Properties

Label 735.1.f.d
Level $735$
Weight $1$
Character orbit 735.f
Self dual yes
Analytic conductor $0.367$
Analytic rank $0$
Dimension $2$
Projective image $D_{4}$
CM discriminant -15
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [735,1,Mod(344,735)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(735, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("735.344");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 735.f (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: yes
Analytic conductor: \(0.366812784285\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.2.15435.1
Artin image: $D_8$
Artin field: Galois closure of 8.2.8338372875.2

$q$-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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{2} + q^{3} + q^{4} - q^{5} - \beta q^{6} + q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{2} + q^{3} + q^{4} - q^{5} - \beta q^{6} + q^{9} + \beta q^{10} + q^{12} - q^{15} - q^{16} - \beta q^{18} + \beta q^{19} - q^{20} + \beta q^{23} + q^{25} + q^{27} + \beta q^{30} - \beta q^{31} + \beta q^{32} + q^{36} - 2 q^{38} - q^{45} - 2 q^{46} - q^{48} - \beta q^{50} + \beta q^{53} - \beta q^{54} + \beta q^{57} - q^{60} - \beta q^{61} + 2 q^{62} - q^{64} + \beta q^{69} + q^{75} + \beta q^{76} + q^{80} + q^{81} + \beta q^{90} + \beta q^{92} - \beta q^{93} - \beta q^{95} + \beta q^{96} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{9} + 2 q^{12} - 2 q^{15} - 2 q^{16} - 2 q^{20} + 2 q^{25} + 2 q^{27} + 2 q^{36} - 4 q^{38} - 2 q^{45} - 4 q^{46} - 2 q^{48} - 2 q^{60} + 4 q^{62} - 2 q^{64} + 2 q^{75} + 2 q^{80} + 2 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/735\mathbb{Z}\right)^\times\).

\(n\) \(346\) \(442\) \(491\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
344.1
1.41421
−1.41421
−1.41421 1.00000 1.00000 −1.00000 −1.41421 0 0 1.00000 1.41421
344.2 1.41421 1.00000 1.00000 −1.00000 1.41421 0 0 1.00000 −1.41421
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 CM by \(\Q(\sqrt{-15}) \)
21.c even 2 1 inner
35.c odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 735.1.f.d yes 2
3.b odd 2 1 735.1.f.c 2
5.b even 2 1 735.1.f.c 2
5.c odd 4 2 3675.1.c.f 4
7.b odd 2 1 735.1.f.c 2
7.c even 3 2 735.1.o.c 4
7.d odd 6 2 735.1.o.d 4
15.d odd 2 1 CM 735.1.f.d yes 2
15.e even 4 2 3675.1.c.f 4
21.c even 2 1 inner 735.1.f.d yes 2
21.g even 6 2 735.1.o.c 4
21.h odd 6 2 735.1.o.d 4
35.c odd 2 1 inner 735.1.f.d yes 2
35.f even 4 2 3675.1.c.f 4
35.i odd 6 2 735.1.o.c 4
35.j even 6 2 735.1.o.d 4
35.k even 12 4 3675.1.u.f 8
35.l odd 12 4 3675.1.u.f 8
105.g even 2 1 735.1.f.c 2
105.k odd 4 2 3675.1.c.f 4
105.o odd 6 2 735.1.o.c 4
105.p even 6 2 735.1.o.d 4
105.w odd 12 4 3675.1.u.f 8
105.x even 12 4 3675.1.u.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
735.1.f.c 2 3.b odd 2 1
735.1.f.c 2 5.b even 2 1
735.1.f.c 2 7.b odd 2 1
735.1.f.c 2 105.g even 2 1
735.1.f.d yes 2 1.a even 1 1 trivial
735.1.f.d yes 2 15.d odd 2 1 CM
735.1.f.d yes 2 21.c even 2 1 inner
735.1.f.d yes 2 35.c odd 2 1 inner
735.1.o.c 4 7.c even 3 2
735.1.o.c 4 21.g even 6 2
735.1.o.c 4 35.i odd 6 2
735.1.o.c 4 105.o odd 6 2
735.1.o.d 4 7.d odd 6 2
735.1.o.d 4 21.h odd 6 2
735.1.o.d 4 35.j even 6 2
735.1.o.d 4 105.p even 6 2
3675.1.c.f 4 5.c odd 4 2
3675.1.c.f 4 15.e even 4 2
3675.1.c.f 4 35.f even 4 2
3675.1.c.f 4 105.k odd 4 2
3675.1.u.f 8 35.k even 12 4
3675.1.u.f 8 35.l odd 12 4
3675.1.u.f 8 105.w odd 12 4
3675.1.u.f 8 105.x even 12 4

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(735, [\chi])\):

\( T_{2}^{2} - 2 \) Copy content Toggle raw display
\( T_{17} \) Copy content Toggle raw display
\( T_{167} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

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