Properties

Label 735.2.p.c
Level $735$
Weight $2$
Character orbit 735.p
Analytic conductor $5.869$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [735,2,Mod(374,735)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(735, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("735.374");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 735.p (of order \(6\), degree \(2\), 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: \(5.86900454856\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 105)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + (\beta_{5} - \beta_1 + 1) q^{3} + (\beta_1 - 1) q^{4} + (\beta_{3} + \beta_{2}) q^{5} + (\beta_{7} + \beta_{4}) q^{6} + \beta_{4} q^{8} + (2 \beta_{3} + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + (\beta_{5} - \beta_1 + 1) q^{3} + (\beta_1 - 1) q^{4} + (\beta_{3} + \beta_{2}) q^{5} + (\beta_{7} + \beta_{4}) q^{6} + \beta_{4} q^{8} + (2 \beta_{3} + \beta_1) q^{9} + (\beta_{6} + 3 \beta_1 - 3) q^{10} - 2 \beta_{5} q^{11} + ( - \beta_{3} + \beta_1) q^{12} - 4 q^{13} + (\beta_{7} - \beta_{5} + \beta_{4} + \beta_{3} + 2) q^{15} + 5 \beta_1 q^{16} - 2 \beta_{5} q^{17} + (2 \beta_{6} - \beta_{4} + \beta_{2}) q^{18} + (\beta_{5} - \beta_{4} - \beta_{3}) q^{20} - 2 \beta_{7} q^{22} + 2 \beta_{2} q^{23} + (\beta_{6} + \beta_{4} - \beta_{2}) q^{24} + (2 \beta_{6} + \beta_1 - 1) q^{25} - 4 \beta_{2} q^{26} + ( - \beta_{5} + \beta_{3} + 5) q^{27} + ( - 4 \beta_{5} + 4 \beta_{3}) q^{29} + ( - \beta_{7} + \beta_{6} - 3 \beta_{3} + 2 \beta_{2} + 3 \beta_1) q^{30} - 4 \beta_{6} q^{31} + ( - 3 \beta_{4} + 3 \beta_{2}) q^{32} + ( - 2 \beta_{3} - 4 \beta_1) q^{33} - 2 \beta_{7} q^{34} + (2 \beta_{5} - 2 \beta_{3} - 1) q^{36} + ( - 4 \beta_{5} + 4 \beta_1 - 4) q^{39} + ( - \beta_{7} + \beta_{6} + 3 \beta_1) q^{40} - 2 \beta_{4} q^{41} + 2 \beta_{7} q^{43} + 2 \beta_{3} q^{44} + (2 \beta_{6} + \beta_{5} - \beta_{4} + \beta_{2} - 4 \beta_1 + 4) q^{45} + (6 \beta_1 - 6) q^{46} + 2 \beta_{3} q^{47} + (5 \beta_{5} - 5 \beta_{3} + 5) q^{48} + (6 \beta_{5} - \beta_{4} - 6 \beta_{3}) q^{50} + ( - 2 \beta_{3} - 4 \beta_1) q^{51} + ( - 4 \beta_1 + 4) q^{52} + ( - \beta_{7} + \beta_{6} + 5 \beta_{2}) q^{54} + ( - 2 \beta_{7} - 4) q^{55} + ( - 4 \beta_{7} + 4 \beta_{6}) q^{58} + ( - 4 \beta_{4} + 4 \beta_{2}) q^{59} + ( - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{2} + 2 \beta_1 - 2) q^{60} + (4 \beta_{7} - 4 \beta_{6}) q^{61} + ( - 12 \beta_{5} + 12 \beta_{3}) q^{62} + q^{64} + ( - 4 \beta_{3} - 4 \beta_{2}) q^{65} + ( - 2 \beta_{6} + 4 \beta_{4} - 4 \beta_{2}) q^{66} + 2 \beta_{6} q^{67} + 2 \beta_{3} q^{68} + (2 \beta_{7} + 2 \beta_{4}) q^{69} + (2 \beta_{5} - 2 \beta_{3}) q^{71} + ( - 2 \beta_{7} + 2 \beta_{6} + \beta_{2}) q^{72} + (8 \beta_1 - 8) q^{73} + ( - 2 \beta_{7} + 2 \beta_{6} - \beta_{3} + 4 \beta_{2} + \beta_1) q^{75} + ( - 4 \beta_{7} - 4 \beta_{4}) q^{78} - 8 \beta_1 q^{79} + (5 \beta_{5} - 5 \beta_{4} + 5 \beta_{2}) q^{80} + (4 \beta_{5} - 7 \beta_1 + 7) q^{81} - 6 \beta_1 q^{82} + (2 \beta_{5} - 2 \beta_{3}) q^{83} + ( - 2 \beta_{7} - 4) q^{85} - 6 \beta_{3} q^{86} + ( - 4 \beta_{5} - 8 \beta_1 + 8) q^{87} - 2 \beta_{6} q^{88} - 6 \beta_{2} q^{89} + (\beta_{7} + 6 \beta_{5} + 4 \beta_{4} - 6 \beta_{3} - 3) q^{90} - 2 \beta_{4} q^{92} + (4 \beta_{7} - 4 \beta_{6} - 8 \beta_{2}) q^{93} + 2 \beta_{6} q^{94} + (3 \beta_{7} - 3 \beta_{6} + 3 \beta_{2}) q^{96} + 8 q^{97} + ( - 2 \beta_{5} + 2 \beta_{3} - 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{3} - 4 q^{4} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{3} - 4 q^{4} + 4 q^{9} - 12 q^{10} + 4 q^{12} - 32 q^{13} + 16 q^{15} + 20 q^{16} - 4 q^{25} + 40 q^{27} + 12 q^{30} - 16 q^{33} - 8 q^{36} - 16 q^{39} + 12 q^{40} + 16 q^{45} - 24 q^{46} + 40 q^{48} - 16 q^{51} + 16 q^{52} - 32 q^{55} - 8 q^{60} + 8 q^{64} - 32 q^{73} + 4 q^{75} - 32 q^{79} + 28 q^{81} - 24 q^{82} - 32 q^{85} + 32 q^{87} - 24 q^{90} + 64 q^{97} - 64 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{24}^{4} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{24}^{6} + \zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{24}^{7} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\zeta_{24}^{6} + 2\zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \zeta_{24}^{7} - \zeta_{24}^{5} + \zeta_{24}^{3} + 2\zeta_{24} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( 2\zeta_{24}^{7} + \zeta_{24}^{5} - \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display
\(\zeta_{24}\)\(=\) \( ( \beta_{7} + \beta_{6} + 3\beta_{3} ) / 6 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( ( \beta_{4} + \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( -\beta_{7} + 2\beta_{6} + 3\beta_{5} - 3\beta_{3} ) / 6 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( 2\beta_{7} - \beta_{6} + 3\beta_{5} ) / 6 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( ( -\beta_{4} + 2\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( \beta_{7} + \beta_{6} - 3\beta_{3} ) / 6 \) Copy content Toggle raw display

Character values

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

\(n\) \(346\) \(442\) \(491\)
\(\chi(n)\) \(\beta_{1}\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
374.1
−0.258819 + 0.965926i
0.258819 0.965926i
−0.965926 0.258819i
0.965926 + 0.258819i
−0.258819 0.965926i
0.258819 + 0.965926i
−0.965926 + 0.258819i
0.965926 0.258819i
−0.866025 1.50000i −0.724745 1.57313i −0.500000 + 0.866025i −2.09077 0.792893i −1.73205 + 2.44949i 0 −1.73205 −1.94949 + 2.28024i 0.621320 + 3.82282i
374.2 −0.866025 1.50000i 1.72474 0.158919i −0.500000 + 0.866025i 0.358719 2.20711i −1.73205 2.44949i 0 −1.73205 2.94949 0.548188i −3.62132 + 1.37333i
374.3 0.866025 + 1.50000i −0.724745 1.57313i −0.500000 + 0.866025i −0.358719 + 2.20711i 1.73205 2.44949i 0 1.73205 −1.94949 + 2.28024i −3.62132 + 1.37333i
374.4 0.866025 + 1.50000i 1.72474 0.158919i −0.500000 + 0.866025i 2.09077 + 0.792893i 1.73205 + 2.44949i 0 1.73205 2.94949 0.548188i 0.621320 + 3.82282i
509.1 −0.866025 + 1.50000i −0.724745 + 1.57313i −0.500000 0.866025i −2.09077 + 0.792893i −1.73205 2.44949i 0 −1.73205 −1.94949 2.28024i 0.621320 3.82282i
509.2 −0.866025 + 1.50000i 1.72474 + 0.158919i −0.500000 0.866025i 0.358719 + 2.20711i −1.73205 + 2.44949i 0 −1.73205 2.94949 + 0.548188i −3.62132 1.37333i
509.3 0.866025 1.50000i −0.724745 + 1.57313i −0.500000 0.866025i −0.358719 2.20711i 1.73205 + 2.44949i 0 1.73205 −1.94949 2.28024i −3.62132 1.37333i
509.4 0.866025 1.50000i 1.72474 + 0.158919i −0.500000 0.866025i 2.09077 0.792893i 1.73205 2.44949i 0 1.73205 2.94949 + 0.548188i 0.621320 3.82282i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 374.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.c even 3 1 inner
21.h odd 6 1 inner
35.c odd 2 1 inner
35.i odd 6 1 inner
105.g even 2 1 inner
105.p even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 735.2.p.c 8
3.b odd 2 1 inner 735.2.p.c 8
5.b even 2 1 735.2.p.a 8
7.b odd 2 1 735.2.p.a 8
7.c even 3 1 105.2.g.a 4
7.c even 3 1 inner 735.2.p.c 8
7.d odd 6 1 105.2.g.c yes 4
7.d odd 6 1 735.2.p.a 8
15.d odd 2 1 735.2.p.a 8
21.c even 2 1 735.2.p.a 8
21.g even 6 1 105.2.g.c yes 4
21.g even 6 1 735.2.p.a 8
21.h odd 6 1 105.2.g.a 4
21.h odd 6 1 inner 735.2.p.c 8
28.f even 6 1 1680.2.k.a 4
28.g odd 6 1 1680.2.k.c 4
35.c odd 2 1 inner 735.2.p.c 8
35.i odd 6 1 105.2.g.a 4
35.i odd 6 1 inner 735.2.p.c 8
35.j even 6 1 105.2.g.c yes 4
35.j even 6 1 735.2.p.a 8
35.k even 12 2 525.2.b.j 8
35.l odd 12 2 525.2.b.j 8
84.j odd 6 1 1680.2.k.a 4
84.n even 6 1 1680.2.k.c 4
105.g even 2 1 inner 735.2.p.c 8
105.o odd 6 1 105.2.g.c yes 4
105.o odd 6 1 735.2.p.a 8
105.p even 6 1 105.2.g.a 4
105.p even 6 1 inner 735.2.p.c 8
105.w odd 12 2 525.2.b.j 8
105.x even 12 2 525.2.b.j 8
140.p odd 6 1 1680.2.k.a 4
140.s even 6 1 1680.2.k.c 4
420.ba even 6 1 1680.2.k.a 4
420.be odd 6 1 1680.2.k.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.g.a 4 7.c even 3 1
105.2.g.a 4 21.h odd 6 1
105.2.g.a 4 35.i odd 6 1
105.2.g.a 4 105.p even 6 1
105.2.g.c yes 4 7.d odd 6 1
105.2.g.c yes 4 21.g even 6 1
105.2.g.c yes 4 35.j even 6 1
105.2.g.c yes 4 105.o odd 6 1
525.2.b.j 8 35.k even 12 2
525.2.b.j 8 35.l odd 12 2
525.2.b.j 8 105.w odd 12 2
525.2.b.j 8 105.x even 12 2
735.2.p.a 8 5.b even 2 1
735.2.p.a 8 7.b odd 2 1
735.2.p.a 8 7.d odd 6 1
735.2.p.a 8 15.d odd 2 1
735.2.p.a 8 21.c even 2 1
735.2.p.a 8 21.g even 6 1
735.2.p.a 8 35.j even 6 1
735.2.p.a 8 105.o odd 6 1
735.2.p.c 8 1.a even 1 1 trivial
735.2.p.c 8 3.b odd 2 1 inner
735.2.p.c 8 7.c even 3 1 inner
735.2.p.c 8 21.h odd 6 1 inner
735.2.p.c 8 35.c odd 2 1 inner
735.2.p.c 8 35.i odd 6 1 inner
735.2.p.c 8 105.g even 2 1 inner
735.2.p.c 8 105.p even 6 1 inner
1680.2.k.a 4 28.f even 6 1
1680.2.k.a 4 84.j odd 6 1
1680.2.k.a 4 140.p odd 6 1
1680.2.k.a 4 420.ba even 6 1
1680.2.k.c 4 28.g odd 6 1
1680.2.k.c 4 84.n even 6 1
1680.2.k.c 4 140.s even 6 1
1680.2.k.c 4 420.be odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(735, [\chi])\):

\( T_{2}^{4} + 3T_{2}^{2} + 9 \) Copy content Toggle raw display
\( T_{13} + 4 \) Copy content Toggle raw display
\( T_{257}^{4} - 200T_{257}^{2} + 40000 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 3 T^{2} + 9)^{2} \) Copy content Toggle raw display
$3$ \( (T^{4} - 2 T^{3} + T^{2} - 6 T + 9)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} + 2 T^{6} - 21 T^{4} + 50 T^{2} + \cdots + 625 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} - 8 T^{2} + 64)^{2} \) Copy content Toggle raw display
$13$ \( (T + 4)^{8} \) Copy content Toggle raw display
$17$ \( (T^{4} - 8 T^{2} + 64)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( (T^{4} + 12 T^{2} + 144)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 32)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} - 96 T^{2} + 9216)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( (T^{2} - 12)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 24)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} - 8 T^{2} + 64)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( (T^{4} + 48 T^{2} + 2304)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 96 T^{2} + 9216)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 24 T^{2} + 576)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 8)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 8 T + 64)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 8 T + 64)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 8)^{4} \) Copy content Toggle raw display
$89$ \( (T^{4} + 108 T^{2} + 11664)^{2} \) Copy content Toggle raw display
$97$ \( (T - 8)^{8} \) Copy content Toggle raw display
show more
show less