Properties

Label 465.2.a.h
Level $465$
Weight $2$
Character orbit 465.a
Self dual yes
Analytic conductor $3.713$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: 4.4.8468.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 5x^{2} + 3x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 5x^{2} + 3x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{3} - 3\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 4\nu + 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{3} - \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{3} + \beta_{2} + \beta _1 + 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{3} - 3\beta_{2} + 5\beta _1 + 2 ) / 2 \) 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
−1.89122
2.27841
−0.704624
1.31743
−2.46793 −1.00000 4.09069 1.00000 2.46793 2.31451 −5.15968 1.00000 −2.46793
1.2 0.0872450 −1.00000 −1.99239 1.00000 −0.0872450 −3.46958 −0.348316 1.00000 0.0872450
1.3 1.79888 −1.00000 1.23597 1.00000 −1.79888 4.20813 −1.37440 1.00000 1.79888
1.4 2.58181 −1.00000 4.66573 1.00000 −2.58181 0.946946 6.88240 1.00000 2.58181
\(n\): 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.h 4
3.b odd 2 1 1395.2.a.k 4
4.b odd 2 1 7440.2.a.bz 4
5.b even 2 1 2325.2.a.v 4
5.c odd 4 2 2325.2.c.p 8
15.d odd 2 1 6975.2.a.bo 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
465.2.a.h 4 1.a even 1 1 trivial
1395.2.a.k 4 3.b odd 2 1
2325.2.a.v 4 5.b even 2 1
2325.2.c.p 8 5.c odd 4 2
6975.2.a.bo 4 15.d odd 2 1
7440.2.a.bz 4 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 2T_{2}^{3} - 6T_{2}^{2} + 12T_{2} - 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(465))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 2 T^{3} - 6 T^{2} + 12 T - 1 \) Copy content Toggle raw display
$3$ \( (T + 1)^{4} \) Copy content Toggle raw display
$5$ \( (T - 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 4 T^{3} - 10 T^{2} + 46 T - 32 \) Copy content Toggle raw display
$11$ \( T^{4} - 6 T^{3} - 12 T^{2} + 56 T + 32 \) Copy content Toggle raw display
$13$ \( T^{4} - 2 T^{3} - 50 T^{2} + 94 T + 4 \) Copy content Toggle raw display
$17$ \( T^{4} + 10 T^{3} + 20 T^{2} + \cdots - 136 \) Copy content Toggle raw display
$19$ \( (T - 4)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} - 6 T^{3} - 84 T^{2} + \cdots + 2048 \) Copy content Toggle raw display
$29$ \( T^{4} - 12 T^{2} + 14 T - 4 \) Copy content Toggle raw display
$31$ \( (T + 1)^{4} \) Copy content Toggle raw display
$37$ \( T^{4} - 8 T^{3} - 26 T^{2} + 174 T + 388 \) Copy content Toggle raw display
$41$ \( T^{4} + 6 T^{3} - 64 T^{2} - 480 T - 704 \) Copy content Toggle raw display
$43$ \( T^{4} - 2 T^{3} - 92 T^{2} + 232 T + 928 \) Copy content Toggle raw display
$47$ \( T^{4} + 2 T^{3} - 108 T^{2} + \cdots - 736 \) Copy content Toggle raw display
$53$ \( T^{4} + 6 T^{3} - 8 T^{2} - 20 T - 8 \) Copy content Toggle raw display
$59$ \( T^{4} - 4 T^{3} - 108 T^{2} + \cdots - 808 \) Copy content Toggle raw display
$61$ \( T^{4} - 168 T^{2} + 608 T + 848 \) Copy content Toggle raw display
$67$ \( T^{4} + 2 T^{3} - 66 T^{2} - 202 T + 184 \) Copy content Toggle raw display
$71$ \( T^{4} - 22 T^{3} + 88 T^{2} + \cdots - 1808 \) Copy content Toggle raw display
$73$ \( T^{4} + 32 T^{3} + 342 T^{2} + \cdots + 788 \) Copy content Toggle raw display
$79$ \( T^{4} - 24 T^{3} - 68 T^{2} + \cdots - 24064 \) Copy content Toggle raw display
$83$ \( T^{4} + 6 T^{3} - 44 T^{2} - 340 T - 464 \) Copy content Toggle raw display
$89$ \( T^{4} + 14 T^{3} - 60 T^{2} - 482 T - 92 \) Copy content Toggle raw display
$97$ \( T^{4} + 14 T^{3} + 48 T^{2} - 64 \) Copy content Toggle raw display
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