Properties

Label 2888.1.bb.a
Level $2888$
Weight $1$
Character orbit 2888.bb
Analytic conductor $1.441$
Analytic rank $0$
Dimension $18$
Projective image $D_{19}$
CM discriminant -8
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2888,1,Mod(115,2888)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2888, base_ring=CyclotomicField(38))
 
chi = DirichletCharacter(H, H._module([19, 19, 4]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2888.115");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2888 = 2^{3} \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2888.bb (of order \(38\), degree \(18\), 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: \(1.44129975648\)
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: \(D_{19}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{19} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{38}^{3} q^{2} + ( - \zeta_{38}^{17} + 1) q^{3} + \zeta_{38}^{6} q^{4} + ( - \zeta_{38}^{3} - \zeta_{38}) q^{6} - \zeta_{38}^{9} q^{8} + ( - \zeta_{38}^{17} - \zeta_{38}^{15} + 1) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{38}^{3} q^{2} + ( - \zeta_{38}^{17} + 1) q^{3} + \zeta_{38}^{6} q^{4} + ( - \zeta_{38}^{3} - \zeta_{38}) q^{6} - \zeta_{38}^{9} q^{8} + ( - \zeta_{38}^{17} - \zeta_{38}^{15} + 1) q^{9} + ( - \zeta_{38}^{15} + \zeta_{38}^{8}) q^{11} + (\zeta_{38}^{6} + \zeta_{38}^{4}) q^{12} + \zeta_{38}^{12} q^{16} + (\zeta_{38}^{6} - \zeta_{38}) q^{17} + (\zeta_{38}^{18} - \zeta_{38}^{3} - \zeta_{38}) q^{18} + \zeta_{38}^{16} q^{19} + (\zeta_{38}^{18} - \zeta_{38}^{11}) q^{22} + ( - \zeta_{38}^{9} - \zeta_{38}^{7}) q^{24} - \zeta_{38}^{5} q^{25} + ( - \zeta_{38}^{17} + \zeta_{38}^{15} + \cdots + 1) q^{27} + \cdots + ( - \zeta_{38}^{15} + \cdots + \zeta_{38}^{4}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - q^{2} + 17 q^{3} - q^{4} - 2 q^{6} - q^{8} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - q^{2} + 17 q^{3} - q^{4} - 2 q^{6} - q^{8} + 16 q^{9} - 2 q^{11} - 2 q^{12} - q^{16} - 2 q^{17} - 3 q^{18} - q^{19} - 2 q^{22} - 2 q^{24} - q^{25} + 15 q^{27} - q^{32} - 4 q^{33} - 2 q^{34} - 3 q^{36} + 18 q^{38} - 2 q^{41} - 2 q^{43} - 2 q^{44} - 2 q^{48} - q^{49} - q^{50} - 4 q^{51} - 4 q^{54} - 2 q^{57} - 2 q^{59} - q^{64} - 4 q^{66} - 2 q^{67} - 2 q^{68} - 3 q^{72} - 2 q^{73} - 2 q^{75} - q^{76} + 14 q^{81} - 2 q^{82} + 17 q^{83} - 2 q^{86} - 2 q^{88} - 2 q^{89} - 2 q^{96} - 2 q^{97} - q^{98} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2888\mathbb{Z}\right)^\times\).

\(n\) \(1445\) \(2167\) \(2529\)
\(\chi(n)\) \(-1\) \(-1\) \(\zeta_{38}^{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
115.1
0.401695 + 0.915773i
0.0825793 0.996584i
−0.546948 + 0.837166i
0.879474 0.475947i
0.879474 + 0.475947i
−0.546948 0.837166i
0.0825793 + 0.996584i
0.401695 0.915773i
−0.789141 + 0.614213i
0.986361 0.164595i
−0.945817 0.324699i
0.677282 + 0.735724i
−0.245485 0.969400i
−0.245485 + 0.969400i
0.677282 0.735724i
−0.945817 + 0.324699i
0.986361 + 0.164595i
−0.789141 0.614213i
0.945817 + 0.324699i 0.322718 0.735724i 0.789141 + 0.614213i 0 0.544122 0.591074i 0 0.546948 + 0.837166i 0.240139 + 0.260861i 0
267.1 0.245485 0.969400i 0.0136387 + 0.164595i −0.879474 0.475947i 0 0.162906 + 0.0271842i 0 −0.677282 + 0.735724i 0.959456 0.160105i 0
419.1 −0.986361 0.164595i 0.598305 + 0.915773i 0.945817 + 0.324699i 0 −0.439413 1.00176i 0 −0.879474 0.475947i −0.0789770 + 0.180049i 0
571.1 −0.0825793 + 0.996584i 1.54695 + 0.837166i −0.986361 0.164595i 0 −0.962053 + 1.47253i 0 0.245485 0.969400i 1.14525 + 1.75294i 0
875.1 −0.0825793 0.996584i 1.54695 0.837166i −0.986361 + 0.164595i 0 −0.962053 1.47253i 0 0.245485 + 0.969400i 1.14525 1.75294i 0
1027.1 −0.986361 + 0.164595i 0.598305 0.915773i 0.945817 0.324699i 0 −0.439413 + 1.00176i 0 −0.879474 + 0.475947i −0.0789770 0.180049i 0
1179.1 0.245485 + 0.969400i 0.0136387 0.164595i −0.879474 + 0.475947i 0 0.162906 0.0271842i 0 −0.677282 0.735724i 0.959456 + 0.160105i 0
1331.1 0.945817 0.324699i 0.322718 + 0.735724i 0.789141 0.614213i 0 0.544122 + 0.591074i 0 0.546948 0.837166i 0.240139 0.260861i 0
1483.1 −0.401695 0.915773i 1.24549 + 0.969400i −0.677282 + 0.735724i 0 0.387445 1.52999i 0 0.945817 + 0.324699i 0.366012 + 1.44535i 0
1635.1 −0.879474 + 0.475947i 1.94582 + 0.324699i 0.546948 0.837166i 0 −1.86584 + 0.640542i 0 −0.0825793 + 0.996584i 2.73496 + 0.938912i 0
1787.1 0.546948 + 0.837166i 1.78914 0.614213i −0.401695 + 0.915773i 0 1.49277 + 1.16187i 0 −0.986361 + 0.164595i 2.03463 1.58361i 0
1939.1 0.789141 0.614213i 0.917421 0.996584i 0.245485 0.969400i 0 0.111859 1.34994i 0 −0.401695 0.915773i −0.0689406 0.831990i 0
2091.1 −0.677282 0.735724i 0.120526 0.475947i −0.0825793 + 0.996584i 0 −0.431796 + 0.233676i 0 0.789141 0.614213i 0.667474 + 0.361219i 0
2243.1 −0.677282 + 0.735724i 0.120526 + 0.475947i −0.0825793 0.996584i 0 −0.431796 0.233676i 0 0.789141 + 0.614213i 0.667474 0.361219i 0
2395.1 0.789141 + 0.614213i 0.917421 + 0.996584i 0.245485 + 0.969400i 0 0.111859 + 1.34994i 0 −0.401695 + 0.915773i −0.0689406 + 0.831990i 0
2547.1 0.546948 0.837166i 1.78914 + 0.614213i −0.401695 0.915773i 0 1.49277 1.16187i 0 −0.986361 0.164595i 2.03463 + 1.58361i 0
2699.1 −0.879474 0.475947i 1.94582 0.324699i 0.546948 + 0.837166i 0 −1.86584 0.640542i 0 −0.0825793 0.996584i 2.73496 0.938912i 0
2851.1 −0.401695 + 0.915773i 1.24549 0.969400i −0.677282 0.735724i 0 0.387445 + 1.52999i 0 0.945817 0.324699i 0.366012 1.44535i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 115.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
361.g even 19 1 inner
2888.bb odd 38 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2888.1.bb.a 18
8.d odd 2 1 CM 2888.1.bb.a 18
361.g even 19 1 inner 2888.1.bb.a 18
2888.bb odd 38 1 inner 2888.1.bb.a 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2888.1.bb.a 18 1.a even 1 1 trivial
2888.1.bb.a 18 8.d odd 2 1 CM
2888.1.bb.a 18 361.g even 19 1 inner
2888.1.bb.a 18 2888.bb odd 38 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(2888, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} + T^{17} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{18} - 17 T^{17} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{18} \) Copy content Toggle raw display
$7$ \( T^{18} \) Copy content Toggle raw display
$11$ \( T^{18} + 2 T^{17} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{18} \) Copy content Toggle raw display
$17$ \( T^{18} + 2 T^{17} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( T^{18} + T^{17} + \cdots + 1 \) Copy content Toggle raw display
$23$ \( T^{18} \) Copy content Toggle raw display
$29$ \( T^{18} \) Copy content Toggle raw display
$31$ \( T^{18} \) Copy content Toggle raw display
$37$ \( T^{18} \) Copy content Toggle raw display
$41$ \( T^{18} + 2 T^{17} + \cdots + 1 \) Copy content Toggle raw display
$43$ \( T^{18} + 2 T^{17} + \cdots + 1 \) Copy content Toggle raw display
$47$ \( T^{18} \) Copy content Toggle raw display
$53$ \( T^{18} \) Copy content Toggle raw display
$59$ \( T^{18} + 2 T^{17} + \cdots + 1 \) Copy content Toggle raw display
$61$ \( T^{18} \) Copy content Toggle raw display
$67$ \( T^{18} + 2 T^{17} + \cdots + 1 \) Copy content Toggle raw display
$71$ \( T^{18} \) Copy content Toggle raw display
$73$ \( T^{18} + 2 T^{17} + \cdots + 1 \) Copy content Toggle raw display
$79$ \( T^{18} \) Copy content Toggle raw display
$83$ \( T^{18} - 17 T^{17} + \cdots + 1 \) Copy content Toggle raw display
$89$ \( T^{18} + 2 T^{17} + \cdots + 1 \) Copy content Toggle raw display
$97$ \( T^{18} + 2 T^{17} + \cdots + 262144 \) Copy content Toggle raw display
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