Properties

Label 1300.2.bb.f
Level $1300$
Weight $2$
Character orbit 1300.bb
Analytic conductor $10.381$
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] = [1300,2,Mod(549,1300)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1300, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1300.549");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1300 = 2^{2} \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1300.bb (of order \(6\), degree \(2\), 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: \(10.3805522628\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 52)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 \zeta_{12} q^{3} + ( - \zeta_{12}^{3} + \zeta_{12}) q^{7} + 6 \zeta_{12}^{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 \zeta_{12} q^{3} + ( - \zeta_{12}^{3} + \zeta_{12}) q^{7} + 6 \zeta_{12}^{2} q^{9} + ( - 5 \zeta_{12}^{2} + 5) q^{11} + ( - \zeta_{12}^{3} + 4 \zeta_{12}) q^{13} + ( - 3 \zeta_{12}^{3} + 3 \zeta_{12}) q^{17} - 3 \zeta_{12}^{2} q^{19} + 3 q^{21} - \zeta_{12} q^{23} + 9 \zeta_{12}^{3} q^{27} + (\zeta_{12}^{2} - 1) q^{29} - 8 q^{31} + ( - 15 \zeta_{12}^{3} + 15 \zeta_{12}) q^{33} - 3 \zeta_{12} q^{37} + (9 \zeta_{12}^{2} + 3) q^{39} + (3 \zeta_{12}^{2} - 3) q^{41} + (\zeta_{12}^{3} - \zeta_{12}) q^{43} + 4 \zeta_{12}^{3} q^{47} + (6 \zeta_{12}^{2} - 6) q^{49} + 9 q^{51} + 6 \zeta_{12}^{3} q^{53} - 9 \zeta_{12}^{3} q^{57} + 5 \zeta_{12}^{2} q^{59} + 5 \zeta_{12}^{2} q^{61} + 6 \zeta_{12} q^{63} - 7 \zeta_{12} q^{67} - 3 \zeta_{12}^{2} q^{69} + 11 \zeta_{12}^{2} q^{71} - 14 \zeta_{12}^{3} q^{73} - 5 \zeta_{12}^{3} q^{77} + 4 q^{79} + (9 \zeta_{12}^{2} - 9) q^{81} - 12 \zeta_{12}^{3} q^{83} + (3 \zeta_{12}^{3} - 3 \zeta_{12}) q^{87} + (9 \zeta_{12}^{2} - 9) q^{89} + ( - \zeta_{12}^{2} + 4) q^{91} - 24 \zeta_{12} q^{93} + (\zeta_{12}^{3} - \zeta_{12}) q^{97} + 30 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 12 q^{9} + 10 q^{11} - 6 q^{19} + 12 q^{21} - 2 q^{29} - 32 q^{31} + 30 q^{39} - 6 q^{41} - 12 q^{49} + 36 q^{51} + 10 q^{59} + 10 q^{61} - 6 q^{69} + 22 q^{71} + 16 q^{79} - 18 q^{81} - 18 q^{89} + 14 q^{91} + 120 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(301\) \(651\) \(677\)
\(\chi(n)\) \(-\zeta_{12}^{2}\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
549.1
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
0 −2.59808 + 1.50000i 0 0 0 −0.866025 0.500000i 0 3.00000 5.19615i 0
549.2 0 2.59808 1.50000i 0 0 0 0.866025 + 0.500000i 0 3.00000 5.19615i 0
1049.1 0 −2.59808 1.50000i 0 0 0 −0.866025 + 0.500000i 0 3.00000 + 5.19615i 0
1049.2 0 2.59808 + 1.50000i 0 0 0 0.866025 0.500000i 0 3.00000 + 5.19615i 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
13.c even 3 1 inner
65.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1300.2.bb.f 4
5.b even 2 1 inner 1300.2.bb.f 4
5.c odd 4 1 52.2.e.a 2
5.c odd 4 1 1300.2.i.f 2
13.c even 3 1 inner 1300.2.bb.f 4
15.e even 4 1 468.2.l.a 2
20.e even 4 1 208.2.i.d 2
35.f even 4 1 2548.2.k.d 2
35.k even 12 1 2548.2.i.h 2
35.k even 12 1 2548.2.l.a 2
35.l odd 12 1 2548.2.i.a 2
35.l odd 12 1 2548.2.l.h 2
40.i odd 4 1 832.2.i.j 2
40.k even 4 1 832.2.i.a 2
60.l odd 4 1 1872.2.t.f 2
65.f even 4 1 676.2.h.b 4
65.h odd 4 1 676.2.e.a 2
65.k even 4 1 676.2.h.b 4
65.n even 6 1 inner 1300.2.bb.f 4
65.o even 12 1 676.2.d.d 2
65.o even 12 1 676.2.h.b 4
65.q odd 12 1 52.2.e.a 2
65.q odd 12 1 676.2.a.e 1
65.q odd 12 1 1300.2.i.f 2
65.r odd 12 1 676.2.a.d 1
65.r odd 12 1 676.2.e.a 2
65.t even 12 1 676.2.d.d 2
65.t even 12 1 676.2.h.b 4
195.bc odd 12 1 6084.2.b.l 2
195.bf even 12 1 6084.2.a.k 1
195.bl even 12 1 468.2.l.a 2
195.bl even 12 1 6084.2.a.f 1
195.bn odd 12 1 6084.2.b.l 2
260.be odd 12 1 2704.2.f.a 2
260.bg even 12 1 2704.2.a.a 1
260.bj even 12 1 208.2.i.d 2
260.bj even 12 1 2704.2.a.b 1
260.bl odd 12 1 2704.2.f.a 2
455.cq odd 12 1 2548.2.l.h 2
455.cs even 12 1 2548.2.l.a 2
455.cx odd 12 1 2548.2.i.a 2
455.db even 12 1 2548.2.i.h 2
455.dc even 12 1 2548.2.k.d 2
520.cm even 12 1 832.2.i.a 2
520.cq odd 12 1 832.2.i.j 2
780.cj odd 12 1 1872.2.t.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
52.2.e.a 2 5.c odd 4 1
52.2.e.a 2 65.q odd 12 1
208.2.i.d 2 20.e even 4 1
208.2.i.d 2 260.bj even 12 1
468.2.l.a 2 15.e even 4 1
468.2.l.a 2 195.bl even 12 1
676.2.a.d 1 65.r odd 12 1
676.2.a.e 1 65.q odd 12 1
676.2.d.d 2 65.o even 12 1
676.2.d.d 2 65.t even 12 1
676.2.e.a 2 65.h odd 4 1
676.2.e.a 2 65.r odd 12 1
676.2.h.b 4 65.f even 4 1
676.2.h.b 4 65.k even 4 1
676.2.h.b 4 65.o even 12 1
676.2.h.b 4 65.t even 12 1
832.2.i.a 2 40.k even 4 1
832.2.i.a 2 520.cm even 12 1
832.2.i.j 2 40.i odd 4 1
832.2.i.j 2 520.cq odd 12 1
1300.2.i.f 2 5.c odd 4 1
1300.2.i.f 2 65.q odd 12 1
1300.2.bb.f 4 1.a even 1 1 trivial
1300.2.bb.f 4 5.b even 2 1 inner
1300.2.bb.f 4 13.c even 3 1 inner
1300.2.bb.f 4 65.n even 6 1 inner
1872.2.t.f 2 60.l odd 4 1
1872.2.t.f 2 780.cj odd 12 1
2548.2.i.a 2 35.l odd 12 1
2548.2.i.a 2 455.cx odd 12 1
2548.2.i.h 2 35.k even 12 1
2548.2.i.h 2 455.db even 12 1
2548.2.k.d 2 35.f even 4 1
2548.2.k.d 2 455.dc even 12 1
2548.2.l.a 2 35.k even 12 1
2548.2.l.a 2 455.cs even 12 1
2548.2.l.h 2 35.l odd 12 1
2548.2.l.h 2 455.cq odd 12 1
2704.2.a.a 1 260.bg even 12 1
2704.2.a.b 1 260.bj even 12 1
2704.2.f.a 2 260.be odd 12 1
2704.2.f.a 2 260.bl odd 12 1
6084.2.a.f 1 195.bl even 12 1
6084.2.a.k 1 195.bf even 12 1
6084.2.b.l 2 195.bc odd 12 1
6084.2.b.l 2 195.bn odd 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1300, [\chi])\):

\( T_{3}^{4} - 9T_{3}^{2} + 81 \) Copy content Toggle raw display
\( T_{7}^{4} - T_{7}^{2} + 1 \) Copy content Toggle raw display
\( T_{19}^{2} + 3T_{19} + 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 9T^{2} + 81 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$11$ \( (T^{2} - 5 T + 25)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} - 22T^{2} + 169 \) Copy content Toggle raw display
$17$ \( T^{4} - 9T^{2} + 81 \) Copy content Toggle raw display
$19$ \( (T^{2} + 3 T + 9)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$29$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$31$ \( (T + 8)^{4} \) Copy content Toggle raw display
$37$ \( T^{4} - 9T^{2} + 81 \) Copy content Toggle raw display
$41$ \( (T^{2} + 3 T + 9)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$47$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 5 T + 25)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 5 T + 25)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 49T^{2} + 2401 \) Copy content Toggle raw display
$71$ \( (T^{2} - 11 T + 121)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 196)^{2} \) Copy content Toggle raw display
$79$ \( (T - 4)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 144)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 9 T + 81)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
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