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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [605,2,Mod(81,605)] 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("605.81"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(605, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 605 = 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 605.g (of order \(5\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,4,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.83094932229\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.159390625.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 6x^{6} - 11x^{5} + 21x^{4} - 5x^{3} + 10x^{2} + 25x + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 55)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{4} + \beta_{2} + \beta_1) q^{2} + \beta_1 q^{3} + (\beta_{7} + 2 \beta_{6} + \beta_{5} + \cdots - 2) q^{4} - \beta_{6} q^{5} + (\beta_{6} - \beta_{5} + \beta_{4} + \cdots - 2) q^{6} + ( - 2 \beta_{6} + \beta_{5} - 3 \beta_{3} + \cdots + 2) q^{7}+ \cdots + ( - 7 \beta_{6} + 2 \beta_{5} + \cdots + 13) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{2} + q^{3} - 6 q^{4} - 2 q^{5} - 13 q^{6} + 3 q^{7} + 2 q^{8} - 5 q^{9} - 6 q^{10} - 28 q^{12} - 4 q^{13} + 16 q^{14} + q^{15} - 20 q^{16} - q^{17} - 14 q^{18} + q^{19} - q^{20} + 12 q^{21}+ \cdots + 84 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 6x^{6} - 11x^{5} + 21x^{4} - 5x^{3} + 10x^{2} + 25x + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 555\nu^{7} - 2159\nu^{6} + 7489\nu^{5} - 18164\nu^{4} + 40069\nu^{3} - 84434\nu^{2} + 43855\nu + 375 ) / 94655 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -970\nu^{7} - 1002\nu^{6} - 6608\nu^{5} + 9063\nu^{4} - 14943\nu^{3} + 27673\nu^{2} - 68120\nu + 35160 ) / 94655 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -1604\nu^{7} + 4159\nu^{6} - 12059\nu^{5} + 28414\nu^{4} - 81659\nu^{3} + 38305\nu^{2} - 13500\nu - 13875 ) / 94655 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -2052\nu^{7} + 2252\nu^{6} - 19912\nu^{5} + 21007\nu^{4} - 82042\nu^{3} + 35785\nu^{2} - 19395\nu - 90925 ) / 94655 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -2667\nu^{7} + 6691\nu^{6} - 17466\nu^{5} + 50856\nu^{4} - 82441\nu^{3} + 72554\nu^{2} - 4035\nu - 12035 ) / 94655 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 4024\nu^{7} - 1464\nu^{6} + 21519\nu^{5} - 26434\nu^{4} + 59219\nu^{3} + 22635\nu^{2} + 54640\nu + 66675 ) / 94655 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{7} - \beta_{6} - \beta_{5} - \beta_{3} - 3\beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{6} + \beta_{5} - 4\beta_{4} - \beta_{3} - \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 7\beta_{7} + 7\beta_{6} + 2\beta_{5} + 13\beta_{3} + 13\beta_{2} - 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -8\beta_{7} - 11\beta_{6} - 20\beta_{5} + 20\beta_{4} - 11\beta_{2} + 8\beta _1 - 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -19\beta_{7} - 19\beta_{4} - 68\beta_{3} - 36\beta_{2} - 24\beta _1 + 36 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 111\beta_{7} + 81\beta_{6} + 111\beta_{5} - 55\beta_{4} + 81\beta_{3} + 148\beta_{2} - 56\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/605\mathbb{Z}\right)^\times\).

\(n\) \(122\) \(486\)
\(\chi(n)\) \(1\) \(-1 + \beta_{2} + \beta_{3} + \beta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
81.1
0.453245 − 1.39494i
−0.762262 + 2.34600i
−0.628998 + 0.456994i
1.43801 − 1.04478i
0.453245 + 1.39494i
−0.762262 − 2.34600i
−0.628998 − 0.456994i
1.43801 + 1.04478i
0.0756511 + 0.0549637i 0.453245 − 1.39494i −0.615332 − 1.89380i −0.809017 + 0.587785i 0.110960 − 0.0806171i −1.39815 − 4.30308i 0.115332 − 0.354955i 0.686611 + 0.498852i −0.0935099
81.2 2.04238 + 1.48388i −0.762262 + 2.34600i 1.35140 + 4.15918i −0.809017 + 0.587785i −5.03801 + 3.66033i −0.646930 − 1.99105i −1.85140 + 5.69802i −2.49563 − 1.81318i −2.52452
251.1 −0.697759 + 2.14748i −0.628998 + 0.456994i −2.50678 − 1.82128i 0.309017 + 0.951057i −0.542497 − 1.66963i 0.100294 + 0.0728678i 2.00678 − 1.45801i −0.740256 + 2.27827i −2.25800
251.2 0.579725 − 1.78421i 1.43801 − 1.04478i −1.22929 − 0.893133i 0.309017 + 0.951057i −1.03045 − 3.17141i 3.44479 + 2.50279i 0.729292 − 0.529862i 0.0492728 − 0.151646i 1.87603
366.1 0.0756511 − 0.0549637i 0.453245 + 1.39494i −0.615332 + 1.89380i −0.809017 − 0.587785i 0.110960 + 0.0806171i −1.39815 + 4.30308i 0.115332 + 0.354955i 0.686611 − 0.498852i −0.0935099
366.2 2.04238 − 1.48388i −0.762262 − 2.34600i 1.35140 − 4.15918i −0.809017 − 0.587785i −5.03801 − 3.66033i −0.646930 + 1.99105i −1.85140 − 5.69802i −2.49563 + 1.81318i −2.52452
511.1 −0.697759 − 2.14748i −0.628998 − 0.456994i −2.50678 + 1.82128i 0.309017 − 0.951057i −0.542497 + 1.66963i 0.100294 − 0.0728678i 2.00678 + 1.45801i −0.740256 − 2.27827i −2.25800
511.2 0.579725 + 1.78421i 1.43801 + 1.04478i −1.22929 + 0.893133i 0.309017 − 0.951057i −1.03045 + 3.17141i 3.44479 − 2.50279i 0.729292 + 0.529862i 0.0492728 + 0.151646i 1.87603
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 81.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 605.2.g.n 8
11.b odd 2 1 55.2.g.a ✓ 8
11.c even 5 1 605.2.a.i 4
11.c even 5 2 605.2.g.g 8
11.c even 5 1 inner 605.2.g.n 8
11.d odd 10 1 55.2.g.a ✓ 8
11.d odd 10 1 605.2.a.l 4
11.d odd 10 2 605.2.g.j 8
33.d even 2 1 495.2.n.f 8
33.f even 10 1 495.2.n.f 8
33.f even 10 1 5445.2.a.bg 4
33.h odd 10 1 5445.2.a.bu 4
44.c even 2 1 880.2.bo.e 8
44.g even 10 1 880.2.bo.e 8
44.g even 10 1 9680.2.a.cs 4
44.h odd 10 1 9680.2.a.cv 4
55.d odd 2 1 275.2.h.b 8
55.e even 4 2 275.2.z.b 16
55.h odd 10 1 275.2.h.b 8
55.h odd 10 1 3025.2.a.v 4
55.j even 10 1 3025.2.a.be 4
55.l even 20 2 275.2.z.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.g.a ✓ 8 11.b odd 2 1
55.2.g.a ✓ 8 11.d odd 10 1
275.2.h.b 8 55.d odd 2 1
275.2.h.b 8 55.h odd 10 1
275.2.z.b 16 55.e even 4 2
275.2.z.b 16 55.l even 20 2
495.2.n.f 8 33.d even 2 1
495.2.n.f 8 33.f even 10 1
605.2.a.i 4 11.c even 5 1
605.2.a.l 4 11.d odd 10 1
605.2.g.g 8 11.c even 5 2
605.2.g.j 8 11.d odd 10 2
605.2.g.n 8 1.a even 1 1 trivial
605.2.g.n 8 11.c even 5 1 inner
880.2.bo.e 8 44.c even 2 1
880.2.bo.e 8 44.g even 10 1
3025.2.a.v 4 55.h odd 10 1
3025.2.a.be 4 55.j even 10 1
5445.2.a.bg 4 33.f even 10 1
5445.2.a.bu 4 33.h odd 10 1
9680.2.a.cs 4 44.g even 10 1
9680.2.a.cv 4 44.h odd 10 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(605, [\chi])\):

\( T_{2}^{8} - 4T_{2}^{7} + 13T_{2}^{6} - 30T_{2}^{5} + 71T_{2}^{4} - 90T_{2}^{3} + 127T_{2}^{2} - 18T_{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{8} - T_{3}^{7} + 6T_{3}^{6} - 11T_{3}^{5} + 21T_{3}^{4} - 5T_{3}^{3} + 10T_{3}^{2} + 25T_{3} + 25 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 4 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{8} - T^{7} + \cdots + 25 \) Copy content Toggle raw display
$5$ \( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} - 3 T^{7} + \cdots + 25 \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} + 4 T^{7} + \cdots + 121 \) Copy content Toggle raw display
$17$ \( T^{8} + T^{7} + \cdots + 121 \) Copy content Toggle raw display
$19$ \( T^{8} - T^{7} + \cdots + 625 \) Copy content Toggle raw display
$23$ \( (T^{4} + 9 T^{3} + \cdots - 1669)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + 19 T^{7} + \cdots + 3025 \) Copy content Toggle raw display
$31$ \( T^{8} - 6 T^{7} + \cdots + 10201 \) Copy content Toggle raw display
$37$ \( T^{8} - 4 T^{7} + \cdots + 22801 \) Copy content Toggle raw display
$41$ \( T^{8} - 4 T^{7} + \cdots + 249001 \) Copy content Toggle raw display
$43$ \( (T^{4} + 21 T^{3} + \cdots - 59)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 4 T^{7} + \cdots + 5041 \) Copy content Toggle raw display
$53$ \( T^{8} - 3 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$59$ \( T^{8} + 19 T^{7} + \cdots + 9150625 \) Copy content Toggle raw display
$61$ \( T^{8} - 2 T^{7} + \cdots + 3025 \) Copy content Toggle raw display
$67$ \( (T^{4} + T^{3} - 82 T^{2} + \cdots - 101)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} - 40 T^{7} + \cdots + 60824401 \) Copy content Toggle raw display
$73$ \( T^{8} - 23 T^{7} + \cdots + 151321 \) Copy content Toggle raw display
$79$ \( T^{8} + 17 T^{7} + \cdots + 4644025 \) Copy content Toggle raw display
$83$ \( T^{8} - 25 T^{7} + \cdots + 841 \) Copy content Toggle raw display
$89$ \( (T^{4} - 150 T^{2} + \cdots + 725)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} - 12 T^{7} + \cdots + 625 \) Copy content Toggle raw display
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