Properties

Label 75.22.a.a
Level $75$
Weight $22$
Character orbit 75.a
Self dual yes
Analytic conductor $209.608$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [75,22,Mod(1,75)]
 
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([0, 0]))
 
N = Newforms(chi, 22, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("75.1");
 
S:= CuspForms(chi, 22);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 75.a (trivial)

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: \(209.608008215\)
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)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - 1728 q^{2} + 59049 q^{3} + 888832 q^{4} - 102036672 q^{6} - 538429808 q^{7} + 2087976960 q^{8} + 3486784401 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 1728 q^{2} + 59049 q^{3} + 888832 q^{4} - 102036672 q^{6} - 538429808 q^{7} + 2087976960 q^{8} + 3486784401 q^{9} - 64113040188 q^{11} + 52484640768 q^{12} + 130980107986 q^{13} + 930406708224 q^{14} - 5472039993344 q^{16} - 8242029723618 q^{17} - 6025163444928 q^{18} + 13492101753020 q^{19} - 31793741732592 q^{21} + 110787333444864 q^{22} + 233184825844776 q^{23} + 123292951511040 q^{24} - 226333626599808 q^{26} + 205891132094649 q^{27} - 478573643104256 q^{28} - 20\!\cdots\!70 q^{29}+ \cdots - 22\!\cdots\!88 q^{99}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
−1728.00 59049.0 888832. 0 −1.02037e8 −5.38430e8 2.08798e9 3.48678e9 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 75.22.a.a 1
5.b even 2 1 3.22.a.b 1
5.c odd 4 2 75.22.b.b 2
15.d odd 2 1 9.22.a.a 1
20.d odd 2 1 48.22.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.22.a.b 1 5.b even 2 1
9.22.a.a 1 15.d odd 2 1
48.22.a.d 1 20.d odd 2 1
75.22.a.a 1 1.a even 1 1 trivial
75.22.b.b 2 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 1728 \) acting on \(S_{22}^{\mathrm{new}}(\Gamma_0(75))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 1728 \) Copy content Toggle raw display
$3$ \( T - 59049 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T + 538429808 \) Copy content Toggle raw display
$11$ \( T + 64113040188 \) Copy content Toggle raw display
$13$ \( T - 130980107986 \) Copy content Toggle raw display
$17$ \( T + 8242029723618 \) Copy content Toggle raw display
$19$ \( T - 13492101753020 \) Copy content Toggle raw display
$23$ \( T - 233184825844776 \) Copy content Toggle raw display
$29$ \( T + 2024562031123770 \) Copy content Toggle raw display
$31$ \( T + 6869194988701768 \) Copy content Toggle raw display
$37$ \( T + 3443998107027638 \) Copy content Toggle raw display
$41$ \( T + 21\!\cdots\!58 \) Copy content Toggle raw display
$43$ \( T - 71\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( T + 28\!\cdots\!48 \) Copy content Toggle raw display
$53$ \( T - 21\!\cdots\!46 \) Copy content Toggle raw display
$59$ \( T - 15\!\cdots\!60 \) Copy content Toggle raw display
$61$ \( T - 43\!\cdots\!62 \) Copy content Toggle raw display
$67$ \( T + 92\!\cdots\!68 \) Copy content Toggle raw display
$71$ \( T + 20\!\cdots\!28 \) Copy content Toggle raw display
$73$ \( T + 16\!\cdots\!74 \) Copy content Toggle raw display
$79$ \( T - 67\!\cdots\!80 \) Copy content Toggle raw display
$83$ \( T + 39\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( T - 41\!\cdots\!90 \) Copy content Toggle raw display
$97$ \( T + 57\!\cdots\!98 \) Copy content Toggle raw display
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