Properties

Label 1300.2.m.b
Level $1300$
Weight $2$
Character orbit 1300.m
Analytic conductor $10.381$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1300,2,Mod(57,1300)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1300, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 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("1300.57");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1300 = 2^{2} \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1300.m (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: \(10.3805522628\)
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: \( 2 \)
Twist minimal: no (minimal twist has level 260)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{9} + (\beta_{3} + 2 \beta_{2} - 2) q^{11} + (3 \beta_{2} + 2) q^{13} + (\beta_{2} + 2 \beta_1 + 1) q^{17} + ( - 3 \beta_{3} - 4 \beta_{2} + 4) q^{19} + ( - \beta_{3} + 4 \beta_{2} - 4) q^{23} + ( - 2 \beta_{3} - 2 \beta_{2} + 2) q^{27} + (\beta_{3} + 4 \beta_{2} + \beta_1) q^{29} + (4 \beta_{2} + 3 \beta_1 + 4) q^{31} + (\beta_{3} - \beta_1 - 2) q^{33} + (3 \beta_{3} + 6 \beta_{2} + 3 \beta_1) q^{37} + (3 \beta_{3} + 2 \beta_1) q^{39} + (5 \beta_{2} - 2 \beta_1 + 5) q^{41} + ( - 3 \beta_{3} - 2 \beta_{2} + 2) q^{43} + ( - 4 \beta_{3} - 4 \beta_{2} - 4 \beta_1) q^{47} + 7 q^{49} + ( - \beta_{3} + 4 \beta_{2} - \beta_1) q^{51} + (\beta_{2} - 4 \beta_1 + 1) q^{53} + ( - \beta_{3} + \beta_1 + 6) q^{57} + ( - 4 \beta_{2} + \beta_1 - 4) q^{59} + ( - 3 \beta_{3} + 3 \beta_1 + 2) q^{61} + (3 \beta_{3} - 3 \beta_1 + 4) q^{67} + (5 \beta_{3} - 5 \beta_1 + 2) q^{69} + (2 \beta_{2} + \beta_1 + 2) q^{71} + (3 \beta_{3} - 3 \beta_1 - 8) q^{73} + (3 \beta_{3} + 3 \beta_1) q^{79} + (3 \beta_{3} - 3 \beta_1 + 1) q^{81} + (2 \beta_{3} + 8 \beta_{2} + 2 \beta_1) q^{83} + (2 \beta_{3} + 2 \beta_{2} - 2) q^{87} + ( - \beta_{2} - 2 \beta_1 - 1) q^{89} + (\beta_{3} + 6 \beta_{2} + \beta_1) q^{93} + (6 \beta_{3} - 6 \beta_1 - 8) q^{97} + (4 \beta_{2} + 3 \beta_1 + 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{3} - 6 q^{11} + 8 q^{13} + 10 q^{19} - 18 q^{23} + 4 q^{27} + 10 q^{31} - 4 q^{33} + 2 q^{39} + 24 q^{41} + 2 q^{43} + 28 q^{49} + 12 q^{53} + 20 q^{57} - 18 q^{59} - 4 q^{61} + 28 q^{67} + 28 q^{69} + 6 q^{71} - 20 q^{73} + 16 q^{81} - 4 q^{87} - 8 q^{97} + 10 q^{99}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( \nu^{2} + \nu + 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} + 2\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{2} + \nu - 1 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{3} + \beta _1 - 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{3} + \beta_{2} - \beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1300\mathbb{Z}\right)^\times\).

\(n\) \(301\) \(651\) \(677\)
\(\chi(n)\) \(\beta_{2}\) \(1\) \(-\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
57.1
1.61803i
0.618034i
1.61803i
0.618034i
0 −1.61803 + 1.61803i 0 0 0 0 0 2.23607i 0
57.2 0 0.618034 0.618034i 0 0 0 0 0 2.23607i 0
593.1 0 −1.61803 1.61803i 0 0 0 0 0 2.23607i 0
593.2 0 0.618034 + 0.618034i 0 0 0 0 0 2.23607i 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
65.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1300.2.m.b 4
5.b even 2 1 260.2.m.b 4
5.c odd 4 1 260.2.r.b yes 4
5.c odd 4 1 1300.2.r.b 4
13.d odd 4 1 1300.2.r.b 4
15.d odd 2 1 2340.2.u.f 4
15.e even 4 1 2340.2.bp.e 4
20.d odd 2 1 1040.2.bg.j 4
20.e even 4 1 1040.2.cd.j 4
65.f even 4 1 260.2.m.b 4
65.g odd 4 1 260.2.r.b yes 4
65.k even 4 1 inner 1300.2.m.b 4
195.n even 4 1 2340.2.bp.e 4
195.u odd 4 1 2340.2.u.f 4
260.l odd 4 1 1040.2.bg.j 4
260.u even 4 1 1040.2.cd.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.2.m.b 4 5.b even 2 1
260.2.m.b 4 65.f even 4 1
260.2.r.b yes 4 5.c odd 4 1
260.2.r.b yes 4 65.g odd 4 1
1040.2.bg.j 4 20.d odd 2 1
1040.2.bg.j 4 260.l odd 4 1
1040.2.cd.j 4 20.e even 4 1
1040.2.cd.j 4 260.u even 4 1
1300.2.m.b 4 1.a even 1 1 trivial
1300.2.m.b 4 65.k even 4 1 inner
1300.2.r.b 4 5.c odd 4 1
1300.2.r.b 4 13.d odd 4 1
2340.2.u.f 4 15.d odd 2 1
2340.2.u.f 4 195.u odd 4 1
2340.2.bp.e 4 15.e even 4 1
2340.2.bp.e 4 195.n even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 2T_{3}^{3} + 2T_{3}^{2} - 4T_{3} + 4 \) acting on \(S_{2}^{\mathrm{new}}(1300, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 2 T^{3} + 2 T^{2} - 4 T + 4 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + 6 T^{3} + 18 T^{2} + 12 T + 4 \) Copy content Toggle raw display
$13$ \( (T^{2} - 4 T + 13)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 100 \) Copy content Toggle raw display
$19$ \( T^{4} - 10 T^{3} + 50 T^{2} + \cdots + 100 \) Copy content Toggle raw display
$23$ \( T^{4} + 18 T^{3} + 162 T^{2} + \cdots + 1444 \) Copy content Toggle raw display
$29$ \( T^{4} + 28T^{2} + 16 \) Copy content Toggle raw display
$31$ \( T^{4} - 10 T^{3} + 50 T^{2} + \cdots + 100 \) Copy content Toggle raw display
$37$ \( T^{4} + 108T^{2} + 1296 \) Copy content Toggle raw display
$41$ \( T^{4} - 24 T^{3} + 288 T^{2} + \cdots + 3844 \) Copy content Toggle raw display
$43$ \( T^{4} - 2 T^{3} + 2 T^{2} + 44 T + 484 \) Copy content Toggle raw display
$47$ \( (T^{2} + 80)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} - 12 T^{3} + 72 T^{2} + \cdots + 484 \) Copy content Toggle raw display
$59$ \( T^{4} + 18 T^{3} + 162 T^{2} + \cdots + 1444 \) Copy content Toggle raw display
$61$ \( (T^{2} + 2 T - 44)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 14 T + 4)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} - 6 T^{3} + 18 T^{2} - 12 T + 4 \) Copy content Toggle raw display
$73$ \( (T^{2} + 10 T - 20)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + 108T^{2} + 1296 \) Copy content Toggle raw display
$83$ \( T^{4} + 112T^{2} + 256 \) Copy content Toggle raw display
$89$ \( T^{4} + 100 \) Copy content Toggle raw display
$97$ \( (T^{2} + 4 T - 176)^{2} \) Copy content Toggle raw display
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