Properties

Label 1350.2.f.b
Level $1350$
Weight $2$
Character orbit 1350.f
Analytic conductor $10.780$
Analytic rank $0$
Dimension $8$
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(107,1350)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1350, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1350.107");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1350.f (of order \(4\), degree \(2\), 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: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{5} q^{2} - \beta_{3} q^{4} + ( - \beta_{2} + 3 \beta_1) q^{7} + \beta_1 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{2} - \beta_{3} q^{4} + ( - \beta_{2} + 3 \beta_1) q^{7} + \beta_1 q^{8} - 3 \beta_{3} q^{11} + 3 \beta_{5} q^{13} + (\beta_{6} - 3) q^{14} - q^{16} + (3 \beta_{7} + 3 \beta_{5}) q^{17} + ( - 3 \beta_{4} - \beta_{3}) q^{19} + 3 \beta_1 q^{22} - 3 \beta_1 q^{23} + 3 \beta_{3} q^{26} + (\beta_{7} + 3 \beta_{5}) q^{28} + ( - 3 \beta_{6} - 3) q^{29} - 2 q^{31} + \beta_{5} q^{32} + ( - 3 \beta_{4} + 3 \beta_{3}) q^{34} + (4 \beta_{2} + 3 \beta_1) q^{37} + (3 \beta_{2} + \beta_1) q^{38} - 6 \beta_{3} q^{41} + ( - \beta_{7} + 9 \beta_{5}) q^{43} - 3 q^{44} + 3 q^{46} - 9 \beta_{5} q^{47} + ( - 6 \beta_{4} + 5 \beta_{3}) q^{49} - 3 \beta_1 q^{52} + ( - 3 \beta_{2} + 3 \beta_1) q^{53} + ( - \beta_{4} + 3 \beta_{3}) q^{56} + ( - 3 \beta_{7} + 3 \beta_{5}) q^{58} + (6 \beta_{6} + 3) q^{59} + ( - 3 \beta_{6} - 4) q^{61} + 2 \beta_{5} q^{62} + \beta_{3} q^{64} + ( - 3 \beta_{2} + 3 \beta_1) q^{67} + (3 \beta_{2} - 3 \beta_1) q^{68} + (3 \beta_{4} - 6 \beta_{3}) q^{71} + (2 \beta_{7} + 12 \beta_{5}) q^{73} + ( - 4 \beta_{6} - 3) q^{74} + ( - 3 \beta_{6} - 1) q^{76} + (3 \beta_{7} + 9 \beta_{5}) q^{77} + ( - 3 \beta_{4} - 5 \beta_{3}) q^{79} + 6 \beta_1 q^{82} + ( - 6 \beta_{2} + 6 \beta_1) q^{83} + (\beta_{4} + 9 \beta_{3}) q^{86} + 3 \beta_{5} q^{88} + (3 \beta_{6} + 3) q^{89} + ( - 3 \beta_{6} + 9) q^{91} - 3 \beta_{5} q^{92} - 9 \beta_{3} q^{94} + ( - 5 \beta_{2} + 6 \beta_1) q^{97} + (6 \beta_{2} - 5 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 24 q^{14} - 8 q^{16} - 24 q^{29} - 16 q^{31} - 24 q^{44} + 24 q^{46} + 24 q^{59} - 32 q^{61} - 24 q^{74} - 8 q^{76} + 24 q^{89} + 72 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{24}^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{24}^{5} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \zeta_{24}^{6} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( 2\zeta_{24}^{4} - 1 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\zeta_{24}^{5} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -\zeta_{24}^{6} + 2\zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( 2\zeta_{24}^{7} - \zeta_{24}^{3} \) Copy content Toggle raw display
\(\zeta_{24}\)\(=\) \( ( \beta_{5} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( ( \beta_{6} + \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( ( \beta_{4} + 1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( -\beta_{5} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( \beta_{3} \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( \beta_{7} + \beta_1 ) / 2 \) Copy content Toggle raw display

Character values

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

\(n\) \(1001\) \(1027\)
\(\chi(n)\) \(-1\) \(\beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
107.1
0.965926 + 0.258819i
−0.258819 0.965926i
0.258819 + 0.965926i
−0.965926 0.258819i
0.965926 0.258819i
−0.258819 + 0.965926i
0.258819 0.965926i
−0.965926 + 0.258819i
−0.707107 + 0.707107i 0 1.00000i 0 0 0.896575 + 0.896575i 0.707107 + 0.707107i 0 0
107.2 −0.707107 + 0.707107i 0 1.00000i 0 0 3.34607 + 3.34607i 0.707107 + 0.707107i 0 0
107.3 0.707107 0.707107i 0 1.00000i 0 0 −3.34607 3.34607i −0.707107 0.707107i 0 0
107.4 0.707107 0.707107i 0 1.00000i 0 0 −0.896575 0.896575i −0.707107 0.707107i 0 0
593.1 −0.707107 0.707107i 0 1.00000i 0 0 0.896575 0.896575i 0.707107 0.707107i 0 0
593.2 −0.707107 0.707107i 0 1.00000i 0 0 3.34607 3.34607i 0.707107 0.707107i 0 0
593.3 0.707107 + 0.707107i 0 1.00000i 0 0 −3.34607 + 3.34607i −0.707107 + 0.707107i 0 0
593.4 0.707107 + 0.707107i 0 1.00000i 0 0 −0.896575 + 0.896575i −0.707107 + 0.707107i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 107.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
15.e even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1350.2.f.b 8
3.b odd 2 1 1350.2.f.e yes 8
5.b even 2 1 inner 1350.2.f.b 8
5.c odd 4 2 1350.2.f.e yes 8
15.d odd 2 1 1350.2.f.e yes 8
15.e even 4 2 inner 1350.2.f.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1350.2.f.b 8 1.a even 1 1 trivial
1350.2.f.b 8 5.b even 2 1 inner
1350.2.f.b 8 15.e even 4 2 inner
1350.2.f.e yes 8 3.b odd 2 1
1350.2.f.e yes 8 5.c odd 4 2
1350.2.f.e yes 8 15.d odd 2 1

Hecke kernels

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

\( T_{7}^{8} + 504T_{7}^{4} + 1296 \) Copy content Toggle raw display
\( T_{29}^{2} + 6T_{29} - 18 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 504T^{4} + 1296 \) Copy content Toggle raw display
$11$ \( (T^{2} + 9)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} + 81)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 4536 T^{4} + 104976 \) Copy content Toggle raw display
$19$ \( (T^{4} + 56 T^{2} + 676)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 81)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 6 T - 18)^{4} \) Copy content Toggle raw display
$31$ \( (T + 2)^{8} \) Copy content Toggle raw display
$37$ \( T^{8} + 9954 T^{4} + \cdots + 2313441 \) Copy content Toggle raw display
$41$ \( (T^{2} + 36)^{4} \) Copy content Toggle raw display
$43$ \( T^{8} + 16056 T^{4} + \cdots + 37015056 \) Copy content Toggle raw display
$47$ \( (T^{4} + 6561)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + 4536 T^{4} + 104976 \) Copy content Toggle raw display
$59$ \( (T^{2} - 6 T - 99)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 8 T - 11)^{4} \) Copy content Toggle raw display
$67$ \( T^{8} + 4536 T^{4} + 104976 \) Copy content Toggle raw display
$71$ \( (T^{4} + 126 T^{2} + 81)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 62496 T^{4} + \cdots + 303595776 \) Copy content Toggle raw display
$79$ \( (T^{4} + 104 T^{2} + 4)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 72576 T^{4} + \cdots + 26873856 \) Copy content Toggle raw display
$89$ \( (T^{2} - 6 T - 18)^{4} \) Copy content Toggle raw display
$97$ \( T^{8} + 46242 T^{4} + \cdots + 2313441 \) Copy content Toggle raw display
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