Properties

Label 2366.2.a.ba
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} - 4) q^{11} + ( - \beta_1 - 1) q^{12} + q^{14} + (\beta_{2} - \beta_1 - 1) q^{15} + q^{16} + (5 \beta_{2} + 1) q^{17} + (\beta_{2} + 2 \beta_1) q^{18} + ( - \beta_1 - 4) q^{19} + ( - \beta_{2} + \beta_1) q^{20} + ( - \beta_1 - 1) q^{21} + ( - \beta_{2} - 4) q^{22} + (\beta_{2} - 3 \beta_1 - 1) q^{23} + ( - \beta_1 - 1) q^{24} + ( - 2 \beta_{2} + \beta_1 - 4) q^{25} + ( - 4 \beta_{2} + \beta_1 - 2) q^{27} + q^{28} + (5 \beta_{2} - \beta_1 + 1) q^{29} + (\beta_{2} - \beta_1 - 1) q^{30} + (3 \beta_{2} + \beta_1) q^{31} + q^{32} + (2 \beta_{2} + 4 \beta_1 + 5) q^{33} + (5 \beta_{2} + 1) q^{34} + ( - \beta_{2} + \beta_1) q^{35} + (\beta_{2} + 2 \beta_1) q^{36} + ( - 3 \beta_{2} + 3 \beta_1 - 2) q^{37} + ( - \beta_1 - 4) q^{38} + ( - \beta_{2} + \beta_1) q^{40} + ( - 7 \beta_{2} + 7 \beta_1 - 7) q^{41} + ( - \beta_1 - 1) q^{42} + (2 \beta_{2} - 3 \beta_1 - 4) q^{43} + ( - \beta_{2} - 4) q^{44} + (2 \beta_{2} - \beta_1 + 2) q^{45} + (\beta_{2} - 3 \beta_1 - 1) q^{46} + (\beta_{2} - 6 \beta_1 + 4) q^{47} + ( - \beta_1 - 1) q^{48} + q^{49} + ( - 2 \beta_{2} + \beta_1 - 4) q^{50} + ( - 10 \beta_{2} - \beta_1 - 6) q^{51} + ( - 4 \beta_{2} - \beta_1 - 1) q^{53} + ( - 4 \beta_{2} + \beta_1 - 2) q^{54} + (2 \beta_{2} - 3 \beta_1) q^{55} + q^{56} + (\beta_{2} + 5 \beta_1 + 6) q^{57} + (5 \beta_{2} - \beta_1 + 1) q^{58} + ( - \beta_{2} - 6) q^{59} + (\beta_{2} - \beta_1 - 1) q^{60} + (\beta_{2} + 4 \beta_1) q^{61} + (3 \beta_{2} + \beta_1) q^{62} + (\beta_{2} + 2 \beta_1) q^{63} + q^{64} + (2 \beta_{2} + 4 \beta_1 + 5) q^{66} + (3 \beta_1 - 8) q^{67} + (5 \beta_{2} + 1) q^{68} + (\beta_{2} + 4 \beta_1 + 6) q^{69} + ( - \beta_{2} + \beta_1) q^{70} + ( - 5 \beta_{2} + 8 \beta_1 + 1) q^{71} + (\beta_{2} + 2 \beta_1) q^{72} + ( - 2 \beta_{2} - \beta_1) q^{73} + ( - 3 \beta_{2} + 3 \beta_1 - 2) q^{74} + (3 \beta_{2} + 3 \beta_1 + 4) q^{75} + ( - \beta_1 - 4) q^{76} + ( - \beta_{2} - 4) q^{77} + ( - 5 \beta_1 + 5) q^{79} + ( - \beta_{2} + \beta_1) q^{80} + (4 \beta_{2} - 5 \beta_1 + 4) q^{81} + ( - 7 \beta_{2} + 7 \beta_1 - 7) q^{82} + ( - \beta_{2} + 7 \beta_1 - 3) q^{83} + ( - \beta_1 - 1) q^{84} + (9 \beta_{2} - 4 \beta_1) q^{85} + (2 \beta_{2} - 3 \beta_1 - 4) q^{86} + ( - 9 \beta_{2} - 4) q^{87} + ( - \beta_{2} - 4) q^{88} + (7 \beta_{2} - 7 \beta_1 + 6) q^{89} + (2 \beta_{2} - \beta_1 + 2) q^{90} + (\beta_{2} - 3 \beta_1 - 1) q^{92} + ( - 7 \beta_{2} - \beta_1 - 5) q^{93} + (\beta_{2} - 6 \beta_1 + 4) q^{94} + (4 \beta_{2} - 4 \beta_1 - 1) q^{95} + ( - \beta_1 - 1) q^{96} + ( - 7 \beta_{2} + \beta_1 - 9) q^{97} + q^{98} + ( - 5 \beta_{2} - 9 \beta_1 - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} - 4 q^{3} + 3 q^{4} + 2 q^{5} - 4 q^{6} + 3 q^{7} + 3 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{2} - 4 q^{3} + 3 q^{4} + 2 q^{5} - 4 q^{6} + 3 q^{7} + 3 q^{8} + q^{9} + 2 q^{10} - 11 q^{11} - 4 q^{12} + 3 q^{14} - 5 q^{15} + 3 q^{16} - 2 q^{17} + q^{18} - 13 q^{19} + 2 q^{20} - 4 q^{21} - 11 q^{22} - 7 q^{23} - 4 q^{24} - 9 q^{25} - q^{27} + 3 q^{28} - 3 q^{29} - 5 q^{30} - 2 q^{31} + 3 q^{32} + 17 q^{33} - 2 q^{34} + 2 q^{35} + q^{36} - 13 q^{38} + 2 q^{40} - 7 q^{41} - 4 q^{42} - 17 q^{43} - 11 q^{44} + 3 q^{45} - 7 q^{46} + 5 q^{47} - 4 q^{48} + 3 q^{49} - 9 q^{50} - 9 q^{51} - q^{54} - 5 q^{55} + 3 q^{56} + 22 q^{57} - 3 q^{58} - 17 q^{59} - 5 q^{60} + 3 q^{61} - 2 q^{62} + q^{63} + 3 q^{64} + 17 q^{66} - 21 q^{67} - 2 q^{68} + 21 q^{69} + 2 q^{70} + 16 q^{71} + q^{72} + q^{73} + 12 q^{75} - 13 q^{76} - 11 q^{77} + 10 q^{79} + 2 q^{80} + 3 q^{81} - 7 q^{82} - q^{83} - 4 q^{84} - 13 q^{85} - 17 q^{86} - 3 q^{87} - 11 q^{88} + 4 q^{89} + 3 q^{90} - 7 q^{92} - 9 q^{93} + 5 q^{94} - 11 q^{95} - 4 q^{96} - 19 q^{97} + 3 q^{98} - 13 q^{99}+O(q^{100}) \) 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 −2.80194 1.00000 0.554958 −2.80194 1.00000 1.00000 4.85086 0.554958
1.2 1.00000 −1.44504 1.00000 2.24698 −1.44504 1.00000 1.00000 −0.911854 2.24698
1.3 1.00000 0.246980 1.00000 −0.801938 0.246980 1.00000 1.00000 −2.93900 −0.801938
\(n\): 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.ba yes 3
13.b even 2 1 2366.2.a.v 3
13.d odd 4 2 2366.2.d.n 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2366.2.a.v 3 13.b even 2 1
2366.2.a.ba yes 3 1.a even 1 1 trivial
2366.2.d.n 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2366))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{3} + 4 T^{2} + 3 T - 1 \) Copy content Toggle raw display
$5$ \( T^{3} - 2T^{2} - T + 1 \) Copy content Toggle raw display
$7$ \( (T - 1)^{3} \) Copy content Toggle raw display
$11$ \( T^{3} + 11 T^{2} + 38 T + 41 \) Copy content Toggle raw display
$13$ \( T^{3} \) Copy content Toggle raw display
$17$ \( T^{3} + 2 T^{2} - 57 T - 71 \) Copy content Toggle raw display
$19$ \( T^{3} + 13 T^{2} + 54 T + 71 \) Copy content Toggle raw display
$23$ \( T^{3} + 7T^{2} - 49 \) Copy content Toggle raw display
$29$ \( T^{3} + 3 T^{2} - 46 T + 1 \) Copy content Toggle raw display
$31$ \( T^{3} + 2 T^{2} - 29 T - 71 \) Copy content Toggle raw display
$37$ \( T^{3} - 21T - 7 \) Copy content Toggle raw display
$41$ \( T^{3} + 7 T^{2} - 98 T - 343 \) Copy content Toggle raw display
$43$ \( T^{3} + 17 T^{2} + 80 T + 71 \) Copy content Toggle raw display
$47$ \( T^{3} - 5 T^{2} - 64 T - 29 \) Copy content Toggle raw display
$53$ \( T^{3} - 49T + 91 \) Copy content Toggle raw display
$59$ \( T^{3} + 17 T^{2} + 94 T + 169 \) Copy content Toggle raw display
$61$ \( T^{3} - 3 T^{2} - 46 T - 1 \) Copy content Toggle raw display
$67$ \( T^{3} + 21 T^{2} + 126 T + 203 \) Copy content Toggle raw display
$71$ \( T^{3} - 16 T^{2} - 29 T + 841 \) Copy content Toggle raw display
$73$ \( T^{3} - T^{2} - 16 T + 29 \) Copy content Toggle raw display
$79$ \( T^{3} - 10 T^{2} - 25 T + 125 \) Copy content Toggle raw display
$83$ \( T^{3} + T^{2} - 100 T + 181 \) Copy content Toggle raw display
$89$ \( T^{3} - 4 T^{2} - 109 T + 239 \) Copy content Toggle raw display
$97$ \( T^{3} + 19 T^{2} + 20 T - 461 \) Copy content Toggle raw display
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