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Properties

Label 810.2.i.h
Level $810$
Weight $2$
Character orbit 810.i
Analytic conductor $6.468$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [810,2,Mod(109,810)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(810, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("810.109");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 810 = 2 \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 810.i (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: \(6.46788256372\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.2702336256.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 9x^{6} + 56x^{4} + 225x^{2} + 625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 270)
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.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} - \beta_{2} q^{4} - \beta_1 q^{5} + (2 \beta_{5} + 2 \beta_{3} + \beta_{2} + 1) q^{7} - \beta_{6} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{2} - \beta_{2} q^{4} - \beta_1 q^{5} + (2 \beta_{5} + 2 \beta_{3} + \beta_{2} + 1) q^{7} - \beta_{6} q^{8} + (\beta_{5} + 1) q^{10} + ( - 2 \beta_{7} - \beta_{4}) q^{11} + ( - \beta_{6} - \beta_{4} - 2 \beta_1) q^{14} + ( - \beta_{2} - 1) q^{16} + 4 \beta_{6} q^{17} - 6 q^{19} + (\beta_{7} + \beta_{4}) q^{20} + (2 \beta_{3} + \beta_{2}) q^{22} + ( - 2 \beta_{6} - 2 \beta_{4}) q^{23} + (\beta_{5} + \beta_{3} - 4 \beta_{2} - 4) q^{25} + (2 \beta_{5} + 1) q^{28} + 7 \beta_{2} q^{31} + ( - \beta_{6} - \beta_{4}) q^{32} + (4 \beta_{2} + 4) q^{34} + ( - \beta_{7} - 10 \beta_{6} + \cdots - \beta_1) q^{35}+ \cdots - 12 \beta_{6} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{4} + 4 q^{10} - 4 q^{16} - 48 q^{19} - 18 q^{25} - 28 q^{31} + 16 q^{34} + 2 q^{40} - 16 q^{46} + 48 q^{49} + 76 q^{55} + 16 q^{61} - 8 q^{64} - 38 q^{70} - 24 q^{76} - 8 q^{85} + 8 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 9x^{6} + 56x^{4} + 225x^{2} + 625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -9\nu^{6} - 56\nu^{4} - 504\nu^{2} - 2025 ) / 1400 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -11\nu^{6} - 224\nu^{4} - 616\nu^{2} - 2475 ) / 1400 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 11\nu^{7} + 224\nu^{5} + 616\nu^{3} + 2475\nu ) / 7000 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{6} - 1 ) / 56 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -9\nu^{7} - 56\nu^{5} - 154\nu^{3} - 625\nu ) / 1750 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{7} - 9\nu^{5} - 56\nu^{3} - 225\nu ) / 125 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + \beta_{3} - 4\beta_{2} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -4\beta_{7} + 5\beta_{6} - 4\beta_{4} - 4\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -9\beta_{3} + 11\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 11\beta_{7} + 56\beta_{4} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -56\beta_{5} - 1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -280\beta_{6} - 280\beta_{4} - \beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/810\mathbb{Z}\right)^\times\).

\(n\) \(487\) \(731\)
\(\chi(n)\) \(-1\) \(\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1
1.52274 − 1.63746i
−0.656712 + 2.13746i
0.656712 − 2.13746i
−1.52274 + 1.63746i
1.52274 + 1.63746i
−0.656712 − 2.13746i
0.656712 + 2.13746i
−1.52274 − 1.63746i
−0.866025 + 0.500000i 0 0.500000 − 0.866025i −1.52274 + 1.63746i 0 3.77492 − 2.17945i 1.00000i 0 0.500000 − 2.17945i
109.2 −0.866025 + 0.500000i 0 0.500000 − 0.866025i 0.656712 − 2.13746i 0 −3.77492 + 2.17945i 1.00000i 0 0.500000 + 2.17945i
109.3 0.866025 − 0.500000i 0 0.500000 − 0.866025i −0.656712 + 2.13746i 0 −3.77492 + 2.17945i − 1.00000i 0 0.500000 + 2.17945i
109.4 0.866025 − 0.500000i 0 0.500000 − 0.866025i 1.52274 − 1.63746i 0 3.77492 − 2.17945i − 1.00000i 0 0.500000 − 2.17945i
379.1 −0.866025 − 0.500000i 0 0.500000 + 0.866025i −1.52274 − 1.63746i 0 3.77492 + 2.17945i − 1.00000i 0 0.500000 + 2.17945i
379.2 −0.866025 − 0.500000i 0 0.500000 + 0.866025i 0.656712 + 2.13746i 0 −3.77492 − 2.17945i − 1.00000i 0 0.500000 − 2.17945i
379.3 0.866025 + 0.500000i 0 0.500000 + 0.866025i −0.656712 − 2.13746i 0 −3.77492 − 2.17945i 1.00000i 0 0.500000 − 2.17945i
379.4 0.866025 + 0.500000i 0 0.500000 + 0.866025i 1.52274 + 1.63746i 0 3.77492 + 2.17945i 1.00000i 0 0.500000 + 2.17945i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 109.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner
15.d odd 2 1 inner
45.h odd 6 1 inner
45.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 810.2.i.h 8
3.b odd 2 1 inner 810.2.i.h 8
5.b even 2 1 inner 810.2.i.h 8
9.c even 3 1 270.2.c.c ✓ 4
9.c even 3 1 inner 810.2.i.h 8
9.d odd 6 1 270.2.c.c ✓ 4
9.d odd 6 1 inner 810.2.i.h 8
15.d odd 2 1 inner 810.2.i.h 8
36.f odd 6 1 2160.2.f.m 4
36.h even 6 1 2160.2.f.m 4
45.h odd 6 1 270.2.c.c ✓ 4
45.h odd 6 1 inner 810.2.i.h 8
45.j even 6 1 270.2.c.c ✓ 4
45.j even 6 1 inner 810.2.i.h 8
45.k odd 12 1 1350.2.a.w 2
45.k odd 12 1 1350.2.a.x 2
45.l even 12 1 1350.2.a.w 2
45.l even 12 1 1350.2.a.x 2
180.n even 6 1 2160.2.f.m 4
180.p odd 6 1 2160.2.f.m 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
270.2.c.c ✓ 4 9.c even 3 1
270.2.c.c ✓ 4 9.d odd 6 1
270.2.c.c ✓ 4 45.h odd 6 1
270.2.c.c ✓ 4 45.j even 6 1
810.2.i.h 8 1.a even 1 1 trivial
810.2.i.h 8 3.b odd 2 1 inner
810.2.i.h 8 5.b even 2 1 inner
810.2.i.h 8 9.c even 3 1 inner
810.2.i.h 8 9.d odd 6 1 inner
810.2.i.h 8 15.d odd 2 1 inner
810.2.i.h 8 45.h odd 6 1 inner
810.2.i.h 8 45.j even 6 1 inner
1350.2.a.w 2 45.k odd 12 1
1350.2.a.w 2 45.l even 12 1
1350.2.a.x 2 45.k odd 12 1
1350.2.a.x 2 45.l even 12 1
2160.2.f.m 4 36.f odd 6 1
2160.2.f.m 4 36.h even 6 1
2160.2.f.m 4 180.n even 6 1
2160.2.f.m 4 180.p odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(810, [\chi])\):

\( T_{7}^{4} - 19T_{7}^{2} + 361 \) Copy content Toggle raw display
\( T_{11}^{4} + 19T_{11}^{2} + 361 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 9 T^{6} + \cdots + 625 \) Copy content Toggle raw display
$7$ \( (T^{4} - 19 T^{2} + 361)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 19 T^{2} + 361)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( (T^{2} + 16)^{4} \) Copy content Toggle raw display
$19$ \( (T + 6)^{8} \) Copy content Toggle raw display
$23$ \( (T^{4} - 4 T^{2} + 16)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( (T^{2} + 7 T + 49)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 76)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} + 76 T^{2} + 5776)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 76 T^{2} + 5776)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 4 T^{2} + 16)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 9)^{4} \) Copy content Toggle raw display
$59$ \( (T^{4} + 76 T^{2} + 5776)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 4 T + 16)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} - 76 T^{2} + 5776)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( (T^{2} + 19)^{4} \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( (T^{4} - 25 T^{2} + 625)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 76)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} - 19 T^{2} + 361)^{2} \) Copy content Toggle raw display
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