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Properties

Label 1584.4.d.a
Level $1584$
Weight $4$
Character orbit 1584.d
Analytic conductor $93.459$
Analytic rank $0$
Dimension $10$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1584,4,Mod(287,1584)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1584.287"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1584, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1584 = 2^{4} \cdot 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1584.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,0,0,0,0,0,0,0,0,0,-110] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(93.4590254491\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2 x^{9} - 363 x^{8} + 504 x^{7} + 45321 x^{6} - 45330 x^{5} - 2130421 x^{4} + 2465240 x^{3} + \cdots + 97834851 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{5} + (\beta_{7} - \beta_{4}) q^{7} - 11 q^{11} + ( - \beta_{2} + 9) q^{13} + ( - \beta_{8} + 2 \beta_{7} + \cdots - 2 \beta_1) q^{17} + (\beta_{9} - \beta_{8} + \cdots + 4 \beta_1) q^{19}+ \cdots + ( - 4 \beta_{6} + 9 \beta_{5} + \cdots - 136) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 110 q^{11} + 88 q^{13} + 168 q^{23} - 250 q^{25} - 720 q^{35} + 524 q^{37} - 576 q^{47} - 1442 q^{49} + 120 q^{59} - 412 q^{61} - 1392 q^{71} + 952 q^{73} - 3528 q^{83} + 1884 q^{85} - 4896 q^{95}+ \cdots - 1352 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2 x^{9} - 363 x^{8} + 504 x^{7} + 45321 x^{6} - 45330 x^{5} - 2130421 x^{4} + 2465240 x^{3} + \cdots + 97834851 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 1071166502 \nu^{9} - 67566855003 \nu^{8} + 557324170153 \nu^{7} + 20278103944215 \nu^{6} + \cdots + 15\!\cdots\!47 ) / 63\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 6425097861 \nu^{9} + 68645714979 \nu^{8} - 2224174102429 \nu^{7} - 33557131283370 \nu^{6} + \cdots + 23\!\cdots\!54 ) / 63\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 153917089 \nu^{9} - 1565200479 \nu^{8} - 55918328871 \nu^{7} + 412012132320 \nu^{6} + \cdots - 56\!\cdots\!04 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2846511697 \nu^{9} - 4621856892 \nu^{8} - 965716891008 \nu^{7} + 877317725135 \nu^{6} + \cdots - 50\!\cdots\!17 ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 2846511697 \nu^{9} + 4621856892 \nu^{8} + 965716891008 \nu^{7} - 877317725135 \nu^{6} + \cdots + 40\!\cdots\!17 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 10607772899 \nu^{9} - 49047683661 \nu^{8} + 4190156428211 \nu^{7} + \cdots - 12\!\cdots\!86 ) / 31\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 6539549349 \nu^{9} - 22244010489 \nu^{8} - 2225885508761 \nu^{7} + 5847757094970 \nu^{6} + \cdots - 78\!\cdots\!14 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 16973196597 \nu^{9} + 158881396833 \nu^{8} - 6680050583983 \nu^{7} - 55474768222740 \nu^{6} + \cdots - 98\!\cdots\!92 ) / 31\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 67801181941 \nu^{9} + 327742606776 \nu^{8} + 26636388278924 \nu^{7} + \cdots + 34\!\cdots\!01 ) / 63\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} + 2\beta_{4} + 1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} + 2\beta_{3} - 4\beta _1 + 147 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3 \beta_{9} - 6 \beta_{8} - 33 \beta_{7} - 19 \beta_{6} + 61 \beta_{5} + 241 \beta_{4} + 17 \beta_{3} + \cdots + 195 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 16 \beta_{9} - 88 \beta_{8} - 24 \beta_{7} + 122 \beta_{6} + 11 \beta_{5} + 160 \beta_{4} + \cdots + 16934 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 450 \beta_{9} - 3000 \beta_{8} - 18900 \beta_{7} - 6163 \beta_{6} + 15456 \beta_{5} + 99998 \beta_{4} + \cdots + 107282 ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 2212 \beta_{9} - 12226 \beta_{8} - 5508 \beta_{7} + 6871 \beta_{6} + 2971 \beta_{5} + 38446 \beta_{4} + \cdots + 1039538 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 28056 \beta_{9} - 651252 \beta_{8} - 3904656 \beta_{7} - 746442 \beta_{6} + 1879147 \beta_{5} + \cdots + 19993189 ) / 6 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 851680 \beta_{9} - 4984000 \beta_{8} - 3382240 \beta_{7} + 1329795 \beta_{6} + 1363846 \beta_{5} + \cdots + 244088103 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 107751 \beta_{9} - 64927392 \beta_{8} - 353179671 \beta_{7} - 38099908 \beta_{6} + 103178212 \beta_{5} + \cdots + 1512913902 ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1584\mathbb{Z}\right)^\times\).

\(n\) \(145\) \(353\) \(991\) \(1189\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
287.1
12.5681 + 1.41421i
−11.6113 − 1.41421i
−6.70099 − 1.41421i
6.07168 + 1.41421i
0.672515 + 1.41421i
0.672515 − 1.41421i
6.07168 − 1.41421i
−6.70099 + 1.41421i
−11.6113 + 1.41421i
12.5681 − 1.41421i
0 0 0 − 17.7740i 0 − 31.8907i 0 0 0
287.2 0 0 0 − 16.4209i 0 23.4972i 0 0 0
287.3 0 0 0 − 9.47663i 0 − 26.7130i 0 0 0
287.4 0 0 0 − 8.58665i 0 7.54616i 0 0 0
287.5 0 0 0 − 0.951079i 0 9.81498i 0 0 0
287.6 0 0 0 0.951079i 0 − 9.81498i 0 0 0
287.7 0 0 0 8.58665i 0 − 7.54616i 0 0 0
287.8 0 0 0 9.47663i 0 26.7130i 0 0 0
287.9 0 0 0 16.4209i 0 − 23.4972i 0 0 0
287.10 0 0 0 17.7740i 0 31.8907i 0 0 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 287.10
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1584.4.d.a ✓ 10
3.b odd 2 1 1584.4.d.b yes 10
4.b odd 2 1 1584.4.d.b yes 10
12.b even 2 1 inner 1584.4.d.a ✓ 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1584.4.d.a ✓ 10 1.a even 1 1 trivial
1584.4.d.a ✓ 10 12.b even 2 1 inner
1584.4.d.b yes 10 3.b odd 2 1
1584.4.d.b yes 10 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(1584, [\chi])\):

\( T_{5}^{10} + 750T_{5}^{8} + 188244T_{5}^{6} + 17977736T_{5}^{4} + 580155312T_{5}^{2} + 510209568 \) Copy content Toggle raw display
\( T_{23}^{5} - 84T_{23}^{4} - 28458T_{23}^{3} + 3480352T_{23}^{2} - 122628660T_{23} + 1364421168 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots + 510209568 \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 2198050284672 \) Copy content Toggle raw display
$11$ \( (T + 11)^{10} \) Copy content Toggle raw display
$13$ \( (T^{5} - 44 T^{4} + \cdots - 10131264)^{2} \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 33\!\cdots\!68 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 68\!\cdots\!08 \) Copy content Toggle raw display
$23$ \( (T^{5} - 84 T^{4} + \cdots + 1364421168)^{2} \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 34\!\cdots\!28 \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{5} - 262 T^{4} + \cdots - 274492992096)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 27\!\cdots\!48 \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 97\!\cdots\!68 \) Copy content Toggle raw display
$47$ \( (T^{5} + 288 T^{4} + \cdots - 150940104480)^{2} \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 11\!\cdots\!88 \) Copy content Toggle raw display
$59$ \( (T^{5} + \cdots + 1452973420800)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} + \cdots - 1067838555400)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 10\!\cdots\!32 \) Copy content Toggle raw display
$71$ \( (T^{5} + \cdots + 2320492147776)^{2} \) Copy content Toggle raw display
$73$ \( (T^{5} + \cdots + 3915334934080)^{2} \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 37\!\cdots\!48 \) Copy content Toggle raw display
$83$ \( (T^{5} + \cdots - 377508064374144)^{2} \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 15\!\cdots\!32 \) Copy content Toggle raw display
$97$ \( (T^{5} + \cdots - 1852064912000)^{2} \) Copy content Toggle raw display
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