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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [75,13,Mod(26,75)] 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("75.26"); S:= CuspForms(chi, 13); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(75, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 13, names="a")
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 13 \)
Character orbit: \([\chi]\) \(=\) 75.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,1350] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(68.5495362957\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-26}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 26 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 3)
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 \(\beta = 18\sqrt{-26}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + ( - 3 \beta + 675) q^{3} - 4328 q^{4} + (675 \beta + 25272) q^{6} - 40250 q^{7} - 232 \beta q^{8} + ( - 4050 \beta + 379809) q^{9} + 12650 \beta q^{11} + (12984 \beta - 2921400) q^{12} + \cdots + (4804583850 \beta + 431582580000) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 1350 q^{3} - 8656 q^{4} + 50544 q^{6} - 80500 q^{7} + 759618 q^{9} - 5842800 q^{12} - 2568100 q^{13} - 31546240 q^{16} + 68234400 q^{18} + 106687156 q^{19} - 54337500 q^{21} - 213127200 q^{22} - 11726208 q^{24}+ \cdots + 863165160000 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).

\(n\) \(26\) \(52\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1
− 5.09902i
5.09902i
− 91.7824i 675.000 + 275.347i −4328.00 0 25272.0 − 61953.1i −40250.0 21293.5i 379809. + 371719.i 0
26.2 91.7824i 675.000 − 275.347i −4328.00 0 25272.0 + 61953.1i −40250.0 − 21293.5i 379809. − 371719.i 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
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 75.13.c.c 2
3.b odd 2 1 inner 75.13.c.c 2
5.b even 2 1 3.13.b.b ✓ 2
5.c odd 4 2 75.13.d.b 4
15.d odd 2 1 3.13.b.b ✓ 2
15.e even 4 2 75.13.d.b 4
20.d odd 2 1 48.13.e.b 2
40.e odd 2 1 192.13.e.c 2
40.f even 2 1 192.13.e.d 2
45.h odd 6 2 81.13.d.c 4
45.j even 6 2 81.13.d.c 4
60.h even 2 1 48.13.e.b 2
120.i odd 2 1 192.13.e.d 2
120.m even 2 1 192.13.e.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.13.b.b ✓ 2 5.b even 2 1
3.13.b.b ✓ 2 15.d odd 2 1
48.13.e.b 2 20.d odd 2 1
48.13.e.b 2 60.h even 2 1
75.13.c.c 2 1.a even 1 1 trivial
75.13.c.c 2 3.b odd 2 1 inner
75.13.d.b 4 5.c odd 4 2
75.13.d.b 4 15.e even 4 2
81.13.d.c 4 45.h odd 6 2
81.13.d.c 4 45.j even 6 2
192.13.e.c 2 40.e odd 2 1
192.13.e.c 2 120.m even 2 1
192.13.e.d 2 40.f even 2 1
192.13.e.d 2 120.i odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{13}^{\mathrm{new}}(75, [\chi])\):

\( T_{2}^{2} + 8424 \) Copy content Toggle raw display
\( T_{7} + 40250 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 8424 \) Copy content Toggle raw display
$3$ \( T^{2} - 1350 T + 531441 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T + 40250)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 1348029540000 \) Copy content Toggle raw display
$13$ \( (T + 1284050)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 220359487855104 \) Copy content Toggle raw display
$19$ \( (T - 53343578)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 11\!\cdots\!44 \) Copy content Toggle raw display
$29$ \( T^{2} + 14\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T - 66526202)^{2} \) Copy content Toggle raw display
$37$ \( (T + 2228726450)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 67\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T + 8977216250)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 11\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( T^{2} + 16\!\cdots\!04 \) Copy content Toggle raw display
$59$ \( T^{2} + 21\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T + 40679935918)^{2} \) Copy content Toggle raw display
$67$ \( (T + 121176846650)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 20\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T - 60956187550)^{2} \) Copy content Toggle raw display
$79$ \( (T + 252324997702)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 16\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( T^{2} + 12\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T + 653817778850)^{2} \) Copy content Toggle raw display
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