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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [567,2,Mod(188,567)] 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("567.188"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(567, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([1, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 567 = 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 567.o (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,0,0,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.52751779461\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4x^{14} + 154x^{12} + 140x^{10} + 7267x^{8} + 24500x^{6} + 234094x^{4} + 553976x^{2} + 1590121 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{11} q^{2} + \beta_{3} q^{4} + ( - \beta_{15} - \beta_{13}) q^{5} + (\beta_{6} + \beta_{5} + \beta_1 + 1) q^{7} + ( - \beta_{12} - \beta_{11} + \cdots + \beta_{4}) q^{8} + ( - \beta_{10} + \beta_{9} + \cdots + \beta_{2}) q^{10}+ \cdots + (\beta_{13} + 6 \beta_{12} + \cdots - 6 \beta_{4}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{7} + 8 q^{16} - 32 q^{22} - 56 q^{25} - 24 q^{28} - 16 q^{37} + 56 q^{43} + 64 q^{46} + 40 q^{49} + 16 q^{58} + 64 q^{64} - 16 q^{67} - 60 q^{70} + 32 q^{79} - 24 q^{85} - 16 q^{88} - 72 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4x^{14} + 154x^{12} + 140x^{10} + 7267x^{8} + 24500x^{6} + 234094x^{4} + 553976x^{2} + 1590121 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 14207147 \nu^{14} + 2072480170 \nu^{12} - 11399320608 \nu^{10} + 325807318103 \nu^{8} + \cdots + 182701188097693 ) / 44829996281640 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 166693 \nu^{14} + 298535 \nu^{12} - 48061362 \nu^{10} + 184832932 \nu^{8} + \cdots + 171968214527 ) / 232279773480 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 24665523 \nu^{14} - 33315975 \nu^{12} - 742056982 \nu^{10} - 37204237473 \nu^{8} + \cdots - 24837394011308 ) / 14943332093880 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 22561694 \nu^{15} - 1700872689 \nu^{13} + 11753123826 \nu^{11} + \cdots - 203784626227917 \nu ) / 116557990332264 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 28116 \nu^{14} - 276290 \nu^{12} + 5123244 \nu^{10} - 16541379 \nu^{8} + 182158676 \nu^{6} + \cdots - 3112502729 ) / 6084418605 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 83560578 \nu^{14} - 253287725 \nu^{12} + 10429868257 \nu^{10} + 35362683078 \nu^{8} + \cdots + 47977243766463 ) / 14943332093880 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 518550309 \nu^{15} - 13073468210 \nu^{13} + 67917750324 \nu^{11} + \cdots - 27\!\cdots\!74 \nu ) / 582789951661320 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 5359 \nu^{15} - 1260 \nu^{13} + 737016 \nu^{11} + 3163814 \nu^{9} + 43043364 \nu^{7} + \cdots + 5254458594 \nu ) / 4917975660 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 214238363 \nu^{14} - 1471720630 \nu^{12} + 38206912232 \nu^{10} - 85510617342 \nu^{8} + \cdots + 37791050649488 ) / 14943332093880 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 233090094 \nu^{14} + 2032076180 \nu^{12} - 39017931461 \nu^{10} + 122239769296 \nu^{8} + \cdots + 74462133697226 ) / 14943332093880 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 200750180 \nu^{15} - 1689846 \nu^{13} + 22450200096 \nu^{11} + 177097734715 \nu^{9} + \cdots + 108381129665949 \nu ) / 116557990332264 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 1920394 \nu^{15} - 26528820 \nu^{13} + 380309901 \nu^{11} - 2604718426 \nu^{9} + \cdots - 1428717108516 \nu ) / 949169302380 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 8089069 \nu^{15} - 34523285 \nu^{13} + 1249597581 \nu^{11} + 507671954 \nu^{9} + \cdots - 134033666201 \nu ) / 3019637055240 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 15481048 \nu^{15} + 194543770 \nu^{13} - 3387105687 \nu^{11} + 17990857762 \nu^{9} + \cdots + 7960469357062 \nu ) / 3019637055240 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 893669594 \nu^{15} - 4634831275 \nu^{13} + 122946773913 \nu^{11} + \cdots + 441300217925144 \nu ) / 116557990332264 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{15} - \beta_{14} - 2\beta_{13} + \beta_{12} + \beta_{11} + 2\beta_{8} - 2\beta_{7} + \beta_{4} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{10} - \beta_{9} + 5\beta_{6} - 3\beta_{5} + 8\beta_{3} + 3\beta_{2} + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -9\beta_{15} - 6\beta_{14} + 3\beta_{13} + 12\beta_{12} + 5\beta_{11} - 14\beta_{8} - 15\beta_{7} - 4\beta_{4} ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 8\beta_{10} + 14\beta_{9} + 8\beta_{6} - 15\beta_{5} + 14\beta_{3} + 42\beta_{2} - 6\beta _1 - 92 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 16 \beta_{15} + 31 \beta_{14} + 119 \beta_{13} + 92 \beta_{12} - 176 \beta_{11} - 109 \beta_{8} + \cdots - 191 \beta_{4} ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 62\beta_{10} + 31\beta_{9} - 46\beta_{6} + 100\beta_{5} - 159\beta_{3} - 59\beta_{2} - 41\beta _1 - 242 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 901 \beta_{15} + 601 \beta_{14} + 782 \beta_{13} - 1381 \beta_{12} - 1777 \beta_{11} + \cdots + 491 \beta_{4} ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 532 \beta_{10} - 1256 \beta_{9} - 1796 \beta_{6} + 5841 \beta_{5} - 3836 \beta_{3} - 4560 \beta_{2} + \cdots + 3101 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 5931 \beta_{15} + 90 \beta_{14} - 5427 \beta_{13} - 21951 \beta_{12} + 5125 \beta_{11} + \cdots + 18814 \beta_{4} ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 14141 \beta_{10} - 11810 \beta_{9} - 7391 \beta_{6} + 6633 \beta_{5} + 26059 \beta_{3} - 13044 \beta_{2} + \cdots + 91331 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 60556 \beta_{15} - 67189 \beta_{14} - 120779 \beta_{13} + 12673 \beta_{12} + 211364 \beta_{11} + \cdots + 26267 \beta_{4} ) / 3 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 53698 \beta_{10} + 23962 \beta_{9} + 49658 \beta_{6} - 239954 \beta_{5} + 163716 \beta_{3} + \cdots + 46509 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 1034245 \beta_{15} - 314383 \beta_{14} - 234680 \beta_{13} + 2475655 \beta_{12} + \cdots - 1745381 \beta_{4} ) / 3 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 183149 \beta_{10} + 1567757 \beta_{9} + 1995071 \beta_{6} - 5696583 \beta_{5} - 139504 \beta_{3} + \cdots - 8062769 ) / 3 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 134811 \beta_{15} + 5561772 \beta_{14} + 10461687 \beta_{13} + 15789516 \beta_{12} + \cdots - 12256330 \beta_{4} ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/567\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(407\)
\(\chi(n)\) \(-1\) \(-\beta_{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
188.1
−2.55995 − 1.85284i
1.14573 + 1.85284i
−2.30929 + 1.60218i
0.895075 − 1.60218i
−0.895075 + 1.60218i
2.30929 − 1.60218i
−1.14573 − 1.85284i
2.55995 + 1.85284i
−2.55995 + 1.85284i
1.14573 − 1.85284i
−2.30929 − 1.60218i
0.895075 + 1.60218i
−0.895075 − 1.60218i
2.30929 + 1.60218i
−1.14573 + 1.85284i
2.55995 − 1.85284i
−1.67303 − 0.965926i 0 0.866025 + 1.50000i −1.85284 − 3.20921i 0 2.08624 + 1.62714i 0.517638i 0 7.15882i
188.2 −1.67303 − 0.965926i 0 0.866025 + 1.50000i 1.85284 + 3.20921i 0 −2.45227 − 0.993168i 0.517638i 0 − 7.15882i
188.3 −0.448288 − 0.258819i 0 −0.866025 − 1.50000i −1.60218 − 2.77506i 0 −1.27925 − 2.31593i 1.93185i 0 1.65870i
188.4 −0.448288 − 0.258819i 0 −0.866025 − 1.50000i 1.60218 + 2.77506i 0 2.64528 − 0.0500989i 1.93185i 0 − 1.65870i
188.5 0.448288 + 0.258819i 0 −0.866025 − 1.50000i −1.60218 − 2.77506i 0 2.64528 − 0.0500989i − 1.93185i 0 − 1.65870i
188.6 0.448288 + 0.258819i 0 −0.866025 − 1.50000i 1.60218 + 2.77506i 0 −1.27925 − 2.31593i − 1.93185i 0 1.65870i
188.7 1.67303 + 0.965926i 0 0.866025 + 1.50000i −1.85284 − 3.20921i 0 −2.45227 − 0.993168i − 0.517638i 0 − 7.15882i
188.8 1.67303 + 0.965926i 0 0.866025 + 1.50000i 1.85284 + 3.20921i 0 2.08624 + 1.62714i − 0.517638i 0 7.15882i
377.1 −1.67303 + 0.965926i 0 0.866025 − 1.50000i −1.85284 + 3.20921i 0 2.08624 − 1.62714i − 0.517638i 0 − 7.15882i
377.2 −1.67303 + 0.965926i 0 0.866025 − 1.50000i 1.85284 − 3.20921i 0 −2.45227 + 0.993168i − 0.517638i 0 7.15882i
377.3 −0.448288 + 0.258819i 0 −0.866025 + 1.50000i −1.60218 + 2.77506i 0 −1.27925 + 2.31593i − 1.93185i 0 − 1.65870i
377.4 −0.448288 + 0.258819i 0 −0.866025 + 1.50000i 1.60218 − 2.77506i 0 2.64528 + 0.0500989i − 1.93185i 0 1.65870i
377.5 0.448288 − 0.258819i 0 −0.866025 + 1.50000i −1.60218 + 2.77506i 0 2.64528 + 0.0500989i 1.93185i 0 1.65870i
377.6 0.448288 − 0.258819i 0 −0.866025 + 1.50000i 1.60218 − 2.77506i 0 −1.27925 + 2.31593i 1.93185i 0 − 1.65870i
377.7 1.67303 − 0.965926i 0 0.866025 − 1.50000i −1.85284 + 3.20921i 0 −2.45227 + 0.993168i 0.517638i 0 7.15882i
377.8 1.67303 − 0.965926i 0 0.866025 − 1.50000i 1.85284 − 3.20921i 0 2.08624 − 1.62714i 0.517638i 0 − 7.15882i
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 188.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner
21.c even 2 1 inner
63.l odd 6 1 inner
63.o even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 567.2.o.g 16
3.b odd 2 1 inner 567.2.o.g 16
7.b odd 2 1 inner 567.2.o.g 16
9.c even 3 1 567.2.c.b ✓ 8
9.c even 3 1 inner 567.2.o.g 16
9.d odd 6 1 567.2.c.b ✓ 8
9.d odd 6 1 inner 567.2.o.g 16
21.c even 2 1 inner 567.2.o.g 16
63.l odd 6 1 567.2.c.b ✓ 8
63.l odd 6 1 inner 567.2.o.g 16
63.o even 6 1 567.2.c.b ✓ 8
63.o even 6 1 inner 567.2.o.g 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
567.2.c.b ✓ 8 9.c even 3 1
567.2.c.b ✓ 8 9.d odd 6 1
567.2.c.b ✓ 8 63.l odd 6 1
567.2.c.b ✓ 8 63.o even 6 1
567.2.o.g 16 1.a even 1 1 trivial
567.2.o.g 16 3.b odd 2 1 inner
567.2.o.g 16 7.b odd 2 1 inner
567.2.o.g 16 9.c even 3 1 inner
567.2.o.g 16 9.d odd 6 1 inner
567.2.o.g 16 21.c even 2 1 inner
567.2.o.g 16 63.l odd 6 1 inner
567.2.o.g 16 63.o even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(567, [\chi])\):

\( T_{2}^{8} - 4T_{2}^{6} + 15T_{2}^{4} - 4T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{13}^{8} - 42T_{13}^{6} + 1623T_{13}^{4} - 5922T_{13}^{2} + 19881 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} - 4 T^{6} + 15 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} + 24 T^{6} + \cdots + 19881)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} - 2 T^{7} + \cdots + 2401)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} - 16 T^{6} + \cdots + 256)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} - 42 T^{6} + \cdots + 19881)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 72 T^{2} + 1269)^{4} \) Copy content Toggle raw display
$19$ \( T^{16} \) Copy content Toggle raw display
$23$ \( (T^{8} - 64 T^{6} + \cdots + 65536)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} - 52 T^{6} + \cdots + 279841)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} - 48 T^{6} + \cdots + 318096)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 2 T - 11)^{8} \) Copy content Toggle raw display
$41$ \( (T^{8} + 84 T^{6} + \cdots + 318096)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 14 T^{3} + \cdots + 2116)^{4} \) Copy content Toggle raw display
$47$ \( (T^{8} + 108 T^{6} + \cdots + 318096)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 28 T^{2} + 4)^{4} \) Copy content Toggle raw display
$59$ \( (T^{8} + 108 T^{6} + \cdots + 318096)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} - 54 T^{6} + \cdots + 19881)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 4 T^{3} + \cdots + 1936)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} + 28 T^{2} + 4)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} + 162 T^{2} + 1269)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} - 8 T^{3} + \cdots + 1024)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} + 84 T^{6} + \cdots + 318096)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 216 T^{2} + 11421)^{4} \) Copy content Toggle raw display
$97$ \( (T^{8} - 216 T^{6} + \cdots + 5089536)^{2} \) Copy content Toggle raw display
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