Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1575,4,Mod(1,1575)] 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("1575.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1575, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1575.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,3,0,9,0,0,-14,51,0,0,-62] 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: yes
Analytic conductor: \(92.9280082590\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{41}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 105)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(\beta = \frac{1}{2}(1 + \sqrt{41})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 1) q^{2} + (3 \beta + 3) q^{4} - 7 q^{7} + (\beta + 25) q^{8} + (2 \beta - 32) q^{11} + (10 \beta - 2) q^{13} + ( - 7 \beta - 7) q^{14} + (3 \beta + 11) q^{16} + ( - 12 \beta + 26) q^{17}+ \cdots + (49 \beta + 49) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{2} + 9 q^{4} - 14 q^{7} + 51 q^{8} - 62 q^{11} + 6 q^{13} - 21 q^{14} + 25 q^{16} + 40 q^{17} - 122 q^{19} - 52 q^{22} + 16 q^{23} + 214 q^{26} - 63 q^{28} - 352 q^{29} + 66 q^{31} - 309 q^{32}+ \cdots + 147 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.70156
3.70156
−1.70156 0 −5.10469 0 0 −7.00000 22.2984 0 0
1.2 4.70156 0 14.1047 0 0 −7.00000 28.7016 0 0
\(n\): e.g. 2-40 or 80-90
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( -1 \)
\(5\) \( +1 \)
\(7\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1575.4.a.y 2
3.b odd 2 1 525.4.a.i 2
5.b even 2 1 315.4.a.g 2
15.d odd 2 1 105.4.a.g ✓ 2
15.e even 4 2 525.4.d.j 4
35.c odd 2 1 2205.4.a.v 2
60.h even 2 1 1680.4.a.y 2
105.g even 2 1 735.4.a.q 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.a.g ✓ 2 15.d odd 2 1
315.4.a.g 2 5.b even 2 1
525.4.a.i 2 3.b odd 2 1
525.4.d.j 4 15.e even 4 2
735.4.a.q 2 105.g even 2 1
1575.4.a.y 2 1.a even 1 1 trivial
1680.4.a.y 2 60.h even 2 1
2205.4.a.v 2 35.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1575))\):

\( T_{2}^{2} - 3T_{2} - 8 \) Copy content Toggle raw display
\( T_{11}^{2} + 62T_{11} + 920 \) Copy content Toggle raw display
\( T_{13}^{2} - 6T_{13} - 1016 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 3T - 8 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T + 7)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 62T + 920 \) Copy content Toggle raw display
$13$ \( T^{2} - 6T - 1016 \) Copy content Toggle raw display
$17$ \( T^{2} - 40T - 1076 \) Copy content Toggle raw display
$19$ \( T^{2} + 122T + 3680 \) Copy content Toggle raw display
$23$ \( T^{2} - 16T - 23552 \) Copy content Toggle raw display
$29$ \( T^{2} + 352T + 29500 \) Copy content Toggle raw display
$31$ \( T^{2} - 66T - 13712 \) Copy content Toggle raw display
$37$ \( T^{2} - 188T - 56764 \) Copy content Toggle raw display
$41$ \( T^{2} + 16T - 119492 \) Copy content Toggle raw display
$43$ \( T^{2} - 396T - 63296 \) Copy content Toggle raw display
$47$ \( T^{2} + 188T - 192064 \) Copy content Toggle raw display
$53$ \( T^{2} - 982T + 206600 \) Copy content Toggle raw display
$59$ \( T^{2} + 516T + 7360 \) Copy content Toggle raw display
$61$ \( T^{2} + 880T + 121276 \) Copy content Toggle raw display
$67$ \( T^{2} - 356T - 501152 \) Copy content Toggle raw display
$71$ \( T^{2} + 310T - 51784 \) Copy content Toggle raw display
$73$ \( T^{2} + 326T + 11768 \) Copy content Toggle raw display
$79$ \( T^{2} - 1832 T + 838400 \) Copy content Toggle raw display
$83$ \( T^{2} + 680T - 398704 \) Copy content Toggle raw display
$89$ \( T^{2} + 796T - 998780 \) Copy content Toggle raw display
$97$ \( T^{2} - 670T - 679936 \) Copy content Toggle raw display
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