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Properties

Label 1350.2.j.c
Level $1350$
Weight $2$
Character orbit 1350.j
Analytic conductor $10.780$
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] = [1350,2,Mod(199,1350)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1350, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1350.199");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1350.j (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: \(10.7798042729\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 90)
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.

\(f(q)\) \(=\) \( q - \zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} + 4 \zeta_{12} q^{7} - \zeta_{12}^{3} q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} + 4 \zeta_{12} q^{7} - \zeta_{12}^{3} q^{8} + ( - 3 \zeta_{12}^{2} + 3) q^{11} + ( - 4 \zeta_{12}^{3} + 4 \zeta_{12}) q^{13} - 4 \zeta_{12}^{2} q^{14} + (\zeta_{12}^{2} - 1) q^{16} - 3 \zeta_{12}^{3} q^{17} - 5 q^{19} + (3 \zeta_{12}^{3} - 3 \zeta_{12}) q^{22} + ( - 6 \zeta_{12}^{3} + 6 \zeta_{12}) q^{23} - 4 q^{26} + 4 \zeta_{12}^{3} q^{28} + (6 \zeta_{12}^{2} - 6) q^{29} - 2 \zeta_{12}^{2} q^{31} + ( - \zeta_{12}^{3} + \zeta_{12}) q^{32} + (3 \zeta_{12}^{2} - 3) q^{34} - 4 \zeta_{12}^{3} q^{37} + 5 \zeta_{12} q^{38} - 3 \zeta_{12}^{2} q^{41} + 11 \zeta_{12} q^{43} + 3 q^{44} - 6 q^{46} + 9 \zeta_{12}^{2} q^{49} + 4 \zeta_{12} q^{52} + 6 \zeta_{12}^{3} q^{53} + ( - 4 \zeta_{12}^{2} + 4) q^{56} + ( - 6 \zeta_{12}^{3} + 6 \zeta_{12}) q^{58} + 3 \zeta_{12}^{2} q^{59} + ( - 10 \zeta_{12}^{2} + 10) q^{61} + 2 \zeta_{12}^{3} q^{62} - q^{64} + ( - 5 \zeta_{12}^{3} + 5 \zeta_{12}) q^{67} + ( - 3 \zeta_{12}^{3} + 3 \zeta_{12}) q^{68} - 6 q^{71} + 7 \zeta_{12}^{3} q^{73} + (4 \zeta_{12}^{2} - 4) q^{74} - 5 \zeta_{12}^{2} q^{76} + ( - 12 \zeta_{12}^{3} + 12 \zeta_{12}) q^{77} + ( - 14 \zeta_{12}^{2} + 14) q^{79} + 3 \zeta_{12}^{3} q^{82} - 12 \zeta_{12} q^{83} - 11 \zeta_{12}^{2} q^{86} - 3 \zeta_{12} q^{88} + 6 q^{89} + 16 q^{91} + 6 \zeta_{12} q^{92} - 11 \zeta_{12} q^{97} - 9 \zeta_{12}^{3} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{4} + 6 q^{11} - 8 q^{14} - 2 q^{16} - 20 q^{19} - 16 q^{26} - 12 q^{29} - 4 q^{31} - 6 q^{34} - 6 q^{41} + 12 q^{44} - 24 q^{46} + 18 q^{49} + 8 q^{56} + 6 q^{59} + 20 q^{61} - 4 q^{64} - 24 q^{71} - 8 q^{74} - 10 q^{76} + 28 q^{79} - 22 q^{86} + 24 q^{89} + 64 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1350\mathbb{Z}\right)^\times\).

\(n\) \(1001\) \(1027\)
\(\chi(n)\) \(-\zeta_{12}^{2}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1
0.866025 − 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 − 0.500000i
−0.866025 + 0.500000i 0 0.500000 − 0.866025i 0 0 3.46410 − 2.00000i 1.00000i 0 0
199.2 0.866025 − 0.500000i 0 0.500000 − 0.866025i 0 0 −3.46410 + 2.00000i − 1.00000i 0 0
1099.1 −0.866025 − 0.500000i 0 0.500000 + 0.866025i 0 0 3.46410 + 2.00000i − 1.00000i 0 0
1099.2 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0 0 −3.46410 − 2.00000i 1.00000i 0 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
9.c even 3 1 inner
45.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1350.2.j.c 4
3.b odd 2 1 450.2.j.a 4
5.b even 2 1 inner 1350.2.j.c 4
5.c odd 4 1 270.2.e.a 2
5.c odd 4 1 1350.2.e.g 2
9.c even 3 1 inner 1350.2.j.c 4
9.c even 3 1 4050.2.c.d 2
9.d odd 6 1 450.2.j.a 4
9.d odd 6 1 4050.2.c.p 2
15.d odd 2 1 450.2.j.a 4
15.e even 4 1 90.2.e.b ✓ 2
15.e even 4 1 450.2.e.d 2
20.e even 4 1 2160.2.q.d 2
45.h odd 6 1 450.2.j.a 4
45.h odd 6 1 4050.2.c.p 2
45.j even 6 1 inner 1350.2.j.c 4
45.j even 6 1 4050.2.c.d 2
45.k odd 12 1 270.2.e.a 2
45.k odd 12 1 810.2.a.e 1
45.k odd 12 1 1350.2.e.g 2
45.k odd 12 1 4050.2.a.q 1
45.l even 12 1 90.2.e.b ✓ 2
45.l even 12 1 450.2.e.d 2
45.l even 12 1 810.2.a.a 1
45.l even 12 1 4050.2.a.bi 1
60.l odd 4 1 720.2.q.c 2
180.v odd 12 1 720.2.q.c 2
180.v odd 12 1 6480.2.a.z 1
180.x even 12 1 2160.2.q.d 2
180.x even 12 1 6480.2.a.l 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.2.e.b ✓ 2 15.e even 4 1
90.2.e.b ✓ 2 45.l even 12 1
270.2.e.a 2 5.c odd 4 1
270.2.e.a 2 45.k odd 12 1
450.2.e.d 2 15.e even 4 1
450.2.e.d 2 45.l even 12 1
450.2.j.a 4 3.b odd 2 1
450.2.j.a 4 9.d odd 6 1
450.2.j.a 4 15.d odd 2 1
450.2.j.a 4 45.h odd 6 1
720.2.q.c 2 60.l odd 4 1
720.2.q.c 2 180.v odd 12 1
810.2.a.a 1 45.l even 12 1
810.2.a.e 1 45.k odd 12 1
1350.2.e.g 2 5.c odd 4 1
1350.2.e.g 2 45.k odd 12 1
1350.2.j.c 4 1.a even 1 1 trivial
1350.2.j.c 4 5.b even 2 1 inner
1350.2.j.c 4 9.c even 3 1 inner
1350.2.j.c 4 45.j even 6 1 inner
2160.2.q.d 2 20.e even 4 1
2160.2.q.d 2 180.x even 12 1
4050.2.a.q 1 45.k odd 12 1
4050.2.a.bi 1 45.l even 12 1
4050.2.c.d 2 9.c even 3 1
4050.2.c.d 2 45.j even 6 1
4050.2.c.p 2 9.d odd 6 1
4050.2.c.p 2 45.h odd 6 1
6480.2.a.l 1 180.x even 12 1
6480.2.a.z 1 180.v odd 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1350, [\chi])\):

\( T_{7}^{4} - 16T_{7}^{2} + 256 \) Copy content Toggle raw display
\( T_{11}^{2} - 3T_{11} + 9 \) Copy content Toggle raw display
\( T_{19} + 5 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 16T^{2} + 256 \) Copy content Toggle raw display
$11$ \( (T^{2} - 3 T + 9)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} - 16T^{2} + 256 \) Copy content Toggle raw display
$17$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$19$ \( (T + 5)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$29$ \( (T^{2} + 6 T + 36)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 3 T + 9)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} - 121 T^{2} + 14641 \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 3 T + 9)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 10 T + 100)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 25T^{2} + 625 \) Copy content Toggle raw display
$71$ \( (T + 6)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 49)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 14 T + 196)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} - 144 T^{2} + 20736 \) Copy content Toggle raw display
$89$ \( (T - 6)^{4} \) Copy content Toggle raw display
$97$ \( T^{4} - 121 T^{2} + 14641 \) Copy content Toggle raw display
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