Properties

Label 225.10.a.f
Level $225$
Weight $10$
Character orbit 225.a
Self dual yes
Analytic conductor $115.883$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [225,10,Mod(1,225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(225, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("225.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 225.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: \(115.883063137\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
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)
 
\(f(q)\) \(=\) \( q + 22 q^{2} - 28 q^{4} + 5988 q^{7} - 11880 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 22 q^{2} - 28 q^{4} + 5988 q^{7} - 11880 q^{8} + 14648 q^{11} - 37906 q^{13} + 131736 q^{14} - 247024 q^{16} - 441098 q^{17} + 441820 q^{19} + 322256 q^{22} + 2264136 q^{23} - 833932 q^{26} - 167664 q^{28} + 1049350 q^{29} - 7910568 q^{31} + 648032 q^{32} - 9704156 q^{34} + 20992558 q^{37} + 9720040 q^{38} - 13285562 q^{41} + 23130764 q^{43} - 410144 q^{44} + 49810992 q^{46} - 13873688 q^{47} - 4497463 q^{49} + 1061368 q^{52} - 57635174 q^{53} - 71137440 q^{56} + 23085700 q^{58} + 32042120 q^{59} + 110664022 q^{61} - 174032496 q^{62} + 140732992 q^{64} + 118568268 q^{67} + 12350744 q^{68} - 276679712 q^{71} + 264023294 q^{73} + 461836276 q^{74} - 12370960 q^{76} + 87712224 q^{77} + 448202760 q^{79} - 292282364 q^{82} + 851015796 q^{83} + 508876808 q^{86} - 174018240 q^{88} - 189894930 q^{89} - 226981128 q^{91} - 63395808 q^{92} - 305221136 q^{94} + 1014149278 q^{97} - 98944186 q^{98}+O(q^{100}) \) 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.

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
22.0000 0 −28.0000 0 0 5988.00 −11880.0 0 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 225.10.a.f 1
3.b odd 2 1 75.10.a.a 1
5.b even 2 1 45.10.a.a 1
5.c odd 4 2 225.10.b.b 2
15.d odd 2 1 15.10.a.b 1
15.e even 4 2 75.10.b.b 2
60.h even 2 1 240.10.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.10.a.b 1 15.d odd 2 1
45.10.a.a 1 5.b even 2 1
75.10.a.a 1 3.b odd 2 1
75.10.b.b 2 15.e even 4 2
225.10.a.f 1 1.a even 1 1 trivial
225.10.b.b 2 5.c odd 4 2
240.10.a.g 1 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(225))\):

\( T_{2} - 22 \) Copy content Toggle raw display
\( T_{7} - 5988 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 22 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T - 5988 \) Copy content Toggle raw display
$11$ \( T - 14648 \) Copy content Toggle raw display
$13$ \( T + 37906 \) Copy content Toggle raw display
$17$ \( T + 441098 \) Copy content Toggle raw display
$19$ \( T - 441820 \) Copy content Toggle raw display
$23$ \( T - 2264136 \) Copy content Toggle raw display
$29$ \( T - 1049350 \) Copy content Toggle raw display
$31$ \( T + 7910568 \) Copy content Toggle raw display
$37$ \( T - 20992558 \) Copy content Toggle raw display
$41$ \( T + 13285562 \) Copy content Toggle raw display
$43$ \( T - 23130764 \) Copy content Toggle raw display
$47$ \( T + 13873688 \) Copy content Toggle raw display
$53$ \( T + 57635174 \) Copy content Toggle raw display
$59$ \( T - 32042120 \) Copy content Toggle raw display
$61$ \( T - 110664022 \) Copy content Toggle raw display
$67$ \( T - 118568268 \) Copy content Toggle raw display
$71$ \( T + 276679712 \) Copy content Toggle raw display
$73$ \( T - 264023294 \) Copy content Toggle raw display
$79$ \( T - 448202760 \) Copy content Toggle raw display
$83$ \( T - 851015796 \) Copy content Toggle raw display
$89$ \( T + 189894930 \) Copy content Toggle raw display
$97$ \( T - 1014149278 \) Copy content Toggle raw display
show more
show less