Properties

Label 945.2.a.n
Level $945$
Weight $2$
Character orbit 945.a
Self dual yes
Analytic conductor $7.546$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [945,2,Mod(1,945)]
 
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, 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("945.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 945 = 3^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 945.a (trivial)

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: yes
Analytic conductor: \(7.54586299101\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.144344.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 8x^{2} + 5x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + 2) q^{4} + q^{5} + q^{7} + (\beta_{3} + 2 \beta_1 + 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} + 2) q^{4} + q^{5} + q^{7} + (\beta_{3} + 2 \beta_1 + 1) q^{8} + \beta_1 q^{10} + ( - \beta_{2} + \beta_1 - 1) q^{11} + ( - \beta_{2} - \beta_1 + 1) q^{13} + \beta_1 q^{14} + (\beta_{3} + 2 \beta_{2} + \beta_1 + 4) q^{16} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{17} + ( - \beta_{3} + 1) q^{19} + (\beta_{2} + 2) q^{20} + ( - \beta_{3} + \beta_{2} - 3 \beta_1 + 3) q^{22} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{23} + q^{25} + ( - \beta_{3} - \beta_{2} - \beta_1 - 5) q^{26} + (\beta_{2} + 2) q^{28} + ( - 2 \beta_{2} - 2) q^{29} + (\beta_{3} - \beta_{2} - \beta_1 + 2) q^{31} + (\beta_{3} + 3 \beta_{2} + 4 \beta_1 + 4) q^{32} + ( - 3 \beta_{2} + 2 \beta_1 - 3) q^{34} + q^{35} + (2 \beta_1 + 2) q^{37} + ( - \beta_{3} - 2 \beta_{2} + \beta_1) q^{38} + (\beta_{3} + 2 \beta_1 + 1) q^{40} + (\beta_{3} + \beta_{2} + \beta_1 - 1) q^{41} + (\beta_{3} - \beta_{2} - \beta_1 + 5) q^{43} + ( - 3 \beta_{2} + 3 \beta_1 - 9) q^{44} + ( - 3 \beta_{2} + 2 \beta_1 - 3) q^{46} + (\beta_{3} + \beta_{2} - 3 \beta_1 + 5) q^{47} + q^{49} + \beta_1 q^{50} + ( - 2 \beta_{3} - \beta_{2} - 5 \beta_1 - 7) q^{52} + (\beta_{3} - 2 \beta_{2} + 2 \beta_1 - 1) q^{53} + ( - \beta_{2} + \beta_1 - 1) q^{55} + (\beta_{3} + 2 \beta_1 + 1) q^{56} + ( - 2 \beta_{3} - 6 \beta_1 - 2) q^{58} - 2 \beta_1 q^{59} + ( - \beta_{3} + \beta_{2} - \beta_1 + 2) q^{61} + (\beta_{2} - 5) q^{62} + (2 \beta_{3} + 2 \beta_{2} + 8 \beta_1 + 11) q^{64} + ( - \beta_{2} - \beta_1 + 1) q^{65} + (\beta_{3} - 3 \beta_{2} + \beta_1 + 3) q^{67} + ( - \beta_{3} - 7 \beta_1 + 5) q^{68} + \beta_1 q^{70} + (4 \beta_{2} - 4 \beta_1 - 2) q^{71} + (\beta_{2} + \beta_1 - 3) q^{73} + (2 \beta_{2} + 2 \beta_1 + 8) q^{74} + ( - \beta_{3} - \beta_{2} - 4 \beta_1) q^{76} + ( - \beta_{2} + \beta_1 - 1) q^{77} + ( - \beta_{3} + \beta_{2} + 3 \beta_1 + 2) q^{79} + (\beta_{3} + 2 \beta_{2} + \beta_1 + 4) q^{80} + (2 \beta_{3} + 3 \beta_{2} + \beta_1 + 5) q^{82} + ( - 2 \beta_{2} - 2 \beta_1 + 7) q^{83} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{85} + (\beta_{2} + 3 \beta_1 - 5) q^{86} + ( - \beta_{3} + \beta_{2} - 9 \beta_1 + 3) q^{88} + (\beta_{3} + \beta_{2} - 5 \beta_1 - 1) q^{89} + ( - \beta_{2} - \beta_1 + 1) q^{91} + ( - \beta_{3} - 7 \beta_1 + 5) q^{92} + (2 \beta_{3} - \beta_{2} + 7 \beta_1 - 11) q^{94} + ( - \beta_{3} + 1) q^{95} + (2 \beta_{3} - 2 \beta_{2} + 4 \beta_1 + 2) q^{97} + \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{2} + 9 q^{4} + 4 q^{5} + 4 q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{2} + 9 q^{4} + 4 q^{5} + 4 q^{7} + 6 q^{8} + q^{10} - 4 q^{11} + 2 q^{13} + q^{14} + 19 q^{16} + 4 q^{19} + 9 q^{20} + 10 q^{22} + 4 q^{25} - 22 q^{26} + 9 q^{28} - 10 q^{29} + 6 q^{31} + 23 q^{32} - 13 q^{34} + 4 q^{35} + 10 q^{37} - q^{38} + 6 q^{40} - 2 q^{41} + 18 q^{43} - 36 q^{44} - 13 q^{46} + 18 q^{47} + 4 q^{49} + q^{50} - 34 q^{52} - 4 q^{53} - 4 q^{55} + 6 q^{56} - 14 q^{58} - 2 q^{59} + 8 q^{61} - 19 q^{62} + 54 q^{64} + 2 q^{65} + 10 q^{67} + 13 q^{68} + q^{70} - 8 q^{71} - 10 q^{73} + 36 q^{74} - 5 q^{76} - 4 q^{77} + 12 q^{79} + 19 q^{80} + 24 q^{82} + 24 q^{83} - 16 q^{86} + 4 q^{88} - 8 q^{89} + 2 q^{91} + 13 q^{92} - 38 q^{94} + 4 q^{95} + 10 q^{97} + q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 8x^{2} + 5x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 6\nu - 1 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 6\beta _1 + 1 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.46608
−0.857589
1.51533
2.80834
−2.46608 0 4.08154 1.00000 0 1.00000 −5.13325 0 −2.46608
1.2 −0.857589 0 −1.26454 1.00000 0 1.00000 2.79964 0 −0.857589
1.3 1.51533 0 0.296215 1.00000 0 1.00000 −2.58179 0 1.51533
1.4 2.80834 0 5.88678 1.00000 0 1.00000 10.9154 0 2.80834
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(-1\)
\(7\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 945.2.a.n yes 4
3.b odd 2 1 945.2.a.m 4
5.b even 2 1 4725.2.a.bo 4
7.b odd 2 1 6615.2.a.bh 4
15.d odd 2 1 4725.2.a.bx 4
21.c even 2 1 6615.2.a.be 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
945.2.a.m 4 3.b odd 2 1
945.2.a.n yes 4 1.a even 1 1 trivial
4725.2.a.bo 4 5.b even 2 1
4725.2.a.bx 4 15.d odd 2 1
6615.2.a.be 4 21.c even 2 1
6615.2.a.bh 4 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(945))\):

\( T_{2}^{4} - T_{2}^{3} - 8T_{2}^{2} + 5T_{2} + 9 \) Copy content Toggle raw display
\( T_{11}^{4} + 4T_{11}^{3} - 13T_{11}^{2} - 18T_{11} + 36 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{3} - 8 T^{2} + 5 T + 9 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T - 1)^{4} \) Copy content Toggle raw display
$7$ \( (T - 1)^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + 4 T^{3} - 13 T^{2} - 18 T + 36 \) Copy content Toggle raw display
$13$ \( T^{4} - 2 T^{3} - 29 T^{2} + 76 T - 48 \) Copy content Toggle raw display
$17$ \( T^{4} - 45 T^{2} + 4 T + 372 \) Copy content Toggle raw display
$19$ \( T^{4} - 4 T^{3} - 32 T^{2} + 16 T + 139 \) Copy content Toggle raw display
$23$ \( T^{4} - 45 T^{2} + 4 T + 372 \) Copy content Toggle raw display
$29$ \( T^{4} + 10 T^{3} - 28 T^{2} + \cdots + 384 \) Copy content Toggle raw display
$31$ \( T^{4} - 6 T^{3} - 39 T^{2} + 36 T + 20 \) Copy content Toggle raw display
$37$ \( T^{4} - 10 T^{3} + 4 T^{2} + 112 T - 32 \) Copy content Toggle raw display
$41$ \( T^{4} + 2 T^{3} - 83 T^{2} - 336 T - 324 \) Copy content Toggle raw display
$43$ \( T^{4} - 18 T^{3} + 69 T^{2} + \cdots - 196 \) Copy content Toggle raw display
$47$ \( T^{4} - 18 T^{3} + T^{2} + 1112 T - 3888 \) Copy content Toggle raw display
$53$ \( T^{4} + 4 T^{3} - 84 T^{2} - 8 T + 111 \) Copy content Toggle raw display
$59$ \( T^{4} + 2 T^{3} - 32 T^{2} - 40 T + 144 \) Copy content Toggle raw display
$61$ \( T^{4} - 8 T^{3} - 21 T^{2} + 152 T + 200 \) Copy content Toggle raw display
$67$ \( T^{4} - 10 T^{3} - 99 T^{2} + \cdots + 500 \) Copy content Toggle raw display
$71$ \( T^{4} + 8 T^{3} - 280 T^{2} + \cdots + 6480 \) Copy content Toggle raw display
$73$ \( T^{4} + 10 T^{3} + 7 T^{2} - 136 T - 284 \) Copy content Toggle raw display
$79$ \( T^{4} - 12 T^{3} - 73 T^{2} + \cdots + 1784 \) Copy content Toggle raw display
$83$ \( T^{4} - 24 T^{3} + 94 T^{2} + \cdots - 5583 \) Copy content Toggle raw display
$89$ \( T^{4} + 8 T^{3} - 215 T^{2} + \cdots + 5004 \) Copy content Toggle raw display
$97$ \( T^{4} - 10 T^{3} - 228 T^{2} + \cdots + 4320 \) Copy content Toggle raw display
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