Properties

Label 65.2.e.a
Level $65$
Weight $2$
Character orbit 65.e
Analytic conductor $0.519$
Analytic rank $0$
Dimension $4$
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(16,65)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(65, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("65.16");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 65 = 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 65.e (of order \(3\), degree \(2\), 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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 2x^{2} + x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 2x^{2} + x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 1 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 2\nu^{2} - 2\nu - 1 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{2} - 1 \) Copy content Toggle raw display

Character values

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

\(n\) \(27\) \(41\)
\(\chi(n)\) \(1\) \(\beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1
0.809017 + 1.40126i
−0.309017 0.535233i
0.809017 1.40126i
−0.309017 + 0.535233i
−0.809017 1.40126i 1.11803 + 1.93649i −0.309017 + 0.535233i 1.00000 1.80902 3.13331i 0.118034 0.204441i −2.23607 −1.00000 + 1.73205i −0.809017 1.40126i
16.2 0.309017 + 0.535233i −1.11803 1.93649i 0.809017 1.40126i 1.00000 0.690983 1.19682i −2.11803 + 3.66854i 2.23607 −1.00000 + 1.73205i 0.309017 + 0.535233i
61.1 −0.809017 + 1.40126i 1.11803 1.93649i −0.309017 0.535233i 1.00000 1.80902 + 3.13331i 0.118034 + 0.204441i −2.23607 −1.00000 1.73205i −0.809017 + 1.40126i
61.2 0.309017 0.535233i −1.11803 + 1.93649i 0.809017 + 1.40126i 1.00000 0.690983 + 1.19682i −2.11803 3.66854i 2.23607 −1.00000 1.73205i 0.309017 0.535233i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 65.2.e.a 4
3.b odd 2 1 585.2.j.e 4
4.b odd 2 1 1040.2.q.n 4
5.b even 2 1 325.2.e.b 4
5.c odd 4 2 325.2.o.a 8
13.b even 2 1 845.2.e.g 4
13.c even 3 1 inner 65.2.e.a 4
13.c even 3 1 845.2.a.e 2
13.d odd 4 2 845.2.m.e 8
13.e even 6 1 845.2.a.b 2
13.e even 6 1 845.2.e.g 4
13.f odd 12 2 845.2.c.c 4
13.f odd 12 2 845.2.m.e 8
39.h odd 6 1 7605.2.a.bf 2
39.i odd 6 1 585.2.j.e 4
39.i odd 6 1 7605.2.a.ba 2
52.j odd 6 1 1040.2.q.n 4
65.l even 6 1 4225.2.a.y 2
65.n even 6 1 325.2.e.b 4
65.n even 6 1 4225.2.a.u 2
65.q odd 12 2 325.2.o.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.e.a 4 1.a even 1 1 trivial
65.2.e.a 4 13.c even 3 1 inner
325.2.e.b 4 5.b even 2 1
325.2.e.b 4 65.n even 6 1
325.2.o.a 8 5.c odd 4 2
325.2.o.a 8 65.q odd 12 2
585.2.j.e 4 3.b odd 2 1
585.2.j.e 4 39.i odd 6 1
845.2.a.b 2 13.e even 6 1
845.2.a.e 2 13.c even 3 1
845.2.c.c 4 13.f odd 12 2
845.2.e.g 4 13.b even 2 1
845.2.e.g 4 13.e even 6 1
845.2.m.e 8 13.d odd 4 2
845.2.m.e 8 13.f odd 12 2
1040.2.q.n 4 4.b odd 2 1
1040.2.q.n 4 52.j odd 6 1
4225.2.a.u 2 65.n even 6 1
4225.2.a.y 2 65.l even 6 1
7605.2.a.ba 2 39.i odd 6 1
7605.2.a.bf 2 39.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + T_{2}^{3} + 2T_{2}^{2} - T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(65, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + T^{3} + 2 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{4} + 5T^{2} + 25 \) Copy content Toggle raw display
$5$ \( (T - 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 4 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{4} + 4 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( (T^{2} + 2 T + 13)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 2 T^{3} + \cdots + 361 \) Copy content Toggle raw display
$19$ \( T^{4} + 4 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$23$ \( T^{4} + 12 T^{3} + \cdots + 961 \) Copy content Toggle raw display
$29$ \( T^{4} - 6 T^{3} + \cdots + 121 \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 3 T + 9)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 6 T^{3} + \cdots + 5041 \) Copy content Toggle raw display
$43$ \( T^{4} - 8 T^{3} + \cdots + 121 \) Copy content Toggle raw display
$47$ \( (T^{2} - 8 T - 64)^{2} \) Copy content Toggle raw display
$53$ \( (T - 6)^{4} \) Copy content Toggle raw display
$59$ \( T^{4} + 12 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$61$ \( T^{4} + 2 T^{3} + \cdots + 32041 \) Copy content Toggle raw display
$67$ \( T^{4} + 8 T^{3} + \cdots + 841 \) Copy content Toggle raw display
$71$ \( T^{4} - 8 T^{3} + \cdots + 121 \) Copy content Toggle raw display
$73$ \( (T + 6)^{4} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 80)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 9 T + 81)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 2 T^{3} + \cdots + 361 \) Copy content Toggle raw display
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