Properties

Label 1368.1.eh.a
Level $1368$
Weight $1$
Character orbit 1368.eh
Analytic conductor $0.683$
Analytic rank $0$
Dimension $6$
Projective image $D_{9}$
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] = [1368,1,Mod(595,1368)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1368, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([9, 9, 0, 14]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1368.595");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1368 = 2^{3} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1368.eh (of order \(18\), degree \(6\), 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: \(0.682720937282\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 152)
Projective image: \(D_{9}\)
Projective field: Galois closure of 9.1.69564674215936.1

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{18}^{5} q^{2} - \zeta_{18} q^{4} - \zeta_{18}^{6} q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{18}^{5} q^{2} - \zeta_{18} q^{4} - \zeta_{18}^{6} q^{8} + ( - \zeta_{18}^{8} - \zeta_{18}^{4}) q^{11} + \zeta_{18}^{2} q^{16} - \zeta_{18}^{5} q^{17} - \zeta_{18} q^{19} + (\zeta_{18}^{4} + 1) q^{22} - \zeta_{18}^{7} q^{25} + \zeta_{18}^{7} q^{32} + \zeta_{18} q^{34} - \zeta_{18}^{6} q^{38} + (\zeta_{18}^{3} + \zeta_{18}) q^{41} - \zeta_{18}^{8} q^{43} + (\zeta_{18}^{5} - 1) q^{44} + \zeta_{18}^{6} q^{49} + \zeta_{18}^{3} q^{50} + (\zeta_{18}^{7} + \zeta_{18}^{3}) q^{59} - \zeta_{18}^{3} q^{64} + ( - \zeta_{18}^{5} - \zeta_{18}^{3}) q^{67} + \zeta_{18}^{6} q^{68} + (\zeta_{18}^{4} + 1) q^{73} + \zeta_{18}^{2} q^{76} + (\zeta_{18}^{8} + \zeta_{18}^{6}) q^{82} + ( - \zeta_{18}^{4} - \zeta_{18}^{2}) q^{83} + \zeta_{18}^{4} q^{86} + ( - \zeta_{18}^{5} - \zeta_{18}) q^{88} - \zeta_{18}^{7} q^{89} + (\zeta_{18}^{6} + \zeta_{18}^{4}) q^{97} - \zeta_{18}^{2} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 3 q^{8} + 6 q^{22} + 3 q^{38} + 3 q^{41} - 6 q^{44} - 3 q^{49} + 3 q^{50} + 3 q^{59} - 3 q^{64} - 3 q^{67} - 3 q^{68} + 6 q^{73} - 3 q^{82} - 3 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1368\mathbb{Z}\right)^\times\).

\(n\) \(343\) \(685\) \(1009\) \(1217\)
\(\chi(n)\) \(-1\) \(-1\) \(-\zeta_{18}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
595.1
−0.173648 + 0.984808i
0.939693 + 0.342020i
0.939693 0.342020i
−0.766044 + 0.642788i
−0.173648 0.984808i
−0.766044 0.642788i
−0.766044 + 0.642788i 0 0.173648 0.984808i 0 0 0 0.500000 + 0.866025i 0 0
739.1 −0.173648 + 0.984808i 0 −0.939693 0.342020i 0 0 0 0.500000 0.866025i 0 0
883.1 −0.173648 0.984808i 0 −0.939693 + 0.342020i 0 0 0 0.500000 + 0.866025i 0 0
955.1 0.939693 0.342020i 0 0.766044 0.642788i 0 0 0 0.500000 0.866025i 0 0
1099.1 −0.766044 0.642788i 0 0.173648 + 0.984808i 0 0 0 0.500000 0.866025i 0 0
1315.1 0.939693 + 0.342020i 0 0.766044 + 0.642788i 0 0 0 0.500000 + 0.866025i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 595.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
19.e even 9 1 inner
152.u odd 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1368.1.eh.a 6
3.b odd 2 1 152.1.u.a 6
8.d odd 2 1 CM 1368.1.eh.a 6
12.b even 2 1 608.1.bg.a 6
15.d odd 2 1 3800.1.cv.c 6
15.e even 4 2 3800.1.cq.b 12
19.e even 9 1 inner 1368.1.eh.a 6
24.f even 2 1 152.1.u.a 6
24.h odd 2 1 608.1.bg.a 6
57.d even 2 1 2888.1.u.e 6
57.f even 6 1 2888.1.u.a 6
57.f even 6 1 2888.1.u.f 6
57.h odd 6 1 2888.1.u.b 6
57.h odd 6 1 2888.1.u.g 6
57.j even 18 1 2888.1.f.c 3
57.j even 18 2 2888.1.k.c 6
57.j even 18 1 2888.1.u.a 6
57.j even 18 1 2888.1.u.e 6
57.j even 18 1 2888.1.u.f 6
57.l odd 18 1 152.1.u.a 6
57.l odd 18 1 2888.1.f.d 3
57.l odd 18 2 2888.1.k.b 6
57.l odd 18 1 2888.1.u.b 6
57.l odd 18 1 2888.1.u.g 6
120.m even 2 1 3800.1.cv.c 6
120.q odd 4 2 3800.1.cq.b 12
152.u odd 18 1 inner 1368.1.eh.a 6
228.v even 18 1 608.1.bg.a 6
285.bd odd 18 1 3800.1.cv.c 6
285.bi even 36 2 3800.1.cq.b 12
456.l odd 2 1 2888.1.u.e 6
456.s odd 6 1 2888.1.u.a 6
456.s odd 6 1 2888.1.u.f 6
456.u even 6 1 2888.1.u.b 6
456.u even 6 1 2888.1.u.g 6
456.bh odd 18 1 608.1.bg.a 6
456.bt odd 18 1 2888.1.f.c 3
456.bt odd 18 2 2888.1.k.c 6
456.bt odd 18 1 2888.1.u.a 6
456.bt odd 18 1 2888.1.u.e 6
456.bt odd 18 1 2888.1.u.f 6
456.bu even 18 1 152.1.u.a 6
456.bu even 18 1 2888.1.f.d 3
456.bu even 18 2 2888.1.k.b 6
456.bu even 18 1 2888.1.u.b 6
456.bu even 18 1 2888.1.u.g 6
2280.eg even 18 1 3800.1.cv.c 6
2280.fl odd 36 2 3800.1.cq.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.1.u.a 6 3.b odd 2 1
152.1.u.a 6 24.f even 2 1
152.1.u.a 6 57.l odd 18 1
152.1.u.a 6 456.bu even 18 1
608.1.bg.a 6 12.b even 2 1
608.1.bg.a 6 24.h odd 2 1
608.1.bg.a 6 228.v even 18 1
608.1.bg.a 6 456.bh odd 18 1
1368.1.eh.a 6 1.a even 1 1 trivial
1368.1.eh.a 6 8.d odd 2 1 CM
1368.1.eh.a 6 19.e even 9 1 inner
1368.1.eh.a 6 152.u odd 18 1 inner
2888.1.f.c 3 57.j even 18 1
2888.1.f.c 3 456.bt odd 18 1
2888.1.f.d 3 57.l odd 18 1
2888.1.f.d 3 456.bu even 18 1
2888.1.k.b 6 57.l odd 18 2
2888.1.k.b 6 456.bu even 18 2
2888.1.k.c 6 57.j even 18 2
2888.1.k.c 6 456.bt odd 18 2
2888.1.u.a 6 57.f even 6 1
2888.1.u.a 6 57.j even 18 1
2888.1.u.a 6 456.s odd 6 1
2888.1.u.a 6 456.bt odd 18 1
2888.1.u.b 6 57.h odd 6 1
2888.1.u.b 6 57.l odd 18 1
2888.1.u.b 6 456.u even 6 1
2888.1.u.b 6 456.bu even 18 1
2888.1.u.e 6 57.d even 2 1
2888.1.u.e 6 57.j even 18 1
2888.1.u.e 6 456.l odd 2 1
2888.1.u.e 6 456.bt odd 18 1
2888.1.u.f 6 57.f even 6 1
2888.1.u.f 6 57.j even 18 1
2888.1.u.f 6 456.s odd 6 1
2888.1.u.f 6 456.bt odd 18 1
2888.1.u.g 6 57.h odd 6 1
2888.1.u.g 6 57.l odd 18 1
2888.1.u.g 6 456.u even 6 1
2888.1.u.g 6 456.bu even 18 1
3800.1.cq.b 12 15.e even 4 2
3800.1.cq.b 12 120.q odd 4 2
3800.1.cq.b 12 285.bi even 36 2
3800.1.cq.b 12 2280.fl odd 36 2
3800.1.cv.c 6 15.d odd 2 1
3800.1.cv.c 6 120.m even 2 1
3800.1.cv.c 6 285.bd odd 18 1
3800.1.cv.c 6 2280.eg even 18 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1368, [\chi])\).

Hecke characteristic polynomials

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