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Properties

Label 735.1.o.b
Level $735$
Weight $1$
Character orbit 735.o
Analytic conductor $0.367$
Analytic rank $0$
Dimension $2$
Projective image $D_{2}$
CM/RM discs -15, -35, 21
Inner twists $8$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [735,1,Mod(569,735)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(735, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 3, 2])) N = Newforms(chi, 1, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("735.569"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Level: \( N \) \(=\) \( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 735.o (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.366812784285\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{2}\)
Projective field: Galois closure of \(\Q(\sqrt{-15}, \sqrt{21})\)
Artin image: $C_3\times D_4$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\)

$q$-expansion

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

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + \zeta_{6} q^{3} + \zeta_{6} q^{4} - \zeta_{6}^{2} q^{5} + \zeta_{6}^{2} q^{9} + \zeta_{6}^{2} q^{12} + q^{15} + \zeta_{6}^{2} q^{16} - 2 \zeta_{6} q^{17} + q^{20} - \zeta_{6} q^{25} - q^{27} - q^{36} + \cdots - 2 q^{85} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} + q^{4} + q^{5} - q^{9} - q^{12} + 2 q^{15} - q^{16} - 2 q^{17} + 2 q^{20} - q^{25} - 2 q^{27} - 2 q^{36} + q^{45} - 2 q^{47} - 2 q^{48} + 2 q^{51} + q^{60} - 2 q^{64} + 2 q^{68} + q^{75}+ \cdots - 4 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(346\) \(442\) \(491\)
\(\chi(n)\) \(\zeta_{6}^{2}\) \(-1\) \(-1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
569.1
0.500000 + 0.866025i
0.500000 − 0.866025i
0 0.500000 + 0.866025i 0.500000 + 0.866025i 0.500000 − 0.866025i 0 0 0 −0.500000 + 0.866025i 0
704.1 0 0.500000 − 0.866025i 0.500000 − 0.866025i 0.500000 + 0.866025i 0 0 0 −0.500000 − 0.866025i 0
\(n\): 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 RM by \(\Q(\sqrt{21}) \)
35.c odd 2 1 CM by \(\Q(\sqrt{-35}) \)
7.c even 3 1 inner
21.g even 6 1 inner
35.i odd 6 1 inner
105.o odd 6 1 inner

Twists

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

Hecke kernels

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

\( T_{2} \) Copy content Toggle raw display
\( T_{17}^{2} + 2T_{17} + 4 \) Copy content Toggle raw display
\( T_{167} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} - T + 1 \) 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
$17$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) 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
$47$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} \) 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
$79$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$83$ \( (T - 2)^{2} \) 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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