Properties

Label 1300.2.m.a
Level $1300$
Weight $2$
Character orbit 1300.m
Analytic conductor $10.381$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (2 i + 2) q^{3} + 4 i q^{7} + 5 i q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (2 i + 2) q^{3} + 4 i q^{7} + 5 i q^{9} + (4 i - 4) q^{11} + ( - 2 i + 3) q^{13} + ( - i - 1) q^{17} + (8 i - 8) q^{21} + ( - 4 i + 4) q^{23} + (4 i - 4) q^{27} - 2 i q^{29} + ( - 2 i - 2) q^{31} - 16 q^{33} + 2 i q^{37} + (2 i + 10) q^{39} + ( - 7 i - 7) q^{41} + (2 i - 2) q^{43} + 8 i q^{47} - 9 q^{49} - 4 i q^{51} + ( - i - 1) q^{53} + 8 q^{61} - 20 q^{63} + 12 q^{67} + 16 q^{69} + (2 i + 2) q^{71} + 12 q^{73} + ( - 16 i - 16) q^{77} + 16 i q^{79} - q^{81} + 12 i q^{83} + ( - 4 i + 4) q^{87} + (7 i + 7) q^{89} + (12 i + 8) q^{91} - 8 i q^{93} + 2 q^{97} + ( - 20 i - 20) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{3}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{3} - 8 q^{11} + 6 q^{13} - 2 q^{17} - 16 q^{21} + 8 q^{23} - 8 q^{27} - 4 q^{31} - 32 q^{33} + 20 q^{39} - 14 q^{41} - 4 q^{43} - 18 q^{49} - 2 q^{53} + 16 q^{61} - 40 q^{63} + 24 q^{67} + 32 q^{69} + 4 q^{71} + 24 q^{73} - 32 q^{77} - 2 q^{81} + 8 q^{87} + 14 q^{89} + 16 q^{91} + 4 q^{97} - 40 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(301\) \(651\) \(677\)
\(\chi(n)\) \(i\) \(1\) \(-i\)

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
1.00000i
1.00000i
0 2.00000 2.00000i 0 0 0 4.00000i 0 5.00000i 0
593.1 0 2.00000 + 2.00000i 0 0 0 4.00000i 0 5.00000i 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.a 2
5.b even 2 1 260.2.m.a 2
5.c odd 4 1 260.2.r.a yes 2
5.c odd 4 1 1300.2.r.a 2
13.d odd 4 1 1300.2.r.a 2
15.d odd 2 1 2340.2.u.c 2
15.e even 4 1 2340.2.bp.b 2
20.d odd 2 1 1040.2.bg.h 2
20.e even 4 1 1040.2.cd.h 2
65.f even 4 1 260.2.m.a 2
65.g odd 4 1 260.2.r.a yes 2
65.k even 4 1 inner 1300.2.m.a 2
195.n even 4 1 2340.2.bp.b 2
195.u odd 4 1 2340.2.u.c 2
260.l odd 4 1 1040.2.bg.h 2
260.u even 4 1 1040.2.cd.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.2.m.a 2 5.b even 2 1
260.2.m.a 2 65.f even 4 1
260.2.r.a yes 2 5.c odd 4 1
260.2.r.a yes 2 65.g odd 4 1
1040.2.bg.h 2 20.d odd 2 1
1040.2.bg.h 2 260.l odd 4 1
1040.2.cd.h 2 20.e even 4 1
1040.2.cd.h 2 260.u even 4 1
1300.2.m.a 2 1.a even 1 1 trivial
1300.2.m.a 2 65.k even 4 1 inner
1300.2.r.a 2 5.c odd 4 1
1300.2.r.a 2 13.d odd 4 1
2340.2.u.c 2 15.d odd 2 1
2340.2.u.c 2 195.u odd 4 1
2340.2.bp.b 2 15.e even 4 1
2340.2.bp.b 2 195.n even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 4T_{3} + 8 \) acting on \(S_{2}^{\mathrm{new}}(1300, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 4T + 8 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 16 \) Copy content Toggle raw display
$11$ \( T^{2} + 8T + 32 \) Copy content Toggle raw display
$13$ \( T^{2} - 6T + 13 \) Copy content Toggle raw display
$17$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 8T + 32 \) Copy content Toggle raw display
$29$ \( T^{2} + 4 \) Copy content Toggle raw display
$31$ \( T^{2} + 4T + 8 \) Copy content Toggle raw display
$37$ \( T^{2} + 4 \) Copy content Toggle raw display
$41$ \( T^{2} + 14T + 98 \) Copy content Toggle raw display
$43$ \( T^{2} + 4T + 8 \) Copy content Toggle raw display
$47$ \( T^{2} + 64 \) Copy content Toggle raw display
$53$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( (T - 8)^{2} \) Copy content Toggle raw display
$67$ \( (T - 12)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} - 4T + 8 \) Copy content Toggle raw display
$73$ \( (T - 12)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 256 \) Copy content Toggle raw display
$83$ \( T^{2} + 144 \) Copy content Toggle raw display
$89$ \( T^{2} - 14T + 98 \) Copy content Toggle raw display
$97$ \( (T - 2)^{2} \) Copy content Toggle raw display
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