Properties

Label 465.2.a.f
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: 3.3.148.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) 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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 3x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 2 \) 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
−1.48119
0.311108
2.17009
−1.48119 1.00000 0.193937 1.00000 −1.48119 2.48119 2.67513 1.00000 −1.48119
1.2 0.311108 1.00000 −1.90321 1.00000 0.311108 0.688892 −1.21432 1.00000 0.311108
1.3 2.17009 1.00000 2.70928 1.00000 2.17009 −1.17009 1.53919 1.00000 2.17009
\(n\): 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.f 3
3.b odd 2 1 1395.2.a.i 3
4.b odd 2 1 7440.2.a.bp 3
5.b even 2 1 2325.2.a.q 3
5.c odd 4 2 2325.2.c.o 6
15.d odd 2 1 6975.2.a.be 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
465.2.a.f 3 1.a even 1 1 trivial
1395.2.a.i 3 3.b odd 2 1
2325.2.a.q 3 5.b even 2 1
2325.2.c.o 6 5.c odd 4 2
6975.2.a.be 3 15.d odd 2 1
7440.2.a.bp 3 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} - T_{2}^{2} - 3T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(465))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - T^{2} - 3T + 1 \) Copy content Toggle raw display
$3$ \( (T - 1)^{3} \) Copy content Toggle raw display
$5$ \( (T - 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} - 2 T^{2} - 2 T + 2 \) Copy content Toggle raw display
$11$ \( (T - 2)^{3} \) Copy content Toggle raw display
$13$ \( T^{3} + 2 T^{2} - 22 T + 2 \) Copy content Toggle raw display
$17$ \( T^{3} - 28T + 52 \) Copy content Toggle raw display
$19$ \( T^{3} \) Copy content Toggle raw display
$23$ \( T^{3} - 2 T^{2} - 4 T + 4 \) Copy content Toggle raw display
$29$ \( T^{3} - 8 T^{2} - 16 T + 130 \) Copy content Toggle raw display
$31$ \( (T - 1)^{3} \) Copy content Toggle raw display
$37$ \( T^{3} - 118T - 358 \) Copy content Toggle raw display
$41$ \( T^{3} - 8 T^{2} - 32 T + 128 \) Copy content Toggle raw display
$43$ \( T^{3} + 10 T^{2} - 4 T - 8 \) Copy content Toggle raw display
$47$ \( T^{3} + 6 T^{2} - 36 T - 68 \) Copy content Toggle raw display
$53$ \( T^{3} - 40T + 76 \) Copy content Toggle raw display
$59$ \( T^{3} - 14 T^{2} - 28 T + 670 \) Copy content Toggle raw display
$61$ \( T^{3} + 2 T^{2} - 36 T - 104 \) Copy content Toggle raw display
$67$ \( T^{3} + 16 T^{2} + 2 T - 302 \) Copy content Toggle raw display
$71$ \( T^{3} - 20 T^{2} + 84 T + 134 \) Copy content Toggle raw display
$73$ \( T^{3} - 70T + 86 \) Copy content Toggle raw display
$79$ \( T^{3} - 8 T^{2} - 92 T + 380 \) Copy content Toggle raw display
$83$ \( T^{3} - 2 T^{2} - 132 T - 4 \) Copy content Toggle raw display
$89$ \( T^{3} + 6 T^{2} - 184 T - 1070 \) Copy content Toggle raw display
$97$ \( (T + 12)^{3} \) Copy content Toggle raw display
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