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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [120,8,Mod(49,120)] 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("120.49"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(120, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 120 = 2^{3} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 120.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(37.4862030581\)
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: \( x^{10} - 2 x^{9} + 2 x^{8} - 26952 x^{7} + 1726045 x^{6} - 37546898 x^{5} + 434846858 x^{4} + \cdots + 25896328200 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{22}\cdot 3^{4}\cdot 5^{5} \)
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.

\(f(q)\) \(=\) \( q - 27 \beta_1 q^{3} + (\beta_{5} - 32 \beta_1 + 38) q^{5} + ( - \beta_{5} + \beta_{3} + 12 \beta_1) q^{7} - 729 q^{9} + (\beta_{8} + 3 \beta_{7} + 3 \beta_{6} + \cdots - 30) q^{11} + (2 \beta_{9} + \beta_{7} + 3 \beta_{6} + \cdots + 3) q^{13}+ \cdots + ( - 729 \beta_{8} - 2187 \beta_{7} + \cdots + 21870) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 376 q^{5} - 7290 q^{9} - 284 q^{11} - 8694 q^{15} + 67552 q^{19} + 3348 q^{21} + 110646 q^{25} - 47560 q^{29} - 573792 q^{31} + 406420 q^{35} + 36396 q^{39} - 1518332 q^{41} - 274104 q^{45} - 912306 q^{49}+ \cdots + 207036 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2 x^{9} + 2 x^{8} - 26952 x^{7} + 1726045 x^{6} - 37546898 x^{5} + 434846858 x^{4} + \cdots + 25896328200 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 15\!\cdots\!89 \nu^{9} + \cdots + 12\!\cdots\!00 ) / 47\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 14\!\cdots\!71 \nu^{9} + \cdots - 17\!\cdots\!40 ) / 25\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 20\!\cdots\!43 \nu^{9} + \cdots + 94\!\cdots\!60 ) / 25\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 14\!\cdots\!63 \nu^{9} + \cdots + 18\!\cdots\!00 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 26\!\cdots\!83 \nu^{9} + \cdots - 19\!\cdots\!00 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 27\!\cdots\!67 \nu^{9} + \cdots + 36\!\cdots\!00 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 58\!\cdots\!23 \nu^{9} + \cdots + 56\!\cdots\!00 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 63\!\cdots\!91 \nu^{9} + \cdots + 11\!\cdots\!00 ) / 42\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 73\!\cdots\!39 \nu^{9} + \cdots + 33\!\cdots\!00 ) / 32\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2 \beta_{9} - 2 \beta_{8} + \beta_{7} + 9 \beta_{6} + 10 \beta_{5} + 16 \beta_{4} + \beta_{3} + \cdots + 97 ) / 480 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -5\beta_{9} + 19\beta_{7} - 30\beta_{6} - 94\beta_{5} - 8\beta_{4} + 15\beta_{3} - 30698\beta _1 - 30 ) / 60 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2146 \beta_{9} + 2146 \beta_{8} - 14151 \beta_{7} + 2015 \beta_{6} + 26864 \beta_{5} - 18758 \beta_{4} + \cdots + 3895707 ) / 480 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 12565 \beta_{8} + 55617 \beta_{7} + 39251 \beta_{6} - 22885 \beta_{5} + 165905 \beta_{4} + \cdots - 40179364 ) / 60 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 3325358 \beta_{9} + 3325358 \beta_{8} - 3788149 \beta_{7} - 23523981 \beta_{6} - 33436450 \beta_{5} + \cdots + 8193501887 ) / 480 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 23310545 \beta_{9} - 70828561 \beta_{7} + 98708520 \beta_{6} + 290224336 \beta_{5} + 42948602 \beta_{4} + \cdots + 98708520 ) / 60 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 5640691834 \beta_{9} - 5640691834 \beta_{8} + 40510330599 \beta_{7} - 6632024735 \beta_{6} + \cdots - 14883081240243 ) / 480 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 41159919085 \beta_{8} - 172945621203 \beta_{7} - 124306630079 \beta_{6} + 75667638955 \beta_{5} + \cdots + 111442868331916 ) / 60 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 9772129623182 \beta_{9} - 9772129623182 \beta_{8} + 11536125246541 \beta_{7} + 70397864261349 \beta_{6} + \cdots - 26\!\cdots\!03 ) / 480 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/120\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(41\) \(61\) \(97\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
13.3885 − 13.3885i
−1.27345 + 1.27345i
−29.5200 + 29.5200i
14.5095 − 14.5095i
3.89551 − 3.89551i
13.3885 + 13.3885i
−1.27345 − 1.27345i
−29.5200 − 29.5200i
14.5095 + 14.5095i
3.89551 + 3.89551i
0 − 27.0000i 0 −275.869 + 44.9613i 0 1.68356i 0 −729.000 0
49.2 0 − 27.0000i 0 −119.909 − 252.481i 0 738.301i 0 −729.000 0
49.3 0 − 27.0000i 0 86.6176 + 265.749i 0 − 671.425i 0 −729.000 0
49.4 0 − 27.0000i 0 222.068 − 169.738i 0 − 1340.80i 0 −729.000 0
49.5 0 − 27.0000i 0 275.092 − 49.4910i 0 1334.24i 0 −729.000 0
49.6 0 27.0000i 0 −275.869 − 44.9613i 0 − 1.68356i 0 −729.000 0
49.7 0 27.0000i 0 −119.909 + 252.481i 0 − 738.301i 0 −729.000 0
49.8 0 27.0000i 0 86.6176 − 265.749i 0 671.425i 0 −729.000 0
49.9 0 27.0000i 0 222.068 + 169.738i 0 1340.80i 0 −729.000 0
49.10 0 27.0000i 0 275.092 + 49.4910i 0 − 1334.24i 0 −729.000 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 49.10
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 120.8.f.a ✓ 10
3.b odd 2 1 360.8.f.c 10
4.b odd 2 1 240.8.f.f 10
5.b even 2 1 inner 120.8.f.a ✓ 10
5.c odd 4 1 600.8.a.t 5
5.c odd 4 1 600.8.a.u 5
15.d odd 2 1 360.8.f.c 10
20.d odd 2 1 240.8.f.f 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.8.f.a ✓ 10 1.a even 1 1 trivial
120.8.f.a ✓ 10 5.b even 2 1 inner
240.8.f.f 10 4.b odd 2 1
240.8.f.f 10 20.d odd 2 1
360.8.f.c 10 3.b odd 2 1
360.8.f.c 10 15.d odd 2 1
600.8.a.t 5 5.c odd 4 1
600.8.a.u 5 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{10} + 4573868 T_{7}^{8} + 7009421753008 T_{7}^{6} + \cdots + 22\!\cdots\!00 \) acting on \(S_{8}^{\mathrm{new}}(120, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( (T^{2} + 729)^{5} \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots + 29\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( (T^{5} + \cdots + 23\!\cdots\!12)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 14\!\cdots\!16 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 14\!\cdots\!44 \) Copy content Toggle raw display
$19$ \( (T^{5} + \cdots + 16\!\cdots\!08)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{5} + \cdots - 20\!\cdots\!48)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} + \cdots - 76\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 30\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{5} + \cdots - 43\!\cdots\!28)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 16\!\cdots\!96 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 31\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 52\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{5} + \cdots + 12\!\cdots\!04)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} + \cdots + 29\!\cdots\!32)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 87\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( (T^{5} + \cdots - 37\!\cdots\!24)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 28\!\cdots\!84 \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots - 78\!\cdots\!52)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 14\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( (T^{5} + \cdots + 48\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 20\!\cdots\!16 \) Copy content Toggle raw display
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