 + 0.245765i 0.615661 0.788011i 0.515869 + 0.764807i 0.669131 + 0.743145i 0.113214 + 0.254282i 0 −0.532196 + 0.0559360i −0.241922 0.970296i −0.270078 + 0.0673380i
2159.1 0.457027 + 1.59384i 0.997564 + 0.0697565i −1.48342 + 0.926942i 0.913545 + 0.406737i 0.344733 + 1.62184i 0 −0.923173 0.831229i 0.990268 + 0.139173i −0.230759 + 1.64194i
2309.1 −1.24871 + 0.609036i −0.961262 + 0.275637i 0.572689 0.733008i −0.104528 0.994522i 1.03246 0.929634i 0 0.0201618 0.0948538i 0.848048 0.529919i 0.736226 + 1.17821i
2339.1 0.457027 1.59384i 0.997564 0.0697565i −1.48342 0.926942i 0.913545 0.406737i 0.344733 1.62184i 0 −0.923173 + 0.831229i 0.990268 0.139173i −0.230759 1.64194i
2459.1 −0.130676 + 0.929805i −0.0348995 0.999391i 0.113800 + 0.0326316i −0.978148 + 0.207912i 0.933799 + 0.0981463i 0 −0.427114 + 0.959315i −0.997564 + 0.0697565i −0.0654974 0.936656i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 359.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 CM by \(\Q(\sqrt{-15}) \)
181.q even 90 1 inner
2715.dx odd 90 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2715.1.dx.b yes 24
3.b odd 2 1 2715.1.dx.a 24
5.b even 2 1 2715.1.dx.a 24
15.d odd 2 1 CM 2715.1.dx.b yes 24
181.q even 90 1 inner 2715.1.dx.b yes 24
543.bg odd 90 1 2715.1.dx.a 24
905.bz even 90 1 2715.1.dx.a 24
2715.dx odd 90 1 inner 2715.1.dx.b yes 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2715.1.dx.a 24 3.b odd 2 1
2715.1.dx.a 24 5.b even 2 1
2715.1.dx.a 24 543.bg odd 90 1
2715.1.dx.a 24 905.bz even 90 1
2715.1.dx.b yes 24 1.a even 1 1 trivial
2715.1.dx.b yes 24 15.d odd 2 1 CM
2715.1.dx.b yes 24 181.q even 90 1 inner
2715.1.dx.b yes 24 2715.dx odd 90 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{24} - 6 T_{2}^{23} + 21 T_{2}^{22} - 52 T_{2}^{21} + 105 T_{2}^{20} - 186 T_{2}^{19} + 321 T_{2}^{18} + \cdots + 1 \) acting on \(S_{1}^{\mathrm{new}}(2715, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{24} - 6 T^{23} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{24} + T^{21} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( (T^{8} - T^{7} + T^{5} + \cdots + 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{24} \) Copy content Toggle raw display
$11$ \( T^{24} \) Copy content Toggle raw display
$13$ \( T^{24} \) Copy content Toggle raw display
$17$ \( T^{24} + 3 T^{23} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( T^{24} + 24 T^{22} + \cdots + 1 \) Copy content Toggle raw display
$23$ \( T^{24} + 3 T^{23} + \cdots + 1 \) Copy content Toggle raw display
$29$ \( T^{24} \) Copy content Toggle raw display
$31$ \( T^{24} - 6 T^{22} + \cdots + 1 \) Copy content Toggle raw display
$37$ \( T^{24} \) Copy content Toggle raw display
$41$ \( T^{24} \) Copy content Toggle raw display
$43$ \( T^{24} \) Copy content Toggle raw display
$47$ \( T^{24} + 6 T^{23} + \cdots + 1 \) Copy content Toggle raw display
$53$ \( T^{24} - 24 T^{23} + \cdots + 1 \) Copy content Toggle raw display
$59$ \( T^{24} \) Copy content Toggle raw display
$61$ \( T^{24} - 50 T^{18} + \cdots + 15625 \) Copy content Toggle raw display
$67$ \( T^{24} \) Copy content Toggle raw display
$71$ \( T^{24} \) Copy content Toggle raw display
$73$ \( T^{24} \) Copy content Toggle raw display
$79$ \( T^{24} - 6 T^{21} + \cdots + 1 \) Copy content Toggle raw display
$83$ \( T^{24} - 21 T^{21} + \cdots + 1 \) Copy content Toggle raw display
$89$ \( T^{24} \) Copy content Toggle raw display
$97$ \( T^{24} \) Copy content Toggle raw display
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