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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [700,5,Mod(349,700)] 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("700.349"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(700, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 700.h (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4] 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: \(72.3589741587\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 28)
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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} + ( - 7 \beta_{3} + 7 \beta_1) q^{7} - 33 q^{9} + 18 q^{11} - 19 \beta_{3} q^{13} + 60 \beta_{3} q^{17} + 13 \beta_{2} q^{19} + (7 \beta_{2} - 336) q^{21} + 738 \beta_1 q^{23} - 114 \beta_{3} q^{27}+ \cdots - 594 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 132 q^{9} + 72 q^{11} - 1344 q^{21} + 3384 q^{29} - 3648 q^{39} + 9212 q^{49} + 11520 q^{51} - 12600 q^{71} + 15928 q^{79} - 11196 q^{81} + 25536 q^{91} - 2376 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 8\zeta_{12}^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -4\zeta_{12}^{3} + 8\zeta_{12} \) Copy content Toggle raw display

Display \(\zeta_{12}^j\) in terms of \(\beta_i\)

\(\zeta_{12}\)\(=\) \( ( \beta_{3} + 4\beta_1 ) / 8 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( \beta_{2} + 4 ) / 8 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( \beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{12}^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/700\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(351\) \(477\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
349.1
−0.866025 − 0.500000i
−0.866025 + 0.500000i
0.866025 − 0.500000i
0.866025 + 0.500000i
0 −6.92820 0 0 0 48.4974 − 7.00000i 0 −33.0000 0
349.2 0 −6.92820 0 0 0 48.4974 + 7.00000i 0 −33.0000 0
349.3 0 6.92820 0 0 0 −48.4974 − 7.00000i 0 −33.0000 0
349.4 0 6.92820 0 0 0 −48.4974 + 7.00000i 0 −33.0000 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
7.b odd 2 1 inner
35.c odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 700.5.h.a 4
5.b even 2 1 inner 700.5.h.a 4
5.c odd 4 1 28.5.b.a ✓ 2
5.c odd 4 1 700.5.d.a 2
7.b odd 2 1 inner 700.5.h.a 4
15.e even 4 1 252.5.d.a 2
20.e even 4 1 112.5.c.b 2
35.c odd 2 1 inner 700.5.h.a 4
35.f even 4 1 28.5.b.a ✓ 2
35.f even 4 1 700.5.d.a 2
35.k even 12 1 196.5.h.a 2
35.k even 12 1 196.5.h.b 2
35.l odd 12 1 196.5.h.a 2
35.l odd 12 1 196.5.h.b 2
40.i odd 4 1 448.5.c.c 2
40.k even 4 1 448.5.c.d 2
60.l odd 4 1 1008.5.f.c 2
105.k odd 4 1 252.5.d.a 2
140.j odd 4 1 112.5.c.b 2
280.s even 4 1 448.5.c.c 2
280.y odd 4 1 448.5.c.d 2
420.w even 4 1 1008.5.f.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.5.b.a ✓ 2 5.c odd 4 1
28.5.b.a ✓ 2 35.f even 4 1
112.5.c.b 2 20.e even 4 1
112.5.c.b 2 140.j odd 4 1
196.5.h.a 2 35.k even 12 1
196.5.h.a 2 35.l odd 12 1
196.5.h.b 2 35.k even 12 1
196.5.h.b 2 35.l odd 12 1
252.5.d.a 2 15.e even 4 1
252.5.d.a 2 105.k odd 4 1
448.5.c.c 2 40.i odd 4 1
448.5.c.c 2 280.s even 4 1
448.5.c.d 2 40.k even 4 1
448.5.c.d 2 280.y odd 4 1
700.5.d.a 2 5.c odd 4 1
700.5.d.a 2 35.f even 4 1
700.5.h.a 4 1.a even 1 1 trivial
700.5.h.a 4 5.b even 2 1 inner
700.5.h.a 4 7.b odd 2 1 inner
700.5.h.a 4 35.c odd 2 1 inner
1008.5.f.c 2 60.l odd 4 1
1008.5.f.c 2 420.w even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 48 \) acting on \(S_{5}^{\mathrm{new}}(700, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - 48)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 4606 T^{2} + 5764801 \) Copy content Toggle raw display
$11$ \( (T - 18)^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} - 17328)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 172800)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 8112)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 544644)^{2} \) Copy content Toggle raw display
$29$ \( (T - 846)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 1354752)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 5692996)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 2904768)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 6300100)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 11619072)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 72900)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 9850032)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 42142512)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 6002500)^{2} \) Copy content Toggle raw display
$71$ \( (T + 3150)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} - 55488)^{2} \) Copy content Toggle raw display
$79$ \( (T - 3982)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 25090992)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 57868992)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 158297088)^{2} \) Copy content Toggle raw display
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