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Properties

Label 1352.2.o.a
Level $1352$
Weight $2$
Character orbit 1352.o
Analytic conductor $10.796$
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] = [1352,2,Mod(361,1352)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1352, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1352.361");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1352 = 2^{3} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1352.o (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: \(10.7957743533\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 104)
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.

\(f(q)\) \(=\) \( q + (\zeta_{12}^{2} - 1) q^{3} + \zeta_{12}^{3} q^{5} + ( - 5 \zeta_{12}^{3} + 5 \zeta_{12}) q^{7} + 2 \zeta_{12}^{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{12}^{2} - 1) q^{3} + \zeta_{12}^{3} q^{5} + ( - 5 \zeta_{12}^{3} + 5 \zeta_{12}) q^{7} + 2 \zeta_{12}^{2} q^{9} + 2 \zeta_{12} q^{11} - \zeta_{12} q^{15} - 3 \zeta_{12}^{2} q^{17} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{19} + 5 \zeta_{12}^{3} q^{21} + ( - 4 \zeta_{12}^{2} + 4) q^{23} + 4 q^{25} - 5 q^{27} + ( - 6 \zeta_{12}^{2} + 6) q^{29} + 4 \zeta_{12}^{3} q^{31} + (2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{33} + 5 \zeta_{12}^{2} q^{35} - 11 \zeta_{12} q^{37} + 8 \zeta_{12} q^{41} - \zeta_{12}^{2} q^{43} + (2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{45} + 9 \zeta_{12}^{3} q^{47} + ( - 18 \zeta_{12}^{2} + 18) q^{49} + 3 q^{51} - 12 q^{53} + (2 \zeta_{12}^{2} - 2) q^{55} + 2 \zeta_{12}^{3} q^{57} + ( - 6 \zeta_{12}^{3} + 6 \zeta_{12}) q^{59} + 10 \zeta_{12} q^{63} + 6 \zeta_{12} q^{67} + 4 \zeta_{12}^{2} q^{69} + (7 \zeta_{12}^{3} - 7 \zeta_{12}) q^{71} - 2 \zeta_{12}^{3} q^{73} + (4 \zeta_{12}^{2} - 4) q^{75} + 10 q^{77} + 12 q^{79} + (\zeta_{12}^{2} - 1) q^{81} + 16 \zeta_{12}^{3} q^{83} + ( - 3 \zeta_{12}^{3} + 3 \zeta_{12}) q^{85} + 6 \zeta_{12}^{2} q^{87} + 10 \zeta_{12} q^{89} - 4 \zeta_{12} q^{93} + 2 \zeta_{12}^{2} q^{95} + ( - 10 \zeta_{12}^{3} + 10 \zeta_{12}) q^{97} + 4 \zeta_{12}^{3} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{3} + 4 q^{9} - 6 q^{17} + 8 q^{23} + 16 q^{25} - 20 q^{27} + 12 q^{29} + 10 q^{35} - 2 q^{43} + 36 q^{49} + 12 q^{51} - 48 q^{53} - 4 q^{55} + 8 q^{69} - 8 q^{75} + 40 q^{77} + 48 q^{79} - 2 q^{81} + 12 q^{87} + 4 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1352\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(1015\) \(1185\)
\(\chi(n)\) \(1\) \(1\) \(\zeta_{12}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
0.866025 − 0.500000i
−0.866025 + 0.500000i
−0.866025 − 0.500000i
0.866025 + 0.500000i
0 −0.500000 − 0.866025i 0 − 1.00000i 0 4.33013 + 2.50000i 0 1.00000 − 1.73205i 0
361.2 0 −0.500000 − 0.866025i 0 1.00000i 0 −4.33013 − 2.50000i 0 1.00000 − 1.73205i 0
1161.1 0 −0.500000 + 0.866025i 0 − 1.00000i 0 −4.33013 + 2.50000i 0 1.00000 + 1.73205i 0
1161.2 0 −0.500000 + 0.866025i 0 1.00000i 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
13.b even 2 1 inner
13.c even 3 1 inner
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1352.2.o.a 4
13.b even 2 1 inner 1352.2.o.a 4
13.c even 3 1 1352.2.f.b 2
13.c even 3 1 inner 1352.2.o.a 4
13.d odd 4 1 1352.2.i.b 2
13.d odd 4 1 1352.2.i.c 2
13.e even 6 1 1352.2.f.b 2
13.e even 6 1 inner 1352.2.o.a 4
13.f odd 12 1 104.2.a.a ✓ 1
13.f odd 12 1 1352.2.a.b 1
13.f odd 12 1 1352.2.i.b 2
13.f odd 12 1 1352.2.i.c 2
39.k even 12 1 936.2.a.f 1
52.i odd 6 1 2704.2.f.e 2
52.j odd 6 1 2704.2.f.e 2
52.l even 12 1 208.2.a.b 1
52.l even 12 1 2704.2.a.d 1
65.o even 12 1 2600.2.d.f 2
65.s odd 12 1 2600.2.a.e 1
65.t even 12 1 2600.2.d.f 2
91.bc even 12 1 5096.2.a.c 1
104.u even 12 1 832.2.a.h 1
104.x odd 12 1 832.2.a.c 1
156.v odd 12 1 1872.2.a.l 1
208.be odd 12 1 3328.2.b.a 2
208.bf even 12 1 3328.2.b.t 2
208.bk even 12 1 3328.2.b.t 2
208.bl odd 12 1 3328.2.b.a 2
260.bc even 12 1 5200.2.a.bb 1
312.bo even 12 1 7488.2.a.x 1
312.bq odd 12 1 7488.2.a.u 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
104.2.a.a ✓ 1 13.f odd 12 1
208.2.a.b 1 52.l even 12 1
832.2.a.c 1 104.x odd 12 1
832.2.a.h 1 104.u even 12 1
936.2.a.f 1 39.k even 12 1
1352.2.a.b 1 13.f odd 12 1
1352.2.f.b 2 13.c even 3 1
1352.2.f.b 2 13.e even 6 1
1352.2.i.b 2 13.d odd 4 1
1352.2.i.b 2 13.f odd 12 1
1352.2.i.c 2 13.d odd 4 1
1352.2.i.c 2 13.f odd 12 1
1352.2.o.a 4 1.a even 1 1 trivial
1352.2.o.a 4 13.b even 2 1 inner
1352.2.o.a 4 13.c even 3 1 inner
1352.2.o.a 4 13.e even 6 1 inner
1872.2.a.l 1 156.v odd 12 1
2600.2.a.e 1 65.s odd 12 1
2600.2.d.f 2 65.o even 12 1
2600.2.d.f 2 65.t even 12 1
2704.2.a.d 1 52.l even 12 1
2704.2.f.e 2 52.i odd 6 1
2704.2.f.e 2 52.j odd 6 1
3328.2.b.a 2 208.be odd 12 1
3328.2.b.a 2 208.bl odd 12 1
3328.2.b.t 2 208.bf even 12 1
3328.2.b.t 2 208.bk even 12 1
5096.2.a.c 1 91.bc even 12 1
5200.2.a.bb 1 260.bc even 12 1
7488.2.a.u 1 312.bq odd 12 1
7488.2.a.x 1 312.bo even 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1352, [\chi])\):

\( T_{3}^{2} + T_{3} + 1 \) Copy content Toggle raw display
\( T_{5}^{2} + 1 \) Copy content Toggle raw display
\( T_{11}^{4} - 4T_{11}^{2} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} - 25T^{2} + 625 \) Copy content Toggle raw display
$11$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + 3 T + 9)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$23$ \( (T^{2} - 4 T + 16)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 6 T + 36)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} - 121 T^{2} + 14641 \) Copy content Toggle raw display
$41$ \( T^{4} - 64T^{2} + 4096 \) Copy content Toggle raw display
$43$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 81)^{2} \) Copy content Toggle raw display
$53$ \( (T + 12)^{4} \) Copy content Toggle raw display
$59$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$71$ \( T^{4} - 49T^{2} + 2401 \) Copy content Toggle raw display
$73$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$79$ \( (T - 12)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 256)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 100 T^{2} + 10000 \) Copy content Toggle raw display
$97$ \( T^{4} - 100 T^{2} + 10000 \) Copy content Toggle raw display
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