Properties

Label 75.13.c.a
Level $75$
Weight $13$
Character orbit 75.c
Self dual yes
Analytic conductor $68.550$
Analytic rank $0$
Dimension $1$
CM discriminant -3
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [75,13,Mod(26,75)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
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")
 
//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);
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 13 \)
Character orbit: \([\chi]\) \(=\) 75.c (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(68.5495362957\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - 729 q^{3} + 4096 q^{4} + 153502 q^{7} + 531441 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 729 q^{3} + 4096 q^{4} + 153502 q^{7} + 531441 q^{9} - 2985984 q^{12} + 9397582 q^{13} + 16777216 q^{16} + 17886962 q^{19} - 111902958 q^{21} - 387420489 q^{27} + 628744192 q^{28} - 530187838 q^{31} + 2176782336 q^{36} - 2826257618 q^{37} - 6850837278 q^{39} + 235885102 q^{43} - 12230590464 q^{48} + 9721576803 q^{49} + 38492495872 q^{52} - 13039595298 q^{57} + 74063873522 q^{61} + 81577256382 q^{63} + 68719476736 q^{64} + 151031344462 q^{67} - 104459767778 q^{73} + 73264996352 q^{76} - 444304748158 q^{79} + 282429536481 q^{81} - 458354515968 q^{84} + 1442547632164 q^{91} + 386506933902 q^{93} + 1662757858942 q^{97}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
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
0
0 −729.000 4096.00 0 0 153502. 0 531441. 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)

Twists

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

Hecke kernels

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

\( T_{2} \) Copy content Toggle raw display
\( T_{7} - 153502 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 729 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T - 153502 \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T - 9397582 \) Copy content Toggle raw display
$17$ \( T \) Copy content Toggle raw display
$19$ \( T - 17886962 \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T \) Copy content Toggle raw display
$31$ \( T + 530187838 \) Copy content Toggle raw display
$37$ \( T + 2826257618 \) Copy content Toggle raw display
$41$ \( T \) Copy content Toggle raw display
$43$ \( T - 235885102 \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T - 74063873522 \) Copy content Toggle raw display
$67$ \( T - 151031344462 \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T + 104459767778 \) Copy content Toggle raw display
$79$ \( T + 444304748158 \) Copy content Toggle raw display
$83$ \( T \) Copy content Toggle raw display
$89$ \( T \) Copy content Toggle raw display
$97$ \( T - 1662757858942 \) Copy content Toggle raw display
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