Properties

Label 945.2.l.a
Level $945$
Weight $2$
Character orbit 945.l
Analytic conductor $7.546$
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] = [945,2,Mod(46,945)]
 
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([4, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("945.46");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 945 = 3^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 945.l (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: \(7.54586299101\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{13})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 4x^{2} + 3x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 315)
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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + (\beta_{3} + 1) q^{4} + \beta_{2} q^{5} + (\beta_{2} + 3) q^{7} + 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} + (\beta_{3} + 1) q^{4} + \beta_{2} q^{5} + (\beta_{2} + 3) q^{7} + 3 q^{8} + ( - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{10} + (3 \beta_{2} - 2 \beta_1 + 3) q^{13} + (2 \beta_{3} + \beta_{2} - \beta_1 + 1) q^{14} + (\beta_{3} - 2) q^{16} + ( - 2 \beta_{3} - \beta_{2} - 2 \beta_1 + 2) q^{17} + ( - \beta_{2} + 2 \beta_1 - 1) q^{19} + ( - \beta_{3} + 2 \beta_{2} - \beta_1 + 1) q^{20} + (4 \beta_{3} - 4 \beta_{2} + 4 \beta_1 - 4) q^{23} + ( - \beta_{2} - 1) q^{25} + (9 \beta_{2} - 3 \beta_1 + 9) q^{26} + (2 \beta_{3} + 2 \beta_{2} - \beta_1 + 4) q^{28} + ( - 4 \beta_{3} + \beta_{2} - 4 \beta_1 + 4) q^{29} + (2 \beta_{3} - 1) q^{31} + ( - \beta_{3} - 3) q^{32} + (\beta_{3} + 5 \beta_{2} + \beta_1 - 1) q^{34} + (2 \beta_{2} - 1) q^{35} + ( - \beta_{2} + 2 \beta_1 - 1) q^{37} + ( - 7 \beta_{2} + \beta_1 - 7) q^{38} + 3 \beta_{2} q^{40} + (3 \beta_{2} + 3) q^{41} + (4 \beta_{3} - 5 \beta_{2} + 4 \beta_1 - 4) q^{43} + (4 \beta_{3} - 16 \beta_{2} + 4 \beta_1 - 4) q^{46} + ( - 4 \beta_{3} + 3) q^{47} + (5 \beta_{2} + 8) q^{49} + ( - \beta_{2} + \beta_1 - 1) q^{50} + (12 \beta_{2} - 5 \beta_1 + 12) q^{52} + (2 \beta_{3} - 5 \beta_{2} + 2 \beta_1 - 2) q^{53} + (3 \beta_{2} + 9) q^{56} + ( - \beta_{3} + 13 \beta_{2} - \beta_1 + 1) q^{58} + ( - 2 \beta_{3} + 9) q^{59} + ( - 4 \beta_{3} - 1) q^{61} + (\beta_{3} + 6) q^{62} + ( - 6 \beta_{3} + 1) q^{64} + ( - 2 \beta_{3} - 1) q^{65} + ( - 4 \beta_{3} - 1) q^{67} + ( - \beta_{3} + 4 \beta_{2} - \beta_1 + 1) q^{68} + ( - 3 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 2) q^{70} + ( - 2 \beta_{3} + \beta_{2} - 2 \beta_1 + 2) q^{73} + ( - 7 \beta_{2} + \beta_1 - 7) q^{74} + ( - 8 \beta_{2} + 3 \beta_1 - 8) q^{76} + ( - 6 \beta_{3} + 5) q^{79} + ( - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{80} + (3 \beta_{2} - 3 \beta_1 + 3) q^{82} + (4 \beta_{3} + 5 \beta_{2} + 4 \beta_1 - 4) q^{83} + (\beta_{2} + 2 \beta_1 + 1) q^{85} + (5 \beta_{3} - 17 \beta_{2} + 5 \beta_1 - 5) q^{86} + (3 \beta_{2} + 3) q^{89} + ( - 2 \beta_{3} + 9 \beta_{2} - 6 \beta_1 + 8) q^{91} + (8 \beta_{3} - 20 \beta_{2} + 8 \beta_1 - 8) q^{92} + ( - \beta_{3} - 12) q^{94} + (2 \beta_{3} - 1) q^{95} + (2 \beta_{3} + 9 \beta_{2} + 2 \beta_1 - 2) q^{97} + (3 \beta_{3} + 5 \beta_{2} - 5 \beta_1 + 5) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} + 6 q^{4} - 2 q^{5} + 10 q^{7} + 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} + 6 q^{4} - 2 q^{5} + 10 q^{7} + 12 q^{8} - q^{10} + 4 q^{13} + 5 q^{14} - 6 q^{16} + 4 q^{17} - 3 q^{20} + 4 q^{23} - 2 q^{25} + 15 q^{26} + 15 q^{28} + 2 q^{29} - 14 q^{32} - 11 q^{34} - 8 q^{35} - 13 q^{38} - 6 q^{40} + 6 q^{41} + 6 q^{43} + 28 q^{46} + 4 q^{47} + 22 q^{49} - q^{50} + 19 q^{52} + 8 q^{53} + 30 q^{56} - 25 q^{58} + 32 q^{59} - 12 q^{61} + 26 q^{62} - 8 q^{64} - 8 q^{65} - 12 q^{67} - 7 q^{68} - 4 q^{70} - 13 q^{74} - 13 q^{76} + 8 q^{79} + 3 q^{80} + 3 q^{82} - 14 q^{83} + 4 q^{85} + 29 q^{86} + 6 q^{89} + 4 q^{91} + 32 q^{92} - 50 q^{94} - 20 q^{97} + 11 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 4x^{2} + 3x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + 4\nu^{2} - 4\nu - 3 ) / 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 7 ) / 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 3\beta_{2} + \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4\beta_{3} - 7 \) Copy content Toggle raw display

Character values

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

\(n\) \(136\) \(596\) \(757\)
\(\chi(n)\) \(-1 - \beta_{2}\) \(-1 - \beta_{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
46.1
1.15139 + 1.99426i
−0.651388 1.12824i
1.15139 1.99426i
−0.651388 + 1.12824i
−1.30278 0 −0.302776 −0.500000 + 0.866025i 0 2.50000 + 0.866025i 3.00000 0 0.651388 1.12824i
46.2 2.30278 0 3.30278 −0.500000 + 0.866025i 0 2.50000 + 0.866025i 3.00000 0 −1.15139 + 1.99426i
226.1 −1.30278 0 −0.302776 −0.500000 0.866025i 0 2.50000 0.866025i 3.00000 0 0.651388 + 1.12824i
226.2 2.30278 0 3.30278 −0.500000 0.866025i 0 2.50000 0.866025i 3.00000 0 −1.15139 1.99426i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.h even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 945.2.l.a 4
3.b odd 2 1 315.2.l.a yes 4
7.c even 3 1 945.2.k.a 4
9.c even 3 1 945.2.k.a 4
9.d odd 6 1 315.2.k.a 4
21.h odd 6 1 315.2.k.a 4
63.h even 3 1 inner 945.2.l.a 4
63.j odd 6 1 315.2.l.a yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.k.a 4 9.d odd 6 1
315.2.k.a 4 21.h odd 6 1
315.2.l.a yes 4 3.b odd 2 1
315.2.l.a yes 4 63.j odd 6 1
945.2.k.a 4 7.c even 3 1
945.2.k.a 4 9.c even 3 1
945.2.l.a 4 1.a even 1 1 trivial
945.2.l.a 4 63.h even 3 1 inner

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T - 3)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - 5 T + 7)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} - 4 T^{3} + 25 T^{2} + 36 T + 81 \) Copy content Toggle raw display
$17$ \( T^{4} - 4 T^{3} + 25 T^{2} + 36 T + 81 \) Copy content Toggle raw display
$19$ \( T^{4} + 13T^{2} + 169 \) Copy content Toggle raw display
$23$ \( T^{4} - 4 T^{3} + 64 T^{2} + \cdots + 2304 \) Copy content Toggle raw display
$29$ \( T^{4} - 2 T^{3} + 55 T^{2} + \cdots + 2601 \) Copy content Toggle raw display
$31$ \( (T^{2} - 13)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 13T^{2} + 169 \) Copy content Toggle raw display
$41$ \( (T^{2} - 3 T + 9)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} - 6 T^{3} + 79 T^{2} + \cdots + 1849 \) Copy content Toggle raw display
$47$ \( (T^{2} - 2 T - 51)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} - 8 T^{3} + 61 T^{2} - 24 T + 9 \) Copy content Toggle raw display
$59$ \( (T^{2} - 16 T + 51)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 6 T - 43)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 6 T - 43)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + 13T^{2} + 169 \) Copy content Toggle raw display
$79$ \( (T^{2} - 4 T - 113)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 14 T^{3} + 199 T^{2} - 42 T + 9 \) Copy content Toggle raw display
$89$ \( (T^{2} - 3 T + 9)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 20 T^{3} + 313 T^{2} + \cdots + 7569 \) Copy content Toggle raw display
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