Properties

Label 465.2.a.e
Level $465$
Weight $2$
Character orbit 465.a
Self dual yes
Analytic conductor $3.713$
Analytic rank $0$
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] = [465,2,Mod(1,465)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(465, 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("465.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 465 = 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 465.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: \(3.71304369399\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.564.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 5x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - q^{3} + (\beta_{2} + 2) q^{4} - q^{5} - \beta_1 q^{6} + ( - \beta_1 + 3) q^{7} + (\beta_{2} + \beta_1 + 1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - q^{3} + (\beta_{2} + 2) q^{4} - q^{5} - \beta_1 q^{6} + ( - \beta_1 + 3) q^{7} + (\beta_{2} + \beta_1 + 1) q^{8} + q^{9} - \beta_1 q^{10} + 2 \beta_1 q^{11} + ( - \beta_{2} - 2) q^{12} + (\beta_1 + 1) q^{13} + ( - \beta_{2} + 3 \beta_1 - 4) q^{14} + q^{15} + (2 \beta_1 + 1) q^{16} + ( - \beta_{2} + 1) q^{17} + \beta_1 q^{18} + ( - 2 \beta_{2} - 2) q^{19} + ( - \beta_{2} - 2) q^{20} + (\beta_1 - 3) q^{21} + (2 \beta_{2} + 8) q^{22} + (\beta_{2} - 2 \beta_1 + 3) q^{23} + ( - \beta_{2} - \beta_1 - 1) q^{24} + q^{25} + (\beta_{2} + \beta_1 + 4) q^{26} - q^{27} + (2 \beta_{2} - 3 \beta_1 + 5) q^{28} + ( - \beta_{2} - 3 \beta_1) q^{29} + \beta_1 q^{30} + q^{31} + ( - \beta_1 + 6) q^{32} - 2 \beta_1 q^{33} + ( - \beta_{2} - 1) q^{34} + (\beta_1 - 3) q^{35} + (\beta_{2} + 2) q^{36} + (\beta_1 + 3) q^{37} + ( - 2 \beta_{2} - 4 \beta_1 - 2) q^{38} + ( - \beta_1 - 1) q^{39} + ( - \beta_{2} - \beta_1 - 1) q^{40} + ( - 2 \beta_1 - 2) q^{41} + (\beta_{2} - 3 \beta_1 + 4) q^{42} + ( - 2 \beta_1 + 4) q^{43} + (2 \beta_{2} + 6 \beta_1 + 2) q^{44} - q^{45} + ( - \beta_{2} + 4 \beta_1 - 7) q^{46} + ( - 3 \beta_{2} + 2 \beta_1 - 5) q^{47} + ( - 2 \beta_1 - 1) q^{48} + (\beta_{2} - 6 \beta_1 + 6) q^{49} + \beta_1 q^{50} + (\beta_{2} - 1) q^{51} + (2 \beta_{2} + 3 \beta_1 + 3) q^{52} + (\beta_{2} - 2 \beta_1 + 1) q^{53} - \beta_1 q^{54} - 2 \beta_1 q^{55} + (\beta_{2} + \beta_1 - 2) q^{56} + (2 \beta_{2} + 2) q^{57} + ( - 4 \beta_{2} - \beta_1 - 13) q^{58} + (\beta_{2} - \beta_1 - 2) q^{59} + (\beta_{2} + 2) q^{60} - 2 q^{61} + \beta_1 q^{62} + ( - \beta_1 + 3) q^{63} + ( - \beta_{2} + 2 \beta_1 - 6) q^{64} + ( - \beta_1 - 1) q^{65} + ( - 2 \beta_{2} - 8) q^{66} + (2 \beta_{2} - \beta_1 + 7) q^{67} + (\beta_{2} - 2 \beta_1 - 3) q^{68} + ( - \beta_{2} + 2 \beta_1 - 3) q^{69} + (\beta_{2} - 3 \beta_1 + 4) q^{70} + (3 \beta_{2} + \beta_1 + 4) q^{71} + (\beta_{2} + \beta_1 + 1) q^{72} + (3 \beta_1 + 5) q^{73} + (\beta_{2} + 3 \beta_1 + 4) q^{74} - q^{75} + ( - 2 \beta_{2} - 4 \beta_1 - 14) q^{76} + ( - 2 \beta_{2} + 6 \beta_1 - 8) q^{77} + ( - \beta_{2} - \beta_1 - 4) q^{78} + ( - \beta_{2} - 2 \beta_1 + 7) q^{79} + ( - 2 \beta_1 - 1) q^{80} + q^{81} + ( - 2 \beta_{2} - 2 \beta_1 - 8) q^{82} + (3 \beta_{2} - 2 \beta_1 + 9) q^{83} + ( - 2 \beta_{2} + 3 \beta_1 - 5) q^{84} + (\beta_{2} - 1) q^{85} + ( - 2 \beta_{2} + 4 \beta_1 - 8) q^{86} + (\beta_{2} + 3 \beta_1) q^{87} + (4 \beta_{2} + 4 \beta_1 + 10) q^{88} + (5 \beta_{2} - 5 \beta_1 + 6) q^{89} - \beta_1 q^{90} + ( - \beta_{2} + 2 \beta_1 - 1) q^{91} + (\beta_{2} - 4 \beta_1 + 9) q^{92} - q^{93} + ( - \beta_{2} - 8 \beta_1 + 5) q^{94} + (2 \beta_{2} + 2) q^{95} + (\beta_1 - 6) q^{96} + ( - 2 \beta_{2} + 6 \beta_1) q^{97} + ( - 5 \beta_{2} + 7 \beta_1 - 23) q^{98} + 2 \beta_1 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{2} - 3 q^{3} + 5 q^{4} - 3 q^{5} - q^{6} + 8 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + q^{2} - 3 q^{3} + 5 q^{4} - 3 q^{5} - q^{6} + 8 q^{7} + 3 q^{8} + 3 q^{9} - q^{10} + 2 q^{11} - 5 q^{12} + 4 q^{13} - 8 q^{14} + 3 q^{15} + 5 q^{16} + 4 q^{17} + q^{18} - 4 q^{19} - 5 q^{20} - 8 q^{21} + 22 q^{22} + 6 q^{23} - 3 q^{24} + 3 q^{25} + 12 q^{26} - 3 q^{27} + 10 q^{28} - 2 q^{29} + q^{30} + 3 q^{31} + 17 q^{32} - 2 q^{33} - 2 q^{34} - 8 q^{35} + 5 q^{36} + 10 q^{37} - 8 q^{38} - 4 q^{39} - 3 q^{40} - 8 q^{41} + 8 q^{42} + 10 q^{43} + 10 q^{44} - 3 q^{45} - 16 q^{46} - 10 q^{47} - 5 q^{48} + 11 q^{49} + q^{50} - 4 q^{51} + 10 q^{52} - q^{54} - 2 q^{55} - 6 q^{56} + 4 q^{57} - 36 q^{58} - 8 q^{59} + 5 q^{60} - 6 q^{61} + q^{62} + 8 q^{63} - 15 q^{64} - 4 q^{65} - 22 q^{66} + 18 q^{67} - 12 q^{68} - 6 q^{69} + 8 q^{70} + 10 q^{71} + 3 q^{72} + 18 q^{73} + 14 q^{74} - 3 q^{75} - 44 q^{76} - 16 q^{77} - 12 q^{78} + 20 q^{79} - 5 q^{80} + 3 q^{81} - 24 q^{82} + 22 q^{83} - 10 q^{84} - 4 q^{85} - 18 q^{86} + 2 q^{87} + 30 q^{88} + 8 q^{89} - q^{90} + 22 q^{92} - 3 q^{93} + 8 q^{94} + 4 q^{95} - 17 q^{96} + 8 q^{97} - 57 q^{98} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 5x + 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) 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
−2.08613
0.571993
2.51414
−2.08613 −1.00000 2.35194 −1.00000 2.08613 5.08613 −0.734191 1.00000 2.08613
1.2 0.571993 −1.00000 −1.67282 −1.00000 −0.571993 2.42801 −2.10083 1.00000 −0.571993
1.3 2.51414 −1.00000 4.32088 −1.00000 −2.51414 0.485863 5.83502 1.00000 −2.51414
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(1\)
\(31\) \(-1\)

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 465.2.a.e 3
3.b odd 2 1 1395.2.a.j 3
4.b odd 2 1 7440.2.a.bs 3
5.b even 2 1 2325.2.a.r 3
5.c odd 4 2 2325.2.c.k 6
15.d odd 2 1 6975.2.a.bf 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
465.2.a.e 3 1.a even 1 1 trivial
1395.2.a.j 3 3.b odd 2 1
2325.2.a.r 3 5.b even 2 1
2325.2.c.k 6 5.c odd 4 2
6975.2.a.bf 3 15.d odd 2 1
7440.2.a.bs 3 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} - T_{2}^{2} - 5T_{2} + 3 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(465))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - T^{2} - 5T + 3 \) Copy content Toggle raw display
$3$ \( (T + 1)^{3} \) Copy content Toggle raw display
$5$ \( (T + 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} - 8 T^{2} + 16 T - 6 \) Copy content Toggle raw display
$11$ \( T^{3} - 2 T^{2} - 20 T + 24 \) Copy content Toggle raw display
$13$ \( T^{3} - 4T^{2} + 6 \) Copy content Toggle raw display
$17$ \( T^{3} - 4 T^{2} - 4 T + 4 \) Copy content Toggle raw display
$19$ \( T^{3} + 4 T^{2} - 32 T - 96 \) Copy content Toggle raw display
$23$ \( T^{3} - 6 T^{2} - 12 T + 4 \) Copy content Toggle raw display
$29$ \( T^{3} + 2 T^{2} - 66 T + 114 \) Copy content Toggle raw display
$31$ \( (T - 1)^{3} \) Copy content Toggle raw display
$37$ \( T^{3} - 10 T^{2} + 28 T - 18 \) Copy content Toggle raw display
$41$ \( T^{3} + 8T^{2} - 48 \) Copy content Toggle raw display
$43$ \( T^{3} - 10 T^{2} + 12 T + 24 \) Copy content Toggle raw display
$47$ \( T^{3} + 10 T^{2} - 52 T - 508 \) Copy content Toggle raw display
$53$ \( T^{3} - 24T - 36 \) Copy content Toggle raw display
$59$ \( T^{3} + 8 T^{2} + 10 T - 6 \) Copy content Toggle raw display
$61$ \( (T + 2)^{3} \) Copy content Toggle raw display
$67$ \( T^{3} - 18 T^{2} + 72 T + 82 \) Copy content Toggle raw display
$71$ \( T^{3} - 10 T^{2} - 66 T + 258 \) Copy content Toggle raw display
$73$ \( T^{3} - 18 T^{2} + 60 T + 106 \) Copy content Toggle raw display
$79$ \( T^{3} - 20 T^{2} + 96 T + 36 \) Copy content Toggle raw display
$83$ \( T^{3} - 22 T^{2} + 76 T + 492 \) Copy content Toggle raw display
$89$ \( T^{3} - 8 T^{2} - 262 T + 1394 \) Copy content Toggle raw display
$97$ \( T^{3} - 8 T^{2} - 168 T + 1488 \) Copy content Toggle raw display
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