Properties

Label 280.2.b.d
Level $280$
Weight $2$
Character orbit 280.b
Analytic conductor $2.236$
Analytic rank $0$
Dimension $12$
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: \(12\)
Coefficient field: 12.0.8272021826830336.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 3x^{10} - 4x^{9} + 4x^{8} - 12x^{7} + 10x^{6} - 24x^{5} + 16x^{4} - 32x^{3} + 48x^{2} + 64 \) 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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{6} q^{2} + (\beta_{7} - \beta_{3} + \beta_1) q^{3} + (\beta_{8} - \beta_{6} - \beta_1) q^{4} - \beta_{7} q^{5} + (\beta_{10} + 2 \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} + \beta_{2} + \beta_1 + 1) q^{6} + q^{7} + ( - \beta_{10} + \beta_{7} + \beta_{4} - \beta_{2} + 1) q^{8} + (\beta_{11} - \beta_{9} + \beta_{6} - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{2} + (\beta_{7} - \beta_{3} + \beta_1) q^{3} + (\beta_{8} - \beta_{6} - \beta_1) q^{4} - \beta_{7} q^{5} + (\beta_{10} + 2 \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} + \beta_{2} + \beta_1 + 1) q^{6} + q^{7} + ( - \beta_{10} + \beta_{7} + \beta_{4} - \beta_{2} + 1) q^{8} + (\beta_{11} - \beta_{9} + \beta_{6} - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{9} - \beta_{4} q^{10} + (\beta_{11} - \beta_{10} + \beta_{9} - \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} - \beta_{2}) q^{11} + ( - \beta_{11} + 2 \beta_{10} - \beta_{9} + \beta_{7} + \beta_{6} - \beta_{4} - \beta_{3} + \beta_1) q^{12} + ( - \beta_{11} + \beta_{7} - \beta_{3} + \beta_{2} + \beta_1) q^{13} + \beta_{6} q^{14} + ( - \beta_{10} - \beta_{5} + 1) q^{15} + ( - \beta_{10} - \beta_{5} + \beta_{4} + \beta_{2} - \beta_1 + 1) q^{16} + (\beta_{10} - \beta_{9} - \beta_{8} + 2 \beta_{6} + \beta_{5} + 2 \beta_{3} + 2 \beta_1 + 1) q^{17} + ( - \beta_{11} + 2 \beta_{10} + \beta_{9} - \beta_{8} + \beta_{7} + 2 \beta_{5} + \beta_{4} + \beta_{3} + \cdots - 2) q^{18}+ \cdots + ( - 4 \beta_{11} + 6 \beta_{10} - 2 \beta_{9} + 2 \beta_{8} + 12 \beta_{7} + \cdots + 4 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 2 q^{2} + 6 q^{4} + 12 q^{7} + 10 q^{8} - 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 2 q^{2} + 6 q^{4} + 12 q^{7} + 10 q^{8} - 20 q^{9} + 16 q^{12} - 2 q^{14} + 2 q^{16} - 2 q^{18} + 4 q^{20} + 12 q^{22} + 8 q^{23} - 24 q^{24} - 12 q^{25} + 6 q^{28} + 12 q^{30} + 24 q^{31} - 2 q^{32} - 24 q^{33} - 20 q^{34} - 18 q^{36} + 12 q^{38} - 48 q^{39} + 12 q^{40} - 16 q^{41} + 16 q^{44} - 48 q^{46} - 16 q^{47} + 20 q^{48} + 12 q^{49} + 2 q^{50} + 4 q^{52} + 44 q^{54} - 8 q^{55} + 10 q^{56} + 40 q^{57} + 4 q^{58} - 8 q^{60} + 8 q^{62} - 20 q^{63} - 6 q^{64} + 8 q^{65} + 64 q^{66} - 56 q^{68} - 32 q^{71} - 46 q^{72} - 8 q^{73} - 32 q^{74} - 12 q^{76} - 24 q^{78} + 8 q^{80} + 60 q^{81} - 28 q^{82} + 16 q^{84} - 76 q^{86} + 48 q^{87} - 40 q^{88} - 48 q^{89} + 24 q^{90} + 12 q^{94} + 28 q^{96} + 32 q^{97} - 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 3x^{10} - 4x^{9} + 4x^{8} - 12x^{7} + 10x^{6} - 24x^{5} + 16x^{4} - 32x^{3} + 48x^{2} + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 2 \nu^{11} - 5 \nu^{10} + 7 \nu^{9} - 17 \nu^{8} + 43 \nu^{7} - 30 \nu^{6} + 92 \nu^{5} - 62 \nu^{4} + 170 \nu^{3} - 140 \nu^{2} + 64 \nu - 384 ) / 136 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 9 \nu^{11} + 116 \nu^{10} - 57 \nu^{9} + 136 \nu^{8} - 440 \nu^{7} + 356 \nu^{6} - 822 \nu^{5} + 704 \nu^{4} - 1496 \nu^{3} + 1344 \nu^{2} - 560 \nu + 3360 ) / 544 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{11} + 3\nu^{9} - 4\nu^{8} + 4\nu^{7} - 12\nu^{6} + 10\nu^{5} - 24\nu^{4} + 16\nu^{3} + 48\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{11} - 3\nu^{9} + 4\nu^{8} - 4\nu^{7} + 12\nu^{6} - 10\nu^{5} + 24\nu^{4} - 16\nu^{3} + 32\nu^{2} - 48\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 23 \nu^{11} + 66 \nu^{10} + 13 \nu^{9} + 34 \nu^{8} - 180 \nu^{7} + 124 \nu^{6} - 242 \nu^{5} - 52 \nu^{4} - 816 \nu^{3} + 80 \nu^{2} + 80 \nu + 2240 ) / 544 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 15 \nu^{11} - 29 \nu^{10} - 7 \nu^{9} + 17 \nu^{8} + 42 \nu^{7} - 4 \nu^{6} + 78 \nu^{5} + 62 \nu^{4} + 340 \nu^{3} - 200 \nu^{2} - 336 \nu - 976 ) / 272 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 29 \nu^{11} - 38 \nu^{10} + 43 \nu^{9} - 102 \nu^{8} + 184 \nu^{7} - 228 \nu^{6} + 298 \nu^{5} - 580 \nu^{4} + 680 \nu^{3} - 384 \nu^{2} + 976 \nu - 960 ) / 544 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 19 \nu^{11} - 22 \nu^{10} + 7 \nu^{9} + 34 \nu^{8} + 60 \nu^{7} + 4 \nu^{6} + 58 \nu^{5} + 108 \nu^{4} + 272 \nu^{3} - 208 \nu^{2} - 752 \nu - 928 ) / 272 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 19 \nu^{11} - 22 \nu^{10} + 7 \nu^{9} + 34 \nu^{8} + 60 \nu^{7} + 4 \nu^{6} + 58 \nu^{5} + 108 \nu^{4} + 272 \nu^{3} - 208 \nu^{2} - 208 \nu - 928 ) / 272 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 45 \nu^{11} - 32 \nu^{10} + 55 \nu^{9} - 204 \nu^{8} + 180 \nu^{7} - 260 \nu^{6} + 242 \nu^{5} - 696 \nu^{4} + 272 \nu^{3} - 80 \nu^{2} + 1552 \nu + 480 ) / 544 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 93 \nu^{11} + 20 \nu^{10} + 91 \nu^{9} - 272 \nu^{8} + 32 \nu^{7} - 356 \nu^{6} + 210 \nu^{5} - 704 \nu^{4} - 136 \nu^{3} + 16 \nu^{2} + 2192 \nu + 2080 ) / 544 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{9} - \beta_{8} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + \beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + \beta_{6} + \beta_{2} + \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{11} - \beta_{10} + \beta_{9} - \beta_{6} - \beta_{5} - \beta_{3} + 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{11} - 2\beta_{10} - \beta_{8} - \beta_{7} + \beta_{5} + 3\beta_{4} + 2\beta_{3} - 2\beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2\beta_{10} + 2\beta_{9} - \beta_{8} + \beta_{7} + 3\beta_{6} + 4\beta_{5} + 4\beta_{4} + \beta_{2} + 3\beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -\beta_{11} + \beta_{9} + 2\beta_{8} + 3\beta_{7} - 8\beta_{6} - 3\beta_{5} - \beta_{4} - 4\beta_{3} + 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -6\beta_{10} + 2\beta_{9} - \beta_{8} + 5\beta_{7} - 5\beta_{6} + 2\beta_{4} + 2\beta_{3} - 3\beta_{2} - 5\beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 3 \beta_{11} + 4 \beta_{10} + 9 \beta_{9} + 2 \beta_{8} - 3 \beta_{7} + 6 \beta_{6} + 9 \beta_{5} + 7 \beta_{4} + 8 \beta_{3} + 2 \beta_{2} + 2 \beta _1 - 5 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 2 \beta_{11} - 4 \beta_{10} - 4 \beta_{9} + 3 \beta_{8} + 5 \beta_{7} - 11 \beta_{6} - 6 \beta_{5} - 6 \beta_{4} - 4 \beta_{3} + 5 \beta_{2} + 11 \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 9 \beta_{11} - 16 \beta_{10} + \beta_{9} + 4 \beta_{8} + 23 \beta_{7} - 10 \beta_{6} - \beta_{5} + 9 \beta_{4} - 12 \beta_{3} - 2 \beta_{2} - 6 \beta _1 + 1 \) Copy content Toggle raw display

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
−0.258252 1.39043i
−0.258252 + 1.39043i
0.832593 + 1.14315i
0.832593 1.14315i
−1.11909 0.864661i
−1.11909 + 0.864661i
1.39608 + 0.225774i
1.39608 0.225774i
−0.722588 + 1.21568i
−0.722588 1.21568i
−0.128739 + 1.40834i
−0.128739 1.40834i
−1.39043 0.258252i 0.861041i 1.86661 + 0.718165i 1.00000i 0.222366 1.19722i 1.00000 −2.40993 1.48062i 2.25861 0.258252 1.39043i
141.2 −1.39043 + 0.258252i 0.861041i 1.86661 0.718165i 1.00000i 0.222366 + 1.19722i 1.00000 −2.40993 + 1.48062i 2.25861 0.258252 + 1.39043i
141.3 −1.14315 0.832593i 3.42822i 0.613577 + 1.90356i 1.00000i −2.85431 + 3.91897i 1.00000 0.883477 2.68691i −8.75270 −0.832593 + 1.14315i
141.4 −1.14315 + 0.832593i 3.42822i 0.613577 1.90356i 1.00000i −2.85431 3.91897i 1.00000 0.883477 + 2.68691i −8.75270 −0.832593 1.14315i
141.5 −0.864661 1.11909i 0.903031i −0.504724 + 1.93527i 1.00000i −1.01057 + 0.780815i 1.00000 2.60215 1.10852i 2.18454 1.11909 0.864661i
141.6 −0.864661 + 1.11909i 0.903031i −0.504724 1.93527i 1.00000i −1.01057 0.780815i 1.00000 2.60215 + 1.10852i 2.18454 1.11909 + 0.864661i
141.7 −0.225774 1.39608i 2.07981i −1.89805 + 0.630396i 1.00000i 2.90357 0.469568i 1.00000 1.30861 + 2.50750i −1.32561 −1.39608 + 0.225774i
141.8 −0.225774 + 1.39608i 2.07981i −1.89805 0.630396i 1.00000i 2.90357 + 0.469568i 1.00000 1.30861 2.50750i −1.32561 −1.39608 0.225774i
141.9 1.21568 0.722588i 1.52755i 0.955734 1.75686i 1.00000i 1.10379 + 1.85701i 1.00000 −0.107626 2.82638i 0.666582 0.722588 + 1.21568i
141.10 1.21568 + 0.722588i 1.52755i 0.955734 + 1.75686i 1.00000i 1.10379 1.85701i 1.00000 −0.107626 + 2.82638i 0.666582 0.722588 1.21568i
141.11 1.40834 0.128739i 2.83397i 1.96685 0.362616i 1.00000i −0.364842 3.99120i 1.00000 2.72332 0.763897i −5.03141 0.128739 + 1.40834i
141.12 1.40834 + 0.128739i 2.83397i 1.96685 + 0.362616i 1.00000i −0.364842 + 3.99120i 1.00000 2.72332 + 0.763897i −5.03141 0.128739 1.40834i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 141.12
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.d 12
4.b odd 2 1 1120.2.b.d 12
8.b even 2 1 inner 280.2.b.d 12
8.d odd 2 1 1120.2.b.d 12
16.e even 4 1 8960.2.a.ca 6
16.e even 4 1 8960.2.a.cf 6
16.f odd 4 1 8960.2.a.cd 6
16.f odd 4 1 8960.2.a.cg 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.b.d 12 1.a even 1 1 trivial
280.2.b.d 12 8.b even 2 1 inner
1120.2.b.d 12 4.b odd 2 1
1120.2.b.d 12 8.d odd 2 1
8960.2.a.ca 6 16.e even 4 1
8960.2.a.cd 6 16.f odd 4 1
8960.2.a.cf 6 16.e even 4 1
8960.2.a.cg 6 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 \(S_{2}^{\mathrm{new}}(280, [\chi])\):

\( T_{3}^{12} + 28T_{3}^{10} + 278T_{3}^{8} + 1212T_{3}^{6} + 2385T_{3}^{4} + 1984T_{3}^{2} + 576 \) Copy content Toggle raw display
\( T_{13}^{12} + 76T_{13}^{10} + 1958T_{13}^{8} + 19860T_{13}^{6} + 65713T_{13}^{4} + 77896T_{13}^{2} + 26896 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 2 T^{11} - T^{10} - 8 T^{9} + \cdots + 64 \) Copy content Toggle raw display
$3$ \( T^{12} + 28 T^{10} + 278 T^{8} + \cdots + 576 \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$7$ \( (T - 1)^{12} \) Copy content Toggle raw display
$11$ \( T^{12} + 76 T^{10} + 2086 T^{8} + \cdots + 138384 \) Copy content Toggle raw display
$13$ \( T^{12} + 76 T^{10} + 1958 T^{8} + \cdots + 26896 \) Copy content Toggle raw display
$17$ \( (T^{6} - 74 T^{4} + 44 T^{3} + 729 T^{2} + \cdots - 396)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + 56 T^{10} + 928 T^{8} + \cdots + 4096 \) Copy content Toggle raw display
$23$ \( (T^{6} - 4 T^{5} - 68 T^{4} + 352 T^{3} + \cdots - 768)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + 124 T^{10} + 4038 T^{8} + \cdots + 123904 \) Copy content Toggle raw display
$31$ \( (T^{6} - 12 T^{5} + 20 T^{4} + 176 T^{3} + \cdots + 1152)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + 280 T^{10} + \cdots + 435306496 \) Copy content Toggle raw display
$41$ \( (T^{6} + 8 T^{5} - 64 T^{4} - 480 T^{3} + \cdots + 7200)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + 416 T^{10} + 59696 T^{8} + \cdots + 6718464 \) Copy content Toggle raw display
$47$ \( (T^{6} + 8 T^{5} - 194 T^{4} + \cdots - 224128)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + 384 T^{10} + \cdots + 6379536384 \) Copy content Toggle raw display
$59$ \( T^{12} + 352 T^{10} + \cdots + 17314349056 \) Copy content Toggle raw display
$61$ \( T^{12} + 208 T^{10} + \cdots + 70829056 \) Copy content Toggle raw display
$67$ \( T^{12} + 504 T^{10} + \cdots + 18939904 \) Copy content Toggle raw display
$71$ \( (T^{6} + 16 T^{5} - 104 T^{4} - 2784 T^{3} + \cdots + 8704)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + 4 T^{5} - 204 T^{4} - 1472 T^{3} + \cdots + 8192)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} - 214 T^{4} - 1176 T^{3} + \cdots + 90784)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + 240 T^{10} + \cdots + 46022656 \) Copy content Toggle raw display
$89$ \( (T^{6} + 24 T^{5} - 80 T^{4} - 4864 T^{3} + \cdots - 51808)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} - 16 T^{5} - 66 T^{4} + 2172 T^{3} + \cdots + 38068)^{2} \) Copy content Toggle raw display
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