Properties

Label 2365.2.a.g
Level $2365$
Weight $2$
Character orbit 2365.a
Self dual yes
Analytic conductor $18.885$
Analytic rank $1$
Dimension $2$
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] = [2365,2,Mod(1,2365)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2365, 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("2365.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2365 = 5 \cdot 11 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2365.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: \(18.8846200780\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) 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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + (\beta - 1) q^{3} - q^{4} + q^{5} + ( - \beta + 1) q^{6} + ( - 2 \beta + 2) q^{7} + 3 q^{8} + ( - \beta + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + (\beta - 1) q^{3} - q^{4} + q^{5} + ( - \beta + 1) q^{6} + ( - 2 \beta + 2) q^{7} + 3 q^{8} + ( - \beta + 2) q^{9} - q^{10} + q^{11} + ( - \beta + 1) q^{12} + 2 \beta q^{13} + (2 \beta - 2) q^{14} + (\beta - 1) q^{15} - q^{16} + (2 \beta - 4) q^{17} + (\beta - 2) q^{18} + ( - 2 \beta - 2) q^{19} - q^{20} + (2 \beta - 10) q^{21} - q^{22} + ( - 2 \beta - 2) q^{23} + (3 \beta - 3) q^{24} + q^{25} - 2 \beta q^{26} + ( - \beta - 3) q^{27} + (2 \beta - 2) q^{28} + (3 \beta - 5) q^{29} + ( - \beta + 1) q^{30} - 5 q^{32} + (\beta - 1) q^{33} + ( - 2 \beta + 4) q^{34} + ( - 2 \beta + 2) q^{35} + (\beta - 2) q^{36} - 2 \beta q^{37} + (2 \beta + 2) q^{38} + 8 q^{39} + 3 q^{40} + 2 \beta q^{41} + ( - 2 \beta + 10) q^{42} - q^{43} - q^{44} + ( - \beta + 2) q^{45} + (2 \beta + 2) q^{46} + ( - 2 \beta + 6) q^{47} + ( - \beta + 1) q^{48} + ( - 4 \beta + 13) q^{49} - q^{50} + ( - 4 \beta + 12) q^{51} - 2 \beta q^{52} + (3 \beta - 1) q^{53} + (\beta + 3) q^{54} + q^{55} + ( - 6 \beta + 6) q^{56} + ( - 2 \beta - 6) q^{57} + ( - 3 \beta + 5) q^{58} + 4 \beta q^{59} + ( - \beta + 1) q^{60} + (3 \beta - 5) q^{61} + ( - 4 \beta + 12) q^{63} + 7 q^{64} + 2 \beta q^{65} + ( - \beta + 1) q^{66} + ( - 2 \beta - 6) q^{67} + ( - 2 \beta + 4) q^{68} + ( - 2 \beta - 6) q^{69} + (2 \beta - 2) q^{70} - 4 q^{71} + ( - 3 \beta + 6) q^{72} + ( - \beta - 9) q^{73} + 2 \beta q^{74} + (\beta - 1) q^{75} + (2 \beta + 2) q^{76} + ( - 2 \beta + 2) q^{77} - 8 q^{78} + ( - 3 \beta - 9) q^{79} - q^{80} - 7 q^{81} - 2 \beta q^{82} + ( - \beta - 7) q^{83} + ( - 2 \beta + 10) q^{84} + (2 \beta - 4) q^{85} + q^{86} + ( - 5 \beta + 17) q^{87} + 3 q^{88} + ( - 6 \beta + 8) q^{89} + (\beta - 2) q^{90} - 16 q^{91} + (2 \beta + 2) q^{92} + (2 \beta - 6) q^{94} + ( - 2 \beta - 2) q^{95} + ( - 5 \beta + 5) q^{96} + ( - \beta + 15) q^{97} + (4 \beta - 13) q^{98} + ( - \beta + 2) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - q^{3} - 2 q^{4} + 2 q^{5} + q^{6} + 2 q^{7} + 6 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - q^{3} - 2 q^{4} + 2 q^{5} + q^{6} + 2 q^{7} + 6 q^{8} + 3 q^{9} - 2 q^{10} + 2 q^{11} + q^{12} + 2 q^{13} - 2 q^{14} - q^{15} - 2 q^{16} - 6 q^{17} - 3 q^{18} - 6 q^{19} - 2 q^{20} - 18 q^{21} - 2 q^{22} - 6 q^{23} - 3 q^{24} + 2 q^{25} - 2 q^{26} - 7 q^{27} - 2 q^{28} - 7 q^{29} + q^{30} - 10 q^{32} - q^{33} + 6 q^{34} + 2 q^{35} - 3 q^{36} - 2 q^{37} + 6 q^{38} + 16 q^{39} + 6 q^{40} + 2 q^{41} + 18 q^{42} - 2 q^{43} - 2 q^{44} + 3 q^{45} + 6 q^{46} + 10 q^{47} + q^{48} + 22 q^{49} - 2 q^{50} + 20 q^{51} - 2 q^{52} + q^{53} + 7 q^{54} + 2 q^{55} + 6 q^{56} - 14 q^{57} + 7 q^{58} + 4 q^{59} + q^{60} - 7 q^{61} + 20 q^{63} + 14 q^{64} + 2 q^{65} + q^{66} - 14 q^{67} + 6 q^{68} - 14 q^{69} - 2 q^{70} - 8 q^{71} + 9 q^{72} - 19 q^{73} + 2 q^{74} - q^{75} + 6 q^{76} + 2 q^{77} - 16 q^{78} - 21 q^{79} - 2 q^{80} - 14 q^{81} - 2 q^{82} - 15 q^{83} + 18 q^{84} - 6 q^{85} + 2 q^{86} + 29 q^{87} + 6 q^{88} + 10 q^{89} - 3 q^{90} - 32 q^{91} + 6 q^{92} - 10 q^{94} - 6 q^{95} + 5 q^{96} + 29 q^{97} - 22 q^{98} + 3 q^{99}+O(q^{100}) \) 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
−1.56155
2.56155
−1.00000 −2.56155 −1.00000 1.00000 2.56155 5.12311 3.00000 3.56155 −1.00000
1.2 −1.00000 1.56155 −1.00000 1.00000 −1.56155 −3.12311 3.00000 −0.561553 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(11\) \(-1\)
\(43\) \(1\)

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 2365.2.a.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2365.2.a.g 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2365))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + T - 4 \) Copy content Toggle raw display
$5$ \( (T - 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 2T - 16 \) Copy content Toggle raw display
$11$ \( (T - 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 2T - 16 \) Copy content Toggle raw display
$17$ \( T^{2} + 6T - 8 \) Copy content Toggle raw display
$19$ \( T^{2} + 6T - 8 \) Copy content Toggle raw display
$23$ \( T^{2} + 6T - 8 \) Copy content Toggle raw display
$29$ \( T^{2} + 7T - 26 \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 2T - 16 \) Copy content Toggle raw display
$41$ \( T^{2} - 2T - 16 \) Copy content Toggle raw display
$43$ \( (T + 1)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 10T + 8 \) Copy content Toggle raw display
$53$ \( T^{2} - T - 38 \) Copy content Toggle raw display
$59$ \( T^{2} - 4T - 64 \) Copy content Toggle raw display
$61$ \( T^{2} + 7T - 26 \) Copy content Toggle raw display
$67$ \( T^{2} + 14T + 32 \) Copy content Toggle raw display
$71$ \( (T + 4)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 19T + 86 \) Copy content Toggle raw display
$79$ \( T^{2} + 21T + 72 \) Copy content Toggle raw display
$83$ \( T^{2} + 15T + 52 \) Copy content Toggle raw display
$89$ \( T^{2} - 10T - 128 \) Copy content Toggle raw display
$97$ \( T^{2} - 29T + 206 \) Copy content Toggle raw display
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