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Properties

Label 1350.2.e.n
Level $1350$
Weight $2$
Character orbit 1350.e
Analytic conductor $10.780$
Analytic rank $0$
Dimension $4$
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] = [1350,2,Mod(451,1350)]
 
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([4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1350.451");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1350.e (of order \(3\), 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 \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 450)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{2} - \beta_1 q^{4} + ( - \beta_{3} + \beta_{2} - 2 \beta_1 + 2) q^{7} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{2} - \beta_1 q^{4} + ( - \beta_{3} + \beta_{2} - 2 \beta_1 + 2) q^{7} - q^{8} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{11} + ( - \beta_{2} + 2 \beta_1) q^{13} + (\beta_{2} - 2 \beta_1) q^{14} + (\beta_1 - 1) q^{16} + 2 \beta_{3} q^{17} + (\beta_{3} + 5) q^{19} + 2 \beta_{2} q^{22} + \beta_{2} q^{23} + ( - \beta_{3} + 2) q^{26} + (\beta_{3} - 2) q^{28} + ( - \beta_{3} + \beta_{2}) q^{29} + ( - \beta_{2} - 2 \beta_1) q^{31} + \beta_1 q^{32} + (2 \beta_{3} - 2 \beta_{2}) q^{34} + (3 \beta_{3} + 4) q^{37} + (\beta_{3} - \beta_{2} - 5 \beta_1 + 5) q^{38} + 9 \beta_1 q^{41} + ( - \beta_{3} + \beta_{2} - 5 \beta_1 + 5) q^{43} + 2 \beta_{3} q^{44} + \beta_{3} q^{46} + (2 \beta_{3} - 2 \beta_{2} - 6 \beta_1 + 6) q^{47} + (4 \beta_{2} - 3 \beta_1) q^{49} + ( - \beta_{3} + \beta_{2} - 2 \beta_1 + 2) q^{52} + (\beta_{3} - 6) q^{53} + (\beta_{3} - \beta_{2} + 2 \beta_1 - 2) q^{56} + \beta_{2} q^{58} + ( - \beta_{2} - 3 \beta_1) q^{59} + (8 \beta_1 - 8) q^{61} + ( - \beta_{3} - 2) q^{62} + q^{64} + (3 \beta_{2} - 7 \beta_1) q^{67} - 2 \beta_{2} q^{68} + ( - 3 \beta_{3} - 6) q^{71} + q^{73} + (3 \beta_{3} - 3 \beta_{2} - 4 \beta_1 + 4) q^{74} + ( - \beta_{2} - 5 \beta_1) q^{76} + (4 \beta_{2} - 12 \beta_1) q^{77} + ( - 6 \beta_{3} + 6 \beta_{2} + \cdots - 2) q^{79}+ \cdots + (4 \beta_{3} - 3) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} - 2 q^{4} + 4 q^{7} - 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} - 2 q^{4} + 4 q^{7} - 4 q^{8} + 4 q^{13} - 4 q^{14} - 2 q^{16} + 20 q^{19} + 8 q^{26} - 8 q^{28} - 4 q^{31} + 2 q^{32} + 16 q^{37} + 10 q^{38} + 18 q^{41} + 10 q^{43} + 12 q^{47} - 6 q^{49} + 4 q^{52} - 24 q^{53} - 4 q^{56} - 6 q^{59} - 16 q^{61} - 8 q^{62} + 4 q^{64} - 14 q^{67} - 24 q^{71} + 4 q^{73} + 8 q^{74} - 10 q^{76} - 24 q^{77} - 4 q^{79} + 36 q^{82} + 6 q^{83} - 10 q^{86} + 36 q^{89} + 40 q^{91} - 12 q^{94} - 2 q^{97} - 12 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 2\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 4\nu ) / 2 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{3} + 4\beta_{2} ) / 3 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(1001\) \(1027\)
\(\chi(n)\) \(-\beta_{1}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
451.1
1.22474 + 0.707107i
−1.22474 − 0.707107i
1.22474 − 0.707107i
−1.22474 + 0.707107i
0.500000 − 0.866025i 0 −0.500000 − 0.866025i 0 0 −0.224745 + 0.389270i −1.00000 0 0
451.2 0.500000 − 0.866025i 0 −0.500000 − 0.866025i 0 0 2.22474 − 3.85337i −1.00000 0 0
901.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0 0 −0.224745 − 0.389270i −1.00000 0 0
901.2 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0 0 2.22474 + 3.85337i −1.00000 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
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1350.2.e.n 4
3.b odd 2 1 450.2.e.l ✓ 4
5.b even 2 1 1350.2.e.k 4
5.c odd 4 2 1350.2.j.g 8
9.c even 3 1 inner 1350.2.e.n 4
9.c even 3 1 4050.2.a.bl 2
9.d odd 6 1 450.2.e.l ✓ 4
9.d odd 6 1 4050.2.a.bu 2
15.d odd 2 1 450.2.e.m yes 4
15.e even 4 2 450.2.j.f 8
45.h odd 6 1 450.2.e.m yes 4
45.h odd 6 1 4050.2.a.br 2
45.j even 6 1 1350.2.e.k 4
45.j even 6 1 4050.2.a.by 2
45.k odd 12 2 1350.2.j.g 8
45.k odd 12 2 4050.2.c.w 4
45.l even 12 2 450.2.j.f 8
45.l even 12 2 4050.2.c.y 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
450.2.e.l ✓ 4 3.b odd 2 1
450.2.e.l ✓ 4 9.d odd 6 1
450.2.e.m yes 4 15.d odd 2 1
450.2.e.m yes 4 45.h odd 6 1
450.2.j.f 8 15.e even 4 2
450.2.j.f 8 45.l even 12 2
1350.2.e.k 4 5.b even 2 1
1350.2.e.k 4 45.j even 6 1
1350.2.e.n 4 1.a even 1 1 trivial
1350.2.e.n 4 9.c even 3 1 inner
1350.2.j.g 8 5.c odd 4 2
1350.2.j.g 8 45.k odd 12 2
4050.2.a.bl 2 9.c even 3 1
4050.2.a.br 2 45.h odd 6 1
4050.2.a.bu 2 9.d odd 6 1
4050.2.a.by 2 45.j even 6 1
4050.2.c.w 4 45.k odd 12 2
4050.2.c.y 4 45.l even 12 2

Hecke kernels

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

\( T_{7}^{4} - 4T_{7}^{3} + 18T_{7}^{2} + 8T_{7} + 4 \) Copy content Toggle raw display
\( T_{11}^{4} + 24T_{11}^{2} + 576 \) Copy content Toggle raw display
\( T_{17}^{2} - 24 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 4 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$11$ \( T^{4} + 24T^{2} + 576 \) Copy content Toggle raw display
$13$ \( T^{4} - 4 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$17$ \( (T^{2} - 24)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 10 T + 19)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 6T^{2} + 36 \) Copy content Toggle raw display
$29$ \( T^{4} + 6T^{2} + 36 \) Copy content Toggle raw display
$31$ \( T^{4} + 4 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$37$ \( (T^{2} - 8 T - 38)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 9 T + 81)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} - 10 T^{3} + \cdots + 361 \) Copy content Toggle raw display
$47$ \( T^{4} - 12 T^{3} + \cdots + 144 \) Copy content Toggle raw display
$53$ \( (T^{2} + 12 T + 30)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 6 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$61$ \( (T^{2} + 8 T + 64)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 14 T^{3} + \cdots + 25 \) Copy content Toggle raw display
$71$ \( (T^{2} + 12 T - 18)^{2} \) Copy content Toggle raw display
$73$ \( (T - 1)^{4} \) Copy content Toggle raw display
$79$ \( T^{4} + 4 T^{3} + \cdots + 44944 \) Copy content Toggle raw display
$83$ \( T^{4} - 6 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$89$ \( (T - 9)^{4} \) Copy content Toggle raw display
$97$ \( T^{4} + 2 T^{3} + \cdots + 9025 \) Copy content Toggle raw display
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