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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [475,4,Mod(324,475)] 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("475.324"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(475, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 475.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,16,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(28.0259072527\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 95)
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 \(\beta = 2i\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 \beta q^{3} + 8 q^{4} - 11 \beta q^{7} + 11 q^{9} - 12 q^{11} - 16 \beta q^{12} - 4 \beta q^{13} + 64 q^{16} - 33 \beta q^{17} - 19 q^{19} - 88 q^{21} + 15 \beta q^{23} - 76 \beta q^{27} - 88 \beta q^{28} + \cdots - 132 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 16 q^{4} + 22 q^{9} - 24 q^{11} + 128 q^{16} - 38 q^{19} - 176 q^{21} + 12 q^{29} - 128 q^{31} + 176 q^{36} - 64 q^{39} + 108 q^{41} - 192 q^{44} - 282 q^{49} - 528 q^{51} + 1128 q^{59} - 1412 q^{61}+ \cdots - 264 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/475\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
324.1
1.00000i
− 1.00000i
0 − 4.00000i 8.00000 0 0 − 22.0000i 0 11.0000 0
324.2 0 4.00000i 8.00000 0 0 22.0000i 0 11.0000 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 475.4.b.e 2
5.b even 2 1 inner 475.4.b.e 2
5.c odd 4 1 95.4.a.a ✓ 1
5.c odd 4 1 475.4.a.d 1
15.e even 4 1 855.4.a.e 1
20.e even 4 1 1520.4.a.b 1
95.g even 4 1 1805.4.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.4.a.a ✓ 1 5.c odd 4 1
475.4.a.d 1 5.c odd 4 1
475.4.b.e 2 1.a even 1 1 trivial
475.4.b.e 2 5.b even 2 1 inner
855.4.a.e 1 15.e even 4 1
1520.4.a.b 1 20.e even 4 1
1805.4.a.f 1 95.g even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(475, [\chi])\):

\( T_{2} \) Copy content Toggle raw display
\( T_{3}^{2} + 16 \) Copy content Toggle raw display
\( T_{7}^{2} + 484 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 16 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 484 \) Copy content Toggle raw display
$11$ \( (T + 12)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 64 \) Copy content Toggle raw display
$17$ \( T^{2} + 4356 \) Copy content Toggle raw display
$19$ \( (T + 19)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 900 \) Copy content Toggle raw display
$29$ \( (T - 6)^{2} \) Copy content Toggle raw display
$31$ \( (T + 64)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 256 \) Copy content Toggle raw display
$41$ \( (T - 54)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 33124 \) Copy content Toggle raw display
$47$ \( T^{2} + 352836 \) Copy content Toggle raw display
$53$ \( T^{2} + 156816 \) Copy content Toggle raw display
$59$ \( (T - 564)^{2} \) Copy content Toggle raw display
$61$ \( (T + 706)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 394384 \) Copy content Toggle raw display
$71$ \( (T + 984)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 196 \) Copy content Toggle raw display
$79$ \( (T - 328)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 86436 \) Copy content Toggle raw display
$89$ \( (T + 918)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 2446096 \) Copy content Toggle raw display
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