Properties

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

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [735,2,Mod(521,735)]
 
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, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("735.521");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 735.s (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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{6} - 2) q^{2} + ( - \zeta_{6} + 2) q^{3} + ( - \zeta_{6} + 1) q^{4} + \zeta_{6} q^{5} + (3 \zeta_{6} - 3) q^{6} + ( - 2 \zeta_{6} + 1) q^{8} + ( - 3 \zeta_{6} + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{6} - 2) q^{2} + ( - \zeta_{6} + 2) q^{3} + ( - \zeta_{6} + 1) q^{4} + \zeta_{6} q^{5} + (3 \zeta_{6} - 3) q^{6} + ( - 2 \zeta_{6} + 1) q^{8} + ( - 3 \zeta_{6} + 3) q^{9} + ( - \zeta_{6} - 1) q^{10} + ( - 2 \zeta_{6} - 2) q^{11} + ( - 2 \zeta_{6} + 1) q^{12} + (4 \zeta_{6} - 2) q^{13} + (\zeta_{6} + 1) q^{15} + 5 \zeta_{6} q^{16} + ( - 6 \zeta_{6} + 6) q^{17} + (6 \zeta_{6} - 3) q^{18} + ( - 4 \zeta_{6} + 8) q^{19} + q^{20} + 6 q^{22} + ( - \zeta_{6} + 2) q^{23} - 3 \zeta_{6} q^{24} + (\zeta_{6} - 1) q^{25} - 6 \zeta_{6} q^{26} + ( - 6 \zeta_{6} + 3) q^{27} + (2 \zeta_{6} - 1) q^{29} - 3 q^{30} + (2 \zeta_{6} + 2) q^{31} + ( - 3 \zeta_{6} - 3) q^{32} - 6 q^{33} + (12 \zeta_{6} - 6) q^{34} - 3 \zeta_{6} q^{36} - 4 \zeta_{6} q^{37} + (12 \zeta_{6} - 12) q^{38} + 6 \zeta_{6} q^{39} + ( - \zeta_{6} + 2) q^{40} + 3 q^{41} + q^{43} + (2 \zeta_{6} - 4) q^{44} + 3 q^{45} + (3 \zeta_{6} - 3) q^{46} + (5 \zeta_{6} + 5) q^{48} + ( - 2 \zeta_{6} + 1) q^{50} + ( - 12 \zeta_{6} + 6) q^{51} + (2 \zeta_{6} + 2) q^{52} + 9 \zeta_{6} q^{54} + ( - 4 \zeta_{6} + 2) q^{55} + ( - 12 \zeta_{6} + 12) q^{57} - 3 \zeta_{6} q^{58} + ( - \zeta_{6} + 2) q^{60} + ( - 3 \zeta_{6} + 6) q^{61} - 6 q^{62} - q^{64} + (2 \zeta_{6} - 4) q^{65} + ( - 6 \zeta_{6} + 12) q^{66} + ( - 13 \zeta_{6} + 13) q^{67} - 6 \zeta_{6} q^{68} + ( - 3 \zeta_{6} + 3) q^{69} + (8 \zeta_{6} - 4) q^{71} + ( - 3 \zeta_{6} - 3) q^{72} + ( - 2 \zeta_{6} - 2) q^{73} + (4 \zeta_{6} + 4) q^{74} + (2 \zeta_{6} - 1) q^{75} + ( - 8 \zeta_{6} + 4) q^{76} + ( - 6 \zeta_{6} - 6) q^{78} + 16 \zeta_{6} q^{79} + (5 \zeta_{6} - 5) q^{80} - 9 \zeta_{6} q^{81} + (3 \zeta_{6} - 6) q^{82} + 9 q^{83} + 6 q^{85} + (\zeta_{6} - 2) q^{86} + 3 \zeta_{6} q^{87} + (6 \zeta_{6} - 6) q^{88} + 3 \zeta_{6} q^{89} + (3 \zeta_{6} - 6) q^{90} + ( - 2 \zeta_{6} + 1) q^{92} + 6 q^{93} + (4 \zeta_{6} + 4) q^{95} - 9 q^{96} + ( - 12 \zeta_{6} + 6) q^{97} + (6 \zeta_{6} - 12) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{2} + 3 q^{3} + q^{4} + q^{5} - 3 q^{6} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{2} + 3 q^{3} + q^{4} + q^{5} - 3 q^{6} + 3 q^{9} - 3 q^{10} - 6 q^{11} + 3 q^{15} + 5 q^{16} + 6 q^{17} + 12 q^{19} + 2 q^{20} + 12 q^{22} + 3 q^{23} - 3 q^{24} - q^{25} - 6 q^{26} - 6 q^{30} + 6 q^{31} - 9 q^{32} - 12 q^{33} - 3 q^{36} - 4 q^{37} - 12 q^{38} + 6 q^{39} + 3 q^{40} + 6 q^{41} + 2 q^{43} - 6 q^{44} + 6 q^{45} - 3 q^{46} + 15 q^{48} + 6 q^{52} + 9 q^{54} + 12 q^{57} - 3 q^{58} + 3 q^{60} + 9 q^{61} - 12 q^{62} - 2 q^{64} - 6 q^{65} + 18 q^{66} + 13 q^{67} - 6 q^{68} + 3 q^{69} - 9 q^{72} - 6 q^{73} + 12 q^{74} - 18 q^{78} + 16 q^{79} - 5 q^{80} - 9 q^{81} - 9 q^{82} + 18 q^{83} + 12 q^{85} - 3 q^{86} + 3 q^{87} - 6 q^{88} + 3 q^{89} - 9 q^{90} + 12 q^{93} + 12 q^{95} - 18 q^{96} - 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
521.1
0.500000 + 0.866025i
0.500000 0.866025i
−1.50000 + 0.866025i 1.50000 0.866025i 0.500000 0.866025i 0.500000 + 0.866025i −1.50000 + 2.59808i 0 1.73205i 1.50000 2.59808i −1.50000 0.866025i
656.1 −1.50000 0.866025i 1.50000 + 0.866025i 0.500000 + 0.866025i 0.500000 0.866025i −1.50000 2.59808i 0 1.73205i 1.50000 + 2.59808i −1.50000 + 0.866025i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.g 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.s.c 2
3.b odd 2 1 735.2.s.e 2
7.b odd 2 1 105.2.s.a 2
7.c even 3 1 105.2.s.b yes 2
7.c even 3 1 735.2.b.a 2
7.d odd 6 1 735.2.b.b 2
7.d odd 6 1 735.2.s.e 2
21.c even 2 1 105.2.s.b yes 2
21.g even 6 1 735.2.b.a 2
21.g even 6 1 inner 735.2.s.c 2
21.h odd 6 1 105.2.s.a 2
21.h odd 6 1 735.2.b.b 2
35.c odd 2 1 525.2.t.e 2
35.f even 4 2 525.2.q.b 4
35.j even 6 1 525.2.t.a 2
35.l odd 12 2 525.2.q.a 4
105.g even 2 1 525.2.t.a 2
105.k odd 4 2 525.2.q.a 4
105.o odd 6 1 525.2.t.e 2
105.x even 12 2 525.2.q.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.s.a 2 7.b odd 2 1
105.2.s.a 2 21.h odd 6 1
105.2.s.b yes 2 7.c even 3 1
105.2.s.b yes 2 21.c even 2 1
525.2.q.a 4 35.l odd 12 2
525.2.q.a 4 105.k odd 4 2
525.2.q.b 4 35.f even 4 2
525.2.q.b 4 105.x even 12 2
525.2.t.a 2 35.j even 6 1
525.2.t.a 2 105.g even 2 1
525.2.t.e 2 35.c odd 2 1
525.2.t.e 2 105.o odd 6 1
735.2.b.a 2 7.c even 3 1
735.2.b.a 2 21.g even 6 1
735.2.b.b 2 7.d odd 6 1
735.2.b.b 2 21.h odd 6 1
735.2.s.c 2 1.a even 1 1 trivial
735.2.s.c 2 21.g even 6 1 inner
735.2.s.e 2 3.b odd 2 1
735.2.s.e 2 7.d odd 6 1

Hecke kernels

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

\( T_{2}^{2} + 3T_{2} + 3 \) Copy content Toggle raw display
\( T_{13}^{2} + 12 \) Copy content Toggle raw display
\( T_{17}^{2} - 6T_{17} + 36 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 3T + 3 \) Copy content Toggle raw display
$3$ \( T^{2} - 3T + 3 \) Copy content Toggle raw display
$5$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 6T + 12 \) Copy content Toggle raw display
$13$ \( T^{2} + 12 \) Copy content Toggle raw display
$17$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$19$ \( T^{2} - 12T + 48 \) Copy content Toggle raw display
$23$ \( T^{2} - 3T + 3 \) Copy content Toggle raw display
$29$ \( T^{2} + 3 \) Copy content Toggle raw display
$31$ \( T^{2} - 6T + 12 \) Copy content Toggle raw display
$37$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$41$ \( (T - 3)^{2} \) Copy content Toggle raw display
$43$ \( (T - 1)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 9T + 27 \) Copy content Toggle raw display
$67$ \( T^{2} - 13T + 169 \) Copy content Toggle raw display
$71$ \( T^{2} + 48 \) Copy content Toggle raw display
$73$ \( T^{2} + 6T + 12 \) Copy content Toggle raw display
$79$ \( T^{2} - 16T + 256 \) Copy content Toggle raw display
$83$ \( (T - 9)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$97$ \( T^{2} + 108 \) Copy content Toggle raw display
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