Properties

Label 363.2.a.j
Level $363$
Weight $2$
Character orbit 363.a
Self dual yes
Analytic conductor $2.899$
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] = [363,2,Mod(1,363)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(363, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("363.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 363 = 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 363.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: \(2.89856959337\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{3}, \sqrt{11})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 7x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + q^{3} + (\beta_{3} + 2) q^{4} - \beta_{3} q^{5} + \beta_1 q^{6} + ( - \beta_{2} + \beta_1) q^{7} + (2 \beta_{2} + \beta_1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + q^{3} + (\beta_{3} + 2) q^{4} - \beta_{3} q^{5} + \beta_1 q^{6} + ( - \beta_{2} + \beta_1) q^{7} + (2 \beta_{2} + \beta_1) q^{8} + q^{9} + ( - 2 \beta_{2} - \beta_1) q^{10} + (\beta_{3} + 2) q^{12} + (2 \beta_{2} - 3 \beta_1) q^{13} + 2 q^{14} - \beta_{3} q^{15} + (\beta_{3} + 4) q^{16} + ( - 2 \beta_{2} - \beta_1) q^{17} + \beta_1 q^{18} + (\beta_{2} + \beta_1) q^{19} + ( - \beta_{3} - 8) q^{20} + ( - \beta_{2} + \beta_1) q^{21} + 2 q^{23} + (2 \beta_{2} + \beta_1) q^{24} + ( - \beta_{3} + 3) q^{25} + ( - \beta_{3} - 8) q^{26} + q^{27} + 2 \beta_{2} q^{28} - \beta_1 q^{29} + ( - 2 \beta_{2} - \beta_1) q^{30} + (\beta_{3} + 5) q^{31} + ( - 2 \beta_{2} + 3 \beta_1) q^{32} + ( - 3 \beta_{3} - 8) q^{34} + ( - 4 \beta_{2} + 2 \beta_1) q^{35} + (\beta_{3} + 2) q^{36} + 5 q^{37} + (2 \beta_{3} + 6) q^{38} + (2 \beta_{2} - 3 \beta_1) q^{39} + (2 \beta_{2} - 7 \beta_1) q^{40} + (4 \beta_{2} - 5 \beta_1) q^{41} + 2 q^{42} + (2 \beta_{2} - 4 \beta_1) q^{43} - \beta_{3} q^{45} + 2 \beta_1 q^{46} + (2 \beta_{3} - 6) q^{47} + (\beta_{3} + 4) q^{48} + ( - \beta_{3} - 4) q^{49} + ( - 2 \beta_{2} + 2 \beta_1) q^{50} + ( - 2 \beta_{2} - \beta_1) q^{51} + ( - 6 \beta_{2} - 3 \beta_1) q^{52} + (3 \beta_{3} + 6) q^{53} + \beta_1 q^{54} + 2 \beta_{3} q^{56} + (\beta_{2} + \beta_1) q^{57} + ( - \beta_{3} - 4) q^{58} - 6 q^{59} + ( - \beta_{3} - 8) q^{60} + ( - 3 \beta_{2} + \beta_1) q^{61} + (2 \beta_{2} + 6 \beta_1) q^{62} + ( - \beta_{2} + \beta_1) q^{63} - \beta_{3} q^{64} + (10 \beta_{2} - 3 \beta_1) q^{65} + (3 \beta_{3} + 9) q^{67} + ( - 2 \beta_{2} - 9 \beta_1) q^{68} + 2 q^{69} - 2 \beta_{3} q^{70} + (2 \beta_{3} - 4) q^{71} + (2 \beta_{2} + \beta_1) q^{72} + ( - 3 \beta_{2} + 5 \beta_1) q^{73} + 5 \beta_1 q^{74} + ( - \beta_{3} + 3) q^{75} + (2 \beta_{2} + 6 \beta_1) q^{76} + ( - \beta_{3} - 8) q^{78} + ( - \beta_{2} + 3 \beta_1) q^{79} + ( - 3 \beta_{3} - 8) q^{80} + q^{81} + ( - \beta_{3} - 12) q^{82} + (2 \beta_{2} + 2 \beta_1) q^{83} + 2 \beta_{2} q^{84} + ( - 2 \beta_{2} + 7 \beta_1) q^{85} + ( - 2 \beta_{3} - 12) q^{86} - \beta_1 q^{87} + ( - \beta_{3} - 4) q^{89} + ( - 2 \beta_{2} - \beta_1) q^{90} + (2 \beta_{3} - 8) q^{91} + (2 \beta_{3} + 4) q^{92} + (\beta_{3} + 5) q^{93} + (4 \beta_{2} - 4 \beta_1) q^{94} - 4 \beta_1 q^{95} + ( - 2 \beta_{2} + 3 \beta_1) q^{96} + ( - 4 \beta_{3} - 3) q^{97} + ( - 2 \beta_{2} - 5 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} + 6 q^{4} + 2 q^{5} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} + 6 q^{4} + 2 q^{5} + 4 q^{9} + 6 q^{12} + 8 q^{14} + 2 q^{15} + 14 q^{16} - 30 q^{20} + 8 q^{23} + 14 q^{25} - 30 q^{26} + 4 q^{27} + 18 q^{31} - 26 q^{34} + 6 q^{36} + 20 q^{37} + 20 q^{38} + 8 q^{42} + 2 q^{45} - 28 q^{47} + 14 q^{48} - 14 q^{49} + 18 q^{53} - 4 q^{56} - 14 q^{58} - 24 q^{59} - 30 q^{60} + 2 q^{64} + 30 q^{67} + 8 q^{69} + 4 q^{70} - 20 q^{71} + 14 q^{75} - 30 q^{78} - 26 q^{80} + 4 q^{81} - 46 q^{82} - 44 q^{86} - 14 q^{89} - 36 q^{91} + 12 q^{92} + 18 q^{93} - 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 7x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 5\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{2} + 5\beta_1 \) 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
−2.52434
−0.792287
0.792287
2.52434
−2.52434 1.00000 4.37228 −2.37228 −2.52434 −0.792287 −5.98844 1.00000 5.98844
1.2 −0.792287 1.00000 −1.37228 3.37228 −0.792287 −2.52434 2.67181 1.00000 −2.67181
1.3 0.792287 1.00000 −1.37228 3.37228 0.792287 2.52434 −2.67181 1.00000 2.67181
1.4 2.52434 1.00000 4.37228 −2.37228 2.52434 0.792287 5.98844 1.00000 −5.98844
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(11\) \(1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 363.2.a.j 4
3.b odd 2 1 1089.2.a.u 4
4.b odd 2 1 5808.2.a.ck 4
5.b even 2 1 9075.2.a.cv 4
11.b odd 2 1 inner 363.2.a.j 4
11.c even 5 4 363.2.e.n 16
11.d odd 10 4 363.2.e.n 16
33.d even 2 1 1089.2.a.u 4
44.c even 2 1 5808.2.a.ck 4
55.d odd 2 1 9075.2.a.cv 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
363.2.a.j 4 1.a even 1 1 trivial
363.2.a.j 4 11.b odd 2 1 inner
363.2.e.n 16 11.c even 5 4
363.2.e.n 16 11.d odd 10 4
1089.2.a.u 4 3.b odd 2 1
1089.2.a.u 4 33.d even 2 1
5808.2.a.ck 4 4.b odd 2 1
5808.2.a.ck 4 44.c even 2 1
9075.2.a.cv 4 5.b even 2 1
9075.2.a.cv 4 55.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 7T_{2}^{2} + 4 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(363))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 7T^{2} + 4 \) Copy content Toggle raw display
$3$ \( (T - 1)^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} - T - 8)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} - 7T^{2} + 4 \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} - 51T^{2} + 576 \) Copy content Toggle raw display
$17$ \( T^{4} - 43T^{2} + 256 \) Copy content Toggle raw display
$19$ \( T^{4} - 19T^{2} + 16 \) Copy content Toggle raw display
$23$ \( (T - 2)^{4} \) Copy content Toggle raw display
$29$ \( T^{4} - 7T^{2} + 4 \) Copy content Toggle raw display
$31$ \( (T^{2} - 9 T + 12)^{2} \) Copy content Toggle raw display
$37$ \( (T - 5)^{4} \) Copy content Toggle raw display
$41$ \( T^{4} - 151T^{2} + 3844 \) Copy content Toggle raw display
$43$ \( (T^{2} - 44)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 14 T + 16)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 9 T - 54)^{2} \) Copy content Toggle raw display
$59$ \( (T + 6)^{4} \) Copy content Toggle raw display
$61$ \( T^{4} - 43T^{2} + 256 \) Copy content Toggle raw display
$67$ \( (T^{2} - 15 T - 18)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 10 T - 8)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} - 139T^{2} + 4624 \) Copy content Toggle raw display
$79$ \( T^{4} - 51T^{2} + 576 \) Copy content Toggle raw display
$83$ \( T^{4} - 76T^{2} + 256 \) Copy content Toggle raw display
$89$ \( (T^{2} + 7 T + 4)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 2 T - 131)^{2} \) Copy content Toggle raw display
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