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Properties

Label 280.4.q.c
Level $280$
Weight $4$
Character orbit 280.q
Analytic conductor $16.521$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [280,4,Mod(81,280)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(280, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("280.81");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 280 = 2^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 280.q (of order \(3\), degree \(2\), 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: \(16.5205348016\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{11} + 97 x^{10} + 136 x^{9} + 6932 x^{8} + 7120 x^{7} + 190192 x^{6} + 97856 x^{5} + \cdots + 199148544 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{14} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{5} + \beta_{3} - 1) q^{3} + 5 \beta_{3} q^{5} + (\beta_{10} + 2 \beta_{3} + \beta_{2} - 1) q^{7} + ( - \beta_{11} - \beta_{10} + \cdots - 4 \beta_1) q^{9} + ( - \beta_{10} - \beta_{9} + \beta_{6} + \cdots + 9) q^{11}+ \cdots + (25 \beta_{11} - 25 \beta_{10} + \cdots - 498) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 7 q^{3} + 30 q^{5} - q^{7} - 39 q^{9} + 51 q^{11} - 54 q^{13} - 70 q^{15} - 6 q^{17} - 9 q^{19} + 123 q^{21} + 72 q^{23} - 150 q^{25} + 1118 q^{27} - 242 q^{29} - 132 q^{31} + 156 q^{33} - 115 q^{35}+ \cdots - 6978 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - x^{11} + 97 x^{10} + 136 x^{9} + 6932 x^{8} + 7120 x^{7} + 190192 x^{6} + 97856 x^{5} + \cdots + 199148544 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 5144955608 \nu^{11} - 63399912935 \nu^{10} - 17165331791883 \nu^{9} + 45225604010075 \nu^{8} + \cdots + 34\!\cdots\!96 ) / 22\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 23266726295 \nu^{11} + 288099243676 \nu^{10} + 1745352572186 \nu^{9} + \cdots + 91\!\cdots\!28 ) / 67\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 79243202735 \nu^{11} - 1106092690847 \nu^{10} + 13009368288059 \nu^{9} + \cdots + 58\!\cdots\!40 ) / 67\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 14826950951 \nu^{11} + 24358089449 \nu^{10} - 1325595039257 \nu^{9} + \cdots + 22\!\cdots\!80 ) / 32\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 18305245152 \nu^{11} + 603360023261 \nu^{10} + 1436579804809 \nu^{9} + \cdots + 25\!\cdots\!92 ) / 37\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 279440238408 \nu^{11} + 383921255709 \nu^{10} + 6378010240201 \nu^{9} + \cdots - 17\!\cdots\!20 ) / 22\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 868684120625 \nu^{11} + 1942888905761 \nu^{10} - 90244855812413 \nu^{9} + \cdots + 61\!\cdots\!12 ) / 33\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 699311292110 \nu^{11} - 3038539372501 \nu^{10} - 56929262300333 \nu^{9} + \cdots + 64\!\cdots\!24 ) / 16\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 2977161987743 \nu^{11} - 10713100087955 \nu^{10} + 314020590874775 \nu^{9} + \cdots - 15\!\cdots\!56 ) / 67\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 1173847773575 \nu^{11} + 1286309184371 \nu^{10} - 118305612912423 \nu^{9} + \cdots - 25\!\cdots\!72 ) / 22\!\cdots\!84 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} + \beta_{10} - \beta_{7} - 2\beta_{5} + 32\beta_{3} + \beta_{2} - 31 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3 \beta_{11} - 3 \beta_{10} - \beta_{9} - 5 \beta_{8} - 11 \beta_{7} - 56 \beta_{5} - 5 \beta_{4} + \cdots - 36 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -83\beta_{11} - 90\beta_{10} - 13\beta_{8} - 7\beta_{7} - 11\beta_{6} - 1613\beta_{3} + 6\beta_{2} - 231\beta _1 + 13 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 602 \beta_{11} - 391 \beta_{10} + 107 \beta_{9} + 800 \beta_{7} - 107 \beta_{6} + 3732 \beta_{5} + \cdots + 5569 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 943 \beta_{11} + 943 \beta_{10} + 1183 \beta_{9} + 1757 \beta_{8} + 8855 \beta_{7} + 23076 \beta_{5} + \cdots + 94192 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 37887 \beta_{11} + 52434 \beta_{10} + 29961 \beta_{8} + 14547 \beta_{7} + 9411 \beta_{6} + 615105 \beta_{3} + \cdots - 29961 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 552202 \beta_{11} + 460771 \beta_{10} - 99543 \beta_{9} - 634792 \beta_{7} + 99543 \beta_{6} + \cdots - 6845173 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1066731 \beta_{11} - 1066731 \beta_{10} - 791131 \beta_{9} - 2221265 \beta_{8} - 6610355 \beta_{7} + \cdots - 46030512 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 35287739 \beta_{11} - 43244826 \beta_{10} - 15457261 \beta_{8} - 7957087 \beta_{7} - 7940495 \beta_{6} + \cdots + 15457261 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 360656498 \beta_{11} - 279055087 \beta_{10} + 65288435 \beta_{9} + 447993608 \beta_{7} + \cdots + 4060966729 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(57\) \(71\) \(141\) \(241\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-\beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
81.1
4.47849 + 7.75697i
2.25227 + 3.90105i
1.45114 + 2.51344i
−1.73749 − 3.00942i
−2.56740 − 4.44687i
−3.37701 − 5.84915i
4.47849 − 7.75697i
2.25227 − 3.90105i
1.45114 − 2.51344i
−1.73749 + 3.00942i
−2.56740 + 4.44687i
−3.37701 + 5.84915i
0 −4.97849 + 8.62300i 0 2.50000 + 4.33013i 0 −14.6186 + 11.3709i 0 −36.0707 − 62.4763i 0
81.2 0 −2.75227 + 4.76708i 0 2.50000 + 4.33013i 0 18.4919 + 1.02408i 0 −1.65001 − 2.85789i 0
81.3 0 −1.95114 + 3.37947i 0 2.50000 + 4.33013i 0 −5.33627 − 17.7348i 0 5.88613 + 10.1951i 0
81.4 0 1.23749 − 2.14339i 0 2.50000 + 4.33013i 0 17.8048 + 5.09810i 0 10.4372 + 18.0778i 0
81.5 0 2.06740 − 3.58085i 0 2.50000 + 4.33013i 0 −17.7560 − 5.26551i 0 4.95169 + 8.57658i 0
81.6 0 2.87701 − 4.98312i 0 2.50000 + 4.33013i 0 0.914156 + 18.4977i 0 −3.05435 − 5.29029i 0
121.1 0 −4.97849 − 8.62300i 0 2.50000 − 4.33013i 0 −14.6186 − 11.3709i 0 −36.0707 + 62.4763i 0
121.2 0 −2.75227 − 4.76708i 0 2.50000 − 4.33013i 0 18.4919 − 1.02408i 0 −1.65001 + 2.85789i 0
121.3 0 −1.95114 − 3.37947i 0 2.50000 − 4.33013i 0 −5.33627 + 17.7348i 0 5.88613 − 10.1951i 0
121.4 0 1.23749 + 2.14339i 0 2.50000 − 4.33013i 0 17.8048 − 5.09810i 0 10.4372 − 18.0778i 0
121.5 0 2.06740 + 3.58085i 0 2.50000 − 4.33013i 0 −17.7560 + 5.26551i 0 4.95169 − 8.57658i 0
121.6 0 2.87701 + 4.98312i 0 2.50000 − 4.33013i 0 0.914156 − 18.4977i 0 −3.05435 + 5.29029i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 81.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 280.4.q.c ✓ 12
4.b odd 2 1 560.4.q.r 12
7.c even 3 1 inner 280.4.q.c ✓ 12
7.c even 3 1 1960.4.a.ba 6
7.d odd 6 1 1960.4.a.x 6
28.g odd 6 1 560.4.q.r 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.4.q.c ✓ 12 1.a even 1 1 trivial
280.4.q.c ✓ 12 7.c even 3 1 inner
560.4.q.r 12 4.b odd 2 1
560.4.q.r 12 28.g odd 6 1
1960.4.a.x 6 7.d odd 6 1
1960.4.a.ba 6 7.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} + 7 T_{3}^{11} + 125 T_{3}^{10} + 136 T_{3}^{9} + 6341 T_{3}^{8} + 4105 T_{3}^{7} + \cdots + 158608836 \) acting on \(S_{4}^{\mathrm{new}}(280, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 158608836 \) Copy content Toggle raw display
$5$ \( (T^{2} - 5 T + 25)^{6} \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 16\!\cdots\!49 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 16\!\cdots\!76 \) Copy content Toggle raw display
$13$ \( (T^{6} + 27 T^{5} + \cdots + 664762112)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 12\!\cdots\!56 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 28\!\cdots\!89 \) Copy content Toggle raw display
$29$ \( (T^{6} + 121 T^{5} + \cdots - 7144947798)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 11\!\cdots\!96 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{6} - 464 T^{5} + \cdots + 445954812315)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots + 62569573723766)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 52\!\cdots\!64 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 25\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 19\!\cdots\!96 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 13\!\cdots\!16 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 56\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 16\!\cdots\!68)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 44\!\cdots\!44 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 96\!\cdots\!04 \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots + 44\!\cdots\!56)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 14\!\cdots\!16 \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots - 25\!\cdots\!52)^{2} \) Copy content Toggle raw display
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