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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [240,8,Mod(49,240)] 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("240.49"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(240, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 240.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,264] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(74.9724061162\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{2641})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1321x^{2} + 435600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 5 \)
Twist minimal: no (minimal twist has level 30)
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.

\(f(q)\) \(=\) \( q + 27 \beta_1 q^{3} + (\beta_{3} + 2 \beta_{2} + 88 \beta_1 + 66) q^{5} + (4 \beta_{3} - 8 \beta_{2} + 282 \beta_1) q^{7} - 729 q^{9} + (28 \beta_{3} + 14 \beta_{2} + 1860) q^{11} + (6 \beta_{3} - 12 \beta_{2} + 3696 \beta_1) q^{13}+ \cdots + ( - 20412 \beta_{3} - 10206 \beta_{2} - 1355940) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 264 q^{5} - 2916 q^{9} + 7440 q^{11} - 9504 q^{15} + 69456 q^{19} - 30456 q^{21} + 60396 q^{25} - 372384 q^{29} - 436192 q^{31} + 534576 q^{35} - 399168 q^{39} + 608616 q^{41} - 192456 q^{45} - 1249524 q^{49}+ \cdots - 5423760 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 1321x^{2} + 435600 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 661\nu ) / 660 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + 660\nu^{2} - 1981\nu + 435930 ) / 330 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 2640\nu^{2} + 1981\nu + 1743720 ) / 660 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{3} - 2\beta_{2} - 5\beta_1 ) / 10 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{3} + \beta_{2} - 6605 ) / 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -661\beta_{3} + 1322\beta_{2} + 9905\beta_1 ) / 10 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/240\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(97\) \(161\) \(181\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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
26.1953i
− 25.1953i
− 26.1953i
25.1953i
0 − 27.0000i 0 −139.563 − 242.172i 0 745.813i 0 −729.000 0
49.2 0 − 27.0000i 0 271.563 + 66.1720i 0 − 1309.81i 0 −729.000 0
49.3 0 27.0000i 0 −139.563 + 242.172i 0 − 745.813i 0 −729.000 0
49.4 0 27.0000i 0 271.563 − 66.1720i 0 1309.81i 0 −729.000 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 240.8.f.d 4
4.b odd 2 1 30.8.c.b ✓ 4
5.b even 2 1 inner 240.8.f.d 4
12.b even 2 1 90.8.c.b 4
20.d odd 2 1 30.8.c.b ✓ 4
20.e even 4 1 150.8.a.r 2
20.e even 4 1 150.8.a.s 2
60.h even 2 1 90.8.c.b 4
60.l odd 4 1 450.8.a.be 2
60.l odd 4 1 450.8.a.bh 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.8.c.b ✓ 4 4.b odd 2 1
30.8.c.b ✓ 4 20.d odd 2 1
90.8.c.b 4 12.b even 2 1
90.8.c.b 4 60.h even 2 1
150.8.a.r 2 20.e even 4 1
150.8.a.s 2 20.e even 4 1
240.8.f.d 4 1.a even 1 1 trivial
240.8.f.d 4 5.b even 2 1 inner
450.8.a.be 2 60.l odd 4 1
450.8.a.bh 2 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} + 2271848T_{7}^{2} + 954286719376 \) acting on \(S_{8}^{\mathrm{new}}(240, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 729)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} + \cdots + 6103515625 \) Copy content Toggle raw display
$7$ \( T^{4} + \cdots + 954286719376 \) Copy content Toggle raw display
$11$ \( (T^{2} - 3720 T - 9481300)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + \cdots + 127317733322256 \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots + 20\!\cdots\!36 \) Copy content Toggle raw display
$19$ \( (T^{2} - 34728 T + 63818496)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{2} + 186192 T - 19057296384)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 218096 T - 6515247296)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 87\!\cdots\!56 \) Copy content Toggle raw display
$41$ \( (T^{2} - 304308 T - 36687881884)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots + 54\!\cdots\!36 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 27\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( (T^{2} + \cdots + 5193452974700)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + \cdots + 8831371838500)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 20\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( (T^{2} + \cdots + 20010069370800)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 29\!\cdots\!36 \) Copy content Toggle raw display
$79$ \( (T^{2} + \cdots - 58253255294976)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 56\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( (T^{2} + \cdots - 25448685498556)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 90\!\cdots\!56 \) Copy content Toggle raw display
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