Properties

Label 75.22.a.c
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)
 
\(f(q)\) \(=\) \( q + 2844 q^{2} + 59049 q^{3} + 5991184 q^{4} + 167935356 q^{6} - 363303920 q^{7} + 11074627008 q^{8} + 3486784401 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2844 q^{2} + 59049 q^{3} + 5991184 q^{4} + 167935356 q^{6} - 363303920 q^{7} + 11074627008 q^{8} + 3486784401 q^{9} + 14581833156 q^{11} + 353773424016 q^{12} - 113350790702 q^{13} - 1033236348480 q^{14} + 18931815702784 q^{16} + 8589389597982 q^{17} + 9916414836444 q^{18} - 29202939273796 q^{19} - 21452733172080 q^{21} + 41470733495664 q^{22} + 155899214954280 q^{23} + 653945650195392 q^{24} - 322369648756488 q^{26} + 205891132094649 q^{27} - 21\!\cdots\!80 q^{28}+ \cdots + 50\!\cdots\!56 q^{99}+O(q^{100}) \) 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
2844.00 59049.0 5.99118e6 0 1.67935e8 −3.63304e8 1.10746e10 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.c 1
5.b even 2 1 3.22.a.a 1
5.c odd 4 2 75.22.b.a 2
15.d odd 2 1 9.22.a.d 1
20.d odd 2 1 48.22.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.22.a.a 1 5.b even 2 1
9.22.a.d 1 15.d odd 2 1
48.22.a.e 1 20.d odd 2 1
75.22.a.c 1 1.a even 1 1 trivial
75.22.b.a 2 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} - 2844 \) acting on \(S_{22}^{\mathrm{new}}(\Gamma_0(75))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 2844 \) Copy content Toggle raw display
$3$ \( T - 59049 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T + 363303920 \) Copy content Toggle raw display
$11$ \( T - 14581833156 \) Copy content Toggle raw display
$13$ \( T + 113350790702 \) Copy content Toggle raw display
$17$ \( T - 8589389597982 \) Copy content Toggle raw display
$19$ \( T + 29202939273796 \) Copy content Toggle raw display
$23$ \( T - 155899214954280 \) Copy content Toggle raw display
$29$ \( T - 2400788707090758 \) Copy content Toggle raw display
$31$ \( T - 2239820676947000 \) Copy content Toggle raw display
$37$ \( T - 30\!\cdots\!90 \) Copy content Toggle raw display
$41$ \( T + 10\!\cdots\!30 \) Copy content Toggle raw display
$43$ \( T - 16\!\cdots\!24 \) Copy content Toggle raw display
$47$ \( T - 66\!\cdots\!08 \) Copy content Toggle raw display
$53$ \( T + 43\!\cdots\!30 \) Copy content Toggle raw display
$59$ \( T - 55\!\cdots\!16 \) Copy content Toggle raw display
$61$ \( T + 71\!\cdots\!02 \) Copy content Toggle raw display
$67$ \( T - 15\!\cdots\!12 \) Copy content Toggle raw display
$71$ \( T - 26\!\cdots\!32 \) Copy content Toggle raw display
$73$ \( T + 13\!\cdots\!50 \) Copy content Toggle raw display
$79$ \( T + 16\!\cdots\!40 \) Copy content Toggle raw display
$83$ \( T - 17\!\cdots\!72 \) Copy content Toggle raw display
$89$ \( T + 31\!\cdots\!86 \) Copy content Toggle raw display
$97$ \( T + 94\!\cdots\!18 \) Copy content Toggle raw display
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