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Properties

Label 810.2.f.b
Level $810$
Weight $2$
Character orbit 810.f
Analytic conductor $6.468$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [810,2,Mod(323,810)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(810, base_ring=CyclotomicField(4))
 
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("810.323");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 810 = 2 \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 810.f (of order \(4\), degree \(2\), 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 \(\Q(i)\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 90)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{3} q^{4} + (\beta_{5} + \beta_{4} + \beta_1) q^{5} + ( - \beta_{7} - 2 \beta_{3} + 2) q^{7} - \beta_{5} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{3} q^{4} + (\beta_{5} + \beta_{4} + \beta_1) q^{5} + ( - \beta_{7} - 2 \beta_{3} + 2) q^{7} - \beta_{5} q^{8} + (\beta_{7} + \beta_{3} + 1) q^{10} + (2 \beta_{5} + 2 \beta_{4} - 2 \beta_1) q^{11} + 2 \beta_{2} q^{13} + (\beta_{6} + 2 \beta_{5} + 2 \beta_1) q^{14} - q^{16} + ( - \beta_{6} - \beta_{4} + 2 \beta_1) q^{17} + ( - \beta_{7} - 4 \beta_{3} - \beta_{2}) q^{19} + ( - \beta_{6} - \beta_{5} + \beta_1) q^{20} + (2 \beta_{7} - 2 \beta_{3} + 2) q^{22} + \beta_{5} q^{23} + (2 \beta_{7} + 2 \beta_{2} - 1) q^{25} + 2 \beta_{4} q^{26} + (2 \beta_{3} + \beta_{2} + 2) q^{28} + ( - \beta_{6} + \beta_{5} + \beta_1) q^{29} + ( - \beta_{7} + \beta_{2} - 2) q^{31} - \beta_1 q^{32} + ( - \beta_{7} + 2 \beta_{3} - \beta_{2}) q^{34} + (3 \beta_{6} + 4 \beta_{5} + \cdots + 3 \beta_1) q^{35}+ \cdots + (4 \beta_{6} + 4 \beta_{5} - 4 \beta_{4}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 16 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 16 q^{7} + 8 q^{10} - 8 q^{16} + 16 q^{22} - 8 q^{25} + 16 q^{28} - 16 q^{31} - 24 q^{37} - 8 q^{40} + 8 q^{46} - 48 q^{55} + 8 q^{58} - 24 q^{61} - 8 q^{67} + 32 q^{70} - 16 q^{73} + 32 q^{76} - 32 q^{82} + 40 q^{85} + 16 q^{88} + 48 q^{91} - 24 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

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

Display \(\zeta_{24}^j\) in terms of \(\beta_i\)

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

Display \(\beta_i\) in terms of \(\zeta_{24}^j\)

Character values

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

\(n\) \(487\) \(731\)
\(\chi(n)\) \(-\beta_{3}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
323.1
0.258819 + 0.965926i
−0.965926 − 0.258819i
−0.258819 − 0.965926i
0.965926 + 0.258819i
−0.965926 + 0.258819i
0.258819 − 0.965926i
0.965926 − 0.258819i
−0.258819 + 0.965926i
−0.707107 − 0.707107i 0 1.00000i −1.41421 − 1.73205i 0 3.22474 − 3.22474i 0.707107 − 0.707107i 0 −0.224745 + 2.22474i
323.2 −0.707107 − 0.707107i 0 1.00000i −1.41421 + 1.73205i 0 0.775255 − 0.775255i 0.707107 − 0.707107i 0 2.22474 − 0.224745i
323.3 0.707107 + 0.707107i 0 1.00000i 1.41421 − 1.73205i 0 0.775255 − 0.775255i −0.707107 + 0.707107i 0 2.22474 − 0.224745i
323.4 0.707107 + 0.707107i 0 1.00000i 1.41421 + 1.73205i 0 3.22474 − 3.22474i −0.707107 + 0.707107i 0 −0.224745 + 2.22474i
647.1 −0.707107 + 0.707107i 0 − 1.00000i −1.41421 − 1.73205i 0 0.775255 + 0.775255i 0.707107 + 0.707107i 0 2.22474 + 0.224745i
647.2 −0.707107 + 0.707107i 0 − 1.00000i −1.41421 + 1.73205i 0 3.22474 + 3.22474i 0.707107 + 0.707107i 0 −0.224745 − 2.22474i
647.3 0.707107 − 0.707107i 0 − 1.00000i 1.41421 − 1.73205i 0 3.22474 + 3.22474i −0.707107 − 0.707107i 0 −0.224745 − 2.22474i
647.4 0.707107 − 0.707107i 0 − 1.00000i 1.41421 + 1.73205i 0 0.775255 + 0.775255i −0.707107 − 0.707107i 0 2.22474 + 0.224745i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 323.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 810.2.f.b 8
3.b odd 2 1 inner 810.2.f.b 8
5.c odd 4 1 inner 810.2.f.b 8
9.c even 3 1 90.2.l.a ✓ 8
9.c even 3 1 270.2.m.a 8
9.d odd 6 1 90.2.l.a ✓ 8
9.d odd 6 1 270.2.m.a 8
15.e even 4 1 inner 810.2.f.b 8
36.f odd 6 1 720.2.cu.a 8
36.h even 6 1 720.2.cu.a 8
45.h odd 6 1 450.2.p.a 8
45.h odd 6 1 1350.2.q.g 8
45.j even 6 1 450.2.p.a 8
45.j even 6 1 1350.2.q.g 8
45.k odd 12 1 90.2.l.a ✓ 8
45.k odd 12 1 270.2.m.a 8
45.k odd 12 1 450.2.p.a 8
45.k odd 12 1 1350.2.q.g 8
45.l even 12 1 90.2.l.a ✓ 8
45.l even 12 1 270.2.m.a 8
45.l even 12 1 450.2.p.a 8
45.l even 12 1 1350.2.q.g 8
180.v odd 12 1 720.2.cu.a 8
180.x even 12 1 720.2.cu.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.2.l.a ✓ 8 9.c even 3 1
90.2.l.a ✓ 8 9.d odd 6 1
90.2.l.a ✓ 8 45.k odd 12 1
90.2.l.a ✓ 8 45.l even 12 1
270.2.m.a 8 9.c even 3 1
270.2.m.a 8 9.d odd 6 1
270.2.m.a 8 45.k odd 12 1
270.2.m.a 8 45.l even 12 1
450.2.p.a 8 45.h odd 6 1
450.2.p.a 8 45.j even 6 1
450.2.p.a 8 45.k odd 12 1
450.2.p.a 8 45.l even 12 1
720.2.cu.a 8 36.f odd 6 1
720.2.cu.a 8 36.h even 6 1
720.2.cu.a 8 180.v odd 12 1
720.2.cu.a 8 180.x even 12 1
810.2.f.b 8 1.a even 1 1 trivial
810.2.f.b 8 3.b odd 2 1 inner
810.2.f.b 8 5.c odd 4 1 inner
810.2.f.b 8 15.e even 4 1 inner
1350.2.q.g 8 45.h odd 6 1
1350.2.q.g 8 45.j even 6 1
1350.2.q.g 8 45.k odd 12 1
1350.2.q.g 8 45.l even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} - 8T_{7}^{3} + 32T_{7}^{2} - 40T_{7} + 25 \) acting on \(S_{2}^{\mathrm{new}}(810, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} + 2 T^{2} + 25)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} - 8 T^{3} + 32 T^{2} + \cdots + 25)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 40 T^{2} + 16)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 144)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 392T^{4} + 16 \) Copy content Toggle raw display
$19$ \( (T^{4} + 44 T^{2} + 100)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 10 T^{2} + 1)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 4 T - 2)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 6 T + 18)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} + 70 T^{2} + 841)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 144)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 6561)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + 8456 T^{4} + 6250000 \) Copy content Toggle raw display
$59$ \( (T^{4} - 220 T^{2} + 11236)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 6 T + 3)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + 4 T^{3} + \cdots + 625)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 40 T^{2} + 16)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 8 T^{3} + \cdots + 1600)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 6)^{4} \) Copy content Toggle raw display
$83$ \( T^{8} + 882T^{4} + 81 \) Copy content Toggle raw display
$89$ \( (T^{4} - 70 T^{2} + 361)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 12 T^{3} + \cdots + 900)^{2} \) Copy content Toggle raw display
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