Properties

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

Related objects

Downloads

Learn more

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: \(4\)
Coefficient field: 4.4.8657424.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 125x^{2} + 126x + 3726 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3^{4} \)
Twist minimal: yes
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)
 

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.

\(f(q)\) \(=\) \( q + 2 q^{2} + 4 q^{4} - 5 q^{5} + (\beta_{3} + \beta_1 + 1) q^{7} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} + 4 q^{4} - 5 q^{5} + (\beta_{3} + \beta_1 + 1) q^{7} + 8 q^{8} - 10 q^{10} + (\beta_{2} - 9) q^{11} + ( - \beta_{3} - \beta_{2} + 2 \beta_1 + 9) q^{13} + (2 \beta_{3} + 2 \beta_1 + 2) q^{14} + 16 q^{16} + ( - 2 \beta_{3} - \beta_{2} - \beta_1 + 22) q^{17} + (4 \beta_{3} + 41) q^{19} - 20 q^{20} + (2 \beta_{2} - 18) q^{22} + (2 \beta_{2} - 5 \beta_1 + 8) q^{23} + 25 q^{25} + ( - 2 \beta_{3} - 2 \beta_{2} + 4 \beta_1 + 18) q^{26} + (4 \beta_{3} + 4 \beta_1 + 4) q^{28} + (2 \beta_{3} - \beta_{2} + 9) q^{29} + ( - 4 \beta_{3} + 2 \beta_{2} - 5 \beta_1 + 109) q^{31} + 32 q^{32} + ( - 4 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 44) q^{34} + ( - 5 \beta_{3} - 5 \beta_1 - 5) q^{35} + (12 \beta_{3} - \beta_{2} + \beta_1 + 172) q^{37} + (8 \beta_{3} + 82) q^{38} - 40 q^{40} + ( - 11 \beta_{3} - 6 \beta_{2} + 2 \beta_1 + 28) q^{41} + ( - 10 \beta_{3} + 7 \beta_{2} - 7 \beta_1 + 114) q^{43} + (4 \beta_{2} - 36) q^{44} + (4 \beta_{2} - 10 \beta_1 + 16) q^{46} + (8 \beta_{3} - 5 \beta_{2} - 2 \beta_1 + 200) q^{47} + ( - 14 \beta_{3} - 12 \beta_{2} + \beta_1 + 335) q^{49} + 50 q^{50} + ( - 4 \beta_{3} - 4 \beta_{2} + 8 \beta_1 + 36) q^{52} + (7 \beta_{2} + 8 \beta_1 + 46) q^{53} + ( - 5 \beta_{2} + 45) q^{55} + (8 \beta_{3} + 8 \beta_1 + 8) q^{56} + (4 \beta_{3} - 2 \beta_{2} + 18) q^{58} + (9 \beta_{3} + 2 \beta_{2} + 20 \beta_1 + 202) q^{59} + ( - 30 \beta_{3} - 6 \beta_{2} + 2 \beta_1 + 150) q^{61} + ( - 8 \beta_{3} + 4 \beta_{2} - 10 \beta_1 + 218) q^{62} + 64 q^{64} + (5 \beta_{3} + 5 \beta_{2} - 10 \beta_1 - 45) q^{65} + ( - 4 \beta_{3} + 18 \beta_{2} - 106) q^{67} + ( - 8 \beta_{3} - 4 \beta_{2} - 4 \beta_1 + 88) q^{68} + ( - 10 \beta_{3} - 10 \beta_1 - 10) q^{70} + (6 \beta_{3} + 3 \beta_{2} + 87) q^{71} + (18 \beta_{3} + 9 \beta_{2} - 17 \beta_1 - 32) q^{73} + (24 \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 344) q^{74} + (16 \beta_{3} + 164) q^{76} + ( - 76 \beta_{3} + 5 \beta_{2} - 27 \beta_1 + 144) q^{77} + (26 \beta_{3} + 4 \beta_{2} + 16 \beta_1 + 280) q^{79} - 80 q^{80} + ( - 22 \beta_{3} - 12 \beta_{2} + 4 \beta_1 + 56) q^{82} + (70 \beta_{3} + 5 \beta_{2} + 11 \beta_1 + 136) q^{83} + (10 \beta_{3} + 5 \beta_{2} + 5 \beta_1 - 110) q^{85} + ( - 20 \beta_{3} + 14 \beta_{2} - 14 \beta_1 + 228) q^{86} + (8 \beta_{2} - 72) q^{88} + ( - 22 \beta_{3} - 21 \beta_{2} + 10 \beta_1 + 287) q^{89} + (58 \beta_{3} - 2 \beta_{2} + 27 \beta_1 + 724) q^{91} + (8 \beta_{2} - 20 \beta_1 + 32) q^{92} + (16 \beta_{3} - 10 \beta_{2} - 4 \beta_1 + 400) q^{94} + ( - 20 \beta_{3} - 205) q^{95} + (16 \beta_{3} - 9 \beta_{2} - 27 \beta_1 + 608) q^{97} + ( - 28 \beta_{3} - 24 \beta_{2} + 2 \beta_1 + 670) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{2} + 16 q^{4} - 20 q^{5} + 2 q^{7} + 32 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{2} + 16 q^{4} - 20 q^{5} + 2 q^{7} + 32 q^{8} - 40 q^{10} - 36 q^{11} + 32 q^{13} + 4 q^{14} + 64 q^{16} + 90 q^{17} + 164 q^{19} - 80 q^{20} - 72 q^{22} + 42 q^{23} + 100 q^{25} + 64 q^{26} + 8 q^{28} + 36 q^{29} + 446 q^{31} + 128 q^{32} + 180 q^{34} - 10 q^{35} + 686 q^{37} + 328 q^{38} - 160 q^{40} + 108 q^{41} + 470 q^{43} - 144 q^{44} + 84 q^{46} + 804 q^{47} + 1338 q^{49} + 200 q^{50} + 128 q^{52} + 168 q^{53} + 180 q^{55} + 16 q^{56} + 72 q^{58} + 768 q^{59} + 596 q^{61} + 892 q^{62} + 256 q^{64} - 160 q^{65} - 424 q^{67} + 360 q^{68} - 20 q^{70} + 348 q^{71} - 94 q^{73} + 1372 q^{74} + 656 q^{76} + 630 q^{77} + 1088 q^{79} - 320 q^{80} + 216 q^{82} + 522 q^{83} - 450 q^{85} + 940 q^{86} - 288 q^{88} + 1128 q^{89} + 2842 q^{91} + 168 q^{92} + 1608 q^{94} - 820 q^{95} + 2486 q^{97} + 2676 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 125x^{2} + 126x + 3726 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{2} + 8\nu - 69 ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 64\nu - 63 ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{2} + \nu + 63 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + 3\beta _1 + 6 ) / 9 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -8\beta_{3} + 3\beta _1 + 573 ) / 9 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 64\beta_{3} + 27\beta_{2} + 192\beta _1 + 951 ) / 9 \) 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
−8.37910
−6.40373
9.37910
7.40373
2.00000 0 4.00000 −5.00000 0 −36.5296 8.00000 0 −10.0000
1.2 2.00000 0 4.00000 −5.00000 0 −9.81890 8.00000 0 −10.0000
1.3 2.00000 0 4.00000 −5.00000 0 16.7450 8.00000 0 −10.0000
1.4 2.00000 0 4.00000 −5.00000 0 31.6035 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.u yes 4
3.b odd 2 1 810.4.a.t 4
9.c even 3 2 810.4.e.bg 8
9.d odd 6 2 810.4.e.bh 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
810.4.a.t 4 3.b odd 2 1
810.4.a.u yes 4 1.a even 1 1 trivial
810.4.e.bg 8 9.c even 3 2
810.4.e.bh 8 9.d odd 6 2

Hecke kernels

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

\( T_{7}^{4} - 2T_{7}^{3} - 1353T_{7}^{2} + 7186T_{7} + 189814 \) Copy content Toggle raw display
\( T_{11}^{4} + 36T_{11}^{3} - 3051T_{11}^{2} - 73872T_{11} + 2137428 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 2)^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T + 5)^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 2 T^{3} - 1353 T^{2} + \cdots + 189814 \) Copy content Toggle raw display
$11$ \( T^{4} + 36 T^{3} - 3051 T^{2} + \cdots + 2137428 \) Copy content Toggle raw display
$13$ \( T^{4} - 32 T^{3} - 6303 T^{2} + \cdots - 3726728 \) Copy content Toggle raw display
$17$ \( T^{4} - 90 T^{3} - 3150 T^{2} + \cdots + 769176 \) Copy content Toggle raw display
$19$ \( (T^{2} - 82 T - 2207)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 42 T^{3} - 31959 T^{2} + \cdots + 50706324 \) Copy content Toggle raw display
$29$ \( T^{4} - 36 T^{3} - 5967 T^{2} + \cdots - 1248048 \) Copy content Toggle raw display
$31$ \( T^{4} - 446 T^{3} + \cdots - 680893352 \) Copy content Toggle raw display
$37$ \( T^{4} - 686 T^{3} + 100950 T^{2} + \cdots - 6422144 \) Copy content Toggle raw display
$41$ \( T^{4} - 108 T^{3} + \cdots + 1284131637 \) Copy content Toggle raw display
$43$ \( T^{4} - 470 T^{3} + \cdots - 10201923248 \) Copy content Toggle raw display
$47$ \( T^{4} - 804 T^{3} + \cdots - 9720884034 \) Copy content Toggle raw display
$53$ \( T^{4} - 168 T^{3} + \cdots + 1171497654 \) Copy content Toggle raw display
$59$ \( T^{4} - 768 T^{3} + \cdots + 26021284425 \) Copy content Toggle raw display
$61$ \( T^{4} - 596 T^{3} + \cdots + 3959540608 \) Copy content Toggle raw display
$67$ \( T^{4} + 424 T^{3} + \cdots + 218849906176 \) Copy content Toggle raw display
$71$ \( T^{4} - 348 T^{3} + \cdots - 247505868 \) Copy content Toggle raw display
$73$ \( T^{4} + 94 T^{3} + \cdots - 38907781064 \) Copy content Toggle raw display
$79$ \( T^{4} - 1088 T^{3} + \cdots - 104956713200 \) Copy content Toggle raw display
$83$ \( T^{4} - 522 T^{3} + \cdots + 598397219352 \) Copy content Toggle raw display
$89$ \( T^{4} - 1128 T^{3} + \cdots + 211674255696 \) Copy content Toggle raw display
$97$ \( T^{4} - 2486 T^{3} + \cdots - 1403907776672 \) Copy content Toggle raw display
show more
show less