Properties

Label 725.2.k.a
Level $725$
Weight $2$
Character orbit 725.k
Analytic conductor $5.789$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [725,2,Mod(146,725)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(725, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([6, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("725.146");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 725 = 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 725.k (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: \(5.78915414654\)
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: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + ( - \zeta_{10}^{3} + \zeta_{10} - 1) q^{3} + 2 \zeta_{10}^{3} q^{4} + ( - 2 \zeta_{10}^{3} + \cdots - 2 \zeta_{10}) q^{5}+ \cdots + (3 \zeta_{10}^{2} - 2 \zeta_{10} + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{10}^{3} + \zeta_{10} - 1) q^{3} + 2 \zeta_{10}^{3} q^{4} + ( - 2 \zeta_{10}^{3} + \cdots - 2 \zeta_{10}) q^{5}+ \cdots + ( - 4 \zeta_{10}^{3} + 4 \zeta_{10}^{2} + 5) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} + 2 q^{4} - 5 q^{5} - 12 q^{7} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{3} + 2 q^{4} - 5 q^{5} - 12 q^{7} + 7 q^{9} + 4 q^{11} - 2 q^{12} + q^{13} + 10 q^{15} - 4 q^{16} + 4 q^{17} + 17 q^{19} + 12 q^{21} + 18 q^{23} - 5 q^{25} + 5 q^{27} - 6 q^{28} - q^{29} + 9 q^{31} + q^{33} + 15 q^{35} - 14 q^{36} + 22 q^{37} - q^{39} - 2 q^{41} + 2 q^{44} - 10 q^{45} + 6 q^{47} + 4 q^{48} + 8 q^{49} - 4 q^{51} - 2 q^{52} - 12 q^{53} - 10 q^{55} - 32 q^{57} + 5 q^{59} + 20 q^{60} + 25 q^{61} - 21 q^{63} + 8 q^{64} + 5 q^{65} + 5 q^{67} - 8 q^{68} - 33 q^{69} + 19 q^{71} - 12 q^{73} - 20 q^{75} - 4 q^{76} - 12 q^{77} - 15 q^{79} - 20 q^{80} + 14 q^{81} + 9 q^{83} + 6 q^{84} + 10 q^{85} + q^{87} - 10 q^{89} - 3 q^{91} + 34 q^{92} - 4 q^{93} + 20 q^{95} - 3 q^{97} + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/725\mathbb{Z}\right)^\times\).

\(n\) \(176\) \(552\)
\(\chi(n)\) \(1\) \(-\zeta_{10}^{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
146.1
−0.309017 0.951057i
0.809017 0.587785i
0.809017 + 0.587785i
−0.309017 + 0.951057i
0 −2.11803 1.53884i 1.61803 + 1.17557i −1.80902 + 1.31433i 0 −3.00000 0 1.19098 + 3.66547i 0
291.1 0 0.118034 + 0.363271i −0.618034 1.90211i −0.690983 + 2.12663i 0 −3.00000 0 2.30902 1.67760i 0
436.1 0 0.118034 0.363271i −0.618034 + 1.90211i −0.690983 2.12663i 0 −3.00000 0 2.30902 + 1.67760i 0
581.1 0 −2.11803 + 1.53884i 1.61803 1.17557i −1.80902 1.31433i 0 −3.00000 0 1.19098 3.66547i 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
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 725.2.k.a 4
25.d even 5 1 inner 725.2.k.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
725.2.k.a 4 1.a even 1 1 trivial
725.2.k.a 4 25.d even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{2}^{\mathrm{new}}(725, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 4 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{4} + 5 T^{3} + \cdots + 25 \) Copy content Toggle raw display
$7$ \( (T + 3)^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - 4 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{4} - T^{3} + T^{2} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{4} - 4 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$19$ \( T^{4} - 17 T^{3} + \cdots + 3721 \) Copy content Toggle raw display
$23$ \( T^{4} - 18 T^{3} + \cdots + 3721 \) Copy content Toggle raw display
$29$ \( T^{4} + T^{3} + T^{2} + \cdots + 1 \) Copy content Toggle raw display
$31$ \( T^{4} - 9 T^{3} + \cdots + 361 \) Copy content Toggle raw display
$37$ \( T^{4} - 22 T^{3} + \cdots + 5776 \) Copy content Toggle raw display
$41$ \( T^{4} + 2 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$43$ \( (T^{2} - 45)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 6 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$53$ \( T^{4} + 12 T^{3} + \cdots + 1296 \) Copy content Toggle raw display
$59$ \( T^{4} - 5 T^{3} + \cdots + 9025 \) Copy content Toggle raw display
$61$ \( T^{4} - 25 T^{3} + \cdots + 21025 \) Copy content Toggle raw display
$67$ \( T^{4} - 5 T^{3} + \cdots + 9025 \) Copy content Toggle raw display
$71$ \( T^{4} - 19 T^{3} + \cdots + 3481 \) Copy content Toggle raw display
$73$ \( T^{4} + 12 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$79$ \( T^{4} + 15 T^{3} + \cdots + 3025 \) Copy content Toggle raw display
$83$ \( T^{4} - 9 T^{3} + \cdots + 6561 \) Copy content Toggle raw display
$89$ \( T^{4} + 10 T^{3} + \cdots + 400 \) Copy content Toggle raw display
$97$ \( T^{4} + 3 T^{3} + \cdots + 81 \) Copy content Toggle raw display
show more
show less