Properties

Label 65.2.d.a
Level $65$
Weight $2$
Character orbit 65.d
Analytic conductor $0.519$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [65,2,Mod(64,65)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(65, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("65.64");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 65 = 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 65.d (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(0.519027613138\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(\beta = 2i\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/65\mathbb{Z}\right)^\times\).

\(n\) \(27\) \(41\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
64.1
1.00000i
1.00000i
−1.00000 2.00000i −1.00000 −1.00000 2.00000i 2.00000i 0 3.00000 −1.00000 1.00000 + 2.00000i
64.2 −1.00000 2.00000i −1.00000 −1.00000 + 2.00000i 2.00000i 0 3.00000 −1.00000 1.00000 2.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 65.2.d.a 2
3.b odd 2 1 585.2.h.c 2
4.b odd 2 1 1040.2.f.a 2
5.b even 2 1 65.2.d.b yes 2
5.c odd 4 1 325.2.c.b 2
5.c odd 4 1 325.2.c.e 2
13.b even 2 1 65.2.d.b yes 2
13.c even 3 2 845.2.l.b 4
13.d odd 4 1 845.2.b.a 2
13.d odd 4 1 845.2.b.b 2
13.e even 6 2 845.2.l.a 4
13.f odd 12 2 845.2.n.a 4
13.f odd 12 2 845.2.n.b 4
15.d odd 2 1 585.2.h.b 2
20.d odd 2 1 1040.2.f.b 2
39.d odd 2 1 585.2.h.b 2
52.b odd 2 1 1040.2.f.b 2
65.d even 2 1 inner 65.2.d.a 2
65.f even 4 1 4225.2.a.k 1
65.f even 4 1 4225.2.a.m 1
65.g odd 4 1 845.2.b.a 2
65.g odd 4 1 845.2.b.b 2
65.h odd 4 1 325.2.c.b 2
65.h odd 4 1 325.2.c.e 2
65.k even 4 1 4225.2.a.e 1
65.k even 4 1 4225.2.a.h 1
65.l even 6 2 845.2.l.b 4
65.n even 6 2 845.2.l.a 4
65.s odd 12 2 845.2.n.a 4
65.s odd 12 2 845.2.n.b 4
195.e odd 2 1 585.2.h.c 2
260.g odd 2 1 1040.2.f.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.d.a 2 1.a even 1 1 trivial
65.2.d.a 2 65.d even 2 1 inner
65.2.d.b yes 2 5.b even 2 1
65.2.d.b yes 2 13.b even 2 1
325.2.c.b 2 5.c odd 4 1
325.2.c.b 2 65.h odd 4 1
325.2.c.e 2 5.c odd 4 1
325.2.c.e 2 65.h odd 4 1
585.2.h.b 2 15.d odd 2 1
585.2.h.b 2 39.d odd 2 1
585.2.h.c 2 3.b odd 2 1
585.2.h.c 2 195.e odd 2 1
845.2.b.a 2 13.d odd 4 1
845.2.b.a 2 65.g odd 4 1
845.2.b.b 2 13.d odd 4 1
845.2.b.b 2 65.g odd 4 1
845.2.l.a 4 13.e even 6 2
845.2.l.a 4 65.n even 6 2
845.2.l.b 4 13.c even 3 2
845.2.l.b 4 65.l even 6 2
845.2.n.a 4 13.f odd 12 2
845.2.n.a 4 65.s odd 12 2
845.2.n.b 4 13.f odd 12 2
845.2.n.b 4 65.s odd 12 2
1040.2.f.a 2 4.b odd 2 1
1040.2.f.a 2 260.g odd 2 1
1040.2.f.b 2 20.d odd 2 1
1040.2.f.b 2 52.b odd 2 1
4225.2.a.e 1 65.k even 4 1
4225.2.a.h 1 65.k even 4 1
4225.2.a.k 1 65.f even 4 1
4225.2.a.m 1 65.f even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(65, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 4 \) Copy content Toggle raw display
$5$ \( T^{2} + 2T + 5 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 4 \) Copy content Toggle raw display
$13$ \( T^{2} - 6T + 13 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 36 \) Copy content Toggle raw display
$23$ \( T^{2} + 36 \) Copy content Toggle raw display
$29$ \( (T - 6)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 36 \) Copy content Toggle raw display
$37$ \( (T - 6)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 64 \) Copy content Toggle raw display
$43$ \( T^{2} + 36 \) Copy content Toggle raw display
$47$ \( (T + 8)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 144 \) Copy content Toggle raw display
$59$ \( T^{2} + 4 \) Copy content Toggle raw display
$61$ \( (T - 6)^{2} \) Copy content Toggle raw display
$67$ \( (T - 12)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 4 \) Copy content Toggle raw display
$73$ \( (T + 6)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( (T + 4)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 64 \) Copy content Toggle raw display
$97$ \( (T + 6)^{2} \) Copy content Toggle raw display
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