Properties

Label 465.2.a.g
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 3x + 1 \) : 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
0.311108
2.17009
−1.48119
−1.21432 1.00000 −0.525428 −1.00000 −1.21432 −1.59210 3.06668 1.00000 1.21432
1.2 1.53919 1.00000 0.369102 −1.00000 1.53919 4.87936 −2.51026 1.00000 −1.53919
1.3 2.67513 1.00000 5.15633 −1.00000 2.67513 −1.28726 8.44358 1.00000 −2.67513
\(n\): 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.g 3
3.b odd 2 1 1395.2.a.h 3
4.b odd 2 1 7440.2.a.bm 3
5.b even 2 1 2325.2.a.p 3
5.c odd 4 2 2325.2.c.l 6
15.d odd 2 1 6975.2.a.bi 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
465.2.a.g 3 1.a even 1 1 trivial
1395.2.a.h 3 3.b odd 2 1
2325.2.a.p 3 5.b even 2 1
2325.2.c.l 6 5.c odd 4 2
6975.2.a.bi 3 15.d odd 2 1
7440.2.a.bm 3 4.b odd 2 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - 3T^{2} - T + 5 \) Copy content Toggle raw display
$3$ \( (T - 1)^{3} \) Copy content Toggle raw display
$5$ \( (T + 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} - 2 T^{2} - 12 T - 10 \) Copy content Toggle raw display
$11$ \( T^{3} - 2 T^{2} - 12 T + 8 \) Copy content Toggle raw display
$13$ \( T^{3} + 6 T^{2} + 8 T - 2 \) Copy content Toggle raw display
$17$ \( T^{3} - 4 T^{2} - 4 T + 20 \) Copy content Toggle raw display
$19$ \( T^{3} + 8T^{2} - 32 \) Copy content Toggle raw display
$23$ \( T^{3} - 14 T^{2} + 60 T - 76 \) Copy content Toggle raw display
$29$ \( T^{3} - 16 T^{2} + 62 T - 10 \) Copy content Toggle raw display
$31$ \( (T + 1)^{3} \) Copy content Toggle raw display
$37$ \( T^{3} + 8T^{2} - 74 \) Copy content Toggle raw display
$41$ \( T^{3} - 4 T^{2} - 88 T - 16 \) Copy content Toggle raw display
$43$ \( T^{3} + 2 T^{2} - 60 T - 200 \) Copy content Toggle raw display
$47$ \( T^{3} - 14 T^{2} - 92 T + 1388 \) Copy content Toggle raw display
$53$ \( T^{3} - 8 T^{2} - 72 T - 100 \) Copy content Toggle raw display
$59$ \( T^{3} + 26 T^{2} + 202 T + 466 \) Copy content Toggle raw display
$61$ \( T^{3} + 18 T^{2} + 44 T - 296 \) Copy content Toggle raw display
$67$ \( T^{3} - 136T - 274 \) Copy content Toggle raw display
$71$ \( T^{3} - 4 T^{2} + 2 T + 2 \) Copy content Toggle raw display
$73$ \( T^{3} + 12 T^{2} - 64 T - 134 \) Copy content Toggle raw display
$79$ \( T^{3} - 4 T^{2} - 144 T - 500 \) Copy content Toggle raw display
$83$ \( T^{3} + 10 T^{2} - 44 T - 388 \) Copy content Toggle raw display
$89$ \( T^{3} + 6 T^{2} - 18 T - 50 \) Copy content Toggle raw display
$97$ \( T^{3} - 8 T^{2} - 240 T + 1712 \) Copy content Toggle raw display
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