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Properties

Label 1352.1.h.a
Level $1352$
Weight $1$
Character orbit 1352.h
Analytic conductor $0.675$
Analytic rank $0$
Dimension $6$
Projective image $D_{7}$
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] = [1352,1,Mod(675,1352)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1352, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 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("1352.675");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1352 = 2^{3} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1352.h (of order \(2\), degree \(1\), 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: \(0.674735897080\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.153664.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 5x^{4} + 6x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{7}\)
Projective field: Galois closure of 7.1.2471326208.1

$q$-expansion

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

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} - \beta_{4} q^{3} - q^{4} - \beta_1 q^{6} - \beta_{5} q^{8} + ( - \beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{2} - \beta_{4} q^{3} - q^{4} - \beta_1 q^{6} - \beta_{5} q^{8} + ( - \beta_{2} + 1) q^{9} - \beta_{3} q^{11} + \beta_{4} q^{12} + q^{16} + ( - \beta_{4} - \beta_{2} + 1) q^{17} + ( - \beta_{3} + \beta_1) q^{18} + ( - \beta_{5} - \beta_{3} + \beta_1) q^{19} + (\beta_{4} + \beta_{2} - 1) q^{22} + \beta_1 q^{24} - q^{25} + (\beta_{2} - 1) q^{27} + \beta_{5} q^{32} + (\beta_{5} - \beta_1) q^{33} - \beta_{3} q^{34} + (\beta_{2} - 1) q^{36} + \beta_{2} q^{38} + ( - \beta_{5} - \beta_{3} + \beta_1) q^{41} + \beta_{2} q^{43} + \beta_{3} q^{44} - \beta_{4} q^{48} - q^{49} - \beta_{5} q^{50} + ( - \beta_{4} + 1) q^{51} + (\beta_{3} - \beta_1) q^{54} + (\beta_{3} - \beta_1) q^{57} + \beta_1 q^{59} - q^{64} + (\beta_{4} - 1) q^{66} - \beta_1 q^{67} + (\beta_{4} + \beta_{2} - 1) q^{68} + (\beta_{3} - \beta_1) q^{72} + \beta_1 q^{73} + \beta_{4} q^{75} + (\beta_{5} + \beta_{3} - \beta_1) q^{76} + \beta_{4} q^{81} + \beta_{2} q^{82} + ( - \beta_{5} - \beta_{3} + \beta_1) q^{83} + (\beta_{5} + \beta_{3} - \beta_1) q^{86} + ( - \beta_{4} - \beta_{2} + 1) q^{88} + \beta_1 q^{89} - \beta_1 q^{96} + \beta_{3} q^{97} - \beta_{5} q^{98} + \beta_{5} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{3} - 6 q^{4} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 2 q^{3} - 6 q^{4} + 4 q^{9} + 2 q^{12} + 6 q^{16} + 2 q^{17} - 2 q^{22} - 6 q^{25} - 4 q^{27} - 4 q^{36} + 2 q^{38} + 2 q^{43} - 2 q^{48} - 6 q^{49} + 4 q^{51} - 6 q^{64} - 4 q^{66} - 2 q^{68} + 2 q^{75} + 2 q^{81} + 2 q^{82} + 2 q^{88}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 5x^{4} + 6x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 3\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} + 3\nu^{2} + 1 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{5} + 4\nu^{3} + 3\nu \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 3\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} - 3\beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{5} - 4\beta_{3} + 9\beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1352\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(1015\) \(1185\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
675.1
− 1.80194i
− 0.445042i
1.24698i
1.80194i
0.445042i
− 1.24698i
− 1.00000i −1.80194 −1.00000 0 1.80194i 0 1.00000i 2.24698 0
675.2 − 1.00000i −0.445042 −1.00000 0 0.445042i 0 1.00000i −0.801938 0
675.3 − 1.00000i 1.24698 −1.00000 0 − 1.24698i 0 1.00000i 0.554958 0
675.4 1.00000i −1.80194 −1.00000 0 − 1.80194i 0 − 1.00000i 2.24698 0
675.5 1.00000i −0.445042 −1.00000 0 − 0.445042i 0 − 1.00000i −0.801938 0
675.6 1.00000i 1.24698 −1.00000 0 1.24698i 0 − 1.00000i 0.554958 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 675.6
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1352.1.h.a 6
8.d odd 2 1 CM 1352.1.h.a 6
13.b even 2 1 inner 1352.1.h.a 6
13.c even 3 2 1352.1.p.c 12
13.d odd 4 1 1352.1.g.b ✓ 3
13.d odd 4 1 1352.1.g.c yes 3
13.e even 6 2 1352.1.p.c 12
13.f odd 12 2 1352.1.n.b 6
13.f odd 12 2 1352.1.n.c 6
104.h odd 2 1 inner 1352.1.h.a 6
104.m even 4 1 1352.1.g.b ✓ 3
104.m even 4 1 1352.1.g.c yes 3
104.n odd 6 2 1352.1.p.c 12
104.p odd 6 2 1352.1.p.c 12
104.u even 12 2 1352.1.n.b 6
104.u even 12 2 1352.1.n.c 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1352.1.g.b ✓ 3 13.d odd 4 1
1352.1.g.b ✓ 3 104.m even 4 1
1352.1.g.c yes 3 13.d odd 4 1
1352.1.g.c yes 3 104.m even 4 1
1352.1.h.a 6 1.a even 1 1 trivial
1352.1.h.a 6 8.d odd 2 1 CM
1352.1.h.a 6 13.b even 2 1 inner
1352.1.h.a 6 104.h odd 2 1 inner
1352.1.n.b 6 13.f odd 12 2
1352.1.n.b 6 104.u even 12 2
1352.1.n.c 6 13.f odd 12 2
1352.1.n.c 6 104.u even 12 2
1352.1.p.c 12 13.c even 3 2
1352.1.p.c 12 13.e even 6 2
1352.1.p.c 12 104.n odd 6 2
1352.1.p.c 12 104.p odd 6 2

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1352, [\chi])\).

Hecke characteristic polynomials

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