Properties

Label 945.2.d.c
Level $945$
Weight $2$
Character orbit 945.d
Analytic conductor $7.546$
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] = [945,2,Mod(379,945)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(945, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("945.379");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 945 = 3^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 945.d (of order \(2\), degree \(1\), 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: \(8\)
Coefficient field: 8.0.49787136.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + ( - \beta_{4} - 1) q^{4} + (\beta_{6} - \beta_{5}) q^{5} + \beta_{3} q^{7} + (\beta_{6} + \beta_{2}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + ( - \beta_{4} - 1) q^{4} + (\beta_{6} - \beta_{5}) q^{5} + \beta_{3} q^{7} + (\beta_{6} + \beta_{2}) q^{8} + (2 \beta_{3} + \beta_1 - 1) q^{10} + (\beta_{7} - \beta_{5}) q^{11} + (2 \beta_{3} + \beta_1) q^{13} + \beta_{5} q^{14} - \beta_{4} q^{16} + (2 \beta_{6} + 3 \beta_{2}) q^{17} + (\beta_{4} + 5) q^{19} + ( - \beta_{7} + 2 \beta_{6} + 3 \beta_{5} + \beta_{2}) q^{20} + (\beta_{3} + \beta_1) q^{22} + (\beta_{6} - 2 \beta_{2}) q^{23} + (2 \beta_{4} + 2 \beta_{3} + 1) q^{25} + ( - \beta_{7} + 5 \beta_{5}) q^{26} - \beta_1 q^{28} + (3 \beta_{7} - 2 \beta_{5}) q^{29} + ( - 2 \beta_{4} - 5) q^{31} + (3 \beta_{6} + 4 \beta_{2}) q^{32} + (3 \beta_{4} + 7) q^{34} + (\beta_{7} - \beta_{2}) q^{35} + ( - \beta_{3} + 4 \beta_1) q^{37} + ( - \beta_{6} - 7 \beta_{2}) q^{38} + (\beta_{4} - \beta_{3} - \beta_1 - 1) q^{40} + ( - 4 \beta_{7} + 5 \beta_{5}) q^{41} - \beta_{3} q^{43} + (\beta_{7} + 2 \beta_{5}) q^{44} + ( - 2 \beta_{4} - 7) q^{46} + ( - \beta_{6} - \beta_{2}) q^{47} - q^{49} + ( - 2 \beta_{6} + 2 \beta_{5} - 5 \beta_{2}) q^{50} + ( - 5 \beta_{3} - 3 \beta_1) q^{52} + 3 \beta_{2} q^{53} + (\beta_{4} - 2 \beta_{3} + \beta_1 + 2) q^{55} + (\beta_{7} - \beta_{5}) q^{56} + (\beta_{3} + 2 \beta_1) q^{58} + ( - 7 \beta_{7} + 5 \beta_{5}) q^{59} + (\beta_{4} + 3) q^{61} + (2 \beta_{6} + 9 \beta_{2}) q^{62} + (2 \beta_{4} + 9) q^{64} + ( - \beta_{6} + \beta_{5} - 5 \beta_{2}) q^{65} + (7 \beta_{3} - 3 \beta_1) q^{67} + (\beta_{6} - 7 \beta_{2}) q^{68} + ( - \beta_{4} - \beta_{3} - 3) q^{70} + (7 \beta_{7} - 2 \beta_{5}) q^{71} + (\beta_{3} - 2 \beta_1) q^{73} + ( - 4 \beta_{7} + 11 \beta_{5}) q^{74} + ( - 5 \beta_{4} - 10) q^{76} + ( - \beta_{6} - \beta_{2}) q^{77} + ( - 3 \beta_{4} + 2) q^{79} + ( - \beta_{7} + 3 \beta_{6} + 2 \beta_{5} + \beta_{2}) q^{80} + ( - 6 \beta_{3} - 5 \beta_1) q^{82} + (5 \beta_{6} + 9 \beta_{2}) q^{83} + (2 \beta_{4} - 4 \beta_{3} - 3 \beta_1 - 1) q^{85} - \beta_{5} q^{86} - 3 \beta_{3} q^{88} + (5 \beta_{7} - 5 \beta_{5}) q^{89} + ( - \beta_{4} - 3) q^{91} + (4 \beta_{6} + 7 \beta_{2}) q^{92} + ( - \beta_{4} - 2) q^{94} + (\beta_{7} + 2 \beta_{6} - 7 \beta_{5} - \beta_{2}) q^{95} + ( - 7 \beta_{3} + 3 \beta_1) q^{97} + \beta_{2} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{4} - 8 q^{10} + 4 q^{16} + 36 q^{19} - 32 q^{31} + 44 q^{34} - 12 q^{40} - 48 q^{46} - 8 q^{49} + 12 q^{55} + 20 q^{61} + 64 q^{64} - 20 q^{70} - 60 q^{76} + 28 q^{79} - 16 q^{85} - 20 q^{91} - 12 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} + 5\nu^{5} + 15\nu^{3} + 42\nu ) / 20 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{6} + 5\nu^{4} + 15\nu^{2} + 8 ) / 20 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{7} - 5\nu^{5} + 5\nu^{3} - 16\nu ) / 40 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{6} - 3\nu^{4} - \nu^{2} - 8 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} - 3\nu^{5} - 5\nu^{3} - 4\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 7\nu^{6} + 5\nu^{4} + 15\nu^{2} + 44 ) / 20 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{7} - \nu^{5} - 3\nu^{3} - 6\nu ) / 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} - \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} + \beta_{4} + 2\beta_{2} - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{7} - \beta_{5} + 5\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -2\beta_{6} - 3\beta_{4} + \beta_{2} - 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 5\beta_{7} - 6\beta_{5} - 6\beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 5\beta_{6} - 5\beta_{2} - 9 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -10\beta_{7} + 3\beta_{5} - 3\beta_{3} - 7\beta_1 ) / 2 \) Copy content Toggle raw display

Character values

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

\(n\) \(136\) \(596\) \(757\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
379.1
1.09445 + 0.895644i
−1.09445 0.895644i
0.228425 1.39564i
−0.228425 + 1.39564i
0.228425 + 1.39564i
−0.228425 1.39564i
1.09445 0.895644i
−1.09445 + 0.895644i
2.18890i 0 −2.79129 −2.18890 0.456850i 0 1.00000i 1.73205i 0 −1.00000 + 4.79129i
379.2 2.18890i 0 −2.79129 2.18890 0.456850i 0 1.00000i 1.73205i 0 −1.00000 4.79129i
379.3 0.456850i 0 1.79129 −0.456850 2.18890i 0 1.00000i 1.73205i 0 −1.00000 + 0.208712i
379.4 0.456850i 0 1.79129 0.456850 2.18890i 0 1.00000i 1.73205i 0 −1.00000 0.208712i
379.5 0.456850i 0 1.79129 −0.456850 + 2.18890i 0 1.00000i 1.73205i 0 −1.00000 0.208712i
379.6 0.456850i 0 1.79129 0.456850 + 2.18890i 0 1.00000i 1.73205i 0 −1.00000 + 0.208712i
379.7 2.18890i 0 −2.79129 −2.18890 + 0.456850i 0 1.00000i 1.73205i 0 −1.00000 4.79129i
379.8 2.18890i 0 −2.79129 2.18890 + 0.456850i 0 1.00000i 1.73205i 0 −1.00000 + 4.79129i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 379.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 945.2.d.c 8
3.b odd 2 1 inner 945.2.d.c 8
5.b even 2 1 inner 945.2.d.c 8
5.c odd 4 1 4725.2.a.br 4
5.c odd 4 1 4725.2.a.bs 4
15.d odd 2 1 inner 945.2.d.c 8
15.e even 4 1 4725.2.a.br 4
15.e even 4 1 4725.2.a.bs 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
945.2.d.c 8 1.a even 1 1 trivial
945.2.d.c 8 3.b odd 2 1 inner
945.2.d.c 8 5.b even 2 1 inner
945.2.d.c 8 15.d odd 2 1 inner
4725.2.a.br 4 5.c odd 4 1
4725.2.a.br 4 15.e even 4 1
4725.2.a.bs 4 5.c odd 4 1
4725.2.a.bs 4 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(945, [\chi])\):

\( T_{2}^{4} + 5T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{11}^{2} - 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 5 T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - 34T^{4} + 625 \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} - 3)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} + 23 T^{2} + 1)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 41 T^{2} + 289)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 9 T + 15)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} + 33 T^{2} + 225)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 41 T^{2} + 289)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 8 T - 5)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + 170 T^{2} + 6889)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - 125 T^{2} + 3481)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} + 3)^{4} \) Copy content Toggle raw display
$53$ \( (T^{4} + 45 T^{2} + 81)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 230 T^{2} + 10201)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 5 T + 1)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + 155 T^{2} + 289)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 209 T^{2} + 289)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 21)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} - 7 T - 35)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 350 T^{2} + 14161)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 75)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 155 T^{2} + 289)^{2} \) Copy content Toggle raw display
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