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Properties

Label 280.2.b.c
Level $280$
Weight $2$
Character orbit 280.b
Analytic conductor $2.236$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [280,2,Mod(141,280)]
 
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, 1, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("280.141");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 280 = 2^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 280.b (of order \(2\), degree \(1\), 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: \(2.23581125660\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.18939904.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 14x^{6} - 28x^{5} + 43x^{4} - 44x^{3} + 30x^{2} - 12x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{6} q^{2} + (\beta_{7} - \beta_{5}) q^{3} + (\beta_{7} - \beta_{5} + \beta_{4} + \cdots - 1) q^{4}+ \cdots - 2 \beta_1 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{6} q^{2} + (\beta_{7} - \beta_{5}) q^{3} + (\beta_{7} - \beta_{5} + \beta_{4} + \cdots - 1) q^{4}+ \cdots + ( - 2 \beta_{7} - 4 \beta_{6} + \cdots + 4 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{4} - 8 q^{7} + 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{4} - 8 q^{7} + 12 q^{8} - 4 q^{10} - 12 q^{12} - 8 q^{15} - 8 q^{17} + 8 q^{18} - 4 q^{20} - 4 q^{22} - 8 q^{23} + 20 q^{24} - 8 q^{25} - 20 q^{26} + 4 q^{28} + 4 q^{30} - 8 q^{31} + 8 q^{33} + 4 q^{34} + 16 q^{36} + 16 q^{38} - 32 q^{39} + 4 q^{40} + 24 q^{41} + 20 q^{44} + 24 q^{48} + 8 q^{49} - 12 q^{52} + 8 q^{54} - 8 q^{55} - 12 q^{56} - 8 q^{57} - 32 q^{58} + 12 q^{60} - 24 q^{62} + 8 q^{64} - 16 q^{65} + 4 q^{66} - 20 q^{68} + 4 q^{70} + 16 q^{71} + 16 q^{72} + 32 q^{73} + 16 q^{74} + 8 q^{76} - 4 q^{78} + 48 q^{79} - 8 q^{80} - 8 q^{81} - 32 q^{82} + 12 q^{84} + 24 q^{86} - 32 q^{87} + 4 q^{88} + 56 q^{89} - 8 q^{90} - 8 q^{94} - 8 q^{95} - 48 q^{96} + 24 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 14x^{6} - 28x^{5} + 43x^{4} - 44x^{3} + 30x^{2} - 12x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{6} - 3\nu^{5} + 10\nu^{4} - 15\nu^{3} + 19\nu^{2} - 12\nu + 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -2\nu^{7} + 7\nu^{6} - 24\nu^{5} + 42\nu^{4} - 59\nu^{3} + 48\nu^{2} - 24\nu + 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -3\nu^{7} + 10\nu^{6} - 35\nu^{5} + 60\nu^{4} - 87\nu^{3} + 73\nu^{2} - 42\nu + 11 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -7\nu^{7} + 25\nu^{6} - 87\nu^{5} + 158\nu^{4} - 231\nu^{3} + 206\nu^{2} - 118\nu + 31 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -8\nu^{7} + 28\nu^{6} - 98\nu^{5} + 175\nu^{4} - 256\nu^{3} + 223\nu^{2} - 126\nu + 31 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( 8\nu^{7} - 28\nu^{6} + 98\nu^{5} - 175\nu^{4} + 257\nu^{3} - 224\nu^{2} + 130\nu - 32 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( 10\nu^{7} - 35\nu^{6} + 123\nu^{5} - 220\nu^{4} + 325\nu^{3} - 285\nu^{2} + 166\nu - 42 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{7} + 3\beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{7} + \beta_{6} + 5\beta_{5} - 3\beta_{4} + 5\beta_{3} - 3\beta_{2} + 4\beta _1 - 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 12\beta_{7} - 11\beta_{6} + 11\beta_{5} - 5\beta_{4} - \beta_{3} - 9\beta_{2} + 2\beta _1 + 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 6\beta_{7} - 17\beta_{6} - 13\beta_{5} + 13\beta_{4} - 23\beta_{3} + 3\beta_{2} - 18\beta _1 + 21 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -46\beta_{7} + 29\beta_{6} - 67\beta_{5} + 37\beta_{4} - 15\beta_{3} + 47\beta_{2} - 24\beta _1 - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -76\beta_{7} + 105\beta_{6} - 7\beta_{5} - 31\beta_{4} + 91\beta_{3} + 39\beta_{2} + 68\beta _1 - 83 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/280\mathbb{Z}\right)^\times\).

\(n\) \(57\) \(71\) \(141\) \(241\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
141.1
0.500000 + 0.0297061i
0.500000 − 0.0297061i
0.500000 + 0.691860i
0.500000 − 0.691860i
0.500000 + 1.44392i
0.500000 − 1.44392i
0.500000 − 2.10607i
0.500000 + 2.10607i
−0.874559 − 1.11137i 2.41421i −0.470294 + 1.94392i 1.00000i 2.68309 − 2.11137i −1.00000 2.57172 − 1.17740i −2.82843 1.11137 − 0.874559i
141.2 −0.874559 + 1.11137i − 2.41421i −0.470294 − 1.94392i − 1.00000i 2.68309 + 2.11137i −1.00000 2.57172 + 1.17740i −2.82843 1.11137 + 0.874559i
141.3 −0.635665 − 1.26330i 0.414214i −1.19186 + 1.60607i − 1.00000i 0.523276 − 0.263301i −1.00000 2.78658 + 0.484753i 2.82843 −1.26330 + 0.635665i
141.4 −0.635665 + 1.26330i − 0.414214i −1.19186 − 1.60607i 1.00000i 0.523276 + 0.263301i −1.00000 2.78658 − 0.484753i 2.82843 −1.26330 − 0.635665i
141.5 0.167452 − 1.40426i − 2.41421i −1.94392 − 0.470294i − 1.00000i −3.39020 − 0.404265i −1.00000 −0.985930 + 2.65103i −2.82843 −1.40426 − 0.167452i
141.6 0.167452 + 1.40426i 2.41421i −1.94392 + 0.470294i 1.00000i −3.39020 + 0.404265i −1.00000 −0.985930 − 2.65103i −2.82843 −1.40426 + 0.167452i
141.7 1.34277 − 0.443806i 0.414214i 1.60607 − 1.19186i − 1.00000i 0.183830 + 0.556194i −1.00000 1.62764 − 2.31318i 2.82843 −0.443806 − 1.34277i
141.8 1.34277 + 0.443806i − 0.414214i 1.60607 + 1.19186i 1.00000i 0.183830 − 0.556194i −1.00000 1.62764 + 2.31318i 2.82843 −0.443806 + 1.34277i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 141.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 280.2.b.c ✓ 8
4.b odd 2 1 1120.2.b.c 8
8.b even 2 1 inner 280.2.b.c ✓ 8
8.d odd 2 1 1120.2.b.c 8
16.e even 4 1 8960.2.a.bt 4
16.e even 4 1 8960.2.a.bu 4
16.f odd 4 1 8960.2.a.bs 4
16.f odd 4 1 8960.2.a.bv 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.b.c ✓ 8 1.a even 1 1 trivial
280.2.b.c ✓ 8 8.b even 2 1 inner
1120.2.b.c 8 4.b odd 2 1
1120.2.b.c 8 8.d odd 2 1
8960.2.a.bs 4 16.f odd 4 1
8960.2.a.bt 4 16.e even 4 1
8960.2.a.bu 4 16.e even 4 1
8960.2.a.bv 4 16.f odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(280, [\chi])\):

\( T_{3}^{4} + 6T_{3}^{2} + 1 \) Copy content Toggle raw display
\( T_{13}^{8} + 52T_{13}^{6} + 166T_{13}^{4} + 52T_{13}^{2} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 2 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( (T^{4} + 6 T^{2} + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$7$ \( (T + 1)^{8} \) Copy content Toggle raw display
$11$ \( T^{8} + 52 T^{6} + \cdots + 1681 \) Copy content Toggle raw display
$13$ \( T^{8} + 52 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( (T^{4} + 4 T^{3} - 22 T^{2} + \cdots + 41)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + 120 T^{6} + \cdots + 602176 \) Copy content Toggle raw display
$23$ \( (T^{4} + 4 T^{3} - 20 T^{2} + \cdots - 56)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + 156 T^{6} + \cdots + 160801 \) Copy content Toggle raw display
$31$ \( (T^{4} + 4 T^{3} - 44 T^{2} + \cdots - 56)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + 88 T^{6} + \cdots + 61504 \) Copy content Toggle raw display
$41$ \( (T^{4} - 12 T^{3} + \cdots + 392)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 104 T^{6} + \cdots + 18496 \) Copy content Toggle raw display
$47$ \( (T^{4} - 74 T^{2} + \cdots + 721)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + 128 T^{6} + \cdots + 4096 \) Copy content Toggle raw display
$59$ \( T^{8} + 416 T^{6} + \cdots + 66064384 \) Copy content Toggle raw display
$61$ \( T^{8} + 424 T^{6} + \cdots + 47004736 \) Copy content Toggle raw display
$67$ \( T^{8} + 128 T^{6} + \cdots + 4096 \) Copy content Toggle raw display
$71$ \( (T^{4} - 8 T^{3} + \cdots + 3728)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 16 T^{3} + \cdots + 2048)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 24 T^{3} + \cdots - 1751)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 176 T^{6} + \cdots + 12544 \) Copy content Toggle raw display
$89$ \( (T^{4} - 28 T^{3} + \cdots - 8696)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 12 T^{3} + \cdots - 10287)^{2} \) Copy content Toggle raw display
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