Properties

Label 368.2.j.b
Level $368$
Weight $2$
Character orbit 368.j
Analytic conductor $2.938$
Analytic rank $1$
Dimension $4$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [368,2,Mod(93,368)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(368, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("368.93");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 368 = 2^{4} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 368.j (of order \(4\), degree \(2\), 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: \(2.93849479438\)
Analytic rank: \(1\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{2} - \beta_1) q^{2} + (\beta_{3} + \beta_{2} - 1) q^{3} + (\beta_{3} + \beta_1) q^{4} + ( - \beta_{2} - 1) q^{5} + (\beta_{3} - 3 \beta_{2} + 2 \beta_1 - 2) q^{6} + ( - \beta_{3} + \beta_{2} + \beta_1 - 1) q^{7} + ( - 2 \beta_{2} - 2) q^{8} + ( - 3 \beta_{3} - 3 \beta_{2} + \cdots - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} - \beta_1) q^{2} + (\beta_{3} + \beta_{2} - 1) q^{3} + (\beta_{3} + \beta_1) q^{4} + ( - \beta_{2} - 1) q^{5} + (\beta_{3} - 3 \beta_{2} + 2 \beta_1 - 2) q^{6} + ( - \beta_{3} + \beta_{2} + \beta_1 - 1) q^{7} + ( - 2 \beta_{2} - 2) q^{8} + ( - 3 \beta_{3} - 3 \beta_{2} + \cdots - 3) q^{9}+ \cdots + ( - 6 \beta_{3} - 12 \beta_{2} + 12) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} - 6 q^{3} - 4 q^{5} - 6 q^{6} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} - 6 q^{3} - 4 q^{5} - 6 q^{6} - 8 q^{8} + 4 q^{10} - 4 q^{11} - 2 q^{13} + 4 q^{14} + 12 q^{15} + 8 q^{16} - 12 q^{17} + 24 q^{18} - 4 q^{19} + 4 q^{20} + 4 q^{22} + 24 q^{24} - 18 q^{26} + 18 q^{27} - 4 q^{28} - 2 q^{29} - 28 q^{31} + 8 q^{32} - 12 q^{33} + 4 q^{35} + 12 q^{36} - 24 q^{37} + 12 q^{38} - 8 q^{43} - 20 q^{44} - 12 q^{45} + 2 q^{46} - 40 q^{47} + 12 q^{49} + 6 q^{50} + 24 q^{51} + 16 q^{52} - 20 q^{53} - 18 q^{54} + 8 q^{56} + 18 q^{58} + 28 q^{59} - 12 q^{60} + 8 q^{61} + 26 q^{62} - 24 q^{63} + 4 q^{65} - 24 q^{66} - 12 q^{67} + 12 q^{68} + 6 q^{69} - 12 q^{70} - 24 q^{72} + 24 q^{74} + 18 q^{75} - 16 q^{76} + 28 q^{77} + 42 q^{78} - 16 q^{79} - 8 q^{80} - 72 q^{81} - 6 q^{82} - 8 q^{83} - 12 q^{84} + 12 q^{85} + 8 q^{86} - 36 q^{90} + 16 q^{91} + 4 q^{92} + 30 q^{93} + 26 q^{94} + 8 q^{95} - 24 q^{96} + 44 q^{97} + 6 q^{98} + 60 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{2} + \zeta_{12} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{12}^{3} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{12}^{2} + \zeta_{12} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( -\beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/368\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(97\) \(277\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
93.1
0.866025 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
−1.36603 + 0.366025i −0.633975 0.633975i 1.73205 1.00000i −1.00000 + 1.00000i 1.09808 + 0.633975i 2.73205i −2.00000 + 2.00000i 2.19615i 1.00000 1.73205i
93.2 0.366025 1.36603i −2.36603 2.36603i −1.73205 1.00000i −1.00000 + 1.00000i −4.09808 + 2.36603i 0.732051i −2.00000 + 2.00000i 8.19615i 1.00000 + 1.73205i
277.1 −1.36603 0.366025i −0.633975 + 0.633975i 1.73205 + 1.00000i −1.00000 1.00000i 1.09808 0.633975i 2.73205i −2.00000 2.00000i 2.19615i 1.00000 + 1.73205i
277.2 0.366025 + 1.36603i −2.36603 + 2.36603i −1.73205 + 1.00000i −1.00000 1.00000i −4.09808 2.36603i 0.732051i −2.00000 2.00000i 8.19615i 1.00000 1.73205i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 368.2.j.b 4
4.b odd 2 1 1472.2.j.b 4
16.e even 4 1 inner 368.2.j.b 4
16.f odd 4 1 1472.2.j.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
368.2.j.b 4 1.a even 1 1 trivial
368.2.j.b 4 16.e even 4 1 inner
1472.2.j.b 4 4.b odd 2 1
1472.2.j.b 4 16.f odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 6T_{3}^{3} + 18T_{3}^{2} + 18T_{3} + 9 \) acting on \(S_{2}^{\mathrm{new}}(368, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 2 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$3$ \( T^{4} + 6 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$5$ \( (T^{2} + 2 T + 2)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 8T^{2} + 4 \) Copy content Toggle raw display
$11$ \( T^{4} + 4 T^{3} + \cdots + 484 \) Copy content Toggle raw display
$13$ \( T^{4} + 2 T^{3} + \cdots + 169 \) Copy content Toggle raw display
$17$ \( (T^{2} + 6 T + 6)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 4 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$23$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} + 2 T^{3} + \cdots + 169 \) Copy content Toggle raw display
$31$ \( (T^{2} + 14 T + 37)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 24 T^{3} + \cdots + 4356 \) Copy content Toggle raw display
$41$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 8 T^{3} + \cdots + 256 \) Copy content Toggle raw display
$47$ \( (T^{2} + 20 T + 97)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 20 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$59$ \( T^{4} - 28 T^{3} + \cdots + 5476 \) Copy content Toggle raw display
$61$ \( (T^{2} - 4 T + 8)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 12 T^{3} + \cdots + 36 \) Copy content Toggle raw display
$71$ \( T^{4} + 186T^{2} + 4761 \) Copy content Toggle raw display
$73$ \( (T^{2} + 49)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 8 T - 32)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 8 T^{3} + \cdots + 20164 \) Copy content Toggle raw display
$89$ \( (T^{2} + 108)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 22 T + 118)^{2} \) Copy content Toggle raw display
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