Properties

Label 945.2.i.b
Level $945$
Weight $2$
Character orbit 945.i
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(316,945)]
 
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, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("945.316");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 945 = 3^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 945.i (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: \(2\)
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: \(\Z[a_1, \ldots, a_{5}]\)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 2 \zeta_{6} + 2) q^{2} - 2 \zeta_{6} q^{4} - \zeta_{6} q^{5} + (\zeta_{6} - 1) q^{7} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \zeta_{6} + 2) q^{2} - 2 \zeta_{6} q^{4} - \zeta_{6} q^{5} + (\zeta_{6} - 1) q^{7} - 2 q^{10} + ( - 3 \zeta_{6} + 3) q^{11} - 6 \zeta_{6} q^{13} + 2 \zeta_{6} q^{14} + ( - 4 \zeta_{6} + 4) q^{16} + 2 q^{17} - 2 q^{19} + (2 \zeta_{6} - 2) q^{20} - 6 \zeta_{6} q^{22} + 4 \zeta_{6} q^{23} + (\zeta_{6} - 1) q^{25} - 12 q^{26} + 2 q^{28} + (\zeta_{6} - 1) q^{29} - 10 \zeta_{6} q^{31} - 8 \zeta_{6} q^{32} + ( - 4 \zeta_{6} + 4) q^{34} + q^{35} - 2 q^{37} + (4 \zeta_{6} - 4) q^{38} - 6 \zeta_{6} q^{41} + (4 \zeta_{6} - 4) q^{43} - 6 q^{44} + 8 q^{46} + ( - 7 \zeta_{6} + 7) q^{47} - \zeta_{6} q^{49} + 2 \zeta_{6} q^{50} + (12 \zeta_{6} - 12) q^{52} + 4 q^{53} - 3 q^{55} + 2 \zeta_{6} q^{58} + 14 \zeta_{6} q^{59} + (4 \zeta_{6} - 4) q^{61} - 20 q^{62} - 8 q^{64} + (6 \zeta_{6} - 6) q^{65} + 2 \zeta_{6} q^{67} - 4 \zeta_{6} q^{68} + ( - 2 \zeta_{6} + 2) q^{70} + 9 q^{71} + 13 q^{73} + (4 \zeta_{6} - 4) q^{74} + 4 \zeta_{6} q^{76} + 3 \zeta_{6} q^{77} + (17 \zeta_{6} - 17) q^{79} - 4 q^{80} - 12 q^{82} + ( - 13 \zeta_{6} + 13) q^{83} - 2 \zeta_{6} q^{85} + 8 \zeta_{6} q^{86} - 6 q^{89} + 6 q^{91} + ( - 8 \zeta_{6} + 8) q^{92} - 14 \zeta_{6} q^{94} + 2 \zeta_{6} q^{95} + ( - 9 \zeta_{6} + 9) q^{97} - 2 q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 2 q^{4} - q^{5} - q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - 2 q^{4} - q^{5} - q^{7} - 4 q^{10} + 3 q^{11} - 6 q^{13} + 2 q^{14} + 4 q^{16} + 4 q^{17} - 4 q^{19} - 2 q^{20} - 6 q^{22} + 4 q^{23} - q^{25} - 24 q^{26} + 4 q^{28} - q^{29} - 10 q^{31} - 8 q^{32} + 4 q^{34} + 2 q^{35} - 4 q^{37} - 4 q^{38} - 6 q^{41} - 4 q^{43} - 12 q^{44} + 16 q^{46} + 7 q^{47} - q^{49} + 2 q^{50} - 12 q^{52} + 8 q^{53} - 6 q^{55} + 2 q^{58} + 14 q^{59} - 4 q^{61} - 40 q^{62} - 16 q^{64} - 6 q^{65} + 2 q^{67} - 4 q^{68} + 2 q^{70} + 18 q^{71} + 26 q^{73} - 4 q^{74} + 4 q^{76} + 3 q^{77} - 17 q^{79} - 8 q^{80} - 24 q^{82} + 13 q^{83} - 2 q^{85} + 8 q^{86} - 12 q^{89} + 12 q^{91} + 8 q^{92} - 14 q^{94} + 2 q^{95} + 9 q^{97} - 4 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
316.1
0.500000 + 0.866025i
0.500000 0.866025i
1.00000 1.73205i 0 −1.00000 1.73205i −0.500000 0.866025i 0 −0.500000 + 0.866025i 0 0 −2.00000
631.1 1.00000 + 1.73205i 0 −1.00000 + 1.73205i −0.500000 + 0.866025i 0 −0.500000 0.866025i 0 0 −2.00000
\(n\): 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 945.2.i.b 2
3.b odd 2 1 315.2.i.a 2
9.c even 3 1 inner 945.2.i.b 2
9.c even 3 1 2835.2.a.b 1
9.d odd 6 1 315.2.i.a 2
9.d odd 6 1 2835.2.a.h 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.i.a 2 3.b odd 2 1
315.2.i.a 2 9.d odd 6 1
945.2.i.b 2 1.a even 1 1 trivial
945.2.i.b 2 9.c even 3 1 inner
2835.2.a.b 1 9.c even 3 1
2835.2.a.h 1 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 2T_{2} + 4 \) acting on \(S_{2}^{\mathrm{new}}(945, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$7$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$11$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$13$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$17$ \( (T - 2)^{2} \) Copy content Toggle raw display
$19$ \( (T + 2)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$29$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$31$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
$37$ \( (T + 2)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$43$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$47$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$53$ \( (T - 4)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 14T + 196 \) Copy content Toggle raw display
$61$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$67$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$71$ \( (T - 9)^{2} \) Copy content Toggle raw display
$73$ \( (T - 13)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 17T + 289 \) Copy content Toggle raw display
$83$ \( T^{2} - 13T + 169 \) Copy content Toggle raw display
$89$ \( (T + 6)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
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