Properties

Label 945.2.be.a
Level $945$
Weight $2$
Character orbit 945.be
Analytic conductor $7.546$
Analytic rank $0$
Dimension $2$
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] = [945,2,Mod(206,945)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(945, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([5, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("945.206");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 945 = 3^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 945.be (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: \(7.54586299101\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 315)
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 \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{6} + 1) q^{2} + \zeta_{6} q^{4} - q^{5} + (\zeta_{6} - 3) q^{7} + ( - 2 \zeta_{6} + 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{6} + 1) q^{2} + \zeta_{6} q^{4} - q^{5} + (\zeta_{6} - 3) q^{7} + ( - 2 \zeta_{6} + 1) q^{8} + ( - \zeta_{6} - 1) q^{10} + ( - 4 \zeta_{6} + 2) q^{11} + ( - 4 \zeta_{6} - 4) q^{13} + ( - \zeta_{6} - 4) q^{14} + ( - 5 \zeta_{6} + 5) q^{16} + ( - 6 \zeta_{6} + 6) q^{17} - \zeta_{6} q^{20} + ( - 6 \zeta_{6} + 6) q^{22} + (6 \zeta_{6} - 3) q^{23} + q^{25} - 12 \zeta_{6} q^{26} + ( - 2 \zeta_{6} - 1) q^{28} + (2 \zeta_{6} - 4) q^{31} + ( - 3 \zeta_{6} + 6) q^{32} + ( - 6 \zeta_{6} + 12) q^{34} + ( - \zeta_{6} + 3) q^{35} + 2 \zeta_{6} q^{37} + (2 \zeta_{6} - 1) q^{40} + ( - 6 \zeta_{6} + 6) q^{41} - \zeta_{6} q^{43} + ( - 2 \zeta_{6} + 4) q^{44} + (9 \zeta_{6} - 9) q^{46} + (9 \zeta_{6} - 9) q^{47} + ( - 5 \zeta_{6} + 8) q^{49} + (\zeta_{6} + 1) q^{50} + ( - 8 \zeta_{6} + 4) q^{52} + (2 \zeta_{6} + 2) q^{53} + (4 \zeta_{6} - 2) q^{55} + (5 \zeta_{6} - 1) q^{56} + ( - 7 \zeta_{6} - 7) q^{61} - 6 q^{62} - q^{64} + (4 \zeta_{6} + 4) q^{65} - 5 \zeta_{6} q^{67} + 6 q^{68} + (\zeta_{6} + 4) q^{70} + ( - 8 \zeta_{6} + 4) q^{71} + ( - 6 \zeta_{6} - 6) q^{73} + (4 \zeta_{6} - 2) q^{74} + (10 \zeta_{6} - 2) q^{77} + ( - 10 \zeta_{6} + 10) q^{79} + (5 \zeta_{6} - 5) q^{80} + ( - 6 \zeta_{6} + 12) q^{82} - 12 \zeta_{6} q^{83} + (6 \zeta_{6} - 6) q^{85} + ( - 2 \zeta_{6} + 1) q^{86} - 6 q^{88} + 15 \zeta_{6} q^{89} + (4 \zeta_{6} + 16) q^{91} + (3 \zeta_{6} - 6) q^{92} + (9 \zeta_{6} - 18) q^{94} + (4 \zeta_{6} - 8) q^{97} + ( - 2 \zeta_{6} + 13) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{2} + q^{4} - 2 q^{5} - 5 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 3 q^{2} + q^{4} - 2 q^{5} - 5 q^{7} - 3 q^{10} - 12 q^{13} - 9 q^{14} + 5 q^{16} + 6 q^{17} - q^{20} + 6 q^{22} + 2 q^{25} - 12 q^{26} - 4 q^{28} - 6 q^{31} + 9 q^{32} + 18 q^{34} + 5 q^{35} + 2 q^{37} + 6 q^{41} - q^{43} + 6 q^{44} - 9 q^{46} - 9 q^{47} + 11 q^{49} + 3 q^{50} + 6 q^{53} + 3 q^{56} - 21 q^{61} - 12 q^{62} - 2 q^{64} + 12 q^{65} - 5 q^{67} + 12 q^{68} + 9 q^{70} - 18 q^{73} + 6 q^{77} + 10 q^{79} - 5 q^{80} + 18 q^{82} - 12 q^{83} - 6 q^{85} - 12 q^{88} + 15 q^{89} + 36 q^{91} - 9 q^{92} - 27 q^{94} - 12 q^{97} + 24 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/945\mathbb{Z}\right)^\times\).

\(n\) \(136\) \(596\) \(757\)
\(\chi(n)\) \(\zeta_{6}\) \(1 - \zeta_{6}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
206.1
0.500000 + 0.866025i
0.500000 0.866025i
1.50000 + 0.866025i 0 0.500000 + 0.866025i −1.00000 0 −2.50000 + 0.866025i 1.73205i 0 −1.50000 0.866025i
656.1 1.50000 0.866025i 0 0.500000 0.866025i −1.00000 0 −2.50000 0.866025i 1.73205i 0 −1.50000 + 0.866025i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.s even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 945.2.be.a 2
3.b odd 2 1 315.2.be.a yes 2
7.d odd 6 1 945.2.t.a 2
9.c even 3 1 315.2.t.a 2
9.d odd 6 1 945.2.t.a 2
21.g even 6 1 315.2.t.a 2
63.k odd 6 1 315.2.be.a yes 2
63.s even 6 1 inner 945.2.be.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.t.a 2 9.c even 3 1
315.2.t.a 2 21.g even 6 1
315.2.be.a yes 2 3.b odd 2 1
315.2.be.a yes 2 63.k odd 6 1
945.2.t.a 2 7.d odd 6 1
945.2.t.a 2 9.d odd 6 1
945.2.be.a 2 1.a even 1 1 trivial
945.2.be.a 2 63.s even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 3T_{2} + 3 \) acting on \(S_{2}^{\mathrm{new}}(945, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 3T + 3 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 5T + 7 \) Copy content Toggle raw display
$11$ \( T^{2} + 12 \) Copy content Toggle raw display
$13$ \( T^{2} + 12T + 48 \) Copy content Toggle raw display
$17$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 27 \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 6T + 12 \) Copy content Toggle raw display
$37$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$41$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$43$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$47$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$53$ \( T^{2} - 6T + 12 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 21T + 147 \) Copy content Toggle raw display
$67$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$71$ \( T^{2} + 48 \) Copy content Toggle raw display
$73$ \( T^{2} + 18T + 108 \) Copy content Toggle raw display
$79$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$83$ \( T^{2} + 12T + 144 \) Copy content Toggle raw display
$89$ \( T^{2} - 15T + 225 \) Copy content Toggle raw display
$97$ \( T^{2} + 12T + 48 \) Copy content Toggle raw display
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