Properties

Label 735.2.i.l
Level $735$
Weight $2$
Character orbit 735.i
Analytic conductor $5.869$
Analytic rank $0$
Dimension $4$
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(226,735)]
 
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([0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("735.226");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 735.i (of order \(3\), 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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 105)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + \beta_1) q^{2} + (\beta_1 - 1) q^{3} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 2) q^{4} + \beta_1 q^{5} + (\beta_{3} - 1) q^{6} + (2 \beta_{3} - 6) q^{8} - \beta_1 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + \beta_1) q^{2} + (\beta_1 - 1) q^{3} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 2) q^{4} + \beta_1 q^{5} + (\beta_{3} - 1) q^{6} + (2 \beta_{3} - 6) q^{8} - \beta_1 q^{9} + (\beta_{3} - \beta_{2} + \beta_1 - 1) q^{10} + (\beta_{3} - \beta_{2} - \beta_1 + 1) q^{11} + (2 \beta_{2} - 2 \beta_1) q^{12} + ( - \beta_{3} - 4) q^{13} - q^{15} + (4 \beta_{2} - 8 \beta_1) q^{16} + (\beta_{3} - \beta_{2} - 5 \beta_1 + 5) q^{17} + ( - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{18} + ( - 2 \beta_{2} + \beta_1) q^{19} + (2 \beta_{3} - 2) q^{20} - 2 q^{22} + ( - \beta_{2} + 3 \beta_1) q^{23} + ( - 2 \beta_{3} + 2 \beta_{2} - 6 \beta_1 + 6) q^{24} + (\beta_1 - 1) q^{25} + (3 \beta_{2} - \beta_1) q^{26} + q^{27} + (3 \beta_{3} + 1) q^{29} + (\beta_{2} - \beta_1) q^{30} + (2 \beta_{3} - 2 \beta_{2} - 3 \beta_1 + 3) q^{31} + ( - 8 \beta_{3} + 8 \beta_{2} - 8 \beta_1 + 8) q^{32} + (\beta_{2} + \beta_1) q^{33} + ( - 4 \beta_{3} + 2) q^{34} + ( - 2 \beta_{3} + 2) q^{36} + ( - 3 \beta_{2} - 2 \beta_1) q^{37} + (3 \beta_{3} - 3 \beta_{2} + 7 \beta_1 - 7) q^{38} + (\beta_{3} - \beta_{2} - 4 \beta_1 + 4) q^{39} + (2 \beta_{2} - 6 \beta_1) q^{40} + ( - \beta_{3} - 1) q^{41} + ( - 3 \beta_{3} - 2) q^{43} - 4 \beta_1 q^{44} + ( - \beta_1 + 1) q^{45} + (4 \beta_{3} - 4 \beta_{2} + 6 \beta_1 - 6) q^{46} + 2 \beta_1 q^{47} + ( - 4 \beta_{3} + 8) q^{48} + (\beta_{3} - 1) q^{50} + (\beta_{2} + 5 \beta_1) q^{51} + ( - 6 \beta_{3} + 6 \beta_{2} - 2 \beta_1 + 2) q^{52} + (6 \beta_{3} - 6 \beta_{2} + 2 \beta_1 - 2) q^{53} + ( - \beta_{2} + \beta_1) q^{54} + (\beta_{3} + 1) q^{55} + (2 \beta_{3} - 1) q^{57} + (2 \beta_{2} - 8 \beta_1) q^{58} + (3 \beta_{3} - 3 \beta_{2} - 5 \beta_1 + 5) q^{59} + ( - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 2) q^{60} + 4 \beta_1 q^{61} + ( - \beta_{3} - 3) q^{62} + ( - 8 \beta_{3} + 16) q^{64} + ( - \beta_{2} - 4 \beta_1) q^{65} + ( - 2 \beta_1 + 2) q^{66} + ( - 5 \beta_{3} + 5 \beta_{2} - 6 \beta_1 + 6) q^{67} + ( - 8 \beta_{2} + 4 \beta_1) q^{68} + (\beta_{3} - 3) q^{69} + ( - 3 \beta_{3} + 1) q^{71} + ( - 2 \beta_{2} + 6 \beta_1) q^{72} + ( - 5 \beta_{3} + 5 \beta_{2} - 4 \beta_1 + 4) q^{73} + (\beta_{3} - \beta_{2} + 7 \beta_1 - 7) q^{74} - \beta_1 q^{75} + (6 \beta_{3} - 14) q^{76} + ( - 3 \beta_{3} + 1) q^{78} + ( - 6 \beta_{2} - 3 \beta_1) q^{79} + ( - 4 \beta_{3} + 4 \beta_{2} - 8 \beta_1 + 8) q^{80} + (\beta_1 - 1) q^{81} + 2 \beta_1 q^{82} + (7 \beta_{3} - 3) q^{83} + (\beta_{3} + 5) q^{85} + ( - \beta_{2} + 7 \beta_1) q^{86} + ( - 3 \beta_{3} + 3 \beta_{2} + \beta_1 - 1) q^{87} + ( - 4 \beta_{3} + 4 \beta_{2}) q^{88} + ( - 7 \beta_{2} + 3 \beta_1) q^{89} + ( - \beta_{3} + 1) q^{90} + (8 \beta_{3} - 12) q^{92} + (2 \beta_{2} + 3 \beta_1) q^{93} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 2) q^{94} + (2 \beta_{3} - 2 \beta_{2} + \beta_1 - 1) q^{95} + ( - 8 \beta_{2} + 8 \beta_1) q^{96} + (4 \beta_{3} - 8) q^{97} + ( - \beta_{3} - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} - 2 q^{3} - 4 q^{4} + 2 q^{5} - 4 q^{6} - 24 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} - 2 q^{3} - 4 q^{4} + 2 q^{5} - 4 q^{6} - 24 q^{8} - 2 q^{9} - 2 q^{10} + 2 q^{11} - 4 q^{12} - 16 q^{13} - 4 q^{15} - 16 q^{16} + 10 q^{17} + 2 q^{18} + 2 q^{19} - 8 q^{20} - 8 q^{22} + 6 q^{23} + 12 q^{24} - 2 q^{25} - 2 q^{26} + 4 q^{27} + 4 q^{29} - 2 q^{30} + 6 q^{31} + 16 q^{32} + 2 q^{33} + 8 q^{34} + 8 q^{36} - 4 q^{37} - 14 q^{38} + 8 q^{39} - 12 q^{40} - 4 q^{41} - 8 q^{43} - 8 q^{44} + 2 q^{45} - 12 q^{46} + 4 q^{47} + 32 q^{48} - 4 q^{50} + 10 q^{51} + 4 q^{52} - 4 q^{53} + 2 q^{54} + 4 q^{55} - 4 q^{57} - 16 q^{58} + 10 q^{59} + 4 q^{60} + 8 q^{61} - 12 q^{62} + 64 q^{64} - 8 q^{65} + 4 q^{66} + 12 q^{67} + 8 q^{68} - 12 q^{69} + 4 q^{71} + 12 q^{72} + 8 q^{73} - 14 q^{74} - 2 q^{75} - 56 q^{76} + 4 q^{78} - 6 q^{79} + 16 q^{80} - 2 q^{81} + 4 q^{82} - 12 q^{83} + 20 q^{85} + 14 q^{86} - 2 q^{87} + 6 q^{89} + 4 q^{90} - 48 q^{92} + 6 q^{93} - 4 q^{94} - 2 q^{95} + 16 q^{96} - 32 q^{97} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{12}^{3} + \zeta_{12} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{12}^{3} + 2\zeta_{12} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( ( -\beta_{3} + 2\beta_{2} ) / 3 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
226.1
0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
−0.366025 + 0.633975i −0.500000 0.866025i 0.732051 + 1.26795i 0.500000 0.866025i 0.732051 0 −2.53590 −0.500000 + 0.866025i 0.366025 + 0.633975i
226.2 1.36603 2.36603i −0.500000 0.866025i −2.73205 4.73205i 0.500000 0.866025i −2.73205 0 −9.46410 −0.500000 + 0.866025i −1.36603 2.36603i
361.1 −0.366025 0.633975i −0.500000 + 0.866025i 0.732051 1.26795i 0.500000 + 0.866025i 0.732051 0 −2.53590 −0.500000 0.866025i 0.366025 0.633975i
361.2 1.36603 + 2.36603i −0.500000 + 0.866025i −2.73205 + 4.73205i 0.500000 + 0.866025i −2.73205 0 −9.46410 −0.500000 0.866025i −1.36603 + 2.36603i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 735.2.i.l 4
7.b odd 2 1 105.2.i.d 4
7.c even 3 1 735.2.a.h 2
7.c even 3 1 inner 735.2.i.l 4
7.d odd 6 1 105.2.i.d 4
7.d odd 6 1 735.2.a.g 2
21.c even 2 1 315.2.j.c 4
21.g even 6 1 315.2.j.c 4
21.g even 6 1 2205.2.a.z 2
21.h odd 6 1 2205.2.a.ba 2
28.d even 2 1 1680.2.bg.o 4
28.f even 6 1 1680.2.bg.o 4
35.c odd 2 1 525.2.i.f 4
35.f even 4 1 525.2.r.a 4
35.f even 4 1 525.2.r.f 4
35.i odd 6 1 525.2.i.f 4
35.i odd 6 1 3675.2.a.bg 2
35.j even 6 1 3675.2.a.be 2
35.k even 12 1 525.2.r.a 4
35.k even 12 1 525.2.r.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.i.d 4 7.b odd 2 1
105.2.i.d 4 7.d odd 6 1
315.2.j.c 4 21.c even 2 1
315.2.j.c 4 21.g even 6 1
525.2.i.f 4 35.c odd 2 1
525.2.i.f 4 35.i odd 6 1
525.2.r.a 4 35.f even 4 1
525.2.r.a 4 35.k even 12 1
525.2.r.f 4 35.f even 4 1
525.2.r.f 4 35.k even 12 1
735.2.a.g 2 7.d odd 6 1
735.2.a.h 2 7.c even 3 1
735.2.i.l 4 1.a even 1 1 trivial
735.2.i.l 4 7.c even 3 1 inner
1680.2.bg.o 4 28.d even 2 1
1680.2.bg.o 4 28.f even 6 1
2205.2.a.z 2 21.g even 6 1
2205.2.a.ba 2 21.h odd 6 1
3675.2.a.be 2 35.j even 6 1
3675.2.a.bg 2 35.i odd 6 1

Hecke kernels

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

\( T_{2}^{4} - 2T_{2}^{3} + 6T_{2}^{2} + 4T_{2} + 4 \) Copy content Toggle raw display
\( T_{13}^{2} + 8T_{13} + 13 \) Copy content Toggle raw display
\( T_{17}^{4} - 10T_{17}^{3} + 78T_{17}^{2} - 220T_{17} + 484 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 2 T^{3} + 6 T^{2} + 4 T + 4 \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - 2 T^{3} + 6 T^{2} + 4 T + 4 \) Copy content Toggle raw display
$13$ \( (T^{2} + 8 T + 13)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 10 T^{3} + 78 T^{2} + \cdots + 484 \) Copy content Toggle raw display
$19$ \( T^{4} - 2 T^{3} + 15 T^{2} + 22 T + 121 \) Copy content Toggle raw display
$23$ \( T^{4} - 6 T^{3} + 30 T^{2} - 36 T + 36 \) Copy content Toggle raw display
$29$ \( (T^{2} - 2 T - 26)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} - 6 T^{3} + 39 T^{2} + 18 T + 9 \) Copy content Toggle raw display
$37$ \( T^{4} + 4 T^{3} + 39 T^{2} - 92 T + 529 \) Copy content Toggle raw display
$41$ \( (T^{2} + 2 T - 2)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 4 T - 23)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 4 T^{3} + 120 T^{2} + \cdots + 10816 \) Copy content Toggle raw display
$59$ \( T^{4} - 10 T^{3} + 102 T^{2} + 20 T + 4 \) Copy content Toggle raw display
$61$ \( (T^{2} - 4 T + 16)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 12 T^{3} + 183 T^{2} + \cdots + 1521 \) Copy content Toggle raw display
$71$ \( (T^{2} - 2 T - 26)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} - 8 T^{3} + 123 T^{2} + \cdots + 3481 \) Copy content Toggle raw display
$79$ \( T^{4} + 6 T^{3} + 135 T^{2} + \cdots + 9801 \) Copy content Toggle raw display
$83$ \( (T^{2} + 6 T - 138)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 6 T^{3} + 174 T^{2} + \cdots + 19044 \) Copy content Toggle raw display
$97$ \( (T^{2} + 16 T + 16)^{2} \) Copy content Toggle raw display
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