Properties

Label 735.1.f.d
Level 735735
Weight 11
Character orbit 735.f
Self dual yes
Analytic conductor 0.3670.367
Analytic rank 00
Dimension 22
Projective image D4D_{4}
CM discriminant -15
Inner twists 44

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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 N == 735=3572 735 = 3 \cdot 5 \cdot 7^{2}
Weight: k k == 1 1
Character orbit: [χ][\chi] == 735.f (of order 22, degree 11, 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: 0.3668127842850.366812784285
Analytic rank: 00
Dimension: 22
Coefficient field: Q(2)\Q(\sqrt{2})
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x22 x^{2} - 2 Copy content Toggle raw display
Coefficient ring: Z[a1,a2]\Z[a_1, a_2]
Coefficient ring index: 1 1
Twist minimal: yes
Projective image: D4D_{4}
Projective field: Galois closure of 4.2.15435.1
Artin image: D8D_8
Artin field: Galois closure of 8.2.8338372875.2

qq-expansion

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

Coefficients of the qq-expansion are expressed in terms of β=2\beta = \sqrt{2}. We also show the integral qq-expansion of the trace form.

f(q)f(q) == qβq2+q3+q4q5βq6+q9+βq10+q12q15q16βq18+βq19q20+βq23+q25+q27+βq30++βq96+O(q100) 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} + \cdots + \beta q^{96} +O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 2q+2q3+2q42q5+2q9+2q122q152q162q20+2q25+2q27+2q364q382q454q462q482q60+4q622q64+2q75++2q81+O(q100) 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}+ \cdots + 2 q^{81}+O(q^{100}) Copy content Toggle raw display

Character values

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

nn 346346 442442 491491
χ(n)\chi(n) 11 1-1 1-1

Embeddings

For each embedding ιm\iota_m of the coefficient field, the values ιm(an)\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   ιm(ν)\iota_m(\nu) a2 a_{2} a3 a_{3} a4 a_{4} a5 a_{5} a6 a_{6} a7 a_{7} a8 a_{8} a9 a_{9} a10 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
nn: 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(15)\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 S1new(735,[χ])S_{1}^{\mathrm{new}}(735, [\chi]):

T222 T_{2}^{2} - 2 Copy content Toggle raw display
T17 T_{17} Copy content Toggle raw display
T167+2 T_{167} + 2 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T22 T^{2} - 2 Copy content Toggle raw display
33 (T1)2 (T - 1)^{2} Copy content Toggle raw display
55 (T+1)2 (T + 1)^{2} Copy content Toggle raw display
77 T2 T^{2} Copy content Toggle raw display
1111 T2 T^{2} Copy content Toggle raw display
1313 T2 T^{2} Copy content Toggle raw display
1717 T2 T^{2} Copy content Toggle raw display
1919 T22 T^{2} - 2 Copy content Toggle raw display
2323 T22 T^{2} - 2 Copy content Toggle raw display
2929 T2 T^{2} Copy content Toggle raw display
3131 T22 T^{2} - 2 Copy content Toggle raw display
3737 T2 T^{2} Copy content Toggle raw display
4141 T2 T^{2} Copy content Toggle raw display
4343 T2 T^{2} Copy content Toggle raw display
4747 T2 T^{2} Copy content Toggle raw display
5353 T22 T^{2} - 2 Copy content Toggle raw display
5959 T2 T^{2} Copy content Toggle raw display
6161 T22 T^{2} - 2 Copy content Toggle raw display
6767 T2 T^{2} Copy content Toggle raw display
7171 T2 T^{2} Copy content Toggle raw display
7373 T2 T^{2} Copy content Toggle raw display
7979 T2 T^{2} Copy content Toggle raw display
8383 T2 T^{2} Copy content Toggle raw display
8989 T2 T^{2} Copy content Toggle raw display
9797 T2 T^{2} Copy content Toggle raw display
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