Properties

Label 810.4.a.b
Level $810$
Weight $4$
Character orbit 810.a
Self dual yes
Analytic conductor $47.792$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [810,4,Mod(1,810)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(810, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("810.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 810 = 2 \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 810.a (trivial)

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: yes
Analytic conductor: \(47.7915471046\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 90)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - 2 q^{2} + 4 q^{4} + 5 q^{5} - 16 q^{7} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{2} + 4 q^{4} + 5 q^{5} - 16 q^{7} - 8 q^{8} - 10 q^{10} - 57 q^{11} - 64 q^{13} + 32 q^{14} + 16 q^{16} - 99 q^{17} - 49 q^{19} + 20 q^{20} + 114 q^{22} + 198 q^{23} + 25 q^{25} + 128 q^{26} - 64 q^{28} + 66 q^{29} + 146 q^{31} - 32 q^{32} + 198 q^{34} - 80 q^{35} - 28 q^{37} + 98 q^{38} - 40 q^{40} + 411 q^{41} - 223 q^{43} - 228 q^{44} - 396 q^{46} + 132 q^{47} - 87 q^{49} - 50 q^{50} - 256 q^{52} + 654 q^{53} - 285 q^{55} + 128 q^{56} - 132 q^{58} + 33 q^{59} + 458 q^{61} - 292 q^{62} + 64 q^{64} - 320 q^{65} - 385 q^{67} - 396 q^{68} + 160 q^{70} + 642 q^{71} - 247 q^{73} + 56 q^{74} - 196 q^{76} + 912 q^{77} - 106 q^{79} + 80 q^{80} - 822 q^{82} + 324 q^{83} - 495 q^{85} + 446 q^{86} + 456 q^{88} - 414 q^{89} + 1024 q^{91} + 792 q^{92} - 264 q^{94} - 245 q^{95} - 1885 q^{97} + 174 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
−2.00000 0 4.00000 5.00000 0 −16.0000 −8.00000 0 −10.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(5\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 810.4.a.b 1
3.b odd 2 1 810.4.a.e 1
9.c even 3 2 270.4.e.a 2
9.d odd 6 2 90.4.e.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.4.e.a 2 9.d odd 6 2
270.4.e.a 2 9.c even 3 2
810.4.a.b 1 1.a even 1 1 trivial
810.4.a.e 1 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(810))\):

\( T_{7} + 16 \) Copy content Toggle raw display
\( T_{11} + 57 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 2 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T - 5 \) Copy content Toggle raw display
$7$ \( T + 16 \) Copy content Toggle raw display
$11$ \( T + 57 \) Copy content Toggle raw display
$13$ \( T + 64 \) Copy content Toggle raw display
$17$ \( T + 99 \) Copy content Toggle raw display
$19$ \( T + 49 \) Copy content Toggle raw display
$23$ \( T - 198 \) Copy content Toggle raw display
$29$ \( T - 66 \) Copy content Toggle raw display
$31$ \( T - 146 \) Copy content Toggle raw display
$37$ \( T + 28 \) Copy content Toggle raw display
$41$ \( T - 411 \) Copy content Toggle raw display
$43$ \( T + 223 \) Copy content Toggle raw display
$47$ \( T - 132 \) Copy content Toggle raw display
$53$ \( T - 654 \) Copy content Toggle raw display
$59$ \( T - 33 \) Copy content Toggle raw display
$61$ \( T - 458 \) Copy content Toggle raw display
$67$ \( T + 385 \) Copy content Toggle raw display
$71$ \( T - 642 \) Copy content Toggle raw display
$73$ \( T + 247 \) Copy content Toggle raw display
$79$ \( T + 106 \) Copy content Toggle raw display
$83$ \( T - 324 \) Copy content Toggle raw display
$89$ \( T + 414 \) Copy content Toggle raw display
$97$ \( T + 1885 \) Copy content Toggle raw display
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