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Properties

Label 2872.1.b.e
Level $2872$
Weight $1$
Character orbit 2872.b
Analytic conductor $1.433$
Analytic rank $0$
Dimension $18$
Projective image $D_{38}$
CM discriminant -359
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2872,1,Mod(717,2872)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2872, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("2872.717");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2872 = 2^{3} \cdot 359 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2872.b (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: no
Analytic conductor: \(1.43331471628\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\Q(\zeta_{38})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - x^{17} + x^{16} - x^{15} + x^{14} - x^{13} + x^{12} - x^{11} + x^{10} - x^{9} + x^{8} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{38}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{38} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{38}^{13} q^{2} + ( - \zeta_{38}^{11} - \zeta_{38}^{8}) q^{3} - \zeta_{38}^{7} q^{4} + ( - \zeta_{38}^{10} - \zeta_{38}^{9}) q^{5} + (\zeta_{38}^{5} + \zeta_{38}^{2}) q^{6} + \zeta_{38} q^{8} + (\zeta_{38}^{16} - \zeta_{38}^{3} - 1) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{38}^{13} q^{2} + ( - \zeta_{38}^{11} - \zeta_{38}^{8}) q^{3} - \zeta_{38}^{7} q^{4} + ( - \zeta_{38}^{10} - \zeta_{38}^{9}) q^{5} + (\zeta_{38}^{5} + \zeta_{38}^{2}) q^{6} + \zeta_{38} q^{8} + (\zeta_{38}^{16} - \zeta_{38}^{3} - 1) q^{9} + (\zeta_{38}^{4} + \zeta_{38}^{3}) q^{10} + ( - \zeta_{38}^{15} - \zeta_{38}^{4}) q^{11} + (\zeta_{38}^{18} + \zeta_{38}^{15}) q^{12} + (\zeta_{38}^{18} + \zeta_{38}^{17} + \cdots - \zeta_{38}) q^{15} + \cdots + (\zeta_{38}^{18} + \zeta_{38}^{15} + \cdots + \zeta_{38}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + q^{2} - q^{4} + q^{8} - 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + q^{2} - q^{4} + q^{8} - 20 q^{9} - q^{16} + 2 q^{17} + q^{18} - 2 q^{23} - 20 q^{25} + q^{32} - 2 q^{34} - q^{36} - 2 q^{41} + 2 q^{46} + 2 q^{47} + 18 q^{49} + q^{50} - q^{64} + 2 q^{68} + q^{72} - 2 q^{73} + 2 q^{79} + 18 q^{81} + 2 q^{82} - 19 q^{90} + 17 q^{92} - 2 q^{94} - 19 q^{96} + q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2872\mathbb{Z}\right)^\times\).

\(n\) \(719\) \(1437\) \(2161\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
717.1
−0.546948 − 0.837166i
−0.546948 + 0.837166i
0.401695 − 0.915773i
0.401695 + 0.915773i
0.986361 − 0.164595i
0.986361 + 0.164595i
0.677282 + 0.735724i
0.677282 − 0.735724i
−0.245485 + 0.969400i
−0.245485 − 0.969400i
−0.945817 + 0.324699i
−0.945817 − 0.324699i
−0.789141 − 0.614213i
−0.789141 + 0.614213i
0.0825793 − 0.996584i
0.0825793 + 0.996584i
0.879474 − 0.475947i
0.879474 + 0.475947i
−0.945817 − 0.324699i − 1.99317i 0.789141 + 0.614213i 0.951895i −0.647181 + 1.88517i 0 −0.546948 − 0.837166i −2.97272 0.309080 − 0.900319i
717.2 −0.945817 + 0.324699i 1.99317i 0.789141 − 0.614213i − 0.951895i −0.647181 − 1.88517i 0 −0.546948 + 0.837166i −2.97272 0.309080 + 0.900319i
717.3 −0.789141 − 0.614213i 0.329189i 0.245485 + 0.969400i − 1.67433i 0.202192 − 0.259777i 0 0.401695 − 0.915773i 0.891634 −1.02840 + 1.32128i
717.4 −0.789141 + 0.614213i − 0.329189i 0.245485 − 0.969400i 1.67433i 0.202192 + 0.259777i 0 0.401695 + 0.915773i 0.891634 −1.02840 − 1.32128i
717.5 −0.546948 − 0.837166i 1.93880i −0.401695 + 0.915773i 1.99317i 1.62310 − 1.06042i 0 0.986361 − 0.164595i −2.75895 1.66861 − 1.09016i
717.6 −0.546948 + 0.837166i − 1.93880i −0.401695 − 0.915773i − 1.99317i 1.62310 + 1.06042i 0 0.986361 + 0.164595i −2.75895 1.66861 + 1.09016i
717.7 −0.245485 − 0.969400i − 0.649399i −0.879474 + 0.475947i − 1.83155i −0.629528 + 0.159418i 0 0.677282 + 0.735724i 0.578281 −1.77550 + 0.449618i
717.8 −0.245485 + 0.969400i 0.649399i −0.879474 − 0.475947i 1.83155i −0.629528 − 0.159418i 0 0.677282 − 0.735724i 0.578281 −1.77550 − 0.449618i
717.9 0.0825793 − 0.996584i − 1.83155i −0.986361 − 0.164595i 1.22843i −1.82529 − 0.151248i 0 −0.245485 + 0.969400i −2.35456 1.22423 + 0.101443i
717.10 0.0825793 + 0.996584i 1.83155i −0.986361 + 0.164595i − 1.22843i −1.82529 + 0.151248i 0 −0.245485 − 0.969400i −2.35456 1.22423 − 0.101443i
717.11 0.401695 − 0.915773i 0.951895i −0.677282 − 0.735724i − 0.329189i 0.871720 + 0.382372i 0 −0.945817 + 0.324699i 0.0938963 −0.301463 − 0.132234i
717.12 0.401695 + 0.915773i − 0.951895i −0.677282 + 0.735724i 0.329189i 0.871720 − 0.382372i 0 −0.945817 − 0.324699i 0.0938963 −0.301463 + 0.132234i
717.13 0.677282 − 0.735724i 1.67433i −0.0825793 − 0.996584i − 0.649399i 1.23185 + 1.13399i 0 −0.789141 − 0.614213i −1.80339 −0.477778 − 0.439826i
717.14 0.677282 + 0.735724i − 1.67433i −0.0825793 + 0.996584i 0.649399i 1.23185 − 1.13399i 0 −0.789141 + 0.614213i −1.80339 −0.477778 + 0.439826i
717.15 0.879474 − 0.475947i − 1.22843i 0.546948 − 0.837166i 1.47145i −0.584666 − 1.08037i 0 0.0825793 − 0.996584i −0.509029 0.700332 + 1.29410i
717.16 0.879474 + 0.475947i 1.22843i 0.546948 + 0.837166i − 1.47145i −0.584666 + 1.08037i 0 0.0825793 + 0.996584i −0.509029 0.700332 − 1.29410i
717.17 0.986361 − 0.164595i − 1.47145i 0.945817 − 0.324699i − 1.93880i −0.242192 − 1.45138i 0 0.879474 − 0.475947i −1.16516 −0.319116 − 1.91236i
717.18 0.986361 + 0.164595i 1.47145i 0.945817 + 0.324699i 1.93880i −0.242192 + 1.45138i 0 0.879474 + 0.475947i −1.16516 −0.319116 + 1.91236i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 717.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
359.b odd 2 1 CM by \(\Q(\sqrt{-359}) \)
8.b even 2 1 inner
2872.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2872.1.b.e ✓ 18
8.b even 2 1 inner 2872.1.b.e ✓ 18
359.b odd 2 1 CM 2872.1.b.e ✓ 18
2872.b odd 2 1 inner 2872.1.b.e ✓ 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2872.1.b.e ✓ 18 1.a even 1 1 trivial
2872.1.b.e ✓ 18 8.b even 2 1 inner
2872.1.b.e ✓ 18 359.b odd 2 1 CM
2872.1.b.e ✓ 18 2872.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 \(S_{1}^{\mathrm{new}}(2872, [\chi])\):

\( T_{3}^{18} + 19 T_{3}^{16} + 152 T_{3}^{14} + 665 T_{3}^{12} + 1729 T_{3}^{10} + 2717 T_{3}^{8} + \cdots + 19 \) Copy content Toggle raw display
\( T_{13} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} - T^{17} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{18} + 19 T^{16} + \cdots + 19 \) Copy content Toggle raw display
$5$ \( T^{18} + 19 T^{16} + \cdots + 19 \) Copy content Toggle raw display
$7$ \( T^{18} \) Copy content Toggle raw display
$11$ \( T^{18} + 19 T^{16} + \cdots + 19 \) Copy content Toggle raw display
$13$ \( T^{18} \) Copy content Toggle raw display
$17$ \( (T^{9} - T^{8} - 8 T^{7} + \cdots - 1)^{2} \) Copy content Toggle raw display
$19$ \( T^{18} \) Copy content Toggle raw display
$23$ \( (T^{9} + T^{8} - 8 T^{7} + \cdots + 1)^{2} \) Copy content Toggle raw display
$29$ \( T^{18} \) Copy content Toggle raw display
$31$ \( T^{18} \) Copy content Toggle raw display
$37$ \( T^{18} + 19 T^{16} + \cdots + 19 \) Copy content Toggle raw display
$41$ \( (T^{9} + T^{8} - 8 T^{7} + \cdots + 1)^{2} \) Copy content Toggle raw display
$43$ \( T^{18} \) Copy content Toggle raw display
$47$ \( (T^{9} - T^{8} - 8 T^{7} + \cdots - 1)^{2} \) Copy content Toggle raw display
$53$ \( T^{18} \) Copy content Toggle raw display
$59$ \( T^{18} \) Copy content Toggle raw display
$61$ \( T^{18} \) Copy content Toggle raw display
$67$ \( T^{18} \) Copy content Toggle raw display
$71$ \( T^{18} \) Copy content Toggle raw display
$73$ \( (T^{9} + T^{8} - 8 T^{7} + \cdots + 1)^{2} \) Copy content Toggle raw display
$79$ \( (T^{9} - T^{8} - 8 T^{7} + \cdots - 1)^{2} \) Copy content Toggle raw display
$83$ \( T^{18} \) Copy content Toggle raw display
$89$ \( T^{18} \) Copy content Toggle raw display
$97$ \( T^{18} \) Copy content Toggle raw display
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