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Properties

Label 560.4.q.p
Level $560$
Weight $4$
Character orbit 560.q
Analytic conductor $33.041$
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] = [560,4,Mod(81,560)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(560, 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("560.81");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 560.q (of order \(3\), degree \(2\), not 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: \(33.0410696032\)
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: \( x^{12} - 2 x^{11} + 95 x^{10} - 258 x^{9} + 7289 x^{8} - 15564 x^{7} + 170984 x^{6} + 88720 x^{5} + \cdots + 6718464 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: no (minimal twist has level 280)
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.

\(f(q)\) \(=\) \( q + (\beta_{3} - \beta_1 - 1) q^{3} + 5 \beta_{3} q^{5} + (\beta_{9} + 3 \beta_{3} + \beta_1) q^{7} + (\beta_{9} + \beta_{8} + \beta_{7} + \cdots + 1) q^{9} + ( - \beta_{8} + \beta_{5} + \beta_{4} + \cdots - 2) q^{11}+ \cdots + ( - 3 \beta_{11} - 12 \beta_{10} + \cdots - 21) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 8 q^{3} + 30 q^{5} + 14 q^{7} - 34 q^{9} - 240 q^{13} - 80 q^{15} + 122 q^{17} - 2 q^{19} + 170 q^{21} + 30 q^{23} - 150 q^{25} + 64 q^{27} + 476 q^{29} - 122 q^{31} + 584 q^{33} - 100 q^{35} + 342 q^{37}+ \cdots + 216 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} + 95 x^{10} - 258 x^{9} + 7289 x^{8} - 15564 x^{7} + 170984 x^{6} + 88720 x^{5} + \cdots + 6718464 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 267705222294445 \nu^{11} + \cdots + 82\!\cdots\!16 ) / 23\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 10\!\cdots\!79 \nu^{11} + \cdots + 20\!\cdots\!28 ) / 92\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 15217828912144 \nu^{11} - 8836236713621 \nu^{10} + \cdots + 68\!\cdots\!32 ) / 85\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 12\!\cdots\!94 \nu^{11} + \cdots + 43\!\cdots\!52 ) / 23\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 59\!\cdots\!01 \nu^{11} + \cdots + 44\!\cdots\!24 ) / 92\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 62\!\cdots\!81 \nu^{11} + \cdots - 56\!\cdots\!76 ) / 92\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 11\!\cdots\!63 \nu^{11} + \cdots - 11\!\cdots\!36 ) / 11\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 57\!\cdots\!95 \nu^{11} + \cdots + 20\!\cdots\!32 ) / 23\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 335263802565144 \nu^{11} - 235819143081583 \nu^{10} + \cdots - 35\!\cdots\!28 ) / 12\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 10\!\cdots\!93 \nu^{11} + \cdots + 19\!\cdots\!44 ) / 92\!\cdots\!04 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} + \beta_{8} + \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} - 32\beta_{3} - 2\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{11} + 3\beta_{10} - 3\beta_{9} - 3\beta_{8} - 4\beta_{6} + 60\beta_{4} + 3\beta_{3} - \beta_{2} + 32 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 71 \beta_{10} + 15 \beta_{9} - 62 \beta_{8} - 71 \beta_{7} + 76 \beta_{6} - 14 \beta_{5} + \cdots - 1847 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 69 \beta_{11} + 84 \beta_{9} + 327 \beta_{8} + 351 \beta_{7} - 84 \beta_{6} + 396 \beta_{5} - 3664 \beta_{4} + \cdots + 327 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 9 \beta_{11} + 4987 \beta_{10} - 5581 \beta_{9} - 1539 \beta_{8} - 1530 \beta_{6} - 4042 \beta_{5} + \cdots + 115942 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 32019 \beta_{10} + 18039 \beta_{9} - 11928 \beta_{8} - 32019 \beta_{7} + 34612 \beta_{6} + \cdots - 573971 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 5231 \beta_{11} + 276254 \beta_{9} + 408389 \beta_{8} + 364103 \beta_{7} - 276254 \beta_{6} + \cdots + 408389 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 315309 \beta_{11} + 2697495 \beta_{10} - 2579607 \beta_{9} - 1356579 \beta_{8} - 1671888 \beta_{6} + \cdots + 49203216 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 27378739 \beta_{10} + 10805643 \beta_{9} - 19582930 \beta_{8} - 27378739 \beta_{7} + 31178404 \beta_{6} + \cdots - 621474251 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 21802933 \beta_{11} + 110900976 \beta_{9} + 214648479 \beta_{8} + 219532347 \beta_{7} - 110900976 \beta_{6} + \cdots + 214648479 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/560\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(337\) \(351\) \(421\)
\(\chi(n)\) \(-\beta_{3}\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
81.1
3.54223 − 6.13533i
3.22348 − 5.58323i
0.889448 − 1.54057i
−0.531189 + 0.920046i
−1.69507 + 2.93594i
−4.42891 + 7.67109i
3.54223 + 6.13533i
3.22348 + 5.58323i
0.889448 + 1.54057i
−0.531189 − 0.920046i
−1.69507 − 2.93594i
−4.42891 − 7.67109i
0 −4.04223 + 7.00136i 0 2.50000 + 4.33013i 0 −18.4932 + 0.999858i 0 −19.1793 − 33.2196i 0
81.2 0 −3.72348 + 6.44926i 0 2.50000 + 4.33013i 0 15.4269 − 10.2474i 0 −14.2286 − 24.6447i 0
81.3 0 −1.38945 + 2.40659i 0 2.50000 + 4.33013i 0 −2.01227 + 18.4106i 0 9.63887 + 16.6950i 0
81.4 0 0.0311889 − 0.0540207i 0 2.50000 + 4.33013i 0 16.7481 + 7.90586i 0 13.4981 + 23.3793i 0
81.5 0 1.19507 − 2.06992i 0 2.50000 + 4.33013i 0 3.21331 − 18.2394i 0 10.6436 + 18.4353i 0
81.6 0 3.92891 − 6.80507i 0 2.50000 + 4.33013i 0 −7.88276 + 16.7589i 0 −17.3726 − 30.0903i 0
401.1 0 −4.04223 − 7.00136i 0 2.50000 − 4.33013i 0 −18.4932 − 0.999858i 0 −19.1793 + 33.2196i 0
401.2 0 −3.72348 − 6.44926i 0 2.50000 − 4.33013i 0 15.4269 + 10.2474i 0 −14.2286 + 24.6447i 0
401.3 0 −1.38945 − 2.40659i 0 2.50000 − 4.33013i 0 −2.01227 − 18.4106i 0 9.63887 − 16.6950i 0
401.4 0 0.0311889 + 0.0540207i 0 2.50000 − 4.33013i 0 16.7481 − 7.90586i 0 13.4981 − 23.3793i 0
401.5 0 1.19507 + 2.06992i 0 2.50000 − 4.33013i 0 3.21331 + 18.2394i 0 10.6436 − 18.4353i 0
401.6 0 3.92891 + 6.80507i 0 2.50000 − 4.33013i 0 −7.88276 − 16.7589i 0 −17.3726 + 30.0903i 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 560.4.q.p 12
4.b odd 2 1 280.4.q.e ✓ 12
7.c even 3 1 inner 560.4.q.p 12
28.f even 6 1 1960.4.a.bb 6
28.g odd 6 1 280.4.q.e ✓ 12
28.g odd 6 1 1960.4.a.w 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.4.q.e ✓ 12 4.b odd 2 1
280.4.q.e ✓ 12 28.g odd 6 1
560.4.q.p 12 1.a even 1 1 trivial
560.4.q.p 12 7.c even 3 1 inner
1960.4.a.w 6 28.g odd 6 1
1960.4.a.bb 6 28.f even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(560, [\chi])\):

\( T_{3}^{12} + 8 T_{3}^{11} + 130 T_{3}^{10} + 560 T_{3}^{9} + 8447 T_{3}^{8} + 34856 T_{3}^{7} + \cdots + 38416 \) Copy content Toggle raw display
\( T_{11}^{12} + 6101 T_{11}^{10} + 211064 T_{11}^{9} + 27970393 T_{11}^{8} + 961017372 T_{11}^{7} + \cdots + 68\!\cdots\!64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 8 T^{11} + \cdots + 38416 \) 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
$11$ \( T^{12} + \cdots + 68\!\cdots\!64 \) Copy content Toggle raw display
$13$ \( (T^{6} + 120 T^{5} + \cdots + 8623951872)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 14\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 10\!\cdots\!36 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 13\!\cdots\!09 \) Copy content Toggle raw display
$29$ \( (T^{6} - 238 T^{5} + \cdots + 931883943676)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 26\!\cdots\!16 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 46\!\cdots\!56 \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots - 2860161422643)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots - 44576110032948)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 63\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 54\!\cdots\!04 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 99\!\cdots\!56 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 69\!\cdots\!84 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 17\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots - 15\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 15\!\cdots\!96 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 61\!\cdots\!44 \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots - 28\!\cdots\!52)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 25\!\cdots\!24 \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots + 32\!\cdots\!68)^{2} \) Copy content Toggle raw display
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