Properties

Label 1300.2.bb.c
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 260)
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.

\(f(q)\) \(=\) \( q + \zeta_{12} q^{3} + (5 \zeta_{12}^{3} - 5 \zeta_{12}) q^{7} - 2 \zeta_{12}^{2} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{12} q^{3} + (5 \zeta_{12}^{3} - 5 \zeta_{12}) q^{7} - 2 \zeta_{12}^{2} q^{9} + ( - 5 \zeta_{12}^{2} + 5) q^{11} + ( - 3 \zeta_{12}^{3} + 4 \zeta_{12}) q^{13} + ( - \zeta_{12}^{3} + \zeta_{12}) q^{17} - 3 \zeta_{12}^{2} q^{19} - 5 q^{21} - 3 \zeta_{12} q^{23} - 5 \zeta_{12}^{3} q^{27} + (\zeta_{12}^{2} - 1) q^{29} + ( - 5 \zeta_{12}^{3} + 5 \zeta_{12}) q^{33} + 7 \zeta_{12} q^{37} + (\zeta_{12}^{2} + 3) q^{39} + ( - 5 \zeta_{12}^{2} + 5) q^{41} + ( - 5 \zeta_{12}^{3} + 5 \zeta_{12}) q^{43} - 12 \zeta_{12}^{3} q^{47} + ( - 18 \zeta_{12}^{2} + 18) q^{49} + q^{51} + 2 \zeta_{12}^{3} q^{53} - 3 \zeta_{12}^{3} q^{57} - 11 \zeta_{12}^{2} q^{59} + 13 \zeta_{12}^{2} q^{61} + 10 \zeta_{12} q^{63} + 3 \zeta_{12} q^{67} - 3 \zeta_{12}^{2} q^{69} - 13 \zeta_{12}^{2} q^{71} - 2 \zeta_{12}^{3} q^{73} + 25 \zeta_{12}^{3} q^{77} + 4 q^{79} + (\zeta_{12}^{2} - 1) q^{81} + 12 \zeta_{12}^{3} q^{83} + (\zeta_{12}^{3} - \zeta_{12}) q^{87} + ( - 7 \zeta_{12}^{2} + 7) q^{89} + (15 \zeta_{12}^{2} - 20) q^{91} + (11 \zeta_{12}^{3} - 11 \zeta_{12}) q^{97} - 10 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{9} + 10 q^{11} - 6 q^{19} - 20 q^{21} - 2 q^{29} + 14 q^{39} + 10 q^{41} + 36 q^{49} + 4 q^{51} - 22 q^{59} + 26 q^{61} - 6 q^{69} - 26 q^{71} + 16 q^{79} - 2 q^{81} + 14 q^{89} - 50 q^{91} - 40 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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 −0.866025 + 0.500000i 0 0 0 4.33013 + 2.50000i 0 −1.00000 + 1.73205i 0
549.2 0 0.866025 0.500000i 0 0 0 −4.33013 2.50000i 0 −1.00000 + 1.73205i 0
1049.1 0 −0.866025 0.500000i 0 0 0 4.33013 2.50000i 0 −1.00000 1.73205i 0
1049.2 0 0.866025 + 0.500000i 0 0 0 −4.33013 + 2.50000i 0 −1.00000 1.73205i 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.c 4
5.b even 2 1 inner 1300.2.bb.c 4
5.c odd 4 1 260.2.i.c 2
5.c odd 4 1 1300.2.i.c 2
13.c even 3 1 inner 1300.2.bb.c 4
15.e even 4 1 2340.2.q.c 2
20.e even 4 1 1040.2.q.f 2
65.n even 6 1 inner 1300.2.bb.c 4
65.o even 12 1 3380.2.f.c 2
65.q odd 12 1 260.2.i.c 2
65.q odd 12 1 1300.2.i.c 2
65.q odd 12 1 3380.2.a.d 1
65.r odd 12 1 3380.2.a.e 1
65.t even 12 1 3380.2.f.c 2
195.bl even 12 1 2340.2.q.c 2
260.bj even 12 1 1040.2.q.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.2.i.c 2 5.c odd 4 1
260.2.i.c 2 65.q odd 12 1
1040.2.q.f 2 20.e even 4 1
1040.2.q.f 2 260.bj even 12 1
1300.2.i.c 2 5.c odd 4 1
1300.2.i.c 2 65.q odd 12 1
1300.2.bb.c 4 1.a even 1 1 trivial
1300.2.bb.c 4 5.b even 2 1 inner
1300.2.bb.c 4 13.c even 3 1 inner
1300.2.bb.c 4 65.n even 6 1 inner
2340.2.q.c 2 15.e even 4 1
2340.2.q.c 2 195.bl even 12 1
3380.2.a.d 1 65.q odd 12 1
3380.2.a.e 1 65.r odd 12 1
3380.2.f.c 2 65.o even 12 1
3380.2.f.c 2 65.t even 12 1

Hecke kernels

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

\( T_{3}^{4} - T_{3}^{2} + 1 \) Copy content Toggle raw display
\( T_{7}^{4} - 25T_{7}^{2} + 625 \) 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
$3$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 25T^{2} + 625 \) 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
$17$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$19$ \( (T^{2} + 3 T + 9)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 9T^{2} + 81 \) Copy content Toggle raw display
$29$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( T^{4} - 49T^{2} + 2401 \) Copy content Toggle raw display
$41$ \( (T^{2} - 5 T + 25)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} - 25T^{2} + 625 \) Copy content Toggle raw display
$47$ \( (T^{2} + 144)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 11 T + 121)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 13 T + 169)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 9T^{2} + 81 \) Copy content Toggle raw display
$71$ \( (T^{2} + 13 T + 169)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$79$ \( (T - 4)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 144)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 7 T + 49)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} - 121 T^{2} + 14641 \) Copy content Toggle raw display
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