Properties

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

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [725,2,Mod(307,725)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(725, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([1, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("725.307");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 725 = 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 725.j (of order \(4\), degree \(2\), 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: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + 3 q^{4} + ( - \beta_{3} + \beta_{2}) q^{7} + \beta_{3} q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} + 3 q^{4} + ( - \beta_{3} + \beta_{2}) q^{7} + \beta_{3} q^{8} + 3 q^{9} + (4 \beta_1 + 4) q^{11} + (2 \beta_{3} - 2 \beta_{2}) q^{13} + (5 \beta_1 - 5) q^{14} - q^{16} - 2 \beta_{3} q^{17} + 3 \beta_{3} q^{18} + ( - 2 \beta_1 + 2) q^{19} + (4 \beta_{3} + 4 \beta_{2}) q^{22} + (\beta_{3} + \beta_{2}) q^{23} + ( - 10 \beta_1 + 10) q^{26} + ( - 3 \beta_{3} + 3 \beta_{2}) q^{28} + (5 \beta_1 - 2) q^{29} + ( - 4 \beta_1 - 4) q^{31} - 3 \beta_{3} q^{32} - 10 q^{34} + 9 q^{36} - 2 \beta_{2} q^{37} + (2 \beta_{3} - 2 \beta_{2}) q^{38} + ( - \beta_1 + 1) q^{41} - 2 \beta_{2} q^{43} + (12 \beta_1 + 12) q^{44} + (5 \beta_1 + 5) q^{46} + 2 \beta_{2} q^{47} - 3 \beta_1 q^{49} + (6 \beta_{3} - 6 \beta_{2}) q^{52} + ( - 4 \beta_{3} - 4 \beta_{2}) q^{53} + (5 \beta_1 - 5) q^{56} + ( - 2 \beta_{3} + 5 \beta_{2}) q^{58} + ( - 9 \beta_1 - 9) q^{61} + ( - 4 \beta_{3} - 4 \beta_{2}) q^{62} + ( - 3 \beta_{3} + 3 \beta_{2}) q^{63} - 13 q^{64} + ( - \beta_{3} - \beta_{2}) q^{67} - 6 \beta_{3} q^{68} + 12 \beta_1 q^{71} + 3 \beta_{3} q^{72} - 2 \beta_{3} q^{73} - 10 \beta_1 q^{74} + ( - 6 \beta_1 + 6) q^{76} - 8 \beta_{3} q^{77} + ( - 2 \beta_1 + 2) q^{79} + 9 q^{81} + (\beta_{3} - \beta_{2}) q^{82} + ( - 3 \beta_{3} - 3 \beta_{2}) q^{83} - 10 \beta_1 q^{86} + (4 \beta_{3} + 4 \beta_{2}) q^{88} + ( - 7 \beta_1 + 7) q^{89} + 20 \beta_1 q^{91} + (3 \beta_{3} + 3 \beta_{2}) q^{92} + 10 \beta_1 q^{94} + 2 \beta_{2} q^{97} - 3 \beta_{2} q^{98} + (12 \beta_1 + 12) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 12 q^{4} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 12 q^{4} + 12 q^{9} + 16 q^{11} - 20 q^{14} - 4 q^{16} + 8 q^{19} + 40 q^{26} - 8 q^{29} - 16 q^{31} - 40 q^{34} + 36 q^{36} + 4 q^{41} + 48 q^{44} + 20 q^{46} - 20 q^{56} - 36 q^{61} - 52 q^{64} + 24 q^{76} + 8 q^{79} + 36 q^{81} + 28 q^{89} + 48 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 3x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{3} + 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} + 4\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{2} + 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{2} + 2\beta_1 \) Copy content Toggle raw display

Character values

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

\(n\) \(176\) \(552\)
\(\chi(n)\) \(\beta_{1}\) \(-\beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
307.1
1.61803i
0.618034i
1.61803i
0.618034i
−2.23607 0 3.00000 0 0 2.23607 + 2.23607i −2.23607 3.00000 0
307.2 2.23607 0 3.00000 0 0 −2.23607 2.23607i 2.23607 3.00000 0
418.1 −2.23607 0 3.00000 0 0 2.23607 2.23607i −2.23607 3.00000 0
418.2 2.23607 0 3.00000 0 0 −2.23607 + 2.23607i 2.23607 3.00000 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
5.b even 2 1 inner
145.e even 4 1 inner
145.j even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 725.2.j.a yes 4
5.b even 2 1 inner 725.2.j.a yes 4
5.c odd 4 2 725.2.e.a 4
29.c odd 4 1 725.2.e.a 4
145.e even 4 1 inner 725.2.j.a yes 4
145.f odd 4 1 725.2.e.a 4
145.j even 4 1 inner 725.2.j.a yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
725.2.e.a 4 5.c odd 4 2
725.2.e.a 4 29.c odd 4 1
725.2.e.a 4 145.f odd 4 1
725.2.j.a yes 4 1.a even 1 1 trivial
725.2.j.a yes 4 5.b even 2 1 inner
725.2.j.a yes 4 145.e even 4 1 inner
725.2.j.a yes 4 145.j even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 5 \) acting on \(S_{2}^{\mathrm{new}}(725, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 5)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 100 \) Copy content Toggle raw display
$11$ \( (T^{2} - 8 T + 32)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 1600 \) Copy content Toggle raw display
$17$ \( (T^{2} - 20)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 4 T + 8)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 100 \) Copy content Toggle raw display
$29$ \( (T^{2} + 4 T + 29)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 8 T + 32)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 20)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 2 T + 2)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 20)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 20)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 25600 \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 18 T + 162)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 100 \) Copy content Toggle raw display
$71$ \( (T^{2} + 144)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 20)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 4 T + 8)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 8100 \) Copy content Toggle raw display
$89$ \( (T^{2} - 14 T + 98)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 20)^{2} \) Copy content Toggle raw display
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