Properties

Label 280.6.a.f
Level $280$
Weight $6$
Character orbit 280.a
Self dual yes
Analytic conductor $44.907$
Analytic rank $0$
Dimension $3$
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] = [280,6,Mod(1,280)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(280, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("280.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 280 = 2^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 280.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: \(44.9074695476\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.996509.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 267x + 1100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} + 5) q^{3} - 25 q^{5} + 49 q^{7} + (8 \beta_{2} + 3 \beta_1 + 4) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + 5) q^{3} - 25 q^{5} + 49 q^{7} + (8 \beta_{2} + 3 \beta_1 + 4) q^{9} + (18 \beta_{2} + 5 \beta_1 - 9) q^{11} + (\beta_{2} + 6 \beta_1 + 181) q^{13} + ( - 25 \beta_{2} - 125) q^{15} + (21 \beta_{2} - 28 \beta_1 + 87) q^{17} + (45 \beta_{2} - 17 \beta_1 - 340) q^{19} + (49 \beta_{2} + 245) q^{21} + (29 \beta_{2} - 17 \beta_1 - 188) q^{23} + 625 q^{25} + ( - 107 \beta_{2} + 42 \beta_1 + 563) q^{27} + (82 \beta_{2} + \beta_1 + 2085) q^{29} + (177 \beta_{2} + 49 \beta_1 - 726) q^{31} + (225 \beta_{2} + 84 \beta_1 + 3921) q^{33} - 1225 q^{35} + (451 \beta_{2} - 157 \beta_1 + 2700) q^{37} + (400 \beta_{2} + 39 \beta_1 + 1091) q^{39} + (277 \beta_{2} + 191 \beta_1 + 4206) q^{41} + ( - 333 \beta_{2} - 203 \beta_1 + 5444) q^{43} + ( - 200 \beta_{2} - 75 \beta_1 - 100) q^{45} + ( - 141 \beta_{2} + 58 \beta_1 + 10049) q^{47} + 2401 q^{49} + ( - 858 \beta_{2} - 105 \beta_1 + 5265) q^{51} + (8 \beta_{2} + 2 \beta_1 + 8624) q^{53} + ( - 450 \beta_{2} - 125 \beta_1 + 225) q^{55} + ( - 817 \beta_{2} + 33 \beta_1 + 8392) q^{57} + ( - 1708 \beta_{2} + 84 \beta_1 + 11576) q^{59} + (1675 \beta_{2} + 137 \beta_1 + 2782) q^{61} + (392 \beta_{2} + 147 \beta_1 + 196) q^{63} + ( - 25 \beta_{2} - 150 \beta_1 - 4525) q^{65} + ( - 1348 \beta_{2} + 532 \beta_1 + 14964) q^{67} + ( - 713 \beta_{2} - 15 \beta_1 + 5600) q^{69} + ( - 504 \beta_{2} - 504 \beta_1 + 18816) q^{71} + (718 \beta_{2} - 298 \beta_1 - 5642) q^{73} + (625 \beta_{2} + 3125) q^{75} + (882 \beta_{2} + 245 \beta_1 - 441) q^{77} + ( - 1938 \beta_{2} + 531 \beta_1 + 6605) q^{79} + ( - 190 \beta_{2} - 798 \beta_1 - 22163) q^{81} + ( - 2258 \beta_{2} - 966 \beta_1 + 41864) q^{83} + ( - 525 \beta_{2} + 700 \beta_1 - 2175) q^{85} + (2367 \beta_{2} + 252 \beta_1 + 28623) q^{87} + ( - 3923 \beta_{2} + 31 \beta_1 + 42798) q^{89} + (49 \beta_{2} + 294 \beta_1 + 8869) q^{91} + (1569 \beta_{2} + 825 \beta_1 + 35370) q^{93} + ( - 1125 \beta_{2} + 425 \beta_1 + 8500) q^{95} + (161 \beta_{2} - 924 \beta_1 - 25037) q^{97} + (3246 \beta_{2} - 36 \beta_1 + 71238) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 14 q^{3} - 75 q^{5} + 147 q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 14 q^{3} - 75 q^{5} + 147 q^{7} + q^{9} - 50 q^{11} + 536 q^{13} - 350 q^{15} + 268 q^{17} - 1048 q^{19} + 686 q^{21} - 576 q^{23} + 1875 q^{25} + 1754 q^{27} + 6172 q^{29} - 2404 q^{31} + 11454 q^{33} - 3675 q^{35} + 7806 q^{37} + 2834 q^{39} + 12150 q^{41} + 16868 q^{43} - 25 q^{45} + 30230 q^{47} + 7203 q^{49} + 16758 q^{51} + 25862 q^{53} + 1250 q^{55} + 25960 q^{57} + 36352 q^{59} + 6534 q^{61} + 49 q^{63} - 13400 q^{65} + 45708 q^{67} + 17528 q^{69} + 57456 q^{71} - 17346 q^{73} + 8750 q^{75} - 2450 q^{77} + 21222 q^{79} - 65501 q^{81} + 128816 q^{83} - 6700 q^{85} + 83250 q^{87} + 132286 q^{89} + 26264 q^{91} + 103716 q^{93} + 26200 q^{95} - 74348 q^{97} + 210504 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 267x + 1100 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{2} + 21\nu + 169 ) / 7 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} + 7\nu - 183 ) / 7 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta _1 + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 21\beta_{2} - 7\beta _1 + 718 ) / 4 \) 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
4.35887
−17.6538
14.2949
0 −14.0697 0 −25.0000 0 49.0000 0 −45.0423 0
1.2 0 5.72563 0 −25.0000 0 49.0000 0 −210.217 0
1.3 0 22.3441 0 −25.0000 0 49.0000 0 256.259 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(1\)
\(7\) \(-1\)

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 280.6.a.f 3
4.b odd 2 1 560.6.a.r 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.6.a.f 3 1.a even 1 1 trivial
560.6.a.r 3 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{3} - 14T_{3}^{2} - 267T_{3} + 1800 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(280))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} - 14 T^{2} - 267 T + 1800 \) Copy content Toggle raw display
$5$ \( (T + 25)^{3} \) Copy content Toggle raw display
$7$ \( (T - 49)^{3} \) Copy content Toggle raw display
$11$ \( T^{3} + 50 T^{2} - 201871 T - 32091540 \) Copy content Toggle raw display
$13$ \( T^{3} - 536 T^{2} + \cdots + 40540670 \) Copy content Toggle raw display
$17$ \( T^{3} - 268 T^{2} + \cdots - 1677666906 \) Copy content Toggle raw display
$19$ \( T^{3} + 1048 T^{2} + \cdots - 339889184 \) Copy content Toggle raw display
$23$ \( T^{3} + 576 T^{2} + \cdots - 470543360 \) Copy content Toggle raw display
$29$ \( T^{3} - 6172 T^{2} + \cdots - 4080790758 \) Copy content Toggle raw display
$31$ \( T^{3} + 2404 T^{2} + \cdots - 42412572288 \) Copy content Toggle raw display
$37$ \( T^{3} - 7806 T^{2} + \cdots + 754109894424 \) Copy content Toggle raw display
$41$ \( T^{3} - 12150 T^{2} + \cdots + 859615169760 \) Copy content Toggle raw display
$43$ \( T^{3} - 16868 T^{2} + \cdots + 770989825360 \) Copy content Toggle raw display
$47$ \( T^{3} - 30230 T^{2} + \cdots - 822133181192 \) Copy content Toggle raw display
$53$ \( T^{3} - 25862 T^{2} + \cdots - 640341284960 \) Copy content Toggle raw display
$59$ \( T^{3} - 36352 T^{2} + \cdots + 2921311866880 \) Copy content Toggle raw display
$61$ \( T^{3} - 6534 T^{2} + \cdots - 5471960282336 \) Copy content Toggle raw display
$67$ \( T^{3} - 45708 T^{2} + \cdots + 17305695495360 \) Copy content Toggle raw display
$71$ \( T^{3} - 57456 T^{2} + \cdots + 5500862115840 \) Copy content Toggle raw display
$73$ \( T^{3} + 17346 T^{2} + \cdots - 2212406609000 \) Copy content Toggle raw display
$79$ \( T^{3} - 21222 T^{2} + \cdots - 14445274057688 \) Copy content Toggle raw display
$83$ \( T^{3} + \cdots + 193963320099072 \) Copy content Toggle raw display
$89$ \( T^{3} + \cdots + 107630146696800 \) Copy content Toggle raw display
$97$ \( T^{3} + \cdots - 147037236625098 \) Copy content Toggle raw display
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