Properties

Label 725.2.d.a
Level $725$
Weight $2$
Character orbit 725.d
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(724,725)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(725, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("725.724");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 725 = 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 725.d (of order \(2\), degree \(1\), not 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: \(\Q(i, \sqrt{5})\)
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: no (minimal twist has level 29)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} - \beta_{3} q^{3} + 3 q^{4} + 5 q^{6} - 2 \beta_1 q^{7} - \beta_{3} q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} - \beta_{3} q^{3} + 3 q^{4} + 5 q^{6} - 2 \beta_1 q^{7} - \beta_{3} q^{8} + 2 q^{9} - \beta_{2} q^{11} - 3 \beta_{3} q^{12} - \beta_1 q^{13} + 2 \beta_{2} q^{14} - q^{16} + 2 \beta_{3} q^{17} - 2 \beta_{3} q^{18} + 2 \beta_{2} q^{21} + 5 \beta_1 q^{22} + 6 \beta_1 q^{23} + 5 q^{24} + \beta_{2} q^{26} + \beta_{3} q^{27} - 6 \beta_1 q^{28} + (2 \beta_{2} + 3) q^{29} - 3 \beta_{2} q^{31} + 3 \beta_{3} q^{32} + 5 \beta_1 q^{33} - 10 q^{34} + 6 q^{36} + \beta_{2} q^{39} + 2 \beta_{2} q^{41} - 10 \beta_1 q^{42} - 3 \beta_{3} q^{43} - 3 \beta_{2} q^{44} - 6 \beta_{2} q^{46} - \beta_{3} q^{47} + \beta_{3} q^{48} + 3 q^{49} - 10 q^{51} - 3 \beta_1 q^{52} - 9 \beta_1 q^{53} - 5 q^{54} + 2 \beta_{2} q^{56} + ( - 3 \beta_{3} - 10 \beta_1) q^{58} - 6 q^{59} - 6 \beta_{2} q^{61} + 15 \beta_1 q^{62} - 4 \beta_1 q^{63} - 13 q^{64} - 5 \beta_{2} q^{66} - 8 \beta_1 q^{67} + 6 \beta_{3} q^{68} - 6 \beta_{2} q^{69} - 2 \beta_{3} q^{72} - 2 \beta_{3} q^{77} - 5 \beta_1 q^{78} - 3 \beta_{2} q^{79} - 11 q^{81} - 10 \beta_1 q^{82} - 6 \beta_1 q^{83} + 6 \beta_{2} q^{84} + 15 q^{86} + ( - 3 \beta_{3} - 10 \beta_1) q^{87} + 5 \beta_1 q^{88} - 2 \beta_{2} q^{89} - 2 q^{91} + 18 \beta_1 q^{92} + 15 \beta_1 q^{93} + 5 q^{94} - 15 q^{96} - 6 \beta_{3} q^{97} - 3 \beta_{3} q^{98} - 2 \beta_{2} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 12 q^{4} + 20 q^{6} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 12 q^{4} + 20 q^{6} + 8 q^{9} - 4 q^{16} + 20 q^{24} + 12 q^{29} - 40 q^{34} + 24 q^{36} + 12 q^{49} - 40 q^{51} - 20 q^{54} - 24 q^{59} - 52 q^{64} - 44 q^{81} + 60 q^{86} - 8 q^{91} + 20 q^{94} - 60 q^{96}+O(q^{100}) \) 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 \(\left(\mathbb{Z}/725\mathbb{Z}\right)^\times\).

\(n\) \(176\) \(552\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
724.1
0.618034i
0.618034i
1.61803i
1.61803i
−2.23607 −2.23607 3.00000 0 5.00000 2.00000i −2.23607 2.00000 0
724.2 −2.23607 −2.23607 3.00000 0 5.00000 2.00000i −2.23607 2.00000 0
724.3 2.23607 2.23607 3.00000 0 5.00000 2.00000i 2.23607 2.00000 0
724.4 2.23607 2.23607 3.00000 0 5.00000 2.00000i 2.23607 2.00000 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
29.b even 2 1 inner
145.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 725.2.d.a 4
5.b even 2 1 inner 725.2.d.a 4
5.c odd 4 1 29.2.b.a 2
5.c odd 4 1 725.2.c.c 2
15.e even 4 1 261.2.c.a 2
20.e even 4 1 464.2.e.a 2
29.b even 2 1 inner 725.2.d.a 4
35.f even 4 1 1421.2.b.b 2
40.i odd 4 1 1856.2.e.g 2
40.k even 4 1 1856.2.e.f 2
60.l odd 4 1 4176.2.o.k 2
145.d even 2 1 inner 725.2.d.a 4
145.e even 4 1 841.2.a.b 2
145.h odd 4 1 29.2.b.a 2
145.h odd 4 1 725.2.c.c 2
145.j even 4 1 841.2.a.b 2
145.o even 28 6 841.2.d.h 12
145.p odd 28 6 841.2.e.g 12
145.q odd 28 6 841.2.e.g 12
145.t even 28 6 841.2.d.h 12
435.i odd 4 1 7569.2.a.i 2
435.p even 4 1 261.2.c.a 2
435.t odd 4 1 7569.2.a.i 2
580.o even 4 1 464.2.e.a 2
1015.l even 4 1 1421.2.b.b 2
1160.bb even 4 1 1856.2.e.f 2
1160.be odd 4 1 1856.2.e.g 2
1740.v odd 4 1 4176.2.o.k 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.2.b.a 2 5.c odd 4 1
29.2.b.a 2 145.h odd 4 1
261.2.c.a 2 15.e even 4 1
261.2.c.a 2 435.p even 4 1
464.2.e.a 2 20.e even 4 1
464.2.e.a 2 580.o even 4 1
725.2.c.c 2 5.c odd 4 1
725.2.c.c 2 145.h odd 4 1
725.2.d.a 4 1.a even 1 1 trivial
725.2.d.a 4 5.b even 2 1 inner
725.2.d.a 4 29.b even 2 1 inner
725.2.d.a 4 145.d even 2 1 inner
841.2.a.b 2 145.e even 4 1
841.2.a.b 2 145.j even 4 1
841.2.d.h 12 145.o even 28 6
841.2.d.h 12 145.t even 28 6
841.2.e.g 12 145.p odd 28 6
841.2.e.g 12 145.q odd 28 6
1421.2.b.b 2 35.f even 4 1
1421.2.b.b 2 1015.l even 4 1
1856.2.e.f 2 40.k even 4 1
1856.2.e.f 2 1160.bb even 4 1
1856.2.e.g 2 40.i odd 4 1
1856.2.e.g 2 1160.be odd 4 1
4176.2.o.k 2 60.l odd 4 1
4176.2.o.k 2 1740.v odd 4 1
7569.2.a.i 2 435.i odd 4 1
7569.2.a.i 2 435.t odd 4 1

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
$3$ \( (T^{2} - 5)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 5)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 20)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 6 T + 29)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 45)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 20)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 45)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 5)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 81)^{2} \) Copy content Toggle raw display
$59$ \( (T + 6)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 180)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 45)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 20)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 180)^{2} \) Copy content Toggle raw display
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