Properties

Label 465.2.a.c
Level $465$
Weight $2$
Character orbit 465.a
Self dual yes
Analytic conductor $3.713$
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] = [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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) 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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta - 1) q^{2} + q^{3} + ( - 2 \beta + 1) q^{4} - q^{5} + (\beta - 1) q^{6} + ( - \beta - 2) q^{7} + (\beta - 3) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta - 1) q^{2} + q^{3} + ( - 2 \beta + 1) q^{4} - q^{5} + (\beta - 1) q^{6} + ( - \beta - 2) q^{7} + (\beta - 3) q^{8} + q^{9} + ( - \beta + 1) q^{10} + 2 \beta q^{11} + ( - 2 \beta + 1) q^{12} + ( - \beta - 4) q^{13} - \beta q^{14} - q^{15} + 3 q^{16} - 4 q^{17} + (\beta - 1) q^{18} - 2 \beta q^{19} + (2 \beta - 1) q^{20} + ( - \beta - 2) q^{21} + ( - 2 \beta + 4) q^{22} - 6 q^{23} + (\beta - 3) q^{24} + q^{25} + ( - 3 \beta + 2) q^{26} + q^{27} + (3 \beta + 2) q^{28} + ( - 3 \beta - 2) q^{29} + ( - \beta + 1) q^{30} + q^{31} + (\beta + 3) q^{32} + 2 \beta q^{33} + ( - 4 \beta + 4) q^{34} + (\beta + 2) q^{35} + ( - 2 \beta + 1) q^{36} + \beta q^{37} + (2 \beta - 4) q^{38} + ( - \beta - 4) q^{39} + ( - \beta + 3) q^{40} + (2 \beta + 2) q^{41} - \beta q^{42} + 8 \beta q^{43} + (2 \beta - 8) q^{44} - q^{45} + ( - 6 \beta + 6) q^{46} + (2 \beta - 2) q^{47} + 3 q^{48} + (4 \beta - 1) q^{49} + (\beta - 1) q^{50} - 4 q^{51} + 7 \beta q^{52} - 4 q^{53} + (\beta - 1) q^{54} - 2 \beta q^{55} + (\beta + 4) q^{56} - 2 \beta q^{57} + (\beta - 4) q^{58} + (3 \beta + 4) q^{59} + (2 \beta - 1) q^{60} + (6 \beta - 2) q^{61} + (\beta - 1) q^{62} + ( - \beta - 2) q^{63} + (2 \beta - 7) q^{64} + (\beta + 4) q^{65} + ( - 2 \beta + 4) q^{66} + ( - 7 \beta - 6) q^{67} + (8 \beta - 4) q^{68} - 6 q^{69} + \beta q^{70} - 7 \beta q^{71} + (\beta - 3) q^{72} + (7 \beta + 4) q^{73} + ( - \beta + 2) q^{74} + q^{75} + ( - 2 \beta + 8) q^{76} + ( - 4 \beta - 4) q^{77} + ( - 3 \beta + 2) q^{78} + (6 \beta - 6) q^{79} - 3 q^{80} + q^{81} + 2 q^{82} + ( - 4 \beta - 6) q^{83} + (3 \beta + 2) q^{84} + 4 q^{85} + ( - 8 \beta + 16) q^{86} + ( - 3 \beta - 2) q^{87} + ( - 6 \beta + 4) q^{88} + ( - 5 \beta - 2) q^{89} + ( - \beta + 1) q^{90} + (6 \beta + 10) q^{91} + (12 \beta - 6) q^{92} + q^{93} + ( - 4 \beta + 6) q^{94} + 2 \beta q^{95} + (\beta + 3) q^{96} + ( - 4 \beta + 10) q^{97} + ( - 5 \beta + 9) q^{98} + 2 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} - 4 q^{7} - 6 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} - 4 q^{7} - 6 q^{8} + 2 q^{9} + 2 q^{10} + 2 q^{12} - 8 q^{13} - 2 q^{15} + 6 q^{16} - 8 q^{17} - 2 q^{18} - 2 q^{20} - 4 q^{21} + 8 q^{22} - 12 q^{23} - 6 q^{24} + 2 q^{25} + 4 q^{26} + 2 q^{27} + 4 q^{28} - 4 q^{29} + 2 q^{30} + 2 q^{31} + 6 q^{32} + 8 q^{34} + 4 q^{35} + 2 q^{36} - 8 q^{38} - 8 q^{39} + 6 q^{40} + 4 q^{41} - 16 q^{44} - 2 q^{45} + 12 q^{46} - 4 q^{47} + 6 q^{48} - 2 q^{49} - 2 q^{50} - 8 q^{51} - 8 q^{53} - 2 q^{54} + 8 q^{56} - 8 q^{58} + 8 q^{59} - 2 q^{60} - 4 q^{61} - 2 q^{62} - 4 q^{63} - 14 q^{64} + 8 q^{65} + 8 q^{66} - 12 q^{67} - 8 q^{68} - 12 q^{69} - 6 q^{72} + 8 q^{73} + 4 q^{74} + 2 q^{75} + 16 q^{76} - 8 q^{77} + 4 q^{78} - 12 q^{79} - 6 q^{80} + 2 q^{81} + 4 q^{82} - 12 q^{83} + 4 q^{84} + 8 q^{85} + 32 q^{86} - 4 q^{87} + 8 q^{88} - 4 q^{89} + 2 q^{90} + 20 q^{91} - 12 q^{92} + 2 q^{93} + 12 q^{94} + 6 q^{96} + 20 q^{97} + 18 q^{98}+O(q^{100}) \) 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
−1.41421
1.41421
−2.41421 1.00000 3.82843 −1.00000 −2.41421 −0.585786 −4.41421 1.00000 2.41421
1.2 0.414214 1.00000 −1.82843 −1.00000 0.414214 −3.41421 −1.58579 1.00000 −0.414214
\(n\): 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.c 2
3.b odd 2 1 1395.2.a.g 2
4.b odd 2 1 7440.2.a.be 2
5.b even 2 1 2325.2.a.n 2
5.c odd 4 2 2325.2.c.i 4
15.d odd 2 1 6975.2.a.u 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
465.2.a.c 2 1.a even 1 1 trivial
1395.2.a.g 2 3.b odd 2 1
2325.2.a.n 2 5.b even 2 1
2325.2.c.i 4 5.c odd 4 2
6975.2.a.u 2 15.d odd 2 1
7440.2.a.be 2 4.b odd 2 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 2T - 1 \) Copy content Toggle raw display
$3$ \( (T - 1)^{2} \) Copy content Toggle raw display
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 4T + 2 \) Copy content Toggle raw display
$11$ \( T^{2} - 8 \) Copy content Toggle raw display
$13$ \( T^{2} + 8T + 14 \) Copy content Toggle raw display
$17$ \( (T + 4)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 8 \) Copy content Toggle raw display
$23$ \( (T + 6)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 4T - 14 \) Copy content Toggle raw display
$31$ \( (T - 1)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 2 \) Copy content Toggle raw display
$41$ \( T^{2} - 4T - 4 \) Copy content Toggle raw display
$43$ \( T^{2} - 128 \) Copy content Toggle raw display
$47$ \( T^{2} + 4T - 4 \) Copy content Toggle raw display
$53$ \( (T + 4)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 8T - 2 \) Copy content Toggle raw display
$61$ \( T^{2} + 4T - 68 \) Copy content Toggle raw display
$67$ \( T^{2} + 12T - 62 \) Copy content Toggle raw display
$71$ \( T^{2} - 98 \) Copy content Toggle raw display
$73$ \( T^{2} - 8T - 82 \) Copy content Toggle raw display
$79$ \( T^{2} + 12T - 36 \) Copy content Toggle raw display
$83$ \( T^{2} + 12T + 4 \) Copy content Toggle raw display
$89$ \( T^{2} + 4T - 46 \) Copy content Toggle raw display
$97$ \( T^{2} - 20T + 68 \) Copy content Toggle raw display
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