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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [450,8,Mod(199,450)] 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("450.199"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(450, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 450.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,-256,0,0,0,0,0,0,4320] 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: \(140.573261468\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 6985x^{2} + 12194064 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 8 \beta_1 q^{2} - 64 q^{4} + ( - \beta_{2} - 350 \beta_1) q^{7} + 512 \beta_1 q^{8} + ( - 16 \beta_{3} + 1080) q^{11} + (22 \beta_{2} + 2315 \beta_1) q^{13} + ( - 8 \beta_{3} - 2800) q^{14} + 4096 q^{16}+ \cdots + (5600 \beta_{2} - 4602576 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 256 q^{4} + 4320 q^{11} - 11200 q^{14} + 16384 q^{16} - 41936 q^{19} + 74080 q^{26} + 163200 q^{29} - 254536 q^{31} - 185856 q^{34} + 1786560 q^{41} - 276480 q^{44} + 579840 q^{46} + 2301288 q^{49}+ \cdots + 6489600 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 6985x^{2} + 12194064 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 3493\nu ) / 3492 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 10477\nu ) / 1164 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 6\nu^{2} + 20955 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{2} - 3\beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 20955 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3493\beta_{2} + 31431\beta_1 ) / 6 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/450\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1
58.5953i
− 59.5953i
59.5953i
− 58.5953i
− 8.00000i 0 −64.0000 0 0 − 704.572i 512.000i 0 0
199.2 − 8.00000i 0 −64.0000 0 0 4.57157i 512.000i 0 0
199.3 8.00000i 0 −64.0000 0 0 − 4.57157i − 512.000i 0 0
199.4 8.00000i 0 −64.0000 0 0 704.572i − 512.000i 0 0
\(n\): e.g. 2-40 or 80-90
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 450.8.c.u 4
3.b odd 2 1 450.8.c.t 4
5.b even 2 1 inner 450.8.c.u 4
5.c odd 4 1 450.8.a.bb ✓ 2
5.c odd 4 1 450.8.a.bk yes 2
15.d odd 2 1 450.8.c.t 4
15.e even 4 1 450.8.a.bf yes 2
15.e even 4 1 450.8.a.bg yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
450.8.a.bb ✓ 2 5.c odd 4 1
450.8.a.bf yes 2 15.e even 4 1
450.8.a.bg yes 2 15.e even 4 1
450.8.a.bk yes 2 5.c odd 4 1
450.8.c.t 4 3.b odd 2 1
450.8.c.t 4 15.d odd 2 1
450.8.c.u 4 1.a even 1 1 trivial
450.8.c.u 4 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{8}^{\mathrm{new}}(450, [\chi])\):

\( T_{7}^{4} + 496442T_{7}^{2} + 10374841 \) Copy content Toggle raw display
\( T_{11}^{2} - 2160T_{11} - 31018176 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 496442 T^{2} + 10374841 \) Copy content Toggle raw display
$11$ \( (T^{2} - 2160 T - 31018176)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + \cdots + 30\!\cdots\!21 \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots + 28\!\cdots\!96 \) Copy content Toggle raw display
$19$ \( (T^{2} + 20968 T - 1552745969)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 83\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{2} - 81600 T + 1270378944)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 127268 T - 2893656269)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 15\!\cdots\!76 \) Copy content Toggle raw display
$41$ \( (T^{2} - 893280 T + 197676907200)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots + 11\!\cdots\!81 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 12\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( (T^{2} + 2025120 T - 211124918016)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + \cdots + 1022728183429)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 98\!\cdots\!41 \) Copy content Toggle raw display
$71$ \( (T^{2} - 1961520 T - 299873939904)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 47\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{2} + \cdots - 7476925113584)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 81\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( (T^{2} + \cdots - 24275679289344)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 53\!\cdots\!41 \) Copy content Toggle raw display
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