Properties

Label 2366.2.a.y
Level $2366$
Weight $2$
Character orbit 2366.a
Self dual yes
Analytic conductor $18.893$
Analytic rank $1$
Dimension $3$
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] = [2366,2,Mod(1,2366)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2366, 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("2366.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2366 = 2 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2366.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: \(18.8926051182\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + ( - \beta_1 + 1) q^{3} + q^{4} + (\beta_{2} + \beta_1) q^{5} + (\beta_1 - 1) q^{6} + q^{7} - q^{8} + (\beta_{2} - 2 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + ( - \beta_1 + 1) q^{3} + q^{4} + (\beta_{2} + \beta_1) q^{5} + (\beta_1 - 1) q^{6} + q^{7} - q^{8} + (\beta_{2} - 2 \beta_1) q^{9} + ( - \beta_{2} - \beta_1) q^{10} + \beta_{2} q^{11} + ( - \beta_1 + 1) q^{12} - q^{14} + ( - \beta_{2} + \beta_1 - 3) q^{15} + q^{16} + ( - 3 \beta_{2} + 2 \beta_1 - 1) q^{17} + ( - \beta_{2} + 2 \beta_1) q^{18} + ( - 2 \beta_{2} + \beta_1 - 6) q^{19} + (\beta_{2} + \beta_1) q^{20} + ( - \beta_1 + 1) q^{21} - \beta_{2} q^{22} + ( - 3 \beta_{2} + \beta_1 - 1) q^{23} + (\beta_1 - 1) q^{24} + (2 \beta_{2} + \beta_1) q^{25} + (2 \beta_{2} + \beta_1) q^{27} + q^{28} + ( - 3 \beta_{2} - \beta_1 + 1) q^{29} + (\beta_{2} - \beta_1 + 3) q^{30} + ( - 3 \beta_{2} + 5 \beta_1 - 4) q^{31} - q^{32} - q^{33} + (3 \beta_{2} - 2 \beta_1 + 1) q^{34} + (\beta_{2} + \beta_1) q^{35} + (\beta_{2} - 2 \beta_1) q^{36} + ( - \beta_{2} + \beta_1 - 10) q^{37} + (2 \beta_{2} - \beta_1 + 6) q^{38} + ( - \beta_{2} - \beta_1) q^{40} + ( - 3 \beta_{2} - \beta_1 - 5) q^{41} + (\beta_1 - 1) q^{42} + ( - 2 \beta_{2} - 3 \beta_1) q^{43} + \beta_{2} q^{44} + ( - 4 \beta_{2} + \beta_1 - 4) q^{45} + (3 \beta_{2} - \beta_1 + 1) q^{46} + (5 \beta_{2} - 8 \beta_1 + 2) q^{47} + ( - \beta_1 + 1) q^{48} + q^{49} + ( - 2 \beta_{2} - \beta_1) q^{50} + ( - 2 \beta_{2} + 3 \beta_1 - 2) q^{51} + (3 \beta_1 - 5) q^{53} + ( - 2 \beta_{2} - \beta_1) q^{54} + (\beta_1 + 2) q^{55} - q^{56} + ( - \beta_{2} + 7 \beta_1 - 6) q^{57} + (3 \beta_{2} + \beta_1 - 1) q^{58} + (3 \beta_{2} - 6 \beta_1 + 4) q^{59} + ( - \beta_{2} + \beta_1 - 3) q^{60} + (7 \beta_{2} - 2 \beta_1 + 2) q^{61} + (3 \beta_{2} - 5 \beta_1 + 4) q^{62} + (\beta_{2} - 2 \beta_1) q^{63} + q^{64} + q^{66} + 5 \beta_1 q^{67} + ( - 3 \beta_{2} + 2 \beta_1 - 1) q^{68} + ( - \beta_{2} + 2 \beta_1) q^{69} + ( - \beta_{2} - \beta_1) q^{70} + (5 \beta_{2} - 4 \beta_1 + 3) q^{71} + ( - \beta_{2} + 2 \beta_1) q^{72} + (4 \beta_{2} - 11 \beta_1 - 2) q^{73} + (\beta_{2} - \beta_1 + 10) q^{74} + ( - \beta_{2} + \beta_1 - 4) q^{75} + ( - 2 \beta_{2} + \beta_1 - 6) q^{76} + \beta_{2} q^{77} + (8 \beta_{2} + 3 \beta_1 + 1) q^{79} + (\beta_{2} + \beta_1) q^{80} + ( - 4 \beta_{2} + 7 \beta_1 - 4) q^{81} + (3 \beta_{2} + \beta_1 + 5) q^{82} + (3 \beta_{2} - \beta_1 + 3) q^{83} + ( - \beta_1 + 1) q^{84} + (3 \beta_{2} - 4 \beta_1) q^{85} + (2 \beta_{2} + 3 \beta_1) q^{86} + (\beta_{2} - 2 \beta_1 + 6) q^{87} - \beta_{2} q^{88} + ( - 7 \beta_{2} + 9 \beta_1 - 10) q^{89} + (4 \beta_{2} - \beta_1 + 4) q^{90} + ( - 3 \beta_{2} + \beta_1 - 1) q^{92} + ( - 5 \beta_{2} + 9 \beta_1 - 11) q^{93} + ( - 5 \beta_{2} + 8 \beta_1 - 2) q^{94} + ( - 4 \beta_{2} - 8 \beta_1 - 1) q^{95} + (\beta_1 - 1) q^{96} + (\beta_{2} + \beta_1 + 5) q^{97} - q^{98} + ( - 3 \beta_{2} + \beta_1 - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} + 2 q^{3} + 3 q^{4} - 2 q^{6} + 3 q^{7} - 3 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{2} + 2 q^{3} + 3 q^{4} - 2 q^{6} + 3 q^{7} - 3 q^{8} - 3 q^{9} - q^{11} + 2 q^{12} - 3 q^{14} - 7 q^{15} + 3 q^{16} + 2 q^{17} + 3 q^{18} - 15 q^{19} + 2 q^{21} + q^{22} + q^{23} - 2 q^{24} - q^{25} - q^{27} + 3 q^{28} + 5 q^{29} + 7 q^{30} - 4 q^{31} - 3 q^{32} - 3 q^{33} - 2 q^{34} - 3 q^{36} - 28 q^{37} + 15 q^{38} - 13 q^{41} - 2 q^{42} - q^{43} - q^{44} - 7 q^{45} - q^{46} - 7 q^{47} + 2 q^{48} + 3 q^{49} + q^{50} - q^{51} - 12 q^{53} + q^{54} + 7 q^{55} - 3 q^{56} - 10 q^{57} - 5 q^{58} + 3 q^{59} - 7 q^{60} - 3 q^{61} + 4 q^{62} - 3 q^{63} + 3 q^{64} + 3 q^{66} + 5 q^{67} + 2 q^{68} + 3 q^{69} + 3 q^{72} - 21 q^{73} + 28 q^{74} - 10 q^{75} - 15 q^{76} - q^{77} - 2 q^{79} - q^{81} + 13 q^{82} + 5 q^{83} + 2 q^{84} - 7 q^{85} + q^{86} + 15 q^{87} + q^{88} - 14 q^{89} + 7 q^{90} + q^{92} - 19 q^{93} + 7 q^{94} - 7 q^{95} - 2 q^{96} + 15 q^{97} - 3 q^{98} + q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of \(\nu = \zeta_{14} + \zeta_{14}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) 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
1.80194
0.445042
−1.24698
−1.00000 −0.801938 1.00000 3.04892 0.801938 1.00000 −1.00000 −2.35690 −3.04892
1.2 −1.00000 0.554958 1.00000 −1.35690 −0.554958 1.00000 −1.00000 −2.69202 1.35690
1.3 −1.00000 2.24698 1.00000 −1.69202 −2.24698 1.00000 −1.00000 2.04892 1.69202
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(7\) \(-1\)
\(13\) \(-1\)

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 2366.2.a.y 3
13.b even 2 1 2366.2.a.bd yes 3
13.d odd 4 2 2366.2.d.p 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2366.2.a.y 3 1.a even 1 1 trivial
2366.2.a.bd yes 3 13.b even 2 1
2366.2.d.p 6 13.d odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2366))\):

\( T_{3}^{3} - 2T_{3}^{2} - T_{3} + 1 \) Copy content Toggle raw display
\( T_{5}^{3} - 7T_{5} - 7 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{3} - 2T^{2} - T + 1 \) Copy content Toggle raw display
$5$ \( T^{3} - 7T - 7 \) Copy content Toggle raw display
$7$ \( (T - 1)^{3} \) Copy content Toggle raw display
$11$ \( T^{3} + T^{2} - 2T - 1 \) Copy content Toggle raw display
$13$ \( T^{3} \) Copy content Toggle raw display
$17$ \( T^{3} - 2 T^{2} - 15 T - 13 \) Copy content Toggle raw display
$19$ \( T^{3} + 15 T^{2} + 68 T + 83 \) Copy content Toggle raw display
$23$ \( T^{3} - T^{2} - 16 T - 13 \) Copy content Toggle raw display
$29$ \( T^{3} - 5 T^{2} - 22 T + 97 \) Copy content Toggle raw display
$31$ \( T^{3} + 4 T^{2} - 39 T + 41 \) Copy content Toggle raw display
$37$ \( T^{3} + 28 T^{2} + 259 T + 791 \) Copy content Toggle raw display
$41$ \( T^{3} + 13 T^{2} + 26 T + 1 \) Copy content Toggle raw display
$43$ \( T^{3} + T^{2} - 44 T + 83 \) Copy content Toggle raw display
$47$ \( T^{3} + 7 T^{2} - 98 T - 637 \) Copy content Toggle raw display
$53$ \( T^{3} + 12 T^{2} + 27 T - 13 \) Copy content Toggle raw display
$59$ \( T^{3} - 3 T^{2} - 60 T - 127 \) Copy content Toggle raw display
$61$ \( T^{3} + 3 T^{2} - 88 T + 113 \) Copy content Toggle raw display
$67$ \( T^{3} - 5 T^{2} - 50 T + 125 \) Copy content Toggle raw display
$71$ \( T^{3} - 49T + 91 \) Copy content Toggle raw display
$73$ \( T^{3} + 21 T^{2} - 70 T - 2359 \) Copy content Toggle raw display
$79$ \( T^{3} + 2 T^{2} - 225 T - 1247 \) Copy content Toggle raw display
$83$ \( T^{3} - 5 T^{2} - 8 T + 41 \) Copy content Toggle raw display
$89$ \( T^{3} + 14 T^{2} - 91 T - 301 \) Copy content Toggle raw display
$97$ \( T^{3} - 15 T^{2} + 68 T - 97 \) Copy content Toggle raw display
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