Properties

Label 225.2.h.a
Level $225$
Weight $2$
Character orbit 225.h
Analytic conductor $1.797$
Analytic rank $0$
Dimension $4$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [225,2,Mod(46,225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(225, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 6]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("225.46");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 225.h (of order \(5\), degree \(4\), 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: no
Analytic conductor: \(1.79663404548\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 75)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{10}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{10}^{3} + \zeta_{10}^{2} + \cdots + 1) q^{2}+ \cdots + 3 \zeta_{10}^{2} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{10}^{3} + \zeta_{10}^{2} + \cdots + 1) q^{2}+ \cdots + ( - 13 \zeta_{10}^{3} + 13 \zeta_{10}^{2} + \cdots + 13) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{2} + q^{4} - 5 q^{5} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{2} + q^{4} - 5 q^{5} - 3 q^{8} - 5 q^{10} + 6 q^{11} - 2 q^{13} + 10 q^{14} + q^{16} + 8 q^{17} - 4 q^{19} + 4 q^{22} - 10 q^{23} - 5 q^{25} - 18 q^{26} - 10 q^{28} + 8 q^{29} - 20 q^{32} + 7 q^{34} + 10 q^{35} - 15 q^{37} - 6 q^{38} + 15 q^{40} + 4 q^{43} + 4 q^{44} - 10 q^{46} + 2 q^{47} + 52 q^{49} + 5 q^{50} + 2 q^{52} + 5 q^{53} - 10 q^{55} + 30 q^{56} - 8 q^{58} - 4 q^{59} + 2 q^{61} + 10 q^{62} - 7 q^{64} - 25 q^{65} + 2 q^{67} + 2 q^{68} - 10 q^{70} - 8 q^{71} - 10 q^{73} - 10 q^{74} + 4 q^{76} + 20 q^{77} + 5 q^{80} - 10 q^{82} + 18 q^{83} + 5 q^{85} - 14 q^{86} + 18 q^{88} + 9 q^{89} - 40 q^{91} - 10 q^{92} - 2 q^{94} - 10 q^{95} + 2 q^{97} + 13 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/225\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-\zeta_{10}^{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
46.1
−0.309017 0.951057i
0.809017 0.587785i
0.809017 + 0.587785i
−0.309017 + 0.951057i
−0.309017 + 0.951057i 0 0.809017 + 0.587785i −1.80902 + 1.31433i 0 −4.47214 −2.42705 + 1.76336i 0 −0.690983 2.12663i
91.1 0.809017 + 0.587785i 0 −0.309017 0.951057i −0.690983 + 2.12663i 0 4.47214 0.927051 2.85317i 0 −1.80902 + 1.31433i
136.1 0.809017 0.587785i 0 −0.309017 + 0.951057i −0.690983 2.12663i 0 4.47214 0.927051 + 2.85317i 0 −1.80902 1.31433i
181.1 −0.309017 0.951057i 0 0.809017 0.587785i −1.80902 1.31433i 0 −4.47214 −2.42705 1.76336i 0 −0.690983 + 2.12663i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.2.h.a 4
3.b odd 2 1 75.2.g.a 4
15.d odd 2 1 375.2.g.a 4
15.e even 4 2 375.2.i.a 8
25.d even 5 1 inner 225.2.h.a 4
25.d even 5 1 5625.2.a.a 2
25.e even 10 1 5625.2.a.h 2
75.h odd 10 1 375.2.g.a 4
75.h odd 10 1 1875.2.a.a 2
75.j odd 10 1 75.2.g.a 4
75.j odd 10 1 1875.2.a.d 2
75.l even 20 2 375.2.i.a 8
75.l even 20 2 1875.2.b.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.2.g.a 4 3.b odd 2 1
75.2.g.a 4 75.j odd 10 1
225.2.h.a 4 1.a even 1 1 trivial
225.2.h.a 4 25.d even 5 1 inner
375.2.g.a 4 15.d odd 2 1
375.2.g.a 4 75.h odd 10 1
375.2.i.a 8 15.e even 4 2
375.2.i.a 8 75.l even 20 2
1875.2.a.a 2 75.h odd 10 1
1875.2.a.d 2 75.j odd 10 1
1875.2.b.b 4 75.l even 20 2
5625.2.a.a 2 25.d even 5 1
5625.2.a.h 2 25.e even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - T_{2}^{3} + T_{2}^{2} - T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(225, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{3} + T^{2} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 5 T^{3} + \cdots + 25 \) Copy content Toggle raw display
$7$ \( (T^{2} - 20)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} - 6 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$13$ \( T^{4} + 2 T^{3} + \cdots + 361 \) Copy content Toggle raw display
$17$ \( T^{4} - 8 T^{3} + \cdots + 121 \) Copy content Toggle raw display
$19$ \( T^{4} + 4 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$23$ \( T^{4} + 10 T^{3} + \cdots + 400 \) Copy content Toggle raw display
$29$ \( T^{4} - 8 T^{3} + \cdots + 841 \) Copy content Toggle raw display
$31$ \( T^{4} + 40 T^{2} + \cdots + 400 \) Copy content Toggle raw display
$37$ \( T^{4} + 15 T^{3} + \cdots + 625 \) Copy content Toggle raw display
$41$ \( T^{4} + 10 T^{2} + \cdots + 25 \) Copy content Toggle raw display
$43$ \( (T^{2} - 2 T - 44)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 2 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$53$ \( T^{4} - 5 T^{3} + \cdots + 25 \) Copy content Toggle raw display
$59$ \( T^{4} + 4 T^{3} + \cdots + 256 \) Copy content Toggle raw display
$61$ \( T^{4} - 2 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$67$ \( T^{4} - 2 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$71$ \( T^{4} + 8 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$73$ \( T^{4} + 10 T^{3} + \cdots + 625 \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} - 18 T^{3} + \cdots + 1936 \) Copy content Toggle raw display
$89$ \( T^{4} - 9 T^{3} + \cdots + 1681 \) Copy content Toggle raw display
$97$ \( T^{4} - 2 T^{3} + \cdots + 361 \) Copy content Toggle raw display
show more
show less